Initial population.

309 files changed, 192093 insertions, 0 deletions

diff --git a/src/hyper/pages/ALIST.ht b/src/hyper/pages/ALIST.ht
new file mode 100644
index 00000000..897d022e
--- /dev/null
+++ b/src/hyper/pages/ALIST.ht
@@ -0,0 +1,90 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\AssociationListXmpTitle}{AssociationList}
+\newcommand{\AssociationListXmpNumber}{9.1}
+%
+% =====================================================================
+\begin{page}{AssociationListXmpPage}{9.1 AssociationList}
+% =====================================================================
+\beginscroll
+%
+
+The \spadtype{AssociationList} constructor provides a general structure for
+associative storage.
+%-% \HDindex{list!association}{AssociationListXmpPage}{9.1}{AssociationList}
+This type provides association lists in which data objects can be saved
+%-% \HDindex{association list}{AssociationListXmpPage}{9.1}{AssociationList}
+according to keys of any type.
+For a given association list, specific types must be chosen for the keys and
+entries.
+You can think of the representation of an association list as a list
+of records with key and entry fields.
+
+Association lists are a form of table and so most of the operations available
+%-% \HDindex{table}{AssociationListXmpPage}{9.1}{AssociationList}
+for \spadtype{Table} are also available for \spadtype{AssociationList}.
+They can also be viewed as lists and can be manipulated accordingly.
+
+\xtc{
+This is a \pspadtype{Record} type with age and gender fields.
+}{
+\spadpaste{Data := Record(monthsOld : Integer, gender : String) \bound{Data}}
+}
+\xtc{
+In this expression, \spad{al} is declared to be an association
+list whose keys are strings and whose entries are the above records.
+}{
+\spadpaste{al : AssociationList(String,Data) \free{Data}\bound{al}}
+}
+\xtc{
+The \spadfunFrom{table}{AssociationList} operation is used to create
+an empty association list.
+}{
+\spadpaste{al := table() \free{al}\bound{al1}}
+}
+\xtc{
+You can use assignment syntax to add things to the association list.
+}{
+\spadpaste{al."bob" := [407,"male"]\$Data \free{al1}\bound{al2}}
+}
+\xtc{
+}{
+\spadpaste{al."judith" := [366,"female"]\$Data \free{al2}\bound{al3}}
+}
+\xtc{
+}{
+\spadpaste{al."katie" := [24,"female"]\$Data \free{al3}\bound{al4}}
+}
+\xtc{
+Perhaps we should have included a species field.
+}{
+\spadpaste{al."smokie" := [200,"female"]\$Data \free{al4}\bound{al5}}
+}
+\xtc{
+Now look at what is in the association list.
+Note that the last-added (key, entry) pair is at the beginning of the list.
+}{
+\spadpaste{al \free{al5}}
+}
+\xtc{
+You can reset the entry for an existing key.
+}{
+\spadpaste{al."katie" := [23,"female"]\$Data \free{al5}\bound{al6}}
+}
+\xtc{
+Use \spadfunFromX{delete}{AssociationList} to destructively remove
+an element of the association list.
+Use \spadfunFrom{delete}{AssociationList} to return a copy of the
+association list with the element deleted.
+The second argument is the index of the element to delete.
+}{
+\spadpaste{delete!(al,1) \free{al6}\bound{al7}}
+}
+
+For more information about tables, see \downlink{`Table'}{TableXmpPage}\ignore{Table}.
+For more information about lists, see \downlink{`List'}{ListXmpPage}\ignore{List}.
+\showBlurb{AssociationList}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ALIST.pht b/src/hyper/pages/ALIST.pht
new file mode 100644
index 00000000..3cf6ce88
--- /dev/null
+++ b/src/hyper/pages/ALIST.pht
@@ -0,0 +1,173 @@
+\begin{patch}{AssociationListXmpPagePatch1}
+\begin{paste}{AssociationListXmpPageFull1}{AssociationListXmpPageEmpty1}
+\pastebutton{AssociationListXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{Data := Record(monthsOld : Integer, gender : String)\bound{Data }}
+\indentrel{3}\begin{verbatim}
+   (1)  Record(monthsOld: Integer,gender: String)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPageEmpty1}
+\begin{paste}{AssociationListXmpPageEmpty1}{AssociationListXmpPagePatch1}
+\pastebutton{AssociationListXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{Data := Record(monthsOld : Integer, gender : String)\bound{Data }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPagePatch2}
+\begin{paste}{AssociationListXmpPageFull2}{AssociationListXmpPageEmpty2}
+\pastebutton{AssociationListXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{al : AssociationList(String,Data)\free{Data }\bound{al }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPageEmpty2}
+\begin{paste}{AssociationListXmpPageEmpty2}{AssociationListXmpPagePatch2}
+\pastebutton{AssociationListXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{al : AssociationList(String,Data)\free{Data }\bound{al }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPagePatch3}
+\begin{paste}{AssociationListXmpPageFull3}{AssociationListXmpPageEmpty3}
+\pastebutton{AssociationListXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{al := table()\free{al }\bound{al1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  table()
+Type: AssociationList(String,Record(monthsOld: Integer,gender: String))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPageEmpty3}
+\begin{paste}{AssociationListXmpPageEmpty3}{AssociationListXmpPagePatch3}
+\pastebutton{AssociationListXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{al := table()\free{al }\bound{al1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPagePatch4}
+\begin{paste}{AssociationListXmpPageFull4}{AssociationListXmpPageEmpty4}
+\pastebutton{AssociationListXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{al."bob" := [407,"male"]$Data\free{al1 }\bound{al2 }}
+\indentrel{3}\begin{verbatim}
+   (4)  [monthsOld= 407,gender= "male"]
+        Type: Record(monthsOld: Integer,gender: String)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPageEmpty4}
+\begin{paste}{AssociationListXmpPageEmpty4}{AssociationListXmpPagePatch4}
+\pastebutton{AssociationListXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{al."bob" := [407,"male"]$Data\free{al1 }\bound{al2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPagePatch5}
+\begin{paste}{AssociationListXmpPageFull5}{AssociationListXmpPageEmpty5}
+\pastebutton{AssociationListXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{al."judith" := [366,"female"]$Data\free{al2 }\bound{al3 }}
+\indentrel{3}\begin{verbatim}
+   (5)  [monthsOld= 366,gender= "female"]
+        Type: Record(monthsOld: Integer,gender: String)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPageEmpty5}
+\begin{paste}{AssociationListXmpPageEmpty5}{AssociationListXmpPagePatch5}
+\pastebutton{AssociationListXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{al."judith" := [366,"female"]$Data\free{al2 }\bound{al3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPagePatch6}
+\begin{paste}{AssociationListXmpPageFull6}{AssociationListXmpPageEmpty6}
+\pastebutton{AssociationListXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{al."katie" := [24,"female"]$Data\free{al3 }\bound{al4 }}
+\indentrel{3}\begin{verbatim}
+   (6)  [monthsOld= 24,gender= "female"]
+        Type: Record(monthsOld: Integer,gender: String)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPageEmpty6}
+\begin{paste}{AssociationListXmpPageEmpty6}{AssociationListXmpPagePatch6}
+\pastebutton{AssociationListXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{al."katie" := [24,"female"]$Data\free{al3 }\bound{al4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPagePatch7}
+\begin{paste}{AssociationListXmpPageFull7}{AssociationListXmpPageEmpty7}
+\pastebutton{AssociationListXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{al."smokie" := [200,"female"]$Data\free{al4 }\bound{al5 }}
+\indentrel{3}\begin{verbatim}
+   (7)  [monthsOld= 200,gender= "female"]
+        Type: Record(monthsOld: Integer,gender: String)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPageEmpty7}
+\begin{paste}{AssociationListXmpPageEmpty7}{AssociationListXmpPagePatch7}
+\pastebutton{AssociationListXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{al."smokie" := [200,"female"]$Data\free{al4 }\bound{al5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPagePatch8}
+\begin{paste}{AssociationListXmpPageFull8}{AssociationListXmpPageEmpty8}
+\pastebutton{AssociationListXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{al\free{al5 }}
+\indentrel{3}\begin{verbatim}
+   (8)
+   table
+      "smokie"= [monthsOld= 200,gender= "female"]
+  ,
+      "katie"= [monthsOld= 24,gender= "female"]
+  ,
+      "judith"= [monthsOld= 366,gender= "female"]
+  ,
+      "bob"= [monthsOld= 407,gender= "male"]
+Type: AssociationList(String,Record(monthsOld: Integer,gender: String))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPageEmpty8}
+\begin{paste}{AssociationListXmpPageEmpty8}{AssociationListXmpPagePatch8}
+\pastebutton{AssociationListXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{al\free{al5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPagePatch9}
+\begin{paste}{AssociationListXmpPageFull9}{AssociationListXmpPageEmpty9}
+\pastebutton{AssociationListXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{al."katie" := [23,"female"]$Data\free{al5 }\bound{al6 }}
+\indentrel{3}\begin{verbatim}
+   (9)  [monthsOld= 23,gender= "female"]
+        Type: Record(monthsOld: Integer,gender: String)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPageEmpty9}
+\begin{paste}{AssociationListXmpPageEmpty9}{AssociationListXmpPagePatch9}
+\pastebutton{AssociationListXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{al."katie" := [23,"female"]$Data\free{al5 }\bound{al6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPagePatch10}
+\begin{paste}{AssociationListXmpPageFull10}{AssociationListXmpPageEmpty10}
+\pastebutton{AssociationListXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{delete!(al,1)\free{al6 }\bound{al7 }}
+\indentrel{3}\begin{verbatim}
+   (10)
+   table
+      "katie"= [monthsOld= 23,gender= "female"]
+  ,
+      "judith"= [monthsOld= 366,gender= "female"]
+  ,
+      "bob"= [monthsOld= 407,gender= "male"]
+Type: AssociationList(String,Record(monthsOld: Integer,gender: String))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AssociationListXmpPageEmpty10}
+\begin{paste}{AssociationListXmpPageEmpty10}{AssociationListXmpPagePatch10}
+\pastebutton{AssociationListXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{delete!(al,1)\free{al6 }\bound{al7 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ANNA-ES.ht b/src/hyper/pages/ANNA-ES.ht
new file mode 100644
index 00000000..87145df6
--- /dev/null
+++ b/src/hyper/pages/ANNA-ES.ht
@@ -0,0 +1,690 @@
+\begin{page}{UXANNA}{AXIOM/NAG Expert System}
+\centerline{\tt{\inputbitmap{\htbmdir{}/anna_logo.xbm}}\rm}
+\newline
+\centerline{This expert system chooses, and uses, NAG numerical routines.}
+\begin{scroll}
+\blankline
+\indent{2}
+\beginmenu
+\menumemolink{Integration}{UXANNAInt}
+\blankline
+\menumemolink{Ordinary Differential Equations}{UXANNAOde}
+\blankline
+\menumemolink{Partial Differential Equations}{UXANNAPde}
+\blankline
+\menumemolink{Optimization}{UXANNAOpt}
+\vspace{10}
+\menumemolink{About the AXIOM/NAG Expert System}{UXANNATxt}
+%\unixcommand{(Postscript)}{ghostview \ \htbmdir{}/anna.ps}
+%\blankline
+%\menumemolink{How to use the NAGLINK}{nagLinkIntroPage}
+%\blankline
+%\menumemolink{Tutorial}{tutorialIntroPage}
+%\blankline
+%\item \menulispdownlink{Interpolation}{(|interp1|}
+\endmenu
+\indent{0}
+\end{scroll}
+%\unixcommand{Netscape}{netscape \ http:\/\/www.bath.ac.uk\/\~masbjd\/anna.html}
+\autobutt{HelpContents}
+\end{page}
+
+\begin{page}{UXANNAInt}{Integration}
+Welcome to the Integration section of {\tt
+\inputbitmap{\htbmdir{}/anna.xbm.tiny}}, the {\em AXIOM/NAG Expert
+System}.  This system chooses, and uses, NAG numerical routines.
+\begin{scroll}
+\blankline
+\indent{2}
+\beginmenu
+\item \menulispdownlink{Integration}{(|annaInt|)}\space{}\newline
+\indent{5} Integrating a function over a finite or infinite range.
+\blankline
+\item \menulispdownlink{Multiple Integration}{(|annaMInt|)}\space{}\newline
+\indent{5} Integrating a multivariate function over a finite space.  
+The dimensions of the space need to be 2 <= n <= 15.
+\blankline
+\menudownlink{Examples}{UXANNAIntEx}\space{}\newline
+\indent{5} Examples of integration.  These examples cover all of the major 
+methods.  Parameters can be changed to investigate the effect 
+on the choice of method.
+\endmenu
+\indent{0}
+\end{scroll}
+\autobutt{MainHelp}
+\end{page}
+
+\begin{page}{UXANNAOde}{Ordinary Differential Equations}
+Welcome to the Ordinary Differential Equations section of {\tt
+\inputbitmap{\htbmdir{}/anna.xbm.tiny}}, the 
+{\em AXIOM/NAG Expert System}.  
+This system chooses, and uses, NAG numerical routines.
+\begin{scroll}
+\blankline
+\blankline
+\indent{2}
+\beginmenu
+\item \menulispdownlink{Ordinary Differential Equations}{(|annaOde|)}\space{}\newline
+\indent{5} Finding a solution to an Initial Value Problem of a set of Ordinary Differential Equations.
+\blankline
+\menudownlink{Examples}{UXANNAOdeEx}\newline
+\indent{5} Examples of ODE problems with various features using both stiff
+and non-stiff methods.  Parameters can be changed to investigate the effect 
+on the choice of method.
+\autobutt{MainHelp}
+\endmenu
+\indent{0}
+\end{scroll}
+\end{page}
+
+\begin{page}{UXANNAOpt}{Optimization}
+Welcome to the Optimization section of {\tt
+\inputbitmap{\htbmdir{}/anna.xbm.tiny}}, the {\em AXIOM/NAG Expert System}.  
+This system chooses, and uses, NAG numerical routines.
+\begin{scroll}
+\blankline
+\indent{2}
+\beginmenu
+\item \menulispdownlink{Optimization of a Single Multivariate Function}
+{(|annaOpt|)}\space{}\newline
+\indent{6} Finding the minimum of a function in n variables. 
+\newline
+\indent{6} Linear Programming and Quadratic Programming problems.
+\blankline
+\indent{4} 
+\beginmenu
+\menudownlink{Examples}{UXANNAOptEx}\newline
+\indent{8} Examples of optimization problems with various constraint features.
+\endmenu
+\blankline
+\item \menulispdownlink{Optimization of a set of observations of a data set}
+{(|annaOpt2|)}\space{}\newline
+\indent{6} Least-squares problems.
+\newline
+\indent{6} Checking the goodness of fit of a least-squares model.
+\blankline
+\indent{4} 
+\beginmenu
+\menudownlink{Examples}{UXANNAOpt2Ex}\newline
+\indent{8} Examples of least squares problems.
+\endmenu
+\endmenu
+\indent{0}
+\end{scroll}
+\autobutt{MainHelp}
+\end{page}
+
+\begin{page}{UXANNAPde}{Partial Differential Equations}
+Welcome to the Partial Differential Equations section of {\tt
+\inputbitmap{\htbmdir{}/anna.xbm.tiny}}, the
+{\em AXIOM/NAG Expert System}.  
+\begin{scroll}
+\indent{2}
+\beginmenu
+\menulispdownlink{Second Order Elliptic Partial Differential Equation}{(|annaPDESolve|)}
+\newline
+\indent{4} Discretizing the PDE:
+\newline
+\centerline{\inputbitmap{\htbmdir{}/d03eef.xbm}}
+defined on a rectangular region with boundary conditions of the form 
+\centerline{\inputbitmap{\htbmdir{}/d03eef1.bitmap}} 
+and solving the resulting
+seven-diagonal finite difference equations using a multigrid technique.
+\blankline
+%\menulispdownlink{Helmholtz Equation in 3-D, Cartesian Coordinates}{(|d03fafVars|)}
+%\newline
+%\indent{4} Descretizing the PDE:
+%\newline
+%\centerline{\inputbitmap{\htbmdir{}/d03faf.xbm}}
+%and solving the resulting
+%seven-diagonal finite difference equations using a method based on the Fast
+%Fourier Transform.
+\endmenu
+\end{scroll}
+\autobutt{MainHelp}
+\end{page}
+
+\begin{page}{UXANNAOptEx}{Examples Using the AXIOM/NAG Expert System}
+\begin{scroll}
+Select any of these examples and you will be presented with a page which
+contains active areas for the function and its parameters.
+\blankline
+These parameters can be altered by selecting the area and replacing the
+default parameters by the new values. 
+\blankline
+\beginmenu
+\item \menulispdownlink{Example 1: \newline 
+\indent{2} Minimize the function: 
+\centerline{\inputbitmap{\htbmdir{}/opt1.xbm}}}{(|annaOptDefaultSolve1|)}\space{}
+\blankline
+\item \menulispdownlink{Example 2: \newline 
+\indent{2} Minimize the function: 
+\centerline{\inputbitmap{\htbmdir{}/opt2.xbm}}}{(|annaOptDefaultSolve2|)}\space{}
+\newline \indent{3} With conditions: 
+\centerline{\inputbitmap{\htbmdir{}/opt2c.xbm}}
+\blankline
+\item \menulispdownlink{Example 3: \newline 
+\indent{2} Minimize the function: 
+\centerline{\inputbitmap{\htbmdir{}/opt3.xbm}}}{(|annaOptDefaultSolve3|)}\space{}
+\newline \indent{3} With conditions: 
+\centerline{\inputbitmap{\htbmdir{}/opt3c1.xbm}}
+\centerline{\inputbitmap{\htbmdir{}/opt3c2.xbm}}
+\blankline
+\item \menulispdownlink{Example 4: \newline 
+\indent{2} Minimize the function: 
+\centerline{\inputbitmap{\htbmdir{}/opt4.xbm}}
+}{(|annaOptDefaultSolve4|)}\space{}
+\newline \indent{3} With conditions: 
+\centerline{\inputbitmap{\htbmdir{}/opt4c1.xbm}}
+\centerline{\inputbitmap{\htbmdir{}/opt4c2.xbm}}
+\centerline{\inputbitmap{\htbmdir{}/opt4c3.xbm}}
+\blankline
+\item \menulispdownlink{Example 5: \newline 
+\indent{2} Minimize the function: 
+\centerline{\inputbitmap{\htbmdir{}/opt5.xbm}}
+}{(|annaOptDefaultSolve5|)}\space{}
+\newline \indent{3} With conditions: 
+\centerline{\inputbitmap{\htbmdir{}/opt4c1.xbm}}
+\centerline{\inputbitmap{\htbmdir{}/opt4c2.xbm}}
+\centerline{\inputbitmap{\htbmdir{}/opt4c3.xbm}}
+\blankline
+\endmenu
+\end{scroll}
+\autobutt{Mainhelp}
+\end{page}
+
+\begin{page}{UXANNAOpt2Ex}{Examples Using the AXIOM/NAG Expert System}
+\begin{scroll}
+Select this example and you will be presented with a page which
+contains active areas for the function and its parameters.
+\blankline
+These parameters can be altered by selecting the area and replacing the
+default parameters by the new values. 
+\blankline
+\beginmenu
+\blankline
+\item \menulispdownlink{Example 1: \newline 
+\indent{2} Calculate a least-squares minimization of the following functions: 
+\centerline{\inputbitmap{\htbmdir{}/opt61.xbm}}
+\centerline{\inputbitmap{\htbmdir{}/opt62.xbm}}
+\centerline{\inputbitmap{\htbmdir{}/opt63.xbm}}}
+{(|annaOpt2DefaultSolve|)}\space{}
+\endmenu
+\end{scroll}
+\autobutt{MainHelp}
+\end{page}
+
+
+\begin{page}{UXANNAIntEx}{Examples Using the AXIOM/NAG Expert System}
+\begin{scroll}
+Select any of these examples and you will be presented with a page which
+contains active areas for the function and its parameters.
+\blankline
+These parameters can be altered by selecting the area and replacing the
+default parameters by the new values.  In this way you can investigate the
+effect of the new parameters on the choice of method.
+\blankline
+\beginmenu
+\item \menulispdownlink{Example 1: \newline 
+\centerline{\inputbitmap{\htbmdir{}/int1.xbm}}}{(|annaFoo|)}\space{}
+\blankline
+\item \menulispdownlink{Example 2: \newline 
+\centerline{\inputbitmap{\htbmdir{}/int2.xbm}}}{(|annaBar|)}\space{}
+\blankline
+\item \menulispdownlink{Example 3: \newline 
+\centerline{\inputbitmap{\htbmdir{}/int3.xbm}}}{(|annaJoe|)}\space{}
+\blankline
+\item \menulispdownlink{Example 4: \newline 
+\centerline{\inputbitmap{\htbmdir{}/int4.xbm}}}{(|annaSue|)}\space{}
+\blankline
+\item \menulispdownlink{Example 5: \newline 
+\centerline{\inputbitmap{\htbmdir{}/int5.xbm}}}{(|annaAnn|)}\space{}
+\blankline
+\item \menulispdownlink{Example 6: \newline 
+\centerline{\inputbitmap{\htbmdir{}/int6.xbm}}}{(|annaBab|)}\space{}
+\blankline
+\item \menulispdownlink{Example 7: \newline 
+\centerline{\inputbitmap{\htbmdir{}/int7.xbm}}}{(|annaFnar|)}\space{}
+\blankline
+\item \menulispdownlink{Example 8: \newline 
+\centerline{\inputbitmap{\htbmdir{}/int8.xbm}}}{(|annaDan|)}\space{}
+\blankline
+\item \menulispdownlink{Example 9: \newline 
+\centerline{\inputbitmap{\htbmdir{}/int9.xbm}}}{(|annaBlah|)}\space{}
+\blankline
+\item \menulispdownlink{Example 10: \newline 
+\centerline{\inputbitmap{\htbmdir{}/int10.xbm}}}{(|annaTub|)}\space{}
+\blankline
+\item \menulispdownlink{Example 11: \newline 
+\centerline{\inputbitmap{\htbmdir{}/int13.xbm}}}{(|annaRats|)}\space{}
+\blankline
+\item \menulispdownlink{Example 12: \newline 
+\centerline{\inputbitmap{\htbmdir{}/int11.xbm}}}{(|annaMInt|)}\space{}
+\endmenu
+\end{scroll}
+\autobutt{MainHelp}
+\end{page}
+
+\begin{page}{UXANNAOdeEx}{Examples Using the AXIOM/NAG Expert System}
+\begin{scroll}
+Analyses the function for various attributes, chooses and 
+then uses a suitable ODE solver to provide a 
+solution to the system of n ODEs \center{\htbitmap{d02gaf},}
+for i = 1,2,...,n.
+\blankline
+Select either of these examples and you will be presented with a page which
+contains active areas for the function and its parameters.
+\blankline
+These parameters can be altered by selecting the area and replacing the
+default parameters by the new values.  In this way you can investigate the
+effect of the new parameters on the choice of method.
+\blankline
+\beginmenu
+\item \menulispdownlink{Example 1: \tab{12}
+\inputbitmap{\htbmdir{}/ode1.xbm}}{(|annaOdeDefaultSolve1|)}
+\blankline with initial conditions: \newline
+\tab{12}\inputbitmap{\htbmdir{}/y1.xbm} \space{1} and \space{1}
+\inputbitmap{\htbmdir{}/x1.xbm}
+\blankline
+\blankline
+\blankline
+\item \menulispdownlink{Example 2: \tab{12}
+\inputbitmap{\htbmdir{}/ode2.xbm}}{(|annaOdeDefaultSolve2|)}
+\blankline with initial conditions: \newline
+\tab{12}\inputbitmap{\htbmdir{}/y2.xbm} \space{1} and \space{1}
+\inputbitmap{\htbmdir{}/x1.xbm}
+\blankline
+\blankline
+\blankline
+\endmenu
+\end{scroll}
+\autobutt{MainHelp}
+\end{page}
+
+\begin{page}{UXANNATxt}{About the AXIOM/NAG Expert System}
+\begin{scroll}
+\centerline{\tt\inputbitmap{\htbmdir{}/anna_logo.xbm}\rm} 
+\vspace{-30}\horizontalline
+In applied mathematics, electronic and chemical engineering, the modelling
+process can produce a number of mathematical problems which require numerical
+solutions for which symbolic methods are either not possible or not obvious. 
+With the plethora of numerical library routines for the solution of these
+problems often the numerical analyst has to answer the question {\em Which
+routine to choose?} 
+\blankline
+Some analysis needs to be carried out before the
+appropriate routine can be identified i.e. {\em How stiff is this ODE?} and
+{\em Is this function continuous?}  It may well be the case that more than
+one routine is applicable to the problem.  So the question may become {\em
+Which is likely to be the best?}  Such a choice may be critical for both
+accuracy and efficiency.
+\blankline
+An expert system is thus required to make this choice based on the result of
+its own analysis of the problem, call the routine and act on the outcome. 
+This may be to put the answer in a relevant form or react to an apparent
+failure of the chosen routine and thus choose and call an alternative.  
+It should also have sufficient explanation mechanisms to inform on the choice
+of routine and the reasons for that choice.
+\blankline
+\end{scroll}
+\downlink{ Examples }{UXANNAEx}
+\downlink{ Introduction }{UXANNAIntro}
+\downlink{ Decision Agents }{UXANNADec}
+\downlink{ Inference Mechanisms }{UXANNAInfer}
+\downlink{ Measure Functions }{UXANNAMeas}
+\end{page}
+
+\begin{page}{UXANNAIntro}{Introduction to the AXIOM/NAG Expert System}
+\begin{scroll}
+\centerline{\tt\inputbitmap{\htbmdir{}/anna_logo.xbm}\rm}
+\vspace{-30}\horizontalline
+Deciding amongst, and then implementing, several possible approaches to
+solving a numerical problem can be daunting for a novice user, or tedious for
+an expert.  Different attributes of the problem need to be
+identified and their possible interactions weighed up before a final decision
+about which method to use can be made. 
+\blankline
+The implementation is then largely an
+automatic, if laborious, process of writing, compiling and linking usually
+Fortran code.  The aim is to build an expert system which will use computer
+algebra to analyse such features of a problem, inference mechanisms and a
+knowledge base to choose a numerical method appropriate to the solution of a
+given problem.  
+\blankline
+Any interactive system is constrained by the need to provide a reasonable
+response time for the user.  Given the complexity of some of the analysis our
+system will need to do, it is clear that we should only aim to select a good
+method, rather than try to identify the best one available.  The overall goal
+is to provide a ``black-box'' interface to numerical software which allows
+non-experts access to its full potential.  It will also provide explanation
+mechanisms commensurate with its role as a teaching aid.
+\blankline
+Given, say, an integration to perform (which may or may not be able to be
+handled symbolically), the system should choose and apply an appropriate
+method, thus mirroring as closely as possible the way that an experienced
+numerical analyst would think so, for example, given an integration to
+perform:\newline
+{\it \centerline{\inputbitmap{\htbmdir{}/int1.xbm}}}
+\newline
+the experienced analyst would see that the integral is semi-infinite and
+transform it by splitting the range and transforming the integral over {\it
+[1,\inputbitmap{\htbmdir{}/infty.xbm}]} into an integral over 
+{\it [0,1] } using the transformation {\it x -> 1/t}.  
+A different numerical routine might be used over each
+sub-region and the results added to give the final answer. 
+\blankline
+It then requires
+the translation of the problem into Fortran code which may be extensive.
+Even with this simple example, the process is quite involved.
+\blankline
+\end{scroll}
+\autobuttons
+\downlink{ Decision Agents }{UXANNADec}
+\downlink{ Inference Mechanisms }{UXANNAInfer}
+\downlink{ Method Domains }{UXANNAMeth}
+\downlink{ Measure Functions }{UXANNAMeas}
+\end{page}
+
+\begin{page}{UXANNAEx}{Example using the AXIOM/NAG Expert System}
+\begin{scroll}
+\xtc{
+{\bf Example 1}: The integral
+{\centerline{\inputbitmap{\htbmdir{}/int1.xbm}}}
+\newline
+is performed as follows:
+\blankline
+}{}
+\xtc{
+}{
+\spadpaste{ans := integrate((exp(-X^3)+exp(-3*X^2))/sqrt(X),0.0..\%plusInfinity)\bound{ans} }
+}
+\blankline
+\xtc{
+It creates a composite structure for which the field containing the result can be 
+expanded as required.\blankline
+}{
+\spadpaste{ans . 'result\free{ans}}
+}
+\blankline
+\xtc{
+}{
+\spadpaste{ans . 'abserr\free{ans}}
+}
+\blankline
+This system has performed the analysis described above, done the necessary
+problem transformation, written any necessary Fortran, called two different
+numerical routines, and amalgamated their
+results.  This whole process was transparent to the user.
+\end{scroll}
+\autobuttons
+\downlink{Example 2}{UXANNAEx2}
+%\downlink{Decision Agents}{UXANNADec}
+\end{page}
+
+\begin{page}{UXANNAEx2}{Example using the AXIOM/NAG Expert System}
+\begin{scroll}
+\xtc{
+{\bf Example 2}: The ODE
+{\centerline{\inputbitmap{\htbmdir{}/ode3.xbm}\space{1}with
+\space{1}
+{\inputbitmap{\htbmdir{}/y3.xbm}}}}
+\newline
+could be solved as follows:
+\blankline
+}{}
+\xtc{
+}{
+\spadpaste{ans2 := solve([Y[2],-1001*Y[2]-1000*Y[1]], 0.0, 10.0,
+[1.0,-1.0], [2,4,6,8], 1.0e-4)\bound{ans2} }
+}
+\blankline
+\xtc{
+It creates a composite structure for which the field containing the result can be 
+expanded as required.\blankline
+}{
+\spadpaste{ans2 . 'result\free{ans2}}
+}
+\blankline
+\xtc{
+}{
+\spadpaste{ans2 . 'y\free{ans2}}
+}
+\blankline
+\end{scroll}
+\autobuttons
+\downlink{Example 3}{UXANNAEx3}
+%\downlink{Decision Agents}{UXANNADec}
+\end{page}
+
+\begin{page}{UXANNAEx3}{Example using the AXIOM/NAG Expert System}
+\begin{scroll}
+\xtc{
+{\bf Example 3}: The function
+{\centerline{\inputbitmap{\htbmdir{}/opt2.xbm}}}
+with simple bounds
+{\centerline{\inputbitmap{\htbmdir{}/opt2c.xbm}}}
+\newline
+could be minimized as follows:
+\blankline
+}{}
+\xtc{
+}{
+\spadpaste{ans3 := optimize((X[1]+10*X[2])**2 + 5*(X[3]-X[4])**2 + 
+(X[2]-2*X[3])**4 + 10*(X[1]-X[4])**4, [3,-1,0,1], [1,-2,\%minusInfinity,1], 
+[3,0,\%plusInfinity,3])\bound{ans3} }
+}
+\blankline
+\xtc{
+It creates a composite structure for which the field containing the minimum can be 
+expanded as required.\blankline
+}{
+\spadpaste{ans3 . objf\free{ans3}}
+}
+\blankline
+\xtc{
+}{
+\spadpaste{ans3 . x\free{ans3}}
+}
+\blankline
+\xtc{
+}{
+\spadpaste{ans3 . attributes\free{ans3}}
+}
+\blankline
+\end{scroll}
+\autobuttons
+\downlink{Decision Agents}{UXANNADec}
+\end{page}
+
+\begin{page}{UXANNADec}{Decision Agents}
+\begin{scroll}
+\blankline
+Some features are either present or absent in a problem.  Examples of such
+binary decisions include {\em is a matrix symmetric?} and {\em is a
+function continuous?}  However in practice many questions are about the {\em
+degree} to which a problem exhibits a property: {\em how much does a
+function oscillate?}, or {\em how stiff are these differential equations?}
+\blankline
+We have therefore created decision agents of two types, reflecting their
+property --- {\em Binary Agents} are Boolean functions returning either true
+or false and {\em Intensity Functions} are quantitative and return a range of
+different values, either numerical or structured types. The framework we are
+developing is able to deal with both these forms of information.  
+\blankline
+
+In any given problem area (for example solving ordinary differential
+equations, optimization etc.) we have a selection of {\em methods}.  These
+might be to use a particular NAG routine, or they might involve employing a
+higher-level strategy such as transforming the problem into an equivalent,
+but easier to solve, form.  
+\blankline
+Associated with every method we define a {\em
+measure function} which assesses the suitability of that method to a
+particular problem.  Each measure function has access to a range of symbolic
+{\em agents} which can answer questions about the various properties of the
+problem in hand.
+\blankline
+\end{scroll}
+\downlink{ Inference Mechanisms }{UXANNAInfer}
+\downlink{ Method Domains }{UXANNAMeth}
+\downlink{ Measure Functions }{UXANNAMeas}
+\downlink{ Computational Agents }{UXANNAAgent}
+
+\end{page}
+
+\begin{page}{UXANNAInfer}{Inference Mechanisms}
+\begin{scroll}
+\blankline
+The inference machine will take the problem description as provided by the
+user and perform an initial analysis to verify its validity.  It will
+consider, in turn, all of the available methods within its knowledge base
+which might solve that problem.  In doing so it analyses the input problem to
+find out about any attributes that could affect the ability of the methods
+under consideration to perform effectively.  
+\blankline
+Some of these
+measures may use lazy evaluation in the sense that, if a method already
+assessed is believed to be a good candidate, and if evaluating the current
+measure will be relatively expensive, then that measure will not be evaluated
+unless later evidence shows that the selected method is not, in fact, a
+successful strategy, for example if it has failed.
+\end{scroll}
+\downlink{ Method Domains }{UXANNAMeth}
+\downlink{ Measure Functions }{UXANNAMeas}
+\downlink{ Computational Agents }{UXANNAAgent}
+\downlink{ Examples }{UXANNAEx}
+\end{page}
+
+\begin{page}{UXANNAMeth}{Method Domains}
+\begin{scroll}
+\blankline
+An AXIOM {\em domain} has been created for each method or strategy for
+solving the problem.  These method domains each implement two functions with
+a uniform (method independant) interface.
+\blankline {\bf measure:} A function which calculates an estimate of suitability of
+this particular method to the problem if there is a possibility that the
+method under consideration is more appropriate than one already investigated.
+\blankline
+If it may be possible to improve on the current favourite method, the function
+will call computational agents to analyse the problem for specific features
+and calculate the measure from the results these agents return, 
+using a variation on the Lucks/Gladwell intensity and compatibility
+model if conflict between attributes, as investigated by these computational
+agents, may be present.
+\blankline
+{\bf implementation:} A function which may be one of two distinct kinds. 
+The first kind uses the interface to the NAG Library to call a particular 
+routine with the required parameters.  Some of the parameters may need to be
+calculated from the data provided before the external function call.  
+\blankline
+The other kind will apply a ``high level'' strategy to try to solve the
+problem e.g.~a transformation of an expression from one that is difficult to
+solve to one which is easier, or a splitting of the problem into several more
+easily solvable parts.  For example, for a solution of the equation above,
+since the integral is semi-infinite we might wish to transform the range by,
+say, using the mapping {\it y -> 1/x} on the section {\it 1
+< x < \inputbitmap{\htbmdir{}/infty.xbm}}) and
+adding the result to the unmapped section {\it 0 < x < 1}.
+\blankline
+\end{scroll}
+\downlink{ Measure Functions }{UXANNAMeas}
+\downlink{ Computational Agents }{UXANNAAgent}
+\downlink{ Examples }{UXANNAEx}
+\end{page}
+
+\begin{page}{UXANNAMeas}{Measure Functions}
+\begin{scroll}
+\blankline
+Each measure function will estimate the ability of a particular method to
+solve a problem.  It will consult whichever agents are needed to perform
+analysis on the problem in order to calculate the measure.  There is a
+parameter which would contain the best compatibility
+value found so far.  
+\blankline
+However, the interpretation we give to the results of some tests is not
+always clear-cut.  If a set of tests give
+conflicting advice as to the appropriateness of a particular method, it
+becomes important to decide not only {\it whether} certain properties are
+present but also their {\it degree}.  This gives us a basis for estimating the
+compatibility of each property.  
+\blankline
+We have taken for our model the system recommended by Lucks and Gladwell
+which uses a system of measurement of compatibility allowing for interaction
+and conflict between a number of attributes.  All of these processes may not
+be required if the choice is clear-cut e.g. we have an integral to calculate
+which has a particular singularity structure for which one particular method
+has been specifically constructed.  However, for more difficult cases a
+composite picture should be built up to calculate a true measurement. 
+\blankline
+How the compatibility functions interpret the measurements of various
+attributes is up to them and may vary between differing methods.  It is this
+area that takes as its basis the {\it judgement} of Numerical Analysis
+`experts' whether that be from the documentation (which may be deficient in
+certain respects) or from alternative sources.  However, its assessment of
+the suitability or otherwise of a particular method is reflected in a single
+normalised value facilitating the direct comparison of the suitability of a
+number of possible methods.
+\blankline
+\end{scroll}
+\downlink{ Computational Agents }{UXANNAAgent}
+\downlink{ Examples }{UXANNAEx}
+\end{page}
+
+\begin{page}{UXANNAAgent}{Computational Agents}
+\begin{scroll}
+\blankline
+Computational Agents are those program segments which investigate the
+attributes of the input function or functions, such as 
+{\bf stiffnessAndStabilityOfODEIF}
+(the {\em IF} indicates that it is an {\em Intensity Function} i.e. one that
+returns a normalised real number or a set of normalised real numbers).  They
+are usually functions or programs written completely in the \Language{}
+language and implemented using computer algebra.
+\blankline
+Some agents will be common to more than one problem domain whereas others
+will be specific to a single domain.  They also vary greatly in their
+complexity.  It is a fairly simple task to return details about the range of
+a function since this information will have been included in the problem
+specification.  It is a different order of complexity to return details of
+its singularity structure.
+\blankline
+\xtc{
+As an example, here is a call to the computational agent {\bf
+singularitiesOf} to obtain the list of singularities of the function 
+{\it tan x} which are in the range 
+{\it 0..12\inputbitmap{\htbmdir{}/pi.xbm}}:
+\blankline
+}{
+}
+\xtc{
+}{
+\spadpaste{s := singularitiesOf(tan x,[x],0..12*\%pi)$ESCONT \free{lib3} }
+}
+\blankline
+Each of these computational agents which may be called by a number of method
+domains retain their output in a dynamic hash-table, so speeding the process
+and retaining efficiency.
+\end{scroll}
+\downlink{ Examples }{UXANNAEx}
+\end{page}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/src/hyper/pages/ANNA-ES.pht b/src/hyper/pages/ANNA-ES.pht
new file mode 100644
index 00000000..67294731
--- /dev/null
+++ b/src/hyper/pages/ANNA-ES.pht
@@ -0,0 +1,205 @@
+\begin{patch}{UXANNAExPatch1}
+\begin{paste}{UXANNAExFull1}{UXANNAExEmpty1}
+\pastebutton{UXANNAExFull1}{\hidepaste}
+\tab{5}\spadcommand{ans := integrate((exp(-X^3)+exp(-3*X^2))/sqrt(X),0.0..\%plusInfinity)\bound{ans }}
+\indentrel{3}\begin{verbatim}
+   (1)
+   [d01ajfAnnaTypeAnswer: Result, abserr: DoubleFloat,
+    method: Result, result: DoubleFloat,
+    explanations: List(String), attributes: List(Any),
+    d01apfAnnaTypeAnswer: Result]
+                                           Type: Result
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UXANNAExEmpty1}
+\begin{paste}{UXANNAExEmpty1}{UXANNAExPatch1}
+\pastebutton{UXANNAExEmpty1}{\showpaste}
+\tab{5}\spadcommand{ans := integrate((exp(-X^3)+exp(-3*X^2))/sqrt(X),0.0..\%plusInfinity)\bound{ans }}
+\end{paste}\end{patch}
+
+\begin{patch}{UXANNAExPatch2}
+\begin{paste}{UXANNAExFull2}{UXANNAExEmpty2}
+\pastebutton{UXANNAExFull2}{\hidepaste}
+\tab{5}\spadcommand{ans . 'result\free{ans }}
+\indentrel{3}\begin{verbatim}
+   (2)  3.23287256251958
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UXANNAExEmpty2}
+\begin{paste}{UXANNAExEmpty2}{UXANNAExPatch2}
+\pastebutton{UXANNAExEmpty2}{\showpaste}
+\tab{5}\spadcommand{ans . 'result\free{ans }}
+\end{paste}\end{patch}
+
+\begin{patch}{UXANNAExPatch3}
+\begin{paste}{UXANNAExFull3}{UXANNAExEmpty3}
+\pastebutton{UXANNAExFull3}{\hidepaste}
+\tab{5}\spadcommand{ans . 'abserr\free{ans }}
+\indentrel{3}\begin{verbatim}
+   (3)  2.69960156338737e-08
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UXANNAExEmpty3}
+\begin{paste}{UXANNAExEmpty3}{UXANNAExPatch3}
+\pastebutton{UXANNAExEmpty3}{\showpaste}
+\tab{5}\spadcommand{ans . 'abserr\free{ans }}
+\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx2Patch1}
+\begin{paste}{UXANNAEx2Full1}{UXANNAEx2Empty1}
+\pastebutton{UXANNAEx2Full1}{\hidepaste}
+\tab{5}\spadcommand{ans2 := solve([Y[2],-1001*Y[2]-1000*Y[1]], 0.0, 10.0, [1.0,-1.0], [2,4,6,8], 1.0e-4)\bound{ans2 }}
+\indentrel{3}\begin{verbatim}
+   (1)
+   [ifail: Integer, intensityFunctions: List(String),
+    tol: DoubleFloat, result: Matrix(DoubleFloat),
+    y: Matrix(DoubleFloat), method: Result,
+    explanations: List(String), x: DoubleFloat]
+                                           Type: Result
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx2Empty1}
+\begin{paste}{UXANNAEx2Empty1}{UXANNAEx2Patch1}
+\pastebutton{UXANNAEx2Empty1}{\showpaste}
+\tab{5}\spadcommand{ans2 := solve([Y[2],-1001*Y[2]-1000*Y[1]], 0.0, 10.0, [1.0,-1.0], [2,4,6,8], 1.0e-4)\bound{ans2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx2Patch2}
+\begin{paste}{UXANNAEx2Full2}{UXANNAEx2Empty2}
+\pastebutton{UXANNAEx2Full2}{\hidepaste}
+\tab{5}\spadcommand{ans2 . 'result\free{ans2 }}
+\indentrel{3}\begin{verbatim}
+        Ú        1.0                   - 1.0         ¿
+        ³                                            ³
+        ³ 0.135329965475963     - 0.135329965475963  ³
+        ³                                            ³
+   (2)  ³ 0.018315370486087     - 0.0183153704860871 ³
+        ³                                            ³
+        ³0.00247869592278824   - 0.00247869592278825 ³
+        ³                                            ³
+        À0.000335497710871914  - 0.000335497710871913Ù
+                               Type: Matrix DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx2Empty2}
+\begin{paste}{UXANNAEx2Empty2}{UXANNAEx2Patch2}
+\pastebutton{UXANNAEx2Empty2}{\showpaste}
+\tab{5}\spadcommand{ans2 . 'result\free{ans2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx2Patch3}
+\begin{paste}{UXANNAEx2Full3}{UXANNAEx2Empty3}
+\pastebutton{UXANNAEx2Full3}{\hidepaste}
+\tab{5}\spadcommand{ans2 . 'y\free{ans2 }}
+\indentrel{3}\begin{verbatim}
+   (3)  [4.54002176643708e-05  - 4.54002176643708e-05]
+                               Type: Matrix DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx2Empty3}
+\begin{paste}{UXANNAEx2Empty3}{UXANNAEx2Patch3}
+\pastebutton{UXANNAEx2Empty3}{\showpaste}
+\tab{5}\spadcommand{ans2 . 'y\free{ans2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx3Patch1}
+\begin{paste}{UXANNAEx3Full1}{UXANNAEx3Empty1}
+\pastebutton{UXANNAEx3Full1}{\hidepaste}
+\tab{5}\spadcommand{ans3 := optimize((X[1]+10*X[2])**2 + 5*(X[3]-X[4])**2 + (X[2]-2*X[3])**4 + 10*(X[1]-X[4])**4, [3,-1,0,1], [1,-2,\%minusInfinity,1], [3,0,\%plusInfinity,3])\bound{ans3 }}
+\indentrel{3}\begin{verbatim}
+   (1)
+   [ifail: Integer, bl: Matrix(DoubleFloat),
+    bu: Matrix(DoubleFloat), method: Result,
+    attributes: List(String), explanations: List(String),
+    x: Matrix(DoubleFloat), objf: DoubleFloat]
+                                           Type: Result
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx3Empty1}
+\begin{paste}{UXANNAEx3Empty1}{UXANNAEx3Patch1}
+\pastebutton{UXANNAEx3Empty1}{\showpaste}
+\tab{5}\spadcommand{ans3 := optimize((X[1]+10*X[2])**2 + 5*(X[3]-X[4])**2 + (X[2]-2*X[3])**4 + 10*(X[1]-X[4])**4, [3,-1,0,1], [1,-2,\%minusInfinity,1], [3,0,\%plusInfinity,3])\bound{ans3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx3Patch2}
+\begin{paste}{UXANNAEx3Full2}{UXANNAEx3Empty2}
+\pastebutton{UXANNAEx3Full2}{\hidepaste}
+\tab{5}\spadcommand{ans3 . objf\free{ans3 }}
+\indentrel{3}\begin{verbatim}
+   (2)  2.43378751212073
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx3Empty2}
+\begin{paste}{UXANNAEx3Empty2}{UXANNAEx3Patch2}
+\pastebutton{UXANNAEx3Empty2}{\showpaste}
+\tab{5}\spadcommand{ans3 . objf\free{ans3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx3Patch3}
+\begin{paste}{UXANNAEx3Full3}{UXANNAEx3Empty3}
+\pastebutton{UXANNAEx3Full3}{\hidepaste}
+\tab{5}\spadcommand{ans3 . x\free{ans3 }}
+\indentrel{3}\begin{verbatim}
+   (3)
+   [1.0  - 0.0852325900999037  0.409303588204477  1.0]
+                               Type: Matrix DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx3Empty3}
+\begin{paste}{UXANNAEx3Empty3}{UXANNAEx3Patch3}
+\pastebutton{UXANNAEx3Empty3}{\showpaste}
+\tab{5}\spadcommand{ans3 . x\free{ans3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx3Patch4}
+\begin{paste}{UXANNAEx3Full4}{UXANNAEx3Empty4}
+\pastebutton{UXANNAEx3Full4}{\hidepaste}
+\tab{5}\spadcommand{ans3 . attributes\free{ans3 }}
+\indentrel{3}\begin{verbatim}
+   (4)
+   ["The object function is non-linear",
+    "There are simple bounds on the variables",
+    "There are no constraint functions"]
+                                      Type: List String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UXANNAEx3Empty4}
+\begin{paste}{UXANNAEx3Empty4}{UXANNAEx3Patch4}
+\pastebutton{UXANNAEx3Empty4}{\showpaste}
+\tab{5}\spadcommand{ans3 . attributes\free{ans3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{UXANNAAgentPatch1}
+\begin{paste}{UXANNAAgentFull1}{UXANNAAgentEmpty1}
+\pastebutton{UXANNAAgentFull1}{\hidepaste}
+\tab{5}\spadcommand{s := singularitiesOf(tan x,[x],0..12*\%pi)$ESCONT\free{lib3 }}
+\indentrel{3}\begin{verbatim}
+   (1)
+   [1.5707963267948966, 4.7123889803846897,
+    7.8539816339744828, 10.995574287564276,
+    14.137166941154069, 17.27875959474386,
+    20.420352248333657, 23.561944901923447,
+    26.703537555513243, 29.845130209103033, ...]
+                               Type: Stream DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UXANNAAgentEmpty1}
+\begin{paste}{UXANNAAgentEmpty1}{UXANNAAgentPatch1}
+\pastebutton{UXANNAAgentEmpty1}{\showpaste}
+\tab{5}\spadcommand{s := singularitiesOf(tan x,[x],0..12*\%pi)$ESCONT\free{lib3 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ARRAY1.ht b/src/hyper/pages/ARRAY1.ht
new file mode 100644
index 00000000..c2199a05
--- /dev/null
+++ b/src/hyper/pages/ARRAY1.ht
@@ -0,0 +1,72 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\OneDimensionalArrayXmpTitle}{OneDimensionalArray}
+\newcommand{\OneDimensionalArrayXmpNumber}{9.57}
+%
+% =====================================================================
+\begin{page}{OneDimensionalArrayXmpPage}{9.57 OneDimensionalArray}
+% =====================================================================
+\beginscroll
+The \spadtype{OneDimensionalArray} domain is used for storing data in a
+one-dimensional indexed data structure.
+Such an array is a homogeneous data structure in that all the entries of
+the array must belong to the same \Language{} domain.
+Each array has a fixed length specified by the user and arrays are not
+extensible.
+The indexing of one-dimensional arrays is one-based.
+This means that the ``first'' element of an array is given the index
+\spad{1}.
+See also \downlink{`Vector'}{VectorXmpPage}\ignore{Vector} and \downlink{`FlexibleArray'}{FlexibleArrayXmpPage}\ignore{FlexibleArray}.
+\xtc{
+To create a one-dimensional array, apply the
+operation \spadfun{oneDimensionalArray} to a list.
+}{
+\spadpaste{oneDimensionalArray [i**2 for i in 1..10]}
+}
+\xtc{
+Another approach is to first create \spad{a}, a one-dimensional array of 10 \spad{0}'s.
+\spadtype{OneDimensionalArray} has the convenient abbreviation \spadtype{ARRAY1}.
+}{
+\spadpaste{a : ARRAY1 INT := new(10,0)\bound{a}}
+}
+\xtc{
+Set each \spad{i}th element to i, then display the result.
+}{
+\spadpaste{for i in 1..10 repeat a.i := i; a\bound{a1}\free{a}}
+}
+\xtc{
+Square each element by mapping the function
+\texht{$i \mapsto i^2$}{i +-> i**2} onto each element.
+}{
+\spadpaste{map!(i +-> i ** 2,a); a\bound{a3}\free{a2}}
+}
+\xtc{
+Reverse the elements in place.
+}{
+\spadpaste{reverse! a\bound{a4}\free{a3}}
+}
+\xtc{
+Swap the \spad{4}th and \spad{5}th element.
+}{
+\spadpaste{swap!(a,4,5); a\bound{a5}\free{a4}}
+}
+\xtc{
+Sort the elements in place.
+}{
+\spadpaste{sort! a \bound{a6}\free{a5}}
+}
+\xtc{
+Create a new one-dimensional array \spad{b} containing the last 5 elements of \spad{a}.
+}{
+\spadpaste{b := a(6..10)\bound{b}\free{a6}}
+}
+\xtc{
+Replace the first 5 elements of \spad{a} with those of \spad{b}.
+}{
+\spadpaste{copyInto!(a,b,1)\free{b}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ARRAY1.pht b/src/hyper/pages/ARRAY1.pht
new file mode 100644
index 00000000..1a2079ad
--- /dev/null
+++ b/src/hyper/pages/ARRAY1.pht
@@ -0,0 +1,144 @@
+\begin{patch}{OneDimensionalArrayXmpPagePatch1}
+\begin{paste}{OneDimensionalArrayXmpPageFull1}{OneDimensionalArrayXmpPageEmpty1}
+\pastebutton{OneDimensionalArrayXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{oneDimensionalArray [i**2 for i in 1..10]}
+\indentrel{3}\begin{verbatim}
+   (1)  [1,4,9,16,25,36,49,64,81,100]
+              Type: OneDimensionalArray PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPageEmpty1}
+\begin{paste}{OneDimensionalArrayXmpPageEmpty1}{OneDimensionalArrayXmpPagePatch1}
+\pastebutton{OneDimensionalArrayXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{oneDimensionalArray [i**2 for i in 1..10]}
+\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPagePatch2}
+\begin{paste}{OneDimensionalArrayXmpPageFull2}{OneDimensionalArrayXmpPageEmpty2}
+\pastebutton{OneDimensionalArrayXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{a : ARRAY1 INT := new(10,0)\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (2)  [0,0,0,0,0,0,0,0,0,0]
+                      Type: OneDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPageEmpty2}
+\begin{paste}{OneDimensionalArrayXmpPageEmpty2}{OneDimensionalArrayXmpPagePatch2}
+\pastebutton{OneDimensionalArrayXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{a : ARRAY1 INT := new(10,0)\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPagePatch3}
+\begin{paste}{OneDimensionalArrayXmpPageFull3}{OneDimensionalArrayXmpPageEmpty3}
+\pastebutton{OneDimensionalArrayXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{for i in 1..10 repeat a.i := i; a\bound{a1 }\free{a }}
+\indentrel{3}\begin{verbatim}
+   (3)  [1,2,3,4,5,6,7,8,9,10]
+                      Type: OneDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPageEmpty3}
+\begin{paste}{OneDimensionalArrayXmpPageEmpty3}{OneDimensionalArrayXmpPagePatch3}
+\pastebutton{OneDimensionalArrayXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{for i in 1..10 repeat a.i := i; a\bound{a1 }\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPagePatch4}
+\begin{paste}{OneDimensionalArrayXmpPageFull4}{OneDimensionalArrayXmpPageEmpty4}
+\pastebutton{OneDimensionalArrayXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{map!(i +-> i ** 2,a); a\bound{a3 }\free{a2 }}
+\indentrel{3}\begin{verbatim}
+   (4)  [1,4,9,16,25,36,49,64,81,100]
+                      Type: OneDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPageEmpty4}
+\begin{paste}{OneDimensionalArrayXmpPageEmpty4}{OneDimensionalArrayXmpPagePatch4}
+\pastebutton{OneDimensionalArrayXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{map!(i +-> i ** 2,a); a\bound{a3 }\free{a2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPagePatch5}
+\begin{paste}{OneDimensionalArrayXmpPageFull5}{OneDimensionalArrayXmpPageEmpty5}
+\pastebutton{OneDimensionalArrayXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{reverse! a\bound{a4 }\free{a3 }}
+\indentrel{3}\begin{verbatim}
+   (5)  [100,81,64,49,36,25,16,9,4,1]
+                      Type: OneDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPageEmpty5}
+\begin{paste}{OneDimensionalArrayXmpPageEmpty5}{OneDimensionalArrayXmpPagePatch5}
+\pastebutton{OneDimensionalArrayXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{reverse! a\bound{a4 }\free{a3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPagePatch6}
+\begin{paste}{OneDimensionalArrayXmpPageFull6}{OneDimensionalArrayXmpPageEmpty6}
+\pastebutton{OneDimensionalArrayXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{swap!(a,4,5); a\bound{a5 }\free{a4 }}
+\indentrel{3}\begin{verbatim}
+   (6)  [100,81,64,36,49,25,16,9,4,1]
+                      Type: OneDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPageEmpty6}
+\begin{paste}{OneDimensionalArrayXmpPageEmpty6}{OneDimensionalArrayXmpPagePatch6}
+\pastebutton{OneDimensionalArrayXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{swap!(a,4,5); a\bound{a5 }\free{a4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPagePatch7}
+\begin{paste}{OneDimensionalArrayXmpPageFull7}{OneDimensionalArrayXmpPageEmpty7}
+\pastebutton{OneDimensionalArrayXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{sort! a\bound{a6 }\free{a5 }}
+\indentrel{3}\begin{verbatim}
+   (7)  [1,4,9,16,25,36,49,64,81,100]
+                      Type: OneDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPageEmpty7}
+\begin{paste}{OneDimensionalArrayXmpPageEmpty7}{OneDimensionalArrayXmpPagePatch7}
+\pastebutton{OneDimensionalArrayXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{sort! a\bound{a6 }\free{a5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPagePatch8}
+\begin{paste}{OneDimensionalArrayXmpPageFull8}{OneDimensionalArrayXmpPageEmpty8}
+\pastebutton{OneDimensionalArrayXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{b := a(6..10)\bound{b }\free{a6 }}
+\indentrel{3}\begin{verbatim}
+   (8)  [36,49,64,81,100]
+                      Type: OneDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPageEmpty8}
+\begin{paste}{OneDimensionalArrayXmpPageEmpty8}{OneDimensionalArrayXmpPagePatch8}
+\pastebutton{OneDimensionalArrayXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{b := a(6..10)\bound{b }\free{a6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPagePatch9}
+\begin{paste}{OneDimensionalArrayXmpPageFull9}{OneDimensionalArrayXmpPageEmpty9}
+\pastebutton{OneDimensionalArrayXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{copyInto!(a,b,1)\free{b }}
+\indentrel{3}\begin{verbatim}
+   (9)  [36,49,64,81,100,36,49,64,81,100]
+                      Type: OneDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OneDimensionalArrayXmpPageEmpty9}
+\begin{paste}{OneDimensionalArrayXmpPageEmpty9}{OneDimensionalArrayXmpPagePatch9}
+\pastebutton{OneDimensionalArrayXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{copyInto!(a,b,1)\free{b }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ARRAY2.ht b/src/hyper/pages/ARRAY2.ht
new file mode 100644
index 00000000..8e53f226
--- /dev/null
+++ b/src/hyper/pages/ARRAY2.ht
@@ -0,0 +1,169 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\TwoDimensionalArrayXmpTitle}{TwoDimensionalArray}
+\newcommand{\TwoDimensionalArrayXmpNumber}{9.82}
+%
+% =====================================================================
+\begin{page}{TwoDimensionalArrayXmpPage}{9.82 TwoDimensionalArray}
+% =====================================================================
+\beginscroll
+
+The \spadtype{TwoDimensionalArray} domain is used for storing data in a
+%-% \HDindex{array!two-dimensional}{TwoDimensionalArrayXmpPage}{9.82}{TwoDimensionalArray}
+\twodim{} data structure indexed by row and by column.
+Such an array is a homogeneous data structure in that all the entries of
+the array must belong to the same \Language{} domain (although see
+\downlink{``\ugTypesAnyNoneTitle''}{ugTypesAnyNonePage} in Section \ugTypesAnyNoneNumber\ignore{ugTypesAnyNone}).
+Each array has a fixed number of rows and columns specified by the user
+and arrays are not extensible.
+In \Language{}, the indexing of two-dimensional arrays is one-based.
+This means that both the ``first'' row of an array and the ``first''
+column of an array are given the index \spad{1}.
+Thus, the entry in the upper left corner of an array is in position
+\spad{(1,1)}.
+
+
+The operation \spadfunFrom{new}{TwoDimensionalArray} creates
+an array with a specified number of rows and columns and fills the components
+of that array with a specified entry.
+The arguments of this operation specify the number of rows, the number
+of columns, and the entry.
+\xtc{
+This creates a five-by-four array of integers, all of whose entries are
+zero.
+}{
+\spadpaste{arr : ARRAY2 INT := new(5,4,0) \bound{arr}}
+}
+The entries of this array can be set to other integers using
+the operation \spadfunFrom{setelt}{TwoDimensionalArray}.
+
+\xtc{
+Issue this to set the element in the upper left corner of this array to
+\spad{17}.
+}{
+\spadpaste{setelt(arr,1,1,17) \free{arr}\bound{arr1}}
+}
+\xtc{
+Now the first element of the array is \spad{17.}
+}{
+\spadpaste{arr \free{arr1}}
+}
+\xtc{
+Likewise, elements of an array are extracted using the operation
+\spadfunFrom{elt}{TwoDimensionalArray}.
+}{
+\spadpaste{elt(arr,1,1) \free{arr1}}
+}
+\xtc{
+Another way to use these two operations is as follows.
+This sets the element in position \spad{(3,2)} of the array to \spad{15}.
+}{
+\spadpaste{arr(3,2) := 15 \free{arr1}\bound{arr2}}
+}
+\xtc{
+This extracts the element in position \spad{(3,2)} of the array.
+}{
+\spadpaste{arr(3,2) \free{arr2}}
+}
+The operations \spadfunFrom{elt}{TwoDimensionalArray} and
+\spadfunFrom{setelt}{TwoDimensionalArray} come equipped with an error
+check which verifies that the indices are in the proper ranges.
+For example, the above array has five rows and four columns, so if you ask
+for the entry in position \spad{(6,2)} with \spad{arr(6,2)} \Language{}
+displays an error message.
+If there is no need for an error check, you can call the operations
+\spadfunFrom{qelt}{TwoDimensionalArray} and
+\spadfunFromX{qsetelt}{TwoDimensionalArray} which provide the same
+functionality but without the error check.
+Typically, these operations are called in well-tested programs.
+
+\xtc{
+The operations \spadfunFrom{row}{TwoDimensionalArray} and
+\spadfunFrom{column}{TwoDimensionalArray} extract rows and columns,
+respectively, and return objects of \spadtype{OneDimensionalArray} with
+the same underlying element type.
+}{
+\spadpaste{row(arr,1) \free{arr2}}
+}
+\xtc{
+}{
+\spadpaste{column(arr,1) \free{arr2}}
+}
+
+\xtc{
+You can determine the dimensions of an array by calling the
+operations \spadfunFrom{nrows}{TwoDimensionalArray} and
+\spadfunFrom{ncols}{TwoDimensionalArray},
+which return the number of rows and columns, respectively.
+}{
+\spadpaste{nrows(arr) \free{arr2}}
+}
+\xtc{
+}{
+\spadpaste{ncols(arr) \free{arr2}}
+}
+\xtc{
+To apply an operation to every element of an array, use
+\spadfunFrom{map}{TwoDimensionalArray}.
+This creates a new array.
+This expression negates every element.
+}{
+\spadpaste{map(-,arr) \free{arr2}}
+}
+\xtc{
+This creates an array where all the elements are doubled.
+}{
+\spadpaste{map((x +-> x + x),arr) \free{arr2}}
+}
+\xtc{
+To change the array destructively, use
+\spadfunFromX{map}{TwoDimensionalArray} instead of
+\spadfunFrom{map}{TwoDimensionalArray}.
+If you need to make a copy of any array, use
+\spadfunFrom{copy}{TwoDimensionalArray}.
+}{
+\spadpaste{arrc := copy(arr) \bound{arrc}\free{arr2}}
+}
+\xtc{
+}{
+\spadpaste{map!(-,arrc) \free{arrc}}
+}
+\xtc{
+}{
+\spadpaste{arrc \free{arrc}}
+}
+\xtc{
+}{
+\spadpaste{arr  \free{arr2}}
+}
+
+\xtc{
+Use \spadfunFrom{member?}{TwoDimensionalArray} to see if a given element
+is in an array.
+}{
+\spadpaste{member?(17,arr) \free{arr2}}
+}
+\xtc{
+}{
+\spadpaste{member?(10317,arr) \free{arr2}}
+}
+\xtc{
+To see how many times an element appears in an array, use
+\spadfunFrom{count}{TwoDimensionalArray}.
+}{
+\spadpaste{count(17,arr) \free{arr2}}
+}
+\xtc{
+}{
+\spadpaste{count(0,arr) \free{arr2}}
+}
+
+For more information about the operations available for
+\spadtype{TwoDimensionalArray}, issue \spadcmd{)show
+TwoDimensionalArray}.
+For information on related topics, see \downlink{`Matrix'}{MatrixXmpPage}\ignore{Matrix} and
+\downlink{`OneDimensionalArray'}{OneDimensionalArrayXmpPage}\ignore{OneDimensionalArray}.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ARRAY2.pht b/src/hyper/pages/ARRAY2.pht
new file mode 100644
index 00000000..bc797ee8
--- /dev/null
+++ b/src/hyper/pages/ARRAY2.pht
@@ -0,0 +1,384 @@
+\begin{patch}{TwoDimensionalArrayXmpPagePatch1}
+\begin{paste}{TwoDimensionalArrayXmpPageFull1}{TwoDimensionalArrayXmpPageEmpty1}
+\pastebutton{TwoDimensionalArrayXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{arr : ARRAY2 INT := new(5,4,0)\bound{arr }}
+\indentrel{3}\begin{verbatim}
+        Ú0  0  0  0¿
+        ³          ³
+        ³0  0  0  0³
+        ³          ³
+   (1)  ³0  0  0  0³
+        ³          ³
+        ³0  0  0  0³
+        ³          ³
+        À0  0  0  0Ù
+                      Type: TwoDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty1}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty1}{TwoDimensionalArrayXmpPagePatch1}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{arr : ARRAY2 INT := new(5,4,0)\bound{arr }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch2}
+\begin{paste}{TwoDimensionalArrayXmpPageFull2}{TwoDimensionalArrayXmpPageEmpty2}
+\pastebutton{TwoDimensionalArrayXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{setelt(arr,1,1,17)\free{arr }\bound{arr1 }}
+\indentrel{3}\begin{verbatim}
+   (2)  17
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty2}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty2}{TwoDimensionalArrayXmpPagePatch2}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{setelt(arr,1,1,17)\free{arr }\bound{arr1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch3}
+\begin{paste}{TwoDimensionalArrayXmpPageFull3}{TwoDimensionalArrayXmpPageEmpty3}
+\pastebutton{TwoDimensionalArrayXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{arr\free{arr1 }}
+\indentrel{3}\begin{verbatim}
+        Ú17  0  0  0¿
+        ³           ³
+        ³0   0  0  0³
+        ³           ³
+   (3)  ³0   0  0  0³
+        ³           ³
+        ³0   0  0  0³
+        ³           ³
+        À0   0  0  0Ù
+                      Type: TwoDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty3}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty3}{TwoDimensionalArrayXmpPagePatch3}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{arr\free{arr1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch4}
+\begin{paste}{TwoDimensionalArrayXmpPageFull4}{TwoDimensionalArrayXmpPageEmpty4}
+\pastebutton{TwoDimensionalArrayXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{elt(arr,1,1)\free{arr1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  17
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty4}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty4}{TwoDimensionalArrayXmpPagePatch4}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{elt(arr,1,1)\free{arr1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch5}
+\begin{paste}{TwoDimensionalArrayXmpPageFull5}{TwoDimensionalArrayXmpPageEmpty5}
+\pastebutton{TwoDimensionalArrayXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{arr(3,2) := 15\free{arr1 }\bound{arr2 }}
+\indentrel{3}\begin{verbatim}
+   (5)  15
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty5}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty5}{TwoDimensionalArrayXmpPagePatch5}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{arr(3,2) := 15\free{arr1 }\bound{arr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch6}
+\begin{paste}{TwoDimensionalArrayXmpPageFull6}{TwoDimensionalArrayXmpPageEmpty6}
+\pastebutton{TwoDimensionalArrayXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{arr(3,2)\free{arr2 }}
+\indentrel{3}\begin{verbatim}
+   (6)  15
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty6}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty6}{TwoDimensionalArrayXmpPagePatch6}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{arr(3,2)\free{arr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch7}
+\begin{paste}{TwoDimensionalArrayXmpPageFull7}{TwoDimensionalArrayXmpPageEmpty7}
+\pastebutton{TwoDimensionalArrayXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{row(arr,1)\free{arr2 }}
+\indentrel{3}\begin{verbatim}
+   (7)  [17,0,0,0]
+                      Type: OneDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty7}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty7}{TwoDimensionalArrayXmpPagePatch7}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{row(arr,1)\free{arr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch8}
+\begin{paste}{TwoDimensionalArrayXmpPageFull8}{TwoDimensionalArrayXmpPageEmpty8}
+\pastebutton{TwoDimensionalArrayXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{column(arr,1)\free{arr2 }}
+\indentrel{3}\begin{verbatim}
+   (8)  [17,0,0,0,0]
+                      Type: OneDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty8}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty8}{TwoDimensionalArrayXmpPagePatch8}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{column(arr,1)\free{arr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch9}
+\begin{paste}{TwoDimensionalArrayXmpPageFull9}{TwoDimensionalArrayXmpPageEmpty9}
+\pastebutton{TwoDimensionalArrayXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{nrows(arr)\free{arr2 }}
+\indentrel{3}\begin{verbatim}
+   (9)  5
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty9}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty9}{TwoDimensionalArrayXmpPagePatch9}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{nrows(arr)\free{arr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch10}
+\begin{paste}{TwoDimensionalArrayXmpPageFull10}{TwoDimensionalArrayXmpPageEmpty10}
+\pastebutton{TwoDimensionalArrayXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{ncols(arr)\free{arr2 }}
+\indentrel{3}\begin{verbatim}
+   (10)  4
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty10}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty10}{TwoDimensionalArrayXmpPagePatch10}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{ncols(arr)\free{arr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch11}
+\begin{paste}{TwoDimensionalArrayXmpPageFull11}{TwoDimensionalArrayXmpPageEmpty11}
+\pastebutton{TwoDimensionalArrayXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{map(-,arr)\free{arr2 }}
+\indentrel{3}\begin{verbatim}
+         Ú- 17   0    0  0¿
+         ³                ³
+         ³ 0     0    0  0³
+         ³                ³
+   (11)  ³ 0    - 15  0  0³
+         ³                ³
+         ³ 0     0    0  0³
+         ³                ³
+         À 0     0    0  0Ù
+                      Type: TwoDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty11}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty11}{TwoDimensionalArrayXmpPagePatch11}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{map(-,arr)\free{arr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch12}
+\begin{paste}{TwoDimensionalArrayXmpPageFull12}{TwoDimensionalArrayXmpPageEmpty12}
+\pastebutton{TwoDimensionalArrayXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{map((x +-> x + x),arr)\free{arr2 }}
+\indentrel{3}\begin{verbatim}
+         Ú34  0   0  0¿
+         ³            ³
+         ³0   0   0  0³
+         ³            ³
+   (12)  ³0   30  0  0³
+         ³            ³
+         ³0   0   0  0³
+         ³            ³
+         À0   0   0  0Ù
+                      Type: TwoDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty12}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty12}{TwoDimensionalArrayXmpPagePatch12}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{map((x +-> x + x),arr)\free{arr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch13}
+\begin{paste}{TwoDimensionalArrayXmpPageFull13}{TwoDimensionalArrayXmpPageEmpty13}
+\pastebutton{TwoDimensionalArrayXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{arrc := copy(arr)\bound{arrc }\free{arr2 }}
+\indentrel{3}\begin{verbatim}
+         Ú17  0   0  0¿
+         ³            ³
+         ³0   0   0  0³
+         ³            ³
+   (13)  ³0   15  0  0³
+         ³            ³
+         ³0   0   0  0³
+         ³            ³
+         À0   0   0  0Ù
+                      Type: TwoDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty13}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty13}{TwoDimensionalArrayXmpPagePatch13}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{arrc := copy(arr)\bound{arrc }\free{arr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch14}
+\begin{paste}{TwoDimensionalArrayXmpPageFull14}{TwoDimensionalArrayXmpPageEmpty14}
+\pastebutton{TwoDimensionalArrayXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{map!(-,arrc)\free{arrc }}
+\indentrel{3}\begin{verbatim}
+         Ú- 17   0    0  0¿
+         ³                ³
+         ³ 0     0    0  0³
+         ³                ³
+   (14)  ³ 0    - 15  0  0³
+         ³                ³
+         ³ 0     0    0  0³
+         ³                ³
+         À 0     0    0  0Ù
+                      Type: TwoDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty14}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty14}{TwoDimensionalArrayXmpPagePatch14}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{map!(-,arrc)\free{arrc }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch15}
+\begin{paste}{TwoDimensionalArrayXmpPageFull15}{TwoDimensionalArrayXmpPageEmpty15}
+\pastebutton{TwoDimensionalArrayXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{arrc\free{arrc }}
+\indentrel{3}\begin{verbatim}
+         Ú- 17   0    0  0¿
+         ³                ³
+         ³ 0     0    0  0³
+         ³                ³
+   (15)  ³ 0    - 15  0  0³
+         ³                ³
+         ³ 0     0    0  0³
+         ³                ³
+         À 0     0    0  0Ù
+                      Type: TwoDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty15}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty15}{TwoDimensionalArrayXmpPagePatch15}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{arrc\free{arrc }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch16}
+\begin{paste}{TwoDimensionalArrayXmpPageFull16}{TwoDimensionalArrayXmpPageEmpty16}
+\pastebutton{TwoDimensionalArrayXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{arr\free{arr2 }}
+\indentrel{3}\begin{verbatim}
+         Ú17  0   0  0¿
+         ³            ³
+         ³0   0   0  0³
+         ³            ³
+   (16)  ³0   15  0  0³
+         ³            ³
+         ³0   0   0  0³
+         ³            ³
+         À0   0   0  0Ù
+                      Type: TwoDimensionalArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty16}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty16}{TwoDimensionalArrayXmpPagePatch16}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{arr\free{arr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch17}
+\begin{paste}{TwoDimensionalArrayXmpPageFull17}{TwoDimensionalArrayXmpPageEmpty17}
+\pastebutton{TwoDimensionalArrayXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{member?(17,arr)\free{arr2 }}
+\indentrel{3}\begin{verbatim}
+   (17)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty17}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty17}{TwoDimensionalArrayXmpPagePatch17}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{member?(17,arr)\free{arr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch18}
+\begin{paste}{TwoDimensionalArrayXmpPageFull18}{TwoDimensionalArrayXmpPageEmpty18}
+\pastebutton{TwoDimensionalArrayXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{member?(10317,arr)\free{arr2 }}
+\indentrel{3}\begin{verbatim}
+   (18)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty18}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty18}{TwoDimensionalArrayXmpPagePatch18}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{member?(10317,arr)\free{arr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch19}
+\begin{paste}{TwoDimensionalArrayXmpPageFull19}{TwoDimensionalArrayXmpPageEmpty19}
+\pastebutton{TwoDimensionalArrayXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{count(17,arr)\free{arr2 }}
+\indentrel{3}\begin{verbatim}
+   (19)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty19}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty19}{TwoDimensionalArrayXmpPagePatch19}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{count(17,arr)\free{arr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPagePatch20}
+\begin{paste}{TwoDimensionalArrayXmpPageFull20}{TwoDimensionalArrayXmpPageEmpty20}
+\pastebutton{TwoDimensionalArrayXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{count(0,arr)\free{arr2 }}
+\indentrel{3}\begin{verbatim}
+   (20)  18
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoDimensionalArrayXmpPageEmpty20}
+\begin{paste}{TwoDimensionalArrayXmpPageEmpty20}{TwoDimensionalArrayXmpPagePatch20}
+\pastebutton{TwoDimensionalArrayXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{count(0,arr)\free{arr2 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/BBTREE.ht b/src/hyper/pages/BBTREE.ht
new file mode 100644
index 00000000..e1e41d68
--- /dev/null
+++ b/src/hyper/pages/BBTREE.ht
@@ -0,0 +1,116 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\BalancedBinaryTreeXmpTitle}{BalancedBinaryTree}
+\newcommand{\BalancedBinaryTreeXmpNumber}{9.2}
+%
+% =====================================================================
+\begin{page}{BalancedBinaryTreeXmpPage}{9.2 BalancedBinaryTree}
+% =====================================================================
+\beginscroll
+\spadtype{BalancedBinaryTrees(S)} is the domain
+of balanced binary trees with elements of type \spad{S} at the nodes.
+%-% \HDindex{tree!balanced binary}{BalancedBinaryTreeXmpPage}{9.2}{BalancedBinaryTree}
+A binary tree is either \spadfun{empty} or else
+%-% \HDindex{balanced binary tree}{BalancedBinaryTreeXmpPage}{9.2}{BalancedBinaryTree}
+consists of a \spadfun{node} having a \spadfun{value}
+and two branches, each branch a binary tree.
+A balanced binary tree is one that is balanced with respect its leaves.
+One with \texht{$2^k$}{2**k} leaves is
+perfectly ``balanced'': the tree has minimum depth, and
+the \spadfun{left} and \spadfun{right}
+branch of every interior node is identical in shape.
+
+Balanced binary trees are useful in algebraic computation for
+so-called ``divide-and-conquer'' algorithms.
+Conceptually, the data for a problem is initially placed at the
+root of the tree.
+The original data is then split into two subproblems, one for each
+subtree.
+And so on.
+Eventually, the problem is solved at the leaves of the tree.
+A solution to the original problem is obtained by some mechanism
+that can reassemble the pieces.
+In fact, an implementation of the Chinese Remainder Algorithm
+%-% \HDindex{Chinese Remainder Algorithm}{BalancedBinaryTreeXmpPage}{9.2}{BalancedBinaryTree}
+using balanced binary trees was first proposed by David Y. Y.
+%-% \HDindex{Yun, David Y. Y.}{BalancedBinaryTreeXmpPage}{9.2}{BalancedBinaryTree}
+Yun at the IBM T. J.
+Watson Research Center in Yorktown Heights, New York, in 1978.
+It served as the prototype for polymorphic algorithms in
+\Language{}.
+
+In what follows, rather than perform a series of computations with
+a single expression, the expression is reduced modulo a number of
+integer primes, a computation is done with modular arithmetic for
+each prime, and the Chinese Remainder Algorithm is used to obtain
+the answer to the original problem.
+We illustrate this principle with the computation of \texht{$12^2
+= 144$}{12 ** 2 = 144}.
+
+\xtc{
+A list of moduli.
+}{
+\spadpaste{lm := [3,5,7,11]\bound{lm}}
+}
+\xtc{
+The expression \spad{modTree(n, lm)}
+creates a balanced binary tree with leaf values
+\spad{n mod m} for each modulus \spad{m} in \spad{lm}.
+}{
+\spadpaste{modTree(12,lm)\free{lm}}
+}
+\xtc{
+Operation \spad{modTree} does this using
+operations on balanced binary trees.
+We trace its steps.
+Create a balanced binary tree \spad{t} of zeros with four leaves.
+}{
+\spadpaste{t := balancedBinaryTree(\#lm, 0)\bound{t}\free{lm}}
+}
+\xtc{
+The leaves of the tree are set to the individual moduli.
+}{
+\spadpaste{setleaves!(t,lm)\bound{t1}\free{t}}
+}
+\xtc{
+Use \spadfunX{mapUp} to do
+a bottom-up traversal of \spad{t}, setting each interior node
+to the product of the values at the nodes of its children.
+}{
+\spadpaste{mapUp!(t,_*)\bound{t2}\free{t1}}
+}
+\xtc{
+The value at the node of every subtree is the product of the moduli
+of the leaves of the subtree.
+}{
+\spadpaste{t \bound{t3}\free{t2}}
+}
+\xtc{
+Operation \spadfunX{mapDown}\spad{(t,a,fn)} replaces the value
+\spad{v} at each node of \spad{t} by \spad{fn(a,v)}.
+}{
+\spadpaste{mapDown!(t,12,_rem)\bound{t4}\free{t3}}
+}
+\xtc{
+The operation \spadfun{leaves} returns the leaves of the resulting
+tree.
+In this case, it returns the list of \spad{12 mod m} for each
+modulus \spad{m}.
+}{
+\spadpaste{leaves \%\bound{t5}\free{t4}}
+}
+\xtc{
+Compute the square of the images of \spad{12} modulo each \spad{m}.
+}{
+\spadpaste{squares := [x**2 rem m for x in \% for m in lm]\bound{t6}\free{t5}}
+}
+\xtc{
+Call the Chinese Remainder Algorithm to get the
+answer for \texht{$12^2$}{12**2}.
+}{
+\spadpaste{chineseRemainder(\%,lm)\free{t6}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/BBTREE.pht b/src/hyper/pages/BBTREE.pht
new file mode 100644
index 00000000..2effaf85
--- /dev/null
+++ b/src/hyper/pages/BBTREE.pht
@@ -0,0 +1,160 @@
+\begin{patch}{BalancedBinaryTreeXmpPagePatch1}
+\begin{paste}{BalancedBinaryTreeXmpPageFull1}{BalancedBinaryTreeXmpPageEmpty1}
+\pastebutton{BalancedBinaryTreeXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{lm := [3,5,7,11]\bound{lm }}
+\indentrel{3}\begin{verbatim}
+   (1)  [3,5,7,11]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPageEmpty1}
+\begin{paste}{BalancedBinaryTreeXmpPageEmpty1}{BalancedBinaryTreeXmpPagePatch1}
+\pastebutton{BalancedBinaryTreeXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{lm := [3,5,7,11]\bound{lm }}
+\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPagePatch2}
+\begin{paste}{BalancedBinaryTreeXmpPageFull2}{BalancedBinaryTreeXmpPageEmpty2}
+\pastebutton{BalancedBinaryTreeXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{modTree(12,lm)\free{lm }}
+\indentrel{3}\begin{verbatim}
+   (2)  [0,2,5,1]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPageEmpty2}
+\begin{paste}{BalancedBinaryTreeXmpPageEmpty2}{BalancedBinaryTreeXmpPagePatch2}
+\pastebutton{BalancedBinaryTreeXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{modTree(12,lm)\free{lm }}
+\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPagePatch3}
+\begin{paste}{BalancedBinaryTreeXmpPageFull3}{BalancedBinaryTreeXmpPageEmpty3}
+\pastebutton{BalancedBinaryTreeXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{t := balancedBinaryTree(\#lm, 0)\bound{t }\free{lm }}
+\indentrel{3}\begin{verbatim}
+   (3)  [[0,0,0],0,[0,0,0]]
+            Type: BalancedBinaryTree NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPageEmpty3}
+\begin{paste}{BalancedBinaryTreeXmpPageEmpty3}{BalancedBinaryTreeXmpPagePatch3}
+\pastebutton{BalancedBinaryTreeXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{t := balancedBinaryTree(\#lm, 0)\bound{t }\free{lm }}
+\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPagePatch4}
+\begin{paste}{BalancedBinaryTreeXmpPageFull4}{BalancedBinaryTreeXmpPageEmpty4}
+\pastebutton{BalancedBinaryTreeXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{setleaves!(t,lm)\bound{t1 }\free{t }}
+\indentrel{3}\begin{verbatim}
+   (4)  [[3,0,5],0,[7,0,11]]
+            Type: BalancedBinaryTree NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPageEmpty4}
+\begin{paste}{BalancedBinaryTreeXmpPageEmpty4}{BalancedBinaryTreeXmpPagePatch4}
+\pastebutton{BalancedBinaryTreeXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{setleaves!(t,lm)\bound{t1 }\free{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPagePatch5}
+\begin{paste}{BalancedBinaryTreeXmpPageFull5}{BalancedBinaryTreeXmpPageEmpty5}
+\pastebutton{BalancedBinaryTreeXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{mapUp!(t,_*)\bound{t2 }\free{t1 }}
+\indentrel{3}\begin{verbatim}
+   (5)  1155
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPageEmpty5}
+\begin{paste}{BalancedBinaryTreeXmpPageEmpty5}{BalancedBinaryTreeXmpPagePatch5}
+\pastebutton{BalancedBinaryTreeXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{mapUp!(t,_*)\bound{t2 }\free{t1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPagePatch6}
+\begin{paste}{BalancedBinaryTreeXmpPageFull6}{BalancedBinaryTreeXmpPageEmpty6}
+\pastebutton{BalancedBinaryTreeXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{t\bound{t3 }\free{t2 }}
+\indentrel{3}\begin{verbatim}
+   (6)  [[3,15,5],1155,[7,77,11]]
+            Type: BalancedBinaryTree NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPageEmpty6}
+\begin{paste}{BalancedBinaryTreeXmpPageEmpty6}{BalancedBinaryTreeXmpPagePatch6}
+\pastebutton{BalancedBinaryTreeXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{t\bound{t3 }\free{t2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPagePatch7}
+\begin{paste}{BalancedBinaryTreeXmpPageFull7}{BalancedBinaryTreeXmpPageEmpty7}
+\pastebutton{BalancedBinaryTreeXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{mapDown!(t,12,_rem)\bound{t4 }\free{t3 }}
+\indentrel{3}\begin{verbatim}
+   (7)  [[0,12,2],12,[5,12,1]]
+            Type: BalancedBinaryTree NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPageEmpty7}
+\begin{paste}{BalancedBinaryTreeXmpPageEmpty7}{BalancedBinaryTreeXmpPagePatch7}
+\pastebutton{BalancedBinaryTreeXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{mapDown!(t,12,_rem)\bound{t4 }\free{t3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPagePatch8}
+\begin{paste}{BalancedBinaryTreeXmpPageFull8}{BalancedBinaryTreeXmpPageEmpty8}
+\pastebutton{BalancedBinaryTreeXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{leaves \%\bound{t5 }\free{t4 }}
+\indentrel{3}\begin{verbatim}
+   (8)  [0,2,5,1]
+                          Type: List NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPageEmpty8}
+\begin{paste}{BalancedBinaryTreeXmpPageEmpty8}{BalancedBinaryTreeXmpPagePatch8}
+\pastebutton{BalancedBinaryTreeXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{leaves \%\bound{t5 }\free{t4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPagePatch9}
+\begin{paste}{BalancedBinaryTreeXmpPageFull9}{BalancedBinaryTreeXmpPageEmpty9}
+\pastebutton{BalancedBinaryTreeXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{squares := [x**2 rem m for x in \% for m in lm]\bound{t6 }\free{t5 }}
+\indentrel{3}\begin{verbatim}
+   (9)  [0,4,4,1]
+                          Type: List NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPageEmpty9}
+\begin{paste}{BalancedBinaryTreeXmpPageEmpty9}{BalancedBinaryTreeXmpPagePatch9}
+\pastebutton{BalancedBinaryTreeXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{squares := [x**2 rem m for x in \% for m in lm]\bound{t6 }\free{t5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPagePatch10}
+\begin{paste}{BalancedBinaryTreeXmpPageFull10}{BalancedBinaryTreeXmpPageEmpty10}
+\pastebutton{BalancedBinaryTreeXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{chineseRemainder(\%,lm)\free{t6 }}
+\indentrel{3}\begin{verbatim}
+   (10)  144
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BalancedBinaryTreeXmpPageEmpty10}
+\begin{paste}{BalancedBinaryTreeXmpPageEmpty10}{BalancedBinaryTreeXmpPagePatch10}
+\pastebutton{BalancedBinaryTreeXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{chineseRemainder(\%,lm)\free{t6 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/BINARY.ht b/src/hyper/pages/BINARY.ht
new file mode 100644
index 00000000..e43dce27
--- /dev/null
+++ b/src/hyper/pages/BINARY.ht
@@ -0,0 +1,56 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\BinaryExpansionXmpTitle}{BinaryExpansion}
+\newcommand{\BinaryExpansionXmpNumber}{9.4}
+%
+% =====================================================================
+\begin{page}{BinaryExpansionXmpPage}{9.4 BinaryExpansion}
+% =====================================================================
+\beginscroll
+ 
+All rational numbers have repeating binary expansions.
+Operations to access the individual bits of a binary expansion can
+be obtained by converting the value to \spadtype{RadixExpansion(2)}.
+More examples of expansions are available in
+\downlink{`DecimalExpansion'}{DecimalExpansionXmpPage}\ignore{DecimalExpansion},
+\downlink{`HexadecimalExpansion'}{HexadecimalExpansionXmpPage}\ignore{HexadecimalExpansion}, and
+\downlink{`RadixExpansion'}{RadixExpansionXmpPage}\ignore{RadixExpansion}.
+ 
+\xtc{
+The expansion (of type \spadtype{BinaryExpansion}) of a rational number
+is returned by the \spadfunFrom{binary}{BinaryExpansion} operation.
+}{
+\spadpaste{r := binary(22/7) \bound{r}}
+}
+\xtc{
+Arithmetic is exact.
+}{
+\spadpaste{r + binary(6/7) \free{r}}
+}
+\xtc{
+The period of the expansion can be short or long \ldots
+}{
+\spadpaste{[binary(1/i) for i in 102..106] }
+}
+\xtc{
+or very long.
+}{
+\spadpaste{binary(1/1007) }
+}
+\xtc{
+These numbers are bona fide algebraic objects.
+}{
+\spadpaste{p := binary(1/4)*x**2 + binary(2/3)*x + binary(4/9)  \bound{p}}
+}
+\xtc{
+}{
+\spadpaste{q := D(p, x) \free{p}\bound{q}}
+}
+\xtc{
+}{
+\spadpaste{g := gcd(p, q)            \free{p q}\bound{g}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/BINARY.pht b/src/hyper/pages/BINARY.pht
new file mode 100644
index 00000000..8590e7a2
--- /dev/null
+++ b/src/hyper/pages/BINARY.pht
@@ -0,0 +1,140 @@
+\begin{patch}{BinaryExpansionXmpPagePatch1}
+\begin{paste}{BinaryExpansionXmpPageFull1}{BinaryExpansionXmpPageEmpty1}
+\pastebutton{BinaryExpansionXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{r := binary(22/7)\bound{r }}
+\indentrel{3}\begin{verbatim}
+           ___
+   (1)  11.001
+                                  Type: BinaryExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinaryExpansionXmpPageEmpty1}
+\begin{paste}{BinaryExpansionXmpPageEmpty1}{BinaryExpansionXmpPagePatch1}
+\pastebutton{BinaryExpansionXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{r := binary(22/7)\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinaryExpansionXmpPagePatch2}
+\begin{paste}{BinaryExpansionXmpPageFull2}{BinaryExpansionXmpPageEmpty2}
+\pastebutton{BinaryExpansionXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{r + binary(6/7)\free{r }}
+\indentrel{3}\begin{verbatim}
+   (2)  100
+                                  Type: BinaryExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinaryExpansionXmpPageEmpty2}
+\begin{paste}{BinaryExpansionXmpPageEmpty2}{BinaryExpansionXmpPagePatch2}
+\pastebutton{BinaryExpansionXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{r + binary(6/7)\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinaryExpansionXmpPagePatch3}
+\begin{paste}{BinaryExpansionXmpPageFull3}{BinaryExpansionXmpPageEmpty3}
+\pastebutton{BinaryExpansionXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{[binary(1/i) for i in 102..106]}
+\indentrel{3}\begin{verbatim}
+   (3)
+       ________
+   [0.000000101,
+      ___________________________________________________
+    0.000000100111110001000101100101111001110010010101001,
+         ____________    ____________
+    0.000000100111011, 0.000000100111,
+
+     0.0
+       OVERBAR
+          00000100110101001000011100111110110010101101111
+            00011
+     ]
+                             Type: List BinaryExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinaryExpansionXmpPageEmpty3}
+\begin{paste}{BinaryExpansionXmpPageEmpty3}{BinaryExpansionXmpPagePatch3}
+\pastebutton{BinaryExpansionXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{[binary(1/i) for i in 102..106]}
+\end{paste}\end{patch}
+
+\begin{patch}{BinaryExpansionXmpPagePatch4}
+\begin{paste}{BinaryExpansionXmpPageFull4}{BinaryExpansionXmpPageEmpty4}
+\pastebutton{BinaryExpansionXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{binary(1/1007)}
+\indentrel{3}\begin{verbatim}
+   (4)
+   0.
+     OVERBAR
+        0000000001000001000101001001011110000011111100001
+          01111110010110001111101000100111001001100110001
+          10010010101011110110100110000000011000011001111
+          01110001101000101111010010001111011000010101110
+          11100111010101110011001010010111000000011100011
+          11001000000100100100110111001010100111010001101
+          11011010111000100100000110010110110000001011001
+          01111100010100000101010101101011000001101101110
+          10010101111111010111010100110010000101001101100
+          0100110001000100001000011000111010011110001
+                                  Type: BinaryExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinaryExpansionXmpPageEmpty4}
+\begin{paste}{BinaryExpansionXmpPageEmpty4}{BinaryExpansionXmpPagePatch4}
+\pastebutton{BinaryExpansionXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{binary(1/1007)}
+\end{paste}\end{patch}
+
+\begin{patch}{BinaryExpansionXmpPagePatch5}
+\begin{paste}{BinaryExpansionXmpPageFull5}{BinaryExpansionXmpPageEmpty5}
+\pastebutton{BinaryExpansionXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{p := binary(1/4)*x**2 + binary(2/3)*x + binary(4/9)\bound{p }}
+\indentrel{3}\begin{verbatim}
+             2     __      ______
+   (5)  0.01x  + 0.10x + 0.011100
+                       Type: Polynomial BinaryExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinaryExpansionXmpPageEmpty5}
+\begin{paste}{BinaryExpansionXmpPageEmpty5}{BinaryExpansionXmpPagePatch5}
+\pastebutton{BinaryExpansionXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{p := binary(1/4)*x**2 + binary(2/3)*x + binary(4/9)\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinaryExpansionXmpPagePatch6}
+\begin{paste}{BinaryExpansionXmpPageFull6}{BinaryExpansionXmpPageEmpty6}
+\pastebutton{BinaryExpansionXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{q := D(p, x)\free{p }\bound{q }}
+\indentrel{3}\begin{verbatim}
+                 __
+   (6)  0.1x + 0.10
+                       Type: Polynomial BinaryExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinaryExpansionXmpPageEmpty6}
+\begin{paste}{BinaryExpansionXmpPageEmpty6}{BinaryExpansionXmpPagePatch6}
+\pastebutton{BinaryExpansionXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{q := D(p, x)\free{p }\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinaryExpansionXmpPagePatch7}
+\begin{paste}{BinaryExpansionXmpPageFull7}{BinaryExpansionXmpPageEmpty7}
+\pastebutton{BinaryExpansionXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{g := gcd(p, q)\free{p q }\bound{g }}
+\indentrel{3}\begin{verbatim}
+              __
+   (7)  x + 1.01
+                       Type: Polynomial BinaryExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinaryExpansionXmpPageEmpty7}
+\begin{paste}{BinaryExpansionXmpPageEmpty7}{BinaryExpansionXmpPagePatch7}
+\pastebutton{BinaryExpansionXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{g := gcd(p, q)\free{p q }\bound{g }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/BOP.ht b/src/hyper/pages/BOP.ht
new file mode 100644
index 00000000..64cf58c7
--- /dev/null
+++ b/src/hyper/pages/BOP.ht
@@ -0,0 +1,144 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\BasicOperatorXmpTitle}{BasicOperator}
+\newcommand{\BasicOperatorXmpNumber}{9.3}
+%
+% =====================================================================
+\begin{page}{BasicOperatorXmpPage}{9.3 BasicOperator}
+% =====================================================================
+\beginscroll
+%
+
+A basic operator is an object that can be symbolically
+applied to a list of arguments from a set, the result being
+a kernel over that set or an expression.
+In addition to this section, please see \downlink{`Expression'}{ExpressionXmpPage}\ignore{Expression}
+and \downlink{`Kernel'}{KernelXmpPage}\ignore{Kernel} for additional information and examples.
+
+You create an object of type \axiomType{BasicOperator} by using the
+\axiomFunFrom{operator}{BasicOperator} operation.
+This first form of this operation has one argument and it
+must be a symbol.
+The symbol should be quoted in case the
+name has been used as an identifier to which a value
+has been assigned.
+
+A frequent application of \axiomType{BasicOperator} is the
+creation of an operator to represent the unknown function when
+solving a differential equation.
+\xtc{
+Let \axiom{y} be the unknown function in terms of \axiom{x}.
+}{
+\spadpaste{y := operator 'y \bound{y}}
+}
+%
+\xtc{
+This is how you enter
+the equation \axiom{y'' + y' + y = 0}.
+}{
+\spadpaste{deq := D(y x, x, 2) + D(y x, x) + y x = 0\bound{e1}\free{y}}
+}
+%
+\xtc{
+To solve the above equation, enter this.
+}{
+\spadpaste{solve(deq, y, x) \free{e1}\free{y}}
+}
+See \downlink{``\ugProblemDEQTitle''}{ugProblemDEQPage} in Section \ugProblemDEQNumber\ignore{ugProblemDEQ}
+for this kind of use of \axiomType{BasicOperator}.
+
+Use the single argument form of
+\axiomFunFrom{operator}{BasicOperator} (as above) when you intend
+to use the operator to create functional expressions with an
+arbitrary number of arguments
+\xtc{
+{\it Nary} means an arbitrary number of arguments can be used
+in the functional expressions.
+}{
+\spadpaste{nary? y \free{y}}
+}
+\xtc{
+}{
+\spadpaste{unary? y \free{y}}
+}
+Use the two-argument form when you want to restrict the number of
+arguments in the functional expressions created with the operator.
+\xtc{
+This operator can only be used to create functional expressions
+with one argument.
+}{
+\spadpaste{opOne := operator('opOne, 1) \bound{opOne}}
+}
+\xtc{
+}{
+\spadpaste{nary? opOne \free{opOne}}
+}
+\xtc{
+}{
+\spadpaste{unary? opOne \free{opOne}}
+}
+\xtc{
+Use \axiomFunFrom{arity}{BasicOperator} to learn the number of
+arguments that can be used.
+It returns {\tt "false"} if the operator is nary.
+}{
+\spadpaste{arity opOne \free{opOne}}
+}
+\xtc{
+Use \axiomFunFrom{name}{BasicOperator} to learn the name of an
+operator.
+}{
+\spadpaste{name opOne \free{opOne}}
+}
+\xtc{
+Use \axiomFunFrom{is?}{BasicOperator} to learn if an operator has a
+particular name.
+}{
+\spadpaste{is?(opOne, 'z2) \free{opOne}}
+}
+\xtc{
+You can also use a string as the name to be tested against.
+}{
+\spadpaste{is?(opOne, "opOne") \free{opOne}}
+}
+
+You can attached named properties to an operator.
+These are rarely used at the top-level of the \Language{}
+interactive environment but are used with \Language{}
+library source code.
+\xtc{
+By default, an operator has no properties.
+}{
+\spadpaste{properties y \free{y}}
+}
+The interface for setting and getting properties is somewhat awkward
+because the property values are stored as values of type
+\axiomType{None}.
+\xtc{
+Attach a property by using \axiomFunFrom{setProperty}{BasicOperator}.
+}{
+\spadpaste{setProperty(y, "use", "unknown function" :: None ) \free{y}\bound{spy}}
+}
+\xtc{
+}{
+\spadpaste{properties y \free{y spy}}
+}
+\xtc{
+We {\it know} the property value has type \axiomType{String}.
+}{
+\spadpaste{property(y, "use") :: None pretend String \free{y spy}}
+}
+\xtc{
+Use \axiomFunFrom{deleteProperty!}{BasicOperator} to destructively
+remove a property.
+}{
+\spadpaste{deleteProperty!(y, "use") \free{y spy}\bound{dpy}}
+}
+\xtc{
+}{
+\spadpaste{properties y \free{dpy}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/BOP.pht b/src/hyper/pages/BOP.pht
new file mode 100644
index 00000000..6156e506
--- /dev/null
+++ b/src/hyper/pages/BOP.pht
@@ -0,0 +1,296 @@
+\begin{patch}{BasicOperatorXmpPagePatch1}
+\begin{paste}{BasicOperatorXmpPageFull1}{BasicOperatorXmpPageEmpty1}
+\pastebutton{BasicOperatorXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{y := operator 'y\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (1)  y
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty1}
+\begin{paste}{BasicOperatorXmpPageEmpty1}{BasicOperatorXmpPagePatch1}
+\pastebutton{BasicOperatorXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{y := operator 'y\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch2}
+\begin{paste}{BasicOperatorXmpPageFull2}{BasicOperatorXmpPageEmpty2}
+\pastebutton{BasicOperatorXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0\bound{e1 }\free{y }}
+\indentrel{3}\begin{verbatim}
+         ,,       ,
+   (2)  y  (x) + y (x) + y(x)= 0
+
+                      Type: Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty2}
+\begin{paste}{BasicOperatorXmpPageEmpty2}{BasicOperatorXmpPagePatch2}
+\pastebutton{BasicOperatorXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0\bound{e1 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch3}
+\begin{paste}{BasicOperatorXmpPageFull3}{BasicOperatorXmpPageEmpty3}
+\pastebutton{BasicOperatorXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e1 }\free{y }}
+\indentrel{3}\begin{verbatim}
+   (3)
+   [particular= 0,
+                          x     x
+                  ÚÄ¿   - Ä   - Ä      ÚÄ¿
+                x\³3      2     2    x\³3
+    basis= [cos(ÄÄÄÄÄ)%e   ,%e   sin(ÄÄÄÄÄ)]]
+                  2                    2
+Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty3}
+\begin{paste}{BasicOperatorXmpPageEmpty3}{BasicOperatorXmpPagePatch3}
+\pastebutton{BasicOperatorXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e1 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch4}
+\begin{paste}{BasicOperatorXmpPageFull4}{BasicOperatorXmpPageEmpty4}
+\pastebutton{BasicOperatorXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{nary? y\free{y }}
+\indentrel{3}\begin{verbatim}
+   (4)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty4}
+\begin{paste}{BasicOperatorXmpPageEmpty4}{BasicOperatorXmpPagePatch4}
+\pastebutton{BasicOperatorXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{nary? y\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch5}
+\begin{paste}{BasicOperatorXmpPageFull5}{BasicOperatorXmpPageEmpty5}
+\pastebutton{BasicOperatorXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{unary? y\free{y }}
+\indentrel{3}\begin{verbatim}
+   (5)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty5}
+\begin{paste}{BasicOperatorXmpPageEmpty5}{BasicOperatorXmpPagePatch5}
+\pastebutton{BasicOperatorXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{unary? y\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch6}
+\begin{paste}{BasicOperatorXmpPageFull6}{BasicOperatorXmpPageEmpty6}
+\pastebutton{BasicOperatorXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{opOne := operator('opOne, 1)\bound{opOne }}
+\indentrel{3}\begin{verbatim}
+   (6)  opOne
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty6}
+\begin{paste}{BasicOperatorXmpPageEmpty6}{BasicOperatorXmpPagePatch6}
+\pastebutton{BasicOperatorXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{opOne := operator('opOne, 1)\bound{opOne }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch7}
+\begin{paste}{BasicOperatorXmpPageFull7}{BasicOperatorXmpPageEmpty7}
+\pastebutton{BasicOperatorXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{nary? opOne\free{opOne }}
+\indentrel{3}\begin{verbatim}
+   (7)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty7}
+\begin{paste}{BasicOperatorXmpPageEmpty7}{BasicOperatorXmpPagePatch7}
+\pastebutton{BasicOperatorXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{nary? opOne\free{opOne }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch8}
+\begin{paste}{BasicOperatorXmpPageFull8}{BasicOperatorXmpPageEmpty8}
+\pastebutton{BasicOperatorXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{unary? opOne\free{opOne }}
+\indentrel{3}\begin{verbatim}
+   (8)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty8}
+\begin{paste}{BasicOperatorXmpPageEmpty8}{BasicOperatorXmpPagePatch8}
+\pastebutton{BasicOperatorXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{unary? opOne\free{opOne }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch9}
+\begin{paste}{BasicOperatorXmpPageFull9}{BasicOperatorXmpPageEmpty9}
+\pastebutton{BasicOperatorXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{arity opOne\free{opOne }}
+\indentrel{3}\begin{verbatim}
+   (9)  1
+                    Type: Union(NonNegativeInteger,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty9}
+\begin{paste}{BasicOperatorXmpPageEmpty9}{BasicOperatorXmpPagePatch9}
+\pastebutton{BasicOperatorXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{arity opOne\free{opOne }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch10}
+\begin{paste}{BasicOperatorXmpPageFull10}{BasicOperatorXmpPageEmpty10}
+\pastebutton{BasicOperatorXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{name opOne\free{opOne }}
+\indentrel{3}\begin{verbatim}
+   (10)  opOne
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty10}
+\begin{paste}{BasicOperatorXmpPageEmpty10}{BasicOperatorXmpPagePatch10}
+\pastebutton{BasicOperatorXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{name opOne\free{opOne }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch11}
+\begin{paste}{BasicOperatorXmpPageFull11}{BasicOperatorXmpPageEmpty11}
+\pastebutton{BasicOperatorXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{is?(opOne, 'z2)\free{opOne }}
+\indentrel{3}\begin{verbatim}
+   (11)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty11}
+\begin{paste}{BasicOperatorXmpPageEmpty11}{BasicOperatorXmpPagePatch11}
+\pastebutton{BasicOperatorXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{is?(opOne, 'z2)\free{opOne }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch12}
+\begin{paste}{BasicOperatorXmpPageFull12}{BasicOperatorXmpPageEmpty12}
+\pastebutton{BasicOperatorXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{is?(opOne, "opOne")\free{opOne }}
+\indentrel{3}\begin{verbatim}
+   (12)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty12}
+\begin{paste}{BasicOperatorXmpPageEmpty12}{BasicOperatorXmpPagePatch12}
+\pastebutton{BasicOperatorXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{is?(opOne, "opOne")\free{opOne }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch13}
+\begin{paste}{BasicOperatorXmpPageFull13}{BasicOperatorXmpPageEmpty13}
+\pastebutton{BasicOperatorXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{properties y\free{y }}
+\indentrel{3}\begin{verbatim}
+   (13)  table()
+                     Type: AssociationList(String,None)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty13}
+\begin{paste}{BasicOperatorXmpPageEmpty13}{BasicOperatorXmpPagePatch13}
+\pastebutton{BasicOperatorXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{properties y\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch14}
+\begin{paste}{BasicOperatorXmpPageFull14}{BasicOperatorXmpPageEmpty14}
+\pastebutton{BasicOperatorXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{setProperty(y, "use", "unknown function" :: None )\free{y }\bound{spy }}
+\indentrel{3}\begin{verbatim}
+   (14)  y
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty14}
+\begin{paste}{BasicOperatorXmpPageEmpty14}{BasicOperatorXmpPagePatch14}
+\pastebutton{BasicOperatorXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{setProperty(y, "use", "unknown function" :: None )\free{y }\bound{spy }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch15}
+\begin{paste}{BasicOperatorXmpPageFull15}{BasicOperatorXmpPageEmpty15}
+\pastebutton{BasicOperatorXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{properties y\free{y spy }}
+\indentrel{3}\begin{verbatim}
+   (15)  table("use"= NONE)
+                     Type: AssociationList(String,None)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty15}
+\begin{paste}{BasicOperatorXmpPageEmpty15}{BasicOperatorXmpPagePatch15}
+\pastebutton{BasicOperatorXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{properties y\free{y spy }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch16}
+\begin{paste}{BasicOperatorXmpPageFull16}{BasicOperatorXmpPageEmpty16}
+\pastebutton{BasicOperatorXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{property(y, "use") :: None pretend String\free{y spy }}
+\indentrel{3}\begin{verbatim}
+   (16)  "unknown function"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty16}
+\begin{paste}{BasicOperatorXmpPageEmpty16}{BasicOperatorXmpPagePatch16}
+\pastebutton{BasicOperatorXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{property(y, "use") :: None pretend String\free{y spy }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch17}
+\begin{paste}{BasicOperatorXmpPageFull17}{BasicOperatorXmpPageEmpty17}
+\pastebutton{BasicOperatorXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{deleteProperty!(y, "use")\free{y spy }\bound{dpy }}
+\indentrel{3}\begin{verbatim}
+   (17)  y
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty17}
+\begin{paste}{BasicOperatorXmpPageEmpty17}{BasicOperatorXmpPagePatch17}
+\pastebutton{BasicOperatorXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{deleteProperty!(y, "use")\free{y spy }\bound{dpy }}
+\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPagePatch18}
+\begin{paste}{BasicOperatorXmpPageFull18}{BasicOperatorXmpPageEmpty18}
+\pastebutton{BasicOperatorXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{properties y\free{dpy }}
+\indentrel{3}\begin{verbatim}
+   (18)  table()
+                     Type: AssociationList(String,None)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BasicOperatorXmpPageEmpty18}
+\begin{paste}{BasicOperatorXmpPageEmpty18}{BasicOperatorXmpPagePatch18}
+\pastebutton{BasicOperatorXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{properties y\free{dpy }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/BSTREE.ht b/src/hyper/pages/BSTREE.ht
new file mode 100644
index 00000000..8ae762dc
--- /dev/null
+++ b/src/hyper/pages/BSTREE.ht
@@ -0,0 +1,104 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\BinarySearchTreeXmpTitle}{BinarySearchTree}
+\newcommand{\BinarySearchTreeXmpNumber}{9.5}
+%
+% =====================================================================
+\begin{page}{BinarySearchTreeXmpPage}{9.5 BinarySearchTree}
+% =====================================================================
+\beginscroll
+\spadtype{BinarySearchTree(R)} is the domain of binary trees with
+%-% \HDindex{tree!binary search}{BinarySearchTreeXmpPage}{9.5}{BinarySearchTree}
+elements of type \spad{R}, ordered across the nodes of the tree.
+%-% \HDindex{binary search tree}{BinarySearchTreeXmpPage}{9.5}{BinarySearchTree}
+A non-empty binary search tree has a value
+of type \spad{R}, and \spadfun{right} and \spadfun{left}
+binary search subtrees.
+If a subtree is empty, it is displayed as a period (``.'').
+
+\xtc{
+Define a list of values to be placed across the tree.
+The resulting tree has \spad{8} at the root;
+all other elements are in the left subtree.
+}{
+\spadpaste{lv := [8,3,5,4,6,2,1,5,7]\bound{lv}}
+}
+\xtc{
+A convenient way to create a binary search tree is to
+apply the operation \spadfun{binarySearchTree} to a list of elements.
+}{
+\spadpaste{t := binarySearchTree lv\free{lv}\bound{t}}
+}
+\xtc{
+Another approach is to first create an empty binary search tree of integers.
+}{
+\spadpaste{emptybst := empty()\$BSTREE(INT)\bound{e}}
+}
+\xtc{
+Insert the value \spad{8}.
+This establishes \spad{8} as the root of the binary search tree.
+Values inserted later that are less than \spad{8} get stored in
+the \spadfun{left} subtree, others in the \spadfun{right} subtree.
+}{
+\spadpaste{t1 := insert!(8,emptybst)\free{e}\bound{t1}}
+}
+\xtc{
+Insert the value \spad{3}. This number becomes the root of the
+\spadfun{left} subtree of \spad{t1}.
+For optimal retrieval, it is thus important to insert the
+middle elements first.
+}{
+\spadpaste{insert!(3,t1)\free{t1}}
+}
+\xtc{
+We go back to the original tree \spad{t}.
+The leaves of the binary search tree are those which have empty
+\spadfun{left} and \spadfun{right} subtrees.
+}{
+\spadpaste{leaves t\free{t}}
+}
+\xtc{
+The operation
+\spadfun{split}\spad{(k,t)} returns a \spadgloss{record} containing
+the two subtrees: one with all elements ``less'' than \spad{k},
+another with elements ``greater'' than \spad{k}.
+}{
+\spadpaste{split(3,t)\free{t}}
+}
+\xtc{
+Define \userfun{insertRoot} to insert new elements by
+creating a new node.
+}{
+\spadpaste{insertRoot: (INT,BSTREE INT) -> BSTREE INT\bound{x}}
+}
+\xtc{
+The new node puts the inserted value between its
+``less'' tree and ``greater'' tree.
+}{
+\begin{spadsrc}[\bound{x1}\free{x}]
+insertRoot(x, t) ==
+    a := split(x, t)
+    node(a.less, x, a.greater)
+\end{spadsrc}
+}
+\xtc{
+Function \userfun{buildFromRoot} builds
+a binary search tree from a list of elements \spad{ls}
+and the empty tree \spad{emptybst}.
+}{
+\spadpaste{buildFromRoot ls == reduce(insertRoot,ls,emptybst)\bound{x2}\free{x1 e}}
+}
+\xtc{
+Apply this to the reverse of the list \spad{lv}.
+}{
+\spadpaste{rt := buildFromRoot reverse lv\bound{rt}\free{lv x2}}
+}
+\xtc{
+Have \Language{} check that these are equal.
+}{
+\spadpaste{(t = rt)@Boolean\free{rt t}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/BSTREE.pht b/src/hyper/pages/BSTREE.pht
new file mode 100644
index 00000000..b18379f6
--- /dev/null
+++ b/src/hyper/pages/BSTREE.pht
@@ -0,0 +1,196 @@
+\begin{patch}{BinarySearchTreeXmpPagePatch1}
+\begin{paste}{BinarySearchTreeXmpPageFull1}{BinarySearchTreeXmpPageEmpty1}
+\pastebutton{BinarySearchTreeXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{lv := [8,3,5,4,6,2,1,5,7]\bound{lv }}
+\indentrel{3}\begin{verbatim}
+   (1)  [8,3,5,4,6,2,1,5,7]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPageEmpty1}
+\begin{paste}{BinarySearchTreeXmpPageEmpty1}{BinarySearchTreeXmpPagePatch1}
+\pastebutton{BinarySearchTreeXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{lv := [8,3,5,4,6,2,1,5,7]\bound{lv }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPagePatch2}
+\begin{paste}{BinarySearchTreeXmpPageFull2}{BinarySearchTreeXmpPageEmpty2}
+\pastebutton{BinarySearchTreeXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{t := binarySearchTree lv\free{lv }\bound{t }}
+\indentrel{3}\begin{verbatim}
+   (2)  [[[1,2,.],3,[4,5,[5,6,7]]],8,.]
+                 Type: BinarySearchTree PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPageEmpty2}
+\begin{paste}{BinarySearchTreeXmpPageEmpty2}{BinarySearchTreeXmpPagePatch2}
+\pastebutton{BinarySearchTreeXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{t := binarySearchTree lv\free{lv }\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPagePatch3}
+\begin{paste}{BinarySearchTreeXmpPageFull3}{BinarySearchTreeXmpPageEmpty3}
+\pastebutton{BinarySearchTreeXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{emptybst := empty()$BSTREE(INT)\bound{e }}
+\indentrel{3}\begin{verbatim}
+   (3)  []
+                         Type: BinarySearchTree Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPageEmpty3}
+\begin{paste}{BinarySearchTreeXmpPageEmpty3}{BinarySearchTreeXmpPagePatch3}
+\pastebutton{BinarySearchTreeXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{emptybst := empty()$BSTREE(INT)\bound{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPagePatch4}
+\begin{paste}{BinarySearchTreeXmpPageFull4}{BinarySearchTreeXmpPageEmpty4}
+\pastebutton{BinarySearchTreeXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{t1 := insert!(8,emptybst)\free{e }\bound{t1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  8
+                         Type: BinarySearchTree Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPageEmpty4}
+\begin{paste}{BinarySearchTreeXmpPageEmpty4}{BinarySearchTreeXmpPagePatch4}
+\pastebutton{BinarySearchTreeXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{t1 := insert!(8,emptybst)\free{e }\bound{t1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPagePatch5}
+\begin{paste}{BinarySearchTreeXmpPageFull5}{BinarySearchTreeXmpPageEmpty5}
+\pastebutton{BinarySearchTreeXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{insert!(3,t1)\free{t1 }}
+\indentrel{3}\begin{verbatim}
+   (5)  [3,8,.]
+                         Type: BinarySearchTree Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPageEmpty5}
+\begin{paste}{BinarySearchTreeXmpPageEmpty5}{BinarySearchTreeXmpPagePatch5}
+\pastebutton{BinarySearchTreeXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{insert!(3,t1)\free{t1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPagePatch6}
+\begin{paste}{BinarySearchTreeXmpPageFull6}{BinarySearchTreeXmpPageEmpty6}
+\pastebutton{BinarySearchTreeXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{leaves t\free{t }}
+\indentrel{3}\begin{verbatim}
+   (6)  [1,4,5,7]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPageEmpty6}
+\begin{paste}{BinarySearchTreeXmpPageEmpty6}{BinarySearchTreeXmpPagePatch6}
+\pastebutton{BinarySearchTreeXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{leaves t\free{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPagePatch7}
+\begin{paste}{BinarySearchTreeXmpPageFull7}{BinarySearchTreeXmpPageEmpty7}
+\pastebutton{BinarySearchTreeXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{split(3,t)\free{t }}
+\indentrel{3}\begin{verbatim}
+   (7)
+   [less= [1,2,.],greater= [[.,3,[4,5,[5,6,7]]],8,.]]
+Type: Record(less: BinarySearchTree PositiveInteger,greater: BinarySearchTree PositiveInteger)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPageEmpty7}
+\begin{paste}{BinarySearchTreeXmpPageEmpty7}{BinarySearchTreeXmpPagePatch7}
+\pastebutton{BinarySearchTreeXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{split(3,t)\free{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPagePatch8}
+\begin{paste}{BinarySearchTreeXmpPageFull8}{BinarySearchTreeXmpPageEmpty8}
+\pastebutton{BinarySearchTreeXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{insertRoot: (INT,BSTREE INT) -> BSTREE INT\bound{x }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPageEmpty8}
+\begin{paste}{BinarySearchTreeXmpPageEmpty8}{BinarySearchTreeXmpPagePatch8}
+\pastebutton{BinarySearchTreeXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{insertRoot: (INT,BSTREE INT) -> BSTREE INT\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPagePatch9}
+\begin{paste}{BinarySearchTreeXmpPageFull9}{BinarySearchTreeXmpPageEmpty9}
+\pastebutton{BinarySearchTreeXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{insertRoot(x, t) ==
+    a := split(x, t)
+    node(a.less, x, a.greater)
+\bound{x1 }\free{x }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPageEmpty9}
+\begin{paste}{BinarySearchTreeXmpPageEmpty9}{BinarySearchTreeXmpPagePatch9}
+\pastebutton{BinarySearchTreeXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{insertRoot(x, t) ==
+    a := split(x, t)
+    node(a.less, x, a.greater)
+\bound{x1 }\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPagePatch10}
+\begin{paste}{BinarySearchTreeXmpPageFull10}{BinarySearchTreeXmpPageEmpty10}
+\pastebutton{BinarySearchTreeXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{buildFromRoot ls == reduce(insertRoot,ls,emptybst)\bound{x2 }\free{x1 e }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPageEmpty10}
+\begin{paste}{BinarySearchTreeXmpPageEmpty10}{BinarySearchTreeXmpPagePatch10}
+\pastebutton{BinarySearchTreeXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{buildFromRoot ls == reduce(insertRoot,ls,emptybst)\bound{x2 }\free{x1 e }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPagePatch11}
+\begin{paste}{BinarySearchTreeXmpPageFull11}{BinarySearchTreeXmpPageEmpty11}
+\pastebutton{BinarySearchTreeXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{rt := buildFromRoot reverse lv\bound{rt }\free{lv x2 }}
+\indentrel{3}\begin{verbatim}
+   (11)  [[[1,2,.],3,[4,5,[5,6,7]]],8,.]
+                         Type: BinarySearchTree Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPageEmpty11}
+\begin{paste}{BinarySearchTreeXmpPageEmpty11}{BinarySearchTreeXmpPagePatch11}
+\pastebutton{BinarySearchTreeXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{rt := buildFromRoot reverse lv\bound{rt }\free{lv x2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPagePatch12}
+\begin{paste}{BinarySearchTreeXmpPageFull12}{BinarySearchTreeXmpPageEmpty12}
+\pastebutton{BinarySearchTreeXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{(t = rt)@Boolean\free{rt t }}
+\indentrel{3}\begin{verbatim}
+   (12)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{BinarySearchTreeXmpPageEmpty12}
+\begin{paste}{BinarySearchTreeXmpPageEmpty12}{BinarySearchTreeXmpPagePatch12}
+\pastebutton{BinarySearchTreeXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{(t = rt)@Boolean\free{rt t }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/CARD.ht b/src/hyper/pages/CARD.ht
new file mode 100644
index 00000000..f3bbc75b
--- /dev/null
+++ b/src/hyper/pages/CARD.ht
@@ -0,0 +1,148 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\CardinalNumberXmpTitle}{CardinalNumber}
+\newcommand{\CardinalNumberXmpNumber}{9.6}
+%
+% =====================================================================
+\begin{page}{CardinalNumberXmpPage}{9.6 CardinalNumber}
+% =====================================================================
+\beginscroll
+
+The \spadtype{CardinalNumber} domain can be used for values indicating
+the cardinality of sets, both finite and infinite.
+For example, the \spadfunFrom{dimension}{VectorSpace} operation in the
+category \spadtype{VectorSpace} returns a cardinal number.
+
+The non-negative integers have a natural construction as cardinals
+\begin{verbatim}
+0 = #{ }, 1 = {0}, 2 = {0, 1}, ..., n = {i | 0 <= i < n}.
+\end{verbatim}
+The fact that \spad{0} acts as a zero for the multiplication of cardinals is
+equivalent to the axiom of choice.
+
+\xtc{
+Cardinal numbers can be created by conversion from non-negative integers.
+}{
+\spadpaste{c0 := 0 :: CardinalNumber \bound{c0}}
+}
+\xtc{
+}{
+\spadpaste{c1 := 1 :: CardinalNumber \bound{c1}}
+}
+\xtc{
+}{
+\spadpaste{c2 := 2 :: CardinalNumber \bound{c2}}
+}
+\xtc{
+}{
+\spadpaste{c3 := 3 :: CardinalNumber \bound{c3}}
+}
+\xtc{
+They can also be obtained as the named cardinal \spad{Aleph(n)}.
+}{
+\spadpaste{A0 := Aleph 0 \bound{A0}}
+}
+\xtc{
+}{
+\spadpaste{A1 := Aleph 1 \bound{A1}}
+}
+\xtc{
+The \spadfunFrom{finite?}{CardinalNumber} operation tests whether a value is a
+finite cardinal, that is, a non-negative integer.
+}{
+\spadpaste{finite? c2 \free{c2}}
+}
+\xtc{
+}{
+\spadpaste{finite? A0 \free{A0}}
+}
+\xtc{
+Similarly, the \spadfunFrom{countable?}{CardinalNumber}
+operation determines whether a value is
+a countable cardinal, that is, finite or \spad{Aleph(0)}.
+}{
+\spadpaste{countable? c2 \free{c2}}
+}
+\xtc{
+}{
+\spadpaste{countable? A0 \free{A0}}
+}
+\xtc{
+}{
+\spadpaste{countable? A1 \free{A1}}
+}
+Arithmetic operations are defined on cardinal numbers as follows:
+If \spad{x = \#X}  and  \spad{y = \#Y} then
+
+\indent{0}
+\spad{x+y  = \#(X+Y)} \tab{20} cardinality of the disjoint union \newline
+\spad{x-y  = \#(X-Y)} \tab{20} cardinality of the relative complement \newline
+\spad{x*y  = \#(X*Y)} \tab{20} cardinality of the Cartesian product \newline
+\spad{x**y = \#(X**Y)}\tab{20} cardinality of the set of maps from \spad{Y} to \spad{X} \newline
+
+\xtc{
+Here are some arithmetic examples.
+}{
+\spadpaste{[c2 + c2, c2 + A1] \free{c2 A1}}
+}
+\xtc{
+}{
+\spadpaste{[c0*c2, c1*c2, c2*c2, c0*A1, c1*A1, c2*A1, A0*A1] \free{c0,c1,c2,A0,A1}}
+}
+\xtc{
+}{
+\spadpaste{[c2**c0, c2**c1, c2**c2, A1**c0, A1**c1, A1**c2] \free{c0,c1,c2,A1}}
+}
+\xtc{
+Subtraction is a partial operation: it is not defined
+when subtracting a larger cardinal from a smaller one, nor
+when subtracting two equal infinite cardinals.
+}{
+\spadpaste{[c2-c1, c2-c2, c2-c3, A1-c2, A1-A0, A1-A1] \free{c1,c2,c3,A0,A1}}
+}
+%-% \HDindex{generalized continuum hypothesis}{CardinalNumberXmpPage}{9.6}{CardinalNumber}
+The generalized continuum hypothesis asserts that
+\begin{verbatim}
+2**Aleph i = Aleph(i+1)
+\end{verbatim}
+and is independent of the axioms of set theory.\footnote{Goedel,
+{\it The consistency of the continuum hypothesis,}
+Ann. Math. Studies, Princeton Univ. Press, 1940.}
+\xtc{
+The \spadtype{CardinalNumber} domain provides an operation to assert
+whether the hypothesis is to be assumed.
+}{
+\spadpaste{generalizedContinuumHypothesisAssumed true \bound{GCH}}
+}
+\xtc{
+When the generalized continuum hypothesis
+is assumed, exponentiation to a transfinite power is allowed.
+}{
+\spadpaste{[c0**A0, c1**A0, c2**A0, A0**A0, A0**A1, A1**A0, A1**A1] \free{c0,c1,c2,A0,A1,GCH}}
+}
+
+Three commonly encountered cardinal numbers are
+
+\indent{0}
+\spad{a = \#}{\bf Z} \tab{20}       countable infinity \newline
+\spad{c = \#}{\bf R} \tab{20}      the continuum       \newline
+\spad{f = \#\{g| g: [0,1] -> {\bf R}\}}                \newline
+
+\xtc{
+In this domain, these values are obtained under the
+generalized continuum hypothesis in this way.
+}{
+\spadpaste{a := Aleph 0 \free{GCH}\bound{a}}
+}
+\xtc{
+}{
+\spadpaste{c := 2**a    \free{a}  \bound{c}}
+}
+\xtc{
+}{
+\spadpaste{f := 2**c    \free{c}  \bound{f}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/CARD.pht b/src/hyper/pages/CARD.pht
new file mode 100644
index 00000000..887e029e
--- /dev/null
+++ b/src/hyper/pages/CARD.pht
@@ -0,0 +1,321 @@
+\begin{patch}{CardinalNumberXmpPagePatch1}
+\begin{paste}{CardinalNumberXmpPageFull1}{CardinalNumberXmpPageEmpty1}
+\pastebutton{CardinalNumberXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{c0 := 0 :: CardinalNumber\bound{c0 }}
+\indentrel{3}\begin{verbatim}
+   (1)  0
+                                   Type: CardinalNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty1}
+\begin{paste}{CardinalNumberXmpPageEmpty1}{CardinalNumberXmpPagePatch1}
+\pastebutton{CardinalNumberXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{c0 := 0 :: CardinalNumber\bound{c0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch2}
+\begin{paste}{CardinalNumberXmpPageFull2}{CardinalNumberXmpPageEmpty2}
+\pastebutton{CardinalNumberXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{c1 := 1 :: CardinalNumber\bound{c1 }}
+\indentrel{3}\begin{verbatim}
+   (2)  1
+                                   Type: CardinalNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty2}
+\begin{paste}{CardinalNumberXmpPageEmpty2}{CardinalNumberXmpPagePatch2}
+\pastebutton{CardinalNumberXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{c1 := 1 :: CardinalNumber\bound{c1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch3}
+\begin{paste}{CardinalNumberXmpPageFull3}{CardinalNumberXmpPageEmpty3}
+\pastebutton{CardinalNumberXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{c2 := 2 :: CardinalNumber\bound{c2 }}
+\indentrel{3}\begin{verbatim}
+   (3)  2
+                                   Type: CardinalNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty3}
+\begin{paste}{CardinalNumberXmpPageEmpty3}{CardinalNumberXmpPagePatch3}
+\pastebutton{CardinalNumberXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{c2 := 2 :: CardinalNumber\bound{c2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch4}
+\begin{paste}{CardinalNumberXmpPageFull4}{CardinalNumberXmpPageEmpty4}
+\pastebutton{CardinalNumberXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{c3 := 3 :: CardinalNumber\bound{c3 }}
+\indentrel{3}\begin{verbatim}
+   (4)  3
+                                   Type: CardinalNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty4}
+\begin{paste}{CardinalNumberXmpPageEmpty4}{CardinalNumberXmpPagePatch4}
+\pastebutton{CardinalNumberXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{c3 := 3 :: CardinalNumber\bound{c3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch5}
+\begin{paste}{CardinalNumberXmpPageFull5}{CardinalNumberXmpPageEmpty5}
+\pastebutton{CardinalNumberXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{A0 := Aleph 0\bound{A0 }}
+\indentrel{3}\begin{verbatim}
+   (5)  Aleph(0)
+                                   Type: CardinalNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty5}
+\begin{paste}{CardinalNumberXmpPageEmpty5}{CardinalNumberXmpPagePatch5}
+\pastebutton{CardinalNumberXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{A0 := Aleph 0\bound{A0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch6}
+\begin{paste}{CardinalNumberXmpPageFull6}{CardinalNumberXmpPageEmpty6}
+\pastebutton{CardinalNumberXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{A1 := Aleph 1\bound{A1 }}
+\indentrel{3}\begin{verbatim}
+   (6)  Aleph(1)
+                                   Type: CardinalNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty6}
+\begin{paste}{CardinalNumberXmpPageEmpty6}{CardinalNumberXmpPagePatch6}
+\pastebutton{CardinalNumberXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{A1 := Aleph 1\bound{A1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch7}
+\begin{paste}{CardinalNumberXmpPageFull7}{CardinalNumberXmpPageEmpty7}
+\pastebutton{CardinalNumberXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{finite? c2\free{c2 }}
+\indentrel{3}\begin{verbatim}
+   (7)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty7}
+\begin{paste}{CardinalNumberXmpPageEmpty7}{CardinalNumberXmpPagePatch7}
+\pastebutton{CardinalNumberXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{finite? c2\free{c2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch8}
+\begin{paste}{CardinalNumberXmpPageFull8}{CardinalNumberXmpPageEmpty8}
+\pastebutton{CardinalNumberXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{finite? A0\free{A0 }}
+\indentrel{3}\begin{verbatim}
+   (8)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty8}
+\begin{paste}{CardinalNumberXmpPageEmpty8}{CardinalNumberXmpPagePatch8}
+\pastebutton{CardinalNumberXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{finite? A0\free{A0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch9}
+\begin{paste}{CardinalNumberXmpPageFull9}{CardinalNumberXmpPageEmpty9}
+\pastebutton{CardinalNumberXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{countable? c2\free{c2 }}
+\indentrel{3}\begin{verbatim}
+   (9)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty9}
+\begin{paste}{CardinalNumberXmpPageEmpty9}{CardinalNumberXmpPagePatch9}
+\pastebutton{CardinalNumberXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{countable? c2\free{c2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch10}
+\begin{paste}{CardinalNumberXmpPageFull10}{CardinalNumberXmpPageEmpty10}
+\pastebutton{CardinalNumberXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{countable? A0\free{A0 }}
+\indentrel{3}\begin{verbatim}
+   (10)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty10}
+\begin{paste}{CardinalNumberXmpPageEmpty10}{CardinalNumberXmpPagePatch10}
+\pastebutton{CardinalNumberXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{countable? A0\free{A0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch11}
+\begin{paste}{CardinalNumberXmpPageFull11}{CardinalNumberXmpPageEmpty11}
+\pastebutton{CardinalNumberXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{countable? A1\free{A1 }}
+\indentrel{3}\begin{verbatim}
+   (11)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty11}
+\begin{paste}{CardinalNumberXmpPageEmpty11}{CardinalNumberXmpPagePatch11}
+\pastebutton{CardinalNumberXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{countable? A1\free{A1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch12}
+\begin{paste}{CardinalNumberXmpPageFull12}{CardinalNumberXmpPageEmpty12}
+\pastebutton{CardinalNumberXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{[c2 + c2, c2 + A1]\free{c2 A1 }}
+\indentrel{3}\begin{verbatim}
+   (12)  [4,Aleph(1)]
+                              Type: List CardinalNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty12}
+\begin{paste}{CardinalNumberXmpPageEmpty12}{CardinalNumberXmpPagePatch12}
+\pastebutton{CardinalNumberXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{[c2 + c2, c2 + A1]\free{c2 A1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch13}
+\begin{paste}{CardinalNumberXmpPageFull13}{CardinalNumberXmpPageEmpty13}
+\pastebutton{CardinalNumberXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{[c0*c2, c1*c2, c2*c2, c0*A1, c1*A1, c2*A1, A0*A1]\free{c0 c1 c2 A0 A1 }}
+\indentrel{3}\begin{verbatim}
+   (13)  [0,2,4,0,Aleph(1),Aleph(1),Aleph(1)]
+                              Type: List CardinalNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty13}
+\begin{paste}{CardinalNumberXmpPageEmpty13}{CardinalNumberXmpPagePatch13}
+\pastebutton{CardinalNumberXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{[c0*c2, c1*c2, c2*c2, c0*A1, c1*A1, c2*A1, A0*A1]\free{c0 c1 c2 A0 A1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch14}
+\begin{paste}{CardinalNumberXmpPageFull14}{CardinalNumberXmpPageEmpty14}
+\pastebutton{CardinalNumberXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{[c2**c0, c2**c1, c2**c2, A1**c0, A1**c1, A1**c2]\free{c0 c1 c2 A1 }}
+\indentrel{3}\begin{verbatim}
+   (14)  [1,2,4,1,Aleph(1),Aleph(1)]
+                              Type: List CardinalNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty14}
+\begin{paste}{CardinalNumberXmpPageEmpty14}{CardinalNumberXmpPagePatch14}
+\pastebutton{CardinalNumberXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{[c2**c0, c2**c1, c2**c2, A1**c0, A1**c1, A1**c2]\free{c0 c1 c2 A1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch15}
+\begin{paste}{CardinalNumberXmpPageFull15}{CardinalNumberXmpPageEmpty15}
+\pastebutton{CardinalNumberXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{[c2-c1, c2-c2, c2-c3, A1-c2, A1-A0, A1-A1]\free{c1 c2 c3 A0 A1 }}
+\indentrel{3}\begin{verbatim}
+   (15)  [1,0,"failed",Aleph(1),Aleph(1),"failed"]
+              Type: List Union(CardinalNumber,"failed")
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty15}
+\begin{paste}{CardinalNumberXmpPageEmpty15}{CardinalNumberXmpPagePatch15}
+\pastebutton{CardinalNumberXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{[c2-c1, c2-c2, c2-c3, A1-c2, A1-A0, A1-A1]\free{c1 c2 c3 A0 A1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch16}
+\begin{paste}{CardinalNumberXmpPageFull16}{CardinalNumberXmpPageEmpty16}
+\pastebutton{CardinalNumberXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{generalizedContinuumHypothesisAssumed true\bound{GCH }}
+\indentrel{3}\begin{verbatim}
+   (16)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty16}
+\begin{paste}{CardinalNumberXmpPageEmpty16}{CardinalNumberXmpPagePatch16}
+\pastebutton{CardinalNumberXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{generalizedContinuumHypothesisAssumed true\bound{GCH }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch17}
+\begin{paste}{CardinalNumberXmpPageFull17}{CardinalNumberXmpPageEmpty17}
+\pastebutton{CardinalNumberXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{[c0**A0, c1**A0, c2**A0, A0**A0, A0**A1, A1**A0, A1**A1]\free{c0 c1 c2 A0 A1 GCH }}
+\indentrel{3}\begin{verbatim}
+   (17)
+   [0,1,Aleph(1),Aleph(1),Aleph(2),Aleph(1),Aleph(2)]
+                              Type: List CardinalNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty17}
+\begin{paste}{CardinalNumberXmpPageEmpty17}{CardinalNumberXmpPagePatch17}
+\pastebutton{CardinalNumberXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{[c0**A0, c1**A0, c2**A0, A0**A0, A0**A1, A1**A0, A1**A1]\free{c0 c1 c2 A0 A1 GCH }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch18}
+\begin{paste}{CardinalNumberXmpPageFull18}{CardinalNumberXmpPageEmpty18}
+\pastebutton{CardinalNumberXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{a := Aleph 0\free{GCH }\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (18)  Aleph(0)
+                                   Type: CardinalNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty18}
+\begin{paste}{CardinalNumberXmpPageEmpty18}{CardinalNumberXmpPagePatch18}
+\pastebutton{CardinalNumberXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{a := Aleph 0\free{GCH }\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch19}
+\begin{paste}{CardinalNumberXmpPageFull19}{CardinalNumberXmpPageEmpty19}
+\pastebutton{CardinalNumberXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{c := 2**a\free{a }\bound{c }}
+\indentrel{3}\begin{verbatim}
+   (19)  Aleph(1)
+                                   Type: CardinalNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty19}
+\begin{paste}{CardinalNumberXmpPageEmpty19}{CardinalNumberXmpPagePatch19}
+\pastebutton{CardinalNumberXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{c := 2**a\free{a }\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPagePatch20}
+\begin{paste}{CardinalNumberXmpPageFull20}{CardinalNumberXmpPageEmpty20}
+\pastebutton{CardinalNumberXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{f := 2**c\free{c }\bound{f }}
+\indentrel{3}\begin{verbatim}
+   (20)  Aleph(2)
+                                   Type: CardinalNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CardinalNumberXmpPageEmpty20}
+\begin{paste}{CardinalNumberXmpPageEmpty20}{CardinalNumberXmpPagePatch20}
+\pastebutton{CardinalNumberXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{f := 2**c\free{c }\bound{f }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/CARTEN.ht b/src/hyper/pages/CARTEN.ht
new file mode 100644
index 00000000..702621f3
--- /dev/null
+++ b/src/hyper/pages/CARTEN.ht
@@ -0,0 +1,418 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\CartesianTensorXmpTitle}{CartesianTensor}
+\newcommand{\CartesianTensorXmpNumber}{9.7}
+%
+% =====================================================================
+\begin{page}{CartesianTensorXmpPage}{9.7 CartesianTensor}
+% =====================================================================
+\beginscroll
+
+\spadtype{CartesianTensor(i0,dim,R)}
+provides Cartesian tensors with
+components belonging to a commutative ring \spad{R}.
+Tensors can be described as a generalization of vectors and matrices.
+%-% \HDindex{tensor!Cartesian}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+This gives a concise {\it tensor algebra} for multilinear objects
+supported by the \spadtype{CartesianTensor} domain.
+You can form the inner or outer product of any two tensors and you can add
+or subtract tensors with the same number of components.
+Additionally, various forms of traces and transpositions are useful.
+
+The \spadtype{CartesianTensor} constructor allows you to specify the
+minimum index for subscripting.
+In what follows we discuss in detail how to manipulate tensors.
+
+\xtc{
+Here we construct the domain of Cartesian tensors of dimension 2 over the
+integers, with indices starting at 1.
+}{
+\spadpaste{CT := CARTEN(i0 := 1, 2, Integer) \bound{CT i0}}
+}
+
+\subsubsection{Forming tensors}
+
+\labelSpace{2pc}
+\xtc{
+Scalars can be converted to tensors of rank zero.
+}{
+\spadpaste{t0: CT := 8 \free{CT}\bound{t0}}
+}
+\xtc{
+}{
+\spadpaste{rank t0 \free{t0}}
+}
+\xtc{
+Vectors (mathematical direct products, rather than one dimensional array
+structures) can be converted to tensors of rank one.
+}{
+\spadpaste{v: DirectProduct(2, Integer) := directProduct [3,4] \bound{v}}
+}
+\xtc{
+}{
+\spadpaste{Tv: CT := v \free{v CT}\bound{Tv}}
+}
+\xtc{
+Matrices can be converted to tensors of rank two.
+}{
+\spadpaste{m: SquareMatrix(2, Integer) := matrix [[1,2],[4,5]] \bound{m}}
+}
+\xtc{
+}{
+\spadpaste{Tm: CT := m \bound{Tm}\free{CT m}}
+}
+\xtc{
+}{
+\spadpaste{n: SquareMatrix(2, Integer) := matrix [[2,3],[0,1]] \bound{n}}
+}
+\xtc{
+}{
+\spadpaste{Tn: CT := n \bound{Tn}\free{CT n}}
+}
+\xtc{
+In general, a tensor of rank \spad{k} can be formed by making a
+list of rank \spad{k-1} tensors or, alternatively, a \spad{k}-deep nested
+list of lists.
+}{
+\spadpaste{t1: CT := [2, 3] \free{CT}\bound{t1}}
+}
+\xtc{
+}{
+\spadpaste{rank t1 \free{t1}}
+}
+\xtc{
+}{
+\spadpaste{t2: CT := [t1, t1] \free{CT t1}\bound{t2}}
+}
+\xtc{
+}{
+\spadpaste{t3: CT := [t2, t2] \free{CT t2}\bound{t3}}
+}
+\xtc{
+}{
+\spadpaste{tt: CT := [t3, t3]; tt := [tt, tt] \free{CT t3}\bound{tt}}
+}
+\xtc{
+}{
+\spadpaste{rank tt \free{tt}}
+}
+
+\subsubsection{Multiplication}
+%\head{subsection}{Multiplication}{ugxCartenMult}
+
+Given two tensors of rank \spad{k1} and \spad{k2}, the outer
+%-% \HDindex{tensor!outer product}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+\spadfunFrom{product}{CartesianTensor} forms a new tensor of rank
+\spad{k1+k2}.
+\xtc{
+Here
+\texht{$T_{mn}(i,j,k,l) = T_m(i,j) \  T_n(k,l).$}{%
+\spad{Tmn(i,j,k,l) = Tm(i,j)*Tn(k,l).}}
+}{
+\spadpaste{Tmn := product(Tm, Tn)\free{Tm Tn}\bound{Tmn}}
+}
+
+The inner product (\spadfunFrom{contract}{CartesianTensor}) forms a tensor
+%-% \HDindex{tensor!inner product}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+of rank \spad{k1+k2-2}.
+This product generalizes the vector dot product and matrix-vector product
+by summing component products along two indices.
+\xtc{
+Here we sum along the second index of \texht{$T_m$}{\spad{Tm}} and the
+first index of \texht{$T_v$}{\spad{Tv}}.
+Here
+\texht{$T_{mv} = \sum_{j=1}^{\hbox{\tiny\rm dim}} T_m(i,j) \  T_v(j)$}{%
+\spad{Tmv(i) = sum Tm(i,j) * Tv(j) for j in 1..dim.}}
+}{
+\spadpaste{Tmv := contract(Tm,2,Tv,1)\free{Tm Tv}\bound{Tmv}}
+}
+
+The multiplication operator \spadopFrom{*}{CartesianTensor} is scalar
+multiplication or an inner product depending on the ranks of the
+arguments.
+\xtc{
+If either argument is rank zero it is treated as scalar multiplication.
+Otherwise, \spad{a*b} is the inner product summing the last index of
+\spad{a} with the first index of \spad{b}.
+}{
+\spadpaste{Tm*Tv \free{Tm Tv}}
+}
+\xtc{
+This definition is consistent with the inner product on matrices
+and vectors.
+}{
+\spadpaste{Tmv = m * v \free{Tmv m v}}
+}
+
+\subsubsection{Selecting Components}
+%\head{subsection}{Selecting Components}{ugxCartenSelect}
+
+\labelSpace{2pc}
+\xtc{
+For tensors of low rank (that is, four or less), components can be selected
+by applying the tensor to its indices.
+}{
+\spadpaste{t0()      \free{t0}}
+}
+\xtc{
+}{
+\spadpaste{t1(1+1)   \free{t1}}
+}
+\xtc{
+}{
+\spadpaste{t2(2,1)   \free{t2}}
+}
+\xtc{
+}{
+\spadpaste{t3(2,1,2) \free{t3}}
+}
+\xtc{
+}{
+\spadpaste{Tmn(2,1,2,1) \free{Tmn}}
+}
+\xtc{
+A general indexing mechanism is provided for a list of indices.
+}{
+\spadpaste{t0[]        \free{t0}}
+}
+\xtc{
+}{
+\spadpaste{t1[2]        \free{t1}}
+}
+\xtc{
+}{
+\spadpaste{t2[2,1]      \free{t2}}
+}
+\xtc{
+The general mechanism works for tensors of arbitrary rank, but is
+somewhat less efficient since the intermediate index list must be created.
+}{
+\spadpaste{t3[2,1,2]    \free{t3}}
+}
+\xtc{
+}{
+\spadpaste{Tmn[2,1,2,1] \free{Tmn}}
+}
+
+\subsubsection{Contraction}
+%\head{subsection}{Contraction}{ugxCartenContract}
+
+A ``contraction'' between two tensors is an inner product, as we have
+%-% \HDindex{tensor!contraction}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+seen above.
+You can also contract a pair of indices of a single tensor.
+This corresponds to a ``trace'' in linear algebra.
+The expression \spad{contract(t,k1,k2)} forms a new tensor by summing the
+diagonal given by indices in position \spad{k1} and \spad{k2}.
+\xtc{
+This is the tensor given by
+\texht{$xT_{mn} = \sum_{k=1}^{\hbox{\tiny\rm dim}} T_{mn}(k,k,i,j).$}{%
+\spad{cTmn(i,j) = sum Tmn(k,k,i,j) for k in 1..dim.}}
+}{
+\spadpaste{cTmn := contract(Tmn,1,2) \free{Tmn}\bound{cTmn}}
+}
+\xtc{
+Since \spad{Tmn} is the outer product of matrix \spad{m} and matrix \spad{n},
+the above is equivalent to this.
+}{
+\spadpaste{trace(m) * n \free{m n}}
+}
+\xtc{
+In this and the next few examples,
+we show all possible contractions of \spad{Tmn} and their matrix
+algebra equivalents.
+}{
+\spadpaste{contract(Tmn,1,2) = trace(m) * n      \free{Tmn m n}}
+}
+\xtc{
+}{
+\spadpaste{contract(Tmn,1,3) = transpose(m) * n  \free{Tmn m n}}
+}
+\xtc{
+}{
+\spadpaste{contract(Tmn,1,4) = transpose(m) * transpose(n) \free{Tmn m n}}
+}
+\xtc{
+}{
+\spadpaste{contract(Tmn,2,3) = m * n             \free{Tmn m n}}
+}
+\xtc{
+}{
+\spadpaste{contract(Tmn,2,4) = m * transpose(n)  \free{Tmn m n}}
+}
+\xtc{
+}{
+\spadpaste{contract(Tmn,3,4) = trace(n) * m      \free{Tmn m n}}
+}
+
+\subsubsection{Transpositions}
+%\head{subsection}{Transpositions}{ugxCartenTranspos}
+
+You can exchange any desired pair of indices using the
+\spadfunFrom{transpose}{CartesianTensor} operation.
+
+\labelSpace{1pc}
+\xtc{
+Here the indices in positions one and three are exchanged, that is,
+\texht{$tT_{mn}(i,j,k,l) = T_{mn}(k,j,i,l).$}{%
+\spad{tTmn(i,j,k,l) = Tmn(k,j,i,l).}}
+}{
+\spadpaste{tTmn := transpose(Tmn,1,3) \free{tTmn}}
+}
+\xtc{
+If no indices are specified, the first and last index are exchanged.
+}{
+\spadpaste{transpose Tmn \free{Tmn}}
+}
+\xtc{
+This is consistent with the matrix transpose.
+}{
+\spadpaste{transpose Tm = transpose m \free{Tm m}}
+}
+
+If a more complicated reordering of the indices is required, then the
+\spadfunFrom{reindex}{CartesianTensor} operation can be used.
+This operation allows the indices to be arbitrarily permuted.
+\xtc{
+This defines
+\texht{$rT_{mn}(i,j,k,l) = \allowbreak T_{mn}(i,l,j,k).$}{%
+\spad{rTmn(i,j,k,l) = Tmn(i,l,j,k).}}
+}{
+\spadpaste{rTmn := reindex(Tmn, [1,4,2,3]) \free{Tmn}\bound{rTmn}}
+}
+
+\subsubsection{Arithmetic}
+%\head{subsection}{Arithmetic}{ugxCartenArith}
+
+\labelSpace{2pc}
+\xtc{
+Tensors of equal rank can be added or subtracted so arithmetic
+expressions can be used to produce new tensors.
+}{
+\spadpaste{tt := transpose(Tm)*Tn - Tn*transpose(Tm) \free{Tm Tn}\bound{tt2}}
+}
+\xtc{
+}{
+\spadpaste{Tv*(tt+Tn) \free{tt2 Tv Tn}}
+}
+\xtc{
+}{
+\spadpaste{reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm) \free{Tm Tn}}
+}
+
+\subsubsection{Specific Tensors}
+%\head{subsection}{Specific Tensors}{ugxCartenSpecific}
+
+Two specific tensors have properties which depend only on the
+dimension.
+
+\labelSpace{1pc}
+\xtc{
+The Kronecker delta satisfies
+%-% \HDindex{Kronecker delta}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+\texht{}{%
+\begin{verbatim}
+             +-
+             |   1  if i  = j
+delta(i,j) = |
+             |   0  if i ^= j
+             +-
+\end{verbatim}
+}%
+}{
+\spadpaste{delta:  CT := kroneckerDelta()   \free{CT}\bound{delta}}
+}
+\xtc{
+This can be used to reindex via contraction.
+}{
+\spadpaste{contract(Tmn, 2, delta, 1) = reindex(Tmn, [1,3,4,2]) \free{Tmn delta}}
+}
+
+\xtc{
+The Levi Civita symbol determines the sign of a permutation of indices.
+}{
+\spadpaste{epsilon:CT := leviCivitaSymbol() \free{CT}\bound{epsilon}}
+}
+Here we have:
+\texht{}{%
+\begin{verbatim}
+epsilon(i1,...,idim)
+     = +1  if i1,...,idim is an even permutation of i0,...,i0+dim-1
+     = -1  if i1,...,idim is an  odd permutation of i0,...,i0+dim-1
+     =  0  if i1,...,idim is not   a permutation of i0,...,i0+dim-1
+\end{verbatim}
+}
+\xtc{
+This property can be used to form determinants.
+}{
+\spadpaste{contract(epsilon*Tm*epsilon, 1,2) = 2 * determinant m \free{epsilon Tm m}}
+}
+
+\subsubsection{Properties of the CartesianTensor domain}
+%\head{subsection}{Properties of the CartesianTensor domain}{ugxCartenProp}
+
+\spadtype{GradedModule(R,E)} denotes ``\spad{E}-graded \spad{R}-module'',
+that is, a collection of \spad{R}-modules indexed by an abelian monoid
+\spad{E.}
+%-% \HDexptypeindex{GradedModule}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+An element \spad{g} of \spad{G[s]} for some specific \spad{s}
+in \spad{E} is said to be an element of \spad{G} with
+\spadfunFrom{degree}{GradedModule} \spad{s}.
+Sums are defined in each module \spad{G[s]} so two elements of
+\spad{G} can be added if they have the same degree.
+Morphisms can be defined and composed by degree to give the
+mathematical category of graded modules.
+
+\spadtype{GradedAlgebra(R,E)} denotes ``\spad{E}-graded \spad{R}-algebra.''
+%-% \HDexptypeindex{GradedAlgebra}{CartesianTensorXmpPage}{9.7}{CartesianTensor}
+A graded algebra is a graded module together with a degree preserving
+\spad{R}-bilinear map, called the \spadfunFrom{product}{GradedAlgebra}.
+
+\begin{verbatim}
+degree(product(a,b))= degree(a) + degree(b)
+
+product(r*a,b) = product(a,r*b) = r*product(a,b)
+product(a1+a2,b) = product(a1,b) + product(a2,b)
+product(a,b1+b2) = product(a,b1) + product(a,b2)
+product(a,product(b,c)) = product(product(a,b),c)
+\end{verbatim}
+
+The domain \spadtype{CartesianTensor(i0, dim, R)} belongs
+to the category \spadtype{GradedAlgebra(R, NonNegativeInteger)}.
+The non-negative integer \spadfunFrom{degree}{GradedAlgebra}
+is the tensor rank
+and the graded algebra \spadfunFrom{product}{GradedAlgebra}
+is the tensor outer product.
+The graded module addition captures the notion that
+only tensors of equal rank can be added.
+
+If \spad{V} is a vector space of dimension \spad{dim} over \spad{R},
+then the tensor module \spad{T[k](V)} is defined as
+\begin{verbatim}
+T[0](V) = R
+T[k](V) = T[k-1](V) * V
+\end{verbatim}
+where \spadop{*} denotes the \spad{R}-module tensor
+\spadfunFrom{product}{GradedAlgebra}.
+\spadtype{CartesianTensor(i0,dim,R)} is the graded algebra in which
+the degree \spad{k} module is \spad{T[k](V)}.
+
+\subsubsection{Tensor Calculus}
+%\head{subsection}{Tensor Calculus}{ugxCartenCalculus}
+
+It should be noted here that often tensors are used in the context of
+tensor-valued manifold maps.
+This leads to the notion of covariant and contravariant bases with tensor
+component functions transforming in specific ways under a change of
+coordinates on the manifold.
+This is no more directly supported by the \spadtype{CartesianTensor}
+domain than it is by the \spadtype{Vector} domain.
+However, it is possible to have the components implicitly represent
+component maps by choosing a polynomial or expression type for the
+components.
+In this case, it is up to the user to satisfy any constraints which arise
+on the basis of this interpretation.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/CARTEN.pht b/src/hyper/pages/CARTEN.pht
new file mode 100644
index 00000000..ed63da81
--- /dev/null
+++ b/src/hyper/pages/CARTEN.pht
@@ -0,0 +1,846 @@
+\begin{patch}{CartesianTensorXmpPagePatch1}
+\begin{paste}{CartesianTensorXmpPageFull1}{CartesianTensorXmpPageEmpty1}
+\pastebutton{CartesianTensorXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{CT := CARTEN(i0 := 1, 2, Integer)\bound{CT i0 }}
+\indentrel{3}\begin{verbatim}
+   (1)  CartesianTensor(1,2,Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty1}
+\begin{paste}{CartesianTensorXmpPageEmpty1}{CartesianTensorXmpPagePatch1}
+\pastebutton{CartesianTensorXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{CT := CARTEN(i0 := 1, 2, Integer)\bound{CT i0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch2}
+\begin{paste}{CartesianTensorXmpPageFull2}{CartesianTensorXmpPageEmpty2}
+\pastebutton{CartesianTensorXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{t0: CT := 8\free{CT }\bound{t0 }}
+\indentrel{3}\begin{verbatim}
+   (2)  8
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty2}
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+\tab{5}\spadcommand{t0: CT := 8\free{CT }\bound{t0 }}
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+
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+\begin{paste}{CartesianTensorXmpPageFull3}{CartesianTensorXmpPageEmpty3}
+\pastebutton{CartesianTensorXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{rank t0\free{t0 }}
+\indentrel{3}\begin{verbatim}
+   (3)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
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+\pastebutton{CartesianTensorXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{v: DirectProduct(2, Integer) := directProduct [3,4]\bound{v }}
+\indentrel{3}\begin{verbatim}
+   (4)  [3,4]
+                         Type: DirectProduct(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
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+\tab{5}\spadcommand{Tv: CT := v\free{v CT }\bound{Tv }}
+\indentrel{3}\begin{verbatim}
+   (5)  [3,4]
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
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+\pastebutton{CartesianTensorXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{m: SquareMatrix(2, Integer) := matrix [[1,2],[4,5]]\bound{m }}
+\indentrel{3}\begin{verbatim}
+        Ú1  2¿
+   (6)  ³    ³
+        À4  5Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
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+
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+\begin{paste}{CartesianTensorXmpPageFull7}{CartesianTensorXmpPageEmpty7}
+\pastebutton{CartesianTensorXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{Tm: CT := m\bound{Tm }\free{CT m }}
+\indentrel{3}\begin{verbatim}
+        Ú1  2¿
+   (7)  ³    ³
+        À4  5Ù
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty7}
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+\tab{5}\spadcommand{Tm: CT := m\bound{Tm }\free{CT m }}
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+\tab{5}\spadcommand{n: SquareMatrix(2, Integer) := matrix [[2,3],[0,1]]\bound{n }}
+\indentrel{3}\begin{verbatim}
+        Ú2  3¿
+   (8)  ³    ³
+        À0  1Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty8}
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+\pastebutton{CartesianTensorXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{Tn: CT := n\bound{Tn }\free{CT n }}
+\indentrel{3}\begin{verbatim}
+        Ú2  3¿
+   (9)  ³    ³
+        À0  1Ù
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty9}
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+\tab{5}\spadcommand{Tn: CT := n\bound{Tn }\free{CT n }}
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+\tab{5}\spadcommand{t1: CT := [2, 3]\free{CT }\bound{t1 }}
+\indentrel{3}\begin{verbatim}
+   (10)  [2,3]
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
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+\begin{paste}{CartesianTensorXmpPageFull11}{CartesianTensorXmpPageEmpty11}
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+\indentrel{3}\begin{verbatim}
+   (11)  1
+                                  Type: PositiveInteger
+\end{verbatim}
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+
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+\tab{5}\spadcommand{t2: CT := [t1, t1]\free{CT t1 }\bound{t2 }}
+\indentrel{3}\begin{verbatim}
+         Ú2  3¿
+   (12)  ³    ³
+         À2  3Ù
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty12}
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+\tab{5}\spadcommand{t2: CT := [t1, t1]\free{CT t1 }\bound{t2 }}
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+\begin{paste}{CartesianTensorXmpPageFull13}{CartesianTensorXmpPageEmpty13}
+\pastebutton{CartesianTensorXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{t3: CT := [t2, t2]\free{CT t2 }\bound{t3 }}
+\indentrel{3}\begin{verbatim}
+          Ú2  3¿ Ú2  3¿
+   (13)  [³    ³,³    ³]
+          À2  3Ù À2  3Ù
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty13}
+\begin{paste}{CartesianTensorXmpPageEmpty13}{CartesianTensorXmpPagePatch13}
+\pastebutton{CartesianTensorXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{t3: CT := [t2, t2]\free{CT t2 }\bound{t3 }}
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+\tab{5}\spadcommand{tt: CT := [t3, t3]; tt := [tt, tt]\free{CT t3 }\bound{tt }}
+\indentrel{3}\begin{verbatim}
+          ÚÚ2  3¿  Ú2  3¿¿ ÚÚ2  3¿  Ú2  3¿¿
+          ³³    ³  ³    ³³ ³³    ³  ³    ³³
+          ³À2  3Ù  À2  3Ù³ ³À2  3Ù  À2  3Ù³
+   (14)  [³              ³,³              ³]
+          ³Ú2  3¿  Ú2  3¿³ ³Ú2  3¿  Ú2  3¿³
+          ³³    ³  ³    ³³ ³³    ³  ³    ³³
+          ÀÀ2  3Ù  À2  3ÙÙ ÀÀ2  3Ù  À2  3ÙÙ
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
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+\tab{5}\spadcommand{tt: CT := [t3, t3]; tt := [tt, tt]\free{CT t3 }\bound{tt }}
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+
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+\begin{paste}{CartesianTensorXmpPageFull15}{CartesianTensorXmpPageEmpty15}
+\pastebutton{CartesianTensorXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{rank tt\free{tt }}
+\indentrel{3}\begin{verbatim}
+   (15)  5
+                                  Type: PositiveInteger
+\end{verbatim}
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+\pastebutton{CartesianTensorXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{Tmn := product(Tm, Tn)\free{Tm Tn }\bound{Tmn }}
+\indentrel{3}\begin{verbatim}
+         ÚÚ2  3¿    Ú4  6¿ ¿
+         ³³    ³    ³    ³ ³
+         ³À0  1Ù    À0  2Ù ³
+   (16)  ³                 ³
+         ³Ú8  12¿  Ú10  15¿³
+         ³³     ³  ³      ³³
+         ÀÀ0  4 Ù  À0   5 ÙÙ
+                     Type: CartesianTensor(1,2,Integer)
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+\tab{5}\spadcommand{Tmv := contract(Tm,2,Tv,1)\free{Tm Tv }\bound{Tmv }}
+\indentrel{3}\begin{verbatim}
+   (17)  [11,32]
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
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+   (18)  [11,32]
+                     Type: CartesianTensor(1,2,Integer)
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+\indentrel{3}\begin{verbatim}
+   (19)  [11,32]= [11,32]
+            Type: Equation CartesianTensor(1,2,Integer)
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+   (20)  8
+                                  Type: PositiveInteger
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+   (21)  3
+                                  Type: PositiveInteger
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+                                  Type: PositiveInteger
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+   (24)  0
+                               Type: NonNegativeInteger
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+   (25)  8
+                                  Type: PositiveInteger
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+                                  Type: PositiveInteger
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+                                  Type: PositiveInteger
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+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty29}
+\begin{paste}{CartesianTensorXmpPageEmpty29}{CartesianTensorXmpPagePatch29}
+\pastebutton{CartesianTensorXmpPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{Tmn[2,1,2,1]\free{Tmn }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch30}
+\begin{paste}{CartesianTensorXmpPageFull30}{CartesianTensorXmpPageEmpty30}
+\pastebutton{CartesianTensorXmpPageFull30}{\hidepaste}
+\tab{5}\spadcommand{cTmn := contract(Tmn,1,2)\free{Tmn }\bound{cTmn }}
+\indentrel{3}\begin{verbatim}
+         Ú12  18¿
+   (30)  ³      ³
+         À0   6 Ù
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty30}
+\begin{paste}{CartesianTensorXmpPageEmpty30}{CartesianTensorXmpPagePatch30}
+\pastebutton{CartesianTensorXmpPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{cTmn := contract(Tmn,1,2)\free{Tmn }\bound{cTmn }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch31}
+\begin{paste}{CartesianTensorXmpPageFull31}{CartesianTensorXmpPageEmpty31}
+\pastebutton{CartesianTensorXmpPageFull31}{\hidepaste}
+\tab{5}\spadcommand{trace(m) * n\free{m n }}
+\indentrel{3}\begin{verbatim}
+         Ú12  18¿
+   (31)  ³      ³
+         À0   6 Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty31}
+\begin{paste}{CartesianTensorXmpPageEmpty31}{CartesianTensorXmpPagePatch31}
+\pastebutton{CartesianTensorXmpPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{trace(m) * n\free{m n }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch32}
+\begin{paste}{CartesianTensorXmpPageFull32}{CartesianTensorXmpPageEmpty32}
+\pastebutton{CartesianTensorXmpPageFull32}{\hidepaste}
+\tab{5}\spadcommand{contract(Tmn,1,2) = trace(m) * n\free{Tmn m n }}
+\indentrel{3}\begin{verbatim}
+         Ú12  18¿  Ú12  18¿
+   (32)  ³      ³= ³      ³
+         À0   6 Ù  À0   6 Ù
+            Type: Equation CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty32}
+\begin{paste}{CartesianTensorXmpPageEmpty32}{CartesianTensorXmpPagePatch32}
+\pastebutton{CartesianTensorXmpPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{contract(Tmn,1,2) = trace(m) * n\free{Tmn m n }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch33}
+\begin{paste}{CartesianTensorXmpPageFull33}{CartesianTensorXmpPageEmpty33}
+\pastebutton{CartesianTensorXmpPageFull33}{\hidepaste}
+\tab{5}\spadcommand{contract(Tmn,1,3) = transpose(m) * n\free{Tmn m n }}
+\indentrel{3}\begin{verbatim}
+         Ú2  7 ¿  Ú2  7 ¿
+   (33)  ³     ³= ³     ³
+         À4  11Ù  À4  11Ù
+            Type: Equation CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty33}
+\begin{paste}{CartesianTensorXmpPageEmpty33}{CartesianTensorXmpPagePatch33}
+\pastebutton{CartesianTensorXmpPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{contract(Tmn,1,3) = transpose(m) * n\free{Tmn m n }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch34}
+\begin{paste}{CartesianTensorXmpPageFull34}{CartesianTensorXmpPageEmpty34}
+\pastebutton{CartesianTensorXmpPageFull34}{\hidepaste}
+\tab{5}\spadcommand{contract(Tmn,1,4) = transpose(m) * transpose(n)\free{Tmn m n }}
+\indentrel{3}\begin{verbatim}
+         Ú14  4¿  Ú14  4¿
+   (34)  ³     ³= ³     ³
+         À19  5Ù  À19  5Ù
+            Type: Equation CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty34}
+\begin{paste}{CartesianTensorXmpPageEmpty34}{CartesianTensorXmpPagePatch34}
+\pastebutton{CartesianTensorXmpPageEmpty34}{\showpaste}
+\tab{5}\spadcommand{contract(Tmn,1,4) = transpose(m) * transpose(n)\free{Tmn m n }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch35}
+\begin{paste}{CartesianTensorXmpPageFull35}{CartesianTensorXmpPageEmpty35}
+\pastebutton{CartesianTensorXmpPageFull35}{\hidepaste}
+\tab{5}\spadcommand{contract(Tmn,2,3) = m * n\free{Tmn m n }}
+\indentrel{3}\begin{verbatim}
+         Ú2  5 ¿  Ú2  5 ¿
+   (35)  ³     ³= ³     ³
+         À8  17Ù  À8  17Ù
+            Type: Equation CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty35}
+\begin{paste}{CartesianTensorXmpPageEmpty35}{CartesianTensorXmpPagePatch35}
+\pastebutton{CartesianTensorXmpPageEmpty35}{\showpaste}
+\tab{5}\spadcommand{contract(Tmn,2,3) = m * n\free{Tmn m n }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch36}
+\begin{paste}{CartesianTensorXmpPageFull36}{CartesianTensorXmpPageEmpty36}
+\pastebutton{CartesianTensorXmpPageFull36}{\hidepaste}
+\tab{5}\spadcommand{contract(Tmn,2,4) = m * transpose(n)\free{Tmn m n }}
+\indentrel{3}\begin{verbatim}
+         Ú8   2¿  Ú8   2¿
+   (36)  ³     ³= ³     ³
+         À23  5Ù  À23  5Ù
+            Type: Equation CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty36}
+\begin{paste}{CartesianTensorXmpPageEmpty36}{CartesianTensorXmpPagePatch36}
+\pastebutton{CartesianTensorXmpPageEmpty36}{\showpaste}
+\tab{5}\spadcommand{contract(Tmn,2,4) = m * transpose(n)\free{Tmn m n }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch37}
+\begin{paste}{CartesianTensorXmpPageFull37}{CartesianTensorXmpPageEmpty37}
+\pastebutton{CartesianTensorXmpPageFull37}{\hidepaste}
+\tab{5}\spadcommand{contract(Tmn,3,4) = trace(n) * m\free{Tmn m n }}
+\indentrel{3}\begin{verbatim}
+         Ú3   6 ¿  Ú3   6 ¿
+   (37)  ³      ³= ³      ³
+         À12  15Ù  À12  15Ù
+            Type: Equation CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty37}
+\begin{paste}{CartesianTensorXmpPageEmpty37}{CartesianTensorXmpPagePatch37}
+\pastebutton{CartesianTensorXmpPageEmpty37}{\showpaste}
+\tab{5}\spadcommand{contract(Tmn,3,4) = trace(n) * m\free{Tmn m n }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch38}
+\begin{paste}{CartesianTensorXmpPageFull38}{CartesianTensorXmpPageEmpty38}
+\pastebutton{CartesianTensorXmpPageFull38}{\hidepaste}
+\tab{5}\spadcommand{tTmn := transpose(Tmn,1,3)\free{tTmn }}
+\indentrel{3}\begin{verbatim}
+         ÚÚ2  3 ¿  Ú4   6 ¿¿
+         ³³     ³  ³      ³³
+         ³À8  12Ù  À10  15Ù³
+   (38)  ³                 ³
+         ³Ú0  1¿    Ú0  2¿ ³
+         ³³    ³    ³    ³ ³
+         ÀÀ0  4Ù    À0  5Ù Ù
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty38}
+\begin{paste}{CartesianTensorXmpPageEmpty38}{CartesianTensorXmpPagePatch38}
+\pastebutton{CartesianTensorXmpPageEmpty38}{\showpaste}
+\tab{5}\spadcommand{tTmn := transpose(Tmn,1,3)\free{tTmn }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch39}
+\begin{paste}{CartesianTensorXmpPageFull39}{CartesianTensorXmpPageEmpty39}
+\pastebutton{CartesianTensorXmpPageFull39}{\hidepaste}
+\tab{5}\spadcommand{transpose Tmn\free{Tmn }}
+\indentrel{3}\begin{verbatim}
+         ÚÚ2  8¿   Ú4  10¿¿
+         ³³    ³   ³     ³³
+         ³À0  0Ù   À0  0 Ù³
+   (39)  ³                ³
+         ³Ú3  12¿  Ú6  15¿³
+         ³³     ³  ³     ³³
+         ÀÀ1  4 Ù  À2  5 ÙÙ
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty39}
+\begin{paste}{CartesianTensorXmpPageEmpty39}{CartesianTensorXmpPagePatch39}
+\pastebutton{CartesianTensorXmpPageEmpty39}{\showpaste}
+\tab{5}\spadcommand{transpose Tmn\free{Tmn }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch40}
+\begin{paste}{CartesianTensorXmpPageFull40}{CartesianTensorXmpPageEmpty40}
+\pastebutton{CartesianTensorXmpPageFull40}{\hidepaste}
+\tab{5}\spadcommand{transpose Tm = transpose m\free{Tm m }}
+\indentrel{3}\begin{verbatim}
+         Ú1  4¿  Ú1  4¿
+   (40)  ³    ³= ³    ³
+         À2  5Ù  À2  5Ù
+            Type: Equation CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty40}
+\begin{paste}{CartesianTensorXmpPageEmpty40}{CartesianTensorXmpPagePatch40}
+\pastebutton{CartesianTensorXmpPageEmpty40}{\showpaste}
+\tab{5}\spadcommand{transpose Tm = transpose m\free{Tm m }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch41}
+\begin{paste}{CartesianTensorXmpPageFull41}{CartesianTensorXmpPageEmpty41}
+\pastebutton{CartesianTensorXmpPageFull41}{\hidepaste}
+\tab{5}\spadcommand{rTmn := reindex(Tmn, [1,4,2,3])\free{Tmn }\bound{rTmn }}
+\indentrel{3}\begin{verbatim}
+         ÚÚ2  0¿   Ú3  1¿ ¿
+         ³³    ³   ³    ³ ³
+         ³À4  0Ù   À6  2Ù ³
+   (41)  ³                ³
+         ³Ú8   0¿  Ú12  4¿³
+         ³³     ³  ³     ³³
+         ÀÀ10  0Ù  À15  5ÙÙ
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty41}
+\begin{paste}{CartesianTensorXmpPageEmpty41}{CartesianTensorXmpPagePatch41}
+\pastebutton{CartesianTensorXmpPageEmpty41}{\showpaste}
+\tab{5}\spadcommand{rTmn := reindex(Tmn, [1,4,2,3])\free{Tmn }\bound{rTmn }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch42}
+\begin{paste}{CartesianTensorXmpPageFull42}{CartesianTensorXmpPageEmpty42}
+\pastebutton{CartesianTensorXmpPageFull42}{\hidepaste}
+\tab{5}\spadcommand{tt := transpose(Tm)*Tn - Tn*transpose(Tm)\free{Tm Tn }\bound{tt2 }}
+\indentrel{3}\begin{verbatim}
+         Ú- 6  - 16¿
+   (42)  ³         ³
+         À 2    6  Ù
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty42}
+\begin{paste}{CartesianTensorXmpPageEmpty42}{CartesianTensorXmpPagePatch42}
+\pastebutton{CartesianTensorXmpPageEmpty42}{\showpaste}
+\tab{5}\spadcommand{tt := transpose(Tm)*Tn - Tn*transpose(Tm)\free{Tm Tn }\bound{tt2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch43}
+\begin{paste}{CartesianTensorXmpPageFull43}{CartesianTensorXmpPageEmpty43}
+\pastebutton{CartesianTensorXmpPageFull43}{\hidepaste}
+\tab{5}\spadcommand{Tv*(tt+Tn)\free{tt2 Tv Tn }}
+\indentrel{3}\begin{verbatim}
+   (43)  [- 4,- 11]
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty43}
+\begin{paste}{CartesianTensorXmpPageEmpty43}{CartesianTensorXmpPagePatch43}
+\pastebutton{CartesianTensorXmpPageEmpty43}{\showpaste}
+\tab{5}\spadcommand{Tv*(tt+Tn)\free{tt2 Tv Tn }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch44}
+\begin{paste}{CartesianTensorXmpPageFull44}{CartesianTensorXmpPageEmpty44}
+\pastebutton{CartesianTensorXmpPageFull44}{\hidepaste}
+\tab{5}\spadcommand{reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm)\free{Tm Tn }}
+\indentrel{3}\begin{verbatim}
+         ÚÚ46   84 ¿  Ú57   114¿¿
+         ³³        ³  ³        ³³
+         ³À174  212Ù  À228  285Ù³
+   (44)  ³                      ³
+         ³ Ú18  24¿    Ú17  30¿ ³
+         ³ ³      ³    ³      ³ ³
+         À À57  63Ù    À63  76Ù Ù
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty44}
+\begin{paste}{CartesianTensorXmpPageEmpty44}{CartesianTensorXmpPagePatch44}
+\pastebutton{CartesianTensorXmpPageEmpty44}{\showpaste}
+\tab{5}\spadcommand{reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm)\free{Tm Tn }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch45}
+\begin{paste}{CartesianTensorXmpPageFull45}{CartesianTensorXmpPageEmpty45}
+\pastebutton{CartesianTensorXmpPageFull45}{\hidepaste}
+\tab{5}\spadcommand{delta: CT := kroneckerDelta()\free{CT }\bound{delta }}
+\indentrel{3}\begin{verbatim}
+         Ú1  0¿
+   (45)  ³    ³
+         À0  1Ù
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty45}
+\begin{paste}{CartesianTensorXmpPageEmpty45}{CartesianTensorXmpPagePatch45}
+\pastebutton{CartesianTensorXmpPageEmpty45}{\showpaste}
+\tab{5}\spadcommand{delta: CT := kroneckerDelta()\free{CT }\bound{delta }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch46}
+\begin{paste}{CartesianTensorXmpPageFull46}{CartesianTensorXmpPageEmpty46}
+\pastebutton{CartesianTensorXmpPageFull46}{\hidepaste}
+\tab{5}\spadcommand{contract(Tmn, 2, delta, 1) = reindex(Tmn, [1,3,4,2])\free{Tmn delta }}
+\indentrel{3}\begin{verbatim}
+         Ú Ú2  4¿   Ú0  0¿¿  Ú Ú2  4¿   Ú0  0¿¿
+         ³ ³    ³   ³    ³³  ³ ³    ³   ³    ³³
+         ³ À3  6Ù   À1  2Ù³  ³ À3  6Ù   À1  2Ù³
+   (46)  ³                ³= ³                ³
+         ³Ú8   10¿  Ú0  0¿³  ³Ú8   10¿  Ú0  0¿³
+         ³³      ³  ³    ³³  ³³      ³  ³    ³³
+         ÀÀ12  15Ù  À4  5ÙÙ  ÀÀ12  15Ù  À4  5ÙÙ
+            Type: Equation CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty46}
+\begin{paste}{CartesianTensorXmpPageEmpty46}{CartesianTensorXmpPagePatch46}
+\pastebutton{CartesianTensorXmpPageEmpty46}{\showpaste}
+\tab{5}\spadcommand{contract(Tmn, 2, delta, 1) = reindex(Tmn, [1,3,4,2])\free{Tmn delta }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch47}
+\begin{paste}{CartesianTensorXmpPageFull47}{CartesianTensorXmpPageEmpty47}
+\pastebutton{CartesianTensorXmpPageFull47}{\hidepaste}
+\tab{5}\spadcommand{epsilon:CT := leviCivitaSymbol()\free{CT }\bound{epsilon }}
+\indentrel{3}\begin{verbatim}
+         Ú 0   1¿
+   (47)  ³      ³
+         À- 1  0Ù
+                     Type: CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty47}
+\begin{paste}{CartesianTensorXmpPageEmpty47}{CartesianTensorXmpPagePatch47}
+\pastebutton{CartesianTensorXmpPageEmpty47}{\showpaste}
+\tab{5}\spadcommand{epsilon:CT := leviCivitaSymbol()\free{CT }\bound{epsilon }}
+\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPagePatch48}
+\begin{paste}{CartesianTensorXmpPageFull48}{CartesianTensorXmpPageEmpty48}
+\pastebutton{CartesianTensorXmpPageFull48}{\hidepaste}
+\tab{5}\spadcommand{contract(epsilon*Tm*epsilon, 1,2) = 2 * determinant m\free{epsilon Tm m }}
+\indentrel{3}\begin{verbatim}
+   (48)  - 6= - 6
+            Type: Equation CartesianTensor(1,2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CartesianTensorXmpPageEmpty48}
+\begin{paste}{CartesianTensorXmpPageEmpty48}{CartesianTensorXmpPagePatch48}
+\pastebutton{CartesianTensorXmpPageEmpty48}{\showpaste}
+\tab{5}\spadcommand{contract(epsilon*Tm*epsilon, 1,2) = 2 * determinant m\free{epsilon Tm m }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/CCLASS.ht b/src/hyper/pages/CCLASS.ht
new file mode 100644
index 00000000..e550aade
--- /dev/null
+++ b/src/hyper/pages/CCLASS.ht
@@ -0,0 +1,96 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\CharacterClassXmpTitle}{CharacterClass}
+\newcommand{\CharacterClassXmpNumber}{9.9}
+%
+% =====================================================================
+\begin{page}{CharacterClassXmpPage}{9.9 CharacterClass}
+% =====================================================================
+\beginscroll
+ 
+The \spadtype{CharacterClass} domain allows classes of characters to be
+defined and manipulated efficiently.
+%-% \HDexptypeindex{Character}{CharacterClassXmpPage}{9.9}{CharacterClass}
+ 
+\xtc{
+Character classes can be created by giving either a string or a list
+of characters.
+}{
+\spadpaste{cl1 := charClass [char "a", char "e", char "i", char "o", char "u", char "y"] \bound{cl1}}
+}
+\xtc{
+}{
+\spadpaste{cl2 := charClass "bcdfghjklmnpqrstvwxyz" \bound{cl2}}
+}
+\xtc{
+A number of character classes are predefined for convenience.
+}{
+\spadpaste{digit()}
+}
+\xtc{
+}{
+\spadpaste{hexDigit()}
+}
+\xtc{
+}{
+\spadpaste{upperCase()}
+}
+\xtc{
+}{
+\spadpaste{lowerCase()}
+}
+\xtc{
+}{
+\spadpaste{alphabetic()}
+}
+\xtc{
+}{
+\spadpaste{alphanumeric()}
+}
+\xtc{
+You can quickly test whether a character belongs to a class.
+}{
+\spadpaste{member?(char "a", cl1) \free{cl1}}
+}
+\xtc{
+}{
+\spadpaste{member?(char "a", cl2) \free{cl2}}
+}
+\xtc{
+Classes have the usual set operations because
+the \spadtype{CharacterClass} domain belongs to the category
+\spadtype{FiniteSetAggregate(Character)}.
+}{
+\spadpaste{intersect(cl1, cl2)  \free{cl1 cl2}}
+}
+\xtc{
+}{
+\spadpaste{union(cl1,cl2)       \free{cl1 cl2}}
+}
+\xtc{
+}{
+\spadpaste{difference(cl1,cl2)  \free{cl1 cl2}}
+}
+\xtc{
+}{
+\spadpaste{intersect(complement(cl1),cl2)  \free{cl1 cl2}}
+}
+\xtc{
+You can modify character classes by adding or removing characters.
+}{
+\spadpaste{insert!(char "a", cl2) \free{cl2}\bound{cl22}}
+}
+\xtc{
+}{
+\spadpaste{remove!(char "b", cl2) \free{cl22}\bound{cl23}}
+}
+ 
+For more information on related topics, see
+\downlink{`Character'}{CharacterXmpPage}\ignore{Character} and
+\downlink{`String'}{StringXmpPage}\ignore{String}.
+%
+\showBlurb{CharacterClass}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/CCLASS.pht b/src/hyper/pages/CCLASS.pht
new file mode 100644
index 00000000..d1f1d5eb
--- /dev/null
+++ b/src/hyper/pages/CCLASS.pht
@@ -0,0 +1,259 @@
+\begin{patch}{CharacterClassXmpPagePatch1}
+\begin{paste}{CharacterClassXmpPageFull1}{CharacterClassXmpPageEmpty1}
+\pastebutton{CharacterClassXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{cl1 := charClass [char "a", char "e", char "i", char "o", char "u", char "y"]\bound{cl1 }}
+\indentrel{3}\begin{verbatim}
+   (1)  "aeiouy"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty1}
+\begin{paste}{CharacterClassXmpPageEmpty1}{CharacterClassXmpPagePatch1}
+\pastebutton{CharacterClassXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{cl1 := charClass [char "a", char "e", char "i", char "o", char "u", char "y"]\bound{cl1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch2}
+\begin{paste}{CharacterClassXmpPageFull2}{CharacterClassXmpPageEmpty2}
+\pastebutton{CharacterClassXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{cl2 := charClass "bcdfghjklmnpqrstvwxyz"\bound{cl2 }}
+\indentrel{3}\begin{verbatim}
+   (2)  "bcdfghjklmnpqrstvwxyz"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty2}
+\begin{paste}{CharacterClassXmpPageEmpty2}{CharacterClassXmpPagePatch2}
+\pastebutton{CharacterClassXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{cl2 := charClass "bcdfghjklmnpqrstvwxyz"\bound{cl2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch3}
+\begin{paste}{CharacterClassXmpPageFull3}{CharacterClassXmpPageEmpty3}
+\pastebutton{CharacterClassXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{digit()}
+\indentrel{3}\begin{verbatim}
+   (3)  "0123456789"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty3}
+\begin{paste}{CharacterClassXmpPageEmpty3}{CharacterClassXmpPagePatch3}
+\pastebutton{CharacterClassXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{digit()}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch4}
+\begin{paste}{CharacterClassXmpPageFull4}{CharacterClassXmpPageEmpty4}
+\pastebutton{CharacterClassXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{hexDigit()}
+\indentrel{3}\begin{verbatim}
+   (4)  "0123456789ABCDEFabcdef"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty4}
+\begin{paste}{CharacterClassXmpPageEmpty4}{CharacterClassXmpPagePatch4}
+\pastebutton{CharacterClassXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{hexDigit()}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch5}
+\begin{paste}{CharacterClassXmpPageFull5}{CharacterClassXmpPageEmpty5}
+\pastebutton{CharacterClassXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{upperCase()}
+\indentrel{3}\begin{verbatim}
+   (5)  "ABCDEFGHIJKLMNOPQRSTUVWXYZ"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty5}
+\begin{paste}{CharacterClassXmpPageEmpty5}{CharacterClassXmpPagePatch5}
+\pastebutton{CharacterClassXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{upperCase()}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch6}
+\begin{paste}{CharacterClassXmpPageFull6}{CharacterClassXmpPageEmpty6}
+\pastebutton{CharacterClassXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{lowerCase()}
+\indentrel{3}\begin{verbatim}
+   (6)  "abcdefghijklmnopqrstuvwxyz"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty6}
+\begin{paste}{CharacterClassXmpPageEmpty6}{CharacterClassXmpPagePatch6}
+\pastebutton{CharacterClassXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{lowerCase()}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch7}
+\begin{paste}{CharacterClassXmpPageFull7}{CharacterClassXmpPageEmpty7}
+\pastebutton{CharacterClassXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{alphabetic()}
+\indentrel{3}\begin{verbatim}
+   (7)
+   "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty7}
+\begin{paste}{CharacterClassXmpPageEmpty7}{CharacterClassXmpPagePatch7}
+\pastebutton{CharacterClassXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{alphabetic()}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch8}
+\begin{paste}{CharacterClassXmpPageFull8}{CharacterClassXmpPageEmpty8}
+\pastebutton{CharacterClassXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{alphanumeric()}
+\indentrel{3}\begin{verbatim}
+   (8)
+  "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqr
+  stuvwxyz"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty8}
+\begin{paste}{CharacterClassXmpPageEmpty8}{CharacterClassXmpPagePatch8}
+\pastebutton{CharacterClassXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{alphanumeric()}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch9}
+\begin{paste}{CharacterClassXmpPageFull9}{CharacterClassXmpPageEmpty9}
+\pastebutton{CharacterClassXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{member?(char "a", cl1)\free{cl1 }}
+\indentrel{3}\begin{verbatim}
+   (9)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty9}
+\begin{paste}{CharacterClassXmpPageEmpty9}{CharacterClassXmpPagePatch9}
+\pastebutton{CharacterClassXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{member?(char "a", cl1)\free{cl1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch10}
+\begin{paste}{CharacterClassXmpPageFull10}{CharacterClassXmpPageEmpty10}
+\pastebutton{CharacterClassXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{member?(char "a", cl2)\free{cl2 }}
+\indentrel{3}\begin{verbatim}
+   (10)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty10}
+\begin{paste}{CharacterClassXmpPageEmpty10}{CharacterClassXmpPagePatch10}
+\pastebutton{CharacterClassXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{member?(char "a", cl2)\free{cl2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch11}
+\begin{paste}{CharacterClassXmpPageFull11}{CharacterClassXmpPageEmpty11}
+\pastebutton{CharacterClassXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{intersect(cl1, cl2)\free{cl1 cl2 }}
+\indentrel{3}\begin{verbatim}
+   (11)  "y"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty11}
+\begin{paste}{CharacterClassXmpPageEmpty11}{CharacterClassXmpPagePatch11}
+\pastebutton{CharacterClassXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{intersect(cl1, cl2)\free{cl1 cl2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch12}
+\begin{paste}{CharacterClassXmpPageFull12}{CharacterClassXmpPageEmpty12}
+\pastebutton{CharacterClassXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{union(cl1,cl2)\free{cl1 cl2 }}
+\indentrel{3}\begin{verbatim}
+   (12)  "abcdefghijklmnopqrstuvwxyz"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty12}
+\begin{paste}{CharacterClassXmpPageEmpty12}{CharacterClassXmpPagePatch12}
+\pastebutton{CharacterClassXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{union(cl1,cl2)\free{cl1 cl2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch13}
+\begin{paste}{CharacterClassXmpPageFull13}{CharacterClassXmpPageEmpty13}
+\pastebutton{CharacterClassXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{difference(cl1,cl2)\free{cl1 cl2 }}
+\indentrel{3}\begin{verbatim}
+   (13)  "aeiou"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty13}
+\begin{paste}{CharacterClassXmpPageEmpty13}{CharacterClassXmpPagePatch13}
+\pastebutton{CharacterClassXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{difference(cl1,cl2)\free{cl1 cl2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch14}
+\begin{paste}{CharacterClassXmpPageFull14}{CharacterClassXmpPageEmpty14}
+\pastebutton{CharacterClassXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{intersect(complement(cl1),cl2)\free{cl1 cl2 }}
+\indentrel{3}\begin{verbatim}
+   (14)  "bcdfghjklmnpqrstvwxz"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty14}
+\begin{paste}{CharacterClassXmpPageEmpty14}{CharacterClassXmpPagePatch14}
+\pastebutton{CharacterClassXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{intersect(complement(cl1),cl2)\free{cl1 cl2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch15}
+\begin{paste}{CharacterClassXmpPageFull15}{CharacterClassXmpPageEmpty15}
+\pastebutton{CharacterClassXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{insert!(char "a", cl2)\free{cl2 }\bound{cl22 }}
+\indentrel{3}\begin{verbatim}
+   (15)  "abcdfghjklmnpqrstvwxyz"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty15}
+\begin{paste}{CharacterClassXmpPageEmpty15}{CharacterClassXmpPagePatch15}
+\pastebutton{CharacterClassXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{insert!(char "a", cl2)\free{cl2 }\bound{cl22 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPagePatch16}
+\begin{paste}{CharacterClassXmpPageFull16}{CharacterClassXmpPageEmpty16}
+\pastebutton{CharacterClassXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{remove!(char "b", cl2)\free{cl22 }\bound{cl23 }}
+\indentrel{3}\begin{verbatim}
+   (16)  "acdfghjklmnpqrstvwxyz"
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterClassXmpPageEmpty16}
+\begin{paste}{CharacterClassXmpPageEmpty16}{CharacterClassXmpPagePatch16}
+\pastebutton{CharacterClassXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{remove!(char "b", cl2)\free{cl22 }\bound{cl23 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/CHAR.ht b/src/hyper/pages/CHAR.ht
new file mode 100644
index 00000000..5bda343e
--- /dev/null
+++ b/src/hyper/pages/CHAR.ht
@@ -0,0 +1,92 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\CharacterXmpTitle}{Character}
+\newcommand{\CharacterXmpNumber}{9.8}
+%
+% =====================================================================
+\begin{page}{CharacterXmpPage}{9.8 Character}
+% =====================================================================
+\beginscroll
+ 
+The members of the domain \spadtype{Character} are values
+representing letters, numerals and other text elements.
+For more information on related topics, see
+\downlink{`CharacterClass'}{CharacterClassXmpPage}\ignore{CharacterClass} and \downlink{`String'}{StringXmpPage}\ignore{String}.
+ 
+\xtc{
+Characters can be obtained using \spadtype{String} notation.
+}{
+\spadpaste{chars := [char "a", char "A", char "X", char "8", char "+"] \bound{chars}}
+}
+\xtc{
+Certain characters are available by name.
+This is the blank character.
+}{
+\spadpaste{space()}
+}
+\xtc{
+This is the quote that is used in strings.
+}{
+\spadpaste{quote()}
+}
+\xtc{
+This is the escape character that allows quotes and other characters
+within strings.
+}{
+\spadpaste{escape()}
+}
+\xtc{
+Characters are represented as integers in a machine-dependent way.
+The integer value can be obtained using the \spadfunFrom{ord}{Character}
+operation.
+It is always true that \spad{char(ord c) = c} and \spad{ord(char i) = i},
+provided that \spad{i} is in the range \spad{0..size()\$Character-1}.
+}{
+\spadpaste{[ord c for c in chars] \free{chars}}
+}
+ 
+\xtc{
+The \spadfunFrom{lowerCase}{Character} operation converts an upper case
+letter to the corresponding lower case letter.
+If the argument is not an upper case letter, then it is returned
+unchanged.
+}{
+\spadpaste{[upperCase c for c in chars] \free{chars}}
+}
+\xtc{
+Likewise, the \spadfunFrom{upperCase}{Character} operation converts lower
+case letters to upper case.
+}{
+\spadpaste{[lowerCase c for c in chars] \free{chars}}
+}
+ 
+\xtc{
+A number of tests are available to determine whether characters
+belong to certain families.
+}{
+\spadpaste{[alphabetic? c for c in chars] \free{chars}}
+}
+\xtc{
+}{
+\spadpaste{[upperCase? c for c in chars] \free{chars}}
+}
+\xtc{
+}{
+\spadpaste{[lowerCase? c for c in chars] \free{chars}}
+}
+\xtc{
+}{
+\spadpaste{[digit? c for c in chars] \free{chars}}
+}
+\xtc{
+}{
+\spadpaste{[hexDigit? c for c in chars] \free{chars}}
+}
+\xtc{
+}{
+\spadpaste{[alphanumeric? c for c in chars] \free{chars}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/CHAR.pht b/src/hyper/pages/CHAR.pht
new file mode 100644
index 00000000..2287bc0c
--- /dev/null
+++ b/src/hyper/pages/CHAR.pht
@@ -0,0 +1,208 @@
+\begin{patch}{CharacterXmpPagePatch1}
+\begin{paste}{CharacterXmpPageFull1}{CharacterXmpPageEmpty1}
+\pastebutton{CharacterXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{chars := [char "a", char "A", char "X", char "8", char "+"]\bound{chars }}
+\indentrel{3}\begin{verbatim}
+   (1)  [a,A,X,8,+]
+                                   Type: List Character
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPageEmpty1}
+\begin{paste}{CharacterXmpPageEmpty1}{CharacterXmpPagePatch1}
+\pastebutton{CharacterXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{chars := [char "a", char "A", char "X", char "8", char "+"]\bound{chars }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPagePatch2}
+\begin{paste}{CharacterXmpPageFull2}{CharacterXmpPageEmpty2}
+\pastebutton{CharacterXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{space()}
+\indentrel{3}\begin{verbatim}
+   (2)
+                                        Type: Character
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPageEmpty2}
+\begin{paste}{CharacterXmpPageEmpty2}{CharacterXmpPagePatch2}
+\pastebutton{CharacterXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{space()}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPagePatch3}
+\begin{paste}{CharacterXmpPageFull3}{CharacterXmpPageEmpty3}
+\pastebutton{CharacterXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{quote()}
+\indentrel{3}\begin{verbatim}
+   (3)  "
+                                        Type: Character
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPageEmpty3}
+\begin{paste}{CharacterXmpPageEmpty3}{CharacterXmpPagePatch3}
+\pastebutton{CharacterXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{quote()}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPagePatch4}
+\begin{paste}{CharacterXmpPageFull4}{CharacterXmpPageEmpty4}
+\pastebutton{CharacterXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{escape()}
+\indentrel{3}\begin{verbatim}
+   (4)  _
+                                        Type: Character
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPageEmpty4}
+\begin{paste}{CharacterXmpPageEmpty4}{CharacterXmpPagePatch4}
+\pastebutton{CharacterXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{escape()}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPagePatch5}
+\begin{paste}{CharacterXmpPageFull5}{CharacterXmpPageEmpty5}
+\pastebutton{CharacterXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{[ord c for c in chars]\free{chars }}
+\indentrel{3}\begin{verbatim}
+   (5)  [97,65,88,56,43]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPageEmpty5}
+\begin{paste}{CharacterXmpPageEmpty5}{CharacterXmpPagePatch5}
+\pastebutton{CharacterXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{[ord c for c in chars]\free{chars }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPagePatch6}
+\begin{paste}{CharacterXmpPageFull6}{CharacterXmpPageEmpty6}
+\pastebutton{CharacterXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{[upperCase c for c in chars]\free{chars }}
+\indentrel{3}\begin{verbatim}
+   (6)  [A,A,X,8,+]
+                                   Type: List Character
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPageEmpty6}
+\begin{paste}{CharacterXmpPageEmpty6}{CharacterXmpPagePatch6}
+\pastebutton{CharacterXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{[upperCase c for c in chars]\free{chars }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPagePatch7}
+\begin{paste}{CharacterXmpPageFull7}{CharacterXmpPageEmpty7}
+\pastebutton{CharacterXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{[lowerCase c for c in chars]\free{chars }}
+\indentrel{3}\begin{verbatim}
+   (7)  [a,a,x,8,+]
+                                   Type: List Character
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPageEmpty7}
+\begin{paste}{CharacterXmpPageEmpty7}{CharacterXmpPagePatch7}
+\pastebutton{CharacterXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{[lowerCase c for c in chars]\free{chars }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPagePatch8}
+\begin{paste}{CharacterXmpPageFull8}{CharacterXmpPageEmpty8}
+\pastebutton{CharacterXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{[alphabetic? c for c in chars]\free{chars }}
+\indentrel{3}\begin{verbatim}
+   (8)  [true,true,true,false,false]
+                                     Type: List Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPageEmpty8}
+\begin{paste}{CharacterXmpPageEmpty8}{CharacterXmpPagePatch8}
+\pastebutton{CharacterXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{[alphabetic? c for c in chars]\free{chars }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPagePatch9}
+\begin{paste}{CharacterXmpPageFull9}{CharacterXmpPageEmpty9}
+\pastebutton{CharacterXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{[upperCase? c for c in chars]\free{chars }}
+\indentrel{3}\begin{verbatim}
+   (9)  [false,true,true,false,false]
+                                     Type: List Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPageEmpty9}
+\begin{paste}{CharacterXmpPageEmpty9}{CharacterXmpPagePatch9}
+\pastebutton{CharacterXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{[upperCase? c for c in chars]\free{chars }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPagePatch10}
+\begin{paste}{CharacterXmpPageFull10}{CharacterXmpPageEmpty10}
+\pastebutton{CharacterXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{[lowerCase? c for c in chars]\free{chars }}
+\indentrel{3}\begin{verbatim}
+   (10)  [true,false,false,false,false]
+                                     Type: List Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPageEmpty10}
+\begin{paste}{CharacterXmpPageEmpty10}{CharacterXmpPagePatch10}
+\pastebutton{CharacterXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{[lowerCase? c for c in chars]\free{chars }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPagePatch11}
+\begin{paste}{CharacterXmpPageFull11}{CharacterXmpPageEmpty11}
+\pastebutton{CharacterXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{[digit? c for c in chars]\free{chars }}
+\indentrel{3}\begin{verbatim}
+   (11)  [false,false,false,true,false]
+                                     Type: List Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPageEmpty11}
+\begin{paste}{CharacterXmpPageEmpty11}{CharacterXmpPagePatch11}
+\pastebutton{CharacterXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{[digit? c for c in chars]\free{chars }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPagePatch12}
+\begin{paste}{CharacterXmpPageFull12}{CharacterXmpPageEmpty12}
+\pastebutton{CharacterXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{[hexDigit? c for c in chars]\free{chars }}
+\indentrel{3}\begin{verbatim}
+   (12)  [true,true,false,true,false]
+                                     Type: List Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPageEmpty12}
+\begin{paste}{CharacterXmpPageEmpty12}{CharacterXmpPagePatch12}
+\pastebutton{CharacterXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{[hexDigit? c for c in chars]\free{chars }}
+\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPagePatch13}
+\begin{paste}{CharacterXmpPageFull13}{CharacterXmpPageEmpty13}
+\pastebutton{CharacterXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{[alphanumeric? c for c in chars]\free{chars }}
+\indentrel{3}\begin{verbatim}
+   (13)  [true,true,true,true,false]
+                                     Type: List Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CharacterXmpPageEmpty13}
+\begin{paste}{CharacterXmpPageEmpty13}{CharacterXmpPagePatch13}
+\pastebutton{CharacterXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{[alphanumeric? c for c in chars]\free{chars }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/CLIF.ht b/src/hyper/pages/CLIF.ht
new file mode 100644
index 00000000..23dbf74b
--- /dev/null
+++ b/src/hyper/pages/CLIF.ht
@@ -0,0 +1,296 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\CliffordAlgebraXmpTitle}{CliffordAlgebra}
+\newcommand{\CliffordAlgebraXmpNumber}{9.10}
+%
+% =====================================================================
+\begin{page}{CliffordAlgebraXmpPage}{9.10 CliffordAlgebra}
+% =====================================================================
+\beginscroll
+
+\noindent
+\spadtype{CliffordAlgebra(n,K,Q)} defines a vector space of dimension
+\texht{$2^n$}{\spad{2**n}} over the field \texht{$K$}{\spad{K}} with a
+given quadratic form \spad{Q}.
+If \texht{$\{e_1, \ldots, e_n\}$}{\spad{\{e(i), 1<=i<=n\}}}
+is a basis for  \texht{$K^n$}{\spad{K**n}} then
+\begin{verbatim}
+{ 1,
+  e(i) 1 <= i <= n,
+  e(i1)*e(i2) 1 <= i1 < i2 <=n,
+  ...,
+  e(1)*e(2)*...*e(n) }
+\end{verbatim}
+is a basis for the Clifford algebra.
+The algebra is defined by the relations
+\begin{verbatim}
+e(i)*e(i) = Q(e(i))
+e(i)*e(j) = -e(j)*e(i),  i ^= j
+\end{verbatim}
+Examples of Clifford Algebras are
+gaussians (complex numbers), quaternions,
+exterior algebras and spin algebras.
+%
+
+\beginmenu
+    \menudownlink{{9.10.1. The Complex Numbers as a Clifford Algebra}}{ugxCliffordComplexPage}
+    \menudownlink{{9.10.2. The Quaternion Numbers as a Clifford Algebra}}{ugxCliffordQuaternPage}
+    \menudownlink{{9.10.3. The Exterior Algebra on a Three Space}}{ugxCliffordExteriorPage}
+    \menudownlink{{9.10.4. The Dirac Spin Algebra}}{ugxCliffordDiracPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxCliffordComplexTitle}{The Complex Numbers as a Clifford Algebra}
+\newcommand{\ugxCliffordComplexNumber}{9.10.1.}
+%
+% =====================================================================
+\begin{page}{ugxCliffordComplexPage}{9.10.1. The Complex Numbers as a Clifford Algebra}
+% =====================================================================
+\beginscroll
+
+\labelSpace{5pc}
+\xtc{
+This is the field over which we will work, rational functions with
+integer coefficients.
+}{
+\spadpaste{K := Fraction Polynomial Integer \bound{K}}
+}
+\xtc{
+We use this matrix for the quadratic form.
+}{
+\spadpaste{m := matrix [[-1]] \bound{m}}
+}
+\xtc{
+We get complex arithmetic by using this domain.
+%-% \HDindex{complex numbers}{ugxCliffordComplexPage}{9.10.1.}{The Complex Numbers as a Clifford Algebra}
+}{
+\spadpaste{C := CliffordAlgebra(1, K, quadraticForm m) \free{K m}\bound{C}}
+}
+\xtc{
+Here is \spad{i}, the usual square root of \spad{-1.}
+}{
+\spadpaste{i: C := e(1)   \bound{i}\free{C}}
+}
+\xtc{
+Here are some examples of the arithmetic.
+}{
+\spadpaste{x := a + b * i \bound{x}\free{i}}
+}
+\xtc{
+}{
+\spadpaste{y := c + d * i \bound{y}\free{i}}
+}
+\xtc{
+See \downlink{`Complex'}{ComplexXmpPage}\ignore{Complex} for examples of \Language{}'s constructor
+implementing complex numbers.
+}{
+\spadpaste{x * y \free{x y}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxCliffordQuaternTitle}{The Quaternion Numbers as a Clifford Algebra}
+\newcommand{\ugxCliffordQuaternNumber}{9.10.2.}
+%
+% =====================================================================
+\begin{page}{ugxCliffordQuaternPage}{9.10.2. The Quaternion Numbers as a Clifford Algebra}
+% =====================================================================
+\beginscroll
+
+\labelSpace{3pc}
+\xtc{
+This is the field over which we will work, rational functions with
+integer coefficients.
+}{
+\spadpaste{K := Fraction Polynomial Integer \bound{K}}
+}
+\xtc{
+We use this matrix for the quadratic form.
+}{
+\spadpaste{m := matrix [[-1,0],[0,-1]] \bound{m}}
+}
+\xtc{
+The resulting domain is the quaternions.
+%-% \HDindex{quaternions}{ugxCliffordQuaternPage}{9.10.2.}{The Quaternion Numbers as a Clifford Algebra}
+}{
+\spadpaste{H  := CliffordAlgebra(2, K, quadraticForm m) \free{K m}\bound{H}}
+}
+\xtc{
+We use Hamilton's notation for \spad{i},\spad{j},\spad{k}.
+}{
+\spadpaste{i: H  := e(1) \free{H}\bound{i}}
+}
+\xtc{
+}{
+\spadpaste{j: H  := e(2) \free{H}\bound{j}}
+}
+\xtc{
+}{
+\spadpaste{k: H  := i * j \free{H,i,j}\bound{k}}
+}
+\xtc{
+}{
+\spadpaste{x := a + b * i + c * j + d * k \free{i j k}\bound{x}}
+}
+\xtc{
+}{
+\spadpaste{y := e + f * i + g * j + h * k \free{i j k}\bound{y}}
+}
+\xtc{
+}{
+\spadpaste{x + y \free{x y}}
+}
+\xtc{
+}{
+\spadpaste{x * y \free{x y}}
+}
+\xtc{
+See \downlink{`Quaternion'}{QuaternionXmpPage}\ignore{Quaternion} for examples of \Language{}'s constructor
+implementing quaternions.
+}{
+\spadpaste{y * x \free{x y}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxCliffordExteriorTitle}{The Exterior Algebra on a Three Space}
+\newcommand{\ugxCliffordExteriorNumber}{9.10.3.}
+%
+% =====================================================================
+\begin{page}{ugxCliffordExteriorPage}{9.10.3. The Exterior Algebra on a Three Space}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+\xtc{
+This is the field over which we will work, rational functions with
+integer coefficients.
+}{
+\spadpaste{K := Fraction Polynomial Integer \bound{K}}
+}
+\xtc{
+If we chose the three by three zero quadratic form, we obtain
+%-% \HDindex{exterior algebra}{ugxCliffordExteriorPage}{9.10.3.}{The Exterior Algebra on a Three Space}
+the exterior algebra on \spad{e(1),e(2),e(3)}.
+%-% \HDindex{algebra!exterior}{ugxCliffordExteriorPage}{9.10.3.}{The Exterior Algebra on a Three Space}
+}{
+\spadpaste{Ext := CliffordAlgebra(3, K, quadraticForm 0) \bound{Ext}\free{K}}
+}
+\xtc{
+This is a three dimensional vector algebra.
+We define \spad{i}, \spad{j}, \spad{k} as the unit vectors.
+}{
+\spadpaste{i: Ext := e(1) \free{Ext}\bound{i}}
+}
+\xtc{
+}{
+\spadpaste{j: Ext := e(2) \free{Ext}\bound{j}}
+}
+\xtc{
+}{
+\spadpaste{k: Ext := e(3) \free{Ext}\bound{k}}
+}
+\xtc{
+Now it is possible to do arithmetic.
+}{
+\spadpaste{x := x1*i + x2*j + x3*k \free{i j k}\bound{x}}
+}
+\xtc{
+}{
+\spadpaste{y := y1*i + y2*j + y3*k \free{i j k}\bound{y}}
+}
+\xtc{
+}{
+\spadpaste{x + y         \free{x y}}
+}
+\xtc{
+}{
+\spadpaste{x * y + y * x \free{x y}}
+}
+\xtc{
+On an \spad{n} space, a grade \spad{p} form has a dual \spad{n-p} form.
+In particular, in three space the dual of a grade two element identifies
+\spad{e1*e2->e3, e2*e3->e1, e3*e1->e2}.
+}{
+\spadpaste{dual2 a == coefficient(a,[2,3]) * i + coefficient(a,[3,1]) * j + coefficient(a,[1,2]) * k \free{i j k}\bound{dual2}}
+}
+\xtc{
+The vector cross product is then given by this.
+}{
+\spadpaste{dual2(x*y) \free{x y dual2}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxCliffordDiracTitle}{The Dirac Spin Algebra}
+\newcommand{\ugxCliffordDiracNumber}{9.10.4.}
+%
+% =====================================================================
+\begin{page}{ugxCliffordDiracPage}{9.10.4. The Dirac Spin Algebra}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+%-% \HDindex{Dirac spin algebra}{ugxCliffordDiracPage}{9.10.4.}{The Dirac Spin Algebra}
+%-% \HDindex{algebra!Dirac spin}{ugxCliffordDiracPage}{9.10.4.}{The Dirac Spin Algebra}
+%
+\xtc{
+In this section we will work over the field of rational numbers.
+}{
+\spadpaste{K := Fraction Integer \bound{K}}
+}
+\xtc{
+We define the quadratic form to be the Minkowski space-time metric.
+}{
+\spadpaste{g := matrix [[1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1]] \bound{g}}
+}
+\xtc{
+We obtain the Dirac spin algebra
+used in Relativistic Quantum Field Theory.
+%-% \HDindex{Relativistic Quantum Field Theory}{ugxCliffordDiracPage}{9.10.4.}{The Dirac Spin Algebra}
+}{
+\spadpaste{D := CliffordAlgebra(4,K, quadraticForm g) \free{K g}\bound{D}}
+}
+\xtc{
+The usual notation for the basis is \texht{$\gamma$}{\spad{gamma}}
+with a superscript.
+For \Language{} input we will use \spad{gam(i)}:
+}{
+\spadpaste{gam := [e(i)\$D for i in 1..4] \free{D}\bound{gam}}
+}
+\noindent
+There are various contraction identities of the form
+\begin{verbatim}
+g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) =
+      2*(gam(s)gam(m)gam(n)gam(r) + gam(r)*gam(n)*gam(m)*gam(s))
+\end{verbatim}
+where a sum over \spad{l} and \spad{t} is implied.
+\xtc{
+Verify this identity for particular values of \spad{m,n,r,s}.
+}{
+\spadpaste{m := 1; n:= 2; r := 3; s := 4; \bound{m,n,r,s}}
+}
+\xtc{
+}{
+\spadpaste{lhs := reduce(+, [reduce(+, [ g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) for l in 1..4]) for t in 1..4]) \bound{lhs}\free{g gam m n r s}}
+}
+\xtc{
+}{
+\spadpaste{rhs := 2*(gam s * gam m*gam n*gam r + gam r*gam n*gam m*gam s) \bound{rhs}\free{lhs g gam m n r s}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/CLIF.pht b/src/hyper/pages/CLIF.pht
new file mode 100644
index 00000000..1331d459
--- /dev/null
+++ b/src/hyper/pages/CLIF.pht
@@ -0,0 +1,622 @@
+\begin{patch}{ugxCliffordDiracPagePatch1}
+\begin{paste}{ugxCliffordDiracPageFull1}{ugxCliffordDiracPageEmpty1}
+\pastebutton{ugxCliffordDiracPageFull1}{\hidepaste}
+\tab{5}\spadcommand{K := Fraction Integer\bound{K }}
+\indentrel{3}\begin{verbatim}
+   (1)  Fraction Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordDiracPageEmpty1}
+\begin{paste}{ugxCliffordDiracPageEmpty1}{ugxCliffordDiracPagePatch1}
+\pastebutton{ugxCliffordDiracPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{K := Fraction Integer\bound{K }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordDiracPagePatch2}
+\begin{paste}{ugxCliffordDiracPageFull2}{ugxCliffordDiracPageEmpty2}
+\pastebutton{ugxCliffordDiracPageFull2}{\hidepaste}
+\tab{5}\spadcommand{g := matrix [[1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1]]\bound{g }}
+\indentrel{3}\begin{verbatim}
+        Ú1   0    0    0 ¿
+        ³                ³
+        ³0  - 1   0    0 ³
+   (2)  ³                ³
+        ³0   0   - 1   0 ³
+        ³                ³
+        À0   0    0   - 1Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordDiracPageEmpty2}
+\begin{paste}{ugxCliffordDiracPageEmpty2}{ugxCliffordDiracPagePatch2}
+\pastebutton{ugxCliffordDiracPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{g := matrix [[1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1]]\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordDiracPagePatch3}
+\begin{paste}{ugxCliffordDiracPageFull3}{ugxCliffordDiracPageEmpty3}
+\pastebutton{ugxCliffordDiracPageFull3}{\hidepaste}
+\tab{5}\spadcommand{D := CliffordAlgebra(4,K, quadraticForm g)\free{K g }\bound{D }}
+\indentrel{3}\begin{verbatim}
+   (3)  CliffordAlgebra(4,Fraction Integer,MATRIX)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordDiracPageEmpty3}
+\begin{paste}{ugxCliffordDiracPageEmpty3}{ugxCliffordDiracPagePatch3}
+\pastebutton{ugxCliffordDiracPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{D := CliffordAlgebra(4,K, quadraticForm g)\free{K g }\bound{D }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordDiracPagePatch4}
+\begin{paste}{ugxCliffordDiracPageFull4}{ugxCliffordDiracPageEmpty4}
+\pastebutton{ugxCliffordDiracPageFull4}{\hidepaste}
+\tab{5}\spadcommand{gam := [e(i)$D for i in 1..4]\free{D }\bound{gam }}
+\indentrel{3}\begin{verbatim}
+   (4)  [e ,e ,e ,e ]
+          1  2  3  4
+  Type: List CliffordAlgebra(4,Fraction Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordDiracPageEmpty4}
+\begin{paste}{ugxCliffordDiracPageEmpty4}{ugxCliffordDiracPagePatch4}
+\pastebutton{ugxCliffordDiracPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{gam := [e(i)$D for i in 1..4]\free{D }\bound{gam }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordDiracPagePatch5}
+\begin{paste}{ugxCliffordDiracPageFull5}{ugxCliffordDiracPageEmpty5}
+\pastebutton{ugxCliffordDiracPageFull5}{\hidepaste}
+\tab{5}\spadcommand{m := 1; n:= 2; r := 3; s := 4;\bound{m n r s }}
+\indentrel{3}\begin{verbatim}
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordDiracPageEmpty5}
+\begin{paste}{ugxCliffordDiracPageEmpty5}{ugxCliffordDiracPagePatch5}
+\pastebutton{ugxCliffordDiracPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{m := 1; n:= 2; r := 3; s := 4;\bound{m n r s }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordDiracPagePatch6}
+\begin{paste}{ugxCliffordDiracPageFull6}{ugxCliffordDiracPageEmpty6}
+\pastebutton{ugxCliffordDiracPageFull6}{\hidepaste}
+\tab{5}\spadcommand{lhs := reduce(+, [reduce(+, [ g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) for l in 1..4]) for t in 1..4])\bound{lhs }\free{g gam m n r s }}
+\indentrel{3}\begin{verbatim}
+   (6)  - 4e e e e
+            1 2 3 4
+       Type: CliffordAlgebra(4,Fraction Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordDiracPageEmpty6}
+\begin{paste}{ugxCliffordDiracPageEmpty6}{ugxCliffordDiracPagePatch6}
+\pastebutton{ugxCliffordDiracPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{lhs := reduce(+, [reduce(+, [ g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) for l in 1..4]) for t in 1..4])\bound{lhs }\free{g gam m n r s }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordDiracPagePatch7}
+\begin{paste}{ugxCliffordDiracPageFull7}{ugxCliffordDiracPageEmpty7}
+\pastebutton{ugxCliffordDiracPageFull7}{\hidepaste}
+\tab{5}\spadcommand{rhs := 2*(gam s * gam m*gam n*gam r + gam r*gam n*gam m*gam s)\bound{rhs }\free{lhs g gam m n r s }}
+\indentrel{3}\begin{verbatim}
+   (7)  - 4e e e e
+            1 2 3 4
+       Type: CliffordAlgebra(4,Fraction Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordDiracPageEmpty7}
+\begin{paste}{ugxCliffordDiracPageEmpty7}{ugxCliffordDiracPagePatch7}
+\pastebutton{ugxCliffordDiracPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{rhs := 2*(gam s * gam m*gam n*gam r + gam r*gam n*gam m*gam s)\bound{rhs }\free{lhs g gam m n r s }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPagePatch1}
+\begin{paste}{ugxCliffordComplexPageFull1}{ugxCliffordComplexPageEmpty1}
+\pastebutton{ugxCliffordComplexPageFull1}{\hidepaste}
+\tab{5}\spadcommand{K := Fraction Polynomial Integer\bound{K }}
+\indentrel{3}\begin{verbatim}
+   (1)  Fraction Polynomial Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPageEmpty1}
+\begin{paste}{ugxCliffordComplexPageEmpty1}{ugxCliffordComplexPagePatch1}
+\pastebutton{ugxCliffordComplexPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{K := Fraction Polynomial Integer\bound{K }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPagePatch2}
+\begin{paste}{ugxCliffordComplexPageFull2}{ugxCliffordComplexPageEmpty2}
+\pastebutton{ugxCliffordComplexPageFull2}{\hidepaste}
+\tab{5}\spadcommand{m := matrix [[-1]]\bound{m }}
+\indentrel{3}\begin{verbatim}
+   (2)  [- 1]
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPageEmpty2}
+\begin{paste}{ugxCliffordComplexPageEmpty2}{ugxCliffordComplexPagePatch2}
+\pastebutton{ugxCliffordComplexPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{m := matrix [[-1]]\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPagePatch3}
+\begin{paste}{ugxCliffordComplexPageFull3}{ugxCliffordComplexPageEmpty3}
+\pastebutton{ugxCliffordComplexPageFull3}{\hidepaste}
+\tab{5}\spadcommand{C := CliffordAlgebra(1, K, quadraticForm m)\free{K m }\bound{C }}
+\indentrel{3}\begin{verbatim}
+   (3)
+   CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPageEmpty3}
+\begin{paste}{ugxCliffordComplexPageEmpty3}{ugxCliffordComplexPagePatch3}
+\pastebutton{ugxCliffordComplexPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{C := CliffordAlgebra(1, K, quadraticForm m)\free{K m }\bound{C }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPagePatch4}
+\begin{paste}{ugxCliffordComplexPageFull4}{ugxCliffordComplexPageEmpty4}
+\pastebutton{ugxCliffordComplexPageFull4}{\hidepaste}
+\tab{5}\spadcommand{i: C := e(1)\bound{i }\free{C }}
+\indentrel{3}\begin{verbatim}
+   (4)  e
+         1
+Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPageEmpty4}
+\begin{paste}{ugxCliffordComplexPageEmpty4}{ugxCliffordComplexPagePatch4}
+\pastebutton{ugxCliffordComplexPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{i: C := e(1)\bound{i }\free{C }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPagePatch5}
+\begin{paste}{ugxCliffordComplexPageFull5}{ugxCliffordComplexPageEmpty5}
+\pastebutton{ugxCliffordComplexPageFull5}{\hidepaste}
+\tab{5}\spadcommand{x := a + b * i\bound{x }\free{i }}
+\indentrel{3}\begin{verbatim}
+   (5)  a + b e
+               1
+Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPageEmpty5}
+\begin{paste}{ugxCliffordComplexPageEmpty5}{ugxCliffordComplexPagePatch5}
+\pastebutton{ugxCliffordComplexPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{x := a + b * i\bound{x }\free{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPagePatch6}
+\begin{paste}{ugxCliffordComplexPageFull6}{ugxCliffordComplexPageEmpty6}
+\pastebutton{ugxCliffordComplexPageFull6}{\hidepaste}
+\tab{5}\spadcommand{y := c + d * i\bound{y }\free{i }}
+\indentrel{3}\begin{verbatim}
+   (6)  c + d e
+               1
+Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPageEmpty6}
+\begin{paste}{ugxCliffordComplexPageEmpty6}{ugxCliffordComplexPagePatch6}
+\pastebutton{ugxCliffordComplexPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{y := c + d * i\bound{y }\free{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPagePatch7}
+\begin{paste}{ugxCliffordComplexPageFull7}{ugxCliffordComplexPageEmpty7}
+\pastebutton{ugxCliffordComplexPageFull7}{\hidepaste}
+\tab{5}\spadcommand{x * y\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (7)  - b d + a c + (a d + b c)e
+                                  1
+Type: CliffordAlgebra(1,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordComplexPageEmpty7}
+\begin{paste}{ugxCliffordComplexPageEmpty7}{ugxCliffordComplexPagePatch7}
+\pastebutton{ugxCliffordComplexPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{x * y\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPagePatch1}
+\begin{paste}{ugxCliffordQuaternPageFull1}{ugxCliffordQuaternPageEmpty1}
+\pastebutton{ugxCliffordQuaternPageFull1}{\hidepaste}
+\tab{5}\spadcommand{K := Fraction Polynomial Integer\bound{K }}
+\indentrel{3}\begin{verbatim}
+   (1)  Fraction Polynomial Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPageEmpty1}
+\begin{paste}{ugxCliffordQuaternPageEmpty1}{ugxCliffordQuaternPagePatch1}
+\pastebutton{ugxCliffordQuaternPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{K := Fraction Polynomial Integer\bound{K }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPagePatch2}
+\begin{paste}{ugxCliffordQuaternPageFull2}{ugxCliffordQuaternPageEmpty2}
+\pastebutton{ugxCliffordQuaternPageFull2}{\hidepaste}
+\tab{5}\spadcommand{m := matrix [[-1,0],[0,-1]]\bound{m }}
+\indentrel{3}\begin{verbatim}
+        Ú- 1   0 ¿
+   (2)  ³        ³
+        À 0   - 1Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPageEmpty2}
+\begin{paste}{ugxCliffordQuaternPageEmpty2}{ugxCliffordQuaternPagePatch2}
+\pastebutton{ugxCliffordQuaternPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{m := matrix [[-1,0],[0,-1]]\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPagePatch3}
+\begin{paste}{ugxCliffordQuaternPageFull3}{ugxCliffordQuaternPageEmpty3}
+\pastebutton{ugxCliffordQuaternPageFull3}{\hidepaste}
+\tab{5}\spadcommand{H := CliffordAlgebra(2, K, quadraticForm m)\free{K m }\bound{H }}
+\indentrel{3}\begin{verbatim}
+   (3)
+   CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPageEmpty3}
+\begin{paste}{ugxCliffordQuaternPageEmpty3}{ugxCliffordQuaternPagePatch3}
+\pastebutton{ugxCliffordQuaternPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{H := CliffordAlgebra(2, K, quadraticForm m)\free{K m }\bound{H }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPagePatch4}
+\begin{paste}{ugxCliffordQuaternPageFull4}{ugxCliffordQuaternPageEmpty4}
+\pastebutton{ugxCliffordQuaternPageFull4}{\hidepaste}
+\tab{5}\spadcommand{i: H := e(1)\free{H }\bound{i }}
+\indentrel{3}\begin{verbatim}
+   (4)  e
+         1
+Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPageEmpty4}
+\begin{paste}{ugxCliffordQuaternPageEmpty4}{ugxCliffordQuaternPagePatch4}
+\pastebutton{ugxCliffordQuaternPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{i: H := e(1)\free{H }\bound{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPagePatch5}
+\begin{paste}{ugxCliffordQuaternPageFull5}{ugxCliffordQuaternPageEmpty5}
+\pastebutton{ugxCliffordQuaternPageFull5}{\hidepaste}
+\tab{5}\spadcommand{j: H := e(2)\free{H }\bound{j }}
+\indentrel{3}\begin{verbatim}
+   (5)  e
+         2
+Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPageEmpty5}
+\begin{paste}{ugxCliffordQuaternPageEmpty5}{ugxCliffordQuaternPagePatch5}
+\pastebutton{ugxCliffordQuaternPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{j: H := e(2)\free{H }\bound{j }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPagePatch6}
+\begin{paste}{ugxCliffordQuaternPageFull6}{ugxCliffordQuaternPageEmpty6}
+\pastebutton{ugxCliffordQuaternPageFull6}{\hidepaste}
+\tab{5}\spadcommand{k: H := i * j\free{H i j }\bound{k }}
+\indentrel{3}\begin{verbatim}
+   (6)  e e
+         1 2
+Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPageEmpty6}
+\begin{paste}{ugxCliffordQuaternPageEmpty6}{ugxCliffordQuaternPagePatch6}
+\pastebutton{ugxCliffordQuaternPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{k: H := i * j\free{H i j }\bound{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPagePatch7}
+\begin{paste}{ugxCliffordQuaternPageFull7}{ugxCliffordQuaternPageEmpty7}
+\pastebutton{ugxCliffordQuaternPageFull7}{\hidepaste}
+\tab{5}\spadcommand{x := a + b * i + c * j + d * k\free{i j k }\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (7)  a + b e  + c e  + d e e
+               1      2      1 2
+Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPageEmpty7}
+\begin{paste}{ugxCliffordQuaternPageEmpty7}{ugxCliffordQuaternPagePatch7}
+\pastebutton{ugxCliffordQuaternPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{x := a + b * i + c * j + d * k\free{i j k }\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPagePatch8}
+\begin{paste}{ugxCliffordQuaternPageFull8}{ugxCliffordQuaternPageEmpty8}
+\pastebutton{ugxCliffordQuaternPageFull8}{\hidepaste}
+\tab{5}\spadcommand{y := e + f * i + g * j + h * k\free{i j k }\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (8)  e + f e  + g e  + h e e
+               1      2      1 2
+Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPageEmpty8}
+\begin{paste}{ugxCliffordQuaternPageEmpty8}{ugxCliffordQuaternPagePatch8}
+\pastebutton{ugxCliffordQuaternPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{y := e + f * i + g * j + h * k\free{i j k }\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPagePatch9}
+\begin{paste}{ugxCliffordQuaternPageFull9}{ugxCliffordQuaternPageEmpty9}
+\pastebutton{ugxCliffordQuaternPageFull9}{\hidepaste}
+\tab{5}\spadcommand{x + y\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (9)  e + a + (f + b)e  + (g + c)e  + (h + d)e e
+                        1           2           1 2
+Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPageEmpty9}
+\begin{paste}{ugxCliffordQuaternPageEmpty9}{ugxCliffordQuaternPagePatch9}
+\pastebutton{ugxCliffordQuaternPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{x + y\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPagePatch10}
+\begin{paste}{ugxCliffordQuaternPageFull10}{ugxCliffordQuaternPageEmpty10}
+\pastebutton{ugxCliffordQuaternPageFull10}{\hidepaste}
+\tab{5}\spadcommand{x * y\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (10)
+     - d h - c g - b f + a e + (c h - d g + a f + b e)e
+                                                       1
+   + 
+     (- b h + a g + d f + c e)e
+                               2
+   + 
+     (a h + b g - c f + d e)e e
+                             1 2
+Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPageEmpty10}
+\begin{paste}{ugxCliffordQuaternPageEmpty10}{ugxCliffordQuaternPagePatch10}
+\pastebutton{ugxCliffordQuaternPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{x * y\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPagePatch11}
+\begin{paste}{ugxCliffordQuaternPageFull11}{ugxCliffordQuaternPageEmpty11}
+\pastebutton{ugxCliffordQuaternPageFull11}{\hidepaste}
+\tab{5}\spadcommand{y * x\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (11)
+     - d h - c g - b f + a e + (- c h + d g + a f + b e)e
+                                                         1
+   + 
+   (b h + a g - d f + c e)e  + (a h - b g + c f + d e)e e
+                           2                           1 2
+Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordQuaternPageEmpty11}
+\begin{paste}{ugxCliffordQuaternPageEmpty11}{ugxCliffordQuaternPagePatch11}
+\pastebutton{ugxCliffordQuaternPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{y * x\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPagePatch1}
+\begin{paste}{ugxCliffordExteriorPageFull1}{ugxCliffordExteriorPageEmpty1}
+\pastebutton{ugxCliffordExteriorPageFull1}{\hidepaste}
+\tab{5}\spadcommand{K := Fraction Polynomial Integer\bound{K }}
+\indentrel{3}\begin{verbatim}
+   (1)  Fraction Polynomial Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPageEmpty1}
+\begin{paste}{ugxCliffordExteriorPageEmpty1}{ugxCliffordExteriorPagePatch1}
+\pastebutton{ugxCliffordExteriorPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{K := Fraction Polynomial Integer\bound{K }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPagePatch2}
+\begin{paste}{ugxCliffordExteriorPageFull2}{ugxCliffordExteriorPageEmpty2}
+\pastebutton{ugxCliffordExteriorPageFull2}{\hidepaste}
+\tab{5}\spadcommand{Ext := CliffordAlgebra(3, K, quadraticForm 0)\bound{Ext }\free{K }}
+\indentrel{3}\begin{verbatim}
+   (2)
+   CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPageEmpty2}
+\begin{paste}{ugxCliffordExteriorPageEmpty2}{ugxCliffordExteriorPagePatch2}
+\pastebutton{ugxCliffordExteriorPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{Ext := CliffordAlgebra(3, K, quadraticForm 0)\bound{Ext }\free{K }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPagePatch3}
+\begin{paste}{ugxCliffordExteriorPageFull3}{ugxCliffordExteriorPageEmpty3}
+\pastebutton{ugxCliffordExteriorPageFull3}{\hidepaste}
+\tab{5}\spadcommand{i: Ext := e(1)\free{Ext }\bound{i }}
+\indentrel{3}\begin{verbatim}
+   (3)  e
+         1
+Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPageEmpty3}
+\begin{paste}{ugxCliffordExteriorPageEmpty3}{ugxCliffordExteriorPagePatch3}
+\pastebutton{ugxCliffordExteriorPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{i: Ext := e(1)\free{Ext }\bound{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPagePatch4}
+\begin{paste}{ugxCliffordExteriorPageFull4}{ugxCliffordExteriorPageEmpty4}
+\pastebutton{ugxCliffordExteriorPageFull4}{\hidepaste}
+\tab{5}\spadcommand{j: Ext := e(2)\free{Ext }\bound{j }}
+\indentrel{3}\begin{verbatim}
+   (4)  e
+         2
+Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPageEmpty4}
+\begin{paste}{ugxCliffordExteriorPageEmpty4}{ugxCliffordExteriorPagePatch4}
+\pastebutton{ugxCliffordExteriorPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{j: Ext := e(2)\free{Ext }\bound{j }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPagePatch5}
+\begin{paste}{ugxCliffordExteriorPageFull5}{ugxCliffordExteriorPageEmpty5}
+\pastebutton{ugxCliffordExteriorPageFull5}{\hidepaste}
+\tab{5}\spadcommand{k: Ext := e(3)\free{Ext }\bound{k }}
+\indentrel{3}\begin{verbatim}
+   (5)  e
+         3
+Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPageEmpty5}
+\begin{paste}{ugxCliffordExteriorPageEmpty5}{ugxCliffordExteriorPagePatch5}
+\pastebutton{ugxCliffordExteriorPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{k: Ext := e(3)\free{Ext }\bound{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPagePatch6}
+\begin{paste}{ugxCliffordExteriorPageFull6}{ugxCliffordExteriorPageEmpty6}
+\pastebutton{ugxCliffordExteriorPageFull6}{\hidepaste}
+\tab{5}\spadcommand{x := x1*i + x2*j + x3*k\free{i j k }\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (6)  x1 e  + x2 e  + x3 e
+            1       2       3
+Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPageEmpty6}
+\begin{paste}{ugxCliffordExteriorPageEmpty6}{ugxCliffordExteriorPagePatch6}
+\pastebutton{ugxCliffordExteriorPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{x := x1*i + x2*j + x3*k\free{i j k }\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPagePatch7}
+\begin{paste}{ugxCliffordExteriorPageFull7}{ugxCliffordExteriorPageEmpty7}
+\pastebutton{ugxCliffordExteriorPageFull7}{\hidepaste}
+\tab{5}\spadcommand{y := y1*i + y2*j + y3*k\free{i j k }\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (7)  y1 e  + y2 e  + y3 e
+            1       2       3
+Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPageEmpty7}
+\begin{paste}{ugxCliffordExteriorPageEmpty7}{ugxCliffordExteriorPagePatch7}
+\pastebutton{ugxCliffordExteriorPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{y := y1*i + y2*j + y3*k\free{i j k }\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPagePatch8}
+\begin{paste}{ugxCliffordExteriorPageFull8}{ugxCliffordExteriorPageEmpty8}
+\pastebutton{ugxCliffordExteriorPageFull8}{\hidepaste}
+\tab{5}\spadcommand{x + y\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (8)  (y1 + x1)e  + (y2 + x2)e  + (y3 + x3)e
+                  1             2             3
+Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPageEmpty8}
+\begin{paste}{ugxCliffordExteriorPageEmpty8}{ugxCliffordExteriorPagePatch8}
+\pastebutton{ugxCliffordExteriorPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{x + y\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPagePatch9}
+\begin{paste}{ugxCliffordExteriorPageFull9}{ugxCliffordExteriorPageEmpty9}
+\pastebutton{ugxCliffordExteriorPageFull9}{\hidepaste}
+\tab{5}\spadcommand{x * y + y * x\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (9)  0
+Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPageEmpty9}
+\begin{paste}{ugxCliffordExteriorPageEmpty9}{ugxCliffordExteriorPagePatch9}
+\pastebutton{ugxCliffordExteriorPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{x * y + y * x\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPagePatch10}
+\begin{paste}{ugxCliffordExteriorPageFull10}{ugxCliffordExteriorPageEmpty10}
+\pastebutton{ugxCliffordExteriorPageFull10}{\hidepaste}
+\tab{5}\spadcommand{dual2 a == coefficient(a,[2,3]) * i + coefficient(a,[3,1]) * j + coefficient(a,[1,2]) * k\free{i j k }\bound{dual2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPageEmpty10}
+\begin{paste}{ugxCliffordExteriorPageEmpty10}{ugxCliffordExteriorPagePatch10}
+\pastebutton{ugxCliffordExteriorPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{dual2 a == coefficient(a,[2,3]) * i + coefficient(a,[3,1]) * j + coefficient(a,[1,2]) * k\free{i j k }\bound{dual2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPagePatch11}
+\begin{paste}{ugxCliffordExteriorPageFull11}{ugxCliffordExteriorPageEmpty11}
+\pastebutton{ugxCliffordExteriorPageFull11}{\hidepaste}
+\tab{5}\spadcommand{dual2(x*y)\free{x y dual2 }}
+\indentrel{3}\begin{verbatim}
+   (11)
+     (x2 y3 - x3 y2)e  + (- x1 y3 + x3 y1)e
+                     1                     2
+   + 
+     (x1 y2 - x2 y1)e
+                     3
+Type: CliffordAlgebra(3,Fraction Polynomial Integer,MATRIX)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxCliffordExteriorPageEmpty11}
+\begin{paste}{ugxCliffordExteriorPageEmpty11}{ugxCliffordExteriorPagePatch11}
+\pastebutton{ugxCliffordExteriorPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{dual2(x*y)\free{x y dual2 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/COMPLEX.ht b/src/hyper/pages/COMPLEX.ht
new file mode 100644
index 00000000..f01b98f1
--- /dev/null
+++ b/src/hyper/pages/COMPLEX.ht
@@ -0,0 +1,105 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\ComplexXmpTitle}{Complex}
+\newcommand{\ComplexXmpNumber}{9.11}
+%
+% =====================================================================
+\begin{page}{ComplexXmpPage}{9.11 Complex}
+% =====================================================================
+\beginscroll
+%
+
+The \spadtype{Complex} constructor implements complex objects over a
+commutative ring \spad{R}.
+Typically, the ring \spad{R} is \spadtype{Integer}, \spadtype{Fraction
+Integer}, \spadtype{Float} or \spadtype{DoubleFloat}.
+\spad{R} can also be a symbolic type, like \spadtype{Polynomial Integer}.
+For more information about the numerical and graphical aspects of complex
+numbers, see \downlink{``\ugProblemNumericTitle''}{ugProblemNumericPage} in Section \ugProblemNumericNumber\ignore{ugProblemNumeric}.
+
+\xtc{
+Complex objects are created by the \spadfunFrom{complex}{Complex} operation.
+}{
+\spadpaste{a := complex(4/3,5/2) \bound{a}}
+}
+\xtc{
+}{
+\spadpaste{b := complex(4/3,-5/2) \bound{b}}
+}
+\xtc{
+The standard arithmetic operations are available.
+}{
+\spadpaste{a + b \free{a b}}
+}
+\xtc{
+}{
+\spadpaste{a - b \free{a b}}
+}
+\xtc{
+}{
+\spadpaste{a * b \free{a b}}
+}
+\xtc{
+If \spad{R} is a field, you can also divide the complex objects.
+}{
+\spadpaste{a / b \free{a b}\bound{adb}}
+}
+\xtc{
+Use a conversion (\downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert}) to view the last
+object as a fraction of complex integers.
+}{
+\spadpaste{\% :: Fraction Complex Integer \free{adb}}
+}
+\xtc{
+The predefined macro \spad{\%i} is defined to be \spad{complex(0,1)}.
+}{
+\spadpaste{3.4 + 6.7 * \%i}
+}
+\xtc{
+You can also compute the \spadfunFrom{conjugate}{Complex} and
+\spadfunFrom{norm}{Complex} of a complex number.
+}{
+\spadpaste{conjugate a \free{a}}
+}
+\xtc{
+}{
+\spadpaste{norm a \free{a}}
+}
+\xtc{
+The \spadfunFrom{real}{Complex} and \spadfunFrom{imag}{Complex} operations
+are provided to extract the real and imaginary parts, respectively.
+}{
+\spadpaste{real a \free{a}}
+}
+\xtc{
+}{
+\spadpaste{imag a \free{a}}
+}
+
+\xtc{
+The domain \spadtype{Complex Integer} is also called the Gaussian
+integers.
+%-% \HDindex{Gaussian integer}{ComplexXmpPage}{9.11}{Complex}
+If \spad{R} is the integers (or, more generally,
+a \spadtype{EuclideanDomain}), you can compute greatest common divisors.
+}{
+\spadpaste{gcd(13 - 13*\%i,31 + 27*\%i)}
+}
+\xtc{
+You can also compute least common multiples.
+}{
+\spadpaste{lcm(13 - 13*\%i,31 + 27*\%i)}
+}
+\xtc{
+You can \spadfunFrom{factor}{Complex} Gaussian integers.
+}{
+\spadpaste{factor(13 - 13*\%i)}
+}
+\xtc{
+}{
+\spadpaste{factor complex(2,0)}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/COMPLEX.pht b/src/hyper/pages/COMPLEX.pht
new file mode 100644
index 00000000..e7c69742
--- /dev/null
+++ b/src/hyper/pages/COMPLEX.pht
@@ -0,0 +1,277 @@
+\begin{patch}{ComplexXmpPagePatch1}
+\begin{paste}{ComplexXmpPageFull1}{ComplexXmpPageEmpty1}
+\pastebutton{ComplexXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{a := complex(4/3,5/2)\bound{a }}
+\indentrel{3}\begin{verbatim}
+        4   5
+   (1)  Ä + Ä %i
+        3   2
+                         Type: Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty1}
+\begin{paste}{ComplexXmpPageEmpty1}{ComplexXmpPagePatch1}
+\pastebutton{ComplexXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{a := complex(4/3,5/2)\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch2}
+\begin{paste}{ComplexXmpPageFull2}{ComplexXmpPageEmpty2}
+\pastebutton{ComplexXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{b := complex(4/3,-5/2)\bound{b }}
+\indentrel{3}\begin{verbatim}
+        4   5
+   (2)  Ä - Ä %i
+        3   2
+                         Type: Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty2}
+\begin{paste}{ComplexXmpPageEmpty2}{ComplexXmpPagePatch2}
+\pastebutton{ComplexXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{b := complex(4/3,-5/2)\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch3}
+\begin{paste}{ComplexXmpPageFull3}{ComplexXmpPageEmpty3}
+\pastebutton{ComplexXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{a + b\free{a b }}
+\indentrel{3}\begin{verbatim}
+        8
+   (3)  Ä
+        3
+                         Type: Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty3}
+\begin{paste}{ComplexXmpPageEmpty3}{ComplexXmpPagePatch3}
+\pastebutton{ComplexXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{a + b\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch4}
+\begin{paste}{ComplexXmpPageFull4}{ComplexXmpPageEmpty4}
+\pastebutton{ComplexXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{a - b\free{a b }}
+\indentrel{3}\begin{verbatim}
+   (4)  5%i
+                         Type: Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty4}
+\begin{paste}{ComplexXmpPageEmpty4}{ComplexXmpPagePatch4}
+\pastebutton{ComplexXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{a - b\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch5}
+\begin{paste}{ComplexXmpPageFull5}{ComplexXmpPageEmpty5}
+\pastebutton{ComplexXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{a * b\free{a b }}
+\indentrel{3}\begin{verbatim}
+        289
+   (5)  ÄÄÄ
+         36
+                         Type: Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty5}
+\begin{paste}{ComplexXmpPageEmpty5}{ComplexXmpPagePatch5}
+\pastebutton{ComplexXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{a * b\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch6}
+\begin{paste}{ComplexXmpPageFull6}{ComplexXmpPageEmpty6}
+\pastebutton{ComplexXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{a / b\free{a b }\bound{adb }}
+\indentrel{3}\begin{verbatim}
+          161   240
+   (6)  - ÄÄÄ + ÄÄÄ %i
+          289   289
+                         Type: Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty6}
+\begin{paste}{ComplexXmpPageEmpty6}{ComplexXmpPagePatch6}
+\pastebutton{ComplexXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{a / b\free{a b }\bound{adb }}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch7}
+\begin{paste}{ComplexXmpPageFull7}{ComplexXmpPageEmpty7}
+\pastebutton{ComplexXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{\% :: Fraction Complex Integer\free{adb }}
+\indentrel{3}\begin{verbatim}
+        - 15 + 8%i
+   (7)  ÄÄÄÄÄÄÄÄÄÄ
+         15 + 8%i
+                         Type: Fraction Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty7}
+\begin{paste}{ComplexXmpPageEmpty7}{ComplexXmpPagePatch7}
+\pastebutton{ComplexXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{\% :: Fraction Complex Integer\free{adb }}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch8}
+\begin{paste}{ComplexXmpPageFull8}{ComplexXmpPageEmpty8}
+\pastebutton{ComplexXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{3.4 + 6.7 * \%i}
+\indentrel{3}\begin{verbatim}
+   (8)  3.4 + 6.7 %i
+                                    Type: Complex Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty8}
+\begin{paste}{ComplexXmpPageEmpty8}{ComplexXmpPagePatch8}
+\pastebutton{ComplexXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{3.4 + 6.7 * \%i}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch9}
+\begin{paste}{ComplexXmpPageFull9}{ComplexXmpPageEmpty9}
+\pastebutton{ComplexXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{conjugate a\free{a }}
+\indentrel{3}\begin{verbatim}
+        4   5
+   (9)  Ä - Ä %i
+        3   2
+                         Type: Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty9}
+\begin{paste}{ComplexXmpPageEmpty9}{ComplexXmpPagePatch9}
+\pastebutton{ComplexXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{conjugate a\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch10}
+\begin{paste}{ComplexXmpPageFull10}{ComplexXmpPageEmpty10}
+\pastebutton{ComplexXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{norm a\free{a }}
+\indentrel{3}\begin{verbatim}
+         289
+   (10)  ÄÄÄ
+          36
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty10}
+\begin{paste}{ComplexXmpPageEmpty10}{ComplexXmpPagePatch10}
+\pastebutton{ComplexXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{norm a\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch11}
+\begin{paste}{ComplexXmpPageFull11}{ComplexXmpPageEmpty11}
+\pastebutton{ComplexXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{real a\free{a }}
+\indentrel{3}\begin{verbatim}
+         4
+   (11)  Ä
+         3
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty11}
+\begin{paste}{ComplexXmpPageEmpty11}{ComplexXmpPagePatch11}
+\pastebutton{ComplexXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{real a\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch12}
+\begin{paste}{ComplexXmpPageFull12}{ComplexXmpPageEmpty12}
+\pastebutton{ComplexXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{imag a\free{a }}
+\indentrel{3}\begin{verbatim}
+         5
+   (12)  Ä
+         2
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty12}
+\begin{paste}{ComplexXmpPageEmpty12}{ComplexXmpPagePatch12}
+\pastebutton{ComplexXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{imag a\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch13}
+\begin{paste}{ComplexXmpPageFull13}{ComplexXmpPageEmpty13}
+\pastebutton{ComplexXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{gcd(13 - 13*\%i,31 + 27*\%i)}
+\indentrel{3}\begin{verbatim}
+   (13)  5 + %i
+                                  Type: Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty13}
+\begin{paste}{ComplexXmpPageEmpty13}{ComplexXmpPagePatch13}
+\pastebutton{ComplexXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{gcd(13 - 13*\%i,31 + 27*\%i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch14}
+\begin{paste}{ComplexXmpPageFull14}{ComplexXmpPageEmpty14}
+\pastebutton{ComplexXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{lcm(13 - 13*\%i,31 + 27*\%i)}
+\indentrel{3}\begin{verbatim}
+   (14)  143 - 39%i
+                                  Type: Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty14}
+\begin{paste}{ComplexXmpPageEmpty14}{ComplexXmpPagePatch14}
+\pastebutton{ComplexXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{lcm(13 - 13*\%i,31 + 27*\%i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch15}
+\begin{paste}{ComplexXmpPageFull15}{ComplexXmpPageEmpty15}
+\pastebutton{ComplexXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{factor(13 - 13*\%i)}
+\indentrel{3}\begin{verbatim}
+   (15)  - (1 + %i)(2 + 3%i)(3 + 2%i)
+                         Type: Factored Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty15}
+\begin{paste}{ComplexXmpPageEmpty15}{ComplexXmpPagePatch15}
+\pastebutton{ComplexXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{factor(13 - 13*\%i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPagePatch16}
+\begin{paste}{ComplexXmpPageFull16}{ComplexXmpPageEmpty16}
+\pastebutton{ComplexXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{factor complex(2,0)}
+\indentrel{3}\begin{verbatim}
+                      2
+   (16)  - %i (1 + %i)
+                         Type: Factored Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ComplexXmpPageEmpty16}
+\begin{paste}{ComplexXmpPageEmpty16}{ComplexXmpPagePatch16}
+\pastebutton{ComplexXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{factor complex(2,0)}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/CONTFRAC.ht b/src/hyper/pages/CONTFRAC.ht
new file mode 100644
index 00000000..241ced2a
--- /dev/null
+++ b/src/hyper/pages/CONTFRAC.ht
@@ -0,0 +1,302 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\ContinuedFractionXmpTitle}{ContinuedFraction}
+\newcommand{\ContinuedFractionXmpNumber}{9.12}
+%
+% =====================================================================
+\begin{page}{ContinuedFractionXmpPage}{9.12 ContinuedFraction}
+% =====================================================================
+\beginscroll
+
+Continued fractions have been a fascinating and useful tool in mathematics
+%-% \HDindex{fraction!continued}{ContinuedFractionXmpPage}{9.12}{ContinuedFraction}
+for well over three hundred years.
+\Language{} implements continued fractions for fractions of any Euclidean
+%-% \HDindex{continued fraction}{ContinuedFractionXmpPage}{9.12}{ContinuedFraction}
+domain.
+In practice, this usually means rational numbers.
+In this section we demonstrate some of the operations available for
+manipulating both finite and infinite continued fractions.
+It may be helpful if you review \downlink{`Stream'}{StreamXmpPage}\ignore{Stream} to remind yourself of some
+of the operations with streams.
+
+The \spadtype{ContinuedFraction} domain is a field and therefore you can
+add, subtract, multiply and divide the fractions.
+\xtc{
+The \spadfunFrom{continuedFraction}{ContinuedFraction} operation
+converts its fractional argument to a continued fraction.
+}{
+\spadpaste{c := continuedFraction(314159/100000) \bound{c}}
+}
+%
+This display is a compact form of the bulkier
+\texht{\narrowDisplay{%
+3 + {\displaystyle 1 \over {\displaystyle
+7 + {1 \over {\displaystyle
+15 + {1 \over {\displaystyle
+1 + {1 \over {\displaystyle
+25 + {1 \over {\displaystyle
+1 + {1 \over {\displaystyle
+7 + {1 \over 4}}}}}}}}}}}}}}%
+}{
+\begin{verbatim}
+        3 +                 1
+            -------------------------------
+            7 +               1
+                ---------------------------
+                15 +            1
+                     ----------------------
+                     1 +          1
+                         ------------------
+                         25 +       1
+                              -------------
+                              1 +     1
+                                  ---------
+                                  7 +   1
+                                      -----
+                                        4
+\end{verbatim}
+}
+You can write any rational number in a similar form.
+The fraction will be finite and you can always take the ``numerators'' to
+be \spad{1}.
+That is, any rational number can be written as a simple, finite continued
+fraction of the form
+\texht{\narrowDisplay{%
+a_1 + {\displaystyle 1 \over {\displaystyle
+a_2 + {1 \over {\displaystyle
+a_3 + {1 \over {\displaystyle \ddots
+a_{n-1} + {1 \over a_n}}}}}}}}%
+}{
+\begin{verbatim}
+        a(1) +           1
+               -------------------------
+               a(2) +          1
+                      --------------------
+                      a(3) +
+                             .
+                              .
+                               .
+                                     1
+                               -------------
+                               a(n-1) +  1
+                                        ----
+                                        a(n)
+\end{verbatim}
+}
+\xtc{
+The \texht{$a_i$}{\spad{a(i)}} are called partial quotients and the operation
+\spadfunFrom{partialQuotients}{ContinuedFraction} creates a stream of them.
+}{
+\spadpaste{partialQuotients c \free{c}}
+}
+\xtc{
+By considering more and more of the fraction, you get the
+\spadfunFrom{convergents}{ContinuedFraction}.
+For example, the first convergent is \texht{$a_1$}{\spad{a(1)}},
+the second is
+\texht{$a_1 + 1/a_2$}{\spad{a(1) + 1/a(2)}} and so on.
+}{
+\spadpaste{convergents c \free{c}}
+}
+%
+\xtc{
+Since this is a finite continued fraction, the last convergent is
+the original rational number, in reduced form.
+The result of \spadfunFrom{approximants}{ContinuedFraction}
+is always an infinite stream, though it may just repeat the ``last''
+value.
+}{
+\spadpaste{approximants c \free{c}}
+}
+\xtc{
+Inverting \spad{c} only changes the partial quotients of its fraction
+by inserting a \spad{0} at the beginning of the list.
+}{
+\spadpaste{pq := partialQuotients(1/c) \free{c}\bound{pq}}
+}
+\xtc{
+Do this to recover the original continued fraction from this list of
+partial quotients.
+The three-argument form of the
+\spadfunFrom{continuedFraction}{ContinuedFraction} operation takes an
+element which is the whole part of the fraction, a stream of elements
+which are the numerators of the fraction, and a stream of elements which
+are the denominators of the fraction.
+}{
+\spadpaste{continuedFraction(first pq,repeating [1],rest pq) \free{pq}}
+}
+\xtc{
+The streams need not be finite for
+\spadfunFrom{continuedFraction}{ContinuedFraction}.
+Can you guess which irrational number has the following continued
+fraction?
+See the end of this section for the answer.
+}{
+\spadpaste{z:=continuedFraction(3,repeating [1],repeating [3,6]) \bound{z}}
+}
+%
+
+In 1737 Euler discovered the infinite continued fraction expansion
+\texht{\narrowDisplay{%
+{{e - 1} \over 2} =
+{1 \over {\displaystyle
+1 + {1 \over {\displaystyle
+6 + {1 \over {\displaystyle
+10 + {1 \over {\displaystyle
+14 + \cdots}}}}}}}}}%
+}{
+\begin{verbatim}
+        e - 1             1
+        ----- = ---------------------
+          2     1 +         1
+                    -----------------
+                    6 +       1
+                        -------------
+                        10 +    1
+                             --------
+                             14 + ...
+\end{verbatim}
+}
+We use this expansion to compute rational and floating point
+approximations of \spad{e}.\footnote{For this and other interesting
+expansions, see C. D. Olds, {\it Continued Fractions,}
+New Mathematical Library, (New York: Random House, 1963), pp.
+134--139.}
+
+\xtc{
+By looking at the above expansion, we see that the whole part is \spad{0}
+and the numerators are all equal to \spad{1}.
+This constructs the stream of denominators.
+}{
+\spadpaste{dens:Stream Integer := cons(1,generate((x+->x+4),6)) \bound{dens}}
+}
+\xtc{
+Therefore this is the continued fraction expansion for
+\texht{$(e - 1) / 2$}{\spad{(e-1)/2}}.
+}{
+\spadpaste{cf := continuedFraction(0,repeating [1],dens) \free{dens}\bound{cf}}
+}
+\xtc{
+These are the rational number convergents.
+}{
+\spadpaste{ccf := convergents cf \free{cf}\bound{ccf}}
+}
+\xtc{
+You can get rational convergents for \spad{e} by multiplying by \spad{2} and
+adding \spad{1}.
+}{
+\spadpaste{eConvergents := [2*e + 1 for e in ccf] \bound{ec}\free{ccf}}
+}
+%
+\xtc{
+You can also compute the floating point approximations to these convergents.
+}{
+\spadpaste{eConvergents :: Stream Float \free{ec}}
+}
+\xtc{
+Compare this to the value of \spad{e} computed by the
+\spadfunFrom{exp}{Float} operation in \spadtype{Float}.
+}{
+\spadpaste{exp 1.0}
+}
+
+In about 1658, Lord Brouncker established the following expansion
+for \texht{$4 / \pi$}{\spad{4/pi}}.
+\texht{\narrowDisplay{%
+1 + {\displaystyle
+1 \over {\displaystyle
+2 + {9 \over {\displaystyle
+2 + {25 \over {\displaystyle
+2 + {49 \over {\displaystyle
+2 + {81 \over {\displaystyle
+2 + \cdots}}}}}}}}}}}%
+}{
+\begin{verbatim}
+        1 +            1
+            -----------------------
+            2 +          9
+                -------------------
+                2 +        25
+                    ---------------
+                    2 +      49
+                        -----------
+                        2 +    81
+                            -------
+                            2 + ...
+\end{verbatim}
+}
+\xtc{
+Let's use this expansion to compute rational and floating point
+approximations for \texht{$\pi$}{\spad{pi}}.
+}{
+\spadpaste{cf := continuedFraction(1,[(2*i+1)**2 for i in 0..],repeating [2])\bound{cf1}}
+}
+\xtc{
+}{
+\spadpaste{ccf := convergents cf \free{cf1}\bound{ccf1}}
+}
+\xtc{
+}{
+\spadpaste{piConvergents := [4/p for p in ccf] \bound{piConvergents}\free{ccf1}}
+}
+\xtc{
+As you can see, the values are converging to
+\texht{$\pi$}{\spad{pi}} = 3.14159265358979323846...,
+but not very quickly.
+}{
+\spadpaste{piConvergents :: Stream Float \free{piConvergents}}
+}
+
+\xtc{
+You need not restrict yourself to continued fractions of integers.
+Here is an expansion for a quotient of Gaussian integers.
+%-% \HDindex{Gaussian integer}{ContinuedFractionXmpPage}{9.12}{ContinuedFraction}
+}{
+\spadpaste{continuedFraction((- 122 + 597*\%i)/(4 - 4*\%i))}
+}
+\xtc{
+This is an expansion for a quotient of polynomials in one variable
+with rational number coefficients.
+}{
+\spadpaste{r : Fraction UnivariatePolynomial(x,Fraction Integer) \bound{rdec}}
+}
+\xtc{
+}{
+\spadpaste{r := ((x - 1) * (x - 2)) / ((x-3) * (x-4)) \free{rdec}\bound{r}}
+}
+\xtc{
+}{
+\spadpaste{continuedFraction r \free{r}}
+}
+
+To conclude this section, we give you evidence that
+\texht{\narrowDisplay{%
+z =
+{3+\zag{1}{3}+\zag{1}{6}+\zag{1}{3}+\zag{1}{6}+\zag{1}{3}+\zag{1}{6}+
+\zag{1}{3}+\zag{1}{6}+\zag{1}{3}+\zag{1}{6}+...}}%
+}{
+\begin{verbatim}
+    z = 3 +            1
+            -----------------------
+            3 +          1
+                -------------------
+                6 +        1
+                    ---------------
+                    3 +      1
+                        -----------
+                        6 +    1
+                            -------
+                            3 + ...
+\end{verbatim}
+}
+is the expansion of \texht{$\sqrt{11}$}{the square root of \spad{11}}.
+%
+\xtc{
+}{
+\spadpaste{[i*i for i in convergents(z) :: Stream Float] \free{z}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/CONTFRAC.pht b/src/hyper/pages/CONTFRAC.pht
new file mode 100644
index 00000000..fb95eff0
--- /dev/null
+++ b/src/hyper/pages/CONTFRAC.pht
@@ -0,0 +1,437 @@
+\begin{patch}{ContinuedFractionXmpPagePatch1}
+\begin{paste}{ContinuedFractionXmpPageFull1}{ContinuedFractionXmpPageEmpty1}
+\pastebutton{ContinuedFractionXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{c := continuedFraction(314159/100000)\bound{c }}
+\indentrel{3}\begin{verbatim}
+   (1)
+           1 ³     1  ³     1 ³     1  ³     1 ³     1 ³
+     3 + ÚÄÄÄÙ + ÚÄÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÙ
+         ³ 7     ³ 15     ³ 1     ³ 25     ³ 1     ³ 7
+   + 
+       1 ³
+     ÚÄÄÄÙ
+     ³ 4
+                        Type: ContinuedFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty1}
+\begin{paste}{ContinuedFractionXmpPageEmpty1}{ContinuedFractionXmpPagePatch1}
+\pastebutton{ContinuedFractionXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{c := continuedFraction(314159/100000)\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch2}
+\begin{paste}{ContinuedFractionXmpPageFull2}{ContinuedFractionXmpPageEmpty2}
+\pastebutton{ContinuedFractionXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{partialQuotients c\free{c }}
+\indentrel{3}\begin{verbatim}
+   (2)  [3,7,15,1,25,1,7,4]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty2}
+\begin{paste}{ContinuedFractionXmpPageEmpty2}{ContinuedFractionXmpPagePatch2}
+\pastebutton{ContinuedFractionXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{partialQuotients c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch3}
+\begin{paste}{ContinuedFractionXmpPageFull3}{ContinuedFractionXmpPageEmpty3}
+\pastebutton{ContinuedFractionXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{convergents c\free{c }}
+\indentrel{3}\begin{verbatim}
+           22 333 355 9208 9563 76149 314159
+   (3)  [3,ÄÄ,ÄÄÄ,ÄÄÄ,ÄÄÄÄ,ÄÄÄÄ,ÄÄÄÄÄ,ÄÄÄÄÄÄ]
+            7 106 113 2931 3044 24239 100000
+                          Type: Stream Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty3}
+\begin{paste}{ContinuedFractionXmpPageEmpty3}{ContinuedFractionXmpPagePatch3}
+\pastebutton{ContinuedFractionXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{convergents c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch4}
+\begin{paste}{ContinuedFractionXmpPageFull4}{ContinuedFractionXmpPageEmpty4}
+\pastebutton{ContinuedFractionXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{approximants c\free{c }}
+\indentrel{3}\begin{verbatim}
+                                      ______
+           22 333 355 9208 9563 76149 314159
+   (4)  [3,ÄÄ,ÄÄÄ,ÄÄÄ,ÄÄÄÄ,ÄÄÄÄ,ÄÄÄÄÄ,ÄÄÄÄÄÄ]
+            7 106 113 2931 3044 24239 100000
+                          Type: Stream Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty4}
+\begin{paste}{ContinuedFractionXmpPageEmpty4}{ContinuedFractionXmpPagePatch4}
+\pastebutton{ContinuedFractionXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{approximants c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch5}
+\begin{paste}{ContinuedFractionXmpPageFull5}{ContinuedFractionXmpPageEmpty5}
+\pastebutton{ContinuedFractionXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{pq := partialQuotients(1/c)\free{c }\bound{pq }}
+\indentrel{3}\begin{verbatim}
+   (5)  [0,3,7,15,1,25,1,7,4]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty5}
+\begin{paste}{ContinuedFractionXmpPageEmpty5}{ContinuedFractionXmpPagePatch5}
+\pastebutton{ContinuedFractionXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{pq := partialQuotients(1/c)\free{c }\bound{pq }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch6}
+\begin{paste}{ContinuedFractionXmpPageFull6}{ContinuedFractionXmpPageEmpty6}
+\pastebutton{ContinuedFractionXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{continuedFraction(first pq,repeating [1],rest pq)\free{pq }}
+\indentrel{3}\begin{verbatim}
+   (6)
+       1 ³     1 ³     1  ³     1 ³     1  ³     1 ³
+     ÚÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÄÙ + ÚÄÄÄÙ
+     ³ 3     ³ 7     ³ 15     ³ 1     ³ 25     ³ 1
+   + 
+       1 ³     1 ³
+     ÚÄÄÄÙ + ÚÄÄÄÙ
+     ³ 7     ³ 4
+                        Type: ContinuedFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty6}
+\begin{paste}{ContinuedFractionXmpPageEmpty6}{ContinuedFractionXmpPagePatch6}
+\pastebutton{ContinuedFractionXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{continuedFraction(first pq,repeating [1],rest pq)\free{pq }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch7}
+\begin{paste}{ContinuedFractionXmpPageFull7}{ContinuedFractionXmpPageEmpty7}
+\pastebutton{ContinuedFractionXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{z:=continuedFraction(3,repeating [1],repeating [3,6])\bound{z }}
+\indentrel{3}\begin{verbatim}
+   (7)
+           1 ³     1 ³     1 ³     1 ³     1 ³     1 ³
+     3 + ÚÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÙ
+         ³ 3     ³ 6     ³ 3     ³ 6     ³ 3     ³ 6
+   + 
+       1 ³     1 ³     1 ³     1 ³
+     ÚÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÙ + ...
+     ³ 3     ³ 6     ³ 3     ³ 6
+                        Type: ContinuedFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty7}
+\begin{paste}{ContinuedFractionXmpPageEmpty7}{ContinuedFractionXmpPagePatch7}
+\pastebutton{ContinuedFractionXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{z:=continuedFraction(3,repeating [1],repeating [3,6])\bound{z }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch8}
+\begin{paste}{ContinuedFractionXmpPageFull8}{ContinuedFractionXmpPageEmpty8}
+\pastebutton{ContinuedFractionXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{dens:Stream Integer := cons(1,generate((x+->x+4),6))\bound{dens }}
+\indentrel{3}\begin{verbatim}
+   (8)  [1,6,10,14,18,22,26,30,34,38,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty8}
+\begin{paste}{ContinuedFractionXmpPageEmpty8}{ContinuedFractionXmpPagePatch8}
+\pastebutton{ContinuedFractionXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{dens:Stream Integer := cons(1,generate((x+->x+4),6))\bound{dens }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch9}
+\begin{paste}{ContinuedFractionXmpPageFull9}{ContinuedFractionXmpPageEmpty9}
+\pastebutton{ContinuedFractionXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{cf := continuedFraction(0,repeating [1],dens)\free{dens }\bound{cf }}
+\indentrel{3}\begin{verbatim}
+   (9)
+       1 ³     1 ³     1  ³     1  ³     1  ³     1  ³
+     ÚÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÄÙ + ÚÄÄÄÄÙ + ÚÄÄÄÄÙ + ÚÄÄÄÄÙ
+     ³ 1     ³ 6     ³ 10     ³ 14     ³ 18     ³ 22
+   + 
+       1  ³     1  ³     1  ³     1  ³
+     ÚÄÄÄÄÙ + ÚÄÄÄÄÙ + ÚÄÄÄÄÙ + ÚÄÄÄÄÙ + ...
+     ³ 26     ³ 30     ³ 34     ³ 38
+                        Type: ContinuedFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty9}
+\begin{paste}{ContinuedFractionXmpPageEmpty9}{ContinuedFractionXmpPagePatch9}
+\pastebutton{ContinuedFractionXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{cf := continuedFraction(0,repeating [1],dens)\free{dens }\bound{cf }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch10}
+\begin{paste}{ContinuedFractionXmpPageFull10}{ContinuedFractionXmpPageEmpty10}
+\pastebutton{ContinuedFractionXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{ccf := convergents cf\free{cf }\bound{ccf }}
+\indentrel{3}\begin{verbatim}
+   (10)
+          6  61   860  15541  342762   8927353
+   [0, 1, Ä, ÄÄ, ÄÄÄÄ, ÄÄÄÄÄ, ÄÄÄÄÄÄ, ÄÄÄÄÄÄÄÄ,
+          7  71  1001  18089  398959  10391023
+    268163352   9126481321
+    ÄÄÄÄÄÄÄÄÄ, ÄÄÄÄÄÄÄÄÄÄÄ, ...]
+    312129649  10622799089
+                          Type: Stream Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty10}
+\begin{paste}{ContinuedFractionXmpPageEmpty10}{ContinuedFractionXmpPagePatch10}
+\pastebutton{ContinuedFractionXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{ccf := convergents cf\free{cf }\bound{ccf }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch11}
+\begin{paste}{ContinuedFractionXmpPageFull11}{ContinuedFractionXmpPageEmpty11}
+\pastebutton{ContinuedFractionXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{eConvergents := [2*e + 1 for e in ccf]\bound{ec }\free{ccf }}
+\indentrel{3}\begin{verbatim}
+   (11)
+          19  193  2721  49171  1084483  28245729
+   [1, 3, ÄÄ, ÄÄÄ, ÄÄÄÄ, ÄÄÄÄÄ, ÄÄÄÄÄÄÄ, ÄÄÄÄÄÄÄÄ,
+           7   71  1001  18089   398959  10391023
+    848456353  28875761731
+    ÄÄÄÄÄÄÄÄÄ, ÄÄÄÄÄÄÄÄÄÄÄ, ...]
+    312129649  10622799089
+                          Type: Stream Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty11}
+\begin{paste}{ContinuedFractionXmpPageEmpty11}{ContinuedFractionXmpPagePatch11}
+\pastebutton{ContinuedFractionXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{eConvergents := [2*e + 1 for e in ccf]\bound{ec }\free{ccf }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch12}
+\begin{paste}{ContinuedFractionXmpPageFull12}{ContinuedFractionXmpPageEmpty12}
+\pastebutton{ContinuedFractionXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{eConvergents :: Stream Float\free{ec }}
+\indentrel{3}\begin{verbatim}
+   (12)
+   [1.0, 3.0, 2.7142857142 857142857,
+    2.7183098591 549295775, 2.7182817182 817182817,
+    2.7182818287 356957267, 2.7182818284 585634113,
+    2.7182818284 590458514, 2.7182818284 590452348,
+    2.7182818284 590452354, ...]
+                                     Type: Stream Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty12}
+\begin{paste}{ContinuedFractionXmpPageEmpty12}{ContinuedFractionXmpPagePatch12}
+\pastebutton{ContinuedFractionXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{eConvergents :: Stream Float\free{ec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch13}
+\begin{paste}{ContinuedFractionXmpPageFull13}{ContinuedFractionXmpPageEmpty13}
+\pastebutton{ContinuedFractionXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{exp 1.0}
+\indentrel{3}\begin{verbatim}
+   (13)  2.7182818284 590452354
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty13}
+\begin{paste}{ContinuedFractionXmpPageEmpty13}{ContinuedFractionXmpPagePatch13}
+\pastebutton{ContinuedFractionXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{exp 1.0}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch14}
+\begin{paste}{ContinuedFractionXmpPageFull14}{ContinuedFractionXmpPageEmpty14}
+\pastebutton{ContinuedFractionXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{cf := continuedFraction(1,[(2*i+1)**2 for i in 0..],repeating [2])\bound{cf1 }}
+\indentrel{3}\begin{verbatim}
+   (14)
+           1 ³     9 ³     25 ³     49 ³     81 ³
+     1 + ÚÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÄÙ + ÚÄÄÄÄÙ + ÚÄÄÄÄÙ
+         ³ 2     ³ 2     ³ 2      ³ 2      ³ 2
+   + 
+     121 ³     169 ³     225 ³     289 ³     361 ³
+   ÚÄÄÄÄÄÙ + ÚÄÄÄÄÄÙ + ÚÄÄÄÄÄÙ + ÚÄÄÄÄÄÙ + ÚÄÄÄÄÄÙ + ...
+   ³  2      ³  2      ³  2      ³  2      ³  2
+                        Type: ContinuedFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty14}
+\begin{paste}{ContinuedFractionXmpPageEmpty14}{ContinuedFractionXmpPagePatch14}
+\pastebutton{ContinuedFractionXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{cf := continuedFraction(1,[(2*i+1)**2 for i in 0..],repeating [2])\bound{cf1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch15}
+\begin{paste}{ContinuedFractionXmpPageFull15}{ContinuedFractionXmpPageEmpty15}
+\pastebutton{ContinuedFractionXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{ccf := convergents cf\free{cf1 }\bound{ccf1 }}
+\indentrel{3}\begin{verbatim}
+   (15)
+      3 15 105 315 3465 45045 45045 765765 14549535
+   [1,Ä,ÄÄ,ÄÄÄ,ÄÄÄ,ÄÄÄÄ,ÄÄÄÄÄ,ÄÄÄÄÄ,ÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄ,...]
+      2 13  76 263 2578 36979 33976 622637 11064338
+                          Type: Stream Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty15}
+\begin{paste}{ContinuedFractionXmpPageEmpty15}{ContinuedFractionXmpPagePatch15}
+\pastebutton{ContinuedFractionXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{ccf := convergents cf\free{cf1 }\bound{ccf1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch16}
+\begin{paste}{ContinuedFractionXmpPageFull16}{ContinuedFractionXmpPageEmpty16}
+\pastebutton{ContinuedFractionXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{piConvergents := [4/p for p in ccf]\bound{piConvergents }\free{ccf1 }}
+\indentrel{3}\begin{verbatim}
+   (16)
+       8  52  304  1052  10312  147916  135904  2490548
+   [4, Ä, ÄÄ, ÄÄÄ, ÄÄÄÄ, ÄÄÄÄÄ, ÄÄÄÄÄÄ, ÄÄÄÄÄÄ, ÄÄÄÄÄÄÄ,
+       3  15  105   315   3465   45045   45045   765765
+    44257352
+    ÄÄÄÄÄÄÄÄ, ...]
+    14549535
+                          Type: Stream Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty16}
+\begin{paste}{ContinuedFractionXmpPageEmpty16}{ContinuedFractionXmpPagePatch16}
+\pastebutton{ContinuedFractionXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{piConvergents := [4/p for p in ccf]\bound{piConvergents }\free{ccf1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch17}
+\begin{paste}{ContinuedFractionXmpPageFull17}{ContinuedFractionXmpPageEmpty17}
+\pastebutton{ContinuedFractionXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{piConvergents :: Stream Float\free{piConvergents }}
+\indentrel{3}\begin{verbatim}
+   (17)
+   [4.0, 2.6666666666 666666667, 3.4666666666 666666667,
+    2.8952380952 380952381, 3.3396825396 825396825,
+    2.9760461760 461760462, 3.2837384837 384837385,
+    3.0170718170 718170718, 3.2523659347 188758953,
+    3.0418396189 294022111, ...]
+                                     Type: Stream Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty17}
+\begin{paste}{ContinuedFractionXmpPageEmpty17}{ContinuedFractionXmpPagePatch17}
+\pastebutton{ContinuedFractionXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{piConvergents :: Stream Float\free{piConvergents }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch18}
+\begin{paste}{ContinuedFractionXmpPageFull18}{ContinuedFractionXmpPageEmpty18}
+\pastebutton{ContinuedFractionXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{continuedFraction((- 122 + 597*\%i)/(4 - 4*\%i))}
+\indentrel{3}\begin{verbatim}
+                            1    ³         1     ³
+   (18)  - 90 + 59%i + ÚÄÄÄÄÄÄÄÄÄÙ + ÚÄÄÄÄÄÄÄÄÄÄÄÙ
+                       ³ 1 - 2%i     ³ - 1 + 2%i
+                Type: ContinuedFraction Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty18}
+\begin{paste}{ContinuedFractionXmpPageEmpty18}{ContinuedFractionXmpPagePatch18}
+\pastebutton{ContinuedFractionXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{continuedFraction((- 122 + 597*\%i)/(4 - 4*\%i))}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch19}
+\begin{paste}{ContinuedFractionXmpPageFull19}{ContinuedFractionXmpPageEmpty19}
+\pastebutton{ContinuedFractionXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{r : Fraction UnivariatePolynomial(x,Fraction Integer)\bound{rdec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty19}
+\begin{paste}{ContinuedFractionXmpPageEmpty19}{ContinuedFractionXmpPagePatch19}
+\pastebutton{ContinuedFractionXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{r : Fraction UnivariatePolynomial(x,Fraction Integer)\bound{rdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch20}
+\begin{paste}{ContinuedFractionXmpPageFull20}{ContinuedFractionXmpPageEmpty20}
+\pastebutton{ContinuedFractionXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{r := ((x - 1) * (x - 2)) / ((x-3) * (x-4))\free{rdec }\bound{r }}
+\indentrel{3}\begin{verbatim}
+           2
+          x  - 3x + 2
+   (20)  ÄÄÄÄÄÄÄÄÄÄÄÄ
+          2
+         x  - 7x + 12
+Type: Fraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty20}
+\begin{paste}{ContinuedFractionXmpPageEmpty20}{ContinuedFractionXmpPagePatch20}
+\pastebutton{ContinuedFractionXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{r := ((x - 1) * (x - 2)) / ((x-3) * (x-4))\free{rdec }\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch21}
+\begin{paste}{ContinuedFractionXmpPageFull21}{ContinuedFractionXmpPageEmpty21}
+\pastebutton{ContinuedFractionXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{continuedFraction r\free{r }}
+\indentrel{3}\begin{verbatim}
+                  1    ³         1     ³
+   (21)  1 + ÚÄÄÄÄÄÄÄÄÄÙ + ÚÄÄÄÄÄÄÄÄÄÄÄÙ
+             ³ 1     9     ³ 16     40
+             ³ Ä x - Ä     ³ ÄÄ x - ÄÄ
+             ³ 4     8     ³  3      3
+Type: ContinuedFraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty21}
+\begin{paste}{ContinuedFractionXmpPageEmpty21}{ContinuedFractionXmpPagePatch21}
+\pastebutton{ContinuedFractionXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{continuedFraction r\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPagePatch22}
+\begin{paste}{ContinuedFractionXmpPageFull22}{ContinuedFractionXmpPageEmpty22}
+\pastebutton{ContinuedFractionXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{[i*i for i in convergents(z) :: Stream Float]\free{z }}
+\indentrel{3}\begin{verbatim}
+   (22)
+   [9.0, 11.1111111111 11111111, 10.9944598337 9501385,
+    11.0002777777 77777778, 10.9999860763 98799786,
+    11.0000006979 29731039, 10.9999999650 15834446,
+    11.0000000017 53603304, 10.9999999999 12099531,
+    11.0000000000 04406066, ...]
+                                     Type: Stream Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ContinuedFractionXmpPageEmpty22}
+\begin{paste}{ContinuedFractionXmpPageEmpty22}{ContinuedFractionXmpPagePatch22}
+\pastebutton{ContinuedFractionXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{[i*i for i in convergents(z) :: Stream Float]\free{z }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/CPHelp.ht b/src/hyper/pages/CPHelp.ht
new file mode 100644
index 00000000..8eadbc31
--- /dev/null
+++ b/src/hyper/pages/CPHelp.ht
@@ -0,0 +1,44 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Control Panel Help Page
+% @(#)CPHelp.ht	1.2 91/06/27 00:44:38
+
+\begin{page}{CPHelp}{Control Panel Bits}
+\beginscroll
+
+Here are some stuff from a Three Dimensional Viewport's Control Panel \newline
+
+Main Control Panel: \newline \newline
+
+%%
+%% These things were generated from the file helpCP3D.dat
+%%
+%% The help comments are hokey and are not the intended 
+%% format.
+%%
+
+Rotate: \helpbit{rotate3D}
+Zoom: \helpbit{zoom3D}
+Translate up/down left/right in the window: \helpbit{translate3D}
+Changing the color of the rendered surface: \helpbit{color3D}
+Turn the axes on and off: \helpbit{axes3D}
+Display surface as a transparent wire mesh: \helpbit{transparent3D}
+Display surface with hidden surface removed and the core dumped: \helpbit{opaque3D}
+Display rendered surface: \helpbit{render3D}
+Show region within which the function is defined: \helpbit{region3D}
+Change position of light source: \helpbit{lighting3D}
+Change Perspective/Clipping of surface: \helpbit{volume3D}
+Reset to original viewpoint: \helpbit{reset3D}
+Show grid lines on the rendered surface: \helpbit{outline3D}
+Hide the menu: \helpbit{hide3D}
+Close the viewport: \helpbit{close3D}
+
+\newline
+\newline
+\endscroll
+\autobuttons
+\end{page}
+
+
diff --git a/src/hyper/pages/CYCLES.ht b/src/hyper/pages/CYCLES.ht
new file mode 100644
index 00000000..4d8a63a1
--- /dev/null
+++ b/src/hyper/pages/CYCLES.ht
@@ -0,0 +1,377 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\CycleIndicatorsXmpTitle}{CycleIndicators}
+\newcommand{\CycleIndicatorsXmpNumber}{9.13}
+%
+% =====================================================================
+\begin{page}{CycleIndicatorsXmpPage}{9.13 CycleIndicators}
+% =====================================================================
+\beginscroll
+This section is based upon the paper
+J. H. Redfield, ``The Theory of Group-Reduced Distributions'',
+%-% \HDindex{Redfield, J. H.}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+American J. Math.,49 (1927) 433-455,
+and is an application of group theory to enumeration problems.
+It is a development of the work by P. A. MacMahon on the
+%-% \HDindex{MacMahon, P. A.}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+application of symmetric functions and Hammond operators to
+%-% \HDindex{function!symmetric}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+combinatorial theory.
+%-% \HDindex{operator!Hammond}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+%-% \HDindex{combinatorics}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+
+The theory is based upon the power sum symmetric functions
+\subscriptIt{s}{i}
+which are the sum of the \eth{\it i} powers of the variables.
+The cycle index of a permutation is an expression that specifies
+the sizes of the cycles of a permutation, and
+may be represented as a partition.
+A partition of a non-negative integer \spad{n} is a collection
+of positive integers called its parts whose sum is \spad{n}.
+%-% \HDindex{cycle index}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+For example, the partition
+%-% \HDindex{partition}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+\texht{$(3^2 \  2 \ 1^2)$}{\spad{3**2 2 1**2)}}
+will be used to represent
+\texht{$s^2_3 s_2 s^2_1$}{\spad{(s_3)**2 s_2 (s_1)**2}}
+and will indicate that the permutation has two cycles of length 3,
+%-% \HDindex{permutation}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+one of length 2 and two of length 1.
+The cycle index of a permutation group is the sum of the cycle indices
+of its permutations divided by the number of permutations.
+The cycle indices of certain groups are provided.
+\xtc{
+We first expose something from the library.
+}{
+\spadpaste{)expose EVALCYC}
+}
+\xtc{
+The operation \spadfun{complete} returns the cycle index of the
+%-% \HDindex{group!symmetric}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+symmetric group of order \spad{n} for argument \spad{n}.
+Alternatively, it is the \eth{\spad{n}} complete homogeneous symmetric
+function expressed in terms of power sum symmetric functions.
+}{
+\spadpaste{complete 1}
+}
+\xtc{
+}{
+\spadpaste{complete 2}
+}
+\xtc{
+}{
+\spadpaste{complete 3}
+}
+\xtc{
+}{
+\spadpaste{complete 7}
+}
+\xtc{
+The operation \spadfun{elementary} computes the \eth{\spad{n}}
+elementary symmetric function for argument \spad{n.}
+}{
+\spadpaste{elementary 7}
+}
+\xtc{
+The operation \spadfun{alternating} returns
+%-% \HDindex{group!alternating}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+the cycle index of the alternating group
+having an even number of even parts in each cycle partition.
+}{
+\spadpaste{alternating 7}
+}
+\xtc{
+The operation \spadfun{cyclic} returns the cycle index of the cyclic group.
+%-% \HDindex{group!cyclic}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+}{
+\spadpaste{cyclic 7}
+}
+\xtc{
+The operation \spadfun{dihedral} is the cycle index of the
+%-% \HDindex{group!dihedral}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+dihedral group.
+}{
+\spadpaste{dihedral 7}
+}
+\xtc{
+The operation \spadfun{graphs} for argument \spad{n} returns
+the cycle index of the group of permutations on
+the edges of the complete graph with \spad{n} nodes induced by
+%-% \HDindex{graph}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+applying the symmetric group to the nodes.
+}{
+\spadpaste{graphs 5}
+}
+
+The cycle index of a direct product of two groups is the product
+of the cycle indices of the groups.
+Redfield provided two operations on two cycle indices which will
+be called ``cup'' and ``cap'' here.
+The \spadfun{cup} of two cycle indices is a kind of scalar product
+that combines monomials for permutations with the same cycles.
+The \spadfun{cap} operation provides the sum of the coefficients
+of the result of the \spadfun{cup} operation which will be an
+integer that enumerates what Redfield called
+group-reduced distributions.
+
+We can, for example, represent  \spad{complete 2 * complete 2}
+as the set of objects \spad{a a b b} and
+\spad{complete 2 * complete 1 * complete 1} as \spad{c c d e.}
+
+\xtc{
+This integer
+is the number of different sets of four pairs.
+}{
+\spadpaste{cap(complete 2**2, complete 2*complete 1**2)}
+}
+For example,
+\begin{verbatim}
+a a b b     a a b b    a a b b   a a b b
+c c d e     c d c e    c e c d   d e c c
+\end{verbatim}
+
+\xtc{
+This integer
+is the number of different sets of four pairs no two pairs being equal.
+}{
+\spadpaste{cap(elementary 2**2, complete 2*complete 1**2)}
+}
+For example,
+\begin{verbatim}
+a a b b    a a b b
+c d c e    c e c d
+\end{verbatim}
+In this case the configurations enumerated are easily constructed,
+however the theory merely enumerates them providing little help in
+actually constructing them.
+\xtc{
+Here are the
+number of 6-pairs, first from \spad{a a a b b c,} second from
+\spad{d d e e f g.}
+}{
+\spadpaste{cap(complete 3*complete 2*complete 1,complete 2**2*complete 1**2)}
+}
+\xtc{
+Here it is again, but with no equal pairs.
+}{
+\spadpaste{cap(elementary 3*elementary 2*elementary 1,complete 2**2*complete 1**2)}
+}
+\xtc{
+}{
+\spadpaste{cap(complete 3*complete 2*complete 1,elementary 2**2*elementary 1**2)}
+}
+\xtc{
+The number of 6-triples, first from \spad{a a a b b c,} second from
+\spad{d d e e f g,} third from \spad{h h i i j j.}
+}{
+\spadpaste{eval(cup(complete 3*complete 2*complete 1, cup(complete 2**2*complete 1**2,complete 2**3)))}
+}
+\xtc{
+The cycle index of vertices of a square is dihedral 4.
+}{
+\spadpaste{square:=dihedral 4}
+}
+\xtc{
+The number of different squares with 2 red vertices and 2 blue vertices.
+}{
+\spadpaste{cap(complete 2**2,square)}
+}
+\xtc{
+The number of necklaces with 3 red beads, 2 blue beads and 2 green beads.
+}{
+\spadpaste{cap(complete 3*complete 2**2,dihedral 7)}
+}
+\xtc{
+The number of graphs with 5 nodes and 7 edges.
+}{
+\spadpaste{cap(graphs 5,complete 7*complete 3)}
+}
+\xtc{
+The cycle index of rotations of vertices of a cube.
+}{
+\spadpaste{s(x) == powerSum(x)}
+}
+\xtc{
+}{
+\spadpaste{cube:=(1/24)*(s 1**8+9*s 2**4 + 8*s 3**2*s 1**2+6*s 4**2)}
+}
+\xtc{
+The number of cubes with 4 red vertices and 4 blue vertices.
+}{
+\spadpaste{cap(complete 4**2,cube)}
+}
+\xtc{
+The number of labeled graphs with degree sequence \spad{2 2 2 1 1}
+with no loops or multiple edges.
+}{
+\spadpaste{cap(complete 2**3*complete 1**2,wreath(elementary 4,elementary 2))}
+}
+\xtc{
+Again, but
+with loops allowed but not multiple edges.
+}{
+\spadpaste{cap(complete 2**3*complete 1**2,wreath(elementary 4,complete 2))}
+}
+\xtc{
+Again, but
+with multiple edges allowed, but not loops
+}{
+\spadpaste{cap(complete 2**3*complete 1**2,wreath(complete 4,elementary 2))}
+}
+\xtc{
+Again, but
+with both multiple edges and loops allowed
+}{
+\spadpaste{cap(complete 2**3*complete 1**2,wreath(complete 4,complete 2))}
+}
+
+Having constructed a cycle index for a configuration
+we are at liberty to evaluate the
+\subscriptIt{s}{i}
+components any way we please.
+For example we can produce enumerating generating functions.
+%-% \HDindex{function!enumerating generating}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+This is done by providing a function \spad{f} on an integer \spad{i} to the
+value required of \subscriptIt{s}{i},
+and then evaluating \spad{eval(f, cycleindex)}.
+\xtc{
+}{
+\spadpaste{x: ULS(FRAC INT,'x,0) := 'x \bound{x}}
+}
+\xtc{
+}{
+\spadpaste{ZeroOrOne: INT -> ULS(FRAC INT, 'x, 0) \bound{zodec}}
+}
+\xtc{
+}{
+\spadpaste{Integers: INT -> ULS(FRAC INT, 'x, 0) \bound{idec}}
+}
+\xtc{
+For the integers 0 and 1, or two colors.
+}{
+\spadpaste{ZeroOrOne n == 1+x**n \free{x zodec}\bound{zo}}
+}
+\xtc{
+}{
+\spadpaste{ZeroOrOne 5 \free{zo}}
+}
+\xtc{
+For the integers \spad{0, 1, 2, ...} we have this.
+}{
+\spadpaste{Integers n == 1/(1-x**n) \free{x idec}\bound{i}}
+}
+\xtc{
+}{
+\spadpaste{Integers 5 \free{i}}
+}
+
+\xtc{
+The coefficient of \texht{$x^n$}{\spad{x**n}}
+is the number of graphs with 5 nodes
+and \spad{n} edges.
+}{
+\spadpaste{eval(ZeroOrOne, graphs 5) \free{zo}}
+}
+\xtc{
+The coefficient of \texht{$x^n$}{\spad{x**n}}  is the number of necklaces with
+\spad{n} red beads and \spad{n-8} green beads.
+}{
+\spadpaste{eval(ZeroOrOne,dihedral 8) \free{zo}}
+}
+\xtc{
+The coefficient of \texht{$x^n$}{\spad{x**n}} is the number of partitions of
+\spad{n} into 4 or fewer parts.
+}{
+\spadpaste{eval(Integers,complete 4) \free{i}}
+}
+\xtc{
+The coefficient of \texht{$x^n$}{\spad{x**n}} is the number of
+partitions of \spad{n} into 4
+boxes containing ordered distinct parts.
+}{
+\spadpaste{eval(Integers,elementary 4) \free{i}}
+}
+\xtc{
+The coefficient of \texht{$x^n$}{\spad{x**n}} is the number of
+different cubes with \spad{n} red vertices and \spad{8-n} green ones.
+}{
+\spadpaste{eval(ZeroOrOne,cube) \free{zo}}
+}
+\xtc{
+The coefficient of \texht{$x^n$}{\spad{x**n}} is the number of different cubes with integers
+on the vertices whose sum is \spad{n.}
+}{
+\spadpaste{eval(Integers,cube) \free{i}}
+}
+\xtc{
+The coefficient of \texht{$x^n$}{\spad{x**n}} is the number of
+graphs with 5 nodes and with integers on the edges whose sum is
+\spad{n.}
+In other words, the enumeration is of multigraphs with 5 nodes and
+\spad{n} edges.
+}{
+\spadpaste{eval(Integers,graphs 5) \free{i}}
+}
+\xtc{
+Graphs with 15 nodes enumerated with respect to number of edges.
+}{
+\spadpaste{eval(ZeroOrOne ,graphs 15) \free{zo}}
+}
+\xtc{
+Necklaces with 7 green beads, 8 white beads, 5 yellow beads and 10
+red beads.
+}{
+\spadpaste{cap(dihedral 30,complete 7*complete 8*complete 5*complete 10)}
+}
+The operation \spadfun{SFunction} is the S-function or Schur function
+of a partition written
+as a descending list of integers expressed in terms of power sum
+symmetric functions.
+\xtc{
+In this case the argument partition represents a tableau shape.
+For example \spad{3,2,2,1} represents a tableau with three boxes in the
+first row, two boxes in the second and third rows, and one box in the
+fourth row.
+\spad{SFunction [3,2,2,1]}
+counts the number of different tableaux of shape \spad{3, 2, 2, 1} filled
+with objects with an ascending order in the columns and a
+non-descending order in the rows.
+}{
+\spadpaste{sf3221:= SFunction [3,2,2,1] \bound{sf3221}}
+}
+\xtc{
+This is the number filled with \spad{a a b b c c d d.}
+}{
+\spadpaste{cap(sf3221,complete 2**4) \free{sf3221}}
+}
+The configurations enumerated above are:
+\begin{verbatim}
+a a b    a a c    a a d
+b c      b b      b b
+c d      c d      c c
+d        d        d
+\end{verbatim}
+\xtc{
+This is the number of tableaux filled with \spad{1..8.}
+%-% \HDindex{tableaux}{CycleIndicatorsXmpPage}{9.13}{CycleIndicators}
+}{
+\spadpaste{cap(sf3221, powerSum 1**8)\free{sf3221}}
+}
+\xtc{
+The coefficient of \texht{$x^n$}{\spad{x**n}} is the number
+of column strict reverse plane partitions of \spad{n} of shape
+\spad{3 2 2 1.}
+}{
+\spadpaste{eval(Integers, sf3221)\free{i sf3221}}
+}
+The smallest is
+\begin{verbatim}
+0 0 0
+1 1
+2 2
+3
+\end{verbatim}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/CYCLES.pht b/src/hyper/pages/CYCLES.pht
new file mode 100644
index 00000000..65fc1f3a
--- /dev/null
+++ b/src/hyper/pages/CYCLES.pht
@@ -0,0 +1,855 @@
+\begin{patch}{CycleIndicatorsXmpPagePatch1}
+\begin{paste}{CycleIndicatorsXmpPageFull1}{CycleIndicatorsXmpPageEmpty1}
+\pastebutton{CycleIndicatorsXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)expose EVALCYC}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty1}
+\begin{paste}{CycleIndicatorsXmpPageEmpty1}{CycleIndicatorsXmpPagePatch1}
+\pastebutton{CycleIndicatorsXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)expose EVALCYC}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch2}
+\begin{paste}{CycleIndicatorsXmpPageFull2}{CycleIndicatorsXmpPageEmpty2}
+\pastebutton{CycleIndicatorsXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{complete 1}
+\indentrel{3}\begin{verbatim}
+   (1)  (1)
+             Type: SymmetricPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty2}
+\begin{paste}{CycleIndicatorsXmpPageEmpty2}{CycleIndicatorsXmpPagePatch2}
+\pastebutton{CycleIndicatorsXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{complete 1}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch3}
+\begin{paste}{CycleIndicatorsXmpPageFull3}{CycleIndicatorsXmpPageEmpty3}
+\pastebutton{CycleIndicatorsXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{complete 2}
+\indentrel{3}\begin{verbatim}
+        1       1   2
+   (2)  Ä (2) + Ä (1 )
+        2       2
+             Type: SymmetricPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty3}
+\begin{paste}{CycleIndicatorsXmpPageEmpty3}{CycleIndicatorsXmpPagePatch3}
+\pastebutton{CycleIndicatorsXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{complete 2}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch4}
+\begin{paste}{CycleIndicatorsXmpPageFull4}{CycleIndicatorsXmpPageEmpty4}
+\pastebutton{CycleIndicatorsXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{complete 3}
+\indentrel{3}\begin{verbatim}
+        1       1         1   3
+   (3)  Ä (3) + Ä (2 1) + Ä (1 )
+        3       2         6
+             Type: SymmetricPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty4}
+\begin{paste}{CycleIndicatorsXmpPageEmpty4}{CycleIndicatorsXmpPagePatch4}
+\pastebutton{CycleIndicatorsXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{complete 3}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch5}
+\begin{paste}{CycleIndicatorsXmpPageFull5}{CycleIndicatorsXmpPageEmpty5}
+\pastebutton{CycleIndicatorsXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{complete 7}
+\indentrel{3}\begin{verbatim}
+   (4)
+     1       1          1          1     2     1
+     Ä (7) + Ä (6 1) + ÄÄ (5 2) + ÄÄ (5 1 ) + ÄÄ (4 3)
+     7       6         10         10          12
+   + 
+     1            1     3     1   2      1     2
+     Ä (4 2 1) + ÄÄ (4 1 ) + ÄÄ (3 1) + ÄÄ (3 2 )
+     8           24          18         24
+   + 
+      1       2     1     4     1   3      1   2 3
+     ÄÄ (3 2 1 ) + ÄÄ (3 1 ) + ÄÄ (2 1) + ÄÄ (2 1 )
+     12            72          48         48
+   + 
+      1      5      1    7
+     ÄÄÄ (2 1 ) + ÄÄÄÄ (1 )
+     240          5040
+             Type: SymmetricPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty5}
+\begin{paste}{CycleIndicatorsXmpPageEmpty5}{CycleIndicatorsXmpPagePatch5}
+\pastebutton{CycleIndicatorsXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{complete 7}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch6}
+\begin{paste}{CycleIndicatorsXmpPageFull6}{CycleIndicatorsXmpPageEmpty6}
+\pastebutton{CycleIndicatorsXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{elementary 7}
+\indentrel{3}\begin{verbatim}
+   (5)
+     1       1          1          1     2     1
+     Ä (7) - Ä (6 1) - ÄÄ (5 2) + ÄÄ (5 1 ) - ÄÄ (4 3)
+     7       6         10         10          12
+   + 
+     1            1     3     1   2      1     2
+     Ä (4 2 1) - ÄÄ (4 1 ) + ÄÄ (3 1) + ÄÄ (3 2 )
+     8           24          18         24
+   + 
+        1       2     1     4     1   3      1   2 3
+     - ÄÄ (3 2 1 ) + ÄÄ (3 1 ) - ÄÄ (2 1) + ÄÄ (2 1 )
+       12            72          48         48
+   + 
+        1      5      1    7
+     - ÄÄÄ (2 1 ) + ÄÄÄÄ (1 )
+       240          5040
+             Type: SymmetricPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty6}
+\begin{paste}{CycleIndicatorsXmpPageEmpty6}{CycleIndicatorsXmpPagePatch6}
+\pastebutton{CycleIndicatorsXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{elementary 7}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch7}
+\begin{paste}{CycleIndicatorsXmpPageFull7}{CycleIndicatorsXmpPageEmpty7}
+\pastebutton{CycleIndicatorsXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{alternating 7}
+\indentrel{3}\begin{verbatim}
+   (6)
+     2       1     2    1           1   2      1     2
+     Ä (7) + Ä (5 1 ) + Ä (4 2 1) + Ä (3 1) + ÄÄ (3 2 )
+     7       5          4           9         12
+   + 
+      1     4     1   2 3      1    7
+     ÄÄ (3 1 ) + ÄÄ (2 1 ) + ÄÄÄÄ (1 )
+     36          24          2520
+             Type: SymmetricPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty7}
+\begin{paste}{CycleIndicatorsXmpPageEmpty7}{CycleIndicatorsXmpPagePatch7}
+\pastebutton{CycleIndicatorsXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{alternating 7}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch8}
+\begin{paste}{CycleIndicatorsXmpPageFull8}{CycleIndicatorsXmpPageEmpty8}
+\pastebutton{CycleIndicatorsXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{cyclic 7}
+\indentrel{3}\begin{verbatim}
+        6       1   7
+   (7)  Ä (7) + Ä (1 )
+        7       7
+             Type: SymmetricPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty8}
+\begin{paste}{CycleIndicatorsXmpPageEmpty8}{CycleIndicatorsXmpPagePatch8}
+\pastebutton{CycleIndicatorsXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{cyclic 7}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch9}
+\begin{paste}{CycleIndicatorsXmpPageFull9}{CycleIndicatorsXmpPageEmpty9}
+\pastebutton{CycleIndicatorsXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{dihedral 7}
+\indentrel{3}\begin{verbatim}
+        3       1   3      1   7
+   (8)  Ä (7) + Ä (2 1) + ÄÄ (1 )
+        7       2         14
+             Type: SymmetricPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty9}
+\begin{paste}{CycleIndicatorsXmpPageEmpty9}{CycleIndicatorsXmpPagePatch9}
+\pastebutton{CycleIndicatorsXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{dihedral 7}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch10}
+\begin{paste}{CycleIndicatorsXmpPageFull10}{CycleIndicatorsXmpPageEmpty10}
+\pastebutton{CycleIndicatorsXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{graphs 5}
+\indentrel{3}\begin{verbatim}
+   (9)
+     1           1   2    1   2     1   3     1   4 2
+     Ä (6 3 1) + Ä (5 ) + Ä (4 2) + Ä (3 1) + Ä (2 1 )
+     6           5        4         6         8
+   + 
+      1   3 4     1    10
+     ÄÄ (2 1 ) + ÄÄÄ (1  )
+     12          120
+             Type: SymmetricPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty10}
+\begin{paste}{CycleIndicatorsXmpPageEmpty10}{CycleIndicatorsXmpPagePatch10}
+\pastebutton{CycleIndicatorsXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{graphs 5}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch11}
+\begin{paste}{CycleIndicatorsXmpPageFull11}{CycleIndicatorsXmpPageEmpty11}
+\pastebutton{CycleIndicatorsXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{cap(complete 2**2, complete 2*complete 1**2)}
+\indentrel{3}\begin{verbatim}
+   (10)  4
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty11}
+\begin{paste}{CycleIndicatorsXmpPageEmpty11}{CycleIndicatorsXmpPagePatch11}
+\pastebutton{CycleIndicatorsXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{cap(complete 2**2, complete 2*complete 1**2)}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch12}
+\begin{paste}{CycleIndicatorsXmpPageFull12}{CycleIndicatorsXmpPageEmpty12}
+\pastebutton{CycleIndicatorsXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{cap(elementary 2**2, complete 2*complete 1**2)}
+\indentrel{3}\begin{verbatim}
+   (11)  2
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty12}
+\begin{paste}{CycleIndicatorsXmpPageEmpty12}{CycleIndicatorsXmpPagePatch12}
+\pastebutton{CycleIndicatorsXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{cap(elementary 2**2, complete 2*complete 1**2)}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch13}
+\begin{paste}{CycleIndicatorsXmpPageFull13}{CycleIndicatorsXmpPageEmpty13}
+\pastebutton{CycleIndicatorsXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{cap(complete 3*complete 2*complete 1,complete 2**2*complete 1**2)}
+\indentrel{3}\begin{verbatim}
+   (12)  24
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty13}
+\begin{paste}{CycleIndicatorsXmpPageEmpty13}{CycleIndicatorsXmpPagePatch13}
+\pastebutton{CycleIndicatorsXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{cap(complete 3*complete 2*complete 1,complete 2**2*complete 1**2)}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch14}
+\begin{paste}{CycleIndicatorsXmpPageFull14}{CycleIndicatorsXmpPageEmpty14}
+\pastebutton{CycleIndicatorsXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{cap(elementary 3*elementary 2*elementary 1,complete 2**2*complete 1**2)}
+\indentrel{3}\begin{verbatim}
+   (13)  8
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty14}
+\begin{paste}{CycleIndicatorsXmpPageEmpty14}{CycleIndicatorsXmpPagePatch14}
+\pastebutton{CycleIndicatorsXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{cap(elementary 3*elementary 2*elementary 1,complete 2**2*complete 1**2)}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch15}
+\begin{paste}{CycleIndicatorsXmpPageFull15}{CycleIndicatorsXmpPageEmpty15}
+\pastebutton{CycleIndicatorsXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{cap(complete 3*complete 2*complete 1,elementary 2**2*elementary 1**2)}
+\indentrel{3}\begin{verbatim}
+   (14)  8
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty15}
+\begin{paste}{CycleIndicatorsXmpPageEmpty15}{CycleIndicatorsXmpPagePatch15}
+\pastebutton{CycleIndicatorsXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{cap(complete 3*complete 2*complete 1,elementary 2**2*elementary 1**2)}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch16}
+\begin{paste}{CycleIndicatorsXmpPageFull16}{CycleIndicatorsXmpPageEmpty16}
+\pastebutton{CycleIndicatorsXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{eval(cup(complete 3*complete 2*complete 1, cup(complete 2**2*complete 1**2,complete 2**3)))}
+\indentrel{3}\begin{verbatim}
+   (15)  1500
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty16}
+\begin{paste}{CycleIndicatorsXmpPageEmpty16}{CycleIndicatorsXmpPagePatch16}
+\pastebutton{CycleIndicatorsXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{eval(cup(complete 3*complete 2*complete 1, cup(complete 2**2*complete 1**2,complete 2**3)))}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch17}
+\begin{paste}{CycleIndicatorsXmpPageFull17}{CycleIndicatorsXmpPageEmpty17}
+\pastebutton{CycleIndicatorsXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{square:=dihedral 4}
+\indentrel{3}\begin{verbatim}
+         1       3   2    1     2    1   4
+   (16)  Ä (4) + Ä (2 ) + Ä (2 1 ) + Ä (1 )
+         4       8        4          8
+             Type: SymmetricPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty17}
+\begin{paste}{CycleIndicatorsXmpPageEmpty17}{CycleIndicatorsXmpPagePatch17}
+\pastebutton{CycleIndicatorsXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{square:=dihedral 4}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch18}
+\begin{paste}{CycleIndicatorsXmpPageFull18}{CycleIndicatorsXmpPageEmpty18}
+\pastebutton{CycleIndicatorsXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{cap(complete 2**2,square)}
+\indentrel{3}\begin{verbatim}
+   (17)  2
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty18}
+\begin{paste}{CycleIndicatorsXmpPageEmpty18}{CycleIndicatorsXmpPagePatch18}
+\pastebutton{CycleIndicatorsXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{cap(complete 2**2,square)}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch19}
+\begin{paste}{CycleIndicatorsXmpPageFull19}{CycleIndicatorsXmpPageEmpty19}
+\pastebutton{CycleIndicatorsXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{cap(complete 3*complete 2**2,dihedral 7)}
+\indentrel{3}\begin{verbatim}
+   (18)  18
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty19}
+\begin{paste}{CycleIndicatorsXmpPageEmpty19}{CycleIndicatorsXmpPagePatch19}
+\pastebutton{CycleIndicatorsXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{cap(complete 3*complete 2**2,dihedral 7)}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch20}
+\begin{paste}{CycleIndicatorsXmpPageFull20}{CycleIndicatorsXmpPageEmpty20}
+\pastebutton{CycleIndicatorsXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{cap(graphs 5,complete 7*complete 3)}
+\indentrel{3}\begin{verbatim}
+   (19)  4
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty20}
+\begin{paste}{CycleIndicatorsXmpPageEmpty20}{CycleIndicatorsXmpPagePatch20}
+\pastebutton{CycleIndicatorsXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{cap(graphs 5,complete 7*complete 3)}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch21}
+\begin{paste}{CycleIndicatorsXmpPageFull21}{CycleIndicatorsXmpPageEmpty21}
+\pastebutton{CycleIndicatorsXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{s(x) == powerSum(x)}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty21}
+\begin{paste}{CycleIndicatorsXmpPageEmpty21}{CycleIndicatorsXmpPagePatch21}
+\pastebutton{CycleIndicatorsXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{s(x) == powerSum(x)}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch22}
+\begin{paste}{CycleIndicatorsXmpPageFull22}{CycleIndicatorsXmpPageEmpty22}
+\pastebutton{CycleIndicatorsXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{cube:=(1/24)*(s 1**8+9*s 2**4 + 8*s 3**2*s 1**2+6*s 4**2)}
+\indentrel{3}\begin{verbatim}
+         1   2    1   2 2    3   4     1   8
+   (21)  Ä (4 ) + Ä (3 1 ) + Ä (2 ) + ÄÄ (1 )
+         4        3          8        24
+             Type: SymmetricPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty22}
+\begin{paste}{CycleIndicatorsXmpPageEmpty22}{CycleIndicatorsXmpPagePatch22}
+\pastebutton{CycleIndicatorsXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{cube:=(1/24)*(s 1**8+9*s 2**4 + 8*s 3**2*s 1**2+6*s 4**2)}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch23}
+\begin{paste}{CycleIndicatorsXmpPageFull23}{CycleIndicatorsXmpPageEmpty23}
+\pastebutton{CycleIndicatorsXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{cap(complete 4**2,cube)}
+\indentrel{3}\begin{verbatim}
+   (22)  7
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty23}
+\begin{paste}{CycleIndicatorsXmpPageEmpty23}{CycleIndicatorsXmpPagePatch23}
+\pastebutton{CycleIndicatorsXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{cap(complete 4**2,cube)}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch24}
+\begin{paste}{CycleIndicatorsXmpPageFull24}{CycleIndicatorsXmpPageEmpty24}
+\pastebutton{CycleIndicatorsXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{cap(complete 2**3*complete 1**2,wreath(elementary 4,elementary 2))}
+\indentrel{3}\begin{verbatim}
+   (23)  7
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty24}
+\begin{paste}{CycleIndicatorsXmpPageEmpty24}{CycleIndicatorsXmpPagePatch24}
+\pastebutton{CycleIndicatorsXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{cap(complete 2**3*complete 1**2,wreath(elementary 4,elementary 2))}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch25}
+\begin{paste}{CycleIndicatorsXmpPageFull25}{CycleIndicatorsXmpPageEmpty25}
+\pastebutton{CycleIndicatorsXmpPageFull25}{\hidepaste}
+\tab{5}\spadcommand{cap(complete 2**3*complete 1**2,wreath(elementary 4,complete 2))}
+\indentrel{3}\begin{verbatim}
+   (24)  17
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty25}
+\begin{paste}{CycleIndicatorsXmpPageEmpty25}{CycleIndicatorsXmpPagePatch25}
+\pastebutton{CycleIndicatorsXmpPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{cap(complete 2**3*complete 1**2,wreath(elementary 4,complete 2))}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch26}
+\begin{paste}{CycleIndicatorsXmpPageFull26}{CycleIndicatorsXmpPageEmpty26}
+\pastebutton{CycleIndicatorsXmpPageFull26}{\hidepaste}
+\tab{5}\spadcommand{cap(complete 2**3*complete 1**2,wreath(complete 4,elementary 2))}
+\indentrel{3}\begin{verbatim}
+   (25)  10
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty26}
+\begin{paste}{CycleIndicatorsXmpPageEmpty26}{CycleIndicatorsXmpPagePatch26}
+\pastebutton{CycleIndicatorsXmpPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{cap(complete 2**3*complete 1**2,wreath(complete 4,elementary 2))}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch27}
+\begin{paste}{CycleIndicatorsXmpPageFull27}{CycleIndicatorsXmpPageEmpty27}
+\pastebutton{CycleIndicatorsXmpPageFull27}{\hidepaste}
+\tab{5}\spadcommand{cap(complete 2**3*complete 1**2,wreath(complete 4,complete 2))}
+\indentrel{3}\begin{verbatim}
+   (26)  23
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty27}
+\begin{paste}{CycleIndicatorsXmpPageEmpty27}{CycleIndicatorsXmpPagePatch27}
+\pastebutton{CycleIndicatorsXmpPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{cap(complete 2**3*complete 1**2,wreath(complete 4,complete 2))}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch28}
+\begin{paste}{CycleIndicatorsXmpPageFull28}{CycleIndicatorsXmpPageEmpty28}
+\pastebutton{CycleIndicatorsXmpPageFull28}{\hidepaste}
+\tab{5}\spadcommand{x: ULS(FRAC INT,'x,0) := 'x\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (27)  x
+    Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty28}
+\begin{paste}{CycleIndicatorsXmpPageEmpty28}{CycleIndicatorsXmpPagePatch28}
+\pastebutton{CycleIndicatorsXmpPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{x: ULS(FRAC INT,'x,0) := 'x\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch29}
+\begin{paste}{CycleIndicatorsXmpPageFull29}{CycleIndicatorsXmpPageEmpty29}
+\pastebutton{CycleIndicatorsXmpPageFull29}{\hidepaste}
+\tab{5}\spadcommand{ZeroOrOne: INT -> ULS(FRAC INT, 'x, 0)\bound{zodec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty29}
+\begin{paste}{CycleIndicatorsXmpPageEmpty29}{CycleIndicatorsXmpPagePatch29}
+\pastebutton{CycleIndicatorsXmpPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{ZeroOrOne: INT -> ULS(FRAC INT, 'x, 0)\bound{zodec }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch30}
+\begin{paste}{CycleIndicatorsXmpPageFull30}{CycleIndicatorsXmpPageEmpty30}
+\pastebutton{CycleIndicatorsXmpPageFull30}{\hidepaste}
+\tab{5}\spadcommand{Integers: INT -> ULS(FRAC INT, 'x, 0)\bound{idec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty30}
+\begin{paste}{CycleIndicatorsXmpPageEmpty30}{CycleIndicatorsXmpPagePatch30}
+\pastebutton{CycleIndicatorsXmpPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{Integers: INT -> ULS(FRAC INT, 'x, 0)\bound{idec }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch31}
+\begin{paste}{CycleIndicatorsXmpPageFull31}{CycleIndicatorsXmpPageEmpty31}
+\pastebutton{CycleIndicatorsXmpPageFull31}{\hidepaste}
+\tab{5}\spadcommand{ZeroOrOne n == 1+x**n\free{x zodec }\bound{zo }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty31}
+\begin{paste}{CycleIndicatorsXmpPageEmpty31}{CycleIndicatorsXmpPagePatch31}
+\pastebutton{CycleIndicatorsXmpPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{ZeroOrOne n == 1+x**n\free{x zodec }\bound{zo }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch32}
+\begin{paste}{CycleIndicatorsXmpPageFull32}{CycleIndicatorsXmpPageEmpty32}
+\pastebutton{CycleIndicatorsXmpPageFull32}{\hidepaste}
+\tab{5}\spadcommand{ZeroOrOne 5\free{zo }}
+\indentrel{3}\begin{verbatim}
+              5
+   (31)  1 + x
+    Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty32}
+\begin{paste}{CycleIndicatorsXmpPageEmpty32}{CycleIndicatorsXmpPagePatch32}
+\pastebutton{CycleIndicatorsXmpPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{ZeroOrOne 5\free{zo }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch33}
+\begin{paste}{CycleIndicatorsXmpPageFull33}{CycleIndicatorsXmpPageEmpty33}
+\pastebutton{CycleIndicatorsXmpPageFull33}{\hidepaste}
+\tab{5}\spadcommand{Integers n == 1/(1-x**n)\free{x idec }\bound{i }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty33}
+\begin{paste}{CycleIndicatorsXmpPageEmpty33}{CycleIndicatorsXmpPagePatch33}
+\pastebutton{CycleIndicatorsXmpPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{Integers n == 1/(1-x**n)\free{x idec }\bound{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch34}
+\begin{paste}{CycleIndicatorsXmpPageFull34}{CycleIndicatorsXmpPageEmpty34}
+\pastebutton{CycleIndicatorsXmpPageFull34}{\hidepaste}
+\tab{5}\spadcommand{Integers 5\free{i }}
+\indentrel{3}\begin{verbatim}
+              5    10      11
+   (33)  1 + x  + x   + O(x  )
+    Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty34}
+\begin{paste}{CycleIndicatorsXmpPageEmpty34}{CycleIndicatorsXmpPagePatch34}
+\pastebutton{CycleIndicatorsXmpPageEmpty34}{\showpaste}
+\tab{5}\spadcommand{Integers 5\free{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch35}
+\begin{paste}{CycleIndicatorsXmpPageFull35}{CycleIndicatorsXmpPageEmpty35}
+\pastebutton{CycleIndicatorsXmpPageFull35}{\hidepaste}
+\tab{5}\spadcommand{eval(ZeroOrOne, graphs 5)\free{zo }}
+\indentrel{3}\begin{verbatim}
+   (34)
+               2     3     4     5     6     7     8    9
+     1 + x + 2x  + 4x  + 6x  + 6x  + 6x  + 4x  + 2x  + x
+   + 
+      10      11
+     x   + O(x  )
+    Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty35}
+\begin{paste}{CycleIndicatorsXmpPageEmpty35}{CycleIndicatorsXmpPagePatch35}
+\pastebutton{CycleIndicatorsXmpPageEmpty35}{\showpaste}
+\tab{5}\spadcommand{eval(ZeroOrOne, graphs 5)\free{zo }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch36}
+\begin{paste}{CycleIndicatorsXmpPageFull36}{CycleIndicatorsXmpPageEmpty36}
+\pastebutton{CycleIndicatorsXmpPageFull36}{\hidepaste}
+\tab{5}\spadcommand{eval(ZeroOrOne,dihedral 8)\free{zo }}
+\indentrel{3}\begin{verbatim}
+                   2     3     4     5     6    7    8
+   (35)  1 + x + 4x  + 5x  + 8x  + 5x  + 4x  + x  + x
+    Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty36}
+\begin{paste}{CycleIndicatorsXmpPageEmpty36}{CycleIndicatorsXmpPagePatch36}
+\pastebutton{CycleIndicatorsXmpPageEmpty36}{\showpaste}
+\tab{5}\spadcommand{eval(ZeroOrOne,dihedral 8)\free{zo }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch37}
+\begin{paste}{CycleIndicatorsXmpPageFull37}{CycleIndicatorsXmpPageEmpty37}
+\pastebutton{CycleIndicatorsXmpPageFull37}{\hidepaste}
+\tab{5}\spadcommand{eval(Integers,complete 4)\free{i }}
+\indentrel{3}\begin{verbatim}
+   (36)
+               2     3     4     5     6      7      8
+     1 + x + 2x  + 3x  + 5x  + 6x  + 9x  + 11x  + 15x
+   + 
+        9      10      11
+     18x  + 23x   + O(x  )
+    Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty37}
+\begin{paste}{CycleIndicatorsXmpPageEmpty37}{CycleIndicatorsXmpPagePatch37}
+\pastebutton{CycleIndicatorsXmpPageEmpty37}{\showpaste}
+\tab{5}\spadcommand{eval(Integers,complete 4)\free{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch38}
+\begin{paste}{CycleIndicatorsXmpPageFull38}{CycleIndicatorsXmpPageEmpty38}
+\pastebutton{CycleIndicatorsXmpPageFull38}{\hidepaste}
+\tab{5}\spadcommand{eval(Integers,elementary 4)\free{i }}
+\indentrel{3}\begin{verbatim}
+   (37)
+      6    7     8     9     10     11     12      13
+     x  + x  + 2x  + 3x  + 5x   + 6x   + 9x   + 11x
+   + 
+        14      15      16      17
+     15x   + 18x   + 23x   + O(x  )
+    Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty38}
+\begin{paste}{CycleIndicatorsXmpPageEmpty38}{CycleIndicatorsXmpPagePatch38}
+\pastebutton{CycleIndicatorsXmpPageEmpty38}{\showpaste}
+\tab{5}\spadcommand{eval(Integers,elementary 4)\free{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch39}
+\begin{paste}{CycleIndicatorsXmpPageFull39}{CycleIndicatorsXmpPageEmpty39}
+\pastebutton{CycleIndicatorsXmpPageFull39}{\hidepaste}
+\tab{5}\spadcommand{eval(ZeroOrOne,cube)\free{zo }}
+\indentrel{3}\begin{verbatim}
+                   2     3     4     5     6    7    8
+   (38)  1 + x + 3x  + 3x  + 7x  + 3x  + 3x  + x  + x
+    Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty39}
+\begin{paste}{CycleIndicatorsXmpPageEmpty39}{CycleIndicatorsXmpPagePatch39}
+\pastebutton{CycleIndicatorsXmpPageEmpty39}{\showpaste}
+\tab{5}\spadcommand{eval(ZeroOrOne,cube)\free{zo }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch40}
+\begin{paste}{CycleIndicatorsXmpPageFull40}{CycleIndicatorsXmpPageEmpty40}
+\pastebutton{CycleIndicatorsXmpPageFull40}{\hidepaste}
+\tab{5}\spadcommand{eval(Integers,cube)\free{i }}
+\indentrel{3}\begin{verbatim}
+   (39)
+               2     3      4      5      6       7
+     1 + x + 4x  + 7x  + 21x  + 37x  + 85x  + 151x
+   + 
+         8       9       10      11
+     292x  + 490x  + 848x   + O(x  )
+    Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty40}
+\begin{paste}{CycleIndicatorsXmpPageEmpty40}{CycleIndicatorsXmpPagePatch40}
+\pastebutton{CycleIndicatorsXmpPageEmpty40}{\showpaste}
+\tab{5}\spadcommand{eval(Integers,cube)\free{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch41}
+\begin{paste}{CycleIndicatorsXmpPageFull41}{CycleIndicatorsXmpPageEmpty41}
+\pastebutton{CycleIndicatorsXmpPageFull41}{\hidepaste}
+\tab{5}\spadcommand{eval(Integers,graphs 5)\free{i }}
+\indentrel{3}\begin{verbatim}
+   (40)
+               2     3      4      5      6       7
+     1 + x + 3x  + 7x  + 17x  + 35x  + 76x  + 149x
+   + 
+         8       9       10      11
+     291x  + 539x  + 974x   + O(x  )
+    Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty41}
+\begin{paste}{CycleIndicatorsXmpPageEmpty41}{CycleIndicatorsXmpPagePatch41}
+\pastebutton{CycleIndicatorsXmpPageEmpty41}{\showpaste}
+\tab{5}\spadcommand{eval(Integers,graphs 5)\free{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch42}
+\begin{paste}{CycleIndicatorsXmpPageFull42}{CycleIndicatorsXmpPageEmpty42}
+\pastebutton{CycleIndicatorsXmpPageFull42}{\hidepaste}
+\tab{5}\spadcommand{eval(ZeroOrOne ,graphs 15)\free{zo }}
+\indentrel{3}\begin{verbatim}
+   (41)
+               2     3      4      5      6       7
+     1 + x + 2x  + 5x  + 11x  + 26x  + 68x  + 177x
+   + 
+         8        9        10      11
+     496x  + 1471x  + 4583x   + O(x  )
+    Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty42}
+\begin{paste}{CycleIndicatorsXmpPageEmpty42}{CycleIndicatorsXmpPagePatch42}
+\pastebutton{CycleIndicatorsXmpPageEmpty42}{\showpaste}
+\tab{5}\spadcommand{eval(ZeroOrOne ,graphs 15)\free{zo }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch43}
+\begin{paste}{CycleIndicatorsXmpPageFull43}{CycleIndicatorsXmpPageEmpty43}
+\pastebutton{CycleIndicatorsXmpPageFull43}{\hidepaste}
+\tab{5}\spadcommand{cap(dihedral 30,complete 7*complete 8*complete 5*complete 10)}
+\indentrel{3}\begin{verbatim}
+   (42)  49958972383320
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty43}
+\begin{paste}{CycleIndicatorsXmpPageEmpty43}{CycleIndicatorsXmpPagePatch43}
+\pastebutton{CycleIndicatorsXmpPageEmpty43}{\showpaste}
+\tab{5}\spadcommand{cap(dihedral 30,complete 7*complete 8*complete 5*complete 10)}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch44}
+\begin{paste}{CycleIndicatorsXmpPageFull44}{CycleIndicatorsXmpPageEmpty44}
+\pastebutton{CycleIndicatorsXmpPageFull44}{\hidepaste}
+\tab{5}\spadcommand{sf3221:= SFunction [3,2,2,1]\bound{sf3221 }}
+\indentrel{3}\begin{verbatim}
+   (43)
+      1          1     2     1   2     1
+     ÄÄ (6 2) - ÄÄ (6 1 ) - ÄÄ (4 ) + ÄÄ (4 3 1)
+     12         12          16        12
+   + 
+      1     4     1   2      1   2 2     1     2
+     ÄÄ (4 1 ) - ÄÄ (3 2) + ÄÄ (3 1 ) - ÄÄ (3 2 1)
+     24          36         36          24
+   + 
+        1       3     1     5     1    4     1   3 2
+     - ÄÄ (3 2 1 ) - ÄÄ (3 1 ) - ÄÄÄ (2 ) + ÄÄ (2 1 )
+       36            72          192        48
+   + 
+      1   2 4     1      6     1    8
+     ÄÄ (2 1 ) - ÄÄÄ (2 1 ) + ÄÄÄ (1 )
+     96          144          576
+             Type: SymmetricPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty44}
+\begin{paste}{CycleIndicatorsXmpPageEmpty44}{CycleIndicatorsXmpPagePatch44}
+\pastebutton{CycleIndicatorsXmpPageEmpty44}{\showpaste}
+\tab{5}\spadcommand{sf3221:= SFunction [3,2,2,1]\bound{sf3221 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch45}
+\begin{paste}{CycleIndicatorsXmpPageFull45}{CycleIndicatorsXmpPageEmpty45}
+\pastebutton{CycleIndicatorsXmpPageFull45}{\hidepaste}
+\tab{5}\spadcommand{cap(sf3221,complete 2**4)\free{sf3221 }}
+\indentrel{3}\begin{verbatim}
+   (44)  3
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty45}
+\begin{paste}{CycleIndicatorsXmpPageEmpty45}{CycleIndicatorsXmpPagePatch45}
+\pastebutton{CycleIndicatorsXmpPageEmpty45}{\showpaste}
+\tab{5}\spadcommand{cap(sf3221,complete 2**4)\free{sf3221 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch46}
+\begin{paste}{CycleIndicatorsXmpPageFull46}{CycleIndicatorsXmpPageEmpty46}
+\pastebutton{CycleIndicatorsXmpPageFull46}{\hidepaste}
+\tab{5}\spadcommand{cap(sf3221, powerSum 1**8)\free{sf3221 }}
+\indentrel{3}\begin{verbatim}
+   (45)  70
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty46}
+\begin{paste}{CycleIndicatorsXmpPageEmpty46}{CycleIndicatorsXmpPagePatch46}
+\pastebutton{CycleIndicatorsXmpPageEmpty46}{\showpaste}
+\tab{5}\spadcommand{cap(sf3221, powerSum 1**8)\free{sf3221 }}
+\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPagePatch47}
+\begin{paste}{CycleIndicatorsXmpPageFull47}{CycleIndicatorsXmpPageEmpty47}
+\pastebutton{CycleIndicatorsXmpPageFull47}{\hidepaste}
+\tab{5}\spadcommand{eval(Integers, sf3221)\free{i sf3221 }}
+\indentrel{3}\begin{verbatim}
+   (46)
+      9     10     11      12      13      14      15
+     x  + 3x   + 7x   + 14x   + 27x   + 47x   + 79x
+   + 
+         16       17       18       19      20
+     126x   + 196x   + 294x   + 432x   + O(x  )
+    Type: UnivariateLaurentSeries(Fraction Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{CycleIndicatorsXmpPageEmpty47}
+\begin{paste}{CycleIndicatorsXmpPageEmpty47}{CycleIndicatorsXmpPagePatch47}
+\pastebutton{CycleIndicatorsXmpPageEmpty47}{\showpaste}
+\tab{5}\spadcommand{eval(Integers, sf3221)\free{i sf3221 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/DECIMAL.ht b/src/hyper/pages/DECIMAL.ht
new file mode 100644
index 00000000..ddf3e72e
--- /dev/null
+++ b/src/hyper/pages/DECIMAL.ht
@@ -0,0 +1,58 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\DecimalExpansionXmpTitle}{DecimalExpansion}
+\newcommand{\DecimalExpansionXmpNumber}{9.15}
+%
+% =====================================================================
+\begin{page}{DecimalExpansionXmpPage}{9.15 DecimalExpansion}
+% =====================================================================
+\beginscroll
+
+
+All rationals have repeating decimal expansions.
+Operations to access the individual digits of a decimal expansion can
+be obtained by converting the value to \spadtype{RadixExpansion(10)}.
+More examples of expansions are available in
+\downlink{`BinaryExpansion'}{BinaryExpansionXmpPage}\ignore{BinaryExpansion},
+\downlink{`HexadecimalExpansion'}{HexadecimalExpansionXmpPage}\ignore{HexadecimalExpansion}, and
+\downlink{`RadixExpansion'}{RadixExpansionXmpPage}\ignore{RadixExpansion}.
+\showBlurb{DecimalExpansion}
+
+\xtc{
+The operation \spadfunFrom{decimal}{DecimalExpansion} is used to create
+this expansion of type \spadtype{DecimalExpansion}.
+}{
+\spadpaste{r := decimal(22/7) \bound{r}}
+}
+\xtc{
+Arithmetic is exact.
+}{
+\spadpaste{r + decimal(6/7) \free{r}}
+}
+\xtc{
+The period of the expansion can be short or long \ldots
+}{
+\spadpaste{[decimal(1/i) for i in 350..354] }
+}
+\xtc{
+or very long.
+}{
+\spadpaste{decimal(1/2049) }
+}
+\xtc{
+These numbers are bona fide algebraic objects.
+}{
+\spadpaste{p := decimal(1/4)*x**2 + decimal(2/3)*x + decimal(4/9)  \bound{p}}
+}
+\xtc{
+}{
+\spadpaste{q := differentiate(p, x) \free{p}\bound{q}}
+}
+\xtc{
+}{
+\spadpaste{g := gcd(p, q)            \free{p q} \bound{g}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/DECIMAL.pht b/src/hyper/pages/DECIMAL.pht
new file mode 100644
index 00000000..38c32dee
--- /dev/null
+++ b/src/hyper/pages/DECIMAL.pht
@@ -0,0 +1,136 @@
+\begin{patch}{DecimalExpansionXmpPagePatch1}
+\begin{paste}{DecimalExpansionXmpPageFull1}{DecimalExpansionXmpPageEmpty1}
+\pastebutton{DecimalExpansionXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{r := decimal(22/7)\bound{r }}
+\indentrel{3}\begin{verbatim}
+          ______
+   (1)  3.142857
+                                 Type: DecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DecimalExpansionXmpPageEmpty1}
+\begin{paste}{DecimalExpansionXmpPageEmpty1}{DecimalExpansionXmpPagePatch1}
+\pastebutton{DecimalExpansionXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{r := decimal(22/7)\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{DecimalExpansionXmpPagePatch2}
+\begin{paste}{DecimalExpansionXmpPageFull2}{DecimalExpansionXmpPageEmpty2}
+\pastebutton{DecimalExpansionXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{r + decimal(6/7)\free{r }}
+\indentrel{3}\begin{verbatim}
+   (2)  4
+                                 Type: DecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DecimalExpansionXmpPageEmpty2}
+\begin{paste}{DecimalExpansionXmpPageEmpty2}{DecimalExpansionXmpPagePatch2}
+\pastebutton{DecimalExpansionXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{r + decimal(6/7)\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{DecimalExpansionXmpPagePatch3}
+\begin{paste}{DecimalExpansionXmpPageFull3}{DecimalExpansionXmpPageEmpty3}
+\pastebutton{DecimalExpansionXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{[decimal(1/i) for i in 350..354]}
+\indentrel{3}\begin{verbatim}
+   (3)
+        ______    ______         __
+   [0.00285714, 0.002849, 0.0028409,
+      ________________________________
+    0.00283286118980169971671388101983,
+
+     0.0
+       OVERBAR
+          02824858757062146892655367231638418079096045197
+            74011299435
+     ]
+                            Type: List DecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DecimalExpansionXmpPageEmpty3}
+\begin{paste}{DecimalExpansionXmpPageEmpty3}{DecimalExpansionXmpPagePatch3}
+\pastebutton{DecimalExpansionXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{[decimal(1/i) for i in 350..354]}
+\end{paste}\end{patch}
+
+\begin{patch}{DecimalExpansionXmpPagePatch4}
+\begin{paste}{DecimalExpansionXmpPageFull4}{DecimalExpansionXmpPageEmpty4}
+\pastebutton{DecimalExpansionXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{decimal(1/2049)}
+\indentrel{3}\begin{verbatim}
+   (4)
+   0.
+     OVERBAR
+        0004880429477794045876037091264031234748657881893
+          60663738408979990239141044411908247925817471937
+          53050268423621278672523182040019521717911176183
+          50414836505612493899463152757442654953635919960
+          95656417764763299170326988775012201073694485114
+          69009272816007808687164470473401659346022449975
+          59785261102977061981454367984382625671059053196
+          6813079551
+                                 Type: DecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DecimalExpansionXmpPageEmpty4}
+\begin{paste}{DecimalExpansionXmpPageEmpty4}{DecimalExpansionXmpPagePatch4}
+\pastebutton{DecimalExpansionXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{decimal(1/2049)}
+\end{paste}\end{patch}
+
+\begin{patch}{DecimalExpansionXmpPagePatch5}
+\begin{paste}{DecimalExpansionXmpPageFull5}{DecimalExpansionXmpPageEmpty5}
+\pastebutton{DecimalExpansionXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{p := decimal(1/4)*x**2 + decimal(2/3)*x + decimal(4/9)\bound{p }}
+\indentrel{3}\begin{verbatim}
+             2     _      _
+   (5)  0.25x  + 0.6x + 0.4
+                      Type: Polynomial DecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DecimalExpansionXmpPageEmpty5}
+\begin{paste}{DecimalExpansionXmpPageEmpty5}{DecimalExpansionXmpPagePatch5}
+\pastebutton{DecimalExpansionXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{p := decimal(1/4)*x**2 + decimal(2/3)*x + decimal(4/9)\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{DecimalExpansionXmpPagePatch6}
+\begin{paste}{DecimalExpansionXmpPageFull6}{DecimalExpansionXmpPageEmpty6}
+\pastebutton{DecimalExpansionXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{q := differentiate(p, x)\free{p }\bound{q }}
+\indentrel{3}\begin{verbatim}
+                 _
+   (6)  0.5x + 0.6
+                      Type: Polynomial DecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DecimalExpansionXmpPageEmpty6}
+\begin{paste}{DecimalExpansionXmpPageEmpty6}{DecimalExpansionXmpPagePatch6}
+\pastebutton{DecimalExpansionXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{q := differentiate(p, x)\free{p }\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{DecimalExpansionXmpPagePatch7}
+\begin{paste}{DecimalExpansionXmpPageFull7}{DecimalExpansionXmpPageEmpty7}
+\pastebutton{DecimalExpansionXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{g := gcd(p, q)\free{p q }\bound{g }}
+\indentrel{3}\begin{verbatim}
+              _
+   (7)  x + 1.3
+                      Type: Polynomial DecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DecimalExpansionXmpPageEmpty7}
+\begin{paste}{DecimalExpansionXmpPageEmpty7}{DecimalExpansionXmpPagePatch7}
+\pastebutton{DecimalExpansionXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{g := gcd(p, q)\free{p q }\bound{g }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/DERHAM.ht b/src/hyper/pages/DERHAM.ht
new file mode 100644
index 00000000..b5a78cae
--- /dev/null
+++ b/src/hyper/pages/DERHAM.ht
@@ -0,0 +1,207 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\DeRhamComplexXmpTitle}{DeRhamComplex}
+\newcommand{\DeRhamComplexXmpNumber}{9.14}
+%
+% =====================================================================
+\begin{page}{DeRhamComplexXmpPage}{9.14 DeRhamComplex}
+% =====================================================================
+\beginscroll
+ 
+The domain constructor \spadtype{DeRhamComplex} creates the
+class of differential forms of arbitrary degree over a coefficient ring.
+The De Rham complex constructor takes two arguments: a ring, \spad{coefRing,}
+and a list of coordinate variables.
+ 
+\xtc{
+This is the ring of coefficients.
+}{
+\spadpaste{coefRing := Integer \bound{coefRing}}
+}
+\xtc{
+These are the coordinate variables.
+}{
+\spadpaste{lv : List Symbol := [x,y,z] \bound{lv}}
+}
+\xtc{
+This is the De Rham complex of Euclidean three-space using
+coordinates \spad{x, y} and \spad{z.}
+}{
+\spadpaste{der := DERHAM(coefRing,lv) \free{coefRing}\free{lv}\bound{der}}
+}
+ 
+This complex allows us to describe differential forms having
+expressions of integers as coefficients.
+These coefficients can involve any number of variables, for example,
+\spad{f(x,t,r,y,u,z).}
+As we've chosen to work with ordinary
+Euclidean three-space, expressions involving these forms
+are treated as functions of
+\spad{x, y} and \spad{z} with the additional arguments
+\spad{t, r} and \spad{u} regarded as symbolic constants.
+\xtc{
+Here are some examples of coefficients.
+}{
+\spadpaste{R := Expression coefRing \free{coefRing}\bound{R}}
+}
+\xtc{
+}{
+\spadpaste{f : R := x**2*y*z-5*x**3*y**2*z**5 \free{R}\bound{f}}
+}
+\xtc{
+}{
+\spadpaste{g : R := z**2*y*cos(z)-7*sin(x**3*y**2)*z**2 \free{R}\bound{g}}
+}
+\xtc{
+}{
+\spadpaste{h : R :=x*y*z-2*x**3*y*z**2 \free{R}\bound{h}}
+}
+\xtc{
+We now define
+the multiplicative basis elements for the exterior algebra over \spad{R}.
+}{
+\spadpaste{dx : der := generator(1) \free{der}\bound{dx}}
+}
+\xtc{
+}{
+\spadpaste{dy : der := generator(2)\free{der}\bound{dy}}
+}
+\xtc{
+}{
+\spadpaste{dz : der := generator(3)\free{der}\bound{dz}}
+}
+\xtc{
+This is an alternative way to give the above assignments.
+}{
+\spadpaste{[dx,dy,dz] := [generator(i)\$der for i in 1..3] \free{der}\bound{dxyz}}
+}
+\xtc{
+Now we define some one-forms.
+}{
+\spadpaste{alpha : der := f*dx + g*dy + h*dz \bound{alpha}\free{der f g h dxyz}}
+}
+\xtc{
+}{
+\spadpaste{beta  : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy \bound{beta}\free{der f g h dxyz}}
+}
+\xtc{
+A well-known theorem states that the composition of
+\spadfunFrom{exteriorDifferential}{DeRhamComplex}
+with itself is the zero map for continuous forms.
+Let's verify this theorem for \spad{alpha}.
+}{
+\spadpaste{exteriorDifferential alpha; \free{alpha}\bound{ed}}
+}
+\xtc{
+We suppressed the lengthy output of the last expression, but nevertheless, the
+composition is zero.
+}{
+\spadpaste{exteriorDifferential \% \free{ed}}
+}
+ 
+\xtc{
+Now we check that \spadfunFrom{exteriorDifferential}{DeRhamComplex}
+is a ``graded derivation'' \spad{D,} that is, \spad{D} satisfies:
+\begin{verbatim}
+D(a*b) = D(a)*b + (-1)**degree(a)*a*D(b)
+\end{verbatim}
+}{
+\spadpaste{gamma := alpha * beta \bound{gamma}\free{alpha}\free{beta}}
+}
+\xtc{
+We try this for the one-forms \spad{alpha} and \spad{beta}.
+}{
+\spadpaste{exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - alpha * exteriorDifferential(beta)) \free{alpha beta gamma}}
+}
+\xtc{
+Now we define some ``basic operators'' (see \downlink{`Operator'}{OperatorXmpPage}\ignore{Operator}).
+%-% \HDindex{operator}{DeRhamComplexXmpPage}{9.14}{DeRhamComplex}
+}{
+\spadpaste{a : BOP := operator('a) \bound{ao}}
+}
+\xtc{
+}{
+\spadpaste{b : BOP := operator('b) \bound{bo}}
+}
+\xtc{
+}{
+\spadpaste{c : BOP := operator('c) \bound{co}}
+}
+\xtc{
+We also define
+some indeterminate one- and two-forms using these operators.
+}{
+\spadpaste{sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz \bound{sigma}\free{ao bo co dxyz}}
+}
+\xtc{
+}{
+\spadpaste{theta  := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz \bound{theta}\free{ao bo co dxyz}}
+}
+ 
+\xtc{
+This allows us to get formal definitions for the ``gradient'' \ldots
+}{
+\spadpaste{totalDifferential(a(x,y,z))\$der \free{ao der}}
+}
+\xtc{
+the ``curl'' \ldots
+}{
+\spadpaste{exteriorDifferential sigma \free{sigma}}
+}
+\xtc{
+and the ``divergence.''
+}{
+\spadpaste{exteriorDifferential theta \free{theta}}
+}
+ 
+\xtc{
+Note that the De Rham complex is an algebra with unity.
+This element \spad{1} is the basis for elements for zero-forms, that is,
+functions in our space.
+}{
+\spadpaste{one : der := 1 \bound{one}\free{der}}
+}
+ 
+\xtc{
+To convert a function to a function lying in the De Rham complex,
+multiply the function by ``one.''
+}{
+\spadpaste{g1 : der := a([x,t,y,u,v,z,e]) * one \free{der one ao}\bound{g1}}
+}
+\xtc{
+A current limitation of \Language{} forces you to write
+functions with more than four arguments using square brackets in this way.
+}{
+\spadpaste{h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one \free{der one ao}\bound{h1}}
+}
+ 
+\xtc{
+Now note how the system keeps track of where your coordinate functions
+are located in expressions.
+}{
+\spadpaste{exteriorDifferential g1 \free{g1}}
+}
+\xtc{
+}{
+\spadpaste{exteriorDifferential h1 \free{h1}}
+}
+ 
+\xtc{
+In this example of Euclidean three-space, the basis for the De Rham complex
+consists of the eight forms: \spad{1}, \spad{dx}, \spad{dy}, \spad{dz},
+\spad{dx*dy}, \spad{dx*dz}, \spad{dy*dz}, and \spad{dx*dy*dz}.
+}{
+\spadpaste{coefficient(gamma, dx*dy) \free{gamma dxyz}}
+}
+\xtc{
+}{
+\spadpaste{coefficient(gamma, one) \free{gamma one}}
+}
+\xtc{
+}{
+\spadpaste{coefficient(g1,one) \free{g1 one}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/DERHAM.pht b/src/hyper/pages/DERHAM.pht
new file mode 100644
index 00000000..8342214e
--- /dev/null
+++ b/src/hyper/pages/DERHAM.pht
@@ -0,0 +1,591 @@
+\begin{patch}{DeRhamComplexXmpPagePatch1}
+\begin{paste}{DeRhamComplexXmpPageFull1}{DeRhamComplexXmpPageEmpty1}
+\pastebutton{DeRhamComplexXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{coefRing := Integer\bound{coefRing }}
+\indentrel{3}\begin{verbatim}
+   (1)  Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty1}
+\begin{paste}{DeRhamComplexXmpPageEmpty1}{DeRhamComplexXmpPagePatch1}
+\pastebutton{DeRhamComplexXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{coefRing := Integer\bound{coefRing }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch2}
+\begin{paste}{DeRhamComplexXmpPageFull2}{DeRhamComplexXmpPageEmpty2}
+\pastebutton{DeRhamComplexXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{lv : List Symbol := [x,y,z]\bound{lv }}
+\indentrel{3}\begin{verbatim}
+   (2)  [x,y,z]
+                                      Type: List Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty2}
+\begin{paste}{DeRhamComplexXmpPageEmpty2}{DeRhamComplexXmpPagePatch2}
+\pastebutton{DeRhamComplexXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{lv : List Symbol := [x,y,z]\bound{lv }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch3}
+\begin{paste}{DeRhamComplexXmpPageFull3}{DeRhamComplexXmpPageEmpty3}
+\pastebutton{DeRhamComplexXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{der := DERHAM(coefRing,lv)\free{coefRing }\free{lv }\bound{der }}
+\indentrel{3}\begin{verbatim}
+   (3)  DeRhamComplex(Integer,[x,y,z])
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty3}
+\begin{paste}{DeRhamComplexXmpPageEmpty3}{DeRhamComplexXmpPagePatch3}
+\pastebutton{DeRhamComplexXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{der := DERHAM(coefRing,lv)\free{coefRing }\free{lv }\bound{der }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch4}
+\begin{paste}{DeRhamComplexXmpPageFull4}{DeRhamComplexXmpPageEmpty4}
+\pastebutton{DeRhamComplexXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{R := Expression coefRing\free{coefRing }\bound{R }}
+\indentrel{3}\begin{verbatim}
+   (4)  Expression Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty4}
+\begin{paste}{DeRhamComplexXmpPageEmpty4}{DeRhamComplexXmpPagePatch4}
+\pastebutton{DeRhamComplexXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{R := Expression coefRing\free{coefRing }\bound{R }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch5}
+\begin{paste}{DeRhamComplexXmpPageFull5}{DeRhamComplexXmpPageEmpty5}
+\pastebutton{DeRhamComplexXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{f : R := x**2*y*z-5*x**3*y**2*z**5\free{R }\bound{f }}
+\indentrel{3}\begin{verbatim}
+            3 2 5    2
+   (5)  - 5x y z  + x y z
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty5}
+\begin{paste}{DeRhamComplexXmpPageEmpty5}{DeRhamComplexXmpPagePatch5}
+\pastebutton{DeRhamComplexXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{f : R := x**2*y*z-5*x**3*y**2*z**5\free{R }\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch6}
+\begin{paste}{DeRhamComplexXmpPageFull6}{DeRhamComplexXmpPageEmpty6}
+\pastebutton{DeRhamComplexXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{g : R := z**2*y*cos(z)-7*sin(x**3*y**2)*z**2\free{R }\bound{g }}
+\indentrel{3}\begin{verbatim}
+            2     3 2       2
+   (6)  - 7z sin(x y ) + y z cos(z)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty6}
+\begin{paste}{DeRhamComplexXmpPageEmpty6}{DeRhamComplexXmpPagePatch6}
+\pastebutton{DeRhamComplexXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{g : R := z**2*y*cos(z)-7*sin(x**3*y**2)*z**2\free{R }\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch7}
+\begin{paste}{DeRhamComplexXmpPageFull7}{DeRhamComplexXmpPageEmpty7}
+\pastebutton{DeRhamComplexXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{h : R :=x*y*z-2*x**3*y*z**2\free{R }\bound{h }}
+\indentrel{3}\begin{verbatim}
+            3   2
+   (7)  - 2x y z  + x y z
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty7}
+\begin{paste}{DeRhamComplexXmpPageEmpty7}{DeRhamComplexXmpPagePatch7}
+\pastebutton{DeRhamComplexXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{h : R :=x*y*z-2*x**3*y*z**2\free{R }\bound{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch8}
+\begin{paste}{DeRhamComplexXmpPageFull8}{DeRhamComplexXmpPageEmpty8}
+\pastebutton{DeRhamComplexXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{dx : der := generator(1)\free{der }\bound{dx }}
+\indentrel{3}\begin{verbatim}
+   (8)  dx
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty8}
+\begin{paste}{DeRhamComplexXmpPageEmpty8}{DeRhamComplexXmpPagePatch8}
+\pastebutton{DeRhamComplexXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{dx : der := generator(1)\free{der }\bound{dx }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch9}
+\begin{paste}{DeRhamComplexXmpPageFull9}{DeRhamComplexXmpPageEmpty9}
+\pastebutton{DeRhamComplexXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{dy : der := generator(2)\free{der }\bound{dy }}
+\indentrel{3}\begin{verbatim}
+   (9)  dy
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty9}
+\begin{paste}{DeRhamComplexXmpPageEmpty9}{DeRhamComplexXmpPagePatch9}
+\pastebutton{DeRhamComplexXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{dy : der := generator(2)\free{der }\bound{dy }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch10}
+\begin{paste}{DeRhamComplexXmpPageFull10}{DeRhamComplexXmpPageEmpty10}
+\pastebutton{DeRhamComplexXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{dz : der := generator(3)\free{der }\bound{dz }}
+\indentrel{3}\begin{verbatim}
+   (10)  dz
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty10}
+\begin{paste}{DeRhamComplexXmpPageEmpty10}{DeRhamComplexXmpPagePatch10}
+\pastebutton{DeRhamComplexXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{dz : der := generator(3)\free{der }\bound{dz }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch11}
+\begin{paste}{DeRhamComplexXmpPageFull11}{DeRhamComplexXmpPageEmpty11}
+\pastebutton{DeRhamComplexXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{[dx,dy,dz] := [generator(i)$der for i in 1..3]\free{der }\bound{dxyz }}
+\indentrel{3}\begin{verbatim}
+   (11)  [dx,dy,dz]
+              Type: List DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty11}
+\begin{paste}{DeRhamComplexXmpPageEmpty11}{DeRhamComplexXmpPagePatch11}
+\pastebutton{DeRhamComplexXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{[dx,dy,dz] := [generator(i)$der for i in 1..3]\free{der }\bound{dxyz }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch12}
+\begin{paste}{DeRhamComplexXmpPageFull12}{DeRhamComplexXmpPageEmpty12}
+\pastebutton{DeRhamComplexXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{alpha : der := f*dx + g*dy + h*dz\bound{alpha }\free{der f g h dxyz }}
+\indentrel{3}\begin{verbatim}
+   (12)
+          3   2
+     (- 2x y z  + x y z)dz
+   + 
+        2     3 2       2                 3 2 5    2
+   (- 7z sin(x y ) + y z cos(z))dy + (- 5x y z  + x y z)dx
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty12}
+\begin{paste}{DeRhamComplexXmpPageEmpty12}{DeRhamComplexXmpPagePatch12}
+\pastebutton{DeRhamComplexXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{alpha : der := f*dx + g*dy + h*dz\bound{alpha }\free{der f g h dxyz }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch13}
+\begin{paste}{DeRhamComplexXmpPageFull13}{DeRhamComplexXmpPageEmpty13}
+\pastebutton{DeRhamComplexXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy\bound{beta }\free{der f g h dxyz }}
+\indentrel{3}\begin{verbatim}
+   (13)  x dy + cos(tan(x y z) + x y z)dx
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty13}
+\begin{paste}{DeRhamComplexXmpPageEmpty13}{DeRhamComplexXmpPagePatch13}
+\pastebutton{DeRhamComplexXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy\bound{beta }\free{der f g h dxyz }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch14}
+\begin{paste}{DeRhamComplexXmpPageFull14}{DeRhamComplexXmpPageEmpty14}
+\pastebutton{DeRhamComplexXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{exteriorDifferential alpha;\free{alpha }\bound{ed }}
+\indentrel{3}\begin{verbatim}
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty14}
+\begin{paste}{DeRhamComplexXmpPageEmpty14}{DeRhamComplexXmpPagePatch14}
+\pastebutton{DeRhamComplexXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{exteriorDifferential alpha;\free{alpha }\bound{ed }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch15}
+\begin{paste}{DeRhamComplexXmpPageFull15}{DeRhamComplexXmpPageEmpty15}
+\pastebutton{DeRhamComplexXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{exteriorDifferential \%\free{ed }}
+\indentrel{3}\begin{verbatim}
+   (15)  0
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty15}
+\begin{paste}{DeRhamComplexXmpPageEmpty15}{DeRhamComplexXmpPagePatch15}
+\pastebutton{DeRhamComplexXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{exteriorDifferential \%\free{ed }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch16}
+\begin{paste}{DeRhamComplexXmpPageFull16}{DeRhamComplexXmpPageEmpty16}
+\pastebutton{DeRhamComplexXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{gamma := alpha * beta\bound{gamma }\free{alpha }\free{beta }}
+\indentrel{3}\begin{verbatim}
+   (16)
+        4   2    2
+     (2x y z  - x y z)dy dz
+   + 
+        3   2
+     (2x y z  - x y z)cos(tan(x y z) + x y z)dx dz
+   + 
+              2     3 2       2
+           (7z sin(x y ) - y z cos(z))
+        *
+           cos(tan(x y z) + x y z)
+       + 
+             4 2 5    3
+         - 5x y z  + x y z
+    *
+       dx dy
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty16}
+\begin{paste}{DeRhamComplexXmpPageEmpty16}{DeRhamComplexXmpPagePatch16}
+\pastebutton{DeRhamComplexXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{gamma := alpha * beta\bound{gamma }\free{alpha }\free{beta }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch17}
+\begin{paste}{DeRhamComplexXmpPageFull17}{DeRhamComplexXmpPageEmpty17}
+\pastebutton{DeRhamComplexXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - alpha * exteriorDifferential(beta))\free{alpha beta gamma }}
+\indentrel{3}\begin{verbatim}
+   (17)  0
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty17}
+\begin{paste}{DeRhamComplexXmpPageEmpty17}{DeRhamComplexXmpPagePatch17}
+\pastebutton{DeRhamComplexXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - alpha * exteriorDifferential(beta))\free{alpha beta gamma }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch18}
+\begin{paste}{DeRhamComplexXmpPageFull18}{DeRhamComplexXmpPageEmpty18}
+\pastebutton{DeRhamComplexXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{a : BOP := operator('a)\bound{ao }}
+\indentrel{3}\begin{verbatim}
+   (18)  a
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty18}
+\begin{paste}{DeRhamComplexXmpPageEmpty18}{DeRhamComplexXmpPagePatch18}
+\pastebutton{DeRhamComplexXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{a : BOP := operator('a)\bound{ao }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch19}
+\begin{paste}{DeRhamComplexXmpPageFull19}{DeRhamComplexXmpPageEmpty19}
+\pastebutton{DeRhamComplexXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{b : BOP := operator('b)\bound{bo }}
+\indentrel{3}\begin{verbatim}
+   (19)  b
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty19}
+\begin{paste}{DeRhamComplexXmpPageEmpty19}{DeRhamComplexXmpPagePatch19}
+\pastebutton{DeRhamComplexXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{b : BOP := operator('b)\bound{bo }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch20}
+\begin{paste}{DeRhamComplexXmpPageFull20}{DeRhamComplexXmpPageEmpty20}
+\pastebutton{DeRhamComplexXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{c : BOP := operator('c)\bound{co }}
+\indentrel{3}\begin{verbatim}
+   (20)  c
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty20}
+\begin{paste}{DeRhamComplexXmpPageEmpty20}{DeRhamComplexXmpPagePatch20}
+\pastebutton{DeRhamComplexXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{c : BOP := operator('c)\bound{co }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch21}
+\begin{paste}{DeRhamComplexXmpPageFull21}{DeRhamComplexXmpPageEmpty21}
+\pastebutton{DeRhamComplexXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz\bound{sigma }\free{ao bo co dxyz }}
+\indentrel{3}\begin{verbatim}
+   (21)  c(x,y,z)dz + b(x,y,z)dy + a(x,y,z)dx
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty21}
+\begin{paste}{DeRhamComplexXmpPageEmpty21}{DeRhamComplexXmpPagePatch21}
+\pastebutton{DeRhamComplexXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz\bound{sigma }\free{ao bo co dxyz }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch22}
+\begin{paste}{DeRhamComplexXmpPageFull22}{DeRhamComplexXmpPageEmpty22}
+\pastebutton{DeRhamComplexXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz\bound{theta }\free{ao bo co dxyz }}
+\indentrel{3}\begin{verbatim}
+   (22)  c(x,y,z)dy dz + b(x,y,z)dx dz + a(x,y,z)dx dy
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty22}
+\begin{paste}{DeRhamComplexXmpPageEmpty22}{DeRhamComplexXmpPagePatch22}
+\pastebutton{DeRhamComplexXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz\bound{theta }\free{ao bo co dxyz }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch23}
+\begin{paste}{DeRhamComplexXmpPageFull23}{DeRhamComplexXmpPageEmpty23}
+\pastebutton{DeRhamComplexXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{totalDifferential(a(x,y,z))$der\free{ao der }}
+\indentrel{3}\begin{verbatim}
+   (23)  a  (x,y,z)dz + a  (x,y,z)dy + a  (x,y,z)dx
+          ,3             ,2             ,1
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty23}
+\begin{paste}{DeRhamComplexXmpPageEmpty23}{DeRhamComplexXmpPagePatch23}
+\pastebutton{DeRhamComplexXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{totalDifferential(a(x,y,z))$der\free{ao der }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch24}
+\begin{paste}{DeRhamComplexXmpPageFull24}{DeRhamComplexXmpPageEmpty24}
+\pastebutton{DeRhamComplexXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{exteriorDifferential sigma\free{sigma }}
+\indentrel{3}\begin{verbatim}
+   (24)
+     (c  (x,y,z) - b  (x,y,z))dy dz
+       ,2           ,3
+   + 
+     (c  (x,y,z) - a  (x,y,z))dx dz
+       ,1           ,3
+   + 
+     (b  (x,y,z) - a  (x,y,z))dx dy
+       ,1           ,2
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty24}
+\begin{paste}{DeRhamComplexXmpPageEmpty24}{DeRhamComplexXmpPagePatch24}
+\pastebutton{DeRhamComplexXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{exteriorDifferential sigma\free{sigma }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch25}
+\begin{paste}{DeRhamComplexXmpPageFull25}{DeRhamComplexXmpPageEmpty25}
+\pastebutton{DeRhamComplexXmpPageFull25}{\hidepaste}
+\tab{5}\spadcommand{exteriorDifferential theta\free{theta }}
+\indentrel{3}\begin{verbatim}
+   (25)  (c  (x,y,z) - b  (x,y,z) + a  (x,y,z))dx dy dz
+           ,1           ,2           ,3
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty25}
+\begin{paste}{DeRhamComplexXmpPageEmpty25}{DeRhamComplexXmpPagePatch25}
+\pastebutton{DeRhamComplexXmpPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{exteriorDifferential theta\free{theta }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch26}
+\begin{paste}{DeRhamComplexXmpPageFull26}{DeRhamComplexXmpPageEmpty26}
+\pastebutton{DeRhamComplexXmpPageFull26}{\hidepaste}
+\tab{5}\spadcommand{one : der := 1\bound{one }\free{der }}
+\indentrel{3}\begin{verbatim}
+   (26)  1
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty26}
+\begin{paste}{DeRhamComplexXmpPageEmpty26}{DeRhamComplexXmpPagePatch26}
+\pastebutton{DeRhamComplexXmpPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{one : der := 1\bound{one }\free{der }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch27}
+\begin{paste}{DeRhamComplexXmpPageFull27}{DeRhamComplexXmpPageEmpty27}
+\pastebutton{DeRhamComplexXmpPageFull27}{\hidepaste}
+\tab{5}\spadcommand{g1 : der := a([x,t,y,u,v,z,e]) * one\free{der one ao }\bound{g1 }}
+\indentrel{3}\begin{verbatim}
+   (27)  a(x,t,y,u,v,z,e)
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty27}
+\begin{paste}{DeRhamComplexXmpPageEmpty27}{DeRhamComplexXmpPagePatch27}
+\pastebutton{DeRhamComplexXmpPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{g1 : der := a([x,t,y,u,v,z,e]) * one\free{der one ao }\bound{g1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch28}
+\begin{paste}{DeRhamComplexXmpPageFull28}{DeRhamComplexXmpPageEmpty28}
+\pastebutton{DeRhamComplexXmpPageFull28}{\hidepaste}
+\tab{5}\spadcommand{h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one\free{der one ao }\bound{h1 }}
+\indentrel{3}\begin{verbatim}
+   (28)  a(x,y,x,t,x,z,y,r,u,x)
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty28}
+\begin{paste}{DeRhamComplexXmpPageEmpty28}{DeRhamComplexXmpPagePatch28}
+\pastebutton{DeRhamComplexXmpPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one\free{der one ao }\bound{h1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch29}
+\begin{paste}{DeRhamComplexXmpPageFull29}{DeRhamComplexXmpPageEmpty29}
+\pastebutton{DeRhamComplexXmpPageFull29}{\hidepaste}
+\tab{5}\spadcommand{exteriorDifferential g1\free{g1 }}
+\indentrel{3}\begin{verbatim}
+   (29)
+     a  (x,t,y,u,v,z,e)dz + a  (x,t,y,u,v,z,e)dy
+      ,6                     ,3
+   + 
+     a  (x,t,y,u,v,z,e)dx
+      ,1
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty29}
+\begin{paste}{DeRhamComplexXmpPageEmpty29}{DeRhamComplexXmpPagePatch29}
+\pastebutton{DeRhamComplexXmpPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{exteriorDifferential g1\free{g1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch30}
+\begin{paste}{DeRhamComplexXmpPageFull30}{DeRhamComplexXmpPageEmpty30}
+\pastebutton{DeRhamComplexXmpPageFull30}{\hidepaste}
+\tab{5}\spadcommand{exteriorDifferential h1\free{h1 }}
+\indentrel{3}\begin{verbatim}
+   (30)
+     a  (x,y,x,t,x,z,y,r,u,x)dz
+      ,6
+   + 
+           a  (x,y,x,t,x,z,y,r,u,x)
+            ,7
+         + 
+           a  (x,y,x,t,x,z,y,r,u,x)
+            ,2
+    *
+       dy
+   + 
+         a   (x,y,x,t,x,z,y,r,u,x)
+          ,10
+       + 
+         a  (x,y,x,t,x,z,y,r,u,x)
+          ,5
+       + 
+       a  (x,y,x,t,x,z,y,r,u,x) + a  (x,y,x,t,x,z,y,r,u,x)
+        ,3                         ,1
+    *
+       dx
+                   Type: DeRhamComplex(Integer,[x,y,z])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty30}
+\begin{paste}{DeRhamComplexXmpPageEmpty30}{DeRhamComplexXmpPagePatch30}
+\pastebutton{DeRhamComplexXmpPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{exteriorDifferential h1\free{h1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch31}
+\begin{paste}{DeRhamComplexXmpPageFull31}{DeRhamComplexXmpPageEmpty31}
+\pastebutton{DeRhamComplexXmpPageFull31}{\hidepaste}
+\tab{5}\spadcommand{coefficient(gamma, dx*dy)\free{gamma dxyz }}
+\indentrel{3}\begin{verbatim}
+   (31)
+        2     3 2       2
+     (7z sin(x y ) - y z cos(z))cos(tan(x y z) + x y z)
+   + 
+         4 2 5    3
+     - 5x y z  + x y z
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty31}
+\begin{paste}{DeRhamComplexXmpPageEmpty31}{DeRhamComplexXmpPagePatch31}
+\pastebutton{DeRhamComplexXmpPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{coefficient(gamma, dx*dy)\free{gamma dxyz }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch32}
+\begin{paste}{DeRhamComplexXmpPageFull32}{DeRhamComplexXmpPageEmpty32}
+\pastebutton{DeRhamComplexXmpPageFull32}{\hidepaste}
+\tab{5}\spadcommand{coefficient(gamma, one)\free{gamma one }}
+\indentrel{3}\begin{verbatim}
+   (32)  0
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty32}
+\begin{paste}{DeRhamComplexXmpPageEmpty32}{DeRhamComplexXmpPagePatch32}
+\pastebutton{DeRhamComplexXmpPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{coefficient(gamma, one)\free{gamma one }}
+\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPagePatch33}
+\begin{paste}{DeRhamComplexXmpPageFull33}{DeRhamComplexXmpPageEmpty33}
+\pastebutton{DeRhamComplexXmpPageFull33}{\hidepaste}
+\tab{5}\spadcommand{coefficient(g1,one)\free{g1 one }}
+\indentrel{3}\begin{verbatim}
+   (33)  a(x,t,y,u,v,z,e)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DeRhamComplexXmpPageEmpty33}
+\begin{paste}{DeRhamComplexXmpPageEmpty33}{DeRhamComplexXmpPagePatch33}
+\pastebutton{DeRhamComplexXmpPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{coefficient(g1,one)\free{g1 one }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/DFLOAT.ht b/src/hyper/pages/DFLOAT.ht
new file mode 100644
index 00000000..c1c2120f
--- /dev/null
+++ b/src/hyper/pages/DFLOAT.ht
@@ -0,0 +1,112 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\DoubleFloatXmpTitle}{DoubleFloat}
+\newcommand{\DoubleFloatXmpNumber}{9.17}
+%
+% =====================================================================
+\begin{page}{DoubleFloatXmpPage}{9.17 DoubleFloat}
+% =====================================================================
+\beginscroll
+%
+
+\Language{} provides two kinds of floating point numbers.
+The domain \spadtype{Float} (abbreviation \spadtype{FLOAT})
+implements a model of arbitrary
+%-% \HDindex{floating point!arbitrary precision}{DoubleFloatXmpPage}{9.17}{DoubleFloat}
+precision floating point numbers.
+%-% \HDexptypeindex{Float}{DoubleFloatXmpPage}{9.17}{DoubleFloat}
+The domain \spadtype{DoubleFloat} (abbreviation \spadtype{DFLOAT})
+is intended to make available
+%-% \HDindex{floating point!hardware}{DoubleFloatXmpPage}{9.17}{DoubleFloat}
+hardware floating point arithmetic in \Language{}.
+The actual model of floating point \spadtype{DoubleFloat} that provides is
+system-dependent.
+For example, on the IBM system 370 \Language{} uses IBM double
+precision which has fourteen hexadecimal digits of precision or roughly
+sixteen decimal digits.
+Arbitrary precision floats allow the user to specify the
+precision at which arithmetic operations are computed.
+Although this is an attractive facility, it comes at a cost.
+Arbitrary-precision floating-point arithmetic typically takes
+twenty to two hundred times more time than hardware floating point.
+
+The usual arithmetic and elementary functions are available for
+\spadtype{DoubleFloat}.
+Use \spadsys{)show DoubleFloat} to get a list of operations or
+the \HyperName{} \Browse{} facility to get more extensive
+documentation about \spadtype{DoubleFloat}.
+
+\xtc{
+By default, floating point numbers that you enter into \Language{}
+are of type \spadtype{Float}.
+}{
+\spadpaste{2.71828}
+}
+You must therefore tell \Language{} that you want to use
+\spadtype{DoubleFloat} values and operations.
+The following are some conservative guidelines for getting
+\Language{} to use \spadtype{DoubleFloat}.
+
+\xtc{
+To get a value of type \spadtype{DoubleFloat}, use a target with
+\spadSyntax{@}, \ldots
+}{
+\spadpaste{2.71828@DoubleFloat}
+}
+\xtc{
+a conversion, \ldots
+}{
+\spadpaste{2.71828 :: DoubleFloat}
+}
+\xtc{
+or an assignment to a declared variable.
+It is more efficient if you use a target rather than an explicit or
+implicit conversion.
+}{
+\spadpaste{eApprox : DoubleFloat := 2.71828 \bound{eApprox}}
+}
+
+\xtc{
+You also need to declare functions that work with
+\spadtype{DoubleFloat}.
+}{
+\spadpaste{avg : List DoubleFloat -> DoubleFloat \bound{avgDec}}
+}
+\xtc{
+}{
+\begin{spadsrc}[\bound{avg}\free{avgDec}]
+avg l ==
+  empty? l => 0 :: DoubleFloat
+  reduce(_+,l) / #l
+\end{spadsrc}
+}
+\xtc{
+}{
+\spadpaste{avg [] \free{avg}}
+}
+\xtc{
+}{
+\spadpaste{avg [3.4,9.7,-6.8] \free{avg}}
+}
+\xtc{
+Use package-calling for operations from \spadtype{DoubleFloat} unless
+the arguments themselves are already of type \spadtype{DoubleFloat}.
+}{
+\spadpaste{cos(3.1415926)\$DoubleFloat}
+}
+\xtc{
+}{
+\spadpaste{cos(3.1415926 :: DoubleFloat)}
+}
+
+By far, the most common usage of \spadtype{DoubleFloat} is for functions
+to be graphed.
+For more information about \Language{}'s numerical and graphical
+facilities, see
+\downlink{``\ugGraphTitle''}{ugGraphPage} in Section \ugGraphNumber\ignore{ugGraph},
+\downlink{``\ugProblemNumericTitle''}{ugProblemNumericPage} in Section \ugProblemNumericNumber\ignore{ugProblemNumeric}, and
+\downlink{`Float'}{FloatXmpPage}\ignore{Float}.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/DFLOAT.pht b/src/hyper/pages/DFLOAT.pht
new file mode 100644
index 00000000..5dbc76be
--- /dev/null
+++ b/src/hyper/pages/DFLOAT.pht
@@ -0,0 +1,164 @@
+\begin{patch}{DoubleFloatXmpPagePatch1}
+\begin{paste}{DoubleFloatXmpPageFull1}{DoubleFloatXmpPageEmpty1}
+\pastebutton{DoubleFloatXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{2.71828}
+\indentrel{3}\begin{verbatim}
+   (1)  2.71828
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPageEmpty1}
+\begin{paste}{DoubleFloatXmpPageEmpty1}{DoubleFloatXmpPagePatch1}
+\pastebutton{DoubleFloatXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{2.71828}
+\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPagePatch2}
+\begin{paste}{DoubleFloatXmpPageFull2}{DoubleFloatXmpPageEmpty2}
+\pastebutton{DoubleFloatXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{2.71828@DoubleFloat}
+\indentrel{3}\begin{verbatim}
+   (2)  2.71828
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPageEmpty2}
+\begin{paste}{DoubleFloatXmpPageEmpty2}{DoubleFloatXmpPagePatch2}
+\pastebutton{DoubleFloatXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{2.71828@DoubleFloat}
+\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPagePatch3}
+\begin{paste}{DoubleFloatXmpPageFull3}{DoubleFloatXmpPageEmpty3}
+\pastebutton{DoubleFloatXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{2.71828 :: DoubleFloat}
+\indentrel{3}\begin{verbatim}
+   (3)  2.71828
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPageEmpty3}
+\begin{paste}{DoubleFloatXmpPageEmpty3}{DoubleFloatXmpPagePatch3}
+\pastebutton{DoubleFloatXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{2.71828 :: DoubleFloat}
+\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPagePatch4}
+\begin{paste}{DoubleFloatXmpPageFull4}{DoubleFloatXmpPageEmpty4}
+\pastebutton{DoubleFloatXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{eApprox : DoubleFloat := 2.71828\bound{eApprox }}
+\indentrel{3}\begin{verbatim}
+   (4)  2.71828
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPageEmpty4}
+\begin{paste}{DoubleFloatXmpPageEmpty4}{DoubleFloatXmpPagePatch4}
+\pastebutton{DoubleFloatXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{eApprox : DoubleFloat := 2.71828\bound{eApprox }}
+\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPagePatch5}
+\begin{paste}{DoubleFloatXmpPageFull5}{DoubleFloatXmpPageEmpty5}
+\pastebutton{DoubleFloatXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{avg : List DoubleFloat -> DoubleFloat\bound{avgDec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPageEmpty5}
+\begin{paste}{DoubleFloatXmpPageEmpty5}{DoubleFloatXmpPagePatch5}
+\pastebutton{DoubleFloatXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{avg : List DoubleFloat -> DoubleFloat\bound{avgDec }}
+\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPagePatch6}
+\begin{paste}{DoubleFloatXmpPageFull6}{DoubleFloatXmpPageEmpty6}
+\pastebutton{DoubleFloatXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{avg l ==
+  empty? l => 0 :: DoubleFloat
+  reduce(_+,l) / \#l
+\bound{avg }\free{avgDec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPageEmpty6}
+\begin{paste}{DoubleFloatXmpPageEmpty6}{DoubleFloatXmpPagePatch6}
+\pastebutton{DoubleFloatXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{avg l ==
+  empty? l => 0 :: DoubleFloat
+  reduce(_+,l) / \#l
+\bound{avg }\free{avgDec }}
+\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPagePatch7}
+\begin{paste}{DoubleFloatXmpPageFull7}{DoubleFloatXmpPageEmpty7}
+\pastebutton{DoubleFloatXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{avg []\free{avg }}
+\indentrel{3}\begin{verbatim}
+   (7)  0.0
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPageEmpty7}
+\begin{paste}{DoubleFloatXmpPageEmpty7}{DoubleFloatXmpPagePatch7}
+\pastebutton{DoubleFloatXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{avg []\free{avg }}
+\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPagePatch8}
+\begin{paste}{DoubleFloatXmpPageFull8}{DoubleFloatXmpPageEmpty8}
+\pastebutton{DoubleFloatXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{avg [3.4,9.7,-6.8]\free{avg }}
+\indentrel{3}\begin{verbatim}
+   (8)  2.1
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPageEmpty8}
+\begin{paste}{DoubleFloatXmpPageEmpty8}{DoubleFloatXmpPagePatch8}
+\pastebutton{DoubleFloatXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{avg [3.4,9.7,-6.8]\free{avg }}
+\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPagePatch9}
+\begin{paste}{DoubleFloatXmpPageFull9}{DoubleFloatXmpPageEmpty9}
+\pastebutton{DoubleFloatXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{cos(3.1415926)$DoubleFloat}
+\indentrel{3}\begin{verbatim}
+   (9)  - 0.999999999999999
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPageEmpty9}
+\begin{paste}{DoubleFloatXmpPageEmpty9}{DoubleFloatXmpPagePatch9}
+\pastebutton{DoubleFloatXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{cos(3.1415926)$DoubleFloat}
+\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPagePatch10}
+\begin{paste}{DoubleFloatXmpPageFull10}{DoubleFloatXmpPageEmpty10}
+\pastebutton{DoubleFloatXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{cos(3.1415926 :: DoubleFloat)}
+\indentrel{3}\begin{verbatim}
+   (10)  - 0.999999999999999
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DoubleFloatXmpPageEmpty10}
+\begin{paste}{DoubleFloatXmpPageEmpty10}{DoubleFloatXmpPagePatch10}
+\pastebutton{DoubleFloatXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{cos(3.1415926 :: DoubleFloat)}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/DMP.ht b/src/hyper/pages/DMP.ht
new file mode 100644
index 00000000..3240ddc1
--- /dev/null
+++ b/src/hyper/pages/DMP.ht
@@ -0,0 +1,87 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\DistributedMultivariatePolynomialXmpTitle}{DistributedMultivariatePolynomial}
+\newcommand{\DistributedMultivariatePolynomialXmpNumber}{9.16}
+%
+% =====================================================================
+\begin{page}{DistributedMultivariatePolynomialXmpPage}{9.16 DistributedMultivariatePolynomial}
+% =====================================================================
+\beginscroll
+
+%-% \HDexptypeindex{DistributedMultivariatePolynomial}{DistributedMultivariatePolynomialXmpPage}{9.16}{DistributedMultivariatePolynomial}
+%-% \HDexptypeindex{HomogeneousDistributedMultivariatePolynomial}{DistributedMultivariatePolynomialXmpPage}{9.16}{DistributedMultivariatePolynomial}
+\texht{\hyphenation{
+Homo-gen-eous-Dis-tributed-Multi-var-i-ate-Pol-y-nomial
+}}{}
+
+\spadtype{DistributedMultivariatePolynomial} and
+\spadtype{HomogeneousDistributedMultivariatePolynomial}, abbreviated
+\spadtype{DMP} and \spadtype{HDMP}, respectively, are very similar to
+\spadtype{MultivariatePolynomial} except that they are represented and
+displayed in a non-recursive manner.
+\xtc{
+}{
+\spadpaste{(d1,d2,d3) : DMP([z,y,x],FRAC INT) \bound{d1dec d2dec d3dec}}
+}
+\xtc{
+The constructor
+\spadtype{DMP} orders its monomials lexicographically while
+\spadtype{HDMP} orders them by total order refined by reverse
+lexicographic order.
+}{
+\spadpaste{d1 := -4*z + 4*y**2*x + 16*x**2 + 1 \bound{d1}\free{d1dec}}
+}
+\xtc{
+}{
+\spadpaste{d2 := 2*z*y**2 + 4*x + 1 \bound{d2}\free{d2dec}}
+}
+\xtc{
+}{
+\spadpaste{d3 := 2*z*x**2 - 2*y**2 - x \bound{d3}\free{d3dec}}
+}
+\xtc{
+These constructors are mostly used in \texht{Gr\"{o}bner}{Groebner}
+%-% \HDindex{Groebner basis@{Gr\protect\"{o}bner basis}}{DistributedMultivariatePolynomialXmpPage}{9.16}{DistributedMultivariatePolynomial}
+basis calculations.
+}{
+\spadpaste{groebner [d1,d2,d3] \free{d1 d2 d3}}
+}
+\xtc{
+}{
+\spadpaste{(n1,n2,n3) : HDMP([z,y,x],FRAC INT) \bound{ndec}}
+}
+\xtc{
+}{
+\spadpaste{(n1,n2,n3) := (d1,d2,d3) \free{ndec}\bound{n}\free{d1 d2 d3}}
+}
+\xtc{
+Note that we get a different
+\texht{Gr\"{o}bner}{Groebner} basis
+when we use the \spadtype{HDMP} polynomials, as expected.
+}{
+\spadpaste{groebner [n1,n2,n3] \free{n}}
+}
+
+\spadtype{GeneralDistributedMultivariatePolynomial} is somewhat
+more flexible in the sense that as well as accepting a list of
+variables to specify the variable ordering, it also takes a
+predicate on exponent vectors to specify the term ordering.
+With this polynomial type the user can experiment with the effect
+of using completely arbitrary term orderings.
+This flexibility is mostly important for algorithms such as
+\texht{Gr\"{o}bner}{Groebner} basis calculations which can be very
+sensitive to term ordering.
+%-% \HDexptypeindex{GeneralDistributedMultivariatePolynomial}{DistributedMultivariatePolynomialXmpPage}{9.16}{DistributedMultivariatePolynomial}
+
+For more information on related topics, see
+\downlink{``\ugIntroVariablesTitle''}{ugIntroVariablesPage} in Section \ugIntroVariablesNumber\ignore{ugIntroVariables},
+\downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert},
+\downlink{`Polynomial'}{PolynomialXmpPage}\ignore{Polynomial},
+\downlink{`UnivariatePolynomial'}{UnivariatePolynomialXmpPage}\ignore{UnivariatePolynomial}, and
+\downlink{`MultivariatePolynomial'}{MultivariatePolynomialXmpPage}\ignore{MultivariatePolynomial}.
+%
+\showBlurb{DistributedMultivariatePolynomial}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/DMP.pht b/src/hyper/pages/DMP.pht
new file mode 100644
index 00000000..75cf5a4d
--- /dev/null
+++ b/src/hyper/pages/DMP.pht
@@ -0,0 +1,163 @@
+\begin{patch}{DistributedMultivariatePolynomialXmpPagePatch1}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageFull1}{DistributedMultivariatePolynomialXmpPageEmpty1}
+\pastebutton{DistributedMultivariatePolynomialXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{(d1,d2,d3) : DMP([z,y,x],FRAC INT)\bound{d1dec d2dec d3dec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPageEmpty1}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageEmpty1}{DistributedMultivariatePolynomialXmpPagePatch1}
+\pastebutton{DistributedMultivariatePolynomialXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{(d1,d2,d3) : DMP([z,y,x],FRAC INT)\bound{d1dec d2dec d3dec }}
+\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPagePatch2}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageFull2}{DistributedMultivariatePolynomialXmpPageEmpty2}
+\pastebutton{DistributedMultivariatePolynomialXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{d1 := -4*z + 4*y**2*x + 16*x**2 + 1\bound{d1 }\free{d1dec }}
+\indentrel{3}\begin{verbatim}
+                 2       2
+   (2)  - 4z + 4y x + 16x  + 1
+Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPageEmpty2}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageEmpty2}{DistributedMultivariatePolynomialXmpPagePatch2}
+\pastebutton{DistributedMultivariatePolynomialXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{d1 := -4*z + 4*y**2*x + 16*x**2 + 1\bound{d1 }\free{d1dec }}
+\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPagePatch3}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageFull3}{DistributedMultivariatePolynomialXmpPageEmpty3}
+\pastebutton{DistributedMultivariatePolynomialXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{d2 := 2*z*y**2 + 4*x + 1\bound{d2 }\free{d2dec }}
+\indentrel{3}\begin{verbatim}
+            2
+   (3)  2z y  + 4x + 1
+Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPageEmpty3}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageEmpty3}{DistributedMultivariatePolynomialXmpPagePatch3}
+\pastebutton{DistributedMultivariatePolynomialXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{d2 := 2*z*y**2 + 4*x + 1\bound{d2 }\free{d2dec }}
+\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPagePatch4}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageFull4}{DistributedMultivariatePolynomialXmpPageEmpty4}
+\pastebutton{DistributedMultivariatePolynomialXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{d3 := 2*z*x**2 - 2*y**2 - x\bound{d3 }\free{d3dec }}
+\indentrel{3}\begin{verbatim}
+            2     2
+   (4)  2z x  - 2y  - x
+Type: DistributedMultivariatePolynomial([z,y,x],Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPageEmpty4}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageEmpty4}{DistributedMultivariatePolynomialXmpPagePatch4}
+\pastebutton{DistributedMultivariatePolynomialXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{d3 := 2*z*x**2 - 2*y**2 - x\bound{d3 }\free{d3dec }}
+\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPagePatch5}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageFull5}{DistributedMultivariatePolynomialXmpPageEmpty5}
+\pastebutton{DistributedMultivariatePolynomialXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{groebner [d1,d2,d3]\free{d1 d2 d3 }}
+\indentrel{3}\begin{verbatim}
+   (5)
+   [
+           1568  6   1264  5    6   4   182  3   2047  2
+       z - ÄÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄ x  + ÄÄÄ x  - ÄÄÄÄ x
+           2745       305      305      549       610
+     + 
+          103      2857
+       - ÄÄÄÄ x - ÄÄÄÄÄ
+         2745     10980
+     ,
+
+        2    112  6    84  5   1264  4    13  3    84  2
+       y  + ÄÄÄÄ x  - ÄÄÄ x  - ÄÄÄÄ x  - ÄÄÄ x  + ÄÄÄ x
+            2745      305       305      549      305
+     + 
+       1772       2
+       ÄÄÄÄ x + ÄÄÄÄ
+       2745     2745
+     ,
+     7   29  6   17  4   11  3    1  2   15     1
+    x  + ÄÄ x  - ÄÄ x  - ÄÄ x  + ÄÄ x  + ÄÄ x + Ä]
+          4      16       8      32      16     4
+Type: List DistributedMultivariatePolynomial([z,y,x],Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPageEmpty5}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageEmpty5}{DistributedMultivariatePolynomialXmpPagePatch5}
+\pastebutton{DistributedMultivariatePolynomialXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{groebner [d1,d2,d3]\free{d1 d2 d3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPagePatch6}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageFull6}{DistributedMultivariatePolynomialXmpPageEmpty6}
+\pastebutton{DistributedMultivariatePolynomialXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{(n1,n2,n3) : HDMP([z,y,x],FRAC INT)\bound{ndec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPageEmpty6}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageEmpty6}{DistributedMultivariatePolynomialXmpPagePatch6}
+\pastebutton{DistributedMultivariatePolynomialXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{(n1,n2,n3) : HDMP([z,y,x],FRAC INT)\bound{ndec }}
+\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPagePatch7}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageFull7}{DistributedMultivariatePolynomialXmpPageEmpty7}
+\pastebutton{DistributedMultivariatePolynomialXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{(n1,n2,n3) := (d1,d2,d3)\free{ndec }\bound{n }\free{d1 d2 d3 }}
+\indentrel{3}\begin{verbatim}
+            2     2
+   (7)  2z x  - 2y  - x
+Type: HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPageEmpty7}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageEmpty7}{DistributedMultivariatePolynomialXmpPagePatch7}
+\pastebutton{DistributedMultivariatePolynomialXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{(n1,n2,n3) := (d1,d2,d3)\free{ndec }\bound{n }\free{d1 d2 d3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPagePatch8}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageFull8}{DistributedMultivariatePolynomialXmpPageEmpty8}
+\pastebutton{DistributedMultivariatePolynomialXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{groebner [n1,n2,n3]\free{n }}
+\indentrel{3}\begin{verbatim}
+   (8)
+     4     3   3  2   1     1
+   [y  + 2x  - Ä x  + Ä z - Ä,
+               2      2     8
+     4   29  3   1  2   7        9     1     2        1
+    x  + ÄÄ x  - Ä y  - Ä z x - ÄÄ x - Ä, z y  + 2x + Ä,
+          4      8      4       16     4              2
+     2      2       1     2    2   1
+    y x + 4x  - z + Ä, z x  - y  - Ä x,
+                    4              2
+     2     2     2   1     3
+    z  - 4y  + 2x  - Ä z - Ä x]
+                     4     2
+Type: List HomogeneousDistributedMultivariatePolynomial([z,y,x],Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{DistributedMultivariatePolynomialXmpPageEmpty8}
+\begin{paste}{DistributedMultivariatePolynomialXmpPageEmpty8}{DistributedMultivariatePolynomialXmpPagePatch8}
+\pastebutton{DistributedMultivariatePolynomialXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{groebner [n1,n2,n3]\free{n }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/EQ.ht b/src/hyper/pages/EQ.ht
new file mode 100644
index 00000000..06cc2818
--- /dev/null
+++ b/src/hyper/pages/EQ.ht
@@ -0,0 +1,84 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\EquationXmpTitle}{Equation}
+\newcommand{\EquationXmpNumber}{9.19}
+%
+% =====================================================================
+\begin{page}{EquationXmpPage}{9.19 Equation}
+% =====================================================================
+\beginscroll
+
+The \spadtype{Equation} domain provides equations as mathematical objects.
+These are used, for example, as the input to various
+\spadfunFrom{solve}{TransSolvePackage} operations.
+
+\xtc{
+Equations are created using the equals symbol, \spadopFrom{=}{Equation}.
+}{
+\spadpaste{eq1 := 3*x + 4*y = 5 \bound{eq1}}
+}
+\xtc{
+}{
+\spadpaste{eq2 := 2*x + 2*y = 3 \bound{eq2}}
+}
+\xtc{
+The left- and right-hand sides of an equation are accessible using
+the operations \spadfunFrom{lhs}{Equation} and \spadfunFrom{rhs}{Equation}.
+}{
+\spadpaste{lhs eq1 \free{eq1}}
+}
+\xtc{
+}{
+\spadpaste{rhs eq1 \free{eq1}}
+}
+
+\xtc{
+Arithmetic operations are supported
+and operate on both sides of the equation.
+}{
+\spadpaste{eq1 + eq2   \free{eq1 eq2}}
+}
+\xtc{
+}{
+\spadpaste{eq1 * eq2   \free{eq1 eq2}}
+}
+\xtc{
+}{
+\spadpaste{2*eq2 - eq1 \free{eq1 eq2}}
+}
+\xtc{
+Equations may be created for any type so the arithmetic operations
+will be defined only when they make sense.  For example,
+exponentiation is not defined for equations involving non-square matrices.
+}{
+\spadpaste{eq1**2 \free{eq1}}
+}
+
+\xtc{
+Note that an equals symbol is also used to {\it test}
+for equality of values in certain contexts.
+For example, \spad{x+1} and \spad{y} are unequal as polynomials.
+}{
+\spadpaste{if x+1 = y then "equal" else "unequal"}
+}
+\xtc{
+}{
+\spadpaste{eqpol := x+1 = y \bound{eqpol}}
+}
+\xtc{
+If an equation is used where a \spadtype{Boolean} value
+is required, then it is evaluated using the equality
+test from the operand type.
+}{
+\spadpaste{if eqpol then "equal" else "unequal" \free{eqpol}}
+}
+\xtc{
+If one wants a \spadtype{Boolean} value rather than an equation,
+all one has to do is ask!
+}{
+\spadpaste{eqpol::Boolean \free{eqpol}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/EQ.pht b/src/hyper/pages/EQ.pht
new file mode 100644
index 00000000..d232a8ea
--- /dev/null
+++ b/src/hyper/pages/EQ.pht
@@ -0,0 +1,194 @@
+\begin{patch}{EquationXmpPagePatch1}
+\begin{paste}{EquationXmpPageFull1}{EquationXmpPageEmpty1}
+\pastebutton{EquationXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{eq1 := 3*x + 4*y = 5\bound{eq1 }}
+\indentrel{3}\begin{verbatim}
+   (1)  4y + 3x= 5
+                      Type: Equation Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPageEmpty1}
+\begin{paste}{EquationXmpPageEmpty1}{EquationXmpPagePatch1}
+\pastebutton{EquationXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{eq1 := 3*x + 4*y = 5\bound{eq1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPagePatch2}
+\begin{paste}{EquationXmpPageFull2}{EquationXmpPageEmpty2}
+\pastebutton{EquationXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{eq2 := 2*x + 2*y = 3\bound{eq2 }}
+\indentrel{3}\begin{verbatim}
+   (2)  2y + 2x= 3
+                      Type: Equation Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPageEmpty2}
+\begin{paste}{EquationXmpPageEmpty2}{EquationXmpPagePatch2}
+\pastebutton{EquationXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{eq2 := 2*x + 2*y = 3\bound{eq2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPagePatch3}
+\begin{paste}{EquationXmpPageFull3}{EquationXmpPageEmpty3}
+\pastebutton{EquationXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{lhs eq1\free{eq1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  4y + 3x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPageEmpty3}
+\begin{paste}{EquationXmpPageEmpty3}{EquationXmpPagePatch3}
+\pastebutton{EquationXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{lhs eq1\free{eq1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPagePatch4}
+\begin{paste}{EquationXmpPageFull4}{EquationXmpPageEmpty4}
+\pastebutton{EquationXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{rhs eq1\free{eq1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  5
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPageEmpty4}
+\begin{paste}{EquationXmpPageEmpty4}{EquationXmpPagePatch4}
+\pastebutton{EquationXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{rhs eq1\free{eq1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPagePatch5}
+\begin{paste}{EquationXmpPageFull5}{EquationXmpPageEmpty5}
+\pastebutton{EquationXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{eq1 + eq2\free{eq1 eq2 }}
+\indentrel{3}\begin{verbatim}
+   (5)  6y + 5x= 8
+                      Type: Equation Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPageEmpty5}
+\begin{paste}{EquationXmpPageEmpty5}{EquationXmpPagePatch5}
+\pastebutton{EquationXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{eq1 + eq2\free{eq1 eq2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPagePatch6}
+\begin{paste}{EquationXmpPageFull6}{EquationXmpPageEmpty6}
+\pastebutton{EquationXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{eq1 * eq2\free{eq1 eq2 }}
+\indentrel{3}\begin{verbatim}
+          2             2
+   (6)  8y  + 14x y + 6x = 15
+                      Type: Equation Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPageEmpty6}
+\begin{paste}{EquationXmpPageEmpty6}{EquationXmpPagePatch6}
+\pastebutton{EquationXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{eq1 * eq2\free{eq1 eq2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPagePatch7}
+\begin{paste}{EquationXmpPageFull7}{EquationXmpPageEmpty7}
+\pastebutton{EquationXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{2*eq2 - eq1\free{eq1 eq2 }}
+\indentrel{3}\begin{verbatim}
+   (7)  x= 1
+                      Type: Equation Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPageEmpty7}
+\begin{paste}{EquationXmpPageEmpty7}{EquationXmpPagePatch7}
+\pastebutton{EquationXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{2*eq2 - eq1\free{eq1 eq2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPagePatch8}
+\begin{paste}{EquationXmpPageFull8}{EquationXmpPageEmpty8}
+\pastebutton{EquationXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{eq1**2\free{eq1 }}
+\indentrel{3}\begin{verbatim}
+           2             2
+   (8)  16y  + 24x y + 9x = 25
+                      Type: Equation Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPageEmpty8}
+\begin{paste}{EquationXmpPageEmpty8}{EquationXmpPagePatch8}
+\pastebutton{EquationXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{eq1**2\free{eq1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPagePatch9}
+\begin{paste}{EquationXmpPageFull9}{EquationXmpPageEmpty9}
+\pastebutton{EquationXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{if x+1 = y then "equal" else "unequal"}
+\indentrel{3}\begin{verbatim}
+   (9)  "unequal"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPageEmpty9}
+\begin{paste}{EquationXmpPageEmpty9}{EquationXmpPagePatch9}
+\pastebutton{EquationXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{if x+1 = y then "equal" else "unequal"}
+\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPagePatch10}
+\begin{paste}{EquationXmpPageFull10}{EquationXmpPageEmpty10}
+\pastebutton{EquationXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{eqpol := x+1 = y\bound{eqpol }}
+\indentrel{3}\begin{verbatim}
+   (10)  x + 1= y
+                      Type: Equation Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPageEmpty10}
+\begin{paste}{EquationXmpPageEmpty10}{EquationXmpPagePatch10}
+\pastebutton{EquationXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{eqpol := x+1 = y\bound{eqpol }}
+\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPagePatch11}
+\begin{paste}{EquationXmpPageFull11}{EquationXmpPageEmpty11}
+\pastebutton{EquationXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{if eqpol then "equal" else "unequal"\free{eqpol }}
+\indentrel{3}\begin{verbatim}
+   (11)  "unequal"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPageEmpty11}
+\begin{paste}{EquationXmpPageEmpty11}{EquationXmpPagePatch11}
+\pastebutton{EquationXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{if eqpol then "equal" else "unequal"\free{eqpol }}
+\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPagePatch12}
+\begin{paste}{EquationXmpPageFull12}{EquationXmpPageEmpty12}
+\pastebutton{EquationXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{eqpol::Boolean\free{eqpol }}
+\indentrel{3}\begin{verbatim}
+   (12)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EquationXmpPageEmpty12}
+\begin{paste}{EquationXmpPageEmpty12}{EquationXmpPagePatch12}
+\pastebutton{EquationXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{eqpol::Boolean\free{eqpol }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/EQTBL.ht b/src/hyper/pages/EQTBL.ht
new file mode 100644
index 00000000..2e5680b9
--- /dev/null
+++ b/src/hyper/pages/EQTBL.ht
@@ -0,0 +1,55 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\EqTableXmpTitle}{EqTable}
+\newcommand{\EqTableXmpNumber}{9.18}
+%
+% =====================================================================
+\begin{page}{EqTableXmpPage}{9.18 EqTable}
+% =====================================================================
+\beginscroll
+%
+
+The \spadtype{EqTable} domain provides tables where the keys are compared
+%-% \HDindex{table}{EqTableXmpPage}{9.18}{EqTable}
+using \spadfunFrom{eq?}{EqTable}.
+Keys are considered equal only if they are the same instance of a
+structure.
+This is useful if the keys are themselves updatable structures.
+Otherwise, all operations are the same as for type \spadtype{Table}.
+See \downlink{`Table'}{TableXmpPage}\ignore{Table} for general information about tables.
+\showBlurb{EqTable}
+
+\xtc{
+The operation \spadfunFrom{table}{EqTable} is here used to create a table
+where the keys are lists of integers.
+}{
+\spadpaste{e: EqTable(List Integer, Integer) := table() \bound{e}}
+}
+\xtc{
+These two lists are equal according to \spadopFrom{=}{List},
+but not according to \spadfunFrom{eq?}{List}.
+}{
+\spadpaste{l1 := [1,2,3] \bound{l1}}
+}
+\xtc{
+}{
+\spadpaste{l2 := [1,2,3] \bound{l2}}
+}
+\xtc{
+Because the two lists are not \spadfunFrom{eq?}{List}, separate values
+can be stored under each.
+}{
+\spadpaste{e.l1 := 111   \free{e l1} \bound{e1}}
+}
+\xtc{
+}{
+\spadpaste{e.l2 := 222   \free{e1 l2} \bound{e2}}
+}
+\xtc{
+}{
+\spadpaste{e.l1          \free{e2 l1}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/EQTBL.pht b/src/hyper/pages/EQTBL.pht
new file mode 100644
index 00000000..c73987b9
--- /dev/null
+++ b/src/hyper/pages/EQTBL.pht
@@ -0,0 +1,96 @@
+\begin{patch}{EqTableXmpPagePatch1}
+\begin{paste}{EqTableXmpPageFull1}{EqTableXmpPageEmpty1}
+\pastebutton{EqTableXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{e: EqTable(List Integer, Integer) := table()\bound{e }}
+\indentrel{3}\begin{verbatim}
+   (1)  table()
+                    Type: EqTable(List Integer,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EqTableXmpPageEmpty1}
+\begin{paste}{EqTableXmpPageEmpty1}{EqTableXmpPagePatch1}
+\pastebutton{EqTableXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{e: EqTable(List Integer, Integer) := table()\bound{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{EqTableXmpPagePatch2}
+\begin{paste}{EqTableXmpPageFull2}{EqTableXmpPageEmpty2}
+\pastebutton{EqTableXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{l1 := [1,2,3]\bound{l1 }}
+\indentrel{3}\begin{verbatim}
+   (2)  [1,2,3]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EqTableXmpPageEmpty2}
+\begin{paste}{EqTableXmpPageEmpty2}{EqTableXmpPagePatch2}
+\pastebutton{EqTableXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{l1 := [1,2,3]\bound{l1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{EqTableXmpPagePatch3}
+\begin{paste}{EqTableXmpPageFull3}{EqTableXmpPageEmpty3}
+\pastebutton{EqTableXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{l2 := [1,2,3]\bound{l2 }}
+\indentrel{3}\begin{verbatim}
+   (3)  [1,2,3]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EqTableXmpPageEmpty3}
+\begin{paste}{EqTableXmpPageEmpty3}{EqTableXmpPagePatch3}
+\pastebutton{EqTableXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{l2 := [1,2,3]\bound{l2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{EqTableXmpPagePatch4}
+\begin{paste}{EqTableXmpPageFull4}{EqTableXmpPageEmpty4}
+\pastebutton{EqTableXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{e.l1 := 111\free{e l1 }\bound{e1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  111
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EqTableXmpPageEmpty4}
+\begin{paste}{EqTableXmpPageEmpty4}{EqTableXmpPagePatch4}
+\pastebutton{EqTableXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{e.l1 := 111\free{e l1 }\bound{e1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{EqTableXmpPagePatch5}
+\begin{paste}{EqTableXmpPageFull5}{EqTableXmpPageEmpty5}
+\pastebutton{EqTableXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{e.l2 := 222\free{e1 l2 }\bound{e2 }}
+\indentrel{3}\begin{verbatim}
+   (5)  222
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EqTableXmpPageEmpty5}
+\begin{paste}{EqTableXmpPageEmpty5}{EqTableXmpPagePatch5}
+\pastebutton{EqTableXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{e.l2 := 222\free{e1 l2 }\bound{e2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{EqTableXmpPagePatch6}
+\begin{paste}{EqTableXmpPageFull6}{EqTableXmpPageEmpty6}
+\pastebutton{EqTableXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{e.l1\free{e2 l1 }}
+\indentrel{3}\begin{verbatim}
+   (6)  111
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{EqTableXmpPageEmpty6}
+\begin{paste}{EqTableXmpPageEmpty6}{EqTableXmpPagePatch6}
+\pastebutton{EqTableXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{e.l1\free{e2 l1 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/EXIT.ht b/src/hyper/pages/EXIT.ht
new file mode 100644
index 00000000..076b880d
--- /dev/null
+++ b/src/hyper/pages/EXIT.ht
@@ -0,0 +1,64 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\ExitXmpTitle}{Exit}
+\newcommand{\ExitXmpNumber}{9.20}
+%
+% =====================================================================
+\begin{page}{ExitXmpPage}{9.20 Exit}
+% =====================================================================
+\beginscroll
+
+A function that does not return directly to its caller has
+\spadtype{Exit} as its return type.
+The operation \spadfun{error} is an example of one which does not return
+to its caller.
+Instead, it causes a return to top-level.
+\xtc{
+}{
+\spadpaste{n := 0 \bound{n}}
+}
+\xtc{
+The function \userfun{gasp} is given return type \spadtype{Exit} since it is
+guaranteed never to return a value to its caller.
+}{
+\begin{spadsrc}[\bound{gasp}\free{n}]
+gasp(): Exit ==
+    free n
+    n := n + 1
+    error "Oh no!"
+\end{spadsrc}
+}
+\xtc{
+The return type of \userfun{half} is determined by resolving the types of
+the two branches of the \spad{if}.
+}{
+\begin{spadsrc}[\bound{half}\free{gasp}]
+half(k) ==
+  if odd? k then gasp()
+  else k quo 2
+\end{spadsrc}
+}
+\xtc{
+Because
+\userfun{gasp} has the return type \spadtype{Exit},
+the type of \spad{if} in \userfun{half} is resolved to be \spadtype{Integer}.
+}{
+\spadpaste{half 4 \free{half}\bound{app1}}
+}
+\xtc{
+}{
+\spadpaste{half 3 \free{half app1}\bound{app2}}
+}
+\xtc{
+}{
+\spadpaste{n \free{app2}}
+}
+
+For functions which return no value at all, use \spadtype{Void}.
+See \downlink{``\ugUserTitle''}{ugUserPage} in Section \ugUserNumber\ignore{ugUser} and \downlink{`Void'}{VoidXmpPage}\ignore{Void} for more information.
+%
+\showBlurb{Exit}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/EXIT.pht b/src/hyper/pages/EXIT.pht
new file mode 100644
index 00000000..c8a261d3
--- /dev/null
+++ b/src/hyper/pages/EXIT.pht
@@ -0,0 +1,106 @@
+\begin{patch}{ExitXmpPagePatch1}
+\begin{paste}{ExitXmpPageFull1}{ExitXmpPageEmpty1}
+\pastebutton{ExitXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{n := 0\bound{n }}
+\indentrel{3}\begin{verbatim}
+   (1)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExitXmpPageEmpty1}
+\begin{paste}{ExitXmpPageEmpty1}{ExitXmpPagePatch1}
+\pastebutton{ExitXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{n := 0\bound{n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExitXmpPagePatch2}
+\begin{paste}{ExitXmpPageFull2}{ExitXmpPageEmpty2}
+\pastebutton{ExitXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{gasp(): Exit ==
+    free n
+    n := n + 1
+    error "Oh no!"
+\bound{gasp }\free{n }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExitXmpPageEmpty2}
+\begin{paste}{ExitXmpPageEmpty2}{ExitXmpPagePatch2}
+\pastebutton{ExitXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{gasp(): Exit ==
+    free n
+    n := n + 1
+    error "Oh no!"
+\bound{gasp }\free{n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExitXmpPagePatch3}
+\begin{paste}{ExitXmpPageFull3}{ExitXmpPageEmpty3}
+\pastebutton{ExitXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{half(k) ==
+  if odd? k then gasp()
+  else k quo 2
+\bound{half }\free{gasp }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExitXmpPageEmpty3}
+\begin{paste}{ExitXmpPageEmpty3}{ExitXmpPagePatch3}
+\pastebutton{ExitXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{half(k) ==
+  if odd? k then gasp()
+  else k quo 2
+\bound{half }\free{gasp }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExitXmpPagePatch4}
+\begin{paste}{ExitXmpPageFull4}{ExitXmpPageEmpty4}
+\pastebutton{ExitXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{half 4\free{half }\bound{app1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExitXmpPageEmpty4}
+\begin{paste}{ExitXmpPageEmpty4}{ExitXmpPagePatch4}
+\pastebutton{ExitXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{half 4\free{half }\bound{app1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExitXmpPagePatch5}
+\begin{paste}{ExitXmpPageFull5}{ExitXmpPageEmpty5}
+\pastebutton{ExitXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{half 3\free{half app1 }\bound{app2 }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExitXmpPageEmpty5}
+\begin{paste}{ExitXmpPageEmpty5}{ExitXmpPagePatch5}
+\pastebutton{ExitXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{half 3\free{half app1 }\bound{app2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExitXmpPagePatch6}
+\begin{paste}{ExitXmpPageFull6}{ExitXmpPageEmpty6}
+\pastebutton{ExitXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{n\free{app2 }}
+\indentrel{3}\begin{verbatim}
+   (5)  1
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExitXmpPageEmpty6}
+\begin{paste}{ExitXmpPageEmpty6}{ExitXmpPagePatch6}
+\pastebutton{ExitXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{n\free{app2 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/EXPR.ht b/src/hyper/pages/EXPR.ht
new file mode 100644
index 00000000..0984b52f
--- /dev/null
+++ b/src/hyper/pages/EXPR.ht
@@ -0,0 +1,205 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\ExpressionXmpTitle}{Expression}
+\newcommand{\ExpressionXmpNumber}{9.21}
+%
+% =====================================================================
+\begin{page}{ExpressionXmpPage}{9.21 Expression}
+% =====================================================================
+\beginscroll
+%
+
+\axiomType{Expression} is a constructor that creates domains whose
+objects can have very general symbolic forms.
+Here are some examples:
+\xtc{
+This is an object of type \axiomType{Expression Integer}.
+}{
+\spadpaste{sin(x) + 3*cos(x)**2}
+}
+\xtc{
+This is an object of type \axiomType{Expression Float}.
+}{
+\spadpaste{tan(x) - 3.45*x}
+}
+\xtc{
+This object contains symbolic function applications, sums,
+products, square roots, and a quotient.
+}{
+\spadpaste{(tan sqrt 7 - sin sqrt 11)**2 / (4 - cos(x - y))}
+}
+As you can see, \axiomType{Expression} actually takes an argument
+domain.
+The {\it coefficients} of the terms within the expression belong
+to the argument domain.
+\axiomType{Integer} and \axiomType{Float}, along with
+\axiomType{Complex Integer} and \axiomType{Complex Float}
+are the most common coefficient domains.
+\xtc{
+The choice of whether
+to use a \axiomType{Complex} coefficient domain or not is
+important since \Language{} can perform some simplifications
+on real-valued objects
+}{
+\spadpaste{log(exp  x)@Expression(Integer)} 
+}
+\xtc{
+... which are not valid on complex ones.
+}{
+\spadpaste{log(exp  x)@Expression(Complex Integer)} 
+}
+\xtc{
+Many potential coefficient domains, such as
+\axiomType{AlgebraicNumber}, are not usually used because
+\axiomType{Expression} can subsume them.
+}{
+\spadpaste{sqrt 3 + sqrt(2 + sqrt(-5)) \bound{algnum1}}
+}
+\xtc{
+}{
+\spadpaste{\% :: Expression Integer \free{algnum1}}
+}
+Note that we sometimes talk about ``an object of type
+\axiomType{Expression}.'' This is not really correct because we
+should say, for example, ``an object of type \axiomType{Expression
+Integer}'' or ``an object of type \axiomType{Expression Float}.''
+By a similar abuse of language, when we refer to an ``expression''
+in this section we will mean an object of type
+\axiomType{Expression R} for some domain {\bf R}.
+
+The \Language{} documentation contains many examples of the use
+of \axiomType{Expression}.
+For the rest of this section, we'll give you some pointers to those
+examples plus give you some idea of how to manipulate expressions.
+
+It is important for you to know that \axiomType{Expression}
+creates domains that have category \axiomType{Field}.
+Thus you can invert any non-zero expression and you shouldn't
+expect an operation like \axiomFun{factor} to give you much
+information.
+You can imagine expressions as being represented as quotients of
+``multivariate'' polynomials where the ``variables'' are kernels
+(see \downlink{`Kernel'}{KernelXmpPage}\ignore{Kernel}).
+A kernel can either be a symbol such as \axiom{x} or a symbolic
+function application like \axiom{sin(x + 4)}.
+The second example is actually a nested kernel since the argument
+to \axiomFun{sin} contains the kernel \axiom{x}.
+\xtc{
+}{
+\spadpaste{height mainKernel sin(x + 4)}
+}
+Actually, the argument to \axiomFun{sin} is an expression, and so
+the structure of \axiomType{Expression} is recursive.
+\downlink{`Kernel'}{KernelXmpPage}\ignore{Kernel} demonstrates how to extract the kernels in an
+expression.
+
+Use the \HyperName{} Browse facility to see what operations are
+applicable to expression.
+At the time of this writing, there were 262 operations with 147
+distinct name in \axiomType{Expression Integer}.
+For example, \axiomFunFrom{numer}{Expression} and
+\axiomFunFrom{denom}{Expression} extract the numerator and
+denominator of an expression.
+\xtc{
+}{
+\spadpaste{e := (sin(x) - 4)**2 / ( 1 - 2*y*sqrt(- y) ) \bound{e}}
+}
+\xtc{
+}{
+\spadpaste{numer e \free{e}}
+}
+\xtc{
+}{
+\spadpaste{denom e \free{e}}
+}
+\xtc{
+Use \axiomFunFrom{D}{Expression} to compute partial derivatives.
+}{
+\spadpaste{D(e, x) \free{e}}
+}
+\xtc{
+See \downlink{``\ugIntroCalcDerivTitle''}{ugIntroCalcDerivPage} in Section \ugIntroCalcDerivNumber\ignore{ugIntroCalcDeriv}
+for more examples of expressions and derivatives.
+}{
+\spadpaste{D(e, [x, y], [1, 2]) \free{e}}
+}
+See \downlink{``\ugIntroCalcLimitsTitle''}{ugIntroCalcLimitsPage} in Section \ugIntroCalcLimitsNumber\ignore{ugIntroCalcLimits} and
+\downlink{``\ugIntroSeriesTitle''}{ugIntroSeriesPage} in Section \ugIntroSeriesNumber\ignore{ugIntroSeries}
+for more examples of expressions and calculus.
+Differential equations involving expressions are discussed in
+\downlink{``\ugProblemDEQTitle''}{ugProblemDEQPage} in Section \ugProblemDEQNumber\ignore{ugProblemDEQ}.
+Chapter 8 has many advanced examples: see
+\downlink{``\ugProblemIntegrationTitle''}{ugProblemIntegrationPage} in Section \ugProblemIntegrationNumber\ignore{ugProblemIntegration}
+for a discussion of \Language{}'s integration facilities.
+
+When an expression involves no ``symbol kernels'' (for example,
+\axiom{x}), it may be possible to numerically evaluate the
+expression.
+\xtc{
+If you suspect the evaluation will create a complex number,
+use \axiomFun{complexNumeric}.
+}{
+\spadpaste{complexNumeric(cos(2 - 3*\%i))}
+}
+\xtc{
+If you know it will be real, use \axiomFun{numeric}.
+}{
+\spadpaste{numeric(tan 3.8)}
+}
+The \axiomFun{numeric} operation will display an error message if
+the evaluation yields a calue with an non-zero imaginary part.
+Both of these operations have an optional second argument
+\axiom{n} which specifies that the accuracy of the approximation
+be up to \axiom{n} decimal places.
+
+When an expression involves no ``symbolic application'' kernels,
+it may be possible to convert it a polynomial or rational
+function in the variables that are present.
+\xtc{
+}{
+\spadpaste{e2 := cos(x**2 - y + 3) \bound{e2}}
+}
+\xtc{
+}{
+\spadpaste{e3 := asin(e2) - \%pi/2 \free{e2}\bound{e3}}
+}
+\xtc{
+}{
+\spadpaste{e3 :: Polynomial Integer \free{e3}}
+}
+\xtc{
+This also works for the polynomial types where specific variables
+and their ordering are given.
+}{
+\spadpaste{e3 :: DMP([x, y], Integer) \free{e3}}
+}
+
+Finally, a certain amount of simplication takes place as
+expressions are constructed.
+\xtc{
+}{
+\spadpaste{sin \%pi}
+}
+\xtc{
+}{
+\spadpaste{cos(\%pi / 4)}
+}
+\xtc{
+For simplications that involve multiple terms of the expression,
+use \axiomFun{simplify}.
+}{
+\spadpaste{tan(x)**6 + 3*tan(x)**4 + 3*tan(x)**2 + 1 \bound{tan6}}
+}
+\xtc{
+}{
+\spadpaste{simplify \% \free{tan6}}
+}
+See \downlink{``\ugUserRulesTitle''}{ugUserRulesPage} in Section \ugUserRulesNumber\ignore{ugUserRules}
+for examples of how to write your own rewrite rules for
+expressions.
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/EXPR.pht b/src/hyper/pages/EXPR.pht
new file mode 100644
index 00000000..9d571ebf
--- /dev/null
+++ b/src/hyper/pages/EXPR.pht
@@ -0,0 +1,423 @@
+\begin{patch}{ExpressionXmpPagePatch1}
+\begin{paste}{ExpressionXmpPageFull1}{ExpressionXmpPageEmpty1}
+\pastebutton{ExpressionXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{sin(x) + 3*cos(x)**2}
+\indentrel{3}\begin{verbatim}
+                        2
+   (1)  sin(x) + 3cos(x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty1}
+\begin{paste}{ExpressionXmpPageEmpty1}{ExpressionXmpPagePatch1}
+\pastebutton{ExpressionXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{sin(x) + 3*cos(x)**2}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch2}
+\begin{paste}{ExpressionXmpPageFull2}{ExpressionXmpPageEmpty2}
+\pastebutton{ExpressionXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{tan(x) - 3.45*x}
+\indentrel{3}\begin{verbatim}
+   (2)  tan(x) - 3.45 x
+                                 Type: Expression Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty2}
+\begin{paste}{ExpressionXmpPageEmpty2}{ExpressionXmpPagePatch2}
+\pastebutton{ExpressionXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{tan(x) - 3.45*x}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch3}
+\begin{paste}{ExpressionXmpPageFull3}{ExpressionXmpPageEmpty3}
+\pastebutton{ExpressionXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{(tan sqrt 7 - sin sqrt 11)**2 / (4 - cos(x - y))}
+\indentrel{3}\begin{verbatim}
+   (3)
+          ÚÄ¿ 2         ÚÄÄ¿      ÚÄ¿         ÚÄÄ¿ 2
+   - tan(\³7 )  + 2sin(\³11 )tan(\³7 ) - sin(\³11 )
+   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                     cos(y - x) - 4
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty3}
+\begin{paste}{ExpressionXmpPageEmpty3}{ExpressionXmpPagePatch3}
+\pastebutton{ExpressionXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{(tan sqrt 7 - sin sqrt 11)**2 / (4 - cos(x - y))}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch4}
+\begin{paste}{ExpressionXmpPageFull4}{ExpressionXmpPageEmpty4}
+\pastebutton{ExpressionXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{log(exp x)@Expression(Integer)}
+\indentrel{3}\begin{verbatim}
+   (4)  x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty4}
+\begin{paste}{ExpressionXmpPageEmpty4}{ExpressionXmpPagePatch4}
+\pastebutton{ExpressionXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{log(exp x)@Expression(Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch5}
+\begin{paste}{ExpressionXmpPageFull5}{ExpressionXmpPageEmpty5}
+\pastebutton{ExpressionXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{log(exp x)@Expression(Complex Integer)}
+\indentrel{3}\begin{verbatim}
+              x
+   (5)  log(%e )
+                       Type: Expression Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty5}
+\begin{paste}{ExpressionXmpPageEmpty5}{ExpressionXmpPagePatch5}
+\pastebutton{ExpressionXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{log(exp x)@Expression(Complex Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch6}
+\begin{paste}{ExpressionXmpPageFull6}{ExpressionXmpPageEmpty6}
+\pastebutton{ExpressionXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{sqrt 3 + sqrt(2 + sqrt(-5))\bound{algnum1 }}
+\indentrel{3}\begin{verbatim}
+         ÚÄÄÄÄÄÄÄÄÄÄ¿
+         ³ ÚÄÄÄ¿         ÚÄ¿
+   (6)  \³\³- 5  + 2  + \³3
+                                  Type: AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty6}
+\begin{paste}{ExpressionXmpPageEmpty6}{ExpressionXmpPagePatch6}
+\pastebutton{ExpressionXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{sqrt 3 + sqrt(2 + sqrt(-5))\bound{algnum1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch7}
+\begin{paste}{ExpressionXmpPageFull7}{ExpressionXmpPageEmpty7}
+\pastebutton{ExpressionXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{\% :: Expression Integer\free{algnum1 }}
+\indentrel{3}\begin{verbatim}
+         ÚÄÄÄÄÄÄÄÄÄÄ¿
+         ³ ÚÄÄÄ¿         ÚÄ¿
+   (7)  \³\³- 5  + 2  + \³3
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty7}
+\begin{paste}{ExpressionXmpPageEmpty7}{ExpressionXmpPagePatch7}
+\pastebutton{ExpressionXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{\% :: Expression Integer\free{algnum1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch8}
+\begin{paste}{ExpressionXmpPageFull8}{ExpressionXmpPageEmpty8}
+\pastebutton{ExpressionXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{height mainKernel sin(x + 4)}
+\indentrel{3}\begin{verbatim}
+   (8)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty8}
+\begin{paste}{ExpressionXmpPageEmpty8}{ExpressionXmpPagePatch8}
+\pastebutton{ExpressionXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{height mainKernel sin(x + 4)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch9}
+\begin{paste}{ExpressionXmpPageFull9}{ExpressionXmpPageEmpty9}
+\pastebutton{ExpressionXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{e := (sin(x) - 4)**2 / ( 1 - 2*y*sqrt(- y) )\bound{e }}
+\indentrel{3}\begin{verbatim}
+                2
+        - sin(x)  + 8sin(x) - 16
+   (9)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                 ÚÄÄÄ¿
+              2y\³- y  - 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty9}
+\begin{paste}{ExpressionXmpPageEmpty9}{ExpressionXmpPagePatch9}
+\pastebutton{ExpressionXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{e := (sin(x) - 4)**2 / ( 1 - 2*y*sqrt(- y) )\bound{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch10}
+\begin{paste}{ExpressionXmpPageFull10}{ExpressionXmpPageEmpty10}
+\pastebutton{ExpressionXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{numer e\free{e }}
+\indentrel{3}\begin{verbatim}
+                 2
+   (10)  - sin(x)  + 8sin(x) - 16
+Type: SparseMultivariatePolynomial(Integer,Kernel Expression Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty10}
+\begin{paste}{ExpressionXmpPageEmpty10}{ExpressionXmpPagePatch10}
+\pastebutton{ExpressionXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{numer e\free{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch11}
+\begin{paste}{ExpressionXmpPageFull11}{ExpressionXmpPageEmpty11}
+\pastebutton{ExpressionXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{denom e\free{e }}
+\indentrel{3}\begin{verbatim}
+            ÚÄÄÄ¿
+   (11)  2y\³- y  - 1
+Type: SparseMultivariatePolynomial(Integer,Kernel Expression Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty11}
+\begin{paste}{ExpressionXmpPageEmpty11}{ExpressionXmpPagePatch11}
+\pastebutton{ExpressionXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{denom e\free{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch12}
+\begin{paste}{ExpressionXmpPageFull12}{ExpressionXmpPageEmpty12}
+\pastebutton{ExpressionXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{D(e, x)\free{e }}
+\indentrel{3}\begin{verbatim}
+   (12)
+                                      ÚÄÄÄ¿
+       (4y cos(x)sin(x) - 16y cos(x))\³- y
+     + 
+       - 2cos(x)sin(x) + 8cos(x)
+  /
+        ÚÄÄÄ¿     3
+     4y\³- y  + 4y  - 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty12}
+\begin{paste}{ExpressionXmpPageEmpty12}{ExpressionXmpPagePatch12}
+\pastebutton{ExpressionXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{D(e, x)\free{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch13}
+\begin{paste}{ExpressionXmpPageFull13}{ExpressionXmpPageEmpty13}
+\pastebutton{ExpressionXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{D(e, [x, y], [1, 2])\free{e }}
+\indentrel{3}\begin{verbatim}
+   (13)
+                   7       4
+           (- 2304y  + 960y )cos(x)sin(x)
+         + 
+                 7        4
+           (9216y  - 3840y )cos(x)
+      *
+          ÚÄÄÄ¿
+         \³- y
+     + 
+              9        6       3
+       (- 960y  + 2160y  - 180y  - 3)cos(x)sin(x)
+     + 
+             9        6       3
+       (3840y  - 8640y  + 720y  + 12)cos(x)
+  /
+            12        9        6       3      ÚÄÄÄ¿
+       (256y   - 1792y  + 1120y  - 112y  + 1)\³- y
+     + 
+              11        8       5      2
+       - 1024y   + 1792y  - 448y  + 16y
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty13}
+\begin{paste}{ExpressionXmpPageEmpty13}{ExpressionXmpPagePatch13}
+\pastebutton{ExpressionXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{D(e, [x, y], [1, 2])\free{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch14}
+\begin{paste}{ExpressionXmpPageFull14}{ExpressionXmpPageEmpty14}
+\pastebutton{ExpressionXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{complexNumeric(cos(2 - 3*\%i))}
+\indentrel{3}\begin{verbatim}
+   (14)
+   - 4.1896256909 688072301 + 9.1092278937 55336598 %i
+                                    Type: Complex Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty14}
+\begin{paste}{ExpressionXmpPageEmpty14}{ExpressionXmpPagePatch14}
+\pastebutton{ExpressionXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{complexNumeric(cos(2 - 3*\%i))}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch15}
+\begin{paste}{ExpressionXmpPageFull15}{ExpressionXmpPageEmpty15}
+\pastebutton{ExpressionXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{numeric(tan 3.8)}
+\indentrel{3}\begin{verbatim}
+   (15)  0.7735560905 0312607286
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty15}
+\begin{paste}{ExpressionXmpPageEmpty15}{ExpressionXmpPagePatch15}
+\pastebutton{ExpressionXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{numeric(tan 3.8)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch16}
+\begin{paste}{ExpressionXmpPageFull16}{ExpressionXmpPageEmpty16}
+\pastebutton{ExpressionXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{e2 := cos(x**2 - y + 3)\bound{e2 }}
+\indentrel{3}\begin{verbatim}
+                  2
+   (16)  cos(y - x  - 3)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty16}
+\begin{paste}{ExpressionXmpPageEmpty16}{ExpressionXmpPagePatch16}
+\pastebutton{ExpressionXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{e2 := cos(x**2 - y + 3)\bound{e2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch17}
+\begin{paste}{ExpressionXmpPageFull17}{ExpressionXmpPageEmpty17}
+\pastebutton{ExpressionXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{e3 := asin(e2) - \%pi/2\free{e2 }\bound{e3 }}
+\indentrel{3}\begin{verbatim}
+                2
+   (17)  - y + x  + 3
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty17}
+\begin{paste}{ExpressionXmpPageEmpty17}{ExpressionXmpPagePatch17}
+\pastebutton{ExpressionXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{e3 := asin(e2) - \%pi/2\free{e2 }\bound{e3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch18}
+\begin{paste}{ExpressionXmpPageFull18}{ExpressionXmpPageEmpty18}
+\pastebutton{ExpressionXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{e3 :: Polynomial Integer\free{e3 }}
+\indentrel{3}\begin{verbatim}
+                2
+   (18)  - y + x  + 3
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty18}
+\begin{paste}{ExpressionXmpPageEmpty18}{ExpressionXmpPagePatch18}
+\pastebutton{ExpressionXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{e3 :: Polynomial Integer\free{e3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch19}
+\begin{paste}{ExpressionXmpPageFull19}{ExpressionXmpPageEmpty19}
+\pastebutton{ExpressionXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{e3 :: DMP([x, y], Integer)\free{e3 }}
+\indentrel{3}\begin{verbatim}
+          2
+   (19)  x  - y + 3
+ Type: DistributedMultivariatePolynomial([x,y],Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty19}
+\begin{paste}{ExpressionXmpPageEmpty19}{ExpressionXmpPagePatch19}
+\pastebutton{ExpressionXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{e3 :: DMP([x, y], Integer)\free{e3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch20}
+\begin{paste}{ExpressionXmpPageFull20}{ExpressionXmpPageEmpty20}
+\pastebutton{ExpressionXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{sin \%pi}
+\indentrel{3}\begin{verbatim}
+   (20)  0
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty20}
+\begin{paste}{ExpressionXmpPageEmpty20}{ExpressionXmpPagePatch20}
+\pastebutton{ExpressionXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{sin \%pi}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch21}
+\begin{paste}{ExpressionXmpPageFull21}{ExpressionXmpPageEmpty21}
+\pastebutton{ExpressionXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{cos(\%pi / 4)}
+\indentrel{3}\begin{verbatim}
+          ÚÄ¿
+         \³2
+   (21)  ÄÄÄÄ
+           2
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty21}
+\begin{paste}{ExpressionXmpPageEmpty21}{ExpressionXmpPagePatch21}
+\pastebutton{ExpressionXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{cos(\%pi / 4)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch22}
+\begin{paste}{ExpressionXmpPageFull22}{ExpressionXmpPageEmpty22}
+\pastebutton{ExpressionXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{tan(x)**6 + 3*tan(x)**4 + 3*tan(x)**2 + 1\bound{tan6 }}
+\indentrel{3}\begin{verbatim}
+               6          4          2
+   (22)  tan(x)  + 3tan(x)  + 3tan(x)  + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty22}
+\begin{paste}{ExpressionXmpPageEmpty22}{ExpressionXmpPagePatch22}
+\pastebutton{ExpressionXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{tan(x)**6 + 3*tan(x)**4 + 3*tan(x)**2 + 1\bound{tan6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPagePatch23}
+\begin{paste}{ExpressionXmpPageFull23}{ExpressionXmpPageEmpty23}
+\pastebutton{ExpressionXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{simplify \%\free{tan6 }}
+\indentrel{3}\begin{verbatim}
+            1
+   (23)  ÄÄÄÄÄÄÄ
+               6
+         cos(x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExpressionXmpPageEmpty23}
+\begin{paste}{ExpressionXmpPageEmpty23}{ExpressionXmpPagePatch23}
+\pastebutton{ExpressionXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{simplify \%\free{tan6 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/FARRAY.ht b/src/hyper/pages/FARRAY.ht
new file mode 100644
index 00000000..a406eb00
--- /dev/null
+++ b/src/hyper/pages/FARRAY.ht
@@ -0,0 +1,127 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\FlexibleArrayXmpTitle}{FlexibleArray}
+\newcommand{\FlexibleArrayXmpNumber}{9.26}
+%
+% =====================================================================
+\begin{page}{FlexibleArrayXmpPage}{9.26 FlexibleArray}
+% =====================================================================
+\beginscroll
+The \spadtype{FlexibleArray} domain constructor creates
+one-dimensional arrays of elements of the same type.
+%-% \HDindex{array!flexible}{FlexibleArrayXmpPage}{9.26}{FlexibleArray}
+Flexible arrays are an attempt to provide a data type that has the
+%-% \HDindex{flexible array}{FlexibleArrayXmpPage}{9.26}{FlexibleArray}
+best features of both one-dimensional arrays (fast, random access
+to elements) and lists (flexibility).
+They are implemented by a fixed block of storage.
+When necessary for expansion, a new, larger block of storage is
+allocated and the elements from the old storage area are copied
+into the new block.
+
+Flexible arrays have available most of the operations provided by
+\spadtype{OneDimensionalArray} (see \downlink{`OneDimensionalArray'}{OneDimensionalArrayXmpPage}\ignore{OneDimensionalArray}
+and \downlink{`Vector'}{VectorXmpPage}\ignore{Vector}).
+Since flexible arrays are also of category
+\spadtype{ExtensibleLinearAggregate}, they have operations
+\spadfunX{concat}, \spadfunX{delete}, \spadfunX{insert},
+\spadfunX{merge}, \spadfunX{remove}, \spadfunX{removeDuplicates},
+and \spadfunX{select}.
+In addition, the operations \spadfun{physicalLength} and
+\spadfunX{physicalLength} provide user-control over expansion and
+contraction.
+
+\xtc{
+A convenient way to create a flexible array is to apply
+the operation \spadfun{flexibleArray} to a list of values.
+}{
+\spadpaste{flexibleArray [i for i in 1..6]}
+}
+\xtc{
+Create a flexible array of six zeroes.
+}{
+\spadpaste{f : FARRAY INT := new(6,0)\bound{f}}
+}
+\xtc{
+For \texht{$i=1\ldots 6$}{i = 1..6},
+set the \eth{\smath{i}} element to \smath{i}.
+Display \spad{f}.
+}{
+\spadpaste{for i in 1..6 repeat f.i := i; f\bound{f1}\free{f}}
+}
+\xtc{
+Initially, the physical length is the same as the number of elements.
+}{
+\spadpaste{physicalLength f\free{f1}}
+}
+\xtc{
+Add an element to the end of \spad{f}.
+}{
+\spadpaste{concat!(f,11)\bound{f2}\free{f1}}
+}
+\xtc{
+See that its physical length has grown.
+}{
+\spadpaste{physicalLength f\free{f2}}
+}
+\xtc{
+Make \spad{f} grow to have room for \spad{15} elements.
+}{
+\spadpaste{physicalLength!(f,15)\bound{f3}\free{f2}}
+}
+\xtc{
+Concatenate the elements of \spad{f} to itself.
+The physical length allows room for three more values at the end.
+}{
+\spadpaste{concat!(f,f)\bound{f4}\free{f3}}
+}
+\xtc{
+Use \spadfunX{insert} to add an element to the front of
+a flexible array.
+}{
+\spadpaste{insert!(22,f,1)\bound{f5}\free{f4}}
+}
+\xtc{
+Create a second flexible array from \spad{f} consisting of the
+elements from index 10 forward.
+}{
+\spadpaste{g := f(10..)\bound{g}\free{f5}}
+}
+\xtc{
+Insert this array at the front of \spad{f}.
+}{
+\spadpaste{insert!(g,f,1)\bound{g1}\free{g f5}}
+}
+\xtc{
+Merge the flexible array \spad{f} into \spad{g} after sorting each in place.
+}{
+\spadpaste{merge!(sort! f, sort! g)\bound{f6}\free{g f5}}
+}
+\xtc{
+Remove duplicates in place.
+}{
+\spadpaste{removeDuplicates! f\bound{f7}\free{f6}}
+}
+\xtc{
+Remove all odd integers.
+}{
+\spadpaste{select!(i +-> even? i,f)\bound{f8}\free{f7}}
+}
+\xtc{
+All these operations have shrunk the physical length of \spad{f}.
+}{
+\spadpaste{physicalLength f\free{b8}}
+}
+\xtc{
+To force \Language{} not to shrink flexible arrays call the
+\spadfun{shrinkable} operation with the argument \axiom{false}.
+You must package call this operation.
+The previous value is returned.
+}{
+\spadpaste{shrinkable(false)\$FlexibleArray(Integer)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/FARRAY.pht b/src/hyper/pages/FARRAY.pht
new file mode 100644
index 00000000..5346e3e5
--- /dev/null
+++ b/src/hyper/pages/FARRAY.pht
@@ -0,0 +1,258 @@
+\begin{patch}{FlexibleArrayXmpPagePatch1}
+\begin{paste}{FlexibleArrayXmpPageFull1}{FlexibleArrayXmpPageEmpty1}
+\pastebutton{FlexibleArrayXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{flexibleArray [i for i in 1..6]}
+\indentrel{3}\begin{verbatim}
+   (1)  [1,2,3,4,5,6]
+                    Type: FlexibleArray PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty1}
+\begin{paste}{FlexibleArrayXmpPageEmpty1}{FlexibleArrayXmpPagePatch1}
+\pastebutton{FlexibleArrayXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{flexibleArray [i for i in 1..6]}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch2}
+\begin{paste}{FlexibleArrayXmpPageFull2}{FlexibleArrayXmpPageEmpty2}
+\pastebutton{FlexibleArrayXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f : FARRAY INT := new(6,0)\bound{f }}
+\indentrel{3}\begin{verbatim}
+   (2)  [0,0,0,0,0,0]
+                            Type: FlexibleArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty2}
+\begin{paste}{FlexibleArrayXmpPageEmpty2}{FlexibleArrayXmpPagePatch2}
+\pastebutton{FlexibleArrayXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f : FARRAY INT := new(6,0)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch3}
+\begin{paste}{FlexibleArrayXmpPageFull3}{FlexibleArrayXmpPageEmpty3}
+\pastebutton{FlexibleArrayXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{for i in 1..6 repeat f.i := i; f\bound{f1 }\free{f }}
+\indentrel{3}\begin{verbatim}
+   (3)  [1,2,3,4,5,6]
+                            Type: FlexibleArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty3}
+\begin{paste}{FlexibleArrayXmpPageEmpty3}{FlexibleArrayXmpPagePatch3}
+\pastebutton{FlexibleArrayXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{for i in 1..6 repeat f.i := i; f\bound{f1 }\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch4}
+\begin{paste}{FlexibleArrayXmpPageFull4}{FlexibleArrayXmpPageEmpty4}
+\pastebutton{FlexibleArrayXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{physicalLength f\free{f1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  6
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty4}
+\begin{paste}{FlexibleArrayXmpPageEmpty4}{FlexibleArrayXmpPagePatch4}
+\pastebutton{FlexibleArrayXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{physicalLength f\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch5}
+\begin{paste}{FlexibleArrayXmpPageFull5}{FlexibleArrayXmpPageEmpty5}
+\pastebutton{FlexibleArrayXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{concat!(f,11)\bound{f2 }\free{f1 }}
+\indentrel{3}\begin{verbatim}
+   (5)  [1,2,3,4,5,6,11]
+                            Type: FlexibleArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty5}
+\begin{paste}{FlexibleArrayXmpPageEmpty5}{FlexibleArrayXmpPagePatch5}
+\pastebutton{FlexibleArrayXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{concat!(f,11)\bound{f2 }\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch6}
+\begin{paste}{FlexibleArrayXmpPageFull6}{FlexibleArrayXmpPageEmpty6}
+\pastebutton{FlexibleArrayXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{physicalLength f\free{f2 }}
+\indentrel{3}\begin{verbatim}
+   (6)  10
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty6}
+\begin{paste}{FlexibleArrayXmpPageEmpty6}{FlexibleArrayXmpPagePatch6}
+\pastebutton{FlexibleArrayXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{physicalLength f\free{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch7}
+\begin{paste}{FlexibleArrayXmpPageFull7}{FlexibleArrayXmpPageEmpty7}
+\pastebutton{FlexibleArrayXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{physicalLength!(f,15)\bound{f3 }\free{f2 }}
+\indentrel{3}\begin{verbatim}
+   (7)  [1,2,3,4,5,6,11]
+                            Type: FlexibleArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty7}
+\begin{paste}{FlexibleArrayXmpPageEmpty7}{FlexibleArrayXmpPagePatch7}
+\pastebutton{FlexibleArrayXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{physicalLength!(f,15)\bound{f3 }\free{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch8}
+\begin{paste}{FlexibleArrayXmpPageFull8}{FlexibleArrayXmpPageEmpty8}
+\pastebutton{FlexibleArrayXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{concat!(f,f)\bound{f4 }\free{f3 }}
+\indentrel{3}\begin{verbatim}
+   (8)  [1,2,3,4,5,6,11,1,2,3,4,5,6,11]
+                            Type: FlexibleArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty8}
+\begin{paste}{FlexibleArrayXmpPageEmpty8}{FlexibleArrayXmpPagePatch8}
+\pastebutton{FlexibleArrayXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{concat!(f,f)\bound{f4 }\free{f3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch9}
+\begin{paste}{FlexibleArrayXmpPageFull9}{FlexibleArrayXmpPageEmpty9}
+\pastebutton{FlexibleArrayXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{insert!(22,f,1)\bound{f5 }\free{f4 }}
+\indentrel{3}\begin{verbatim}
+   (9)  [22,1,2,3,4,5,6,11,1,2,3,4,5,6,11]
+                            Type: FlexibleArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty9}
+\begin{paste}{FlexibleArrayXmpPageEmpty9}{FlexibleArrayXmpPagePatch9}
+\pastebutton{FlexibleArrayXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{insert!(22,f,1)\bound{f5 }\free{f4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch10}
+\begin{paste}{FlexibleArrayXmpPageFull10}{FlexibleArrayXmpPageEmpty10}
+\pastebutton{FlexibleArrayXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{g := f(10..)\bound{g }\free{f5 }}
+\indentrel{3}\begin{verbatim}
+   (10)  [2,3,4,5,6,11]
+                            Type: FlexibleArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty10}
+\begin{paste}{FlexibleArrayXmpPageEmpty10}{FlexibleArrayXmpPagePatch10}
+\pastebutton{FlexibleArrayXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{g := f(10..)\bound{g }\free{f5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch11}
+\begin{paste}{FlexibleArrayXmpPageFull11}{FlexibleArrayXmpPageEmpty11}
+\pastebutton{FlexibleArrayXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{insert!(g,f,1)\bound{g1 }\free{g f5 }}
+\indentrel{3}\begin{verbatim}
+   (11)  [2,3,4,5,6,11,22,1,2,3,4,5,6,11,1,2,3,4,5,6,11]
+                            Type: FlexibleArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty11}
+\begin{paste}{FlexibleArrayXmpPageEmpty11}{FlexibleArrayXmpPagePatch11}
+\pastebutton{FlexibleArrayXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{insert!(g,f,1)\bound{g1 }\free{g f5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch12}
+\begin{paste}{FlexibleArrayXmpPageFull12}{FlexibleArrayXmpPageEmpty12}
+\pastebutton{FlexibleArrayXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{merge!(sort! f, sort! g)\bound{f6 }\free{g f5 }}
+\indentrel{3}\begin{verbatim}
+   (12)
+   [1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5,
+    6, 6, 6, 6, 11, 11, 11, 11, 22]
+                            Type: FlexibleArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty12}
+\begin{paste}{FlexibleArrayXmpPageEmpty12}{FlexibleArrayXmpPagePatch12}
+\pastebutton{FlexibleArrayXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{merge!(sort! f, sort! g)\bound{f6 }\free{g f5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch13}
+\begin{paste}{FlexibleArrayXmpPageFull13}{FlexibleArrayXmpPageEmpty13}
+\pastebutton{FlexibleArrayXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{removeDuplicates! f\bound{f7 }\free{f6 }}
+\indentrel{3}\begin{verbatim}
+   (13)  [1,2,3,4,5,6,11,22]
+                            Type: FlexibleArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty13}
+\begin{paste}{FlexibleArrayXmpPageEmpty13}{FlexibleArrayXmpPagePatch13}
+\pastebutton{FlexibleArrayXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{removeDuplicates! f\bound{f7 }\free{f6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch14}
+\begin{paste}{FlexibleArrayXmpPageFull14}{FlexibleArrayXmpPageEmpty14}
+\pastebutton{FlexibleArrayXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{select!(i +-> even? i,f)\bound{f8 }\free{f7 }}
+\indentrel{3}\begin{verbatim}
+   (14)  [2,4,6,22]
+                            Type: FlexibleArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty14}
+\begin{paste}{FlexibleArrayXmpPageEmpty14}{FlexibleArrayXmpPagePatch14}
+\pastebutton{FlexibleArrayXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{select!(i +-> even? i,f)\bound{f8 }\free{f7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch15}
+\begin{paste}{FlexibleArrayXmpPageFull15}{FlexibleArrayXmpPageEmpty15}
+\pastebutton{FlexibleArrayXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{physicalLength f\free{b8 }}
+\indentrel{3}\begin{verbatim}
+   (15)  8
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty15}
+\begin{paste}{FlexibleArrayXmpPageEmpty15}{FlexibleArrayXmpPagePatch15}
+\pastebutton{FlexibleArrayXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{physicalLength f\free{b8 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPagePatch16}
+\begin{paste}{FlexibleArrayXmpPageFull16}{FlexibleArrayXmpPageEmpty16}
+\pastebutton{FlexibleArrayXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{shrinkable(false)$FlexibleArray(Integer)}
+\indentrel{3}\begin{verbatim}
+   (16)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FlexibleArrayXmpPageEmpty16}
+\begin{paste}{FlexibleArrayXmpPageEmpty16}{FlexibleArrayXmpPagePatch16}
+\pastebutton{FlexibleArrayXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{shrinkable(false)$FlexibleArray(Integer)}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/FILE.ht b/src/hyper/pages/FILE.ht
new file mode 100644
index 00000000..93fa12b6
--- /dev/null
+++ b/src/hyper/pages/FILE.ht
@@ -0,0 +1,103 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\FileXmpTitle}{File}
+\newcommand{\FileXmpNumber}{9.24}
+%
+% =====================================================================
+\begin{page}{FileXmpPage}{9.24 File}
+% =====================================================================
+\beginscroll
+
+The \spadtype{File(S)} domain provides a basic interface to read and
+write values of type \spad{S} in files.
+\xtc{
+Before working with a file, it must be made accessible to \Language{} with
+the \spadfunFrom{open}{File} operation.
+}{
+\spadpaste{ifile:File List Integer:=open("/tmp/jazz1","output")  \bound{ifile}}
+}
+The \spadfunFrom{open}{File} function arguments are a \spadtype{FileName}
+and a \spadtype{String} specifying the mode.
+If a full pathname is not specified, the current default directory is
+assumed.
+The mode must be one of \spad{"input"} or \spad{"output"}.
+If it is not specified, \spad{"input"} is assumed.
+Once the file has been opened, you can read or write data.
+\xtc{
+The operations \spadfunFromX{read}{File} and \spadfunFromX{write}{File} are
+provided.
+}{
+\spadpaste{write!(ifile, [-1,2,3])        \free{ifile}\bound{ifile1}}
+}
+\xtc{
+}{
+\spadpaste{write!(ifile, [10,-10,0,111])  \free{ifile1}\bound{ifile2}}
+}
+\xtc{
+}{
+\spadpaste{write!(ifile, [7])             \free{ifile2}\bound{ifile3}}
+}
+\xtc{
+You can change from writing to reading (or vice versa)
+by reopening a file.
+}{
+\spadpaste{reopen!(ifile, "input")        \free{ifile3}\bound{ifile4}}
+}
+\xtc{
+}{
+\spadpaste{read! ifile                    \free{ifile4}\bound{ifile5}}
+}
+\xtc{
+}{
+\spadpaste{read! ifile                    \free{ifile5}\bound{ifile6}}
+}
+\xtc{
+The \spadfunFromX{read}{File} operation can cause an error if one tries to
+read more data than is in the file.
+To guard against this possibility the \spadfunFromX{readIfCan}{File}
+operation should be used.
+}{
+\spadpaste{readIfCan! ifile  \free{ifile6}\bound{ifile7}}
+}
+\xtc{
+}{
+\spadpaste{readIfCan! ifile  \free{ifile7}\bound{ifile8}}
+}
+\xtc{
+You can find the current mode of the file, and the file's name.
+}{
+\spadpaste{iomode ifile           \free{ifile}}
+}
+\xtc{
+}{
+\spadpaste{name ifile             \free{ifile}}
+}
+\xtc{
+When you are finished with a file, you should close it.
+}{
+\spadpaste{close! ifile           \free{ifile}\bound{ifileA}}
+}
+\noOutputXtc{
+}{
+\spadpaste{)system rm /tmp/jazz1  \free{ifileA}}
+}
+%\xtc{
+%}{
+%\spadcommand{)clear all  \free{}\bound{}}
+%}
+
+A limitation of the underlying LISP system is that not all values can be
+represented in a file.
+In particular, delayed values containing compiled functions cannot be
+saved.
+
+For more information on related topics, see
+\downlink{`TextFile'}{TextFileXmpPage}\ignore{TextFile},
+\downlink{`KeyedAccessFile'}{KeyedAccessFileXmpPage}\ignore{KeyedAccessFile},
+\downlink{`Library'}{LibraryXmpPage}\ignore{Library}, and
+\downlink{`FileName'}{FileNameXmpPage}\ignore{FileName}.
+\showBlurb{File}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/FILE.pht b/src/hyper/pages/FILE.pht
new file mode 100644
index 00000000..19cd4b2d
--- /dev/null
+++ b/src/hyper/pages/FILE.pht
@@ -0,0 +1,206 @@
+\begin{patch}{FileXmpPagePatch1}
+\begin{paste}{FileXmpPageFull1}{FileXmpPageEmpty1}
+\pastebutton{FileXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{ifile:File List Integer:=open("/tmp/jazz1","output")\bound{ifile }}
+\indentrel{3}\begin{verbatim}
+   (1)  "/tmp/jazz1"
+                                Type: File List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPageEmpty1}
+\begin{paste}{FileXmpPageEmpty1}{FileXmpPagePatch1}
+\pastebutton{FileXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{ifile:File List Integer:=open("/tmp/jazz1","output")\bound{ifile }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPagePatch2}
+\begin{paste}{FileXmpPageFull2}{FileXmpPageEmpty2}
+\pastebutton{FileXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{write!(ifile, [-1,2,3])\free{ifile }\bound{ifile1 }}
+\indentrel{3}\begin{verbatim}
+   (2)  [- 1,2,3]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPageEmpty2}
+\begin{paste}{FileXmpPageEmpty2}{FileXmpPagePatch2}
+\pastebutton{FileXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{write!(ifile, [-1,2,3])\free{ifile }\bound{ifile1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPagePatch3}
+\begin{paste}{FileXmpPageFull3}{FileXmpPageEmpty3}
+\pastebutton{FileXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{write!(ifile, [10,-10,0,111])\free{ifile1 }\bound{ifile2 }}
+\indentrel{3}\begin{verbatim}
+   (3)  [10,- 10,0,111]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPageEmpty3}
+\begin{paste}{FileXmpPageEmpty3}{FileXmpPagePatch3}
+\pastebutton{FileXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{write!(ifile, [10,-10,0,111])\free{ifile1 }\bound{ifile2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPagePatch4}
+\begin{paste}{FileXmpPageFull4}{FileXmpPageEmpty4}
+\pastebutton{FileXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{write!(ifile, [7])\free{ifile2 }\bound{ifile3 }}
+\indentrel{3}\begin{verbatim}
+   (4)  [7]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPageEmpty4}
+\begin{paste}{FileXmpPageEmpty4}{FileXmpPagePatch4}
+\pastebutton{FileXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{write!(ifile, [7])\free{ifile2 }\bound{ifile3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPagePatch5}
+\begin{paste}{FileXmpPageFull5}{FileXmpPageEmpty5}
+\pastebutton{FileXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{reopen!(ifile, "input")\free{ifile3 }\bound{ifile4 }}
+\indentrel{3}\begin{verbatim}
+   (5)  "/tmp/jazz1"
+                                Type: File List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPageEmpty5}
+\begin{paste}{FileXmpPageEmpty5}{FileXmpPagePatch5}
+\pastebutton{FileXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{reopen!(ifile, "input")\free{ifile3 }\bound{ifile4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPagePatch6}
+\begin{paste}{FileXmpPageFull6}{FileXmpPageEmpty6}
+\pastebutton{FileXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{read! ifile\free{ifile4 }\bound{ifile5 }}
+\indentrel{3}\begin{verbatim}
+   (6)  [- 1,2,3]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPageEmpty6}
+\begin{paste}{FileXmpPageEmpty6}{FileXmpPagePatch6}
+\pastebutton{FileXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{read! ifile\free{ifile4 }\bound{ifile5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPagePatch7}
+\begin{paste}{FileXmpPageFull7}{FileXmpPageEmpty7}
+\pastebutton{FileXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{read! ifile\free{ifile5 }\bound{ifile6 }}
+\indentrel{3}\begin{verbatim}
+   (7)  [10,- 10,0,111]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPageEmpty7}
+\begin{paste}{FileXmpPageEmpty7}{FileXmpPagePatch7}
+\pastebutton{FileXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{read! ifile\free{ifile5 }\bound{ifile6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPagePatch8}
+\begin{paste}{FileXmpPageFull8}{FileXmpPageEmpty8}
+\pastebutton{FileXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{readIfCan! ifile\free{ifile6 }\bound{ifile7 }}
+\indentrel{3}\begin{verbatim}
+   (8)  [7]
+                          Type: Union(List Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPageEmpty8}
+\begin{paste}{FileXmpPageEmpty8}{FileXmpPagePatch8}
+\pastebutton{FileXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{readIfCan! ifile\free{ifile6 }\bound{ifile7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPagePatch9}
+\begin{paste}{FileXmpPageFull9}{FileXmpPageEmpty9}
+\pastebutton{FileXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{readIfCan! ifile\free{ifile7 }\bound{ifile8 }}
+\indentrel{3}\begin{verbatim}
+   (9)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPageEmpty9}
+\begin{paste}{FileXmpPageEmpty9}{FileXmpPagePatch9}
+\pastebutton{FileXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{readIfCan! ifile\free{ifile7 }\bound{ifile8 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPagePatch10}
+\begin{paste}{FileXmpPageFull10}{FileXmpPageEmpty10}
+\pastebutton{FileXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{iomode ifile\free{ifile }}
+\indentrel{3}\begin{verbatim}
+   (10)  "input"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPageEmpty10}
+\begin{paste}{FileXmpPageEmpty10}{FileXmpPagePatch10}
+\pastebutton{FileXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{iomode ifile\free{ifile }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPagePatch11}
+\begin{paste}{FileXmpPageFull11}{FileXmpPageEmpty11}
+\pastebutton{FileXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{name ifile\free{ifile }}
+\indentrel{3}\begin{verbatim}
+   (11)  "/tmp/jazz1"
+                                         Type: FileName
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPageEmpty11}
+\begin{paste}{FileXmpPageEmpty11}{FileXmpPagePatch11}
+\pastebutton{FileXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{name ifile\free{ifile }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPagePatch12}
+\begin{paste}{FileXmpPageFull12}{FileXmpPageEmpty12}
+\pastebutton{FileXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{close! ifile\free{ifile }\bound{ifileA }}
+\indentrel{3}\begin{verbatim}
+   (12)  "/tmp/jazz1"
+                                Type: File List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPageEmpty12}
+\begin{paste}{FileXmpPageEmpty12}{FileXmpPagePatch12}
+\pastebutton{FileXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{close! ifile\free{ifile }\bound{ifileA }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPagePatch13}
+\begin{paste}{FileXmpPageFull13}{FileXmpPageEmpty13}
+\pastebutton{FileXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{)system rm /tmp/jazz1\free{ifileA }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileXmpPageEmpty13}
+\begin{paste}{FileXmpPageEmpty13}{FileXmpPagePatch13}
+\pastebutton{FileXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{)system rm /tmp/jazz1\free{ifileA }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/FLOAT.ht b/src/hyper/pages/FLOAT.ht
new file mode 100644
index 00000000..82879931
--- /dev/null
+++ b/src/hyper/pages/FLOAT.ht
@@ -0,0 +1,336 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\FloatXmpTitle}{Float}
+\newcommand{\FloatXmpNumber}{9.27}
+%
+% =====================================================================
+\begin{page}{FloatXmpPage}{9.27 Float}
+% =====================================================================
+\beginscroll
+%
+
+\Language{} provides two kinds of floating point numbers.
+The domain \spadtype{Float} (abbreviation \spadtype{FLOAT})
+implements a model of arbitrary
+%-% \HDindex{floating point!arbitrary precision}{FloatXmpPage}{9.27}{Float}
+precision floating point numbers.
+The domain \spadtype{DoubleFloat} (abbreviation \spadtype{DFLOAT})
+is intended to make available
+%-% \HDindex{floating point!hardware}{FloatXmpPage}{9.27}{Float}
+hardware floating point arithmetic in \Language{}.
+%-% \HDexptypeindex{DoubleFloat}{FloatXmpPage}{9.27}{Float}
+The actual model of floating point that \spadtype{DoubleFloat} provides is
+system-dependent.
+For example, on the IBM system 370 \Language{} uses IBM double
+precision which has fourteen hexadecimal digits of precision or roughly
+sixteen decimal digits.
+Arbitrary precision floats allow the user to specify the
+precision at which arithmetic operations are computed.
+Although this is an attractive facility, it comes at a cost.
+Arbitrary-precision floating-point arithmetic typically takes
+twenty to two hundred times more time than hardware floating point.
+
+For more information about \Language{}'s numeric and graphic
+facilities,  see
+\downlink{``\ugGraphTitle''}{ugGraphPage} in Section \ugGraphNumber\ignore{ugGraph},
+\downlink{``\ugProblemNumericTitle''}{ugProblemNumericPage} in Section \ugProblemNumericNumber\ignore{ugProblemNumeric}, and
+\downlink{`DoubleFloat'}{DoubleFloatXmpPage}\ignore{DoubleFloat}.
+
+\beginmenu
+    \menudownlink{{9.27.1. Introduction to Float}}{ugxFloatIntroPage}
+    \menudownlink{{9.27.2. Conversion Functions}}{ugxFloatConvertPage}
+    \menudownlink{{9.27.3. Output Functions}}{ugxFloatOutputPage}
+    \menudownlink{{9.27.4. An Example: Determinant of a Hilbert Matrix}}{ugxFloatHilbertPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFloatIntroTitle}{Introduction to Float}
+\newcommand{\ugxFloatIntroNumber}{9.27.1.}
+%
+% =====================================================================
+\begin{page}{ugxFloatIntroPage}{9.27.1. Introduction to Float}
+% =====================================================================
+\beginscroll
+
+Scientific notation is supported for input and output
+%-% \HDindex{floating point!input}{ugxFloatIntroPage}{9.27.1.}{Introduction to Float}
+of floating point numbers.
+%-% \HDindex{scientific notation}{ugxFloatIntroPage}{9.27.1.}{Introduction to Float}
+A floating point number is written as a string of digits containing a
+decimal point optionally followed by the letter ``{\tt E}'', and then
+the exponent.
+\xtc{
+We begin by doing some calculations using arbitrary precision floats.
+The default precision is twenty decimal digits.
+}{
+\spadpaste{1.234}
+}
+\xtc{
+A decimal base for the exponent is assumed, so the number
+\spad{1.234E2}  denotes
+\texht{$1.234 \cdot  10^2$}{\spad{1.234 * 10**2}}.
+}{
+\spadpaste{1.234E2}
+}
+\xtc{
+The normal arithmetic operations are available for floating point
+numbers.
+}{
+\spadpaste{sqrt(1.2 + 2.3 / 3.4 ** 4.5)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFloatConvertTitle}{Conversion Functions}
+\newcommand{\ugxFloatConvertNumber}{9.27.2.}
+%
+% =====================================================================
+\begin{page}{ugxFloatConvertPage}{9.27.2. Conversion Functions}
+% =====================================================================
+\beginscroll
+
+\labelSpace{3pc}
+\xtc{
+You can use conversion
+(\downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert})
+to go back and forth between \spadtype{Integer}, \spadtype{Fraction Integer}
+and \spadtype{Float}, as appropriate.
+}{
+\spadpaste{i := 3 :: Float \bound{i}}
+}
+\xtc{
+}{
+\spadpaste{i :: Integer \free{i}}
+}
+\xtc{
+}{
+\spadpaste{i :: Fraction Integer \free{i}}
+}
+\xtc{
+Since you are explicitly asking for a conversion, you must take
+responsibility for any loss of exactness.
+}{
+\spadpaste{r := 3/7 :: Float \bound{r}}
+}
+\xtc{
+}{
+\spadpaste{r :: Fraction Integer \free{r}}
+}
+\xtc{
+This conversion cannot be performed: use
+\spadfunFrom{truncate}{Float} or \spadfunFrom{round}{Float} if that
+is what you intend.
+}{
+\spadpaste{r :: Integer \free{r}}
+}
+
+\xtc{
+The operations \spadfunFrom{truncate}{Float} and \spadfunFrom{round}{Float}
+truncate  \ldots
+}{
+\spadpaste{truncate 3.6}
+}
+\xtc{
+and round
+to the nearest integral \spadtype{Float} respectively.
+}{
+\spadpaste{round 3.6}
+}
+\xtc{
+}{
+\spadpaste{truncate(-3.6)}
+}
+\xtc{
+}{
+\spadpaste{round(-3.6)}
+}
+\xtc{
+The operation \spadfunFrom{fractionPart}{Float} computes the fractional part of
+\spad{x}, that is, \spad{x - truncate x}.
+}{
+\spadpaste{fractionPart 3.6}
+}
+\xtc{
+The operation \spadfunFrom{digits}{Float} allows the user to set the
+precision.
+It returns the previous value it was using.
+}{
+\spadpaste{digits 40 \bound{d40}}
+}
+\xtc{
+}{
+\spadpaste{sqrt 0.2}
+}
+\xtc{
+}{
+\spadpaste{pi()\$Float \free{d40}}
+}
+\xtc{
+The precision is only limited by the computer memory available.
+Calculations at 500 or more digits of precision are not difficult.
+}{
+\spadpaste{digits 500 \bound{d1000}}
+}
+\xtc{
+}{
+\spadpaste{pi()\$Float \free{d1000}}
+}
+\xtc{
+Reset \spadfunFrom{digits}{Float} to its default value.
+}{
+\spadpaste{digits 20}
+}
+Numbers of type \spadtype{Float} are represented as a record of two
+integers, namely, the mantissa and the exponent where the base of the
+exponent is binary.
+That is, the floating point number \spad{(m,e)} represents the number
+\texht{$m \cdot 2^e$}{\spad{m * 2**e}}.
+A consequence of using a binary base is that decimal numbers can not, in
+general, be represented exactly.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFloatOutputTitle}{Output Functions}
+\newcommand{\ugxFloatOutputNumber}{9.27.3.}
+%
+% =====================================================================
+\begin{page}{ugxFloatOutputPage}{9.27.3. Output Functions}
+% =====================================================================
+\beginscroll
+
+A number of operations exist for specifying how numbers of type \spadtype{Float} are to be
+%-% \HDindex{floating point!output}{ugxFloatOutputPage}{9.27.3.}{Output Functions}
+displayed.
+By default, spaces are inserted every ten digits in the
+output for readability.\footnote{Note that you cannot include spaces
+in the input form of a floating point number, though you can use
+underscores.}
+
+
+\xtc{
+Output spacing can be modified with the \spadfunFrom{outputSpacing}{Float}
+operation.
+This inserts no spaces and then displays the value of \spad{x}.
+}{
+\spadpaste{outputSpacing 0; x := sqrt 0.2 \bound{x}\bound{os0}}
+}
+\xtc{
+Issue this to have the spaces inserted every \spad{5} digits.
+}{
+\spadpaste{outputSpacing 5; x \bound{os5}\free{x}}
+}
+\xtc{
+By default, the system displays floats in either fixed format
+or scientific format, depending on the magnitude of the number.
+}{
+\spadpaste{y := x/10**10 \bound{y}\free{x os5}}
+}
+\xtc{
+A particular format may be requested with the operations
+\spadfunFrom{outputFloating}{Float} and \spadfunFrom{outputFixed}{Float}.
+}{
+\spadpaste{outputFloating(); x \bound{of} \free{os5 x}}
+}
+\xtc{
+}{
+\spadpaste{outputFixed(); y \bound{ox} \free{os5 y}}
+}
+\xtc{
+Additionally, you can ask for \spad{n} digits to be displayed after the
+decimal point.
+}{
+\spadpaste{outputFloating 2; y \bound{of2} \free{os5 y}}
+}
+\xtc{
+}{
+\spadpaste{outputFixed 2; x \bound{ox2} \free{os5 x}}
+}
+\xtc{
+This resets the output printing to the default behavior.
+}{
+\spadpaste{outputGeneral()}
+}
+%
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFloatHilbertTitle}{An Example: Determinant of a Hilbert Matrix}
+\newcommand{\ugxFloatHilbertNumber}{9.27.4.}
+%
+% =====================================================================
+\begin{page}{ugxFloatHilbertPage}{9.27.4. An Example: Determinant of a Hilbert Matrix}
+% =====================================================================
+\beginscroll
+
+Consider the problem of computing the determinant of a \spad{10} by \spad{10}
+%-% \HDindex{Hilbert matrix}{ugxFloatHilbertPage}{9.27.4.}{An Example: Determinant of a Hilbert Matrix}
+Hilbert matrix.
+The \eth{\spad{(i,j)}} entry of a Hilbert matrix is given by
+\spad{1/(i+j+1)}.
+
+\labelSpace{2pc}
+\xtc{
+First do the computation using rational numbers to obtain the
+exact result.
+}{
+\spadpaste{a: Matrix Fraction Integer := matrix [[1/(i+j+1) for j in 0..9] for i in 0..9] \bound{a}}
+}
+\xtc{
+This version of \spadfunFrom{determinant}{Matrix} uses Gaussian
+elimination.
+}{
+\spadpaste{d:= determinant a \free{a}\bound{d}}
+}
+\xtc{
+}{
+\spadpaste{d :: Float \free{d}}
+}
+\xtc{
+Now use hardware floats. Note that a semicolon (;) is used
+to prevent the display of the matrix.
+}{
+\spadpaste{b: Matrix DoubleFloat := matrix [[1/(i+j+1\$DoubleFloat) for j in 0..9] for i in 0..9]; \bound{b}}
+}
+\xtc{
+The result given by hardware floats is correct only to four significant
+digits of precision.
+In the jargon of numerical analysis, the Hilbert matrix is said to be
+``ill-conditioned.''
+%-% \HDindex{matrix!ill-conditioned}{ugxFloatHilbertPage}{9.27.4.}{An Example: Determinant of a Hilbert Matrix}
+}{
+\spadpaste{determinant b \free{b}}
+}
+\xtc{
+Now repeat the computation at a higher precision using \spadtype{Float}.
+}{
+\spadpaste{digits 40 \bound{d40}}
+}
+\xtc{
+}{
+\spadpaste{c: Matrix Float := matrix [[1/(i+j+1\$Float) for j in 0..9] for i in 0..9]; \free{d40} \bound{c}}
+}
+\xtc{
+}{
+\spadpaste{determinant c \free{c}}
+}
+\xtc{
+Reset \spadfunFrom{digits}{Float} to its default value.
+}{
+\spadpaste{digits 20}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/FLOAT.pht b/src/hyper/pages/FLOAT.pht
new file mode 100644
index 00000000..d1181af9
--- /dev/null
+++ b/src/hyper/pages/FLOAT.pht
@@ -0,0 +1,645 @@
+\begin{patch}{ugxFloatIntroPagePatch1}
+\begin{paste}{ugxFloatIntroPageFull1}{ugxFloatIntroPageEmpty1}
+\pastebutton{ugxFloatIntroPageFull1}{\hidepaste}
+\tab{5}\spadcommand{1.234}
+\indentrel{3}\begin{verbatim}
+   (1)  1.234
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatIntroPageEmpty1}
+\begin{paste}{ugxFloatIntroPageEmpty1}{ugxFloatIntroPagePatch1}
+\pastebutton{ugxFloatIntroPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{1.234}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatIntroPagePatch2}
+\begin{paste}{ugxFloatIntroPageFull2}{ugxFloatIntroPageEmpty2}
+\pastebutton{ugxFloatIntroPageFull2}{\hidepaste}
+\tab{5}\spadcommand{1.234E2}
+\indentrel{3}\begin{verbatim}
+   (2)  123.4
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatIntroPageEmpty2}
+\begin{paste}{ugxFloatIntroPageEmpty2}{ugxFloatIntroPagePatch2}
+\pastebutton{ugxFloatIntroPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{1.234E2}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatIntroPagePatch3}
+\begin{paste}{ugxFloatIntroPageFull3}{ugxFloatIntroPageEmpty3}
+\pastebutton{ugxFloatIntroPageFull3}{\hidepaste}
+\tab{5}\spadcommand{sqrt(1.2 + 2.3 / 3.4 ** 4.5)}
+\indentrel{3}\begin{verbatim}
+   (3)  1.0996972790 671286226
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatIntroPageEmpty3}
+\begin{paste}{ugxFloatIntroPageEmpty3}{ugxFloatIntroPagePatch3}
+\pastebutton{ugxFloatIntroPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{sqrt(1.2 + 2.3 / 3.4 ** 4.5)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPagePatch1}
+\begin{paste}{ugxFloatOutputPageFull1}{ugxFloatOutputPageEmpty1}
+\pastebutton{ugxFloatOutputPageFull1}{\hidepaste}
+\tab{5}\spadcommand{outputSpacing 0; x := sqrt 0.2\bound{x }\bound{os0 }}
+\indentrel{3}\begin{verbatim}
+   (1)  0.44721359549995793928
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPageEmpty1}
+\begin{paste}{ugxFloatOutputPageEmpty1}{ugxFloatOutputPagePatch1}
+\pastebutton{ugxFloatOutputPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{outputSpacing 0; x := sqrt 0.2\bound{x }\bound{os0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPagePatch2}
+\begin{paste}{ugxFloatOutputPageFull2}{ugxFloatOutputPageEmpty2}
+\pastebutton{ugxFloatOutputPageFull2}{\hidepaste}
+\tab{5}\spadcommand{outputSpacing 5; x\bound{os5 }\free{x }}
+\indentrel{3}\begin{verbatim}
+   (2)  0.44721 35954 99957 93928
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPageEmpty2}
+\begin{paste}{ugxFloatOutputPageEmpty2}{ugxFloatOutputPagePatch2}
+\pastebutton{ugxFloatOutputPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{outputSpacing 5; x\bound{os5 }\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPagePatch3}
+\begin{paste}{ugxFloatOutputPageFull3}{ugxFloatOutputPageEmpty3}
+\pastebutton{ugxFloatOutputPageFull3}{\hidepaste}
+\tab{5}\spadcommand{y := x/10**10\bound{y }\free{x os5 }}
+\indentrel{3}\begin{verbatim}
+   (3)  0.44721 35954 99957 93928 E -10
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPageEmpty3}
+\begin{paste}{ugxFloatOutputPageEmpty3}{ugxFloatOutputPagePatch3}
+\pastebutton{ugxFloatOutputPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{y := x/10**10\bound{y }\free{x os5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPagePatch4}
+\begin{paste}{ugxFloatOutputPageFull4}{ugxFloatOutputPageEmpty4}
+\pastebutton{ugxFloatOutputPageFull4}{\hidepaste}
+\tab{5}\spadcommand{outputFloating(); x\bound{of }\free{os5 x }}
+\indentrel{3}\begin{verbatim}
+   (4)  0.44721 35954 99957 93928 E 0
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPageEmpty4}
+\begin{paste}{ugxFloatOutputPageEmpty4}{ugxFloatOutputPagePatch4}
+\pastebutton{ugxFloatOutputPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{outputFloating(); x\bound{of }\free{os5 x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPagePatch5}
+\begin{paste}{ugxFloatOutputPageFull5}{ugxFloatOutputPageEmpty5}
+\pastebutton{ugxFloatOutputPageFull5}{\hidepaste}
+\tab{5}\spadcommand{outputFixed(); y\bound{ox }\free{os5 y }}
+\indentrel{3}\begin{verbatim}
+   (5)  0.00000 00000 44721 35954 99957 93928
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPageEmpty5}
+\begin{paste}{ugxFloatOutputPageEmpty5}{ugxFloatOutputPagePatch5}
+\pastebutton{ugxFloatOutputPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{outputFixed(); y\bound{ox }\free{os5 y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPagePatch6}
+\begin{paste}{ugxFloatOutputPageFull6}{ugxFloatOutputPageEmpty6}
+\pastebutton{ugxFloatOutputPageFull6}{\hidepaste}
+\tab{5}\spadcommand{outputFloating 2; y\bound{of2 }\free{os5 y }}
+\indentrel{3}\begin{verbatim}
+   (6)  0.45 E -10
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPageEmpty6}
+\begin{paste}{ugxFloatOutputPageEmpty6}{ugxFloatOutputPagePatch6}
+\pastebutton{ugxFloatOutputPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{outputFloating 2; y\bound{of2 }\free{os5 y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPagePatch7}
+\begin{paste}{ugxFloatOutputPageFull7}{ugxFloatOutputPageEmpty7}
+\pastebutton{ugxFloatOutputPageFull7}{\hidepaste}
+\tab{5}\spadcommand{outputFixed 2; x\bound{ox2 }\free{os5 x }}
+\indentrel{3}\begin{verbatim}
+   (7)  0.45
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPageEmpty7}
+\begin{paste}{ugxFloatOutputPageEmpty7}{ugxFloatOutputPagePatch7}
+\pastebutton{ugxFloatOutputPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{outputFixed 2; x\bound{ox2 }\free{os5 x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPagePatch8}
+\begin{paste}{ugxFloatOutputPageFull8}{ugxFloatOutputPageEmpty8}
+\pastebutton{ugxFloatOutputPageFull8}{\hidepaste}
+\tab{5}\spadcommand{outputGeneral()}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatOutputPageEmpty8}
+\begin{paste}{ugxFloatOutputPageEmpty8}{ugxFloatOutputPagePatch8}
+\pastebutton{ugxFloatOutputPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{outputGeneral()}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPagePatch1}
+\begin{paste}{ugxFloatHilbertPageFull1}{ugxFloatHilbertPageEmpty1}
+\pastebutton{ugxFloatHilbertPageFull1}{\hidepaste}
+\tab{5}\spadcommand{a: Matrix Fraction Integer := matrix [[1/(i+j+1) for j in 0..9] for i in 0..9]\bound{a }}
+\indentrel{3}\begin{verbatim}
+        Ú    1   1   1   1   1   1   1   1    1¿
+        ³1   Ä   Ä   Ä   Ä   Ä   Ä   Ä   Ä   Äij
+        ³    2   3   4   5   6   7   8   9   10³
+        ³                                      ³
+        ³1   1   1   1   1   1   1   1    1   1³
+        ³Ä   Ä   Ä   Ä   Ä   Ä   Ä   Ä   ÄÄ  Äij
+        ³2   3   4   5   6   7   8   9   10  11³
+        ³                                      ³
+        ³1   1   1   1   1   1   1    1   1   1³
+        ³Ä   Ä   Ä   Ä   Ä   Ä   Ä   ÄÄ  ÄÄ  Äij
+        ³3   4   5   6   7   8   9   10  11  12³
+        ³                                      ³
+        ³1   1   1   1   1   1    1   1   1   1³
+        ³Ä   Ä   Ä   Ä   Ä   Ä   ÄÄ  ÄÄ  ÄÄ  Äij
+        ³4   5   6   7   8   9   10  11  12  13³
+        ³                                      ³
+        ³1   1   1   1   1    1   1   1   1   1³
+        ³Ä   Ä   Ä   Ä   Ä   ÄÄ  ÄÄ  ÄÄ  ÄÄ  Äij
+        ³5   6   7   8   9   10  11  12  13  14³
+   (1)  ³                                      ³
+        ³1   1   1   1    1   1   1   1   1   1³
+        ³Ä   Ä   Ä   Ä   ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  Äij
+        ³6   7   8   9   10  11  12  13  14  15³
+        ³                                      ³
+        ³1   1   1    1   1   1   1   1   1   1³
+        ³Ä   Ä   Ä   ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  Äij
+        ³7   8   9   10  11  12  13  14  15  16³
+        ³                                      ³
+        ³1   1    1   1   1   1   1   1   1   1³
+        ³Ä   Ä   ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  Äij
+        ³8   9   10  11  12  13  14  15  16  17³
+        ³                                      ³
+        ³1    1   1   1   1   1   1   1   1   1³
+        ³Ä   ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  Äij
+        ³9   10  11  12  13  14  15  16  17  18³
+        ³                                      ³
+        ³ 1   1   1   1   1   1   1   1   1   1³
+        ³ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  ÄÄ  Äij
+        À10  11  12  13  14  15  16  17  18  19Ù
+                          Type: Matrix Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPageEmpty1}
+\begin{paste}{ugxFloatHilbertPageEmpty1}{ugxFloatHilbertPagePatch1}
+\pastebutton{ugxFloatHilbertPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{a: Matrix Fraction Integer := matrix [[1/(i+j+1) for j in 0..9] for i in 0..9]\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPagePatch2}
+\begin{paste}{ugxFloatHilbertPageFull2}{ugxFloatHilbertPageEmpty2}
+\pastebutton{ugxFloatHilbertPageFull2}{\hidepaste}
+\tab{5}\spadcommand{d:= determinant a\free{a }\bound{d }}
+\indentrel{3}\begin{verbatim}
+   (2)
+                             1
+   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+   46206893947914691316295628839036278726983680000000000
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPageEmpty2}
+\begin{paste}{ugxFloatHilbertPageEmpty2}{ugxFloatHilbertPagePatch2}
+\pastebutton{ugxFloatHilbertPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{d:= determinant a\free{a }\bound{d }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPagePatch3}
+\begin{paste}{ugxFloatHilbertPageFull3}{ugxFloatHilbertPageEmpty3}
+\pastebutton{ugxFloatHilbertPageFull3}{\hidepaste}
+\tab{5}\spadcommand{d :: Float\free{d }}
+\indentrel{3}\begin{verbatim}
+   (3)  0.21641 79226 43149 18691 E -52
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPageEmpty3}
+\begin{paste}{ugxFloatHilbertPageEmpty3}{ugxFloatHilbertPagePatch3}
+\pastebutton{ugxFloatHilbertPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{d :: Float\free{d }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPagePatch4}
+\begin{paste}{ugxFloatHilbertPageFull4}{ugxFloatHilbertPageEmpty4}
+\pastebutton{ugxFloatHilbertPageFull4}{\hidepaste}
+\tab{5}\spadcommand{b: Matrix DoubleFloat := matrix [[1/(i+j+1$DoubleFloat) for j in 0..9] for i in 0..9];\bound{b }}
+\indentrel{3}\begin{verbatim}
+                               Type: Matrix DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPageEmpty4}
+\begin{paste}{ugxFloatHilbertPageEmpty4}{ugxFloatHilbertPagePatch4}
+\pastebutton{ugxFloatHilbertPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{b: Matrix DoubleFloat := matrix [[1/(i+j+1$DoubleFloat) for j in 0..9] for i in 0..9];\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPagePatch5}
+\begin{paste}{ugxFloatHilbertPageFull5}{ugxFloatHilbertPageEmpty5}
+\pastebutton{ugxFloatHilbertPageFull5}{\hidepaste}
+\tab{5}\spadcommand{determinant b\free{b }}
+\indentrel{3}\begin{verbatim}
+   (5)  2.1643677945721411e-53
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPageEmpty5}
+\begin{paste}{ugxFloatHilbertPageEmpty5}{ugxFloatHilbertPagePatch5}
+\pastebutton{ugxFloatHilbertPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{determinant b\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPagePatch6}
+\begin{paste}{ugxFloatHilbertPageFull6}{ugxFloatHilbertPageEmpty6}
+\pastebutton{ugxFloatHilbertPageFull6}{\hidepaste}
+\tab{5}\spadcommand{digits 40\bound{d40 }}
+\indentrel{3}\begin{verbatim}
+   (6)  20
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPageEmpty6}
+\begin{paste}{ugxFloatHilbertPageEmpty6}{ugxFloatHilbertPagePatch6}
+\pastebutton{ugxFloatHilbertPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{digits 40\bound{d40 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPagePatch7}
+\begin{paste}{ugxFloatHilbertPageFull7}{ugxFloatHilbertPageEmpty7}
+\pastebutton{ugxFloatHilbertPageFull7}{\hidepaste}
+\tab{5}\spadcommand{c: Matrix Float := matrix [[1/(i+j+1$Float) for j in 0..9] for i in 0..9];\free{d40 }\bound{c }}
+\indentrel{3}\begin{verbatim}
+                                     Type: Matrix Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPageEmpty7}
+\begin{paste}{ugxFloatHilbertPageEmpty7}{ugxFloatHilbertPagePatch7}
+\pastebutton{ugxFloatHilbertPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{c: Matrix Float := matrix [[1/(i+j+1$Float) for j in 0..9] for i in 0..9];\free{d40 }\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPagePatch8}
+\begin{paste}{ugxFloatHilbertPageFull8}{ugxFloatHilbertPageEmpty8}
+\pastebutton{ugxFloatHilbertPageFull8}{\hidepaste}
+\tab{5}\spadcommand{determinant c\free{c }}
+\indentrel{3}\begin{verbatim}
+   (8)
+   0.21641 79226 43149 18690 60594 98362 26174 36159 E -52
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPageEmpty8}
+\begin{paste}{ugxFloatHilbertPageEmpty8}{ugxFloatHilbertPagePatch8}
+\pastebutton{ugxFloatHilbertPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{determinant c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPagePatch9}
+\begin{paste}{ugxFloatHilbertPageFull9}{ugxFloatHilbertPageEmpty9}
+\pastebutton{ugxFloatHilbertPageFull9}{\hidepaste}
+\tab{5}\spadcommand{digits 20}
+\indentrel{3}\begin{verbatim}
+   (9)  40
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatHilbertPageEmpty9}
+\begin{paste}{ugxFloatHilbertPageEmpty9}{ugxFloatHilbertPagePatch9}
+\pastebutton{ugxFloatHilbertPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{digits 20}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch1}
+\begin{paste}{ugxFloatConvertPageFull1}{ugxFloatConvertPageEmpty1}
+\pastebutton{ugxFloatConvertPageFull1}{\hidepaste}
+\tab{5}\spadcommand{i := 3 :: Float\bound{i }}
+\indentrel{3}\begin{verbatim}
+   (1)  3.0
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty1}
+\begin{paste}{ugxFloatConvertPageEmpty1}{ugxFloatConvertPagePatch1}
+\pastebutton{ugxFloatConvertPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{i := 3 :: Float\bound{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch2}
+\begin{paste}{ugxFloatConvertPageFull2}{ugxFloatConvertPageEmpty2}
+\pastebutton{ugxFloatConvertPageFull2}{\hidepaste}
+\tab{5}\spadcommand{i :: Integer\free{i }}
+\indentrel{3}\begin{verbatim}
+   (2)  3
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty2}
+\begin{paste}{ugxFloatConvertPageEmpty2}{ugxFloatConvertPagePatch2}
+\pastebutton{ugxFloatConvertPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{i :: Integer\free{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch3}
+\begin{paste}{ugxFloatConvertPageFull3}{ugxFloatConvertPageEmpty3}
+\pastebutton{ugxFloatConvertPageFull3}{\hidepaste}
+\tab{5}\spadcommand{i :: Fraction Integer\free{i }}
+\indentrel{3}\begin{verbatim}
+   (3)  3
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty3}
+\begin{paste}{ugxFloatConvertPageEmpty3}{ugxFloatConvertPagePatch3}
+\pastebutton{ugxFloatConvertPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{i :: Fraction Integer\free{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch4}
+\begin{paste}{ugxFloatConvertPageFull4}{ugxFloatConvertPageEmpty4}
+\pastebutton{ugxFloatConvertPageFull4}{\hidepaste}
+\tab{5}\spadcommand{r := 3/7 :: Float\bound{r }}
+\indentrel{3}\begin{verbatim}
+   (4)  0.42857 14285 71428 57143
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty4}
+\begin{paste}{ugxFloatConvertPageEmpty4}{ugxFloatConvertPagePatch4}
+\pastebutton{ugxFloatConvertPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{r := 3/7 :: Float\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch5}
+\begin{paste}{ugxFloatConvertPageFull5}{ugxFloatConvertPageEmpty5}
+\pastebutton{ugxFloatConvertPageFull5}{\hidepaste}
+\tab{5}\spadcommand{r :: Fraction Integer\free{r }}
+\indentrel{3}\begin{verbatim}
+        3
+   (5)  Ä
+        7
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty5}
+\begin{paste}{ugxFloatConvertPageEmpty5}{ugxFloatConvertPagePatch5}
+\pastebutton{ugxFloatConvertPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{r :: Fraction Integer\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch6}
+\begin{paste}{ugxFloatConvertPageFull6}{ugxFloatConvertPageEmpty6}
+\pastebutton{ugxFloatConvertPageFull6}{\hidepaste}
+\tab{5}\spadcommand{r :: Integer\free{r }}
+\indentrel{3}\begin{verbatim}
+   0.42857 14285 71428 57143
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty6}
+\begin{paste}{ugxFloatConvertPageEmpty6}{ugxFloatConvertPagePatch6}
+\pastebutton{ugxFloatConvertPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{r :: Integer\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch7}
+\begin{paste}{ugxFloatConvertPageFull7}{ugxFloatConvertPageEmpty7}
+\pastebutton{ugxFloatConvertPageFull7}{\hidepaste}
+\tab{5}\spadcommand{truncate 3.6}
+\indentrel{3}\begin{verbatim}
+   (6)  3.0
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty7}
+\begin{paste}{ugxFloatConvertPageEmpty7}{ugxFloatConvertPagePatch7}
+\pastebutton{ugxFloatConvertPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{truncate 3.6}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch8}
+\begin{paste}{ugxFloatConvertPageFull8}{ugxFloatConvertPageEmpty8}
+\pastebutton{ugxFloatConvertPageFull8}{\hidepaste}
+\tab{5}\spadcommand{round 3.6}
+\indentrel{3}\begin{verbatim}
+   (7)  4.0
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty8}
+\begin{paste}{ugxFloatConvertPageEmpty8}{ugxFloatConvertPagePatch8}
+\pastebutton{ugxFloatConvertPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{round 3.6}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch9}
+\begin{paste}{ugxFloatConvertPageFull9}{ugxFloatConvertPageEmpty9}
+\pastebutton{ugxFloatConvertPageFull9}{\hidepaste}
+\tab{5}\spadcommand{truncate(-3.6)}
+\indentrel{3}\begin{verbatim}
+   (8)  - 3.0
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty9}
+\begin{paste}{ugxFloatConvertPageEmpty9}{ugxFloatConvertPagePatch9}
+\pastebutton{ugxFloatConvertPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{truncate(-3.6)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch10}
+\begin{paste}{ugxFloatConvertPageFull10}{ugxFloatConvertPageEmpty10}
+\pastebutton{ugxFloatConvertPageFull10}{\hidepaste}
+\tab{5}\spadcommand{round(-3.6)}
+\indentrel{3}\begin{verbatim}
+   (9)  - 4.0
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty10}
+\begin{paste}{ugxFloatConvertPageEmpty10}{ugxFloatConvertPagePatch10}
+\pastebutton{ugxFloatConvertPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{round(-3.6)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch11}
+\begin{paste}{ugxFloatConvertPageFull11}{ugxFloatConvertPageEmpty11}
+\pastebutton{ugxFloatConvertPageFull11}{\hidepaste}
+\tab{5}\spadcommand{fractionPart 3.6}
+\indentrel{3}\begin{verbatim}
+   (10)  0.6
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty11}
+\begin{paste}{ugxFloatConvertPageEmpty11}{ugxFloatConvertPagePatch11}
+\pastebutton{ugxFloatConvertPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{fractionPart 3.6}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch12}
+\begin{paste}{ugxFloatConvertPageFull12}{ugxFloatConvertPageEmpty12}
+\pastebutton{ugxFloatConvertPageFull12}{\hidepaste}
+\tab{5}\spadcommand{digits 40\bound{d40 }}
+\indentrel{3}\begin{verbatim}
+   (11)  20
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty12}
+\begin{paste}{ugxFloatConvertPageEmpty12}{ugxFloatConvertPagePatch12}
+\pastebutton{ugxFloatConvertPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{digits 40\bound{d40 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch13}
+\begin{paste}{ugxFloatConvertPageFull13}{ugxFloatConvertPageEmpty13}
+\pastebutton{ugxFloatConvertPageFull13}{\hidepaste}
+\tab{5}\spadcommand{sqrt 0.2}
+\indentrel{3}\begin{verbatim}
+   (12)
+   0.44721 35954 99957 93928 18347 33746 25524 70881
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty13}
+\begin{paste}{ugxFloatConvertPageEmpty13}{ugxFloatConvertPagePatch13}
+\pastebutton{ugxFloatConvertPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{sqrt 0.2}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch14}
+\begin{paste}{ugxFloatConvertPageFull14}{ugxFloatConvertPageEmpty14}
+\pastebutton{ugxFloatConvertPageFull14}{\hidepaste}
+\tab{5}\spadcommand{pi()$Float\free{d40 }}
+\indentrel{3}\begin{verbatim}
+   (13)
+   3.14159 26535 89793 23846 26433 83279 50288 4197
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty14}
+\begin{paste}{ugxFloatConvertPageEmpty14}{ugxFloatConvertPagePatch14}
+\pastebutton{ugxFloatConvertPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{pi()$Float\free{d40 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch15}
+\begin{paste}{ugxFloatConvertPageFull15}{ugxFloatConvertPageEmpty15}
+\pastebutton{ugxFloatConvertPageFull15}{\hidepaste}
+\tab{5}\spadcommand{digits 500\bound{d1000 }}
+\indentrel{3}\begin{verbatim}
+   (14)  40
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty15}
+\begin{paste}{ugxFloatConvertPageEmpty15}{ugxFloatConvertPagePatch15}
+\pastebutton{ugxFloatConvertPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{digits 500\bound{d1000 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch16}
+\begin{paste}{ugxFloatConvertPageFull16}{ugxFloatConvertPageEmpty16}
+\pastebutton{ugxFloatConvertPageFull16}{\hidepaste}
+\tab{5}\spadcommand{pi()$Float\free{d1000 }}
+\indentrel{3}\begin{verbatim}
+   (15)
+  3.14159 26535 89793 23846 26433 83279 50288 41971 69399
+   37510 58209 74944 59230 78164 06286 20899 86280 34825 
+  34211 70679 82148 08651 32823 06647 09384 46095 50582 2
+  3172 53594 08128 48111 74502 84102 70193 85211 05559 64
+  462 29489 54930 38196 44288 10975 66593 34461 28475 648
+  23 37867 83165 27120 19091 45648 56692 34603 48610 4543
+  2 66482 13393 60726 02491 41273 72458 70066 06315 58817
+   48815 20920 96282 92540 91715 36436 78925 90360 01133 
+  05305 48820 46652 13841 46951 94151 16094 33057 27036 5
+  7595 91953 09218 61173 81932 61179 31051 18548 07446 23
+  799 62749 56735 18857 52724 89122 79381 83011 9491
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty16}
+\begin{paste}{ugxFloatConvertPageEmpty16}{ugxFloatConvertPagePatch16}
+\pastebutton{ugxFloatConvertPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{pi()$Float\free{d1000 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPagePatch17}
+\begin{paste}{ugxFloatConvertPageFull17}{ugxFloatConvertPageEmpty17}
+\pastebutton{ugxFloatConvertPageFull17}{\hidepaste}
+\tab{5}\spadcommand{digits 20}
+\indentrel{3}\begin{verbatim}
+   (16)  500
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFloatConvertPageEmpty17}
+\begin{paste}{ugxFloatConvertPageEmpty17}{ugxFloatConvertPagePatch17}
+\pastebutton{ugxFloatConvertPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{digits 20}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/FNAME.ht b/src/hyper/pages/FNAME.ht
new file mode 100644
index 00000000..60f652e7
--- /dev/null
+++ b/src/hyper/pages/FNAME.ht
@@ -0,0 +1,130 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\FileNameXmpTitle}{FileName}
+\newcommand{\FileNameXmpNumber}{9.25}
+%
+% =====================================================================
+\begin{page}{FileNameXmpPage}{9.25 FileName}
+% =====================================================================
+\beginscroll
+ 
+The \spadtype{FileName} domain provides an interface to the computer's
+file system.
+Functions are provided to manipulate file names and to test properties of
+files.
+ 
+The simplest way to use file names in the \Language{} interpreter is to
+rely on conversion to and from strings.
+The syntax of these strings depends on the operating system.
+\xtc{
+}{
+\spadpaste{fn: FileName \bound{fndecl}}
+}
+\xtc{
+On AIX, this is a proper file syntax:
+}{
+\spadpaste{fn := "/spad/src/input/fname.input" \free{fndecl}\bound{fn}}
+}
+ 
+Although it is very convenient to be able to use string notation
+for file names in the interpreter, it is desirable to have a portable
+way of creating and manipulating file names from within programs.
+\xtc{
+A measure of portability is obtained by considering a file name
+to consist of three parts: the {\it directory}, the {\it name},
+and the {\it extension}.
+}{
+\spadpaste{directory fn \free{fn}}
+}
+\xtc{
+}{
+\spadpaste{name fn \free{fn}}
+}
+\xtc{
+}{
+\spadpaste{extension fn \free{fn}}
+}
+The meaning of these three parts depends on the operating system.
+For example, on CMS the file \spad{"SPADPROF INPUT M"}
+would have directory \spad{"M"}, name \spad{"SPADPROF"} and
+extension \spad{"INPUT"}.
+ 
+\xtc{
+It is possible to create a filename from its parts.
+}{
+\spadpaste{fn := filename("/u/smwatt/work", "fname", "input") \free{fndecl}\bound{fn1}}
+}
+\xtc{
+When writing programs, it is helpful to refer to directories via
+variables.
+}{
+\spadpaste{objdir := "/tmp" \bound{objdir}}
+}
+\xtc{
+}{
+\spadpaste{fn := filename(objdir, "table", "spad") \free{fndecl,objdir}\bound{fn2}}
+}
+\xtc{
+If the directory or the extension is given as an empty string, then
+a default is used.  On AIX, the defaults are the current directory
+and no extension.
+}{
+\spadpaste{fn := filename("", "letter", "") \free{fndecl}\bound{fn3}}
+}
+ 
+Three tests provide information about names in the file system.
+\xtc{
+The \spadfunFrom{exists?}{FileName} operation tests whether the named file exists.
+}{
+\spadpaste{exists? "/etc/passwd"}
+}
+\xtc{
+The operation \spadfunFrom{readable?}{FileName} tells whether the named file
+can be read.  If the file does not exist, then it cannot be read.
+}{
+\spadpaste{readable? "/etc/passwd"}
+}
+\xtc{
+}{
+\spadpaste{readable? "/etc/security/passwd"}
+}
+\xtc{
+}{
+\spadpaste{readable? "/ect/passwd"}
+}
+\xtc{
+Likewise, the operation \spadfunFrom{writable?}{FileName} tells whether the named file
+can be written.
+If the file does not exist, the test is determined
+by the properties of the directory.
+}{
+\spadpaste{writable? "/etc/passwd"}
+}
+\xtc{
+}{
+\spadpaste{writable? "/dev/null"}
+}
+\xtc{
+}{
+\spadpaste{writable? "/etc/DoesNotExist"}
+}
+\xtc{
+}{
+\spadpaste{writable? "/tmp/DoesNotExist"}
+}
+ 
+The \spadfunFrom{new}{FileName} operation constructs the name of a new
+writable file.
+The argument sequence is the same as for \spadfunFrom{filename}{FileName},
+except that the name part is actually a prefix for a constructed
+unique name.
+\xtc{
+The resulting file is in the specified directory
+with the given extension, and the same defaults are used.
+}{
+\spadpaste{fn := new(objdir, "xxx", "yy") \free{objdir,fndecl}\bound{fn4}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/FNAME.pht b/src/hyper/pages/FNAME.pht
new file mode 100644
index 00000000..06831d3e
--- /dev/null
+++ b/src/hyper/pages/FNAME.pht
@@ -0,0 +1,287 @@
+\begin{patch}{FileNameXmpPagePatch1}
+\begin{paste}{FileNameXmpPageFull1}{FileNameXmpPageEmpty1}
+\pastebutton{FileNameXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{fn: FileName\bound{fndecl }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty1}
+\begin{paste}{FileNameXmpPageEmpty1}{FileNameXmpPagePatch1}
+\pastebutton{FileNameXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{fn: FileName\bound{fndecl }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch2}
+\begin{paste}{FileNameXmpPageFull2}{FileNameXmpPageEmpty2}
+\pastebutton{FileNameXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{fn := "/spad/src/input/fname.input"\free{fndecl }\bound{fn }}
+\indentrel{3}\begin{verbatim}
+   (2)  "/spad/src/input/fname.input"
+                                         Type: FileName
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty2}
+\begin{paste}{FileNameXmpPageEmpty2}{FileNameXmpPagePatch2}
+\pastebutton{FileNameXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{fn := "/spad/src/input/fname.input"\free{fndecl }\bound{fn }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch3}
+\begin{paste}{FileNameXmpPageFull3}{FileNameXmpPageEmpty3}
+\pastebutton{FileNameXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{directory fn\free{fn }}
+\indentrel{3}\begin{verbatim}
+   (3)  "/spad/src/input"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty3}
+\begin{paste}{FileNameXmpPageEmpty3}{FileNameXmpPagePatch3}
+\pastebutton{FileNameXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{directory fn\free{fn }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch4}
+\begin{paste}{FileNameXmpPageFull4}{FileNameXmpPageEmpty4}
+\pastebutton{FileNameXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{name fn\free{fn }}
+\indentrel{3}\begin{verbatim}
+   (4)  "fname"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty4}
+\begin{paste}{FileNameXmpPageEmpty4}{FileNameXmpPagePatch4}
+\pastebutton{FileNameXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{name fn\free{fn }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch5}
+\begin{paste}{FileNameXmpPageFull5}{FileNameXmpPageEmpty5}
+\pastebutton{FileNameXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{extension fn\free{fn }}
+\indentrel{3}\begin{verbatim}
+   (5)  "input"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty5}
+\begin{paste}{FileNameXmpPageEmpty5}{FileNameXmpPagePatch5}
+\pastebutton{FileNameXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{extension fn\free{fn }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch6}
+\begin{paste}{FileNameXmpPageFull6}{FileNameXmpPageEmpty6}
+\pastebutton{FileNameXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{fn := filename("/u/smwatt/work", "fname", "input")\free{fndecl }\bound{fn1 }}
+\indentrel{3}\begin{verbatim}
+   (6)  "/u/smwatt/work/fname.input"
+                                         Type: FileName
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty6}
+\begin{paste}{FileNameXmpPageEmpty6}{FileNameXmpPagePatch6}
+\pastebutton{FileNameXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{fn := filename("/u/smwatt/work", "fname", "input")\free{fndecl }\bound{fn1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch7}
+\begin{paste}{FileNameXmpPageFull7}{FileNameXmpPageEmpty7}
+\pastebutton{FileNameXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{objdir := "/tmp"\bound{objdir }}
+\indentrel{3}\begin{verbatim}
+   (7)  "/tmp"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty7}
+\begin{paste}{FileNameXmpPageEmpty7}{FileNameXmpPagePatch7}
+\pastebutton{FileNameXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{objdir := "/tmp"\bound{objdir }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch8}
+\begin{paste}{FileNameXmpPageFull8}{FileNameXmpPageEmpty8}
+\pastebutton{FileNameXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{fn := filename(objdir, "table", "spad")\free{fndecl objdir }\bound{fn2 }}
+\indentrel{3}\begin{verbatim}
+   (8)  "/tmp/table.spad"
+                                         Type: FileName
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty8}
+\begin{paste}{FileNameXmpPageEmpty8}{FileNameXmpPagePatch8}
+\pastebutton{FileNameXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{fn := filename(objdir, "table", "spad")\free{fndecl objdir }\bound{fn2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch9}
+\begin{paste}{FileNameXmpPageFull9}{FileNameXmpPageEmpty9}
+\pastebutton{FileNameXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{fn := filename("", "letter", "")\free{fndecl }\bound{fn3 }}
+\indentrel{3}\begin{verbatim}
+   (9)  "letter"
+                                         Type: FileName
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty9}
+\begin{paste}{FileNameXmpPageEmpty9}{FileNameXmpPagePatch9}
+\pastebutton{FileNameXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{fn := filename("", "letter", "")\free{fndecl }\bound{fn3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch10}
+\begin{paste}{FileNameXmpPageFull10}{FileNameXmpPageEmpty10}
+\pastebutton{FileNameXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{exists? "/etc/passwd"}
+\indentrel{3}\begin{verbatim}
+   (10)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty10}
+\begin{paste}{FileNameXmpPageEmpty10}{FileNameXmpPagePatch10}
+\pastebutton{FileNameXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{exists? "/etc/passwd"}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch11}
+\begin{paste}{FileNameXmpPageFull11}{FileNameXmpPageEmpty11}
+\pastebutton{FileNameXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{readable? "/etc/passwd"}
+\indentrel{3}\begin{verbatim}
+   (11)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty11}
+\begin{paste}{FileNameXmpPageEmpty11}{FileNameXmpPagePatch11}
+\pastebutton{FileNameXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{readable? "/etc/passwd"}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch12}
+\begin{paste}{FileNameXmpPageFull12}{FileNameXmpPageEmpty12}
+\pastebutton{FileNameXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{readable? "/etc/security/passwd"}
+\indentrel{3}\begin{verbatim}
+   (12)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty12}
+\begin{paste}{FileNameXmpPageEmpty12}{FileNameXmpPagePatch12}
+\pastebutton{FileNameXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{readable? "/etc/security/passwd"}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch13}
+\begin{paste}{FileNameXmpPageFull13}{FileNameXmpPageEmpty13}
+\pastebutton{FileNameXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{readable? "/ect/passwd"}
+\indentrel{3}\begin{verbatim}
+   (13)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty13}
+\begin{paste}{FileNameXmpPageEmpty13}{FileNameXmpPagePatch13}
+\pastebutton{FileNameXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{readable? "/ect/passwd"}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch14}
+\begin{paste}{FileNameXmpPageFull14}{FileNameXmpPageEmpty14}
+\pastebutton{FileNameXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{writable? "/etc/passwd"}
+\indentrel{3}\begin{verbatim}
+   (14)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty14}
+\begin{paste}{FileNameXmpPageEmpty14}{FileNameXmpPagePatch14}
+\pastebutton{FileNameXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{writable? "/etc/passwd"}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch15}
+\begin{paste}{FileNameXmpPageFull15}{FileNameXmpPageEmpty15}
+\pastebutton{FileNameXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{writable? "/dev/null"}
+\indentrel{3}\begin{verbatim}
+   (15)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty15}
+\begin{paste}{FileNameXmpPageEmpty15}{FileNameXmpPagePatch15}
+\pastebutton{FileNameXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{writable? "/dev/null"}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch16}
+\begin{paste}{FileNameXmpPageFull16}{FileNameXmpPageEmpty16}
+\pastebutton{FileNameXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{writable? "/etc/DoesNotExist"}
+\indentrel{3}\begin{verbatim}
+   (16)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty16}
+\begin{paste}{FileNameXmpPageEmpty16}{FileNameXmpPagePatch16}
+\pastebutton{FileNameXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{writable? "/etc/DoesNotExist"}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch17}
+\begin{paste}{FileNameXmpPageFull17}{FileNameXmpPageEmpty17}
+\pastebutton{FileNameXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{writable? "/tmp/DoesNotExist"}
+\indentrel{3}\begin{verbatim}
+   (17)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty17}
+\begin{paste}{FileNameXmpPageEmpty17}{FileNameXmpPagePatch17}
+\pastebutton{FileNameXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{writable? "/tmp/DoesNotExist"}
+\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPagePatch18}
+\begin{paste}{FileNameXmpPageFull18}{FileNameXmpPageEmpty18}
+\pastebutton{FileNameXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{fn := new(objdir, "xxx", "yy")\free{objdir fndecl }\bound{fn4 }}
+\indentrel{3}\begin{verbatim}
+   (18)  "/tmp/xxx82222.yy"
+                                         Type: FileName
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FileNameXmpPageEmpty18}
+\begin{paste}{FileNameXmpPageEmpty18}{FileNameXmpPagePatch18}
+\pastebutton{FileNameXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{fn := new(objdir, "xxx", "yy")\free{objdir fndecl }\bound{fn4 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/FPARFRAC.ht b/src/hyper/pages/FPARFRAC.ht
new file mode 100644
index 00000000..2076f958
--- /dev/null
+++ b/src/hyper/pages/FPARFRAC.ht
@@ -0,0 +1,101 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\FullPartialFractionExpansionXmpTitle}{FullPartialFractionExpansion}
+\newcommand{\FullPartialFractionExpansionXmpNumber}{9.29}
+%
+% =====================================================================
+\begin{page}{FullPartialFractionExpansionXmpPage}{9.29 FullPartialFractionExpansion}
+% =====================================================================
+\beginscroll
+
+The domain \spadtype{FullPartialFractionExpansion} implements
+factor-free conversion of quotients to full partial fractions.
+
+\xtc{
+Our examples will all involve quotients of univariate polynomials
+with rational number coefficients.
+}{
+\spadpaste{Fx := FRAC UP(x, FRAC INT) \bound{Fx}}
+}
+\xtc{
+Here is a simple-looking rational function.
+}{
+\spadpaste{f : Fx := 36 / (x**5-2*x**4-2*x**3+4*x**2+x-2) \bound{f}\free{Fx}}
+}
+\xtc{
+We use \spadfunFrom{fullPartialFraction}{FullPartialFractionExpansion}
+to convert it to an object of type
+\spadtype{FullPartialFractionExpansion}.
+}{
+\spadpaste{g := fullPartialFraction f \bound{g}\free{f}}
+}
+\xtc{
+Use a coercion to change it back into a quotient.
+}{
+\spadpaste{g :: Fx \free{g}}
+}
+\xtc{
+Full partial fractions differentiate faster than rational
+functions.
+}{
+\spadpaste{g5 := D(g, 5) \free{g}\bound{g5}}
+}
+\xtc{
+}{
+\spadpaste{f5 := D(f, 5) \free{f}\bound{f5}}
+}
+\xtc{
+We can check that the two forms represent the same function.
+}{
+\spadpaste{g5::Fx - f5 \free{Fx g5 f5}}
+}
+\xtc{
+Here are some examples that are more complicated.
+}{
+\spadpaste{f : Fx := (x**5 * (x-1)) / ((x**2 + x + 1)**2 * (x-2)**3) \free{Fx}\bound{f2}}
+}
+\xtc{
+}{
+\spadpaste{g := fullPartialFraction f \free{f2}\bound{g2}}
+}
+\xtc{
+}{
+\spadpaste{g :: Fx - f \free{f2 g2 Fx}}
+}
+\xtc{
+}{
+\spadpaste{f : Fx := (2*x**7-7*x**5+26*x**3+8*x) / (x**8-5*x**6+6*x**4+4*x**2-8) \free{Fx}\bound{f3}}
+}
+\xtc{
+}{
+\spadpaste{g := fullPartialFraction f \free{f3}\bound{g3}}
+}
+\xtc{
+}{
+\spadpaste{g :: Fx - f \free{f3 g3 Fx}}
+}
+\xtc{
+}{
+\spadpaste{f:Fx := x**3 / (x**21 + 2*x**20 + 4*x**19 + 7*x**18 + 10*x**17 + 17*x**16 + 22*x**15 + 30*x**14 + 36*x**13 + 40*x**12 + 47*x**11 + 46*x**10 + 49*x**9 + 43*x**8 + 38*x**7 + 32*x**6 + 23*x**5 + 19*x**4 + 10*x**3 + 7*x**2 + 2*x + 1)\free{Fx}\bound{f4}}
+}
+\xtc{
+}{
+\spadpaste{g := fullPartialFraction f \free{f4}\bound{g4}}
+}
+\xtc{
+This verification takes much longer than the conversion to
+partial fractions.
+}{
+\spadpaste{g :: Fx - f \free{f4 g4 Fx}}
+}
+
+For more information, see the paper:
+Bronstein, M and Salvy, B.
+``Full Partial Fraction Decomposition of Rational Functions,''
+{\it Proceedings of ISSAC'93, Kiev}, ACM Press.
+All see \downlink{`PartialFraction'}{PartialFractionXmpPage}\ignore{PartialFraction} for standard partial fraction
+decompositions.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/FPARFRAC.pht b/src/hyper/pages/FPARFRAC.pht
new file mode 100644
index 00000000..805d6731
--- /dev/null
+++ b/src/hyper/pages/FPARFRAC.pht
@@ -0,0 +1,411 @@
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch1}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull1}{FullPartialFractionExpansionXmpPageEmpty1}
+\pastebutton{FullPartialFractionExpansionXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{Fx := FRAC UP(x, FRAC INT)\bound{Fx }}
+\indentrel{3}\begin{verbatim}
+   (1)
+   Fraction UnivariatePolynomial(x,Fraction Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty1}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty1}{FullPartialFractionExpansionXmpPagePatch1}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{Fx := FRAC UP(x, FRAC INT)\bound{Fx }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch2}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull2}{FullPartialFractionExpansionXmpPageEmpty2}
+\pastebutton{FullPartialFractionExpansionXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f : Fx := 36 / (x**5-2*x**4-2*x**3+4*x**2+x-2)\bound{f }\free{Fx }}
+\indentrel{3}\begin{verbatim}
+                     36
+   (2)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+         5     4     3     2
+        x  - 2x  - 2x  + 4x  + x - 2
+Type: Fraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty2}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty2}{FullPartialFractionExpansionXmpPagePatch2}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f : Fx := 36 / (x**5-2*x**4-2*x**3+4*x**2+x-2)\bound{f }\free{Fx }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch3}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull3}{FullPartialFractionExpansionXmpPageEmpty3}
+\pastebutton{FullPartialFractionExpansionXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{g := fullPartialFraction f\bound{g }\free{f }}
+\indentrel{3}\begin{verbatim}
+          4       4        ÄÄ¿      - 3%A - 6
+   (3)  ÄÄÄÄÄ - ÄÄÄÄÄ +    >        ÄÄÄÄÄÄÄÄÄ
+        x - 2   x + 1      ÄÄÙ              2
+                          2         (x - %A)
+                        %A  - 1= 0
+Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty3}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty3}{FullPartialFractionExpansionXmpPagePatch3}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{g := fullPartialFraction f\bound{g }\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch4}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull4}{FullPartialFractionExpansionXmpPageEmpty4}
+\pastebutton{FullPartialFractionExpansionXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{g :: Fx\free{g }}
+\indentrel{3}\begin{verbatim}
+                     36
+   (4)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+         5     4     3     2
+        x  - 2x  - 2x  + 4x  + x - 2
+Type: Fraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty4}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty4}{FullPartialFractionExpansionXmpPagePatch4}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{g :: Fx\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch5}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull5}{FullPartialFractionExpansionXmpPageEmpty5}
+\pastebutton{FullPartialFractionExpansionXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{g5 := D(g, 5)\free{g }\bound{g5 }}
+\indentrel{3}\begin{verbatim}
+   (5)
+        480        480        ÄÄ¿      2160%A + 4320
+   - ÄÄÄÄÄÄÄÄ + ÄÄÄÄÄÄÄÄ +    >        ÄÄÄÄÄÄÄÄÄÄÄÄÄ
+            6          6      ÄÄÙ                7
+     (x - 2)    (x + 1)      2           (x - %A)
+                           %A  - 1= 0
+Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty5}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty5}{FullPartialFractionExpansionXmpPagePatch5}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{g5 := D(g, 5)\free{g }\bound{g5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch6}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull6}{FullPartialFractionExpansionXmpPageEmpty6}
+\pastebutton{FullPartialFractionExpansionXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{f5 := D(f, 5)\free{f }\bound{f5 }}
+\indentrel{3}\begin{verbatim}
+   (6)
+                10           9            8            7
+       - 544320x   + 4354560x  - 14696640x  + 28615680x
+     + 
+                  6            5            4            3
+       - 40085280x  + 46656000x  - 39411360x  + 18247680x
+     + 
+                 2
+       - 5870880x  + 3317760x + 246240
+  /
+        20      19      18      17       16       15
+       x   - 12x   + 53x   - 76x   - 159x   + 676x
+     + 
+             14        13        12        11        10
+       - 391x   - 1596x   + 2527x   + 1148x   - 4977x
+     + 
+            9        8        7        6        5       4
+       1372x  + 4907x  - 3444x  - 2381x  + 2924x  + 276x
+     + 
+              3       2
+       - 1184x  + 208x  + 192x - 64
+Type: Fraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty6}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty6}{FullPartialFractionExpansionXmpPagePatch6}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{f5 := D(f, 5)\free{f }\bound{f5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch7}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull7}{FullPartialFractionExpansionXmpPageEmpty7}
+\pastebutton{FullPartialFractionExpansionXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{g5::Fx - f5\free{Fx g5 f5 }}
+\indentrel{3}\begin{verbatim}
+   (7)  0
+Type: Fraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty7}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty7}{FullPartialFractionExpansionXmpPagePatch7}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{g5::Fx - f5\free{Fx g5 f5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch8}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull8}{FullPartialFractionExpansionXmpPageEmpty8}
+\pastebutton{FullPartialFractionExpansionXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{f : Fx := (x**5 * (x-1)) / ((x**2 + x + 1)**2 * (x-2)**3)\free{Fx }\bound{f2 }}
+\indentrel{3}\begin{verbatim}
+                       6    5
+                      x  - x
+   (8)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+         7     6     5     3     2
+        x  - 4x  + 3x  + 9x  - 6x  - 4x - 8
+Type: Fraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty8}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty8}{FullPartialFractionExpansionXmpPagePatch8}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{f : Fx := (x**5 * (x-1)) / ((x**2 + x + 1)**2 * (x-2)**3)\free{Fx }\bound{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch9}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull9}{FullPartialFractionExpansionXmpPageEmpty9}
+\pastebutton{FullPartialFractionExpansionXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{g := fullPartialFraction f\free{f2 }\bound{g2 }}
+\indentrel{3}\begin{verbatim}
+   (9)
+      1952       464        32
+      ÄÄÄÄ       ÄÄÄ        ÄÄ
+      2401       343        49
+     ÄÄÄÄÄÄ + ÄÄÄÄÄÄÄÄ + ÄÄÄÄÄÄÄÄ
+      x - 2          2          3
+              (x - 2)    (x - 2)
+   + 
+                         179       135
+                      - ÄÄÄÄ %A + ÄÄÄÄ
+           ÄÄ¿          2401      2401
+           >          ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+           ÄÄÙ             x - %A
+       2
+     %A  + %A + 1= 0
+   + 
+                       37        20
+                      ÄÄÄÄ %A + ÄÄÄÄ
+           ÄÄ¿        1029      1029
+           >          ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+           ÄÄÙ                   2
+       2                 (x - %A)
+     %A  + %A + 1= 0
+Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty9}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty9}{FullPartialFractionExpansionXmpPagePatch9}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{g := fullPartialFraction f\free{f2 }\bound{g2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch10}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull10}{FullPartialFractionExpansionXmpPageEmpty10}
+\pastebutton{FullPartialFractionExpansionXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{g :: Fx - f\free{f2 g2 Fx }}
+\indentrel{3}\begin{verbatim}
+   (10)  0
+Type: Fraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty10}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty10}{FullPartialFractionExpansionXmpPagePatch10}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{g :: Fx - f\free{f2 g2 Fx }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch11}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull11}{FullPartialFractionExpansionXmpPageEmpty11}
+\pastebutton{FullPartialFractionExpansionXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{f : Fx := (2*x**7-7*x**5+26*x**3+8*x) / (x**8-5*x**6+6*x**4+4*x**2-8)\free{Fx }\bound{f3 }}
+\indentrel{3}\begin{verbatim}
+             7     5      3
+           2x  - 7x  + 26x  + 8x
+   (11)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+          8     6     4     2
+         x  - 5x  + 6x  + 4x  - 8
+Type: Fraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty11}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty11}{FullPartialFractionExpansionXmpPagePatch11}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{f : Fx := (2*x**7-7*x**5+26*x**3+8*x) / (x**8-5*x**6+6*x**4+4*x**2-8)\free{Fx }\bound{f3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch12}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull12}{FullPartialFractionExpansionXmpPageEmpty12}
+\pastebutton{FullPartialFractionExpansionXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{g := fullPartialFraction f\free{f3 }\bound{g3 }}
+\indentrel{3}\begin{verbatim}
+   (12)
+                    1
+                    Ä
+        ÄÄ¿         2        ÄÄ¿          1
+        >        ÄÄÄÄÄÄ +    >        ÄÄÄÄÄÄÄÄÄ
+        ÄÄÙ      x - %A      ÄÄÙ              3
+       2                    2         (x - %A)
+     %A  - 2= 0           %A  - 2= 0
+   + 
+                    1
+                    Ä
+        ÄÄ¿         2
+        >        ÄÄÄÄÄÄ
+        ÄÄÙ      x - %A
+       2
+     %A  + 1= 0
+Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty12}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty12}{FullPartialFractionExpansionXmpPagePatch12}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{g := fullPartialFraction f\free{f3 }\bound{g3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch13}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull13}{FullPartialFractionExpansionXmpPageEmpty13}
+\pastebutton{FullPartialFractionExpansionXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{g :: Fx - f\free{f3 g3 Fx }}
+\indentrel{3}\begin{verbatim}
+   (13)  0
+Type: Fraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty13}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty13}{FullPartialFractionExpansionXmpPagePatch13}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{g :: Fx - f\free{f3 g3 Fx }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch14}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull14}{FullPartialFractionExpansionXmpPageEmpty14}
+\pastebutton{FullPartialFractionExpansionXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{f:Fx := x**3 / (x**21 + 2*x**20 + 4*x**19 + 7*x**18 + 10*x**17 + 17*x**16 + 22*x**15 + 30*x**14 + 36*x**13 + 40*x**12 + 47*x**11 + 46*x**10 + 49*x**9 + 43*x**8 + 38*x**7 + 32*x**6 + 23*x**5 + 19*x**4 + 10*x**3 + 7*x**2 + 2*x + 1)\free{Fx }\bound{f4 }}
+\indentrel{3}\begin{verbatim}
+   (14)
+      3
+     x
+  /
+        21     20     19     18      17      16      15
+       x   + 2x   + 4x   + 7x   + 10x   + 17x   + 22x
+     + 
+          14      13      12      11      10      9      8
+       30x   + 36x   + 40x   + 47x   + 46x   + 49x  + 43x
+     + 
+          7      6      5      4      3     2
+       38x  + 32x  + 23x  + 19x  + 10x  + 7x  + 2x + 1
+Type: Fraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty14}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty14}{FullPartialFractionExpansionXmpPagePatch14}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{f:Fx := x**3 / (x**21 + 2*x**20 + 4*x**19 + 7*x**18 + 10*x**17 + 17*x**16 + 22*x**15 + 30*x**14 + 36*x**13 + 40*x**12 + 47*x**11 + 46*x**10 + 49*x**9 + 43*x**8 + 38*x**7 + 32*x**6 + 23*x**5 + 19*x**4 + 10*x**3 + 7*x**2 + 2*x + 1)\free{Fx }\bound{f4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch15}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull15}{FullPartialFractionExpansionXmpPageEmpty15}
+\pastebutton{FullPartialFractionExpansionXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{g := fullPartialFraction f\free{f4 }\bound{g4 }}
+\indentrel{3}\begin{verbatim}
+   (15)
+                  1                        1      19
+                  Ä %A                     Ä %A - ÄÄ
+        ÄÄ¿       2             ÄÄ¿        9      27
+        >        ÄÄÄÄÄÄ +       >          ÄÄÄÄÄÄÄÄÄ
+        ÄÄÙ      x - %A         ÄÄÙ          x - %A
+       2                    2
+     %A  + 1= 0           %A  + %A + 1= 0
+   + 
+                       1       1
+                      ÄÄ %A - ÄÄ
+           ÄÄ¿        27      27
+           >          ÄÄÄÄÄÄÄÄÄÄ
+           ÄÄÙ                 2
+       2               (x - %A)
+     %A  + %A + 1= 0
+   + 
+     SIGMA
+          5     2
+        %A  + %A  + 1= 0
+    ,
+               96556567040   4   420961732891   3
+            - ÄÄÄÄÄÄÄÄÄÄÄÄ %A  + ÄÄÄÄÄÄÄÄÄÄÄÄ %A
+              912390759099       912390759099
+          + 
+               59101056149   2   373545875923
+            - ÄÄÄÄÄÄÄÄÄÄÄÄ %A  - ÄÄÄÄÄÄÄÄÄÄÄÄ %A
+              912390759099       912390759099
+          + 
+            529673492498
+            ÄÄÄÄÄÄÄÄÄÄÄÄ
+            912390759099
+       /
+          x - %A
+   + 
+     SIGMA
+          5     2
+        %A  + %A  + 1= 0
+    ,
+               5580868   4    2024443   3    4321919   2
+            - ÄÄÄÄÄÄÄÄ %A  - ÄÄÄÄÄÄÄÄ %A  + ÄÄÄÄÄÄÄÄ %A
+              94070601       94070601       94070601
+          + 
+               84614        5070620
+            - ÄÄÄÄÄÄÄ %A - ÄÄÄÄÄÄÄÄ
+              1542141      94070601
+       /
+                  2
+          (x - %A)
+   + 
+     SIGMA
+          5     2
+        %A  + %A  + 1= 0
+    ,
+             1610957   4    2763014   3    2016775   2
+            ÄÄÄÄÄÄÄÄ %A  + ÄÄÄÄÄÄÄÄ %A  - ÄÄÄÄÄÄÄÄ %A
+            94070601       94070601       94070601
+          + 
+             266953        4529359
+            ÄÄÄÄÄÄÄÄ %A + ÄÄÄÄÄÄÄÄ
+            94070601      94070601
+       /
+                  3
+          (x - %A)
+Type: FullPartialFractionExpansion(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty15}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty15}{FullPartialFractionExpansionXmpPagePatch15}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{g := fullPartialFraction f\free{f4 }\bound{g4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPagePatch16}
+\begin{paste}{FullPartialFractionExpansionXmpPageFull16}{FullPartialFractionExpansionXmpPageEmpty16}
+\pastebutton{FullPartialFractionExpansionXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{g :: Fx - f\free{f4 g4 Fx }}
+\indentrel{3}\begin{verbatim}
+   (16)  0
+Type: Fraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FullPartialFractionExpansionXmpPageEmpty16}
+\begin{paste}{FullPartialFractionExpansionXmpPageEmpty16}{FullPartialFractionExpansionXmpPagePatch16}
+\pastebutton{FullPartialFractionExpansionXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{g :: Fx - f\free{f4 g4 Fx }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/FR.ht b/src/hyper/pages/FR.ht
new file mode 100644
index 00000000..9087f607
--- /dev/null
+++ b/src/hyper/pages/FR.ht
@@ -0,0 +1,316 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\FactoredXmpTitle}{Factored}
+\newcommand{\FactoredXmpNumber}{9.22}
+%
+% =====================================================================
+\begin{page}{FactoredXmpPage}{9.22 Factored}
+% =====================================================================
+\beginscroll
+
+\spadtype{Factored} creates a domain whose objects are kept in factored
+%-% \HDindex{factorization}{FactoredXmpPage}{9.22}{Factored}
+form as long as possible.
+Thus certain operations like \spadopFrom{*}{Factored}
+(multiplication) and \spadfunFrom{gcd}{Factored} are relatively
+easy to do.
+Others, such as addition, require somewhat more work, and
+the result may not be completely factored
+unless the argument domain \spad{R} provides a
+\spadfunFrom{factor}{Factored} operation.
+Each object consists of a unit and a list of factors, where each
+factor consists of a member of \spad{R} (the {\em base}), an
+exponent, and a flag indicating what is known about the base.
+A flag may be one of \spad{"nil"}, \spad{"sqfr"}, \spad{"irred"}
+or \spad{"prime"}, which mean that nothing is known about the
+base, it is square-free, it is irreducible, or it is prime,
+respectively.
+The current restriction to factored objects of integral domains
+allows simplification to be performed without worrying about
+multiplication order.
+
+\beginmenu
+    \menudownlink{{9.22.1. Decomposing Factored Objects}}{ugxFactoredDecompPage}
+    \menudownlink{{9.22.2. Expanding Factored Objects}}{ugxFactoredExpandPage}
+    \menudownlink{{9.22.3. Arithmetic with Factored Objects}}{ugxFactoredArithPage}
+    \menudownlink{{9.22.4. Creating New Factored Objects}}{ugxFactoredNewPage}
+    \menudownlink{{9.22.5. Factored Objects with Variables}}{ugxFactoredVarPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFactoredDecompTitle}{Decomposing Factored Objects}
+\newcommand{\ugxFactoredDecompNumber}{9.22.1.}
+%
+% =====================================================================
+\begin{page}{ugxFactoredDecompPage}{9.22.1. Decomposing Factored Objects}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+\xtc{
+In this section we will work with a factored integer.
+}{
+\spadpaste{g := factor(4312) \bound{g}}
+}
+\xtc{
+Let's begin by decomposing \spad{g} into pieces.
+The only possible units for integers are \spad{1} and \spad{-1}.
+}{
+\spadpaste{unit(g) \free{g}}
+}
+\xtc{
+There are three factors.
+}{
+\spadpaste{numberOfFactors(g) \free{g}}
+}
+\xtc{
+We can make a list of the bases, \ldots
+}{
+\spadpaste{[nthFactor(g,i) for i in 1..numberOfFactors(g)] \free{g}}
+}
+\xtc{
+and the exponents, \ldots
+}{
+\spadpaste{[nthExponent(g,i) for i in 1..numberOfFactors(g)] \free{g}}
+}
+\xtc{
+and the flags.
+You can see that all the bases (factors) are prime.
+}{
+\spadpaste{[nthFlag(g,i) for i in 1..numberOfFactors(g)] \free{g}}
+}
+\xtc{
+A useful operation for pulling apart a factored object into a list
+of records of the components is \spadfunFrom{factorList}{Factored}.
+}{
+\spadpaste{factorList(g) \free{g}}
+}
+\xtc{
+If you don't care about the flags, use \spadfunFrom{factors}{Factored}.
+}{
+\spadpaste{factors(g) \free{g}\bound{prev1}}
+}
+\xtc{
+Neither of these operations returns the unit.
+}{
+\spadpaste{first(\%).factor \free{prev1}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFactoredExpandTitle}{Expanding Factored Objects}
+\newcommand{\ugxFactoredExpandNumber}{9.22.2.}
+%
+% =====================================================================
+\begin{page}{ugxFactoredExpandPage}{9.22.2. Expanding Factored Objects}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+\xtc{
+Recall that we are working with this factored integer.
+}{
+\spadpaste{g := factor(4312) \bound{g}}
+}
+\xtc{
+To multiply out the factors with their multiplicities, use
+\spadfunFrom{expand}{Factored}.
+}{
+\spadpaste{expand(g) \free{g}}
+}
+\xtc{
+If you would like, say, the distinct factors multiplied together
+but with multiplicity one, you could do it this way.
+}{
+\spadpaste{reduce(*,[t.factor for t in factors(g)]) \free{g}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFactoredArithTitle}{Arithmetic with Factored Objects}
+\newcommand{\ugxFactoredArithNumber}{9.22.3.}
+%
+% =====================================================================
+\begin{page}{ugxFactoredArithPage}{9.22.3. Arithmetic with Factored Objects}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+\xtc{
+We're still working with this factored integer.
+}{
+\spadpaste{g := factor(4312) \bound{g}}
+}
+\xtc{
+We'll also define this factored integer.
+}{
+\spadpaste{f := factor(246960) \bound{f}}
+}
+\xtc{
+Operations involving multiplication and division are particularly
+easy with factored objects.
+}{
+\spadpaste{f * g \free{f g}}
+}
+\xtc{
+}{
+\spadpaste{f**500 \free{f}}
+}
+\xtc{
+}{
+\spadpaste{gcd(f,g) \free{f g}}
+}
+\xtc{
+}{
+\spadpaste{lcm(f,g) \free{f g}}
+}
+\xtc{
+If we use addition and subtraction things can slow down because
+we may need to compute greatest common divisors.
+}{
+\spadpaste{f + g \free{f g}}
+}
+\xtc{
+}{
+\spadpaste{f - g \free{f g}}
+}
+\xtc{
+Test for equality with \spad{0} and \spad{1} by using
+\spadfunFrom{zero?}{Factored} and \spadfunFrom{one?}{Factored},
+respectively.
+}{
+\spadpaste{zero?(factor(0))}
+}
+\xtc{
+}{
+\spadpaste{zero?(g) \free{g}}
+}
+\xtc{
+}{
+\spadpaste{one?(factor(1))}
+}
+\xtc{
+}{
+\spadpaste{one?(f) \free{f}}
+}
+\xtc{
+Another way to get the zero and one factored objects is to use
+package calling (see \downlink{``\ugTypesPkgCallTitle''}{ugTypesPkgCallPage} in Section \ugTypesPkgCallNumber\ignore{ugTypesPkgCall}).
+}{
+\spadpaste{0\$Factored(Integer)}
+}
+\xtc{
+}{
+\spadpaste{1\$Factored(Integer)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFactoredNewTitle}{Creating New Factored Objects}
+\newcommand{\ugxFactoredNewNumber}{9.22.4.}
+%
+% =====================================================================
+\begin{page}{ugxFactoredNewPage}{9.22.4. Creating New Factored Objects}
+% =====================================================================
+\beginscroll
+
+The \spadfunFrom{map}{Factored} operation is used to iterate across the
+unit and bases of a factored object.
+See \downlink{`FactoredFunctions2'}{FactoredFunctionsTwoXmpPage}\ignore{FactoredFunctions2} for a discussion of
+\spadfunFrom{map}{Factored}.
+
+\labelSpace{1pc}
+\xtc{
+The following four operations take a base and an exponent and create
+a factored object.
+They differ in handling the flag component.
+}{
+\spadpaste{nilFactor(24,2) \bound{prev2}}
+}
+\xtc{
+This factor has no associated information.
+}{
+\spadpaste{nthFlag(\%,1) \free{prev2}}
+}
+\xtc{
+This factor is asserted to be square-free.
+}{
+\spadpaste{sqfrFactor(30,2) \bound{prev3}}
+}
+\xtc{
+This factor is asserted to be irreducible.
+}{
+\spadpaste{irreducibleFactor(13,10) \bound{prev4}}
+}
+\xtc{
+This factor is asserted to be prime.
+}{
+\spadpaste{primeFactor(11,5) \bound{prev5}}
+}
+
+\xtc{
+A partial inverse to \spadfunFrom{factorList}{Factored} is
+\spadfunFrom{makeFR}{Factored}.
+}{
+\spadpaste{h := factor(-720) \bound{h}}
+}
+\xtc{
+The first argument is the unit and the second is a list of records as
+returned by \spadfunFrom{factorList}{Factored}.
+}{
+\spadpaste{h - makeFR(unit(h),factorList(h)) \free{h}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxFactoredVarTitle}{Factored Objects with Variables}
+\newcommand{\ugxFactoredVarNumber}{9.22.5.}
+%
+% =====================================================================
+\begin{page}{ugxFactoredVarPage}{9.22.5. Factored Objects with Variables}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+\xtc{
+Some of the operations available for polynomials are also available
+for factored polynomials.
+}{
+\spadpaste{p := (4*x*x-12*x+9)*y*y + (4*x*x-12*x+9)*y + 28*x*x - 84*x + 63 \bound{p}}
+}
+\xtc{
+}{
+\spadpaste{fp := factor(p) \free{p}\bound{fp}}
+}
+\xtc{
+You can differentiate with respect to a variable.
+}{
+\spadpaste{D(p,x) \free{p}}
+}
+\xtc{
+}{
+\spadpaste{D(fp,x) \free{fp}\bound{prev1}}
+}
+\xtc{
+}{
+\spadpaste{numberOfFactors(\%) \free{prev1}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/FR.pht b/src/hyper/pages/FR.pht
new file mode 100644
index 00000000..c72f4e0c
--- /dev/null
+++ b/src/hyper/pages/FR.pht
@@ -0,0 +1,633 @@
+\begin{patch}{ugxFactoredArithPagePatch1}
+\begin{paste}{ugxFactoredArithPageFull1}{ugxFactoredArithPageEmpty1}
+\pastebutton{ugxFactoredArithPageFull1}{\hidepaste}
+\tab{5}\spadcommand{g := factor(4312)\bound{g }}
+\indentrel{3}\begin{verbatim}
+         3 2
+   (1)  2 7 11
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty1}
+\begin{paste}{ugxFactoredArithPageEmpty1}{ugxFactoredArithPagePatch1}
+\pastebutton{ugxFactoredArithPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{g := factor(4312)\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPagePatch2}
+\begin{paste}{ugxFactoredArithPageFull2}{ugxFactoredArithPageEmpty2}
+\pastebutton{ugxFactoredArithPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f := factor(246960)\bound{f }}
+\indentrel{3}\begin{verbatim}
+         4 2   3
+   (2)  2 3 5 7
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty2}
+\begin{paste}{ugxFactoredArithPageEmpty2}{ugxFactoredArithPagePatch2}
+\pastebutton{ugxFactoredArithPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f := factor(246960)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPagePatch3}
+\begin{paste}{ugxFactoredArithPageFull3}{ugxFactoredArithPageEmpty3}
+\pastebutton{ugxFactoredArithPageFull3}{\hidepaste}
+\tab{5}\spadcommand{f * g\free{f g }}
+\indentrel{3}\begin{verbatim}
+         7 2   5
+   (3)  2 3 5 7 11
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty3}
+\begin{paste}{ugxFactoredArithPageEmpty3}{ugxFactoredArithPagePatch3}
+\pastebutton{ugxFactoredArithPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f * g\free{f g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPagePatch4}
+\begin{paste}{ugxFactoredArithPageFull4}{ugxFactoredArithPageEmpty4}
+\pastebutton{ugxFactoredArithPageFull4}{\hidepaste}
+\tab{5}\spadcommand{f**500\free{f }}
+\indentrel{3}\begin{verbatim}
+         2000 1000 500 1500
+   (4)  2    3    5   7
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty4}
+\begin{paste}{ugxFactoredArithPageEmpty4}{ugxFactoredArithPagePatch4}
+\pastebutton{ugxFactoredArithPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{f**500\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPagePatch5}
+\begin{paste}{ugxFactoredArithPageFull5}{ugxFactoredArithPageEmpty5}
+\pastebutton{ugxFactoredArithPageFull5}{\hidepaste}
+\tab{5}\spadcommand{gcd(f,g)\free{f g }}
+\indentrel{3}\begin{verbatim}
+         3 2
+   (5)  2 7
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty5}
+\begin{paste}{ugxFactoredArithPageEmpty5}{ugxFactoredArithPagePatch5}
+\pastebutton{ugxFactoredArithPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{gcd(f,g)\free{f g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPagePatch6}
+\begin{paste}{ugxFactoredArithPageFull6}{ugxFactoredArithPageEmpty6}
+\pastebutton{ugxFactoredArithPageFull6}{\hidepaste}
+\tab{5}\spadcommand{lcm(f,g)\free{f g }}
+\indentrel{3}\begin{verbatim}
+         4 2   3
+   (6)  2 3 5 7 11
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty6}
+\begin{paste}{ugxFactoredArithPageEmpty6}{ugxFactoredArithPagePatch6}
+\pastebutton{ugxFactoredArithPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{lcm(f,g)\free{f g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPagePatch7}
+\begin{paste}{ugxFactoredArithPageFull7}{ugxFactoredArithPageEmpty7}
+\pastebutton{ugxFactoredArithPageFull7}{\hidepaste}
+\tab{5}\spadcommand{f + g\free{f g }}
+\indentrel{3}\begin{verbatim}
+         3 2
+   (7)  2 7 641
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty7}
+\begin{paste}{ugxFactoredArithPageEmpty7}{ugxFactoredArithPagePatch7}
+\pastebutton{ugxFactoredArithPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{f + g\free{f g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPagePatch8}
+\begin{paste}{ugxFactoredArithPageFull8}{ugxFactoredArithPageEmpty8}
+\pastebutton{ugxFactoredArithPageFull8}{\hidepaste}
+\tab{5}\spadcommand{f - g\free{f g }}
+\indentrel{3}\begin{verbatim}
+         3 2
+   (8)  2 7 619
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty8}
+\begin{paste}{ugxFactoredArithPageEmpty8}{ugxFactoredArithPagePatch8}
+\pastebutton{ugxFactoredArithPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{f - g\free{f g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPagePatch9}
+\begin{paste}{ugxFactoredArithPageFull9}{ugxFactoredArithPageEmpty9}
+\pastebutton{ugxFactoredArithPageFull9}{\hidepaste}
+\tab{5}\spadcommand{zero?(factor(0))}
+\indentrel{3}\begin{verbatim}
+   (9)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty9}
+\begin{paste}{ugxFactoredArithPageEmpty9}{ugxFactoredArithPagePatch9}
+\pastebutton{ugxFactoredArithPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{zero?(factor(0))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPagePatch10}
+\begin{paste}{ugxFactoredArithPageFull10}{ugxFactoredArithPageEmpty10}
+\pastebutton{ugxFactoredArithPageFull10}{\hidepaste}
+\tab{5}\spadcommand{zero?(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (10)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty10}
+\begin{paste}{ugxFactoredArithPageEmpty10}{ugxFactoredArithPagePatch10}
+\pastebutton{ugxFactoredArithPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{zero?(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPagePatch11}
+\begin{paste}{ugxFactoredArithPageFull11}{ugxFactoredArithPageEmpty11}
+\pastebutton{ugxFactoredArithPageFull11}{\hidepaste}
+\tab{5}\spadcommand{one?(factor(1))}
+\indentrel{3}\begin{verbatim}
+   (11)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty11}
+\begin{paste}{ugxFactoredArithPageEmpty11}{ugxFactoredArithPagePatch11}
+\pastebutton{ugxFactoredArithPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{one?(factor(1))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPagePatch12}
+\begin{paste}{ugxFactoredArithPageFull12}{ugxFactoredArithPageEmpty12}
+\pastebutton{ugxFactoredArithPageFull12}{\hidepaste}
+\tab{5}\spadcommand{one?(f)\free{f }}
+\indentrel{3}\begin{verbatim}
+   (12)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty12}
+\begin{paste}{ugxFactoredArithPageEmpty12}{ugxFactoredArithPagePatch12}
+\pastebutton{ugxFactoredArithPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{one?(f)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPagePatch13}
+\begin{paste}{ugxFactoredArithPageFull13}{ugxFactoredArithPageEmpty13}
+\pastebutton{ugxFactoredArithPageFull13}{\hidepaste}
+\tab{5}\spadcommand{0$Factored(Integer)}
+\indentrel{3}\begin{verbatim}
+   (13)  0
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty13}
+\begin{paste}{ugxFactoredArithPageEmpty13}{ugxFactoredArithPagePatch13}
+\pastebutton{ugxFactoredArithPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{0$Factored(Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPagePatch14}
+\begin{paste}{ugxFactoredArithPageFull14}{ugxFactoredArithPageEmpty14}
+\pastebutton{ugxFactoredArithPageFull14}{\hidepaste}
+\tab{5}\spadcommand{1$Factored(Integer)}
+\indentrel{3}\begin{verbatim}
+   (14)  1
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredArithPageEmpty14}
+\begin{paste}{ugxFactoredArithPageEmpty14}{ugxFactoredArithPagePatch14}
+\pastebutton{ugxFactoredArithPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{1$Factored(Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPagePatch1}
+\begin{paste}{ugxFactoredDecompPageFull1}{ugxFactoredDecompPageEmpty1}
+\pastebutton{ugxFactoredDecompPageFull1}{\hidepaste}
+\tab{5}\spadcommand{g := factor(4312)\bound{g }}
+\indentrel{3}\begin{verbatim}
+         3 2
+   (1)  2 7 11
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPageEmpty1}
+\begin{paste}{ugxFactoredDecompPageEmpty1}{ugxFactoredDecompPagePatch1}
+\pastebutton{ugxFactoredDecompPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{g := factor(4312)\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPagePatch2}
+\begin{paste}{ugxFactoredDecompPageFull2}{ugxFactoredDecompPageEmpty2}
+\pastebutton{ugxFactoredDecompPageFull2}{\hidepaste}
+\tab{5}\spadcommand{unit(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (2)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPageEmpty2}
+\begin{paste}{ugxFactoredDecompPageEmpty2}{ugxFactoredDecompPagePatch2}
+\pastebutton{ugxFactoredDecompPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{unit(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPagePatch3}
+\begin{paste}{ugxFactoredDecompPageFull3}{ugxFactoredDecompPageEmpty3}
+\pastebutton{ugxFactoredDecompPageFull3}{\hidepaste}
+\tab{5}\spadcommand{numberOfFactors(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (3)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPageEmpty3}
+\begin{paste}{ugxFactoredDecompPageEmpty3}{ugxFactoredDecompPagePatch3}
+\pastebutton{ugxFactoredDecompPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{numberOfFactors(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPagePatch4}
+\begin{paste}{ugxFactoredDecompPageFull4}{ugxFactoredDecompPageEmpty4}
+\pastebutton{ugxFactoredDecompPageFull4}{\hidepaste}
+\tab{5}\spadcommand{[nthFactor(g,i) for i in 1..numberOfFactors(g)]\free{g }}
+\indentrel{3}\begin{verbatim}
+   (4)  [2,7,11]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPageEmpty4}
+\begin{paste}{ugxFactoredDecompPageEmpty4}{ugxFactoredDecompPagePatch4}
+\pastebutton{ugxFactoredDecompPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{[nthFactor(g,i) for i in 1..numberOfFactors(g)]\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPagePatch5}
+\begin{paste}{ugxFactoredDecompPageFull5}{ugxFactoredDecompPageEmpty5}
+\pastebutton{ugxFactoredDecompPageFull5}{\hidepaste}
+\tab{5}\spadcommand{[nthExponent(g,i) for i in 1..numberOfFactors(g)]\free{g }}
+\indentrel{3}\begin{verbatim}
+   (5)  [3,2,1]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPageEmpty5}
+\begin{paste}{ugxFactoredDecompPageEmpty5}{ugxFactoredDecompPagePatch5}
+\pastebutton{ugxFactoredDecompPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{[nthExponent(g,i) for i in 1..numberOfFactors(g)]\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPagePatch6}
+\begin{paste}{ugxFactoredDecompPageFull6}{ugxFactoredDecompPageEmpty6}
+\pastebutton{ugxFactoredDecompPageFull6}{\hidepaste}
+\tab{5}\spadcommand{[nthFlag(g,i) for i in 1..numberOfFactors(g)]\free{g }}
+\indentrel{3}\begin{verbatim}
+   (6)  ["prime","prime","prime"]
+         Type: List Union("nil","sqfr","irred","prime")
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPageEmpty6}
+\begin{paste}{ugxFactoredDecompPageEmpty6}{ugxFactoredDecompPagePatch6}
+\pastebutton{ugxFactoredDecompPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{[nthFlag(g,i) for i in 1..numberOfFactors(g)]\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPagePatch7}
+\begin{paste}{ugxFactoredDecompPageFull7}{ugxFactoredDecompPageEmpty7}
+\pastebutton{ugxFactoredDecompPageFull7}{\hidepaste}
+\tab{5}\spadcommand{factorList(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (7)
+   [[flg= "prime",fctr= 2,xpnt= 3],
+    [flg= "prime",fctr= 7,xpnt= 2],
+    [flg= "prime",fctr= 11,xpnt= 1]]
+Type: List Record(flg: Union("nil","sqfr","irred","prime"),fctr: Integer,xpnt: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPageEmpty7}
+\begin{paste}{ugxFactoredDecompPageEmpty7}{ugxFactoredDecompPagePatch7}
+\pastebutton{ugxFactoredDecompPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{factorList(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPagePatch8}
+\begin{paste}{ugxFactoredDecompPageFull8}{ugxFactoredDecompPageEmpty8}
+\pastebutton{ugxFactoredDecompPageFull8}{\hidepaste}
+\tab{5}\spadcommand{factors(g)\free{g }\bound{prev1 }}
+\indentrel{3}\begin{verbatim}
+   (8)
+   [[factor= 2,exponent= 3], [factor= 7,exponent= 2],
+    [factor= 11,exponent= 1]]
+   Type: List Record(factor: Integer,exponent: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPageEmpty8}
+\begin{paste}{ugxFactoredDecompPageEmpty8}{ugxFactoredDecompPagePatch8}
+\pastebutton{ugxFactoredDecompPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{factors(g)\free{g }\bound{prev1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPagePatch9}
+\begin{paste}{ugxFactoredDecompPageFull9}{ugxFactoredDecompPageEmpty9}
+\pastebutton{ugxFactoredDecompPageFull9}{\hidepaste}
+\tab{5}\spadcommand{first(\%).factor\free{prev1 }}
+\indentrel{3}\begin{verbatim}
+   (9)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredDecompPageEmpty9}
+\begin{paste}{ugxFactoredDecompPageEmpty9}{ugxFactoredDecompPagePatch9}
+\pastebutton{ugxFactoredDecompPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{first(\%).factor\free{prev1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredExpandPagePatch1}
+\begin{paste}{ugxFactoredExpandPageFull1}{ugxFactoredExpandPageEmpty1}
+\pastebutton{ugxFactoredExpandPageFull1}{\hidepaste}
+\tab{5}\spadcommand{g := factor(4312)\bound{g }}
+\indentrel{3}\begin{verbatim}
+         3 2
+   (1)  2 7 11
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredExpandPageEmpty1}
+\begin{paste}{ugxFactoredExpandPageEmpty1}{ugxFactoredExpandPagePatch1}
+\pastebutton{ugxFactoredExpandPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{g := factor(4312)\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredExpandPagePatch2}
+\begin{paste}{ugxFactoredExpandPageFull2}{ugxFactoredExpandPageEmpty2}
+\pastebutton{ugxFactoredExpandPageFull2}{\hidepaste}
+\tab{5}\spadcommand{expand(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (2)  4312
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredExpandPageEmpty2}
+\begin{paste}{ugxFactoredExpandPageEmpty2}{ugxFactoredExpandPagePatch2}
+\pastebutton{ugxFactoredExpandPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{expand(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredExpandPagePatch3}
+\begin{paste}{ugxFactoredExpandPageFull3}{ugxFactoredExpandPageEmpty3}
+\pastebutton{ugxFactoredExpandPageFull3}{\hidepaste}
+\tab{5}\spadcommand{reduce(*,[t.factor for t in factors(g)])\free{g }}
+\indentrel{3}\begin{verbatim}
+   (3)  154
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredExpandPageEmpty3}
+\begin{paste}{ugxFactoredExpandPageEmpty3}{ugxFactoredExpandPagePatch3}
+\pastebutton{ugxFactoredExpandPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{reduce(*,[t.factor for t in factors(g)])\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredVarPagePatch1}
+\begin{paste}{ugxFactoredVarPageFull1}{ugxFactoredVarPageEmpty1}
+\pastebutton{ugxFactoredVarPageFull1}{\hidepaste}
+\tab{5}\spadcommand{p := (4*x*x-12*x+9)*y*y + (4*x*x-12*x+9)*y + 28*x*x - 84*x + 63\bound{p }}
+\indentrel{3}\begin{verbatim}
+   (1)
+      2            2      2                  2
+   (4x  - 12x + 9)y  + (4x  - 12x + 9)y + 28x  - 84x + 63
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredVarPageEmpty1}
+\begin{paste}{ugxFactoredVarPageEmpty1}{ugxFactoredVarPagePatch1}
+\pastebutton{ugxFactoredVarPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{p := (4*x*x-12*x+9)*y*y + (4*x*x-12*x+9)*y + 28*x*x - 84*x + 63\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredVarPagePatch2}
+\begin{paste}{ugxFactoredVarPageFull2}{ugxFactoredVarPageEmpty2}
+\pastebutton{ugxFactoredVarPageFull2}{\hidepaste}
+\tab{5}\spadcommand{fp := factor(p)\free{p }\bound{fp }}
+\indentrel{3}\begin{verbatim}
+                2  2
+   (2)  (2x - 3) (y  + y + 7)
+                      Type: Factored Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredVarPageEmpty2}
+\begin{paste}{ugxFactoredVarPageEmpty2}{ugxFactoredVarPagePatch2}
+\pastebutton{ugxFactoredVarPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{fp := factor(p)\free{p }\bound{fp }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredVarPagePatch3}
+\begin{paste}{ugxFactoredVarPageFull3}{ugxFactoredVarPageEmpty3}
+\pastebutton{ugxFactoredVarPageFull3}{\hidepaste}
+\tab{5}\spadcommand{D(p,x)\free{p }}
+\indentrel{3}\begin{verbatim}
+                  2
+   (3)  (8x - 12)y  + (8x - 12)y + 56x - 84
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredVarPageEmpty3}
+\begin{paste}{ugxFactoredVarPageEmpty3}{ugxFactoredVarPagePatch3}
+\pastebutton{ugxFactoredVarPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{D(p,x)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredVarPagePatch4}
+\begin{paste}{ugxFactoredVarPageFull4}{ugxFactoredVarPageEmpty4}
+\pastebutton{ugxFactoredVarPageFull4}{\hidepaste}
+\tab{5}\spadcommand{D(fp,x)\free{fp }\bound{prev1 }}
+\indentrel{3}\begin{verbatim}
+                   2
+   (4)  4(2x - 3)(y  + y + 7)
+                      Type: Factored Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredVarPageEmpty4}
+\begin{paste}{ugxFactoredVarPageEmpty4}{ugxFactoredVarPagePatch4}
+\pastebutton{ugxFactoredVarPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{D(fp,x)\free{fp }\bound{prev1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredVarPagePatch5}
+\begin{paste}{ugxFactoredVarPageFull5}{ugxFactoredVarPageEmpty5}
+\pastebutton{ugxFactoredVarPageFull5}{\hidepaste}
+\tab{5}\spadcommand{numberOfFactors(\%)\free{prev1 }}
+\indentrel{3}\begin{verbatim}
+   (5)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredVarPageEmpty5}
+\begin{paste}{ugxFactoredVarPageEmpty5}{ugxFactoredVarPagePatch5}
+\pastebutton{ugxFactoredVarPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{numberOfFactors(\%)\free{prev1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPagePatch1}
+\begin{paste}{ugxFactoredNewPageFull1}{ugxFactoredNewPageEmpty1}
+\pastebutton{ugxFactoredNewPageFull1}{\hidepaste}
+\tab{5}\spadcommand{nilFactor(24,2)\bound{prev2 }}
+\indentrel{3}\begin{verbatim}
+          2
+   (1)  24
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPageEmpty1}
+\begin{paste}{ugxFactoredNewPageEmpty1}{ugxFactoredNewPagePatch1}
+\pastebutton{ugxFactoredNewPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{nilFactor(24,2)\bound{prev2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPagePatch2}
+\begin{paste}{ugxFactoredNewPageFull2}{ugxFactoredNewPageEmpty2}
+\pastebutton{ugxFactoredNewPageFull2}{\hidepaste}
+\tab{5}\spadcommand{nthFlag(\%,1)\free{prev2 }}
+\indentrel{3}\begin{verbatim}
+   (2)  "nil"
+                                 Type: Union("nil",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPageEmpty2}
+\begin{paste}{ugxFactoredNewPageEmpty2}{ugxFactoredNewPagePatch2}
+\pastebutton{ugxFactoredNewPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{nthFlag(\%,1)\free{prev2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPagePatch3}
+\begin{paste}{ugxFactoredNewPageFull3}{ugxFactoredNewPageEmpty3}
+\pastebutton{ugxFactoredNewPageFull3}{\hidepaste}
+\tab{5}\spadcommand{sqfrFactor(30,2)\bound{prev3 }}
+\indentrel{3}\begin{verbatim}
+          2
+   (3)  30
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPageEmpty3}
+\begin{paste}{ugxFactoredNewPageEmpty3}{ugxFactoredNewPagePatch3}
+\pastebutton{ugxFactoredNewPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{sqfrFactor(30,2)\bound{prev3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPagePatch4}
+\begin{paste}{ugxFactoredNewPageFull4}{ugxFactoredNewPageEmpty4}
+\pastebutton{ugxFactoredNewPageFull4}{\hidepaste}
+\tab{5}\spadcommand{irreducibleFactor(13,10)\bound{prev4 }}
+\indentrel{3}\begin{verbatim}
+          10
+   (4)  13
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPageEmpty4}
+\begin{paste}{ugxFactoredNewPageEmpty4}{ugxFactoredNewPagePatch4}
+\pastebutton{ugxFactoredNewPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{irreducibleFactor(13,10)\bound{prev4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPagePatch5}
+\begin{paste}{ugxFactoredNewPageFull5}{ugxFactoredNewPageEmpty5}
+\pastebutton{ugxFactoredNewPageFull5}{\hidepaste}
+\tab{5}\spadcommand{primeFactor(11,5)\bound{prev5 }}
+\indentrel{3}\begin{verbatim}
+          5
+   (5)  11
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPageEmpty5}
+\begin{paste}{ugxFactoredNewPageEmpty5}{ugxFactoredNewPagePatch5}
+\pastebutton{ugxFactoredNewPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{primeFactor(11,5)\bound{prev5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPagePatch6}
+\begin{paste}{ugxFactoredNewPageFull6}{ugxFactoredNewPageEmpty6}
+\pastebutton{ugxFactoredNewPageFull6}{\hidepaste}
+\tab{5}\spadcommand{h := factor(-720)\bound{h }}
+\indentrel{3}\begin{verbatim}
+           4 2
+   (6)  - 2 3 5
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPageEmpty6}
+\begin{paste}{ugxFactoredNewPageEmpty6}{ugxFactoredNewPagePatch6}
+\pastebutton{ugxFactoredNewPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{h := factor(-720)\bound{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPagePatch7}
+\begin{paste}{ugxFactoredNewPageFull7}{ugxFactoredNewPageEmpty7}
+\pastebutton{ugxFactoredNewPageFull7}{\hidepaste}
+\tab{5}\spadcommand{h - makeFR(unit(h),factorList(h))\free{h }}
+\indentrel{3}\begin{verbatim}
+   (7)  0
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxFactoredNewPageEmpty7}
+\begin{paste}{ugxFactoredNewPageEmpty7}{ugxFactoredNewPagePatch7}
+\pastebutton{ugxFactoredNewPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{h - makeFR(unit(h),factorList(h))\free{h }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/FR2.ht b/src/hyper/pages/FR2.ht
new file mode 100644
index 00000000..f2c00049
--- /dev/null
+++ b/src/hyper/pages/FR2.ht
@@ -0,0 +1,66 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\FactoredFunctionsTwoXmpTitle}{FactoredFunctions2}
+\newcommand{\FactoredFunctionsTwoXmpNumber}{9.23}
+%
+% =====================================================================
+\begin{page}{FactoredFunctionsTwoXmpPage}{9.23 FactoredFunctions2}
+% =====================================================================
+\beginscroll
+
+%%
+%% fr2.htex
+%%
+%% FactoredFunctions2
+%%
+
+The \spadtype{FactoredFunctions2} package implements one operation,
+\spadfunFrom{map}{FactoredFunctions2}, for applying an operation to every
+base in a factored object and to the unit.
+\xtc{
+}{
+\spadpaste{double(x) == x + x \bound{double}}
+}
+\xtc{
+}{
+\spadpaste{f := factor(720) \bound{f}}
+}
+\xtc{
+Actually, the \spadfunFrom{map}{FactoredFunctions2} operation used
+in this example comes from \spadtype{Factored} itself, since
+\userfun{double} takes an integer argument and returns an integer
+result.
+}{
+\spadpaste{map(double,f) \free{f}\free{double}}
+}
+\xtc{
+If we want to use an operation that returns an object that has a type
+different from the operation's argument,
+the \spadfunFrom{map}{FactoredFunctions2} in \spadtype{Factored}
+cannot be used and we use the one in \spadtype{FactoredFunctions2}.
+}{
+\spadpaste{makePoly(b) == x + b \bound{makePoly}}
+}
+\xtc{
+In fact, the ``2'' in the name of the package means that we might
+be using factored objects of two different types.
+}{
+\spadpaste{g := map(makePoly,f) \free{f}\free{makePoly}\bound{g}}
+}
+It is important to note that both versions of \spadfunFrom{map}{FactoredFunctions2}
+destroy any information known about the bases (the fact that they are prime,
+for instance).
+\xtc{
+The flags for each base are set to ``nil'' in the object returned
+by \spadfunFrom{map}{FactoredFunctions2}.
+}{
+\spadpaste{nthFlag(g,1) \free{g}}
+}
+
+For more information about factored objects and their use, see
+\downlink{`Factored'}{FactoredXmpPage}\ignore{Factored} and
+\downlink{``\ugProblemGaloisTitle''}{ugProblemGaloisPage} in Section \ugProblemGaloisNumber\ignore{ugProblemGalois}.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/FR2.pht b/src/hyper/pages/FR2.pht
new file mode 100644
index 00000000..5b26c40f
--- /dev/null
+++ b/src/hyper/pages/FR2.pht
@@ -0,0 +1,97 @@
+\begin{patch}{FactoredFunctionsTwoXmpPagePatch1}
+\begin{paste}{FactoredFunctionsTwoXmpPageFull1}{FactoredFunctionsTwoXmpPageEmpty1}
+\pastebutton{FactoredFunctionsTwoXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{double(x) == x + x\bound{double }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FactoredFunctionsTwoXmpPageEmpty1}
+\begin{paste}{FactoredFunctionsTwoXmpPageEmpty1}{FactoredFunctionsTwoXmpPagePatch1}
+\pastebutton{FactoredFunctionsTwoXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{double(x) == x + x\bound{double }}
+\end{paste}\end{patch}
+
+\begin{patch}{FactoredFunctionsTwoXmpPagePatch2}
+\begin{paste}{FactoredFunctionsTwoXmpPageFull2}{FactoredFunctionsTwoXmpPageEmpty2}
+\pastebutton{FactoredFunctionsTwoXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f := factor(720)\bound{f }}
+\indentrel{3}\begin{verbatim}
+         4 2
+   (2)  2 3 5
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FactoredFunctionsTwoXmpPageEmpty2}
+\begin{paste}{FactoredFunctionsTwoXmpPageEmpty2}{FactoredFunctionsTwoXmpPagePatch2}
+\pastebutton{FactoredFunctionsTwoXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f := factor(720)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{FactoredFunctionsTwoXmpPagePatch3}
+\begin{paste}{FactoredFunctionsTwoXmpPageFull3}{FactoredFunctionsTwoXmpPageEmpty3}
+\pastebutton{FactoredFunctionsTwoXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{map(double,f)\free{f }\free{double }}
+\indentrel{3}\begin{verbatim}
+           4 2
+   (3)  2 4 6 10
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FactoredFunctionsTwoXmpPageEmpty3}
+\begin{paste}{FactoredFunctionsTwoXmpPageEmpty3}{FactoredFunctionsTwoXmpPagePatch3}
+\pastebutton{FactoredFunctionsTwoXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{map(double,f)\free{f }\free{double }}
+\end{paste}\end{patch}
+
+\begin{patch}{FactoredFunctionsTwoXmpPagePatch4}
+\begin{paste}{FactoredFunctionsTwoXmpPageFull4}{FactoredFunctionsTwoXmpPageEmpty4}
+\pastebutton{FactoredFunctionsTwoXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{makePoly(b) == x + b\bound{makePoly }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FactoredFunctionsTwoXmpPageEmpty4}
+\begin{paste}{FactoredFunctionsTwoXmpPageEmpty4}{FactoredFunctionsTwoXmpPagePatch4}
+\pastebutton{FactoredFunctionsTwoXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{makePoly(b) == x + b\bound{makePoly }}
+\end{paste}\end{patch}
+
+\begin{patch}{FactoredFunctionsTwoXmpPagePatch5}
+\begin{paste}{FactoredFunctionsTwoXmpPageFull5}{FactoredFunctionsTwoXmpPageEmpty5}
+\pastebutton{FactoredFunctionsTwoXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{g := map(makePoly,f)\free{f }\free{makePoly }\bound{g }}
+\indentrel{3}\begin{verbatim}
+                      4       2
+   (5)  (x + 1)(x + 2) (x + 3) (x + 5)
+                      Type: Factored Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FactoredFunctionsTwoXmpPageEmpty5}
+\begin{paste}{FactoredFunctionsTwoXmpPageEmpty5}{FactoredFunctionsTwoXmpPagePatch5}
+\pastebutton{FactoredFunctionsTwoXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{g := map(makePoly,f)\free{f }\free{makePoly }\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{FactoredFunctionsTwoXmpPagePatch6}
+\begin{paste}{FactoredFunctionsTwoXmpPageFull6}{FactoredFunctionsTwoXmpPageEmpty6}
+\pastebutton{FactoredFunctionsTwoXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{nthFlag(g,1)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (6)  "nil"
+                                 Type: Union("nil",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FactoredFunctionsTwoXmpPageEmpty6}
+\begin{paste}{FactoredFunctionsTwoXmpPageEmpty6}{FactoredFunctionsTwoXmpPagePatch6}
+\pastebutton{FactoredFunctionsTwoXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{nthFlag(g,1)\free{g }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/FRAC.ht b/src/hyper/pages/FRAC.ht
new file mode 100644
index 00000000..eec3322f
--- /dev/null
+++ b/src/hyper/pages/FRAC.ht
@@ -0,0 +1,106 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\FractionXmpTitle}{Fraction}
+\newcommand{\FractionXmpNumber}{9.28}
+%
+% =====================================================================
+\begin{page}{FractionXmpPage}{9.28 Fraction}
+% =====================================================================
+\beginscroll
+%
+
+The \spadtype{Fraction} domain implements quotients.
+%-% \HDindex{fraction}{FractionXmpPage}{9.28}{Fraction}
+The elements must belong to a domain of category \spadtype{IntegralDomain}:
+multiplication must be commutative and the product of two non-zero
+elements must not be zero.
+This allows you to make fractions of most things you would think of,
+but don't expect to create a fraction of two matrices!
+The abbreviation for \spadtype{Fraction} is \spadtype{FRAC}.
+
+\xtc{
+Use \spadopFrom{/}{Fraction} to create a fraction.
+}{
+\spadpaste{a := 11/12 \bound{a}}
+}
+\xtc{
+}{
+\spadpaste{b := 23/24 \bound{b}}
+}
+\xtc{
+The standard arithmetic operations are available.
+}{
+\spadpaste{3 - a*b**2 + a + b/a \free{a}\free{b}}
+}
+\xtc{
+Extract the numerator and denominator by using
+%-% \HDindex{numerator}{FractionXmpPage}{9.28}{Fraction}
+\spadfunFrom{numer}{Fraction} and \spadfunFrom{denom}{Fraction},
+%-% \HDindex{denominator}{FractionXmpPage}{9.28}{Fraction}
+respectively.
+}{
+\spadpaste{numer(a) \free{a}}
+}
+\xtc{
+}{
+\spadpaste{denom(b) \free{b}}
+}
+Operations like \spadfunFrom{max}{Fraction}, \spadfunFrom{min}{Fraction},
+\spadfunFrom{negative?}{Fraction}, \spadfunFrom{positive?}{Fraction} and
+\spadfunFrom{zero?}{Fraction} are all available if they are provided for
+the numerators and denominators.
+See \downlink{`Integer'}{IntegerXmpPage}\ignore{Integer} for examples.
+
+Don't expect a useful answer from \spadfunFrom{factor}{Fraction},
+\spadfunFrom{gcd}{Fraction} or \spadfunFrom{lcm}{Fraction} if you apply
+them to fractions.
+\xtc{
+}{
+\spadpaste{r := (x**2 + 2*x + 1)/(x**2 - 2*x + 1) \bound{r}}
+}
+\xtc{
+Since all non-zero fractions are invertible, these operations have trivial
+definitions.
+}{
+\spadpaste{factor(r) \free{r}}
+}
+\xtc{
+Use \spadfunFrom{map}{Fraction} to apply \spadfunFrom{factor}{Fraction} to
+the numerator and denominator, which is probably what you mean.
+}{
+\spadpaste{map(factor,r) \free{r}}
+}
+
+\xtc{
+Other forms of fractions are available.
+%-% \HDindex{fraction!continued}{FractionXmpPage}{9.28}{Fraction}
+Use \spadfun{continuedFraction} to create a continued fraction.
+%-% \HDindex{continued fraction}{FractionXmpPage}{9.28}{Fraction}
+}{
+\spadpaste{continuedFraction(7/12)}
+}
+\xtc{
+Use \spadfun{partialFraction} to create a partial fraction.
+%-% \HDindex{fraction!partial}{FractionXmpPage}{9.28}{Fraction}
+See \downlink{`ContinuedFraction'}{ContinuedFractionXmpPage}\ignore{ContinuedFraction} and \downlink{`PartialFraction'}{PartialFractionXmpPage}\ignore{PartialFraction} for
+additional information and examples.
+%-% \HDindex{partial fraction}{FractionXmpPage}{9.28}{Fraction}
+}{
+\spadpaste{partialFraction(7,12)}
+}
+
+\xtc{
+Use conversion to create alternative views of fractions with objects
+moved in and out of the numerator and denominator.
+}{
+\spadpaste{g := 2/3 + 4/5*\%i \bound{g}}
+}
+\xtc{
+Conversion is discussed in detail in \downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert}.
+}{
+\spadpaste{g :: FRAC COMPLEX INT \free{g}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/FRAC.pht b/src/hyper/pages/FRAC.pht
new file mode 100644
index 00000000..f5fa2f87
--- /dev/null
+++ b/src/hyper/pages/FRAC.pht
@@ -0,0 +1,219 @@
+\begin{patch}{FractionXmpPagePatch1}
+\begin{paste}{FractionXmpPageFull1}{FractionXmpPageEmpty1}
+\pastebutton{FractionXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{a := 11/12\bound{a }}
+\indentrel{3}\begin{verbatim}
+        11
+   (1)  ÄÄ
+        12
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPageEmpty1}
+\begin{paste}{FractionXmpPageEmpty1}{FractionXmpPagePatch1}
+\pastebutton{FractionXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{a := 11/12\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPagePatch2}
+\begin{paste}{FractionXmpPageFull2}{FractionXmpPageEmpty2}
+\pastebutton{FractionXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{b := 23/24\bound{b }}
+\indentrel{3}\begin{verbatim}
+        23
+   (2)  ÄÄ
+        24
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPageEmpty2}
+\begin{paste}{FractionXmpPageEmpty2}{FractionXmpPagePatch2}
+\pastebutton{FractionXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{b := 23/24\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPagePatch3}
+\begin{paste}{FractionXmpPageFull3}{FractionXmpPageEmpty3}
+\pastebutton{FractionXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{3 - a*b**2 + a + b/a\free{a }\free{b }}
+\indentrel{3}\begin{verbatim}
+        313271
+   (3)  ÄÄÄÄÄÄ
+         76032
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPageEmpty3}
+\begin{paste}{FractionXmpPageEmpty3}{FractionXmpPagePatch3}
+\pastebutton{FractionXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{3 - a*b**2 + a + b/a\free{a }\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPagePatch4}
+\begin{paste}{FractionXmpPageFull4}{FractionXmpPageEmpty4}
+\pastebutton{FractionXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{numer(a)\free{a }}
+\indentrel{3}\begin{verbatim}
+   (4)  11
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPageEmpty4}
+\begin{paste}{FractionXmpPageEmpty4}{FractionXmpPagePatch4}
+\pastebutton{FractionXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{numer(a)\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPagePatch5}
+\begin{paste}{FractionXmpPageFull5}{FractionXmpPageEmpty5}
+\pastebutton{FractionXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{denom(b)\free{b }}
+\indentrel{3}\begin{verbatim}
+   (5)  24
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPageEmpty5}
+\begin{paste}{FractionXmpPageEmpty5}{FractionXmpPagePatch5}
+\pastebutton{FractionXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{denom(b)\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPagePatch6}
+\begin{paste}{FractionXmpPageFull6}{FractionXmpPageEmpty6}
+\pastebutton{FractionXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{r := (x**2 + 2*x + 1)/(x**2 - 2*x + 1)\bound{r }}
+\indentrel{3}\begin{verbatim}
+         2
+        x  + 2x + 1
+   (6)  ÄÄÄÄÄÄÄÄÄÄÄ
+         2
+        x  - 2x + 1
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPageEmpty6}
+\begin{paste}{FractionXmpPageEmpty6}{FractionXmpPagePatch6}
+\pastebutton{FractionXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{r := (x**2 + 2*x + 1)/(x**2 - 2*x + 1)\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPagePatch7}
+\begin{paste}{FractionXmpPageFull7}{FractionXmpPageEmpty7}
+\pastebutton{FractionXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{factor(r)\free{r }}
+\indentrel{3}\begin{verbatim}
+         2
+        x  + 2x + 1
+   (7)  ÄÄÄÄÄÄÄÄÄÄÄ
+         2
+        x  - 2x + 1
+             Type: Factored Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPageEmpty7}
+\begin{paste}{FractionXmpPageEmpty7}{FractionXmpPagePatch7}
+\pastebutton{FractionXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{factor(r)\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPagePatch8}
+\begin{paste}{FractionXmpPageFull8}{FractionXmpPageEmpty8}
+\pastebutton{FractionXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{map(factor,r)\free{r }}
+\indentrel{3}\begin{verbatim}
+               2
+        (x + 1)
+   (8)  ÄÄÄÄÄÄÄÄ
+               2
+        (x - 1)
+             Type: Fraction Factored Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPageEmpty8}
+\begin{paste}{FractionXmpPageEmpty8}{FractionXmpPagePatch8}
+\pastebutton{FractionXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{map(factor,r)\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPagePatch9}
+\begin{paste}{FractionXmpPageFull9}{FractionXmpPageEmpty9}
+\pastebutton{FractionXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{continuedFraction(7/12)}
+\indentrel{3}\begin{verbatim}
+          1 ³     1 ³     1 ³     1 ³
+   (9)  ÚÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÙ
+        ³ 1     ³ 1     ³ 2     ³ 2
+                        Type: ContinuedFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPageEmpty9}
+\begin{paste}{FractionXmpPageEmpty9}{FractionXmpPagePatch9}
+\pastebutton{FractionXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{continuedFraction(7/12)}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPagePatch10}
+\begin{paste}{FractionXmpPageFull10}{FractionXmpPageEmpty10}
+\pastebutton{FractionXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{partialFraction(7,12)}
+\indentrel{3}\begin{verbatim}
+              3   1
+   (10)  1 - ÄÄ + Ä
+              2   3
+             2
+                          Type: PartialFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPageEmpty10}
+\begin{paste}{FractionXmpPageEmpty10}{FractionXmpPagePatch10}
+\pastebutton{FractionXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{partialFraction(7,12)}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPagePatch11}
+\begin{paste}{FractionXmpPageFull11}{FractionXmpPageEmpty11}
+\pastebutton{FractionXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{g := 2/3 + 4/5*\%i\bound{g }}
+\indentrel{3}\begin{verbatim}
+         2   4
+   (11)  Ä + Ä %i
+         3   5
+                         Type: Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPageEmpty11}
+\begin{paste}{FractionXmpPageEmpty11}{FractionXmpPagePatch11}
+\pastebutton{FractionXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{g := 2/3 + 4/5*\%i\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPagePatch12}
+\begin{paste}{FractionXmpPageFull12}{FractionXmpPageEmpty12}
+\pastebutton{FractionXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{g :: FRAC COMPLEX INT\free{g }}
+\indentrel{3}\begin{verbatim}
+         10 + 12%i
+   (12)  ÄÄÄÄÄÄÄÄÄ
+             15
+                         Type: Fraction Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionXmpPageEmpty12}
+\begin{paste}{FractionXmpPageEmpty12}{FractionXmpPagePatch12}
+\pastebutton{FractionXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{g :: FRAC COMPLEX INT\free{g }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/GBF.ht b/src/hyper/pages/GBF.ht
new file mode 100644
index 00000000..64213963
--- /dev/null
+++ b/src/hyper/pages/GBF.ht
@@ -0,0 +1,103 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\GroebnerFactorizationPackageXmpTitle}{GroebnerFactorizationPackage}
+\newcommand{\GroebnerFactorizationPackageXmpNumber}{9.31}
+%
+% =====================================================================
+\begin{page}{GroebnerFactorizationPackageXmpPage}{9.31 GroebnerFactorizationPackage}
+% =====================================================================
+\beginscroll
+%%
+%%  GBF
+%%
+%%   GroebnerFactorizationPackage
+%%
+%%  J. Grabmeier 26 04 91
+
+Solving systems of polynomial equations with the
+\texht{Gr\"{o}bner}{Groebner}
+%-% \HDindex{Groebner basis@{Gr\protect\"{o}bner basis}}{GroebnerFactorizationPackageXmpPage}{9.31}{GroebnerFactorizationPackage}
+basis algorithm can often be very time consuming because, in general,
+%-% \HDindex{basis!Groebner@{Gr\protect\"{o}bner}}{GroebnerFactorizationPackageXmpPage}{9.31}{GroebnerFactorizationPackage}
+the algorithm has exponential run-time.
+These systems, which often come from concrete applications,
+frequently have symmetries which are not taken advantage of by the
+algorithm.
+However, it often happens in this case
+that the polynomials which occur during the
+\texht{Gr\"{o}bner}{Groebner} calculations are reducible.
+Since \Language{} has an excellent polynomial factorization algorithm, it is
+very natural to combine the
+\texht{Gr\"{o}bner}{Groebner} and factorization algorithms.
+
+\spadtype{GroebnerFactorizationPackage} exports the
+\axiomFunFrom{groebnerFactorize}{GroebnerFactorizationPackage}
+operation which implements a modified
+\texht{Gr\"{o}bner}{Groebner} basis algorithm.
+In this algorithm, each polynomial that is to be put into the
+partial list of the basis is first factored.
+The remaining calculation is split into as many parts as there are
+irreducible factors.
+Call these factors \texht{$p_1, \ldots,p_n.$}{\spad{p1, ..., pn.}}
+In the branches corresponding to \texht{$p_2,
+\ldots,p_n,$}{\spad{p2, ..., pn,}} the factor
+\texht{$p_1$}{\spad{p1}} can be divided out, and so on.
+This package also contains operations that allow you to specify the
+polynomials that are not zero on the common roots of the final
+\texht{Gr\"{o}bner}{Groebner} basis.
+
+Here is an example from chemistry.
+%-% \HDindex{chemistry}{GroebnerFactorizationPackageXmpPage}{9.31}{GroebnerFactorizationPackage}
+In a theoretical model of the cyclohexan \texht{${\rm C}_6{\rm H}_{12}$}{C6H12},
+the six carbon atoms each sit in the center of gravity of a
+%-% \HDindex{cyclohexan}{GroebnerFactorizationPackageXmpPage}{9.31}{GroebnerFactorizationPackage}
+tetrahedron that has two hydrogen atoms and two carbon atoms at its
+corners.
+We first normalize and set the length of each edge to 1.
+Hence, the distances of one fixed carbon atom to each of its
+immediate neighbours is 1.
+We will denote the distances to the other three carbon atoms by
+\smath{x}, \smath{y} and \smath{z}.
+
+\texht{A.~Dress}{A. Dress} developed a theory to decide whether a set of points
+%% reference?
+and distances between them can be realized in an \smath{n}-dimensional space.
+Here, of course, we have \smath{n = 3}.
+\xtc{
+}{
+\spadpaste{mfzn : SQMATRIX(6,DMP([x,y,z],Fraction INT)) := [[0,1,1,1,1,1], [1,0,1,8/3,x,8/3], [1,1,0,1,8/3,y], [1,8/3,1,0,1,8/3], [1,x,8/3,1,0,1], [1,8/3,y,8/3,1,0]] \bound{mfzn}}
+}
+\xtc{
+For the cyclohexan, the distances have to satisfy this equation.
+}{
+\spadpaste{eq := determinant mfzn \free{mfzn}\bound{eq}}
+}
+\xtc{
+They also must satisfy the equations
+given by cyclic shifts of the indeterminates.
+}{
+\spadpaste{groebnerFactorize [eq, eval(eq, [x,y,z], [y,z,x]), eval(eq, [x,y,z], [z,x,y])] \free{eq}}
+}
+
+The union of the solutions of this list is the
+solution of our original problem.
+If we impose positivity conditions, we get two relevant ideals.
+One ideal is
+zero-dimensional, namely \smath{x = y = z = 11/3}, and this
+determines the ``boat'' form of the cyclohexan.
+The other ideal is one-dimensional,
+which means that we have a solution space given by one parameter.
+This gives the ``chair'' form of the cyclohexan.
+The parameter describes the angle of the ``back of the chair.''
+
+\axiomFunFrom{groebnerFactorize}{GroebnerFactorizationPackage}
+has an optional \axiomType{Boolean}-valued second argument.
+When it is \spad{true} partial results are displayed, since it may happen that the
+calculation does not terminate in a reasonable time.
+See the source code for \spadtype{GroebnerFactorizationPackage}
+in {\bf groebf\spadFileExt{}} for more details about the algorithms
+used.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/GBF.pht b/src/hyper/pages/GBF.pht
new file mode 100644
index 00000000..ea39b58e
--- /dev/null
+++ b/src/hyper/pages/GBF.pht
@@ -0,0 +1,110 @@
+\begin{patch}{GroebnerFactorizationPackageXmpPagePatch1}
+\begin{paste}{GroebnerFactorizationPackageXmpPageFull1}{GroebnerFactorizationPackageXmpPageEmpty1}
+\pastebutton{GroebnerFactorizationPackageXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{mfzn : SQMATRIX(6,DMP([x,y,z],Fraction INT)) := [[0,1,1,1,1,1], [1,0,1,8/3,x,8/3], [1,1,0,1,8/3,y], [1,8/3,1,0,1,8/3], [1,x,8/3,1,0,1], [1,8/3,y,8/3,1,0]]\bound{mfzn }}
+\indentrel{3}\begin{verbatim}
+        Ú0  1  1  1  1  1¿
+        ³                ³
+        ³         8     8³
+        ³1  0  1  Ä  x  ij
+        ³         3     3³
+        ³                ³
+        ³            8   ³
+        ³1  1  0  1  Ä  y³
+        ³            3   ³
+        ³                ³
+   (1)  ³   8           8³
+        ³1  Ä  1  0  1  ij
+        ³   3           3³
+        ³                ³
+        ³      8         ³
+        ³1  x  Ä  1  0  1³
+        ³      3         ³
+        ³                ³
+        ³   8     8      ³
+        ³1  Ä  y  Ä  1  0³
+        À   3     3      Ù
+Type: SquareMatrix(6,DistributedMultivariatePolynomial([x,y,z],Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{GroebnerFactorizationPackageXmpPageEmpty1}
+\begin{paste}{GroebnerFactorizationPackageXmpPageEmpty1}{GroebnerFactorizationPackageXmpPagePatch1}
+\pastebutton{GroebnerFactorizationPackageXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{mfzn : SQMATRIX(6,DMP([x,y,z],Fraction INT)) := [[0,1,1,1,1,1], [1,0,1,8/3,x,8/3], [1,1,0,1,8/3,y], [1,8/3,1,0,1,8/3], [1,x,8/3,1,0,1], [1,8/3,y,8/3,1,0]]\bound{mfzn }}
+\end{paste}\end{patch}
+
+\begin{patch}{GroebnerFactorizationPackageXmpPagePatch2}
+\begin{paste}{GroebnerFactorizationPackageXmpPageFull2}{GroebnerFactorizationPackageXmpPageEmpty2}
+\pastebutton{GroebnerFactorizationPackageXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{eq := determinant mfzn\free{mfzn }\bound{eq }}
+\indentrel{3}\begin{verbatim}
+   (2)
+        2 2   22  2    25  2   22    2   388       250
+     - x y  + ÄÄ x y - ÄÄ x  + ÄÄ x y  - ÄÄÄ x y - ÄÄÄ x
+               3        9       3         9         27
+   + 
+       25  2   250     14575
+     - ÄÄ y  - ÄÄÄ y + ÄÄÄÄÄ
+        9       27       81
+Type: DistributedMultivariatePolynomial([x,y,z],Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{GroebnerFactorizationPackageXmpPageEmpty2}
+\begin{paste}{GroebnerFactorizationPackageXmpPageEmpty2}{GroebnerFactorizationPackageXmpPagePatch2}
+\pastebutton{GroebnerFactorizationPackageXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{eq := determinant mfzn\free{mfzn }\bound{eq }}
+\end{paste}\end{patch}
+
+\begin{patch}{GroebnerFactorizationPackageXmpPagePatch3}
+\begin{paste}{GroebnerFactorizationPackageXmpPageFull3}{GroebnerFactorizationPackageXmpPageEmpty3}
+\pastebutton{GroebnerFactorizationPackageXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{groebnerFactorize [eq, eval(eq, [x,y,z], [y,z,x]), eval(eq, [x,y,z], [z,x,y])]\free{eq }}
+\indentrel{3}\begin{verbatim}
+   (3)
+   [
+                  22           22     22     121
+     [x y + x z - ÄÄ x + y z - ÄÄ y - ÄÄ z + ÄÄÄ,
+                   3            3      3      3
+
+            2   22       25        2   22       25
+         x z  - ÄÄ x z + ÄÄ x + y z  - ÄÄ y z + ÄÄ y
+                 3        9             3        9
+       + 
+           22  2   388     250
+         - ÄÄ z  + ÄÄÄ z + ÄÄÄ
+            3       9       27
+       ,
+
+          2 2   22  2    25  2   22    2   388       250
+         y z  - ÄÄ y z + ÄÄ y  - ÄÄ y z  + ÄÄÄ y z + ÄÄÄ y
+                 3        9       3         9         27
+       + 
+         25  2   250     14575
+         ÄÄ z  + ÄÄÄ z - ÄÄÄÄÄ
+          9       27       81
+       ]
+     ,
+             21994  2   21994     4427     463
+    [x + y - ÄÄÄÄÄ,y  - ÄÄÄÄÄ y + ÄÄÄÄ,z - ÄÄÄ],
+              5625       5625      675      87
+      2   1       11     5     265        2   38     265
+    [x  - Ä x z - ÄÄ x - Ä z + ÄÄÄ,y - z,z  - ÄÄ z + ÄÄÄ],
+          2        2     6      18             3      9
+         25     11     11        11     11     11
+    [x - ÄÄ,y - ÄÄ,z - ÄÄ], [x - ÄÄ,y - ÄÄ,z - ÄÄ],
+          9      3      3         3      3      3
+         5     5     5        19     5     5
+    [x + Ä,y + Ä,z + Ä], [x - ÄÄ,y + Ä,z + Ä]]
+         3     3     3         3     3     3
+Type: List List DistributedMultivariatePolynomial([x,y,z],Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{GroebnerFactorizationPackageXmpPageEmpty3}
+\begin{paste}{GroebnerFactorizationPackageXmpPageEmpty3}{GroebnerFactorizationPackageXmpPagePatch3}
+\pastebutton{GroebnerFactorizationPackageXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{groebnerFactorize [eq, eval(eq, [x,y,z], [y,z,x]), eval(eq, [x,y,z], [z,x,y])]\free{eq }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/GSTBL.ht b/src/hyper/pages/GSTBL.ht
new file mode 100644
index 00000000..6d5aaede
--- /dev/null
+++ b/src/hyper/pages/GSTBL.ht
@@ -0,0 +1,61 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\GeneralSparseTableXmpTitle}{GeneralSparseTable}
+\newcommand{\GeneralSparseTableXmpNumber}{9.30}
+%
+% =====================================================================
+\begin{page}{GeneralSparseTableXmpPage}{9.30 GeneralSparseTable}
+% =====================================================================
+\beginscroll
+%
+Sometimes when working with tables there is a natural value to use
+as the entry in all but a few cases.
+The \spadtype{GeneralSparseTable} constructor can be used to provide any
+table type with a default value for entries.
+See \downlink{`Table'}{TableXmpPage}\ignore{Table} for general information about tables.
+\showBlurb{GeneralSparseTable}
+
+Suppose we launched a fund-raising campaign to raise fifty thousand dollars.
+To record the contributions, we want a table with strings as keys
+(for the names) and integer entries (for the amount).
+In a data base of cash contributions, unless someone
+has been explicitly entered, it is reasonable to assume they have made
+a zero dollar contribution.
+\xtc{
+This creates a keyed access file with default entry \spad{0}.
+}{
+\spadpaste{patrons: GeneralSparseTable(String, Integer, KeyedAccessFile(Integer), 0) := table() ; \bound{patrons}}
+}
+\xtc{
+Now \spad{patrons} can be used just as any other table.
+Here we record two gifts.
+}{
+\spadpaste{patrons."Smith" := 10500 \free{patrons}\bound{smith}}
+}
+\xtc{
+}{
+\spadpaste{patrons."Jones" := 22000 \free{smith}\bound{jones}}
+}
+\xtc{
+Now let us look up the size of the contributions from Jones and Stingy.
+}{
+\spadpaste{patrons."Jones"  \free{jones}}
+}
+\xtc{
+}{
+\spadpaste{patrons."Stingy" \free{jones}}
+}
+\xtc{
+Have we met our seventy thousand dollar goal?
+}{
+\spadpaste{reduce(+, entries patrons) \free{jones}}
+}
+\noOutputXtc{
+So the project is cancelled and we can delete the data base:
+}{
+\spadpaste{)system rm -r kaf*.sdata \free{patrons}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/GSTBL.pht b/src/hyper/pages/GSTBL.pht
new file mode 100644
index 00000000..ff0b5cb4
--- /dev/null
+++ b/src/hyper/pages/GSTBL.pht
@@ -0,0 +1,109 @@
+\begin{patch}{GeneralSparseTableXmpPagePatch1}
+\begin{paste}{GeneralSparseTableXmpPageFull1}{GeneralSparseTableXmpPageEmpty1}
+\pastebutton{GeneralSparseTableXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{patrons: GeneralSparseTable(String, Integer, KeyedAccessFile(Integer), 0) := table() ;\bound{patrons }}
+\indentrel{3}\begin{verbatim}
+Type: GeneralSparseTable(String,Integer,KeyedAccessFile Integer,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{GeneralSparseTableXmpPageEmpty1}
+\begin{paste}{GeneralSparseTableXmpPageEmpty1}{GeneralSparseTableXmpPagePatch1}
+\pastebutton{GeneralSparseTableXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{patrons: GeneralSparseTable(String, Integer, KeyedAccessFile(Integer), 0) := table() ;\bound{patrons }}
+\end{paste}\end{patch}
+
+\begin{patch}{GeneralSparseTableXmpPagePatch2}
+\begin{paste}{GeneralSparseTableXmpPageFull2}{GeneralSparseTableXmpPageEmpty2}
+\pastebutton{GeneralSparseTableXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{patrons."Smith" := 10500\free{patrons }\bound{smith }}
+\indentrel{3}\begin{verbatim}
+   (2)  10500
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{GeneralSparseTableXmpPageEmpty2}
+\begin{paste}{GeneralSparseTableXmpPageEmpty2}{GeneralSparseTableXmpPagePatch2}
+\pastebutton{GeneralSparseTableXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{patrons."Smith" := 10500\free{patrons }\bound{smith }}
+\end{paste}\end{patch}
+
+\begin{patch}{GeneralSparseTableXmpPagePatch3}
+\begin{paste}{GeneralSparseTableXmpPageFull3}{GeneralSparseTableXmpPageEmpty3}
+\pastebutton{GeneralSparseTableXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{patrons."Jones" := 22000\free{smith }\bound{jones }}
+\indentrel{3}\begin{verbatim}
+   (3)  22000
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{GeneralSparseTableXmpPageEmpty3}
+\begin{paste}{GeneralSparseTableXmpPageEmpty3}{GeneralSparseTableXmpPagePatch3}
+\pastebutton{GeneralSparseTableXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{patrons."Jones" := 22000\free{smith }\bound{jones }}
+\end{paste}\end{patch}
+
+\begin{patch}{GeneralSparseTableXmpPagePatch4}
+\begin{paste}{GeneralSparseTableXmpPageFull4}{GeneralSparseTableXmpPageEmpty4}
+\pastebutton{GeneralSparseTableXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{patrons."Jones"\free{jones }}
+\indentrel{3}\begin{verbatim}
+   (4)  22000
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{GeneralSparseTableXmpPageEmpty4}
+\begin{paste}{GeneralSparseTableXmpPageEmpty4}{GeneralSparseTableXmpPagePatch4}
+\pastebutton{GeneralSparseTableXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{patrons."Jones"\free{jones }}
+\end{paste}\end{patch}
+
+\begin{patch}{GeneralSparseTableXmpPagePatch5}
+\begin{paste}{GeneralSparseTableXmpPageFull5}{GeneralSparseTableXmpPageEmpty5}
+\pastebutton{GeneralSparseTableXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{patrons."Stingy"\free{jones }}
+\indentrel{3}\begin{verbatim}
+   (5)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{GeneralSparseTableXmpPageEmpty5}
+\begin{paste}{GeneralSparseTableXmpPageEmpty5}{GeneralSparseTableXmpPagePatch5}
+\pastebutton{GeneralSparseTableXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{patrons."Stingy"\free{jones }}
+\end{paste}\end{patch}
+
+\begin{patch}{GeneralSparseTableXmpPagePatch6}
+\begin{paste}{GeneralSparseTableXmpPageFull6}{GeneralSparseTableXmpPageEmpty6}
+\pastebutton{GeneralSparseTableXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{reduce(+, entries patrons)\free{jones }}
+\indentrel{3}\begin{verbatim}
+   (6)  32500
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{GeneralSparseTableXmpPageEmpty6}
+\begin{paste}{GeneralSparseTableXmpPageEmpty6}{GeneralSparseTableXmpPagePatch6}
+\pastebutton{GeneralSparseTableXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{reduce(+, entries patrons)\free{jones }}
+\end{paste}\end{patch}
+
+\begin{patch}{GeneralSparseTableXmpPagePatch7}
+\begin{paste}{GeneralSparseTableXmpPageFull7}{GeneralSparseTableXmpPageEmpty7}
+\pastebutton{GeneralSparseTableXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{)system rm -r kaf*.sdata\free{patrons }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{GeneralSparseTableXmpPageEmpty7}
+\begin{paste}{GeneralSparseTableXmpPageEmpty7}{GeneralSparseTableXmpPagePatch7}
+\pastebutton{GeneralSparseTableXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{)system rm -r kaf*.sdata\free{patrons }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/HEAP.ht b/src/hyper/pages/HEAP.ht
new file mode 100644
index 00000000..974e5dca
--- /dev/null
+++ b/src/hyper/pages/HEAP.ht
@@ -0,0 +1,69 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\HeapXmpTitle}{Heap}
+\newcommand{\HeapXmpNumber}{9.32}
+%
+% =====================================================================
+\begin{page}{HeapXmpPage}{9.32 Heap}
+% =====================================================================
+\beginscroll
+The domain \spadtype{Heap(S)} implements a priority queue of
+objects of type \spad{S} such that
+%-% \HDindex{priority queue}{HeapXmpPage}{9.32}{Heap}
+the operation \spadfunX{extract} removes and returns
+the maximum element.
+%-% \HDindex{heap}{HeapXmpPage}{9.32}{Heap}
+The implementation represents heaps as flexible arrays
+(see \downlink{`FlexibleArray'}{FlexibleArrayXmpPage}\ignore{FlexibleArray}).
+The representation and algorithms give complexity
+of \texht{$O(\log(n))$}{O(log n)} for insertion and extractions,
+and \texht{$O(n)$}{O(n)} for construction.
+
+\xtc{
+Create a heap of six elements.
+}{
+\spadpaste{h := heap [-4,9,11,2,7,-7]\bound{h}}
+}
+\xtc{
+Use \spadfunX{insert} to add an element.
+}{
+\spadpaste{insert!(3,h)\bound{h1}\free{h}}
+}
+\xtc{
+The operation \spadfunX{extract} removes and returns
+the maximum element.
+}{
+\spadpaste{extract! h\bound{h2}\free{h1}}
+}
+\xtc{
+The internal structure of \spad{h} has been
+appropriately adjusted.
+}{
+\spadpaste{h\free{h2}}
+}
+\xtc{
+Now \spadfunX{extract} elements repeatedly
+until none are left, collecting the elements in a list.
+}{
+\spadpaste{[extract!(h) while not empty?(h)]\bound{h2}}
+}
+\xtc{
+Another way to produce the same result is by defining
+a \userfun{heapsort} function.
+}{
+\spadpaste{heapsort(x) == (empty? x => []; cons(extract!(x),heapsort x))\bound{f}}
+}
+\xtc{
+Create another sample heap.
+}{
+\spadpaste{h1 := heap [17,-4,9,-11,2,7,-7]\bound{h1}}
+}
+\xtc{
+Apply \spadfun{heapsort} to present elements in order.
+}{
+\spadpaste{heapsort h1\free{f}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/HEAP.pht b/src/hyper/pages/HEAP.pht
new file mode 100644
index 00000000..f7a95088
--- /dev/null
+++ b/src/hyper/pages/HEAP.pht
@@ -0,0 +1,127 @@
+\begin{patch}{HeapXmpPagePatch1}
+\begin{paste}{HeapXmpPageFull1}{HeapXmpPageEmpty1}
+\pastebutton{HeapXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{h := heap [-4,9,11,2,7,-7]\bound{h }}
+\indentrel{3}\begin{verbatim}
+   (1)  [11,7,9,- 4,2,- 7]
+                                     Type: Heap Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPageEmpty1}
+\begin{paste}{HeapXmpPageEmpty1}{HeapXmpPagePatch1}
+\pastebutton{HeapXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{h := heap [-4,9,11,2,7,-7]\bound{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPagePatch2}
+\begin{paste}{HeapXmpPageFull2}{HeapXmpPageEmpty2}
+\pastebutton{HeapXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{insert!(3,h)\bound{h1 }\free{h }}
+\indentrel{3}\begin{verbatim}
+   (2)  [11,7,9,- 4,2,- 7,3]
+                                     Type: Heap Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPageEmpty2}
+\begin{paste}{HeapXmpPageEmpty2}{HeapXmpPagePatch2}
+\pastebutton{HeapXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{insert!(3,h)\bound{h1 }\free{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPagePatch3}
+\begin{paste}{HeapXmpPageFull3}{HeapXmpPageEmpty3}
+\pastebutton{HeapXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{extract! h\bound{h2 }\free{h1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  11
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPageEmpty3}
+\begin{paste}{HeapXmpPageEmpty3}{HeapXmpPagePatch3}
+\pastebutton{HeapXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{extract! h\bound{h2 }\free{h1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPagePatch4}
+\begin{paste}{HeapXmpPageFull4}{HeapXmpPageEmpty4}
+\pastebutton{HeapXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{h\free{h2 }}
+\indentrel{3}\begin{verbatim}
+   (4)  [9,7,3,- 4,2,- 7]
+                                     Type: Heap Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPageEmpty4}
+\begin{paste}{HeapXmpPageEmpty4}{HeapXmpPagePatch4}
+\pastebutton{HeapXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{h\free{h2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPagePatch5}
+\begin{paste}{HeapXmpPageFull5}{HeapXmpPageEmpty5}
+\pastebutton{HeapXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{[extract!(h) while not empty?(h)]\bound{h2 }}
+\indentrel{3}\begin{verbatim}
+   (5)  [9,7,3,2,- 4,- 7]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPageEmpty5}
+\begin{paste}{HeapXmpPageEmpty5}{HeapXmpPagePatch5}
+\pastebutton{HeapXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{[extract!(h) while not empty?(h)]\bound{h2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPagePatch6}
+\begin{paste}{HeapXmpPageFull6}{HeapXmpPageEmpty6}
+\pastebutton{HeapXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{heapsort(x) == (empty? x => []; cons(extract!(x),heapsort x))\bound{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPageEmpty6}
+\begin{paste}{HeapXmpPageEmpty6}{HeapXmpPagePatch6}
+\pastebutton{HeapXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{heapsort(x) == (empty? x => []; cons(extract!(x),heapsort x))\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPagePatch7}
+\begin{paste}{HeapXmpPageFull7}{HeapXmpPageEmpty7}
+\pastebutton{HeapXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{h1 := heap [17,-4,9,-11,2,7,-7]\bound{h1 }}
+\indentrel{3}\begin{verbatim}
+   (7)  [17,2,9,- 11,- 4,7,- 7]
+                                     Type: Heap Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPageEmpty7}
+\begin{paste}{HeapXmpPageEmpty7}{HeapXmpPagePatch7}
+\pastebutton{HeapXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{h1 := heap [17,-4,9,-11,2,7,-7]\bound{h1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPagePatch8}
+\begin{paste}{HeapXmpPageFull8}{HeapXmpPageEmpty8}
+\pastebutton{HeapXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{heapsort h1\free{f }}
+\indentrel{3}\begin{verbatim}
+   (8)  [17,9,7,2,- 4,- 7,- 11]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HeapXmpPageEmpty8}
+\begin{paste}{HeapXmpPageEmpty8}{HeapXmpPagePatch8}
+\pastebutton{HeapXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{heapsort h1\free{f }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/HEXADEC.ht b/src/hyper/pages/HEXADEC.ht
new file mode 100644
index 00000000..e807f91f
--- /dev/null
+++ b/src/hyper/pages/HEXADEC.ht
@@ -0,0 +1,59 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\HexadecimalExpansionXmpTitle}{HexadecimalExpansion}
+\newcommand{\HexadecimalExpansionXmpNumber}{9.33}
+%
+% =====================================================================
+\begin{page}{HexadecimalExpansionXmpPage}{9.33 HexadecimalExpansion}
+% =====================================================================
+\beginscroll
+
+All rationals have repeating hexadecimal expansions.
+The operation \spadfunFrom{hex}{HexadecimalExpansion} returns these
+expansions of type \spadtype{HexadecimalExpansion}.
+Operations to access the individual numerals of a hexadecimal expansion can
+be obtained by converting the value to \spadtype{RadixExpansion(16)}.
+More examples of expansions are available in the
+\downlink{`DecimalExpansion'}{DecimalExpansionXmpPage}\ignore{DecimalExpansion},
+\downlink{`BinaryExpansion'}{BinaryExpansionXmpPage}\ignore{BinaryExpansion}, and
+\downlink{`RadixExpansion'}{RadixExpansionXmpPage}\ignore{RadixExpansion}.
+
+\showBlurb{HexadecimalExpansion}
+
+\xtc{
+This is a hexadecimal expansion of a rational number.
+}{
+\spadpaste{r := hex(22/7) \bound{r}}
+}
+\xtc{
+Arithmetic is exact.
+}{
+\spadpaste{r + hex(6/7) \free{r}}
+}
+\xtc{
+The period of the expansion can be short or long \ldots
+}{
+\spadpaste{[hex(1/i) for i in 350..354] }
+}
+\xtc{
+or very long!
+}{
+\spadpaste{hex(1/1007) }
+}
+\xtc{
+These numbers are bona fide algebraic objects.
+}{
+\spadpaste{p := hex(1/4)*x**2 + hex(2/3)*x + hex(4/9)  \bound{p}}
+}
+\xtc{
+}{
+\spadpaste{q := D(p, x) \free{p}\bound{q}}
+}
+\xtc{
+}{
+\spadpaste{g := gcd(p, q)            \free{p}\free{q}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/HEXADEC.pht b/src/hyper/pages/HEXADEC.pht
new file mode 100644
index 00000000..09985963
--- /dev/null
+++ b/src/hyper/pages/HEXADEC.pht
@@ -0,0 +1,127 @@
+\begin{patch}{HexadecimalExpansionXmpPagePatch1}
+\begin{paste}{HexadecimalExpansionXmpPageFull1}{HexadecimalExpansionXmpPageEmpty1}
+\pastebutton{HexadecimalExpansionXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{r := hex(22/7)\bound{r }}
+\indentrel{3}\begin{verbatim}
+          ___
+   (1)  3.249
+                             Type: HexadecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HexadecimalExpansionXmpPageEmpty1}
+\begin{paste}{HexadecimalExpansionXmpPageEmpty1}{HexadecimalExpansionXmpPagePatch1}
+\pastebutton{HexadecimalExpansionXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{r := hex(22/7)\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{HexadecimalExpansionXmpPagePatch2}
+\begin{paste}{HexadecimalExpansionXmpPageFull2}{HexadecimalExpansionXmpPageEmpty2}
+\pastebutton{HexadecimalExpansionXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{r + hex(6/7)\free{r }}
+\indentrel{3}\begin{verbatim}
+   (2)  4
+                             Type: HexadecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HexadecimalExpansionXmpPageEmpty2}
+\begin{paste}{HexadecimalExpansionXmpPageEmpty2}{HexadecimalExpansionXmpPagePatch2}
+\pastebutton{HexadecimalExpansionXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{r + hex(6/7)\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{HexadecimalExpansionXmpPagePatch3}
+\begin{paste}{HexadecimalExpansionXmpPageFull3}{HexadecimalExpansionXmpPageEmpty3}
+\pastebutton{HexadecimalExpansionXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{[hex(1/i) for i in 350..354]}
+\indentrel{3}\begin{verbatim}
+   (3)
+       _______________    _________      _____
+   [0.00BB3EE721A54D88, 0.00BAB6561, 0.00BA2E8,
+      ______________________
+    0.00B9A7862A0FF465879D5F,
+       _____________________________
+    0.00B92143FA36F5E02E4850FE8DBD78]
+                        Type: List HexadecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HexadecimalExpansionXmpPageEmpty3}
+\begin{paste}{HexadecimalExpansionXmpPageEmpty3}{HexadecimalExpansionXmpPagePatch3}
+\pastebutton{HexadecimalExpansionXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{[hex(1/i) for i in 350..354]}
+\end{paste}\end{patch}
+
+\begin{patch}{HexadecimalExpansionXmpPagePatch4}
+\begin{paste}{HexadecimalExpansionXmpPageFull4}{HexadecimalExpansionXmpPageEmpty4}
+\pastebutton{HexadecimalExpansionXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{hex(1/1007)}
+\indentrel{3}\begin{verbatim}
+   (4)
+   0.
+     OVERBAR
+        0041149783F0BF2C7D13933192AF6980619EE345E91EC2BB9
+          D5CCA5C071E40926E54E8DDAE24196C0B2F8A0AAD60DBA5
+          7F5D4C8536262210C74F1
+                             Type: HexadecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HexadecimalExpansionXmpPageEmpty4}
+\begin{paste}{HexadecimalExpansionXmpPageEmpty4}{HexadecimalExpansionXmpPagePatch4}
+\pastebutton{HexadecimalExpansionXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{hex(1/1007)}
+\end{paste}\end{patch}
+
+\begin{patch}{HexadecimalExpansionXmpPagePatch5}
+\begin{paste}{HexadecimalExpansionXmpPageFull5}{HexadecimalExpansionXmpPageEmpty5}
+\pastebutton{HexadecimalExpansionXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{p := hex(1/4)*x**2 + hex(2/3)*x + hex(4/9)\bound{p }}
+\indentrel{3}\begin{verbatim}
+            2     _      ___
+   (5)  0.4x  + 0.Ax + 0.71C
+                  Type: Polynomial HexadecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HexadecimalExpansionXmpPageEmpty5}
+\begin{paste}{HexadecimalExpansionXmpPageEmpty5}{HexadecimalExpansionXmpPagePatch5}
+\pastebutton{HexadecimalExpansionXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{p := hex(1/4)*x**2 + hex(2/3)*x + hex(4/9)\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{HexadecimalExpansionXmpPagePatch6}
+\begin{paste}{HexadecimalExpansionXmpPageFull6}{HexadecimalExpansionXmpPageEmpty6}
+\pastebutton{HexadecimalExpansionXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{q := D(p, x)\free{p }\bound{q }}
+\indentrel{3}\begin{verbatim}
+                 _
+   (6)  0.8x + 0.A
+                  Type: Polynomial HexadecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HexadecimalExpansionXmpPageEmpty6}
+\begin{paste}{HexadecimalExpansionXmpPageEmpty6}{HexadecimalExpansionXmpPagePatch6}
+\pastebutton{HexadecimalExpansionXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{q := D(p, x)\free{p }\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{HexadecimalExpansionXmpPagePatch7}
+\begin{paste}{HexadecimalExpansionXmpPageFull7}{HexadecimalExpansionXmpPageEmpty7}
+\pastebutton{HexadecimalExpansionXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{g := gcd(p, q)\free{p }\free{q }}
+\indentrel{3}\begin{verbatim}
+              _
+   (7)  x + 1.5
+                  Type: Polynomial HexadecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{HexadecimalExpansionXmpPageEmpty7}
+\begin{paste}{HexadecimalExpansionXmpPageEmpty7}{HexadecimalExpansionXmpPagePatch7}
+\pastebutton{HexadecimalExpansionXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{g := gcd(p, q)\free{p }\free{q }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/HTXAdvPage1.ht b/src/hyper/pages/HTXAdvPage1.ht
new file mode 100644
index 00000000..0e209ad8
--- /dev/null
+++ b/src/hyper/pages/HTXAdvPage1.ht
@@ -0,0 +1,104 @@
+\begin{page}{HTXAdvPage1}{Input areas}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+You have probably seen input areas in other \HyperName{}
+pages. They provide {\it dynamic link} capabilities.
+Instead of having a choice between certain actions,
+they allow you to specify an action on--the--fly.
+To use them, you need the following commands:
+\beginImportant
+\newline
+{\tt \\inputstring\{{\it label}\}\{{\it length}\}\{{\it default value}\}}
+\newline
+{\tt \\stringvalue\{{\it label}\}}
+\endImportant
+
+The first command puts up an input area of the {\it length}
+specified. The {\it default value} is placed in it.
+The first argument, {\it label} gives a name to the
+contents of the input area.
+You can refer to those contents by using
+the second command. Never place a {\tt \\stringvalue} command
+in an "exposed" part of the page. It is only meant
+to be used as an argument to an {\it action}.
+Here are some examples.
+
+
+
+
+
+\beginImportant
+\begin{paste}{HTXAdvPage1xPaste1}{HTXAdvPage1xPatch1}
+\pastebutton{HTXAdvPage1xPaste1}{Interpret}
+\newline
+{\tt Page name \\tab\{16\} }
+{\tt \\inputstring\{pagetogo\}\{30\}\{RootPage\}}\newline
+{\tt \\newline}\newline
+{\tt \\downlink\{GO!\}\{\\stringvalue\{pagetogo\}\}}\newline
+\end{paste}
+\endImportant
+
+\beginImportant
+\begin{paste}{HTXAdvPage1xPaste2}{HTXAdvPage1xPatch2}
+\pastebutton{HTXAdvPage1xPaste2}{Interpret}
+\newline
+{\tt File to edit \\tab\{16\}}\newline
+{\tt \\inputstring\{filetoedit\}\{30\}\{/etc/passwd\}}\newline
+{\tt \\newline}\newline
+{\tt \\unixcommand\{Ready!\}\{xterm  -e vi \\stringvalue\{filetoedit\}\}}
+\end{paste}
+\endImportant
+
+
+\end{scroll}
+\beginmenu
+\menulink{Next Page --- Radio boxes}{HTXAdvPage2}
+\endmenu
+
+\end{page}
+
+
+
+\begin{patch}{HTXAdvPage1xPatch1}
+\begin{paste}{HTXAdvPage1xPaste1A}{HTXAdvPage1xPatch1A}
+\pastebutton{HTXAdvPage1xPaste1A}{Source}
+\newline
+Page name \tab{16}
+\inputstring{pagetogo}{30}{RootPage}
+\newline
+\downlink{GO!}{\stringvalue{pagetogo}}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXAdvPage1xPatch1A}
+\begin{paste}{HTXAdvPage1xPaste1B}{HTXAdvPage1xPatch1}
+\pastebutton{HTXAdvPage1xPaste1B}{Interpret}
+\newline
+{\tt Page name \\tab\{16\} }
+{\tt \\inputstring\{pagetogo\}\{30\}\{RootPage\}}\newline
+{\tt \\newline}\newline
+{\tt \\downlink\{GO!\}\{\\stringvalue\{pagetogo\}\}}\newline
+\end{paste}
+\end{patch}
+
+
+\begin{patch}{HTXAdvPage1xPatch2}
+\begin{paste}{HTXAdvPage1xPaste2A}{HTXAdvPage1xPatch2A}
+\pastebutton{HTXAdvPage1xPaste2A}{Source}
+\newline
+File to edit \tab{16}
+\inputstring{filetoedit}{30}{/etc/passwd}
+\newline
+\unixcommand{Ready!}{xterm  -e vi \stringvalue{filetoedit}}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXAdvPage1xPatch2A}
+\begin{paste}{HTXAdvPage1xPaste2B}{HTXAdvPage1xPatch2}
+\pastebutton{HTXAdvPage1xPaste2B}{Interpret}
+\newline
+{\tt File to edit \\tab\{16\}}\newline
+{\tt \\inputstring\{filetoedit\}\{30\}\{/etc/passwd\}}\newline
+{\tt \\newline}\newline
+{\tt \\unixcommand\{Ready!\}\{xterm  -e vi \\stringvalue\{filetoedit\}\}}
+\end{paste}
+\end{patch}
diff --git a/src/hyper/pages/HTXAdvPage2.ht b/src/hyper/pages/HTXAdvPage2.ht
new file mode 100644
index 00000000..fbf571cf
--- /dev/null
+++ b/src/hyper/pages/HTXAdvPage2.ht
@@ -0,0 +1,127 @@
+\begin{page}{HTXAdvPage2}{Radio buttons}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+
+If you just want to make a multiple-choice
+type selection, why not use the {\it radio buttons}.
+
+You need to use bitmaps for the active areas (the buttons) but
+\HyperName{} will keep track of the currently activated button.
+You can use this boolean information somewhere else on your page.
+The commands to use are:
+\beginImportant
+\newline
+{\tt \\radioboxes\{{\it group name}\}\{{\it bitmap file1}\}\{{\it bitmap file0}\}}
+\newline
+{\tt \\radiobox[{\it initial state}]\{{\it label}\}\{{\it group name}\}}
+\newline
+{\tt \\boxvalue\{{\it label}\}}
+\endImportant
+
+The {\tt \\radioboxes} command sets up a group of {\tt \\radiobox}
+buttons. The {\it group name} is a label for the group. The filenames
+for the bitmaps are specified in {\it bitmap file1} and {\it bitmap file0}.
+The first one should denote an activated button and the second
+a de-activated one.
+
+To display each button in a group, use {\tt \\radiobox}.
+The {\it initial state} should be either {\tt 1} or {\tt 0}
+depending on whether the button should first be displayed as activated or not.
+The second {\it label} argument defines the name by which the
+current state of the button can be referred to.
+The third argument specifies which group this button belongs to.
+
+The {\tt \\boxvalue} command can then be used in various
+actions. The value of it will
+be either {\tt t} or {\tt nil}.
+
+In the example below, we use the {\tt \\htbmfile} macro
+defined in {\bf util.ht} so that we do not have to write
+the full bitmap file pathnames.
+
+This is how we set up the group. The {\tt \\radioboxes} command does not display
+anything.
+Note that these commands cannot be included in a {\it patch}.
+This is why we display this time the source and the result
+at the same time.
+\beginImportant
+\newline
+{\tt \\radioboxes\{group\}\{\\htbmfile\{pick\}\}\{\\htbmfile\{unpick\}\}}\newline
+{\tt \\newline \\table\{}\newline
+{\tt \{\\radiobox[1]\{b1\}\{group\}\}}\newline
+{\tt \{\\radiobox[0]\{b2\}\{group\}\}}\newline
+{\tt \{\\radiobox[0]\{b3\}\{group\}\}\}}\newline
+{\tt \\newline}\newline
+{\tt \\lispcommand\{lisp\}\{(pprint (list}\newline
+{\tt \\boxvalue\{b1\} \\boxvalue\{b2\} \\boxvalue\{b3\}))\}}\newline
+{\tt \\newline}\newline
+{\tt \\unixcommand\{unix\}\{echo '\\boxvalue\{b1\}}\newline
+{\tt \\boxvalue\{b2\} \\boxvalue\{b3\}'\}}
+\endImportant
+\radioboxes{group}{\htbmfile{pick}}{\htbmfile{unpick}}
+\table{
+{\radiobox[1]{b1}{group}}
+{\radiobox[0]{b2}{group}}
+{\radiobox[0]{b3}{group}}}
+\newline
+\lispcommand{lisp}{(pprint (list
+\boxvalue{b1} \boxvalue{b2} \boxvalue{b3}))}
+\newline
+\unixcommand{unix}{echo '\boxvalue{b1}
+\boxvalue{b2} \boxvalue{b3}'}
+\endImportant
+
+
+
+
+You can only set one radio button at a time. If you want
+a non--exclusive selection, try {\tt \\inputbox}.
+The syntax for this command is
+\beginImportant
+\newline
+{\tt \\inputbox[{\it initial state}]\{{\it label}\}\{{\it bitmap file1}\}\{{\it bitmap file0}\}}
+\endImportant
+
+There is no group command for these.
+\beginImportant
+\newline
+{\tt \\table\{}\newline
+{\tt \{\\inputbox[1]\{c1\}\{\\htbmfile\{pick\}\}\{\\htbmfile\{unpick\}\}\}}\newline
+{\tt \{\\inputbox\{c2\}\{\\htbmfile\{pick\}\}\{\\htbmfile\{unpick\}\}\}}\newline
+{\tt \{\\inputbox[1]\{c3\}\{\\htbmfile\{pick\}\}\{\\htbmfile\{unpick\}\}\}\}}\newline
+{\tt \\newline}\newline
+{\tt \\lispcommand\{lisp\}\{(pprint (list}\newline
+{\tt \\boxvalue\{c1\} \\boxvalue\{c2\} \\boxvalue\{c3\}))\}}\newline
+{\tt \\newline}\newline
+{\tt \\unixcommand\{unix\}\{echo }\newline
+{\tt '\\boxvalue\{c1\} \\boxvalue\{c2\} \\boxvalue\{c3\}'\}}\newline
+\endImportant
+\table{
+{\inputbox[1]{c1}{\htbmfile{pick}}{\htbmfile{unpick}}}
+{\inputbox{c2}{\htbmfile{pick}}{\htbmfile{unpick}}}
+{\inputbox[1]{c3}{\htbmfile{pick}}{\htbmfile{unpick}}}}
+\newline
+\lispcommand{lisp}{(pprint (list
+\boxvalue{c1} \boxvalue{c2} \boxvalue{c3}))}
+\newline
+\unixcommand{unix}{echo
+'\boxvalue{c1} \boxvalue{c2} \boxvalue{c3}'}
+\endImportant
+
+
+Note that the {\it initial state} is an
+optional argument. If omitted
+the button will initially
+be deactivated.
+
+\end{scroll}
+\beginmenu
+\menulink{Next Page --- Macros}{HTXAdvPage3}
+\endmenu
+
+\end{page}
+
+
+
+
diff --git a/src/hyper/pages/HTXAdvPage3.ht b/src/hyper/pages/HTXAdvPage3.ht
new file mode 100644
index 00000000..9db9209c
--- /dev/null
+++ b/src/hyper/pages/HTXAdvPage3.ht
@@ -0,0 +1,71 @@
+\begin{page}{HTXAdvPage3}{Macros}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+
+Sometimes you may find yourself having to
+write
+almost the same piece of \HyperName{}
+text many times. Thankfully, there is a command to ease
+the work.
+It is the {\tt \\newcommand} command and provides
+a macro facility for \HyperName{}.
+In this way, you can give a short name to a sequence of \HyperName{}
+text and use that name to include the sequence in your pages.
+The way this works is the following
+\beginImportant
+\newline
+\centerline{{\tt \\newcommand\{\\{\it name}\}[{\it number of arguments}]\{{\it \HyperName{} text}\}}}
+\endImportant
+and here is an example from {\bf util.ht}
+\beginImportant
+\newline
+{\tt \\newcommand\{\\axiomSig\}[2]\{\\axiomType\{\#1\} \{\\tt ->\} \\axiomType\{\#2\}\}}
+\newline
+{\tt \\newcommand\{\\axiomType\}[1]\{\\lispdownlink\{\#1\}\{(|spadType| '|\#1|)\}\}}
+\endImportant
+
+You see that a macro's definition can invoke another.
+Don't create a circular definition though!
+Notice how the arguments of the macro are used
+in the definition. The {\tt \#{\it n}} construct
+is the place--holder of the {\it n}'th argument.
+
+To use the macro, just treat it as an ordinary command.
+For instance
+\beginImportant
+\newline
+{\tt \\axiomSig\{Integer\}\{List Integer\}}
+\endImportant
+displays and acts like this
+\beginImportant
+\newline
+\axiomSig{Integer}{List Integer}
+\endImportant
+
+The best way to familiarise yourself to
+macros is to study the macros defined in
+\centerline{
+{\bf \env{AXIOM}/doc/hypertex/pages/util.ht}
+}
+It is highly probable that a good many of them
+will prove useful to you.
+Clever use of macros will allow you to
+create \HyperName{} text that can be
+formatted by other programs (such as TeX).
+The \Language{} User Guide was written
+in such a way as to make translation in
+\HyperName{} form and TeX form a mechanical process.
+
+
+
+
+\end{scroll}
+\beginmenu
+\menulink{Next Page --- Patch and Paste}{HTXAdvPage4}
+\endmenu
+
+\end{page}
+
+
+
diff --git a/src/hyper/pages/HTXAdvPage4.ht b/src/hyper/pages/HTXAdvPage4.ht
new file mode 100644
index 00000000..34db7810
--- /dev/null
+++ b/src/hyper/pages/HTXAdvPage4.ht
@@ -0,0 +1,157 @@
+\begin{page}{HTXAdvPage4}{Patch and Paste}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+
+A powerful \HyperName{} feature is
+the ability to {\it replace}
+part of a displayed page with another part
+when an active area is clicked. The group commands
+{\it patch} and {\it paste} offer this facility.
+A {\it paste} region can appear anywhere
+within a page or a {\it patch}. A {\it patch} region must be defined
+outside a page definition.
+
+We need a few  objects to define the {\it paste}
+region. These are a {\it name} with which to
+refer to it, some way of specifying what it
+is to be replaced by and a {\it trigger} for the
+replacement. A {\it patch} is how we specify the
+second of these objects.
+The {\it patch} is generally a sequence of \HyperName{}
+text.
+
+If we want to have the option of returning to the original
+(or ,indeed, proceeding to a {\it third} alternative)
+we clearly must include a {\it paste} in the {\it patch}.
+
+Let us start with a simple example. We wish to
+have the word {\tt initial} somewhere on the page replaced by the
+word {\tt final} at a click of a button.
+Let us first define the {\it patch}. It will just contain
+the word {\tt final}. Here is a definition of a
+patch called {\tt patch1} (note that
+the actual definition must be outside this page's definition).
+\beginImportant
+\newline
+{\tt \\begin\{patch\}\{patch1\}} \newline
+{\tt final}\newline
+{\tt \\end\{patch\}}
+\endImportant
+We now define a {\it paste} region exactly where we
+want the word {\tt initial} to appear.
+\beginImportant
+\newline
+{\tt \\begin\{paste\}\{paste1\}\{patch1\}}\newline
+{\tt initial}\newline
+{\tt \\end\{paste\}}
+\centerline{{\it results in}}
+\begin{paste}{paste1}{patch1}
+initial
+\end{paste}
+\endImportant
+We have specified first the name of the {\it paste} region
+which is {\tt paste1} and then the name of the
+replacement {\it patch} which is {\tt patch1}.
+Something is missing -- the trigger.
+To include a trigger we write
+\beginImportant
+\newline
+{\tt \\pastebutton\{paste1\}\{trigger\}
+\centerline{{\it results in}}
+\pastebutton{paste1}{trigger}
+\endImportant
+This new command {\tt \\pastebutton} displays the second argument
+as an active area. The first argument specifies the {\it paste}
+region it refers to. Clicking on {\tt trigger} above will
+replace the word {\tt initial} with the word {\tt final}.
+
+We can, if we like, include the {\tt \\pastebutton} in the {\it paste}
+region.
+Let us improve on the situation by providing a way
+of going back to the original word {\tt initial} on the page.
+The {\it patch} must itself include a {\it paste}.
+What will the replacement {\it patch} for this new {\it paste}
+be ? Well, we  have to define a second {\it patch}
+that contains all the stuff in the original {\it paste}
+region. Here is the updated {\it patch} for the first replacement.
+The {\tt \\MenuDotBitmap} macro is defined in {\bf util.ht}.
+It displays a button bitmap.
+This time we put the {\tt \\pastebutton}
+inside the {\it paste}.
+\beginImportant
+\newline
+{\tt \\begin\{patch\}\{Patch1\}}\newline
+{\tt \\begin\{paste\}\{Paste2\}\{Patch2\}}\newline
+{\tt \\pastebutton\{Paste2\}\{\\MenuDotBitmap\}}\newline
+{\tt final}\newline
+{\tt \\end\{paste\}}\newline
+{\tt \\end\{patch\}}\newline
+\endImportant
+and the new {\tt Patch2} {\it patch}
+\beginImportant
+\newline
+{\tt \\begin\{patch\}\{Patch2\}}\newline
+{\tt \\begin\{paste\}\{Paste3\}\{Patch1\}}\newline
+{\tt \\pastebutton\{Paste3\}\{\\MenuDotBitmap\}}\newline
+{\tt initial}\newline
+{\tt \\end\{paste\}}\newline
+{\tt \\end\{patch\}}\newline
+\endImportant
+
+Remember that these {\it patch} definitons must
+occur outside a {\tt \\begin\{page\} - \\end\{page\}} group.
+What is left now is to define the starting {\it paste}
+region.
+\beginImportant
+\newline
+{\tt \\begin\{paste\}\{Paste1\}\{Patch1\}}\newline
+{\tt \\pastebutton\{Paste1\}\{\\MenuDotBitmap\}}\newline
+{\tt initial}\newline
+{\tt \\end\{paste\}}
+\centerline{{\it results in}}
+\begin{paste}{Paste1}{Patch1}
+\pastebutton{Paste1}{\MenuDotBitmap}
+initial
+\end{paste}
+\endImportant
+
+Clicking on the button above next to {\tt initial}
+will replace the {\tt Paste1} region with
+{\tt Patch1}. That {\it patch}
+also contains a {\it paste} region ({\tt Paste2}).
+Clicking on {\it its} button will put up
+{\tt Patch2} which has a {\it paste} region ({\tt Paste3}).
+Clicking on {\it its} button will put up {\tt Patch1}
+again. In that way, we close the chain of replacements.
+
+
+
+
+\end{scroll}
+\beginmenu
+\menulink{Next Page --- \Language{} paste-ins}{HTXAdvPage5}
+\endmenu
+
+\end{page}
+
+
+
+
+\begin{patch}{patch1}
+final
+\end{patch}
+
+\begin{patch}{Patch1}
+\begin{paste}{Paste2}{Patch2}
+\pastebutton{Paste2}{\MenuDotBitmap}
+final
+\end{paste}
+\end{patch}
+
+\begin{patch}{Patch2}
+\begin{paste}{Paste3}{Patch1}
+\pastebutton{Paste3}{\MenuDotBitmap}
+initial
+\end{paste}
+\end{patch}
diff --git a/src/hyper/pages/HTXAdvPage5.ht b/src/hyper/pages/HTXAdvPage5.ht
new file mode 100644
index 00000000..7c99f389
--- /dev/null
+++ b/src/hyper/pages/HTXAdvPage5.ht
@@ -0,0 +1,129 @@
+\begin{page}{HTXAdvPage5}{\Language{} paste-ins}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+The {\it paste} and {\it patch} facility (see \downlink{previous page}{HTXAdvPage4})
+is used to display (or hide) the \Language{} output
+of an \Language{} command ({\tt\\axiomcommand})
+included in a
+\HyperName{} page.
+
+A mechanism has been set up to {\it automatically} generate
+these paste-ins. It amounts to replacing an
+{\tt \\axiomcommand} by a {\tt \\pastecommand} in the
+\HyperName{} page.
+
+In the case of a \axiomOp{draw} \Language{} command
+, where the result is to create an interactive viewport,
+the appropriate command to use is {\tt \\pastegraph}.
+The effect of this is to include (as the output)
+the \Language{} generated {\it image} of the graph
+as an active area. Clicking on it will put up an
+interactive viewport.
+The {\tt \\pastegraph} command should be used only when
+the result of the associated \Language{} operation is
+to {\it create} an interactive viewport. It is {\it not} necessarily
+appropriate for all commands whose result
+is a \axiomType{Two{}Dimensional{}Viewport} or \axiomType{Three{}Dimensional{}Viewport}.
+The {\tt \\pastecommand} and {\tt \\pastegraph}
+are  macros defined in {\bf util.ht}.
+
+
+
+There is no automatic paste-in generation facility
+for \Language{} piles (the {\tt \\begin\{axiomsrc\}} command).
+
+
+The automatic paste-in generation mechanism works
+by invoking \Language{} with a particular option.
+\HyperName{} is also started automatically. It reads
+the {\tt \\pastecommand} and {\tt \\pastegraph}
+commands in all pages in a specified
+{\bf somefile.ht}  and passes them to
+\Language{} for evaluation. \HyperName{}
+captures the output and writes it out (to a
+file called {\bf somefile.pht}) as the body of
+some {\it patch} definitions. The commands
+encountered are written to a file
+{\bf somefile.input} which you can {\tt )read} from an \Language{} session.
+It also creates directories for the graphics
+images encountered. Those files and directories
+will be written under the {\it current} directory.
+The idea is that you then include the {\it patch} definitions
+in {\bf somefile.pht} in your local database using {\bf htadd}.
+
+
+You can try this feature now. Edit a file called, say, {\bf trypaste.ht}
+in a directory, say {\bf /tmp}. Put the following \HyperName{} text
+in it.
+\beginImportant
+\newline
+{\tt \\begin\{page\}\{TryPaste\}\{Trying out paste-in generation\}}\newline
+{\tt \\begin\{scroll\}}\newline
+{\tt \\pastecommand\{f z == z**2 \\bound\{f\}\}}\newline
+{\tt \\pastegraph\{draw(f,-1..1) \\free\{f\}\}}\newline
+{\tt \\pastecommand\{x:= f 3 \\free\{f\}\}}\newline
+{\tt \\end\{scroll\}}\newline
+{\tt \\end\{page\}}\newline
+\endImportant
+
+From the directory that contains the {\bf trypaste.ht},
+issue
+\centerline{
+{\tt htadd -l ./trypaste.ht}
+}
+You will get the
+{\bf ht.db} database file.
+Set the environment variable {\tt HTPATH} so that
+it points first to your directory and then the system directory.
+In the {\bf /bin/csh}, you might use
+\centerline{
+{\tt setenv HTPATH /tmp:\$AXIOM/doc/hypertex/pages}
+}
+Make sure that no {\bf trypaste.input} or {\bf trypaste.pht}
+files exist in the directory.
+Then issue
+\centerline{
+{\tt axiom -paste trypaste.ht}
+}
+ and wait for
+\Language{} to finish.
+
+There is a modification you will wish to make to
+the {\bf trypaste.pht} file.
+This is because the generated \HyperName{} text will assume that
+the {\it viewport} data will be located in the
+{\it system} directory.
+\centerline{{\bf \env{AXIOM}/doc/viewports}}
+You may want to place your images in a different directory,
+say, {\bf /u/sugar/viewports}.
+If so, then change all occurences of
+\beginImportant
+\newline
+{\tt \\env\{AXIOM\}/doc/viewports/}
+\centerline{{\it by}}
+{\tt /u/sugar/viewports/}
+\endImportant
+in the file {\bf trypaste.pht}. The last step
+is to include the {\it patch} definitions
+in {\bf trypaste.pht} to your local database.
+Issue
+\centerline{
+{\tt htadd -l ./trypaste.pht}
+}
+to update the database. If you have provided
+a link to your pages from the {\tt RootPage}
+via the {\tt \\localinfo} macro, you should now
+be able to start \Language{} or \HyperName{}
+and see the computed \Language{} output whenever you
+click on the buttons to the left of each command.
+
+
+
+\end{scroll}
+\beginmenu
+\menulink{Next Page --- Miscellaneous}{HTXAdvPage6}
+\endmenu
+
+\end{page}
+
diff --git a/src/hyper/pages/HTXAdvPage6.ht b/src/hyper/pages/HTXAdvPage6.ht
new file mode 100644
index 00000000..2966dd78
--- /dev/null
+++ b/src/hyper/pages/HTXAdvPage6.ht
@@ -0,0 +1,128 @@
+\begin{page}{HTXAdvPage6}{Miscellaneous}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+
+We present here a few commands that may be of some use
+to you.
+You may want to know certain parameters so that you can pass
+them to one of the action {\tt \\command}s.
+
+The {\tt \\thispage} command shows the name of the
+current page.
+
+\beginImportant
+\begin{paste}{HTXAdvPage6xPaste1}{HTXAdvPage6xPatch1}
+\pastebutton{HTXAdvPage6xPaste1}{Interpret}
+\newline
+{\tt \\thispage}\newline
+{\tt \\newline}\newline
+{\tt \\lispcommand\{Lisp\}\{(pprint "\\thispage")\}}\newline
+\end{paste}
+\endImportant
+
+
+The {\tt \\windowid} command shows the X Windows
+{\it WindowID} of the current window.
+
+\beginImportant
+\begin{paste}{HTXAdvPage6xPaste2}{HTXAdvPage6xPatch2}
+\pastebutton{HTXAdvPage6xPaste2}{Interpret}
+\newline
+{\tt \\windowid}\newline
+{\tt \\newline}\newline
+{\tt \\lispcommand\{Lisp\}\{(pprint \\windowid)\}}\newline
+\end{paste}
+\endImportant
+
+% \examplenumber not documented
+
+The {\tt \\env} command gets the value of an environment
+variable. It is an error to use {\tt \\env} with an undefined
+environment variable.
+
+\beginImportant
+\begin{paste}{HTXAdvPage6xPaste3}{HTXAdvPage6xPatch3}
+\pastebutton{HTXAdvPage6xPaste3}{Interpret}
+\newline
+{\tt \\env\{AXIOM\}}
+\end{paste}
+\endImportant
+
+
+The {\tt \\nolines} command, if included somewhere
+in the page, eliminates the horizontal lines that
+delimit the header and footer regions.
+
+
+% \beep not documented
+
+%\returnbutton{homebutton}{ReturnPage}
+
+%\upbutton{upbutton}{UpPage}
+
+
+\end{scroll}
+\beginmenu
+\menulink{Back to Menu}{HTXAdvTopPage}
+\endmenu
+
+\end{page}
+
+
+
+\begin{patch}{HTXAdvPage6xPatch1}
+\begin{paste}{HTXAdvPage6xPaste1A}{HTXAdvPage6xPatch1A}
+\pastebutton{HTXAdvPage6xPaste1A}{Source}
+\newline
+\thispage
+\newline
+\lispcommand{Lisp}{(pprint "\thispage")}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXAdvPage6xPatch1A}
+\begin{paste}{HTXAdvPage6xPaste1B}{HTXAdvPage6xPatch1}
+\pastebutton{HTXAdvPage6xPaste1B}{Interpret}
+\newline
+{\tt \\thispage}\newline
+{\tt \\newline}\newline
+{\tt \\lispcommand\{Lisp\}\{(pprint "\\thispage")\}}\newline
+\end{paste}
+\end{patch}
+
+
+\begin{patch}{HTXAdvPage6xPatch2}
+\begin{paste}{HTXAdvPage6xPaste2A}{HTXAdvPage6xPatch2A}
+\pastebutton{HTXAdvPage6xPaste2A}{Source}
+\newline
+\windowid
+\newline
+\lispcommand{Lisp}{(pprint \windowid)}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXAdvPage6xPatch2A}
+\begin{paste}{HTXAdvPage6xPaste2B}{HTXAdvPage6xPatch2}
+\pastebutton{HTXAdvPage6xPaste2B}{Interpret}
+\newline
+{\tt \\windowid}\newline
+{\tt \\newline}\newline
+{\tt \\lispcommand\{Lisp\}\{(pprint \\windowid)\}}\newline
+\end{paste}
+\end{patch}
+
+
+\begin{patch}{HTXAdvPage6xPatch3}
+\begin{paste}{HTXAdvPage6xPaste3A}{HTXAdvPage6xPatch3A}
+\pastebutton{HTXAdvPage6xPaste3A}{Source}
+\newline
+\env{AXIOM}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXAdvPage6xPatch3A}
+\begin{paste}{HTXAdvPage6xPaste3B}{HTXAdvPage6xPatch3}
+\pastebutton{HTXAdvPage6xPaste3B}{Interpret}
+\newline
+{\tt \\env\{AXIOM\}}\newline
+\end{paste}
+\end{patch}
+
diff --git a/src/hyper/pages/HTXAdvTopPage.ht b/src/hyper/pages/HTXAdvTopPage.ht
new file mode 100644
index 00000000..aa3cebbc
--- /dev/null
+++ b/src/hyper/pages/HTXAdvTopPage.ht
@@ -0,0 +1,17 @@
+\begin{page}{HTXAdvTopPage}{Advanced features in\ \HyperName{}}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\beginscroll
+\beginmenu
+\menudownlink{Creating input areas}{HTXAdvPage1}
+\menudownlink{Creating radio boxes}{HTXAdvPage2}
+\menudownlink{Define new macros     }{HTXAdvPage3}
+\menudownlink{Using patch and paste}{HTXAdvPage4}
+\menudownlink{Generate paste-ins for \Language{} commands}{HTXAdvPage5}
+\menudownlink{Miscellaneous}{HTXAdvPage6}
+\endmenu
+\endscroll
+\end{page}
+
+
+
+
diff --git a/src/hyper/pages/HTXFormatPage1.ht b/src/hyper/pages/HTXFormatPage1.ht
new file mode 100644
index 00000000..d8addbcc
--- /dev/null
+++ b/src/hyper/pages/HTXFormatPage1.ht
@@ -0,0 +1,75 @@
+\begin{page}{HTXFormatPage1}{Using the special characters}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+You can display the special characters ({\tt \$ \\ \{ \{ \[ \] \% \#})
+by simply inserting the backslash {\tt \\} character just before any one of them.
+So,
+\beginImportant
+\begin{paste}{HTXFormatPage1xPaste1}{HTXFormatPage1xPatch1}
+\pastebutton{HTXFormatPage1xPaste1}{Interpret}
+\newline
+{\tt the characters \\\$, \\\\, \\\{, \\\}, \\\[, \\\], \\\%, \\\# }
+\end{paste}
+\endImportant
+
+
+The {\tt \%} character is used in \HyperName{} as a comment marker.
+If it is encountered on any line (without being preceded by the {\tt \\}
+ character,
+of course), \HyperName{} will ignore all remaining characters on that line.
+\beginImportant
+\begin{paste}{HTXFormatPage1xPaste2}{HTXFormatPage1xPatch2}
+\pastebutton{HTXFormatPage1xPaste2}{Interpret}
+\newline
+{\tt  the latest figures indicate \% GET THE LATEST FIGURES}\newline
+{\tt a steady rise}\indent{0}
+\end{paste}
+\endImportant
+
+Earlier versions of \HyperName{} merged the words "indicate" and "a"
+into one word in the example above. This no longer occurs.
+
+%The two lines below are from Release 1.x
+%Note that you must leave a space at the beginning of the line after the comment,
+%otherwise \HyperName{} would treat "indicate" and "a" as one word.
+
+
+\end{scroll}
+\beginmenu
+\menulink{Next -- Formatting without commands}{HTXFormatPage2}
+\endmenu
+
+\end{page}
+\begin{patch}{HTXFormatPage1xPatch1}
+\begin{paste}{HTXFormatPage1xPaste1A}{HTXFormatPage1xPatch1A}
+\pastebutton{HTXFormatPage1xPaste1A}{Source}
+\newline
+the characters \$, \\, \{, \}, \[, \],  \%, \#
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage1xPatch1A}
+\begin{paste}{HTXFormatPage1xPaste1B}{HTXFormatPage1xPatch1}
+\pastebutton{HTXFormatPage1xPaste1B}{Interpret}
+\newline
+{\tt the characters \\\$, \\\\, \\\{, \\\}, \\\[, \\\], \\\%, \\\# }
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage1xPatch2}
+\begin{paste}{HTXFormatPage1xPaste2A}{HTXFormatPage1xPatch2A}
+\pastebutton{HTXFormatPage1xPaste2A}{Source}
+\newline
+the latest figures indicate % GET THE LATEST FIGURES
+a steady rise
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage1xPatch2A}
+\begin{paste}{HTXFormatPage1xPaste2B}{HTXFormatPage1xPatch2}
+\pastebutton{HTXFormatPage1xPaste2B}{Interpret}
+\newline
+{\tt  the latest figures indicate \% GET THE LATEST FIGURES}\newline
+{\tt a steady rise}\indent{0}
+\end{paste}
+\end{patch}
+
diff --git a/src/hyper/pages/HTXFormatPage2.ht b/src/hyper/pages/HTXFormatPage2.ht
new file mode 100644
index 00000000..6c3e1979
--- /dev/null
+++ b/src/hyper/pages/HTXFormatPage2.ht
@@ -0,0 +1,169 @@
+\begin{page}{HTXFormatPage2}{Formatting without commands}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+\HyperName{} will interpret normal text in a {\em source file}
+according to the following rules. \newline
+\menuitemstyle{\indentrel{4}
+Spaces mark the ends of words. The number of spaces between
+words is {\em not} significant, that is, you cannot control word spacing
+by inserting or removing extra space characters.
+\indentrel{-4}}
+\beginImportant
+\begin{paste}{HTXFormatPage2xxPaste1}{HTXFormatPage2xPatch1}
+\pastebutton{HTXFormatPage2xxPaste1}{Interpret}
+\begin{verbatim}
+word    spacing is       not  important
+\end{verbatim}
+\end{paste}
+\endImportant
+\menuitemstyle{\indentrel{4}
+End-of-line  characters are not significant.
+You can break up lines in the source file as you like. The end-of-line
+will be interpreted as a space. 
+Take advantage of this feature to improve the readability of the
+source file.
+\indentrel{-4}}
+\beginImportant
+\begin{paste}{HTXFormatPage2xxPaste2}{HTXFormatPage2xPatch2}
+\pastebutton{HTXFormatPage2xxPaste2}{Interpret}
+\begin{verbatim}
+This is
+one
+sentence.
+\end{verbatim}
+\end{paste}
+\endImportant
+\menuitemstyle{\indentrel{4}
+A blank line marks the end of a paragraph. 
+Leaving more blank lines that necessary has no effect.
+\indentrel{-4}}
+\beginImportant
+\begin{paste}{HTXFormatPage2xxPaste3}{HTXFormatPage2xPatch3}
+\pastebutton{HTXFormatPage2xxPaste3}{Interpret}
+\begin{verbatim}
+some end.% A COMMENT
+
+
+Start a paragraph
+
+Start another paragraph.
+\end{verbatim}
+\end{paste}
+\endImportant
+\menuitemstyle{\indentrel{4}
+The two-character combination {\tt \{\}} can be used to indicate
+possible breaking of long words. It does not affect the formatting
+in any other way.
+\indentrel{-4}}
+\beginImportant
+\begin{paste}{HTXFormatPage2xxPaste4}{HTXFormatPage2xPatch4}
+\pastebutton{HTXFormatPage2xxPaste4}{Interpret}
+\begin{verbatim}
+Generalized{}Multivariate{}Factorize
+One{}Dimensional{}Array{}Aggregate
+Elementary{}Function{}Definite{}Integration
+Elementary{}Functions{}Univariate{}Puiseux{}Series
+Finite{}Field{}Cyclic{}Group{}Extension{}ByPolynomial
+\end{verbatim}
+\end{paste}
+\endImportant
+
+
+
+
+\end{scroll}
+\beginmenu
+\menulink{Next -- Using different fonts}{HTXFormatPage3}
+\endmenu
+
+\end{page}
+
+\begin{patch}{HTXFormatPage2xPatch1}
+\begin{paste}{HTXFormatPage2xxPaste1A}{HTXFormatPage2xPatch1A}
+\pastebutton{HTXFormatPage2xxPaste1A}{Source}
+\newline
+word    spacing is       not  important
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage2xPatch1A}
+\begin{paste}{HTXFormatPage2xxPaste1B}{HTXFormatPage2xPatch1}
+\pastebutton{HTXFormatPage2xxPaste1B}{Interpret}
+\begin{verbatim}
+word    spacing is       not  important
+\end{verbatim}
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage2xPatch2}
+\begin{paste}{HTXFormatPage2xxPaste2A}{HTXFormatPage2xPatch2A}
+\pastebutton{HTXFormatPage2xxPaste2A}{Source}
+\newline
+This is
+one
+sentence.
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage2xPatch2A}
+\begin{paste}{HTXFormatPage2xxPaste2B}{HTXFormatPage2xPatch2}
+\pastebutton{HTXFormatPage2xxPaste2B}{Interpret}
+\begin{verbatim}
+This is
+one
+sentence.
+\end{verbatim}
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage2xPatch3}
+\begin{paste}{HTXFormatPage2xxPaste3A}{HTXFormatPage2xPatch3A}
+\pastebutton{HTXFormatPage2xxPaste3A}{Source}
+\newline
+some end.% A COMMENT
+
+
+Start a paragraph.
+
+Start another paragraph.
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage2xPatch3A}
+\begin{paste}{HTXFormatPage2xxPaste3B}{HTXFormatPage2xPatch3}
+\pastebutton{HTXFormatPage2xxPaste3B}{Interpret}
+\begin{verbatim}
+some end.% A COMMENT
+
+
+Start a paragraph
+
+Start another paragraph.
+\end{verbatim}
+\end{paste}
+\end{patch}
+
+
+
+\begin{patch}{HTXFormatPage2xPatch4}
+\begin{paste}{HTXFormatPage2xxPaste4A}{HTXFormatPage2xPatch4A}
+\pastebutton{HTXFormatPage2xxPaste4A}{Source}
+\newline
+Generalized{}Multivariate{}Factorize
+One{}Dimensional{}Array{}Aggregate
+Elementary{}Function{}Definite{}Integration
+Elementary{}Functions{}Univariate{}Puiseux{}Series
+Finite{}Field{}Cyclic{}Group{}Extension{}ByPolynomial
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage2xPatch4A}
+\begin{paste}{HTXFormatPage2xxPaste4B}{HTXFormatPage2xPatch4}
+\pastebutton{HTXFormatPage2xxPaste4B}{Interpret}
+\begin{verbatim}
+Generalized{}Multivariate{}Factorize
+One{}Dimensional{}Array{}Aggregate
+Elementary{}Function{}Definite{}Integration
+Elementary{}Functions{}Univariate{}Puiseux{}Series
+Finite{}Field{}Cyclic{}Group{}Extension{}ByPolynomial
+\end{verbatim}
+\end{paste}
+\end{patch}
+
diff --git a/src/hyper/pages/HTXFormatPage3.ht b/src/hyper/pages/HTXFormatPage3.ht
new file mode 100644
index 00000000..7320d2a1
--- /dev/null
+++ b/src/hyper/pages/HTXFormatPage3.ht
@@ -0,0 +1,146 @@
+\begin{page}{HTXFormatPage3}{Using different fonts}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+You can use various fonts for the text. \HyperName{} makes
+four {\em logical} fonts available to you: a {\em roman} font, an
+{\em emphasised} font, a {\em bold} font and a
+{\em typewriter} font. The actual font that corresponds to
+each logical font is determined by the end user via a
+defaults file. The colour for each of these fonts can also
+be specified.
+
+Normal text is displayed with the roman font.
+If you want to emphasize some text, use the {\tt \\em}
+command in the following way.
+\beginImportant
+\begin{paste}{HTXFormatPage3xPaste1}{HTXFormatPage3xPatch1}
+\pastebutton{HTXFormatPage3xPaste1}{Interpret}
+\newline
+{\tt this is \{\\em emphasised\} text}
+\end{paste}
+\endImportant
+
+Note the use of the braces to enclose command and "arguments".
+All font commands are specified in the same way. The {\tt \\em} command
+will in fact {\em switch} between roman and emphasised
+font every time it is used.
+\beginImportant
+\begin{paste}{HTXFormatPage3xPaste2}{HTXFormatPage3xPatch2}
+\pastebutton{HTXFormatPage3xPaste2}{Interpret}
+\newline
+{\tt \{\\em this is \{\\em emphasised\} text\}}
+\end{paste}
+\endImportant
+
+If you want to be sure that the emphasized font will be used, specify
+the {\tt \\it} command. Similarly, you can explicitly select the roman font
+with the {\tt \\rm} command.
+\beginImportant
+\begin{paste}{HTXFormatPage3xPaste3}{HTXFormatPage3xPatch3}
+\pastebutton{HTXFormatPage3xPaste3}{Interpret}
+\newline
+{\tt \{\\em this is \{\\it emphasised\} text and this is \{\\rm roman\}\}}
+\end{paste}
+\endImportant
+
+
+The bold font is selected with the {\tt \\bf} command and the typewriter
+font with the {\tt \\tt} command. All these commands can be applied to
+individual characters, words, sentences etc.
+\beginImportant
+\begin{paste}{HTXFormatPage3xPaste4}{HTXFormatPage3xPatch4}
+\pastebutton{HTXFormatPage3xPaste4}{Interpret}
+\newline
+{\tt \{\\bf U\}\{\\tt g\}\{\\it l\}\{\\rm y\}}
+\end{paste}
+\endImportant
+
+
+Currently, \HyperName{} does not adjust its internal spacing rules
+to each font individually. This means that, for consistent results,
+users are encouraged to specify (in the defaults file)
+"character-cell" fonts that are not
+too small or too large for \HyperName{}. Here is the correspondence
+between the above font commands and the defaults names:\newline
+\menuitemstyle{RmFont \tab{26} {\tt \\rm} or {\tt \\em} }\newline
+\menuitemstyle{BoldFont \tab{26} {\tt \\bf} }\newline
+\menuitemstyle{EmphasizeFont \tab{26} {\tt \\it} or {\tt \\em} }\newline
+\menuitemstyle{Ttfont \tab{26} {\tt \\tt} }\newline
+
+\HyperName{} uses two more logical fonts that can be specified by
+the end user : AxiomFont and ActiveFont. However, you cannot
+explicitly use these fonts in your text. The ActiveFont is automatically
+used for active area text and the AxiomFont is reserved for
+active \Language{} commands.
+
+
+
+
+\end{scroll}
+\beginmenu
+\menulink{Next -- Indentation}{HTXFormatPage4}
+\endmenu
+
+\end{page}
+
+\begin{patch}{HTXFormatPage3xPatch1}
+\begin{paste}{HTXFormatPage3xPaste1A}{HTXFormatPage3xPatch1A}
+\pastebutton{HTXFormatPage3xPaste1A}{Source}
+\newline
+this is {\em emphasised} text
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage3xPatch1A}
+\begin{paste}{HTXFormatPage3xPaste1B}{HTXFormatPage3xPatch1}
+\pastebutton{HTXFormatPage3xPaste1B}{Interpret}
+\newline
+{\tt this is \{\\em emphasised\} text}
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage3xPatch2}
+\begin{paste}{HTXFormatPage3xPaste2A}{HTXFormatPage3xPatch2A}
+\pastebutton{HTXFormatPage3xPaste2A}{Source}
+\newline
+{\em this is {\em emphasised} text}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage3xPatch2A}
+\begin{paste}{HTXFormatPage3xPaste2B}{HTXFormatPage3xPatch2}
+\pastebutton{HTXFormatPage3xPaste2B}{Interpret}
+\newline
+{\tt \{\\em this is \{\\em emphasised\} text\}}
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage3xPatch3}
+\begin{paste}{HTXFormatPage3xPaste3A}{HTXFormatPage3xPatch3A}
+\pastebutton{HTXFormatPage3xPaste3A}{Source}
+\newline
+{\em this is {\it emphasised} text and this is {\rm roman}}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage3xPatch3A}
+\begin{paste}{HTXFormatPage3xPaste3B}{HTXFormatPage3xPatch3}
+\pastebutton{HTXFormatPage3xPaste3B}{Interpret}
+\newline
+{\tt \{\\em this is \{\\it emphasised\} text and this is \{\\rm roman\}\}}
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage3xPatch4}
+\begin{paste}{HTXFormatPage3xPaste4A}{HTXFormatPage3xPatch4A}
+\pastebutton{HTXFormatPage3xPaste4A}{Source}
+\newline
+{\bf U}{\tt g}{\it l}{\rm y}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage3xPatch4A}
+\begin{paste}{HTXFormatPage3xPaste4B}{HTXFormatPage3xPatch4}
+\pastebutton{HTXFormatPage3xPaste4B}{Interpret}
+\newline
+{\tt \{\\bf U\}\{\\tt g\}\{\\it l\}\{\\rm y\}}
+\end{paste}
+\end{patch}
+
diff --git a/src/hyper/pages/HTXFormatPage4.ht b/src/hyper/pages/HTXFormatPage4.ht
new file mode 100644
index 00000000..bf320141
--- /dev/null
+++ b/src/hyper/pages/HTXFormatPage4.ht
@@ -0,0 +1,199 @@
+\begin{page}{HTXFormatPage4}{Indentation}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+You can control the indentation of lines of text in \HyperName{} with
+some useful commands.
+Use the command {\tt \\par} to force a new paragraph if you don't want
+to use the blank-line rule. The first line of a new paragraph
+will normally be indented by a standard small amount. If you
+just want to start on a new line, use the {\tt \\newline}
+command.
+\beginImportant
+\begin{paste}{HTXFormatPage4xPaste1}{HTXFormatPage4xPatch1}
+\pastebutton{HTXFormatPage4xPaste1}{Interpret}
+\newline
+{\tt let us start a new line \\newline here }
+\end{paste}
+\endImportant
+
+The  command {\tt \\indent\{{\it value}\}} will
+set the left-page-margin {\it value} characters
+to the right of the standard left-page-margin.
+The initial (standard) state of a page can be reset by the
+{\tt \\indent\{0\}} command.
+The first lines of paragraphs will be indented
+by the {\it extra} standard  amount. The {\tt \\indent\{{\it value}\}}
+command does {\em not} force a new line or paragraph.
+
+You can also use the {\tt \\indentrel\{{\it value}\}} command.
+Here, the {\it value} argument is a {\em relative} indentation
+which can be positive or negative.
+Otherwise, it behaves in the same way as the {\tt \\indent} command.
+\beginImportant
+\begin{paste}{HTXFormatPage4xPaste2}{HTXFormatPage4xPatch2}
+\pastebutton{HTXFormatPage4xPaste2}{Interpret}
+\newline
+{\tt let us start a new line \\newline \\indent\{10\} here }\newline
+{\tt \\newline \\indentrel\{-5\} there}\newline
+{\tt \\newline \\indentrel\{-5\} back}
+\end{paste}
+\endImportant
+
+The {\tt \\centerline\{{\it some text}\}} command will center its
+argument between the current left and right margins. The argument of
+the command should not be more than a paragraph of text and should not contain
+any commands that change the left margin. The centered text will
+start on a new line.
+\beginImportant
+\begin{paste}{HTXFormatPage4xPaste3}{HTXFormatPage4xPatch3}
+\pastebutton{HTXFormatPage4xPaste3}{Interpret}
+\newline
+{\tt previous text. \\centerline\{This could}\newline
+{\tt be some heading.\} Carry on}
+\end{paste}
+\endImportant
+
+Placing text in vertically aligned columns is easily done with the
+{\tt \\tab\{{\it value}\}} command. The {\tt \\tab} command has the
+immediate effect of placing the next word {\it value} characters to
+the right of the current left margin.
+\beginImportant
+\begin{paste}{HTXFormatPage4xPaste4}{HTXFormatPage4xPatch4}
+\pastebutton{HTXFormatPage4xPaste4}{Interpret}
+\newline
+{\tt \\indent\{5\}\\newline}\newline
+{\tt Team A \\tab\{17\}Score\\tab\{25\}Team B\\tab\{42\}Score\\newline}\newline
+{\tt 012345678901234567890123456789012345678901234567890\\newline}\newline
+{\tt Green-Red\\tab\{17\}4\\tab\{25\}Blue-Black\\tab\{42\}6\\newline}\newline
+{\tt \\indent\{0\}}
+\end{paste}
+\endImportant
+
+If you wish to preserve the indentation of a piece of text
+you can use the {\tt verbatim} group command. Simply place
+a {\tt \\begin\{verbatim\}} and {\tt \\end\{verbatim\}} around
+the text. Note that \HyperName{} commands will
+not be interpreted within the {\tt verbatim} group.
+\beginImportant
+\begin{paste}{HTXFormatPage4xPaste5}{HTXFormatPage4xPatch5}
+\pastebutton{HTXFormatPage4xPaste5}{Interpret}
+\newline
+{\tt \\begin\{verbatim\}}
+\begin{verbatim}
+This    spacing     will be      preserved
+                    {\bf is}     preserved
+\end{verbatim}
+{\tt \\end\{verbatim\}}\newline
+\end{paste}
+
+
+
+
+
+
+
+\end{scroll}
+\beginmenu
+\menulink{Next -- Creating Lists and Tables}{HTXFormatPage5}
+\endmenu
+
+\end{page}
+
+\begin{patch}{HTXFormatPage4xPatch1}
+\begin{paste}{HTXFormatPage4xPaste1A}{HTXFormatPage4xPatch1A}
+\pastebutton{HTXFormatPage4xPaste1A}{Source}
+\newline
+let us start a new line \newline here
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage4xPatch1A}
+\begin{paste}{HTXFormatPage4xPaste1B}{HTXFormatPage4xPatch1}
+\pastebutton{HTXFormatPage4xPaste1B}{Interpret}
+\newline
+{\tt let us start a new line \\newline here }
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage4xPatch2}
+\begin{paste}{HTXFormatPage4xPaste2A}{HTXFormatPage4xPatch2A}
+\pastebutton{HTXFormatPage4xPaste2A}{Source}
+\newline
+let us start a new line\newline\indent{10} here
+\newline\indentrel{-5} there
+\newline\indentrel{-5} back
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage4xPatch2A}
+\begin{paste}{HTXFormatPage4xPaste2B}{HTXFormatPage4xPatch2}
+\pastebutton{HTXFormatPage4xPaste2B}{Interpret}
+\newline
+{\tt let us start a new line \\newline \\indent\{10\} here }\newline
+{\tt \\newline \\indentrel\{-5\} there}\newline
+{\tt \\newline \\indentrel\{-5\} back}
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage4xPatch3}
+\begin{paste}{HTXFormatPage4xPaste3A}{HTXFormatPage4xPatch3A}
+\pastebutton{HTXFormatPage4xPaste3A}{Source}
+\newline
+previous text. \centerline{This could
+be some heading.} Carry on
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage4xPatch3A}
+\begin{paste}{HTXFormatPage4xPaste3B}{HTXFormatPage4xPatch3}
+\pastebutton{HTXFormatPage4xPaste3B}{Interpret}
+\newline
+{\tt previous text. \\centerline\{This could}\newline
+{\tt be some heading.\} Carry on}
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage4xPatch4}
+\begin{paste}{HTXFormatPage4xPaste4A}{HTXFormatPage4xPatch4A}
+\pastebutton{HTXFormatPage4xPaste4A}{Source}
+\newline
+\indent{5}\newline
+Team A \tab{17}Score\tab{25}Team B\tab{42}Score\newline
+012345678901234567890123456789012345678901234567890\newline
+Green-Red\tab{17}4\tab{25}Blue-Black\tab{42}6\newline
+\indent{0}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage4xPatch4A}
+\begin{paste}{HTXFormatPage4xPaste4B}{HTXFormatPage4xPatch4}
+\pastebutton{HTXFormatPage4xPaste4B}{Interpret}
+\newline
+{\tt \\indent\{5\}\\newline}\newline
+{\tt Team A \\tab\{17\}Score\\tab\{25\}Team B\\tab\{42\}Score\\newline}\newline
+{\tt 012345678901234567890123456789012345678901234567890\\newline}\newline
+{\tt Green-Red\\tab\{17\}4\\tab\{25\}Blue-Black\\tab\{42\}6\\newline}\newline
+{\tt \\indent\{0\}}
+\end{paste}
+\end{patch}
+
+
+\begin{patch}{HTXFormatPage4xPatch5}
+\begin{paste}{HTXFormatPage4xPaste5A}{HTXFormatPage4xPatch5A}
+\pastebutton{HTXFormatPage4xPaste5A}{Source}
+\begin{verbatim}
+This    spacing     will be      preserved
+                    {\bf is}     preserved
+\end{verbatim}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage4xPatch5A}
+\begin{paste}{HTXFormatPage4xPaste5B}{HTXFormatPage4xPatch5}
+\pastebutton{HTXFormatPage4xPaste5B}{Interpret}
+\newline
+{\tt \\begin\{verbatim\}}
+\begin{verbatim}
+This    spacing     will be      preserved
+                    {\bf is}     preserved
+\end{verbatim}
+{\tt \\end\{verbatim\}}\newline
+\end{paste}
+\end{patch}
+
diff --git a/src/hyper/pages/HTXFormatPage5.ht b/src/hyper/pages/HTXFormatPage5.ht
new file mode 100644
index 00000000..689c7a33
--- /dev/null
+++ b/src/hyper/pages/HTXFormatPage5.ht
@@ -0,0 +1,203 @@
+\begin{page}{HTXFormatPage5}{Creating Lists and Tables}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+The {\tt \\begin\{items\} {\rm -}  \\end\{items\}} group command constructs itemized lists.
+The {\tt \\item} command separates the items in the list.
+The indentation rules for the list group are different from those
+of a paragraph. The first line of an item will
+normally extend further to the left than the rest of the lines.
+Both commands accept {\em optional} arguments.
+Optional arguments are enclosed in square brackets ({\tt \[ \]}) rather
+than braces.
+
+The indentation of subsequent lines in an item is determined by the
+optional argument {\it some text} in the {\tt \\begin\{items\}\[{\it some text}\]}
+command. The optional argument is {\em not} displayed. Its width is calculated
+and used to indent all subsequent lines in the group except from the
+first line of each new item. This indentation rule applies to all text
+{\em before} the first {\tt \\item} command as well.
+
+The {\tt \\item\[{\it some text}\]} command specifies the start of a new item.
+The {\it some text} optional argument will be displayed in {\em bold}
+font at the current left-page-margin. Then, the text following the command
+will be displayed in normal fashion with the above indentation rule.
+\beginImportant
+\begin{paste}{HTXFormatPage5xPaste1}{HTXFormatPage5xPatch1}
+\pastebutton{HTXFormatPage5xPaste1}{Interpret}
+\newline
+{\tt \\indent\{5\}}\newline
+{\tt \\begin\{items\}\[how wide am I\]}\newline
+{\tt Here we carry on but a \\newline} \newline
+{\tt new line will be indented } \newline
+{\tt \\item\[how wide am I\] fits nicely. Here is a \\newline new line in an item.}\newline
+{\tt \\item\[again\] to show another item}\newline
+{\tt \\item\[\\\\tab\]\\tab\{0\} can be used \\tab\{15\} effectively}\newline
+{\tt \\end\{items\}}\newline
+{\tt \\indent\{0\}}\newline
+\end{paste}
+\endImportant
+
+
+Note that the {\tt \\begin\{items\}} command immediately sets the left-
+page-margin to a new value. Subsequent {\tt \\tab} or {\tt \\centerline} commands
+refer to this new margin.
+Any explicit margin setting commands included
+in the group {\em will} have the normal effect.
+The {\tt \\par} command does not produce
+the standard paragraph indentation within a list group --- it behaves
+instead like {\tt \\newline}.
+
+
+You can nest list groups like the following example suggests.
+\beginImportant
+\begin{paste}{HTXFormatPage5xPaste2}{HTXFormatPage5xPatch2}
+\pastebutton{HTXFormatPage5xPaste2}{Interpret}
+\newline
+{\tt \\begin\{items\}\[quitealot\]}\newline
+{\tt A nested list:}\newline
+{\tt \\item\[The first\] item of an itemized list is on this line.}\newline
+{\tt \\item\[The second\] item of the list starts here.  It contains another}\newline
+{\tt list nested inside it.}\newline
+{\tt \\begin\{items\}\[somuchmore\]}\newline
+{\tt \\item \[First\]\\tab\{0\}This is the first item of an enumerated}\newline
+{\tt list that is nested within the itemized list.}\newline
+{\tt \\item \[Second\]\\tab\{0\}This is the second item of the inner list.}\newline
+{\tt \\end\{items\}}\newline
+{\tt This is the rest of the second item of the outer list.  It}\newline
+{\tt is no more interesting than any other part of the item.}\newline
+{\tt \\item\[The third\] item of the list.}\newline
+{\tt \\end\{items\}}\newline
+\end{paste}
+\endImportant
+
+Another facility for presenting lists is the {\tt \\table} command.
+The correct syntax for it is : {\tt \\table\{\{{\it item a}\} \{{\it item b}\} {\it ..}\}}.
+The items in the braces will be placed in as many aligned columns
+as is possible for the current window dimensions or page width.
+If one item is particularly long there will probably be only one column
+in the table. Here is a table of color names.
+\beginImportant
+\begin{paste}{HTXFormatPage5xPaste3}{HTXFormatPage5xPatch3}
+\pastebutton{HTXFormatPage5xPaste3}{Interpret}
+\newline
+{\tt
+\\table\{
+\{Dark Orchid\} \{Dark Salmon\} \{Dark Sea Green\} \{Dark Slate Blue\}
+\{Dark Slate Gray\} \{Dark Turquoise\} \{Dark Violet\} \{Deep Pink\}
+\{Deep Sky Blue\} \{Dodger Blue\} \{Floral White\} \{Forest Green\}
+\{Ghost White\} \{Hot Pink\} \{Indian Red\} \{Lavender Blush\}
+\}
+}
+\end{paste}
+\endImportant
+
+
+
+
+\end{scroll}
+\beginmenu
+\menulink{Next -- Boxes and Lines}{HTXFormatPage6}
+\endmenu
+
+\end{page}
+
+\begin{patch}{HTXFormatPage5xPatch1}
+\begin{paste}{HTXFormatPage5xPaste1A}{HTXFormatPage5xPatch1A}
+\pastebutton{HTXFormatPage5xPaste1A}{Source}
+\newline
+\indent{5}
+\begin{items}[how wide am I]
+Here we carry on but a \newline
+new line will be indented
+\item[how wide am I] fits nicely. Here is a \newline new line in an item.
+\item[again] to show another item
+\item[\\tab]\tab{0} can be used \tab{15} effectively
+\end{items}
+\indent{0}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage5xPatch1A}
+\begin{paste}{HTXFormatPage5xPaste1B}{HTXFormatPage5xPatch1}
+\pastebutton{HTXFormatPage5xPaste1B}{Interpret}
+\newline
+{\tt \\indent\{5\}}\newline
+{\tt \\begin\{items\}\[how wide am I\]}\newline
+{\tt Here we carry on but a \\newline} \newline
+{\tt new line will be indented } \newline
+{\tt \\item\[how wide am I\] fits nicely. Here is a \\newline new line in an item.}\newline
+{\tt \\item\[again\] to show another item}\newline
+{\tt \\item\[\\\\tab\]\\tab\{0\} can be used \\tab\{15\} effectively}\newline
+{\tt \\end\{items\}}\newline
+{\tt \\indent\{0\}}\newline
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage5xPatch2}
+\begin{paste}{HTXFormatPage5xPaste2A}{HTXFormatPage5xPatch2A}
+\pastebutton{HTXFormatPage5xPaste2A}{Source}
+\newline
+\begin{items}[quitealot]
+A nested list:
+\item[The first] item of an itemized list is on this line.
+\item[The second] item of the list starts here.  It contains another
+list nested inside it.
+\begin{items}[somuchmore]
+\item [First]\tab{0}This is the first item of the
+list that is nested within the itemized list.
+\item [Second]\tab{0}This is the second item of the inner list.
+\end{items}
+This is the rest of the second item of the outer list.  It
+is no more interesting than any other part of the item.
+\item [The third] item of the list.
+\end{items}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage5xPatch2A}
+\begin{paste}{HTXFormatPage5xPaste2B}{HTXFormatPage5xPatch2}
+\pastebutton{HTXFormatPage5xPaste2B}{Interpret}
+\newline
+{\tt \\begin\{items\}\[quitealot\]}\newline
+{\tt A nested list:}\newline
+{\tt \\item\[The first\] item of an itemized list is on this line.}\newline
+{\tt \\item\[The second\] item of the list starts here.  It contains another}\newline
+{\tt list nested inside it.}\newline
+{\tt \\begin\{items\}\[somuchmore\]}\newline
+{\tt \\item \[First\]\\tab\{0\}This is the first item of an enumerated}\newline
+{\tt list that is nested within the itemized list.}\newline
+{\tt \\item \[Second\]\\tab\{0\}This is the second item of the inner list.}\newline
+{\tt \\end\{items\}}\newline
+{\tt This is the rest of the second item of the outer list.  It}\newline
+{\tt is no more interesting than any other part of the item.}\newline
+{\tt \\item\[The third\] item of the list.}\newline
+{\tt \\end\{items\}}\newline
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage5xPatch3}
+\begin{paste}{HTXFormatPage5xPaste3A}{HTXFormatPage5xPatch3A}
+\pastebutton{HTXFormatPage5xPaste3A}{Source}
+\newline
+\table{
+{Dark Orchid} {Dark Salmon} {Dark Sea Green} {Dark Slate Blue} {Dark Slate Gray}
+{Dark Turquoise} {Dark Violet} {Deep Pink} {Deep Sky Blue} {Dodger Blue}
+{Floral White} {Forest Green} {Ghost White} {Hot Pink} {Indian Red}
+{Lavender Blush}
+}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage5xPatch3A}
+\begin{paste}{HTXFormatPage5xPaste3B}{HTXFormatPage5xPatch3}
+\pastebutton{HTXFormatPage5xPaste3B}{Interpret}
+\newline
+{\tt
+\\table\{
+\{Dark Orchid\} \{Dark Salmon\} \{Dark Sea Green\} \{Dark Slate Blue\}
+\{Dark Slate Gray\} \{Dark Turquoise\} \{Dark Violet\} \{Deep Pink\}
+\{Deep Sky Blue\} \{Dodger Blue\} \{Floral White\} \{Forest Green\}
+\{Ghost White\} \{Hot Pink\} \{Indian Red\} \{Lavender Blush\}
+\}
+}
+\end{paste}
+\end{patch}
+
diff --git a/src/hyper/pages/HTXFormatPage6.ht b/src/hyper/pages/HTXFormatPage6.ht
new file mode 100644
index 00000000..b94200ca
--- /dev/null
+++ b/src/hyper/pages/HTXFormatPage6.ht
@@ -0,0 +1,67 @@
+\begin{page}{HTXFormatPage6}{Boxes and Lines}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+The {\tt \\fbox} command can be used to place a box around one or more
+words. The argument of the {\tt \\fbox} command is the text that will be
+placed in the box. This command should only be used for text that can fit
+in one line.
+
+\beginImportant
+\begin{paste}{HTXFormatPage6xPaste1}{HTXFormatPage6xPatch1}
+\pastebutton{HTXFormatPage6xPaste1}{Interpret}
+\newline
+{\tt \\fbox\{Boxed!\}}\newline
+\end{paste}
+\endImportant
+
+
+Use the {\tt \\horizontalline} command to draw a horizontal line
+across the window. This might be useful for added emphasis.
+
+\beginImportant
+\begin{paste}{HTXFormatPage6xPaste2}{HTXFormatPage6xPatch2}
+\pastebutton{HTXFormatPage6xPaste2}{Interpret}
+\newline
+{\tt \\horizontalline}\newline
+\end{paste}
+\endImportant
+
+\end{scroll}
+\beginmenu
+\menulink{Next -- Micro-Spacing}{HTXFormatPage7}
+\endmenu
+
+\end{page}
+
+
+\begin{patch}{HTXFormatPage6xPatch1}
+\begin{paste}{HTXFormatPage6xPaste1A}{HTXFormatPage6xPatch1A}
+\pastebutton{HTXFormatPage6xPaste1A}{Source}
+\newline
+\fbox{Boxed!}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage6xPatch1A}
+\begin{paste}{HTXFormatPage6xPaste1B}{HTXFormatPage6xPatch1}
+\pastebutton{HTXFormatPage6xPaste1B}{Interpret}
+\newline
+{\tt \\fbox\{Boxed!\}}\newline
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage6xPatch2}
+\begin{paste}{HTXFormatPage6xPaste2A}{HTXFormatPage6xPatch2A}
+\pastebutton{HTXFormatPage6xPaste2A}{Source}
+\newline
+\horizontalline
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage6xPatch2A}
+\begin{paste}{HTXFormatPage6xPaste2B}{HTXFormatPage6xPatch2}
+\pastebutton{HTXFormatPage6xPaste2B}{Interpret}
+\newline
+{\tt \\horizontalline}\newline
+\end{paste}
+\end{patch}
+
diff --git a/src/hyper/pages/HTXFormatPage7.ht b/src/hyper/pages/HTXFormatPage7.ht
new file mode 100644
index 00000000..5717b043
--- /dev/null
+++ b/src/hyper/pages/HTXFormatPage7.ht
@@ -0,0 +1,148 @@
+\begin{page}{HTXFormatPage7}{Micro-Spacing}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+There are three commands that one can use to exercise finer control
+over the appearance of text on a page: {\tt \\space}, {\tt \\hspace}
+and {\tt \\vspace}.
+
+The {\tt \\space\{{\it value}\}} command accepts an integer argument and simply
+changes the position of the next character to the right or to the left.
+A negative argument will move the next character to the left and a
+positive one to the right. The unit of movement is {\it the width
+of a character}. In this way one can overstrike characters to produce
+various effects.
+
+\beginImportant
+\begin{paste}{HTXFormatPage7xPaste1}{HTXFormatPage7xPatch1}
+\pastebutton{HTXFormatPage7xPaste1}{Interpret}
+\newline
+{\tt 0\\space\{-1\}\\}\newline
+{\tt underlined\\space\{-10\}__________}\newline
+\end{paste}
+\endImportant
+
+
+The {\tt \\hspace\{{\it value}\}} command
+is similar to the {\tt \\space\{{\it value}\}} command.
+It also accepts an integer argument and
+changes the position of the next character to the right or to the left.
+A negative argument will move the next character to the left and a
+positive one to the right. The unit of movement is {\it a pixel}.
+The {\it value} argument specifies an offset from the default placement
+of the character.
+
+\beginImportant
+\begin{paste}{HTXFormatPage7xPaste2}{HTXFormatPage7xPatch2}
+\pastebutton{HTXFormatPage7xPaste2}{Interpret}
+\newline
+{\tt x\\hspace\{-4\}x\\hspace\{-3\}x\\hspace\{-2\}x\\hspace\{-1\}x\%}\newline
+{\tt x\\hspace\{1\}x\\hspace\{2\}x\\hspace\{3\}x\\hspace\{4\}x}
+\end{paste}
+\endImportant
+
+The {\tt \\vspace\{{\it value}\}} command is similar to the
+{\tt \\hspace\{{\it value}\}} command but (as the name suggests)
+works in the vertical direction. The unit of movement is {\it a
+pixel}. The {\it value} argument specifies an offset from {\it
+the next line}. A negative argument moves the next character up
+and a positive down. This command can be used for subscripts and
+superscripts. One drawback in the use of {\tt \\vspace} is that
+it can only work with a particular font at a time. This is
+because the inter-line spacing depends on the font being used
+and the value of it is needed to get "back" on the line.
+In general, the command {\tt \\vspace\{{\it - ils}\}} will
+have a null effect when {\it ils} = ( font ascent + font descent + 5 ).
+The example below assumes that {\it ils} = 25 e.g. the Rom14 font
+on the RISC System/6000.
+
+\beginImportant
+\begin{paste}{HTXFormatPage7xPaste3}{HTXFormatPage7xPatch3}
+\pastebutton{HTXFormatPage7xPaste3}{Interpret}
+\newline
+{\tt CO\\vspace\{-18\}2\\vspace\{-32\} + CaO ->}\newline
+{\tt CaCO\\vspace\{-18\}3\\vspace\{-32\}\\newline}\newline
+{\tt R\\space\{-1\}~\\vspace\{-18\}æv\\vspace\{-32\}}\newline
+{\tt \\hspace\{4\}-\\hspace\{8\}---\\hspace\{-12\}}\newline
+{\tt \\vspace\{-32\}1\\space\{-1\}\\vspace\{-7\}2}\newline
+{\tt \\vspace\{-36\}\\hspace\{8\}g\\space\{-1\}~}\newline
+{\tt \\vspace\{-18\}æv\\vspace\{-32\}\\hspace\{2\}R}\newline
+{\tt \\space\{-1\}~ = T\\vspace\{-18\}æv\\vspace\{-32\}}\newline
+{\tt \\vspace\{-25\}}
+\end{paste}
+\endImportant
+
+
+\end{scroll}
+\beginmenu
+\menulink{Next -- Bitmaps and Images}{HTXFormatPage8}
+\endmenu
+
+\end{page}
+
+%%%%%%%%%%%%%%%%%%%PATCHES%%%%%%%%%%%%%%
+\begin{patch}{HTXFormatPage7xPatch1}
+\begin{paste}{HTXFormatPage7xPaste1A}{HTXFormatPage7xPatch1A}
+\pastebutton{HTXFormatPage7xPaste1A}{Source}
+\newline
+0\space{-1}\\
+underlined\space{-10}__________
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage7xPatch1A}
+\begin{paste}{HTXFormatPage7xPaste1B}{HTXFormatPage7xPatch1}
+\pastebutton{HTXFormatPage7xPaste1B}{Interpret}
+\newline
+{\tt 0\\space\{-1\}\\}\newline
+{\tt underlined\\space\{-10\}__________}\newline
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage7xPatch2}
+\begin{paste}{HTXFormatPage7xPaste2A}{HTXFormatPage7xPatch2A}
+\pastebutton{HTXFormatPage7xPaste2A}{Source}
+\newline
+x\hspace{-4}x\hspace{-3}x\hspace{-2}x\hspace{-1}x%
+x\hspace{1}x\hspace{2}x\hspace{3}x\hspace{4}x
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage7xPatch2A}
+\begin{paste}{HTXFormatPage7xPaste2B}{HTXFormatPage7xPatch2}
+\pastebutton{HTXFormatPage7xPaste2B}{Interpret}
+\newline
+{\tt x\\hspace\{-4\}x\\hspace\{-3\}x\\hspace\{-2\}x\\hspace\{-1\}x\%}\newline
+{\tt x\\hspace\{1\}x\\hspace\{2\}x\\hspace\{3\}x\\hspace\{4\}x}
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage7xPatch3}
+\begin{paste}{HTXFormatPage7xPaste3A}{HTXFormatPage7xPatch3A}
+\pastebutton{HTXFormatPage7xPaste3A}{Source}
+\newline
+CO\vspace{-18}2\vspace{-32} + CaO ->
+CaCO\vspace{-18}3\vspace{-32}\newline
+R\space{-1}~\vspace{-18}æv\vspace{-32}
+\hspace{4}-\hspace{8}---\hspace{-12}
+\vspace{-32}1\space{-1}\vspace{-7}2
+\vspace{-36}\hspace{8}g\space{-1}~
+\vspace{-18}æv\vspace{-32}\hspace{2}R
+\space{-1}~ = T\vspace{-18}æv\vspace{-32}
+\vspace{-25}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage7xPatch3A}
+\begin{paste}{HTXFormatPage7xPaste3B}{HTXFormatPage7xPatch3}
+\pastebutton{HTXFormatPage7xPaste3B}{Interpret}
+\newline
+{\tt CO\\vspace\{-18\}2\\vspace\{-32\} + CaO ->}\newline
+{\tt CaCO\\vspace\{-18\}3\\vspace\{-32\}\\newline}\newline
+{\tt R\\space\{-1\}~\\vspace\{-18\}æv\\vspace\{-32\}}\newline
+{\tt \\hspace\{4\}-\\hspace\{8\}---\\hspace\{-12\}}\newline
+{\tt \\vspace\{-32\}1\\space\{-1\}\\vspace\{-7\}2}\newline
+{\tt \\vspace\{-36\}\\hspace\{8\}g\\space\{-1\}~}\newline
+{\tt \\vspace\{-18\}æv\\vspace\{-32\}\\hspace\{2\}R}\newline
+{\tt \\space\{-1\}~ = T\\vspace\{-18\}æv\\vspace\{-32\}}\newline
+{\tt \\vspace\{-25\}}
+\end{paste}
+\end{patch}
+
diff --git a/src/hyper/pages/HTXFormatPage8.ht b/src/hyper/pages/HTXFormatPage8.ht
new file mode 100644
index 00000000..f0de82a0
--- /dev/null
+++ b/src/hyper/pages/HTXFormatPage8.ht
@@ -0,0 +1,98 @@
+\begin{page}{HTXFormatPage8}{Bitmaps and Images}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+The commands {\tt \\inputbitmap\{{\it filename}\}}
+and {\tt \\inputimage\{{\it filename}\}}
+allow you to include an X11 bitmap or an
+\Language{}-generated viewport in a \HyperName{}
+page.
+
+In the case of the {\tt \\inputbitmap} command
+the {\it filename} parameter must be the full pathname
+of an X11 bitmap file.
+
+\beginImportant
+\begin{paste}{HTXFormatPage8xPaste1}{HTXFormatPage8xPatch1}
+\pastebutton{HTXFormatPage8xPaste1}{Interpret}
+\newline
+{\tt \\inputbitmap\{\env{AXIOM}/doc/hypertex/bitmaps/sup.bitmap\} }
+\end{paste}
+\endImportant
+
+The {\it filename} parameter of the {\tt \\inputimage}
+command must be the full pathname of a {\it compressed XPM image} file without the name extensions.
+\HyperName{} always adds ".xpm.Z" to whatever filename you give and looks for the augmented filename.
+Such files can be generated by \Language{} command
+\axiomOp{write} with the {\tt "image"} or {\tt "pixmap"}
+options.
+
+\beginImportant
+\begin{paste}{HTXFormatPage8xPaste2}{HTXFormatPage8xPatch2}
+\pastebutton{HTXFormatPage8xPaste2}{Interpret}
+\newline
+{\tt \\inputimage\{\env{AXIOM}/doc/viewports/ugProblemNumericPage30.VIEW/image\}}
+\end{paste}
+\endImportant
+
+Be careful not to break the pathname across lines.
+
+The {\tt \\inputimage} command will automatically select
+the {\it image.xpm} or the {\it image.bm} file for you
+based on the capabilities of your X server. 
+
+For your convenience, there are two macros defined
+in \centerline{ {\bf \env{AXIOM}{}/doc/hypertex/pages/util.ht}.}
+The {\tt \\viewport} macro eliminates the need to specify
+the {\tt .VIEW/image} part and the
+{\tt \\axiomViewport} macro automatically selects viewport
+files in the system directories. The above {\tt \\inputimage}
+could have been written
+\beginImportant
+{\tt \\viewport\{\env{AXIOM}/doc/viewports/ugProblemNumericPage30\}}
+\endImportant
+or
+\beginImportant
+{\tt \\axiomViewport\{ugProblemNumericPage30\}}
+\endImportant
+
+
+
+\end{scroll}
+\beginmenu
+\menulink{Back to Formatting menu}{HTXFormatTopPage}
+\endmenu
+
+\end{page}
+
+
+\begin{patch}{HTXFormatPage8xPatch1}
+\begin{paste}{HTXFormatPage8xPaste1A}{HTXFormatPage8xPatch1A}
+\pastebutton{HTXFormatPage8xPaste1A}{Source}
+\newline
+\inputbitmap{\env{AXIOM}/doc/hypertex/bitmaps/sup.bitmap}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage8xPatch1A}
+\begin{paste}{HTXFormatPage8xPaste1B}{HTXFormatPage8xPatch1}
+\pastebutton{HTXFormatPage8xPaste1B}{Interpret}
+\newline
+{\tt \\inputbitmap\{\env{AXIOM}/doc/hypertex/bitmaps/sup.bitmap\} }
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXFormatPage8xPatch2}
+\begin{paste}{HTXFormatPage8xPaste2A}{HTXFormatPage8xPatch2A}
+\pastebutton{HTXFormatPage8xPaste2A}{Source}
+\newline
+\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage30.VIEW/image}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXFormatPage8xPatch2A}
+\begin{paste}{HTXFormatPage8xPaste2B}{HTXFormatPage8xPatch2}
+\pastebutton{HTXFormatPage8xPaste2B}{Interpret}
+\newline
+{\tt \\inputimage\{\env{AXIOM}/doc/viewports/ugProblemNumericPage30.VIEW/image\}}
+\end{paste}
+\end{patch}
+
diff --git a/src/hyper/pages/HTXFormatTopPage.ht b/src/hyper/pages/HTXFormatTopPage.ht
new file mode 100644
index 00000000..9b2adbca
--- /dev/null
+++ b/src/hyper/pages/HTXFormatTopPage.ht
@@ -0,0 +1,19 @@
+\begin{page}{HTXFormatTopPage}{Formatting in \ \HyperName{}}
+\centerline{\fbox{{\tt \thispage}}}\newline
+
+\HyperName{} offers various facilities for formatting text and images.
+You can learn about these facilities by clicking on the topics below.
+\begin{scroll}
+\beginmenu
+\menudownlink{Special Characters}{HTXFormatPage1}
+\menudownlink{Formatting without commands}{HTXFormatPage2}
+\menudownlink{Using different fonts}{HTXFormatPage3}
+\menudownlink{Indentation}{HTXFormatPage4}
+\menudownlink{Creating Lists and Tables}{HTXFormatPage5}
+\menudownlink{Boxes and Lines}{HTXFormatPage6}
+\menudownlink{Micro-Spacing}{HTXFormatPage7}
+\menudownlink{Bitmaps and Images}{HTXFormatPage8}
+\endmenu
+\end{scroll}
+\end{page}
+
diff --git a/src/hyper/pages/HTXIntroPage1.ht b/src/hyper/pages/HTXIntroPage1.ht
new file mode 100644
index 00000000..31601b91
--- /dev/null
+++ b/src/hyper/pages/HTXIntroPage1.ht
@@ -0,0 +1,40 @@
+\begin{page}{HTXIntroPage1}{What \ \HyperName{} does}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\beginscroll
+
+Take a close look at the objects in the \HyperName{} window you are now reading.
+Most of them are text. Resize the window using the window manager
+facilities. The text is reformatted to fit the window
+border. This action is performed by \HyperName{}. At the simplest
+level, it provides a method for {\em formatting} text in a window.
+In fact, it can place other things on the window as well, such as
+bitmaps or color images. The {\em buttons} you see at either
+side at the top of the window are bitmaps.
+
+Move the cursor so that it rests on one of those buttons. You notice that
+the cursor has changed appearance. This indicates that there is an action associated
+with the button. This action will be performed when you click the mouse button
+over the {\em active area}. If you are familiar with \HyperName{}, you know
+that the active area can be words, bitmaps, images or {\em input areas}. In
+fact, anything that can be displayed in a \HyperName{} window can be an
+active area.
+
+So, what can the action associated with an active area be? \HyperName{}
+allows quite a bit of freedom in defining that action. We will have a close
+look at this issue \downlink{later on}{HTXLinkTopPage}. For now, recall
+the various actions that you have encountered so far --- executing \Language{}
+commands, popping up new windows, providing parameters for other active areas,
+and replacing or changing the contents of the window. The most common action
+is to bring up some \HyperName{} text in the same or a different window.
+This lets us create {\em links} between pieces of text and images. A
+system with such capability is usually called a {\em hypertext} system.
+\HyperName{} is in fact much more.
+
+\endscroll
+\beginmenu
+\menulink{Next -- How \HyperName{} does it}{HTXIntroPage2}
+\menuwindowlink{Review some features of \HyperName{}}{ugHyperPage}
+\endmenu
+
+\helppage{ugHyperPage}
+\end{page}
diff --git a/src/hyper/pages/HTXIntroPage2.ht b/src/hyper/pages/HTXIntroPage2.ht
new file mode 100644
index 00000000..cb2d21c1
--- /dev/null
+++ b/src/hyper/pages/HTXIntroPage2.ht
@@ -0,0 +1,52 @@
+\begin{page}{HTXIntroPage2}{How \ \HyperName{} does it}
+{\centerline{\fbox{{\tt \thispage}}}\newline}
+\beginscroll
+
+\HyperName{} can read the {\em hypertext} information from standard text
+files. This means that you can create or change this information with any
+text editor. Once this information has been entered into the files, a
+special program, called {\bf htadd}, scans these files and produces
+a database (another file called {\bf ht.db}) of {\em objects}
+encountered in the files. \HyperName{}
+consults this database when it first starts and so knows where it might
+find the definitions of these objects. You can maintain several such
+databases on different directories. You indicate which database you
+want \HyperName{} to consult by setting an {\em environment variable}
+called {\bf HTPATH}.
+
+In general, hypertext must obviously use some kind of special (that is,
+non-textual) marks for all the  extra functionality it provides. In
+\HyperName{}, these marks are some special characters --- special
+in the sense that they are not interpreted as ordinary displayable text.
+These characters, however, are part of the standard ASCII set.
+There is also a way to display these special characters as text .
+The \HyperName{} special characters are :
+
+\beginImportant
+\noindent{\em Special Characters}: {\tt \table{{\$}{\\}{\{}{\}}{\[}{\]}{\%}{\#}}}
+\endImportant
+
+
+
+\HyperName{} uses the special characters to distinguish between
+{\em text} and {\em commands} (by {\em text}, we mean here anything
+displayable). The commands are instructions to
+\HyperName{} to treat some text in a particular way. Some commands
+define special \HyperName{} objects. The most important objects
+are {\em pages}, {\em patches}, and {\em macros}.
+A {\em page} is a description of the contents of a
+\HyperName{} window. A {\em patch} is a portion of a page.
+A {\em macro} is a user-defined new \HyperName{} command.
+Some commands allow special text {\em formatting} and others
+associate some text with an action.
+
+In order to display anything at all in \HyperName, you must define a
+{\em page}. The next section explains how to define a {\em page} and put
+some simple text into it.
+\endscroll
+\beginmenu
+\menudownlink{Next -- Define a simple text page}{HTXIntroPage3}
+\endmenu
+
+\end{page}
+
diff --git a/src/hyper/pages/HTXIntroPage3.ht b/src/hyper/pages/HTXIntroPage3.ht
new file mode 100644
index 00000000..7c3a3b77
--- /dev/null
+++ b/src/hyper/pages/HTXIntroPage3.ht
@@ -0,0 +1,95 @@
+\begin{page}{HTXIntroPage3}{A simple text page}
+{\centerline{\fbox{{\tt \thispage}}}\newline}
+\begin{scroll}
+
+
+A page is defined by a {\em group} command. Group commands are used to
+delimit a group, that is, to declare where a group starts and where it
+ends. The proper syntax for a page definition is as follows:
+\beginImportant
+{\tt \\begin\{page\}\{{\it name}\}\{{\it a title}\}}
+\newline
+.
+\newline
+.
+\newline
+.
+\newline
+{\tt \\end\{page\}}
+\beginImportant
+
+Note the use of the special characters {\tt \\}, {\tt \{} and {\tt \}}.
+The {\tt \\} (backslash) character introduces a command, in this case, {\tt begin}.
+The {\tt \{ \}} (braces) delimit the {\em parameters} to the command.
+The first parameter (the word {\tt page}) specifies this as a page definition
+command.
+
+The second parameter can be any single unbroken word consisting of
+alphanumeric characters only, and specifies the name of the page by which
+it can be referred to by other commands. You should choose
+this internal name with care so as to avoid potential conflict with
+page names that are defined by the \Language{} system. This caveat only
+applies in the case where you have started \HyperName{} with the \Language{}
+database --- see \downlink{later on}{HTXLinkPage6}. It is suggested that
+the page names you define start with the letters {\tt UX} (standing for
+{\tt U}ser e{\tt X}tensions). You can have a look at the \Language{}
+system database file {\centerline{\bf \env{AXIOM}/doc/hypertex/pages/ht.db} }
+which contains the names of all pages, macros and patches used by \Language{}.
+
+The third parameter specifies a title for the page.
+The title of a page is the area at the very top
+of the window, between the buttons. Virtually anything
+that can be put in the main page can also be put in the
+title. As an example, {\em this} page's
+declaration is like this:\newline
+{\tt \\begin\{page\}\{\thispage\}\{A simple text page\}}
+
+Everything you type between the {\tt \\begin\{page\}} command and the next
+{\tt \\end\{page\}} command will become the body of the page. It is
+an error to insert another {\tt \\begin\{page\}} between the two, that is,
+this group command cannot be nested.
+
+There is another useful group command that should be mentioned here --- the
+{\em scroll} command. It controls the portion of the
+page that will be scrollable. \HyperName{} will split a page in three
+sections: a {\em header}, a {\em scroll region} and a {\em footer}. \HyperName{}
+will always try to keep the header and footer regions visible
+on the page; the header at the top and the footer at the bottom. The middle
+scroll region will be truncated and a scroll bar will be
+automatically provided if the window becomes too small for the whole contents
+of the page. Only one scroll region can be defined in a page and the correct
+syntax is as follows:
+\beginImportant
+{\tt \\begin\{scroll\}}
+\newline
+.
+\newline
+.
+\newline
+.
+\newline
+{\tt \\end\{scroll\}}
+\beginImportant
+
+This group should be placed inside the relevant page group. The text between the
+{\tt \\begin\{page\}} and {\tt \\begin\{scroll\}} commands defines the header
+region, the text inside the scroll group defines the scroll region and the text
+between the {\tt \\end\{scroll\}} and {\tt \\end\{page\}} commands defines the
+footer region. It is important to keep the header and footer areas small.
+Use them to display information that might be needed at any time by the user.
+If you don't define a scroll region in your page, you may find that a portion
+of the page is truncated.
+
+You are now ready to experiment with a page of your own. If you just want
+to display some text on a page, you don't need any other \HyperName{}
+commands. Just make sure that the text you type for the title, header,
+scroll and footer regions does not contain (for the moment) any of the \HyperName{}
+special characters.
+
+\end{scroll}
+\beginmenu
+\menuwindowlink{Try out what you learned}{HTXTryPage}
+\menudownlink{Next -- Learn how to format text}{HTXFormatTopPage}
+\endmenu
+\end{page}
+
diff --git a/src/hyper/pages/HTXIntroTopPage.ht b/src/hyper/pages/HTXIntroTopPage.ht
new file mode 100644
index 00000000..b0306963
--- /dev/null
+++ b/src/hyper/pages/HTXIntroTopPage.ht
@@ -0,0 +1,27 @@
+\begin{page}{HTXIntroTopPage}{First Steps}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\beginscroll
+
+\HyperName{} is both a way of presenting information and
+a customisable front-end. \Language{} uses
+it for its own purpose as a front-end and documentation system.
+\HyperName{} has special facilities that allow it to interact
+very closely with \Language{}.  The \Browse{} facility, the Basic
+Commands section and the ability to execute \Language{} commands
+by clicking on \HyperName{} text are witness to this.
+
+These pages will show you the features of \HyperName{} that might
+make it appropriate for your own use in, for example, providing
+documentation for \Language{} code that you write or some other purpose.
+
+It is recommended that you get familiar with the {\em use} of
+\HyperName{} before proceeding.
+
+\endscroll
+\beginmenu
+\menudownlink{What \HyperName{} does}{HTXIntroPage1}
+\menudownlink{How \HyperName{} does it}{HTXIntroPage2}
+\menudownlink{Define a simple text page}{HTXIntroPage3}
+\endmenu
+
+\end{page}
diff --git a/src/hyper/pages/HTXLinkPage1.ht b/src/hyper/pages/HTXLinkPage1.ht
new file mode 100644
index 00000000..9c13fb61
--- /dev/null
+++ b/src/hyper/pages/HTXLinkPage1.ht
@@ -0,0 +1,104 @@
+\begin{page}{HTXLinkPage1}{Linking to a named page}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+In \HyperName{}, hypertext links are specified by different
+flavors of the {\tt \\link} command. These commands take two
+arguments. One argument specifies the active area, that
+is, the {\it trigger} of the link. The second argument specifies the
+{\it target} of the link, that is, a page. The trigger can be quite arbitrary
+\HyperName{} text and can include images or whole paragraphs. The trigger
+text will be formatted in the normal fashion but its default font will be
+the font specified by the ActiveFont resource.
+
+The simplest kind of \HyperName{} link is a link to a named page.
+Clicking on the trigger will cause the named page to appear in a
+\HyperName{} window.
+There are three flavors for such a link.
+\begin{items}[123456]
+\item\menuitemstyle{{\tt \\windowlink\{{\it trigger}\}\{{\it page name}\}}}
+\newline
+This link command, when activated, will create a new window for the named page.
+\newline
+There will be no \centerline{\UpBitmap{}  or \ReturnBitmap{}}  buttons on the new page.
+The new page will have a \centerline{\ExitBitmap{}} button, however.
+The original page containing the {\tt \\windowlink} command will be unaffected.
+\item\menuitemstyle{{\tt \\downlink\{{\it trigger}\}\{{\it page name}\}} }
+\newline This link command, when activated, will cause the
+current page to be replaced by the target page
+in the same \HyperName{} window.
+A \centerline{\UpBitmap{}}  button will automatically be placed
+on the new page allowing you to get back to the page
+containing the {\tt \\downlink} command.
+If the current page has a \centerline{\ReturnBitmap{}} button then
+the target page will also carry it. The associated
+target page of that button will be the same as it is
+in the current page.
+\item\menuitemstyle{{\tt \\memolink\{{\it trigger}\}\{{\it page name}\}}}
+\newline This link command is similar to the {\tt \\downlink} command.
+In addition, it will cause a \centerline{\ReturnBitmap{}}  button to be included in
+the target page and all pages {\tt \\downlink}ed from it. This button will act as a
+direct link to the page containing the {\tt \\memolink} command allowing
+a short-cut to be taken.
+\end{items}
+
+\beginImportant
+\begin{paste}{HTXLinkPage1xPaste1}{HTXLinkPage1xPatch1}
+\pastebutton{HTXLinkPage1xPaste1}{Interpret}
+\newline
+{\tt \\windowlink\{windowlink to Actions menu\}\{HTXLinkTopPage\}\\newline}\newline
+{\tt \\downlink\{downlink to Actions menu\}\{HTXLinkTopPage\}\\newline}\newline
+{\tt \\memolink\{memolink to Actions menu\}\{HTXLinkTopPage\}}
+\end{paste}
+\endImportant
+
+
+There is a fourth button that can appear at the top of the page
+next to the \centerline{\ExitBitmap{}} button.
+Its purpose is to provide access to a particular {\it help page}
+associated with the current page.
+That is the \centerline{\HelpBitmap{}} button. The command to use
+is
+\centerline{{\tt \\helppage\{{\it help page name}\}}}
+The {\tt \\helppage} command {\it must } be placed
+just before the {\tt \\end\{page\}} command.
+For instance, to get a help button on this page
+the following command is used.
+\centerline{{\tt {\\helppage\{TestHelpPage\}}}}
+Clicking on the help button at the top
+will display the {\tt TestHelpPage} page in a new window.
+
+\end{scroll}
+\beginmenu
+\menulink{Next -- Standard Pages}{HTXLinkPage2}
+\endmenu
+
+\helppage{TestHelpPage}
+\end{page}
+\begin{patch}{HTXLinkPage1xPatch1}
+\begin{paste}{HTXLinkPage1xPaste1A}{HTXLinkPage1xPatch1A}
+\pastebutton{HTXLinkPage1xPaste1A}{Source}
+\newline
+\windowlink{windowlink to Actions
+menu}{HTXLinkTopPage}\newline
+\downlink{downlink to Actions menu}{HTXLinkTopPage}\newline
+\memolink{memolink to Actions menu}{HTXLinkTopPage}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage1xPatch1A}
+\begin{paste}{HTXLinkPage1xPaste1B}{HTXLinkPage1xPatch1}
+\pastebutton{HTXLinkPage1xPaste1B}{Interpret}
+\newline
+{\tt \\windowlink\{windowlink to Actions menu\}\{HTXLinkTopPage\}\\newline}\newline
+{\tt \\downlink\{downlink to Actions menu\}\{HTXLinkTopPage\}\\newline}\newline
+{\tt \\memolink\{memolink to Actions menu\}\{HTXLinkTopPage\}}
+\end{paste}
+\end{patch}
+
+\begin{page}{TestHelpPage}{Test Help Page}
+\begin{scroll}
+
+\vspace{100}
+\centerline{Is this any help?}
+\end{scroll}
+\end{page}
diff --git a/src/hyper/pages/HTXLinkPage2.ht b/src/hyper/pages/HTXLinkPage2.ht
new file mode 100644
index 00000000..7e54ce5d
--- /dev/null
+++ b/src/hyper/pages/HTXLinkPage2.ht
@@ -0,0 +1,110 @@
+\begin{page}{HTXLinkPage2}{Standard Pages}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+You have reached this page after performing
+a series of mouse clicks on \HyperName{}
+active areas. Each time, a {\tt \\link}
+command was activated. Well, how does it all
+start?
+The answer is that \HyperName{} always puts up
+a particular page called {\tt RootPage} when
+it starts up. If this page is not found in the database,
+\HyperName{} will immediately exit.
+It is, of course, desirable that the {\tt RootPage}
+contains links to other pages!
+It is possible to override
+\Language{}'s choice of {\tt RootPage} and provide your own
+to \HyperName{}. This is done in the same way as
+you would override any \Language{}-defined page and is
+discussed in \downlink{How to use your pages with \HyperName{}}{HTXLinkPage6}.
+
+
+
+You may have noticed that \HyperName{}
+uses some pages when certain events occur.
+There is a page that is put up, for instance,
+whenever \HyperName{} cannot connect to \Language{}.
+Another page is put up whenever there is a formatting
+error and yet another when a request for an unknown page
+is made. Finally, there is a page that prompts
+for confirmation when you press the
+exit button on the initial page.
+
+
+These pages have standard names and must be provided
+in the \HyperName{} page database.
+They are already defined in the \Language{} system
+\HyperName{} page database so that you do not have to
+define them yourself.
+
+Here are the pages required by \HyperName{}. You can click on any of these
+to see their contents. Click on their exit buttons when you are finished.
+
+\beginImportant
+\begin{paste}{HTXLinkPage2xPaste1}{HTXLinkPage2xPatch1}
+\pastebutton{HTXLinkPage2xPaste1}{Interpret}
+\newline
+{\tt \\table\{}\newline
+{\tt \{\\windowlink\{SpadNotConnectedPage\}\{SpadNotConnectedPage\}\}}\newline
+{\tt \{\\windowlink\{UnknownPage\}\{UnknownPage\}\}}\newline
+{\tt \{\\windowlink\{ErrorPage\}\{ErrorPage\}\}}\newline
+{\tt \{\\windowlink\{ProtectedQuitPage\}\{ProtectedQuitPage\}\}}\newline
+{\tt \}}\newline
+\end{paste}
+\endImportant
+
+
+In addition, \HyperName{} uses certain bitmaps for its buttons.
+They are also provided in the \Language{} system
+bitmap directory and \HyperName{} knows where to find them.
+
+The bitmap files required by \HyperName{} are the following.
+\newline
+\tab{7}{\it exit.bitmap}\tab{22} = \tab{25}{\ExitBitmap{}} \newline
+\tab{7}{\it help2.bitmap}\tab{22} = \tab{25}{\HelpBitmap{}} \newline
+\tab{7}{\it up3.bitmap}\tab{22} = \tab{25}{\UpBitmap{}}\newline
+\tab{7}{\it return3.bitmap}\tab{22} = \tab{25}{\ReturnBitmap{}}\newline
+\tab{7}{\it noop.bitmap}\tab{22} = \tab{25}{\NoopBitmap{}}
+
+These files must exist in your current directory if
+the {\tt AXIOM} environment variable is not set.
+If it is, then \HyperName{} will assume that it points
+to the \Language{} system directory and will look for
+these files in
+{\bf \$AXIOM/doc/hypertex/bitmaps}.
+
+
+
+
+\end{scroll}
+\beginmenu
+\menulink{Next -- Active \Language{} commands}{HTXLinkPage3}
+\endmenu
+
+\end{page}
+\begin{patch}{HTXLinkPage2xPatch1}
+\begin{paste}{HTXLinkPage2xPaste1A}{HTXLinkPage2xPatch1A}
+\pastebutton{HTXLinkPage2xPaste1A}{Source}
+\newline
+\table{
+{\windowlink{SpadNotConnectedPage}{SpadNotConnectedPage}}
+{\windowlink{UnknownPage}{UnknownPage}}
+{\windowlink{ErrorPage}{ErrorPage}}
+{\windowlink{ProtectedQuitPage}{ProtectedQuitPage}}
+}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage2xPatch1A}
+\begin{paste}{HTXLinkPage2xPaste1B}{HTXLinkPage2xPatch1}
+\pastebutton{HTXLinkPage2xPaste1B}{Interpret}
+\newline
+{\tt \\table\{}\newline
+{\tt \{\\windowlink\{SpadNotConnectedPage\}\{SpadNotConnectedPage\}\}}\newline
+{\tt \{\\windowlink\{UnknownPage\}\{UnknownPage\}\}}\newline
+{\tt \{\\windowlink\{ErrorPage\}\{ErrorPage\}\}}\newline
+{\tt \{\\windowlink\{ProtectedQuitPage\}\{ProtectedQuitPage\}\}}\newline
+{\tt \}}\newline
+\end{paste}
+\end{patch}
+
diff --git a/src/hyper/pages/HTXLinkPage3.ht b/src/hyper/pages/HTXLinkPage3.ht
new file mode 100644
index 00000000..2f782e83
--- /dev/null
+++ b/src/hyper/pages/HTXLinkPage3.ht
@@ -0,0 +1,218 @@
+\begin{page}{HTXLinkPage3}{Active\ \Language{} commands}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+This section explains how to include \Language{}
+commands in your page. The commands we will
+introduce are actually {\it macros} that are defined
+in
+\centerline{{\bf \env{AXIOM}/doc/hypertex/pages/util.ht}}
+This means that you can use them only if you include
+this file in your \HyperName{} database.
+
+The first command to learn is
+\horizontalline
+{\tt \\axiomcommand\{ {\it command }{\tt \ \\free\{}{\it var1 var2 ...}{\tt \}\ \\bound\{}{\it var}{\tt \}\ \}} }
+\horizontalline
+
+
+The {\tt \\free\{\}} and {\tt \\bound\{\}} directives are optional.
+We will come to them in a minute. The {\it command} above is the
+text of the \Language{} command. Only single line commands are allowed
+here.
+This text will be displayed in the reserved AxiomFont logical
+font.  The area of the text will be active and clicking on it
+will attempt to send the command to \Language{} for evaluation.
+A new \Language{} interpreter window (and \Language{} frame)
+will be created if this was the first \Language{} command
+activated in the current page. If not, the command will be sent
+to the already opened \Language{} interpreter window for the current page.
+Note that it {\it is} necessary to escape special
+\HyperName{} characters with the {\tt '\\'} backslash character.
+The exceptions are the characters {\tt \[\]}; they do not
+need to be escaped in this context.
+
+\beginImportant
+\begin{paste}{HTXLinkPage3xPaste1}{HTXLinkPage3xPatch1}
+\pastebutton{HTXLinkPage3xPaste1}{Interpret}
+\newline
+{\tt \\axiomcommand\{ l:=brace[1,2,3] ; length:=\\\# l ; m:=[1,2]\}}
+\end{paste}
+\endImportant
+
+
+The optional {\tt \\free\{\}} and {\tt \\bound\{\}} directives
+provide dependency control. The reader of a \HyperName{}
+page is not forced to click on the commands in the
+order in which in they appear on the page.  If the
+correct {\tt \\free\{\}} and {\tt \\bound\{\}}
+specifications are made, clicking on a command
+will result in execution of all those other
+commands that should be executed before it.
+This will {\it only} happen the first time the command is
+clicked.
+
+So, how are the dependencies specified?
+The arguments of the {\tt \\free\{\}} directive must
+be space-separated words (labels). The argument of {\tt \\bound\{\}}
+must be a single (unique for the page) label. Each label in the {\tt \\free\{\}} list
+must exist as an argument to one (and only one) {\tt \\bound\{\}} directive
+somewhere in the current page.
+When the command is activated, \HyperName{} will look
+in the {\tt \\free\{\}} list and check each label.
+For each label, it will find the command that specifies that label
+in its {\tt \\bound\{\}} directive and
+execute it if it has not been already executed.
+The order of labels in the {\tt \\free\{\}} directive list
+is respected. \HyperName{} will follow all
+dependency links recursively.
+
+Here is  an example.
+Clicking on the third command will automatically
+execute all of them in the correct sequence.
+Note that in this case the order of labels in the last
+line is immaterial since {\tt v2} explicitly depends on {\tt v1}.
+
+\beginImportant
+\begin{paste}{HTXLinkPage3xPaste2}{HTXLinkPage3xPatch2}
+\pastebutton{HTXLinkPage3xPaste2}{Interpret}
+\newline
+{\tt \\axiomcommand\{a:=1;d:=4 \\bound\{v1\}\}}\newline
+{\tt \\newline}\newline
+{\tt \\axiomcommand\{b:=a+3 \\free\{v1\} \\bound\{v2\}\}}\newline
+{\tt \\newline}\newline
+{\tt \\axiomcommand\{c:=b+d \\free\{v1 v2\} \\bound\{v3\}\}}\newline
+\end{paste}
+\endImportant
+
+The second command deals with multi-line \Language{}
+code. This is the command to use for execution of
+an \Language{} {\it pile}. It is a {\it group}
+command. The proper syntax for it is as follows:
+\horizontalline
+{\tt \\begin\{spadsrc\}\ [\\free\{{\it var1 var2} ...\}\ \\bound\{{\it var}\}]}
+\newline
+.
+\newline
+.
+\newline
+{\tt \\end\{spadsrc\}}
+\horizontalline
+
+Again, the {\tt \\free} and {\tt \\bound} directives are
+optional. If they are specified (in exactly the same way
+as {\tt \\axiomcommand}), they must be enclosed in
+square brackets {\tt []}.
+The lines between the {\tt \\begin} and {\tt \\end}
+contain the \Language{} statements. Indentation
+will be respected. \HyperName{} will
+actually save this part in a temporary file
+and instruct \Language{} to read the file
+with the {\tt )read} system command.
+
+Here is an example. The execution of the following
+fragment is dependent on the {\tt v3} label.
+Make sure that previous commands are active (and
+hence the label {\tt v3} is "visible") before
+trying to execute it. If the label {\tt v3}
+is not seen in the page, \HyperName{} will
+print an error message on standard output
+and ignore the dependency.
+
+
+\beginImportant
+\begin{paste}{HTXLinkPage3xPaste3}{HTXLinkPage3xPatch3}
+\pastebutton{HTXLinkPage3xPaste3}{Interpret}
+\newline
+{\tt \\begin\{spadsrc\}\ [\\free\{v3\}\ \\bound\{v4\}]}\newline
+{\tt f\ x\ ==}\newline
+{\tt \ \ \ x+c}\newline
+{\tt f\ 3}\newline
+{\tt \\end\{spadsrc\}}
+\end{paste}
+\endImportant
+
+There is, in fact, more that one can do
+with \Language{} commands. In pages elsewhere
+in the system, \Language{} commands appear next
+to button like this \ \MenuDotBitmap{}.
+Clicking on this button, one can see the output
+for that command. The output has been
+pre-computed and is also stored in
+\HyperName{} files. This is done using
+{\it patch} and {\it paste}.
+It is the same mechanism that
+is used to alternatively display
+\HyperName{} source and interpreted
+result in this and other pages.
+It is explained \downlink{later on}{HTXAdvPage5}.
+
+
+\end{scroll}
+\beginmenu
+\menulink{Next -- Linking to Lisp}{HTXLinkPage4}
+\endmenu
+
+\end{page}
+\begin{patch}{HTXLinkPage3xPatch1}
+\begin{paste}{HTXLinkPage3xPaste1A}{HTXLinkPage3xPatch1A}
+\pastebutton{HTXLinkPage3xPaste1A}{Source}
+\newline
+\axiomcommand{ l:=brace[1,2,3] ; length:=\# l ; m:=[1,2]}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage3xPatch1A}
+\begin{paste}{HTXLinkPage3xPaste1B}{HTXLinkPage3xPatch1}
+\pastebutton{HTXLinkPage3xPaste1B}{Interpret}
+\newline
+{\tt \\axiomcommand\{ l:=brace[1,2,3] ; length:=\\\# l ; m:=[1,2]\}}
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXLinkPage3xPatch2}
+\begin{paste}{HTXLinkPage3xPaste2A}{HTXLinkPage3xPatch2A}
+\pastebutton{HTXLinkPage3xPaste2A}{Source}
+\newline
+\axiomcommand{a:=1;d:=4 \bound{v1}}
+\newline
+\axiomcommand{b:=a+3 \free{v1} \bound{v2}}
+\newline
+\axiomcommand{c:=b+d \free{v1 v2} \bound{v3}}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage3xPatch2A}
+\begin{paste}{HTXLinkPage3xPaste2B}{HTXLinkPage3xPatch2}
+\pastebutton{HTXLinkPage3xPaste2B}{Interpret}
+\newline
+{\tt \\axiomcommand\{a:=1;d:=4 \\bound\{v1\}\}}\newline
+{\tt \\newline}\newline
+{\tt \\axiomcommand\{b:=a+3 \\free\{v1\} \\bound\{v2\}\}}\newline
+{\tt \\newline}\newline
+{\tt \\axiomcommand\{c:=b+d \\free\{v1 v2\} \\bound\{v3\}\}}\newline
+\end{paste}
+\end{patch}
+
+
+\begin{patch}{HTXLinkPage3xPatch3}
+\begin{paste}{HTXLinkPage3xPaste3A}{HTXLinkPage3xPatch3A}
+\pastebutton{HTXLinkPage3xPaste3A}{Source}
+\newline
+\begin{spadsrc} [\free{v3} \bound{v4}]
+f x ==
+   x+c
+f 3
+\end{spadsrc}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage3xPatch3A}
+\begin{paste}{HTXLinkPage3xPaste3B}{HTXLinkPage3xPatch3}
+\pastebutton{HTXLinkPage3xPaste3B}{Interpret}
+\newline
+{\tt \\begin\{spadsrc\}\ [\\free\{v3\}\ \\bound\{v4\}]}\newline
+{\tt f\ x\ ==}\newline
+{\tt \ \ \ x+c}\newline
+{\tt f\ 3}\newline
+{\tt \\end\{spadsrc\}}
+\end{paste}
+\end{patch}
+
diff --git a/src/hyper/pages/HTXLinkPage4.ht b/src/hyper/pages/HTXLinkPage4.ht
new file mode 100644
index 00000000..7d35621e
--- /dev/null
+++ b/src/hyper/pages/HTXLinkPage4.ht
@@ -0,0 +1,328 @@
+\begin{page}{HTXLinkPage4}{Linking to Lisp}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+Another feature of the \Language{}\hspace{2}--\HyperName{}
+link is the ability to execute {\it Lisp}
+code at a click of a button.
+There are two things one can do.
+
+The first is to cause the evaluation
+of a {\it Lisp} form and ignore (as far as \HyperName{}
+is concerned) its value. The evaluation of the function
+might have an effect however on your \Language{} session.
+
+The command for this is
+\horizontalline
+\centerline{ {\tt \\lispcommand\{{\it text}\}\{{\it Lisp form}\}}}
+\horizontalline
+
+Here is an example. We will first define a {\it Lisp} function
+and then execute it.  Notice that the \HyperName{}
+special characters must be escaped (this is on top
+of {\it Lisp} escaping conventions).
+
+
+\beginImportant
+\begin{paste}{HTXLinkPage4xPaste1}{HTXLinkPage4xPatch1}
+\pastebutton{HTXLinkPage4xPaste1}{Interpret}
+\newline
+{\tt \\lispcommand\{Definition\}\{(defun HTXTESTFUNCTION ()}\newline
+{\tt  (print "Hello from HyperDoc \\\\\\\\ \\\% \\\{ \\\}"))\}} \newline
+{\tt \\newline}\newline
+{\tt \\lispcommand\{Execution\}\{(HTXTESTFUNCTION)\}} \newline
+\end{paste}
+\endImportant
+
+Your command will be executed as soon as
+\Language{} completes any computation it might be
+carrying out.
+
+
+%\axiomcommand{)lisp (defun f () (pprint "hello"))}
+%\lispcommand{f}{(|f|)}
+
+
+The second thing you can do is quite powerful. It allows you
+to delegate to a {\it Lisp} function
+the {\it dynamic} creation of a page. This is used
+in \Browse{} to present
+the \Language{} Library in a hypertext form.
+
+The command to use is a lot like the {\tt link} commands
+you encountered \downlink{earlier}{HTXLinkPage1} and comes in three flavours.
+\centerline{{\tt \\lispwindowlink\{{\it trigger}\}\{{\it Lisp form}\}}}
+\centerline{{\tt \\lispdownlink\{{\it trigger}\}\{{\it Lisp form}\}}}
+\centerline{{\tt \\lispmemolink\{{\it trigger}\}\{{\it Lisp form}\}}}
+
+The difference between the three versions is the same as before.
+When such a link is activated, \HyperName{} issues the
+{\it Lisp form} to \Language{} and waits for a full
+page definition. An important point to note is that
+\HyperName{} does {\it not} use
+the value of the {\it Lisp form} but, instead, it
+depends on its {\it side-effects}.
+What {\it must} happen during evaluation
+of the form is enough evaluations of a special {\it Lisp}
+function called {\bf issueHT} to define a page.
+The argument of {\bf issueHT} is a string
+containing \HyperName{} text. Perhaps an example will clarify
+matters.
+
+First we will define a {\it Lisp} function that accepts
+a string argument and calls {\bf issueHT} a few times.
+The strings that are passed to {\bf issueHT} construct
+a \HyperName{} page that would just contain our
+original argument centered roughly on the page.
+Then we write the {\tt \\lisplink} with a call to
+the function. Finally, we execute a {\it Lisp}
+command that just pretty--prints the function's definition.
+
+
+
+\beginImportant
+\begin{paste}{HTXLinkPage4xPaste2}{HTXLinkPage4xPatch2}
+\pastebutton{HTXLinkPage4xPaste2}{Interpret}
+\newline
+{\tt \\lispcommand\{Definition\}\{(defun HTXTESTPAGE (x) (|issueHT|}\newline
+{\tt  "\\\\\\\\begin\\\{page\\\}\\\{LispTestPage\\\}\\\{Lisp Test Page\\\}}\newline
+{\tt  \\\\\\\\vspace\\\{150\\\} \\\\\\\\centerline\\\{") (|issueHT| x) (|issueHT|}\newline
+{\tt  "\\\} \\\\\\\\end\\\{page\\\}" ) ) \}}\newline
+{\tt \\newline}\newline
+{\tt \\lispwindowlink\{Link to it\}\{(HTXTESTPAGE "Hi there")\}}\newline
+{\tt \\newline}\newline
+{\tt \\lispcommand\{Show Lisp definition\}\{(pprint (symbol-function 'HTXTESTPAGE))\}}\newline
+\end{paste}
+\endImportant
+
+The {\tt '\\\{'} and {\tt '\\\}'} is required to escape
+\HyperName{}'s special characters {\tt '\{'} and {\tt  '\}'}.
+The {\tt '\\\\\\\\'} has the following rationale.
+We need to send to \HyperName{} (from {\it Lisp}) the sequence
+{\tt \\begin}. But {\tt '\\'} is a special {\it Lisp}
+character. Therefore the {\it Lisp} string must be
+{\tt '\\\\begin'}. But to specify this
+in \HyperName{} we need to escape the two {\tt '\\'}.
+Therefore, we write {\tt '\\\\\\\\begin'}.
+
+
+The definition of {\tt HTXTESTPAGE} would have been written in {\it Lisp}
+as follows.
+\begin{verbatim}
+(defun HTXTESTPAGE (X)
+   (|issueHT|
+      "\\begin{page}{LispTestPage}{Lisp Test Page} \\vspace{200} \\centerline{")
+   (|issueHT| X)
+   (|issueHT| "} \\end{page}"))
+\end{verbatim}
+
+
+
+You should not execute {\tt HTXTESTPAGE} in the
+{\it Lisp} environment manually. It is meant to
+be executed {\it only} in response to a
+\HyperName{} request.
+
+Can you pop-up a named page from {\it Lisp} regardless of
+user action? Yes --- use {\it Lisp} function {\bf linkToHTPage}
+with the page name as a string argument. Click on the
+{\tt \\axiomcommand} below. Then, in your \Language{}
+session, you can repeat it if you like.
+
+\beginImportant
+\begin{paste}{HTXLinkPage4xPaste3}{HTXLinkPage4xPatch3}
+\pastebutton{HTXLinkPage4xPaste3}{Interpret}
+\newline
+{\tt \\axiomcommand\{)lisp (|linkToHTPage| "RootPage")\}}
+\end{paste}
+\endImportant
+
+You can also pop-up a {\it dynamic} page regardless of user action.
+To do this, make sure you evaluate the {\it Lisp form}
+{\bf (|startHTPage| 50)} before using {\bf issueHT}.
+The example below requires the {\tt HTXTESTPAGE} function
+to be defined in {\it Lisp} so you should make sure
+you have executed the command above that defines it.
+
+\beginImportant
+\begin{paste}{HTXLinkPage4xPaste4}{HTXLinkPage4xPatch4}
+\pastebutton{HTXLinkPage4xPaste4}{Interpret}
+\newline
+{\tt \\axiomcommand\{)lisp (progn (|startHTPage| 50)(HTXTESTPAGE "Immediately"))\}}
+\end{paste}
+\endImportant
+
+Now, the most important use of this facility
+so far has been in the \Browse{} and Basic Commands components of
+\HyperName{}. Instead of giving you details of the various
+\Browse{} {\it Lisp} functions, a few macros are defined in
+\centerline{{\bf \$AXIOM/doc/hypertex/pages/util.ht}}
+
+The most important defined macros are
+\beginImportant
+\table{
+{ {\tt \\axiomType\{{\it constructor}\}} }
+{ {\tt \\axiomOp\{{\it operation}\}} }
+{ {\tt \\axiomOpFrom\{{\it operation }\}\{{\it constructor}\}}}
+}
+\endImportant
+
+Here are some examples of their use.
+\beginImportant
+\begin{paste}{HTXLinkPage4xPaste5}{HTXLinkPage4xPatch5}
+\pastebutton{HTXLinkPage4xPaste5}{Interpret}
+\newline
+{\tt \\axiomType\{Expression Integer\}}\newline
+{\tt \newline}\newline
+{\tt \\axiomType\{Expression\}}\newline
+{\tt \newline}\newline
+{\tt \\axiomType\{EXPR\}}\newline
+{\tt \newline}\newline
+{\tt \\axiomOp\{reduce\}}\newline
+{\tt \newline}\newline
+{\tt \\axiomOp\{as*\}}\newline
+{\tt \newline}\newline
+{\tt \\axiomOpFrom\{reduce\}\{Expression\}}\newline
+\end{paste}
+\endImportant
+
+The macro {\tt \\axiomType} brings up the \Browse{}
+constructor page for the constructor specified.
+You can specify a full name, or an abbreviation
+or just the top level name.
+The macro {\tt \\axiomOp} brings up a list of operations
+matching the argument.
+The macro {\tt \\axiomOpFrom} shows documentation
+about the specified operation whose origin is
+constructor. No wildcard in the operation name
+or type abbreviation is
+allowed here.  You should also specify just the top level type.
+
+
+
+
+
+
+
+\end{scroll}
+\beginmenu
+\menulink{Next -- Linking to Unix}{HTXLinkPage5}
+\endmenu
+
+\end{page}
+\begin{patch}{HTXLinkPage4xPatch1}
+\begin{paste}{HTXLinkPage4xPaste1A}{HTXLinkPage4xPatch1A}
+\pastebutton{HTXLinkPage4xPaste1A}{Source}
+\newline
+\lispcommand{Definition}{(defun HTXTESTFUNCTION ()
+(print "Hello from HyperDoc \\\\ \% \{ \}"))}
+\newline
+\lispcommand{Execution}{(HTXTESTFUNCTION)}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage4xPatch1A}
+\begin{paste}{HTXLinkPage4xPaste1B}{HTXLinkPage4xPatch1}
+\pastebutton{HTXLinkPage4xPaste1B}{Interpret}
+\newline
+{\tt \\lispcommand\{Definition\}\{(defun HTXTESTFUNCTION () (print "Hello from HyperDoc \\\\\\\\ \\\% \\\{ \\\}"))\}} \newline
+{\tt \\newline}\newline
+{\tt \\lispcommand\{Execution\}\{(HTXTESTFUNCTION)\}} \newline
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXLinkPage4xPatch2}
+\begin{paste}{HTXLinkPage4xPaste2A}{HTXLinkPage4xPatch2A}
+\pastebutton{HTXLinkPage4xPaste2A}{Source}
+\newline
+\lispcommand{Definition}{(defun HTXTESTPAGE (x) (|issueHT|
+"\\\\begin\{page\}\{LispTestPage\}\{Lisp Test Page\}
+\\\\vspace\{150\} \\\\centerline\{") (|issueHT| x) (|issueHT|
+"\} \\\\end\{page\}" ) ) }
+\newline
+\lispwindowlink{Link to it}{(HTXTESTPAGE "Hi there")}
+\newline
+\lispcommand{Show Lisp definition}{(pprint (symbol-function 'HTXTESTPAGE))}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage4xPatch2A}
+\begin{paste}{HTXLinkPage4xPaste2B}{HTXLinkPage4xPatch2}
+\pastebutton{HTXLinkPage4xPaste2B}{Interpret}
+\newline
+{\tt \\lispcommand\{Definition\}\{(defun HTXTESTPAGE (x) (|issueHT|}\newline
+{\tt  "\\\\\\\\begin\\\{page\\\}\\\{LispTestPage\\\}\\\{Lisp Test Page\\\}}\newline
+{\tt  \\\\\\\\vspace\\\{150\\\} \\\\\\\\centerline\\\{") (|issueHT| x) (|issueHT|}\newline
+{\tt  "\\\} \\\\\\\\end\\\{page\\\}" ) ) \}}\newline
+{\tt \\newline}\newline
+{\tt \\lispwindowlink\{Link to it\}\{(HTXTESTPAGE "Hi there")\}}\newline
+{\tt \\newline}\newline
+{\tt \\lispcommand\{Show Lisp definition\}\{(pprint (symbol-function 'HTXTESTPAGE))\}}\newline
+\end{paste}
+\end{patch}
+
+
+\begin{patch}{HTXLinkPage4xPatch3}
+\begin{paste}{HTXLinkPage4xPaste3A}{HTXLinkPage4xPatch3A}
+\pastebutton{HTXLinkPage4xPaste3A}{Source}
+\newline
+\axiomcommand{)lisp (|linkToHTPage| "RootPage")}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage4xPatch3A}
+\begin{paste}{HTXLinkPage4xPaste3B}{HTXLinkPage4xPatch3}
+\pastebutton{HTXLinkPage4xPaste3B}{Interpret}
+\newline
+{\tt \\axiomcommand\{)lisp (|linkToHTPage| "RootPage")\}}
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXLinkPage4xPatch4}
+\begin{paste}{HTXLinkPage4xPaste4A}{HTXLinkPage4xPatch4A}
+\pastebutton{HTXLinkPage4xPaste4A}{Source}
+\newline
+\axiomcommand{)lisp (progn (|startHTPage| 50)(HTXTESTPAGE "Immediately"))}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage4xPatch4A}
+\begin{paste}{HTXLinkPage4xPaste4B}{HTXLinkPage4xPatch4}
+\pastebutton{HTXLinkPage4xPaste4B}{Interpret}
+\newline
+{\tt \\axiomcommand\{)lisp (progn (|startHTPage| 50)(HTXTESTPAGE "Immediately"))\}}
+\end{paste}
+\end{patch}
+
+
+\begin{patch}{HTXLinkPage4xPatch5}
+\begin{paste}{HTXLinkPage4xPaste5A}{HTXLinkPage4xPatch5A}
+\pastebutton{HTXLinkPage4xPaste5A}{Source}
+\newline
+\axiomType{Expression Integer}
+\newline
+\axiomType{Expression}
+\newline
+\axiomType{EXPR}
+\newline
+\axiomOp{reduce}
+\newline
+\axiomOp{as*}
+\newline
+\axiomOpFrom{reduce}{Expression}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage4xPatch5A}
+\begin{paste}{HTXLinkPage4xPaste5B}{HTXLinkPage4xPatch5}
+\pastebutton{HTXLinkPage4xPaste5B}{Interpret}
+\newline
+{\tt \\axiomType\{Expression Integer\}}\newline
+{\tt \newline}\newline
+{\tt \\axiomType\{Expression\}}\newline
+{\tt \newline}\newline
+{\tt \\axiomType\{EXPR\}}\newline
+{\tt \newline}\newline
+{\tt \\axiomOp\{reduce\}}\newline
+{\tt \newline}\newline
+{\tt \\axiomOp\{as*\}}\newline
+{\tt \newline}\newline
+{\tt \\axiomOpFrom\{reduce\}\{Expression\}}\newline
+\end{paste}
+\end{patch}
+
diff --git a/src/hyper/pages/HTXLinkPage5.ht b/src/hyper/pages/HTXLinkPage5.ht
new file mode 100644
index 00000000..aaa3fab0
--- /dev/null
+++ b/src/hyper/pages/HTXLinkPage5.ht
@@ -0,0 +1,104 @@
+\begin{page}{HTXLinkPage5}{Linking to Unix}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+Let us conclude the tour of \HyperName{}
+actions that can be triggered with a click of a button
+with two more facilities. These are
+\beginImportant
+\table{
+{ {\tt \\unixcommand\{{\it trigger text}\}\{{\it unix command}\}}}
+{ {\tt \\unixlink\{{\it trigger text}\}\{{\it unix command}\}}}
+}
+\endImportant
+
+
+The first one, {\tt \\unixcommand}, is very much like
+{\tt \\axiomcommand} and {\tt \\lispcommand}.
+The trigger text becomes an active area. Clicking on it
+will force \HyperName{} to pass the second argument
+to the system as a shell command to be executed.
+The shell used is {\bf /bin/sh}.
+\HyperName{} ignores the output of the command.
+
+
+\beginImportant
+\begin{paste}{HTXLinkPage5xPaste1}{HTXLinkPage5xPatch1}
+\pastebutton{HTXLinkPage5xPaste1}{Interpret}
+\newline
+{\tt \\unixcommand\{List \\\$HOME directory\}\{ls \\\$HOME\}}\newline
+\end{paste}
+\endImportant
+
+The {\tt \\unixlink} command delegates to a another
+program the creation of a dynamic page. When the trigger
+text is activated, \HyperName{} will invoke the command
+specified in the second argument. It will then start reading
+the {\it standard output} of the command until
+a complete page has been received. It is important that
+a single page and nothing more is written by the command.
+This command is essentially a {\tt \\downlink}, i.e.
+the new page replaces the current page in the window.
+There aren't any other flavours of {\tt \\unixlink}.
+A trivial example is to use {\bf cat} on a \HyperName{}
+file known to contain just one page.
+
+\beginImportant
+\begin{paste}{HTXLinkPage5xPaste2}{HTXLinkPage5xPatch2}
+\pastebutton{HTXLinkPage5xPaste2}{Interpret}
+\newline
+{\tt \\unixlink\{Some file\}} \newline
+{\tt \{cat\\ \\env\{AXIOM\}/doc/hypertex/pages/HTXplay.ht\}}
+\end{paste}
+\endImportant
+
+
+Two things to notice in the second argument of
+{\tt \\unixlink}: You must use a {\it hard space}
+{\tt '\\\ '} to preserve the spacing in the command.
+Also, the {\tt \\env} command allows you to use
+an environment variable in \HyperName{} text.
+
+With a little ingenuity (and maybe some shell and {\bf awk} scripts !)
+, one can use these
+facilities to create, say, a point-and-click
+directory viewer which allows you to edit
+a file by clicking on its name.
+
+\end{scroll}
+\beginmenu
+\menulink{Next -- How to use your pages with \HyperName{}}{HTXLinkPage6}
+\endmenu
+
+\end{page}
+\begin{patch}{HTXLinkPage5xPatch1}
+\begin{paste}{HTXLinkPage5xPaste1A}{HTXLinkPage5xPatch1A}
+\pastebutton{HTXLinkPage5xPaste1A}{Source}
+\newline
+\unixcommand{List \$HOME directory}{ls \$HOME}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage5xPatch1A}
+\begin{paste}{HTXLinkPage5xPaste1B}{HTXLinkPage5xPatch1}
+\pastebutton{HTXLinkPage5xPaste1B}{Interpret}
+\newline
+{\tt \\unixcommand\{List \\\$HOME directory\}\{ls \\\$HOME\}}\newline
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXLinkPage5xPatch2}
+\begin{paste}{HTXLinkPage5xPaste2A}{HTXLinkPage5xPatch2A}
+\pastebutton{HTXLinkPage5xPaste2A}{Source}
+\newline
+\unixlink{Some file}
+{cat\ \env{AXIOM}/doc/hypertex/pages/HTXplay.ht}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage5xPatch2A}
+\begin{paste}{HTXLinkPage5xPaste2B}{HTXLinkPage5xPatch2}
+\pastebutton{HTXLinkPage5xPaste2B}{Interpret}
+\newline
+{\tt \\unixlink\{Some file\}} \newline
+{\tt \{cat\\ \\env\{AXIOM\}/doc/hypertex/pages/HTXplay.ht\}}
+\end{paste}
+\end{patch}
diff --git a/src/hyper/pages/HTXLinkPage6.ht b/src/hyper/pages/HTXLinkPage6.ht
new file mode 100644
index 00000000..eb9e4b23
--- /dev/null
+++ b/src/hyper/pages/HTXLinkPage6.ht
@@ -0,0 +1,218 @@
+\begin{page}{HTXLinkPage6}{How to use your pages with \ \HyperName{}}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\begin{scroll}
+
+Let us say that you have written a few \HyperName{}
+pages and you would like to incorporate them in the system.
+Here is what you should do.
+
+Put all your files in some directory and make sure that
+they all have the {\bf .ht} extension.
+
+You will need a way of "hooking" into a system--defined
+\HyperName{} page. The proper way to do this is to use
+the {\tt \\localinfo} macro. The \Language{} system
+\HyperName{} page database includes, as it should,
+a {\tt RootPage}. This is the page that first comes up
+when you start \HyperName{}. This page contains
+a line like this.
+\beginImportant
+\newline
+{\tt \\localinfo}
+\endImportant
+
+This macro is defined in
+\centerline{ {\bf \env{AXIOM}/doc/hypertex/pages/util.ht}}
+to be (see \downlink{Macros}{HTXAdvPage3} to learn how to define macros):
+\beginImportant
+\newline
+{\tt \\newcommand\{\\localinfo\}\{\}}
+\endImportant
+which is an empty definition (the second argument of {\tt \\newcommand}).
+The idea then is that you {\it override} this definition of the macro
+with your own.
+To do that, include a definition like the following in one (possibly the
+one that contains your top--level page) of your files. You can
+put this definition in its own file if you like.
+\beginImportant
+\newline
+{\tt \\newcommand\{\\localinfo\}\{\\menuwindowlink\{{\it active text}\}} \newline
+{\tt \{{\it page name}\} \\tab\{16\}{\it short description}\}}
+\endImportant
+
+If you have a look at the initial \HyperName{} page, you will
+probably be able to decipher what this does. The macro
+{\tt \\menuwindowlink} is defined (again in {\bf util.ht})
+and is responsible for putting the little square to the left of the
+active area.
+Specify a word or two for {\it active text}. That will become the
+trigger of the {\tt \\link}. Specify the page name of your top--level page
+in {\it page name}. Finally, you can give a comment about the topic
+under {\it short description}. That will appear to the right of the
+{\it active text}.
+
+The next thing you need to do is to create a {\it local database}
+for your files. You will use the {\bf \env{AXIOM}/bin/htadd} program.
+This program will create an {\bf ht.db} file that summarises your
+definitions and acts as an index. Let us present an example
+of its use. Suppose you have two files {\bf user1.ht} and {\bf user2.ht}
+in directory {\bf /u/sugar/\HyperName{}}. You should create the {\bf ht.db}
+in that same directory. To create the {\bf ht.db} file you issue to
+the unix shell:
+\beginImportant
+\newline
+{\tt htadd -f /u/sugar/\HyperName{} /u/sugar/\HyperName{}/user1.ht /u/sugar/\HyperName{}/user2.ht}
+\centerline{or ,if you are already in /u/sugar/\HyperName{}}
+{\tt htadd -l ./user1.ht ./user2.ht}
+\endImportant
+
+
+The options and conventions for {\bf htadd} will be explained below.
+To start \HyperName{} with your own pages, you now need to tell
+it where to search for {\bf ht.db} files and \HyperName{} {\bf .ht}
+files. To do this, define the shell environment variable
+{\bf HTPATH}. The value should be a colon {\tt ':'} separated
+list of directory full pathnames.
+The order of the directories is respected with earlier entries overriding
+later ones. Since we want all the \Language{} pages but need to override the
+{\tt \\localinfo} macro, we should use the value
+\centerline{{\bf /u/sugar/\HyperName{}:\env{AXIOM}/doc/hypertex/pages}}
+The way that you define environment variables depends on the shell
+you are using. In the {\bf /bin/csh}, it would be
+\newline
+{\bf setenv HTPATH /u/sugar/\HyperName{}:\env{AXIOM}{}/doc{}/hypertex{}/pages}
+
+
+
+\beginImportant
+\begin{paste}{HTXLinkPage6xPaste1}{HTXLinkPage6xPatch1}
+\pastebutton{HTXLinkPage6xPaste1}{Options for {\bf htadd}}
+\newline
+\end{paste}
+\endImportant
+
+
+\beginImportant
+\begin{paste}{HTXLinkPage6xPaste2}{HTXLinkPage6xPatch2}
+\pastebutton{HTXLinkPage6xPaste2}{Where does \HyperName{} look for files}
+\newline
+\end{paste}
+\endImportant
+
+
+
+\end{scroll}
+\beginmenu
+\menulink{Back to Actions menu}{HTXLinkTopPage}
+\endmenu
+
+\end{page}
+\begin{patch}{HTXLinkPage6xPatch1}
+\begin{paste}{HTXLinkPage6xPaste1A}{HTXLinkPage6xPatch1A}
+\pastebutton{HTXLinkPage6xPaste1A}{Hide}
+\newline
+Name:
+
+{\tt htadd - create or modify a \HyperName{} database}
+\vspace{}
+\newline
+Syntax:
+
+{\tt htadd [ -l | -s | -f\ }{\it path}{\tt ] [ -d | -n ]\ }{\it filename ...}
+\vspace{}
+\newline
+Options:\indentrel{4}\newline
+{\tt -l}\tab{8}\indentrel{8}
+build {\bf ht.db} database in current working directory.
+This is the default behaviour if no {\tt -l}, {\tt -s} or {\tt -f}
+is specified.
+
+\indentrel{-8}\newline
+{\tt -s}\tab{8}\indentrel{8}
+build {\bf ht.db} database in {\it system} directory. The
+system directory is built as follows. If the {\tt AXIOM}
+variable is defined, the {\bf \$AXIOM/doc/hypertex/pages} directory
+is used. If {\tt AXIOM} is not defined, the
+{\bf /usr/local/axiom/doc/hypertex/pages} directory is used.
+
+
+\indentrel{-8}\newline
+{\tt -f\ }{\it path}\newline\tab{8}\indentrel{8}
+build {\bf ht.db} database in specified {\it path}.
+
+\indentrel{-8}\newline
+{\tt -d}\tab{8}\indentrel{8}
+delete the entries in the specified files from {\bf ht.db}.
+
+\indentrel{-8}\newline
+{\tt -n}\tab{8}\indentrel{8}
+delete {\bf ht.db} and create a new one using only the files
+specified.
+
+If none of {\tt -n} and {\tt -d} is specified, the {\bf ht.db}
+is updated with the entries in the file specified.
+
+
+\indentrel{-8}
+\indentrel{-4}
+\vspace{}\newline
+Filename interpretation :
+\indentrel{12}\newline
+A full pathname (i.e. anything that has a {\tt '/'} in it)
+will be taken do be a completely specified file.
+Otherwise, the following interpretation will occur:
+If the {\tt HTPATH} variable is defined, the directories
+specified in it will be tried in order. If {\tt HTPATH}
+is not defined, then, if {\tt AXIOM} is defined, the
+{\bf \$AXIOM/doc/hypertex/pages} will be tried, else
+the file will be deemed missing and {\bf htadd} will fail.
+\indentrel{-12}
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage6xPatch1A}
+\begin{paste}{HTXLinkPage6xPaste1B}{HTXLinkPage6xPatch1}
+\pastebutton{HTXLinkPage6xPaste1B}{Options for {\bf htadd}}
+\newline
+\end{paste}
+\end{patch}
+
+\begin{patch}{HTXLinkPage6xPatch2}
+\begin{paste}{HTXLinkPage6xPaste2A}{HTXLinkPage6xPatch2A}
+\pastebutton{HTXLinkPage6xPaste2A}{Hide}
+\indentrel{12}\newline
+The \HyperName{} program is
+\centerline{{\bf \env{AXIOM}/lib/hypertex}}
+If {\tt AXIOM} is defined and {\tt HTPATH} is not
+(this is the case when \Language{} starts \HyperName{})
+\HyperName{} will look in
+\centerline{{\bf \env{AXIOM}/doc/hypertex/pages}}
+for the {\bf ht.db} file and all \HyperName{} pages.
+If {\tt HTPATH} is defined, it is assumed that
+it alone points to the directories to be searched
+(the above default will NOT be searched unless
+explicitly specified in {\tt HTPATH}).
+For each directory in {\tt HTPATH}, the {\bf ht.db}
+file, if there, will be read.
+Each file listed in {\bf ht.db} will
+then be searched for in the complete sequence
+of directories in {\tt HTPATH}. Note that
+the {\bf ht.db} does not keep full pathnames
+ of files.
+If a {\it page}, {\it macro} or {\it patch}
+(specified in some {\bf ht.db}) happens
+to be (in a file) in more than one of the directories
+specified in {\tt HTPATH}, \HyperName{}
+will print a warning and explain which version
+in which file is ignored. Generally, earlier
+directories in {\tt HTPATH} are preferred over later
+ones.
+\indentrel{-12}\newline
+\end{paste}
+\end{patch}
+\begin{patch}{HTXLinkPage6xPatch2A}
+\begin{paste}{HTXLinkPage6xPaste2B}{HTXLinkPage6xPatch2}
+\pastebutton{HTXLinkPage6xPaste2B}{Where does \HyperName{} look for files}}
+\newline
+\end{paste}
+\end{patch}
+
diff --git a/src/hyper/pages/HTXLinkTopPage.ht b/src/hyper/pages/HTXLinkTopPage.ht
new file mode 100644
index 00000000..76f8d045
--- /dev/null
+++ b/src/hyper/pages/HTXLinkTopPage.ht
@@ -0,0 +1,19 @@
+\begin{page}{HTXLinkTopPage}{Actions in \ \HyperName{}}
+\centerline{\fbox{{\tt \thispage}}}\newline
+\HyperName{} offers various types of hypertext links.
+You can learn about these facilities by clicking on the topics below.
+\begin{scroll}
+\beginmenu
+\menudownlink{Linking to a named page}{HTXLinkPage1}
+\menudownlink{Standard pages}{HTXLinkPage2}
+\menudownlink{Active \Language{} commands}{HTXLinkPage3}
+\menudownlink{Linking to Lisp}{HTXLinkPage4}
+\menudownlink{Linking to Unix}{HTXLinkPage5}
+\menudownlink{How to use your pages with \HyperName{}}{HTXLinkPage6}
+\endmenu
+\end{scroll}
+\end{page}
+
+
+
+
diff --git a/src/hyper/pages/HTXTopPage.ht b/src/hyper/pages/HTXTopPage.ht
new file mode 100644
index 00000000..d9146ffc
--- /dev/null
+++ b/src/hyper/pages/HTXTopPage.ht
@@ -0,0 +1,15 @@
+\begin{page}{HTXTopPage}{Extending\  \HyperName{}}
+\centerline{\fbox{{\tt \thispage}}}\newline
+This is a guide to extending \HyperName{}. You can learn
+how to write your own \HyperName{} pages and link them to the
+\HyperName{} page database that \Language{} uses.
+\begin{scroll}
+\beginmenu
+\menumemolink{Introduction}{HTXIntroTopPage} \tab{20} An easy start.
+\menumemolink{Formatting}{HTXFormatTopPage} \tab{20} Learn how to format text.
+\menumemolink{Actions}{HTXLinkTopPage} \tab{20} Learn how to define actions.
+\menumemolink{Advanced features}{HTXAdvTopPage} \tab{20} More effects.
+\menuwindowlink{Try it!}{HTXTryPage} \tab{20} Try out what you learn.
+\endmenu
+\end{scroll}
+\end{page}
diff --git a/src/hyper/pages/HTXTryPage.ht b/src/hyper/pages/HTXTryPage.ht
new file mode 100644
index 00000000..ba876a80
--- /dev/null
+++ b/src/hyper/pages/HTXTryPage.ht
@@ -0,0 +1,44 @@
+\begin{page}{HTXTryPage}{Try out \ \HyperName{}}
+\centerline{\fbox{{\tt \thispage}}}\newline
+
+This page allows you to quickly experiment with \HyperName{}.
+It is a good idea to keep it handy as you learn about various commands.
+
+\beginscroll
+
+We are going to use here the \HyperName{} facilities that allow
+us to communicate with external programs and files. For more
+information see \downlink{later on}{HTXLinkPage5}.
+\beginmenu
+\item\menuitemstyle{
+In order to use the buttons at the bottom of this page, you must
+first specify a name for the file you are going to use to hold
+\HyperName{} commands. Edit the input area below to change the
+name of the file.}
+\item\menuitemstyle{
+If the file you specified does not yet exist, click on the
+{\bf Initialize} button below. This action will fill the file
+with the minimum of \HyperName{} commands necessary to define a page.}
+\item\menuitemstyle{
+If you want to edit the file, just click on the {\bf Edit} button.
+This action will pop up a window, and invoke the {\it vi}
+editor on the file. Alternatively, use an editor of your choice.}
+\item\menuitemstyle{
+Once you have finished making the changes to the file, update it and
+click on the {\bf Link} button. \HyperName{} will then read
+the file, interpret it as a new page, and display the page on
+this window. If you change the file and want to display it again,
+just get back to this page and click on {\bf Link} again. }
+\endmenu
+\endscroll
+\beginmenu
+{\it Filename: }{\inputstring{filename}{40}{\env{HOME}/HTXplay.ht}}
+\menuunixcommand{Initialize}{cp\space{1}\env{AXIOM}/doc/hypertex/pages/HTXplay.ht \stringvalue{filename}} \tab{20} Get a fresh copy from the system.
+\menuunixcommand{Edit}{xterm -T "\stringvalue{filename}" -e vi \stringvalue{filename}} \tab{20} Edit the file.
+\menuunixwindow{Link}{cat \space{1}\stringvalue{filename}} \tab{20} Link to the page defined in the file.
+\endmenu
+{\it Important : The file must contain
+one and only one page definition and must not contain any macro or patch
+definitions.}
+\end{page}
+
diff --git a/src/hyper/pages/HTXplay.ht b/src/hyper/pages/HTXplay.ht
new file mode 100644
index 00000000..5426fd53
--- /dev/null
+++ b/src/hyper/pages/HTXplay.ht
@@ -0,0 +1,7 @@
+\begin{page}{HTXPlayPage}{Play Page}
+HEADER
+\begin{scroll}
+MAIN TEXT
+\end{scroll}
+FOOTER
+\end{page}
diff --git a/src/hyper/pages/INT.ht b/src/hyper/pages/INT.ht
new file mode 100644
index 00000000..ac092b41
--- /dev/null
+++ b/src/hyper/pages/INT.ht
@@ -0,0 +1,387 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\IntegerXmpTitle}{Integer}
+\newcommand{\IntegerXmpNumber}{9.34}
+%
+% =====================================================================
+\begin{page}{IntegerXmpPage}{9.34 Integer}
+% =====================================================================
+\beginscroll
+
+%% int.htex
+%%
+%% Integer examples
+%%
+
+\Language{} provides many operations for manipulating arbitrary
+precision integers.
+In this section we will show some of those that come from \spadtype{Integer}
+itself plus some that are implemented in other packages.
+More examples of using integers are in the following sections:
+\downlink{``\ugIntroNumbersTitle''}{ugIntroNumbersPage} in section \ugIntroNumbersNumber
+\downlink{`IntegerNumberTheoryFunctions'}{IntegerNumberTheoryFunctionsXmpPage}\ignore{IntegerNumberTheoryFunctions},
+\downlink{`DecimalExpansion'}{DecimalExpansionXmpPage}\ignore{DecimalExpansion},
+\downlink{`BinaryExpansion'}{BinaryExpansionXmpPage}\ignore{BinaryExpansion},
+\downlink{`HexadecimalExpansion'}{HexadecimalExpansionXmpPage}\ignore{HexadecimalExpansion}, and
+\downlink{`RadixExpansion'}{RadixExpansionXmpPage}\ignore{RadixExpansion}.
+
+\beginmenu
+    \menudownlink{{9.34.1. Basic Functions}}{ugxIntegerBasicPage}
+    \menudownlink{{9.34.2. Primes and Factorization}}{ugxIntegerPrimesPage}
+    \menudownlink{{9.34.3. Some Number Theoretic Functions}}{ugxIntegerNTPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxIntegerBasicTitle}{Basic Functions}
+\newcommand{\ugxIntegerBasicNumber}{9.34.1.}
+%
+% =====================================================================
+\begin{page}{ugxIntegerBasicPage}{9.34.1. Basic Functions}
+% =====================================================================
+\beginscroll
+
+\labelSpace{3pc}
+\xtc{
+The size of an integer in \Language{} is only limited by the amount of
+computer storage you have available.
+The usual arithmetic operations are available.
+}{
+\spadpaste{2**(5678 - 4856 + 2 * 17)}
+}
+
+\xtc{
+There are a number of ways of working with the sign of an integer.
+Let's use this \spad{x} as an example.
+}{
+\spadpaste{x := -101 \bound{x}}
+}
+\xtc{
+First of all, there is the absolute value function.
+}{
+\spadpaste{abs(x) \free{x}}
+}
+\xtc{
+The \spadfunFrom{sign}{Integer} operation returns \spad{-1} if its argument
+is negative, \spad{0} if zero and \spad{1} if positive.
+}{
+\spadpaste{sign(x) \free{x}}
+}
+%
+\xtc{
+You can determine if an integer is negative in several other ways.
+}{
+\spadpaste{x < 0 \free{x}}
+}
+\xtc{
+}{
+\spadpaste{x <= -1 \free{x}}
+}
+\xtc{
+}{
+\spadpaste{negative?(x) \free{x}}
+}
+%
+\xtc{
+Similarly, you can find out if it is positive.
+}{
+\spadpaste{x > 0 \free{x}}
+}
+\xtc{
+}{
+\spadpaste{x >= 1 \free{x}}
+}
+\xtc{
+}{
+\spadpaste{positive?(x) \free{x}}
+}
+\xtc{
+This is the recommended way of determining whether an integer is zero.
+}{
+\spadpaste{zero?(x) \free{x}}
+}
+%
+
+\beginImportant
+Use the \spadfunFrom{zero?}{Integer} operation whenever you are testing any
+mathematical object for equality with zero.
+This is usually more efficient that using \spadop{=} (think of matrices:
+it is easier to tell if a matrix is zero by just checking term by term
+than constructing another ``zero'' matrix and comparing the two
+matrices term by term) and also avoids the problem that \spadop{=} is
+usually used for creating equations.
+\endImportant
+
+\xtc{
+This is the recommended way of determining
+whether an integer is equal to one.
+}{
+\spadpaste{one?(x) \free{x}}
+}
+
+\xtc{
+This syntax is used to test equality using \spadopFrom{=}{Integer}.
+It says that you want a \spadtype{Boolean} (\spad{true} or
+\spad{false}) answer rather than an equation.
+}{
+\spadpaste{(x = -101)@Boolean \free{x}}
+}
+
+\xtc{
+The operations \spadfunFrom{odd?}{Integer} and
+\spadfunFrom{even?}{Integer} determine whether an integer is odd or even,
+respectively.
+They each return a \spadtype{Boolean} object.
+}{
+\spadpaste{odd?(x) \free{x}}
+}
+\xtc{
+}{
+\spadpaste{even?(x) \free{x}}
+}
+\xtc{
+The operation \spadfunFrom{gcd}{Integer} computes the greatest common divisor of
+two integers.
+}{
+\spadpaste{gcd(56788,43688)}
+}
+\xtc{
+The operation
+\spadfunFrom{lcm}{Integer} computes their least common multiple.
+}{
+\spadpaste{lcm(56788,43688)}
+}
+\xtc{
+To determine the maximum of two integers, use \spadfunFrom{max}{Integer}.
+}{
+\spadpaste{max(678,567)}
+}
+\xtc{
+To determine the minimum, use \spadfunFrom{min}{Integer}.
+}{
+\spadpaste{min(678,567)}
+}
+
+\xtc{
+The \spadfun{reduce} operation is used to extend
+binary operations to more than two arguments.
+For example, you can use \spadfun{reduce} to find the maximum integer in
+a list or compute the least common multiple of all integers in the list.
+}{
+\spadpaste{reduce(max,[2,45,-89,78,100,-45])}
+}
+\xtc{
+}{
+\spadpaste{reduce(min,[2,45,-89,78,100,-45])}
+}
+\xtc{
+}{
+\spadpaste{reduce(gcd,[2,45,-89,78,100,-45])}
+}
+\xtc{
+}{
+\spadpaste{reduce(lcm,[2,45,-89,78,100,-45])}
+}
+
+\xtc{
+The infix operator ``/'' is {\it not} used to compute the quotient
+of integers.
+Rather, it is used to create rational numbers as described in
+\downlink{`Fraction'}{FractionXmpPage}\ignore{Fraction}.
+}{
+\spadpaste{13 / 4}
+}
+\xtc{
+The infix operation \spadfunFrom{quo}{Integer} computes the integer
+quotient.
+}{
+\spadpaste{13 quo 4}
+}
+\xtc{
+The infix operation \spadfunFrom{rem}{Integer} computes the integer
+remainder.
+}{
+\spadpaste{13 rem 4}
+}
+\xtc{
+One integer is evenly divisible by another if the remainder is zero.
+The operation \spadfunFrom{exquo}{Integer} can also be used.
+See \downlink{``\ugTypesUnionsTitle''}{ugTypesUnionsPage} in Section \ugTypesUnionsNumber\ignore{ugTypesUnions} for an example.
+}{
+\spadpaste{zero?(167604736446952 rem 2003644)}
+}
+
+\xtc{
+The operation \spadfunFrom{divide}{Integer} returns a record of the
+quotient and remainder and thus is more efficient when both are needed.
+}{
+\spadpaste{d := divide(13,4) \bound{d}}
+}
+\xtc{
+}{
+\spadpaste{d.quotient \free{d}}
+}
+\xtc{
+Records are discussed in detail in \downlink{``\ugTypesRecordsTitle''}{ugTypesRecordsPage} in Section \ugTypesRecordsNumber\ignore{ugTypesRecords}.
+}{
+\spadpaste{d.remainder \free{d}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxIntegerPrimesTitle}{Primes and Factorization}
+\newcommand{\ugxIntegerPrimesNumber}{9.34.2.}
+%
+% =====================================================================
+\begin{page}{ugxIntegerPrimesPage}{9.34.2. Primes and Factorization}
+% =====================================================================
+\beginscroll
+
+\labelSpace{3pc}
+\xtc{
+Use the operation \spadfunFrom{factor}{Integer} to factor integers.
+It returns an object of type \spadtype{Factored Integer}.
+%-% \HDindex{factorization}{ugxIntegerPrimesPage}{9.34.2.}{Primes and Factorization}
+See \downlink{`Factored'}{FactoredXmpPage}\ignore{Factored} for a discussion of the
+manipulation of factored objects.
+}{
+\spadpaste{factor 102400}
+}
+
+\xtc{
+The operation \spadfunFrom{prime?}{Integer} returns \spad{true} or \spad{false} depending
+on whether its argument is a prime.
+%-% \HDindex{prime}{ugxIntegerPrimesPage}{9.34.2.}{Primes and Factorization}
+}{
+\spadpaste{prime? 7}
+}
+\xtc{
+}{
+\spadpaste{prime? 8}
+}
+\xtc{
+The operation \spadfunFrom{nextPrime}{IntegerPrimesPackage} returns the
+least prime number greater than its argument.
+}{
+\spadpaste{nextPrime 100}
+}
+\xtc{
+The operation
+\spadfunFrom{prevPrime}{IntegerPrimesPackage} returns the greatest prime
+number less than its argument.
+}{
+\spadpaste{prevPrime 100}
+}
+\xtc{
+To compute all primes between two integers (inclusively), use the
+operation \spadfunFrom{primes}{IntegerPrimesPackage}.
+}{
+\spadpaste{primes(100,175)}
+}
+\xtc{
+You might sometimes want to see the factorization of an integer
+when it is considered a {\it Gaussian integer}.
+%-% \HDindex{Gaussian integer}{ugxIntegerPrimesPage}{9.34.2.}{Primes and Factorization}
+See \downlink{`Complex'}{ComplexXmpPage}\ignore{Complex} for more details.
+}{
+\spadpaste{factor(2 :: Complex Integer)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxIntegerNTTitle}{Some Number Theoretic Functions}
+\newcommand{\ugxIntegerNTNumber}{9.34.3.}
+%
+% =====================================================================
+\begin{page}{ugxIntegerNTPage}{9.34.3. Some Number Theoretic Functions}
+% =====================================================================
+\beginscroll
+
+\Language{} provides several number theoretic operations for integers.
+More examples are in \downlink{`IntegerNumberTheoryFunctions'}{IntegerNumberTheoryFunctionsXmpPage}\ignore{IntegerNumberTheoryFunctions}.
+
+\labelSpace{1pc}
+\xtc{
+The operation \spadfunFrom{fibonacci}{IntegerNumberTheoryFunctions}
+computes the Fibonacci numbers.
+%-% \HDindex{Fibonacci numbers}{ugxIntegerNTPage}{9.34.3.}{Some Number Theoretic Functions}
+The algorithm has running time
+\texht{$O\,(\log^3(n))$}{O(\spad{log(n)**3})} for argument \spad{n}.
+}{
+\spadpaste{[fibonacci(k) for k in 0..]}
+}
+\xtc{
+The operation \spadfunFrom{legendre}{IntegerNumberTheoryFunctions}
+computes the Legendre symbol for its two integer arguments where the
+second one is prime.
+If you know the second argument to be prime, use
+\spadfunFrom{jacobi}{IntegerNumberTheoryFunctions} instead where no check
+is made.
+}{
+\spadpaste{[legendre(i,11) for i in 0..10]}
+}
+\xtc{
+The operation \spadfunFrom{jacobi}{IntegerNumberTheoryFunctions} computes
+the Jacobi symbol for its two integer arguments.
+%-% \HDindex{Jacobi symbol}{ugxIntegerNTPage}{9.34.3.}{Some Number Theoretic Functions}
+By convention, \spad{0} is returned if the greatest common divisor of the
+numerator and denominator is not \spad{1}.
+}{
+\spadpaste{[jacobi(i,15) for i in 0..9]}
+}
+\xtc{
+The operation \spadfunFrom{eulerPhi}{IntegerNumberTheoryFunctions} computes
+%-% \HDindex{Euler!phi function@{$\varphi$ function}}{ugxIntegerNTPage}{9.34.3.}{Some Number Theoretic Functions}
+the values of Euler's \texht{$\phi$}{phi}-function where
+\texht{$\phi(n)$}{\spad{phi(n)}}
+equals the number of positive integers
+less than or equal to \spad{n} that are relatively prime to
+the positive integer \spad{n}.
+%-% \HDindex{phi@{$\varphi$}}{ugxIntegerNTPage}{9.34.3.}{Some Number Theoretic Functions}
+}{
+\spadpaste{[eulerPhi i for i in 1..]}
+}
+\xtc{
+The operation \spadfunFrom{moebiusMu}{IntegerNumberTheoryFunctions}
+%-% \HDindex{mu@{$\mu$}}{ugxIntegerNTPage}{9.34.3.}{Some Number Theoretic Functions}
+computes the \texht{M\"{o}bius $\mu$}{Moebius mu} function.
+%-% \HDindex{Moebius@{M\"{o}bius}!mu function@{$\mu$ function}}{ugxIntegerNTPage}{9.34.3.}{Some Number Theoretic Functions}
+}{
+\spadpaste{[moebiusMu i for i in 1..]}
+}
+
+\xtc{
+Although they have somewhat limited utility, \Language{} provides
+%-% \HDindex{Roman numerals}{ugxIntegerNTPage}{9.34.3.}{Some Number Theoretic Functions}
+Roman numerals.
+}{
+\spadpaste{a := roman(78) \bound{a}}
+}
+\xtc{
+}{
+\spadpaste{b := roman(87) \bound{b}}
+}
+\xtc{
+}{
+\spadpaste{a + b \free{a}\free{b}}
+}
+\xtc{
+}{
+\spadpaste{a * b \free{a}\free{b}}
+}
+\xtc{
+}{
+\spadpaste{b rem a \free{a}\free{b}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/INT.pht b/src/hyper/pages/INT.pht
new file mode 100644
index 00000000..4264dc2e
--- /dev/null
+++ b/src/hyper/pages/INT.pht
@@ -0,0 +1,763 @@
+\begin{patch}{ugxIntegerPrimesPagePatch1}
+\begin{paste}{ugxIntegerPrimesPageFull1}{ugxIntegerPrimesPageEmpty1}
+\pastebutton{ugxIntegerPrimesPageFull1}{\hidepaste}
+\tab{5}\spadcommand{factor 102400}
+\indentrel{3}\begin{verbatim}
+         12 2
+   (1)  2  5
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerPrimesPageEmpty1}
+\begin{paste}{ugxIntegerPrimesPageEmpty1}{ugxIntegerPrimesPagePatch1}
+\pastebutton{ugxIntegerPrimesPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{factor 102400}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerPrimesPagePatch2}
+\begin{paste}{ugxIntegerPrimesPageFull2}{ugxIntegerPrimesPageEmpty2}
+\pastebutton{ugxIntegerPrimesPageFull2}{\hidepaste}
+\tab{5}\spadcommand{prime? 7}
+\indentrel{3}\begin{verbatim}
+   (2)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerPrimesPageEmpty2}
+\begin{paste}{ugxIntegerPrimesPageEmpty2}{ugxIntegerPrimesPagePatch2}
+\pastebutton{ugxIntegerPrimesPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{prime? 7}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerPrimesPagePatch3}
+\begin{paste}{ugxIntegerPrimesPageFull3}{ugxIntegerPrimesPageEmpty3}
+\pastebutton{ugxIntegerPrimesPageFull3}{\hidepaste}
+\tab{5}\spadcommand{prime? 8}
+\indentrel{3}\begin{verbatim}
+   (3)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerPrimesPageEmpty3}
+\begin{paste}{ugxIntegerPrimesPageEmpty3}{ugxIntegerPrimesPagePatch3}
+\pastebutton{ugxIntegerPrimesPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{prime? 8}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerPrimesPagePatch4}
+\begin{paste}{ugxIntegerPrimesPageFull4}{ugxIntegerPrimesPageEmpty4}
+\pastebutton{ugxIntegerPrimesPageFull4}{\hidepaste}
+\tab{5}\spadcommand{nextPrime 100}
+\indentrel{3}\begin{verbatim}
+   (4)  101
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerPrimesPageEmpty4}
+\begin{paste}{ugxIntegerPrimesPageEmpty4}{ugxIntegerPrimesPagePatch4}
+\pastebutton{ugxIntegerPrimesPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{nextPrime 100}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerPrimesPagePatch5}
+\begin{paste}{ugxIntegerPrimesPageFull5}{ugxIntegerPrimesPageEmpty5}
+\pastebutton{ugxIntegerPrimesPageFull5}{\hidepaste}
+\tab{5}\spadcommand{prevPrime 100}
+\indentrel{3}\begin{verbatim}
+   (5)  97
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerPrimesPageEmpty5}
+\begin{paste}{ugxIntegerPrimesPageEmpty5}{ugxIntegerPrimesPagePatch5}
+\pastebutton{ugxIntegerPrimesPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{prevPrime 100}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerPrimesPagePatch6}
+\begin{paste}{ugxIntegerPrimesPageFull6}{ugxIntegerPrimesPageEmpty6}
+\pastebutton{ugxIntegerPrimesPageFull6}{\hidepaste}
+\tab{5}\spadcommand{primes(100,175)}
+\indentrel{3}\begin{verbatim}
+   (6)
+   [173, 167, 163, 157, 151, 149, 139, 137, 131, 127,
+    113, 109, 107, 103, 101]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerPrimesPageEmpty6}
+\begin{paste}{ugxIntegerPrimesPageEmpty6}{ugxIntegerPrimesPagePatch6}
+\pastebutton{ugxIntegerPrimesPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{primes(100,175)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerPrimesPagePatch7}
+\begin{paste}{ugxIntegerPrimesPageFull7}{ugxIntegerPrimesPageEmpty7}
+\pastebutton{ugxIntegerPrimesPageFull7}{\hidepaste}
+\tab{5}\spadcommand{factor(2 :: Complex Integer)}
+\indentrel{3}\begin{verbatim}
+                     2
+   (7)  - %i (1 + %i)
+                         Type: Factored Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerPrimesPageEmpty7}
+\begin{paste}{ugxIntegerPrimesPageEmpty7}{ugxIntegerPrimesPagePatch7}
+\pastebutton{ugxIntegerPrimesPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{factor(2 :: Complex Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPagePatch1}
+\begin{paste}{ugxIntegerNTPageFull1}{ugxIntegerNTPageEmpty1}
+\pastebutton{ugxIntegerNTPageFull1}{\hidepaste}
+\tab{5}\spadcommand{[fibonacci(k) for k in 0..]}
+\indentrel{3}\begin{verbatim}
+   (1)  [0,1,1,2,3,5,8,13,21,34,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPageEmpty1}
+\begin{paste}{ugxIntegerNTPageEmpty1}{ugxIntegerNTPagePatch1}
+\pastebutton{ugxIntegerNTPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{[fibonacci(k) for k in 0..]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPagePatch2}
+\begin{paste}{ugxIntegerNTPageFull2}{ugxIntegerNTPageEmpty2}
+\pastebutton{ugxIntegerNTPageFull2}{\hidepaste}
+\tab{5}\spadcommand{[legendre(i,11) for i in 0..10]}
+\indentrel{3}\begin{verbatim}
+   (2)  [0,1,- 1,1,1,1,- 1,- 1,- 1,1,- 1]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPageEmpty2}
+\begin{paste}{ugxIntegerNTPageEmpty2}{ugxIntegerNTPagePatch2}
+\pastebutton{ugxIntegerNTPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{[legendre(i,11) for i in 0..10]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPagePatch3}
+\begin{paste}{ugxIntegerNTPageFull3}{ugxIntegerNTPageEmpty3}
+\pastebutton{ugxIntegerNTPageFull3}{\hidepaste}
+\tab{5}\spadcommand{[jacobi(i,15) for i in 0..9]}
+\indentrel{3}\begin{verbatim}
+   (3)  [0,1,1,0,1,0,0,- 1,1,0]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPageEmpty3}
+\begin{paste}{ugxIntegerNTPageEmpty3}{ugxIntegerNTPagePatch3}
+\pastebutton{ugxIntegerNTPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{[jacobi(i,15) for i in 0..9]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPagePatch4}
+\begin{paste}{ugxIntegerNTPageFull4}{ugxIntegerNTPageEmpty4}
+\pastebutton{ugxIntegerNTPageFull4}{\hidepaste}
+\tab{5}\spadcommand{[eulerPhi i for i in 1..]}
+\indentrel{3}\begin{verbatim}
+   (4)  [1,1,2,2,4,2,6,4,6,4,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPageEmpty4}
+\begin{paste}{ugxIntegerNTPageEmpty4}{ugxIntegerNTPagePatch4}
+\pastebutton{ugxIntegerNTPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{[eulerPhi i for i in 1..]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPagePatch5}
+\begin{paste}{ugxIntegerNTPageFull5}{ugxIntegerNTPageEmpty5}
+\pastebutton{ugxIntegerNTPageFull5}{\hidepaste}
+\tab{5}\spadcommand{[moebiusMu i for i in 1..]}
+\indentrel{3}\begin{verbatim}
+   (5)  [1,- 1,- 1,0,- 1,1,- 1,0,0,1,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPageEmpty5}
+\begin{paste}{ugxIntegerNTPageEmpty5}{ugxIntegerNTPagePatch5}
+\pastebutton{ugxIntegerNTPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{[moebiusMu i for i in 1..]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPagePatch6}
+\begin{paste}{ugxIntegerNTPageFull6}{ugxIntegerNTPageEmpty6}
+\pastebutton{ugxIntegerNTPageFull6}{\hidepaste}
+\tab{5}\spadcommand{a := roman(78)\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (6)  LXXVIII
+                                     Type: RomanNumeral
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPageEmpty6}
+\begin{paste}{ugxIntegerNTPageEmpty6}{ugxIntegerNTPagePatch6}
+\pastebutton{ugxIntegerNTPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{a := roman(78)\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPagePatch7}
+\begin{paste}{ugxIntegerNTPageFull7}{ugxIntegerNTPageEmpty7}
+\pastebutton{ugxIntegerNTPageFull7}{\hidepaste}
+\tab{5}\spadcommand{b := roman(87)\bound{b }}
+\indentrel{3}\begin{verbatim}
+   (7)  LXXXVII
+                                     Type: RomanNumeral
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPageEmpty7}
+\begin{paste}{ugxIntegerNTPageEmpty7}{ugxIntegerNTPagePatch7}
+\pastebutton{ugxIntegerNTPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{b := roman(87)\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPagePatch8}
+\begin{paste}{ugxIntegerNTPageFull8}{ugxIntegerNTPageEmpty8}
+\pastebutton{ugxIntegerNTPageFull8}{\hidepaste}
+\tab{5}\spadcommand{a + b\free{a }\free{b }}
+\indentrel{3}\begin{verbatim}
+   (8)  CLXV
+                                     Type: RomanNumeral
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPageEmpty8}
+\begin{paste}{ugxIntegerNTPageEmpty8}{ugxIntegerNTPagePatch8}
+\pastebutton{ugxIntegerNTPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{a + b\free{a }\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPagePatch9}
+\begin{paste}{ugxIntegerNTPageFull9}{ugxIntegerNTPageEmpty9}
+\pastebutton{ugxIntegerNTPageFull9}{\hidepaste}
+\tab{5}\spadcommand{a * b\free{a }\free{b }}
+\indentrel{3}\begin{verbatim}
+   (9)  MMMMMMDCCLXXXVI
+                                     Type: RomanNumeral
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPageEmpty9}
+\begin{paste}{ugxIntegerNTPageEmpty9}{ugxIntegerNTPagePatch9}
+\pastebutton{ugxIntegerNTPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{a * b\free{a }\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPagePatch10}
+\begin{paste}{ugxIntegerNTPageFull10}{ugxIntegerNTPageEmpty10}
+\pastebutton{ugxIntegerNTPageFull10}{\hidepaste}
+\tab{5}\spadcommand{b rem a\free{a }\free{b }}
+\indentrel{3}\begin{verbatim}
+   (10)  IX
+                                     Type: RomanNumeral
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerNTPageEmpty10}
+\begin{paste}{ugxIntegerNTPageEmpty10}{ugxIntegerNTPagePatch10}
+\pastebutton{ugxIntegerNTPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{b rem a\free{a }\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch1}
+\begin{paste}{ugxIntegerBasicPageFull1}{ugxIntegerBasicPageEmpty1}
+\pastebutton{ugxIntegerBasicPageFull1}{\hidepaste}
+\tab{5}\spadcommand{2**(5678 - 4856 + 2 * 17)}
+\indentrel{3}\begin{verbatim}
+   (1)
+  480481077043500814718154092512592439123952613987168226_
+   34738556100880842000763082930863425270914120837430745_
+   72278211496076276922026433435687527334980249539302425_
+   42523045817764949544214392905306388478705146745768073_
+   877141698859815495632935288783334250628775936
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty1}
+\begin{paste}{ugxIntegerBasicPageEmpty1}{ugxIntegerBasicPagePatch1}
+\pastebutton{ugxIntegerBasicPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{2**(5678 - 4856 + 2 * 17)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch2}
+\begin{paste}{ugxIntegerBasicPageFull2}{ugxIntegerBasicPageEmpty2}
+\pastebutton{ugxIntegerBasicPageFull2}{\hidepaste}
+\tab{5}\spadcommand{x := -101\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (2)  - 101
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty2}
+\begin{paste}{ugxIntegerBasicPageEmpty2}{ugxIntegerBasicPagePatch2}
+\pastebutton{ugxIntegerBasicPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{x := -101\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch3}
+\begin{paste}{ugxIntegerBasicPageFull3}{ugxIntegerBasicPageEmpty3}
+\pastebutton{ugxIntegerBasicPageFull3}{\hidepaste}
+\tab{5}\spadcommand{abs(x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (3)  101
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty3}
+\begin{paste}{ugxIntegerBasicPageEmpty3}{ugxIntegerBasicPagePatch3}
+\pastebutton{ugxIntegerBasicPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{abs(x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch4}
+\begin{paste}{ugxIntegerBasicPageFull4}{ugxIntegerBasicPageEmpty4}
+\pastebutton{ugxIntegerBasicPageFull4}{\hidepaste}
+\tab{5}\spadcommand{sign(x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (4)  - 1
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty4}
+\begin{paste}{ugxIntegerBasicPageEmpty4}{ugxIntegerBasicPagePatch4}
+\pastebutton{ugxIntegerBasicPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{sign(x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch5}
+\begin{paste}{ugxIntegerBasicPageFull5}{ugxIntegerBasicPageEmpty5}
+\pastebutton{ugxIntegerBasicPageFull5}{\hidepaste}
+\tab{5}\spadcommand{x < 0\free{x }}
+\indentrel{3}\begin{verbatim}
+   (5)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty5}
+\begin{paste}{ugxIntegerBasicPageEmpty5}{ugxIntegerBasicPagePatch5}
+\pastebutton{ugxIntegerBasicPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{x < 0\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch6}
+\begin{paste}{ugxIntegerBasicPageFull6}{ugxIntegerBasicPageEmpty6}
+\pastebutton{ugxIntegerBasicPageFull6}{\hidepaste}
+\tab{5}\spadcommand{x <= -1\free{x }}
+\indentrel{3}\begin{verbatim}
+   (6)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty6}
+\begin{paste}{ugxIntegerBasicPageEmpty6}{ugxIntegerBasicPagePatch6}
+\pastebutton{ugxIntegerBasicPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{x <= -1\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch7}
+\begin{paste}{ugxIntegerBasicPageFull7}{ugxIntegerBasicPageEmpty7}
+\pastebutton{ugxIntegerBasicPageFull7}{\hidepaste}
+\tab{5}\spadcommand{negative?(x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (7)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty7}
+\begin{paste}{ugxIntegerBasicPageEmpty7}{ugxIntegerBasicPagePatch7}
+\pastebutton{ugxIntegerBasicPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{negative?(x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch8}
+\begin{paste}{ugxIntegerBasicPageFull8}{ugxIntegerBasicPageEmpty8}
+\pastebutton{ugxIntegerBasicPageFull8}{\hidepaste}
+\tab{5}\spadcommand{x > 0\free{x }}
+\indentrel{3}\begin{verbatim}
+   (8)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty8}
+\begin{paste}{ugxIntegerBasicPageEmpty8}{ugxIntegerBasicPagePatch8}
+\pastebutton{ugxIntegerBasicPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{x > 0\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch9}
+\begin{paste}{ugxIntegerBasicPageFull9}{ugxIntegerBasicPageEmpty9}
+\pastebutton{ugxIntegerBasicPageFull9}{\hidepaste}
+\tab{5}\spadcommand{x >= 1\free{x }}
+\indentrel{3}\begin{verbatim}
+   (9)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty9}
+\begin{paste}{ugxIntegerBasicPageEmpty9}{ugxIntegerBasicPagePatch9}
+\pastebutton{ugxIntegerBasicPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{x >= 1\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch10}
+\begin{paste}{ugxIntegerBasicPageFull10}{ugxIntegerBasicPageEmpty10}
+\pastebutton{ugxIntegerBasicPageFull10}{\hidepaste}
+\tab{5}\spadcommand{positive?(x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (10)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty10}
+\begin{paste}{ugxIntegerBasicPageEmpty10}{ugxIntegerBasicPagePatch10}
+\pastebutton{ugxIntegerBasicPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{positive?(x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch11}
+\begin{paste}{ugxIntegerBasicPageFull11}{ugxIntegerBasicPageEmpty11}
+\pastebutton{ugxIntegerBasicPageFull11}{\hidepaste}
+\tab{5}\spadcommand{zero?(x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (11)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty11}
+\begin{paste}{ugxIntegerBasicPageEmpty11}{ugxIntegerBasicPagePatch11}
+\pastebutton{ugxIntegerBasicPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{zero?(x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch12}
+\begin{paste}{ugxIntegerBasicPageFull12}{ugxIntegerBasicPageEmpty12}
+\pastebutton{ugxIntegerBasicPageFull12}{\hidepaste}
+\tab{5}\spadcommand{one?(x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (12)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty12}
+\begin{paste}{ugxIntegerBasicPageEmpty12}{ugxIntegerBasicPagePatch12}
+\pastebutton{ugxIntegerBasicPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{one?(x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch13}
+\begin{paste}{ugxIntegerBasicPageFull13}{ugxIntegerBasicPageEmpty13}
+\pastebutton{ugxIntegerBasicPageFull13}{\hidepaste}
+\tab{5}\spadcommand{(x = -101)@Boolean\free{x }}
+\indentrel{3}\begin{verbatim}
+   (13)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty13}
+\begin{paste}{ugxIntegerBasicPageEmpty13}{ugxIntegerBasicPagePatch13}
+\pastebutton{ugxIntegerBasicPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{(x = -101)@Boolean\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch14}
+\begin{paste}{ugxIntegerBasicPageFull14}{ugxIntegerBasicPageEmpty14}
+\pastebutton{ugxIntegerBasicPageFull14}{\hidepaste}
+\tab{5}\spadcommand{odd?(x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (14)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty14}
+\begin{paste}{ugxIntegerBasicPageEmpty14}{ugxIntegerBasicPagePatch14}
+\pastebutton{ugxIntegerBasicPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{odd?(x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch15}
+\begin{paste}{ugxIntegerBasicPageFull15}{ugxIntegerBasicPageEmpty15}
+\pastebutton{ugxIntegerBasicPageFull15}{\hidepaste}
+\tab{5}\spadcommand{even?(x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (15)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty15}
+\begin{paste}{ugxIntegerBasicPageEmpty15}{ugxIntegerBasicPagePatch15}
+\pastebutton{ugxIntegerBasicPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{even?(x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch16}
+\begin{paste}{ugxIntegerBasicPageFull16}{ugxIntegerBasicPageEmpty16}
+\pastebutton{ugxIntegerBasicPageFull16}{\hidepaste}
+\tab{5}\spadcommand{gcd(56788,43688)}
+\indentrel{3}\begin{verbatim}
+   (16)  4
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty16}
+\begin{paste}{ugxIntegerBasicPageEmpty16}{ugxIntegerBasicPagePatch16}
+\pastebutton{ugxIntegerBasicPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{gcd(56788,43688)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch17}
+\begin{paste}{ugxIntegerBasicPageFull17}{ugxIntegerBasicPageEmpty17}
+\pastebutton{ugxIntegerBasicPageFull17}{\hidepaste}
+\tab{5}\spadcommand{lcm(56788,43688)}
+\indentrel{3}\begin{verbatim}
+   (17)  620238536
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty17}
+\begin{paste}{ugxIntegerBasicPageEmpty17}{ugxIntegerBasicPagePatch17}
+\pastebutton{ugxIntegerBasicPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{lcm(56788,43688)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch18}
+\begin{paste}{ugxIntegerBasicPageFull18}{ugxIntegerBasicPageEmpty18}
+\pastebutton{ugxIntegerBasicPageFull18}{\hidepaste}
+\tab{5}\spadcommand{max(678,567)}
+\indentrel{3}\begin{verbatim}
+   (18)  678
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty18}
+\begin{paste}{ugxIntegerBasicPageEmpty18}{ugxIntegerBasicPagePatch18}
+\pastebutton{ugxIntegerBasicPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{max(678,567)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch19}
+\begin{paste}{ugxIntegerBasicPageFull19}{ugxIntegerBasicPageEmpty19}
+\pastebutton{ugxIntegerBasicPageFull19}{\hidepaste}
+\tab{5}\spadcommand{min(678,567)}
+\indentrel{3}\begin{verbatim}
+   (19)  567
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty19}
+\begin{paste}{ugxIntegerBasicPageEmpty19}{ugxIntegerBasicPagePatch19}
+\pastebutton{ugxIntegerBasicPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{min(678,567)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch20}
+\begin{paste}{ugxIntegerBasicPageFull20}{ugxIntegerBasicPageEmpty20}
+\pastebutton{ugxIntegerBasicPageFull20}{\hidepaste}
+\tab{5}\spadcommand{reduce(max,[2,45,-89,78,100,-45])}
+\indentrel{3}\begin{verbatim}
+   (20)  100
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty20}
+\begin{paste}{ugxIntegerBasicPageEmpty20}{ugxIntegerBasicPagePatch20}
+\pastebutton{ugxIntegerBasicPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{reduce(max,[2,45,-89,78,100,-45])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch21}
+\begin{paste}{ugxIntegerBasicPageFull21}{ugxIntegerBasicPageEmpty21}
+\pastebutton{ugxIntegerBasicPageFull21}{\hidepaste}
+\tab{5}\spadcommand{reduce(min,[2,45,-89,78,100,-45])}
+\indentrel{3}\begin{verbatim}
+   (21)  - 89
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty21}
+\begin{paste}{ugxIntegerBasicPageEmpty21}{ugxIntegerBasicPagePatch21}
+\pastebutton{ugxIntegerBasicPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{reduce(min,[2,45,-89,78,100,-45])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch22}
+\begin{paste}{ugxIntegerBasicPageFull22}{ugxIntegerBasicPageEmpty22}
+\pastebutton{ugxIntegerBasicPageFull22}{\hidepaste}
+\tab{5}\spadcommand{reduce(gcd,[2,45,-89,78,100,-45])}
+\indentrel{3}\begin{verbatim}
+   (22)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty22}
+\begin{paste}{ugxIntegerBasicPageEmpty22}{ugxIntegerBasicPagePatch22}
+\pastebutton{ugxIntegerBasicPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{reduce(gcd,[2,45,-89,78,100,-45])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch23}
+\begin{paste}{ugxIntegerBasicPageFull23}{ugxIntegerBasicPageEmpty23}
+\pastebutton{ugxIntegerBasicPageFull23}{\hidepaste}
+\tab{5}\spadcommand{reduce(lcm,[2,45,-89,78,100,-45])}
+\indentrel{3}\begin{verbatim}
+   (23)  1041300
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty23}
+\begin{paste}{ugxIntegerBasicPageEmpty23}{ugxIntegerBasicPagePatch23}
+\pastebutton{ugxIntegerBasicPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{reduce(lcm,[2,45,-89,78,100,-45])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch24}
+\begin{paste}{ugxIntegerBasicPageFull24}{ugxIntegerBasicPageEmpty24}
+\pastebutton{ugxIntegerBasicPageFull24}{\hidepaste}
+\tab{5}\spadcommand{13 / 4}
+\indentrel{3}\begin{verbatim}
+         13
+   (24)  ÄÄ
+          4
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty24}
+\begin{paste}{ugxIntegerBasicPageEmpty24}{ugxIntegerBasicPagePatch24}
+\pastebutton{ugxIntegerBasicPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{13 / 4}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch25}
+\begin{paste}{ugxIntegerBasicPageFull25}{ugxIntegerBasicPageEmpty25}
+\pastebutton{ugxIntegerBasicPageFull25}{\hidepaste}
+\tab{5}\spadcommand{13 quo 4}
+\indentrel{3}\begin{verbatim}
+   (25)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty25}
+\begin{paste}{ugxIntegerBasicPageEmpty25}{ugxIntegerBasicPagePatch25}
+\pastebutton{ugxIntegerBasicPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{13 quo 4}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch26}
+\begin{paste}{ugxIntegerBasicPageFull26}{ugxIntegerBasicPageEmpty26}
+\pastebutton{ugxIntegerBasicPageFull26}{\hidepaste}
+\tab{5}\spadcommand{13 rem 4}
+\indentrel{3}\begin{verbatim}
+   (26)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty26}
+\begin{paste}{ugxIntegerBasicPageEmpty26}{ugxIntegerBasicPagePatch26}
+\pastebutton{ugxIntegerBasicPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{13 rem 4}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch27}
+\begin{paste}{ugxIntegerBasicPageFull27}{ugxIntegerBasicPageEmpty27}
+\pastebutton{ugxIntegerBasicPageFull27}{\hidepaste}
+\tab{5}\spadcommand{zero?(167604736446952 rem 2003644)}
+\indentrel{3}\begin{verbatim}
+   (27)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty27}
+\begin{paste}{ugxIntegerBasicPageEmpty27}{ugxIntegerBasicPagePatch27}
+\pastebutton{ugxIntegerBasicPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{zero?(167604736446952 rem 2003644)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch28}
+\begin{paste}{ugxIntegerBasicPageFull28}{ugxIntegerBasicPageEmpty28}
+\pastebutton{ugxIntegerBasicPageFull28}{\hidepaste}
+\tab{5}\spadcommand{d := divide(13,4)\bound{d }}
+\indentrel{3}\begin{verbatim}
+   (28)  [quotient= 3,remainder= 1]
+     Type: Record(quotient: Integer,remainder: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty28}
+\begin{paste}{ugxIntegerBasicPageEmpty28}{ugxIntegerBasicPagePatch28}
+\pastebutton{ugxIntegerBasicPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{d := divide(13,4)\bound{d }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch29}
+\begin{paste}{ugxIntegerBasicPageFull29}{ugxIntegerBasicPageEmpty29}
+\pastebutton{ugxIntegerBasicPageFull29}{\hidepaste}
+\tab{5}\spadcommand{d.quotient\free{d }}
+\indentrel{3}\begin{verbatim}
+   (29)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty29}
+\begin{paste}{ugxIntegerBasicPageEmpty29}{ugxIntegerBasicPagePatch29}
+\pastebutton{ugxIntegerBasicPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{d.quotient\free{d }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPagePatch30}
+\begin{paste}{ugxIntegerBasicPageFull30}{ugxIntegerBasicPageEmpty30}
+\pastebutton{ugxIntegerBasicPageFull30}{\hidepaste}
+\tab{5}\spadcommand{d.remainder\free{d }}
+\indentrel{3}\begin{verbatim}
+   (30)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxIntegerBasicPageEmpty30}
+\begin{paste}{ugxIntegerBasicPageEmpty30}{ugxIntegerBasicPagePatch30}
+\pastebutton{ugxIntegerBasicPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{d.remainder\free{d }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/INTHEORY.ht b/src/hyper/pages/INTHEORY.ht
new file mode 100644
index 00000000..d96b703d
--- /dev/null
+++ b/src/hyper/pages/INTHEORY.ht
@@ -0,0 +1,226 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\IntegerNumberTheoryFunctionsXmpTitle}{IntegerNumberTheoryFunctions}
+\newcommand{\IntegerNumberTheoryFunctionsXmpNumber}{9.36}
+%
+% =====================================================================
+\begin{page}{IntegerNumberTheoryFunctionsXmpPage}{9.36 IntegerNumberTheoryFunctions}
+% =====================================================================
+\beginscroll
+
+
+The \spadtype{IntegerNumberTheoryFunctions} package contains a variety of
+operations of interest to number theorists.
+%-% \HDindex{number theory}{IntegerNumberTheoryFunctionsXmpPage}{9.36}{IntegerNumberTheoryFunctions}
+Many of these operations deal with divisibility properties of integers.
+(Recall that an integer \spad{a} divides an integer \spad{b} if there is
+an integer \spad{c} such that \spad{b = a * c}.)
+
+\xtc{
+The operation \spadfunFrom{divisors}{IntegerNumberTheoryFunctions}
+returns a list of the divisors of an integer.
+}{
+\spadpaste{div144 := divisors(144) \bound{div144}}
+}
+\xtc{
+You can now compute the number of divisors of \spad{144} and the sum of
+the divisors of \spad{144} by counting and summing the elements of the
+list we just created.
+}{
+\spadpaste{\#(div144) \free{div144}}
+}
+\xtc{
+}{
+\spadpaste{reduce(+,div144) \free{div144}}
+}
+
+Of course, you can compute the number of divisors of an integer \spad{n},
+usually denoted \spad{d(n)}, and the sum of the divisors of an integer
+\spad{n}, usually denoted \spad{\texht{$\sigma$}{sigma}(n)},
+%-% \HDindex{sigma@{$\sigma$}}{IntegerNumberTheoryFunctionsXmpPage}{9.36}{IntegerNumberTheoryFunctions}
+without ever listing the divisors of \spad{n}.
+
+\xtc{
+In \Language{}, you can simply call the operations
+\spadfunFrom{numberOfDivisors}{IntegerNumberTheoryFunctions} and
+\spadfunFrom{sumOfDivisors}{IntegerNumberTheoryFunctions}.
+}{
+\spadpaste{numberOfDivisors(144)}
+}
+\xtc{
+}{
+\spadpaste{sumOfDivisors(144)}
+}
+
+The key is that \spad{d(n)} and
+\spad{\texht{$\sigma$}{sigma}(n)}
+are ``multiplicative functions.''
+This means that when \spad{n} and \spad{m} are relatively prime, that is, when
+\spad{n} and \spad{m} have no prime factor in common, then
+\spad{d(nm) = d(n) d(m)} and
+\spad{\texht{$\sigma$}{sigma}(nm) = \texht{$\sigma$}{sigma}(n)
+\texht{$\sigma$}{sigma}(m)}.
+Note that these functions are trivial to compute when \spad{n} is a prime
+power and are computed for general \spad{n} from the prime factorization
+of \spad{n}.
+Other examples of multiplicative functions are
+\spad{\texht{$\sigma_k$}{sigma_k}(n)}, the sum of the \eth{\spad{k}} powers of
+the divisors of \spad{n} and \texht{$\varphi(n)$}{\spad{phi(n)}}, the
+number of integers between 1 and \spad{n} which are prime to \spad{n}.
+The corresponding \Language{} operations are called
+\spadfunFrom{sumOfKthPowerDivisors}{IntegerNumberTheoryFunctions} and
+\spadfunFrom{eulerPhi}{IntegerNumberTheoryFunctions}.
+%-% \HDindex{phi@{$\varphi$}}{IntegerNumberTheoryFunctionsXmpPage}{9.36}{IntegerNumberTheoryFunctions}
+%-% \HDindex{Euler!phi function@{$\varphi$ function}}{IntegerNumberTheoryFunctionsXmpPage}{9.36}{IntegerNumberTheoryFunctions}
+
+An interesting function is \spad{\texht{$\mu$}{mu}(n)},
+%-% \HDindex{mu@{$\mu$}}{IntegerNumberTheoryFunctionsXmpPage}{9.36}{IntegerNumberTheoryFunctions}
+the \texht{M\"{o}bius $\mu$}{Moebius mu} function, defined
+%-% \HDindex{Moebius@{M\"{o}bius}!mu function@{$\mu$ function}}{IntegerNumberTheoryFunctionsXmpPage}{9.36}{IntegerNumberTheoryFunctions}
+as follows:
+\spad{\texht{$\mu$}{mu}(1) = 1}, \spad{\texht{$\mu$}{mu}(n) = 0},
+when \spad{n} is divisible by a
+square, and
+\spad{\texht{$\mu = {(-1)}^k$}{mu(n) = (-1) ** k}}, when \spad{n}
+is the product of \spad{k} distinct primes.
+The corresponding \Language{} operation is
+\spadfunFrom{moebiusMu}{IntegerNumberTheoryFunctions}.
+This function occurs in the following theorem:
+
+\noindent
+{\bf Theorem} (\texht{M\"{o}bius}{Moebius} Inversion Formula): \newline
+%\texht{\begin{quotation}\noindent}{\newline\indent{5}}
+Let \spad{f(n)} be a function on the positive integers and let \spad{F(n)}
+be defined by
+\texht{\narrowDisplay{F(n) = \sum_{d \mid n} f(n)}}{\spad{F(n) =}
+sum of \spad{f(n)} over \spad{d | n}}
+where the sum is taken over the positive divisors of \spad{n}.
+Then the values of \spad{f(n)} can be recovered from the values of
+\spad{F(n)}:
+\texht{\narrowDisplay{f(n) = \sum_{d \mid n} \mu(n) F({{n}\over {d}})}}{\spad{f(n) =}
+sum of \spad{mu(n) F(n/d)} over \spad{d | n},}
+where again the sum is taken over the positive divisors of \spad{n}.
+
+\xtc{
+When \spad{f(n) = 1}, then \spad{F(n) = d(n)}.
+Thus, if you sum \spad{\texht{$\mu$}{mu}(d) \texht{$\cdot$}{*} d(n/d)}
+over the positive divisors
+\spad{d} of \spad{n}, you should always get \spad{1}.
+}{
+\spadpaste{f1(n) == reduce(+,[moebiusMu(d) * numberOfDivisors(quo(n,d)) for d in divisors(n)]) \bound{f1}}
+}
+\xtc{
+}{
+\spadpaste{f1(200) \free{f1}}
+}
+\xtc{
+}{
+\spadpaste{f1(846) \free{f1}}
+}
+\xtc{
+Similarly, when \spad{f(n) = n}, then \spad{F(n) = \texht{$\sigma$}{sigma}(n)}.
+Thus, if you sum \spad{\texht{$\mu$}{mu}(d) \texht{$\cdot$}{*}
+\texht{$\sigma$}{sigma}(n/d)} over the positive divisors
+\spad{d} of \spad{n}, you should always get \spad{n}.
+}{
+\spadpaste{f2(n) == reduce(+,[moebiusMu(d) * sumOfDivisors(quo(n,d)) for d in divisors(n)]) \bound{f2}}
+}
+\xtc{
+}{
+\spadpaste{f2(200) \free{f2}}
+}
+\xtc{
+}{
+\spadpaste{f2(846) \free{f2}}
+}
+
+
+The Fibonacci numbers are defined by \spad{F(1) = F(2) = 1} and
+%-% \HDindex{Fibonacci numbers}{IntegerNumberTheoryFunctionsXmpPage}{9.36}{IntegerNumberTheoryFunctions}
+\spad{F(n) = F(n-1) + F(n-2)} for \spad{n = 3,4, ...}.
+\xtc{
+The operation \spadfunFrom{fibonacci}{IntegerNumberTheoryFunctions}
+computes the \eth{\spad{n}} Fibonacci number.
+}{
+\spadpaste{fibonacci(25)}
+}
+\xtc{
+}{
+\spadpaste{[fibonacci(n) for n in 1..15]}
+}
+\xtc{
+Fibonacci numbers can also be expressed as sums of binomial coefficients.
+}{
+\spadpaste{fib(n) == reduce(+,[binomial(n-1-k,k) for k in 0..quo(n-1,2)]) \bound{fib}}
+}
+\xtc{
+}{
+\spadpaste{fib(25) \free{fib}}
+}
+\xtc{
+}{
+\spadpaste{[fib(n) for n in 1..15] \free{fib}}
+}
+
+Quadratic symbols can be computed with the operations
+\spadfunFrom{legendre}{IntegerNumberTheoryFunctions} and
+\spadfunFrom{jacobi}{IntegerNumberTheoryFunctions}.
+The Legendre symbol
+%-% \HDindex{Legendre!symbol}{IntegerNumberTheoryFunctionsXmpPage}{9.36}{IntegerNumberTheoryFunctions}
+\texht{$\left({a \over p}\right)$}{\spad{(a/p)}}
+is defined for integers \spad{a} and
+\spad{p} with \spad{p} an odd prime number.
+By definition, \texht{$\left({a \over p}\right)$}{\spad{(a/p) = +1}},
+when \spad{a} is a square \spad{(mod p)},
+\texht{$\left({a \over p}\right)$}{\spad{(a/p) = -1}},
+when \spad{a} is not a square \spad{(mod p)}, and
+\texht{$\left({a \over p}\right)$}{\spad{(a/p) = 0}},
+when \spad{a} is divisible by \spad{p}.
+\xtc{
+You compute \texht{$\left({a \over p}\right)$}{\spad{(a/p)}}
+via the command \spad{legendre(a,p)}.
+}{
+\spadpaste{legendre(3,5)}
+}
+\xtc{
+}{
+\spadpaste{legendre(23,691)}
+}
+The Jacobi symbol \texht{$\left({a \over n}\right)$}{\spad{(a/n)}}
+is the usual extension of the Legendre
+symbol, where \spad{n} is an arbitrary integer.
+The most important property of the Jacobi symbol is the following:
+if \spad{K} is a quadratic field with discriminant \spad{d} and quadratic
+character \texht{$\chi$}{\spad{chi}},
+then \texht{$\chi$}{\spad{chi}}\spad{(n) = (d/n)}.
+Thus, you can use the Jacobi symbol
+%-% \HDindex{Jacobi symbol}{IntegerNumberTheoryFunctionsXmpPage}{9.36}{IntegerNumberTheoryFunctions}
+to compute, say, the class numbers of
+%-% \HDindex{class number}{IntegerNumberTheoryFunctionsXmpPage}{9.36}{IntegerNumberTheoryFunctions}
+imaginary quadratic fields from a standard class number formula.
+%-% \HDindex{field!imaginary quadratic}{IntegerNumberTheoryFunctionsXmpPage}{9.36}{IntegerNumberTheoryFunctions}
+\xtc{
+This function computes the class number of the imaginary
+quadratic field with discriminant \spad{d}.
+}{
+\spadpaste{h(d) == quo(reduce(+, [jacobi(d,k) for k in 1..quo(-d, 2)]), 2 - jacobi(d,2)) \bound{h}}
+}
+\xtc{
+}{
+\spadpaste{h(-163) \free{h}}   
+}
+\xtc{
+}{
+\spadpaste{h(-499) \free{h}}   
+}
+\xtc{
+}{
+\spadpaste{h(-1832) \free{h}}  
+}
+
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/INTHEORY.pht b/src/hyper/pages/INTHEORY.pht
new file mode 100644
index 00000000..7456ecbf
--- /dev/null
+++ b/src/hyper/pages/INTHEORY.pht
@@ -0,0 +1,348 @@
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch1}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull1}{IntegerNumberTheoryFunctionsXmpPageEmpty1}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{div144 := divisors(144)\bound{div144 }}
+\indentrel{3}\begin{verbatim}
+   (1)  [1,2,3,4,6,8,9,12,16,18,24,36,48,72,144]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty1}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty1}{IntegerNumberTheoryFunctionsXmpPagePatch1}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{div144 := divisors(144)\bound{div144 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch2}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull2}{IntegerNumberTheoryFunctionsXmpPageEmpty2}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{\#(div144)\free{div144 }}
+\indentrel{3}\begin{verbatim}
+   (2)  15
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty2}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty2}{IntegerNumberTheoryFunctionsXmpPagePatch2}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{\#(div144)\free{div144 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch3}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull3}{IntegerNumberTheoryFunctionsXmpPageEmpty3}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{reduce(+,div144)\free{div144 }}
+\indentrel{3}\begin{verbatim}
+   (3)  403
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty3}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty3}{IntegerNumberTheoryFunctionsXmpPagePatch3}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{reduce(+,div144)\free{div144 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch4}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull4}{IntegerNumberTheoryFunctionsXmpPageEmpty4}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{numberOfDivisors(144)}
+\indentrel{3}\begin{verbatim}
+   (4)  15
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty4}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty4}{IntegerNumberTheoryFunctionsXmpPagePatch4}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{numberOfDivisors(144)}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch5}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull5}{IntegerNumberTheoryFunctionsXmpPageEmpty5}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{sumOfDivisors(144)}
+\indentrel{3}\begin{verbatim}
+   (5)  403
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty5}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty5}{IntegerNumberTheoryFunctionsXmpPagePatch5}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{sumOfDivisors(144)}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch6}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull6}{IntegerNumberTheoryFunctionsXmpPageEmpty6}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{f1(n) == reduce(+,[moebiusMu(d) * numberOfDivisors(quo(n,d)) for d in divisors(n)])\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty6}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty6}{IntegerNumberTheoryFunctionsXmpPagePatch6}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{f1(n) == reduce(+,[moebiusMu(d) * numberOfDivisors(quo(n,d)) for d in divisors(n)])\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch7}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull7}{IntegerNumberTheoryFunctionsXmpPageEmpty7}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{f1(200)\free{f1 }}
+\indentrel{3}\begin{verbatim}
+   (7)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty7}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty7}{IntegerNumberTheoryFunctionsXmpPagePatch7}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{f1(200)\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch8}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull8}{IntegerNumberTheoryFunctionsXmpPageEmpty8}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{f1(846)\free{f1 }}
+\indentrel{3}\begin{verbatim}
+   (8)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty8}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty8}{IntegerNumberTheoryFunctionsXmpPagePatch8}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{f1(846)\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch9}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull9}{IntegerNumberTheoryFunctionsXmpPageEmpty9}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{f2(n) == reduce(+,[moebiusMu(d) * sumOfDivisors(quo(n,d)) for d in divisors(n)])\bound{f2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty9}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty9}{IntegerNumberTheoryFunctionsXmpPagePatch9}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{f2(n) == reduce(+,[moebiusMu(d) * sumOfDivisors(quo(n,d)) for d in divisors(n)])\bound{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch10}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull10}{IntegerNumberTheoryFunctionsXmpPageEmpty10}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{f2(200)\free{f2 }}
+\indentrel{3}\begin{verbatim}
+   (10)  200
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty10}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty10}{IntegerNumberTheoryFunctionsXmpPagePatch10}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{f2(200)\free{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch11}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull11}{IntegerNumberTheoryFunctionsXmpPageEmpty11}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{f2(846)\free{f2 }}
+\indentrel{3}\begin{verbatim}
+   (11)  846
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty11}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty11}{IntegerNumberTheoryFunctionsXmpPagePatch11}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{f2(846)\free{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch12}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull12}{IntegerNumberTheoryFunctionsXmpPageEmpty12}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{fibonacci(25)}
+\indentrel{3}\begin{verbatim}
+   (12)  75025
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty12}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty12}{IntegerNumberTheoryFunctionsXmpPagePatch12}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{fibonacci(25)}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch13}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull13}{IntegerNumberTheoryFunctionsXmpPageEmpty13}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{[fibonacci(n) for n in 1..15]}
+\indentrel{3}\begin{verbatim}
+   (13)  [1,1,2,3,5,8,13,21,34,55,89,144,233,377,610]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty13}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty13}{IntegerNumberTheoryFunctionsXmpPagePatch13}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{[fibonacci(n) for n in 1..15]}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch14}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull14}{IntegerNumberTheoryFunctionsXmpPageEmpty14}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{fib(n) == reduce(+,[binomial(n-1-k,k) for k in 0..quo(n-1,2)])\bound{fib }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty14}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty14}{IntegerNumberTheoryFunctionsXmpPagePatch14}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{fib(n) == reduce(+,[binomial(n-1-k,k) for k in 0..quo(n-1,2)])\bound{fib }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch15}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull15}{IntegerNumberTheoryFunctionsXmpPageEmpty15}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{fib(25)\free{fib }}
+\indentrel{3}\begin{verbatim}
+   (15)  75025
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty15}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty15}{IntegerNumberTheoryFunctionsXmpPagePatch15}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{fib(25)\free{fib }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch16}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull16}{IntegerNumberTheoryFunctionsXmpPageEmpty16}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{[fib(n) for n in 1..15]\free{fib }}
+\indentrel{3}\begin{verbatim}
+   (16)  [1,1,2,3,5,8,13,21,34,55,89,144,233,377,610]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty16}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty16}{IntegerNumberTheoryFunctionsXmpPagePatch16}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{[fib(n) for n in 1..15]\free{fib }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch17}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull17}{IntegerNumberTheoryFunctionsXmpPageEmpty17}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{legendre(3,5)}
+\indentrel{3}\begin{verbatim}
+   (17)  - 1
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty17}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty17}{IntegerNumberTheoryFunctionsXmpPagePatch17}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{legendre(3,5)}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch18}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull18}{IntegerNumberTheoryFunctionsXmpPageEmpty18}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{legendre(23,691)}
+\indentrel{3}\begin{verbatim}
+   (18)  - 1
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty18}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty18}{IntegerNumberTheoryFunctionsXmpPagePatch18}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{legendre(23,691)}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch19}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull19}{IntegerNumberTheoryFunctionsXmpPageEmpty19}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{h(d) == quo(reduce(+, [jacobi(d,k) for k in 1..quo(-d, 2)]), 2 - jacobi(d,2))\bound{h }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty19}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty19}{IntegerNumberTheoryFunctionsXmpPagePatch19}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{h(d) == quo(reduce(+, [jacobi(d,k) for k in 1..quo(-d, 2)]), 2 - jacobi(d,2))\bound{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch20}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull20}{IntegerNumberTheoryFunctionsXmpPageEmpty20}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{h(-163)\free{h }}
+\indentrel{3}\begin{verbatim}
+   (20)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty20}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty20}{IntegerNumberTheoryFunctionsXmpPagePatch20}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{h(-163)\free{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch21}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull21}{IntegerNumberTheoryFunctionsXmpPageEmpty21}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{h(-499)\free{h }}
+\indentrel{3}\begin{verbatim}
+   (21)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty21}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty21}{IntegerNumberTheoryFunctionsXmpPagePatch21}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{h(-499)\free{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPagePatch22}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageFull22}{IntegerNumberTheoryFunctionsXmpPageEmpty22}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{h(-1832)\free{h }}
+\indentrel{3}\begin{verbatim}
+   (22)  26
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerNumberTheoryFunctionsXmpPageEmpty22}
+\begin{paste}{IntegerNumberTheoryFunctionsXmpPageEmpty22}{IntegerNumberTheoryFunctionsXmpPagePatch22}
+\pastebutton{IntegerNumberTheoryFunctionsXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{h(-1832)\free{h }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/KAFILE.ht b/src/hyper/pages/KAFILE.ht
new file mode 100644
index 00000000..d47acee6
--- /dev/null
+++ b/src/hyper/pages/KAFILE.ht
@@ -0,0 +1,144 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\KeyedAccessFileXmpTitle}{KeyedAccessFile}
+\newcommand{\KeyedAccessFileXmpNumber}{9.38}
+%
+% =====================================================================
+\begin{page}{KeyedAccessFileXmpPage}{9.38 KeyedAccessFile}
+% =====================================================================
+\beginscroll
+
+The domain \spadtype{KeyedAccessFile(S)} provides files which can be used as
+%-% \HDindex{file!keyed access}{KeyedAccessFileXmpPage}{9.38}{KeyedAccessFile}
+associative tables.
+Data values are stored in these files and can be retrieved according to
+their keys.
+The keys must be strings so this type behaves very much like the
+\spadtype{StringTable(S)} domain.
+The difference is that keyed access files reside in secondary storage while
+string tables are kept in memory.
+For more information on table-oriented operations, see the description of
+\spadtype{Table}.
+
+\xtc{
+Before a keyed access file can be used, it must first be opened.
+A new file can be created by opening it for output.
+}{
+\spadpaste{ey: KeyedAccessFile(Integer) := open("/tmp/editor.year", "output")  \bound{ey}}
+}
+\xtc{
+Just as for vectors, tables or lists, values are saved in a keyed access file
+by setting elements.
+}{
+\spadpaste{ey."Char"     := 1986 \free{ey}\bound{eya}}
+}
+\xtc{
+}{
+\spadpaste{ey."Caviness" := 1985 \free{ey}\bound{eyb}}
+}
+\xtc{
+}{
+\spadpaste{ey."Fitch"    := 1984 \free{ey}\bound{eyc}}
+}
+\xtc{
+Values are retrieved using application, in any of its syntactic forms.
+}{
+\spadpaste{ey."Char"     \free{eya}}
+}
+\xtc{
+}{
+\spadpaste{ey("Char")    \free{eya}}
+}
+\xtc{
+}{
+\spadpaste{ey "Char"     \free{eya}}
+}
+\xtc{
+Attempting to retrieve a non-existent element in this way causes an error.
+If it is not known whether a key exists, you should use the
+\spadfunFrom{search}{KeyedAccessFile} operation.
+}{
+\spadpaste{search("Char", ey)   \free{eya,eyb,eyc}\bound{eyaa}}
+}
+\xtc{
+}{
+\spadpaste{search("Smith", ey)  \free{eyaa}}
+}
+\xtc{
+When an entry is no longer needed, it can be removed from the file.
+}{
+\spadpaste{remove!("Char", ey)  \free{eyaa}\bound{eybb}}
+}
+\xtc{
+The \spadfunFrom{keys}{KeyedAccessFile} operation returns a list of all the
+keys for a given file.
+}{
+\spadpaste{keys ey  \free{eybb}}
+}
+\xtc{
+The \spadfunFrom{\#}{KeyedAccessFile} operation gives the
+number of entries.
+}{
+\spadpaste{\#ey     \free{eybb}}
+}
+
+The table view of keyed access files provides safe operations.
+That is, if the \Language{} program is terminated between file operations,
+the file is left in a consistent, current state.
+This means, however, that the operations are somewhat costly.
+For example, after each update the file is closed.
+\xtc{
+Here we add several more items to the file, then check its contents.
+}{
+\spadpaste{KE := Record(key: String, entry: Integer)  \bound{KE}}
+}
+\xtc{
+}{
+\spadpaste{reopen!(ey, "output")  \free{eybb,KE}\bound{eycc}}
+}
+\xtc{
+If many items are to be added to a file at the same time, then
+it is more efficient to use the \spadfunFromX{write}{KeyedAccessFile} operation.
+}{
+\spadpaste{write!(ey, ["van Hulzen", 1983]\$KE)  \bound{eyccA}\free{eycc}}
+}
+\xtc{
+}{
+\spadpaste{write!(ey, ["Calmet", 1982]\$KE)      \bound{eyccB}\free{eycc}}
+}
+\xtc{
+}{
+\spadpaste{write!(ey, ["Wang", 1981]\$KE)        \bound{eyccC}\free{eycc}}
+}
+\xtc{
+}{
+\spadpaste{close! ey   \free{eyccA,eyccB,eyccC}\bound{eydd}}
+}
+\xtc{
+The \spadfunFromX{read}{KeyedAccessFile} operation is also available from
+the file view, but it returns elements in a random order.
+It is generally clearer and more efficient to use the
+\spadfunFrom{keys}{KeyedAccessFile} operation and to extract elements by
+key.
+}{
+\spadpaste{keys ey     \free{eydd}}
+}
+\xtc{
+}{
+\spadpaste{members ey  \free{eydd}}
+}
+\noOutputXtc{
+}{
+\spadpaste{)system rm -r /tmp/editor.year  \free{ey}}
+}
+
+For more information on related topics,  see
+\downlink{`File'}{FileXmpPage}\ignore{File},
+\downlink{`TextFile'}{TextFileXmpPage}\ignore{TextFile}, and
+\downlink{`Library'}{LibraryXmpPage}\ignore{Library}.
+%
+\showBlurb{KeyedAccessFile}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/KAFILE.pht b/src/hyper/pages/KAFILE.pht
new file mode 100644
index 00000000..ea8d9238
--- /dev/null
+++ b/src/hyper/pages/KAFILE.pht
@@ -0,0 +1,335 @@
+\begin{patch}{KeyedAccessFileXmpPagePatch1}
+\begin{paste}{KeyedAccessFileXmpPageFull1}{KeyedAccessFileXmpPageEmpty1}
+\pastebutton{KeyedAccessFileXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{ey: KeyedAccessFile(Integer) := open("/tmp/editor.year", "output")\bound{ey }}
+\indentrel{3}\begin{verbatim}
+   (1)  "/tmp/editor.year"
+                          Type: KeyedAccessFile Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty1}
+\begin{paste}{KeyedAccessFileXmpPageEmpty1}{KeyedAccessFileXmpPagePatch1}
+\pastebutton{KeyedAccessFileXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{ey: KeyedAccessFile(Integer) := open("/tmp/editor.year", "output")\bound{ey }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch2}
+\begin{paste}{KeyedAccessFileXmpPageFull2}{KeyedAccessFileXmpPageEmpty2}
+\pastebutton{KeyedAccessFileXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{ey."Char" := 1986\free{ey }\bound{eya }}
+\indentrel{3}\begin{verbatim}
+   (2)  1986
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty2}
+\begin{paste}{KeyedAccessFileXmpPageEmpty2}{KeyedAccessFileXmpPagePatch2}
+\pastebutton{KeyedAccessFileXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{ey."Char" := 1986\free{ey }\bound{eya }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch3}
+\begin{paste}{KeyedAccessFileXmpPageFull3}{KeyedAccessFileXmpPageEmpty3}
+\pastebutton{KeyedAccessFileXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{ey."Caviness" := 1985\free{ey }\bound{eyb }}
+\indentrel{3}\begin{verbatim}
+   (3)  1985
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty3}
+\begin{paste}{KeyedAccessFileXmpPageEmpty3}{KeyedAccessFileXmpPagePatch3}
+\pastebutton{KeyedAccessFileXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{ey."Caviness" := 1985\free{ey }\bound{eyb }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch4}
+\begin{paste}{KeyedAccessFileXmpPageFull4}{KeyedAccessFileXmpPageEmpty4}
+\pastebutton{KeyedAccessFileXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{ey."Fitch" := 1984\free{ey }\bound{eyc }}
+\indentrel{3}\begin{verbatim}
+   (4)  1984
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty4}
+\begin{paste}{KeyedAccessFileXmpPageEmpty4}{KeyedAccessFileXmpPagePatch4}
+\pastebutton{KeyedAccessFileXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{ey."Fitch" := 1984\free{ey }\bound{eyc }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch5}
+\begin{paste}{KeyedAccessFileXmpPageFull5}{KeyedAccessFileXmpPageEmpty5}
+\pastebutton{KeyedAccessFileXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{ey."Char"\free{eya }}
+\indentrel{3}\begin{verbatim}
+   (5)  1986
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty5}
+\begin{paste}{KeyedAccessFileXmpPageEmpty5}{KeyedAccessFileXmpPagePatch5}
+\pastebutton{KeyedAccessFileXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{ey."Char"\free{eya }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch6}
+\begin{paste}{KeyedAccessFileXmpPageFull6}{KeyedAccessFileXmpPageEmpty6}
+\pastebutton{KeyedAccessFileXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{ey("Char")\free{eya }}
+\indentrel{3}\begin{verbatim}
+   (6)  1986
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty6}
+\begin{paste}{KeyedAccessFileXmpPageEmpty6}{KeyedAccessFileXmpPagePatch6}
+\pastebutton{KeyedAccessFileXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{ey("Char")\free{eya }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch7}
+\begin{paste}{KeyedAccessFileXmpPageFull7}{KeyedAccessFileXmpPageEmpty7}
+\pastebutton{KeyedAccessFileXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{ey "Char"\free{eya }}
+\indentrel{3}\begin{verbatim}
+   (7)  1986
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty7}
+\begin{paste}{KeyedAccessFileXmpPageEmpty7}{KeyedAccessFileXmpPagePatch7}
+\pastebutton{KeyedAccessFileXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{ey "Char"\free{eya }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch8}
+\begin{paste}{KeyedAccessFileXmpPageFull8}{KeyedAccessFileXmpPageEmpty8}
+\pastebutton{KeyedAccessFileXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{search("Char", ey)\free{eya eyb eyc }\bound{eyaa }}
+\indentrel{3}\begin{verbatim}
+   (8)  1986
+                               Type: Union(Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty8}
+\begin{paste}{KeyedAccessFileXmpPageEmpty8}{KeyedAccessFileXmpPagePatch8}
+\pastebutton{KeyedAccessFileXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{search("Char", ey)\free{eya eyb eyc }\bound{eyaa }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch9}
+\begin{paste}{KeyedAccessFileXmpPageFull9}{KeyedAccessFileXmpPageEmpty9}
+\pastebutton{KeyedAccessFileXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{search("Smith", ey)\free{eyaa }}
+\indentrel{3}\begin{verbatim}
+   (9)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty9}
+\begin{paste}{KeyedAccessFileXmpPageEmpty9}{KeyedAccessFileXmpPagePatch9}
+\pastebutton{KeyedAccessFileXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{search("Smith", ey)\free{eyaa }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch10}
+\begin{paste}{KeyedAccessFileXmpPageFull10}{KeyedAccessFileXmpPageEmpty10}
+\pastebutton{KeyedAccessFileXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{remove!("Char", ey)\free{eyaa }\bound{eybb }}
+\indentrel{3}\begin{verbatim}
+   (10)  1986
+                               Type: Union(Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty10}
+\begin{paste}{KeyedAccessFileXmpPageEmpty10}{KeyedAccessFileXmpPagePatch10}
+\pastebutton{KeyedAccessFileXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{remove!("Char", ey)\free{eyaa }\bound{eybb }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch11}
+\begin{paste}{KeyedAccessFileXmpPageFull11}{KeyedAccessFileXmpPageEmpty11}
+\pastebutton{KeyedAccessFileXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{keys ey\free{eybb }}
+\indentrel{3}\begin{verbatim}
+   (11)  ["Fitch","Caviness"]
+                                      Type: List String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty11}
+\begin{paste}{KeyedAccessFileXmpPageEmpty11}{KeyedAccessFileXmpPagePatch11}
+\pastebutton{KeyedAccessFileXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{keys ey\free{eybb }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch12}
+\begin{paste}{KeyedAccessFileXmpPageFull12}{KeyedAccessFileXmpPageEmpty12}
+\pastebutton{KeyedAccessFileXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{\#ey\free{eybb }}
+\indentrel{3}\begin{verbatim}
+   (12)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty12}
+\begin{paste}{KeyedAccessFileXmpPageEmpty12}{KeyedAccessFileXmpPagePatch12}
+\pastebutton{KeyedAccessFileXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{\#ey\free{eybb }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch13}
+\begin{paste}{KeyedAccessFileXmpPageFull13}{KeyedAccessFileXmpPageEmpty13}
+\pastebutton{KeyedAccessFileXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{KE := Record(key: String, entry: Integer)\bound{KE }}
+\indentrel{3}\begin{verbatim}
+   (13)  Record(key: String,entry: Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty13}
+\begin{paste}{KeyedAccessFileXmpPageEmpty13}{KeyedAccessFileXmpPagePatch13}
+\pastebutton{KeyedAccessFileXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{KE := Record(key: String, entry: Integer)\bound{KE }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch14}
+\begin{paste}{KeyedAccessFileXmpPageFull14}{KeyedAccessFileXmpPageEmpty14}
+\pastebutton{KeyedAccessFileXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{reopen!(ey, "output")\free{eybb KE }\bound{eycc }}
+\indentrel{3}\begin{verbatim}
+   (14)  "/tmp/editor.year"
+                          Type: KeyedAccessFile Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty14}
+\begin{paste}{KeyedAccessFileXmpPageEmpty14}{KeyedAccessFileXmpPagePatch14}
+\pastebutton{KeyedAccessFileXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{reopen!(ey, "output")\free{eybb KE }\bound{eycc }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch15}
+\begin{paste}{KeyedAccessFileXmpPageFull15}{KeyedAccessFileXmpPageEmpty15}
+\pastebutton{KeyedAccessFileXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{write!(ey, ["van Hulzen", 1983]$KE)\bound{eyccA }\free{eycc }}
+\indentrel{3}\begin{verbatim}
+   (15)  [key= "van Hulzen",entry= 1983]
+               Type: Record(key: String,entry: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty15}
+\begin{paste}{KeyedAccessFileXmpPageEmpty15}{KeyedAccessFileXmpPagePatch15}
+\pastebutton{KeyedAccessFileXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{write!(ey, ["van Hulzen", 1983]$KE)\bound{eyccA }\free{eycc }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch16}
+\begin{paste}{KeyedAccessFileXmpPageFull16}{KeyedAccessFileXmpPageEmpty16}
+\pastebutton{KeyedAccessFileXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{write!(ey, ["Calmet", 1982]$KE)\bound{eyccB }\free{eycc }}
+\indentrel{3}\begin{verbatim}
+   (16)  [key= "Calmet",entry= 1982]
+               Type: Record(key: String,entry: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty16}
+\begin{paste}{KeyedAccessFileXmpPageEmpty16}{KeyedAccessFileXmpPagePatch16}
+\pastebutton{KeyedAccessFileXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{write!(ey, ["Calmet", 1982]$KE)\bound{eyccB }\free{eycc }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch17}
+\begin{paste}{KeyedAccessFileXmpPageFull17}{KeyedAccessFileXmpPageEmpty17}
+\pastebutton{KeyedAccessFileXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{write!(ey, ["Wang", 1981]$KE)\bound{eyccC }\free{eycc }}
+\indentrel{3}\begin{verbatim}
+   (17)  [key= "Wang",entry= 1981]
+               Type: Record(key: String,entry: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty17}
+\begin{paste}{KeyedAccessFileXmpPageEmpty17}{KeyedAccessFileXmpPagePatch17}
+\pastebutton{KeyedAccessFileXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{write!(ey, ["Wang", 1981]$KE)\bound{eyccC }\free{eycc }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch18}
+\begin{paste}{KeyedAccessFileXmpPageFull18}{KeyedAccessFileXmpPageEmpty18}
+\pastebutton{KeyedAccessFileXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{close! ey\free{eyccA eyccB eyccC }\bound{eydd }}
+\indentrel{3}\begin{verbatim}
+   (18)  "/tmp/editor.year"
+                          Type: KeyedAccessFile Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty18}
+\begin{paste}{KeyedAccessFileXmpPageEmpty18}{KeyedAccessFileXmpPagePatch18}
+\pastebutton{KeyedAccessFileXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{close! ey\free{eyccA eyccB eyccC }\bound{eydd }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch19}
+\begin{paste}{KeyedAccessFileXmpPageFull19}{KeyedAccessFileXmpPageEmpty19}
+\pastebutton{KeyedAccessFileXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{keys ey\free{eydd }}
+\indentrel{3}\begin{verbatim}
+   (19)
+   ["Wang","Calmet","van Hulzen","Fitch","Caviness"]
+                                      Type: List String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty19}
+\begin{paste}{KeyedAccessFileXmpPageEmpty19}{KeyedAccessFileXmpPagePatch19}
+\pastebutton{KeyedAccessFileXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{keys ey\free{eydd }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch20}
+\begin{paste}{KeyedAccessFileXmpPageFull20}{KeyedAccessFileXmpPageEmpty20}
+\pastebutton{KeyedAccessFileXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{members ey\free{eydd }}
+\indentrel{3}\begin{verbatim}
+   (20)  [1981,1982,1983,1984,1985]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty20}
+\begin{paste}{KeyedAccessFileXmpPageEmpty20}{KeyedAccessFileXmpPagePatch20}
+\pastebutton{KeyedAccessFileXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{members ey\free{eydd }}
+\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPagePatch21}
+\begin{paste}{KeyedAccessFileXmpPageFull21}{KeyedAccessFileXmpPageEmpty21}
+\pastebutton{KeyedAccessFileXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{)system rm -r /tmp/editor.year\free{ey }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KeyedAccessFileXmpPageEmpty21}
+\begin{paste}{KeyedAccessFileXmpPageEmpty21}{KeyedAccessFileXmpPagePatch21}
+\pastebutton{KeyedAccessFileXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{)system rm -r /tmp/editor.year\free{ey }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/KERNEL.ht b/src/hyper/pages/KERNEL.ht
new file mode 100644
index 00000000..90120e68
--- /dev/null
+++ b/src/hyper/pages/KERNEL.ht
@@ -0,0 +1,143 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\KernelXmpTitle}{Kernel}
+\newcommand{\KernelXmpNumber}{9.37}
+%
+% =====================================================================
+\begin{page}{KernelXmpPage}{9.37 Kernel}
+% =====================================================================
+\beginscroll
+%
+
+A {\it kernel} is a symbolic function application (such as
+\spad{sin(x + y)}) or a symbol (such as \spad{x}).
+More precisely, a non-symbol kernel over a set {\it S} is an operator applied
+to a given list of arguments from {\it S}.
+The operator has type \axiomType{BasicOperator}
+(see \downlink{`BasicOperator'}{BasicOperatorXmpPage}\ignore{BasicOperator}) and the kernel object is usually part of
+an expression object (see \downlink{`Expression'}{ExpressionXmpPage}\ignore{Expression}).
+
+Kernels are created implicitly for you when you
+create expressions.
+\xtc{
+}{
+\spadpaste{x :: Expression Integer}
+}
+\xtc{
+You can directly create a ``symbol'' kernel by using the
+\axiomFunFrom{kernel}{Kernel} operation.
+}{
+\spadpaste{kernel x}
+}
+\xtc{
+This expression has two different kernels.
+}{
+\spadpaste{sin(x) + cos(x) \bound{sincos}}
+}
+\xtc{
+The operator \axiomFunFrom{kernels}{Expression} returns a list
+of the kernels in an object of type \axiomType{Expression}.
+}{
+\spadpaste{kernels \% \free{sincos}}
+}
+\xtc{
+This expression also has two different kernels.
+}{
+\spadpaste{sin(x)**2 + sin(x) + cos(x) \bound{sincos2}}
+}
+\xtc{
+The \spad{sin(x)} kernel is used twice.
+}{
+\spadpaste{kernels \% \free{sincos2}}
+}
+\xtc{
+An expression need not contain any kernels.
+}{
+\spadpaste{kernels(1 :: Expression Integer)}
+}
+\xtc{
+If one or more kernels are present, one of them is
+designated the {\it main} kernel.
+}{
+\spadpaste{mainKernel(cos(x) + tan(x))}
+}
+\xtc{
+Kernels can be nested. Use \axiomFunFrom{height}{Kernel} to determine
+the nesting depth.
+}{
+\spadpaste{height kernel x}
+}
+\xtc{
+This has height 2 because the \spad{x} has height 1 and then we apply
+an operator to that.
+}{
+\spadpaste{height mainKernel(sin x)}
+}
+\xtc{
+}{
+\spadpaste{height mainKernel(sin cos x)}
+}
+\xtc{
+}{
+\spadpaste{height mainKernel(sin cos (tan x + sin x))}
+}
+\xtc{
+Use the \axiomFunFrom{operator}{Kernel} operation to extract the
+operator component of the kernel.
+The operator has type \axiomType{BasicOperator}.
+}{
+\spadpaste{operator mainKernel(sin cos (tan x + sin x))}
+}
+\xtc{
+Use the \axiomFunFrom{name}{Kernel} operation to extract the name of the
+operator component of the kernel.
+The name has type \axiomType{Symbol}.
+This is really just a shortcut for a two-step process of extracting the
+operator and then calling \axiomFunFrom{name}{BasicOperator} on the
+operator.
+}{
+\spadpaste{name mainKernel(sin cos (tan x + sin x))}
+}
+\Language{} knows about functions such as \axiomFun{sin}, \axiomFun{cos}
+and so on and can make kernels and then expressions using them.
+To create a kernel and expression using an arbitrary operator, use
+\axiomFunFrom{operator}{BasicOperator}.
+\xtc{
+Now \spad{f} can be used to create symbolic function applications.
+}{
+\spadpaste{f := operator 'f \bound{f}}
+}
+\xtc{
+}{
+\spadpaste{e := f(x, y, 10) \free{f}\bound{e}}
+}
+\xtc{
+Use the \axiomFunFrom{is?}{Kernel} operation to learn if the
+operator component of a kernel is equal to a given operator.
+}{
+\spadpaste{is?(e, f) \free{f e}}
+}
+\xtc{
+You can also use a symbol or a string as the second
+argument to \axiomFunFrom{is?}{Kernel}.
+}{
+\spadpaste{is?(e, 'f) \free{e}}
+}
+\xtc{
+Use the \axiomFunFrom{argument}{Kernel} operation to get a list containing
+the argument component of a kernel.
+}{
+\spadpaste{argument mainKernel e \free{f}\free{e}}
+}
+
+Conceptually, an object of type \axiomType{Expression} can be thought
+of a quotient of multivariate polynomials, where the ``variables''
+are kernels.
+The arguments of the kernels are again expressions and so the
+structure recurses.
+See \downlink{`Expression'}{ExpressionXmpPage}\ignore{Expression} for examples of using kernels to
+take apart expression objects.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/KERNEL.pht b/src/hyper/pages/KERNEL.pht
new file mode 100644
index 00000000..c005dd68
--- /dev/null
+++ b/src/hyper/pages/KERNEL.pht
@@ -0,0 +1,305 @@
+\begin{patch}{KernelXmpPagePatch1}
+\begin{paste}{KernelXmpPageFull1}{KernelXmpPageEmpty1}
+\pastebutton{KernelXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{x :: Expression Integer}
+\indentrel{3}\begin{verbatim}
+   (1)  x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty1}
+\begin{paste}{KernelXmpPageEmpty1}{KernelXmpPagePatch1}
+\pastebutton{KernelXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{x :: Expression Integer}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch2}
+\begin{paste}{KernelXmpPageFull2}{KernelXmpPageEmpty2}
+\pastebutton{KernelXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{kernel x}
+\indentrel{3}\begin{verbatim}
+   (2)  x
+                        Type: Kernel Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty2}
+\begin{paste}{KernelXmpPageEmpty2}{KernelXmpPagePatch2}
+\pastebutton{KernelXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{kernel x}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch3}
+\begin{paste}{KernelXmpPageFull3}{KernelXmpPageEmpty3}
+\pastebutton{KernelXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{sin(x) + cos(x)\bound{sincos }}
+\indentrel{3}\begin{verbatim}
+   (3)  sin(x) + cos(x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty3}
+\begin{paste}{KernelXmpPageEmpty3}{KernelXmpPagePatch3}
+\pastebutton{KernelXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{sin(x) + cos(x)\bound{sincos }}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch4}
+\begin{paste}{KernelXmpPageFull4}{KernelXmpPageEmpty4}
+\pastebutton{KernelXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{kernels \%\free{sincos }}
+\indentrel{3}\begin{verbatim}
+   (4)  [sin(x),cos(x)]
+                   Type: List Kernel Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty4}
+\begin{paste}{KernelXmpPageEmpty4}{KernelXmpPagePatch4}
+\pastebutton{KernelXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{kernels \%\free{sincos }}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch5}
+\begin{paste}{KernelXmpPageFull5}{KernelXmpPageEmpty5}
+\pastebutton{KernelXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{sin(x)**2 + sin(x) + cos(x)\bound{sincos2 }}
+\indentrel{3}\begin{verbatim}
+              2
+   (5)  sin(x)  + sin(x) + cos(x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty5}
+\begin{paste}{KernelXmpPageEmpty5}{KernelXmpPagePatch5}
+\pastebutton{KernelXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{sin(x)**2 + sin(x) + cos(x)\bound{sincos2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch6}
+\begin{paste}{KernelXmpPageFull6}{KernelXmpPageEmpty6}
+\pastebutton{KernelXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{kernels \%\free{sincos2 }}
+\indentrel{3}\begin{verbatim}
+   (6)  [sin(x),cos(x)]
+                   Type: List Kernel Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty6}
+\begin{paste}{KernelXmpPageEmpty6}{KernelXmpPagePatch6}
+\pastebutton{KernelXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{kernels \%\free{sincos2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch7}
+\begin{paste}{KernelXmpPageFull7}{KernelXmpPageEmpty7}
+\pastebutton{KernelXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{kernels(1 :: Expression Integer)}
+\indentrel{3}\begin{verbatim}
+   (7)  []
+                   Type: List Kernel Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty7}
+\begin{paste}{KernelXmpPageEmpty7}{KernelXmpPagePatch7}
+\pastebutton{KernelXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{kernels(1 :: Expression Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch8}
+\begin{paste}{KernelXmpPageFull8}{KernelXmpPageEmpty8}
+\pastebutton{KernelXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{mainKernel(cos(x) + tan(x))}
+\indentrel{3}\begin{verbatim}
+   (8)  tan(x)
+             Type: Union(Kernel Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty8}
+\begin{paste}{KernelXmpPageEmpty8}{KernelXmpPagePatch8}
+\pastebutton{KernelXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{mainKernel(cos(x) + tan(x))}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch9}
+\begin{paste}{KernelXmpPageFull9}{KernelXmpPageEmpty9}
+\pastebutton{KernelXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{height kernel x}
+\indentrel{3}\begin{verbatim}
+   (9)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty9}
+\begin{paste}{KernelXmpPageEmpty9}{KernelXmpPagePatch9}
+\pastebutton{KernelXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{height kernel x}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch10}
+\begin{paste}{KernelXmpPageFull10}{KernelXmpPageEmpty10}
+\pastebutton{KernelXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{height mainKernel(sin x)}
+\indentrel{3}\begin{verbatim}
+   (10)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty10}
+\begin{paste}{KernelXmpPageEmpty10}{KernelXmpPagePatch10}
+\pastebutton{KernelXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{height mainKernel(sin x)}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch11}
+\begin{paste}{KernelXmpPageFull11}{KernelXmpPageEmpty11}
+\pastebutton{KernelXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{height mainKernel(sin cos x)}
+\indentrel{3}\begin{verbatim}
+   (11)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty11}
+\begin{paste}{KernelXmpPageEmpty11}{KernelXmpPagePatch11}
+\pastebutton{KernelXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{height mainKernel(sin cos x)}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch12}
+\begin{paste}{KernelXmpPageFull12}{KernelXmpPageEmpty12}
+\pastebutton{KernelXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{height mainKernel(sin cos (tan x + sin x))}
+\indentrel{3}\begin{verbatim}
+   (12)  4
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty12}
+\begin{paste}{KernelXmpPageEmpty12}{KernelXmpPagePatch12}
+\pastebutton{KernelXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{height mainKernel(sin cos (tan x + sin x))}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch13}
+\begin{paste}{KernelXmpPageFull13}{KernelXmpPageEmpty13}
+\pastebutton{KernelXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{operator mainKernel(sin cos (tan x + sin x))}
+\indentrel{3}\begin{verbatim}
+   (13)  sin
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty13}
+\begin{paste}{KernelXmpPageEmpty13}{KernelXmpPagePatch13}
+\pastebutton{KernelXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{operator mainKernel(sin cos (tan x + sin x))}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch14}
+\begin{paste}{KernelXmpPageFull14}{KernelXmpPageEmpty14}
+\pastebutton{KernelXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{name mainKernel(sin cos (tan x + sin x))}
+\indentrel{3}\begin{verbatim}
+   (14)  sin
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty14}
+\begin{paste}{KernelXmpPageEmpty14}{KernelXmpPagePatch14}
+\pastebutton{KernelXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{name mainKernel(sin cos (tan x + sin x))}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch15}
+\begin{paste}{KernelXmpPageFull15}{KernelXmpPageEmpty15}
+\pastebutton{KernelXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{f := operator 'f\bound{f }}
+\indentrel{3}\begin{verbatim}
+   (15)  f
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty15}
+\begin{paste}{KernelXmpPageEmpty15}{KernelXmpPagePatch15}
+\pastebutton{KernelXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{f := operator 'f\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch16}
+\begin{paste}{KernelXmpPageFull16}{KernelXmpPageEmpty16}
+\pastebutton{KernelXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{e := f(x, y, 10)\free{f }\bound{e }}
+\indentrel{3}\begin{verbatim}
+   (16)  f(x,y,10)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty16}
+\begin{paste}{KernelXmpPageEmpty16}{KernelXmpPagePatch16}
+\pastebutton{KernelXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{e := f(x, y, 10)\free{f }\bound{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch17}
+\begin{paste}{KernelXmpPageFull17}{KernelXmpPageEmpty17}
+\pastebutton{KernelXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{is?(e, f)\free{f e }}
+\indentrel{3}\begin{verbatim}
+   (17)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty17}
+\begin{paste}{KernelXmpPageEmpty17}{KernelXmpPagePatch17}
+\pastebutton{KernelXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{is?(e, f)\free{f e }}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch18}
+\begin{paste}{KernelXmpPageFull18}{KernelXmpPageEmpty18}
+\pastebutton{KernelXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{is?(e, 'f)\free{e }}
+\indentrel{3}\begin{verbatim}
+   (18)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty18}
+\begin{paste}{KernelXmpPageEmpty18}{KernelXmpPagePatch18}
+\pastebutton{KernelXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{is?(e, 'f)\free{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPagePatch19}
+\begin{paste}{KernelXmpPageFull19}{KernelXmpPageEmpty19}
+\pastebutton{KernelXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{argument mainKernel e\free{f }\free{e }}
+\indentrel{3}\begin{verbatim}
+   (19)  [x,y,10]
+                          Type: List Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{KernelXmpPageEmpty19}
+\begin{paste}{KernelXmpPageEmpty19}{KernelXmpPagePatch19}
+\pastebutton{KernelXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{argument mainKernel e\free{f }\free{e }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/LAZM3PK.ht b/src/hyper/pages/LAZM3PK.ht
new file mode 100644
index 00000000..e4eea5a8
--- /dev/null
+++ b/src/hyper/pages/LAZM3PK.ht
@@ -0,0 +1,273 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\LazardSetSolvingPackageXmpTitle}{LazardSetSolvingPackage}
+\newcommand{\LazardSetSolvingPackageXmpNumber}{9.40}
+%
+% =====================================================================
+\begin{page}{LazardSetSolvingPackageXmpPage}{9.40 LazardSetSolvingPackage}
+% =====================================================================
+\beginscroll
+The \spadtype{LazardSetSolvingPackage} package constructor solves 
+polynomial systems by means of Lazard triangular sets.
+However one condition is relaxed:
+Regular triangular sets whose saturated ideals have positive dimension
+are not necessarily normalized.
+
+The decompositions are computed in two steps.
+First the algorithm of Moreno Maza (implemented in 
+the \spadtype{RegularTriangularSet} domain constructor)
+is called.
+Then the resulting decompositions are converted into lists 
+of square-free regular triangular sets
+and the redundant components are removed.
+Moreover, zero-dimensional regular triangular sets are normalized.
+
+Note that the way of understanding triangular decompositions 
+is detailed in the example of the \spadtype{RegularTriangularSet}
+constructor.
+
+The \spadtype{LazardSetSolvingPackage} constructor takes six arguments.
+The first one, {\bf R}, is the coefficient ring of the polynomials;
+it must belong to the category \spadtype{GcdDomain}.
+The second one, {\bf E}, is the exponent monoid of the polynomials;
+it must belong to the category \spadtype{OrderedAbelianMonoidSup}.
+the third one, {\bf V}, is the ordered set of variables;
+it must belong to the category \spadtype{OrderedSet}.
+The fourth one is the polynomial ring;
+it must belong to the category \spadtype{RecursivePolynomialCategory(R,E,V)}.
+The fifth one is a domain of the category \spadtype{RegularTriangularSetCategory(R,E,V,P)}
+and the last one is a domain of the category \spadtype{SquareFreeRegularTriangularSetCategory(R,E,V,P)}.
+The abbreviation for \spadtype{LazardSetSolvingPackage} is \spad{LAZM3PK}.
+
+{\bf N.B.} For the purpose of solving zero-dimensional algebraic systems,
+see also \spadtype{LexTriangularPackage} and \spadtype{ZeroDimensionalSolvePackage}.
+These packages are easier to call than \spad{LAZM3PK}.
+Moreover, the \spadtype{ZeroDimensionalSolvePackage} 
+package  provides operations
+to compute either the complex roots or the real roots.
+
+We illustrate now the use of the \spadtype{LazardSetSolvingPackage} package 
+constructor with two examples (Butcher and Vermeer).
+
+
+\xtc{
+Define the coefficient ring.
+}{
+\spadpaste{R := Integer \bound{R}}
+}
+
+
+\xtc{
+Define the list of variables,
+}{
+\spadpaste{ls : List Symbol := [b1,x,y,z,t,v,u,w] \bound{ls}}
+} 
+
+\xtc{
+and make it an ordered set;
+}{
+\spadpaste{V := OVAR(ls) \free{ls} \bound{V}}
+}
+
+\xtc{
+then define the exponent monoid.
+}{
+\spadpaste{E := IndexedExponents V \free{V} \bound{E}}
+}
+
+\xtc{
+Define the polynomial ring.
+}{
+\spadpaste{P := NSMP(R, V) \free{R} \free{V} \bound{P}}
+}
+
+\xtc{
+Let the variables be polynomial.
+}{
+\spadpaste{b1: P := 'b1  \free{P} \bound{b1}}
+}
+\xtc{
+}{
+\spadpaste{x: P := 'x \free{P} \bound{x}}
+}
+\xtc{
+}{
+\spadpaste{y: P := 'y \free{P} \bound{y}}
+}
+\xtc{
+}{
+\spadpaste{z: P := 'z \free{P} \bound{z}}
+}
+\xtc{
+}{
+\spadpaste{t: P := 't \free{P} \bound{t}}
+}
+\xtc{
+}{
+\spadpaste{u: P := 'u  \free{P} \bound{u}}
+}
+\xtc{
+}{
+\spadpaste{v: P := 'v  \free{P} \bound{v}}
+}
+\xtc{
+}{
+\spadpaste{w: P := 'w  \free{P} \bound{w}}
+}
+
+\xtc{
+Now call the \spadtype{RegularTriangularSet} domain constructor.
+}{
+\spadpaste{T := REGSET(R,E,V,P) \free{R} \free{E} \free{V} \free{P} \bound{T} }
+}
+
+\xtc{
+Define a polynomial system (the Butcher example).
+}{
+\spadpaste{p0 := b1 + y + z - t - w \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p0}}
+}
+\xtc{
+}{
+\spadpaste{p1 := 2*z*u + 2*y*v + 2*t*w - 2*w**2 - w - 1 \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p1}}
+}
+\xtc{
+}{
+\spadpaste{p2 := 3*z*u**2 + 3*y*v**2 - 3*t*w**2 + 3*w**3 + 3*w**2 - t + 4*w \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p2}}
+}
+\xtc{
+}{
+\spadpaste{p3 := 6*x*z*v - 6*t*w**2 + 6*w**3 - 3*t*w + 6*w**2 - t + 4*w \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p3}}
+}
+\xtc{
+}{
+\spadpaste{p4 := 4*z*u**3+ 4*y*v**3+ 4*t*w**3- 4*w**4 - 6*w**3+ 4*t*w- 10*w**2- w- 1 \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p4}}
+}
+\xtc{
+}{
+\spadpaste{p5 := 8*x*z*u*v +8*t*w**3 -8*w**4 +4*t*w**2 -12*w**3 +4*t*w -14*w**2 -3*w -1  \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p5}}
+}
+\xtc{
+}{
+\spadpaste{p6 := 12*x*z*v**2+12*t*w**3 -12*w**4 +12*t*w**2 -18*w**3 +8*t*w -14*w**2 -w -1 \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p6}}
+}
+\xtc{
+}{
+\spadpaste{p7 := -24*t*w**3 + 24*w**4 - 24*t*w**2 + 36*w**3 - 8*t*w + 26*w**2 + 7*w + 1 \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p7}}
+}
+\xtc{
+}{
+\spadpaste{lp := [p0, p1, p2, p3, p4, p5, p6, p7] \free{p0} \free{p1} \free{p2} \free{p3} \free{p4} \free{p5} \free{p6} \free{p7} \bound{lp}}
+}
+
+\xtc{
+First of all, let us solve this system in the sense of Lazard by means of the \spadtype{REGSET}
+constructor:
+}{
+\spadpaste{lts := zeroSetSplit(lp,false)$T \free{lp} \free{T} \bound{lts}}
+}
+
+\xtc{
+We can get the dimensions of each component
+of a decomposition as follows.
+}{
+\spadpaste{[coHeight(ts) for ts in lts] \free{lts}}
+}
+
+The first five sets have a simple shape.
+However, the last one, which has dimension zero, can be simplified
+by using Lazard triangular sets.
+
+\xtc{
+Thus we call the \spadtype{SquareFreeRegularTriangularSet} domain constructor,
+}{
+\spadpaste{ST := SREGSET(R,E,V,P) \free{R} \free{E} \free{V} \free{P} \bound{ST} }
+}
+
+\xtc{
+and set the \spadtype{LAZM3PK} package constructor to our situation.
+}{
+\spadpaste{pack := LAZM3PK(R,E,V,P,T,ST) \free{R} \free{E} \free{V} \free{P} \free{T} \free{ST} \bound{pack} }
+}
+
+\xtc{
+We are ready to solve the system by means of Lazard triangular sets:
+}{
+\spadpaste{zeroSetSplit(lp,false)$pack \free{lp} \free{pack}}
+}
+
+We see the sixth triangular set is {\em nicer} now: 
+each one of its polynomials has a constant initial.
+
+We follow with the Vermeer example. The ordering is the usual one
+for this system.
+
+\xtc{
+Define the polynomial system.
+}{
+\spadpaste{f0 := (w - v) ** 2 + (u - t) ** 2 - 1 \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{f0}}
+}
+\xtc{
+}{
+\spadpaste{f1 := t ** 2 - v ** 3 \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{f1}}
+}
+\xtc{
+}{
+\spadpaste{f2 := 2 * t * (w - v) + 3 * v ** 2 * (u - t) \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{f2}}
+}
+\xtc{
+}{
+\spadpaste{f3 := (3 * z * v ** 2 - 1) * (2 * z * t - 1) \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{f3}}
+}
+\xtc{
+}{
+\spadpaste{lf := [f0, f1, f2, f3] \free{f0} \free{f1} \free{f2} \free{f3} \bound{lf}}
+}
+
+
+\xtc{
+First of all, let us solve this system in the sense of Kalkbrener by means of the \spadtype{REGSET}
+constructor:
+}{
+\spadpaste{zeroSetSplit(lf,true)$T \free{lf} \free{T}}
+}
+
+We have obtained one regular chain (i.e. regular triangular set) with dimension 1.
+This set is in fact a characterist set of the (radical of) of the ideal generated
+by the input system {\bf lf}.
+Thus we have only the {\em generic points} of the variety associated with {\bf lf}
+(for the elimination ordering given by {\bf ls}).
+
+So let us get now a full description of this variety.
+\xtc{
+Hence, we solve this system in the sense of Lazard by means of the \spadtype{REGSET}
+constructor:
+}{
+\spadpaste{zeroSetSplit(lf,false)$T \free{lf} \free{T}}
+}
+
+We retrieve our regular chain of dimension 1 and we get three regular chains
+of dimension 0 corresponding to the {\em degenerated cases}.
+We want now to simplify these zero-dimensional regular chains
+by using Lazard triangular sets.
+Moreover, this will allow us to prove that the above decomposition has no redundant component.
+{\bf N.B.} Generally, decompositions computed by the \spadtype{REGSET}
+constructor do not have redundant components.
+However, to be sure that no redundant component occurs one needs to use
+the \spadtype{SREGSET} or \spadtype{LAZM3PK} constructors.
+
+\xtc{
+So let us solve the input system  in the sense of Lazard by means of the \spadtype{LAZM3PK} constructor:
+}{
+\spadpaste{zeroSetSplit(lf,false)$pack \free{lf} \free{pack}}
+}
+Due to square-free factorization, we obtained now four zero-dimensional regular chains.
+Moreover, each of them is normalized (the initials are constant).
+Note that these zero-dimensional components may be investigated further
+with the \spadtype{ZeroDimensionalSolvePackage} package constructor.
+
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/LAZM3PK.pht b/src/hyper/pages/LAZM3PK.pht
new file mode 100644
index 00000000..5afd80fe
--- /dev/null
+++ b/src/hyper/pages/LAZM3PK.pht
@@ -0,0 +1,955 @@
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch1}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull1}{LazardSetSolvingPackageXmpPageEmpty1}
+\pastebutton{LazardSetSolvingPackageXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{R := Integer\bound{R }}
+\indentrel{3}\begin{verbatim}
+   (1)  Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty1}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty1}{LazardSetSolvingPackageXmpPagePatch1}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{R := Integer\bound{R }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch2}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull2}{LazardSetSolvingPackageXmpPageEmpty2}
+\pastebutton{LazardSetSolvingPackageXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{ls : List Symbol := [b1,x,y,z,t,v,u,w]\bound{ls }}
+\indentrel{3}\begin{verbatim}
+   (2)  [b1,x,y,z,t,v,u,w]
+                                      Type: List Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty2}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty2}{LazardSetSolvingPackageXmpPagePatch2}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{ls : List Symbol := [b1,x,y,z,t,v,u,w]\bound{ls }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch3}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull3}{LazardSetSolvingPackageXmpPageEmpty3}
+\pastebutton{LazardSetSolvingPackageXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
+\indentrel{3}\begin{verbatim}
+   (3)  OrderedVariableList [b1,x,y,z,t,v,u,w]
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty3}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty3}{LazardSetSolvingPackageXmpPagePatch3}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch4}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull4}{LazardSetSolvingPackageXmpPageEmpty4}
+\pastebutton{LazardSetSolvingPackageXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{E := IndexedExponents V\free{V }\bound{E }}
+\indentrel{3}\begin{verbatim}
+   (4)
+   IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w]
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty4}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty4}{LazardSetSolvingPackageXmpPagePatch4}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{E := IndexedExponents V\free{V }\bound{E }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch5}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull5}{LazardSetSolvingPackageXmpPageEmpty5}
+\pastebutton{LazardSetSolvingPackageXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
+\indentrel{3}\begin{verbatim}
+   (5)
+  NewSparseMultivariatePolynomial(Integer,OrderedVariable
+  List [b1,x,y,z,t,v,u,w])
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty5}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty5}{LazardSetSolvingPackageXmpPagePatch5}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch6}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull6}{LazardSetSolvingPackageXmpPageEmpty6}
+\pastebutton{LazardSetSolvingPackageXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{b1: P := 'b1\free{P }\bound{b1 }}
+\indentrel{3}\begin{verbatim}
+   (6)  b1
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty6}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty6}{LazardSetSolvingPackageXmpPagePatch6}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{b1: P := 'b1\free{P }\bound{b1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch7}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull7}{LazardSetSolvingPackageXmpPageEmpty7}
+\pastebutton{LazardSetSolvingPackageXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{x: P := 'x\free{P }\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (7)  x
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty7}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty7}{LazardSetSolvingPackageXmpPagePatch7}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{x: P := 'x\free{P }\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch8}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull8}{LazardSetSolvingPackageXmpPageEmpty8}
+\pastebutton{LazardSetSolvingPackageXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{y: P := 'y\free{P }\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (8)  y
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty8}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty8}{LazardSetSolvingPackageXmpPagePatch8}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{y: P := 'y\free{P }\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch9}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull9}{LazardSetSolvingPackageXmpPageEmpty9}
+\pastebutton{LazardSetSolvingPackageXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{z: P := 'z\free{P }\bound{z }}
+\indentrel{3}\begin{verbatim}
+   (9)  z
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty9}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty9}{LazardSetSolvingPackageXmpPagePatch9}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{z: P := 'z\free{P }\bound{z }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch10}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull10}{LazardSetSolvingPackageXmpPageEmpty10}
+\pastebutton{LazardSetSolvingPackageXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{t: P := 't\free{P }\bound{t }}
+\indentrel{3}\begin{verbatim}
+   (10)  t
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty10}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty10}{LazardSetSolvingPackageXmpPagePatch10}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{t: P := 't\free{P }\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch11}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull11}{LazardSetSolvingPackageXmpPageEmpty11}
+\pastebutton{LazardSetSolvingPackageXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{u: P := 'u\free{P }\bound{u }}
+\indentrel{3}\begin{verbatim}
+   (11)  u
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty11}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty11}{LazardSetSolvingPackageXmpPagePatch11}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{u: P := 'u\free{P }\bound{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch12}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull12}{LazardSetSolvingPackageXmpPageEmpty12}
+\pastebutton{LazardSetSolvingPackageXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{v: P := 'v\free{P }\bound{v }}
+\indentrel{3}\begin{verbatim}
+   (12)  v
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty12}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty12}{LazardSetSolvingPackageXmpPagePatch12}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{v: P := 'v\free{P }\bound{v }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch13}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull13}{LazardSetSolvingPackageXmpPageEmpty13}
+\pastebutton{LazardSetSolvingPackageXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{w: P := 'w\free{P }\bound{w }}
+\indentrel{3}\begin{verbatim}
+   (13)  w
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty13}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty13}{LazardSetSolvingPackageXmpPagePatch13}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{w: P := 'w\free{P }\bound{w }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch14}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull14}{LazardSetSolvingPackageXmpPageEmpty14}
+\pastebutton{LazardSetSolvingPackageXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{T := REGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{T }}
+\indentrel{3}\begin{verbatim}
+   (14)
+  RegularTriangularSet(Integer,IndexedExponents OrderedVa
+  riableList [b1,x,y,z,t,v,u,w],OrderedVariableList [b1,x
+  ,y,z,t,v,u,w],NewSparseMultivariatePolynomial(Integer,O
+  rderedVariableList [b1,x,y,z,t,v,u,w]))
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty14}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty14}{LazardSetSolvingPackageXmpPagePatch14}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{T := REGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{T }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch15}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull15}{LazardSetSolvingPackageXmpPageEmpty15}
+\pastebutton{LazardSetSolvingPackageXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{p0 := b1 + y + z - t - w\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p0 }}
+\indentrel{3}\begin{verbatim}
+   (15)  b1 + y + z - t - w
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty15}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty15}{LazardSetSolvingPackageXmpPagePatch15}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{p0 := b1 + y + z - t - w\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch16}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull16}{LazardSetSolvingPackageXmpPageEmpty16}
+\pastebutton{LazardSetSolvingPackageXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{p1 := 2*z*u + 2*y*v + 2*t*w - 2*w**2 - w - 1\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p1 }}
+\indentrel{3}\begin{verbatim}
+                                2
+   (16)  2v y + 2u z + 2w t - 2w  - w - 1
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty16}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty16}{LazardSetSolvingPackageXmpPagePatch16}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{p1 := 2*z*u + 2*y*v + 2*t*w - 2*w**2 - w - 1\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch17}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull17}{LazardSetSolvingPackageXmpPageEmpty17}
+\pastebutton{LazardSetSolvingPackageXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{p2 := 3*z*u**2 + 3*y*v**2 - 3*t*w**2 + 3*w**3 + 3*w**2 - t + 4*w\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p2 }}
+\indentrel{3}\begin{verbatim}
+           2      2         2           3     2
+   (17)  3v y + 3u z + (- 3w  - 1)t + 3w  + 3w  + 4w
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty17}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty17}{LazardSetSolvingPackageXmpPagePatch17}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{p2 := 3*z*u**2 + 3*y*v**2 - 3*t*w**2 + 3*w**3 + 3*w**2 - t + 4*w\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch18}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull18}{LazardSetSolvingPackageXmpPageEmpty18}
+\pastebutton{LazardSetSolvingPackageXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{p3 := 6*x*z*v - 6*t*w**2 + 6*w**3 - 3*t*w + 6*w**2 - t + 4*w\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p3 }}
+\indentrel{3}\begin{verbatim}
+                       2                3     2
+   (18)  6v z x + (- 6w  - 3w - 1)t + 6w  + 6w  + 4w
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty18}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty18}{LazardSetSolvingPackageXmpPagePatch18}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{p3 := 6*x*z*v - 6*t*w**2 + 6*w**3 - 3*t*w + 6*w**2 - t + 4*w\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch19}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull19}{LazardSetSolvingPackageXmpPageEmpty19}
+\pastebutton{LazardSetSolvingPackageXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{p4 := 4*z*u**3+ 4*y*v**3+ 4*t*w**3- 4*w**4 - 6*w**3+ 4*t*w- 10*w**2- w- 1\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p4 }}
+\indentrel{3}\begin{verbatim}
+   (19)
+     3      3       3            4     3      2
+   4v y + 4u z + (4w  + 4w)t - 4w  - 6w  - 10w  - w - 1
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty19}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty19}{LazardSetSolvingPackageXmpPagePatch19}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{p4 := 4*z*u**3+ 4*y*v**3+ 4*t*w**3- 4*w**4 - 6*w**3+ 4*t*w- 10*w**2- w- 1\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch20}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull20}{LazardSetSolvingPackageXmpPageEmpty20}
+\pastebutton{LazardSetSolvingPackageXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{p5 := 8*x*z*u*v +8*t*w**3 -8*w**4 +4*t*w**2 -12*w**3 +4*t*w -14*w**2 -3*w -1\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p5 }}
+\indentrel{3}\begin{verbatim}
+   (20)
+                   3     2            4      3      2
+     8u v z x + (8w  + 4w  + 4w)t - 8w  - 12w  - 14w  - 3w
+   + 
+     - 1
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty20}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty20}{LazardSetSolvingPackageXmpPagePatch20}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{p5 := 8*x*z*u*v +8*t*w**3 -8*w**4 +4*t*w**2 -12*w**3 +4*t*w -14*w**2 -3*w -1\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch21}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull21}{LazardSetSolvingPackageXmpPageEmpty21}
+\pastebutton{LazardSetSolvingPackageXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{p6 := 12*x*z*v**2+12*t*w**3 -12*w**4 +12*t*w**2 -18*w**3 +8*t*w -14*w**2 -w -1\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p6 }}
+\indentrel{3}\begin{verbatim}
+   (21)
+        2          3      2             4      3      2
+     12v z x + (12w  + 12w  + 8w)t - 12w  - 18w  - 14w
+   + 
+     - w - 1
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty21}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty21}{LazardSetSolvingPackageXmpPagePatch21}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{p6 := 12*x*z*v**2+12*t*w**3 -12*w**4 +12*t*w**2 -18*w**3 +8*t*w -14*w**2 -w -1\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch22}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull22}{LazardSetSolvingPackageXmpPageEmpty22}
+\pastebutton{LazardSetSolvingPackageXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{p7 := -24*t*w**3 + 24*w**4 - 24*t*w**2 + 36*w**3 - 8*t*w + 26*w**2 + 7*w + 1\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p7 }}
+\indentrel{3}\begin{verbatim}
+   (22)
+         3      2             4      3      2
+   (- 24w  - 24w  - 8w)t + 24w  + 36w  + 26w  + 7w + 1
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty22}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty22}{LazardSetSolvingPackageXmpPagePatch22}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{p7 := -24*t*w**3 + 24*w**4 - 24*t*w**2 + 36*w**3 - 8*t*w + 26*w**2 + 7*w + 1\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{p7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch23}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull23}{LazardSetSolvingPackageXmpPageEmpty23}
+\pastebutton{LazardSetSolvingPackageXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{lp := [p0, p1, p2, p3, p4, p5, p6, p7]\free{p0 }\free{p1 }\free{p2 }\free{p3 }\free{p4 }\free{p5 }\free{p6 }\free{p7 }\bound{lp }}
+\indentrel{3}\begin{verbatim}
+   (23)
+                                               2
+   [b1 + y + z - t - w, 2v y + 2u z + 2w t - 2w  - w - 1,
+      2      2         2           3     2
+    3v y + 3u z + (- 3w  - 1)t + 3w  + 3w  + 4w,
+                  2                3     2
+    6v z x + (- 6w  - 3w - 1)t + 6w  + 6w  + 4w,
+      3      3       3            4     3      2
+    4v y + 4u z + (4w  + 4w)t - 4w  - 6w  - 10w  - w - 1,
+
+                     3     2            4      3      2
+       8u v z x + (8w  + 4w  + 4w)t - 8w  - 12w  - 14w
+     + 
+       - 3w - 1
+     ,
+
+          2          3      2             4      3      2
+       12v z x + (12w  + 12w  + 8w)t - 12w  - 18w  - 14w
+     + 
+       - w - 1
+     ,
+          3      2             4      3      2
+    (- 24w  - 24w  - 8w)t + 24w  + 36w  + 26w  + 7w + 1]
+Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty23}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty23}{LazardSetSolvingPackageXmpPagePatch23}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{lp := [p0, p1, p2, p3, p4, p5, p6, p7]\free{p0 }\free{p1 }\free{p2 }\free{p3 }\free{p4 }\free{p5 }\free{p6 }\free{p7 }\bound{lp }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch24}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull24}{LazardSetSolvingPackageXmpPageEmpty24}
+\pastebutton{LazardSetSolvingPackageXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{lts := zeroSetSplit(lp,false)$T\free{lp }\free{T }\bound{lts }}
+\indentrel{3}\begin{verbatim}
+   (24)
+   [{w + 1,u,v,t + 1,b1 + y + z + 2},
+    {w + 1,v,t + 1,z,b1 + y + 2},
+    {w + 1,t + 1,z,y,b1 + 2},
+    {w + 1,v - u,t + 1,y + z,x,b1 + 2},
+    {w + 1,u,t + 1,y,x,b1 + z + 2},
+
+          5       4      3     2
+     {144w  + 216w  + 96w  + 6w  - 11w - 1,
+          2                 5       4      3     2
+      (12w  + 9w + 1)u - 72w  - 108w  - 42w  - 9w  - 3w,
+          2                 4      3      2
+      (12w  + 9w + 1)v + 36w  + 54w  + 18w ,
+          3      2             4      3      2
+      (24w  + 24w  + 8w)t - 24w  - 36w  - 26w  - 7w - 1,
+
+                     2                 2
+         (12u v - 12u )z + (12w v + 12w  + 4)t + (3w - 5)v
+       + 
+            4      3     2
+         36w  + 42w  + 6w  - 16w
+       ,
+                             2
+      2v y + 2u z + 2w t - 2w  - w - 1,
+                    2                3     2
+      6v z x + (- 6w  - 3w - 1)t + 6w  + 6w  + 4w,
+      b1 + y + z - t - w}
+     ]
+Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty24}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty24}{LazardSetSolvingPackageXmpPagePatch24}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{lts := zeroSetSplit(lp,false)$T\free{lp }\free{T }\bound{lts }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch25}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull25}{LazardSetSolvingPackageXmpPageEmpty25}
+\pastebutton{LazardSetSolvingPackageXmpPageFull25}{\hidepaste}
+\tab{5}\spadcommand{[coHeight(ts) for ts in lts]\free{lts }}
+\indentrel{3}\begin{verbatim}
+   (25)  [3,3,3,2,2,0]
+                          Type: List NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty25}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty25}{LazardSetSolvingPackageXmpPagePatch25}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{[coHeight(ts) for ts in lts]\free{lts }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch26}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull26}{LazardSetSolvingPackageXmpPageEmpty26}
+\pastebutton{LazardSetSolvingPackageXmpPageFull26}{\hidepaste}
+\tab{5}\spadcommand{ST := SREGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{ST }}
+\indentrel{3}\begin{verbatim}
+   (26)
+  SquareFreeRegularTriangularSet(Integer,IndexedExponents
+   OrderedVariableList [b1,x,y,z,t,v,u,w],OrderedVariable
+  List [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomial
+  (Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]))
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty26}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty26}{LazardSetSolvingPackageXmpPagePatch26}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{ST := SREGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{ST }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch27}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull27}{LazardSetSolvingPackageXmpPageEmpty27}
+\pastebutton{LazardSetSolvingPackageXmpPageFull27}{\hidepaste}
+\tab{5}\spadcommand{pack := LAZM3PK(R,E,V,P,T,ST)\free{R }\free{E }\free{V }\free{P }\free{T }\free{ST }\bound{pack }}
+\indentrel{3}\begin{verbatim}
+   (27)
+  LazardSetSolvingPackage(Integer,IndexedExponents Ordere
+  dVariableList [b1,x,y,z,t,v,u,w],OrderedVariableList [b
+  1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomial(Intege
+  r,OrderedVariableList [b1,x,y,z,t,v,u,w]),RegularTriang
+  ularSet(Integer,IndexedExponents OrderedVariableList [b
+  1,x,y,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w]
+  ,NewSparseMultivariatePolynomial(Integer,OrderedVariabl
+  eList [b1,x,y,z,t,v,u,w])),SquareFreeRegularTriangularS
+  et(Integer,IndexedExponents OrderedVariableList [b1,x,y
+  ,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewS
+  parseMultivariatePolynomial(Integer,OrderedVariableList
+   [b1,x,y,z,t,v,u,w])))
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty27}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty27}{LazardSetSolvingPackageXmpPagePatch27}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{pack := LAZM3PK(R,E,V,P,T,ST)\free{R }\free{E }\free{V }\free{P }\free{T }\free{ST }\bound{pack }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch28}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull28}{LazardSetSolvingPackageXmpPageEmpty28}
+\pastebutton{LazardSetSolvingPackageXmpPageFull28}{\hidepaste}
+\tab{5}\spadcommand{zeroSetSplit(lp,false)$pack\free{lp }\free{pack }}
+\indentrel{3}\begin{verbatim}
+   (28)
+   [{w + 1,t + 1,z,y,b1 + 2},
+    {w + 1,v,t + 1,z,b1 + y + 2},
+    {w + 1,u,v,t + 1,b1 + y + z + 2},
+    {w + 1,v - u,t + 1,y + z,x,b1 + 2},
+    {w + 1,u,t + 1,y,x,b1 + z + 2},
+
+          5       4      3     2
+     {144w  + 216w  + 96w  + 6w  - 11w - 1,
+             4      3      2
+      u - 24w  - 36w  - 14w  + w + 1,
+              4      3      2
+      3v - 48w  - 60w  - 10w  + 8w + 2,
+             4      3      2
+      t - 24w  - 36w  - 14w  - w + 1,
+                  4        3        2
+      486z - 2772w  - 4662w  - 2055w  + 30w + 127,
+                    4         3        2
+      2916y - 22752w  - 30312w  - 8220w  + 2064w + 1561,
+                  4        3       2
+      356x - 3696w  - 4536w  - 968w  + 822w + 371,
+                     4         3         2
+      2916b1 - 30600w  - 46692w  - 20274w  - 8076w + 593}
+     ]
+Type: List SquareFreeRegularTriangularSet(Integer,IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty28}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty28}{LazardSetSolvingPackageXmpPagePatch28}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{zeroSetSplit(lp,false)$pack\free{lp }\free{pack }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch29}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull29}{LazardSetSolvingPackageXmpPageEmpty29}
+\pastebutton{LazardSetSolvingPackageXmpPageFull29}{\hidepaste}
+\tab{5}\spadcommand{f0 := (w - v) ** 2 + (u - t) ** 2 - 1\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{f0 }}
+\indentrel{3}\begin{verbatim}
+          2           2           2    2
+   (29)  t  - 2u t + v  - 2w v + u  + w  - 1
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty29}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty29}{LazardSetSolvingPackageXmpPagePatch29}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{f0 := (w - v) ** 2 + (u - t) ** 2 - 1\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{f0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch30}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull30}{LazardSetSolvingPackageXmpPageEmpty30}
+\pastebutton{LazardSetSolvingPackageXmpPageFull30}{\hidepaste}
+\tab{5}\spadcommand{f1 := t ** 2 - v ** 3\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+          2    3
+   (30)  t  - v
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty30}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty30}{LazardSetSolvingPackageXmpPagePatch30}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{f1 := t ** 2 - v ** 3\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch31}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull31}{LazardSetSolvingPackageXmpPageEmpty31}
+\pastebutton{LazardSetSolvingPackageXmpPageFull31}{\hidepaste}
+\tab{5}\spadcommand{f2 := 2 * t * (w - v) + 3 * v ** 2 * (u - t)\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{f2 }}
+\indentrel{3}\begin{verbatim}
+              2                   2
+   (31)  (- 3v  - 2v + 2w)t + 3u v
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty31}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty31}{LazardSetSolvingPackageXmpPagePatch31}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{f2 := 2 * t * (w - v) + 3 * v ** 2 * (u - t)\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch32}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull32}{LazardSetSolvingPackageXmpPageEmpty32}
+\pastebutton{LazardSetSolvingPackageXmpPageFull32}{\hidepaste}
+\tab{5}\spadcommand{f3 := (3 * z * v ** 2 - 1) * (2 * z * t - 1)\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{f3 }}
+\indentrel{3}\begin{verbatim}
+           2   2             2
+   (32)  6v t z  + (- 2t - 3v )z + 1
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty32}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty32}{LazardSetSolvingPackageXmpPagePatch32}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{f3 := (3 * z * v ** 2 - 1) * (2 * z * t - 1)\free{b1 }\free{x }\free{y }\free{z }\free{t }\free{u }\free{v }\free{w }\bound{f3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch33}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull33}{LazardSetSolvingPackageXmpPageEmpty33}
+\pastebutton{LazardSetSolvingPackageXmpPageFull33}{\hidepaste}
+\tab{5}\spadcommand{lf := [f0, f1, f2, f3]\free{f0 }\free{f1 }\free{f2 }\free{f3 }\bound{lf }}
+\indentrel{3}\begin{verbatim}
+   (33)
+     2           2           2    2       2    3
+   [t  - 2u t + v  - 2w v + u  + w  - 1, t  - v ,
+         2                   2
+    (- 3v  - 2v + 2w)t + 3u v ,
+      2   2             2
+    6v t z  + (- 2t - 3v )z + 1]
+Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty33}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty33}{LazardSetSolvingPackageXmpPagePatch33}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{lf := [f0, f1, f2, f3]\free{f0 }\free{f1 }\free{f2 }\free{f3 }\bound{lf }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch34}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull34}{LazardSetSolvingPackageXmpPageEmpty34}
+\pastebutton{LazardSetSolvingPackageXmpPageFull34}{\hidepaste}
+\tab{5}\spadcommand{zeroSetSplit(lf,true)$T\free{lf }\free{T }}
+\indentrel{3}\begin{verbatim}
+   (34)
+   [
+     {
+             6           3       2                 4
+         729u  + (- 1458w  + 729w  - 4158w - 1685)u
+       + 
+                 6        5        4        3       2
+             729w  - 1458w  - 2619w  - 4892w  - 297w
+           + 
+             5814w + 427
+        *
+            2
+           u
+       + 
+             8       7        6        5        4        3
+         729w  + 216w  - 2900w  - 2376w  + 3870w  + 4072w
+       + 
+                2
+         - 1188w  - 1656w + 529
+       ,
+
+                  4           3       2                  2
+             2187u  + (- 4374w  - 972w  - 12474w - 2868)u
+           + 
+                  6        5         4        3        2
+             2187w  - 1944w  - 10125w  - 4800w  + 2501w
+           + 
+             4968w - 1587
+        *
+           v
+       + 
+               3       2  2       6        5        4
+         (1944w  - 108w )u  + 972w  + 3024w  - 1080w
+       + 
+             3        2
+         496w  + 1116w
+       ,
+         2                   2
+      (3v  + 2v - 2w)t - 3u v ,
+                        2  2           2
+      ((4v - 4w)t - 6u v )z  + (2t + 3v )z - 1}
+     ]
+Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty34}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty34}{LazardSetSolvingPackageXmpPagePatch34}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty34}{\showpaste}
+\tab{5}\spadcommand{zeroSetSplit(lf,true)$T\free{lf }\free{T }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch35}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull35}{LazardSetSolvingPackageXmpPageEmpty35}
+\pastebutton{LazardSetSolvingPackageXmpPageFull35}{\hidepaste}
+\tab{5}\spadcommand{zeroSetSplit(lf,false)$T\free{lf }\free{T }}
+\indentrel{3}\begin{verbatim}
+   (35)
+   [
+     {
+             6           3       2                 4
+         729u  + (- 1458w  + 729w  - 4158w - 1685)u
+       + 
+                 6        5        4        3       2
+             729w  - 1458w  - 2619w  - 4892w  - 297w
+           + 
+             5814w + 427
+        *
+            2
+           u
+       + 
+             8       7        6        5        4        3
+         729w  + 216w  - 2900w  - 2376w  + 3870w  + 4072w
+       + 
+                2
+         - 1188w  - 1656w + 529
+       ,
+
+                  4           3       2                  2
+             2187u  + (- 4374w  - 972w  - 12474w - 2868)u
+           + 
+                  6        5         4        3        2
+             2187w  - 1944w  - 10125w  - 4800w  + 2501w
+           + 
+             4968w - 1587
+        *
+           v
+       + 
+               3       2  2       6        5        4
+         (1944w  - 108w )u  + 972w  + 3024w  - 1080w
+       + 
+             3        2
+         496w  + 1116w
+       ,
+         2                   2
+      (3v  + 2v - 2w)t - 3u v ,
+                        2  2           2
+      ((4v - 4w)t - 6u v )z  + (2t + 3v )z - 1}
+     ,
+
+         4     3      2
+     {27w  + 4w  - 54w  - 36w + 23, u,
+                     2             2          2
+      (12w + 2)v - 9w  - 2w + 9, 6t  - 2v - 3w  + 2w + 3,
+      2t z - 1}
+     ,
+
+     {
+               6         5         4          3         2
+         59049w  + 91854w  - 45198w  + 145152w  + 63549w
+       + 
+         60922w + 21420
+       ,
+
+                            5                  4
+             31484448266904w  - 18316865522574w
+           + 
+                            3                 2
+             23676995746098w  + 6657857188965w
+           + 
+             8904703998546w + 3890631403260
+        *
+            2
+           u
+       + 
+                        5                  4
+         94262810316408w  - 82887296576616w
+       + 
+                        3                  2
+         89801831438784w  + 28141734167208w
+       + 
+         38070359425432w + 16003865949120
+       ,
+
+              2             2
+         (243w  + 36w + 85)v
+       + 
+             2       3      2                    3     2
+       (- 81u  - 162w  + 36w  + 154w + 72)v - 72w  + 4w
+       ,
+         2                   2
+      (3v  + 2v - 2w)t - 3u v ,
+                        2  2           2
+      ((4v - 4w)t - 6u v )z  + (2t + 3v )z - 1}
+     ,
+
+         4     3      2
+     {27w  + 4w  - 54w  - 36w + 23, u,
+                     2             2          2
+      (12w + 2)v - 9w  - 2w + 9, 6t  - 2v - 3w  + 2w + 3,
+        2
+      3v z - 1}
+     ]
+Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty35}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty35}{LazardSetSolvingPackageXmpPagePatch35}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty35}{\showpaste}
+\tab{5}\spadcommand{zeroSetSplit(lf,false)$T\free{lf }\free{T }}
+\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPagePatch36}
+\begin{paste}{LazardSetSolvingPackageXmpPageFull36}{LazardSetSolvingPackageXmpPageEmpty36}
+\pastebutton{LazardSetSolvingPackageXmpPageFull36}{\hidepaste}
+\tab{5}\spadcommand{zeroSetSplit(lf,false)$pack\free{lf }\free{pack }}
+\indentrel{3}\begin{verbatim}
+   (36)
+   [
+     {
+             6           3       2                 4
+         729u  + (- 1458w  + 729w  - 4158w - 1685)u
+       + 
+                 6        5        4        3       2
+             729w  - 1458w  - 2619w  - 4892w  - 297w
+           + 
+             5814w + 427
+        *
+            2
+           u
+       + 
+             8       7        6        5        4        3
+         729w  + 216w  - 2900w  - 2376w  + 3870w  + 4072w
+       + 
+                2
+         - 1188w  - 1656w + 529
+       ,
+
+                  4           3       2                  2
+             2187u  + (- 4374w  - 972w  - 12474w - 2868)u
+           + 
+                  6        5         4        3        2
+             2187w  - 1944w  - 10125w  - 4800w  + 2501w
+           + 
+             4968w - 1587
+        *
+           v
+       + 
+               3       2  2       6        5        4
+         (1944w  - 108w )u  + 972w  + 3024w  - 1080w
+       + 
+             3        2
+         496w  + 1116w
+       ,
+         2                   2
+      (3v  + 2v - 2w)t - 3u v ,
+                        2  2           2
+      ((4v - 4w)t - 6u v )z  + (2t + 3v )z - 1}
+     ,
+
+         2                 2
+     {81w  + 18w + 28, 729u  - 1890w - 533,
+         2
+      81v  + (- 162w + 27)v - 72w - 112,
+      11881t + (972w + 2997)u v + (- 11448w - 11536)u,
+
+                         2
+         641237934604288z
+       + 
+                 (78614584763904w + 26785578742272)u
+               + 
+                 236143618655616w + 70221988585728
+            *
+               v
+           + 
+             (358520253138432w + 101922133759488)u
+           + 
+             142598803536000w + 54166419595008
+        *
+           z
+       + 
+         (32655103844499w - 44224572465882)u v
+       + 
+         (43213900115457w - 32432039102070)u
+       }
+     ,
+
+         4     3      2
+     {27w  + 4w  - 54w  - 36w + 23, u,
+                 3     2
+      218v - 162w  + 3w  + 160w + 153,
+          2      3      2
+      109t  - 27w  - 54w  + 63w + 80,
+                      3      2
+      1744z + (- 1458w  + 27w  + 1440w + 505)t}
+     ,
+
+         4     3      2
+     {27w  + 4w  - 54w  - 36w + 23, u,
+                 3     2
+      218v - 162w  + 3w  + 160w + 153,
+          2      3      2
+      109t  - 27w  - 54w  + 63w + 80,
+                  3     2
+      1308z + 162w  - 3w  - 814w - 153}
+     ,
+
+          4       3        2
+     {729w  + 972w  - 1026w  + 1684w + 765,
+         2      2
+      81u  + 72w  + 16w - 72,
+                 3       2
+      702v - 162w  - 225w  + 40w - 99,
+                    3       2
+      11336t + (324w  - 603w  - 1718w - 1557)u,
+
+                   2
+         595003968z
+       + 
+                             3             2
+                 - 963325386w  - 898607682w  + 1516286466w
+               + 
+                 - 3239166186
+            *
+               u
+           + 
+                          3              2
+             - 1579048992w  - 1796454288w  + 2428328160w
+           + 
+             - 4368495024
+        *
+           z
+       + 
+                        3              2
+             9713133306w  + 9678670317w  - 16726834476w
+           + 
+             28144233593
+        *
+           u
+       }
+     ]
+Type: List SquareFreeRegularTriangularSet(Integer,IndexedExponents OrderedVariableList [b1,x,y,z,t,v,u,w],OrderedVariableList [b1,x,y,z,t,v,u,w],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [b1,x,y,z,t,v,u,w]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LazardSetSolvingPackageXmpPageEmpty36}
+\begin{paste}{LazardSetSolvingPackageXmpPageEmpty36}{LazardSetSolvingPackageXmpPagePatch36}
+\pastebutton{LazardSetSolvingPackageXmpPageEmpty36}{\showpaste}
+\tab{5}\spadcommand{zeroSetSplit(lf,false)$pack\free{lf }\free{pack }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/LEXP.ht b/src/hyper/pages/LEXP.ht
new file mode 100644
index 00000000..317fc724
--- /dev/null
+++ b/src/hyper/pages/LEXP.ht
@@ -0,0 +1,70 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\LieExponentialsXmpTitle}{LieExponentials}
+\newcommand{\LieExponentialsXmpNumber}{9.42}
+%
+% =====================================================================
+\begin{page}{LieExponentialsXmpPage}{9.42 LieExponentials}
+% =====================================================================
+\beginscroll
+\xtc{
+}{
+\spadpaste{ a: Symbol := 'a \bound{a}}
+}
+\xtc{
+}{
+\spadpaste{ b: Symbol := 'b \bound{b}}
+}
+
+Declarations of domains
+\xtc{
+}{
+\spadpaste{ coef     := Fraction(Integer) \bound{coef}}
+}
+\xtc{
+}{
+\spadpaste{ group    := LieExponentials(Symbol, coef, 3) \free{coef} \bound{group}}
+}
+\xtc{
+}{
+\spadpaste{ lpoly    := LiePolynomial(Symbol, coef) \free{coef} \bound{lpoly}}
+}
+\xtc{
+}{
+\spadpaste{ poly     := XPBWPolynomial(Symbol, coef) \free{coef} \bound{poly}}
+}
+
+Calculations
+\xtc{
+}{
+\spadpaste{ ea := exp(a::lpoly)$group \free{a} \free{lpoly} \free{group} \bound{ea}}
+}
+\xtc{
+}{
+\spadpaste{ eb := exp(b::lpoly)$group \free{b} \free{lpoly} \free{group}  \bound{eb}}
+}
+\xtc{
+}{
+\spadpaste{ g: group := ea*eb \free{ea} \free{eb} \bound{g}}
+}
+\xtc{
+}{
+\spadpaste{ g :: poly \free{g} \free{poly}}
+}
+\xtc{
+}{
+\spadpaste{ log(g)$group \free{g} \free{group}}
+}
+\xtc{
+}{
+\spadpaste{ g1: group := inv(g) \free{g} \free{group} \bound{g1}}
+}
+\xtc{
+}{
+\spadpaste{ g*g1 \free{g} \free{g1}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/LEXP.pht b/src/hyper/pages/LEXP.pht
new file mode 100644
index 00000000..0e873943
--- /dev/null
+++ b/src/hyper/pages/LEXP.pht
@@ -0,0 +1,227 @@
+\begin{patch}{LieExponentialsXmpPagePatch1}
+\begin{paste}{LieExponentialsXmpPageFull1}{LieExponentialsXmpPageEmpty1}
+\pastebutton{LieExponentialsXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{ a: Symbol := 'a\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (1)  a
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPageEmpty1}
+\begin{paste}{LieExponentialsXmpPageEmpty1}{LieExponentialsXmpPagePatch1}
+\pastebutton{LieExponentialsXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{ a: Symbol := 'a\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPagePatch2}
+\begin{paste}{LieExponentialsXmpPageFull2}{LieExponentialsXmpPageEmpty2}
+\pastebutton{LieExponentialsXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{ b: Symbol := 'b\bound{b }}
+\indentrel{3}\begin{verbatim}
+   (2)  b
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPageEmpty2}
+\begin{paste}{LieExponentialsXmpPageEmpty2}{LieExponentialsXmpPagePatch2}
+\pastebutton{LieExponentialsXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{ b: Symbol := 'b\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPagePatch3}
+\begin{paste}{LieExponentialsXmpPageFull3}{LieExponentialsXmpPageEmpty3}
+\pastebutton{LieExponentialsXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{ coef := Fraction(Integer)\bound{coef }}
+\indentrel{3}\begin{verbatim}
+   (3)  Fraction Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPageEmpty3}
+\begin{paste}{LieExponentialsXmpPageEmpty3}{LieExponentialsXmpPagePatch3}
+\pastebutton{LieExponentialsXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{ coef := Fraction(Integer)\bound{coef }}
+\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPagePatch4}
+\begin{paste}{LieExponentialsXmpPageFull4}{LieExponentialsXmpPageEmpty4}
+\pastebutton{LieExponentialsXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{ group := LieExponentials(Symbol, coef, 3)\free{coef }\bound{group }}
+\indentrel{3}\begin{verbatim}
+   (4)  LieExponentials(Symbol,Fraction Integer,3)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPageEmpty4}
+\begin{paste}{LieExponentialsXmpPageEmpty4}{LieExponentialsXmpPagePatch4}
+\pastebutton{LieExponentialsXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{ group := LieExponentials(Symbol, coef, 3)\free{coef }\bound{group }}
+\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPagePatch5}
+\begin{paste}{LieExponentialsXmpPageFull5}{LieExponentialsXmpPageEmpty5}
+\pastebutton{LieExponentialsXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{ lpoly := LiePolynomial(Symbol, coef)\free{coef }\bound{lpoly }}
+\indentrel{3}\begin{verbatim}
+   (5)  LiePolynomial(Symbol,Fraction Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPageEmpty5}
+\begin{paste}{LieExponentialsXmpPageEmpty5}{LieExponentialsXmpPagePatch5}
+\pastebutton{LieExponentialsXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{ lpoly := LiePolynomial(Symbol, coef)\free{coef }\bound{lpoly }}
+\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPagePatch6}
+\begin{paste}{LieExponentialsXmpPageFull6}{LieExponentialsXmpPageEmpty6}
+\pastebutton{LieExponentialsXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{ poly := XPBWPolynomial(Symbol, coef)\free{coef }\bound{poly }}
+\indentrel{3}\begin{verbatim}
+   (6)  XPBWPolynomial(Symbol,Fraction Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPageEmpty6}
+\begin{paste}{LieExponentialsXmpPageEmpty6}{LieExponentialsXmpPagePatch6}
+\pastebutton{LieExponentialsXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{ poly := XPBWPolynomial(Symbol, coef)\free{coef }\bound{poly }}
+\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPagePatch7}
+\begin{paste}{LieExponentialsXmpPageFull7}{LieExponentialsXmpPageEmpty7}
+\pastebutton{LieExponentialsXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{ ea := exp(a::lpoly)$group\free{a }\free{lpoly }\free{group }\bound{ea }}
+\indentrel{3}\begin{verbatim}
+         [a]
+   (7)  e
+       Type: LieExponentials(Symbol,Fraction Integer,3)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPageEmpty7}
+\begin{paste}{LieExponentialsXmpPageEmpty7}{LieExponentialsXmpPagePatch7}
+\pastebutton{LieExponentialsXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{ ea := exp(a::lpoly)$group\free{a }\free{lpoly }\free{group }\bound{ea }}
+\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPagePatch8}
+\begin{paste}{LieExponentialsXmpPageFull8}{LieExponentialsXmpPageEmpty8}
+\pastebutton{LieExponentialsXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{ eb := exp(b::lpoly)$group\free{b }\free{lpoly }\free{group }\bound{eb }}
+\indentrel{3}\begin{verbatim}
+         [b]
+   (8)  e
+       Type: LieExponentials(Symbol,Fraction Integer,3)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPageEmpty8}
+\begin{paste}{LieExponentialsXmpPageEmpty8}{LieExponentialsXmpPagePatch8}
+\pastebutton{LieExponentialsXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{ eb := exp(b::lpoly)$group\free{b }\free{lpoly }\free{group }\bound{eb }}
+\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPagePatch9}
+\begin{paste}{LieExponentialsXmpPageFull9}{LieExponentialsXmpPageEmpty9}
+\pastebutton{LieExponentialsXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{ g: group := ea*eb\free{ea }\free{eb }\bound{g }}
+\indentrel{3}\begin{verbatim}
+             1     2        1   2
+             Ä [a b ]       Ä [a b]
+         [b] 2        [a b] 2       [a]
+   (9)  e   e        e     e       e
+       Type: LieExponentials(Symbol,Fraction Integer,3)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPageEmpty9}
+\begin{paste}{LieExponentialsXmpPageEmpty9}{LieExponentialsXmpPagePatch9}
+\pastebutton{LieExponentialsXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{ g: group := ea*eb\free{ea }\free{eb }\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPagePatch10}
+\begin{paste}{LieExponentialsXmpPageFull10}{LieExponentialsXmpPageEmpty10}
+\pastebutton{LieExponentialsXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{ g :: poly\free{g }\free{poly }}
+\indentrel{3}\begin{verbatim}
+   (10)
+                     1                           1
+     1 + [a] + [b] + Ä [a][a] + [a b] + [b][a] + Ä [b][b]
+                     2                           2
+   + 
+     1             1   2                1     2
+     Ä [a][a][a] + Ä [a b] + [a b][a] + Ä [a b ]
+     6             2                    2
+   + 
+     1                        1             1
+     Ä [b][a][a] + [b][a b] + Ä [b][b][a] + Ä [b][b][b]
+     2                        2             6
+          Type: XPBWPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPageEmpty10}
+\begin{paste}{LieExponentialsXmpPageEmpty10}{LieExponentialsXmpPagePatch10}
+\pastebutton{LieExponentialsXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{ g :: poly\free{g }\free{poly }}
+\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPagePatch11}
+\begin{paste}{LieExponentialsXmpPageFull11}{LieExponentialsXmpPageEmpty11}
+\pastebutton{LieExponentialsXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{ log(g)$group\free{g }\free{group }}
+\indentrel{3}\begin{verbatim}
+                     1          1   2      1     2
+   (11)  [a] + [b] + Ä [a b] + ÄÄ [a b] + ÄÄ [a b ]
+                     2         12         12
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPageEmpty11}
+\begin{paste}{LieExponentialsXmpPageEmpty11}{LieExponentialsXmpPagePatch11}
+\pastebutton{LieExponentialsXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{ log(g)$group\free{g }\free{group }}
+\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPagePatch12}
+\begin{paste}{LieExponentialsXmpPageFull12}{LieExponentialsXmpPageEmpty12}
+\pastebutton{LieExponentialsXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{ g1: group := inv(g)\free{g }\free{group }\bound{g1 }}
+\indentrel{3}\begin{verbatim}
+          - [b] - [a]
+   (12)  e     e
+       Type: LieExponentials(Symbol,Fraction Integer,3)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPageEmpty12}
+\begin{paste}{LieExponentialsXmpPageEmpty12}{LieExponentialsXmpPagePatch12}
+\pastebutton{LieExponentialsXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{ g1: group := inv(g)\free{g }\free{group }\bound{g1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPagePatch13}
+\begin{paste}{LieExponentialsXmpPageFull13}{LieExponentialsXmpPageEmpty13}
+\pastebutton{LieExponentialsXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{ g*g1\free{g }\free{g1 }}
+\indentrel{3}\begin{verbatim}
+   (13)  1
+       Type: LieExponentials(Symbol,Fraction Integer,3)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LieExponentialsXmpPageEmpty13}
+\begin{paste}{LieExponentialsXmpPageEmpty13}{LieExponentialsXmpPagePatch13}
+\pastebutton{LieExponentialsXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{ g*g1\free{g }\free{g1 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/LEXTRIPK.ht b/src/hyper/pages/LEXTRIPK.ht
new file mode 100644
index 00000000..f3ecda2c
--- /dev/null
+++ b/src/hyper/pages/LEXTRIPK.ht
@@ -0,0 +1,251 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\LexTriangularPackageXmpTitle}{LexTriangularPackage}
+\newcommand{\LexTriangularPackageXmpNumber}{9.39}
+%
+% =====================================================================
+\begin{page}{LexTriangularPackageXmpPage}{9.39 LexTriangularPackage}
+% =====================================================================
+\beginscroll
+The \spadtype{LexTriangularPackage} package constructor provides an implementation
+of the {\em lexTriangular} algorithm 
+(D. Lazard "Solving Zero-dimensional Algebraic Systems", J. of Symbol. Comput., 1992).
+This algorithm decomposes a zero-dimensional variety into zero-sets of regular
+triangular sets.
+Thus the input system must have a finite number of complex solutions.
+Moreover, this system needs to be a lexicographical Groebner basis.
+
+This package takes two arguments: the coefficient-ring {\bf R} of the polynomials,
+which must be a \spadtype{GcdDomain}
+and their set of variables given by {\bf ls} a \spadtype{List Symbol}.
+The type of the input polynomials must be \spadtype{NewSparseMultivariatePolynomial(R,V)}
+where {\bf V} is \spadtype{OrderedVariableList(ls)}.
+The abbreviation for \spadtype{LexTriangularPackage} is \spadtype{LEXTRIPK}.
+The main operations are \axiomOpFrom{lexTriangular}{LexTriangularPackage}
+and \axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage}.
+The later provide decompositions by means of square-free regular triangular sets,
+built with the \spadtype{SREGSET} constructor,
+whereas the former uses the \spadtype{REGSET} constructor.
+Note that these constructors also implement another algorithm
+for solving algebraic systems by means of regular triangular sets;
+in that case no computations of Groebner bases are needed and the input
+system may have any dimension (i.e. it may have an infinite number
+of solutions).
+
+The implementation of the {\em lexTriangular} algorithm
+provided in the \spadtype{LexTriangularPackage} constructor
+differs from that reported in "Computations of gcd over
+algebraic towers of simple extensions" by M. Moreno Maza and R. Rioboo
+(in proceedings of AAECC11, Paris, 1995).
+Indeed, the \axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage} operation
+removes all multiplicities of the solutions (i.e. the computed solutions
+are pairwise different) and the \axiomOpFrom{lexTriangular}{LexTriangularPackage}
+operation may keep some multiplicities; this later operation runs generally faster
+than the former. 
+
+The interest of the {\em lexTriangular} algorithm is due
+to the following experimental remark.
+For some examples, a triangular decomposition
+of a zero-dimensional variety can be computed faster 
+via a lexicographical Groebner basis computation than
+by using a direct method (like that of \spadtype{SREGSET}
+and \spadtype{REGSET}).
+This happens typically when the total degree of the system
+relies essentially on its smallest variable (like in the
+{\em Katsura} systems).
+When this is not the case, the direct method may give better timings
+(like in the {\em Rose} system).
+
+Of course, the direct method can also be applied to a
+lexicographical Groebner basis.
+However, the {\em lexTriangular} algorithm takes advantage
+of the structure of this basis and avoids many unnecessary
+computations which are performed by the direct method.
+
+For this purpose of solving algebraic systems with a finite number of solutions,
+see also the \spadtype{ZeroDimensionalSolvePackage}.
+It allows to use both strategies (the lexTriangular algorithm and the direct
+method) for computing either the complex or real roots of a system.
+
+Note that the way of understanding triangular decompositions 
+is detailed in the example of the \spadtype{RegularTriangularSet}
+constructor.
+
+Since the \spadtype{LEXTRIPK} package constructor is limited 
+to zero-dimensional systems, it provides a
+\axiomOpFrom{zeroDimensional?}{LexTriangularPackage}
+operation to check whether this requirement holds.
+There is also a \axiomOpFrom{groebner}{LexTriangularPackage}
+operation to compute the lexicographical Groebner basis
+of a set of polynomials with type \spadtype{NewSparseMultivariatePolynomial(R,V)}.
+The elimination ordering is that given by {\bf ls}
+(the greatest variable being the first element of {\bf ls}).
+This basis is computed by the {\em FLGM} algorithm 
+(Faugere et al. "Efficient Computation of Zero-Dimensional Groebner Bases by 
+        Change of Ordering" , J. of Symbol. Comput., 1993)
+implemented in the \spadtype{LinGroebnerPackage} package constructor.
+Once a lexicographical Groebner basis is computed,
+then one can call the operations \axiomOpFrom{lexTriangular}{LexTriangularPackage}
+and \axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage}.
+Note that these operations admit an optional argument
+to produce normalized triangular sets.
+There is also a \axiomOpFrom{zeroSetSplit}{LexTriangularPackage} operation
+which does all the job from the input system;
+an error is produced if this system is not zero-dimensional.
+
+Let us illustrate the facilities of the \spadtype{LEXTRIPK} constructor
+by a famous example, the {\em cyclic-6 root} system.
+\xtc{
+Define the coefficient ring.
+}{
+\spadpaste{R := Integer \bound{R}}
+}
+\xtc{
+Define the list of variables,
+}{
+\spadpaste{ls : List Symbol := [a,b,c,d,e,f] \bound{ls}}
+}
+\xtc{
+and make it an ordered set.
+}{
+\spadpaste{V := OVAR(ls) \free{ls} \bound{V}}
+}
+\xtc{
+Define the polynomial ring.
+}{
+\spadpaste{P := NSMP(R, V) \free{R} \free{V} \bound{P}}
+}
+\xtc{
+Define the polynomials.
+}{
+\spadpaste{p1: P :=  a*b*c*d*e*f - 1 \free{P} \bound{p1}}
+}
+\xtc{
+}{
+\spadpaste{p2: P := a*b*c*d*e +a*b*c*d*f +a*b*c*e*f +a*b*d*e*f +a*c*d*e*f +b*c*d*e*f  \free{P} \bound{p2}}
+}
+\xtc{
+}{
+\spadpaste{p3: P :=  a*b*c*d + a*b*c*f + a*b*e*f + a*d*e*f + b*c*d*e + c*d*e*f \free{P} \bound{p3}}
+}
+\xtc{
+}{
+\spadpaste{p4: P := a*b*c + a*b*f + a*e*f + b*c*d + c*d*e + d*e*f  \free{P} \bound{p4}}
+}
+\xtc{
+}{
+\spadpaste{p5: P := a*b + a*f + b*c + c*d + d*e + e*f \free{P} \bound{p5}}
+}
+\xtc{
+}{
+\spadpaste{p6: P := a + b + c + d + e + f  \free{P} \bound{p6}}
+}
+\xtc{
+}{
+\spadpaste{lp := [p1, p2, p3, p4, p5, p6] \free{p1} \free{p2} \free{p3} \free{p4} \free{p5} \free{p6} \bound{lp}}
+}
+\xtc{
+Now call \spadtype{LEXTRIPK} .
+}{
+\spadpaste{lextripack :=  LEXTRIPK(R,ls) \free{R} \free{ls} \bound{lextripack}}
+}
+\xtc{
+Compute the lexicographical Groebner basis of the system.
+This may take between 5 minutes and one hour, depending on your machine.
+}{
+\spadpaste{lg := groebner(lp)$lextripack \free{lp} \free{lextripack} \bound{lg}}
+}
+\xtc{
+Apply lexTriangular to compute a decomposition into regular triangular sets.
+This should not take more than 5 seconds.
+}{
+\spadpaste{lexTriangular(lg,false)$lextripack \free{lg} \free{lextripack}}
+}
+Note that the first set of the decomposition is normalized
+(all initials are integer numbers) but not the second one
+(normalized triangular sets are defined in the
+description of the \spadtype{NormalizedTriangularSetCategory} constructor).
+\xtc{
+So apply now lexTriangular to produce normalized triangular sets.
+}{
+\spadpaste{lts := lexTriangular(lg,true)$lextripack \free{lg} \free{lextripack} \bound{lts}}
+}
+\xtc{
+We check that all initials are constant.
+}{
+\spadpaste{[[init(p) for p in (ts :: List(P))] for ts in lts]  \free{lts}}
+}
+Note that each triangular set in {\bf lts} is a lexicographical Groebner basis.
+Recall that a point belongs to the variety associated with {\bf lp} if and only if
+it belongs to that associated with one triangular set {\bf ts} in {\bf lts}.
+
+By running the \axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage} operation,
+we retrieve the above decomposition.
+\xtc{
+}{
+\spadpaste{squareFreeLexTriangular(lg,true)$lextripack \free{lg} \free{lextripack}}
+}
+Thus the solutions given by {\bf lts} are pairwise different.
+\xtc{
+We count them as follows.
+}{
+\spadpaste{reduce(+,[degree(ts) for ts in lts]) \free{lts}}
+}
+
+We can investigate the triangular decomposition {\bf lts} by
+using the \spadtype{ZeroDimensionalSolvePackage}.
+\xtc{
+This requires to add an extra variable (smaller than the others) as follows.
+}{
+\spadpaste{ls2 : List Symbol := concat(ls,new()$Symbol) \free{ls} \bound{ls2}}
+}
+
+\xtc{
+Then we call the package.
+}{
+\spadpaste{zdpack := ZDSOLVE(R,ls,ls2) \free{R} \free{ls} \free{ls2} \bound{zdpack}}
+}
+
+
+\xtc{
+We compute a univariate representation of the variety associated with the input
+system as follows.
+}{
+\spadpaste{concat [univariateSolve(ts)$zdpack for ts in lts] \free{lts} \free{zdpack}}
+}
+Since the \axiomOpFrom{univariateSolve}{ZeroDimensionalSolvePackage} operation may
+split a regular set, it returns a list. This explains the use
+of \axiomOpFrom{concat}{List}.
+
+Look at the last item of the result. It consists of two parts.
+For any complex root {\bf ?} of the univariate polynomial in the first part,
+we get a tuple of univariate polynomials (in {\bf a}, ..., {\bf f} respectively)
+by replacing {\bf \%A} by {\bf ?} in the second part.
+Each of these tuples {\bf t} describes a point of the variety associated with {\bf lp}
+by equaling to zero the polynomials in {\bf t}.
+
+Note that the way of reading these univariate representations is explained also
+in the example illustrating the \spadtype{ZeroDimensionalSolvePackage} constructor.
+
+\xtc{
+Now, we compute the points of the variety with real coordinates.
+}{
+\spadpaste{concat [realSolve(ts)$zdpack for ts in lts] \free{lts} \free{zdpack}}
+}
+We obtain 24 points given by lists of elements in the \spadtype{RealClosure}
+of \spadtype{Fraction} of {\bf R}.
+In each list, the first value corresponds to the indeterminate {\bf f},
+the second to {\bf e} and so on.
+See \spadtype{ZeroDimensionalSolvePackage} to learn more about
+the \axiomOpFrom{realSolve}{ZeroDimensionalSolvePackage} operation.
+
+
+
+
+
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/LEXTRIPK.pht b/src/hyper/pages/LEXTRIPK.pht
new file mode 100644
index 00000000..18c3b7b9
--- /dev/null
+++ b/src/hyper/pages/LEXTRIPK.pht
@@ -0,0 +1,2553 @@
+\begin{patch}{LexTriangularPackageXmpPagePatch1}
+\begin{paste}{LexTriangularPackageXmpPageFull1}{LexTriangularPackageXmpPageEmpty1}
+\pastebutton{LexTriangularPackageXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{R := Integer\bound{R }}
+\indentrel{3}\begin{verbatim}
+   (1)  Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty1}
+\begin{paste}{LexTriangularPackageXmpPageEmpty1}{LexTriangularPackageXmpPagePatch1}
+\pastebutton{LexTriangularPackageXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{R := Integer\bound{R }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch2}
+\begin{paste}{LexTriangularPackageXmpPageFull2}{LexTriangularPackageXmpPageEmpty2}
+\pastebutton{LexTriangularPackageXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{ls : List Symbol := [a,b,c,d,e,f]\bound{ls }}
+\indentrel{3}\begin{verbatim}
+   (2)  [a,b,c,d,e,f]
+                                      Type: List Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty2}
+\begin{paste}{LexTriangularPackageXmpPageEmpty2}{LexTriangularPackageXmpPagePatch2}
+\pastebutton{LexTriangularPackageXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{ls : List Symbol := [a,b,c,d,e,f]\bound{ls }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch3}
+\begin{paste}{LexTriangularPackageXmpPageFull3}{LexTriangularPackageXmpPageEmpty3}
+\pastebutton{LexTriangularPackageXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
+\indentrel{3}\begin{verbatim}
+   (3)  OrderedVariableList [a,b,c,d,e,f]
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty3}
+\begin{paste}{LexTriangularPackageXmpPageEmpty3}{LexTriangularPackageXmpPagePatch3}
+\pastebutton{LexTriangularPackageXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch4}
+\begin{paste}{LexTriangularPackageXmpPageFull4}{LexTriangularPackageXmpPageEmpty4}
+\pastebutton{LexTriangularPackageXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
+\indentrel{3}\begin{verbatim}
+   (4)
+  NewSparseMultivariatePolynomial(Integer,OrderedVariable
+  List [a,b,c,d,e,f])
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty4}
+\begin{paste}{LexTriangularPackageXmpPageEmpty4}{LexTriangularPackageXmpPagePatch4}
+\pastebutton{LexTriangularPackageXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch5}
+\begin{paste}{LexTriangularPackageXmpPageFull5}{LexTriangularPackageXmpPageEmpty5}
+\pastebutton{LexTriangularPackageXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{p1: P := a*b*c*d*e*f - 1\free{P }\bound{p1 }}
+\indentrel{3}\begin{verbatim}
+   (5)  f e d c b a - 1
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty5}
+\begin{paste}{LexTriangularPackageXmpPageEmpty5}{LexTriangularPackageXmpPagePatch5}
+\pastebutton{LexTriangularPackageXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{p1: P := a*b*c*d*e*f - 1\free{P }\bound{p1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch6}
+\begin{paste}{LexTriangularPackageXmpPageFull6}{LexTriangularPackageXmpPageEmpty6}
+\pastebutton{LexTriangularPackageXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{p2: P := a*b*c*d*e +a*b*c*d*f +a*b*c*e*f +a*b*d*e*f +a*c*d*e*f +b*c*d*e*f\free{P }\bound{p2 }}
+\indentrel{3}\begin{verbatim}
+   (6)
+   ((((e + f)d + f e)c + f e d)b + f e d c)a + f e d c b
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty6}
+\begin{paste}{LexTriangularPackageXmpPageEmpty6}{LexTriangularPackageXmpPagePatch6}
+\pastebutton{LexTriangularPackageXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{p2: P := a*b*c*d*e +a*b*c*d*f +a*b*c*e*f +a*b*d*e*f +a*c*d*e*f +b*c*d*e*f\free{P }\bound{p2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch7}
+\begin{paste}{LexTriangularPackageXmpPageFull7}{LexTriangularPackageXmpPageEmpty7}
+\pastebutton{LexTriangularPackageXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{p3: P := a*b*c*d + a*b*c*f + a*b*e*f + a*d*e*f + b*c*d*e + c*d*e*f\free{P }\bound{p3 }}
+\indentrel{3}\begin{verbatim}
+   (7)  (((d + f)c + f e)b + f e d)a + e d c b + f e d c
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty7}
+\begin{paste}{LexTriangularPackageXmpPageEmpty7}{LexTriangularPackageXmpPagePatch7}
+\pastebutton{LexTriangularPackageXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{p3: P := a*b*c*d + a*b*c*f + a*b*e*f + a*d*e*f + b*c*d*e + c*d*e*f\free{P }\bound{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch8}
+\begin{paste}{LexTriangularPackageXmpPageFull8}{LexTriangularPackageXmpPageEmpty8}
+\pastebutton{LexTriangularPackageXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{p4: P := a*b*c + a*b*f + a*e*f + b*c*d + c*d*e + d*e*f\free{P }\bound{p4 }}
+\indentrel{3}\begin{verbatim}
+   (8)  ((c + f)b + f e)a + d c b + e d c + f e d
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty8}
+\begin{paste}{LexTriangularPackageXmpPageEmpty8}{LexTriangularPackageXmpPagePatch8}
+\pastebutton{LexTriangularPackageXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{p4: P := a*b*c + a*b*f + a*e*f + b*c*d + c*d*e + d*e*f\free{P }\bound{p4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch9}
+\begin{paste}{LexTriangularPackageXmpPageFull9}{LexTriangularPackageXmpPageEmpty9}
+\pastebutton{LexTriangularPackageXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{p5: P := a*b + a*f + b*c + c*d + d*e + e*f\free{P }\bound{p5 }}
+\indentrel{3}\begin{verbatim}
+   (9)  (b + f)a + c b + d c + e d + f e
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty9}
+\begin{paste}{LexTriangularPackageXmpPageEmpty9}{LexTriangularPackageXmpPagePatch9}
+\pastebutton{LexTriangularPackageXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{p5: P := a*b + a*f + b*c + c*d + d*e + e*f\free{P }\bound{p5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch10}
+\begin{paste}{LexTriangularPackageXmpPageFull10}{LexTriangularPackageXmpPageEmpty10}
+\pastebutton{LexTriangularPackageXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{p6: P := a + b + c + d + e + f\free{P }\bound{p6 }}
+\indentrel{3}\begin{verbatim}
+   (10)  a + b + c + d + e + f
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty10}
+\begin{paste}{LexTriangularPackageXmpPageEmpty10}{LexTriangularPackageXmpPagePatch10}
+\pastebutton{LexTriangularPackageXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{p6: P := a + b + c + d + e + f\free{P }\bound{p6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch11}
+\begin{paste}{LexTriangularPackageXmpPageFull11}{LexTriangularPackageXmpPageEmpty11}
+\pastebutton{LexTriangularPackageXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{lp := [p1, p2, p3, p4, p5, p6]\free{p1 }\free{p2 }\free{p3 }\free{p4 }\free{p5 }\free{p6 }\bound{lp }}
+\indentrel{3}\begin{verbatim}
+   (11)
+   [f e d c b a - 1,
+    ((((e + f)d + f e)c + f e d)b + f e d c)a + f e d c b,
+    (((d + f)c + f e)b + f e d)a + e d c b + f e d c,
+    ((c + f)b + f e)a + d c b + e d c + f e d,
+    (b + f)a + c b + d c + e d + f e,
+    a + b + c + d + e + f]
+Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty11}
+\begin{paste}{LexTriangularPackageXmpPageEmpty11}{LexTriangularPackageXmpPagePatch11}
+\pastebutton{LexTriangularPackageXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{lp := [p1, p2, p3, p4, p5, p6]\free{p1 }\free{p2 }\free{p3 }\free{p4 }\free{p5 }\free{p6 }\bound{lp }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch12}
+\begin{paste}{LexTriangularPackageXmpPageFull12}{LexTriangularPackageXmpPageEmpty12}
+\pastebutton{LexTriangularPackageXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{lextripack := LEXTRIPK(R,ls)\free{R }\free{ls }\bound{lextripack }}
+\indentrel{3}\begin{verbatim}
+   (12)  LexTriangularPackage(Integer,[a,b,c,d,e,f])
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty12}
+\begin{paste}{LexTriangularPackageXmpPageEmpty12}{LexTriangularPackageXmpPagePatch12}
+\pastebutton{LexTriangularPackageXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{lextripack := LEXTRIPK(R,ls)\free{R }\free{ls }\bound{lextripack }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch13}
+\begin{paste}{LexTriangularPackageXmpPageFull13}{LexTriangularPackageXmpPageEmpty13}
+\pastebutton{LexTriangularPackageXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{lg := groebner(lp)$lextripack\free{lp }\free{lextripack }\bound{lg }}
+\indentrel{3}\begin{verbatim}
+   (13)
+   [a + b + c + d + e + f,
+
+                        2
+       3968379498283200b  + 15873517993132800f b
+     + 
+                        2
+       3968379498283200d  + 15873517993132800f d
+     + 
+                        3 5                     4 4
+       3968379498283200f e  - 15873517993132800f e
+     + 
+                         5 3
+       23810276989699200f e
+     + 
+                           6                       2
+       (206355733910726400f  + 230166010900425600)e
+     + 
+                             43                       37
+           - 729705987316687f   + 1863667496867205421f
+         + 
+                                 31
+           291674853771731104461f
+         + 
+                                 25
+           365285994691106921745f
+         + 
+                              19
+           549961185828911895f
+         + 
+                                   13
+           - 365048404038768439269f
+         + 
+                                 7
+         - 292382820431504027669f  - 2271898467631865497f
+      *
+         e
+     + 
+                          44                        38
+       - 3988812642545399f   + 10187423878429609997f
+     + 
+                              32
+       1594377523424314053637f
+     + 
+                              26                       20
+       1994739308439916238065f   + 1596840088052642815f
+     + 
+                                14
+       - 1993494118301162145413f
+     + 
+                              8                        2
+     - 1596049742289689815053f  - 11488171330159667449f
+     ,
+
+       (23810276989699200c - 23810276989699200f)b
+     + 
+                         2
+       23810276989699200c  + 71430830969097600f c
+     + 
+                           2
+       - 23810276989699200d  - 95241107958796800f d
+     + 
+                           3 5                      4 4
+       - 55557312975964800f e  + 174608697924460800f e
+     + 
+                            5 3
+       - 174608697924460800f e
+     + 
+                              6                        2
+       (- 2428648252949318400f  - 2611193709870345600)e
+     + 
+                            43                        37
+           8305444561289527f   - 21212087151945459641f
+         + 
+                                    31
+           - 3319815883093451385381f
+         + 
+                                    25
+           - 4157691646261657136445f
+         + 
+                                 19
+           - 6072721607510764095f
+         + 
+                                  13
+           4154986709036460221649f
+         + 
+                                7
+         3327761311138587096749f  + 25885340608290841637f
+      *
+         e
+     + 
+                         44                         38
+       45815897629010329f   - 117013765582151891207f
+     + 
+                                 32
+       - 18313166848970865074187f
+     + 
+                                 26
+       - 22909971239649297438915f
+     + 
+                              20
+       - 16133250761305157265f
+     + 
+                               14
+       22897305857636178256623f
+     + 
+                             8                         2
+     18329944781867242497923f  + 130258531002020420699f
+     ,
+
+       (7936758996566400d - 7936758996566400f)b
+     + 
+                                                3 5
+       - 7936758996566400f d - 7936758996566400f e
+     + 
+                         4 4                     5 3
+       23810276989699200f e  - 23810276989699200f e
+     + 
+                             6                       2
+       (- 337312257354072000f  - 369059293340337600)e
+     + 
+                            43                       37
+           1176345388640471f   - 3004383582891473073f
+         + 
+                                   31
+           - 470203502707246105653f
+         + 
+                                   25
+           - 588858183402644348085f
+         + 
+                                19
+           - 856939308623513535f
+         + 
+                                 13
+           588472674242340526377f
+         + 
+                                 7
+           471313241958371103517f  + 3659742549078552381f
+      *
+         e
+     + 
+                        44                        38
+       6423170513956901f   - 16404772137036480803f
+     + 
+                                32
+       - 2567419165227528774463f
+     + 
+                                26
+       - 3211938090825682172335f
+     + 
+                             20
+       - 2330490332697587485f
+     + 
+                              14
+       3210100109444754864587f
+     + 
+                              8                        2
+       2569858315395162617847f  + 18326089487427735751f
+     ,
+
+       (11905138494849600e - 11905138494849600f)b
+     + 
+                          3 5                     4 4
+       - 3968379498283200f e  + 15873517993132800f e
+     + 
+                           5 3
+       - 27778656487982400f e
+     + 
+                             6                       2
+       (- 208339923659868000f  - 240086959646133600)e
+     + 
+                           43                       37
+           786029984751110f   - 2007519008182245250f
+         + 
+                                   31
+           - 314188062908073807090f
+         + 
+                                   25
+           - 393423667537929575250f
+         + 
+                                19
+           - 550329120654394950f
+         + 
+                                 13
+           393196408728889612770f
+         + 
+                                 7
+           314892372799176495730f  + 2409386515146668530f
+      *
+         e
+     + 
+                        44                        38
+       4177638546747827f   - 10669685294602576381f
+     + 
+                                32
+       - 1669852980419949524601f
+     + 
+                                26
+       - 2089077057287904170745f
+     + 
+                             20
+       - 1569899763580278795f
+     + 
+                              14
+       2087864026859015573349f
+     + 
+                              8                        2
+       1671496085945199577969f  + 11940257226216280177f
+     ,
+
+                          6
+       (11905138494849600f  - 11905138494849600)b
+     + 
+                           2 5                     3 4
+       - 15873517993132800f e  + 39683794982832000f e
+     + 
+                           4 3
+       - 39683794982832000f e
+     + 
+                             11                      5  2
+       (- 686529653202993600f   - 607162063237329600f )e
+     + 
+                          42                      36
+           65144531306704f   - 166381280901088652f
+         + 
+                                  30
+           - 26033434502470283472f
+         + 
+                                  24
+           - 31696259583860650140f
+         + 
+                              18                        12
+           971492093167581360f   + 32220085033691389548f
+         + 
+                                6
+           25526177666070529808f  + 138603268355749244
+      *
+         e
+     + 
+                       43                      37
+       167620036074811f   - 428102417974791473f
+     + 
+                              31                        25
+       - 66997243801231679313f   - 83426716722148750485f
+     + 
+                          19                        13
+       203673895369980765f   + 83523056326010432457f
+     + 
+                            7
+       66995789640238066937f  + 478592855549587901f
+     ,
+
+                    3                   2
+       801692827936c  + 2405078483808f c
+     + 
+                       2                45
+       - 2405078483808f c - 13752945467f
+     + 
+                      39                    33
+       35125117815561f   + 5496946957826433f
+     + 
+                        27                  21
+       6834659447749117f   - 44484880462461f
+     + 
+                          15                    9
+       - 6873406230093057f   - 5450844938762633f
+     + 
+                     3
+       1216586044571f
+     ,
+
+       (23810276989699200d - 23810276989699200f)c
+     + 
+                         2
+       23810276989699200d  + 71430830969097600f d
+     + 
+                        3 5                     4 4
+       7936758996566400f e  - 31747035986265600f e
+     + 
+                         5 3
+       31747035986265600f e
+     + 
+                           6                       2
+       (404774708824886400f  + 396837949828320000)e
+     + 
+                              43                       37
+           - 1247372229446701f   + 3185785654596621203f
+         + 
+                                 31
+           498594866849974751463f
+         + 
+                                 25
+           624542545845791047935f
+         + 
+                              19
+           931085755769682885f
+         + 
+                                   13
+           - 624150663582417063387f
+         + 
+                                 7
+         - 499881859388360475647f  - 3926885313819527351f
+      *
+         e
+     + 
+                          44                        38
+       - 7026011547118141f   + 17944427051950691243f
+     + 
+                              32
+       2808383522593986603543f
+     + 
+                              26                       20
+       3513624142354807530135f   + 2860757006705537685f
+     + 
+                                14
+       - 3511356735642190737267f
+     + 
+                              8                        2
+     - 2811332494697103819887f  - 20315011631522847311f
+     ,
+
+       (7936758996566400e - 7936758996566400f)c
+     + 
+                           43                     37
+           - 4418748183673f   + 11285568707456559f
+         + 
+                               31                       25
+           1765998617294451019f   + 2173749283622606155f
+         + 
+                               19                       13
+           - 55788292195402895f   - 2215291421788292951f
+         + 
+                                 7
+           - 1718142665347430851f  + 30256569458230237f
+      *
+         e
+     + 
+                     44                     38
+       4418748183673f   - 11285568707456559f
+     + 
+                             32                       26
+       - 1765998617294451019f   - 2173749283622606155f
+     + 
+                         20                       14
+       55788292195402895f   + 2215291421788292951f
+     + 
+                           8                     2
+       1718142665347430851f  - 30256569458230237f
+     ,
+
+                       6
+       (72152354514240f  - 72152354514240)c
+     + 
+                   43                   37
+       40950859449f   - 104588980990367f
+     + 
+                           31                     25
+       - 16367227395575307f   - 20268523416527355f
+     + 
+                       19                     13
+       442205002259535f   + 20576059935789063f
+     + 
+                         7
+       15997133796970563f  - 275099892785581f
+     ,
+
+                        3                      2
+       1984189749141600d  + 5952569247424800f d
+     + 
+                          2                     4 5
+       - 5952569247424800f d - 3968379498283200f e
+     + 
+                         5 4                     3
+       15873517993132800f e  + 17857707742274400e
+     + 
+                             7                        2
+       (- 148814231185620000f  - 162703559429611200f)e
+     + 
+                             44                      38
+           - 390000914678878f   + 996062704593756434f
+         + 
+                                 32
+           155886323972034823914f
+         + 
+                                 26                    20
+           194745956143985421330f   + 6205077595574430f
+         + 
+                                   14
+           - 194596512653299068786f
+         + 
+                                 8                       2
+         - 155796897940756922666f  - 1036375759077320978f
+      *
+         e
+     + 
+                         45                      39
+       - 374998630035991f   + 957747106595453993f
+     + 
+                             33                         27
+       149889155566764891693f   + 187154171443494641685f
+     + 
+                            21                         15
+       - 127129015426348065f   - 187241533243115040417f
+     + 
+                               9                      3
+       - 149719983567976534037f  - 836654081239648061f
+     ,
+
+       (5952569247424800e - 5952569247424800f)d
+     + 
+                          3 5                    4 4
+       - 3968379498283200f e  + 9920948745708000f e
+     + 
+                          5 3
+       - 3968379498283200f e
+     + 
+                             6                       2
+       (- 148814231185620000f  - 150798420934761600)e
+     + 
+                           43                       37
+           492558110242553f   - 1257992359608074599f
+         + 
+                                   31
+           - 196883094539368513959f
+         + 
+                                   25
+           - 246562115745735428055f
+         + 
+                                19
+           - 325698701993885505f
+         + 
+                                 13
+           246417769883651808111f
+         + 
+                                 7
+           197327352068200652911f  + 1523373796389332143f
+      *
+         e
+     + 
+                        44                       38
+       2679481081803026f   - 6843392695421906608f
+     + 
+                                32
+       - 1071020459642646913578f
+     + 
+                                26                      20
+       - 1339789169692041240060f   - 852746750910750210f
+     + 
+                              14
+       1339105101971878401312f
+     + 
+                              8                       2
+       1071900289758712984762f  + 7555239072072727756f
+     ,
+
+                          6
+       (11905138494849600f  - 11905138494849600)d
+     + 
+                          2 5                     3 4
+       - 7936758996566400f e  + 31747035986265600f e
+     + 
+                           4 3
+       - 31747035986265600f e
+     + 
+                             11                      5  2
+       (- 420648226818019200f   - 404774708824886400f )e
+     + 
+                          42                     36
+           15336187600889f   - 39169739565161107f
+         + 
+                                 30
+           - 6127176127489690827f
+         + 
+                                 24
+           - 7217708742310509615f
+         + 
+                              18                       12
+           538628483890722735f   + 7506804353843507643f
+         + 
+                               6
+           5886160769782607203f  + 63576108396535879
+      *
+         e
+     + 
+                      43                      37
+       71737781777066f   - 183218856207557938f
+     + 
+                              31                        25
+       - 28672874271132276078f   - 35625223686939812010f
+     + 
+                          19                        13
+       164831339634084390f   + 35724160423073052642f
+     + 
+                            7
+       28627022578664910622f  + 187459987029680506f
+     ,
+
+                        6                      5
+       1322793166094400e  - 3968379498283200f e
+     + 
+                        2 4                    3 3
+       3968379498283200f e  - 5291172664377600f e
+     + 
+                             10                      4  2
+       (- 230166010900425600f   - 226197631402142400f )e
+     + 
+                                47
+           - 152375364610443885f
+         + 
+                                 41
+           389166626064854890415f
+         + 
+                                   35
+           60906097841360558987335f
+         + 
+                                   29
+           76167367934608798697275f
+         + 
+                                23
+           27855066785995181125f
+         + 
+                                     17
+           - 76144952817052723145495f
+         + 
+                                     11
+           - 60933629892463517546975f
+         + 
+                                   5
+           - 411415071682002547795f
+      *
+         e
+     + 
+                         42                      36
+       - 209493533143822f   + 535045979490560586f
+     + 
+                            30                         24
+       83737947964973553146f   + 104889507084213371570f
+     + 
+                          18                         12
+       167117997269207870f   - 104793725781390615514f
+     + 
+                              6
+       - 83842685189903180394f  - 569978796672974242
+     ,
+
+                       6                   3
+       (25438330117200f  + 25438330117200)e
+     + 
+                       7                    2
+       (76314990351600f  + 76314990351600f)e
+     + 
+                           44                    38
+           - 1594966552735f   + 4073543370415745f
+         + 
+                              32                      26
+           637527159231148925f   + 797521176113606525f
+         + 
+                           20                      14
+           530440941097175f   - 797160527306433145f
+         + 
+                                8                    2
+           - 638132320196044965f  - 4510507167940725f
+      *
+         e
+     + 
+                       45                     39
+       - 6036376800443f   + 15416903421476909f
+     + 
+                           33                       27
+       2412807646192304449f   + 3017679923028013705f
+     + 
+                        21                       15
+       1422320037411955f   - 3016560402417843941f
+     + 
+                             9                     3
+       - 2414249368183033161f  - 16561862361763873f
+     ,
+
+                      12                  2
+       (1387545279120f   - 1387545279120)e
+     + 
+                      43                  37
+           4321823003f   - 11037922310209f
+         + 
+                              31                    25
+           - 1727510711947989f   - 2165150991154425f
+         + 
+                           19                    13
+           - 5114342560755f   + 2162682824948601f
+         + 
+                            7
+           1732620732685741f  + 13506088516033f
+      *
+         e
+     + 
+                   44                  38
+       24177661775f   - 61749727185325f
+     + 
+                          32                     26
+       - 9664106795754225f   - 12090487758628245f
+     + 
+                       20                     14
+       - 8787672733575f   + 12083693383005045f
+     + 
+                        8                  2
+       9672870290826025f  + 68544102808525f
+     ,
+
+        48        42          36          30          18
+       f   - 2554f   - 399710f   - 499722f   + 499722f
+     + 
+              12        6
+       399710f   + 2554f  - 1
+     ]
+Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty13}
+\begin{paste}{LexTriangularPackageXmpPageEmpty13}{LexTriangularPackageXmpPagePatch13}
+\pastebutton{LexTriangularPackageXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{lg := groebner(lp)$lextripack\free{lp }\free{lextripack }\bound{lg }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch14}
+\begin{paste}{LexTriangularPackageXmpPageFull14}{LexTriangularPackageXmpPageEmpty14}
+\pastebutton{LexTriangularPackageXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{lexTriangular(lg,false)$lextripack\free{lg }\free{lextripack }}
+\indentrel{3}\begin{verbatim}
+   (14)
+   [
+       6
+     {f  + 1,
+       6       5     2 4     3 3     4 2     5
+      e  - 3f e  + 3f e  - 4f e  + 3f e  - 3f e - 1,
+            2 5     3 4     4 3     5 2
+      3d + f e  - 4f e  + 4f e  - 2f e  - 2e + 2f, c + f,
+             2 5     3 4     4 3      5 2
+      3b + 2f e  - 5f e  + 5f e  - 10f e  - 4e + 7f,
+           2 5     3 4     4 3     5 2
+      a - f e  + 3f e  - 3f e  + 4f e  + 3e - 3f}
+     ,
+
+       6                     2           2
+     {f  - 1, e - f, d - f, c  + 4f c + f ,
+                         2
+      (c - f)b - f c - 5f , a + b + c + 3f}
+     ,
+      6                        2           2
+    {f  - 1,e - f,d - f,c - f,b  + 4f b + f ,a + b + 4f},
+
+       6              2           2
+     {f  - 1, e - f, d  + 4f d + f ,
+                         2
+      (d - f)c - f d - 5f , b - f, a + c + d + 3f}
+     ,
+
+     {
+          36        30          24          18          12
+         f   - 2554f   - 399709f   - 502276f   - 399709f
+       + 
+                6
+         - 2554f  + 1
+       ,
+
+                    12              2
+         (161718564f   - 161718564)e
+       + 
+                      31              25                19
+             - 504205f   + 1287737951f   + 201539391380f
+           + 
+                        13                7
+           253982817368f   + 201940704665f  + 1574134601f
+        *
+           e
+       + 
+                   32              26                 20
+         - 2818405f   + 7198203911f   + 1126548149060f
+       + 
+                     14                 8              2
+       1416530563364f   + 1127377589345f  + 7988820725f
+       ,
+
+                       6
+         (693772639560f  - 693772639560)d
+       + 
+                        2 5                 3 4
+         - 462515093040f e  + 1850060372160f e
+       + 
+                         4 3
+         - 1850060372160f e
+       + 
+                           11                  5  2
+         (- 24513299931120f   - 23588269745040f )e
+       + 
+                         30                 24
+             - 890810428f   + 2275181044754f
+           + 
+                             18                   12
+             355937263869776f   + 413736880104344f
+           + 
+                             6
+             342849304487996f  + 3704966481878
+        *
+           e
+       + 
+                      31                  25
+         - 4163798003f   + 10634395752169f
+       + 
+                          19                    13
+         1664161760192806f   + 2079424391370694f
+       + 
+                          7
+         1668153650635921f  + 10924274392693f
+       ,
+
+                      6                           31
+         (12614047992f  - 12614047992)c - 7246825f
+       + 
+                     25                 19
+         18508536599f   + 2896249516034f
+       + 
+                     13                 7
+       3581539649666f   + 2796477571739f  - 48094301893f
+       ,
+
+                       6
+         (693772639560f  - 693772639560)b
+       + 
+                        2 5                 3 4
+         - 925030186080f e  + 2312575465200f e
+       + 
+                         4 3
+         - 2312575465200f e
+       + 
+                           11                  5  2
+         (- 40007555547960f   - 35382404617560f )e
+       + 
+                          30                 24
+             - 3781280823f   + 9657492291789f
+           + 
+                              18                    12
+             1511158913397906f   + 1837290892286154f
+           + 
+                              6
+             1487216006594361f  + 8077238712093
+        *
+           e
+       + 
+                      31                  25
+         - 9736390478f   + 24866827916734f
+       + 
+                          19                    13
+         3891495681905296f   + 4872556418871424f
+       + 
+                          7
+         3904047887269606f  + 27890075838538f
+       ,
+      a + b + c + d + e + f}
+     ,
+
+       6       2           2                     2
+     {f  - 1, e  + 4f e + f , (e - f)d - f e - 5f ,
+      c - f, b - f, a + d + e + 3f}
+     ]
+         Type: List RegularChain(Integer,[a,b,c,d,e,f])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty14}
+\begin{paste}{LexTriangularPackageXmpPageEmpty14}{LexTriangularPackageXmpPagePatch14}
+\pastebutton{LexTriangularPackageXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{lexTriangular(lg,false)$lextripack\free{lg }\free{lextripack }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch15}
+\begin{paste}{LexTriangularPackageXmpPageFull15}{LexTriangularPackageXmpPageEmpty15}
+\pastebutton{LexTriangularPackageXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{lts := lexTriangular(lg,true)$lextripack\free{lg }\free{lextripack }\bound{lts }}
+\indentrel{3}\begin{verbatim}
+   (15)
+   [
+       6
+     {f  + 1,
+       6       5     2 4     3 3     4 2     5
+      e  - 3f e  + 3f e  - 4f e  + 3f e  - 3f e - 1,
+            2 5     3 4     4 3     5 2
+      3d + f e  - 4f e  + 4f e  - 2f e  - 2e + 2f, c + f,
+             2 5     3 4     4 3      5 2
+      3b + 2f e  - 5f e  + 5f e  - 10f e  - 4e + 7f,
+           2 5     3 4     4 3     5 2
+      a - f e  + 3f e  - 3f e  + 4f e  + 3e - 3f}
+     ,
+      6                  2           2
+    {f  - 1,e - f,d - f,c  + 4f c + f ,b + c + 4f,a - f},
+      6                        2           2
+    {f  - 1,e - f,d - f,c - f,b  + 4f b + f ,a + b + 4f},
+      6            2           2
+    {f  - 1,e - f,d  + 4f d + f ,c + d + 4f,b - f,a - f},
+
+     {
+          36        30          24          18          12
+         f   - 2554f   - 399709f   - 502276f   - 399709f
+       + 
+                6
+         - 2554f  + 1
+       ,
+
+                       2
+         1387545279120e
+       + 
+                        31                  25
+             4321823003f   - 11037922310209f
+           + 
+                                19                    13
+             - 1727506390124986f   - 2176188913464634f
+           + 
+                                7
+             - 1732620732685741f  - 13506088516033f
+        *
+           e
+       + 
+                     32                  26
+         24177661775f   - 61749727185325f
+       + 
+                            20                     14
+         - 9664082618092450f   - 12152237485813570f
+       + 
+                            8                  2
+         - 9672870290826025f  - 68544102808525f
+       ,
+
+         1387545279120d
+       + 
+                          30                 24
+             - 1128983050f   + 2883434331830f
+           + 
+                             18                   12
+             451234998755840f   + 562426491685760f
+           + 
+                             6
+             447129055314890f  - 165557857270
+        *
+           e
+       + 
+                      31                 25
+         - 1816935351f   + 4640452214013f
+       + 
+                         19                   13
+         726247129626942f   + 912871801716798f
+       + 
+                         7
+         726583262666877f  + 4909358645961f
+       ,
+
+                                    31                 25
+         1387545279120c + 778171189f   - 1987468196267f
+       + 
+                           19                   13
+         - 310993556954378f   - 383262822316802f
+       + 
+                           7
+         - 300335488637543f  + 5289595037041f
+       ,
+
+         1387545279120b
+       + 
+                        30                 24
+             1128983050f   - 2883434331830f
+           + 
+                               18                   12
+             - 451234998755840f   - 562426491685760f
+           + 
+                               6
+             - 447129055314890f  + 165557857270
+        *
+           e
+       + 
+                      31                 25
+         - 3283058841f   + 8384938292463f
+       + 
+                          19                    13
+         1312252817452422f   + 1646579934064638f
+       + 
+                          7
+         1306372958656407f  + 4694680112151f
+       ,
+
+                                                      31
+         1387545279120a + 1387545279120e + 4321823003f
+       + 
+                          25                    19
+         - 11037922310209f   - 1727506390124986f
+       + 
+                            13                    7
+         - 2176188913464634f   - 1732620732685741f
+       + 
+         - 13506088516033f
+       }
+     ,
+      6      2           2
+    {f  - 1,e  + 4f e + f ,d + e + 4f,c - f,b - f,a - f}]
+         Type: List RegularChain(Integer,[a,b,c,d,e,f])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty15}
+\begin{paste}{LexTriangularPackageXmpPageEmpty15}{LexTriangularPackageXmpPagePatch15}
+\pastebutton{LexTriangularPackageXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{lts := lexTriangular(lg,true)$lextripack\free{lg }\free{lextripack }\bound{lts }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch16}
+\begin{paste}{LexTriangularPackageXmpPageFull16}{LexTriangularPackageXmpPageEmpty16}
+\pastebutton{LexTriangularPackageXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{[[init(p) for p in (ts :: List(P))] for ts in lts]\free{lts }}
+\indentrel{3}\begin{verbatim}
+   (16)
+   [[1,3,1,3,1,1], [1,1,1,1,1,1], [1,1,1,1,1,1],
+    [1,1,1,1,1,1],
+
+     [1387545279120, 1387545279120, 1387545279120,
+      1387545279120, 1387545279120, 1]
+     ,
+    [1,1,1,1,1,1]]
+Type: List List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty16}
+\begin{paste}{LexTriangularPackageXmpPageEmpty16}{LexTriangularPackageXmpPagePatch16}
+\pastebutton{LexTriangularPackageXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{[[init(p) for p in (ts :: List(P))] for ts in lts]\free{lts }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch17}
+\begin{paste}{LexTriangularPackageXmpPageFull17}{LexTriangularPackageXmpPageEmpty17}
+\pastebutton{LexTriangularPackageXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{squareFreeLexTriangular(lg,true)$lextripack\free{lg }\free{lextripack }}
+\indentrel{3}\begin{verbatim}
+   (17)
+   [
+       6
+     {f  + 1,
+       6       5     2 4     3 3     4 2     5
+      e  - 3f e  + 3f e  - 4f e  + 3f e  - 3f e - 1,
+            2 5     3 4     4 3     5 2
+      3d + f e  - 4f e  + 4f e  - 2f e  - 2e + 2f, c + f,
+             2 5     3 4     4 3      5 2
+      3b + 2f e  - 5f e  + 5f e  - 10f e  - 4e + 7f,
+           2 5     3 4     4 3     5 2
+      a - f e  + 3f e  - 3f e  + 4f e  + 3e - 3f}
+     ,
+      6                  2           2
+    {f  - 1,e - f,d - f,c  + 4f c + f ,b + c + 4f,a - f},
+      6                        2           2
+    {f  - 1,e - f,d - f,c - f,b  + 4f b + f ,a + b + 4f},
+      6            2           2
+    {f  - 1,e - f,d  + 4f d + f ,c + d + 4f,b - f,a - f},
+
+     {
+          36        30          24          18          12
+         f   - 2554f   - 399709f   - 502276f   - 399709f
+       + 
+                6
+         - 2554f  + 1
+       ,
+
+                       2
+         1387545279120e
+       + 
+                        31                  25
+             4321823003f   - 11037922310209f
+           + 
+                                19                    13
+             - 1727506390124986f   - 2176188913464634f
+           + 
+                                7
+             - 1732620732685741f  - 13506088516033f
+        *
+           e
+       + 
+                     32                  26
+         24177661775f   - 61749727185325f
+       + 
+                            20                     14
+         - 9664082618092450f   - 12152237485813570f
+       + 
+                            8                  2
+         - 9672870290826025f  - 68544102808525f
+       ,
+
+         1387545279120d
+       + 
+                          30                 24
+             - 1128983050f   + 2883434331830f
+           + 
+                             18                   12
+             451234998755840f   + 562426491685760f
+           + 
+                             6
+             447129055314890f  - 165557857270
+        *
+           e
+       + 
+                      31                 25
+         - 1816935351f   + 4640452214013f
+       + 
+                         19                   13
+         726247129626942f   + 912871801716798f
+       + 
+                         7
+         726583262666877f  + 4909358645961f
+       ,
+
+                                    31                 25
+         1387545279120c + 778171189f   - 1987468196267f
+       + 
+                           19                   13
+         - 310993556954378f   - 383262822316802f
+       + 
+                           7
+         - 300335488637543f  + 5289595037041f
+       ,
+
+         1387545279120b
+       + 
+                        30                 24
+             1128983050f   - 2883434331830f
+           + 
+                               18                   12
+             - 451234998755840f   - 562426491685760f
+           + 
+                               6
+             - 447129055314890f  + 165557857270
+        *
+           e
+       + 
+                      31                 25
+         - 3283058841f   + 8384938292463f
+       + 
+                          19                    13
+         1312252817452422f   + 1646579934064638f
+       + 
+                          7
+         1306372958656407f  + 4694680112151f
+       ,
+
+                                                      31
+         1387545279120a + 1387545279120e + 4321823003f
+       + 
+                          25                    19
+         - 11037922310209f   - 1727506390124986f
+       + 
+                            13                    7
+         - 2176188913464634f   - 1732620732685741f
+       + 
+         - 13506088516033f
+       }
+     ,
+      6      2           2
+    {f  - 1,e  + 4f e + f ,d + e + 4f,c - f,b - f,a - f}]
+Type: List SquareFreeRegularTriangularSet(Integer,IndexedExponents OrderedVariableList [a,b,c,d,e,f],OrderedVariableList [a,b,c,d,e,f],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [a,b,c,d,e,f]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty17}
+\begin{paste}{LexTriangularPackageXmpPageEmpty17}{LexTriangularPackageXmpPagePatch17}
+\pastebutton{LexTriangularPackageXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{squareFreeLexTriangular(lg,true)$lextripack\free{lg }\free{lextripack }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch18}
+\begin{paste}{LexTriangularPackageXmpPageFull18}{LexTriangularPackageXmpPageEmpty18}
+\pastebutton{LexTriangularPackageXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{reduce(+,[degree(ts) for ts in lts])\free{lts }}
+\indentrel{3}\begin{verbatim}
+   (18)  156
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty18}
+\begin{paste}{LexTriangularPackageXmpPageEmpty18}{LexTriangularPackageXmpPagePatch18}
+\pastebutton{LexTriangularPackageXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{reduce(+,[degree(ts) for ts in lts])\free{lts }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch19}
+\begin{paste}{LexTriangularPackageXmpPageFull19}{LexTriangularPackageXmpPageEmpty19}
+\pastebutton{LexTriangularPackageXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{ls2 : List Symbol := concat(ls,new()$Symbol)\free{ls }\bound{ls2 }}
+\indentrel{3}\begin{verbatim}
+   (19)  [a,b,c,d,e,f,%A]
+                                      Type: List Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty19}
+\begin{paste}{LexTriangularPackageXmpPageEmpty19}{LexTriangularPackageXmpPagePatch19}
+\pastebutton{LexTriangularPackageXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{ls2 : List Symbol := concat(ls,new()$Symbol)\free{ls }\bound{ls2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch20}
+\begin{paste}{LexTriangularPackageXmpPageFull20}{LexTriangularPackageXmpPageEmpty20}
+\pastebutton{LexTriangularPackageXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{zdpack := ZDSOLVE(R,ls,ls2)\free{R }\free{ls }\free{ls2 }\bound{zdpack }}
+\indentrel{3}\begin{verbatim}
+   (20)
+  ZeroDimensionalSolvePackage(Integer,[a,b,c,d,e,f],[a,b,
+  c,d,e,f,%A])
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty20}
+\begin{paste}{LexTriangularPackageXmpPageEmpty20}{LexTriangularPackageXmpPagePatch20}
+\pastebutton{LexTriangularPackageXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{zdpack := ZDSOLVE(R,ls,ls2)\free{R }\free{ls }\free{ls2 }\bound{zdpack }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch21}
+\begin{paste}{LexTriangularPackageXmpPageFull21}{LexTriangularPackageXmpPageEmpty21}
+\pastebutton{LexTriangularPackageXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{concat [univariateSolve(ts)$zdpack for ts in lts]\free{lts }\free{zdpack }}
+\indentrel{3}\begin{verbatim}
+   (21)
+   [
+                     4      2
+     [complexRoots= ?  - 13?  + 49,
+
+       coordinates =
+                 3                3
+         [7a + %A  - 6%A, 21b + %A  + %A,
+                   3                3
+          21c - 2%A  + 19%A, 7d - %A  + 6%A,
+                  3                3
+          21e - %A  - %A, 21f + 2%A  - 19%A]
+       ]
+     ,
+
+                     4      2
+     [complexRoots= ?  + 11?  + 49,
+
+       coordinates =
+                   3                 3
+         [35a + 3%A  + 19%A, 35b + %A  + 18%A,
+                   3                3
+          35c - 2%A  - %A, 35d - 3%A  - 19%A,
+                  3                  3
+          35e - %A  - 18%A, 35f + 2%A  + %A]
+       ]
+     ,
+
+     [
+       complexRoots =
+            8      7      6       5       4       3
+           ?  - 12?  + 58?  - 120?  + 207?  - 360?
+         + 
+               2
+           802?  - 1332? + 1369
+       ,
+
+       coordinates =
+         [
+                                7           6            5
+             43054532a + 33782%A  - 546673%A  + 3127348%A
+           + 
+                        4            3             2
+             - 6927123%A  + 4365212%A  - 25086957%A
+           + 
+             39582814%A - 107313172
+           ,
+
+                                7           6            5
+             43054532b - 33782%A  + 546673%A  - 3127348%A
+           + 
+                      4            3             2
+             6927123%A  - 4365212%A  + 25086957%A
+           + 
+             - 39582814%A + 107313172
+           ,
+
+                                7           6            5
+             21527266c - 22306%A  + 263139%A  - 1166076%A
+           + 
+                      4            3             2
+             1821805%A  - 2892788%A  + 10322663%A
+           + 
+             - 9026596%A + 12950740
+           ,
+
+                                7           6            5
+             43054532d + 22306%A  - 263139%A  + 1166076%A
+           + 
+                        4            3             2
+             - 1821805%A  + 2892788%A  - 10322663%A
+           + 
+             30553862%A - 12950740
+           ,
+
+                                7           6            5
+             43054532e - 22306%A  + 263139%A  - 1166076%A
+           + 
+                      4            3             2
+             1821805%A  - 2892788%A  + 10322663%A
+           + 
+             - 30553862%A + 12950740
+           ,
+
+                                7           6            5
+             21527266f + 22306%A  - 263139%A  + 1166076%A
+           + 
+                        4            3             2
+             - 1821805%A  + 2892788%A  - 10322663%A
+           + 
+             9026596%A - 12950740
+           ]
+       ]
+     ,
+
+     [
+       complexRoots =
+            8      7      6       5       4       3
+           ?  + 12?  + 58?  + 120?  + 207?  + 360?
+         + 
+               2
+           802?  + 1332? + 1369
+       ,
+
+       coordinates =
+         [
+                                7           6            5
+             43054532a + 33782%A  + 546673%A  + 3127348%A
+           + 
+                      4            3             2
+             6927123%A  + 4365212%A  + 25086957%A
+           + 
+             39582814%A + 107313172
+           ,
+
+                                7           6            5
+             43054532b - 33782%A  - 546673%A  - 3127348%A
+           + 
+                        4            3             2
+             - 6927123%A  - 4365212%A  - 25086957%A
+           + 
+             - 39582814%A - 107313172
+           ,
+
+                                7           6            5
+             21527266c - 22306%A  - 263139%A  - 1166076%A
+           + 
+                        4            3             2
+             - 1821805%A  - 2892788%A  - 10322663%A
+           + 
+             - 9026596%A - 12950740
+           ,
+
+                                7           6            5
+             43054532d + 22306%A  + 263139%A  + 1166076%A
+           + 
+                      4            3             2
+             1821805%A  + 2892788%A  + 10322663%A
+           + 
+             30553862%A + 12950740
+           ,
+
+                                7           6            5
+             43054532e - 22306%A  - 263139%A  - 1166076%A
+           + 
+                        4            3             2
+             - 1821805%A  - 2892788%A  - 10322663%A
+           + 
+             - 30553862%A - 12950740
+           ,
+
+                                7           6            5
+             21527266f + 22306%A  + 263139%A  + 1166076%A
+           + 
+                      4            3             2
+             1821805%A  + 2892788%A  + 10322663%A
+           + 
+             9026596%A + 12950740
+           ]
+       ]
+     ,
+
+                     4    2
+     [complexRoots= ?  - ?  + 1,
+
+       coordinates =
+                        3             3
+         [a - %A, b + %A  - %A, c + %A , d + %A,
+                3             3
+          e - %A  + %A, f - %A ]
+       ]
+     ,
+
+                     8     6      4      2
+     [complexRoots= ?  + 4?  + 12?  + 16?  + 4,
+
+       coordinates =
+                  7      5       3
+         [4a - 2%A  - 7%A  - 20%A  - 22%A,
+                  7      5       3
+          4b + 2%A  + 7%A  + 20%A  + 22%A,
+                 7      5       3
+          4c + %A  + 3%A  + 10%A  + 10%A,
+                 7      5       3
+          4d + %A  + 3%A  + 10%A  + 6%A,
+                 7      5       3
+          4e - %A  - 3%A  - 10%A  - 6%A,
+                 7      5       3
+          4f - %A  - 3%A  - 10%A  - 10%A]
+       ]
+     ,
+
+                     4     3      2
+     [complexRoots= ?  + 6?  + 30?  + 36? + 36,
+
+       coordinates =
+                  3      2
+         [30a - %A  - 5%A  - 30%A - 6,
+                 3      2
+          6b + %A  + 5%A  + 24%A + 6,
+                  3      2
+          30c - %A  - 5%A  - 6,
+                  3      2
+          30d - %A  - 5%A  - 30%A - 6,
+                  3      2
+          30e - %A  - 5%A  - 30%A - 6,
+                  3      2
+          30f - %A  - 5%A  - 30%A - 6]
+       ]
+     ,
+
+                     4     3      2
+     [complexRoots= ?  - 6?  + 30?  - 36? + 36,
+
+       coordinates =
+                  3      2
+         [30a - %A  + 5%A  - 30%A + 6,
+                 3      2
+          6b + %A  - 5%A  + 24%A - 6,
+                  3      2
+          30c - %A  + 5%A  + 6,
+                  3      2
+          30d - %A  + 5%A  - 30%A + 6,
+                  3      2
+          30e - %A  + 5%A  - 30%A + 6,
+                  3      2
+          30f - %A  + 5%A  - 30%A + 6]
+       ]
+     ,
+
+                     2
+     [complexRoots= ?  + 6? + 6,
+
+       coordinates =
+         [a + 1,b - %A - 5,c + %A + 1,d + 1,e + 1,f + 1]
+       ]
+     ,
+
+                     2
+     [complexRoots= ?  - 6? + 6,
+
+       coordinates =
+         [a - 1,b - %A + 5,c + %A - 1,d - 1,e - 1,f - 1]
+       ]
+     ,
+
+                     4     3      2
+     [complexRoots= ?  + 6?  + 30?  + 36? + 36,
+
+       coordinates =
+                 3      2
+         [6a + %A  + 5%A  + 24%A + 6,
+                  3      2
+          30b - %A  - 5%A  - 6,
+                  3      2
+          30c - %A  - 5%A  - 30%A - 6,
+                  3      2
+          30d - %A  - 5%A  - 30%A - 6,
+                  3      2
+          30e - %A  - 5%A  - 30%A - 6,
+                  3      2
+          30f - %A  - 5%A  - 30%A - 6]
+       ]
+     ,
+
+                     4     3      2
+     [complexRoots= ?  - 6?  + 30?  - 36? + 36,
+
+       coordinates =
+                 3      2
+         [6a + %A  - 5%A  + 24%A - 6,
+                  3      2
+          30b - %A  + 5%A  + 6,
+                  3      2
+          30c - %A  + 5%A  - 30%A + 6,
+                  3      2
+          30d - %A  + 5%A  - 30%A + 6,
+                  3      2
+          30e - %A  + 5%A  - 30%A + 6,
+                  3      2
+          30f - %A  + 5%A  - 30%A + 6]
+       ]
+     ,
+
+                     2
+     [complexRoots= ?  + 6? + 6,
+
+       coordinates =
+         [a - %A - 5,b + %A + 1,c + 1,d + 1,e + 1,f + 1]
+       ]
+     ,
+
+                     2
+     [complexRoots= ?  - 6? + 6,
+
+       coordinates =
+         [a - %A + 5,b + %A - 1,c - 1,d - 1,e - 1,f - 1]
+       ]
+     ,
+
+                     4     3      2
+     [complexRoots= ?  + 6?  + 30?  + 36? + 36,
+
+       coordinates =
+                  3      2
+         [30a - %A  - 5%A  - 30%A - 6,
+                  3      2
+          30b - %A  - 5%A  - 30%A - 6,
+                 3      2
+          6c + %A  + 5%A  + 24%A + 6,
+                  3      2
+          30d - %A  - 5%A  - 6,
+                  3      2
+          30e - %A  - 5%A  - 30%A - 6,
+                  3      2
+          30f - %A  - 5%A  - 30%A - 6]
+       ]
+     ,
+
+                     4     3      2
+     [complexRoots= ?  - 6?  + 30?  - 36? + 36,
+
+       coordinates =
+                  3      2
+         [30a - %A  + 5%A  - 30%A + 6,
+                  3      2
+          30b - %A  + 5%A  - 30%A + 6,
+                 3      2
+          6c + %A  - 5%A  + 24%A - 6,
+                  3      2
+          30d - %A  + 5%A  + 6,
+                  3      2
+          30e - %A  + 5%A  - 30%A + 6,
+                  3      2
+          30f - %A  + 5%A  - 30%A + 6]
+       ]
+     ,
+
+                     2
+     [complexRoots= ?  + 6? + 6,
+
+       coordinates =
+         [a + 1,b + 1,c - %A - 5,d + %A + 1,e + 1,f + 1]
+       ]
+     ,
+
+                     2
+     [complexRoots= ?  - 6? + 6,
+
+       coordinates =
+         [a - 1,b - 1,c - %A + 5,d + %A - 1,e - 1,f - 1]
+       ]
+     ,
+
+     [
+       complexRoots =
+          8     7      6      5      4     2
+         ?  + 6?  + 16?  + 24?  + 18?  - 8?  + 4
+       ,
+
+       coordinates =
+         [
+                     7      6       5       4      3
+             2a + 2%A  + 9%A  + 18%A  + 19%A  + 4%A
+           + 
+                   2
+             - 10%A  - 2%A + 4
+           ,
+
+                     7      6       5       4      3
+             2b + 2%A  + 9%A  + 18%A  + 19%A  + 4%A
+           + 
+                   2
+             - 10%A  - 4%A + 4
+           ,
+                 7      6      5      4      3
+          2c - %A  - 4%A  - 8%A  - 9%A  - 4%A  - 2%A - 4,
+                 7      6      5      4      3
+          2d + %A  + 4%A  + 8%A  + 9%A  + 4%A  + 2%A + 4,
+
+                     7      6       5       4      3
+             2e - 2%A  - 9%A  - 18%A  - 19%A  - 4%A
+           + 
+                 2
+             10%A  + 4%A - 4
+           ,
+
+                     7      6       5       4      3
+             2f - 2%A  - 9%A  - 18%A  - 19%A  - 4%A
+           + 
+                 2
+             10%A  + 2%A - 4
+           ]
+       ]
+     ,
+
+     [
+       complexRoots =
+            8      7      6       5       4       3
+           ?  + 12?  + 64?  + 192?  + 432?  + 768?
+         + 
+                2
+           1024?  + 768? + 256
+       ,
+
+       coordinates =
+         [
+                         7        6        5         4
+             1408a - 19%A  - 200%A  - 912%A  - 2216%A
+           + 
+                     3         2
+             - 4544%A  - 6784%A  - 6976%A - 1792
+           ,
+
+                         7        6         5         4
+             1408b - 37%A  - 408%A  - 1952%A  - 5024%A
+           + 
+                      3          2
+             - 10368%A  - 16768%A  - 17920%A - 5120
+           ,
+
+                         7        6         5         4
+             1408c + 37%A  + 408%A  + 1952%A  + 5024%A
+           + 
+                    3          2
+             10368%A  + 16768%A  + 17920%A + 5120
+           ,
+
+                         7        6        5         4
+             1408d + 19%A  + 200%A  + 912%A  + 2216%A
+           + 
+                   3         2
+             4544%A  + 6784%A  + 6976%A + 1792
+           ,
+          2e + %A, 2f - %A]
+       ]
+     ,
+
+                     8     6      4      2
+     [complexRoots= ?  + 4?  + 12?  + 16?  + 4,
+
+       coordinates =
+                 7      5       3
+         [4a - %A  - 3%A  - 10%A  - 6%A,
+                 7      5       3
+          4b - %A  - 3%A  - 10%A  - 10%A,
+                  7      5       3
+          4c - 2%A  - 7%A  - 20%A  - 22%A,
+                  7      5       3
+          4d + 2%A  + 7%A  + 20%A  + 22%A,
+                 7      5       3
+          4e + %A  + 3%A  + 10%A  + 10%A,
+                 7      5       3
+          4f + %A  + 3%A  + 10%A  + 6%A]
+       ]
+     ,
+
+                     8      6      4       2
+     [complexRoots= ?  + 16?  - 96?  + 256?  + 256,
+
+       coordinates =
+                   7       5        3
+         [512a - %A  - 12%A  + 176%A  - 448%A,
+                   7       5       3
+          128b - %A  - 16%A  + 96%A  - 256%A,
+                   7       5       3
+          128c + %A  + 16%A  - 96%A  + 256%A,
+                   7       5        3
+          512d + %A  + 12%A  - 176%A  + 448%A, 2e + %A,
+          2f - %A]
+       ]
+     ,
+
+     [
+       complexRoots =
+            8      7      6       5       4       3
+           ?  - 12?  + 64?  - 192?  + 432?  - 768?
+         + 
+                2
+           1024?  - 768? + 256
+       ,
+
+       coordinates =
+         [
+                         7        6        5         4
+             1408a - 19%A  + 200%A  - 912%A  + 2216%A
+           + 
+                     3         2
+             - 4544%A  + 6784%A  - 6976%A + 1792
+           ,
+
+                         7        6         5         4
+             1408b - 37%A  + 408%A  - 1952%A  + 5024%A
+           + 
+                      3          2
+             - 10368%A  + 16768%A  - 17920%A + 5120
+           ,
+
+                         7        6         5         4
+             1408c + 37%A  - 408%A  + 1952%A  - 5024%A
+           + 
+                    3          2
+             10368%A  - 16768%A  + 17920%A - 5120
+           ,
+
+                         7        6        5         4
+             1408d + 19%A  - 200%A  + 912%A  - 2216%A
+           + 
+                   3         2
+             4544%A  - 6784%A  + 6976%A - 1792
+           ,
+          2e + %A, 2f - %A]
+       ]
+     ,
+
+     [
+       complexRoots =
+          8     7      6      5      4     2
+         ?  - 6?  + 16?  - 24?  + 18?  - 8?  + 4
+       ,
+
+       coordinates =
+         [
+                     7      6       5       4      3
+             2a + 2%A  - 9%A  + 18%A  - 19%A  + 4%A
+           + 
+                 2
+             10%A  - 2%A - 4
+           ,
+
+                     7      6       5       4      3
+             2b + 2%A  - 9%A  + 18%A  - 19%A  + 4%A
+           + 
+                 2
+             10%A  - 4%A - 4
+           ,
+                 7      6      5      4      3
+          2c - %A  + 4%A  - 8%A  + 9%A  - 4%A  - 2%A + 4,
+                 7      6      5      4      3
+          2d + %A  - 4%A  + 8%A  - 9%A  + 4%A  + 2%A - 4,
+
+                     7      6       5       4      3
+             2e - 2%A  + 9%A  - 18%A  + 19%A  - 4%A
+           + 
+                   2
+             - 10%A  + 4%A + 4
+           ,
+
+                     7      6       5       4      3
+             2f - 2%A  + 9%A  - 18%A  + 19%A  - 4%A
+           + 
+                   2
+             - 10%A  + 2%A + 4
+           ]
+       ]
+     ,
+
+                     4      2
+     [complexRoots= ?  + 12?  + 144,
+
+       coordinates =
+                  2               2               2
+         [12a - %A  - 12, 12b - %A  - 12, 12c - %A  - 12,
+                  2              2
+          12d - %A  - 12, 6e + %A  + 3%A + 12,
+                 2
+          6f + %A  - 3%A + 12]
+       ]
+     ,
+
+                     4     3      2
+     [complexRoots= ?  + 6?  + 30?  + 36? + 36,
+
+       coordinates =
+                 3      2
+         [6a - %A  - 5%A  - 24%A - 6,
+                  3      2
+          30b + %A  + 5%A  + 30%A + 6,
+                  3      2
+          30c + %A  + 5%A  + 30%A + 6,
+                  3      2
+          30d + %A  + 5%A  + 30%A + 6,
+                  3      2
+          30e + %A  + 5%A  + 30%A + 6,
+                  3      2
+          30f + %A  + 5%A  + 6]
+       ]
+     ,
+
+                     4     3      2
+     [complexRoots= ?  - 6?  + 30?  - 36? + 36,
+
+       coordinates =
+                 3      2
+         [6a - %A  + 5%A  - 24%A + 6,
+                  3      2
+          30b + %A  - 5%A  + 30%A - 6,
+                  3      2
+          30c + %A  - 5%A  + 30%A - 6,
+                  3      2
+          30d + %A  - 5%A  + 30%A - 6,
+                  3      2
+          30e + %A  - 5%A  + 30%A - 6,
+                  3      2
+          30f + %A  - 5%A  - 6]
+       ]
+     ,
+
+                     4      2
+     [complexRoots= ?  + 12?  + 144,
+
+       coordinates =
+                  2               2               2
+         [12a + %A  + 12, 12b + %A  + 12, 12c + %A  + 12,
+                  2              2
+          12d + %A  + 12, 6e - %A  + 3%A - 12,
+                 2
+          6f - %A  - 3%A - 12]
+       ]
+     ,
+
+                     2
+     [complexRoots= ?  - 12,
+
+       coordinates =
+         [a - 1,b - 1,c - 1,d - 1,2e + %A + 4,2f - %A + 4]
+       ]
+     ,
+
+                     2
+     [complexRoots= ?  + 6? + 6,
+
+       coordinates =
+         [a + %A + 5,b - 1,c - 1,d - 1,e - 1,f - %A - 1]
+       ]
+     ,
+
+                     2
+     [complexRoots= ?  - 6? + 6,
+
+       coordinates =
+         [a + %A - 5,b + 1,c + 1,d + 1,e + 1,f - %A + 1]
+       ]
+     ,
+
+                     2
+     [complexRoots= ?  - 12,
+
+       coordinates =
+         [a + 1,b + 1,c + 1,d + 1,2e + %A - 4,2f - %A - 4]
+       ]
+     ,
+
+                     4     3      2
+     [complexRoots= ?  + 6?  + 30?  + 36? + 36,
+
+       coordinates =
+                  3      2
+         [30a - %A  - 5%A  - 30%A - 6,
+                  3      2
+          30b - %A  - 5%A  - 30%A - 6,
+                  3      2
+          30c - %A  - 5%A  - 30%A - 6,
+                 3      2
+          6d + %A  + 5%A  + 24%A + 6,
+                  3      2
+          30e - %A  - 5%A  - 6,
+                  3      2
+          30f - %A  - 5%A  - 30%A - 6]
+       ]
+     ,
+
+                     4     3      2
+     [complexRoots= ?  - 6?  + 30?  - 36? + 36,
+
+       coordinates =
+                  3      2
+         [30a - %A  + 5%A  - 30%A + 6,
+                  3      2
+          30b - %A  + 5%A  - 30%A + 6,
+                  3      2
+          30c - %A  + 5%A  - 30%A + 6,
+                 3      2
+          6d + %A  - 5%A  + 24%A - 6,
+                  3      2
+          30e - %A  + 5%A  + 6,
+                  3      2
+          30f - %A  + 5%A  - 30%A + 6]
+       ]
+     ,
+
+                     2
+     [complexRoots= ?  + 6? + 6,
+
+       coordinates =
+         [a + 1,b + 1,c + 1,d - %A - 5,e + %A + 1,f + 1]
+       ]
+     ,
+
+                     2
+     [complexRoots= ?  - 6? + 6,
+
+       coordinates =
+         [a - 1,b - 1,c - 1,d - %A + 5,e + %A - 1,f - 1]
+       ]
+     ]
+Type: List Record(complexRoots: SparseUnivariatePolynomial Integer,coordinates: List Polynomial Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty21}
+\begin{paste}{LexTriangularPackageXmpPageEmpty21}{LexTriangularPackageXmpPagePatch21}
+\pastebutton{LexTriangularPackageXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{concat [univariateSolve(ts)$zdpack for ts in lts]\free{lts }\free{zdpack }}
+\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPagePatch22}
+\begin{paste}{LexTriangularPackageXmpPageFull22}{LexTriangularPackageXmpPageEmpty22}
+\pastebutton{LexTriangularPackageXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{concat [realSolve(ts)$zdpack for ts in lts]\free{lts }\free{zdpack }}
+\indentrel{3}\begin{verbatim}
+   (22)
+   [[%R1,%R1,%R1,%R5,- %R5 - 4%R1,%R1],
+    [%R1,%R1,%R1,%R6,- %R6 - 4%R1,%R1],
+    [%R2,%R2,%R2,%R3,- %R3 - 4%R2,%R2],
+    [%R2,%R2,%R2,%R4,- %R4 - 4%R2,%R2],
+    [%R7,%R7,%R7,%R7,%R11,- %R11 - 4%R7],
+    [%R7,%R7,%R7,%R7,%R12,- %R12 - 4%R7],
+    [%R8,%R8,%R8,%R8,%R9,- %R9 - 4%R8],
+    [%R8,%R8,%R8,%R8,%R10,- %R10 - 4%R8],
+    [%R13,%R13,%R17,- %R17 - 4%R13,%R13,%R13],
+    [%R13,%R13,%R18,- %R18 - 4%R13,%R13,%R13],
+    [%R14,%R14,%R15,- %R15 - 4%R14,%R14,%R14],
+    [%R14,%R14,%R16,- %R16 - 4%R14,%R14,%R14],
+
+     [%R19, %R29,
+
+           7865521      31   6696179241     25
+         ÄÄÄÄÄÄÄÄÄÄ %R19   - ÄÄÄÄÄÄÄÄÄÄ %R19
+         6006689520          2002229840
+       + 
+           25769893181     19   1975912990729     13
+         - ÄÄÄÄÄÄÄÄÄÄÄ %R19   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+             49235160             3003344760
+       + 
+           1048460696489     7   21252634831
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19  - ÄÄÄÄÄÄÄÄÄÄÄ %R19
+             2002229840           6006689520
+       ,
+
+             778171189       31   1987468196267     25
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19   + ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+           1387545279120          1387545279120
+       + 
+         155496778477189     19   191631411158401     13
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+           693772639560             693772639560
+       + 
+         300335488637543     7   755656433863
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19  - ÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+          1387545279120          198220754160
+       ,
+
+          1094352947      31   2794979430821     25
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R19   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+         462515093040           462515093040
+       + 
+           218708802908737     19   91476663003591     13
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19   - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+             231257546520             77085848840
+       + 
+           145152550961823     7   1564893370717
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19  - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+             154171697680           462515093040
+       ,
+
+                    4321823003      31
+         - %R29 - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+                  1387545279120
+       + 
+         180949546069     25   863753195062493     19
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R19   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+          22746643920            693772639560
+       + 
+         1088094456732317     13   1732620732685741     7
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+           693772639560              1387545279120
+       + 
+         13506088516033
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+          1387545279120
+       ]
+     ,
+
+     [%R19, %R30,
+
+           7865521      31   6696179241     25
+         ÄÄÄÄÄÄÄÄÄÄ %R19   - ÄÄÄÄÄÄÄÄÄÄ %R19
+         6006689520          2002229840
+       + 
+           25769893181     19   1975912990729     13
+         - ÄÄÄÄÄÄÄÄÄÄÄ %R19   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+             49235160             3003344760
+       + 
+           1048460696489     7   21252634831
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19  - ÄÄÄÄÄÄÄÄÄÄÄ %R19
+             2002229840           6006689520
+       ,
+
+             778171189       31   1987468196267     25
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19   + ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+           1387545279120          1387545279120
+       + 
+         155496778477189     19   191631411158401     13
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+           693772639560             693772639560
+       + 
+         300335488637543     7   755656433863
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19  - ÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+          1387545279120          198220754160
+       ,
+
+          1094352947      31   2794979430821     25
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R19   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+         462515093040           462515093040
+       + 
+           218708802908737     19   91476663003591     13
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19   - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+             231257546520             77085848840
+       + 
+           145152550961823     7   1564893370717
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19  - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+             154171697680           462515093040
+       ,
+
+                    4321823003      31
+         - %R30 - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+                  1387545279120
+       + 
+         180949546069     25   863753195062493     19
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R19   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+          22746643920            693772639560
+       + 
+         1088094456732317     13   1732620732685741     7
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+           693772639560              1387545279120
+       + 
+         13506088516033
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R19
+          1387545279120
+       ]
+     ,
+
+     [%R20, %R27,
+
+           7865521      31   6696179241     25
+         ÄÄÄÄÄÄÄÄÄÄ %R20   - ÄÄÄÄÄÄÄÄÄÄ %R20
+         6006689520          2002229840
+       + 
+           25769893181     19   1975912990729     13
+         - ÄÄÄÄÄÄÄÄÄÄÄ %R20   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+             49235160             3003344760
+       + 
+           1048460696489     7   21252634831
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20  - ÄÄÄÄÄÄÄÄÄÄÄ %R20
+             2002229840           6006689520
+       ,
+
+             778171189       31   1987468196267     25
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20   + ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+           1387545279120          1387545279120
+       + 
+         155496778477189     19   191631411158401     13
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+           693772639560             693772639560
+       + 
+         300335488637543     7   755656433863
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20  - ÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+          1387545279120          198220754160
+       ,
+
+          1094352947      31   2794979430821     25
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R20   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+         462515093040           462515093040
+       + 
+           218708802908737     19   91476663003591     13
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20   - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+             231257546520             77085848840
+       + 
+           145152550961823     7   1564893370717
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20  - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+             154171697680           462515093040
+       ,
+
+                    4321823003      31
+         - %R27 - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+                  1387545279120
+       + 
+         180949546069     25   863753195062493     19
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R20   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+          22746643920            693772639560
+       + 
+         1088094456732317     13   1732620732685741     7
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+           693772639560              1387545279120
+       + 
+         13506088516033
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+          1387545279120
+       ]
+     ,
+
+     [%R20, %R28,
+
+           7865521      31   6696179241     25
+         ÄÄÄÄÄÄÄÄÄÄ %R20   - ÄÄÄÄÄÄÄÄÄÄ %R20
+         6006689520          2002229840
+       + 
+           25769893181     19   1975912990729     13
+         - ÄÄÄÄÄÄÄÄÄÄÄ %R20   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+             49235160             3003344760
+       + 
+           1048460696489     7   21252634831
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20  - ÄÄÄÄÄÄÄÄÄÄÄ %R20
+             2002229840           6006689520
+       ,
+
+             778171189       31   1987468196267     25
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20   + ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+           1387545279120          1387545279120
+       + 
+         155496778477189     19   191631411158401     13
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+           693772639560             693772639560
+       + 
+         300335488637543     7   755656433863
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20  - ÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+          1387545279120          198220754160
+       ,
+
+          1094352947      31   2794979430821     25
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R20   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+         462515093040           462515093040
+       + 
+           218708802908737     19   91476663003591     13
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20   - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+             231257546520             77085848840
+       + 
+           145152550961823     7   1564893370717
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20  - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+             154171697680           462515093040
+       ,
+
+                    4321823003      31
+         - %R28 - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+                  1387545279120
+       + 
+         180949546069     25   863753195062493     19
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R20   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+          22746643920            693772639560
+       + 
+         1088094456732317     13   1732620732685741     7
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+           693772639560              1387545279120
+       + 
+         13506088516033
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R20
+          1387545279120
+       ]
+     ,
+
+     [%R21, %R25,
+
+           7865521      31   6696179241     25
+         ÄÄÄÄÄÄÄÄÄÄ %R21   - ÄÄÄÄÄÄÄÄÄÄ %R21
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+       + 
+           25769893181     19   1975912990729     13
+         - ÄÄÄÄÄÄÄÄÄÄÄ %R21   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+             49235160             3003344760
+       + 
+           1048460696489     7   21252634831
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21  - ÄÄÄÄÄÄÄÄÄÄÄ %R21
+             2002229840           6006689520
+       ,
+
+             778171189       31   1987468196267     25
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21   + ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+           1387545279120          1387545279120
+       + 
+         155496778477189     19   191631411158401     13
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+           693772639560             693772639560
+       + 
+         300335488637543     7   755656433863
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21  - ÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+          1387545279120          198220754160
+       ,
+
+          1094352947      31   2794979430821     25
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R21   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+         462515093040           462515093040
+       + 
+           218708802908737     19   91476663003591     13
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21   - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+             231257546520             77085848840
+       + 
+           145152550961823     7   1564893370717
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21  - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+             154171697680           462515093040
+       ,
+
+                    4321823003      31
+         - %R25 - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+                  1387545279120
+       + 
+         180949546069     25   863753195062493     19
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R21   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+          22746643920            693772639560
+       + 
+         1088094456732317     13   1732620732685741     7
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+           693772639560              1387545279120
+       + 
+         13506088516033
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+          1387545279120
+       ]
+     ,
+
+     [%R21, %R26,
+
+           7865521      31   6696179241     25
+         ÄÄÄÄÄÄÄÄÄÄ %R21   - ÄÄÄÄÄÄÄÄÄÄ %R21
+         6006689520          2002229840
+       + 
+           25769893181     19   1975912990729     13
+         - ÄÄÄÄÄÄÄÄÄÄÄ %R21   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+             49235160             3003344760
+       + 
+           1048460696489     7   21252634831
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21  - ÄÄÄÄÄÄÄÄÄÄÄ %R21
+             2002229840           6006689520
+       ,
+
+             778171189       31   1987468196267     25
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21   + ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+           1387545279120          1387545279120
+       + 
+         155496778477189     19   191631411158401     13
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+           693772639560             693772639560
+       + 
+         300335488637543     7   755656433863
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21  - ÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+          1387545279120          198220754160
+       ,
+
+          1094352947      31   2794979430821     25
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R21   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+         462515093040           462515093040
+       + 
+           218708802908737     19   91476663003591     13
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21   - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+             231257546520             77085848840
+       + 
+           145152550961823     7   1564893370717
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21  - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+             154171697680           462515093040
+       ,
+
+                    4321823003      31
+         - %R26 - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+                  1387545279120
+       + 
+         180949546069     25   863753195062493     19
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R21   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+          22746643920            693772639560
+       + 
+         1088094456732317     13   1732620732685741     7
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+           693772639560              1387545279120
+       + 
+         13506088516033
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R21
+          1387545279120
+       ]
+     ,
+
+     [%R22, %R23,
+
+           7865521      31   6696179241     25
+         ÄÄÄÄÄÄÄÄÄÄ %R22   - ÄÄÄÄÄÄÄÄÄÄ %R22
+         6006689520          2002229840
+       + 
+           25769893181     19   1975912990729     13
+         - ÄÄÄÄÄÄÄÄÄÄÄ %R22   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+             49235160             3003344760
+       + 
+           1048460696489     7   21252634831
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22  - ÄÄÄÄÄÄÄÄÄÄÄ %R22
+             2002229840           6006689520
+       ,
+
+             778171189       31   1987468196267     25
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22   + ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+           1387545279120          1387545279120
+       + 
+         155496778477189     19   191631411158401     13
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+           693772639560             693772639560
+       + 
+         300335488637543     7   755656433863
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22  - ÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+          1387545279120          198220754160
+       ,
+
+          1094352947      31   2794979430821     25
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R22   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+         462515093040           462515093040
+       + 
+           218708802908737     19   91476663003591     13
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22   - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+             231257546520             77085848840
+       + 
+           145152550961823     7   1564893370717
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22  - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+             154171697680           462515093040
+       ,
+
+                    4321823003      31
+         - %R23 - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+                  1387545279120
+       + 
+         180949546069     25   863753195062493     19
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R22   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+          22746643920            693772639560
+       + 
+         1088094456732317     13   1732620732685741     7
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+           693772639560              1387545279120
+       + 
+         13506088516033
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+          1387545279120
+       ]
+     ,
+
+     [%R22, %R24,
+
+           7865521      31   6696179241     25
+         ÄÄÄÄÄÄÄÄÄÄ %R22   - ÄÄÄÄÄÄÄÄÄÄ %R22
+         6006689520          2002229840
+       + 
+           25769893181     19   1975912990729     13
+         - ÄÄÄÄÄÄÄÄÄÄÄ %R22   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+             49235160             3003344760
+       + 
+           1048460696489     7   21252634831
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22  - ÄÄÄÄÄÄÄÄÄÄÄ %R22
+             2002229840           6006689520
+       ,
+
+             778171189       31   1987468196267     25
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22   + ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+           1387545279120          1387545279120
+       + 
+         155496778477189     19   191631411158401     13
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+           693772639560             693772639560
+       + 
+         300335488637543     7   755656433863
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22  - ÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+          1387545279120          198220754160
+       ,
+
+          1094352947      31   2794979430821     25
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R22   - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+         462515093040           462515093040
+       + 
+           218708802908737     19   91476663003591     13
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22   - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+             231257546520             77085848840
+       + 
+           145152550961823     7   1564893370717
+         - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22  - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+             154171697680           462515093040
+       ,
+
+                    4321823003      31
+         - %R24 - ÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+                  1387545279120
+       + 
+         180949546069     25   863753195062493     19
+         ÄÄÄÄÄÄÄÄÄÄÄÄ %R22   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+          22746643920            693772639560
+       + 
+         1088094456732317     13   1732620732685741     7
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22   + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+           693772639560              1387545279120
+       + 
+         13506088516033
+         ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ %R22
+          1387545279120
+       ]
+     ,
+    [%R31,%R35,- %R35 - 4%R31,%R31,%R31,%R31],
+    [%R31,%R36,- %R36 - 4%R31,%R31,%R31,%R31],
+    [%R32,%R33,- %R33 - 4%R32,%R32,%R32,%R32],
+    [%R32,%R34,- %R34 - 4%R32,%R32,%R32,%R32]]
+           Type: List List RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LexTriangularPackageXmpPageEmpty22}
+\begin{paste}{LexTriangularPackageXmpPageEmpty22}{LexTriangularPackageXmpPagePatch22}
+\pastebutton{LexTriangularPackageXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{concat [realSolve(ts)$zdpack for ts in lts]\free{lts }\free{zdpack }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/LIB.ht b/src/hyper/pages/LIB.ht
new file mode 100644
index 00000000..e14b397c
--- /dev/null
+++ b/src/hyper/pages/LIB.ht
@@ -0,0 +1,67 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\LibraryXmpTitle}{Library}
+\newcommand{\LibraryXmpNumber}{9.41}
+%
+% =====================================================================
+\begin{page}{LibraryXmpPage}{9.41 Library}
+% =====================================================================
+\beginscroll
+ 
+The \spadtype{Library} domain provides a simple way to store \Language{}
+values in a file.
+This domain is similar to \spadtype{KeyedAccessFile} but fewer declarations
+are needed and items of different types can be saved together in the same
+file.
+ 
+\xtc{
+To create a library, you supply a file name.
+}{
+\spadpaste{stuff := library "/tmp/Neat.stuff" \bound{stuff}}
+}
+\xtc{
+Now values can be saved by key in the file.
+The keys should be mnemonic, just as the field names are for records.
+They can be given either as strings or symbols.
+}{
+\spadpaste{stuff.int    := 32**2     \free{stuff}\bound{stuffa}}
+}
+\xtc{
+}{
+\spadpaste{stuff."poly" := x**2 + 1  \free{stuffa}\bound{stuffb}}
+}
+\xtc{
+}{
+\spadpaste{stuff.str    := "Hello"   \free{stuffb}\bound{stuffc}}
+}
+\xtc{
+You obtain
+the set of available keys using the \spadfunFrom{keys}{Library} operation.
+}{
+\spadpaste{keys stuff \free{stuffa,stuffb,stuffc}\bound{stuffabc}}
+}
+\xtc{
+You extract values  by giving the desired key in this way.
+}{
+\spadpaste{stuff.poly     \free{stuffb}}
+}
+\xtc{
+}{
+\spadpaste{stuff("poly")  \free{stuffb}}
+}
+\noOutputXtc{
+When the file is no longer needed, you should remove it from the
+file system.
+}{
+\spadpaste{)system rm -rf /tmp/Neat.stuff  \free{stuff}\bound{rmstuff}}
+}
+ 
+For more information on related topics, see
+\downlink{`File'}{FileXmpPage}\ignore{File},
+\downlink{`TextFile'}{TextFileXmpPage}\ignore{TextFile}, and
+\downlink{`KeyedAccessFile'}{KeyedAccessFileXmpPage}\ignore{KeyedAccessFile}.
+\showBlurb{Library}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/LIB.pht b/src/hyper/pages/LIB.pht
new file mode 100644
index 00000000..27593b08
--- /dev/null
+++ b/src/hyper/pages/LIB.pht
@@ -0,0 +1,129 @@
+\begin{patch}{LibraryXmpPagePatch1}
+\begin{paste}{LibraryXmpPageFull1}{LibraryXmpPageEmpty1}
+\pastebutton{LibraryXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{stuff := library "/tmp/Neat.stuff"\bound{stuff }}
+\indentrel{3}\begin{verbatim}
+   (1)  "/tmp/Neat.stuff"
+                                          Type: Library
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPageEmpty1}
+\begin{paste}{LibraryXmpPageEmpty1}{LibraryXmpPagePatch1}
+\pastebutton{LibraryXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{stuff := library "/tmp/Neat.stuff"\bound{stuff }}
+\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPagePatch2}
+\begin{paste}{LibraryXmpPageFull2}{LibraryXmpPageEmpty2}
+\pastebutton{LibraryXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{stuff.int := 32**2\free{stuff }\bound{stuffa }}
+\indentrel{3}\begin{verbatim}
+   (2)  1024
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPageEmpty2}
+\begin{paste}{LibraryXmpPageEmpty2}{LibraryXmpPagePatch2}
+\pastebutton{LibraryXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{stuff.int := 32**2\free{stuff }\bound{stuffa }}
+\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPagePatch3}
+\begin{paste}{LibraryXmpPageFull3}{LibraryXmpPageEmpty3}
+\pastebutton{LibraryXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{stuff."poly" := x**2 + 1\free{stuffa }\bound{stuffb }}
+\indentrel{3}\begin{verbatim}
+         2
+   (3)  x  + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPageEmpty3}
+\begin{paste}{LibraryXmpPageEmpty3}{LibraryXmpPagePatch3}
+\pastebutton{LibraryXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{stuff."poly" := x**2 + 1\free{stuffa }\bound{stuffb }}
+\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPagePatch4}
+\begin{paste}{LibraryXmpPageFull4}{LibraryXmpPageEmpty4}
+\pastebutton{LibraryXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{stuff.str := "Hello"\free{stuffb }\bound{stuffc }}
+\indentrel{3}\begin{verbatim}
+   (4)  "Hello"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPageEmpty4}
+\begin{paste}{LibraryXmpPageEmpty4}{LibraryXmpPagePatch4}
+\pastebutton{LibraryXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{stuff.str := "Hello"\free{stuffb }\bound{stuffc }}
+\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPagePatch5}
+\begin{paste}{LibraryXmpPageFull5}{LibraryXmpPageEmpty5}
+\pastebutton{LibraryXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{keys stuff\free{stuffa stuffb stuffc }\bound{stuffabc }}
+\indentrel{3}\begin{verbatim}
+   (5)  ["str","poly","int"]
+                                      Type: List String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPageEmpty5}
+\begin{paste}{LibraryXmpPageEmpty5}{LibraryXmpPagePatch5}
+\pastebutton{LibraryXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{keys stuff\free{stuffa stuffb stuffc }\bound{stuffabc }}
+\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPagePatch6}
+\begin{paste}{LibraryXmpPageFull6}{LibraryXmpPageEmpty6}
+\pastebutton{LibraryXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{stuff.poly\free{stuffb }}
+\indentrel{3}\begin{verbatim}
+         2
+   (6)  x  + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPageEmpty6}
+\begin{paste}{LibraryXmpPageEmpty6}{LibraryXmpPagePatch6}
+\pastebutton{LibraryXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{stuff.poly\free{stuffb }}
+\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPagePatch7}
+\begin{paste}{LibraryXmpPageFull7}{LibraryXmpPageEmpty7}
+\pastebutton{LibraryXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{stuff("poly")\free{stuffb }}
+\indentrel{3}\begin{verbatim}
+         2
+   (7)  x  + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPageEmpty7}
+\begin{paste}{LibraryXmpPageEmpty7}{LibraryXmpPagePatch7}
+\pastebutton{LibraryXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{stuff("poly")\free{stuffb }}
+\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPagePatch8}
+\begin{paste}{LibraryXmpPageFull8}{LibraryXmpPageEmpty8}
+\pastebutton{LibraryXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{)system rm -rf /tmp/Neat.stuff\free{stuff }\bound{rmstuff }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LibraryXmpPageEmpty8}
+\begin{paste}{LibraryXmpPageEmpty8}{LibraryXmpPagePatch8}
+\pastebutton{LibraryXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{)system rm -rf /tmp/Neat.stuff\free{stuff }\bound{rmstuff }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/LIST.ht b/src/hyper/pages/LIST.ht
new file mode 100644
index 00000000..645cf57b
--- /dev/null
+++ b/src/hyper/pages/LIST.ht
@@ -0,0 +1,338 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\ListXmpTitle}{List}
+\newcommand{\ListXmpNumber}{9.47}
+%
+% =====================================================================
+\begin{page}{ListXmpPage}{9.47 List}
+% =====================================================================
+\beginscroll
+A \spadgloss{list} is a finite collection of elements in a specified
+order that can contain duplicates.
+A list is a convenient structure to work with because it is easy
+to add or remove elements and the length need not be constant.
+There are many different kinds of lists in \Language{}, but the
+default types (and those used most often) are created by the
+\spadtype{List} constructor.
+For example, there are objects of type \spadtype{List Integer},
+\spadtype{List Float} and \spadtype{List Polynomial Fraction Integer}.
+Indeed, you can even have \spadtype{List List List Boolean}
+(that is, lists of lists of lists of Boolean values).
+You can have lists of any type of \Language{} object.
+
+\beginmenu
+    \menudownlink{{9.47.1. Creating Lists}}{ugxListCreatePage}
+    \menudownlink{{9.47.2. Accessing List Elements}}{ugxListAccessPage}
+    \menudownlink{{9.47.3. Changing List Elements}}{ugxListChangePage}
+    \menudownlink{{9.47.4. Other Functions}}{ugxListOtherPage}
+    \menudownlink{{9.47.5. Dot, Dot}}{ugxListDotPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxListCreateTitle}{Creating Lists}
+\newcommand{\ugxListCreateNumber}{9.47.1.}
+%
+% =====================================================================
+\begin{page}{ugxListCreatePage}{9.47.1. Creating Lists}
+% =====================================================================
+\beginscroll
+
+The easiest way to create a list with, for example, the elements
+\spad{2, 4, 5, 6} is to enclose the elements with square
+brackets and separate the elements with commas.
+\xtc{
+The spaces after the commas are optional, but they do improve the
+readability.
+}{
+\spadpaste{[2, 4, 5, 6]}
+}
+\xtc{
+To create a list with the single element \spad{1}, you can use
+either \spad{[1]} or the operation \spadfunFrom{list}{List}.
+}{
+\spadpaste{[1]}
+}
+\xtc{
+}{
+\spadpaste{list(1)}
+}
+\xtc{
+Once created, two lists \spad{k} and \spad{m} can be
+concatenated by issuing \spad{append(k,m)}.
+\spadfunFrom{append}{List} does {\it not} physically join the lists,
+but rather produces a new list with the elements coming from the two
+arguments.
+}{
+\spadpaste{append([1,2,3],[5,6,7])}
+}
+\xtc{
+Use \spadfunFrom{cons}{List} to append an element onto the front of a
+list.
+}{
+\spadpaste{cons(10,[9,8,7])}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxListAccessTitle}{Accessing List Elements}
+\newcommand{\ugxListAccessNumber}{9.47.2.}
+%
+% =====================================================================
+\begin{page}{ugxListAccessPage}{9.47.2. Accessing List Elements}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+\xtc{
+To determine whether a list has any elements, use the operation
+\spadfunFrom{empty?}{List}.
+}{
+\spadpaste{empty? [x+1]}
+}
+\xtc{
+Alternatively, equality with the list constant \spadfunFrom{nil}{List} can
+be tested.
+}{
+\spadpaste{([] = nil)@Boolean}
+}
+
+\xtc{
+We'll use this in some of the following examples.
+}{
+\spadpaste{k := [4,3,7,3,8,5,9,2] \bound{k}}
+}
+\xtc{
+Each of the next four expressions extracts the \spadfunFrom{first}{List}
+element of \spad{k}.
+}{
+\spadpaste{first k \free{k}}
+}
+\xtc{
+}{
+\spadpaste{k.first \free{k}}
+}
+\xtc{
+}{
+\spadpaste{k.1 \free{k}}
+}
+\xtc{
+}{
+\spadpaste{k(1) \free{k}}
+}
+The last two forms generalize to \spad{k.i} and \spad{k(i)},
+respectively, where
+\texht{$ 1 \leq i \leq n$}{\spad{1 <= i <= n}}
+and
+\spad{n} equals the length of \spad{k}.
+\xtc{
+This length is calculated by \spadopFrom{\#}{List}.
+}{
+\spadpaste{n := \#k \free{k}}
+}
+Performing an operation such as \spad{k.i} is sometimes
+referred to as {\it indexing into k} or
+{\it elting into k}.
+The latter phrase comes about because the name of the operation
+that extracts elements is called \spadfunFrom{elt}{List}.
+That is, \spad{k.3} is just alternative syntax for \spad{elt(k,3)}.
+It is important to remember that list indices
+begin with 1.
+If we issue \spad{k := [1,3,2,9,5]} then \spad{k.4}
+returns \spad{9}.
+It is an error to use an index that is not in the range from
+\spad{1} to the length of the list.
+
+\xtc{
+The last element of a list is extracted by any of the
+following three expressions.
+}{
+\spadpaste{last k \free{k}}
+}
+\xtc{
+}{
+\spadpaste{k.last \free{k}}
+}
+\xtc{
+This form computes the index of the last element and then
+extracts the element from the list.
+}{
+\spadpaste{k.(\#k) \free{k}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxListChangeTitle}{Changing List Elements}
+\newcommand{\ugxListChangeNumber}{9.47.3.}
+%
+% =====================================================================
+\begin{page}{ugxListChangePage}{9.47.3. Changing List Elements}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+\xtc{
+We'll use this in some of the following examples.
+}{
+\spadpaste{k := [4,3,7,3,8,5,9,2] \bound{k}}
+}
+\xtc{
+List elements are reset by using the \spad{k.i} form on
+the left-hand side of an assignment.
+This expression resets the first element of \spad{k} to \spad{999}.
+}{
+\spadpaste{k.1 := 999 \free{k}\bound{k1}}
+}
+\xtc{
+As with indexing into a list, it is an error to use an index
+that is not within the proper bounds.
+Here you see that \spad{k} was modified.
+}{
+\spadpaste{k \free{k1}}
+}
+
+The operation that performs the assignment of an element to a particular
+position in a list is called \spadfunFrom{setelt}{List}.
+This operation is {\it destructive} in that it changes the list.
+In the above example, the assignment returned the value \spad{999} and
+\spad{k} was modified.
+For this reason, lists are called \spadglos{mutable} objects: it is
+possible to change part of a list (mutate it) rather than always returning
+a new list reflecting the intended modifications.
+\xtc{
+Moreover, since lists can share structure, changes to one list can
+sometimes affect others.
+}{
+\spadpaste{k := [1,2] \bound{k2}}
+}
+\xtc{
+}{
+\spadpaste{m := cons(0,k) \free{k2}\bound{m}}
+}
+\xtc{
+Change the second element of \spad{m}.
+}{
+\spadpaste{m.2 := 99 \free{m}\bound{m2}}
+}
+\xtc{
+See, \spad{m} was altered.
+}{
+\spadpaste{m \free{m2}}
+}
+\xtc{
+But what about \spad{k}?
+It changed too!
+}{
+\spadpaste{k  \free{m2 k2}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxListOtherTitle}{Other Functions}
+\newcommand{\ugxListOtherNumber}{9.47.4.}
+%
+% =====================================================================
+\begin{page}{ugxListOtherPage}{9.47.4. Other Functions}
+% =====================================================================
+\beginscroll
+
+\labelSpace{3pc}
+\xtc{
+An operation that is used frequently in list processing is that
+which returns all elements in a list after the first element.
+}{
+\spadpaste{k := [1,2,3] \bound{k}}
+}
+\xtc{
+Use the \spadfunFrom{rest}{List} operation to do this.
+}{
+\spadpaste{rest k \free{k}}
+}
+
+\xtc{
+To remove duplicate elements in a list \spad{k}, use
+\spadfunFrom{removeDuplicates}{List}.
+}{
+\spadpaste{removeDuplicates [4,3,4,3,5,3,4]}
+}
+\xtc{
+To get a list with elements in the order opposite to those in
+a list \spad{k}, use \spadfunFrom{reverse}{List}.
+}{
+\spadpaste{reverse [1,2,3,4,5,6]}
+}
+\xtc{
+To test whether an element is in a list, use \spadfunFrom{member?}{List}:
+\spad{member?(a,k)} returns \spad{true} or \spad{false}
+depending on whether \spad{a} is in \spad{k} or not.
+}{
+\spadpaste{member?(1/2,[3/4,5/6,1/2])}
+}
+\xtc{
+}{
+\spadpaste{member?(1/12,[3/4,5/6,1/2])}
+}
+
+As an exercise, the reader should determine how to get a
+list containing all but the last of the elements in a given non-empty
+list \spad{k}.\footnote{\spad{reverse(rest(reverse(k)))} works.}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxListDotTitle}{Dot, Dot}
+\newcommand{\ugxListDotNumber}{9.47.5.}
+%
+% =====================================================================
+\begin{page}{ugxListDotPage}{9.47.5. Dot, Dot}
+% =====================================================================
+\beginscroll
+
+Certain lists are used so often that \Language{} provides an easy way
+of constructing them.
+If \spad{n} and \spad{m} are integers, then \spad{expand [n..m]}
+creates a list containing \spad{n, n+1, ... m}.
+If \spad{n > m} then the list is empty.
+It is actually permissible to leave off the \spad{m} in the
+dot-dot construction (see below).
+
+\xtc{
+The dot-dot notation can be used more than once in a list
+construction and with specific elements being given.
+Items separated by dots are called {\it segments.}
+%-% \HDindex{segment}{ugxListDotPage}{9.47.5.}{Dot, Dot}
+}{
+\spadpaste{[1..3,10,20..23]}
+}
+\xtc{
+Segments can be expanded into the range of items between the
+endpoints by using \spadfunFrom{expand}{Segment}.
+}{
+\spadpaste{expand [1..3,10,20..23]}
+}
+\xtc{
+What happens if we leave off a number on the right-hand side of
+\spadopFrom{..}{UniversalSegment}?
+}{
+\spadpaste{expand [1..]}
+}
+What is created in this case is a \spadtype{Stream} which is a
+generalization of a list.
+See \downlink{`Stream'}{StreamXmpPage}\ignore{Stream} for more information.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/LIST.pht b/src/hyper/pages/LIST.pht
new file mode 100644
index 00000000..59ecb371
--- /dev/null
+++ b/src/hyper/pages/LIST.pht
@@ -0,0 +1,528 @@
+\begin{patch}{ugxListDotPagePatch1}
+\begin{paste}{ugxListDotPageFull1}{ugxListDotPageEmpty1}
+\pastebutton{ugxListDotPageFull1}{\hidepaste}
+\tab{5}\spadcommand{[1..3,10,20..23]}
+\indentrel{3}\begin{verbatim}
+   (1)  [1..3,10..10,20..23]
+                     Type: List Segment PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListDotPageEmpty1}
+\begin{paste}{ugxListDotPageEmpty1}{ugxListDotPagePatch1}
+\pastebutton{ugxListDotPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{[1..3,10,20..23]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListDotPagePatch2}
+\begin{paste}{ugxListDotPageFull2}{ugxListDotPageEmpty2}
+\pastebutton{ugxListDotPageFull2}{\hidepaste}
+\tab{5}\spadcommand{expand [1..3,10,20..23]}
+\indentrel{3}\begin{verbatim}
+   (2)  [1,2,3,10,20,21,22,23]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListDotPageEmpty2}
+\begin{paste}{ugxListDotPageEmpty2}{ugxListDotPagePatch2}
+\pastebutton{ugxListDotPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{expand [1..3,10,20..23]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListDotPagePatch3}
+\begin{paste}{ugxListDotPageFull3}{ugxListDotPageEmpty3}
+\pastebutton{ugxListDotPageFull3}{\hidepaste}
+\tab{5}\spadcommand{expand [1..]}
+\indentrel{3}\begin{verbatim}
+   (3)  [1,2,3,4,5,6,7,8,9,10,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListDotPageEmpty3}
+\begin{paste}{ugxListDotPageEmpty3}{ugxListDotPagePatch3}
+\pastebutton{ugxListDotPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{expand [1..]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListOtherPagePatch1}
+\begin{paste}{ugxListOtherPageFull1}{ugxListOtherPageEmpty1}
+\pastebutton{ugxListOtherPageFull1}{\hidepaste}
+\tab{5}\spadcommand{k := [1,2,3]\bound{k }}
+\indentrel{3}\begin{verbatim}
+   (1)  [1,2,3]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListOtherPageEmpty1}
+\begin{paste}{ugxListOtherPageEmpty1}{ugxListOtherPagePatch1}
+\pastebutton{ugxListOtherPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{k := [1,2,3]\bound{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListOtherPagePatch2}
+\begin{paste}{ugxListOtherPageFull2}{ugxListOtherPageEmpty2}
+\pastebutton{ugxListOtherPageFull2}{\hidepaste}
+\tab{5}\spadcommand{rest k\free{k }}
+\indentrel{3}\begin{verbatim}
+   (2)  [2,3]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListOtherPageEmpty2}
+\begin{paste}{ugxListOtherPageEmpty2}{ugxListOtherPagePatch2}
+\pastebutton{ugxListOtherPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{rest k\free{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListOtherPagePatch3}
+\begin{paste}{ugxListOtherPageFull3}{ugxListOtherPageEmpty3}
+\pastebutton{ugxListOtherPageFull3}{\hidepaste}
+\tab{5}\spadcommand{removeDuplicates [4,3,4,3,5,3,4]}
+\indentrel{3}\begin{verbatim}
+   (3)  [4,3,5]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListOtherPageEmpty3}
+\begin{paste}{ugxListOtherPageEmpty3}{ugxListOtherPagePatch3}
+\pastebutton{ugxListOtherPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{removeDuplicates [4,3,4,3,5,3,4]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListOtherPagePatch4}
+\begin{paste}{ugxListOtherPageFull4}{ugxListOtherPageEmpty4}
+\pastebutton{ugxListOtherPageFull4}{\hidepaste}
+\tab{5}\spadcommand{reverse [1,2,3,4,5,6]}
+\indentrel{3}\begin{verbatim}
+   (4)  [6,5,4,3,2,1]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListOtherPageEmpty4}
+\begin{paste}{ugxListOtherPageEmpty4}{ugxListOtherPagePatch4}
+\pastebutton{ugxListOtherPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{reverse [1,2,3,4,5,6]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListOtherPagePatch5}
+\begin{paste}{ugxListOtherPageFull5}{ugxListOtherPageEmpty5}
+\pastebutton{ugxListOtherPageFull5}{\hidepaste}
+\tab{5}\spadcommand{member?(1/2,[3/4,5/6,1/2])}
+\indentrel{3}\begin{verbatim}
+   (5)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListOtherPageEmpty5}
+\begin{paste}{ugxListOtherPageEmpty5}{ugxListOtherPagePatch5}
+\pastebutton{ugxListOtherPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{member?(1/2,[3/4,5/6,1/2])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListOtherPagePatch6}
+\begin{paste}{ugxListOtherPageFull6}{ugxListOtherPageEmpty6}
+\pastebutton{ugxListOtherPageFull6}{\hidepaste}
+\tab{5}\spadcommand{member?(1/12,[3/4,5/6,1/2])}
+\indentrel{3}\begin{verbatim}
+   (6)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListOtherPageEmpty6}
+\begin{paste}{ugxListOtherPageEmpty6}{ugxListOtherPagePatch6}
+\pastebutton{ugxListOtherPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{member?(1/12,[3/4,5/6,1/2])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePagePatch1}
+\begin{paste}{ugxListChangePageFull1}{ugxListChangePageEmpty1}
+\pastebutton{ugxListChangePageFull1}{\hidepaste}
+\tab{5}\spadcommand{k := [4,3,7,3,8,5,9,2]\bound{k }}
+\indentrel{3}\begin{verbatim}
+   (1)  [4,3,7,3,8,5,9,2]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePageEmpty1}
+\begin{paste}{ugxListChangePageEmpty1}{ugxListChangePagePatch1}
+\pastebutton{ugxListChangePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{k := [4,3,7,3,8,5,9,2]\bound{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePagePatch2}
+\begin{paste}{ugxListChangePageFull2}{ugxListChangePageEmpty2}
+\pastebutton{ugxListChangePageFull2}{\hidepaste}
+\tab{5}\spadcommand{k.1 := 999\free{k }\bound{k1 }}
+\indentrel{3}\begin{verbatim}
+   (2)  999
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePageEmpty2}
+\begin{paste}{ugxListChangePageEmpty2}{ugxListChangePagePatch2}
+\pastebutton{ugxListChangePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{k.1 := 999\free{k }\bound{k1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePagePatch3}
+\begin{paste}{ugxListChangePageFull3}{ugxListChangePageEmpty3}
+\pastebutton{ugxListChangePageFull3}{\hidepaste}
+\tab{5}\spadcommand{k\free{k1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  [999,3,7,3,8,5,9,2]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePageEmpty3}
+\begin{paste}{ugxListChangePageEmpty3}{ugxListChangePagePatch3}
+\pastebutton{ugxListChangePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{k\free{k1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePagePatch4}
+\begin{paste}{ugxListChangePageFull4}{ugxListChangePageEmpty4}
+\pastebutton{ugxListChangePageFull4}{\hidepaste}
+\tab{5}\spadcommand{k := [1,2]\bound{k2 }}
+\indentrel{3}\begin{verbatim}
+   (4)  [1,2]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePageEmpty4}
+\begin{paste}{ugxListChangePageEmpty4}{ugxListChangePagePatch4}
+\pastebutton{ugxListChangePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{k := [1,2]\bound{k2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePagePatch5}
+\begin{paste}{ugxListChangePageFull5}{ugxListChangePageEmpty5}
+\pastebutton{ugxListChangePageFull5}{\hidepaste}
+\tab{5}\spadcommand{m := cons(0,k)\free{k2 }\bound{m }}
+\indentrel{3}\begin{verbatim}
+   (5)  [0,1,2]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePageEmpty5}
+\begin{paste}{ugxListChangePageEmpty5}{ugxListChangePagePatch5}
+\pastebutton{ugxListChangePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{m := cons(0,k)\free{k2 }\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePagePatch6}
+\begin{paste}{ugxListChangePageFull6}{ugxListChangePageEmpty6}
+\pastebutton{ugxListChangePageFull6}{\hidepaste}
+\tab{5}\spadcommand{m.2 := 99\free{m }\bound{m2 }}
+\indentrel{3}\begin{verbatim}
+   (6)  99
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePageEmpty6}
+\begin{paste}{ugxListChangePageEmpty6}{ugxListChangePagePatch6}
+\pastebutton{ugxListChangePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{m.2 := 99\free{m }\bound{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePagePatch7}
+\begin{paste}{ugxListChangePageFull7}{ugxListChangePageEmpty7}
+\pastebutton{ugxListChangePageFull7}{\hidepaste}
+\tab{5}\spadcommand{m\free{m2 }}
+\indentrel{3}\begin{verbatim}
+   (7)  [0,99,2]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePageEmpty7}
+\begin{paste}{ugxListChangePageEmpty7}{ugxListChangePagePatch7}
+\pastebutton{ugxListChangePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{m\free{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePagePatch8}
+\begin{paste}{ugxListChangePageFull8}{ugxListChangePageEmpty8}
+\pastebutton{ugxListChangePageFull8}{\hidepaste}
+\tab{5}\spadcommand{k\free{m2 k2 }}
+\indentrel{3}\begin{verbatim}
+   (8)  [99,2]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListChangePageEmpty8}
+\begin{paste}{ugxListChangePageEmpty8}{ugxListChangePagePatch8}
+\pastebutton{ugxListChangePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{k\free{m2 k2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPagePatch1}
+\begin{paste}{ugxListAccessPageFull1}{ugxListAccessPageEmpty1}
+\pastebutton{ugxListAccessPageFull1}{\hidepaste}
+\tab{5}\spadcommand{empty? [x+1]}
+\indentrel{3}\begin{verbatim}
+   (1)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPageEmpty1}
+\begin{paste}{ugxListAccessPageEmpty1}{ugxListAccessPagePatch1}
+\pastebutton{ugxListAccessPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{empty? [x+1]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPagePatch2}
+\begin{paste}{ugxListAccessPageFull2}{ugxListAccessPageEmpty2}
+\pastebutton{ugxListAccessPageFull2}{\hidepaste}
+\tab{5}\spadcommand{([] = nil)@Boolean}
+\indentrel{3}\begin{verbatim}
+   (2)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPageEmpty2}
+\begin{paste}{ugxListAccessPageEmpty2}{ugxListAccessPagePatch2}
+\pastebutton{ugxListAccessPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{([] = nil)@Boolean}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPagePatch3}
+\begin{paste}{ugxListAccessPageFull3}{ugxListAccessPageEmpty3}
+\pastebutton{ugxListAccessPageFull3}{\hidepaste}
+\tab{5}\spadcommand{k := [4,3,7,3,8,5,9,2]\bound{k }}
+\indentrel{3}\begin{verbatim}
+   (3)  [4,3,7,3,8,5,9,2]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPageEmpty3}
+\begin{paste}{ugxListAccessPageEmpty3}{ugxListAccessPagePatch3}
+\pastebutton{ugxListAccessPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{k := [4,3,7,3,8,5,9,2]\bound{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPagePatch4}
+\begin{paste}{ugxListAccessPageFull4}{ugxListAccessPageEmpty4}
+\pastebutton{ugxListAccessPageFull4}{\hidepaste}
+\tab{5}\spadcommand{first k\free{k }}
+\indentrel{3}\begin{verbatim}
+   (4)  4
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPageEmpty4}
+\begin{paste}{ugxListAccessPageEmpty4}{ugxListAccessPagePatch4}
+\pastebutton{ugxListAccessPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{first k\free{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPagePatch5}
+\begin{paste}{ugxListAccessPageFull5}{ugxListAccessPageEmpty5}
+\pastebutton{ugxListAccessPageFull5}{\hidepaste}
+\tab{5}\spadcommand{k.first\free{k }}
+\indentrel{3}\begin{verbatim}
+   (5)  4
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPageEmpty5}
+\begin{paste}{ugxListAccessPageEmpty5}{ugxListAccessPagePatch5}
+\pastebutton{ugxListAccessPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{k.first\free{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPagePatch6}
+\begin{paste}{ugxListAccessPageFull6}{ugxListAccessPageEmpty6}
+\pastebutton{ugxListAccessPageFull6}{\hidepaste}
+\tab{5}\spadcommand{k.1\free{k }}
+\indentrel{3}\begin{verbatim}
+   (6)  4
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPageEmpty6}
+\begin{paste}{ugxListAccessPageEmpty6}{ugxListAccessPagePatch6}
+\pastebutton{ugxListAccessPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{k.1\free{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPagePatch7}
+\begin{paste}{ugxListAccessPageFull7}{ugxListAccessPageEmpty7}
+\pastebutton{ugxListAccessPageFull7}{\hidepaste}
+\tab{5}\spadcommand{k(1)\free{k }}
+\indentrel{3}\begin{verbatim}
+   (7)  4
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPageEmpty7}
+\begin{paste}{ugxListAccessPageEmpty7}{ugxListAccessPagePatch7}
+\pastebutton{ugxListAccessPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{k(1)\free{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPagePatch8}
+\begin{paste}{ugxListAccessPageFull8}{ugxListAccessPageEmpty8}
+\pastebutton{ugxListAccessPageFull8}{\hidepaste}
+\tab{5}\spadcommand{n := \#k\free{k }}
+\indentrel{3}\begin{verbatim}
+   (8)  8
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPageEmpty8}
+\begin{paste}{ugxListAccessPageEmpty8}{ugxListAccessPagePatch8}
+\pastebutton{ugxListAccessPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{n := \#k\free{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPagePatch9}
+\begin{paste}{ugxListAccessPageFull9}{ugxListAccessPageEmpty9}
+\pastebutton{ugxListAccessPageFull9}{\hidepaste}
+\tab{5}\spadcommand{last k\free{k }}
+\indentrel{3}\begin{verbatim}
+   (9)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPageEmpty9}
+\begin{paste}{ugxListAccessPageEmpty9}{ugxListAccessPagePatch9}
+\pastebutton{ugxListAccessPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{last k\free{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPagePatch10}
+\begin{paste}{ugxListAccessPageFull10}{ugxListAccessPageEmpty10}
+\pastebutton{ugxListAccessPageFull10}{\hidepaste}
+\tab{5}\spadcommand{k.last\free{k }}
+\indentrel{3}\begin{verbatim}
+   (10)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPageEmpty10}
+\begin{paste}{ugxListAccessPageEmpty10}{ugxListAccessPagePatch10}
+\pastebutton{ugxListAccessPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{k.last\free{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPagePatch11}
+\begin{paste}{ugxListAccessPageFull11}{ugxListAccessPageEmpty11}
+\pastebutton{ugxListAccessPageFull11}{\hidepaste}
+\tab{5}\spadcommand{k.(\#k)\free{k }}
+\indentrel{3}\begin{verbatim}
+   (11)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListAccessPageEmpty11}
+\begin{paste}{ugxListAccessPageEmpty11}{ugxListAccessPagePatch11}
+\pastebutton{ugxListAccessPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{k.(\#k)\free{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListCreatePagePatch1}
+\begin{paste}{ugxListCreatePageFull1}{ugxListCreatePageEmpty1}
+\pastebutton{ugxListCreatePageFull1}{\hidepaste}
+\tab{5}\spadcommand{[2, 4, 5, 6]}
+\indentrel{3}\begin{verbatim}
+   (1)  [2,4,5,6]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListCreatePageEmpty1}
+\begin{paste}{ugxListCreatePageEmpty1}{ugxListCreatePagePatch1}
+\pastebutton{ugxListCreatePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{[2, 4, 5, 6]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListCreatePagePatch2}
+\begin{paste}{ugxListCreatePageFull2}{ugxListCreatePageEmpty2}
+\pastebutton{ugxListCreatePageFull2}{\hidepaste}
+\tab{5}\spadcommand{[1]}
+\indentrel{3}\begin{verbatim}
+   (2)  [1]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListCreatePageEmpty2}
+\begin{paste}{ugxListCreatePageEmpty2}{ugxListCreatePagePatch2}
+\pastebutton{ugxListCreatePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{[1]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListCreatePagePatch3}
+\begin{paste}{ugxListCreatePageFull3}{ugxListCreatePageEmpty3}
+\pastebutton{ugxListCreatePageFull3}{\hidepaste}
+\tab{5}\spadcommand{list(1)}
+\indentrel{3}\begin{verbatim}
+   (3)  [1]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListCreatePageEmpty3}
+\begin{paste}{ugxListCreatePageEmpty3}{ugxListCreatePagePatch3}
+\pastebutton{ugxListCreatePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{list(1)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListCreatePagePatch4}
+\begin{paste}{ugxListCreatePageFull4}{ugxListCreatePageEmpty4}
+\pastebutton{ugxListCreatePageFull4}{\hidepaste}
+\tab{5}\spadcommand{append([1,2,3],[5,6,7])}
+\indentrel{3}\begin{verbatim}
+   (4)  [1,2,3,5,6,7]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListCreatePageEmpty4}
+\begin{paste}{ugxListCreatePageEmpty4}{ugxListCreatePagePatch4}
+\pastebutton{ugxListCreatePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{append([1,2,3],[5,6,7])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxListCreatePagePatch5}
+\begin{paste}{ugxListCreatePageFull5}{ugxListCreatePageEmpty5}
+\pastebutton{ugxListCreatePageFull5}{\hidepaste}
+\tab{5}\spadcommand{cons(10,[9,8,7])}
+\indentrel{3}\begin{verbatim}
+   (5)  [10,9,8,7]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxListCreatePageEmpty5}
+\begin{paste}{ugxListCreatePageEmpty5}{ugxListCreatePagePatch5}
+\pastebutton{ugxListCreatePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{cons(10,[9,8,7])}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/LODO.ht b/src/hyper/pages/LODO.ht
new file mode 100644
index 00000000..02174a75
--- /dev/null
+++ b/src/hyper/pages/LODO.ht
@@ -0,0 +1,118 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\LinearOrdinaryDifferentialOperatorXmpTitle}{LinearOrdinaryDifferentialOperator}
+\newcommand{\LinearOrdinaryDifferentialOperatorXmpNumber}{9.44}
+%
+% =====================================================================
+\begin{page}{LinearOrdinaryDifferentialOperatorXmpPage}{9.44 LinearOrdinaryDifferentialOperator}
+% =====================================================================
+\beginscroll
+
+\spadtype{LinearOrdinaryDifferentialOperator(A, diff)} is the domain of linear
+%-% \HDindex{operator!linear ordinary differential}{LinearOrdinaryDifferentialOperatorXmpPage}{9.44}{LinearOrdinaryDifferentialOperator}
+ordinary differential operators with coefficients in a ring
+\spad{A} with a given derivation.
+%This includes the cases of operators which are polynomials in \spad{D}
+%acting upon scalar or vector expressions of a single variable.
+%The coefficients of the operator polynomials can be integers, rational
+%functions, matrices or elements of other domains.
+\showBlurb{LinearOrdinaryDifferentialOperator}
+
+\beginmenu
+    \menudownlink{{9.44.1. Differential Operators with Series Coefficients}}{ugxLinearOrdinaryDifferentialOperatorSeriesPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxLinearOrdinaryDifferentialOperatorSeriesTitle}{Differential Operators with Series Coefficients}
+\newcommand{\ugxLinearOrdinaryDifferentialOperatorSeriesNumber}{9.44.1.}
+%
+% =====================================================================
+\begin{page}{ugxLinearOrdinaryDifferentialOperatorSeriesPage}{9.44.1. Differential Operators with Series Coefficients}
+% =====================================================================
+\beginscroll
+
+\noindent
+{\bf Problem:}
+Find the first few coefficients of \spad{exp(x)/x**i} of \spad{Dop phi} where
+\begin{verbatim}
+Dop := D**3 + G/x**2 * D + H/x**3 - 1
+phi := sum(s[i]*exp(x)/x**i, i = 0..)
+\end{verbatim}
+
+\noindent
+{\bf Solution:}
+\xtc{
+Define the differential.
+}{
+\spadpaste{Dx: LODO(EXPR INT, f +-> D(f, x)) \bound{Dxd}}
+}
+\xtc{
+}{
+\spadpaste{Dx := D() \free{Dxd}\bound{Dx}}
+}
+\xtc{
+Now define the differential operator \spad{Dop}.
+}{
+\spadpaste{Dop:= Dx**3 + G/x**2*Dx + H/x**3 - 1 \free{Dx}\bound{Dop}}
+}
+\xtc{
+}{
+\spadpaste{n == 3 \bound{n3}}
+}
+\xtc{
+}{
+\spadpaste{phi == reduce(+,[subscript(s,[i])*exp(x)/x**i for i in 0..n]) \bound{phi}}
+}
+\xtc{
+}{
+\spadpaste{phi1 ==  Dop(phi) / exp x \bound{phi1}\free{Dop phi}}
+}
+\xtc{
+}{
+\spadpaste{phi2 == phi1 *x**(n+3) \bound{phi2}\free{phi1}}
+}
+\xtc{
+}{
+\spadpaste{phi3 == retract(phi2)@(POLY INT) \bound{phi3}\free{phi2}}
+}
+\xtc{
+}{
+\spadpaste{pans == phi3 ::UP(x,POLY INT) \free{phi3}\bound{pans}}
+}
+\xtc{
+}{
+\spadpaste{pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1] \bound{pans1}\free{pans}}
+}
+\xtc{
+}{
+\spadpaste{leq == solve(pans1,[subscript(s,[i]) for i in 1..n]) \bound{leq}\free{pans1}}
+}
+\xtc{
+Evaluate this for several values of \spad{n}.
+}{
+\spadpaste{leq \free{n3 leq}}
+}
+\xtc{
+}{
+\spadpaste{n==4 \bound{n4}}
+}
+\xtc{
+}{
+\spadpaste{leq \free{n4 leq}}
+}
+\xtc{
+}{
+\spadpaste{n==7 \bound{n7}}
+}
+\xtc{
+}{
+\spadpaste{leq \free{n7 leq}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/LODO.pht b/src/hyper/pages/LODO.pht
new file mode 100644
index 00000000..22e701b6
--- /dev/null
+++ b/src/hyper/pages/LODO.pht
@@ -0,0 +1,391 @@
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch1}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull1}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty1}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull1}{\hidepaste}
+\tab{5}\spadcommand{Dx: LODO(EXPR INT, f +-> D(f, x))\bound{Dxd }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty1}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty1}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch1}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{Dx: LODO(EXPR INT, f +-> D(f, x))\bound{Dxd }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch2}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull2}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty2}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull2}{\hidepaste}
+\tab{5}\spadcommand{Dx := D()\free{Dxd }\bound{Dx }}
+\indentrel{3}\begin{verbatim}
+   (2)  D
+Type: LinearOrdinaryDifferentialOperator(Expression Integer,theMap NIL)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty2}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty2}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch2}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{Dx := D()\free{Dxd }\bound{Dx }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch3}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull3}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty3}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull3}{\hidepaste}
+\tab{5}\spadcommand{Dop:= Dx**3 + G/x**2*Dx + H/x**3 - 1\free{Dx }\bound{Dop }}
+\indentrel{3}\begin{verbatim}
+                       3
+         3    G     - x  + H
+   (3)  D  + ÄÄ D + ÄÄÄÄÄÄÄÄ
+              2         3
+             x         x
+Type: LinearOrdinaryDifferentialOperator(Expression Integer,theMap NIL)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty3}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty3}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch3}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{Dop:= Dx**3 + G/x**2*Dx + H/x**3 - 1\free{Dx }\bound{Dop }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch4}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull4}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty4}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull4}{\hidepaste}
+\tab{5}\spadcommand{n == 3\bound{n3 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty4}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty4}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch4}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{n == 3\bound{n3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch5}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull5}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty5}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull5}{\hidepaste}
+\tab{5}\spadcommand{phi == reduce(+,[subscript(s,[i])*exp(x)/x**i for i in 0..n])\bound{phi }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty5}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty5}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch5}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{phi == reduce(+,[subscript(s,[i])*exp(x)/x**i for i in 0..n])\bound{phi }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch6}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull6}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty6}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull6}{\hidepaste}
+\tab{5}\spadcommand{phi1 == Dop(phi) / exp x\bound{phi1 }\free{Dop phi }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty6}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty6}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch6}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{phi1 == Dop(phi) / exp x\bound{phi1 }\free{Dop phi }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch7}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull7}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty7}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull7}{\hidepaste}
+\tab{5}\spadcommand{phi2 == phi1 *x**(n+3)\bound{phi2 }\free{phi1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty7}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty7}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch7}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{phi2 == phi1 *x**(n+3)\bound{phi2 }\free{phi1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch8}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull8}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty8}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull8}{\hidepaste}
+\tab{5}\spadcommand{phi3 == retract(phi2)@(POLY INT)\bound{phi3 }\free{phi2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty8}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty8}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch8}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{phi3 == retract(phi2)@(POLY INT)\bound{phi3 }\free{phi2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch9}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull9}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty9}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull9}{\hidepaste}
+\tab{5}\spadcommand{pans == phi3 ::UP(x,POLY INT)\free{phi3 }\bound{pans }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty9}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty9}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch9}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{pans == phi3 ::UP(x,POLY INT)\free{phi3 }\bound{pans }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch10}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull10}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty10}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull10}{\hidepaste}
+\tab{5}\spadcommand{pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1]\bound{pans1 }\free{pans }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty10}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty10}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch10}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1]\bound{pans1 }\free{pans }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch11}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull11}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty11}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull11}{\hidepaste}
+\tab{5}\spadcommand{leq == solve(pans1,[subscript(s,[i]) for i in 1..n])\bound{leq }\free{pans1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty11}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty11}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch11}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{leq == solve(pans1,[subscript(s,[i]) for i in 1..n])\bound{leq }\free{pans1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch12}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull12}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty12}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull12}{\hidepaste}
+\tab{5}\spadcommand{leq\free{n3 leq }}
+\indentrel{3}\begin{verbatim}
+   (12)
+   [
+                             2
+          s G      3s H + s G  + 6s G
+           0         0     0       0
+     [s = ÄÄÄ, s = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,
+       1   3    2          18
+                              3        2
+          (9s G + 54s )H + s G  + 18s G  + 72s G
+             0       0      0        0        0
+      s = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ]
+       3                    162
+     ]
+   Type: List List Equation Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty12}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty12}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch12}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{leq\free{n3 leq }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch13}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull13}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty13}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull13}{\hidepaste}
+\tab{5}\spadcommand{n==4\bound{n4 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty13}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty13}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch13}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{n==4\bound{n4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch14}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull14}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty14}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull14}{\hidepaste}
+\tab{5}\spadcommand{leq\free{n4 leq }}
+\indentrel{3}\begin{verbatim}
+   (14)
+   [
+                             2
+          s G      3s H + s G  + 6s G
+           0         0     0       0
+     [s = ÄÄÄ, s = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,
+       1   3    2          18
+                              3        2
+          (9s G + 54s )H + s G  + 18s G  + 72s G
+             0       0      0        0        0
+      s = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,
+       3                    162
+
+       s  =
+        4
+                  2         2                          4
+             27s H  + (18s G  + 378s G + 1296s )H + s G
+                0         0         0         0      0
+           + 
+                  3         2
+             36s G  + 396s G  + 1296s G
+                0         0          0
+        /
+           1944
+       ]
+     ]
+   Type: List List Equation Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty14}
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+\tab{5}\spadcommand{leq\free{n4 leq }}
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+
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+\tab{5}\spadcommand{n==7\bound{n7 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty15}
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+\tab{5}\spadcommand{n==7\bound{n7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch16}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull16}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty16}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageFull16}{\hidepaste}
+\tab{5}\spadcommand{leq\free{n7 leq }}
+\indentrel{3}\begin{verbatim}
+   (16)
+   [
+                             2
+          s G      3s H + s G  + 6s G
+           0         0     0       0
+     [s = ÄÄÄ, s = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,
+       1   3    2          18
+                              3        2
+          (9s G + 54s )H + s G  + 18s G  + 72s G
+             0       0      0        0        0
+      s = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,
+       3                    162
+
+       s  =
+        4
+                  2         2                          4
+             27s H  + (18s G  + 378s G + 1296s )H + s G
+                0         0         0         0      0
+           + 
+                  3         2
+             36s G  + 396s G  + 1296s G
+                0         0          0
+        /
+           1944
+       ,
+
+       s  =
+        5
+                               2
+             (135s G + 2268s )H
+                  0         0
+           + 
+                   3          2
+             (30s G  + 1350s G  + 16416s G + 38880s )H
+                 0          0           0          0
+           + 
+              5        4          3          2
+           s G  + 60s G  + 1188s G  + 9504s G  + 25920s G
+            0        0          0          0           0
+        /
+           29160
+       ,
+
+       s  =
+        6
+                   3          2                        2
+             405s H  + (405s G  + 18468s G + 174960s )H
+                 0          0           0           0
+           + 
+                      4          3           2
+                 45s G  + 3510s G  + 88776s G  + 777600s G
+                    0          0           0            0
+               + 
+                 1166400s
+                         0
+            *
+               H
+           + 
+                6        5          4           3
+             s G  + 90s G  + 2628s G  + 27864s G
+              0        0          0           0
+           + 
+                     2
+             90720s G
+                   0
+        /
+           524880
+       ,
+
+       s  =
+        7
+                                 3
+             (2835s G + 91854s )H
+                   0          0
+           + 
+                       3           2
+                 945s G  + 81648s G  + 2082996s G
+                     0           0             0
+               + 
+                 14171760s
+                          0
+            *
+                2
+               H
+           + 
+                      5          4            3
+                 63s G  + 7560s G  + 317520s G
+                    0          0            0
+               + 
+                           2
+                 5554008s G  + 34058880s G
+                         0              0
+            *
+               H
+           + 
+                7         6          5           4
+             s G  + 126s G  + 4788s G  + 25272s G
+              0         0          0           0
+           + 
+                         3              2
+             - 1744416s G  - 26827200s G  - 97977600s G
+                       0              0              0
+        /
+           11022480
+       ]
+     ]
+   Type: List List Equation Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty16}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty16}{ugxLinearOrdinaryDifferentialOperatorSeriesPagePatch16}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorSeriesPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{leq\free{n7 leq }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/LODO1.ht b/src/hyper/pages/LODO1.ht
new file mode 100644
index 00000000..527ee37f
--- /dev/null
+++ b/src/hyper/pages/LODO1.ht
@@ -0,0 +1,184 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\LinearOrdinaryDifferentialOperatorOneXmpTitle}{LinearOrdinaryDifferentialOperator1}
+\newcommand{\LinearOrdinaryDifferentialOperatorOneXmpNumber}{9.45}
+%
+% =====================================================================
+\begin{page}{LinearOrdinaryDifferentialOperatorOneXmpPage}{9.45 LinearOrdinaryDifferentialOperator1}
+% =====================================================================
+\beginscroll
+
+\spadtype{LinearOrdinaryDifferentialOperator1(A)} is the domain of linear
+%-% \HDindex{operator!linear ordinary differential}{LinearOrdinaryDifferentialOperatorOneXmpPage}{9.45}{LinearOrdinaryDifferentialOperator1}
+ordinary differential operators with coefficients in the differential ring
+\spad{A}.
+%This includes the cases of operators which are polynomials in \spad{D}
+%acting upon scalar or vector expressions of a single variable.
+%The coefficients of the operator polynomials can be integers, rational
+%functions, matrices or elements of other domains.
+\showBlurb{LinearOrdinaryDifferentialOperator1}
+
+\beginmenu
+    \menudownlink{{9.45.1. Differential Operators with Rational Function Coefficients}}{ugxLinearOrdinaryDifferentialOperatorOneRatPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxLinearOrdinaryDifferentialOperatorOneRatTitle}{Differential Operators with Rational Function Coefficients}
+\newcommand{\ugxLinearOrdinaryDifferentialOperatorOneRatNumber}{9.45.1.}
+%
+% =====================================================================
+\begin{page}{ugxLinearOrdinaryDifferentialOperatorOneRatPage}{9.45.1. Differential Operators with Rational Function Coefficients}
+% =====================================================================
+\beginscroll
+
+This example shows differential operators with rational function
+coefficients.  In this case operator multiplication is non-commutative and,
+since the coefficients form a field, an operator division algorithm exists.
+
+\labelSpace{2pc}
+\xtc{
+We begin by defining \spad{RFZ} to be the rational functions in
+\spad{x} with integer coefficients and \spad{Dx} to be the differential
+operator for \spad{d/dx}.
+}{
+\spadpaste{RFZ := Fraction UnivariatePolynomial('x, Integer) \bound{RFZ0}}
+}
+\xtc{
+}{
+\spadpaste{x : RFZ := 'x \free{RFZ0}\bound{RFZ}}
+}
+\xtc{
+}{
+\spadpaste{Dx : LODO1 RFZ := D()\bound{Dx}\free{RFZ}}
+}
+\xtc{
+Operators are created using the usual arithmetic operations.
+}{
+\spadpaste{b : LODO1 RFZ := 3*x**2*Dx**2 + 2*Dx + 1/x  \free{Dx}\bound{b}}
+}
+\xtc{
+}{
+\spadpaste{a : LODO1 RFZ := b*(5*x*Dx + 7)             \free{b Dx}\bound{a}}
+}
+\xtc{
+Operator multiplication corresponds to functional composition.
+}{
+\spadpaste{p := x**2 + 1/x**2 \bound{p}\free{RFZ}}
+}
+\xtc{
+Since operator coefficients depend on \spad{x}, the multiplication is
+not commutative.
+}{
+\spadpaste{(a*b - b*a) p \free{a b p}}
+}
+
+When the coefficients of operator polynomials come from a field, as in this
+case, it is possible to define operator division.
+Division on the left and division on the right yield different results when
+the multiplication is non-commutative.
+
+The results of \spadfunFrom{leftDivide}{LinearOrdinaryDifferentialOperator1} and
+\spadfunFrom{rightDivide}{LinearOrdinaryDifferentialOperator1} are
+quotient-remainder pairs satisfying: \newline
+%
+\spad{leftDivide(a,b) = [q, r]} such that  \spad{a = b*q + r} \newline
+\spad{rightDivide(a,b) = [q, r]} such that  \spad{a = q*b + r} \newline
+%
+\xtc{
+In both cases, the \spadfunFrom{degree}{LinearOrdinaryDifferentialOperator1}
+of the remainder, \spad{r}, is less than
+the degree of \spad{b}.
+}{
+\spadpaste{ld := leftDivide(a,b) \bound{ld}\free{a b}}
+}
+\xtc{
+}{
+\spadpaste{a = b * ld.quotient + ld.remainder \free{a b ld}}
+}
+\xtc{
+The operations of left and right division
+are so-called because the quotient is obtained by dividing
+\spad{a} on that side by \spad{b}.
+}{
+\spadpaste{rd := rightDivide(a,b) \bound{rd}\free{a b}}
+}
+\xtc{
+}{
+\spadpaste{a = rd.quotient * b + rd.remainder \free{a b rd}}
+}
+
+\xtc{
+Operations \spadfunFrom{rightQuotient}{LinearOrdinaryDifferentialOperator1}
+and \spadfunFrom{rightRemainder}{LinearOrdinaryDifferentialOperator1}
+are available if only one of the quotient or remainder are of
+interest to you.
+This is the quotient from right division.
+}{
+\spadpaste{rightQuotient(a,b) \free{a b}}
+}
+\xtc{
+This is the remainder from right division.
+The corresponding ``left'' functions
+\spadfunFrom{leftQuotient}{LinearOrdinaryDifferentialOperator1} and
+\spadfunFrom{leftRemainder}{LinearOrdinaryDifferentialOperator1}
+are also available.
+}{
+\spadpaste{rightRemainder(a,b) \free{a b}}
+}
+
+\xtc{
+For exact division, the operations
+\spadfunFrom{leftExactQuotient}{LinearOrdinaryDifferentialOperator1} and
+\spadfunFrom{rightExactQuotient}{LinearOrdinaryDifferentialOperator1} are supplied.
+These return the quotient but only if the remainder is zero.
+The call \spad{rightExactQuotient(a,b)} would yield an error.
+}{
+\spadpaste{leftExactQuotient(a,b) \free{a b}}
+}
+
+\xtc{
+The division operations allow the computation of left and right greatest
+common divisors (\spadfunFrom{leftGcd}{LinearOrdinaryDifferentialOperator1} and
+\spadfunFrom{rightGcd}{LinearOrdinaryDifferentialOperator1}) via remainder
+sequences, and consequently the computation of left and right least common
+multiples (\spadfunFrom{rightLcm}{LinearOrdinaryDifferentialOperator1} and
+\spadfunFrom{leftLcm}{LinearOrdinaryDifferentialOperator1}).
+}{
+\spadpaste{e := leftGcd(a,b) \bound{e}\free{a b}}
+}
+\xtc{
+Note that a greatest common divisor doesn't necessarily divide \spad{a} and
+%-% \HDindex{greatest common divisor}{ugxLinearOrdinaryDifferentialOperatorOneRatPage}{9.45.1.}{Differential Operators with Rational Function Coefficients}
+\spad{b} on both sides.
+Here the left greatest common divisor does not divide \spad{a} on the right.
+}{
+\spadpaste{leftRemainder(a, e) \free{a e}}
+}
+\xtc{
+}{
+\spadpaste{rightRemainder(a, e) \free{a e}}
+}
+
+\xtc{
+Similarly, a least common multiple
+%-% \HDindex{least common multiple}{ugxLinearOrdinaryDifferentialOperatorOneRatPage}{9.45.1.}{Differential Operators with Rational Function Coefficients}
+is not necessarily divisible from both sides.
+}{
+\spadpaste{f := rightLcm(a,b) \bound{f}\free{a b}}
+}
+\xtc{
+}{
+\spadpaste{rightRemainder(f, b) \free{f b}}
+}
+\xtc{
+}{
+\spadpaste{leftRemainder(f, b) \free{f b}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/LODO1.pht b/src/hyper/pages/LODO1.pht
new file mode 100644
index 00000000..0c7ae652
--- /dev/null
+++ b/src/hyper/pages/LODO1.pht
@@ -0,0 +1,356 @@
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch1}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull1}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty1}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull1}{\hidepaste}
+\tab{5}\spadcommand{RFZ := Fraction UnivariatePolynomial('x, Integer)\bound{RFZ0 }}
+\indentrel{3}\begin{verbatim}
+   (1)  Fraction UnivariatePolynomial(x,Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty1}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty1}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch1}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{RFZ := Fraction UnivariatePolynomial('x, Integer)\bound{RFZ0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch2}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull2}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty2}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull2}{\hidepaste}
+\tab{5}\spadcommand{x : RFZ := 'x\free{RFZ0 }\bound{RFZ }}
+\indentrel{3}\begin{verbatim}
+   (2)  x
+         Type: Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty2}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty2}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch2}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{x : RFZ := 'x\free{RFZ0 }\bound{RFZ }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch3}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull3}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty3}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull3}{\hidepaste}
+\tab{5}\spadcommand{Dx : LODO1 RFZ := D()\bound{Dx }\free{RFZ }}
+\indentrel{3}\begin{verbatim}
+   (3)  D
+Type: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty3}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty3}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch3}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{Dx : LODO1 RFZ := D()\bound{Dx }\free{RFZ }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch4}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull4}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty4}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull4}{\hidepaste}
+\tab{5}\spadcommand{b : LODO1 RFZ := 3*x**2*Dx**2 + 2*Dx + 1/x\free{Dx }\bound{b }}
+\indentrel{3}\begin{verbatim}
+          2 2        1
+   (4)  3x D  + 2D + Ä
+                     x
+Type: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty4}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty4}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch4}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{b : LODO1 RFZ := 3*x**2*Dx**2 + 2*Dx + 1/x\free{Dx }\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch5}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull5}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty5}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull5}{\hidepaste}
+\tab{5}\spadcommand{a : LODO1 RFZ := b*(5*x*Dx + 7)\free{b Dx }\bound{a }}
+\indentrel{3}\begin{verbatim}
+           3 3       2        2         7
+   (5)  15x D  + (51x  + 10x)D  + 29D + Ä
+                                        x
+Type: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty5}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty5}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch5}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{a : LODO1 RFZ := b*(5*x*Dx + 7)\free{b Dx }\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch6}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull6}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty6}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull6}{\hidepaste}
+\tab{5}\spadcommand{p := x**2 + 1/x**2\bound{p }\free{RFZ }}
+\indentrel{3}\begin{verbatim}
+         4
+        x  + 1
+   (6)  ÄÄÄÄÄÄ
+           2
+          x
+         Type: Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty6}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty6}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch6}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{p := x**2 + 1/x**2\bound{p }\free{RFZ }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch7}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull7}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty7}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull7}{\hidepaste}
+\tab{5}\spadcommand{(a*b - b*a) p\free{a b p }}
+\indentrel{3}\begin{verbatim}
+             4
+        - 75x  + 540x - 75
+   (7)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                 4
+                x
+         Type: Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty7}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty7}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch7}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{(a*b - b*a) p\free{a b p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch8}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull8}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty8}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull8}{\hidepaste}
+\tab{5}\spadcommand{ld := leftDivide(a,b)\bound{ld }\free{a b }}
+\indentrel{3}\begin{verbatim}
+   (8)  [quotient= 5x D + 7,remainder= 0]
+Type: Record(quotient: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer),remainder: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty8}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty8}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch8}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{ld := leftDivide(a,b)\bound{ld }\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch9}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull9}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty9}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull9}{\hidepaste}
+\tab{5}\spadcommand{a = b * ld.quotient + ld.remainder\free{a b ld }}
+\indentrel{3}\begin{verbatim}
+   (9)
+      3 3       2        2         7
+   15x D  + (51x  + 10x)D  + 29D + Ä =
+                                   x
+        3 3       2        2         7
+     15x D  + (51x  + 10x)D  + 29D + Ä
+                                     x
+Type: Equation LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty9}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty9}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch9}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{a = b * ld.quotient + ld.remainder\free{a b ld }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch10}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull10}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty10}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull10}{\hidepaste}
+\tab{5}\spadcommand{rd := rightDivide(a,b)\bound{rd }\free{a b }}
+\indentrel{3}\begin{verbatim}
+                                              5
+   (10)  [quotient= 5x D + 7,remainder= 10D + Ä]
+                                              x
+Type: Record(quotient: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer),remainder: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty10}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty10}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch10}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{rd := rightDivide(a,b)\bound{rd }\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch11}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull11}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty11}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull11}{\hidepaste}
+\tab{5}\spadcommand{a = rd.quotient * b + rd.remainder\free{a b rd }}
+\indentrel{3}\begin{verbatim}
+   (11)
+      3 3       2        2         7
+   15x D  + (51x  + 10x)D  + 29D + Ä =
+                                   x
+        3 3       2        2         7
+     15x D  + (51x  + 10x)D  + 29D + Ä
+                                     x
+Type: Equation LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty11}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty11}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch11}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{a = rd.quotient * b + rd.remainder\free{a b rd }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch12}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull12}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty12}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull12}{\hidepaste}
+\tab{5}\spadcommand{rightQuotient(a,b)\free{a b }}
+\indentrel{3}\begin{verbatim}
+   (12)  5x D + 7
+Type: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty12}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty12}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch12}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{rightQuotient(a,b)\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch13}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull13}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty13}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull13}{\hidepaste}
+\tab{5}\spadcommand{rightRemainder(a,b)\free{a b }}
+\indentrel{3}\begin{verbatim}
+               5
+   (13)  10D + Ä
+               x
+Type: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty13}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty13}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch13}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{rightRemainder(a,b)\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch14}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull14}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty14}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull14}{\hidepaste}
+\tab{5}\spadcommand{leftExactQuotient(a,b)\free{a b }}
+\indentrel{3}\begin{verbatim}
+   (14)  5x D + 7
+Type: Union(LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty14}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty14}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch14}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{leftExactQuotient(a,b)\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch15}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull15}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty15}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull15}{\hidepaste}
+\tab{5}\spadcommand{e := leftGcd(a,b)\bound{e }\free{a b }}
+\indentrel{3}\begin{verbatim}
+           2 2        1
+   (15)  3x D  + 2D + Ä
+                      x
+Type: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty15}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty15}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch15}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{e := leftGcd(a,b)\bound{e }\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch16}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull16}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty16}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull16}{\hidepaste}
+\tab{5}\spadcommand{leftRemainder(a, e)\free{a e }}
+\indentrel{3}\begin{verbatim}
+   (16)  0
+Type: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty16}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty16}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch16}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{leftRemainder(a, e)\free{a e }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch17}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull17}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty17}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull17}{\hidepaste}
+\tab{5}\spadcommand{rightRemainder(a, e)\free{a e }}
+\indentrel{3}\begin{verbatim}
+               5
+   (17)  10D + Ä
+               x
+Type: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty17}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty17}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch17}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{rightRemainder(a, e)\free{a e }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch18}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull18}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty18}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull18}{\hidepaste}
+\tab{5}\spadcommand{f := rightLcm(a,b)\bound{f }\free{a b }}
+\indentrel{3}\begin{verbatim}
+            3 3       2        2         7
+   (18)  15x D  + (51x  + 10x)D  + 29D + Ä
+                                         x
+Type: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty18}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty18}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch18}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{f := rightLcm(a,b)\bound{f }\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch19}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull19}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty19}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull19}{\hidepaste}
+\tab{5}\spadcommand{rightRemainder(f, b)\free{f b }}
+\indentrel{3}\begin{verbatim}
+               5
+   (19)  10D + Ä
+               x
+Type: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty19}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty19}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch19}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{rightRemainder(f, b)\free{f b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch20}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull20}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty20}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageFull20}{\hidepaste}
+\tab{5}\spadcommand{leftRemainder(f, b)\free{f b }}
+\indentrel{3}\begin{verbatim}
+   (20)  0
+Type: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty20}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty20}{ugxLinearOrdinaryDifferentialOperatorOneRatPagePatch20}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorOneRatPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{leftRemainder(f, b)\free{f b }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/LODO2.ht b/src/hyper/pages/LODO2.ht
new file mode 100644
index 00000000..e5021a48
--- /dev/null
+++ b/src/hyper/pages/LODO2.ht
@@ -0,0 +1,189 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\LinearOrdinaryDifferentialOperatorTwoXmpTitle}{LinearOrdinaryDifferentialOperator2}
+\newcommand{\LinearOrdinaryDifferentialOperatorTwoXmpNumber}{9.46}
+%
+% =====================================================================
+\begin{page}{LinearOrdinaryDifferentialOperatorTwoXmpPage}{9.46 LinearOrdinaryDifferentialOperator2}
+% =====================================================================
+\beginscroll
+
+\spadtype{LinearOrdinaryDifferentialOperator2(A, M)} is the domain of linear
+%-% \HDindex{operator!linear ordinary differential}{LinearOrdinaryDifferentialOperatorTwoXmpPage}{9.46}{LinearOrdinaryDifferentialOperator2}
+ordinary differential operators with coefficients in the differential ring
+\spad{A} and operating on \spad{M}, an \spad{A}-module.
+This includes the cases of operators which are polynomials in \spad{D}
+acting upon scalar or vector expressions of a single variable.
+The coefficients of the operator polynomials can be integers, rational
+functions, matrices or elements of other domains.
+\showBlurb{LinearOrdinaryDifferentialOperator2}
+
+\beginmenu
+    \menudownlink{{9.46.1. Differential Operators with Constant Coefficients}}{ugxLinearOrdinaryDifferentialOperatorTwoConstPage}
+    \menudownlink{{9.46.2. Differential Operators with Matrix Coefficients Operating on Vectors}}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxLinearOrdinaryDifferentialOperatorTwoConstTitle}{Differential Operators with Constant Coefficients}
+\newcommand{\ugxLinearOrdinaryDifferentialOperatorTwoConstNumber}{9.46.1.}
+%
+% =====================================================================
+\begin{page}{ugxLinearOrdinaryDifferentialOperatorTwoConstPage}{9.46.1. Differential Operators with Constant Coefficients}
+% =====================================================================
+\beginscroll
+
+This example shows differential operators with rational
+number coefficients operating on univariate polynomials.
+
+\labelSpace{3pc}
+\xtc{
+We begin by making type assignments so we can conveniently refer
+to univariate polynomials in \spad{x} over the rationals.
+}{
+\spadpaste{Q  := Fraction Integer \bound{Q}}
+}
+\xtc{
+}{
+\spadpaste{PQ := UnivariatePolynomial('x, Q) \free{Q}\bound{PQ0}}
+}
+\xtc{
+}{
+\spadpaste{x: PQ := 'x \free{PQ0}\bound{x}}
+}
+\xtc{
+Now we assign \spad{Dx} to be the differential operator
+\spadfunFrom{D}{LinearOrdinaryDifferentialOperator2}
+corresponding to \spad{d/dx}.
+}{
+\spadpaste{Dx: LODO2(Q, PQ) := D() \free{Q PQ0}\bound{Dx}}
+}
+\xtc{
+New operators are created as polynomials in \spad{D()}.
+}{
+\spadpaste{a := Dx  + 1 \bound{a}\free{Dx}}
+}
+\xtc{
+}{
+\spadpaste{b := a + 1/2*Dx**2 - 1/2 \bound{b}\free{Dx}}
+}
+\xtc{
+To apply the operator \spad{a} to the value \spad{p} the usual function
+call syntax is used.
+}{
+\spadpaste{p := 4*x**2 + 2/3 \free{x}\bound{p}}
+}
+\xtc{
+}{
+\spadpaste{a p \free{a p}}
+}
+\xtc{
+Operator multiplication is defined by the identity \spad{(a*b) p = a(b(p))}
+}{
+\spadpaste{(a * b) p = a b p \free{a b p}}
+}
+\xtc{
+Exponentiation follows from multiplication.
+}{
+\spadpaste{c := (1/9)*b*(a + b)**2 \free{a b}\bound{c}}
+}
+\xtc{
+Finally, note that operator expressions may be applied directly.
+}{
+\spadpaste{(a**2 - 3/4*b + c) (p + 1) \free{a b c p}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxLinearOrdinaryDifferentialOperatorTwoMatrixTitle}{Differential Operators with Matrix Coefficients Operating on Vectors}
+\newcommand{\ugxLinearOrdinaryDifferentialOperatorTwoMatrixNumber}{9.46.2.}
+%
+% =====================================================================
+\begin{page}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPage}{9.46.2. Differential Operators with Matrix Coefficients Operating on Vectors}
+% =====================================================================
+\beginscroll
+
+This is another example of linear ordinary differential
+operators with non-commutative multiplication.
+Unlike the rational function case, the differential ring of
+square matrices (of a given dimension) with univariate
+polynomial entries does not form a field.
+Thus the number of operations available is more limited.
+
+\labelSpace{1pc}
+\xtc{
+In this section, the operators have three by three
+matrix coefficients with polynomial entries.
+}{
+\spadpaste{PZ   := UnivariatePolynomial(x,Integer)\bound{PZ0}}
+}
+\xtc{
+}{
+\spadpaste{x:PZ := 'x \free{PZ0}\bound{PZ}}
+}
+\xtc{
+}{
+\spadpaste{Mat  := SquareMatrix(3,PZ)\free{PZ}\bound{Mat}}
+}
+\xtc{
+The operators act on the vectors considered as a \spad{Mat}-module.
+}{
+\spadpaste{Vect := DPMM(3, PZ, Mat, PZ);\free{PZ,Mat}\bound{Vect}}
+}
+\xtc{
+}{
+\spadpaste{Modo := LODO2(Mat, Vect);\free{Mat Vect}\bound{Modo}}
+}
+\xtc{
+The matrix \spad{m} is used as a coefficient and the vectors \spad{p}
+and \spad{q} are operated upon.
+}{
+\spadpaste{m:Mat := matrix [[x**2,1,0],[1,x**4,0],[0,0,4*x**2]]\free{Mat}\bound{m}}
+}
+\xtc{
+}{
+\spadpaste{p:Vect := directProduct [3*x**2+1,2*x,7*x**3+2*x]\free{Vect}\bound{p}}
+}
+\xtc{
+}{
+\spadpaste{q: Vect := m * p\free{m p Vect}\bound{q}}
+}
+\xtc{
+Now form a few operators.
+}{
+\spadpaste{Dx : Modo := D()\bound{Dx}\free{Modo}}
+}
+\xtc{
+}{
+\spadpaste{a : Modo := Dx  + m\bound{a}\free{m Dx}}
+}
+\xtc{
+}{
+\spadpaste{b : Modo := m*Dx  + 1\bound{b}\free{m Dx}}
+}
+\xtc{
+}{
+\spadpaste{c := a*b \bound{c}\free{a b}}
+}
+\xtc{
+These operators can be applied to vector values.
+}{
+\spadpaste{a p \free{p a}}
+}
+\xtc{
+}{
+\spadpaste{b p \free{p b}}
+}
+\xtc{
+}{
+\spadpaste{(a + b + c) (p + q) \free{a b c p q}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/LODO2.pht b/src/hyper/pages/LODO2.pht
new file mode 100644
index 00000000..9c267563
--- /dev/null
+++ b/src/hyper/pages/LODO2.pht
@@ -0,0 +1,499 @@
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch1}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull1}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty1}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull1}{\hidepaste}
+\tab{5}\spadcommand{Q := Fraction Integer\bound{Q }}
+\indentrel{3}\begin{verbatim}
+   (1)  Fraction Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty1}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty1}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch1}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{Q := Fraction Integer\bound{Q }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch2}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull2}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty2}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull2}{\hidepaste}
+\tab{5}\spadcommand{PQ := UnivariatePolynomial('x, Q)\free{Q }\bound{PQ0 }}
+\indentrel{3}\begin{verbatim}
+   (2)  UnivariatePolynomial(x,Fraction Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty2}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty2}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch2}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{PQ := UnivariatePolynomial('x, Q)\free{Q }\bound{PQ0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch3}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull3}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty3}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull3}{\hidepaste}
+\tab{5}\spadcommand{x: PQ := 'x\free{PQ0 }\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (3)  x
+         Type: UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty3}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty3}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch3}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{x: PQ := 'x\free{PQ0 }\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch4}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull4}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty4}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull4}{\hidepaste}
+\tab{5}\spadcommand{Dx: LODO2(Q, PQ) := D()\free{Q PQ0 }\bound{Dx }}
+\indentrel{3}\begin{verbatim}
+   (4)  D
+Type: LinearOrdinaryDifferentialOperator2(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty4}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty4}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch4}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{Dx: LODO2(Q, PQ) := D()\free{Q PQ0 }\bound{Dx }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch5}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull5}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty5}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull5}{\hidepaste}
+\tab{5}\spadcommand{a := Dx + 1\bound{a }\free{Dx }}
+\indentrel{3}\begin{verbatim}
+   (5)  D + 1
+Type: LinearOrdinaryDifferentialOperator2(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty5}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty5}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch5}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{a := Dx + 1\bound{a }\free{Dx }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch6}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull6}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty6}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull6}{\hidepaste}
+\tab{5}\spadcommand{b := a + 1/2*Dx**2 - 1/2\bound{b }\free{Dx }}
+\indentrel{3}\begin{verbatim}
+        1  2       1
+   (6)  Ä D  + D + Ä
+        2          2
+Type: LinearOrdinaryDifferentialOperator2(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty6}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty6}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch6}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{b := a + 1/2*Dx**2 - 1/2\bound{b }\free{Dx }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch7}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull7}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty7}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull7}{\hidepaste}
+\tab{5}\spadcommand{p := 4*x**2 + 2/3\free{x }\bound{p }}
+\indentrel{3}\begin{verbatim}
+          2   2
+   (7)  4x  + Ä
+              3
+         Type: UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty7}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty7}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch7}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{p := 4*x**2 + 2/3\free{x }\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch8}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull8}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty8}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull8}{\hidepaste}
+\tab{5}\spadcommand{a p\free{a p }}
+\indentrel{3}\begin{verbatim}
+          2        2
+   (8)  4x  + 8x + Ä
+                   3
+         Type: UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty8}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty8}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch8}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{a p\free{a p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch9}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull9}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty9}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull9}{\hidepaste}
+\tab{5}\spadcommand{(a * b) p = a b p\free{a b p }}
+\indentrel{3}\begin{verbatim}
+          2         37    2         37
+   (9)  2x  + 12x + ÄÄ= 2x  + 12x + ÄÄ
+                     3               3
+Type: Equation UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty9}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty9}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch9}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{(a * b) p = a b p\free{a b p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch10}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull10}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty10}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull10}{\hidepaste}
+\tab{5}\spadcommand{c := (1/9)*b*(a + b)**2\free{a b }\bound{c }}
+\indentrel{3}\begin{verbatim}
+   (10)
+    1  6    5  5   13  4   19  3   79  2    7     1
+   ÄÄ D  + ÄÄ D  + ÄÄ D  + ÄÄ D  + ÄÄ D  + ÄÄ D + Ä
+   72      36      24      18      72      12     8
+Type: LinearOrdinaryDifferentialOperator2(Fraction Integer,UnivariatePolynomial(x,Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty10}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty10}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch10}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{c := (1/9)*b*(a + b)**2\free{a b }\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch11}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull11}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty11}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageFull11}{\hidepaste}
+\tab{5}\spadcommand{(a**2 - 3/4*b + c) (p + 1)\free{a b c p }}
+\indentrel{3}\begin{verbatim}
+           2   44     541
+   (11)  3x  + ÄÄ x + ÄÄÄ
+                3      36
+         Type: UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty11}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty11}{ugxLinearOrdinaryDifferentialOperatorTwoConstPagePatch11}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoConstPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{(a**2 - 3/4*b + c) (p + 1)\free{a b c p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch1}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull1}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty1}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull1}{\hidepaste}
+\tab{5}\spadcommand{PZ := UnivariatePolynomial(x,Integer)\bound{PZ0 }}
+\indentrel{3}\begin{verbatim}
+   (1)  UnivariatePolynomial(x,Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty1}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty1}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch1}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{PZ := UnivariatePolynomial(x,Integer)\bound{PZ0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch2}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull2}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty2}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull2}{\hidepaste}
+\tab{5}\spadcommand{x:PZ := 'x\free{PZ0 }\bound{PZ }}
+\indentrel{3}\begin{verbatim}
+   (2)  x
+                  Type: UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty2}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty2}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch2}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{x:PZ := 'x\free{PZ0 }\bound{PZ }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch3}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull3}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty3}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull3}{\hidepaste}
+\tab{5}\spadcommand{Mat := SquareMatrix(3,PZ)\free{PZ }\bound{Mat }}
+\indentrel{3}\begin{verbatim}
+   (3)  SquareMatrix(3,UnivariatePolynomial(x,Integer))
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty3}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty3}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch3}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{Mat := SquareMatrix(3,PZ)\free{PZ }\bound{Mat }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch4}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull4}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty4}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull4}{\hidepaste}
+\tab{5}\spadcommand{Vect := DPMM(3, PZ, Mat, PZ);\free{PZ Mat }\bound{Vect }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty4}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty4}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch4}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{Vect := DPMM(3, PZ, Mat, PZ);\free{PZ Mat }\bound{Vect }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch5}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull5}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty5}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull5}{\hidepaste}
+\tab{5}\spadcommand{Modo := LODO2(Mat, Vect);\free{Mat Vect }\bound{Modo }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty5}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty5}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch5}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{Modo := LODO2(Mat, Vect);\free{Mat Vect }\bound{Modo }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch6}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull6}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty6}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull6}{\hidepaste}
+\tab{5}\spadcommand{m:Mat := matrix [[x**2,1,0],[1,x**4,0],[0,0,4*x**2]]\free{Mat }\bound{m }}
+\indentrel{3}\begin{verbatim}
+        Ú 2         ¿
+        ³x   1    0 ³
+        ³           ³
+   (6)  ³     4     ³
+        ³1   x    0 ³
+        ³           ³
+        ³          2³
+        À0   0   4x Ù
+  Type: SquareMatrix(3,UnivariatePolynomial(x,Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty6}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty6}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch6}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{m:Mat := matrix [[x**2,1,0],[1,x**4,0],[0,0,4*x**2]]\free{Mat }\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch7}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull7}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty7}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull7}{\hidepaste}
+\tab{5}\spadcommand{p:Vect := directProduct [3*x**2+1,2*x,7*x**3+2*x]\free{Vect }\bound{p }}
+\indentrel{3}\begin{verbatim}
+           2          3
+   (7)  [3x  + 1,2x,7x  + 2x]
+Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty7}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty7}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch7}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{p:Vect := directProduct [3*x**2+1,2*x,7*x**3+2*x]\free{Vect }\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch8}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull8}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty8}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull8}{\hidepaste}
+\tab{5}\spadcommand{q: Vect := m * p\free{m p Vect }\bound{q }}
+\indentrel{3}\begin{verbatim}
+           4    2        5     2        5     3
+   (8)  [3x  + x  + 2x,2x  + 3x  + 1,28x  + 8x ]
+Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty8}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty8}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch8}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{q: Vect := m * p\free{m p Vect }\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch9}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull9}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty9}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull9}{\hidepaste}
+\tab{5}\spadcommand{Dx : Modo := D()\bound{Dx }\free{Modo }}
+\indentrel{3}\begin{verbatim}
+   (9)  D
+Type: LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Integer)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty9}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty9}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch9}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{Dx : Modo := D()\bound{Dx }\free{Modo }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch10}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull10}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty10}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull10}{\hidepaste}
+\tab{5}\spadcommand{a : Modo := Dx + m\bound{a }\free{m Dx }}
+\indentrel{3}\begin{verbatim}
+             Ú 2         ¿
+             ³x   1    0 ³
+             ³           ³
+   (10)  D + ³     4     ³
+             ³1   x    0 ³
+             ³           ³
+             ³          2³
+             À0   0   4x Ù
+Type: LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Integer)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty10}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty10}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch10}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{a : Modo := Dx + m\bound{a }\free{m Dx }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch11}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull11}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty11}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull11}{\hidepaste}
+\tab{5}\spadcommand{b : Modo := m*Dx + 1\bound{b }\free{m Dx }}
+\indentrel{3}\begin{verbatim}
+         Ú 2         ¿
+         ³x   1    0 ³    Ú1  0  0¿
+         ³           ³    ³       ³
+   (11)  ³     4     ³D + ³0  1  0³
+         ³1   x    0 ³    ³       ³
+         ³           ³    À0  0  1Ù
+         ³          2³
+         À0   0   4x Ù
+Type: LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Integer)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty11}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty11}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch11}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{b : Modo := m*Dx + 1\bound{b }\free{m Dx }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch12}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull12}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty12}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull12}{\hidepaste}
+\tab{5}\spadcommand{c := a*b\bound{c }\free{a b }}
+\indentrel{3}\begin{verbatim}
+   (12)
+     Ú 2         ¿
+     ³x   1    0 ³
+     ³           ³ 2
+     ³     4     ³D
+     ³1   x    0 ³
+     ³           ³
+     ³          2³
+     À0   0   4x Ù
+   + 
+     Ú 4              4    2                  ¿
+     ³x  + 2x + 2    x  + x            0      ³
+     ³                                        ³
+     ³   4    2     8     3                   ³D
+     ³  x  + x     x  + 4x  + 2        0      ³
+     ³                                        ³
+     ³                              4         ³
+     À     0            0        16x  + 8x + 1Ù
+   + 
+     Ú 2         ¿
+     ³x   1    0 ³
+     ³           ³
+     ³     4     ³
+     ³1   x    0 ³
+     ³           ³
+     ³          2³
+     À0   0   4x Ù
+Type: LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Integer)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty12}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty12}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch12}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{c := a*b\bound{c }\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch13}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull13}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty13}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull13}{\hidepaste}
+\tab{5}\spadcommand{a p\free{p a }}
+\indentrel{3}\begin{verbatim}
+   (13)
+      4    2        5     2        5     3      2
+   [3x  + x  + 8x,2x  + 3x  + 3,28x  + 8x  + 21x  + 2]
+Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty13}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty13}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch13}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{a p\free{p a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch14}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull14}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty14}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull14}{\hidepaste}
+\tab{5}\spadcommand{b p\free{p b }}
+\indentrel{3}\begin{verbatim}
+            3     2       4         4     3     2
+   (14)  [6x  + 3x  + 3,2x  + 8x,84x  + 7x  + 8x  + 2x]
+Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty14}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty14}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch14}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{b p\free{p b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch15}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull15}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty15}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageFull15}{\hidepaste}
+\tab{5}\spadcommand{(a + b + c) (p + q)\free{a b c p q }}
+\indentrel{3}\begin{verbatim}
+   (15)
+   [
+          8      7      6      5      4      3      2
+       10x  + 12x  + 16x  + 30x  + 85x  + 94x  + 40x
+     + 
+       40x + 17
+     ,
+
+          12      9      8      7     6      5      4
+       10x   + 10x  + 12x  + 92x  + 6x  + 32x  + 72x
+     + 
+          3      2
+       28x  + 49x  + 32x + 19
+     ,
+
+            8       7        6        5       4       3
+       2240x  + 224x  + 1280x  + 3508x  + 492x  + 751x
+     + 
+          2
+       98x  + 18x + 4
+     ]
+Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty15}
+\begin{paste}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty15}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPagePatch15}
+\pastebutton{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{(a + b + c) (p + q)\free{a b c p q }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/LPOLY.ht b/src/hyper/pages/LPOLY.ht
new file mode 100644
index 00000000..52616fef
--- /dev/null
+++ b/src/hyper/pages/LPOLY.ht
@@ -0,0 +1,136 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\LiePolynomialXmpTitle}{LiePolynomial}
+\newcommand{\LiePolynomialXmpNumber}{9.43}
+%
+% =====================================================================
+\begin{page}{LiePolynomialXmpPage}{9.43 LiePolynomial}
+% =====================================================================
+\beginscroll
+Declaration of domains
+\xtc{
+}{
+\spadpaste{RN    := Fraction Integer \bound{RN}}
+}
+\xtc{
+}{
+\spadpaste{Lpoly := LiePolynomial(Symbol,RN) \bound{Lpoly} \free{RN}}
+}
+\xtc{
+}{
+\spadpaste{Dpoly := XDPOLY(Symbol,RN) \bound{Dpoly} \free{RN}}
+}
+\xtc{
+}{
+\spadpaste{Lword := LyndonWord Symbol \bound{Lword}}
+}
+
+Initialisation
+\xtc{
+}{
+\spadpaste{a:Symbol := 'a \bound{a}}
+}
+\xtc{
+}{
+\spadpaste{b:Symbol := 'b \bound{b}}
+}
+\xtc{
+}{
+\spadpaste{c:Symbol := 'c \bound{c}}
+}
+\xtc{
+}{
+\spadpaste{aa: Lpoly := a \bound{aa} \free{Lpoly} \free{a}}
+}
+\xtc{
+}{
+\spadpaste{bb: Lpoly := b \bound{bb} \free{Lpoly} \free{b}}
+}
+\xtc{
+}{
+\spadpaste{cc: Lpoly := c \bound{cc} \free{Lpoly} \free{c}}
+}
+\xtc{
+}{
+\spadpaste{p : Lpoly := [aa,bb] \bound{p} \free{aa} \free{bb} \free{Lpoly}}
+}
+\xtc{
+}{
+\spadpaste{q : Lpoly := [p,bb] \bound{q} \free{p} \free{bb} \free{Lpoly}}
+}
+\xtc{
+All the Lyndon words of order 4
+}{
+\spadpaste{liste : List Lword := LyndonWordsList([a,b], 4) \free{a} \free{b} \free{Lword} \bound{liste}}
+}
+\xtc{
+}{
+\spadpaste{r: Lpoly := p + q + 3*LiePoly(liste.4)$Lpoly \bound{r} \free{Lpoly} \free{p} \free{q} \free{liste}}
+}
+\xtc{
+}{
+\spadpaste{s:Lpoly := [p,r] \bound{s} \free{Lpoly} \free{p} \free{r}}
+}
+\xtc{
+}{
+\spadpaste{t:Lpoly  := s  + 2*LiePoly(liste.3) - 5*LiePoly(liste.5) \bound{t} \free{Lpoly} \free{s} \free{liste} }
+}
+\xtc{
+}{
+\spadpaste{degree t \free{t}}
+}
+\xtc{
+}{
+\spadpaste{mirror t \free{t}}
+}
+
+Jacobi Relation
+\xtc{
+}{
+\spadpaste{Jacobi(p: Lpoly, q: Lpoly, r: Lpoly): Lpoly == [[p,q]$Lpoly, r] + [[q,r]$Lpoly, p] + [[r,p]$Lpoly, q] \free{Lpoly} \bound{J}}
+}
+
+Tests
+\xtc{
+}{
+\spadpaste{test: Lpoly := Jacobi(a,b,b) \free{J Lpoly a b} \bound{test1}}
+}
+\xtc{
+}{
+\spadpaste{test: Lpoly := Jacobi(p,q,r) \free{J p q r Lpoly} \bound{test2}}
+}
+\xtc{
+}{
+\spadpaste{test: Lpoly := Jacobi(r,s,t) \free{J r s t Lpoly} \bound{test3}}
+}
+
+Evaluation
+\xtc{
+}{
+\spadpaste{eval(p, a, p)$Lpoly}
+}
+\xtc{
+}{
+\spadpaste{eval(p, [a,b], [2*bb, 3*aa])$Lpoly \free{p a b bb aa Lpoly}}
+}
+\xtc{
+}{
+\spadpaste{r: Lpoly := [p,c] \free{p c Lpoly} \bound{rr}}
+}
+\xtc{
+}{
+\spadpaste{r1: Lpoly := eval(r, [a,b,c], [bb, cc, aa])$Lpoly \free{rr a b c aa bb cc Lpoly} \bound{r1}}
+}
+\xtc{
+}{
+\spadpaste{r2: Lpoly := eval(r, [a,b,c], [cc, aa, bb])$Lpoly \free{rr a b c cc bb aa Lpoly} \bound{r2}}
+}
+\xtc{
+}{
+\spadpaste{r + r1 + r2 \free{rr r1 r2}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/LPOLY.pht b/src/hyper/pages/LPOLY.pht
new file mode 100644
index 00000000..ac44b36a
--- /dev/null
+++ b/src/hyper/pages/LPOLY.pht
@@ -0,0 +1,455 @@
+\begin{patch}{LiePolynomialXmpPagePatch1}
+\begin{paste}{LiePolynomialXmpPageFull1}{LiePolynomialXmpPageEmpty1}
+\pastebutton{LiePolynomialXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{RN := Fraction Integer\bound{RN }}
+\indentrel{3}\begin{verbatim}
+   (1)  Fraction Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty1}
+\begin{paste}{LiePolynomialXmpPageEmpty1}{LiePolynomialXmpPagePatch1}
+\pastebutton{LiePolynomialXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{RN := Fraction Integer\bound{RN }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch2}
+\begin{paste}{LiePolynomialXmpPageFull2}{LiePolynomialXmpPageEmpty2}
+\pastebutton{LiePolynomialXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{Lpoly := LiePolynomial(Symbol,RN)\bound{Lpoly }\free{RN }}
+\indentrel{3}\begin{verbatim}
+   (2)  LiePolynomial(Symbol,Fraction Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty2}
+\begin{paste}{LiePolynomialXmpPageEmpty2}{LiePolynomialXmpPagePatch2}
+\pastebutton{LiePolynomialXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{Lpoly := LiePolynomial(Symbol,RN)\bound{Lpoly }\free{RN }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch3}
+\begin{paste}{LiePolynomialXmpPageFull3}{LiePolynomialXmpPageEmpty3}
+\pastebutton{LiePolynomialXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{Dpoly := XDPOLY(Symbol,RN)\bound{Dpoly }\free{RN }}
+\indentrel{3}\begin{verbatim}
+   (3)  XDistributedPolynomial(Symbol,Fraction Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty3}
+\begin{paste}{LiePolynomialXmpPageEmpty3}{LiePolynomialXmpPagePatch3}
+\pastebutton{LiePolynomialXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{Dpoly := XDPOLY(Symbol,RN)\bound{Dpoly }\free{RN }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch4}
+\begin{paste}{LiePolynomialXmpPageFull4}{LiePolynomialXmpPageEmpty4}
+\pastebutton{LiePolynomialXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{Lword := LyndonWord Symbol\bound{Lword }}
+\indentrel{3}\begin{verbatim}
+   (4)  LyndonWord Symbol
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty4}
+\begin{paste}{LiePolynomialXmpPageEmpty4}{LiePolynomialXmpPagePatch4}
+\pastebutton{LiePolynomialXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{Lword := LyndonWord Symbol\bound{Lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch5}
+\begin{paste}{LiePolynomialXmpPageFull5}{LiePolynomialXmpPageEmpty5}
+\pastebutton{LiePolynomialXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{a:Symbol := 'a\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (5)  a
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty5}
+\begin{paste}{LiePolynomialXmpPageEmpty5}{LiePolynomialXmpPagePatch5}
+\pastebutton{LiePolynomialXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{a:Symbol := 'a\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch6}
+\begin{paste}{LiePolynomialXmpPageFull6}{LiePolynomialXmpPageEmpty6}
+\pastebutton{LiePolynomialXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{b:Symbol := 'b\bound{b }}
+\indentrel{3}\begin{verbatim}
+   (6)  b
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty6}
+\begin{paste}{LiePolynomialXmpPageEmpty6}{LiePolynomialXmpPagePatch6}
+\pastebutton{LiePolynomialXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{b:Symbol := 'b\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch7}
+\begin{paste}{LiePolynomialXmpPageFull7}{LiePolynomialXmpPageEmpty7}
+\pastebutton{LiePolynomialXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{c:Symbol := 'c\bound{c }}
+\indentrel{3}\begin{verbatim}
+   (7)  c
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty7}
+\begin{paste}{LiePolynomialXmpPageEmpty7}{LiePolynomialXmpPagePatch7}
+\pastebutton{LiePolynomialXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{c:Symbol := 'c\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch8}
+\begin{paste}{LiePolynomialXmpPageFull8}{LiePolynomialXmpPageEmpty8}
+\pastebutton{LiePolynomialXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{aa: Lpoly := a\bound{aa }\free{Lpoly }\free{a }}
+\indentrel{3}\begin{verbatim}
+   (8)  [a]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty8}
+\begin{paste}{LiePolynomialXmpPageEmpty8}{LiePolynomialXmpPagePatch8}
+\pastebutton{LiePolynomialXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{aa: Lpoly := a\bound{aa }\free{Lpoly }\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch9}
+\begin{paste}{LiePolynomialXmpPageFull9}{LiePolynomialXmpPageEmpty9}
+\pastebutton{LiePolynomialXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{bb: Lpoly := b\bound{bb }\free{Lpoly }\free{b }}
+\indentrel{3}\begin{verbatim}
+   (9)  [b]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty9}
+\begin{paste}{LiePolynomialXmpPageEmpty9}{LiePolynomialXmpPagePatch9}
+\pastebutton{LiePolynomialXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{bb: Lpoly := b\bound{bb }\free{Lpoly }\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch10}
+\begin{paste}{LiePolynomialXmpPageFull10}{LiePolynomialXmpPageEmpty10}
+\pastebutton{LiePolynomialXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{cc: Lpoly := c\bound{cc }\free{Lpoly }\free{c }}
+\indentrel{3}\begin{verbatim}
+   (10)  [c]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty10}
+\begin{paste}{LiePolynomialXmpPageEmpty10}{LiePolynomialXmpPagePatch10}
+\pastebutton{LiePolynomialXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{cc: Lpoly := c\bound{cc }\free{Lpoly }\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch11}
+\begin{paste}{LiePolynomialXmpPageFull11}{LiePolynomialXmpPageEmpty11}
+\pastebutton{LiePolynomialXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{p : Lpoly := [aa,bb]\bound{p }\free{aa }\free{bb }\free{Lpoly }}
+\indentrel{3}\begin{verbatim}
+   (11)  [a b]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty11}
+\begin{paste}{LiePolynomialXmpPageEmpty11}{LiePolynomialXmpPagePatch11}
+\pastebutton{LiePolynomialXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{p : Lpoly := [aa,bb]\bound{p }\free{aa }\free{bb }\free{Lpoly }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch12}
+\begin{paste}{LiePolynomialXmpPageFull12}{LiePolynomialXmpPageEmpty12}
+\pastebutton{LiePolynomialXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{q : Lpoly := [p,bb]\bound{q }\free{p }\free{bb }\free{Lpoly }}
+\indentrel{3}\begin{verbatim}
+             2
+   (12)  [a b ]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty12}
+\begin{paste}{LiePolynomialXmpPageEmpty12}{LiePolynomialXmpPagePatch12}
+\pastebutton{LiePolynomialXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{q : Lpoly := [p,bb]\bound{q }\free{p }\free{bb }\free{Lpoly }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch13}
+\begin{paste}{LiePolynomialXmpPageFull13}{LiePolynomialXmpPageEmpty13}
+\pastebutton{LiePolynomialXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{liste : List Lword := LyndonWordsList([a,b], 4)\free{a }\free{b }\free{Lword }\bound{liste }}
+\indentrel{3}\begin{verbatim}
+   (13)
+                    2       2    3     2 2      3
+   [[a],[b],[a b],[a b],[a b ],[a b],[a b ],[a b ]]
+                           Type: List LyndonWord Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty13}
+\begin{paste}{LiePolynomialXmpPageEmpty13}{LiePolynomialXmpPagePatch13}
+\pastebutton{LiePolynomialXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{liste : List Lword := LyndonWordsList([a,b], 4)\free{a }\free{b }\free{Lword }\bound{liste }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch14}
+\begin{paste}{LiePolynomialXmpPageFull14}{LiePolynomialXmpPageEmpty14}
+\pastebutton{LiePolynomialXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{r: Lpoly := p + q + 3*LiePoly(liste.4)$Lpoly\bound{r }\free{Lpoly }\free{p }\free{q }\free{liste }}
+\indentrel{3}\begin{verbatim}
+                    2         2
+   (14)  [a b] + 3[a b] + [a b ]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty14}
+\begin{paste}{LiePolynomialXmpPageEmpty14}{LiePolynomialXmpPagePatch14}
+\pastebutton{LiePolynomialXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{r: Lpoly := p + q + 3*LiePoly(liste.4)$Lpoly\bound{r }\free{Lpoly }\free{p }\free{q }\free{liste }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch15}
+\begin{paste}{LiePolynomialXmpPageFull15}{LiePolynomialXmpPageEmpty15}
+\pastebutton{LiePolynomialXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{s:Lpoly := [p,r]\bound{s }\free{Lpoly }\free{p }\free{r }}
+\indentrel{3}\begin{verbatim}
+              2                 2
+   (15)  - 3[a b a b] + [a b a b ]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty15}
+\begin{paste}{LiePolynomialXmpPageEmpty15}{LiePolynomialXmpPagePatch15}
+\pastebutton{LiePolynomialXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{s:Lpoly := [p,r]\bound{s }\free{Lpoly }\free{p }\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch16}
+\begin{paste}{LiePolynomialXmpPageFull16}{LiePolynomialXmpPageEmpty16}
+\pastebutton{LiePolynomialXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{t:Lpoly := s + 2*LiePoly(liste.3) - 5*LiePoly(liste.5)\bound{t }\free{Lpoly }\free{s }\free{liste }}
+\indentrel{3}\begin{verbatim}
+                       2       2                 2
+   (16)  2[a b] - 5[a b ] - 3[a b a b] + [a b a b ]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty16}
+\begin{paste}{LiePolynomialXmpPageEmpty16}{LiePolynomialXmpPagePatch16}
+\pastebutton{LiePolynomialXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{t:Lpoly := s + 2*LiePoly(liste.3) - 5*LiePoly(liste.5)\bound{t }\free{Lpoly }\free{s }\free{liste }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch17}
+\begin{paste}{LiePolynomialXmpPageFull17}{LiePolynomialXmpPageEmpty17}
+\pastebutton{LiePolynomialXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{degree t\free{t }}
+\indentrel{3}\begin{verbatim}
+   (17)  5
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty17}
+\begin{paste}{LiePolynomialXmpPageEmpty17}{LiePolynomialXmpPagePatch17}
+\pastebutton{LiePolynomialXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{degree t\free{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch18}
+\begin{paste}{LiePolynomialXmpPageFull18}{LiePolynomialXmpPageEmpty18}
+\pastebutton{LiePolynomialXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{mirror t\free{t }}
+\indentrel{3}\begin{verbatim}
+                         2       2                 2
+   (18)  - 2[a b] - 5[a b ] - 3[a b a b] + [a b a b ]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty18}
+\begin{paste}{LiePolynomialXmpPageEmpty18}{LiePolynomialXmpPagePatch18}
+\pastebutton{LiePolynomialXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{mirror t\free{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch19}
+\begin{paste}{LiePolynomialXmpPageFull19}{LiePolynomialXmpPageEmpty19}
+\pastebutton{LiePolynomialXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{Jacobi(p: Lpoly, q: Lpoly, r: Lpoly): Lpoly == [[p,q]$Lpoly, r] + [[q,r]$Lpoly, p] + [[r,p]$Lpoly, q]\free{Lpoly }\bound{J }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty19}
+\begin{paste}{LiePolynomialXmpPageEmpty19}{LiePolynomialXmpPagePatch19}
+\pastebutton{LiePolynomialXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{Jacobi(p: Lpoly, q: Lpoly, r: Lpoly): Lpoly == [[p,q]$Lpoly, r] + [[q,r]$Lpoly, p] + [[r,p]$Lpoly, q]\free{Lpoly }\bound{J }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch20}
+\begin{paste}{LiePolynomialXmpPageFull20}{LiePolynomialXmpPageEmpty20}
+\pastebutton{LiePolynomialXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{test: Lpoly := Jacobi(a,b,b)\free{J Lpoly a b }\bound{test1 }}
+\indentrel{3}\begin{verbatim}
+   (20)  0
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty20}
+\begin{paste}{LiePolynomialXmpPageEmpty20}{LiePolynomialXmpPagePatch20}
+\pastebutton{LiePolynomialXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{test: Lpoly := Jacobi(a,b,b)\free{J Lpoly a b }\bound{test1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch21}
+\begin{paste}{LiePolynomialXmpPageFull21}{LiePolynomialXmpPageEmpty21}
+\pastebutton{LiePolynomialXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{test: Lpoly := Jacobi(p,q,r)\free{J p q r Lpoly }\bound{test2 }}
+\indentrel{3}\begin{verbatim}
+   (21)  0
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty21}
+\begin{paste}{LiePolynomialXmpPageEmpty21}{LiePolynomialXmpPagePatch21}
+\pastebutton{LiePolynomialXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{test: Lpoly := Jacobi(p,q,r)\free{J p q r Lpoly }\bound{test2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch22}
+\begin{paste}{LiePolynomialXmpPageFull22}{LiePolynomialXmpPageEmpty22}
+\pastebutton{LiePolynomialXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{test: Lpoly := Jacobi(r,s,t)\free{J r s t Lpoly }\bound{test3 }}
+\indentrel{3}\begin{verbatim}
+   (22)  0
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty22}
+\begin{paste}{LiePolynomialXmpPageEmpty22}{LiePolynomialXmpPagePatch22}
+\pastebutton{LiePolynomialXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{test: Lpoly := Jacobi(r,s,t)\free{J r s t Lpoly }\bound{test3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch23}
+\begin{paste}{LiePolynomialXmpPageFull23}{LiePolynomialXmpPageEmpty23}
+\pastebutton{LiePolynomialXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{eval(p, a, p)$Lpoly}
+\indentrel{3}\begin{verbatim}
+             2
+   (23)  [a b ]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty23}
+\begin{paste}{LiePolynomialXmpPageEmpty23}{LiePolynomialXmpPagePatch23}
+\pastebutton{LiePolynomialXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{eval(p, a, p)$Lpoly}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch24}
+\begin{paste}{LiePolynomialXmpPageFull24}{LiePolynomialXmpPageEmpty24}
+\pastebutton{LiePolynomialXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{eval(p, [a,b], [2*bb, 3*aa])$Lpoly\free{p a b bb aa Lpoly }}
+\indentrel{3}\begin{verbatim}
+   (24)  - 6[a b]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty24}
+\begin{paste}{LiePolynomialXmpPageEmpty24}{LiePolynomialXmpPagePatch24}
+\pastebutton{LiePolynomialXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{eval(p, [a,b], [2*bb, 3*aa])$Lpoly\free{p a b bb aa Lpoly }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch25}
+\begin{paste}{LiePolynomialXmpPageFull25}{LiePolynomialXmpPageEmpty25}
+\pastebutton{LiePolynomialXmpPageFull25}{\hidepaste}
+\tab{5}\spadcommand{r: Lpoly := [p,c]\free{p c Lpoly }\bound{rr }}
+\indentrel{3}\begin{verbatim}
+   (25)  [a b c] + [a c b]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty25}
+\begin{paste}{LiePolynomialXmpPageEmpty25}{LiePolynomialXmpPagePatch25}
+\pastebutton{LiePolynomialXmpPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{r: Lpoly := [p,c]\free{p c Lpoly }\bound{rr }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch26}
+\begin{paste}{LiePolynomialXmpPageFull26}{LiePolynomialXmpPageEmpty26}
+\pastebutton{LiePolynomialXmpPageFull26}{\hidepaste}
+\tab{5}\spadcommand{r1: Lpoly := eval(r, [a,b,c], [bb, cc, aa])$Lpoly\free{rr a b c aa bb cc Lpoly }\bound{r1 }}
+\indentrel{3}\begin{verbatim}
+   (26)  - [a b c]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty26}
+\begin{paste}{LiePolynomialXmpPageEmpty26}{LiePolynomialXmpPagePatch26}
+\pastebutton{LiePolynomialXmpPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{r1: Lpoly := eval(r, [a,b,c], [bb, cc, aa])$Lpoly\free{rr a b c aa bb cc Lpoly }\bound{r1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch27}
+\begin{paste}{LiePolynomialXmpPageFull27}{LiePolynomialXmpPageEmpty27}
+\pastebutton{LiePolynomialXmpPageFull27}{\hidepaste}
+\tab{5}\spadcommand{r2: Lpoly := eval(r, [a,b,c], [cc, aa, bb])$Lpoly\free{rr a b c cc bb aa Lpoly }\bound{r2 }}
+\indentrel{3}\begin{verbatim}
+   (27)  - [a c b]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty27}
+\begin{paste}{LiePolynomialXmpPageEmpty27}{LiePolynomialXmpPagePatch27}
+\pastebutton{LiePolynomialXmpPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{r2: Lpoly := eval(r, [a,b,c], [cc, aa, bb])$Lpoly\free{rr a b c cc bb aa Lpoly }\bound{r2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPagePatch28}
+\begin{paste}{LiePolynomialXmpPageFull28}{LiePolynomialXmpPageEmpty28}
+\pastebutton{LiePolynomialXmpPageFull28}{\hidepaste}
+\tab{5}\spadcommand{r + r1 + r2\free{rr r1 r2 }}
+\indentrel{3}\begin{verbatim}
+   (28)  0
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LiePolynomialXmpPageEmpty28}
+\begin{paste}{LiePolynomialXmpPageEmpty28}{LiePolynomialXmpPagePatch28}
+\pastebutton{LiePolynomialXmpPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{r + r1 + r2\free{rr r1 r2 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/LWORD.ht b/src/hyper/pages/LWORD.ht
new file mode 100644
index 00000000..5d065a57
--- /dev/null
+++ b/src/hyper/pages/LWORD.ht
@@ -0,0 +1,109 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\LyndonWordXmpTitle}{LyndonWord}
+\newcommand{\LyndonWordXmpNumber}{9.48}
+%
+% =====================================================================
+\begin{page}{LyndonWordXmpPage}{9.48 LyndonWord}
+% =====================================================================
+\beginscroll
+Initialisations
+\xtc{
+}{
+\spadpaste{a:Symbol :='a \bound{a}}
+}
+\xtc{
+}{
+\spadpaste{b:Symbol :='b \bound{b}}
+}
+\xtc{
+}{
+\spadpaste{c:Symbol :='c \bound{c}}
+}
+\xtc{
+}{
+\spadpaste{lword:= LyndonWord(Symbol) \bound{lword}}
+}
+\xtc{
+}{
+\spadpaste{magma := Magma(Symbol) \bound{magma}}
+}
+\xtc{
+}{
+\spadpaste{word   := OrderedFreeMonoid(Symbol) \bound{word}}
+}
+\xtc{
+All Lyndon words of with a, b, c to order 3
+}{
+\spadpaste{LyndonWordsList1([a,b,c],3)$lword \free{lword} \free{a} \free{b} \free{c} }
+}
+\xtc{
+All Lyndon words of with a, b, c to order 3 in flat list
+}{
+\spadpaste{LyndonWordsList([a,b,c],3)$lword \free{a} \free{b} \free{c} \free{lword}}
+}
+\xtc{
+All Lyndon words of with a, b to order 5
+}{
+\spadpaste{lw := LyndonWordsList([a,b],5)$lword \free{a} \free{b} \free{lword} \bound{lw}}
+}
+\xtc{
+}{
+\spadpaste{w1 : word := lw.4 :: word \free{word} \free{lw} \bound{w1}}
+}
+\xtc{
+}{
+\spadpaste{w2 : word := lw.5 :: word \free{word} \free{lw} \bound{w2}}
+}
+
+Let's try factoring
+\xtc{
+}{
+\spadpaste{factor(a::word)$lword \free{a word lword}}
+}
+\xtc{
+}{
+\spadpaste{factor(w1*w2)$lword \free{ w1 w2 lword}}
+}
+\xtc{
+}{
+\spadpaste{factor(w2*w2)$lword \free{w2 lword}}
+}
+\xtc{
+}{
+\spadpaste{factor(w2*w1)$lword \free{w1 w2 lword}}
+}
+
+Checks and coercions
+\xtc{
+}{
+\spadpaste{lyndon?(w1)$lword \free{w1 lword}}
+}
+\xtc{
+}{
+\spadpaste{lyndon?(w1*w2)$lword \free{w1 w2 lword}}
+}
+\xtc{
+}{
+\spadpaste{lyndon?(w2*w1)$lword \free{w1 w2 lword}}
+}
+\xtc{
+}{
+\spadpaste{lyndonIfCan(w1)$lword \free{w1 lword}}
+}
+\xtc{
+}{
+\spadpaste{lyndonIfCan(w2*w1)$lword \free{w1 w2 lword}}
+}
+\xtc{
+}{
+\spadpaste{lyndon(w1)$lword \free{w1 lword}}
+}
+\xtc{
+}{
+\spadpaste{lyndon(w1*w2)$lword \free{w1 w2 lword}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/LWORD.pht b/src/hyper/pages/LWORD.pht
new file mode 100644
index 00000000..9c3675e5
--- /dev/null
+++ b/src/hyper/pages/LWORD.pht
@@ -0,0 +1,377 @@
+\begin{patch}{LyndonWordXmpPagePatch1}
+\begin{paste}{LyndonWordXmpPageFull1}{LyndonWordXmpPageEmpty1}
+\pastebutton{LyndonWordXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{a:Symbol :='a\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (1)  a
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty1}
+\begin{paste}{LyndonWordXmpPageEmpty1}{LyndonWordXmpPagePatch1}
+\pastebutton{LyndonWordXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{a:Symbol :='a\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch2}
+\begin{paste}{LyndonWordXmpPageFull2}{LyndonWordXmpPageEmpty2}
+\pastebutton{LyndonWordXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{b:Symbol :='b\bound{b }}
+\indentrel{3}\begin{verbatim}
+   (2)  b
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty2}
+\begin{paste}{LyndonWordXmpPageEmpty2}{LyndonWordXmpPagePatch2}
+\pastebutton{LyndonWordXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{b:Symbol :='b\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch3}
+\begin{paste}{LyndonWordXmpPageFull3}{LyndonWordXmpPageEmpty3}
+\pastebutton{LyndonWordXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{c:Symbol :='c\bound{c }}
+\indentrel{3}\begin{verbatim}
+   (3)  c
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty3}
+\begin{paste}{LyndonWordXmpPageEmpty3}{LyndonWordXmpPagePatch3}
+\pastebutton{LyndonWordXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{c:Symbol :='c\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch4}
+\begin{paste}{LyndonWordXmpPageFull4}{LyndonWordXmpPageEmpty4}
+\pastebutton{LyndonWordXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{lword:= LyndonWord(Symbol)\bound{lword }}
+\indentrel{3}\begin{verbatim}
+   (4)  LyndonWord Symbol
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty4}
+\begin{paste}{LyndonWordXmpPageEmpty4}{LyndonWordXmpPagePatch4}
+\pastebutton{LyndonWordXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{lword:= LyndonWord(Symbol)\bound{lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch5}
+\begin{paste}{LyndonWordXmpPageFull5}{LyndonWordXmpPageEmpty5}
+\pastebutton{LyndonWordXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{magma := Magma(Symbol)\bound{magma }}
+\indentrel{3}\begin{verbatim}
+   (5)  Magma Symbol
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty5}
+\begin{paste}{LyndonWordXmpPageEmpty5}{LyndonWordXmpPagePatch5}
+\pastebutton{LyndonWordXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{magma := Magma(Symbol)\bound{magma }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch6}
+\begin{paste}{LyndonWordXmpPageFull6}{LyndonWordXmpPageEmpty6}
+\pastebutton{LyndonWordXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{word := OrderedFreeMonoid(Symbol)\bound{word }}
+\indentrel{3}\begin{verbatim}
+   (6)  OrderedFreeMonoid Symbol
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty6}
+\begin{paste}{LyndonWordXmpPageEmpty6}{LyndonWordXmpPagePatch6}
+\pastebutton{LyndonWordXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{word := OrderedFreeMonoid(Symbol)\bound{word }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch7}
+\begin{paste}{LyndonWordXmpPageFull7}{LyndonWordXmpPageEmpty7}
+\pastebutton{LyndonWordXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{LyndonWordsList1([a,b,c],3)$lword\free{lword }\free{a }\free{b }\free{c }}
+\indentrel{3}\begin{verbatim}
+   (7)
+   [[[a],[b],[c]], [[a b],[a c],[b c]],
+
+        2      2        2                         2
+     [[a b], [a c], [a b ], [a b c], [a c b], [a c ],
+        2        2
+      [b c], [b c ]]
+     ]
+       Type: OneDimensionalArray List LyndonWord Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty7}
+\begin{paste}{LyndonWordXmpPageEmpty7}{LyndonWordXmpPagePatch7}
+\pastebutton{LyndonWordXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{LyndonWordsList1([a,b,c],3)$lword\free{lword }\free{a }\free{b }\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch8}
+\begin{paste}{LyndonWordXmpPageFull8}{LyndonWordXmpPageEmpty8}
+\pastebutton{LyndonWordXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{LyndonWordsList([a,b,c],3)$lword\free{a }\free{b }\free{c }\free{lword }}
+\indentrel{3}\begin{verbatim}
+   (8)
+                                          2      2
+   [[a], [b], [c], [a b], [a c], [b c], [a b], [a c],
+        2                         2     2        2
+    [a b ], [a b c], [a c b], [a c ], [b c], [b c ]]
+                           Type: List LyndonWord Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty8}
+\begin{paste}{LyndonWordXmpPageEmpty8}{LyndonWordXmpPagePatch8}
+\pastebutton{LyndonWordXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{LyndonWordsList([a,b,c],3)$lword\free{a }\free{b }\free{c }\free{lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch9}
+\begin{paste}{LyndonWordXmpPageFull9}{LyndonWordXmpPageEmpty9}
+\pastebutton{LyndonWordXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{lw := LyndonWordsList([a,b],5)$lword\free{a }\free{b }\free{lword }\bound{lw }}
+\indentrel{3}\begin{verbatim}
+   (9)
+                       2        2     3      2 2
+   [[a], [b], [a b], [a b], [a b ], [a b], [a b ],
+        3     4      3 2     2          2 3           2
+    [a b ], [a b], [a b ], [a b a b], [a b ], [a b a b ],
+        4
+    [a b ]]
+                           Type: List LyndonWord Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty9}
+\begin{paste}{LyndonWordXmpPageEmpty9}{LyndonWordXmpPagePatch9}
+\pastebutton{LyndonWordXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{lw := LyndonWordsList([a,b],5)$lword\free{a }\free{b }\free{lword }\bound{lw }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch10}
+\begin{paste}{LyndonWordXmpPageFull10}{LyndonWordXmpPageEmpty10}
+\pastebutton{LyndonWordXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{w1 : word := lw.4 :: word\free{word }\free{lw }\bound{w1 }}
+\indentrel{3}\begin{verbatim}
+          2
+   (10)  a b
+                         Type: OrderedFreeMonoid Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty10}
+\begin{paste}{LyndonWordXmpPageEmpty10}{LyndonWordXmpPagePatch10}
+\pastebutton{LyndonWordXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{w1 : word := lw.4 :: word\free{word }\free{lw }\bound{w1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch11}
+\begin{paste}{LyndonWordXmpPageFull11}{LyndonWordXmpPageEmpty11}
+\pastebutton{LyndonWordXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{w2 : word := lw.5 :: word\free{word }\free{lw }\bound{w2 }}
+\indentrel{3}\begin{verbatim}
+            2
+   (11)  a b
+                         Type: OrderedFreeMonoid Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty11}
+\begin{paste}{LyndonWordXmpPageEmpty11}{LyndonWordXmpPagePatch11}
+\pastebutton{LyndonWordXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{w2 : word := lw.5 :: word\free{word }\free{lw }\bound{w2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch12}
+\begin{paste}{LyndonWordXmpPageFull12}{LyndonWordXmpPageEmpty12}
+\pastebutton{LyndonWordXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{factor(a::word)$lword\free{a word lword }}
+\indentrel{3}\begin{verbatim}
+   (12)  [[a]]
+                           Type: List LyndonWord Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty12}
+\begin{paste}{LyndonWordXmpPageEmpty12}{LyndonWordXmpPagePatch12}
+\pastebutton{LyndonWordXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{factor(a::word)$lword\free{a word lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch13}
+\begin{paste}{LyndonWordXmpPageFull13}{LyndonWordXmpPageEmpty13}
+\pastebutton{LyndonWordXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{factor(w1*w2)$lword\free{w1 w2 lword }}
+\indentrel{3}\begin{verbatim}
+            2     2
+   (13)  [[a b a b ]]
+                           Type: List LyndonWord Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty13}
+\begin{paste}{LyndonWordXmpPageEmpty13}{LyndonWordXmpPagePatch13}
+\pastebutton{LyndonWordXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{factor(w1*w2)$lword\free{w1 w2 lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch14}
+\begin{paste}{LyndonWordXmpPageFull14}{LyndonWordXmpPageEmpty14}
+\pastebutton{LyndonWordXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{factor(w2*w2)$lword\free{w2 lword }}
+\indentrel{3}\begin{verbatim}
+              2      2
+   (14)  [[a b ],[a b ]]
+                           Type: List LyndonWord Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty14}
+\begin{paste}{LyndonWordXmpPageEmpty14}{LyndonWordXmpPagePatch14}
+\pastebutton{LyndonWordXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{factor(w2*w2)$lword\free{w2 lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch15}
+\begin{paste}{LyndonWordXmpPageFull15}{LyndonWordXmpPageEmpty15}
+\pastebutton{LyndonWordXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{factor(w2*w1)$lword\free{w1 w2 lword }}
+\indentrel{3}\begin{verbatim}
+              2    2
+   (15)  [[a b ],[a b]]
+                           Type: List LyndonWord Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty15}
+\begin{paste}{LyndonWordXmpPageEmpty15}{LyndonWordXmpPagePatch15}
+\pastebutton{LyndonWordXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{factor(w2*w1)$lword\free{w1 w2 lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch16}
+\begin{paste}{LyndonWordXmpPageFull16}{LyndonWordXmpPageEmpty16}
+\pastebutton{LyndonWordXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{lyndon?(w1)$lword\free{w1 lword }}
+\indentrel{3}\begin{verbatim}
+   (16)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty16}
+\begin{paste}{LyndonWordXmpPageEmpty16}{LyndonWordXmpPagePatch16}
+\pastebutton{LyndonWordXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{lyndon?(w1)$lword\free{w1 lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch17}
+\begin{paste}{LyndonWordXmpPageFull17}{LyndonWordXmpPageEmpty17}
+\pastebutton{LyndonWordXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{lyndon?(w1*w2)$lword\free{w1 w2 lword }}
+\indentrel{3}\begin{verbatim}
+   (17)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty17}
+\begin{paste}{LyndonWordXmpPageEmpty17}{LyndonWordXmpPagePatch17}
+\pastebutton{LyndonWordXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{lyndon?(w1*w2)$lword\free{w1 w2 lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch18}
+\begin{paste}{LyndonWordXmpPageFull18}{LyndonWordXmpPageEmpty18}
+\pastebutton{LyndonWordXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{lyndon?(w2*w1)$lword\free{w1 w2 lword }}
+\indentrel{3}\begin{verbatim}
+   (18)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty18}
+\begin{paste}{LyndonWordXmpPageEmpty18}{LyndonWordXmpPagePatch18}
+\pastebutton{LyndonWordXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{lyndon?(w2*w1)$lword\free{w1 w2 lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch19}
+\begin{paste}{LyndonWordXmpPageFull19}{LyndonWordXmpPageEmpty19}
+\pastebutton{LyndonWordXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{lyndonIfCan(w1)$lword\free{w1 lword }}
+\indentrel{3}\begin{verbatim}
+           2
+   (19)  [a b]
+                     Type: Union(LyndonWord Symbol,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty19}
+\begin{paste}{LyndonWordXmpPageEmpty19}{LyndonWordXmpPagePatch19}
+\pastebutton{LyndonWordXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{lyndonIfCan(w1)$lword\free{w1 lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch20}
+\begin{paste}{LyndonWordXmpPageFull20}{LyndonWordXmpPageEmpty20}
+\pastebutton{LyndonWordXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{lyndonIfCan(w2*w1)$lword\free{w1 w2 lword }}
+\indentrel{3}\begin{verbatim}
+   (20)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty20}
+\begin{paste}{LyndonWordXmpPageEmpty20}{LyndonWordXmpPagePatch20}
+\pastebutton{LyndonWordXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{lyndonIfCan(w2*w1)$lword\free{w1 w2 lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch21}
+\begin{paste}{LyndonWordXmpPageFull21}{LyndonWordXmpPageEmpty21}
+\pastebutton{LyndonWordXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{lyndon(w1)$lword\free{w1 lword }}
+\indentrel{3}\begin{verbatim}
+           2
+   (21)  [a b]
+                                Type: LyndonWord Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty21}
+\begin{paste}{LyndonWordXmpPageEmpty21}{LyndonWordXmpPagePatch21}
+\pastebutton{LyndonWordXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{lyndon(w1)$lword\free{w1 lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPagePatch22}
+\begin{paste}{LyndonWordXmpPageFull22}{LyndonWordXmpPageEmpty22}
+\pastebutton{LyndonWordXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{lyndon(w1*w2)$lword\free{w1 w2 lword }}
+\indentrel{3}\begin{verbatim}
+           2     2
+   (22)  [a b a b ]
+                                Type: LyndonWord Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{LyndonWordXmpPageEmpty22}
+\begin{paste}{LyndonWordXmpPageEmpty22}{LyndonWordXmpPagePatch22}
+\pastebutton{LyndonWordXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{lyndon(w1*w2)$lword\free{w1 w2 lword }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/Link.ht b/src/hyper/pages/Link.ht
new file mode 100644
index 00000000..6cedca06
--- /dev/null
+++ b/src/hyper/pages/Link.ht
@@ -0,0 +1,628 @@
+
+%insert a pointer to reference section
+
+% Page pointed to from top level menu
+\begin{page}{htxl}{The AXIOM Link to NAG Software}
+\beginscroll
+\beginmenu
+\menumemolink{Introduction to the NAG Library Link}{nagLinkIntroPage}
+\menumemolink{Access the Link from HyperDoc}{htxl1}
+\menulispmemolink{Browser pages for individual routines}{(|kSearch| "Nag*")}
+\menumemolink{NAG Library Documentation}{FoundationLibraryDocPage}
+\endmenu
+\endscroll
+\end{page}
+
+
+
+\begin{page}{htxl1}{Use of the Link from HyperDoc}
+Click on the chapter of routines that you would like to use.
+\beginscroll
+\beginmenu
+\menumemolink{C02}{c02}\tab{8} Zeros of Polynomials
+\menumemolink{C05}{c05}\tab{8} Roots of One or More Transcendental Equations
+\menumemolink{C06}{c06}\tab{8} Summation of Series
+\menumemolink{D01}{d01}\tab{8} Quadrature
+\menumemolink{D02}{d02}\tab{8} Ordinary Differential Equations
+\menumemolink{D03}{d03}\tab{8} Partial Differential Equations
+\menumemolink{E01}{e01}\tab{8} Interpolation
+\menumemolink{E02}{e02}\tab{8} Curve and Surface Fitting
+\menumemolink{E04}{e04}\tab{8} Minimizing or Maximizing a Function
+\menumemolink{F01}{f01}\tab{8} Matrix Operations, Including Inversion
+\menumemolink{F02}{f02}\tab{8} Eigenvalues and Eigenvectors
+\menumemolink{F04}{f04}\tab{8} Simultaneous Linear Equations
+\menumemolink{F07}{f07}\tab{8} Linear Equations (LAPACK)
+\menumemolink{S}{s}\tab{8} Approximations of Special Functions
+\endmenu
+\endscroll
+\end{page}
+
+\begin{page}{c02}{C02 Zeros of Polynomials}
+\beginscroll
+\centerline{What would you like to do?}
+\newline
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter c02 Manual Page}{manpageXXc02}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagPolynomialRootsPackage")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{C02AFF}{(|c02aff|)}\space{}
+\tab{10}  All zeros of a complex polynomial 
+\menulispdownlink{C02AGF}{(|c02agf|)}\space{}
+\tab{10}  All zeros of a real polynomial
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
+\begin{page}{c05}{C05 Roots of One or More Transcendental Equations}
+\beginscroll
+\centerline{What would you like to do?}
+\newline
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter c05 Manual Page}{manpageXXc05}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagRootFindingPackage")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{C05ADF}{(|c05adf|)}\space{}
+\tab{10}  Zero of continuous function in given interval, Bus and Dekker algorithm
+\menulispdownlink{C05NBF}{(|c05nbf|)}\space{}
+\tab{10}  Solution of system of nonlinear equations using function values only
+\menulispdownlink{C05PBF}{(|c05pbf|)}\space{}
+\tab{10}  Solution of system of nonlinear equations using 1st derivatives
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
+\begin{page}{c06}{C06 Summation of Series}
+\beginscroll
+\centerline{What would you like to do?}
+\newline
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter c06 Manual Page}{manpageXXc06}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagSeriesSummationPackage")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{C06EAF}{(|c06eaf|)}\space{}
+\tab{10}  Single 1-D real discrete Fourier transform, no extra workspace
+\menulispdownlink{C06EBF}{(|c06ebf|)}\space{}
+\tab{10}  Single 1-D Hermitian discrete Fourier transform, no extra workspace
+\menulispdownlink{C06ECF}{(|c06ecf|)}\space{}
+\tab{10}  Single 1-D complex discrete Fourier transform, no extra workspace
+\menulispdownlink{C06EKF}{(|c06ekf|)}\space{}
+\tab{10}  Circular convolution or correlation of two real vectors, no extra
+workspace
+\menulispdownlink{C06FPF}{(|c06fpf|)}\space{}
+\tab{10}  Multiple 1-D real discrete Fourier transforms
+\menulispdownlink{C06FQF}{(|c06fqf|)}\space{}
+\tab{10}  Multiple 1-D Hermitian discrete Fourier transforms
+\menulispdownlink{C06FRF}{(|c06frf|)}\space{}
+\tab{10}  Multiple 1-D complex discrete Fourier transforms
+\menulispdownlink{C06FUF}{(|c06fuf|)}\space{}
+\tab{10}  2-D complex discrete Fourier transforms
+\menulispdownlink{C06GBF}{(|c06gbf|)}\space{}
+\tab{10}  Complex conjugate of Hermitian sequence
+\menulispdownlink{C06GCF}{(|c06gcf|)}\space{}
+\tab{10}  Complex conjugate of complex sequence
+\menulispdownlink{C06GQF}{(|c06gqf|)}\space{}
+\tab{10}  Complex conjugate of multiple Hermitian sequences
+\menulispdownlink{C06GSF}{(|c06gsf|)}\space{}
+\tab{10}  Convert Hermitian sequences to general complex sequences
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
+\begin{page}{d01}{D01 Quadrature}
+\beginscroll
+\centerline{What would you like to do?}
+\newline
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter d01 Manual Page}{manpageXXd01}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagIntegrationPackage")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{D01AJF}{(|d01ajf|)}\space{}
+\tab{10}  1-D quadrature, adaptive, finite interval, strategy due to Plessens
+and de Doncker, allowing for badly-behaved integrands
+\menulispdownlink{D01AKF}{(|d01akf|)}\space{}
+\tab{10}  1-D quadrature, adaptive, finite interval, method suitable for
+oscillating functions
+\menulispdownlink{D01ALF}{(|d01alf|)}\space{}
+\tab{10}  1-D quadrature, adaptive, finite interval, allowing for 
+singularities at user specified points
+\menulispdownlink{D01AMF}{(|d01amf|)}\space{}
+\tab{10}  1-D quadrature, adaptive, infinite or semi-finite interval
+\menulispdownlink{D01ANF}{(|d01anf|)}\space{}
+\tab{10}  1-D quadrature, adaptive, finite interval, weight function
+ cos(\omega x) or sin(\omega x)
+\menulispdownlink{D01APF}{(|d01apf|)}\space{}
+\tab{10}  1-D quadrature, adaptive, finite interval, weight function
+with end point singularities of algebraico-logarithmic type
+\menulispdownlink{D01AQF}{(|d01aqf|)}\space{}
+\tab{10}  1-D quadrature, adaptive, finite interval, weight function
+1/(x-c), Cauchy principle value (Hilbert transform)
+\menulispdownlink{D01ASF}{(|d01asf|)}\space{}
+\tab{10}  1-D quadrature, adaptive, semi-infinite interval, weight function
+cos(\omega x) or sin(\omega x)
+\menulispdownlink{D01BBF}{(|d01bbf|)}\space{}
+\tab{10} Pre-computed weights and abscissae for Gaussian quadrature rules,
+restricted choice of rule
+\menulispdownlink{D01FCF}{(|d01fcf|)}\space{}
+\tab{10} Multi-dimensional adaptive quadrature over hyper-rectangle
+\menulispdownlink{D01GAF}{(|d01gaf|)}\space{}
+\tab{10} 1-D quadrature, integration of function defined by data values,
+Gill-Miller method
+\menulispdownlink{D01GBF}{(|d01gbf|)}\space{}
+\tab{10} Multi-dimensional quadrature over hyper-rectangle, Monte Carlo method
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
+\begin{page}{d02}{D02 Ordinary Differential Equations}
+\beginscroll
+\centerline{What would you like to do?}
+\newline
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter d02 Manual Page}{manpageXXd02}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagOrdinaryDifferentialEquationsPackage")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{D02BBF}{(|d02bbf|)}\space{}
+\tab{10}  ODEs, IVP, Runge-Kutta-Merson method, over a range, 
+intermediate output
+\menulispdownlink{D02BHF}{(|d02bhf|)}\space{}
+\tab{10}  ODEs, IVP, Runge-Kutta-Merson method, until function of 
+solution is zero
+\menulispdownlink{D02CJF}{(|d02cjf|)}\space{}
+\tab{10}  ODEs, IVP, Adams method, until function of solution is zero,
+intermediate output
+\menulispdownlink{D02EJF}{(|d02ejf|)}\space{}
+\tab{10}  ODEs, stiff IVP, BDF method, until function of solution is zero,
+intermediate output
+\menulispdownlink{D02GAF}{(|d02gaf|)}\space{}
+\tab{10}  ODEs, boundary value problem, finite difference technique with
+deferred correction, simple nonlinear problem
+\menulispdownlink{D02GBF}{(|d02gbf|)}\space{}
+\tab{10}  ODEs, boundary value problem, finite difference technique with
+deferred correction, general nonlinear problem
+\menulispdownlink{D02KEF}{(|d02kef|)}\space{}
+\tab{10}  2nd order Sturm-Liouville problem, regular/singular system,
+finite/infinite range, eigenvalue and eigenfunction, user-specified 
+break-points
+\menulispdownlink{D02RAF}{(|d02raf|)}\space{}
+\tab{10}  ODEs, general nonlinear boundary value problem, finite difference
+technique with deferred correction, continuation facility
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
+\begin{page}{d03}{D03 Partial Differential Equations}
+\beginscroll
+\centerline{What would you like to do?}
+\newline
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter d03 Manual Page}{manpageXXd03}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagPartialDifferentialEquationsPackage")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{D03EDF}{(|d03edf|)}\space{}
+\tab{10}  Elliptic PDE, solution of finite difference equations by a multigrid 
+technique
+\menulispdownlink{D03EEF}{(|d03eef|)}\space{}
+\tab{10}  Discretize a 2nd order elliptic PDE on a rectangle
+\menulispdownlink{D03FAF}{(|d03faf|)}\space{}
+\tab{10}  Elliptic PDE, Helmholtz equation, 3-D Cartesian co-ordinates
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
+\begin{page}{e01}{E01 Interpolation}
+\beginscroll
+\centerline{What would you like to do?}
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter e01 Manual Page}{manpageXXe01}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagInterpolationPackage")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{E01BAF}{(|e01baf|)}\space{}
+\tab{10}  Interpolating functions, cubic spline interpolant, one variable
+\menulispdownlink{E01BEF}{(|e01bef|)}\space{}
+\tab{10}  Interpolating functions, monotonicity-preserving, piecewise
+cubic Hermite, one variable
+\menulispdownlink{E01BFF}{(|e01bff|)}\space{}
+\tab{10}  Interpolated values, interpolant computed by E01BEF, function
+only, one variable
+\menulispdownlink{E01BGF}{(|e01bgf|)}\space{}
+\tab{10}  Interpolated values, interpolant computed by E01BEF, function
+and 1st derivative, one variable
+\menulispdownlink{E01BHF}{(|e01bhf|)}\space{}
+\tab{10}  Interpolated values, interpolant computed by E01BEF, definite
+integral, one variable
+\menulispdownlink{E01DAF}{(|e01daf|)}\space{}
+\tab{10}  Interpolating functions, fitting bicubic spline, data on a 
+rectangular grid
+\menulispdownlink{E01SAF}{(|e01saf|)}\space{}
+\tab{10}  Interpolating functions, method of Renka and Cline, two variables
+\menulispdownlink{E01SEF}{(|e01sef|)}\space{}
+\tab{10}  Interpolating functions, modified Shepherd's method, two variables
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
+\begin{page}{e02}{E02 Curve and Surface Fitting}
+\beginscroll
+\centerline{What would you like to do?}
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter e02 Manual Page}{manpageXXe02}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagFittingPackage")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{E02ADF}{(|e02adf|)}\space{}
+\tab{10}  Least-squares curve fit, by polynomials, arbitrary data points
+\menulispdownlink{E02AEF}{(|e02aef|)}\space{}
+\tab{10}  Evaluation of fitted polynomial in one variable from Chebyshev series
+form (simplified parameter list)
+\menulispdownlink{E02AGF}{(|e02agf|)}\space{}
+\tab{10}  Least-squares polynomial fit, values and derivatives may be
+constrained, arbitrary data points
+\menulispdownlink{E02AHF}{(|e02ahf|)}\space{}
+\tab{10}  Derivative of fitted polynomial in Chebyshev series form
+\menulispdownlink{E02AJF}{(|e02ajf|)}\space{}
+\tab{10}  Integral of fitted polynomial in Chebyshev series form
+\menulispdownlink{E02AKF}{(|e02akf|)}\space{}
+\tab{10}  Evaluation of fitted polynomial in one variable, from Chebyshev 
+series form
+\menulispdownlink{E02BAF}{(|e02baf|)}\space{}
+\tab{10}  Least-squares curve cubic spline fit (including interpolation)
+\menulispdownlink{E02BBF}{(|e02bbf|)}\space{}
+\tab{10}  Evaluation of fitted cubic spline, function only
+\menulispdownlink{E02BCF}{(|e02bcf|)}\space{}
+\tab{10}  Evaluation of fitted cubic spline, function and derivatives
+\menulispdownlink{E02BDF}{(|e02bdf|)}\space{}
+\tab{10}  Evaluation of fitted cubic spline, definite integral
+\menulispdownlink{E02BEF}{(|e02bef|)}\space{}
+\tab{10}  Least-squares curve cubic spline fit, automatic knot placement
+\menulispdownlink{E02DAF}{(|e02daf|)}\space{}
+\tab{10}  Least-squares surface fit, bicubic splines
+\menulispdownlink{E02DCF}{(|e02dcf|)}\space{}
+\tab{10}  Least-squares surface fit by bicubic splines with automatic knot
+placement, data on a rectangular grid
+\menulispdownlink{E02DDF}{(|e02ddf|)}\space{}
+\tab{10}  Least-squares surface fit by bicubic splines with automatic knot
+placement, scattered data
+\menulispdownlink{E02DEF}{(|e02def|)}\space{}
+\tab{10}  Evaluation of a fitted bicubic spline at a vector of points
+\menulispdownlink{E02DFF}{(|e02dff|)}\space{}
+\tab{10}  Evaluation of a fitted bicubic spline at a mesh of points
+\menulispdownlink{E02GAF}{(|e02gaf|)}\space{}
+\tab{10}  \htbitmap{l1}-approximation by general linear function
+\menulispdownlink{E02ZAF}{(|e02zaf|)}\space{}
+\tab{10}  Sort 2-D sata into panels for fitting bicubic splines
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
+\begin{page}{e04}{E04 Minimizing or Maximizing a Function}
+\beginscroll
+\centerline{What would you like to do?}
+\newline
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter e04 Manual Page}{manpageXXe04}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagOptimisationPackage")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{E04DGF}{(|e04dgf|)}\space{}
+\tab{10}  Unconstrained minimum, pre-conditioned conjugate gradient algorithm,
+function of several variables using 1st derivatives
+\menulispdownlink{E04FDF}{(|e04fdf|)}\space{}
+\tab{10}  Unconstrained minimum of a sum of squares, combined Gauss-Newton
+and modified Newton algorithm using function values only
+\menulispdownlink{E04GCF}{(|e04gcf|)}\space{}
+\tab{10}  Unconstrained minimum, of a sum of squares, combined Gauss-Newton
+and modified Newton algorithm using 1st derivatives
+\menulispdownlink{E04JAF}{(|e04jaf|)}\space{}
+\tab{10}  Minimum, function of several variables, quasi-Newton algorithm,
+simple bounds, using function values only
+\menulispdownlink{E04MBF}{(|e04mbf|)}\space{}
+\tab{10}  Linear programming problem
+\menulispdownlink{E04NAF}{(|e04naf|)}\space{}
+\tab{10}  Quadratic programming problem
+\menulispdownlink{E04UCF}{(|e04ucf|)}\space{}
+\tab{10}  Minimum, function of several variables, sequential QP method,
+nonlinear constraints, using function values and optionally 1st derivatives
+\menulispdownlink{E04YCF}{(|e04ycf|)}\space{}
+\tab{10}  Covariance matrix for non-linear least-squares problem
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
+\begin{page}{f01}{F01 Matrix Operations - Including Inversion}
+\beginscroll
+\centerline{What would you like to do?}
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter f Manual Page}{manpageXXf}
+\menuwindowlink{Foundation Library Chapter f01 Manual Page}{manpageXXf01}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagMatrixOperationsPackage")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{F01BRF}{(|f01brf|)}\space{}
+\tab{10}  {\it LU} factorization of real sparse matrix
+\menulispdownlink{F01BSF}{(|f01bsf|)}\space{}
+\tab{10}  {\it LU} factorization of real sparse matrix with known sparsity
+pattern
+\menulispdownlink{F01MAF}{(|f01maf|)}\space{}
+\tab{10}  \htbitmap{llt} factorization of real sparse
+symmetric positive-definite matrix
+\menulispdownlink{F01MCF}{(|f01mcf|)}\space{}
+\tab{10}  \htbitmap{ldlt} factorization of real 
+symmetric positive-definite variable-bandwith matrix
+\menulispdownlink{F01QCF}{(|f01qcf|)}\space{}
+\tab{10}  {\it QR} factorization of real {\it m} by {\it n} matrix 
+(m \htbitmap{great=} n)
+\menulispdownlink{F01QDF}{(|f01qdf|)}\space{}
+\tab{10} Operations with orthogonal matrices, compute {\it QB} or
+\htbitmap{f01qdf} after factorization by F01QCF or F01QFF
+\menulispdownlink{F01QEF}{(|f01qef|)}\space{}
+\tab{10} Operations with orthogonal matrices, form columns of {\it Q}
+after factorization by F01QCF or F01QFF 
+\menulispdownlink{F01RCF}{(|f01rcf|)}\space{}
+\tab{10} {\it QR} factorization of complex {\it m} by {\it n} matrix 
+(m \htbitmap{great=} n) 
+\menulispdownlink{F01RDF}{(|f01rdf|)}\space{}
+\tab{10} Operations with unitary matrices, compute {\it QB} or
+\htbitmap{f01rdf} after factorization by F01RCF
+\menulispdownlink{F01REF}{(|f01ref|)}\space{}
+\tab{10} Operations with unitary matrices, form columns of {\it Q}
+after factorization by F01RCF
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
+\begin{page}{f02}{F02 Eigenvalues and Eigenvectors}
+\beginscroll
+\centerline{What would you like to do?}
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter f Manual Page}{manpageXXf}
+\menuwindowlink{Foundation Library Chapter f02 Manual Page}{manpageXXf02}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagEigenPackage")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{F02AAF}{(|f02aaf|)}\space{}
+\tab{10}  All eigenvalues of real symmetric matrix (Black box)
+\menulispdownlink{F02ABF}{(|f02abf|)}\space{}
+\tab{10}  All eigenvalues and eigenvectors of real symmetric matrix (Black box)
+\menulispdownlink{F02ADF}{(|f02adf|)}\space{}
+\tab{10}  All eigenvalues of generalized real eigenproblem of the form
+Ax = \lambda Bx where A and B are symmetric and B is positive definite
+\menulispdownlink{F02AEF}{(|f02aef|)}\space{}
+\tab{10}  All eigenvalues and eigenvectors of generalized real eigenproblem 
+of the form Ax = \lambda Bx where A and B are symmetric and B is positive 
+definite
+\menulispdownlink{F02AFF}{(|f02aff|)}\space{}
+\tab{10}  All eigenvalues of real matrix (Black box)
+\menulispdownlink{F02AGF}{(|f02agf|)}\space{}
+\tab{10}  All eigenvalues and eigenvectors of real matrix (Black box)
+\menulispdownlink{F02AJF}{(|f02ajf|)}\space{}
+\tab{10}  All eigenvalues of complex matrix (Black box)
+\menulispdownlink{F02AKF}{(|f02akf|)}\space{}
+\tab{10}  All eigenvalues and eigenvectors of complex matrix (Black box)
+\menulispdownlink{F02AWF}{(|f02awf|)}\space{}
+\tab{10}  All eigenvalues of complex Hermitian matrix (Black box)
+\menulispdownlink{F02AXF}{(|f02axf|)}\space{}
+\tab{10}  All eigenvalues and eigenvectors of complex Hermitian 
+matrix (Black box)
+\menulispdownlink{F02BBF}{(|f02bbf|)}\space{}
+\tab{10}  Selected eigenvalues and eigenvectors of real symmetric
+matrix (Black box)
+\menulispdownlink{F02BJF}{(|f02bjf|)}\space{}
+\tab{10}  All eigenvalues and optionally eigenvectors of generalized
+eigenproblem by {\it QZ} algorithm, real matrices (Black box)
+\menulispdownlink{F02FJF}{(|f02fjf|)}\space{}
+\tab{10}  Selected eigenvalues and eigenvectors of sparse symmetric 
+eigenproblem
+\menulispdownlink{F02WEF}{(|f02wef|)}\space{}
+\tab{10}  SVD of real matrix
+\menulispdownlink{F02XEF}{(|f02xef|)}\space{}
+\tab{10}  SVD of complex matrix
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
+\begin{page}{f04}{F04 Simultaneous Linear Equations}
+\beginscroll
+\centerline{What would you like to do?}
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter f Manual Page}{manpageXXf}
+\menuwindowlink{Foundation Library Chapter f04 Manual Page}{manpageXXf04}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagLinearEquationSolvingPackage")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{F04ADF}{(|f04adf|)}\space{}
+\tab{10}  Solution of complex simultaneous linear equations, with multiple
+right-hand sides (Black box)
+\menulispdownlink{F04ARF}{(|f04arf|)}\space{}
+\tab{10}  Solution of real simultaneous linear equations, one right-hand side
+(Black box)
+\menulispdownlink{F04ASF}{(|f04asf|)}\space{}
+\tab{10}  Solution of real symmetric positive-definite simultaneous linear
+equations, one right-hand side using iterative refinement (Black box)
+\menulispdownlink{F04ATF}{(|f04atf|)}\space{}
+\tab{10}  Solution of real simultaneous linear equations, one right-hand side 
+using iterative refinement (Black box)
+\menulispdownlink{F04AXF}{(|f04axf|)}\space{}
+\tab{10}  Approximate solution of real sparse simultaneous linear equations
+(coefficient matrix already factorized by F01BRF or F01BSF)
+\menulispdownlink{F04FAF}{(|f04faf|)}\space{}
+\tab{10}  Solution of real symmetric positive-definite tridiagonal 
+simultaneous linear equations, one right-hand side (Black box)
+\menulispdownlink{F04JGF}{(|f04jgf|)}\space{}
+\tab{10}  Least-squares (if rank = n) or minimal least-squares (if rank < n) 
+solution of m real equations in n unknowns, rank \htbitmap{less=} n,
+ m \htbitmap{great=} n
+\menulispdownlink{F04MAF}{(|f04maf|)}\space{}
+\tab{10}  Real sparse symmetric positive-definite simultaneous linear
+equations(coefficient matrix already factorized)
+\menulispdownlink{F04MBF}{(|f04mbf|)}\space{}
+\tab{10}  Real sparse symmetric simultaneous linear equations
+\menulispdownlink{F04MCF}{(|f04mcf|)}\space{}
+\tab{10}  Approximate solution of real symmetric positive-definite 
+variable-bandwidth simultaneous linear equations (coefficient matrix 
+already factorized)
+\menulispdownlink{F04QAF}{(|f04qaf|)}\space{}
+\tab{10}  Sparse linear least-squares problem, {\it m} real equations
+in {\it n} unknowns
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
+\begin{page}{f07}{F07 Linear Equations (LAPACK)}
+\beginscroll
+\centerline{What would you like to do?}
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter f Manual Page}{manpageXXf}
+\menuwindowlink{Foundation Library Chapter f07 Manual Page}{manpageXXf07}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagLapack")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{F07ADF}{(|f07adf|)}\space{}
+\tab{10}  (DGETRF) {\it LU} factorization of real {\it m} by {\it n} matrix
+\menulispdownlink{F07AEF}{(|f07aef|)}\space{}
+\tab{10}  (DGETRS) Solution of real system of linear equations, multiple 
+right hand sides, matrix factorized by F07ADF
+\menulispdownlink{F07FDF}{(|f07fdf|)}\space{}
+\tab{10}  (DPOTRF) Cholesky factorization of real symmetric positive-definite
+matrix
+\menulispdownlink{F07FEF}{(|f07fef|)}\space{}
+\tab{10}  (DPOTRS) Solution of real symmetric positive-definite system of
+linear equations, multiple right-hand sides, matrix already factorized by 
+F07FDF
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
+
+\begin{page}{s}{S \space{2} Approximations of Special Functions}
+\beginscroll
+\centerline{What would you like to do?}
+\beginmenu
+\item Read
+\menuwindowlink{Foundation Library Chapter s Manual Page}{manpageXXs}
+\item or
+\menulispwindowlink{Browse}{(|kSearch| "NagSpecialFunctionsPackage")}\tab{10} through this chapter
+\item or use the routines:
+\menulispdownlink{S01EAF}{(|s01eaf|)}\space{}
+\tab{10}  Complex exponential {\em exp(z)}
+\menulispdownlink{S13AAF}{(|s13aaf|)}\space{}
+\tab{10}  Exponential integral \htbitmap{s13aaf2}
+\menulispdownlink{S13ACF}{(|s13acf|)}\space{}
+\tab{10}  Cosine integral {\em Ci(x)}
+\menulispdownlink{S13ADF}{(|s13adf|)}\space{}
+\tab{10}  Sine integral {\em Si(x)}
+\menulispdownlink{S14AAF}{(|s14aaf|)}\space{}
+\tab{10}  Gamma function \Gamma
+\menulispdownlink{S14ABF}{(|s14abf|)}\space{}
+\tab{10}  Log Gamma function {\em ln \Gamma}
+\menulispdownlink{S14BAF}{(|s14baf|)}\space{}
+\tab{10}  Incomplete gamma functions P(a,x) and Q(a,x)
+\menulispdownlink{S15ADF}{(|s15adf|)}\space{}
+\tab{10}  Complement of error function {\em erfc x }
+\menulispdownlink{S15AEF}{(|s15aef|)}\space{}
+\tab{10}  Error function {\em erf x}
+\menulispdownlink{S17ACF}{(|s17acf|)}\space{}
+\tab{10}  Bessel function \space{1} \htbitmap{s17acf} 
+\menulispdownlink{S17ADF}{(|s17adf|)}\space{}
+\tab{10}  Bessel function \space{1} \htbitmap{s17adf} 
+\menulispdownlink{S17AEF}{(|s17aef|)}\space{}
+\tab{10}  Bessel function \space{1} \htbitmap{s17aef1} 
+\menulispdownlink{S17AFF}{(|s17aff|)}\space{}
+\tab{10}  Bessel function \space{1} \htbitmap{s17aff1} 
+\menulispdownlink{S17AGF}{(|s17agf|)}
+\tab{10}  Airy function {\em Ai(x)}
+\menulispdownlink{S17AHF}{(|s17ahf|)}
+\tab{10}  Airy function {\em Bi(x)}
+\menulispdownlink{S17AJF}{(|s17ajf|)}
+\tab{10}  Airy function {\em Ai'(x)}
+\menulispdownlink{S17AKF}{(|s17akf|)}
+\tab{10}  Airy function {\em Bi'(x)}
+\menulispdownlink{S17DCF}{(|s17dcf|)}
+\tab{10} Bessel function \htbitmap{s17dcf}, real a \space{1}
+\htbitmap{great=} 0, complex z, v = 0,1,2,...
+\menulispdownlink{S17DEF}{(|s17def|)}
+\tab{10} Bessel function \htbitmap{s17def}, real a \space{1}
+\htbitmap{great=} 0, complex z, v = 0,1,2,...
+\menulispdownlink{S17DGF}{(|s17dgf|)}
+\tab{10} Airy function {\em Ai(z)} and {\em Ai'(z)}, complex z
+\menulispdownlink{S17DHF}{(|s17dhf|)}
+\tab{10} Airy function {\em Bi(z)} and {\em Bi'(z)}, complex z
+\menulispdownlink{S17DLF}{(|s17dlf|)}
+\tab{10} Hankel function \vspace{-32} \htbitmap{s17dlf}
+\vspace{-37}, j = 1,2, real a \space{1} \htbitmap{great=} 0, 
+complex z, v = 0,1,2,... \newline
+\menulispdownlink{S18ACF}{(|s18acf|)}
+\tab{10} Modified Bessel function \space{1} \htbitmap{s18acf1} 
+\menulispdownlink{S18ADF}{(|s18adf|)}
+\tab{10} Modified Bessel function \space{1} \htbitmap{s18adf1}
+\menulispdownlink{S18AEF}{(|s18aef|)}
+\tab{10} Modified Bessel function \space{1} \htbitmap{s18aef1} 
+\menulispdownlink{S18AFF}{(|s18aff|)}
+\tab{10} Modified Bessel function \space{1} \htbitmap{s18aff1} 
+\menulispdownlink{S18DCF}{(|s18dcf|)}
+\tab{10} Modified bessel function \htbitmap{s18dcf}, real a \space{1}
+\htbitmap{great=} 0, complex z, v = 0,1,2,...
+\menulispdownlink{S18DEF}{(|s18def|)}
+\tab{10} Modified bessel function \htbitmap{s18def}, real a \space{1}
+\htbitmap{great=} 0, complex z, v = 0,1,2,...
+\menulispdownlink{S19AAF}{(|s19aaf|)}
+\tab{10} Kelvin function {\em ber x}
+\menulispdownlink{S19ABF}{(|s19abf|)}
+\tab{10} Kelvin function {\em bei x}
+\menulispdownlink{S19ACF}{(|s19acf|)}
+\tab{10} Kelvin function {\em ker x}
+\menulispdownlink{S19ADF}{(|s19adf|)}
+\tab{10} Kelvin function {\em kei x}
+\menulispdownlink{S20ACF}{(|s20acf|)}
+\tab{10} Fresnel integral {\em S(x)}
+\menulispdownlink{S20ADF}{(|s20adf|)}
+\tab{10} Fresnel integral {\em C(x)}
+\menulispdownlink{S21BAF}{(|s21baf|)}
+\tab{10} Degenerate symmetrised elliptic integral of 1st kind 
+\space{1} \htbitmap{s21baf1} 
+\menulispdownlink{S21BBF}{(|s21bbf|)}
+\tab{10} Symmetrised elliptic integral of 1st kind \space{1} 
+\vspace{-28} \htbitmap{s21bbf1} \vspace{-40} 
+\menulispdownlink{S21BCF}{(|s21bcf|)}
+\tab{10} Symmetrised elliptic integral of 2nd kind \space{1} 
+\vspace{-28} \htbitmap{s21bcf1} \vspace{-40} 
+\menulispdownlink{S21BDF}{(|s21bdf|)}
+\tab{10} Symmetrised elliptic integral of 3rd kind \space{1} 
+\vspace{-26} \htbitmap{s21bdf1} \vspace{-40} 
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+
diff --git a/src/hyper/pages/MAGMA.ht b/src/hyper/pages/MAGMA.ht
new file mode 100644
index 00000000..1876426e
--- /dev/null
+++ b/src/hyper/pages/MAGMA.ht
@@ -0,0 +1,113 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\MagmaXmpTitle}{Magma}
+\newcommand{\MagmaXmpNumber}{9.49}
+%
+% =====================================================================
+\begin{page}{MagmaXmpPage}{9.49 Magma}
+% =====================================================================
+\beginscroll
+Initialisations
+\xtc{
+}{
+\spadpaste{x:Symbol :='x \bound{x}}
+}
+\xtc{
+}{
+\spadpaste{y:Symbol :='y \bound{y}}
+}
+\xtc{
+}{
+\spadpaste{z:Symbol :='z \bound{z}}
+}
+\xtc{
+}{
+\spadpaste{word := OrderedFreeMonoid(Symbol) \bound{word}}
+}
+\xtc{
+}{
+\spadpaste{tree := Magma(Symbol) \bound{tree}}
+}
+
+Let's make some trees
+\xtc{
+}{
+\spadpaste{a:tree := x*x \free{x tree} \bound{a}}
+}
+\xtc{
+}{
+\spadpaste{b:tree := y*y \free{y tree} \bound{b}}
+}
+\xtc{
+}{
+\spadpaste{c:tree := a*b \free{a b tree} \bound{c}}
+}
+
+Query the trees
+\xtc{
+}{
+\spadpaste{left c \free{c}}
+}
+\xtc{
+}{
+\spadpaste{right c \free{c}}
+}
+\xtc{
+}{
+\spadpaste{length c \free{c}}
+}
+\xtc{
+Coerce to the monoid
+}{
+\spadpaste{c::word \free{c word}}
+}
+
+Check ordering
+\xtc{
+}{
+\spadpaste{a < b \free{a b}}
+}
+\xtc{
+}{
+\spadpaste{a < c \free{a c}}
+}
+\xtc{
+}{
+\spadpaste{b < c \free{b c}}
+}
+
+Navigate the tree
+\xtc{
+}{
+\spadpaste{first c \free{c}}
+}
+\xtc{
+}{
+\spadpaste{rest c \free{c}}
+}
+\xtc{
+}{
+\spadpaste{rest rest c  \free{c}}
+}
+
+Check ordering
+\xtc{
+}{
+\spadpaste{ax:tree := a*x \free{a x tree} \bound{ax}}
+}
+\xtc{
+}{
+\spadpaste{xa:tree := x*a \free{a x tree} \bound{xa}}
+}
+\xtc{
+}{
+\spadpaste{xa < ax \free{ax xa}}
+}
+\xtc{
+}{
+\spadpaste{lexico(xa,ax) \free{ax xa}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/MAGMA.pht b/src/hyper/pages/MAGMA.pht
new file mode 100644
index 00000000..f6cbc6a2
--- /dev/null
+++ b/src/hyper/pages/MAGMA.pht
@@ -0,0 +1,353 @@
+\begin{patch}{MagmaXmpPagePatch1}
+\begin{paste}{MagmaXmpPageFull1}{MagmaXmpPageEmpty1}
+\pastebutton{MagmaXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{x:Symbol :='x\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (1)  x
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty1}
+\begin{paste}{MagmaXmpPageEmpty1}{MagmaXmpPagePatch1}
+\pastebutton{MagmaXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{x:Symbol :='x\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch2}
+\begin{paste}{MagmaXmpPageFull2}{MagmaXmpPageEmpty2}
+\pastebutton{MagmaXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{y:Symbol :='y\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (2)  y
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty2}
+\begin{paste}{MagmaXmpPageEmpty2}{MagmaXmpPagePatch2}
+\pastebutton{MagmaXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{y:Symbol :='y\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch3}
+\begin{paste}{MagmaXmpPageFull3}{MagmaXmpPageEmpty3}
+\pastebutton{MagmaXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{z:Symbol :='z\bound{z }}
+\indentrel{3}\begin{verbatim}
+   (3)  z
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty3}
+\begin{paste}{MagmaXmpPageEmpty3}{MagmaXmpPagePatch3}
+\pastebutton{MagmaXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{z:Symbol :='z\bound{z }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch4}
+\begin{paste}{MagmaXmpPageFull4}{MagmaXmpPageEmpty4}
+\pastebutton{MagmaXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{word := OrderedFreeMonoid(Symbol)\bound{word }}
+\indentrel{3}\begin{verbatim}
+   (4)  OrderedFreeMonoid Symbol
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty4}
+\begin{paste}{MagmaXmpPageEmpty4}{MagmaXmpPagePatch4}
+\pastebutton{MagmaXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{word := OrderedFreeMonoid(Symbol)\bound{word }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch5}
+\begin{paste}{MagmaXmpPageFull5}{MagmaXmpPageEmpty5}
+\pastebutton{MagmaXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{tree := Magma(Symbol)\bound{tree }}
+\indentrel{3}\begin{verbatim}
+   (5)  Magma Symbol
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty5}
+\begin{paste}{MagmaXmpPageEmpty5}{MagmaXmpPagePatch5}
+\pastebutton{MagmaXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{tree := Magma(Symbol)\bound{tree }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch6}
+\begin{paste}{MagmaXmpPageFull6}{MagmaXmpPageEmpty6}
+\pastebutton{MagmaXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{a:tree := x*x\free{x tree }\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (6)  [x,x]
+                                     Type: Magma Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty6}
+\begin{paste}{MagmaXmpPageEmpty6}{MagmaXmpPagePatch6}
+\pastebutton{MagmaXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{a:tree := x*x\free{x tree }\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch7}
+\begin{paste}{MagmaXmpPageFull7}{MagmaXmpPageEmpty7}
+\pastebutton{MagmaXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{b:tree := y*y\free{y tree }\bound{b }}
+\indentrel{3}\begin{verbatim}
+   (7)  [y,y]
+                                     Type: Magma Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty7}
+\begin{paste}{MagmaXmpPageEmpty7}{MagmaXmpPagePatch7}
+\pastebutton{MagmaXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{b:tree := y*y\free{y tree }\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch8}
+\begin{paste}{MagmaXmpPageFull8}{MagmaXmpPageEmpty8}
+\pastebutton{MagmaXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{c:tree := a*b\free{a b tree }\bound{c }}
+\indentrel{3}\begin{verbatim}
+   (8)  [[x,x],[y,y]]
+                                     Type: Magma Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty8}
+\begin{paste}{MagmaXmpPageEmpty8}{MagmaXmpPagePatch8}
+\pastebutton{MagmaXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{c:tree := a*b\free{a b tree }\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch9}
+\begin{paste}{MagmaXmpPageFull9}{MagmaXmpPageEmpty9}
+\pastebutton{MagmaXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{left c\free{c }}
+\indentrel{3}\begin{verbatim}
+   (9)  [x,x]
+                                     Type: Magma Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty9}
+\begin{paste}{MagmaXmpPageEmpty9}{MagmaXmpPagePatch9}
+\pastebutton{MagmaXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{left c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch10}
+\begin{paste}{MagmaXmpPageFull10}{MagmaXmpPageEmpty10}
+\pastebutton{MagmaXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{right c\free{c }}
+\indentrel{3}\begin{verbatim}
+   (10)  [y,y]
+                                     Type: Magma Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty10}
+\begin{paste}{MagmaXmpPageEmpty10}{MagmaXmpPagePatch10}
+\pastebutton{MagmaXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{right c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch11}
+\begin{paste}{MagmaXmpPageFull11}{MagmaXmpPageEmpty11}
+\pastebutton{MagmaXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{length c\free{c }}
+\indentrel{3}\begin{verbatim}
+   (11)  4
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty11}
+\begin{paste}{MagmaXmpPageEmpty11}{MagmaXmpPagePatch11}
+\pastebutton{MagmaXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{length c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch12}
+\begin{paste}{MagmaXmpPageFull12}{MagmaXmpPageEmpty12}
+\pastebutton{MagmaXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{c::word\free{c word }}
+\indentrel{3}\begin{verbatim}
+          2 2
+   (12)  x y
+                         Type: OrderedFreeMonoid Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty12}
+\begin{paste}{MagmaXmpPageEmpty12}{MagmaXmpPagePatch12}
+\pastebutton{MagmaXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{c::word\free{c word }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch13}
+\begin{paste}{MagmaXmpPageFull13}{MagmaXmpPageEmpty13}
+\pastebutton{MagmaXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{a < b\free{a b }}
+\indentrel{3}\begin{verbatim}
+   (13)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty13}
+\begin{paste}{MagmaXmpPageEmpty13}{MagmaXmpPagePatch13}
+\pastebutton{MagmaXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{a < b\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch14}
+\begin{paste}{MagmaXmpPageFull14}{MagmaXmpPageEmpty14}
+\pastebutton{MagmaXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{a < c\free{a c }}
+\indentrel{3}\begin{verbatim}
+   (14)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty14}
+\begin{paste}{MagmaXmpPageEmpty14}{MagmaXmpPagePatch14}
+\pastebutton{MagmaXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{a < c\free{a c }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch15}
+\begin{paste}{MagmaXmpPageFull15}{MagmaXmpPageEmpty15}
+\pastebutton{MagmaXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{b < c\free{b c }}
+\indentrel{3}\begin{verbatim}
+   (15)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty15}
+\begin{paste}{MagmaXmpPageEmpty15}{MagmaXmpPagePatch15}
+\pastebutton{MagmaXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{b < c\free{b c }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch16}
+\begin{paste}{MagmaXmpPageFull16}{MagmaXmpPageEmpty16}
+\pastebutton{MagmaXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{first c\free{c }}
+\indentrel{3}\begin{verbatim}
+   (16)  x
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty16}
+\begin{paste}{MagmaXmpPageEmpty16}{MagmaXmpPagePatch16}
+\pastebutton{MagmaXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{first c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch17}
+\begin{paste}{MagmaXmpPageFull17}{MagmaXmpPageEmpty17}
+\pastebutton{MagmaXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{rest c\free{c }}
+\indentrel{3}\begin{verbatim}
+   (17)  [x,[y,y]]
+                                     Type: Magma Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty17}
+\begin{paste}{MagmaXmpPageEmpty17}{MagmaXmpPagePatch17}
+\pastebutton{MagmaXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{rest c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch18}
+\begin{paste}{MagmaXmpPageFull18}{MagmaXmpPageEmpty18}
+\pastebutton{MagmaXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{rest rest c\free{c }}
+\indentrel{3}\begin{verbatim}
+   (18)  [y,y]
+                                     Type: Magma Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty18}
+\begin{paste}{MagmaXmpPageEmpty18}{MagmaXmpPagePatch18}
+\pastebutton{MagmaXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{rest rest c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch19}
+\begin{paste}{MagmaXmpPageFull19}{MagmaXmpPageEmpty19}
+\pastebutton{MagmaXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{ax:tree := a*x\free{a x tree }\bound{ax }}
+\indentrel{3}\begin{verbatim}
+   (19)  [[x,x],x]
+                                     Type: Magma Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty19}
+\begin{paste}{MagmaXmpPageEmpty19}{MagmaXmpPagePatch19}
+\pastebutton{MagmaXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{ax:tree := a*x\free{a x tree }\bound{ax }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch20}
+\begin{paste}{MagmaXmpPageFull20}{MagmaXmpPageEmpty20}
+\pastebutton{MagmaXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{xa:tree := x*a\free{a x tree }\bound{xa }}
+\indentrel{3}\begin{verbatim}
+   (20)  [x,[x,x]]
+                                     Type: Magma Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty20}
+\begin{paste}{MagmaXmpPageEmpty20}{MagmaXmpPagePatch20}
+\pastebutton{MagmaXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{xa:tree := x*a\free{a x tree }\bound{xa }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch21}
+\begin{paste}{MagmaXmpPageFull21}{MagmaXmpPageEmpty21}
+\pastebutton{MagmaXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{xa < ax\free{ax xa }}
+\indentrel{3}\begin{verbatim}
+   (21)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty21}
+\begin{paste}{MagmaXmpPageEmpty21}{MagmaXmpPagePatch21}
+\pastebutton{MagmaXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{xa < ax\free{ax xa }}
+\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPagePatch22}
+\begin{paste}{MagmaXmpPageFull22}{MagmaXmpPageEmpty22}
+\pastebutton{MagmaXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{lexico(xa,ax)\free{ax xa }}
+\indentrel{3}\begin{verbatim}
+   (22)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MagmaXmpPageEmpty22}
+\begin{paste}{MagmaXmpPageEmpty22}{MagmaXmpPagePatch22}
+\pastebutton{MagmaXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{lexico(xa,ax)\free{ax xa }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/MAPPKG1.ht b/src/hyper/pages/MAPPKG1.ht
new file mode 100644
index 00000000..33553bc3
--- /dev/null
+++ b/src/hyper/pages/MAPPKG1.ht
@@ -0,0 +1,174 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\MappingPackageOneXmpTitle}{MappingPackage1}
+\newcommand{\MappingPackageOneXmpNumber}{9.51}
+%
+% =====================================================================
+\begin{page}{MappingPackageOneXmpPage}{9.51 MappingPackage1}
+% =====================================================================
+\beginscroll
+
+Function are objects of type \pspadtype{Mapping}.
+In this section we demonstrate some library operations from the
+packages \spadtype{MappingPackage1}, \spadtype{MappingPackage2}, and
+\spadtype{MappingPackage3} that manipulate and create functions.
+Some terminology: a {\it nullary} function takes no arguments,
+%-% \HDindex{function!nullary}{MappingPackageOneXmpPage}{9.51}{MappingPackage1}
+a {\it unary} function takes one argument, and
+%-% \HDindex{function!unary}{MappingPackageOneXmpPage}{9.51}{MappingPackage1}
+a {\it binary} function takes two arguments.
+%-% \HDindex{function!binary}{MappingPackageOneXmpPage}{9.51}{MappingPackage1}
+
+\xtc{
+We begin by creating an example function that raises a
+rational number to an integer exponent.
+}{
+\spadpaste{power(q: FRAC INT, n: INT): FRAC INT == q**n \bound{power}}
+}
+\xtc{
+}{
+\spadpaste{power(2,3) \free{power}}
+}
+\xtc{
+The \spadfunFrom{twist}{MappingPackage3} operation
+transposes the arguments of a binary function.
+Here \spad{rewop(a, b)} is \spad{power(b, a)}.
+}{
+\spadpaste{rewop := twist power \free{power}\bound{rewop}}
+}
+\xtc{
+This is \texht{$2^3.$}{\spad{2**3.}}
+}{
+\spadpaste{rewop(3, 2) \free{rewop}}
+}
+\xtc{
+Now we define \userfun{square} in terms of \userfun{power}.
+}{
+\spadpaste{square: FRAC INT -> FRAC INT \bound{squaredec}}
+}
+\xtc{
+The \spadfunFrom{curryRight}{MappingPackage3} operation creates a unary
+function from a binary one by providing a constant
+argument on the right.
+}{
+\spadpaste{square:= curryRight(power, 2) \free{squaredec poswer}\bound{square}}
+}
+\xtc{
+Likewise, the
+\spadfunFrom{curryLeft}{MappingPackage3} operation provides a constant
+argument on the left.
+}{
+\spadpaste{square 4 \free{square}}
+}
+\xtc{
+The \spadfunFrom{constantRight}{MappingPackage3} operation creates
+(in a trivial way) a binary function from a unary one:
+\spad{constantRight(f)} is the function \spad{g} such that
+\spad{g(a,b)= f(a).}
+}{
+\spadpaste{squirrel:= constantRight(square)\$MAPPKG3(FRAC INT,FRAC INT,FRAC INT) \free{square}\bound{squirrel}}
+}
+\xtc{
+Likewise,
+\spad{constantLeft(f)} is the function \spad{g} such that \spad{g(a,b)= f(b).}
+}{
+\spadpaste{squirrel(1/2, 1/3) \free{squirrel}}
+}
+\xtc{
+The \spadfunFrom{curry}{MappingPackage2} operation makes a unary function nullary.
+}{
+\spadpaste{sixteen := curry(square, 4/1) \free{square}\bound{sixteen}}
+}
+\xtc{
+}{
+\spadpaste{sixteen() \free{sixteen}}
+}
+\xtc{
+The \spadopFrom{*}{MappingPackage3} operation
+constructs composed functions.
+}{
+\spadpaste{square2:=square*square \free{square}\bound{square2}}
+}
+\xtc{
+}{
+\spadpaste{square2  3 \free{square2}}
+}
+\xtc{
+Use the \spadopFrom{**}{MappingPackage1} operation to create
+functions that are \spad{n}-fold iterations of other functions.
+}{
+\spadpaste{sc(x: FRAC INT): FRAC INT == x + 1 \bound{sc}}
+}
+\xtc{
+This is a list of \pspadtype{Mapping} objects.
+}{
+\spadpaste{incfns := [sc**i for i in 0..10] \free{sc}\bound{incfns}}
+}
+\xtc{
+This is a list of applications of those functions.
+}{
+\spadpaste{[f 4 for f in incfns] \free{incfns}}
+}
+\xtc{
+Use the \spadfunFrom{recur}{MappingPackage1}
+operation for recursion:
+\spad{g := recur f} means \spad{g(n,x) == f(n,f(n-1,...f(1,x))).}
+}{
+\spadpaste{times(n:NNI, i:INT):INT == n*i \bound{rdec}}
+}
+\xtc{
+}{
+\spadpaste{r := recur(times) \free{rdec}\bound{r}}
+}
+\xtc{
+This is a factorial function.
+%-% \HDindex{factorial}{MappingPackageOneXmpPage}{9.51}{MappingPackage1}
+}{
+\spadpaste{fact := curryRight(r, 1) \free{r}\bound{fact}}
+}
+\xtc{
+}{
+\spadpaste{fact 4 \free{fact}}
+}
+\xtc{
+Constructed functions can be used within other functions.
+}{
+\begin{spadsrc}[\free{square}\bound{mto2ton}]
+mto2ton(m, n) ==
+  raiser := square**n
+  raiser m
+\end{spadsrc}
+}
+\xtc{
+This is \texht{$3^{2^3}.$}{\spad{3**(2**3).}}
+}{
+\spadpaste{mto2ton(3, 3) \free{mto2ton}}
+}
+\xtc{
+Here \userfun{shiftfib} is a unary function that modifies its argument.
+}{
+\begin{spadsrc}[\bound{shiftfib}]
+shiftfib(r: List INT) : INT ==
+  t := r.1
+  r.1 := r.2
+  r.2 := r.2 + t
+  t
+\end{spadsrc}
+}
+\xtc{
+By currying over the argument we get a function with private state.
+}{
+\spadpaste{fibinit: List INT := [0, 1] \bound{fibinitdec}}
+}
+\xtc{
+}{
+\spadpaste{fibs := curry(shiftfib, fibinit) \free{shiftfib fibinit}\bound{fibs}}
+}
+\xtc{
+}{
+\spadpaste{[fibs() for i in 0..30] \free{fibs}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/MAPPKG1.pht b/src/hyper/pages/MAPPKG1.pht
new file mode 100644
index 00000000..f17e04a8
--- /dev/null
+++ b/src/hyper/pages/MAPPKG1.pht
@@ -0,0 +1,443 @@
+\begin{patch}{MappingPackageOneXmpPagePatch1}
+\begin{paste}{MappingPackageOneXmpPageFull1}{MappingPackageOneXmpPageEmpty1}
+\pastebutton{MappingPackageOneXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{power(q: FRAC INT, n: INT): FRAC INT == q**n\bound{power }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty1}
+\begin{paste}{MappingPackageOneXmpPageEmpty1}{MappingPackageOneXmpPagePatch1}
+\pastebutton{MappingPackageOneXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{power(q: FRAC INT, n: INT): FRAC INT == q**n\bound{power }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch2}
+\begin{paste}{MappingPackageOneXmpPageFull2}{MappingPackageOneXmpPageEmpty2}
+\pastebutton{MappingPackageOneXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{power(2,3)\free{power }}
+\indentrel{3}\begin{verbatim}
+   (2)  8
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty2}
+\begin{paste}{MappingPackageOneXmpPageEmpty2}{MappingPackageOneXmpPagePatch2}
+\pastebutton{MappingPackageOneXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{power(2,3)\free{power }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch3}
+\begin{paste}{MappingPackageOneXmpPageFull3}{MappingPackageOneXmpPageEmpty3}
+\pastebutton{MappingPackageOneXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{rewop := twist power\free{power }\bound{rewop }}
+\indentrel{3}\begin{verbatim}
+   (3)  theMap(MAPPKG3;twist;MM;5!0)
+ Type: ((Integer,Fraction Integer) -> Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty3}
+\begin{paste}{MappingPackageOneXmpPageEmpty3}{MappingPackageOneXmpPagePatch3}
+\pastebutton{MappingPackageOneXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{rewop := twist power\free{power }\bound{rewop }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch4}
+\begin{paste}{MappingPackageOneXmpPageFull4}{MappingPackageOneXmpPageEmpty4}
+\pastebutton{MappingPackageOneXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{rewop(3, 2)\free{rewop }}
+\indentrel{3}\begin{verbatim}
+   (4)  8
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty4}
+\begin{paste}{MappingPackageOneXmpPageEmpty4}{MappingPackageOneXmpPagePatch4}
+\pastebutton{MappingPackageOneXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{rewop(3, 2)\free{rewop }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch5}
+\begin{paste}{MappingPackageOneXmpPageFull5}{MappingPackageOneXmpPageEmpty5}
+\pastebutton{MappingPackageOneXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{square: FRAC INT -> FRAC INT\bound{squaredec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty5}
+\begin{paste}{MappingPackageOneXmpPageEmpty5}{MappingPackageOneXmpPagePatch5}
+\pastebutton{MappingPackageOneXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{square: FRAC INT -> FRAC INT\bound{squaredec }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch6}
+\begin{paste}{MappingPackageOneXmpPageFull6}{MappingPackageOneXmpPageEmpty6}
+\pastebutton{MappingPackageOneXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{square:= curryRight(power, 2)\free{squaredec poswer }\bound{square }}
+\indentrel{3}\begin{verbatim}
+   (6)  theMap(MAPPKG3;curryRight;MBM;1!0,430)
+           Type: (Fraction Integer -> Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty6}
+\begin{paste}{MappingPackageOneXmpPageEmpty6}{MappingPackageOneXmpPagePatch6}
+\pastebutton{MappingPackageOneXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{square:= curryRight(power, 2)\free{squaredec poswer }\bound{square }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch7}
+\begin{paste}{MappingPackageOneXmpPageFull7}{MappingPackageOneXmpPageEmpty7}
+\pastebutton{MappingPackageOneXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{square 4\free{square }}
+\indentrel{3}\begin{verbatim}
+   (7)  16
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty7}
+\begin{paste}{MappingPackageOneXmpPageEmpty7}{MappingPackageOneXmpPagePatch7}
+\pastebutton{MappingPackageOneXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{square 4\free{square }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch8}
+\begin{paste}{MappingPackageOneXmpPageFull8}{MappingPackageOneXmpPageEmpty8}
+\pastebutton{MappingPackageOneXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{squirrel:= constantRight(square)$MAPPKG3(FRAC INT,FRAC INT,FRAC INT)\free{square }\bound{squirrel }}
+\indentrel{3}\begin{verbatim}
+   (8)  theMap(MAPPKG3;constantRight;MM;3!0)
+Type: ((Fraction Integer,Fraction Integer) -> Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty8}
+\begin{paste}{MappingPackageOneXmpPageEmpty8}{MappingPackageOneXmpPagePatch8}
+\pastebutton{MappingPackageOneXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{squirrel:= constantRight(square)$MAPPKG3(FRAC INT,FRAC INT,FRAC INT)\free{square }\bound{squirrel }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch9}
+\begin{paste}{MappingPackageOneXmpPageFull9}{MappingPackageOneXmpPageEmpty9}
+\pastebutton{MappingPackageOneXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{squirrel(1/2, 1/3)\free{squirrel }}
+\indentrel{3}\begin{verbatim}
+        1
+   (9)  Ä
+        4
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty9}
+\begin{paste}{MappingPackageOneXmpPageEmpty9}{MappingPackageOneXmpPagePatch9}
+\pastebutton{MappingPackageOneXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{squirrel(1/2, 1/3)\free{squirrel }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch10}
+\begin{paste}{MappingPackageOneXmpPageFull10}{MappingPackageOneXmpPageEmpty10}
+\pastebutton{MappingPackageOneXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{sixteen := curry(square, 4/1)\free{square }\bound{sixteen }}
+\indentrel{3}\begin{verbatim}
+   (10)  theMap(MAPPKG2;curry;MAM;2!0,488)
+                         Type: (() -> Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty10}
+\begin{paste}{MappingPackageOneXmpPageEmpty10}{MappingPackageOneXmpPagePatch10}
+\pastebutton{MappingPackageOneXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{sixteen := curry(square, 4/1)\free{square }\bound{sixteen }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch11}
+\begin{paste}{MappingPackageOneXmpPageFull11}{MappingPackageOneXmpPageEmpty11}
+\pastebutton{MappingPackageOneXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{sixteen()\free{sixteen }}
+\indentrel{3}\begin{verbatim}
+   (11)  16
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty11}
+\begin{paste}{MappingPackageOneXmpPageEmpty11}{MappingPackageOneXmpPagePatch11}
+\pastebutton{MappingPackageOneXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{sixteen()\free{sixteen }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch12}
+\begin{paste}{MappingPackageOneXmpPageFull12}{MappingPackageOneXmpPageEmpty12}
+\pastebutton{MappingPackageOneXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{square2:=square*square\free{square }\bound{square2 }}
+\indentrel{3}\begin{verbatim}
+   (12)  theMap(MAPPKG3;*;MMM;6!0,589)
+           Type: (Fraction Integer -> Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty12}
+\begin{paste}{MappingPackageOneXmpPageEmpty12}{MappingPackageOneXmpPagePatch12}
+\pastebutton{MappingPackageOneXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{square2:=square*square\free{square }\bound{square2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch13}
+\begin{paste}{MappingPackageOneXmpPageFull13}{MappingPackageOneXmpPageEmpty13}
+\pastebutton{MappingPackageOneXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{square2 3\free{square2 }}
+\indentrel{3}\begin{verbatim}
+   (13)  81
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty13}
+\begin{paste}{MappingPackageOneXmpPageEmpty13}{MappingPackageOneXmpPagePatch13}
+\pastebutton{MappingPackageOneXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{square2 3\free{square2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch14}
+\begin{paste}{MappingPackageOneXmpPageFull14}{MappingPackageOneXmpPageEmpty14}
+\pastebutton{MappingPackageOneXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{sc(x: FRAC INT): FRAC INT == x + 1\bound{sc }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty14}
+\begin{paste}{MappingPackageOneXmpPageEmpty14}{MappingPackageOneXmpPagePatch14}
+\pastebutton{MappingPackageOneXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{sc(x: FRAC INT): FRAC INT == x + 1\bound{sc }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch15}
+\begin{paste}{MappingPackageOneXmpPageFull15}{MappingPackageOneXmpPageEmpty15}
+\pastebutton{MappingPackageOneXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{incfns := [sc**i for i in 0..10]\free{sc }\bound{incfns }}
+\indentrel{3}\begin{verbatim}
+   (15)
+   [theMap(MAPPKG1;**;MNniM;6!0,314),
+    theMap(MAPPKG1;**;MNniM;6!0,963),
+    theMap(MAPPKG1;**;MNniM;6!0,810),
+    theMap(MAPPKG1;**;MNniM;6!0,546),
+    theMap(MAPPKG1;**;MNniM;6!0,338),
+    theMap(MAPPKG1;**;MNniM;6!0,989),
+    theMap(MAPPKG1;**;MNniM;6!0,218),
+    theMap(MAPPKG1;**;MNniM;6!0,20),
+    theMap(MAPPKG1;**;MNniM;6!0,533),
+    theMap(MAPPKG1;**;MNniM;6!0,437),
+    theMap(MAPPKG1;**;MNniM;6!0,900)]
+      Type: List (Fraction Integer -> Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty15}
+\begin{paste}{MappingPackageOneXmpPageEmpty15}{MappingPackageOneXmpPagePatch15}
+\pastebutton{MappingPackageOneXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{incfns := [sc**i for i in 0..10]\free{sc }\bound{incfns }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch16}
+\begin{paste}{MappingPackageOneXmpPageFull16}{MappingPackageOneXmpPageEmpty16}
+\pastebutton{MappingPackageOneXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{[f 4 for f in incfns]\free{incfns }}
+\indentrel{3}\begin{verbatim}
+   (16)  [4,5,6,7,8,9,10,11,12,13,14]
+                            Type: List Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty16}
+\begin{paste}{MappingPackageOneXmpPageEmpty16}{MappingPackageOneXmpPagePatch16}
+\pastebutton{MappingPackageOneXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{[f 4 for f in incfns]\free{incfns }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch17}
+\begin{paste}{MappingPackageOneXmpPageFull17}{MappingPackageOneXmpPageEmpty17}
+\pastebutton{MappingPackageOneXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{times(n:NNI, i:INT):INT == n*i\bound{rdec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty17}
+\begin{paste}{MappingPackageOneXmpPageEmpty17}{MappingPackageOneXmpPagePatch17}
+\pastebutton{MappingPackageOneXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{times(n:NNI, i:INT):INT == n*i\bound{rdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch18}
+\begin{paste}{MappingPackageOneXmpPageFull18}{MappingPackageOneXmpPageEmpty18}
+\pastebutton{MappingPackageOneXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{r := recur(times)\free{rdec }\bound{r }}
+\indentrel{3}\begin{verbatim}
+   (18)  theMap(MAPPKG1;recur;2M;7!0,161)
+        Type: ((NonNegativeInteger,Integer) -> Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty18}
+\begin{paste}{MappingPackageOneXmpPageEmpty18}{MappingPackageOneXmpPagePatch18}
+\pastebutton{MappingPackageOneXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{r := recur(times)\free{rdec }\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch19}
+\begin{paste}{MappingPackageOneXmpPageFull19}{MappingPackageOneXmpPageEmpty19}
+\pastebutton{MappingPackageOneXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{fact := curryRight(r, 1)\free{r }\bound{fact }}
+\indentrel{3}\begin{verbatim}
+   (19)  theMap(MAPPKG3;curryRight;MBM;1!0,541)
+                  Type: (NonNegativeInteger -> Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty19}
+\begin{paste}{MappingPackageOneXmpPageEmpty19}{MappingPackageOneXmpPagePatch19}
+\pastebutton{MappingPackageOneXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{fact := curryRight(r, 1)\free{r }\bound{fact }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch20}
+\begin{paste}{MappingPackageOneXmpPageFull20}{MappingPackageOneXmpPageEmpty20}
+\pastebutton{MappingPackageOneXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{fact 4\free{fact }}
+\indentrel{3}\begin{verbatim}
+   (20)  24
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty20}
+\begin{paste}{MappingPackageOneXmpPageEmpty20}{MappingPackageOneXmpPagePatch20}
+\pastebutton{MappingPackageOneXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{fact 4\free{fact }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch21}
+\begin{paste}{MappingPackageOneXmpPageFull21}{MappingPackageOneXmpPageEmpty21}
+\pastebutton{MappingPackageOneXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{mto2ton(m, n) ==
+  raiser := square**n
+  raiser m
+\free{square }\bound{mto2ton }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty21}
+\begin{paste}{MappingPackageOneXmpPageEmpty21}{MappingPackageOneXmpPagePatch21}
+\pastebutton{MappingPackageOneXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{mto2ton(m, n) ==
+  raiser := square**n
+  raiser m
+\free{square }\bound{mto2ton }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch22}
+\begin{paste}{MappingPackageOneXmpPageFull22}{MappingPackageOneXmpPageEmpty22}
+\pastebutton{MappingPackageOneXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{mto2ton(3, 3)\free{mto2ton }}
+\indentrel{3}\begin{verbatim}
+   (22)  6561
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty22}
+\begin{paste}{MappingPackageOneXmpPageEmpty22}{MappingPackageOneXmpPagePatch22}
+\pastebutton{MappingPackageOneXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{mto2ton(3, 3)\free{mto2ton }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch23}
+\begin{paste}{MappingPackageOneXmpPageFull23}{MappingPackageOneXmpPageEmpty23}
+\pastebutton{MappingPackageOneXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{shiftfib(r: List INT) : INT ==
+  t := r.1
+  r.1 := r.2
+  r.2 := r.2 + t
+  t
+\bound{shiftfib }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty23}
+\begin{paste}{MappingPackageOneXmpPageEmpty23}{MappingPackageOneXmpPagePatch23}
+\pastebutton{MappingPackageOneXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{shiftfib(r: List INT) : INT ==
+  t := r.1
+  r.1 := r.2
+  r.2 := r.2 + t
+  t
+\bound{shiftfib }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch24}
+\begin{paste}{MappingPackageOneXmpPageFull24}{MappingPackageOneXmpPageEmpty24}
+\pastebutton{MappingPackageOneXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{fibinit: List INT := [0, 1]\bound{fibinitdec }}
+\indentrel{3}\begin{verbatim}
+   (24)  [0,1]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty24}
+\begin{paste}{MappingPackageOneXmpPageEmpty24}{MappingPackageOneXmpPagePatch24}
+\pastebutton{MappingPackageOneXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{fibinit: List INT := [0, 1]\bound{fibinitdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch25}
+\begin{paste}{MappingPackageOneXmpPageFull25}{MappingPackageOneXmpPageEmpty25}
+\pastebutton{MappingPackageOneXmpPageFull25}{\hidepaste}
+\tab{5}\spadcommand{fibs := curry(shiftfib, fibinit)\free{shiftfib fibinit }\bound{fibs }}
+\indentrel{3}\begin{verbatim}
+   (25)  theMap(MAPPKG2;curry;MAM;2!0,91)
+                                  Type: (() -> Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty25}
+\begin{paste}{MappingPackageOneXmpPageEmpty25}{MappingPackageOneXmpPagePatch25}
+\pastebutton{MappingPackageOneXmpPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{fibs := curry(shiftfib, fibinit)\free{shiftfib fibinit }\bound{fibs }}
+\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPagePatch26}
+\begin{paste}{MappingPackageOneXmpPageFull26}{MappingPackageOneXmpPageEmpty26}
+\pastebutton{MappingPackageOneXmpPageFull26}{\hidepaste}
+\tab{5}\spadcommand{[fibs() for i in 0..30]\free{fibs }}
+\indentrel{3}\begin{verbatim}
+   (26)
+   [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,
+    377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711,
+    28657, 46368, 75025, 121393, 196418, 317811, 514229,
+    832040]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MappingPackageOneXmpPageEmpty26}
+\begin{paste}{MappingPackageOneXmpPageEmpty26}{MappingPackageOneXmpPagePatch26}
+\pastebutton{MappingPackageOneXmpPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{[fibs() for i in 0..30]\free{fibs }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/MATRIX.ht b/src/hyper/pages/MATRIX.ht
new file mode 100644
index 00000000..0390526d
--- /dev/null
+++ b/src/hyper/pages/MATRIX.ht
@@ -0,0 +1,319 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\MatrixXmpTitle}{Matrix}
+\newcommand{\MatrixXmpNumber}{9.52}
+%
+% =====================================================================
+\begin{page}{MatrixXmpPage}{9.52 Matrix}
+% =====================================================================
+\beginscroll
+
+The \spadtype{Matrix} domain provides arithmetic operations on matrices
+and standard functions from linear algebra.
+This domain is similar to the \spadtype{TwoDimensionalArray} domain, except
+that the entries for \spadtype{Matrix} must belong to a \spadtype{Ring}.
+
+\beginmenu
+    \menudownlink{{9.52.1. Creating Matrices}}{ugxMatrixCreatePage}
+    \menudownlink{{9.52.2. Operations on Matrices}}{ugxMatrixOpsPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxMatrixCreateTitle}{Creating Matrices}
+\newcommand{\ugxMatrixCreateNumber}{9.52.1.}
+%
+% =====================================================================
+\begin{page}{ugxMatrixCreatePage}{9.52.1. Creating Matrices}
+% =====================================================================
+\beginscroll
+
+There are many ways to create a matrix from a collection of
+values or from existing matrices.
+
+\xtc{
+If the matrix has almost all items equal to the same value,
+use \spadfunFrom{new}{Matrix} to create a matrix filled with that
+value and then reset the entries that are different.
+}{
+\spadpaste{m : Matrix(Integer) := new(3,3,0) \bound{m}}
+}
+\xtc{
+To change the entry in the second row, third column to \spad{5}, use
+\spadfunFrom{setelt}{Matrix}.
+}{
+\spadpaste{setelt(m,2,3,5) \free{m}\bound{m1}}
+}
+\xtc{
+An alternative syntax is to use assignment.
+}{
+\spadpaste{m(1,2) := 10 \free{m1}\bound{m2}}
+}
+\xtc{
+The matrix was {\it destructively modified}.
+%-% \HDindex{matrix!destructive change to}{ugxMatrixCreatePage}{9.52.1.}{Creating Matrices}
+}{
+\spadpaste{m \free{m2}}
+}
+
+\xtc{
+If you already have the matrix entries as a list of lists,
+use \spadfunFrom{matrix}{Matrix}.
+}{
+\spadpaste{matrix [[1,2,3,4],[0,9,8,7]]}
+}
+
+\xtc{
+If the matrix is diagonal, use
+%-% \HDindex{matrix!diagonal}{ugxMatrixCreatePage}{9.52.1.}{Creating Matrices}
+\spadfunFrom{diagonalMatrix}{Matrix}.
+}{
+\spadpaste{dm := diagonalMatrix [1,x**2,x**3,x**4,x**5] \bound{dm}}
+}
+\xtc{
+Use \spadfunFromX{setRow}{Matrix} and
+\spadfunFromX{setColumn}{Matrix} to change a row or column of a matrix.
+}{
+\spadpaste{setRow!(dm,5,vector [1,1,1,1,1]) \free{dm}\bound{dm1}}
+}
+\xtc{
+}{
+\spadpaste{setColumn!(dm,2,vector [y,y,y,y,y]) \free{dm1}\bound{dm2}}
+}
+
+%
+\xtc{
+Use \spadfunFrom{copy}{Matrix} to make a copy of a matrix.
+%-% \HDindex{matrix!copying}{ugxMatrixCreatePage}{9.52.1.}{Creating Matrices}
+}{
+\spadpaste{cdm := copy(dm) \free{dm2}\bound{cdm}}
+}
+\xtc{
+This is useful if you intend to modify a matrix destructively but
+want a copy of the original.
+}{
+\spadpaste{setelt(dm,4,1,1-x**7) \free{dm2}\bound{setdm}}
+}
+\xtc{
+}{
+\spadpaste{[dm,cdm] \free{setdm cdm}}
+}
+
+%
+\xtc{
+Use \spadfunFrom{subMatrix}{Matrix} to extract part of an
+%-% \HDindex{matrix!submatrix of}{ugxMatrixCreatePage}{9.52.1.}{Creating Matrices}
+existing matrix.
+The syntax is \spad{subMatrix({\it m, firstrow, lastrow, firstcol,
+lastcol})}.
+}{
+\spadpaste{subMatrix(dm,2,3,2,4) \free{setdm}}
+}
+
+%
+\xtc{
+To change a submatrix, use \spadfunFromX{setsubMatrix}{Matrix}.
+}{
+\spadpaste{d := diagonalMatrix [1.2,-1.3,1.4,-1.5] \bound{d}}
+}
+\xtc{
+If \spad{e} is too big to fit where you specify, an error message
+is displayed.
+Use \spadfunFrom{subMatrix}{Matrix} to extract part of \spad{e}, if
+necessary.
+}{
+\spadpaste{e := matrix [[6.7,9.11],[-31.33,67.19]] \bound{e}}
+}
+\xtc{
+This changes the submatrix of \spad{d} whose upper left corner is
+at the first row and second column and whose size is that of \spad{e}.
+}{
+\spadpaste{setsubMatrix!(d,1,2,e) \free{d e}\bound{d1}}
+}
+\xtc{
+}{
+\spadpaste{d \free{d1}}
+}
+%
+
+%
+\xtc{
+Matrices can be joined either horizontally or vertically to make
+%-% \HDindex{matrix!concatenating}{ugxMatrixCreatePage}{9.52.1.}{Creating Matrices}
+new matrices.
+}{
+\spadpaste{a := matrix [[1/2,1/3,1/4],[1/5,1/6,1/7]] \bound{a}}
+}
+\xtc{
+}{
+\spadpaste{b := matrix [[3/5,3/7,3/11],[3/13,3/17,3/19]] \bound{b}}
+}
+\xtc{
+Use \spadfunFrom{horizConcat}{Matrix} to append them side to side.
+The two matrices must have the same number of rows.
+}{
+\spadpaste{horizConcat(a,b) \free{a b}}
+}
+\xtc{
+Use \spadfunFrom{vertConcat}{Matrix} to stack one upon the other.
+The two matrices must have the same number of columns.
+}{
+\spadpaste{vab := vertConcat(a,b) \free{a b}\bound{vab}}
+}
+
+%
+\xtc{
+The operation
+\spadfunFrom{transpose}{Matrix} is used to create a new matrix by reflection
+%-% \HDindex{matrix!transposing}{ugxMatrixCreatePage}{9.52.1.}{Creating Matrices}
+across the main diagonal.
+}{
+\spadpaste{transpose vab \free{vab}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxMatrixOpsTitle}{Operations on Matrices}
+\newcommand{\ugxMatrixOpsNumber}{9.52.2.}
+%
+% =====================================================================
+\begin{page}{ugxMatrixOpsPage}{9.52.2. Operations on Matrices}
+% =====================================================================
+\beginscroll
+
+\labelSpace{3pc}
+\xtc{
+\Language{} provides both left and right scalar multiplication.
+}{
+\spadpaste{m := matrix [[1,2],[3,4]] \bound{m}}
+}
+\xtc{
+}{
+\spadpaste{4 * m * (-5)\free{m}}
+}
+\xtc{
+You can add, subtract, and multiply matrices provided, of course, that
+the matrices have compatible dimensions.
+If not, an error message is displayed.
+}{
+\spadpaste{n := matrix([[1,0,-2],[-3,5,1]]) \bound{n}}
+}
+\xtc{
+This following product is defined but \spad{n * m} is not.
+}{
+\spadpaste{m * n \free{m n}}
+}
+
+The operations \spadfunFrom{nrows}{Matrix} and \spadfunFrom{ncols}{Matrix}
+return the number of rows and columns of a matrix.
+You can extract a row or a column of a matrix using the operations
+\spadfunFrom{row}{Matrix} and \spadfunFrom{column}{Matrix}.
+The object returned is a \spadtype{Vector}.
+\xtc{
+Here is the third column of the matrix \spad{n}.
+}{
+\spadpaste{vec := column(n,3) \free{n} \bound{vec}}
+}
+\xtc{
+You can multiply a matrix on the left by a ``row vector'' and on the right
+by a ``column vector.''
+}{
+\spadpaste{vec * m \free{vec m}}
+}
+\xtc{
+Of course, the dimensions of the vector and the matrix must be compatible
+or an error message is returned.
+}{
+\spadpaste{m * vec \free{vec m}}
+}
+
+The operation \spadfunFrom{inverse}{Matrix} computes the inverse of a
+matrix if
+%-% \HDindex{matrix!inverse of}{ugxMatrixOpsPage}{9.52.2.}{Operations on Matrices}
+the matrix is invertible, and returns \spad{"failed"} if not.
+\xtc{
+This Hilbert matrix is invertible.
+}{
+\spadpaste{hilb := matrix([[1/(i + j) for i in 1..3] for j in 1..3]) \bound{hilb}}
+}
+\xtc{
+}{
+\spadpaste{inverse(hilb) \free{hilb}}
+}
+\xtc{
+This matrix is not invertible.
+}{
+\spadpaste{mm := matrix([[1,2,3,4], [5,6,7,8], [9,10,11,12], [13,14,15,16]]) \bound{mm}}
+}
+\xtc{
+}{
+\spadpaste{inverse(mm) \free{mm}}
+}
+
+The operation \spadfunFrom{determinant}{Matrix} computes the determinant of a matrix
+%-% \HDindex{matrix!determinant of}{ugxMatrixOpsPage}{9.52.2.}{Operations on Matrices}
+provided that the entries of the matrix belong to a \spadtype{CommutativeRing}.
+\xtc{
+The above matrix \spad{mm} is not invertible and, hence, must have
+determinant \spad{0}.
+}{
+\spadpaste{determinant(mm) \free{mm}}
+}
+\xtc{
+The operation
+\spadfunFrom{trace}{SquareMatrix} computes the trace of a {\em square} matrix.
+%-% \HDindex{matrix!trace of}{ugxMatrixOpsPage}{9.52.2.}{Operations on Matrices}
+}{
+\spadpaste{trace(mm) \free{mm}}
+}
+
+\xtc{
+The operation \spadfunFrom{rank}{Matrix} computes the {\it rank} of a
+matrix:
+%-% \HDindex{matrix!rank of}{ugxMatrixOpsPage}{9.52.2.}{Operations on Matrices}
+the maximal number of linearly independent rows or columns.
+}{
+\spadpaste{rank(mm) \free{mm}}
+}
+\xtc{
+The operation \spadfunFrom{nullity}{Matrix} computes the {\it nullity} of
+a matrix: the dimension of its null space.
+}{
+\spadpaste{nullity(mm) \free{mm}}
+}
+\xtc{
+The operation \spadfunFrom{nullSpace}{Matrix} returns a list containing a basis for the
+null space of a matrix.
+Note that the nullity is the number of elements in a basis for the null space.
+}{
+\spadpaste{nullSpace(mm) \free{mm}}
+}
+\xtc{
+The operation \spadfunFrom{rowEchelon}{Matrix} returns the row echelon form of a
+%-% \HDindex{matrix!row echelon form of}{ugxMatrixOpsPage}{9.52.2.}{Operations on Matrices}
+matrix.
+It is easy to see that the rank of this matrix is two and that its nullity
+is also two.
+}{
+\spadpaste{rowEchelon(mm) \free{mm}}
+}
+
+For more information on related topics, see
+\downlink{``\ugIntroTwoDimTitle''}{ugIntroTwoDimPage} in Section \ugIntroTwoDimNumber\ignore{ugIntroTwoDim},
+\downlink{``\ugProblemEigenTitle''}{ugProblemEigenPage} in Section \ugProblemEigenNumber\ignore{ugProblemEigen},
+\downlink{``\ugxFloatHilbertTitle''}{ugxFloatHilbertPage} in Section \ugxFloatHilbertNumber\ignore{ugxFloatHilbert},
+\downlink{`Permanent'}{PermanentXmpPage}\ignore{Permanent},
+\downlink{`Vector'}{VectorXmpPage}\ignore{Vector},
+\downlink{`OneDimensionalArray'}{OneDimensionalArrayXmpPage}\ignore{OneDimensionalArray}, and
+\downlink{`TwoDimensionalArray'}{TwoDimensionalArrayXmpPage}\ignore{TwoDimensionalArray}.
+%
+\showBlurb{Matrix}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/MATRIX.pht b/src/hyper/pages/MATRIX.pht
new file mode 100644
index 00000000..4c5d611a
--- /dev/null
+++ b/src/hyper/pages/MATRIX.pht
@@ -0,0 +1,771 @@
+\begin{patch}{ugxMatrixOpsPagePatch1}
+\begin{paste}{ugxMatrixOpsPageFull1}{ugxMatrixOpsPageEmpty1}
+\pastebutton{ugxMatrixOpsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{m := matrix [[1,2],[3,4]]\bound{m }}
+\indentrel{3}\begin{verbatim}
+        Ú1  2¿
+   (1)  ³    ³
+        À3  4Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty1}
+\begin{paste}{ugxMatrixOpsPageEmpty1}{ugxMatrixOpsPagePatch1}
+\pastebutton{ugxMatrixOpsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{m := matrix [[1,2],[3,4]]\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch2}
+\begin{paste}{ugxMatrixOpsPageFull2}{ugxMatrixOpsPageEmpty2}
+\pastebutton{ugxMatrixOpsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{4 * m * (-5)\free{m }}
+\indentrel{3}\begin{verbatim}
+        Ú- 20  - 40¿
+   (2)  ³          ³
+        À- 60  - 80Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty2}
+\begin{paste}{ugxMatrixOpsPageEmpty2}{ugxMatrixOpsPagePatch2}
+\pastebutton{ugxMatrixOpsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{4 * m * (-5)\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch3}
+\begin{paste}{ugxMatrixOpsPageFull3}{ugxMatrixOpsPageEmpty3}
+\pastebutton{ugxMatrixOpsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{n := matrix([[1,0,-2],[-3,5,1]])\bound{n }}
+\indentrel{3}\begin{verbatim}
+        Ú 1   0  - 2¿
+   (3)  ³           ³
+        À- 3  5   1 Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty3}
+\begin{paste}{ugxMatrixOpsPageEmpty3}{ugxMatrixOpsPagePatch3}
+\pastebutton{ugxMatrixOpsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{n := matrix([[1,0,-2],[-3,5,1]])\bound{n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch4}
+\begin{paste}{ugxMatrixOpsPageFull4}{ugxMatrixOpsPageEmpty4}
+\pastebutton{ugxMatrixOpsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{m * n\free{m n }}
+\indentrel{3}\begin{verbatim}
+        Ú- 5  10   0 ¿
+   (4)  ³            ³
+        À- 9  20  - 2Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty4}
+\begin{paste}{ugxMatrixOpsPageEmpty4}{ugxMatrixOpsPagePatch4}
+\pastebutton{ugxMatrixOpsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{m * n\free{m n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch5}
+\begin{paste}{ugxMatrixOpsPageFull5}{ugxMatrixOpsPageEmpty5}
+\pastebutton{ugxMatrixOpsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{vec := column(n,3)\free{n }\bound{vec }}
+\indentrel{3}\begin{verbatim}
+   (5)  [- 2,1]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty5}
+\begin{paste}{ugxMatrixOpsPageEmpty5}{ugxMatrixOpsPagePatch5}
+\pastebutton{ugxMatrixOpsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{vec := column(n,3)\free{n }\bound{vec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch6}
+\begin{paste}{ugxMatrixOpsPageFull6}{ugxMatrixOpsPageEmpty6}
+\pastebutton{ugxMatrixOpsPageFull6}{\hidepaste}
+\tab{5}\spadcommand{vec * m\free{vec m }}
+\indentrel{3}\begin{verbatim}
+   (6)  [1,0]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty6}
+\begin{paste}{ugxMatrixOpsPageEmpty6}{ugxMatrixOpsPagePatch6}
+\pastebutton{ugxMatrixOpsPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{vec * m\free{vec m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch7}
+\begin{paste}{ugxMatrixOpsPageFull7}{ugxMatrixOpsPageEmpty7}
+\pastebutton{ugxMatrixOpsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{m * vec\free{vec m }}
+\indentrel{3}\begin{verbatim}
+   (7)  [0,- 2]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty7}
+\begin{paste}{ugxMatrixOpsPageEmpty7}{ugxMatrixOpsPagePatch7}
+\pastebutton{ugxMatrixOpsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{m * vec\free{vec m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch8}
+\begin{paste}{ugxMatrixOpsPageFull8}{ugxMatrixOpsPageEmpty8}
+\pastebutton{ugxMatrixOpsPageFull8}{\hidepaste}
+\tab{5}\spadcommand{hilb := matrix([[1/(i + j) for i in 1..3] for j in 1..3])\bound{hilb }}
+\indentrel{3}\begin{verbatim}
+        Ú1  1  1¿
+        ³Ä  Ä  ij
+        ³2  3  4³
+        ³       ³
+        ³1  1  1³
+   (8)  ³Ä  Ä  ij
+        ³3  4  5³
+        ³       ³
+        ³1  1  1³
+        ³Ä  Ä  ij
+        À4  5  6Ù
+                          Type: Matrix Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty8}
+\begin{paste}{ugxMatrixOpsPageEmpty8}{ugxMatrixOpsPagePatch8}
+\pastebutton{ugxMatrixOpsPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{hilb := matrix([[1/(i + j) for i in 1..3] for j in 1..3])\bound{hilb }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch9}
+\begin{paste}{ugxMatrixOpsPageFull9}{ugxMatrixOpsPageEmpty9}
+\pastebutton{ugxMatrixOpsPageFull9}{\hidepaste}
+\tab{5}\spadcommand{inverse(hilb)\free{hilb }}
+\indentrel{3}\begin{verbatim}
+        Ú 72    - 240   180 ¿
+        ³                   ³
+   (9)  ³- 240   900   - 720³
+        ³                   ³
+        À 180   - 720   600 Ù
+               Type: Union(Matrix Fraction Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty9}
+\begin{paste}{ugxMatrixOpsPageEmpty9}{ugxMatrixOpsPagePatch9}
+\pastebutton{ugxMatrixOpsPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{inverse(hilb)\free{hilb }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch10}
+\begin{paste}{ugxMatrixOpsPageFull10}{ugxMatrixOpsPageEmpty10}
+\pastebutton{ugxMatrixOpsPageFull10}{\hidepaste}
+\tab{5}\spadcommand{mm := matrix([[1,2,3,4], [5,6,7,8], [9,10,11,12], [13,14,15,16]])\bound{mm }}
+\indentrel{3}\begin{verbatim}
+         Ú1   2   3   4 ¿
+         ³              ³
+         ³5   6   7   8 ³
+   (10)  ³              ³
+         ³9   10  11  12³
+         ³              ³
+         À13  14  15  16Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty10}
+\begin{paste}{ugxMatrixOpsPageEmpty10}{ugxMatrixOpsPagePatch10}
+\pastebutton{ugxMatrixOpsPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{mm := matrix([[1,2,3,4], [5,6,7,8], [9,10,11,12], [13,14,15,16]])\bound{mm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch11}
+\begin{paste}{ugxMatrixOpsPageFull11}{ugxMatrixOpsPageEmpty11}
+\pastebutton{ugxMatrixOpsPageFull11}{\hidepaste}
+\tab{5}\spadcommand{inverse(mm)\free{mm }}
+\indentrel{3}\begin{verbatim}
+   (11)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty11}
+\begin{paste}{ugxMatrixOpsPageEmpty11}{ugxMatrixOpsPagePatch11}
+\pastebutton{ugxMatrixOpsPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{inverse(mm)\free{mm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch12}
+\begin{paste}{ugxMatrixOpsPageFull12}{ugxMatrixOpsPageEmpty12}
+\pastebutton{ugxMatrixOpsPageFull12}{\hidepaste}
+\tab{5}\spadcommand{determinant(mm)\free{mm }}
+\indentrel{3}\begin{verbatim}
+   (12)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty12}
+\begin{paste}{ugxMatrixOpsPageEmpty12}{ugxMatrixOpsPagePatch12}
+\pastebutton{ugxMatrixOpsPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{determinant(mm)\free{mm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch13}
+\begin{paste}{ugxMatrixOpsPageFull13}{ugxMatrixOpsPageEmpty13}
+\pastebutton{ugxMatrixOpsPageFull13}{\hidepaste}
+\tab{5}\spadcommand{trace(mm)\free{mm }}
+\indentrel{3}\begin{verbatim}
+   (13)  34
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty13}
+\begin{paste}{ugxMatrixOpsPageEmpty13}{ugxMatrixOpsPagePatch13}
+\pastebutton{ugxMatrixOpsPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{trace(mm)\free{mm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch14}
+\begin{paste}{ugxMatrixOpsPageFull14}{ugxMatrixOpsPageEmpty14}
+\pastebutton{ugxMatrixOpsPageFull14}{\hidepaste}
+\tab{5}\spadcommand{rank(mm)\free{mm }}
+\indentrel{3}\begin{verbatim}
+   (14)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty14}
+\begin{paste}{ugxMatrixOpsPageEmpty14}{ugxMatrixOpsPagePatch14}
+\pastebutton{ugxMatrixOpsPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{rank(mm)\free{mm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch15}
+\begin{paste}{ugxMatrixOpsPageFull15}{ugxMatrixOpsPageEmpty15}
+\pastebutton{ugxMatrixOpsPageFull15}{\hidepaste}
+\tab{5}\spadcommand{nullity(mm)\free{mm }}
+\indentrel{3}\begin{verbatim}
+   (15)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty15}
+\begin{paste}{ugxMatrixOpsPageEmpty15}{ugxMatrixOpsPagePatch15}
+\pastebutton{ugxMatrixOpsPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{nullity(mm)\free{mm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch16}
+\begin{paste}{ugxMatrixOpsPageFull16}{ugxMatrixOpsPageEmpty16}
+\pastebutton{ugxMatrixOpsPageFull16}{\hidepaste}
+\tab{5}\spadcommand{nullSpace(mm)\free{mm }}
+\indentrel{3}\begin{verbatim}
+   (16)  [[1,- 2,1,0],[2,- 3,0,1]]
+                              Type: List Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty16}
+\begin{paste}{ugxMatrixOpsPageEmpty16}{ugxMatrixOpsPagePatch16}
+\pastebutton{ugxMatrixOpsPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{nullSpace(mm)\free{mm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPagePatch17}
+\begin{paste}{ugxMatrixOpsPageFull17}{ugxMatrixOpsPageEmpty17}
+\pastebutton{ugxMatrixOpsPageFull17}{\hidepaste}
+\tab{5}\spadcommand{rowEchelon(mm)\free{mm }}
+\indentrel{3}\begin{verbatim}
+         Ú1  2  3  4 ¿
+         ³           ³
+         ³0  4  8  12³
+   (17)  ³           ³
+         ³0  0  0  0 ³
+         ³           ³
+         À0  0  0  0 Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixOpsPageEmpty17}
+\begin{paste}{ugxMatrixOpsPageEmpty17}{ugxMatrixOpsPagePatch17}
+\pastebutton{ugxMatrixOpsPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{rowEchelon(mm)\free{mm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch1}
+\begin{paste}{ugxMatrixCreatePageFull1}{ugxMatrixCreatePageEmpty1}
+\pastebutton{ugxMatrixCreatePageFull1}{\hidepaste}
+\tab{5}\spadcommand{m : Matrix(Integer) := new(3,3,0)\bound{m }}
+\indentrel{3}\begin{verbatim}
+        Ú0  0  0¿
+        ³       ³
+   (1)  ³0  0  0³
+        ³       ³
+        À0  0  0Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty1}
+\begin{paste}{ugxMatrixCreatePageEmpty1}{ugxMatrixCreatePagePatch1}
+\pastebutton{ugxMatrixCreatePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{m : Matrix(Integer) := new(3,3,0)\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch2}
+\begin{paste}{ugxMatrixCreatePageFull2}{ugxMatrixCreatePageEmpty2}
+\pastebutton{ugxMatrixCreatePageFull2}{\hidepaste}
+\tab{5}\spadcommand{setelt(m,2,3,5)\free{m }\bound{m1 }}
+\indentrel{3}\begin{verbatim}
+   (2)  5
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty2}
+\begin{paste}{ugxMatrixCreatePageEmpty2}{ugxMatrixCreatePagePatch2}
+\pastebutton{ugxMatrixCreatePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{setelt(m,2,3,5)\free{m }\bound{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch3}
+\begin{paste}{ugxMatrixCreatePageFull3}{ugxMatrixCreatePageEmpty3}
+\pastebutton{ugxMatrixCreatePageFull3}{\hidepaste}
+\tab{5}\spadcommand{m(1,2) := 10\free{m1 }\bound{m2 }}
+\indentrel{3}\begin{verbatim}
+   (3)  10
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty3}
+\begin{paste}{ugxMatrixCreatePageEmpty3}{ugxMatrixCreatePagePatch3}
+\pastebutton{ugxMatrixCreatePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{m(1,2) := 10\free{m1 }\bound{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch4}
+\begin{paste}{ugxMatrixCreatePageFull4}{ugxMatrixCreatePageEmpty4}
+\pastebutton{ugxMatrixCreatePageFull4}{\hidepaste}
+\tab{5}\spadcommand{m\free{m2 }}
+\indentrel{3}\begin{verbatim}
+        Ú0  10  0¿
+        ³        ³
+   (4)  ³0  0   5³
+        ³        ³
+        À0  0   0Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty4}
+\begin{paste}{ugxMatrixCreatePageEmpty4}{ugxMatrixCreatePagePatch4}
+\pastebutton{ugxMatrixCreatePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{m\free{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch5}
+\begin{paste}{ugxMatrixCreatePageFull5}{ugxMatrixCreatePageEmpty5}
+\pastebutton{ugxMatrixCreatePageFull5}{\hidepaste}
+\tab{5}\spadcommand{matrix [[1,2,3,4],[0,9,8,7]]}
+\indentrel{3}\begin{verbatim}
+        Ú1  2  3  4¿
+   (5)  ³          ³
+        À0  9  8  7Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty5}
+\begin{paste}{ugxMatrixCreatePageEmpty5}{ugxMatrixCreatePagePatch5}
+\pastebutton{ugxMatrixCreatePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{matrix [[1,2,3,4],[0,9,8,7]]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch6}
+\begin{paste}{ugxMatrixCreatePageFull6}{ugxMatrixCreatePageEmpty6}
+\pastebutton{ugxMatrixCreatePageFull6}{\hidepaste}
+\tab{5}\spadcommand{dm := diagonalMatrix [1,x**2,x**3,x**4,x**5]\bound{dm }}
+\indentrel{3}\begin{verbatim}
+        Ú1  0   0   0   0 ¿
+        ³                 ³
+        ³    2            ³
+        ³0  x   0   0   0 ³
+        ³                 ³
+        ³        3        ³
+   (6)  ³0  0   x   0   0 ³
+        ³                 ³
+        ³            4    ³
+        ³0  0   0   x   0 ³
+        ³                 ³
+        ³                5³
+        À0  0   0   0   x Ù
+                        Type: Matrix Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty6}
+\begin{paste}{ugxMatrixCreatePageEmpty6}{ugxMatrixCreatePagePatch6}
+\pastebutton{ugxMatrixCreatePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{dm := diagonalMatrix [1,x**2,x**3,x**4,x**5]\bound{dm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch7}
+\begin{paste}{ugxMatrixCreatePageFull7}{ugxMatrixCreatePageEmpty7}
+\pastebutton{ugxMatrixCreatePageFull7}{\hidepaste}
+\tab{5}\spadcommand{setRow!(dm,5,vector [1,1,1,1,1])\free{dm }\bound{dm1 }}
+\indentrel{3}\begin{verbatim}
+        Ú1  0   0   0   0¿
+        ³                ³
+        ³    2           ³
+        ³0  x   0   0   0³
+        ³                ³
+   (7)  ³        3       ³
+        ³0  0   x   0   0³
+        ³                ³
+        ³            4   ³
+        ³0  0   0   x   0³
+        ³                ³
+        À1  1   1   1   1Ù
+                        Type: Matrix Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty7}
+\begin{paste}{ugxMatrixCreatePageEmpty7}{ugxMatrixCreatePagePatch7}
+\pastebutton{ugxMatrixCreatePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{setRow!(dm,5,vector [1,1,1,1,1])\free{dm }\bound{dm1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch8}
+\begin{paste}{ugxMatrixCreatePageFull8}{ugxMatrixCreatePageEmpty8}
+\pastebutton{ugxMatrixCreatePageFull8}{\hidepaste}
+\tab{5}\spadcommand{setColumn!(dm,2,vector [y,y,y,y,y])\free{dm1 }\bound{dm2 }}
+\indentrel{3}\begin{verbatim}
+        Ú1  y  0   0   0¿
+        ³               ³
+        ³0  y  0   0   0³
+        ³               ³
+        ³       3       ³
+   (8)  ³0  y  x   0   0³
+        ³               ³
+        ³           4   ³
+        ³0  y  0   x   0³
+        ³               ³
+        À1  y  1   1   1Ù
+                        Type: Matrix Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty8}
+\begin{paste}{ugxMatrixCreatePageEmpty8}{ugxMatrixCreatePagePatch8}
+\pastebutton{ugxMatrixCreatePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{setColumn!(dm,2,vector [y,y,y,y,y])\free{dm1 }\bound{dm2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch9}
+\begin{paste}{ugxMatrixCreatePageFull9}{ugxMatrixCreatePageEmpty9}
+\pastebutton{ugxMatrixCreatePageFull9}{\hidepaste}
+\tab{5}\spadcommand{cdm := copy(dm)\free{dm2 }\bound{cdm }}
+\indentrel{3}\begin{verbatim}
+        Ú1  y  0   0   0¿
+        ³               ³
+        ³0  y  0   0   0³
+        ³               ³
+        ³       3       ³
+   (9)  ³0  y  x   0   0³
+        ³               ³
+        ³           4   ³
+        ³0  y  0   x   0³
+        ³               ³
+        À1  y  1   1   1Ù
+                        Type: Matrix Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty9}
+\begin{paste}{ugxMatrixCreatePageEmpty9}{ugxMatrixCreatePagePatch9}
+\pastebutton{ugxMatrixCreatePageEmpty9}{\showpaste}
+\tab{5}\spadcommand{cdm := copy(dm)\free{dm2 }\bound{cdm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch10}
+\begin{paste}{ugxMatrixCreatePageFull10}{ugxMatrixCreatePageEmpty10}
+\pastebutton{ugxMatrixCreatePageFull10}{\hidepaste}
+\tab{5}\spadcommand{setelt(dm,4,1,1-x**7)\free{dm2 }\bound{setdm }}
+\indentrel{3}\begin{verbatim}
+            7
+   (10)  - x  + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty10}
+\begin{paste}{ugxMatrixCreatePageEmpty10}{ugxMatrixCreatePagePatch10}
+\pastebutton{ugxMatrixCreatePageEmpty10}{\showpaste}
+\tab{5}\spadcommand{setelt(dm,4,1,1-x**7)\free{dm2 }\bound{setdm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch11}
+\begin{paste}{ugxMatrixCreatePageFull11}{ugxMatrixCreatePageEmpty11}
+\pastebutton{ugxMatrixCreatePageFull11}{\hidepaste}
+\tab{5}\spadcommand{[dm,cdm]\free{setdm cdm }}
+\indentrel{3}\begin{verbatim}
+          Ú   1      y  0   0   0¿ Ú1  y  0   0   0¿
+          ³                      ³ ³               ³
+          ³   0      y  0   0   0³ ³0  y  0   0   0³
+          ³                      ³ ³               ³
+          ³              3       ³ ³       3       ³
+   (11)  [³   0      y  x   0   0³,³0  y  x   0   0³]
+          ³                      ³ ³               ³
+          ³   7              4   ³ ³           4   ³
+          ³- x  + 1  y  0   x   0³ ³0  y  0   x   0³
+          ³                      ³ ³               ³
+          À   1      y  1   1   1Ù À1  y  1   1   1Ù
+                   Type: List Matrix Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty11}
+\begin{paste}{ugxMatrixCreatePageEmpty11}{ugxMatrixCreatePagePatch11}
+\pastebutton{ugxMatrixCreatePageEmpty11}{\showpaste}
+\tab{5}\spadcommand{[dm,cdm]\free{setdm cdm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch12}
+\begin{paste}{ugxMatrixCreatePageFull12}{ugxMatrixCreatePageEmpty12}
+\pastebutton{ugxMatrixCreatePageFull12}{\hidepaste}
+\tab{5}\spadcommand{subMatrix(dm,2,3,2,4)\free{setdm }}
+\indentrel{3}\begin{verbatim}
+         Úy  0   0¿
+   (12)  ³        ³
+         ³    3   ³
+         Ày  x   0Ù
+                        Type: Matrix Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty12}
+\begin{paste}{ugxMatrixCreatePageEmpty12}{ugxMatrixCreatePagePatch12}
+\pastebutton{ugxMatrixCreatePageEmpty12}{\showpaste}
+\tab{5}\spadcommand{subMatrix(dm,2,3,2,4)\free{setdm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch13}
+\begin{paste}{ugxMatrixCreatePageFull13}{ugxMatrixCreatePageEmpty13}
+\pastebutton{ugxMatrixCreatePageFull13}{\hidepaste}
+\tab{5}\spadcommand{d := diagonalMatrix [1.2,-1.3,1.4,-1.5]\bound{d }}
+\indentrel{3}\begin{verbatim}
+         Ú1.2   0.0   0.0   0.0 ¿
+         ³                      ³
+         ³0.0  - 1.3  0.0   0.0 ³
+   (13)  ³                      ³
+         ³0.0   0.0   1.4   0.0 ³
+         ³                      ³
+         À0.0   0.0   0.0  - 1.5Ù
+                                     Type: Matrix Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty13}
+\begin{paste}{ugxMatrixCreatePageEmpty13}{ugxMatrixCreatePagePatch13}
+\pastebutton{ugxMatrixCreatePageEmpty13}{\showpaste}
+\tab{5}\spadcommand{d := diagonalMatrix [1.2,-1.3,1.4,-1.5]\bound{d }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch14}
+\begin{paste}{ugxMatrixCreatePageFull14}{ugxMatrixCreatePageEmpty14}
+\pastebutton{ugxMatrixCreatePageFull14}{\hidepaste}
+\tab{5}\spadcommand{e := matrix [[6.7,9.11],[-31.33,67.19]]\bound{e }}
+\indentrel{3}\begin{verbatim}
+         Ú  6.7    9.11 ¿
+   (14)  ³              ³
+         À- 31.33  67.19Ù
+                                     Type: Matrix Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty14}
+\begin{paste}{ugxMatrixCreatePageEmpty14}{ugxMatrixCreatePagePatch14}
+\pastebutton{ugxMatrixCreatePageEmpty14}{\showpaste}
+\tab{5}\spadcommand{e := matrix [[6.7,9.11],[-31.33,67.19]]\bound{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch15}
+\begin{paste}{ugxMatrixCreatePageFull15}{ugxMatrixCreatePageEmpty15}
+\pastebutton{ugxMatrixCreatePageFull15}{\hidepaste}
+\tab{5}\spadcommand{setsubMatrix!(d,1,2,e)\free{d e }\bound{d1 }}
+\indentrel{3}\begin{verbatim}
+         Ú1.2    6.7    9.11    0.0 ¿
+         ³                          ³
+         ³0.0  - 31.33  67.19   0.0 ³
+   (15)  ³                          ³
+         ³0.0    0.0     1.4    0.0 ³
+         ³                          ³
+         À0.0    0.0     0.0   - 1.5Ù
+                                     Type: Matrix Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty15}
+\begin{paste}{ugxMatrixCreatePageEmpty15}{ugxMatrixCreatePagePatch15}
+\pastebutton{ugxMatrixCreatePageEmpty15}{\showpaste}
+\tab{5}\spadcommand{setsubMatrix!(d,1,2,e)\free{d e }\bound{d1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch16}
+\begin{paste}{ugxMatrixCreatePageFull16}{ugxMatrixCreatePageEmpty16}
+\pastebutton{ugxMatrixCreatePageFull16}{\hidepaste}
+\tab{5}\spadcommand{d\free{d1 }}
+\indentrel{3}\begin{verbatim}
+         Ú1.2    6.7    9.11    0.0 ¿
+         ³                          ³
+         ³0.0  - 31.33  67.19   0.0 ³
+   (16)  ³                          ³
+         ³0.0    0.0     1.4    0.0 ³
+         ³                          ³
+         À0.0    0.0     0.0   - 1.5Ù
+                                     Type: Matrix Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty16}
+\begin{paste}{ugxMatrixCreatePageEmpty16}{ugxMatrixCreatePagePatch16}
+\pastebutton{ugxMatrixCreatePageEmpty16}{\showpaste}
+\tab{5}\spadcommand{d\free{d1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch17}
+\begin{paste}{ugxMatrixCreatePageFull17}{ugxMatrixCreatePageEmpty17}
+\pastebutton{ugxMatrixCreatePageFull17}{\hidepaste}
+\tab{5}\spadcommand{a := matrix [[1/2,1/3,1/4],[1/5,1/6,1/7]]\bound{a }}
+\indentrel{3}\begin{verbatim}
+         Ú1  1  1¿
+         ³Ä  Ä  ij
+         ³2  3  4³
+   (17)  ³       ³
+         ³1  1  1³
+         ³Ä  Ä  ij
+         À5  6  7Ù
+                          Type: Matrix Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty17}
+\begin{paste}{ugxMatrixCreatePageEmpty17}{ugxMatrixCreatePagePatch17}
+\pastebutton{ugxMatrixCreatePageEmpty17}{\showpaste}
+\tab{5}\spadcommand{a := matrix [[1/2,1/3,1/4],[1/5,1/6,1/7]]\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch18}
+\begin{paste}{ugxMatrixCreatePageFull18}{ugxMatrixCreatePageEmpty18}
+\pastebutton{ugxMatrixCreatePageFull18}{\hidepaste}
+\tab{5}\spadcommand{b := matrix [[3/5,3/7,3/11],[3/13,3/17,3/19]]\bound{b }}
+\indentrel{3}\begin{verbatim}
+         Ú3   3    3¿
+         ³Ä   Ä   Äij
+         ³5   7   11³
+   (18)  ³          ³
+         ³ 3   3   3³
+         ³ÄÄ  ÄÄ  Äij
+         À13  17  19Ù
+                          Type: Matrix Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty18}
+\begin{paste}{ugxMatrixCreatePageEmpty18}{ugxMatrixCreatePagePatch18}
+\pastebutton{ugxMatrixCreatePageEmpty18}{\showpaste}
+\tab{5}\spadcommand{b := matrix [[3/5,3/7,3/11],[3/13,3/17,3/19]]\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch19}
+\begin{paste}{ugxMatrixCreatePageFull19}{ugxMatrixCreatePageEmpty19}
+\pastebutton{ugxMatrixCreatePageFull19}{\hidepaste}
+\tab{5}\spadcommand{horizConcat(a,b)\free{a b }}
+\indentrel{3}\begin{verbatim}
+         Ú1  1  1  3   3    3¿
+         ³Ä  Ä  Ä  Ä   Ä   Äij
+         ³2  3  4  5   7   11³
+   (19)  ³                   ³
+         ³1  1  1   3   3   3³
+         ³Ä  Ä  Ä  ÄÄ  ÄÄ  Äij
+         À5  6  7  13  17  19Ù
+                          Type: Matrix Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty19}
+\begin{paste}{ugxMatrixCreatePageEmpty19}{ugxMatrixCreatePagePatch19}
+\pastebutton{ugxMatrixCreatePageEmpty19}{\showpaste}
+\tab{5}\spadcommand{horizConcat(a,b)\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch20}
+\begin{paste}{ugxMatrixCreatePageFull20}{ugxMatrixCreatePageEmpty20}
+\pastebutton{ugxMatrixCreatePageFull20}{\hidepaste}
+\tab{5}\spadcommand{vab := vertConcat(a,b)\free{a b }\bound{vab }}
+\indentrel{3}\begin{verbatim}
+         Ú1   1   1 ¿
+         ³Ä   Ä   Ä ³
+         ³2   3   4 ³
+         ³          ³
+         ³1   1   1 ³
+         ³Ä   Ä   Ä ³
+         ³5   6   7 ³
+   (20)  ³          ³
+         ³3   3    3³
+         ³Ä   Ä   Äij
+         ³5   7   11³
+         ³          ³
+         ³ 3   3   3³
+         ³ÄÄ  ÄÄ  Äij
+         À13  17  19Ù
+                          Type: Matrix Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty20}
+\begin{paste}{ugxMatrixCreatePageEmpty20}{ugxMatrixCreatePagePatch20}
+\pastebutton{ugxMatrixCreatePageEmpty20}{\showpaste}
+\tab{5}\spadcommand{vab := vertConcat(a,b)\free{a b }\bound{vab }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePagePatch21}
+\begin{paste}{ugxMatrixCreatePageFull21}{ugxMatrixCreatePageEmpty21}
+\pastebutton{ugxMatrixCreatePageFull21}{\hidepaste}
+\tab{5}\spadcommand{transpose vab\free{vab }}
+\indentrel{3}\begin{verbatim}
+         Ú1  1  3    3¿
+         ³Ä  Ä  Ä   Äij
+         ³2  5  5   13³
+         ³            ³
+         ³1  1  3    3³
+   (21)  ³Ä  Ä  Ä   Äij
+         ³3  6  7   17³
+         ³            ³
+         ³1  1   3   3³
+         ³Ä  Ä  ÄÄ  Äij
+         À4  7  11  19Ù
+                          Type: Matrix Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxMatrixCreatePageEmpty21}
+\begin{paste}{ugxMatrixCreatePageEmpty21}{ugxMatrixCreatePagePatch21}
+\pastebutton{ugxMatrixCreatePageEmpty21}{\showpaste}
+\tab{5}\spadcommand{transpose vab\free{vab }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/MKFUNC.ht b/src/hyper/pages/MKFUNC.ht
new file mode 100644
index 00000000..c8ae31d1
--- /dev/null
+++ b/src/hyper/pages/MKFUNC.ht
@@ -0,0 +1,86 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\MakeFunctionXmpTitle}{MakeFunction}
+\newcommand{\MakeFunctionXmpNumber}{9.50}
+%
+% =====================================================================
+\begin{page}{MakeFunctionXmpPage}{9.50 MakeFunction}
+% =====================================================================
+\beginscroll
+It is sometimes useful to be able to define a function given by
+the result of a calculation.
+%
+\xtc{
+Suppose that you have obtained the following expression
+after several computations
+and that you now want to tabulate the numerical values of \spad{f}
+for \spad{x} between \spad{-1} and \spad{+1} with increment
+\spad{0.1}.
+}{
+\spadpaste{expr := (x - exp x + 1)**2 * (sin(x**2) * x + 1)**3 \bound{expr}}
+}
+%
+You could, of course, use the function
+\spadfunFrom{eval}{Expression} within a loop and evaluate
+\spad{expr} twenty-one times, but this would be quite slow.
+A better way is to create a numerical function \spad{f} such that
+\spad{f(x)} is defined by the expression \spad{expr} above,
+but without retyping \spad{expr}!
+The package \spadtype{MakeFunction} provides the operation
+\spadfunFrom{function}{MakeFunction} which does exactly this.
+%
+\xtc{
+Issue this to create the function \spad{f(x)} given by \spad{expr}.
+}{
+\spadpaste{function(expr, f, x) \bound{f}\free{expr}}
+}
+\xtc{
+To tabulate \spad{expr}, we can now quickly evaluate \spad{f} 21 times.
+}{
+\spadpaste{tbl := [f(0.1 * i - 1) for i in 0..20]; \free{f}\bound{tbl}}
+}
+%
+%
+\xtc{
+Use the list \spad{[x1,...,xn]} as the
+third argument to \spadfunFrom{function}{MakeFunction}
+to create a multivariate function \spad{f(x1,...,xn)}.
+}{
+\spadpaste{e := (x - y + 1)**2 * (x**2 * y + 1)**2 \bound{e}}
+}
+\xtc{
+}{
+\spadpaste{function(e, g, [x, y]) \free{e}}
+}
+%
+%
+\xtc{
+In the case of just two
+variables, they can be given as arguments without making them into a list.
+}{
+\spadpaste{function(e, h, x, y) \free{e}\bound{h}}
+}
+%
+%
+\xtc{
+Note that the functions created by \spadfunFrom{function}{MakeFunction}
+are not limited to floating point numbers, but can be applied to any type
+for which they are defined.
+}{
+\spadpaste{m1 := squareMatrix [[1, 2], [3, 4]] \bound{m1}}
+}
+\xtc{
+}{
+\spadpaste{m2 := squareMatrix [[1, 0], [-1, 1]] \bound{m2}}
+}
+\xtc{
+}{
+\spadpaste{h(m1, m2) \free{h m1 m2}}
+}
+%
+For more information, see \downlink{``\ugUserMakeTitle''}{ugUserMakePage} in Section \ugUserMakeNumber\ignore{ugUserMake}.
+\showBlurb{MakeFunction}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/MKFUNC.pht b/src/hyper/pages/MKFUNC.pht
new file mode 100644
index 00000000..8760d488
--- /dev/null
+++ b/src/hyper/pages/MKFUNC.pht
@@ -0,0 +1,171 @@
+\begin{patch}{MakeFunctionXmpPagePatch1}
+\begin{paste}{MakeFunctionXmpPageFull1}{MakeFunctionXmpPageEmpty1}
+\pastebutton{MakeFunctionXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{expr := (x - exp x + 1)**2 * (sin(x**2) * x + 1)**3\bound{expr }}
+\indentrel{3}\begin{verbatim}
+   (1)
+       3   x 2        4     3   x    5     4    3      2 3
+     (x (%e )  + (- 2x  - 2x )%e  + x  + 2x  + x )sin(x )
+   + 
+          2   x 2        3     2   x     4     3     2
+       (3x (%e )  + (- 6x  - 6x )%e  + 3x  + 6x  + 3x )
+    *
+            2 2
+       sin(x )
+   + 
+            x 2        2        x     3     2           2
+     (3x (%e )  + (- 6x  - 6x)%e  + 3x  + 6x  + 3x)sin(x )
+   + 
+        x 2               x    2
+     (%e )  + (- 2x - 2)%e  + x  + 2x + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPageEmpty1}
+\begin{paste}{MakeFunctionXmpPageEmpty1}{MakeFunctionXmpPagePatch1}
+\pastebutton{MakeFunctionXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{expr := (x - exp x + 1)**2 * (sin(x**2) * x + 1)**3\bound{expr }}
+\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPagePatch2}
+\begin{paste}{MakeFunctionXmpPageFull2}{MakeFunctionXmpPageEmpty2}
+\pastebutton{MakeFunctionXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{function(expr, f, x)\bound{f }\free{expr }}
+\indentrel{3}\begin{verbatim}
+   (2)  f
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPageEmpty2}
+\begin{paste}{MakeFunctionXmpPageEmpty2}{MakeFunctionXmpPagePatch2}
+\pastebutton{MakeFunctionXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{function(expr, f, x)\bound{f }\free{expr }}
+\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPagePatch3}
+\begin{paste}{MakeFunctionXmpPageFull3}{MakeFunctionXmpPageEmpty3}
+\pastebutton{MakeFunctionXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{tbl := [f(0.1 * i - 1) for i in 0..20];\free{f }\bound{tbl }}
+\indentrel{3}\begin{verbatim}
+                                       Type: List Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPageEmpty3}
+\begin{paste}{MakeFunctionXmpPageEmpty3}{MakeFunctionXmpPagePatch3}
+\pastebutton{MakeFunctionXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{tbl := [f(0.1 * i - 1) for i in 0..20];\free{f }\bound{tbl }}
+\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPagePatch4}
+\begin{paste}{MakeFunctionXmpPageFull4}{MakeFunctionXmpPageEmpty4}
+\pastebutton{MakeFunctionXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{e := (x - y + 1)**2 * (x**2 * y + 1)**2\bound{e }}
+\indentrel{3}\begin{verbatim}
+   (4)
+      4 4        5     4     2  3
+     x y  + (- 2x  - 2x  + 2x )y
+   + 
+       6     5    4     3     2      2
+     (x  + 2x  + x  - 4x  - 4x  + 1)y
+   + 
+        4     3     2               2
+     (2x  + 4x  + 2x  - 2x - 2)y + x  + 2x + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPageEmpty4}
+\begin{paste}{MakeFunctionXmpPageEmpty4}{MakeFunctionXmpPagePatch4}
+\pastebutton{MakeFunctionXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{e := (x - y + 1)**2 * (x**2 * y + 1)**2\bound{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPagePatch5}
+\begin{paste}{MakeFunctionXmpPageFull5}{MakeFunctionXmpPageEmpty5}
+\pastebutton{MakeFunctionXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{function(e, g, [x, y])\free{e }}
+\indentrel{3}\begin{verbatim}
+   (5)  g
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPageEmpty5}
+\begin{paste}{MakeFunctionXmpPageEmpty5}{MakeFunctionXmpPagePatch5}
+\pastebutton{MakeFunctionXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{function(e, g, [x, y])\free{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPagePatch6}
+\begin{paste}{MakeFunctionXmpPageFull6}{MakeFunctionXmpPageEmpty6}
+\pastebutton{MakeFunctionXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{function(e, h, x, y)\free{e }\bound{h }}
+\indentrel{3}\begin{verbatim}
+   (6)  h
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPageEmpty6}
+\begin{paste}{MakeFunctionXmpPageEmpty6}{MakeFunctionXmpPagePatch6}
+\pastebutton{MakeFunctionXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{function(e, h, x, y)\free{e }\bound{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPagePatch7}
+\begin{paste}{MakeFunctionXmpPageFull7}{MakeFunctionXmpPageEmpty7}
+\pastebutton{MakeFunctionXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{m1 := squareMatrix [[1, 2], [3, 4]]\bound{m1 }}
+\indentrel{3}\begin{verbatim}
+        Ú1  2¿
+   (7)  ³    ³
+        À3  4Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPageEmpty7}
+\begin{paste}{MakeFunctionXmpPageEmpty7}{MakeFunctionXmpPagePatch7}
+\pastebutton{MakeFunctionXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{m1 := squareMatrix [[1, 2], [3, 4]]\bound{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPagePatch8}
+\begin{paste}{MakeFunctionXmpPageFull8}{MakeFunctionXmpPageEmpty8}
+\pastebutton{MakeFunctionXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{m2 := squareMatrix [[1, 0], [-1, 1]]\bound{m2 }}
+\indentrel{3}\begin{verbatim}
+        Ú 1   0¿
+   (8)  ³      ³
+        À- 1  1Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPageEmpty8}
+\begin{paste}{MakeFunctionXmpPageEmpty8}{MakeFunctionXmpPagePatch8}
+\pastebutton{MakeFunctionXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{m2 := squareMatrix [[1, 0], [-1, 1]]\bound{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPagePatch9}
+\begin{paste}{MakeFunctionXmpPageFull9}{MakeFunctionXmpPageEmpty9}
+\pastebutton{MakeFunctionXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{h(m1, m2)\free{h m1 m2 }}
+\indentrel{3}\begin{verbatim}
+        Ú- 7836   8960 ¿
+   (9)  ³              ³
+        À- 17132  19588Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MakeFunctionXmpPageEmpty9}
+\begin{paste}{MakeFunctionXmpPageEmpty9}{MakeFunctionXmpPagePatch9}
+\pastebutton{MakeFunctionXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{h(m1, m2)\free{h m1 m2 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/MPOLY.ht b/src/hyper/pages/MPOLY.ht
new file mode 100644
index 00000000..1619aae4
--- /dev/null
+++ b/src/hyper/pages/MPOLY.ht
@@ -0,0 +1,110 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\MultivariatePolynomialXmpTitle}{MultivariatePolynomial}
+\newcommand{\MultivariatePolynomialXmpNumber}{9.54}
+%
+% =====================================================================
+\begin{page}{MultivariatePolynomialXmpPage}{9.54 MultivariatePolynomial}
+% =====================================================================
+\beginscroll
+
+The domain constructor \spadtype{MultivariatePolynomial} is similar to
+\spadtype{Polynomial} except that it specifies the variables to be used.
+%-% \HDindex{polynomial!multiple variables} Most functions available for{MultivariatePolynomialXmpPage}{9.54}{MultivariatePolynomial}
+\spadtype{Polynomial} are available for \spadtype{MultivariatePolynomial}.
+The abbreviation for \spadtype{MultivariatePolynomial} is
+\spadtype{MPOLY}.
+The type expressions
+\centerline{{\spadtype{MultivariatePolynomial([x,y],Integer)}}}
+and
+\centerline{{\spadtype{MPOLY([x,y],INT)}}}
+refer to the domain of multivariate polynomials in the variables
+\spad{x} and \spad{y} where the coefficients are restricted to be integers.
+The first
+variable specified is the main variable and the display of the
+polynomial reflects this.
+\xtc{
+This polynomial appears with
+terms in descending powers of the variable \spad{x}.
+}{
+\spadpaste{m : MPOLY([x,y],INT) := (x**2 - x*y**3 +3*y)**2 \bound{m}}
+}
+\xtc{
+It is easy to see a different variable ordering by doing a conversion.
+}{
+\spadpaste{m :: MPOLY([y,x],INT) \free{m}}
+}
+\xtc{
+You can use other, unspecified variables, by using
+\spadtype{Polynomial} in the coefficient type of \spadtype{MPOLY}.
+}{
+\spadpaste{p : MPOLY([x,y],POLY INT) \bound{pdec}}
+}
+\xtc{
+}{
+\spadpaste{p := (a**2*x - b*y**2 + 1)**2 \free{pdec}\bound{p}}
+}
+\xtc{
+Conversions can be used to re-express such polynomials in terms of
+the other variables.  For example, you can first push all the
+variables into a polynomial with integer coefficients.
+}{
+\spadpaste{p :: POLY INT \free{p}\bound{prev}}
+}
+\xtc{
+Now pull out the variables of interest.
+}{
+\spadpaste{\% :: MPOLY([a,b],POLY INT) \free{prev}}
+}
+
+\beginImportant
+\noindent {\bf Restriction:}
+\texht{\begin{quotation}\noindent}{\newline\indent{5}}
+\Language{} does not allow you to create types where
+\spadtype{MultivariatePolynomial} is contained in the coefficient type of
+\spadtype{Polynomial}. Therefore,
+\spad{MPOLY([x,y],POLY INT)} is legal but
+\spad{POLY MPOLY([x,y],INT)} is not.
+\texht{\end{quotation}}{\indent{0}}
+\endImportant
+
+\xtc{
+Multivariate polynomials may be combined with univariate polynomials
+to create types with special structures.
+}{
+\spadpaste{q : UP(x, FRAC MPOLY([y,z],INT)) \bound{qdec}}
+}
+\xtc{
+This is a polynomial in \spad{x} whose coefficients are
+quotients of polynomials in \spad{y} and \spad{z}.
+}{
+\spadpaste{q := (x**2 - x*(z+1)/y +2)**2 \free{qdec}\bound{q}}
+}
+\xtc{
+Use conversions for structural rearrangements.
+\spad{z} does not appear in a denominator and so it can be made
+the main variable.
+}{
+\spadpaste{q :: UP(z, FRAC MPOLY([x,y],INT)) \free{q}}
+}
+\xtc{
+Or you can make a multivariate polynomial in \spad{x} and \spad{z}
+whose coefficients are fractions in polynomials in \spad{y}.
+}{
+\spadpaste{q :: MPOLY([x,z], FRAC UP(y,INT)) \free{q}}
+}
+A conversion like \spad{q :: MPOLY([x,y], FRAC UP(z,INT))} is not
+possible in this example
+because \spad{y} appears in the denominator of a fraction.
+As you can see, \Language{} provides extraordinary flexibility in the
+manipulation and display of expressions via its conversion facility.
+
+For more information on related topics, see
+\downlink{`Polynomial'}{PolynomialXmpPage}\ignore{Polynomial},
+\downlink{`UnivariatePolynomial'}{UnivariatePolynomialXmpPage}\ignore{UnivariatePolynomial}, and
+\downlink{`DistributedMultivariatePolynomial'}{DistributedMultivariatePolynomialXmpPage}\ignore{DistributedMultivariatePolynomial}.
+\showBlurb{MultivariatePolynomial}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/MPOLY.pht b/src/hyper/pages/MPOLY.pht
new file mode 100644
index 00000000..310a655c
--- /dev/null
+++ b/src/hyper/pages/MPOLY.pht
@@ -0,0 +1,190 @@
+\begin{patch}{MultivariatePolynomialXmpPagePatch1}
+\begin{paste}{MultivariatePolynomialXmpPageFull1}{MultivariatePolynomialXmpPageEmpty1}
+\pastebutton{MultivariatePolynomialXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{m : MPOLY([x,y],INT) := (x**2 - x*y**3 +3*y)**2\bound{m }}
+\indentrel{3}\begin{verbatim}
+         4     3 3     6       2     4      2
+   (1)  x  - 2y x  + (y  + 6y)x  - 6y x + 9y
+            Type: MultivariatePolynomial([x,y],Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPageEmpty1}
+\begin{paste}{MultivariatePolynomialXmpPageEmpty1}{MultivariatePolynomialXmpPagePatch1}
+\pastebutton{MultivariatePolynomialXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{m : MPOLY([x,y],INT) := (x**2 - x*y**3 +3*y)**2\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPagePatch2}
+\begin{paste}{MultivariatePolynomialXmpPageFull2}{MultivariatePolynomialXmpPageEmpty2}
+\pastebutton{MultivariatePolynomialXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{m :: MPOLY([y,x],INT)\free{m }}
+\indentrel{3}\begin{verbatim}
+         2 6       4     3 3     2     2     4
+   (2)  x y  - 6x y  - 2x y  + 9y  + 6x y + x
+            Type: MultivariatePolynomial([y,x],Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPageEmpty2}
+\begin{paste}{MultivariatePolynomialXmpPageEmpty2}{MultivariatePolynomialXmpPagePatch2}
+\pastebutton{MultivariatePolynomialXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{m :: MPOLY([y,x],INT)\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPagePatch3}
+\begin{paste}{MultivariatePolynomialXmpPageFull3}{MultivariatePolynomialXmpPageEmpty3}
+\pastebutton{MultivariatePolynomialXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{p : MPOLY([x,y],POLY INT)\bound{pdec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPageEmpty3}
+\begin{paste}{MultivariatePolynomialXmpPageEmpty3}{MultivariatePolynomialXmpPagePatch3}
+\pastebutton{MultivariatePolynomialXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{p : MPOLY([x,y],POLY INT)\bound{pdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPagePatch4}
+\begin{paste}{MultivariatePolynomialXmpPageFull4}{MultivariatePolynomialXmpPageEmpty4}
+\pastebutton{MultivariatePolynomialXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{p := (a**2*x - b*y**2 + 1)**2\free{pdec }\bound{p }}
+\indentrel{3}\begin{verbatim}
+         4 2        2   2     2      2 4       2
+   (4)  a x  + (- 2a b y  + 2a )x + b y  - 2b y  + 1
+ Type: MultivariatePolynomial([x,y],Polynomial Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPageEmpty4}
+\begin{paste}{MultivariatePolynomialXmpPageEmpty4}{MultivariatePolynomialXmpPagePatch4}
+\pastebutton{MultivariatePolynomialXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{p := (a**2*x - b*y**2 + 1)**2\free{pdec }\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPagePatch5}
+\begin{paste}{MultivariatePolynomialXmpPageFull5}{MultivariatePolynomialXmpPageEmpty5}
+\pastebutton{MultivariatePolynomialXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{p :: POLY INT\free{p }\bound{prev }}
+\indentrel{3}\begin{verbatim}
+         2 4        2          2    4 2     2
+   (5)  b y  + (- 2a b x - 2b)y  + a x  + 2a x + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPageEmpty5}
+\begin{paste}{MultivariatePolynomialXmpPageEmpty5}{MultivariatePolynomialXmpPagePatch5}
+\pastebutton{MultivariatePolynomialXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{p :: POLY INT\free{p }\bound{prev }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPagePatch6}
+\begin{paste}{MultivariatePolynomialXmpPageFull6}{MultivariatePolynomialXmpPageEmpty6}
+\pastebutton{MultivariatePolynomialXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{\% :: MPOLY([a,b],POLY INT)\free{prev }}
+\indentrel{3}\begin{verbatim}
+         2 4          2        2    4 2     2
+   (6)  x a  + (- 2x y b + 2x)a  + y b  - 2y b + 1
+ Type: MultivariatePolynomial([a,b],Polynomial Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPageEmpty6}
+\begin{paste}{MultivariatePolynomialXmpPageEmpty6}{MultivariatePolynomialXmpPagePatch6}
+\pastebutton{MultivariatePolynomialXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{\% :: MPOLY([a,b],POLY INT)\free{prev }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPagePatch7}
+\begin{paste}{MultivariatePolynomialXmpPageFull7}{MultivariatePolynomialXmpPageEmpty7}
+\pastebutton{MultivariatePolynomialXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{q : UP(x, FRAC MPOLY([y,z],INT))\bound{qdec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPageEmpty7}
+\begin{paste}{MultivariatePolynomialXmpPageEmpty7}{MultivariatePolynomialXmpPagePatch7}
+\pastebutton{MultivariatePolynomialXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{q : UP(x, FRAC MPOLY([y,z],INT))\bound{qdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPagePatch8}
+\begin{paste}{MultivariatePolynomialXmpPageFull8}{MultivariatePolynomialXmpPageEmpty8}
+\pastebutton{MultivariatePolynomialXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{q := (x**2 - x*(z+1)/y +2)**2\free{qdec }\bound{q }}
+\indentrel{3}\begin{verbatim}
+   (8)
+                          2    2
+      4   - 2z - 2  3   4y  + z  + 2z + 1  2   - 4z - 4
+     x  + ÄÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄÄ x
+              y                  2                 y
+                                y
+   + 
+     4
+Type: UnivariatePolynomial(x,Fraction MultivariatePolynomial([y,z],Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPageEmpty8}
+\begin{paste}{MultivariatePolynomialXmpPageEmpty8}{MultivariatePolynomialXmpPagePatch8}
+\pastebutton{MultivariatePolynomialXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{q := (x**2 - x*(z+1)/y +2)**2\free{qdec }\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPagePatch9}
+\begin{paste}{MultivariatePolynomialXmpPageFull9}{MultivariatePolynomialXmpPageEmpty9}
+\pastebutton{MultivariatePolynomialXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{q :: UP(z, FRAC MPOLY([x,y],INT))\free{q }}
+\indentrel{3}\begin{verbatim}
+   (9)
+      2            3     2
+     x   2   - 2y x  + 2x  - 4y x
+     ÄÄ z  + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ z
+      2                2
+     y                y
+   + 
+      2 4       3      2      2            2
+     y x  - 2y x  + (4y  + 1)x  - 4y x + 4y
+     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                         2
+                        y
+Type: UnivariatePolynomial(z,Fraction MultivariatePolynomial([x,y],Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPageEmpty9}
+\begin{paste}{MultivariatePolynomialXmpPageEmpty9}{MultivariatePolynomialXmpPagePatch9}
+\pastebutton{MultivariatePolynomialXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{q :: UP(z, FRAC MPOLY([x,y],INT))\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPagePatch10}
+\begin{paste}{MultivariatePolynomialXmpPageFull10}{MultivariatePolynomialXmpPageEmpty10}
+\pastebutton{MultivariatePolynomialXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{q :: MPOLY([x,z], FRAC UP(y,INT))\free{q }}
+\indentrel{3}\begin{verbatim}
+   (10)
+                                            2
+      4      2     2  3     1  2    2     4y  + 1  2
+     x  + (- Ä z - Ä)x  + (ÄÄ z  + ÄÄ z + ÄÄÄÄÄÄÄ)x
+             y     y        2       2         2
+                           y       y         y
+   + 
+        4     4
+     (- Ä z - Ä)x + 4
+        y     y
+Type: MultivariatePolynomial([x,z],Fraction UnivariatePolynomial(y,Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultivariatePolynomialXmpPageEmpty10}
+\begin{paste}{MultivariatePolynomialXmpPageEmpty10}{MultivariatePolynomialXmpPagePatch10}
+\pastebutton{MultivariatePolynomialXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{q :: MPOLY([x,z], FRAC UP(y,INT))\free{q }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/MSET.ht b/src/hyper/pages/MSET.ht
new file mode 100644
index 00000000..381d9356
--- /dev/null
+++ b/src/hyper/pages/MSET.ht
@@ -0,0 +1,95 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\MultiSetXmpTitle}{MultiSet}
+\newcommand{\MultiSetXmpNumber}{9.53}
+%
+% =====================================================================
+\begin{page}{MultiSetXmpPage}{9.53 MultiSet}
+% =====================================================================
+\beginscroll
+The domain \spadtype{Multiset(R)} is similar to \spadtype{Set(R)}
+except that multiplicities
+(counts of duplications) are maintained and displayed.
+Use the operation \spadfunFrom{multiset}{Multiset} to create
+%-% \HDindex{set!vs. multiset}{MultiSetXmpPage}{9.53}{MultiSet}
+multisets from lists.
+%-% \HDindex{multiset}{MultiSetXmpPage}{9.53}{MultiSet}
+All the standard operations from sets are available for
+multisets.
+An element with multiplicity greater than one has the
+multiplicity displayed first, then a colon, and then the element.
+
+\xtc{
+Create a multiset of integers.
+}{
+\spadpaste{s := multiset [1,2,3,4,5,4,3,2,3,4,5,6,7,4,10]\bound{s}}
+}
+\xtc{
+The operation \spadfunX{insert} adds an element to a multiset.
+}{
+\spadpaste{insert!(3,s)\bound{s1}\free{s}}
+}
+\xtc{
+Use \spadfunX{remove} to remove an element.
+If a third argument is present, it specifies how many instances
+to remove. Otherwise all instances of the element are removed.
+Display the resulting multiset.
+}{
+\spadpaste{remove!(3,s,1); s\bound{s2}\free{s1}}
+}
+\xtc{
+}{
+\spadpaste{remove!(5,s); s\bound{s2}\free{s1}}
+}
+\xtc{
+The operation \spadfun{count} returns the number of copies
+of a given value.
+}{
+\spadpaste{count(5,s)\free{s2}}
+}
+\xtc{
+A second multiset.
+}{
+\spadpaste{t := multiset [2,2,2,-9]\bound{t}}
+}
+\xtc{
+The \spadfun{union} of two multisets is additive.
+}{
+\spadpaste{U := union(s,t)\bound{U}}
+}
+\xtc{
+The \spadfun{intersect} operation gives the elements that are in
+common, with additive multiplicity.
+}{
+\spadpaste{I := intersect(s,t)\bound{I}}
+}
+\xtc{
+The \spadfun{difference} of \spad{s} and \spad{t} consists of
+the elements that \spad{s} has but \spad{t} does not.
+Elements are regarded as indistinguishable, so that if \spad{s}
+and \spad{t} have any element in common, the \spadfun{difference}
+does not contain that element.
+}{
+\spadpaste{difference(s,t)\free{s2 t}}
+}
+\xtc{
+The \spadfun{symmetricDifference} is the \spadfun{union}
+of \spad{difference(s, t)} and \spad{difference(t, s)}.
+}{
+\spadpaste{S := symmetricDifference(s,t)\bound{S}\free{s2 t}}
+}
+\xtc{
+Check that the \spadfun{union} of the \spadfun{symmetricDifference} and
+the \spadfun{intersect} equals the \spadfun{union} of the elements.
+}{
+\spadpaste{(U = union(S,I))@Boolean\free{S I U}}
+}
+\xtc{
+Check some inclusion relations.
+}{
+\spadpaste{t1 := multiset [1,2,2,3]; [t1 < t, t1 < s, t < s, t1 <= s]\free{t s2}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/MSET.pht b/src/hyper/pages/MSET.pht
new file mode 100644
index 00000000..09fc5390
--- /dev/null
+++ b/src/hyper/pages/MSET.pht
@@ -0,0 +1,192 @@
+\begin{patch}{MultiSetXmpPagePatch1}
+\begin{paste}{MultiSetXmpPageFull1}{MultiSetXmpPageEmpty1}
+\pastebutton{MultiSetXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{s := multiset [1,2,3,4,5,4,3,2,3,4,5,6,7,4,10]\bound{s }}
+\indentrel{3}\begin{verbatim}
+   (1)  {7,2: 5,3: 3,1,10,6,4: 4,2: 2}
+                         Type: Multiset PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPageEmpty1}
+\begin{paste}{MultiSetXmpPageEmpty1}{MultiSetXmpPagePatch1}
+\pastebutton{MultiSetXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{s := multiset [1,2,3,4,5,4,3,2,3,4,5,6,7,4,10]\bound{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPagePatch2}
+\begin{paste}{MultiSetXmpPageFull2}{MultiSetXmpPageEmpty2}
+\pastebutton{MultiSetXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{insert!(3,s)\bound{s1 }\free{s }}
+\indentrel{3}\begin{verbatim}
+   (2)  {7,2: 5,4: 3,1,10,6,4: 4,2: 2}
+                         Type: Multiset PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPageEmpty2}
+\begin{paste}{MultiSetXmpPageEmpty2}{MultiSetXmpPagePatch2}
+\pastebutton{MultiSetXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{insert!(3,s)\bound{s1 }\free{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPagePatch3}
+\begin{paste}{MultiSetXmpPageFull3}{MultiSetXmpPageEmpty3}
+\pastebutton{MultiSetXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{remove!(3,s,1); s\bound{s2 }\free{s1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  {7,2: 5,3: 3,1,10,6,4: 4,2: 2}
+                         Type: Multiset PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPageEmpty3}
+\begin{paste}{MultiSetXmpPageEmpty3}{MultiSetXmpPagePatch3}
+\pastebutton{MultiSetXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{remove!(3,s,1); s\bound{s2 }\free{s1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPagePatch4}
+\begin{paste}{MultiSetXmpPageFull4}{MultiSetXmpPageEmpty4}
+\pastebutton{MultiSetXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{remove!(5,s); s\bound{s2 }\free{s1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  {7,3: 3,1,10,6,4: 4,2: 2}
+                         Type: Multiset PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPageEmpty4}
+\begin{paste}{MultiSetXmpPageEmpty4}{MultiSetXmpPagePatch4}
+\pastebutton{MultiSetXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{remove!(5,s); s\bound{s2 }\free{s1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPagePatch5}
+\begin{paste}{MultiSetXmpPageFull5}{MultiSetXmpPageEmpty5}
+\pastebutton{MultiSetXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{count(5,s)\free{s2 }}
+\indentrel{3}\begin{verbatim}
+   (5)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPageEmpty5}
+\begin{paste}{MultiSetXmpPageEmpty5}{MultiSetXmpPagePatch5}
+\pastebutton{MultiSetXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{count(5,s)\free{s2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPagePatch6}
+\begin{paste}{MultiSetXmpPageFull6}{MultiSetXmpPageEmpty6}
+\pastebutton{MultiSetXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{t := multiset [2,2,2,-9]\bound{t }}
+\indentrel{3}\begin{verbatim}
+   (6)  {- 9,3: 2}
+                                 Type: Multiset Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPageEmpty6}
+\begin{paste}{MultiSetXmpPageEmpty6}{MultiSetXmpPagePatch6}
+\pastebutton{MultiSetXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{t := multiset [2,2,2,-9]\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPagePatch7}
+\begin{paste}{MultiSetXmpPageFull7}{MultiSetXmpPageEmpty7}
+\pastebutton{MultiSetXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{U := union(s,t)\bound{U }}
+\indentrel{3}\begin{verbatim}
+   (7)  {7,3: 3,1,- 9,10,6,4: 4,5: 2}
+                                 Type: Multiset Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPageEmpty7}
+\begin{paste}{MultiSetXmpPageEmpty7}{MultiSetXmpPagePatch7}
+\pastebutton{MultiSetXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{U := union(s,t)\bound{U }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPagePatch8}
+\begin{paste}{MultiSetXmpPageFull8}{MultiSetXmpPageEmpty8}
+\pastebutton{MultiSetXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{I := intersect(s,t)\bound{I }}
+\indentrel{3}\begin{verbatim}
+   (8)  {5: 2}
+                                 Type: Multiset Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPageEmpty8}
+\begin{paste}{MultiSetXmpPageEmpty8}{MultiSetXmpPagePatch8}
+\pastebutton{MultiSetXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{I := intersect(s,t)\bound{I }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPagePatch9}
+\begin{paste}{MultiSetXmpPageFull9}{MultiSetXmpPageEmpty9}
+\pastebutton{MultiSetXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{difference(s,t)\free{s2 t }}
+\indentrel{3}\begin{verbatim}
+   (9)  {7,3: 3,1,10,6,4: 4}
+                                 Type: Multiset Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPageEmpty9}
+\begin{paste}{MultiSetXmpPageEmpty9}{MultiSetXmpPagePatch9}
+\pastebutton{MultiSetXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{difference(s,t)\free{s2 t }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPagePatch10}
+\begin{paste}{MultiSetXmpPageFull10}{MultiSetXmpPageEmpty10}
+\pastebutton{MultiSetXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{S := symmetricDifference(s,t)\bound{S }\free{s2 t }}
+\indentrel{3}\begin{verbatim}
+   (10)  {7,3: 3,1,- 9,10,6,4: 4}
+                                 Type: Multiset Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPageEmpty10}
+\begin{paste}{MultiSetXmpPageEmpty10}{MultiSetXmpPagePatch10}
+\pastebutton{MultiSetXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{S := symmetricDifference(s,t)\bound{S }\free{s2 t }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPagePatch11}
+\begin{paste}{MultiSetXmpPageFull11}{MultiSetXmpPageEmpty11}
+\pastebutton{MultiSetXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{(U = union(S,I))@Boolean\free{S I U }}
+\indentrel{3}\begin{verbatim}
+   (11)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPageEmpty11}
+\begin{paste}{MultiSetXmpPageEmpty11}{MultiSetXmpPagePatch11}
+\pastebutton{MultiSetXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{(U = union(S,I))@Boolean\free{S I U }}
+\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPagePatch12}
+\begin{paste}{MultiSetXmpPageFull12}{MultiSetXmpPageEmpty12}
+\pastebutton{MultiSetXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{t1 := multiset [1,2,2,3]; [t1 < t, t1 < s, t < s, t1 <= s]\free{t s2 }}
+\indentrel{3}\begin{verbatim}
+   (12)  [false,true,false,true]
+                                     Type: List Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MultiSetXmpPageEmpty12}
+\begin{paste}{MultiSetXmpPageEmpty12}{MultiSetXmpPagePatch12}
+\pastebutton{MultiSetXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{t1 := multiset [1,2,2,3]; [t1 < t, t1 < s, t < s, t1 <= s]\free{t s2 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/NONE.ht b/src/hyper/pages/NONE.ht
new file mode 100644
index 00000000..ba32ed88
--- /dev/null
+++ b/src/hyper/pages/NONE.ht
@@ -0,0 +1,35 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\NoneXmpTitle}{None}
+\newcommand{\NoneXmpNumber}{9.55}
+%
+% =====================================================================
+\begin{page}{NoneXmpPage}{9.55 None}
+% =====================================================================
+\beginscroll
+The \spadtype{None} domain is not very useful for interactive
+work but it is provided nevertheless for completeness of the
+\Language{} type system.
+\xtc{
+Probably the only place you will ever see it is if you enter an
+%-% \HDindex{list!empty}{NoneXmpPage}{9.55}{None}
+empty list with no type information.
+}{
+\spadpaste{[]}
+}
+\xtc{
+Such an empty list can be converted into an empty list
+of any other type.
+}{
+\spadpaste{[] :: List Float}
+}
+\xtc{
+If you wish to produce an empty list of a particular
+type directly, such as \spadtype{List NonNegativeInteger}, do it this way.
+}{
+\spadpaste{[]\$List(NonNegativeInteger)}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/NONE.pht b/src/hyper/pages/NONE.pht
new file mode 100644
index 00000000..7e748c0d
--- /dev/null
+++ b/src/hyper/pages/NONE.pht
@@ -0,0 +1,48 @@
+\begin{patch}{NoneXmpPagePatch1}
+\begin{paste}{NoneXmpPageFull1}{NoneXmpPageEmpty1}
+\pastebutton{NoneXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{[]}
+\indentrel{3}\begin{verbatim}
+   (1)  []
+                                        Type: List None
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{NoneXmpPageEmpty1}
+\begin{paste}{NoneXmpPageEmpty1}{NoneXmpPagePatch1}
+\pastebutton{NoneXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{[]}
+\end{paste}\end{patch}
+
+\begin{patch}{NoneXmpPagePatch2}
+\begin{paste}{NoneXmpPageFull2}{NoneXmpPageEmpty2}
+\pastebutton{NoneXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{[] :: List Float}
+\indentrel{3}\begin{verbatim}
+   (2)  []
+                                       Type: List Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{NoneXmpPageEmpty2}
+\begin{paste}{NoneXmpPageEmpty2}{NoneXmpPagePatch2}
+\pastebutton{NoneXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{[] :: List Float}
+\end{paste}\end{patch}
+
+\begin{patch}{NoneXmpPagePatch3}
+\begin{paste}{NoneXmpPageFull3}{NoneXmpPageEmpty3}
+\pastebutton{NoneXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{[]$List(NonNegativeInteger)}
+\indentrel{3}\begin{verbatim}
+   (3)  []
+                          Type: List NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{NoneXmpPageEmpty3}
+\begin{paste}{NoneXmpPageEmpty3}{NoneXmpPagePatch3}
+\pastebutton{NoneXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{[]$List(NonNegativeInteger)}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/OCT.ht b/src/hyper/pages/OCT.ht
new file mode 100644
index 00000000..b6109fce
--- /dev/null
+++ b/src/hyper/pages/OCT.ht
@@ -0,0 +1,121 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\OctonionXmpTitle}{Octonion}
+\newcommand{\OctonionXmpNumber}{9.56}
+%
+% =====================================================================
+\begin{page}{OctonionXmpPage}{9.56 Octonion}
+% =====================================================================
+\beginscroll
+
+The Octonions, also called the Cayley-Dixon algebra, defined over a
+commutative ring are an eight-dimensional non-associative algebra.
+Their construction from quaternions is similar to the construction
+of quaternions from complex numbers (see \downlink{`Quaternion'}{QuaternionXmpPage}\ignore{Quaternion}).
+%-% \HDexptypeindex{Quaternion}{OctonionXmpPage}{9.56}{Octonion}
+%
+\xtc{
+As \spadtype{Octonion} creates an eight-dimensional algebra, you have to
+give eight components to construct an octonion.
+}{
+\spadpaste{oci1 := octon(1,2,3,4,5,6,7,8) \bound{oci1}}
+}
+\xtc{
+}{
+\spadpaste{oci2 := octon(7,2,3,-4,5,6,-7,0) \bound{oci2}}
+}
+%
+%
+\xtc{
+Or you can use two quaternions to create an octonion.
+}{
+\spadpaste{oci3 := octon(quatern(-7,-12,3,-10), quatern(5,6,9,0)) \bound{oci3}}
+}
+%
+%
+\xtc{
+You can easily demonstrate the non-associativity of multiplication.
+}{
+\spadpaste{(oci1 * oci2) * oci3 - oci1 * (oci2 * oci3) \free{oci1 oci2 oci3}}
+}
+%
+As with the quaternions, we have a real part, the imaginary
+parts \spad{i}, \spad{j}, \spad{k}, and four
+additional imaginary parts \spad{E}, \spad{I}, \spad{J} and \spad{K}.
+These parts correspond to the canonical basis
+\spad{(1,i,j,k,E,I,J,K)}.
+\xtc{
+For each basis element there is a component operation to extract
+the coefficient of the basis element for a given octonion.
+%\spadfunFrom{real}{Octonion},
+%\spadfunFrom{imagi}{Octonion},
+%\spadfunFrom{imagj}{Octonion},
+%\spadfunFrom{imagk}{Octonion},
+%\spadfunFrom{imagE}{Octonion},
+%\spadfunFrom{imagI}{Octonion},
+%\spadfunFrom{imagJ}{Octonion}, and
+%\spadfunFrom{imagK}{Octonion}.
+}{
+\spadpaste{[real oci1, imagi oci1, imagj oci1, imagk oci1, imagE oci1, imagI oci1, imagJ oci1, imagK oci1] \free{oci1}}
+}
+%
+A basis with respect to the
+quaternions is given by \spad{(1,E)}.
+However, you might ask, what then are the commuting rules?
+To answer this, we create some generic elements.
+%
+\xtc{
+We do this in \Language{}
+by simply changing the ground ring from \spadtype{Integer} to
+\spadtype{Polynomial Integer}.
+}{
+\spadpaste{q : Quaternion Polynomial Integer := quatern(q1, qi, qj, qk) \bound{q}}
+}
+\xtc{
+}{
+\spadpaste{E : Octonion Polynomial Integer:= octon(0,0,0,0,1,0,0,0) \bound{E}}
+}
+%
+\xtc{
+Note that quaternions are automatically converted to octonions in the
+obvious way.
+}{
+\spadpaste{q * E \free{q E}}
+}
+\xtc{
+}{
+\spadpaste{E * q \free{E q}}
+}
+\xtc{
+}{
+\spadpaste{q * 1\$(Octonion Polynomial Integer) \free{q}}
+}
+\xtc{
+}{
+\spadpaste{1\$(Octonion Polynomial Integer) * q \free{q}}
+}
+\xtc{
+Finally, we check that the \spadfunFrom{norm}{Octonion},
+defined as the sum of the squares of the coefficients,
+is a multiplicative map.
+}{
+\spadpaste{o : Octonion Polynomial Integer := octon(o1, oi, oj, ok, oE, oI, oJ, oK) \bound{o}}
+}
+\xtc{
+}{
+\spadpaste{norm o \free{o}}
+}
+\xtc{
+}{
+\spadpaste{p : Octonion Polynomial Integer := octon(p1, pi, pj, pk, pE, pI, pJ, pK) \bound{p}}
+}
+\xtc{
+Since the result is \spad{0}, the norm is multiplicative.
+}{
+\spadpaste{norm(o*p)-norm(p)*norm(p)\free{o p} }
+}
+\showBlurb{Octonion}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/OCT.pht b/src/hyper/pages/OCT.pht
new file mode 100644
index 00000000..e29fb1bd
--- /dev/null
+++ b/src/hyper/pages/OCT.pht
@@ -0,0 +1,333 @@
+\begin{patch}{OctonionXmpPagePatch1}
+\begin{paste}{OctonionXmpPageFull1}{OctonionXmpPageEmpty1}
+\pastebutton{OctonionXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{oci1 := octon(1,2,3,4,5,6,7,8)\bound{oci1 }}
+\indentrel{3}\begin{verbatim}
+   (1)  1 + 2i + 3j + 4k + 5E + 6I + 7J + 8K
+                                 Type: Octonion Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty1}
+\begin{paste}{OctonionXmpPageEmpty1}{OctonionXmpPagePatch1}
+\pastebutton{OctonionXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{oci1 := octon(1,2,3,4,5,6,7,8)\bound{oci1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch2}
+\begin{paste}{OctonionXmpPageFull2}{OctonionXmpPageEmpty2}
+\pastebutton{OctonionXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{oci2 := octon(7,2,3,-4,5,6,-7,0)\bound{oci2 }}
+\indentrel{3}\begin{verbatim}
+   (2)  7 + 2i + 3j - 4k + 5E + 6I - 7J
+                                 Type: Octonion Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty2}
+\begin{paste}{OctonionXmpPageEmpty2}{OctonionXmpPagePatch2}
+\pastebutton{OctonionXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{oci2 := octon(7,2,3,-4,5,6,-7,0)\bound{oci2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch3}
+\begin{paste}{OctonionXmpPageFull3}{OctonionXmpPageEmpty3}
+\pastebutton{OctonionXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{oci3 := octon(quatern(-7,-12,3,-10), quatern(5,6,9,0))\bound{oci3 }}
+\indentrel{3}\begin{verbatim}
+   (3)  - 7 - 12i + 3j - 10k + 5E + 6I + 9J
+                                 Type: Octonion Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty3}
+\begin{paste}{OctonionXmpPageEmpty3}{OctonionXmpPagePatch3}
+\pastebutton{OctonionXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{oci3 := octon(quatern(-7,-12,3,-10), quatern(5,6,9,0))\bound{oci3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch4}
+\begin{paste}{OctonionXmpPageFull4}{OctonionXmpPageEmpty4}
+\pastebutton{OctonionXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{(oci1 * oci2) * oci3 - oci1 * (oci2 * oci3)\free{oci1 oci2 oci3 }}
+\indentrel{3}\begin{verbatim}
+   (4)
+   2696i - 2928j - 4072k + 16E - 1192I + 832J + 2616K
+                                 Type: Octonion Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty4}
+\begin{paste}{OctonionXmpPageEmpty4}{OctonionXmpPagePatch4}
+\pastebutton{OctonionXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{(oci1 * oci2) * oci3 - oci1 * (oci2 * oci3)\free{oci1 oci2 oci3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch5}
+\begin{paste}{OctonionXmpPageFull5}{OctonionXmpPageEmpty5}
+\pastebutton{OctonionXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{[real oci1, imagi oci1, imagj oci1, imagk oci1, imagE oci1, imagI oci1, imagJ oci1, imagK oci1]\free{oci1 }}
+\indentrel{3}\begin{verbatim}
+   (5)  [1,2,3,4,5,6,7,8]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty5}
+\begin{paste}{OctonionXmpPageEmpty5}{OctonionXmpPagePatch5}
+\pastebutton{OctonionXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{[real oci1, imagi oci1, imagj oci1, imagk oci1, imagE oci1, imagI oci1, imagJ oci1, imagK oci1]\free{oci1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch6}
+\begin{paste}{OctonionXmpPageFull6}{OctonionXmpPageEmpty6}
+\pastebutton{OctonionXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{q : Quaternion Polynomial Integer := quatern(q1, qi, qj, qk)\bound{q }}
+\indentrel{3}\begin{verbatim}
+   (6)  q1 + qi i + qj j + qk k
+                    Type: Quaternion Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty6}
+\begin{paste}{OctonionXmpPageEmpty6}{OctonionXmpPagePatch6}
+\pastebutton{OctonionXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{q : Quaternion Polynomial Integer := quatern(q1, qi, qj, qk)\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch7}
+\begin{paste}{OctonionXmpPageFull7}{OctonionXmpPageEmpty7}
+\pastebutton{OctonionXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{E : Octonion Polynomial Integer:= octon(0,0,0,0,1,0,0,0)\bound{E }}
+\indentrel{3}\begin{verbatim}
+   (7)  E
+                      Type: Octonion Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty7}
+\begin{paste}{OctonionXmpPageEmpty7}{OctonionXmpPagePatch7}
+\pastebutton{OctonionXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{E : Octonion Polynomial Integer:= octon(0,0,0,0,1,0,0,0)\bound{E }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch8}
+\begin{paste}{OctonionXmpPageFull8}{OctonionXmpPageEmpty8}
+\pastebutton{OctonionXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{q * E\free{q E }}
+\indentrel{3}\begin{verbatim}
+   (8)  q1 E + qi I + qj J + qk K
+                      Type: Octonion Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty8}
+\begin{paste}{OctonionXmpPageEmpty8}{OctonionXmpPagePatch8}
+\pastebutton{OctonionXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{q * E\free{q E }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch9}
+\begin{paste}{OctonionXmpPageFull9}{OctonionXmpPageEmpty9}
+\pastebutton{OctonionXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{E * q\free{E q }}
+\indentrel{3}\begin{verbatim}
+   (9)  q1 E - qi I - qj J - qk K
+                      Type: Octonion Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty9}
+\begin{paste}{OctonionXmpPageEmpty9}{OctonionXmpPagePatch9}
+\pastebutton{OctonionXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{E * q\free{E q }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch10}
+\begin{paste}{OctonionXmpPageFull10}{OctonionXmpPageEmpty10}
+\pastebutton{OctonionXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{q * 1$(Octonion Polynomial Integer)\free{q }}
+\indentrel{3}\begin{verbatim}
+   (10)  q1 + qi i + qj j + qk k
+                      Type: Octonion Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty10}
+\begin{paste}{OctonionXmpPageEmpty10}{OctonionXmpPagePatch10}
+\pastebutton{OctonionXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{q * 1$(Octonion Polynomial Integer)\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch11}
+\begin{paste}{OctonionXmpPageFull11}{OctonionXmpPageEmpty11}
+\pastebutton{OctonionXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{1$(Octonion Polynomial Integer) * q\free{q }}
+\indentrel{3}\begin{verbatim}
+   (11)  q1 + qi i + qj j + qk k
+                      Type: Octonion Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty11}
+\begin{paste}{OctonionXmpPageEmpty11}{OctonionXmpPagePatch11}
+\pastebutton{OctonionXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{1$(Octonion Polynomial Integer) * q\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch12}
+\begin{paste}{OctonionXmpPageFull12}{OctonionXmpPageEmpty12}
+\pastebutton{OctonionXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{o : Octonion Polynomial Integer := octon(o1, oi, oj, ok, oE, oI, oJ, oK)\bound{o }}
+\indentrel{3}\begin{verbatim}
+   (12)
+   o1 + oi i + oj j + ok k + oE E + oI I + oJ J + oK K
+                      Type: Octonion Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty12}
+\begin{paste}{OctonionXmpPageEmpty12}{OctonionXmpPagePatch12}
+\pastebutton{OctonionXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{o : Octonion Polynomial Integer := octon(o1, oi, oj, ok, oE, oI, oJ, oK)\bound{o }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch13}
+\begin{paste}{OctonionXmpPageFull13}{OctonionXmpPageEmpty13}
+\pastebutton{OctonionXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{norm o\free{o }}
+\indentrel{3}\begin{verbatim}
+           2     2     2     2     2     2     2     2
+   (13)  ok  + oj  + oi  + oK  + oJ  + oI  + oE  + o1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty13}
+\begin{paste}{OctonionXmpPageEmpty13}{OctonionXmpPagePatch13}
+\pastebutton{OctonionXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{norm o\free{o }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch14}
+\begin{paste}{OctonionXmpPageFull14}{OctonionXmpPageEmpty14}
+\pastebutton{OctonionXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{p : Octonion Polynomial Integer := octon(p1, pi, pj, pk, pE, pI, pJ, pK)\bound{p }}
+\indentrel{3}\begin{verbatim}
+   (14)
+   p1 + pi i + pj j + pk k + pE E + pI I + pJ J + pK K
+                      Type: Octonion Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty14}
+\begin{paste}{OctonionXmpPageEmpty14}{OctonionXmpPagePatch14}
+\pastebutton{OctonionXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{p : Octonion Polynomial Integer := octon(p1, pi, pj, pk, pE, pI, pJ, pK)\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPagePatch15}
+\begin{paste}{OctonionXmpPageFull15}{OctonionXmpPageEmpty15}
+\pastebutton{OctonionXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{norm(o*p)-norm(p)*norm(p)\free{o p }}
+\indentrel{3}\begin{verbatim}
+   (15)
+         4
+     - pk
+   + 
+              2      2      2      2      2      2      2
+         - 2pj  - 2pi  - 2pK  - 2pJ  - 2pI  - 2pE  - 2p1
+       + 
+           2     2     2     2     2     2     2     2
+         ok  + oj  + oi  + oK  + oJ  + oI  + oE  + o1
+    *
+         2
+       pk
+   + 
+         4
+     - pj
+   + 
+              2      2      2      2      2      2     2
+         - 2pi  - 2pK  - 2pJ  - 2pI  - 2pE  - 2p1  + ok
+       + 
+           2     2     2     2     2     2     2
+         oj  + oi  + oK  + oJ  + oI  + oE  + o1
+    *
+         2
+       pj
+   + 
+         4
+     - pi
+   + 
+              2      2      2      2      2     2     2
+         - 2pK  - 2pJ  - 2pI  - 2pE  - 2p1  + ok  + oj
+       + 
+           2     2     2     2     2     2
+         oi  + oK  + oJ  + oI  + oE  + o1
+    *
+         2
+       pi
+   + 
+         4
+     - pK
+   + 
+              2      2      2      2     2     2     2
+         - 2pJ  - 2pI  - 2pE  - 2p1  + ok  + oj  + oi
+       + 
+           2     2     2     2     2
+         oK  + oJ  + oI  + oE  + o1
+    *
+         2
+       pK
+   + 
+         4
+     - pJ
+   + 
+              2      2      2     2     2     2     2
+         - 2pI  - 2pE  - 2p1  + ok  + oj  + oi  + oK
+       + 
+           2     2     2     2
+         oJ  + oI  + oE  + o1
+    *
+         2
+       pJ
+   + 
+         4
+     - pI
+   + 
+              2      2     2     2     2     2     2     2
+         - 2pE  - 2p1  + ok  + oj  + oi  + oK  + oJ  + oI
+       + 
+           2     2
+         oE  + o1
+    *
+         2
+       pI
+   + 
+         4
+     - pE
+   + 
+              2     2     2     2     2     2     2     2
+         - 2p1  + ok  + oj  + oi  + oK  + oJ  + oI  + oE
+       + 
+           2
+         o1
+    *
+         2
+       pE
+   + 
+         4
+     - p1
+   + 
+        2     2     2     2     2     2     2     2   2
+     (ok  + oj  + oi  + oK  + oJ  + oI  + oE  + o1 )p1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OctonionXmpPageEmpty15}
+\begin{paste}{OctonionXmpPageEmpty15}{OctonionXmpPagePatch15}
+\pastebutton{OctonionXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{norm(o*p)-norm(p)*norm(p)\free{o p }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ODPOL.ht b/src/hyper/pages/ODPOL.ht
new file mode 100644
index 00000000..855a40b9
--- /dev/null
+++ b/src/hyper/pages/ODPOL.ht
@@ -0,0 +1,272 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\OrderlyDifferentialPolynomialXmpTitle}{OrderlyDifferentialPolynomial}
+\newcommand{\OrderlyDifferentialPolynomialXmpNumber}{9.60}
+%
+% =====================================================================
+\begin{page}{OrderlyDifferentialPolynomialXmpPage}{9.60 OrderlyDifferentialPolynomial}
+% =====================================================================
+\beginscroll
+
+Many systems of differential equations may be transformed to equivalent
+%-% \HDindex{differential equation}{OrderlyDifferentialPolynomialXmpPage}{9.60}{OrderlyDifferentialPolynomial}
+systems of ordinary differential equations where the equations are
+%-% \HDindex{equation!differential}{OrderlyDifferentialPolynomialXmpPage}{9.60}{OrderlyDifferentialPolynomial}
+expressed polynomially in terms of the unknown functions.
+%-% \HDindex{polynomial!differential polynomial}{OrderlyDifferentialPolynomialXmpPage}{9.60}{OrderlyDifferentialPolynomial}
+In \Language{}, the domain constructors
+\spadtype{OrderlyDifferentialPolynomial}
+%-% \HDindex{differential polynomial}{OrderlyDifferentialPolynomialXmpPage}{9.60}{OrderlyDifferentialPolynomial}
+(abbreviated \spadtype{ODPOL}) and
+\spadtype{SequentialDifferentialPolynomial} (abbreviation
+\spadtype{SDPOL}) implement two domains of ordinary differential
+polynomials over any differential ring.
+In the simplest case, this differential ring is usually either the ring of
+integers, or the field of rational numbers.
+However, \Language{} can handle ordinary differential polynomials over a
+field of rational functions in a single indeterminate.
+%-% \HDexptypeindex{OrderlyDifferentialPolynomial}{OrderlyDifferentialPolynomialXmpPage}{9.60}{OrderlyDifferentialPolynomial}
+%-% \HDexptypeindex{SequentialDifferentialPolynomial}{OrderlyDifferentialPolynomialXmpPage}{9.60}{OrderlyDifferentialPolynomial}
+
+The two domains \spadtype{ODPOL} and \spadtype{SDPOL} are almost
+identical, the only difference being the choice of a different ranking,
+which is an ordering of the derivatives of the indeterminates.
+The first domain uses an orderly ranking, that is, derivatives of higher
+order are ranked higher, and derivatives of the same order are ranked
+alphabetically.
+The second domain uses a sequential ranking, where derivatives are ordered
+first alphabetically by the differential indeterminates, and then by
+order.
+A more general domain constructor,
+\spadtype{DifferentialSparseMultivariatePolynomial} (abbreviation
+\spadtype{DSMP}) allows both a user-provided list of differential
+indeterminates as well as a user-defined ranking.
+We shall illustrate \spadtype{ODPOL(FRAC INT)}, which constructs a domain
+of ordinary differential polynomials in an arbitrary number of
+differential indeterminates with rational numbers as coefficients.
+\xtc{
+}{
+\spadpaste{dpol:= ODPOL(FRAC INT) \bound{dpol}}
+}
+
+\xtc{
+A differential indeterminate \spad{w} may be viewed as an infinite
+sequence of algebraic indeterminates, which are the derivatives of
+\spad{w}.
+To facilitate referencing these, \Language{} provides the operation
+\spadfunFrom{makeVariable}{OrderlyDifferentialPolynomial} to convert an
+element of type \spadtype{Symbol} to a map from the natural numbers to the
+differential polynomial ring.
+}{
+\spadpaste{w := makeVariable('w)\$dpol \free{dpol}\bound{w}}
+}
+\xtc{
+}{
+\spadpaste{z := makeVariable('z)\$dpol \free{dpol}\bound{z}}
+}
+\xtc{
+The fifth derivative of \spad{w} can be obtained by applying the map
+\spad{w} to the number \spad{5.}
+Note that the order of differentiation is given as a subscript (except
+when the order is 0).
+}{
+\spadpaste{w.5 \free{w}}
+}
+\xtc{
+}{
+\spadpaste{w 0 \free{w}}
+}
+\xtc{
+The first five derivatives of \spad{z} can be generated by a list.
+}{
+\spadpaste{[z.i for i in 1..5] \free{z}}
+}
+\xtc{
+The usual arithmetic can be used to form a differential polynomial from
+the derivatives.
+}{
+\spadpaste{f:= w.4 - w.1 * w.1 * z.3 \free{w}\free{z}\bound{f}}
+}
+\xtc{
+}{
+\spadpaste{g:=(z.1)**3 * (z.2)**2 - w.2 \free{z}\free{w}\bound{g}}
+}
+\xtc{
+The operation \spadfunFrom{D}{OrderlyDifferentialPolynomial}
+computes the derivative of any differential polynomial.
+}{
+\spadpaste{D(f) \free{f}}
+}
+\xtc{
+The same operation can compute higher derivatives, like the
+fourth derivative.
+}{
+\spadpaste{D(f,4) \free{f}}
+}
+\xtc{
+The operation \spadfunFrom{makeVariable}{OrderlyDifferentialPolynomial}
+creates a map to facilitate referencing the derivatives of \spad{f},
+similar to the map \spad{w}.
+}{
+\spadpaste{df:=makeVariable(f)\$dpol \free{f}\bound{df}}
+}
+\xtc{
+The fourth derivative of f may be referenced easily.
+}{
+\spadpaste{df.4 \free{df}}
+}
+\xtc{
+The operation \spadfunFrom{order}{OrderlyDifferentialPolynomial}
+returns the order of a differential polynomial, or the order
+in a specified differential indeterminate.
+}{
+\spadpaste{order(g)  \free{g}}
+}
+\xtc{
+}{
+\spadpaste{order(g, 'w)  \free{g}}
+}
+\xtc{
+The operation
+\spadfunFrom{differentialVariables}{OrderlyDifferentialPolynomial} returns
+a list of differential indeterminates occurring in a differential
+polynomial.
+}{
+\spadpaste{differentialVariables(g)  \free{g}}
+}
+\xtc{
+The operation \spadfunFrom{degree}{OrderlyDifferentialPolynomial} returns
+the degree, or the degree in the differential indeterminate specified.
+}{
+\spadpaste{degree(g) \free{g}}
+}
+\xtc{
+}{
+\spadpaste{degree(g, 'w)  \free{g}}
+}
+\xtc{
+The operation \spadfunFrom{weights}{OrderlyDifferentialPolynomial} returns
+a list of weights of differential monomials appearing in differential
+polynomial, or a list of weights in a specified differential
+indeterminate.
+}{
+\spadpaste{weights(g)  \free{g}}
+}
+\xtc{
+}{
+\spadpaste{weights(g,'w) \free{g}}
+}
+\xtc{
+The operation \spadfunFrom{weight}{OrderlyDifferentialPolynomial} returns
+the maximum weight of all differential monomials appearing in the
+differential polynomial.
+}{
+\spadpaste{weight(g)  \free{g}}
+}
+\xtc{
+A differential polynomial is {\em isobaric} if the weights of all
+differential monomials appearing in it are equal.
+}{
+\spadpaste{isobaric?(g) \free{g}}
+}
+\xtc{
+To substitute {\em differentially}, use
+\spadfunFrom{eval}{OrderlyDifferentialPolynomial}.
+Note that we must coerce \spad{'w} to \spadtype{Symbol}, since in
+\spadtype{ODPOL}, differential indeterminates belong to the domain
+\spadtype{Symbol}.
+Compare this result to the next, which substitutes {\em algebraically} (no
+substitution is done since \spad{w.0} does not appear in \spad{g}).
+}{
+\spadpaste{eval(g,['w::Symbol],[f]) \free{f}\free{g}}
+}
+\xtc{
+}{
+\spadpaste{eval(g,variables(w.0),[f]) \free{f}\free{g}}
+}
+\xtc{
+Since \spadtype{OrderlyDifferentialPolynomial} belongs to
+\spadtype{PolynomialCategory}, all the operations defined in the latter
+category, or in packages for the latter category, are available.
+}{
+\spadpaste{monomials(g) \free{g}}
+}
+\xtc{
+}{
+\spadpaste{variables(g) \free{g}}
+}
+\xtc{
+}{
+\spadpaste{gcd(f,g) \free{f}\free{g}}
+}
+\xtc{
+}{
+\spadpaste{groebner([f,g]) \free{f}\free{g}}
+}
+\xtc{
+The next three operations are essential for elimination procedures in
+differential polynomial rings.
+The operation \spadfunFrom{leader}{OrderlyDifferentialPolynomial} returns
+the leader of a differential polynomial, which is the highest ranked
+derivative of the differential indeterminates that occurs.
+}{
+\spadpaste{lg:=leader(g)  \free{g}\bound{lg}}
+}
+\xtc{
+The operation \spadfunFrom{separant}{OrderlyDifferentialPolynomial} returns
+the separant of a differential polynomial, which is the partial derivative
+with respect to the leader.
+}{
+\spadpaste{sg:=separant(g)  \free{g}\bound{sg}}
+}
+\xtc{
+The operation \spadfunFrom{initial}{OrderlyDifferentialPolynomial} returns
+the initial, which is the leading coefficient when the given differential
+polynomial is expressed as a polynomial in the leader.
+}{
+\spadpaste{ig:=initial(g)  \free{g}\bound{ig}}
+}
+\xtc{
+Using these three operations, it is possible to reduce \spad{f} modulo the
+differential ideal generated by \spad{g}.
+The general scheme is to first reduce the order, then reduce the degree in
+the leader.
+First, eliminate \spad{z.3} using the derivative of \spad{g}.
+}{
+\spadpaste{g1 := D g \free{g}\bound{g1}}
+}
+\xtc{
+Find its leader.
+}{
+\spadpaste{lg1:= leader g1 \free{g1}\bound{lg1}}
+}
+\xtc{
+Differentiate \spad{f} partially with respect to this leader.
+}{
+\spadpaste{pdf:=D(f, lg1) \free{f}\free{lg1}\bound{pdf}}
+}
+\xtc{
+Compute the partial remainder of \spad{f} with respect to \spad{g}.
+}{
+\spadpaste{prf:=sg * f- pdf * g1 \free{f}\free{sg}\free{pdf}\free{g1}\bound{prf}}
+}
+\xtc{
+Note that high powers of \spad{lg} still appear in \spad{prf}.
+Compute the leading coefficient of \spad{prf}
+as a polynomial in the leader of \spad{g}.
+}{
+\spadpaste{lcf:=leadingCoefficient univariate(prf, lg) \free{prf}\free{lg}\bound{lcf}}
+}
+\xtc{
+Finally, continue eliminating the high powers of \spad{lg} appearing in
+\spad{prf} to obtain the (pseudo) remainder of \spad{f} modulo \spad{g}
+and its derivatives.
+}{
+\spadpaste{ig * prf - lcf * g * lg \free{ig}\free{prf}\free{lcf}\free{g}\free{lg}}
+}
+\showBlurb{OrderlyDifferentialPolyomial}
+\showBlurb{SequentialDifferentialPolynomial}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ODPOL.pht b/src/hyper/pages/ODPOL.pht
new file mode 100644
index 00000000..2c37bab9
--- /dev/null
+++ b/src/hyper/pages/ODPOL.pht
@@ -0,0 +1,626 @@
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch1}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull1}{OrderlyDifferentialPolynomialXmpPageEmpty1}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{dpol:= ODPOL(FRAC INT)\bound{dpol }}
+\indentrel{3}\begin{verbatim}
+   (1)  OrderlyDifferentialPolynomial Fraction Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty1}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty1}{OrderlyDifferentialPolynomialXmpPagePatch1}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{dpol:= ODPOL(FRAC INT)\bound{dpol }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch2}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull2}{OrderlyDifferentialPolynomialXmpPageEmpty2}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{w := makeVariable('w)$dpol\free{dpol }\bound{w }}
+\indentrel{3}\begin{verbatim}
+   (2)  theMap(DPOLCAT-;makeVariable;AM;17!0,62)
+Type: (NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty2}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty2}{OrderlyDifferentialPolynomialXmpPagePatch2}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{w := makeVariable('w)$dpol\free{dpol }\bound{w }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch3}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull3}{OrderlyDifferentialPolynomialXmpPageEmpty3}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{z := makeVariable('z)$dpol\free{dpol }\bound{z }}
+\indentrel{3}\begin{verbatim}
+   (3)  theMap(DPOLCAT-;makeVariable;AM;17!0,187)
+Type: (NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty3}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty3}{OrderlyDifferentialPolynomialXmpPagePatch3}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{z := makeVariable('z)$dpol\free{dpol }\bound{z }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch4}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull4}{OrderlyDifferentialPolynomialXmpPageEmpty4}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{w.5\free{w }}
+\indentrel{3}\begin{verbatim}
+   (4)  w
+         5
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty4}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty4}{OrderlyDifferentialPolynomialXmpPagePatch4}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{w.5\free{w }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch5}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull5}{OrderlyDifferentialPolynomialXmpPageEmpty5}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{w 0\free{w }}
+\indentrel{3}\begin{verbatim}
+   (5)  w
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty5}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty5}{OrderlyDifferentialPolynomialXmpPagePatch5}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{w 0\free{w }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch6}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull6}{OrderlyDifferentialPolynomialXmpPageEmpty6}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{[z.i for i in 1..5]\free{z }}
+\indentrel{3}\begin{verbatim}
+   (6)  [z ,z ,z ,z ,z ]
+          1  2  3  4  5
+Type: List OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty6}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty6}{OrderlyDifferentialPolynomialXmpPagePatch6}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{[z.i for i in 1..5]\free{z }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch7}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull7}{OrderlyDifferentialPolynomialXmpPageEmpty7}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{f:= w.4 - w.1 * w.1 * z.3\free{w }\free{z }\bound{f }}
+\indentrel{3}\begin{verbatim}
+               2
+   (7)  w  - w  z
+         4    1  3
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty7}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty7}{OrderlyDifferentialPolynomialXmpPagePatch7}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{f:= w.4 - w.1 * w.1 * z.3\free{w }\free{z }\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch8}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull8}{OrderlyDifferentialPolynomialXmpPageEmpty8}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{g:=(z.1)**3 * (z.2)**2 - w.2\free{z }\free{w }\bound{g }}
+\indentrel{3}\begin{verbatim}
+          3  2
+   (8)  z  z   - w
+         1  2     2
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty8}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty8}{OrderlyDifferentialPolynomialXmpPagePatch8}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{g:=(z.1)**3 * (z.2)**2 - w.2\free{z }\free{w }\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch9}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull9}{OrderlyDifferentialPolynomialXmpPageEmpty9}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{D(f)\free{f }}
+\indentrel{3}\begin{verbatim}
+               2
+   (9)  w  - w  z  - 2w w z
+         5    1  4     1 2 3
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty9}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty9}{OrderlyDifferentialPolynomialXmpPagePatch9}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{D(f)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch10}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull10}{OrderlyDifferentialPolynomialXmpPageEmpty10}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{D(f,4)\free{f }}
+\indentrel{3}\begin{verbatim}
+   (10)
+            2                               2
+     w  - w  z  - 8w w z  + (- 12w w  - 12w  )z  - 2w z w
+      8    1  7     1 2 6         1 3      2   5     1 3 5
+   + 
+                                         2
+     (- 8w w  - 24w w )z  - 8w z w  - 6w  z
+          1 4      2 3  4     2 3 4     3  3
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty10}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty10}{OrderlyDifferentialPolynomialXmpPagePatch10}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{D(f,4)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch11}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull11}{OrderlyDifferentialPolynomialXmpPageEmpty11}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{df:=makeVariable(f)$dpol\free{f }\bound{df }}
+\indentrel{3}\begin{verbatim}
+   (11)  theMap(DPOLCAT-;makeVariable;AM;17!0,488)
+Type: (NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty11}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty11}{OrderlyDifferentialPolynomialXmpPagePatch11}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{df:=makeVariable(f)$dpol\free{f }\bound{df }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch12}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull12}{OrderlyDifferentialPolynomialXmpPageEmpty12}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{df.4\free{df }}
+\indentrel{3}\begin{verbatim}
+   (12)
+            2                               2
+     w  - w  z  - 8w w z  + (- 12w w  - 12w  )z  - 2w z w
+      8    1  7     1 2 6         1 3      2   5     1 3 5
+   + 
+                                         2
+     (- 8w w  - 24w w )z  - 8w z w  - 6w  z
+          1 4      2 3  4     2 3 4     3  3
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty12}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty12}{OrderlyDifferentialPolynomialXmpPagePatch12}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{df.4\free{df }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch13}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull13}{OrderlyDifferentialPolynomialXmpPageEmpty13}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{order(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (13)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty13}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty13}{OrderlyDifferentialPolynomialXmpPagePatch13}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{order(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch14}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull14}{OrderlyDifferentialPolynomialXmpPageEmpty14}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{order(g, 'w)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (14)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty14}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty14}{OrderlyDifferentialPolynomialXmpPagePatch14}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{order(g, 'w)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch15}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull15}{OrderlyDifferentialPolynomialXmpPageEmpty15}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{differentialVariables(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (15)  [z,w]
+                                      Type: List Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty15}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty15}{OrderlyDifferentialPolynomialXmpPagePatch15}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{differentialVariables(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch16}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull16}{OrderlyDifferentialPolynomialXmpPageEmpty16}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{degree(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+           2  3
+   (16)  z  z
+          2  1
+Type: IndexedExponents OrderlyDifferentialVariable Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty16}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty16}{OrderlyDifferentialPolynomialXmpPagePatch16}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{degree(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch17}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull17}{OrderlyDifferentialPolynomialXmpPageEmpty17}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{degree(g, 'w)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (17)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty17}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty17}{OrderlyDifferentialPolynomialXmpPagePatch17}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{degree(g, 'w)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch18}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull18}{OrderlyDifferentialPolynomialXmpPageEmpty18}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{weights(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (18)  [7,2]
+                          Type: List NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty18}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty18}{OrderlyDifferentialPolynomialXmpPagePatch18}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{weights(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch19}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull19}{OrderlyDifferentialPolynomialXmpPageEmpty19}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{weights(g,'w)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (19)  [2]
+                          Type: List NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty19}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty19}{OrderlyDifferentialPolynomialXmpPagePatch19}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{weights(g,'w)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch20}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull20}{OrderlyDifferentialPolynomialXmpPageEmpty20}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{weight(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (20)  7
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty20}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty20}{OrderlyDifferentialPolynomialXmpPagePatch20}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{weight(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch21}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull21}{OrderlyDifferentialPolynomialXmpPageEmpty21}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{isobaric?(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (21)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty21}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty21}{OrderlyDifferentialPolynomialXmpPagePatch21}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{isobaric?(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch22}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull22}{OrderlyDifferentialPolynomialXmpPageEmpty22}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{eval(g,['w::Symbol],[f])\free{f }\free{g }}
+\indentrel{3}\begin{verbatim}
+   (22)
+            2                           2        3  2
+   - w  + w  z  + 4w w z  + (2w w  + 2w  )z  + z  z
+      6    1  5     1 2 4      1 3     2   3    1  2
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty22}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty22}{OrderlyDifferentialPolynomialXmpPagePatch22}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{eval(g,['w::Symbol],[f])\free{f }\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch23}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull23}{OrderlyDifferentialPolynomialXmpPageEmpty23}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{eval(g,variables(w.0),[f])\free{f }\free{g }}
+\indentrel{3}\begin{verbatim}
+           3  2
+   (23)  z  z   - w
+          1  2     2
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty23}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty23}{OrderlyDifferentialPolynomialXmpPagePatch23}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{eval(g,variables(w.0),[f])\free{f }\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch24}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull24}{OrderlyDifferentialPolynomialXmpPageEmpty24}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{monomials(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+            3  2
+   (24)  [z  z  ,- w ]
+           1  2     2
+Type: List OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty24}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty24}{OrderlyDifferentialPolynomialXmpPagePatch24}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{monomials(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch25}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull25}{OrderlyDifferentialPolynomialXmpPageEmpty25}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull25}{\hidepaste}
+\tab{5}\spadcommand{variables(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (25)  [z ,w ,z ]
+           2  2  1
+          Type: List OrderlyDifferentialVariable Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty25}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty25}{OrderlyDifferentialPolynomialXmpPagePatch25}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{variables(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch26}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull26}{OrderlyDifferentialPolynomialXmpPageEmpty26}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull26}{\hidepaste}
+\tab{5}\spadcommand{gcd(f,g)\free{f }\free{g }}
+\indentrel{3}\begin{verbatim}
+   (26)  1
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty26}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty26}{OrderlyDifferentialPolynomialXmpPagePatch26}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{gcd(f,g)\free{f }\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch27}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull27}{OrderlyDifferentialPolynomialXmpPageEmpty27}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull27}{\hidepaste}
+\tab{5}\spadcommand{groebner([f,g])\free{f }\free{g }}
+\indentrel{3}\begin{verbatim}
+                 2     3  2
+   (27)  [w  - w  z ,z  z   - w ]
+           4    1  3  1  2     2
+Type: List OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty27}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty27}{OrderlyDifferentialPolynomialXmpPagePatch27}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{groebner([f,g])\free{f }\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch28}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull28}{OrderlyDifferentialPolynomialXmpPageEmpty28}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull28}{\hidepaste}
+\tab{5}\spadcommand{lg:=leader(g)\free{g }\bound{lg }}
+\indentrel{3}\begin{verbatim}
+   (28)  z
+          2
+               Type: OrderlyDifferentialVariable Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty28}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty28}{OrderlyDifferentialPolynomialXmpPagePatch28}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{lg:=leader(g)\free{g }\bound{lg }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch29}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull29}{OrderlyDifferentialPolynomialXmpPageEmpty29}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull29}{\hidepaste}
+\tab{5}\spadcommand{sg:=separant(g)\free{g }\bound{sg }}
+\indentrel{3}\begin{verbatim}
+            3
+   (29)  2z  z
+           1  2
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty29}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty29}{OrderlyDifferentialPolynomialXmpPagePatch29}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{sg:=separant(g)\free{g }\bound{sg }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch30}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull30}{OrderlyDifferentialPolynomialXmpPageEmpty30}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull30}{\hidepaste}
+\tab{5}\spadcommand{ig:=initial(g)\free{g }\bound{ig }}
+\indentrel{3}\begin{verbatim}
+           3
+   (30)  z
+          1
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty30}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty30}{OrderlyDifferentialPolynomialXmpPagePatch30}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{ig:=initial(g)\free{g }\bound{ig }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch31}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull31}{OrderlyDifferentialPolynomialXmpPageEmpty31}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull31}{\hidepaste}
+\tab{5}\spadcommand{g1 := D g\free{g }\bound{g1 }}
+\indentrel{3}\begin{verbatim}
+            3               2  3
+   (31)  2z  z z  - w  + 3z  z
+           1  2 3    3     1  2
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty31}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty31}{OrderlyDifferentialPolynomialXmpPagePatch31}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{g1 := D g\free{g }\bound{g1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch32}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull32}{OrderlyDifferentialPolynomialXmpPageEmpty32}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull32}{\hidepaste}
+\tab{5}\spadcommand{lg1:= leader g1\free{g1 }\bound{lg1 }}
+\indentrel{3}\begin{verbatim}
+   (32)  z
+          3
+               Type: OrderlyDifferentialVariable Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty32}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty32}{OrderlyDifferentialPolynomialXmpPagePatch32}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{lg1:= leader g1\free{g1 }\bound{lg1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch33}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull33}{OrderlyDifferentialPolynomialXmpPageEmpty33}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull33}{\hidepaste}
+\tab{5}\spadcommand{pdf:=D(f, lg1)\free{f }\free{lg1 }\bound{pdf }}
+\indentrel{3}\begin{verbatim}
+             2
+   (33)  - w
+            1
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty33}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty33}{OrderlyDifferentialPolynomialXmpPagePatch33}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{pdf:=D(f, lg1)\free{f }\free{lg1 }\bound{pdf }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch34}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull34}{OrderlyDifferentialPolynomialXmpPageEmpty34}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull34}{\hidepaste}
+\tab{5}\spadcommand{prf:=sg * f- pdf * g1\free{f }\free{sg }\free{pdf }\free{g1 }\bound{prf }}
+\indentrel{3}\begin{verbatim}
+            3         2        2  2  3
+   (34)  2z  z w  - w  w  + 3w  z  z
+           1  2 4    1  3     1  1  2
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty34}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty34}{OrderlyDifferentialPolynomialXmpPagePatch34}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty34}{\showpaste}
+\tab{5}\spadcommand{prf:=sg * f- pdf * g1\free{f }\free{sg }\free{pdf }\free{g1 }\bound{prf }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch35}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull35}{OrderlyDifferentialPolynomialXmpPageEmpty35}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull35}{\hidepaste}
+\tab{5}\spadcommand{lcf:=leadingCoefficient univariate(prf, lg)\free{prf }\free{lg }\bound{lcf }}
+\indentrel{3}\begin{verbatim}
+            2  2
+   (35)  3w  z
+           1  1
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty35}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty35}{OrderlyDifferentialPolynomialXmpPagePatch35}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty35}{\showpaste}
+\tab{5}\spadcommand{lcf:=leadingCoefficient univariate(prf, lg)\free{prf }\free{lg }\bound{lcf }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPagePatch36}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageFull36}{OrderlyDifferentialPolynomialXmpPageEmpty36}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageFull36}{\hidepaste}
+\tab{5}\spadcommand{ig * prf - lcf * g * lg\free{ig }\free{prf }\free{lcf }\free{g }\free{lg }}
+\indentrel{3}\begin{verbatim}
+            6         2  3        2  2
+   (36)  2z  z w  - w  z  w  + 3w  z  w z
+           1  2 4    1  1  3     1  1  2 2
+   Type: OrderlyDifferentialPolynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderlyDifferentialPolynomialXmpPageEmpty36}
+\begin{paste}{OrderlyDifferentialPolynomialXmpPageEmpty36}{OrderlyDifferentialPolynomialXmpPagePatch36}
+\pastebutton{OrderlyDifferentialPolynomialXmpPageEmpty36}{\showpaste}
+\tab{5}\spadcommand{ig * prf - lcf * g * lg\free{ig }\free{prf }\free{lcf }\free{g }\free{lg }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/OP.ht b/src/hyper/pages/OP.ht
new file mode 100644
index 00000000..ef706a12
--- /dev/null
+++ b/src/hyper/pages/OP.ht
@@ -0,0 +1,173 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\OperatorXmpTitle}{Operator}
+\newcommand{\OperatorXmpNumber}{9.58}
+%
+% =====================================================================
+\begin{page}{OperatorXmpPage}{9.58 Operator}
+% =====================================================================
+\beginscroll
+Given any ring \spad{R}, the ring of the \spadtype{Integer}-linear operators
+over \spad{R} is called \spadtype{Operator(R)}.
+%-% \HDindex{operator}{OperatorXmpPage}{9.58}{Operator}
+To create an operator over \spad{R}, first create a basic operator using the
+operation \spadfun{operator}, and then convert it to \spadtype{Operator(R)}
+for the \spad{R} you want.
+%
+\xtc{
+We choose \spad{R} to be the two by two matrices over the integers.
+}{
+\spadpaste{R := SQMATRIX(2, INT)\bound{r}}
+}
+\xtc{
+Create the operator \spad{tilde} on \spad{R}.
+}{
+\spadpaste{t := operator("tilde") :: OP(R) \free{r}\bound{t}}
+}
+%
+Since \spadtype{Operator} is unexposed we must either package-call operations
+from it, or expose it explicitly.  For convenience we will do the latter.
+%
+\noOutputXtc{
+Expose \spad{Operator}.
+}{
+\spadpaste{)set expose add constructor Operator \free{t}\bound{expose}}
+}
+%
+To attach an evaluation function (from \spad{R} to \spad{R}) to an
+operator over \spad{R}, use \spad{evaluate(op, f)} where \spad{op}
+is an operator over \spad{R} and \spad{f} is a function \spad{R ->
+R}.
+This needs to be done only once when the operator is defined.
+Note that \spad{f} must be \spadtype{Integer}-linear (that is,
+\spad{f(ax+y) = a f(x) + f(y)} for any integer \spad{a}, and any
+\spad{x} and \spad{y} in \spad{R}).
+%
+\xtc{
+We now attach the transpose map to the above operator \spad{t}.
+}{
+\spadpaste{evaluate(t, m +-> transpose m)\free{expose}\free{t}\bound{evt}}
+}
+%
+Operators can be manipulated formally as in any ring: \spadop{+} is the
+pointwise addition and \spadop{*} is composition.
+Any element \spad{x} of \spad{R} can be converted to an operator
+\subscriptText{\tt op}{\tt x}
+over \spad{R}, and the evaluation function of
+\subscriptText{\tt op}{\tt x}
+is left-multiplication by \spad{x}.
+%
+\xtc{
+Multiplying on the
+left by this matrix swaps the two rows.
+}{
+\spadpaste{s : R := matrix [[0, 1], [1, 0]]\bound{s}}
+}
+%
+\xtc{
+Can you guess what is the action of the following operator?
+}{
+\spadpaste{rho := t * s\free{evt s}\bound{rho}}
+}
+%
+%
+\xtc{
+Hint: applying \spad{rho} four times gives the identity, so
+\spad{rho**4-1} should return 0 when applied to any two by two matrix.
+}{
+\spadpaste{z := rho**4 - 1\free{rho}\bound{z}}
+}
+%
+%
+\xtc{
+Now check with this matrix.
+}{
+\spadpaste{m:R := matrix [[1, 2], [3, 4]]\bound{m}}
+}
+\xtc{
+}{
+\spadpaste{z m\free{z m}}
+}
+%
+%
+\xtc{
+As you have probably guessed by now, \spad{rho} acts on matrices
+by rotating the elements clockwise.
+}{
+\spadpaste{rho m\free{rho m}}
+}
+\xtc{
+}{
+\spadpaste{rho rho m\free{rho m}}
+}
+\xtc{
+}{
+\spadpaste{(rho**3) m\free{rho m}}
+}
+%
+%
+\xtc{
+Do the swapping of rows and transposition commute?
+We can check by computing their bracket.
+}{
+\spadpaste{b := t * s - s * t\free{s evt}\bound{b}}
+}
+%
+%
+\xtc{
+Now apply it to \spad{m}.
+}{
+\spadpaste{b m \free{b m}}
+}
+%
+ 
+Next we demonstrate how to define a differential operator
+on a polynomial ring.
+\xtc{
+This is the recursive definition of the \spad{n}-th Legendre polynomial.
+%-% \HDindex{polynomial!Legendre}{OperatorXmpPage}{9.58}{Operator}
+}{
+\begin{spadsrc}[\bound{l}]
+L n ==
+  n = 0 => 1
+  n = 1 => x
+  (2*n-1)/n * x * L(n-1) - (n-1)/n * L(n-2)
+\end{spadsrc}
+}
+\xtc{
+Create the differential operator \texht{$d \over {dx}$}{\spad{d/dx}} on
+polynomials in \spad{x} over the rational numbers.
+}{
+\spadpaste{dx := operator("D") :: OP(POLY FRAC INT) \bound{dx}}
+}
+\xtc{
+Now attach the map to it.
+}{
+\spadpaste{evaluate(dx, p +-> D(p, 'x)) \free{dx}\bound{edx}}
+}
+\xtc{
+This is the differential equation satisfied by the \spad{n}-th
+Legendre polynomial.
+}{
+\spadpaste{E n == (1 - x**2) * dx**2 - 2 * x * dx + n*(n+1) \free{edx}\bound{E}}
+}
+\xtc{
+Now we verify this for \spad{n = 15}.
+Here is the polynomial.
+}{
+\spadpaste{L 15 \free{L}}
+}
+\xtc{
+Here is the operator.
+}{
+\spadpaste{E 15 \free{E}}
+}
+\xtc{
+Here is the evaluation.
+}{
+\spadpaste{(E 15)(L 15) \free{L E}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/OP.pht b/src/hyper/pages/OP.pht
new file mode 100644
index 00000000..4b553864
--- /dev/null
+++ b/src/hyper/pages/OP.pht
@@ -0,0 +1,373 @@
+\begin{patch}{OperatorXmpPagePatch1}
+\begin{paste}{OperatorXmpPageFull1}{OperatorXmpPageEmpty1}
+\pastebutton{OperatorXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{R := SQMATRIX(2, INT)\bound{r }}
+\indentrel{3}\begin{verbatim}
+   (1)  SquareMatrix(2,Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty1}
+\begin{paste}{OperatorXmpPageEmpty1}{OperatorXmpPagePatch1}
+\pastebutton{OperatorXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{R := SQMATRIX(2, INT)\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch2}
+\begin{paste}{OperatorXmpPageFull2}{OperatorXmpPageEmpty2}
+\pastebutton{OperatorXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{t := operator("tilde") :: OP(R)\free{r }\bound{t }}
+\indentrel{3}\begin{verbatim}
+   (2)  tilde
+                 Type: Operator SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty2}
+\begin{paste}{OperatorXmpPageEmpty2}{OperatorXmpPagePatch2}
+\pastebutton{OperatorXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{t := operator("tilde") :: OP(R)\free{r }\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch3}
+\begin{paste}{OperatorXmpPageFull3}{OperatorXmpPageEmpty3}
+\pastebutton{OperatorXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{)set expose add constructor Operator\free{t }\bound{expose }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty3}
+\begin{paste}{OperatorXmpPageEmpty3}{OperatorXmpPagePatch3}
+\pastebutton{OperatorXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{)set expose add constructor Operator\free{t }\bound{expose }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch4}
+\begin{paste}{OperatorXmpPageFull4}{OperatorXmpPageEmpty4}
+\pastebutton{OperatorXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{evaluate(t, m +-> transpose m)\free{expose }\free{t }\bound{evt }}
+\indentrel{3}\begin{verbatim}
+   (3)  tilde
+                 Type: Operator SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty4}
+\begin{paste}{OperatorXmpPageEmpty4}{OperatorXmpPagePatch4}
+\pastebutton{OperatorXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{evaluate(t, m +-> transpose m)\free{expose }\free{t }\bound{evt }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch5}
+\begin{paste}{OperatorXmpPageFull5}{OperatorXmpPageEmpty5}
+\pastebutton{OperatorXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{s : R := matrix [[0, 1], [1, 0]]\bound{s }}
+\indentrel{3}\begin{verbatim}
+        Ú0  1¿
+   (4)  ³    ³
+        À1  0Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty5}
+\begin{paste}{OperatorXmpPageEmpty5}{OperatorXmpPagePatch5}
+\pastebutton{OperatorXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{s : R := matrix [[0, 1], [1, 0]]\bound{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch6}
+\begin{paste}{OperatorXmpPageFull6}{OperatorXmpPageEmpty6}
+\pastebutton{OperatorXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{rho := t * s\free{evt s }\bound{rho }}
+\indentrel{3}\begin{verbatim}
+             Ú0  1¿
+   (5)  tilde³    ³
+             À1  0Ù
+                 Type: Operator SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty6}
+\begin{paste}{OperatorXmpPageEmpty6}{OperatorXmpPagePatch6}
+\pastebutton{OperatorXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{rho := t * s\free{evt s }\bound{rho }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch7}
+\begin{paste}{OperatorXmpPageFull7}{OperatorXmpPageEmpty7}
+\pastebutton{OperatorXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{z := rho**4 - 1\free{rho }\bound{z }}
+\indentrel{3}\begin{verbatim}
+   (6)
+              Ú0  1¿     Ú0  1¿     Ú0  1¿     Ú0  1¿
+   - 1 + tilde³    ³tilde³    ³tilde³    ³tilde³    ³
+              À1  0Ù     À1  0Ù     À1  0Ù     À1  0Ù
+                 Type: Operator SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty7}
+\begin{paste}{OperatorXmpPageEmpty7}{OperatorXmpPagePatch7}
+\pastebutton{OperatorXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{z := rho**4 - 1\free{rho }\bound{z }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch8}
+\begin{paste}{OperatorXmpPageFull8}{OperatorXmpPageEmpty8}
+\pastebutton{OperatorXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{m:R := matrix [[1, 2], [3, 4]]\bound{m }}
+\indentrel{3}\begin{verbatim}
+        Ú1  2¿
+   (7)  ³    ³
+        À3  4Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty8}
+\begin{paste}{OperatorXmpPageEmpty8}{OperatorXmpPagePatch8}
+\pastebutton{OperatorXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{m:R := matrix [[1, 2], [3, 4]]\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch9}
+\begin{paste}{OperatorXmpPageFull9}{OperatorXmpPageEmpty9}
+\pastebutton{OperatorXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{z m\free{z m }}
+\indentrel{3}\begin{verbatim}
+        Ú0  0¿
+   (8)  ³    ³
+        À0  0Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty9}
+\begin{paste}{OperatorXmpPageEmpty9}{OperatorXmpPagePatch9}
+\pastebutton{OperatorXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{z m\free{z m }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch10}
+\begin{paste}{OperatorXmpPageFull10}{OperatorXmpPageEmpty10}
+\pastebutton{OperatorXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{rho m\free{rho m }}
+\indentrel{3}\begin{verbatim}
+        Ú3  1¿
+   (9)  ³    ³
+        À4  2Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty10}
+\begin{paste}{OperatorXmpPageEmpty10}{OperatorXmpPagePatch10}
+\pastebutton{OperatorXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{rho m\free{rho m }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch11}
+\begin{paste}{OperatorXmpPageFull11}{OperatorXmpPageEmpty11}
+\pastebutton{OperatorXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{rho rho m\free{rho m }}
+\indentrel{3}\begin{verbatim}
+         Ú4  3¿
+   (10)  ³    ³
+         À2  1Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty11}
+\begin{paste}{OperatorXmpPageEmpty11}{OperatorXmpPagePatch11}
+\pastebutton{OperatorXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{rho rho m\free{rho m }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch12}
+\begin{paste}{OperatorXmpPageFull12}{OperatorXmpPageEmpty12}
+\pastebutton{OperatorXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{(rho**3) m\free{rho m }}
+\indentrel{3}\begin{verbatim}
+         Ú2  4¿
+   (11)  ³    ³
+         À1  3Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty12}
+\begin{paste}{OperatorXmpPageEmpty12}{OperatorXmpPagePatch12}
+\pastebutton{OperatorXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{(rho**3) m\free{rho m }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch13}
+\begin{paste}{OperatorXmpPageFull13}{OperatorXmpPageEmpty13}
+\pastebutton{OperatorXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{b := t * s - s * t\free{s evt }\bound{b }}
+\indentrel{3}\begin{verbatim}
+           Ú0  1¿             Ú0  1¿
+   (12)  - ³    ³tilde + tilde³    ³
+           À1  0Ù             À1  0Ù
+                 Type: Operator SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty13}
+\begin{paste}{OperatorXmpPageEmpty13}{OperatorXmpPagePatch13}
+\pastebutton{OperatorXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{b := t * s - s * t\free{s evt }\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch14}
+\begin{paste}{OperatorXmpPageFull14}{OperatorXmpPageEmpty14}
+\pastebutton{OperatorXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{b m\free{b m }}
+\indentrel{3}\begin{verbatim}
+         Ú1  - 3¿
+   (13)  ³      ³
+         À3  - 1Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty14}
+\begin{paste}{OperatorXmpPageEmpty14}{OperatorXmpPagePatch14}
+\pastebutton{OperatorXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{b m\free{b m }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch15}
+\begin{paste}{OperatorXmpPageFull15}{OperatorXmpPageEmpty15}
+\pastebutton{OperatorXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{L n ==
+  n = 0 => 1
+  n = 1 => x
+  (2*n-1)/n * x * L(n-1) - (n-1)/n * L(n-2)
+\bound{l }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty15}
+\begin{paste}{OperatorXmpPageEmpty15}{OperatorXmpPagePatch15}
+\pastebutton{OperatorXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{L n ==
+  n = 0 => 1
+  n = 1 => x
+  (2*n-1)/n * x * L(n-1) - (n-1)/n * L(n-2)
+\bound{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch16}
+\begin{paste}{OperatorXmpPageFull16}{OperatorXmpPageEmpty16}
+\pastebutton{OperatorXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{dx := operator("D") :: OP(POLY FRAC INT)\bound{dx }}
+\indentrel{3}\begin{verbatim}
+   (15)  D
+             Type: Operator Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty16}
+\begin{paste}{OperatorXmpPageEmpty16}{OperatorXmpPagePatch16}
+\pastebutton{OperatorXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{dx := operator("D") :: OP(POLY FRAC INT)\bound{dx }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch17}
+\begin{paste}{OperatorXmpPageFull17}{OperatorXmpPageEmpty17}
+\pastebutton{OperatorXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{evaluate(dx, p +-> D(p, 'x))\free{dx }\bound{edx }}
+\indentrel{3}\begin{verbatim}
+   (16)  D
+             Type: Operator Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty17}
+\begin{paste}{OperatorXmpPageEmpty17}{OperatorXmpPagePatch17}
+\pastebutton{OperatorXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{evaluate(dx, p +-> D(p, 'x))\free{dx }\bound{edx }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch18}
+\begin{paste}{OperatorXmpPageFull18}{OperatorXmpPageEmpty18}
+\pastebutton{OperatorXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{E n == (1 - x**2) * dx**2 - 2 * x * dx + n*(n+1)\free{edx }\bound{E }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty18}
+\begin{paste}{OperatorXmpPageEmpty18}{OperatorXmpPagePatch18}
+\pastebutton{OperatorXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{E n == (1 - x**2) * dx**2 - 2 * x * dx + n*(n+1)\free{edx }\bound{E }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch19}
+\begin{paste}{OperatorXmpPageFull19}{OperatorXmpPageEmpty19}
+\pastebutton{OperatorXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{L 15\free{L }}
+\indentrel{3}\begin{verbatim}
+   (18)
+     9694845  15   35102025  13   50702925  11
+     ÄÄÄÄÄÄÄ x   - ÄÄÄÄÄÄÄÄ x   + ÄÄÄÄÄÄÄÄ x
+       2048          2048           2048
+   + 
+       37182145  9   14549535  7   2909907  5   255255  3
+     - ÄÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄÄ x  - ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄ x
+         2048          2048          2048        2048
+   + 
+       6435
+     - ÄÄÄÄ x
+       2048
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty19}
+\begin{paste}{OperatorXmpPageEmpty19}{OperatorXmpPagePatch19}
+\pastebutton{OperatorXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{L 15\free{L }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch20}
+\begin{paste}{OperatorXmpPageFull20}{OperatorXmpPageEmpty20}
+\pastebutton{OperatorXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{E 15\free{E }}
+\indentrel{3}\begin{verbatim}
+                        2      2
+   (19)  240 - 2x D - (x  - 1)D
+             Type: Operator Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty20}
+\begin{paste}{OperatorXmpPageEmpty20}{OperatorXmpPagePatch20}
+\pastebutton{OperatorXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{E 15\free{E }}
+\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPagePatch21}
+\begin{paste}{OperatorXmpPageFull21}{OperatorXmpPageEmpty21}
+\pastebutton{OperatorXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{(E 15)(L 15)\free{L E }}
+\indentrel{3}\begin{verbatim}
+   (20)  0
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OperatorXmpPageEmpty21}
+\begin{paste}{OperatorXmpPageEmpty21}{OperatorXmpPagePatch21}
+\pastebutton{OperatorXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{(E 15)(L 15)\free{L E }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/OVAR.ht b/src/hyper/pages/OVAR.ht
new file mode 100644
index 00000000..c6ccb3ad
--- /dev/null
+++ b/src/hyper/pages/OVAR.ht
@@ -0,0 +1,44 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\OrderedVariableListXmpTitle}{OrderedVariableList}
+\newcommand{\OrderedVariableListXmpNumber}{9.59}
+%
+% =====================================================================
+\begin{page}{OrderedVariableListXmpPage}{9.59 OrderedVariableList}
+% =====================================================================
+\beginscroll
+%
+
+The domain \spadtype{OrderedVariableList} provides symbols
+which are restricted to a particular list and have a definite 
+ordering. Those two features are specified by a \spadtype{List Symbol}
+object that is the argument to the domain.
+\xtc{
+This is a sample ordering of three symbols.
+}{
+\spadpaste{ls:List Symbol:=['x,'a,'z] \bound{ls}}
+}
+\xtc{
+Let's build the domain
+}{
+\spadpaste{Z:=OVAR ls \bound{Z} \free{ls}}
+}
+\xtc{
+How many variables does it have?
+}{
+\spadpaste{size()$Z \free{Z}}
+}
+\xtc{
+They are (in the imposed order)
+}{
+\spadpaste{lv:=[index(i::PI)$Z for i in 1..size()$Z] \bound{lv}\free{Z}}
+}
+\xtc{
+Check that the ordering is right
+}{
+\spadpaste{sorted?(>,lv) \free{lv}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/OVAR.pht b/src/hyper/pages/OVAR.pht
new file mode 100644
index 00000000..fee0ca6c
--- /dev/null
+++ b/src/hyper/pages/OVAR.pht
@@ -0,0 +1,80 @@
+\begin{patch}{OrderedVariableListXmpPagePatch1}
+\begin{paste}{OrderedVariableListXmpPageFull1}{OrderedVariableListXmpPageEmpty1}
+\pastebutton{OrderedVariableListXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{ls:List Symbol:=['x,'a,'z]\bound{ls }}
+\indentrel{3}\begin{verbatim}
+   (1)  [x,a,z]
+                                      Type: List Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderedVariableListXmpPageEmpty1}
+\begin{paste}{OrderedVariableListXmpPageEmpty1}{OrderedVariableListXmpPagePatch1}
+\pastebutton{OrderedVariableListXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{ls:List Symbol:=['x,'a,'z]\bound{ls }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderedVariableListXmpPagePatch2}
+\begin{paste}{OrderedVariableListXmpPageFull2}{OrderedVariableListXmpPageEmpty2}
+\pastebutton{OrderedVariableListXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{Z:=OVAR ls\bound{Z }\free{ls }}
+\indentrel{3}\begin{verbatim}
+   (2)  OrderedVariableList [x,a,z]
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderedVariableListXmpPageEmpty2}
+\begin{paste}{OrderedVariableListXmpPageEmpty2}{OrderedVariableListXmpPagePatch2}
+\pastebutton{OrderedVariableListXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{Z:=OVAR ls\bound{Z }\free{ls }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderedVariableListXmpPagePatch3}
+\begin{paste}{OrderedVariableListXmpPageFull3}{OrderedVariableListXmpPageEmpty3}
+\pastebutton{OrderedVariableListXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{size()$Z\free{Z }}
+\indentrel{3}\begin{verbatim}
+   (3)  3
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderedVariableListXmpPageEmpty3}
+\begin{paste}{OrderedVariableListXmpPageEmpty3}{OrderedVariableListXmpPagePatch3}
+\pastebutton{OrderedVariableListXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{size()$Z\free{Z }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderedVariableListXmpPagePatch4}
+\begin{paste}{OrderedVariableListXmpPageFull4}{OrderedVariableListXmpPageEmpty4}
+\pastebutton{OrderedVariableListXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{lv:=[index(i::PI)$Z for i in 1..size()$Z]\bound{lv }\free{Z }}
+\indentrel{3}\begin{verbatim}
+   (4)  [x,a,z]
+                 Type: List OrderedVariableList [x,a,z]
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderedVariableListXmpPageEmpty4}
+\begin{paste}{OrderedVariableListXmpPageEmpty4}{OrderedVariableListXmpPagePatch4}
+\pastebutton{OrderedVariableListXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{lv:=[index(i::PI)$Z for i in 1..size()$Z]\bound{lv }\free{Z }}
+\end{paste}\end{patch}
+
+\begin{patch}{OrderedVariableListXmpPagePatch5}
+\begin{paste}{OrderedVariableListXmpPageFull5}{OrderedVariableListXmpPageEmpty5}
+\pastebutton{OrderedVariableListXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{sorted?(>,lv)\free{lv }}
+\indentrel{3}\begin{verbatim}
+   (5)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OrderedVariableListXmpPageEmpty5}
+\begin{paste}{OrderedVariableListXmpPageEmpty5}{OrderedVariableListXmpPagePatch5}
+\pastebutton{OrderedVariableListXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{sorted?(>,lv)\free{lv }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/PERMAN.ht b/src/hyper/pages/PERMAN.ht
new file mode 100644
index 00000000..607de71e
--- /dev/null
+++ b/src/hyper/pages/PERMAN.ht
@@ -0,0 +1,51 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\PermanentXmpTitle}{Permanent}
+\newcommand{\PermanentXmpNumber}{9.62}
+%
+% =====================================================================
+\begin{page}{PermanentXmpPage}{9.62 Permanent}
+% =====================================================================
+\beginscroll
+The package \spadtype{Permanent} provides the function
+\spadfunFrom{permanent}{Permanent} for square matrices.
+%-% \HDindex{matrix!permanent of}{PermanentXmpPage}{9.62}{Permanent}
+The \spadfunFrom{permanent}{Permanent} of a square matrix can be computed
+in the same way as the determinant by expansion of minors except that for
+the permanent the sign for each element is \spad{1}, rather than being
+\spad{1} if the row plus column indices is positive and \spad{-1}
+otherwise.
+This function is much more difficult to compute efficiently than the
+\spadfunFrom{determinant}{Matrix}.
+An example of the use of \spadfunFrom{permanent}{Permanent} is the
+calculation of the \eth{\spad{n}} derangement number, defined to be the number
+of different possibilities for \spad{n} couples to dance but never with
+their own spouse.
+\xtc{
+Consider an \spad{n} by \spad{n} matrix with entries \spad{0} on the diagonal and
+\spad{1} elsewhere.
+Think of the rows as one-half of each couple (for example, the males) and the
+columns the other half.
+The permanent of such a matrix gives the desired derangement number.
+}{
+\begin{spadsrc}[\bound{kn}]
+kn n ==
+  r : MATRIX INT := new(n,n,1)
+  for i in 1..n repeat
+    r.i.i := 0
+  r
+\end{spadsrc}
+}
+\xtc{
+Here are some derangement numbers, which you see grow quite fast.
+}{
+\spadpaste{permanent(kn(5) :: SQMATRIX(5,INT)) \free{kn}}
+}
+\xtc{
+}{
+\spadpaste{[permanent(kn(n) :: SQMATRIX(n,INT)) for n in 1..13] \free{kn}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/PERMAN.pht b/src/hyper/pages/PERMAN.pht
new file mode 100644
index 00000000..684dd7fe
--- /dev/null
+++ b/src/hyper/pages/PERMAN.pht
@@ -0,0 +1,59 @@
+\begin{patch}{PermanentXmpPagePatch1}
+\begin{paste}{PermanentXmpPageFull1}{PermanentXmpPageEmpty1}
+\pastebutton{PermanentXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{kn n ==
+  r : MATRIX INT := new(n,n,1)
+  for i in 1..n repeat
+    r.i.i := 0
+  r
+\bound{kn }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PermanentXmpPageEmpty1}
+\begin{paste}{PermanentXmpPageEmpty1}{PermanentXmpPagePatch1}
+\pastebutton{PermanentXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{kn n ==
+  r : MATRIX INT := new(n,n,1)
+  for i in 1..n repeat
+    r.i.i := 0
+  r
+\bound{kn }}
+\end{paste}\end{patch}
+
+\begin{patch}{PermanentXmpPagePatch2}
+\begin{paste}{PermanentXmpPageFull2}{PermanentXmpPageEmpty2}
+\pastebutton{PermanentXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{permanent(kn(5) :: SQMATRIX(5,INT))\free{kn }}
+\indentrel{3}\begin{verbatim}
+   (2)  44
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PermanentXmpPageEmpty2}
+\begin{paste}{PermanentXmpPageEmpty2}{PermanentXmpPagePatch2}
+\pastebutton{PermanentXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{permanent(kn(5) :: SQMATRIX(5,INT))\free{kn }}
+\end{paste}\end{patch}
+
+\begin{patch}{PermanentXmpPagePatch3}
+\begin{paste}{PermanentXmpPageFull3}{PermanentXmpPageEmpty3}
+\pastebutton{PermanentXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{[permanent(kn(n) :: SQMATRIX(n,INT)) for n in 1..13]\free{kn }}
+\indentrel{3}\begin{verbatim}
+   (3)
+   [0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961,
+    14684570, 176214841, 2290792932]
+                          Type: List NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PermanentXmpPageEmpty3}
+\begin{paste}{PermanentXmpPageEmpty3}{PermanentXmpPagePatch3}
+\pastebutton{PermanentXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{[permanent(kn(n) :: SQMATRIX(n,INT)) for n in 1..13]\free{kn }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/PFR.ht b/src/hyper/pages/PFR.ht
new file mode 100644
index 00000000..b50ae3d4
--- /dev/null
+++ b/src/hyper/pages/PFR.ht
@@ -0,0 +1,143 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\PartialFractionXmpTitle}{PartialFraction}
+\newcommand{\PartialFractionXmpNumber}{9.61}
+%
+% =====================================================================
+\begin{page}{PartialFractionXmpPage}{9.61 PartialFraction}
+% =====================================================================
+\beginscroll
+%%
+%% pfr.htex
+%%
+
+A {\it partial fraction} is a decomposition of a quotient into
+%-% \HDindex{partial fraction}{PartialFractionXmpPage}{9.61}{PartialFraction}
+a sum
+%-% \HDindex{fraction!partial}{PartialFractionXmpPage}{9.61}{PartialFraction}
+of quotients where the denominators of the summands are powers of
+primes.\footnote{Most people first encounter partial fractions when they
+are learning integral calculus.
+For a technical discussion of partial fractions, see, for example, Lang's {\it
+Algebra.}} For example, the rational number \spad{1/6} is decomposed into
+\spad{1/2 -1/3}.
+You can compute partial fractions of quotients of objects from domains
+belonging to the category \spadtype{EuclideanDomain}.
+For example, \spadtype{Integer}, \spadtype{Complex Integer}, and
+\spadtype{UnivariatePolynomial(x, Fraction Integer)} all belong to
+\spadtype{EuclideanDomain}.
+In the examples following, we demonstrate how to decompose quotients of
+each of these kinds of object into partial fractions.
+Issue the system command
+\spadcmd{)show PartialFraction}
+to display the full list of operations defined by \spadtype{PartialFraction}.
+
+It is necessary that we know how to factor the denominator when we want to
+compute a partial fraction.
+Although the interpreter can often do this automatically, it may be
+necessary for you to include a call to \spadfun{factor}.
+In these examples, it is not necessary to factor the
+denominators explicitly.
+%
+\xtc{
+The main operation for computing partial fractions is called
+\spadfunFrom{partialFraction}{PartialFraction} and we use this to
+compute a decomposition of \spad{1 / 10!}.
+The first argument to \spadfunFrom{partialFraction}{PartialFraction} is
+the numerator of the quotient and the second argument is the factored
+denominator.
+}{
+\spadpaste{partialFraction(1,factorial 10) \bound{prev1}}
+}
+\xtc{
+Since the denominators are powers of primes, it may be possible
+to expand the numerators further with respect to those primes. Use the
+operation \spadfunFrom{padicFraction}{PartialFraction} to do this.
+}{
+\spadpaste{f := padicFraction(\%) \free{prev1}\bound{f}}
+}
+%
+%
+\xtc{
+The operation \spadfunFrom{compactFraction}{PartialFraction} returns an
+expanded fraction into the usual form.
+The compacted version is used internally for computational efficiency.
+}{
+\spadpaste{compactFraction(f) \free{f}}
+}
+%
+\xtc{
+You can add, subtract, multiply
+and divide partial fractions.  In addition, you can extract the parts
+of the decomposition.
+\spadfunFrom{numberOfFractionalTerms}{PartialFraction} computes
+the number of terms in the fractional part.
+This does not include the whole part of the fraction,
+which you get by calling \spadfunFrom{wholePart}{PartialFraction}.
+In this example, the whole part is just \spad{0}.
+}{
+\spadpaste{numberOfFractionalTerms(f) \free{f}}
+}
+\xtc{
+The operation
+\spadfunFrom{nthFractionalTerm}{PartialFraction} returns the individual terms in the
+decomposition.
+Notice that the object returned is a partial fraction itself.
+\spadfunFrom{firstNumer}{PartialFraction} and
+\spadfunFrom{firstDenom}{PartialFraction} extract the numerator and
+denominator of the first term of the fraction.
+}{
+\spadpaste{nthFractionalTerm(f,3) \free{f}}
+}
+%
+
+%
+\xtc{
+Given two gaussian integers (see \downlink{`Complex'}{ComplexXmpPage}\ignore{Complex}), you can
+decompose their quotient into a partial fraction.
+}{
+\spadpaste{partialFraction(1,- 13 + 14 * \%i) \bound{prev2}}
+}
+%
+\xtc{
+To convert back to a quotient, simply use a conversion.
+}{
+\spadpaste{\% :: Fraction Complex Integer \free{prev2}}
+}
+
+To conclude this section, we compute the decomposition of
+\texht{\narrowDisplay{1 \over {{(x + 1)}{(x + 2)}^2{(x + 3)}^3{(x + 4)}^4}}}{
+\begin{verbatim}
+                   1
+     -------------------------------
+                   2       3       4
+     (x + 1)(x + 2) (x + 3) (x + 4)
+\end{verbatim}
+}
+The polynomials in this object have type
+\spadtype{UnivariatePolynomial(x, Fraction Integer)}.
+%
+\xtc{
+We use the \spadfunFrom{primeFactor}{Factored} operation (see
+\downlink{`Factored'}{FactoredXmpPage}\ignore{Factored}) to create the denominator in factored form directly.
+}{
+\spadpaste{u : FR UP(x, FRAC INT) := reduce(*,[primeFactor(x+i,i) for i in 1..4]) \bound{u}}
+}
+%
+%
+\xtc{
+These are the compact and expanded partial fractions for the quotient.
+}{
+\spadpaste{partialFraction(1,u) \free{u}\bound{prev3}}
+}
+\xtc{
+}{
+\spadpaste{padicFraction \% \free{prev3}}
+}
+
+All see \downlink{`FullPartialFractionExpansion'}{FullPartialFractionExpansionXmpPage}\ignore{FullPartialFractionExpansion} for examples of
+factor-free conversion of quotients to full partial fractions.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/PFR.pht b/src/hyper/pages/PFR.pht
new file mode 100644
index 00000000..89339199
--- /dev/null
+++ b/src/hyper/pages/PFR.pht
@@ -0,0 +1,204 @@
+\begin{patch}{PartialFractionXmpPagePatch1}
+\begin{paste}{PartialFractionXmpPageFull1}{PartialFractionXmpPageEmpty1}
+\pastebutton{PartialFractionXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{partialFraction(1,factorial 10)\bound{prev1 }}
+\indentrel{3}\begin{verbatim}
+        159   23   12   1
+   (1)  ÄÄÄ - ÄÄ - ÄÄ + Ä
+          8    4    2   7
+         2    3    5
+                          Type: PartialFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPageEmpty1}
+\begin{paste}{PartialFractionXmpPageEmpty1}{PartialFractionXmpPagePatch1}
+\pastebutton{PartialFractionXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{partialFraction(1,factorial 10)\bound{prev1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPagePatch2}
+\begin{paste}{PartialFractionXmpPageFull2}{PartialFractionXmpPageEmpty2}
+\pastebutton{PartialFractionXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f := padicFraction(\%)\free{prev1 }\bound{f }}
+\indentrel{3}\begin{verbatim}
+   (2)
+   1    1    1    1    1    1    2    1    2   2    2   1
+   Ä + ÄÄ + ÄÄ + ÄÄ + ÄÄ + ÄÄ - ÄÄ - ÄÄ - ÄÄ - Ä - ÄÄ + Ä
+   2    4    5    6    7    8    2    3    4   5    2   7
+       2    2    2    2    2    3    3    3        5
+                          Type: PartialFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPageEmpty2}
+\begin{paste}{PartialFractionXmpPageEmpty2}{PartialFractionXmpPagePatch2}
+\pastebutton{PartialFractionXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f := padicFraction(\%)\free{prev1 }\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPagePatch3}
+\begin{paste}{PartialFractionXmpPageFull3}{PartialFractionXmpPageEmpty3}
+\pastebutton{PartialFractionXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{compactFraction(f)\free{f }}
+\indentrel{3}\begin{verbatim}
+        159   23   12   1
+   (3)  ÄÄÄ - ÄÄ - ÄÄ + Ä
+          8    4    2   7
+         2    3    5
+                          Type: PartialFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPageEmpty3}
+\begin{paste}{PartialFractionXmpPageEmpty3}{PartialFractionXmpPagePatch3}
+\pastebutton{PartialFractionXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{compactFraction(f)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPagePatch4}
+\begin{paste}{PartialFractionXmpPageFull4}{PartialFractionXmpPageEmpty4}
+\pastebutton{PartialFractionXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{numberOfFractionalTerms(f)\free{f }}
+\indentrel{3}\begin{verbatim}
+   (4)  12
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPageEmpty4}
+\begin{paste}{PartialFractionXmpPageEmpty4}{PartialFractionXmpPagePatch4}
+\pastebutton{PartialFractionXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{numberOfFractionalTerms(f)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPagePatch5}
+\begin{paste}{PartialFractionXmpPageFull5}{PartialFractionXmpPageEmpty5}
+\pastebutton{PartialFractionXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{nthFractionalTerm(f,3)\free{f }}
+\indentrel{3}\begin{verbatim}
+         1
+   (5)  ÄÄ
+         5
+        2
+                          Type: PartialFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPageEmpty5}
+\begin{paste}{PartialFractionXmpPageEmpty5}{PartialFractionXmpPagePatch5}
+\pastebutton{PartialFractionXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{nthFractionalTerm(f,3)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPagePatch6}
+\begin{paste}{PartialFractionXmpPageFull6}{PartialFractionXmpPageEmpty6}
+\pastebutton{PartialFractionXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{partialFraction(1,- 13 + 14 * \%i)\bound{prev2 }}
+\indentrel{3}\begin{verbatim}
+             1         4
+   (6)  - ÄÄÄÄÄÄÄ + ÄÄÄÄÄÄÄ
+          1 + 2%i   3 + 8%i
+                  Type: PartialFraction Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPageEmpty6}
+\begin{paste}{PartialFractionXmpPageEmpty6}{PartialFractionXmpPagePatch6}
+\pastebutton{PartialFractionXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{partialFraction(1,- 13 + 14 * \%i)\bound{prev2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPagePatch7}
+\begin{paste}{PartialFractionXmpPageFull7}{PartialFractionXmpPageEmpty7}
+\pastebutton{PartialFractionXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{\% :: Fraction Complex Integer\free{prev2 }}
+\indentrel{3}\begin{verbatim}
+              %i
+   (7)  - ÄÄÄÄÄÄÄÄÄ
+          14 + 13%i
+                         Type: Fraction Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPageEmpty7}
+\begin{paste}{PartialFractionXmpPageEmpty7}{PartialFractionXmpPagePatch7}
+\pastebutton{PartialFractionXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{\% :: Fraction Complex Integer\free{prev2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPagePatch8}
+\begin{paste}{PartialFractionXmpPageFull8}{PartialFractionXmpPageEmpty8}
+\pastebutton{PartialFractionXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{u : FR UP(x, FRAC INT) := reduce(*,[primeFactor(x+i,i) for i in 1..4])\bound{u }}
+\indentrel{3}\begin{verbatim}
+                      2       3       4
+   (8)  (x + 1)(x + 2) (x + 3) (x + 4)
+Type: Factored UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPageEmpty8}
+\begin{paste}{PartialFractionXmpPageEmpty8}{PartialFractionXmpPagePatch8}
+\pastebutton{PartialFractionXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{u : FR UP(x, FRAC INT) := reduce(*,[primeFactor(x+i,i) for i in 1..4])\bound{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPagePatch9}
+\begin{paste}{PartialFractionXmpPageFull9}{PartialFractionXmpPageEmpty9}
+\pastebutton{PartialFractionXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{partialFraction(1,u)\free{u }\bound{prev3 }}
+\indentrel{3}\begin{verbatim}
+   (9)
+       1     1      7     17  2         139
+      ÄÄÄ    Ä x + ÄÄ   - ÄÄ x  - 12x - ÄÄÄ
+      648    4     16      8             8
+     ÄÄÄÄÄ + ÄÄÄÄÄÄÄÄ + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+     x + 1          2                3
+             (x + 2)          (x + 3)
+   + 
+     607  3   10115  2   391     44179
+     ÄÄÄ x  + ÄÄÄÄÄ x  + ÄÄÄ x + ÄÄÄÄÄ
+     324       432        4       324
+     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                         4
+                  (x + 4)
+Type: PartialFraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPageEmpty9}
+\begin{paste}{PartialFractionXmpPageEmpty9}{PartialFractionXmpPagePatch9}
+\pastebutton{PartialFractionXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{partialFraction(1,u)\free{u }\bound{prev3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPagePatch10}
+\begin{paste}{PartialFractionXmpPageFull10}{PartialFractionXmpPageEmpty10}
+\pastebutton{PartialFractionXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{padicFraction \%\free{prev3 }}
+\indentrel{3}\begin{verbatim}
+   (10)
+       1       1         1        17        3
+      ÄÄÄ      Ä        ÄÄ        ÄÄ        Ä
+      648      4        16         8        4
+     ÄÄÄÄÄ + ÄÄÄÄÄ - ÄÄÄÄÄÄÄÄ - ÄÄÄÄÄ + ÄÄÄÄÄÄÄÄ
+     x + 1   x + 2          2   x + 3          2
+                     (x + 2)            (x + 3)
+   + 
+           1       607       403        13          1
+           Ä       ÄÄÄ       ÄÄÄ        ÄÄ         ÄÄ
+           2       324       432        36         12
+     - ÄÄÄÄÄÄÄÄ + ÄÄÄÄÄ + ÄÄÄÄÄÄÄÄ + ÄÄÄÄÄÄÄÄ + ÄÄÄÄÄÄÄÄ
+              3   x + 4          2          3          4
+       (x + 3)            (x + 4)    (x + 4)    (x + 4)
+Type: PartialFraction UnivariatePolynomial(x,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PartialFractionXmpPageEmpty10}
+\begin{paste}{PartialFractionXmpPageEmpty10}{PartialFractionXmpPagePatch10}
+\pastebutton{PartialFractionXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{padicFraction \%\free{prev3 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/POLY.ht b/src/hyper/pages/POLY.ht
new file mode 100644
index 00000000..088986f7
--- /dev/null
+++ b/src/hyper/pages/POLY.ht
@@ -0,0 +1,350 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\PolynomialXmpTitle}{Polynomial}
+\newcommand{\PolynomialXmpNumber}{9.63}
+%
+% =====================================================================
+\begin{page}{PolynomialXmpPage}{9.63 Polynomial}
+% =====================================================================
+\beginscroll
+
+The domain constructor \spadtype{Polynomial} (abbreviation: \spadtype{POLY})
+provides polynomials with an arbitrary number of unspecified
+variables.
+
+\xtc{
+It is used to create the default polynomial domains
+in \Language{}.
+Here the coefficients are integers.
+}{
+\spadpaste{x + 1}
+}
+\xtc{
+Here the coefficients have type \spadtype{Float}.
+}{
+\spadpaste{z - 2.3}
+}
+\xtc{
+And here we have a polynomial in two variables with coefficients which
+have type \spadtype{Fraction Integer}.
+}{
+\spadpaste{y**2 - z + 3/4}
+}
+
+The representation of objects of domains created by \spadtype{Polynomial}
+is that of recursive univariate polynomials.\footnote{The term
+\spad{univariate} means ``one variable.'' \spad{multivariate} means
+``possibly more than one variable.''}
+\xtc{
+This recursive structure is sometimes obvious from the display of
+a polynomial.
+}{
+\spadpaste{y **2 + x*y + y \bound{prev}}
+}
+In this example, you see that the polynomial is stored as a polynomial in
+\spad{y} with coefficients that are polynomials in \spad{x} with integer
+coefficients.
+In fact, you really don't need to worry about the representation unless
+you are working on an advanced application where it is critical.
+The polynomial types created from
+\spadtype{DistributedMultivariatePolynomial} and
+\spadtype{NewDistributedMultivariatePolynomial} (discussed in
+\downlink{`DistributedMultivariatePolynomial'}{DistributedMultivariatePolynomialXmpPage}\ignore{DistributedMultivariatePolynomial}) are stored and displayed in a
+non-recursive manner.
+\xtc{
+You see a ``flat'' display of the above
+polynomial by converting to one of those types.
+}{
+\spadpaste{\% :: DMP([y,x],INT) \free{prev}}
+}
+
+We will demonstrate many of the polynomial facilities by using two
+polynomials with integer coefficients.
+\xtc{
+By default, the interpreter expands polynomial expressions, even if they
+are written in a factored format.
+}{
+\spadpaste{p := (y-1)**2 * x * z \bound{p}}
+}
+\xtc{
+See \downlink{`Factored'}{FactoredXmpPage}\ignore{Factored} to see
+how to create objects in factored form directly.
+}{
+\spadpaste{q := (y-1) * x * (z+5) \bound{q}}
+}
+\xtc{
+The fully factored form can be recovered by using
+\spadfunFrom{factor}{Polynomial}.
+}{
+\spadpaste{factor(q) \free{q}}
+}
+This is the same name used for the operation to factor integers.
+Such reuse of names is called \spadglos{overloading} and makes it much easier
+to think of solving problems in general ways.
+\Language{} facilities for factoring polynomials created with
+\spadtype{Polynomial} are currently restricted to
+the integer and rational number coefficient cases.
+There are more complete facilities for factoring univariate polynomials:
+see \downlink{``\ugProblemFactorTitle''}{ugProblemFactorPage} in Section \ugProblemFactorNumber\ignore{ugProblemFactor}.
+
+\xtc{
+The standard arithmetic operations are available for polynomials.
+}{
+\spadpaste{p - q**2\free{p q}}
+}
+\xtc{
+The operation \spadfunFrom{gcd}{Polynomial} is used to compute the
+greatest common divisor of two polynomials.
+}{
+\spadpaste{gcd(p,q) \free{p q}\bound{prev4}}
+}
+\xtc{
+In the case of \spad{p} and \spad{q}, the gcd is obvious from their
+definitions.
+We factor the gcd to show this relationship better.
+}{
+\spadpaste{factor \% \free{prev4}}
+}
+\xtc{
+The least common multiple is computed by using \spadfunFrom{lcm}{Polynomial}.
+}{
+\spadpaste{lcm(p,q) \free{p q}}
+}
+\xtc{
+Use \spadfunFrom{content}{Polynomial} to compute the greatest common divisor of the
+coefficients of the polynomial.
+}{
+\spadpaste{content p \free{p}}
+}
+
+Many of the operations on polynomials require you to specify a variable.
+For example, \spadfunFrom{resultant}{Polynomial} requires you to give the
+variable in which the polynomials should be expressed.
+\xtc{
+This computes the resultant of the values of \spad{p} and
+\spad{q}, considering them as polynomials in the variable \spad{z}.
+They do not share a root when thought of as polynomials in \spad{z}.
+}{
+\spadpaste{resultant(p,q,z) \free{p q}}
+}
+\xtc{
+This value is \spad{0} because as polynomials in \spad{x} the polynomials
+have a common root.
+}{
+\spadpaste{resultant(p,q,x) \free{p}\free{q}}
+}
+The data type used for the variables created by \spadtype{Polynomial} is
+\spadtype{Symbol}.
+As mentioned above, the representation used by \spadtype{Polynomial} is
+recursive and so there is a main variable for nonconstant polynomials.
+\xtc{
+The operation \spadfunFrom{mainVariable}{Polynomial} returns this
+variable.
+The return type is actually a union of \spadtype{Symbol} and
+\spad{"failed"}.
+}{
+\spadpaste{mainVariable p \free{p}}
+}
+\xtc{
+The latter branch of the union is be used if the polynomial has no
+variables, that is, is a constant.
+}{
+\spadpaste{mainVariable(1 :: POLY INT)}
+}
+\xtc{
+You can also use the predicate \spadfunFrom{ground?}{Polynomial} to test
+whether a polynomial is in fact a member of its ground ring.
+}{
+\spadpaste{ground? p \free{p}}
+}
+\xtc{
+}{
+\spadpaste{ground?(1 :: POLY INT)}
+}
+\xtc{
+The complete list of variables actually used in a particular polynomial is
+returned by \spadfunFrom{variables}{Polynomial}.
+For constant polynomials, this list is empty.
+}{
+\spadpaste{variables p \free{p}}
+}
+
+\xtc{
+The \spadfunFrom{degree}{Polynomial} operation returns the
+degree of a polynomial in a specific variable.
+}{
+\spadpaste{degree(p,x) \free{p}}
+}
+\xtc{
+}{
+\spadpaste{degree(p,y) \free{p}}
+}
+\xtc{
+}{
+\spadpaste{degree(p,z) \free{p}}
+}
+\xtc{
+If you give a list of variables for the second argument, a list
+of the degrees in those variables is returned.
+}{
+\spadpaste{degree(p,[x,y,z]) \free{p}}
+}
+\xtc{
+The minimum degree of a variable in a polynomial is computed using
+\spadfunFrom{minimumDegree}{Polynomial}.
+}{
+\spadpaste{minimumDegree(p,z) \free{p}}
+}
+\xtc{
+The total degree of a polynomial is returned by
+\spadfunFrom{totalDegree}{Polynomial}.
+}{
+\spadpaste{totalDegree p \free{p}}
+}
+
+\xtc{
+It is often convenient to think of a polynomial as a leading monomial plus
+the remaining terms.
+}{
+\spadpaste{leadingMonomial p \free{p}}
+}
+\xtc{
+The \spadfunFrom{reductum}{Polynomial} operation returns a polynomial
+consisting of the sum of the monomials after the first.
+}{
+\spadpaste{reductum p \free{p}}
+}
+\xtc{
+These have the obvious relationship that the original polynomial
+is equal to the leading monomial plus the reductum.
+}{
+\spadpaste{p - leadingMonomial p - reductum p \free{p}}
+}
+\xtc{
+The value returned by \spadfunFrom{leadingMonomial}{Polynomial} includes
+the coefficient of that term.
+This is extracted by using
+\spadfunFrom{leadingCoefficient}{Polynomial} on the original polynomial.
+}{
+\spadpaste{leadingCoefficient p \free{p}}
+}
+\xtc{
+The operation \spadfunFrom{eval}{Polynomial} is used to substitute a value
+for a variable in a polynomial.
+}{
+\spadpaste{p  \free{p}}
+}
+\xtc{
+This value may be another variable, a constant or a polynomial.
+}{
+\spadpaste{eval(p,x,w) \free{p}}
+}
+\xtc{
+}{
+\spadpaste{eval(p,x,1) \free{p}}
+}
+\xtc{
+Actually, all the things being substituted are just polynomials,
+some more trivial than others.
+}{
+\spadpaste{eval(p,x,y**2 - 1) \free{p}}
+}
+
+\xtc{
+Derivatives are computed using the \spadfunFrom{D}{Polynomial}
+operation.
+}{
+\spadpaste{D(p,x) \free{p}}
+}
+\xtc{
+The first argument is the polynomial and the second is the variable.
+}{
+\spadpaste{D(p,y) \free{p}}
+}
+\xtc{
+Even if the polynomial has only one variable, you must specify it.
+}{
+\spadpaste{D(p,z) \free{p}}
+}
+
+Integration of polynomials is similar and the
+\spadfunFrom{integrate}{Polynomial} operation is used.
+
+\xtc{
+Integration requires that the coefficients support division.
+Consequently,
+\Language{} converts polynomials over the integers to polynomials over
+the rational numbers before integrating them.
+}{
+\spadpaste{integrate(p,y) \free{p}}
+}
+
+It is not possible, in general, to divide two polynomials.
+In our example using polynomials over the integers, the operation
+\spadfunFrom{monicDivide}{Polynomial} divides a polynomial by a monic
+polynomial (that is, a polynomial with leading coefficient equal to 1).
+The result is a record of the quotient and remainder of the
+division.
+\xtc{
+You must specify the variable in which to express the polynomial.
+}{
+\spadpaste{qr := monicDivide(p,x+1,x) \free{p}\bound{qr}}
+}
+\xtc{
+The selectors of the components of the record are
+\spad{quotient} and \spad{remainder}.
+Issue this to extract the remainder.
+}{
+\spadpaste{qr.remainder \free{qr}}
+}
+\xtc{
+Now that we can extract the components, we can demonstrate the
+relationship among them and the arguments to our original expression
+\spad{qr := monicDivide(p,x+1,x)}.
+}{
+\spadpaste{p - ((x+1) * qr.quotient + qr.remainder) \free{p}\free{qr}}
+}
+
+\xtc{
+If the \spadopFrom{/}{Fraction} operator is used with polynomials, a
+fraction object is created.
+In this example, the result is an object of type \spadtype{Fraction
+Polynomial Integer}.
+}{
+\spadpaste{p/q \free{p}\free{q}}
+}
+\xtc{
+If you use rational numbers as polynomial coefficients, the
+resulting object is of type \spadtype{Polynomial Fraction Integer}.
+}{
+\spadpaste{(2/3) * x**2 - y + 4/5 \bound{prev1}}
+}
+\xtc{
+This can be converted to a fraction of polynomials and back again, if
+required.
+}{
+\spadpaste{\% :: FRAC POLY INT \free{prev1}\bound{prev2}}
+}
+\xtc{
+}{
+\spadpaste{\% :: POLY FRAC INT \free{prev2}\bound{prev3}}
+}
+\xtc{
+To convert the coefficients to floating point,
+map the \spadfun{numeric} operation on the coefficients of the polynomial.
+}{
+\spadpaste{map(numeric,\%) \free{prev3}}
+}
+
+For more information on related topics, see
+\downlink{`UnivariatePolynomial'}{UnivariatePolynomialXmpPage}\ignore{UnivariatePolynomial},
+\downlink{`MultivariatePolynomial'}{MultivariatePolynomialXmpPage}\ignore{MultivariatePolynomial}, and
+\downlink{`DistributedMultivariatePolynomial'}{DistributedMultivariatePolynomialXmpPage}\ignore{DistributedMultivariatePolynomial}.
+You can also issue the system command
+\spadcmd{)show Polynomial}
+to display the full list of operations defined by
+\spadtype{Polynomial}.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/POLY.pht b/src/hyper/pages/POLY.pht
new file mode 100644
index 00000000..c3067fbb
--- /dev/null
+++ b/src/hyper/pages/POLY.pht
@@ -0,0 +1,773 @@
+\begin{patch}{PolynomialXmpPagePatch1}
+\begin{paste}{PolynomialXmpPageFull1}{PolynomialXmpPageEmpty1}
+\pastebutton{PolynomialXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{x + 1}
+\indentrel{3}\begin{verbatim}
+   (1)  x + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty1}
+\begin{paste}{PolynomialXmpPageEmpty1}{PolynomialXmpPagePatch1}
+\pastebutton{PolynomialXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{x + 1}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch2}
+\begin{paste}{PolynomialXmpPageFull2}{PolynomialXmpPageEmpty2}
+\pastebutton{PolynomialXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{z - 2.3}
+\indentrel{3}\begin{verbatim}
+   (2)  z - 2.3
+                                 Type: Polynomial Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty2}
+\begin{paste}{PolynomialXmpPageEmpty2}{PolynomialXmpPagePatch2}
+\pastebutton{PolynomialXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{z - 2.3}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch3}
+\begin{paste}{PolynomialXmpPageFull3}{PolynomialXmpPageEmpty3}
+\pastebutton{PolynomialXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{y**2 - z + 3/4}
+\indentrel{3}\begin{verbatim}
+               2   3
+   (3)  - z + y  + Ä
+                   4
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty3}
+\begin{paste}{PolynomialXmpPageEmpty3}{PolynomialXmpPagePatch3}
+\pastebutton{PolynomialXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{y**2 - z + 3/4}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch4}
+\begin{paste}{PolynomialXmpPageFull4}{PolynomialXmpPageEmpty4}
+\pastebutton{PolynomialXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{y **2 + x*y + y\bound{prev }}
+\indentrel{3}\begin{verbatim}
+         2
+   (4)  y  + (x + 1)y
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty4}
+\begin{paste}{PolynomialXmpPageEmpty4}{PolynomialXmpPagePatch4}
+\pastebutton{PolynomialXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{y **2 + x*y + y\bound{prev }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch5}
+\begin{paste}{PolynomialXmpPageFull5}{PolynomialXmpPageEmpty5}
+\pastebutton{PolynomialXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{\% :: DMP([y,x],INT)\free{prev }}
+\indentrel{3}\begin{verbatim}
+         2
+   (5)  y  + y x + y
+ Type: DistributedMultivariatePolynomial([y,x],Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty5}
+\begin{paste}{PolynomialXmpPageEmpty5}{PolynomialXmpPagePatch5}
+\pastebutton{PolynomialXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{\% :: DMP([y,x],INT)\free{prev }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch6}
+\begin{paste}{PolynomialXmpPageFull6}{PolynomialXmpPageEmpty6}
+\pastebutton{PolynomialXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{p := (y-1)**2 * x * z\bound{p }}
+\indentrel{3}\begin{verbatim}
+            2
+   (6)  (x y  - 2x y + x)z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty6}
+\begin{paste}{PolynomialXmpPageEmpty6}{PolynomialXmpPagePatch6}
+\pastebutton{PolynomialXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{p := (y-1)**2 * x * z\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch7}
+\begin{paste}{PolynomialXmpPageFull7}{PolynomialXmpPageEmpty7}
+\pastebutton{PolynomialXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{q := (y-1) * x * (z+5)\bound{q }}
+\indentrel{3}\begin{verbatim}
+   (7)  (x y - x)z + 5x y - 5x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty7}
+\begin{paste}{PolynomialXmpPageEmpty7}{PolynomialXmpPagePatch7}
+\pastebutton{PolynomialXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{q := (y-1) * x * (z+5)\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch8}
+\begin{paste}{PolynomialXmpPageFull8}{PolynomialXmpPageEmpty8}
+\pastebutton{PolynomialXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{factor(q)\free{q }}
+\indentrel{3}\begin{verbatim}
+   (8)  x(y - 1)(z + 5)
+                      Type: Factored Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty8}
+\begin{paste}{PolynomialXmpPageEmpty8}{PolynomialXmpPagePatch8}
+\pastebutton{PolynomialXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{factor(q)\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch9}
+\begin{paste}{PolynomialXmpPageFull9}{PolynomialXmpPageEmpty9}
+\pastebutton{PolynomialXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{p - q**2\free{p q }}
+\indentrel{3}\begin{verbatim}
+   (9)
+         2 2     2     2  2
+     (- x y  + 2x y - x )z
+   + 
+            2      2       2             2            2 2
+     ((- 10x  + x)y  + (20x  - 2x)y - 10x  + x)z - 25x y
+   + 
+        2       2
+     50x y - 25x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty9}
+\begin{paste}{PolynomialXmpPageEmpty9}{PolynomialXmpPagePatch9}
+\pastebutton{PolynomialXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{p - q**2\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch10}
+\begin{paste}{PolynomialXmpPageFull10}{PolynomialXmpPageEmpty10}
+\pastebutton{PolynomialXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{gcd(p,q)\free{p q }\bound{prev4 }}
+\indentrel{3}\begin{verbatim}
+   (10)  x y - x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty10}
+\begin{paste}{PolynomialXmpPageEmpty10}{PolynomialXmpPagePatch10}
+\pastebutton{PolynomialXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{gcd(p,q)\free{p q }\bound{prev4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch11}
+\begin{paste}{PolynomialXmpPageFull11}{PolynomialXmpPageEmpty11}
+\pastebutton{PolynomialXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{factor \%\free{prev4 }}
+\indentrel{3}\begin{verbatim}
+   (11)  x(y - 1)
+                      Type: Factored Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty11}
+\begin{paste}{PolynomialXmpPageEmpty11}{PolynomialXmpPagePatch11}
+\pastebutton{PolynomialXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{factor \%\free{prev4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch12}
+\begin{paste}{PolynomialXmpPageFull12}{PolynomialXmpPageEmpty12}
+\pastebutton{PolynomialXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{lcm(p,q)\free{p q }}
+\indentrel{3}\begin{verbatim}
+             2             2        2
+   (12)  (x y  - 2x y + x)z  + (5x y  - 10x y + 5x)z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty12}
+\begin{paste}{PolynomialXmpPageEmpty12}{PolynomialXmpPagePatch12}
+\pastebutton{PolynomialXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{lcm(p,q)\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch13}
+\begin{paste}{PolynomialXmpPageFull13}{PolynomialXmpPageEmpty13}
+\pastebutton{PolynomialXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{content p\free{p }}
+\indentrel{3}\begin{verbatim}
+   (13)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty13}
+\begin{paste}{PolynomialXmpPageEmpty13}{PolynomialXmpPagePatch13}
+\pastebutton{PolynomialXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{content p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch14}
+\begin{paste}{PolynomialXmpPageFull14}{PolynomialXmpPageEmpty14}
+\pastebutton{PolynomialXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{resultant(p,q,z)\free{p q }}
+\indentrel{3}\begin{verbatim}
+           2 3      2 2      2      2
+   (14)  5x y  - 15x y  + 15x y - 5x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty14}
+\begin{paste}{PolynomialXmpPageEmpty14}{PolynomialXmpPagePatch14}
+\pastebutton{PolynomialXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{resultant(p,q,z)\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch15}
+\begin{paste}{PolynomialXmpPageFull15}{PolynomialXmpPageEmpty15}
+\pastebutton{PolynomialXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{resultant(p,q,x)\free{p }\free{q }}
+\indentrel{3}\begin{verbatim}
+   (15)  0
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty15}
+\begin{paste}{PolynomialXmpPageEmpty15}{PolynomialXmpPagePatch15}
+\pastebutton{PolynomialXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{resultant(p,q,x)\free{p }\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch16}
+\begin{paste}{PolynomialXmpPageFull16}{PolynomialXmpPageEmpty16}
+\pastebutton{PolynomialXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{mainVariable p\free{p }}
+\indentrel{3}\begin{verbatim}
+   (16)  z
+                                Type: Union(Symbol,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty16}
+\begin{paste}{PolynomialXmpPageEmpty16}{PolynomialXmpPagePatch16}
+\pastebutton{PolynomialXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{mainVariable p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch17}
+\begin{paste}{PolynomialXmpPageFull17}{PolynomialXmpPageEmpty17}
+\pastebutton{PolynomialXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{mainVariable(1 :: POLY INT)}
+\indentrel{3}\begin{verbatim}
+   (17)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty17}
+\begin{paste}{PolynomialXmpPageEmpty17}{PolynomialXmpPagePatch17}
+\pastebutton{PolynomialXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{mainVariable(1 :: POLY INT)}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch18}
+\begin{paste}{PolynomialXmpPageFull18}{PolynomialXmpPageEmpty18}
+\pastebutton{PolynomialXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{ground? p\free{p }}
+\indentrel{3}\begin{verbatim}
+   (18)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty18}
+\begin{paste}{PolynomialXmpPageEmpty18}{PolynomialXmpPagePatch18}
+\pastebutton{PolynomialXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{ground? p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch19}
+\begin{paste}{PolynomialXmpPageFull19}{PolynomialXmpPageEmpty19}
+\pastebutton{PolynomialXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{ground?(1 :: POLY INT)}
+\indentrel{3}\begin{verbatim}
+   (19)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty19}
+\begin{paste}{PolynomialXmpPageEmpty19}{PolynomialXmpPagePatch19}
+\pastebutton{PolynomialXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{ground?(1 :: POLY INT)}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch20}
+\begin{paste}{PolynomialXmpPageFull20}{PolynomialXmpPageEmpty20}
+\pastebutton{PolynomialXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{variables p\free{p }}
+\indentrel{3}\begin{verbatim}
+   (20)  [z,y,x]
+                                      Type: List Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty20}
+\begin{paste}{PolynomialXmpPageEmpty20}{PolynomialXmpPagePatch20}
+\pastebutton{PolynomialXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{variables p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch21}
+\begin{paste}{PolynomialXmpPageFull21}{PolynomialXmpPageEmpty21}
+\pastebutton{PolynomialXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{degree(p,x)\free{p }}
+\indentrel{3}\begin{verbatim}
+   (21)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty21}
+\begin{paste}{PolynomialXmpPageEmpty21}{PolynomialXmpPagePatch21}
+\pastebutton{PolynomialXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{degree(p,x)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch22}
+\begin{paste}{PolynomialXmpPageFull22}{PolynomialXmpPageEmpty22}
+\pastebutton{PolynomialXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{degree(p,y)\free{p }}
+\indentrel{3}\begin{verbatim}
+   (22)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty22}
+\begin{paste}{PolynomialXmpPageEmpty22}{PolynomialXmpPagePatch22}
+\pastebutton{PolynomialXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{degree(p,y)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch23}
+\begin{paste}{PolynomialXmpPageFull23}{PolynomialXmpPageEmpty23}
+\pastebutton{PolynomialXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{degree(p,z)\free{p }}
+\indentrel{3}\begin{verbatim}
+   (23)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty23}
+\begin{paste}{PolynomialXmpPageEmpty23}{PolynomialXmpPagePatch23}
+\pastebutton{PolynomialXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{degree(p,z)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch24}
+\begin{paste}{PolynomialXmpPageFull24}{PolynomialXmpPageEmpty24}
+\pastebutton{PolynomialXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{degree(p,[x,y,z])\free{p }}
+\indentrel{3}\begin{verbatim}
+   (24)  [1,2,1]
+                          Type: List NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty24}
+\begin{paste}{PolynomialXmpPageEmpty24}{PolynomialXmpPagePatch24}
+\pastebutton{PolynomialXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{degree(p,[x,y,z])\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch25}
+\begin{paste}{PolynomialXmpPageFull25}{PolynomialXmpPageEmpty25}
+\pastebutton{PolynomialXmpPageFull25}{\hidepaste}
+\tab{5}\spadcommand{minimumDegree(p,z)\free{p }}
+\indentrel{3}\begin{verbatim}
+   (25)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty25}
+\begin{paste}{PolynomialXmpPageEmpty25}{PolynomialXmpPagePatch25}
+\pastebutton{PolynomialXmpPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{minimumDegree(p,z)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch26}
+\begin{paste}{PolynomialXmpPageFull26}{PolynomialXmpPageEmpty26}
+\pastebutton{PolynomialXmpPageFull26}{\hidepaste}
+\tab{5}\spadcommand{totalDegree p\free{p }}
+\indentrel{3}\begin{verbatim}
+   (26)  4
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty26}
+\begin{paste}{PolynomialXmpPageEmpty26}{PolynomialXmpPagePatch26}
+\pastebutton{PolynomialXmpPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{totalDegree p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch27}
+\begin{paste}{PolynomialXmpPageFull27}{PolynomialXmpPageEmpty27}
+\pastebutton{PolynomialXmpPageFull27}{\hidepaste}
+\tab{5}\spadcommand{leadingMonomial p\free{p }}
+\indentrel{3}\begin{verbatim}
+            2
+   (27)  x y z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty27}
+\begin{paste}{PolynomialXmpPageEmpty27}{PolynomialXmpPagePatch27}
+\pastebutton{PolynomialXmpPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{leadingMonomial p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch28}
+\begin{paste}{PolynomialXmpPageFull28}{PolynomialXmpPageEmpty28}
+\pastebutton{PolynomialXmpPageFull28}{\hidepaste}
+\tab{5}\spadcommand{reductum p\free{p }}
+\indentrel{3}\begin{verbatim}
+   (28)  (- 2x y + x)z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty28}
+\begin{paste}{PolynomialXmpPageEmpty28}{PolynomialXmpPagePatch28}
+\pastebutton{PolynomialXmpPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{reductum p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch29}
+\begin{paste}{PolynomialXmpPageFull29}{PolynomialXmpPageEmpty29}
+\pastebutton{PolynomialXmpPageFull29}{\hidepaste}
+\tab{5}\spadcommand{p - leadingMonomial p - reductum p\free{p }}
+\indentrel{3}\begin{verbatim}
+   (29)  0
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty29}
+\begin{paste}{PolynomialXmpPageEmpty29}{PolynomialXmpPagePatch29}
+\pastebutton{PolynomialXmpPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{p - leadingMonomial p - reductum p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch30}
+\begin{paste}{PolynomialXmpPageFull30}{PolynomialXmpPageEmpty30}
+\pastebutton{PolynomialXmpPageFull30}{\hidepaste}
+\tab{5}\spadcommand{leadingCoefficient p\free{p }}
+\indentrel{3}\begin{verbatim}
+   (30)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty30}
+\begin{paste}{PolynomialXmpPageEmpty30}{PolynomialXmpPagePatch30}
+\pastebutton{PolynomialXmpPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{leadingCoefficient p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch31}
+\begin{paste}{PolynomialXmpPageFull31}{PolynomialXmpPageEmpty31}
+\pastebutton{PolynomialXmpPageFull31}{\hidepaste}
+\tab{5}\spadcommand{p\free{p }}
+\indentrel{3}\begin{verbatim}
+             2
+   (31)  (x y  - 2x y + x)z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty31}
+\begin{paste}{PolynomialXmpPageEmpty31}{PolynomialXmpPagePatch31}
+\pastebutton{PolynomialXmpPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch32}
+\begin{paste}{PolynomialXmpPageFull32}{PolynomialXmpPageEmpty32}
+\pastebutton{PolynomialXmpPageFull32}{\hidepaste}
+\tab{5}\spadcommand{eval(p,x,w)\free{p }}
+\indentrel{3}\begin{verbatim}
+             2
+   (32)  (w y  - 2w y + w)z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty32}
+\begin{paste}{PolynomialXmpPageEmpty32}{PolynomialXmpPagePatch32}
+\pastebutton{PolynomialXmpPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{eval(p,x,w)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch33}
+\begin{paste}{PolynomialXmpPageFull33}{PolynomialXmpPageEmpty33}
+\pastebutton{PolynomialXmpPageFull33}{\hidepaste}
+\tab{5}\spadcommand{eval(p,x,1)\free{p }}
+\indentrel{3}\begin{verbatim}
+           2
+   (33)  (y  - 2y + 1)z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty33}
+\begin{paste}{PolynomialXmpPageEmpty33}{PolynomialXmpPagePatch33}
+\pastebutton{PolynomialXmpPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{eval(p,x,1)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch34}
+\begin{paste}{PolynomialXmpPageFull34}{PolynomialXmpPageEmpty34}
+\pastebutton{PolynomialXmpPageFull34}{\hidepaste}
+\tab{5}\spadcommand{eval(p,x,y**2 - 1)\free{p }}
+\indentrel{3}\begin{verbatim}
+           4     3
+   (34)  (y  - 2y  + 2y - 1)z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty34}
+\begin{paste}{PolynomialXmpPageEmpty34}{PolynomialXmpPagePatch34}
+\pastebutton{PolynomialXmpPageEmpty34}{\showpaste}
+\tab{5}\spadcommand{eval(p,x,y**2 - 1)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch35}
+\begin{paste}{PolynomialXmpPageFull35}{PolynomialXmpPageEmpty35}
+\pastebutton{PolynomialXmpPageFull35}{\hidepaste}
+\tab{5}\spadcommand{D(p,x)\free{p }}
+\indentrel{3}\begin{verbatim}
+           2
+   (35)  (y  - 2y + 1)z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty35}
+\begin{paste}{PolynomialXmpPageEmpty35}{PolynomialXmpPagePatch35}
+\pastebutton{PolynomialXmpPageEmpty35}{\showpaste}
+\tab{5}\spadcommand{D(p,x)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch36}
+\begin{paste}{PolynomialXmpPageFull36}{PolynomialXmpPageEmpty36}
+\pastebutton{PolynomialXmpPageFull36}{\hidepaste}
+\tab{5}\spadcommand{D(p,y)\free{p }}
+\indentrel{3}\begin{verbatim}
+   (36)  (2x y - 2x)z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty36}
+\begin{paste}{PolynomialXmpPageEmpty36}{PolynomialXmpPagePatch36}
+\pastebutton{PolynomialXmpPageEmpty36}{\showpaste}
+\tab{5}\spadcommand{D(p,y)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch37}
+\begin{paste}{PolynomialXmpPageFull37}{PolynomialXmpPageEmpty37}
+\pastebutton{PolynomialXmpPageFull37}{\hidepaste}
+\tab{5}\spadcommand{D(p,z)\free{p }}
+\indentrel{3}\begin{verbatim}
+            2
+   (37)  x y  - 2x y + x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty37}
+\begin{paste}{PolynomialXmpPageEmpty37}{PolynomialXmpPagePatch37}
+\pastebutton{PolynomialXmpPageEmpty37}{\showpaste}
+\tab{5}\spadcommand{D(p,z)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch38}
+\begin{paste}{PolynomialXmpPageFull38}{PolynomialXmpPageEmpty38}
+\pastebutton{PolynomialXmpPageFull38}{\hidepaste}
+\tab{5}\spadcommand{integrate(p,y)\free{p }}
+\indentrel{3}\begin{verbatim}
+          1    3      2
+   (38)  (Ä x y  - x y  + x y)z
+          3
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty38}
+\begin{paste}{PolynomialXmpPageEmpty38}{PolynomialXmpPagePatch38}
+\pastebutton{PolynomialXmpPageEmpty38}{\showpaste}
+\tab{5}\spadcommand{integrate(p,y)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch39}
+\begin{paste}{PolynomialXmpPageFull39}{PolynomialXmpPageEmpty39}
+\pastebutton{PolynomialXmpPageFull39}{\hidepaste}
+\tab{5}\spadcommand{qr := monicDivide(p,x+1,x)\free{p }\bound{qr }}
+\indentrel{3}\begin{verbatim}
+   (39)
+                2                           2
+   [quotient= (y  - 2y + 1)z,remainder= (- y  + 2y - 1)z]
+Type: Record(quotient: Polynomial Integer,remainder: Polynomial Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty39}
+\begin{paste}{PolynomialXmpPageEmpty39}{PolynomialXmpPagePatch39}
+\pastebutton{PolynomialXmpPageEmpty39}{\showpaste}
+\tab{5}\spadcommand{qr := monicDivide(p,x+1,x)\free{p }\bound{qr }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch40}
+\begin{paste}{PolynomialXmpPageFull40}{PolynomialXmpPageEmpty40}
+\pastebutton{PolynomialXmpPageFull40}{\hidepaste}
+\tab{5}\spadcommand{qr.remainder\free{qr }}
+\indentrel{3}\begin{verbatim}
+             2
+   (40)  (- y  + 2y - 1)z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty40}
+\begin{paste}{PolynomialXmpPageEmpty40}{PolynomialXmpPagePatch40}
+\pastebutton{PolynomialXmpPageEmpty40}{\showpaste}
+\tab{5}\spadcommand{qr.remainder\free{qr }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch41}
+\begin{paste}{PolynomialXmpPageFull41}{PolynomialXmpPageEmpty41}
+\pastebutton{PolynomialXmpPageFull41}{\hidepaste}
+\tab{5}\spadcommand{p - ((x+1) * qr.quotient + qr.remainder)\free{p }\free{qr }}
+\indentrel{3}\begin{verbatim}
+   (41)  0
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty41}
+\begin{paste}{PolynomialXmpPageEmpty41}{PolynomialXmpPagePatch41}
+\pastebutton{PolynomialXmpPageEmpty41}{\showpaste}
+\tab{5}\spadcommand{p - ((x+1) * qr.quotient + qr.remainder)\free{p }\free{qr }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch42}
+\begin{paste}{PolynomialXmpPageFull42}{PolynomialXmpPageEmpty42}
+\pastebutton{PolynomialXmpPageFull42}{\hidepaste}
+\tab{5}\spadcommand{p/q\free{p }\free{q }}
+\indentrel{3}\begin{verbatim}
+         (y - 1)z
+   (42)  ÄÄÄÄÄÄÄÄ
+           z + 5
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty42}
+\begin{paste}{PolynomialXmpPageEmpty42}{PolynomialXmpPagePatch42}
+\pastebutton{PolynomialXmpPageEmpty42}{\showpaste}
+\tab{5}\spadcommand{p/q\free{p }\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch43}
+\begin{paste}{PolynomialXmpPageFull43}{PolynomialXmpPageEmpty43}
+\pastebutton{PolynomialXmpPageFull43}{\hidepaste}
+\tab{5}\spadcommand{(2/3) * x**2 - y + 4/5\bound{prev1 }}
+\indentrel{3}\begin{verbatim}
+               2  2   4
+   (43)  - y + Ä x  + Ä
+               3      5
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty43}
+\begin{paste}{PolynomialXmpPageEmpty43}{PolynomialXmpPagePatch43}
+\pastebutton{PolynomialXmpPageEmpty43}{\showpaste}
+\tab{5}\spadcommand{(2/3) * x**2 - y + 4/5\bound{prev1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch44}
+\begin{paste}{PolynomialXmpPageFull44}{PolynomialXmpPageEmpty44}
+\pastebutton{PolynomialXmpPageFull44}{\hidepaste}
+\tab{5}\spadcommand{\% :: FRAC POLY INT\free{prev1 }\bound{prev2 }}
+\indentrel{3}\begin{verbatim}
+                    2
+         - 15y + 10x  + 12
+   (44)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                 15
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty44}
+\begin{paste}{PolynomialXmpPageEmpty44}{PolynomialXmpPagePatch44}
+\pastebutton{PolynomialXmpPageEmpty44}{\showpaste}
+\tab{5}\spadcommand{\% :: FRAC POLY INT\free{prev1 }\bound{prev2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch45}
+\begin{paste}{PolynomialXmpPageFull45}{PolynomialXmpPageEmpty45}
+\pastebutton{PolynomialXmpPageFull45}{\hidepaste}
+\tab{5}\spadcommand{\% :: POLY FRAC INT\free{prev2 }\bound{prev3 }}
+\indentrel{3}\begin{verbatim}
+               2  2   4
+   (45)  - y + Ä x  + Ä
+               3      5
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty45}
+\begin{paste}{PolynomialXmpPageEmpty45}{PolynomialXmpPagePatch45}
+\pastebutton{PolynomialXmpPageEmpty45}{\showpaste}
+\tab{5}\spadcommand{\% :: POLY FRAC INT\free{prev2 }\bound{prev3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPagePatch46}
+\begin{paste}{PolynomialXmpPageFull46}{PolynomialXmpPageEmpty46}
+\pastebutton{PolynomialXmpPageFull46}{\hidepaste}
+\tab{5}\spadcommand{map(numeric,\%)\free{prev3 }}
+\indentrel{3}\begin{verbatim}
+                                            2
+   (46)  - 1.0 y + 0.6666666666 6666666667 x  + 0.8
+                                 Type: Polynomial Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialXmpPageEmpty46}
+\begin{paste}{PolynomialXmpPageEmpty46}{PolynomialXmpPagePatch46}
+\pastebutton{PolynomialXmpPageEmpty46}{\showpaste}
+\tab{5}\spadcommand{map(numeric,\%)\free{prev3 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/QUAT.ht b/src/hyper/pages/QUAT.ht
new file mode 100644
index 00000000..eaae2c84
--- /dev/null
+++ b/src/hyper/pages/QUAT.ht
@@ -0,0 +1,81 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\QuaternionXmpTitle}{Quaternion}
+\newcommand{\QuaternionXmpNumber}{9.64}
+%
+% =====================================================================
+\begin{page}{QuaternionXmpPage}{9.64 Quaternion}
+% =====================================================================
+\beginscroll
+The domain constructor \spadtype{Quaternion} implements quaternions over
+commutative rings.
+For information on related topics, see
+%\menuxmpref{CliffordAlgebra}
+\downlink{`Complex'}{ComplexXmpPage}\ignore{Complex} and
+\downlink{`Octonion'}{OctonionXmpPage}\ignore{Octonion}.
+You can also issue the system command
+\spadcmd{)show Quaternion}
+to display the full list of operations defined by
+\spadtype{Quaternion}.
+
+\xtc{
+The basic operation for creating quaternions is
+\spadfunFrom{quatern}{Quaternion}.
+This is a quaternion over the rational numbers.
+}{
+\spadpaste{q := quatern(2/11,-8,3/4,1) \bound{q}}
+}
+\xtc{
+The four arguments are the real part, the \spad{i} imaginary part, the
+\spad{j} imaginary part, and the \spad{k} imaginary part, respectively.
+}{
+\spadpaste{[real q, imagI q, imagJ q, imagK q] \free{q}}
+}
+\xtc{
+Because \spad{q} is over the rationals (and nonzero), you can invert it.
+}{
+\spadpaste{inv q \free{q}}
+}
+\xtc{
+The usual arithmetic (ring) operations are available
+}{
+\spadpaste{q**6 \free{q}}
+}
+\xtc{
+}{
+\spadpaste{r := quatern(-2,3,23/9,-89); q + r \bound{r}\free{q}}
+}
+%
+\xtc{
+In general, multiplication is not commutative.
+}{
+\spadpaste{q * r - r * q\free{q r}}
+}
+\xtc{
+There are no predefined constants for the imaginary \spad{i, j},
+and \spad{k} parts, but you can easily define them.
+}{
+\spadpaste{i:=quatern(0,1,0,0); j:=quatern(0,0,1,0); k:=quatern(0,0,0,1) \bound{i j k}}
+}
+\xtc{
+These satisfy the normal identities.
+}{
+\spadpaste{[i*i, j*j, k*k, i*j, j*k, k*i, q*i] \free{i j k q}}
+}
+\xtc{
+The norm is the quaternion times its conjugate.
+}{
+\spadpaste{norm q \free{q}}
+}
+\xtc{
+}{
+\spadpaste{conjugate q \free{q} \bound{prev}}
+}
+\xtc{
+}{
+\spadpaste{q * \% \free{q prev}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/QUAT.pht b/src/hyper/pages/QUAT.pht
new file mode 100644
index 00000000..5d872fe0
--- /dev/null
+++ b/src/hyper/pages/QUAT.pht
@@ -0,0 +1,201 @@
+\begin{patch}{QuaternionXmpPagePatch1}
+\begin{paste}{QuaternionXmpPageFull1}{QuaternionXmpPageEmpty1}
+\pastebutton{QuaternionXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{q := quatern(2/11,-8,3/4,1)\bound{q }}
+\indentrel{3}\begin{verbatim}
+         2        3
+   (1)  ÄÄ - 8i + Ä j + k
+        11        4
+                      Type: Quaternion Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPageEmpty1}
+\begin{paste}{QuaternionXmpPageEmpty1}{QuaternionXmpPagePatch1}
+\pastebutton{QuaternionXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{q := quatern(2/11,-8,3/4,1)\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPagePatch2}
+\begin{paste}{QuaternionXmpPageFull2}{QuaternionXmpPageEmpty2}
+\pastebutton{QuaternionXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{[real q, imagI q, imagJ q, imagK q]\free{q }}
+\indentrel{3}\begin{verbatim}
+          2     3
+   (2)  [ÄÄ,- 8,Ä,1]
+         11     4
+                            Type: List Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPageEmpty2}
+\begin{paste}{QuaternionXmpPageEmpty2}{QuaternionXmpPagePatch2}
+\pastebutton{QuaternionXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{[real q, imagI q, imagJ q, imagK q]\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPagePatch3}
+\begin{paste}{QuaternionXmpPageFull3}{QuaternionXmpPageEmpty3}
+\pastebutton{QuaternionXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{inv q\free{q }}
+\indentrel{3}\begin{verbatim}
+          352     15488      484       1936
+   (3)  ÄÄÄÄÄÄ + ÄÄÄÄÄÄ i - ÄÄÄÄÄ j - ÄÄÄÄÄÄ k
+        126993   126993     42331     126993
+                      Type: Quaternion Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPageEmpty3}
+\begin{paste}{QuaternionXmpPageEmpty3}{QuaternionXmpPagePatch3}
+\pastebutton{QuaternionXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{inv q\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPagePatch4}
+\begin{paste}{QuaternionXmpPageFull4}{QuaternionXmpPageEmpty4}
+\pastebutton{QuaternionXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{q**6\free{q }}
+\indentrel{3}\begin{verbatim}
+   (4)
+       2029490709319345   48251690851     144755072553
+     - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ - ÄÄÄÄÄÄÄÄÄÄÄ i + ÄÄÄÄÄÄÄÄÄÄÄÄ j
+          7256313856        1288408         41229056
+   + 
+     48251690851
+     ÄÄÄÄÄÄÄÄÄÄÄ k
+       10307264
+                      Type: Quaternion Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPageEmpty4}
+\begin{paste}{QuaternionXmpPageEmpty4}{QuaternionXmpPagePatch4}
+\pastebutton{QuaternionXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{q**6\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPagePatch5}
+\begin{paste}{QuaternionXmpPageFull5}{QuaternionXmpPageEmpty5}
+\pastebutton{QuaternionXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{r := quatern(-2,3,23/9,-89); q + r\bound{r }\free{q }}
+\indentrel{3}\begin{verbatim}
+          20        119
+   (5)  - ÄÄ - 5i + ÄÄÄ j - 88k
+          11         36
+                      Type: Quaternion Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPageEmpty5}
+\begin{paste}{QuaternionXmpPageEmpty5}{QuaternionXmpPagePatch5}
+\pastebutton{QuaternionXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{r := quatern(-2,3,23/9,-89); q + r\bound{r }\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPagePatch6}
+\begin{paste}{QuaternionXmpPageFull6}{QuaternionXmpPageEmpty6}
+\pastebutton{QuaternionXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{q * r - r * q\free{q r }}
+\indentrel{3}\begin{verbatim}
+          2495             817
+   (6)  - ÄÄÄÄ i - 1418j - ÄÄÄ k
+           18               18
+                      Type: Quaternion Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPageEmpty6}
+\begin{paste}{QuaternionXmpPageEmpty6}{QuaternionXmpPagePatch6}
+\pastebutton{QuaternionXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{q * r - r * q\free{q r }}
+\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPagePatch7}
+\begin{paste}{QuaternionXmpPageFull7}{QuaternionXmpPageEmpty7}
+\pastebutton{QuaternionXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{i:=quatern(0,1,0,0); j:=quatern(0,0,1,0); k:=quatern(0,0,0,1)\bound{i j k }}
+\indentrel{3}\begin{verbatim}
+   (7)  k
+                               Type: Quaternion Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPageEmpty7}
+\begin{paste}{QuaternionXmpPageEmpty7}{QuaternionXmpPagePatch7}
+\pastebutton{QuaternionXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{i:=quatern(0,1,0,0); j:=quatern(0,0,1,0); k:=quatern(0,0,0,1)\bound{i j k }}
+\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPagePatch8}
+\begin{paste}{QuaternionXmpPageFull8}{QuaternionXmpPageEmpty8}
+\pastebutton{QuaternionXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{[i*i, j*j, k*k, i*j, j*k, k*i, q*i]\free{i j k q }}
+\indentrel{3}\begin{verbatim}
+                                2         3
+   (8)  [- 1,- 1,- 1,k,i,j,8 + ÄÄ i + j - Ä k]
+                               11         4
+                 Type: List Quaternion Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPageEmpty8}
+\begin{paste}{QuaternionXmpPageEmpty8}{QuaternionXmpPagePatch8}
+\pastebutton{QuaternionXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{[i*i, j*j, k*k, i*j, j*k, k*i, q*i]\free{i j k q }}
+\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPagePatch9}
+\begin{paste}{QuaternionXmpPageFull9}{QuaternionXmpPageEmpty9}
+\pastebutton{QuaternionXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{norm q\free{q }}
+\indentrel{3}\begin{verbatim}
+        126993
+   (9)  ÄÄÄÄÄÄ
+         1936
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPageEmpty9}
+\begin{paste}{QuaternionXmpPageEmpty9}{QuaternionXmpPagePatch9}
+\pastebutton{QuaternionXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{norm q\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPagePatch10}
+\begin{paste}{QuaternionXmpPageFull10}{QuaternionXmpPageEmpty10}
+\pastebutton{QuaternionXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{conjugate q\free{q }\bound{prev }}
+\indentrel{3}\begin{verbatim}
+          2        3
+   (10)  ÄÄ + 8i - Ä j - k
+         11        4
+                      Type: Quaternion Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPageEmpty10}
+\begin{paste}{QuaternionXmpPageEmpty10}{QuaternionXmpPagePatch10}
+\pastebutton{QuaternionXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{conjugate q\free{q }\bound{prev }}
+\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPagePatch11}
+\begin{paste}{QuaternionXmpPageFull11}{QuaternionXmpPageEmpty11}
+\pastebutton{QuaternionXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{q * \%\free{q prev }}
+\indentrel{3}\begin{verbatim}
+         126993
+   (11)  ÄÄÄÄÄÄ
+          1936
+                      Type: Quaternion Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{QuaternionXmpPageEmpty11}
+\begin{paste}{QuaternionXmpPageEmpty11}{QuaternionXmpPagePatch11}
+\pastebutton{QuaternionXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{q * \%\free{q prev }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/RADIX.ht b/src/hyper/pages/RADIX.ht
new file mode 100644
index 00000000..cffd3630
--- /dev/null
+++ b/src/hyper/pages/RADIX.ht
@@ -0,0 +1,116 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\RadixExpansionXmpTitle}{RadixExpansion}
+\newcommand{\RadixExpansionXmpNumber}{9.65}
+%
+% =====================================================================
+\begin{page}{RadixExpansionXmpPage}{9.65 RadixExpansion}
+% =====================================================================
+\beginscroll
+
+It possible to expand numbers in general bases.
+
+\labelSpace{2pc}
+\xtc{
+Here we expand \spad{111} in base \spad{5}.
+This means
+\texht{$10^2+10^1+10^0 = 4 \cdot 5^2+2 \cdot 5^1 + 5^0.$}{%
+\spad{10**2+10**1+10**0 = 4*5**2+2*5**1+5**0.}}
+}{
+\spadpaste{111::RadixExpansion(5)}
+}
+
+\xtc{
+You can expand fractions to form repeating expansions.
+}{
+\spadpaste{(5/24)::RadixExpansion(2)}
+}
+\xtc{
+}{
+\spadpaste{(5/24)::RadixExpansion(3)}
+}
+\xtc{
+}{
+\spadpaste{(5/24)::RadixExpansion(8)}
+}
+\xtc{
+}{
+\spadpaste{(5/24)::RadixExpansion(10)}
+}
+\xtc{
+For bases from 11 to 36 the letters A through Z are used.
+}{
+\spadpaste{(5/24)::RadixExpansion(12)}
+}
+\xtc{
+}{
+\spadpaste{(5/24)::RadixExpansion(16)}
+}
+\xtc{
+}{
+\spadpaste{(5/24)::RadixExpansion(36)}
+}
+\xtc{
+For bases greater than 36, the ragits are separated by blanks.
+}{
+\spadpaste{(5/24)::RadixExpansion(38)}
+}
+\xtc{
+The \spadtype{RadixExpansion} type provides operations to obtain the
+individual ragits.
+Here is a rational number in base \spad{8}.
+}{
+\spadpaste{a := (76543/210)::RadixExpansion(8) \bound{a}}
+}
+\xtc{
+The operation \spadfunFrom{wholeRagits}{RadixExpansion} returns a list of the
+ragits for the integral part of the number.
+}{
+\spadpaste{w := wholeRagits a \free{a}\bound{w}}
+}
+\xtc{
+The operations \spadfunFrom{prefixRagits}{RadixExpansion} and \spadfunFrom{cycleRagits}{RadixExpansion}
+return lists of the initial and repeating ragits in the
+fractional part of the number.
+}{
+\spadpaste{f0 := prefixRagits a \free{a}\bound{f0}}
+}
+\xtc{
+}{
+\spadpaste{f1 := cycleRagits a \free{a}\bound{f1}}
+}
+\xtc{
+You can construct any radix expansion by giving the
+whole, prefix and cycle parts.
+The declaration is necessary to let \Language{}
+know the base of the ragits.
+}{
+\spadpaste{u:RadixExpansion(8):=wholeRadix(w)+fractRadix(f0,f1) \free{w f0 f1}\bound{u}}
+}
+\xtc{
+If there is no repeating part, then the list \spad{[0]} should be used.
+}{
+\spadpaste{v: RadixExpansion(12) := fractRadix([1,2,3,11], [0]) \bound{v}}
+}
+\xtc{
+If you are not interested in the repeating nature of the expansion,
+an infinite stream of ragits can be obtained using
+\spadfunFrom{fractRagits}{RadixExpansion}.
+}{
+\spadpaste{fractRagits(u) \free{u}}
+}
+\xtc{
+Of course, it's possible to recover the fraction representation:
+}{
+\spadpaste{a :: Fraction(Integer) \free{a}}
+}
+
+\showBlurb{RadixExpansion}
+More examples of expansions are available in
+\downlink{`DecimalExpansion'}{DecimalExpansionXmpPage}\ignore{DecimalExpansion},
+\downlink{`BinaryExpansion'}{BinaryExpansionXmpPage}\ignore{BinaryExpansion}, and
+\downlink{`HexadecimalExpansion'}{HexadecimalExpansionXmpPage}\ignore{HexadecimalExpansion}.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/RADIX.pht b/src/hyper/pages/RADIX.pht
new file mode 100644
index 00000000..dc7b4dcc
--- /dev/null
+++ b/src/hyper/pages/RADIX.pht
@@ -0,0 +1,284 @@
+\begin{patch}{RadixExpansionXmpPagePatch1}
+\begin{paste}{RadixExpansionXmpPageFull1}{RadixExpansionXmpPageEmpty1}
+\pastebutton{RadixExpansionXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{111::RadixExpansion(5)}
+\indentrel{3}\begin{verbatim}
+   (1)  421
+                                 Type: RadixExpansion 5
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty1}
+\begin{paste}{RadixExpansionXmpPageEmpty1}{RadixExpansionXmpPagePatch1}
+\pastebutton{RadixExpansionXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{111::RadixExpansion(5)}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch2}
+\begin{paste}{RadixExpansionXmpPageFull2}{RadixExpansionXmpPageEmpty2}
+\pastebutton{RadixExpansionXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(2)}
+\indentrel{3}\begin{verbatim}
+             __
+   (2)  0.00110
+                                 Type: RadixExpansion 2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty2}
+\begin{paste}{RadixExpansionXmpPageEmpty2}{RadixExpansionXmpPagePatch2}
+\pastebutton{RadixExpansionXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(2)}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch3}
+\begin{paste}{RadixExpansionXmpPageFull3}{RadixExpansionXmpPageEmpty3}
+\pastebutton{RadixExpansionXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(3)}
+\indentrel{3}\begin{verbatim}
+           __
+   (3)  0.012
+                                 Type: RadixExpansion 3
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty3}
+\begin{paste}{RadixExpansionXmpPageEmpty3}{RadixExpansionXmpPagePatch3}
+\pastebutton{RadixExpansionXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(3)}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch4}
+\begin{paste}{RadixExpansionXmpPageFull4}{RadixExpansionXmpPageEmpty4}
+\pastebutton{RadixExpansionXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(8)}
+\indentrel{3}\begin{verbatim}
+           __
+   (4)  0.152
+                                 Type: RadixExpansion 8
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty4}
+\begin{paste}{RadixExpansionXmpPageEmpty4}{RadixExpansionXmpPagePatch4}
+\pastebutton{RadixExpansionXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(8)}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch5}
+\begin{paste}{RadixExpansionXmpPageFull5}{RadixExpansionXmpPageEmpty5}
+\pastebutton{RadixExpansionXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(10)}
+\indentrel{3}\begin{verbatim}
+             _
+   (5)  0.2083
+                                Type: RadixExpansion 10
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty5}
+\begin{paste}{RadixExpansionXmpPageEmpty5}{RadixExpansionXmpPagePatch5}
+\pastebutton{RadixExpansionXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(10)}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch6}
+\begin{paste}{RadixExpansionXmpPageFull6}{RadixExpansionXmpPageEmpty6}
+\pastebutton{RadixExpansionXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(12)}
+\indentrel{3}\begin{verbatim}
+   (6)  0.26
+                                Type: RadixExpansion 12
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty6}
+\begin{paste}{RadixExpansionXmpPageEmpty6}{RadixExpansionXmpPagePatch6}
+\pastebutton{RadixExpansionXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(12)}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch7}
+\begin{paste}{RadixExpansionXmpPageFull7}{RadixExpansionXmpPageEmpty7}
+\pastebutton{RadixExpansionXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(16)}
+\indentrel{3}\begin{verbatim}
+           _
+   (7)  0.35
+                                Type: RadixExpansion 16
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty7}
+\begin{paste}{RadixExpansionXmpPageEmpty7}{RadixExpansionXmpPagePatch7}
+\pastebutton{RadixExpansionXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(16)}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch8}
+\begin{paste}{RadixExpansionXmpPageFull8}{RadixExpansionXmpPageEmpty8}
+\pastebutton{RadixExpansionXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(36)}
+\indentrel{3}\begin{verbatim}
+   (8)  0.7I
+                                Type: RadixExpansion 36
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty8}
+\begin{paste}{RadixExpansionXmpPageEmpty8}{RadixExpansionXmpPagePatch8}
+\pastebutton{RadixExpansionXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(36)}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch9}
+\begin{paste}{RadixExpansionXmpPageFull9}{RadixExpansionXmpPageEmpty9}
+\pastebutton{RadixExpansionXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(38)}
+\indentrel{3}\begin{verbatim}
+                    _____
+   (9)  0 . 7 34 31 25 12
+                                Type: RadixExpansion 38
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty9}
+\begin{paste}{RadixExpansionXmpPageEmpty9}{RadixExpansionXmpPagePatch9}
+\pastebutton{RadixExpansionXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{(5/24)::RadixExpansion(38)}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch10}
+\begin{paste}{RadixExpansionXmpPageFull10}{RadixExpansionXmpPageEmpty10}
+\pastebutton{RadixExpansionXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{a := (76543/210)::RadixExpansion(8)\bound{a }}
+\indentrel{3}\begin{verbatim}
+              ____
+   (10)  554.37307
+                                 Type: RadixExpansion 8
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty10}
+\begin{paste}{RadixExpansionXmpPageEmpty10}{RadixExpansionXmpPagePatch10}
+\pastebutton{RadixExpansionXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{a := (76543/210)::RadixExpansion(8)\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch11}
+\begin{paste}{RadixExpansionXmpPageFull11}{RadixExpansionXmpPageEmpty11}
+\pastebutton{RadixExpansionXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{w := wholeRagits a\free{a }\bound{w }}
+\indentrel{3}\begin{verbatim}
+   (11)  [5,5,4]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty11}
+\begin{paste}{RadixExpansionXmpPageEmpty11}{RadixExpansionXmpPagePatch11}
+\pastebutton{RadixExpansionXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{w := wholeRagits a\free{a }\bound{w }}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch12}
+\begin{paste}{RadixExpansionXmpPageFull12}{RadixExpansionXmpPageEmpty12}
+\pastebutton{RadixExpansionXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{f0 := prefixRagits a\free{a }\bound{f0 }}
+\indentrel{3}\begin{verbatim}
+   (12)  [3]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty12}
+\begin{paste}{RadixExpansionXmpPageEmpty12}{RadixExpansionXmpPagePatch12}
+\pastebutton{RadixExpansionXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{f0 := prefixRagits a\free{a }\bound{f0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch13}
+\begin{paste}{RadixExpansionXmpPageFull13}{RadixExpansionXmpPageEmpty13}
+\pastebutton{RadixExpansionXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{f1 := cycleRagits a\free{a }\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+   (13)  [7,3,0,7]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty13}
+\begin{paste}{RadixExpansionXmpPageEmpty13}{RadixExpansionXmpPagePatch13}
+\pastebutton{RadixExpansionXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{f1 := cycleRagits a\free{a }\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch14}
+\begin{paste}{RadixExpansionXmpPageFull14}{RadixExpansionXmpPageEmpty14}
+\pastebutton{RadixExpansionXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{u:RadixExpansion(8):=wholeRadix(w)+fractRadix(f0,f1)\free{w f0 f1 }\bound{u }}
+\indentrel{3}\begin{verbatim}
+              ____
+   (14)  554.37307
+                                 Type: RadixExpansion 8
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty14}
+\begin{paste}{RadixExpansionXmpPageEmpty14}{RadixExpansionXmpPagePatch14}
+\pastebutton{RadixExpansionXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{u:RadixExpansion(8):=wholeRadix(w)+fractRadix(f0,f1)\free{w f0 f1 }\bound{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch15}
+\begin{paste}{RadixExpansionXmpPageFull15}{RadixExpansionXmpPageEmpty15}
+\pastebutton{RadixExpansionXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{v: RadixExpansion(12) := fractRadix([1,2,3,11], [0])\bound{v }}
+\indentrel{3}\begin{verbatim}
+               _
+   (15)  0.123B0
+                                Type: RadixExpansion 12
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty15}
+\begin{paste}{RadixExpansionXmpPageEmpty15}{RadixExpansionXmpPagePatch15}
+\pastebutton{RadixExpansionXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{v: RadixExpansion(12) := fractRadix([1,2,3,11], [0])\bound{v }}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch16}
+\begin{paste}{RadixExpansionXmpPageFull16}{RadixExpansionXmpPageEmpty16}
+\pastebutton{RadixExpansionXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{fractRagits(u)\free{u }}
+\indentrel{3}\begin{verbatim}
+              _______
+   (16)  [3,7,3,0,7,7]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty16}
+\begin{paste}{RadixExpansionXmpPageEmpty16}{RadixExpansionXmpPagePatch16}
+\pastebutton{RadixExpansionXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{fractRagits(u)\free{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPagePatch17}
+\begin{paste}{RadixExpansionXmpPageFull17}{RadixExpansionXmpPageEmpty17}
+\pastebutton{RadixExpansionXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{a :: Fraction(Integer)\free{a }}
+\indentrel{3}\begin{verbatim}
+         76543
+   (17)  ÄÄÄÄÄ
+          210
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RadixExpansionXmpPageEmpty17}
+\begin{paste}{RadixExpansionXmpPageEmpty17}{RadixExpansionXmpPagePatch17}
+\pastebutton{RadixExpansionXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{a :: Fraction(Integer)\free{a }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/RECLOS.ht b/src/hyper/pages/RECLOS.ht
new file mode 100644
index 00000000..02f1145e
--- /dev/null
+++ b/src/hyper/pages/RECLOS.ht
@@ -0,0 +1,382 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\RealClosureXmpTitle}{RealClosure}
+\newcommand{\RealClosureXmpNumber}{9.66}
+%
+% =====================================================================
+\begin{page}{RealClosureXmpPage}{9.66 RealClosure}
+% =====================================================================
+\beginscroll
+
+
+The Real Closure 1.0 package provided by Renaud Rioboo (Renaud.Rioboo@lip6.fr) consists of different
+packages, categories and domains :
+
+\begin{items}
+\item The package \axiomType{RealPolynomialUtilitiesPackage} which needs a \axiomType{Field} {\em F} and a
+\axiomType{UnivariatePolynomialCategory} domain with coefficients in {\em F}. It computes some
+simple functions such as Sturm and Sylvester sequences 
+(\axiomOpFrom{sturmSequence}{RealPolynomialUtilitiesPackage}, 
+\axiomOpFrom{sylvesterSequence}{RealPolynomialUtilitiesPackage}).
+
+\item The category \axiomType{RealRootCharacterizationCategory} provides abstract
+functions to work with "real roots" of univariate polynomials. These
+resemble variables with some functionality needed to compute important
+operations.
+
+\item The category \axiomType{RealClosedField} provides common operations available over
+real closed fiels. These include finding all the roots of a univariate
+polynomial, taking square (and higher) roots, ...
+
+\item The domain \axiomType{RightOpenIntervalRootCharacterization} is the main code that
+provides the functionality of \axiomType{RealRootCharacterizationCategory} for the case
+of archimedean fields. Abstract roots are encoded with a left closed right
+open interval containing the root together with a defining polynomial for the
+root.
+
+\item  The \axiomType{RealClosure} domain is the end-user code. It provides usual arithmetic
+with real algebraic numbers, along with the functionality of a real closed
+field. It also provides functions to approximate a real algebraic number by an
+element of the base field. This approximation may either be absolute
+(\axiomOpFrom{approximate}{RealClosure}) or relative (\axiomOpFrom{relativeApprox}{RealClosure}).
+
+
+\end{items}
+
+\centerline{CAVEATS}
+
+Since real algebraic expressions are stored as depending on "real roots" which
+are managed like variables, there is an ordering on these. This ordering is
+dynamical in the sense that any new algebraic takes precedence over older
+ones. In particular every creation function raises a new "real root". This has
+the effect that when you type something like \spad{sqrt(2) + sqrt(2)} you have two
+new variables which happen to be equal. To avoid this name the expression such
+as in \spad{s2 := sqrt(2) ; s2 + s2}
+
+Also note that computing times depend strongly on the ordering you implicitly
+provide. Please provide algebraics in the order which seems most natural to you.
+
+\centerline{LIMITATIONS}
+
+This packages uses
+algorithms which are published in [1] and [2] which are based on field
+arithmetics, in particular for polynomial gcd related algorithms. This can be
+quite slow for high degree polynomials and subresultants methods usually work
+best. Beta versions of the package try to use these techniques in a better
+way and work significantly faster. These are mostly based on unpublished
+algorithms and cannot be distributed. Please contact the author if you have a
+particular problem to solve or want to use these versions.
+
+Be aware that approximations behave as post-processing and that all
+computations are done excatly. They can thus be quite time consuming when
+depending on several "real roots".
+
+\centerline{REFERENCES}
+
+
+[1]  R. Rioboo : Real Algebraic Closure of an ordered Field : Implementation 
+     in Axiom. 
+     In proceedings of the ISSAC'92 Conference, Berkeley 1992 pp. 206-215.
+
+[2]  Z. Ligatsikas, R. Rioboo, M. F. Roy : Generic computation of the real
+     closure of an ordered field.
+     In Mathematics and Computers in Simulation Volume 42, Issue 4-6,
+     November 1996.
+
+\centerline{EXAMPLES}
+\xtc{
+We shall work with the real closure of the ordered field of 
+rational numbers.
+}{
+\spadpaste{Ran := RECLOS(FRAC INT) \bound{Ran}}
+}
+\xtc{
+Some simple signs for square roots, these correspond to an extension
+of degree 16 of the rational numbers. Examples provided by J. Abbot.
+}{
+\spadpaste{fourSquares(a:Ran,b:Ran,c:Ran,d:Ran):Ran == sqrt(a)+sqrt(b) - sqrt(c)-sqrt(d) \free{Ran} \bound{fs}}
+}
+\xtc{
+These produce values very close to zero.
+}{
+\spadpaste{squareDiff1 := fourSquares(73,548,60,586) \free{fs}\bound{sd1}}
+}
+\xtc{
+}{
+\spadpaste{recip(squareDiff1)\free{sd1}}
+}
+\xtc{
+}{
+\spadpaste{sign(squareDiff1)\free{sd1}}
+}
+\xtc{
+}{
+\spadpaste{squareDiff2 := fourSquares(165,778,86,990) \free{fs}\bound{sd2}}
+}
+\xtc{
+}{
+\spadpaste{recip(squareDiff2)\free{sd2}}
+}
+\xtc{
+}{
+\spadpaste{sign(squareDiff2)\free{sd2}}
+}
+\xtc{
+}{
+\spadpaste{squareDiff3 := fourSquares(217,708,226,692) \free{fs}\bound{sd3}}
+}
+\xtc{
+}{
+\spadpaste{recip(squareDiff3)\free{sd3}}
+}
+\xtc{
+}{
+\spadpaste{sign(squareDiff3)\free{sd3}}
+}
+\xtc{
+}{
+\spadpaste{squareDiff4 := fourSquares(155,836,162,820)  \free{fs}\bound{sd4}}
+}
+\xtc{
+}{
+\spadpaste{recip(squareDiff4)\free{sd4}}
+}
+\xtc{
+}{
+\spadpaste{sign(squareDiff4)\free{sd4}}
+}
+\xtc{
+}{
+\spadpaste{squareDiff5 := fourSquares(591,772,552,818) \free{fs}\bound{sd5}}
+}
+\xtc{
+}{
+\spadpaste{recip(squareDiff5)\free{sd5}}
+}
+\xtc{
+}{
+\spadpaste{sign(squareDiff5)\free{sd5}}
+}
+\xtc{
+}{
+\spadpaste{squareDiff6 := fourSquares(434,1053,412,1088) \free{fs}\bound{sd6}}
+}
+\xtc{
+}{
+\spadpaste{recip(squareDiff6)\free{sd6}}
+}
+\xtc{
+}{
+\spadpaste{sign(squareDiff6)\free{sd6}}
+}
+\xtc{
+}{
+\spadpaste{squareDiff7 := fourSquares(514,1049,446,1152) \free{fs}\bound{sd7}}
+}
+\xtc{
+}{
+\spadpaste{recip(squareDiff7)\free{sd7}}
+}
+\xtc{
+}{
+\spadpaste{sign(squareDiff7)\free{sd7}}
+}
+\xtc{
+}{
+\spadpaste{squareDiff8 := fourSquares(190,1751,208,1698) \free{fs}\bound{sd8}}
+}
+\xtc{
+}{
+\spadpaste{recip(squareDiff8)\free{sd8}}
+}
+\xtc{
+}{
+\spadpaste{sign(squareDiff8)\free{sd8}}
+}
+\xtc{
+This should give three digits of precision
+}{
+\spadpaste{relativeApprox(squareDiff8,10**(-3))::Float \free{sd8}}
+}
+\xtc{
+The sum of these 4 roots is 0
+}{
+\spadpaste{l := allRootsOf((x**2-2)**2-2)$Ran \free{Ran} \bound{l}}
+}
+\xtc{
+Check that they are all roots of the same polynomial
+}{
+\spadpaste{removeDuplicates map(mainDefiningPolynomial,l) \free{l}}
+}
+\xtc{
+We can see at a glance that they are separate roots
+}{
+\spadpaste{map(mainCharacterization,l) \free{l}}
+}
+\xtc{
+Check the sum and product
+}{
+\spadpaste{[reduce(+,l),reduce(*,l)-2] \free{l}}
+}
+\xtc{
+A more complicated test that involve an extension of degree 256.
+This is a way of checking nested radical identities.
+}{
+\spadpaste{(s2, s5, s10) := (sqrt(2)$Ran, sqrt(5)$Ran, sqrt(10)$Ran) \free{Ran}\bound{s2}\bound{s5}\bound{s10}}
+}
+\xtc{
+}{
+\spadpaste{eq1:=sqrt(s10+3)*sqrt(s5+2) - sqrt(s10-3)*sqrt(s5-2) = sqrt(10*s2+10) \free{s2}\free{s5}\free{s10}\bound{eq1}}
+}
+\xtc{
+}{
+\spadpaste{eq1::Boolean \free{eq1}}
+}
+\xtc{
+}{
+\spadpaste{eq2:=sqrt(s5+2)*sqrt(s2+1) - sqrt(s5-2)*sqrt(s2-1) = sqrt(2*s10+2)\free{s2}\free{s5}\free{s10}\bound{eq2}}
+}
+\xtc{
+}{
+\spadpaste{eq2::Boolean \free{eq2}}
+}
+\xtc{
+Some more examples from J. M. Arnaudies
+}{
+\spadpaste{s3 := sqrt(3)$Ran \free{Ran}\bound{s3}}
+}
+\xtc{
+}{
+\spadpaste{s7:= sqrt(7)$Ran \free{Ran}\bound{s7}}
+}
+\xtc{
+}{
+\spadpaste{e1 := sqrt(2*s7-3*s3,3) \free{s7} \free{s3} \bound{e1}}
+}
+\xtc{
+}{
+\spadpaste{e2 := sqrt(2*s7+3*s3,3) \free{s7} \free{s3} \bound{e2}}
+}
+\xtc{
+This should be null
+}{
+\spadpaste{e2-e1-s3 \free{e2} \free{e1} \free{s3}}
+}
+\xtc{
+A quartic polynomial
+}{
+\spadpaste{pol : UP(x,Ran) := x**4+(7/3)*x**2+30*x-(100/3) \free{Ran} \bound{pol}}
+}
+\xtc{
+Add some cubic roots
+}{
+\spadpaste{r1 := sqrt(7633)$Ran \free{Ran}\bound{r1}}
+}
+\xtc{
+}{
+\spadpaste{alpha := sqrt(5*r1-436,3)/3 \free{r1} \bound{alpha}}
+}
+\xtc{
+}{
+\spadpaste{beta := -sqrt(5*r1+436,3)/3 \free{r1} \bound{beta}}
+}
+\xtc{
+this should be null
+}{
+\spadpaste{pol.(alpha+beta-1/3) \free{pol} \free{alpha} \free{beta}}
+}
+\xtc{
+A quintic polynomial
+}{
+\spadpaste{qol : UP(x,Ran) := x**5+10*x**3+20*x+22 \free{Ran}\bound{qol}}
+}
+\xtc{
+Add some cubic roots
+}{
+\spadpaste{r2 := sqrt(153)$Ran \free{Ran}\bound{r2}}
+}
+\xtc{
+}{
+\spadpaste{alpha2 := sqrt(r2-11,5) \free{r2}\bound{alpha2}}
+}
+\xtc{
+}{
+\spadpaste{beta2 := -sqrt(r2+11,5) \free{r2}\bound{beta2}}
+}
+\xtc{
+this should be null
+}{
+\spadpaste{qol(alpha2+beta2) \free{qol}\free{alpha2}\free{beta2}}
+}
+\xtc{
+Finally, some examples from the book Computer Algebra by 
+Davenport, Siret and Tournier (page 77).
+The last one is due to Ramanujan.
+}{
+\spadpaste{dst1:=sqrt(9+4*s2)=1+2*s2 \free{s2}\bound{dst1}}
+}
+\xtc{
+}{
+\spadpaste{dst1::Boolean \free{dst1}}
+}
+\xtc{
+}{
+\spadpaste{s6:Ran:=sqrt 6 \free{Ran}\bound{s6}}
+}
+\xtc{
+}{
+\spadpaste{dst2:=sqrt(5+2*s6)+sqrt(5-2*s6) = 2*s3 \free{s6}\free{s3}\bound{dst2}}
+}
+\xtc{
+}{
+\spadpaste{dst2::Boolean \free{dst2}}
+}
+\xtc{
+}{
+\spadpaste{s29:Ran:=sqrt 29 \free{Ran}\bound{s29}}
+}
+\xtc{
+}{
+\spadpaste{dst4:=sqrt(16-2*s29+2*sqrt(55-10*s29)) = sqrt(22+2*s5)-sqrt(11+2*s29)+s5 \free{s29}\free{s5}\bound{dst4}}
+}
+\xtc{
+}{
+\spadpaste{dst4::Boolean \free{dst4}}
+}
+\xtc{
+}{
+\spadpaste{dst6:=sqrt((112+70*s2)+(46+34*s2)*s5) = (5+4*s2)+(3+s2)*s5 \free{s2}\free{s5}\bound{dst6}}
+}
+\xtc{
+}{
+\spadpaste{dst6::Boolean \free{dst6}}
+}
+\xtc{
+}{
+\spadpaste{f3:Ran:=sqrt(3,5) \free{Ran}\bound{f3}}
+}
+\xtc{
+}{
+\spadpaste{f25:Ran:=sqrt(1/25,5) \free{Ran}\bound{f25}}
+}
+\xtc{
+}{
+\spadpaste{f32:Ran:=sqrt(32/5,5) \free{Ran}\bound{f32}}
+}
+\xtc{
+}{
+\spadpaste{f27:Ran:=sqrt(27/5,5) \free{Ran}\bound{f27}}
+}
+\xtc{
+}{
+\spadpaste{dst5:=sqrt((f32-f27,3)) = f25*(1+f3-f3**2)\free{f32}\free{f27}\free{f25}\free{f3}\bound{dst5}}
+}
+\xtc{
+}{
+\spadpaste{dst5::Boolean \free{dst5}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/RECLOS.pht b/src/hyper/pages/RECLOS.pht
new file mode 100644
index 00000000..2e3ed9cc
--- /dev/null
+++ b/src/hyper/pages/RECLOS.pht
@@ -0,0 +1,1258 @@
+\begin{patch}{RealClosureXmpPagePatch1}
+\begin{paste}{RealClosureXmpPageFull1}{RealClosureXmpPageEmpty1}
+\pastebutton{RealClosureXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{Ran := RECLOS(FRAC INT)\bound{Ran }}
+\indentrel{3}\begin{verbatim}
+   (1)  RealClosure Fraction Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty1}
+\begin{paste}{RealClosureXmpPageEmpty1}{RealClosureXmpPagePatch1}
+\pastebutton{RealClosureXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{Ran := RECLOS(FRAC INT)\bound{Ran }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch2}
+\begin{paste}{RealClosureXmpPageFull2}{RealClosureXmpPageEmpty2}
+\pastebutton{RealClosureXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{fourSquares(a:Ran,b:Ran,c:Ran,d:Ran):Ran == sqrt(a)+sqrt(b) - sqrt(c)-sqrt(d)\free{Ran }\bound{fs }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty2}
+\begin{paste}{RealClosureXmpPageEmpty2}{RealClosureXmpPagePatch2}
+\pastebutton{RealClosureXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{fourSquares(a:Ran,b:Ran,c:Ran,d:Ran):Ran == sqrt(a)+sqrt(b) - sqrt(c)-sqrt(d)\free{Ran }\bound{fs }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch3}
+\begin{paste}{RealClosureXmpPageFull3}{RealClosureXmpPageEmpty3}
+\pastebutton{RealClosureXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{squareDiff1 := fourSquares(73,548,60,586)\free{fs }\bound{sd1 }}
+\indentrel{3}\begin{verbatim}
+           ÚÄÄÄ¿    ÚÄÄ¿    ÚÄÄÄ¿    ÚÄÄ¿
+   (3)  - \³586  - \³60  + \³548  + \³73
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty3}
+\begin{paste}{RealClosureXmpPageEmpty3}{RealClosureXmpPagePatch3}
+\pastebutton{RealClosureXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{squareDiff1 := fourSquares(73,548,60,586)\free{fs }\bound{sd1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch4}
+\begin{paste}{RealClosureXmpPageFull4}{RealClosureXmpPageEmpty4}
+\pastebutton{RealClosureXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{recip(squareDiff1)\free{sd1 }}
+\indentrel{3}\begin{verbatim}
+   (4)
+                ÚÄÄÄ¿          ÚÄÄ¿  ÚÄÄ¿
+         (54602\³548  + 149602\³73 )\³60
+       + 
+               ÚÄÄ¿ ÚÄÄÄ¿
+         49502\³73 \³548  + 9900895
+    *
+        ÚÄÄÄ¿
+       \³586
+   + 
+             ÚÄÄ¿ ÚÄÄÄ¿             ÚÄÄ¿            ÚÄÄÄ¿
+     (154702\³73 \³548  + 30941947)\³60  + 10238421\³548
+   + 
+              ÚÄÄ¿
+     28051871\³73
+          Type: Union(RealClosure Fraction Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty4}
+\begin{paste}{RealClosureXmpPageEmpty4}{RealClosureXmpPagePatch4}
+\pastebutton{RealClosureXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{recip(squareDiff1)\free{sd1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch5}
+\begin{paste}{RealClosureXmpPageFull5}{RealClosureXmpPageEmpty5}
+\pastebutton{RealClosureXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{sign(squareDiff1)\free{sd1 }}
+\indentrel{3}\begin{verbatim}
+   (5)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty5}
+\begin{paste}{RealClosureXmpPageEmpty5}{RealClosureXmpPagePatch5}
+\pastebutton{RealClosureXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{sign(squareDiff1)\free{sd1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch6}
+\begin{paste}{RealClosureXmpPageFull6}{RealClosureXmpPageEmpty6}
+\pastebutton{RealClosureXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{squareDiff2 := fourSquares(165,778,86,990)\free{fs }\bound{sd2 }}
+\indentrel{3}\begin{verbatim}
+           ÚÄÄÄ¿    ÚÄÄ¿    ÚÄÄÄ¿    ÚÄÄÄ¿
+   (6)  - \³990  - \³86  + \³778  + \³165
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty6}
+\begin{paste}{RealClosureXmpPageEmpty6}{RealClosureXmpPagePatch6}
+\pastebutton{RealClosureXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{squareDiff2 := fourSquares(165,778,86,990)\free{fs }\bound{sd2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch7}
+\begin{paste}{RealClosureXmpPageFull7}{RealClosureXmpPageEmpty7}
+\pastebutton{RealClosureXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{recip(squareDiff2)\free{sd2 }}
+\indentrel{3}\begin{verbatim}
+   (7)
+                 ÚÄÄÄ¿           ÚÄÄÄ¿  ÚÄÄ¿
+         (556778\³778  + 1209010\³165 )\³86
+       + 
+                ÚÄÄÄ¿ ÚÄÄÄ¿
+         401966\³165 \³778  + 144019431
+    *
+        ÚÄÄÄ¿
+       \³990
+   + 
+              ÚÄÄÄ¿ ÚÄÄÄ¿              ÚÄÄ¿
+     (1363822\³165 \³778  + 488640503)\³86
+   + 
+               ÚÄÄÄ¿             ÚÄÄÄ¿
+     162460913\³778  + 352774119\³165
+          Type: Union(RealClosure Fraction Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty7}
+\begin{paste}{RealClosureXmpPageEmpty7}{RealClosureXmpPagePatch7}
+\pastebutton{RealClosureXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{recip(squareDiff2)\free{sd2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch8}
+\begin{paste}{RealClosureXmpPageFull8}{RealClosureXmpPageEmpty8}
+\pastebutton{RealClosureXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{sign(squareDiff2)\free{sd2 }}
+\indentrel{3}\begin{verbatim}
+   (8)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty8}
+\begin{paste}{RealClosureXmpPageEmpty8}{RealClosureXmpPagePatch8}
+\pastebutton{RealClosureXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{sign(squareDiff2)\free{sd2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch9}
+\begin{paste}{RealClosureXmpPageFull9}{RealClosureXmpPageEmpty9}
+\pastebutton{RealClosureXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{squareDiff3 := fourSquares(217,708,226,692)\free{fs }\bound{sd3 }}
+\indentrel{3}\begin{verbatim}
+           ÚÄÄÄ¿    ÚÄÄÄ¿    ÚÄÄÄ¿    ÚÄÄÄ¿
+   (9)  - \³692  - \³226  + \³708  + \³217
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty9}
+\begin{paste}{RealClosureXmpPageEmpty9}{RealClosureXmpPagePatch9}
+\pastebutton{RealClosureXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{squareDiff3 := fourSquares(217,708,226,692)\free{fs }\bound{sd3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch10}
+\begin{paste}{RealClosureXmpPageFull10}{RealClosureXmpPageEmpty10}
+\pastebutton{RealClosureXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{recip(squareDiff3)\free{sd3 }}
+\indentrel{3}\begin{verbatim}
+   (10)
+                  ÚÄÄÄ¿         ÚÄÄÄ¿  ÚÄÄÄ¿
+         (- 34102\³708  - 61598\³217 )\³226
+       + 
+                 ÚÄÄÄ¿ ÚÄÄÄ¿
+         - 34802\³217 \³708  - 13641141
+    *
+        ÚÄÄÄ¿
+       \³692
+   + 
+              ÚÄÄÄ¿ ÚÄÄÄ¿             ÚÄÄÄ¿
+     (- 60898\³217 \³708  - 23869841)\³226
+   + 
+                ÚÄÄÄ¿            ÚÄÄÄ¿
+     - 13486123\³708  - 24359809\³217
+          Type: Union(RealClosure Fraction Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty10}
+\begin{paste}{RealClosureXmpPageEmpty10}{RealClosureXmpPagePatch10}
+\pastebutton{RealClosureXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{recip(squareDiff3)\free{sd3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch11}
+\begin{paste}{RealClosureXmpPageFull11}{RealClosureXmpPageEmpty11}
+\pastebutton{RealClosureXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{sign(squareDiff3)\free{sd3 }}
+\indentrel{3}\begin{verbatim}
+   (11)  - 1
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty11}
+\begin{paste}{RealClosureXmpPageEmpty11}{RealClosureXmpPagePatch11}
+\pastebutton{RealClosureXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{sign(squareDiff3)\free{sd3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch12}
+\begin{paste}{RealClosureXmpPageFull12}{RealClosureXmpPageEmpty12}
+\pastebutton{RealClosureXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{squareDiff4 := fourSquares(155,836,162,820)\free{fs }\bound{sd4 }}
+\indentrel{3}\begin{verbatim}
+            ÚÄÄÄ¿    ÚÄÄÄ¿    ÚÄÄÄ¿    ÚÄÄÄ¿
+   (12)  - \³820  - \³162  + \³836  + \³155
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty12}
+\begin{paste}{RealClosureXmpPageEmpty12}{RealClosureXmpPagePatch12}
+\pastebutton{RealClosureXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{squareDiff4 := fourSquares(155,836,162,820)\free{fs }\bound{sd4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch13}
+\begin{paste}{RealClosureXmpPageFull13}{RealClosureXmpPageEmpty13}
+\pastebutton{RealClosureXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{recip(squareDiff4)\free{sd4 }}
+\indentrel{3}\begin{verbatim}
+   (13)
+                  ÚÄÄÄ¿         ÚÄÄÄ¿  ÚÄÄÄ¿
+         (- 37078\³836  - 86110\³155 )\³162
+       + 
+                 ÚÄÄÄ¿ ÚÄÄÄ¿
+         - 37906\³155 \³836  - 13645107
+    *
+        ÚÄÄÄ¿
+       \³820
+   + 
+              ÚÄÄÄ¿ ÚÄÄÄ¿             ÚÄÄÄ¿
+     (- 85282\³155 \³836  - 30699151)\³162
+   + 
+                ÚÄÄÄ¿            ÚÄÄÄ¿
+     - 13513901\³836  - 31384703\³155
+          Type: Union(RealClosure Fraction Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty13}
+\begin{paste}{RealClosureXmpPageEmpty13}{RealClosureXmpPagePatch13}
+\pastebutton{RealClosureXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{recip(squareDiff4)\free{sd4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch14}
+\begin{paste}{RealClosureXmpPageFull14}{RealClosureXmpPageEmpty14}
+\pastebutton{RealClosureXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{sign(squareDiff4)\free{sd4 }}
+\indentrel{3}\begin{verbatim}
+   (14)  - 1
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty14}
+\begin{paste}{RealClosureXmpPageEmpty14}{RealClosureXmpPagePatch14}
+\pastebutton{RealClosureXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{sign(squareDiff4)\free{sd4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch15}
+\begin{paste}{RealClosureXmpPageFull15}{RealClosureXmpPageEmpty15}
+\pastebutton{RealClosureXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{squareDiff5 := fourSquares(591,772,552,818)\free{fs }\bound{sd5 }}
+\indentrel{3}\begin{verbatim}
+            ÚÄÄÄ¿    ÚÄÄÄ¿    ÚÄÄÄ¿    ÚÄÄÄ¿
+   (15)  - \³818  - \³552  + \³772  + \³591
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty15}
+\begin{paste}{RealClosureXmpPageEmpty15}{RealClosureXmpPagePatch15}
+\pastebutton{RealClosureXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{squareDiff5 := fourSquares(591,772,552,818)\free{fs }\bound{sd5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch16}
+\begin{paste}{RealClosureXmpPageFull16}{RealClosureXmpPageEmpty16}
+\pastebutton{RealClosureXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{recip(squareDiff5)\free{sd5 }}
+\indentrel{3}\begin{verbatim}
+   (16)
+                ÚÄÄÄ¿         ÚÄÄÄ¿  ÚÄÄÄ¿
+         (70922\³772  + 81058\³591 )\³552
+       + 
+               ÚÄÄÄ¿ ÚÄÄÄ¿
+         68542\³591 \³772  + 46297673
+    *
+        ÚÄÄÄ¿
+       \³818
+   + 
+            ÚÄÄÄ¿ ÚÄÄÄ¿             ÚÄÄÄ¿            ÚÄÄÄ¿
+     (83438\³591 \³772  + 56359389)\³552  + 47657051\³772
+   + 
+              ÚÄÄÄ¿
+     54468081\³591
+          Type: Union(RealClosure Fraction Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty16}
+\begin{paste}{RealClosureXmpPageEmpty16}{RealClosureXmpPagePatch16}
+\pastebutton{RealClosureXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{recip(squareDiff5)\free{sd5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch17}
+\begin{paste}{RealClosureXmpPageFull17}{RealClosureXmpPageEmpty17}
+\pastebutton{RealClosureXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{sign(squareDiff5)\free{sd5 }}
+\indentrel{3}\begin{verbatim}
+   (17)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty17}
+\begin{paste}{RealClosureXmpPageEmpty17}{RealClosureXmpPagePatch17}
+\pastebutton{RealClosureXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{sign(squareDiff5)\free{sd5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch18}
+\begin{paste}{RealClosureXmpPageFull18}{RealClosureXmpPageEmpty18}
+\pastebutton{RealClosureXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{squareDiff6 := fourSquares(434,1053,412,1088)\free{fs }\bound{sd6 }}
+\indentrel{3}\begin{verbatim}
+            ÚÄÄÄÄ¿    ÚÄÄÄ¿    ÚÄÄÄÄ¿    ÚÄÄÄ¿
+   (18)  - \³1088  - \³412  + \³1053  + \³434
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty18}
+\begin{paste}{RealClosureXmpPageEmpty18}{RealClosureXmpPagePatch18}
+\pastebutton{RealClosureXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{squareDiff6 := fourSquares(434,1053,412,1088)\free{fs }\bound{sd6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch19}
+\begin{paste}{RealClosureXmpPageFull19}{RealClosureXmpPageEmpty19}
+\pastebutton{RealClosureXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{recip(squareDiff6)\free{sd6 }}
+\indentrel{3}\begin{verbatim}
+   (19)
+                 ÚÄÄÄÄ¿          ÚÄÄÄ¿  ÚÄÄÄ¿
+         (115442\³1053  + 179818\³434 )\³412
+       + 
+                ÚÄÄÄ¿ ÚÄÄÄÄ¿
+         112478\³434 \³1053  + 76037291
+    *
+        ÚÄÄÄÄ¿
+       \³1088
+   + 
+             ÚÄÄÄ¿ ÚÄÄÄÄ¿              ÚÄÄÄ¿
+     (182782\³434 \³1053  + 123564147)\³412
+   + 
+              ÚÄÄÄÄ¿             ÚÄÄÄ¿
+     77290639\³1053  + 120391609\³434
+          Type: Union(RealClosure Fraction Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty19}
+\begin{paste}{RealClosureXmpPageEmpty19}{RealClosureXmpPagePatch19}
+\pastebutton{RealClosureXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{recip(squareDiff6)\free{sd6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch20}
+\begin{paste}{RealClosureXmpPageFull20}{RealClosureXmpPageEmpty20}
+\pastebutton{RealClosureXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{sign(squareDiff6)\free{sd6 }}
+\indentrel{3}\begin{verbatim}
+   (20)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty20}
+\begin{paste}{RealClosureXmpPageEmpty20}{RealClosureXmpPagePatch20}
+\pastebutton{RealClosureXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{sign(squareDiff6)\free{sd6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch21}
+\begin{paste}{RealClosureXmpPageFull21}{RealClosureXmpPageEmpty21}
+\pastebutton{RealClosureXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{squareDiff7 := fourSquares(514,1049,446,1152)\free{fs }\bound{sd7 }}
+\indentrel{3}\begin{verbatim}
+            ÚÄÄÄÄ¿    ÚÄÄÄ¿    ÚÄÄÄÄ¿    ÚÄÄÄ¿
+   (21)  - \³1152  - \³446  + \³1049  + \³514
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty21}
+\begin{paste}{RealClosureXmpPageEmpty21}{RealClosureXmpPagePatch21}
+\pastebutton{RealClosureXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{squareDiff7 := fourSquares(514,1049,446,1152)\free{fs }\bound{sd7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch22}
+\begin{paste}{RealClosureXmpPageFull22}{RealClosureXmpPageEmpty22}
+\pastebutton{RealClosureXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{recip(squareDiff7)\free{sd7 }}
+\indentrel{3}\begin{verbatim}
+   (22)
+                 ÚÄÄÄÄ¿          ÚÄÄÄ¿  ÚÄÄÄ¿
+         (349522\³1049  + 499322\³514 )\³446
+       + 
+                ÚÄÄÄ¿ ÚÄÄÄÄ¿
+         325582\³514 \³1049  + 239072537
+    *
+        ÚÄÄÄÄ¿
+       \³1152
+   + 
+             ÚÄÄÄ¿ ÚÄÄÄÄ¿              ÚÄÄÄ¿
+     (523262\³514 \³1049  + 384227549)\³446
+   + 
+               ÚÄÄÄÄ¿             ÚÄÄÄ¿
+     250534873\³1049  + 357910443\³514
+          Type: Union(RealClosure Fraction Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty22}
+\begin{paste}{RealClosureXmpPageEmpty22}{RealClosureXmpPagePatch22}
+\pastebutton{RealClosureXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{recip(squareDiff7)\free{sd7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch23}
+\begin{paste}{RealClosureXmpPageFull23}{RealClosureXmpPageEmpty23}
+\pastebutton{RealClosureXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{sign(squareDiff7)\free{sd7 }}
+\indentrel{3}\begin{verbatim}
+   (23)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty23}
+\begin{paste}{RealClosureXmpPageEmpty23}{RealClosureXmpPagePatch23}
+\pastebutton{RealClosureXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{sign(squareDiff7)\free{sd7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch24}
+\begin{paste}{RealClosureXmpPageFull24}{RealClosureXmpPageEmpty24}
+\pastebutton{RealClosureXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{squareDiff8 := fourSquares(190,1751,208,1698)\free{fs }\bound{sd8 }}
+\indentrel{3}\begin{verbatim}
+            ÚÄÄÄÄ¿    ÚÄÄÄ¿    ÚÄÄÄÄ¿    ÚÄÄÄ¿
+   (24)  - \³1698  - \³208  + \³1751  + \³190
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty24}
+\begin{paste}{RealClosureXmpPageEmpty24}{RealClosureXmpPagePatch24}
+\pastebutton{RealClosureXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{squareDiff8 := fourSquares(190,1751,208,1698)\free{fs }\bound{sd8 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch25}
+\begin{paste}{RealClosureXmpPageFull25}{RealClosureXmpPageEmpty25}
+\pastebutton{RealClosureXmpPageFull25}{\hidepaste}
+\tab{5}\spadcommand{recip(squareDiff8)\free{sd8 }}
+\indentrel{3}\begin{verbatim}
+   (25)
+                   ÚÄÄÄÄ¿          ÚÄÄÄ¿  ÚÄÄÄ¿
+         (- 214702\³1751  - 651782\³190 )\³208
+       + 
+                  ÚÄÄÄ¿ ÚÄÄÄÄ¿
+         - 224642\³190 \³1751  - 129571901
+    *
+        ÚÄÄÄÄ¿
+       \³1698
+   + 
+               ÚÄÄÄ¿ ÚÄÄÄÄ¿              ÚÄÄÄ¿
+     (- 641842\³190 \³1751  - 370209881)\³208
+   + 
+                 ÚÄÄÄÄ¿             ÚÄÄÄ¿
+     - 127595865\³1751  - 387349387\³190
+          Type: Union(RealClosure Fraction Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty25}
+\begin{paste}{RealClosureXmpPageEmpty25}{RealClosureXmpPagePatch25}
+\pastebutton{RealClosureXmpPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{recip(squareDiff8)\free{sd8 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch26}
+\begin{paste}{RealClosureXmpPageFull26}{RealClosureXmpPageEmpty26}
+\pastebutton{RealClosureXmpPageFull26}{\hidepaste}
+\tab{5}\spadcommand{sign(squareDiff8)\free{sd8 }}
+\indentrel{3}\begin{verbatim}
+   (26)  - 1
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty26}
+\begin{paste}{RealClosureXmpPageEmpty26}{RealClosureXmpPagePatch26}
+\pastebutton{RealClosureXmpPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{sign(squareDiff8)\free{sd8 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch27}
+\begin{paste}{RealClosureXmpPageFull27}{RealClosureXmpPageEmpty27}
+\pastebutton{RealClosureXmpPageFull27}{\hidepaste}
+\tab{5}\spadcommand{relativeApprox(squareDiff8,10**(-3))::Float\free{sd8 }}
+\indentrel{3}\begin{verbatim}
+   (27)  - 0.2340527771 5937700123 E -10
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty27}
+\begin{paste}{RealClosureXmpPageEmpty27}{RealClosureXmpPagePatch27}
+\pastebutton{RealClosureXmpPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{relativeApprox(squareDiff8,10**(-3))::Float\free{sd8 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch28}
+\begin{paste}{RealClosureXmpPageFull28}{RealClosureXmpPageEmpty28}
+\pastebutton{RealClosureXmpPageFull28}{\hidepaste}
+\tab{5}\spadcommand{l := allRootsOf((x**2-2)**2-2)$Ran\free{Ran }\bound{l }}
+\indentrel{3}\begin{verbatim}
+   (28)  [%R33,%R34,%R35,%R36]
+                Type: List RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty28}
+\begin{paste}{RealClosureXmpPageEmpty28}{RealClosureXmpPagePatch28}
+\pastebutton{RealClosureXmpPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{l := allRootsOf((x**2-2)**2-2)$Ran\free{Ran }\bound{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch29}
+\begin{paste}{RealClosureXmpPageFull29}{RealClosureXmpPageEmpty29}
+\pastebutton{RealClosureXmpPageFull29}{\hidepaste}
+\tab{5}\spadcommand{removeDuplicates map(mainDefiningPolynomial,l)\free{l }}
+\indentrel{3}\begin{verbatim}
+           4     2
+   (29)  [?  - 4?  + 2]
+Type: List Union(SparseUnivariatePolynomial RealClosure Fraction Integer,"failed")
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty29}
+\begin{paste}{RealClosureXmpPageEmpty29}{RealClosureXmpPagePatch29}
+\pastebutton{RealClosureXmpPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{removeDuplicates map(mainDefiningPolynomial,l)\free{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch30}
+\begin{paste}{RealClosureXmpPageFull30}{RealClosureXmpPageEmpty30}
+\pastebutton{RealClosureXmpPageFull30}{\hidepaste}
+\tab{5}\spadcommand{map(mainCharacterization,l)\free{l }}
+\indentrel{3}\begin{verbatim}
+   (30)  [[- 2,- 1[,[- 1,0[,[0,1[,[1,2[]
+Type: List Union(RightOpenIntervalRootCharacterization(RealClosure Fraction Integer,SparseUnivariatePolynomial RealClosure Fraction Integer),"failed")
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty30}
+\begin{paste}{RealClosureXmpPageEmpty30}{RealClosureXmpPagePatch30}
+\pastebutton{RealClosureXmpPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{map(mainCharacterization,l)\free{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch31}
+\begin{paste}{RealClosureXmpPageFull31}{RealClosureXmpPageEmpty31}
+\pastebutton{RealClosureXmpPageFull31}{\hidepaste}
+\tab{5}\spadcommand{[reduce(+,l),reduce(*,l)-2]\free{l }}
+\indentrel{3}\begin{verbatim}
+   (31)  [0,0]
+                Type: List RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty31}
+\begin{paste}{RealClosureXmpPageEmpty31}{RealClosureXmpPagePatch31}
+\pastebutton{RealClosureXmpPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{[reduce(+,l),reduce(*,l)-2]\free{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch32}
+\begin{paste}{RealClosureXmpPageFull32}{RealClosureXmpPageEmpty32}
+\pastebutton{RealClosureXmpPageFull32}{\hidepaste}
+\tab{5}\spadcommand{(s2, s5, s10) := (sqrt(2)$Ran, sqrt(5)$Ran, sqrt(10)$Ran)\free{Ran }\bound{s2 }\bound{s5 }\bound{s10 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄ¿
+   (32)  \³10
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty32}
+\begin{paste}{RealClosureXmpPageEmpty32}{RealClosureXmpPagePatch32}
+\pastebutton{RealClosureXmpPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{(s2, s5, s10) := (sqrt(2)$Ran, sqrt(5)$Ran, sqrt(10)$Ran)\free{Ran }\bound{s2 }\bound{s5 }\bound{s10 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch33}
+\begin{paste}{RealClosureXmpPageFull33}{RealClosureXmpPageEmpty33}
+\pastebutton{RealClosureXmpPageFull33}{\hidepaste}
+\tab{5}\spadcommand{eq1:=sqrt(s10+3)*sqrt(s5+2) - sqrt(s10-3)*sqrt(s5-2) = sqrt(10*s2+10)\free{s2 }\free{s5 }\free{s10 }\bound{eq1 }}
+\indentrel{3}\begin{verbatim}
+   (33)
+      ÚÄÄÄÄÄÄÄÄÄ¿ ÚÄÄÄÄÄÄÄÄ¿    ÚÄÄÄÄÄÄÄÄÄ¿ ÚÄÄÄÄÄÄÄÄ¿
+      ³ ÚÄÄ¿      ³ ÚÄ¿         ³ ÚÄÄ¿      ³ ÚÄ¿
+   - \³\³10  - 3 \³\³5  - 2  + \³\³10  + 3 \³\³5  + 2  =
+      ÚÄÄÄÄÄÄÄÄÄÄÄ¿
+      ³   ÚÄ¿
+     \³10\³2  + 10
+            Type: Equation RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty33}
+\begin{paste}{RealClosureXmpPageEmpty33}{RealClosureXmpPagePatch33}
+\pastebutton{RealClosureXmpPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{eq1:=sqrt(s10+3)*sqrt(s5+2) - sqrt(s10-3)*sqrt(s5-2) = sqrt(10*s2+10)\free{s2 }\free{s5 }\free{s10 }\bound{eq1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch34}
+\begin{paste}{RealClosureXmpPageFull34}{RealClosureXmpPageEmpty34}
+\pastebutton{RealClosureXmpPageFull34}{\hidepaste}
+\tab{5}\spadcommand{eq1::Boolean\free{eq1 }}
+\indentrel{3}\begin{verbatim}
+   (34)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty34}
+\begin{paste}{RealClosureXmpPageEmpty34}{RealClosureXmpPagePatch34}
+\pastebutton{RealClosureXmpPageEmpty34}{\showpaste}
+\tab{5}\spadcommand{eq1::Boolean\free{eq1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch35}
+\begin{paste}{RealClosureXmpPageFull35}{RealClosureXmpPageEmpty35}
+\pastebutton{RealClosureXmpPageFull35}{\hidepaste}
+\tab{5}\spadcommand{eq2:=sqrt(s5+2)*sqrt(s2+1) - sqrt(s5-2)*sqrt(s2-1) = sqrt(2*s10+2)\free{s2 }\free{s5 }\free{s10 }\bound{eq2 }}
+\indentrel{3}\begin{verbatim}
+   (35)
+      ÚÄÄÄÄÄÄÄÄ¿ ÚÄÄÄÄÄÄÄÄ¿    ÚÄÄÄÄÄÄÄÄ¿ ÚÄÄÄÄÄÄÄÄ¿
+      ³ ÚÄ¿      ³ ÚÄ¿         ³ ÚÄ¿      ³ ÚÄ¿
+   - \³\³5  - 2 \³\³2  - 1  + \³\³5  + 2 \³\³2  + 1  =
+      ÚÄÄÄÄÄÄÄÄÄÄ¿
+      ³  ÚÄÄ¿
+     \³2\³10  + 2
+            Type: Equation RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty35}
+\begin{paste}{RealClosureXmpPageEmpty35}{RealClosureXmpPagePatch35}
+\pastebutton{RealClosureXmpPageEmpty35}{\showpaste}
+\tab{5}\spadcommand{eq2:=sqrt(s5+2)*sqrt(s2+1) - sqrt(s5-2)*sqrt(s2-1) = sqrt(2*s10+2)\free{s2 }\free{s5 }\free{s10 }\bound{eq2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch36}
+\begin{paste}{RealClosureXmpPageFull36}{RealClosureXmpPageEmpty36}
+\pastebutton{RealClosureXmpPageFull36}{\hidepaste}
+\tab{5}\spadcommand{eq2::Boolean\free{eq2 }}
+\indentrel{3}\begin{verbatim}
+   (36)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty36}
+\begin{paste}{RealClosureXmpPageEmpty36}{RealClosureXmpPagePatch36}
+\pastebutton{RealClosureXmpPageEmpty36}{\showpaste}
+\tab{5}\spadcommand{eq2::Boolean\free{eq2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch37}
+\begin{paste}{RealClosureXmpPageFull37}{RealClosureXmpPageEmpty37}
+\pastebutton{RealClosureXmpPageFull37}{\hidepaste}
+\tab{5}\spadcommand{s3 := sqrt(3)$Ran\free{Ran }\bound{s3 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄ¿
+   (37)  \³3
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty37}
+\begin{paste}{RealClosureXmpPageEmpty37}{RealClosureXmpPagePatch37}
+\pastebutton{RealClosureXmpPageEmpty37}{\showpaste}
+\tab{5}\spadcommand{s3 := sqrt(3)$Ran\free{Ran }\bound{s3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch38}
+\begin{paste}{RealClosureXmpPageFull38}{RealClosureXmpPageEmpty38}
+\pastebutton{RealClosureXmpPageFull38}{\hidepaste}
+\tab{5}\spadcommand{s7:= sqrt(7)$Ran\free{Ran }\bound{s7 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄ¿
+   (38)  \³7
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty38}
+\begin{paste}{RealClosureXmpPageEmpty38}{RealClosureXmpPagePatch38}
+\pastebutton{RealClosureXmpPageEmpty38}{\showpaste}
+\tab{5}\spadcommand{s7:= sqrt(7)$Ran\free{Ran }\bound{s7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch39}
+\begin{paste}{RealClosureXmpPageFull39}{RealClosureXmpPageEmpty39}
+\pastebutton{RealClosureXmpPageFull39}{\hidepaste}
+\tab{5}\spadcommand{e1 := sqrt(2*s7-3*s3,3)\free{s7 }\free{s3 }\bound{e1 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+         3³  ÚÄ¿     ÚÄ¿
+   (39)  \³2\³7  - 3\³3
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty39}
+\begin{paste}{RealClosureXmpPageEmpty39}{RealClosureXmpPagePatch39}
+\pastebutton{RealClosureXmpPageEmpty39}{\showpaste}
+\tab{5}\spadcommand{e1 := sqrt(2*s7-3*s3,3)\free{s7 }\free{s3 }\bound{e1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch40}
+\begin{paste}{RealClosureXmpPageFull40}{RealClosureXmpPageEmpty40}
+\pastebutton{RealClosureXmpPageFull40}{\hidepaste}
+\tab{5}\spadcommand{e2 := sqrt(2*s7+3*s3,3)\free{s7 }\free{s3 }\bound{e2 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+         3³  ÚÄ¿     ÚÄ¿
+   (40)  \³2\³7  + 3\³3
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty40}
+\begin{paste}{RealClosureXmpPageEmpty40}{RealClosureXmpPagePatch40}
+\pastebutton{RealClosureXmpPageEmpty40}{\showpaste}
+\tab{5}\spadcommand{e2 := sqrt(2*s7+3*s3,3)\free{s7 }\free{s3 }\bound{e2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch41}
+\begin{paste}{RealClosureXmpPageFull41}{RealClosureXmpPageEmpty41}
+\pastebutton{RealClosureXmpPageFull41}{\hidepaste}
+\tab{5}\spadcommand{e2-e1-s3\free{e2 }\free{e1 }\free{s3 }}
+\indentrel{3}\begin{verbatim}
+   (41)  0
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty41}
+\begin{paste}{RealClosureXmpPageEmpty41}{RealClosureXmpPagePatch41}
+\pastebutton{RealClosureXmpPageEmpty41}{\showpaste}
+\tab{5}\spadcommand{e2-e1-s3\free{e2 }\free{e1 }\free{s3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch42}
+\begin{paste}{RealClosureXmpPageFull42}{RealClosureXmpPageEmpty42}
+\pastebutton{RealClosureXmpPageFull42}{\hidepaste}
+\tab{5}\spadcommand{pol : UP(x,Ran) := x**4+(7/3)*x**2+30*x-(100/3)\free{Ran }\bound{pol }}
+\indentrel{3}\begin{verbatim}
+          4   7  2         100
+   (42)  x  + Ä x  + 30x - ÄÄÄ
+              3             3
+Type: UnivariatePolynomial(x,RealClosure Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty42}
+\begin{paste}{RealClosureXmpPageEmpty42}{RealClosureXmpPagePatch42}
+\pastebutton{RealClosureXmpPageEmpty42}{\showpaste}
+\tab{5}\spadcommand{pol : UP(x,Ran) := x**4+(7/3)*x**2+30*x-(100/3)\free{Ran }\bound{pol }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch43}
+\begin{paste}{RealClosureXmpPageFull43}{RealClosureXmpPageEmpty43}
+\pastebutton{RealClosureXmpPageFull43}{\hidepaste}
+\tab{5}\spadcommand{r1 := sqrt(7633)$Ran\free{Ran }\bound{r1 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄÄÄ¿
+   (43)  \³7633
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty43}
+\begin{paste}{RealClosureXmpPageEmpty43}{RealClosureXmpPagePatch43}
+\pastebutton{RealClosureXmpPageEmpty43}{\showpaste}
+\tab{5}\spadcommand{r1 := sqrt(7633)$Ran\free{Ran }\bound{r1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch44}
+\begin{paste}{RealClosureXmpPageFull44}{RealClosureXmpPageEmpty44}
+\pastebutton{RealClosureXmpPageFull44}{\hidepaste}
+\tab{5}\spadcommand{alpha := sqrt(5*r1-436,3)/3\free{r1 }\bound{alpha }}
+\indentrel{3}\begin{verbatim}
+            ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+         1 3³  ÚÄÄÄÄ¿
+   (44)  Ä \³5\³7633  - 436
+         3
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty44}
+\begin{paste}{RealClosureXmpPageEmpty44}{RealClosureXmpPagePatch44}
+\pastebutton{RealClosureXmpPageEmpty44}{\showpaste}
+\tab{5}\spadcommand{alpha := sqrt(5*r1-436,3)/3\free{r1 }\bound{alpha }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch45}
+\begin{paste}{RealClosureXmpPageFull45}{RealClosureXmpPageEmpty45}
+\pastebutton{RealClosureXmpPageFull45}{\hidepaste}
+\tab{5}\spadcommand{beta := -sqrt(5*r1+436,3)/3\free{r1 }\bound{beta }}
+\indentrel{3}\begin{verbatim}
+              ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+           1 3³  ÚÄÄÄÄ¿
+   (45)  - Ä \³5\³7633  + 436
+           3
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty45}
+\begin{paste}{RealClosureXmpPageEmpty45}{RealClosureXmpPagePatch45}
+\pastebutton{RealClosureXmpPageEmpty45}{\showpaste}
+\tab{5}\spadcommand{beta := -sqrt(5*r1+436,3)/3\free{r1 }\bound{beta }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch46}
+\begin{paste}{RealClosureXmpPageFull46}{RealClosureXmpPageEmpty46}
+\pastebutton{RealClosureXmpPageFull46}{\hidepaste}
+\tab{5}\spadcommand{pol.(alpha+beta-1/3)\free{pol }\free{alpha }\free{beta }}
+\indentrel{3}\begin{verbatim}
+   (46)  0
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty46}
+\begin{paste}{RealClosureXmpPageEmpty46}{RealClosureXmpPagePatch46}
+\pastebutton{RealClosureXmpPageEmpty46}{\showpaste}
+\tab{5}\spadcommand{pol.(alpha+beta-1/3)\free{pol }\free{alpha }\free{beta }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch47}
+\begin{paste}{RealClosureXmpPageFull47}{RealClosureXmpPageEmpty47}
+\pastebutton{RealClosureXmpPageFull47}{\hidepaste}
+\tab{5}\spadcommand{qol : UP(x,Ran) := x**5+10*x**3+20*x+22\free{Ran }\bound{qol }}
+\indentrel{3}\begin{verbatim}
+          5      3
+   (47)  x  + 10x  + 20x + 22
+Type: UnivariatePolynomial(x,RealClosure Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty47}
+\begin{paste}{RealClosureXmpPageEmpty47}{RealClosureXmpPagePatch47}
+\pastebutton{RealClosureXmpPageEmpty47}{\showpaste}
+\tab{5}\spadcommand{qol : UP(x,Ran) := x**5+10*x**3+20*x+22\free{Ran }\bound{qol }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch48}
+\begin{paste}{RealClosureXmpPageFull48}{RealClosureXmpPageEmpty48}
+\pastebutton{RealClosureXmpPageFull48}{\hidepaste}
+\tab{5}\spadcommand{r2 := sqrt(153)$Ran\free{Ran }\bound{r2 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄÄ¿
+   (48)  \³153
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty48}
+\begin{paste}{RealClosureXmpPageEmpty48}{RealClosureXmpPagePatch48}
+\pastebutton{RealClosureXmpPageEmpty48}{\showpaste}
+\tab{5}\spadcommand{r2 := sqrt(153)$Ran\free{Ran }\bound{r2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch49}
+\begin{paste}{RealClosureXmpPageFull49}{RealClosureXmpPageEmpty49}
+\pastebutton{RealClosureXmpPageFull49}{\hidepaste}
+\tab{5}\spadcommand{alpha2 := sqrt(r2-11,5)\free{r2 }\bound{alpha2 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄÄÄÄÄÄÄÄÄÄ¿
+         5³ ÚÄÄÄ¿
+   (49)  \³\³153  - 11
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty49}
+\begin{paste}{RealClosureXmpPageEmpty49}{RealClosureXmpPagePatch49}
+\pastebutton{RealClosureXmpPageEmpty49}{\showpaste}
+\tab{5}\spadcommand{alpha2 := sqrt(r2-11,5)\free{r2 }\bound{alpha2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch50}
+\begin{paste}{RealClosureXmpPageFull50}{RealClosureXmpPageEmpty50}
+\pastebutton{RealClosureXmpPageFull50}{\hidepaste}
+\tab{5}\spadcommand{beta2 := -sqrt(r2+11,5)\free{r2 }\bound{beta2 }}
+\indentrel{3}\begin{verbatim}
+            ÚÄÄÄÄÄÄÄÄÄÄÄ¿
+           5³ ÚÄÄÄ¿
+   (50)  - \³\³153  + 11
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty50}
+\begin{paste}{RealClosureXmpPageEmpty50}{RealClosureXmpPagePatch50}
+\pastebutton{RealClosureXmpPageEmpty50}{\showpaste}
+\tab{5}\spadcommand{beta2 := -sqrt(r2+11,5)\free{r2 }\bound{beta2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch51}
+\begin{paste}{RealClosureXmpPageFull51}{RealClosureXmpPageEmpty51}
+\pastebutton{RealClosureXmpPageFull51}{\hidepaste}
+\tab{5}\spadcommand{qol(alpha2+beta2)\free{qol }\free{alpha2 }\free{beta2 }}
+\indentrel{3}\begin{verbatim}
+   (51)  0
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty51}
+\begin{paste}{RealClosureXmpPageEmpty51}{RealClosureXmpPagePatch51}
+\pastebutton{RealClosureXmpPageEmpty51}{\showpaste}
+\tab{5}\spadcommand{qol(alpha2+beta2)\free{qol }\free{alpha2 }\free{beta2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch52}
+\begin{paste}{RealClosureXmpPageFull52}{RealClosureXmpPageEmpty52}
+\pastebutton{RealClosureXmpPageFull52}{\hidepaste}
+\tab{5}\spadcommand{dst1:=sqrt(9+4*s2)=1+2*s2\free{s2 }\bound{dst1 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄÄÄÄÄÄÄÄ¿
+          ³  ÚÄ¿         ÚÄ¿
+   (52)  \³4\³2  + 9 = 2\³2  + 1
+            Type: Equation RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty52}
+\begin{paste}{RealClosureXmpPageEmpty52}{RealClosureXmpPagePatch52}
+\pastebutton{RealClosureXmpPageEmpty52}{\showpaste}
+\tab{5}\spadcommand{dst1:=sqrt(9+4*s2)=1+2*s2\free{s2 }\bound{dst1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch53}
+\begin{paste}{RealClosureXmpPageFull53}{RealClosureXmpPageEmpty53}
+\pastebutton{RealClosureXmpPageFull53}{\hidepaste}
+\tab{5}\spadcommand{dst1::Boolean\free{dst1 }}
+\indentrel{3}\begin{verbatim}
+   (53)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty53}
+\begin{paste}{RealClosureXmpPageEmpty53}{RealClosureXmpPagePatch53}
+\pastebutton{RealClosureXmpPageEmpty53}{\showpaste}
+\tab{5}\spadcommand{dst1::Boolean\free{dst1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch54}
+\begin{paste}{RealClosureXmpPageFull54}{RealClosureXmpPageEmpty54}
+\pastebutton{RealClosureXmpPageFull54}{\hidepaste}
+\tab{5}\spadcommand{s6:Ran:=sqrt 6\free{Ran }\bound{s6 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄ¿
+   (54)  \³6
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty54}
+\begin{paste}{RealClosureXmpPageEmpty54}{RealClosureXmpPagePatch54}
+\pastebutton{RealClosureXmpPageEmpty54}{\showpaste}
+\tab{5}\spadcommand{s6:Ran:=sqrt 6\free{Ran }\bound{s6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch55}
+\begin{paste}{RealClosureXmpPageFull55}{RealClosureXmpPageEmpty55}
+\pastebutton{RealClosureXmpPageFull55}{\hidepaste}
+\tab{5}\spadcommand{dst2:=sqrt(5+2*s6)+sqrt(5-2*s6) = 2*s3\free{s6 }\free{s3 }\bound{dst2 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄÄÄÄÄÄÄÄÄÄ¿    ÚÄÄÄÄÄÄÄÄÄ¿
+          ³    ÚÄ¿         ³  ÚÄ¿         ÚÄ¿
+   (55)  \³- 2\³6  + 5  + \³2\³6  + 5 = 2\³3
+            Type: Equation RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty55}
+\begin{paste}{RealClosureXmpPageEmpty55}{RealClosureXmpPagePatch55}
+\pastebutton{RealClosureXmpPageEmpty55}{\showpaste}
+\tab{5}\spadcommand{dst2:=sqrt(5+2*s6)+sqrt(5-2*s6) = 2*s3\free{s6 }\free{s3 }\bound{dst2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch56}
+\begin{paste}{RealClosureXmpPageFull56}{RealClosureXmpPageEmpty56}
+\pastebutton{RealClosureXmpPageFull56}{\hidepaste}
+\tab{5}\spadcommand{dst2::Boolean\free{dst2 }}
+\indentrel{3}\begin{verbatim}
+   (56)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty56}
+\begin{paste}{RealClosureXmpPageEmpty56}{RealClosureXmpPagePatch56}
+\pastebutton{RealClosureXmpPageEmpty56}{\showpaste}
+\tab{5}\spadcommand{dst2::Boolean\free{dst2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch57}
+\begin{paste}{RealClosureXmpPageFull57}{RealClosureXmpPageEmpty57}
+\pastebutton{RealClosureXmpPageFull57}{\hidepaste}
+\tab{5}\spadcommand{s29:Ran:=sqrt 29\free{Ran }\bound{s29 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄ¿
+   (57)  \³29
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty57}
+\begin{paste}{RealClosureXmpPageEmpty57}{RealClosureXmpPagePatch57}
+\pastebutton{RealClosureXmpPageEmpty57}{\showpaste}
+\tab{5}\spadcommand{s29:Ran:=sqrt 29\free{Ran }\bound{s29 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch58}
+\begin{paste}{RealClosureXmpPageFull58}{RealClosureXmpPageEmpty58}
+\pastebutton{RealClosureXmpPageFull58}{\hidepaste}
+\tab{5}\spadcommand{dst4:=sqrt(16-2*s29+2*sqrt(55-10*s29)) = sqrt(22+2*s5)-sqrt(11+2*s29)+s5\free{s29 }\free{s5 }\bound{dst4 }}
+\indentrel{3}\begin{verbatim}
+   (58)
+    ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+    ³  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+    ³  ³     ÚÄÄ¿           ÚÄÄ¿
+   \³2\³- 10\³29  + 55  - 2\³29  + 16  =
+        ÚÄÄÄÄÄÄÄÄÄÄÄ¿    ÚÄÄÄÄÄÄÄÄÄÄ¿
+        ³  ÚÄÄ¿          ³  ÚÄ¿          ÚÄ¿
+     - \³2\³29  + 11  + \³2\³5  + 22  + \³5
+            Type: Equation RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty58}
+\begin{paste}{RealClosureXmpPageEmpty58}{RealClosureXmpPagePatch58}
+\pastebutton{RealClosureXmpPageEmpty58}{\showpaste}
+\tab{5}\spadcommand{dst4:=sqrt(16-2*s29+2*sqrt(55-10*s29)) = sqrt(22+2*s5)-sqrt(11+2*s29)+s5\free{s29 }\free{s5 }\bound{dst4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch59}
+\begin{paste}{RealClosureXmpPageFull59}{RealClosureXmpPageEmpty59}
+\pastebutton{RealClosureXmpPageFull59}{\hidepaste}
+\tab{5}\spadcommand{dst4::Boolean\free{dst4 }}
+\indentrel{3}\begin{verbatim}
+   (59)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty59}
+\begin{paste}{RealClosureXmpPageEmpty59}{RealClosureXmpPagePatch59}
+\pastebutton{RealClosureXmpPageEmpty59}{\showpaste}
+\tab{5}\spadcommand{dst4::Boolean\free{dst4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch60}
+\begin{paste}{RealClosureXmpPageFull60}{RealClosureXmpPageEmpty60}
+\pastebutton{RealClosureXmpPageFull60}{\hidepaste}
+\tab{5}\spadcommand{dst6:=sqrt((112+70*s2)+(46+34*s2)*s5) = (5+4*s2)+(3+s2)*s5\free{s2 }\free{s5 }\bound{dst6 }}
+\indentrel{3}\begin{verbatim}
+   (60)
+    ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+    ³    ÚÄ¿       ÚÄ¿      ÚÄ¿
+   \³(34\³2  + 46)\³5  + 70\³2  + 112  =
+       ÚÄ¿      ÚÄ¿     ÚÄ¿
+     (\³2  + 3)\³5  + 4\³2  + 5
+            Type: Equation RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty60}
+\begin{paste}{RealClosureXmpPageEmpty60}{RealClosureXmpPagePatch60}
+\pastebutton{RealClosureXmpPageEmpty60}{\showpaste}
+\tab{5}\spadcommand{dst6:=sqrt((112+70*s2)+(46+34*s2)*s5) = (5+4*s2)+(3+s2)*s5\free{s2 }\free{s5 }\bound{dst6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch61}
+\begin{paste}{RealClosureXmpPageFull61}{RealClosureXmpPageEmpty61}
+\pastebutton{RealClosureXmpPageFull61}{\hidepaste}
+\tab{5}\spadcommand{dst6::Boolean\free{dst6 }}
+\indentrel{3}\begin{verbatim}
+   (61)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty61}
+\begin{paste}{RealClosureXmpPageEmpty61}{RealClosureXmpPagePatch61}
+\pastebutton{RealClosureXmpPageEmpty61}{\showpaste}
+\tab{5}\spadcommand{dst6::Boolean\free{dst6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch62}
+\begin{paste}{RealClosureXmpPageFull62}{RealClosureXmpPageEmpty62}
+\pastebutton{RealClosureXmpPageFull62}{\hidepaste}
+\tab{5}\spadcommand{f3:Ran:=sqrt(3,5)\free{Ran }\bound{f3 }}
+\indentrel{3}\begin{verbatim}
+         5ÚÄ¿
+   (62)  \³3
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty62}
+\begin{paste}{RealClosureXmpPageEmpty62}{RealClosureXmpPagePatch62}
+\pastebutton{RealClosureXmpPageEmpty62}{\showpaste}
+\tab{5}\spadcommand{f3:Ran:=sqrt(3,5)\free{Ran }\bound{f3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch63}
+\begin{paste}{RealClosureXmpPageFull63}{RealClosureXmpPageEmpty63}
+\pastebutton{RealClosureXmpPageFull63}{\hidepaste}
+\tab{5}\spadcommand{f25:Ran:=sqrt(1/25,5)\free{Ran }\bound{f25 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄ¿
+          ³ 1
+   (63)  5³ÄÄ
+         \³25
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty63}
+\begin{paste}{RealClosureXmpPageEmpty63}{RealClosureXmpPagePatch63}
+\pastebutton{RealClosureXmpPageEmpty63}{\showpaste}
+\tab{5}\spadcommand{f25:Ran:=sqrt(1/25,5)\free{Ran }\bound{f25 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch64}
+\begin{paste}{RealClosureXmpPageFull64}{RealClosureXmpPageEmpty64}
+\pastebutton{RealClosureXmpPageFull64}{\hidepaste}
+\tab{5}\spadcommand{f32:Ran:=sqrt(32/5,5)\free{Ran }\bound{f32 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄ¿
+          ³32
+   (64)  5³ÄÄ
+         \³ 5
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty64}
+\begin{paste}{RealClosureXmpPageEmpty64}{RealClosureXmpPagePatch64}
+\pastebutton{RealClosureXmpPageEmpty64}{\showpaste}
+\tab{5}\spadcommand{f32:Ran:=sqrt(32/5,5)\free{Ran }\bound{f32 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch65}
+\begin{paste}{RealClosureXmpPageFull65}{RealClosureXmpPageEmpty65}
+\pastebutton{RealClosureXmpPageFull65}{\hidepaste}
+\tab{5}\spadcommand{f27:Ran:=sqrt(27/5,5)\free{Ran }\bound{f27 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄ¿
+          ³27
+   (65)  5³ÄÄ
+         \³ 5
+                     Type: RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty65}
+\begin{paste}{RealClosureXmpPageEmpty65}{RealClosureXmpPagePatch65}
+\pastebutton{RealClosureXmpPageEmpty65}{\showpaste}
+\tab{5}\spadcommand{f27:Ran:=sqrt(27/5,5)\free{Ran }\bound{f27 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch66}
+\begin{paste}{RealClosureXmpPageFull66}{RealClosureXmpPageEmpty66}
+\pastebutton{RealClosureXmpPageFull66}{\hidepaste}
+\tab{5}\spadcommand{dst5:=sqrt((f32-f27,3)) = f25*(1+f3-f3**2)\free{f32 }\free{f27 }\free{f25 }\free{f3 }\bound{dst5 }}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+          ³   ÚÄÄ¿    ÚÄÄ¿                         ÚÄÄ¿
+          ³   ³27     ³32       5ÚÄ¿2   5ÚÄ¿       ³ 1
+   (66)  3³- 5³ÄÄ  + 5³ÄÄ  = (- \³3   + \³3  + 1) 5³ÄÄ
+         \³  \³ 5    \³ 5                         \³25
+            Type: Equation RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty66}
+\begin{paste}{RealClosureXmpPageEmpty66}{RealClosureXmpPagePatch66}
+\pastebutton{RealClosureXmpPageEmpty66}{\showpaste}
+\tab{5}\spadcommand{dst5:=sqrt((f32-f27,3)) = f25*(1+f3-f3**2)\free{f32 }\free{f27 }\free{f25 }\free{f3 }\bound{dst5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPagePatch67}
+\begin{paste}{RealClosureXmpPageFull67}{RealClosureXmpPageEmpty67}
+\pastebutton{RealClosureXmpPageFull67}{\hidepaste}
+\tab{5}\spadcommand{dst5::Boolean\free{dst5 }}
+\indentrel{3}\begin{verbatim}
+   (67)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RealClosureXmpPageEmpty67}
+\begin{paste}{RealClosureXmpPageEmpty67}{RealClosureXmpPagePatch67}
+\pastebutton{RealClosureXmpPageEmpty67}{\showpaste}
+\tab{5}\spadcommand{dst5::Boolean\free{dst5 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/REGSET.ht b/src/hyper/pages/REGSET.ht
new file mode 100644
index 00000000..90a6674c
--- /dev/null
+++ b/src/hyper/pages/REGSET.ht
@@ -0,0 +1,432 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\RegularTriangularSetXmpTitle}{RegularTriangularSet}
+\newcommand{\RegularTriangularSetXmpNumber}{9.67}
+%
+% =====================================================================
+\begin{page}{RegularTriangularSetXmpPage}{9.67 RegularTriangularSet}
+% =====================================================================
+\beginscroll
+The \spadtype{RegularTriangularSet} domain constructor implements
+regular triangular sets.
+These particular triangular sets were introduced by M. Kalkbrener (1991) 
+in his PhD Thesis under the name regular chains.
+Regular chains and their related concepts are presented in
+the paper "On the Theories of Triangular sets" By P. Aubry, D. Lazard
+and M. Moreno Maza (to appear in the Journal of Symbolic Computation).
+The \spadtype{RegularTriangularSet} constructor also provides a new method (by the third author) 
+for solving polynomial system by means of regular chains.
+This method has two ways of solving.
+One has the same specifications as Kalkbrener's algorithm (1991)
+and the other is closer to Lazard's method (Discr. App. Math, 1991).
+Moreover, this new method removes redundant component from the 
+decompositions when this is not {\em too expensive}.
+This is always the case with square-free regular chains.
+So if you want to obtain decompositions without redundant components
+just use the \spadtype{SquareFreeRegularTriangularSet} domain constructor 
+or the \spadtype{LazardSetSolvingPackage} package constructor.
+See also the \spadtype{LexTriangularPackage} and \spadtype{ZeroDimensionalSolvePackage} for the case of 
+algebraic systems with a finite number of (complex) solutions.
+
+One of the main features of regular triangular sets is that they
+naturally define towers of simple extensions of a field.
+This allows to perform with multivariate polynomials the 
+same kind of operations as one can do in an \spadtype{EuclideanDomain}.
+
+The \spadtype{RegularTriangularSet} constructor takes four arguments.
+The first one, {\bf R}, is the coefficient ring of the polynomials;
+it must belong to the category \spadtype{GcdDomain}.
+The second one, {\bf E}, is the exponent monoid of the polynomials;
+it must belong to the category \spadtype{OrderedAbelianMonoidSup}.
+the third one, {\bf V}, is the ordered set of variables;
+it must belong to the category \spadtype{OrderedSet}.
+The last one is the polynomial ring;
+it must belong to the category \spadtype{RecursivePolynomialCategory(R,E,V)}.
+The abbreviation for \spadtype{RegularTriangularSet} is
+\spadtype{REGSET}.
+See also the constructor \spadtype{RegularChain} which only takes
+two arguments, the coefficient ring and the ordered set of variables;
+in that case, polynomials are necessarily built with the
+\spadtype{NewSparseMultivariatePolynomial}  domain constructor.
+
+We shall explain now how to use the constructor \spadtype{REGSET} 
+and how to read the decomposition of a polynomial system by means of regular sets.
+
+Let us give some examples.
+We start with an easy one (Donati-Traverso) 
+in order to understand the two ways of
+solving polynomial systems provided by the \spadtype{REGSET} constructor.
+\xtc{
+Define the coefficient ring.
+}{
+\spadpaste{R := Integer \bound{R}}
+}
+\xtc{
+Define the list of variables,
+}{
+\spadpaste{ls : List Symbol := [x,y,z,t] \bound{ls}}
+}
+\xtc{
+and make it an ordered set;
+}{
+\spadpaste{V := OVAR(ls) \free{ls} \bound{V}}
+}
+\xtc{
+then define the exponent monoid.
+}{
+\spadpaste{E := IndexedExponents V \free{V} \bound{E}}
+}
+\xtc{
+Define the polynomial ring.
+}{
+\spadpaste{P := NSMP(R, V) \free{R} \free{V} \bound{P}}
+}
+\xtc{
+Let the variables be polynomial.
+}{
+\spadpaste{x: P := 'x \free{P} \bound{x}}
+}
+\xtc{
+}{
+\spadpaste{y: P := 'y \free{P} \bound{y}}
+}
+\xtc{
+}{
+\spadpaste{z: P := 'z \free{P} \bound{z}}
+}
+\xtc{
+}{
+\spadpaste{t: P := 't \free{P} \bound{t}}
+}
+\xtc{
+Now call the \spadtype{RegularTriangularSet} domain constructor.
+}{
+\spadpaste{T := REGSET(R,E,V,P) \free{R} \free{E} \free{V} \free{P} \bound{T} }
+}
+\xtc{
+Define a polynomial system.
+}{
+\spadpaste{p1 := x ** 31 - x ** 6 - x - y \free{x} \free{y} \bound{p1}}
+}
+\xtc{
+}{
+\spadpaste{p2 := x ** 8  - z \free{x} \free{z} \bound{p2}}
+}
+\xtc{
+}{
+\spadpaste{p3 := x ** 10 - t \free{x} \free{t} \bound{p3}}
+}
+\xtc{
+}{
+\spadpaste{lp := [p1, p2, p3] \free{p1} \free{p2} \free{p3} \bound{lp}}
+}
+\xtc{
+First of all, let us solve this system in the sense of Kalkbrener.
+}{
+\spadpaste{zeroSetSplit(lp)$T \free{lp} \free{T}}
+}
+\xtc{
+And now in the sense of Lazard (or Wu and other authors).
+}{
+\spadpaste{lts := zeroSetSplit(lp,false)$T \free{lp} \free{T} \bound{lts}}
+}
+
+We can see that the first decomposition is a subset of the second.
+So how can both be correct ?
+
+Recall first that polynomials from a domain of the category
+\spadtype{RecursivePolynomialCategory} are regarded
+as univariate polynomials in their main variable.
+For instance the second polynomial in the first set
+of each decomposition has main variable {\bf y}
+and its initial (i.e. its leading coefficient w.r.t. its main
+variable) is {\bf t z}.
+
+Now let us explain how to read the second decomposition.
+Note that the non-constant initials of the first set are
+\texht{$t^4-t$}{{\bf t^4 - t}} and \texht{$t z$}{{\bf t z}}.
+Then the solutions described by this first set are the common
+zeros of its polynomials that do not cancel the polynomials
+\texht{$t^4-t$}{{\bf t^4 - t}} and \texht{$ty z$}{{\bf t z}}.
+Now the solutions of the input system {\bf lp} satisfying
+these equations are described by the second and the third
+sets of the decomposition.
+Thus, in some sense, they can be considered as degenerated
+solutions.
+The solutions given by the first set are called the generic
+points of the system; they give the general form of the
+solutions.
+The first decomposition only provides these generic points.
+This latter decomposition is useful when they are many degenerated 
+solutions (which is sometimes hard to compute) and when
+one is only interested in general informations, like
+the dimension of the input system.
+
+\xtc{
+We can get the dimensions of each component
+of a decomposition as follows.
+}{
+\spadpaste{[coHeight(ts) for ts in lts] \free{lts}}
+}
+
+Thus the first set has dimension one.
+Indeed {\bf t} can take any value, except {\bf 0}
+or any third root of {\bf 1}, whereas {\bf z}
+is completely determined from {\bf t},
+{\bf y} is given by {\bf z} and {\bf t},
+and finally {\bf x} is given by the other three variables.
+In the second and the third sets of the second decomposition
+the four variables are completely determined and thus
+these sets have dimension zero.
+
+We give now the precise specifications of each decomposition. 
+This assume some mathematical knowledge.
+However, for the non-expert user, the above explanations will 
+be sufficient to understand the other features of the
+\spadtype{RSEGSET} constructor.
+
+The input system {\bf lp} is decomposed in the sense
+of Kalkbrener as finitely many regular sets {\bf T1,...,Ts}
+such that the radical ideal generated by {\bf lp}
+is the intersection of the radicals of the
+saturated ideals of {\bf T1,...,Ts}.
+In other words, the affine variety associated with {\bf lp} 
+is the union of the closures (w.r.t. Zarisky topology)
+of the regular-zeros sets of {\bf T1,...,Ts}.
+
+{\bf N. B.} The prime ideals associated with the 
+radical of the saturated ideal of
+a regular triangular set have all the same dimension;
+moreover these prime ideals can be given by characteristic
+sets with the same main variables. 
+Thus a decomposition in the sense of Kalkbrener
+is unmixed dimensional.
+Then it can be viewed as a {\em lazy}
+decomposition into prime ideals (some of these 
+prime ideals being merged into unmixed dimensional ideals).
+
+Now we explain the other way of solving by means of regular 
+triangular sets.
+The input system {\bf lp} is decomposed in the sense
+of Lazard as finitely many regular triangular sets {\bf T1,...,Ts}
+such that the affine variety associated with {\bf lp}
+is the union of the regular-zeros sets of {\bf T1,...,Ts}.
+Thus a decomposition in the sense of Lazard is also
+a decomposition in the sense of Kalkbrener; the converse
+is false as we have seen before.
+
+When the input system has a finite number of solutions,
+both ways of solving provide similar decompositions as
+we shall see with this second example (Caprasse).
+
+
+\xtc{
+Define a polynomial system.
+}{
+\spadpaste{f1 := y**2*z+2*x*y*t-2*x-z \free{z} \free{x} \free{y} \free{t} \bound{f1}}
+}
+\xtc{
+}{
+\spadpaste{f2 :=   -x**3*z+ 4*x*y**2*z+ 4*x**2*y*t+ 2*y**3*t+ 4*x**2- 10*y**2+ 4*x*z- 10*y*t+ 2 \free{z} \free{x} \free{y} \free{t} \bound{f2}}
+}
+\xtc{
+}{
+\spadpaste{f3 :=  2*y*z*t+x*t**2-x-2*z \free{z} \free{x} \free{y} \free{t} \bound{f3}}
+}
+\xtc{
+}{
+\spadpaste{f4 :=   -x*z**3+ 4*y*z**2*t+ 4*x*z*t**2+ 2*y*t**3+ 4*x*z+ 4*z**2-10*y*t- 10*t**2+2 \free{z} \free{x} \free{y} \free{t} \bound{f4}}
+}
+\xtc{
+}{
+\spadpaste{lf := [f1, f2, f3, f4] \free{f1} \free{f2} \free{f3}  \free{f4} \bound{lf}}
+}
+
+\xtc{
+First of all, let us solve this system in the sense of Kalkbrener.
+}{
+\spadpaste{zeroSetSplit(lf)$T \free{lf} \free{T}}
+}
+\xtc{
+And now in the sense of Lazard (or Wu and other authors).
+}{
+\spadpaste{lts2 := zeroSetSplit(lf,false)$T \free{lf} \free{T} \bound{lts2}}
+}
+
+Up to the ordering of the components, both decompositions are identical.
+
+\xtc{
+Let us check that each component has a finite number of solutions.
+}{
+\spadpaste{[coHeight(ts) for ts in lts2] \free{lts2}}
+}
+
+\xtc{
+Let us count the degrees of each component,
+}{
+\spadpaste{degrees := [degree(ts) for ts in lts2] \free{lts2} \bound{degrees}}
+}
+\xtc{
+and compute their sum.
+}{
+\spadpaste{reduce(+,degrees) \free{degrees}}
+}
+
+We study now the options of the \spadfun{zeroSetSplit} operation.
+As we have seen yet, there is an optional second argument 
+which is a boolean value. If this value is true (this
+is the default) then the decomposition is computed
+in the sense of Kalkbrener, otherwise it is computed
+in the sense of Lazard.
+
+There is a second boolean optional argument that
+can be used (in that case the first optional
+argument must be present).
+This second option allows you to get some information
+during the computations.
+
+Therefore, we need to understand a little what is 
+going on during the computations.
+An important feature of the algorithm is that 
+the intermediate computations are managed in some sense 
+like the processes of a Unix system.
+Indeed, each intermediate computation may generate
+other intermediate computations and the management
+of all these computations is a crucial task for
+the efficiency.
+Thus any intermediate computation may be suspended,
+killed or resumed, depending on algebraic considerations
+that determine priorities for these processes.
+The goal is of course to go as fast as possible 
+towards the final decomposition which means to avoid
+as much as possible unnecessary computations.
+
+To follow the computations, one needs to set to
+\spad{true} the second argument.
+Then a lot of numbers and letters are displayed.
+Between a \spad{[} and a \spad{]} one has 
+the state of the processes at a given time.
+Just after \spad{[} one can see the number of
+processes. 
+Then each process is represented by two numbers
+between \spad{<} and \spad{>}.
+A process consists of a list of polynomial {\bf ps}
+and a triangular set {\bf ts}; its goal is to compute
+the common zeros of {\bf ps} that belong to the
+regular-zeros set of {\bf ts}.
+After the processes, the number between pipes
+gives the total number of polynomials
+in all the sets \spad{ps}.
+Finally, the number between braces gives the number
+of components of a decomposition that are already
+computed. This number may decrease.
+
+Let us take a third example (Czapor-Geddes-Wang) to see how these
+informations are displayed.
+
+\xtc{
+Define a polynomial system.
+}{
+\spadpaste{u : R := 2  \free{R} \bound{u}}
+}
+\xtc{
+}{
+\spadpaste{q1 := 2*(u-1)**2+ 2*(x-z*x+z**2)+ y**2*(x-1)**2- 2*u*x+ 2*y*t*(1-x)*(x-z)+ 2*u*z*t*(t-y)+ u**2*t**2*(1-2*z)+ 2*u*t**2*(z-x)+ 2*u*t*y*(z-1)+ 2*u*z*x*(y+1)+ (u**2-2*u)*z**2*t**2+ 2*u**2*z**2+ 4*u*(1-u)*z+ t**2*(z-x)**2 \free{z} \free{x} \free{y} \free{t} \free{u} \bound{q1}}
+}
+\xtc{
+}{
+\spadpaste{q2 := t*(2*z+1)*(x-z)+ y*(z+2)*(1-x)+ u*(u-2)*t+ u*(1-2*u)*z*t+ u*y*(x+u-z*x-1)+ u*(u+1)*z**2*t \free{z} \free{x} \free{y} \free{t} \free{u} \bound{q2}}
+}
+\xtc{
+}{
+\spadpaste{q3 := -u**2*(z-1)**2+ 2*z*(z-x)-2*(x-1) \free{z} \free{x} \free{y} \free{t} \free{u}  \bound{q3}}
+}
+\xtc{
+}{
+\spadpaste{q4 :=   u**2+4*(z-x**2)+3*y**2*(x-1)**2- 3*t**2*(z-x)**2 +3*u**2*t**2*(z-1)**2+u**2*z*(z-2)+6*u*t*y*(z+x+z*x-1) \free{z} \free{x} \free{y} \free{t} \free{u} \bound{q4}}
+}
+\xtc{
+}{
+\spadpaste{lq := [q1, q2, q3, q4] \free{q1} \free{q2} \free{q3}  \free{q4} \bound{lq}}
+}
+
+
+\xtc{
+Let us try the information option.
+N.B. The timing should be between 1 and 10 minutes, depending on your machine.
+}{
+\spadpaste{zeroSetSplit(lq,true,true)$T \free{lq} \free{T}}
+}
+
+Between a sequence of processes, thus between a \spad{]} and a \spad{[}
+you can see capital letters \spad{W, G, I} and lower case letters
+\spad{i, w}. Each time a capital letter appears a non-trivial computation
+has be performed and its result is put in a hash-table.
+Each time a lower case letter appears a needed result has been 
+found in an hash-table.
+The use of these hash-tables generally speed up the computations.
+However, on very large systems, it may happen that these hash-tables
+become too  big  to be handle by your AXIOM configuration.
+Then in these exceptional cases, you may prefer getting a result
+(even if it takes a long time) than getting nothing.
+Hence you need to know how to prevent the \spadtype{RSEGSET} constructor
+from using these hash-tables.
+In that case you will be using the \spadfun{zeroSetSplit} with five arguments.
+The first one is the input system {\bf lp} as above.
+The second one is a boolean value \spad{hash?} which is \spad{true}
+iff you want to use hash-tables.
+The third one is boolean value \spad{clos?} which is \spad{true}
+iff you want to solve your system in the sense of Kalkbrener,
+the other way remaining that of Lazard.
+The fourth argument is boolean value \spad{info?} which is \spad{true}
+iff you want to display information during the computations.
+The last one is boolean value \spad{prep?} which is \spad{true}
+iff you want to use some heuristics that are performed on the
+input system before starting the real algorithm.
+The value of this flag is \spad{true} when you are using \spadfun{zeroSetSplit} 
+with less than five arguments.
+Note that there is no available signature for \spadfun{zeroSetSplit} with 
+four arguments.
+
+We finish this section by some remarks about both ways of
+solving, in the sense of Kalkbrener or in the sense of Lazard.
+For problems with a finite number of solutions, there are
+theoretically equivalent and the resulting decompositions
+are identical, up to the ordering of the components.
+However, when solving in the sense of Lazard, the algorithm
+behaves differently.
+In that case, it becomes more incremental than in the sense
+of Kalkbrener. That means the polynomials of the input system
+are considered one after another whereas in the sense of Kalkbrener
+the input system is treated more globally.
+
+This makes an important difference in positive dimension.
+Indeed when solving in the sense of Kalkbrener, the
+{\em Primeidealkettensatz} of Krull is used.
+That means any regular triangular containing more polynomials
+than the input system can be deleted.
+This is not possible when solving in the sense of Lazard.
+This explains why Kalkbrener's decompositions
+usually contain less components than those of Lazard.
+However, it may happen with some examples that the incremental process
+(that cannot be used when solving in the sense of Kalkbrener)
+provide a more efficient way of solving than the global one
+even if the {\em Primeidealkettensatz} is used.
+Thus just try both, with the various options, before concluding 
+that you cannot solve your favorite system with \spadfun{zeroSetSplit}.
+There exist more options at the development level that are not
+currently available in this public version.
+So you are welcome to contact {\em marc@nag.co.uk} for more
+information and help.
+
+
+
+
+
+
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/REGSET.pht b/src/hyper/pages/REGSET.pht
new file mode 100644
index 00000000..9c620785
--- /dev/null
+++ b/src/hyper/pages/REGSET.pht
@@ -0,0 +1,1012 @@
+\begin{patch}{RegularTriangularSetXmpPagePatch1}
+\begin{paste}{RegularTriangularSetXmpPageFull1}{RegularTriangularSetXmpPageEmpty1}
+\pastebutton{RegularTriangularSetXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{R := Integer\bound{R }}
+\indentrel{3}\begin{verbatim}
+   (1)  Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty1}
+\begin{paste}{RegularTriangularSetXmpPageEmpty1}{RegularTriangularSetXmpPagePatch1}
+\pastebutton{RegularTriangularSetXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{R := Integer\bound{R }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch2}
+\begin{paste}{RegularTriangularSetXmpPageFull2}{RegularTriangularSetXmpPageEmpty2}
+\pastebutton{RegularTriangularSetXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{ls : List Symbol := [x,y,z,t]\bound{ls }}
+\indentrel{3}\begin{verbatim}
+   (2)  [x,y,z,t]
+                                      Type: List Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty2}
+\begin{paste}{RegularTriangularSetXmpPageEmpty2}{RegularTriangularSetXmpPagePatch2}
+\pastebutton{RegularTriangularSetXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{ls : List Symbol := [x,y,z,t]\bound{ls }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch3}
+\begin{paste}{RegularTriangularSetXmpPageFull3}{RegularTriangularSetXmpPageEmpty3}
+\pastebutton{RegularTriangularSetXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
+\indentrel{3}\begin{verbatim}
+   (3)  OrderedVariableList [x,y,z,t]
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty3}
+\begin{paste}{RegularTriangularSetXmpPageEmpty3}{RegularTriangularSetXmpPagePatch3}
+\pastebutton{RegularTriangularSetXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch4}
+\begin{paste}{RegularTriangularSetXmpPageFull4}{RegularTriangularSetXmpPageEmpty4}
+\pastebutton{RegularTriangularSetXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{E := IndexedExponents V\free{V }\bound{E }}
+\indentrel{3}\begin{verbatim}
+   (4)  IndexedExponents OrderedVariableList [x,y,z,t]
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty4}
+\begin{paste}{RegularTriangularSetXmpPageEmpty4}{RegularTriangularSetXmpPagePatch4}
+\pastebutton{RegularTriangularSetXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{E := IndexedExponents V\free{V }\bound{E }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch5}
+\begin{paste}{RegularTriangularSetXmpPageFull5}{RegularTriangularSetXmpPageEmpty5}
+\pastebutton{RegularTriangularSetXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
+\indentrel{3}\begin{verbatim}
+   (5)
+  NewSparseMultivariatePolynomial(Integer,OrderedVariable
+  List [x,y,z,t])
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty5}
+\begin{paste}{RegularTriangularSetXmpPageEmpty5}{RegularTriangularSetXmpPagePatch5}
+\pastebutton{RegularTriangularSetXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch6}
+\begin{paste}{RegularTriangularSetXmpPageFull6}{RegularTriangularSetXmpPageEmpty6}
+\pastebutton{RegularTriangularSetXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{x: P := 'x\free{P }\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (6)  x
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty6}
+\begin{paste}{RegularTriangularSetXmpPageEmpty6}{RegularTriangularSetXmpPagePatch6}
+\pastebutton{RegularTriangularSetXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{x: P := 'x\free{P }\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch7}
+\begin{paste}{RegularTriangularSetXmpPageFull7}{RegularTriangularSetXmpPageEmpty7}
+\pastebutton{RegularTriangularSetXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{y: P := 'y\free{P }\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (7)  y
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty7}
+\begin{paste}{RegularTriangularSetXmpPageEmpty7}{RegularTriangularSetXmpPagePatch7}
+\pastebutton{RegularTriangularSetXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{y: P := 'y\free{P }\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch8}
+\begin{paste}{RegularTriangularSetXmpPageFull8}{RegularTriangularSetXmpPageEmpty8}
+\pastebutton{RegularTriangularSetXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{z: P := 'z\free{P }\bound{z }}
+\indentrel{3}\begin{verbatim}
+   (8)  z
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty8}
+\begin{paste}{RegularTriangularSetXmpPageEmpty8}{RegularTriangularSetXmpPagePatch8}
+\pastebutton{RegularTriangularSetXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{z: P := 'z\free{P }\bound{z }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch9}
+\begin{paste}{RegularTriangularSetXmpPageFull9}{RegularTriangularSetXmpPageEmpty9}
+\pastebutton{RegularTriangularSetXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{t: P := 't\free{P }\bound{t }}
+\indentrel{3}\begin{verbatim}
+   (9)  t
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty9}
+\begin{paste}{RegularTriangularSetXmpPageEmpty9}{RegularTriangularSetXmpPagePatch9}
+\pastebutton{RegularTriangularSetXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{t: P := 't\free{P }\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch10}
+\begin{paste}{RegularTriangularSetXmpPageFull10}{RegularTriangularSetXmpPageEmpty10}
+\pastebutton{RegularTriangularSetXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{T := REGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{T }}
+\indentrel{3}\begin{verbatim}
+   (10)
+  RegularTriangularSet(Integer,IndexedExponents OrderedVa
+  riableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewS
+  parseMultivariatePolynomial(Integer,OrderedVariableList
+   [x,y,z,t]))
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty10}
+\begin{paste}{RegularTriangularSetXmpPageEmpty10}{RegularTriangularSetXmpPagePatch10}
+\pastebutton{RegularTriangularSetXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{T := REGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{T }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch11}
+\begin{paste}{RegularTriangularSetXmpPageFull11}{RegularTriangularSetXmpPageEmpty11}
+\pastebutton{RegularTriangularSetXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{p1 := x ** 31 - x ** 6 - x - y\free{x }\free{y }\bound{p1 }}
+\indentrel{3}\begin{verbatim}
+          31    6
+   (11)  x   - x  - x - y
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty11}
+\begin{paste}{RegularTriangularSetXmpPageEmpty11}{RegularTriangularSetXmpPagePatch11}
+\pastebutton{RegularTriangularSetXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{p1 := x ** 31 - x ** 6 - x - y\free{x }\free{y }\bound{p1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch12}
+\begin{paste}{RegularTriangularSetXmpPageFull12}{RegularTriangularSetXmpPageEmpty12}
+\pastebutton{RegularTriangularSetXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{p2 := x ** 8 - z\free{x }\free{z }\bound{p2 }}
+\indentrel{3}\begin{verbatim}
+          8
+   (12)  x  - z
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty12}
+\begin{paste}{RegularTriangularSetXmpPageEmpty12}{RegularTriangularSetXmpPagePatch12}
+\pastebutton{RegularTriangularSetXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{p2 := x ** 8 - z\free{x }\free{z }\bound{p2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch13}
+\begin{paste}{RegularTriangularSetXmpPageFull13}{RegularTriangularSetXmpPageEmpty13}
+\pastebutton{RegularTriangularSetXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{p3 := x ** 10 - t\free{x }\free{t }\bound{p3 }}
+\indentrel{3}\begin{verbatim}
+          10
+   (13)  x   - t
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty13}
+\begin{paste}{RegularTriangularSetXmpPageEmpty13}{RegularTriangularSetXmpPagePatch13}
+\pastebutton{RegularTriangularSetXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{p3 := x ** 10 - t\free{x }\free{t }\bound{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch14}
+\begin{paste}{RegularTriangularSetXmpPageFull14}{RegularTriangularSetXmpPageEmpty14}
+\pastebutton{RegularTriangularSetXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
+\indentrel{3}\begin{verbatim}
+           31    6          8      10
+   (14)  [x   - x  - x - y,x  - z,x   - t]
+Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty14}
+\begin{paste}{RegularTriangularSetXmpPageEmpty14}{RegularTriangularSetXmpPagePatch14}
+\pastebutton{RegularTriangularSetXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch15}
+\begin{paste}{RegularTriangularSetXmpPageFull15}{RegularTriangularSetXmpPageEmpty15}
+\pastebutton{RegularTriangularSetXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{zeroSetSplit(lp)$T\free{lp }\free{T }}
+\indentrel{3}\begin{verbatim}
+   (15)
+   [
+       5    4       2     3     8     5    3    2
+     {z  - t , t z y  + 2z y - t  + 2t  + t  - t ,
+        4                2
+      (t  - t)x - t y - z }
+     ]
+Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty15}
+\begin{paste}{RegularTriangularSetXmpPageEmpty15}{RegularTriangularSetXmpPagePatch15}
+\pastebutton{RegularTriangularSetXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{zeroSetSplit(lp)$T\free{lp }\free{T }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch16}
+\begin{paste}{RegularTriangularSetXmpPageFull16}{RegularTriangularSetXmpPageEmpty16}
+\pastebutton{RegularTriangularSetXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{lts := zeroSetSplit(lp,false)$T\free{lp }\free{T }\bound{lts }}
+\indentrel{3}\begin{verbatim}
+   (16)
+   [
+       5    4       2     3     8     5    3    2
+     {z  - t , t z y  + 2z y - t  + 2t  + t  - t ,
+        4                2
+      (t  - t)x - t y - z }
+     ,
+      3      5          2     3         2
+    {t  - 1,z  - t,t z y  + 2z y + 1,z x  - t},
+    {t,z,y,x}]
+Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty16}
+\begin{paste}{RegularTriangularSetXmpPageEmpty16}{RegularTriangularSetXmpPagePatch16}
+\pastebutton{RegularTriangularSetXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{lts := zeroSetSplit(lp,false)$T\free{lp }\free{T }\bound{lts }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch17}
+\begin{paste}{RegularTriangularSetXmpPageFull17}{RegularTriangularSetXmpPageEmpty17}
+\pastebutton{RegularTriangularSetXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{[coHeight(ts) for ts in lts]\free{lts }}
+\indentrel{3}\begin{verbatim}
+   (17)  [1,0,0]
+                          Type: List NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty17}
+\begin{paste}{RegularTriangularSetXmpPageEmpty17}{RegularTriangularSetXmpPagePatch17}
+\pastebutton{RegularTriangularSetXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{[coHeight(ts) for ts in lts]\free{lts }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch18}
+\begin{paste}{RegularTriangularSetXmpPageFull18}{RegularTriangularSetXmpPageEmpty18}
+\pastebutton{RegularTriangularSetXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{f1 := y**2*z+2*x*y*t-2*x-z\free{z }\free{x }\free{y }\free{t }\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+                          2
+   (18)  (2t y - 2)x + z y  - z
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty18}
+\begin{paste}{RegularTriangularSetXmpPageEmpty18}{RegularTriangularSetXmpPagePatch18}
+\pastebutton{RegularTriangularSetXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{f1 := y**2*z+2*x*y*t-2*x-z\free{z }\free{x }\free{y }\free{t }\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch19}
+\begin{paste}{RegularTriangularSetXmpPageFull19}{RegularTriangularSetXmpPageEmpty19}
+\pastebutton{RegularTriangularSetXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{f2 := -x**3*z+ 4*x*y**2*z+ 4*x**2*y*t+ 2*y**3*t+ 4*x**2- 10*y**2+ 4*x*z- 10*y*t+ 2\free{z }\free{x }\free{y }\free{t }\bound{f2 }}
+\indentrel{3}\begin{verbatim}
+   (19)
+          3              2        2              3      2
+     - z x  + (4t y + 4)x  + (4z y  + 4z)x + 2t y  - 10y
+   + 
+     - 10t y + 2
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty19}
+\begin{paste}{RegularTriangularSetXmpPageEmpty19}{RegularTriangularSetXmpPagePatch19}
+\pastebutton{RegularTriangularSetXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{f2 := -x**3*z+ 4*x*y**2*z+ 4*x**2*y*t+ 2*y**3*t+ 4*x**2- 10*y**2+ 4*x*z- 10*y*t+ 2\free{z }\free{x }\free{y }\free{t }\bound{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch20}
+\begin{paste}{RegularTriangularSetXmpPageFull20}{RegularTriangularSetXmpPageEmpty20}
+\pastebutton{RegularTriangularSetXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{f3 := 2*y*z*t+x*t**2-x-2*z\free{z }\free{x }\free{y }\free{t }\bound{f3 }}
+\indentrel{3}\begin{verbatim}
+           2
+   (20)  (t  - 1)x + 2t z y - 2z
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty20}
+\begin{paste}{RegularTriangularSetXmpPageEmpty20}{RegularTriangularSetXmpPagePatch20}
+\pastebutton{RegularTriangularSetXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{f3 := 2*y*z*t+x*t**2-x-2*z\free{z }\free{x }\free{y }\free{t }\bound{f3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch21}
+\begin{paste}{RegularTriangularSetXmpPageFull21}{RegularTriangularSetXmpPageEmpty21}
+\pastebutton{RegularTriangularSetXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{f4 := -x*z**3+ 4*y*z**2*t+ 4*x*z*t**2+ 2*y*t**3+ 4*x*z+ 4*z**2-10*y*t- 10*t**2+2\free{z }\free{x }\free{y }\free{t }\bound{f4 }}
+\indentrel{3}\begin{verbatim}
+   (21)
+         3      2                2     3             2
+     (- z  + (4t  + 4)z)x + (4t z  + 2t  - 10t)y + 4z
+   + 
+          2
+     - 10t  + 2
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty21}
+\begin{paste}{RegularTriangularSetXmpPageEmpty21}{RegularTriangularSetXmpPagePatch21}
+\pastebutton{RegularTriangularSetXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{f4 := -x*z**3+ 4*y*z**2*t+ 4*x*z*t**2+ 2*y*t**3+ 4*x*z+ 4*z**2-10*y*t- 10*t**2+2\free{z }\free{x }\free{y }\free{t }\bound{f4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch22}
+\begin{paste}{RegularTriangularSetXmpPageFull22}{RegularTriangularSetXmpPageEmpty22}
+\pastebutton{RegularTriangularSetXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{lf := [f1, f2, f3, f4]\free{f1 }\free{f2 }\free{f3 }\free{f4 }\bound{lf }}
+\indentrel{3}\begin{verbatim}
+   (22)
+                     2
+   [(2t y - 2)x + z y  - z,
+
+            3              2        2              3
+       - z x  + (4t y + 4)x  + (4z y  + 4z)x + 2t y
+     + 
+            2
+       - 10y  - 10t y + 2
+     ,
+      2
+    (t  - 1)x + 2t z y - 2z,
+
+           3      2                2     3             2
+       (- z  + (4t  + 4)z)x + (4t z  + 2t  - 10t)y + 4z
+     + 
+            2
+       - 10t  + 2
+     ]
+Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty22}
+\begin{paste}{RegularTriangularSetXmpPageEmpty22}{RegularTriangularSetXmpPagePatch22}
+\pastebutton{RegularTriangularSetXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{lf := [f1, f2, f3, f4]\free{f1 }\free{f2 }\free{f3 }\free{f4 }\bound{lf }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch23}
+\begin{paste}{RegularTriangularSetXmpPageFull23}{RegularTriangularSetXmpPageEmpty23}
+\pastebutton{RegularTriangularSetXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{zeroSetSplit(lf)$T\free{lf }\free{T }}
+\indentrel{3}\begin{verbatim}
+   (23)
+   [
+       2       8      6       2
+     {t  - 1, z  - 16z  + 256z  - 256, t y - 1,
+        3            2
+      (z  - 8z)x - 8z  + 16}
+     ,
+       2      2     2
+    {3t  + 1,z  - 7t  - 1,y + t,x + z},
+      8      6      2         3            2
+    {t  - 10t  + 10t  - 1,z,(t  - 5t)y - 5t  + 1,x},
+      2      2
+    {t  + 3,z  - 4,y + t,x - z}]
+Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty23}
+\begin{paste}{RegularTriangularSetXmpPageEmpty23}{RegularTriangularSetXmpPagePatch23}
+\pastebutton{RegularTriangularSetXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{zeroSetSplit(lf)$T\free{lf }\free{T }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch24}
+\begin{paste}{RegularTriangularSetXmpPageFull24}{RegularTriangularSetXmpPageEmpty24}
+\pastebutton{RegularTriangularSetXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{lts2 := zeroSetSplit(lf,false)$T\free{lf }\free{T }\bound{lts2 }}
+\indentrel{3}\begin{verbatim}
+   (24)
+      8      6      2         3            2
+   [{t  - 10t  + 10t  - 1,z,(t  - 5t)y - 5t  + 1,x},
+
+       2       8      6       2
+     {t  - 1, z  - 16z  + 256z  - 256, t y - 1,
+        3            2
+      (z  - 8z)x - 8z  + 16}
+     ,
+       2      2     2
+    {3t  + 1,z  - 7t  - 1,y + t,x + z},
+      2      2
+    {t  + 3,z  - 4,y + t,x - z}]
+Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty24}
+\begin{paste}{RegularTriangularSetXmpPageEmpty24}{RegularTriangularSetXmpPagePatch24}
+\pastebutton{RegularTriangularSetXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{lts2 := zeroSetSplit(lf,false)$T\free{lf }\free{T }\bound{lts2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch25}
+\begin{paste}{RegularTriangularSetXmpPageFull25}{RegularTriangularSetXmpPageEmpty25}
+\pastebutton{RegularTriangularSetXmpPageFull25}{\hidepaste}
+\tab{5}\spadcommand{[coHeight(ts) for ts in lts2]\free{lts2 }}
+\indentrel{3}\begin{verbatim}
+   (25)  [0,0,0,0]
+                          Type: List NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty25}
+\begin{paste}{RegularTriangularSetXmpPageEmpty25}{RegularTriangularSetXmpPagePatch25}
+\pastebutton{RegularTriangularSetXmpPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{[coHeight(ts) for ts in lts2]\free{lts2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch26}
+\begin{paste}{RegularTriangularSetXmpPageFull26}{RegularTriangularSetXmpPageEmpty26}
+\pastebutton{RegularTriangularSetXmpPageFull26}{\hidepaste}
+\tab{5}\spadcommand{degrees := [degree(ts) for ts in lts2]\free{lts2 }\bound{degrees }}
+\indentrel{3}\begin{verbatim}
+   (26)  [8,16,4,4]
+                          Type: List NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty26}
+\begin{paste}{RegularTriangularSetXmpPageEmpty26}{RegularTriangularSetXmpPagePatch26}
+\pastebutton{RegularTriangularSetXmpPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{degrees := [degree(ts) for ts in lts2]\free{lts2 }\bound{degrees }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch27}
+\begin{paste}{RegularTriangularSetXmpPageFull27}{RegularTriangularSetXmpPageEmpty27}
+\pastebutton{RegularTriangularSetXmpPageFull27}{\hidepaste}
+\tab{5}\spadcommand{reduce(+,degrees)\free{degrees }}
+\indentrel{3}\begin{verbatim}
+   (27)  32
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty27}
+\begin{paste}{RegularTriangularSetXmpPageEmpty27}{RegularTriangularSetXmpPagePatch27}
+\pastebutton{RegularTriangularSetXmpPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{reduce(+,degrees)\free{degrees }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch28}
+\begin{paste}{RegularTriangularSetXmpPageFull28}{RegularTriangularSetXmpPageEmpty28}
+\pastebutton{RegularTriangularSetXmpPageFull28}{\hidepaste}
+\tab{5}\spadcommand{u : R := 2\free{R }\bound{u }}
+\indentrel{3}\begin{verbatim}
+   (28)  2
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty28}
+\begin{paste}{RegularTriangularSetXmpPageEmpty28}{RegularTriangularSetXmpPagePatch28}
+\pastebutton{RegularTriangularSetXmpPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{u : R := 2\free{R }\bound{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch29}
+\begin{paste}{RegularTriangularSetXmpPageFull29}{RegularTriangularSetXmpPageEmpty29}
+\pastebutton{RegularTriangularSetXmpPageFull29}{\hidepaste}
+\tab{5}\spadcommand{q1 := 2*(u-1)**2+ 2*(x-z*x+z**2)+ y**2*(x-1)**2- 2*u*x+ 2*y*t*(1-x)*(x-z)+ 2*u*z*t*(t-y)+ u**2*t**2*(1-2*z)+ 2*u*t**2*(z-x)+ 2*u*t*y*(z-1)+ 2*u*z*x*(y+1)+ (u**2-2*u)*z**2*t**2+ 2*u**2*z**2+ 4*u*(1-u)*z+ t**2*(z-x)**2\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q1 }}
+\indentrel{3}\begin{verbatim}
+   (29)
+       2           2  2
+     (y  - 2t y + t )x
+   + 
+          2                            2           2
+     (- 2y  + ((2t + 4)z + 2t)y + (- 2t  + 2)z - 4t  - 2)x
+   + 
+      2                      2       2          2
+     y  + (- 2t z - 4t)y + (t  + 10)z  - 8z + 4t  + 2
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty29}
+\begin{paste}{RegularTriangularSetXmpPageEmpty29}{RegularTriangularSetXmpPagePatch29}
+\pastebutton{RegularTriangularSetXmpPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{q1 := 2*(u-1)**2+ 2*(x-z*x+z**2)+ y**2*(x-1)**2- 2*u*x+ 2*y*t*(1-x)*(x-z)+ 2*u*z*t*(t-y)+ u**2*t**2*(1-2*z)+ 2*u*t**2*(z-x)+ 2*u*t*y*(z-1)+ 2*u*z*x*(y+1)+ (u**2-2*u)*z**2*t**2+ 2*u**2*z**2+ 4*u*(1-u)*z+ t**2*(z-x)**2\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch30}
+\begin{paste}{RegularTriangularSetXmpPageFull30}{RegularTriangularSetXmpPageEmpty30}
+\pastebutton{RegularTriangularSetXmpPageFull30}{\hidepaste}
+\tab{5}\spadcommand{q2 := t*(2*z+1)*(x-z)+ y*(z+2)*(1-x)+ u*(u-2)*t+ u*(1-2*u)*z*t+ u*y*(x+u-z*x-1)+ u*(u+1)*z**2*t\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q2 }}
+\indentrel{3}\begin{verbatim}
+                                               2
+   (30)  (- 3z y + 2t z + t)x + (z + 4)y + 4t z  - 7t z
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty30}
+\begin{paste}{RegularTriangularSetXmpPageEmpty30}{RegularTriangularSetXmpPagePatch30}
+\pastebutton{RegularTriangularSetXmpPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{q2 := t*(2*z+1)*(x-z)+ y*(z+2)*(1-x)+ u*(u-2)*t+ u*(1-2*u)*z*t+ u*y*(x+u-z*x-1)+ u*(u+1)*z**2*t\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch31}
+\begin{paste}{RegularTriangularSetXmpPageFull31}{RegularTriangularSetXmpPageEmpty31}
+\pastebutton{RegularTriangularSetXmpPageFull31}{\hidepaste}
+\tab{5}\spadcommand{q3 := -u**2*(z-1)**2+ 2*z*(z-x)-2*(x-1)\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q3 }}
+\indentrel{3}\begin{verbatim}
+                         2
+   (31)  (- 2z - 2)x - 2z  + 8z - 2
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty31}
+\begin{paste}{RegularTriangularSetXmpPageEmpty31}{RegularTriangularSetXmpPagePatch31}
+\pastebutton{RegularTriangularSetXmpPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{q3 := -u**2*(z-1)**2+ 2*z*(z-x)-2*(x-1)\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch32}
+\begin{paste}{RegularTriangularSetXmpPageFull32}{RegularTriangularSetXmpPageEmpty32}
+\pastebutton{RegularTriangularSetXmpPageFull32}{\hidepaste}
+\tab{5}\spadcommand{q4 := u**2+4*(z-x**2)+3*y**2*(x-1)**2- 3*t**2*(z-x)**2 +3*u**2*t**2*(z-1)**2+u**2*z*(z-2)+6*u*t*y*(z+x+z*x-1)\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q4 }}
+\indentrel{3}\begin{verbatim}
+   (32)
+        2     2      2        2                      2
+     (3y  - 3t  - 4)x  + (- 6y  + (12t z + 12t)y + 6t z)x
+   + 
+       2                       2      2         2
+     3y  + (12t z - 12t)y + (9t  + 4)z  + (- 24t  - 4)z
+   + 
+        2
+     12t  + 4
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty32}
+\begin{paste}{RegularTriangularSetXmpPageEmpty32}{RegularTriangularSetXmpPagePatch32}
+\pastebutton{RegularTriangularSetXmpPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{q4 := u**2+4*(z-x**2)+3*y**2*(x-1)**2- 3*t**2*(z-x)**2 +3*u**2*t**2*(z-1)**2+u**2*z*(z-2)+6*u*t*y*(z+x+z*x-1)\free{z }\free{x }\free{y }\free{t }\free{u }\bound{q4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch33}
+\begin{paste}{RegularTriangularSetXmpPageFull33}{RegularTriangularSetXmpPageEmpty33}
+\pastebutton{RegularTriangularSetXmpPageFull33}{\hidepaste}
+\tab{5}\spadcommand{lq := [q1, q2, q3, q4]\free{q1 }\free{q2 }\free{q3 }\free{q4 }\bound{lq }}
+\indentrel{3}\begin{verbatim}
+   (33)
+   [
+         2           2  2
+       (y  - 2t y + t )x
+     + 
+               2                            2           2
+           - 2y  + ((2t + 4)z + 2t)y + (- 2t  + 2)z - 4t
+         + 
+           - 2
+      *
+         x
+     + 
+        2                      2       2          2
+       y  + (- 2t z - 4t)y + (t  + 10)z  - 8z + 4t  + 2
+     ,
+                                          2
+    (- 3z y + 2t z + t)x + (z + 4)y + 4t z  - 7t z,
+                    2
+    (- 2z - 2)x - 2z  + 8z - 2,
+
+          2     2      2
+       (3y  - 3t  - 4)x
+     + 
+            2                      2        2
+       (- 6y  + (12t z + 12t)y + 6t z)x + 3y
+     + 
+                           2      2         2            2
+       (12t z - 12t)y + (9t  + 4)z  + (- 24t  - 4)z + 12t
+     + 
+       4
+     ]
+Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty33}
+\begin{paste}{RegularTriangularSetXmpPageEmpty33}{RegularTriangularSetXmpPagePatch33}
+\pastebutton{RegularTriangularSetXmpPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{lq := [q1, q2, q3, q4]\free{q1 }\free{q2 }\free{q3 }\free{q4 }\bound{lq }}
+\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPagePatch34}
+\begin{paste}{RegularTriangularSetXmpPageFull34}{RegularTriangularSetXmpPageEmpty34}
+\pastebutton{RegularTriangularSetXmpPageFull34}{\hidepaste}
+\tab{5}\spadcommand{zeroSetSplit(lq,true,true)$T\free{lq }\free{T }}
+\indentrel{3}\begin{verbatim}
+
+   *** QCMPACK Statistics ***
+      Table     size:  36
+      Entries reused:  255
+
+   *** REGSETGCD: Gcd Statistics ***
+      Table     size:  125
+      Entries reused:  0
+
+   *** REGSETGCD: Inv Set Statistics ***
+      Table     size:  30
+      Entries reused:  0
+   (34)
+   [
+     {
+                         24                   23
+         960725655771966t   + 386820897948702t
+       + 
+                          22                    21
+         8906817198608181t   + 2704966893949428t
+       + 
+                           20                    19
+         37304033340228264t   + 7924782817170207t
+       + 
+                           18                     17
+         93126799040354990t   + 13101273653130910t
+       + 
+                            16                     15
+         156146250424711858t   + 16626490957259119t
+       + 
+                            14                     13
+         190699288479805763t   + 24339173367625275t
+       + 
+                            12                     11
+         180532313014960135t   + 35288089030975378t
+       + 
+                            10                     9
+         135054975747656285t   + 34733736952488540t
+       + 
+                           8                     7
+         75947600354493972t  + 19772555692457088t
+       + 
+                           6                    5
+         28871558573755428t  + 5576152439081664t
+       + 
+                          4                   3
+         6321711820352976t  + 438314209312320t
+       + 
+                       2
+       581105748367008t  - 60254467992576t + 1449115951104
+       ,
+
+              266042108694913023855152657370520823616684_
+               74181372891857784
+            *
+                23
+               t
+           + 
+              443104378424686086067294899528296664238693_
+               556855017735265295
+            *
+                22
+               t
+           + 
+              279078393286701234679141342358988327155321_
+               305829547090310242
+            *
+                21
+               t
+           + 
+              339027636141323246510761717661554305462062_
+               6391823613392185226
+            *
+                20
+               t
+           + 
+              941478179503540575554198645220352803719793_
+               196473813837434129
+            *
+                19
+               t
+           + 
+              115478551946794752422116967496739493525857_
+               47674184320988144390
+            *
+                18
+               t
+           + 
+              134360956676559778988170165669941321646721_
+               5660333356417241432
+            *
+                17
+               t
+           + 
+              232338138681478735039335516171756408598991_
+               02987800663566699334
+            *
+                16
+               t
+           + 
+              869574020537672336950845440508790740850931_
+               336484983573386433
+            *
+                15
+               t
+           + 
+              315615543058769348754194614869699265542417_
+               50065103460820476969
+            *
+                14
+               t
+           + 
+              127140099028771748744206595254773187955482_
+               3889855386072264931
+            *
+                13
+               t
+           + 
+              319450899138637360448025269640795401983370_
+               49550503295825160523
+            *
+                12
+               t
+           + 
+              373873570428814450987137156023284588443910_
+               2270778010470931960
+            *
+                11
+               t
+           + 
+              252939975123914120261446014357711315875619_
+               05532992045692885927
+            *
+                10
+               t
+           + 
+              521023900984606712346926279987005277341047_
+               1135950175008046524
+            *
+                9
+               t
+           + 
+              150838879869302971662598705686082704274031_
+               87606238713491129188
+            *
+                8
+               t
+           + 
+              352208723469293012638368627077577955348176_
+               9125670839075109000
+            *
+                7
+               t
+           + 
+              607994520039568101308653379256888649110124_
+               4247440034969288588
+            *
+                6
+               t
+           + 
+              109063485243390088819991375624798602319698_
+               7723469934933603680
+            *
+                5
+               t
+           + 
+              140581943087190710229443253753833540210283_
+               8994019667487458352
+            *
+                4
+               t
+           + 
+              880715279503204500725366712655077488783478_
+               28884933605202432
+            *
+                3
+               t
+           + 
+              135882489433640933229781177155977768016065_
+               765482378657129440
+            *
+                2
+               t
+           + 
+             -
+                 139572834428822622305598946074003140825_
+                  16690749975646520320
+               *
+                  t
+           + 
+            33463769297318929927725832570930847259211711_
+             2855749713920
+        *
+           z
+       + 
+          8567175484043952879756725964506833932149637101_
+           090521164936
+        *
+            23
+           t
+       + 
+          1497923928642017918457083740327289424987975192_
+           51667250945721
+        *
+            22
+           t
+       + 
+          7725837178364582215741086158215976413812300307_
+           4190374021550
+        *
+            21
+           t
+       + 
+          1108862254126854214498918940708612211184560556_
+           764334742191654
+        *
+            20
+           t
+       + 
+          2132504944606788652197744801068260537838157896_
+           21501732672327
+        *
+            19
+           t
+       + 
+          3668929075160666195729177894178343514501987898_
+           410131431699882
+        *
+            18
+           t
+       + 
+          1713889064710018728794901243687482363147654590_
+           39567820048872
+        *
+            17
+           t
+       + 
+          7192430746914602166660233477331022483144921771_
+           645523139658986
+        *
+            16
+           t
+       + 
+         -
+             1287986746896900728128799656330902919596631_
+              43108437362453385
+           *
+               15
+              t
+       + 
+          9553010858341425909306423132921134040856028790_
+           803526430270671
+        *
+            14
+           t
+       + 
+         -
+             1329609624567549287453868764630043782465845_
+              8709144441096603
+           *
+               13
+              t
+       + 
+          9475806805814145326383085518325333106881690568_
+           644274964864413
+        *
+            12
+           t
+       + 
+          8032346879251334588616598556640849276062987947_
+           99856265539336
+        *
+            11
+           t
+       + 
+          7338202759292865165994622349207516400662174302_
+           614595173333825
+        *
+            10
+           t
+       + 
+          1308004628480367351164369613111971668880538855_
+           640917200187108
+        *
+            9
+           t
+       + 
+          4268059455741255498880229598973705747098216067_
+           697754352634748
+        *
+            8
+           t
+       + 
+          8928935268585140957913187759040933001030456015_
+           14470613580600
+        *
+            7
+           t
+       + 
+          1679152575460683956631925852181341501981598137_
+           465328797013652
+        *
+            6
+           t
+       + 
+          2697574157679229803789671541433578355441131582_
+           80591408043936
+        *
+            5
+           t
+       + 
+          3809515278646575290335808298012827240813453726_
+           80202920198224
+        *
+            4
+           t
+       + 
+          1978554529422849503299882693760134113272503533_
+           9452913286656
+        *
+            3
+           t
+       + 
+          3647741205738478294236663530339663776330392817_
+           4935079178528
+        *
+            2
+           t
+       + 
+         -
+             3722212879279038648713080422224976273210890_
+              229485838670848
+           *
+              t
+       + 
+        890797248531143483612306344840138620247285999068_
+         74105856
+       ,
+         3      2                  3       2
+      (3z  - 11z  + 8z + 4)y + 2t z  + 4t z  - 5t z - t,
+                  2
+      (z + 1)x + z  - 4z + 1}
+     ]
+Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RegularTriangularSetXmpPageEmpty34}
+\begin{paste}{RegularTriangularSetXmpPageEmpty34}{RegularTriangularSetXmpPagePatch34}
+\pastebutton{RegularTriangularSetXmpPageEmpty34}{\showpaste}
+\tab{5}\spadcommand{zeroSetSplit(lq,true,true)$T\free{lq }\free{T }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ROMAN.ht b/src/hyper/pages/ROMAN.ht
new file mode 100644
index 00000000..44c58a64
--- /dev/null
+++ b/src/hyper/pages/ROMAN.ht
@@ -0,0 +1,84 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\RomanNumeralXmpTitle}{RomanNumeral}
+\newcommand{\RomanNumeralXmpNumber}{9.68}
+%
+% =====================================================================
+\begin{page}{RomanNumeralXmpPage}{9.68 RomanNumeral}
+% =====================================================================
+\beginscroll
+
+The Roman numeral package was added to \Language{} in MCMLXXXVI
+%-% \HDindex{Roman numerals}{RomanNumeralXmpPage}{9.68}{RomanNumeral}
+for use in denoting higher order derivatives.
+
+\xtc{
+For example, let \spad{f} be a symbolic operator.
+}{
+\spadpaste{f := operator 'f \bound{f}}
+}
+\xtc{
+This is the seventh derivative of \spad{f} with respect to \spad{x}.
+}{
+\spadpaste{D(f x,x,7) \free{f}}
+}
+\xtc{
+You can have integers printed as Roman numerals by declaring variables to
+be of type \spadtype{RomanNumeral} (abbreviation \spadtype{ROMAN}).
+}{
+\spadpaste{a := roman(1978 - 1965) \bound{a}}
+}
+
+This package now has a small but devoted group of followers that claim
+this domain has shown its efficacy in many other contexts.
+They claim that Roman numerals are every bit as useful as ordinary
+integers.
+\xtc{
+In a sense, they are correct, because Roman numerals form a ring and you
+can therefore construct polynomials with Roman numeral coefficients,
+matrices over Roman numerals, etc..
+}{
+\spadpaste{x : UTS(ROMAN,'x,0) := x \bound{x}}
+}
+\xtc{
+Was Fibonacci Italian or ROMAN?
+%-% \HDindex{Fibonacci numbers}{RomanNumeralXmpPage}{9.68}{RomanNumeral}
+}{
+\spadpaste{recip(1 - x - x**2) \free{x}}
+}
+\xtc{
+You can also construct fractions with Roman numeral numerators and
+denominators, as this matrix Hilberticus illustrates.
+}{
+\spadpaste{m : MATRIX FRAC ROMAN \bound{m}}
+}
+\xtc{
+}{
+\spadpaste{m := matrix [[1/(i + j) for i in 1..3] for j in 1..3] \free{m} \bound{m1}}
+}
+\xtc{
+Note that the inverse of the matrix has integral \spadtype{ROMAN} entries.
+}{
+\spadpaste{inverse m \free{m1}}
+}
+\xtc{
+Unfortunately, the spoil-sports say that the fun stops when
+the numbers get big---mostly
+because the Romans didn't establish conventions about representing
+very large numbers.
+}{
+\spadpaste{y := factorial 10 \bound{y}}
+}
+\xtc{
+You work it out!
+}{
+\spadpaste{roman y \free{y}}
+}
+Issue the system command
+\spadcmd{)show RomanNumeral}
+to display the full list of operations defined by
+\spadtype{RomanNumeral}.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ROMAN.pht b/src/hyper/pages/ROMAN.pht
new file mode 100644
index 00000000..7983f3bf
--- /dev/null
+++ b/src/hyper/pages/ROMAN.pht
@@ -0,0 +1,182 @@
+\begin{patch}{RomanNumeralXmpPagePatch1}
+\begin{paste}{RomanNumeralXmpPageFull1}{RomanNumeralXmpPageEmpty1}
+\pastebutton{RomanNumeralXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{f := operator 'f\bound{f }}
+\indentrel{3}\begin{verbatim}
+   (1)  f
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPageEmpty1}
+\begin{paste}{RomanNumeralXmpPageEmpty1}{RomanNumeralXmpPagePatch1}
+\pastebutton{RomanNumeralXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{f := operator 'f\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPagePatch2}
+\begin{paste}{RomanNumeralXmpPageFull2}{RomanNumeralXmpPageEmpty2}
+\pastebutton{RomanNumeralXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{D(f x,x,7)\free{f }}
+\indentrel{3}\begin{verbatim}
+         (vii)
+   (2)  f     (x)
+
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPageEmpty2}
+\begin{paste}{RomanNumeralXmpPageEmpty2}{RomanNumeralXmpPagePatch2}
+\pastebutton{RomanNumeralXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{D(f x,x,7)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPagePatch3}
+\begin{paste}{RomanNumeralXmpPageFull3}{RomanNumeralXmpPageEmpty3}
+\pastebutton{RomanNumeralXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{a := roman(1978 - 1965)\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (3)  XIII
+                                     Type: RomanNumeral
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPageEmpty3}
+\begin{paste}{RomanNumeralXmpPageEmpty3}{RomanNumeralXmpPagePatch3}
+\pastebutton{RomanNumeralXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{a := roman(1978 - 1965)\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPagePatch4}
+\begin{paste}{RomanNumeralXmpPageFull4}{RomanNumeralXmpPageEmpty4}
+\pastebutton{RomanNumeralXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{x : UTS(ROMAN,'x,0) := x\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (4)  x
+         Type: UnivariateTaylorSeries(RomanNumeral,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPageEmpty4}
+\begin{paste}{RomanNumeralXmpPageEmpty4}{RomanNumeralXmpPagePatch4}
+\pastebutton{RomanNumeralXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{x : UTS(ROMAN,'x,0) := x\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPagePatch5}
+\begin{paste}{RomanNumeralXmpPageFull5}{RomanNumeralXmpPageEmpty5}
+\pastebutton{RomanNumeralXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{recip(1 - x - x**2)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (5)
+                 2        3      4         5         6
+     I + x + II x  + III x  + V x  + VIII x  + XIII x
+   + 
+          7          8       9           10      11
+     XXI x  + XXXIV x  + LV x  + LXXXIX x   + O(x  )
+Type: Union(UnivariateTaylorSeries(RomanNumeral,x,0),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPageEmpty5}
+\begin{paste}{RomanNumeralXmpPageEmpty5}{RomanNumeralXmpPagePatch5}
+\pastebutton{RomanNumeralXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{recip(1 - x - x**2)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPagePatch6}
+\begin{paste}{RomanNumeralXmpPageFull6}{RomanNumeralXmpPageEmpty6}
+\pastebutton{RomanNumeralXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{m : MATRIX FRAC ROMAN\bound{m }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPageEmpty6}
+\begin{paste}{RomanNumeralXmpPageEmpty6}{RomanNumeralXmpPagePatch6}
+\pastebutton{RomanNumeralXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{m : MATRIX FRAC ROMAN\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPagePatch7}
+\begin{paste}{RomanNumeralXmpPageFull7}{RomanNumeralXmpPageEmpty7}
+\pastebutton{RomanNumeralXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{m := matrix [[1/(i + j) for i in 1..3] for j in 1..3]\free{m }\bound{m1 }}
+\indentrel{3}\begin{verbatim}
+        Ú I    I    I¿
+        ³ÄÄ   ÄÄÄ  Äij
+        ³II   III  IV³
+        ³            ³
+        ³ I    I   I ³
+   (7)  ³ÄÄÄ  ÄÄ   Ä ³
+        ³III  IV   V ³
+        ³            ³
+        ³ I    I    I³
+        ³ÄÄ    Ä   Äij
+        ÀIV    V   VIÙ
+                     Type: Matrix Fraction RomanNumeral
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPageEmpty7}
+\begin{paste}{RomanNumeralXmpPageEmpty7}{RomanNumeralXmpPagePatch7}
+\pastebutton{RomanNumeralXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{m := matrix [[1/(i + j) for i in 1..3] for j in 1..3]\free{m }\bound{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPagePatch8}
+\begin{paste}{RomanNumeralXmpPageFull8}{RomanNumeralXmpPageEmpty8}
+\pastebutton{RomanNumeralXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{inverse m\free{m1 }}
+\indentrel{3}\begin{verbatim}
+        ÚLXXII   - CCXL    CLXXX ¿
+        ³                        ³
+   (8)  ³- CCXL    CM     - DCCXX³
+        ³                        ³
+        ÀCLXXX   - DCCXX    DC   Ù
+          Type: Union(Matrix Fraction RomanNumeral,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPageEmpty8}
+\begin{paste}{RomanNumeralXmpPageEmpty8}{RomanNumeralXmpPagePatch8}
+\pastebutton{RomanNumeralXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{inverse m\free{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPagePatch9}
+\begin{paste}{RomanNumeralXmpPageFull9}{RomanNumeralXmpPageEmpty9}
+\pastebutton{RomanNumeralXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{y := factorial 10\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (9)  3628800
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPageEmpty9}
+\begin{paste}{RomanNumeralXmpPageEmpty9}{RomanNumeralXmpPagePatch9}
+\pastebutton{RomanNumeralXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{y := factorial 10\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPagePatch10}
+\begin{paste}{RomanNumeralXmpPageFull10}{RomanNumeralXmpPageEmpty10}
+\pastebutton{RomanNumeralXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{roman y\free{y }}
+\indentrel{3}\begin{verbatim}
+   (10)
+  ((((I))))((((I))))((((I)))) (((I)))(((I)))(((I)))(((I))
+  )(((I)))(((I))) ((I))((I)) MMMMMMMMDCCC
+                                     Type: RomanNumeral
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RomanNumeralXmpPageEmpty10}
+\begin{paste}{RomanNumeralXmpPageEmpty10}{RomanNumeralXmpPagePatch10}
+\pastebutton{RomanNumeralXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{roman y\free{y }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/SEG.ht b/src/hyper/pages/SEG.ht
new file mode 100644
index 00000000..03b5cbf7
--- /dev/null
+++ b/src/hyper/pages/SEG.ht
@@ -0,0 +1,89 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\SegmentXmpTitle}{Segment}
+\newcommand{\SegmentXmpNumber}{9.69}
+%
+% =====================================================================
+\begin{page}{SegmentXmpPage}{9.69 Segment}
+% =====================================================================
+\beginscroll
+
+The \spadtype{Segment} domain provides a generalized interval type.
+
+\labelSpace{2pc}
+\xtc{
+Segments are created using the \spadSyntax{..} construct
+by indicating the (included) end points.
+}{
+\spadpaste{s := 3..10 \bound{s}}
+}
+\xtc{
+The first end point is called the \spadfunFrom{lo}{Segment}
+and the second is called \spadfunFrom{hi}{Segment}.
+}{
+\spadpaste{lo s \free{s}}
+}
+\xtc{
+These names are used even though the end points might belong to an
+unordered set.
+}{
+\spadpaste{hi s \free{s}}
+}
+
+\xtc{
+In addition to the end points, each segment has an integer ``increment.''
+An increment can be specified using the ``\spad{by}'' construct.
+\spadkey{by}
+}{
+\spadpaste{t := 10..3 by -2 \bound{t}}
+}
+\xtc{
+This part can be obtained using the \spadfunFrom{incr}{Segment} function.
+}{
+\spadpaste{incr s \free{s}}
+}
+\xtc{
+Unless otherwise specified, the increment is \spad{1}.
+}{
+\spadpaste{incr t \free{t}}
+}
+
+\xtc{
+A single value can be converted to a segment with equal end points.
+This happens if segments and single values are mixed in a list.
+}{
+\spadpaste{l := [1..3, 5, 9, 15..11 by -1] \bound{l}}
+}
+
+\xtc{
+If the underlying type is an ordered ring, it is possible to perform
+additional operations.
+The \spadfunFrom{expand}{Segment} operation creates a list of
+points in a segment.
+}{
+\spadpaste{expand s \free{s}}
+}
+\xtc{
+If \spad{k > 0}, then \spad{expand(l..h by k)} creates the list
+\spad{[l, l+k, ..., lN]} where \spad{lN <= h < lN+k}.
+If \spad{k < 0}, then \spad{lN >= h > lN+k}.
+}{
+\spadpaste{expand t \free{t}}
+}
+
+\xtc{
+It is also possible to expand a list of segments.  This is equivalent
+to appending lists obtained by expanding each segment individually.
+}{
+\spadpaste{expand l \free{l}}
+}
+
+For more information on related topics, see
+\downlink{`SegmentBinding'}{SegmentBindingXmpPage}\ignore{SegmentBinding} and
+\downlink{`UniversalSegment'}{UniversalSegmentXmpPage}\ignore{UniversalSegment}.
+%
+\showBlurb{Segment}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/SEG.pht b/src/hyper/pages/SEG.pht
new file mode 100644
index 00000000..abaf399c
--- /dev/null
+++ b/src/hyper/pages/SEG.pht
@@ -0,0 +1,160 @@
+\begin{patch}{SegmentXmpPagePatch1}
+\begin{paste}{SegmentXmpPageFull1}{SegmentXmpPageEmpty1}
+\pastebutton{SegmentXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{s := 3..10\bound{s }}
+\indentrel{3}\begin{verbatim}
+   (1)  3..10
+                          Type: Segment PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPageEmpty1}
+\begin{paste}{SegmentXmpPageEmpty1}{SegmentXmpPagePatch1}
+\pastebutton{SegmentXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{s := 3..10\bound{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPagePatch2}
+\begin{paste}{SegmentXmpPageFull2}{SegmentXmpPageEmpty2}
+\pastebutton{SegmentXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{lo s\free{s }}
+\indentrel{3}\begin{verbatim}
+   (2)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPageEmpty2}
+\begin{paste}{SegmentXmpPageEmpty2}{SegmentXmpPagePatch2}
+\pastebutton{SegmentXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{lo s\free{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPagePatch3}
+\begin{paste}{SegmentXmpPageFull3}{SegmentXmpPageEmpty3}
+\pastebutton{SegmentXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{hi s\free{s }}
+\indentrel{3}\begin{verbatim}
+   (3)  10
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPageEmpty3}
+\begin{paste}{SegmentXmpPageEmpty3}{SegmentXmpPagePatch3}
+\pastebutton{SegmentXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{hi s\free{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPagePatch4}
+\begin{paste}{SegmentXmpPageFull4}{SegmentXmpPageEmpty4}
+\pastebutton{SegmentXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{t := 10..3 by -2\bound{t }}
+\indentrel{3}\begin{verbatim}
+   (4)  10..3 by - 2
+                          Type: Segment PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPageEmpty4}
+\begin{paste}{SegmentXmpPageEmpty4}{SegmentXmpPagePatch4}
+\pastebutton{SegmentXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{t := 10..3 by -2\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPagePatch5}
+\begin{paste}{SegmentXmpPageFull5}{SegmentXmpPageEmpty5}
+\pastebutton{SegmentXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{incr s\free{s }}
+\indentrel{3}\begin{verbatim}
+   (5)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPageEmpty5}
+\begin{paste}{SegmentXmpPageEmpty5}{SegmentXmpPagePatch5}
+\pastebutton{SegmentXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{incr s\free{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPagePatch6}
+\begin{paste}{SegmentXmpPageFull6}{SegmentXmpPageEmpty6}
+\pastebutton{SegmentXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{incr t\free{t }}
+\indentrel{3}\begin{verbatim}
+   (6)  - 2
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPageEmpty6}
+\begin{paste}{SegmentXmpPageEmpty6}{SegmentXmpPagePatch6}
+\pastebutton{SegmentXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{incr t\free{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPagePatch7}
+\begin{paste}{SegmentXmpPageFull7}{SegmentXmpPageEmpty7}
+\pastebutton{SegmentXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{l := [1..3, 5, 9, 15..11 by -1]\bound{l }}
+\indentrel{3}\begin{verbatim}
+   (7)  [1..3,5..5,9..9,15..11 by - 1]
+                     Type: List Segment PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPageEmpty7}
+\begin{paste}{SegmentXmpPageEmpty7}{SegmentXmpPagePatch7}
+\pastebutton{SegmentXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{l := [1..3, 5, 9, 15..11 by -1]\bound{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPagePatch8}
+\begin{paste}{SegmentXmpPageFull8}{SegmentXmpPageEmpty8}
+\pastebutton{SegmentXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{expand s\free{s }}
+\indentrel{3}\begin{verbatim}
+   (8)  [3,4,5,6,7,8,9,10]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPageEmpty8}
+\begin{paste}{SegmentXmpPageEmpty8}{SegmentXmpPagePatch8}
+\pastebutton{SegmentXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{expand s\free{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPagePatch9}
+\begin{paste}{SegmentXmpPageFull9}{SegmentXmpPageEmpty9}
+\pastebutton{SegmentXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{expand t\free{t }}
+\indentrel{3}\begin{verbatim}
+   (9)  [10,8,6,4]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPageEmpty9}
+\begin{paste}{SegmentXmpPageEmpty9}{SegmentXmpPagePatch9}
+\pastebutton{SegmentXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{expand t\free{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPagePatch10}
+\begin{paste}{SegmentXmpPageFull10}{SegmentXmpPageEmpty10}
+\pastebutton{SegmentXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{expand l\free{l }}
+\indentrel{3}\begin{verbatim}
+   (10)  [1,2,3,5,9,15,14,13,12,11]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentXmpPageEmpty10}
+\begin{paste}{SegmentXmpPageEmpty10}{SegmentXmpPagePatch10}
+\pastebutton{SegmentXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{expand l\free{l }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/SEGBIND.ht b/src/hyper/pages/SEGBIND.ht
new file mode 100644
index 00000000..bec87f27
--- /dev/null
+++ b/src/hyper/pages/SEGBIND.ht
@@ -0,0 +1,54 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\SegmentBindingXmpTitle}{SegmentBinding}
+\newcommand{\SegmentBindingXmpNumber}{9.70}
+%
+% =====================================================================
+\begin{page}{SegmentBindingXmpPage}{9.70 SegmentBinding}
+% =====================================================================
+\beginscroll
+
+The \spadtype{SegmentBinding} type is used
+to indicate a range for a named symbol.
+
+\labelSpace{2pc}
+\xtc{
+First give the symbol, then an \spadSyntax{=} and finally a
+segment of values.
+}{
+\spadpaste{x = a..b}
+}
+\xtc{
+This is used to provide a convenient
+syntax for arguments to certain operations.
+}{
+\spadpaste{sum(i**2, i = 0..n)}
+}
+\graphpaste{draw(x**2, x = -2..2)}
+
+\xtc{
+The left-hand side must be of type \spadtype{Symbol} but the
+right-hand side can be a segment over any type.
+}{
+\spadpaste{sb := y = 1/2..3/2 \bound{sb}}
+}
+\xtc{
+The left- and right-hand sides can be obtained using the
+\spadfunFrom{variable}{SegmentBinding} and
+\spadfunFrom{segment}{SegmentBinding} operations.
+}{
+\spadpaste{variable(sb) \free{sb}}
+}
+\xtc{
+}{
+\spadpaste{segment(sb)  \free{sb}}
+}
+
+For more information on related topics, see
+\downlink{`Segment'}{SegmentXmpPage}\ignore{Segment} and \downlink{`UniversalSegment'}{UniversalSegmentXmpPage}\ignore{UniversalSegment}.
+%
+\showBlurb{SegmentBinding}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/SEGBIND.pht b/src/hyper/pages/SEGBIND.pht
new file mode 100644
index 00000000..0b5c62a5
--- /dev/null
+++ b/src/hyper/pages/SEGBIND.pht
@@ -0,0 +1,100 @@
+\begin{patch}{SegmentBindingXmpPagePatch1}
+\begin{paste}{SegmentBindingXmpPageFull1}{SegmentBindingXmpPageEmpty1}
+\pastebutton{SegmentBindingXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{x = a..b}
+\indentrel{3}\begin{verbatim}
+   (1)  x= a..b
+                            Type: SegmentBinding Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentBindingXmpPageEmpty1}
+\begin{paste}{SegmentBindingXmpPageEmpty1}{SegmentBindingXmpPagePatch1}
+\pastebutton{SegmentBindingXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{x = a..b}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentBindingXmpPagePatch2}
+\begin{paste}{SegmentBindingXmpPageFull2}{SegmentBindingXmpPageEmpty2}
+\pastebutton{SegmentBindingXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{sum(i**2, i = 0..n)}
+\indentrel{3}\begin{verbatim}
+          3     2
+        2n  + 3n  + n
+   (2)  ÄÄÄÄÄÄÄÄÄÄÄÄÄ
+              6
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentBindingXmpPageEmpty2}
+\begin{paste}{SegmentBindingXmpPageEmpty2}{SegmentBindingXmpPagePatch2}
+\pastebutton{SegmentBindingXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{sum(i**2, i = 0..n)}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentBindingXmpPagePatch3}
+\begin{paste}{SegmentBindingXmpPageFull3}{SegmentBindingXmpPageEmpty3}
+\pastebutton{SegmentBindingXmpPageFull3}{\hidepaste}
+\tab{5}\spadgraph{draw(x**2, x = -2..2)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/SegmentBindingXmpPage3.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/SegmentBindingXmpPage3}}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentBindingXmpPageEmpty3}
+\begin{paste}{SegmentBindingXmpPageEmpty3}{SegmentBindingXmpPagePatch3}
+\pastebutton{SegmentBindingXmpPageEmpty3}{\showpaste}
+\tab{5}\spadgraph{draw(x**2, x = -2..2)}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentBindingXmpPagePatch4}
+\begin{paste}{SegmentBindingXmpPageFull4}{SegmentBindingXmpPageEmpty4}
+\pastebutton{SegmentBindingXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{sb := y = 1/2..3/2\bound{sb }}
+\indentrel{3}\begin{verbatim}
+            1    3
+   (4)  y= (Ä)..(Ä)
+            2    2
+                  Type: SegmentBinding Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentBindingXmpPageEmpty4}
+\begin{paste}{SegmentBindingXmpPageEmpty4}{SegmentBindingXmpPagePatch4}
+\pastebutton{SegmentBindingXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{sb := y = 1/2..3/2\bound{sb }}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentBindingXmpPagePatch5}
+\begin{paste}{SegmentBindingXmpPageFull5}{SegmentBindingXmpPageEmpty5}
+\pastebutton{SegmentBindingXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{variable(sb)\free{sb }}
+\indentrel{3}\begin{verbatim}
+   (5)  y
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentBindingXmpPageEmpty5}
+\begin{paste}{SegmentBindingXmpPageEmpty5}{SegmentBindingXmpPagePatch5}
+\pastebutton{SegmentBindingXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{variable(sb)\free{sb }}
+\end{paste}\end{patch}
+
+\begin{patch}{SegmentBindingXmpPagePatch6}
+\begin{paste}{SegmentBindingXmpPageFull6}{SegmentBindingXmpPageEmpty6}
+\pastebutton{SegmentBindingXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{segment(sb)\free{sb }}
+\indentrel{3}\begin{verbatim}
+         1    3
+   (6)  (Ä)..(Ä)
+         2    2
+                         Type: Segment Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SegmentBindingXmpPageEmpty6}
+\begin{paste}{SegmentBindingXmpPageEmpty6}{SegmentBindingXmpPagePatch6}
+\pastebutton{SegmentBindingXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{segment(sb)\free{sb }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/SET.ht b/src/hyper/pages/SET.ht
new file mode 100644
index 00000000..2a1a988a
--- /dev/null
+++ b/src/hyper/pages/SET.ht
@@ -0,0 +1,130 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\SetXmpTitle}{Set}
+\newcommand{\SetXmpNumber}{9.71}
+%
+% =====================================================================
+\begin{page}{SetXmpPage}{9.71 Set}
+% =====================================================================
+\beginscroll
+%
+
+The \spadtype{Set} domain allows one to represent explicit finite sets of values.
+These are similar to lists, but duplicate elements are not allowed.
+\xtc{
+Sets can be created by giving a fixed set of values \ldots
+}{
+\spadpaste{s := set [x**2-1, y**2-1, z**2-1] \bound{s}}
+}
+\xtc{
+or by using a collect form, just as for lists.
+In either case, the set is formed from a finite collection of values.
+}{
+\spadpaste{t := set [x**i - i+1 for i in 2..10 | prime? i] \bound{t}}
+}
+
+\xtc{
+The basic operations on sets are
+\spadfunFrom{intersect}{Set}, \spadfunFrom{union}{Set},
+\spadfunFrom{difference}{Set},
+and \spadfunFrom{symmetricDifference}{Set}.
+}{
+\spadpaste{i := intersect(s,t)      \free{s t}\bound{i}}
+}
+\xtc{
+}{
+\spadpaste{u := union(s,t)          \free{s t}\bound{u}}
+}
+\xtc{
+The set \spad{difference(s,t)} contains those members of \spad{s} which
+are not in \spad{t}.
+}{
+\spadpaste{difference(s,t)          \free{s t}}
+}
+\xtc{
+The set \spad{symmetricDifference(s,t)} contains those elements which are
+in \spad{s} or \spad{t} but not in both.
+}{
+\spadpaste{symmetricDifference(s,t)          \free{s t}}
+}
+
+\xtc{
+Set membership is tested using the \spadfunFrom{member?}{Set} operation.
+}{
+\spadpaste{member?(y, s)               \free{s}}
+}
+\xtc{
+}{
+\spadpaste{member?((y+1)*(y-1), s)     \free{s}}
+}
+\xtc{
+The \spadfunFrom{subset?}{Set} function determines whether one set is a subset
+of another.
+}{
+\spadpaste{subset?(i, s)               \free{i s}}
+}
+\xtc{
+}{
+\spadpaste{subset?(u, s)               \free{u s}}
+}
+
+\xtc{
+When the base type is finite, the absolute complement of a set is
+defined.
+This finds the set of all multiplicative generators of
+\spadtype{PrimeField 11}---the integers mod \spad{11.}
+}{
+\spadpaste{gs := set [g for i in 1..11 | primitive?(g := i::PF 11)] \bound{gs}}
+}
+\xtc{
+The following values are not generators.
+}{
+\spadpaste{complement gs \free{gs}}
+}
+
+Often the members of a set are computed individually; in addition,
+values can be inserted or removed from a set over the course of a
+computation.
+\xtc{
+There are two ways to do this:
+}{
+\spadpaste{a := set [i**2 for i in 1..5] \bound{a}}
+}
+\xtc{
+One is to view a set as a data structure and to apply updating operations.
+}{
+\spadpaste{insert!(32, a) \free{a}\bound{ainsert}}
+}
+\xtc{
+}{
+\spadpaste{remove!(25, a) \free{a}\bound{aremove}}
+}
+\xtc{
+}{
+\spadpaste{a \free{aremove ainsert}}
+}
+\xtc{
+The other way is to view a set as a mathematical entity and to
+create new sets from old.
+}{
+\spadpaste{b := b0 := set [i**2 for i in 1..5] \bound{b}}
+}
+\xtc{
+}{
+\spadpaste{b := union(b, {32})         \free{b}\bound{binsert}}
+}
+\xtc{
+}{
+\spadpaste{b := difference(b, {25})    \free{binsert}\bound{bremove}}
+}
+\xtc{
+}{
+\spadpaste{b0 \free{bremove}}
+}
+
+For more information about lists, see \downlink{`List'}{ListXmpPage}\ignore{List}.
+\showBlurb{Set}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/SET.pht b/src/hyper/pages/SET.pht
new file mode 100644
index 00000000..aef8d01e
--- /dev/null
+++ b/src/hyper/pages/SET.pht
@@ -0,0 +1,326 @@
+\begin{patch}{SetXmpPagePatch1}
+\begin{paste}{SetXmpPageFull1}{SetXmpPageEmpty1}
+\pastebutton{SetXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{s := set [x**2-1, y**2-1, z**2-1]\bound{s }}
+\indentrel{3}\begin{verbatim}
+          2      2      2
+   (1)  {x  - 1,y  - 1,z  - 1}
+                           Type: Set Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty1}
+\begin{paste}{SetXmpPageEmpty1}{SetXmpPagePatch1}
+\pastebutton{SetXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{s := set [x**2-1, y**2-1, z**2-1]\bound{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch2}
+\begin{paste}{SetXmpPageFull2}{SetXmpPageEmpty2}
+\pastebutton{SetXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{t := set [x**i - i+1 for i in 2..10 | prime? i]\bound{t }}
+\indentrel{3}\begin{verbatim}
+          2      3      5      7
+   (2)  {x  - 1,x  - 2,x  - 4,x  - 6}
+                           Type: Set Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty2}
+\begin{paste}{SetXmpPageEmpty2}{SetXmpPagePatch2}
+\pastebutton{SetXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{t := set [x**i - i+1 for i in 2..10 | prime? i]\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch3}
+\begin{paste}{SetXmpPageFull3}{SetXmpPageEmpty3}
+\pastebutton{SetXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{i := intersect(s,t)\free{s t }\bound{i }}
+\indentrel{3}\begin{verbatim}
+          2
+   (3)  {x  - 1}
+                           Type: Set Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty3}
+\begin{paste}{SetXmpPageEmpty3}{SetXmpPagePatch3}
+\pastebutton{SetXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{i := intersect(s,t)\free{s t }\bound{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch4}
+\begin{paste}{SetXmpPageFull4}{SetXmpPageEmpty4}
+\pastebutton{SetXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{u := union(s,t)\free{s t }\bound{u }}
+\indentrel{3}\begin{verbatim}
+          2      3      5      7      2      2
+   (4)  {x  - 1,x  - 2,x  - 4,x  - 6,y  - 1,z  - 1}
+                           Type: Set Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty4}
+\begin{paste}{SetXmpPageEmpty4}{SetXmpPagePatch4}
+\pastebutton{SetXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{u := union(s,t)\free{s t }\bound{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch5}
+\begin{paste}{SetXmpPageFull5}{SetXmpPageEmpty5}
+\pastebutton{SetXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{difference(s,t)\free{s t }}
+\indentrel{3}\begin{verbatim}
+          2      2
+   (5)  {y  - 1,z  - 1}
+                           Type: Set Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty5}
+\begin{paste}{SetXmpPageEmpty5}{SetXmpPagePatch5}
+\pastebutton{SetXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{difference(s,t)\free{s t }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch6}
+\begin{paste}{SetXmpPageFull6}{SetXmpPageEmpty6}
+\pastebutton{SetXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{symmetricDifference(s,t)\free{s t }}
+\indentrel{3}\begin{verbatim}
+          3      5      7      2      2
+   (6)  {x  - 2,x  - 4,x  - 6,y  - 1,z  - 1}
+                           Type: Set Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty6}
+\begin{paste}{SetXmpPageEmpty6}{SetXmpPagePatch6}
+\pastebutton{SetXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{symmetricDifference(s,t)\free{s t }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch7}
+\begin{paste}{SetXmpPageFull7}{SetXmpPageEmpty7}
+\pastebutton{SetXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{member?(y, s)\free{s }}
+\indentrel{3}\begin{verbatim}
+   (7)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty7}
+\begin{paste}{SetXmpPageEmpty7}{SetXmpPagePatch7}
+\pastebutton{SetXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{member?(y, s)\free{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch8}
+\begin{paste}{SetXmpPageFull8}{SetXmpPageEmpty8}
+\pastebutton{SetXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{member?((y+1)*(y-1), s)\free{s }}
+\indentrel{3}\begin{verbatim}
+   (8)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty8}
+\begin{paste}{SetXmpPageEmpty8}{SetXmpPagePatch8}
+\pastebutton{SetXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{member?((y+1)*(y-1), s)\free{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch9}
+\begin{paste}{SetXmpPageFull9}{SetXmpPageEmpty9}
+\pastebutton{SetXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{subset?(i, s)\free{i s }}
+\indentrel{3}\begin{verbatim}
+   (9)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty9}
+\begin{paste}{SetXmpPageEmpty9}{SetXmpPagePatch9}
+\pastebutton{SetXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{subset?(i, s)\free{i s }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch10}
+\begin{paste}{SetXmpPageFull10}{SetXmpPageEmpty10}
+\pastebutton{SetXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{subset?(u, s)\free{u s }}
+\indentrel{3}\begin{verbatim}
+   (10)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty10}
+\begin{paste}{SetXmpPageEmpty10}{SetXmpPagePatch10}
+\pastebutton{SetXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{subset?(u, s)\free{u s }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch11}
+\begin{paste}{SetXmpPageFull11}{SetXmpPageEmpty11}
+\pastebutton{SetXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{gs := set [g for i in 1..11 | primitive?(g := i::PF 11)]\bound{gs }}
+\indentrel{3}\begin{verbatim}
+   (11)  {2,6,7,8}
+                                Type: Set PrimeField 11
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty11}
+\begin{paste}{SetXmpPageEmpty11}{SetXmpPagePatch11}
+\pastebutton{SetXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{gs := set [g for i in 1..11 | primitive?(g := i::PF 11)]\bound{gs }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch12}
+\begin{paste}{SetXmpPageFull12}{SetXmpPageEmpty12}
+\pastebutton{SetXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{complement gs\free{gs }}
+\indentrel{3}\begin{verbatim}
+   (12)  {1,3,4,5,9,10,0}
+                                Type: Set PrimeField 11
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty12}
+\begin{paste}{SetXmpPageEmpty12}{SetXmpPagePatch12}
+\pastebutton{SetXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{complement gs\free{gs }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch13}
+\begin{paste}{SetXmpPageFull13}{SetXmpPageEmpty13}
+\pastebutton{SetXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{a := set [i**2 for i in 1..5]\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (13)  {1,4,9,16,25}
+                              Type: Set PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty13}
+\begin{paste}{SetXmpPageEmpty13}{SetXmpPagePatch13}
+\pastebutton{SetXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{a := set [i**2 for i in 1..5]\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch14}
+\begin{paste}{SetXmpPageFull14}{SetXmpPageEmpty14}
+\pastebutton{SetXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{insert!(32, a)\free{a }\bound{ainsert }}
+\indentrel{3}\begin{verbatim}
+   (14)  {1,4,9,16,25,32}
+                              Type: Set PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty14}
+\begin{paste}{SetXmpPageEmpty14}{SetXmpPagePatch14}
+\pastebutton{SetXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{insert!(32, a)\free{a }\bound{ainsert }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch15}
+\begin{paste}{SetXmpPageFull15}{SetXmpPageEmpty15}
+\pastebutton{SetXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{remove!(25, a)\free{a }\bound{aremove }}
+\indentrel{3}\begin{verbatim}
+   (15)  {1,4,9,16,32}
+                              Type: Set PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty15}
+\begin{paste}{SetXmpPageEmpty15}{SetXmpPagePatch15}
+\pastebutton{SetXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{remove!(25, a)\free{a }\bound{aremove }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch16}
+\begin{paste}{SetXmpPageFull16}{SetXmpPageEmpty16}
+\pastebutton{SetXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{a\free{aremove ainsert }}
+\indentrel{3}\begin{verbatim}
+   (16)  {1,4,9,16,32}
+                              Type: Set PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty16}
+\begin{paste}{SetXmpPageEmpty16}{SetXmpPagePatch16}
+\pastebutton{SetXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{a\free{aremove ainsert }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch17}
+\begin{paste}{SetXmpPageFull17}{SetXmpPageEmpty17}
+\pastebutton{SetXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{b := b0 := set [i**2 for i in 1..5]\bound{b }}
+\indentrel{3}\begin{verbatim}
+   (17)  {1,4,9,16,25}
+                              Type: Set PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty17}
+\begin{paste}{SetXmpPageEmpty17}{SetXmpPagePatch17}
+\pastebutton{SetXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{b := b0 := set [i**2 for i in 1..5]\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch18}
+\begin{paste}{SetXmpPageFull18}{SetXmpPageEmpty18}
+\pastebutton{SetXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{b := union(b,32)\free{b }\bound{binsert }}
+\indentrel{3}\begin{verbatim}
+   (18)  {1,4,9,16,25,32}
+                              Type: Set PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty18}
+\begin{paste}{SetXmpPageEmpty18}{SetXmpPagePatch18}
+\pastebutton{SetXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{b := union(b,32)\free{b }\bound{binsert }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch19}
+\begin{paste}{SetXmpPageFull19}{SetXmpPageEmpty19}
+\pastebutton{SetXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{b := difference(b,25)\free{binsert }\bound{bremove }}
+\indentrel{3}\begin{verbatim}
+   (19)  {1,4,9,16,32}
+                              Type: Set PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty19}
+\begin{paste}{SetXmpPageEmpty19}{SetXmpPagePatch19}
+\pastebutton{SetXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{b := difference(b,25)\free{binsert }\bound{bremove }}
+\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPagePatch20}
+\begin{paste}{SetXmpPageFull20}{SetXmpPageEmpty20}
+\pastebutton{SetXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{b0\free{bremove }}
+\indentrel{3}\begin{verbatim}
+   (20)  {1,4,9,16,25}
+                              Type: Set PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SetXmpPageEmpty20}
+\begin{paste}{SetXmpPageEmpty20}{SetXmpPagePatch20}
+\pastebutton{SetXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{b0\free{bremove }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/SINT.ht b/src/hyper/pages/SINT.ht
new file mode 100644
index 00000000..fdadf644
--- /dev/null
+++ b/src/hyper/pages/SINT.ht
@@ -0,0 +1,116 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\SingleIntegerXmpTitle}{SingleInteger}
+\newcommand{\SingleIntegerXmpNumber}{9.72}
+%
+% =====================================================================
+\begin{page}{SingleIntegerXmpPage}{9.72 SingleInteger}
+% =====================================================================
+\beginscroll
+
+%%
+%% SINT.htex
+%%
+%%
+The \axiomType{SingleInteger} domain is intended to provide support
+%-% \HDindex{integer!machine}{SingleIntegerXmpPage}{9.72}{SingleInteger}
+in \Language{} for machine integer arithmetic.
+It is generally much faster than (bignum) \axiomType{Integer} arithmetic
+but suffers from a limited range of values.
+Since \Language{} can be implemented on top of various
+dialects of Lisp, the actual representation of small integers
+may not correspond exactly to the host machines integer representation.
+
+In the CCL implementation of AXIOM (Release 2.1 onwards) the underlying
+representation of \axiomType{SingleInteger} is the same as \axiomType{Integer}.  
+The underlying Lisp primitives treat machine-word sized computations
+specially.
+
+%Under \spadgloss{AKCL} on the IBM Risc System/6000,
+%small integers are restricted
+%to the range \texht{$-{2^{26}}$}{\spad{-2**26}} to
+%\texht{${2^{26}}-1$}{\spad{2**26 - 1}},
+%allowing 1 bit for overflow detection.
+
+\xtc{
+You can discover the minimum and maximum values in your implementation
+by using \spadfunFrom{min}{SingleInteger} and \spadfunFrom{max}{SingleInteger}.
+}{
+\spadpaste{min()\$SingleInteger}
+}
+\xtc{
+}{
+\spadpaste{max()\$SingleInteger}
+}
+\xtc{
+To avoid confusion with \axiomType{Integer}, which is the default
+type for integers, you usually need to work with declared variables
+(\downlink{``\ugTypesDeclareTitle''}{ugTypesDeclarePage} in Section \ugTypesDeclareNumber\ignore{ugTypesDeclare}) \ldots
+}{
+\spadpaste{a := 1234 :: SingleInteger \bound{a}}
+}
+\xtc{
+or use package calling
+(\downlink{``\ugTypesPkgCallTitle''}{ugTypesPkgCallPage} in Section \ugTypesPkgCallNumber\ignore{ugTypesPkgCall}).
+}{
+\spadpaste{b := 124\$SingleInteger \bound{b}}
+}
+\xtc{
+You can add, multiply and subtract \axiomType{SingleInteger} objects,
+and ask for the greatest common divisor (\spadfun{gcd}).
+}{
+\spadpaste{gcd(a,b) \free{a}\free{b}}
+}
+\xtc{
+The least common multiple (\spadfun{lcm}) is also available.
+}{
+\spadpaste{lcm(a,b) \free{a}\free{b}}
+}
+
+\xtc{
+Operations \spadfunFrom{mulmod}{SingleInteger},
+\spadfunFrom{addmod}{SingleInteger},
+\spadfunFrom{submod}{SingleInteger}, and
+\spadfunFrom{invmod}{SingleInteger} are similar---they provide
+arithmetic modulo a given small integer.
+Here is \spad{5 * 6 {\tt mod} 13}.
+}{
+\spadpaste{mulmod(5,6,13)\$SingleInteger}
+}
+\xtc{
+To reduce a small integer modulo a prime, use \spadfunFrom{positiveRemainder}{SingleInteger}.
+}{
+\spadpaste{positiveRemainder(37,13)\$SingleInteger}
+}
+\xtc{
+Operations
+\spadfunFrom{And}{SingleInteger},
+\spadfunFrom{Or}{SingleInteger},
+\spadfunFrom{xor}{SingleInteger},
+and \spadfunFrom{Not}{SingleInteger}
+provide bit level operations on small integers.
+}{
+\spadpaste{And(3,4)\$SingleInteger}
+}
+\xtc{
+Use
+\spad{shift(int,numToShift)} to shift bits, where
+\spad{i} is shifted left if \spad{numToShift} is positive, right
+if negative.
+}{
+\spadpaste{shift(1,4)\$SingleInteger}
+}
+\xtc{
+}{
+\spadpaste{shift(31,-1)\$SingleInteger}
+}
+
+Many other operations are available for small integers, including
+many of those provided for \axiomType{Integer}.
+To see the other operations, use the Browse \HyperName{} facility
+(\downlink{``\ugBrowseTitle''}{ugBrowsePage} in Section \ugBrowseNumber\ignore{ugBrowse}).
+\showBlurb{SingleInteger}.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/SINT.pht b/src/hyper/pages/SINT.pht
new file mode 100644
index 00000000..82df8c24
--- /dev/null
+++ b/src/hyper/pages/SINT.pht
@@ -0,0 +1,176 @@
+\begin{patch}{SingleIntegerXmpPagePatch1}
+\begin{paste}{SingleIntegerXmpPageFull1}{SingleIntegerXmpPageEmpty1}
+\pastebutton{SingleIntegerXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{min()$SingleInteger}
+\indentrel{3}\begin{verbatim}
+   (1)  - 134217728
+                                    Type: SingleInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPageEmpty1}
+\begin{paste}{SingleIntegerXmpPageEmpty1}{SingleIntegerXmpPagePatch1}
+\pastebutton{SingleIntegerXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{min()$SingleInteger}
+\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPagePatch2}
+\begin{paste}{SingleIntegerXmpPageFull2}{SingleIntegerXmpPageEmpty2}
+\pastebutton{SingleIntegerXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{max()$SingleInteger}
+\indentrel{3}\begin{verbatim}
+   (2)  134217727
+                                    Type: SingleInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPageEmpty2}
+\begin{paste}{SingleIntegerXmpPageEmpty2}{SingleIntegerXmpPagePatch2}
+\pastebutton{SingleIntegerXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{max()$SingleInteger}
+\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPagePatch3}
+\begin{paste}{SingleIntegerXmpPageFull3}{SingleIntegerXmpPageEmpty3}
+\pastebutton{SingleIntegerXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{a := 1234 :: SingleInteger\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (3)  1234
+                                    Type: SingleInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPageEmpty3}
+\begin{paste}{SingleIntegerXmpPageEmpty3}{SingleIntegerXmpPagePatch3}
+\pastebutton{SingleIntegerXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{a := 1234 :: SingleInteger\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPagePatch4}
+\begin{paste}{SingleIntegerXmpPageFull4}{SingleIntegerXmpPageEmpty4}
+\pastebutton{SingleIntegerXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{b := 124$SingleInteger\bound{b }}
+\indentrel{3}\begin{verbatim}
+   (4)  124
+                                    Type: SingleInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPageEmpty4}
+\begin{paste}{SingleIntegerXmpPageEmpty4}{SingleIntegerXmpPagePatch4}
+\pastebutton{SingleIntegerXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{b := 124$SingleInteger\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPagePatch5}
+\begin{paste}{SingleIntegerXmpPageFull5}{SingleIntegerXmpPageEmpty5}
+\pastebutton{SingleIntegerXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{gcd(a,b)\free{a }\free{b }}
+\indentrel{3}\begin{verbatim}
+   (5)  2
+                                    Type: SingleInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPageEmpty5}
+\begin{paste}{SingleIntegerXmpPageEmpty5}{SingleIntegerXmpPagePatch5}
+\pastebutton{SingleIntegerXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{gcd(a,b)\free{a }\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPagePatch6}
+\begin{paste}{SingleIntegerXmpPageFull6}{SingleIntegerXmpPageEmpty6}
+\pastebutton{SingleIntegerXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{lcm(a,b)\free{a }\free{b }}
+\indentrel{3}\begin{verbatim}
+   (6)  76508
+                                    Type: SingleInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPageEmpty6}
+\begin{paste}{SingleIntegerXmpPageEmpty6}{SingleIntegerXmpPagePatch6}
+\pastebutton{SingleIntegerXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{lcm(a,b)\free{a }\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPagePatch7}
+\begin{paste}{SingleIntegerXmpPageFull7}{SingleIntegerXmpPageEmpty7}
+\pastebutton{SingleIntegerXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{mulmod(5,6,13)$SingleInteger}
+\indentrel{3}\begin{verbatim}
+   (7)  4
+                                    Type: SingleInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPageEmpty7}
+\begin{paste}{SingleIntegerXmpPageEmpty7}{SingleIntegerXmpPagePatch7}
+\pastebutton{SingleIntegerXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{mulmod(5,6,13)$SingleInteger}
+\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPagePatch8}
+\begin{paste}{SingleIntegerXmpPageFull8}{SingleIntegerXmpPageEmpty8}
+\pastebutton{SingleIntegerXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{positiveRemainder(37,13)$SingleInteger}
+\indentrel{3}\begin{verbatim}
+   (8)  11
+                                    Type: SingleInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPageEmpty8}
+\begin{paste}{SingleIntegerXmpPageEmpty8}{SingleIntegerXmpPagePatch8}
+\pastebutton{SingleIntegerXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{positiveRemainder(37,13)$SingleInteger}
+\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPagePatch9}
+\begin{paste}{SingleIntegerXmpPageFull9}{SingleIntegerXmpPageEmpty9}
+\pastebutton{SingleIntegerXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{And(3,4)$SingleInteger}
+\indentrel{3}\begin{verbatim}
+   (9)  0
+                                    Type: SingleInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPageEmpty9}
+\begin{paste}{SingleIntegerXmpPageEmpty9}{SingleIntegerXmpPagePatch9}
+\pastebutton{SingleIntegerXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{And(3,4)$SingleInteger}
+\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPagePatch10}
+\begin{paste}{SingleIntegerXmpPageFull10}{SingleIntegerXmpPageEmpty10}
+\pastebutton{SingleIntegerXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{shift(1,4)$SingleInteger}
+\indentrel{3}\begin{verbatim}
+   (10)  16
+                                    Type: SingleInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPageEmpty10}
+\begin{paste}{SingleIntegerXmpPageEmpty10}{SingleIntegerXmpPagePatch10}
+\pastebutton{SingleIntegerXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{shift(1,4)$SingleInteger}
+\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPagePatch11}
+\begin{paste}{SingleIntegerXmpPageFull11}{SingleIntegerXmpPageEmpty11}
+\pastebutton{SingleIntegerXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{shift(31,-1)$SingleInteger}
+\indentrel{3}\begin{verbatim}
+   (11)  15
+                                    Type: SingleInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SingleIntegerXmpPageEmpty11}
+\begin{paste}{SingleIntegerXmpPageEmpty11}{SingleIntegerXmpPagePatch11}
+\pastebutton{SingleIntegerXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{shift(31,-1)$SingleInteger}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/SQMATRIX.ht b/src/hyper/pages/SQMATRIX.ht
new file mode 100644
index 00000000..b5db8bcf
--- /dev/null
+++ b/src/hyper/pages/SQMATRIX.ht
@@ -0,0 +1,67 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\SquareMatrixXmpTitle}{SquareMatrix}
+\newcommand{\SquareMatrixXmpNumber}{9.74}
+%
+% =====================================================================
+\begin{page}{SquareMatrixXmpPage}{9.74 SquareMatrix}
+% =====================================================================
+\beginscroll
+ 
+The top level matrix type in \Language{} is \spadtype{Matrix}
+(see \downlink{`Matrix'}{MatrixXmpPage}\ignore{Matrix}), which
+provides basic arithmetic and linear algebra functions.
+However, since the matrices can be of any size it is not true that any pair
+can be added or multiplied.
+Thus \spadtype{Matrix} has little algebraic structure.
+ 
+Sometimes you want to use matrices as coefficients for polynomials
+or in other algebraic contexts.  In this case, \spadtype{SquareMatrix}
+%-% \HDindex{matrix!square}{SquareMatrixXmpPage}{9.74}{SquareMatrix}
+should be used.  The domain \spadtype{SquareMatrix(n,R)} gives the ring of
+\spad{n} by \spad{n} square matrices over \spad{R}.
+ 
+\xtc{
+Since \spadtype{SquareMatrix} is not normally exposed at the top level,
+you must expose it before it can be used.
+}{
+\spadpaste{)set expose add constructor SquareMatrix \bound{SQ}}
+}
+\xtc{
+Once \spad{SQMATRIX} has been exposed,
+values can be created using the \spadfunFrom{squareMatrix}{SquareMatrix}
+function.
+}{
+\spadpaste{m := squareMatrix [[1,-\%i],[\%i,4]] \bound{m}\free{SQ}}
+}
+\xtc{
+The usual arithmetic operations are available.
+}{
+\spadpaste{m*m - m \free{m}}
+}
+\xtc{
+Square matrices can be used where ring elements are required.
+For example, here is a matrix with matrix entries.
+}{
+\spadpaste{mm := squareMatrix [[m, 1], [1-m, m**2]] \free{m}\bound{mm}}
+}
+\xtc{
+Or you can construct a polynomial with  square matrix coefficients.
+}{
+\spadpaste{p := (x + m)**2 \free{m}\bound{p}}
+}
+\xtc{
+This value can be converted to a square matrix with polynomial coefficients.
+}{
+\spadpaste{p::SquareMatrix(2, ?) \free{p}}
+}
+ 
+For more information on related topics, see
+\downlink{``\ugTypesWritingModesTitle''}{ugTypesWritingModesPage} in Section \ugTypesWritingModesNumber\ignore{ugTypesWritingModes},
+\downlink{``\ugTypesExposeTitle''}{ugTypesExposePage} in Section \ugTypesExposeNumber\ignore{ugTypesExpose}, and
+\downlink{`Matrix'}{MatrixXmpPage}\ignore{Matrix}.
+\showBlurb{SquareMatrix}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/SQMATRIX.pht b/src/hyper/pages/SQMATRIX.pht
new file mode 100644
index 00000000..e58b3c60
--- /dev/null
+++ b/src/hyper/pages/SQMATRIX.pht
@@ -0,0 +1,110 @@
+\begin{patch}{SquareMatrixXmpPagePatch1}
+\begin{paste}{SquareMatrixXmpPageFull1}{SquareMatrixXmpPageEmpty1}
+\pastebutton{SquareMatrixXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)set expose add constructor SquareMatrix\bound{SQ }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareMatrixXmpPageEmpty1}
+\begin{paste}{SquareMatrixXmpPageEmpty1}{SquareMatrixXmpPagePatch1}
+\pastebutton{SquareMatrixXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)set expose add constructor SquareMatrix\bound{SQ }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareMatrixXmpPagePatch2}
+\begin{paste}{SquareMatrixXmpPageFull2}{SquareMatrixXmpPageEmpty2}
+\pastebutton{SquareMatrixXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{m := squareMatrix [[1,-\%i],[\%i,4]]\bound{m }\free{SQ }}
+\indentrel{3}\begin{verbatim}
+        Ú1   - %i¿
+   (1)  ³        ³
+        À%i   4  Ù
+                  Type: SquareMatrix(2,Complex Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareMatrixXmpPageEmpty2}
+\begin{paste}{SquareMatrixXmpPageEmpty2}{SquareMatrixXmpPagePatch2}
+\pastebutton{SquareMatrixXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{m := squareMatrix [[1,-\%i],[\%i,4]]\bound{m }\free{SQ }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareMatrixXmpPagePatch3}
+\begin{paste}{SquareMatrixXmpPageFull3}{SquareMatrixXmpPageEmpty3}
+\pastebutton{SquareMatrixXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{m*m - m\free{m }}
+\indentrel{3}\begin{verbatim}
+        Ú 1   - 4%i¿
+   (2)  ³          ³
+        À4%i   13  Ù
+                  Type: SquareMatrix(2,Complex Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareMatrixXmpPageEmpty3}
+\begin{paste}{SquareMatrixXmpPageEmpty3}{SquareMatrixXmpPagePatch3}
+\pastebutton{SquareMatrixXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{m*m - m\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareMatrixXmpPagePatch4}
+\begin{paste}{SquareMatrixXmpPageFull4}{SquareMatrixXmpPageEmpty4}
+\pastebutton{SquareMatrixXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{mm := squareMatrix [[m, 1], [1-m, m**2]]\free{m }\bound{mm }}
+\indentrel{3}\begin{verbatim}
+        ÚÚ1   - %i¿      Ú1  0¿   ¿
+        ³³        ³      ³    ³   ³
+        ³À%i   4  Ù      À0  1Ù   ³
+   (3)  ³                         ³
+        ³Ú 0    %i ¿  Ú 2   - 5%i¿³
+        ³³         ³  ³          ³³
+        ÀÀ- %i  - 3Ù  À5%i   17  ÙÙ
+  Type: SquareMatrix(2,SquareMatrix(2,Complex Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareMatrixXmpPageEmpty4}
+\begin{paste}{SquareMatrixXmpPageEmpty4}{SquareMatrixXmpPagePatch4}
+\pastebutton{SquareMatrixXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{mm := squareMatrix [[m, 1], [1-m, m**2]]\free{m }\bound{mm }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareMatrixXmpPagePatch5}
+\begin{paste}{SquareMatrixXmpPageFull5}{SquareMatrixXmpPageEmpty5}
+\pastebutton{SquareMatrixXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{p := (x + m)**2\free{m }\bound{p }}
+\indentrel{3}\begin{verbatim}
+         2   Ú 2   - 2%i¿    Ú 2   - 5%i¿
+   (4)  x  + ³          ³x + ³          ³
+             À2%i    8  Ù    À5%i   17  Ù
+       Type: Polynomial SquareMatrix(2,Complex Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareMatrixXmpPageEmpty5}
+\begin{paste}{SquareMatrixXmpPageEmpty5}{SquareMatrixXmpPagePatch5}
+\pastebutton{SquareMatrixXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{p := (x + m)**2\free{m }\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareMatrixXmpPagePatch6}
+\begin{paste}{SquareMatrixXmpPageFull6}{SquareMatrixXmpPageEmpty6}
+\pastebutton{SquareMatrixXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{p::SquareMatrix(2, ?)\free{p }}
+\indentrel{3}\begin{verbatim}
+        Ú 2                        ¿
+        ³x  + 2x + 2  - 2%i x - 5%i³
+   (5)  ³                          ³
+        ³              2           ³
+        À2%i x + 5%i  x  + 8x + 17 Ù
+       Type: SquareMatrix(2,Polynomial Complex Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareMatrixXmpPageEmpty6}
+\begin{paste}{SquareMatrixXmpPageEmpty6}{SquareMatrixXmpPagePatch6}
+\pastebutton{SquareMatrixXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{p::SquareMatrix(2, ?)\free{p }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/SREGSET.ht b/src/hyper/pages/SREGSET.ht
new file mode 100644
index 00000000..c7f67796
--- /dev/null
+++ b/src/hyper/pages/SREGSET.ht
@@ -0,0 +1,186 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\SquareFreeRegularTriangularSetXmpTitle}{SquareFreeRegularTriangularSet}
+\newcommand{\SquareFreeRegularTriangularSetXmpNumber}{9.75}
+%
+% =====================================================================
+\begin{page}{SquareFreeRegularTriangularSetXmpPage}{9.75 SquareFreeRegularTriangularSet}
+% =====================================================================
+\beginscroll
+The \spadtype{SquareFreeRegularTriangularSet} domain constructor implements
+square-free regular triangular sets.
+See the \spadtype{RegularTriangularSet} domain constructor
+for general regular triangular sets.
+Let {\em T} be a regular triangular set consisting of polynomials
+{\em t1, ..., tm} ordered by increasing main variables.
+The regular triangular set {\em T} is square-free if {\em T} 
+is empty or if {\em t1, ..., tm-1} is square-free  and if
+the polynomial {\em tm} is square-free as
+a univariate polynomial with coefficients in the tower
+of simple extensions associated with {\em t1, ..., tm-1}.
+
+The main interest of square-free regular triangular sets
+is that their associated towers of simple extensions
+are product of fields.
+Consequently, the saturated ideal of a square-free regular triangular set
+is radical.
+This property simplifies some of the operations related
+to regular triangular sets.
+However, building square-free regular triangular sets
+is generally more  expensive than building
+general regular triangular sets.
+
+As the \spadtype{RegularTriangularSet} domain constructor,
+the \spadtype{SquareFreeRegularTriangularSet} domain constructor also implements
+a method for solving polynomial systems by means of regular triangular sets.
+This is in fact the same method with some adaptations to take into
+account the fact that the computed regular chains are square-free.
+Note that it is also possible to pass from a decomposition
+into general regular triangular sets to a decomposition into
+square-free regular triangular sets.
+This conversion is used internally by the 
+\spadtype{LazardSetSolvingPackage} package constructor.
+
+{\bf N.B.} When solving polynomial systems with the 
+\spadtype{SquareFreeRegularTriangularSet} domain constructor
+or the \spadtype{LazardSetSolvingPackage} package constructor,
+decompositions have no redundant components.
+See also \spadtype{LexTriangularPackage} and \spadtype{ZeroDimensionalSolvePackage} for the case of 
+algebraic systems with a finite number of (complex) solutions.
+
+We shall explain now how to use the constructor \spadtype{SquareFreeRegularTriangularSet}.
+
+This constructor takes four arguments.
+The first one, {\bf R}, is the coefficient ring of the polynomials;
+it must belong to the category \spadtype{GcdDomain}.
+The second one, {\bf E}, is the exponent monoid of the polynomials;
+it must belong to the category \spadtype{OrderedAbelianMonoidSup}.
+the third one, {\bf V}, is the ordered set of variables;
+it must belong to the category \spadtype{OrderedSet}.
+The last one is the polynomial ring;
+it must belong to the category \spadtype{RecursivePolynomialCategory(R,E,V)}.
+The abbreviation for \spadtype{SquareFreeRegularTriangularSet} is
+\spadtype{SREGSET}.
+
+Note that the way of understanding triangular decompositions 
+is detailed in the example of the \spadtype{RegularTriangularSet}
+constructor.
+
+\xtc{
+Let us illustrate the use of this constructor with one example (Donati-Traverso).
+Define the coefficient ring.
+}{
+\spadpaste{R := Integer \bound{R}}
+}
+\xtc{
+Define the list of variables,
+}{
+\spadpaste{ls : List Symbol := [x,y,z,t] \bound{ls}}
+}
+\xtc{
+and make it an ordered set;
+}{
+\spadpaste{V := OVAR(ls) \free{ls} \bound{V}}
+}
+\xtc{
+then define the exponent monoid.
+}{
+\spadpaste{E := IndexedExponents V \free{V} \bound{E}}
+}
+\xtc{
+Define the polynomial ring.
+}{
+\spadpaste{P := NSMP(R, V) \free{R} \free{V} \bound{P}}
+}
+\xtc{
+Let the variables be polynomial.
+}{
+\spadpaste{x: P := 'x \free{P} \bound{x}}
+}
+\xtc{
+}{
+\spadpaste{y: P := 'y \free{P} \bound{y}}
+}
+\xtc{
+}{
+\spadpaste{z: P := 'z \free{P} \bound{z}}
+}
+\xtc{
+}{
+\spadpaste{t: P := 't \free{P} \bound{t}}
+}
+\xtc{
+Now call the \spadtype{SquareFreeRegularTriangularSet} domain constructor.
+}{
+\spadpaste{ST := SREGSET(R,E,V,P) \free{R} \free{E} \free{V} \free{P} \bound{ST} }
+}
+\xtc{
+Define a polynomial system.
+}{
+\spadpaste{p1 := x ** 31 - x ** 6 - x - y \free{x} \free{y} \bound{p1}}
+}
+\xtc{
+}{
+\spadpaste{p2 := x ** 8  - z \free{x} \free{z} \bound{p2}}
+}
+\xtc{
+}{
+\spadpaste{p3 := x ** 10 - t \free{x} \free{t} \bound{p3}}
+}
+\xtc{
+}{
+\spadpaste{lp := [p1, p2, p3] \free{p1} \free{p2} \free{p3} \bound{lp}}
+}
+
+\xtc{
+First of all, let us solve this system in the sense of Kalkbrener.
+}{
+\spadpaste{zeroSetSplit(lp)$ST \free{lp} \free{ST}}
+}
+\xtc{
+And now in the sense of Lazard (or Wu and other authors).
+}{
+\spadpaste{zeroSetSplit(lp,false)$ST \free{lp} \free{ST} \bound{lts}}
+}
+
+Now to see the difference with the \spadtype{RegularTriangularSet} domain constructor,
+\xtc{
+we define:
+}{
+\spadpaste{T := REGSET(R,E,V,P) \free{R} \free{E} \free{V} \free{P} \bound{T} }
+}
+\xtc{
+and compute:
+}{
+\spadpaste{lts := zeroSetSplit(lp,false)$T \free{lp} \free{T} \bound{lts}}
+}
+If you look at the second set in both decompositions in the sense of Lazard,
+you will see that the polynomial with main variable {\bf y} is not the same.
+
+Let us understand what has happened.
+\xtc{
+We define:
+}{
+\spadpaste{ts := lts.2  \free{lts}  \bound{ts}}
+}
+\xtc{
+}{
+\spadpaste{pol := select(ts,'y)$T \free{ts} \free{y} \free{T} \bound{pol}}
+}
+\xtc{
+}{
+\spadpaste{tower := collectUnder(ts,'y)$T \free{ts} \free{y} \free{T} \bound{tower}}
+}
+\xtc{
+}{
+\spadpaste{pack := RegularTriangularSetGcdPackage(R,E,V,P,T) \free{R} \free{E} \free{V} \free{P} \free{T} \bound{pack}}
+}
+\xtc{
+Then we compute:
+}{
+\spadpaste{toseSquareFreePart(pol,tower)$pack \free{pol} \free{tower} \free{pack}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/SREGSET.pht b/src/hyper/pages/SREGSET.pht
new file mode 100644
index 00000000..ea96c7b1
--- /dev/null
+++ b/src/hyper/pages/SREGSET.pht
@@ -0,0 +1,416 @@
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch1}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull1}{SquareFreeRegularTriangularSetXmpPageEmpty1}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{R := Integer\bound{R }}
+\indentrel{3}\begin{verbatim}
+   (1)  Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty1}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty1}{SquareFreeRegularTriangularSetXmpPagePatch1}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{R := Integer\bound{R }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch2}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull2}{SquareFreeRegularTriangularSetXmpPageEmpty2}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{ls : List Symbol := [x,y,z,t]\bound{ls }}
+\indentrel{3}\begin{verbatim}
+   (2)  [x,y,z,t]
+                                      Type: List Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty2}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty2}{SquareFreeRegularTriangularSetXmpPagePatch2}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{ls : List Symbol := [x,y,z,t]\bound{ls }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch3}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull3}{SquareFreeRegularTriangularSetXmpPageEmpty3}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
+\indentrel{3}\begin{verbatim}
+   (3)  OrderedVariableList [x,y,z,t]
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty3}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty3}{SquareFreeRegularTriangularSetXmpPagePatch3}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch4}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull4}{SquareFreeRegularTriangularSetXmpPageEmpty4}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{E := IndexedExponents V\free{V }\bound{E }}
+\indentrel{3}\begin{verbatim}
+   (4)  IndexedExponents OrderedVariableList [x,y,z,t]
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty4}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty4}{SquareFreeRegularTriangularSetXmpPagePatch4}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{E := IndexedExponents V\free{V }\bound{E }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch5}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull5}{SquareFreeRegularTriangularSetXmpPageEmpty5}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
+\indentrel{3}\begin{verbatim}
+   (5)
+  NewSparseMultivariatePolynomial(Integer,OrderedVariable
+  List [x,y,z,t])
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty5}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty5}{SquareFreeRegularTriangularSetXmpPagePatch5}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch6}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull6}{SquareFreeRegularTriangularSetXmpPageEmpty6}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{x: P := 'x\free{P }\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (6)  x
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty6}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty6}{SquareFreeRegularTriangularSetXmpPagePatch6}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{x: P := 'x\free{P }\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch7}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull7}{SquareFreeRegularTriangularSetXmpPageEmpty7}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{y: P := 'y\free{P }\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (7)  y
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty7}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty7}{SquareFreeRegularTriangularSetXmpPagePatch7}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{y: P := 'y\free{P }\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch8}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull8}{SquareFreeRegularTriangularSetXmpPageEmpty8}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{z: P := 'z\free{P }\bound{z }}
+\indentrel{3}\begin{verbatim}
+   (8)  z
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty8}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty8}{SquareFreeRegularTriangularSetXmpPagePatch8}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{z: P := 'z\free{P }\bound{z }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch9}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull9}{SquareFreeRegularTriangularSetXmpPageEmpty9}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{t: P := 't\free{P }\bound{t }}
+\indentrel{3}\begin{verbatim}
+   (9)  t
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty9}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty9}{SquareFreeRegularTriangularSetXmpPagePatch9}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{t: P := 't\free{P }\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch10}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull10}{SquareFreeRegularTriangularSetXmpPageEmpty10}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{ST := SREGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{ST }}
+\indentrel{3}\begin{verbatim}
+   (10)
+  SquareFreeRegularTriangularSet(Integer,IndexedExponents
+   OrderedVariableList [x,y,z,t],OrderedVariableList [x,y
+  ,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVa
+  riableList [x,y,z,t]))
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty10}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty10}{SquareFreeRegularTriangularSetXmpPagePatch10}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{ST := SREGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{ST }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch11}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull11}{SquareFreeRegularTriangularSetXmpPageEmpty11}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{p1 := x ** 31 - x ** 6 - x - y\free{x }\free{y }\bound{p1 }}
+\indentrel{3}\begin{verbatim}
+          31    6
+   (11)  x   - x  - x - y
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty11}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty11}{SquareFreeRegularTriangularSetXmpPagePatch11}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{p1 := x ** 31 - x ** 6 - x - y\free{x }\free{y }\bound{p1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch12}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull12}{SquareFreeRegularTriangularSetXmpPageEmpty12}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{p2 := x ** 8 - z\free{x }\free{z }\bound{p2 }}
+\indentrel{3}\begin{verbatim}
+          8
+   (12)  x  - z
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty12}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty12}{SquareFreeRegularTriangularSetXmpPagePatch12}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{p2 := x ** 8 - z\free{x }\free{z }\bound{p2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch13}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull13}{SquareFreeRegularTriangularSetXmpPageEmpty13}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{p3 := x ** 10 - t\free{x }\free{t }\bound{p3 }}
+\indentrel{3}\begin{verbatim}
+          10
+   (13)  x   - t
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty13}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty13}{SquareFreeRegularTriangularSetXmpPagePatch13}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{p3 := x ** 10 - t\free{x }\free{t }\bound{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch14}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull14}{SquareFreeRegularTriangularSetXmpPageEmpty14}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
+\indentrel{3}\begin{verbatim}
+           31    6          8      10
+   (14)  [x   - x  - x - y,x  - z,x   - t]
+Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty14}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty14}{SquareFreeRegularTriangularSetXmpPagePatch14}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch15}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull15}{SquareFreeRegularTriangularSetXmpPageEmpty15}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{zeroSetSplit(lp)$ST\free{lp }\free{ST }}
+\indentrel{3}\begin{verbatim}
+   (15)
+   [
+       5    4       2     3     8     5    3    2
+     {z  - t , t z y  + 2z y - t  + 2t  + t  - t ,
+        4                2
+      (t  - t)x - t y - z }
+     ]
+Type: List SquareFreeRegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty15}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty15}{SquareFreeRegularTriangularSetXmpPagePatch15}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{zeroSetSplit(lp)$ST\free{lp }\free{ST }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch16}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull16}{SquareFreeRegularTriangularSetXmpPageEmpty16}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{zeroSetSplit(lp,false)$ST\free{lp }\free{ST }\bound{lts }}
+\indentrel{3}\begin{verbatim}
+   (16)
+   [
+       5    4       2     3     8     5    3    2
+     {z  - t , t z y  + 2z y - t  + 2t  + t  - t ,
+        4                2
+      (t  - t)x - t y - z }
+     ,
+      3      5            2    2
+    {t  - 1,z  - t,t y + z ,z x  - t}, {t,z,y,x}]
+Type: List SquareFreeRegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty16}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty16}{SquareFreeRegularTriangularSetXmpPagePatch16}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{zeroSetSplit(lp,false)$ST\free{lp }\free{ST }\bound{lts }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch17}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull17}{SquareFreeRegularTriangularSetXmpPageEmpty17}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{T := REGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{T }}
+\indentrel{3}\begin{verbatim}
+   (17)
+  RegularTriangularSet(Integer,IndexedExponents OrderedVa
+  riableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewS
+  parseMultivariatePolynomial(Integer,OrderedVariableList
+   [x,y,z,t]))
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty17}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty17}{SquareFreeRegularTriangularSetXmpPagePatch17}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{T := REGSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{T }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch18}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull18}{SquareFreeRegularTriangularSetXmpPageEmpty18}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{lts := zeroSetSplit(lp,false)$T\free{lp }\free{T }\bound{lts }}
+\indentrel{3}\begin{verbatim}
+   (18)
+   [
+       5    4       2     3     8     5    3    2
+     {z  - t , t z y  + 2z y - t  + 2t  + t  - t ,
+        4                2
+      (t  - t)x - t y - z }
+     ,
+      3      5          2     3         2
+    {t  - 1,z  - t,t z y  + 2z y + 1,z x  - t},
+    {t,z,y,x}]
+Type: List RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty18}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty18}{SquareFreeRegularTriangularSetXmpPagePatch18}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{lts := zeroSetSplit(lp,false)$T\free{lp }\free{T }\bound{lts }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch19}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull19}{SquareFreeRegularTriangularSetXmpPageEmpty19}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{ts := lts.2\free{lts }\bound{ts }}
+\indentrel{3}\begin{verbatim}
+           3      5          2     3         2
+   (19)  {t  - 1,z  - t,t z y  + 2z y + 1,z x  - t}
+Type: RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty19}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty19}{SquareFreeRegularTriangularSetXmpPagePatch19}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{ts := lts.2\free{lts }\bound{ts }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch20}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull20}{SquareFreeRegularTriangularSetXmpPageEmpty20}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{pol := select(ts,'y)$T\free{ts }\free{y }\free{T }\bound{pol }}
+\indentrel{3}\begin{verbatim}
+              2     3
+   (20)  t z y  + 2z y + 1
+Type: Union(NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty20}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty20}{SquareFreeRegularTriangularSetXmpPagePatch20}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{pol := select(ts,'y)$T\free{ts }\free{y }\free{T }\bound{pol }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch21}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull21}{SquareFreeRegularTriangularSetXmpPageEmpty21}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{tower := collectUnder(ts,'y)$T\free{ts }\free{y }\free{T }\bound{tower }}
+\indentrel{3}\begin{verbatim}
+           3      5
+   (21)  {t  - 1,z  - t}
+Type: RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty21}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty21}{SquareFreeRegularTriangularSetXmpPagePatch21}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{tower := collectUnder(ts,'y)$T\free{ts }\free{y }\free{T }\bound{tower }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch22}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull22}{SquareFreeRegularTriangularSetXmpPageEmpty22}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{pack := RegularTriangularSetGcdPackage(R,E,V,P,T)\free{R }\free{E }\free{V }\free{P }\free{T }\bound{pack }}
+\indentrel{3}\begin{verbatim}
+   (22)
+  RegularTriangularSetGcdPackage(Integer,IndexedExponents
+   OrderedVariableList [x,y,z,t],OrderedVariableList [x,y
+  ,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVa
+  riableList [x,y,z,t]),RegularTriangularSet(Integer,Inde
+  xedExponents OrderedVariableList [x,y,z,t],OrderedVaria
+  bleList [x,y,z,t],NewSparseMultivariatePolynomial(Integ
+  er,OrderedVariableList [x,y,z,t])))
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty22}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty22}{SquareFreeRegularTriangularSetXmpPagePatch22}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{pack := RegularTriangularSetGcdPackage(R,E,V,P,T)\free{R }\free{E }\free{V }\free{P }\free{T }\bound{pack }}
+\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPagePatch23}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageFull23}{SquareFreeRegularTriangularSetXmpPageEmpty23}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{toseSquareFreePart(pol,tower)$pack\free{pol }\free{tower }\free{pack }}
+\indentrel{3}\begin{verbatim}
+                       2          3      5
+   (23)  [[val= t y + z ,tower= {t  - 1,z  - t}]]
+Type: List Record(val: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]),tower: RegularTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SquareFreeRegularTriangularSetXmpPageEmpty23}
+\begin{paste}{SquareFreeRegularTriangularSetXmpPageEmpty23}{SquareFreeRegularTriangularSetXmpPagePatch23}
+\pastebutton{SquareFreeRegularTriangularSetXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{toseSquareFreePart(pol,tower)$pack\free{pol }\free{tower }\free{pack }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/STBL.ht b/src/hyper/pages/STBL.ht
new file mode 100644
index 00000000..c3b1f79b
--- /dev/null
+++ b/src/hyper/pages/STBL.ht
@@ -0,0 +1,62 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\SparseTableXmpTitle}{SparseTable}
+\newcommand{\SparseTableXmpNumber}{9.73}
+%
+% =====================================================================
+\begin{page}{SparseTableXmpPage}{9.73 SparseTable}
+% =====================================================================
+\beginscroll
+%
+The \spadtype{SparseTable} domain provides a general purpose
+table type with default entries.
+\xtc{
+Here we create a table to save strings under integer keys.
+The value \spad{"Try again!"} is returned if no other value has been
+stored for a key.
+}{
+\spadpaste{t: SparseTable(Integer, String, "Try again!") := table() \bound{t}}
+}
+\xtc{
+Entries can be stored in the table.
+}{
+\spadpaste{t.3 := "Number three" \free{t}\bound{t1}}
+}
+\xtc{
+}{
+\spadpaste{t.4 := "Number four" \free{t}\bound{t2}}
+}
+\xtc{
+These values can be retrieved as usual, but if a look up fails
+the default entry will be returned.
+}{
+\spadpaste{t.3 \free{t1}}
+}
+\xtc{
+}{
+\spadpaste{t.2 \free{t}}
+}
+\xtc{
+To see which values are explicitly stored, the
+\spadfunFrom{keys}{SparseTable} and \spadfunFrom{entries}{SparseTable}
+functions can be used.
+}{
+\spadpaste{keys t \free{t1 t2}}
+}
+\xtc{
+}{
+\spadpaste{entries t \free{t1 t2}}
+}
+If a specific table representation
+is required, the \spadtype{GeneralSparseTable} constructor should be used.
+The domain \spadtype{SparseTable(K, E, dflt)} is equivalent to
+\spadtype{GeneralSparseTable(K,E, Table(K,E), dflt)}.
+%-% \HDexptypeindex{GeneralSparseTable}{SparseTableXmpPage}{9.73}{SparseTable}
+For more information, see
+\downlink{`Table'}{TableXmpPage}\ignore{Table} and
+\downlink{`GeneralSparseTable'}{GeneralSparseTableXmpPage}\ignore{GeneralSparseTable}.
+\showBlurb{SparseTable}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/STBL.pht b/src/hyper/pages/STBL.pht
new file mode 100644
index 00000000..a4e15625
--- /dev/null
+++ b/src/hyper/pages/STBL.pht
@@ -0,0 +1,112 @@
+\begin{patch}{SparseTableXmpPagePatch1}
+\begin{paste}{SparseTableXmpPageFull1}{SparseTableXmpPageEmpty1}
+\pastebutton{SparseTableXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{t: SparseTable(Integer, String, "Try again!") := table()\bound{t }}
+\indentrel{3}\begin{verbatim}
+   (1)  table()
+           Type: SparseTable(Integer,String,Try again!)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SparseTableXmpPageEmpty1}
+\begin{paste}{SparseTableXmpPageEmpty1}{SparseTableXmpPagePatch1}
+\pastebutton{SparseTableXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{t: SparseTable(Integer, String, "Try again!") := table()\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{SparseTableXmpPagePatch2}
+\begin{paste}{SparseTableXmpPageFull2}{SparseTableXmpPageEmpty2}
+\pastebutton{SparseTableXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{t.3 := "Number three"\free{t }\bound{t1 }}
+\indentrel{3}\begin{verbatim}
+   (2)  "Number three"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SparseTableXmpPageEmpty2}
+\begin{paste}{SparseTableXmpPageEmpty2}{SparseTableXmpPagePatch2}
+\pastebutton{SparseTableXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{t.3 := "Number three"\free{t }\bound{t1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{SparseTableXmpPagePatch3}
+\begin{paste}{SparseTableXmpPageFull3}{SparseTableXmpPageEmpty3}
+\pastebutton{SparseTableXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{t.4 := "Number four"\free{t }\bound{t2 }}
+\indentrel{3}\begin{verbatim}
+   (3)  "Number four"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SparseTableXmpPageEmpty3}
+\begin{paste}{SparseTableXmpPageEmpty3}{SparseTableXmpPagePatch3}
+\pastebutton{SparseTableXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{t.4 := "Number four"\free{t }\bound{t2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{SparseTableXmpPagePatch4}
+\begin{paste}{SparseTableXmpPageFull4}{SparseTableXmpPageEmpty4}
+\pastebutton{SparseTableXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{t.3\free{t1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  "Number three"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SparseTableXmpPageEmpty4}
+\begin{paste}{SparseTableXmpPageEmpty4}{SparseTableXmpPagePatch4}
+\pastebutton{SparseTableXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{t.3\free{t1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{SparseTableXmpPagePatch5}
+\begin{paste}{SparseTableXmpPageFull5}{SparseTableXmpPageEmpty5}
+\pastebutton{SparseTableXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{t.2\free{t }}
+\indentrel{3}\begin{verbatim}
+   (5)  "Try again!"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SparseTableXmpPageEmpty5}
+\begin{paste}{SparseTableXmpPageEmpty5}{SparseTableXmpPagePatch5}
+\pastebutton{SparseTableXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{t.2\free{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{SparseTableXmpPagePatch6}
+\begin{paste}{SparseTableXmpPageFull6}{SparseTableXmpPageEmpty6}
+\pastebutton{SparseTableXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{keys t\free{t1 t2 }}
+\indentrel{3}\begin{verbatim}
+   (6)  [4,3]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SparseTableXmpPageEmpty6}
+\begin{paste}{SparseTableXmpPageEmpty6}{SparseTableXmpPagePatch6}
+\pastebutton{SparseTableXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{keys t\free{t1 t2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{SparseTableXmpPagePatch7}
+\begin{paste}{SparseTableXmpPageFull7}{SparseTableXmpPageEmpty7}
+\pastebutton{SparseTableXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{entries t\free{t1 t2 }}
+\indentrel{3}\begin{verbatim}
+   (7)  ["Number four","Number three"]
+                                      Type: List String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SparseTableXmpPageEmpty7}
+\begin{paste}{SparseTableXmpPageEmpty7}{SparseTableXmpPagePatch7}
+\pastebutton{SparseTableXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{entries t\free{t1 t2 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/STREAM.ht b/src/hyper/pages/STREAM.ht
new file mode 100644
index 00000000..3be1441a
--- /dev/null
+++ b/src/hyper/pages/STREAM.ht
@@ -0,0 +1,94 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\StreamXmpTitle}{Stream}
+\newcommand{\StreamXmpNumber}{9.76}
+%
+% =====================================================================
+\begin{page}{StreamXmpPage}{9.76 Stream}
+% =====================================================================
+\beginscroll
+
+A \spadtype{Stream} object is represented as a list whose last element contains the
+wherewithal to create the next element, should it ever be required.
+\xtc{
+Let \spad{ints} be the infinite stream of non-negative integers.
+}{
+\spadpaste{ints := [i for i in 0..] \bound{ints}}
+}
+By default, ten stream elements are calculated.
+This number may be changed to something else by the system command
+\spadcmd{)set streams calculate}.
+%-% \HDsyscmdindex{set streams calculate}{StreamXmpPage}{9.76}{Stream}
+For the display purposes of this book, we have chosen a smaller value.
+\xtc{
+More generally, you can construct a stream by specifying its initial
+value and a function which, when given an element, creates the next element.
+}{
+\spadpaste{f : List INT -> List INT \bound{fdec}}
+}
+\xtc{
+}{
+\spadpaste{f x == [x.1 + x.2, x.1] \bound{f}\free{fdec}}
+}
+\xtc{
+}{
+\spadpaste{fibs := [i.2 for i in [generate(f,[1,1])]] \bound{fibs}\free{f}}
+}
+\xtc{
+You can create the stream of odd non-negative integers by either filtering
+them from the integers, or by evaluating an expression for each integer.
+}{
+\spadpaste{[i for i in ints | odd? i] \free{ints}}
+}
+\xtc{
+}{
+\spadpaste{odds := [2*i+1 for i in ints]\bound{odds}\free{ints}}
+}
+\xtc{
+You can accumulate the initial segments of a stream using the
+\spadfunFrom{scan}{StreamFunctions2} operation.
+}{
+\spadpaste{scan(0,+,odds) \free{odds}}
+}
+\xtc{
+The corresponding elements of
+two or more streams can be combined in this way.
+}{
+\spadpaste{[i*j for i in ints for j in odds]\free{ints} \free{odds}}
+}
+\xtc{
+}{
+\spadpaste{map(*,ints,odds)\free{ints odds}}
+}
+\xtc{
+Many operations similar to those applicable to lists are available for
+streams.
+}{
+\spadpaste{first ints \free{ints}}
+}
+\xtc{
+}{
+\spadpaste{rest ints \free{ints}}
+}
+\xtc{
+}{
+\spadpaste{fibs 20 \free{fibs}}
+}
+The packages \spadtype{StreamFunctions1},
+%-% \HDexptypeindex{StreamFunctions1}{StreamXmpPage}{9.76}{Stream}
+\spadtype{StreamFunctions2} and
+%-% \HDexptypeindex{StreamFunctions2}{StreamXmpPage}{9.76}{Stream}
+\spadtype{StreamFunctions3} export some useful stream manipulation
+operations.
+%-% \HDexptypeindex{StreamFunctions3}{StreamXmpPage}{9.76}{Stream}
+For more information, see
+\downlink{``\ugLangItsTitle''}{ugLangItsPage} in Section \ugLangItsNumber\ignore{ugLangIts},
+\downlink{``\ugProblemSeriesTitle''}{ugProblemSeriesPage} in Section \ugProblemSeriesNumber\ignore{ugProblemSeries},
+\downlink{`ContinuedFraction'}{ContinuedFractionXmpPage}\ignore{ContinuedFraction}, and
+\downlink{`List'}{ListXmpPage}\ignore{List}.
+%
+\showBlurb{Stream}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/STREAM.pht b/src/hyper/pages/STREAM.pht
new file mode 100644
index 00000000..98216cea
--- /dev/null
+++ b/src/hyper/pages/STREAM.pht
@@ -0,0 +1,190 @@
+\begin{patch}{StreamXmpPagePatch1}
+\begin{paste}{StreamXmpPageFull1}{StreamXmpPageEmpty1}
+\pastebutton{StreamXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{ints := [i for i in 0..]\bound{ints }}
+\indentrel{3}\begin{verbatim}
+   (1)  [0,1,2,3,4,5,6,7,8,9,...]
+                        Type: Stream NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPageEmpty1}
+\begin{paste}{StreamXmpPageEmpty1}{StreamXmpPagePatch1}
+\pastebutton{StreamXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{ints := [i for i in 0..]\bound{ints }}
+\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPagePatch2}
+\begin{paste}{StreamXmpPageFull2}{StreamXmpPageEmpty2}
+\pastebutton{StreamXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f : List INT -> List INT\bound{fdec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPageEmpty2}
+\begin{paste}{StreamXmpPageEmpty2}{StreamXmpPagePatch2}
+\pastebutton{StreamXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f : List INT -> List INT\bound{fdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPagePatch3}
+\begin{paste}{StreamXmpPageFull3}{StreamXmpPageEmpty3}
+\pastebutton{StreamXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{f x == [x.1 + x.2, x.1]\bound{f }\free{fdec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPageEmpty3}
+\begin{paste}{StreamXmpPageEmpty3}{StreamXmpPagePatch3}
+\pastebutton{StreamXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f x == [x.1 + x.2, x.1]\bound{f }\free{fdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPagePatch4}
+\begin{paste}{StreamXmpPageFull4}{StreamXmpPageEmpty4}
+\pastebutton{StreamXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{fibs := [i.2 for i in [generate(f,[1,1])]]\bound{fibs }\free{f }}
+\indentrel{3}\begin{verbatim}
+   (4)  [1,1,2,3,5,8,13,21,34,55,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPageEmpty4}
+\begin{paste}{StreamXmpPageEmpty4}{StreamXmpPagePatch4}
+\pastebutton{StreamXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{fibs := [i.2 for i in [generate(f,[1,1])]]\bound{fibs }\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPagePatch5}
+\begin{paste}{StreamXmpPageFull5}{StreamXmpPageEmpty5}
+\pastebutton{StreamXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{[i for i in ints | odd? i]\free{ints }}
+\indentrel{3}\begin{verbatim}
+   (5)  [1,3,5,7,9,11,13,15,17,19,...]
+                        Type: Stream NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPageEmpty5}
+\begin{paste}{StreamXmpPageEmpty5}{StreamXmpPagePatch5}
+\pastebutton{StreamXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{[i for i in ints | odd? i]\free{ints }}
+\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPagePatch6}
+\begin{paste}{StreamXmpPageFull6}{StreamXmpPageEmpty6}
+\pastebutton{StreamXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{odds := [2*i+1 for i in ints]\bound{odds }\free{ints }}
+\indentrel{3}\begin{verbatim}
+   (6)  [1,3,5,7,9,11,13,15,17,19,...]
+                        Type: Stream NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPageEmpty6}
+\begin{paste}{StreamXmpPageEmpty6}{StreamXmpPagePatch6}
+\pastebutton{StreamXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{odds := [2*i+1 for i in ints]\bound{odds }\free{ints }}
+\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPagePatch7}
+\begin{paste}{StreamXmpPageFull7}{StreamXmpPageEmpty7}
+\pastebutton{StreamXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{scan(0,+,odds)\free{odds }}
+\indentrel{3}\begin{verbatim}
+   (7)  [1,4,9,16,25,36,49,64,81,100,...]
+                        Type: Stream NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPageEmpty7}
+\begin{paste}{StreamXmpPageEmpty7}{StreamXmpPagePatch7}
+\pastebutton{StreamXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{scan(0,+,odds)\free{odds }}
+\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPagePatch8}
+\begin{paste}{StreamXmpPageFull8}{StreamXmpPageEmpty8}
+\pastebutton{StreamXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{[i*j for i in ints for j in odds]\free{ints }\free{odds }}
+\indentrel{3}\begin{verbatim}
+   (8)  [0,3,10,21,36,55,78,105,136,171,...]
+                        Type: Stream NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPageEmpty8}
+\begin{paste}{StreamXmpPageEmpty8}{StreamXmpPagePatch8}
+\pastebutton{StreamXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{[i*j for i in ints for j in odds]\free{ints }\free{odds }}
+\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPagePatch9}
+\begin{paste}{StreamXmpPageFull9}{StreamXmpPageEmpty9}
+\pastebutton{StreamXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{map(*,ints,odds)\free{ints odds }}
+\indentrel{3}\begin{verbatim}
+   (9)  [0,3,10,21,36,55,78,105,136,171,...]
+                        Type: Stream NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPageEmpty9}
+\begin{paste}{StreamXmpPageEmpty9}{StreamXmpPagePatch9}
+\pastebutton{StreamXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{map(*,ints,odds)\free{ints odds }}
+\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPagePatch10}
+\begin{paste}{StreamXmpPageFull10}{StreamXmpPageEmpty10}
+\pastebutton{StreamXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{first ints\free{ints }}
+\indentrel{3}\begin{verbatim}
+   (10)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPageEmpty10}
+\begin{paste}{StreamXmpPageEmpty10}{StreamXmpPagePatch10}
+\pastebutton{StreamXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{first ints\free{ints }}
+\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPagePatch11}
+\begin{paste}{StreamXmpPageFull11}{StreamXmpPageEmpty11}
+\pastebutton{StreamXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{rest ints\free{ints }}
+\indentrel{3}\begin{verbatim}
+   (11)  [1,2,3,4,5,6,7,8,9,10,...]
+                        Type: Stream NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPageEmpty11}
+\begin{paste}{StreamXmpPageEmpty11}{StreamXmpPagePatch11}
+\pastebutton{StreamXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{rest ints\free{ints }}
+\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPagePatch12}
+\begin{paste}{StreamXmpPageFull12}{StreamXmpPageEmpty12}
+\pastebutton{StreamXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{fibs 20\free{fibs }}
+\indentrel{3}\begin{verbatim}
+   (12)  6765
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StreamXmpPageEmpty12}
+\begin{paste}{StreamXmpPageEmpty12}{StreamXmpPagePatch12}
+\pastebutton{StreamXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{fibs 20\free{fibs }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/STRING.ht b/src/hyper/pages/STRING.ht
new file mode 100644
index 00000000..3a29d765
--- /dev/null
+++ b/src/hyper/pages/STRING.ht
@@ -0,0 +1,216 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\StringXmpTitle}{String}
+\newcommand{\StringXmpNumber}{9.77}
+%
+% =====================================================================
+\begin{page}{StringXmpPage}{9.77 String}
+% =====================================================================
+\beginscroll
+
+The type \spadtype{String} provides character strings.
+Character strings provide all the operations for a one-dimensional array
+of characters, plus additional operations for manipulating text.
+For more information on related topics, see \downlink{`Character'}{CharacterXmpPage}\ignore{Character} and
+\downlink{`CharacterClass'}{CharacterClassXmpPage}\ignore{CharacterClass}.
+You can also issue the system command \spadcmd{)show String} to display
+the full list of operations defined by \spadtype{String}.
+
+\xtc{
+String values can be created using double quotes.
+}{
+\spadpaste{hello := "Hello, I'm AXIOM!" \bound{hello}}
+}
+\xtc{
+Note, however, that double quotes and underscores must be preceded by
+an extra underscore.
+}{
+\spadpaste{said  := "Jane said, _"Look!_"" \bound{said}}
+}
+\xtc{
+}{
+\spadpaste{saw   := "She saw exactly one underscore: __." \bound{saw}}
+}
+\xtc{
+It is also possible to use \spadfunFrom{new}{String} to create a string of any size
+filled with a given character.
+Since there are many \spadfun{new} functions
+it is necessary to indicate the desired type.
+}{
+\spadpaste{gasp: String := new(32, char "x") \bound{gasp}}
+}
+\xtc{
+The length of a string is given by \spadopFrom{\#}{List}.
+}{
+\spadpaste{\#gasp \free{gasp}}
+}
+\xtc{
+Indexing operations allow characters to be extracted or replaced in strings.
+For any string \spad{s}, indices lie in the range \spad{1..\#s}.
+}{
+\spadpaste{hello.2 \free{hello}}
+}
+\xtc{
+Indexing is really just the application of a string to a subscript,
+so any application syntax works.
+}{
+\spadpaste{hello 2  \free{hello}}
+}
+\xtc{
+}{
+\spadpaste{hello(2) \free{hello}}
+}
+\xtc{
+If it is important not to modify a given string, it should be copied
+before any updating operations are used.
+}{
+\spadpaste{hullo := copy hello \free{hello}\bound{hullo0}}
+}
+\xtc{
+}{
+\spadpaste{hullo.2 := char "u"; [hello, hullo] \free{hullo0 hello}\bound{hullo}}
+}
+
+\xtc{
+Operations are provided to split and join strings.
+The \spadfunFrom{concat}{String} operation allows several strings to be joined
+together.
+}{
+\spadpaste{saidsaw := concat ["alpha","---","omega"] \bound{saidsaw}}
+}
+\xtc{
+There is a version of \spadfunFrom{concat}{String} that works with
+two strings.
+}{
+\spadpaste{concat("hello ","goodbye")}
+}
+\xtc{
+Juxtaposition can also be used to concatenate strings.
+}{
+\spadpaste{"This " "is " "several " "strings " "concatenated."}
+}
+\xtc{
+Substrings are obtained by giving an index range.
+}{
+\spadpaste{hello(1..5) \free{hello}}
+}
+\xtc{
+}{
+\spadpaste{hello(8..) \free{hello}}
+}
+\xtc{
+A string can be split into several substrings by giving a separation character
+or character class.
+}{
+\spadpaste{split(hello, char " ")              \free{hello}}
+}
+\xtc{
+}{
+\spadpaste{other := complement alphanumeric(); \bound{other}}
+}
+\xtc{
+}{
+\spadpaste{split(saidsaw, other)               \free{saidsaw other}}
+}
+\xtc{
+Unwanted characters can be trimmed from the beginning or end of a string
+using the operations \spadfunFrom{trim}{String}, \spadfunFrom{leftTrim}{String}
+and \spadfunFrom{rightTrim}{String}.
+}{
+\spadpaste{trim     ("\#\# ++ relax ++ \#\#", char "\#")}
+}
+\xtc{
+Each of these functions takes a string and a second argument to specify
+the characters to be discarded.
+}{
+\spadpaste{trim     ("\#\# ++ relax ++ \#\#", other) \free{other}}
+}
+\xtc{
+The second argument can be given
+either as a single character or as a character class.
+}{
+\spadpaste{leftTrim ("\#\# ++ relax ++ \#\#", other) \free{other}}
+}
+\xtc{
+}{
+\spadpaste{rightTrim("\#\# ++ relax ++ \#\#", other) \free{other}}
+}
+
+\xtc{
+Strings can be changed to upper case or lower case using the operations
+\spadfunFrom{upperCase}{String}, \spadfunFromX{upperCase}{String}, \spadfunFrom{lowerCase}{String} and
+\spadfunFromX{lowerCase}{String}.
+}{
+\spadpaste{upperCase hello \free{hello}}
+}
+\xtc{
+The versions with the exclamation mark
+change the original string, while the others produce a copy.
+}{
+\spadpaste{lowerCase hello \free{hello}}
+}
+
+\xtc{
+Some basic string matching is provided.
+The function \spadfunFrom{prefix?}{String}
+tests whether one string is an initial prefix of another.
+}{
+\spadpaste{prefix?("He", "Hello")}
+}
+\xtc{
+}{
+\spadpaste{prefix?("Her", "Hello")}
+}
+\xtc{
+A similar function, \spadfunFrom{suffix?}{String}, tests for suffixes.
+}{
+\spadpaste{suffix?("", "Hello")}
+}
+\xtc{
+}{
+\spadpaste{suffix?("LO", "Hello")}
+}
+\xtc{
+The function \spadfunFrom{substring?}{String} tests for a substring given a starting
+position.
+}{
+\spadpaste{substring?("ll", "Hello", 3)}
+}
+\xtc{
+}{
+\spadpaste{substring?("ll", "Hello", 4)}
+}
+
+\xtc{
+A number of \spadfunFrom{position}{String} functions locate things in strings.
+If the first argument to position is a string, then \spad{position(s,t,i)}
+finds the location of \spad{s} as a substring of \spad{t} starting the
+search at position \spad{i}.
+}{
+\spadpaste{n := position("nd", "underground",   1) \bound{n}}
+}
+\xtc{
+}{
+\spadpaste{n := position("nd", "underground", n+1) \free{n} \bound{n1}}
+}
+\xtc{
+If \spad{s} is not found, then \spad{0} is returned (\spad{minIndex(s)-1}
+in \spadtype{IndexedString}).
+}{
+\spadpaste{n := position("nd", "underground", n+1) \free{n1}\bound{n2}}
+}
+\xtc{
+To search for a specific character or a member of a character class,
+a different first argument is used.
+}{
+\spadpaste{position(char "d", "underground", 1)}
+}
+\xtc{
+}{
+\spadpaste{position(hexDigit(), "underground", 1)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/STRING.pht b/src/hyper/pages/STRING.pht
new file mode 100644
index 00000000..db078225
--- /dev/null
+++ b/src/hyper/pages/STRING.pht
@@ -0,0 +1,559 @@
+\begin{patch}{StringXmpPagePatch1}
+\begin{paste}{StringXmpPageFull1}{StringXmpPageEmpty1}
+\pastebutton{StringXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{hello := "Hello, I'm AXIOM!"\bound{hello }}
+\indentrel{3}\begin{verbatim}
+   (1)  "Hello, I'm AXIOM!"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty1}
+\begin{paste}{StringXmpPageEmpty1}{StringXmpPagePatch1}
+\pastebutton{StringXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{hello := "Hello, I'm AXIOM!"\bound{hello }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch2}
+\begin{paste}{StringXmpPageFull2}{StringXmpPageEmpty2}
+\pastebutton{StringXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{said := "Jane said, _"Look!_""\bound{said }}
+\indentrel{3}\begin{verbatim}
+   (2)  "Jane said, "Look!""
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty2}
+\begin{paste}{StringXmpPageEmpty2}{StringXmpPagePatch2}
+\pastebutton{StringXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{said := "Jane said, _"Look!_""\bound{said }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch3}
+\begin{paste}{StringXmpPageFull3}{StringXmpPageEmpty3}
+\pastebutton{StringXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{saw := "She saw exactly one underscore: __."\bound{saw }}
+\indentrel{3}\begin{verbatim}
+   (3)  "She saw exactly one underscore: _."
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty3}
+\begin{paste}{StringXmpPageEmpty3}{StringXmpPagePatch3}
+\pastebutton{StringXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{saw := "She saw exactly one underscore: __."\bound{saw }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch4}
+\begin{paste}{StringXmpPageFull4}{StringXmpPageEmpty4}
+\pastebutton{StringXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{gasp: String := new(32, char "x")\bound{gasp }}
+\indentrel{3}\begin{verbatim}
+   (4)  "xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty4}
+\begin{paste}{StringXmpPageEmpty4}{StringXmpPagePatch4}
+\pastebutton{StringXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{gasp: String := new(32, char "x")\bound{gasp }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch5}
+\begin{paste}{StringXmpPageFull5}{StringXmpPageEmpty5}
+\pastebutton{StringXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{\#gasp\free{gasp }}
+\indentrel{3}\begin{verbatim}
+   (5)  32
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty5}
+\begin{paste}{StringXmpPageEmpty5}{StringXmpPagePatch5}
+\pastebutton{StringXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{\#gasp\free{gasp }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch6}
+\begin{paste}{StringXmpPageFull6}{StringXmpPageEmpty6}
+\pastebutton{StringXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{hello.2\free{hello }}
+\indentrel{3}\begin{verbatim}
+   (6)  e
+                                        Type: Character
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty6}
+\begin{paste}{StringXmpPageEmpty6}{StringXmpPagePatch6}
+\pastebutton{StringXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{hello.2\free{hello }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch7}
+\begin{paste}{StringXmpPageFull7}{StringXmpPageEmpty7}
+\pastebutton{StringXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{hello 2\free{hello }}
+\indentrel{3}\begin{verbatim}
+   (7)  e
+                                        Type: Character
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty7}
+\begin{paste}{StringXmpPageEmpty7}{StringXmpPagePatch7}
+\pastebutton{StringXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{hello 2\free{hello }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch8}
+\begin{paste}{StringXmpPageFull8}{StringXmpPageEmpty8}
+\pastebutton{StringXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{hello(2)\free{hello }}
+\indentrel{3}\begin{verbatim}
+   (8)  e
+                                        Type: Character
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty8}
+\begin{paste}{StringXmpPageEmpty8}{StringXmpPagePatch8}
+\pastebutton{StringXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{hello(2)\free{hello }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch9}
+\begin{paste}{StringXmpPageFull9}{StringXmpPageEmpty9}
+\pastebutton{StringXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{hullo := copy hello\free{hello }\bound{hullo0 }}
+\indentrel{3}\begin{verbatim}
+   (9)  "Hello, I'm AXIOM!"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty9}
+\begin{paste}{StringXmpPageEmpty9}{StringXmpPagePatch9}
+\pastebutton{StringXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{hullo := copy hello\free{hello }\bound{hullo0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch10}
+\begin{paste}{StringXmpPageFull10}{StringXmpPageEmpty10}
+\pastebutton{StringXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{hullo.2 := char "u"; [hello, hullo]\free{hullo0 hello }\bound{hullo }}
+\indentrel{3}\begin{verbatim}
+   (10)  ["Hello, I'm AXIOM!","Hullo, I'm AXIOM!"]
+                                      Type: List String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty10}
+\begin{paste}{StringXmpPageEmpty10}{StringXmpPagePatch10}
+\pastebutton{StringXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{hullo.2 := char "u"; [hello, hullo]\free{hullo0 hello }\bound{hullo }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch11}
+\begin{paste}{StringXmpPageFull11}{StringXmpPageEmpty11}
+\pastebutton{StringXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{saidsaw := concat ["alpha","---","omega"]\bound{saidsaw }}
+\indentrel{3}\begin{verbatim}
+   (11)  "alpha---omega"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty11}
+\begin{paste}{StringXmpPageEmpty11}{StringXmpPagePatch11}
+\pastebutton{StringXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{saidsaw := concat ["alpha","---","omega"]\bound{saidsaw }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch12}
+\begin{paste}{StringXmpPageFull12}{StringXmpPageEmpty12}
+\pastebutton{StringXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{concat("hello ","goodbye")}
+\indentrel{3}\begin{verbatim}
+   (12)  "hello goodbye"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty12}
+\begin{paste}{StringXmpPageEmpty12}{StringXmpPagePatch12}
+\pastebutton{StringXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{concat("hello ","goodbye")}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch13}
+\begin{paste}{StringXmpPageFull13}{StringXmpPageEmpty13}
+\pastebutton{StringXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{"This " "is " "several " "strings " "concatenated."}
+\indentrel{3}\begin{verbatim}
+   (13)  "This is several strings concatenated."
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty13}
+\begin{paste}{StringXmpPageEmpty13}{StringXmpPagePatch13}
+\pastebutton{StringXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{"This " "is " "several " "strings " "concatenated."}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch14}
+\begin{paste}{StringXmpPageFull14}{StringXmpPageEmpty14}
+\pastebutton{StringXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{hello(1..5)\free{hello }}
+\indentrel{3}\begin{verbatim}
+   (14)  "Hello"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty14}
+\begin{paste}{StringXmpPageEmpty14}{StringXmpPagePatch14}
+\pastebutton{StringXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{hello(1..5)\free{hello }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch15}
+\begin{paste}{StringXmpPageFull15}{StringXmpPageEmpty15}
+\pastebutton{StringXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{hello(8..)\free{hello }}
+\indentrel{3}\begin{verbatim}
+   (15)  "I'm AXIOM!"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty15}
+\begin{paste}{StringXmpPageEmpty15}{StringXmpPagePatch15}
+\pastebutton{StringXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{hello(8..)\free{hello }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch16}
+\begin{paste}{StringXmpPageFull16}{StringXmpPageEmpty16}
+\pastebutton{StringXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{split(hello, char " ")\free{hello }}
+\indentrel{3}\begin{verbatim}
+   (16)  ["Hello,","I'm","AXIOM!"]
+                                      Type: List String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty16}
+\begin{paste}{StringXmpPageEmpty16}{StringXmpPagePatch16}
+\pastebutton{StringXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{split(hello, char " ")\free{hello }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch17}
+\begin{paste}{StringXmpPageFull17}{StringXmpPageEmpty17}
+\pastebutton{StringXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{other := complement alphanumeric();\bound{other }}
+\indentrel{3}\begin{verbatim}
+                                   Type: CharacterClass
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty17}
+\begin{paste}{StringXmpPageEmpty17}{StringXmpPagePatch17}
+\pastebutton{StringXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{other := complement alphanumeric();\bound{other }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch18}
+\begin{paste}{StringXmpPageFull18}{StringXmpPageEmpty18}
+\pastebutton{StringXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{split(saidsaw, other)\free{saidsaw other }}
+\indentrel{3}\begin{verbatim}
+   (18)  ["alpha","omega"]
+                                      Type: List String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty18}
+\begin{paste}{StringXmpPageEmpty18}{StringXmpPagePatch18}
+\pastebutton{StringXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{split(saidsaw, other)\free{saidsaw other }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch19}
+\begin{paste}{StringXmpPageFull19}{StringXmpPageEmpty19}
+\pastebutton{StringXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{trim ("\#\# ++ relax ++ \#\#", char "\#")}
+\indentrel{3}\begin{verbatim}
+   (19)  " ++ relax ++ "
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty19}
+\begin{paste}{StringXmpPageEmpty19}{StringXmpPagePatch19}
+\pastebutton{StringXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{trim ("\#\# ++ relax ++ \#\#", char "\#")}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch20}
+\begin{paste}{StringXmpPageFull20}{StringXmpPageEmpty20}
+\pastebutton{StringXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{trim ("\#\# ++ relax ++ \#\#", other)\free{other }}
+\indentrel{3}\begin{verbatim}
+   (20)  "relax"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty20}
+\begin{paste}{StringXmpPageEmpty20}{StringXmpPagePatch20}
+\pastebutton{StringXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{trim ("\#\# ++ relax ++ \#\#", other)\free{other }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch21}
+\begin{paste}{StringXmpPageFull21}{StringXmpPageEmpty21}
+\pastebutton{StringXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{leftTrim ("\#\# ++ relax ++ \#\#", other)\free{other }}
+\indentrel{3}\begin{verbatim}
+   (21)  "relax ++ ##"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty21}
+\begin{paste}{StringXmpPageEmpty21}{StringXmpPagePatch21}
+\pastebutton{StringXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{leftTrim ("\#\# ++ relax ++ \#\#", other)\free{other }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch22}
+\begin{paste}{StringXmpPageFull22}{StringXmpPageEmpty22}
+\pastebutton{StringXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{rightTrim("\#\# ++ relax ++ \#\#", other)\free{other }}
+\indentrel{3}\begin{verbatim}
+   (22)  "## ++ relax"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty22}
+\begin{paste}{StringXmpPageEmpty22}{StringXmpPagePatch22}
+\pastebutton{StringXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{rightTrim("\#\# ++ relax ++ \#\#", other)\free{other }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch23}
+\begin{paste}{StringXmpPageFull23}{StringXmpPageEmpty23}
+\pastebutton{StringXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{upperCase hello\free{hello }}
+\indentrel{3}\begin{verbatim}
+   (23)  "HELLO, I'M AXIOM!"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty23}
+\begin{paste}{StringXmpPageEmpty23}{StringXmpPagePatch23}
+\pastebutton{StringXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{upperCase hello\free{hello }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch24}
+\begin{paste}{StringXmpPageFull24}{StringXmpPageEmpty24}
+\pastebutton{StringXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{lowerCase hello\free{hello }}
+\indentrel{3}\begin{verbatim}
+   (24)  "hello, i'm axiom!"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty24}
+\begin{paste}{StringXmpPageEmpty24}{StringXmpPagePatch24}
+\pastebutton{StringXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{lowerCase hello\free{hello }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch25}
+\begin{paste}{StringXmpPageFull25}{StringXmpPageEmpty25}
+\pastebutton{StringXmpPageFull25}{\hidepaste}
+\tab{5}\spadcommand{prefix?("He", "Hello")}
+\indentrel{3}\begin{verbatim}
+   (25)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty25}
+\begin{paste}{StringXmpPageEmpty25}{StringXmpPagePatch25}
+\pastebutton{StringXmpPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{prefix?("He", "Hello")}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch26}
+\begin{paste}{StringXmpPageFull26}{StringXmpPageEmpty26}
+\pastebutton{StringXmpPageFull26}{\hidepaste}
+\tab{5}\spadcommand{prefix?("Her", "Hello")}
+\indentrel{3}\begin{verbatim}
+   (26)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty26}
+\begin{paste}{StringXmpPageEmpty26}{StringXmpPagePatch26}
+\pastebutton{StringXmpPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{prefix?("Her", "Hello")}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch27}
+\begin{paste}{StringXmpPageFull27}{StringXmpPageEmpty27}
+\pastebutton{StringXmpPageFull27}{\hidepaste}
+\tab{5}\spadcommand{suffix?("", "Hello")}
+\indentrel{3}\begin{verbatim}
+   (27)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty27}
+\begin{paste}{StringXmpPageEmpty27}{StringXmpPagePatch27}
+\pastebutton{StringXmpPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{suffix?("", "Hello")}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch28}
+\begin{paste}{StringXmpPageFull28}{StringXmpPageEmpty28}
+\pastebutton{StringXmpPageFull28}{\hidepaste}
+\tab{5}\spadcommand{suffix?("LO", "Hello")}
+\indentrel{3}\begin{verbatim}
+   (28)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty28}
+\begin{paste}{StringXmpPageEmpty28}{StringXmpPagePatch28}
+\pastebutton{StringXmpPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{suffix?("LO", "Hello")}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch29}
+\begin{paste}{StringXmpPageFull29}{StringXmpPageEmpty29}
+\pastebutton{StringXmpPageFull29}{\hidepaste}
+\tab{5}\spadcommand{substring?("ll", "Hello", 3)}
+\indentrel{3}\begin{verbatim}
+   (29)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty29}
+\begin{paste}{StringXmpPageEmpty29}{StringXmpPagePatch29}
+\pastebutton{StringXmpPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{substring?("ll", "Hello", 3)}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch30}
+\begin{paste}{StringXmpPageFull30}{StringXmpPageEmpty30}
+\pastebutton{StringXmpPageFull30}{\hidepaste}
+\tab{5}\spadcommand{substring?("ll", "Hello", 4)}
+\indentrel{3}\begin{verbatim}
+   (30)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty30}
+\begin{paste}{StringXmpPageEmpty30}{StringXmpPagePatch30}
+\pastebutton{StringXmpPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{substring?("ll", "Hello", 4)}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch31}
+\begin{paste}{StringXmpPageFull31}{StringXmpPageEmpty31}
+\pastebutton{StringXmpPageFull31}{\hidepaste}
+\tab{5}\spadcommand{n := position("nd", "underground", 1)\bound{n }}
+\indentrel{3}\begin{verbatim}
+   (31)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty31}
+\begin{paste}{StringXmpPageEmpty31}{StringXmpPagePatch31}
+\pastebutton{StringXmpPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{n := position("nd", "underground", 1)\bound{n }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch32}
+\begin{paste}{StringXmpPageFull32}{StringXmpPageEmpty32}
+\pastebutton{StringXmpPageFull32}{\hidepaste}
+\tab{5}\spadcommand{n := position("nd", "underground", n+1)\free{n }\bound{n1 }}
+\indentrel{3}\begin{verbatim}
+   (32)  10
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty32}
+\begin{paste}{StringXmpPageEmpty32}{StringXmpPagePatch32}
+\pastebutton{StringXmpPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{n := position("nd", "underground", n+1)\free{n }\bound{n1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch33}
+\begin{paste}{StringXmpPageFull33}{StringXmpPageEmpty33}
+\pastebutton{StringXmpPageFull33}{\hidepaste}
+\tab{5}\spadcommand{n := position("nd", "underground", n+1)\free{n1 }\bound{n2 }}
+\indentrel{3}\begin{verbatim}
+   (33)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty33}
+\begin{paste}{StringXmpPageEmpty33}{StringXmpPagePatch33}
+\pastebutton{StringXmpPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{n := position("nd", "underground", n+1)\free{n1 }\bound{n2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch34}
+\begin{paste}{StringXmpPageFull34}{StringXmpPageEmpty34}
+\pastebutton{StringXmpPageFull34}{\hidepaste}
+\tab{5}\spadcommand{position(char "d", "underground", 1)}
+\indentrel{3}\begin{verbatim}
+   (34)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty34}
+\begin{paste}{StringXmpPageEmpty34}{StringXmpPagePatch34}
+\pastebutton{StringXmpPageEmpty34}{\showpaste}
+\tab{5}\spadcommand{position(char "d", "underground", 1)}
+\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPagePatch35}
+\begin{paste}{StringXmpPageFull35}{StringXmpPageEmpty35}
+\pastebutton{StringXmpPageFull35}{\hidepaste}
+\tab{5}\spadcommand{position(hexDigit(), "underground", 1)}
+\indentrel{3}\begin{verbatim}
+   (35)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringXmpPageEmpty35}
+\begin{paste}{StringXmpPageEmpty35}{StringXmpPagePatch35}
+\pastebutton{StringXmpPageEmpty35}{\showpaste}
+\tab{5}\spadcommand{position(hexDigit(), "underground", 1)}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/STRTBL.ht b/src/hyper/pages/STRTBL.ht
new file mode 100644
index 00000000..ec4f6786
--- /dev/null
+++ b/src/hyper/pages/STRTBL.ht
@@ -0,0 +1,40 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\StringTableXmpTitle}{StringTable}
+\newcommand{\StringTableXmpNumber}{9.78}
+%
+% =====================================================================
+\begin{page}{StringTableXmpPage}{9.78 StringTable}
+% =====================================================================
+\beginscroll
+%
+This domain provides a table type in which the keys are known to
+be strings so special techniques can be used.
+Other than performance, the type \spadtype{StringTable(S)} should
+behave exactly the same way as \spadtype{Table(String,S)}.
+See \downlink{`Table'}{TableXmpPage}\ignore{Table} for general information about tables.
+\showBlurb{StringTable}
+
+\xtc{
+This creates a new table whose keys are strings.
+}{
+\spadpaste{t: StringTable(Integer) := table()  \bound{t}}
+}
+\xtc{
+The value associated with each string key is the number of
+characters in the string.
+}{
+\begin{spadsrc}[\free{t}\bound{h}]
+for s in split("My name is Ian Watt.",char " ")
+  repeat
+    t.s := #s
+\end{spadsrc}
+}
+\xtc{
+}{
+\spadpaste{for key in keys t repeat output [key, t.key] \free{t h}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/STRTBL.pht b/src/hyper/pages/STRTBL.pht
new file mode 100644
index 00000000..e68e2167
--- /dev/null
+++ b/src/hyper/pages/STRTBL.pht
@@ -0,0 +1,57 @@
+\begin{patch}{StringTableXmpPagePatch1}
+\begin{paste}{StringTableXmpPageFull1}{StringTableXmpPageEmpty1}
+\pastebutton{StringTableXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{t: StringTable(Integer) := table()\bound{t }}
+\indentrel{3}\begin{verbatim}
+   (1)  table()
+                              Type: StringTable Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringTableXmpPageEmpty1}
+\begin{paste}{StringTableXmpPageEmpty1}{StringTableXmpPagePatch1}
+\pastebutton{StringTableXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{t: StringTable(Integer) := table()\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringTableXmpPagePatch2}
+\begin{paste}{StringTableXmpPageFull2}{StringTableXmpPageEmpty2}
+\pastebutton{StringTableXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{for s in split("My name is Ian Watt.",char " ")
+  repeat
+    t.s := \#s
+\free{t }\bound{h }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringTableXmpPageEmpty2}
+\begin{paste}{StringTableXmpPageEmpty2}{StringTableXmpPagePatch2}
+\pastebutton{StringTableXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{for s in split("My name is Ian Watt.",char " ")
+  repeat
+    t.s := \#s
+\free{t }\bound{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{StringTableXmpPagePatch3}
+\begin{paste}{StringTableXmpPageFull3}{StringTableXmpPageEmpty3}
+\pastebutton{StringTableXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{for key in keys t repeat output [key, t.key]\free{t h }}
+\indentrel{3}\begin{verbatim}
+   ["Ian",3]
+   ["My",2]
+   ["Watt.",5]
+   ["name",4]
+   ["is",2]
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{StringTableXmpPageEmpty3}
+\begin{paste}{StringTableXmpPageEmpty3}{StringTableXmpPagePatch3}
+\pastebutton{StringTableXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{for key in keys t repeat output [key, t.key]\free{t h }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/SYMBOL.ht b/src/hyper/pages/SYMBOL.ht
new file mode 100644
index 00000000..da25d20c
--- /dev/null
+++ b/src/hyper/pages/SYMBOL.ht
@@ -0,0 +1,151 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\SymbolXmpTitle}{Symbol}
+\newcommand{\SymbolXmpNumber}{9.79}
+%
+% =====================================================================
+\begin{page}{SymbolXmpPage}{9.79 Symbol}
+% =====================================================================
+\beginscroll
+ 
+Symbols are one of the basic types manipulated by \Language{}.
+The \spadtype{Symbol} domain provides ways to create
+symbols of many varieties.
+\showBlurb{Symbol}
+ 
+\xtc{
+The simplest way to create a symbol is to ``single quote'' an identifier.
+%-% \HDindex{quote}{SymbolXmpPage}{9.79}{Symbol}
+}{
+\spadpaste{X: Symbol := 'x \bound{X}}
+}
+\xtc{
+This gives the symbol even if \spad{x} has been assigned a value.
+If \spad{x} has not been assigned a value, then it is possible to omit
+the quote.
+}{
+\spadpaste{XX: Symbol := x}
+}
+\xtc{
+Declarations must be used when working
+with symbols, because otherwise the interpreter tries to place
+values in a more specialized type \spadtype{Variable}.
+}{
+\spadpaste{A := 'a}
+}
+\xtc{
+}{
+\spadpaste{B := b}
+}
+\xtc{
+The normal way of entering polynomials uses this fact.
+}{
+\spadpaste{x**2 + 1}
+}
+ 
+\xtc{
+Another convenient way to create symbols is to convert a string.
+This is useful when the name is to be constructed by a program.
+}{
+\spadpaste{"Hello"::Symbol}
+}
+\xtc{
+Sometimes it is necessary to generate new unique symbols, for example, to
+name constants of integration.
+The expression \spad{new()} generates a symbol starting with \spad{\%}.
+}{
+\spadpaste{new()\$Symbol}
+}
+\xtc{
+Successive calls to \spadfunFrom{new}{Symbol} produce different symbols.
+}{
+\spadpaste{new()\$Symbol}
+}
+\xtc{
+The expression \spad{new("s")} produces a symbol starting with \spad{\%s}.
+}{
+\spadpaste{new("xyz")\$Symbol}
+}
+ 
+\xtc{
+A symbol can be adorned in various ways.
+The most basic thing is applying a symbol to a list
+of subscripts.
+}{
+\spadpaste{X[i,j] \free{X}}
+}
+ 
+\xtc{
+Somewhat less pretty is to attach subscripts, superscripts or arguments.
+}{
+\spadpaste{U := subscript(u, [1,2,1,2]) \bound{U}}
+}
+\xtc{
+}{
+\spadpaste{V := superscript(v, [n]) \bound{V}}
+}
+\xtc{
+}{
+\spadpaste{P := argscript(p, [t]) \bound{P}}
+}
+ 
+\xtc{
+It is possible to test whether a symbol has scripts using the
+\spadfunFrom{scripted?}{Symbol} test.
+}{
+\spadpaste{scripted? U \free{U}}
+}
+\xtc{
+}{
+\spadpaste{scripted? X \free{X}}
+}
+\xtc{
+If a symbol is not scripted, then it may be converted to a string.
+}{
+\spadpaste{string X \free{X}}
+}
+\xtc{
+The basic parts can always be extracted using the
+\spadfunFrom{name}{Symbol} and \spadfunFrom{scripts}{Symbol} operations.
+}{
+\spadpaste{name U \free{U}}
+}
+\xtc{
+}{
+\spadpaste{scripts U \free{U}}
+}
+\xtc{
+}{
+\spadpaste{name X \free{X}}
+}
+\xtc{
+}{
+\spadpaste{scripts X \free{X}}
+}
+ 
+\xtc{
+The most general form is obtained using the \spadfunFrom{script}{Symbol}
+operation.
+This operation takes an argument which is a list containing, in this order,
+lists of subscripts, superscripts, presuperscripts, presubscripts and
+arguments to a symbol.
+}{
+\spadpaste{M := script(Mammoth, [[i,j],[k,l],[0,1],[2],[u,v,w]]) \bound{M}}
+}
+\xtc{
+}{
+\spadpaste{scripts M \free{M}}
+}
+\xtc{
+If trailing lists of scripts are omitted, they are assumed to be empty.
+}{
+\spadpaste{N := script(Nut, [[i,j],[k,l],[0,1]]) \bound{N}}
+}
+\xtc{
+}{
+\spadpaste{scripts N \free{N}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/SYMBOL.pht b/src/hyper/pages/SYMBOL.pht
new file mode 100644
index 00000000..86909eb7
--- /dev/null
+++ b/src/hyper/pages/SYMBOL.pht
@@ -0,0 +1,398 @@
+\begin{patch}{SymbolXmpPagePatch1}
+\begin{paste}{SymbolXmpPageFull1}{SymbolXmpPageEmpty1}
+\pastebutton{SymbolXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{X: Symbol := 'x\bound{X }}
+\indentrel{3}\begin{verbatim}
+   (1)  x
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty1}
+\begin{paste}{SymbolXmpPageEmpty1}{SymbolXmpPagePatch1}
+\pastebutton{SymbolXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{X: Symbol := 'x\bound{X }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch2}
+\begin{paste}{SymbolXmpPageFull2}{SymbolXmpPageEmpty2}
+\pastebutton{SymbolXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{XX: Symbol := x}
+\indentrel{3}\begin{verbatim}
+   (2)  x
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty2}
+\begin{paste}{SymbolXmpPageEmpty2}{SymbolXmpPagePatch2}
+\pastebutton{SymbolXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{XX: Symbol := x}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch3}
+\begin{paste}{SymbolXmpPageFull3}{SymbolXmpPageEmpty3}
+\pastebutton{SymbolXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{A := 'a}
+\indentrel{3}\begin{verbatim}
+   (3)  a
+                                       Type: Variable a
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty3}
+\begin{paste}{SymbolXmpPageEmpty3}{SymbolXmpPagePatch3}
+\pastebutton{SymbolXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{A := 'a}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch4}
+\begin{paste}{SymbolXmpPageFull4}{SymbolXmpPageEmpty4}
+\pastebutton{SymbolXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{B := b}
+\indentrel{3}\begin{verbatim}
+   (4)  b
+                                       Type: Variable b
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty4}
+\begin{paste}{SymbolXmpPageEmpty4}{SymbolXmpPagePatch4}
+\pastebutton{SymbolXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{B := b}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch5}
+\begin{paste}{SymbolXmpPageFull5}{SymbolXmpPageEmpty5}
+\pastebutton{SymbolXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{x**2 + 1}
+\indentrel{3}\begin{verbatim}
+         2
+   (5)  x  + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty5}
+\begin{paste}{SymbolXmpPageEmpty5}{SymbolXmpPagePatch5}
+\pastebutton{SymbolXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{x**2 + 1}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch6}
+\begin{paste}{SymbolXmpPageFull6}{SymbolXmpPageEmpty6}
+\pastebutton{SymbolXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{"Hello"::Symbol}
+\indentrel{3}\begin{verbatim}
+   (6)  Hello
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty6}
+\begin{paste}{SymbolXmpPageEmpty6}{SymbolXmpPagePatch6}
+\pastebutton{SymbolXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{"Hello"::Symbol}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch7}
+\begin{paste}{SymbolXmpPageFull7}{SymbolXmpPageEmpty7}
+\pastebutton{SymbolXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{new()$Symbol}
+\indentrel{3}\begin{verbatim}
+   (7)  %A
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty7}
+\begin{paste}{SymbolXmpPageEmpty7}{SymbolXmpPagePatch7}
+\pastebutton{SymbolXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{new()$Symbol}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch8}
+\begin{paste}{SymbolXmpPageFull8}{SymbolXmpPageEmpty8}
+\pastebutton{SymbolXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{new()$Symbol}
+\indentrel{3}\begin{verbatim}
+   (8)  %B
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty8}
+\begin{paste}{SymbolXmpPageEmpty8}{SymbolXmpPagePatch8}
+\pastebutton{SymbolXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{new()$Symbol}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch9}
+\begin{paste}{SymbolXmpPageFull9}{SymbolXmpPageEmpty9}
+\pastebutton{SymbolXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{new("xyz")$Symbol}
+\indentrel{3}\begin{verbatim}
+   (9)  %xyz0
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty9}
+\begin{paste}{SymbolXmpPageEmpty9}{SymbolXmpPagePatch9}
+\pastebutton{SymbolXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{new("xyz")$Symbol}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch10}
+\begin{paste}{SymbolXmpPageFull10}{SymbolXmpPageEmpty10}
+\pastebutton{SymbolXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{X[i,j]\free{X }}
+\indentrel{3}\begin{verbatim}
+   (10)  x
+          i,j
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty10}
+\begin{paste}{SymbolXmpPageEmpty10}{SymbolXmpPagePatch10}
+\pastebutton{SymbolXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{X[i,j]\free{X }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch11}
+\begin{paste}{SymbolXmpPageFull11}{SymbolXmpPageEmpty11}
+\pastebutton{SymbolXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{U := subscript(u, [1,2,1,2])\bound{U }}
+\indentrel{3}\begin{verbatim}
+   (11)  u
+          1,2,1,2
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty11}
+\begin{paste}{SymbolXmpPageEmpty11}{SymbolXmpPagePatch11}
+\pastebutton{SymbolXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{U := subscript(u, [1,2,1,2])\bound{U }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch12}
+\begin{paste}{SymbolXmpPageFull12}{SymbolXmpPageEmpty12}
+\pastebutton{SymbolXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{V := superscript(v, [n])\bound{V }}
+\indentrel{3}\begin{verbatim}
+          n
+   (12)  v
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty12}
+\begin{paste}{SymbolXmpPageEmpty12}{SymbolXmpPagePatch12}
+\pastebutton{SymbolXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{V := superscript(v, [n])\bound{V }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch13}
+\begin{paste}{SymbolXmpPageFull13}{SymbolXmpPageEmpty13}
+\pastebutton{SymbolXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{P := argscript(p, [t])\bound{P }}
+\indentrel{3}\begin{verbatim}
+   (13)  p(t)
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty13}
+\begin{paste}{SymbolXmpPageEmpty13}{SymbolXmpPagePatch13}
+\pastebutton{SymbolXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{P := argscript(p, [t])\bound{P }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch14}
+\begin{paste}{SymbolXmpPageFull14}{SymbolXmpPageEmpty14}
+\pastebutton{SymbolXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{scripted? U\free{U }}
+\indentrel{3}\begin{verbatim}
+   (14)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty14}
+\begin{paste}{SymbolXmpPageEmpty14}{SymbolXmpPagePatch14}
+\pastebutton{SymbolXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{scripted? U\free{U }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch15}
+\begin{paste}{SymbolXmpPageFull15}{SymbolXmpPageEmpty15}
+\pastebutton{SymbolXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{scripted? X\free{X }}
+\indentrel{3}\begin{verbatim}
+   (15)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty15}
+\begin{paste}{SymbolXmpPageEmpty15}{SymbolXmpPagePatch15}
+\pastebutton{SymbolXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{scripted? X\free{X }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch16}
+\begin{paste}{SymbolXmpPageFull16}{SymbolXmpPageEmpty16}
+\pastebutton{SymbolXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{string X\free{X }}
+\indentrel{3}\begin{verbatim}
+   (16)  "x"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty16}
+\begin{paste}{SymbolXmpPageEmpty16}{SymbolXmpPagePatch16}
+\pastebutton{SymbolXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{string X\free{X }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch17}
+\begin{paste}{SymbolXmpPageFull17}{SymbolXmpPageEmpty17}
+\pastebutton{SymbolXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{name U\free{U }}
+\indentrel{3}\begin{verbatim}
+   (17)  u
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty17}
+\begin{paste}{SymbolXmpPageEmpty17}{SymbolXmpPagePatch17}
+\pastebutton{SymbolXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{name U\free{U }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch18}
+\begin{paste}{SymbolXmpPageFull18}{SymbolXmpPageEmpty18}
+\pastebutton{SymbolXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{scripts U\free{U }}
+\indentrel{3}\begin{verbatim}
+   (18)
+   [sub= [1,2,1,2],sup= [],presup= [],presub= [],args= []]
+Type: Record(sub: List OutputForm,sup: List OutputForm,presup: List OutputForm,presub: List OutputForm,args: List OutputForm)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty18}
+\begin{paste}{SymbolXmpPageEmpty18}{SymbolXmpPagePatch18}
+\pastebutton{SymbolXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{scripts U\free{U }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch19}
+\begin{paste}{SymbolXmpPageFull19}{SymbolXmpPageEmpty19}
+\pastebutton{SymbolXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{name X\free{X }}
+\indentrel{3}\begin{verbatim}
+   (19)  x
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty19}
+\begin{paste}{SymbolXmpPageEmpty19}{SymbolXmpPagePatch19}
+\pastebutton{SymbolXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{name X\free{X }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch20}
+\begin{paste}{SymbolXmpPageFull20}{SymbolXmpPageEmpty20}
+\pastebutton{SymbolXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{scripts X\free{X }}
+\indentrel{3}\begin{verbatim}
+   (20)
+   [sub= [],sup= [],presup= [],presub= [],args= []]
+Type: Record(sub: List OutputForm,sup: List OutputForm,presup: List OutputForm,presub: List OutputForm,args: List OutputForm)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty20}
+\begin{paste}{SymbolXmpPageEmpty20}{SymbolXmpPagePatch20}
+\pastebutton{SymbolXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{scripts X\free{X }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch21}
+\begin{paste}{SymbolXmpPageFull21}{SymbolXmpPageEmpty21}
+\pastebutton{SymbolXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{M := script(Mammoth, [[i,j],[k,l],[0,1],[2],[u,v,w]])\bound{M }}
+\indentrel{3}\begin{verbatim}
+         0,1       k,l
+   (21)     Mammoth   (u,v,w)
+           2       i,j
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty21}
+\begin{paste}{SymbolXmpPageEmpty21}{SymbolXmpPagePatch21}
+\pastebutton{SymbolXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{M := script(Mammoth, [[i,j],[k,l],[0,1],[2],[u,v,w]])\bound{M }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch22}
+\begin{paste}{SymbolXmpPageFull22}{SymbolXmpPageEmpty22}
+\pastebutton{SymbolXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{scripts M\free{M }}
+\indentrel{3}\begin{verbatim}
+   (22)
+   [sub= [i,j], sup= [k,l], presup= [0,1], presub= [2],
+    args= [u,v,w]]
+Type: Record(sub: List OutputForm,sup: List OutputForm,presup: List OutputForm,presub: List OutputForm,args: List OutputForm)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty22}
+\begin{paste}{SymbolXmpPageEmpty22}{SymbolXmpPagePatch22}
+\pastebutton{SymbolXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{scripts M\free{M }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch23}
+\begin{paste}{SymbolXmpPageFull23}{SymbolXmpPageEmpty23}
+\pastebutton{SymbolXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{N := script(Nut, [[i,j],[k,l],[0,1]])\bound{N }}
+\indentrel{3}\begin{verbatim}
+         0,1   k,l
+   (23)     Nut
+               i,j
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty23}
+\begin{paste}{SymbolXmpPageEmpty23}{SymbolXmpPagePatch23}
+\pastebutton{SymbolXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{N := script(Nut, [[i,j],[k,l],[0,1]])\bound{N }}
+\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPagePatch24}
+\begin{paste}{SymbolXmpPageFull24}{SymbolXmpPageEmpty24}
+\pastebutton{SymbolXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{scripts N\free{N }}
+\indentrel{3}\begin{verbatim}
+   (24)
+   [sub= [i,j], sup= [k,l], presup= [0,1], presub= [],
+    args= []]
+Type: Record(sub: List OutputForm,sup: List OutputForm,presup: List OutputForm,presub: List OutputForm,args: List OutputForm)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SymbolXmpPageEmpty24}
+\begin{paste}{SymbolXmpPageEmpty24}{SymbolXmpPagePatch24}
+\pastebutton{SymbolXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{scripts N\free{N }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/TABLE.ht b/src/hyper/pages/TABLE.ht
new file mode 100644
index 00000000..0d50f223
--- /dev/null
+++ b/src/hyper/pages/TABLE.ht
@@ -0,0 +1,157 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\TableXmpTitle}{Table}
+\newcommand{\TableXmpNumber}{9.80}
+%
+% =====================================================================
+\begin{page}{TableXmpPage}{9.80 Table}
+% =====================================================================
+\beginscroll
+%
+
+The \spadtype{Table} constructor provides a general structure
+for associative storage.
+This type provides hash tables in which data objects
+can be saved according to keys of any type.
+For a given table, specific types must be chosen for the keys and entries.
+
+\xtc{
+In this example the keys to the table are polynomials with
+integer coefficients.
+The entries in the table are strings.
+}{
+\spadpaste{t: Table(Polynomial Integer, String) := table() \bound{t}}
+}
+\xtc{
+To save an entry in the table, the \spadfunFrom{setelt}{Table} operation is used.
+This can be called directly, giving the table a key and an entry.
+}{
+\spadpaste{setelt(t, x**2 - 1, "Easy to factor") \bound{p1}\free{t}}
+}
+\xtc{
+Alternatively, you can use assignment syntax.
+}{
+\spadpaste{t(x**3 + 1) := "Harder to factor" \bound{p2}\free{p1}}
+}
+\xtc{
+}{
+\spadpaste{t(x)        := "The easiest to factor" \bound{p3}\free{p2}}
+}
+\xtc{
+Entries are retrieved from the table by calling the
+\spadfunFrom{elt}{Table} operation.
+}{
+\spadpaste{elt(t, x) \free{p3}}
+}
+\xtc{
+This operation is called when a table is ``applied'' to a key using
+this or the following syntax.
+}{
+\spadpaste{t.x \free{p3}}
+}
+\xtc{
+}{
+\spadpaste{t x \free{p3}}
+}
+\xtc{
+Parentheses are used only for grouping.  They are needed if the key is
+an infixed expression.
+}{
+\spadpaste{t.(x**2 - 1) \free{p3}}
+}
+\xtc{
+Note that the \spadfunFrom{elt}{Table} operation is used only when the
+key is known to be in the table---otherwise an error is generated.
+}{
+\spadpaste{t (x**3 + 1) \free{p3}}
+}
+
+\xtc{
+You can get a list of all the keys to a table using the
+\spadfunFrom{keys}{Table} operation.
+}{
+\spadpaste{keys t \free{p3}}
+}
+\xtc{
+If you wish to test whether a key is in a table, the
+\spadfunFrom{search}{Table} operation is used.
+This operation returns either an entry or \spad{"failed"}.
+}{
+\spadpaste{search(x, t) \free{p3}}
+}
+\xtc{
+}{
+\spadpaste{search(x**2, t) \free{p3}}
+}
+\xtc{
+The return type is a union so the success of the search can be tested
+using \spad{case}.
+\spadkey{case}
+}{
+\spadpaste{search(x**2, t) case "failed" \free{p3}}
+}
+\xtc{
+The \spadfunFromX{remove}{Table} operation is used to delete values from a
+table.
+}{
+\spadpaste{remove!(x**2-1, t) \free{p3} \bound{p4}}
+}
+\xtc{
+If an entry exists under the key, then it is returned.  Otherwise
+\spadfunFromX{remove}{Table} returns \spad{"failed"}.
+}{
+\spadpaste{remove!(x-1, t) \free{p4}\bound{p5}}
+}
+
+\xtc{
+The number of key-entry pairs can be found using the
+\spadfunFrom{\#}{Table} operation.
+}{
+\spadpaste{\#t \free{p5}}
+}
+\xtc{
+Just as \spadfunFrom{keys}{Table} returns a list of keys to the table, a
+list of all the entries can be obtained using the
+\spadfunFrom{members}{Table} operation.
+}{
+\spadpaste{members t \free{p5}}
+}
+\xtc{
+A number of useful operations take functions and map them on to the
+table to compute the result.  Here we count the entries which
+have \spad{"Hard"} as a prefix.
+}{
+\spadpaste{count(s: String +-> prefix?("Hard", s), t) \free{p5}}
+}
+
+Other table types are provided to support various needs.
+\indent{4}
+\beginitems
+\item[-] \spadtype{AssociationList} gives a list with a table view.
+This allows new entries to be appended onto the front of the list
+to cover up old entries.
+This is useful when table entries need to be stacked or when
+frequent list traversals are required.
+See \downlink{`AssociationList'}{AssociationListXmpPage}\ignore{AssociationList} for more information.
+\item[-] \spadtype{EqTable} gives tables in which keys are considered
+equal only when they are in fact the same instance of a structure.
+See \downlink{`EqTable'}{EqTableXmpPage}\ignore{EqTable} for more information.
+\item[-] \spadtype{StringTable} should be used when the keys are known to
+be strings.
+See \downlink{`StringTable'}{StringTableXmpPage}\ignore{StringTable} for more information.
+\item[-] \spadtype{SparseTable} provides tables with default
+entries, so
+lookup never fails.  The \spadtype{GeneralSparseTable} constructor
+can be used to make any table type behave this way.
+See \downlink{`SparseTable'}{SparseTableXmpPage}\ignore{SparseTable} for more information.
+\item[-] \spadtype{KeyedAccessFile} allows values to be saved in a file,
+accessed as a table.
+See \downlink{`KeyedAccessFile'}{KeyedAccessFileXmpPage}\ignore{KeyedAccessFile} for more information.
+\enditems
+\indent{0}
+%
+\showBlurb{Table}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/TABLE.pht b/src/hyper/pages/TABLE.pht
new file mode 100644
index 00000000..a7a88d25
--- /dev/null
+++ b/src/hyper/pages/TABLE.pht
@@ -0,0 +1,289 @@
+\begin{patch}{TableXmpPagePatch1}
+\begin{paste}{TableXmpPageFull1}{TableXmpPageEmpty1}
+\pastebutton{TableXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{t: Table(Polynomial Integer, String) := table()\bound{t }}
+\indentrel{3}\begin{verbatim}
+   (1)  table()
+                 Type: Table(Polynomial Integer,String)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty1}
+\begin{paste}{TableXmpPageEmpty1}{TableXmpPagePatch1}
+\pastebutton{TableXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{t: Table(Polynomial Integer, String) := table()\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch2}
+\begin{paste}{TableXmpPageFull2}{TableXmpPageEmpty2}
+\pastebutton{TableXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{setelt(t, x**2 - 1, "Easy to factor")\bound{p1 }\free{t }}
+\indentrel{3}\begin{verbatim}
+   (2)  "Easy to factor"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty2}
+\begin{paste}{TableXmpPageEmpty2}{TableXmpPagePatch2}
+\pastebutton{TableXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{setelt(t, x**2 - 1, "Easy to factor")\bound{p1 }\free{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch3}
+\begin{paste}{TableXmpPageFull3}{TableXmpPageEmpty3}
+\pastebutton{TableXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{t(x**3 + 1) := "Harder to factor"\bound{p2 }\free{p1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  "Harder to factor"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty3}
+\begin{paste}{TableXmpPageEmpty3}{TableXmpPagePatch3}
+\pastebutton{TableXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{t(x**3 + 1) := "Harder to factor"\bound{p2 }\free{p1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch4}
+\begin{paste}{TableXmpPageFull4}{TableXmpPageEmpty4}
+\pastebutton{TableXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{t(x) := "The easiest to factor"\bound{p3 }\free{p2 }}
+\indentrel{3}\begin{verbatim}
+   (4)  "The easiest to factor"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty4}
+\begin{paste}{TableXmpPageEmpty4}{TableXmpPagePatch4}
+\pastebutton{TableXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{t(x) := "The easiest to factor"\bound{p3 }\free{p2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch5}
+\begin{paste}{TableXmpPageFull5}{TableXmpPageEmpty5}
+\pastebutton{TableXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{elt(t, x)\free{p3 }}
+\indentrel{3}\begin{verbatim}
+   (5)  "The easiest to factor"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty5}
+\begin{paste}{TableXmpPageEmpty5}{TableXmpPagePatch5}
+\pastebutton{TableXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{elt(t, x)\free{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch6}
+\begin{paste}{TableXmpPageFull6}{TableXmpPageEmpty6}
+\pastebutton{TableXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{t.x\free{p3 }}
+\indentrel{3}\begin{verbatim}
+   (6)  "The easiest to factor"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty6}
+\begin{paste}{TableXmpPageEmpty6}{TableXmpPagePatch6}
+\pastebutton{TableXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{t.x\free{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch7}
+\begin{paste}{TableXmpPageFull7}{TableXmpPageEmpty7}
+\pastebutton{TableXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{t x\free{p3 }}
+\indentrel{3}\begin{verbatim}
+   (7)  "The easiest to factor"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty7}
+\begin{paste}{TableXmpPageEmpty7}{TableXmpPagePatch7}
+\pastebutton{TableXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{t x\free{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch8}
+\begin{paste}{TableXmpPageFull8}{TableXmpPageEmpty8}
+\pastebutton{TableXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{t.(x**2 - 1)\free{p3 }}
+\indentrel{3}\begin{verbatim}
+   (8)  "Easy to factor"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty8}
+\begin{paste}{TableXmpPageEmpty8}{TableXmpPagePatch8}
+\pastebutton{TableXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{t.(x**2 - 1)\free{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch9}
+\begin{paste}{TableXmpPageFull9}{TableXmpPageEmpty9}
+\pastebutton{TableXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{t (x**3 + 1)\free{p3 }}
+\indentrel{3}\begin{verbatim}
+   (9)  "Harder to factor"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty9}
+\begin{paste}{TableXmpPageEmpty9}{TableXmpPagePatch9}
+\pastebutton{TableXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{t (x**3 + 1)\free{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch10}
+\begin{paste}{TableXmpPageFull10}{TableXmpPageEmpty10}
+\pastebutton{TableXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{keys t\free{p3 }}
+\indentrel{3}\begin{verbatim}
+             3      2
+   (10)  [x,x  + 1,x  - 1]
+                          Type: List Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty10}
+\begin{paste}{TableXmpPageEmpty10}{TableXmpPagePatch10}
+\pastebutton{TableXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{keys t\free{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch11}
+\begin{paste}{TableXmpPageFull11}{TableXmpPageEmpty11}
+\pastebutton{TableXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{search(x, t)\free{p3 }}
+\indentrel{3}\begin{verbatim}
+   (11)  "The easiest to factor"
+                                Type: Union(String,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty11}
+\begin{paste}{TableXmpPageEmpty11}{TableXmpPagePatch11}
+\pastebutton{TableXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{search(x, t)\free{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch12}
+\begin{paste}{TableXmpPageFull12}{TableXmpPageEmpty12}
+\pastebutton{TableXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{search(x**2, t)\free{p3 }}
+\indentrel{3}\begin{verbatim}
+   (12)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty12}
+\begin{paste}{TableXmpPageEmpty12}{TableXmpPagePatch12}
+\pastebutton{TableXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{search(x**2, t)\free{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch13}
+\begin{paste}{TableXmpPageFull13}{TableXmpPageEmpty13}
+\pastebutton{TableXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{search(x**2, t) case "failed"\free{p3 }}
+\indentrel{3}\begin{verbatim}
+   (13)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty13}
+\begin{paste}{TableXmpPageEmpty13}{TableXmpPagePatch13}
+\pastebutton{TableXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{search(x**2, t) case "failed"\free{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch14}
+\begin{paste}{TableXmpPageFull14}{TableXmpPageEmpty14}
+\pastebutton{TableXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{remove!(x**2-1, t)\free{p3 }\bound{p4 }}
+\indentrel{3}\begin{verbatim}
+   (14)  "Easy to factor"
+                                Type: Union(String,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty14}
+\begin{paste}{TableXmpPageEmpty14}{TableXmpPagePatch14}
+\pastebutton{TableXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{remove!(x**2-1, t)\free{p3 }\bound{p4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch15}
+\begin{paste}{TableXmpPageFull15}{TableXmpPageEmpty15}
+\pastebutton{TableXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{remove!(x-1, t)\free{p4 }\bound{p5 }}
+\indentrel{3}\begin{verbatim}
+   (15)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty15}
+\begin{paste}{TableXmpPageEmpty15}{TableXmpPagePatch15}
+\pastebutton{TableXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{remove!(x-1, t)\free{p4 }\bound{p5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch16}
+\begin{paste}{TableXmpPageFull16}{TableXmpPageEmpty16}
+\pastebutton{TableXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{\#t\free{p5 }}
+\indentrel{3}\begin{verbatim}
+   (16)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty16}
+\begin{paste}{TableXmpPageEmpty16}{TableXmpPagePatch16}
+\pastebutton{TableXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{\#t\free{p5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch17}
+\begin{paste}{TableXmpPageFull17}{TableXmpPageEmpty17}
+\pastebutton{TableXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{members t\free{p5 }}
+\indentrel{3}\begin{verbatim}
+   (17)  ["The easiest to factor","Harder to factor"]
+                                      Type: List String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty17}
+\begin{paste}{TableXmpPageEmpty17}{TableXmpPagePatch17}
+\pastebutton{TableXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{members t\free{p5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPagePatch18}
+\begin{paste}{TableXmpPageFull18}{TableXmpPageEmpty18}
+\pastebutton{TableXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{count(s: String +-> prefix?("Hard", s), t)\free{p5 }}
+\indentrel{3}\begin{verbatim}
+   (18)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TableXmpPageEmpty18}
+\begin{paste}{TableXmpPageEmpty18}{TableXmpPagePatch18}
+\pastebutton{TableXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{count(s: String +-> prefix?("Hard", s), t)\free{p5 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/TEXTFILE.ht b/src/hyper/pages/TEXTFILE.ht
new file mode 100644
index 00000000..24fa3611
--- /dev/null
+++ b/src/hyper/pages/TEXTFILE.ht
@@ -0,0 +1,87 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\TextFileXmpTitle}{TextFile}
+\newcommand{\TextFileXmpNumber}{9.81}
+%
+% =====================================================================
+\begin{page}{TextFileXmpPage}{9.81 TextFile}
+% =====================================================================
+\beginscroll
+
+The domain \spadtype{TextFile} allows \Language{} to read and write
+character data and exchange text with other programs.
+This type behaves in \Language{} much like a \spadtype{File} of strings,
+with additional operations to cause new lines.
+We give an example of how to produce an upper case copy of a file.
+\xtc{
+This is the file from which we read the text.
+}{
+\spadpaste{f1: TextFile := open("/etc/group", "input")   \bound{f1}}
+}
+\xtc{
+This is the file to which we read the text.
+}{
+\spadpaste{f2: TextFile := open("/tmp/MOTD", "output")  \bound{f2}}
+}
+\xtc{
+Entire lines are handled using the \spadfunFromX{readLine}{TextFile} and
+\spadfunFromX{writeLine}{TextFile} operations.
+}{
+\spadpaste{l := readLine! f1 \free{f1}\bound{l}}
+}
+\xtc{
+}{
+\spadpaste{writeLine!(f2, upperCase l) \free{f2 l}}
+}
+\xtc{
+Use the
+\spadfunFrom{endOfFile?}{TextFile} operation to check if you have
+reached the end of the file.
+}{
+\begin{spadsrc}[\free{f1 f2}\bound{Copied}]
+while not endOfFile? f1 repeat
+    s := readLine! f1
+    writeLine!(f2, upperCase s)
+\end{spadsrc}
+}
+\xtc{
+The file \spad{f1} is exhausted and should be closed.
+}{
+\spadpaste{close! f1  \free{Copied}\bound{closed1}}
+}
+
+\xtc{
+It is sometimes useful to write lines a bit at a time.
+The \spadfunFromX{write}{TextFile} operation allows this.
+}{
+\spadpaste{write!(f2, "-The-")  \free{Copied}\bound{tthhee}}
+}
+\xtc{
+}{
+\spadpaste{write!(f2, "-End-")  \free{tthhee}\bound{eenndd}}
+}
+\xtc{
+This ends the line.
+This is done in a machine-dependent manner.
+}{
+\spadpaste{writeLine! f2        \free{eenndd}\bound{LastLine}}
+}
+\xtc{
+}{
+\spadpaste{close! f2              \free{LastLine}\bound{closed2}}
+}
+\noOutputXtc{
+Finally, clean up.
+}{
+\spadpaste{)system rm /tmp/MOTD   \free{closed2}}
+}
+
+For more information on related topics,  see
+\downlink{`File'}{FileXmpPage}\ignore{File},
+\downlink{`KeyedAccessFile'}{KeyedAccessFileXmpPage}\ignore{KeyedAccessFile}, and
+\downlink{`Library'}{LibraryXmpPage}\ignore{Library}.
+\showBlurb{TextFile}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/TEXTFILE.pht b/src/hyper/pages/TEXTFILE.pht
new file mode 100644
index 00000000..ebf5125e
--- /dev/null
+++ b/src/hyper/pages/TEXTFILE.pht
@@ -0,0 +1,179 @@
+\begin{patch}{TextFileXmpPagePatch1}
+\begin{paste}{TextFileXmpPageFull1}{TextFileXmpPageEmpty1}
+\pastebutton{TextFileXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{f1: TextFile := open("/etc/group", "input")\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+   (1)  "/etc/group"
+                                         Type: TextFile
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPageEmpty1}
+\begin{paste}{TextFileXmpPageEmpty1}{TextFileXmpPagePatch1}
+\pastebutton{TextFileXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{f1: TextFile := open("/etc/group", "input")\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPagePatch2}
+\begin{paste}{TextFileXmpPageFull2}{TextFileXmpPageEmpty2}
+\pastebutton{TextFileXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f2: TextFile := open("/tmp/MOTD", "output")\bound{f2 }}
+\indentrel{3}\begin{verbatim}
+   (2)  "/tmp/MOTD"
+                                         Type: TextFile
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPageEmpty2}
+\begin{paste}{TextFileXmpPageEmpty2}{TextFileXmpPagePatch2}
+\pastebutton{TextFileXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f2: TextFile := open("/tmp/MOTD", "output")\bound{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPagePatch3}
+\begin{paste}{TextFileXmpPageFull3}{TextFileXmpPageEmpty3}
+\pastebutton{TextFileXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{l := readLine! f1\free{f1 }\bound{l }}
+\indentrel{3}\begin{verbatim}
+   (3)  "system:*:0:root"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPageEmpty3}
+\begin{paste}{TextFileXmpPageEmpty3}{TextFileXmpPagePatch3}
+\pastebutton{TextFileXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{l := readLine! f1\free{f1 }\bound{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPagePatch4}
+\begin{paste}{TextFileXmpPageFull4}{TextFileXmpPageEmpty4}
+\pastebutton{TextFileXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{writeLine!(f2, upperCase l)\free{f2 l }}
+\indentrel{3}\begin{verbatim}
+   (4)  "SYSTEM:*:0:ROOT"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPageEmpty4}
+\begin{paste}{TextFileXmpPageEmpty4}{TextFileXmpPagePatch4}
+\pastebutton{TextFileXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{writeLine!(f2, upperCase l)\free{f2 l }}
+\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPagePatch5}
+\begin{paste}{TextFileXmpPageFull5}{TextFileXmpPageEmpty5}
+\pastebutton{TextFileXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{while not endOfFile? f1 repeat
+    s := readLine! f1
+    writeLine!(f2, upperCase s)
+\free{f1 f2 }\bound{Copied }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPageEmpty5}
+\begin{paste}{TextFileXmpPageEmpty5}{TextFileXmpPagePatch5}
+\pastebutton{TextFileXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{while not endOfFile? f1 repeat
+    s := readLine! f1
+    writeLine!(f2, upperCase s)
+\free{f1 f2 }\bound{Copied }}
+\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPagePatch6}
+\begin{paste}{TextFileXmpPageFull6}{TextFileXmpPageEmpty6}
+\pastebutton{TextFileXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{close! f1\free{Copied }\bound{closed1 }}
+\indentrel{3}\begin{verbatim}
+   (6)  "/etc/group"
+                                         Type: TextFile
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPageEmpty6}
+\begin{paste}{TextFileXmpPageEmpty6}{TextFileXmpPagePatch6}
+\pastebutton{TextFileXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{close! f1\free{Copied }\bound{closed1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPagePatch7}
+\begin{paste}{TextFileXmpPageFull7}{TextFileXmpPageEmpty7}
+\pastebutton{TextFileXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{write!(f2, "-The-")\free{Copied }\bound{tthhee }}
+\indentrel{3}\begin{verbatim}
+   (7)  "-The-"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPageEmpty7}
+\begin{paste}{TextFileXmpPageEmpty7}{TextFileXmpPagePatch7}
+\pastebutton{TextFileXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{write!(f2, "-The-")\free{Copied }\bound{tthhee }}
+\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPagePatch8}
+\begin{paste}{TextFileXmpPageFull8}{TextFileXmpPageEmpty8}
+\pastebutton{TextFileXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{write!(f2, "-End-")\free{tthhee }\bound{eenndd }}
+\indentrel{3}\begin{verbatim}
+   (8)  "-End-"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPageEmpty8}
+\begin{paste}{TextFileXmpPageEmpty8}{TextFileXmpPagePatch8}
+\pastebutton{TextFileXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{write!(f2, "-End-")\free{tthhee }\bound{eenndd }}
+\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPagePatch9}
+\begin{paste}{TextFileXmpPageFull9}{TextFileXmpPageEmpty9}
+\pastebutton{TextFileXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{writeLine! f2\free{eenndd }\bound{LastLine }}
+\indentrel{3}\begin{verbatim}
+   (9)  ""
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPageEmpty9}
+\begin{paste}{TextFileXmpPageEmpty9}{TextFileXmpPagePatch9}
+\pastebutton{TextFileXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{writeLine! f2\free{eenndd }\bound{LastLine }}
+\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPagePatch10}
+\begin{paste}{TextFileXmpPageFull10}{TextFileXmpPageEmpty10}
+\pastebutton{TextFileXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{close! f2\free{LastLine }\bound{closed2 }}
+\indentrel{3}\begin{verbatim}
+   (10)  "/tmp/MOTD"
+                                         Type: TextFile
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPageEmpty10}
+\begin{paste}{TextFileXmpPageEmpty10}{TextFileXmpPagePatch10}
+\pastebutton{TextFileXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{close! f2\free{LastLine }\bound{closed2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPagePatch11}
+\begin{paste}{TextFileXmpPageFull11}{TextFileXmpPageEmpty11}
+\pastebutton{TextFileXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{)system rm /tmp/MOTD\free{closed2 }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TextFileXmpPageEmpty11}
+\begin{paste}{TextFileXmpPageEmpty11}{TextFileXmpPagePatch11}
+\pastebutton{TextFileXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{)system rm /tmp/MOTD\free{closed2 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/UNISEG.ht b/src/hyper/pages/UNISEG.ht
new file mode 100644
index 00000000..229c77c7
--- /dev/null
+++ b/src/hyper/pages/UNISEG.ht
@@ -0,0 +1,68 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\UniversalSegmentXmpTitle}{UniversalSegment}
+\newcommand{\UniversalSegmentXmpNumber}{9.84}
+%
+% =====================================================================
+\begin{page}{UniversalSegmentXmpPage}{9.84 UniversalSegment}
+% =====================================================================
+\beginscroll
+
+The \spadtype{UniversalSegment} domain generalizes \spadtype{Segment}
+by allowing segments without a ``hi'' end point.
+\xtc{
+}{
+\spadpaste{pints  := 1..         \bound{pints}}
+}
+\xtc{
+}{
+\spadpaste{nevens := (0..) by -2 \bound{nevens}}
+}
+\xtc{
+Values of type \spadtype{Segment} are automatically converted to
+type \spadtype{UniversalSegment} when appropriate.
+%-% \HDexptypeindex{Segment}{UniversalSegmentXmpPage}{9.84}{UniversalSegment}
+}{
+\spadpaste{useg: UniversalSegment(Integer) := 3..10 \bound{useg}}
+}
+\xtc{
+The operation \spadfunFrom{hasHi}{UniversalSegment} is used to test
+whether a segment has a \spad{hi} end point.
+}{
+\spadpaste{hasHi pints  \free{pints}}
+}
+\xtc{
+}{
+\spadpaste{hasHi nevens \free{nevens}}
+}
+\xtc{
+}{
+\spadpaste{hasHi useg   \free{useg}}
+}
+\xtc{
+All operations available on type \spadtype{Segment} apply to
+\spadtype{UniversalSegment}, with the proviso that expansions produce
+streams rather than lists.
+This is to accommodate infinite expansions.
+}{
+\spadpaste{expand pints \free{pints}}
+}
+\xtc{
+}{
+\spadpaste{expand nevens \free{nevens}}
+}
+\xtc{
+}{
+\spadpaste{expand [1, 3, 10..15, 100..]}
+}
+
+For more information on related topics,  see
+\downlink{`Segment'}{SegmentXmpPage}\ignore{Segment},
+\downlink{`SegmentBinding'}{SegmentBindingXmpPage}\ignore{SegmentBinding},
+\downlink{`List'}{ListXmpPage}\ignore{List}, and
+\downlink{`Stream'}{StreamXmpPage}\ignore{Stream}.
+\showBlurb{UniversalSegment}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/UNISEG.pht b/src/hyper/pages/UNISEG.pht
new file mode 100644
index 00000000..bbaed12e
--- /dev/null
+++ b/src/hyper/pages/UNISEG.pht
@@ -0,0 +1,144 @@
+\begin{patch}{UniversalSegmentXmpPagePatch1}
+\begin{paste}{UniversalSegmentXmpPageFull1}{UniversalSegmentXmpPageEmpty1}
+\pastebutton{UniversalSegmentXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{pints := 1..\bound{pints }}
+\indentrel{3}\begin{verbatim}
+   (1)  1..
+                 Type: UniversalSegment PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPageEmpty1}
+\begin{paste}{UniversalSegmentXmpPageEmpty1}{UniversalSegmentXmpPagePatch1}
+\pastebutton{UniversalSegmentXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{pints := 1..\bound{pints }}
+\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPagePatch2}
+\begin{paste}{UniversalSegmentXmpPageFull2}{UniversalSegmentXmpPageEmpty2}
+\pastebutton{UniversalSegmentXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{nevens := (0..) by -2\bound{nevens }}
+\indentrel{3}\begin{verbatim}
+   (2)  0.. by - 2
+              Type: UniversalSegment NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPageEmpty2}
+\begin{paste}{UniversalSegmentXmpPageEmpty2}{UniversalSegmentXmpPagePatch2}
+\pastebutton{UniversalSegmentXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{nevens := (0..) by -2\bound{nevens }}
+\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPagePatch3}
+\begin{paste}{UniversalSegmentXmpPageFull3}{UniversalSegmentXmpPageEmpty3}
+\pastebutton{UniversalSegmentXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{useg: UniversalSegment(Integer) := 3..10\bound{useg }}
+\indentrel{3}\begin{verbatim}
+   (3)  3..10
+                         Type: UniversalSegment Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPageEmpty3}
+\begin{paste}{UniversalSegmentXmpPageEmpty3}{UniversalSegmentXmpPagePatch3}
+\pastebutton{UniversalSegmentXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{useg: UniversalSegment(Integer) := 3..10\bound{useg }}
+\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPagePatch4}
+\begin{paste}{UniversalSegmentXmpPageFull4}{UniversalSegmentXmpPageEmpty4}
+\pastebutton{UniversalSegmentXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{hasHi pints\free{pints }}
+\indentrel{3}\begin{verbatim}
+   (4)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPageEmpty4}
+\begin{paste}{UniversalSegmentXmpPageEmpty4}{UniversalSegmentXmpPagePatch4}
+\pastebutton{UniversalSegmentXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{hasHi pints\free{pints }}
+\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPagePatch5}
+\begin{paste}{UniversalSegmentXmpPageFull5}{UniversalSegmentXmpPageEmpty5}
+\pastebutton{UniversalSegmentXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{hasHi nevens\free{nevens }}
+\indentrel{3}\begin{verbatim}
+   (5)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPageEmpty5}
+\begin{paste}{UniversalSegmentXmpPageEmpty5}{UniversalSegmentXmpPagePatch5}
+\pastebutton{UniversalSegmentXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{hasHi nevens\free{nevens }}
+\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPagePatch6}
+\begin{paste}{UniversalSegmentXmpPageFull6}{UniversalSegmentXmpPageEmpty6}
+\pastebutton{UniversalSegmentXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{hasHi useg\free{useg }}
+\indentrel{3}\begin{verbatim}
+   (6)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPageEmpty6}
+\begin{paste}{UniversalSegmentXmpPageEmpty6}{UniversalSegmentXmpPagePatch6}
+\pastebutton{UniversalSegmentXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{hasHi useg\free{useg }}
+\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPagePatch7}
+\begin{paste}{UniversalSegmentXmpPageFull7}{UniversalSegmentXmpPageEmpty7}
+\pastebutton{UniversalSegmentXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{expand pints\free{pints }}
+\indentrel{3}\begin{verbatim}
+   (7)  [1,2,3,4,5,6,7,8,9,10,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPageEmpty7}
+\begin{paste}{UniversalSegmentXmpPageEmpty7}{UniversalSegmentXmpPagePatch7}
+\pastebutton{UniversalSegmentXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{expand pints\free{pints }}
+\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPagePatch8}
+\begin{paste}{UniversalSegmentXmpPageFull8}{UniversalSegmentXmpPageEmpty8}
+\pastebutton{UniversalSegmentXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{expand nevens\free{nevens }}
+\indentrel{3}\begin{verbatim}
+   (8)  [0,- 2,- 4,- 6,- 8,- 10,- 12,- 14,- 16,- 18,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPageEmpty8}
+\begin{paste}{UniversalSegmentXmpPageEmpty8}{UniversalSegmentXmpPagePatch8}
+\pastebutton{UniversalSegmentXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{expand nevens\free{nevens }}
+\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPagePatch9}
+\begin{paste}{UniversalSegmentXmpPageFull9}{UniversalSegmentXmpPageEmpty9}
+\pastebutton{UniversalSegmentXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{expand [1, 3, 10..15, 100..]}
+\indentrel{3}\begin{verbatim}
+   (9)  [1,3,10,11,12,13,14,15,100,101,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UniversalSegmentXmpPageEmpty9}
+\begin{paste}{UniversalSegmentXmpPageEmpty9}{UniversalSegmentXmpPagePatch9}
+\pastebutton{UniversalSegmentXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{expand [1, 3, 10..15, 100..]}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/UP.ht b/src/hyper/pages/UP.ht
new file mode 100644
index 00000000..b3afe6e0
--- /dev/null
+++ b/src/hyper/pages/UP.ht
@@ -0,0 +1,274 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\UnivariatePolynomialXmpTitle}{UnivariatePolynomial}
+\newcommand{\UnivariatePolynomialXmpNumber}{9.83}
+%
+% =====================================================================
+\begin{page}{UnivariatePolynomialXmpPage}{9.83 UnivariatePolynomial}
+% =====================================================================
+\beginscroll
+
+The domain constructor \spadtype{UnivariatePolynomial}
+%-% \HDindex{polynomial!one variable}{UnivariatePolynomialXmpPage}{9.83}{UnivariatePolynomial}
+(abbreviated \spadtype{UP})
+creates domains of univariate polynomials in a specified variable.
+For example, the domain
+\spadtype{UP(a1,POLY FRAC INT)} provides polynomials in the single variable
+\spad{a1} whose coefficients are general polynomials with rational
+number coefficients.
+
+\beginImportant
+\noindent {\bf Restriction:}
+\texht{\begin{quotation}\noindent}{\newline\indent{5}}
+\Language{} does not allow you to create types where
+\spadtype{UnivariatePolynomial} is contained in the coefficient type of
+\spadtype{Polynomial}. Therefore,
+\spadtype{UP(x,POLY INT)} is legal but \spadtype{POLY UP(x,INT)} is not.
+\texht{\end{quotation}}{\indent{0}}
+\endImportant
+
+\xtc{
+\spadtype{UP(x,INT)} is the domain of polynomials in the single
+variable \spad{x} with integer coefficients.
+}{
+\spadpaste{(p,q) : UP(x,INT) \bound{pdec}\bound{qdec}}
+}
+\xtc{
+}{
+\spadpaste{p := (3*x-1)**2 * (2*x + 8) \free{pdec}\bound{p}}
+}
+\xtc{
+}{
+\spadpaste{q := (1 - 6*x + 9*x**2)**2 \free{qdec}\bound{q}}
+}
+\xtc{
+The usual arithmetic operations are available for univariate
+polynomials.
+}{
+\spadpaste{p**2 + p*q  \free{p q}}
+}
+\xtc{
+The operation \spadfunFrom{leadingCoefficient}{UnivariatePolynomial}
+extracts the coefficient of the term of highest degree.
+}{
+\spadpaste{leadingCoefficient p \free{p}}
+}
+\xtc{
+The operation \spadfunFrom{degree}{UnivariatePolynomial} returns
+the degree of the polynomial.
+Since the polynomial has only one variable, the variable is not supplied
+to operations like \spadfunFrom{degree}{UnivariatePolynomial}.
+}{
+\spadpaste{degree p \free{p}}
+}
+\xtc{
+The reductum of the polynomial, the polynomial obtained by
+subtracting the term of highest order, is returned by
+\spadfunFrom{reductum}{UnivariatePolynomial}.
+}{
+\spadpaste{reductum p \free{p}}
+}
+\xtc{
+The operation \spadfunFrom{gcd}{UnivariatePolynomial} computes the
+greatest common divisor of two polynomials.
+}{
+\spadpaste{gcd(p,q) \free{p q}}
+}
+\xtc{
+The operation \spadfunFrom{lcm}{UnivariatePolynomial} computes the
+least common multiple.
+}{
+\spadpaste{lcm(p,q) \free{p q}}
+}
+\xtc{
+The operation \spadfunFrom{resultant}{UnivariatePolynomial}
+computes the resultant of two univariate polynomials.
+In the case of \spad{p} and \spad{q}, the resultant is \spad{0} because they
+share a common root.
+}{
+\spadpaste{resultant(p,q) \free{p q}}
+}
+\xtc{
+To compute the derivative of a univariate polynomial with respect to its
+variable, use \spadfunFrom{D}{UnivariatePolynomial}.
+}{
+\spadpaste{D p \free{p}}
+}
+\xtc{
+Univariate polynomials can also be used as if they were functions.
+To evaluate a univariate polynomial at some point, apply
+the polynomial to the point.
+}{
+\spadpaste{p(2) \free{p}}
+}
+\xtc{
+The same syntax is used for composing two univariate polynomials, i.e.
+substituting one polynomial for the variable in another.
+This substitutes \spad{q} for the variable in \spad{p}.
+}{
+\spadpaste{p(q) \free{p q}}
+}
+\xtc{
+This substitutes \spad{p} for the variable in \spad{q}.
+}{
+\spadpaste{q(p) \free{p q}}
+}
+\xtc{
+To obtain a list of coefficients of the polynomial, use
+\spadfunFrom{coefficients}{UnivariatePolynomial}.
+}{
+\spadpaste{l := coefficients p \free{p}\bound{l}}
+}
+\xtc{
+From this you can use \spadfunFrom{gcd}{UnivariatePolynomial}
+and \spadfunFrom{reduce}{List}
+to compute the content of the polynomial.
+}{
+\spadpaste{reduce(gcd,l) \free{l}}
+}
+\xtc{
+Alternatively (and more easily),
+you can just call \spadfunFrom{content}{UnivariatePolynomial}.
+}{
+\spadpaste{content p \free{p}}
+}
+
+Note that the operation \spadfunFrom{coefficients}{UnivariatePolynomial}
+omits the zero coefficients from the list.
+Sometimes it is useful to convert a univariate polynomial
+to a vector whose \eth{\spad{i }} position contains the degree \spad{i-1}
+coefficient of the polynomial.
+\xtc{
+}{
+\spadpaste{ux := (x**4+2*x+3)::UP(x,INT) \bound{ux}}
+}
+\xtc{
+To get a complete vector of coefficients, use the operation
+\spadfunFrom{vectorise}{UnivariatePolynomial}, which takes a
+univariate polynomial and an integer denoting the length of the
+desired vector.
+}{
+\spadpaste{vectorise(ux,5) \free{ux}}
+}
+
+It is common to want to do something to every term of a polynomial,
+creating a new polynomial in the process.
+\xtc{
+This is a function for iterating across the terms of a polynomial,
+squaring each term.
+}{
+\begin{spadsrc}[\bound{squareTerms}]
+squareTerms(p) ==
+  reduce(+,[t**2 for t in monomials p])
+\end{spadsrc}
+}
+\xtc{
+Recall what \spad{p} looked like.
+}{
+\spadpaste{p \free{p}}
+}
+\xtc{
+We can demonstrate \userfun{squareTerms} on \spad{p}.
+}{
+\spadpaste{squareTerms p \free{p}\free{squareTerms}}
+}
+
+When the coefficients of the univariate polynomial belong to a
+field,\footnote{For example, when the coefficients are rational
+numbers, as opposed to integers.  The important property of
+a field is that non-zero elements can be divided and produce
+another element. The quotient of the integers 2 and 3 is not
+another integer.}
+it is possible to compute quotients and remainders.
+\xtc{
+}{
+\spadpaste{(r,s) : UP(a1,FRAC INT) \bound{rdec}\bound{sdec}}
+}
+\xtc{
+}{
+\spadpaste{r := a1**2 - 2/3  \free{rdec}\bound{r}}
+}
+\xtc{
+}{
+\spadpaste{s := a1 + 4       \free{sdec}\bound{s}}
+}
+\xtc{
+When the coefficients are rational numbers or rational expressions, the
+operation \spadfunFrom{quo}{UnivariatePolynomial} computes the quotient
+of two polynomials.
+}{
+\spadpaste{r quo s \free{r s}}
+}
+\xtc{
+The operation
+\spadfunFrom{rem}{UnivariatePolynomial} computes the remainder.
+}{
+\spadpaste{r rem s \free{r s}}
+}
+\xtc{
+The operation \spadfunFrom{divide}{UnivariatePolynomial} can be used to
+return a record of both components.
+}{
+\spadpaste{d := divide(r, s) \free{r s}\bound{d}}
+}
+\xtc{
+Now we check the arithmetic!
+}{
+\spadpaste{r - (d.quotient * s + d.remainder) \free{r s d}}
+}
+\xtc{
+It is also possible to integrate univariate polynomials when the
+coefficients belong to a field.
+}{
+\spadpaste{integrate r \free{r}}
+}
+\xtc{
+}{
+\spadpaste{integrate s \free{s}}
+}
+
+One application of univariate polynomials is to see expressions in terms
+of a specific variable.
+%
+\xtc{
+We start with a polynomial in \spad{a1} whose coefficients
+are quotients of polynomials in \spad{b1} and \spad{b2}.
+}{
+\spadpaste{t : UP(a1,FRAC POLY INT) \bound{tdec}}
+}
+\xtc{
+Since in this case we are not talking about using multivariate
+polynomials in only two variables, we use \spadtype{Polynomial}.
+We also use \spadtype{Fraction} because we want fractions.
+}{
+\spadpaste{t := a1**2 - a1/b2 + (b1**2-b1)/(b2+3) \free{tdec}\bound{t}}
+}
+\xtc{
+We push all the variables into a single quotient of polynomials.
+}{
+\spadpaste{u : FRAC POLY INT := t \bound{u}\free{t}}
+}
+\xtc{
+Alternatively, we can view this as a polynomial in the variable
+This is a {\it mode-directed} conversion: you indicate
+as much of the structure as you care about and let \Language{}
+decide on the full type and how to do the transformation.
+}{
+\spadpaste{u :: UP(b1,?) \free{u}}
+}
+
+See \downlink{``\ugProblemFactorTitle''}{ugProblemFactorPage} in Section \ugProblemFactorNumber\ignore{ugProblemFactor}
+for a discussion of the factorization facilities
+in \Language{} for univariate polynomials.
+For more information on related topics, see
+\downlink{``\ugIntroVariablesTitle''}{ugIntroVariablesPage} in Section \ugIntroVariablesNumber\ignore{ugIntroVariables},
+\downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert},
+\downlink{`Polynomial'}{PolynomialXmpPage}\ignore{Polynomial},
+\downlink{`MultivariatePolynomial'}{MultivariatePolynomialXmpPage}\ignore{MultivariatePolynomial}, and
+\downlink{`DistributedMultivariatePolynomial'}{DistributedMultivariatePolynomialXmpPage}\ignore{DistributedMultivariatePolynomial}.
+%
+\showBlurb{UnivariatePolynomial}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/UP.pht b/src/hyper/pages/UP.pht
new file mode 100644
index 00000000..15f62786
--- /dev/null
+++ b/src/hyper/pages/UP.pht
@@ -0,0 +1,612 @@
+\begin{patch}{UnivariatePolynomialXmpPagePatch1}
+\begin{paste}{UnivariatePolynomialXmpPageFull1}{UnivariatePolynomialXmpPageEmpty1}
+\pastebutton{UnivariatePolynomialXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{(p,q) : UP(x,INT)\bound{pdec }\bound{qdec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty1}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty1}{UnivariatePolynomialXmpPagePatch1}
+\pastebutton{UnivariatePolynomialXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{(p,q) : UP(x,INT)\bound{pdec }\bound{qdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch2}
+\begin{paste}{UnivariatePolynomialXmpPageFull2}{UnivariatePolynomialXmpPageEmpty2}
+\pastebutton{UnivariatePolynomialXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{p := (3*x-1)**2 * (2*x + 8)\free{pdec }\bound{p }}
+\indentrel{3}\begin{verbatim}
+           3      2
+   (2)  18x  + 60x  - 46x + 8
+                  Type: UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty2}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty2}{UnivariatePolynomialXmpPagePatch2}
+\pastebutton{UnivariatePolynomialXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{p := (3*x-1)**2 * (2*x + 8)\free{pdec }\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch3}
+\begin{paste}{UnivariatePolynomialXmpPageFull3}{UnivariatePolynomialXmpPageEmpty3}
+\pastebutton{UnivariatePolynomialXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{q := (1 - 6*x + 9*x**2)**2\free{qdec }\bound{q }}
+\indentrel{3}\begin{verbatim}
+           4       3      2
+   (3)  81x  - 108x  + 54x  - 12x + 1
+                  Type: UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty3}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty3}{UnivariatePolynomialXmpPagePatch3}
+\pastebutton{UnivariatePolynomialXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{q := (1 - 6*x + 9*x**2)**2\free{qdec }\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch4}
+\begin{paste}{UnivariatePolynomialXmpPageFull4}{UnivariatePolynomialXmpPageEmpty4}
+\pastebutton{UnivariatePolynomialXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{p**2 + p*q\free{p q }}
+\indentrel{3}\begin{verbatim}
+   (4)
+          7        6        5         4        3        2
+     1458x  + 3240x  - 7074x  + 10584x  - 9282x  + 4120x
+   + 
+     - 878x + 72
+                  Type: UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty4}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty4}{UnivariatePolynomialXmpPagePatch4}
+\pastebutton{UnivariatePolynomialXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{p**2 + p*q\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch5}
+\begin{paste}{UnivariatePolynomialXmpPageFull5}{UnivariatePolynomialXmpPageEmpty5}
+\pastebutton{UnivariatePolynomialXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{leadingCoefficient p\free{p }}
+\indentrel{3}\begin{verbatim}
+   (5)  18
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty5}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty5}{UnivariatePolynomialXmpPagePatch5}
+\pastebutton{UnivariatePolynomialXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{leadingCoefficient p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch6}
+\begin{paste}{UnivariatePolynomialXmpPageFull6}{UnivariatePolynomialXmpPageEmpty6}
+\pastebutton{UnivariatePolynomialXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{degree p\free{p }}
+\indentrel{3}\begin{verbatim}
+   (6)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty6}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty6}{UnivariatePolynomialXmpPagePatch6}
+\pastebutton{UnivariatePolynomialXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{degree p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch7}
+\begin{paste}{UnivariatePolynomialXmpPageFull7}{UnivariatePolynomialXmpPageEmpty7}
+\pastebutton{UnivariatePolynomialXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{reductum p\free{p }}
+\indentrel{3}\begin{verbatim}
+           2
+   (7)  60x  - 46x + 8
+                  Type: UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty7}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty7}{UnivariatePolynomialXmpPagePatch7}
+\pastebutton{UnivariatePolynomialXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{reductum p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch8}
+\begin{paste}{UnivariatePolynomialXmpPageFull8}{UnivariatePolynomialXmpPageEmpty8}
+\pastebutton{UnivariatePolynomialXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{gcd(p,q)\free{p q }}
+\indentrel{3}\begin{verbatim}
+          2
+   (8)  9x  - 6x + 1
+                  Type: UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty8}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty8}{UnivariatePolynomialXmpPagePatch8}
+\pastebutton{UnivariatePolynomialXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{gcd(p,q)\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch9}
+\begin{paste}{UnivariatePolynomialXmpPageFull9}{UnivariatePolynomialXmpPageEmpty9}
+\pastebutton{UnivariatePolynomialXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{lcm(p,q)\free{p q }}
+\indentrel{3}\begin{verbatim}
+            5       4       3       2
+   (9)  162x  + 432x  - 756x  + 408x  - 94x + 8
+                  Type: UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty9}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty9}{UnivariatePolynomialXmpPagePatch9}
+\pastebutton{UnivariatePolynomialXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{lcm(p,q)\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch10}
+\begin{paste}{UnivariatePolynomialXmpPageFull10}{UnivariatePolynomialXmpPageEmpty10}
+\pastebutton{UnivariatePolynomialXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{resultant(p,q)\free{p q }}
+\indentrel{3}\begin{verbatim}
+   (10)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty10}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty10}{UnivariatePolynomialXmpPagePatch10}
+\pastebutton{UnivariatePolynomialXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{resultant(p,q)\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch11}
+\begin{paste}{UnivariatePolynomialXmpPageFull11}{UnivariatePolynomialXmpPageEmpty11}
+\pastebutton{UnivariatePolynomialXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{D p\free{p }}
+\indentrel{3}\begin{verbatim}
+            2
+   (11)  54x  + 120x - 46
+                  Type: UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty11}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty11}{UnivariatePolynomialXmpPagePatch11}
+\pastebutton{UnivariatePolynomialXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{D p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch12}
+\begin{paste}{UnivariatePolynomialXmpPageFull12}{UnivariatePolynomialXmpPageEmpty12}
+\pastebutton{UnivariatePolynomialXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{p(2)\free{p }}
+\indentrel{3}\begin{verbatim}
+   (12)  300
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty12}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty12}{UnivariatePolynomialXmpPagePatch12}
+\pastebutton{UnivariatePolynomialXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{p(2)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch13}
+\begin{paste}{UnivariatePolynomialXmpPageFull13}{UnivariatePolynomialXmpPageEmpty13}
+\pastebutton{UnivariatePolynomialXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{p(q)\free{p q }}
+\indentrel{3}\begin{verbatim}
+   (13)
+             12            11            10            9
+     9565938x   - 38263752x   + 70150212x   - 77944680x
+   + 
+              8            7            6           5
+     58852170x  - 32227632x  + 13349448x  - 4280688x
+   + 
+             4          3         2
+     1058184x  - 192672x  + 23328x  - 1536x + 40
+                  Type: UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty13}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty13}{UnivariatePolynomialXmpPagePatch13}
+\pastebutton{UnivariatePolynomialXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{p(q)\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch14}
+\begin{paste}{UnivariatePolynomialXmpPageFull14}{UnivariatePolynomialXmpPageEmpty14}
+\pastebutton{UnivariatePolynomialXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{q(p)\free{p q }}
+\indentrel{3}\begin{verbatim}
+   (14)
+             12             11             10
+     8503056x   + 113374080x   + 479950272x
+   + 
+               9              8             7
+     404997408x  - 1369516896x  - 626146848x
+   + 
+                6              5              4
+     2939858712x  - 2780728704x  + 1364312160x
+   + 
+                 3            2
+     - 396838872x  + 69205896x  - 6716184x + 279841
+                  Type: UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty14}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty14}{UnivariatePolynomialXmpPagePatch14}
+\pastebutton{UnivariatePolynomialXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{q(p)\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch15}
+\begin{paste}{UnivariatePolynomialXmpPageFull15}{UnivariatePolynomialXmpPageEmpty15}
+\pastebutton{UnivariatePolynomialXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{l := coefficients p\free{p }\bound{l }}
+\indentrel{3}\begin{verbatim}
+   (15)  [18,60,- 46,8]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty15}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty15}{UnivariatePolynomialXmpPagePatch15}
+\pastebutton{UnivariatePolynomialXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{l := coefficients p\free{p }\bound{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch16}
+\begin{paste}{UnivariatePolynomialXmpPageFull16}{UnivariatePolynomialXmpPageEmpty16}
+\pastebutton{UnivariatePolynomialXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{reduce(gcd,l)\free{l }}
+\indentrel{3}\begin{verbatim}
+   (16)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty16}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty16}{UnivariatePolynomialXmpPagePatch16}
+\pastebutton{UnivariatePolynomialXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{reduce(gcd,l)\free{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch17}
+\begin{paste}{UnivariatePolynomialXmpPageFull17}{UnivariatePolynomialXmpPageEmpty17}
+\pastebutton{UnivariatePolynomialXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{content p\free{p }}
+\indentrel{3}\begin{verbatim}
+   (17)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty17}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty17}{UnivariatePolynomialXmpPagePatch17}
+\pastebutton{UnivariatePolynomialXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{content p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch18}
+\begin{paste}{UnivariatePolynomialXmpPageFull18}{UnivariatePolynomialXmpPageEmpty18}
+\pastebutton{UnivariatePolynomialXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{ux := (x**4+2*x+3)::UP(x,INT)\bound{ux }}
+\indentrel{3}\begin{verbatim}
+          4
+   (18)  x  + 2x + 3
+                  Type: UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty18}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty18}{UnivariatePolynomialXmpPagePatch18}
+\pastebutton{UnivariatePolynomialXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{ux := (x**4+2*x+3)::UP(x,INT)\bound{ux }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch19}
+\begin{paste}{UnivariatePolynomialXmpPageFull19}{UnivariatePolynomialXmpPageEmpty19}
+\pastebutton{UnivariatePolynomialXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{vectorise(ux,5)\free{ux }}
+\indentrel{3}\begin{verbatim}
+   (19)  [3,2,0,0,1]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty19}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty19}{UnivariatePolynomialXmpPagePatch19}
+\pastebutton{UnivariatePolynomialXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{vectorise(ux,5)\free{ux }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch20}
+\begin{paste}{UnivariatePolynomialXmpPageFull20}{UnivariatePolynomialXmpPageEmpty20}
+\pastebutton{UnivariatePolynomialXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{squareTerms(p) ==
+  reduce(+,[t**2 for t in monomials p])
+\bound{squareTerms }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty20}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty20}{UnivariatePolynomialXmpPagePatch20}
+\pastebutton{UnivariatePolynomialXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{squareTerms(p) ==
+  reduce(+,[t**2 for t in monomials p])
+\bound{squareTerms }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch21}
+\begin{paste}{UnivariatePolynomialXmpPageFull21}{UnivariatePolynomialXmpPageEmpty21}
+\pastebutton{UnivariatePolynomialXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{p\free{p }}
+\indentrel{3}\begin{verbatim}
+            3      2
+   (21)  18x  + 60x  - 46x + 8
+                  Type: UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty21}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty21}{UnivariatePolynomialXmpPagePatch21}
+\pastebutton{UnivariatePolynomialXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{p\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch22}
+\begin{paste}{UnivariatePolynomialXmpPageFull22}{UnivariatePolynomialXmpPageEmpty22}
+\pastebutton{UnivariatePolynomialXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{squareTerms p\free{p }\free{squareTerms }}
+\indentrel{3}\begin{verbatim}
+             6        4        2
+   (22)  324x  + 3600x  + 2116x  + 64
+                  Type: UnivariatePolynomial(x,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty22}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty22}{UnivariatePolynomialXmpPagePatch22}
+\pastebutton{UnivariatePolynomialXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{squareTerms p\free{p }\free{squareTerms }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch23}
+\begin{paste}{UnivariatePolynomialXmpPageFull23}{UnivariatePolynomialXmpPageEmpty23}
+\pastebutton{UnivariatePolynomialXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{(r,s) : UP(a1,FRAC INT)\bound{rdec }\bound{sdec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty23}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty23}{UnivariatePolynomialXmpPagePatch23}
+\pastebutton{UnivariatePolynomialXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{(r,s) : UP(a1,FRAC INT)\bound{rdec }\bound{sdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch24}
+\begin{paste}{UnivariatePolynomialXmpPageFull24}{UnivariatePolynomialXmpPageEmpty24}
+\pastebutton{UnivariatePolynomialXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{r := a1**2 - 2/3\free{rdec }\bound{r }}
+\indentrel{3}\begin{verbatim}
+           2   2
+   (24)  a1  - Ä
+               3
+        Type: UnivariatePolynomial(a1,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty24}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty24}{UnivariatePolynomialXmpPagePatch24}
+\pastebutton{UnivariatePolynomialXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{r := a1**2 - 2/3\free{rdec }\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch25}
+\begin{paste}{UnivariatePolynomialXmpPageFull25}{UnivariatePolynomialXmpPageEmpty25}
+\pastebutton{UnivariatePolynomialXmpPageFull25}{\hidepaste}
+\tab{5}\spadcommand{s := a1 + 4\free{sdec }\bound{s }}
+\indentrel{3}\begin{verbatim}
+   (25)  a1 + 4
+        Type: UnivariatePolynomial(a1,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty25}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty25}{UnivariatePolynomialXmpPagePatch25}
+\pastebutton{UnivariatePolynomialXmpPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{s := a1 + 4\free{sdec }\bound{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch26}
+\begin{paste}{UnivariatePolynomialXmpPageFull26}{UnivariatePolynomialXmpPageEmpty26}
+\pastebutton{UnivariatePolynomialXmpPageFull26}{\hidepaste}
+\tab{5}\spadcommand{r quo s\free{r s }}
+\indentrel{3}\begin{verbatim}
+   (26)  a1 - 4
+        Type: UnivariatePolynomial(a1,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty26}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty26}{UnivariatePolynomialXmpPagePatch26}
+\pastebutton{UnivariatePolynomialXmpPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{r quo s\free{r s }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch27}
+\begin{paste}{UnivariatePolynomialXmpPageFull27}{UnivariatePolynomialXmpPageEmpty27}
+\pastebutton{UnivariatePolynomialXmpPageFull27}{\hidepaste}
+\tab{5}\spadcommand{r rem s\free{r s }}
+\indentrel{3}\begin{verbatim}
+         46
+   (27)  ÄÄ
+          3
+        Type: UnivariatePolynomial(a1,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty27}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty27}{UnivariatePolynomialXmpPagePatch27}
+\pastebutton{UnivariatePolynomialXmpPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{r rem s\free{r s }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch28}
+\begin{paste}{UnivariatePolynomialXmpPageFull28}{UnivariatePolynomialXmpPageEmpty28}
+\pastebutton{UnivariatePolynomialXmpPageFull28}{\hidepaste}
+\tab{5}\spadcommand{d := divide(r, s)\free{r s }\bound{d }}
+\indentrel{3}\begin{verbatim}
+                                      46
+   (28)  [quotient= a1 - 4,remainder= ÄÄ]
+                                       3
+Type: Record(quotient: UnivariatePolynomial(a1,Fraction Integer),remainder: UnivariatePolynomial(a1,Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty28}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty28}{UnivariatePolynomialXmpPagePatch28}
+\pastebutton{UnivariatePolynomialXmpPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{d := divide(r, s)\free{r s }\bound{d }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch29}
+\begin{paste}{UnivariatePolynomialXmpPageFull29}{UnivariatePolynomialXmpPageEmpty29}
+\pastebutton{UnivariatePolynomialXmpPageFull29}{\hidepaste}
+\tab{5}\spadcommand{r - (d.quotient * s + d.remainder)\free{r s d }}
+\indentrel{3}\begin{verbatim}
+   (29)  0
+        Type: UnivariatePolynomial(a1,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty29}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty29}{UnivariatePolynomialXmpPagePatch29}
+\pastebutton{UnivariatePolynomialXmpPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{r - (d.quotient * s + d.remainder)\free{r s d }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch30}
+\begin{paste}{UnivariatePolynomialXmpPageFull30}{UnivariatePolynomialXmpPageEmpty30}
+\pastebutton{UnivariatePolynomialXmpPageFull30}{\hidepaste}
+\tab{5}\spadcommand{integrate r\free{r }}
+\indentrel{3}\begin{verbatim}
+         1   3   2
+   (30)  Ä a1  - Ä a1
+         3       3
+        Type: UnivariatePolynomial(a1,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty30}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty30}{UnivariatePolynomialXmpPagePatch30}
+\pastebutton{UnivariatePolynomialXmpPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{integrate r\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch31}
+\begin{paste}{UnivariatePolynomialXmpPageFull31}{UnivariatePolynomialXmpPageEmpty31}
+\pastebutton{UnivariatePolynomialXmpPageFull31}{\hidepaste}
+\tab{5}\spadcommand{integrate s\free{s }}
+\indentrel{3}\begin{verbatim}
+         1   2
+   (31)  Ä a1  + 4a1
+         2
+        Type: UnivariatePolynomial(a1,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty31}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty31}{UnivariatePolynomialXmpPagePatch31}
+\pastebutton{UnivariatePolynomialXmpPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{integrate s\free{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch32}
+\begin{paste}{UnivariatePolynomialXmpPageFull32}{UnivariatePolynomialXmpPageEmpty32}
+\pastebutton{UnivariatePolynomialXmpPageFull32}{\hidepaste}
+\tab{5}\spadcommand{t : UP(a1,FRAC POLY INT)\bound{tdec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty32}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty32}{UnivariatePolynomialXmpPagePatch32}
+\pastebutton{UnivariatePolynomialXmpPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{t : UP(a1,FRAC POLY INT)\bound{tdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch33}
+\begin{paste}{UnivariatePolynomialXmpPageFull33}{UnivariatePolynomialXmpPageEmpty33}
+\pastebutton{UnivariatePolynomialXmpPageFull33}{\hidepaste}
+\tab{5}\spadcommand{t := a1**2 - a1/b2 + (b1**2-b1)/(b2+3)\free{tdec }\bound{t }}
+\indentrel{3}\begin{verbatim}
+                         2
+           2    1      b1  - b1
+   (33)  a1  - ÄÄ a1 + ÄÄÄÄÄÄÄÄ
+               b2       b2 + 3
+Type: UnivariatePolynomial(a1,Fraction Polynomial Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty33}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty33}{UnivariatePolynomialXmpPagePatch33}
+\pastebutton{UnivariatePolynomialXmpPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{t := a1**2 - a1/b2 + (b1**2-b1)/(b2+3)\free{tdec }\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch34}
+\begin{paste}{UnivariatePolynomialXmpPageFull34}{UnivariatePolynomialXmpPageEmpty34}
+\pastebutton{UnivariatePolynomialXmpPageFull34}{\hidepaste}
+\tab{5}\spadcommand{u : FRAC POLY INT := t\bound{u }\free{t }}
+\indentrel{3}\begin{verbatim}
+           2  2      2           2
+         a1 b2  + (b1  - b1 + 3a1  - a1)b2 - 3a1
+   (34)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                          2
+                        b2  + 3b2
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty34}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty34}{UnivariatePolynomialXmpPagePatch34}
+\pastebutton{UnivariatePolynomialXmpPageEmpty34}{\showpaste}
+\tab{5}\spadcommand{u : FRAC POLY INT := t\bound{u }\free{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPagePatch35}
+\begin{paste}{UnivariatePolynomialXmpPageFull35}{UnivariatePolynomialXmpPageEmpty35}
+\pastebutton{UnivariatePolynomialXmpPageFull35}{\hidepaste}
+\tab{5}\spadcommand{u :: UP(b1,?)\free{u }}
+\indentrel{3}\begin{verbatim}
+                                    2
+            1     2      1        a1 b2 - a1
+   (35)  ÄÄÄÄÄÄ b1  - ÄÄÄÄÄÄ b1 + ÄÄÄÄÄÄÄÄÄÄ
+         b2 + 3       b2 + 3          b2
+Type: UnivariatePolynomial(b1,Fraction Polynomial Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{UnivariatePolynomialXmpPageEmpty35}
+\begin{paste}{UnivariatePolynomialXmpPageEmpty35}{UnivariatePolynomialXmpPagePatch35}
+\pastebutton{UnivariatePolynomialXmpPageEmpty35}{\showpaste}
+\tab{5}\spadcommand{u :: UP(b1,?)\free{u }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/VECTOR.ht b/src/hyper/pages/VECTOR.ht
new file mode 100644
index 00000000..13be70a1
--- /dev/null
+++ b/src/hyper/pages/VECTOR.ht
@@ -0,0 +1,117 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\VectorXmpTitle}{Vector}
+\newcommand{\VectorXmpNumber}{9.85}
+%
+% =====================================================================
+\begin{page}{VectorXmpPage}{9.85 Vector}
+% =====================================================================
+\beginscroll
+
+
+The \spadtype{Vector} domain is used for storing data in a one-dimensional
+indexed data structure.
+A vector is a homogeneous data structure in that all the components of the
+vector must belong to the same \Language{} domain.
+Each vector has a fixed length specified by the user; vectors are not
+extensible.
+This domain is similar to the \spadtype{OneDimensionalArray} domain,
+except that when the components of a \spadtype{Vector} belong to a
+\spadtype{Ring}, arithmetic operations are provided.
+For more examples of operations that are defined for both
+\spadtype{Vector} and \spadtype{OneDimensionalArray}, see
+\downlink{`OneDimensionalArray'}{OneDimensionalArrayXmpPage}\ignore{OneDimensionalArray}.
+
+As with the \spadtype{OneDimensionalArray} domain, a \spadtype{Vector} can
+be created by calling the operation \spadfunFrom{new}{Vector}, its components
+can be accessed by calling the operations \spadfunFrom{elt}{Vector} and
+\spadfunFrom{qelt}{Vector}, and its components can be reset by calling the
+operations \spadfunFrom{setelt}{Vector} and
+\spadfunFromX{qsetelt}{Vector}.
+\xtc{
+This creates a vector of integers of length
+\spad{5} all of whose components are \spad{12}.
+}{
+\spadpaste{u : VECTOR INT := new(5,12) \bound{u}}
+}
+\xtc{
+This is how you create a vector from a list of its components.
+}{
+\spadpaste{v : VECTOR INT := vector([1,2,3,4,5]) \bound{v}}
+}
+
+\xtc{
+Indexing for vectors begins at \spad{1}.
+The last element has index equal to the length of the vector,
+which is computed by \spadopFrom{\#}{Vector}.
+}{
+\spadpaste{\#(v) \free{v}}
+}
+\xtc{
+This is the standard way to use \spadfunFrom{elt}{Vector} to extract an
+element.
+Functionally, it is the same as if you had typed \spad{elt(v,2)}.
+}{
+\spadpaste{v.2 \free{v}}
+}
+\xtc{
+This is the standard way to use \spadfunFrom{setelt}{Vector} to change an
+element.
+It is the same as if you had typed \spad{setelt(v,3,99)}.
+}{
+\spadpaste{v.3 := 99 \free{v}\bound{vdelta}}
+}
+\xtc{
+Now look at \spad{v} to see the change.
+You can
+use \spadfunFrom{qelt}{Vector} and \spadfunFromX{qsetelt}{Vector} (instead
+of \spadfunFrom{elt}{Vector} and \spadfunFrom{setelt}{Vector},
+respectively) but {\it only} when you know that the index is within the valid
+range.
+}{
+\spadpaste{v \free{vdelta}}
+}
+
+\xtc{
+When the components belong to a \spadtype{Ring}, \Language{}
+provides arithmetic operations for \spadtype{Vector}.
+These include left and right scalar multiplication.
+}{
+\spadpaste{5 * v \free{vdelta}}
+}
+\xtc{
+}{
+\spadpaste{v * 7 \free{vdelta}}
+}
+\xtc{
+}{
+\spadpaste{w : VECTOR INT := vector([2,3,4,5,6]) \bound{w}}
+}
+\xtc{
+Addition and subtraction are also available.
+}{
+\spadpaste{v + w \free{vdelta w}}
+}
+\xtc{
+Of course, when adding or subtracting, the two vectors must have the same
+length or an error message is displayed.
+}{
+\spadpaste{v - w \free{vdelta w}}
+}
+
+For more information about other aggregate domains,
+see the following:
+\downlink{`List'}{ListXmpPage}\ignore{List},
+\downlink{`Matrix'}{MatrixXmpPage}\ignore{Matrix},
+\downlink{`OneDimensionalArray'}{OneDimensionalArrayXmpPage}\ignore{OneDimensionalArray},
+\downlink{`Set'}{SetXmpPage}\ignore{Set},
+\downlink{`Table'}{TableXmpPage}\ignore{Table}, and
+\downlink{`TwoDimensionalArray'}{TwoDimensionalArrayXmpPage}\ignore{TwoDimensionalArray}.
+Issue the system command \spadcmd{)show Vector}
+to display the full list of operations defined by
+\spadtype{Vector}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/VECTOR.pht b/src/hyper/pages/VECTOR.pht
new file mode 100644
index 00000000..9c8394f0
--- /dev/null
+++ b/src/hyper/pages/VECTOR.pht
@@ -0,0 +1,176 @@
+\begin{patch}{VectorXmpPagePatch1}
+\begin{paste}{VectorXmpPageFull1}{VectorXmpPageEmpty1}
+\pastebutton{VectorXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{u : VECTOR INT := new(5,12)\bound{u }}
+\indentrel{3}\begin{verbatim}
+   (1)  [12,12,12,12,12]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPageEmpty1}
+\begin{paste}{VectorXmpPageEmpty1}{VectorXmpPagePatch1}
+\pastebutton{VectorXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{u : VECTOR INT := new(5,12)\bound{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPagePatch2}
+\begin{paste}{VectorXmpPageFull2}{VectorXmpPageEmpty2}
+\pastebutton{VectorXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{v : VECTOR INT := vector([1,2,3,4,5])\bound{v }}
+\indentrel{3}\begin{verbatim}
+   (2)  [1,2,3,4,5]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPageEmpty2}
+\begin{paste}{VectorXmpPageEmpty2}{VectorXmpPagePatch2}
+\pastebutton{VectorXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{v : VECTOR INT := vector([1,2,3,4,5])\bound{v }}
+\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPagePatch3}
+\begin{paste}{VectorXmpPageFull3}{VectorXmpPageEmpty3}
+\pastebutton{VectorXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{\#(v)\free{v }}
+\indentrel{3}\begin{verbatim}
+   (3)  5
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPageEmpty3}
+\begin{paste}{VectorXmpPageEmpty3}{VectorXmpPagePatch3}
+\pastebutton{VectorXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{\#(v)\free{v }}
+\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPagePatch4}
+\begin{paste}{VectorXmpPageFull4}{VectorXmpPageEmpty4}
+\pastebutton{VectorXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{v.2\free{v }}
+\indentrel{3}\begin{verbatim}
+   (4)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPageEmpty4}
+\begin{paste}{VectorXmpPageEmpty4}{VectorXmpPagePatch4}
+\pastebutton{VectorXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{v.2\free{v }}
+\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPagePatch5}
+\begin{paste}{VectorXmpPageFull5}{VectorXmpPageEmpty5}
+\pastebutton{VectorXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{v.3 := 99\free{v }\bound{vdelta }}
+\indentrel{3}\begin{verbatim}
+   (5)  99
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPageEmpty5}
+\begin{paste}{VectorXmpPageEmpty5}{VectorXmpPagePatch5}
+\pastebutton{VectorXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{v.3 := 99\free{v }\bound{vdelta }}
+\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPagePatch6}
+\begin{paste}{VectorXmpPageFull6}{VectorXmpPageEmpty6}
+\pastebutton{VectorXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{v\free{vdelta }}
+\indentrel{3}\begin{verbatim}
+   (6)  [1,2,99,4,5]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPageEmpty6}
+\begin{paste}{VectorXmpPageEmpty6}{VectorXmpPagePatch6}
+\pastebutton{VectorXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{v\free{vdelta }}
+\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPagePatch7}
+\begin{paste}{VectorXmpPageFull7}{VectorXmpPageEmpty7}
+\pastebutton{VectorXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{5 * v\free{vdelta }}
+\indentrel{3}\begin{verbatim}
+   (7)  [5,10,495,20,25]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPageEmpty7}
+\begin{paste}{VectorXmpPageEmpty7}{VectorXmpPagePatch7}
+\pastebutton{VectorXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{5 * v\free{vdelta }}
+\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPagePatch8}
+\begin{paste}{VectorXmpPageFull8}{VectorXmpPageEmpty8}
+\pastebutton{VectorXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{v * 7\free{vdelta }}
+\indentrel{3}\begin{verbatim}
+   (8)  [7,14,693,28,35]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPageEmpty8}
+\begin{paste}{VectorXmpPageEmpty8}{VectorXmpPagePatch8}
+\pastebutton{VectorXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{v * 7\free{vdelta }}
+\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPagePatch9}
+\begin{paste}{VectorXmpPageFull9}{VectorXmpPageEmpty9}
+\pastebutton{VectorXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{w : VECTOR INT := vector([2,3,4,5,6])\bound{w }}
+\indentrel{3}\begin{verbatim}
+   (9)  [2,3,4,5,6]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPageEmpty9}
+\begin{paste}{VectorXmpPageEmpty9}{VectorXmpPagePatch9}
+\pastebutton{VectorXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{w : VECTOR INT := vector([2,3,4,5,6])\bound{w }}
+\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPagePatch10}
+\begin{paste}{VectorXmpPageFull10}{VectorXmpPageEmpty10}
+\pastebutton{VectorXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{v + w\free{vdelta w }}
+\indentrel{3}\begin{verbatim}
+   (10)  [3,5,103,9,11]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPageEmpty10}
+\begin{paste}{VectorXmpPageEmpty10}{VectorXmpPagePatch10}
+\pastebutton{VectorXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{v + w\free{vdelta w }}
+\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPagePatch11}
+\begin{paste}{VectorXmpPageFull11}{VectorXmpPageEmpty11}
+\pastebutton{VectorXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{v - w\free{vdelta w }}
+\indentrel{3}\begin{verbatim}
+   (11)  [- 1,- 1,95,- 1,- 1]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VectorXmpPageEmpty11}
+\begin{paste}{VectorXmpPageEmpty11}{VectorXmpPagePatch11}
+\pastebutton{VectorXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{v - w\free{vdelta w }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/VOID.ht b/src/hyper/pages/VOID.ht
new file mode 100644
index 00000000..4dc87ca3
--- /dev/null
+++ b/src/hyper/pages/VOID.ht
@@ -0,0 +1,57 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\VoidXmpTitle}{Void}
+\newcommand{\VoidXmpNumber}{9.86}
+%
+% =====================================================================
+\begin{page}{VoidXmpPage}{9.86 Void}
+% =====================================================================
+\beginscroll
+
+When an expression is not in a value context, it is given type \spadtype{Void}.
+For example, in the expression
+\begin{verbatim}
+r := (a; b; if c then d else e; f)
+\end{verbatim}
+values are used only from the subexpressions \spad{c} and \spad{f}: all
+others are thrown away.
+The subexpressions \spad{a}, \spad{b}, \spad{d} and \spad{e}
+are evaluated for side-effects only and have type \spadtype{Void}.
+There is a unique value of type \spadtype{Void}.
+
+\xtc{
+You will most often see results of type \spadtype{Void} when you
+declare a variable.
+}{
+\spadpaste{a : Integer}
+}
+\noOutputXtc{
+Usually no output is displayed for \spadtype{Void} results.
+You can force the display of a rather ugly object by issuing
+\spadcmd{)set message void on}.
+%-% \HDsyscmdindex{set message void}{VoidXmpPage}{9.86}{Void}
+}{
+\spadpaste{)set message void on}
+}
+\xtc{
+}{
+\spadpaste{b : Fraction Integer}
+}
+\noOutputXtc{
+}{
+\spadpaste{)set message void off}
+}
+\xtc{
+All values can be converted to type \spadtype{Void}.
+}{
+\spadpaste{3::Void \bound{prev}}
+}
+\xtc{
+Once a value has been converted to \spadtype{Void}, it cannot be recovered.
+}{
+\spadpaste{\% :: PositiveInteger \free{prev}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/VOID.pht b/src/hyper/pages/VOID.pht
new file mode 100644
index 00000000..7603a350
--- /dev/null
+++ b/src/hyper/pages/VOID.pht
@@ -0,0 +1,89 @@
+\begin{patch}{VoidXmpPagePatch1}
+\begin{paste}{VoidXmpPageFull1}{VoidXmpPageEmpty1}
+\pastebutton{VoidXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{a : Integer}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VoidXmpPageEmpty1}
+\begin{paste}{VoidXmpPageEmpty1}{VoidXmpPagePatch1}
+\pastebutton{VoidXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{a : Integer}
+\end{paste}\end{patch}
+
+\begin{patch}{VoidXmpPagePatch2}
+\begin{paste}{VoidXmpPageFull2}{VoidXmpPageEmpty2}
+\pastebutton{VoidXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{)set message void on}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VoidXmpPageEmpty2}
+\begin{paste}{VoidXmpPageEmpty2}{VoidXmpPagePatch2}
+\pastebutton{VoidXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{)set message void on}
+\end{paste}\end{patch}
+
+\begin{patch}{VoidXmpPagePatch3}
+\begin{paste}{VoidXmpPageFull3}{VoidXmpPageEmpty3}
+\pastebutton{VoidXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{b : Fraction Integer}
+\indentrel{3}\begin{verbatim}
+   (2)  "()"
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VoidXmpPageEmpty3}
+\begin{paste}{VoidXmpPageEmpty3}{VoidXmpPagePatch3}
+\pastebutton{VoidXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{b : Fraction Integer}
+\end{paste}\end{patch}
+
+\begin{patch}{VoidXmpPagePatch4}
+\begin{paste}{VoidXmpPageFull4}{VoidXmpPageEmpty4}
+\pastebutton{VoidXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{)set message void off}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VoidXmpPageEmpty4}
+\begin{paste}{VoidXmpPageEmpty4}{VoidXmpPagePatch4}
+\pastebutton{VoidXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{)set message void off}
+\end{paste}\end{patch}
+
+\begin{patch}{VoidXmpPagePatch5}
+\begin{paste}{VoidXmpPageFull5}{VoidXmpPageEmpty5}
+\pastebutton{VoidXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{3::Void\bound{prev }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VoidXmpPageEmpty5}
+\begin{paste}{VoidXmpPageEmpty5}{VoidXmpPagePatch5}
+\pastebutton{VoidXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{3::Void\bound{prev }}
+\end{paste}\end{patch}
+
+\begin{patch}{VoidXmpPagePatch6}
+\begin{paste}{VoidXmpPageFull6}{VoidXmpPageEmpty6}
+\pastebutton{VoidXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{\% :: PositiveInteger\free{prev }}
+\indentrel{3}\begin{verbatim}
+   "()"
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{VoidXmpPageEmpty6}
+\begin{paste}{VoidXmpPageEmpty6}{VoidXmpPagePatch6}
+\pastebutton{VoidXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{\% :: PositiveInteger\free{prev }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/WUTSET.ht b/src/hyper/pages/WUTSET.ht
new file mode 100644
index 00000000..d41587a8
--- /dev/null
+++ b/src/hyper/pages/WUTSET.ht
@@ -0,0 +1,128 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\WuWenTsunTriangularSetXmpTitle}{WuWenTsunTriangularSet}
+\newcommand{\WuWenTsunTriangularSetXmpNumber}{9.87}
+%
+% =====================================================================
+\begin{page}{WuWenTsunTriangularSetXmpPage}{9.87 WuWenTsunTriangularSet}
+% =====================================================================
+\beginscroll
+The \spadtype{WuWenTsunTriangularSet} domain constructor implements
+the characteristic set method of Wu Wen Tsun.
+This algorithm computes a list of triangular sets from a list
+of polynomials such that the algebraic variety defined by the 
+given list of polynomials decomposes into the union of the regular-zero sets 
+of the computed triangular sets.
+The constructor takes four arguments.
+The first one, {\bf R}, is the coefficient ring of the polynomials;
+it must belong to the category \spadtype{IntegralDomain}.
+The second one, {\bf E}, is the exponent monoid of the polynomials;
+it must belong to the category \spadtype{OrderedAbelianMonoidSup}.
+The third one, {\bf V}, is the ordered set of variables;
+it must belong to the category \spadtype{OrderedSet}.
+The last one is the polynomial ring;
+it must belong to the category \spadtype{RecursivePolynomialCategory(R,E,V)}.
+The abbreviation for \spadtype{WuWenTsunTriangularSet} is
+\spadtype{WUTSET}.
+
+Let us illustrate the facilities by an example.
+
+\xtc{
+Define the coefficient ring.
+}{
+\spadpaste{R := Integer \bound{R}}
+}
+\xtc{
+Define the list of variables,
+}{
+\spadpaste{ls : List Symbol := [x,y,z,t] \bound{ls}}
+}
+\xtc{
+and make it an ordered set;
+}{
+\spadpaste{V := OVAR(ls) \free{ls} \bound{V}}
+}
+\xtc{
+then define the exponent monoid.
+}{
+\spadpaste{E := IndexedExponents V \free{V} \bound{E}}
+}
+\xtc{
+Define the polynomial ring.
+}{
+\spadpaste{P := NSMP(R, V) \free{R} \free{V} \bound{P}}
+}
+\xtc{
+Let the variables be polynomial.
+}{
+\spadpaste{x: P := 'x \free{P} \bound{x}}
+}
+\xtc{
+}{
+\spadpaste{y: P := 'y \free{P} \bound{y}}
+}
+\xtc{
+}{
+\spadpaste{z: P := 'z \free{P} \bound{z}}
+}
+\xtc{
+}{
+\spadpaste{t: P := 't \free{P} \bound{t}}
+}
+\xtc{
+Now call the \spadtype{WuWenTsunTriangularSet} domain constructor.
+}{
+\spadpaste{T := WUTSET(R,E,V,P) \free{R} \free{E} \free{V} \free{P} \bound{T} }
+}
+\xtc{
+Define a polynomial system.
+}{
+\spadpaste{p1 := x ** 31 - x ** 6 - x - y \free{x} \free{y} \bound{p1}}
+}
+\xtc{
+}{
+\spadpaste{p2 := x ** 8  - z \free{x} \free{z} \bound{p2}}
+}
+\xtc{
+}{
+\spadpaste{p3 := x ** 10 - t \free{x} \free{t} \bound{p3}}
+}
+\xtc{
+}{
+\spadpaste{lp := [p1, p2, p3] \free{p1} \free{p2} \free{p3} \bound{lp}}
+}
+\xtc{
+Compute a characteristic set of the system.
+}{
+\spadpaste{characteristicSet(lp)$T \free{lp} \free{T}}
+}
+\xtc{
+Solve the system.
+}{
+\spadpaste{zeroSetSplit(lp)$T \free{lp} \free{T}}
+}
+
+
+The \spadtype{RegularTriangularSet} and \spadtype{SquareFreeRegularTriangularSet} domain constructors,
+and the  \spadtype{LazardSetSolvingPackage}, \spadtype{SquareFreeRegularTriangularSet} 
+and \spadtype{ZeroDimensionalSolvePackage} package constructors
+also provide operations to compute triangular decompositions of algebraic varieties.
+These five constructor use a special kind of characteristic sets, called regular triangular sets.
+These special characteristic sets have better properties than the general ones.
+Regular triangular sets and their related concepts are presented in
+the paper "On the Theories of Triangular sets" By P. Aubry, D. Lazard
+and M. Moreno Maza (to appear in the Journal of Symbolic Computation).
+The decomposition algorithm (due to the third author) available in the 
+four above constructors provide generally better timings than 
+the characteristic set method.
+In fact, the \spadtype{WUTSET} constructor remains interesting 
+for the purpose of manipulating characteristic sets whereas
+the other constructors are more convenient for solving polynomial systems.
+
+Note that the way of understanding triangular decompositions 
+is detailed in the example of the \spadtype{RegularTriangularSet}
+constructor.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/WUTSET.pht b/src/hyper/pages/WUTSET.pht
new file mode 100644
index 00000000..75ce4321
--- /dev/null
+++ b/src/hyper/pages/WUTSET.pht
@@ -0,0 +1,280 @@
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch1}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull1}{WuWenTsunTriangularSetXmpPageEmpty1}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{R := Integer\bound{R }}
+\indentrel{3}\begin{verbatim}
+   (1)  Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty1}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty1}{WuWenTsunTriangularSetXmpPagePatch1}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{R := Integer\bound{R }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch2}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull2}{WuWenTsunTriangularSetXmpPageEmpty2}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{ls : List Symbol := [x,y,z,t]\bound{ls }}
+\indentrel{3}\begin{verbatim}
+   (2)  [x,y,z,t]
+                                      Type: List Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty2}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty2}{WuWenTsunTriangularSetXmpPagePatch2}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{ls : List Symbol := [x,y,z,t]\bound{ls }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch3}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull3}{WuWenTsunTriangularSetXmpPageEmpty3}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
+\indentrel{3}\begin{verbatim}
+   (3)  OrderedVariableList [x,y,z,t]
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty3}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty3}{WuWenTsunTriangularSetXmpPagePatch3}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch4}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull4}{WuWenTsunTriangularSetXmpPageEmpty4}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{E := IndexedExponents V\free{V }\bound{E }}
+\indentrel{3}\begin{verbatim}
+   (4)  IndexedExponents OrderedVariableList [x,y,z,t]
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty4}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty4}{WuWenTsunTriangularSetXmpPagePatch4}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{E := IndexedExponents V\free{V }\bound{E }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch5}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull5}{WuWenTsunTriangularSetXmpPageEmpty5}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
+\indentrel{3}\begin{verbatim}
+   (5)
+  NewSparseMultivariatePolynomial(Integer,OrderedVariable
+  List [x,y,z,t])
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty5}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty5}{WuWenTsunTriangularSetXmpPagePatch5}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch6}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull6}{WuWenTsunTriangularSetXmpPageEmpty6}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{x: P := 'x\free{P }\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (6)  x
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty6}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty6}{WuWenTsunTriangularSetXmpPagePatch6}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{x: P := 'x\free{P }\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch7}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull7}{WuWenTsunTriangularSetXmpPageEmpty7}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{y: P := 'y\free{P }\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (7)  y
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty7}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty7}{WuWenTsunTriangularSetXmpPagePatch7}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{y: P := 'y\free{P }\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch8}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull8}{WuWenTsunTriangularSetXmpPageEmpty8}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{z: P := 'z\free{P }\bound{z }}
+\indentrel{3}\begin{verbatim}
+   (8)  z
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty8}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty8}{WuWenTsunTriangularSetXmpPagePatch8}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{z: P := 'z\free{P }\bound{z }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch9}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull9}{WuWenTsunTriangularSetXmpPageEmpty9}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{t: P := 't\free{P }\bound{t }}
+\indentrel{3}\begin{verbatim}
+   (9)  t
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty9}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty9}{WuWenTsunTriangularSetXmpPagePatch9}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{t: P := 't\free{P }\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch10}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull10}{WuWenTsunTriangularSetXmpPageEmpty10}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{T := WUTSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{T }}
+\indentrel{3}\begin{verbatim}
+   (10)
+  WuWenTsunTriangularSet(Integer,IndexedExponents Ordered
+  VariableList [x,y,z,t],OrderedVariableList [x,y,z,t],Ne
+  wSparseMultivariatePolynomial(Integer,OrderedVariableLi
+  st [x,y,z,t]))
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty10}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty10}{WuWenTsunTriangularSetXmpPagePatch10}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{T := WUTSET(R,E,V,P)\free{R }\free{E }\free{V }\free{P }\bound{T }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch11}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull11}{WuWenTsunTriangularSetXmpPageEmpty11}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{p1 := x ** 31 - x ** 6 - x - y\free{x }\free{y }\bound{p1 }}
+\indentrel{3}\begin{verbatim}
+          31    6
+   (11)  x   - x  - x - y
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty11}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty11}{WuWenTsunTriangularSetXmpPagePatch11}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{p1 := x ** 31 - x ** 6 - x - y\free{x }\free{y }\bound{p1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch12}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull12}{WuWenTsunTriangularSetXmpPageEmpty12}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{p2 := x ** 8 - z\free{x }\free{z }\bound{p2 }}
+\indentrel{3}\begin{verbatim}
+          8
+   (12)  x  - z
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty12}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty12}{WuWenTsunTriangularSetXmpPagePatch12}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{p2 := x ** 8 - z\free{x }\free{z }\bound{p2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch13}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull13}{WuWenTsunTriangularSetXmpPageEmpty13}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{p3 := x ** 10 - t\free{x }\free{t }\bound{p3 }}
+\indentrel{3}\begin{verbatim}
+          10
+   (13)  x   - t
+Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty13}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty13}{WuWenTsunTriangularSetXmpPagePatch13}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{p3 := x ** 10 - t\free{x }\free{t }\bound{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch14}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull14}{WuWenTsunTriangularSetXmpPageEmpty14}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
+\indentrel{3}\begin{verbatim}
+           31    6          8      10
+   (14)  [x   - x  - x - y,x  - z,x   - t]
+Type: List NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty14}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty14}{WuWenTsunTriangularSetXmpPagePatch14}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch15}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull15}{WuWenTsunTriangularSetXmpPageEmpty15}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{characteristicSet(lp)$T\free{lp }\free{T }}
+\indentrel{3}\begin{verbatim}
+   (15)
+     5    4   4 2 2     3 4        7     4      6    6
+   {z  - t , t z y  + 2t z y + (- t  + 2t  - t)z  + t z,
+      3      3     3     3
+    (t  - 1)z x - z y - t }
+Type: Union(WuWenTsunTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t])),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty15}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty15}{WuWenTsunTriangularSetXmpPagePatch15}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{characteristicSet(lp)$T\free{lp }\free{T }}
+\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPagePatch16}
+\begin{paste}{WuWenTsunTriangularSetXmpPageFull16}{WuWenTsunTriangularSetXmpPageEmpty16}
+\pastebutton{WuWenTsunTriangularSetXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{zeroSetSplit(lp)$T\free{lp }\free{T }}
+\indentrel{3}\begin{verbatim}
+   (16)
+                 3      5    4  3     3    2
+   [{t,z,y,x}, {t  - 1,z  - t ,z y + t ,z x  - t},
+
+       5    4
+     {z  - t ,
+       4 2 2     3 4        7     4      6    6
+      t z y  + 2t z y + (- t  + 2t  - t)z  + t z,
+        3      3     3     3
+      (t  - 1)z x - z y - t }
+     ]
+Type: List WuWenTsunTriangularSet(Integer,IndexedExponents OrderedVariableList [x,y,z,t],OrderedVariableList [x,y,z,t],NewSparseMultivariatePolynomial(Integer,OrderedVariableList [x,y,z,t]))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{WuWenTsunTriangularSetXmpPageEmpty16}
+\begin{paste}{WuWenTsunTriangularSetXmpPageEmpty16}{WuWenTsunTriangularSetXmpPagePatch16}
+\pastebutton{WuWenTsunTriangularSetXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{zeroSetSplit(lp)$T\free{lp }\free{T }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/XPBWPOLY.ht b/src/hyper/pages/XPBWPOLY.ht
new file mode 100644
index 00000000..3352a675
--- /dev/null
+++ b/src/hyper/pages/XPBWPOLY.ht
@@ -0,0 +1,179 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\XPBWPolynomialXmpTitle}{XPBWPolynomial}
+\newcommand{\XPBWPolynomialXmpNumber}{9.88}
+%
+% =====================================================================
+\begin{page}{XPBWPolynomialXmpPage}{9.88 XPBWPolynomial}
+% =====================================================================
+\beginscroll
+Initialisations
+\xtc{
+}{
+\spadpaste{a:Symbol := 'a \bound{a}}
+}
+\xtc{
+}{
+\spadpaste{b:Symbol := 'b \bound{b}}
+}
+\xtc{
+}{
+\spadpaste{RN    := Fraction(Integer) \bound{RN}}
+}
+\xtc{
+}{
+\spadpaste{word   := OrderedFreeMonoid Symbol \bound{word}}
+}
+\xtc{
+}{
+\spadpaste{lword := LyndonWord(Symbol) \bound{lword}}
+}
+\xtc{
+}{
+\spadpaste{base  := PoincareBirkhoffWittLyndonBasis Symbol \bound{base}}
+}
+\xtc{
+}{
+\spadpaste{dpoly := XDistributedPolynomial(Symbol, RN) \bound{dpoly} \free{RN}}
+}
+\xtc{
+}{
+\spadpaste{rpoly := XRecursivePolynomial(Symbol, RN) \bound{rpoly} \free{RN}}
+}
+\xtc{
+}{
+\spadpaste{lpoly := LiePolynomial(Symbol, RN) \bound{lpoly} \free{RN}}
+}
+\xtc{
+}{
+\spadpaste{poly  := XPBWPolynomial(Symbol, RN) \bound{poly} \free{RN}}
+}
+\xtc{
+}{
+\spadpaste{liste : List lword := LyndonWordsList([a,b], 6) \bound{liste} \free{lword a b }}
+}
+
+Let's make some polynomials
+\xtc{
+}{
+\spadpaste{0$poly \free{poly}}
+}
+\xtc{
+}{
+\spadpaste{1$poly \free{poly}}
+}
+\xtc{
+}{
+\spadpaste{p : poly := a \free{a poly} \bound{p}}
+}
+\xtc{
+}{
+\spadpaste{q : poly := b \free{b poly} \bound{q}}
+}
+\xtc{
+}{
+\spadpaste{pq: poly := p*q \free{p q poly} \bound{pq}}
+}
+\xtc{
+Coerce to distributed polynomial
+}{
+\spadpaste{pq :: dpoly \free{pq dpoly}}
+}
+
+Check some polynomial operations
+\xtc{
+}{
+\spadpaste{mirror pq \free{pq}}
+}
+\xtc{
+}{
+\spadpaste{ListOfTerms pq \free{pq}}
+}
+\xtc{
+}{
+\spadpaste{reductum pq \free{pq}}
+}
+\xtc{
+}{
+\spadpaste{leadingMonomial pq \free{pq}}
+}
+\xtc{
+}{
+\spadpaste{coefficients pq \free{pq}}
+}
+\xtc{
+}{
+\spadpaste{leadingTerm pq \free{pq}}
+}
+\xtc{
+}{
+\spadpaste{degree pq \free{pq}}
+}
+\xtc{
+}{
+\spadpaste{pq4:=exp(pq,4) \bound{pq4} \free{pq}}
+}
+\xtc{
+}{
+\spadpaste{log(pq4,4) - pq \free{pq4 pq} }
+}
+
+Calculations with verification in \axiomType{XDistributedPolynomial}.
+\xtc{
+}{
+\spadpaste{lp1 :lpoly := LiePoly liste.10 \free{liste lpoly} \bound{lp1}}
+}
+\xtc{
+}{
+\spadpaste{lp2 :lpoly := LiePoly liste.11 \free{liste lpoly} \bound{lp2}}
+}
+\xtc{
+}{
+\spadpaste{lp  :lpoly := [lp1, lp2] \free{lp1 lp2 lpoly} \bound{lp}}
+}
+\xtc{
+}{
+\spadpaste{lpd1: dpoly := lp1 \free{lp1 dpoly} \bound{lpd1}}
+}
+\xtc{
+}{
+\spadpaste{lpd2: dpoly := lp2 \free{lp2 dpoly} \bound{lpd2}}
+}
+\xtc{
+}{
+\spadpaste{lpd : dpoly := lpd1 * lpd2 - lpd2 * lpd1 \free{dpoly lpd1 lpd2} \bound{lpd}}
+}
+\xtc{
+}{
+\spadpaste{lp :: dpoly - lpd \free{lpd dpoly lp}}
+}
+
+Calculations with verification in \axiomType{XRecursivePolynomial}.
+\xtc{
+}{
+\spadpaste{p := 3 * lp \free{lp} \bound{pp}}
+}
+\xtc{
+}{
+\spadpaste{q := lp1 \free{lp1} \bound{qq}}
+}
+\xtc{
+}{
+\spadpaste{pq:= p * q \free{pp qq} \bound{pqpq}}
+}
+\xtc{
+}{
+\spadpaste{pr:rpoly := p :: rpoly \free{rpoly pp} \bound{pr}}
+}
+\xtc{
+}{
+\spadpaste{qr:rpoly := q :: rpoly \free{rpoly qq} \bound{qr}}
+}
+\xtc{
+}{
+\spadpaste{pq :: rpoly - pr*qr \free{pr qr rpoly pqpq} }
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/XPBWPOLY.pht b/src/hyper/pages/XPBWPOLY.pht
new file mode 100644
index 00000000..b56a1d9e
--- /dev/null
+++ b/src/hyper/pages/XPBWPOLY.pht
@@ -0,0 +1,877 @@
+\begin{patch}{XPBWPolynomialXmpPagePatch1}
+\begin{paste}{XPBWPolynomialXmpPageFull1}{XPBWPolynomialXmpPageEmpty1}
+\pastebutton{XPBWPolynomialXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{a:Symbol := 'a\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (1)  a
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty1}
+\begin{paste}{XPBWPolynomialXmpPageEmpty1}{XPBWPolynomialXmpPagePatch1}
+\pastebutton{XPBWPolynomialXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{a:Symbol := 'a\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch2}
+\begin{paste}{XPBWPolynomialXmpPageFull2}{XPBWPolynomialXmpPageEmpty2}
+\pastebutton{XPBWPolynomialXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{b:Symbol := 'b\bound{b }}
+\indentrel{3}\begin{verbatim}
+   (2)  b
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty2}
+\begin{paste}{XPBWPolynomialXmpPageEmpty2}{XPBWPolynomialXmpPagePatch2}
+\pastebutton{XPBWPolynomialXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{b:Symbol := 'b\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch3}
+\begin{paste}{XPBWPolynomialXmpPageFull3}{XPBWPolynomialXmpPageEmpty3}
+\pastebutton{XPBWPolynomialXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{RN := Fraction(Integer)\bound{RN }}
+\indentrel{3}\begin{verbatim}
+   (3)  Fraction Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty3}
+\begin{paste}{XPBWPolynomialXmpPageEmpty3}{XPBWPolynomialXmpPagePatch3}
+\pastebutton{XPBWPolynomialXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{RN := Fraction(Integer)\bound{RN }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch4}
+\begin{paste}{XPBWPolynomialXmpPageFull4}{XPBWPolynomialXmpPageEmpty4}
+\pastebutton{XPBWPolynomialXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{word := OrderedFreeMonoid Symbol\bound{word }}
+\indentrel{3}\begin{verbatim}
+   (4)  OrderedFreeMonoid Symbol
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty4}
+\begin{paste}{XPBWPolynomialXmpPageEmpty4}{XPBWPolynomialXmpPagePatch4}
+\pastebutton{XPBWPolynomialXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{word := OrderedFreeMonoid Symbol\bound{word }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch5}
+\begin{paste}{XPBWPolynomialXmpPageFull5}{XPBWPolynomialXmpPageEmpty5}
+\pastebutton{XPBWPolynomialXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{lword := LyndonWord(Symbol)\bound{lword }}
+\indentrel{3}\begin{verbatim}
+   (5)  LyndonWord Symbol
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty5}
+\begin{paste}{XPBWPolynomialXmpPageEmpty5}{XPBWPolynomialXmpPagePatch5}
+\pastebutton{XPBWPolynomialXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{lword := LyndonWord(Symbol)\bound{lword }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch6}
+\begin{paste}{XPBWPolynomialXmpPageFull6}{XPBWPolynomialXmpPageEmpty6}
+\pastebutton{XPBWPolynomialXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{base := PoincareBirkhoffWittLyndonBasis Symbol\bound{base }}
+\indentrel{3}\begin{verbatim}
+   (6)  PoincareBirkhoffWittLyndonBasis Symbol
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty6}
+\begin{paste}{XPBWPolynomialXmpPageEmpty6}{XPBWPolynomialXmpPagePatch6}
+\pastebutton{XPBWPolynomialXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{base := PoincareBirkhoffWittLyndonBasis Symbol\bound{base }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch7}
+\begin{paste}{XPBWPolynomialXmpPageFull7}{XPBWPolynomialXmpPageEmpty7}
+\pastebutton{XPBWPolynomialXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{dpoly := XDistributedPolynomial(Symbol, RN)\bound{dpoly }\free{RN }}
+\indentrel{3}\begin{verbatim}
+   (7)  XDistributedPolynomial(Symbol,Fraction Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty7}
+\begin{paste}{XPBWPolynomialXmpPageEmpty7}{XPBWPolynomialXmpPagePatch7}
+\pastebutton{XPBWPolynomialXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{dpoly := XDistributedPolynomial(Symbol, RN)\bound{dpoly }\free{RN }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch8}
+\begin{paste}{XPBWPolynomialXmpPageFull8}{XPBWPolynomialXmpPageEmpty8}
+\pastebutton{XPBWPolynomialXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{rpoly := XRecursivePolynomial(Symbol, RN)\bound{rpoly }\free{RN }}
+\indentrel{3}\begin{verbatim}
+   (8)  XRecursivePolynomial(Symbol,Fraction Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty8}
+\begin{paste}{XPBWPolynomialXmpPageEmpty8}{XPBWPolynomialXmpPagePatch8}
+\pastebutton{XPBWPolynomialXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{rpoly := XRecursivePolynomial(Symbol, RN)\bound{rpoly }\free{RN }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch9}
+\begin{paste}{XPBWPolynomialXmpPageFull9}{XPBWPolynomialXmpPageEmpty9}
+\pastebutton{XPBWPolynomialXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{lpoly := LiePolynomial(Symbol, RN)\bound{lpoly }\free{RN }}
+\indentrel{3}\begin{verbatim}
+   (9)  LiePolynomial(Symbol,Fraction Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty9}
+\begin{paste}{XPBWPolynomialXmpPageEmpty9}{XPBWPolynomialXmpPagePatch9}
+\pastebutton{XPBWPolynomialXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{lpoly := LiePolynomial(Symbol, RN)\bound{lpoly }\free{RN }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch10}
+\begin{paste}{XPBWPolynomialXmpPageFull10}{XPBWPolynomialXmpPageEmpty10}
+\pastebutton{XPBWPolynomialXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{poly := XPBWPolynomial(Symbol, RN)\bound{poly }\free{RN }}
+\indentrel{3}\begin{verbatim}
+   (10)  XPBWPolynomial(Symbol,Fraction Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty10}
+\begin{paste}{XPBWPolynomialXmpPageEmpty10}{XPBWPolynomialXmpPagePatch10}
+\pastebutton{XPBWPolynomialXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{poly := XPBWPolynomial(Symbol, RN)\bound{poly }\free{RN }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch11}
+\begin{paste}{XPBWPolynomialXmpPageFull11}{XPBWPolynomialXmpPageEmpty11}
+\pastebutton{XPBWPolynomialXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{liste : List lword := LyndonWordsList([a,b], 6)\bound{liste }\free{lword a b }}
+\indentrel{3}\begin{verbatim}
+   (11)
+                       2        2     3      2 2
+   [[a], [b], [a b], [a b], [a b ], [a b], [a b ],
+        3     4      3 2     2          2 3           2
+    [a b ], [a b], [a b ], [a b a b], [a b ], [a b a b ],
+        4     5      4 2     3          3 3     2     2
+    [a b ], [a b], [a b ], [a b a b], [a b ], [a b a b ],
+      2 2        2 4           3       5
+    [a b a b], [a b ], [a b a b ], [a b ]]
+                           Type: List LyndonWord Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty11}
+\begin{paste}{XPBWPolynomialXmpPageEmpty11}{XPBWPolynomialXmpPagePatch11}
+\pastebutton{XPBWPolynomialXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{liste : List lword := LyndonWordsList([a,b], 6)\bound{liste }\free{lword a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch12}
+\begin{paste}{XPBWPolynomialXmpPageFull12}{XPBWPolynomialXmpPageEmpty12}
+\pastebutton{XPBWPolynomialXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{0$poly\free{poly }}
+\indentrel{3}\begin{verbatim}
+   (12)  0
+          Type: XPBWPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty12}
+\begin{paste}{XPBWPolynomialXmpPageEmpty12}{XPBWPolynomialXmpPagePatch12}
+\pastebutton{XPBWPolynomialXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{0$poly\free{poly }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch13}
+\begin{paste}{XPBWPolynomialXmpPageFull13}{XPBWPolynomialXmpPageEmpty13}
+\pastebutton{XPBWPolynomialXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{1$poly\free{poly }}
+\indentrel{3}\begin{verbatim}
+   (13)  1
+          Type: XPBWPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty13}
+\begin{paste}{XPBWPolynomialXmpPageEmpty13}{XPBWPolynomialXmpPagePatch13}
+\pastebutton{XPBWPolynomialXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{1$poly\free{poly }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch14}
+\begin{paste}{XPBWPolynomialXmpPageFull14}{XPBWPolynomialXmpPageEmpty14}
+\pastebutton{XPBWPolynomialXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{p : poly := a\free{a poly }\bound{p }}
+\indentrel{3}\begin{verbatim}
+   (14)  [a]
+          Type: XPBWPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty14}
+\begin{paste}{XPBWPolynomialXmpPageEmpty14}{XPBWPolynomialXmpPagePatch14}
+\pastebutton{XPBWPolynomialXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{p : poly := a\free{a poly }\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch15}
+\begin{paste}{XPBWPolynomialXmpPageFull15}{XPBWPolynomialXmpPageEmpty15}
+\pastebutton{XPBWPolynomialXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{q : poly := b\free{b poly }\bound{q }}
+\indentrel{3}\begin{verbatim}
+   (15)  [b]
+          Type: XPBWPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty15}
+\begin{paste}{XPBWPolynomialXmpPageEmpty15}{XPBWPolynomialXmpPagePatch15}
+\pastebutton{XPBWPolynomialXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{q : poly := b\free{b poly }\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch16}
+\begin{paste}{XPBWPolynomialXmpPageFull16}{XPBWPolynomialXmpPageEmpty16}
+\pastebutton{XPBWPolynomialXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{pq: poly := p*q\free{p q poly }\bound{pq }}
+\indentrel{3}\begin{verbatim}
+   (16)  [a b] + [b][a]
+          Type: XPBWPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty16}
+\begin{paste}{XPBWPolynomialXmpPageEmpty16}{XPBWPolynomialXmpPagePatch16}
+\pastebutton{XPBWPolynomialXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{pq: poly := p*q\free{p q poly }\bound{pq }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch17}
+\begin{paste}{XPBWPolynomialXmpPageFull17}{XPBWPolynomialXmpPageEmpty17}
+\pastebutton{XPBWPolynomialXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{pq :: dpoly\free{pq dpoly }}
+\indentrel{3}\begin{verbatim}
+   (17)  a b
+  Type: XDistributedPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty17}
+\begin{paste}{XPBWPolynomialXmpPageEmpty17}{XPBWPolynomialXmpPagePatch17}
+\pastebutton{XPBWPolynomialXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{pq :: dpoly\free{pq dpoly }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch18}
+\begin{paste}{XPBWPolynomialXmpPageFull18}{XPBWPolynomialXmpPageEmpty18}
+\pastebutton{XPBWPolynomialXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{mirror pq\free{pq }}
+\indentrel{3}\begin{verbatim}
+   (18)  [b][a]
+          Type: XPBWPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty18}
+\begin{paste}{XPBWPolynomialXmpPageEmpty18}{XPBWPolynomialXmpPagePatch18}
+\pastebutton{XPBWPolynomialXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{mirror pq\free{pq }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch19}
+\begin{paste}{XPBWPolynomialXmpPageFull19}{XPBWPolynomialXmpPageEmpty19}
+\pastebutton{XPBWPolynomialXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{ListOfTerms pq\free{pq }}
+\indentrel{3}\begin{verbatim}
+   (19)  [[k= [b][a],c= 1],[k= [a b],c= 1]]
+Type: List Record(k: PoincareBirkhoffWittLyndonBasis Symbol,c: Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty19}
+\begin{paste}{XPBWPolynomialXmpPageEmpty19}{XPBWPolynomialXmpPagePatch19}
+\pastebutton{XPBWPolynomialXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{ListOfTerms pq\free{pq }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch20}
+\begin{paste}{XPBWPolynomialXmpPageFull20}{XPBWPolynomialXmpPageEmpty20}
+\pastebutton{XPBWPolynomialXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{reductum pq\free{pq }}
+\indentrel{3}\begin{verbatim}
+   (20)  [a b]
+          Type: XPBWPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty20}
+\begin{paste}{XPBWPolynomialXmpPageEmpty20}{XPBWPolynomialXmpPagePatch20}
+\pastebutton{XPBWPolynomialXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{reductum pq\free{pq }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch21}
+\begin{paste}{XPBWPolynomialXmpPageFull21}{XPBWPolynomialXmpPageEmpty21}
+\pastebutton{XPBWPolynomialXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{leadingMonomial pq\free{pq }}
+\indentrel{3}\begin{verbatim}
+   (21)  [b][a]
+           Type: PoincareBirkhoffWittLyndonBasis Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty21}
+\begin{paste}{XPBWPolynomialXmpPageEmpty21}{XPBWPolynomialXmpPagePatch21}
+\pastebutton{XPBWPolynomialXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{leadingMonomial pq\free{pq }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch22}
+\begin{paste}{XPBWPolynomialXmpPageFull22}{XPBWPolynomialXmpPageEmpty22}
+\pastebutton{XPBWPolynomialXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{coefficients pq\free{pq }}
+\indentrel{3}\begin{verbatim}
+   (22)  [1,1]
+                            Type: List Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty22}
+\begin{paste}{XPBWPolynomialXmpPageEmpty22}{XPBWPolynomialXmpPagePatch22}
+\pastebutton{XPBWPolynomialXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{coefficients pq\free{pq }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch23}
+\begin{paste}{XPBWPolynomialXmpPageFull23}{XPBWPolynomialXmpPageEmpty23}
+\pastebutton{XPBWPolynomialXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{leadingTerm pq\free{pq }}
+\indentrel{3}\begin{verbatim}
+   (23)  [k= [b][a],c= 1]
+Type: Record(k: PoincareBirkhoffWittLyndonBasis Symbol,c: Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty23}
+\begin{paste}{XPBWPolynomialXmpPageEmpty23}{XPBWPolynomialXmpPagePatch23}
+\pastebutton{XPBWPolynomialXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{leadingTerm pq\free{pq }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch24}
+\begin{paste}{XPBWPolynomialXmpPageFull24}{XPBWPolynomialXmpPageEmpty24}
+\pastebutton{XPBWPolynomialXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{degree pq\free{pq }}
+\indentrel{3}\begin{verbatim}
+   (24)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty24}
+\begin{paste}{XPBWPolynomialXmpPageEmpty24}{XPBWPolynomialXmpPagePatch24}
+\pastebutton{XPBWPolynomialXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{degree pq\free{pq }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch25}
+\begin{paste}{XPBWPolynomialXmpPageFull25}{XPBWPolynomialXmpPageEmpty25}
+\pastebutton{XPBWPolynomialXmpPageFull25}{\hidepaste}
+\tab{5}\spadcommand{pq4:=exp(pq,4)\bound{pq4 }\free{pq }}
+\indentrel{3}\begin{verbatim}
+   (25)
+                          1              1     2
+     1 + [a b] + [b][a] + Ä [a b][a b] + Ä [a b ][a]
+                          2              2
+   + 
+     1      2     3               1
+     Ä [b][a b] + Ä [b][a b][a] + Ä [b][b][a][a]
+     2            2               2
+          Type: XPBWPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty25}
+\begin{paste}{XPBWPolynomialXmpPageEmpty25}{XPBWPolynomialXmpPagePatch25}
+\pastebutton{XPBWPolynomialXmpPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{pq4:=exp(pq,4)\bound{pq4 }\free{pq }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch26}
+\begin{paste}{XPBWPolynomialXmpPageFull26}{XPBWPolynomialXmpPageEmpty26}
+\pastebutton{XPBWPolynomialXmpPageFull26}{\hidepaste}
+\tab{5}\spadcommand{log(pq4,4) - pq\free{pq4 pq }}
+\indentrel{3}\begin{verbatim}
+   (26)  0
+          Type: XPBWPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty26}
+\begin{paste}{XPBWPolynomialXmpPageEmpty26}{XPBWPolynomialXmpPagePatch26}
+\pastebutton{XPBWPolynomialXmpPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{log(pq4,4) - pq\free{pq4 pq }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch27}
+\begin{paste}{XPBWPolynomialXmpPageFull27}{XPBWPolynomialXmpPageEmpty27}
+\pastebutton{XPBWPolynomialXmpPageFull27}{\hidepaste}
+\tab{5}\spadcommand{lp1 :lpoly := LiePoly liste.10\free{liste lpoly }\bound{lp1 }}
+\indentrel{3}\begin{verbatim}
+           3 2
+   (27)  [a b ]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty27}
+\begin{paste}{XPBWPolynomialXmpPageEmpty27}{XPBWPolynomialXmpPagePatch27}
+\pastebutton{XPBWPolynomialXmpPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{lp1 :lpoly := LiePoly liste.10\free{liste lpoly }\bound{lp1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch28}
+\begin{paste}{XPBWPolynomialXmpPageFull28}{XPBWPolynomialXmpPageEmpty28}
+\pastebutton{XPBWPolynomialXmpPageFull28}{\hidepaste}
+\tab{5}\spadcommand{lp2 :lpoly := LiePoly liste.11\free{liste lpoly }\bound{lp2 }}
+\indentrel{3}\begin{verbatim}
+           2
+   (28)  [a b a b]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty28}
+\begin{paste}{XPBWPolynomialXmpPageEmpty28}{XPBWPolynomialXmpPagePatch28}
+\pastebutton{XPBWPolynomialXmpPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{lp2 :lpoly := LiePoly liste.11\free{liste lpoly }\bound{lp2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch29}
+\begin{paste}{XPBWPolynomialXmpPageFull29}{XPBWPolynomialXmpPageEmpty29}
+\pastebutton{XPBWPolynomialXmpPageFull29}{\hidepaste}
+\tab{5}\spadcommand{lp :lpoly := [lp1, lp2]\free{lp1 lp2 lpoly }\bound{lp }}
+\indentrel{3}\begin{verbatim}
+           3 2 2
+   (29)  [a b a b a b]
+           Type: LiePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty29}
+\begin{paste}{XPBWPolynomialXmpPageEmpty29}{XPBWPolynomialXmpPagePatch29}
+\pastebutton{XPBWPolynomialXmpPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{lp :lpoly := [lp1, lp2]\free{lp1 lp2 lpoly }\bound{lp }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch30}
+\begin{paste}{XPBWPolynomialXmpPageFull30}{XPBWPolynomialXmpPageEmpty30}
+\pastebutton{XPBWPolynomialXmpPageFull30}{\hidepaste}
+\tab{5}\spadcommand{lpd1: dpoly := lp1\free{lp1 dpoly }\bound{lpd1 }}
+\indentrel{3}\begin{verbatim}
+   (30)
+      3 2     2         2 2                    2 2
+     a b  - 2a b a b - a b a + 4a b a b a - a b a
+   + 
+               2    2 3
+     - 2b a b a  + b a
+  Type: XDistributedPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty30}
+\begin{paste}{XPBWPolynomialXmpPageEmpty30}{XPBWPolynomialXmpPagePatch30}
+\pastebutton{XPBWPolynomialXmpPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{lpd1: dpoly := lp1\free{lp1 dpoly }\bound{lpd1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch31}
+\begin{paste}{XPBWPolynomialXmpPageFull31}{XPBWPolynomialXmpPageEmpty31}
+\pastebutton{XPBWPolynomialXmpPageFull31}{\hidepaste}
+\tab{5}\spadcommand{lpd2: dpoly := lp2\free{lp2 dpoly }\bound{lpd2 }}
+\indentrel{3}\begin{verbatim}
+   (31)
+      2         2 2          2                    2 2
+     a b a b - a b a - 3a b a b + 4a b a b a - a b a
+   + 
+         3        2             2
+     2b a b - 3b a b a + b a b a
+  Type: XDistributedPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty31}
+\begin{paste}{XPBWPolynomialXmpPageEmpty31}{XPBWPolynomialXmpPagePatch31}
+\pastebutton{XPBWPolynomialXmpPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{lpd2: dpoly := lp2\free{lp2 dpoly }\bound{lpd2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch32}
+\begin{paste}{XPBWPolynomialXmpPageFull32}{XPBWPolynomialXmpPageEmpty32}
+\pastebutton{XPBWPolynomialXmpPageFull32}{\hidepaste}
+\tab{5}\spadcommand{lpd : dpoly := lpd1 * lpd2 - lpd2 * lpd1\free{dpoly lpd1 lpd2 }\bound{lpd }}
+\indentrel{3}\begin{verbatim}
+   (32)
+      3 2 2         3 2 2 2      3 2     2
+     a b a b a b - a b a b a - 3a b a b a b
+   + 
+       3 2             3 2   2 2     3 3 3      3 3 2
+     4a b a b a b a - a b a b a  + 2a b a b - 3a b a b a
+   + 
+      3 3     2    2       3 2     2       2 2
+     a b a b a  - a b a b a b  + 3a b a b a b a
+   + 
+       2           2       2
+     6a b a b a b a b - 12a b a b a b a b a
+   + 
+       2         2 2     2     2 3      2     2 2
+     3a b a b a b a  - 4a b a b a b + 6a b a b a b a
+   + 
+        2     3 3    2 2 4 2     2 2 3          2 2 2   2
+     - a b a b a  + a b a b  - 3a b a b a b + 3a b a b a b
+   + 
+         2 2     3      2 2     2        2 2         2
+     - 2a b a b a b + 3a b a b a b a - 3a b a b a b a
+   + 
+      2 2   2 3         2   3 2         2   2
+     a b a b a  + 3a b a b a b  - 6a b a b a b a b
+   + 
+             2   2 2           2
+     - 3a b a b a b a + 12a b a b a b a b a
+   + 
+             2     2 2         2 2     2         2 3 3
+     - 3a b a b a b a  - 6a b a b a b a  + 3a b a b a
+   + 
+                 4 2              3
+     - 4a b a b a b  + 12a b a b a b a b
+   + 
+                  2   2                  3
+     - 12a b a b a b a b + 8a b a b a b a b
+   + 
+                      2                         2
+     - 12a b a b a b a b a + 12a b a b a b a b a
+   + 
+                   2 3      2 5 2       2 4
+     - 4a b a b a b a  + a b a b  - 3a b a b a b
+   + 
+         2 3   2        2 2   3        2 2   2
+     3a b a b a b - 2a b a b a b + 3a b a b a b a
+   + 
+           2 2       2      2 2 2 3       3   3 2
+     - 3a b a b a b a  + a b a b a  - 2b a b a b
+   + 
+         3   2            3   2 2        3
+     4b a b a b a b + 2b a b a b a - 8b a b a b a b a
+   + 
+         3     2 2       3 2     2       3 3 3
+     2b a b a b a  + 4b a b a b a  - 2b a b a
+   + 
+         2   4 2       2   3            2   3 2
+     3b a b a b  - 6b a b a b a b - 3b a b a b a
+   + 
+          2   2              2   2 2 2       2           2
+     12b a b a b a b a - 3b a b a b a  - 6b a b a b a b a
+   + 
+         2     2 3          5 2           4 2
+     3b a b a b a  - b a b a b  + 3b a b a b a
+   + 
+             3   2             3                  3 2 2
+     6b a b a b a b - 12b a b a b a b a + 3b a b a b a
+   + 
+               2   3            2   2             2 2 3
+     - 4b a b a b a b + 6b a b a b a b a - b a b a b a
+   + 
+      2 5         2 5 2      2 4   2      2 4
+     b a b a b - b a b a - 3b a b a b + 4b a b a b a
+   + 
+      2 4 2 2     2 3   3      2 3   2       2 3       2
+   - b a b a  + 2b a b a b - 3b a b a b a + b a b a b a
+  Type: XDistributedPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty32}
+\begin{paste}{XPBWPolynomialXmpPageEmpty32}{XPBWPolynomialXmpPagePatch32}
+\pastebutton{XPBWPolynomialXmpPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{lpd : dpoly := lpd1 * lpd2 - lpd2 * lpd1\free{dpoly lpd1 lpd2 }\bound{lpd }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch33}
+\begin{paste}{XPBWPolynomialXmpPageFull33}{XPBWPolynomialXmpPageEmpty33}
+\pastebutton{XPBWPolynomialXmpPageFull33}{\hidepaste}
+\tab{5}\spadcommand{lp :: dpoly - lpd\free{lpd dpoly lp }}
+\indentrel{3}\begin{verbatim}
+   (33)  0
+  Type: XDistributedPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty33}
+\begin{paste}{XPBWPolynomialXmpPageEmpty33}{XPBWPolynomialXmpPagePatch33}
+\pastebutton{XPBWPolynomialXmpPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{lp :: dpoly - lpd\free{lpd dpoly lp }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch34}
+\begin{paste}{XPBWPolynomialXmpPageFull34}{XPBWPolynomialXmpPageEmpty34}
+\pastebutton{XPBWPolynomialXmpPageFull34}{\hidepaste}
+\tab{5}\spadcommand{p := 3 * lp\free{lp }\bound{pp }}
+\indentrel{3}\begin{verbatim}
+            3 2 2
+   (34)  3[a b a b a b]
+          Type: XPBWPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty34}
+\begin{paste}{XPBWPolynomialXmpPageEmpty34}{XPBWPolynomialXmpPagePatch34}
+\pastebutton{XPBWPolynomialXmpPageEmpty34}{\showpaste}
+\tab{5}\spadcommand{p := 3 * lp\free{lp }\bound{pp }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch35}
+\begin{paste}{XPBWPolynomialXmpPageFull35}{XPBWPolynomialXmpPageEmpty35}
+\pastebutton{XPBWPolynomialXmpPageFull35}{\hidepaste}
+\tab{5}\spadcommand{q := lp1\free{lp1 }\bound{qq }}
+\indentrel{3}\begin{verbatim}
+           3 2
+   (35)  [a b ]
+          Type: XPBWPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty35}
+\begin{paste}{XPBWPolynomialXmpPageEmpty35}{XPBWPolynomialXmpPagePatch35}
+\pastebutton{XPBWPolynomialXmpPageEmpty35}{\showpaste}
+\tab{5}\spadcommand{q := lp1\free{lp1 }\bound{qq }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch36}
+\begin{paste}{XPBWPolynomialXmpPageFull36}{XPBWPolynomialXmpPageEmpty36}
+\pastebutton{XPBWPolynomialXmpPageFull36}{\hidepaste}
+\tab{5}\spadcommand{pq:= p * q\free{pp qq }\bound{pqpq }}
+\indentrel{3}\begin{verbatim}
+            3 2 2        3 2
+   (36)  3[a b a b a b][a b ]
+          Type: XPBWPolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty36}
+\begin{paste}{XPBWPolynomialXmpPageEmpty36}{XPBWPolynomialXmpPagePatch36}
+\pastebutton{XPBWPolynomialXmpPageEmpty36}{\showpaste}
+\tab{5}\spadcommand{pq:= p * q\free{pp qq }\bound{pqpq }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch37}
+\begin{paste}{XPBWPolynomialXmpPageFull37}{XPBWPolynomialXmpPageEmpty37}
+\pastebutton{XPBWPolynomialXmpPageFull37}{\hidepaste}
+\tab{5}\spadcommand{pr:rpoly := p :: rpoly\free{rpoly pp }\bound{pr }}
+\indentrel{3}\begin{verbatim}
+   (37)
+       a
+    *
+           a
+        *
+               a b b
+            *
+                   a
+                *
+                     a b(a b 3 + b a(- 3))
+                   + 
+                     b(a(a b(- 9) + b a 12) + b a a(- 3))
+               + 
+                 b a(a(a b 6 + b a(- 9)) + b a a 3)
+           + 
+               b
+            *
+                   a b
+                *
+                       a
+                    *
+                         a(a b b(- 3) + b b a 9)
+                       + 
+                       b(a(a b 18 + b a(- 36)) + b a a 9)
+                   + 
+                       b
+                    *
+                         a a(a b(- 12) + b a 18)
+                       + 
+                         b a a a(- 3)
+               + 
+                   b a
+                *
+                       a
+                    *
+                     (a(a b b 3 + b a b(- 9)) + b a a b 9)
+                   + 
+                       b
+                    *
+                           a
+                        *
+                             a(a b(- 6) + b a 9)
+                           + 
+                             b a a(- 9)
+                       + 
+                         b a a a 3
+       + 
+           b
+        *
+               a
+            *
+                   a b
+                *
+                       a
+                    *
+                           a
+                        *
+                             a b b 9
+                           + 
+                             b(a b(- 18) + b a(- 9))
+                       + 
+                         b(a b a 36 + b a a(- 9))
+                   + 
+                     b(a b a a(- 18) + b a a a 9)
+               + 
+                   b a
+                *
+                       a
+                    *
+                         a(a b b(- 12) + b a b 36)
+                       + 
+                         b a a b(- 36)
+                   + 
+                       b
+                    *
+                           a
+                        *
+                             a(a b 24 + b a(- 36))
+                           + 
+                             b a a 36
+                       + 
+                         b a a a(- 12)
+           + 
+               b a a
+            *
+                 a(a(a b b 3 + b a b(- 9)) + b a a b 9)
+               + 
+                   b
+                *
+                     a(a(a b(- 6) + b a 9) + b a a(- 9))
+                   + 
+                     b a a a 3
+   + 
+       b
+    *
+           a
+        *
+               a
+            *
+                   a b
+                *
+                       a
+                    *
+                         a(a b b(- 6) + b(a b 12 + b a 6))
+                       + 
+                         b(a b a(- 24) + b a a 6)
+                   + 
+                     b(a b a a 12 + b a a a(- 6))
+               + 
+                   b a
+                *
+                       a
+                    *
+                           a
+                        *
+                             a b b 9
+                           + 
+                             b(a b(- 18) + b a(- 9))
+                       + 
+                         b(a b a 36 + b a a(- 9))
+                   + 
+                     b(a b a a(- 18) + b a a a 9)
+           + 
+               b a a
+            *
+                   a
+                *
+                     a(a b b(- 3) + b b a 9)
+                   + 
+                     b(a(a b 18 + b a(- 36)) + b a a 9)
+               + 
+               b(a a(a b(- 12) + b a 18) + b a a a(- 3))
+       + 
+           b a a a
+        *
+               a
+            *
+                 a b(a b 3 + b a(- 3))
+               + 
+                 b(a(a b(- 9) + b a 12) + b a a(- 3))
+           + 
+             b a(a(a b 6 + b a(- 9)) + b a a 3)
+    Type: XRecursivePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty37}
+\begin{paste}{XPBWPolynomialXmpPageEmpty37}{XPBWPolynomialXmpPagePatch37}
+\pastebutton{XPBWPolynomialXmpPageEmpty37}{\showpaste}
+\tab{5}\spadcommand{pr:rpoly := p :: rpoly\free{rpoly pp }\bound{pr }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch38}
+\begin{paste}{XPBWPolynomialXmpPageFull38}{XPBWPolynomialXmpPageEmpty38}
+\pastebutton{XPBWPolynomialXmpPageFull38}{\hidepaste}
+\tab{5}\spadcommand{qr:rpoly := q :: rpoly\free{rpoly qq }\bound{qr }}
+\indentrel{3}\begin{verbatim}
+   (38)
+       a
+    *
+         a(a b b 1 + b(a b(- 2) + b a(- 1)))
+       + 
+         b(a b a 4 + b a a(- 1))
+   + 
+     b(a b a a(- 2) + b a a a 1)
+    Type: XRecursivePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty38}
+\begin{paste}{XPBWPolynomialXmpPageEmpty38}{XPBWPolynomialXmpPagePatch38}
+\pastebutton{XPBWPolynomialXmpPageEmpty38}{\showpaste}
+\tab{5}\spadcommand{qr:rpoly := q :: rpoly\free{rpoly qq }\bound{qr }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPagePatch39}
+\begin{paste}{XPBWPolynomialXmpPageFull39}{XPBWPolynomialXmpPageEmpty39}
+\pastebutton{XPBWPolynomialXmpPageFull39}{\hidepaste}
+\tab{5}\spadcommand{pq :: rpoly - pr*qr\free{pr qr rpoly pqpq }}
+\indentrel{3}\begin{verbatim}
+   (39)  0
+    Type: XRecursivePolynomial(Symbol,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPBWPolynomialXmpPageEmpty39}
+\begin{paste}{XPBWPolynomialXmpPageEmpty39}{XPBWPolynomialXmpPagePatch39}
+\pastebutton{XPBWPolynomialXmpPageEmpty39}{\showpaste}
+\tab{5}\spadcommand{pq :: rpoly - pr*qr\free{pr qr rpoly pqpq }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/XPOLY.ht b/src/hyper/pages/XPOLY.ht
new file mode 100644
index 00000000..7278e432
--- /dev/null
+++ b/src/hyper/pages/XPOLY.ht
@@ -0,0 +1,117 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\XPolynomialXmpTitle}{XPolynomial}
+\newcommand{\XPolynomialXmpNumber}{9.89}
+%
+% =====================================================================
+\begin{page}{XPolynomialXmpPage}{9.89 XPolynomial}
+% =====================================================================
+\beginscroll
+The \spadtype{XPolynomial} domain constructor implements multivariate polynomials
+whose set of variables is \spadtype{Symbol}.
+These variables do not commute.
+The only parameter of this construtor is 
+the coefficient ring which may be non-commutative.
+However, coefficients and variables commute.
+The representation of the polynomials is recursive.
+The abbreviation for \spadtype{XPolynomial} is \spadtype{XPOLY}.
+
+Other constructors like \spadtype{XPolynomialRing}, \spadtype{XRecursivePolynomial} 
+\spadtype{XDistributedPolynomial},
+\spadtype{LiePolynomial} and
+\spadtype{XPBWPolynomial}
+implement multivariate polynomials
+in non-commutative variables.
+
+We illustrate now some of the facilities of the \spadtype{XPOLY} domain constructor.
+
+
+\xtc{
+Define a polynomial ring over the integers.
+}{
+\spadpaste{poly := XPolynomial(Integer) \bound{poly}}
+}
+
+\xtc{
+Define a first polynomial,
+}{
+\spadpaste{pr: poly := 2*x + 3*y-5  \free{poly} \bound{pr}}
+}
+
+
+\xtc{
+and a second one.
+}{
+\spadpaste{pr2: poly := pr*pr  \free{poly} \bound{pr2}}
+}
+
+\xtc{
+Rewrite {\bf pr} in a distributive way,
+}{
+\spadpaste{pd  := expand pr   \free{pr} \bound{pd}}
+}
+
+\xtc{
+compute its square,
+}{
+\spadpaste{pd2 := pd*pd   \free{pd} \bound{pd2}}
+}
+
+\xtc{
+and checks that:
+}{
+\spadpaste{expand(pr2) - pd2 \free{pr2} \free{pd2}}
+}
+
+
+\xtc{
+We define:
+}{
+\spadpaste{qr :=  pr**3 \free{pr} \bound{qr}}
+}
+
+\xtc{
+and:
+}{
+\spadpaste{qd :=  pd**3 \free{pd} \bound{qd}}
+}
+
+\xtc{
+We truncate {\bf qd} at degree {\bf 3}:
+}{
+\spadpaste{ trunc(qd,2) \free{qd}}
+}
+
+\xtc{
+The same for {\bf qr}:
+}{
+\spadpaste{trunc(qr,2) \free{qr}}
+}
+
+\xtc{
+We define:
+}{
+\spadpaste{Word := OrderedFreeMonoid Symbol \bound{Word}}
+}
+
+\xtc{
+and:
+}{
+\spadpaste{w: Word := x*y**2 \free{Word} \bound{w}}
+}
+
+\xtc{
+The we can compute the right-quotient of {\bf qr} by {\bf r}:
+}{
+\spadpaste{rquo(qr,w) \free{qr} \free{w}}
+}
+
+\xtc{
+and the shuffle-product of {\bf pr} by {\bf r}:
+}{
+\spadpaste{sh(pr,w::poly) \free{pr} \free{w}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/XPOLY.pht b/src/hyper/pages/XPOLY.pht
new file mode 100644
index 00000000..738fc889
--- /dev/null
+++ b/src/hyper/pages/XPOLY.pht
@@ -0,0 +1,245 @@
+\begin{patch}{XPolynomialXmpPagePatch1}
+\begin{paste}{XPolynomialXmpPageFull1}{XPolynomialXmpPageEmpty1}
+\pastebutton{XPolynomialXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{poly := XPolynomial(Integer)\bound{poly }}
+\indentrel{3}\begin{verbatim}
+   (1)  XPolynomial Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty1}
+\begin{paste}{XPolynomialXmpPageEmpty1}{XPolynomialXmpPagePatch1}
+\pastebutton{XPolynomialXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{poly := XPolynomial(Integer)\bound{poly }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPagePatch2}
+\begin{paste}{XPolynomialXmpPageFull2}{XPolynomialXmpPageEmpty2}
+\pastebutton{XPolynomialXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{pr: poly := 2*x + 3*y-5\free{poly }\bound{pr }}
+\indentrel{3}\begin{verbatim}
+   (2)  - 5 + x 2 + y 3
+                              Type: XPolynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty2}
+\begin{paste}{XPolynomialXmpPageEmpty2}{XPolynomialXmpPagePatch2}
+\pastebutton{XPolynomialXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{pr: poly := 2*x + 3*y-5\free{poly }\bound{pr }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPagePatch3}
+\begin{paste}{XPolynomialXmpPageFull3}{XPolynomialXmpPageEmpty3}
+\pastebutton{XPolynomialXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{pr2: poly := pr*pr\free{poly }\bound{pr2 }}
+\indentrel{3}\begin{verbatim}
+   (3)  25 + x(- 20 + x 4 + y 6) + y(- 30 + x 6 + y 9)
+                              Type: XPolynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty3}
+\begin{paste}{XPolynomialXmpPageEmpty3}{XPolynomialXmpPagePatch3}
+\pastebutton{XPolynomialXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{pr2: poly := pr*pr\free{poly }\bound{pr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPagePatch4}
+\begin{paste}{XPolynomialXmpPageFull4}{XPolynomialXmpPageEmpty4}
+\pastebutton{XPolynomialXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{pd := expand pr\free{pr }\bound{pd }}
+\indentrel{3}\begin{verbatim}
+   (4)  - 5 + 2x + 3y
+           Type: XDistributedPolynomial(Symbol,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty4}
+\begin{paste}{XPolynomialXmpPageEmpty4}{XPolynomialXmpPagePatch4}
+\pastebutton{XPolynomialXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{pd := expand pr\free{pr }\bound{pd }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPagePatch5}
+\begin{paste}{XPolynomialXmpPageFull5}{XPolynomialXmpPageEmpty5}
+\pastebutton{XPolynomialXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{pd2 := pd*pd\free{pd }\bound{pd2 }}
+\indentrel{3}\begin{verbatim}
+                           2                   2
+   (5)  25 - 20x - 30y + 4x  + 6x y + 6y x + 9y
+           Type: XDistributedPolynomial(Symbol,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty5}
+\begin{paste}{XPolynomialXmpPageEmpty5}{XPolynomialXmpPagePatch5}
+\pastebutton{XPolynomialXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{pd2 := pd*pd\free{pd }\bound{pd2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPagePatch6}
+\begin{paste}{XPolynomialXmpPageFull6}{XPolynomialXmpPageEmpty6}
+\pastebutton{XPolynomialXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{expand(pr2) - pd2\free{pr2 }\free{pd2 }}
+\indentrel{3}\begin{verbatim}
+   (6)  0
+           Type: XDistributedPolynomial(Symbol,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty6}
+\begin{paste}{XPolynomialXmpPageEmpty6}{XPolynomialXmpPagePatch6}
+\pastebutton{XPolynomialXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{expand(pr2) - pd2\free{pr2 }\free{pd2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPagePatch7}
+\begin{paste}{XPolynomialXmpPageFull7}{XPolynomialXmpPageEmpty7}
+\pastebutton{XPolynomialXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{qr := pr**3\free{pr }\bound{qr }}
+\indentrel{3}\begin{verbatim}
+   (7)
+     - 125
+   + 
+     x(150 + x(- 60 + x 8 + y 12) + y(- 90 + x 12 + y 18))
+   + 
+   y(225 + x(- 90 + x 12 + y 18) + y(- 135 + x 18 + y 27))
+                              Type: XPolynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty7}
+\begin{paste}{XPolynomialXmpPageEmpty7}{XPolynomialXmpPagePatch7}
+\pastebutton{XPolynomialXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{qr := pr**3\free{pr }\bound{qr }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPagePatch8}
+\begin{paste}{XPolynomialXmpPageFull8}{XPolynomialXmpPageEmpty8}
+\pastebutton{XPolynomialXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{qd := pd**3\free{pd }\bound{qd }}
+\indentrel{3}\begin{verbatim}
+   (8)
+                              2                       2
+     - 125 + 150x + 225y - 60x  - 90x y - 90y x - 135y
+   + 
+       3      2                   2        2
+     8x  + 12x y + 12x y x + 18x y  + 12y x  + 18y x y
+   + 
+        2       3
+     18y x + 27y
+           Type: XDistributedPolynomial(Symbol,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty8}
+\begin{paste}{XPolynomialXmpPageEmpty8}{XPolynomialXmpPagePatch8}
+\pastebutton{XPolynomialXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{qd := pd**3\free{pd }\bound{qd }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPagePatch9}
+\begin{paste}{XPolynomialXmpPageFull9}{XPolynomialXmpPageEmpty9}
+\pastebutton{XPolynomialXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{ trunc(qd,2)\free{qd }}
+\indentrel{3}\begin{verbatim}
+   (9)
+                            2                       2
+   - 125 + 150x + 225y - 60x  - 90x y - 90y x - 135y
+           Type: XDistributedPolynomial(Symbol,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty9}
+\begin{paste}{XPolynomialXmpPageEmpty9}{XPolynomialXmpPagePatch9}
+\pastebutton{XPolynomialXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{ trunc(qd,2)\free{qd }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPagePatch10}
+\begin{paste}{XPolynomialXmpPageFull10}{XPolynomialXmpPageEmpty10}
+\pastebutton{XPolynomialXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{trunc(qr,2)\free{qr }}
+\indentrel{3}\begin{verbatim}
+   (10)
+     - 125 + x(150 + x(- 60) + y(- 90))
+   + 
+     y(225 + x(- 90) + y(- 135))
+                              Type: XPolynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty10}
+\begin{paste}{XPolynomialXmpPageEmpty10}{XPolynomialXmpPagePatch10}
+\pastebutton{XPolynomialXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{trunc(qr,2)\free{qr }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPagePatch11}
+\begin{paste}{XPolynomialXmpPageFull11}{XPolynomialXmpPageEmpty11}
+\pastebutton{XPolynomialXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{Word := OrderedFreeMonoid Symbol\bound{Word }}
+\indentrel{3}\begin{verbatim}
+   (11)  OrderedFreeMonoid Symbol
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty11}
+\begin{paste}{XPolynomialXmpPageEmpty11}{XPolynomialXmpPagePatch11}
+\pastebutton{XPolynomialXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{Word := OrderedFreeMonoid Symbol\bound{Word }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPagePatch12}
+\begin{paste}{XPolynomialXmpPageFull12}{XPolynomialXmpPageEmpty12}
+\pastebutton{XPolynomialXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{w: Word := x*y**2\free{Word }\bound{w }}
+\indentrel{3}\begin{verbatim}
+            2
+   (12)  x y
+                         Type: OrderedFreeMonoid Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty12}
+\begin{paste}{XPolynomialXmpPageEmpty12}{XPolynomialXmpPagePatch12}
+\pastebutton{XPolynomialXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{w: Word := x*y**2\free{Word }\bound{w }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPagePatch13}
+\begin{paste}{XPolynomialXmpPageFull13}{XPolynomialXmpPageEmpty13}
+\pastebutton{XPolynomialXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{rquo(qr,w)\free{qr }\free{w }}
+\indentrel{3}\begin{verbatim}
+   (13)  18
+                              Type: XPolynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty13}
+\begin{paste}{XPolynomialXmpPageEmpty13}{XPolynomialXmpPagePatch13}
+\pastebutton{XPolynomialXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{rquo(qr,w)\free{qr }\free{w }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPagePatch14}
+\begin{paste}{XPolynomialXmpPageFull14}{XPolynomialXmpPageEmpty14}
+\pastebutton{XPolynomialXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{sh(pr,w::poly)\free{pr }\free{w }}
+\indentrel{3}\begin{verbatim}
+   (14)
+   x(x y y 4 + y(x y 2 + y(- 5 + x 2 + y 9))) + y x y y 3
+                              Type: XPolynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialXmpPageEmpty14}
+\begin{paste}{XPolynomialXmpPageEmpty14}{XPolynomialXmpPagePatch14}
+\pastebutton{XPolynomialXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{sh(pr,w::poly)\free{pr }\free{w }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/XPR.ht b/src/hyper/pages/XPR.ht
new file mode 100644
index 00000000..267ae9a9
--- /dev/null
+++ b/src/hyper/pages/XPR.ht
@@ -0,0 +1,132 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\XPolynomialRingXmpTitle}{XPolynomialRing}
+\newcommand{\XPolynomialRingXmpNumber}{9.90}
+%
+% =====================================================================
+\begin{page}{XPolynomialRingXmpPage}{9.90 XPolynomialRing}
+% =====================================================================
+\beginscroll
+The \spadtype{XPolynomialRing} domain constructor implements
+generalized polynomials with coefficients from an arbitrary \spadtype{Ring}
+(not necessarily commutative) and whose exponents are
+words from an arbitrary \spadtype{OrderedMonoid}
+(not necessarily commutative too).
+Thus these polynomials are (finite) linear combinations of words.
+
+This constructor takes two arguments.
+The first one is a \spadtype{Ring}
+and the second is an \spadtype{OrderedMonoid}.
+The abbreviation for \spadtype{XPolynomialRing} is \spadtype{XPR}.
+
+Other constructors like \spadtype{XPolynomial}, \spadtype{XRecursivePolynomial} 
+\spadtype{XDistributedPolynomial},
+\spadtype{LiePolynomial} and
+\spadtype{XPBWPolynomial}
+implement multivariate polynomials
+in non-commutative variables.
+
+We illustrate now some of the facilities of the \spadtype{XPR} domain constructor.
+
+\xtc{
+Define the free ordered monoid generated by the symbols.
+}{
+\spadpaste{Word := OrderedFreeMonoid(Symbol) \bound{Word}}
+}
+
+\xtc{
+Define the linear combinations of these words with integer coefficients.
+}{
+\spadpaste{poly:= XPR(Integer,Word) \free{Word} \bound{poly}}
+}
+
+
+\xtc{
+Then we define a first element from {\bf poly}.
+}{
+\spadpaste{p:poly := 2 * x - 3 * y + 1 \free{poly} \bound{p}}
+}
+
+\xtc{
+And a second one.
+}{
+\spadpaste{q:poly := 2 * x + 1 \free{poly} \bound{q}}
+}
+
+
+\xtc{
+We compute their sum,
+}{
+\spadpaste{p + q  \free{p}\free{q} }
+}
+
+\xtc{
+their product,
+}{
+\spadpaste{p * q  \free{p}\free{q} }
+}
+
+\xtc{
+and see that variables do not commute.
+}{
+\spadpaste{(p +q)^2 -p^2 -q^2 - 2*p*q  \free{p}\free{q} }
+}
+
+
+
+\xtc{
+Now we define a ring of square matrices,
+}{
+\spadpaste{M := SquareMatrix(2,Fraction Integer)  \bound{M}}
+}
+
+\xtc{
+and the linear combinations of words with these  matrices as coefficients.
+}{
+\spadpaste{poly1:= XPR(M,Word) \free{Word} \free{M} \bound{poly1}}
+}
+
+
+\xtc{
+Define a first matrix,
+}{
+\spadpaste{m1:M := matrix [[i*j**2 for i in 1..2] for j in 1..2] \free{M} \bound{m1}}
+}
+
+\xtc{
+a second one,
+}{
+\spadpaste{m2:M := m1 - 5/4 \free{M} \free{m1} \bound{m2}}
+}
+
+\xtc{
+and a third one.
+}{
+\spadpaste{m3: M := m2**2 \free{M} \free{m2} \bound{m3}}
+}
+
+\xtc{
+Define a polynomial,
+}{
+\spadpaste{pm:poly1   := m1*x + m2*y + m3*z - 2/3 \free{poly1} \free{m1} \free{m2} \free{m3} \bound{pm}}
+}
+
+
+\xtc{
+a second one,
+}{
+\spadpaste{qm:poly1 := pm - m1*x \free{m1} \free{pm} \bound{qm}}
+}
+
+\xtc{
+and the following power.
+}{
+\spadpaste{qm**3 \bound{qm}}
+}
+
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/XPR.pht b/src/hyper/pages/XPR.pht
new file mode 100644
index 00000000..0d97a8f4
--- /dev/null
+++ b/src/hyper/pages/XPR.pht
@@ -0,0 +1,325 @@
+\begin{patch}{XPolynomialRingXmpPagePatch1}
+\begin{paste}{XPolynomialRingXmpPageFull1}{XPolynomialRingXmpPageEmpty1}
+\pastebutton{XPolynomialRingXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{Word := OrderedFreeMonoid(Symbol)\bound{Word }}
+\indentrel{3}\begin{verbatim}
+   (1)  OrderedFreeMonoid Symbol
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty1}
+\begin{paste}{XPolynomialRingXmpPageEmpty1}{XPolynomialRingXmpPagePatch1}
+\pastebutton{XPolynomialRingXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{Word := OrderedFreeMonoid(Symbol)\bound{Word }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch2}
+\begin{paste}{XPolynomialRingXmpPageFull2}{XPolynomialRingXmpPageEmpty2}
+\pastebutton{XPolynomialRingXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{poly:= XPR(Integer,Word)\free{Word }\bound{poly }}
+\indentrel{3}\begin{verbatim}
+   (2)
+   XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty2}
+\begin{paste}{XPolynomialRingXmpPageEmpty2}{XPolynomialRingXmpPagePatch2}
+\pastebutton{XPolynomialRingXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{poly:= XPR(Integer,Word)\free{Word }\bound{poly }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch3}
+\begin{paste}{XPolynomialRingXmpPageFull3}{XPolynomialRingXmpPageEmpty3}
+\pastebutton{XPolynomialRingXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{p:poly := 2 * x - 3 * y + 1\free{poly }\bound{p }}
+\indentrel{3}\begin{verbatim}
+   (3)  1 + 2x - 3y
+Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty3}
+\begin{paste}{XPolynomialRingXmpPageEmpty3}{XPolynomialRingXmpPagePatch3}
+\pastebutton{XPolynomialRingXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{p:poly := 2 * x - 3 * y + 1\free{poly }\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch4}
+\begin{paste}{XPolynomialRingXmpPageFull4}{XPolynomialRingXmpPageEmpty4}
+\pastebutton{XPolynomialRingXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{q:poly := 2 * x + 1\free{poly }\bound{q }}
+\indentrel{3}\begin{verbatim}
+   (4)  1 + 2x
+Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty4}
+\begin{paste}{XPolynomialRingXmpPageEmpty4}{XPolynomialRingXmpPagePatch4}
+\pastebutton{XPolynomialRingXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{q:poly := 2 * x + 1\free{poly }\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch5}
+\begin{paste}{XPolynomialRingXmpPageFull5}{XPolynomialRingXmpPageEmpty5}
+\pastebutton{XPolynomialRingXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{p + q\free{p }\free{q }}
+\indentrel{3}\begin{verbatim}
+   (5)  2 + 4x - 3y
+Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty5}
+\begin{paste}{XPolynomialRingXmpPageEmpty5}{XPolynomialRingXmpPagePatch5}
+\pastebutton{XPolynomialRingXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{p + q\free{p }\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch6}
+\begin{paste}{XPolynomialRingXmpPageFull6}{XPolynomialRingXmpPageEmpty6}
+\pastebutton{XPolynomialRingXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{p * q\free{p }\free{q }}
+\indentrel{3}\begin{verbatim}
+                        2
+   (6)  1 + 4x - 3y + 4x  - 6y x
+Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty6}
+\begin{paste}{XPolynomialRingXmpPageEmpty6}{XPolynomialRingXmpPagePatch6}
+\pastebutton{XPolynomialRingXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{p * q\free{p }\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch7}
+\begin{paste}{XPolynomialRingXmpPageFull7}{XPolynomialRingXmpPageEmpty7}
+\pastebutton{XPolynomialRingXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{(p +q)^2 -p^2 -q^2 - 2*p*q\free{p }\free{q }}
+\indentrel{3}\begin{verbatim}
+   (7)  - 6x y + 6y x
+Type: XPolynomialRing(Integer,OrderedFreeMonoid Symbol)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty7}
+\begin{paste}{XPolynomialRingXmpPageEmpty7}{XPolynomialRingXmpPagePatch7}
+\pastebutton{XPolynomialRingXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{(p +q)^2 -p^2 -q^2 - 2*p*q\free{p }\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch8}
+\begin{paste}{XPolynomialRingXmpPageFull8}{XPolynomialRingXmpPageEmpty8}
+\pastebutton{XPolynomialRingXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{M := SquareMatrix(2,Fraction Integer)\bound{M }}
+\indentrel{3}\begin{verbatim}
+   (8)  SquareMatrix(2,Fraction Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty8}
+\begin{paste}{XPolynomialRingXmpPageEmpty8}{XPolynomialRingXmpPagePatch8}
+\pastebutton{XPolynomialRingXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{M := SquareMatrix(2,Fraction Integer)\bound{M }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch9}
+\begin{paste}{XPolynomialRingXmpPageFull9}{XPolynomialRingXmpPageEmpty9}
+\pastebutton{XPolynomialRingXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{poly1:= XPR(M,Word)\free{Word }\free{M }\bound{poly1 }}
+\indentrel{3}\begin{verbatim}
+   (9)
+  XPolynomialRing(SquareMatrix(2,Fraction Integer),Ordere
+  dFreeMonoid Symbol)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty9}
+\begin{paste}{XPolynomialRingXmpPageEmpty9}{XPolynomialRingXmpPagePatch9}
+\pastebutton{XPolynomialRingXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{poly1:= XPR(M,Word)\free{Word }\free{M }\bound{poly1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch10}
+\begin{paste}{XPolynomialRingXmpPageFull10}{XPolynomialRingXmpPageEmpty10}
+\pastebutton{XPolynomialRingXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{m1:M := matrix [[i*j**2 for i in 1..2] for j in 1..2]\free{M }\bound{m1 }}
+\indentrel{3}\begin{verbatim}
+         Ú1  2¿
+   (10)  ³    ³
+         À4  8Ù
+                 Type: SquareMatrix(2,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty10}
+\begin{paste}{XPolynomialRingXmpPageEmpty10}{XPolynomialRingXmpPagePatch10}
+\pastebutton{XPolynomialRingXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{m1:M := matrix [[i*j**2 for i in 1..2] for j in 1..2]\free{M }\bound{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch11}
+\begin{paste}{XPolynomialRingXmpPageFull11}{XPolynomialRingXmpPageEmpty11}
+\pastebutton{XPolynomialRingXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{m2:M := m1 - 5/4\free{M }\free{m1 }\bound{m2 }}
+\indentrel{3}\begin{verbatim}
+         Ú  1    ¿
+         ³- Ä  2 ³
+         ³  4    ³
+   (11)  ³       ³
+         ³     27³
+         ³ 4   Äij
+         À      4Ù
+                 Type: SquareMatrix(2,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty11}
+\begin{paste}{XPolynomialRingXmpPageEmpty11}{XPolynomialRingXmpPagePatch11}
+\pastebutton{XPolynomialRingXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{m2:M := m1 - 5/4\free{M }\free{m1 }\bound{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch12}
+\begin{paste}{XPolynomialRingXmpPageFull12}{XPolynomialRingXmpPageEmpty12}
+\pastebutton{XPolynomialRingXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{m3: M := m2**2\free{M }\free{m2 }\bound{m3 }}
+\indentrel{3}\begin{verbatim}
+         Ú129     ¿
+         ³ÄÄÄ  13 ³
+         ³ 16     ³
+   (12)  ³        ³
+         ³     857³
+         ³26   ÄÄij
+         À      16Ù
+                 Type: SquareMatrix(2,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty12}
+\begin{paste}{XPolynomialRingXmpPageEmpty12}{XPolynomialRingXmpPagePatch12}
+\pastebutton{XPolynomialRingXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{m3: M := m2**2\free{M }\free{m2 }\bound{m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch13}
+\begin{paste}{XPolynomialRingXmpPageFull13}{XPolynomialRingXmpPageEmpty13}
+\pastebutton{XPolynomialRingXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{pm:poly1 := m1*x + m2*y + m3*z - 2/3\free{poly1 }\free{m1 }\free{m2 }\free{m3 }\bound{pm }}
+\indentrel{3}\begin{verbatim}
+         Ú  2     ¿             Ú  1    ¿    Ú129     ¿
+         ³- Ä   0 ³             ³- Ä  2 ³    ³ÄÄÄ  13 ³
+         ³  3     ³   Ú1  2¿    ³  4    ³    ³ 16     ³
+   (13)  ³        ³ + ³    ³x + ³       ³y + ³        ³z
+         ³       2³   À4  8Ù    ³     27³    ³     857³
+         ³ 0   - ij             ³ 4   Äij    ³26   ÄÄij
+         À       3Ù             À      4Ù    À      16Ù
+Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty13}
+\begin{paste}{XPolynomialRingXmpPageEmpty13}{XPolynomialRingXmpPagePatch13}
+\pastebutton{XPolynomialRingXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{pm:poly1 := m1*x + m2*y + m3*z - 2/3\free{poly1 }\free{m1 }\free{m2 }\free{m3 }\bound{pm }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch14}
+\begin{paste}{XPolynomialRingXmpPageFull14}{XPolynomialRingXmpPageEmpty14}
+\pastebutton{XPolynomialRingXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{qm:poly1 := pm - m1*x\free{m1 }\free{pm }\bound{qm }}
+\indentrel{3}\begin{verbatim}
+         Ú  2     ¿   Ú  1    ¿    Ú129     ¿
+         ³- Ä   0 ³   ³- Ä  2 ³    ³ÄÄÄ  13 ³
+         ³  3     ³   ³  4    ³    ³ 16     ³
+   (14)  ³        ³ + ³       ³y + ³        ³z
+         ³       2³   ³     27³    ³     857³
+         ³ 0   - ij   ³ 4   Äij    ³26   ÄÄij
+         À       3Ù   À      4Ù    À      16Ù
+Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty14}
+\begin{paste}{XPolynomialRingXmpPageEmpty14}{XPolynomialRingXmpPagePatch14}
+\pastebutton{XPolynomialRingXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{qm:poly1 := pm - m1*x\free{m1 }\free{pm }\bound{qm }}
+\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPagePatch15}
+\begin{paste}{XPolynomialRingXmpPageFull15}{XPolynomialRingXmpPageEmpty15}
+\pastebutton{XPolynomialRingXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{qm**3\bound{qm }}
+\indentrel{3}\begin{verbatim}
+   (15)
+     Ú   8      ¿   Ú  1  8¿    Ú43   52 ¿
+     ³- ÄÄ   0  ³   ³- Ä  ij    ³ÄÄ   ÄÄ ³
+     ³  27      ³   ³  3  3³    ³ 4    3 ³
+     ³          ³ + ³      ³y + ³        ³z
+     ³         8³   ³16    ³    ³104  857³
+     ³ 0    - Äij   ³ÄÄ   9³    ³ÄÄÄ  ÄÄij
+     À        27Ù   À 3    Ù    À 3    12Ù
+   + 
+     Ú  129       ¿     Ú  3199     831 ¿
+     ³- ÄÄÄ  - 26 ³     ³- ÄÄÄÄ   - ÄÄÄ ³
+     ³   8        ³ 2   ³   32       4  ³
+     ³            ³y  + ³               ³y z
+     ³         857³     ³  831     26467³
+     ³- 52   - ÄÄij     ³- ÄÄÄ   - ÄÄÄÄij
+     À          8 Ù     À   2        32 Ù
+   + 
+     Ú  3199     831 ¿      Ú  103169     6409 ¿
+     ³- ÄÄÄÄ   - ÄÄÄ ³      ³- ÄÄÄÄÄÄ   - ÄÄÄÄ ³
+     ³   32       4  ³      ³    128        4  ³ 2
+     ³               ³z y + ³                  ³z
+     ³  831     26467³      ³   6409     820977³
+     ³- ÄÄÄ   - ÄÄÄÄij      ³ - ÄÄÄÄ   - ÄÄÄÄÄij
+     À   2        32 Ù      À     2        128 Ù
+   + 
+     Ú3199   831 ¿     Ú103169   6409 ¿
+     ³ÄÄÄÄ   ÄÄÄ ³     ³ÄÄÄÄÄÄ   ÄÄÄÄ ³
+     ³ 64     8  ³ 3   ³  256      8  ³ 2
+     ³           ³y  + ³              ³y z
+     ³831   26467³     ³ 6409   820977³
+     ³ÄÄÄ   ÄÄÄÄij     ³ ÄÄÄÄ   ÄÄÄÄÄij
+     À 4      64 Ù     À   4      256 Ù
+   + 
+     Ú103169   6409 ¿        Ú3178239   795341 ¿
+     ³ÄÄÄÄÄÄ   ÄÄÄÄ ³        ³ÄÄÄÄÄÄÄ   ÄÄÄÄÄÄ ³
+     ³  256      8  ³        ³  1024      128  ³   2
+     ³              ³y z y + ³                 ³y z
+     ³ 6409   820977³        ³795341   25447787³
+     ³ ÄÄÄÄ   ÄÄÄÄÄij        ³ÄÄÄÄÄÄ   ÄÄÄÄÄÄÄij
+     À   4      256 Ù        À  64       1024  Ù
+   + 
+     Ú103169   6409 ¿       Ú3178239   795341 ¿
+     ³ÄÄÄÄÄÄ   ÄÄÄÄ ³       ³ÄÄÄÄÄÄÄ   ÄÄÄÄÄÄ ³
+     ³  256      8  ³   2   ³  1024      128  ³
+     ³              ³z y  + ³                 ³z y z
+     ³ 6409   820977³       ³795341   25447787³
+     ³ ÄÄÄÄ   ÄÄÄÄÄij       ³ÄÄÄÄÄÄ   ÄÄÄÄÄÄÄij
+     À   4      256 Ù       À  64       1024  Ù
+   + 
+     Ú3178239   795341 ¿      Ú98625409  12326223 ¿
+     ³ÄÄÄÄÄÄÄ   ÄÄÄÄÄÄ ³      ³ÄÄÄÄÄÄÄÄ  ÄÄÄÄÄÄÄÄ ³
+     ³  1024      128  ³ 2    ³  4096       256   ³ 3
+     ³                 ³z y + ³                   ³z
+     ³795341   25447787³      ³12326223  788893897³
+     ³ÄÄÄÄÄÄ   ÄÄÄÄÄÄÄij      ³ÄÄÄÄÄÄÄÄ  ÄÄÄÄÄÄÄÄij
+     À  64       1024  Ù      À   128       4096  Ù
+Type: XPolynomialRing(SquareMatrix(2,Fraction Integer),OrderedFreeMonoid Symbol)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{XPolynomialRingXmpPageEmpty15}
+\begin{paste}{XPolynomialRingXmpPageEmpty15}{XPolynomialRingXmpPagePatch15}
+\pastebutton{XPolynomialRingXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{qm**3\bound{qm }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ZDSOLVE.ht b/src/hyper/pages/ZDSOLVE.ht
new file mode 100644
index 00000000..fb7ece3f
--- /dev/null
+++ b/src/hyper/pages/ZDSOLVE.ht
@@ -0,0 +1,269 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\ZeroDimensionalSolvePackageXmpTitle}{ZeroDimensionalSolvePackage}
+\newcommand{\ZeroDimensionalSolvePackageXmpNumber}{9.91}
+%
+% =====================================================================
+\begin{page}{ZeroDimensionalSolvePackageXmpPage}{9.91 ZeroDimensionalSolvePackage}
+% =====================================================================
+\beginscroll
+The \spadtype{ZeroDimensionalSolvePackage} package constructor 
+provides operations for computing symbolically the complex or real roots of 
+zero-dimensional algebraic systems.
+
+The package provides {\bf no} multiplicity information (i.e. some returned
+roots may be double or higher) but only distinct roots are returned. 
+
+Complex roots are given by means of univariate representations
+of irreducible regular chains. 
+These representations are computed by the \axiomOpFrom{univariateSolve}{ZeroDimensionalSolvePackage}
+operation (by calling the \spadtype{InternalRationalUnivariateRepresentationPackage} 
+package constructor which does the job).
+
+Real roots are given by means of tuples
+of coordinates lying in the \spadtype{RealClosure} of the coefficient ring.
+They are computed by the \axiomOpFrom{realSolve}{ZeroDimensionalSolvePackage}
+and \axiomOpFrom{positiveSolve}{ZeroDimensionalSolvePackage} operations.
+The former computes all the solutions of the input system with real coordinates
+whereas the later concentrate on the solutions with (strictly) positive coordinates.
+In both cases, the computations are performed by the \spadtype{RealClosure} constructor.
+
+Both computations of complex roots and real roots rely on triangular decompositions.
+These decompositions can be computed in two different ways.
+First, by a applying the \axiomOpFrom{zeroSetSplit}{RegularTriangularSet} 
+operation from the \spadtype{REGSET} domain constructor.
+In that case, no Groebner bases are computed.
+This strategy is used by default.
+Secondly, by applying the \axiomOpFrom{zeroSetSplit}{LexTriangularPackage}
+from \spadtype{LEXTRIPK}.
+To use this later strategy with the operations
+\axiomOpFrom{univariateSolve}{ZeroDimensionalSolvePackage},
+\axiomOpFrom{realSolve}{ZeroDimensionalSolvePackage}
+and \axiomOpFrom{positiveSolve}{ZeroDimensionalSolvePackage} 
+one just needs to use an extra boolean argument.
+
+Note that the way of understanding triangular decompositions 
+is detailed in the example of the \spadtype{RegularTriangularSet}
+constructor.
+
+The \spadtype{ZeroDimensionalSolvePackage} constructor takes three arguments. 
+The first one {\bf R} is the coefficient ring;
+it must belong to the categories
+\spadtype{OrderedRing}, \spadtype{EuclideanDomain},  \spadtype{CharacteristicZero}
+and \spadtype{RealConstant}.
+This means essentially that {\bf R} is \spadtype{Integer} or \spadtype{Fraction(Integer)}.
+The second argument {\bf ls} is the list of variables involved 
+in the systems to solve. 
+The third one MUST BE {\bf concat(ls,s)} where
+{\bf s} is an additional symbol used for the univariate representations.
+The abbreviation for \spadtype{ZeroDimensionalSolvePackage} is \spadtype{ZDSOLVE}.
+
+We illustrate now how to use the constructor \spadtype{ZDSOLVE}
+by two examples: the {\em Arnborg and Lazard} system and the {\em L-3} system (Aubry and Moreno Maza).
+Note that the use of this package is also demonstrated in the example
+of the \spadtype{LexTriangularPackage} constructor.
+
+\xtc{
+Define the coefficient ring.
+}{
+\spadpaste{R := Integer \bound{R}}
+}
+
+\xtc{
+Define the lists of variables:
+}{
+\spadpaste{ls : List Symbol := [x,y,z,t] \bound{ls}}
+}
+
+\xtc{
+and:
+}{
+\spadpaste{ls2 : List Symbol := [x,y,z,t,new()$Symbol] \bound{ls2}}
+}
+
+\xtc{
+Call the package:
+}{
+\spadpaste{pack := ZDSOLVE(R,ls,ls2) \free{ls} \free{ls2} \free{R} \bound{pack}}
+}
+
+\xtc{
+Define a polynomial system (Arnborg-Lazard)
+}{
+\spadpaste{p1 := x**2*y*z + x*y**2*z + x*y*z**2 + x*y*z + x*y + x*z + y*z \bound{p1}}
+}
+\xtc{
+}{
+\spadpaste{p2 := x**2*y**2*z + x*y**2*z**2 + x**2*y*z + x*y*z + y*z + x + z \bound{p2}}
+}
+\xtc{
+}{
+\spadpaste{p3 := x**2*y**2*z**2 + x**2*y**2*z + x*y**2*z + x*y*z + x*z + z + 1 \bound{p3}}
+}
+\xtc{
+}{
+\spadpaste{lp := [p1, p2, p3] \free{p1} \free{p2} \free{p3} \bound{lp}}
+}
+Note that these polynomials do not involve the variable {\bf t};
+we will use it in the second example.
+
+\xtc{
+First compute a decomposition into regular chains (i.e. regular triangular sets).
+}{
+\spadpaste{triangSolve(lp)$pack \free{lp} \free{pack}}
+}
+
+We can see easily from this decomposition (consisting of a single
+regular chain) that the input system has 20 complex roots.
+
+\xtc{
+Then we compute a univariate representation of this regular chain.
+}{
+\spadpaste{univariateSolve(lp)$pack \free{lp} \free{pack}}
+}
+
+We see that the zeros of our regular chain are split into three components.
+This is due to the use of univariate polynomial factorization.
+
+Each of these components consist of two parts.
+The first one is an irreducible univariate polynomial {\bf p(?)} which defines
+a simple algebraic extension of the field of fractions of {\bf R}.
+The second one consists of multivariate polynomials {\bf pol1(x,\%A)},
+{\bf pol2(y,\%A)} and {\bf pol3(z,\%A)}.
+Each of these polynomials involve two variables: one is an indeterminate
+{\bf x}, {\bf y} or {\bf z}
+of the input system {\bf lp} and the other is {\bf \%A} which represents any root of {\bf p(?)}.
+Recall that this {\bf \%A} is the last element of the third parameter of 
+\spadtype{ZDSOLVE}.
+Thus any complex root {\bf ?} of {\bf p(?)} leads to a solution of the input system {\bf lp} 
+by replacing {\bf \%A} by this {\bf ?} in {\bf pol1(x,\%A)},
+{\bf pol2(y,\%A)} and {\bf pol3(z,\%A)}.
+Note that the polynomials {\bf pol1(x,\%A)},
+{\bf pol2(y,\%A)} and {\bf pol3(z,\%A)} have degree one 
+w.r.t. {\bf x}, {\bf y} or {\bf z} respectively.
+This is always the case for all univariate representations.
+Hence the operation {\bf univariateSolve} replaces a
+system of multivariate polynomials by a list of univariate
+polynomials, what justifies its name.
+Another example of univariate representations illustrates
+the \spadtype{LexTriangularPackage} package constructor.
+
+\xtc{
+We now compute the solutions with real coordinates:
+}{
+\spadpaste{lr := realSolve(lp)$pack \free{lp} \free{pack} \bound{lr}}
+}
+
+\xtc{
+The number of real solutions for the input system is:
+}{
+\spadpaste{\# lr \free{lr}}
+}
+
+Each of these real solutions is given by a list of elements in  \spadtype{RealClosure(R)}.
+In these 8 lists, the first element is a value of {\bf z}, 
+the second of {\bf y} and the last of {\bf x}.
+This is logical since by setting the list of variables of the package
+to {\bf [x,y,z,t]} we mean that the elimination ordering on the
+variables is {\bf t < z < y < x }.
+Note that each system treated by the \spadtype{ZDSOLVE} package constructor
+needs only to be zero-dimensional w.r.t. the variables involved in the system it-self
+and not necessarily w.r.t. all the variables used to define the package.
+
+\xtc{
+We can approximate these real numbers as follows. 
+This computation takes between 30 sec. and 5 min, depending on your machine.
+}{
+\spadpaste{[[approximate(r,1/1000000) for r in point] for point in lr] \free{lr}}
+}
+
+\xtc{
+We can also concentrate on the solutions with real (strictly) positive coordinates:
+}{
+\spadpaste{lpr := positiveSolve(lp)$pack \free{lp} \free{pack} \bound{lpr}}
+}
+
+Thus we have checked that the input system has no solution with strictly positive coordinates.
+
+
+\xtc{
+Let us define another polynomial system ({\em L-3}).
+}{
+\spadpaste{f0 := x**3 + y + z + t- 1 \bound{f0}}
+}
+\xtc{
+}{
+\spadpaste{f1 := x + y**3 + z + t -1 \bound{f1}}
+}
+\xtc{
+}{
+\spadpaste{f2 := x + y + z**3 + t-1 \bound{f2}}
+}
+\xtc{
+}{
+\spadpaste{f3 := x + y + z + t**3 -1 \bound{f3}}
+}
+\xtc{
+}{
+\spadpaste{lf := [f0, f1, f2, f3] \free{f0} \free{f1} \free{f2} \free{f3} \bound{lf}}
+}
+
+
+\xtc{
+First compute a decomposition into regular chains (i.e. regular triangular sets).
+}{
+\spadpaste{lts := triangSolve(lf)$pack \free{lf} \free{pack} \bound{lts}}
+}
+
+
+\xtc{
+Then we compute a univariate representation.
+}{
+\spadpaste{univariateSolve(lf)$pack \free{lf} \free{pack}}
+}
+
+Note that this computation is made from the input system {\bf lf}.
+\xtc{
+However it is possible to reuse a pre-computed regular chain as follows:
+}{
+\spadpaste{ts := lts.1 \free{lts} \bound{ts}}
+}
+\xtc{
+}{
+\spadpaste{univariateSolve(ts)$pack \free{ts} \free{pack}}
+}
+\xtc{
+}{
+\spadpaste{realSolve(ts)$pack  \free{ts} \free{pack}}
+}
+
+\xtc{
+We compute now the full set of points with real coordinates:
+}{
+\spadpaste{lr2 := realSolve(lf)$pack \free{lf} \free{pack}  \bound{lr2}}
+}
+
+
+\xtc{
+The number of real solutions for the input system is:
+}{
+\spadpaste{\#lr2 \free{lr2}}
+}
+Another example of computation of real solutions illustrates
+the \spadtype{LexTriangularPackage} package constructor.
+
+\xtc{
+We concentrate now on the solutions with real (strictly) positive coordinates:
+}{
+\spadpaste{lpr2 := positiveSolve(lf)$pack \free{lf} \free{pack} \bound{lpr2}}
+}
+
+\xtc{
+Finally, we approximate the coordinates of this point with 20 exact digits:
+}{
+\spadpaste{[approximate(r,1/10**21)::Float for r in lpr2.1] \free{lpr2}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ZDSOLVE.pht b/src/hyper/pages/ZDSOLVE.pht
new file mode 100644
index 00000000..fccef013
--- /dev/null
+++ b/src/hyper/pages/ZDSOLVE.pht
@@ -0,0 +1,2274 @@
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch1}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull1}{ZeroDimensionalSolvePackageXmpPageEmpty1}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{R := Integer\bound{R }}
+\indentrel{3}\begin{verbatim}
+   (1)  Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty1}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty1}{ZeroDimensionalSolvePackageXmpPagePatch1}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{R := Integer\bound{R }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch2}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull2}{ZeroDimensionalSolvePackageXmpPageEmpty2}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{ls : List Symbol := [x,y,z,t]\bound{ls }}
+\indentrel{3}\begin{verbatim}
+   (2)  [x,y,z,t]
+                                      Type: List Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty2}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty2}{ZeroDimensionalSolvePackageXmpPagePatch2}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{ls : List Symbol := [x,y,z,t]\bound{ls }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch3}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull3}{ZeroDimensionalSolvePackageXmpPageEmpty3}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{ls2 : List Symbol := [x,y,z,t,new()$Symbol]\bound{ls2 }}
+\indentrel{3}\begin{verbatim}
+   (3)  [x,y,z,t,%A]
+                                      Type: List Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty3}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty3}{ZeroDimensionalSolvePackageXmpPagePatch3}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{ls2 : List Symbol := [x,y,z,t,new()$Symbol]\bound{ls2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch4}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull4}{ZeroDimensionalSolvePackageXmpPageEmpty4}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{pack := ZDSOLVE(R,ls,ls2)\free{ls }\free{ls2 }\free{R }\bound{pack }}
+\indentrel{3}\begin{verbatim}
+   (4)
+  ZeroDimensionalSolvePackage(Integer,[x,y,z,t],[x,y,z,t,
+  %A])
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty4}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty4}{ZeroDimensionalSolvePackageXmpPagePatch4}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{pack := ZDSOLVE(R,ls,ls2)\free{ls }\free{ls2 }\free{R }\bound{pack }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch5}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull5}{ZeroDimensionalSolvePackageXmpPageEmpty5}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{p1 := x**2*y*z + x*y**2*z + x*y*z**2 + x*y*z + x*y + x*z + y*z\bound{p1 }}
+\indentrel{3}\begin{verbatim}
+             2       2     2
+   (5)  x y z  + (x y  + (x  + x + 1)y + x)z + x y
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty5}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty5}{ZeroDimensionalSolvePackageXmpPagePatch5}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{p1 := x**2*y*z + x*y**2*z + x*y*z**2 + x*y*z + x*y + x*z + y*z\bound{p1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch6}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull6}{ZeroDimensionalSolvePackageXmpPageEmpty6}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{p2 := x**2*y**2*z + x*y**2*z**2 + x**2*y*z + x*y*z + y*z + x + z\bound{p2 }}
+\indentrel{3}\begin{verbatim}
+           2 2     2 2     2
+   (6)  x y z  + (x y  + (x  + x + 1)y + 1)z + x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty6}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty6}{ZeroDimensionalSolvePackageXmpPagePatch6}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{p2 := x**2*y**2*z + x*y**2*z**2 + x**2*y*z + x*y*z + y*z + x + z\bound{p2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch7}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull7}{ZeroDimensionalSolvePackageXmpPageEmpty7}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{p3 := x**2*y**2*z**2 + x**2*y**2*z + x*y**2*z + x*y*z + x*z + z + 1\bound{p3 }}
+\indentrel{3}\begin{verbatim}
+         2 2 2      2      2
+   (7)  x y z  + ((x  + x)y  + x y + x + 1)z + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty7}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty7}{ZeroDimensionalSolvePackageXmpPagePatch7}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{p3 := x**2*y**2*z**2 + x**2*y**2*z + x*y**2*z + x*y*z + x*z + z + 1\bound{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch8}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull8}{ZeroDimensionalSolvePackageXmpPageEmpty8}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
+\indentrel{3}\begin{verbatim}
+   (8)
+         2       2     2
+   [x y z  + (x y  + (x  + x + 1)y + x)z + x y,
+       2 2     2 2     2
+    x y z  + (x y  + (x  + x + 1)y + 1)z + x,
+     2 2 2      2      2
+    x y z  + ((x  + x)y  + x y + x + 1)z + 1]
+                          Type: List Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty8}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty8}{ZeroDimensionalSolvePackageXmpPagePatch8}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{lp := [p1, p2, p3]\free{p1 }\free{p2 }\free{p3 }\bound{lp }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch9}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull9}{ZeroDimensionalSolvePackageXmpPageEmpty9}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull9}{\hidepaste}
+\tab{5}\spadcommand{triangSolve(lp)$pack\free{lp }\free{pack }}
+\indentrel{3}\begin{verbatim}
+   (9)
+   [
+     {
+          20     19      18      17       16      15
+         z   - 6z   - 41z   + 71z   + 106z   + 92z
+       + 
+             14       13       12       11       10
+         197z   + 145z   + 257z   + 278z   + 201z
+       + 
+             9       8       7       6      5       4
+         278z  + 257z  + 145z  + 197z  + 92z  + 106z
+       + 
+            3      2
+         71z  - 41z  - 6z + 1
+       ,
+
+                      19            18             17
+             14745844z   + 50357474z   - 130948857z
+           + 
+                         16             15             14
+             - 185261586z   - 180077775z   - 338007307z
+           + 
+                         13             12             11
+             - 275379623z   - 453190404z   - 474597456z
+           + 
+                         10             9             8
+             - 366147695z   - 481433567z  - 430613166z
+           + 
+                         7             6             5
+             - 261878358z  - 326073537z  - 163008796z
+           + 
+                         4             3            2
+             - 177213227z  - 104356755z  + 65241699z
+           + 
+             9237732z - 1567348
+        *
+           y
+       + 
+                 19           18            17
+         1917314z   + 6508991z   - 16973165z
+       + 
+                    16            15            14
+         - 24000259z   - 23349192z   - 43786426z
+       + 
+                    13            12            11
+         - 35696474z   - 58724172z   - 61480792z
+       + 
+                    10            9            8
+         - 47452440z   - 62378085z  - 55776527z
+       + 
+                    7            6            5
+         - 33940618z  - 42233406z  - 21122875z
+       + 
+                    4            3           2
+         - 22958177z  - 13504569z  + 8448317z  + 1195888z
+       + 
+         - 202934
+       ,
+
+               3       2       3    2               2
+             (z  - 2z)y  + (- z  - z  - 2z - 1)y - z  - z
+           + 
+             1
+        *
+           x
+       + 
+          2
+         z  - 1
+       }
+     ]
+             Type: List RegularChain(Integer,[x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty9}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty9}{ZeroDimensionalSolvePackageXmpPagePatch9}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{triangSolve(lp)$pack\free{lp }\free{pack }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch10}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull10}{ZeroDimensionalSolvePackageXmpPageEmpty10}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull10}{\hidepaste}
+\tab{5}\spadcommand{univariateSolve(lp)$pack\free{lp }\free{pack }}
+\indentrel{3}\begin{verbatim}
+   (10)
+   [
+     [
+       complexRoots =
+            12      11      10     9     8      7      6
+           ?   - 12?   + 24?   + 4?  - 9?  + 27?  - 21?
+         + 
+              5     4     3      2
+           27?  - 9?  + 4?  + 24?  - 12? + 1
+       ,
+
+       coordinates =
+         [
+                       11        10         9        8
+             63x + 62%A   - 721%A   + 1220%A  + 705%A
+           + 
+                    7         6        5         4       3
+             - 285%A  + 1512%A  - 735%A  + 1401%A  - 21%A
+           + 
+                  2
+             215%A  + 1577%A - 142
+           ,
+
+                       11        10         9        8
+             63y - 75%A   + 890%A   - 1682%A  - 516%A
+           + 
+                  7         6         5         4        3
+             588%A  - 1953%A  + 1323%A  - 1815%A  + 426%A
+           + 
+                    2
+             - 243%A  - 1801%A + 679
+           ,
+          z - %A]
+       ]
+     ,
+
+                     6    5    4    3    2
+     [complexRoots= ?  + ?  + ?  + ?  + ?  + ? + 1,
+                          5       3
+      coordinates= [x - %A ,y - %A ,z - %A]]
+     ,
+
+                     2
+     [complexRoots= ?  + 5? + 1,
+      coordinates= [x - 1,y - 1,z - %A]]
+     ]
+Type: List Record(complexRoots: SparseUnivariatePolynomial Integer,coordinates: List Polynomial Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty10}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty10}{ZeroDimensionalSolvePackageXmpPagePatch10}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{univariateSolve(lp)$pack\free{lp }\free{pack }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch11}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull11}{ZeroDimensionalSolvePackageXmpPageEmpty11}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull11}{\hidepaste}
+\tab{5}\spadcommand{lr := realSolve(lp)$pack\free{lp }\free{pack }\bound{lr }}
+\indentrel{3}\begin{verbatim}
+   (11)
+   [
+     [%R1,
+
+         1184459    19   2335702    18   5460230    17
+         ÄÄÄÄÄÄÄ %R1   - ÄÄÄÄÄÄÄ %R1   - ÄÄÄÄÄÄÄ %R1
+         1645371          548457          182819
+       + 
+         79900378    16   43953929    15   13420192    14
+         ÄÄÄÄÄÄÄÄ %R1   + ÄÄÄÄÄÄÄÄ %R1   + ÄÄÄÄÄÄÄÄ %R1
+          1645371          548457           182819
+       + 
+         553986    13   193381378    12   35978916    11
+         ÄÄÄÄÄÄ %R1   + ÄÄÄÄÄÄÄÄÄ %R1   + ÄÄÄÄÄÄÄÄ %R1
+          3731           1645371           182819
+       + 
+         358660781    10   271667666    9   118784873    8
+         ÄÄÄÄÄÄÄÄÄ %R1   + ÄÄÄÄÄÄÄÄÄ %R1  + ÄÄÄÄÄÄÄÄÄ %R1
+          1645371           1645371           548457
+       + 
+         337505020    7   1389370    6   688291    5
+         ÄÄÄÄÄÄÄÄÄ %R1  + ÄÄÄÄÄÄÄ %R1  + ÄÄÄÄÄÄ %R1
+          1645371          11193          4459
+       + 
+         3378002    4   140671876    3   32325724    2
+         ÄÄÄÄÄÄÄ %R1  + ÄÄÄÄÄÄÄÄÄ %R1  + ÄÄÄÄÄÄÄÄ %R1
+          42189          1645371          548457
+       + 
+           8270       9741532
+         - ÄÄÄÄ %R1 - ÄÄÄÄÄÄÄ
+            343       1645371
+       ,
+
+            91729    19   487915    18   4114333    17
+         - ÄÄÄÄÄÄ %R1   + ÄÄÄÄÄÄ %R1   + ÄÄÄÄÄÄÄ %R1
+           705159         705159          705159
+       + 
+           1276987    16   13243117    15   16292173    14
+         - ÄÄÄÄÄÄÄ %R1   - ÄÄÄÄÄÄÄÄ %R1   - ÄÄÄÄÄÄÄÄ %R1
+            235053          705159           705159
+       + 
+           26536060    13   722714    12   5382578    11
+         - ÄÄÄÄÄÄÄÄ %R1   - ÄÄÄÄÄÄ %R1   - ÄÄÄÄÄÄÄ %R1
+            705159           18081          100737
+       + 
+           15449995    10   14279770    9   6603890    8
+         - ÄÄÄÄÄÄÄÄ %R1   - ÄÄÄÄÄÄÄÄ %R1  - ÄÄÄÄÄÄÄ %R1
+            235053           235053          100737
+       + 
+           409930    7   37340389    6   34893715    5
+         - ÄÄÄÄÄÄ %R1  - ÄÄÄÄÄÄÄÄ %R1  - ÄÄÄÄÄÄÄÄ %R1
+            6027          705159          705159
+       + 
+           26686318    4   801511    3   17206178    2
+         - ÄÄÄÄÄÄÄÄ %R1  - ÄÄÄÄÄÄ %R1  - ÄÄÄÄÄÄÄÄ %R1
+            705159          26117         705159
+       + 
+           4406102       377534
+         - ÄÄÄÄÄÄÄ %R1 + ÄÄÄÄÄÄ
+            705159       705159
+       ]
+     ,
+
+     [%R2,
+
+         1184459    19   2335702    18   5460230    17
+         ÄÄÄÄÄÄÄ %R2   - ÄÄÄÄÄÄÄ %R2   - ÄÄÄÄÄÄÄ %R2
+         1645371          548457          182819
+       + 
+         79900378    16   43953929    15   13420192    14
+         ÄÄÄÄÄÄÄÄ %R2   + ÄÄÄÄÄÄÄÄ %R2   + ÄÄÄÄÄÄÄÄ %R2
+          1645371          548457           182819
+       + 
+         553986    13   193381378    12   35978916    11
+         ÄÄÄÄÄÄ %R2   + ÄÄÄÄÄÄÄÄÄ %R2   + ÄÄÄÄÄÄÄÄ %R2
+          3731           1645371           182819
+       + 
+         358660781    10   271667666    9   118784873    8
+         ÄÄÄÄÄÄÄÄÄ %R2   + ÄÄÄÄÄÄÄÄÄ %R2  + ÄÄÄÄÄÄÄÄÄ %R2
+          1645371           1645371           548457
+       + 
+         337505020    7   1389370    6   688291    5
+         ÄÄÄÄÄÄÄÄÄ %R2  + ÄÄÄÄÄÄÄ %R2  + ÄÄÄÄÄÄ %R2
+          1645371          11193          4459
+       + 
+         3378002    4   140671876    3   32325724    2
+         ÄÄÄÄÄÄÄ %R2  + ÄÄÄÄÄÄÄÄÄ %R2  + ÄÄÄÄÄÄÄÄ %R2
+          42189          1645371          548457
+       + 
+           8270       9741532
+         - ÄÄÄÄ %R2 - ÄÄÄÄÄÄÄ
+            343       1645371
+       ,
+
+            91729    19   487915    18   4114333    17
+         - ÄÄÄÄÄÄ %R2   + ÄÄÄÄÄÄ %R2   + ÄÄÄÄÄÄÄ %R2
+           705159         705159          705159
+       + 
+           1276987    16   13243117    15   16292173    14
+         - ÄÄÄÄÄÄÄ %R2   - ÄÄÄÄÄÄÄÄ %R2   - ÄÄÄÄÄÄÄÄ %R2
+            235053          705159           705159
+       + 
+           26536060    13   722714    12   5382578    11
+         - ÄÄÄÄÄÄÄÄ %R2   - ÄÄÄÄÄÄ %R2   - ÄÄÄÄÄÄÄ %R2
+            705159           18081          100737
+       + 
+           15449995    10   14279770    9   6603890    8
+         - ÄÄÄÄÄÄÄÄ %R2   - ÄÄÄÄÄÄÄÄ %R2  - ÄÄÄÄÄÄÄ %R2
+            235053           235053          100737
+       + 
+           409930    7   37340389    6   34893715    5
+         - ÄÄÄÄÄÄ %R2  - ÄÄÄÄÄÄÄÄ %R2  - ÄÄÄÄÄÄÄÄ %R2
+            6027          705159          705159
+       + 
+           26686318    4   801511    3   17206178    2
+         - ÄÄÄÄÄÄÄÄ %R2  - ÄÄÄÄÄÄ %R2  - ÄÄÄÄÄÄÄÄ %R2
+            705159          26117         705159
+       + 
+           4406102       377534
+         - ÄÄÄÄÄÄÄ %R2 + ÄÄÄÄÄÄ
+            705159       705159
+       ]
+     ,
+
+     [%R3,
+
+         1184459    19   2335702    18   5460230    17
+         ÄÄÄÄÄÄÄ %R3   - ÄÄÄÄÄÄÄ %R3   - ÄÄÄÄÄÄÄ %R3
+         1645371          548457          182819
+       + 
+         79900378    16   43953929    15   13420192    14
+         ÄÄÄÄÄÄÄÄ %R3   + ÄÄÄÄÄÄÄÄ %R3   + ÄÄÄÄÄÄÄÄ %R3
+          1645371          548457           182819
+       + 
+         553986    13   193381378    12   35978916    11
+         ÄÄÄÄÄÄ %R3   + ÄÄÄÄÄÄÄÄÄ %R3   + ÄÄÄÄÄÄÄÄ %R3
+          3731           1645371           182819
+       + 
+         358660781    10   271667666    9   118784873    8
+         ÄÄÄÄÄÄÄÄÄ %R3   + ÄÄÄÄÄÄÄÄÄ %R3  + ÄÄÄÄÄÄÄÄÄ %R3
+          1645371           1645371           548457
+       + 
+         337505020    7   1389370    6   688291    5
+         ÄÄÄÄÄÄÄÄÄ %R3  + ÄÄÄÄÄÄÄ %R3  + ÄÄÄÄÄÄ %R3
+          1645371          11193          4459
+       + 
+         3378002    4   140671876    3   32325724    2
+         ÄÄÄÄÄÄÄ %R3  + ÄÄÄÄÄÄÄÄÄ %R3  + ÄÄÄÄÄÄÄÄ %R3
+          42189          1645371          548457
+       + 
+           8270       9741532
+         - ÄÄÄÄ %R3 - ÄÄÄÄÄÄÄ
+            343       1645371
+       ,
+
+            91729    19   487915    18   4114333    17
+         - ÄÄÄÄÄÄ %R3   + ÄÄÄÄÄÄ %R3   + ÄÄÄÄÄÄÄ %R3
+           705159         705159          705159
+       + 
+           1276987    16   13243117    15   16292173    14
+         - ÄÄÄÄÄÄÄ %R3   - ÄÄÄÄÄÄÄÄ %R3   - ÄÄÄÄÄÄÄÄ %R3
+            235053          705159           705159
+       + 
+           26536060    13   722714    12   5382578    11
+         - ÄÄÄÄÄÄÄÄ %R3   - ÄÄÄÄÄÄ %R3   - ÄÄÄÄÄÄÄ %R3
+            705159           18081          100737
+       + 
+           15449995    10   14279770    9   6603890    8
+         - ÄÄÄÄÄÄÄÄ %R3   - ÄÄÄÄÄÄÄÄ %R3  - ÄÄÄÄÄÄÄ %R3
+            235053           235053          100737
+       + 
+           409930    7   37340389    6   34893715    5
+         - ÄÄÄÄÄÄ %R3  - ÄÄÄÄÄÄÄÄ %R3  - ÄÄÄÄÄÄÄÄ %R3
+            6027          705159          705159
+       + 
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+     ]
+           Type: List List RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty11}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty11}{ZeroDimensionalSolvePackageXmpPagePatch11}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{lr := realSolve(lp)$pack\free{lp }\free{pack }\bound{lr }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch12}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull12}{ZeroDimensionalSolvePackageXmpPageEmpty12}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull12}{\hidepaste}
+\tab{5}\spadcommand{\# lr\free{lr }}
+\indentrel{3}\begin{verbatim}
+   (12)  8
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty12}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty12}{ZeroDimensionalSolvePackageXmpPagePatch12}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{\# lr\free{lr }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch13}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull13}{ZeroDimensionalSolvePackageXmpPageEmpty13}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull13}{\hidepaste}
+\tab{5}\spadcommand{[[approximate(r,1/1000000) for r in point] for point in lr]\free{lr }}
+\indentrel{3}\begin{verbatim}
+   (13)
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+     ,
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+       -
+           899488488040242826510759512197069142713604569_
+            25419782755730018652137599215881377166961263_
+            49101655220195142994932299137183241705867672_
+            383477
+         /
+           116788999866502637217776510069188858270896996_
+            02299347696908357524570777794164352094737678_
+            66507769405888942764587718542434255625992456_
+            372224
+       ,
+
+       -
+           238970488813315687832080154437380839561277150_
+            92084910198474529918855095465195254678390166_
+            13593999693886640036283570552321155037871291_
+            458703265
+         /
+           535548727364509632609040328668993190598822544_
+            46854114332215938336811929575628336714686542_
+            90340746993656285925599117602120446183443145_
+            479421952
+       ]
+     ,
+
+       765693
+     [ÄÄÄÄÄÄÄ,
+      2097152
+
+        855896921981671626787324476117819808872469895861_
+         66701402137657543220023032516857861186783308402_
+         03328837654339523418704917749518340772512899000_
+         391009630373148561
+      /
+        294144244553301079097642841137639349981558021594_
+         58569179064525354957230138568189417023302287798_
+         90141296236721138154231997238917322156711965244_
+         4639331719460159488
+       ,
+
+       -
+           205761823058257210124765032486024256111130258_
+            15435888088439236627675493822416593627122907_
+            77612800192921420574408948085193743688582762_
+            2246433251878894899015
+         /
+           267159820332573553809795235350145022057631375_
+            98908350970917225206427101987719026671839489_
+            06289863714759678360292483949204616471537777_
+            775324180661095366656
+       ]
+     ,
+
+      5743879
+     [ÄÄÄÄÄÄÄ,
+      2097152
+
+        107628881696890684795554639477357020817145672494_
+         26186140236631235747689608504342639713980725465_
+         92772662158833449797698617455397887562900072984_
+         76800060834355318980169340872720504761255988923_
+         27575638305286889535354218094827710589175426028_
+         90060941949620874083007858366669453501766248414_
+         88732463225
+      /
+        313176895708031794664846194002355204419037661345_
+         85849862285496319161966016162197817656155325322_
+         94746529648276430583810894079374566460757823146_
+         88858119555602920851521883888320031865840746939_
+         94260632605898286123092315966691297079864813198_
+         51571942927230340622934023923486703042068153044_
+         0845099008
+       ,
+
+       -
+           211328669918575091836412047556545843787017248_
+            98654859943898281353352644446652845575264927_
+            34931691731407872701432935503473348172076098_
+            72054584900878007756416053431789468836611952_
+            97399805029441626685500981279619504962102219_
+            42878089359674925850594427768502251789758706_
+            752831632503615
+         /
+           162761558493798758024290662434710458088914446_
+            61684597180431538394083725255333098080703636_
+            99585502216011211087103263609551026027769414_
+            08739114812622116813978168258743807532259146_
+            61319399754572005223498385689642856344480185_
+            62038272378787354460106106141518010935617205_
+            1706396253618176
+       ]
+     ,
+
+      19739877
+     [ÄÄÄÄÄÄÄÄ,
+       2097152
+
+       -
+           299724993683270330379901580486152094921504038_
+            75007071777012857667201925305794224789535660_
+            24359860143101547801638082771611160372212874_
+            84777803580987284314922548423836585801362934_
+            17053217025823333509180096017899370239859353_
+            04900460493389873837030853410347089908880814_
+            85398113201846458245880061539477074169948729_
+            58759602107502158919488144768548710315309312_
+            95467332190133702671098200902282300510751860_
+            71859284570302778073977965258138627622392869_
+            96106809728023675
+         /
+           230843327485227859072891008119181102390650414_
+            13214326461239367948739333192706089607021381_
+            93417647898360620229519176632937631786851455_
+            01476602720625902225250555174182368889688380_
+            66366025744317604722402920931967294751602472_
+            68834121141893318848728661844434927287285112_
+            89708076755286489505658586403317856591038706_
+            50061128015164035227410373609905560544769495_
+            27059227070809593049491257519554708879259595_
+            52929920110858560812556635485429471554031675_
+            979542656381353984
+       ,
+
+       -
+           512818926354822848909627639786894008060093841_
+            06630804594079663358450092641094905204598253_
+            16250084723010047035024497436523038925818959_
+            28931293158470135392762143543439867426304729_
+            39091228501338519906964902315660943719943337_
+            95070782624011727587749989296611277318372294_
+            62420711653791043655457414608288470130554391_
+            26204193548854107359401577758966028223645758_
+            64611831512943973974715166920465061850603762_
+            87516256195847052412587282839139194642913955
+         /
+           228828193977843933053120879318129047118363109_
+            24553689903863908242435094636442362497730806_
+            47438987739144921607794682653851741189091711_
+            74186814511497833728419182249767586835872948_
+            66447308566225526872092037244118004814057028_
+            37198310642291275676195774614443815996713502_
+            62939174978359004147086012775237299648862774_
+            26724876224800632688088893248918508424949343_
+            47337603075939980268208482904859678177751444_
+            65749979827872616963053217673201717237252096
+       ]
+     ]
+                       Type: List List Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty13}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty13}{ZeroDimensionalSolvePackageXmpPagePatch13}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{[[approximate(r,1/1000000) for r in point] for point in lr]\free{lr }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch14}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull14}{ZeroDimensionalSolvePackageXmpPageEmpty14}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull14}{\hidepaste}
+\tab{5}\spadcommand{lpr := positiveSolve(lp)$pack\free{lp }\free{pack }\bound{lpr }}
+\indentrel{3}\begin{verbatim}
+   (14)  []
+           Type: List List RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty14}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty14}{ZeroDimensionalSolvePackageXmpPagePatch14}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{lpr := positiveSolve(lp)$pack\free{lp }\free{pack }\bound{lpr }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch15}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull15}{ZeroDimensionalSolvePackageXmpPageEmpty15}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull15}{\hidepaste}
+\tab{5}\spadcommand{f0 := x**3 + y + z + t- 1\bound{f0 }}
+\indentrel{3}\begin{verbatim}
+                  3
+   (15)  z + y + x  + t - 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty15}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty15}{ZeroDimensionalSolvePackageXmpPagePatch15}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{f0 := x**3 + y + z + t- 1\bound{f0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch16}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull16}{ZeroDimensionalSolvePackageXmpPageEmpty16}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull16}{\hidepaste}
+\tab{5}\spadcommand{f1 := x + y**3 + z + t -1\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+              3
+   (16)  z + y  + x + t - 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty16}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty16}{ZeroDimensionalSolvePackageXmpPagePatch16}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{f1 := x + y**3 + z + t -1\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch17}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull17}{ZeroDimensionalSolvePackageXmpPageEmpty17}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull17}{\hidepaste}
+\tab{5}\spadcommand{f2 := x + y + z**3 + t-1\bound{f2 }}
+\indentrel{3}\begin{verbatim}
+          3
+   (17)  z  + y + x + t - 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty17}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty17}{ZeroDimensionalSolvePackageXmpPagePatch17}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{f2 := x + y + z**3 + t-1\bound{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch18}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull18}{ZeroDimensionalSolvePackageXmpPageEmpty18}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull18}{\hidepaste}
+\tab{5}\spadcommand{f3 := x + y + z + t**3 -1\bound{f3 }}
+\indentrel{3}\begin{verbatim}
+                      3
+   (18)  z + y + x + t  - 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty18}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty18}{ZeroDimensionalSolvePackageXmpPagePatch18}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{f3 := x + y + z + t**3 -1\bound{f3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch19}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull19}{ZeroDimensionalSolvePackageXmpPageEmpty19}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull19}{\hidepaste}
+\tab{5}\spadcommand{lf := [f0, f1, f2, f3]\free{f0 }\free{f1 }\free{f2 }\free{f3 }\bound{lf }}
+\indentrel{3}\begin{verbatim}
+   (19)
+             3               3
+   [z + y + x  + t - 1, z + y  + x + t - 1,
+     3                               3
+    z  + y + x + t - 1, z + y + x + t  - 1]
+                          Type: List Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty19}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty19}{ZeroDimensionalSolvePackageXmpPagePatch19}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{lf := [f0, f1, f2, f3]\free{f0 }\free{f1 }\free{f2 }\free{f3 }\bound{lf }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch20}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull20}{ZeroDimensionalSolvePackageXmpPageEmpty20}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull20}{\hidepaste}
+\tab{5}\spadcommand{lts := triangSolve(lf)$pack\free{lf }\free{pack }\bound{lts }}
+\indentrel{3}\begin{verbatim}
+   (20)
+   [
+       2           3        3
+     {t  + t + 1, z  - z - t  + t,
+
+                 3      2
+         (3z + 3t  - 3)y
+       + 
+            2      3           6     3            3      2
+         (3z  + (6t  - 6)z + 3t  - 6t  + 3)y + (3t  - 3)z
+       + 
+            6     3          9     6     3
+         (3t  - 6t  + 3)z + t  - 3t  + 5t  - 3t
+       ,
+      x + y + z}
+     ,
+
+       16     13     10     7      4      2
+     {t   - 6t   + 9t   + 4t  + 15t  - 54t  + 27,
+
+                     15            14             13
+             4907232t   + 40893984t   - 115013088t
+           + 
+                      12            11             10
+             22805712t   + 36330336t   + 162959040t
+           + 
+                         9             8             7
+             - 159859440t  - 156802608t  + 117168768t
+           + 
+                       6             5             4
+             126282384t  - 129351600t  + 306646992t
+           + 
+                       3              2
+             475302816t  - 1006837776t  - 237269088t
+           + 
+             480716208
+        *
+           z
+       + 
+            54       51        48      46         45
+         48t   - 912t   + 8232t   - 72t   - 46848t
+       + 
+              43          42        40          39
+         1152t   + 186324t   - 3780t   - 543144t
+       + 
+                38         37           36         35
+         - 3168t   - 21384t   + 1175251t   + 41184t
+       + 
+                34           33          32           31
+         278003t   - 1843242t   - 301815t   - 1440726t
+       + 
+                 30           29           28          27
+         1912012t   + 1442826t   + 4696262t   - 922481t
+       + 
+                   26            25          24
+         - 4816188t   - 10583524t   - 208751t
+       + 
+                  23            22          21
+         11472138t   + 16762859t   - 857663t
+       + 
+                    20            19           18
+         - 19328175t   - 18270421t   + 4914903t
+       + 
+                  17            16           15
+         22483044t   + 12926517t   - 8605511t
+       + 
+                    14           13           12
+         - 17455518t   - 5014597t   + 8108814t
+       + 
+                 11          10           9           8
+         8465535t   + 190542t   - 4305624t  - 2226123t
+       + 
+                7           6          5          4
+         661905t  + 1169775t  + 226260t  - 209952t
+       + 
+                  3
+         - 141183t  + 27216t
+       ,
+
+                 3      2
+         (3z + 3t  - 3)y
+       + 
+            2      3           6     3            3      2
+         (3z  + (6t  - 6)z + 3t  - 6t  + 3)y + (3t  - 3)z
+       + 
+            6     3          9     6     3
+         (3t  - 6t  + 3)z + t  - 3t  + 5t  - 3t
+       ,
+                   3
+      x + y + z + t  - 1}
+     ,
+              2                       2
+    {t,z - 1,y  - 1,x + y}, {t - 1,z,y  - 1,x + y},
+            2
+    {t - 1,z  - 1,z y + 1,x},
+
+       16     13     10     7      4      2
+     {t   - 6t   + 9t   + 4t  + 15t  - 54t  + 27,
+
+                     29            28             27
+             4907232t   + 40893984t   - 115013088t
+           + 
+                       26             25             24
+             - 1730448t   - 168139584t   + 738024480t
+           + 
+                         23             22              21
+             - 195372288t   + 315849456t   - 2567279232t
+           + 
+                       20              19              18
+             937147968t   + 1026357696t   + 4780488240t
+           + 
+                          17              16
+             - 2893767696t   - 5617160352t
+           + 
+                          15              14
+             - 3427651728t   + 5001100848t
+           + 
+                        13              12             11
+             8720098416t   + 2331732960t   - 499046544t
+           + 
+                           10              9
+             - 16243306272t   - 9748123200t
+           + 
+                        8               7               6
+             3927244320t  + 25257280896t  + 10348032096t
+           + 
+                           5               4             3
+             - 17128672128t  - 14755488768t  + 544086720t
+           + 
+                         2
+             10848188736t  + 1423614528t - 2884297248
+        *
+           z
+       + 
+              68        65         62       60          59
+         - 48t   + 1152t   - 13560t   + 360t   + 103656t
+       + 
+                57          56         54           53
+         - 7560t   - 572820t   + 71316t   + 2414556t
+       + 
+              52          51           50         49
+         2736t   - 402876t   - 7985131t   - 49248t
+       + 
+                 48            47          46           45
+         1431133t   + 20977409t   + 521487t   - 2697635t
+       + 
+                    44           43           42
+         - 43763654t   - 3756573t   - 2093410t
+       + 
+                  41            40            39
+         71546495t   + 19699032t   + 35025028t
+       + 
+                    38            37             36
+         - 89623786t   - 77798760t   - 138654191t
+       + 
+                  35             34             33
+         87596128t   + 235642497t   + 349607642t
+       + 
+                    32             31             30
+         - 93299834t   - 551563167t   - 630995176t
+       + 
+                   29             28             27
+         186818962t   + 995427468t   + 828416204t
+       + 
+                     26              25              24
+         - 393919231t   - 1076617485t   - 1609479791t
+       + 
+                   23              22              21
+         595738126t   + 1198787136t   + 4342832069t
+       + 
+                      20              19              18
+         - 2075938757t   - 4390835799t   - 4822843033t
+       + 
+                    17              16              15
+         6932747678t   + 6172196808t   + 1141517740t
+       + 
+                      14              13              12
+         - 4981677585t   - 9819815280t   - 7404299976t
+       + 
+                     11               10               9
+         - 157295760t   + 29124027630t   + 14856038208t
+       + 
+                       8               7              6
+         - 16184101410t  - 26935440354t  - 3574164258t
+       + 
+                     5               4              3
+         10271338974t  + 11191425264t  + 6869861262t
+       + 
+                      2
+         - 9780477840t  - 3586674168t + 2884297248
+       ,
+
+               3      3      2      6      3           9
+             3z  + (6t  - 6)z  + (6t  - 12t  + 3)z + 2t
+           + 
+                 6    3
+             - 6t  + t  + 3t
+        *
+           y
+       + 
+            3      3      6      3      2
+         (3t  - 3)z  + (6t  - 12t  + 6)z
+       + 
+            9      6      3          12     9     6     3
+         (4t  - 12t  + 11t  - 3)z + t   - 4t  + 5t  - 2t
+       ,
+                   3
+      x + y + z + t  - 1}
+     ,
+            2
+    {t - 1,z  - 1,y,x + z},
+
+       8    7    6     5     4     3      2
+     {t  + t  + t  - 2t  - 2t  - 2t  + 19t  + 19t - 8,
+
+                     7           6           5
+             2395770t  + 3934440t  - 3902067t
+           + 
+                        4           3            2
+             - 10084164t  - 1010448t  + 32386932t
+           + 
+             22413225t - 10432368
+        *
+           z
+       + 
+                  7           6           5           4
+         - 463519t  + 3586833t  + 9494955t  - 8539305t
+       + 
+                    3            2
+         - 33283098t  + 35479377t  + 46263256t - 17419896
+       ,
+
+               4      3      3       6      3      2
+             3z  + (9t  - 9)z  + (12t  - 24t  + 9)z
+           + 
+                    3                    6      4      3
+             (- 152t  + 219t - 67)z - 41t  + 57t  + 25t
+           + 
+             - 57t + 16
+        *
+           y
+       + 
+            3      4      6      3      3
+         (3t  - 3)z  + (9t  - 18t  + 9)z
+       + 
+                3              2
+         (- 181t  + 270t - 89)z
+       + 
+               6       4      3                    7
+         (- 92t  + 135t  + 49t  - 135t + 43)z + 27t
+       + 
+              6      4       3
+         - 27t  - 54t  + 396t  - 486t + 144
+       ,
+                   3
+      x + y + z + t  - 1}
+     ,
+            3
+    {t,z - t  + 1,y - 1,x - 1}, {t - 1,z,y,x},
+    {t,z - 1,y,x}, {t,z,y - 1,x}, {t,z,y,x - 1}]
+             Type: List RegularChain(Integer,[x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty20}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty20}{ZeroDimensionalSolvePackageXmpPagePatch20}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{lts := triangSolve(lf)$pack\free{lf }\free{pack }\bound{lts }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch21}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull21}{ZeroDimensionalSolvePackageXmpPageEmpty21}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull21}{\hidepaste}
+\tab{5}\spadcommand{univariateSolve(lf)$pack\free{lf }\free{pack }}
+\indentrel{3}\begin{verbatim}
+   (21)
+   [
+     [complexRoots= ?,
+      coordinates= [x - 1,y - 1,z + 1,t - %A]]
+     ,
+    [complexRoots= ?,coordinates= [x,y - 1,z,t - %A]],
+    [complexRoots= ? - 1,coordinates= [x,y,z,t - %A]],
+    [complexRoots= ?,coordinates= [x - 1,y,z,t - %A]],
+    [complexRoots= ?,coordinates= [x,y,z - 1,t - %A]],
+
+     [complexRoots= ? - 2,
+      coordinates= [x - 1,y + 1,z,t - 1]]
+     ,
+    [complexRoots= ?,coordinates= [x + 1,y - 1,z,t - 1]],
+
+     [complexRoots= ? - 1,
+      coordinates= [x - 1,y + 1,z - 1,t]]
+     ,
+
+     [complexRoots= ? + 1,
+      coordinates= [x + 1,y - 1,z - 1,t]]
+     ,
+
+                     6     3     2
+     [complexRoots= ?  - 2?  + 3?  - 3,
+
+       coordinates =
+                 3                  3
+         [2x + %A  + %A - 1, 2y + %A  + %A - 1, z - %A,
+          t - %A]
+       ]
+     ,
+
+                     5     3     2
+     [complexRoots= ?  + 3?  - 2?  + 3? - 3,
+
+       coordinates =
+                              3
+         [x - %A,y - %A,z + %A  + 2%A - 1,t - %A]
+       ]
+     ,
+
+                     4    3     2
+     [complexRoots= ?  - ?  - 2?  + 3,
+
+       coordinates =
+                3                 3
+         [x + %A  - %A - 1, y + %A  - %A - 1,
+                3
+          z - %A  + 2%A + 1, t - %A]
+       ]
+     ,
+
+     [complexRoots= ? + 1,
+      coordinates= [x - 1,y - 1,z,t - %A]]
+     ,
+
+                     6     3     2
+     [complexRoots= ?  + 2?  + 3?  - 3,
+
+       coordinates =
+                 3                          3
+         [2x - %A  - %A - 1, y + %A, 2z - %A  - %A - 1,
+          t + %A]
+       ]
+     ,
+
+                     6      4      3      2
+     [complexRoots= ?  + 12?  + 20?  - 45?  - 42? - 953,
+
+       coordinates =
+         [
+                          5       4       3        2
+             12609x + 23%A  + 49%A  - 46%A  + 362%A
+           + 
+             - 5015%A - 8239
+           ,
+
+                          5       4       3        2
+             25218y + 23%A  + 49%A  - 46%A  + 362%A
+           + 
+             7594%A - 8239
+           ,
+
+                          5       4       3        2
+             25218z + 23%A  + 49%A  - 46%A  + 362%A
+           + 
+             7594%A - 8239
+           ,
+
+                          5       4       3        2
+             12609t + 23%A  + 49%A  - 46%A  + 362%A
+           + 
+             - 5015%A - 8239
+           ]
+       ]
+     ,
+
+                     5      3      2
+     [complexRoots= ?  + 12?  - 16?  + 48? - 96,
+
+       coordinates =
+                 3
+         [8x + %A  + 8%A - 8,2y - %A,2z - %A,2t - %A]
+       ]
+     ,
+
+                     5    4     3     2
+     [complexRoots= ?  + ?  - 5?  - 3?  + 9? + 3,
+
+       coordinates =
+                 3                   3
+         [2x - %A  + 2%A - 1, 2y + %A  - 4%A + 1,
+                 3                   3
+          2z - %A  + 2%A - 1, 2t - %A  + 2%A - 1]
+       ]
+     ,
+
+                     4     3     2
+     [complexRoots= ?  - 3?  + 4?  - 6? + 13,
+
+       coordinates =
+                  3      2
+         [9x - 2%A  + 4%A  - %A + 2,
+                 3      2
+          9y + %A  - 2%A  + 5%A - 1,
+                 3      2
+          9z + %A  - 2%A  + 5%A - 1,
+                 3      2
+          9t + %A  - 2%A  - 4%A - 1]
+       ]
+     ,
+
+                     4      2
+     [complexRoots= ?  - 11?  + 37,
+
+       coordinates =
+                 2             2                   2
+         [3x - %A  + 7, 6y + %A  + 3%A - 7, 3z - %A  + 7,
+                 2
+          6t + %A  - 3%A - 7]
+       ]
+     ,
+
+     [complexRoots= ? + 1,
+      coordinates= [x - 1,y,z - 1,t + 1]]
+     ,
+
+     [complexRoots= ? + 2,
+      coordinates= [x,y - 1,z - 1,t + 1]]
+     ,
+
+     [complexRoots= ? - 2,
+      coordinates= [x,y - 1,z + 1,t - 1]]
+     ,
+    [complexRoots= ?,coordinates= [x,y + 1,z - 1,t - 1]],
+
+     [complexRoots= ? - 2,
+      coordinates= [x - 1,y,z + 1,t - 1]]
+     ,
+    [complexRoots= ?,coordinates= [x + 1,y,z - 1,t - 1]],
+
+                     4     3      2
+     [complexRoots= ?  + 5?  + 16?  + 30? + 57,
+
+       coordinates =
+                     3       2
+         [151x + 15%A  + 54%A  + 104%A + 93,
+                     3       2
+          151y - 10%A  - 36%A  - 19%A - 62,
+                    3       2
+          151z - 5%A  - 18%A  - 85%A - 31,
+                    3       2
+          151t - 5%A  - 18%A  - 85%A - 31]
+       ]
+     ,
+
+                     4    3     2
+     [complexRoots= ?  - ?  - 2?  + 3,
+
+       coordinates =
+                3                  3
+         [x - %A  + 2%A + 1, y + %A  - %A - 1, z - %A,
+                3
+          t + %A  - %A - 1]
+       ]
+     ,
+
+                     4     3     2
+     [complexRoots= ?  + 2?  - 8?  + 48,
+
+       coordinates =
+                 3
+         [8x - %A  + 4%A - 8, 2y + %A,
+                 3                   3
+          8z + %A  - 8%A + 8, 8t - %A  + 4%A - 8]
+       ]
+     ,
+
+                     5    4     3     2
+     [complexRoots= ?  + ?  - 2?  - 4?  + 5? + 8,
+
+       coordinates =
+                 3            3            3
+         [3x + %A  - 1,3y + %A  - 1,3z + %A  - 1,t - %A]
+       ]
+     ,
+
+                     3
+     [complexRoots= ?  + 3? - 1,
+      coordinates= [x - %A,y - %A,z - %A,t - %A]]
+     ]
+Type: List Record(complexRoots: SparseUnivariatePolynomial Integer,coordinates: List Polynomial Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty21}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty21}{ZeroDimensionalSolvePackageXmpPagePatch21}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{univariateSolve(lf)$pack\free{lf }\free{pack }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch22}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull22}{ZeroDimensionalSolvePackageXmpPageEmpty22}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull22}{\hidepaste}
+\tab{5}\spadcommand{ts := lts.1\free{lts }\bound{ts }}
+\indentrel{3}\begin{verbatim}
+   (22)
+     2           3        3
+   {t  + t + 1, z  - z - t  + t,
+
+               3      2
+       (3z + 3t  - 3)y
+     + 
+          2      3           6     3            3      2
+       (3z  + (6t  - 6)z + 3t  - 6t  + 3)y + (3t  - 3)z
+     + 
+          6     3          9     6     3
+       (3t  - 6t  + 3)z + t  - 3t  + 5t  - 3t
+     ,
+    x + y + z}
+                  Type: RegularChain(Integer,[x,y,z,t])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty22}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty22}{ZeroDimensionalSolvePackageXmpPagePatch22}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{ts := lts.1\free{lts }\bound{ts }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch23}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull23}{ZeroDimensionalSolvePackageXmpPageEmpty23}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull23}{\hidepaste}
+\tab{5}\spadcommand{univariateSolve(ts)$pack\free{ts }\free{pack }}
+\indentrel{3}\begin{verbatim}
+   (23)
+   [
+                     4     3      2
+     [complexRoots= ?  + 5?  + 16?  + 30? + 57,
+
+       coordinates =
+                     3       2
+         [151x + 15%A  + 54%A  + 104%A + 93,
+                     3       2
+          151y - 10%A  - 36%A  - 19%A - 62,
+                    3       2
+          151z - 5%A  - 18%A  - 85%A - 31,
+                    3       2
+          151t - 5%A  - 18%A  - 85%A - 31]
+       ]
+     ,
+
+                     4    3     2
+     [complexRoots= ?  - ?  - 2?  + 3,
+
+       coordinates =
+                3                  3
+         [x - %A  + 2%A + 1, y + %A  - %A - 1, z - %A,
+                3
+          t + %A  - %A - 1]
+       ]
+     ,
+
+                     4     3     2
+     [complexRoots= ?  + 2?  - 8?  + 48,
+
+       coordinates =
+                 3
+         [8x - %A  + 4%A - 8, 2y + %A,
+                 3                   3
+          8z + %A  - 8%A + 8, 8t - %A  + 4%A - 8]
+       ]
+     ]
+Type: List Record(complexRoots: SparseUnivariatePolynomial Integer,coordinates: List Polynomial Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty23}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty23}{ZeroDimensionalSolvePackageXmpPagePatch23}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{univariateSolve(ts)$pack\free{ts }\free{pack }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch24}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull24}{ZeroDimensionalSolvePackageXmpPageEmpty24}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull24}{\hidepaste}
+\tab{5}\spadcommand{realSolve(ts)$pack\free{ts }\free{pack }}
+\indentrel{3}\begin{verbatim}
+   (24)  []
+           Type: List List RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty24}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty24}{ZeroDimensionalSolvePackageXmpPagePatch24}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{realSolve(ts)$pack\free{ts }\free{pack }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch25}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull25}{ZeroDimensionalSolvePackageXmpPageEmpty25}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull25}{\hidepaste}
+\tab{5}\spadcommand{lr2 := realSolve(lf)$pack\free{lf }\free{pack }\bound{lr2 }}
+\indentrel{3}\begin{verbatim}
+   (25)
+   [[0,- 1,1,1], [0,0,1,0], [1,0,0,0], [0,0,0,1],
+    [0,1,0,0], [1,0,%R37,- %R37], [1,0,%R38,- %R38],
+    [0,1,%R35,- %R35], [0,1,%R36,- %R36], [- 1,0,1,1],
+
+     [%R32,
+
+          1     15    2     14    1     13    4     12
+         ÄÄ %R32   + ÄÄ %R32   + ÄÄ %R32   - ÄÄ %R32
+         27          27          27          27
+       + 
+           11     11    4     10    1     9   14     8
+         - ÄÄ %R32   - ÄÄ %R32   + ÄÄ %R32  + ÄÄ %R32
+           27          27          27         27
+       + 
+          1     7   2     6   1     5   2     4       3
+         ÄÄ %R32  + Ä %R32  + Ä %R32  + Ä %R32  + %R32
+         27         9         3         9
+       + 
+         4     2
+         Ä %R32  - %R32 - 2
+         3
+       ,
+
+            1     15    1     14    1     13    2     12
+         - ÄÄ %R32   - ÄÄ %R32   - ÄÄ %R32   + ÄÄ %R32
+           54          27          54          27
+       + 
+         11     11    2     10    1     9    7     8
+         ÄÄ %R32   + ÄÄ %R32   - ÄÄ %R32  - ÄÄ %R32
+         54          27          54         27
+       + 
+            1     7   1     6   1     5   1     4       3
+         - ÄÄ %R32  - Ä %R32  - Ä %R32  - Ä %R32  - %R32
+           54         9         6         9
+       + 
+           2     2   1        3
+         - Ä %R32  + Ä %R32 + Ä
+           3         2        2
+       ,
+
+            1     15    1     14    1     13    2     12
+         - ÄÄ %R32   - ÄÄ %R32   - ÄÄ %R32   + ÄÄ %R32
+           54          27          54          27
+       + 
+         11     11    2     10    1     9    7     8
+         ÄÄ %R32   + ÄÄ %R32   - ÄÄ %R32  - ÄÄ %R32
+         54          27          54         27
+       + 
+            1     7   1     6   1     5   1     4       3
+         - ÄÄ %R32  - Ä %R32  - Ä %R32  - Ä %R32  - %R32
+           54         9         6         9
+       + 
+           2     2   1        3
+         - Ä %R32  + Ä %R32 + Ä
+           3         2        2
+       ]
+     ,
+
+     [%R33,
+
+          1     15    2     14    1     13    4     12
+         ÄÄ %R33   + ÄÄ %R33   + ÄÄ %R33   - ÄÄ %R33
+         27          27          27          27
+       + 
+           11     11    4     10    1     9   14     8
+         - ÄÄ %R33   - ÄÄ %R33   + ÄÄ %R33  + ÄÄ %R33
+           27          27          27         27
+       + 
+          1     7   2     6   1     5   2     4       3
+         ÄÄ %R33  + Ä %R33  + Ä %R33  + Ä %R33  + %R33
+         27         9         3         9
+       + 
+         4     2
+         Ä %R33  - %R33 - 2
+         3
+       ,
+
+            1     15    1     14    1     13    2     12
+         - ÄÄ %R33   - ÄÄ %R33   - ÄÄ %R33   + ÄÄ %R33
+           54          27          54          27
+       + 
+         11     11    2     10    1     9    7     8
+         ÄÄ %R33   + ÄÄ %R33   - ÄÄ %R33  - ÄÄ %R33
+         54          27          54         27
+       + 
+            1     7   1     6   1     5   1     4       3
+         - ÄÄ %R33  - Ä %R33  - Ä %R33  - Ä %R33  - %R33
+           54         9         6         9
+       + 
+           2     2   1        3
+         - Ä %R33  + Ä %R33 + Ä
+           3         2        2
+       ,
+
+            1     15    1     14    1     13    2     12
+         - ÄÄ %R33   - ÄÄ %R33   - ÄÄ %R33   + ÄÄ %R33
+           54          27          54          27
+       + 
+         11     11    2     10    1     9    7     8
+         ÄÄ %R33   + ÄÄ %R33   - ÄÄ %R33  - ÄÄ %R33
+         54          27          54         27
+       + 
+            1     7   1     6   1     5   1     4       3
+         - ÄÄ %R33  - Ä %R33  - Ä %R33  - Ä %R33  - %R33
+           54         9         6         9
+       + 
+           2     2   1        3
+         - Ä %R33  + Ä %R33 + Ä
+           3         2        2
+       ]
+     ,
+
+     [%R34,
+
+          1     15    2     14    1     13    4     12
+         ÄÄ %R34   + ÄÄ %R34   + ÄÄ %R34   - ÄÄ %R34
+         27          27          27          27
+       + 
+           11     11    4     10    1     9   14     8
+         - ÄÄ %R34   - ÄÄ %R34   + ÄÄ %R34  + ÄÄ %R34
+           27          27          27         27
+       + 
+          1     7   2     6   1     5   2     4       3
+         ÄÄ %R34  + Ä %R34  + Ä %R34  + Ä %R34  + %R34
+         27         9         3         9
+       + 
+         4     2
+         Ä %R34  - %R34 - 2
+         3
+       ,
+
+            1     15    1     14    1     13    2     12
+         - ÄÄ %R34   - ÄÄ %R34   - ÄÄ %R34   + ÄÄ %R34
+           54          27          54          27
+       + 
+         11     11    2     10    1     9    7     8
+         ÄÄ %R34   + ÄÄ %R34   - ÄÄ %R34  - ÄÄ %R34
+         54          27          54         27
+       + 
+            1     7   1     6   1     5   1     4       3
+         - ÄÄ %R34  - Ä %R34  - Ä %R34  - Ä %R34  - %R34
+           54         9         6         9
+       + 
+           2     2   1        3
+         - Ä %R34  + Ä %R34 + Ä
+           3         2        2
+       ,
+
+            1     15    1     14    1     13    2     12
+         - ÄÄ %R34   - ÄÄ %R34   - ÄÄ %R34   + ÄÄ %R34
+           54          27          54          27
+       + 
+         11     11    2     10    1     9    7     8
+         ÄÄ %R34   + ÄÄ %R34   - ÄÄ %R34  - ÄÄ %R34
+         54          27          54         27
+       + 
+            1     7   1     6   1     5   1     4       3
+         - ÄÄ %R34  - Ä %R34  - Ä %R34  - Ä %R34  - %R34
+           54         9         6         9
+       + 
+           2     2   1        3
+         - Ä %R34  + Ä %R34 + Ä
+           3         2        2
+       ]
+     ,
+    [- 1,1,0,1], [- 1,1,1,0],
+
+     [%R23,
+
+            1     15    1     14    1     13    2     12
+         - ÄÄ %R23   - ÄÄ %R23   - ÄÄ %R23   + ÄÄ %R23
+           54          27          54          27
+       + 
+         11     11    2     10    1     9    7     8
+         ÄÄ %R23   + ÄÄ %R23   - ÄÄ %R23  - ÄÄ %R23
+         54          27          54         27
+       + 
+            1     7   1     6   1     5   1     4       3
+         - ÄÄ %R23  - Ä %R23  - Ä %R23  - Ä %R23  - %R23
+           54         9         6         9
+       + 
+           2     2   1        3
+         - Ä %R23  + Ä %R23 + Ä
+           3         2        2
+       ,
+      %R30,
+
+                   1     15    1     14    1     13
+         - %R30 + ÄÄ %R23   + ÄÄ %R23   + ÄÄ %R23
+                  54          27          54
+       + 
+            2     12   11     11    2     10    1     9
+         - ÄÄ %R23   - ÄÄ %R23   - ÄÄ %R23   + ÄÄ %R23
+           27          54          27          54
+       + 
+          7     8    1     7   1     6   1     5   1     4
+         ÄÄ %R23  + ÄÄ %R23  + Ä %R23  + Ä %R23  + Ä %R23
+         27         54         9         6         9
+       + 
+         2     2   1        1
+         Ä %R23  - Ä %R23 - Ä
+         3         2        2
+       ]
+     ,
+
+     [%R23,
+
+            1     15    1     14    1     13    2     12
+         - ÄÄ %R23   - ÄÄ %R23   - ÄÄ %R23   + ÄÄ %R23
+           54          27          54          27
+       + 
+         11     11    2     10    1     9    7     8
+         ÄÄ %R23   + ÄÄ %R23   - ÄÄ %R23  - ÄÄ %R23
+         54          27          54         27
+       + 
+            1     7   1     6   1     5   1     4       3
+         - ÄÄ %R23  - Ä %R23  - Ä %R23  - Ä %R23  - %R23
+           54         9         6         9
+       + 
+           2     2   1        3
+         - Ä %R23  + Ä %R23 + Ä
+           3         2        2
+       ,
+      %R31,
+
+                   1     15    1     14    1     13
+         - %R31 + ÄÄ %R23   + ÄÄ %R23   + ÄÄ %R23
+                  54          27          54
+       + 
+            2     12   11     11    2     10    1     9
+         - ÄÄ %R23   - ÄÄ %R23   - ÄÄ %R23   + ÄÄ %R23
+           27          54          27          54
+       + 
+          7     8    1     7   1     6   1     5   1     4
+         ÄÄ %R23  + ÄÄ %R23  + Ä %R23  + Ä %R23  + Ä %R23
+         27         54         9         6         9
+       + 
+         2     2   1        1
+         Ä %R23  - Ä %R23 - Ä
+         3         2        2
+       ]
+     ,
+
+     [%R24,
+
+            1     15    1     14    1     13    2     12
+         - ÄÄ %R24   - ÄÄ %R24   - ÄÄ %R24   + ÄÄ %R24
+           54          27          54          27
+       + 
+         11     11    2     10    1     9    7     8
+         ÄÄ %R24   + ÄÄ %R24   - ÄÄ %R24  - ÄÄ %R24
+         54          27          54         27
+       + 
+            1     7   1     6   1     5   1     4       3
+         - ÄÄ %R24  - Ä %R24  - Ä %R24  - Ä %R24  - %R24
+           54         9         6         9
+       + 
+           2     2   1        3
+         - Ä %R24  + Ä %R24 + Ä
+           3         2        2
+       ,
+      %R28,
+
+                   1     15    1     14    1     13
+         - %R28 + ÄÄ %R24   + ÄÄ %R24   + ÄÄ %R24
+                  54          27          54
+       + 
+            2     12   11     11    2     10    1     9
+         - ÄÄ %R24   - ÄÄ %R24   - ÄÄ %R24   + ÄÄ %R24
+           27          54          27          54
+       + 
+          7     8    1     7   1     6   1     5   1     4
+         ÄÄ %R24  + ÄÄ %R24  + Ä %R24  + Ä %R24  + Ä %R24
+         27         54         9         6         9
+       + 
+         2     2   1        1
+         Ä %R24  - Ä %R24 - Ä
+         3         2        2
+       ]
+     ,
+
+     [%R24,
+
+            1     15    1     14    1     13    2     12
+         - ÄÄ %R24   - ÄÄ %R24   - ÄÄ %R24   + ÄÄ %R24
+           54          27          54          27
+       + 
+         11     11    2     10    1     9    7     8
+         ÄÄ %R24   + ÄÄ %R24   - ÄÄ %R24  - ÄÄ %R24
+         54          27          54         27
+       + 
+            1     7   1     6   1     5   1     4       3
+         - ÄÄ %R24  - Ä %R24  - Ä %R24  - Ä %R24  - %R24
+           54         9         6         9
+       + 
+           2     2   1        3
+         - Ä %R24  + Ä %R24 + Ä
+           3         2        2
+       ,
+      %R29,
+
+                   1     15    1     14    1     13
+         - %R29 + ÄÄ %R24   + ÄÄ %R24   + ÄÄ %R24
+                  54          27          54
+       + 
+            2     12   11     11    2     10    1     9
+         - ÄÄ %R24   - ÄÄ %R24   - ÄÄ %R24   + ÄÄ %R24
+           27          54          27          54
+       + 
+          7     8    1     7   1     6   1     5   1     4
+         ÄÄ %R24  + ÄÄ %R24  + Ä %R24  + Ä %R24  + Ä %R24
+         27         54         9         6         9
+       + 
+         2     2   1        1
+         Ä %R24  - Ä %R24 - Ä
+         3         2        2
+       ]
+     ,
+
+     [%R25,
+
+            1     15    1     14    1     13    2     12
+         - ÄÄ %R25   - ÄÄ %R25   - ÄÄ %R25   + ÄÄ %R25
+           54          27          54          27
+       + 
+         11     11    2     10    1     9    7     8
+         ÄÄ %R25   + ÄÄ %R25   - ÄÄ %R25  - ÄÄ %R25
+         54          27          54         27
+       + 
+            1     7   1     6   1     5   1     4       3
+         - ÄÄ %R25  - Ä %R25  - Ä %R25  - Ä %R25  - %R25
+           54         9         6         9
+       + 
+           2     2   1        3
+         - Ä %R25  + Ä %R25 + Ä
+           3         2        2
+       ,
+      %R26,
+
+                   1     15    1     14    1     13
+         - %R26 + ÄÄ %R25   + ÄÄ %R25   + ÄÄ %R25
+                  54          27          54
+       + 
+            2     12   11     11    2     10    1     9
+         - ÄÄ %R25   - ÄÄ %R25   - ÄÄ %R25   + ÄÄ %R25
+           27          54          27          54
+       + 
+          7     8    1     7   1     6   1     5   1     4
+         ÄÄ %R25  + ÄÄ %R25  + Ä %R25  + Ä %R25  + Ä %R25
+         27         54         9         6         9
+       + 
+         2     2   1        1
+         Ä %R25  - Ä %R25 - Ä
+         3         2        2
+       ]
+     ,
+
+     [%R25,
+
+            1     15    1     14    1     13    2     12
+         - ÄÄ %R25   - ÄÄ %R25   - ÄÄ %R25   + ÄÄ %R25
+           54          27          54          27
+       + 
+         11     11    2     10    1     9    7     8
+         ÄÄ %R25   + ÄÄ %R25   - ÄÄ %R25  - ÄÄ %R25
+         54          27          54         27
+       + 
+            1     7   1     6   1     5   1     4       3
+         - ÄÄ %R25  - Ä %R25  - Ä %R25  - Ä %R25  - %R25
+           54         9         6         9
+       + 
+           2     2   1        3
+         - Ä %R25  + Ä %R25 + Ä
+           3         2        2
+       ,
+      %R27,
+
+                   1     15    1     14    1     13
+         - %R27 + ÄÄ %R25   + ÄÄ %R25   + ÄÄ %R25
+                  54          27          54
+       + 
+            2     12   11     11    2     10    1     9
+         - ÄÄ %R25   - ÄÄ %R25   - ÄÄ %R25   + ÄÄ %R25
+           27          54          27          54
+       + 
+          7     8    1     7   1     6   1     5   1     4
+         ÄÄ %R25  + ÄÄ %R25  + Ä %R25  + Ä %R25  + Ä %R25
+         27         54         9         6         9
+       + 
+         2     2   1        1
+         Ä %R25  - Ä %R25 - Ä
+         3         2        2
+       ]
+     ,
+    [1,%R21,- %R21,0], [1,%R22,- %R22,0],
+    [1,%R19,0,- %R19], [1,%R20,0,- %R20],
+            1     3   1   1     3   1   1     3   1
+    [%R17,- Ä %R17  + Ä,- Ä %R17  + Ä,- Ä %R17  + Ä],
+            3         3   3         3   3         3
+            1     3   1   1     3   1   1     3   1
+    [%R18,- Ä %R18  + Ä,- Ä %R18  + Ä,- Ä %R18  + Ä]]
+            3         3   3         3   3         3
+           Type: List List RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty25}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty25}{ZeroDimensionalSolvePackageXmpPagePatch25}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{lr2 := realSolve(lf)$pack\free{lf }\free{pack }\bound{lr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch26}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull26}{ZeroDimensionalSolvePackageXmpPageEmpty26}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull26}{\hidepaste}
+\tab{5}\spadcommand{\#lr2\free{lr2 }}
+\indentrel{3}\begin{verbatim}
+   (26)  27
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty26}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty26}{ZeroDimensionalSolvePackageXmpPagePatch26}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{\#lr2\free{lr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch27}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull27}{ZeroDimensionalSolvePackageXmpPageEmpty27}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull27}{\hidepaste}
+\tab{5}\spadcommand{lpr2 := positiveSolve(lf)$pack\free{lf }\free{pack }\bound{lpr2 }}
+\indentrel{3}\begin{verbatim}
+   (27)
+            1     3   1   1     3   1   1     3   1
+   [[%R40,- Ä %R40  + Ä,- Ä %R40  + Ä,- Ä %R40  + Ä]]
+            3         3   3         3   3         3
+           Type: List List RealClosure Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty27}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty27}{ZeroDimensionalSolvePackageXmpPagePatch27}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{lpr2 := positiveSolve(lf)$pack\free{lf }\free{pack }\bound{lpr2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPagePatch28}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageFull28}{ZeroDimensionalSolvePackageXmpPageEmpty28}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageFull28}{\hidepaste}
+\tab{5}\spadcommand{[approximate(r,1/10**21)::Float for r in lpr2.1]\free{lpr2 }}
+\indentrel{3}\begin{verbatim}
+   (28)
+   [0.3221853546 2608559291, 0.3221853546 2608559291,
+    0.3221853546 2608559291, 0.3221853546 2608559291]
+                                       Type: List Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ZeroDimensionalSolvePackageXmpPageEmpty28}
+\begin{paste}{ZeroDimensionalSolvePackageXmpPageEmpty28}{ZeroDimensionalSolvePackageXmpPagePatch28}
+\pastebutton{ZeroDimensionalSolvePackageXmpPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{[approximate(r,1/10**21)::Float for r in lpr2.1]\free{lpr2 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ZLINDEP.ht b/src/hyper/pages/ZLINDEP.ht
new file mode 100644
index 00000000..b3b5601f
--- /dev/null
+++ b/src/hyper/pages/ZLINDEP.ht
@@ -0,0 +1,86 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\IntegerLinearDependenceXmpTitle}{IntegerLinearDependence}
+\newcommand{\IntegerLinearDependenceXmpNumber}{9.35}
+%
+% =====================================================================
+\begin{page}{IntegerLinearDependenceXmpPage}{9.35 IntegerLinearDependence}
+% =====================================================================
+\beginscroll
+
+
+The elements \texht{$v_1, \dots,v_n$}{\spad{v1,...,vn}}
+of a module \spad{M} over a ring \spad{R} are
+%-% \HDindex{linear dependence}{IntegerLinearDependenceXmpPage}{9.35}{IntegerLinearDependence}
+said to be {\it linearly dependent over \spad{R}} if there exist
+\texht{$c_1,\dots,c_n$}{\spad{c1, ..., cn}} in \spad{R},
+not all \smath{0},
+such that \texht{$c_1 v_1 + \dots c_n v_n = 0$}{\spad{c1*v1 + ... + cn*vn = 0}}.
+If such \texht{$c_i$}{\spad{ci}}'s exist,
+they form what is called a {\it linear dependence
+relation over \spad{R}} for the \texht{$v_i$}{\spad{vi}}'s.
+
+The package \spadtype{IntegerLinearDependence} provides functions
+for testing whether some elements of a module over the integers are
+linearly dependent over the integers, and to find the linear
+dependence relations, if any.
+%
+\xtc{
+Consider the domain of two by two square matrices with integer entries.
+}{
+\spadpaste{M := SQMATRIX(2,INT) \bound{M}}
+}
+%
+%
+\xtc{
+Now create three such matrices.
+}{
+\spadpaste{m1: M := squareMatrix matrix [[1, 2], [0, -1]] \free{M}\bound{m1}}
+}
+\xtc{
+}{
+\spadpaste{m2: M := squareMatrix matrix [[2, 3], [1, -2]] \free{M}\bound{m2}}
+}
+\xtc{
+}{
+\spadpaste{m3: M := squareMatrix matrix [[3, 4], [2, -3]] \free{M}\bound{m3}}
+}
+%
+%
+\xtc{
+This tells you whether \spad{m1}, \spad{m2} and \spad{m3} are linearly
+dependent over the integers.
+}{
+\spadpaste{linearlyDependentOverZ? vector [m1, m2, m3] \free{m1 m2 m3}}
+}
+%
+%
+\xtc{
+Since they are linearly dependent, you can ask for the dependence
+relation.
+}{
+\spadpaste{c := linearDependenceOverZ vector [m1, m2, m3] \free{m1 m2 m3}\bound{c}}
+}
+%
+%
+\xtc{
+This means that the following linear combination should be \spad{0}.
+}{
+\spadpaste{c.1 * m1 + c.2 * m2 + c.3 * m3 \free{c m1 m2 m3}}
+}
+%
+When a given set of elements are linearly dependent over \spad{R}, this
+also means that at least one of them can be rewritten as a linear
+combination of the others with coefficients in the quotient field of
+\spad{R}.
+%
+\xtc{
+To express a given element in terms of other elements, use the operation
+\spadfunFrom{solveLinearlyOverQ}{IntegerLinearDependence}.
+}{
+\spadpaste{solveLinearlyOverQ(vector [m1, m3], m2) \free{m1 m2 m3}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ZLINDEP.pht b/src/hyper/pages/ZLINDEP.pht
new file mode 100644
index 00000000..d60b9a96
--- /dev/null
+++ b/src/hyper/pages/ZLINDEP.pht
@@ -0,0 +1,138 @@
+\begin{patch}{IntegerLinearDependenceXmpPagePatch1}
+\begin{paste}{IntegerLinearDependenceXmpPageFull1}{IntegerLinearDependenceXmpPageEmpty1}
+\pastebutton{IntegerLinearDependenceXmpPageFull1}{\hidepaste}
+\tab{5}\spadcommand{M := SQMATRIX(2,INT)\bound{M }}
+\indentrel{3}\begin{verbatim}
+   (1)  SquareMatrix(2,Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPageEmpty1}
+\begin{paste}{IntegerLinearDependenceXmpPageEmpty1}{IntegerLinearDependenceXmpPagePatch1}
+\pastebutton{IntegerLinearDependenceXmpPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{M := SQMATRIX(2,INT)\bound{M }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPagePatch2}
+\begin{paste}{IntegerLinearDependenceXmpPageFull2}{IntegerLinearDependenceXmpPageEmpty2}
+\pastebutton{IntegerLinearDependenceXmpPageFull2}{\hidepaste}
+\tab{5}\spadcommand{m1: M := squareMatrix matrix [[1, 2], [0, -1]]\free{M }\bound{m1 }}
+\indentrel{3}\begin{verbatim}
+        Ú1   2 ¿
+   (2)  ³      ³
+        À0  - 1Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPageEmpty2}
+\begin{paste}{IntegerLinearDependenceXmpPageEmpty2}{IntegerLinearDependenceXmpPagePatch2}
+\pastebutton{IntegerLinearDependenceXmpPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{m1: M := squareMatrix matrix [[1, 2], [0, -1]]\free{M }\bound{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPagePatch3}
+\begin{paste}{IntegerLinearDependenceXmpPageFull3}{IntegerLinearDependenceXmpPageEmpty3}
+\pastebutton{IntegerLinearDependenceXmpPageFull3}{\hidepaste}
+\tab{5}\spadcommand{m2: M := squareMatrix matrix [[2, 3], [1, -2]]\free{M }\bound{m2 }}
+\indentrel{3}\begin{verbatim}
+        Ú2   3 ¿
+   (3)  ³      ³
+        À1  - 2Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPageEmpty3}
+\begin{paste}{IntegerLinearDependenceXmpPageEmpty3}{IntegerLinearDependenceXmpPagePatch3}
+\pastebutton{IntegerLinearDependenceXmpPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{m2: M := squareMatrix matrix [[2, 3], [1, -2]]\free{M }\bound{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPagePatch4}
+\begin{paste}{IntegerLinearDependenceXmpPageFull4}{IntegerLinearDependenceXmpPageEmpty4}
+\pastebutton{IntegerLinearDependenceXmpPageFull4}{\hidepaste}
+\tab{5}\spadcommand{m3: M := squareMatrix matrix [[3, 4], [2, -3]]\free{M }\bound{m3 }}
+\indentrel{3}\begin{verbatim}
+        Ú3   4 ¿
+   (4)  ³      ³
+        À2  - 3Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPageEmpty4}
+\begin{paste}{IntegerLinearDependenceXmpPageEmpty4}{IntegerLinearDependenceXmpPagePatch4}
+\pastebutton{IntegerLinearDependenceXmpPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{m3: M := squareMatrix matrix [[3, 4], [2, -3]]\free{M }\bound{m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPagePatch5}
+\begin{paste}{IntegerLinearDependenceXmpPageFull5}{IntegerLinearDependenceXmpPageEmpty5}
+\pastebutton{IntegerLinearDependenceXmpPageFull5}{\hidepaste}
+\tab{5}\spadcommand{linearlyDependentOverZ? vector [m1, m2, m3]\free{m1 m2 m3 }}
+\indentrel{3}\begin{verbatim}
+   (5)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPageEmpty5}
+\begin{paste}{IntegerLinearDependenceXmpPageEmpty5}{IntegerLinearDependenceXmpPagePatch5}
+\pastebutton{IntegerLinearDependenceXmpPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{linearlyDependentOverZ? vector [m1, m2, m3]\free{m1 m2 m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPagePatch6}
+\begin{paste}{IntegerLinearDependenceXmpPageFull6}{IntegerLinearDependenceXmpPageEmpty6}
+\pastebutton{IntegerLinearDependenceXmpPageFull6}{\hidepaste}
+\tab{5}\spadcommand{c := linearDependenceOverZ vector [m1, m2, m3]\free{m1 m2 m3 }\bound{c }}
+\indentrel{3}\begin{verbatim}
+   (6)  [1,- 2,1]
+                        Type: Union(Vector Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPageEmpty6}
+\begin{paste}{IntegerLinearDependenceXmpPageEmpty6}{IntegerLinearDependenceXmpPagePatch6}
+\pastebutton{IntegerLinearDependenceXmpPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{c := linearDependenceOverZ vector [m1, m2, m3]\free{m1 m2 m3 }\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPagePatch7}
+\begin{paste}{IntegerLinearDependenceXmpPageFull7}{IntegerLinearDependenceXmpPageEmpty7}
+\pastebutton{IntegerLinearDependenceXmpPageFull7}{\hidepaste}
+\tab{5}\spadcommand{c.1 * m1 + c.2 * m2 + c.3 * m3\free{c m1 m2 m3 }}
+\indentrel{3}\begin{verbatim}
+        Ú0  0¿
+   (7)  ³    ³
+        À0  0Ù
+                          Type: SquareMatrix(2,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPageEmpty7}
+\begin{paste}{IntegerLinearDependenceXmpPageEmpty7}{IntegerLinearDependenceXmpPagePatch7}
+\pastebutton{IntegerLinearDependenceXmpPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{c.1 * m1 + c.2 * m2 + c.3 * m3\free{c m1 m2 m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPagePatch8}
+\begin{paste}{IntegerLinearDependenceXmpPageFull8}{IntegerLinearDependenceXmpPageEmpty8}
+\pastebutton{IntegerLinearDependenceXmpPageFull8}{\hidepaste}
+\tab{5}\spadcommand{solveLinearlyOverQ(vector [m1, m3], m2)\free{m1 m2 m3 }}
+\indentrel{3}\begin{verbatim}
+         1 1
+   (8)  [Ä,Ä]
+         2 2
+               Type: Union(Vector Fraction Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerLinearDependenceXmpPageEmpty8}
+\begin{paste}{IntegerLinearDependenceXmpPageEmpty8}{IntegerLinearDependenceXmpPagePatch8}
+\pastebutton{IntegerLinearDependenceXmpPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{solveLinearlyOverQ(vector [m1, m3], m2)\free{m1 m2 m3 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/algebra.ht b/src/hyper/pages/algebra.ht
new file mode 100644
index 00000000..984ec071
--- /dev/null
+++ b/src/hyper/pages/algebra.ht
@@ -0,0 +1,40 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Algebra page
+% @(#)algebra.ht        1.3 91/06/27 00:44:40
+
+
+\begin{page}{AlgebraPage}{Abstract Algebra}
+\beginscroll
+\Language{} provides various facilities for treating topics in abstract
+algebra.
+\beginmenu
+\menulink{Number Theory}{NumberTheoryPage} \newline
+Topics in algebraic number theory.
+%\menulink{Algebraic Geometry}{AlgebraicGeometryPage} \newline
+%Computational algebraic geometry: Groebner bases, integral bases,
+%divisors on curves.
+\menulink{Group Theory}{GroupTheoryPage} \newline
+Permutation groups; representation theory.
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+
+% Number Theory Page
+
+\begin{page}{NumberTheoryPage}{Number Theory}
+\beginscroll
+Here are some sample computations using \Language{}'s algebraic number
+facilities.
+\beginmenu
+\menulink{Galois Groups}{ugProblemGaloisPage} \newline
+Computation of Galois groups using factorizations over number fields.
+\menulink{Number Theory Functions}{IntegerNumberTheoryFunctionsXmpPage}\newline
+Some functions of interest to number theorists.
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
diff --git a/src/hyper/pages/aspex.ht b/src/hyper/pages/aspex.ht
new file mode 100644
index 00000000..c25d4526
--- /dev/null
+++ b/src/hyper/pages/aspex.ht
@@ -0,0 +1,795 @@
+\begin{page}{Asp1ExampleCode}{Asp1 Example Code}
+\begin{verbatim}
+      DOUBLE PRECISION FUNCTION F(X)
+      DOUBLE PRECISION X
+      F=DSIN(X)
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+
+\begin{page}{Asp10ExampleCode}{Asp10 Example Code}
+\begin{verbatim}
+      SUBROUTINE COEFFN(P,Q,DQDL,X,ELAM,JINT)
+      DOUBLE PRECISION ELAM,P,Q,X,DQDL
+      INTEGER JINT
+      P=1.0D0
+      Q=((-1.0D0*X**3)+ELAM*X*X-2.0D0)/(X*X)
+      DQDL=1.0D0
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+
+
+\begin{page}{Asp12ExampleCode}{Asp12 Example Code}
+\begin{verbatim}
+      SUBROUTINE MONIT (MAXIT,IFLAG,ELAM,FINFO)
+      DOUBLE PRECISION ELAM,FINFO(15)
+      INTEGER MAXIT,IFLAG
+      IF(MAXIT.EQ.-1)THEN
+        PRINT*,"Output from Monit"
+      ENDIF
+      PRINT*,MAXIT,IFLAG,ELAM,(FINFO(I),I=1,4)
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp19ExampleCode}{Asp19 Example Code}
+\begin{verbatim}
+      SUBROUTINE LSFUN2(M,N,XC,FVECC,FJACC,LJC)
+      DOUBLE PRECISION FVECC(M),FJACC(LJC,N),XC(N)
+      INTEGER M,N,LJC
+      INTEGER I,J
+      DO 25003 I=1,LJC
+        DO 25004 J=1,N
+          FJACC(I,J)=0.0D0
+25004   CONTINUE
+25003 CONTINUE
+      FVECC(1)=((XC(1)-0.14D0)*XC(3)+(15.0D0*XC(1)-2.1D0)*XC(2)+1.0D0)/(
+     &XC(3)+15.0D0*XC(2))
+      FVECC(2)=((XC(1)-0.18D0)*XC(3)+(7.0D0*XC(1)-1.26D0)*XC(2)+1.0D0)/(
+     &XC(3)+7.0D0*XC(2))
+      FVECC(3)=((XC(1)-0.22D0)*XC(3)+(4.333333333333333D0*XC(1)-0.953333
+     &3333333333D0)*XC(2)+1.0D0)/(XC(3)+4.333333333333333D0*XC(2))
+      FVECC(4)=((XC(1)-0.25D0)*XC(3)+(3.0D0*XC(1)-0.75D0)*XC(2)+1.0D0)/(
+     &XC(3)+3.0D0*XC(2))
+      FVECC(5)=((XC(1)-0.29D0)*XC(3)+(2.2D0*XC(1)-0.6379999999999999D0)*
+     &XC(2)+1.0D0)/(XC(3)+2.2D0*XC(2))
+      FVECC(6)=((XC(1)-0.32D0)*XC(3)+(1.666666666666667D0*XC(1)-0.533333
+     &3333333333D0)*XC(2)+1.0D0)/(XC(3)+1.666666666666667D0*XC(2))
+      FVECC(7)=((XC(1)-0.35D0)*XC(3)+(1.285714285714286D0*XC(1)-0.45D0)*
+     &XC(2)+1.0D0)/(XC(3)+1.285714285714286D0*XC(2))
+      FVECC(8)=((XC(1)-0.39D0)*XC(3)+(XC(1)-0.39D0)*XC(2)+1.0D0)/(XC(3)+
+     &XC(2))
+      FVECC(9)=((XC(1)-0.37D0)*XC(3)+(XC(1)-0.37D0)*XC(2)+1.285714285714
+     &286D0)/(XC(3)+XC(2))
+      FVECC(10)=((XC(1)-0.58D0)*XC(3)+(XC(1)-0.58D0)*XC(2)+1.66666666666
+     &6667D0)/(XC(3)+XC(2))
+      FVECC(11)=((XC(1)-0.73D0)*XC(3)+(XC(1)-0.73D0)*XC(2)+2.2D0)/(XC(3)
+     &+XC(2))
+      FVECC(12)=((XC(1)-0.96D0)*XC(3)+(XC(1)-0.96D0)*XC(2)+3.0D0)/(XC(3)
+     &+XC(2))
+      FVECC(13)=((XC(1)-1.34D0)*XC(3)+(XC(1)-1.34D0)*XC(2)+4.33333333333
+     &3333D0)/(XC(3)+XC(2))
+      FVECC(14)=((XC(1)-2.1D0)*XC(3)+(XC(1)-2.1D0)*XC(2)+7.0D0)/(XC(3)+X
+     &C(2))
+      FVECC(15)=((XC(1)-4.39D0)*XC(3)+(XC(1)-4.39D0)*XC(2)+15.0D0)/(XC(3
+     &)+XC(2))
+      FJACC(1,1)=1.0D0
+      FJACC(1,2)=-15.0D0/(XC(3)**2+30.0D0*XC(2)*XC(3)+225.0D0*XC(2)**2)
+      FJACC(1,3)=-1.0D0/(XC(3)**2+30.0D0*XC(2)*XC(3)+225.0D0*XC(2)**2)
+      FJACC(2,1)=1.0D0
+      FJACC(2,2)=-7.0D0/(XC(3)**2+14.0D0*XC(2)*XC(3)+49.0D0*XC(2)**2)
+      FJACC(2,3)=-1.0D0/(XC(3)**2+14.0D0*XC(2)*XC(3)+49.0D0*XC(2)**2)
+      FJACC(3,1)=1.0D0
+      FJACC(3,2)=((-0.1110223024625157D-15*XC(3))-4.333333333333333D0)/(
+     &XC(3)**2+8.666666666666666D0*XC(2)*XC(3)+18.77777777777778D0*XC(2)
+     &**2)
+      FJACC(3,3)=(0.1110223024625157D-15*XC(2)-1.0D0)/(XC(3)**2+8.666666
+     &666666666D0*XC(2)*XC(3)+18.77777777777778D0*XC(2)**2)
+      FJACC(4,1)=1.0D0
+      FJACC(4,2)=-3.0D0/(XC(3)**2+6.0D0*XC(2)*XC(3)+9.0D0*XC(2)**2)
+      FJACC(4,3)=-1.0D0/(XC(3)**2+6.0D0*XC(2)*XC(3)+9.0D0*XC(2)**2)
+      FJACC(5,1)=1.0D0
+      FJACC(5,2)=((-0.1110223024625157D-15*XC(3))-2.2D0)/(XC(3)**2+4.399
+     &999999999999D0*XC(2)*XC(3)+4.839999999999998D0*XC(2)**2)
+      FJACC(5,3)=(0.1110223024625157D-15*XC(2)-1.0D0)/(XC(3)**2+4.399999
+     &999999999D0*XC(2)*XC(3)+4.839999999999998D0*XC(2)**2)
+      FJACC(6,1)=1.0D0
+      FJACC(6,2)=((-0.2220446049250313D-15*XC(3))-1.666666666666667D0)/(
+     &XC(3)**2+3.333333333333333D0*XC(2)*XC(3)+2.777777777777777D0*XC(2)
+     &**2)
+      FJACC(6,3)=(0.2220446049250313D-15*XC(2)-1.0D0)/(XC(3)**2+3.333333
+     &333333333D0*XC(2)*XC(3)+2.777777777777777D0*XC(2)**2)
+      FJACC(7,1)=1.0D0
+      FJACC(7,2)=((-0.5551115123125783D-16*XC(3))-1.285714285714286D0)/(
+     &XC(3)**2+2.571428571428571D0*XC(2)*XC(3)+1.653061224489796D0*XC(2)
+     &**2)
+      FJACC(7,3)=(0.5551115123125783D-16*XC(2)-1.0D0)/(XC(3)**2+2.571428
+     &571428571D0*XC(2)*XC(3)+1.653061224489796D0*XC(2)**2)
+      FJACC(8,1)=1.0D0
+      FJACC(8,2)=-1.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)
+      FJACC(8,3)=-1.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)
+      FJACC(9,1)=1.0D0
+      FJACC(9,2)=-1.285714285714286D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)*
+     &*2)
+      FJACC(9,3)=-1.285714285714286D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)*
+     &*2)
+      FJACC(10,1)=1.0D0
+      FJACC(10,2)=-1.666666666666667D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)
+     &**2)
+      FJACC(10,3)=-1.666666666666667D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)
+     &**2)
+      FJACC(11,1)=1.0D0
+      FJACC(11,2)=-2.2D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)
+      FJACC(11,3)=-2.2D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)
+      FJACC(12,1)=1.0D0
+      FJACC(12,2)=-3.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)
+      FJACC(12,3)=-3.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)
+      FJACC(13,1)=1.0D0
+      FJACC(13,2)=-4.333333333333333D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)
+     &**2)
+      FJACC(13,3)=-4.333333333333333D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)
+     &**2)
+      FJACC(14,1)=1.0D0
+      FJACC(14,2)=-7.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)
+      FJACC(14,3)=-7.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)
+      FJACC(15,1)=1.0D0
+      FJACC(15,2)=-15.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)
+      FJACC(15,3)=-15.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2)
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp20ExampleCode}{Asp20 Example Code}
+\begin{verbatim}
+      SUBROUTINE QPHESS(N,NROWH,NCOLH,JTHCOL,HESS,X,HX)
+      DOUBLE PRECISION HX(N),X(N),HESS(NROWH,NCOLH)
+      INTEGER JTHCOL,N,NROWH,NCOLH
+      HX(1)=2.0D0*X(1)
+      HX(2)=2.0D0*X(2)
+      HX(3)=2.0D0*X(4)+2.0D0*X(3)
+      HX(4)=2.0D0*X(4)+2.0D0*X(3)
+      HX(5)=2.0D0*X(5)
+      HX(6)=(-2.0D0*X(7))+(-2.0D0*X(6))
+      HX(7)=(-2.0D0*X(7))+(-2.0D0*X(6))
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp24ExampleCode}{Asp24 Example Code}
+\begin{verbatim}
+      SUBROUTINE FUNCT1(N,XC,FC)
+      DOUBLE PRECISION FC,XC(N)
+      INTEGER N
+      FC=10.0D0*XC(4)**4+(-40.0D0*XC(1)*XC(4)**3)+(60.0D0*XC(1)**2+5
+     &.0D0)*XC(4)**2+((-10.0D0*XC(3))+(-40.0D0*XC(1)**3))*XC(4)+16.0D0*X
+     &C(3)**4+(-32.0D0*XC(2)*XC(3)**3)+(24.0D0*XC(2)**2+5.0D0)*XC(3)**2+
+     &(-8.0D0*XC(2)**3*XC(3))+XC(2)**4+100.0D0*XC(2)**2+20.0D0*XC(1)*XC(
+     &2)+10.0D0*XC(1)**4+XC(1)**2
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp27ExampleCode}{Asp27 Example Code}
+\begin{verbatim}
+      FUNCTION DOT(IFLAG,N,Z,W,RWORK,LRWORK,IWORK,LIWORK)
+      DOUBLE PRECISION W(N),Z(N),RWORK(LRWORK)
+      INTEGER N,LIWORK,IFLAG,LRWORK,IWORK(LIWORK)
+      DOT=(W(16)+(-0.5D0*W(15)))*Z(16)+((-0.5D0*W(16))+W(15)+(-0.5D0*W(1
+     &4)))*Z(15)+((-0.5D0*W(15))+W(14)+(-0.5D0*W(13)))*Z(14)+((-0.5D0*W(
+     &14))+W(13)+(-0.5D0*W(12)))*Z(13)+((-0.5D0*W(13))+W(12)+(-0.5D0*W(1
+     &1)))*Z(12)+((-0.5D0*W(12))+W(11)+(-0.5D0*W(10)))*Z(11)+((-0.5D0*W(
+     &11))+W(10)+(-0.5D0*W(9)))*Z(10)+((-0.5D0*W(10))+W(9)+(-0.5D0*W(8))
+     &)*Z(9)+((-0.5D0*W(9))+W(8)+(-0.5D0*W(7)))*Z(8)+((-0.5D0*W(8))+W(7)
+     &+(-0.5D0*W(6)))*Z(7)+((-0.5D0*W(7))+W(6)+(-0.5D0*W(5)))*Z(6)+((-0.
+     &5D0*W(6))+W(5)+(-0.5D0*W(4)))*Z(5)+((-0.5D0*W(5))+W(4)+(-0.5D0*W(3
+     &)))*Z(4)+((-0.5D0*W(4))+W(3)+(-0.5D0*W(2)))*Z(3)+((-0.5D0*W(3))+W(
+     &2)+(-0.5D0*W(1)))*Z(2)+((-0.5D0*W(2))+W(1))*Z(1)
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp28ExampleCode}{Asp28 Example Code}
+\begin{verbatim}
+      SUBROUTINE IMAGE(IFLAG,N,Z,W,RWORK,LRWORK,IWORK,LIWORK)
+      DOUBLE PRECISION Z(N),W(N),IWORK(LRWORK),RWORK(LRWORK)
+      INTEGER N,LIWORK,IFLAG,LRWORK
+      W(1)=0.01707454969713436D0*Z(16)+0.001747395874954051D0*Z(15)+0.00
+     &2106973900813502D0*Z(14)+0.002957434991769087D0*Z(13)+(-0.00700554
+     &0882865317D0*Z(12))+(-0.01219194009813166D0*Z(11))+0.0037230647365
+     &3087D0*Z(10)+0.04932374658377151D0*Z(9)+(-0.03586220812223305D0*Z(
+     &8))+(-0.04723268012114625D0*Z(7))+(-0.02434652144032987D0*Z(6))+0.
+     &2264766947290192D0*Z(5)+(-0.1385343580686922D0*Z(4))+(-0.116530050
+     &8238904D0*Z(3))+(-0.2803531651057233D0*Z(2))+1.019463911841327D0*Z
+     &(1)
+      W(2)=0.0227345011107737D0*Z(16)+0.008812321197398072D0*Z(15)+0.010
+     &94012210519586D0*Z(14)+(-0.01764072463999744D0*Z(13))+(-0.01357136
+     &72105995D0*Z(12))+0.00157466157362272D0*Z(11)+0.05258889186338282D
+     &0*Z(10)+(-0.01981532388243379D0*Z(9))+(-0.06095390688679697D0*Z(8)
+     &)+(-0.04153119955569051D0*Z(7))+0.2176561076571465D0*Z(6)+(-0.0532
+     &5555586632358D0*Z(5))+(-0.1688977368984641D0*Z(4))+(-0.32440166056
+     &67343D0*Z(3))+0.9128222941872173D0*Z(2)+(-0.2419652703415429D0*Z(1
+     &))
+      W(3)=0.03371198197190302D0*Z(16)+0.02021603150122265D0*Z(15)+(-0.0
+     &06607305534689702D0*Z(14))+(-0.03032392238968179D0*Z(13))+0.002033
+     &305231024948D0*Z(12)+0.05375944956767728D0*Z(11)+(-0.0163213312502
+     &9967D0*Z(10))+(-0.05483186562035512D0*Z(9))+(-0.04901428822579872D
+     &0*Z(8))+0.2091097927887612D0*Z(7)+(-0.05760560341383113D0*Z(6))+(-
+     &0.1236679206156403D0*Z(5))+(-0.3523683853026259D0*Z(4))+0.88929961
+     &32269974D0*Z(3)+(-0.2995429545781457D0*Z(2))+(-0.02986582812574917
+     &D0*Z(1))
+      W(4)=0.05141563713660119D0*Z(16)+0.005239165960779299D0*Z(15)+(-0.
+     &01623427735779699D0*Z(14))+(-0.01965809746040371D0*Z(13))+0.054688
+     &97337339577D0*Z(12)+(-0.014224695935687D0*Z(11))+(-0.0505181779315
+     &6355D0*Z(10))+(-0.04353074206076491D0*Z(9))+0.2012230497530726D0*Z
+     &(8)+(-0.06630874514535952D0*Z(7))+(-0.1280829963720053D0*Z(6))+(-0
+     &.305169742604165D0*Z(5))+0.8600427128450191D0*Z(4)+(-0.32415033802
+     &68184D0*Z(3))+(-0.09033531980693314D0*Z(2))+0.09089205517109111D0*
+     &Z(1)
+      W(5)=0.04556369767776375D0*Z(16)+(-0.001822737697581869D0*Z(15))+(
+     &-0.002512226501941856D0*Z(14))+0.02947046460707379D0*Z(13)+(-0.014
+     &45079632086177D0*Z(12))+(-0.05034242196614937D0*Z(11))+(-0.0376966
+     &3291725935D0*Z(10))+0.2171103102175198D0*Z(9)+(-0.0824949256021352
+     &4D0*Z(8))+(-0.1473995209288945D0*Z(7))+(-0.315042193418466D0*Z(6))
+     &+0.9591623347824002D0*Z(5)+(-0.3852396953763045D0*Z(4))+(-0.141718
+     &5427288274D0*Z(3))+(-0.03423495461011043D0*Z(2))+0.319820917706851
+     &6D0*Z(1)
+      W(6)=0.04015147277405744D0*Z(16)+0.01328585741341559D0*Z(15)+0.048
+     &26082005465965D0*Z(14)+(-0.04319641116207706D0*Z(13))+(-0.04931323
+     &319055762D0*Z(12))+(-0.03526886317505474D0*Z(11))+0.22295383396730
+     &01D0*Z(10)+(-0.07375317649315155D0*Z(9))+(-0.1589391311991561D0*Z(
+     &8))+(-0.328001910890377D0*Z(7))+0.952576555482747D0*Z(6)+(-0.31583
+     &09975786731D0*Z(5))+(-0.1846882042225383D0*Z(4))+(-0.0703762046700
+     &4427D0*Z(3))+0.2311852964327382D0*Z(2)+0.04254083491825025D0*Z(1)
+      W(7)=0.06069778964023718D0*Z(16)+0.06681263884671322D0*Z(15)+(-0.0
+     &2113506688615768D0*Z(14))+(-0.083996867458326D0*Z(13))+(-0.0329843
+     &8523869648D0*Z(12))+0.2276878326327734D0*Z(11)+(-0.067356038933017
+     &95D0*Z(10))+(-0.1559813965382218D0*Z(9))+(-0.3363262957694705D0*Z(
+     &8))+0.9442791158560948D0*Z(7)+(-0.3199955249404657D0*Z(6))+(-0.136
+     &2463839920727D0*Z(5))+(-0.1006185171570586D0*Z(4))+0.2057504515015
+     &423D0*Z(3)+(-0.02065879269286707D0*Z(2))+0.03160990266745513D0*Z(1
+     &)
+      W(8)=0.126386868896738D0*Z(16)+0.002563370039476418D0*Z(15)+(-0.05
+     &581757739455641D0*Z(14))+(-0.07777893205900685D0*Z(13))+0.23117338
+     &45834199D0*Z(12)+(-0.06031581134427592D0*Z(11))+(-0.14805474755869
+     &52D0*Z(10))+(-0.3364014128402243D0*Z(9))+0.9364014128402244D0*Z(8)
+     &+(-0.3269452524413048D0*Z(7))+(-0.1396841886557241D0*Z(6))+(-0.056
+     &1733845834199D0*Z(5))+0.1777789320590069D0*Z(4)+(-0.04418242260544
+     &359D0*Z(3))+(-0.02756337003947642D0*Z(2))+0.07361313110326199D0*Z(
+     &1)
+      W(9)=0.07361313110326199D0*Z(16)+(-0.02756337003947642D0*Z(15))+(-
+     &0.04418242260544359D0*Z(14))+0.1777789320590069D0*Z(13)+(-0.056173
+     &3845834199D0*Z(12))+(-0.1396841886557241D0*Z(11))+(-0.326945252441
+     &3048D0*Z(10))+0.9364014128402244D0*Z(9)+(-0.3364014128402243D0*Z(8
+     &))+(-0.1480547475586952D0*Z(7))+(-0.06031581134427592D0*Z(6))+0.23
+     &11733845834199D0*Z(5)+(-0.07777893205900685D0*Z(4))+(-0.0558175773
+     &9455641D0*Z(3))+0.002563370039476418D0*Z(2)+0.126386868896738D0*Z(
+     &1)
+      W(10)=0.03160990266745513D0*Z(16)+(-0.02065879269286707D0*Z(15))+0
+     &.2057504515015423D0*Z(14)+(-0.1006185171570586D0*Z(13))+(-0.136246
+     &3839920727D0*Z(12))+(-0.3199955249404657D0*Z(11))+0.94427911585609
+     &48D0*Z(10)+(-0.3363262957694705D0*Z(9))+(-0.1559813965382218D0*Z(8
+     &))+(-0.06735603893301795D0*Z(7))+0.2276878326327734D0*Z(6)+(-0.032
+     &98438523869648D0*Z(5))+(-0.083996867458326D0*Z(4))+(-0.02113506688
+     &615768D0*Z(3))+0.06681263884671322D0*Z(2)+0.06069778964023718D0*Z(
+     &1)
+      W(11)=0.04254083491825025D0*Z(16)+0.2311852964327382D0*Z(15)+(-0.0
+     &7037620467004427D0*Z(14))+(-0.1846882042225383D0*Z(13))+(-0.315830
+     &9975786731D0*Z(12))+0.952576555482747D0*Z(11)+(-0.328001910890377D
+     &0*Z(10))+(-0.1589391311991561D0*Z(9))+(-0.07375317649315155D0*Z(8)
+     &)+0.2229538339673001D0*Z(7)+(-0.03526886317505474D0*Z(6))+(-0.0493
+     &1323319055762D0*Z(5))+(-0.04319641116207706D0*Z(4))+0.048260820054
+     &65965D0*Z(3)+0.01328585741341559D0*Z(2)+0.04015147277405744D0*Z(1)
+      W(12)=0.3198209177068516D0*Z(16)+(-0.03423495461011043D0*Z(15))+(-
+     &0.1417185427288274D0*Z(14))+(-0.3852396953763045D0*Z(13))+0.959162
+     &3347824002D0*Z(12)+(-0.315042193418466D0*Z(11))+(-0.14739952092889
+     &45D0*Z(10))+(-0.08249492560213524D0*Z(9))+0.2171103102175198D0*Z(8
+     &)+(-0.03769663291725935D0*Z(7))+(-0.05034242196614937D0*Z(6))+(-0.
+     &01445079632086177D0*Z(5))+0.02947046460707379D0*Z(4)+(-0.002512226
+     &501941856D0*Z(3))+(-0.001822737697581869D0*Z(2))+0.045563697677763
+     &75D0*Z(1)
+      W(13)=0.09089205517109111D0*Z(16)+(-0.09033531980693314D0*Z(15))+(
+     &-0.3241503380268184D0*Z(14))+0.8600427128450191D0*Z(13)+(-0.305169
+     &742604165D0*Z(12))+(-0.1280829963720053D0*Z(11))+(-0.0663087451453
+     &5952D0*Z(10))+0.2012230497530726D0*Z(9)+(-0.04353074206076491D0*Z(
+     &8))+(-0.05051817793156355D0*Z(7))+(-0.014224695935687D0*Z(6))+0.05
+     &468897337339577D0*Z(5)+(-0.01965809746040371D0*Z(4))+(-0.016234277
+     &35779699D0*Z(3))+0.005239165960779299D0*Z(2)+0.05141563713660119D0
+     &*Z(1)
+      W(14)=(-0.02986582812574917D0*Z(16))+(-0.2995429545781457D0*Z(15))
+     &+0.8892996132269974D0*Z(14)+(-0.3523683853026259D0*Z(13))+(-0.1236
+     &679206156403D0*Z(12))+(-0.05760560341383113D0*Z(11))+0.20910979278
+     &87612D0*Z(10)+(-0.04901428822579872D0*Z(9))+(-0.05483186562035512D
+     &0*Z(8))+(-0.01632133125029967D0*Z(7))+0.05375944956767728D0*Z(6)+0
+     &.002033305231024948D0*Z(5)+(-0.03032392238968179D0*Z(4))+(-0.00660
+     &7305534689702D0*Z(3))+0.02021603150122265D0*Z(2)+0.033711981971903
+     &02D0*Z(1)
+      W(15)=(-0.2419652703415429D0*Z(16))+0.9128222941872173D0*Z(15)+(-0
+     &.3244016605667343D0*Z(14))+(-0.1688977368984641D0*Z(13))+(-0.05325
+     &555586632358D0*Z(12))+0.2176561076571465D0*Z(11)+(-0.0415311995556
+     &9051D0*Z(10))+(-0.06095390688679697D0*Z(9))+(-0.01981532388243379D
+     &0*Z(8))+0.05258889186338282D0*Z(7)+0.00157466157362272D0*Z(6)+(-0.
+     &0135713672105995D0*Z(5))+(-0.01764072463999744D0*Z(4))+0.010940122
+     &10519586D0*Z(3)+0.008812321197398072D0*Z(2)+0.0227345011107737D0*Z
+     &(1)
+      W(16)=1.019463911841327D0*Z(16)+(-0.2803531651057233D0*Z(15))+(-0.
+     &1165300508238904D0*Z(14))+(-0.1385343580686922D0*Z(13))+0.22647669
+     &47290192D0*Z(12)+(-0.02434652144032987D0*Z(11))+(-0.04723268012114
+     &625D0*Z(10))+(-0.03586220812223305D0*Z(9))+0.04932374658377151D0*Z
+     &(8)+0.00372306473653087D0*Z(7)+(-0.01219194009813166D0*Z(6))+(-0.0
+     &07005540882865317D0*Z(5))+0.002957434991769087D0*Z(4)+0.0021069739
+     &00813502D0*Z(3)+0.001747395874954051D0*Z(2)+0.01707454969713436D0*
+     &Z(1)
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp29ExampleCode}{Asp29 Example Code}
+\begin{verbatim}
+      SUBROUTINE MONIT(ISTATE,NEXTIT,NEVALS,NEVECS,K,F,D)
+      DOUBLE PRECISION D(K),F(K)
+      INTEGER K,NEXTIT,NEVALS,NVECS,ISTATE
+      CALL F02FJZ(ISTATE,NEXTIT,NEVALS,NEVECS,K,F,D)
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp30ExampleCode}{Asp30 Example Code}
+\begin{verbatim}
+      SUBROUTINE APROD(MODE,M,N,X,Y,RWORK,LRWORK,IWORK,LIWORK)
+      DOUBLE PRECISION X(N),Y(M),RWORK(LRWORK)
+      INTEGER M,N,LIWORK,IFAIL,LRWORK,IWORK(LIWORK),MODE
+      DOUBLE PRECISION A(5,5)
+      EXTERNAL F06PAF
+      A(1,1)=1.0D0
+      A(1,2)=0.0D0
+      A(1,3)=0.0D0
+      A(1,4)=-1.0D0
+      A(1,5)=0.0D0
+      A(2,1)=0.0D0
+      A(2,2)=1.0D0
+      A(2,3)=0.0D0
+      A(2,4)=0.0D0
+      A(2,5)=-1.0D0
+      A(3,1)=0.0D0
+      A(3,2)=0.0D0
+      A(3,3)=1.0D0
+      A(3,4)=-1.0D0
+      A(3,5)=0.0D0
+      A(4,1)=-1.0D0
+      A(4,2)=0.0D0
+      A(4,3)=-1.0D0
+      A(4,4)=4.0D0
+      A(4,5)=-1.0D0
+      A(5,1)=0.0D0
+      A(5,2)=-1.0D0
+      A(5,3)=0.0D0
+      A(5,4)=-1.0D0
+      A(5,5)=4.0D0
+      IF(MODE.EQ.1)THEN
+        CALL F06PAF('N',M,N,1.0D0,A,M,X,1,1.0D0,Y,1)
+      ELSEIF(MODE.EQ.2)THEN
+        CALL F06PAF('T',M,N,1.0D0,A,M,Y,1,1.0D0,X,1)
+      ENDIF
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp31ExampleCode}{Asp31 Example Code}
+\begin{verbatim}
+      SUBROUTINE PEDERV(X,Y,PW)
+      DOUBLE PRECISION X,Y(*)
+      DOUBLE PRECISION PW(3,3)
+      PW(1,1)=-0.03999999999999999D0
+      PW(1,2)=10000.0D0*Y(3)
+      PW(1,3)=10000.0D0*Y(2)
+      PW(2,1)=0.03999999999999999D0
+      PW(2,2)=(-10000.0D0*Y(3))+(-60000000.0D0*Y(2))
+      PW(2,3)=-10000.0D0*Y(2)
+      PW(3,1)=0.0D0
+      PW(3,2)=60000000.0D0*Y(2)
+      PW(3,3)=0.0D0
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp33ExampleCode}{Asp33 Example Code}
+\begin{verbatim}
+      SUBROUTINE REPORT(X,V,JINT)
+      DOUBLE PRECISION V(3),X
+      INTEGER JINT
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp34ExampleCode}{Asp34 Example Code}
+\begin{verbatim}
+      SUBROUTINE MSOLVE(IFLAG,N,X,Y,RWORK,LRWORK,IWORK,LIWORK)
+      DOUBLE PRECISION RWORK(LRWORK),X(N),Y(N)
+      INTEGER I,J,N,LIWORK,IFLAG,LRWORK,IWORK(LIWORK)
+      DOUBLE PRECISION W1(3),W2(3),MS(3,3)
+      IFLAG=-1
+      MS(1,1)=2.0D0
+      MS(1,2)=1.0D0
+      MS(1,3)=0.0D0
+      MS(2,1)=1.0D0
+      MS(2,2)=2.0D0
+      MS(2,3)=1.0D0
+      MS(3,1)=0.0D0
+      MS(3,2)=1.0D0
+      MS(3,3)=2.0D0
+      CALL F04ASF(MS,N,X,N,Y,W1,W2,IFLAG)
+      IFLAG=-IFLAG
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp35ExampleCode}{Asp35 Example Code}
+\begin{verbatim}
+      SUBROUTINE FCN(N,X,FVEC,FJAC,LDFJAC,IFLAG)
+      DOUBLE PRECISION X(N),FVEC(N),FJAC(LDFJAC,N)
+      INTEGER LDFJAC,N,IFLAG
+      IF(IFLAG.EQ.1)THEN
+        FVEC(1)=(-1.0D0*X(2))+X(1)
+        FVEC(2)=(-1.0D0*X(3))+2.0D0*X(2)
+        FVEC(3)=3.0D0*X(3)
+      ELSEIF(IFLAG.EQ.2)THEN
+        FJAC(1,1)=1.0D0
+        FJAC(1,2)=-1.0D0
+        FJAC(1,3)=0.0D0
+        FJAC(2,1)=0.0D0
+        FJAC(2,2)=2.0D0
+        FJAC(2,3)=-1.0D0
+        FJAC(3,1)=0.0D0
+        FJAC(3,2)=0.0D0
+        FJAC(3,3)=3.0D0
+      ENDIF
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp4ExampleCode}{Asp4 Example Code}
+\begin{verbatim}
+      DOUBLE PRECISION FUNCTION FUNCTN(NDIM,X)
+      DOUBLE PRECISION X(NDIM)
+      INTEGER NDIM
+      FUNCTN=(4.0D0*X(1)*X(3)**2*DEXP(2.0D0*X(1)*X(3)))/(X(4)**2+(2.0D0*
+     &X(2)+2.0D0)*X(4)+X(2)**2+2.0D0*X(2)+1.0D0)
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp41ExampleCode}{Asp41 Example Code}
+\begin{verbatim}
+      SUBROUTINE FCN(X,EPS,Y,F,N)
+      DOUBLE PRECISION EPS,F(N),X,Y(N)
+      INTEGER N
+      F(1)=Y(2)
+      F(2)=Y(3)
+      F(3)=(-1.0D0*Y(1)*Y(3))+2.0D0*EPS*Y(2)**2+(-2.0D0*EPS)
+      RETURN
+      END
+      SUBROUTINE JACOBF(X,EPS,Y,F,N)
+      DOUBLE PRECISION EPS,F(N,N),X,Y(N)
+      INTEGER N
+      F(1,1)=0.0D0
+      F(1,2)=1.0D0
+      F(1,3)=0.0D0
+      F(2,1)=0.0D0
+      F(2,2)=0.0D0
+      F(2,3)=1.0D0
+      F(3,1)=-1.0D0*Y(3)
+      F(3,2)=4.0D0*EPS*Y(2)
+      F(3,3)=-1.0D0*Y(1)
+      RETURN
+      END
+      SUBROUTINE JACEPS(X,EPS,Y,F,N)
+      DOUBLE PRECISION EPS,F(N),X,Y(N)
+      INTEGER N
+      F(1)=0.0D0
+      F(2)=0.0D0
+      F(3)=2.0D0*Y(2)**2-2.0D0
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp42ExampleCode}{Asp42 Example Code}
+\begin{verbatim}
+      SUBROUTINE G(EPS,YA,YB,BC,N)
+      DOUBLE PRECISION EPS,YA(N),YB(N),BC(N)
+      INTEGER N
+      BC(1)=YA(1)
+      BC(2)=YA(2)
+      BC(3)=YB(2)-1.0D0
+      RETURN
+      END
+      SUBROUTINE JACOBG(EPS,YA,YB,AJ,BJ,N)
+      DOUBLE PRECISION EPS,YA(N),AJ(N,N),BJ(N,N),YB(N)
+      INTEGER N
+      AJ(1,1)=1.0D0
+      AJ(1,2)=0.0D0
+      AJ(1,3)=0.0D0
+      AJ(2,1)=0.0D0
+      AJ(2,2)=1.0D0
+      AJ(2,3)=0.0D0
+      AJ(3,1)=0.0D0
+      AJ(3,2)=0.0D0
+      AJ(3,3)=0.0D0
+      BJ(1,1)=0.0D0
+      BJ(1,2)=0.0D0
+      BJ(1,3)=0.0D0
+      BJ(2,1)=0.0D0
+      BJ(2,2)=0.0D0
+      BJ(2,3)=0.0D0
+      BJ(3,1)=0.0D0
+      BJ(3,2)=1.0D0
+      BJ(3,3)=0.0D0
+      RETURN
+      END
+      SUBROUTINE JACGEP(EPS,YA,YB,BCEP,N)
+      DOUBLE PRECISION EPS,YA(N),YB(N),BCEP(N)
+      INTEGER N
+      BCEP(1)=0.0D0
+      BCEP(2)=0.0D0
+      BCEP(3)=0.0D0
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp49ExampleCode}{Asp49 Example Code}
+\begin{verbatim}
+      SUBROUTINE OBJFUN(MODE,N,X,OBJF,OBJGRD,NSTATE,IUSER,USER)
+      DOUBLE PRECISION X(N),OBJF,OBJGRD(N),USER(*)
+      INTEGER N,IUSER(*),MODE,NSTATE
+      OBJF=X(4)*X(9)+((-1.0D0*X(5))+X(3))*X(8)+((-1.0D0*X(3))+X(1))*X(7)
+     &+(-1.0D0*X(2)*X(6))
+      OBJGRD(1)=X(7)
+      OBJGRD(2)=-1.0D0*X(6)
+      OBJGRD(3)=X(8)+(-1.0D0*X(7))
+      OBJGRD(4)=X(9)
+      OBJGRD(5)=-1.0D0*X(8)
+      OBJGRD(6)=-1.0D0*X(2)
+      OBJGRD(7)=(-1.0D0*X(3))+X(1)
+      OBJGRD(8)=(-1.0D0*X(5))+X(3)
+      OBJGRD(9)=X(4)
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp50ExampleCode}{Asp50 Example Code}
+\begin{verbatim}
+      SUBROUTINE LSFUN1(M,N,XC,FVECC)
+      DOUBLE PRECISION FVECC(M),XC(N)
+      INTEGER I,M,N
+      FVECC(1)=((XC(1)-2.4D0)*XC(3)+(15.0D0*XC(1)-36.0D0)*XC(2)+1.0D0)/(
+     &XC(3)+15.0D0*XC(2))
+      FVECC(2)=((XC(1)-2.8D0)*XC(3)+(7.0D0*XC(1)-19.6D0)*XC(2)+1.0D0)/(X
+     &C(3)+7.0D0*XC(2))
+      FVECC(3)=((XC(1)-3.2D0)*XC(3)+(4.333333333333333D0*XC(1)-13.866666
+     &66666667D0)*XC(2)+1.0D0)/(XC(3)+4.333333333333333D0*XC(2))
+      FVECC(4)=((XC(1)-3.5D0)*XC(3)+(3.0D0*XC(1)-10.5D0)*XC(2)+1.0D0)/(X
+     &C(3)+3.0D0*XC(2))
+      FVECC(5)=((XC(1)-3.9D0)*XC(3)+(2.2D0*XC(1)-8.579999999999998D0)*XC
+     &(2)+1.0D0)/(XC(3)+2.2D0*XC(2))
+      FVECC(6)=((XC(1)-4.199999999999999D0)*XC(3)+(1.666666666666667D0*X
+     &C(1)-7.0D0)*XC(2)+1.0D0)/(XC(3)+1.666666666666667D0*XC(2))
+      FVECC(7)=((XC(1)-4.5D0)*XC(3)+(1.285714285714286D0*XC(1)-5.7857142
+     &85714286D0)*XC(2)+1.0D0)/(XC(3)+1.285714285714286D0*XC(2))
+      FVECC(8)=((XC(1)-4.899999999999999D0)*XC(3)+(XC(1)-4.8999999999999
+     &99D0)*XC(2)+1.0D0)/(XC(3)+XC(2))
+      FVECC(9)=((XC(1)-4.699999999999999D0)*XC(3)+(XC(1)-4.6999999999999
+     &99D0)*XC(2)+1.285714285714286D0)/(XC(3)+XC(2))
+      FVECC(10)=((XC(1)-6.8D0)*XC(3)+(XC(1)-6.8D0)*XC(2)+1.6666666666666
+     &67D0)/(XC(3)+XC(2))
+      FVECC(11)=((XC(1)-8.299999999999999D0)*XC(3)+(XC(1)-8.299999999999
+     &999D0)*XC(2)+2.2D0)/(XC(3)+XC(2))
+      FVECC(12)=((XC(1)-10.6D0)*XC(3)+(XC(1)-10.6D0)*XC(2)+3.0D0)/(XC(3)
+     &+XC(2))
+      FVECC(13)=((XC(1)-1.34D0)*XC(3)+(XC(1)-1.34D0)*XC(2)+4.33333333333
+     &3333D0)/(XC(3)+XC(2))
+      FVECC(14)=((XC(1)-2.1D0)*XC(3)+(XC(1)-2.1D0)*XC(2)+7.0D0)/(XC(3)+X
+     &C(2))
+      FVECC(15)=((XC(1)-4.39D0)*XC(3)+(XC(1)-4.39D0)*XC(2)+15.0D0)/(XC(3
+     &)+XC(2))
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp55ExampleCode}{Asp55 Example Code}
+\begin{verbatim}
+      SUBROUTINE CONFUN(MODE,NCNLN,N,NROWJ,NEEDC,X,C,CJAC,NSTATE,IUSER
+     &,USER)
+      DOUBLE PRECISION C(NCNLN),X(N),CJAC(NROWJ,N),USER(*)
+      INTEGER N,IUSER(*),NEEDC(NCNLN),NROWJ,MODE,NCNLN,NSTATE
+      IF(NEEDC(1).GT.0)THEN
+        C(1)=X(6)**2+X(1)**2
+        CJAC(1,1)=2.0D0*X(1)
+        CJAC(1,2)=0.0D0
+        CJAC(1,3)=0.0D0
+        CJAC(1,4)=0.0D0
+        CJAC(1,5)=0.0D0
+        CJAC(1,6)=2.0D0*X(6)
+      ENDIF
+      IF(NEEDC(2).GT.0)THEN
+        C(2)=X(2)**2+(-2.0D0*X(1)*X(2))+X(1)**2
+        CJAC(2,1)=(-2.0D0*X(2))+2.0D0*X(1)
+        CJAC(2,2)=2.0D0*X(2)+(-2.0D0*X(1))
+        CJAC(2,3)=0.0D0
+        CJAC(2,4)=0.0D0
+        CJAC(2,5)=0.0D0
+        CJAC(2,6)=0.0D0
+      ENDIF
+      IF(NEEDC(3).GT.0)THEN
+        C(3)=X(3)**2+(-2.0D0*X(1)*X(3))+X(2)**2+X(1)**2
+        CJAC(3,1)=(-2.0D0*X(3))+2.0D0*X(1)
+        CJAC(3,2)=2.0D0*X(2)
+        CJAC(3,3)=2.0D0*X(3)+(-2.0D0*X(1))
+        CJAC(3,4)=0.0D0
+        CJAC(3,5)=0.0D0
+        CJAC(3,6)=0.0D0
+      ENDIF
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp6ExampleCode}{Asp6 Example Code}
+\begin{verbatim}
+      SUBROUTINE FCN(N,X,FVEC,IFLAG)
+      DOUBLE PRECISION X(N),FVEC(N)
+      INTEGER N,IFLAG
+      FVEC(1)=(-2.0D0*X(2))+(-2.0D0*X(1)**2)+3.0D0*X(1)+1.0D0
+      FVEC(2)=(-2.0D0*X(3))+(-2.0D0*X(2)**2)+3.0D0*X(2)+(-1.0D0*X(1))+1.
+     &0D0
+      FVEC(3)=(-2.0D0*X(4))+(-2.0D0*X(3)**2)+3.0D0*X(3)+(-1.0D0*X(2))+1.
+     &0D0
+      FVEC(4)=(-2.0D0*X(5))+(-2.0D0*X(4)**2)+3.0D0*X(4)+(-1.0D0*X(3))+1.
+     &0D0
+      FVEC(5)=(-2.0D0*X(6))+(-2.0D0*X(5)**2)+3.0D0*X(5)+(-1.0D0*X(4))+1.
+     &0D0
+      FVEC(6)=(-2.0D0*X(7))+(-2.0D0*X(6)**2)+3.0D0*X(6)+(-1.0D0*X(5))+1.
+     &0D0
+      FVEC(7)=(-2.0D0*X(8))+(-2.0D0*X(7)**2)+3.0D0*X(7)+(-1.0D0*X(6))+1.
+     &0D0
+      FVEC(8)=(-2.0D0*X(9))+(-2.0D0*X(8)**2)+3.0D0*X(8)+(-1.0D0*X(7))+1.
+     &0D0
+      FVEC(9)=(-2.0D0*X(9)**2)+3.0D0*X(9)+(-1.0D0*X(8))+1.0D0
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp7ExampleCode}{Asp7 Example Code}
+\begin{verbatim}
+      SUBROUTINE FCN(X,Z,F)
+      DOUBLE PRECISION F(*),X,Z(*)
+      F(1)=DTAN(Z(3))
+      F(2)=((-0.03199999999999999D0*DCOS(Z(3))*DTAN(Z(3)))+(-0.02D0*Z(2)
+     &**2))/(Z(2)*DCOS(Z(3)))
+      F(3)=-0.03199999999999999D0/(X*Z(2)**2)
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp73ExampleCode}{Asp73 Example Code}
+\begin{verbatim}
+      SUBROUTINE PDEF(X,Y,ALPHA,BETA,GAMMA,DELTA,EPSOLN,PHI,PSI)
+      DOUBLE PRECISION ALPHA,EPSOLN,PHI,X,Y,BETA,DELTA,GAMMA,PSI
+      ALPHA=DSIN(X)
+      BETA=Y
+      GAMMA=X*Y
+      DELTA=DCOS(X)*DSIN(Y)
+      EPSOLN=Y+X
+      PHI=X
+      PSI=Y
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp74ExampleCode}{Asp74 Example Code}
+\begin{verbatim}
+      SUBROUTINE BNDY(X,Y,A,B,C,IBND)
+      DOUBLE PRECISION A,B,C,X,Y
+      INTEGER IBND
+      IF(IBND.EQ.0)THEN
+        A=0.0D0
+        B=1.0D0
+        C=-1.0D0*DSIN(X)
+      ELSEIF(IBND.EQ.1)THEN
+        A=1.0D0
+        B=0.0D0
+        C=DSIN(X)*DSIN(Y)
+      ELSEIF(IBND.EQ.2)THEN
+        A=1.0D0
+        B=0.0D0
+        C=DSIN(X)*DSIN(Y)
+      ELSEIF(IBND.EQ.3)THEN
+        A=0.0D0
+        B=1.0D0
+        C=-1.0D0*DSIN(Y)
+      ENDIF
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp77ExampleCode}{Asp77 Example Code}
+\begin{verbatim}
+      SUBROUTINE FCNF(X,F)
+      DOUBLE PRECISION X
+      DOUBLE PRECISION F(2,2)
+      F(1,1)=0.0D0
+      F(1,2)=1.0D0
+      F(2,1)=0.0D0
+      F(2,2)=-10.0D0
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp78ExampleCode}{Asp78 Example Code}
+\begin{verbatim}
+      SUBROUTINE FCNG(X,G)
+      DOUBLE PRECISION G(*),X
+      G(1)=0.0D0
+      G(2)=0.0D0
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp8ExampleCode}{Asp8 Example Code}
+\begin{verbatim}
+      SUBROUTINE OUTPUT(XSOL,Y,COUNT,M,N,RESULT,FORWRD)
+      DOUBLE PRECISION Y(N),RESULT(M,N),XSOL
+      INTEGER M,N,COUNT
+      LOGICAL FORWRD
+      DOUBLE PRECISION X02ALF,POINTS(8)
+      EXTERNAL X02ALF
+      INTEGER I
+      POINTS(1)=1.0D0
+      POINTS(2)=2.0D0
+      POINTS(3)=3.0D0
+      POINTS(4)=4.0D0
+      POINTS(5)=5.0D0
+      POINTS(6)=6.0D0
+      POINTS(7)=7.0D0
+      POINTS(8)=8.0D0
+      COUNT=COUNT+1
+      DO 25001 I=1,N
+        RESULT(COUNT,I)=Y(I)
+25001 CONTINUE
+      IF(COUNT.EQ.M)THEN
+        IF(FORWRD)THEN
+          XSOL=X02ALF()
+        ELSE
+          XSOL=-X02ALF()
+        ENDIF
+      ELSE
+        XSOL=POINTS(COUNT)
+      ENDIF
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp80ExampleCode}{Asp80 Example Code}
+\begin{verbatim}
+      SUBROUTINE BDYVAL(XL,XR,ELAM,YL,YR)
+      DOUBLE PRECISION ELAM,XL,YL(3),XR,YR(3)
+      YL(1)=XL
+      YL(2)=2.0D0
+      YR(1)=1.0D0
+      YR(2)=-1.0D0*DSQRT(XR+(-1.0D0*ELAM))
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+\begin{page}{Asp9ExampleCode}{Asp9 Example Code}
+\begin{verbatim}
+      DOUBLE PRECISION FUNCTION G(X,Y)
+      DOUBLE PRECISION X,Y(*)
+      G=X+Y(1)
+      RETURN
+      END
+\end{verbatim}
+\end{page}
+
+
+
+
+
+
+
+
diff --git a/src/hyper/pages/basic.ht b/src/hyper/pages/basic.ht
new file mode 100644
index 00000000..2829267b
--- /dev/null
+++ b/src/hyper/pages/basic.ht
@@ -0,0 +1,41 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% @(#)basic.ht	1.1 90/07/12 18:31:12
+
+\begin{page}{BasicCommand}{Basic Commands}
+\beginscroll
+\beginmenu
+\menumemolink{Calculus}{Calculus}\tab{10}
+  Compute integrals, derivatives, or limits
+\menulispmemolink{Matrix}{(|bcMatrix|)}\tab{10}
+  Create a matrix
+%\menulispmemolink{Operations}{(|bcExpand|)}
+%  Expand, factor, simplify, substitute, etc.
+\menulispmemolink{Draw}{(|bcDraw|)}\tab{10}
+  Create 2D or 3D plots.
+\menulispmemolink{Series}{(|bcSeries|)}\tab{10}
+  Create a power series
+\menulispmemolink{Solve}{(|bcSolve|)}\tab{10}
+  Solve an equation or system of equations.
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+ 
+\begin{page}{Calculus}{Calculus}
+\beginscroll
+What would you like to do?
+\beginmenu
+\menulispdownlink{Differentiate}{(|bcDifferentiate|)}\space{}
+\menulispdownlink{Do an Indefinite Integral}{(|bcIndefiniteIntegrate|)}\space{}
+\menulispdownlink{Do a Definite Integral}{(|bcDefiniteIntegrate|)}\space{}
+\menulispdownlink{Find a limit}{(|bcLimit|)}\space{}
+\menulispdownlink{Do a summation}{(|bcSum|)}\space{}
+%\menulispdownlink{Compute a product}{(|bcProduct|)}\space{}
+\endmenu
+\endscroll
+\autobuttons 
+\end{page}
+ 
diff --git a/src/hyper/pages/bmcat.ht b/src/hyper/pages/bmcat.ht
new file mode 100644
index 00000000..0ff5e65e
--- /dev/null
+++ b/src/hyper/pages/bmcat.ht
@@ -0,0 +1,102 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+
+% @(#)bmcat.ht	1.3 91/06/27 00:44:46
+\begin{page}{BitMaps}{Bit Map Catalog}
+\beginscroll
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/1x1}}   1x1
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/2x2}}   2x2
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/black}}   black
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/boxes}}   boxes
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/cntr_ptr}}   cntr_ptr
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/cntr_ptrmsk}}   cntr_ptrmsk
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/cross_weave}}   cross_weave
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/dimple1}}   dimple1
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/dimple3}}   dimple3
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/dot}}   dot
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/flipped_gray}}   flipped_gray
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/gray}}   gray
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/gray1}}   gray1
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/gray3}}   gray3
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/icon}}   icon
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/left_ptr}}   left_ptr
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/left_ptrmsk}}   left_ptrmsk
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/light_gray}}   light_gray
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/opendot}}   opendot
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/opendotMask}}   opendotMask
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/right_ptr}}   right_ptr
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/right_ptrmsk}}   right_ptrmsk
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/root_weave}}   root_weave
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/scales}}   scales
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/sipb}}   sipb
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/star}}   star
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/starMask}}   starMask
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/stipple}}   stipple
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/target}}   target
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/tie_fighter}}   tie_fighter
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/wide_weave}}   wide_weave
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/wierd_size}}   wierd_size
+%\space{} {\inputbitmap{/usr/include/X11/bitmaps/wingdogs}}   wingdogs
+%\horizontalline
+\space{} {\inputbitmap{\htbmdir{}/Xfbox.bitmap}}      Xfbox    \space{4}
+\space{} {\inputbitmap{\htbmdir{}/Xfcirc.bitmap}}     Xfcirc   \space{4}
+\space{} {\inputbitmap{\htbmdir{}/Xnobox.bitmap}}     Xnobox   \space{2}
+\space{} {\inputbitmap{\htbmdir{}/Xnocirc.bitmap}}    Xnocirc  \space{2}
+\newline
+\space{} {\inputbitmap{\htbmdir{}/Xfullfbox.bitmap}}  Xfullfbox 
+\space{} {\inputbitmap{\htbmdir{}/Xfullfcirc.bitmap}} Xfullfcirc
+\space{} {\inputbitmap{\htbmdir{}/Xfullbox.bitmap}}   Xfullbox  
+\space{} {\inputbitmap{\htbmdir{}/Xfullcirc.bitmap}}  Xfullcirc
+\newline
+\space{} {\inputbitmap{\htbmdir{}/Xgreyfbox.bitmap}}  Xgreyfbox 
+\space{} {\inputbitmap{\htbmdir{}/Xgreyfcirc.bitmap}} Xgreyfcirc
+\space{} {\inputbitmap{\htbmdir{}/Xgreybox.bitmap}}   Xgreybox  
+\space{} {\inputbitmap{\htbmdir{}/Xgreycirc.bitmap}}  Xgreycirc
+\newline
+\space{} {\inputbitmap{\htbmdir{}/Xopenfbox.bitmap}}  Xopenfbox
+\space{} {\inputbitmap{\htbmdir{}/Xopenfcirc.bitmap}} Xopenfcirc
+\space{} {\inputbitmap{\htbmdir{}/Xopenbox.bitmap}}   Xopenbox  
+\space{} {\inputbitmap{\htbmdir{}/Xopencirc.bitmap}}  Xopencirc
+\newline
+\space{} {\inputbitmap{\htbmdir{}/Xtickfbox.bitmap}}  Xtickfbox 
+\space{} {\inputbitmap{\htbmdir{}/Xtickfcirc.bitmap}} Xtickfcirc
+\space{} {\inputbitmap{\htbmdir{}/Xtickbox.bitmap}}   Xtickbox  
+\space{} {\inputbitmap{\htbmdir{}/Xtickcirc.bitmap}}  Xtickcirc
+\newline
+\space{} {\inputbitmap{\htbmdir{}/Xxfbox.bitmap}}     Xxfbox    \space{3}
+\space{} {\inputbitmap{\htbmdir{}/Xxfcirc.bitmap}}    Xxfcirc   \space{3}
+\space{} {\inputbitmap{\htbmdir{}/Xxbox.bitmap}}      Xxbox     \space{3}
+\space{} {\inputbitmap{\htbmdir{}/Xxcirc.bitmap}}     Xfcirc    \space{3}
+\horizontalline
+\space{} {\inputbitmap{\htbmdir{}/Xnoface.bitmap}}    Xnoface
+\space{} {\inputbitmap{\htbmdir{}/Xhappy.bitmap}}     Xhappy
+\space{} {\inputbitmap{\htbmdir{}/Xsad.bitmap}}       Xsad
+\space{} {\inputbitmap{\htbmdir{}/Xdesp.bitmap}}      Xdesp
+\space{} {\inputbitmap{\htbmdir{}/Xperv.bitmap}}      Xperv
+\horizontalline
+\space{} {\inputbitmap{\htbmdir{}/exit.bitmap}} exit
+\space{} {\inputbitmap{\htbmdir{}/help3.bitmap}} help3
+\newline
+\space{} {\inputbitmap{\htbmdir{}/down3.bitmap}} down3
+\space{} {\inputbitmap{\htbmdir{}/up3.bitmap}} up3
+\space{} {\inputbitmap{\htbmdir{}/return3.bitmap}} return3
+\newline
+\space{} {\inputbitmap{\htbmdir{}/tear.bitmap}} tear
+\space{} {\inputbitmap{\htbmdir{}/eqpage.bitmap}} eqpage
+\horizontalline
+\space{} {\inputbitmap{\htbmdir{}/sup.bm}} sup.bm
+\space{} {\inputbitmap{\htbmdir{}/sup.bitmap}} sup
+\space{} {\inputbitmap{\htbmdir{}/sdown.bm}} sdown.bm
+\newline
+\space{} {\inputbitmap{\htbmdir{}/ht_icon}} ht_icon
+\space{} {\inputbitmap{\htbmdir{}/smile.bitmap}} smile
+\newline
+\space{} {\inputbitmap{\htbmdir{}/drown.bm}} drown.bm
+\space{} {\inputbitmap{\htbmdir{}/help2.bitmap}} help2
+\space{} {\inputbitmap{\htbmdir{}/return.bitmap}} return
+\space{} {\inputbitmap{\htbmdir{}/back.bitmap}} back
+\endscroll
+\autobuttons
+\end{page}
diff --git a/src/hyper/pages/coverex.ht b/src/hyper/pages/coverex.ht
new file mode 100644
index 00000000..d4988c1e
--- /dev/null
+++ b/src/hyper/pages/coverex.ht
@@ -0,0 +1,195 @@
+% DO NOT EDIT! Created by ex2ht.
+
+\begin{page}{ExampleCoverPage}{Examples Of AXIOM Commands}
+\beginscroll\table{
+{\downlink{Differentiation}{Menuexdiff}}
+{\downlink{Integration}{Menuexint}}
+{\downlink{Laplace Transforms}{Menuexlap}}
+{\downlink{Limits}{Menuexlimit}}
+{\downlink{Matrices}{Menuexmatrix}}
+{\downlink{2-D Graphics}{Menuexplot2d}}
+{\downlink{3-D Graphics}{Menuexplot3d}}
+{\downlink{Series}{Menuexseries}}
+{\downlink{Summations}{Menuexsum}}
+}\endscroll\end{page}
+
+\begin{page}{Menuexdiff}{Differentiation}
+\beginscroll\beginmenu
+\menudownlink{Computing Derivatives}{ExDiffBasic}
+\spadpaste{differentiate(sin(x) * exp(x**2),x)}
+\menudownlink{Derivatives of Functions of Several Variables}{ExDiffSeveralVariables}
+\spadpaste{differentiate(sin(x) * tan(y)/(x**2 + y**2),x)}
+\spadpaste{differentiate(sin(x) * tan(y)/(x**2 + y**2),y)}
+\menudownlink{Derivatives of Higher Order}{ExDiffHigherOrder}
+\spadpaste{differentiate(exp(x**2),x,4)}
+\menudownlink{Multiple Derivatives I}{ExDiffMultipleI}
+\spadpaste{differentiate(sin(x)/(x**2 + y**2),[x,y])}
+\spadpaste{differentiate(sin(x)/(x**2 + y**2),[x,y,y])}
+\menudownlink{Multiple Derivatives II}{ExDiffMultipleII}
+\spadpaste{differentiate(cos(z)/(x**2 + y**3),[x,y,z],[1,2,3])}
+\menudownlink{Derivatives of Functions Involving Formal Integrals}{ExDiffFormalIntegral}
+\spadpaste{f := integrate(sqrt(1 + t**3),t) \bound{f}}
+\spadpaste{differentiate(f,t) \free{f}}
+\spadpaste{differentiate(f * t**2,t) \free{f}}
+\endmenu\endscroll\end{page}
+
+\begin{page}{Menuexint}{Integration}
+\beginscroll\beginmenu
+\menudownlink{Integral of a Rational Function}{ExIntRationalFunction}
+\spadpaste{integrate((x**2+2*x+1)/((x+1)**6+1),x)}
+\spadpaste{integrate(1/(x**3+x+1),x) \bound{i}}
+\spadpaste{definingPolynomial(tower(\%).2::EXPR INT) \free{i}}
+\menudownlink{Integral of a Rational Function with a Real Parameter}{ExIntRationalWithRealParameter}
+\spadpaste{integrate(1/(x**2 + a),x)}
+\menudownlink{Integral of a Rational Function with a Complex Parameter}{ExIntRationalWithComplexParameter}
+\spadpaste{complexIntegrate(1/(x**2 + a),x)}
+\menudownlink{Two Similar Integrands Producing Very Different Results}{ExIntTwoSimilarIntegrands}
+\spadpaste{integrate(x**3 / (a+b*x)**(1/3),x)}
+\spadpaste{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)}
+\menudownlink{An Integral Which Does Not Exist}{ExIntNoSolution}
+\spadpaste{integrate(log(1 + sqrt(a*x + b)) / x,x)}
+\menudownlink{A Trigonometric Function of a Quadratic}{ExIntTrig}
+\spadpaste{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b))/(sqrt(x+b)*(x+cosh(1+sqrt(x+b)))),x)}
+\menudownlink{Integrating a Function with a Hidden Algebraic Relation}{ExIntAlgebraicRelation}
+\spadpaste{integrate(tan(atan(x)/3),x)}
+\menudownlink{Details for integrating a function wiht a Hidden Algebraic Relation}{ExIntAlgebraicRelationExplain}
+\menudownlink{An Integral Involving a Root of a Transcendental Function}{ExIntRadicalOfTranscendental}
+\spadpaste{integrate((x + 1) / (x * (x + log x)**(3/2)),x)}
+\menudownlink{An Integral of a Non-elementary Function}{ExIntNonElementary}
+\spadpaste{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)}
+\endmenu\endscroll\end{page}
+
+\begin{page}{Menuexlap}{Laplace Transforms}
+\beginscroll\beginmenu
+\menudownlink{Laplace transform with a single pole}{ExLapSimplePole}
+\spadpaste{laplace(t**4 * exp(-a*t) / factorial(4), t, s)}
+\menudownlink{Laplace transform of a trigonometric function}{ExLapTrigTrigh}
+\spadpaste{laplace(sin(a*t) * cosh(a*t) - cos(a*t) * sinh(a*t), t, s)}
+\menudownlink{Laplace transform requiring a definite integration}{ExLapDefInt}
+\spadpaste{laplace(2/t * (1 - cos(a*t)), t, s)}
+\menudownlink{Laplace transform of exponentials}{ExLapExpExp}
+\spadpaste{laplace((exp(a*t) - exp(b*t))/t, t, s)}
+\menudownlink{Laplace transform of an exponential integral}{ExLapSpecial1}
+\spadpaste{laplace(exp(a*t+b)*Ei(c*t), t, s)}
+\menudownlink{Laplace transform of special functions}{ExLapSpecial2}
+\spadpaste{laplace(a*Ci(b*t) + c*Si(d*t), t, s)}
+\endmenu\endscroll\end{page}
+
+\begin{page}{Menuexlimit}{Limits}
+\beginscroll\beginmenu
+\menudownlink{Computing Limits}{ExLimitBasic}
+\spadpaste{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)}
+\menudownlink{Limits of Functions with Parameters}{ExLimitParameter}
+\spadpaste{limit(sinh(a*x)/tan(b*x),x = 0)}
+\menudownlink{One-sided Limits}{ExLimitOneSided}
+\spadpaste{limit(x * log(x),x = 0,"right")}
+\spadpaste{limit(x * log(x),x = 0)}
+\menudownlink{Two-sided Limits}{ExLimitTwoSided}
+\spadpaste{limit(sqrt(y**2)/y,y = 0)}
+\spadpaste{limit(sqrt(1 - cos(t))/t,t = 0)}
+\menudownlink{Limits at Infinity}{ExLimitInfinite}
+\spadpaste{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)}
+\spadpaste{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)}
+\menudownlink{Real Limits vs. Complex Limits}{ExLimitRealComplex}
+\spadpaste{limit(z * sin(1/z),z = 0)}
+\spadpaste{complexLimit(z * sin(1/z),z = 0)}
+\menudownlink{Complex Limits at Infinity}{ExLimitComplexInfinite}
+\spadpaste{complexLimit((2 + z)/(1 - z),z = \%infinity)}
+\spadpaste{limit(sin(x)/x,x = \%plusInfinity)}
+\spadpaste{complexLimit(sin(x)/x,x = \%infinity)}
+\endmenu\endscroll\end{page}
+
+\begin{page}{Menuexmatrix}{Matrices}
+\beginscroll\beginmenu
+\menudownlink{Basic Arithmetic Operations on Matrices}{ExMatrixBasicFunction}
+\spadpaste{m1 := matrix([[1,-2,1],[4,2,-4]]) \bound{m1}}
+\spadpaste{m2 := matrix([[1,0,2],[20,30,10],[0,200,100]]) \bound{m2}}
+\spadpaste{m3 := matrix([[1,2,3],[2,4,6]]) \bound{m3}}
+\spadpaste{m1 + m3 \free{m1} \free{m3}}
+\spadpaste{100 * m1 \free{m1}}
+\spadpaste{m1 * m2 \free{m1} \free{m2}}
+\spadpaste{-m1 + m3 * m2 \free{m1} \free{m2} \free{m3}}
+\spadpaste{m3 *vector([1,0,1]) \free{m3}}
+\menudownlink{Constructing new Matrices}{ExConstructMatrix}
+\spadpaste{diagonalMatrix([1,2,3,2,1])}
+\spadpaste{subMatrix(matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]]), 1,3,2,4)}
+\spadpaste{horizConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]])) }
+\spadpaste{vertConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]])) }
+\spadpaste{b:=matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]]) \bound{b}}
+\spadpaste{setsubMatrix!(b,1,1,transpose(subMatrix(b,1,3,1,3)))\free{b}}
+\menudownlink{Trace of a Matrix}{ExTraceMatrix}
+\spadpaste{trace( matrix([[1,x,x**2,x**3],[1,y,y**2,y**3],[1,z,z**2,z**3],[1,u,u**2,u**3]]) )}
+\menudownlink{Determinant of a Matrix}{ExDeterminantMatrix}
+\spadpaste{determinant(matrix([[1,2,3,4],[2,3,2,5],[3,4,5,6],[4,1,6,7]]))}
+\menudownlink{Inverse of a Matrix}{ExInverseMatrix}
+\spadpaste{inverse(matrix([[1,2,1],[-2,3,4],[-1,5,6]])) }
+\menudownlink{Rank of a Matrix}{ExRankMatrix}
+\spadpaste{rank(matrix([[0,4,1],[5,3,-7],[-5,5,9]]))}
+\endmenu\endscroll\end{page}
+
+\begin{page}{Menuexplot2d}{2-D Graphics}
+\beginscroll\beginmenu
+\menudownlink{Plotting Functions of One Variable}{ExPlot2DFunctions}
+\graphpaste{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)}
+\menudownlink{Plotting Parametric Curves}{ExPlot2DParametric}
+\graphpaste{draw(curve(9 * sin(3*t/4),8 * sin(t)),t = -4*\%pi..4*\%pi)}
+\menudownlink{Plotting Using Polar Coordinates}{ExPlot2DPolar}
+\graphpaste{draw(sin(4*t/7),t = 0..14*\%pi,coordinates == polar)}
+\menudownlink{Plotting Plane Algebraic Curves}{ExPlot2DAlgebraic}
+\graphpaste{draw(y**2 + y - (x**3 - x) = 0, x, y, range == [-2..2,-2..1])}
+\endmenu\endscroll\end{page}
+
+\begin{page}{Menuexplot3d}{3-D Graphics}
+\beginscroll\beginmenu
+\menudownlink{Plotting Functions of Two Variables}{ExPlot3DFunctions}
+\graphpaste{draw(cos(x*y),x = -3..3,y = -3..3)}
+\menudownlink{Plotting Parametric Surfaces}{ExPlot3DParametricSurface}
+\graphpaste{draw(surface(5*sin(u)*cos(v),4*sin(u)*sin(v),3*cos(u)),u=0..\%pi,v=0..2*\%pi)}
+\graphpaste{draw(surface(u*cos(v),u*sin(v),u),u=0..4,v=0..2*\%pi)}
+\menudownlink{Plotting Parametric Curves}{ExPlot3DParametricCurve}
+\graphpaste{draw(curve(cos(t),sin(t),t),t=0..6)}
+\graphpaste{draw(curve(t,t**2,t**3),t=-3..3)}
+\endmenu\endscroll\end{page}
+
+\begin{page}{Menuexseries}{Series}
+\beginscroll\beginmenu
+\menudownlink{Converting Expressions to Series}{ExSeriesConvert}
+\spadpaste{series(sin(a*x),x = 0)}
+\spadpaste{series(sin(a*x),a = \%pi/4)}
+\menudownlink{Manipulating Power Series}{ExSeriesManipulate}
+\spadpaste{f := series(1/(1-x),x = 0) \bound{f}}
+\spadpaste{f ** 2 \free{f}}
+\menudownlink{Functions on Power Series}{ExSeriesFunctions}
+\spadpaste{f := series(1/(1-x),x = 0) \bound{f1}}
+\spadpaste{g := log(f) \free{f1} \bound{g}}
+\spadpaste{exp(g) \free{g}}
+\menudownlink{Substituting Numerical Values in Power Series}{ExSeriesSubstitution}
+\spadpaste{f := taylor(exp(x)) \bound{f2}}
+\spadpaste{eval(f,1.0) \free{f2}}
+\endmenu\endscroll\end{page}
+
+\begin{page}{Menuexsum}{Summations}
+\beginscroll\beginmenu
+\menudownlink{Summing the Entries of a List I}{ExSumListEntriesI}
+\spadpaste{[i for i in 1..15]}
+\spadpaste{reduce(+,[i for i in 1..15])}
+\menudownlink{Summing the Entries of a List II}{ExSumListEntriesII}
+\spadpaste{[n**2 for n in 5..20]}
+\spadpaste{reduce(+,[n**2 for n in 5..20])}
+\menudownlink{Approximating e}{ExSumApproximateE}
+\spadpaste{reduce(+,[1.0/factorial(n) for n in 0..20])}
+\menudownlink{Closed Form Summations}{ExSumClosedForm}
+\spadpaste{s := sum(k**2,k = a..b) \bound{s}}
+\spadpaste{eval(s,[a,b],[1,25]) \free{s}}
+\spadpaste{reduce(+,[i**2 for i in 1..25])}
+\menudownlink{Sums of Cubes}{ExSumCubes}
+\spadpaste{sum(k**3,k = 1..n)}
+\spadpaste{sum(k,k = 1..n) ** 2}
+\menudownlink{Sums of Polynomials}{ExSumPolynomial}
+\spadpaste{sum(3*k**2/(c**2 + 1) + 12*k/d,k = (3*a)..(4*b))}
+\menudownlink{Sums of General Functions}{ExSumGeneralFunction}
+\spadpaste{sum(k * x**k,k = 1..n)}
+\menudownlink{Infinite Sums}{ExSumInfinite}
+\spadpaste{limit( sum(1/(k * (k + 2)),k = 1..n) ,n = \%plusInfinity)}
+\endmenu\endscroll\end{page}
+
diff --git a/src/hyper/pages/coverex.pht b/src/hyper/pages/coverex.pht
new file mode 100644
index 00000000..8b1a6db5
--- /dev/null
+++ b/src/hyper/pages/coverex.pht
@@ -0,0 +1,1748 @@
+\begin{patch}{MenuexdiffPatch1}
+\begin{paste}{MenuexdiffFull1}{MenuexdiffEmpty1}
+\pastebutton{MenuexdiffFull1}{\hidepaste}
+\tab{5}\spadcommand{differentiate(sin(x) * exp(x**2),x)}
+\indentrel{3}\begin{verbatim}
+              2                  2
+             x                  x
+   (1)  2x %e  sin(x) + cos(x)%e
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffEmpty1}
+\begin{paste}{MenuexdiffEmpty1}{MenuexdiffPatch1}
+\pastebutton{MenuexdiffEmpty1}{\showpaste}
+\tab{5}\spadcommand{differentiate(sin(x) * exp(x**2),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffPatch2}
+\begin{paste}{MenuexdiffFull2}{MenuexdiffEmpty2}
+\pastebutton{MenuexdiffFull2}{\hidepaste}
+\tab{5}\spadcommand{differentiate(sin(x) * tan(y)/(x**2 + y**2),x)}
+\indentrel{3}\begin{verbatim}
+                         2    2
+        (- 2x sin(x) + (y  + x )cos(x))tan(y)
+   (2)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                    4     2 2    4
+                   y  + 2x y  + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffEmpty2}
+\begin{paste}{MenuexdiffEmpty2}{MenuexdiffPatch2}
+\pastebutton{MenuexdiffEmpty2}{\showpaste}
+\tab{5}\spadcommand{differentiate(sin(x) * tan(y)/(x**2 + y**2),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffPatch3}
+\begin{paste}{MenuexdiffFull3}{MenuexdiffEmpty3}
+\pastebutton{MenuexdiffFull3}{\hidepaste}
+\tab{5}\spadcommand{differentiate(sin(x) * tan(y)/(x**2 + y**2),y)}
+\indentrel{3}\begin{verbatim}
+   (3)
+         2    2             2
+       (y  + x )sin(x)tan(y)  - 2y sin(x)tan(y)
+     + 
+         2    2
+       (y  + x )sin(x)
+  /
+      4     2 2    4
+     y  + 2x y  + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffEmpty3}
+\begin{paste}{MenuexdiffEmpty3}{MenuexdiffPatch3}
+\pastebutton{MenuexdiffEmpty3}{\showpaste}
+\tab{5}\spadcommand{differentiate(sin(x) * tan(y)/(x**2 + y**2),y)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffPatch4}
+\begin{paste}{MenuexdiffFull4}{MenuexdiffEmpty4}
+\pastebutton{MenuexdiffFull4}{\hidepaste}
+\tab{5}\spadcommand{differentiate(exp(x**2),x,4)}
+\indentrel{3}\begin{verbatim}
+                             2
+            4      2        x
+   (4)  (16x  + 48x  + 12)%e
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffEmpty4}
+\begin{paste}{MenuexdiffEmpty4}{MenuexdiffPatch4}
+\pastebutton{MenuexdiffEmpty4}{\showpaste}
+\tab{5}\spadcommand{differentiate(exp(x**2),x,4)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffPatch5}
+\begin{paste}{MenuexdiffFull5}{MenuexdiffEmpty5}
+\pastebutton{MenuexdiffFull5}{\hidepaste}
+\tab{5}\spadcommand{differentiate(sin(x)/(x**2 + y**2),[x,y])}
+\indentrel{3}\begin{verbatim}
+                           3     2
+        8x y sin(x) + (- 2y  - 2x y)cos(x)
+   (5)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+               6     2 4     4 2    6
+              y  + 3x y  + 3x y  + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffEmpty5}
+\begin{paste}{MenuexdiffEmpty5}{MenuexdiffPatch5}
+\pastebutton{MenuexdiffEmpty5}{\showpaste}
+\tab{5}\spadcommand{differentiate(sin(x)/(x**2 + y**2),[x,y])}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffPatch6}
+\begin{paste}{MenuexdiffFull6}{MenuexdiffEmpty6}
+\pastebutton{MenuexdiffFull6}{\hidepaste}
+\tab{5}\spadcommand{differentiate(sin(x)/(x**2 + y**2),[x,y,y])}
+\indentrel{3}\begin{verbatim}
+   (6)
+           2     3             4     2 2     4
+   (- 40x y  + 8x )sin(x) + (6y  + 4x y  - 2x )cos(x)
+   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+              8     2 6     4 4     6 2    8
+             y  + 4x y  + 6x y  + 4x y  + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffEmpty6}
+\begin{paste}{MenuexdiffEmpty6}{MenuexdiffPatch6}
+\pastebutton{MenuexdiffEmpty6}{\showpaste}
+\tab{5}\spadcommand{differentiate(sin(x)/(x**2 + y**2),[x,y,y])}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffPatch7}
+\begin{paste}{MenuexdiffFull7}{MenuexdiffEmpty7}
+\pastebutton{MenuexdiffFull7}{\hidepaste}
+\tab{5}\spadcommand{differentiate(cos(z)/(x**2 + y**3),[x,y,z],[1,2,3])}
+\indentrel{3}\begin{verbatim}
+                    4      3
+            (- 84x y  + 24x y)sin(z)
+   (7)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+         12     2 9     4 6     6 3    8
+        y   + 4x y  + 6x y  + 4x y  + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffEmpty7}
+\begin{paste}{MenuexdiffEmpty7}{MenuexdiffPatch7}
+\pastebutton{MenuexdiffEmpty7}{\showpaste}
+\tab{5}\spadcommand{differentiate(cos(z)/(x**2 + y**3),[x,y,z],[1,2,3])}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffPatch8}
+\begin{paste}{MenuexdiffFull8}{MenuexdiffEmpty8}
+\pastebutton{MenuexdiffFull8}{\hidepaste}
+\tab{5}\spadcommand{f := integrate(sqrt(1 + t**3),t)\bound{f }}
+\indentrel{3}\begin{verbatim}
+           t  ÚÄÄÄÄÄÄÄ¿
+         Ú¿   ³  3
+   (8)   ³   \³%N  + 1 d%N
+        ÀÙ
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffEmpty8}
+\begin{paste}{MenuexdiffEmpty8}{MenuexdiffPatch8}
+\pastebutton{MenuexdiffEmpty8}{\showpaste}
+\tab{5}\spadcommand{f := integrate(sqrt(1 + t**3),t)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffPatch9}
+\begin{paste}{MenuexdiffFull9}{MenuexdiffEmpty9}
+\pastebutton{MenuexdiffFull9}{\hidepaste}
+\tab{5}\spadcommand{differentiate(f,t)\free{f }}
+\indentrel{3}\begin{verbatim}
+         ÚÄÄÄÄÄÄ¿
+         ³ 3
+   (9)  \³t  + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffEmpty9}
+\begin{paste}{MenuexdiffEmpty9}{MenuexdiffPatch9}
+\pastebutton{MenuexdiffEmpty9}{\showpaste}
+\tab{5}\spadcommand{differentiate(f,t)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffPatch10}
+\begin{paste}{MenuexdiffFull10}{MenuexdiffEmpty10}
+\pastebutton{MenuexdiffFull10}{\hidepaste}
+\tab{5}\spadcommand{differentiate(f * t**2,t)\free{f }}
+\indentrel{3}\begin{verbatim}
+              t  ÚÄÄÄÄÄÄÄ¿          ÚÄÄÄÄÄÄ¿
+            Ú¿   ³  3             2 ³ 3
+   (10)  2t ³   \³%N  + 1 d%N  + t \³t  + 1
+           ÀÙ
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexdiffEmpty10}
+\begin{paste}{MenuexdiffEmpty10}{MenuexdiffPatch10}
+\pastebutton{MenuexdiffEmpty10}{\showpaste}
+\tab{5}\spadcommand{differentiate(f * t**2,t)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitPatch1}
+\begin{paste}{MenuexlimitFull1}{MenuexlimitEmpty1}
+\pastebutton{MenuexlimitFull1}{\hidepaste}
+\tab{5}\spadcommand{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)}
+\indentrel{3}\begin{verbatim}
+          1
+   (1)  - Ä
+          2
+Type: Union(OrderedCompletion Fraction Polynomial Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitEmpty1}
+\begin{paste}{MenuexlimitEmpty1}{MenuexlimitPatch1}
+\pastebutton{MenuexlimitEmpty1}{\showpaste}
+\tab{5}\spadcommand{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitPatch2}
+\begin{paste}{MenuexlimitFull2}{MenuexlimitEmpty2}
+\pastebutton{MenuexlimitFull2}{\hidepaste}
+\tab{5}\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)}
+\indentrel{3}\begin{verbatim}
+        a
+   (2)  Ä
+        b
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitEmpty2}
+\begin{paste}{MenuexlimitEmpty2}{MenuexlimitPatch2}
+\pastebutton{MenuexlimitEmpty2}{\showpaste}
+\tab{5}\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitPatch3}
+\begin{paste}{MenuexlimitFull3}{MenuexlimitEmpty3}
+\pastebutton{MenuexlimitFull3}{\hidepaste}
+\tab{5}\spadcommand{limit(x * log(x),x = 0,"right")}
+\indentrel{3}\begin{verbatim}
+   (3)  0
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitEmpty3}
+\begin{paste}{MenuexlimitEmpty3}{MenuexlimitPatch3}
+\pastebutton{MenuexlimitEmpty3}{\showpaste}
+\tab{5}\spadcommand{limit(x * log(x),x = 0,"right")}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitPatch4}
+\begin{paste}{MenuexlimitFull4}{MenuexlimitEmpty4}
+\pastebutton{MenuexlimitFull4}{\hidepaste}
+\tab{5}\spadcommand{limit(x * log(x),x = 0)}
+\indentrel{3}\begin{verbatim}
+   (4)  [leftHandLimit= "failed",rightHandLimit= 0]
+Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitEmpty4}
+\begin{paste}{MenuexlimitEmpty4}{MenuexlimitPatch4}
+\pastebutton{MenuexlimitEmpty4}{\showpaste}
+\tab{5}\spadcommand{limit(x * log(x),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitPatch5}
+\begin{paste}{MenuexlimitFull5}{MenuexlimitEmpty5}
+\pastebutton{MenuexlimitFull5}{\hidepaste}
+\tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)}
+\indentrel{3}\begin{verbatim}
+   (5)  [leftHandLimit= - 1,rightHandLimit= 1]
+Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitEmpty5}
+\begin{paste}{MenuexlimitEmpty5}{MenuexlimitPatch5}
+\pastebutton{MenuexlimitEmpty5}{\showpaste}
+\tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitPatch6}
+\begin{paste}{MenuexlimitFull6}{MenuexlimitEmpty6}
+\pastebutton{MenuexlimitFull6}{\hidepaste}
+\tab{5}\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)}
+\indentrel{3}\begin{verbatim}
+                            1                    1
+   (6)  [leftHandLimit= - ÄÄÄÄ,rightHandLimit= ÄÄÄÄ]
+                           ÚÄ¿                  ÚÄ¿
+                          \³2                  \³2
+Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitEmpty6}
+\begin{paste}{MenuexlimitEmpty6}{MenuexlimitPatch6}
+\pastebutton{MenuexlimitEmpty6}{\showpaste}
+\tab{5}\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitPatch7}
+\begin{paste}{MenuexlimitFull7}{MenuexlimitEmpty7}
+\pastebutton{MenuexlimitFull7}{\hidepaste}
+\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)}
+\indentrel{3}\begin{verbatim}
+         ÚÄ¿
+        \³3
+   (7)  ÄÄÄÄ
+          5
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitEmpty7}
+\begin{paste}{MenuexlimitEmpty7}{MenuexlimitPatch7}
+\pastebutton{MenuexlimitEmpty7}{\showpaste}
+\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitPatch8}
+\begin{paste}{MenuexlimitFull8}{MenuexlimitEmpty8}
+\pastebutton{MenuexlimitFull8}{\hidepaste}
+\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)}
+\indentrel{3}\begin{verbatim}
+           ÚÄ¿
+          \³3
+   (8)  - ÄÄÄÄ
+            5
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitEmpty8}
+\begin{paste}{MenuexlimitEmpty8}{MenuexlimitPatch8}
+\pastebutton{MenuexlimitEmpty8}{\showpaste}
+\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitPatch9}
+\begin{paste}{MenuexlimitFull9}{MenuexlimitEmpty9}
+\pastebutton{MenuexlimitFull9}{\hidepaste}
+\tab{5}\spadcommand{limit(z * sin(1/z),z = 0)}
+\indentrel{3}\begin{verbatim}
+   (9)  0
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitEmpty9}
+\begin{paste}{MenuexlimitEmpty9}{MenuexlimitPatch9}
+\pastebutton{MenuexlimitEmpty9}{\showpaste}
+\tab{5}\spadcommand{limit(z * sin(1/z),z = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitPatch10}
+\begin{paste}{MenuexlimitFull10}{MenuexlimitEmpty10}
+\pastebutton{MenuexlimitFull10}{\hidepaste}
+\tab{5}\spadcommand{complexLimit(z * sin(1/z),z = 0)}
+\indentrel{3}\begin{verbatim}
+   (10)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitEmpty10}
+\begin{paste}{MenuexlimitEmpty10}{MenuexlimitPatch10}
+\pastebutton{MenuexlimitEmpty10}{\showpaste}
+\tab{5}\spadcommand{complexLimit(z * sin(1/z),z = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitPatch11}
+\begin{paste}{MenuexlimitFull11}{MenuexlimitEmpty11}
+\pastebutton{MenuexlimitFull11}{\hidepaste}
+\tab{5}\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)}
+\indentrel{3}\begin{verbatim}
+   (11)  - 1
+   Type: OnePointCompletion Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitEmpty11}
+\begin{paste}{MenuexlimitEmpty11}{MenuexlimitPatch11}
+\pastebutton{MenuexlimitEmpty11}{\showpaste}
+\tab{5}\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitPatch12}
+\begin{paste}{MenuexlimitFull12}{MenuexlimitEmpty12}
+\pastebutton{MenuexlimitFull12}{\hidepaste}
+\tab{5}\spadcommand{limit(sin(x)/x,x = \%plusInfinity)}
+\indentrel{3}\begin{verbatim}
+   (12)  0
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitEmpty12}
+\begin{paste}{MenuexlimitEmpty12}{MenuexlimitPatch12}
+\pastebutton{MenuexlimitEmpty12}{\showpaste}
+\tab{5}\spadcommand{limit(sin(x)/x,x = \%plusInfinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitPatch13}
+\begin{paste}{MenuexlimitFull13}{MenuexlimitEmpty13}
+\pastebutton{MenuexlimitFull13}{\hidepaste}
+\tab{5}\spadcommand{complexLimit(sin(x)/x,x = \%infinity)}
+\indentrel{3}\begin{verbatim}
+   (13)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlimitEmpty13}
+\begin{paste}{MenuexlimitEmpty13}{MenuexlimitPatch13}
+\pastebutton{MenuexlimitEmpty13}{\showpaste}
+\tab{5}\spadcommand{complexLimit(sin(x)/x,x = \%infinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot2dPatch1}
+\begin{paste}{Menuexplot2dFull1}{Menuexplot2dEmpty1}
+\pastebutton{Menuexplot2dFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/Menuexplot2d1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/Menuexplot2d1}}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot2dEmpty1}
+\begin{paste}{Menuexplot2dEmpty1}{Menuexplot2dPatch1}
+\pastebutton{Menuexplot2dEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot2dPatch2}
+\begin{paste}{Menuexplot2dFull2}{Menuexplot2dEmpty2}
+\pastebutton{Menuexplot2dFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(9 * sin(3*t/4),8 * sin(t)),t = -4*\%pi..4*\%pi)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/Menuexplot2d2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/Menuexplot2d2}}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot2dEmpty2}
+\begin{paste}{Menuexplot2dEmpty2}{Menuexplot2dPatch2}
+\pastebutton{Menuexplot2dEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(curve(9 * sin(3*t/4),8 * sin(t)),t = -4*\%pi..4*\%pi)}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot2dPatch3}
+\begin{paste}{Menuexplot2dFull3}{Menuexplot2dEmpty3}
+\pastebutton{Menuexplot2dFull3}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(4*t/7),t = 0..14*\%pi,coordinates == polar)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/Menuexplot2d3.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/Menuexplot2d3}}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot2dEmpty3}
+\begin{paste}{Menuexplot2dEmpty3}{Menuexplot2dPatch3}
+\pastebutton{Menuexplot2dEmpty3}{\showpaste}
+\tab{5}\spadgraph{draw(sin(4*t/7),t = 0..14*\%pi,coordinates == polar)}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot2dPatch4}
+\begin{paste}{Menuexplot2dFull4}{Menuexplot2dEmpty4}
+\pastebutton{Menuexplot2dFull4}{\hidepaste}
+\tab{5}\spadgraph{draw(y**2 + y - (x**3 - x) = 0, x, y, range == [-2..2,-2..1])}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/Menuexplot2d4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/Menuexplot2d4}}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot2dEmpty4}
+\begin{paste}{Menuexplot2dEmpty4}{Menuexplot2dPatch4}
+\pastebutton{Menuexplot2dEmpty4}{\showpaste}
+\tab{5}\spadgraph{draw(y**2 + y - (x**3 - x) = 0, x, y, range == [-2..2,-2..1])}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot3dPatch1}
+\begin{paste}{Menuexplot3dFull1}{Menuexplot3dEmpty1}
+\pastebutton{Menuexplot3dFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(cos(x*y),x = -3..3,y = -3..3)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/Menuexplot3d1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/Menuexplot3d1}}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot3dEmpty1}
+\begin{paste}{Menuexplot3dEmpty1}{Menuexplot3dPatch1}
+\pastebutton{Menuexplot3dEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(cos(x*y),x = -3..3,y = -3..3)}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot3dPatch2}
+\begin{paste}{Menuexplot3dFull2}{Menuexplot3dEmpty2}
+\pastebutton{Menuexplot3dFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(surface(5*sin(u)*cos(v),4*sin(u)*sin(v),3*cos(u)),u=0..\%pi,v=0..2*\%pi)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/Menuexplot3d2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/Menuexplot3d2}}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot3dEmpty2}
+\begin{paste}{Menuexplot3dEmpty2}{Menuexplot3dPatch2}
+\pastebutton{Menuexplot3dEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(surface(5*sin(u)*cos(v),4*sin(u)*sin(v),3*cos(u)),u=0..\%pi,v=0..2*\%pi)}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot3dPatch3}
+\begin{paste}{Menuexplot3dFull3}{Menuexplot3dEmpty3}
+\pastebutton{Menuexplot3dFull3}{\hidepaste}
+\tab{5}\spadgraph{draw(surface(u*cos(v),u*sin(v),u),u=0..4,v=0..2*\%pi)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/Menuexplot3d3.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/Menuexplot3d3}}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot3dEmpty3}
+\begin{paste}{Menuexplot3dEmpty3}{Menuexplot3dPatch3}
+\pastebutton{Menuexplot3dEmpty3}{\showpaste}
+\tab{5}\spadgraph{draw(surface(u*cos(v),u*sin(v),u),u=0..4,v=0..2*\%pi)}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot3dPatch4}
+\begin{paste}{Menuexplot3dFull4}{Menuexplot3dEmpty4}
+\pastebutton{Menuexplot3dFull4}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(cos(t),sin(t),t),t=0..6)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/Menuexplot3d4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/Menuexplot3d4}}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot3dEmpty4}
+\begin{paste}{Menuexplot3dEmpty4}{Menuexplot3dPatch4}
+\pastebutton{Menuexplot3dEmpty4}{\showpaste}
+\tab{5}\spadgraph{draw(curve(cos(t),sin(t),t),t=0..6)}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot3dPatch5}
+\begin{paste}{Menuexplot3dFull5}{Menuexplot3dEmpty5}
+\pastebutton{Menuexplot3dFull5}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(t,t**2,t**3),t=-3..3)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/Menuexplot3d5.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/Menuexplot3d5}}
+\end{paste}\end{patch}
+
+\begin{patch}{Menuexplot3dEmpty5}
+\begin{paste}{Menuexplot3dEmpty5}{Menuexplot3dPatch5}
+\pastebutton{Menuexplot3dEmpty5}{\showpaste}
+\tab{5}\spadgraph{draw(curve(t,t**2,t**3),t=-3..3)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesPatch1}
+\begin{paste}{MenuexseriesFull1}{MenuexseriesEmpty1}
+\pastebutton{MenuexseriesFull1}{\hidepaste}
+\tab{5}\spadcommand{series(sin(a*x),x = 0)}
+\indentrel{3}\begin{verbatim}
+   (1)
+            3        5        7          9
+           a   3    a   5    a    7     a     9
+     a x - ÄÄ x  + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x
+            6      120      5040      362880
+   + 
+           11
+          a      11      12
+     - ÄÄÄÄÄÄÄÄ x   + O(x  )
+       39916800
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesEmpty1}
+\begin{paste}{MenuexseriesEmpty1}{MenuexseriesPatch1}
+\pastebutton{MenuexseriesEmpty1}{\showpaste}
+\tab{5}\spadcommand{series(sin(a*x),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesPatch2}
+\begin{paste}{MenuexseriesFull2}{MenuexseriesEmpty2}
+\pastebutton{MenuexseriesFull2}{\hidepaste}
+\tab{5}\spadcommand{series(sin(a*x),a = \%pi/4)}
+\indentrel{3}\begin{verbatim}
+   (2)
+         %pi x          %pi x      %pi
+     sin(ÄÄÄÄÄ) + x cos(ÄÄÄÄÄ)(a - ÄÄÄ)
+           4              4         4
+   + 
+        2    %pi x                3    %pi x
+       x sin(ÄÄÄÄÄ)              x cos(ÄÄÄÄÄ)
+               4         %pi 2           4         %pi 3
+     - ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)  - ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)
+             2            4            6            4
+   + 
+      4    %pi x                5    %pi x
+     x sin(ÄÄÄÄÄ)              x cos(ÄÄÄÄÄ)
+             4         %pi 4           4         %pi 5
+     ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)  + ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)
+          24            4           120           4
+   + 
+        6    %pi x                7    %pi x
+       x sin(ÄÄÄÄÄ)              x cos(ÄÄÄÄÄ)
+               4         %pi 6           4         %pi 7
+     - ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)  - ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)
+            720           4          5040           4
+   + 
+      8    %pi x                9    %pi x
+     x sin(ÄÄÄÄÄ)              x cos(ÄÄÄÄÄ)
+             4         %pi 8           4         %pi 9
+     ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)  + ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)
+         40320          4         362880          4
+   + 
+        10    %pi x
+       x  sin(ÄÄÄÄÄ)
+                4         %pi 10          %pi 11
+     - ÄÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)   + O((a - ÄÄÄ)  )
+          3628800          4               4
+Type: UnivariatePuiseuxSeries(Expression Integer,a,pi/4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesEmpty2}
+\begin{paste}{MenuexseriesEmpty2}{MenuexseriesPatch2}
+\pastebutton{MenuexseriesEmpty2}{\showpaste}
+\tab{5}\spadcommand{series(sin(a*x),a = \%pi/4)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesPatch3}
+\begin{paste}{MenuexseriesFull3}{MenuexseriesEmpty3}
+\pastebutton{MenuexseriesFull3}{\hidepaste}
+\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }}
+\indentrel{3}\begin{verbatim}
+   (3)
+              2    3    4    5    6    7    8    9    10
+     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
+   + 
+        11
+     O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesEmpty3}
+\begin{paste}{MenuexseriesEmpty3}{MenuexseriesPatch3}
+\pastebutton{MenuexseriesEmpty3}{\showpaste}
+\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesPatch4}
+\begin{paste}{MenuexseriesFull4}{MenuexseriesEmpty4}
+\pastebutton{MenuexseriesFull4}{\hidepaste}
+\tab{5}\spadcommand{f ** 2\free{f }}
+\indentrel{3}\begin{verbatim}
+   (4)
+                2     3     4     5     6     7     8
+     1 + 2x + 3x  + 4x  + 5x  + 6x  + 7x  + 8x  + 9x
+   + 
+        9      10      11
+     10x  + 11x   + O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesEmpty4}
+\begin{paste}{MenuexseriesEmpty4}{MenuexseriesPatch4}
+\pastebutton{MenuexseriesEmpty4}{\showpaste}
+\tab{5}\spadcommand{f ** 2\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesPatch5}
+\begin{paste}{MenuexseriesFull5}{MenuexseriesEmpty5}
+\pastebutton{MenuexseriesFull5}{\hidepaste}
+\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+   (5)
+              2    3    4    5    6    7    8    9    10
+     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
+   + 
+        11
+     O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesEmpty5}
+\begin{paste}{MenuexseriesEmpty5}{MenuexseriesPatch5}
+\pastebutton{MenuexseriesEmpty5}{\showpaste}
+\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesPatch6}
+\begin{paste}{MenuexseriesFull6}{MenuexseriesEmpty6}
+\pastebutton{MenuexseriesFull6}{\hidepaste}
+\tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }}
+\indentrel{3}\begin{verbatim}
+   (6)
+         1  2   1  3   1  4   1  5   1  6   1  7   1  8
+     x + Ä x  + Ä x  + Ä x  + Ä x  + Ä x  + Ä x  + Ä x
+         2      3      4      5      6      7      8
+   + 
+     1  9    1  10    1  11      12
+     Ä x  + ÄÄ x   + ÄÄ x   + O(x  )
+     9      10       11
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesEmpty6}
+\begin{paste}{MenuexseriesEmpty6}{MenuexseriesPatch6}
+\pastebutton{MenuexseriesEmpty6}{\showpaste}
+\tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesPatch7}
+\begin{paste}{MenuexseriesFull7}{MenuexseriesEmpty7}
+\pastebutton{MenuexseriesFull7}{\hidepaste}
+\tab{5}\spadcommand{exp(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (7)
+              2    3    4    5    6    7    8    9    10
+     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
+   + 
+        11
+     O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesEmpty7}
+\begin{paste}{MenuexseriesEmpty7}{MenuexseriesPatch7}
+\pastebutton{MenuexseriesEmpty7}{\showpaste}
+\tab{5}\spadcommand{exp(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesPatch8}
+\begin{paste}{MenuexseriesFull8}{MenuexseriesEmpty8}
+\pastebutton{MenuexseriesFull8}{\hidepaste}
+\tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }}
+\indentrel{3}\begin{verbatim}
+   (8)
+             1  2   1  3    1  4    1   5    1   6
+     1 + x + Ä x  + Ä x  + ÄÄ x  + ÄÄÄ x  + ÄÄÄ x
+             2      6      24      120      720
+   + 
+     1   7     1    8      1    9      1     10      11
+   ÄÄÄÄ x  + ÄÄÄÄÄ x  + ÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x   + O(x  )
+   5040      40320      362880      3628800
+   Type: UnivariateTaylorSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesEmpty8}
+\begin{paste}{MenuexseriesEmpty8}{MenuexseriesPatch8}
+\pastebutton{MenuexseriesEmpty8}{\showpaste}
+\tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesPatch9}
+\begin{paste}{MenuexseriesFull9}{MenuexseriesEmpty9}
+\pastebutton{MenuexseriesFull9}{\hidepaste}
+\tab{5}\spadcommand{eval(f,1.0)\free{f2 }}
+\indentrel{3}\begin{verbatim}
+   (9)
+   [1.0, 2.0, 2.5, 2.6666666666 666666667,
+    2.7083333333 333333333, 2.7166666666 666666667,
+    2.7180555555 555555556, 2.7182539682 53968254,
+    2.7182787698 412698413, 2.7182815255 731922399, ...]
+                          Type: Stream Expression Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexseriesEmpty9}
+\begin{paste}{MenuexseriesEmpty9}{MenuexseriesPatch9}
+\pastebutton{MenuexseriesEmpty9}{\showpaste}
+\tab{5}\spadcommand{eval(f,1.0)\free{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch1}
+\begin{paste}{MenuexmatrixFull1}{MenuexmatrixEmpty1}
+\pastebutton{MenuexmatrixFull1}{\hidepaste}
+\tab{5}\spadcommand{m1 := matrix([[1,-2,1],[4,2,-4]])\bound{m1 }}
+\indentrel{3}\begin{verbatim}
+        Ú1  - 2   1 ¿
+   (1)  ³           ³
+        À4   2   - 4Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty1}
+\begin{paste}{MenuexmatrixEmpty1}{MenuexmatrixPatch1}
+\pastebutton{MenuexmatrixEmpty1}{\showpaste}
+\tab{5}\spadcommand{m1 := matrix([[1,-2,1],[4,2,-4]])\bound{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch2}
+\begin{paste}{MenuexmatrixFull2}{MenuexmatrixEmpty2}
+\pastebutton{MenuexmatrixFull2}{\hidepaste}
+\tab{5}\spadcommand{m2 := matrix([[1,0,2],[20,30,10],[0,200,100]])\bound{m2 }}
+\indentrel{3}\begin{verbatim}
+        Ú1    0    2 ¿
+        ³            ³
+   (2)  ³20  30   10 ³
+        ³            ³
+        À0   200  100Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty2}
+\begin{paste}{MenuexmatrixEmpty2}{MenuexmatrixPatch2}
+\pastebutton{MenuexmatrixEmpty2}{\showpaste}
+\tab{5}\spadcommand{m2 := matrix([[1,0,2],[20,30,10],[0,200,100]])\bound{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch3}
+\begin{paste}{MenuexmatrixFull3}{MenuexmatrixEmpty3}
+\pastebutton{MenuexmatrixFull3}{\hidepaste}
+\tab{5}\spadcommand{m3 := matrix([[1,2,3],[2,4,6]])\bound{m3 }}
+\indentrel{3}\begin{verbatim}
+        Ú1  2  3¿
+   (3)  ³       ³
+        À2  4  6Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty3}
+\begin{paste}{MenuexmatrixEmpty3}{MenuexmatrixPatch3}
+\pastebutton{MenuexmatrixEmpty3}{\showpaste}
+\tab{5}\spadcommand{m3 := matrix([[1,2,3],[2,4,6]])\bound{m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch4}
+\begin{paste}{MenuexmatrixFull4}{MenuexmatrixEmpty4}
+\pastebutton{MenuexmatrixFull4}{\hidepaste}
+\tab{5}\spadcommand{m1 + m3\free{m1 }\free{m3 }}
+\indentrel{3}\begin{verbatim}
+        Ú2  0  4¿
+   (4)  ³       ³
+        À6  6  2Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty4}
+\begin{paste}{MenuexmatrixEmpty4}{MenuexmatrixPatch4}
+\pastebutton{MenuexmatrixEmpty4}{\showpaste}
+\tab{5}\spadcommand{m1 + m3\free{m1 }\free{m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch5}
+\begin{paste}{MenuexmatrixFull5}{MenuexmatrixEmpty5}
+\pastebutton{MenuexmatrixFull5}{\hidepaste}
+\tab{5}\spadcommand{100 * m1\free{m1 }}
+\indentrel{3}\begin{verbatim}
+        Ú100  - 200   100 ¿
+   (5)  ³                 ³
+        À400   200   - 400Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty5}
+\begin{paste}{MenuexmatrixEmpty5}{MenuexmatrixPatch5}
+\pastebutton{MenuexmatrixEmpty5}{\showpaste}
+\tab{5}\spadcommand{100 * m1\free{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch6}
+\begin{paste}{MenuexmatrixFull6}{MenuexmatrixEmpty6}
+\pastebutton{MenuexmatrixFull6}{\hidepaste}
+\tab{5}\spadcommand{m1 * m2\free{m1 }\free{m2 }}
+\indentrel{3}\begin{verbatim}
+        Ú- 39   140    82  ¿
+   (6)  ³                  ³
+        À 44   - 740  - 372Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty6}
+\begin{paste}{MenuexmatrixEmpty6}{MenuexmatrixPatch6}
+\pastebutton{MenuexmatrixEmpty6}{\showpaste}
+\tab{5}\spadcommand{m1 * m2\free{m1 }\free{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch7}
+\begin{paste}{MenuexmatrixFull7}{MenuexmatrixEmpty7}
+\pastebutton{MenuexmatrixFull7}{\hidepaste}
+\tab{5}\spadcommand{-m1 + m3 * m2\free{m1 }\free{m2 }\free{m3 }}
+\indentrel{3}\begin{verbatim}
+        Ú40  662   321¿
+   (7)  ³             ³
+        À78  1318  648Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty7}
+\begin{paste}{MenuexmatrixEmpty7}{MenuexmatrixPatch7}
+\pastebutton{MenuexmatrixEmpty7}{\showpaste}
+\tab{5}\spadcommand{-m1 + m3 * m2\free{m1 }\free{m2 }\free{m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch8}
+\begin{paste}{MenuexmatrixFull8}{MenuexmatrixEmpty8}
+\pastebutton{MenuexmatrixFull8}{\hidepaste}
+\tab{5}\spadcommand{m3 *vector([1,0,1])\free{m3 }}
+\indentrel{3}\begin{verbatim}
+   (8)  [4,8]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty8}
+\begin{paste}{MenuexmatrixEmpty8}{MenuexmatrixPatch8}
+\pastebutton{MenuexmatrixEmpty8}{\showpaste}
+\tab{5}\spadcommand{m3 *vector([1,0,1])\free{m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch9}
+\begin{paste}{MenuexmatrixFull9}{MenuexmatrixEmpty9}
+\pastebutton{MenuexmatrixFull9}{\hidepaste}
+\tab{5}\spadcommand{diagonalMatrix([1,2,3,2,1])}
+\indentrel{3}\begin{verbatim}
+        Ú1  0  0  0  0¿
+        ³             ³
+        ³0  2  0  0  0³
+        ³             ³
+   (9)  ³0  0  3  0  0³
+        ³             ³
+        ³0  0  0  2  0³
+        ³             ³
+        À0  0  0  0  1Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty9}
+\begin{paste}{MenuexmatrixEmpty9}{MenuexmatrixPatch9}
+\pastebutton{MenuexmatrixEmpty9}{\showpaste}
+\tab{5}\spadcommand{diagonalMatrix([1,2,3,2,1])}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch10}
+\begin{paste}{MenuexmatrixFull10}{MenuexmatrixEmpty10}
+\pastebutton{MenuexmatrixFull10}{\hidepaste}
+\tab{5}\spadcommand{subMatrix(matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]]), 1,3,2,4)}
+\indentrel{3}\begin{verbatim}
+         Ú1   2   3 ¿
+         ³          ³
+   (10)  ³6   7   8 ³
+         ³          ³
+         À11  12  13Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty10}
+\begin{paste}{MenuexmatrixEmpty10}{MenuexmatrixPatch10}
+\pastebutton{MenuexmatrixEmpty10}{\showpaste}
+\tab{5}\spadcommand{subMatrix(matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]]), 1,3,2,4)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch11}
+\begin{paste}{MenuexmatrixFull11}{MenuexmatrixEmpty11}
+\pastebutton{MenuexmatrixFull11}{\hidepaste}
+\tab{5}\spadcommand{horizConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]]))}
+\indentrel{3}\begin{verbatim}
+         Ú1  2  3  11  12  13¿
+   (11)  ³                   ³
+         À6  7  8  55  77  88Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty11}
+\begin{paste}{MenuexmatrixEmpty11}{MenuexmatrixPatch11}
+\pastebutton{MenuexmatrixEmpty11}{\showpaste}
+\tab{5}\spadcommand{horizConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]]))}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch12}
+\begin{paste}{MenuexmatrixFull12}{MenuexmatrixEmpty12}
+\pastebutton{MenuexmatrixFull12}{\hidepaste}
+\tab{5}\spadcommand{vertConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]]))}
+\indentrel{3}\begin{verbatim}
+         Ú1   2   3 ¿
+         ³          ³
+         ³6   7   8 ³
+   (12)  ³          ³
+         ³11  12  13³
+         ³          ³
+         À55  77  88Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty12}
+\begin{paste}{MenuexmatrixEmpty12}{MenuexmatrixPatch12}
+\pastebutton{MenuexmatrixEmpty12}{\showpaste}
+\tab{5}\spadcommand{vertConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]]))}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch13}
+\begin{paste}{MenuexmatrixFull13}{MenuexmatrixEmpty13}
+\pastebutton{MenuexmatrixFull13}{\hidepaste}
+\tab{5}\spadcommand{b:=matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]])\bound{b }}
+\indentrel{3}\begin{verbatim}
+         Ú0   1   2   3   4 ¿
+         ³                  ³
+   (13)  ³5   6   7   8   9 ³
+         ³                  ³
+         À10  11  12  13  14Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty13}
+\begin{paste}{MenuexmatrixEmpty13}{MenuexmatrixPatch13}
+\pastebutton{MenuexmatrixEmpty13}{\showpaste}
+\tab{5}\spadcommand{b:=matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]])\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch14}
+\begin{paste}{MenuexmatrixFull14}{MenuexmatrixEmpty14}
+\pastebutton{MenuexmatrixFull14}{\hidepaste}
+\tab{5}\spadcommand{setsubMatrix!(b,1,1,transpose(subMatrix(b,1,3,1,3)))\free{b }}
+\indentrel{3}\begin{verbatim}
+         Ú0  5  10  3   4 ¿
+         ³                ³
+   (14)  ³1  6  11  8   9 ³
+         ³                ³
+         À2  7  12  13  14Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty14}
+\begin{paste}{MenuexmatrixEmpty14}{MenuexmatrixPatch14}
+\pastebutton{MenuexmatrixEmpty14}{\showpaste}
+\tab{5}\spadcommand{setsubMatrix!(b,1,1,transpose(subMatrix(b,1,3,1,3)))\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch15}
+\begin{paste}{MenuexmatrixFull15}{MenuexmatrixEmpty15}
+\pastebutton{MenuexmatrixFull15}{\hidepaste}
+\tab{5}\spadcommand{trace( matrix([[1,x,x**2,x**3],[1,y,y**2,y**3],[1,z,z**2,z**3],[1,u,u**2,u**3]]) )}
+\indentrel{3}\begin{verbatim}
+          2        3
+   (15)  z  + y + u  + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty15}
+\begin{paste}{MenuexmatrixEmpty15}{MenuexmatrixPatch15}
+\pastebutton{MenuexmatrixEmpty15}{\showpaste}
+\tab{5}\spadcommand{trace( matrix([[1,x,x**2,x**3],[1,y,y**2,y**3],[1,z,z**2,z**3],[1,u,u**2,u**3]]) )}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch16}
+\begin{paste}{MenuexmatrixFull16}{MenuexmatrixEmpty16}
+\pastebutton{MenuexmatrixFull16}{\hidepaste}
+\tab{5}\spadcommand{determinant(matrix([[1,2,3,4],[2,3,2,5],[3,4,5,6],[4,1,6,7]]))}
+\indentrel{3}\begin{verbatim}
+   (16)  - 48
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty16}
+\begin{paste}{MenuexmatrixEmpty16}{MenuexmatrixPatch16}
+\pastebutton{MenuexmatrixEmpty16}{\showpaste}
+\tab{5}\spadcommand{determinant(matrix([[1,2,3,4],[2,3,2,5],[3,4,5,6],[4,1,6,7]]))}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch17}
+\begin{paste}{MenuexmatrixFull17}{MenuexmatrixEmpty17}
+\pastebutton{MenuexmatrixFull17}{\hidepaste}
+\tab{5}\spadcommand{inverse(matrix([[1,2,1],[-2,3,4],[-1,5,6]]))}
+\indentrel{3}\begin{verbatim}
+         Ú  2        5 ¿
+         ³- Ä  - 1   Ä ³
+         ³  7        7 ³
+         ³             ³
+   (17)  ³ 8          6³
+         ³ Ä    1   - ij
+         ³ 7          7³
+         ³             ³
+         À- 1  - 1   1 Ù
+               Type: Union(Matrix Fraction Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty17}
+\begin{paste}{MenuexmatrixEmpty17}{MenuexmatrixPatch17}
+\pastebutton{MenuexmatrixEmpty17}{\showpaste}
+\tab{5}\spadcommand{inverse(matrix([[1,2,1],[-2,3,4],[-1,5,6]]))}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixPatch18}
+\begin{paste}{MenuexmatrixFull18}{MenuexmatrixEmpty18}
+\pastebutton{MenuexmatrixFull18}{\hidepaste}
+\tab{5}\spadcommand{rank(matrix([[0,4,1],[5,3,-7],[-5,5,9]]))}
+\indentrel{3}\begin{verbatim}
+   (18)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexmatrixEmpty18}
+\begin{paste}{MenuexmatrixEmpty18}{MenuexmatrixPatch18}
+\pastebutton{MenuexmatrixEmpty18}{\showpaste}
+\tab{5}\spadcommand{rank(matrix([[0,4,1],[5,3,-7],[-5,5,9]]))}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlapPatch1}
+\begin{paste}{MenuexlapFull1}{MenuexlapEmpty1}
+\pastebutton{MenuexlapFull1}{\hidepaste}
+\tab{5}\spadcommand{laplace(t**4 * exp(-a*t) / factorial(4), t, s)}
+\indentrel{3}\begin{verbatim}
+                            1
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+         5       4      2 3      3 2     4     5
+        s  + 5a s  + 10a s  + 10a s  + 5a s + a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlapEmpty1}
+\begin{paste}{MenuexlapEmpty1}{MenuexlapPatch1}
+\pastebutton{MenuexlapEmpty1}{\showpaste}
+\tab{5}\spadcommand{laplace(t**4 * exp(-a*t) / factorial(4), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlapPatch2}
+\begin{paste}{MenuexlapFull2}{MenuexlapEmpty2}
+\pastebutton{MenuexlapFull2}{\hidepaste}
+\tab{5}\spadcommand{laplace(sin(a*t) * cosh(a*t) - cos(a*t) * sinh(a*t), t, s)}
+\indentrel{3}\begin{verbatim}
+             3
+           4a
+   (2)  ÄÄÄÄÄÄÄÄ
+         4     4
+        s  + 4a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlapEmpty2}
+\begin{paste}{MenuexlapEmpty2}{MenuexlapPatch2}
+\pastebutton{MenuexlapEmpty2}{\showpaste}
+\tab{5}\spadcommand{laplace(sin(a*t) * cosh(a*t) - cos(a*t) * sinh(a*t), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlapPatch3}
+\begin{paste}{MenuexlapFull3}{MenuexlapEmpty3}
+\pastebutton{MenuexlapFull3}{\hidepaste}
+\tab{5}\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)}
+\indentrel{3}\begin{verbatim}
+             2    2
+   (3)  log(s  + a ) - 2log(s)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlapEmpty3}
+\begin{paste}{MenuexlapEmpty3}{MenuexlapPatch3}
+\pastebutton{MenuexlapEmpty3}{\showpaste}
+\tab{5}\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlapPatch4}
+\begin{paste}{MenuexlapFull4}{MenuexlapEmpty4}
+\pastebutton{MenuexlapFull4}{\hidepaste}
+\tab{5}\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)}
+\indentrel{3}\begin{verbatim}
+   (4)  - log(s - a) + log(s - b)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlapEmpty4}
+\begin{paste}{MenuexlapEmpty4}{MenuexlapPatch4}
+\pastebutton{MenuexlapEmpty4}{\showpaste}
+\tab{5}\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlapPatch5}
+\begin{paste}{MenuexlapFull5}{MenuexlapEmpty5}
+\pastebutton{MenuexlapFull5}{\hidepaste}
+\tab{5}\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)}
+\indentrel{3}\begin{verbatim}
+          b    s + c - a
+        %e log(ÄÄÄÄÄÄÄÄÄ)
+                   c
+   (5)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+              s - a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlapEmpty5}
+\begin{paste}{MenuexlapEmpty5}{MenuexlapPatch5}
+\pastebutton{MenuexlapEmpty5}{\showpaste}
+\tab{5}\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexlapPatch6}
+\begin{paste}{MenuexlapFull6}{MenuexlapEmpty6}
+\pastebutton{MenuexlapFull6}{\hidepaste}
+\tab{5}\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)}
+\indentrel{3}\begin{verbatim}
+               2    2
+              s  + b             d
+        a log(ÄÄÄÄÄÄÄ) + 2c atan(Ä)
+                  2              s
+                 b
+   (6)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                     2s
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexlapEmpty6}
+\begin{paste}{MenuexlapEmpty6}{MenuexlapPatch6}
+\pastebutton{MenuexlapEmpty6}{\showpaste}
+\tab{5}\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexintPatch1}
+\begin{paste}{MenuexintFull1}{MenuexintEmpty1}
+\pastebutton{MenuexintFull1}{\hidepaste}
+\tab{5}\spadcommand{integrate((x**2+2*x+1)/((x+1)**6+1),x)}
+\indentrel{3}\begin{verbatim}
+              3     2
+        atan(x  + 3x  + 3x + 1)
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                   3
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexintEmpty1}
+\begin{paste}{MenuexintEmpty1}{MenuexintPatch1}
+\pastebutton{MenuexintEmpty1}{\showpaste}
+\tab{5}\spadcommand{integrate((x**2+2*x+1)/((x+1)**6+1),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexintPatch2}
+\begin{paste}{MenuexintFull2}{MenuexintEmpty2}
+\pastebutton{MenuexintFull2}{\hidepaste}
+\tab{5}\spadcommand{integrate(1/(x**3+x+1),x)\bound{i }}
+\indentrel{3}\begin{verbatim}
+   (2)
+           ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+           ³         2
+           ³- 93%%CE0  + 12
+         ( ³ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ  - %%CE0)
+          \³       31
+      *
+         log
+                             ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+                             ³         2
+                             ³- 93%%CE0  + 12           2
+              (62%%CE0 + 31) ³ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ  + 62%%CE0
+                            \³       31
+            + 
+              - 31%%CE0 + 18x - 4
+     + 
+             ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+             ³         2
+             ³- 93%%CE0  + 12
+         (-  ³ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ  - %%CE0)
+            \³       31
+      *
+         log
+                               ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+                               ³         2
+                               ³- 93%%CE0  + 12
+              (- 62%%CE0 - 31) ³ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                              \³       31
+            + 
+                     2
+              62%%CE0  - 31%%CE0 + 18x - 4
+     + 
+                           2
+       2%%CE0 log(- 62%%CE0  + 31%%CE0 + 9x + 4)
+  /
+     2
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexintEmpty2}
+\begin{paste}{MenuexintEmpty2}{MenuexintPatch2}
+\pastebutton{MenuexintEmpty2}{\showpaste}
+\tab{5}\spadcommand{integrate(1/(x**3+x+1),x)\bound{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexintPatch3}
+\begin{paste}{MenuexintFull3}{MenuexintEmpty3}
+\pastebutton{MenuexintFull3}{\hidepaste}
+\tab{5}\spadcommand{definingPolynomial(tower(\%).2::EXPR INT)\free{i }}
+\indentrel{3}\begin{verbatim}
+               3
+        31%%CE0  - 3%%CE0 - 1
+   (3)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                  31
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexintEmpty3}
+\begin{paste}{MenuexintEmpty3}{MenuexintPatch3}
+\pastebutton{MenuexintEmpty3}{\showpaste}
+\tab{5}\spadcommand{definingPolynomial(tower(\%).2::EXPR INT)\free{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexintPatch4}
+\begin{paste}{MenuexintFull4}{MenuexintEmpty4}
+\pastebutton{MenuexintFull4}{\hidepaste}
+\tab{5}\spadcommand{integrate(1/(x**2 + a),x)}
+\indentrel{3}\begin{verbatim}
+               2      ÚÄÄÄ¿
+             (x  - a)\³- a  + 2a x         ÚÄ¿
+         log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)      x\³a
+                      2             atan(ÄÄÄÄÄ)
+                     x  + a                a
+   (4)  [ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄ]
+                     ÚÄÄÄ¿               ÚÄ¿
+                   2\³- a               \³a
+               Type: Union(List Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexintEmpty4}
+\begin{paste}{MenuexintEmpty4}{MenuexintPatch4}
+\pastebutton{MenuexintEmpty4}{\showpaste}
+\tab{5}\spadcommand{integrate(1/(x**2 + a),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexintPatch5}
+\begin{paste}{MenuexintFull5}{MenuexintEmpty5}
+\pastebutton{MenuexintFull5}{\hidepaste}
+\tab{5}\spadcommand{complexIntegrate(1/(x**2 + a),x)}
+\indentrel{3}\begin{verbatim}
+              ÚÄÄÄ¿              ÚÄÄÄ¿
+            x\³- a  + a        x\³- a  - a
+        log(ÄÄÄÄÄÄÄÄÄÄÄ) - log(ÄÄÄÄÄÄÄÄÄÄÄ)
+                ÚÄÄÄ¿              ÚÄÄÄ¿
+               \³- a              \³- a
+   (5)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                        ÚÄÄÄ¿
+                      2\³- a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexintEmpty5}
+\begin{paste}{MenuexintEmpty5}{MenuexintPatch5}
+\pastebutton{MenuexintEmpty5}{\showpaste}
+\tab{5}\spadcommand{complexIntegrate(1/(x**2 + a),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexintPatch6}
+\begin{paste}{MenuexintFull6}{MenuexintEmpty6}
+\pastebutton{MenuexintFull6}{\hidepaste}
+\tab{5}\spadcommand{integrate(x**3 / (a+b*x)**(1/3),x)}
+\indentrel{3}\begin{verbatim}
+   (6)
+        3 3         2 2       2          3 3ÚÄÄÄÄÄÄÄ¿2
+   (120b x  - 135a b x  + 162a b x - 243a )\³b x + a
+   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                              4
+                          440b
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexintEmpty6}
+\begin{paste}{MenuexintEmpty6}{MenuexintPatch6}
+\pastebutton{MenuexintEmpty6}{\showpaste}
+\tab{5}\spadcommand{integrate(x**3 / (a+b*x)**(1/3),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexintPatch7}
+\begin{paste}{MenuexintFull7}{MenuexintEmpty7}
+\pastebutton{MenuexintFull7}{\hidepaste}
+\tab{5}\spadcommand{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)}
+\indentrel{3}\begin{verbatim}
+   (7)
+       -
+              2 2 ÚÄ¿
+            2b x \³3
+         *
+                3ÚÄ¿3ÚÄÄÄÄÄÄÄ¿2   3ÚÄ¿2 3ÚÄÄÄÄÄÄÄ¿
+            log(\³a \³b x + a   + \³a   \³b x + a + a)
+     + 
+         2 2 ÚÄ¿    3ÚÄ¿2 3ÚÄÄÄÄÄÄÄ¿
+       4b x \³3 log(\³a   \³b x + a - a)
+     + 
+                    ÚÄ¿3ÚÄ¿2 3ÚÄÄÄÄÄÄÄ¿    ÚÄ¿
+          2 2     2\³3 \³a   \³b x + a + a\³3
+       12b x atan(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
+                               3a
+     + 
+                    ÚÄ¿3ÚÄ¿3ÚÄÄÄÄÄÄÄ¿2
+       (12b x - 9a)\³3 \³a \³b x + a
+  /
+        2 2 ÚÄ¿3ÚÄ¿
+     18a x \³3 \³a
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexintEmpty7}
+\begin{paste}{MenuexintEmpty7}{MenuexintPatch7}
+\pastebutton{MenuexintEmpty7}{\showpaste}
+\tab{5}\spadcommand{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexintPatch8}
+\begin{paste}{MenuexintFull8}{MenuexintEmpty8}
+\pastebutton{MenuexintFull8}{\hidepaste}
+\tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)}
+\indentrel{3}\begin{verbatim}
+           x      ÚÄÄÄÄÄÄÄÄ¿
+         Ú¿  log(\³b + %N a  + 1)
+   (8)   ³   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ d%N
+        ÀÙ            %N
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexintEmpty8}
+\begin{paste}{MenuexintEmpty8}{MenuexintPatch8}
+\pastebutton{MenuexintEmpty8}{\showpaste}
+\tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexintPatch9}
+\begin{paste}{MenuexintFull9}{MenuexintEmpty9}
+\pastebutton{MenuexintFull9}{\hidepaste}
+\tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b))/(sqrt(x+b)*(x+cosh(1+sqrt(x+b)))),x)}
+\indentrel{3}\begin{verbatim}
+   (9)
+                          ÚÄÄÄÄÄ¿
+                 - 2cosh(\³x + b  + 1) - 2x
+     2log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
+                ÚÄÄÄÄÄ¿              ÚÄÄÄÄÄ¿
+          sinh(\³x + b  + 1) - cosh(\³x + b  + 1)
+   + 
+         ÚÄÄÄÄÄ¿
+     - 2\³x + b
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexintEmpty9}
+\begin{paste}{MenuexintEmpty9}{MenuexintPatch9}
+\pastebutton{MenuexintEmpty9}{\showpaste}
+\tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b))/(sqrt(x+b)*(x+cosh(1+sqrt(x+b)))),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexintPatch10}
+\begin{paste}{MenuexintFull10}{MenuexintEmpty10}
+\pastebutton{MenuexintFull10}{\hidepaste}
+\tab{5}\spadcommand{integrate(tan(atan(x)/3),x)}
+\indentrel{3}\begin{verbatim}
+   (10)
+                 atan(x) 2             atan(x) 2
+       8log(3tan(ÄÄÄÄÄÄÄ)  - 1) - 3tan(ÄÄÄÄÄÄÄ)
+                    3                     3
+     + 
+               atan(x)
+       18x tan(ÄÄÄÄÄÄÄ)
+                  3
+  /
+     18
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexintEmpty10}
+\begin{paste}{MenuexintEmpty10}{MenuexintPatch10}
+\pastebutton{MenuexintEmpty10}{\showpaste}
+\tab{5}\spadcommand{integrate(tan(atan(x)/3),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexintPatch11}
+\begin{paste}{MenuexintFull11}{MenuexintEmpty11}
+\pastebutton{MenuexintFull11}{\hidepaste}
+\tab{5}\spadcommand{integrate((x + 1) / (x * (x + log x)**(3/2)),x)}
+\indentrel{3}\begin{verbatim}
+             ÚÄÄÄÄÄÄÄÄÄÄ¿
+           2\³log(x) + x
+   (11)  - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+             log(x) + x
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexintEmpty11}
+\begin{paste}{MenuexintEmpty11}{MenuexintPatch11}
+\pastebutton{MenuexintEmpty11}{\showpaste}
+\tab{5}\spadcommand{integrate((x + 1) / (x * (x + log x)**(3/2)),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexintPatch12}
+\begin{paste}{MenuexintFull12}{MenuexintEmpty12}
+\pastebutton{MenuexintFull12}{\hidepaste}
+\tab{5}\spadcommand{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)}
+\indentrel{3}\begin{verbatim}
+                      ÚÄÄÄ¿    erf(x) - 1      ÚÄÄÄ¿
+         (erf(x) - 1)\³%pi log(ÄÄÄÄÄÄÄÄÄÄ) - 2\³%pi
+                               erf(x) + 1
+   (12)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                         8erf(x) - 8
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexintEmpty12}
+\begin{paste}{MenuexintEmpty12}{MenuexintPatch12}
+\pastebutton{MenuexintEmpty12}{\showpaste}
+\tab{5}\spadcommand{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumPatch1}
+\begin{paste}{MenuexsumFull1}{MenuexsumEmpty1}
+\pastebutton{MenuexsumFull1}{\hidepaste}
+\tab{5}\spadcommand{[i for i in 1..15]}
+\indentrel{3}\begin{verbatim}
+   (1)  [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumEmpty1}
+\begin{paste}{MenuexsumEmpty1}{MenuexsumPatch1}
+\pastebutton{MenuexsumEmpty1}{\showpaste}
+\tab{5}\spadcommand{[i for i in 1..15]}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumPatch2}
+\begin{paste}{MenuexsumFull2}{MenuexsumEmpty2}
+\pastebutton{MenuexsumFull2}{\hidepaste}
+\tab{5}\spadcommand{reduce(+,[i for i in 1..15])}
+\indentrel{3}\begin{verbatim}
+   (2)  120
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumEmpty2}
+\begin{paste}{MenuexsumEmpty2}{MenuexsumPatch2}
+\pastebutton{MenuexsumEmpty2}{\showpaste}
+\tab{5}\spadcommand{reduce(+,[i for i in 1..15])}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumPatch3}
+\begin{paste}{MenuexsumFull3}{MenuexsumEmpty3}
+\pastebutton{MenuexsumFull3}{\hidepaste}
+\tab{5}\spadcommand{[n**2 for n in 5..20]}
+\indentrel{3}\begin{verbatim}
+   (3)
+   [25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225,
+    256, 289, 324, 361, 400]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumEmpty3}
+\begin{paste}{MenuexsumEmpty3}{MenuexsumPatch3}
+\pastebutton{MenuexsumEmpty3}{\showpaste}
+\tab{5}\spadcommand{[n**2 for n in 5..20]}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumPatch4}
+\begin{paste}{MenuexsumFull4}{MenuexsumEmpty4}
+\pastebutton{MenuexsumFull4}{\hidepaste}
+\tab{5}\spadcommand{reduce(+,[n**2 for n in 5..20])}
+\indentrel{3}\begin{verbatim}
+   (4)  2840
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumEmpty4}
+\begin{paste}{MenuexsumEmpty4}{MenuexsumPatch4}
+\pastebutton{MenuexsumEmpty4}{\showpaste}
+\tab{5}\spadcommand{reduce(+,[n**2 for n in 5..20])}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumPatch5}
+\begin{paste}{MenuexsumFull5}{MenuexsumEmpty5}
+\pastebutton{MenuexsumFull5}{\hidepaste}
+\tab{5}\spadcommand{reduce(+,[1.0/factorial(n) for n in 0..20])}
+\indentrel{3}\begin{verbatim}
+   (5)  2.7182818284 590452354
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumEmpty5}
+\begin{paste}{MenuexsumEmpty5}{MenuexsumPatch5}
+\pastebutton{MenuexsumEmpty5}{\showpaste}
+\tab{5}\spadcommand{reduce(+,[1.0/factorial(n) for n in 0..20])}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumPatch6}
+\begin{paste}{MenuexsumFull6}{MenuexsumEmpty6}
+\pastebutton{MenuexsumFull6}{\hidepaste}
+\tab{5}\spadcommand{s := sum(k**2,k = a..b)\bound{s }}
+\indentrel{3}\begin{verbatim}
+          3     2         3     2
+        2b  + 3b  + b - 2a  + 3a  - a
+   (6)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                      6
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumEmpty6}
+\begin{paste}{MenuexsumEmpty6}{MenuexsumPatch6}
+\pastebutton{MenuexsumEmpty6}{\showpaste}
+\tab{5}\spadcommand{s := sum(k**2,k = a..b)\bound{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumPatch7}
+\begin{paste}{MenuexsumFull7}{MenuexsumEmpty7}
+\pastebutton{MenuexsumFull7}{\hidepaste}
+\tab{5}\spadcommand{eval(s,[a,b],[1,25])\free{s }}
+\indentrel{3}\begin{verbatim}
+   (7)  5525
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumEmpty7}
+\begin{paste}{MenuexsumEmpty7}{MenuexsumPatch7}
+\pastebutton{MenuexsumEmpty7}{\showpaste}
+\tab{5}\spadcommand{eval(s,[a,b],[1,25])\free{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumPatch8}
+\begin{paste}{MenuexsumFull8}{MenuexsumEmpty8}
+\pastebutton{MenuexsumFull8}{\hidepaste}
+\tab{5}\spadcommand{reduce(+,[i**2 for i in 1..25])}
+\indentrel{3}\begin{verbatim}
+   (8)  5525
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumEmpty8}
+\begin{paste}{MenuexsumEmpty8}{MenuexsumPatch8}
+\pastebutton{MenuexsumEmpty8}{\showpaste}
+\tab{5}\spadcommand{reduce(+,[i**2 for i in 1..25])}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumPatch9}
+\begin{paste}{MenuexsumFull9}{MenuexsumEmpty9}
+\pastebutton{MenuexsumFull9}{\hidepaste}
+\tab{5}\spadcommand{sum(k**3,k = 1..n)}
+\indentrel{3}\begin{verbatim}
+         4     3    2
+        n  + 2n  + n
+   (9)  ÄÄÄÄÄÄÄÄÄÄÄÄÄ
+              4
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumEmpty9}
+\begin{paste}{MenuexsumEmpty9}{MenuexsumPatch9}
+\pastebutton{MenuexsumEmpty9}{\showpaste}
+\tab{5}\spadcommand{sum(k**3,k = 1..n)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumPatch10}
+\begin{paste}{MenuexsumFull10}{MenuexsumEmpty10}
+\pastebutton{MenuexsumFull10}{\hidepaste}
+\tab{5}\spadcommand{sum(k,k = 1..n) ** 2}
+\indentrel{3}\begin{verbatim}
+          4     3    2
+         n  + 2n  + n
+   (10)  ÄÄÄÄÄÄÄÄÄÄÄÄÄ
+               4
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumEmpty10}
+\begin{paste}{MenuexsumEmpty10}{MenuexsumPatch10}
+\pastebutton{MenuexsumEmpty10}{\showpaste}
+\tab{5}\spadcommand{sum(k,k = 1..n) ** 2}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumPatch11}
+\begin{paste}{MenuexsumFull11}{MenuexsumEmpty11}
+\pastebutton{MenuexsumFull11}{\hidepaste}
+\tab{5}\spadcommand{sum(3*k**2/(c**2 + 1) + 12*k/d,k = (3*a)..(4*b))}
+\indentrel{3}\begin{verbatim}
+   (11)
+            3      2           3      2
+       (128b  + 48b  + 4b - 54a  + 27a  - 3a)d
+     + 
+            2             2        2       2             2
+       (192b  + 48b - 108a  + 36a)c  + 192b  + 48b - 108a
+     + 
+       36a
+  /
+        2
+     (2c  + 2)d
+           Type: Union(Fraction Polynomial Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumEmpty11}
+\begin{paste}{MenuexsumEmpty11}{MenuexsumPatch11}
+\pastebutton{MenuexsumEmpty11}{\showpaste}
+\tab{5}\spadcommand{sum(3*k**2/(c**2 + 1) + 12*k/d,k = (3*a)..(4*b))}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumPatch12}
+\begin{paste}{MenuexsumFull12}{MenuexsumEmpty12}
+\pastebutton{MenuexsumFull12}{\hidepaste}
+\tab{5}\spadcommand{sum(k * x**k,k = 1..n)}
+\indentrel{3}\begin{verbatim}
+             2               n
+         (n x  + (- n - 1)x)x  + x
+   (12)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                 2
+                x  - 2x + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumEmpty12}
+\begin{paste}{MenuexsumEmpty12}{MenuexsumPatch12}
+\pastebutton{MenuexsumEmpty12}{\showpaste}
+\tab{5}\spadcommand{sum(k * x**k,k = 1..n)}
+\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumPatch13}
+\begin{paste}{MenuexsumFull13}{MenuexsumEmpty13}
+\pastebutton{MenuexsumFull13}{\hidepaste}
+\tab{5}\spadcommand{limit( sum(1/(k * (k + 2)),k = 1..n) ,n = \%plusInfinity)}
+\indentrel{3}\begin{verbatim}
+         3
+   (13)  Ä
+         4
+Type: Union(OrderedCompletion Fraction Polynomial Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{MenuexsumEmpty13}
+\begin{paste}{MenuexsumEmpty13}{MenuexsumPatch13}
+\pastebutton{MenuexsumEmpty13}{\showpaste}
+\tab{5}\spadcommand{limit( sum(1/(k * (k + 2)),k = 1..n) ,n = \%plusInfinity)}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/evalex.ht b/src/hyper/pages/evalex.ht
new file mode 100644
index 00000000..1a2eef3f
--- /dev/null
+++ b/src/hyper/pages/evalex.ht
@@ -0,0 +1,68 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+\begin{page}{PrefixEval}{Example of Standard Evaluation}
+\beginscroll
+We illustrate the general evaluation of {\em op a} for some
+prefix operator {\em op} and operand {\em a}
+by the example: {\em cos(2)}.
+The evaluation steps are as follows:
+\vspace{1}\newline
+1.\tab{3}{\em a} evaluates to a value of some type.
+\newline\tab{3}{\em Example:} {\em 2} evaluates to {\em 2} of type \spadtype{Integer}
+\newline
+2.\tab{3}\Language{} then chooses a function {\em op} based on the type of {\em a}.
+\newline\tab{3}{\em Example:} The function {\em cos:} \spadtype{Float} {\em ->}
+\spadtype{Float} is chosen.
+\newline
+3.\tab{3}If the argument type of the function is different from that of {\em a},
+then the system coerces
+%\downlink{coerces}{Coercion}
+the value of {\em a} to the
+argument type.
+\newline\tab{3}{\em Example:} The integer {\em 2} is coerced to the float {\em 2.0}.
+\newline
+4.\tab{3}The function is then applied to the value of {\em a} to produce the value
+for {\em op a}.
+\newline\tab{3}{\em Example:} The function {\em cos} is applied to {\em 2.0}.
+\vspace{1}\newline
+Try it:
+\example{cos(2)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{InfixEval}{Example of Standard Evaluation}
+\beginscroll
+We illustrate the general evaluation of {\em a op b} for some
+infix operator {\em op} with operands {\em a} and {\em b}
+by the example: {\em 2 + 3.4}.
+The evaluation steps are as follows:
+\vspace{1}\newline
+1.\tab{3}{\em a} and {\em b} are evaluated, each producing a value and a type.
+\newline\tab{3}{\em Example:} {\em 2} evaluates to {\em 2} of type \spadtype{Integer};
+{\em 3.4} evaluates to {\em 3.4} of type \spadtype{Float}.
+\vspace{1}\newline
+2.\tab{3}\Language{} then chooses a function {\em op} based on the types of {\em a} and {\em b}.
+\newline\tab{3}{\em Example:} The function {\em +: (D,D) -> D}
+is chosen requiring a common type {\em D} for both arguments to {\em +}.
+An operation called {\em resolve} determines the `smallest common type' \spadtype{Float}.
+\vspace{1}\newline
+3.\tab{3}If the argument types for the function are different from
+those of {\em a} and {\em b},
+then the system coerces
+%\downlink{coerces}{Coercion}
+the values to the argument types.
+\newline\tab{3}{\em Example:} The integer {\em 2} is coerced to the float {\em 2.0}.
+\vspace{1}\newline
+4.\tab{3}The function is then applied to the values of {\em a} and {\em b}
+to produce the value for {\em a op b}.
+\newline\tab{3}{\em Example:} The function {\em +: (D,D) -> D}, where
+{\em D} = \spadtype{Float} is applied to {\em 2.0} and {\em 3.4} to produce {\em 5.4}.
+\vspace{1}\newline
+Try it:
+\example{2 + 3.4}
+\endscroll
+\autobuttons\end{page}
+
+
diff --git a/src/hyper/pages/evalex.pht b/src/hyper/pages/evalex.pht
new file mode 100644
index 00000000..c44d2710
--- /dev/null
+++ b/src/hyper/pages/evalex.pht
@@ -0,0 +1,32 @@
+\begin{patch}{PrefixEvalPatch1}
+\begin{paste}{PrefixEvalFull1}{PrefixEvalEmpty1}
+\pastebutton{PrefixEvalFull1}{\hidepaste}
+\tab{5}\spadcommand{cos(2)}
+\indentrel{3}\begin{verbatim}
+   (1)  cos(2)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PrefixEvalEmpty1}
+\begin{paste}{PrefixEvalEmpty1}{PrefixEvalPatch1}
+\pastebutton{PrefixEvalEmpty1}{\showpaste}
+\tab{5}\spadcommand{cos(2)}
+\end{paste}\end{patch}
+
+\begin{patch}{InfixEvalPatch1}
+\begin{paste}{InfixEvalFull1}{InfixEvalEmpty1}
+\pastebutton{InfixEvalFull1}{\hidepaste}
+\tab{5}\spadcommand{2 + 3.4}
+\indentrel{3}\begin{verbatim}
+   (1)  5.4
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{InfixEvalEmpty1}
+\begin{paste}{InfixEvalEmpty1}{InfixEvalPatch1}
+\pastebutton{InfixEvalEmpty1}{\showpaste}
+\tab{5}\spadcommand{2 + 3.4}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/exdiff.ht b/src/hyper/pages/exdiff.ht
new file mode 100644
index 00000000..f2f30262
--- /dev/null
+++ b/src/hyper/pages/exdiff.ht
@@ -0,0 +1,80 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Title: Differentiation
+
+%  Author: Clifton J. Williamson
+%  Date created: 1 November 1989
+%  Date last updated: 1 November 1989
+
+\begin{page}{ExDiffBasic}{Computing Derivatives}
+\beginscroll
+To compute a derivative, you must specify an expression and a variable
+of differentiation.
+For example, to compute the derivative of {\em sin(x) * exp(x**2)} with respect to the
+variable {\em x}, issue the following command:
+\spadpaste{differentiate(sin(x) * exp(x**2),x)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExDiffSeveralVariables}{Derivatives of Functions of Several Variables}
+\beginscroll
+Partial derivatives are computed in the same way as derivatives of functions
+of one variable: you specify the function and a variable of differentiation.
+For example:
+\spadpaste{differentiate(sin(x) * tan(y)/(x**2 + y**2),x)}
+\spadpaste{differentiate(sin(x) * tan(y)/(x**2 + y**2),y)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExDiffHigherOrder}{Derivatives of Higher Order}
+\beginscroll
+To compute a derivative of higher order (e.g. a second or third derivative),
+pass the order as the third argument of the function 'differentiate'.
+For example, to compute the fourth derivative of {\em exp(x**2)}, issue the
+following command:
+\spadpaste{differentiate(exp(x**2),x,4)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExDiffMultipleI}{Multiple Derivatives I}
+\beginscroll
+When given a function of several variables, you may take derivatives repeatedly
+and with respect to different variables.
+The following command differentiates the function {\em sin(x)/(x**2 + y**2)}
+first with respect to {\em x} and then with respect to {\em y}:
+\spadpaste{differentiate(sin(x)/(x**2 + y**2),[x,y])}
+As you can see, we first specify the function and then a list of the variables
+of differentiation.
+Variables may appear on the list more than once.
+For example, the following command differentiates the same function with
+respect to {\em x} and then twice with respect to {\em y}.
+\spadpaste{differentiate(sin(x)/(x**2 + y**2),[x,y,y])}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExDiffMultipleII}{Multiple Derivatives II}
+\beginscroll
+You may also compute multiple derivatives by specifying a list of variables
+together with a list of multiplicities.
+For example, to differentiate {\em cos(z)/(x**2 + y**3)}
+first with respect to {\em x},
+then twice with respect to {\em y}, then three times with respect to {\em z},
+issue the following command:
+\spadpaste{differentiate(cos(z)/(x**2 + y**3),[x,y,z],[1,2,3])}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExDiffFormalIntegral}{Derivatives of Functions Involving Formal Integrals}
+\beginscroll
+When a function does not have a closed-form antiderivative, \Language{}
+returns a formal integral.
+A typical example is
+\spadpaste{f := integrate(sqrt(1 + t**3),t) \bound{f}}
+This formal integral may be differentiated, either by itself or in any
+combination with other functions:
+\spadpaste{differentiate(f,t) \free{f}}
+\spadpaste{differentiate(f * t**2,t) \free{f}}
+\endscroll
+\autobuttons\end{page}
diff --git a/src/hyper/pages/exdiff.pht b/src/hyper/pages/exdiff.pht
new file mode 100644
index 00000000..f8f5ca51
--- /dev/null
+++ b/src/hyper/pages/exdiff.pht
@@ -0,0 +1,197 @@
+\begin{patch}{ExDiffBasicPatch1}
+\begin{paste}{ExDiffBasicFull1}{ExDiffBasicEmpty1}
+\pastebutton{ExDiffBasicFull1}{\hidepaste}
+\tab{5}\spadcommand{differentiate(sin(x) * exp(x**2),x)}
+\indentrel{3}\begin{verbatim}
+              2                  2
+             x                  x
+   (1)  2x %e  sin(x) + cos(x)%e
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExDiffBasicEmpty1}
+\begin{paste}{ExDiffBasicEmpty1}{ExDiffBasicPatch1}
+\pastebutton{ExDiffBasicEmpty1}{\showpaste}
+\tab{5}\spadcommand{differentiate(sin(x) * exp(x**2),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExDiffSeveralVariablesPatch1}
+\begin{paste}{ExDiffSeveralVariablesFull1}{ExDiffSeveralVariablesEmpty1}
+\pastebutton{ExDiffSeveralVariablesFull1}{\hidepaste}
+\tab{5}\spadcommand{differentiate(sin(x) * tan(y)/(x**2 + y**2),x)}
+\indentrel{3}\begin{verbatim}
+                         2    2
+        (- 2x sin(x) + (y  + x )cos(x))tan(y)
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                    4     2 2    4
+                   y  + 2x y  + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExDiffSeveralVariablesEmpty1}
+\begin{paste}{ExDiffSeveralVariablesEmpty1}{ExDiffSeveralVariablesPatch1}
+\pastebutton{ExDiffSeveralVariablesEmpty1}{\showpaste}
+\tab{5}\spadcommand{differentiate(sin(x) * tan(y)/(x**2 + y**2),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExDiffSeveralVariablesPatch2}
+\begin{paste}{ExDiffSeveralVariablesFull2}{ExDiffSeveralVariablesEmpty2}
+\pastebutton{ExDiffSeveralVariablesFull2}{\hidepaste}
+\tab{5}\spadcommand{differentiate(sin(x) * tan(y)/(x**2 + y**2),y)}
+\indentrel{3}\begin{verbatim}
+   (2)
+         2    2             2
+       (y  + x )sin(x)tan(y)  - 2y sin(x)tan(y)
+     + 
+         2    2
+       (y  + x )sin(x)
+  /
+      4     2 2    4
+     y  + 2x y  + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExDiffSeveralVariablesEmpty2}
+\begin{paste}{ExDiffSeveralVariablesEmpty2}{ExDiffSeveralVariablesPatch2}
+\pastebutton{ExDiffSeveralVariablesEmpty2}{\showpaste}
+\tab{5}\spadcommand{differentiate(sin(x) * tan(y)/(x**2 + y**2),y)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExDiffMultipleIPatch1}
+\begin{paste}{ExDiffMultipleIFull1}{ExDiffMultipleIEmpty1}
+\pastebutton{ExDiffMultipleIFull1}{\hidepaste}
+\tab{5}\spadcommand{differentiate(sin(x)/(x**2 + y**2),[x,y])}
+\indentrel{3}\begin{verbatim}
+                           3     2
+        8x y sin(x) + (- 2y  - 2x y)cos(x)
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+               6     2 4     4 2    6
+              y  + 3x y  + 3x y  + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExDiffMultipleIEmpty1}
+\begin{paste}{ExDiffMultipleIEmpty1}{ExDiffMultipleIPatch1}
+\pastebutton{ExDiffMultipleIEmpty1}{\showpaste}
+\tab{5}\spadcommand{differentiate(sin(x)/(x**2 + y**2),[x,y])}
+\end{paste}\end{patch}
+
+\begin{patch}{ExDiffMultipleIPatch2}
+\begin{paste}{ExDiffMultipleIFull2}{ExDiffMultipleIEmpty2}
+\pastebutton{ExDiffMultipleIFull2}{\hidepaste}
+\tab{5}\spadcommand{differentiate(sin(x)/(x**2 + y**2),[x,y,y])}
+\indentrel{3}\begin{verbatim}
+   (2)
+           2     3             4     2 2     4
+   (- 40x y  + 8x )sin(x) + (6y  + 4x y  - 2x )cos(x)
+   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+              8     2 6     4 4     6 2    8
+             y  + 4x y  + 6x y  + 4x y  + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExDiffMultipleIEmpty2}
+\begin{paste}{ExDiffMultipleIEmpty2}{ExDiffMultipleIPatch2}
+\pastebutton{ExDiffMultipleIEmpty2}{\showpaste}
+\tab{5}\spadcommand{differentiate(sin(x)/(x**2 + y**2),[x,y,y])}
+\end{paste}\end{patch}
+
+\begin{patch}{ExDiffMultipleIIPatch1}
+\begin{paste}{ExDiffMultipleIIFull1}{ExDiffMultipleIIEmpty1}
+\pastebutton{ExDiffMultipleIIFull1}{\hidepaste}
+\tab{5}\spadcommand{differentiate(cos(z)/(x**2 + y**3),[x,y,z],[1,2,3])}
+\indentrel{3}\begin{verbatim}
+                    4      3
+            (- 84x y  + 24x y)sin(z)
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+         12     2 9     4 6     6 3    8
+        y   + 4x y  + 6x y  + 4x y  + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExDiffMultipleIIEmpty1}
+\begin{paste}{ExDiffMultipleIIEmpty1}{ExDiffMultipleIIPatch1}
+\pastebutton{ExDiffMultipleIIEmpty1}{\showpaste}
+\tab{5}\spadcommand{differentiate(cos(z)/(x**2 + y**3),[x,y,z],[1,2,3])}
+\end{paste}\end{patch}
+
+\begin{patch}{ExDiffHigherOrderPatch1}
+\begin{paste}{ExDiffHigherOrderFull1}{ExDiffHigherOrderEmpty1}
+\pastebutton{ExDiffHigherOrderFull1}{\hidepaste}
+\tab{5}\spadcommand{differentiate(exp(x**2),x,4)}
+\indentrel{3}\begin{verbatim}
+                             2
+            4      2        x
+   (1)  (16x  + 48x  + 12)%e
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExDiffHigherOrderEmpty1}
+\begin{paste}{ExDiffHigherOrderEmpty1}{ExDiffHigherOrderPatch1}
+\pastebutton{ExDiffHigherOrderEmpty1}{\showpaste}
+\tab{5}\spadcommand{differentiate(exp(x**2),x,4)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExDiffFormalIntegralPatch1}
+\begin{paste}{ExDiffFormalIntegralFull1}{ExDiffFormalIntegralEmpty1}
+\pastebutton{ExDiffFormalIntegralFull1}{\hidepaste}
+\tab{5}\spadcommand{f := integrate(sqrt(1 + t**3),t)\bound{f }}
+\indentrel{3}\begin{verbatim}
+           t  ÚÄÄÄÄÄÄÄ¿
+         Ú¿   ³  3
+   (1)   ³   \³%N  + 1 d%N
+        ÀÙ
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExDiffFormalIntegralEmpty1}
+\begin{paste}{ExDiffFormalIntegralEmpty1}{ExDiffFormalIntegralPatch1}
+\pastebutton{ExDiffFormalIntegralEmpty1}{\showpaste}
+\tab{5}\spadcommand{f := integrate(sqrt(1 + t**3),t)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExDiffFormalIntegralPatch2}
+\begin{paste}{ExDiffFormalIntegralFull2}{ExDiffFormalIntegralEmpty2}
+\pastebutton{ExDiffFormalIntegralFull2}{\hidepaste}
+\tab{5}\spadcommand{differentiate(f,t)\free{f }}
+\indentrel{3}\begin{verbatim}
+         ÚÄÄÄÄÄÄ¿
+         ³ 3
+   (2)  \³t  + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExDiffFormalIntegralEmpty2}
+\begin{paste}{ExDiffFormalIntegralEmpty2}{ExDiffFormalIntegralPatch2}
+\pastebutton{ExDiffFormalIntegralEmpty2}{\showpaste}
+\tab{5}\spadcommand{differentiate(f,t)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExDiffFormalIntegralPatch3}
+\begin{paste}{ExDiffFormalIntegralFull3}{ExDiffFormalIntegralEmpty3}
+\pastebutton{ExDiffFormalIntegralFull3}{\hidepaste}
+\tab{5}\spadcommand{differentiate(f * t**2,t)\free{f }}
+\indentrel{3}\begin{verbatim}
+             t  ÚÄÄÄÄÄÄÄ¿          ÚÄÄÄÄÄÄ¿
+           Ú¿   ³  3             2 ³ 3
+   (3)  2t ³   \³%N  + 1 d%N  + t \³t  + 1
+          ÀÙ
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExDiffFormalIntegralEmpty3}
+\begin{paste}{ExDiffFormalIntegralEmpty3}{ExDiffFormalIntegralPatch3}
+\pastebutton{ExDiffFormalIntegralEmpty3}{\showpaste}
+\tab{5}\spadcommand{differentiate(f * t**2,t)\free{f }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/exint.ht b/src/hyper/pages/exint.ht
new file mode 100644
index 00000000..63964bcf
--- /dev/null
+++ b/src/hyper/pages/exint.ht
@@ -0,0 +1,149 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Title: Integration
+
+\begin{page}{ExIntRationalFunction}{Integral of a Rational Function}
+\beginscroll
+The following fraction has a denominator which factors into
+a quadratic and a quartic irreducible polynomial. The usual
+partial fraction approach used by most other computer algebra
+systems either fails or introduces expensive unneeded algebraic
+numbers.
+We use a factorization-free algorithm.
+\spadpaste{integrate((x**2+2*x+1)/((x+1)**6+1),x)}
+There are cases where algebraic numbers are absolutely necessary.
+In that case the answer to the \spadfun{integrate} command will contain
+symbols like \spad{\%\%F0} denoting the algebraic numbers used.
+To find out what the definitions for these numbers is use the
+\spadfun{definingPolynomial} operation on these numbers.
+\spadpaste{integrate(1/(x**3+x+1),x) \bound{i}}
+For example, if a symbol like \spad{\%\%F0} appears in the result of this last
+integration, then \spad{definingPolynomial \%\%F0} will return the
+polynomial that \spad{\%\%F0} is a root of. The next command isolates the
+algebraic number from the expression and displays its defining polynomial.
+\spadpaste{definingPolynomial(tower(\%).2::EXPR INT) \free{i}}
+
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExIntRationalWithRealParameter}{Integral of a Rational Function with a Real Parameter}
+\beginscroll
+When real parameters are present, the form of the integral can depend on
+the signs of some expressions. Rather than query the user or make
+sign assumptions, \Language{} returns all possible answers.
+\spadpaste{integrate(1/(x**2 + a),x)}
+The integrate command generally assumes that all parameters are real.
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExIntRationalWithComplexParameter}{Integral of a Rational Function with a Complex Parameter}
+\beginscroll
+If the parameter is complex instead of real, then the notion of
+sign is undefined and there is a unique answer.
+You can request
+this answer by prepending the word `complex' to the command name:
+\spadpaste{complexIntegrate(1/(x**2 + a),x)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExIntTwoSimilarIntegrands}{Two Similar Integrands Producing Very Different Results}
+\beginscroll
+The following two examples illustrate the limitations of table based
+approaches. The two integrands are very similar, but the answer to one
+of them requires the addition of two new algebraic numbers.
+This one is the easy one:
+\spadpaste{integrate(x**3 / (a+b*x)**(1/3),x)}
+The next one looks very similar
+but the answer is much more complicated. Only an algorithmic approach
+is guaranteed to find what new constants must be added in order to
+find a solution:
+\spadpaste{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExIntNoSolution}{An Integral Which Does Not Exist}
+\beginscroll
+Most computer algebra systems use heuristics or table-driven
+approaches to integration. When these systems cannot determine
+the answer to an integration problem, they reply "I don't know".
+\Language{} uses a complete algorithm for integration.
+It will conclusively prove that an integral
+cannot be expressed in terms of elementary functions.
+When \Language{} returns an integral sign, it has proved
+that no answer exists as an elementary function.
+\spadpaste{integrate(log(1 + sqrt(a*x + b)) / x,x)}
+\endscroll
+\autobuttons\end{page}
+
+%% This example is broken
+%\begin{page}{ExIntChangeOfVariables}{No Change of Variables is Required}
+%\beginscroll
+%Unlike computer algebra systems which rely on heuristics and
+%table-lookup, the algorithmic integration facility
+%of \Language{} never requires you to make a change of variables
+%in order to integrate a function.
+%\spadpaste{integrate(sec(x)**(4/5)*csc(x)**(6/5),x)}
+%\endscroll
+%\autobuttons\end{page}
+
+\begin{page}{ExIntTrig}{A Trigonometric Function of a Quadratic}
+\beginscroll
+\Language{} can handle complicated mixed functions way beyond what you can
+find in tables:
+\spadpaste{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b))/(sqrt(x+b)*(x+cosh(1+sqrt(x+b)))),x)}
+Whenever possible, \Language{} tries to express the answer using the functions
+present in the integrand.
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExIntAlgebraicRelation}{Integrating a Function with a Hidden Algebraic Relation}
+\beginscroll
+A strong structure checking algorithm in \Language{} finds hidden algebraic
+relationships between functions.
+\spadpaste{integrate(tan(atan(x)/3),x)}
+The discovery of this algebraic relationship is necessary
+for correctly integrating this function.
+\downlink{Details.}{ExIntAlgebraicRelationExplain} \space{1}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExIntAlgebraicRelationExplain}{Details for integrating a function wiht a Hidden Algebraic Relation}
+\beginscroll
+Steps taken for integration of:
+\centerline{{\em f := tan(atan(x)/3)}}
+\beginitems
+\item
+1. Replace {\em f} by an equivalent algebraic function {\em g}
+satisfying the algebraic relation:
+\centerline{{\em g**3 - 3*x*g - 3*g + x = 0}}
+\item
+2. Integrate {\em g} using using this algebraic relation; this produces:
+\centerline{{\em (24g**2 - 8)log(3g**2 - 1) + (81x**2 + 24)g**2 + 72xg - 27x**2 - 16}}
+\centerline{{\em /   (54g**2 - 18)}}
+\item
+3. Rationalize the denominator, producing:
+\centerline{{\em (8log(3g**2-1) - 3g**2 + 18xg + 15)/18}}
+\item
+4. Replace {\em g} by the initial {\em f} to produce the final result:
+\centerline{{\em (8log(3tan(atan(x/3))**2-1) - 3tan(atan(x/3))**2 - 18xtan(atan(x/3) + 16)/18)}}
+\enditems
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExIntRadicalOfTranscendental}{An Integral Involving a Root of a Transcendental Function}
+\beginscroll
+This is an example of a mixed function where
+the algebraic layer is over the transcendental one.
+\spadpaste{integrate((x + 1) / (x * (x + log x)**(3/2)),x)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExIntNonElementary}{An Integral of a Non-elementary Function}
+\beginscroll
+While incomplete for non-elementary functions, \Language{} can
+handle some of them:
+\spadpaste{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)}
+\endscroll
+\autobuttons\end{page}
diff --git a/src/hyper/pages/exint.pht b/src/hyper/pages/exint.pht
new file mode 100644
index 00000000..401aab12
--- /dev/null
+++ b/src/hyper/pages/exint.pht
@@ -0,0 +1,298 @@
+\begin{patch}{ExIntRationalWithRealParameterPatch1}
+\begin{paste}{ExIntRationalWithRealParameterFull1}{ExIntRationalWithRealParameterEmpty1}
+\pastebutton{ExIntRationalWithRealParameterFull1}{\hidepaste}
+\tab{5}\spadcommand{integrate(1/(x**2 + a),x)}
+\indentrel{3}\begin{verbatim}
+               2      ÚÄÄÄ¿
+             (x  - a)\³- a  + 2a x         ÚÄ¿
+         log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)      x\³a
+                      2             atan(ÄÄÄÄÄ)
+                     x  + a                a
+   (1)  [ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄ]
+                     ÚÄÄÄ¿               ÚÄ¿
+                   2\³- a               \³a
+               Type: Union(List Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExIntRationalWithRealParameterEmpty1}
+\begin{paste}{ExIntRationalWithRealParameterEmpty1}{ExIntRationalWithRealParameterPatch1}
+\pastebutton{ExIntRationalWithRealParameterEmpty1}{\showpaste}
+\tab{5}\spadcommand{integrate(1/(x**2 + a),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExIntRationalFunctionPatch1}
+\begin{paste}{ExIntRationalFunctionFull1}{ExIntRationalFunctionEmpty1}
+\pastebutton{ExIntRationalFunctionFull1}{\hidepaste}
+\tab{5}\spadcommand{integrate((x**2+2*x+1)/((x+1)**6+1),x)}
+\indentrel{3}\begin{verbatim}
+              3     2
+        atan(x  + 3x  + 3x + 1)
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                   3
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExIntRationalFunctionEmpty1}
+\begin{paste}{ExIntRationalFunctionEmpty1}{ExIntRationalFunctionPatch1}
+\pastebutton{ExIntRationalFunctionEmpty1}{\showpaste}
+\tab{5}\spadcommand{integrate((x**2+2*x+1)/((x+1)**6+1),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExIntRationalFunctionPatch2}
+\begin{paste}{ExIntRationalFunctionFull2}{ExIntRationalFunctionEmpty2}
+\pastebutton{ExIntRationalFunctionFull2}{\hidepaste}
+\tab{5}\spadcommand{integrate(1/(x**3+x+1),x)\bound{i }}
+\indentrel{3}\begin{verbatim}
+   (2)
+           ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+           ³        2
+           ³- 93%%H0  + 12
+         ( ³ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ  - %%H0)
+          \³      31
+      *
+         log
+                            ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+                            ³        2
+                            ³- 93%%H0  + 12          2
+              (62%%H0 + 31) ³ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ  + 62%%H0
+                           \³      31
+            + 
+              - 31%%H0 + 18x - 4
+     + 
+             ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+             ³        2
+             ³- 93%%H0  + 12
+         (-  ³ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ  - %%H0)
+            \³      31
+      *
+         log
+                              ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+                              ³        2
+                              ³- 93%%H0  + 12          2
+              (- 62%%H0 - 31) ³ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ  + 62%%H0
+                             \³      31
+            + 
+              - 31%%H0 + 18x - 4
+     + 
+                         2
+       2%%H0 log(- 62%%H0  + 31%%H0 + 9x + 4)
+  /
+     2
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExIntRationalFunctionEmpty2}
+\begin{paste}{ExIntRationalFunctionEmpty2}{ExIntRationalFunctionPatch2}
+\pastebutton{ExIntRationalFunctionEmpty2}{\showpaste}
+\tab{5}\spadcommand{integrate(1/(x**3+x+1),x)\bound{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExIntRationalFunctionPatch3}
+\begin{paste}{ExIntRationalFunctionFull3}{ExIntRationalFunctionEmpty3}
+\pastebutton{ExIntRationalFunctionFull3}{\hidepaste}
+\tab{5}\spadcommand{definingPolynomial(tower(\%).2::EXPR INT)\free{i }}
+\indentrel{3}\begin{verbatim}
+              3
+        31%%H0  - 3%%H0 - 1
+   (3)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                 31
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExIntRationalFunctionEmpty3}
+\begin{paste}{ExIntRationalFunctionEmpty3}{ExIntRationalFunctionPatch3}
+\pastebutton{ExIntRationalFunctionEmpty3}{\showpaste}
+\tab{5}\spadcommand{definingPolynomial(tower(\%).2::EXPR INT)\free{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExIntAlgebraicRelationPatch1}
+\begin{paste}{ExIntAlgebraicRelationFull1}{ExIntAlgebraicRelationEmpty1}
+\pastebutton{ExIntAlgebraicRelationFull1}{\hidepaste}
+\tab{5}\spadcommand{integrate(tan(atan(x)/3),x)}
+\indentrel{3}\begin{verbatim}
+   (1)
+                 atan(x) 2             atan(x) 2
+       8log(3tan(ÄÄÄÄÄÄÄ)  - 1) - 3tan(ÄÄÄÄÄÄÄ)
+                    3                     3
+     + 
+               atan(x)
+       18x tan(ÄÄÄÄÄÄÄ)
+                  3
+  /
+     18
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExIntAlgebraicRelationEmpty1}
+\begin{paste}{ExIntAlgebraicRelationEmpty1}{ExIntAlgebraicRelationPatch1}
+\pastebutton{ExIntAlgebraicRelationEmpty1}{\showpaste}
+\tab{5}\spadcommand{integrate(tan(atan(x)/3),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExIntRationalWithComplexParameterPatch1}
+\begin{paste}{ExIntRationalWithComplexParameterFull1}{ExIntRationalWithComplexParameterEmpty1}
+\pastebutton{ExIntRationalWithComplexParameterFull1}{\hidepaste}
+\tab{5}\spadcommand{complexIntegrate(1/(x**2 + a),x)}
+\indentrel{3}\begin{verbatim}
+              ÚÄÄÄ¿              ÚÄÄÄ¿
+            x\³- a  + a        x\³- a  - a
+        log(ÄÄÄÄÄÄÄÄÄÄÄ) - log(ÄÄÄÄÄÄÄÄÄÄÄ)
+                ÚÄÄÄ¿              ÚÄÄÄ¿
+               \³- a              \³- a
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                        ÚÄÄÄ¿
+                      2\³- a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExIntRationalWithComplexParameterEmpty1}
+\begin{paste}{ExIntRationalWithComplexParameterEmpty1}{ExIntRationalWithComplexParameterPatch1}
+\pastebutton{ExIntRationalWithComplexParameterEmpty1}{\showpaste}
+\tab{5}\spadcommand{complexIntegrate(1/(x**2 + a),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExIntNoSolutionPatch1}
+\begin{paste}{ExIntNoSolutionFull1}{ExIntNoSolutionEmpty1}
+\pastebutton{ExIntNoSolutionFull1}{\hidepaste}
+\tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)}
+\indentrel{3}\begin{verbatim}
+           x      ÚÄÄÄÄÄÄÄÄ¿
+         Ú¿  log(\³b + %T a  + 1)
+   (1)   ³   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ d%T
+        ÀÙ            %T
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExIntNoSolutionEmpty1}
+\begin{paste}{ExIntNoSolutionEmpty1}{ExIntNoSolutionPatch1}
+\pastebutton{ExIntNoSolutionEmpty1}{\showpaste}
+\tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExIntTwoSimilarIntegrandsPatch1}
+\begin{paste}{ExIntTwoSimilarIntegrandsFull1}{ExIntTwoSimilarIntegrandsEmpty1}
+\pastebutton{ExIntTwoSimilarIntegrandsFull1}{\hidepaste}
+\tab{5}\spadcommand{integrate(x**3 / (a+b*x)**(1/3),x)}
+\indentrel{3}\begin{verbatim}
+   (1)
+        3 3         2 2       2          3 3ÚÄÄÄÄÄÄÄ¿2
+   (120b x  - 135a b x  + 162a b x - 243a )\³b x + a
+   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                              4
+                          440b
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExIntTwoSimilarIntegrandsEmpty1}
+\begin{paste}{ExIntTwoSimilarIntegrandsEmpty1}{ExIntTwoSimilarIntegrandsPatch1}
+\pastebutton{ExIntTwoSimilarIntegrandsEmpty1}{\showpaste}
+\tab{5}\spadcommand{integrate(x**3 / (a+b*x)**(1/3),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExIntTwoSimilarIntegrandsPatch2}
+\begin{paste}{ExIntTwoSimilarIntegrandsFull2}{ExIntTwoSimilarIntegrandsEmpty2}
+\pastebutton{ExIntTwoSimilarIntegrandsFull2}{\hidepaste}
+\tab{5}\spadcommand{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)}
+\indentrel{3}\begin{verbatim}
+   (2)
+       -
+              2 2 ÚÄ¿
+            2b x \³3
+         *
+                3ÚÄ¿3ÚÄÄÄÄÄÄÄ¿2   3ÚÄ¿2 3ÚÄÄÄÄÄÄÄ¿
+            log(\³a \³b x + a   + \³a   \³b x + a + a)
+     + 
+         2 2 ÚÄ¿    3ÚÄ¿2 3ÚÄÄÄÄÄÄÄ¿
+       4b x \³3 log(\³a   \³b x + a - a)
+     + 
+                    ÚÄ¿3ÚÄ¿2 3ÚÄÄÄÄÄÄÄ¿    ÚÄ¿
+          2 2     2\³3 \³a   \³b x + a + a\³3
+       12b x atan(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
+                               3a
+     + 
+                    ÚÄ¿3ÚÄ¿3ÚÄÄÄÄÄÄÄ¿2
+       (12b x - 9a)\³3 \³a \³b x + a
+  /
+        2 2 ÚÄ¿3ÚÄ¿
+     18a x \³3 \³a
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExIntTwoSimilarIntegrandsEmpty2}
+\begin{paste}{ExIntTwoSimilarIntegrandsEmpty2}{ExIntTwoSimilarIntegrandsPatch2}
+\pastebutton{ExIntTwoSimilarIntegrandsEmpty2}{\showpaste}
+\tab{5}\spadcommand{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExIntRadicalOfTranscendentalPatch1}
+\begin{paste}{ExIntRadicalOfTranscendentalFull1}{ExIntRadicalOfTranscendentalEmpty1}
+\pastebutton{ExIntRadicalOfTranscendentalFull1}{\hidepaste}
+\tab{5}\spadcommand{integrate((x + 1) / (x * (x + log x)**(3/2)),x)}
+\indentrel{3}\begin{verbatim}
+            ÚÄÄÄÄÄÄÄÄÄÄ¿
+          2\³log(x) + x
+   (1)  - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+            log(x) + x
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExIntRadicalOfTranscendentalEmpty1}
+\begin{paste}{ExIntRadicalOfTranscendentalEmpty1}{ExIntRadicalOfTranscendentalPatch1}
+\pastebutton{ExIntRadicalOfTranscendentalEmpty1}{\showpaste}
+\tab{5}\spadcommand{integrate((x + 1) / (x * (x + log x)**(3/2)),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExIntNonElementaryPatch1}
+\begin{paste}{ExIntNonElementaryFull1}{ExIntNonElementaryEmpty1}
+\pastebutton{ExIntNonElementaryFull1}{\hidepaste}
+\tab{5}\spadcommand{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)}
+\indentrel{3}\begin{verbatim}
+                     ÚÄÄÄ¿    erf(x) - 1      ÚÄÄÄ¿
+        (erf(x) - 1)\³%pi log(ÄÄÄÄÄÄÄÄÄÄ) - 2\³%pi
+                              erf(x) + 1
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                        8erf(x) - 8
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExIntNonElementaryEmpty1}
+\begin{paste}{ExIntNonElementaryEmpty1}{ExIntNonElementaryPatch1}
+\pastebutton{ExIntNonElementaryEmpty1}{\showpaste}
+\tab{5}\spadcommand{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExIntTrigPatch1}
+\begin{paste}{ExIntTrigFull1}{ExIntTrigEmpty1}
+\pastebutton{ExIntTrigFull1}{\hidepaste}
+\tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b))/(sqrt(x+b)*(x+cosh(1+sqrt(x+b)))),x)}
+\indentrel{3}\begin{verbatim}
+   (1)
+                          ÚÄÄÄÄÄ¿
+                 - 2cosh(\³x + b  + 1) - 2x
+     2log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
+                ÚÄÄÄÄÄ¿              ÚÄÄÄÄÄ¿
+          sinh(\³x + b  + 1) - cosh(\³x + b  + 1)
+   + 
+         ÚÄÄÄÄÄ¿
+     - 2\³x + b
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExIntTrigEmpty1}
+\begin{paste}{ExIntTrigEmpty1}{ExIntTrigPatch1}
+\pastebutton{ExIntTrigEmpty1}{\showpaste}
+\tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b))/(sqrt(x+b)*(x+cosh(1+sqrt(x+b)))),x)}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/exlap.ht b/src/hyper/pages/exlap.ht
new file mode 100644
index 00000000..a1e3b8e7
--- /dev/null
+++ b/src/hyper/pages/exlap.ht
@@ -0,0 +1,58 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Title: Laplace Transforms
+
+\begin{page}{ExLapSimplePole}{Laplace transform with a single pole}
+\beginscroll
+The Laplace transform of t^n e^(a t) has a pole of order n+1 at x = a
+and no other pole. We divide by n! to get a monic denominator in the
+answer.
+\spadpaste{laplace(t**4 * exp(-a*t) / factorial(4), t, s)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLapTrigTrigh}{Laplace transform of a trigonometric function}
+\beginscroll
+Rather than looking up into a table, we use the normalizer to rewrite
+the trigs and hyperbolic trigs to complex exponentials and
+logarithms.
+\spadpaste{laplace(sin(a*t) * cosh(a*t) - cos(a*t) * sinh(a*t), t, s)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLapDefInt}{Laplace transform requiring a definite integration}
+\beginscroll
+When powers of t appear in the denominator, computing the laplace transform
+requires integrating the result of another laplace transform between a
+symbol and infinity. We use the full power of \Language{}'s integrator
+in such cases.
+\spadpaste{laplace(2/t * (1 - cos(a*t)), t, s)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLapExpExp}{Laplace transform of exponentials}
+\beginscroll
+This is another example where it is necessary to
+integrate the result of another laplace transform.
+\spadpaste{laplace((exp(a*t) - exp(b*t))/t, t, s)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLapSpecial1}{Laplace transform of an exponential integral}
+We can handle some restricted cases of special functions, linear exponential
+integrals among them.
+\beginscroll
+\spadpaste{laplace(exp(a*t+b)*Ei(c*t), t, s)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLapSpecial2}{Laplace transform of special functions}
+\beginscroll
+An example with some interesting special functions.
+\spadpaste{laplace(a*Ci(b*t) + c*Si(d*t), t, s)}
+\endscroll
+\autobuttons\end{page}
+
+
diff --git a/src/hyper/pages/exlap.pht b/src/hyper/pages/exlap.pht
new file mode 100644
index 00000000..d7a965ec
--- /dev/null
+++ b/src/hyper/pages/exlap.pht
@@ -0,0 +1,114 @@
+\begin{patch}{ExLapDefIntPatch1}
+\begin{paste}{ExLapDefIntFull1}{ExLapDefIntEmpty1}
+\pastebutton{ExLapDefIntFull1}{\hidepaste}
+\tab{5}\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)}
+\indentrel{3}\begin{verbatim}
+             2    2
+   (1)  log(s  + a ) - 2log(s)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLapDefIntEmpty1}
+\begin{paste}{ExLapDefIntEmpty1}{ExLapDefIntPatch1}
+\pastebutton{ExLapDefIntEmpty1}{\showpaste}
+\tab{5}\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLapExpExpPatch1}
+\begin{paste}{ExLapExpExpFull1}{ExLapExpExpEmpty1}
+\pastebutton{ExLapExpExpFull1}{\hidepaste}
+\tab{5}\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)}
+\indentrel{3}\begin{verbatim}
+   (1)  - log(s - a) + log(s - b)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLapExpExpEmpty1}
+\begin{paste}{ExLapExpExpEmpty1}{ExLapExpExpPatch1}
+\pastebutton{ExLapExpExpEmpty1}{\showpaste}
+\tab{5}\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLapSpecial1Patch1}
+\begin{paste}{ExLapSpecial1Full1}{ExLapSpecial1Empty1}
+\pastebutton{ExLapSpecial1Full1}{\hidepaste}
+\tab{5}\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)}
+\indentrel{3}\begin{verbatim}
+          b    s + c - a
+        %e log(ÄÄÄÄÄÄÄÄÄ)
+                   c
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+              s - a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLapSpecial1Empty1}
+\begin{paste}{ExLapSpecial1Empty1}{ExLapSpecial1Patch1}
+\pastebutton{ExLapSpecial1Empty1}{\showpaste}
+\tab{5}\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLapSpecial2Patch1}
+\begin{paste}{ExLapSpecial2Full1}{ExLapSpecial2Empty1}
+\pastebutton{ExLapSpecial2Full1}{\hidepaste}
+\tab{5}\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)}
+\indentrel{3}\begin{verbatim}
+               2    2
+              s  + b             d
+        a log(ÄÄÄÄÄÄÄ) + 2c atan(Ä)
+                  2              s
+                 b
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                     2s
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLapSpecial2Empty1}
+\begin{paste}{ExLapSpecial2Empty1}{ExLapSpecial2Patch1}
+\pastebutton{ExLapSpecial2Empty1}{\showpaste}
+\tab{5}\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLapTrigTrighPatch1}
+\begin{paste}{ExLapTrigTrighFull1}{ExLapTrigTrighEmpty1}
+\pastebutton{ExLapTrigTrighFull1}{\hidepaste}
+\tab{5}\spadcommand{laplace(sin(a*t) * cosh(a*t) - cos(a*t) * sinh(a*t), t, s)}
+\indentrel{3}\begin{verbatim}
+             3
+           4a
+   (1)  ÄÄÄÄÄÄÄÄ
+         4     4
+        s  + 4a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLapTrigTrighEmpty1}
+\begin{paste}{ExLapTrigTrighEmpty1}{ExLapTrigTrighPatch1}
+\pastebutton{ExLapTrigTrighEmpty1}{\showpaste}
+\tab{5}\spadcommand{laplace(sin(a*t) * cosh(a*t) - cos(a*t) * sinh(a*t), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLapSimplePolePatch1}
+\begin{paste}{ExLapSimplePoleFull1}{ExLapSimplePoleEmpty1}
+\pastebutton{ExLapSimplePoleFull1}{\hidepaste}
+\tab{5}\spadcommand{laplace(t**4 * exp(-a*t) / factorial(4), t, s)}
+\indentrel{3}\begin{verbatim}
+                            1
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+         5       4      2 3      3 2     4     5
+        s  + 5a s  + 10a s  + 10a s  + 5a s + a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLapSimplePoleEmpty1}
+\begin{paste}{ExLapSimplePoleEmpty1}{ExLapSimplePolePatch1}
+\pastebutton{ExLapSimplePoleEmpty1}{\showpaste}
+\tab{5}\spadcommand{laplace(t**4 * exp(-a*t) / factorial(4), t, s)}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/exlimit.ht b/src/hyper/pages/exlimit.ht
new file mode 100644
index 00000000..18f40630
--- /dev/null
+++ b/src/hyper/pages/exlimit.ht
@@ -0,0 +1,126 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Title: Limits
+
+%  Author: Richard Jenks, Clifton J. Williamson
+%  Date created: 1 November 1989
+%  Date last updated: 25 May 1990
+
+\begin{page}{ExLimitBasic}{Computing Limits}
+\beginscroll
+To compute a limit, you must specify a functional expression,
+a variable, and a limiting value for that variable.
+For example, to compute the limit of (x**2 - 3*x + 2)/(x**2 - 1)
+as x approaches 1, issue the following command:
+\spadpaste{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)}
+% answer := -1/2
+Since you have not specified a direction, \Language{} will attempt
+to compute a two-sided limit. Sometimes the limit when approached from
+the left is different from the limit from the right.
+\downlink{Example}{ExLimitTwoSided}. In this case, you may
+wish to ask for a one-sided limit.
+\downlink{How.  }{ExLimitOneSided}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLimitParameter}{Limits of Functions with Parameters}
+\beginscroll
+You may also take limits of functions with parameters. The limit
+will be expressed in terms of those parameters.
+Here's an example:
+\spadpaste{limit(sinh(a*x)/tan(b*x),x = 0)}
+% answer := a/b
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLimitOneSided}{One-sided Limits}
+\beginscroll
+If you have a function which is only defined on one side of a particular value,
+you may wish to compute a one-sided limit.
+For instance, the function \spad{log(x)} is only defined to the right of zero,
+i.e. for \spad{x > 0}.
+Thus, when computing limits of functions involving \spad{log(x)}, you probably
+will want a 'right-hand' limit.
+Here's an example:
+\spadpaste{limit(x * log(x),x = 0,"right")}
+% answer := 0
+When you do not specify \spad{right} or \spad{left} as an optional fourth
+argument, the function \spadfun{limit} will try to compute a two-sided limit.
+In the above case, the limit from the left does not exist, as \Language{}
+will indicate when you try to take a two-sided limit:
+\spadpaste{limit(x * log(x),x = 0)}
+% answer := [left = "failed",right = 0]
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLimitTwoSided}{Two-sided Limits}
+\beginscroll
+A function may be defined on both sides of a particular value, but will
+tend to different limits as its variable tends to that value from the
+left and from the right.
+We can construct an example of this as follows:
+Since { \em sqrt(y**2)} is simply the absolute value of \spad{y},
+the function \spad{sqrt(y**2)/y}
+is simply the sign (+1 or -1) of the real number \spad{y}.
+Therefore, \spad{sqrt(y**2)/y = -1} for \spad{y < 0} and
+\spad{sqrt(y**2)/y = +1} for \spad{y > 0}.
+Watch what happens when we take the limit at \spad{y = 0}.
+\spadpaste{limit(sqrt(y**2)/y,y = 0)}
+% answer := [left = -1,right = 1]
+The answer returned by \Language{} gives both a 'left-hand' and a 'right-hand'
+limit.
+Here's another example, this time using a more complicated function:
+\spadpaste{limit(sqrt(1 - cos(t))/t,t = 0)}
+% answer := [left = -sqrt(1/2),right = sqrt(1/2)]
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLimitInfinite}{Limits at Infinity}
+\beginscroll
+You can compute limits at infinity by passing either 'plus infinity'
+or 'minus infinity' as the third argument of the function \spadfun{limit}.
+To do this, use the constants \spad{\%plusInfinity} and \spad{\%minusInfinity}.
+Here are two examples:
+\spadpaste{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)}
+\spadpaste{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLimitRealComplex}{Real Limits vs. Complex Limits}
+\beginscroll
+When you use the function \spadfun{limit}, you will be taking the limit of a real
+function of a real variable.
+For example, you can compute
+\spadpaste{limit(z * sin(1/z),z = 0)}
+\Language{} returns \spad{0} because as a function of a real variable
+\spad{sin(1/z)} is always between \spad{-1} and \spad{1}, so \spad{z * sin(1/z)}
+tends to \spad{0} as \spad{z} tends to \spad{0}.
+However, as a function of a complex variable, \spad{sin(1/z)} is badly
+behaved around \spad{0}
+(one says that \spad{sin(1/z)} has an 'essential singularity' at \spad{z = 0}).
+When viewed as a function of a complex variable, \spad{z * sin(1/z)}
+does not approach any limit as \spad{z} tends to \spad{0} in the complex plane.
+\Language{} indicates this when we call the function \spadfun{complexLimit}:
+\spadpaste{complexLimit(z * sin(1/z),z = 0)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExLimitComplexInfinite}{Complex Limits at Infinity}
+\beginscroll
+You may also take complex limits at infinity, i.e. limits of a function of
+\spad{z} as \spad{z} approaches infinity on the Riemann sphere.
+Use the symbol \spad{\%infinity} to denote `complex infinity'.
+Also, to compute complex limits rather than real limits, use the
+function \spadfun{complexLimit}.
+Here is an example:
+\spadpaste{complexLimit((2 + z)/(1 - z),z = \%infinity)}
+In many cases, a limit of a real function of a real variable will exist
+when the corresponding complex limit does not.
+For example:
+\spadpaste{limit(sin(x)/x,x = \%plusInfinity)}
+\spadpaste{complexLimit(sin(x)/x,x = \%infinity)}
+\endscroll
+\autobuttons\end{page}
+
diff --git a/src/hyper/pages/exlimit.pht b/src/hyper/pages/exlimit.pht
new file mode 100644
index 00000000..d15d79f6
--- /dev/null
+++ b/src/hyper/pages/exlimit.pht
@@ -0,0 +1,221 @@
+\begin{patch}{ExLimitBasicPatch1}
+\begin{paste}{ExLimitBasicFull1}{ExLimitBasicEmpty1}
+\pastebutton{ExLimitBasicFull1}{\hidepaste}
+\tab{5}\spadcommand{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)}
+\indentrel{3}\begin{verbatim}
+          1
+   (1)  - Ä
+          2
+Type: Union(OrderedCompletion Fraction Polynomial Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLimitBasicEmpty1}
+\begin{paste}{ExLimitBasicEmpty1}{ExLimitBasicPatch1}
+\pastebutton{ExLimitBasicEmpty1}{\showpaste}
+\tab{5}\spadcommand{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLimitComplexInfinitePatch1}
+\begin{paste}{ExLimitComplexInfiniteFull1}{ExLimitComplexInfiniteEmpty1}
+\pastebutton{ExLimitComplexInfiniteFull1}{\hidepaste}
+\tab{5}\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)}
+\indentrel{3}\begin{verbatim}
+   (1)  - 1
+   Type: OnePointCompletion Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLimitComplexInfiniteEmpty1}
+\begin{paste}{ExLimitComplexInfiniteEmpty1}{ExLimitComplexInfinitePatch1}
+\pastebutton{ExLimitComplexInfiniteEmpty1}{\showpaste}
+\tab{5}\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLimitComplexInfinitePatch2}
+\begin{paste}{ExLimitComplexInfiniteFull2}{ExLimitComplexInfiniteEmpty2}
+\pastebutton{ExLimitComplexInfiniteFull2}{\hidepaste}
+\tab{5}\spadcommand{limit(sin(x)/x,x = \%plusInfinity)}
+\indentrel{3}\begin{verbatim}
+   (2)  0
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLimitComplexInfiniteEmpty2}
+\begin{paste}{ExLimitComplexInfiniteEmpty2}{ExLimitComplexInfinitePatch2}
+\pastebutton{ExLimitComplexInfiniteEmpty2}{\showpaste}
+\tab{5}\spadcommand{limit(sin(x)/x,x = \%plusInfinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLimitComplexInfinitePatch3}
+\begin{paste}{ExLimitComplexInfiniteFull3}{ExLimitComplexInfiniteEmpty3}
+\pastebutton{ExLimitComplexInfiniteFull3}{\hidepaste}
+\tab{5}\spadcommand{complexLimit(sin(x)/x,x = \%infinity)}
+\indentrel{3}\begin{verbatim}
+   (3)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLimitComplexInfiniteEmpty3}
+\begin{paste}{ExLimitComplexInfiniteEmpty3}{ExLimitComplexInfinitePatch3}
+\pastebutton{ExLimitComplexInfiniteEmpty3}{\showpaste}
+\tab{5}\spadcommand{complexLimit(sin(x)/x,x = \%infinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLimitOneSidedPatch1}
+\begin{paste}{ExLimitOneSidedFull1}{ExLimitOneSidedEmpty1}
+\pastebutton{ExLimitOneSidedFull1}{\hidepaste}
+\tab{5}\spadcommand{limit(x * log(x),x = 0,"right")}
+\indentrel{3}\begin{verbatim}
+   (1)  0
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLimitOneSidedEmpty1}
+\begin{paste}{ExLimitOneSidedEmpty1}{ExLimitOneSidedPatch1}
+\pastebutton{ExLimitOneSidedEmpty1}{\showpaste}
+\tab{5}\spadcommand{limit(x * log(x),x = 0,"right")}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLimitOneSidedPatch2}
+\begin{paste}{ExLimitOneSidedFull2}{ExLimitOneSidedEmpty2}
+\pastebutton{ExLimitOneSidedFull2}{\hidepaste}
+\tab{5}\spadcommand{limit(x * log(x),x = 0)}
+\indentrel{3}\begin{verbatim}
+   (2)  [leftHandLimit= "failed",rightHandLimit= 0]
+Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLimitOneSidedEmpty2}
+\begin{paste}{ExLimitOneSidedEmpty2}{ExLimitOneSidedPatch2}
+\pastebutton{ExLimitOneSidedEmpty2}{\showpaste}
+\tab{5}\spadcommand{limit(x * log(x),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLimitTwoSidedPatch1}
+\begin{paste}{ExLimitTwoSidedFull1}{ExLimitTwoSidedEmpty1}
+\pastebutton{ExLimitTwoSidedFull1}{\hidepaste}
+\tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)}
+\indentrel{3}\begin{verbatim}
+   (1)  [leftHandLimit= - 1,rightHandLimit= 1]
+Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLimitTwoSidedEmpty1}
+\begin{paste}{ExLimitTwoSidedEmpty1}{ExLimitTwoSidedPatch1}
+\pastebutton{ExLimitTwoSidedEmpty1}{\showpaste}
+\tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLimitTwoSidedPatch2}
+\begin{paste}{ExLimitTwoSidedFull2}{ExLimitTwoSidedEmpty2}
+\pastebutton{ExLimitTwoSidedFull2}{\hidepaste}
+\tab{5}\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)}
+\indentrel{3}\begin{verbatim}
+                            1                    1
+   (2)  [leftHandLimit= - ÄÄÄÄ,rightHandLimit= ÄÄÄÄ]
+                           ÚÄ¿                  ÚÄ¿
+                          \³2                  \³2
+Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLimitTwoSidedEmpty2}
+\begin{paste}{ExLimitTwoSidedEmpty2}{ExLimitTwoSidedPatch2}
+\pastebutton{ExLimitTwoSidedEmpty2}{\showpaste}
+\tab{5}\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLimitInfinitePatch1}
+\begin{paste}{ExLimitInfiniteFull1}{ExLimitInfiniteEmpty1}
+\pastebutton{ExLimitInfiniteFull1}{\hidepaste}
+\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)}
+\indentrel{3}\begin{verbatim}
+         ÚÄ¿
+        \³3
+   (1)  ÄÄÄÄ
+          5
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLimitInfiniteEmpty1}
+\begin{paste}{ExLimitInfiniteEmpty1}{ExLimitInfinitePatch1}
+\pastebutton{ExLimitInfiniteEmpty1}{\showpaste}
+\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLimitInfinitePatch2}
+\begin{paste}{ExLimitInfiniteFull2}{ExLimitInfiniteEmpty2}
+\pastebutton{ExLimitInfiniteFull2}{\hidepaste}
+\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)}
+\indentrel{3}\begin{verbatim}
+           ÚÄ¿
+          \³3
+   (2)  - ÄÄÄÄ
+            5
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLimitInfiniteEmpty2}
+\begin{paste}{ExLimitInfiniteEmpty2}{ExLimitInfinitePatch2}
+\pastebutton{ExLimitInfiniteEmpty2}{\showpaste}
+\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLimitParameterPatch1}
+\begin{paste}{ExLimitParameterFull1}{ExLimitParameterEmpty1}
+\pastebutton{ExLimitParameterFull1}{\hidepaste}
+\tab{5}\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)}
+\indentrel{3}\begin{verbatim}
+        a
+   (1)  Ä
+        b
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLimitParameterEmpty1}
+\begin{paste}{ExLimitParameterEmpty1}{ExLimitParameterPatch1}
+\pastebutton{ExLimitParameterEmpty1}{\showpaste}
+\tab{5}\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLimitRealComplexPatch1}
+\begin{paste}{ExLimitRealComplexFull1}{ExLimitRealComplexEmpty1}
+\pastebutton{ExLimitRealComplexFull1}{\hidepaste}
+\tab{5}\spadcommand{limit(z * sin(1/z),z = 0)}
+\indentrel{3}\begin{verbatim}
+   (1)  0
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLimitRealComplexEmpty1}
+\begin{paste}{ExLimitRealComplexEmpty1}{ExLimitRealComplexPatch1}
+\pastebutton{ExLimitRealComplexEmpty1}{\showpaste}
+\tab{5}\spadcommand{limit(z * sin(1/z),z = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExLimitRealComplexPatch2}
+\begin{paste}{ExLimitRealComplexFull2}{ExLimitRealComplexEmpty2}
+\pastebutton{ExLimitRealComplexFull2}{\hidepaste}
+\tab{5}\spadcommand{complexLimit(z * sin(1/z),z = 0)}
+\indentrel{3}\begin{verbatim}
+   (2)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExLimitRealComplexEmpty2}
+\begin{paste}{ExLimitRealComplexEmpty2}{ExLimitRealComplexPatch2}
+\pastebutton{ExLimitRealComplexEmpty2}{\showpaste}
+\tab{5}\spadcommand{complexLimit(z * sin(1/z),z = 0)}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/exmatrix.ht b/src/hyper/pages/exmatrix.ht
new file mode 100644
index 00000000..de7ff9ce
--- /dev/null
+++ b/src/hyper/pages/exmatrix.ht
@@ -0,0 +1,98 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Title: Matrices
+
+\begin{page}{ExMatrixBasicFunction}{Basic Arithmetic Operations on Matrices}
+\beginscroll
+You can create a matrix using the function \spadfun{matrix}.
+The function takes a list of lists of elements of the ring and produces a
+matrix whose \spad{i}th row contains the elements of the \spad{i}th list.
+For example:
+\spadpaste{m1 := matrix([[1,-2,1],[4,2,-4]]) \bound{m1}}
+\spadpaste{m2 := matrix([[1,0,2],[20,30,10],[0,200,100]]) \bound{m2}}
+\spadpaste{m3 := matrix([[1,2,3],[2,4,6]]) \bound{m3}}
+Some of the basic arithmetic operations on matrices are:
+\spadpaste{m1 + m3 \free{m1} \free{m3}}
+\spadpaste{100 * m1 \free{m1}}
+\spadpaste{m1 * m2 \free{m1} \free{m2}}
+\spadpaste{-m1 + m3 * m2 \free{m1} \free{m2} \free{m3}}
+You can also multiply a matrix and a vector provided
+that the matrix and vector have compatible dimensions.
+\spadpaste{m3 *vector([1,0,1]) \free{m3}}
+However, the dimensions of the matrices must be compatible in order for
+\Language{} to perform an operation - otherwise an error message will occur.
+\endscroll
+\autobuttons\end{page}
+
+
+\begin{page}{ExConstructMatrix}{Constructing new Matrices}
+\beginscroll
+A number of functions exist for constructing new matrices from existing ones.
+
+If you want to create a matrix whose entries are 0 except on the main
+diagonal you can use \spadfun{diagonalMatrix}.
+This function takes a list of ring elements as an argument and returns a
+square matrix which has these elements on the main diagonal.
+Consider the following example:
+\spadpaste{diagonalMatrix([1,2,3,2,1])}
+
+The function \spadfun{subMatrix}(\spad{a},\spad{i},\spad{j},\spad{k},\spad{l}) 
+constructs a new matrix
+consisting of rows \spad{i} through \spad{j} and columns \spad{k} through 
+\spad{l} of \spad{a} , inclusive.
+\spadpaste{subMatrix(matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]]), 1,3,2,4)}
+
+
+The functions \spadfun{horizConcat} and \spadfun{vertConcat} 
+concatenate matrices
+horizontally and vertically, respectively.
+\spadpaste{horizConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]])) }
+\spadpaste{vertConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]])) }
+
+
+The function \spadfunX{setsubMatrix}(\spad{a},\spad{i},\spad{k},\spad{b}) 
+replaces the submatrix of \spad{a}
+starting at row \spad{i} and column \spad{k} with the elements of the matrix i\spad{b}.
+\spadpaste{b:=matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]]) \bound{b}}
+\spadpaste{setsubMatrix!(b,1,1,transpose(subMatrix(b,1,3,1,3)))\free{b}}
+changes the submatrix of \spad{b} consisting of the first 3 rows and columns
+to its transpose.
+\endscroll
+\autobuttons\end{page}
+
+
+\begin{page}{ExTraceMatrix}{Trace of a Matrix}
+\beginscroll
+If you have a square matrix, then you can compute its `trace'.
+The function \spadfun{trace} computes the sum of all elements on the diagonal
+of a matrix.
+For example `trace' for a four by four Vandermonde matrix.
+\spadpaste{trace( matrix([[1,x,x**2,x**3],[1,y,y**2,y**3],[1,z,z**2,z**3],[1,u,u**2,u**3]]) )}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExDeterminantMatrix}{Determinant of a Matrix}
+\beginscroll
+The function \spadfun{determinant} computes the determinant of a matrix over a
+commutative ring, that is a ring whose multiplication is commutative.
+\spadpaste{determinant(matrix([[1,2,3,4],[2,3,2,5],[3,4,5,6],[4,1,6,7]]))}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExInverseMatrix}{Inverse of a Matrix}
+\beginscroll
+The function \spadfun{inverse} computes the inverse of a square matrix.
+\spadpaste{inverse(matrix([[1,2,1],[-2,3,4],[-1,5,6]])) }
+(If the inverse doesn't exist, then \Language{} returns `failed'.)
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExRankMatrix}{Rank of a Matrix}
+\beginscroll
+The function \spadfun{rank} gives you the rank of a matrix:
+\spadpaste{rank(matrix([[0,4,1],[5,3,-7],[-5,5,9]]))}
+\endscroll
+\autobuttons\end{page}
+
diff --git a/src/hyper/pages/exmatrix.pht b/src/hyper/pages/exmatrix.pht
new file mode 100644
index 00000000..2ce8a797
--- /dev/null
+++ b/src/hyper/pages/exmatrix.pht
@@ -0,0 +1,341 @@
+\begin{patch}{ExMatrixBasicFunctionPatch1}
+\begin{paste}{ExMatrixBasicFunctionFull1}{ExMatrixBasicFunctionEmpty1}
+\pastebutton{ExMatrixBasicFunctionFull1}{\hidepaste}
+\tab{5}\spadcommand{m1 := matrix([[1,-2,1],[4,2,-4]])\bound{m1 }}
+\indentrel{3}\begin{verbatim}
+        Ú1  - 2   1 ¿
+   (1)  ³           ³
+        À4   2   - 4Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionEmpty1}
+\begin{paste}{ExMatrixBasicFunctionEmpty1}{ExMatrixBasicFunctionPatch1}
+\pastebutton{ExMatrixBasicFunctionEmpty1}{\showpaste}
+\tab{5}\spadcommand{m1 := matrix([[1,-2,1],[4,2,-4]])\bound{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionPatch2}
+\begin{paste}{ExMatrixBasicFunctionFull2}{ExMatrixBasicFunctionEmpty2}
+\pastebutton{ExMatrixBasicFunctionFull2}{\hidepaste}
+\tab{5}\spadcommand{m2 := matrix([[1,0,2],[20,30,10],[0,200,100]])\bound{m2 }}
+\indentrel{3}\begin{verbatim}
+        Ú1    0    2 ¿
+        ³            ³
+   (2)  ³20  30   10 ³
+        ³            ³
+        À0   200  100Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionEmpty2}
+\begin{paste}{ExMatrixBasicFunctionEmpty2}{ExMatrixBasicFunctionPatch2}
+\pastebutton{ExMatrixBasicFunctionEmpty2}{\showpaste}
+\tab{5}\spadcommand{m2 := matrix([[1,0,2],[20,30,10],[0,200,100]])\bound{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionPatch3}
+\begin{paste}{ExMatrixBasicFunctionFull3}{ExMatrixBasicFunctionEmpty3}
+\pastebutton{ExMatrixBasicFunctionFull3}{\hidepaste}
+\tab{5}\spadcommand{m3 := matrix([[1,2,3],[2,4,6]])\bound{m3 }}
+\indentrel{3}\begin{verbatim}
+        Ú1  2  3¿
+   (3)  ³       ³
+        À2  4  6Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionEmpty3}
+\begin{paste}{ExMatrixBasicFunctionEmpty3}{ExMatrixBasicFunctionPatch3}
+\pastebutton{ExMatrixBasicFunctionEmpty3}{\showpaste}
+\tab{5}\spadcommand{m3 := matrix([[1,2,3],[2,4,6]])\bound{m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionPatch4}
+\begin{paste}{ExMatrixBasicFunctionFull4}{ExMatrixBasicFunctionEmpty4}
+\pastebutton{ExMatrixBasicFunctionFull4}{\hidepaste}
+\tab{5}\spadcommand{m1 + m3\free{m1 }\free{m3 }}
+\indentrel{3}\begin{verbatim}
+        Ú2  0  4¿
+   (4)  ³       ³
+        À6  6  2Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionEmpty4}
+\begin{paste}{ExMatrixBasicFunctionEmpty4}{ExMatrixBasicFunctionPatch4}
+\pastebutton{ExMatrixBasicFunctionEmpty4}{\showpaste}
+\tab{5}\spadcommand{m1 + m3\free{m1 }\free{m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionPatch5}
+\begin{paste}{ExMatrixBasicFunctionFull5}{ExMatrixBasicFunctionEmpty5}
+\pastebutton{ExMatrixBasicFunctionFull5}{\hidepaste}
+\tab{5}\spadcommand{100 * m1\free{m1 }}
+\indentrel{3}\begin{verbatim}
+        Ú100  - 200   100 ¿
+   (5)  ³                 ³
+        À400   200   - 400Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionEmpty5}
+\begin{paste}{ExMatrixBasicFunctionEmpty5}{ExMatrixBasicFunctionPatch5}
+\pastebutton{ExMatrixBasicFunctionEmpty5}{\showpaste}
+\tab{5}\spadcommand{100 * m1\free{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionPatch6}
+\begin{paste}{ExMatrixBasicFunctionFull6}{ExMatrixBasicFunctionEmpty6}
+\pastebutton{ExMatrixBasicFunctionFull6}{\hidepaste}
+\tab{5}\spadcommand{m1 * m2\free{m1 }\free{m2 }}
+\indentrel{3}\begin{verbatim}
+        Ú- 39   140    82  ¿
+   (6)  ³                  ³
+        À 44   - 740  - 372Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionEmpty6}
+\begin{paste}{ExMatrixBasicFunctionEmpty6}{ExMatrixBasicFunctionPatch6}
+\pastebutton{ExMatrixBasicFunctionEmpty6}{\showpaste}
+\tab{5}\spadcommand{m1 * m2\free{m1 }\free{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionPatch7}
+\begin{paste}{ExMatrixBasicFunctionFull7}{ExMatrixBasicFunctionEmpty7}
+\pastebutton{ExMatrixBasicFunctionFull7}{\hidepaste}
+\tab{5}\spadcommand{-m1 + m3 * m2\free{m1 }\free{m2 }\free{m3 }}
+\indentrel{3}\begin{verbatim}
+        Ú40  662   321¿
+   (7)  ³             ³
+        À78  1318  648Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionEmpty7}
+\begin{paste}{ExMatrixBasicFunctionEmpty7}{ExMatrixBasicFunctionPatch7}
+\pastebutton{ExMatrixBasicFunctionEmpty7}{\showpaste}
+\tab{5}\spadcommand{-m1 + m3 * m2\free{m1 }\free{m2 }\free{m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionPatch8}
+\begin{paste}{ExMatrixBasicFunctionFull8}{ExMatrixBasicFunctionEmpty8}
+\pastebutton{ExMatrixBasicFunctionFull8}{\hidepaste}
+\tab{5}\spadcommand{m3 *vector([1,0,1])\free{m3 }}
+\indentrel{3}\begin{verbatim}
+   (8)  [4,8]
+                                   Type: Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExMatrixBasicFunctionEmpty8}
+\begin{paste}{ExMatrixBasicFunctionEmpty8}{ExMatrixBasicFunctionPatch8}
+\pastebutton{ExMatrixBasicFunctionEmpty8}{\showpaste}
+\tab{5}\spadcommand{m3 *vector([1,0,1])\free{m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExRankMatrixPatch1}
+\begin{paste}{ExRankMatrixFull1}{ExRankMatrixEmpty1}
+\pastebutton{ExRankMatrixFull1}{\hidepaste}
+\tab{5}\spadcommand{rank(matrix([[0,4,1],[5,3,-7],[-5,5,9]]))}
+\indentrel{3}\begin{verbatim}
+   (1)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExRankMatrixEmpty1}
+\begin{paste}{ExRankMatrixEmpty1}{ExRankMatrixPatch1}
+\pastebutton{ExRankMatrixEmpty1}{\showpaste}
+\tab{5}\spadcommand{rank(matrix([[0,4,1],[5,3,-7],[-5,5,9]]))}
+\end{paste}\end{patch}
+
+\begin{patch}{ExTraceMatrixPatch1}
+\begin{paste}{ExTraceMatrixFull1}{ExTraceMatrixEmpty1}
+\pastebutton{ExTraceMatrixFull1}{\hidepaste}
+\tab{5}\spadcommand{trace( matrix([[1,x,x**2,x**3],[1,y,y**2,y**3],[1,z,z**2,z**3],[1,u,u**2,u**3]]) )}
+\indentrel{3}\begin{verbatim}
+         2        3
+   (1)  z  + y + u  + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExTraceMatrixEmpty1}
+\begin{paste}{ExTraceMatrixEmpty1}{ExTraceMatrixPatch1}
+\pastebutton{ExTraceMatrixEmpty1}{\showpaste}
+\tab{5}\spadcommand{trace( matrix([[1,x,x**2,x**3],[1,y,y**2,y**3],[1,z,z**2,z**3],[1,u,u**2,u**3]]) )}
+\end{paste}\end{patch}
+
+\begin{patch}{ExInverseMatrixPatch1}
+\begin{paste}{ExInverseMatrixFull1}{ExInverseMatrixEmpty1}
+\pastebutton{ExInverseMatrixFull1}{\hidepaste}
+\tab{5}\spadcommand{inverse(matrix([[1,2,1],[-2,3,4],[-1,5,6]]))}
+\indentrel{3}\begin{verbatim}
+        Ú  2        5 ¿
+        ³- Ä  - 1   Ä ³
+        ³  7        7 ³
+        ³             ³
+   (1)  ³ 8          6³
+        ³ Ä    1   - ij
+        ³ 7          7³
+        ³             ³
+        À- 1  - 1   1 Ù
+               Type: Union(Matrix Fraction Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExInverseMatrixEmpty1}
+\begin{paste}{ExInverseMatrixEmpty1}{ExInverseMatrixPatch1}
+\pastebutton{ExInverseMatrixEmpty1}{\showpaste}
+\tab{5}\spadcommand{inverse(matrix([[1,2,1],[-2,3,4],[-1,5,6]]))}
+\end{paste}\end{patch}
+
+\begin{patch}{ExConstructMatrixPatch1}
+\begin{paste}{ExConstructMatrixFull1}{ExConstructMatrixEmpty1}
+\pastebutton{ExConstructMatrixFull1}{\hidepaste}
+\tab{5}\spadcommand{diagonalMatrix([1,2,3,2,1])}
+\indentrel{3}\begin{verbatim}
+        Ú1  0  0  0  0¿
+        ³             ³
+        ³0  2  0  0  0³
+        ³             ³
+   (1)  ³0  0  3  0  0³
+        ³             ³
+        ³0  0  0  2  0³
+        ³             ³
+        À0  0  0  0  1Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExConstructMatrixEmpty1}
+\begin{paste}{ExConstructMatrixEmpty1}{ExConstructMatrixPatch1}
+\pastebutton{ExConstructMatrixEmpty1}{\showpaste}
+\tab{5}\spadcommand{diagonalMatrix([1,2,3,2,1])}
+\end{paste}\end{patch}
+
+\begin{patch}{ExConstructMatrixPatch2}
+\begin{paste}{ExConstructMatrixFull2}{ExConstructMatrixEmpty2}
+\pastebutton{ExConstructMatrixFull2}{\hidepaste}
+\tab{5}\spadcommand{subMatrix(matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]]), 1,3,2,4)}
+\indentrel{3}\begin{verbatim}
+        Ú1   2   3 ¿
+        ³          ³
+   (2)  ³6   7   8 ³
+        ³          ³
+        À11  12  13Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExConstructMatrixEmpty2}
+\begin{paste}{ExConstructMatrixEmpty2}{ExConstructMatrixPatch2}
+\pastebutton{ExConstructMatrixEmpty2}{\showpaste}
+\tab{5}\spadcommand{subMatrix(matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]]), 1,3,2,4)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExConstructMatrixPatch3}
+\begin{paste}{ExConstructMatrixFull3}{ExConstructMatrixEmpty3}
+\pastebutton{ExConstructMatrixFull3}{\hidepaste}
+\tab{5}\spadcommand{horizConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]]))}
+\indentrel{3}\begin{verbatim}
+        Ú1  2  3  11  12  13¿
+   (3)  ³                   ³
+        À6  7  8  55  77  88Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExConstructMatrixEmpty3}
+\begin{paste}{ExConstructMatrixEmpty3}{ExConstructMatrixPatch3}
+\pastebutton{ExConstructMatrixEmpty3}{\showpaste}
+\tab{5}\spadcommand{horizConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]]))}
+\end{paste}\end{patch}
+
+\begin{patch}{ExConstructMatrixPatch4}
+\begin{paste}{ExConstructMatrixFull4}{ExConstructMatrixEmpty4}
+\pastebutton{ExConstructMatrixFull4}{\hidepaste}
+\tab{5}\spadcommand{vertConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]]))}
+\indentrel{3}\begin{verbatim}
+        Ú1   2   3 ¿
+        ³          ³
+        ³6   7   8 ³
+   (4)  ³          ³
+        ³11  12  13³
+        ³          ³
+        À55  77  88Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExConstructMatrixEmpty4}
+\begin{paste}{ExConstructMatrixEmpty4}{ExConstructMatrixPatch4}
+\pastebutton{ExConstructMatrixEmpty4}{\showpaste}
+\tab{5}\spadcommand{vertConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]]))}
+\end{paste}\end{patch}
+
+\begin{patch}{ExConstructMatrixPatch5}
+\begin{paste}{ExConstructMatrixFull5}{ExConstructMatrixEmpty5}
+\pastebutton{ExConstructMatrixFull5}{\hidepaste}
+\tab{5}\spadcommand{b:=matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]])\bound{b }}
+\indentrel{3}\begin{verbatim}
+        Ú0   1   2   3   4 ¿
+        ³                  ³
+   (5)  ³5   6   7   8   9 ³
+        ³                  ³
+        À10  11  12  13  14Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExConstructMatrixEmpty5}
+\begin{paste}{ExConstructMatrixEmpty5}{ExConstructMatrixPatch5}
+\pastebutton{ExConstructMatrixEmpty5}{\showpaste}
+\tab{5}\spadcommand{b:=matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]])\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExConstructMatrixPatch6}
+\begin{paste}{ExConstructMatrixFull6}{ExConstructMatrixEmpty6}
+\pastebutton{ExConstructMatrixFull6}{\hidepaste}
+\tab{5}\spadcommand{setsubMatrix!(b,1,1,transpose(subMatrix(b,1,3,1,3)))\free{b }}
+\indentrel{3}\begin{verbatim}
+        Ú0  5  10  3   4 ¿
+        ³                ³
+   (6)  ³1  6  11  8   9 ³
+        ³                ³
+        À2  7  12  13  14Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExConstructMatrixEmpty6}
+\begin{paste}{ExConstructMatrixEmpty6}{ExConstructMatrixPatch6}
+\pastebutton{ExConstructMatrixEmpty6}{\showpaste}
+\tab{5}\spadcommand{setsubMatrix!(b,1,1,transpose(subMatrix(b,1,3,1,3)))\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExDeterminantMatrixPatch1}
+\begin{paste}{ExDeterminantMatrixFull1}{ExDeterminantMatrixEmpty1}
+\pastebutton{ExDeterminantMatrixFull1}{\hidepaste}
+\tab{5}\spadcommand{determinant(matrix([[1,2,3,4],[2,3,2,5],[3,4,5,6],[4,1,6,7]]))}
+\indentrel{3}\begin{verbatim}
+   (1)  - 48
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExDeterminantMatrixEmpty1}
+\begin{paste}{ExDeterminantMatrixEmpty1}{ExDeterminantMatrixPatch1}
+\pastebutton{ExDeterminantMatrixEmpty1}{\showpaste}
+\tab{5}\spadcommand{determinant(matrix([[1,2,3,4],[2,3,2,5],[3,4,5,6],[4,1,6,7]]))}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/explot2d.ht b/src/hyper/pages/explot2d.ht
new file mode 100644
index 00000000..803e4d77
--- /dev/null
+++ b/src/hyper/pages/explot2d.ht
@@ -0,0 +1,50 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Title: 2-D Graphics
+
+%  Author: Clifton J. Williamson
+%  Date created: 3 November 1989
+%  Date last updated: 3 November 1989
+
+\begin{page}{ExPlot2DFunctions}{Plotting Functions of One Variable}
+\beginscroll
+To plot a function {\em y = f(x)}, you need only specify the function and the
+interval on which it is to be plotted.
+\graphpaste{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExPlot2DParametric}{Plotting Parametric Curves}
+\beginscroll
+To plot a parametric curve defined by {\em x = f(t)},
+{\em y = g(t)}, specify the functions
+{\em f(t)} and {\em g(t)}
+as arguments of the function `curve', then give
+the interval over which {\em t} is to range.
+\graphpaste{draw(curve(9 * sin(3*t/4),8 * sin(t)),t = -4*\%pi..4*\%pi)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExPlot2DPolar}{Plotting Using Polar Coordinates}
+\beginscroll
+To plot the function {\em r = f(theta)} in polar coordinates,
+use the option {\em coordinates == polar}.
+As usual,
+call the function 'draw' and specify the function {\em f(theta)} and the
+interval over which {\em theta} is to range.
+\graphpaste{draw(sin(4*t/7),t = 0..14*\%pi,coordinates == polar)}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExPlot2DAlgebraic}{Plotting Plane Algebraic Curves}
+\beginscroll
+\Language{} can also plot plane algebraic curves (i.e. curves defined by
+an equation {\em f(x,y) = 0}) provided that the curve is non-singular in the
+region to be sketched.
+Here's an example:
+\graphpaste{draw(y**2 + y - (x**3 - x) = 0, x, y, range == [-2..2,-2..1])}
+Here the region of the sketch is {\em -2 <= x <= 2, -2 <= y <= 1}.
+\endscroll
+\autobuttons\end{page}
diff --git a/src/hyper/pages/explot2d.pht b/src/hyper/pages/explot2d.pht
new file mode 100644
index 00000000..1405f88a
--- /dev/null
+++ b/src/hyper/pages/explot2d.pht
@@ -0,0 +1,52 @@
+\begin{patch}{ExPlot2DPolarPatch1}
+\begin{paste}{ExPlot2DPolarFull1}{ExPlot2DPolarEmpty1}
+\pastebutton{ExPlot2DPolarFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(4*t/7),t = 0..14*\%pi,coordinates == polar)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ExPlot2DPolar1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ExPlot2DPolar1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot2DPolarEmpty1}
+\begin{paste}{ExPlot2DPolarEmpty1}{ExPlot2DPolarPatch1}
+\pastebutton{ExPlot2DPolarEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(sin(4*t/7),t = 0..14*\%pi,coordinates == polar)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot2DAlgebraicPatch1}
+\begin{paste}{ExPlot2DAlgebraicFull1}{ExPlot2DAlgebraicEmpty1}
+\pastebutton{ExPlot2DAlgebraicFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(y**2 + y - (x**3 - x) = 0, x, y, range == [-2..2,-2..1])}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ExPlot2DAlgebraic1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ExPlot2DAlgebraic1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot2DAlgebraicEmpty1}
+\begin{paste}{ExPlot2DAlgebraicEmpty1}{ExPlot2DAlgebraicPatch1}
+\pastebutton{ExPlot2DAlgebraicEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(y**2 + y - (x**3 - x) = 0, x, y, range == [-2..2,-2..1])}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot2DFunctionsPatch1}
+\begin{paste}{ExPlot2DFunctionsFull1}{ExPlot2DFunctionsEmpty1}
+\pastebutton{ExPlot2DFunctionsFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ExPlot2DFunctions1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ExPlot2DFunctions1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot2DFunctionsEmpty1}
+\begin{paste}{ExPlot2DFunctionsEmpty1}{ExPlot2DFunctionsPatch1}
+\pastebutton{ExPlot2DFunctionsEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot2DParametricPatch1}
+\begin{paste}{ExPlot2DParametricFull1}{ExPlot2DParametricEmpty1}
+\pastebutton{ExPlot2DParametricFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(9 * sin(3*t/4),8 * sin(t)),t = -4*\%pi..4*\%pi)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ExPlot2DParametric1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ExPlot2DParametric1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot2DParametricEmpty1}
+\begin{paste}{ExPlot2DParametricEmpty1}{ExPlot2DParametricPatch1}
+\pastebutton{ExPlot2DParametricEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(curve(9 * sin(3*t/4),8 * sin(t)),t = -4*\%pi..4*\%pi)}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/explot3d.ht b/src/hyper/pages/explot3d.ht
new file mode 100644
index 00000000..846a4b08
--- /dev/null
+++ b/src/hyper/pages/explot3d.ht
@@ -0,0 +1,53 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Title: 3-D Graphics
+ 
+% Address comments and questions to the
+%   Computer Algebra Group, Mathematical Sciences Department
+%   IBM Thomas J. Watson Research Center, Box 218
+%   Yorktown Heights, New York  10598  USA
+ 
+%  Author: Clifton J. Williamson
+%  Date created: Memorial Day 1990
+%  Date last updated: same
+ 
+\begin{page}{ExPlot3DFunctions}{Plotting Functions of Two Variables}
+\beginscroll
+To plot a function {\em z = f(x,y)}, you need only specify the function and the
+intervals over which the dependent variables will range.
+For example, here's how you plot the function {\em z = cos(x*y)} as the 
+variables {\em x} and {\em y} both range between -3 and 3:
+\graphpaste{draw(cos(x*y),x = -3..3,y = -3..3)}
+\endscroll
+\autobuttons\end{page}
+ 
+\begin{page}{ExPlot3DParametricSurface}{Plotting Parametric Surfaces}
+\beginscroll
+To plot a parametric surface defined by {\em x = f(u,v)}, 
+{\em y = g(u,v)}, {\em z = h(u,v)}, specify the functions
+{\em f(u,v)}, {\em g(u,v)}, and {\em h(u,v)}
+as arguments of the function `surface', then give
+the intervals over which {\em u} and {\em v} are to range.
+With parametric surfaces, we can create some interesting graphs.
+Here's an egg:
+\graphpaste{draw(surface(5*sin(u)*cos(v),4*sin(u)*sin(v),3*cos(u)),u=0..\%pi,v=0..2*\%pi)}
+Here's a cone:
+\graphpaste{draw(surface(u*cos(v),u*sin(v),u),u=0..4,v=0..2*\%pi)}
+\endscroll
+\autobuttons\end{page}
+ 
+\begin{page}{ExPlot3DParametricCurve}{Plotting Parametric Curves}
+\beginscroll
+To plot a parametric curve defined by {\em x = f(t)}, 
+{\em y = g(t)}, {\em z = h(t)}, specify the functions
+{\em f(t)}, {\em g(t)}, and {\em h(t)}
+as arguments of the function `curve', then give
+the interval over which {\em t} is to range.
+Here is a spiral:
+\graphpaste{draw(curve(cos(t),sin(t),t),t=0..6)}
+Here is the {\em twisted cubic curve}:
+\graphpaste{draw(curve(t,t**2,t**3),t=-3..3)}
+\endscroll
+\autobuttons\end{page}
diff --git a/src/hyper/pages/explot3d.pht b/src/hyper/pages/explot3d.pht
new file mode 100644
index 00000000..7c2a7d75
--- /dev/null
+++ b/src/hyper/pages/explot3d.pht
@@ -0,0 +1,65 @@
+\begin{patch}{ExPlot3DParametricCurvePatch1}
+\begin{paste}{ExPlot3DParametricCurveFull1}{ExPlot3DParametricCurveEmpty1}
+\pastebutton{ExPlot3DParametricCurveFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(cos(t),sin(t),t),t=0..6)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ExPlot3DParametricCurve1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ExPlot3DParametricCurve1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot3DParametricCurveEmpty1}
+\begin{paste}{ExPlot3DParametricCurveEmpty1}{ExPlot3DParametricCurvePatch1}
+\pastebutton{ExPlot3DParametricCurveEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(curve(cos(t),sin(t),t),t=0..6)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot3DParametricCurvePatch2}
+\begin{paste}{ExPlot3DParametricCurveFull2}{ExPlot3DParametricCurveEmpty2}
+\pastebutton{ExPlot3DParametricCurveFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(t,t**2,t**3),t=-3..3)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ExPlot3DParametricCurve2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ExPlot3DParametricCurve2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot3DParametricCurveEmpty2}
+\begin{paste}{ExPlot3DParametricCurveEmpty2}{ExPlot3DParametricCurvePatch2}
+\pastebutton{ExPlot3DParametricCurveEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(curve(t,t**2,t**3),t=-3..3)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot3DParametricSurfacePatch1}
+\begin{paste}{ExPlot3DParametricSurfaceFull1}{ExPlot3DParametricSurfaceEmpty1}
+\pastebutton{ExPlot3DParametricSurfaceFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(surface(5*sin(u)*cos(v),4*sin(u)*sin(v),3*cos(u)),u=0..\%pi,v=0..2*\%pi)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ExPlot3DParametricSurface1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ExPlot3DParametricSurface1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot3DParametricSurfaceEmpty1}
+\begin{paste}{ExPlot3DParametricSurfaceEmpty1}{ExPlot3DParametricSurfacePatch1}
+\pastebutton{ExPlot3DParametricSurfaceEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(surface(5*sin(u)*cos(v),4*sin(u)*sin(v),3*cos(u)),u=0..\%pi,v=0..2*\%pi)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot3DParametricSurfacePatch2}
+\begin{paste}{ExPlot3DParametricSurfaceFull2}{ExPlot3DParametricSurfaceEmpty2}
+\pastebutton{ExPlot3DParametricSurfaceFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(surface(u*cos(v),u*sin(v),u),u=0..4,v=0..2*\%pi)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ExPlot3DParametricSurface2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ExPlot3DParametricSurface2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot3DParametricSurfaceEmpty2}
+\begin{paste}{ExPlot3DParametricSurfaceEmpty2}{ExPlot3DParametricSurfacePatch2}
+\pastebutton{ExPlot3DParametricSurfaceEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(surface(u*cos(v),u*sin(v),u),u=0..4,v=0..2*\%pi)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot3DFunctionsPatch1}
+\begin{paste}{ExPlot3DFunctionsFull1}{ExPlot3DFunctionsEmpty1}
+\pastebutton{ExPlot3DFunctionsFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(cos(x*y),x = -3..3,y = -3..3)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ExPlot3DFunctions1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ExPlot3DFunctions1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ExPlot3DFunctionsEmpty1}
+\begin{paste}{ExPlot3DFunctionsEmpty1}{ExPlot3DFunctionsPatch1}
+\pastebutton{ExPlot3DFunctionsEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(cos(x*y),x = -3..3,y = -3..3)}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/expose.ht b/src/hyper/pages/expose.ht
new file mode 100644
index 00000000..bf55a2c9
--- /dev/null
+++ b/src/hyper/pages/expose.ht
@@ -0,0 +1,115 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+\begin{page}{helpExpose}{Exposure}
+\beginscroll
+Exposure determines what part of the \Language{} library
+is available to you when using \Language{}.
+At any time during your interactive \Language{} session,
+each constructor is either {\em exposed} or {\em unexposed}.
+If a constructor is exposed, its operations are available to
+the interpreter and therefore to you.
+If a constructor is unexposed, the operations are not seen
+by the interpreter and thus not available to you.
+\par
+If you are a beginner, you may only want
+basic parts of the library exposed.
+If you are an expert, you may want to have all parts of the
+library exposed.
+If you have an application that requires
+only a segment of the library, you may want to arrange to expose
+only that segment for your use.
+\par
+\endscroll
+Additional Information:
+\beginmenu
+\menulink{What}{ExposureDef}\tab{8}What is an exposure group?
+\menulink{System}{ExposureSystem}\tab{8}What exposure groups are system defined?
+% \menulink{User}{ExposureUser}\tab{8}How can I define my own?
+\menulink{Details}{ExposureDetails}\tab{8}Some details on exposure
+\endmenu
+\end{page}
+
+\begin{page}{ExposureSystem}{System Defined Exposure Groups}
+\beginscroll
+Exposure is defined by {\em groups}.
+Groups have names.
+Seven exposure groups are system-defined:\beginmenu
+\item\tab{3}{\em current}\tab{12}The currently active exposure group
+\item\tab{3}{\em basic}\tab{12}The default value of {\em current}
+\item\tab{3}{\em category}\tab{13}Category constructors not in {\em basic}
+\item\tab{3}{\em domain}\tab{13}Domain constructors not in {\em basic}
+\item\tab{3}{\em package}\tab{13}Package constructors not in {\em basic}
+\item\tab{3}{\em default}\tab{13}Default constructors not in {\em basic}
+\item\tab{3}{\em hidden}\tab{13}All constructors not in {\em basic}
+\item\tab{3}{\em naglink}\tab{13}All constructors used in the AXIOM NAG Link
+\endmenu
+\par
+When you first use \Language{}, the {\em current} exposure group is
+set to {\em basic} and {\em naglink}. Using \HyperName{} or the system command
+{\em expose}, you may
+change the current exposure group by
+adding or dropping constructors or by setting {\em current}
+to an exposure group you have created.
+\endscroll
+Additional Information:
+\beginmenu
+\menulink{What}{ExposureDef}\tab{10}What is an exposure group?
+%\menulink{User}{ExposureUser}\tab{10}How you can define your own exposure group
+\endmenu
+\end{page}
+
+\begin{page}{ExposureDef}{What is an Exposure Group?}
+\beginscroll
+\par
+An exposure group is a list of constructors to be exposed.
+Those constructors on the list are exposed;
+those not on the list are not exposed.
+The library contains 4 kinds of constructors intuitively described as follows:
+\beginmenu
+\item\menuitemstyle{}{\em domain}\tab{10}Describes computational objects and functions defined on these objects
+\item\menuitemstyle{}{\em package}\tab{10}Describes functions which will work over a variety of domains
+\item\menuitemstyle{}{\em category}\tab{10}Names a set of operations
+\item\menuitemstyle{}{\em default}\tab{10}Provides default functions for a cateogry
+\endmenu
+
+An exposure group is defined by three lists:
+\beginmenu
+\item\menuitemstyle{}{\em groups}\tab{13}A list of other (more basic) groups
+\item\menuitemstyle{}{\em additions}\tab{13}A list of explicit constructors to be included
+\item\menuitemstyle{}{\em subtractions}\tab{13}A list of explicit constructors to be dropped
+\endmenu
+You can define your own exposure groups: give them names and define the
+three above lists to be anything you like.
+Using \HyperName{}, you can conveniently edit your exposure groups,
+install them as the {\em current} exposure, and so on.
+\endscroll
+\end{page}
+
+\begin{page}{ExposureDetails}{Details on Exposure}
+\beginscroll
+Exposure is generally defined by the set of domain and package constructors
+you want to have available.
+Category and default constructors are generally implied.
+A category constructor is exposed if mentioned by {\em any} other constructor
+(including another category).
+A default constructor is exposed if its corresponding category constructor is exposed.
+\par
+If you explicitly add a domain or package
+constructor, its name will be put in an {\em Additions} list.
+The system will also add automatically to the {\em Additions} list
+any category explicity exported by that domain or package.
+If that category has a corresponding default constructor, that
+default constructor will also be added as well.
+\par
+If you like, you can explicitly drop a constructor.
+Any such name is added to the {\em Subtractions} list.
+The system will drop this name from the {\em Additions} list if it appears.
+\par
+If the package or domain takes arguments from an unexported
+domain or declares that its arguments can come from a domain
+which is a member of an unexported category, these constructors
+will {\em not} be added.
+\endscroll
+\end{page}
diff --git a/src/hyper/pages/exseries.ht b/src/hyper/pages/exseries.ht
new file mode 100644
index 00000000..0746c2b0
--- /dev/null
+++ b/src/hyper/pages/exseries.ht
@@ -0,0 +1,67 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Title: Series
+ 
+% Address comments and questions to the
+%   Computer Algebra Group, Mathematical Sciences Department
+%   IBM Thomas J. Watson Research Center, Box 218
+%   Yorktown Heights, New York  10598  USA
+ 
+%  Author: Clifton J. Williamson
+%  Date created: 2 November 1989
+%  Date last updated: 2 November 1989
+ 
+\begin{page}{ExSeriesConvert}{Converting Expressions to Series}
+\beginscroll
+You can convert a functional expression to a power series by using the
+function 'series'.
+Here's an example:
+\spadpaste{series(sin(a*x),x = 0)}
+This causes {\em sin(a*x)} to be expanded in powers of {\em (x - 0)}, that is, in powers
+of {\em x}.
+You can have {\em sin(a*x)} expanded in powers of {\em (a - \%pi/4)} by
+issuing the following command:
+\spadpaste{series(sin(a*x),a = \%pi/4)}
+\endscroll
+\autobuttons\end{page}
+ 
+\begin{page}{ExSeriesManipulate}{Manipulating Power Series}
+\beginscroll
+Once you have created a power series, you can perform arithmetic operations
+on that series.
+First compute the Taylor expansion of {\em 1/(1-x)}:
+\spadpaste{f := series(1/(1-x),x = 0) \bound{f}}
+Now compute the square of that series:
+\spadpaste{f ** 2 \free{f}}
+It's as easy as 1, 2, 3,...
+\endscroll
+\autobuttons\end{page}
+ 
+\begin{page}{ExSeriesFunctions}{Functions on Power Series}
+\beginscroll
+The usual elementary functions ({\em log}, {\em exp}, trigonometric functions, etc.)
+are defined for power series.
+You can create a power series:
+% Warning: currently there are (interpretor) problems with converting
+% rational functions and polynomials to power series.
+\spadpaste{f := series(1/(1-x),x = 0) \bound{f1}}
+and then apply these functions to the series:
+\spadpaste{g := log(f) \free{f1} \bound{g}}
+\spadpaste{exp(g) \free{g}}
+\endscroll
+\autobuttons\end{page}
+ 
+\begin{page}{ExSeriesSubstitution}{Substituting Numerical Values in Power Series}
+\beginscroll
+Here's a way to obtain numerical approximations of {\em e} from the Taylor series
+expansion of {\em exp(x)}.
+First you create the desired Taylor expansion:
+\spadpaste{f := taylor(exp(x)) \bound{f2}}
+Now you evaluate the series at the value {\em 1.0}:
+% Warning: syntax for evaluating power series may change.
+\spadpaste{eval(f,1.0) \free{f2}}
+You get a sequence of partial sums.
+\endscroll
+\autobuttons\end{page}
diff --git a/src/hyper/pages/exseries.pht b/src/hyper/pages/exseries.pht
new file mode 100644
index 00000000..df10df9f
--- /dev/null
+++ b/src/hyper/pages/exseries.pht
@@ -0,0 +1,224 @@
+\begin{patch}{ExSeriesSubstitutionPatch1}
+\begin{paste}{ExSeriesSubstitutionFull1}{ExSeriesSubstitutionEmpty1}
+\pastebutton{ExSeriesSubstitutionFull1}{\hidepaste}
+\tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }}
+\indentrel{3}\begin{verbatim}
+   (1)
+             1  2   1  3    1  4    1   5    1   6
+     1 + x + Ä x  + Ä x  + ÄÄ x  + ÄÄÄ x  + ÄÄÄ x
+             2      6      24      120      720
+   + 
+     1   7     1    8      1    9      1     10      11
+   ÄÄÄÄ x  + ÄÄÄÄÄ x  + ÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x   + O(x  )
+   5040      40320      362880      3628800
+   Type: UnivariateTaylorSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesSubstitutionEmpty1}
+\begin{paste}{ExSeriesSubstitutionEmpty1}{ExSeriesSubstitutionPatch1}
+\pastebutton{ExSeriesSubstitutionEmpty1}{\showpaste}
+\tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesSubstitutionPatch2}
+\begin{paste}{ExSeriesSubstitutionFull2}{ExSeriesSubstitutionEmpty2}
+\pastebutton{ExSeriesSubstitutionFull2}{\hidepaste}
+\tab{5}\spadcommand{eval(f,1.0)\free{f2 }}
+\indentrel{3}\begin{verbatim}
+   (2)
+   [1.0, 2.0, 2.5, 2.6666666666 666666667,
+    2.7083333333 333333333, 2.7166666666 666666667,
+    2.7180555555 555555556, 2.7182539682 53968254,
+    2.7182787698 412698413, 2.7182815255 731922399, ...]
+                          Type: Stream Expression Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesSubstitutionEmpty2}
+\begin{paste}{ExSeriesSubstitutionEmpty2}{ExSeriesSubstitutionPatch2}
+\pastebutton{ExSeriesSubstitutionEmpty2}{\showpaste}
+\tab{5}\spadcommand{eval(f,1.0)\free{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesConvertPatch1}
+\begin{paste}{ExSeriesConvertFull1}{ExSeriesConvertEmpty1}
+\pastebutton{ExSeriesConvertFull1}{\hidepaste}
+\tab{5}\spadcommand{series(sin(a*x),x = 0)}
+\indentrel{3}\begin{verbatim}
+   (1)
+            3        5        7          9
+           a   3    a   5    a    7     a     9
+     a x - ÄÄ x  + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x
+            6      120      5040      362880
+   + 
+           11
+          a      11      12
+     - ÄÄÄÄÄÄÄÄ x   + O(x  )
+       39916800
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesConvertEmpty1}
+\begin{paste}{ExSeriesConvertEmpty1}{ExSeriesConvertPatch1}
+\pastebutton{ExSeriesConvertEmpty1}{\showpaste}
+\tab{5}\spadcommand{series(sin(a*x),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesConvertPatch2}
+\begin{paste}{ExSeriesConvertFull2}{ExSeriesConvertEmpty2}
+\pastebutton{ExSeriesConvertFull2}{\hidepaste}
+\tab{5}\spadcommand{series(sin(a*x),a = \%pi/4)}
+\indentrel{3}\begin{verbatim}
+   (2)
+         %pi x          %pi x      %pi
+     sin(ÄÄÄÄÄ) + x cos(ÄÄÄÄÄ)(a - ÄÄÄ)
+           4              4         4
+   + 
+        2    %pi x                3    %pi x
+       x sin(ÄÄÄÄÄ)              x cos(ÄÄÄÄÄ)
+               4         %pi 2           4         %pi 3
+     - ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)  - ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)
+             2            4            6            4
+   + 
+      4    %pi x                5    %pi x
+     x sin(ÄÄÄÄÄ)              x cos(ÄÄÄÄÄ)
+             4         %pi 4           4         %pi 5
+     ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)  + ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)
+          24            4           120           4
+   + 
+        6    %pi x                7    %pi x
+       x sin(ÄÄÄÄÄ)              x cos(ÄÄÄÄÄ)
+               4         %pi 6           4         %pi 7
+     - ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)  - ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)
+            720           4          5040           4
+   + 
+      8    %pi x                9    %pi x
+     x sin(ÄÄÄÄÄ)              x cos(ÄÄÄÄÄ)
+             4         %pi 8           4         %pi 9
+     ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)  + ÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)
+         40320          4         362880          4
+   + 
+        10    %pi x
+       x  sin(ÄÄÄÄÄ)
+                4         %pi 10          %pi 11
+     - ÄÄÄÄÄÄÄÄÄÄÄÄÄ (a - ÄÄÄ)   + O((a - ÄÄÄ)  )
+          3628800          4               4
+Type: UnivariatePuiseuxSeries(Expression Integer,a,pi/4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesConvertEmpty2}
+\begin{paste}{ExSeriesConvertEmpty2}{ExSeriesConvertPatch2}
+\pastebutton{ExSeriesConvertEmpty2}{\showpaste}
+\tab{5}\spadcommand{series(sin(a*x),a = \%pi/4)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesFunctionsPatch1}
+\begin{paste}{ExSeriesFunctionsFull1}{ExSeriesFunctionsEmpty1}
+\pastebutton{ExSeriesFunctionsFull1}{\hidepaste}
+\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+   (1)
+              2    3    4    5    6    7    8    9    10
+     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
+   + 
+        11
+     O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesFunctionsEmpty1}
+\begin{paste}{ExSeriesFunctionsEmpty1}{ExSeriesFunctionsPatch1}
+\pastebutton{ExSeriesFunctionsEmpty1}{\showpaste}
+\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesFunctionsPatch2}
+\begin{paste}{ExSeriesFunctionsFull2}{ExSeriesFunctionsEmpty2}
+\pastebutton{ExSeriesFunctionsFull2}{\hidepaste}
+\tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }}
+\indentrel{3}\begin{verbatim}
+   (2)
+         1  2   1  3   1  4   1  5   1  6   1  7   1  8
+     x + Ä x  + Ä x  + Ä x  + Ä x  + Ä x  + Ä x  + Ä x
+         2      3      4      5      6      7      8
+   + 
+     1  9    1  10    1  11      12
+     Ä x  + ÄÄ x   + ÄÄ x   + O(x  )
+     9      10       11
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesFunctionsEmpty2}
+\begin{paste}{ExSeriesFunctionsEmpty2}{ExSeriesFunctionsPatch2}
+\pastebutton{ExSeriesFunctionsEmpty2}{\showpaste}
+\tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesFunctionsPatch3}
+\begin{paste}{ExSeriesFunctionsFull3}{ExSeriesFunctionsEmpty3}
+\pastebutton{ExSeriesFunctionsFull3}{\hidepaste}
+\tab{5}\spadcommand{exp(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (3)
+              2    3    4    5    6    7    8    9    10
+     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
+   + 
+        11
+     O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesFunctionsEmpty3}
+\begin{paste}{ExSeriesFunctionsEmpty3}{ExSeriesFunctionsPatch3}
+\pastebutton{ExSeriesFunctionsEmpty3}{\showpaste}
+\tab{5}\spadcommand{exp(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesManipulatePatch1}
+\begin{paste}{ExSeriesManipulateFull1}{ExSeriesManipulateEmpty1}
+\pastebutton{ExSeriesManipulateFull1}{\hidepaste}
+\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }}
+\indentrel{3}\begin{verbatim}
+   (1)
+              2    3    4    5    6    7    8    9    10
+     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
+   + 
+        11
+     O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesManipulateEmpty1}
+\begin{paste}{ExSeriesManipulateEmpty1}{ExSeriesManipulatePatch1}
+\pastebutton{ExSeriesManipulateEmpty1}{\showpaste}
+\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesManipulatePatch2}
+\begin{paste}{ExSeriesManipulateFull2}{ExSeriesManipulateEmpty2}
+\pastebutton{ExSeriesManipulateFull2}{\hidepaste}
+\tab{5}\spadcommand{f ** 2\free{f }}
+\indentrel{3}\begin{verbatim}
+   (2)
+                2     3     4     5     6     7     8
+     1 + 2x + 3x  + 4x  + 5x  + 6x  + 7x  + 8x  + 9x
+   + 
+        9      10      11
+     10x  + 11x   + O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSeriesManipulateEmpty2}
+\begin{paste}{ExSeriesManipulateEmpty2}{ExSeriesManipulatePatch2}
+\pastebutton{ExSeriesManipulateEmpty2}{\showpaste}
+\tab{5}\spadcommand{f ** 2\free{f }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/exsum.ht b/src/hyper/pages/exsum.ht
new file mode 100644
index 00000000..f5e7f8b1
--- /dev/null
+++ b/src/hyper/pages/exsum.ht
@@ -0,0 +1,120 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Title: Summations
+
+% Address comments and questions to the
+%   Computer Algebra Group, Mathematical Sciences Department
+%   IBM Thomas J. Watson Research Center, Box 218
+%   Yorktown Heights, New York  10598  USA
+
+%  Author: Clifton J. Williamson
+%  Date created: 2 November 1989
+%  Date last updated: 2 November 1989
+
+\begin{page}{ExSumListEntriesI}{Summing the Entries of a List I}
+\beginscroll
+In \Language{}, you can create lists of consecutive integers by giving the
+first and last entries of the list.
+Here's how you create a list of the integers between {\em 1} and {\em 15}:
+\spadpaste{[i for i in 1..15]}
+To sum the entries of a list, simply put {\em +/} in front of the list.
+For example, the following command will sum the integers from 1 to 15:
+\spadpaste{reduce(+,[i for i in 1..15])}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExSumListEntriesII}{Summing the Entries of a List II}
+\beginscroll
+In \Language{}, you can also create lists whose elements are some expression
+{\em f(n)} as the parameter n ranges between two integers.
+For example, the following command will create a list of the squares of
+the integers between {\em 5} and {\em 20}:
+\spadpaste{[n**2 for n in 5..20]}
+You can also compute the sum of the entries of this list:
+\spadpaste{reduce(+,[n**2 for n in 5..20])}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExSumApproximateE}{Approximating e}
+\beginscroll
+You can obtain a numerical approximation of the number {\em e} by summing the
+entries of the following list:
+\spadpaste{reduce(+,[1.0/factorial(n) for n in 0..20])}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExSumClosedForm}{Closed Form Summations}
+\beginscroll
+In a previous example, we found the sum of the squares of the integers
+between {\em 5} and {\em 20}.
+We can also use \Language{} to find a formula for the sum of the squares of
+the integers between {\em a} and {\em b}, where {\em a} and {\em b}
+are integers which will remain
+unspecified:
+\spadpaste{s := sum(k**2,k = a..b) \bound{s}}
+{\em sum(k**2,k = a..b)} returns the sum of {\em k**2} as the index {\em k}
+runs from {\em a} to {\em b}.
+Let's check our answer in one particular case by substituting specific values
+for {\em a} and {\em b} in our formula:
+% Warning: syntax for polynomial evaluation will probably change.
+\spadpaste{eval(s,[a,b],[1,25]) \free{s}}
+\spadpaste{reduce(+,[i**2 for i in 1..25])}
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExSumCubes}{Sums of Cubes}
+\beginscroll
+Here's a cute example.
+First compute the sum of the cubes from {\em 1} to {\em n}:
+\spadpaste{sum(k**3,k = 1..n)}
+Then compute the square of the sum of the integers from {\em 1} to {\em n}:
+\spadpaste{sum(k,k = 1..n) ** 2}
+The answers are the same.
+\endscroll
+\autobuttons\end{page}
+
+\begin{page}{ExSumPolynomial}{Sums of Polynomials}
+\beginscroll
+\Language{} can compute {\em sum(f,k = a..b)}
+when {\em f} is any polynomial in {\em k}, even
+one with parameters.
+\spadpaste{sum(3*k**2/(c**2 + 1) + 12*k/d,k = (3*a)..(4*b))}
+\endscroll
+\autobuttons\end{page}
+
+%\begin{page}{ExSumRationalFunction}{Sums of Rational Functions}
+%\beginscroll
+%\Language{} can compute {\em sum(f,k = a..b)} for some rational functions (quotients
+%of polynomials) in {\em k}.
+%\spadpaste{sum(1/(k * (k + 2)),k = 1..n)}
+%However, the method used (Gosper's method) does not guarantee an answer
+%in every case.
+%Here's an example of a sum that the method cannot compute:
+%\spadpaste{sum(1/(k**2 + 1),k = 1..n)}
+%\endscroll
+%\autobuttons\end{page}
+
+\begin{page}{ExSumGeneralFunction}{Sums of General Functions}
+\beginscroll
+Gosper's method can also be used to compute {\em sum(f,k = a..b)}
+for some functions
+f which are not rational functions in {\em k}.
+Here's an example:
+\spadpaste{sum(k * x**k,k = 1..n)}
+\endscroll
+\autobuttons\end{page}
+
+% provide a package for infinite sums
+\begin{page}{ExSumInfinite}{Infinite Sums}
+\beginscroll
+In a few cases, we can compute infinite sums by taking limits of finite
+sums.
+For instance, you can compute the sum of {\em 1/(k * (k + 2))} as {\em k}
+ranges from
+{\em 1} to {\em infinity}.
+Use {\em \%plusInfinity} to denote `plus infinity'.
+\spadpaste{limit( sum(1/(k * (k + 2)),k = 1..n) ,n = \%plusInfinity)}
+\endscroll
+\autobuttons\end{page}
diff --git a/src/hyper/pages/exsum.pht b/src/hyper/pages/exsum.pht
new file mode 100644
index 00000000..7d6c7658
--- /dev/null
+++ b/src/hyper/pages/exsum.pht
@@ -0,0 +1,235 @@
+\begin{patch}{ExSumGeneralFunctionPatch1}
+\begin{paste}{ExSumGeneralFunctionFull1}{ExSumGeneralFunctionEmpty1}
+\pastebutton{ExSumGeneralFunctionFull1}{\hidepaste}
+\tab{5}\spadcommand{sum(k * x**k,k = 1..n)}
+\indentrel{3}\begin{verbatim}
+            2               n
+        (n x  + (- n - 1)x)x  + x
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                2
+               x  - 2x + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSumGeneralFunctionEmpty1}
+\begin{paste}{ExSumGeneralFunctionEmpty1}{ExSumGeneralFunctionPatch1}
+\pastebutton{ExSumGeneralFunctionEmpty1}{\showpaste}
+\tab{5}\spadcommand{sum(k * x**k,k = 1..n)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSumInfinitePatch1}
+\begin{paste}{ExSumInfiniteFull1}{ExSumInfiniteEmpty1}
+\pastebutton{ExSumInfiniteFull1}{\hidepaste}
+\tab{5}\spadcommand{limit( sum(1/(k * (k + 2)),k = 1..n) ,n = \%plusInfinity)}
+\indentrel{3}\begin{verbatim}
+        3
+   (1)  Ä
+        4
+Type: Union(OrderedCompletion Fraction Polynomial Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSumInfiniteEmpty1}
+\begin{paste}{ExSumInfiniteEmpty1}{ExSumInfinitePatch1}
+\pastebutton{ExSumInfiniteEmpty1}{\showpaste}
+\tab{5}\spadcommand{limit( sum(1/(k * (k + 2)),k = 1..n) ,n = \%plusInfinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSumClosedFormPatch1}
+\begin{paste}{ExSumClosedFormFull1}{ExSumClosedFormEmpty1}
+\pastebutton{ExSumClosedFormFull1}{\hidepaste}
+\tab{5}\spadcommand{s := sum(k**2,k = a..b)\bound{s }}
+\indentrel{3}\begin{verbatim}
+          3     2         3     2
+        2b  + 3b  + b - 2a  + 3a  - a
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                      6
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSumClosedFormEmpty1}
+\begin{paste}{ExSumClosedFormEmpty1}{ExSumClosedFormPatch1}
+\pastebutton{ExSumClosedFormEmpty1}{\showpaste}
+\tab{5}\spadcommand{s := sum(k**2,k = a..b)\bound{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSumClosedFormPatch2}
+\begin{paste}{ExSumClosedFormFull2}{ExSumClosedFormEmpty2}
+\pastebutton{ExSumClosedFormFull2}{\hidepaste}
+\tab{5}\spadcommand{eval(s,[a,b],[1,25])\free{s }}
+\indentrel{3}\begin{verbatim}
+   (2)  5525
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSumClosedFormEmpty2}
+\begin{paste}{ExSumClosedFormEmpty2}{ExSumClosedFormPatch2}
+\pastebutton{ExSumClosedFormEmpty2}{\showpaste}
+\tab{5}\spadcommand{eval(s,[a,b],[1,25])\free{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSumClosedFormPatch3}
+\begin{paste}{ExSumClosedFormFull3}{ExSumClosedFormEmpty3}
+\pastebutton{ExSumClosedFormFull3}{\hidepaste}
+\tab{5}\spadcommand{reduce(+,[i**2 for i in 1..25])}
+\indentrel{3}\begin{verbatim}
+   (3)  5525
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSumClosedFormEmpty3}
+\begin{paste}{ExSumClosedFormEmpty3}{ExSumClosedFormPatch3}
+\pastebutton{ExSumClosedFormEmpty3}{\showpaste}
+\tab{5}\spadcommand{reduce(+,[i**2 for i in 1..25])}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSumPolynomialPatch1}
+\begin{paste}{ExSumPolynomialFull1}{ExSumPolynomialEmpty1}
+\pastebutton{ExSumPolynomialFull1}{\hidepaste}
+\tab{5}\spadcommand{sum(3*k**2/(c**2 + 1) + 12*k/d,k = (3*a)..(4*b))}
+\indentrel{3}\begin{verbatim}
+   (1)
+            3      2           3      2
+       (128b  + 48b  + 4b - 54a  + 27a  - 3a)d
+     + 
+            2             2        2       2             2
+       (192b  + 48b - 108a  + 36a)c  + 192b  + 48b - 108a
+     + 
+       36a
+  /
+        2
+     (2c  + 2)d
+           Type: Union(Fraction Polynomial Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSumPolynomialEmpty1}
+\begin{paste}{ExSumPolynomialEmpty1}{ExSumPolynomialPatch1}
+\pastebutton{ExSumPolynomialEmpty1}{\showpaste}
+\tab{5}\spadcommand{sum(3*k**2/(c**2 + 1) + 12*k/d,k = (3*a)..(4*b))}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSumListEntriesIPatch1}
+\begin{paste}{ExSumListEntriesIFull1}{ExSumListEntriesIEmpty1}
+\pastebutton{ExSumListEntriesIFull1}{\hidepaste}
+\tab{5}\spadcommand{[i for i in 1..15]}
+\indentrel{3}\begin{verbatim}
+   (1)  [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSumListEntriesIEmpty1}
+\begin{paste}{ExSumListEntriesIEmpty1}{ExSumListEntriesIPatch1}
+\pastebutton{ExSumListEntriesIEmpty1}{\showpaste}
+\tab{5}\spadcommand{[i for i in 1..15]}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSumListEntriesIPatch2}
+\begin{paste}{ExSumListEntriesIFull2}{ExSumListEntriesIEmpty2}
+\pastebutton{ExSumListEntriesIFull2}{\hidepaste}
+\tab{5}\spadcommand{reduce(+,[i for i in 1..15])}
+\indentrel{3}\begin{verbatim}
+   (2)  120
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSumListEntriesIEmpty2}
+\begin{paste}{ExSumListEntriesIEmpty2}{ExSumListEntriesIPatch2}
+\pastebutton{ExSumListEntriesIEmpty2}{\showpaste}
+\tab{5}\spadcommand{reduce(+,[i for i in 1..15])}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSumApproximateEPatch1}
+\begin{paste}{ExSumApproximateEFull1}{ExSumApproximateEEmpty1}
+\pastebutton{ExSumApproximateEFull1}{\hidepaste}
+\tab{5}\spadcommand{reduce(+,[1.0/factorial(n) for n in 0..20])}
+\indentrel{3}\begin{verbatim}
+   (1)  2.7182818284 590452354
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSumApproximateEEmpty1}
+\begin{paste}{ExSumApproximateEEmpty1}{ExSumApproximateEPatch1}
+\pastebutton{ExSumApproximateEEmpty1}{\showpaste}
+\tab{5}\spadcommand{reduce(+,[1.0/factorial(n) for n in 0..20])}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSumListEntriesIIPatch1}
+\begin{paste}{ExSumListEntriesIIFull1}{ExSumListEntriesIIEmpty1}
+\pastebutton{ExSumListEntriesIIFull1}{\hidepaste}
+\tab{5}\spadcommand{[n**2 for n in 5..20]}
+\indentrel{3}\begin{verbatim}
+   (1)
+   [25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225,
+    256, 289, 324, 361, 400]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSumListEntriesIIEmpty1}
+\begin{paste}{ExSumListEntriesIIEmpty1}{ExSumListEntriesIIPatch1}
+\pastebutton{ExSumListEntriesIIEmpty1}{\showpaste}
+\tab{5}\spadcommand{[n**2 for n in 5..20]}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSumListEntriesIIPatch2}
+\begin{paste}{ExSumListEntriesIIFull2}{ExSumListEntriesIIEmpty2}
+\pastebutton{ExSumListEntriesIIFull2}{\hidepaste}
+\tab{5}\spadcommand{reduce(+,[n**2 for n in 5..20])}
+\indentrel{3}\begin{verbatim}
+   (2)  2840
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSumListEntriesIIEmpty2}
+\begin{paste}{ExSumListEntriesIIEmpty2}{ExSumListEntriesIIPatch2}
+\pastebutton{ExSumListEntriesIIEmpty2}{\showpaste}
+\tab{5}\spadcommand{reduce(+,[n**2 for n in 5..20])}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSumCubesPatch1}
+\begin{paste}{ExSumCubesFull1}{ExSumCubesEmpty1}
+\pastebutton{ExSumCubesFull1}{\hidepaste}
+\tab{5}\spadcommand{sum(k**3,k = 1..n)}
+\indentrel{3}\begin{verbatim}
+         4     3    2
+        n  + 2n  + n
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄ
+              4
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSumCubesEmpty1}
+\begin{paste}{ExSumCubesEmpty1}{ExSumCubesPatch1}
+\pastebutton{ExSumCubesEmpty1}{\showpaste}
+\tab{5}\spadcommand{sum(k**3,k = 1..n)}
+\end{paste}\end{patch}
+
+\begin{patch}{ExSumCubesPatch2}
+\begin{paste}{ExSumCubesFull2}{ExSumCubesEmpty2}
+\pastebutton{ExSumCubesFull2}{\hidepaste}
+\tab{5}\spadcommand{sum(k,k = 1..n) ** 2}
+\indentrel{3}\begin{verbatim}
+         4     3    2
+        n  + 2n  + n
+   (2)  ÄÄÄÄÄÄÄÄÄÄÄÄÄ
+              4
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ExSumCubesEmpty2}
+\begin{paste}{ExSumCubesEmpty2}{ExSumCubesPatch2}
+\pastebutton{ExSumCubesEmpty2}{\showpaste}
+\tab{5}\spadcommand{sum(k,k = 1..n) ** 2}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/function.ht b/src/hyper/pages/function.ht
new file mode 100644
index 00000000..b0da8ec2
--- /dev/null
+++ b/src/hyper/pages/function.ht
@@ -0,0 +1,157 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved.
+
+\begin{page}{FunctionPage}{Functions in \Language{}}
+%
+In \Language{}, a function is an expression in one or more variables.
+(Think of it as a function of those variables).
+You can also define a function by rules or use a built-in function
+\Language{} lets you convert expressions to compiled functions.
+\beginscroll
+\beginmenu
+\menulink{Rational Functions}{RationatFunctionPage} \tab{22}
+Quotients of polynomials.
+
+\menulink{Algebraic Functions}{AlgebraicFunctionPage} \tab{22}
+Those defined by polynomial equations.
+
+\menulink{Elementary Functions}{ElementaryFunctionPage} \tab{22}
+The elementary functions of calculus.
+
+\menulink{Simplification}{FunctionSimplificationPage} \tab{22}
+How to simplify expressions.
+
+\menulink{Pattern Matching}{ugUserRulesPage} \tab{22}
+How to use the pattern matcher.
+\endmenu
+\endscroll
+\newline
+Additional Topics:
+\beginmenu
+
+\menulink{Operator Algebra}{OperatorXmpPage}\tab{22}
+The operator algebra facility.
+
+\endmenu
+\autobuttons \end{page}
+
+\begin{page}{RationatFunctionPage}{Rational Functions}
+\beginscroll
+To create a rational function, just compute the
+quotient of two polynomials:
+\spadpaste{f := (x - y) / (x + y)\bound{f}}
+Use the functions \spadfun{numer} and \spadfun{denom}:
+to recover the numerator and denominator of a fraction:
+%
+\spadpaste{numer f\free{f}}
+\spadpaste{denom f\free{f}}
+%
+Since these are polynomials, you can apply all the
+\downlink{polynomial operations}{PolynomialPage}
+to them.
+You can substitute values or
+other rational functions for the variables using
+the function \spadfun{eval}. The syntax for \spadfun{eval} is
+similar to the one for polynomials:
+\spadpaste{eval(f, x = 1/x)\free{f}}
+\spadpaste{eval(f, [x = y, y = x])\free{f}}
+\endscroll
+\autobuttons
+\end{page}
+
+\begin{page}{AlgebraicFunctionPage}{Algebraic Functions}
+\beginscroll
+Algebraic functions are functions defined by algebraic equations. There
+are two ways of constructing them: using rational powers, or implicitly.
+For rational powers, use \spadopFrom{**}{RadicalCategory}
+(or the system functions \spadfun{sqrt} and
+\spadfun{nthRoot} for square and nth roots):
+\spadpaste{f := sqrt(1 + x ** (1/3))\bound{f}}
+To define an algebraic function implicitly
+use \spadfun{rootOf}. The following
+line defines a function \spad{y} of \spad{x} satisfying the equation
+\spad{y**3 = x*y - y**2 - x**3 + 1}:
+\spadpaste{y := rootOf(y**3 + y**2 - x*y + x**3 - 1, y)\bound{y}}
+You can manipulate, differentiate or integrate an implicitly defined
+algebraic function like any other \Language{} function:
+\spadpaste{differentiate(y, x)\free{y}}
+Higher powers of algebraic functions are automatically reduced during
+calculations:
+\spadpaste{(y + 1) ** 3\free{y}}
+But denominators, are not automatically rationalized:
+\spadpaste{g := inv f\bound{g}\free{y}}
+Use \spadfun{ratDenom} to remove the algebraic quantities from the denominator:
+\spadpaste{ratDenom g\free{g}}
+\endscroll
+\autobuttons \end{page}
+
+\begin{page}{ElementaryFunctionPage}{Elementary Functions}
+\beginscroll
+\Language{} has most of the usual functions from calculus built-in:
+\spadpaste{f := x * log y * sin(1/(x+y))\bound{f}}
+You can substitute values or another elementary functions for the variables
+with the function \spadfun{eval}:
+\spadpaste{eval(f, [x = y, y = x])\free{f}}
+As you can see, the substitutions are made 'in parallel' as in the case
+of polynomials. It's also possible to substitute expressions for kernels
+instead of variables:
+\spadpaste{eval(f, log y = acosh(x + sqrt y))\free{f}}
+\endscroll
+\autobuttons \end{page}
+
+\begin{page}{FunctionSimplificationPage}{Simplification}
+\beginscroll
+Simplifying an expression often means different things at
+different times, so \Language{} offers a large number of
+`simplification' functions.
+The most common one, which performs the usual trigonometric
+simplifications is \spadfun{simplify}:
+\spadpaste{f := cos(x)/sec(x) * log(sin(x)**2/(cos(x)**2+sin(x)**2)) \bound{f}}
+\spadpaste{g := simplify f\bound{g}\free{f}}
+If the result of \spadfun{simplify} is not satisfactory, specific
+transformations are available.
+For example, to rewrite \spad{g} in terms of secants and
+cosecants instead of sines and cosines, issue:
+%
+\spadpaste{h := sin2csc cos2sec g\bound{h}\free{g}}
+%
+To apply the logarithm simplification rules to \spad{h}, issue:
+\spadpaste{expandLog h\free{h}}
+Since the square root of \spad{x**2} is the absolute value of
+\spad{x} and not \spad{x} itself, algebraic radicals are not
+automatically simplified, but you can specifically request it by
+calling \spadfun{rootSimp}:
+%
+\spadpaste{f1 := sqrt((x+1)**3)\bound{f1}}
+\spadpaste{rootSimp f1\free{f1}}
+%
+There are other transformations which are sometimes useful.
+Use the functions \spadfun{complexElementary} and \spadfun{trigs}
+to go back and forth between the complex exponential and
+trigonometric forms of an elementary function:
+%
+\spadpaste{g1 := sin(x + cos x)\bound{g1}}
+\spadpaste{g2 := complexElementary g1\bound{g2}\free{g1}}
+\spadpaste{trigs g2\free{g2}}
+%
+Similarly, the functions \spadfun{realElementary} and
+\spadfun{htrigs} convert hyperbolic functions in and out of their
+exponential form:
+%
+\spadpaste{h1 := sinh(x + cosh x)\bound{h1}}
+\spadpaste{h2 := realElementary h1\bound{h2}\free{h1}}
+\spadpaste{htrigs h2\free{h2}}
+%
+\Language{} has other transformations, most of which
+are in the packages
+\spadtype{ElementaryFunctionStructurePackage},
+\spadtype{TrigonometricManipulations},
+\spadtype{AlgebraicManipulations},
+and \spadtype{TranscendentalManipulations}.
+If you need to apply a simplification rule not built into the
+system, you can use \Language{}'s \downlink{pattern
+matcher}{ugUserRulesPage}.
+\endscroll
+\autobuttons
+\end{page}
diff --git a/src/hyper/pages/function.pht b/src/hyper/pages/function.pht
new file mode 100644
index 00000000..eeb51b20
--- /dev/null
+++ b/src/hyper/pages/function.pht
@@ -0,0 +1,506 @@
+\begin{patch}{RationatFunctionPagePatch1}
+\begin{paste}{RationatFunctionPageFull1}{RationatFunctionPageEmpty1}
+\pastebutton{RationatFunctionPageFull1}{\hidepaste}
+\tab{5}\spadcommand{f := (x - y) / (x + y)\bound{f }}
+\indentrel{3}\begin{verbatim}
+        - y + x
+   (1)  ÄÄÄÄÄÄÄ
+         y + x
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RationatFunctionPageEmpty1}
+\begin{paste}{RationatFunctionPageEmpty1}{RationatFunctionPagePatch1}
+\pastebutton{RationatFunctionPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{f := (x - y) / (x + y)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{RationatFunctionPagePatch2}
+\begin{paste}{RationatFunctionPageFull2}{RationatFunctionPageEmpty2}
+\pastebutton{RationatFunctionPageFull2}{\hidepaste}
+\tab{5}\spadcommand{numer f\free{f }}
+\indentrel{3}\begin{verbatim}
+   (2)  - y + x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RationatFunctionPageEmpty2}
+\begin{paste}{RationatFunctionPageEmpty2}{RationatFunctionPagePatch2}
+\pastebutton{RationatFunctionPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{numer f\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{RationatFunctionPagePatch3}
+\begin{paste}{RationatFunctionPageFull3}{RationatFunctionPageEmpty3}
+\pastebutton{RationatFunctionPageFull3}{\hidepaste}
+\tab{5}\spadcommand{denom f\free{f }}
+\indentrel{3}\begin{verbatim}
+   (3)  y + x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RationatFunctionPageEmpty3}
+\begin{paste}{RationatFunctionPageEmpty3}{RationatFunctionPagePatch3}
+\pastebutton{RationatFunctionPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{denom f\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{RationatFunctionPagePatch4}
+\begin{paste}{RationatFunctionPageFull4}{RationatFunctionPageEmpty4}
+\pastebutton{RationatFunctionPageFull4}{\hidepaste}
+\tab{5}\spadcommand{eval(f, x = 1/x)\free{f }}
+\indentrel{3}\begin{verbatim}
+        - x y + 1
+   (4)  ÄÄÄÄÄÄÄÄÄ
+         x y + 1
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RationatFunctionPageEmpty4}
+\begin{paste}{RationatFunctionPageEmpty4}{RationatFunctionPagePatch4}
+\pastebutton{RationatFunctionPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{eval(f, x = 1/x)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{RationatFunctionPagePatch5}
+\begin{paste}{RationatFunctionPageFull5}{RationatFunctionPageEmpty5}
+\pastebutton{RationatFunctionPageFull5}{\hidepaste}
+\tab{5}\spadcommand{eval(f, [x = y, y = x])\free{f }}
+\indentrel{3}\begin{verbatim}
+        y - x
+   (5)  ÄÄÄÄÄ
+        y + x
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RationatFunctionPageEmpty5}
+\begin{paste}{RationatFunctionPageEmpty5}{RationatFunctionPagePatch5}
+\pastebutton{RationatFunctionPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{eval(f, [x = y, y = x])\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{AlgebraicFunctionPagePatch1}
+\begin{paste}{AlgebraicFunctionPageFull1}{AlgebraicFunctionPageEmpty1}
+\pastebutton{AlgebraicFunctionPageFull1}{\hidepaste}
+\tab{5}\spadcommand{f := sqrt(1 + x ** (1/3))\bound{f }}
+\indentrel{3}\begin{verbatim}
+         ÚÄÄÄÄÄÄÄÄ¿
+         ³3ÚÄ¿
+   (1)  \³\³x  + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AlgebraicFunctionPageEmpty1}
+\begin{paste}{AlgebraicFunctionPageEmpty1}{AlgebraicFunctionPagePatch1}
+\pastebutton{AlgebraicFunctionPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{f := sqrt(1 + x ** (1/3))\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{AlgebraicFunctionPagePatch2}
+\begin{paste}{AlgebraicFunctionPageFull2}{AlgebraicFunctionPageEmpty2}
+\pastebutton{AlgebraicFunctionPageFull2}{\hidepaste}
+\tab{5}\spadcommand{y := rootOf(y**3 + y**2 - x*y + x**3 - 1, y)\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (2)  y
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AlgebraicFunctionPageEmpty2}
+\begin{paste}{AlgebraicFunctionPageEmpty2}{AlgebraicFunctionPagePatch2}
+\pastebutton{AlgebraicFunctionPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{y := rootOf(y**3 + y**2 - x*y + x**3 - 1, y)\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{AlgebraicFunctionPagePatch3}
+\begin{paste}{AlgebraicFunctionPageFull3}{AlgebraicFunctionPageEmpty3}
+\pastebutton{AlgebraicFunctionPageFull3}{\hidepaste}
+\tab{5}\spadcommand{differentiate(y, x)\free{y }}
+\indentrel{3}\begin{verbatim}
+                 2
+           y - 3x
+   (3)  ÄÄÄÄÄÄÄÄÄÄÄÄ
+          2
+        3y  + 2y - x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AlgebraicFunctionPageEmpty3}
+\begin{paste}{AlgebraicFunctionPageEmpty3}{AlgebraicFunctionPagePatch3}
+\pastebutton{AlgebraicFunctionPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{differentiate(y, x)\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{AlgebraicFunctionPagePatch4}
+\begin{paste}{AlgebraicFunctionPageFull4}{AlgebraicFunctionPageEmpty4}
+\pastebutton{AlgebraicFunctionPageFull4}{\hidepaste}
+\tab{5}\spadcommand{(y + 1) ** 3\free{y }}
+\indentrel{3}\begin{verbatim}
+          2               3
+   (4)  2y  + (x + 3)y - x  + 2
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AlgebraicFunctionPageEmpty4}
+\begin{paste}{AlgebraicFunctionPageEmpty4}{AlgebraicFunctionPagePatch4}
+\pastebutton{AlgebraicFunctionPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{(y + 1) ** 3\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{AlgebraicFunctionPagePatch5}
+\begin{paste}{AlgebraicFunctionPageFull5}{AlgebraicFunctionPageEmpty5}
+\pastebutton{AlgebraicFunctionPageFull5}{\hidepaste}
+\tab{5}\spadcommand{g := inv f\bound{g }\free{y }}
+\indentrel{3}\begin{verbatim}
+             1
+   (5)  ÄÄÄÄÄÄÄÄÄÄÄ
+         ÚÄÄÄÄÄÄÄÄ¿
+         ³3ÚÄ¿
+        \³\³x  + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AlgebraicFunctionPageEmpty5}
+\begin{paste}{AlgebraicFunctionPageEmpty5}{AlgebraicFunctionPagePatch5}
+\pastebutton{AlgebraicFunctionPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{g := inv f\bound{g }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{AlgebraicFunctionPagePatch6}
+\begin{paste}{AlgebraicFunctionPageFull6}{AlgebraicFunctionPageEmpty6}
+\pastebutton{AlgebraicFunctionPageFull6}{\hidepaste}
+\tab{5}\spadcommand{ratDenom g\free{g }}
+\indentrel{3}\begin{verbatim}
+                           ÚÄÄÄÄÄÄÄÄ¿
+         3ÚÄ¿2   3ÚÄ¿      ³3ÚÄ¿
+        (\³x   - \³x  + 1)\³\³x  + 1
+   (6)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                    x + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AlgebraicFunctionPageEmpty6}
+\begin{paste}{AlgebraicFunctionPageEmpty6}{AlgebraicFunctionPagePatch6}
+\pastebutton{AlgebraicFunctionPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{ratDenom g\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ElementaryFunctionPagePatch1}
+\begin{paste}{ElementaryFunctionPageFull1}{ElementaryFunctionPageEmpty1}
+\pastebutton{ElementaryFunctionPageFull1}{\hidepaste}
+\tab{5}\spadcommand{f := x * log y * sin(1/(x+y))\bound{f }}
+\indentrel{3}\begin{verbatim}
+                      1
+   (1)  x log(y)sin(ÄÄÄÄÄ)
+                    y + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ElementaryFunctionPageEmpty1}
+\begin{paste}{ElementaryFunctionPageEmpty1}{ElementaryFunctionPagePatch1}
+\pastebutton{ElementaryFunctionPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{f := x * log y * sin(1/(x+y))\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ElementaryFunctionPagePatch2}
+\begin{paste}{ElementaryFunctionPageFull2}{ElementaryFunctionPageEmpty2}
+\pastebutton{ElementaryFunctionPageFull2}{\hidepaste}
+\tab{5}\spadcommand{eval(f, [x = y, y = x])\free{f }}
+\indentrel{3}\begin{verbatim}
+                      1
+   (2)  y log(x)sin(ÄÄÄÄÄ)
+                    y + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ElementaryFunctionPageEmpty2}
+\begin{paste}{ElementaryFunctionPageEmpty2}{ElementaryFunctionPagePatch2}
+\pastebutton{ElementaryFunctionPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{eval(f, [x = y, y = x])\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ElementaryFunctionPagePatch3}
+\begin{paste}{ElementaryFunctionPageFull3}{ElementaryFunctionPageEmpty3}
+\pastebutton{ElementaryFunctionPageFull3}{\hidepaste}
+\tab{5}\spadcommand{eval(f, log y = acosh(x + sqrt y))\free{f }}
+\indentrel{3}\begin{verbatim}
+                1          ÚÄ¿
+   (3)  x sin(ÄÄÄÄÄ)acosh(\³y  + x)
+              y + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ElementaryFunctionPageEmpty3}
+\begin{paste}{ElementaryFunctionPageEmpty3}{ElementaryFunctionPagePatch3}
+\pastebutton{ElementaryFunctionPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{eval(f, log y = acosh(x + sqrt y))\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPagePatch1}
+\begin{paste}{FunctionSimplificationPageFull1}{FunctionSimplificationPageEmpty1}
+\pastebutton{FunctionSimplificationPageFull1}{\hidepaste}
+\tab{5}\spadcommand{f := cos(x)/sec(x) * log(sin(x)**2/(cos(x)**2+sin(x)**2))\bound{f }}
+\indentrel{3}\begin{verbatim}
+                             2
+                       sin(x)
+        cos(x)log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
+                        2         2
+                  sin(x)  + cos(x)
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                   sec(x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPageEmpty1}
+\begin{paste}{FunctionSimplificationPageEmpty1}{FunctionSimplificationPagePatch1}
+\pastebutton{FunctionSimplificationPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{f := cos(x)/sec(x) * log(sin(x)**2/(cos(x)**2+sin(x)**2))\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPagePatch2}
+\begin{paste}{FunctionSimplificationPageFull2}{FunctionSimplificationPageEmpty2}
+\pastebutton{FunctionSimplificationPageFull2}{\hidepaste}
+\tab{5}\spadcommand{g := simplify f\bound{g }\free{f }}
+\indentrel{3}\begin{verbatim}
+              2            2
+   (2)  cos(x) log(- cos(x)  + 1)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPageEmpty2}
+\begin{paste}{FunctionSimplificationPageEmpty2}{FunctionSimplificationPagePatch2}
+\pastebutton{FunctionSimplificationPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{g := simplify f\bound{g }\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPagePatch3}
+\begin{paste}{FunctionSimplificationPageFull3}{FunctionSimplificationPageEmpty3}
+\pastebutton{FunctionSimplificationPageFull3}{\hidepaste}
+\tab{5}\spadcommand{h := sin2csc cos2sec g\bound{h }\free{g }}
+\indentrel{3}\begin{verbatim}
+                  2
+            sec(x)  - 1
+        log(ÄÄÄÄÄÄÄÄÄÄÄ)
+                    2
+              sec(x)
+   (3)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                   2
+             sec(x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPageEmpty3}
+\begin{paste}{FunctionSimplificationPageEmpty3}{FunctionSimplificationPagePatch3}
+\pastebutton{FunctionSimplificationPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{h := sin2csc cos2sec g\bound{h }\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPagePatch4}
+\begin{paste}{FunctionSimplificationPageFull4}{FunctionSimplificationPageEmpty4}
+\pastebutton{FunctionSimplificationPageFull4}{\hidepaste}
+\tab{5}\spadcommand{expandLog h\free{h }}
+\indentrel{3}\begin{verbatim}
+                  2
+        log(sec(x)  - 1) - 2log(sec(x))
+   (4)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                          2
+                    sec(x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPageEmpty4}
+\begin{paste}{FunctionSimplificationPageEmpty4}{FunctionSimplificationPagePatch4}
+\pastebutton{FunctionSimplificationPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{expandLog h\free{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPagePatch5}
+\begin{paste}{FunctionSimplificationPageFull5}{FunctionSimplificationPageEmpty5}
+\pastebutton{FunctionSimplificationPageFull5}{\hidepaste}
+\tab{5}\spadcommand{f1 := sqrt((x+1)**3)\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+         ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+         ³ 3     2
+   (5)  \³x  + 3x  + 3x + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPageEmpty5}
+\begin{paste}{FunctionSimplificationPageEmpty5}{FunctionSimplificationPagePatch5}
+\pastebutton{FunctionSimplificationPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{f1 := sqrt((x+1)**3)\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPagePatch6}
+\begin{paste}{FunctionSimplificationPageFull6}{FunctionSimplificationPageEmpty6}
+\pastebutton{FunctionSimplificationPageFull6}{\hidepaste}
+\tab{5}\spadcommand{rootSimp f1\free{f1 }}
+\indentrel{3}\begin{verbatim}
+                ÚÄÄÄÄÄ¿
+   (6)  (x + 1)\³x + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPageEmpty6}
+\begin{paste}{FunctionSimplificationPageEmpty6}{FunctionSimplificationPagePatch6}
+\pastebutton{FunctionSimplificationPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{rootSimp f1\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPagePatch7}
+\begin{paste}{FunctionSimplificationPageFull7}{FunctionSimplificationPageEmpty7}
+\pastebutton{FunctionSimplificationPageFull7}{\hidepaste}
+\tab{5}\spadcommand{g1 := sin(x + cos x)\bound{g1 }}
+\indentrel{3}\begin{verbatim}
+   (7)  sin(cos(x) + x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPageEmpty7}
+\begin{paste}{FunctionSimplificationPageEmpty7}{FunctionSimplificationPagePatch7}
+\pastebutton{FunctionSimplificationPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{g1 := sin(x + cos x)\bound{g1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPagePatch8}
+\begin{paste}{FunctionSimplificationPageFull8}{FunctionSimplificationPageEmpty8}
+\pastebutton{FunctionSimplificationPageFull8}{\hidepaste}
+\tab{5}\spadcommand{g2 := complexElementary g1\bound{g2 }\free{g1 }}
+\indentrel{3}\begin{verbatim}
+   (8)
+       -
+             ÚÄÄÄ¿
+            \³- 1
+         *
+                %e
+             **
+                               ÚÄÄÄ¿ 2               ÚÄÄÄ¿
+                     ÚÄÄÄ¿   x\³- 1         ÚÄÄÄ¿  x\³- 1
+                    \³- 1 (%e       )  + 2x\³- 1 %e
+                  + 
+                     ÚÄÄÄ¿
+                    \³- 1
+               /
+                       ÚÄÄÄ¿
+                     x\³- 1
+                  2%e
+           **
+              2
+     + 
+        ÚÄÄÄ¿
+       \³- 1
+  /
+                   ÚÄÄÄ¿ 2               ÚÄÄÄ¿
+         ÚÄÄÄ¿   x\³- 1         ÚÄÄÄ¿  x\³- 1     ÚÄÄÄ¿
+        \³- 1 (%e       )  + 2x\³- 1 %e        + \³- 1
+        ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                                ÚÄÄÄ¿
+                              x\³- 1
+                           2%e
+     2%e
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPageEmpty8}
+\begin{paste}{FunctionSimplificationPageEmpty8}{FunctionSimplificationPagePatch8}
+\pastebutton{FunctionSimplificationPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{g2 := complexElementary g1\bound{g2 }\free{g1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPagePatch9}
+\begin{paste}{FunctionSimplificationPageFull9}{FunctionSimplificationPageEmpty9}
+\pastebutton{FunctionSimplificationPageFull9}{\hidepaste}
+\tab{5}\spadcommand{trigs g2\free{g2 }}
+\indentrel{3}\begin{verbatim}
+   (9)  sin(cos(x) + x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPageEmpty9}
+\begin{paste}{FunctionSimplificationPageEmpty9}{FunctionSimplificationPagePatch9}
+\pastebutton{FunctionSimplificationPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{trigs g2\free{g2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPagePatch10}
+\begin{paste}{FunctionSimplificationPageFull10}{FunctionSimplificationPageEmpty10}
+\pastebutton{FunctionSimplificationPageFull10}{\hidepaste}
+\tab{5}\spadcommand{h1 := sinh(x + cosh x)\bound{h1 }}
+\indentrel{3}\begin{verbatim}
+   (10)  sinh(cosh(x) + x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPageEmpty10}
+\begin{paste}{FunctionSimplificationPageEmpty10}{FunctionSimplificationPagePatch10}
+\pastebutton{FunctionSimplificationPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{h1 := sinh(x + cosh x)\bound{h1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPagePatch11}
+\begin{paste}{FunctionSimplificationPageFull11}{FunctionSimplificationPageEmpty11}
+\pastebutton{FunctionSimplificationPageFull11}{\hidepaste}
+\tab{5}\spadcommand{h2 := realElementary h1\bound{h2 }\free{h1 }}
+\indentrel{3}\begin{verbatim}
+               x 2        x     2
+            (%e )  + 2x %e  + 1
+            ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                       x
+                    2%e
+         (%e                   )  - 1
+   (11)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                  x 2        x
+               (%e )  + 2x %e  + 1
+               ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                          x
+                       2%e
+            2%e
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPageEmpty11}
+\begin{paste}{FunctionSimplificationPageEmpty11}{FunctionSimplificationPagePatch11}
+\pastebutton{FunctionSimplificationPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{h2 := realElementary h1\bound{h2 }\free{h1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPagePatch12}
+\begin{paste}{FunctionSimplificationPageFull12}{FunctionSimplificationPageEmpty12}
+\pastebutton{FunctionSimplificationPageFull12}{\hidepaste}
+\tab{5}\spadcommand{htrigs h2\free{h2 }}
+\indentrel{3}\begin{verbatim}
+   (12)  sinh(cosh(x) + x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FunctionSimplificationPageEmpty12}
+\begin{paste}{FunctionSimplificationPageEmpty12}{FunctionSimplificationPagePatch12}
+\pastebutton{FunctionSimplificationPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{htrigs h2\free{h2 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/gloss.ht b/src/hyper/pages/gloss.ht
new file mode 100644
index 00000000..1af01735
--- /dev/null
+++ b/src/hyper/pages/gloss.ht
@@ -0,0 +1,413 @@
+\begin{page}{GlossaryPage}{G l o s s a r y}\beginscroll\beginmenu\item\newline{\em \menuitemstyle{}}{\em !}\space{}
+{\em (syntax)} Suffix character for \spadgloss{destructive operations}.
+\item\newline{\em \menuitemstyle{}}{\em ,}\space{}
+{\em (syntax)} a separator for items in a \spadgloss{tuple},{} \spadignore{e.g.} to separate arguments of a function \spad{f(x,{}y)}.
+\item\newline{\em \menuitemstyle{}}{\em =>}\space{}
+{\em (syntax)} the expression \spad{a => b} is equivalent to \spad{if a then} \spadgloss{exit} \spad{b}.
+\item\newline{\em \menuitemstyle{}}{\em ?}\space{}
+1. {\em (syntax)} a suffix character for Boolean-valued \spadfun{function} names,{} \spadignore{e.g.} \spadfun{odd?}. 2. Suffix character for pattern variables. 3. The special type \spad{?} means {\em don't care}. For example,{} the declaration \newline\center{\spad{x : Polynomial ?}}\newline,{} means that values assigned to \spad{x} must be polynomials over an arbitrary \spadgloss{underlying domain}.
+\item\newline{\em \menuitemstyle{}}{\em abstract datatype}\space{}
+a programming language principle used in \Language{} where a datatype is defined in two parts: (1) a {\em public} part describing a set of \spadgloss{exports},{} principally operations that apply to objects of that type,{} and (2) a {\em private} part describing the implementation of the datatype usually in terms of a \spadgloss{representation} for objects of the type. Programs which create and otherwise manipulate objects of the type may only do so through its exports. The representation and other implementation information is specifically hidden.
+\item\newline{\em \menuitemstyle{}}{\em abstraction}\space{}
+described functionally or conceptually without regard to implementation
+\item\newline{\em \menuitemstyle{}}{\em accuracy}\space{}
+the degree of exactness of an approximation or measurement. In computer algebra systems,{} computations are typically carried out with complete accuracy using integers or rational numbers of indefinite size. Domain \spadtype{Float} provides a function \spadfunFrom{precision}{Float} to change the precision for floating point computations. Computations using \spadtype{DoubleFloat} have a fixed precision but uncertain accuracy.
+\item\newline{\em \menuitemstyle{}}{\em add-chain}\space{}
+a hierarchy formed by \spadgloss{domain extensions}. If domain \spad{A} extends domain \spad{B} and domain \spad{B} extends domain \spad{C},{} then \spad{A} has {\em add-chain} \spad{B}-\spad{C}.
+\item\newline{\em \menuitemstyle{}}{\em aggregate}\space{}
+a data structure designed to hold multiple values. Examples of aggregates are \spadtype{List},{} \spadtype{Set},{} \spadtype{Matrix} and \spadtype{Bits}.
+\item\newline{\em \menuitemstyle{}}{\em AKCL}\space{}
+Austin Kyoto Common LISP,{} a version of \spadgloss{\spad{KCL}} produced by William Schelter,{} Austin,{} Texas.
+\item\newline{\em \menuitemstyle{}}{\em algorithm}\space{}
+a step-by-step procedure for a solution of a problem; a program
+\item\newline{\em \menuitemstyle{}}{\em ancestor}\space{}
+(of a domain) a category which is a \spadgloss{parent} of the domain,{} or a \spadgloss{parent} of a \spadgloss{parent},{} and so on.
+\item\newline{\em \menuitemstyle{}}{\em application}\space{}
+{\em (syntax)} an expression denoting "application" of a function to a set of \spadgloss{argument} parameters. Applications are written as a \spadgloss{parameterized form}. For example,{} the form \spad{f(x,{}y)} indicates the "application of the function \spad{f} to the tuple of arguments \spad{x} and \spad{y}". See also \spadgloss{evaluation} and \spadgloss{invocation}.
+\item\newline{\em \menuitemstyle{}}{\em apply}\space{}
+See \spadgloss{application}.
+\item\newline{\em \menuitemstyle{}}{\em argument}\space{}
+1. (actual argument) a value passed to a function at the time of a \spadglossSee{function call}{application}; also called an {\em actual parameter}. 2. (formal argument) a variable used in the definition of a function to denote the actual argument passed when the function is called.
+\item\newline{\em \menuitemstyle{}}{\em arity}\space{}
+1. (function) the number of arguments. 2. (operator or operation) corresponds to the arity of a function implementing the operator or operation.
+\item\newline{\em \menuitemstyle{}}{\em assignment}\space{}
+{\em (syntax)} an expression of the form \spad{x := e},{} meaning "assign the value of \spad{e} to \spad{x"}. After \spadgloss{evaluation},{} the \spadgloss{variable} \spad{x} \spadglossSee{points}{pointer} to an object obtained by evaluating the expression \spad{e}. If \spad{x} has a \spadgloss{type} as a result of a previous \spadgloss{declaration},{} the object assigned to \spad{x} must have that type. An interpreter must often \spadglossSee{coerce}{coercion} the value of \spad{e} to make that happen. For example,{} in the interpreter,{} \center{\spad{x : Float := 11}} first \spadglossSee{declares}{declaration} \spad{x} to be a float. This declaration causes the interpreter to coerce 11 to 11.0 in order to assign a floating point value to \spad{x}.
+\item\newline{\em \menuitemstyle{}}{\em attribute}\space{}
+a name or functional form denoting {\em any} useful computational property. For example,{} \spadatt{commutative(\spad{"*"})} asserts that "\spadfun{*} is commutative". Also,{} \spadatt{finiteAggregate} is used to assert that an aggregate has a finite number of immediate components.
+\item\newline{\em \menuitemstyle{}}{\em basis}\space{}
+{\em (algebra)} \spad{S} is a basis of a module \spad{M} over a \spadgloss{ring} if \spad{S} generates \spad{M},{} and \spad{S} is linearly independent
+\item\newline{\em \menuitemstyle{}}{\em benefactor}\space{}
+(of a given domain) a domain or package that the given domain explicitly references (for example,{} calls functions from) in its implementation
+\item\newline{\em \menuitemstyle{}}{\em binary}\space{}
+operation or function with \spadgloss{arity} 2
+\item\newline{\em \menuitemstyle{}}{\em binding}\space{}
+the association of a variable with properties such as \spadgloss{value} and \spadgloss{type}. The top-level \spadgloss{environment} in the interpreter consists of bindings for all user variables and functions. Every \spadgloss{function} has an associated set of bindings,{} one for each formal \spadgloss{argument} and \spadgloss{local variable}.
+\item\newline{\em \menuitemstyle{}}{\em block}\space{}
+{\em (syntax)} a control structure where expressions are sequentially \spadglossSee{evaluated}{evaluation}.
+\item\newline{\em \menuitemstyle{}}{\em body}\space{}
+a \spadgloss{function body} or \spadgloss{loop body}.
+\item\newline{\em \menuitemstyle{}}{\em boolean}\space{}
+objects denoted by the \spadgloss{literals} \spad{true} and \spad{false}; elements of domain \spadtype{Boolean}. See also \spadtype{Bits}.
+\item\newline{\em \menuitemstyle{}}{\em built-in function}\space{}
+a \spadgloss{function} in the standard \Language{} library. Contrast \spadgloss{user function}.
+\item\newline{\em \menuitemstyle{}}{\em cache}\space{}
+1. (noun) a mechanism for immediate retrieval of previously computed data. For example,{} a function which does a lengthy computation might store its values in a \spadgloss{hash table} using argument as a key. The hash table then serves as a cache for the function (see also \spadsyscom{)set function cache}). Also,{} when \spadgloss{recurrence relations} which depend upon \spad{n} previous values are compiled,{} the previous \spad{n} values are normally cached (use \spadsyscom{)set functions recurrence} to change this). 2. (verb) to save values in a cache.
+\item\newline{\em \menuitemstyle{}}{\em capsule}\space{}
+the part of the \spadglossSee{body}{function body} of a \spadgloss{domain constructor} that defines the functions implemented by the constructor.
+\item\newline{\em \menuitemstyle{}}{\em case}\space{}
+{\em (syntax)} an operator used to conditionally evaluate code based on the branch of a \spadgloss{Union}. For example,{} if value \spad{u} is \spad{Union(Integer,{}"failed")},{} the conditional expression \spad{if u case Integer then A else B} evaluate \spad{A} if \spad{u} is an integer and \spad{B} otherwise.
+\item\newline{\em \menuitemstyle{}}{\em Category}\space{}
+the distinguished object denoting the type of a category; the class of all categories.
+\item\newline{\em \menuitemstyle{}}{\em category}\space{}
+{\em (basic concept)} second-order types which serve to define useful "classification worlds" for domains,{} such as algebraic constructs (\spadignore{e.g.} groups,{} rings,{} fields),{} and data structures (\spadignore{e.g.} homogeneous aggregates,{} collections,{} dictionaries). Examples of categories are \spadtype{Ring} ("the class of all rings") and \spadtype{Aggregate} ("the class of all aggregates"). The categories of a given world are arranged in a hierarchy (formally,{} a directed acyclic graph). Each category inherits the properties of all its ancestors. Thus,{} for example,{} the category of ordered rings (\spadtype{OrderedRing}) inherits the properties of the category of rings (\spadtype{Ring}) and those of the ordered sets (\spadtype{OrderedSet}). Categories provide a database of algebraic knowledge and ensure mathematical correctness,{} \spadignore{e.g.} that "matrices of polynomials" is correct but "polynomials of hash tables" is not,{} that the multiply operation for "polynomials of continued fractions" is commutative,{} but that for "matrices of power series" is not. optionally provide "default definitions" for operations they export. Categories are defined in \Language{} by functions called \spadgloss{category constructors}. Technically,{} a category designates a class of domains with common \spadgloss{operations} and \spadgloss{attributes} but usually with different \spadgloss{functions} and \spadgloss{representations} for its constituent \spadgloss{objects}. Categories are always defined using the \Language{} library language (see also \spadgloss{category extension}). See also file {\em catdef.spad} for definitions of basic algebraic categories in \Language{}.
+\item\newline{\em \menuitemstyle{}}{\em category constructor}\space{}
+a function that creates categories,{} described by an abstract datatype in the \Language{} programming language. For example,{} the category constructor \spadtype{Module} is a function which takes a domain parameter \spad{R} and creates the category "modules over \spad{R}".
+\item\newline{\em \menuitemstyle{}}{\em category extension}\space{}
+created by a category definition,{} an expression usually of the form \spad{A == B with ...}. In English,{} this means "category A is a \spad{B} with the new operations and attributes as given by ... . See,{} for example,{} file {\em catdef.spad} for a definitions of the algebra categories in \Language{},{} {\em aggcat.spad} for data structure categories.
+\item\newline{\em \menuitemstyle{}}{\em category hierarchy}\space{}
+hierarchy formed by category extensions. The root category is \spadtype{Object}. A category can be defined as a \spadgloss{Join} of two or more categories so as to have multiple \spadgloss{parents}. Categories may also have parameterized so as to allow conditional inheritance.
+\item\newline{\em \menuitemstyle{}}{\em character}\space{}
+1. an element of a character set,{} as represented by a keyboard key. 2. a component of a string. For example,{} the 0th element of the string \spad{"hello there"} is the character {\em h}.
+\item\newline{\em \menuitemstyle{}}{\em client}\space{}
+(of a given domain) any domain or package that explicitly calls functions from the given domain
+\item\newline{\em \menuitemstyle{}}{\em coercion}\space{}
+an automatic transformation of an object of one \spadgloss{type} to an object of a similar or desired target type. In the interpreter,{} coercions and \spadgloss{retractions} are done automatically by the interpreter when a type mismatch occurs. Compare \spadgloss{conversion}.
+\item\newline{\em \menuitemstyle{}}{\em comment}\space{}
+textual remarks imbedded in code. Comments are preceded by a double dash ({\em --}). For \Language{} library code,{} stylized comments for on-line documentation are preceded by a two plus signs ({\em ++}).
+\item\newline{\em \menuitemstyle{}}{\em Common LISP}\space{}
+A version of \spadgloss{LISP} adopted as an informal standard by major users and suppliers of LISP
+\item\newline{\em \menuitemstyle{}}{\em compile-time}\space{}
+the time when category or domain constructors are compiled. Contrast \spadgloss{run-time}.
+\item\newline{\em \menuitemstyle{}}{\em compiler}\space{}
+a program that generates low-level code from a higher-level source language. \Language{} has three compilers. A {\em graphics compiler} converts graphical formulas to a compiled subroutine so that points can be rapidly produced for graphics commands. An {\em interpreter compiler} optionally compiles \spadgloss{user functions} when first \spadglossSee{invoked}{invocation} (use \spadsyscom{)set functions compile} to turn this feature on). A {\em library compiler} compiles all constructors (not available in Release 1)
+\item\newline{\em \menuitemstyle{}}{\em computational object}\space{}
+In \Language{},{} domains are objects. This term is used to distinquish the objects which are members of domains rather than domains themselves.
+\item\newline{\em \menuitemstyle{}}{\em conditional}\space{}
+a \spadgloss{control structure} of the form \spad{if A then B else C}; The \spadgloss{evaluation} of \spad{A} produces \spad{true} or \spad{false}. If \spad{true},{} \spad{B} evaluates to produce a value; otherwise \spad{C} evaluates to produce a value. When the value is not used,{} \spad{else C} part can be omitted.
+\item\newline{\em \menuitemstyle{}}{\em constant}\space{}
+{\em (syntax)} a reserved word used in \spadgloss{signatures} in \Language{} programming language to signify that mark an operation always returns the same value. For example,{} the signature \spad{0: constant -> \$} in the source code of \spadtype{AbelianMonoid} tells the \Language{} compiler that \spad{0} is a constant so that suitable optimizations might be performed.
+\item\newline{\em \menuitemstyle{}}{\em constructor}\space{}
+a \spadgloss{function} which creates a \spadgloss{category},{} \spadgloss{domain},{} or \spadgloss{package}.
+\item\newline{\em \menuitemstyle{}}{\em continuation}\space{}
+when a line of a program is so long that it must be broken into several lines,{} then all but the first line are called {\em continuation lines}. If such a line is given interactively,{} then each incomplete line must end with an underscore.
+\item\newline{\em \menuitemstyle{}}{\em control structure}\space{}
+program structures which can specify a departure from normal sequential execution. \Language{} has four kinds of control structures: \spadgloss{blocks},{} \spadgloss{case} statements,{} \spadgloss{conditionals},{} and \spadgloss{loops}.
+\item\newline{\em \menuitemstyle{}}{\em conversion}\space{}
+the transformation of an object on one \spadgloss{type} to one of another type. Conversions performed automatically are called \spadgloss{coercions}. These happen when the interpreter has a type mismatch and a similar or declared target type is needed. In general,{} the user must use the infix operation \spadSyntax{\spad{::}} to cause this transformation.
+\item\newline{\em \menuitemstyle{}}{\em copying semantics}\space{}
+the programming language semantics used in Pascal but {\em not} in \Language{}. See also \spadgloss{pointer semantics} for details.
+\item\newline{\em \menuitemstyle{}}{\em data structure}\space{}
+a structure for storing data in the computer. Examples are \spadgloss{lists} and \spadgloss{hash tables}.
+\item\newline{\em \menuitemstyle{}}{\em datatype}\space{}
+equivalent to \spadgloss{domain} in \Language{}.
+\item\newline{\em \menuitemstyle{}}{\em declaration}\space{}
+{\em (syntax)} an expression of the form \spad{x : T} where \spad{T} is some \spad{type}. A declaration forces all values \spadglossSee{assigned}{assignment} to \spad{T} to be of that type. If a value is of a different type,{} the interpreter will try to \spadglossSee{coerce}{coercion} the value to type \spad{T}. Declarations are necessary in case of ambiguity or when a user wants to introduce an an \spadglossSee{unexposed}{expose} domain.
+\item\newline{\em \menuitemstyle{}}{\em default definition}\space{}
+a function defined by a \spadgloss{category}. Such definitions appear category definitions of the form \spad{C: Category == T add I} in an optional implmentation part \spad{I} to the right of the keyword \spad{add}.
+\item\newline{\em \menuitemstyle{}}{\em default package}\space{}
+a optional \spadgloss{package} of \spadgloss{functions} associated with a category. Such functions are necessarily defined in terms over other functions exported by the category.
+\item\newline{\em \menuitemstyle{}}{\em definition}\space{}
+{\em (syntax)} 1. An expression of the form \spad{f(a) == b} defining function \spad{f} with \spadgloss{formal arguments} \spad{a} and \spadgloss{body} \spad{b}; equivalent to the statement \spad{f == (a) +-> b}. 2. An expression of the form \spad{a == b} where \spad{a} is a \spadgloss{symbol},{} equivalent to \spad{a() == b}. See also \spadgloss{macro} where a similar substitution is done at \spadgloss{parse} time.
+\item\newline{\em \menuitemstyle{}}{\em delimiter}\space{}
+a \spadgloss{character} which marks the beginning or end of some syntactically correct unit in the language,{} \spadignore{e.g.} \spadSyntax{"} for strings,{} blanks for identifiers.
+\item\newline{\em \menuitemstyle{}}{\em destructive operation}\space{}
+An operation which changes a component or structure of a value. In \Language{},{} all destructive operations have names which end with an exclamation mark ({\em !}). For example,{} domain \spadtype{List} has two operations to reverse the elements of a list,{} one named \spadfunFrom{reverse}{List} which returns a copy of the original list with the elements reversed,{} another named \spadfunFrom{reverse!}{List} which reverses the elements {\em in place} thus destructively changing the original list.
+\item\newline{\em \menuitemstyle{}}{\em documentation}\space{}
+1. on-line or hard copy descriptions of \Language{}; 2. text in library code preceded by {\em ++} comments as opposed to general comments preceded by {\em --}.
+\item\newline{\em \menuitemstyle{}}{\em domain}\space{}
+{\em (basic concept)} a domain corresponds to the usual notion of abstract datatypes: that of a set of values and a set of "exported operations" for the creation and manipulation of these values. Datatypes are parameterized,{} dynamically constructed,{} and can combine with others in any meaningful way,{} \spadignore{e.g.} "lists of floats" (\spadtype{List Float}),{} "fractions of polynomials with integer coefficients" (\spadtype{Fraction Polynomial Integer}),{} "matrices of infinite \spadgloss{streams} of cardinal numbers" (\spadtype{Matrix Stream CardinalNumber}). The term {\em domain} is actually abbreviates {\em domain of computation}. Technically,{} a domain denotes a class of objects,{} a class of \spadgloss{operations} for creating and other manipulating these objects,{} and a class of \spadgloss{attributes} describing computationally useful properties. Domains also provide \spadgloss{functions} for each operation often in terms of some \spadgloss{representation} for the objects. A domain itself is an \spadgloss{object} created by a \spadgloss{function} called a \spadgloss{domain constructor}.
+\item\newline{\em \menuitemstyle{}}{\em domain constructor}\space{}
+a function that creates domains,{} described by an abstract datatype in the \Language{} programming language. Simple domains like \spadtype{Integer} and \spadtype{Boolean} are created by domain constructors with no arguments. Most domain constructors take one or more parameters,{} one usually denoting an \spadgloss{underlying domain}. For example,{} the domain \spadtype{Matrix(R)} denotes "matrices over \spad{R"}. Domains {\em Mapping},{} {\em Record},{} and {\em Union} are primitive domains. All other domains are written in the \Language{} programming language and can be modified by users with access to the library source code.
+\item\newline{\em \menuitemstyle{}}{\em domain extension}\space{}
+a domain constructor \spad{A} is said to {\em extend} a domain constructor \spad{B} if \spad{A}\spad{'s} definition has the form \spad{A == B add ...}. This intuitively means "functions not defined by \spad{A} are assumed to come from \spad{B}". Successive domain extensions form \spadgloss{add-chains} affecting the the \spadglossSee{search order}{lineage} for functions not implemented directly by the domain during \spadgloss{dynamic lookup}.
+\item\newline{\em \menuitemstyle{}}{\em dot notation}\space{}
+using an infix dot ({\em .}) for function application. If \spad{u} is the list \spad{[7,{}4,{}-11]} then both \spad{u(2)} and \spad{u.2} return 4. Dot notation nests to left. Thus \spad{f . g . h} is equivalent to \spad{(f . g) . h}.
+\item\newline{\em \menuitemstyle{}}{\em dynamic}\space{}
+that which is done at \spadgloss{run-time} as opposed to \spadgloss{compile-time}. For example,{} the interpreter will build the domain "matrices over integers" dynamically in response to user input. However,{} the compilation of all functions for matrices and integers is done during \spadgloss{compile-time}. Constrast \spadgloss{static}.
+\item\newline{\em \menuitemstyle{}}{\em dynamic lookup}\space{}
+In \Language{},{} a \spadgloss{domain} may or may not explicitly provide \spadgloss{function} definitions for all of its exported \spadgloss{operations}. These definitions may instead come from domains in the \spadgloss{add-chain} or from \spadgloss{default packages}. When a \spadglossSee{function call}{application} is made for an operation in the domain,{} up to five steps are carried out.\begin{items} \item (1) If the domain itself implements a function for the operation,{} that function is returned. \item (2) Each of the domains in the \spadgloss{add-chain} are searched for one which implements the function; if found,{} the function is returned. \item (3) Each of the \spadgloss{default packages} for the domain are searched in order of the \spadgloss{lineage}. If any of the default packages implements the function,{} the first one found is returned. \item (4) Each of the \spadgloss{default packages} for each of the domains in the \spadgloss{add-chain} are searched in the order of their \spadgloss{lineage}. If any of the default packages implements the function,{} the first one found is returned. \item (5) If all of the above steps fail,{} an error message is reported. \end{items}
+\item\newline{\em \menuitemstyle{}}{\em empty}\space{}
+the unique value of objects with type \spadtype{Void}.
+\item\newline{\em \menuitemstyle{}}{\em environment}\space{}
+a set of \spadgloss{bindings}.
+\item\newline{\em \menuitemstyle{}}{\em evaluation}\space{}
+a systematic process which transforms an \spadgloss{expression} into an object called the \spadgloss{value} of the expression. Evaluation may produce \spadgloss{side effects}.
+\item\newline{\em \menuitemstyle{}}{\em exit}\space{}
+{\em (reserved word)} an \spadgloss{operator} which forces an exit from the current block. For example,{} the \spadgloss{block} \spad{(a := 1; if i > 0 then exit a; a := 2)} will prematurely exit at the second statement with value 1 if the value of \spad{i} is greater than 0. See \spadgloss{\spad{=>}} for an alternate syntax.
+\item\newline{\em \menuitemstyle{}}{\em explicit export}\space{}
+1. (of a domain \spad{D}) any \spadgloss{attribute},{} \spadgloss{operation},{} or \spadgloss{category} explicitly mentioned in the \spadgloss{type} specification part \spad{T} for the domain constructor definition \spad{D: T == I} 2. (of a category \spad{C}) any \spadgloss{attribute},{} \spadgloss{operation},{} or \spadgloss{category} explicitly mentioned in the \spadgloss{type} specification part \spad{T} for the domain constructor definition \spad{C: \spadgloss{Category} == T}
+\item\newline{\em \menuitemstyle{}}{\em export}\space{}
+\spadgloss{explicit export} or \spadgloss{implicit export} of a domain or category
+\item\newline{\em \menuitemstyle{}}{\em expose}\space{}
+some constructors are {\em exposed},{} others {\em unexposed}. Exposed domains and packages are recognized by the interpreter. Use \spadsys{)set expose} to control change what is exposed. To see both exposed and unexposed constructors,{} use \Browse{} with give the system command \spadsyscom{)set hyperdoc browse exposure on}. Unexposed constructors will now appear prefixed by star(\spad{*}).
+\item\newline{\em \menuitemstyle{}}{\em expression}\space{}
+1. any syntactically correct program fragment. 2. an element of domain \spadtype{Expression}
+\item\newline{\em \menuitemstyle{}}{\em extend}\space{}
+see \spadgloss{category extension} or \spadgloss{domain extension}
+\item\newline{\em \menuitemstyle{}}{\em field}\space{}
+{\em (algebra)} a \spadgloss{domain} which is \spadgloss{ring} where every non-zero element is invertible and where \spad{xy=yx}; a member of category \spadtype{Field}. For a complete list of fields,{} click on {\em Domains} under {\em Cross Reference} for \spadtype{Field}.
+\item\newline{\em \menuitemstyle{}}{\em file}\space{}
+a program or collection of data stored on disk,{} tape or other medium.
+\item\newline{\em \menuitemstyle{}}{\em float}\space{}
+a floating-point number with user-specified precision; an element of domain \spadtype{Float}. Floats are \spadgloss{literals} which are written two ways: without an exponent (\spadignore{e.g.} \spad{3.1416}),{} or with an exponent (\spadignore{e.g.} \spad{3.12E-12}). Use function \spadgloss{precision} to change the precision of the mantissage (20 digits by default). See also \spadgloss{small float}.
+\item\newline{\em \menuitemstyle{}}{\em formal parameter}\space{}
+(of a function) an identifier \spadglossSee{bound}{binding} to the value of an actual \spadgloss{argument} on \spadgloss{invocation}. In the function definition \spad{f(x,{}y) == u},{} for example,{} \spad{x} and \spad{y} are the formal parameter.
+\item\newline{\em \menuitemstyle{}}{\em frame}\space{}
+the basic unit of an interactive session; each frame has its own \spadgloss{step number},{} \spadgloss{environment},{} and \spadgloss{history}. In one interactive session,{} users can can create and drop frames,{} and have several active frames simultaneously.
+\item\newline{\em \menuitemstyle{}}{\em free}\space{}
+{\em (syntax)} A keyword used in user-defined functions to declare that a variable is a \spadgloss{free variable} of that function. For example,{} the statement \spad{free x} declares the variable \spad{x} within the body of a function \spad{f} to be a free variable in \spad{f}. Without such a declaration,{} any variable \spad{x} which appears on the left hand side of an assignment is regarded as a \spadgloss{local variable} of that function. If the intention of the assignment is to give an value to a \spadgloss{global variable} \spad{x},{} the body of that function must contain the statement \spad{free x}.
+\item\newline{\em \menuitemstyle{}}{\em free variable}\space{}
+(of a function) a variable which appears in a body of a function but is not \spadglossSee{bound}{binding} by that function. See \spadgloss{local variable} by default.
+\item\newline{\em \menuitemstyle{}}{\em function}\space{}
+implementation of \spadgloss{operation}; it takes zero or more \spadgloss{argument} parameters and produces zero or more values. Functions are objects which can be passed as parameters to functions and can be returned as values of functions. Functions can also create other functions (see also \spadtype{InputForm}). See also \spadgloss{application} and \spadgloss{invocation}. The terms {\em operation} and {\em function} are distinct notions in \Language{}. An operation is an abstraction of a function,{} described by declaring a \spadgloss{signature}. A function is created by providing an implementation of that operation by some piece of \Language{} code. Consider the example of defining a user-function \spad{fact} to compute the \spadfun{factorial} of a nonnegative integer. The \Language{} statement \spad{fact: Integer -> Integer} describes the operation,{} whereas the statement \spad{fact(n) = reduce(*,{}[1..n])} defines the functions. See also \spadgloss{generic function}.
+\item\newline{\em \menuitemstyle{}}{\em function body}\space{}
+the part of a \spadgloss{function}\spad{'s} definition which is evaluated when the function is called at \spadgloss{run-time}; the part of the function definition to the right of the \spadSyntax{\spad{==}}.
+\item\newline{\em \menuitemstyle{}}{\em garbage collection}\space{}
+a system function that automatically recycles memory cells from the \spadgloss{heap}. \Language{} is built upon \spadgloss{Common LISP} which provides this facility.
+\item\newline{\em \menuitemstyle{}}{\em garbage collector}\space{}
+a mechanism for reclaiming storage in the \spadgloss{heap}.
+\item\newline{\em \menuitemstyle{}}{\em Gaussian}\space{}
+a complex-valued expression,{} \spadignore{e.g.} one with both a real and imaginary part; a member of a \spadtype{Complex} domain.
+\item\newline{\em \menuitemstyle{}}{\em generic function}\space{}
+the use of one function to operate on objects of different types; One might regard \Language{} as supporting generic \spadgloss{operations} but not generic functions. One operation \spad{+: (D,{}D) -> D} exists for adding elements in a ring; each ring however provides its own type-specific function for implementing this operation.
+\item\newline{\em \menuitemstyle{}}{\em global variable}\space{}
+A variable which can be referenced freely by functions. In \Language{},{} all top-level user-defined variables defined during an interactive user session are global variables. \Language{} does not allow {\em fluid variables},{} that is,{} variables \spadglossSee{bound}{binding} by functions which can be referenced by functions those functions call.
+\item\newline{\em \menuitemstyle{}}{\em Groebner basis}\space{}
+{\em (algebra)} a special basis for a polynomial ideal that allows a simple test for membership. It is useful in solving systems of polynomial equations.
+\item\newline{\em \menuitemstyle{}}{\em group}\space{}
+{\em (algebra)} a monoid where every element has a multiplicative inverse.
+\item\newline{\em \menuitemstyle{}}{\em hash table}\space{}
+A data structure that efficiency maps a given object to another. A hash table consists of a set of {\em entries},{} each of which associates a {\em key} with a {\em value}. Finding the object stored under a key can be very fast even if there are a large number of entries since keys are {\em hashed} into numerical codes for fast lookup.
+\item\newline{\em \menuitemstyle{}}{\em heap}\space{}
+an area of storage used by data in programs. For example,{} AXIOM will use the heap to hold the partial results of symbolic computations. When cancellations occur,{} these results remain in the heap until \spadglossSee{garbage collected}{garbage collector}.
+\item\newline{\em \menuitemstyle{}}{\em history}\space{}
+a mechanism which records the results for an interactive computation. Using the history facility,{} users can save computations,{} review previous steps of a computation,{} and restore a previous interactive session at some later time. For details,{} issue the system command {\em )history ?} to the interpreter. See also \spadgloss{frame}.
+\item\newline{\em \menuitemstyle{}}{\em ideal}\space{}
+{\em (algebra)} a subset of a ring that is closed under addition and multiplication by arbitrary ring elements,{} \spadignore{i.e.} it\spad{'s} a module over the ring.
+\item\newline{\em \menuitemstyle{}}{\em identifier}\space{}
+{\em (syntax)} an \Language{} name; a \spadgloss{literal} of type \spadtype{Symbol}. An identifier begins with an alphabetical character or \% and may be followed by alphabetic characters,{} digits,{} ? or !. Certain distinquished \spadgloss{reserved words} are not allowed as identifiers but have special meaning in the \Language{}.
+\item\newline{\em \menuitemstyle{}}{\em immutable}\space{}
+an object is immutable if it cannot be changed by an \spadgloss{operation}; not a \spadglossSee{mutable object}{mutable}. Algebraic objects generally immutable: changing an algebraic expression involves copying parts of the original object. One exception is a matrix object of type \spadtype{Matrix}. Examples of mutable objects are data structures such as those of type \spadtype{List}. See also \spadgloss{pointer semantics}.
+\item\newline{\em \menuitemstyle{}}{\em implicit export}\space{}
+(of a domain or category) any \spadgloss{attribute} or \spadgloss{operation} which is either an explicit export or else an explicit export of some category which an explicit category export \spadglossSee{extends}{category extension}.
+\item\newline{\em \menuitemstyle{}}{\em index}\space{}
+1. a variable that counts the number of times a \spadgloss{loop} is repeated. 2. the "address" of an element in a data structure (see also category \spadtype{LinearAggregate}).
+\item\newline{\em \menuitemstyle{}}{\em infix}\space{}
+{\em (syntax)} an \spadgloss{operator} placed between two \spadgloss{operands}; also called a {\em binary operator},{} \spadignore{e.g.} \spad{a + b}. An infix operator may also be used as a \spadgloss{prefix},{} \spadignore{e.g.} \spad{+(a,{}b)} is also permissable in the \Language{} language. Infix operators have a relative \spadgloss{precedence}.
+\item\newline{\em \menuitemstyle{}}{\em input area}\space{}
+a rectangular area on a \HyperName{} screen into which users can enter text.
+\item\newline{\em \menuitemstyle{}}{\em instantiate}\space{}
+to build a \spadgloss{category},{} \spadgloss{domain},{} or \spadgloss{package} at run-time
+\item\newline{\em \menuitemstyle{}}{\em integer}\space{}
+a \spadgloss{literal} object of domain \spadtype{Integer},{} the class of integers with an unbounded number of digits. Integer literals consist of one or more consecutive digits (0-9) with no embedded blanks. Underscores can be used to separate digits in long integers if desirable.
+\item\newline{\em \menuitemstyle{}}{\em interactive}\space{}
+a system where the user interacts with the computer step-by-step
+\item\newline{\em \menuitemstyle{}}{\em interpreter}\space{}
+the subsysystem of \Language{} responsible for handling user input during an interactive session. The following somewhat simplified description of the typical action of the interpreter. The interpreter parsers the user\spad{'s} input expression to create an expression tree then does a bottom-up traversal of the tree. Each subtree encountered which is not a value consists of a root node denoting an operation name and one or more leaf nodes denoting \spadgloss{operands}. The interpreter resolves type mismatches and uses type-inferencing and a library database to determine appropriate types of the operands and the result,{} and an operation to be performed. The interpreter then builds a domain to perform the indicated operation,{} then invokes a function from the domain to compute a value. The subtree is then replaced by that value and the process continues. Once the entire tree has been processed,{} the value replacing the top node of the tree is displayed back to the user as the value of the expression.
+\item\newline{\em \menuitemstyle{}}{\em invocation}\space{}
+(of a function) the run-time process involved in \spadglossSee{evaluating}{evaluation} a a \spadgloss{function} \spadgloss{application}. This process has two steps. First,{} a local \spadgloss{environment} is created where \spadgloss{formal arguments} are locally \spadglossSee{bound}{binding} by \spadgloss{assignment} to their respective actual \spadgloss{argument}. Second,{} the \spadgloss{function body} is evaluated in that local environment. The evaluation of a function is terminated either by completely evaluating the function body or by the evaluation of a \spadSyntax{return} expression.
+\item\newline{\em \menuitemstyle{}}{\em iteration}\space{}
+repeated evaluation of an expression or a sequence of expressions. Iterations use the reserved words \spadSyntax{for},{} \spadSyntax{while},{} and \spadSyntax{repeat}.
+\item\newline{\em \menuitemstyle{}}{\em Join}\space{}
+a primitive \Language{} function taking two or more categories as arguments and producing a category containing all of the operations and attributes from the respective categories.
+\item\newline{\em \menuitemstyle{}}{\em KCL}\space{}
+Kyoto Common LISP,{} a version of \spadgloss{Common LISP} which features compilation of the compilation of LISP into the \spad{C} Programming Language
+\item\newline{\em \menuitemstyle{}}{\em library}\space{}
+In \Language{},{} a coolection of compiled modules respresenting the a \spadgloss{category} or \spadgloss{domain} constructor.
+\item\newline{\em \menuitemstyle{}}{\em lineage}\space{}
+the sequence of \spadgloss{default packages} for a given domain to be searched during \spadgloss{dynamic lookup}. This sequence is computed first by ordering the category \spadgloss{ancestors} of the domain according to their {\em level number},{} an integer equal to to the minimum distance of the domain from the category. Parents have level 1,{} parents of parents have level 2,{} and so on. Among categories with equal level numbers,{} ones which appear in the left-most branches of {\em Join}\spad{s} in the source code come first. See also \spadgloss{dynamic lookup}.
+\item\newline{\em \menuitemstyle{}}{\em LISP}\space{}
+acronymn for List Processing Language,{} a language designed for the manipulation of nonnumerical data. The \Language{} library is translated into LISP then compiled into machine code by an underlying LISP.
+\item\newline{\em \menuitemstyle{}}{\em list}\space{}
+an object of a \spadtype{List} domain.
+\item\newline{\em \menuitemstyle{}}{\em literal}\space{}
+an object with a special syntax in the language. In \Language{},{} there are five types of literals: \spadgloss{booleans},{} \spadgloss{integers},{} \spadgloss{floats},{} \spadgloss{strings},{} and \spadgloss{symbols}.
+\item\newline{\em \menuitemstyle{}}{\em local}\space{}
+{\em (syntax)} A keyword used in user-defined functions to declare that a variable is a \spadgloss{local variable} of that function. Because of default assumptions on variables,{} such a declaration is not necessary but is available to the user for clarity when appropriate.
+\item\newline{\em \menuitemstyle{}}{\em local variable}\space{}
+(of a function) a variable \spadglossSee{bound}{binding} by that function and such that its binding is invisible to any function that function calls. Also called a {\em lexical} variable. By default in the interpreter:\begin{items} \item 1. any variable \spad{x} which appears on the left hand side of an assignment is regarded a local variable of that function. If the intention of an assignment is to change the value of a \spadgloss{global variable} \spad{x},{} the body of the function must then contain the statement \spad{free x}. \item 2. any other variable is regarded as a \spadgloss{free variable}. \end{items} An optional declaration \spad{local x} is available to explicitly declare a variable to be a local variable. All \spadgloss{formal parameters} to the function can be regarded as local variables to the function.
+\item\newline{\em \menuitemstyle{}}{\em loop}\space{}
+1. an expression containing a \spadSyntax{repeat} 2. a collection expression having a \spadSyntax{for} or a \spadSyntax{while},{} \spadignore{e.g.} \spad{[f(i) for i in S]}.
+\item\newline{\em \menuitemstyle{}}{\em loop body}\space{}
+the part of a loop following the \spadSyntax{repeat} that tells what to do each iteration. For example,{} the body of the loop \spad{for x in S repeat B} is \spad{B}. For a collection expression,{} the body of the loop precedes the initial \spadSyntax{for} or \spadSyntax{while}.
+\item\newline{\em \menuitemstyle{}}{\em macro}\space{}
+{\em (syntax)} 1. An expression of the form \spad{macro a == b} where \spad{a} is a \spadgloss{symbol} causes \spad{a} to be textually replaced by the expression \spad{b} at \spadgloss{parse} time. 2. An expression of the form \spad{macro f(a) == b} defines a parameterized macro expansion for a parameterized form \spad{f} This macro causes a form \spad{f}(\spad{x}) to be textually replaced by the expression \spad{c} at parse time,{} where \spad{c} is the expression obtained by replacing \spad{a} by \spad{x} everywhere in \spad{b}. See also \spadgloss{definition} where a similar substitution is done during \spadgloss{evaluation}.
+\item\newline{\em \menuitemstyle{}}{\em mode}\space{}
+a type expression containing a question-mark ({\em ?}). For example,{} the mode {\em P ?} designates {\em the class of all polynomials over an arbitrary ring}.
+\item\newline{\em \menuitemstyle{}}{\em mutable}\space{}
+objects which contain \spadgloss{pointers} to other objects and which have operations defined on them which alter these pointers. Contrast \spadgloss{immutable}. \Language{} uses \spadgloss{pointer semantics} as does \spadgloss{LISP} in contrast with many other languages such as Pascal which use \spadgloss{copying semantics}. See \spadgloss{pointer semantics} for details.
+\item\newline{\em \menuitemstyle{}}{\em name}\space{}
+1. a \spadgloss{symbol} denoting a \spadgloss{variable},{} \spadignore{i.e.} the variable \spad{x}. 2. a \spadgloss{symbol} denoting an \spadgloss{operation},{} \spadignore{i.e.} the operation \spad{divide: (Integer,{}Integer) -> Integer}.
+\item\newline{\em \menuitemstyle{}}{\em nullary}\space{}
+a function with no arguments,{} \spadignore{e.g.} \spadfun{characteristic}.
+\item\newline{\em \menuitemstyle{}}{\em nullary}\space{}
+operation or function with \spadgloss{arity} 0
+\item\newline{\em \menuitemstyle{}}{\em Object}\space{}
+a category with no operations or attributes,{} from which most categories in \Language{} are \spadglossSee{extensions}{category extension}.
+\item\newline{\em \menuitemstyle{}}{\em object}\space{}
+a data entity created or manipulated by programs. Elements of domains,{} functions,{} and domains themselves are objects. Whereas categories are created by functions,{} they cannot be dynamically manipulated in the current system and are thus not considered as objects. The most basic objects are \spadgloss{literals}; all other objects must be created \spadgloss{functions}. Objects can refer to other objects using \spadgloss{pointers}. \Language{} language uses \spadgloss{pointer semantics} when dealing with \spadgloss{mutable} objects.
+\item\newline{\em \menuitemstyle{}}{\em object code}\space{}
+code which can be directly executed by hardware; also known as {\em machine language}.
+\item\newline{\em \menuitemstyle{}}{\em operand}\space{}
+an argument of an \spadgloss{operator} (regarding an operator as a \spadgloss{function}).
+\item\newline{\em \menuitemstyle{}}{\em operation}\space{}
+an abstraction of a \spadgloss{function},{} described by a \spadgloss{signature}. For example,{} \center{\spad{fact: NonNegativeInteger -> NonNegativeInteger}} describes an operation for "the factorial of a (non-negative) integer".
+\item\newline{\em \menuitemstyle{}}{\em operator}\space{}
+special reserved words in the language such as \spadop{+} and \spadop{*}; operators can be either \spadgloss{prefix} or \spadgloss{infix} and have a relative \spadgloss{precedence}.
+\item\newline{\em \menuitemstyle{}}{\em overloading}\space{}
+the use of the same name to denote distinct functions; a function is identified by a \spadgloss{signature} identifying its name,{} the number and types of its arguments,{} and its return types. If two functions can have identical signatures,{} a \spadgloss{package call} must be made to distinquish the two.
+\item\newline{\em \menuitemstyle{}}{\em package}\space{}
+a domain whose exported operations depend solely on the parameters and other explicit domains,{} \spadignore{e.g.} a package for solving systems of equations of polynomials over any field,{} \spadignore{e.g.} floats,{} rational numbers,{} complex rational functions,{} or power series. Facilities for integration,{} differential equations,{} solution of linear or polynomial equations,{} and group theory are provided by "packages". Technically,{} a package is a domain which has no \spadgloss{signature} containing the symbol \$. While domains intuitively provide computational objects you can compute with,{} packages intuitively provide functions (\spadgloss{polymorphic} functions) which will work over a variety of datatypes.
+\item\newline{\em \menuitemstyle{}}{\em package call}\space{}
+{\em (syntax)} an expression of the form \spad{e \$ D} where \spad{e} is an \spadgloss{application} and \spad{D} denotes some \spadgloss{package} (or \spadgloss{domain}).
+\item\newline{\em \menuitemstyle{}}{\em package call}\space{}
+{\em (syntax)} an expression of the form \spad{f(x,{}y)\$D} used to identify that the function \spad{f} is to be one from \spad{D}.
+\item\newline{\em \menuitemstyle{}}{\em package constructor}\space{}
+same as \spadgloss{domain constructor}.
+\item\newline{\em \menuitemstyle{}}{\em parameter}\space{}
+see \spadgloss{argument}
+\item\newline{\em \menuitemstyle{}}{\em parameterized datatype}\space{}
+a domain that is built on another,{} for example,{} polynomials with integer coefficients.
+\item\newline{\em \menuitemstyle{}}{\em parameterized form}\space{}
+a expression of the form \spad{f(x,{}y)},{} an \spadgloss{application} of a function.
+\item\newline{\em \menuitemstyle{}}{\em parent}\space{}
+(of a domain) a category which is explicitly declared in the source code definition for the domain to be an \spadgloss{export} of the domain.
+\item\newline{\em \menuitemstyle{}}{\em parse}\space{}
+1. (verb) to produce an internal representation of a user input string; the resultant internal representation is then "interpreted" by \Language{} to perform some indicated action.
+\item\newline{\em \menuitemstyle{}}{\em parse}\space{}
+the transformation of a user input string representing a valid \Language{} expression into an internal representation as a tree-structure.
+\item\newline{\em \menuitemstyle{}}{\em partially ordered set}\space{}
+a set with a reflexive,{} transitive and antisymetric \spadgloss{binary} operation.
+\item\newline{\em \menuitemstyle{}}{\em pattern match}\space{}
+1. (on expressions) Given a expression called a "subject" \spad{u},{} the attempt to rewrite \spad{u} using a set of "rewrite rules". Each rule has the form \spad{A == B} where \spad{A} indicates a expression called a "pattern" and \spad{B} denotes a "replacement". The meaning of this rule is "replace \spad{A} by \spad{B"}. If a given pattern \spad{A} matches a subexpression of \spad{u},{} that subexpression is replaced by \spad{B}. Once rewritten,{} pattern matching continues until no further changes occur. 2. (on strings) the attempt to match a string indicating a "pattern" to another string called a "subject",{} for example,{} for the purpose of identifying a list of names. In \Browse{},{} users may enter \spadgloss{search strings} for the purpose of identifying constructors,{} operations,{} and attributes.
+\item\newline{\em \menuitemstyle{}}{\em pile}\space{}
+alternate syntax for a block,{} using indentation and column alignment (see also \spadgloss{block}).
+\item\newline{\em \menuitemstyle{}}{\em pointer}\space{}
+a reference implemented by a link directed from one object to another in the computer memory. An object is said to {\em refer} to another if it has a pointer to that other object. Objects can also refer to themselves (cyclic references are legal). Also more than one object can refer to the same object. See also \spadgloss{pointer semantics}.
+\item\newline{\em \menuitemstyle{}}{\em pointer semantics}\space{}
+the programming language semantics used in languages such as LISP which allow objects to be \spadgloss{mutable}. Consider the following sequence of \Language{} statements:\begin{items} \item \spad{x : Vector Integer := [1,{}4,{}7]} \item \spad{y := x} \item \spad{swap!(x,{}2,{}3)} \end{items} The function \spadfunFrom{swap!}{Vector} is used to interchange the 2nd and 3rd value in the list \spad{x} producing the value \spad{[1,{}7,{}4]}. What value does \spad{y} have after evaluation of the third statement? The answer is different in \Language{} than it is in a language with \spadgloss{copying semantics}. In \Language{},{} first the vector [1,{}2,{}3] is created and the variable \spad{x} set to \spadglossSee{point}{pointer} to this object. Let\spad{'s} call this object \spad{V}. Now \spad{V} refers to its \spadgloss{immutable} components 1,{}2,{} and 3. Next,{} the variable \spad{y} is made to point to \spad{V} just as \spad{x} does. Now the third statement interchanges the last 2 elements of \spad{V} (the {\em !} at the end of the name \spadfunFrom{swap!}{Vector} tells you that this operation is destructive,{} that is,{} it changes the elements {\em in place}). Both \spad{x} and \spad{y} perceive this change to \spad{V}. Thus both \spad{x} and \spad{y} then have the value \spad{[1,{}7,{}4]}. In Pascal,{} the second statement causes a copy of \spad{V} to be stored under \spad{y}. Thus the change to \spad{V} made by the third statement does not affect \spad{y}.
+\item\newline{\em \menuitemstyle{}}{\em polymorphic}\space{}
+a \spadgloss{function} parameterized by one or more \spadgloss{domains}; a \spadgloss{algorithm} defined \spadglossSee{categorically}{category}. Every function defined in a domain or package constructor with a domain-valued parameter is polymorphic. For example,{} the same matrix \spadSyntax{*} function is used to multiply "matrices over integers" as "matrices over matrices over integers" Likewise,{} various \ignore{\spad{???}} in \ignore{\spad{???}} solves polynomial equations over any \spadgloss{field}.
+\item\newline{\em \menuitemstyle{}}{\em postfix}\space{}
+an \spadgloss{operator} that follows its single \spadgloss{operand}. Postfix operators are not available in \Language{}.
+\item\newline{\em \menuitemstyle{}}{\em precedence}\space{}
+{\em (syntax)} refers to the so-called {\em binding power} of an operator. For example,{} \spad{*} has higher binding power than \spad{+} so that the expression \spad{a + b * c} is equivalent to \spad{a + (b * c)}.
+\item\newline{\em \menuitemstyle{}}{\em precision}\space{}
+the number of digits in the specification of a number,{} \spadignore{e.g.} as set by \spadfunFrom{precision}{Float}.
+\item\newline{\em \menuitemstyle{}}{\em predicate}\space{}
+1. a Boolean valued function,{} \spadignore{e.g.} \spad{odd: Integer -> Boolean}. 2. an Boolean valued expression
+\item\newline{\em \menuitemstyle{}}{\em prefix}\space{}
+{\em (syntax)} an \spadgloss{operator} such as \spad{-} and \spad{not} that is written {\em before} its single \spadgloss{operand}. Every function of one argument can be used as a prefix operator. For example,{} all of the following have equivalent meaning in \Language{}: \spad{f(x)},{} \spad{f x},{} and \spad{f.x}. See also \spadgloss{dot notation}.
+\item\newline{\em \menuitemstyle{}}{\em quote}\space{}
+the prefix \spadgloss{operator} \spadSyntax{'} meaning {\em do not evaluate}.
+\item\newline{\em \menuitemstyle{}}{\em Record}\space{}
+(basic domain constructor) a domain constructor used to create a inhomogeneous aggregate composed of pairs of "selectors" and \spadgloss{values}. A Record domain is written in the form \spad{Record(a1:D1,{}...,{}an:Dn)} (\spad{n} > 0) where \spad{a1},{}...,{}\spad{an} are identifiers called the {\em selectors} of the record,{} and \spad{D1},{}...,{}\spad{Dn} are domains indicating the type of the component stored under selector \spad{an}.
+\item\newline{\em \menuitemstyle{}}{\em recurrence relation}\space{}
+A relation which can be expressed as a function \spad{f} with some argument \spad{n} which depends on the value of \spad{f} at \spad{k} previous values. In many cases,{} \Language{} will rewrite a recurrence relation on compilation so as to \spadgloss{cache} its previous \spad{k} values and therefore make the computation significantly more efficient.
+\item\newline{\em \menuitemstyle{}}{\em recursion}\space{}
+use of a self-reference within the body of a function. Indirect recursion is when a function uses a function below it in the call chain.
+\item\newline{\em \menuitemstyle{}}{\em recursive}\space{}
+1. A function that calls itself,{} either directly or indirectly through another function. 2. self-referential. See also \spadgloss{recursive}.
+\item\newline{\em \menuitemstyle{}}{\em reference}\space{}
+see \spadgloss{pointer}
+\item\newline{\em \menuitemstyle{}}{\em Rep}\space{}
+a special identifier used as \spadgloss{local variable} of a domain constructor body to denote the representation domain for objects of a domain.
+\item\newline{\em \menuitemstyle{}}{\em representation}\space{}
+a \spadgloss{domain} providing a data structure for elements of a domain; generally denoted by the special identifier \spadgloss{Rep} in the \Language{} programming language. As domains are \spadgloss{abstract datatypes},{} this representation is not available to users of the domain,{} only to functions defined in the \spadgloss{function body} for a domain constructor. Any domain can be used as a representation.
+\item\newline{\em \menuitemstyle{}}{\em reserved word}\space{}
+a special sequence of non-blank characters with special meaning in the \Language{} language. Examples of reserved words are names such as \spadSyntax{for},{} \spadSyntax{if},{} and \spadSyntax{free},{} operator names such as \spadSyntax{+} and \spad{mod},{} special character strings such as spadSyntax{\spad{==}} and spadSyntax{\spad{:=}}.
+\item\newline{\em \menuitemstyle{}}{\em retraction}\space{}
+to move an object in a parameterized domain back to the underlying domain,{} for example to move the object \spad{7} from a "fraction of integers" (domain \spadtype{Fraction Integer}) to "the integers" (domain \spadtype{Integer}).
+\item\newline{\em \menuitemstyle{}}{\em return}\space{}
+when leaving a function,{} the value of the expression following \spadSyntax{return} becomes the value of the function.
+\item\newline{\em \menuitemstyle{}}{\em ring}\space{}
+a set with a commutative addition,{} associative multiplication,{} a unit element,{} and multiplication distributes over addition and subtraction.
+\item\newline{\em \menuitemstyle{}}{\em rule}\space{}
+{\em (syntax)} 1. An expression of the form \spad{rule A == B} indicating a "rewrite rule". 2. An expression of the form \spad{rule (R1;...;Rn)} indicating a set of "rewrite rules" \spad{R1},{}...,{}\spad{Rn}. See \spadgloss{pattern matching} for details.
+\item\newline{\em \menuitemstyle{}}{\em run-time}\space{}
+the time of doing a computation. Contrast \spadgloss{compile-time}. rather than prior to it; \spadgloss{dynamic} as opposed to \spadgloss{static}. For example,{} the decision of the intepreter to build a structure such as "matrices with power series entries" in response to user input is made at run-time.
+\item\newline{\em \menuitemstyle{}}{\em run-time check}\space{}
+an error-checking which can be done only when the program receives user input; for example,{} confirming that a value is in the proper range for a computation.
+\item\newline{\em \menuitemstyle{}}{\em search string}\space{}
+a string entered into an \spadgloss{input area} on a \HyperName{} screen
+\item\newline{\em \menuitemstyle{}}{\em selector}\space{}
+an identifier used to address a component value of a gloss{Record} datatype.
+\item\newline{\em \menuitemstyle{}}{\em semantics}\space{}
+the relationships between symbols and their meanings. The rules for obtaining the {\em meaning} of any syntactically valid expression.
+\item\newline{\em \menuitemstyle{}}{\em semigroup}\space{}
+{\em (algebra)} a \spadgloss{monoid} which need not have an identity; it is closed and associative.
+\item\newline{\em \menuitemstyle{}}{\em side effect}\space{}
+action which changes a component or structure of a value. See \spadgloss{destructive operation} for details.
+\item\newline{\em \menuitemstyle{}}{\em signature}\space{}
+{\em (syntax)} an expression describing an \spadgloss{operation}. A signature has the form as \spad{name : source -> target},{} where \spad{source} gives the type of the arguments of the operation,{} and \spad{target} gives the type of the result.
+\item\newline{\em \menuitemstyle{}}{\em small float}\space{}
+the domain for hardware floating point arithmetic as provided by the computer hardware.
+\item\newline{\em \menuitemstyle{}}{\em small integer}\space{}
+the domain for hardware integer arithmetic. as provided by the computer hardware.
+\item\newline{\em \menuitemstyle{}}{\em source}\space{}
+the \spadgloss{type} of the argument of a \spadgloss{function}; the type expression before the \spad{->} in a \spadgloss{signature}. For example,{} the source of \spad{f : (Integer,{}Integer) -> Integer} is \spad{(Integer,{}Integer)}.
+\item\newline{\em \menuitemstyle{}}{\em sparse}\space{}
+data structure whose elements are mostly identical (a sparse matrix is one filled with mostly zeroes).
+\item\newline{\em \menuitemstyle{}}{\em static}\space{}
+that computation done before run-time,{} such as compilation. Contrast \spadgloss{dynamic}.
+\item\newline{\em \menuitemstyle{}}{\em step number}\space{}
+the number which precedes user input lines in an interactive session; the output of user results is also labeled by this number.
+\item\newline{\em \menuitemstyle{}}{\em stream}\space{}
+an object of \spadtype{Stream(R)},{} a generalization of a \spadgloss{list} to allow an infinite number of elements. Elements of a stream are computed "on demand". Strings are used to implement various forms of power series (\ignore{\spad{???}}).
+\item\newline{\em \menuitemstyle{}}{\em string}\space{}
+an object of domain \spadtype{String}. Strings are \spadgloss{literals} consisting of an arbitrary sequence of \spadgloss{characters} surrounded by double-quotes (\spadSyntax{"}),{} \spadignore{e.g.} \spad{"Look here!"}.
+\item\newline{\em \menuitemstyle{}}{\em subdomain}\space{}
+{\em (basic concept)} a \spadgloss{domain} together with a \spadgloss{predicate} characterizing which members of the domain belong to the subdomain. The exports of a subdomain are usually distinct from the domain itself. A fundamental assumption however is that values in the subdomain are automatically \spadglossSee{coerceable}{coercion} to values in the domain. For example,{} if \spad{n} and \spad{m} are declared to be members of a subdomain of the integers,{} then {\em any} \spadgloss{binary} operation from \spadtype{Integer} is available on \spad{n} and \spad{m}. On the other hand,{} if the result of that operation is to be assigned to,{} say,{} \spad{k},{} also declared to be of that subdomain,{} a \spadgloss{run-time} check is generally necessary to ensure that the result belongs to the subdomain.
+\item\newline{\em \menuitemstyle{}}{\em such that clause}\space{}
+the use of \spadSyntax{|} followed by an expression to filter an iteration.
+\item\newline{\em \menuitemstyle{}}{\em suffix}\space{}
+{\em (syntax)} an \spadgloss{operator} which placed after its operand. Suffix operators are not allowed in the \Language{} language.
+\item\newline{\em \menuitemstyle{}}{\em symbol}\space{}
+objects denoted by \spadgloss{identifier} \spadgloss{literals}; an element of domain \spadtype{Symbol}. The interpreter defaultly converts a symbol \spad{x} into \spadtype{Variable(x)}.
+\item\newline{\em \menuitemstyle{}}{\em syntax}\space{}
+rules of grammar,{} punctuation etc. for forming correct expressions.
+\item\newline{\em \menuitemstyle{}}{\em system commands}\space{}
+top-level \Language{} statements that begin with {\em )}. System commands allow users to query the database,{} read files,{} trace functions,{} and so on.
+\item\newline{\em \menuitemstyle{}}{\em tag}\space{}
+an identifier used to discriminate a branch of a \spadgloss{Union} type.
+\item\newline{\em \menuitemstyle{}}{\em target}\space{}
+the \spadgloss{type} of the result of a \spadgloss{function}; the type expression following the \spad{->} in a \spadgloss{signature}.
+\item\newline{\em \menuitemstyle{}}{\em top-level}\space{}
+refers to direct user interactions with the \Language{} interpreter.
+\item\newline{\em \menuitemstyle{}}{\em totally ordered set}\space{}
+{\em (algebra)} a partially ordered set where any two elements are comparable.
+\item\newline{\em \menuitemstyle{}}{\em trace}\space{}
+use of system function \spadsys{)trace} to track the arguments passed to a function and the values returned.
+\item\newline{\em \menuitemstyle{}}{\em tuple}\space{}
+an expression of two or more other expressions separated by commas,{} \spadignore{e.g.} \spad{4,{}7,{}11}. Tuples are also used for multiple arguments both for \spadgloss{applications} (\spadignore{e.g.} \spad{f(x,{}y)}) and in \spadgloss{signatures} (\spadignore{e.g.} \spad{(Integer,{}Integer) -> Integer}). A tuple is not a data structure,{} rather a syntax mechanism for grouping expressions.
+\item\newline{\em \menuitemstyle{}}{\em type}\space{}
+The type of any \spadgloss{subdomain} is the unique symbol {\em Category}. The type of a \spadgloss{domain} is any \spadgloss{category} that domain belongs to. The type of any other object is either the (unique) domain that object belongs to or any \spadgloss{subdomain} of that domain. The type of objects is in general not unique.
+\item\newline{\em \menuitemstyle{}}{\em type checking}\space{}
+a system function which determines whether the datatype of an object is appropriate for a given operation.
+\item\newline{\em \menuitemstyle{}}{\em type constructor}\space{}
+a \spadgloss{domain constructor} or \spadgloss{category constructor}.
+\item\newline{\em \menuitemstyle{}}{\em type inference}\space{}
+when the interpreter chooses the type for an object based on context. For example,{} if the user interactively issues the definition \center{\spad{f(x) == (x + \%i)**2}} then issues \spad{f(2)},{} the interpreter will infer the type of \spad{f} to be \spad{Integer -> Complex Integer}.
+\item\newline{\em \menuitemstyle{}}{\em unary}\space{}
+operation or function with \spadgloss{arity} 1
+\item\newline{\em \menuitemstyle{}}{\em underlying domain}\space{}
+for a \spadgloss{domain} that has a single domain-valued parameter,{} the {\em underlying domain} refers to that parameter. For example,{} the domain "matrices of integers" (\spadtype{Matrix Integer}) has underlying domain \spadtype{Integer}.
+\item\newline{\em \menuitemstyle{}}{\em Union}\space{}
+(basic domain constructor) a domain constructor used to combine any set of domains into a single domain. A Union domain is written in the form \spad{Union(a1:D1,{}...,{}an:Dn)} (\spad{n} > 0) where \spad{a1},{}...,{}\spad{an} are identifiers called the {\em tags} of the union,{} and \spad{D1},{}...,{}\spad{Dn} are domains called the {\em branches} of the union. The tags \spad{\spad{ai}} are optional,{} but required when two of the \spad{\spad{Di}} are equal,{} \spadignore{e.g.} \spad{Union(inches:Integer,{}centimeters:Integer)}. In the interpreter,{} values of union domains are automatically coerced to values in the branches and vice-versa as appropriate. See also \spadgloss{case}.
+\item\newline{\em \menuitemstyle{}}{\em unit}\space{}
+{\em (algebra)} an invertible element.
+\item\newline{\em \menuitemstyle{}}{\em user function}\space{}
+a function defined by a user during an interactive session. Contrast \spadgloss{built-in function}.
+\item\newline{\em \menuitemstyle{}}{\em user variable}\space{}
+a variable created by the user at top-level during an interactive session
+\item\newline{\em \menuitemstyle{}}{\em value}\space{}
+1. the result of \spadglossSee{evaluating}{evaluation} an expression. 2. a property associated with a \spadgloss{variable} in a \spadgloss{binding} in an \spadgloss{environment}.
+\item\newline{\em \menuitemstyle{}}{\em variable}\space{}
+a means of referring to an object but itself {\spad{\bf} not} an object. A variable has a name and an associated \spadgloss{binding} created by \spadgloss{evaluation} of \Language{} expressions such as \spadgloss{declarations},{} \spadgloss{assignments},{} and \spadgloss{definitions}. In the top-level \spadgloss{environment} of the interpreter,{} variables are \spadgloss{global variables}. Such variables can be freely referenced in user-defined functions although a \spadgloss{free} declaration is needed to assign values to them. See \spadgloss{local variable} for details.
+\item\newline{\em \menuitemstyle{}}{\em Void}\space{}
+the type given when the \spadgloss{value} and \spadgloss{type} of an expression are not needed. Also used when there is no guarantee at run-time that a value and predictable mode will result.
+\item\newline{\em \menuitemstyle{}}{\em wild card}\space{}
+a symbol which matches any substring including the empty string; for example,{} the search string {\em *an*} matches an word containing the consecutive letters {\em a} and {\em n}
+\item\newline{\em \menuitemstyle{}}{\em workspace}\space{}
+an interactive record of the user input and output held in an interactive history file. Each user input and corresponding output expression in the workspace has a corresponding \spadgloss{step number}. The current output expression in the workspace is referred to as \spad{\%}. The output expression associated with step number \spad{n} is referred to by \spad{\%\%(n)}. The \spad{k}-th previous output expression relative to the current step number \spad{n} is referred to by \spad{\%\%(- k)}. Each interactive \spadgloss{frame} has its own workspace.
+\endmenu\endscroll\lispdownlink{Search}{(|htGloss| "\stringvalue{pattern}")} for glossary entry matching \inputstring{pattern}{24}{*}\end{page}
\ No newline at end of file
diff --git a/src/hyper/pages/graphics.ht b/src/hyper/pages/graphics.ht
new file mode 100644
index 00000000..9dfa9ca2
--- /dev/null
+++ b/src/hyper/pages/graphics.ht
@@ -0,0 +1,801 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Graphics Page
+% @(#)graphics.ht       1.9 91/05/29 15:31:14
+%%%%%%%%%%%
+%% Pages %%
+%%%%%%%%%%%
+
+\begin{page}{GraphicsPage}{Graphics}
+\Language{} can plot curves and surfaces of
+various types, as well as lists of points in the plane.
+% sequences, and complex functions.
+\beginscroll
+\beginmenu
+\menulink{Examples}{GraphicsExamplePage} \tab{13}
+See examples of \Language{} graphics
+\menulink{2D Graphics}{TwoDimensionalGraphicsPage} \tab{13}
+Graphics in the real and complex plane
+\menulink{3D Graphics}{ThreeDimensionalGraphicsPage} \tab{13}
+Plot surfaces, curves, or tubes around curves
+\menulink{Viewports}{ViewportPage} \tab{13}
+Customize graphics using Viewports
+\endmenu
+\endscroll
+\autobuttons \end{page}
+
+% Graphics Example Page
+
+\begin{page}{GraphicsExamplePage}{Examples}
+\beginscroll
+Here are some examples of \Language{} graphics.
+Choose a specific type of graph or choose Assorted Examples.
+\beginmenu
+\menulink{Assorted Examples}{AssortedGraphicsExamplePage} \newline
+Examples of each type of \Language{} graphics.
+\menulink{Three Dimensional Graphics}{ThreeDimensionalGraphicsExamplePage} \newline
+Plot parametrically defined surfaces of three functions.
+\menulink{Functions of One Variable}{OneVariableGraphicsExamplePage} \newline
+Plot curves defined by an equation y = f(x).
+\menulink{Parametric Curves}{ParametricCurveGraphicsExamplePage} \newline
+Plot curves defined by parametric equations x = f(t), y = g(t).
+\menulink{Polar Coordinates}{PolarGraphicsExamplePage} \newline
+Plot curves given in polar form by an equation r = f(theta).
+\menulink{Implicit Curves}{ImplicitCurveGraphicsExamplePage} \newline
+Plot non-singular curves defined by a polynomial equation
+\menulink{Lists of Points}{ListPointsGraphicsExamplePage} \newline
+Plot lists of points in the (x,y)-plane.
+% \menulink{Sequences}{SequenceGraphicsExamplePage}
+% Plot a sequence a1, a2, a3,...
+% \menulink{Complex Functions}{ComplexFunctionGraphicsExamplePage}
+% Plot complex functions of a complex variable by means of grid plots.
+\endmenu
+\endscroll
+\autobuttons \end{page}
+
+% Assorted Graphics Example Page
+
+\begin{page}{AssortedGraphicsExamplePage}{Assorted Examples}
+\beginscroll
+Pick a specific example or choose 'All' to see all the examples.\newline
+Function of two variables: z = f(x,y).
+\graphpaste{draw(sin(x * y), x = -2.5..2.5, y = -2.5..2.5) \bound{example1}}
+Function of one variable: y = f(x).
+\graphpaste{draw(sin tan x - tan sin x,x = 0..6) \bound{example2} }
+Plane parametric curve: x = f(t), y = g(t).
+\graphpaste{draw(curve(sin(t)*sin(2*t), sin(3*t)*sin(4*t)), t = 0..2*\%pi) \bound{example3}}
+Space parametric curve: x = f(t), y = g(t), z = h(t).
+\graphpaste{draw(curve(sin(t)*sin(2*t), sin(3*t)*sin(4*t), sin(5*t)*sin(6*t)), t = 0..2*\%pi) \bound{example4}}
+Polar coordinates: r = f(theta).
+\graphpaste{draw(sin(17*t), t = 0..2*\%pi, coordinates == polar) \bound{example5}}
+Implicit curves: p(x,y) = 0.
+\graphpaste{draw(y**2 + y = x**3 - x, x, y,range == \[-2..2,-2..1\]) \bound{example6}}
+Run all examples.
+\spadpaste{All \free{example1 example2 example3 example4 example5 example6}}
+\endscroll
+\autobuttons \end{page}
+
+% Three Dimensional Graphics Example Page
+
+\begin{page}{ThreeDimensionalGraphicsExamplePage}{Three Dimensional Graphics}
+Plots of parametric surfaces defined by functions f(u,v), g(u,v), and h(u,v).
+Choose a particular example or choose 'All' to see all the examples.
+\beginscroll
+Pear Surface.
+\graphpaste{draw(surface((1+exp(-100*u*u))*sin(\%pi*u)*sin(\%pi*v), (1+exp(-100*u*u))*sin(\%pi*u)*cos(\%pi*v), (1+exp(-100*u*u))*cos(\%pi*u)), u=0..1, v=0..2, title=="Pear") \bound{example1}}
+Trigonometric Screw.
+\graphpaste{draw(surface(x*cos(y),x*sin(y),y*cos(x)), x=-4..4, y=0..2*\%pi, var1Steps==40, var2Steps==40, title=="Trig Screw") \bound{example2}}
+Etruscan Venus. \newline
+(click on the draw button to execute this example)
+\spadpaste{a := 1.3 * cos(2*x) * cos(y) + sin(y) * cos(x)\bound{a}} 
+\newline
+\spadpaste{b := 1.3 * sin(2*x) * cos(y) - sin(y) * sin(x)\bound{b}} 
+\newline
+\spadpaste{c := 2.5 * cos(y) \bound{c}} 
+\newline
+\graphpaste{draw(surface(a,b,c), x=0..\%pi, y=-\%pi..\%pi, var1Steps==40, var2Steps==40, title=="Etruscan Venus") \free{a b c} \bound{example3}}
+Banchoff Klein Bottle. \newline
+(click on the draw button to execute this example)
+\spadpaste{f:=cos(x)*(cos(x/2)*(sqrt(2) + cos(y))+(sin(x/2)*sin(y)*cos(y)))\bound{f}} 
+\newline
+\spadpaste{g:=sin(x)*(cos(x/2)*(sqrt(2) + cos(y))+(sin(x/2)*sin(y)*cos(y)))\bound{g}} 
+\newline
+\spadpaste{h:=-sin(x/2)*(sqrt(2)+cos(y)) + cos(x/2)*sin(y)*cos(y) \bound{h}}
+\newline
+\graphpaste{draw(surface(f,g,h), x=0..4*\%pi, y=0..2*\%pi, var1Steps==50, var2Steps==50, title=="Banchoff Klein Bottle") \free{f g h} \bound{example4}}
+\newline
+\spadpaste{All \free{example1 example2 example3 example4}}
+\endscroll
+\autobuttons \end{page}
+
+% Functions of One Variable Example Page
+
+\begin{page}{OneVariableGraphicsExamplePage}{Functions of One Variable}
+\beginscroll
+Plots of functions y = f(x).
+Choose a particular example or choose 'All' to see all the examples.
+\graphpaste{draw(sin tan x - tan sin x, x = 0..6) \bound{example1}}
+\newline
+\graphpaste{draw(sin x + cos x, x = 0..2*\%pi) \bound{example2}}
+\newline
+\graphpaste{draw(sin(1/x), x = -1..1) \bound{example3}}
+\newline
+\graphpaste{draw(x * sin(1/x), x = -1..1) \bound{example4}}
+\newline
+\spadpaste{All \free{example1 example2 example3 example4}}
+\endscroll
+\autobuttons \end{page}
+
+% Parametric Curve Example Page
+
+\begin{page}{ParametricCurveGraphicsExamplePage}{Parametric Curves}
+Plots of parametric curves x = f(t), y = g(t).
+Pick a particular example or choose 'All' to see all the examples.
+\beginscroll
+The Lemniscate of Bernoulli.
+\graphpaste{draw(curve(cos(t/(1+sin(t)**2)), sin(t)*cos(t)/(1+sin(t)**2)), t = -\%pi..\%pi) \bound{example1}}
+Lissajous curve.
+\graphpaste{draw(curve(9*sin(3*t/4), 8*sin(t)), t = -4*\%pi..4*\%pi) \bound{example2}}
+A gnarly closed curve.
+\graphpaste{draw(curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)),t = 0..2*\%pi)
+             \bound{example3}}
+Another closed curve.
+\graphpaste{draw(curve(cos(4*t)*cos(7*t), cos(4*t)*sin(7*t)), t = 0..2*\%pi) \bound{example4}}
+Run all examples on this page.
+\spadpaste{All \free{example1 example2 example3 example4}}
+\endscroll
+\autobuttons \end{page}
+
+% Polar Coordinates Example Page
+
+\begin{page}{PolarGraphicsExamplePage}{Polar Coordinates}
+Plots of curves given by an equation in polar coordinates, r = f(theta).
+Pick a particular example or choose 'All' to see all the examples.
+\beginscroll
+A Circle.
+\graphpaste{draw(1,t = 0..2*\%pi, coordinates == polar) \bound{example1} }
+A Spiral.
+\graphpaste{draw(t,t = 0..100, coordinates == polar) \bound{example2} }
+A Petal Curve.
+\graphpaste{draw(sin(4*t), t = 0..2*\%pi, coordinates == polar) \bound{example3} }
+A Limacon.
+\graphpaste{draw(2 + 3 * sin t, t = 0..2*\%pi, coordinates == polar) \bound{example4} }
+Run all examples on this page.
+\spadpaste{All \free{
+%example1
+example2 example3 example4}}
+\endscroll
+\autobuttons \end{page}
+
+% Implicit Curve Example Page
+
+\begin{page}{ImplicitCurveGraphicsExamplePage}{Implicit Curves}
+Non-singular curves defined by a polynomial equation p(x,y) = 0
+in a rectangular region in the plane.
+Pick a particular example or choose 'All' to see all the examples.
+\beginscroll
+A Conic Section (Hyperbola).
+\graphpaste{draw(x * y = 1, x, y, range == \[-3..3, -3..3\]) \bound{example1} }
+An Elliptic Curve.
+\graphpaste{draw(y**2 + y = x**3 - x, x, y, range == \[-2..2, -2..1\]) \bound{example2} }
+Cartesian Ovals.
+\spadpaste{p := ((x**2 + y**2 + 1) - 8*x)**2 - (8*(x**2 + y**2 + 1) - 4*x - 1) \bound{p} }
+\graphpaste{draw(p = 0, x, y, range == \[-1..11, -7..7\], title == "Cartesian Ovals") \free{p} \bound{example3} }
+Cassinian Ovals: two loops.
+\spadpaste{q := (x**2 + y**2 + 7**2)**2 - (6**4 + 4*7**2*x**2) \bound{q} }
+\graphpaste{draw(q = 0, x, y, range == \[-10..10, -4..4\], title == "Cassinian oval with two loops") \free{q} \bound{example4} }
+Run all examples on this page.
+\spadpaste{All \free{example1 example2 example3 example4}}
+\endscroll
+\autobuttons \end{page}
+
+% Lists of Points Example Page
+
+\menulink{Lists of Points}{ListPointsGraphicsExamplePage} \newline
+
+\begin{page}{ListPointsGraphicsExamplePage}{Lists of Points}
+\Language{} has the ability to create lists of points in a two dimensional
+graphics viewport.  This is done by utilizing the \spadtype{GraphImage} and
+\spadtype{TwoDimensionalViewport} domain facilities.
+\beginscroll
+\indent{5}\newline
+{\em NOTE:  It is only necessary to click on the makeViewport2D command button to plot this curve example}.
+\indent{0}\newline
+\spadpaste{p1 := point [1::SF,1::SF]\$(Point SF) \bound{p1}} 
+\newline
+\spadpaste{p2 := point [0::SF,1::SF]\$(Point SF) \bound{p2}} 
+\newline
+\spadpaste{p3 := point [0::SF,0::SF]\$(Point SF) \bound{p3}} 
+\newline
+\spadpaste{p4 := point [1::SF,0::SF]\$(Point SF) \bound{p4}} 
+\newline
+\spadpaste{p5 := point [1::SF,.5::SF]\$(Point SF) \bound{p5}} 
+\newline
+\spadpaste{p6 := point [.5::SF,0::SF]\$(Point SF) \bound{p6}} 
+\newline
+\spadpaste{p7 := point [0::SF,0.5::SF]\$(Point SF) \bound{p7}} 
+\newline
+\spadpaste{p8 := point [.5::SF,1::SF]\$(Point SF) \bound{p8}} 
+\newline
+\spadpaste{p9 := point [.25::SF,.25::SF]\$(Point SF) \bound{p9}} 
+\newline
+\spadpaste{p10 := point [.25::SF,.75::SF]\$(Point SF) \bound{p10}} 
+\newline
+\spadpaste{p11 := point [.75::SF,.75::SF]\$(Point SF) \bound{p11}} 
+\newline
+\spadpaste{p12 := point [.75::SF,.25::SF]\$(Point SF) \bound{p12}} 
+\newline
+\spadpaste{llp := [[p1,p2],[p2,p3],[p3,p4],[p4,p1],[p5,p6],[p6,p7],[p7,p8],[p8,p5],[p9,p10],[p10,p11],[p11,p12],[p12,p9]] \bound{llp} \free{p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12}} 
+\newline
+\spadpaste{size1 := 6::PositiveInteger \bound{size1}} 
+\newline
+\spadpaste{size2 := 8::PositiveInteger \bound{size2}} 
+\newline
+\spadpaste{size3 := 10::PositiveInteger \bound{size3}} 
+\newline
+\spadpaste{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3] \bound{lsize} \free{size1 size2 size3}} 
+\newline
+\spadpaste{pc1 := pastel red() \bound{pc1}} 
+\newline
+\spadpaste{pc2 := dim green() \bound{pc2}} 
+\newline
+\spadpaste{pc3 := pastel yellow() \bound{pc3}} 
+\newline
+\spadpaste{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3] \bound{lpc} \free{pc1 pc2 pc3}} 
+\newline
+\spadpaste{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()] \bound{lc}} 
+\newline
+\spadpaste{g := makeGraphImage(llp,lpc,lc,lsize)\$GRIMAGE \bound{g} \free{llp lpc lc lsize}} 
+\newline
+\graphpaste{makeViewport2D(g,[title("Lines")])\$VIEW2D \free{g}}
+The \spadfun{makeViewport2D} command takes a list of options as a parameter
+in this example.  The string "Lines" is designated as the viewport's title.
+\endscroll
+\autobuttons \end{page}
+
+% Three Dimensional Graphics Page
+
+\begin{page}{ThreeDimensionalGraphicsPage}{Three Dimensional Graphing}
+\beginscroll
+\beginmenu
+\menulink{Functions of Two Variables}{TwoVariableGraphicsPage} \newline
+Plot surfaces defined by an equation z = f(x,y).
+\menulink{Parametric Curves}{SpaceCurveGraphicsPage} \newline
+Plot curves defined by equations x = f(t), y = g(t), z = g(t).
+\menulink{Parametric Tube Plots}{ParametricTubeGraphicsPage} \newline
+Plot a tube around a parametric space curve.
+\menulink{Parametric Surfaces}{ParametricSurfaceGraphicsPage} \newline
+Plot surfaces defined by x = f(u,v), y = g(u,v), z = h(u,v).
+\menulink{Building Objects}{ugGraphThreeDBuildPage} \newline
+Create objects constructed from geometric primitives.
+\endmenu
+\endscroll
+\autobuttons \end{page}
+
+% Functions of Two Variables Graphics Page
+
+\begin{page}{TwoVariableGraphicsPage}{Functions of Two Variables}
+\beginscroll
+This page describes the plotting of surfaces defined by an equation
+of two variables, z = f(x,y), for which the ranges of x and y are explicitly
+defined.  The basic draw command for this function utilizes either the
+uncompiled function or compiled function format.  The general format for an
+uncompiled function is:
+\indent{5}\newline
+{\em draw(f(x,y), x = a..b, y = c..d)}
+\indent{0}\newline
+where a..b and c..d are segments defining the intervals [a,b] and [c,d] over
+which the variables x and y span.  In this case the function is not compiled
+until the draw command is executed.  Here is an example:
+\graphpaste{draw(cos(x*y),x=-3..3,y=-3..3)}
+In the case of a compiled function, the function is named and compiled
+independently.  This is useful if you intend to use a function often, or
+if the function is long and complex.  The following line shows a function
+whose parameters are of the type Small Float. The function is compiled and
+stored by \Language{} when it is entered.
+\indent{5}\newline
+{\em NOTE:  It is only necessary to click on the draw command button to plot
+this example}.
+\indent{0}\newline
+\spadpaste{f(x:SF,y:SF):SF == sin(x)*cos(y) \bound{f}}
+\newline
+Once the function is compiled the draw command only needs the name of the
+function to execute.  Here is a compiled function example:
+\graphpaste{draw(f,-\%pi..\%pi,-\%pi..\%pi) \free{f}}
+Note that the parameter ranges do not take the variable names as in the
+case of uncompiled functions.  The variables are entered in the order in
+which they are defined in the function specification.  In this case the
+first range specifies the x-variable and the second range specifies the
+y-variable.
+\endscroll
+\autobuttons \end{page}
+
+% Parametric Space Curves Graphics Page
+
+\begin{page}{SpaceCurveGraphicsPage}{Parametric Space Curves}
+\beginscroll
+This page describes the plotting in three dimensional space of a curve
+defined by the parametric equations x = f(t), y = g(t), z = h(t), where
+f, g, and h are functions of the parameter t which ranges over a specified
+interval.  The basic draw command for this function utilizes either the
+uncompiled functions or compiled functions format and uses the \spadfun{curve}
+command to specify the three functions for the x, y, and z components of
+the curve.  The general format for uncompiled functions is:
+\indent{5}\newline
+{\em draw(curve(f(t),g(t),h(t)), t = a..b)}
+\indent{0}\newline
+where a..b is the segment defining the interval [a,b] over which the
+parameter t ranges.  In this case the functions are not compiled until the
+draw command is executed.  Here is an example:
+\graphpaste{draw(curve(cos(t),sin(t),t), t=-12..12)}
+In the case of compiled functions, the functions are named and compiled
+independently.  This is useful if you intend to use the functions often, or
+if the functions are long and complex.  The following lines show functions
+whose parameters are of the type Small Float. The functions are compiled and
+stored by \Language{} when entered.
+\indent{5}\newline
+{\em NOTE:  It is only necessary to click on the draw command button to plot
+this example}.
+\indent{0}\newline
+\spadpaste{i1(t:SF):SF == sin(t)*cos(3*t/5) \bound{i1}}
+\spadpaste{i2(t:SF):SF == cos(t)*cos(3*t/5) \bound{i2}}
+\spadpaste{i3(t:SF):SF == cos(t)*sin(3*t/5) \bound{i3}}
+Once the functions are compiled the draw command only needs the names of
+the functions to execute.  Here is a compiled functions example:
+\graphpaste{draw(curve(i1,i2,i3),0..15*\%pi) \free{i1 i2 i3}}
+Note that the parameter range does not take the variable name as in the
+case of uncompiled functions. It is understood that the indicated range
+applies to the parameter of the functions, which in this case is t.
+\endscroll
+\autobuttons \end{page}
+
+% Parametric Tube Plots Graphics Page
+
+\begin{page}{ParametricTubeGraphicsPage}{Parametric Tube Plots}
+\beginscroll
+This page describes the plotting in three dimensional space of a tube
+around a parametric space curve defined by the parametric equations
+x = f(t), y = g(t), z = h(t), where f, g, and h are functions of the
+parameter t which ranges over a specified interval.  The basic draw command
+for this function utilizes either the uncompiled functions or compiled
+functions format and uses the \spadfun{curve} command to specify the three
+functions for the x, y, and z components of the curve.  This uses the same
+format as that for space curves except that it requires a specification for
+the radius of the tube.  If the radius of the tube is 0, then the result is
+the space curve itself.  The general format for uncompiled functions is:
+\indent{5}\newline
+{\em draw(curve(f(t),g(t),h(t)), t = a..b, tubeRadius == r)}
+\indent{0}\newline
+where a..b is the segment defining the interval [a,b] over which the
+parameter t ranges, and the tubeRadius is indicated by the variable r.
+In this case the functions are not compiled until the draw command is
+executed.  Here is an example:
+\graphpaste{draw(curve(sin(t)*cos(3*t/5), cos(t)*cos(3*t/5), cos(t)*sin(3*t/5)), t=0..15*\%pi,tubeRadius == .15)}
+In the case of compiled functions, the functions are named and compiled
+independently.  This is useful if you intend to use the functions often, or
+if the functions are long and complex.  The following lines show functions
+whose parameters are of the type Small Float. The functions are compiled and
+stored by \Language{} when entered.
+\indent{5}\newline
+{\em NOTE:  It is only necessary to click on the draw command button to plot
+this example}.
+\indent{0}\newline
+\spadpaste{t1(t:SF):SF == 4/(2-sin(3*t))*cos(2*t) \bound{t1}} 
+\newline
+\spadpaste{t2(t:SF):SF == 4/(2-sin(3*t))*sin(2*t) \bound{t2}} 
+\newline
+\spadpaste{t3(t:SF):SF == 4/(2-sin(3*t))*cos(3*t) \bound{t3}} 
+\newline
+Once the function is compiled the draw command only needs the names of the
+functions to execute.  Here is a compiled functions example of a trefoil knot:
+\graphpaste{draw(curve(t1,t2,t3),0..2*\%pi,tubeRadius == .2) \free{t1 t2 t3}}
+Note that the parameter range does not take the variable name as in the
+case of uncompiled functions. It is understood that the indicated range
+applies to the parameter of the functions, which in this case is t.
+Typically, the radius of the tube should be set between 0 and 1.  A radius
+of less than 0 results in it's positive counterpart and a radius of greater
+than one causes self intersection.
+\endscroll
+\autobuttons \end{page}
+
+% Parametric Surfaces Graphics Page
+
+\begin{page}{ParametricSurfaceGraphicsPage}{Parametric Surfaces}
+\beginscroll
+Graphing a surface defined by x = f(u,v), y = g(u,v), z = h(u,v). \newline
+This page describes the plotting of surfaces defined by the parametric
+equations of two variables, x = f(u,v), y = g(u,v), and z = h(u,v),
+for which the ranges of u and v are explicitly defined.  The basic draw
+command for this function utilizes either the uncompiled function or
+compiled function format and uses the \spadfun{surface} command to specify the
+three functions for the x, y and z components of the surface.  The general
+format for uncompiled functions is:
+\indent{5}\newline
+{\em draw(surface(f(u,v),g(u,v),h(u,v)), u = a..b, v = c..d)}
+\indent{0}\newline
+where a..b and c..d are segments defining the intervals [a,b] and [c,d] over
+which the parameters u and v span.  In this case the functions are not
+compiled until the draw command is executed.  Here is an example of a
+surface plotted using the parabolic cylindrical coordinate system option:
+\graphpaste{draw(surface(u*cos(v), u*sin(v),v*cos(u)),u=-4..4,v=0..2*\%pi,
+                  coordinates== parabolicCylindrical)}
+In the case of compiled functions, the functions are named and compiled
+independently.  This is useful if you intend to use the functions often, or
+if the functions are long and complex.  The following lines show functions
+whose parameters are of the type Small Float. The functions are compiled and
+stored by \Language{} when entered.
+\indent{5}\newline
+{\em NOTE:  It is only necessary to click on the draw command button to plot
+this example}.
+\indent{0}\newline
+\spadpaste{n1(u:SF,v:SF):SF == u*cos(v) \bound{n1}} 
+\newline
+\spadpaste{n2(u:SF,v:SF):SF == u*sin(v) \bound{n2}} 
+\newline
+\spadpaste{n3(u:SF,v:SF):SF == u \bound{n3}}
+Once the function is compiled the draw command only needs the names of the
+functions to execute.  Here is a compiled functions example plotted using
+the toroidal coordinate system option: \newline
+\graphpaste{draw(surface(n1,n2,n3), 1.0..4.0, 1.0..4*\%pi,
+             coordinates == toroidal(1\$SF)) \free{n1 n2 n3}}
+Note that the parameter ranges do not take the variable names as in the
+case of uncompiled functions.  The variables are entered in the order in
+which they are defined in the function specification.  In this case the
+first range specifies the u-variable and the second range specifies the
+v-variable.
+\endscroll
+\autobuttons \end{page}
+
+% Building 3D Objects Graphics Page
+
+\begin{page}{3DObjectGraphicsPage}{Building 3D Objects}
+\beginscroll
+This page describes the \Language{} facilities for creating three dimensional
+objects constructed from geometric primitives.  The \Language{} operation
+\spadfun{create3Space()} creates a space to which points, curves, and
+polygons can be added using the operations from the \spadtype{ThreeSpace}
+domain.  The contents of this space can then be displayed in a viewport
+using the \spadfun{makeViewport3D()} command.  It will be necessary to
+have these operations exposed in order to use them. \indent{5}\newline
+{\em NOTE:  It is only necessary to click on the makeViewport3D command button
+to plot this curve example}.
+\indent{0}\newline
+Initially, the space which will hold the objects must be defined and
+compiled, as in the following example:
+\spadpaste{space := create3Space()\$(ThreeSpace SF) \bound{space}}
+Now objects can be sent to this {\em space} as per the operations allowed by
+the \spadtype{ThreeSpace} domain.  The following examples place curves into
+{\em space}.
+\spadpaste{curve(space,[[0,20,20],[0,20,30],[0,30,30],[0,30,100], [0,20,100],[0,20,110],[0,50,110],[0,50,100],[0,40,100], [0,40,30],[0,50,30],[0,50,20],[0,20,20]]) \bound{curveI}} 
+\newline
+\spadpaste{curve(space,[[0,80,20],[0,70,20],[0,70,110],[0,110,110], [0,120,100],[0,120,70],[0,115,65],[0,120,60],[0,120,30], [0,110,20],[0,80,20],[0,80,30],[0,105,30],[0,110,35]]) \bound{curveB1}} 
+\newline
+\spadpaste{curve(space,[[0,110,35],[0,110,55],[0,105,60],[0,80,60],[0,80,70], [0,105,70],[0,110,75],[0,110,95],[0,105,100],[0,80,100], [0,80,30]]) \bound{curveB2}} 
+\newline
+\spadpaste{closedCurve(space,[[0,140,20],[0,140,110],[0,150,110],[0,170,50], [0,190,110],[0,200,110],[0,200,20],[0,190,20],[0,190,75], [0,175,35],[0,165,35],[0,150,75],[0,150,20]]) \bound{curveM}}
+\spadpaste{closedCurve(space,[[200,0,20], [200,0,110], [185,0,110], [160,0,45], [160,0,110], [150,0,110], [150,0,20], [165,0,20], [190,0,85], [190,0,20]]) \bound{curveN}}
+\spadpaste{closedCurve(space,[[140,0,20], [120,0,110], [110,0,110], [90,0,20], [100,0,20], [108,0,50], [123,0,50], [121,0,60], [110,0,60], [115,0,90], [130,0,20]]) \bound{curveA}}
+\spadpaste{closedCurve(space,[[80,0,30], [80,0,100], [70,0,110], [40,0,110], [30,0,100], [30,0,90], [40,0,90], [40,0,95], [45,0,100], [65,0,100], [70,0,95], [70,0,35], [65,0,30], [45,0,30], [40,0,35], [40,0,60], [50,0,60], [50,0,70], [30,0,70], [30,0,30], [40,0,20], [70,0,20]]) \bound{curveG}}
+Once {\em space} contains the desired elements a viewport is created and
+displayed with the following command:
+\graphpaste{makeViewport3D(space,[title("Curves")])\$VIEW3D \free{space curveI curveB1 curveB2 curveM curveN curveA curveG}}
+The parameters for \spadfun{makeViewport3D()} in this example are {\em space},
+which is the name of the three dimensional space that was defined, and a
+string, "curve", which is the title for the viewport.  The tailing string
+{\em \$VIEW3D} exposes the command \spadfun{makeViewport3D()} from the domain
+\spadtype{ThreeDimensionalViewport} if these commands are unexposed.
+\endscroll
+\autobuttons \end{page}
+
+% Two Dimensional Graphics Page
+
+\begin{page}{TwoDimensionalGraphicsPage}{Two Dimensional Graphics}
+\beginscroll
+\beginmenu
+\menulink{Functions of One Variable}{OneVariableGraphicsPage} \newline
+Plot curves defined by an equation y = f(x).
+\menulink{Parametric Curves}{ParametricCurveGraphicsPage} \newline
+Plot curves defined by parametric equations x = f(t), y = g(t).
+\menulink{Polar Coordinates}{PolarGraphicsPage} \newline
+Plot curves given in polar form by an equation r = f(theta).
+\menulink{Implicit Curves}{ImplicitCurveGraphicsPage} \newline
+Plot non-singular curves defined by a polynomial equation
+\menulink{Lists of Points}{ListPointsGraphicsPage} \newline
+Plot lists of points in the (x,y)-plane.
+% \menulink{Sequences}{SeqGraphicsPage} \newline
+% Plot a sequence a1, a2, a3,...
+% \menulink{Complex Functions}{CxFuncGraphicsPage} \newline
+% Plot functions of a complex variable using grid plots.
+\endmenu
+\endscroll
+\autobuttons \end{page}
+
+% Functions of One Variable Graphics Page
+
+\begin{page}{OneVariableGraphicsPage}{Functions of One Variable}
+\beginscroll
+Here we wish to plot a function y = f(x) on an interval [a,b].
+As an example, let's take the function y = sin(tan(x)) - tan(sin(x))
+on the interval [0,6].
+Here is the simplest command that will do this:
+\graphpaste{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)}
+Notice that \Language{} compiled a function before the graph was put
+on the screen.
+The expression sin(tan(x)) - tan(sin(x)) was converted to a compiled
+function so that it's value for various values of x could be computed
+quickly and efficiently.
+Let's graph the same function on a different interval and this time
+we'll give the graph a title.
+The title is a String, which is an optional argument of the command 'draw'.
+\graphpaste{draw(sin(tan(x)) - tan(sin(x)),x = 10..16,title == "y = sin tan x - tan sin x")}
+Once again the expression sin(tan(x)) - tan(sin(x)) was converted to a
+compiled function before any points were computed.
+If you want to graph the same function on a number of intervals, it's
+a good idea to write down a function definition so that the function
+only has to be compiled once.
+Here's an example:
+\spadpaste{f(x) == (x-1)*(x-2)*(x-3) \bound{f}}
+\newline
+\graphpaste{draw(f, 0..2, title == "y = f(x) on \[0,2\]") \free{f}}
+\newline
+\graphpaste{draw(f, 0..4,title == "y = f(x) on \[0,4\]") \free{f}}
+Notice that our titles can be whatever we want, as long as they are
+enclosed by double quotes.  However, a title which is too long to fit
+within the viewport title window will be clipped.
+\endscroll
+\autobuttons \end{page}
+
+% Parametric Curve Graphics Page
+
+\begin{page}{ParametricCurveGraphicsPage}{Parametric Curves}
+\beginscroll
+One way of producing interesting curves is by using parametric equations.
+Let x = f(t) and y = g(t) for two functions f and g as the parameter
+t ranges over an interval \[a,b\].
+Here's an example:
+\graphpaste{draw(curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)), t = 0..2*\%pi)}
+Here 0..2*\%pi represents the interval over which the variable t ranges.
+In the case of parametric curves, \Language{} will compile two functions,
+one for each of the functions f and g.
+You may also put a title on a graph.
+The title may be an arbitrary string and is an optional argument
+to the command 'draw'.
+For example:
+\graphpaste{draw(curve(cos(t), sin(t)), t = 0..2*\%pi, title == "The Unit Circle")}
+If you plan on plotting x = f(t), y = g(t) as t ranges over several intervals,
+you may want to define functions f and g, so that they need not be
+recompiled every time you create a new graph.
+Here's an example:
+\spadpaste{f(t:SF):SF == sin(3*t/4) \bound{f}} 
+\newline
+\spadpaste{g(t:SF):SF == sin(t) \bound{g}} 
+\newline
+\graphpaste{draw(curve(f,g), 0..\%pi) \free{f g}} 
+\newline
+\graphpaste{draw(curve(f,g) ,\%pi..2*\%pi) \free{f g}} 
+\newline
+\graphpaste{draw(curve(f,g), -4*\%pi..4*\%pi) \free{f g}}
+These examples show how the curve changes as the range of parameter t varies.
+\endscroll
+\autobuttons \end{page}
+
+% Polar Coordinates Graphics Page
+
+\begin{page}{PolarGraphicsPage}{Polar Coordinates}
+\beginscroll
+Graphs in polar coordinates are given by an equation r = f(theta) as
+theta ranges over an interval.
+This is equivalent to the parametric curve x = f(theta) * cos(theta),
+y = f(theta) * sin(theta) as theta ranges over the same interval.
+You may create such curves using the command 'draw', with the optional
+argument 'coordinates == polar'.
+Here are some examples:
+\graphpaste{draw(1,t = 0..2*\%pi,coordinates == polar, title == "The Unit Circle")} 
+\newline
+\graphpaste{draw(sin(17*t), t = 0..2*\%pi, coordinates == polar, title == "A Petal Curve")}
+%When you don't specify an interval, \Language{} will assume that you
+%mean 0..2*\%pi.
+You may also define your own functions, when you plan on plotting the
+same curve as theta varies over several intervals.
+\spadpaste{f(t) == cos(4*t/7) \bound{f}} 
+\newline
+\graphpaste{draw(f, 0..2*\%pi, coordinates == polar) \free{f}} 
+\newline
+\graphpaste{draw(f, 0..14*\%pi, coordinates == polar) \free{f}}
+For information on plotting graphs in other coordinate systems see the
+pages for the \spadtype{CoordinateSystems} domain.
+\endscroll
+\autobuttons \end{page}
+
+% Implicit Curve Graphics Page
+
+\begin{page}{ImplicitCurveGraphicsPage}{Implicit Curves}
+\beginscroll
+\Language{} has facilities for graphing a non-singular algebraic curve
+in a rectangular region of the plane.
+An algebraic curve is a curve defined by a polynomial equation
+p(x,y) = 0.
+Non-singular means that the curve is "smooth" in that it does not
+cross itself or come to a point (cusp).
+Algebraically, this means that for any point (a,b) on the curve
+(i.e. a point such that p(a,b) = 0), the partial derivatives
+dp/dx(a,b) and dp/dy(a,b) are not both zero.
+We require that the polynomial have rational or integral coefficients.
+Here is a Cartesian ovals algebraic curve example:
+(click on the draw button to execute this example)
+\spadpaste{p := ((x**2 + y**2 + 1) - 8*x)**2 - (8*(x**2 + y**2 + 1) - 4*x - 1) \bound{p} }
+\graphpaste{draw(p = 0, x, y, range == \[-1..11, -7..7\], title == "Cartesian Ovals") \free{p}}
+{\em A range must be declared for each variable specified in the algebraic
+curve equation}.
+\endscroll
+\autobuttons \end{page}
+
+% List of Points Graphics Page
+
+\begin{page}{ListPointsGraphicsPage}{Lists of Points}
+\beginscroll
+\Language{} has the ability to create lists of points in a two dimensional
+graphics viewport.  This is done by utilizing the \spadtype{GraphImage} and
+\spadtype{TwoDimensionalViewport} domain facilities.
+\indent{5}\newline
+{\em NOTE:  It is only necessary to click on the makeViewport2D command button
+to plot this curve example}.
+\indent{0}\newline
+In this example, the \spadfun{makeGraphImage} command takes a list of lists of
+points parameter, a list of colors for each point in the graph, a list of
+colors for each line in the graph, and a list of numbers which indicate the
+size of each point in the graph.  The following lines create list of lists of
+points which can be read be made into two dimensional graph images.
+\spadpaste{p1 := point [1::SF,1::SF]\$(Point SF) \bound{p1}} 
+\newline
+\spadpaste{p2 := point [0::SF,1::SF]\$(Point SF) \bound{p2}} 
+\newline
+\spadpaste{p3 := point [0::SF,0::SF]\$(Point SF) \bound{p3}} 
+\newline
+\spadpaste{p4 := point [1::SF,0::SF]\$(Point SF) \bound{p4}} 
+\newline
+\spadpaste{p5 := point [1::SF,.5::SF]\$(Point SF) \bound{p5}} 
+\newline
+\spadpaste{p6 := point [.5::SF,0::SF]\$(Point SF) \bound{p6}} 
+\newline
+\spadpaste{p7 := point [0::SF,0.5::SF]\$(Point SF) \bound{p7}} 
+\newline
+\spadpaste{p8 := point [.5::SF,1::SF]\$(Point SF) \bound{p8}} 
+\newline
+\spadpaste{p9 := point [.25::SF,.25::SF]\$(Point SF) \bound{p9}} 
+\newline
+\spadpaste{p10 := point [.25::SF,.75::SF]\$(Point SF) \bound{p10}} 
+\newline
+\spadpaste{p11 := point [.75::SF,.75::SF]\$(Point SF) \bound{p11}} 
+\newline
+\spadpaste{p12 := point [.75::SF,.25::SF]\$(Point SF) \bound{p12}} 
+\newline
+\spadpaste{llp := [[p1,p2],[p2,p3],[p3,p4],[p4,p1],[p5,p6],[p6,p7],[p7,p8],[p8,p5],[p9,p10],[p10,p11],[p11,p12],[p12,p9]] \bound{llp} \free{p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12}} 
+\newline
+These lines set the point color and size, and the line color for all components
+of the graph.
+\spadpaste{size1 := 6::PositiveInteger \bound{size1}} 
+\newline
+\spadpaste{size2 := 8::PositiveInteger \bound{size2}} 
+\newline
+\spadpaste{size3 := 10::PositiveInteger \bound{size3}} 
+\newline
+\spadpaste{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3] \bound{lsize} \free{size1 size2 size3}} 
+\newline
+\spadpaste{pc1 := pastel red() \bound{pc1}} 
+\newline
+\spadpaste{pc2 := dim green() \bound{pc2}} 
+\newline
+\spadpaste{pc3 := pastel yellow() \bound{pc3}} 
+\newline
+\spadpaste{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3] \bound{lpc} \free{pc1 pc2 pc3}} 
+\newline
+\spadpaste{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()] \bound{lc}} 
+\newline
+Now the graph image is created and named according to the component
+specifications indicated above.  The \spadfun{makeViewport2D} command then
+creates a two dimensional viewport for this graph according to the list of
+options specified within the brackets.
+\spadpaste{g := makeGraphImage(llp,lpc,lc,lsize)\$GRIMAGE \bound{g} \free{llp lpc lc lsize}} 
+\newline
+\graphpaste{makeViewport2D(g,[title("Lines")])\$VIEW2D \free{g}}
+The \spadfun{makeViewport2D} command takes a list of options as a parameter.
+In this example the string "Lines" is designated as the viewport's title.
+\endscroll
+\autobuttons \end{page}
+
+% \begin{page}{SequenceGraphicsPage}{Sequences}
+% \beginscroll
+% \endscroll
+% \autobuttons \end{page}
+
+% \begin{page}{ComplexFunctionGraphicsPage}{Complex Functions}
+% \beginscroll
+% \endscroll
+% \autobuttons \end{page}
+
+\begin{page}{ViewportPage}{Stand-alone Viewport}
+\beginscroll
+To get a viewport on a \HyperName{} page, you first need to
+create one in \Language{} and write it out to a
+file that \HyperName{} can call up. \newline
+For example, here we draw a saddle function and assign
+the result to the variable \spad{v}:
+\newline
+\graphpaste{v := draw(x*x-y*y,x=-1..1,y=-1..1) \bound{v}} 
+\newline
+Now that we've created the viewport, we want to write
+the data out to a file. \newline
+To do this, we use the \spadfunFrom{write}{ThreeDimensionalViewport} command which takes
+the following arguments: the viewport to write out,
+the title of the file to write it out to, and an optional
+argument telling the write command what type (or types) of
+data you want to write in additional to the one \Language{} will
+always write out. The optional argument could be
+a string, like "pixmap", or a list of strings, like \["postscript","pixmap"\].
+\HyperName{} needs a "pixmap" data type to include a graph in a page
+so in this case, we write the viewport and tell it to
+also write a "pixmap" file, as well:
+\newline
+\spadpaste{write(v,"saddle","pixmap") \free{v}}
+\newline
+Now we want to put this viewport into a \HyperName{} page.
+Say you've created a viewport and written it out
+to a file called "/tmp/mobius". (\Language{} actually
+tags a ".VIEW" at the end of a viewport data file to
+make it easier to spot when you're rummaging through
+your file system, but you needn't worry about that here
+since \Language{} will always automatically add on a
+".VIEW" for you.)  \newline
+
+{\bf Including Viewports} \newline
+To put a viewport in a
+\HyperName{} page, include the following line in your \HyperName{}
+source code: \newline
+\space{5}\\viewport\{/tmp/mobius\} \newline
+You will get this on your page: \newline
+%%%\space{4}\viewport{/tmp/mobius}\newline
+%\space{4}\spadviewport{mobius}\newline
+\centerline{\spadviewport{mobius}}\newline
+
+{\bf Creating Viewport Buttons} \newline
+To make an active button that would make this viewport
+come to life, include the following: \newline
+\space{5}\\viewportbutton\{ViewButton\}\{/tmp/mobius\} \newline
+this creates this button...\newline
+%%%\centerline{\viewportbutton{ViewButton}{/tmp/mobius}}\newline
+\centerline{\spadviewportbutton{ViewButton}{mobius}}\newline
+
+{\bf Creating Active Viewports} \newline
+To merge the two things descibed above, namely, getting a picture of a
+viewport and creating a button to invoke a live viewport, you can do
+the following: \newline
+
+%\space{5}\\viewportasbutton\{/tmp/mobius\} \newline
+\centerline{\\viewportasbutton\{/tmp/mobius\}}\newline
+
+This would create a picture of a viewport that is an active
+button as well. Try it:
+
+%%%\space{5}\viewportasbutton{/tmp/mobius} \newline
+%\space{5}\spadviewportasbutton{mobius} \newline
+\centerline{\spadviewportasbutton{mobius}}\newline
+
+{\bf Including Viewports Distributed with \Language{}} \newline
+All the above commands have counterparts that allow you to
+access viewports that are already packaged with \Language{}.
+To include those viewports, just add on an "axiom" prefix
+to the above commands: \newline
+
+\centerline{\\axiomviewport\{vA\} for \\viewport\{vA\}}
+\centerline{\\axiomviewportbutton\{vB\} for \\viewportbutton\{vB\}}
+\centerline{\\axiomviewportasbutton\{vC\} for \\viewportasbutton\{vC\}} \newline
+
+All these macros really do is include some path that
+indicates where \Language{} stores the viewports.
+\newline
+\endscroll
+\autobuttons \end{page}
+
diff --git a/src/hyper/pages/graphics.pht b/src/hyper/pages/graphics.pht
new file mode 100644
index 00000000..0447e5f1
--- /dev/null
+++ b/src/hyper/pages/graphics.pht
@@ -0,0 +1,2087 @@
+\begin{patch}{AssortedGraphicsExamplePagePatch1}
+\begin{paste}{AssortedGraphicsExamplePageFull1}{AssortedGraphicsExamplePageEmpty1}
+\pastebutton{AssortedGraphicsExamplePageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(x * y), x = -2.5..2.5, y = -2.5..2.5)\bound{example1 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/AssortedGraphicsExamplePage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/AssortedGraphicsExamplePage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{AssortedGraphicsExamplePageEmpty1}
+\begin{paste}{AssortedGraphicsExamplePageEmpty1}{AssortedGraphicsExamplePagePatch1}
+\pastebutton{AssortedGraphicsExamplePageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(sin(x * y), x = -2.5..2.5, y = -2.5..2.5)\bound{example1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssortedGraphicsExamplePagePatch2}
+\begin{paste}{AssortedGraphicsExamplePageFull2}{AssortedGraphicsExamplePageEmpty2}
+\pastebutton{AssortedGraphicsExamplePageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(sin tan x - tan sin x,x = 0..6)\bound{example2 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/AssortedGraphicsExamplePage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/AssortedGraphicsExamplePage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{AssortedGraphicsExamplePageEmpty2}
+\begin{paste}{AssortedGraphicsExamplePageEmpty2}{AssortedGraphicsExamplePagePatch2}
+\pastebutton{AssortedGraphicsExamplePageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(sin tan x - tan sin x,x = 0..6)\bound{example2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssortedGraphicsExamplePagePatch3}
+\begin{paste}{AssortedGraphicsExamplePageFull3}{AssortedGraphicsExamplePageEmpty3}
+\pastebutton{AssortedGraphicsExamplePageFull3}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(sin(t)*sin(2*t), sin(3*t)*sin(4*t)), t = 0..2*\%pi)\bound{example3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/AssortedGraphicsExamplePage3.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/AssortedGraphicsExamplePage3}}
+\end{paste}\end{patch}
+
+\begin{patch}{AssortedGraphicsExamplePageEmpty3}
+\begin{paste}{AssortedGraphicsExamplePageEmpty3}{AssortedGraphicsExamplePagePatch3}
+\pastebutton{AssortedGraphicsExamplePageEmpty3}{\showpaste}
+\tab{5}\spadgraph{draw(curve(sin(t)*sin(2*t), sin(3*t)*sin(4*t)), t = 0..2*\%pi)\bound{example3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssortedGraphicsExamplePagePatch4}
+\begin{paste}{AssortedGraphicsExamplePageFull4}{AssortedGraphicsExamplePageEmpty4}
+\pastebutton{AssortedGraphicsExamplePageFull4}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(sin(t)*sin(2*t), sin(3*t)*sin(4*t), sin(5*t)*sin(6*t)), t = 0..2*\%pi)\bound{example4 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/AssortedGraphicsExamplePage4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/AssortedGraphicsExamplePage4}}
+\end{paste}\end{patch}
+
+\begin{patch}{AssortedGraphicsExamplePageEmpty4}
+\begin{paste}{AssortedGraphicsExamplePageEmpty4}{AssortedGraphicsExamplePagePatch4}
+\pastebutton{AssortedGraphicsExamplePageEmpty4}{\showpaste}
+\tab{5}\spadgraph{draw(curve(sin(t)*sin(2*t), sin(3*t)*sin(4*t), sin(5*t)*sin(6*t)), t = 0..2*\%pi)\bound{example4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssortedGraphicsExamplePagePatch5}
+\begin{paste}{AssortedGraphicsExamplePageFull5}{AssortedGraphicsExamplePageEmpty5}
+\pastebutton{AssortedGraphicsExamplePageFull5}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(17*t), t = 0..2*\%pi, coordinates == polar)\bound{example5 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/AssortedGraphicsExamplePage5.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/AssortedGraphicsExamplePage5}}
+\end{paste}\end{patch}
+
+\begin{patch}{AssortedGraphicsExamplePageEmpty5}
+\begin{paste}{AssortedGraphicsExamplePageEmpty5}{AssortedGraphicsExamplePagePatch5}
+\pastebutton{AssortedGraphicsExamplePageEmpty5}{\showpaste}
+\tab{5}\spadgraph{draw(sin(17*t), t = 0..2*\%pi, coordinates == polar)\bound{example5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssortedGraphicsExamplePagePatch6}
+\begin{paste}{AssortedGraphicsExamplePageFull6}{AssortedGraphicsExamplePageEmpty6}
+\pastebutton{AssortedGraphicsExamplePageFull6}{\hidepaste}
+\tab{5}\spadgraph{draw(y**2 + y = x**3 - x, x, y,range == [-2..2,-2..1])\bound{example6 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/AssortedGraphicsExamplePage6.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/AssortedGraphicsExamplePage6}}
+\end{paste}\end{patch}
+
+\begin{patch}{AssortedGraphicsExamplePageEmpty6}
+\begin{paste}{AssortedGraphicsExamplePageEmpty6}{AssortedGraphicsExamplePagePatch6}
+\pastebutton{AssortedGraphicsExamplePageEmpty6}{\showpaste}
+\tab{5}\spadgraph{draw(y**2 + y = x**3 - x, x, y,range == [-2..2,-2..1])\bound{example6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{AssortedGraphicsExamplePagePatch7}
+\begin{paste}{AssortedGraphicsExamplePageFull7}{AssortedGraphicsExamplePageEmpty7}
+\pastebutton{AssortedGraphicsExamplePageFull7}{\hidepaste}
+\tab{5}\spadcommand{All\free{example1 example2 example3 example4 example5 example6 }}
+\indentrel{3}\begin{verbatim}
+   (7)  All
+                                     Type: Variable All
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{AssortedGraphicsExamplePageEmpty7}
+\begin{paste}{AssortedGraphicsExamplePageEmpty7}{AssortedGraphicsExamplePagePatch7}
+\pastebutton{AssortedGraphicsExamplePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{All\free{example1 example2 example3 example4 example5 example6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePagePatch1}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageFull1}{ThreeDimensionalGraphicsExamplePageEmpty1}
+\pastebutton{ThreeDimensionalGraphicsExamplePageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(surface((1+exp(-100*u*u))*sin(\%pi*u)*sin(\%pi*v), (1+exp(-100*u*u))*sin(\%pi*u)*cos(\%pi*v), (1+exp(-100*u*u))*cos(\%pi*u)), u=0..1, v=0..2, title=="Pear")\bound{example1 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ThreeDimensionalGraphicsExamplePage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ThreeDimensionalGraphicsExamplePage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePageEmpty1}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageEmpty1}{ThreeDimensionalGraphicsExamplePagePatch1}
+\pastebutton{ThreeDimensionalGraphicsExamplePageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(surface((1+exp(-100*u*u))*sin(\%pi*u)*sin(\%pi*v), (1+exp(-100*u*u))*sin(\%pi*u)*cos(\%pi*v), (1+exp(-100*u*u))*cos(\%pi*u)), u=0..1, v=0..2, title=="Pear")\bound{example1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePagePatch2}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageFull2}{ThreeDimensionalGraphicsExamplePageEmpty2}
+\pastebutton{ThreeDimensionalGraphicsExamplePageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(surface(x*cos(y),x*sin(y),y*cos(x)), x=-4..4, y=0..2*\%pi, var1Steps==40, var2Steps==40, title=="Trig Screw")\bound{example2 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ThreeDimensionalGraphicsExamplePage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ThreeDimensionalGraphicsExamplePage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePageEmpty2}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageEmpty2}{ThreeDimensionalGraphicsExamplePagePatch2}
+\pastebutton{ThreeDimensionalGraphicsExamplePageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(surface(x*cos(y),x*sin(y),y*cos(x)), x=-4..4, y=0..2*\%pi, var1Steps==40, var2Steps==40, title=="Trig Screw")\bound{example2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePagePatch3}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageFull3}{ThreeDimensionalGraphicsExamplePageEmpty3}
+\pastebutton{ThreeDimensionalGraphicsExamplePageFull3}{\hidepaste}
+\tab{5}\spadcommand{a := 1.3 * cos(2*x) * cos(y) + sin(y) * cos(x)\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (3)  cos(x)sin(y) + 1.3 cos(2.0 x)cos(y)
+                                 Type: Expression Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePageEmpty3}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageEmpty3}{ThreeDimensionalGraphicsExamplePagePatch3}
+\pastebutton{ThreeDimensionalGraphicsExamplePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{a := 1.3 * cos(2*x) * cos(y) + sin(y) * cos(x)\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePagePatch4}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageFull4}{ThreeDimensionalGraphicsExamplePageEmpty4}
+\pastebutton{ThreeDimensionalGraphicsExamplePageFull4}{\hidepaste}
+\tab{5}\spadcommand{b := 1.3 * sin(2*x) * cos(y) - sin(y) * sin(x)\bound{b }}
+\indentrel{3}\begin{verbatim}
+   (4)  - 1.0 sin(x)sin(y) + 1.3 cos(y)sin(2.0 x)
+                                 Type: Expression Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePageEmpty4}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageEmpty4}{ThreeDimensionalGraphicsExamplePagePatch4}
+\pastebutton{ThreeDimensionalGraphicsExamplePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{b := 1.3 * sin(2*x) * cos(y) - sin(y) * sin(x)\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePagePatch5}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageFull5}{ThreeDimensionalGraphicsExamplePageEmpty5}
+\pastebutton{ThreeDimensionalGraphicsExamplePageFull5}{\hidepaste}
+\tab{5}\spadcommand{c := 2.5 * cos(y)\bound{c }}
+\indentrel{3}\begin{verbatim}
+   (5)  2.5 cos(y)
+                                 Type: Expression Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePageEmpty5}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageEmpty5}{ThreeDimensionalGraphicsExamplePagePatch5}
+\pastebutton{ThreeDimensionalGraphicsExamplePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{c := 2.5 * cos(y)\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePagePatch6}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageFull6}{ThreeDimensionalGraphicsExamplePageEmpty6}
+\pastebutton{ThreeDimensionalGraphicsExamplePageFull6}{\hidepaste}
+\tab{5}\spadgraph{draw(surface(a,b,c), x=0..\%pi, y=-\%pi..\%pi, var1Steps==40, var2Steps==40, title=="Etruscan Venus")\free{a b c }\bound{example3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ThreeDimensionalGraphicsExamplePage6.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ThreeDimensionalGraphicsExamplePage6}}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePageEmpty6}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageEmpty6}{ThreeDimensionalGraphicsExamplePagePatch6}
+\pastebutton{ThreeDimensionalGraphicsExamplePageEmpty6}{\showpaste}
+\tab{5}\spadgraph{draw(surface(a,b,c), x=0..\%pi, y=-\%pi..\%pi, var1Steps==40, var2Steps==40, title=="Etruscan Venus")\free{a b c }\bound{example3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePagePatch7}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageFull7}{ThreeDimensionalGraphicsExamplePageEmpty7}
+\pastebutton{ThreeDimensionalGraphicsExamplePageFull7}{\hidepaste}
+\tab{5}\spadcommand{f:=cos(x)*(cos(x/2)*(sqrt(2) + cos(y))+(sin(x/2)*sin(y)*cos(y)))\bound{f }}
+\indentrel{3}\begin{verbatim}
+   (7)
+                     x              x
+     cos(x)cos(y)sin(Ä)sin(y) + cos(Ä)cos(x)cos(y)
+                     2              2
+   + 
+      ÚÄ¿    x
+     \³2 cos(Ä)cos(x)
+             2
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePageEmpty7}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageEmpty7}{ThreeDimensionalGraphicsExamplePagePatch7}
+\pastebutton{ThreeDimensionalGraphicsExamplePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{f:=cos(x)*(cos(x/2)*(sqrt(2) + cos(y))+(sin(x/2)*sin(y)*cos(y)))\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePagePatch8}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageFull8}{ThreeDimensionalGraphicsExamplePageEmpty8}
+\pastebutton{ThreeDimensionalGraphicsExamplePageFull8}{\hidepaste}
+\tab{5}\spadcommand{g:=sin(x)*(cos(x/2)*(sqrt(2) + cos(y))+(sin(x/2)*sin(y)*cos(y)))\bound{g }}
+\indentrel{3}\begin{verbatim}
+   (8)
+               x
+     cos(y)sin(Ä)sin(x)sin(y)
+               2
+   + 
+          x           ÚÄ¿    x
+     (cos(Ä)cos(y) + \³2 cos(Ä))sin(x)
+          2                  2
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePageEmpty8}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageEmpty8}{ThreeDimensionalGraphicsExamplePagePatch8}
+\pastebutton{ThreeDimensionalGraphicsExamplePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{g:=sin(x)*(cos(x/2)*(sqrt(2) + cos(y))+(sin(x/2)*sin(y)*cos(y)))\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePagePatch9}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageFull9}{ThreeDimensionalGraphicsExamplePageEmpty9}
+\pastebutton{ThreeDimensionalGraphicsExamplePageFull9}{\hidepaste}
+\tab{5}\spadcommand{h:=-sin(x/2)*(sqrt(2)+cos(y)) + cos(x/2)*sin(y)*cos(y)\bound{h }}
+\indentrel{3}\begin{verbatim}
+            x                             ÚÄ¿     x
+   (9)  cos(Ä)cos(y)sin(y) + (- cos(y) - \³2 )sin(Ä)
+            2                                     2
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePageEmpty9}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageEmpty9}{ThreeDimensionalGraphicsExamplePagePatch9}
+\pastebutton{ThreeDimensionalGraphicsExamplePageEmpty9}{\showpaste}
+\tab{5}\spadcommand{h:=-sin(x/2)*(sqrt(2)+cos(y)) + cos(x/2)*sin(y)*cos(y)\bound{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePagePatch10}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageFull10}{ThreeDimensionalGraphicsExamplePageEmpty10}
+\pastebutton{ThreeDimensionalGraphicsExamplePageFull10}{\hidepaste}
+\tab{5}\spadgraph{draw(surface(f,g,h), x=0..4*\%pi, y=0..2*\%pi, var1Steps==50, var2Steps==50, title=="Banchoff Klein Bottle")\free{f g h }\bound{example4 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ThreeDimensionalGraphicsExamplePage10.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ThreeDimensionalGraphicsExamplePage10}}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePageEmpty10}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageEmpty10}{ThreeDimensionalGraphicsExamplePagePatch10}
+\pastebutton{ThreeDimensionalGraphicsExamplePageEmpty10}{\showpaste}
+\tab{5}\spadgraph{draw(surface(f,g,h), x=0..4*\%pi, y=0..2*\%pi, var1Steps==50, var2Steps==50, title=="Banchoff Klein Bottle")\free{f g h }\bound{example4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePagePatch11}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageFull11}{ThreeDimensionalGraphicsExamplePageEmpty11}
+\pastebutton{ThreeDimensionalGraphicsExamplePageFull11}{\hidepaste}
+\tab{5}\spadcommand{All\free{example1 example2 example3 example4 }}
+\indentrel{3}\begin{verbatim}
+   (11)  All
+                                     Type: Variable All
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ThreeDimensionalGraphicsExamplePageEmpty11}
+\begin{paste}{ThreeDimensionalGraphicsExamplePageEmpty11}{ThreeDimensionalGraphicsExamplePagePatch11}
+\pastebutton{ThreeDimensionalGraphicsExamplePageEmpty11}{\showpaste}
+\tab{5}\spadcommand{All\free{example1 example2 example3 example4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsExamplePagePatch1}
+\begin{paste}{OneVariableGraphicsExamplePageFull1}{OneVariableGraphicsExamplePageEmpty1}
+\pastebutton{OneVariableGraphicsExamplePageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(sin tan x - tan sin x, x = 0..6)\bound{example1 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/OneVariableGraphicsExamplePage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/OneVariableGraphicsExamplePage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsExamplePageEmpty1}
+\begin{paste}{OneVariableGraphicsExamplePageEmpty1}{OneVariableGraphicsExamplePagePatch1}
+\pastebutton{OneVariableGraphicsExamplePageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(sin tan x - tan sin x, x = 0..6)\bound{example1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsExamplePagePatch2}
+\begin{paste}{OneVariableGraphicsExamplePageFull2}{OneVariableGraphicsExamplePageEmpty2}
+\pastebutton{OneVariableGraphicsExamplePageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(sin x + cos x, x = 0..2*\%pi)\bound{example2 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/OneVariableGraphicsExamplePage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/OneVariableGraphicsExamplePage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsExamplePageEmpty2}
+\begin{paste}{OneVariableGraphicsExamplePageEmpty2}{OneVariableGraphicsExamplePagePatch2}
+\pastebutton{OneVariableGraphicsExamplePageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(sin x + cos x, x = 0..2*\%pi)\bound{example2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsExamplePagePatch3}
+\begin{paste}{OneVariableGraphicsExamplePageFull3}{OneVariableGraphicsExamplePageEmpty3}
+\pastebutton{OneVariableGraphicsExamplePageFull3}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(1/x), x = -1..1)\bound{example3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/OneVariableGraphicsExamplePage3.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/OneVariableGraphicsExamplePage3}}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsExamplePageEmpty3}
+\begin{paste}{OneVariableGraphicsExamplePageEmpty3}{OneVariableGraphicsExamplePagePatch3}
+\pastebutton{OneVariableGraphicsExamplePageEmpty3}{\showpaste}
+\tab{5}\spadgraph{draw(sin(1/x), x = -1..1)\bound{example3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsExamplePagePatch4}
+\begin{paste}{OneVariableGraphicsExamplePageFull4}{OneVariableGraphicsExamplePageEmpty4}
+\pastebutton{OneVariableGraphicsExamplePageFull4}{\hidepaste}
+\tab{5}\spadgraph{draw(x * sin(1/x), x = -1..1)\bound{example4 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/OneVariableGraphicsExamplePage4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/OneVariableGraphicsExamplePage4}}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsExamplePageEmpty4}
+\begin{paste}{OneVariableGraphicsExamplePageEmpty4}{OneVariableGraphicsExamplePagePatch4}
+\pastebutton{OneVariableGraphicsExamplePageEmpty4}{\showpaste}
+\tab{5}\spadgraph{draw(x * sin(1/x), x = -1..1)\bound{example4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsExamplePagePatch5}
+\begin{paste}{OneVariableGraphicsExamplePageFull5}{OneVariableGraphicsExamplePageEmpty5}
+\pastebutton{OneVariableGraphicsExamplePageFull5}{\hidepaste}
+\tab{5}\spadcommand{All\free{example1 example2 example3 example4 }}
+\indentrel{3}\begin{verbatim}
+   (5)  All
+                                     Type: Variable All
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsExamplePageEmpty5}
+\begin{paste}{OneVariableGraphicsExamplePageEmpty5}{OneVariableGraphicsExamplePagePatch5}
+\pastebutton{OneVariableGraphicsExamplePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{All\free{example1 example2 example3 example4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsExamplePagePatch1}
+\begin{paste}{ParametricCurveGraphicsExamplePageFull1}{ParametricCurveGraphicsExamplePageEmpty1}
+\pastebutton{ParametricCurveGraphicsExamplePageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(cos(t/(1+sin(t)**2)), sin(t)*cos(t)/(1+sin(t)**2)), t = -\%pi..\%pi)\bound{example1 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ParametricCurveGraphicsExamplePage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ParametricCurveGraphicsExamplePage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsExamplePageEmpty1}
+\begin{paste}{ParametricCurveGraphicsExamplePageEmpty1}{ParametricCurveGraphicsExamplePagePatch1}
+\pastebutton{ParametricCurveGraphicsExamplePageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(curve(cos(t/(1+sin(t)**2)), sin(t)*cos(t)/(1+sin(t)**2)), t = -\%pi..\%pi)\bound{example1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsExamplePagePatch2}
+\begin{paste}{ParametricCurveGraphicsExamplePageFull2}{ParametricCurveGraphicsExamplePageEmpty2}
+\pastebutton{ParametricCurveGraphicsExamplePageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(9*sin(3*t/4), 8*sin(t)), t = -4*\%pi..4*\%pi)\bound{example2 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ParametricCurveGraphicsExamplePage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ParametricCurveGraphicsExamplePage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsExamplePageEmpty2}
+\begin{paste}{ParametricCurveGraphicsExamplePageEmpty2}{ParametricCurveGraphicsExamplePagePatch2}
+\pastebutton{ParametricCurveGraphicsExamplePageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(curve(9*sin(3*t/4), 8*sin(t)), t = -4*\%pi..4*\%pi)\bound{example2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsExamplePagePatch3}
+\begin{paste}{ParametricCurveGraphicsExamplePageFull3}{ParametricCurveGraphicsExamplePageEmpty3}
+\pastebutton{ParametricCurveGraphicsExamplePageFull3}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)),t = 0..2*\%pi)\bound{example3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ParametricCurveGraphicsExamplePage3.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ParametricCurveGraphicsExamplePage3}}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsExamplePageEmpty3}
+\begin{paste}{ParametricCurveGraphicsExamplePageEmpty3}{ParametricCurveGraphicsExamplePagePatch3}
+\pastebutton{ParametricCurveGraphicsExamplePageEmpty3}{\showpaste}
+\tab{5}\spadgraph{draw(curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)),t = 0..2*\%pi)\bound{example3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsExamplePagePatch4}
+\begin{paste}{ParametricCurveGraphicsExamplePageFull4}{ParametricCurveGraphicsExamplePageEmpty4}
+\pastebutton{ParametricCurveGraphicsExamplePageFull4}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(cos(4*t)*cos(7*t), cos(4*t)*sin(7*t)), t = 0..2*\%pi)\bound{example4 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ParametricCurveGraphicsExamplePage4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ParametricCurveGraphicsExamplePage4}}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsExamplePageEmpty4}
+\begin{paste}{ParametricCurveGraphicsExamplePageEmpty4}{ParametricCurveGraphicsExamplePagePatch4}
+\pastebutton{ParametricCurveGraphicsExamplePageEmpty4}{\showpaste}
+\tab{5}\spadgraph{draw(curve(cos(4*t)*cos(7*t), cos(4*t)*sin(7*t)), t = 0..2*\%pi)\bound{example4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsExamplePagePatch5}
+\begin{paste}{ParametricCurveGraphicsExamplePageFull5}{ParametricCurveGraphicsExamplePageEmpty5}
+\pastebutton{ParametricCurveGraphicsExamplePageFull5}{\hidepaste}
+\tab{5}\spadcommand{All\free{example1 example2 example3 example4 }}
+\indentrel{3}\begin{verbatim}
+   (5)  All
+                                     Type: Variable All
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsExamplePageEmpty5}
+\begin{paste}{ParametricCurveGraphicsExamplePageEmpty5}{ParametricCurveGraphicsExamplePagePatch5}
+\pastebutton{ParametricCurveGraphicsExamplePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{All\free{example1 example2 example3 example4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsExamplePagePatch1}
+\begin{paste}{PolarGraphicsExamplePageFull1}{PolarGraphicsExamplePageEmpty1}
+\pastebutton{PolarGraphicsExamplePageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(1,t = 0..2*\%pi, coordinates == polar)\bound{example1 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/PolarGraphicsExamplePage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/PolarGraphicsExamplePage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsExamplePageEmpty1}
+\begin{paste}{PolarGraphicsExamplePageEmpty1}{PolarGraphicsExamplePagePatch1}
+\pastebutton{PolarGraphicsExamplePageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(1,t = 0..2*\%pi, coordinates == polar)\bound{example1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsExamplePagePatch2}
+\begin{paste}{PolarGraphicsExamplePageFull2}{PolarGraphicsExamplePageEmpty2}
+\pastebutton{PolarGraphicsExamplePageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(t,t = 0..100, coordinates == polar)\bound{example2 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/PolarGraphicsExamplePage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/PolarGraphicsExamplePage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsExamplePageEmpty2}
+\begin{paste}{PolarGraphicsExamplePageEmpty2}{PolarGraphicsExamplePagePatch2}
+\pastebutton{PolarGraphicsExamplePageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(t,t = 0..100, coordinates == polar)\bound{example2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsExamplePagePatch3}
+\begin{paste}{PolarGraphicsExamplePageFull3}{PolarGraphicsExamplePageEmpty3}
+\pastebutton{PolarGraphicsExamplePageFull3}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(4*t), t = 0..2*\%pi, coordinates == polar)\bound{example3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/PolarGraphicsExamplePage3.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/PolarGraphicsExamplePage3}}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsExamplePageEmpty3}
+\begin{paste}{PolarGraphicsExamplePageEmpty3}{PolarGraphicsExamplePagePatch3}
+\pastebutton{PolarGraphicsExamplePageEmpty3}{\showpaste}
+\tab{5}\spadgraph{draw(sin(4*t), t = 0..2*\%pi, coordinates == polar)\bound{example3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsExamplePagePatch4}
+\begin{paste}{PolarGraphicsExamplePageFull4}{PolarGraphicsExamplePageEmpty4}
+\pastebutton{PolarGraphicsExamplePageFull4}{\hidepaste}
+\tab{5}\spadgraph{draw(2 + 3 * sin t, t = 0..2*\%pi, coordinates == polar)\bound{example4 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/PolarGraphicsExamplePage4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/PolarGraphicsExamplePage4}}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsExamplePageEmpty4}
+\begin{paste}{PolarGraphicsExamplePageEmpty4}{PolarGraphicsExamplePagePatch4}
+\pastebutton{PolarGraphicsExamplePageEmpty4}{\showpaste}
+\tab{5}\spadgraph{draw(2 + 3 * sin t, t = 0..2*\%pi, coordinates == polar)\bound{example4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsExamplePagePatch5}
+\begin{paste}{PolarGraphicsExamplePageFull5}{PolarGraphicsExamplePageEmpty5}
+\pastebutton{PolarGraphicsExamplePageFull5}{\hidepaste}
+\tab{5}\spadcommand{All\free{example2 example3 example4 }}
+\indentrel{3}\begin{verbatim}
+   (5)  All
+                                     Type: Variable All
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsExamplePageEmpty5}
+\begin{paste}{PolarGraphicsExamplePageEmpty5}{PolarGraphicsExamplePagePatch5}
+\pastebutton{PolarGraphicsExamplePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{All\free{example2 example3 example4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePagePatch1}
+\begin{paste}{ImplicitCurveGraphicsExamplePageFull1}{ImplicitCurveGraphicsExamplePageEmpty1}
+\pastebutton{ImplicitCurveGraphicsExamplePageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(x * y = 1, x, y, range == [-3..3, -3..3])\bound{example1 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ImplicitCurveGraphicsExamplePage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ImplicitCurveGraphicsExamplePage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePageEmpty1}
+\begin{paste}{ImplicitCurveGraphicsExamplePageEmpty1}{ImplicitCurveGraphicsExamplePagePatch1}
+\pastebutton{ImplicitCurveGraphicsExamplePageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(x * y = 1, x, y, range == [-3..3, -3..3])\bound{example1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePagePatch2}
+\begin{paste}{ImplicitCurveGraphicsExamplePageFull2}{ImplicitCurveGraphicsExamplePageEmpty2}
+\pastebutton{ImplicitCurveGraphicsExamplePageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(y**2 + y = x**3 - x, x, y, range == [-2..2, -2..1])\bound{example2 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ImplicitCurveGraphicsExamplePage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ImplicitCurveGraphicsExamplePage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePageEmpty2}
+\begin{paste}{ImplicitCurveGraphicsExamplePageEmpty2}{ImplicitCurveGraphicsExamplePagePatch2}
+\pastebutton{ImplicitCurveGraphicsExamplePageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(y**2 + y = x**3 - x, x, y, range == [-2..2, -2..1])\bound{example2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePagePatch3}
+\begin{paste}{ImplicitCurveGraphicsExamplePageFull3}{ImplicitCurveGraphicsExamplePageEmpty3}
+\pastebutton{ImplicitCurveGraphicsExamplePageFull3}{\hidepaste}
+\tab{5}\spadcommand{p := ((x**2 + y**2 + 1) - 8*x)**2 - (8*(x**2 + y**2 + 1) - 4*x - 1)\bound{p }}
+\indentrel{3}\begin{verbatim}
+   (3)
+    4      2            2    4      3      2
+   y  + (2x  - 16x - 6)y  + x  - 16x  + 58x  - 12x - 6
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePageEmpty3}
+\begin{paste}{ImplicitCurveGraphicsExamplePageEmpty3}{ImplicitCurveGraphicsExamplePagePatch3}
+\pastebutton{ImplicitCurveGraphicsExamplePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{p := ((x**2 + y**2 + 1) - 8*x)**2 - (8*(x**2 + y**2 + 1) - 4*x - 1)\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePagePatch4}
+\begin{paste}{ImplicitCurveGraphicsExamplePageFull4}{ImplicitCurveGraphicsExamplePageEmpty4}
+\pastebutton{ImplicitCurveGraphicsExamplePageFull4}{\hidepaste}
+\tab{5}\spadgraph{draw(p = 0, x, y, range == [-1..11, -7..7], title == "Cartesian Ovals")\free{p }\bound{example3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ImplicitCurveGraphicsExamplePage4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ImplicitCurveGraphicsExamplePage4}}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePageEmpty4}
+\begin{paste}{ImplicitCurveGraphicsExamplePageEmpty4}{ImplicitCurveGraphicsExamplePagePatch4}
+\pastebutton{ImplicitCurveGraphicsExamplePageEmpty4}{\showpaste}
+\tab{5}\spadgraph{draw(p = 0, x, y, range == [-1..11, -7..7], title == "Cartesian Ovals")\free{p }\bound{example3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePagePatch5}
+\begin{paste}{ImplicitCurveGraphicsExamplePageFull5}{ImplicitCurveGraphicsExamplePageEmpty5}
+\pastebutton{ImplicitCurveGraphicsExamplePageFull5}{\hidepaste}
+\tab{5}\spadcommand{q := (x**2 + y**2 + 7**2)**2 - (6**4 + 4*7**2*x**2)\bound{q }}
+\indentrel{3}\begin{verbatim}
+         4      2       2    4      2
+   (5)  y  + (2x  + 98)y  + x  - 98x  + 1105
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePageEmpty5}
+\begin{paste}{ImplicitCurveGraphicsExamplePageEmpty5}{ImplicitCurveGraphicsExamplePagePatch5}
+\pastebutton{ImplicitCurveGraphicsExamplePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{q := (x**2 + y**2 + 7**2)**2 - (6**4 + 4*7**2*x**2)\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePagePatch6}
+\begin{paste}{ImplicitCurveGraphicsExamplePageFull6}{ImplicitCurveGraphicsExamplePageEmpty6}
+\pastebutton{ImplicitCurveGraphicsExamplePageFull6}{\hidepaste}
+\tab{5}\spadgraph{draw(q = 0, x, y, range == [-10..10, -4..4], title == "Cassinian oval with two loops")\free{q }\bound{example4 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ImplicitCurveGraphicsExamplePage6.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ImplicitCurveGraphicsExamplePage6}}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePageEmpty6}
+\begin{paste}{ImplicitCurveGraphicsExamplePageEmpty6}{ImplicitCurveGraphicsExamplePagePatch6}
+\pastebutton{ImplicitCurveGraphicsExamplePageEmpty6}{\showpaste}
+\tab{5}\spadgraph{draw(q = 0, x, y, range == [-10..10, -4..4], title == "Cassinian oval with two loops")\free{q }\bound{example4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePagePatch7}
+\begin{paste}{ImplicitCurveGraphicsExamplePageFull7}{ImplicitCurveGraphicsExamplePageEmpty7}
+\pastebutton{ImplicitCurveGraphicsExamplePageFull7}{\hidepaste}
+\tab{5}\spadcommand{All\free{example1 example2 example3 example4 }}
+\indentrel{3}\begin{verbatim}
+   (7)  All
+                                     Type: Variable All
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsExamplePageEmpty7}
+\begin{paste}{ImplicitCurveGraphicsExamplePageEmpty7}{ImplicitCurveGraphicsExamplePagePatch7}
+\pastebutton{ImplicitCurveGraphicsExamplePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{All\free{example1 example2 example3 example4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch1}
+\begin{paste}{ListPointsGraphicsExamplePageFull1}{ListPointsGraphicsExamplePageEmpty1}
+\pastebutton{ListPointsGraphicsExamplePageFull1}{\hidepaste}
+\tab{5}\spadcommand{p1 := point [1::SF,1::SF]$(Point SF)\bound{p1 }}
+\indentrel{3}\begin{verbatim}
+   (1)  [1.0,1.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty1}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty1}{ListPointsGraphicsExamplePagePatch1}
+\pastebutton{ListPointsGraphicsExamplePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{p1 := point [1::SF,1::SF]$(Point SF)\bound{p1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch2}
+\begin{paste}{ListPointsGraphicsExamplePageFull2}{ListPointsGraphicsExamplePageEmpty2}
+\pastebutton{ListPointsGraphicsExamplePageFull2}{\hidepaste}
+\tab{5}\spadcommand{p2 := point [0::SF,1::SF]$(Point SF)\bound{p2 }}
+\indentrel{3}\begin{verbatim}
+   (2)  [0.0,1.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty2}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty2}{ListPointsGraphicsExamplePagePatch2}
+\pastebutton{ListPointsGraphicsExamplePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{p2 := point [0::SF,1::SF]$(Point SF)\bound{p2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch3}
+\begin{paste}{ListPointsGraphicsExamplePageFull3}{ListPointsGraphicsExamplePageEmpty3}
+\pastebutton{ListPointsGraphicsExamplePageFull3}{\hidepaste}
+\tab{5}\spadcommand{p3 := point [0::SF,0::SF]$(Point SF)\bound{p3 }}
+\indentrel{3}\begin{verbatim}
+   (3)  [0.0,0.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty3}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty3}{ListPointsGraphicsExamplePagePatch3}
+\pastebutton{ListPointsGraphicsExamplePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{p3 := point [0::SF,0::SF]$(Point SF)\bound{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch4}
+\begin{paste}{ListPointsGraphicsExamplePageFull4}{ListPointsGraphicsExamplePageEmpty4}
+\pastebutton{ListPointsGraphicsExamplePageFull4}{\hidepaste}
+\tab{5}\spadcommand{p4 := point [1::SF,0::SF]$(Point SF)\bound{p4 }}
+\indentrel{3}\begin{verbatim}
+   (4)  [1.0,0.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty4}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty4}{ListPointsGraphicsExamplePagePatch4}
+\pastebutton{ListPointsGraphicsExamplePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{p4 := point [1::SF,0::SF]$(Point SF)\bound{p4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch5}
+\begin{paste}{ListPointsGraphicsExamplePageFull5}{ListPointsGraphicsExamplePageEmpty5}
+\pastebutton{ListPointsGraphicsExamplePageFull5}{\hidepaste}
+\tab{5}\spadcommand{p5 := point [1::SF,.5::SF]$(Point SF)\bound{p5 }}
+\indentrel{3}\begin{verbatim}
+   (5)  [1.0,0.5]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty5}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty5}{ListPointsGraphicsExamplePagePatch5}
+\pastebutton{ListPointsGraphicsExamplePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{p5 := point [1::SF,.5::SF]$(Point SF)\bound{p5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch6}
+\begin{paste}{ListPointsGraphicsExamplePageFull6}{ListPointsGraphicsExamplePageEmpty6}
+\pastebutton{ListPointsGraphicsExamplePageFull6}{\hidepaste}
+\tab{5}\spadcommand{p6 := point [.5::SF,0::SF]$(Point SF)\bound{p6 }}
+\indentrel{3}\begin{verbatim}
+   (6)  [0.5,0.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty6}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty6}{ListPointsGraphicsExamplePagePatch6}
+\pastebutton{ListPointsGraphicsExamplePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{p6 := point [.5::SF,0::SF]$(Point SF)\bound{p6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch7}
+\begin{paste}{ListPointsGraphicsExamplePageFull7}{ListPointsGraphicsExamplePageEmpty7}
+\pastebutton{ListPointsGraphicsExamplePageFull7}{\hidepaste}
+\tab{5}\spadcommand{p7 := point [0::SF,0.5::SF]$(Point SF)\bound{p7 }}
+\indentrel{3}\begin{verbatim}
+   (7)  [0.0,0.5]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty7}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty7}{ListPointsGraphicsExamplePagePatch7}
+\pastebutton{ListPointsGraphicsExamplePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{p7 := point [0::SF,0.5::SF]$(Point SF)\bound{p7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch8}
+\begin{paste}{ListPointsGraphicsExamplePageFull8}{ListPointsGraphicsExamplePageEmpty8}
+\pastebutton{ListPointsGraphicsExamplePageFull8}{\hidepaste}
+\tab{5}\spadcommand{p8 := point [.5::SF,1::SF]$(Point SF)\bound{p8 }}
+\indentrel{3}\begin{verbatim}
+   (8)  [0.5,1.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty8}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty8}{ListPointsGraphicsExamplePagePatch8}
+\pastebutton{ListPointsGraphicsExamplePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{p8 := point [.5::SF,1::SF]$(Point SF)\bound{p8 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch9}
+\begin{paste}{ListPointsGraphicsExamplePageFull9}{ListPointsGraphicsExamplePageEmpty9}
+\pastebutton{ListPointsGraphicsExamplePageFull9}{\hidepaste}
+\tab{5}\spadcommand{p9 := point [.25::SF,.25::SF]$(Point SF)\bound{p9 }}
+\indentrel{3}\begin{verbatim}
+   (9)  [0.25,0.25]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty9}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty9}{ListPointsGraphicsExamplePagePatch9}
+\pastebutton{ListPointsGraphicsExamplePageEmpty9}{\showpaste}
+\tab{5}\spadcommand{p9 := point [.25::SF,.25::SF]$(Point SF)\bound{p9 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch10}
+\begin{paste}{ListPointsGraphicsExamplePageFull10}{ListPointsGraphicsExamplePageEmpty10}
+\pastebutton{ListPointsGraphicsExamplePageFull10}{\hidepaste}
+\tab{5}\spadcommand{p10 := point [.25::SF,.75::SF]$(Point SF)\bound{p10 }}
+\indentrel{3}\begin{verbatim}
+   (10)  [0.25,0.75]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty10}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty10}{ListPointsGraphicsExamplePagePatch10}
+\pastebutton{ListPointsGraphicsExamplePageEmpty10}{\showpaste}
+\tab{5}\spadcommand{p10 := point [.25::SF,.75::SF]$(Point SF)\bound{p10 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch11}
+\begin{paste}{ListPointsGraphicsExamplePageFull11}{ListPointsGraphicsExamplePageEmpty11}
+\pastebutton{ListPointsGraphicsExamplePageFull11}{\hidepaste}
+\tab{5}\spadcommand{p11 := point [.75::SF,.75::SF]$(Point SF)\bound{p11 }}
+\indentrel{3}\begin{verbatim}
+   (11)  [0.75,0.75]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty11}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty11}{ListPointsGraphicsExamplePagePatch11}
+\pastebutton{ListPointsGraphicsExamplePageEmpty11}{\showpaste}
+\tab{5}\spadcommand{p11 := point [.75::SF,.75::SF]$(Point SF)\bound{p11 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch12}
+\begin{paste}{ListPointsGraphicsExamplePageFull12}{ListPointsGraphicsExamplePageEmpty12}
+\pastebutton{ListPointsGraphicsExamplePageFull12}{\hidepaste}
+\tab{5}\spadcommand{p12 := point [.75::SF,.25::SF]$(Point SF)\bound{p12 }}
+\indentrel{3}\begin{verbatim}
+   (12)  [0.75,0.25]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty12}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty12}{ListPointsGraphicsExamplePagePatch12}
+\pastebutton{ListPointsGraphicsExamplePageEmpty12}{\showpaste}
+\tab{5}\spadcommand{p12 := point [.75::SF,.25::SF]$(Point SF)\bound{p12 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch13}
+\begin{paste}{ListPointsGraphicsExamplePageFull13}{ListPointsGraphicsExamplePageEmpty13}
+\pastebutton{ListPointsGraphicsExamplePageFull13}{\hidepaste}
+\tab{5}\spadcommand{llp := [[p1,p2],[p2,p3],[p3,p4],[p4,p1],[p5,p6],[p6,p7],[p7,p8],[p8,p5],[p9,p10],[p10,p11],[p11,p12],[p12,p9]]\bound{llp }\free{p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 }}
+\indentrel{3}\begin{verbatim}
+   (13)
+   [[[1.0,1.0],[0.0,1.0]], [[0.0,1.0],[0.0,0.0]],
+    [[0.0,0.0],[1.0,0.0]], [[1.0,0.0],[1.0,1.0]],
+    [[1.0,0.5],[0.5,0.0]], [[0.5,0.0],[0.0,0.5]],
+    [[0.0,0.5],[0.5,1.0]], [[0.5,1.0],[1.0,0.5]],
+    [[0.25,0.25],[0.25,0.75]], [[0.25,0.75],[0.75,0.75]],
+    [[0.75,0.75],[0.75,0.25]], [[0.75,0.25],[0.25,0.25]]]
+                      Type: List List Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty13}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty13}{ListPointsGraphicsExamplePagePatch13}
+\pastebutton{ListPointsGraphicsExamplePageEmpty13}{\showpaste}
+\tab{5}\spadcommand{llp := [[p1,p2],[p2,p3],[p3,p4],[p4,p1],[p5,p6],[p6,p7],[p7,p8],[p8,p5],[p9,p10],[p10,p11],[p11,p12],[p12,p9]]\bound{llp }\free{p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch14}
+\begin{paste}{ListPointsGraphicsExamplePageFull14}{ListPointsGraphicsExamplePageEmpty14}
+\pastebutton{ListPointsGraphicsExamplePageFull14}{\hidepaste}
+\tab{5}\spadcommand{size1 := 6::PositiveInteger\bound{size1 }}
+\indentrel{3}\begin{verbatim}
+   (14)  6
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty14}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty14}{ListPointsGraphicsExamplePagePatch14}
+\pastebutton{ListPointsGraphicsExamplePageEmpty14}{\showpaste}
+\tab{5}\spadcommand{size1 := 6::PositiveInteger\bound{size1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch15}
+\begin{paste}{ListPointsGraphicsExamplePageFull15}{ListPointsGraphicsExamplePageEmpty15}
+\pastebutton{ListPointsGraphicsExamplePageFull15}{\hidepaste}
+\tab{5}\spadcommand{size2 := 8::PositiveInteger\bound{size2 }}
+\indentrel{3}\begin{verbatim}
+   (15)  8
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty15}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty15}{ListPointsGraphicsExamplePagePatch15}
+\pastebutton{ListPointsGraphicsExamplePageEmpty15}{\showpaste}
+\tab{5}\spadcommand{size2 := 8::PositiveInteger\bound{size2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch16}
+\begin{paste}{ListPointsGraphicsExamplePageFull16}{ListPointsGraphicsExamplePageEmpty16}
+\pastebutton{ListPointsGraphicsExamplePageFull16}{\hidepaste}
+\tab{5}\spadcommand{size3 := 10::PositiveInteger\bound{size3 }}
+\indentrel{3}\begin{verbatim}
+   (16)  10
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty16}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty16}{ListPointsGraphicsExamplePagePatch16}
+\pastebutton{ListPointsGraphicsExamplePageEmpty16}{\showpaste}
+\tab{5}\spadcommand{size3 := 10::PositiveInteger\bound{size3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch17}
+\begin{paste}{ListPointsGraphicsExamplePageFull17}{ListPointsGraphicsExamplePageEmpty17}
+\pastebutton{ListPointsGraphicsExamplePageFull17}{\hidepaste}
+\tab{5}\spadcommand{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3]\bound{lsize }\free{size1 size2 size3 }}
+\indentrel{3}\begin{verbatim}
+   (17)  [6,6,6,6,8,8,8,8,10,10,10,10]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty17}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty17}{ListPointsGraphicsExamplePagePatch17}
+\pastebutton{ListPointsGraphicsExamplePageEmpty17}{\showpaste}
+\tab{5}\spadcommand{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3]\bound{lsize }\free{size1 size2 size3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch18}
+\begin{paste}{ListPointsGraphicsExamplePageFull18}{ListPointsGraphicsExamplePageEmpty18}
+\pastebutton{ListPointsGraphicsExamplePageFull18}{\hidepaste}
+\tab{5}\spadcommand{pc1 := pastel red()\bound{pc1 }}
+\indentrel{3}\begin{verbatim}
+   (18)  [Hue: 1  Weight: 1.0] from the Pastel palette
+                                          Type: Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty18}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty18}{ListPointsGraphicsExamplePagePatch18}
+\pastebutton{ListPointsGraphicsExamplePageEmpty18}{\showpaste}
+\tab{5}\spadcommand{pc1 := pastel red()\bound{pc1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch19}
+\begin{paste}{ListPointsGraphicsExamplePageFull19}{ListPointsGraphicsExamplePageEmpty19}
+\pastebutton{ListPointsGraphicsExamplePageFull19}{\hidepaste}
+\tab{5}\spadcommand{pc2 := dim green()\bound{pc2 }}
+\indentrel{3}\begin{verbatim}
+   (19)  [Hue: 14  Weight: 1.0] from the Dim palette
+                                          Type: Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty19}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty19}{ListPointsGraphicsExamplePagePatch19}
+\pastebutton{ListPointsGraphicsExamplePageEmpty19}{\showpaste}
+\tab{5}\spadcommand{pc2 := dim green()\bound{pc2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch20}
+\begin{paste}{ListPointsGraphicsExamplePageFull20}{ListPointsGraphicsExamplePageEmpty20}
+\pastebutton{ListPointsGraphicsExamplePageFull20}{\hidepaste}
+\tab{5}\spadcommand{pc3 := pastel yellow()\bound{pc3 }}
+\indentrel{3}\begin{verbatim}
+   (20)  [Hue: 11  Weight: 1.0] from the Pastel palette
+                                          Type: Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty20}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty20}{ListPointsGraphicsExamplePagePatch20}
+\pastebutton{ListPointsGraphicsExamplePageEmpty20}{\showpaste}
+\tab{5}\spadcommand{pc3 := pastel yellow()\bound{pc3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch21}
+\begin{paste}{ListPointsGraphicsExamplePageFull21}{ListPointsGraphicsExamplePageEmpty21}
+\pastebutton{ListPointsGraphicsExamplePageFull21}{\hidepaste}
+\tab{5}\spadcommand{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3]\bound{lpc }\free{pc1 pc2 pc3 }}
+\indentrel{3}\begin{verbatim}
+   (21)
+   [[Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 11  Weight: 1.0] from the Pastel palette,
+    [Hue: 11  Weight: 1.0] from the Pastel palette,
+    [Hue: 11  Weight: 1.0] from the Pastel palette,
+    [Hue: 11  Weight: 1.0] from the Pastel palette]
+                                     Type: List Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty21}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty21}{ListPointsGraphicsExamplePagePatch21}
+\pastebutton{ListPointsGraphicsExamplePageEmpty21}{\showpaste}
+\tab{5}\spadcommand{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3]\bound{lpc }\free{pc1 pc2 pc3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch22}
+\begin{paste}{ListPointsGraphicsExamplePageFull22}{ListPointsGraphicsExamplePageEmpty22}
+\pastebutton{ListPointsGraphicsExamplePageFull22}{\hidepaste}
+\tab{5}\spadcommand{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()]\bound{lc }}
+\indentrel{3}\begin{verbatim}
+   (22)
+   [[Hue: 22  Weight: 1.0] from the Pastel palette,
+    [Hue: 11  Weight: 1.0] from the Light palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 1  Weight: 1.0] from the Bright palette,
+    [Hue: 14  Weight: 1.0] from the Light palette,
+    [Hue: 11  Weight: 1.0] from the Dim palette,
+    [Hue: 22  Weight: 1.0] from the Bright palette,
+    [Hue: 1  Weight: 1.0] from the Dark palette,
+    [Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 22  Weight: 1.0] from the Light palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 11  Weight: 1.0] from the Light palette]
+                                     Type: List Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty22}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty22}{ListPointsGraphicsExamplePagePatch22}
+\pastebutton{ListPointsGraphicsExamplePageEmpty22}{\showpaste}
+\tab{5}\spadcommand{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()]\bound{lc }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch23}
+\begin{paste}{ListPointsGraphicsExamplePageFull23}{ListPointsGraphicsExamplePageEmpty23}
+\pastebutton{ListPointsGraphicsExamplePageFull23}{\hidepaste}
+\tab{5}\spadcommand{g := makeGraphImage(llp,lpc,lc,lsize)$GRIMAGE\bound{g }\free{llp lpc lc lsize }}
+\indentrel{3}\begin{verbatim}
+   (23)  Graph with 12 point lists
+                                       Type: GraphImage
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty23}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty23}{ListPointsGraphicsExamplePagePatch23}
+\pastebutton{ListPointsGraphicsExamplePageEmpty23}{\showpaste}
+\tab{5}\spadcommand{g := makeGraphImage(llp,lpc,lc,lsize)$GRIMAGE\bound{g }\free{llp lpc lc lsize }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePagePatch24}
+\begin{paste}{ListPointsGraphicsExamplePageFull24}{ListPointsGraphicsExamplePageEmpty24}
+\pastebutton{ListPointsGraphicsExamplePageFull24}{\hidepaste}
+\tab{5}\spadgraph{makeViewport2D(g,[title("Lines")])$VIEW2D\free{g }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ListPointsGraphicsExamplePage24.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ListPointsGraphicsExamplePage24}}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsExamplePageEmpty24}
+\begin{paste}{ListPointsGraphicsExamplePageEmpty24}{ListPointsGraphicsExamplePagePatch24}
+\pastebutton{ListPointsGraphicsExamplePageEmpty24}{\showpaste}
+\tab{5}\spadgraph{makeViewport2D(g,[title("Lines")])$VIEW2D\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoVariableGraphicsPagePatch1}
+\begin{paste}{TwoVariableGraphicsPageFull1}{TwoVariableGraphicsPageEmpty1}
+\pastebutton{TwoVariableGraphicsPageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/TwoVariableGraphicsPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/TwoVariableGraphicsPage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoVariableGraphicsPageEmpty1}
+\begin{paste}{TwoVariableGraphicsPageEmpty1}{TwoVariableGraphicsPagePatch1}
+\pastebutton{TwoVariableGraphicsPageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3)}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoVariableGraphicsPagePatch2}
+\begin{paste}{TwoVariableGraphicsPageFull2}{TwoVariableGraphicsPageEmpty2}
+\pastebutton{TwoVariableGraphicsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f(x:SF,y:SF):SF == sin(x)*cos(y)\bound{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{TwoVariableGraphicsPageEmpty2}
+\begin{paste}{TwoVariableGraphicsPageEmpty2}{TwoVariableGraphicsPagePatch2}
+\pastebutton{TwoVariableGraphicsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f(x:SF,y:SF):SF == sin(x)*cos(y)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoVariableGraphicsPagePatch3}
+\begin{paste}{TwoVariableGraphicsPageFull3}{TwoVariableGraphicsPageEmpty3}
+\pastebutton{TwoVariableGraphicsPageFull3}{\hidepaste}
+\tab{5}\spadgraph{draw(f,-\%pi..\%pi,-\%pi..\%pi)\free{f }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/TwoVariableGraphicsPage3.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/TwoVariableGraphicsPage3}}
+\end{paste}\end{patch}
+
+\begin{patch}{TwoVariableGraphicsPageEmpty3}
+\begin{paste}{TwoVariableGraphicsPageEmpty3}{TwoVariableGraphicsPagePatch3}
+\pastebutton{TwoVariableGraphicsPageEmpty3}{\showpaste}
+\tab{5}\spadgraph{draw(f,-\%pi..\%pi,-\%pi..\%pi)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{SpaceCurveGraphicsPagePatch1}
+\begin{paste}{SpaceCurveGraphicsPageFull1}{SpaceCurveGraphicsPageEmpty1}
+\pastebutton{SpaceCurveGraphicsPageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(cos(t),sin(t),t), t=-12..12)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/SpaceCurveGraphicsPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/SpaceCurveGraphicsPage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{SpaceCurveGraphicsPageEmpty1}
+\begin{paste}{SpaceCurveGraphicsPageEmpty1}{SpaceCurveGraphicsPagePatch1}
+\pastebutton{SpaceCurveGraphicsPageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(curve(cos(t),sin(t),t), t=-12..12)}
+\end{paste}\end{patch}
+
+\begin{patch}{SpaceCurveGraphicsPagePatch2}
+\begin{paste}{SpaceCurveGraphicsPageFull2}{SpaceCurveGraphicsPageEmpty2}
+\pastebutton{SpaceCurveGraphicsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{i1(t:SF):SF == sin(t)*cos(3*t/5)\bound{i1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SpaceCurveGraphicsPageEmpty2}
+\begin{paste}{SpaceCurveGraphicsPageEmpty2}{SpaceCurveGraphicsPagePatch2}
+\pastebutton{SpaceCurveGraphicsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{i1(t:SF):SF == sin(t)*cos(3*t/5)\bound{i1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{SpaceCurveGraphicsPagePatch3}
+\begin{paste}{SpaceCurveGraphicsPageFull3}{SpaceCurveGraphicsPageEmpty3}
+\pastebutton{SpaceCurveGraphicsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{i2(t:SF):SF == cos(t)*cos(3*t/5)\bound{i2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SpaceCurveGraphicsPageEmpty3}
+\begin{paste}{SpaceCurveGraphicsPageEmpty3}{SpaceCurveGraphicsPagePatch3}
+\pastebutton{SpaceCurveGraphicsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{i2(t:SF):SF == cos(t)*cos(3*t/5)\bound{i2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{SpaceCurveGraphicsPagePatch4}
+\begin{paste}{SpaceCurveGraphicsPageFull4}{SpaceCurveGraphicsPageEmpty4}
+\pastebutton{SpaceCurveGraphicsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{i3(t:SF):SF == cos(t)*sin(3*t/5)\bound{i3 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{SpaceCurveGraphicsPageEmpty4}
+\begin{paste}{SpaceCurveGraphicsPageEmpty4}{SpaceCurveGraphicsPagePatch4}
+\pastebutton{SpaceCurveGraphicsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{i3(t:SF):SF == cos(t)*sin(3*t/5)\bound{i3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{SpaceCurveGraphicsPagePatch5}
+\begin{paste}{SpaceCurveGraphicsPageFull5}{SpaceCurveGraphicsPageEmpty5}
+\pastebutton{SpaceCurveGraphicsPageFull5}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(i1,i2,i3),0..15*\%pi)\free{i1 i2 i3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/SpaceCurveGraphicsPage5.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/SpaceCurveGraphicsPage5}}
+\end{paste}\end{patch}
+
+\begin{patch}{SpaceCurveGraphicsPageEmpty5}
+\begin{paste}{SpaceCurveGraphicsPageEmpty5}{SpaceCurveGraphicsPagePatch5}
+\pastebutton{SpaceCurveGraphicsPageEmpty5}{\showpaste}
+\tab{5}\spadgraph{draw(curve(i1,i2,i3),0..15*\%pi)\free{i1 i2 i3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricTubeGraphicsPagePatch1}
+\begin{paste}{ParametricTubeGraphicsPageFull1}{ParametricTubeGraphicsPageEmpty1}
+\pastebutton{ParametricTubeGraphicsPageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(sin(t)*cos(3*t/5), cos(t)*cos(3*t/5), cos(t)*sin(3*t/5)), t=0..15*\%pi,tubeRadius == .15)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ParametricTubeGraphicsPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ParametricTubeGraphicsPage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricTubeGraphicsPageEmpty1}
+\begin{paste}{ParametricTubeGraphicsPageEmpty1}{ParametricTubeGraphicsPagePatch1}
+\pastebutton{ParametricTubeGraphicsPageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(curve(sin(t)*cos(3*t/5), cos(t)*cos(3*t/5), cos(t)*sin(3*t/5)), t=0..15*\%pi,tubeRadius == .15)}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricTubeGraphicsPagePatch2}
+\begin{paste}{ParametricTubeGraphicsPageFull2}{ParametricTubeGraphicsPageEmpty2}
+\pastebutton{ParametricTubeGraphicsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{t1(t:SF):SF == 4/(2-sin(3*t))*cos(2*t)\bound{t1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ParametricTubeGraphicsPageEmpty2}
+\begin{paste}{ParametricTubeGraphicsPageEmpty2}{ParametricTubeGraphicsPagePatch2}
+\pastebutton{ParametricTubeGraphicsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{t1(t:SF):SF == 4/(2-sin(3*t))*cos(2*t)\bound{t1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricTubeGraphicsPagePatch3}
+\begin{paste}{ParametricTubeGraphicsPageFull3}{ParametricTubeGraphicsPageEmpty3}
+\pastebutton{ParametricTubeGraphicsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{t2(t:SF):SF == 4/(2-sin(3*t))*sin(2*t)\bound{t2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ParametricTubeGraphicsPageEmpty3}
+\begin{paste}{ParametricTubeGraphicsPageEmpty3}{ParametricTubeGraphicsPagePatch3}
+\pastebutton{ParametricTubeGraphicsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{t2(t:SF):SF == 4/(2-sin(3*t))*sin(2*t)\bound{t2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricTubeGraphicsPagePatch4}
+\begin{paste}{ParametricTubeGraphicsPageFull4}{ParametricTubeGraphicsPageEmpty4}
+\pastebutton{ParametricTubeGraphicsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{t3(t:SF):SF == 4/(2-sin(3*t))*cos(3*t)\bound{t3 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ParametricTubeGraphicsPageEmpty4}
+\begin{paste}{ParametricTubeGraphicsPageEmpty4}{ParametricTubeGraphicsPagePatch4}
+\pastebutton{ParametricTubeGraphicsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{t3(t:SF):SF == 4/(2-sin(3*t))*cos(3*t)\bound{t3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricTubeGraphicsPagePatch5}
+\begin{paste}{ParametricTubeGraphicsPageFull5}{ParametricTubeGraphicsPageEmpty5}
+\pastebutton{ParametricTubeGraphicsPageFull5}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(t1,t2,t3),0..2*\%pi,tubeRadius == .2)\free{t1 t2 t3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ParametricTubeGraphicsPage5.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ParametricTubeGraphicsPage5}}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricTubeGraphicsPageEmpty5}
+\begin{paste}{ParametricTubeGraphicsPageEmpty5}{ParametricTubeGraphicsPagePatch5}
+\pastebutton{ParametricTubeGraphicsPageEmpty5}{\showpaste}
+\tab{5}\spadgraph{draw(curve(t1,t2,t3),0..2*\%pi,tubeRadius == .2)\free{t1 t2 t3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricSurfaceGraphicsPagePatch1}
+\begin{paste}{ParametricSurfaceGraphicsPageFull1}{ParametricSurfaceGraphicsPageEmpty1}
+\pastebutton{ParametricSurfaceGraphicsPageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(surface(u*cos(v), u*sin(v),v*cos(u)),u=-4..4,v=0..2*\%pi, coordinates== parabolicCylindrical)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ParametricSurfaceGraphicsPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ParametricSurfaceGraphicsPage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricSurfaceGraphicsPageEmpty1}
+\begin{paste}{ParametricSurfaceGraphicsPageEmpty1}{ParametricSurfaceGraphicsPagePatch1}
+\pastebutton{ParametricSurfaceGraphicsPageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(surface(u*cos(v), u*sin(v),v*cos(u)),u=-4..4,v=0..2*\%pi, coordinates== parabolicCylindrical)}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricSurfaceGraphicsPagePatch2}
+\begin{paste}{ParametricSurfaceGraphicsPageFull2}{ParametricSurfaceGraphicsPageEmpty2}
+\pastebutton{ParametricSurfaceGraphicsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{n1(u:SF,v:SF):SF == u*cos(v)\bound{n1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ParametricSurfaceGraphicsPageEmpty2}
+\begin{paste}{ParametricSurfaceGraphicsPageEmpty2}{ParametricSurfaceGraphicsPagePatch2}
+\pastebutton{ParametricSurfaceGraphicsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{n1(u:SF,v:SF):SF == u*cos(v)\bound{n1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricSurfaceGraphicsPagePatch3}
+\begin{paste}{ParametricSurfaceGraphicsPageFull3}{ParametricSurfaceGraphicsPageEmpty3}
+\pastebutton{ParametricSurfaceGraphicsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{n2(u:SF,v:SF):SF == u*sin(v)\bound{n2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ParametricSurfaceGraphicsPageEmpty3}
+\begin{paste}{ParametricSurfaceGraphicsPageEmpty3}{ParametricSurfaceGraphicsPagePatch3}
+\pastebutton{ParametricSurfaceGraphicsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{n2(u:SF,v:SF):SF == u*sin(v)\bound{n2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricSurfaceGraphicsPagePatch4}
+\begin{paste}{ParametricSurfaceGraphicsPageFull4}{ParametricSurfaceGraphicsPageEmpty4}
+\pastebutton{ParametricSurfaceGraphicsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{n3(u:SF,v:SF):SF == u\bound{n3 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ParametricSurfaceGraphicsPageEmpty4}
+\begin{paste}{ParametricSurfaceGraphicsPageEmpty4}{ParametricSurfaceGraphicsPagePatch4}
+\pastebutton{ParametricSurfaceGraphicsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{n3(u:SF,v:SF):SF == u\bound{n3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricSurfaceGraphicsPagePatch5}
+\begin{paste}{ParametricSurfaceGraphicsPageFull5}{ParametricSurfaceGraphicsPageEmpty5}
+\pastebutton{ParametricSurfaceGraphicsPageFull5}{\hidepaste}
+\tab{5}\spadgraph{draw(surface(n1,n2,n3), 1.0..4.0, 1.0..4*\%pi, coordinates == toroidal(1$SF))\free{n1 n2 n3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ParametricSurfaceGraphicsPage5.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ParametricSurfaceGraphicsPage5}}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricSurfaceGraphicsPageEmpty5}
+\begin{paste}{ParametricSurfaceGraphicsPageEmpty5}{ParametricSurfaceGraphicsPagePatch5}
+\pastebutton{ParametricSurfaceGraphicsPageEmpty5}{\showpaste}
+\tab{5}\spadgraph{draw(surface(n1,n2,n3), 1.0..4.0, 1.0..4*\%pi, coordinates == toroidal(1$SF))\free{n1 n2 n3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPagePatch1}
+\begin{paste}{3DObjectGraphicsPageFull1}{3DObjectGraphicsPageEmpty1}
+\pastebutton{3DObjectGraphicsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{space := create3Space()$(ThreeSpace SF)\bound{space }}
+\indentrel{3}\begin{verbatim}
+   (1)  3-Space with 0 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPageEmpty1}
+\begin{paste}{3DObjectGraphicsPageEmpty1}{3DObjectGraphicsPagePatch1}
+\pastebutton{3DObjectGraphicsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{space := create3Space()$(ThreeSpace SF)\bound{space }}
+\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPagePatch2}
+\begin{paste}{3DObjectGraphicsPageFull2}{3DObjectGraphicsPageEmpty2}
+\pastebutton{3DObjectGraphicsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{curve(space,[[0,20,20],[0,20,30],[0,30,30],[0,30,100], [0,20,100],[0,20,110],[0,50,110],[0,50,100],[0,40,100], [0,40,30],[0,50,30],[0,50,20],[0,20,20]])\bound{curveI }}
+\indentrel{3}\begin{verbatim}
+   (2)  3-Space with 1 component
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPageEmpty2}
+\begin{paste}{3DObjectGraphicsPageEmpty2}{3DObjectGraphicsPagePatch2}
+\pastebutton{3DObjectGraphicsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{curve(space,[[0,20,20],[0,20,30],[0,30,30],[0,30,100], [0,20,100],[0,20,110],[0,50,110],[0,50,100],[0,40,100], [0,40,30],[0,50,30],[0,50,20],[0,20,20]])\bound{curveI }}
+\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPagePatch3}
+\begin{paste}{3DObjectGraphicsPageFull3}{3DObjectGraphicsPageEmpty3}
+\pastebutton{3DObjectGraphicsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{curve(space,[[0,80,20],[0,70,20],[0,70,110],[0,110,110], [0,120,100],[0,120,70],[0,115,65],[0,120,60],[0,120,30], [0,110,20],[0,80,20],[0,80,30],[0,105,30],[0,110,35]])\bound{curveB1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  3-Space with 2 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPageEmpty3}
+\begin{paste}{3DObjectGraphicsPageEmpty3}{3DObjectGraphicsPagePatch3}
+\pastebutton{3DObjectGraphicsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{curve(space,[[0,80,20],[0,70,20],[0,70,110],[0,110,110], [0,120,100],[0,120,70],[0,115,65],[0,120,60],[0,120,30], [0,110,20],[0,80,20],[0,80,30],[0,105,30],[0,110,35]])\bound{curveB1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPagePatch4}
+\begin{paste}{3DObjectGraphicsPageFull4}{3DObjectGraphicsPageEmpty4}
+\pastebutton{3DObjectGraphicsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{curve(space,[[0,110,35],[0,110,55],[0,105,60],[0,80,60],[0,80,70], [0,105,70],[0,110,75],[0,110,95],[0,105,100],[0,80,100], [0,80,30]])\bound{curveB2 }}
+\indentrel{3}\begin{verbatim}
+   (4)  3-Space with 3 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPageEmpty4}
+\begin{paste}{3DObjectGraphicsPageEmpty4}{3DObjectGraphicsPagePatch4}
+\pastebutton{3DObjectGraphicsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{curve(space,[[0,110,35],[0,110,55],[0,105,60],[0,80,60],[0,80,70], [0,105,70],[0,110,75],[0,110,95],[0,105,100],[0,80,100], [0,80,30]])\bound{curveB2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPagePatch5}
+\begin{paste}{3DObjectGraphicsPageFull5}{3DObjectGraphicsPageEmpty5}
+\pastebutton{3DObjectGraphicsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{closedCurve(space,[[0,140,20],[0,140,110],[0,150,110],[0,170,50], [0,190,110],[0,200,110],[0,200,20],[0,190,20],[0,190,75], [0,175,35],[0,165,35],[0,150,75],[0,150,20]])\bound{curveM }}
+\indentrel{3}\begin{verbatim}
+   (5)  3-Space with 4 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPageEmpty5}
+\begin{paste}{3DObjectGraphicsPageEmpty5}{3DObjectGraphicsPagePatch5}
+\pastebutton{3DObjectGraphicsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{closedCurve(space,[[0,140,20],[0,140,110],[0,150,110],[0,170,50], [0,190,110],[0,200,110],[0,200,20],[0,190,20],[0,190,75], [0,175,35],[0,165,35],[0,150,75],[0,150,20]])\bound{curveM }}
+\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPagePatch6}
+\begin{paste}{3DObjectGraphicsPageFull6}{3DObjectGraphicsPageEmpty6}
+\pastebutton{3DObjectGraphicsPageFull6}{\hidepaste}
+\tab{5}\spadcommand{closedCurve(space,[[200,0,20], [200,0,110], [185,0,110], [160,0,45], [160,0,110], [150,0,110], [150,0,20], [165,0,20], [190,0,85], [190,0,20]])\bound{curveN }}
+\indentrel{3}\begin{verbatim}
+   (6)  3-Space with 5 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPageEmpty6}
+\begin{paste}{3DObjectGraphicsPageEmpty6}{3DObjectGraphicsPagePatch6}
+\pastebutton{3DObjectGraphicsPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{closedCurve(space,[[200,0,20], [200,0,110], [185,0,110], [160,0,45], [160,0,110], [150,0,110], [150,0,20], [165,0,20], [190,0,85], [190,0,20]])\bound{curveN }}
+\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPagePatch7}
+\begin{paste}{3DObjectGraphicsPageFull7}{3DObjectGraphicsPageEmpty7}
+\pastebutton{3DObjectGraphicsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{closedCurve(space,[[140,0,20], [120,0,110], [110,0,110], [90,0,20], [100,0,20], [108,0,50], [123,0,50], [121,0,60], [110,0,60], [115,0,90], [130,0,20]])\bound{curveA }}
+\indentrel{3}\begin{verbatim}
+   (7)  3-Space with 6 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPageEmpty7}
+\begin{paste}{3DObjectGraphicsPageEmpty7}{3DObjectGraphicsPagePatch7}
+\pastebutton{3DObjectGraphicsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{closedCurve(space,[[140,0,20], [120,0,110], [110,0,110], [90,0,20], [100,0,20], [108,0,50], [123,0,50], [121,0,60], [110,0,60], [115,0,90], [130,0,20]])\bound{curveA }}
+\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPagePatch8}
+\begin{paste}{3DObjectGraphicsPageFull8}{3DObjectGraphicsPageEmpty8}
+\pastebutton{3DObjectGraphicsPageFull8}{\hidepaste}
+\tab{5}\spadcommand{closedCurve(space,[[80,0,30], [80,0,100], [70,0,110], [40,0,110], [30,0,100], [30,0,90], [40,0,90], [40,0,95], [45,0,100], [65,0,100], [70,0,95], [70,0,35], [65,0,30], [45,0,30], [40,0,35], [40,0,60], [50,0,60], [50,0,70], [30,0,70], [30,0,30], [40,0,20], [70,0,20]])\bound{curveG }}
+\indentrel{3}\begin{verbatim}
+   (8)  3-Space with 7 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPageEmpty8}
+\begin{paste}{3DObjectGraphicsPageEmpty8}{3DObjectGraphicsPagePatch8}
+\pastebutton{3DObjectGraphicsPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{closedCurve(space,[[80,0,30], [80,0,100], [70,0,110], [40,0,110], [30,0,100], [30,0,90], [40,0,90], [40,0,95], [45,0,100], [65,0,100], [70,0,95], [70,0,35], [65,0,30], [45,0,30], [40,0,35], [40,0,60], [50,0,60], [50,0,70], [30,0,70], [30,0,30], [40,0,20], [70,0,20]])\bound{curveG }}
+\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPagePatch9}
+\begin{paste}{3DObjectGraphicsPageFull9}{3DObjectGraphicsPageEmpty9}
+\pastebutton{3DObjectGraphicsPageFull9}{\hidepaste}
+\tab{5}\spadgraph{makeViewport3D(space,[title("Curves")])$VIEW3D\free{space curveI curveB1 curveB2 curveM curveN curveA curveG }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/3DObjectGraphicsPage9.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/3DObjectGraphicsPage9}}
+\end{paste}\end{patch}
+
+\begin{patch}{3DObjectGraphicsPageEmpty9}
+\begin{paste}{3DObjectGraphicsPageEmpty9}{3DObjectGraphicsPagePatch9}
+\pastebutton{3DObjectGraphicsPageEmpty9}{\showpaste}
+\tab{5}\spadgraph{makeViewport3D(space,[title("Curves")])$VIEW3D\free{space curveI curveB1 curveB2 curveM curveN curveA curveG }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsPagePatch1}
+\begin{paste}{OneVariableGraphicsPageFull1}{OneVariableGraphicsPageEmpty1}
+\pastebutton{OneVariableGraphicsPageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/OneVariableGraphicsPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/OneVariableGraphicsPage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsPageEmpty1}
+\begin{paste}{OneVariableGraphicsPageEmpty1}{OneVariableGraphicsPagePatch1}
+\pastebutton{OneVariableGraphicsPageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsPagePatch2}
+\begin{paste}{OneVariableGraphicsPageFull2}{OneVariableGraphicsPageEmpty2}
+\pastebutton{OneVariableGraphicsPageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 10..16,title == "y = sin tan x - tan sin x")}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/OneVariableGraphicsPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/OneVariableGraphicsPage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsPageEmpty2}
+\begin{paste}{OneVariableGraphicsPageEmpty2}{OneVariableGraphicsPagePatch2}
+\pastebutton{OneVariableGraphicsPageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 10..16,title == "y = sin tan x - tan sin x")}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsPagePatch3}
+\begin{paste}{OneVariableGraphicsPageFull3}{OneVariableGraphicsPageEmpty3}
+\pastebutton{OneVariableGraphicsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{f(x) == (x-1)*(x-2)*(x-3)\bound{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsPageEmpty3}
+\begin{paste}{OneVariableGraphicsPageEmpty3}{OneVariableGraphicsPagePatch3}
+\pastebutton{OneVariableGraphicsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f(x) == (x-1)*(x-2)*(x-3)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsPagePatch4}
+\begin{paste}{OneVariableGraphicsPageFull4}{OneVariableGraphicsPageEmpty4}
+\pastebutton{OneVariableGraphicsPageFull4}{\hidepaste}
+\tab{5}\spadgraph{draw(f, 0..2, title == "y = f(x) on [0,2]")\free{f }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/OneVariableGraphicsPage4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/OneVariableGraphicsPage4}}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsPageEmpty4}
+\begin{paste}{OneVariableGraphicsPageEmpty4}{OneVariableGraphicsPagePatch4}
+\pastebutton{OneVariableGraphicsPageEmpty4}{\showpaste}
+\tab{5}\spadgraph{draw(f, 0..2, title == "y = f(x) on [0,2]")\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsPagePatch5}
+\begin{paste}{OneVariableGraphicsPageFull5}{OneVariableGraphicsPageEmpty5}
+\pastebutton{OneVariableGraphicsPageFull5}{\hidepaste}
+\tab{5}\spadgraph{draw(f, 0..4,title == "y = f(x) on [0,4]")\free{f }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/OneVariableGraphicsPage5.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/OneVariableGraphicsPage5}}
+\end{paste}\end{patch}
+
+\begin{patch}{OneVariableGraphicsPageEmpty5}
+\begin{paste}{OneVariableGraphicsPageEmpty5}{OneVariableGraphicsPagePatch5}
+\pastebutton{OneVariableGraphicsPageEmpty5}{\showpaste}
+\tab{5}\spadgraph{draw(f, 0..4,title == "y = f(x) on [0,4]")\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPagePatch1}
+\begin{paste}{ParametricCurveGraphicsPageFull1}{ParametricCurveGraphicsPageEmpty1}
+\pastebutton{ParametricCurveGraphicsPageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)), t = 0..2*\%pi)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ParametricCurveGraphicsPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ParametricCurveGraphicsPage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPageEmpty1}
+\begin{paste}{ParametricCurveGraphicsPageEmpty1}{ParametricCurveGraphicsPagePatch1}
+\pastebutton{ParametricCurveGraphicsPageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)), t = 0..2*\%pi)}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPagePatch2}
+\begin{paste}{ParametricCurveGraphicsPageFull2}{ParametricCurveGraphicsPageEmpty2}
+\pastebutton{ParametricCurveGraphicsPageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(cos(t), sin(t)), t = 0..2*\%pi, title == "The Unit Circle")}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ParametricCurveGraphicsPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ParametricCurveGraphicsPage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPageEmpty2}
+\begin{paste}{ParametricCurveGraphicsPageEmpty2}{ParametricCurveGraphicsPagePatch2}
+\pastebutton{ParametricCurveGraphicsPageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(curve(cos(t), sin(t)), t = 0..2*\%pi, title == "The Unit Circle")}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPagePatch3}
+\begin{paste}{ParametricCurveGraphicsPageFull3}{ParametricCurveGraphicsPageEmpty3}
+\pastebutton{ParametricCurveGraphicsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{f(t:SF):SF == sin(3*t/4)\bound{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPageEmpty3}
+\begin{paste}{ParametricCurveGraphicsPageEmpty3}{ParametricCurveGraphicsPagePatch3}
+\pastebutton{ParametricCurveGraphicsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f(t:SF):SF == sin(3*t/4)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPagePatch4}
+\begin{paste}{ParametricCurveGraphicsPageFull4}{ParametricCurveGraphicsPageEmpty4}
+\pastebutton{ParametricCurveGraphicsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{g(t:SF):SF == sin(t)\bound{g }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPageEmpty4}
+\begin{paste}{ParametricCurveGraphicsPageEmpty4}{ParametricCurveGraphicsPagePatch4}
+\pastebutton{ParametricCurveGraphicsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{g(t:SF):SF == sin(t)\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPagePatch5}
+\begin{paste}{ParametricCurveGraphicsPageFull5}{ParametricCurveGraphicsPageEmpty5}
+\pastebutton{ParametricCurveGraphicsPageFull5}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(f,g), 0..\%pi)\free{f g }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ParametricCurveGraphicsPage5.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ParametricCurveGraphicsPage5}}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPageEmpty5}
+\begin{paste}{ParametricCurveGraphicsPageEmpty5}{ParametricCurveGraphicsPagePatch5}
+\pastebutton{ParametricCurveGraphicsPageEmpty5}{\showpaste}
+\tab{5}\spadgraph{draw(curve(f,g), 0..\%pi)\free{f g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPagePatch6}
+\begin{paste}{ParametricCurveGraphicsPageFull6}{ParametricCurveGraphicsPageEmpty6}
+\pastebutton{ParametricCurveGraphicsPageFull6}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(f,g) ,\%pi..2*\%pi)\free{f g }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ParametricCurveGraphicsPage6.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ParametricCurveGraphicsPage6}}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPageEmpty6}
+\begin{paste}{ParametricCurveGraphicsPageEmpty6}{ParametricCurveGraphicsPagePatch6}
+\pastebutton{ParametricCurveGraphicsPageEmpty6}{\showpaste}
+\tab{5}\spadgraph{draw(curve(f,g) ,\%pi..2*\%pi)\free{f g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPagePatch7}
+\begin{paste}{ParametricCurveGraphicsPageFull7}{ParametricCurveGraphicsPageEmpty7}
+\pastebutton{ParametricCurveGraphicsPageFull7}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(f,g), -4*\%pi..4*\%pi)\free{f g }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ParametricCurveGraphicsPage7.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ParametricCurveGraphicsPage7}}
+\end{paste}\end{patch}
+
+\begin{patch}{ParametricCurveGraphicsPageEmpty7}
+\begin{paste}{ParametricCurveGraphicsPageEmpty7}{ParametricCurveGraphicsPagePatch7}
+\pastebutton{ParametricCurveGraphicsPageEmpty7}{\showpaste}
+\tab{5}\spadgraph{draw(curve(f,g), -4*\%pi..4*\%pi)\free{f g }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsPagePatch1}
+\begin{paste}{PolarGraphicsPageFull1}{PolarGraphicsPageEmpty1}
+\pastebutton{PolarGraphicsPageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(1,t = 0..2*\%pi,coordinates == polar, title == "The Unit Circle")}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/PolarGraphicsPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/PolarGraphicsPage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsPageEmpty1}
+\begin{paste}{PolarGraphicsPageEmpty1}{PolarGraphicsPagePatch1}
+\pastebutton{PolarGraphicsPageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(1,t = 0..2*\%pi,coordinates == polar, title == "The Unit Circle")}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsPagePatch2}
+\begin{paste}{PolarGraphicsPageFull2}{PolarGraphicsPageEmpty2}
+\pastebutton{PolarGraphicsPageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(17*t), t = 0..2*\%pi, coordinates == polar, title == "A Petal Curve")}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/PolarGraphicsPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/PolarGraphicsPage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsPageEmpty2}
+\begin{paste}{PolarGraphicsPageEmpty2}{PolarGraphicsPagePatch2}
+\pastebutton{PolarGraphicsPageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(sin(17*t), t = 0..2*\%pi, coordinates == polar, title == "A Petal Curve")}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsPagePatch3}
+\begin{paste}{PolarGraphicsPageFull3}{PolarGraphicsPageEmpty3}
+\pastebutton{PolarGraphicsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{f(t) == cos(4*t/7)\bound{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsPageEmpty3}
+\begin{paste}{PolarGraphicsPageEmpty3}{PolarGraphicsPagePatch3}
+\pastebutton{PolarGraphicsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f(t) == cos(4*t/7)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsPagePatch4}
+\begin{paste}{PolarGraphicsPageFull4}{PolarGraphicsPageEmpty4}
+\pastebutton{PolarGraphicsPageFull4}{\hidepaste}
+\tab{5}\spadgraph{draw(f, 0..2*\%pi, coordinates == polar)\free{f }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/PolarGraphicsPage4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/PolarGraphicsPage4}}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsPageEmpty4}
+\begin{paste}{PolarGraphicsPageEmpty4}{PolarGraphicsPagePatch4}
+\pastebutton{PolarGraphicsPageEmpty4}{\showpaste}
+\tab{5}\spadgraph{draw(f, 0..2*\%pi, coordinates == polar)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsPagePatch5}
+\begin{paste}{PolarGraphicsPageFull5}{PolarGraphicsPageEmpty5}
+\pastebutton{PolarGraphicsPageFull5}{\hidepaste}
+\tab{5}\spadgraph{draw(f, 0..14*\%pi, coordinates == polar)\free{f }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/PolarGraphicsPage5.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/PolarGraphicsPage5}}
+\end{paste}\end{patch}
+
+\begin{patch}{PolarGraphicsPageEmpty5}
+\begin{paste}{PolarGraphicsPageEmpty5}{PolarGraphicsPagePatch5}
+\pastebutton{PolarGraphicsPageEmpty5}{\showpaste}
+\tab{5}\spadgraph{draw(f, 0..14*\%pi, coordinates == polar)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsPagePatch1}
+\begin{paste}{ImplicitCurveGraphicsPageFull1}{ImplicitCurveGraphicsPageEmpty1}
+\pastebutton{ImplicitCurveGraphicsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{p := ((x**2 + y**2 + 1) - 8*x)**2 - (8*(x**2 + y**2 + 1) - 4*x - 1)\bound{p }}
+\indentrel{3}\begin{verbatim}
+   (1)
+    4      2            2    4      3      2
+   y  + (2x  - 16x - 6)y  + x  - 16x  + 58x  - 12x - 6
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsPageEmpty1}
+\begin{paste}{ImplicitCurveGraphicsPageEmpty1}{ImplicitCurveGraphicsPagePatch1}
+\pastebutton{ImplicitCurveGraphicsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{p := ((x**2 + y**2 + 1) - 8*x)**2 - (8*(x**2 + y**2 + 1) - 4*x - 1)\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsPagePatch2}
+\begin{paste}{ImplicitCurveGraphicsPageFull2}{ImplicitCurveGraphicsPageEmpty2}
+\pastebutton{ImplicitCurveGraphicsPageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(p = 0, x, y, range == [-1..11, -7..7], title == "Cartesian Ovals")\free{p }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ImplicitCurveGraphicsPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ImplicitCurveGraphicsPage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ImplicitCurveGraphicsPageEmpty2}
+\begin{paste}{ImplicitCurveGraphicsPageEmpty2}{ImplicitCurveGraphicsPagePatch2}
+\pastebutton{ImplicitCurveGraphicsPageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(p = 0, x, y, range == [-1..11, -7..7], title == "Cartesian Ovals")\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch1}
+\begin{paste}{ListPointsGraphicsPageFull1}{ListPointsGraphicsPageEmpty1}
+\pastebutton{ListPointsGraphicsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{p1 := point [1::SF,1::SF]$(Point SF)\bound{p1 }}
+\indentrel{3}\begin{verbatim}
+   (1)  [1.0,1.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty1}
+\begin{paste}{ListPointsGraphicsPageEmpty1}{ListPointsGraphicsPagePatch1}
+\pastebutton{ListPointsGraphicsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{p1 := point [1::SF,1::SF]$(Point SF)\bound{p1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch2}
+\begin{paste}{ListPointsGraphicsPageFull2}{ListPointsGraphicsPageEmpty2}
+\pastebutton{ListPointsGraphicsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{p2 := point [0::SF,1::SF]$(Point SF)\bound{p2 }}
+\indentrel{3}\begin{verbatim}
+   (2)  [0.0,1.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty2}
+\begin{paste}{ListPointsGraphicsPageEmpty2}{ListPointsGraphicsPagePatch2}
+\pastebutton{ListPointsGraphicsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{p2 := point [0::SF,1::SF]$(Point SF)\bound{p2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch3}
+\begin{paste}{ListPointsGraphicsPageFull3}{ListPointsGraphicsPageEmpty3}
+\pastebutton{ListPointsGraphicsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{p3 := point [0::SF,0::SF]$(Point SF)\bound{p3 }}
+\indentrel{3}\begin{verbatim}
+   (3)  [0.0,0.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty3}
+\begin{paste}{ListPointsGraphicsPageEmpty3}{ListPointsGraphicsPagePatch3}
+\pastebutton{ListPointsGraphicsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{p3 := point [0::SF,0::SF]$(Point SF)\bound{p3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch4}
+\begin{paste}{ListPointsGraphicsPageFull4}{ListPointsGraphicsPageEmpty4}
+\pastebutton{ListPointsGraphicsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{p4 := point [1::SF,0::SF]$(Point SF)\bound{p4 }}
+\indentrel{3}\begin{verbatim}
+   (4)  [1.0,0.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty4}
+\begin{paste}{ListPointsGraphicsPageEmpty4}{ListPointsGraphicsPagePatch4}
+\pastebutton{ListPointsGraphicsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{p4 := point [1::SF,0::SF]$(Point SF)\bound{p4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch5}
+\begin{paste}{ListPointsGraphicsPageFull5}{ListPointsGraphicsPageEmpty5}
+\pastebutton{ListPointsGraphicsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{p5 := point [1::SF,.5::SF]$(Point SF)\bound{p5 }}
+\indentrel{3}\begin{verbatim}
+   (5)  [1.0,0.5]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty5}
+\begin{paste}{ListPointsGraphicsPageEmpty5}{ListPointsGraphicsPagePatch5}
+\pastebutton{ListPointsGraphicsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{p5 := point [1::SF,.5::SF]$(Point SF)\bound{p5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch6}
+\begin{paste}{ListPointsGraphicsPageFull6}{ListPointsGraphicsPageEmpty6}
+\pastebutton{ListPointsGraphicsPageFull6}{\hidepaste}
+\tab{5}\spadcommand{p6 := point [.5::SF,0::SF]$(Point SF)\bound{p6 }}
+\indentrel{3}\begin{verbatim}
+   (6)  [0.5,0.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty6}
+\begin{paste}{ListPointsGraphicsPageEmpty6}{ListPointsGraphicsPagePatch6}
+\pastebutton{ListPointsGraphicsPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{p6 := point [.5::SF,0::SF]$(Point SF)\bound{p6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch7}
+\begin{paste}{ListPointsGraphicsPageFull7}{ListPointsGraphicsPageEmpty7}
+\pastebutton{ListPointsGraphicsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{p7 := point [0::SF,0.5::SF]$(Point SF)\bound{p7 }}
+\indentrel{3}\begin{verbatim}
+   (7)  [0.0,0.5]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty7}
+\begin{paste}{ListPointsGraphicsPageEmpty7}{ListPointsGraphicsPagePatch7}
+\pastebutton{ListPointsGraphicsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{p7 := point [0::SF,0.5::SF]$(Point SF)\bound{p7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch8}
+\begin{paste}{ListPointsGraphicsPageFull8}{ListPointsGraphicsPageEmpty8}
+\pastebutton{ListPointsGraphicsPageFull8}{\hidepaste}
+\tab{5}\spadcommand{p8 := point [.5::SF,1::SF]$(Point SF)\bound{p8 }}
+\indentrel{3}\begin{verbatim}
+   (8)  [0.5,1.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty8}
+\begin{paste}{ListPointsGraphicsPageEmpty8}{ListPointsGraphicsPagePatch8}
+\pastebutton{ListPointsGraphicsPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{p8 := point [.5::SF,1::SF]$(Point SF)\bound{p8 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch9}
+\begin{paste}{ListPointsGraphicsPageFull9}{ListPointsGraphicsPageEmpty9}
+\pastebutton{ListPointsGraphicsPageFull9}{\hidepaste}
+\tab{5}\spadcommand{p9 := point [.25::SF,.25::SF]$(Point SF)\bound{p9 }}
+\indentrel{3}\begin{verbatim}
+   (9)  [0.25,0.25]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty9}
+\begin{paste}{ListPointsGraphicsPageEmpty9}{ListPointsGraphicsPagePatch9}
+\pastebutton{ListPointsGraphicsPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{p9 := point [.25::SF,.25::SF]$(Point SF)\bound{p9 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch10}
+\begin{paste}{ListPointsGraphicsPageFull10}{ListPointsGraphicsPageEmpty10}
+\pastebutton{ListPointsGraphicsPageFull10}{\hidepaste}
+\tab{5}\spadcommand{p10 := point [.25::SF,.75::SF]$(Point SF)\bound{p10 }}
+\indentrel{3}\begin{verbatim}
+   (10)  [0.25,0.75]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty10}
+\begin{paste}{ListPointsGraphicsPageEmpty10}{ListPointsGraphicsPagePatch10}
+\pastebutton{ListPointsGraphicsPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{p10 := point [.25::SF,.75::SF]$(Point SF)\bound{p10 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch11}
+\begin{paste}{ListPointsGraphicsPageFull11}{ListPointsGraphicsPageEmpty11}
+\pastebutton{ListPointsGraphicsPageFull11}{\hidepaste}
+\tab{5}\spadcommand{p11 := point [.75::SF,.75::SF]$(Point SF)\bound{p11 }}
+\indentrel{3}\begin{verbatim}
+   (11)  [0.75,0.75]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty11}
+\begin{paste}{ListPointsGraphicsPageEmpty11}{ListPointsGraphicsPagePatch11}
+\pastebutton{ListPointsGraphicsPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{p11 := point [.75::SF,.75::SF]$(Point SF)\bound{p11 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch12}
+\begin{paste}{ListPointsGraphicsPageFull12}{ListPointsGraphicsPageEmpty12}
+\pastebutton{ListPointsGraphicsPageFull12}{\hidepaste}
+\tab{5}\spadcommand{p12 := point [.75::SF,.25::SF]$(Point SF)\bound{p12 }}
+\indentrel{3}\begin{verbatim}
+   (12)  [0.75,0.25]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty12}
+\begin{paste}{ListPointsGraphicsPageEmpty12}{ListPointsGraphicsPagePatch12}
+\pastebutton{ListPointsGraphicsPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{p12 := point [.75::SF,.25::SF]$(Point SF)\bound{p12 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch13}
+\begin{paste}{ListPointsGraphicsPageFull13}{ListPointsGraphicsPageEmpty13}
+\pastebutton{ListPointsGraphicsPageFull13}{\hidepaste}
+\tab{5}\spadcommand{llp := [[p1,p2],[p2,p3],[p3,p4],[p4,p1],[p5,p6],[p6,p7],[p7,p8],[p8,p5],[p9,p10],[p10,p11],[p11,p12],[p12,p9]]\bound{llp }\free{p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 }}
+\indentrel{3}\begin{verbatim}
+   (13)
+   [[[1.0,1.0],[0.0,1.0]], [[0.0,1.0],[0.0,0.0]],
+    [[0.0,0.0],[1.0,0.0]], [[1.0,0.0],[1.0,1.0]],
+    [[1.0,0.5],[0.5,0.0]], [[0.5,0.0],[0.0,0.5]],
+    [[0.0,0.5],[0.5,1.0]], [[0.5,1.0],[1.0,0.5]],
+    [[0.25,0.25],[0.25,0.75]], [[0.25,0.75],[0.75,0.75]],
+    [[0.75,0.75],[0.75,0.25]], [[0.75,0.25],[0.25,0.25]]]
+                      Type: List List Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty13}
+\begin{paste}{ListPointsGraphicsPageEmpty13}{ListPointsGraphicsPagePatch13}
+\pastebutton{ListPointsGraphicsPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{llp := [[p1,p2],[p2,p3],[p3,p4],[p4,p1],[p5,p6],[p6,p7],[p7,p8],[p8,p5],[p9,p10],[p10,p11],[p11,p12],[p12,p9]]\bound{llp }\free{p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch14}
+\begin{paste}{ListPointsGraphicsPageFull14}{ListPointsGraphicsPageEmpty14}
+\pastebutton{ListPointsGraphicsPageFull14}{\hidepaste}
+\tab{5}\spadcommand{size1 := 6::PositiveInteger\bound{size1 }}
+\indentrel{3}\begin{verbatim}
+   (14)  6
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty14}
+\begin{paste}{ListPointsGraphicsPageEmpty14}{ListPointsGraphicsPagePatch14}
+\pastebutton{ListPointsGraphicsPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{size1 := 6::PositiveInteger\bound{size1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch15}
+\begin{paste}{ListPointsGraphicsPageFull15}{ListPointsGraphicsPageEmpty15}
+\pastebutton{ListPointsGraphicsPageFull15}{\hidepaste}
+\tab{5}\spadcommand{size2 := 8::PositiveInteger\bound{size2 }}
+\indentrel{3}\begin{verbatim}
+   (15)  8
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty15}
+\begin{paste}{ListPointsGraphicsPageEmpty15}{ListPointsGraphicsPagePatch15}
+\pastebutton{ListPointsGraphicsPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{size2 := 8::PositiveInteger\bound{size2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch16}
+\begin{paste}{ListPointsGraphicsPageFull16}{ListPointsGraphicsPageEmpty16}
+\pastebutton{ListPointsGraphicsPageFull16}{\hidepaste}
+\tab{5}\spadcommand{size3 := 10::PositiveInteger\bound{size3 }}
+\indentrel{3}\begin{verbatim}
+   (16)  10
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty16}
+\begin{paste}{ListPointsGraphicsPageEmpty16}{ListPointsGraphicsPagePatch16}
+\pastebutton{ListPointsGraphicsPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{size3 := 10::PositiveInteger\bound{size3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch17}
+\begin{paste}{ListPointsGraphicsPageFull17}{ListPointsGraphicsPageEmpty17}
+\pastebutton{ListPointsGraphicsPageFull17}{\hidepaste}
+\tab{5}\spadcommand{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3]\bound{lsize }\free{size1 size2 size3 }}
+\indentrel{3}\begin{verbatim}
+   (17)  [6,6,6,6,8,8,8,8,10,10,10,10]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty17}
+\begin{paste}{ListPointsGraphicsPageEmpty17}{ListPointsGraphicsPagePatch17}
+\pastebutton{ListPointsGraphicsPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3]\bound{lsize }\free{size1 size2 size3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch18}
+\begin{paste}{ListPointsGraphicsPageFull18}{ListPointsGraphicsPageEmpty18}
+\pastebutton{ListPointsGraphicsPageFull18}{\hidepaste}
+\tab{5}\spadcommand{pc1 := pastel red()\bound{pc1 }}
+\indentrel{3}\begin{verbatim}
+   (18)  [Hue: 1  Weight: 1.0] from the Pastel palette
+                                          Type: Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty18}
+\begin{paste}{ListPointsGraphicsPageEmpty18}{ListPointsGraphicsPagePatch18}
+\pastebutton{ListPointsGraphicsPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{pc1 := pastel red()\bound{pc1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch19}
+\begin{paste}{ListPointsGraphicsPageFull19}{ListPointsGraphicsPageEmpty19}
+\pastebutton{ListPointsGraphicsPageFull19}{\hidepaste}
+\tab{5}\spadcommand{pc2 := dim green()\bound{pc2 }}
+\indentrel{3}\begin{verbatim}
+   (19)  [Hue: 14  Weight: 1.0] from the Dim palette
+                                          Type: Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty19}
+\begin{paste}{ListPointsGraphicsPageEmpty19}{ListPointsGraphicsPagePatch19}
+\pastebutton{ListPointsGraphicsPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{pc2 := dim green()\bound{pc2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch20}
+\begin{paste}{ListPointsGraphicsPageFull20}{ListPointsGraphicsPageEmpty20}
+\pastebutton{ListPointsGraphicsPageFull20}{\hidepaste}
+\tab{5}\spadcommand{pc3 := pastel yellow()\bound{pc3 }}
+\indentrel{3}\begin{verbatim}
+   (20)  [Hue: 11  Weight: 1.0] from the Pastel palette
+                                          Type: Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty20}
+\begin{paste}{ListPointsGraphicsPageEmpty20}{ListPointsGraphicsPagePatch20}
+\pastebutton{ListPointsGraphicsPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{pc3 := pastel yellow()\bound{pc3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch21}
+\begin{paste}{ListPointsGraphicsPageFull21}{ListPointsGraphicsPageEmpty21}
+\pastebutton{ListPointsGraphicsPageFull21}{\hidepaste}
+\tab{5}\spadcommand{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3]\bound{lpc }\free{pc1 pc2 pc3 }}
+\indentrel{3}\begin{verbatim}
+   (21)
+   [[Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 11  Weight: 1.0] from the Pastel palette,
+    [Hue: 11  Weight: 1.0] from the Pastel palette,
+    [Hue: 11  Weight: 1.0] from the Pastel palette,
+    [Hue: 11  Weight: 1.0] from the Pastel palette]
+                                     Type: List Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty21}
+\begin{paste}{ListPointsGraphicsPageEmpty21}{ListPointsGraphicsPagePatch21}
+\pastebutton{ListPointsGraphicsPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3]\bound{lpc }\free{pc1 pc2 pc3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch22}
+\begin{paste}{ListPointsGraphicsPageFull22}{ListPointsGraphicsPageEmpty22}
+\pastebutton{ListPointsGraphicsPageFull22}{\hidepaste}
+\tab{5}\spadcommand{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()]\bound{lc }}
+\indentrel{3}\begin{verbatim}
+   (22)
+   [[Hue: 22  Weight: 1.0] from the Pastel palette,
+    [Hue: 11  Weight: 1.0] from the Light palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 1  Weight: 1.0] from the Bright palette,
+    [Hue: 14  Weight: 1.0] from the Light palette,
+    [Hue: 11  Weight: 1.0] from the Dim palette,
+    [Hue: 22  Weight: 1.0] from the Bright palette,
+    [Hue: 1  Weight: 1.0] from the Dark palette,
+    [Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 22  Weight: 1.0] from the Light palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 11  Weight: 1.0] from the Light palette]
+                                     Type: List Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty22}
+\begin{paste}{ListPointsGraphicsPageEmpty22}{ListPointsGraphicsPagePatch22}
+\pastebutton{ListPointsGraphicsPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()]\bound{lc }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch23}
+\begin{paste}{ListPointsGraphicsPageFull23}{ListPointsGraphicsPageEmpty23}
+\pastebutton{ListPointsGraphicsPageFull23}{\hidepaste}
+\tab{5}\spadcommand{g := makeGraphImage(llp,lpc,lc,lsize)$GRIMAGE\bound{g }\free{llp lpc lc lsize }}
+\indentrel{3}\begin{verbatim}
+   (23)  Graph with 12 point lists
+                                       Type: GraphImage
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty23}
+\begin{paste}{ListPointsGraphicsPageEmpty23}{ListPointsGraphicsPagePatch23}
+\pastebutton{ListPointsGraphicsPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{g := makeGraphImage(llp,lpc,lc,lsize)$GRIMAGE\bound{g }\free{llp lpc lc lsize }}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPagePatch24}
+\begin{paste}{ListPointsGraphicsPageFull24}{ListPointsGraphicsPageEmpty24}
+\pastebutton{ListPointsGraphicsPageFull24}{\hidepaste}
+\tab{5}\spadgraph{makeViewport2D(g,[title("Lines")])$VIEW2D\free{g }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ListPointsGraphicsPage24.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ListPointsGraphicsPage24}}
+\end{paste}\end{patch}
+
+\begin{patch}{ListPointsGraphicsPageEmpty24}
+\begin{paste}{ListPointsGraphicsPageEmpty24}{ListPointsGraphicsPagePatch24}
+\pastebutton{ListPointsGraphicsPageEmpty24}{\showpaste}
+\tab{5}\spadgraph{makeViewport2D(g,[title("Lines")])$VIEW2D\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ViewportPagePatch1}
+\begin{paste}{ViewportPageFull1}{ViewportPageEmpty1}
+\pastebutton{ViewportPageFull1}{\hidepaste}
+\tab{5}\spadgraph{v := draw(x*x-y*y,x=-1..1,y=-1..1)\bound{v }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ViewportPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ViewportPage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ViewportPageEmpty1}
+\begin{paste}{ViewportPageEmpty1}{ViewportPagePatch1}
+\pastebutton{ViewportPageEmpty1}{\showpaste}
+\tab{5}\spadgraph{v := draw(x*x-y*y,x=-1..1,y=-1..1)\bound{v }}
+\end{paste}\end{patch}
+
+\begin{patch}{ViewportPagePatch2}
+\begin{paste}{ViewportPageFull2}{ViewportPageEmpty2}
+\pastebutton{ViewportPageFull2}{\hidepaste}
+\tab{5}\spadcommand{write(v,"saddle","pixmap")\free{v }}
+\indentrel{3}\begin{verbatim}
+   (2)  "NIL"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ViewportPageEmpty2}
+\begin{paste}{ViewportPageEmpty2}{ViewportPagePatch2}
+\pastebutton{ViewportPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{write(v,"saddle","pixmap")\free{v }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/grpthry.ht b/src/hyper/pages/grpthry.ht
new file mode 100644
index 00000000..d75c0e4e
--- /dev/null
+++ b/src/hyper/pages/grpthry.ht
@@ -0,0 +1,207 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% @(#)grpthry.ht        1.2 90/08/22 14:54:14
+
+% Group Theory Page
+% authors: H. Gollan, J. Grabmeier, August 1989
+% @(#)grpthry.ht        2.6 90/05/21 14:44:11
+\begin{page}{GroupTheoryPage}{Group Theory}
+\beginscroll
+\Language{} can work with individual permutations, permutation
+groups and do representation theory.
+\horizontalline
+\newline
+\beginmenu
+\menulink{Info on Group Theory}{InfoGroupTheoryPage}
+
+
+%\menulink{Permutation}{PermutationXmpPage}
+%Calculating within symmetric groups.
+
+%\menulink{Permutation Groups}{PermutationGroupXmpPage}
+%Working with subgroups of a symmetric group.
+
+%\menulink{Permutation Group Examples}{PermutationGroupExampleXmpPage}
+%Working with permutation groups, predefined in the system as Rubik's group.
+
+\menulink{Info on Representation Theory}{InfoRepTheoryPage}
+
+%\menulink{Irreducible Representations of Symmetric Groups}{IrrRepSymNatXmpPage}
+%Alfred Young's natural form for these representations.
+
+%\menulink{Representations of Higher Degree}{RepresentationPackage1XmpPage}
+%Constructing new representations by symmetric and antisymmetric
+%tensors.
+
+%\menulink{Decomposing Representations}{RepresentationPackage2XmpPage}
+%Parker's `Meat-Axe', working in prime characteristics.
+
+\menulink{Representations of \texht{$A_6$}{A6}}{RepA6Page}
+The irreducible representations of the alternating group \texht{$A_6$}{A6} over fields
+of characteristic 2.
+\endmenu
+\endscroll
+\autobuttons \end{page}
+
+% RepA6Page
+% author: J. Grabmeier,  08/08/89
+
+\begin{page}{RepA6Page}{Representations of \texht{$A_6$}{A6}}
+\beginscroll
+In what follows you'll see how to use \Language{} to get all the irreducible
+representations of the alternating group \texht{$A_6$}{A6} over the field with two
+elements (GF 2).
+First, we generate \texht{$A_6$}{A6} by a three-cycle: x = (1,2,3)
+and a 5-cycle: y = (2,3,4,5,6).  Next we have \Language{} calculate
+the permutation representation over the integers and over GF 2:
+\spadpaste{genA6 : LIST PERM INT := [cycle [1,2,3],cycle [2,3,4,5,6]] \bound{genA6}}
+\spadpaste{pRA6 := permutationRepresentation (genA6, 6) \bound{pRA6} \free{genA6}
+}
+Now we apply Parker's 'Meat-Axe' and split it:
+\spadpaste{sp0 := meatAxe (pRA6::(LIST MATRIX PF 2)) \free{pRA6} \bound{sp0}}
+We have found the trivial module as a quotient module
+and a 5-dimensional sub-module.
+Try to split again:
+\spadpaste{sp1 := meatAxe sp0.1 \bound{sp1}}
+and we find a 4-dimensional sub-module and the trivial one again.
+Now we can test if this representaton is absolutely irreducible:
+\spadpaste{isAbsolutelyIrreducible? sp1.2 }
+and we see that this 4-dimensional representation is absolutely irreducible.
+So, we have found a second irreducible representation.
+Now, we construct a representation by reducing an irreducible one
+of the symmetric group S_6 over the integers mod 2.
+We take the one labelled by the partition [2,2,1,1] and
+restrict it to \texht{$A_6$}{A6}:
+\spadpaste{d2211 := irreducibleRepresentation ([2,2,1,1],genA6) \bound{d2211} }
+Now split it:
+\spadpaste{d2211m2 := d2211:: (LIST MATRIX PF 2); sp2 := meatAxe d2211m2 \free{d2211}
+\bound{sp2}}
+This gave both a five and a four dimensional representation.
+Now we take the 4-dimensional one
+and we shall see that it is absolutely irreducible:
+\spadpaste{isAbsolutelyIrreducible? sp2.1}
+The two 4-dimensional representations are not equivalent:
+\spadpaste{areEquivalent? (sp1.2, sp2.1)}
+So we have found a third irreducible representation.
+Now we construct a new representation using the tensor product
+and try to split it:
+\spadpaste{dA6d16 := tensorProduct(sp1.2,sp2.1); meatAxe dA6d16 \bound{dA6d16}}
+The representation is irreducible, but it may be not absolutely irreducible.
+\spadpaste{isAbsolutelyIrreducible? dA6d16}
+So let's try the same procedure over the field with 4 elements:
+\spadpaste{sp3 := meatAxe (dA6d16 :: (LIST MATRIX FF(2,2))) \bound{sp3}}
+Now we find two 8-dimensional representations, dA6d8a and dA6d8b.
+Both are absolutely irreducible...
+\spadpaste{isAbsolutelyIrreducible?  sp3.1}
+\spadpaste{isAbsolutelyIrreducible?  sp3.2}
+and they are not equivalent:
+\spadpaste{areEquivalent? (sp3.1,sp3.2)}
+So we have found five absolutely irreducible representations of \texht{$A_6$}{A6}
+in characteristic 2.
+General theory now tells us that there are no more irreducible ones.
+Here, for future reference are all the absolutely irreducible 2-modular
+representations of \texht{$A_6$}{A6}
+\spadpaste{sp0.2 \free{sp0}}
+\spadpaste{sp1.2 \free{sp1}}
+\spadpaste{sp2.1 \free{sp2}}
+\spadpaste{sp3.1 \free{sp3}}
+\spadpaste{sp3.2 \free{sp3}}
+And here again is the irreducible, but not absolutely irreducible
+representations of \texht{$A_6$}{A6} over GF 2
+\spadpaste{dA6d16 \free{dA6d16}}
+\endscroll
+\autobuttons \end{page}
+
+
+
+
+
+\begin{page}{InfoRepTheoryPage}{Representation Theory}
+\beginscroll
+\horizontalline
+Representation theory for finite groups studies finite groups by
+embedding them in a general linear group over a field or an
+integral domain.
+Hence, we are representing each element of the group by
+an invertible matrix.
+Two matrix representations of a given group are equivalent, if, by changing the
+basis of the underlying
+space, you can go from one to the other.  When you change bases, you
+transform the matrices that are the images of elements by
+conjugating them by an invertible matrix.
+\newline
+\newline
+If we can find a subspace which is fixed under the image
+of the group, then there exists a `base change' after which all the representing
+ matrices
+are in upper triangular block form. The block matrices on
+the main diagonal give a new representation of the group of lower degree.
+Such a representation is said to be `reducible'.
+\newline
+\beginmenu
+%\menulink{Irreducible Representations of Symmetric Groups}{IrrRepSymNatXmpPage}
+
+%Alfred Young's natural form for these representations.
+
+%\menulink{Representations of Higher Degree}{RepresentationPackage1XmpPage}
+%Constructing new representations by symmetric and antisymmetric
+%tensors.
+
+%\menulink{Decomposing Representations}{RepresentationPackage2XmpPage}
+%Parker's `Meat-Axe', working in prime characteristics.
+
+\menulink{Representations of \texht{$A_6$}{A6}}{RepA6Page}
+The irreducible representations of the alternating group \texht{$A_6$}{A6} over fields
+of characteristic 2.
+\endmenu
+\endscroll
+\autobuttons \end{page}
+
+
+
+\begin{page}{InfoGroupTheoryPage}{Group Theory}
+%%
+%% Johannes Grabmeier 03/02/90
+%%
+\beginscroll
+A {\it group} is a set G together with an associative operation
+* satisfying the axioms of existence
+of a unit element and an inverse of every element of the group.
+The \Language{} category \spadtype{Group} represents this setting.
+Many data structures in \Language{} are groups and therefore there
+is a large variety of examples as fields and polynomials,
+although the main interest there is not the group structure.
+
+To work with and in groups in a concrete manner some way of
+representing groups has to be chosen. A group can be given
+as a list of generators and a set of relations. If there
+are no relations, then  we have a {\it free group}, realized
+in the domain \spadtype{FreeMonoid} which won't be discussed here.
+We consider {\it permutation groups}, where a group
+is realized as a subgroup of the symmetric group of a set, i.e.
+the group of all bijections of a set, the operation being the
+composition of maps.
+Indeed,  every group can be realized this way, although
+this may not be practical.
+
+Furthermore group elements can be given as invertible matrices.
+The group operation is reflected by matrix multiplication.
+More precise in representation theory group homomophisms
+from a group to general linear groups are contructed.
+Some algorithms are implemented in \Language{}.
+\newline
+%\beginmenu
+%\menulink{Permutation}{PermutationXmpPage}
+%Calculating within symmetric groups.
+
+%\menulink{Permutation Groups}{PermutationGroupXmpPage}
+%Working with subgroups of a symmetric group.
+
+%\menulink{Permutation Group Examples}{PermutationGroupExampleXmpPage}
+%Working with permutation groups, predefined in the system as Rubik's group.
+%\endmenu
+\endscroll
+\autobuttons \end{page}
+
diff --git a/src/hyper/pages/grpthry.pht b/src/hyper/pages/grpthry.pht
new file mode 100644
index 00000000..b04349ef
--- /dev/null
+++ b/src/hyper/pages/grpthry.pht
@@ -0,0 +1,866 @@
+\begin{patch}{RepA6PagePatch1}
+\begin{paste}{RepA6PageFull1}{RepA6PageEmpty1}
+\pastebutton{RepA6PageFull1}{\hidepaste}
+\tab{5}\spadcommand{genA6 : LIST PERM INT := [cycle [1,2,3],cycle [2,3,4,5,6]]\bound{genA6 }}
+\indentrel{3}\begin{verbatim}
+   (1)  [(1 2 3),(2 3 4 5 6)]
+                         Type: List Permutation Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty1}
+\begin{paste}{RepA6PageEmpty1}{RepA6PagePatch1}
+\pastebutton{RepA6PageEmpty1}{\showpaste}
+\tab{5}\spadcommand{genA6 : LIST PERM INT := [cycle [1,2,3],cycle [2,3,4,5,6]]\bound{genA6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch2}
+\begin{paste}{RepA6PageFull2}{RepA6PageEmpty2}
+\pastebutton{RepA6PageFull2}{\hidepaste}
+\tab{5}\spadcommand{pRA6 := permutationRepresentation (genA6, 6)\bound{pRA6 }\free{genA6 }}
+\indentrel{3}\begin{verbatim}
+         Ú0  0  1  0  0  0¿ Ú1  0  0  0  0  0¿
+         ³                ³ ³                ³
+         ³1  0  0  0  0  0³ ³0  0  0  0  0  1³
+         ³                ³ ³                ³
+         ³0  1  0  0  0  0³ ³0  1  0  0  0  0³
+   (2)  [³                ³,³                ³]
+         ³0  0  0  1  0  0³ ³0  0  1  0  0  0³
+         ³                ³ ³                ³
+         ³0  0  0  0  1  0³ ³0  0  0  1  0  0³
+         ³                ³ ³                ³
+         À0  0  0  0  0  1Ù À0  0  0  0  1  0Ù
+                              Type: List Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty2}
+\begin{paste}{RepA6PageEmpty2}{RepA6PagePatch2}
+\pastebutton{RepA6PageEmpty2}{\showpaste}
+\tab{5}\spadcommand{pRA6 := permutationRepresentation (genA6, 6)\bound{pRA6 }\free{genA6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch3}
+\begin{paste}{RepA6PageFull3}{RepA6PageEmpty3}
+\pastebutton{RepA6PageFull3}{\hidepaste}
+\tab{5}\spadcommand{sp0 := meatAxe (pRA6::(LIST MATRIX PF 2))\free{pRA6 }\bound{sp0 }}
+\indentrel{3}\begin{verbatim}
+   Fingerprint element in generated algebra is singular
+     A proper cyclic submodule is found.
+     Transition matrix computed
+     The inverse of the transition matrix computed
+     Now transform the matrices
+          Ú0  0  1  0  0¿ Ú1  0  0  0  0¿
+          ³             ³ ³             ³
+          ³1  0  0  0  0³ ³1  1  1  1  1³
+          ³             ³ ³             ³
+   (3)  [[³0  1  0  0  0³,³0  1  0  0  0³],[[1],[1]]]
+          ³             ³ ³             ³
+          ³0  0  0  1  0³ ³0  0  1  0  0³
+          ³             ³ ³             ³
+          À0  0  0  0  1Ù À0  0  0  1  0Ù
+                    Type: List List Matrix PrimeField 2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty3}
+\begin{paste}{RepA6PageEmpty3}{RepA6PagePatch3}
+\pastebutton{RepA6PageEmpty3}{\showpaste}
+\tab{5}\spadcommand{sp0 := meatAxe (pRA6::(LIST MATRIX PF 2))\free{pRA6 }\bound{sp0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch4}
+\begin{paste}{RepA6PageFull4}{RepA6PageEmpty4}
+\pastebutton{RepA6PageFull4}{\hidepaste}
+\tab{5}\spadcommand{sp1 := meatAxe sp0.1\bound{sp1 }}
+\indentrel{3}\begin{verbatim}
+   Fingerprint element in generated algebra is singular
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+    We know that all the cyclic submodules generated by a
+  ll
+      non-trivial element of the singular matrix under vi
+  ew are
+      not proper, hence Norton's irreducibility test can 
+  be done:
+     A proper cyclic submodule is found.
+     Transition matrix computed
+     The inverse of the transition matrix computed
+     Now transform the matrices
+    Representation is not irreducible and it will be spli
+  t:
+                    Ú0  1  0  0¿ Ú0  1  1  1¿
+                    ³          ³ ³          ³
+                    ³0  0  1  0³ ³1  1  0  1³
+   (4)  [[[1],[1]],[³          ³,³          ³]]
+                    ³1  0  0  0³ ³1  1  1  0³
+                    ³          ³ ³          ³
+                    À0  0  0  1Ù À1  1  1  1Ù
+                    Type: List List Matrix PrimeField 2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty4}
+\begin{paste}{RepA6PageEmpty4}{RepA6PagePatch4}
+\pastebutton{RepA6PageEmpty4}{\showpaste}
+\tab{5}\spadcommand{sp1 := meatAxe sp0.1\bound{sp1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch5}
+\begin{paste}{RepA6PageFull5}{RepA6PageEmpty5}
+\pastebutton{RepA6PageFull5}{\hidepaste}
+\tab{5}\spadcommand{isAbsolutelyIrreducible? sp1.2}
+\indentrel{3}\begin{verbatim}
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra has
+     one-dimensional kernel
+    We know that all the cyclic submodules generated by a
+  ll
+      non-trivial element of the singular matrix under vi
+  ew are
+      not proper, hence Norton's irreducibility test can 
+  be done:
+     The generated cyclic submodule was not proper
+     Representation is absolutely irreducible
+   (5)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty5}
+\begin{paste}{RepA6PageEmpty5}{RepA6PagePatch5}
+\pastebutton{RepA6PageEmpty5}{\showpaste}
+\tab{5}\spadcommand{isAbsolutelyIrreducible? sp1.2}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch6}
+\begin{paste}{RepA6PageFull6}{RepA6PageEmpty6}
+\pastebutton{RepA6PageFull6}{\hidepaste}
+\tab{5}\spadcommand{d2211 := irreducibleRepresentation ([2,2,1,1],genA6)\bound{d2211 }}
+\indentrel{3}\begin{verbatim}
+   (6)
+    Ú1  0  0  - 1   1    0    0    0    0 ¿
+    ³                                     ³
+    ³0  1  0   1    0    1    0    0    0 ³
+    ³                                     ³
+    ³0  0  1   0    1   - 1   0    0    0 ³
+    ³                                     ³
+    ³0  0  0  - 1   0    0   - 1   0    0 ³
+    ³                                     ³
+   [³0  0  0   0   - 1   0    0   - 1   0 ³,
+    ³                                     ³
+    ³0  0  0   0    0   - 1   0    0   - 1³
+    ³                                     ³
+    ³0  0  0   1    0    0    0    0    0 ³
+    ³                                     ³
+    ³0  0  0   0    1    0    0    0    0 ³
+    ³                                     ³
+    À0  0  0   0    0    1    0    0    0 Ù
+    Ú 0    0   1   0   0  0   1   0  0¿
+    ³                                 ³
+    ³ 0    0   0   0   1  0  - 1  0  0³
+    ³                                 ³
+    ³ 0    0   0   0   0  1   1   0  0³
+    ³                                 ³
+    ³ 0    0   0   0   0  0   1   1  0³
+    ³                                 ³
+    ³ 0    0   0   0   0  0  - 1  0  1³]
+    ³                                 ³
+    ³ 0    0   0   0   0  0   1   0  0³
+    ³                                 ³
+    ³- 1   0   0   0   0  0  - 1  0  0³
+    ³                                 ³
+    ³ 0   - 1  0   0   0  0   1   0  0³
+    ³                                 ³
+    À 0    0   0  - 1  0  0  - 1  0  0Ù
+                              Type: List Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty6}
+\begin{paste}{RepA6PageEmpty6}{RepA6PagePatch6}
+\pastebutton{RepA6PageEmpty6}{\showpaste}
+\tab{5}\spadcommand{d2211 := irreducibleRepresentation ([2,2,1,1],genA6)\bound{d2211 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch7}
+\begin{paste}{RepA6PageFull7}{RepA6PageEmpty7}
+\pastebutton{RepA6PageFull7}{\hidepaste}
+\tab{5}\spadcommand{d2211m2 := d2211:: (LIST MATRIX PF 2); sp2 := meatAxe d2211m2\free{d2211 }\bound{sp2 }}
+\indentrel{3}\begin{verbatim}
+   Fingerprint element in generated algebra is singular
+     A proper cyclic submodule is found.
+     Transition matrix computed
+     The inverse of the transition matrix computed
+     Now transform the matrices
+   (7)
+     Ú1  0  1  1¿ Ú0  0  1  0¿
+     ³          ³ ³          ³
+     ³0  1  0  1³ ³1  1  1  1³
+   [[³          ³,³          ³],
+     ³1  1  0  0³ ³1  0  1  1³
+     ³          ³ ³          ³
+     À0  1  0  0Ù À0  1  0  1Ù
+     Ú1  0  0  0  0¿ Ú1  1  1  0  0¿
+     ³             ³ ³             ³
+     ³0  1  1  1  1³ ³0  0  1  1  1³
+     ³             ³ ³             ³
+    [³0  1  1  0  0³,³1  0  0  1  0³]]
+     ³             ³ ³             ³
+     ³0  1  0  1  0³ ³0  0  1  0  1³
+     ³             ³ ³             ³
+     À0  1  1  1  0Ù À1  0  0  1  1Ù
+                    Type: List List Matrix PrimeField 2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty7}
+\begin{paste}{RepA6PageEmpty7}{RepA6PagePatch7}
+\pastebutton{RepA6PageEmpty7}{\showpaste}
+\tab{5}\spadcommand{d2211m2 := d2211:: (LIST MATRIX PF 2); sp2 := meatAxe d2211m2\free{d2211 }\bound{sp2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch8}
+\begin{paste}{RepA6PageFull8}{RepA6PageEmpty8}
+\pastebutton{RepA6PageFull8}{\hidepaste}
+\tab{5}\spadcommand{isAbsolutelyIrreducible? sp2.1}
+\indentrel{3}\begin{verbatim}
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra has
+     one-dimensional kernel
+    We know that all the cyclic submodules generated by a
+  ll
+      non-trivial element of the singular matrix under vi
+  ew are
+      not proper, hence Norton's irreducibility test can 
+  be done:
+     The generated cyclic submodule was not proper
+     Representation is absolutely irreducible
+   (8)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty8}
+\begin{paste}{RepA6PageEmpty8}{RepA6PagePatch8}
+\pastebutton{RepA6PageEmpty8}{\showpaste}
+\tab{5}\spadcommand{isAbsolutelyIrreducible? sp2.1}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch9}
+\begin{paste}{RepA6PageFull9}{RepA6PageEmpty9}
+\pastebutton{RepA6PageFull9}{\hidepaste}
+\tab{5}\spadcommand{areEquivalent? (sp1.2, sp2.1)}
+\indentrel{3}\begin{verbatim}
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra has
+     one-dimensional kernel
+     There is no isomorphism, as the only possible one
+       fails to do the necessary base change
+
+   Representations are not equivalent.
+   (9)  [0]
+                              Type: Matrix PrimeField 2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty9}
+\begin{paste}{RepA6PageEmpty9}{RepA6PagePatch9}
+\pastebutton{RepA6PageEmpty9}{\showpaste}
+\tab{5}\spadcommand{areEquivalent? (sp1.2, sp2.1)}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch10}
+\begin{paste}{RepA6PageFull10}{RepA6PageEmpty10}
+\pastebutton{RepA6PageFull10}{\hidepaste}
+\tab{5}\spadcommand{dA6d16 := tensorProduct(sp1.2,sp2.1); meatAxe dA6d16\bound{dA6d16 }}
+\indentrel{3}\begin{verbatim}
+  Fingerprint element in generated algebra is non-singula
+  r
+   Fingerprint element in generated algebra is singular
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+  Fingerprint element in generated algebra is non-singula
+  r
+   Fingerprint element in generated algebra is singular
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+   Fingerprint element in generated algebra is singular
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+    We know that all the cyclic submodules generated by a
+  ll
+      non-trivial element of the singular matrix under vi
+  ew are
+      not proper, hence Norton's irreducibility test can 
+  be done:
+     The generated cyclic submodule was not proper
+     Representation is irreducible, but we don't know
+       whether it is absolutely irreducible
+   (10)
+   [
+      Ú0  0  0  0  0  0  0  0  1  0  1  0  0  0  0  0¿
+      ³                                              ³
+      ³0  0  0  0  0  0  0  0  0  1  1  1  0  0  0  0³
+      ³                                              ³
+      ³0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0³
+      ³                                              ³
+      ³0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0³
+      ³                                              ³
+      ³1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0³
+      ³                                              ³
+      ³0  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0³
+      ³                                              ³
+      ³1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0³
+      ³                                              ³
+      ³1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0³
+     [³                                              ³,
+      ³0  0  0  0  1  0  1  0  0  0  0  0  0  0  0  0³
+      ³                                              ³
+      ³0  0  0  0  0  1  1  1  0  0  0  0  0  0  0  0³
+      ³                                              ³
+      ³0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0³
+      ³                                              ³
+      ³0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0³
+      ³                                              ³
+      ³0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  0³
+      ³                                              ³
+      ³0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1³
+      ³                                              ³
+      ³0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0³
+      ³                                              ³
+      À0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  0Ù
+      Ú0  0  0  0  0  1  1  0  0  1  1  0  0  1  1  0¿
+      ³                                              ³
+      ³0  0  0  0  0  1  0  1  0  1  0  1  0  1  0  1³
+      ³                                              ³
+      ³0  0  0  0  1  1  1  0  1  1  1  0  1  1  1  0³
+      ³                                              ³
+      ³0  0  0  0  0  1  1  1  0  1  1  1  0  1  1  1³
+      ³                                              ³
+      ³0  1  1  0  0  1  1  0  0  1  1  0  0  1  1  0³
+      ³                                              ³
+      ³0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1³
+      ³                                              ³
+      ³1  1  1  0  1  1  1  0  1  1  1  0  1  1  1  0³
+      ³                                              ³
+      ³0  1  1  1  0  1  1  1  0  1  1  1  0  1  1  1³
+      ³                                              ³]
+      ³0  1  1  0  0  0  0  0  0  1  1  0  0  1  1  0³
+      ³                                              ³
+      ³0  1  0  1  0  0  0  0  0  1  0  1  0  1  0  1³
+      ³                                              ³
+      ³1  1  1  0  0  0  0  0  1  1  1  0  1  1  1  0³
+      ³                                              ³
+      ³0  1  1  1  0  0  0  0  0  1  1  1  0  1  1  1³
+      ³                                              ³
+      ³0  1  1  0  0  1  1  0  0  0  0  0  0  1  1  0³
+      ³                                              ³
+      ³0  1  0  1  0  1  0  1  0  0  0  0  0  1  0  1³
+      ³                                              ³
+      ³1  1  1  0  1  1  1  0  0  0  0  0  1  1  1  0³
+      ³                                              ³
+      À0  1  1  1  0  1  1  1  0  0  0  0  0  1  1  1Ù
+     ]
+                    Type: List List Matrix PrimeField 2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty10}
+\begin{paste}{RepA6PageEmpty10}{RepA6PagePatch10}
+\pastebutton{RepA6PageEmpty10}{\showpaste}
+\tab{5}\spadcommand{dA6d16 := tensorProduct(sp1.2,sp2.1); meatAxe dA6d16\bound{dA6d16 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch11}
+\begin{paste}{RepA6PageFull11}{RepA6PageEmpty11}
+\pastebutton{RepA6PageFull11}{\hidepaste}
+\tab{5}\spadcommand{isAbsolutelyIrreducible? dA6d16}
+\indentrel{3}\begin{verbatim}
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   We have not found a one-dimensional kernel so far,
+     as we do a random search you could try again
+   (11)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty11}
+\begin{paste}{RepA6PageEmpty11}{RepA6PagePatch11}
+\pastebutton{RepA6PageEmpty11}{\showpaste}
+\tab{5}\spadcommand{isAbsolutelyIrreducible? dA6d16}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch12}
+\begin{paste}{RepA6PageFull12}{RepA6PageEmpty12}
+\pastebutton{RepA6PageFull12}{\hidepaste}
+\tab{5}\spadcommand{sp3 := meatAxe (dA6d16 :: (LIST MATRIX FF(2,2)))\bound{sp3 }}
+\indentrel{3}\begin{verbatim}
+  Fingerprint element in generated algebra is non-singula
+  r
+   Fingerprint element in generated algebra is singular
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+  Fingerprint element in generated algebra is non-singula
+  r
+   Fingerprint element in generated algebra is singular
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+   Fingerprint element in generated algebra is singular
+     The generated cyclic submodule was not proper
+     The generated cyclic submodule was not proper
+     A proper cyclic submodule is found.
+     Transition matrix computed
+     The inverse of the transition matrix computed
+     Now transform the matrices
+   (12)
+   [
+     [
+   [[%A,%A + 1,0,%A,1,%A + 1,0,0],
+    [0,0,%A,%A + 1,%A,%A,0,0],
+    [%A,%A + 1,%A,1,%A + 1,0,0,0],
+    [%A,%A + 1,%A,1,%A,0,0,0],
+    [%A + 1,1,1,1,0,0,%A + 1,%A],
+    [0,0,%A + 1,1,0,0,%A,0], [1,0,1,1,0,0,0,0],
+    [1,1,0,0,0,0,0,0]]
+       ,
+
+   [[1,0,%A,0,1,1,%A,%A + 1],
+    [1,%A + 1,0,0,0,%A + 1,1,%A + 1],
+    [%A,1,%A + 1,%A + 1,%A + 1,1,%A,0],
+    [%A + 1,%A + 1,0,0,1,%A + 1,1,1],
+    [1,0,%A + 1,0,1,1,%A,%A],
+    [0,0,%A + 1,%A + 1,%A + 1,1,1,%A], [0,0,1,0,0,1,0,1],
+    [0,%A,0,%A,1,%A + 1,%A + 1,%A]]
+       ]
+     ,
+
+      Ú0     1       1     %A + 1  0  0  0  0¿
+      ³                                      ³
+      ³1     1     %A + 1    0     0  0  0  0³
+      ³                                      ³
+      ³%A    0       0       0     0  0  0  0³
+      ³                                      ³
+      ³1     %A      0       0     0  0  0  0³
+     [³                                      ³,
+      ³%A  %A + 1    1       1     1  0  1  1³
+      ³                                      ³
+      ³0     0       %A      1     0  1  0  1³
+      ³                                      ³
+      ³%A    1       0       1     1  1  0  0³
+      ³                                      ³
+      À1     %A    %A + 1    %A    0  1  0  0Ù
+
+   [[%A + 1,1,%A,0,0,%A + 1,0,1],
+    [0,%A,1,1,1,0,%A + 1,%A],
+    [0,%A + 1,0,%A + 1,%A + 1,1,%A + 1,%A],
+    [1,%A + 1,1,%A + 1,0,0,%A + 1,1],
+    [0,%A,0,%A + 1,%A + 1,0,0,%A + 1],
+    [%A + 1,0,%A + 1,%A,0,%A + 1,0,%A + 1],
+    [0,1,0,1,%A + 1,0,%A + 1,%A + 1],
+    [%A,%A,%A,1,%A,%A,1,%A + 1]]
+       ]
+     ]
+                Type: List List Matrix FiniteField(2,2)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty12}
+\begin{paste}{RepA6PageEmpty12}{RepA6PagePatch12}
+\pastebutton{RepA6PageEmpty12}{\showpaste}
+\tab{5}\spadcommand{sp3 := meatAxe (dA6d16 :: (LIST MATRIX FF(2,2)))\bound{sp3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch13}
+\begin{paste}{RepA6PageFull13}{RepA6PageEmpty13}
+\pastebutton{RepA6PageFull13}{\hidepaste}
+\tab{5}\spadcommand{isAbsolutelyIrreducible? sp3.1}
+\indentrel{3}\begin{verbatim}
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra has
+     one-dimensional kernel
+    We know that all the cyclic submodules generated by a
+  ll
+      non-trivial element of the singular matrix under vi
+  ew are
+      not proper, hence Norton's irreducibility test can 
+  be done:
+     The generated cyclic submodule was not proper
+     Representation is absolutely irreducible
+   (13)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty13}
+\begin{paste}{RepA6PageEmpty13}{RepA6PagePatch13}
+\pastebutton{RepA6PageEmpty13}{\showpaste}
+\tab{5}\spadcommand{isAbsolutelyIrreducible? sp3.1}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch14}
+\begin{paste}{RepA6PageFull14}{RepA6PageEmpty14}
+\pastebutton{RepA6PageFull14}{\hidepaste}
+\tab{5}\spadcommand{isAbsolutelyIrreducible? sp3.2}
+\indentrel{3}\begin{verbatim}
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra has
+     one-dimensional kernel
+    We know that all the cyclic submodules generated by a
+  ll
+      non-trivial element of the singular matrix under vi
+  ew are
+      not proper, hence Norton's irreducibility test can 
+  be done:
+     The generated cyclic submodule was not proper
+     Representation is absolutely irreducible
+   (14)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty14}
+\begin{paste}{RepA6PageEmpty14}{RepA6PagePatch14}
+\pastebutton{RepA6PageEmpty14}{\showpaste}
+\tab{5}\spadcommand{isAbsolutelyIrreducible? sp3.2}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch15}
+\begin{paste}{RepA6PageFull15}{RepA6PageEmpty15}
+\pastebutton{RepA6PageFull15}{\hidepaste}
+\tab{5}\spadcommand{areEquivalent? (sp3.1,sp3.2)}
+\indentrel{3}\begin{verbatim}
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra does
+     not have a one-dimensional kernel
+   Random element in generated algebra has
+     one-dimensional kernel
+     There is no isomorphism, as the only possible one
+       fails to do the necessary base change
+
+   Representations are not equivalent.
+   (15)  [0]
+                          Type: Matrix FiniteField(2,2)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty15}
+\begin{paste}{RepA6PageEmpty15}{RepA6PagePatch15}
+\pastebutton{RepA6PageEmpty15}{\showpaste}
+\tab{5}\spadcommand{areEquivalent? (sp3.1,sp3.2)}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch16}
+\begin{paste}{RepA6PageFull16}{RepA6PageEmpty16}
+\pastebutton{RepA6PageFull16}{\hidepaste}
+\tab{5}\spadcommand{sp0.2\free{sp0 }}
+\indentrel{3}\begin{verbatim}
+   (16)  [[1],[1]]
+                         Type: List Matrix PrimeField 2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty16}
+\begin{paste}{RepA6PageEmpty16}{RepA6PagePatch16}
+\pastebutton{RepA6PageEmpty16}{\showpaste}
+\tab{5}\spadcommand{sp0.2\free{sp0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch17}
+\begin{paste}{RepA6PageFull17}{RepA6PageEmpty17}
+\pastebutton{RepA6PageFull17}{\hidepaste}
+\tab{5}\spadcommand{sp1.2\free{sp1 }}
+\indentrel{3}\begin{verbatim}
+          Ú0  1  0  0¿ Ú0  1  1  1¿
+          ³          ³ ³          ³
+          ³0  0  1  0³ ³1  1  0  1³
+   (17)  [³          ³,³          ³]
+          ³1  0  0  0³ ³1  1  1  0³
+          ³          ³ ³          ³
+          À0  0  0  1Ù À1  1  1  1Ù
+                         Type: List Matrix PrimeField 2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty17}
+\begin{paste}{RepA6PageEmpty17}{RepA6PagePatch17}
+\pastebutton{RepA6PageEmpty17}{\showpaste}
+\tab{5}\spadcommand{sp1.2\free{sp1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch18}
+\begin{paste}{RepA6PageFull18}{RepA6PageEmpty18}
+\pastebutton{RepA6PageFull18}{\hidepaste}
+\tab{5}\spadcommand{sp2.1\free{sp2 }}
+\indentrel{3}\begin{verbatim}
+          Ú1  0  1  1¿ Ú0  0  1  0¿
+          ³          ³ ³          ³
+          ³0  1  0  1³ ³1  1  1  1³
+   (18)  [³          ³,³          ³]
+          ³1  1  0  0³ ³1  0  1  1³
+          ³          ³ ³          ³
+          À0  1  0  0Ù À0  1  0  1Ù
+                         Type: List Matrix PrimeField 2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty18}
+\begin{paste}{RepA6PageEmpty18}{RepA6PagePatch18}
+\pastebutton{RepA6PageEmpty18}{\showpaste}
+\tab{5}\spadcommand{sp2.1\free{sp2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch19}
+\begin{paste}{RepA6PageFull19}{RepA6PageEmpty19}
+\pastebutton{RepA6PageFull19}{\hidepaste}
+\tab{5}\spadcommand{sp3.1\free{sp3 }}
+\indentrel{3}\begin{verbatim}
+   (19)
+   [
+   [[%A,%A + 1,0,%A,1,%A + 1,0,0],
+    [0,0,%A,%A + 1,%A,%A,0,0],
+    [%A,%A + 1,%A,1,%A + 1,0,0,0],
+    [%A,%A + 1,%A,1,%A,0,0,0],
+    [%A + 1,1,1,1,0,0,%A + 1,%A],
+    [0,0,%A + 1,1,0,0,%A,0], [1,0,1,1,0,0,0,0],
+    [1,1,0,0,0,0,0,0]]
+     ,
+
+   [[1,0,%A,0,1,1,%A,%A + 1],
+    [1,%A + 1,0,0,0,%A + 1,1,%A + 1],
+    [%A,1,%A + 1,%A + 1,%A + 1,1,%A,0],
+    [%A + 1,%A + 1,0,0,1,%A + 1,1,1],
+    [1,0,%A + 1,0,1,1,%A,%A],
+    [0,0,%A + 1,%A + 1,%A + 1,1,1,%A], [0,0,1,0,0,1,0,1],
+    [0,%A,0,%A,1,%A + 1,%A + 1,%A]]
+     ]
+                     Type: List Matrix FiniteField(2,2)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty19}
+\begin{paste}{RepA6PageEmpty19}{RepA6PagePatch19}
+\pastebutton{RepA6PageEmpty19}{\showpaste}
+\tab{5}\spadcommand{sp3.1\free{sp3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch20}
+\begin{paste}{RepA6PageFull20}{RepA6PageEmpty20}
+\pastebutton{RepA6PageFull20}{\hidepaste}
+\tab{5}\spadcommand{sp3.2\free{sp3 }}
+\indentrel{3}\begin{verbatim}
+   (20)
+    Ú0     1       1     %A + 1  0  0  0  0¿
+    ³                                      ³
+    ³1     1     %A + 1    0     0  0  0  0³
+    ³                                      ³
+    ³%A    0       0       0     0  0  0  0³
+    ³                                      ³
+    ³1     %A      0       0     0  0  0  0³
+   [³                                      ³,
+    ³%A  %A + 1    1       1     1  0  1  1³
+    ³                                      ³
+    ³0     0       %A      1     0  1  0  1³
+    ³                                      ³
+    ³%A    1       0       1     1  1  0  0³
+    ³                                      ³
+    À1     %A    %A + 1    %A    0  1  0  0Ù
+
+   [[%A + 1,1,%A,0,0,%A + 1,0,1],
+    [0,%A,1,1,1,0,%A + 1,%A],
+    [0,%A + 1,0,%A + 1,%A + 1,1,%A + 1,%A],
+    [1,%A + 1,1,%A + 1,0,0,%A + 1,1],
+    [0,%A,0,%A + 1,%A + 1,0,0,%A + 1],
+    [%A + 1,0,%A + 1,%A,0,%A + 1,0,%A + 1],
+    [0,1,0,1,%A + 1,0,%A + 1,%A + 1],
+    [%A,%A,%A,1,%A,%A,1,%A + 1]]
+     ]
+                     Type: List Matrix FiniteField(2,2)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty20}
+\begin{paste}{RepA6PageEmpty20}{RepA6PagePatch20}
+\pastebutton{RepA6PageEmpty20}{\showpaste}
+\tab{5}\spadcommand{sp3.2\free{sp3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{RepA6PagePatch21}
+\begin{paste}{RepA6PageFull21}{RepA6PageEmpty21}
+\pastebutton{RepA6PageFull21}{\hidepaste}
+\tab{5}\spadcommand{dA6d16\free{dA6d16 }}
+\indentrel{3}\begin{verbatim}
+   (21)
+    Ú0  0  0  0  1  0  1  1  0  0  0  0  0  0  0  0¿
+    ³                                              ³
+    ³0  0  0  0  0  1  0  1  0  0  0  0  0  0  0  0³
+    ³                                              ³
+    ³0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0³
+    ³                                              ³
+    ³0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0³
+    ³                                              ³
+    ³0  0  0  0  0  0  0  0  1  0  1  1  0  0  0  0³
+    ³                                              ³
+    ³0  0  0  0  0  0  0  0  0  1  0  1  0  0  0  0³
+    ³                                              ³
+    ³0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0³
+    ³                                              ³
+    ³0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0³
+   [³                                              ³,
+    ³1  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0³
+    ³                                              ³
+    ³0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0³
+    ³                                              ³
+    ³1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0³
+    ³                                              ³
+    ³0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0³
+    ³                                              ³
+    ³0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1³
+    ³                                              ³
+    ³0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1³
+    ³                                              ³
+    ³0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  0³
+    ³                                              ³
+    À0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0Ù
+    Ú0  0  0  0  0  0  1  0  0  0  1  0  0  0  1  0¿
+    ³                                              ³
+    ³0  0  0  0  1  1  1  1  1  1  1  1  1  1  1  1³
+    ³                                              ³
+    ³0  0  0  0  1  0  1  1  1  0  1  1  1  0  1  1³
+    ³                                              ³
+    ³0  0  0  0  0  1  0  1  0  1  0  1  0  1  0  1³
+    ³                                              ³
+    ³0  0  1  0  0  0  1  0  0  0  0  0  0  0  1  0³
+    ³                                              ³
+    ³1  1  1  1  1  1  1  1  0  0  0  0  1  1  1  1³
+    ³                                              ³
+    ³1  0  1  1  1  0  1  1  0  0  0  0  1  0  1  1³
+    ³                                              ³
+    ³0  1  0  1  0  1  0  1  0  0  0  0  0  1  0  1³
+    ³                                              ³]
+    ³0  0  1  0  0  0  1  0  0  0  1  0  0  0  0  0³
+    ³                                              ³
+    ³1  1  1  1  1  1  1  1  1  1  1  1  0  0  0  0³
+    ³                                              ³
+    ³1  0  1  1  1  0  1  1  1  0  1  1  0  0  0  0³
+    ³                                              ³
+    ³0  1  0  1  0  1  0  1  0  1  0  1  0  0  0  0³
+    ³                                              ³
+    ³0  0  1  0  0  0  1  0  0  0  1  0  0  0  1  0³
+    ³                                              ³
+    ³1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1³
+    ³                                              ³
+    ³1  0  1  1  1  0  1  1  1  0  1  1  1  0  1  1³
+    ³                                              ³
+    À0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1Ù
+                         Type: List Matrix PrimeField 2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RepA6PageEmpty21}
+\begin{paste}{RepA6PageEmpty21}{RepA6PagePatch21}
+\pastebutton{RepA6PageEmpty21}{\showpaste}
+\tab{5}\spadcommand{dA6d16\free{dA6d16 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/help.ht b/src/hyper/pages/help.ht
new file mode 100644
index 00000000..7820cb95
--- /dev/null
+++ b/src/hyper/pages/help.ht
@@ -0,0 +1,142 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% @(#)help.ht	1.9 92/06/24 17:09:42
+% Help page
+
+%%  % ----------------------------------------------------------------------
+%%  \begin{page}{TopHelpPage}{Help Menu}
+%%  % ----------------------------------------------------------------------
+%%
+%%  You are in ``\HyperName,'' \Language{}'s hypertext document browser.
+%%  This section describes how to make \HyperName do what you want.
+%%  \pp
+%%  This document is organized into pages.
+%%  Pages usually have active areas, marked in \downlink{this font.}{YouTriedIt}
+%%  An active area is usually linked to another page: clicking the mouse
+%%  in the active area moves you to the linked page.
+%%  The mouse cursor changes from a filled dot to an open circle when
+%%  it is in an active area.
+%%  You can try \downlink{this}{YouTriedIt} out now.
+%%  \beginscroll
+%%  You can select any one of the items below:
+%%  \beginmenu
+%%  \menulink{Quit Help}
+%%  {\ifcond\hasreturn ReturnPage\else UpPage\fi}
+%%	Go back to  what you were doing.
+%%
+%%  \item
+%%	\searchwindow{\menuitemstyle{Simple Search}}
+%%		     {\stringvalue{FindTopic}}
+%%	Find references to \inputstring{FindTopic}{15}{series}
+%%	\searchwindow{Start Search} {\stringvalue{FindTopic}}.
+%%
+%%  \menulink{Buttons}{ugHyperHeadingsPage}
+%%  Learn about the buttons at the top of the page.
+%%
+%%  \menulink{Menus}{MenuHelp}
+%%  Learn about menus.
+%%
+%%  %\menulink{Commands}{CommandHelp}
+%%  %	 Learn how to run \Language{} commands from \HyperName.
+%%
+%%  \endmenu
+%%  \endscroll
+%%  \end{page}
+
+%%  % ----------------------------------------------------------------------
+%%  \begin{page}{ButtonHelp}{Know These Buttons}
+%%  % ----------------------------------------------------------------------
+%%  \beginscroll
+%%  All pages have a standard set of buttons at the top.
+%%  \newline
+%%  This is what they mean:
+%%  \par \ExitBitmap   \space{} Leave \HyperName.
+%%  \par \HelpBitmap   \space{} Get help, if there is any.
+%%  \par \UpBitmap     \space{} Go back to the previous page.
+%%  \par \ReturnBitmap \space{} Return to the home menu.
+%%  \newline
+%%  \pp
+%%  \HyperName remembers where you came from.
+%%  You can always make forays into new topics
+%%  without worrying about how to get back.
+%%  \pp
+%%  The buttons at the top of the page can be used at any time
+%%  to return to what you were doing.
+%%  \pp
+%%  The buttons will not be displayed if they are not applicable to the page
+%%  you are currently viewing.	So, for example, there is no ``HOME'' button
+%%  on the top-level menu.
+%%  \endscroll
+%%  \end{page}
+
+%%  % ----------------------------------------------------------------------
+%%  \begin{page}{MenuHelp}{Menus}
+%%  % ----------------------------------------------------------------------
+%%  \pp
+%%  \beginscroll
+%%  A `menu' is a list of topics.
+%%  You went through at least one to get here without even trying.
+%%  \newline
+%%  To choose a topic from a menu, you click on the active area
+%%  for the topic.
+%%  \horizontalline
+%%  Here is a menu to practice:
+%%  \newline
+%%  \beginmenu
+%%  \menulink{Hypertext}{WhatIsHyperText}  What is hypertext?
+%%  \menulink{\HyperName}{WhatIsHyperDoc}  What is \HyperName?
+%%  \menulink{Buttons}{ugHyperHeadingsPage}  Headings in HyperName{}.
+%%  \endmenu
+%%  \endscroll
+%%  %\autobutt{HelpHelp}
+%%  \end{page}
+
+%%  % ----------------------------------------------------------------------
+%%  \begin{page}{WhatIsHyperText}{What is hypertext?}
+%%  % ----------------------------------------------------------------------
+%%  \beginscroll
+%%  A ``hypertext'' document is a structured entity which is part book
+%%  and part program.
+%%  You can think of it as a museum with many exhibits you can try.
+%%  \newline
+%%  ``\HyperName'' is a computer program for presenting \Language{}'s \HyperName
+%%  documents.
+%%  \endscroll
+%%  %\autobutt{HelpHelp}
+%%  \end{page}
+%%
+%%  \begin{page}{WhatIsHyperDoc}{What is \HyperName?}
+%%  \beginscroll
+%%  This is \HyperName, a program for reading hypertext documents.
+%%  The name comes from the fact that the pages of the document are
+%%  composed using a variant of the ``TeX'' typesetting language.
+%%  \endscroll
+%%  %\autobutt{HelpHelp}
+%%  \end{page}
+
+%%  % ----------------------------------------------------------------------
+%%  \begin{page}{ExamplesHelp}{\Language{} Examples}
+%%  % ----------------------------------------------------------------------
+%%  \pp
+%%  \beginscroll
+%%  Many pages have \Language{} examples.  These are ``active''
+%%  also but if you click here, the example is copied to an
+%%  interactive \Language{} ``buffer.''
+%%  You can then edit it or use it as a command for the \Language{} interpreter.
+%%  Try these: \newline
+%%  \indent{5}
+%%  \spadcommand{a:= x**2 + 1 \bound{a}} \newline
+%%  \spadcommand{(a - 2)**2 \free{a}} \newline
+%%  \indent{0}
+%%  Notice that the \Language{} examples are printed in a different font.
+%%  \pp
+%%  The interactive \Language{} buffers will disappear when you leave
+%%  \HyperName.
+%%  If you want to get rid of them beforehand,
+%%  you can always use the ``Cancel'' button of the X window manager.
+%%  \endscroll
+%%  %\autobutt{HelpHelp}
+%%  \end{page}
+
diff --git a/src/hyper/pages/ht.db b/src/hyper/pages/ht.db
new file mode 100644
index 00000000..74d37cb5
--- /dev/null
+++ b/src/hyper/pages/ht.db
@@ -0,0 +1,9543 @@
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+\patch XPBWPolynomialXmpPageEmpty38 32207 856
+\patch XPBWPolynomialXmpPagePatch39 32475 862
+\patch XPBWPolynomialXmpPageEmpty39 32861 872
+	XPOLY.ht 1104291828
+\newcommand XPolynomialXmpTitle 140 3
+\newcommand XPolynomialXmpNumber 187 4
+\page XPolynomialXmpPage 302 7
+	XPOLY.pht 1104291828
+\patch XPolynomialXmpPagePatch1 0 1
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+\patch XPolynomialXmpPageEmpty14 9037 240
+	XPR.ht 1104291828
+\newcommand XPolynomialRingXmpTitle 140 3
+\newcommand XPolynomialRingXmpNumber 195 4
+\page XPolynomialRingXmpPage 314 7
+	XPR.pht 1104291828
+\patch XPolynomialRingXmpPagePatch1 0 1
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+\patch XPolynomialRingXmpPageEmpty15 13269 320
+	ZDSOLVE.ht 1104291828
+\newcommand ZeroDimensionalSolvePackageXmpTitle 140 3
+\newcommand ZeroDimensionalSolvePackageXmpNumber 219 4
+\page ZeroDimensionalSolvePackageXmpPage 350 7
+	ZDSOLVE.pht 1104291828
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+	ZLINDEP.ht 1104291828
+\newcommand IntegerLinearDependenceXmpTitle 140 3
+\newcommand IntegerLinearDependenceXmpNumber 211 4
+\page IntegerLinearDependenceXmpPage 338 7
+	ZLINDEP.pht 1104291828
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+\patch IntegerLinearDependenceXmpPageEmpty3 1986 47
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+\patch IntegerLinearDependenceXmpPageEmpty4 2784 65
+\patch IntegerLinearDependenceXmpPagePatch5 3101 71
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+\patch IntegerLinearDependenceXmpPagePatch6 3847 87
+\patch IntegerLinearDependenceXmpPageEmpty6 4301 97
+\patch IntegerLinearDependenceXmpPagePatch7 4624 103
+\patch IntegerLinearDependenceXmpPageEmpty7 5081 115
+\patch IntegerLinearDependenceXmpPagePatch8 5380 121
+\patch IntegerLinearDependenceXmpPageEmpty8 5839 133
diff --git a/src/hyper/pages/hyperdoc.ht b/src/hyper/pages/hyperdoc.ht
new file mode 100644
index 00000000..1046a6ae
--- /dev/null
+++ b/src/hyper/pages/hyperdoc.ht
@@ -0,0 +1,23 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% @(#)hyperdoc.ht	1.4 92/05/21 16:59:29
+
+\begin{page}{HyperName}{Creating \HyperName Pages}
+
+\beginscroll
+This document tells how to create \HyperName pages.
+To start with, it is rather meager but it will grow with time.
+\beginmenu
+\menulink{Viewports}{ViewportPage} Including live graphics in documents.
+\menulink{Gadjets}{BitMaps} Bitmaps for use in macros.
+\menulink{Control Panel Bits}{CPHelp} Development page for help 
+facility for viewports. yuck.
+%\menulink{Test Pages}{TestPage} Some test pages left by J.M.
+%\menulink{Paste Pages}{PastePage} Examples of how to use paste in areas.
+\endmenu
+
+\endscroll
+\autobuttons
+\end{page}
diff --git a/src/hyper/pages/man0.ht b/src/hyper/pages/man0.ht
new file mode 100644
index 00000000..8583f4ea
--- /dev/null
+++ b/src/hyper/pages/man0.ht
@@ -0,0 +1,73 @@
+% Copyright The Numerical Algorithms Group Limited 1991-94.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+\begin{page}{RefSearchPage}{Reference Search}
+\beginscroll
+Enter search string :
+\inputstring{pattern}{40}{}
+\newline
+\beginmenu
+\menuunixlink{Search}
+  {htsearch "\stringvalue{pattern}"}
+  \tab{15} Reference documentation ({\em *} wild card is not accepted).
+\endmenu
+\endscroll
+\end{page}
+
+\begin{page}{Man0Page}{A X I O M \   B r o w s e r}
+
+\beginscroll
+Enter search string (use {\em *} for wild card unless counter-indicated):
+\inputstring{pattern}{40}{}
+\newline
+\beginmenu
+\menulispmemolink{Constructors}
+  { (|kSearch| '|\stringvalue{pattern}|) }
+  \tab{15} Search for \lispmemolink{categories}{(|cSearch| '|\stringvalue{pattern}|)}, \lispmemolink{domains}{(|dSearch| '|\stringvalue{pattern}|)}, or \lispmemolink{packages}{(|pSearch| '|\stringvalue{pattern}|)}
+\menulispmemolink{Operations}
+  { (|oSearch| '|\stringvalue{pattern}|) }
+  \tab{15} Search for operations.
+\menulispmemolink{Attributes}
+  { (|aSearch| '|\stringvalue{pattern}|) }
+  \tab{15} Search for attributes.
+\menulispmemolink{General}
+  { (|aokSearch| '|\stringvalue{pattern}|) }
+  \tab{15} Search for all three of the above.
+\menulispmemolink{Documentation}
+  { (|docSearch| '|\stringvalue{pattern}|) }
+  \tab{15} Search library documentation.
+\menulispmemolink{Complete}
+  { (|genSearch| '|\stringvalue{pattern}|) }
+  \tab{15} All of the above.
+\menulispmemolink{Selectable}
+  { (|detailedSearch| '|\stringvalue{pattern}|) }
+  \tab{15} Detailed search with selectable options.
+\horizontalline
+\menuunixlink{Reference}
+  {htsearch "\stringvalue{pattern}"}
+  \tab{15} Search Reference documentation ({\em *} wild card is not accepted).
+\menumemolink{Commands}{ugSysCmdPage}
+\tab{15} View system command documentation.
+\endmenu
+\endscroll
+\autobutt{BROWSEhelp}
+\end{page}
+
+
+
+\begin{page}{BROWSEhelp}{The \Language{} BROWSE Facility}
+
+\beginscroll
+\beginmenu
+    \menudownlink{{The Front Page: Searching the Library}}{ugBrowseStartPage}
+    \menudownlink{{The Constructor Page}}{ugBrowseDomainPage}
+    \menudownlink{{Miscellaneous Features of Browse}}{ugBrowseMiscellaneousFeaturesPage}
+\endmenu
+\endscroll
+\newline
+\end{page}
+
+
+
+
diff --git a/src/hyper/pages/mapping.ht b/src/hyper/pages/mapping.ht
new file mode 100644
index 00000000..bfe7d32d
--- /dev/null
+++ b/src/hyper/pages/mapping.ht
@@ -0,0 +1,46 @@
+%---------------------------------------------------------------------------
+%                   Mapping Constructor Page
+%---------------------------------------------------------------------------
+ 
+\begin{page}{DomainMapping}{Domain {\em Mapping(T,S,...)}}
+\beginscroll
+{\em Mapping} takes any number of arguments of the form:
+\indentrel{2}
+\newline \spad{T}, a domain of category \spadtype{SetCategory}
+\newline \spad{S}, a domain of category \spadtype{SetCategory}
+\newline\tab{10}...
+\indentrel{-2}\newline
+This constructor is a primitive in \Language{}.
+\newline
+\beginmenu
+\item\menulispdownlink{Description}{(|dbSpecialDescription| '|Mapping|)}\tab{19}General description
+\item\menulispdownlink{Operations} {(|dbSpecialOperations| '|Mapping|)}\tab{19}All exported operations of \spad{Mapping(T,S)}
+%\item\menudownlink{Examples}   {MappingExamples}   \tab{19}Examples illustrating use
+%\item\menudownlink{Exports}    {MappingExports}    \tab{19}Explicit categories and operations
+\endmenu
+\endscroll\end{page}
+ 
+\begin{page}{MappingDescription}{Domain Constructor {\em Mapping}}	
+\beginscroll
+\newline\menuitemstyle{}\tab{2}Mapping({\em T},{\em S,...})
+\newline\tab{2}{\em Arguments:}\indent{17}\tab{-2}
+{\em T}, a domain of category \spadtype{SetCategory}
+\newline\tab{-2}
+{\em S}, a domain of category \spadtype{SetCategory}
+\newline\tab{10}...
+\indent{0}\newline\tab{2}{\em Returns:}\indent{15}\tab{0}
+the class of mappings from domain ({\em S,...}) 
+into domain {\em T} as described below.
+\indent{0}\newline\tab{2}{\em Description:}\indent{15}\tab{0}
+{\em Mapping(T,S,...)} denotes the class of objects which are
+mappings from a source domain ({\em S,...}) into a target domain {\em T}.
+The {\em Mapping} constructor can take any number of arguments.
+All but the first argument is regarded as part of a source tuple
+for the mapping. 
+For example, {\em Mapping(T,A,B)} denotes the
+class of mappings from {\em (A,B)} into {\em T}.
+{\em Mapping} is a primitive domain of \Language{} which cannot be
+defined in the \Language{} language. 
+\endscroll
+\end{page}
+
diff --git a/src/hyper/pages/nagaux.ht b/src/hyper/pages/nagaux.ht
new file mode 100644
index 00000000..0ff0b666
--- /dev/null
+++ b/src/hyper/pages/nagaux.ht
@@ -0,0 +1,7051 @@
+\begin{page}{manpageXXonline}{NAG On-line Documentation: online}
+\beginscroll
+\begin{verbatim}
+
+
+
+     DOC INTRO(3NAG)   Foundation Library (12/10/92)   DOC INTRO(3NAG)
+
+
+
+                    Introduction to NAG On-Line Documentation
+
+
+          The on-line documentation for the NAG Foundation Library has been
+          generated automatically from the same base material used to
+          create the printed Reference Manual. To make the documentation
+          readable on the widest range of machines, only the basic set of
+          ascii characters has been used.
+
+          Certain mathematical symbols have been constructed using plain
+          ascii characters:
+
+          integral signs                /
+                                        |
+
+                                        /
+
+          summation signs               --
+                                        >
+                                        --
+
+          square root signs                ----
+                                          /
+                                        \/
+
+
+          Large brackets are constructed using vertical stacks of the
+          equivalent ascii character:
+
+                              (  )  [  ]  {  }  |
+                              (  )  [  ]  {  }  |
+                              (  )  [  ]  {  }  |
+
+
+          Fractions are represented as:
+
+                                       a
+                                      ---
+                                      x+1
+
+
+          Greek letters are represented by their names enclosed in round
+          brackets:
+
+                  (alpha)  (beta)  (gamma) .....
+
+                  (Alpha)  (Beta)  (Gamma) .....
+
+
+          Some characters are accented using:
+
+                  ^     ~     
+                  X     X     X
+
+          Other mathematical symbols are represented as follows:
+
+                 *           times
+
+                 <=>         left-right arrow
+
+                 <-          left arrow
+
+                 ~           similar to
+
+                 ~=          similar or equal to
+
+                 ==          equivalent to
+
+                 >=          greater than or equal to
+
+                 <=          less than or equal to
+
+                 >>          much greater than
+
+                 <<          much less than
+
+                 >~          greater than or similar to
+
+                 /=          not equal to
+
+                 dd          partial derivative
+
+                 +-          plus or minus
+
+                 (nabla)     Nabla
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXsummary}{NAG On-line Documentation: summary}
+\beginscroll
+\begin{verbatim}
+
+
+
+     SUMMARY(3NAG)     Foundation Library (12/10/92)     SUMMARY(3NAG)
+
+
+
+          Introduction                                     List of Routines
+          List of Routines
+
+          The NAG Foundation Library contains three categories of routines
+          which can be called by users. They are listed separately in the
+          three sections below.
+
+          Fully Documented Routines
+                254 routines, for each of which an individual routine
+                document is provided. These are regarded as the primary
+                contents of the Foundation Library.
+
+          Fundamental Support Routines
+                83 comparatively simple routines which are documented in
+                compact form in the relevant Chapter Introductions (F06,
+                X01, X02).
+
+          Routines from the NAG Fortran Library
+                An additional 167 routines from the NAG Fortran Library,
+                which are used as auxiliaries in the Foundation Library.
+                They are not documented in this publication, but can be
+                called if you are already familiar with their use in the
+                Fortran Library. Only their names are given here.
+
+
+          Note: all the routines in the above categories have names ending
+          in 'F'. Occasionally this publication may refer to routines whose
+          names end in some other letter (e.g. 'Z', 'Y', 'X'). These are
+          auxiliary routines whose names may be passed as parameters to a
+          Foundation Library routine; you only need to know their names,
+          not how to call them directly.
+
+          Fully Documented Routines
+
+          The Foundation Library contains 254 user-callable routines, for
+          each of which an individual routine document is provided, in the
+          following chapters:
+
+          C02 -- Zeros of Polynomials
+
+          C02AFF  All zeros of complex polynomial, modified Laguerre method
+
+          C02AGF  All zeros of real polynomial, modified Laguerre method
+
+          C05 -- Roots of One or More Transcendental Equations
+
+          C05ADF  Zero of continuous function in given interval, Bus and
+                  Dekker algorithm
+
+          C05NBF  Solution of system of nonlinear equations using function
+                  values only
+
+          C05PBF  Solution of system of nonlinear equations using 1st
+                  derivatives
+
+          C05ZAF  Check user's routine for calculating 1st derivatives
+
+          C06 -- Summation of Series
+
+          C06EAF  Single 1-D real discrete Fourier transform, no extra
+                  workspace
+
+          C06EBF  Single 1-D Hermitian discrete Fourier transform, no extra
+                  workspace
+
+          C06ECF  Single 1-D complex discrete Fourier transform, no extra
+                  workspace
+
+          C06EKF  Circular convolution or correlation of two real vectors,
+                  no extra workspace
+
+          C06FPF  Multiple 1-D real discrete Fourier transforms
+
+          C06FQF  Multiple 1-D Hermitian discrete Fourier transforms
+
+          C06FRF  Multiple 1-D complex discrete Fourier transforms
+
+          C06FUF  2-D complex discrete Fourier transform
+
+          C06GBF  Complex conjugate of Hermitian sequence
+
+          C06GCF  Complex conjugate of complex sequence
+
+          C06GQF  Complex conjugate of multiple Hermitian sequences
+
+          C06GSF  Convert Hermitian sequences to general complex sequences
+
+          D01 -- Quadrature
+
+          D01AJF  1-D quadrature, adaptive, finite interval, strategy due
+                  to Piessens and de Doncker, allowing for badly-behaved
+                  integrands
+
+          D01AKF  1-D quadrature, adaptive, finite interval, method
+                  suitable for oscillating functions
+
+          D01ALF  1-D quadrature, adaptive, finite interval, allowing for
+                  singularities at user-specified break-points
+
+          D01AMF  1-D quadrature, adaptive, infinite or semi-infinite
+                  interval
+
+          D01ANF  1-D quadrature, adaptive, finite interval, weight
+                  function cos((omega)x) or sin((omega)x)
+
+          D01APF  1-D quadrature, adaptive, finite interval, weight
+                  function with end-point singularities of algebraico-
+                  logarithmic type
+
+          D01AQF  1-D quadrature, adaptive, finite interval, weight
+                  function 1/(x-c), Cauchy principal value (Hilbert
+                  transform)
+
+          D01ASF  1-D quadrature, adaptive, semi-infinite interval, weight
+                  function cos((omega)x) or sin((omega)x)
+
+          D01BBF  Weights and abscissae for Gaussian quadrature rules
+
+          D01FCF  Multi-dimensional adaptive quadrature over hyper-
+                  rectangle
+
+          D01GAF  1-D quadrature, integration of function defined by data
+                  values, Gill-Miller method
+
+          D01GBF  Multi-dimensional quadrature over hyper-rectangle, Monte
+                  Carlo method
+
+          D02 -- Ordinary Differential Equations
+
+          D02BBF  ODEs, IVP, Runge-Kutta-Merson method, over a range,
+                  intermediate output
+
+          D02BHF  ODEs, IVP, Runge-Kutta-Merson method, until function of
+                  solution is zero
+
+          D02CJF  ODEs, IVP, Adams method, until function of solution is
+                  zero, intermediate output
+
+          D02EJF  ODEs, stiff IVP, BDF method, until function of solution
+                  is zero, intermediate output
+
+          D02GAF  ODEs, boundary value problem, finite difference technique
+                  with deferred correction, simple nonlinear problem
+
+          D02GBF  ODEs, boundary value problem, finite difference technique
+                  with deferred correction, general linear problem
+
+          D02KEF  2nd order Sturm-Liouville problem, regular/singular
+                  system, finite/infinite range, eigenvalue and
+                  eigenfunction, user-specified break-points
+
+          D02RAF  ODEs, general nonlinear boundary value problem, finite
+                  difference technique with deferred correction,
+                  continuation facility
+
+          D03 -- Partial Differential Equations
+
+          D03EDF  Elliptic PDE, solution of finite difference equations by
+                  a multigrid technique
+
+          D03EEF  Discretize a 2nd order elliptic PDE on a rectangle
+
+          D03FAF  Elliptic PDE, Helmholtz equation, 3-D Cartesian co-
+                  ordinates
+
+          E01 -- Interpolation
+
+          E01BAF  Interpolating functions, cubic spline interpolant, one
+                  variable
+
+          E01BEF  Interpolating functions, monotonicity-preserving,
+                  piecewise cubic Hermite, one variable
+
+          E01BFF  Interpolated values, interpolant computed by E01BEF,
+                  function only, one variable
+
+          E01BGF  Interpolated values, interpolant computed by E01BEF,
+                  function and 1st derivative, one variable
+
+          E01BHF  Interpolated values, interpolant computed by E01BEF,
+                  definite integral, one variable
+
+          E01DAF  Interpolating functions, fitting bicubic spline, data on
+                  rectangular grid
+
+          E01SAF  Interpolating functions, method of Renka and Cline, two
+                  variables
+
+          E01SBF  Interpolated values, evaluate interpolant computed by
+                  E01SAF, two variables
+
+          E01SEF  Interpolating functions, modified Shepard's method, two
+                  variables
+
+          E01SFF  Interpolated values, evaluate interpolant computed by
+                  E01SEF, two variables
+
+          E02 -- Curve and Surface Fitting
+
+          E02ADF  Least-squares curve fit, by polynomials, arbitrary data
+                  points
+
+          E02AEF  Evaluation of fitted polynomial in one variable from
+                  Chebyshev series form (simplified parameter list)
+
+          E02AGF  Least-squares polynomial fit, values and derivatives may
+                  be constrained, arbitrary data points,
+
+          E02AHF  Derivative of fitted polynomial in Chebyshev series form
+
+          E02AJF  Integral of fitted polynomial in Chebyshev series form
+
+          E02AKF  Evaluation of fitted polynomial in one variable, from
+                  Chebyshev series form
+
+          E02BAF  Least-squares curve cubic spline fit (including
+                  interpolation)
+
+          E02BBF  Evaluation of fitted cubic spline, function only
+
+          E02BCF  Evaluation of fitted cubic spline, function and
+                  derivatives
+
+          E02BDF  Evaluation of fitted cubic spline, definite integral
+
+          E02BEF  Least-squares cubic spline curve fit, automatic knot
+                  placement
+
+          E02DAF  Least-squares surface fit, bicubic splines
+
+          E02DCF  Least-squares surface fit by bicubic splines with
+                  automatic knot placement, data on rectangular grid
+
+          E02DDF  Least-squares surface fit by bicubic splines with
+                  automatic knot placement, scattered data
+
+          E02DEF  Evaluation of a fitted bicubic spline at a vector of
+                  points
+
+          E02DFF  Evaluation of a fitted bicubic spline at a mesh of points
+
+          E02GAF  L -approximation by general linear function
+                   1
+
+          E02ZAF  Sort 2-D data into panels for fitting bicubic splines
+
+          E04 -- Minimizing or Maximizing a Function
+
+          E04DGF  Unconstrained minimum, pre-conditioned conjugate gradient
+                  algorithm, function of several variables using 1st
+                  derivatives
+
+          E04DJF  Read optional parameter values for E04DGF from external
+                  file
+
+          E04DKF  Supply optional parameter values to E04DGF
+
+          E04FDF  Unconstrained minimum of a sum of squares, combined
+                  Gauss-Newton and modified Newton algorithm using function
+                  values only
+
+          E04GCF  Unconstrained minimum of a sum of squares, combined
+                  Gauss-Newton and quasi-Newton algorithm, using 1st
+                  derivatives
+
+          E04JAF  Minimum, function of several variables, quasi-Newton
+                  algorithm, simple bounds, using function values only
+
+          E04MBF  Linear programming problem
+
+          E04NAF  Quadratic programming problem
+
+          E04UCF  Minimum, function of several variables, sequential QP
+                  method, nonlinear constraints, using function values and
+                  optionally 1st derivatives
+
+          E04UDF  Read optional parameter values for E04UCF from external
+                  file
+
+          E04UEF  Supply optional parameter values to E04UCF
+
+          E04YCF  Covariance matrix for nonlinear least-squares problem
+
+          F01 -- Matrix Factorizations
+
+          F01BRF  LU factorization of real sparse matrix
+
+          F01BSF  LU factorization of real sparse matrix with known
+                  sparsity pattern
+
+                    T
+          F01MAF  LL  factorization of real sparse symmetric positive-
+                  definite matrix
+
+                     T
+          F01MCF  LDL  factorization of real symmetric positive-definite
+                  variable-bandwidth matrix
+
+          F01QCF  QR factorization of real m by n matrix (m>=n)
+
+                                                                      T
+          F01QDF  Operations with orthogonal matrices, compute QB or Q B
+                  after factorization by F01QCF
+
+          F01QEF  Operations with orthogonal matrices, form columns of Q
+                  after factorization by F01QCF
+
+          F01RCF  QR factorization of complex m by n matrix (m>=n)
+
+                                                                   H
+          F01RDF  Operations with unitary matrices, compute QB or Q B after
+                  factorization by F01RCF
+
+          F01REF  Operations with unitary matrices, form columns of Q after
+                  factorization by F01RCF
+
+          F02 -- Eigenvalues and Eigenvectors
+
+          F02AAF  All eigenvalues of real symmetric matrix
+
+          F02ABF  All eigenvalues and eigenvectors of real symmetric matrix
+
+          F02ADF  All eigenvalues of generalized real symmetric-definite
+                  eigenproblem
+
+          F02AEF  All eigenvalues and eigenvectors of generalized real
+                  symmetric-definite eigenproblem
+
+          F02AFF  All eigenvalues of real matrix
+
+          F02AGF  All eigenvalues and eigenvectors of real matrix
+
+          F02AJF  All eigenvalues of complex matrix
+
+          F02AKF  All eigenvalues and eigenvectors of complex matrix
+
+          F02AWF  All eigenvalues of complex Hermitian matrix
+
+          F02AXF  All eigenvalues and eigenvectors of complex Hermitian
+                  matrix
+
+          F02BBF  Selected eigenvalues and eigenvectors of real symmetric
+                  matrix
+
+          F02BJF  All eigenvalues and optionally eigenvectors of
+                  generalized eigenproblem by QZ algorithm, real matrices
+
+          F02FJF  Selected eigenvalues and eigenvectors of sparse symmetric
+                  eigenproblem
+
+          F02WEF  SVD of real matrix
+
+          F02XEF  SVD of complex matrix
+
+          F04 -- Simultaneous Linear Equations
+
+          F04ADF  Approximate solution of complex simultaneous linear
+                  equations with multiple right-hand sides
+
+          F04ARF  Approximate solution of real simultaneous linear
+                  equations, one right-hand side
+
+          F04ASF  Accurate solution of real symmetric positive-definite
+                  simultaneous linear equations, one right-hand side
+
+          F04ATF  Accurate solution of real simultaneous linear equations,
+                  one right-hand side
+
+          F04AXF  Approximate solution of real sparse simultaneous linear
+                  equations (coefficient matrix already factorized by
+                  F01BRF or F01BSF)
+
+          F04FAF  Approximate solution of real symmetric positive-definite
+                  tridiagonal simultaneous linear equations, one right-hand
+                  side
+
+          F04JGF  Least-squares (if rank = n) or minimal least-squares (if
+                  rank <n) solution of m real equations in n unknowns, rank
+                  <=n, m>=n
+
+          F04MAF  Real sparse symmetric positive-definite simultaneous
+                  linear equations (coefficient matrix already factorized)
+
+          F04MBF  Real sparse symmetric simultaneous linear equations
+
+          F04MCF  Approximate solution of real symmetric positive-definite
+                  variable-bandwidth simultaneous linear equations
+                  (coefficient matrix already factorized)
+
+          F04QAF  Sparse linear least-squares problem, m real equations in
+                  n unknowns
+
+          F07 -- Linear Equations (LAPACK)
+
+          F07ADF  (DGETRF) LU factorization of real m by n matrix
+
+          F07AEF  (DGETRS) Solution of real system of linear equations,
+                  multiple right-hand sides, matrix already factorized by
+                  F07ADF
+
+          F07FDF  (DPOTRF) Cholesky factorization of real symmetric
+                  positive-definite matrix
+
+          F07FEF  (DPOTRS) Solution of real symmetric positive-definite
+                  system of linear equations, multiple right-hand sides,
+                  matrix already factorized by F07FDF
+
+          G01 -- Simple Calculations on Statistical Data
+
+          G01AAF  Mean, variance, skewness, kurtosis etc, one variable,
+                  from raw data
+
+          G01ADF  Mean, variance, skewness, kurtosis etc, one variable,
+                  from frequency table
+
+          G01AEF  Frequency table from raw data
+
+
+          G01AFF  Two-way contingency table analysis, with (chi) /Fisher's
+                  exact test
+
+          G01ALF  Computes a five-point summary (median, hinges and
+                  extremes)
+
+          G01ARF  Constructs a stem and leaf plot
+
+          G01EAF  Computes probabilities for the standard Normal
+                  distribution
+
+          G01EBF  Computes probabilities for Student's t-distribution
+
+                                                  2
+          G01ECF  Computes probabilities for (chi)  distribution
+
+          G01EDF  Computes probabilities for F-distribution
+
+          G01EEF  Computes upper and lower tail probabilities and
+                  probability density function for the beta distribution
+
+          G01EFF  Computes probabilities for the gamma distribution
+
+          G01FAF  Computes deviates for the standard Normal distribution
+
+          G01FBF  Computes deviates for Student's t-distribution
+
+                                                 2
+          G01FCF  Computes deviates for the (chi)  distribution
+
+          G01FDF  Computes deviates for the F-distribution
+
+          G01FEF  Computes deviates for the beta distribution
+
+          G01FFF  Computes deviates for the gamma distribution
+
+          G01HAF  Computes probabilities for the bivariate Normal
+                  distribution
+
+          G02 -- Correlation and Regression Analysis
+
+          G02BNF  Kendall/Spearman non-parametric rank correlation
+                  coefficients, no missing values, overwriting input data
+
+          G02BQF  Kendall/Spearman non-parametric rank correlation
+                  coefficients, no missing values, preserving input data
+
+          G02BXF  Computes (optionally weighted) correlation and covariance
+                  matrices
+
+          G02CAF  Simple linear regression with constant term, no missing
+                  values
+
+          G02DAF  Fits a general (multiple) linear regression model
+
+          G02DGF  Fits a general linear regression model for new dependent
+                  variable
+
+          G02DNF  Computes estimable function of a general linear
+                  regression model and its standard error
+
+          G02FAF  Calculates standardized residuals and influence
+                  statistics
+
+          G02GBF  Fits a generalized linear model with binomial errors
+
+          G02GCF  Fits a generalized linear model with Poisson errors
+
+          G03 -- Multivariate Methods
+
+          G03AAF  Performs principal component analysis
+
+          G03ADF  Performs canonical correlation analysis
+
+          G03BAF  Computes orthogonal rotations for loading matrix,
+                  generalized orthomax criterion
+
+          G05 -- Random Number Generators
+
+          G05CAF  Pseudo-random double precision numbers, uniform
+                  distribution over (0,1)
+
+          G05CBF  Initialise random number generating routines to give
+                  repeatable sequence
+
+          G05CCF  Initialise random number generating routines to give non-
+                  repeatable sequence
+
+          G05CFF  Save state of random number generating routines
+
+          G05CGF  Restore state of random number generating routines
+
+          G05DDF  Pseudo-random double precision numbers, Normal
+                  distribution
+
+          G05DFF  Pseudo-random double precision numbers, Cauchy
+                  distribution
+
+          G05DPF  Pseudo-random double precision numbers, Weibull
+                  distribution
+
+          G05DYF  Pseudo-random integer from uniform distribution
+
+          G05DZF  Pseudo-random logical (boolean) value
+
+          G05EAF  Set up reference vector for multivariate Normal
+                  distribution
+
+          G05ECF  Set up reference vector for generating pseudo-random
+                  integers, Poisson distribution
+
+          G05EDF  Set up reference vector for generating pseudo-random
+                  integers, binomial distribution
+
+          G05EHF  Pseudo-random permutation of an integer vector
+
+          G05EJF  Pseudo-random sample from an integer vector
+
+          G05EXF  Set up reference vector from supplied cumulative
+                  distribution function or probability distribution
+                  function
+
+          G05EYF  Pseudo-random integer from reference vector
+
+          G05EZF  Pseudo-random multivariate Normal vector from reference
+                  vector
+
+          G05FAF  Generates a vector of pseudo-random numbers from a
+                  uniform distribution
+
+          G05FBF  Generates a vector of pseudo-random numbers from a
+                  (negative) exponential distribution
+
+          G05FDF  Generates a vector of pseudo-random numbers from a Normal
+                  distribution
+
+          G05FEF  Generates a vector of pseudo-random numbers from a beta
+                  distribution
+
+          G05FFF  Generates a vector of pseudo-random numbers from a gamma
+                  distribution
+
+          G05HDF  Generates a realisation of a multivariate time series
+                  from a VARMA model
+
+          G08 -- Nonparameteric Statistics
+
+          G08AAF  Sign test on two paired samples
+
+          G08ACF  Median test on two samples of unequal size
+
+          G08AEF  Friedman two-way analysis of variance on k matched
+                  samples
+
+          G08AFF  Kruskal-Wallis one-way analysis of variance on k samples
+                  of unequal size
+
+          G08AGF  Performs the Wilcoxon one sample (matched pairs) signed
+                  rank test
+
+          G08AHF  Performs the Mann-Whitney U test on two independent
+                  samples
+
+          G08AJF  Computes the exact probabilities for the Mann-Whitney U
+                  statistic, no ties in pooled sample
+
+          G08AKF  Computes the exact probabilities for the Mann-Whitney U
+                  statistic, ties in pooled sample
+
+                                    2
+          G08CGF  Performs the (chi)  goodness of fit test, for standard
+                  continuous distributions
+
+          G13 -- Time Series Analysis
+
+          G13AAF  Univariate time series, seasonal and non-seasonal
+                  differencing
+
+          G13ABF  Univariate time series, sample autocorrelation function
+
+          G13ACF  Univariate time series, partial autocorrelations from
+                  autocorrelations
+
+          G13ADF  Univariate time series, preliminary estimation, seasonal
+                  ARIMA model
+
+          G13AFF  Univariate time series, estimation, seasonal ARIMA model
+
+          G13AGF  Univariate time series, update state set for forecasting
+
+          G13AHF  Univariate time series, forecasting from state set
+
+          G13AJF  Univariate time series, state set and forecasts, from
+                  fully specified seasonal ARIMA model
+
+          G13ASF  Univariate time series, diagnostic checking of residuals,
+                  following G13AFF
+
+          G13BAF  Multivariate time series, filtering (pre-whitening) by an
+                  ARIMA model
+
+          G13BCF  Multivariate time series, cross correlations
+
+          G13BDF  Multivariate time series, preliminary estimation of
+                  transfer function model
+
+          G13BEF  Multivariate time series, estimation of multi-input model
+
+          G13BJF  Multivariate time series, state set and forecasts from
+                  fully specified multi-input model
+
+          G13CBF  Univariate time series, smoothed sample spectrum using
+                  spectral smoothing by the trapezium frequency (Daniell)
+                  window
+
+          G13CDF  Multivariate time series, smoothed sample cross spectrum
+                  using spectral smoothing by the trapezium frequency
+                  (Daniell) window
+
+          M01 -- Sorting
+
+          M01CAF  Sort a vector, double precision numbers
+
+          M01DAF  Rank a vector, double precision numbers
+
+          M01DEF  Rank rows of a matrix, double precision numbers
+
+          M01DJF  Rank columns of a matrix, double precision numbers
+
+          M01EAF  Rearrange a vector according to given ranks, double
+                  precision numbers
+
+          M01ZAF  Invert a permutation
+
+          S -- Approximations of Special Functions
+
+                                        z
+          S01EAF  Complex exponential, e
+
+          S13AAF  Exponential integral E (x)
+                                        1
+
+          S13ACF  Cosine integral Ci(x)
+
+          S13ADF  Sine integral Si(x)
+
+          S14AAF  Gamma function
+
+          S14ABF  Log Gamma function
+
+          S14BAF  Incomplete gamma functions P(a,x) and Q(a,x)
+
+          S15ADF  Complement of error function erfc x
+
+          S15AEF  Error function erf x
+
+          S17ACF  Bessel function Y (x)
+                                   0
+
+          S17ADF  Bessel function Y (x)
+                                   1
+
+          S17AEF  Bessel function J (x)
+                                   0
+
+          S17AFF  Bessel function J (x)
+                                   1
+
+          S17AGF  Airy function Ai(x)
+
+          S17AHF  Airy function Bi(x)
+
+          S17AJF  Airy function Ai'(x)
+
+          S17AKF  Airy function Bi'(x)
+
+          S17DCF  Bessel functions Y      (z), real a>=0, complex z,
+                                    (nu)+a
+                  (nu)=0,1,2,...
+
+          S17DEF  Bessel functions J      (z), real a>=0, complex z,
+                                    (nu)+a
+                  (nu)=0,1,2,...
+
+          S17DGF  Airy functions Ai(z) and Ai'(z), complex z
+
+          S17DHF  Airy functions Bi(z) and Bi'(z), complex z
+
+                                    (j)
+          S17DLF  Hankel functions H      (z), j=1,2, real a>=0, complex z,
+                                    (nu)+a
+                  (nu)=0,1,2,...
+
+          S18ACF  Modified Bessel function K (x)
+                                            0
+
+          S18ADF  Modified Bessel function K (x)
+                                            1
+
+          S18AEF  Modified Bessel function I (x)
+                                            0
+
+          S18AFF  Modified Bessel function I (x)
+                                            1
+
+          S18DCF  Modified Bessel functions K      (z), real a>=0, complex
+                                             (nu)+a
+                  z, (nu)=0,1,2,...
+
+          S18DEF  Modified Bessel functions I      (z), real a>=0, complex
+                                             (nu)+a
+                  z, (nu)=0,1,2,...
+
+          S19AAF  Kelvin function ber x
+
+          S19ABF  Kelvin function bei x
+
+          S19ACF  Kelvin function ker x
+
+          S19ADF  Kelvin function kei x
+
+          S20ACF  Fresnel integral S(x)
+
+          S20ADF  Fresnel integral C(x)
+
+          S21BAF  Degenerate symmetrised elliptic integral of 1st kind
+                  R (x,y)
+                   C
+
+          S21BBF  Symmetrised elliptic integral of 1st kind R (x,y,z)
+                                                             F
+
+          S21BCF  Symmetrised elliptic integral of 2nd kind R (x,y,z)
+                                                             D
+
+          S21BDF  Symmetrised elliptic integral of 3rd kind R (x,y,z,r)
+                                                             J
+
+          X04 -- Input/Output Utilities
+
+          X04AAF  Return or set unit number for error messages
+
+          X04ABF  Return or set unit number for advisory messages
+
+          X04CAF  Print a real general matrix
+
+          X04DAF  Print a complex general matrix
+
+          X05 -- Date and Time Utilities
+
+          X05AAF  Return date and time as an array of integers
+
+          X05ABF  Convert array of integers representing date and time to
+                  character string
+
+          X05ACF  Compare two character strings representing date and time
+
+          X05BAF  Return the CPU time
+
+
+
+          Fundamental Support Routines
+
+          The following fundamental support routines are provided and are
+          documented in compact form in the relevant chapter introductory
+          material:
+
+          F06 -- Linear Algebra Support Routines
+
+          F06AAF  (DROTG) Generate real plane rotation
+
+          F06EAF  (DDOT) Dot product of two real vectors
+
+          F06ECF  (DAXPY) Add scalar times real vector to real vector
+
+          F06EDF  (DSCAL) Multiply real vector by scalar
+
+          F06EFF  (DCOPY) Copy real vector
+
+          F06EGF  (DSWAP) Swap two real vectors
+
+          F06EJF  (DNRM2) Compute Euclidean norm of real vector
+
+          F06EKF  (DASUM) Sum the absolute values of real vector elements
+
+          F06EPF  (DROT) Apply real plane rotation
+
+          F06GAF  (ZDOTU) Dot product of two complex vectors, unconjugated
+
+          F06GBF  (ZDOTC) Dot product of two complex vectors, conjugated
+
+          F06GCF  (ZAXPY) Add scalar times complex vector to complex vector
+
+          F06GDF  (ZSCAL) Multiply complex vector by complex scalar
+
+          F06GFF  (ZCOPY) Copy complex vector
+
+          F06GGF  (ZSWAP) Swap two complex vectors
+
+          F06JDF  (ZDSCAL) Multiply complex vector by real scalar
+
+          F06JJF  (DZNRM2) Compute Euclidean norm of complex vector
+
+          F06JKF  (DZASUM) Sum the absolute values of complex vector
+                  elements
+
+          F06JLF  (IDAMAX) Index, real vector element with largest absolute
+                  value
+
+          F06JMF  (IZAMAX) Index, complex vector element with largest
+                  absolute value
+
+          F06PAF  (DGEMV) Matrix-vector product, real rectangular matrix
+
+          F06PBF  (DGBMV) Matrix-vector product, real rectangular band
+                  matrix
+
+          F06PCF  (DSYMV) Matrix-vector product, real symmetric matrix
+
+          F06PDF  (DSBMV) Matrix-vector product, real symmetric band matrix
+
+          F06PEF  (DSPMV) Matrix-vector product, real symmetric packed
+                  matrix
+
+          F06PFF  (DTRMV) Matrix-vector product, real triangular matrix
+
+          F06PGF  (DTBMV) Matrix-vector product, real triangular band
+                  matrix
+
+          F06PHF  (DTPMV) Matrix-vector product, real triangular packed
+                  matrix
+
+          F06PJF  (DTRSV) System of equations, real triangular matrix
+
+          F06PKF  (DTBSV) System of equations, real triangular band matrix
+
+          F06PLF  (DTPSV) System of equations, real triangular packed
+                  matrix
+
+          F06PMF  (DGER) Rank-1 update, real rectangular matrix
+
+          F06PPF  (DSYR) Rank-1 update, real symmetric matrix
+
+          F06PQF  (DSPR) Rank-1 update, real symmetric packed matrix
+
+          F06PRF  (DSYR2) Rank-2 update, real symmetric matrix
+
+          F06PSF  (DSPR2) Rank-2 update, real symmetric packed matrix
+
+          F06SAF  (ZGEMV) Matrix-vector product, complex rectangular matrix
+
+          F06SBF  (ZGBMV) Matrix-vector product, complex rectangular band
+                  matrix
+
+          F06SCF  (ZHEMV) Matrix-vector product, complex Hermitian matrix
+
+          F06SDF  (ZHBMV) Matrix-vector product, complex Hermitian band
+                  matrix
+
+          F06SEF  (ZHPMV) Matrix-vector product, complex Hermitian packed
+                  matrix
+
+          F06SFF  (ZTRMV) Matrix-vector product, complex triangular matrix
+
+          F06SGF  (ZTBMV) Matrix-vector product, complex triangular band
+                  matrix
+
+          F06SHF  (ZTPMV) Matrix-vector product, complex triangular packed
+                  matrix
+
+          F06SJF  (ZTRSV) System of equations, complex triangular matrix
+
+          F06SKF  (ZTBSV) System of equations, complex triangular band
+                  matrix
+
+          F06SLF  (ZTPSV) System of equations, complex triangular packed
+                  matrix
+
+          F06SMF  (ZGERU) Rank-1 update, complex rectangular matrix,
+                  unconjugated vector
+
+          F06SNF  (ZGERC) Rank-1 update, complex rectangular matrix,
+                  conjugated vector
+
+          F06SPF  (ZHER) Rank-1 update, complex Hermitian matrix
+
+          F06SQF  (ZHPR) Rank-1 update, complex Hermitian packed matrix
+
+          F06SRF  (ZHER2) Rank-2 update, complex Hermitian matrix
+
+          F06SSF  (ZHPR2) Rank-2 update, complex Hermitian packed matrix
+
+          F06YAF  (DGEMM) Matrix-matrix product, two real rectangular
+                  matrices
+
+          F06YCF  (DSYMM) Matrix-matrix product, one real symmetric matrix,
+                  one real rectangular matrix
+
+          F06YFF  (DTRMM) Matrix-matrix product, one real triangular
+                  matrix, one real rectangular matrix
+
+          F06YJF  (DTRSM) Solves a system of equations with multiple right-
+                  hand sides, real triangular coefficient matrix
+
+          F06YPF  (DSYRK) Rank-k update of a real symmetric matrix
+
+          F06YRF  (DSYR2K) Rank-2k update of a real symmetric matrix
+
+          F06ZAF  (ZGEMM) Matrix-matrix product, two complex rectangular
+                  matrices
+
+          F06ZCF  (ZHEMM) Matrix-matrix product, one complex Hermitian
+                  matrix, one complex rectangular matrix
+
+          F06ZFF  (ZTRMM) Matrix-matrix product, one complex triangular
+                  matrix, one complex rectangular matrix
+
+          F06ZJF  (ZTRSM) Solves system of equations with multiple right-
+                  hand sides, complex triangular coefficient matrix
+
+          F06ZPF  (ZHERK) Rank-k update of a complex Hermitian matrix
+
+          F06ZRF  (ZHER2K) Rank-2k update of a complex Hermitian matrix
+
+          F06ZTF  (ZSYMM) Matrix-matrix product, one complex symmetric
+                  matrix, one complex rectangular matrix
+
+          F06ZUF  (ZSYRK) Rank-k update of a complex symmetric matrix
+
+          F06ZWF  (ZSYR2K) Rank-2k update of a complex symmetric matrix
+
+          X01 -- Mathematical Constants
+
+          X01AAF  (pi)
+
+          X01ABF  Euler's constant, (gamma)
+
+          X02 -- Machine Constants
+
+          X02AHF  Largest permissible argument for SIN and COS
+
+          X02AJF  Machine precision
+
+          X02AKF  Smallest positive model number
+
+          X02ALF  Largest positive model number
+
+          X02AMF  Safe range of floating-point arithmetic
+
+          X02ANF  Safe range of complex floating-point arithmetic
+
+          X02BBF  Largest representable integer
+
+          X02BEF  Maximum number of decimal digits that can be represented
+
+          X02BHF  Parameter of floating-point arithmetic model, b
+
+          X02BJF  Parameter of floating-point arithmetic model, p
+
+          X02BKF  Parameter of floating-point arithmetic model, e
+                                                                 min
+
+          X02BLF  Parameter of floating-point arithmetic model, e
+                                                                 max
+
+          X02DJF  Parameter of floating-point arithmetic model, ROUNDS
+
+
+
+          Routines from the NAG Fortran Library
+
+          A number of routines from the NAG Fortran Library are used in the
+          Foundation Library as auxiliaries and are not documented here:
+
+          A00AAF
+
+          A02AAF  A02ABF  A02ACF
+
+          C02AJF
+
+          C05AZF  C05NCF  C05PCF
+
+          C06FAF  C06FBF  C06FCF  C06FFF  C06FJF  C06FKF
+
+          C06HAF  C06HBF  C06HCF  C06HDF
+
+          D02CBF  D02CHF  D02NMF  D02NSF  D02NVF  D02PAF
+
+          D02XAF  D02XKF  D02YAF  D02ZAF
+
+          E02AFF
+
+          E04GBF  E04GEF  E04YAF
+
+          F01ADF  F01AEF  F01AFF  F01AGF  F01AHF  F01AJF
+
+          F01AKF  F01AMF  F01APF  F01ATF  F01AUF  F01AVF
+
+          F01AWF  F01AXF  F01BCF  F01BTF  F01CRF  F01LBF
+
+          F01LZF  F01QAF  F01QFF  F01QGF  F01QJF  F01QKF
+
+          F01RFF  F01RGF  F01RJF  F01RKF
+
+          F02AMF  F02ANF  F02APF  F02AQF  F02AVF  F02AYF
+
+          F02BEF  F02SWF  F02SXF  F02SYF  F02SZF  F02UWF
+
+          F02UXF  F02UYF  F02WDF  F02WUF  F02XUF
+
+          F03AAF  F03ABF  F03AEF  F03AFF
+
+          F04AAF  F04AEF  F04AFF  F04AGF  F04AHF  F04AJF
+
+          F04AMF  F04ANF  F04AYF  F04LDF  F04YAF  F04YCF
+
+          F06BAF  F06BCF  F06BLF  F06BMF  F06BNF  F06CAF
+
+          F06CCF  F06CLF  F06DBF  F06DFF  F06FBF  F06FCF
+
+          F06FDF  F06FGF  F06FJF  F06FLF  F06FPF  F06FQF
+
+          F06FRF  F06FSF  F06HBF  F06HGF  F06HQF  F06HRF
+
+          F06KFF  F06KJF  F06KLF  F06QFF  F06QHF  F06QKF
+
+          F06QRF  F06QSF  F06QTF  F06QVF  F06QWF  F06QXF
+
+          F06RAF  F06RJF  F06TFF  F06THF  F06TTF  F06TXF
+
+          F06VJF  F06VKF  F06VXF
+
+          F07AGF  F07AHF  F07AJF  F07FGF  F07FHF  F07FJF
+
+          F07TJF
+
+          G01CEF
+
+          G02BAF  G02BUF  G02BWF  G02DDF
+
+          G13AEF
+
+          M01CBF  M01CCF  M01DBF  M01DCF  M01DFF  M01ZBF
+
+          P01ABF  P01ACF
+
+          S01BAF  S07AAF  S15ABF
+
+          X03AAF
+
+          X04BAF  X04BBF  X04CBF  X04DBF
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXintro}{NAG On-line Documentation: intro}
+\beginscroll
+\begin{verbatim}
+
+
+
+     INTRO(3NAG)       Foundation Library (12/10/92)       INTRO(3NAG)
+
+
+
+          Introduction                               Essential Introduction
+          Essential Introduction to the NAG Foundation Library
+
+          This document is essential reading for any prospective user of
+          the Library.
+
+          This document appears in both the Handbook and the Reference
+          Manual for the NAG Foundation Library, but with a different
+          Section 3 to describe the different forms of routine
+          documentation in the two publications.
+
+          1. The Library and its Documentation
+
+          1.1. Structure of the Library
+
+          1.2. Structure of the Documentation
+
+          1.3. On-line Documentation
+
+          1.4. Implementations of the Library
+
+          1.5. Library Identification
+
+          1.6. Fortran Language Standards
+
+          2. Using the Library
+
+          2.1. General Advice
+
+          2.2. Programming Advice
+
+          2.3. Error handling and the Parameter IFAIL
+
+          2.4. Input/output in the Library
+
+          2.5. Auxiliary Routines
+
+          3. Using the Reference Manual
+
+          3.1. General Guidance
+
+          3.2. Structure of Routine Documents
+
+          3.3. Specifications of Parameters
+
+          3.3.1. Classification of Parameters
+
+          3.3.2. Constraints and Suggested Values
+
+          3.3.3. Array Parameters
+
+          3.4. Implementation-dependent Information
+
+          3.5. Example Programs and Results
+
+          3.6. Summary for New Users
+
+          4. Relationship between the Foundation Library and other NAG Libraries
+
+          4.1. NAG Fortran Library
+
+          4.2. NAG Workstation Library
+
+          4.3. NAG C Library
+
+          5. Contact between Users and NAG
+
+          6. General Information about NAG
+
+          7. References
+
+
+
+          1. The Library and its Documentation
+
+          1.1. Structure of the Library
+
+          The NAG Foundation Library is a comprehensive collection of
+          Fortran 77 routines for the solution of numerical and statistical
+          problems. The word 'routine' is used to denote 'subroutine' or '
+          function'.
+
+          The Library is divided into chapters, each devoted to a branch of
+          numerical analysis or statistics. Each chapter has a three-
+          character name and a title, e.g.
+
+               D01 -- Quadrature
+
+          Exceptionally one chapter (S) has a one-character name. (The
+          chapters and their names are based on the ACM modified SHARE
+          classification index [1].)
+
+          All documented routines in the Library have six-character names,
+          beginning with the characters of the chapter name, e.g.
+
+               D01AJF
+
+          Note that the second and third characters are digits, not
+          letters; e.g. 0 is the digit zero, not the letter O. The last
+          letter of each routine name is always 'F'.
+
+          1.2. Structure of the Documentation
+
+          There are two types of manual for the NAG Foundation Library: a
+          Handbook and a Reference Manual.
+
+          The Handbook has the same chapter structure as the Library: each
+          chapter of routines in the Library has a corresponding chapter
+          (of the same name) in the Handbook. The chapters occur in
+          alphanumeric order. General introductory documents and indexes
+          are placed at the beginning of the Handbook.
+
+          Each chapter in the Handbook contains a Chapter Introduction,
+          followed by concise summaries of the functionality and parameter
+          specifications of each routine in the chapter. Exceptionally, in
+          some chapters (F06, X01, X02) which contain simple support
+          routines, there are no concise summaries: all the routines are
+          described together in the Chapter Introduction.
+
+          The Reference Manual provides complete reference documentation
+          for the NAG Foundation Library. In the Reference Manual, each
+          chapter consists of the following documents:
+
+               Chapter Introduction, e.g. Introduction -- D01;
+
+               Chapter Contents, e.g. Contents -- D01;
+
+               routine documents, one for each documented routine in the
+               chapter.
+
+          A routine document has the same name as the routine which it
+          describes. Within each chapter, routine documents occur in
+          alphanumeric order. As in the Handbook, chapters F06, X01 and X02
+          do not contain separate documentation for individual routines.
+
+          The general introductory documents, indexes and chapter
+          introductions are the same in the Reference Manual as in the
+          Handbook. The only exception is that the Essential Introduction
+          contains a different Section 3 in the two publications, to
+          describe the different forms of routine documentation.
+
+          1.3. On-line Documentation
+
+          Extensive on-line documentation is included as an integral part
+          of the Foundation Library product. This consists of a number of
+          components:
+
+               -- general introductory material, including the Essential
+               Introduction
+
+               -- a summary list of all documented routines
+
+               -- a KWIC Index
+
+               -- Chapter Introductions
+
+               -- routine documents
+
+               -- example programs, data and results.
+
+          The material has been derived in a number of forms to cater for
+          different user requirements, e.g. UNIX man pages, plain text,
+          RICH TEXT format etc, and the appropriate version is included on
+          the distribution media. For each implementation of the Foundation
+          Library the specific documentation (Installers' Note, Users' Note
+          etc) gives details of what is provided.
+
+          1.4. Implementations of the Library
+
+          The NAG Foundation Library is available on many different
+          computer systems. For each distinct system, an implementation of
+          the Library is prepared by NAG, e.g. the IBM RISC System/6000
+          implementation. The implementation is distributed as a tested
+          compiled library.
+
+          An implementation is usually specific to a range of machines; it
+          may also be specific to a particular operating system or
+          compilation system.
+
+          Essentially the same facilities are provided in all
+          implementations of the Library, but, because of differences in
+          arithmetic behaviour and in the compilation system, routines
+          cannot be expected to give identical results on different
+          systems, especially for sensitive numerical problems.
+
+          The documentation supports all implementations of the Library,
+          with the help of a few simple conventions, and a small amount of
+          implementation-dependent information, which is published in a
+          separate Users' Note for each implementation (see Section 3.4).
+
+          1.5. Library Identification
+
+          You must know which implementation of the Library you are using
+          or intend to use. To find out which implementation of the Library
+          is available on your machine, you can run a program which calls
+          the NAG Foundation Library routine A00AAF. This routine has no
+          parameters; it simply outputs text to the advisory message unit
+          (see Section 2.4). An example of the output is:
+
+
+          *** Start of NAG Foundation Library implementation details ***
+          Implementation title: IBM RISC System/6000
+          Precision: FORTRAN double precision
+          Product Code: FFIB601D
+          Release: 1
+          *** End of NAG Foundation Library implementation details ***
+
+          (The product code can be ignored, except possibly when
+          communicating with NAG; see Section 4.)
+
+          1.6. Fortran Language Standards
+
+          All routines in the Library conform to ANSI Standard Fortran 90
+          [8].
+
+          Most of the routines in the Library were originally written to
+          conform to the earlier Fortran 66 [6] and Fortran 77 [7]
+          standards, and their calling sequences contain some parameters
+          which are not strictly necessary in Fortran 90.
+
+          2. Using the Library
+
+          2.1. General Advice
+
+          A NAG Foundation Library routine cannot be guaranteed to return
+          meaningful results, irrespective of the data supplied to it. Care
+          and thought must be exercised in:
+
+          (a) formulating the problem;
+
+          (b) programming the use of library routines;
+
+          (c) assessing the significance of the results.
+
+          2.2. Programming Advice
+
+          The NAG Foundation Library and its documentation are designed on
+          the assumption that users know how to write a calling program in
+          Fortran.
+
+          When programming a call to a routine, read the routine document
+          carefully, especially the description of the Parameters. This
+          states clearly which parameters must have values assigned to them
+          on entry to the routine, and which return useful values on exit.
+          See Section 3.3 for further guidance.
+
+          If a call to a Library routine results in an unexpected error
+          message from the system (or possibly from within the Library),
+          check the following:
+
+
+              Has the NAG routine been called with the correct number of
+              parameters?
+
+              Do the parameters all have the correct type?
+
+              Have all array parameters been dimensioned correctly?
+
+          Remember that all floating-point parameters must be declared to
+          be double precision, either with an explicit DOUBLE PRECISION
+          declaration (or COMPLEX(KIND(1.0D0)) if they are complex), or by
+          using a suitable IMPLICIT statement.
+
+          Avoid the use of NAG-type names for your own program units or
+          COMMON blocks: in general, do not use names which contain a
+          three-character NAG chapter name embedded in them; they may clash
+          with the names of an auxiliary routine or COMMON block used by
+          the NAG Library.
+
+          2.3. Error handling and the Parameter IFAIL
+
+          NAG Foundation Library routines may detect various kinds of
+          error, failure or warning conditions. Such conditions are handled
+          in a systematic way by the Library. They fall roughly into three
+          classes:
+
+          (i)   an invalid value of a parameter on entry to a routine;
+
+          (ii)  a numerical failure during computation (e.g. approximate
+                singularity of a matrix, failure of an iteration to
+                converge);
+
+          (iii) a warning that although the computation has been completed,
+                the results cannot be guaranteed to be completely reliable.
+
+          All three classes are handled in the same way by the Library, and
+          are all referred to here simply as 'errors'.
+
+          The error-handling mechanism uses the parameter IFAIL, which is
+          the last parameter in the calling sequence of most NAG Foundation
+          Library routines. IFAIL serves two purposes:
+
+          (i)   it allows users to specify what action a Library routine
+                should take if it detects an error;
+
+          (ii)  it reports the outcome of a call to a Library routine,
+                either success (IFAIL = 0) or failure (IFAIL /= 0, with
+                different values indicating different reasons for the
+                failure, as explained in Section 6 of the routine document)
+                .
+
+          For the first purpose IFAIL must be assigned a value before
+          calling the routine; since IFAIL is reset by the routine, it must
+          be passed as a variable, not as an integer constant. Allowed
+          values on entry are:
+
+              IFAIL=0: an error message is output, and execution is
+              terminated ('hard failure');
+
+              IFAIL=+1: execution continues without any error message;
+
+              IFAIL=-1: an error message is output, and execution
+              continues.
+
+          The settings IFAIL =+-1 are referred to as 'soft failure'.
+          The safest choice is to set IFAIL to 0, but this is not always
+          convenient: some routines return useful results even though a
+          failure (in some cases merely a warning) is indicated. However,
+          if IFAIL is set to +- 1 on entry, it is essential for the program
+          to test its value on exit from the routine, and to take
+          appropriate action.
+
+          The specification of IFAIL in Section 5 of a routine document
+          suggests a suitable setting of IFAIL for that routine.
+
+          2.4. Input/output in the Library
+
+          Most NAG Foundation Library routines perform no output to an
+          external file, except possibly to output an error message. All
+          error messages are written to a logical error message unit. This
+          unit number (which is set by default to 6 in most
+          implementations) can be changed by calling the Library routine
+          X04AAF.
+
+          Some NAG Foundation Library routines may optionally output their
+          final results, or intermediate results to monitor the course of
+          computation. All output other than error messages is written to a
+          logical advisory message unit. This unit number (which is also
+          set by default to 6 in most implementations) can be changed by
+          calling the Library routine X04ABF. Although it is logically
+          distinct from the error message unit, in practice the two unit
+          numbers may be the same.
+
+          All output from the Library is formatted.
+
+          The only Library routines which perform input from an external
+          file are a few 'option-setting' routines in Chapter E04: the unit
+          number is a parameter to the routine, and all input is formatted.
+
+          You must ensure that the relevant Fortran unit numbers are
+          associated with the desired external files, either by an OPEN
+          statement in your calling program, or by operating system
+          commands.
+
+          2.5. Auxiliary Routines
+
+          In addition to those Library routines which are documented and
+          are intended to be called by users, the Library also contains
+          many auxiliary routines.
+
+          In general, you need not be concerned with them at all, although
+          you may be made aware of their existence if, for example, you
+          examine a memory map of an executable program which calls NAG
+          routines. The only exception is that when calling some NAG
+          Foundation Library routines, you may be required or allowed to
+          supply the name of an auxiliary routine from the Library as an
+          external procedure parameter. The routine documents give the
+          necessary details. In such cases, you only need to supply the
+          name of the routine; you never need to know details of its
+          parameter-list.
+
+          NAG auxiliary routines have names which are similar to the name
+          of the documented routine(s) to which they are related, but with
+          last letter 'Z', 'Y', and so on, e.g. D01AJZ is an auxiliary
+          routine called by D01AJF.
+
+          3. Using the Reference Manual
+
+          3.1. General Guidance
+
+          The Reference Manual is designed to serve the following
+          functions:
+
+              -- to give background information about different areas of
+              numerical and statistical computation;
+
+              -- to advise on the choice of the most suitable NAG
+              Foundation Library routine or routines to solve a particular
+              problem;
+
+              -- to give all the information needed to call a NAG
+              Foundation Library routine correctly from a Fortran program,
+              and to assess the results.
+
+          At the beginning of the Manual are some general introductory
+          documents. The following may help you to find the chapter, and
+          possibly the routine, which you need to solve your problem:
+
+              Contents     -- a list of routines in the Library, by
+              Summary      chapter;
+
+              KWIC Index   -- a keyword index to chapters and routines.
+
+          Having found a likely chapter or routine, you should read the
+          corresponding Chapter Introduction, which gives background
+          information about that area of numerical computation, and
+          recommendations on the choice of a routine, including indexes,
+          tables or decision trees.
+
+          When you have chosen a routine, you must consult the routine
+          document. Each routine document is essentially self-contained (it
+          may contain references to related documents). It includes a
+          description of the method, detailed specifications of each
+          parameter, explanations of each error exit, and remarks on
+          accuracy.
+
+          Example programs which illustrate the use of each routine are
+          distributed with the Library in machine-readable form.
+
+          3.2. Structure of Routine Documents
+
+          All routine documents have the same structure, consisting of nine
+          numbered sections:
+
+               1.  Purpose
+
+               2.  Specification
+
+               3.  Description
+
+               4.  References
+
+               5.  Parameters (see Section 3.3 below)
+
+               6.  Error Indicators
+
+               7.  Accuracy
+
+               8.  Further Comments
+
+               9.  Example (see Section 3.5 below)
+
+          In a few documents, Section 5 also includes a description of
+          printed output which may optionally be produced by the routine.
+
+          3.3. Specifications of Parameters
+
+          Section 5 of each routine document contains the specification of
+          the parameters, in the order of their appearance in the parameter
+          list.
+
+          3.3.1.  Classification of Parameters
+
+          Parameters are classified as follows:
+
+          Input : you must assign values to these parameters on or before
+          entry to the routine, and these values are unchanged on exit from
+          the routine.
+
+          Output : you need not assign values to these parameters on or
+          before entry to the routine; the routine may assign values to
+          them.
+
+          Input/Output : you must assign values to these parameters on or
+          before entry to the routine, and the routine may then change
+          these values.
+
+          Workspace: array parameters which are used as workspace by the
+          routine. You must supply arrays of the correct type and
+          dimension, but you need not be concerned with their contents.
+
+          External Procedure: a subroutine or function which must be
+          supplied (e.g. to evaluate an integrand or to print intermediate
+          output). Usually it must be supplied as part of your calling
+          program, in which case its specification includes full details of
+          its parameter-list and specifications of its parameters (all
+          enclosed in a box). Its parameters are classified in the same way
+          as those of the Library routine, but because you must write the
+          procedure rather than call it, the significance of the
+          classification is different:
+
+              Input : values may be supplied on entry, which your procedure
+              must not change.
+
+              Output : you may or must assign values to these parameters
+              before exit from your procedure.
+
+              Input/Output : values may be supplied on entry, and you may
+              or must assign values to them before exit from your
+              procedure.
+
+          Occasionally, as mentioned in Section 2.5, the procedure can be
+          supplied from the NAG Library, and then you only need to know its
+          name.
+
+          User Workspace: array parameters which are passed by the Library
+          routine to an external procedure parameter. They are not used by
+          the routine, but you may use them to pass information between
+          your calling program and the external procedure.
+
+          3.3.2.  Constraints and Suggested Values
+
+          The word 'Constraint:' or 'Constraints:' in the specification of
+          an Input parameter introduces a statement of the range of valid
+          values for that parameter, e.g.
+
+              Constraint: N > 0.
+
+          If the routine is called with an invalid value for the parameter
+          (e.g. N = 0), the routine will usually take an error exit,
+          returning a non-zero value of IFAIL (see Section 2.3).
+
+          In newer documents constraints on parameters of type CHARACTER
+          only list uppercase alphabetic characters, e.g.
+
+              Constraint: STRING = 'A' or 'B'.
+
+          In practice all routines with CHARACTER parameters will permit
+          the use of lower case characters.
+
+          The phrase 'Suggested Value:' introduces a suggestion for a
+          reasonable initial setting for an Input parameter (e.g. accuracy
+          or maximum number of iterations) in case you are unsure what
+          value to use; you should be prepared to use a different setting
+          if the suggested value turns out to be unsuitable for your
+          problem.
+
+          3.3.3.  Array Parameters
+
+          Most array parameters have dimensions which depend on the size of
+          the problem. In Fortran terminology they have 'adjustable
+          dimensions': the dimensions occurring in their declarations are
+          integer variables which are also parameters of the Library
+          routine.
+
+          For example, a Library routine might have the specification:
+
+
+               SUBROUTINE <name> (M, N, A, B, LDB)
+               INTEGER       M, N, A(N), B(LDB,N), LDB
+
+          For a one-dimensional array parameter, such as A in this example,
+          the specification would begin:
+
+           3:  A(N) -- DOUBLE PRECISION array                         Input
+
+          You must ensure that the dimension of the array, as declared in
+          your calling (sub)program, is at least as large as the value you
+          supply for N. It may be larger; but the routine uses only the
+          first N elements.
+
+          For a two-dimensional array parameter, such as B in the example,
+          the specification might be:
+
+           4:  B(LDB,N) -- DOUBLE PRECISION array              Input/Output
+               On entry: the m by n matrix B.
+
+          and the parameter LDB might be described as follows:
+
+           5:  LDB -- INTEGER                                         Input
+               On entry: the first dimension of the array B as declared in
+               the (sub)program from which <name> is called. Constraint:
+               LDB >= M.
+
+          You must supply the first dimension of the array B, as declared
+          in your calling (sub)program, through the parameter LDB, even
+          though the number of rows actually used by the routine is
+          determined by the parameter M. You must ensure that the first
+          dimension of the array is at least as large as the value you
+          supply for M. The extra parameter LDB is needed because Fortran
+          does not allow information about the dimensions of array
+          parameters to be passed automatically to a routine.
+
+          You must also ensure that the second dimension of the array, as
+          declared in your calling (sub)program, is at least as large as
+          the value you supply for N. It may be larger, but the routine
+          only uses the first N columns.
+
+          A program to call the hypothetical routine used as an example in
+          this section might include the statements:
+
+
+               INTEGER AA(100), BB(100,50)
+               LDB = 100
+               .
+               .
+               .
+               M = 80
+               N = 20
+               CALL <name>(M,N,AA,BB,LDB)
+
+          Fortran requires that the dimensions which occur in array
+          declarations, must be greater than zero. Many NAG routines are
+          designed so that they can be called with a parameter like N in
+          the above example set to 0 (in which case they would usually exit
+          immediately without doing anything). If so, the declarations in
+          the Library routine would use the 'assumed size' array dimension,
+          and would be given as:
+
+
+               INTEGER        M, N, A(*), B(LDB,*), LDB
+
+          However, the original declaration of an array in your calling
+          program must always have constant dimensions, greater than or
+          equal to 1.
+
+          Consult an expert or a textbook on Fortran, if you have
+          difficulty in calling NAG routines with array parameters.
+
+          3.4. Implementation-dependent Information
+
+          In order to support all implementations of the Foundation
+          Library, the Manual has adopted a convention of using bold
+          italics to distinguish terms which have different interpretations
+          in different implementations.
+
+          For example, machine precision denotes the relative precision to
+          which double precision floating-point numbers are stored in the
+          computer, e.g. in an implementation with approximately 16 decimal
+          digits of precision, machine precision has a value of
+                          - 16
+          approximately 10    .
+
+          The precise value of machine precision is given by the function
+          X02AJF. Other functions in Chapter X02 return the values of other
+          implementation-dependent constants, such as the overflow
+          threshold, or the largest representable integer. Refer to the X02
+
+          Chapter Introduction for more details.
+
+          For each implementation of the Library, a separate Users' Note is
+          provided. This is a short document, revised at each Mark. At most
+          installations it is available in machine-readable form. It gives
+          any necessary additional information which applies specifically
+          to that implementation, in particular:
+
+              -- the interpretation of bold italicised terms;
+
+              -- the values returned by X02 routines;
+
+              -- the default unit numbers for output (see Section 2.4).
+
+          3.5. Example Programs and Results
+
+          The last section of each routine document describes an example
+          problem which can be solved by simple use of the routine. The
+          example programs themselves, together with data and results, are
+          not printed in the routine document, but are distributed in
+          machine-readable form with the Library. The programs are designed
+          so that they can fairly easily be modified, and so serve as the
+          basis for a simple program to solve a user's own problem.
+
+          The results distributed with each implementation were obtained
+          using that implementation of the Library; they may not be
+          identical to the results obtained with other implementations.
+
+          3.6. Summary for New Users
+
+          If you are unfamiliar with the NAG Foundation Library and are
+          thinking of using a routine from it, please follow these
+          instructions:
+
+          (a) read the whole of the Essential Introduction;
+
+          (b) consult the Contents Summary or KWIC Index to choose an
+              appropriate chapter or routine;
+
+          (c) read the relevant Chapter Introduction;
+
+          (d) choose a routine, and read the routine document. If the
+              routine does not after all meet your needs, return to steps
+              (b) or (c);
+
+          (e) read the Users' Note for your implementation;
+
+          (f) consult local documentation, which should be provided by your
+              local support staff, about access to the NAG Library on your
+              computing system.
+
+          You should now be in a position to include a call to the routine
+          in a program, and to attempt to run it. You may of course need to
+          refer back to the relevant documentation in the case of
+          difficulties, for advice on assessment of results, and so on.
+
+          As you become familiar with the Library, some of steps (a) to (f)
+          can be omitted, but it is always essential to:
+
+              -- be familiar with the Chapter Introduction;
+
+              -- read the routine document;
+
+              -- be aware of the Users' Note for your implementation.
+
+          4. Relationship between the Foundation Library and other NAG Libraries
+
+          4.1. NAG Fortran Library
+
+          The Foundation Library is a strict subset of the full NAG Fortran
+          Library (Mark 15 or later). Routines in both libraries have
+          identical source code (apart from any modifications necessary for
+          implementation on a specific system) and hence can be called in
+          exactly the same way, though you should consult the relevant
+          implementation-specific documentation for details such as values
+          of machine constants.
+
+          By its very nature, the Foundation Library cannot contain the
+          same extensive range of routines as the full Fortran Library. If
+          your application requires a routine which is not in the
+          Foundation Library, then please consult NAG for information on
+          relevant material available in the Fortran Library.
+
+          Some routines which occur as user-callable routines in the full
+          Fortran Library are included as auxiliary routines in the
+          Foundation Library but they are not documented in this
+          publication and direct calls to them should only be made if you
+          are already familiar with their use in the Fortran Library. A
+          list of all such auxiliary routines is given at the end of the
+          Foundation Library Contents Summary.
+
+          Whereas the full Fortran Library may be provided in either a
+          single precision or a double precision version, the Foundation
+          Library is always provided in double precision.
+
+          4.2. NAG Workstation Library
+
+          The Foundation Library is a successor product to an earlier,
+          smaller subset of the full NAG Fortran Library which was called
+          the NAG Workstation Library. The Foundation Library has greater
+          functionality than the Workstation Library but is not strictly
+          upwards compatible, i.e., a number of routines in the earlier
+          product have been replaced by new material to reflect recent
+          algorithmic developments.
+          If you have used the Workstation Library and wish to convert your
+          programs to call routines from the Foundation Library, please
+          consult the document 'Converting from the Workstation Library' in
+          this Manual.
+
+          4.3. NAG C Library
+
+          NAG has also developed a library of numerical and statistical
+          software for use by C programmers. This now contains over 200
+          user-callable functions and provides similar (but not identical)
+          coverage to that of the Foundation Library. Please contact NAG
+          for further details if you have a requirement for similar quality
+          library code in C.
+
+          5. Contact between Users and NAG
+
+          If you are using the NAG Foundation Library in a multi-user
+          environment and require further advice please consult your local
+          support staff who will be receiving regular information from NAG.
+          This covers such matters as:
+
+               -- obtaining a copy of the Users' Note for your
+               implementation;
+
+               -- obtaining information about local access to the Library;
+
+               -- seeking advice about using the Library;
+
+               -- reporting suspected errors in routines or documents;
+
+               -- making suggestions for new routines or features;
+
+               -- purchasing NAG documentation.
+
+          If you are unable to make contact with a local source of support
+          or are in a single-user environment then please contact NAG
+          directly at any one of the addresses given at the beginning of
+          this publication.
+
+          6. General Information about NAG
+
+          NAG produces and distributes numerical, symbolic, statistical and
+          graphical software for the solution of problems in a wide range
+          of applications in such areas as science, engineering, financial
+          analysis and research.
+
+          For users who write programs and build packages NAG produces sub-
+          program libraries in a range of computer languages (Ada, C,
+          Fortran, Pascal, Turbo Pascal). NAG also provides a number of
+          Fortran programming support products in the NAGWare range --
+          Fortran 77 programming tools, Fortran 90 compilers for a number
+          of machine platforms (including PC-DOS) and VecPar 77 for
+          restructuring and tuning programs for execution on vector or
+          parallel computers.
+
+          For users who do not wish to program in the traditional sense but
+          want the same reliability and qualities offered by our libraries,
+          NAG provides several powerful mathematical and statistical
+          packages for interactive use. A major addition to this range of
+          packages is AXIOM -- the powerful symbolic solver which includes
+          a Hypertext system and graphical capabilities.
+
+          For further details of any of these products, please contact NAG
+          at one of the addresses given at the beginning of this
+          publication.
+
+          References [2], [3], [4], and [5] discuss various aspects of the
+          design and development of the NAG Library, and NAG's technical
+          policies and organisation.
+
+          7. References
+
+          [1]   (1960--1976) Collected Algorithms from ACM Index by subject
+                to algorithms.
+
+          [2]   Ford B (1982) Transportable Numerical Software. Lecture
+                Notes in Computer Science. 142 128--140.
+
+          [3]   Ford B, Bentley J, Du Croz J J and Hague S J (1979) The NAG
+                Library 'machine'. Software Practice and Experience. 9(1)
+                65--72.
+
+          [4]   Ford B and Pool J C T (1984) The Evolving NAG Library
+                Service. Sources and Development of Mathematical Software.
+                (ed W Cowell) Prentice-Hall. 375--397.
+
+          [5]   Hague S J, Nugent S M and Ford B (1982) Computer-based
+                Documentation for the NAG Library. Lecture Notes in Computer
+                Science. 142 91--127.
+
+          [6]   (1966) USA Standard Fortran. Publication X3.9. American
+                National Standards Institute.
+
+          [7]   (1978) American National Standard Fortran. Publication X3.9.
+                American National Standards Institute.
+
+          [8]   (1991) American National Standard Programming Language
+                Fortran 90. Publication X3.198.
+                American National Standards Institute.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXkwic}{NAG On-line Documentation: kwic}
+\beginscroll
+\begin{verbatim}
+
+
+
+     KWIC(3NAG)        Foundation Library (12/10/92)        KWIC(3NAG)
+
+
+
+          Introduction                                  Keywords in Context
+          Keywords in Context
+
+          Pre-computed weights and                                 D01BBF
+          abscissae
+          for Gaussian quadrature rules, restricted choice of ...
+
+          Sum the                                                  F06EKF
+          absolute
+          values of real vector elements (DASUM)
+
+          Sum the                                                  F06JKF
+          absolute
+          values of complex vector elements (DZASUM)
+
+          Index, real vector element with largest                  F06JLF
+          absolute
+          value (IDAMAX)
+
+          Index, complex vector element with largest               F06JMF
+          absolute
+          value (IZAMAX)
+
+          ODEs, IVP,                                               D02CJF
+          Adams
+          method, until function of solution is zero, ...
+
+          1-D quadrature,                                          D01AJF
+          adaptive
+          , finite interval, strategy due to Piessens and de ...
+
+          1-D quadrature,                                          D01AKF
+          adaptive
+          , finite interval, method suitable for oscillating ...
+
+          1-D quadrature,                                          D01ALF
+          adaptive
+          , finite interval, allowing for singularities at ...
+
+          1-D quadrature,                                          D01AMF
+          adaptive
+          , infinite or semi-infinite interval
+
+          1-D quadrature,                                          D01ANF
+          adaptive
+          , finite interval, weight function cos((omega)x) ...
+
+          1-D quadrature,                                          D01APF
+          adaptive
+          , finite interval, weight function with end-point ...
+
+          1-D quadrature,                                          D01AQF
+          adaptive
+          , finite interval, weight function 1/(x-c), ...
+
+          1-D quadrature,                                          D01ASF
+          adaptive
+          , semi-infinite interval, weight function cos((omega)x)
+
+          Multi-dimensional                                        D01FCF
+          adaptive
+          quadrature over hyper-rectangle
+
+          Add                                                      F06ECF
+          scalar times real vector to real vector (DAXPY)
+
+          Add                                                      F06GCF
+          scalar times complex vector to complex vector (ZAXPY)
+
+          Return or set unit number for                            X04ABF
+          advisory
+          messages
+
+          Airy                                                     S17AGF
+          function Ai(x)
+
+          Airy                                                     S17AHF
+          function Bi(x)
+
+          Airy                                                     S17AJF
+          function Ai'(x)
+
+          Airy                                                     S17AKF
+          function Bi'(x)
+
+          Airy                                                     S17DGF
+          functions Ai(z) and Ai('z), complex z
+
+          Airy                                                     S17DHF
+          functions Bi(z) and Bi'(z), complex z
+
+          Airy function                                            S17AGF
+          Ai(x)
+
+          Airy function                                            S17AJF
+          Ai'(x)
+
+          Airy functions                                           S17DGF
+          Ai(z)
+          and Ai'(z), complex z
+
+          Airy functions Ai(z) and                                 S17DGF
+          Ai'(z)
+          , complex z
+          algebraico-logarithmic
+          type
+
+          Two-way contingency table                                G01AFF
+          analysis
+                      2
+          , with (chi) /Fisher's exact test
+
+          Performs principal component                             G03AAF
+          analysis
+
+          Performs canonical correlation                           G03ADF
+          analysis
+
+          Friedman two-way                                         G08AEF
+          analysis
+          of variance on k matched samples
+
+          Kruskal-Wallis one-way                                   G08AFF
+          analysis
+          of variance on k samples of unequal size
+
+          L -                                                      E02GAF
+           1
+          approximation
+          by general linear function
+
+          Approximation                                            E02
+
+          Approximation                                            S
+          of special functions
+
+          ARIMA
+          model
+
+          Univariate time series, estimation, seasonal             G13AFF
+          ARIMA
+          model
+
+          ARIMA
+          model
+
+          ARIMA
+          model
+
+          Safe range of floating-point                             X02AMF
+          arithmetic
+
+          Safe range of complex floating-point                     X02ANF
+          arithmetic
+
+          Parameter of floating-point                              X02BHF
+          arithmetic
+          model, b
+
+          Parameter of floating-point                              X02BJF
+          arithmetic
+          model, p
+
+          Parameter of floating-point                              X02BKF
+          arithmetic
+          model, e
+                  min
+
+          Parameter of floating-point                              X02BLF
+          arithmetic
+          model, e
+                  max
+
+          Parameter of floating-point                              X02DJF
+          arithmetic
+          model, ROUNDS
+
+          Univariate time series, sample                           G13ABF
+          autocorrelation
+          function
+
+          Univariate time series, partial                          G13ACF
+          autocorrelations
+          from autocorrelations
+
+          Univariate time series, partial autocorrelations from    G13ACF
+          autocorrelations
+
+          Least-squares cubic spline curve fit,                    E02BEF
+          automatic
+          knot placement
+
+          Least-squares surface fit by bicubic splines with        E02DCF
+          automatic
+          knot placement, data on rectangular grid
+
+          Least-squares surface fit by bicubic splines with        E02DDF
+          automatic
+          knot placement, scattered data
+
+          B-splines                                                E02
+
+          Matrix-vector product, real rectangular                  F06PBF
+          band
+          matrix (DGBMV)
+
+          Matrix-vector product, real symmetric                    F06PDF
+          band
+          matrix (DSBMV)
+
+          Matrix-vector product, real triangular                   F06PGF
+          band
+          matrix (DTBMV)
+
+          System of equations, real triangular                     F06PKF
+          band
+          matrix (DTBSV)
+
+          Matrix-vector product, complex rectangular               F06SBF
+          band
+          matrix (ZGBMV)
+
+          Matrix-vector product, complex Hermitian                 F06SDF
+          band
+          matrix (ZHBMV)
+
+          Matrix-vector product, complex triangular                F06SGF
+          band
+          matrix (ZTBMV)
+
+          System of equations, complex triangular                  F06SKF
+          band
+          matrix (ZTBSV)
+
+          bandwidth
+          matrix
+
+          Solution of real symmetric positive-definite variable-   F04MCF
+          bandwidth
+          simultaneous linear equations (coefficient matrix ...
+
+          Basic                                                    F06
+          Linear Algebra Subprograms
+
+          ODEs, stiff IVP,                                         D02EJF
+          BDF
+          method, until function of solution is zero,
+          intermediate ...
+
+          Kelvin function                                          S19ABF
+          bei
+          x
+
+          Kelvin function                                          S19AAF
+          ber
+          x
+
+          Bessel                                                   S17ACF
+          function Y (x)
+                    0
+
+          Bessel                                                   S17ADF
+          function Y (x)
+                    1
+
+          Bessel                                                   S17AEF
+          function J (x)
+                    0
+
+          Bessel                                                   S17AFF
+          function J (x)
+                    1
+
+          Bessel                                                   S17DCF
+          functions Y      (z), complex z, real (nu)>=0, ...
+                     (nu)+n
+
+          Bessel                                                   S17DEF
+          functions J      (z), complex z, real (nu)>=0, ...
+                     (nu)+n
+
+          Modified                                                 S18ACF
+          Bessel
+          function K (x)
+                    0
+
+          Modified                                                 S18ADF
+          Bessel
+          function K (x)
+                    1
+
+          Modified                                                 S18AEF
+          Bessel
+          function I (x)
+                    0
+
+          Modified                                                 S18AFF
+          Bessel
+          function I (x)
+                    1
+
+          Modified                                                 S18DCF
+          Bessel
+          functions K      (z), complex z, real (nu)>=0, ...
+                     (nu)+n
+
+          Modified                                                 S18DEF
+          Bessel
+          functions I      (z), complex z, real (nu)>=0, ...
+                     (nu)+n
+
+          beta
+          distribution
+
+          Computes deviates for the                                G01FEF
+          beta
+          distribution
+
+          Generates a vector of pseudo-random numbers from a       G05FEF
+          beta
+          distribution
+
+          Interpolating functions, fitting                         E01DAF
+          bicubic
+          spline, data on rectangular grid
+
+          Least-squares surface fit,                               E02DAF
+          bicubic
+          splines
+
+          Least-squares surface fit by                             E02DCF
+          bicubic
+          splines with automatic knot placement, data on ...
+
+          Least-squares surface fit by                             E02DDF
+          bicubic
+          splines with automatic knot placement, scattered data
+
+          Evaluation of a fitted                                   E02DEF
+          bicubic
+          spline at a vector of points
+
+          Evaluation of a fitted                                   E02DFF
+          bicubic
+          spline at a mesh of points
+
+          Sort 2-D data into panels for fitting                    E02ZAF
+          bicubic
+          splines
+
+          Fits a generalized linear model with                     G02GBF
+          binomial
+          errors
+
+          binomial
+          distribution
+
+          Computes probability for the                             G01HAF
+          bivariate
+          Normal distribution
+
+          Airy function                                            S17AHF
+          Bi(x)
+
+          Airy function                                            S17AKF
+          Bi'(x)
+
+          Airy functions                                           S17DHF
+          Bi(z)
+          and Bi'(z), complex z
+
+          Airy functions Bi(z) and                                 S17DHF
+          Bi'(z)
+          , complex z
+
+          BLAS                                                     F06
+
+          Pseudo-random logical                                    G05DZF
+          (boolean)
+          value
+
+          ODEs,                                                    D02GAF
+          boundary
+          value problem, finite difference technique with ...
+
+          ODEs,                                                    D02GBF
+          boundary
+          value problem, finite difference technique with ...
+
+          ODEs, general nonlinear                                  D02RAF
+          boundary
+          value problem, finite difference technique with ...
+
+          bounds
+          , using function values only
+
+          break-points
+
+          break-points
+
+          Zero of continuous function in given interval,           C05ADF
+          Bus
+          and Dekker algorithm
+
+          Performs                                                 G03ADF
+          canonical
+          correlation analysis
+
+          Carlo
+          method
+
+          Elliptic PDE, Helmholtz equation, 3-D                    D03FAF
+          Cartesian
+          co-ordinates
+
+          Cauchy
+          principal value (Hilbert transform)
+
+          Pseudo-random real numbers,                              G05DFF
+
+          Cauchy
+          distribution
+
+          character
+          string
+
+          Compare two                                              X05ACF
+          character
+          strings representing date and time
+
+          Evaluation of fitted polynomial in one variable from     E02AEF
+          Chebyshev
+          series form (simplified parameter list)
+
+          Derivative of fitted polynomial in                       E02AHF
+          Chebyshev
+          series form
+
+          Integral of fitted polynomial in                         E02AJF
+          Chebyshev
+          series form
+
+          Evaluation of fitted polynomial in one variable, from    E02AKF
+          Chebyshev
+          series form
+
+          Check                                                    C05ZAF
+          user's routine for calculating 1st derivatives
+
+          Univariate time series, diagnostic                       G13ASF
+          checking
+          of residuals, following G13AFF
+
+          Cholesky                                                 F07FDF
+          factorization of real symmetric positive-definite ...
+
+          Circular                                                 C06EKF
+          convolution or correlation of two real vectors, no ...
+
+          Cosine integral                                          S13ACF
+          Ci(x)
+
+          Interpolating functions, method of Renka and             E01SAF
+          Cline
+          , two variables
+
+          Elliptic PDE, Helmholtz equation, 3-D Cartesian          D03FAF
+          co-ordinates
+
+          coefficient
+          matrix already factorized by F01MCF)
+
+          coefficient
+          matrix (DTRSM)
+
+          coefficient
+          matrix (ZTRSM)
+
+          Kendall/Spearman non-parametric rank correlation         G02BNF
+          coefficients
+          , no missing values, overwriting input data
+
+          Kendall/Spearman non-parametric rank correlation         G02BQF
+          coefficients
+          , no missing values, preserving input data
+
+          Operations with orthogonal matrices, form                F01QEF
+          columns
+          of Q after factorization by F01QCF
+
+          Operations with unitary matrices, form                   F01REF
+          columns
+          of Q after factorization by F01RCF
+
+          Rank                                                     M01DJF
+          columns
+          of a matrix, real numbers
+
+          Compare                                                  X05ACF
+          two character strings representing date and time
+
+          Complement                                               S15ADF
+          of error function erfcx
+
+          Unconstrained minimum, pre-                              E04DGF
+          conditioned
+          conjugate gradient algorithm, function of several ...
+
+          Complex                                                  C06GBF
+          conjugate
+          of Hermitian sequence
+
+          Complex                                                  C06GCF
+          conjugate
+          of complex sequence
+
+          Complex                                                  C06GQF
+          conjugate
+          of multiple Hermitian sequences
+
+          Unconstrained minimum, pre-conditioned                   E04DGF
+          conjugate
+          gradient algorithm, function of several variables ...
+
+          Dot product of two complex vectors,                      F06GBF
+          conjugated
+          (ZDOTC)
+
+          Rank-1 update, complex rectangular matrix,               F06SNF
+          conjugated
+          vector (ZGERC)
+
+          Mathematical                                             X01
+          Constants
+
+          Machine                                                  X02
+          Constants
+
+          constrained
+          , arbitrary data points
+
+          constraints
+          , using function values and optionally 1st ...
+
+          Two-way                                                  G01AFF
+          contingency
+                                    2
+          table analysis, with (chi) /Fisher's ...
+
+          continuation
+          facility
+
+          Zero of                                                  C05ADF
+          continuous
+          function in given interval, Bus and Dekker algorithm
+
+                   2
+          continuous
+          distributions
+
+          Convert                                                  C06GSF
+          Hermitian sequences to general complex sequences
+
+          Convert                                                  X05ABF
+          array of integers representing date and time to ...
+
+          Circular                                                 C06EKF
+          convolution
+          or correlation of two real vectors, no extra ...
+
+          Copy                                                     F06EFF
+          real vector (DCOPY)
+
+          Copy                                                     F06GFF
+          complex vector (ZCOPY)
+
+          correction
+          , simple nonlinear problem
+
+          correction
+          , general linear problem
+
+          correction
+          , continuation facility
+
+          Circular convolution or                                  C06EKF
+          correlation
+          of two real vectors, no extra workspace
+
+          Kendall/Spearman non-parametric rank                     G02BNF
+          correlation
+          coefficients, no missing values, overwriting ...
+
+          Kendall/Spearman non-parametric rank                     G02BQF
+          correlation
+          coefficients, no missing values, preserving input ...
+
+          Computes (optionally weighted)                           G02BXF
+          correlation
+          and covariance matrices
+
+          Performs canonical                                       G03ADF
+          correlation
+          analysis
+
+          Multivariate time series, cross-                         G13BCF
+          correlations
+
+          cos
+          ((omega)x) or sin((omega)x)
+
+          cos
+          ((omega)x) or sin((omega)x)
+
+          Cosine                                                   S13ACF
+          integral Ci(x)
+
+          Covariance                                               E04YCF
+          matrix for nonlinear least-squares problem
+
+          Computes (optionally weighted) correlation and           G02BXF
+          covariance
+          matrices
+
+          Return the                                               X05BAF
+          CPU
+          time
+
+          Multivariate time series,                                G13BCF
+          cross-correlations
+
+          Multivariate time series, smoothed sample                G13CDF
+          cross
+          spectrum using spectral smoothing by the trapezium ...
+
+          Interpolating functions,                                 E01BAF
+          cubic
+          spline interpolant, one variable
+
+          cubic
+          Hermite, one variable
+
+          Least-squares curve                                      E02BAF
+          cubic
+          spline fit (including interpolation)
+
+          Evaluation of fitted                                     E02BBF
+          cubic
+          spline, function only
+
+          Evaluation of fitted                                     E02BCF
+          cubic
+          spline, function and derivatives
+
+          Evaluation of fitted                                     E02BDF
+          cubic
+          spline, definite integral
+
+          Least-squares                                            E02BEF
+          cubic
+          spline curve fit, automatic knot placement
+
+          Set up reference vector from supplied                    G05EXF
+          cumulative
+          distribution function or probability distribution ...
+
+          Least-squares                                            E02ADF
+          curve
+          fit, by polynomials, arbitrary data points
+
+          Least-squares                                            E02BAF
+          curve
+          cubic spline fit (including interpolation)
+
+          Least-squares cubic spline                               E02BEF
+          curve
+          fit, automatic knot placement
+
+          Fresnel integral                                         S20ADF
+          C(x)
+
+          Daniell)
+          window
+
+          Daniell)
+          window
+
+          Return                                                   X05AAF
+          date
+          and time as an array of integers
+
+          Convert array of integers representing                   X05ABF
+          date
+          and time to character string
+
+          Compare two character strings representing               X05ACF
+          date
+          and time
+
+          deferred
+          correction, simple nonlinear problem
+
+          deferred
+          correction, general linear problem
+
+          deferred
+          correction, continuation facility
+
+          Interpolated values, interpolant computed by E01BEF,     E01BHF
+          definite
+          integral, one variable
+
+          Evaluation of fitted cubic spline,                       E02BDF
+          definite
+          integral
+
+          definite
+          matrix
+
+             T
+          LDL  factorization of real symmetric positive-           F01MCF
+          definite
+          variable-bandwidth matrix
+
+          definite
+
+          definite
+
+          Solution of real symmetric positive-                     F04ASF
+          definite
+          simultaneous linear equations, one right-hand side ...
+
+          Solution of real symmetric positive-                     F04FAF
+          definite
+          tridiagonal simultaneous linear equations, one ...
+
+          Real sparse symmetric positive-                          F04MAF
+          definite
+          simultaneous linear equations (coefficient matrix ...
+
+          Solution of real symmetric positive-                     F04MCF
+          definite
+          variable-bandwidth simultaneous linear equations ...
+
+          Cholesky factorization of real symmetric positive-       F07FDF
+          definite
+          matrix (DPOTRF)
+
+          Solution of real symmetric positive-                     F07FEF
+          definite
+          system of linear equations, multiple right-hand ...
+
+          Degenerate                                               S21BAF
+          symmetrised elliptic integral of 1st kind R  ...
+                                                     C
+
+          Dekker
+          algorithm
+
+          Computes upper and lower tail and probability            G01EEF
+          density
+          function probabilities for the beta distribution
+
+          Solution of system of nonlinear equations using 1st      C05PBF
+          derivatives
+
+          Check user's routine for calculating 1st                 C05ZAF
+          derivatives
+
+          derivative
+          , one variable
+
+          Least-squares polynomial fit, values and                 E02AGF
+          derivatives
+          may be constrained, arbitrary data points
+
+          Derivative                                               E02AHF
+          of fitted polynomial in Chebyshev series form
+
+          Evaluation of fitted cubic spline, function and          E02BCF
+          derivatives
+
+          derivatives
+
+          derivatives
+
+          derivatives
+
+          Computes                                                 G01FAF
+          deviates
+          for the standard Normal distribution
+
+          Computes                                                 G01FBF
+          deviates
+          for Student's t-distribution
+
+          Computes                                                 G01FCF
+          deviates
+                       2
+          for the (chi)  distribution
+
+          Computes                                                 G01FDF
+          deviates
+          for the F-distribution
+
+          Computes                                                 G01FEF
+          deviates
+          for the beta distribution
+
+          Computes                                                 G01FFF
+          deviates
+          for the gamma distribution
+
+          Univariate time series,                                  G13ASF
+          diagnostic
+          checking of residuals, following G13AFF
+
+          ODEs, boundary value problem, finite                     D02GAF
+          difference
+          technique with deferred correction, simple ...
+
+          ODEs, boundary value problem, finite                     D02GBF
+          difference
+          technique with deferred correction, general linear ...
+
+          difference
+          technique with deferred correction, continuation ...
+
+          Elliptic PDE, solution of finite                         D03EDF
+          difference
+          equations by a multigrid technique
+
+          Univariate time series, seasonal and non-seasonal        G13AAF
+          differencing
+
+          Single 1-D real                                          C06EAF
+          discrete
+          Fourier transform, no extra workspace
+
+          Single 1-D Hermitian                                     C06EBF
+          discrete
+          Fourier transform, no extra workspace
+
+          Single 1-D complex                                       C06ECF
+          discrete
+          Fourier transform, no extra workspace
+
+          Multiple 1-D real                                        C06FPF
+          discrete
+          Fourier transforms
+
+          Multiple 1-D Hermitian                                   C06FQF
+          discrete
+          Fourier transforms
+
+          Multiple 1-D complex                                     C06FRF
+          discrete
+          Fourier transforms
+
+          2-D complex                                              C06FUF
+          discrete
+          Fourier transform
+
+          Discretize                                               D03EEF
+          a 2nd order elliptic PDE on a rectangle
+
+          Computes probabilities for the standard Normal           G01EAF
+          distribution
+
+          Computes probabilities for Student's t-                  G01EBF
+          distribution
+
+                                          2
+          Computes probabilities for (chi)                         G01ECF
+          distribution
+
+          Computes probabilities for F-                            G01EDF
+          distribution
+
+          distribution
+
+          Computes probabilities for the gamma                     G01EFF
+          distribution
+
+          Computes deviates for the standard Normal                G01FAF
+          distribution
+
+          Computes deviates for Student's t-                       G01FBF
+          distribution
+
+                                         2
+          Computes deviates for the (chi)                          G01FCF
+          distribution
+
+          Computes deviates for the F-                             G01FDF
+          distribution
+
+          Computes deviates for the beta                           G01FEF
+          distribution
+
+          Computes deviates for the gamma                          G01FFF
+          distribution
+
+          Computes probability for the bivariate Normal            G01HAF
+          distribution
+
+          Pseudo-random real numbers, uniform                      G05CAF
+          distribution
+          over (0,1)
+
+          Pseudo-random real numbers, Normal                       G05DDF
+          distribution
+
+          Pseudo-random real numbers, Cauchy                       G05DFF
+          distribution
+
+          Pseudo-random real numbers, Weibull                      G05DPF
+          distribution
+
+          Pseudo-random integer from uniform                       G05DYF
+          distribution
+
+          Set up reference vector for multivariate Normal          G05EAF
+          distribution
+
+          distribution
+
+          distribution
+
+          Set up reference vector from supplied cumulative         G05EXF
+          distribution
+          function or probability distribution function
+
+          distribution
+          function
+
+          Generates a vector of random numbers from a uniform      G05FAF
+          distribution
+
+          distribution
+
+          Generates a vector of random numbers from a Normal       G05FDF
+          distribution
+
+          distribution
+
+          distribution
+
+               2
+          (chi)  goodness of fit test, for standard continuous     G08CGF
+          distributions
+
+          Inverse                                                  G01F
+          distributions
+
+          Doncker
+          , allowing for badly-behaved integrands
+
+          Dot                                                      F06EAF
+          product of two real vectors (DDOT)
+
+          Dot                                                      F06GAF
+          product of two complex vectors, unconjugated (ZDOTU)
+
+          Dot                                                      F06GBF
+          product of two complex vectors, conjugated (ZDOTC)
+
+          eigenfunction
+          , user-specified break-points
+
+          All eigenvalues of generalized real                      F02ADF
+          eigenproblem
+          of the form Ax=(lambda)Bx where A and B are ...
+
+          All eigenvalues and eigenvectors of generalized real     F02AEF
+          eigenproblem
+          of the form Ax=(lambda)Bx where A and B are ...
+
+          eigenproblem
+          by QZ algorithm, real matrices
+
+          eigenproblem
+
+          eigenvalue
+          and eigenfunction, user-specified break-points
+
+          All                                                      F02AAF
+          eigenvalues
+          of real symmetric matrix
+
+          All                                                      F02ABF
+          eigenvalues
+          and eigenvectors of real symmetric matrix
+
+          All                                                      F02ADF
+          eigenvalues
+          of generalized real eigenproblem of the form
+          Ax=(lambda)Bx
+
+          All                                                      F02AEF
+          eigenvalues
+          and eigenvectors of generalized real ...
+
+          All                                                      F02AFF
+          eigenvalues
+          of real matrix
+
+          All                                                      F02AGF
+          eigenvalues
+          and eigenvectors of real matrix
+
+          All                                                      F02AJF
+          eigenvalues
+          of complex matrix
+
+          All                                                      F02AKF
+          eigenvalues
+          and eigenvectors of complex matrix
+
+          All                                                      F02AWF
+          eigenvalues
+          of complex Hermitian matrix
+
+          All                                                      F02AXF
+          eigenvalues
+          and eigenvectors of complex Hermitian matrix
+
+          Selected                                                 F02BBF
+          eigenvalues
+          and eigenvectors of real symmetric matrix
+
+          All                                                      F02BJF
+          eigenvalues
+          and optionally eigenvectors of generalized ...
+
+          Selected                                                 F02FJF
+          eigenvalues
+          and eigenvectors of sparse symmetric eigenproblem
+
+          All eigenvalues and                                      F02ABF
+          eigenvectors
+          of real symmetric matrix
+
+          All eigenvalues and                                      F02AEF
+          eigenvectors
+          of generalized real eigenproblem of the form ...
+
+          All eigenvalues and                                      F02AGF
+          eigenvectors
+          of real matrix
+
+          All eigenvalues and                                      F02AKF
+          eigenvectors
+          of complex matrix
+
+          All eigenvalues and                                      F02AXF
+          eigenvectors
+          of complex Hermitian matrix
+
+          Selected eigenvalues and                                 F02BBF
+          eigenvectors
+          of real symmetric matrix
+
+          All eigenvalues and optionally                           F02BJF
+          eigenvectors
+          of generalized eigenproblem by QZ algorithm, ...
+
+          Selected eigenvalues and                                 F02FJF
+          eigenvectors
+          of sparse symmetric eigenproblem
+
+          Elliptic                                                 D03EDF
+          PDE, solution of finite difference equations by a ...
+
+          Discretize a 2nd order                                   D03EEF
+          elliptic
+          PDE on a rectangle
+
+          Elliptic                                                 D03FAF
+          PDE, Helmholtz equation, 3-D Cartesian co-ordinates
+
+          Degenerate symmetrised                                   S21BAF
+          elliptic
+          integral of 1st kind R (x,y)
+                                C
+
+          Symmetrised                                              S21BBF
+          elliptic
+          integral of 1st kind R (x,y,z)
+                                F
+
+          Symmetrised                                              S21BCF
+          elliptic
+          integral of 2nd kind R (x,y,z)
+                                D
+
+          Symmetrised                                              S21BDF
+          elliptic
+          integral of 3rd kind R (x,y,z,r)
+                                J
+
+          end-point
+          singularities of algebraico-logarithmic type
+
+          error
+
+          Fits a generalized linear model with binomial            G02GBF
+          errors
+
+          Fits a generalized linear model with Poisson             G02GCF
+          errors
+
+          Complement of                                            S15ADF
+          error
+          function erfc x
+
+          Error                                                    S15AEF
+          function erf x
+
+          Return or set unit number for                            X04AAF
+          error
+          messages
+
+          Computes                                                 G02DNF
+          estimable
+          function of a general linear regression model and ...
+
+          Univariate time series, preliminary                      G13ADF
+          estimation
+          , seasonal ARIMA model
+
+          Univariate time series,                                  G13AFF
+          estimation
+          , seasonal ARIMA model
+
+          Multivariate time series, preliminary                    G13BDF
+          estimation
+          of transfer function model
+
+          Multivariate time series,                                G13BEF
+          estimation
+          of multi-input model
+
+          Compute                                                  F06EJF
+          Euclidean
+          norm of real vector (DNRM2)
+
+          Compute                                                  F06JJF
+          Euclidean
+          norm of complex vector (DZNRM2)
+
+          Evaluation                                               E02AEF
+          of fitted polynomial in one variable from ...
+
+          Evaluation                                               E02AKF
+          of fitted polynomial in one variable, from ...
+
+          Evaluation                                               E02BBF
+          of fitted cubic spline, function only
+
+          Evaluation                                               E02BCF
+          of fitted cubic spline, function and derivatives
+
+          Evaluation                                               E02BDF
+          of fitted cubic spline, definite integral
+
+          Evaluation                                               E02DEF
+          of a fitted bicubic spline at a vector of points
+
+          Evaluation                                               E02DFF
+          of a fitted bicubic spline at a mesh of points
+
+                                        2
+          exact
+          test
+
+          Computes the                                             G08AJF
+          exact
+          probabilities for the Mann-Whitney U statistic, no ...
+
+          Computes the                                             G08AKF
+          exact
+          probabilities for the Mann-Whitney U statistic, ties ..
+
+          exponential
+          distribution
+
+          Complex                                                  S01EAF
+          exponential
+             z
+          , e
+
+          Exponential                                              S13AAF
+          integral E (x)
+                    1
+
+          Computes a five-point summary (median, hinges and        G01ALF
+          extremes)
+
+          Computes probabilities for                               G01EDF
+          F
+          -distribution
+
+          Computes deviates for the                                G01FDF
+          F
+          -distribution
+
+          LU                                                       F01BRF
+          factorization
+          of real sparse matrix
+
+          LU                                                       F01BSF
+          factorization
+          of real sparse matrix with known sparsity pattern
+
+            T
+          LL                                                       F01MAF
+          factorization
+          of real sparse symmetric positive-definite matrix
+
+             T
+          LDL                                                      F01MCF
+          factorization
+          of real symmetric positive-definite ...
+
+          QR                                                       F01QCF
+          factorization
+          of real m by n matrix (m>=n)
+
+                                       T
+          factorization
+          by F01QCF
+
+          factorization
+          by F01QCF
+
+          QR                                                       F01RCF
+          factorization
+          of complex m by n matrix (m>=n)
+
+                                               H
+          factorization
+          by F01RCF
+
+          factorization
+          by F01RCF
+
+          LU                                                       F07ADF
+          factorization
+          of real m by n matrix (DGETRF)
+
+          Cholesky                                                 F07FDF
+          factorization
+          of real symmetric positive-definite matrix ...
+
+          Multivariate time series,                                G13BAF
+          filtering
+          (pre-whitening) by an ARIMA model
+
+          1-D quadrature, adaptive,                                D01AJF
+          finite
+          interval, strategy due to Piessens and de Doncker, ...
+
+          1-D quadrature, adaptive,                                D01AKF
+          finite
+          interval, method suitable for oscillating functions
+
+          1-D quadrature, adaptive,                                D01ALF
+          finite
+          interval, allowing for singularities at user-specified
+
+          1-D quadrature, adaptive,                                D01ANF
+          finite
+          interval, weight function cos((omega)x) or sin...
+
+          1-D quadrature, adaptive,                                D01APF
+          finite
+          interval, weight function with end-point ...
+
+          1-D quadrature, adaptive,                                D01AQF
+          finite
+          interval, weight function 1/(x-c), Cauchy ...
+
+          ODEs, boundary value problem,                            D02GAF
+          finite
+          difference technique with deferred correction, simple .
+
+          ODEs, boundary value problem,                            D02GBF
+          finite
+          difference technique with deferred correction, general
+
+          finite/infinite
+          range, eigenvalue and eigenfunction, ...
+
+          ODEs, general nonlinear boundary value problem,          D02RAF
+          finite
+          difference technique with deferred correction, ...
+
+          Elliptic PDE, solution of                                D03EDF
+          finite
+          difference equations by a multigrid technique
+
+                                                    2
+          Fisher's
+          exact test
+
+          Interpolating functions,                                 E01DAF
+          fitting
+          bicubic spline, data on rectangular grid
+
+          Least-squares curve                                      E02ADF
+          fit
+          , by polynomials, arbitrary data points
+
+          Evaluation of                                            E02AEF
+          fitted
+          polynomial in one variable from Chebyshev series form .
+
+          Least-squares polynomial                                 E02AGF
+          fit
+          , values and derivatives may be constrained, arbitrary
+
+          Derivative of                                            E02AHF
+          fitted
+          polynomial in Chebyshev series form
+
+          Integral of                                              E02AJF
+          fitted
+          polynomial in Chebyshev series form
+
+          Evaluation of                                            E02AKF
+          fitted
+          polynomial in one variable, from Chebyshev series form
+
+          Least-squares curve cubic spline                         E02BAF
+          fit
+          (including interpolation)
+
+          Evaluation of                                            E02BBF
+          fitted
+          cubic spline, function only
+
+          Evaluation of                                            E02BCF
+          fitted
+          cubic spline, function and derivatives
+
+          Evaluation of                                            E02BDF
+          fitted
+          cubic spline, definite integral
+
+          Least-squares cubic spline curve                         E02BEF
+          fit
+          , automatic knot placement
+
+          Least-squares surface                                    E02DAF
+          fit
+          , bicubic splines
+
+          Least-squares surface                                    E02DCF
+          fit
+          by bicubic splines with automatic knot placement, data
+          on ...
+
+          Least-squares surface                                    E02DDF
+          fit
+          by bicubic splines with automatic knot placement, ...
+
+          Evaluation of a                                          E02DEF
+          fitted
+          bicubic spline at a vector of points
+
+          Evaluation of a                                          E02DFF
+          fitted
+          bicubic spline at a mesh of points
+
+          Sort 2-D data into panels for                            E02ZAF
+          fitting
+          bicubic splines
+
+          Fits                                                     G02DAF
+          a general (multiple) linear regression model
+
+          Fits                                                     G02DGF
+          a general linear regression model for new dependent ...
+
+          Fits                                                     G02GBF
+          a generalized linear model with binomial errors
+
+          Fits                                                     G02GCF
+          a generalized linear model with Poisson errors
+
+                            2
+          Performs the (chi)  goodness of                          G08CGF
+          fit
+          test, for standard continuous distributions
+
+          Goodness of                                              G08
+          fit
+          tests
+
+          Computes a                                               G01ALF
+          five-point
+          summary (median, hinges and extremes)
+
+          Safe range of                                            X02AMF
+          floating-point
+          arithmetic
+
+          Safe range of complex                                    X02ANF
+          floating-point
+          arithmetic
+
+          Parameter of                                             X02BHF
+          floating-point
+          arithmetic model, b
+
+          Parameter of                                             X02BJF
+          floating-point
+          arithmetic model, p
+
+          Parameter of                                             X02BKF
+          floating-point
+          arithmetic model, e
+                             min
+
+          Parameter of                                             X02BLF
+          floating-point
+          arithmetic model, e
+                             max
+
+          Parameter of                                             X02DJF
+          floating-point
+          arithmetic model, ROUNDS
+
+          Univariate time series, update state set for             G13AGF
+          forecasting
+
+          Univariate time series,                                  G13AHF
+          forecasting
+          from state set
+
+          Univariate time series, state set and                    G13AJF
+          forecasts
+          , from fully specified seasonal ARIMA model
+
+          Multivariate time series, state set and                  G13BJF
+          forecasts
+          from fully specified multi-input model
+
+          Single 1-D real discrete                                 C06EAF
+          Fourier
+          transform, no extra workspace
+
+          Single 1-D Hermitian discrete                            C06EBF
+          Fourier
+          transform, no extra workspace
+
+          Single 1-D complex discrete                              C06ECF
+          Fourier
+          transform, no extra workspace
+
+          Multiple 1-D real discrete                               C06FPF
+          Fourier
+          transforms
+
+          Multiple 1-D Hermitian discrete                          C06FQF
+          Fourier
+          transforms
+
+          Multiple 1-D complex discrete                            C06FRF
+          Fourier
+          transforms
+
+          2-D complex discrete                                     C06FUF
+          Fourier
+          transform
+
+          frequency
+          table
+
+          Frequency                                                G01AEF
+          table from raw data
+
+          frequency
+          (Daniell) window
+
+          frequency
+          (Daniell) window
+
+          Fresnel                                                  S20ACF
+          integral S(x)
+
+          Fresnel                                                  S20ADF
+          integral C(x)
+
+          Friedman                                                 G08AEF
+          two-way analysis of variance on k matched samples
+
+          Computes probabilities for the                           G01EFF
+          gamma
+          distribution
+
+          Computes deviates for the                                G01FFF
+          gamma
+          distribution
+
+          Generates a vector of pseudo-random numbers from a       G05FFF
+          gamma
+          distribution
+
+          Gamma                                                    S14AAF
+          function
+
+          Log                                                      S14ABF
+          Gamma
+          function
+
+          Incomplete                                               S14BAF
+          gamma
+          functions P(a,x) and Q(a,x)
+
+          Unconstrained minimum of a sum of squares, combined      E04FDF
+          Gauss-Newton
+          and modified Newton algorithm using function ...
+
+          Unconstrained minimum of a sum of squares, combined      E04GCF
+          Gauss-Newton
+          and quasi-Newton algorithm, using 1st derivatives
+
+          Pre-computed weights and abscissae for                   D01BBF
+          Gaussian
+          quadrature rules, restricted choice of rule
+
+          All eigenvalues of                                       F02ADF
+          generalized
+          real eigenproblem of the form Ax=(lambda)Bx where ...
+
+          All eigenvalues and eigenvectors of                      F02AEF
+          generalized
+          real eigenproblem of the form Ax=(lambda)Bx where ...
+
+          All eigenvalues and optionally eigenvectors of           F02BJF
+          generalized
+          eigenproblem by QZ algorithm, real matrices
+
+          Fits a                                                   G02GBF
+          generalized
+          linear model with binomial errors
+
+          Fits a                                                   G02GCF
+          generalized
+          linear model with Poisson errors
+
+          Computes orthogonal rotations for loading matrix,        G03BAF
+          generalized
+          orthomax criterion
+
+          Generate                                                 F06AAF
+          real plane rotation (DROTG)
+
+          Initialise random number                                 G05CBF
+          generating
+          routines to give repeatable sequence
+
+          Initialise random number                                 G05CCF
+          generating
+          routines to give non-repeatable sequence
+
+          Save state of random number                              G05CFF
+          generating
+          routines
+
+          Restore state of random number                           G05CGF
+          generating
+          routines
+
+          Set up reference vector for                              G05ECF
+          generating
+          pseudo-random integers, Poisson distribution
+
+          Set up reference vector for                              G05EDF
+          generating
+          pseudo-random integers, binomial distribution
+
+          Generates                                                G05FAF
+          a vector of random numbers from a uniform ...
+
+          Generates                                                G05FBF
+          a vector of random numbers from an (negative) ...
+
+          Generates                                                G05FDF
+          a vector of random numbers from a Normal distribution
+
+          Generates                                                G05FEF
+          a vector of pseudo-random numbers from a beta ...
+
+          Generates                                                G05FFF
+          a vector of pseudo-random numbers from a gamma ...
+
+          Generates                                                G05HDF
+          a realisation of a multivariate time series from a ...
+
+          Gill-Miller
+          method
+
+                            2
+          Performs the (chi)                                       G08CGF
+          goodness
+          of fit test, for standard continuous distributions
+
+          Goodness                                                 G08
+          of fit tests
+
+          Unconstrained minimum, pre-conditioned conjugate         E04DGF
+          gradient
+          algorithm, function of several variables using 1st ...
+
+          Hankel                                                   S17DLF
+                     (j)
+          functions H      (z), j=1,2, ...
+                     (nu)+n
+
+          Elliptic PDE,                                            D03FAF
+          Helmholtz
+          equation, 3-D Cartesian co-ordinates
+
+          Hermite
+          , one variable
+
+          Single 1-D                                               C06EBF
+          Hermitian
+          discrete Fourier transform, no extra workspace
+
+          Multiple 1-D                                             C06FQF
+          Hermitian
+          discrete Fourier transforms
+
+          Complex conjugate of                                     C06GBF
+          Hermitian
+          sequence
+
+          Complex conjugate of multiple                            C06GQF
+          Hermitian
+          sequences
+
+          Convert                                                  C06GSF
+          Hermitian
+          sequences to general complex sequences
+
+          All eigenvalues of complex                               F02AWF
+          Hermitian
+          matrix
+
+          All eigenvalues and eigenvectors of complex              F02AXF
+          Hermitian
+          matrix
+
+          Matrix-vector product, complex                           F06SCF
+          Hermitian
+          matrix (ZHEMV)
+
+          Matrix-vector product, complex                           F06SDF
+          Hermitian
+          band matrix (ZHBMV)
+
+          Matrix-vector product, complex                           F06SEF
+          Hermitian
+          packed matrix (ZHPMV)
+
+          Rank-1 update, complex                                   F06SPF
+          Hermitian
+          matrix (ZHER)
+
+          Rank-1 update, complex                                   F06SQF
+          Hermitian
+          packed matrix (ZHPR)
+
+          Rank-2 update, complex                                   F06SRF
+          Hermitian
+          matrix (ZHER2)
+
+          Rank-2 update, complex                                   F06SSF
+          Hermitian
+          packed matrix (ZHPR2)
+
+          Matrix-matrix product, one complex                       F06ZCF
+          Hermitian
+          matrix, one complex rectangular matrix (ZHEMM)
+
+          Rank-k update of a complex                               F06ZPF
+          Hermitian
+          matrix (ZHERK)
+
+          Rank-2k update of a complex                              F06ZRF
+          Hermitian
+          matrix (ZHER2K)
+
+          Hilbert
+          transform)
+
+          Computes a five-point summary (median,                   G01ALF
+          hinges
+          and extremes)
+
+          Multi-dimensional adaptive quadrature over               D01FCF
+          hyper-rectangle
+
+          Multi-dimensional quadrature over                        D01GBF
+          hyper-rectangle
+          , Monte Carlo method
+
+          Incomplete                                               S14BAF
+          gamma functions P(a,x) and Q(a,x)
+
+          Index                                                    F06JLF
+          , real vector element with largest absolute value
+          (IDAMAX)
+
+          Index                                                    F06JMF
+          , complex vector element with largest absolute value ..
+
+          1-D quadrature, adaptive,                                D01AMF
+          infinite
+          or semi-infinite interval
+
+          1-D quadrature, adaptive, infinite or semi-              D01AMF
+          infinite
+          interval
+
+          1-D quadrature, adaptive, semi-                          D01ASF
+          infinite
+          interval, weight function cos((omega)x) or ...
+
+          infinite
+          range, eigenvalue and eigenfunction, user-specified ...
+
+          Calculates standardized residuals and                    G02FAF
+          influence
+          statistics
+
+          Initialise                                               G05CBF
+          random number generating routines to give ...
+
+          Initialise                                               G05CCF
+          random number generating routines to give ...
+
+          input
+          data
+
+          input
+          data
+
+          Multivariate time series, estimation of multi-           G13BEF
+          input
+          model
+
+          input
+          model
+
+          Input/output                                             X04
+          utilities
+
+          Pseudo-random                                            G05DYF
+          integer
+          from uniform distribution
+
+          Set up reference vector for generating pseudo-random     G05ECF
+          integers
+          , Poisson distribution
+
+          Set up reference vector for generating pseudo-random     G05EDF
+          integers
+          , binomial distribution
+
+          Pseudo-random permutation of an                          G05EHF
+          integer
+          vector
+
+          Pseudo-random sample from an                             G05EJF
+          integer
+          vector
+
+          Pseudo-random                                            G05EYF
+          integer
+          from reference vector
+
+          Largest representable                                    X02BBF
+          integer
+
+          Return date and time as an array of                      X05AAF
+          integers
+
+          Convert array of                                         X05ABF
+          integers
+          representing date and time to character string
+
+          integral
+          , one variable
+
+          Integral                                                 E02AJF
+          of fitted polynomial in Chebyshev series form
+
+          Evaluation of fitted cubic spline, definite              E02BDF
+          integral
+
+          Exponential                                              S13AAF
+          integral
+          E (x)
+           1
+
+          Cosine                                                   S13ACF
+          integral
+          Ci(x)
+
+          Sine                                                     S13ADF
+          integral
+          Si(x)
+
+          Fresnel                                                  S20ACF
+          integral
+          S(x)
+
+          Fresnel                                                  S20ADF
+          integral
+          C(x)
+
+          Degenerate symmetrised elliptic                          S21BAF
+          integral
+          of 1st kind R (x,y)
+                       C
+
+          Symmetrised elliptic                                     S21BBF
+          integral
+          of 1st kind R (x,y,z)
+                       F
+
+          Symmetrised elliptic                                     S21BCF
+          integral
+          of 2nd kind R (x,y,z)
+                       D
+
+          Symmetrised elliptic                                     S21BDF
+          integral
+          of 3rd kind R (x,y,z,r)
+                       J
+
+          1-D quadrature,                                          D01GAF
+          integration
+          of function defined by data values, Gill-Miller ...
+
+          Numerical                                                D01
+          integration
+
+          Interpolating                                            E01BAF
+          functions, cubic spline interpolant, one variable
+
+          Interpolating functions, cubic spline                    E01BAF
+          interpolant
+          , one variable
+
+          Interpolating                                            E01BEF
+          functions, monotonicity-preserving, piecewise ...
+
+          Interpolated                                             E01BFF
+          values, interpolant computed by E01BEF, function ...
+
+          Interpolated values,                                     E01BFF
+          interpolant
+          computed by E01BEF, function only, one variable
+
+          Interpolated                                             E01BGF
+          values, interpolant computed by E01BEF, function ...
+
+          Interpolated values,                                     E01BGF
+          interpolant
+          computed by E01BEF, function and 1st derivative, ...
+
+          Interpolated                                             E01BHF
+          values, interpolant computed by E01BEF, definite ...
+
+          Interpolated values,                                     E01BHF
+          interpolant
+          computed by E01BEF, definite integral, one variable
+
+          Interpolating                                            E01DAF
+          functions, fitting bicubic spline, data on ...
+
+          Interpolating                                            E01SAF
+          functions, method of Renka and Cline, two ...
+
+          Interpolating                                            E01SEF
+          functions, modified Shepard's method, two ...
+
+          Least-squares curve cubic spline fit (including          E02BAF
+          interpolation)
+
+          Inverse                                                  G01F
+          distributions
+
+          Invert                                                   M01ZAF
+          a permutation
+
+          iterative
+          refinement
+
+          iterative
+          refinement
+
+          ODEs,                                                    D02BBF
+          IVP
+          , Runge-Kutta-Merson method, over a range, intermediate
+
+          ODEs,                                                    D02BHF
+          IVP
+          , Runge-Kutta-Merson method, until function of solution
+          is ...
+
+          ODEs,                                                    D02CJF
+          IVP
+          , Adams method, until function of solution is zero, ...
+
+          ODEs, stiff                                              D02EJF
+          IVP
+          , BDF method, until function of solution is zero, ...
+
+          Kelvin function                                          S19ADF
+          kei
+          x
+
+          Kelvin                                                   S19AAF
+          function ber x
+
+          Kelvin                                                   S19ABF
+          function bei x
+
+          Kelvin                                                   S19ACF
+          function ker x
+
+          Kelvin                                                   S19ADF
+          function kei x
+
+          Kendall/Spearman                                         G02BNF
+          non-parametric rank correlation ...
+
+          Kendall/Spearman                                         G02BQF
+          non-parametric rank correlation ...
+
+          Kelvin function                                          S19ACF
+          ker
+          x
+
+          Least-squares cubic spline curve fit, automatic          E02BEF
+          knot
+          placement
+
+          knot
+          placement, data on rectangular grid
+
+          knot
+          placement, scattered data
+
+          Kruskal-Wallis                                           G08AFF
+          one-way analysis of variance on k samples of ...
+
+          Mean, variance, skewness,                                G01AAF
+          kurtosis
+          etc, one variable, from raw data
+
+          Mean, variance, skewness,                                G01ADF
+          kurtosis
+          etc, one variable, from frequency table
+
+          All zeros of complex polynomial, modified                C02AFF
+          Laguerre
+          method
+
+          All zeros of real polynomial, modified                   C02AGF
+          Laguerre
+          method
+
+          Index, real vector element with                          F06JLF
+          largest
+          absolute value (IDAMAX)
+
+          Index, complex vector element with                       F06JMF
+          largest
+          absolute value (IZAMAX)
+
+          Largest                                                  X02ALF
+          positive model number
+
+          Largest                                                  X02BBF
+          representable integer
+
+          LDL                                                      F01MCF
+          T
+            factorization of real symmetric positive-definite ...
+
+          Constructs a stem and                                    G01ARF
+          leaf
+          plot
+
+          Least-squares                                            E02ADF
+          curve fit, by polynomials, arbitrary data points
+
+          Least-squares                                            E02AGF
+          polynomial fit, values and derivatives may be ...
+
+          Least-squares                                            E02BAF
+          curve cubic spline fit (including interpolation)
+
+          Least-squares                                            E02BEF
+          cubic spline curve fit, automatic knot placement
+
+          Least-squares                                            E02DAF
+          surface fit, bicubic splines
+
+          Least-squares                                            E02DCF
+          surface fit by bicubic splines with automatic ...
+
+          Least-squares                                            E02DDF
+          surface fit by bicubic splines with automatic ...
+
+          Covariance matrix for nonlinear                          E04YCF
+          least-squares
+          problem
+
+          Least-squares                                            F04JGF
+          (if rank=n) or minimal least-squares (if ...
+
+          Least-squares (if rank=n) or minimal                     F04JGF
+          least-squares
+          (if rank<n) solution of m real equations ...
+
+          Sparse linear                                            F04QAF
+          least-squares
+          problem, m real equations in n unknowns
+
+          linear
+          problem
+
+          L -approximation by general                              E02GAF
+           1
+          linear
+          function
+
+          Linear                                                   E04MBF
+          programming problem
+
+          Solution of complex simultaneous                         F04ADF
+          linear
+          equations with multiple right-hand sides
+
+          Solution of real simultaneous                            F04ARF
+          linear
+          equations, one right-hand side
+
+          linear
+          equations, one right-hand side using iterative ...
+
+          Solution of real simultaneous                            F04ATF
+          linear
+          equations, one right-hand side using iterative ...
+
+          Solution of real sparse simultaneous                     F04AXF
+          linear
+          equations (coefficient matrix already factorized)
+
+          linear
+          equations, one right-hand side
+
+          Real sparse symmetric positive-definite simultaneous     F04MAF
+          linear
+          equations (coefficient matrix already factorized by ...
+
+          Real sparse symmetric simultaneous                       F04MBF
+          linear
+          equations
+
+          linear
+          equations (coefficient matrix already factorized by ...
+
+          Sparse                                                   F04QAF
+          linear
+          least-squares problem, m real equations in n ...
+
+          Solution of real system of                               F07AEF
+          linear
+          equations, multiple right-hand sides, matrix already ..
+
+          linear
+          equations, multiple right-hand sides, matrix already ..
+
+          Simple                                                   G02CAF
+          linear
+          regression with constant term, no missing values
+
+          Fits a general (multiple)                                G02DAF
+          linear
+          regression model
+
+          Fits a general                                           G02DGF
+          linear
+          regression model for new dependent variable
+
+          Computes estimable function of a general                 G02DNF
+          linear
+          regression model and its standard error
+
+          Fits a generalized                                       G02GBF
+          linear
+          model with binomial errors
+
+          Fits a generalized                                       G02GCF
+          linear
+          model with Poisson errors
+
+          Basic                                                    F06
+          Linear
+          Algebra Subprograms
+
+          2nd order Sturm-                                         D02KEF
+          Liouville
+          problem, regular/singular system, finite/infinite ...
+
+          Computes orthogonal rotations for                        G03BAF
+          loading
+          matrix, generalized orthomax criterion
+
+          Location                                                 G08
+          tests
+
+          Log                                                      S14ABF
+          Gamma function
+
+          algebraico-
+          logarithmic
+          type
+
+          Computes upper and                                       G01EEF
+          lower
+          tail and probability density function probabilities for
+
+          LU                                                       F01BRF
+          factorization of real sparse matrix
+
+          LU                                                       F01BSF
+          factorization of real sparse matrix with known sparsity
+
+          LU                                                       F07ADF
+          factorization of real m by n matrix (DGETRF)
+
+          Machine                                                  X02AJF
+          precision
+
+          Machine                                                  X02
+          Constants
+
+          Performs the                                             G08AHF
+          Mann-Whitney
+          U test on two independent samples
+
+          Computes the exact probabilities for the                 G08AJF
+          Mann-Whitney
+          U statistic, no ties in pooled sample
+
+          Computes the exact probabilities for the                 G08AKF
+          Mann-Whitney
+          U statistic, ties in pooled sample
+
+          Friedman two-way analysis of variance on k               G08AEF
+          matched
+          samples
+
+          Performs the Wilcoxon one-sample (                       G08AGF
+          matched
+          pairs) signed rank test
+
+          Mathematical                                             X01
+          Constants
+
+          Maximization                                             E04
+
+          Maximum                                                  X02BEF
+          number of decimal digits that can be represented
+
+          Mean                                                     G01AAF
+          , variance, skewness, kurtosis etc, one variable, from
+
+          Mean                                                     G01ADF
+          , variance, skewness, kurtosis etc, one variable, from
+
+          Computes a five-point summary (                          G01ALF
+          median
+          , hinges and extremes)
+
+          Median                                                   G08ACF
+          test on two samples of unequal size
+
+          ODEs, IVP, Runge-Kutta-                                  D02BBF
+          Merson
+          method, over a range, intermediate output
+
+          ODEs, IVP, Runge-Kutta-                                  D02BHF
+          Merson
+          method, until function of solution is zero (simple ...
+
+          Evaluation of a fitted bicubic spline at a               E02DFF
+          mesh
+          of points
+
+          Miller
+          method
+
+          Least-squares (if rank=n) or                             F04JGF
+          minimal
+          least-squares (if rank<n) solution of m real ...
+
+          Minimization                                             E04
+
+          Unconstrained                                            E04DGF
+          minimum
+          , pre-conditioned conjugate gradient algorithm, ...
+
+          Unconstrained                                            E04FDF
+          minimum
+          of a sum of squares, combined Gauss-Newton and ...
+
+          Unconstrained                                            E04GCF
+          minimum
+          of a sum of squares, combined Gauss-Newton and ...
+
+          Minimum                                                  E04JAF
+          , function of several variables, quasi-Newton ...
+
+          Minimum                                                  E04UCF
+          , function of several variables, sequential QP method,
+
+          missing
+          values, overwriting input data
+
+          missing
+          values, preserving input data
+
+          Simple linear regression with constant term, no          G02CAF
+          missing
+          values
+
+          Fits a general (multiple) linear regression              G02DAF
+          model
+
+          Fits a general linear regression                         G02DGF
+          model
+          for new dependent variable
+
+          model
+          and its standard error
+
+          Fits a generalized linear                                G02GBF
+          model
+          with binomial errors
+
+          Fits a generalized linear                                G02GCF
+          model
+          with Poisson errors
+
+          model
+
+          model
+
+          Univariate time series, estimation, seasonal ARIMA       G13AFF
+          model
+
+          model
+
+          model
+
+          model
+
+          Multivariate time series, estimation of multi-input      G13BEF
+          model
+
+          model
+
+          Smallest positive                                        X02AKF
+          model
+          number
+
+          Largest positive                                         X02ALF
+          model
+          number
+
+          Parameter of floating-point arithmetic                   X02BHF
+          model
+          , b
+
+          Parameter of floating-point arithmetic                   X02BJF
+          model
+          , p
+
+          Parameter of floating-point arithmetic                   X02BKF
+          model
+          , e
+             min
+
+          Parameter of floating-point arithmetic                   X02BLF
+          model
+          , e
+             max
+
+          Parameter of floating-point arithmetic                   X02DJF
+          model
+          , ROUNDS
+
+          All zeros of complex polynomial,                         C02AFF
+          modified
+          Laguerre method
+
+          All zeros of real polynomial,                            C02AGF
+          modified
+          Laguerre method
+
+          Interpolating functions,                                 E01SEF
+          modified
+          Shepard's method, two variables
+
+          modified
+          Newton algorithm using function values only ...
+
+          Modified                                                 S18ACF
+          Bessel function K (x)
+                           0
+
+          Modified                                                 S18ADF
+          Bessel function K (x)
+                           1
+
+          Modified                                                 S18AEF
+          Bessel function I (x)
+                           0
+
+          Modified                                                 S18AFF
+          Bessel function I (x)
+                           1
+
+          Modified                                                 S18DCF
+          Bessel functions K      (z), real ...
+                            (nu)+n
+
+          Modified                                                 S18DEF
+          Bessel functions I      (z), real ...
+                            (nu)+n
+
+          Interpolating functions,                                 E01BEF
+          monotonicity-preserving
+          , piecewise cubic Hermite, one variable
+
+          Multi-dimensional quadrature over hyper-rectangle,       D01GBF
+          Monte
+          Carlo method
+
+          Multi-dimensional                                        D01FCF
+          adaptive quadrature over hyper-rectangle
+
+          Multi-dimensional                                        D01GBF
+          quadrature over hyper-rectangle, Monte ...
+
+          Multivariate time series, estimation of                  G13BEF
+          multi-input
+          model
+
+          multi-input
+          model
+
+          multigrid
+          technique
+
+          Multiple                                                 C06FPF
+          1-D real discrete Fourier transforms
+
+          Multiple                                                 C06FQF
+          1-D Hermitian discrete Fourier transforms
+
+          Multiple                                                 C06FRF
+          1-D complex discrete Fourier transforms
+
+          Complex conjugate of                                     C06GQF
+          multiple
+          Hermitian sequences
+
+          multiple
+          right-hand sides
+
+          Solves a system of equations with                        F06YJF
+          multiple
+          right-hand sides, real triangular coefficient matrix ..
+
+          Solves system of equations with                          F06ZJF
+          multiple
+          right-hand sides, complex triangular coefficient ...
+
+          Solution of real system of linear equations,             F07AEF
+          multiple
+          right-hand sides, matrix already factorized by ...
+
+          multiple
+          right-hand sides, matrix already factorized by ...
+          Fits a general (                                         G02DAF
+          multiple)
+          linear regression model
+
+          Multiply                                                 F06EDF
+          real vector by scalar (DSCAL)
+
+          Multiply                                                 F06GDF
+          complex vector by complex scalar (ZSCAL)
+
+          Multiply                                                 F06JDF
+          complex vector by real scalar (ZDSCAL)
+
+          Set up reference vector for                              G05EAF
+          multivariate
+          Normal distribution
+
+          Pseudo-random                                            G05EZF
+          multivariate
+          Normal vector from reference vector
+
+          Generates a realisation of a                             G05HDF
+          multivariate
+          time series from a VARMA model
+
+          Multivariate                                             G13BAF
+          time series, filtering (pre-whitening) by an ...
+
+          Multivariate                                             G13BCF
+          time series, cross-correlations
+
+          Multivariate                                             G13BDF
+          time series, preliminary estimation of transfer ...
+
+          Multivariate                                             G13BEF
+          time series, estimation of multi-input model
+
+          Multivariate                                             G13BJF
+          time series, state set and forecasts from fully ...
+
+          Multivariate                                             G13CDF
+          time series, smoothed sample cross spectrum ...
+
+          Generates a vector of random numbers from an (           G05FBF
+          negative)
+          exponential distribution
+
+          Newton
+          and modified Newton algorithm using function values ...
+
+          Newton
+          algorithm using function values only
+          Newton
+          and quasi-Newton algorithm, using 1st derivatives
+
+          Newton
+          algorithm, using 1st derivatives
+
+          Minimum, function of several variables, quasi-           E04JAF
+          Newton
+          algorithm, simple bounds, using function values only
+
+          Kendall/Spearman                                         G02BNF
+          non-parametric
+          rank correlation coefficients, no missing ...
+
+          Kendall/Spearman                                         G02BQF
+          non-parametric
+          rank correlation coefficients, no missing ...
+
+          Initialise random number generating routines to give     G05CCF
+          non-repeatable
+          sequence
+
+          Univariate time series, seasonal and                     G13AAF
+          non-seasonal
+          differencing
+
+          Non-parametric                                           G08
+          tests
+
+          Solution of system of                                    C05NBF
+          nonlinear
+          equations using function values only
+
+          Solution of system of                                    C05PBF
+          nonlinear
+          equations using 1st derivatives
+
+          nonlinear
+          problem
+
+          ODEs, general                                            D02RAF
+          nonlinear
+          boundary value problem, finite difference technique ...
+
+          nonlinear
+          constraints, using function values and optionally ...
+
+          Covariance matrix for                                    E04YCF
+          nonlinear
+          least-squares problem
+
+          Nonlinear                                                E04
+          optimization
+
+          Nonlinear                                                E04
+          regression
+
+          Compute Euclidean                                        F06EJF
+          norm
+          of real vector (DNRM2)
+
+          Compute Euclidean                                        F06JJF
+          norm
+          of complex vector (DZNRM2)
+
+          Computes probabilities for the standard                  G01EAF
+          Normal
+          distribution
+
+          Computes deviates for the standard                       G01FAF
+          Normal
+          distribution
+
+          Computes probability for the bivariate                   G01HAF
+          Normal
+          distribution
+
+          Pseudo-random real numbers,                              G05DDF
+          Normal
+          distribution
+
+          Set up reference vector for multivariate                 G05EAF
+          Normal
+          distribution
+
+          Pseudo-random multivariate                               G05EZF
+          Normal
+          vector from reference vector
+
+          Generates a vector of random numbers from a              G05FDF
+          Normal
+          distribution
+
+          Numerical                                                D01
+          integration
+
+          ODEs                                                     D02BBF
+          , IVP, Runge-Kutta-Merson method, over a range, ...
+
+          ODEs                                                     D02BHF
+          , IVP, Runge-Kutta-Merson method, until function of ...
+
+          ODEs                                                     D02CJF
+          , IVP, Adams method, until function of solution is
+          zero, ...
+
+          ODEs                                                     D02EJF
+          , stiff IVP, BDF method, until function of solution is
+
+          ODEs                                                     D02GAF
+          , boundary value problem, finite difference technique .
+
+          ODEs                                                     D02GBF
+          , boundary value problem, finite difference technique .
+
+          ODEs                                                     D02RAF
+          , general nonlinear boundary value problem, finite ...
+
+          Kruskal-Wallis                                           G08AFF
+          one-way
+          analysis of variance on k samples of unequal size
+
+          Performs the Wilcoxon                                    G08AGF
+          one-sample
+          (matched pairs) signed rank test
+
+          Operations                                               F01QDF
+                                                   H
+          with orthogonal matrices, compute QB or Q B ...
+
+          Operations                                               F01QEF
+          with orthogonal matrices, form columns of Q ...
+
+          Operations                                               F01RDF
+                                                H
+          with unitary matrices, compute QB or Q B ...
+
+          Operations                                               F01REF
+          with unitary matrices, form columns of Q after ...
+
+          Nonlinear                                                E04
+          optimization
+
+          Operations with                                          F01QDF
+          orthogonal
+                                   T
+          matrices, compute QB or Q B after ...
+
+          Operations with                                          F01QEF
+          orthogonal
+          matrices, form columns of Q after factorization ...
+
+          Computes                                                 G03BAF
+          orthogonal
+          rotations for loading matrix, generalized orthomax ...
+
+          orthomax
+          criterion
+
+          oscillating
+          functions
+
+          Incomplete gamma functions                               S14BAF
+          P(a,x)
+          and Q(a,x)
+
+          Matrix-vector product, real symmetric                    F06PEF
+          packed
+          matrix (DSPMV)
+
+          Matrix-vector product, real triangular                   F06PHF
+          packed
+          matrix (DTPMV)
+
+          System of equations, real triangular                     F06PLF
+          packed
+          matrix (DTPSV)
+
+          Rank-1 update, real symmetric                            F06PQF
+          packed
+          matrix (DSPR)
+
+          Rank-2 update, real symmetric                            F06PSF
+          packed
+          matrix (DSPR2)
+
+          Matrix-vector product, complex Hermitian                 F06SEF
+          packed
+          matrix (ZHPMV)
+
+          Matrix-vector product, complex triangular                F06SHF
+          packed
+          matrix (ZTPMV)
+
+          System of equations, complex triangular                  F06SLF
+          packed
+          matrix (ZTPSV)
+
+          Rank-1 update, complex Hermitian                         F06SQF
+          packed
+          matrix (ZHPR)
+
+          Rank-2 update, complex Hermitian                         F06SSF
+          packed
+          matrix (ZHPR2)
+
+          Sign test on two                                         G08AAF
+          paired
+          samples
+
+          Performs the Wilcoxon one-sample (matched                G08AGF
+          pairs)
+          signed rank test
+
+          Kendall/Spearman non-                                    G02BNF
+          parametric
+          rank correlation coefficients, no missing values, ...
+
+          Kendall/Spearman non-                                    G02BQF
+          parametric
+          rank correlation coefficients, no missing values, ...
+
+          Non-                                                     G08
+          parametric
+          tests
+
+          Univariate time series,                                  G13ACF
+          partial
+          autocorrelations from autocorrelations
+
+          Elliptic                                                 D03EDF
+          PDE
+          , solution of finite difference equations by a
+          multigrid ...
+
+          Discretize a 2nd order elliptic                          D03EEF
+          PDE
+          on a rectangle
+
+          Elliptic                                                 D03FAF
+          PDE
+          , Helmholtz equation, 3-D Cartesian co-ordinates
+
+          Pseudo-random                                            G05EHF
+          permutation
+          of an integer vector
+
+          Invert a                                                 M01ZAF
+          permutation
+
+          Interpolating functions, monotonicity-preserving,        E01BEF
+          piecewise
+          cubic Hermite, one variable
+
+          Piessens
+          and de Doncker, allowing for badly-behaved integrands
+
+          Generate real                                            F06AAF
+          plane
+          rotation (DROTG)
+          Apply real                                               F06EPF
+          plane
+          rotation (DROT)
+
+          Constructs a stem and leaf                               G01ARF
+          plot
+
+          Fits a generalized linear model with                     G02GCF
+          Poisson
+          errors
+
+          Poisson
+          distribution
+
+          All zeros of complex                                     C02AFF
+          polynomial
+          , modified Laguerre method
+
+          All zeros of real                                        C02AGF
+          polynomial
+          , modified Laguerre method
+
+          Least-squares curve fit, by                              E02ADF
+          polynomials
+          , arbitrary data points
+
+          Evaluation of fitted                                     E02AEF
+          polynomial
+          in one variable from Chebyshev series form ...
+
+          Least-squares                                            E02AGF
+          polynomial
+          fit, values and derivatives may be constrained, ...
+
+          Derivative of fitted                                     E02AHF
+          polynomial
+          in Chebyshev series form
+
+          Integral of fitted                                       E02AJF
+          polynomial
+          in Chebyshev series form
+
+          Evaluation of fitted                                     E02AKF
+          polynomial
+          in one variable, from Chebyshev series form
+
+          pooled
+          sample
+
+          pooled
+          sample
+
+          Pre-computed                                             D01BBF
+          weights and abscissae for Gaussian quadrature ...
+
+          Unconstrained minimum,                                   E04DGF
+          pre-conditioned
+          conjugate gradient algorithm, function of ...
+
+          Multivariate time series, filtering (                    G13BAF
+          pre-whitening)
+          by an ARIMA model
+
+          Machine                                                  X02AJF
+          precision
+
+          Univariate time series,                                  G13ADF
+          preliminary
+          estimation, seasonal ARIMA model
+
+          Multivariate time series,                                G13BDF
+          preliminary
+          estimation of transfer function model
+
+          principal
+          value (Hilbert transform)
+
+          Performs                                                 G03AAF
+          principal
+          component analysis
+
+          Print                                                    X04CAF
+          a real general matrix
+
+          Print                                                    X04DAF
+          a complex general matrix
+
+          Computes                                                 G01EAF
+          probabilities
+          for the standard Normal distribution
+
+          Computes                                                 G01EBF
+          probabilities
+          for Student's t-distribution
+
+          Computes                                                 G01ECF
+          probabilities
+                   2
+          for (chi)  distribution
+
+          Computes                                                 G01EDF
+          probabilities
+          for F-distribution
+
+          Computes upper and lower tail and                        G01EEF
+          probability
+          density function probabilities for the beta ...
+
+          probabilities
+          for the beta distribution
+
+          Computes                                                 G01EFF
+          probabilities
+          for the gamma distribution
+
+          Computes                                                 G01HAF
+          probability
+          for the bivariate Normal distribution
+
+          probability
+          distribution function
+
+          Computes the exact                                       G08AJF
+          probabilities
+          for the Mann-Whitney U statistic, no ties in ...
+
+          Computes the exact                                       G08AKF
+          probabilities
+          for the Mann-Whitney U statistic, ties in ...
+
+          Dot                                                      F06EAF
+          product
+          of two real vectors (DDOT)
+
+          Dot                                                      F06GAF
+          product
+          of two complex vectors, unconjugated (ZDOTU)
+
+          Dot                                                      F06GBF
+          product
+          of two complex vectors, conjugated (ZDOTC)
+
+          Matrix-vector                                            F06PAF
+          product
+          , real rectangular matrix (DGEMV)
+
+          Matrix-vector                                            F06PBF
+          product
+          , real rectangular band matrix (DGBMV)
+
+          Matrix-vector                                            F06PCF
+          product
+          , real symmetric matrix (DSYMV)
+
+          Matrix-vector                                            F06PDF
+          product
+          , real symmetric band matrix (DSBMV)
+
+          Matrix-vector                                            F06PEF
+          product
+          , real symmetric packed matrix (DSPMV)
+
+          Matrix-vector                                            F06PFF
+          product
+          , real triangular matrix (DTRMV)
+
+          Matrix-vector                                            F06PGF
+          product
+          , real triangular band matrix (DTBMV)
+
+          Matrix-vector                                            F06PHF
+          product
+          , real triangular packed matrix (DTPMV)
+
+          Matrix-vector                                            F06SAF
+          product
+          , complex rectangular matrix (ZGEMV)
+
+          Matrix-vector                                            F06SBF
+          product
+          , complex rectangular band matrix (ZGBMV)
+
+          Matrix-vector                                            F06SCF
+          product
+          , complex Hermitian matrix (ZHEMV)
+
+          Matrix-vector                                            F06SDF
+          product
+          , complex Hermitian band matrix (ZHBMV)
+
+          Matrix-vector                                            F06SEF
+          product
+          , complex Hermitian packed matrix (ZHPMV)
+
+          Matrix-vector                                            F06SFF
+          product
+          , complex triangular matrix (ZTRMV)
+
+          Matrix-vector                                            F06SGF
+          product
+          , complex triangular band matrix (ZTBMV)
+
+          Matrix-vector                                            F06SHF
+          product
+          , complex triangular packed matrix (ZTPMV)
+
+          Matrix-matrix                                            F06YAF
+          product
+          , two real rectangular matrices (DGEMM)
+
+          Matrix-matrix                                            F06YCF
+          product
+          , one real symmetric matrix, one real rectangular ...
+
+          Matrix-matrix                                            F06YFF
+          product
+          , one real triangular matrix, one real rectangular ...
+
+          Matrix-matrix                                            F06ZAF
+          product
+          , two complex rectangular matrices (ZGEMM)
+
+          Matrix-matrix                                            F06ZCF
+          product
+          , one complex Hermitian matrix, one complex ...
+
+          Matrix-matrix                                            F06ZFF
+          product
+          , one complex triangular matrix, one complex ...
+
+          Matrix-matrix                                            F06ZTF
+          product
+          , one complex symmetric matrix, one complex ...
+
+          Linear                                                   E04MBF
+          programming
+          problem
+
+          Pseudo-random                                            G05CAF
+          real numbers, uniform distribution over (0,1)
+
+          Pseudo-random                                            G05DDF
+          real numbers, Normal distribution
+
+          Pseudo-random                                            G05DFF
+          real numbers, Cauchy distribution
+
+          Pseudo-random                                            G05DPF
+          real numbers, Weibull distribution
+
+          Pseudo-random                                            G05DYF
+          integer from uniform distribution
+
+          Pseudo-random                                            G05DZF
+          logical (boolean) value
+
+          Set up reference vector for generating                   G05ECF
+          pseudo-random
+          integers, Poisson distribution
+
+          Set up reference vector for generating                   G05EDF
+          pseudo-random
+          integers, binomial distribution
+
+          Pseudo-random                                            G05EHF
+          permutation of an integer vector
+
+          Pseudo-random                                            G05EJF
+          sample from an integer vector
+
+          Pseudo-random                                            G05EYF
+          integer from reference vector
+
+          Pseudo-random                                            G05EZF
+          multivariate Normal vector from reference vector
+
+          Generates a vector of                                    G05FEF
+          pseudo-random
+          numbers from a beta distribution
+
+          Generates a vector of                                    G05FFF
+          pseudo-random
+          numbers from a gamma distribution
+
+          Incomplete gamma functions P(a,x) and                    S14BAF
+          Q(a,x)
+
+          QP                                                       E04NAF
+          problem
+
+          Minimum, function of several variables, sequential       E04UCF
+          QP
+          method, nonlinear constraints, using function values
+          and ...
+
+          QR                                                       F01QCF
+          factorization of real m by n matrix (m<=n)
+
+          QR                                                       F01RCF
+          factorization of complex m by n matrix (m<=n)
+
+          1-D                                                      D01AJF
+          quadrature
+          , adaptive, finite interval, strategy due to ...
+
+          1-D                                                      D01AKF
+          quadrature
+          , adaptive, finite interval, method suitable for ...
+
+          1-D                                                      D01ALF
+          quadrature
+          , adaptive, finite interval, allowing for ...
+          1-D                                                      D01AMF
+          quadrature
+          , adaptive, infinite or semi-infinite interval
+
+          1-D                                                      D01ANF
+          quadrature
+          , adaptive, finite interval, weight function
+          cos((omega)x) ...
+
+          1-D                                                      D01APF
+          quadrature
+          , adaptive, finite interval, weight function with ...
+
+          1-D                                                      D01AQF
+          quadrature
+          , adaptive, finite interval, weight function ...
+
+          1-D                                                      D01ASF
+          quadrature
+          , adaptive, semi-infinite interval, weight function ...
+
+          Pre-computed weights and abscissae for Gaussian          D01BBF
+          quadrature
+          rules, restricted choice of rule
+
+          Multi-dimensional adaptive                               D01FCF
+          quadrature
+          over hyper-rectangle
+
+          1-D                                                      D01GAF
+          quadrature
+          , integration of function defined by data values, ...
+
+          Multi-dimensional                                        D01GBF
+          quadrature
+          over hyper-rectangle, Monte Carlo method
+
+          quasi-Newton
+          algorithm, using 1st derivatives
+
+          Minimum, function of several variables,                  E04JAF
+          quasi-Newton
+          algorithm, simple bounds, using function values ...
+
+          QZ
+          algorithm, real matrices
+
+          Pseudo-                                                  G05CAF
+          random
+          real numbers, uniform distribution over (0,1)
+
+          Initialise                                               G05CBF
+          random
+          number generating routines to give repeatable sequence
+
+          Initialise                                               G05CCF
+          random
+          number generating routines to give non-repeatable ...
+
+          Save state of                                            G05CFF
+          random
+          number generating routines
+
+          Restore state of                                         G05CGF
+          random
+          number generating routines
+
+          Pseudo-                                                  G05DDF
+          random
+          real numbers, Normal distribution
+
+          Pseudo-                                                  G05DFF
+          random
+          real numbers, Cauchy distribution
+
+          Pseudo-                                                  G05DPF
+          random
+          real numbers, Weibull distribution
+
+          Pseudo-                                                  G05DYF
+          random
+          integer from uniform distribution
+
+          Pseudo-                                                  G05DZF
+          random
+          logical (boolean) value
+
+          Set up reference vector for generating pseudo-           G05ECF
+          random
+          integers, Poisson distribution
+
+          Set up reference vector for generating pseudo-           G05EDF
+          random
+          integers, binomial distribution
+
+          Pseudo-                                                  G05EHF
+          random
+          permutation of an integer vector
+
+          Pseudo-                                                  G05EJF
+          random
+          sample from an integer vector
+
+          Pseudo-                                                  G05EYF
+          random
+          integer from reference vector
+
+          Pseudo-                                                  G05EZF
+          random
+          multivariate Normal vector from reference vector
+
+          Generates a vector of                                    G05FAF
+          random
+          numbers from a uniform distribution
+
+          Generates a vector of                                    G05FBF
+          random
+          numbers from an (negative) exponential distribution
+
+          Generates a vector of                                    G05FDF
+          random
+          numbers from a Normal distribution
+
+          Generates a vector of pseudo-                            G05FEF
+          random
+          numbers from a beta distribution
+
+          Generates a vector of pseudo-                            G05FFF
+          random
+          numbers from a gamma distribution
+
+          ODEs, IVP, Runge-Kutta-Merson method, over a             D02BBF
+          range
+          , intermediate output
+
+          range
+          , eigenvalue and eigenfunction, user-specified ...
+
+          Safe                                                     X02AMF
+          range
+          of floating-point arithmetic
+
+          Safe                                                     X02ANF
+          range
+          of complex floating-point arithmetic
+
+          Least-squares (if                                        F04JGF
+          rank=n)
+          or minimal least-squares (if rank<n) ...
+
+          rank
+          <n) solution of m real equations in n unknowns, ...
+
+          rank
+          <=n,m>=n
+
+          Rank-1                                                   F06PMF
+          update, real rectangular matrix (DGER)
+
+          Rank-1                                                   F06PPF
+          update, real symmetric matrix (DSYR)
+
+          Rank-1                                                   F06PQF
+          update, real symmetric packed matrix (DSPR)
+
+          Rank-2                                                   F06PRF
+          update, real symmetric matrix (DSYR2)
+
+          Rank-2                                                   F06PSF
+          update, real symmetric packed matrix (DSPR2)
+
+          Rank-1                                                   F06SMF
+          update, complex rectangular matrix, unconjugated ...
+
+          Rank-1                                                   F06SNF
+          update, complex rectangular matrix, conjugated vector .
+
+          Rank-1                                                   F06SPF
+          update, complex Hermitian matrix (ZHER)
+
+          Rank-1                                                   F06SQF
+          update, complex Hermitian packed matrix (ZHPR)
+
+          Rank-2                                                   F06SRF
+          update, complex Hermitian matrix (ZHER2)
+
+          Rank-2                                                   F06SSF
+          update, complex Hermitian packed matrix (ZHPR2)
+
+          Rank-k                                                   F06YPF
+          update of a real symmetric matrix (DSYRK)
+
+          Rank-2k                                                  F06YRF
+          update of a real symmetric matrix (DSYR2K)
+
+          Rank-k                                                   F06ZPF
+          update of a complex Hermitian matrix (ZHERK)
+
+          Rank-2k                                                  F06ZRF
+          update of a complex Hermitian matrix (ZHER2K)
+
+          Rank-k                                                   F06ZUF
+          update of a complex symmetric matrix (ZSYRK)
+
+          Rank-2k                                                  F06ZWF
+          update of a complex symmetric matrix (ZHER2K)
+
+          Kendall/Spearman non-parametric                          G02BNF
+          rank
+          correlation coefficients, no missing values,
+          overwriting ...
+
+          Kendall/Spearman non-parametric                          G02BQF
+          rank
+          correlation coefficients, no missing values, preserving
+
+          rank
+          test
+
+          Rank                                                     M01DAF
+          a vector, real numbers
+
+          Rank                                                     M01DEF
+          rows of a matrix, real numbers
+
+          Rank                                                     M01DJF
+          columns of a matrix, real numbers
+
+          Rearrange a vector according to given                    M01EAF
+          ranks
+          , real numbers
+
+          Generates a                                              G05HDF
+          realisation
+          of a multivariate time series from a VARMA model
+
+          Rearrange                                                M01EAF
+          a vector according to given ranks, real numbers
+
+          Multi-dimensional adaptive quadrature over hyper-        D01FCF
+          rectangle
+
+          Multi-dimensional quadrature over hyper-                 D01GBF
+          rectangle
+          , Monte Carlo method
+
+          Discretize a 2nd order elliptic PDE on a                 D03EEF
+          rectangle
+
+          rectangular
+          grid
+
+          rectangular
+          grid
+
+          Matrix-vector product, real                              F06PAF
+          rectangular
+          matrix (DGEMV)
+
+          Matrix-vector product, real                              F06PBF
+          rectangular
+          band matrix (DGBMV)
+
+          Rank-1 update, real                                      F06PMF
+          rectangular
+          matrix (DGER)
+
+          Matrix-vector product, complex                           F06SAF
+          rectangular
+          matrix (ZGEMV)
+
+          Matrix-vector product, complex                           F06SBF
+          rectangular
+          band matrix (ZGBMV)
+
+          Rank-1 update, complex                                   F06SMF
+          rectangular
+          matrix, unconjugated vector (ZGERU)
+
+          Rank-1 update, complex                                   F06SNF
+          rectangular
+          matrix, conjugated vector (ZGERC)
+
+          Matrix-matrix product, two real                          F06YAF
+          rectangular
+          matrices (DGEMM)
+
+          rectangular
+          matrix (DSYMM)
+
+          rectangular
+          matrix (DTRMM)
+
+          Matrix-matrix product, two complex                       F06ZAF
+          rectangular
+          matrices (ZGEMM)
+
+          rectangular
+          matrix (ZHEMM)
+
+          rectangular
+          matrix (ZTRMM)
+
+          rectangular
+          matrix (ZSYMM)
+
+          Set up                                                   G05EAF
+          reference
+          vector for multivariate Normal distribution
+
+          Set up                                                   G05ECF
+          reference
+          vector for generating pseudo-random integers, ...
+
+          Set up                                                   G05EDF
+          reference
+          vector for generating pseudo-random integers, ...
+
+          Set up                                                   G05EXF
+          reference
+          vector from supplied cumulative distribution ...
+
+          Pseudo-random integer from                               G05EYF
+          reference
+          vector
+
+          Pseudo-random multivariate Normal vector from            G05EZF
+          reference
+          vector
+
+          refinement
+
+          refinement
+
+          Simple linear                                            G02CAF
+          regression
+          with constant term, no missing values
+
+          Fits a general (multiple) linear                         G02DAF
+          regression
+          model
+
+          Fits a general linear                                    G02DGF
+          regression
+          model for new dependent variable
+
+          Computes estimable function of a general linear          G02DNF
+          regression
+          model and its standard error
+
+          Nonlinear                                                E04
+          regression
+
+          2nd order Sturm-Liouville problem,                       D02KEF
+          regular/singular
+          system, finite/infinite range, eigenvalue ...
+
+          Interpolating functions, method of                       E01SAF
+          Renka
+          and Cline, two variables
+
+          Calculates standardized                                  G02FAF
+          residuals
+          and influence statistics
+          Univariate time series, diagnostic checking of           G13ASF
+          residuals
+          , following G13AFF
+
+          right-hand
+          sides
+
+          Solution of real simultaneous linear equations, one      F04ARF
+          right-hand
+          side
+
+          right-hand
+          side using iterative refinement
+
+          Solution of real simultaneous linear equations, one      F04ATF
+          right-hand
+          side using iterative refinement
+
+          right-hand
+          side
+
+          Solves a system of equations with multiple               F06YJF
+          right-hand
+          sides, real triangular coefficient matrix (DTRSM)
+
+          Solves system of equations with multiple                 F06ZJF
+          right-hand
+          sides, complex triangular coefficient matrix (ZTRSM)
+
+          Solution of real system of linear equations, multiple    F07AEF
+          right-hand
+          sides, matrix already factorized by F07ADF (DGETRS)
+
+          right-hand
+          sides, matrix already factorized by F07FDF (DPOTRS)
+
+          Generate real plane                                      F06AAF
+          rotation
+          (DROTG)
+
+          Apply real plane                                         F06EPF
+          rotation
+          (DROT)
+
+          Computes orthogonal                                      G03BAF
+          rotations
+          for loading matrix, generalized orthomax criterion
+
+          rules
+          , restricted choice of rule
+
+          rule
+          ODEs, IVP,                                               D02BBF
+          Runge-Kutta-Merson
+          method, over a range, intermediate output
+
+          ODEs, IVP,                                               D02BHF
+          Runge-Kutta-Merson
+          method, until function of solution is zero ...
+
+          Safe                                                     X02AMF
+          range of floating-point arithmetic
+
+          Safe                                                     X02ANF
+          range of complex floating-point arithmetic
+
+          Pseudo-random                                            G05EJF
+          sample
+          from an integer vector
+
+          Sign test on two paired                                  G08AAF
+          samples
+
+          Median test on two                                       G08ACF
+          samples
+          of unequal size
+
+          Friedman two-way analysis of variance on k matched       G08AEF
+          samples
+
+          Kruskal-Wallis one-way analysis of variance on k         G08AFF
+          samples
+          of unequal size
+
+          Performs the Wilcoxon one-                               G08AGF
+          sample
+          (matched pairs) signed rank test
+
+          Performs the Mann-Whitney U test on two independent      G08AHF
+          samples
+
+          sample
+
+          sample
+
+          Univariate time series,                                  G13ABF
+          sample
+          autocorrelation function
+
+          Univariate time series, smoothed                         G13CBF
+          sample
+          spectrum using spectral smoothing by the trapezium ...
+
+          Multivariate time series, smoothed                       G13CDF
+          sample
+          cross spectrum using spectral smoothing by the ...
+
+          Add                                                      F06ECF
+          scalar
+          times real vector to real vector (DAXPY)
+
+          Multiply real vector by                                  F06EDF
+          scalar
+          (DSCAL)
+
+          Add                                                      F06GCF
+          scalar
+          times complex vector to complex vector (ZAXPY)
+
+          Multiply complex vector by complex                       F06GDF
+          scalar
+          (ZSCAL)
+
+          Multiply complex vector by real                          F06JDF
+          scalar
+          (ZDSCAL)
+
+          scattered
+          data
+
+          Univariate time series,                                  G13AAF
+          seasonal
+          and non-seasonal differencing
+
+          Univariate time series, seasonal and non-                G13AAF
+          seasonal
+          differencing
+
+          Univariate time series, preliminary estimation,          G13ADF
+          seasonal
+          ARIMA model
+
+          Univariate time series, estimation,                      G13AFF
+          seasonal
+          ARIMA model
+
+          seasonal
+          ARIMA model
+
+          1-D quadrature, adaptive, infinite or                    D01AMF
+          semi-infinite
+          interval
+
+          1-D quadrature, adaptive,                                D01ASF
+          semi-infinite
+          interval, weight function cos((omega)x) ...
+          Complex conjugate of Hermitian                           C06GBF
+          sequence
+
+          Complex conjugate of complex                             C06GCF
+          sequence
+
+          Complex conjugate of multiple Hermitian                  C06GQF
+          sequences
+
+          Convert Hermitian                                        C06GSF
+          sequences
+          to general complex sequences
+
+          Convert Hermitian sequences to general complex           C06GSF
+          sequences
+
+          sequence
+
+          sequence
+
+          Minimum, function of several variables,                  E04UCF
+          sequential
+          QP method, nonlinear constraints, using function ...
+
+          Interpolating functions, modified                        E01SEF
+          Shepard's
+          method, two variables
+
+          Sign                                                     G08AAF
+          test on two paired samples
+
+          Performs the Wilcoxon one-sample (matched pairs)         G08AGF
+          signed
+          rank test
+
+          Solution of complex                                      F04ADF
+          simultaneous
+          linear equations with multiple right-hand sides
+
+          Solution of real                                         F04ARF
+          simultaneous
+          linear equations, one right-hand side
+
+          Solution of real symmetric positive-definite             F04ASF
+          simultaneous
+          linear equations, one right-hand side using ...
+
+          Solution of real                                         F04ATF
+          simultaneous
+          linear equations, one right-hand side using ...
+
+          Solution of real sparse                                  F04AXF
+          simultaneous
+          linear equations (coefficient matrix already ...
+
+          simultaneous
+          linear equations, one right-hand side
+
+          Real sparse symmetric positive-definite                  F04MAF
+          simultaneous
+          linear equations (coefficient matrix already ...
+
+          Real sparse symmetric                                    F04MBF
+          simultaneous
+          linear equations
+
+          simultaneous
+          linear equations (coefficient matrix already ...
+
+          sin
+          ((omega)x)
+
+          sin
+          ((omega)x)
+
+          Sine                                                     S13ADF
+          integral Si(x)
+
+          2nd order Sturm-Liouville problem, regular/              D02KEF
+          singular
+          system, finite/infinite range, eigenvalue and ...
+
+          singularities
+          at user-specified break-points
+
+          singularities
+          of algebraico-logarithmic type
+
+          Mean, variance,                                          G01AAF
+          skewness
+          , kurtosis etc, one variable, from raw data
+
+          Mean, variance,                                          G01ADF
+          skewness
+          , kurtosis etc, one variable, from frequency table
+
+          Smallest                                                 X02AKF
+          positive model number
+
+          Univariate time series,                                  G13CBF
+          smoothed
+          sample spectrum using spectral smoothing by the ...
+
+          smoothing
+          by the trapezium frequency (Daniell) window
+
+          Multivariate time series,                                G13CDF
+          smoothed
+          sample cross spectrum using spectral smoothing by ...
+
+          smoothing
+          by the trapezium frequency (Daniell) window
+
+          Sort                                                     E02ZAF
+          2-D data into panels for fitting bicubic splines
+
+          Sort                                                     M01CAF
+          a vector, real numbers
+
+          LU factorization of real                                 F01BRF
+          sparse
+          matrix
+
+          LU factorization of real                                 F01BSF
+          sparse
+          matrix with known sparsity pattern
+
+          LU factorization of real sparse matrix with known        F01BSF
+          sparsity
+          pattern
+
+            T
+          LL  factorization of real                                F01MAF
+          sparse
+          symmetric positive-definite matrix
+
+          Selected eigenvalues and eigenvectors of                 F02FJF
+          sparse
+          symmetric eigenproblem
+
+          Solution of real                                         F04AXF
+          sparse
+          simultaneous linear equations (coefficient matrix ...
+
+          Real                                                     F04MAF
+          sparse
+          symmetric positive-definite simultaneous linear ...
+
+          Real                                                     F04MBF
+          sparse
+          symmetric simultaneous linear equations
+
+          Sparse                                                   F04QAF
+          linear least-squares problem, m real equations in ...
+
+          Kendall/                                                 G02BNF
+          Spearman
+          non-parametric rank correlation coefficients, no ...
+
+          Kendall/                                                 G02BQF
+          Spearman
+          non-parametric rank correlation coefficients, no ...
+
+          Approximation of                                         S
+          special
+          functions
+
+          Univariate time series, smoothed sample                  G13CBF
+          spectrum
+          using spectral smoothing by the trapezium frequency ...
+
+          spectral
+          smoothing by the trapezium frequency (Daniell) window
+
+          Multivariate time series, smoothed sample cross          G13CDF
+          spectrum
+          using spectral smoothing by the trapezium frequency ...
+
+          spectral
+          smoothing by the trapezium frequency (Daniell) window
+
+          Interpolating functions, cubic                           E01BAF
+          spline
+          interpolant, one variable
+
+          Interpolating functions, fitting bicubic                 E01DAF
+          spline
+          , data on rectangular grid
+
+          Least-squares curve cubic                                E02BAF
+          spline
+          fit (including interpolation)
+
+          Evaluation of fitted cubic                               E02BBF
+          spline
+          , function only
+
+          Evaluation of fitted cubic                               E02BCF
+          spline
+          , function and derivatives
+
+          Evaluation of fitted cubic                               E02BDF
+          spline
+          , definite integral
+
+          Least-squares cubic                                      E02BEF
+          spline
+          curve fit, automatic knot placement
+          Least-squares surface fit, bicubic                       E02DAF
+          splines
+
+          Least-squares surface fit by bicubic                     E02DCF
+          splines
+          with automatic knot placement, data on rectangular grid
+
+          Least-squares surface fit by bicubic                     E02DDF
+          splines
+          with automatic knot placement, scattered data
+
+          Evaluation of a fitted bicubic                           E02DEF
+          spline
+          at a vector of points
+
+          Evaluation of a fitted bicubic                           E02DFF
+          spline
+          at a mesh of points
+
+          Sort 2-D data into panels for fitting bicubic            E02ZAF
+          splines
+
+          B-                                                       E02
+          splines
+
+          Least-                                                   E02ADF
+          squares
+          curve fit, by polynomials, arbitrary data points
+
+          Least-                                                   E02AGF
+          squares
+          polynomial fit, values and derivatives may be ...
+
+          Least-                                                   E02BAF
+          squares
+          curve cubic spline fit (including interpolation)
+
+          Least-                                                   E02BEF
+          squares
+          cubic spline curve fit, automatic knot placement
+
+          Least-                                                   E02DAF
+          squares
+          surface fit, bicubic splines
+
+          Least-                                                   E02DCF
+          squares
+          surface fit by bicubic splines with automatic knot ...
+
+          Least-                                                   E02DDF
+          squares
+          surface fit by bicubic splines with automatic knot ...
+          Unconstrained minimum of a sum of                        E04FDF
+          squares
+          , combined Gauss-Newton and modified Newton algorithm .
+
+          Unconstrained minimum of a sum of                        E04GCF
+          squares
+          , combined Gauss-Newton and quasi-Newton algorithm, ...
+
+          Covariance matrix for nonlinear least-                   E04YCF
+          squares
+          problem
+
+          Least-                                                   F04JGF
+          squares
+          (if rank=n) or minimal least-squares (if ...
+
+          Least-squares (if rank=n) or minimal least-              F04JGF
+          squares
+          (if rank<n) solution of m real equations in n ...
+
+          Sparse linear least-                                     F04QAF
+          squares
+          problem, m real equations in n unknowns
+
+          Computes probabilities for the                           G01EAF
+          standard
+          Normal distribution
+
+          Computes deviates for the                                G01FAF
+          standard
+          Normal distribution
+
+          standard
+          error
+
+                            2
+          Performs the (chi)  goodness of fit test, for            G08CGF
+          standard
+          continuous distributions
+
+          Calculates                                               G02FAF
+          standardized
+          residuals and influence statistics
+
+          Calculates standardized residuals and influence          G02FAF
+          statistics
+
+          statistic
+          , no ties in pooled sample
+
+          statistic
+          , ties in pooled sample
+          Constructs a                                             G01ARF
+          stem
+          and leaf plot
+
+          ODEs,                                                    D02EJF
+          stiff
+          IVP, BDF method, until function of solution is zero, ..
+
+          Computes probabilities for                               G01EBF
+          Student's
+          t-distribution
+
+          Computes deviates for                                    G01FBF
+          Student's
+          t-distribution
+
+          2nd order                                                D02KEF
+          Sturm-Liouville
+          problem, regular/singular system, ...
+
+          Basic Linear Algebra                                     F06
+          Subprograms
+
+          Unconstrained minimum of a                               E04FDF
+          sum
+          of squares, combined Gauss-Newton and modified Newton .
+
+          Unconstrained minimum of a                               E04GCF
+          sum
+          of squares, combined Gauss-Newton and quasi-Newton ...
+
+          Sum                                                      F06EKF
+          the absolute values of real vector elements (DASUM)
+
+          Sum                                                      F06JKF
+          the absolute values of complex vector elements (DZASUM)
+
+          Computes a five-point                                    G01ALF
+          summary
+          (median, hinges and extremes)
+
+          Least-squares                                            E02DAF
+          surface
+          fit, bicubic splines
+
+          Least-squares                                            E02DCF
+          surface
+          fit by bicubic splines with automatic knot placement, .
+
+          Least-squares                                            E02DDF
+          surface
+          fit by bicubic splines with automatic knot placement, .
+          SVD                                                      F02WEF
+          of real matrix
+
+          SVD                                                      F02XEF
+          of complex matrix
+
+          Swap                                                     F06EGF
+          two real vectors (DSWAP)
+
+          Swap                                                     F06GGF
+          two complex vectors (ZSWAP)
+
+          Fresnel integral                                         S20ACF
+          S(x)
+
+            T
+          LL  factorization of real sparse                         F01MAF
+          symmetric
+          positive-definite matrix
+
+             T
+          LDL  factorization of real                               F01MCF
+          symmetric
+          positive-definite variable-bandwidth matrix
+
+          All eigenvalues of real                                  F02AAF
+          symmetric
+          matrix
+
+          All eigenvalues and eigenvectors of real                 F02ABF
+          symmetric
+          matrix
+
+          symmetric
+          and B is positive-definite
+
+          symmetric
+          and B is positive-definite
+
+          Selected eigenvalues and eigenvectors of real            F02BBF
+          symmetric
+          matrix
+
+          Selected eigenvalues and eigenvectors of sparse          F02FJF
+          symmetric
+          eigenproblem
+
+          Solution of real                                         F04ASF
+          symmetric
+          positive-definite simultaneous linear equations, ...
+
+          Solution of real                                         F04FAF
+          symmetric
+          positive-definite tridiagonal simultaneous linear ...
+
+          Real sparse                                              F04MAF
+          symmetric
+          positive-definite simultaneous linear equations ...
+
+          Real sparse                                              F04MBF
+          symmetric
+          simultaneous linear equations
+
+          Solution of real                                         F04MCF
+          symmetric
+          positive-definite variable-bandwidth simultaneous ...
+
+          Matrix-vector product, real                              F06PCF
+          symmetric
+          matrix (DSYMV)
+
+          Matrix-vector product, real                              F06PDF
+          symmetric
+          band matrix (DSBMV)
+
+          Matrix-vector product, real                              F06PEF
+          symmetric
+          packed matrix (DSPMV)
+
+          Rank-1 update, real                                      F06PPF
+          symmetric
+          matrix (DSYR)
+
+          Rank-1 update, real                                      F06PQF
+          symmetric
+          packed matrix (DSPR)
+
+          Rank-2 update, real                                      F06PRF
+          symmetric
+          matrix (DSYR2)
+
+          Rank-2 update, real                                      F06PSF
+          symmetric
+          packed matrix (DSPR2)
+
+          Matrix-matrix product, one real                          F06YCF
+          symmetric
+          matrix, one real rectangular matrix (DSYMM)
+
+          Rank-k update of a real                                  F06YPF
+          symmetric
+          matrix (DSYRK)
+
+          Rank-2k update of a real                                 F06YRF
+          symmetric
+          matrix (DSYR2K)
+
+          Matrix-matrix product, one complex                       F06ZTF
+          symmetric
+          matrix, one complex rectangular matrix (ZSYMM)
+
+          Rank-k update of a complex                               F06ZUF
+          symmetric
+          matrix (ZSYRK)
+
+          Rank-2k update of a complex                              F06ZWF
+          symmetric
+          matrix (ZHER2K)
+
+          Cholesky factorization of real                           F07FDF
+          symmetric
+          positive-definite matrix (DPOTRF)
+
+          Solution of real                                         F07FEF
+          symmetric
+          positive-definite system of linear equations, ...
+
+          Degenerate                                               S21BAF
+          symmetrised
+          elliptic integral of 1st kind R (x,y)
+                                         C
+
+          Symmetrised                                              S21BBF
+          elliptic integral of 1st kind R (x,y,z)
+                                         F
+
+          Symmetrised                                              S21BCF
+          elliptic integral of 2nd kind R (x,y,z)
+                                         D
+
+          Symmetrised                                              S21BDF
+          elliptic integral of 3rd kind R (x,y,z,r)
+                                         J
+
+          Solution of                                              C05NBF
+          system
+          of nonlinear equations using function values only
+
+          Solution of                                              C05PBF
+          system
+          of nonlinear equations using 1st derivatives
+
+          2nd order Sturm-Liouville problem, regular/singular      D02KEF
+          system
+          , finite/infinite range, eigenvalue and eigenfunction,
+
+          System                                                   F06PJF
+          of equations, real triangular matrix (DTRSV)
+
+          System                                                   F06PKF
+          of equations, real triangular band matrix (DTBSV)
+
+          System                                                   F06PLF
+          of equations, real triangular packed matrix (DTPSV)
+
+          System                                                   F06SJF
+          of equations, complex triangular matrix (ZTRSV)
+
+          System                                                   F06SKF
+          of equations, complex triangular band matrix (ZTBSV)
+
+          System                                                   F06SLF
+          of equations, complex triangular packed matrix (ZTPSV)
+
+          Solves a                                                 F06YJF
+          system
+          of equations with multiple right-hand sides, real ...
+
+          Solves                                                   F06ZJF
+          system
+          of equations with multiple right-hand sides, complex ..
+
+          Solution of real                                         F07AEF
+          system
+          of linear equations, multiple right-hand sides, matrix
+
+          Solution of real symmetric positive-definite             F07FEF
+          system
+          of linear equations, multiple right-hand sides, matrix
+
+          Computes probabilities for Student's                     G01EBF
+          t-distribution
+
+          Computes deviates for Student's                          G01FBF
+          t-distribution
+
+          table
+
+          Frequency                                                G01AEF
+          table
+          from raw data
+
+          Two-way contingency                                      G01AFF
+          table
+                              2
+          analysis, with (chi) /Fisher's exact test
+
+          Computes upper and lower                                 G01EEF
+          tail
+          and probability density function probabilities for the
+
+                                        2
+          test
+
+          Sign                                                     G08AAF
+          test
+          on two paired samples
+
+          Median                                                   G08ACF
+          test
+          on two samples of unequal size
+
+          test
+
+          Performs the Mann-Whitney U                              G08AHF
+          test
+          on two independent samples
+
+                            2
+          Performs the (chi)  goodness of fit                      G08CGF
+          test
+          , for standard continuous distributions
+
+          Goodness of fit                                          G08
+          tests
+
+          Location                                                 G08
+          tests
+
+          Non-parametric                                           G08
+          tests
+
+          ties
+          in pooled sample
+
+          ties
+          in pooled sample
+
+          Generates a realisation of a multivariate                G05HDF
+          time
+          series from a VARMA model
+
+          Univariate                                               G13AAF
+          time
+          series, seasonal and non-seasonal differencing
+
+          Univariate                                               G13ABF
+          time
+          series, sample autocorrelation function
+
+          Univariate                                               G13ACF
+          time
+          series, partial autocorrelations from autocorrelations
+
+          Univariate                                               G13ADF
+          time
+          series, preliminary estimation, seasonal ARIMA model
+
+          Univariate                                               G13AFF
+          time
+          series, estimation, seasonal ARIMA model
+
+          Univariate                                               G13AGF
+          time
+          series, update state set for forecasting
+
+          Univariate                                               G13AHF
+          time
+          series, forecasting from state set
+
+          Univariate                                               G13AJF
+          time
+          series, state set and forecasts, from fully specified .
+
+          Univariate                                               G13ASF
+          time
+          series, diagnostic checking of residuals, following
+          G13AFF
+
+          Multivariate                                             G13BAF
+          time
+          series, filtering (pre-whitening) by an ARIMA model
+
+          Multivariate                                             G13BCF
+          time
+          series, cross-correlations
+
+          Multivariate                                             G13BDF
+          time
+          series, preliminary estimation of transfer function
+          model
+
+          Multivariate                                             G13BEF
+          time
+          series, estimation of multi-input model
+
+          Multivariate                                             G13BJF
+          time
+          series, state set and forecasts from fully specified ..
+
+          Univariate                                               G13CBF
+          time
+          series, smoothed sample spectrum using spectral ...
+
+          Multivariate                                             G13CDF
+          time
+          series, smoothed sample cross spectrum using spectral .
+
+          Return date and                                          X05AAF
+          time
+          as an array of integers
+
+          Convert array of integers representing date and          X05ABF
+          time
+          to character string
+
+          Compare two character strings representing date and      X05ACF
+          time
+
+          Return the CPU                                           X05BAF
+          time
+
+          Multivariate time series, preliminary estimation of      G13BDF
+          transfer
+          function model
+
+          Single 1-D real discrete Fourier                         C06EAF
+          transform
+          , no extra workspace
+
+          Single 1-D Hermitian discrete Fourier                    C06EBF
+          transform
+          , no extra workspace
+
+          Single 1-D complex discrete Fourier                      C06ECF
+          transform
+          , no extra workspace
+
+          Multiple 1-D real discrete Fourier                       C06FPF
+          transforms
+
+          Multiple 1-D Hermitian discrete Fourier                  C06FQF
+          transforms
+
+          Multiple 1-D complex discrete Fourier                    C06FRF
+          transforms
+
+          2-D complex discrete Fourier                             C06FUF
+          transform
+
+          transform)
+
+          trapezium
+          frequency (Daniell) window
+          trapezium
+          frequency (Daniell) window
+
+          Matrix-vector product, real                              F06PFF
+          triangular
+          matrix (DTRMV)
+
+          Matrix-vector product, real                              F06PGF
+          triangular
+          band matrix (DTBMV)
+
+          Matrix-vector product, real                              F06PHF
+          triangular
+          packed matrix (DTPMV)
+
+          System of equations, real                                F06PJF
+          triangular
+          matrix (DTRSV)
+
+          System of equations, real                                F06PKF
+          triangular
+          band matrix (DTBSV)
+
+          System of equations, real                                F06PLF
+          triangular
+          packed matrix (DTPSV)
+
+          Matrix-vector product, complex                           F06SFF
+          triangular
+          matrix (ZTRMV)
+
+          Matrix-vector product, complex                           F06SGF
+          triangular
+          band matrix (ZTBMV)
+
+          Matrix-vector product, complex                           F06SHF
+          triangular
+          packed matrix (ZTPMV)
+
+          System of equations, complex                             F06SJF
+          triangular
+          matrix (ZTRSV)
+
+          System of equations, complex                             F06SKF
+          triangular
+          band matrix (ZTBSV)
+
+          System of equations, complex                             F06SLF
+          triangular
+          packed matrix (ZTPSV)
+
+          Matrix-matrix product, one real                          F06YFF
+          triangular
+          matrix, one real rectangular matrix (DTRMM)
+
+          triangular
+          coefficient matrix (DTRSM)
+
+          Matrix-matrix product, one complex                       F06ZFF
+          triangular
+          matrix, one complex rectangular matrix (ZTRMM)
+
+          triangular
+          coefficient matrix (ZTRSM)
+
+          Solution of real symmetric positive-definite             F04FAF
+          tridiagonal
+          simultaneous linear equations, one right-hand side
+
+          Two-way                                                  G01AFF
+                                                2
+          contingency table analysis, with (chi)  ...
+
+          Sign test on                                             G08AAF
+          two
+          paired samples
+
+          Median test on                                           G08ACF
+          two
+          samples of unequal size
+
+          Friedman                                                 G08AEF
+          two-way
+          analysis of variance on k matched samples
+
+          Performs the Mann-Whitney U test on                      G08AHF
+          two
+          independent samples
+
+          Compare                                                  X05ACF
+          two
+          character strings representing date and time
+
+          Dot product of two complex vectors,                      F06GAF
+          unconjugated
+          (ZDOTU)
+
+          Rank-1 update, complex rectangular matrix,               F06SMF
+          unconjugated
+          vector (ZGERU)
+
+          Unconstrained                                            E04DGF
+          minimum, pre-conditioned conjugate gradient ...
+
+          Unconstrained                                            E04FDF
+          minimum of a sum of squares, combined ...
+
+          Unconstrained                                            E04GCF
+          minimum of a sum of squares, combined ...
+
+          Switch for taking precautions to avoid                   X02DAF
+          underflow
+
+          Pseudo-random real numbers,                              G05CAF
+          uniform
+          distribution over (0,1)
+
+          Pseudo-random integer from                               G05DYF
+          uniform
+          distribution
+
+          Generates a vector of random numbers from a              G05FAF
+          uniform
+          distribution
+
+          Operations with                                          F01RDF
+          unitary
+                                   H
+          matrices, compute QB or Q B after ...
+
+          Operations with                                          F01REF
+          unitary
+          matrices, form columns of Q after factorization by ...
+
+          Univariate                                               G13AAF
+          time series, seasonal and non-seasonal differencing
+
+          Univariate                                               G13ABF
+          time series, sample autocorrelation function
+
+          Univariate                                               G13ACF
+          time series, partial autocorrelations from ...
+
+          Univariate                                               G13ADF
+          time series, preliminary estimation, seasonal ...
+
+          Univariate                                               G13AFF
+          time series, estimation, seasonal ARIMA model
+
+          Univariate                                               G13AGF
+          time series, update state set for forecasting
+
+          Univariate                                               G13AHF
+          time series, forecasting from state set
+
+          Univariate                                               G13AJF
+          time series, state set and forecasts, from fully ...
+
+          Univariate                                               G13ASF
+          time series, diagnostic checking of residuals, ...
+
+          Univariate                                               G13CBF
+          time series, smoothed sample spectrum using ...
+
+          Rank-1                                                   F06PMF
+          update
+          , real rectangular matrix (DGER)
+
+          Rank-1                                                   F06PPF
+          update
+          , real symmetric matrix (DSYR)
+
+          Rank-1                                                   F06PQF
+          update
+          , real symmetric packed matrix (DSPR)
+
+          Rank-2                                                   F06PRF
+          update
+          , real symmetric matrix (DSYR2)
+
+          Rank-2                                                   F06PSF
+          update
+          , real symmetric packed matrix (DSPR2)
+
+          Rank-1                                                   F06SMF
+          update
+          , complex rectangular matrix, unconjugated vector
+          (ZGERU)
+
+          Rank-1                                                   F06SNF
+          update
+          , complex rectangular matrix, conjugated vector (ZGERC)
+
+          Rank-1                                                   F06SPF
+          update
+          , complex Hermitian matrix (ZHER)
+
+          Rank-1                                                   F06SQF
+          update
+          , complex Hermitian packed matrix (ZHPR)
+
+          Rank-2                                                   F06SRF
+          update
+          , complex Hermitian matrix (ZHER2)
+
+          Rank-2                                                   F06SSF
+          update
+          , complex Hermitian packed matrix (ZHPR2)
+          Rank-k                                                   F06YPF
+          update
+          of a real symmetric matrix (DSYRK)
+
+          Rank-2k                                                  F06YRF
+          update
+          of a real symmetric matrix (DSYR2K)
+
+          Rank-k                                                   F06ZPF
+          update
+          of a complex Hermitian matrix (ZHERK)
+
+          Rank-2k                                                  F06ZRF
+          update
+          of a complex Hermitian matrix (ZHER2K)
+
+          Rank-k                                                   F06ZUF
+          update
+          of a complex symmetric matrix (ZSYRK)
+
+          Rank-2k                                                  F06ZWF
+          update
+          of a complex symmetric matrix (ZHER2K)
+
+          Univariate time series,                                  G13AGF
+          update
+          state set for forecasting
+
+          Computes                                                 G01EEF
+          upper
+          and lower tail and probability density function ...
+
+          Input/output                                             X04
+          utilities
+
+          Mean,                                                    G01AAF
+          variance
+          , skewness, kurtosis etc, one variable, from raw data
+
+          Mean,                                                    G01ADF
+          variance
+          , skewness, kurtosis etc, one variable, from ...
+
+          Friedman two-way analysis of                             G08AEF
+          variance
+          on k matched samples
+
+          Kruskal-Wallis one-way analysis of                       G08AFF
+          variance
+          on k samples of unequal size
+
+          VARMA
+          model
+
+          Circular convolution or correlation of two real          C06EKF
+          vectors
+          , no extra workspace
+
+          Evaluation of a fitted bicubic spline at a               E02DEF
+          vector
+          of points
+
+          Dot product of two real                                  F06EAF
+          vectors
+          (DDOT)
+
+          Add scalar times real                                    F06ECF
+          vector
+          to real vector (DAXPY)
+
+          Add scalar times real vector to real                     F06ECF
+          vector
+          (DAXPY)
+
+          Multiply real                                            F06EDF
+          vector
+          by scalar (DSCAL)
+
+          Copy real                                                F06EFF
+          vector
+          (DCOPY)
+
+          Swap two real                                            F06EGF
+          vectors
+          (DSWAP)
+
+          Compute Euclidean norm of real                           F06EJF
+          vector
+          (DNRM2)
+
+          Sum the absolute values of real                          F06EKF
+          vector
+          elements (DASUM)
+
+          Dot product of two complex                               F06GAF
+          vectors
+          , unconjugated (ZDOTU)
+
+          Dot product of two complex                               F06GBF
+          vectors
+          , conjugated (ZDOTC)
+
+          Add scalar times complex                                 F06GCF
+          vector
+          to complex vector (ZAXPY)
+
+          Add scalar times complex vector to complex               F06GCF
+          vector
+          (ZAXPY)
+
+          Multiply complex                                         F06GDF
+          vector
+          by complex scalar (ZSCAL)
+
+          Copy complex                                             F06GFF
+          vector
+          (ZCOPY)
+
+          Swap two complex                                         F06GGF
+          vectors
+          (ZSWAP)
+
+          Multiply complex                                         F06JDF
+          vector
+          by real scalar (ZDSCAL)
+
+          Compute Euclidean norm of complex                        F06JJF
+          vector
+          (DZNRM2)
+
+          Sum the absolute values of complex                       F06JKF
+          vector
+          elements (DZASUM)
+
+          Index, real                                              F06JLF
+          vector
+          element with largest absolute value (IDAMAX)
+
+          Index, complex                                           F06JMF
+          vector
+          element with largest absolute value (IZAMAX)
+
+          Matrix-                                                  F06PAF
+          vector
+          product, real rectangular matrix (DGEMV)
+
+          Matrix-                                                  F06PBF
+          vector
+          product, real rectangular band matrix (DGBMV)
+
+          Matrix-                                                  F06PCF
+          vector
+          product, real symmetric matrix (DSYMV)
+
+          Matrix-                                                  F06PDF
+          vector
+          product, real symmetric band matrix (DSBMV)
+
+          Matrix-                                                  F06PEF
+          vector
+          product, real symmetric packed matrix (DSPMV)
+
+          Matrix-                                                  F06PFF
+          vector
+          product, real triangular matrix (DTRMV)
+
+          Matrix-                                                  F06PGF
+          vector
+          product, real triangular band matrix (DTBMV)
+
+          Matrix-                                                  F06PHF
+          vector
+          product, real triangular packed matrix (DTPMV)
+
+          Matrix-                                                  F06SAF
+          vector
+          product, complex rectangular matrix (ZGEMV)
+
+          Matrix-                                                  F06SBF
+          vector
+          product, complex rectangular band matrix (ZGBMV)
+
+          Matrix-                                                  F06SCF
+          vector
+          product, complex Hermitian matrix (ZHEMV)
+
+          Matrix-                                                  F06SDF
+          vector
+          product, complex Hermitian band matrix (ZHBMV)
+
+          Matrix-                                                  F06SEF
+          vector
+          product, complex Hermitian packed matrix (ZHPMV)
+
+          Matrix-                                                  F06SFF
+          vector
+          product, complex triangular matrix (ZTRMV)
+
+          Matrix-                                                  F06SGF
+          vector
+          product, complex triangular band matrix (ZTBMV)
+
+          Matrix-                                                  F06SHF
+          vector
+          product, complex triangular packed matrix (ZTPMV)
+
+          vector
+          (ZGERU)
+          Rank-1 update, complex rectangular matrix, conjugated    F06SNF
+          vector
+          (ZGERC)
+
+          Set up reference                                         G05EAF
+          vector
+          for multivariate Normal distribution
+
+          Set up reference                                         G05ECF
+          vector
+          for generating pseudo-random integers, Poisson ...
+
+          Set up reference                                         G05EDF
+          vector
+          for generating pseudo-random integers, binomial ...
+
+          Pseudo-random permutation of an integer                  G05EHF
+          vector
+
+          Pseudo-random sample from an integer                     G05EJF
+          vector
+
+          Set up reference                                         G05EXF
+          vector
+          from supplied cumulative distribution function or ...
+
+          Pseudo-random integer from reference                     G05EYF
+          vector
+
+          Pseudo-random multivariate Normal                        G05EZF
+          vector
+          from reference vector
+
+          vector
+
+          Generates a                                              G05FAF
+          vector
+          of random numbers from a uniform distribution
+
+          Generates a                                              G05FBF
+          vector
+          of random numbers from an (negative) exponential ...
+
+          Generates a                                              G05FDF
+          vector
+          of random numbers from a Normal distribution
+
+          Generates a                                              G05FEF
+          vector
+          of pseudo-random numbers from a beta distribution
+
+          Generates a                                              G05FFF
+          vector
+          of pseudo-random numbers from a gamma distribution
+
+          Sort a                                                   M01CAF
+          vector
+          , real numbers
+
+          Rank a                                                   M01DAF
+          vector
+          , real numbers
+
+          Rearrange a                                              M01EAF
+          vector
+          according to given ranks, real numbers
+
+          Kruskal-                                                 G08AFF
+          Wallis
+          one-way analysis of variance on k samples of unequal ..
+
+          Pseudo-random real numbers,                              G05DPF
+          Weibull
+          distribution
+
+          1-D quadrature, adaptive, finite interval,               D01ANF
+          weight
+          function cos((omega)x) or sin((omega)x)
+
+          1-D quadrature, adaptive, finite interval,               D01APF
+          weight
+          function with end-point singularities of ...
+
+          1-D quadrature, adaptive, finite interval,               D01AQF
+          weight
+          function 1/(x-c), Cauchy principal value ...
+
+          1-D quadrature, adaptive, semi-infinite interval,        D01ASF
+          weight
+          function cos((omega)x) or sin((omega)x)
+
+          Pre-computed                                             D01BBF
+          weights
+          and abscissae for Gaussian quadrature rules, ...
+
+          Computes (optionally                                     G02BXF
+          weighted)
+          correlation and covariance matrices
+
+          Multivariate time series, filtering (pre-                G13BAF
+          whitening)
+          by an ARIMA model
+
+          Performs the Mann-                                       G08AHF
+          Whitney
+          U test on two independent samples
+
+          Computes the exact probabilities for the Mann-           G08AJF
+          Whitney
+          U statistic, no ties in pooled sample
+
+          Computes the exact probabilities for the Mann-           G08AKF
+          Whitney
+          U statistic, ties in pooled sample
+
+          Performs the                                             G08AGF
+          Wilcoxon
+          one-sample (matched pairs) signed rank test
+
+          window
+
+          window
+
+          Two-way contingency table analysis, with                 G01AFF
+               2
+          (chi)
+          /Fisher's exact test
+
+          Computes probabilities for                               G01ECF
+               2
+          (chi)
+          distribution
+
+          Computes deviates for the                                G01FCF
+               2
+          (chi)
+          distribution
+
+          Performs the                                             G08CGF
+               2
+          (chi)
+          goodness of fit test, for standard continuous ...
+
+          All                                                      C02AFF
+          zeros
+          of complex polynomial, modified Laguerre method
+
+          All                                                      C02AGF
+          zeros
+          of real polynomial, modified Laguerre method
+
+          Zero                                                     C05ADF
+          of continuous function in given interval, Bus and
+          Dekker ...
+
+          zero
+          (simple driver)
+
+          zero
+          , intermediate output (simple driver)
+
+          zero
+          , intermediate output (simple driver)
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXconvert}{NAG On-line Documentation: convert}
+\beginscroll
+\begin{verbatim}
+
+
+
+     CONVERSION(3NAG)  Foundation Library (12/10/92)  CONVERSION(3NAG)
+
+
+
+          Introduction              Converting from the Workstation Library
+          Converting from the Workstation Library
+
+          The NAG Foundation Library is a successor product to an earlier,
+          smaller subset of the full NAG Fortran Library which was called
+          the NAG Workstation Library. The Foundation Library has been
+          designed to be upwards compatible, in terms of functionality,
+          with the Workstation Library. However some routines that were
+          present in the Workstation Library have been replaced by more up-
+          to-date routines from the NAG Fortran Library, which provide
+          improved algorithms or software design.
+
+          The list below gives the names of those routines which were
+          available in the Workstation Library, but are not included in the
+          Foundation Library. For each such routine, it also gives the name
+          of the routine in the Foundation Library which best covers the
+          same functionality.
+
+              Workstation    Foundation
+              Library        Library
+
+              C02AEF         C02AGF
+
+              D02CBF         D02CJF
+
+              D02CHF         D02CJF
+
+              D02EBF         D02EJF
+
+              D02EHF         D02EJF
+
+              D02HAF         D02GAF
+
+              D02HBF         D02RAF
+
+              D02SAF         D02RAF
+
+              E02DBF         E02DEF
+
+              E04VDF         E04UCF
+
+              E04ZCF         E04UCF (see Note 1)
+
+              F01BTF         F07ADF
+
+              F01BXF         F07FDF
+
+              F02WAF         F02WEF
+
+              F04AYF         F07AEF
+
+              F04AZF         F07FEF
+
+              F04YAF         G02DAF
+
+              G01ABF         G02BXF (with M = 2)
+
+              G01BAF         G01EBF
+
+              G01BBF         G01EDF
+
+              G01BCF         G01ECF
+
+              G01BDF         G01EEF
+
+              G01CAF         G01FBF
+
+              G01CBF         G01FDF
+
+              G01CCF         G01FCF
+
+              G01CDF         G01FEF
+
+              G01CEF         G01FAF
+
+              G02BAF         G02BXF
+
+              G02BGF         G02BXF
+
+              G02CEF         G02DAF (see Note 2)
+
+              G02CGF         G02DAF
+
+              G02CJF         G02DAF
+
+              G05DBF         G05FBF
+
+              G05DCF         G05CAF (see Note 3)
+
+              G05DEF         G05FFF
+
+              G05DHF         G05FFF (see Note 4)
+
+              G05EGF         G05HDF
+
+              G05EWF         G05HDF
+
+              G08ABF         G08AGF
+
+              G08ADF         G08AHF
+
+              M01AKF         M01DAF
+
+              M01APF         M01CAF
+
+              S15ABF         G01EAF
+
+              S15ACF         G01EAF
+
+              X02AAF         X02AJF
+
+              X02ACF         X02ALF
+
+          Notes:
+
+          1.    E04ZCF checks user-supplied routines for evaluating the
+                first derivatives of the objective function and constraint
+                functions supplied to E04VDF. This functionality is now
+                provided by E04UCF, using the optional parameters Verify
+                Objective Gradients and Verify Constraint Gradients.
+
+          2.    G02CEF selects variables to be included in a linear
+                regression performed by G02CGF. This functionality is now
+                provided by the parameter ISX of G02DAF.
+
+          3.    A call to G05DCF can be replaced by a simple transformation
+                of the result of a call to G05CAF. The statement
+                     X = G05DCF(A,B)
+                can be replaced by the statements
+
+                     X = G05CAF(X)
+                     X = A + B*LOG(X/(1.0D0-X))
+
+                                                            2
+          4.    G05DHF generates random numbers from a (chi)  distribution
+                with N degrees of freedom. This can be achieved by calling
+                G05FFF with the values DBLE(N)/2.0D0 and 2.0D0 for the
+                parameters A and B respectively.
+
+\end{verbatim}
+\endscroll
+\end{page}
diff --git a/src/hyper/pages/nagc.ht b/src/hyper/pages/nagc.ht
new file mode 100644
index 00000000..ccd60349
--- /dev/null
+++ b/src/hyper/pages/nagc.ht
@@ -0,0 +1,4018 @@
+\begin{page}{manpageXXc02}{NAG On-line Documentation: c02}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C02(3NAG)         Foundation Library (12/10/92)         C02(3NAG)
+
+
+
+          C02 -- Zeros of Polynomials                   Introduction -- C02
+                                    Chapter C02
+                               Zeros of Polynomials
+
+          1. Scope of the Chapter
+
+          This chapter is concerned with computing the zeros of a
+          polynomial with real or complex coefficients.
+
+          2. Background to the Problems
+
+          Let f(z) be a polynomial of degree n with complex coefficients
+          a :
+           i
+
+                            n    n-1    n-2
+                   f(z)==a z +a z   +a z   +...+a   z+a ,  a /=0.
+                          0    1      2          n-1   n    0
+
+          A complex number z  is called a zero of f(z) (or equivalently a
+                            1
+          root of the equation f(z)=0), if:
+
+                                      f(z )=0.
+                                         1
+
+          If z  is a zero, then f(z) can be divided by a factor (z-z ):
+              1                                                     1
+
+                                f(z)=(z-z )f (z)                        (1)
+                                         1  1
+
+          where f (z) is a polynomial of degree n-1. By the Fundamental
+                 1
+          Theorem of Algebra, a polynomial f(z) always has a zero, and so
+          the process of dividing out factors (z-z ) can be continued until
+                                                  i
+          we have a complete factorization of f(z)
+
+                           f(z)==a (z-z )(z-z )...(z-z ).
+                                  0    1     2        n
+
+          Here the complex numbers z ,z ,...,z  are the zeros of f(z); they
+                                    1  2      n
+          may not all be distinct, so it is sometimes more convenient to
+          write:
+
+                                   m       m          m
+                                    1       2          k
+                     f(z)==a (z-z )  (z-z )  ...(z-z )  ,  k<=n,
+                            0    1       2          k
+
+          with distinct zeros z ,z ,...,z  and multiplicities m >=1. If
+                               1  2      k                     i
+          m =1, z  is called a single zero, if m >1, z  is called a
+           i     i                              i     i
+          multiple or repeated zero; a multiple zero is also a zero of the
+          derivative of f(z).
+
+          If the coefficients of f(z) are all real, then the zeros of f(z)
+          are either real or else occur as pairs of conjugate complex
+          numbers x+iy and x-iy. A pair of complex conjugate zeros are the
+                                                 2
+          zeros of a quadratic factor of f(z), (z +rz+s), with real
+          coefficients r and s.
+
+          Mathematicians are accustomed to thinking of polynomials as
+          pleasantly simple functions to work with. However the problem of
+          numerically computing the zeros of an arbitrary polynomial is far
+          from simple. A great variety of algorithms have been proposed, of
+          which a number have been widely used in practice; for a fairly
+          comprehensive survey, see Householder [1]. All general algorithms
+          are iterative. Most converge to one zero at a time; the
+          corresponding factor can then be divided out as in equation (1)
+          above - this process is called deflation or, loosely, dividing
+          out the zero - and the algorithm can be applied again to the
+          polynomial f (z). A pair of complex conjugate zeros can be
+                      1
+          divided out together - this corresponds to dividing f(z) by a
+          quadratic factor.
+
+          Whatever the theoretical basis of the algorithm, a number of
+          practical problems arise: for a thorough discussion of some of
+          them see Peters and Wilkinson [2] and Wilkinson [3]. The most
+          elementary point is that, even if z  is mathematically an exact
+                                             1
+          zero of f(z), because of the fundamental limitations of computer
+          arithmetic the computed value of f(z ) will not necessarily be
+                                              1
+          exactly 0.0. In practice there is usually a small region of
+          values of z about the exact zero at which the computed value of
+          f(z) becomes swamped by rounding errors. Moreover in many
+          algorithms this inaccuracy in the computed value of f(z) results
+          in a similar inaccuracy in the computed step from one iterate to
+          the next. This limits the precision with which any zero can be
+          computed. Deflation is another potential cause of trouble, since,
+          in the notation of equation (1), the computed coefficients of
+          f (z) will not be completely accurate, especially if z  is not an
+           1                                                    1
+          exact zero of f(z); so the zeros of the computed f (z) will
+                                                            1
+          deviate from the zeros of f(z).
+
+          A zero is called ill-conditioned if it is sensitive to small
+          changes in the coefficients of the polynomial. An ill-conditioned
+          zero is likewise sensitive to the computational inaccuracies just
+          mentioned. Conversely a zero is called well-conditioned if it is
+          comparatively insensitive to such perturbations. Roughly speaking
+          a zero which is well separated from other zeros is well-
+          conditioned, while zeros which are close together are ill-
+          conditioned, but in talking about 'closeness' the decisive factor
+          is not the absolute distance between neighbouring zeros but their
+          ratio: if the ratio is close to 1 the zeros are ill-conditioned.
+          In particular, multiple zeros are ill-conditioned. A multiple
+          zero is usually split into a cluster of zeros by perturbations in
+          the polynomial or computational inaccuracies.
+
+          2.1. References
+
+          [1]   Householder A S (1970) The Numerical Treatment of a Single
+                Nonlinear Equation. McGraw-Hill.
+
+          [2]   Peters G and Wilkinson J H (1971) Practical Problems Arising
+                in the Solution of Polynomial Equations. J. Inst. Maths
+                Applics. 8 16--35.
+
+          [3]   Wilkinson J H (1963) Rounding Errors in Algebraic Processes,
+                Chapter 2. HMSO.
+
+          3. Recommendations on Choice and Use of Routines
+
+          3.1. Discussion
+
+          Two routines are available: C02AFF for polynomials with complex
+          coefficients and C02AGF for polynomials with real coefficients.
+
+          C02AFF and C02AGF both use a variant of Laguerre's Method due to
+          BT Smith to calculate each zero until the degree of the deflated
+          polynomial is less than 3, whereupon the remaining zeros are
+          obtained using the 'standard' closed formulae for a quadratic or
+          linear equation.
+
+          The accuracy of the roots will depend on how ill-conditioned they
+          are. Peters and Wilkinson [2] describe techniques for estimating
+          the errors in the zeros after they have been computed.
+
+          3.2. Index
+
+           Zeros of a complex polynomial                             C02AFF
+           Zeros of a real polynomial                                C02AGF
+
+
+          C02 -- Zeros of Polynomials                       Contents -- C02
+          Chapter C02
+
+          Zeros of Polynomials
+
+          C02AFF  All zeros of complex polynomial, modified Laguerre method
+
+          C02AGF  All zeros of real polynomial, modified Laguerre method
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc02aff}{NAG On-line Documentation: c02aff}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C02AFF(3NAG)      Foundation Library (12/10/92)      C02AFF(3NAG)
+
+
+
+          C02 -- Zeros of Polynomials                                C02AFF
+                  C02AFF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C02AFF finds all the roots of a complex polynomial equation,
+          using a variant of Laguerre's Method.
+
+          2. Specification
+
+                 SUBROUTINE C02AFF (A, N, SCALE, Z, W, IFAIL)
+                 INTEGER          N, IFAIL
+                 DOUBLE PRECISION A(2,N+1), Z(2,N), W(4*(N+1))
+                 LOGICAL          SCALE
+
+          3. Description
+
+          The routine attempts to find all the roots of the nth degree
+          complex polynomial equation
+
+                               n    n-1    n-2
+                       P(z)=a z +a z   +a z   +...+a   z+a =0.
+                             0    1      2          n-1   n
+
+          The roots are located using a modified form of Laguerre's Method,
+          originally proposed by Smith [2].
+
+          The method of Laguerre [3] can be described by the iterative
+          scheme
+
+                                             -n*P(z )
+                                                   k
+                          L(z )=z   -z = ----------------,
+                             k   k+1  k             
+                                         P'(z )+-  /H(z )
+                                             k   \/    k
+
+                                           2
+          where H(z )=(n-1)*[(n-1)*(P'(z )) -n*P(z )P''(z )], and z  is
+                   k                    k         k      k         0
+          specified.
+
+          The sign in the denominator is chosen so that the modulus of the
+          Laguerre step at z , viz. |L(z )|, is as small as possible. The
+                            k           k
+          method can be shown to be cubically convergent for isolated roots
+          (real or complex) and linearly convergent for multiple roots.
+          The routine generates a sequence of iterates z , z , z ,..., such
+                                                        1   2   3
+          that |P(z   )|<|P(z )| and ensures that z   +L(z   ) 'roughly'
+                   k+1       k                     k+1    k+1
+          lies inside a circular region of radius |F| about z  known to
+                                                             k
+          contain a zero of P(z); that is, |L(z   )|<=|F|, where F denotes
+                                               k+1
+          the Fejer bound (see Marden [1]) at the point z . Following Smith
+                                                         k
+          [2], F is taken to be min(B,1.445*n*R), where B is an upper bound
+          for the magnitude of the smallest zero given by
+
+                                                         1/n
+                      B=1.0001*min(\/n*L(z ),|r |,|a /a |   ),
+                                          k    1    n  0
+
+          r  is the zero X of smaller magnitude of the quadratic equation
+           1
+
+                                           2
+                    2(P''(z )/(2*n*(n-1)))X +2(P'(z )/n)X+P(z )=0
+                           k                       k         k
+
+          and the Cauchy lower bound R for the smallest zero is computed
+          (using Newton's Method) as the positive root of the polynomial
+          equation
+
+                         n      n-1      n-2
+                    |a |z +|a |z   +|a |z   +...+|a   |z-|a |=0.
+                      0      1        2            n-1     n
+
+          Starting from the origin, successive iterates are generated
+          according to the rule z   =z +L(z ) for k = 1,2,3,... and L(z )
+                                 k+1  k    k                           k
+          is 'adjusted' so that |P(z   )|<|P(z )| and |L(z   )|<=|F|. The
+                                    k+1       k           k+1
+          iterative procedure terminates if P(z   ) is smaller in absolute
+                                               k+1
+          value than the bound on the rounding error in P(z   ) and the
+                                                           k+1
+          current iterate z =z    is taken to be a zero of P(z). The
+                           p  k+1
+                              ~
+          deflated polynomial P(z)=P(z)/(z-z ) of degree n-1 is then
+                                            p
+          formed, and the above procedure is repeated on the deflated
+          polynomial until n<3, whereupon the remaining roots are obtained
+          via the 'standard' closed formulae for a linear (n = 1) or
+          quadratic (n = 2) equation.
+
+          To obtain the roots of a quadratic polynomial, C02AHF(*) can be
+          used.
+
+          4. References
+
+          [1]   Marden M (1966) Geometry of Polynomials. Mathematical
+                Surveys. 3 Am. Math. Soc., Providence, RI.
+
+          [2]   Smith B T (1967) ZERPOL: A Zero Finding Algorithm for
+                Polynomials Using Laguerre's Method. Technical Report.
+                Department of Computer Science, University of Toronto,
+                Canada.
+
+          [3]   Wilkinson J H (1965) The Algebraic Eigenvalue Problem.
+                Clarendon Press.
+
+          5. Parameters
+
+           1:  A(2,N+1) -- DOUBLE PRECISION array                     Input
+               On entry: if A is declared with bounds (2,0:N), then A(1,i)
+               and A(2,i) must contain the real and imaginary parts of a
+                                                                        i
+                                          n-i
+               (i.e., the coefficient of z   ), for i=0,1,...,n.
+               Constraint: A(1,0) /= 0.0 or A(2,0) /= 0.0.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the degree of the polynomial, n. Constraint: N >=
+               1.
+
+           3:  SCALE -- LOGICAL                                       Input
+               On entry: indicates whether or not the polynomial is to be
+               scaled. See Section 8 for advice on when it may be
+               preferable to set SCALE = .FALSE. and for a description of
+               the scaling strategy. Suggested value: SCALE = .TRUE..
+
+           4:  Z(2,N) -- DOUBLE PRECISION array                      Output
+               On exit: the real and imaginary parts of the roots are
+               stored in Z(1,i) and Z(2,i) respectively, for i=1,2,...,n.
+
+           5:  W(4*(N+1)) -- DOUBLE PRECISION array               Workspace
+
+           6:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry A(1,0) = 0.0 and A(2,0) = 0.0,
+
+               or       N < 1.
+
+          IFAIL= 2
+               The iterative procedure has failed to converge. This error
+               is very unlikely to occur. If it does, please contact NAG
+               immediately, as some basic assumption for the arithmetic has
+               been violated. See also Section 8.
+
+          IFAIL= 3
+               Either overflow or underflow prevents the evaluation of P(z)
+               near some of its zeros. This error is very unlikely to
+               occur. If it does, please contact NAG immediately. See also
+               Section 8.
+
+          7. Accuracy
+
+          All roots are evaluated as accurately as possible, but because of
+          the inherent nature of the problem complete accuracy cannot be
+          guaranteed.
+
+          8. Further Comments
+
+          If SCALE = .TRUE., then a scaling factor for the coefficients is
+          chosen as a power of the base B of the machine so that the
+                                                                EMAX-P
+          largest coefficient in magnitude approaches THRESH = B      .
+          Users should note that no scaling is performed if the largest
+          coefficient in magnitude exceeds THRESH, even if SCALE = .TRUE..
+          (For definition of B, EMAX and P see the Chapter Introduction
+          X02.)
+
+          However, with SCALE = .TRUE., overflow may be encountered when
+          the input coefficients a ,a ,a ,...,a  vary widely in magnitude,
+                                  0  1  2      n
+                                                    (4*P)
+          particularly on those machines for which B      overflows. In
+          such cases, SCALE should be set to .FALSE. and the coefficients
+          scaled so that the largest coefficient in magnitude does not
+                  (EMAX-2*P)
+          exceed B          .
+
+          Even so, the scaling strategy used in C02AFF is sometimes
+          insufficient to avoid overflow and/or underflow conditions. In
+          such cases, the user is recommended to scale the independent
+          variable (z) so that the disparity between the largest and
+          smallest coefficient in magnitude is reduced. That is, use the
+          routine to locate the zeros of the polynomial d*P(cz) for some
+          suitable values of c and d. For example, if the original
+                               -100   100 20                   -10
+          polynomial was P(z)=2    i+2   z  , then choosing c=2    and
+             100                                                     20
+          d=2   , for instance, would yield the scaled polynomial i+z  ,
+          which is well-behaved relative to overflow and underflow and has
+                           10
+          zeros which are 2   times those of P(z).
+
+          If the routine fails with IFAIL = 2 or 3, then the real and
+          imaginary parts of any roots obtained before the failure occurred
+          are stored in Z in the reverse order in which they were found.
+          Let n  denote the number of roots found before the failure
+               R
+          occurred. Then Z(1,n) and Z(2,n) contain the real and imaginary
+          parts of the 1st root found, Z(1,n-1) and Z(2,n-1) contain the
+          real and imaginary parts of the 2nd root found, ..., Z(1,n ) and
+                                                                    R
+          Z(2,n ) contain the real and imaginary parts of the n th root
+               R                                               R
+          found. After the failure has occurred, the remaining 2*(n-n )
+                                                                     R
+          elements of Z contain a large negative number (equal to
+                         
+
+          -1/(X02AMF().\/2)).
+
+          9. Example
+
+                                                 5    4    3    2
+          To find the roots of the polynomial a z +a z +a z +a z +a z+a =0,
+                                               0    1    2    3    4   5
+          where a =(5.0+6.0i), a =(30.0+20.0i), a =-(0.2+6.0i),
+                 0              1                2
+          a =(50.0+100000.0i), a =-(2.0-40.0i) and a =(10.0+1.0i).
+           3                    4                   5
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc02agf}{NAG On-line Documentation: c02agf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C02AGF(3NAG)      Foundation Library (12/10/92)      C02AGF(3NAG)
+
+
+
+          C02 -- Zeros of Polynomials                                C02AGF
+                  C02AGF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C02AGF finds all the roots of a real polynomial equation, using a
+          variant of Laguerre's Method.
+
+          2. Specification
+
+                 SUBROUTINE C02AGF (A, N, SCALE, Z, W, IFAIL)
+                 INTEGER          N, IFAIL
+                 DOUBLE PRECISION A(N+1), Z(2,N), W(2*(N+1))
+                 LOGICAL          SCALE
+
+          3. Description
+
+          The routine attempts to find all the roots of the nth degree real
+          polynomial equation
+
+                               n    n-1    n-2
+                       P(z)=a z +a z   +a z   +...+a   z+a =0.
+                             0    1      2          n-1   n
+
+          The roots are located using a modified form of Laguerre's Method,
+          originally proposed by Smith [2].
+
+          The method of Laguerre [3] can be described by the iterative
+          scheme
+
+                                             -n*P(z )
+                                                   k
+                          L(z )=z   -z = ----------------,
+                             k   k+1  k             
+                                         P'(z )+-  /H(z )
+                                             k   \/    k
+
+                                           2
+          where H(z )=(n-1)*[(n-1)*(P'(z )) -n*P(z )P''(z )], and z  is
+                   k                    k         k      k         0
+          specified.
+
+          The sign in the denominator is chosen so that the modulus of the
+          Laguerre step at z , viz. |L(z )|, is as small as possible. The
+                            k           k
+          method can be shown to be cubically convergent for isolated roots
+          (real or complex) and linearly convergent for multiple roots.
+          The routine generates a sequence of iterates z , z , z ,..., such
+                                                        1   2   3
+          that |P(z +1)|<|P(z )| and ensures that z   +L(z   ) 'roughly'
+                   k         k                     k+1    k+1
+          lies inside a circular region of radius |F| about z  known to
+                                                             k
+          contain a zero of P(z); that is, |L(z   )|<=|F|, where F denotes
+                                               k+1
+          the Fejer bound (see Marden [1]) at the point z . Following Smith
+                                                         k
+          [2], F is taken to be min(B,1.445*n*R), where B is an upper bound
+          for the magnitude of the smallest zero given by
+
+                                                         1/n
+                      B=1.0001*min(\/n*L(z ),|r |,|a /a |   ),
+                                          k    1    n  0
+
+          r  is the zero X of smaller magnitude of the quadratic equation
+           1
+
+                                           2
+                    2(P''(z )/(2*n*(n-1)))X +2(P'(z )/n)X+P(z )=0
+                           k                       k         k
+
+          and the Cauchy lower bound R for the smallest zero is computed
+          (using Newton's Method) as the positive root of the polynomial
+          equation
+
+                         n      n-1      n-2
+                    |a |z +|a |z   +|a |z   +...+|a   |z-|a |=0.
+                      0      1        2            n-1     n
+
+          Starting from the origin, successive iterates are generated
+          according to the rule z   =z +L(z ) for k=1,2,3,... and L(z ) is
+                                 k+1  k    k                         k
+                                 k+1       k           k+1
+          iterative procedure terminates if P(z   ) is smaller in absolute
+                                               k+1
+          value than the bound on the rounding error in P(z   ) and the
+                                                           k+1
+          current iterate z =z    is taken to be a zero of P(z) (as is its
+                           p  k-1
+                    
+
+          conjugate z  if z  is complex). The deflated polynomial
+                     p     p
+          ~
+          P(z)=P(z)/(z-z ) of degree n-1 if z  is real
+                        p                    p
+           ~                   
+          (P(z)=P(z)/((z-z )(z-z )) of degree n-2 if z  is complex) is then
+                          p     p                     p
+          formed, and the above procedure is repeated on the deflated
+          polynomial until n<3, whereupon the remaining roots are obtained
+          via the 'standard' closed formulae for a linear (n = 1) or
+          quadratic (n = 2) equation.
+
+          To obtain the roots of a quadratic polynomial, C02AJF(*) can be
+          used.
+
+          4. References
+
+          [1]   Marden M (1966) Geometry of Polynomials. Mathematical
+                Surveys. 3 Am. Math. Soc., Providence, RI.
+
+          [2]   Smith B T (1967) ZERPOL: A Zero Finding Algorithm for
+                Polynomials Using Laguerre's Method. Technical Report.
+                Department of Computer Science, University of Toronto,
+                Canada.
+
+          [3]   Wilkinson J H (1965) The Algebraic Eigenvalue Problem.
+                Clarendon Press.
+
+          5. Parameters
+
+           1:  A(N+1) -- DOUBLE PRECISION array                       Input
+               On entry: if A is declared with bounds (0:N), then A(i)
+                                                          n-i
+               must contain a  (i.e., the coefficient of z   ), for
+                             i
+               i=0,1,...,n. Constraint: A(0) /= 0.0.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the degree of the polynomial, n. Constraint: N >=
+               1.
+
+           3:  SCALE -- LOGICAL                                       Input
+               On entry: indicates whether or not the polynomial is to be
+               scaled. See Section 8 for advice on when it may be
+               preferable to set SCALE = .FALSE. and for a description of
+               the scaling strategy. Suggested value: SCALE = .TRUE..
+
+           4:  Z(2,N) -- DOUBLE PRECISION array                      Output
+               On exit: the real and imaginary parts of the roots are
+               stored in Z(1,i) and Z(2,i) respectively, for i=1,2,...,n.
+               Complex conjugate pairs of roots are stored in consecutive
+               pairs of elements of Z; that is, Z(1,i+1) = Z(1,i) and
+               Z(2,i+1)=-Z(2,i).
+
+           5:  W(2*(N+1)) -- DOUBLE PRECISION array               Workspace
+
+           6:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry A(0) = 0.0,
+
+               or       N < 1.
+
+          IFAIL= 2
+               The iterative procedure has failed to converge. This error
+               is very unlikely to occur. If it does, please contact NAG
+               immediately, as some basic assumption for the arithmetic has
+               been violated. See also Section 8.
+
+          IFAIL= 3
+               Either overflow or underflow prevents the evaluation of P(z)
+               near some of its zeros. This error is very unlikely to
+               occur. If it does, please contact NAG immediately. See also
+               Section 8.
+
+          7. Accuracy
+
+          All roots are evaluated as accurately as possible, but because of
+          the inherent nature of the problem complete accuracy cannot be
+          guaranteed.
+
+          8. Further Comments
+
+          If SCALE = .TRUE., then a scaling factor for the coefficients is
+          chosen as a power of the base B of the machine so that the
+                                                                EMAX-P
+          largest coefficient in magnitude approaches THRESH = B      .
+          Users should note that no scaling is performed if the largest
+          coefficient in magnitude exceeds THRESH, even if SCALE = .TRUE..
+          (For definition of B, EMAX and P see the Chapter Introduction
+          X02.)
+
+          However, with SCALE = .TRUE., overflow may be encountered when
+          the input coefficients a ,a ,a ,...,a  vary widely in magnitude,
+                                  0  1  2      n
+                                                    (4*P)
+          particularly on those machines for which B      overflows. In
+          such cases, SCALE should be set to .FALSE. and the coefficients
+          scaled so that the largest coefficient in magnitude does not
+                  (EMAX-2*P)
+          exceed B          .
+
+          Even so, the scaling strategy used in C02AGF is sometimes
+          insufficient to avoid overflow and/or underflow conditions. In
+          such cases, the user is recommended to scale the independent
+          variable (z) so that the disparity between the largest and
+          smallest coefficient in magnitude is reduced. That is, use the
+          routine to locate the zeros of the polynomial d*P(cz) for some
+          suitable values of c and d. For example, if the original
+                               -100  100 20                   -10
+          polynomial was P(z)=2    +2   z  , then choosing c=2    and
+             100                                                     20
+          d=2   , for instance, would yield the scaled polynomial 1+z  ,
+          which is well-behaved relative to overflow and underflow and has
+                           10
+          zeros which are 2   times those of P(z).
+
+          If the routine fails with IFAIL = 2 or 3, then the real and
+          imaginary parts of any roots obtained before the failure occurred
+          are stored in Z in the reverse order in which they were found.
+          Let n  denote the number of roots found before the failure
+               R
+          occurred. Then Z(1,n) and Z(2,n) contain the real and imaginary
+          parts of the 1st root found, Z(1,n-1) and Z(2,n-1) contain the
+          real and imaginary parts of the 2nd root found, ..., Z(1,n ) and
+                                                                    R
+          Z(2,n ) contain the real and imaginary parts of the n th root
+               R                                               R
+          found. After the failure has occurred, the remaining 2*(n-n )
+                                                                     R
+          elements of Z contain a large negative number (equal to
+                         
+
+          -1/(X02AMF().\/2)).
+
+          9. Example
+
+          To find the roots of the 5th degree polynomial
+           5   4   3   2
+          z +2z +3z +4z +5z+6=0.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc05}{NAG On-line Documentation: c05}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C05(3NAG)         Foundation Library (12/10/92)         C05(3NAG)
+
+
+
+          C05 -- Roots of One or More Transcendental Equations
+                                                         Introduction -- C05
+                                    Chapter C05
+                   Roots of One or More Transcendental Equations
+
+          1. Scope of the Chapter
+
+          This chapter is concerned with the calculation of real zeros of
+          continuous real functions of one or more variables. (Complex
+          equations must be expressed in terms of the equivalent larger
+          system of real equations.)
+
+          2. Background to the Problems
+
+          The chapter divides naturally into two parts.
+
+          2.1. A Single Equation
+
+          The first deals with the real zeros of a real function of a
+          single variable f(x).
+
+          At present, there is only one routine with a simple calling
+          sequence. This routine assumes that the user can determine an
+          initial interval [a,b] within which the desired zero lies, that
+          is f(a)*f(b)<0, and outside which all other zeros lie. The
+          routine then systematically subdivides the interval to produce a
+          final interval containing the zero. This final interval has a
+          length bounded by the user's specified error requirements; the
+          end of the interval where the function has smallest magnitude is
+          returned as the zero. This routine is guaranteed to converge to a
+          simple zero of the function. (Here we define a simple zero as a
+          zero corresponding to a sign-change of the function.) The
+          algorithm used is due to Bus and Dekker.
+
+          2.2. Systems of Equations
+
+          The routines in the second part of this chapter are designed to
+          solve a set of nonlinear equations in n unknowns
+
+
+                                                           T
+                   f (x)=0,  i=1,2,...,n,  x=(x ,x ,...,x )             (1)
+                    i                          1  2      n
+
+          where T stands for transpose.
+
+          It is assumed that the functions are continuous and
+          differentiable so that the matrix of first partial derivatives of
+          the functions, the Jacobian matrix J  (x)=ddf /ddx  evaluated at
+                                              ij       i    j
+          the point x, exists, though it may not be possible to calculate
+          it directly.
+
+          The functions f  must be independent, otherwise there will be an
+                         i
+          infinity of solutions and the methods will fail. However, even
+          when the functions are independent the solutions may not be
+          unique. Since the methods are iterative, an initial guess at the
+          solution has to be supplied, and the solution located will
+          usually be the one closest to this initial guess.
+
+          2.3. References
+
+          [1]   Gill P E and Murray W (1976) Algorithms for the Solution of
+                the Nonlinear Least-squares Problem. NAC 71 National
+                Physical Laboratory.
+
+          [2]   More J J, Garbow B S and Hillstrom K E (1974) User Guide for
+                Minpack-1. ANL-80-74 Argonne National Laboratory.
+
+          [3]   Ortega J M and Rheinboldt W C (1970) Iterative Solution of
+                Nonlinear Equations in Several Variables. Academic Press.
+
+          [4]   Rabinowitz P (1970) Numerical Methods for Nonlinear
+                Algebraic Equations. Gordon and Breach.
+
+          3. Recommendations on Choice and Use of Routines
+
+          3.1. Zeros of Functions of One Variable
+
+          There is only one routine (C05ADF) for solving a single nonlinear
+          equation. This routine is designed for solving problems where the
+          function f(x) whose zero is to be calculated, can be coded as a
+          user-supplied routine.
+
+          C05ADF may only be used when the user can supply an interval
+          [a,b] containing the zero, that is f(a)*f(b)<0.
+
+          3.2. Solution of Sets of Nonlinear Equations
+
+          The solution of a set of nonlinear equations
+
+                        f (x ,x ,...,x )=0,  i=1,2,...,n                (2)
+                         i  1  2      n
+
+          can be regarded as a special case of the problem of finding a
+          minimum of a sum of squares
+
+                           m
+                           /                    2
+                     s(x)= |  [f (x ,x ,...,x )]   (m>=n).              (3)
+                           /    i  1  2      n
+                           i=1
+
+          So the routines in Chapter E04 of the Library are relevant as
+          well as the special nonlinear equations routines.
+
+          There are two routines (C05NBF and C05PBF) for solving a set of
+          nonlinear equations. These routines require the f  (and possibly
+                                                           i
+          their derivatives) to be calculated in user-supplied routines.
+          These should be set up carefully so the Library routines can work
+          as efficiently as possible.
+
+          The main decision which has to be made by the user is whether to
+                                  ddf
+                                     i
+          supply the derivatives  ----. It is advisable to do so if
+                                  ddx
+                                     j
+          possible, since the results obtained by algorithms which use
+          derivatives are generally more reliable than those obtained by
+          algorithms which do not use derivatives.
+
+          C05PBF requires the user to provide the derivatives, whilst
+          C05NBF does not. C05NBF and C05PBF are easy-to-use routines. A
+          routine, C05ZAF, is provided for use in conjunction with C05PBF
+          to check the user-provided derivatives for consistency with the
+          functions themselves. The user is strongly advised to make use of
+          this routine whenever C05PBF is used.
+
+          Firstly, the calculation of the functions and their derivatives
+          should be ordered so that cancellation errors are avoided. This
+          is particularly important in a routine that uses these quantities
+          to build up estimates of higher derivatives.
+
+          Secondly, scaling of the variables has a considerable effect on
+          the efficiency of a routine. The problem should be designed so
+          that the elements of x are of similar magnitude. The same comment
+          applies to the functions, all the f  should be of comparable
+                                             i
+          size.
+
+          The accuracy is usually determined by the accuracy parameters of
+          the routines, but the following points may be useful:
+
+          (i)   Greater accuracy in the solution may be requested by
+                choosing smaller input values for the accuracy parameters.
+                However, if unreasonable accuracy is demanded, rounding
+                errors may become important and cause a failure.
+
+          (ii)  Some idea of the accuracies of the x  may be obtained by
+                                                    i
+                monitoring the progress of the routine to see how many
+                figures remain unchanged during the last few iterations.
+
+          (iii) An approximation to the error in the solution x, given by e
+                where e is the solution to the set of linear equations
+
+                J(x)e=-f(x)
+
+                                                  T
+                where f(x)=(f (x),f (x),...,f (x))  (see Chapter F04).
+                             1     2         n
+
+          (iv)  If the functions f (x) are changed by small amounts
+                                  i
+                (epsilon) , for i=1,2,...,n, then the corresponding change
+                         i
+                in the solution x is given approximately by (sigma), where
+                (sigma) is the solution of the set of linear equations
+
+                J(x)(sigma)=-(epsilon), (see Chapter F04).
+
+                Thus one can estimate the sensitivity of x to any
+                uncertainties in the specification of f (x), for
+                                                       i
+                i=1,2,...,n.
+
+          3.3. Index
+
+          Zeros of functions of one variable:
+               Bus and Dekker algorithm                              C05ADF
+          Zeros of functions of several variables:
+               easy-to-use                                           C05NBF
+               easy-to-use, derivatives required                     C05PBF
+          Checking Routine:
+               Checks user-supplied Jacobian                         C05ZAF
+
+
+          C05 -- Roots of One or More Transcendental Equations
+                                                             Contents -- C05
+          Chapter C05
+
+          Roots of One or More Transcendental Equations
+
+          C05ADF  Zero of continuous function in given interval, Bus and
+                  Dekker algorithm
+
+          C05NBF  Solution of system of nonlinear equations using function
+                  values only
+
+          C05PBF  Solution of system of nonlinear equations using 1st
+                  derivatives
+
+          C05ZAF  Check user's routine for calculating 1st derivatives
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc05adf}{NAG On-line Documentation: c05adf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C05ADF(3NAG)      Foundation Library (12/10/92)      C05ADF(3NAG)
+
+
+
+          C05 -- Roots of One or More Transcendental Equations       C05ADF
+                  C05ADF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C05ADF locates a zero of a continuous function in a given
+          interval by a combination of the methods of linear interpolation,
+          extrapolation and bisection.
+
+          2. Specification
+
+                 SUBROUTINE C05ADF (A, B, EPS, ETA, F, X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION A, B, EPS, ETA, F, X
+                 EXTERNAL         F
+
+          3. Description
+
+          The routine attempts to obtain an approximation to a simple zero
+          of the function f(x) given an initial interval [a,b] such that
+          f(a)*f(b)<=0. The zero is found by calls to C05AZF(*) whose
+          specification should be consulted for details of the method used.
+
+          The approximation x to the zero (alpha) is determined so that one
+          or both of the following criteria are satisfied:
+
+               (i) |x-(alpha)|<EPS,
+
+               (ii) |f(x)|<ETA.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  A -- DOUBLE PRECISION                                  Input
+               On entry: the lower bound of the interval, a.
+
+           2:  B -- DOUBLE PRECISION                                  Input
+               On entry: the upper bound of the interval, b. Constraint: B
+               /= A.
+
+           3:  EPS -- DOUBLE PRECISION                                Input
+               On entry: the absolute tolerance to which the zero is
+               required (see Section 3). Constraint: EPS > 0.0.
+
+           4:  ETA -- DOUBLE PRECISION                                Input
+               On entry: a value such that if |f(x)|<ETA, x is accepted as
+               the zero. ETA may be specified as 0.0 (see Section 7).
+
+           5:  F -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                                    External Procedure
+               F must evaluate the function f whose zero is to be
+               determined.
+
+               Its specification is:
+
+                      DOUBLE PRECISION FUNCTION F (XX)
+                      DOUBLE PRECISION XX
+
+                1:  XX -- DOUBLE PRECISION                            Input
+                    On entry: the point at which the function must be
+                    evaluated.
+               F must be declared as EXTERNAL in the (sub)program from
+               which C05ADF is called. Parameters denoted as Input
+               must not be changed by this procedure.
+
+           6:  X -- DOUBLE PRECISION                                 Output
+               On exit: the approximation to the zero.
+
+           7:  IFAIL -- INTEGER                                Input/Output
+               Before entry, IFAIL must be assigned a value. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               Unless the routine detects an error (see Section 6), IFAIL
+               contains 0 on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry EPS <= 0.0,
+
+               or       A = B,
+
+               or       F(A)*F(B)>0.0.
+
+          IFAIL= 2
+               Too much accuracy has been requested in the computation,
+               that is, EPS is too small for the computer being used. The
+               final value of X is an accurate approximation to the zero.
+
+          IFAIL= 3
+               A change in sign of f(x) has been determined as occurring
+               near the point defined by the final value of X. However,
+               there is some evidence that this sign-change corresponds to
+               a pole of f(x).
+
+          IFAIL= 4
+               Indicates that a serious error has occurred in C05AZF(*).
+               Check all routine calls. Seek expert help.
+
+          7. Accuracy
+
+          This depends on the value of EPS and ETA. If full machine
+          accuracy is required, they may be set very small, resulting in an
+          error exit with IFAIL = 2, although this may involve more
+          iterations than a lesser accuracy. The user is recommended to set
+          ETA = 0.0 and to use EPS to control the accuracy, unless he has
+          considerable knowledge of the size of f(x) for values of x near
+          the zero.
+
+          8. Further Comments
+
+          The time taken by the routine depends primarily on the time spent
+          evaluating F (see Section 5).
+
+          If it is important to determine an interval of length less than
+          EPS containing the zero, or if the function F is expensive to
+          evaluate and the number of calls to F is to be restricted, then
+          use of C05AZF(*) is recommended. Use of C05AZF(*) is also
+          recommended when the structure of the problem to be solved does
+          not permit a simple function F to be written: the reverse
+          communication facilities of C05AZF(*) are more flexible than the
+          direct communication of F required by C05ADF.
+
+          9. Example
+
+                                                            -x
+          The example program below calculates the zero of e  -x within the
+          interval [0,1] to approximately 5 decimal places.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc05nbf}{NAG On-line Documentation: c05nbf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C05NBF(3NAG)      Foundation Library (12/10/92)      C05NBF(3NAG)
+
+
+
+          C05 -- Roots of One or More Transcendental Equations       C05NBF
+                  C05NBF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C05NBF is an easy-to-use routine to find a solution of a system
+          of nonlinear equations by a modification of the Powell hybrid
+          method.
+
+          2. Specification
+
+                 SUBROUTINE C05NBF (FCN, N, X, FVEC, XTOL, WA, LWA, IFAIL)
+                 INTEGER          N, LWA, IFAIL
+                 DOUBLE PRECISION X(N), FVEC(N), XTOL, WA(LWA)
+                 EXTERNAL         FCN
+
+          3. Description
+
+          The system of equations is defined as:
+
+                        f (x ,x ,...,x )=0, for i=1,2,...,n.
+                         i  1  2      n
+
+          C05NBF is based upon the MINPACK routine HYBRD1 (More et al [1]).
+          It chooses the correction at each step as a convex combination of
+          the Newton and scaled gradient directions. Under reasonable
+          conditions this guarantees global convergence for starting points
+          far from the solution and a fast rate of convergence. The
+          Jacobian is updated by the rank-1 method of Broyden. At the
+          starting point the Jacobian is approximated by forward
+          differences, but these are not used again until the rank-1 method
+          fails to produce satisfactory progress. For more details see
+          Powell [2].
+
+          4. References
+
+          [1]   More J J, Garbow B S and Hillstrom K E User Guide for
+                MINPACK-1. Technical Report ANL-80-74. Argonne National
+                Laboratory.
+
+          [2]   Powell M J D (1970) A Hybrid Method for Nonlinear Algebraic
+                Equations. Numerical Methods for Nonlinear Algebraic
+                Equations. (ed P Rabinowitz) Gordon and Breach.
+
+          5. Parameters
+
+           1:  FCN -- SUBROUTINE, supplied by the user.
+                                                    External Procedure
+               FCN must return the values of the functions f  at a point x.
+                                                            i
+
+               Its specification is:
+
+                      SUBROUTINE FCN (N, X, FVEC, IFLAG)
+                      INTEGER          N, IFLAG
+                      DOUBLE PRECISION X(N), FVEC(N)
+
+                1:  N -- INTEGER                                      Input
+                    On entry: the number of equations, n.
+
+                2:  X(N) -- DOUBLE PRECISION array                    Input
+                    On entry: the components of the point x at which the
+                    functions must be evaluated.
+
+                3:  FVEC(N) -- DOUBLE PRECISION array                Output
+                    On exit: the function values f (x) (unless IFLAG is
+                                                  i
+                    set to a negative value by FCN).
+
+                4:  IFLAG -- INTEGER                           Input/Output
+                    On entry: IFLAG > 0. On exit: in general, IFLAG should
+                    not be reset by FCN. If, however, the user wishes to
+                    terminate execution (perhaps because some illegal point
+                    X has been reached), then IFLAG should be set to a
+                    negative integer. This value will be returned through
+                    IFAIL.
+               FCN must be declared as EXTERNAL in the (sub)program
+               from which C05NBF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of equations, n. Constraint: N > 0.
+
+           3:  X(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: an initial guess at the solution vector. On
+               exit: the final estimate of the solution vector.
+
+           4:  FVEC(N) -- DOUBLE PRECISION array                     Output
+               On exit: the function values at the final point, X.
+
+           5:  XTOL -- DOUBLE PRECISION                               Input
+               On entry: the accuracy in X to which the solution is
+               required. Suggested value: the square root of the machine
+               precision. Constraint: XTOL >= 0.0.
+
+           6:  WA(LWA) -- DOUBLE PRECISION array                  Workspace
+
+           7:  LWA -- INTEGER                                         Input
+               On entry: the dimension of the array WA. Constraint:
+               LWA>=N*(3*N+13)/2.
+
+           8:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL< 0
+               The user has set IFLAG negative in FCN. The value of IFAIL
+               will be the same as the user's setting of IFLAG.
+
+          IFAIL= 1
+               On entry N <= 0,
+
+               or       XTOL < 0.0,
+
+               or       LWA<N*(3*N+13)/2.
+
+          IFAIL= 2
+               There have been at least 200*(N+1) evaluations of FCN.
+               Consider restarting the calculation from the final point
+               held in X.
+
+          IFAIL= 3
+               No further improvement in the approximate solution X is
+               possible; XTOL is too small.
+
+          IFAIL= 4
+               The iteration is not making good progress. This failure exit
+               may indicate that the system does not have a zero, or that
+               the solution is very close to the origin (see Section 7).
+               Otherwise, rerunning C05NBF from a different starting point
+               may avoid the region of difficulty.
+
+          7. Accuracy
+
+             ^
+          If x is the true solution, C05NBF tries to ensure that
+
+                                   ^             ^
+                              ||x-x||<=XTOL*||x||.
+
+                                                     -k
+          If this condition is satisfied with XTOL=10  , then the larger
+          components of x have k significant decimal digits. There is a
+          danger that the smaller components of x may have large relative
+          errors, but the fast rate of convergence of C05NBF usually avoids
+          this possibility.
+
+          If XTOL is less than machine precision, and the above test is
+          satisfied with the machine precision in place of XTOL, then the
+          routine exits with IFAIL = 3.
+
+          Note: this convergence test is based purely on relative error,
+          and may not indicate convergence if the solution is very close to
+          the origin.
+
+          The test assumes that the functions are reasonably well behaved.
+          If this condition is not satisfied, then C05NBF may incorrectly
+          indicate convergence. The validity of the answer can be checked,
+          for example, by rerunning C05NBF with a tighter tolerance.
+
+          8. Further Comments
+
+          The time required by C05NBF to solve a given problem depends on n
+          , the behaviour of the functions, the accuracy requested and the
+          starting point. The number of arithmetic operations executed by
+                                                            2
+          C05NBF to process each call of FCN is about 11.5*n . Unless FCN
+          can be evaluated quickly, the timing of C05NBF will be strongly
+          influenced by the time spent in FCN.
+
+          Ideally the problem should be scaled so that at the solution the
+          function values are of comparable magnitude.
+
+          9. Example
+
+          To determine the values x ,..., x  which satisfy the tridiagonal
+                                   1       9
+          equations:
+
+                                  (3-2x )x -2x =-1
+                                       1  1   2
+
+                       -x -1+(3-2x )x -2x   =-1,  i=2,3,...,8
+                         i        i  i   i+1
+
+                                  -x +(3-2x )x =-1.
+                                    8      9  9
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc05pbf}{NAG On-line Documentation: c05pbf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C05PBF(3NAG)      Foundation Library (12/10/92)      C05PBF(3NAG)
+
+
+
+          C05 -- Roots of One or More Transcendental Equations       C05PBF
+                  C05PBF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C05PBF is an easy-to-use routine to find a solution of a system
+          of nonlinear equations by a modification of the Powell hybrid
+          method. The user must provide the Jacobian.
+
+          2. Specification
+
+                 SUBROUTINE C05PBF (FCN, N, X, FVEC, FJAC, LDFJAC, XTOL,
+                1                   WA, LWA, IFAIL)
+                 INTEGER          N, LDFJAC, LWA, IFAIL
+                 DOUBLE PRECISION X(N), FVEC(N), FJAC(LDFJAC,N), XTOL, WA
+                1                 (LWA)
+                 EXTERNAL         FCN
+
+          3. Description
+
+          The system of equations is defined as:
+
+                          f (x ,x ,...,x )=0, i=1,2,...,n.
+                           i  1  2      n
+
+          C05PBF is based upon the MINPACK routine HYBRJ1 (More et al [1]).
+          It chooses the correction at each step as a convex combination of
+          the Newton and scaled gradient directions. Under reasonable
+          conditions this guarantees global convergence for starting points
+          far from the solution and a fast rate of convergence. The
+          Jacobian is updated by the rank-1 method of Broyden. At the
+          starting point the Jacobian is calculated, but it is not
+          recalculated until the rank-1 method fails to produce
+          satisfactory progress. For more details see Powell [2].
+
+          4. References
+
+          [1]   More J J, Garbow B S and Hillstrom K E User Guide for
+                MINPACK-1. Technical Report ANL-80-74. Argonne National
+                Laboratory.
+
+          [2]   Powell M J D (1970) A Hybrid Method for Nonlinear Algebraic
+                Equations. Numerical Methods for Nonlinear Algebraic
+                Equations. (ed P Rabinowitz) Gordon and Breach.
+
+          5. Parameters
+
+           1:  FCN -- SUBROUTINE, supplied by the user.
+                                                    External Procedure
+               Depending upon the value of IFLAG, FCN must either return
+               the values of the functions f  at a point x or return the
+                                            i
+               Jacobian at x.
+
+               Its specification is:
+
+                      SUBROUTINE FCN (N, X, FVEC, FJAC, LDFJAC, IFLAG)
+                      INTEGER          N, LDFJAC, IFLAG
+                      DOUBLE PRECISION X(N), FVEC(N), FJAC(LDFJAC,N)
+
+                1:  N -- INTEGER                                      Input
+                    On entry: the number of equations, n.
+
+                2:  X(N) -- DOUBLE PRECISION array                    Input
+                    On entry: the components of the point x at which the
+                    functions or the Jacobian must be evaluated.
+
+                3:  FVEC(N) -- DOUBLE PRECISION array                Output
+                    On exit: if IFLAG = 1 on entry, FVEC must contain the
+                    function values f (x) (unless IFLAG is set to a
+                                     i
+                    negative value by FCN). If IFLAG = 2 on entry, FVEC
+                    must not be changed.
+
+                4:  FJAC(LDFJAC,N) -- DOUBLE PRECISION array         Output
+                    On exit: if IFLAG = 2 on entry, FJAC(i,j) must contain
+                                  ddf
+                                     i
+                    the value of  ---- at the point x, for i,j=1,2,...,n
+                                  ddx
+                                     j
+                    (unless IFLAG is set to a negative value by FCN).
+
+                    If IFLAG = 1 on entry, FJAC must not be changed.
+
+                5:  LDFJAC -- INTEGER                                 Input
+                    On entry: the first dimension of FJAC.
+
+                6:  IFLAG -- INTEGER                           Input/Output
+                    On entry: IFLAG = 1 or 2:
+                         if IFLAG = 1, FVEC is to be updated;
+
+                         if IFLAG = 2, FJAC is to be updated.
+                    On exit: in general, IFLAG should not be reset by FCN.
+                    If, however, the user wishes to terminate execution
+                    (perhaps because some illegal point x has been reached)
+                    then IFLAG should be set to a negative integer. This
+                    value will be returned through IFAIL.
+               FCN must be declared as EXTERNAL in the (sub)program
+               from which C05PBF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of equations, n. Constraint: N > 0.
+
+           3:  X(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: an initial guess at the solution vector. On
+               exit: the final estimate of the solution vector.
+
+           4:  FVEC(N) -- DOUBLE PRECISION array                     Output
+               On exit: the function values at the final point, X.
+
+           5:  FJAC(LDFJAC,N) -- DOUBLE PRECISION array              Output
+               On exit: the orthogonal matrix Q produced by the QR
+               factorization of the final approximate Jacobian.
+
+           6:  LDFJAC -- INTEGER                                      Input
+               On entry:
+               the first dimension of the array FJAC as declared in the
+               (sub)program from which C05PBF is called.
+               Constraint: LDFJAC >= N.
+
+           7:  XTOL -- DOUBLE PRECISION                               Input
+               On entry: the accuracy in X to which the solution is
+               required. Suggested value: the square root of the machine
+               precision. Constraint: XTOL >= 0.0.
+
+           8:  WA(LWA) -- DOUBLE PRECISION array                  Workspace
+
+           9:  LWA -- INTEGER                                         Input
+               On entry: the dimension of the array WA. Constraint:
+               LWA>=N*(N+13)/2.
+
+          10:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL< 0
+               A negative value of IFAIL indicates an exit from C05PBF
+               because the user has set IFLAG negative in FCN. The value of
+               IFAIL will be the same as the user's setting of IFLAG.
+
+          IFAIL= 1
+               On entry N <= 0,
+
+               or       LDFJAC < N,
+
+               or       XTOL < 0.0,
+
+               or       LWA<N*(N+13)/2.
+
+          IFAIL= 2
+               There have been 100*(N+1) evaluations of the functions.
+               Consider restarting the calculation from the final point
+               held in X.
+
+          IFAIL= 3
+               No further improvement in the approximate solution X is
+               possible; XTOL is too small.
+
+          IFAIL= 4
+               The iteration is not making good progress. This failure exit
+               may indicate that the system does not have a zero or that
+               the solution is very close to the origin (see Section 7).
+               Otherwise, rerunning C05PBF from a different starting point
+               may avoid the region of difficulty.
+
+          7. Accuracy
+
+             ^
+          If x is the true solution, C05PBF tries to ensure that
+
+                                  ^              ^
+                             ||x-x|| <=XTOL*||x|| .
+                                    2            2
+
+                                                     -k
+          If this condition is satisfied with XTOL=10  , then the larger
+          components of x have k significant decimal digits. There is a
+          danger that the smaller components of x may have large relative
+          errors, but the fast rate of convergence of C05PBF usually avoids
+          the possibility.
+
+          If XTOL is less than machine precision and the above test is
+          satisfied with the machine precision in place of XTOL, then the
+          routine exits with IFAIL = 3.
+
+          Note: this convergence test is based purely on relative error,
+          and may not indicate convergence if the solution is very close to
+          the origin.
+
+          The test assumes that the functions and Jacobian are coded
+          consistently and that the functions are reasonably well behaved.
+          If these conditions are not satisfied then C05PBF may incorrectly
+          indicate convergence. The coding of the Jacobian can be checked
+          using C05ZAF. If the Jacobian is coded correctly, then the
+          validity of the answer can be checked by rerunning C05PBF with a
+          tighter tolerance.
+
+          8. Further Comments
+
+          The time required by C05PBF to solve a given problem depends on n
+          , the behaviour of the functions, the accuracy requested and the
+          starting point. The number of arithmetic operations executed by
+                                2
+          C05PBF is about 11.5*n  to process each evaluation of the
+                                   3
+          functions and about 1.3*n  to process each evaluation of the
+          Jacobian. Unless FCN can be evaluated quickly, the timing of
+          C05PBF will be strongly influenced by the time spent in FCN.
+
+          Ideally the problem should be scaled so that, at the solution,
+          the function values are of comparable magnitude.
+
+          9. Example
+
+          To determine the values x ,..., x  which satisfy the tridiagonal
+                                   1       9
+          equations:
+
+                                  (3-2x )x -2x =-1
+                                       1  1   2
+
+                        -x   +(3-2x )x -2x   =-1,  i=2,3,...,8.
+                          i-1      i  i   i+1
+
+                                  -x +(3-2x )x =-1.
+                                    8      9  9
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc05zaf}{NAG On-line Documentation: c05zaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C05ZAF(3NAG)      Foundation Library (12/10/92)      C05ZAF(3NAG)
+
+
+
+          C05 -- Roots of One or More Transcendental Equations       C05ZAF
+                  C05ZAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C05ZAF checks the user-provided gradients of a set of non-linear
+          functions in several variables, for consistency with the
+          functions themselves. The routine must be called twice.
+
+          2. Specification
+
+                 SUBROUTINE C05ZAF (M, N, X, FVEC, FJAC, LDFJAC, XP, FVECP,
+                1                   MODE, ERR)
+                 INTEGER          M, N, LDFJAC, MODE
+                 DOUBLE PRECISION X(N), FVEC(M), FJAC(LDFJAC,N), XP(N),
+                1                 FVECP(M), ERR(M)
+
+          3. Description
+
+          C05ZAF is based upon the MINPACK routine CHKDER (More et al [1]).
+          It checks the ith gradient for consistency with the ith function
+          by computing a forward-difference approximation along a suitably
+          chosen direction and comparing this approximation with the user-
+          supplied gradient along the same direction. The principal
+          characteristic of C05ZAF is its invariance under changes in scale
+          of the variables or functions.
+
+          4. References
+
+          [1]   More J J, Garbow B S and Hillstrom K E User Guide for
+                MINPACK-1. Technical Report ANL-80-74. Argonne National
+                Laboratory.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number of functions.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of variables. For use with C05PBF and
+               C05PCF(*), M = N.
+
+           3:  X(N) -- DOUBLE PRECISION array                         Input
+               On entry: the components of a point x, at which the
+               consistency check is to be made. (See Section 8.)
+
+           4:  FVEC(M) -- DOUBLE PRECISION array                      Input
+               On entry: when MODE = 2, FVEC must contain the functions
+               evaluated at x.
+
+           5:  FJAC(LDFJAC,N) -- DOUBLE PRECISION array               Input
+               On entry: when MODE = 2, FJAC must contain the user-
+               supplied gradients. (The ith row of FJAC must contain the
+               gradient of the ith function evaluated at the point x.)
+
+           6:  LDFJAC -- INTEGER                                      Input
+               On entry:
+               the first dimension of the array FJAC as declared in the
+               (sub)program from which C05ZAF is called.
+               Constraint: LDFJAC >= M.
+
+           7:  XP(N) -- DOUBLE PRECISION array                       Output
+               On exit: when MODE = 1, XP is set to a neighbouring point
+               to X.
+
+           8:  FVECP(M) -- DOUBLE PRECISION array                     Input
+               On entry: when MODE = 2, FVECP must contain the functions
+               evaluated at XP.
+
+           9:  MODE -- INTEGER                                        Input
+               On entry: the value 1 on the first call and the value 2 on
+               the second call of C05ZAF.
+
+          10:  ERR(M) -- DOUBLE PRECISION array                      Output
+               On exit: when MODE = 2, ERR contains measures of
+               correctness of the respective gradients. If there is no loss
+               of significance (see Section 8), then if ERR(i) is 1.0 the i
+               th user-supplied gradient is correct, whilst if ERR(i) is 0.
+               0 the ith gradient is incorrect. For values of ERR(i)
+               between 0.0 and 1.0 the categorisation is less certain. In
+               general, a value of ERR(i)>0.5 indicates that the ith
+               gradient is probably correct.
+
+          6. Error Indicators and Warnings
+
+          None.
+
+          7. Accuracy
+
+          See below.
+
+          8. Further Comments
+
+          The time required by C05ZAF increases with M and N.
+
+          C05ZAF does not perform reliably if cancellation or rounding
+          errors cause a severe loss of significance in the evaluation of a
+          function. Therefore, none of the components of x should be
+          unusually small (in particular, zero) or any other value which
+          may cause loss of significance. The relative differences between
+          corresponding elements of FVECP and FVEC should be at least two
+          orders of magnitude greater than the machine precision.
+
+          9. Example
+
+          This example checks the Jacobian matrix for a problem with 15
+          functions of 3 variables. The results indicate that the first 7
+          gradients are probably incorrect (this is caused by a deliberate
+          error in the code to calculate the Jacobian).
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc06}{NAG On-line Documentation: c06}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C06(3NAG)         Foundation Library (12/10/92)         C06(3NAG)
+
+
+
+          C06 -- Summation of Series                    Introduction -- C06
+                                    Chapter C06
+                                Summation of Series
+
+          1. Scope of the Chapter
+
+          This chapter is concerned with calculating the discrete Fourier
+          transform of a sequence of real or complex data values, and
+          applying it to calculate convolutions and correlations.
+
+          2. Background to the Problems
+
+          2.1. Discrete Fourier Transforms
+
+          2.1.1.  Complex transforms
+
+          Most of the routines in this chapter calculate the finite
+          discrete Fourier transform  (DFT) of a sequence of n complex
+          numbers z , for j=0,1,...,n-1. The transform is defined by:
+                   j
+
+                                  n-1
+                          ^    1  --      (   2(pi)jk)
+                          z = --- >  z exp(-i -------)                  (1)
+                           k      --  j   (      n   )
+                              \/n j=0
+
+          for k=0,1,...,n-1. Note that equation (1) makes sense for all
+                                             ^
+          integral k and with this extension z  is periodic with period n,
+                                              k
+               ^  ^                        ^   ^
+          i.e. z =z    , and in particular z  =z   .
+                k  k+-n                     -k  n-k
+
+                                    ^                                 ^
+          If we write z =x +iy  and z =a +ib , then the definition of z
+                       j  j   j      k  k   k                          k
+          may be written in terms of sines and cosines as:
+
+                            n-1
+                         1  -- (     ( 2(pi)jk)      ( 2(pi)jk))
+                    a = --- >  (x cos( -------)+y sin( -------))
+                     k      -- ( j   (    n   )  j   (    n   ))
+                        \/n j=0
+
+                            n-1
+                         1  -- (     ( 2(pi)jk)      ( 2(pi)jk))
+                    b = --- >  (y cos( -------)-x sin( -------)).
+                     k      -- ( j   (    n   )  j   (    n   ))
+                        \/n j=0
+
+          The original data values z  may conversely be recovered from the
+                                    j
+                    ^
+          transform z  by an  inverse discrete Fourier transform:
+                     k
+
+
+                                  n-1
+                               1  -- ^    (   2(pi)jk)
+                          z = --- >  z exp(+i -------)                  (2)
+                           j      --  k   (      n   )
+                              \/n k=0
+
+          for j=0,1,...,n-1. If we take the complex conjugate of (2), we
+                                                               
+
+                                                               ^
+          find that the sequence z  is the DFT of the sequence z . Hence
+                                  j                             k
+                                          ^
+          the inverse DFT of the sequence z  may be obtained by: taking the
+                                           k
+                                    ^
+          complex conjugates of the z ; performing a DFT; and taking the
+                                     k
+          complex conjugates of the result.
+
+          Notes: definitions of the discrete Fourier transform vary.
+          Sometimes (2) is used as the definition of the DFT, and (1) as
+                                                                      
+
+          the definition of the inverse. Also the scale-factor of 1/\/n may
+          be omitted in the definition of the DFT, and replaced by 1/n in
+          the definition of the inverse.
+
+          2.1.2. Real transforms
+
+          If the original sequence is purely real valued, i.e. z =x , then
+                                                                j  j
+
+                                        n-1
+                         ^           1  --      (   2(pi)jk)
+                         z =a +ib = --- >  x exp(-i -------)
+                          k  k   k      --  j   (      n   )
+                                    \/n j=0
+
+              ^                                ^
+          and z    is the complex conjugate of z . Thus the DFT of a real
+               n-k                              k
+          sequence is a particular type of complex sequence, called a
+          Hermitian sequence, or  half-complex or  conjugate symmetric with
+          the properties:
+                         a   =a  b   =-b  b =0 and, if n is even, b   =0.
+                          n-k  k  n-k   k  0                       n/2
+
+          Thus a Hermitian sequence of n complex data values can be
+          represented by only n, rather than 2n, independent real values.
+          This can obviously lead to economies in storage, the following
+          scheme being used in this chapter: the real parts a  for
+                                                             k
+          0<=k<=n/2 are stored in normal order in the first n/2+1 locations
+          of an array X of length n; the corresponding non-zero imaginary
+          parts are stored in reverse order in the remaining locations of
+          X. In other words, if X is declared with bounds (0:n-1) in the
+                                                               ^
+          user's (sub)program, the real and imaginary parts of z  are
+                                                                k
+          stored as follows:
+
+                         if n=2s    if n=2s-1
+
+              X(0)       a          a
+                          0          0
+
+              X(1)       a          a
+                          1          1
+
+              X(2)       a          a
+                          2          2
+
+              .          .          .
+
+              .          .          .
+
+              .          .          .
+
+              X(s-1)     a          a
+                          s-1        s-1
+
+              X(s)       a          b
+                          s          s-1
+
+              X(s+1)     b          b
+                          s-1        s-2
+
+              .          .          .
+
+              .          .          .
+
+              .          .          .
+
+              X(n-2)     b          b
+                          2          2
+
+              X(n-1)     b          b
+                          1          1
+
+
+                         (     n/2-1                                      )
+                       1 (     --   (     ( 2(pi)jk)      ( 2(pi)jk))     )
+          Hence   x = ---(a +2 >    (a cos( -------)-b sin( -------))+a   )
+                   j     ( 0   --   ( k   (    n   )  k   (    n   ))  n/2)
+                      \/n(     k=0                                        )
+
+          where a    = 0 if n is odd.
+                 n/2
+
+          2.1.3.  Fourier integral transforms
+
+          The usual application of the discrete Fourier transform is that
+          of obtaining an approximation of the  Fourier integral transform
+
+
+                                +infty
+                                /
+                          F(s)= |     f(t)exp(-i2(pi)st)dt
+                                /
+                                -infty
+
+          when f(t) is negligible outside some region (0,c). Dividing the
+          region into n equal intervals we have
+
+                                    n-1
+                                  c --
+                           F(s)~= - >  f exp(-i2(pi)sjc/n)
+                                  n --  j
+                                    j=0
+
+          and so
+
+                                   n-1
+                                 c --
+                            F ~= - >  f exp(-i2(pi)jk/n)
+                             k   n --  j
+                                   j=0
+
+          for k=0,1,...,n-1, where f =f(jc/n) and F =F(k/c).
+                                    j              k
+
+          Hence the discrete Fourier transform gives an approximation to
+          the Fourier integral transform in the region s=0 to s=n/c.
+
+          If the function f(t) is defined over some more general interval
+          (a,b), then the integral transform can still be approximated by
+          the discrete transform provided a shift is applied to move the
+          point a to the origin.
+
+          2.1.4.  Convolutions and correlations
+
+          One of the most important applications of the discrete Fourier
+          transform is to the computation of the discrete convolution or
+          correlation of two vectors x and y defined (as in Brigham [1])
+          by:
+
+                                n-1
+                                --
+               convolution: z = >  x y
+                             k  --  j k-j
+                                j=0
+
+                                n-1
+                                -- 
+               correlation: w = >  x y
+                             k  --  j k+j
+                                j=0
+
+          (Here x and y are assumed to be periodic with period n.)
+
+          Under certain circumstances (see Brigham [1]) these can be used
+          as approximations to the convolution or correlation integrals
+          defined by:
+
+                                    +infty
+                                    /
+                              z(s)= |     x(t)y(s-t)dt
+                                    /
+                                    -infty
+
+          and
+
+                          +infty
+                          /     
+                    w(s)= |     x(t)y(s+t)dt,   -infty<s<+infty.
+                          /
+                          -infty
+
+          For more general advice on the use of Fourier transforms, see
+          Hamming [2]; more detailed information on the fast Fourier
+          transform algorithm can be found in Van Loan [3] and Brigham [1].
+
+          2.2. References
+
+          [1]   Brigham E O (1973) The Fast Fourier Transform. Prentice-
+                Hall.
+
+          [2]   Hamming R W (1962) Numerical Methods for Scientists and
+                Engineers. McGraw-Hill.
+
+          [3]   Van Loan C (1992) Computational Frameworks for the Fast
+                Fourier Transform. SIAM Philadelphia.
+
+          3. Recommendations on Choice and Use of Routines
+
+          3.1. One-dimensional Fourier Transforms
+
+          The choice of routine is determined first of all by whether the
+          data values constitute a real, Hermitian or general complex
+          sequence. It is wasteful of time and storage to use an
+          inappropriate routine.
+
+          Two groups, each of three routines, are provided
+
+                                   Group 1      Group 2
+
+              Real sequences       C06EAF       C06FPF
+
+              Hermitian sequences  C06EBF       C06FQF
+
+              General complex      C06ECF       C06FRF
+              sequences
+
+          Group 1 routines each compute a single transform of length n,
+          without requiring any extra working storage. The Group 1 routines
+          impose some restrictions on the value of n, namely that no prime
+          factor of n may exceed 19 and the total number of prime factors
+          (including repetitions) may not exceed 20 (though the latter
+                                                     6
+          restriction only becomes relevant when n>10 ).
+
+          Group 2 routines are designed to perform several transforms in a
+          single call, all with the same value of n. They do however
+          require more working storage. Even on scalar processors, they may
+          be somewhat faster than repeated calls to Group 1 routines
+          because of reduced overheads and because they pre-compute and
+          store the required values of trigonometric functions. Group 2
+          routines impose no practical restrictions on the value of n;
+          however the fast Fourier transform algorithm ceases to be 'fast'
+          if applied to values of n which cannot be expressed as a product
+          of small prime factors. All the above routines are particularly
+          efficient if the only prime factors of n are 2, 3 or 5.
+
+          If extensive use is to be made of these routines, users who are
+          concerned about efficiency are advised to conduct their own
+          timing tests.
+
+          To compute inverse discrete Fourier transforms the above routines
+          should be used in conjunction with the utility routines C06GBF,
+          C06GCF and C06GQF which form the complex conjugate of a Hermitian
+          or general sequence of complex data values.
+
+          3.2. Multi-dimensional Fourier Transforms
+
+          C06FUF computes a 2-dimensional discrete Fourier transform of a
+          2-dimensional sequence of complex data values. This is defined by
+
+                         n -1 n -1
+                          1    2          (   2(pi)j k )   (   2(pi)j k )
+          ^         1    --   --          (         1 1)   (         2 2)
+          z    = ------- >    >   z    exp(-i ---------)exp(-i ---------).
+                         --   --          (      n     )   (      n     )
+           k k     /n n  j =0 j =0 j j    (       1    )   (       2    )
+            1 2  \/  1 2  1    2    1 2
+
+          3.3. Convolution and Correlation
+
+          C06EKF computes either the discrete convolution or the discrete
+          correlation of two real vectors.
+
+          3.4. Index
+
+          Complex conjugate,
+               complex sequence                                      C06GCF
+               Hermitian sequence                                    C06GBF
+               multiple Hermitian sequences                          C06GQF
+          Complex sequence from Hermitian sequences                  C06GSF
+          Convolution or Correlation
+               real vectors                                          C06EKF
+          Discrete Fourier Transform
+               two-dimensional
+                    complex sequence                                 C06FUF
+               one-dimensional, multiple transforms
+                    complex sequence                                 C06FRF
+                    Hermitian sequence                               C06FQF
+                    real sequence                                    C06FPF
+               one-dimensional, single transforms
+                    complex sequence                                 C06ECF
+                    Hermitian sequence                               C06EBF
+                    real sequence                                    C06EAF
+
+
+
+          C06 -- Summation of Series                        Contents -- C06
+          Chapter C06
+
+          Summation of Series
+
+          C06EAF  Single 1-D real discrete Fourier transform, no extra
+                  workspace
+
+          C06EBF  Single 1-D Hermitian discrete Fourier transform, no extra
+                  workspace
+
+          C06ECF  Single 1-D complex discrete Fourier transform, no extra
+                  workspace
+
+          C06EKF  Circular convolution or correlation of two real vectors,
+                  no extra workspace
+
+          C06FPF  Multiple 1-D real discrete Fourier transforms
+
+          C06FQF  Multiple 1-D Hermitian discrete Fourier transforms
+
+          C06FRF  Multiple 1-D complex discrete Fourier transforms
+
+          C06FUF  2-D complex discrete Fourier transform
+
+          C06GBF  Complex conjugate of Hermitian sequence
+
+          C06GCF  Complex conjugate of complex sequence
+
+          C06GQF  Complex conjugate of multiple Hermitian sequences
+
+          C06GSF  Convert Hermitian sequences to general complex sequences
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc06eaf}{NAG On-line Documentation: c06eaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C06EAF(3NAG)      Foundation Library (12/10/92)      C06EAF(3NAG)
+
+
+
+          C06 -- Summation of Series                                 C06EAF
+                  C06EAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C06EAF calculates the discrete Fourier transform of a sequence of
+          n real data values. (No extra workspace required.)
+
+          2. Specification
+
+                 SUBROUTINE C06EAF (X, N, IFAIL)
+                 INTEGER          N, IFAIL
+                 DOUBLE PRECISION X(N)
+
+          3. Description
+
+          Given a sequence of n real data values x , for j=0,1,...,n-1,
+                                                  j
+          this routine calculates their discrete Fourier transform defined
+          by:
+
+                            n-1
+                    ^    1  --       (   2(pi)jk)
+                    z = --- >  x *exp(-i -------), k=0,1,...,n-1.
+                     k      --  j    (      n   )
+                        \/n j=0
+
+                                      1
+          (Note the scale factor of  --- in this definition.) The
+                                       
+
+                                     \/n
+                             ^
+          transformed values z  are complex, but they form a Hermitian
+                              k
+                          ^                                ^
+          sequence (i.e., z    is the complex conjugate of z ), so they are
+                           n-k                              k
+          completely determined by n real numbers (see also the Chapter
+          Introduction).
+
+          To compute the inverse discrete Fourier transform defined by:
+
+                                   n-1
+                           ^    1  --       (   2(pi)jk)
+                           w = --- >  x *exp(+i -------),
+                            k      --  j    (      n   )
+                               \/n j=0
+
+          this routine should be followed by a call of C06GBF to form the
+                                    ^
+          complex conjugates of the z .
+                                     k
+
+          The routine uses the fast Fourier transform (FFT) algorithm
+          (Brigham [1]). There are some restrictions on the value of n (see
+          Section 5).
+
+          4. References
+
+          [1]   Brigham E O (1973) The Fast Fourier Transform. Prentice-
+                Hall.
+
+          5. Parameters
+
+           1:  X(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: if X is declared with bounds (0:N-1) in the (sub)
+               program from which C06EAF is called, then X(j) must contain
+               x , for j=0,1,...,n-1. On exit: the discrete Fourier
+                j
+               transform stored in Hermitian form. If the components of the
+                         ^
+               transform z  are written as a +ib , and if X is declared
+                          k                 k   k
+               with bounds (0:N-1) in the (sub)program from which C06EAF is
+               called, then for 0<=k<=n/2, a  is contained in X(k), and for
+                                            k
+               1<=k<=(n-1)/2, b  is contained in X(n-k). (See also Section
+                               k
+               2.1.2 of the Chapter Introduction, and the Example Program.)
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of data values, n. The largest prime
+               factor of N must not exceed 19, and the total number of
+               prime factors of N, counting repetitions, must not exceed
+               20. Constraint: N > 1.
+
+           3:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               At least one of the prime factors of N is greater than 19.
+
+          IFAIL= 2
+               N has more than 20 prime factors.
+
+          IFAIL= 3
+               N <= 1.
+
+          7. Accuracy
+
+          Some indication of accuracy can be obtained by performing a
+          subsequent inverse transform and comparing the results with the
+          original sequence (in exact arithmetic they would be identical).
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          n*logn, but also depends on the factorization of n. The routine
+          is somewhat faster than average if the only prime factors of n
+          are 2, 3 or 5; and fastest of all if n is a power of 2.
+
+          On the other hand, the routine is particularly slow if n has
+          several unpaired prime factors, i.e., if the 'square-free' part
+          of n has several factors. For such values of n, routine C06FAF(*)
+          (which requires an additional n elements of workspace) is
+          considerably faster.
+
+          9. Example
+
+          This program reads in a sequence of real data values, and prints
+          their discrete Fourier transform (as computed by C06EAF), after
+          expanding it from Hermitian form into a full complex sequence.
+
+          It then performs an inverse transform using C06GBF and C06EBF,
+          and prints the sequence so obtained alongside the original data
+          values.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc06ebf}{NAG On-line Documentation: c06ebf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C06EBF(3NAG)      Foundation Library (12/10/92)      C06EBF(3NAG)
+
+
+
+          C06 -- Summation of Series                                 C06EBF
+                  C06EBF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C06EBF calculates the discrete Fourier transform of a Hermitian
+          sequence of n complex data values. (No extra workspace required.)
+
+          2. Specification
+
+                 SUBROUTINE C06EBF (X, N, IFAIL)
+                 INTEGER          N, IFAIL
+                 DOUBLE PRECISION X(N)
+
+          3. Description
+
+          Given a Hermitian sequence of n complex data values z  (i.e., a
+                                                               j
+          sequence such that z  is real and z    is the complex conjugate
+                              0              n-j
+          of z , for j=1,2,...,n-1) this routine calculates their discrete
+              j
+          Fourier transform defined by:
+
+                            n-1
+                    ^    1  --       (   2(pi)jk)
+                    x = --- >  z *exp(-i -------), k=0,1,...,n-1.
+                     k      --  j    (      n   )
+                        \/n j=0
+
+                                      1
+          (Note the scale factor of  --- in this definition.) The
+                                       
+
+                                     \/n
+                             ^
+          transformed values x  are purely real (see also the the Chapter
+                              k
+          Introduction).
+
+          To compute the inverse discrete Fourier transform defined by:
+
+                                   n-1
+                           ^    1  --       (   2(pi)jk)
+                           y = --- >  z *exp(+i -------),
+                            k      --  j    (      n   )
+                               \/n j=0
+
+          this routine should be preceded by a call of C06GBF to form the
+          complex conjugates of the z .
+                                     j
+
+          The routine uses the fast Fourier transform (FFT) algorithm
+          (Brigham [1]). There are some restrictions on the value of n (see
+          Section 5).
+
+          4. References
+
+          [1]   Brigham E O (1973) The Fast Fourier Transform. Prentice-
+                Hall.
+
+          5. Parameters
+
+           1:  X(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: the sequence to be transformed stored in
+               Hermitian form. If the data values z  are written as x +iy ,
+                                                   j                 j   j
+               and if X is declared with bounds (0:N-1) in the subroutine
+               from which C06EBF is called, then for 0<=j<=n/2, x  is
+                                                                 j
+               contained in X(j), and for 1<=j<=(n-1)/2, y  is contained in
+                                                          j
+               X(n-j). (See also Section 2.1.2 of the Chapter Introduction
+               and the Example Program.) On exit: the components of the
+                                          ^
+               discrete Fourier transform x . If X is declared with bounds
+                                           k
+               (0:N-1) in the (sub)program from which C06EBF is called,
+                    ^
+               then x  is stored in X(k), for k=0,1,...,n-1.
+                     k
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of data values, n. The largest prime
+               factor of N must not exceed 19, and the total number of
+               prime factors of N, counting repetitions, must not exceed
+               20. Constraint: N > 1.
+
+           3:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               At least one of the prime factors of N is greater than 19.
+
+          IFAIL= 2
+               N has more than 20 prime factors.
+
+          IFAIL= 3
+               N <= 1.
+
+          7. Accuracy
+
+          Some indication of accuracy can be obtained by performing a
+          subsequent inverse transform and comparing the results with the
+          original sequence (in exact arithmetic they would be identical).
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          n*logn, but also depends on the factorization of n. The routine
+          is somewhat faster than average if the only prime factors of n
+          are 2, 3 or 5; and fastest of all if n is a power of 2.
+
+          On the other hand, the routine is particularly slow if n has
+          several unpaired prime factors, i.e., if the 'square-free' part
+          of n has several factors. For such values of n, routine C06FBF(*)
+          (which requires an additional n elements of workspace) is
+          considerably faster.
+
+          9. Example
+
+          This program reads in a sequence of real data values which is
+          assumed to be a Hermitian sequence of complex data values stored
+          in Hermitian form. The input sequence is expanded into a full
+          complex sequence and printed alongside the original sequence. The
+          discrete Fourier transform (as computed by C06EBF) is printed
+          out.
+
+          The program then performs an inverse transform using C06EAF and
+          C06GBF, and prints the sequence so obtained alongside the
+          original data values.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc06ecf}{NAG On-line Documentation: c06ecf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C06ECF(3NAG)      Foundation Library (12/10/92)      C06ECF(3NAG)
+
+
+
+          C06 -- Summation of Series                                 C06ECF
+                  C06ECF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C06ECF calculates the discrete Fourier transform of a sequence of
+          n complex data values. (No extra workspace required.)
+
+          2. Specification
+
+                 SUBROUTINE C06ECF (X, Y, N, IFAIL)
+                 INTEGER          N, IFAIL
+                 DOUBLE PRECISION X(N), Y(N)
+
+          3. Description
+
+          Given a sequence of n complex data values z , for j=0,1,...,n-1,
+                                                     j
+          this routine calculates their discrete Fourier transform defined
+          by:
+
+                            n-1
+                    ^    1  --       (   2(pi)jk)
+                    z = --- >  z *exp(-i -------), k=0,1,...,n-1.
+                     k      --  j    (      n   )
+                        \/n j=0
+
+                                      1
+          (Note the scale factor of  --- in this definition.)
+                                       
+
+                                     \/n
+
+          To compute the inverse discrete Fourier transform defined by:
+
+                                   n-1
+                           ^    1  --       (   2(pi)jk)
+                           w = --- >  z *exp(+i -------),
+                            k      --  j    (      n   )
+                               \/n j=0
+
+          this routine should be preceded and followed by calls of C06GCF
+                                                           ^
+          to form the complex conjugates of the z  and the z .
+                                                 j          k
+
+          The routine uses the fast Fourier transform (FFT) algorithm
+          (Brigham [1]). There are some restrictions on the value of n (see
+          Section 5).
+
+          4. References
+
+          [1]   Brigham E O (1973) The Fast Fourier Transform. Prentice-
+                Hall.
+
+          5. Parameters
+
+           1:  X(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: if X is declared with bounds (0:N-1) in the (sub)
+               program from which C06ECF is called, then X(j) must contain
+               x , the real part of z , for j=0,1,...,n-1. On exit: the
+                j                    j
+               real parts a  of the components of the discrete Fourier
+                           k
+               transform. If X is declared with bounds (0:N-1) in the (sub)
+               program from which C06ECF is called, then a  is contained in
+                                                          k
+               X(k), for k=0,1,...,n-1.
+
+           2:  Y(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: if Y is declared with bounds (0:N-1) in the (sub)
+               program from which C06ECF is called, then Y(j) must contain
+               y , the imaginary part of z , for j=0,1,...,n-1. On exit:
+                j                         j
+               the imaginary parts b  of the components of the discrete
+                                    k
+               Fourier transform. If Y is declared with bounds (0:N-1) in
+               the (sub)program from which C06ECF is called, then b  is
+                                                                   k
+               contained in Y(k), for k=0,1,...,n-1.
+
+           3:  N -- INTEGER                                           Input
+               On entry: the number of data values, n. The largest prime
+               factor of N must not exceed 19, and the total number of
+               prime factors of N, counting repetitions, must not exceed
+               20. Constraint: N > 1.
+
+           4:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               At least one of the prime factors of N is greater than 19.
+
+          IFAIL= 2
+               N has more than 20 prime factors.
+
+          IFAIL= 3
+               N <= 1.
+
+          7. Accuracy
+
+          Some indication of accuracy can be obtained by performing a
+          subsequent inverse transform and comparing the results with the
+          original sequence (in exact arithmetic they would be identical).
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          n*logn, but also depends on the factorization of n. The routine
+          is somewhat faster than average if the only prime factors of n
+          are 2, 3 or 5; and fastest of all if n is a power of 2.
+
+          On the other hand, the routine is particularly slow if n has
+          several unpaired prime factors, i.e., if the 'square-free' part
+          of n has several factors. For such values of n, routine C06FCF(*)
+          (which requires an additional n real elements of workspace) is
+          considerably faster.
+
+          9. Example
+
+          This program reads in a sequence of complex data values and
+          prints their discrete Fourier transform.
+
+          It then performs an inverse transform using C06GCF and C06ECF,
+          and prints the sequence so obtained alongside the original data
+          values.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc06ekf}{NAG On-line Documentation: c06ekf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C06EKF(3NAG)      Foundation Library (12/10/92)      C06EKF(3NAG)
+
+
+
+          C06 -- Summation of Series                                 C06EKF
+                  C06EKF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C06EKF calculates the circular convolution or correlation of two
+          real vectors of period n. No extra workspace is required.
+
+          2. Specification
+
+                 SUBROUTINE C06EKF (JOB, X, Y, N, IFAIL)
+                 INTEGER          JOB, N, IFAIL
+                 DOUBLE PRECISION X(N), Y(N)
+
+          3. Description
+
+          This routine computes:
+
+          if JOB =1, the discrete convolution of x and y, defined by:
+
+                                  n-1        n-1
+                                  --         --
+                              z = >  x y   = >  x   y ;
+                               k  --  j k-j  --  k-j j
+                                  j=0        j=0
+
+          if JOB =2, the discrete correlation of x and y defined by:
+
+                                       n-1
+                                       --
+                                   w = >  x y   .
+                                    k  --  j k+j
+                                       j=0
+
+          Here x and y are real vectors, assumed to be periodic, with
+          period n, i.e., x =x    =x     =...; z and w are then also
+                           j  j+-n  j+-2n
+          periodic with period n.
+
+          Note: this usage of the terms 'convolution' and 'correlation' is
+          taken from Brigham [1]. The term 'convolution' is sometimes used
+          to denote both these computations.
+
+             ^  ^  ^     ^
+          If x, y, z and w are the discrete Fourier transforms of these
+          sequences,
+
+                        n-1
+                ^    1  --       (   2(pi)jk)
+          i.e., x = --- >  x *exp(-i -------), etc,
+                 k      --  j    (      n   )
+                    \/n j=0
+
+                ^      ^ ^
+          then  z =\/n.x y
+                 k      k k
+
+                       
+
+                ^      ^ ^
+          and   w =\/n.x y
+                 k      k k
+
+          (the bar denoting complex conjugate).
+
+          This routine calls the same auxiliary routines as C06EAF and
+          C06EBF to compute discrete Fourier transforms, and there are some
+          restrictions on the value of n.
+
+          4. References
+
+          [1]   Brigham E O (1973) The Fast Fourier Transform. Prentice-
+                Hall.
+
+          5. Parameters
+
+           1:  JOB -- INTEGER                                         Input
+               On entry: the computation to be performed:
+                                    n-1
+                                    --
+                    if JOB = 1, z = >  x y    (convolution);
+                                 k  --  j k-j
+                                    j=0
+
+                                    n-1
+                                    --
+                    if JOB = 2, w = >  x y    (correlation).
+                                 k  --  j k+j
+                                    j=0
+               Constraint: JOB = 1 or 2.
+
+           2:  X(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: the elements of one period of the vector x. If X
+               is declared with bounds (0:N-1) in the (sub)program from
+               which C06EKF is called, then X(j) must contain x , for
+                                                               j
+               j=0,1,...,n-1. On exit: the corresponding elements of the
+               discrete convolution or correlation.
+
+           3:  Y(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: the elements of one period of the vector y. If Y
+               is declared with bounds (0:N-1) in the (sub)program from
+               which C06EKF is called, then Y(j) must contain y , for
+                                                               j
+               j=0,1,...,n-1. On exit: the discrete Fourier transform of
+               the convolution or correlation returned in the array X; the
+               transform is stored in Hermitian form, exactly as described
+               in the document  C06EAF.
+
+           4:  N -- INTEGER                                           Input
+               On entry: the number of values, n, in one period of the
+               vectors X and Y. The largest prime factor of N must not
+               exceed 19, and the total number of prime factors of N,
+               counting repetitions, must not exceed 20. Constraint: N > 1.
+
+           5:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               At least one of the prime factors of N is greater than 19.
+
+          IFAIL= 2
+               N has more than 20 prime factors.
+
+          IFAIL= 3
+               N <= 1.
+
+          IFAIL= 4
+               JOB /= 1 or 2.
+
+          7. Accuracy
+
+          The results should be accurate to within a small multiple of the
+          machine precision.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          n*logn, but also depends on the factorization of n. The routine
+          is faster than average if the only prime factors are 2, 3 or 5;
+          and fastest of all if n is a power of 2.
+
+          The routine is particularly slow if n has several unpaired prime
+          factors, i.e., if the 'square free' part of n has several
+          factors. For such values of n, routine C06FKF(*) is considerably
+          faster (but requires an additional workspace of n elements).
+
+          9. Example
+
+          This program reads in the elements of one period of two real
+          vectors x and y and prints their discrete convolution and
+          correlation (as computed by C06EKF). In realistic computations
+          the number of data values would be much larger.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc06fpf}{NAG On-line Documentation: c06fpf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C06FPF(3NAG)      Foundation Library (12/10/92)      C06FPF(3NAG)
+
+
+
+          C06 -- Summation of Series                                 C06FPF
+                  C06FPF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C06FPF computes the discrete Fourier transforms of m sequences,
+          each containing n real data values. This routine is designed to
+          be particularly efficient on vector processors.
+
+          2. Specification
+
+                 SUBROUTINE C06FPF (M, N, X, INIT, TRIG, WORK, IFAIL)
+                 INTEGER          M, N, IFAIL
+                 DOUBLE PRECISION X(M*N), TRIG(2*N), WORK(M*N)
+                 CHARACTER*1      INIT
+
+          3. Description
+
+                                                   p
+          Given m sequences of n real data values x , for j=0,1,...,n-1;
+                                                   j
+          p=1,2,...,m, this routine simultaneously calculates the Fourier
+          transforms of all the sequences defined by:
+
+                     n-1
+             ^p   1  --  p    (   2(pi)jk)
+             z = --- >  x *exp(-i -------), k=0,1,...,n-1; p=1,2,...,m.
+              k      --  j    (      n   )
+                 \/n j=0
+
+                                   1
+          (Note the scale factor  --- in this definition.)
+                                    
+
+                                  \/n
+
+                                 ^p
+          The transformed values z  are complex, but for each value of p
+                                  k
+              ^p                                 ^p
+          the z  form a Hermitian sequence (i.e.,z    is the complex
+               k                                  n-k
+                       ^p
+          conjugate of z ), so they are completely determined by mn real
+                        k
+          numbers (see also the Chapter Introduction).
+
+          The discrete Fourier transform is sometimes defined using a
+          positive sign in the exponential term:
+
+                                   n-1
+                           ^p   1  --  p    (   2(pi)jk)
+                           z = --- >  x *exp(+i -------).
+                            k      --  j    (      n   )
+                               \/n j=0
+
+          To compute this form, this routine should be followed by a call
+                                                          ^p
+          to C06GQF to form the complex conjugates of the z .
+                                                           k
+
+          The routine uses a variant of the fast Fourier transform (FFT)
+          algorithm (Brigham [1]) known as the Stockham self-sorting
+          algorithm, which is described in Temperton [2]. Special coding is
+          provided for the factors 2, 3, 4, 5 and 6. This routine is
+          designed to be particularly efficient on vector processors, and
+          it becomes especially fast as M, the number of transforms to be
+          computed in parallel, increases.
+
+          4. References
+
+          [1]   Brigham E O (1973) The Fast Fourier Transform. Prentice-
+                Hall.
+
+          [2]   Temperton C (1983) Fast Mixed-Radix Real Fourier Transforms.
+                J. Comput. Phys. 52 340--350.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number of sequences to be transformed, m.
+               Constraint: M >= 1.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of real values in each sequence, n.
+               Constraint: N >= 1.
+
+           3:  X(M,N) -- DOUBLE PRECISION array                Input/Output
+               On entry: the data must be stored in X as if in a two-
+               dimensional array of dimension (1:M,0:N-1); each of the m
+               sequences is stored in a row of the array. In other words,
+               if the data values of the pth sequence to be transformed are
+                           p
+               denoted by x , for j=0,1,...,n-1, then the mn elements of
+                           j
+               the array X must contain the values
+                    1  2      m   1  2       m      1    2        m
+                   x ,x ,...,x , x ,x ,..., x ,...,x   ,x   ,...,x   .
+                    0  0      0   1  1       1      n-1  n-1      n-1
+               On exit: the m discrete Fourier transforms stored as if in
+               a two-dimensional array of dimension (1:M,0:N-1). Each of
+               the m transforms is stored in a row of the array in
+               Hermitian form, overwriting the corresponding original
+               sequence. If the n components of the discrete Fourier
+                         ^p                 p   p                       p
+               transform z  are written as a +ib , then for 0<=k<=n/2, a
+                          k                 k   k                       k
+                                                               p
+               is contained in X(p,k), and for 1<=k<=(n-1)/2, b  is
+                                                               k
+               contained in X(p,n-k). (See also Section 2.1.2 of the
+               Chapter Introduction.)
+
+           4:  INIT -- CHARACTER*1                                    Input
+               On entry: if the trigonometric coefficients required to
+               compute the transforms are to be calculated by the routine
+               and stored in the array TRIG, then INIT must be set equal to
+               'I' (Initial call).
+
+               If INIT contains 'S' (Subsequent call), then the routine
+               assumes that trigonometric coefficients for the specified
+               value of n are supplied in the array TRIG, having been
+               calculated in a previous call to one of C06FPF, C06FQF or
+               C06FRF.
+
+               If INIT contains 'R' (Restart then the routine assumes that
+               trigonometric coefficients for the particular value of n are
+               supplied in the array TRIG, but does not check that C06FPF,
+               C06FQF or C06FRF have previously been called. This option
+               allows the TRIG array to be stored in an external file, read
+               in and re-used without the need for a call with INIT equal
+               to 'I'. The routine carries out a simple test to check that
+               the current value of n is consistent with the array TRIG.
+               Constraint: INIT = 'I', 'S' or 'R'.
+
+           5:  TRIG(2*N) -- DOUBLE PRECISION array             Input/Output
+               On entry: if INIT = 'S' or 'R', TRIG must contain the
+               required coefficients calculated in a previous call of the
+               routine. Otherwise TRIG need not be set. On exit: TRIG
+               contains the required coefficients (computed by the routine
+               if INIT = 'I').
+
+           6:  WORK(M*N) -- DOUBLE PRECISION array                Workspace
+
+           7:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M < 1.
+
+          IFAIL= 2
+               N < 1.
+
+          IFAIL= 3
+               INIT is not one of 'I', 'S' or 'R'.
+
+          IFAIL= 4
+               INIT = 'S', but none of C06FPF, C06FQF or C06FRF has
+               previously been called.
+
+          IFAIL= 5
+               INIT = 'S' or 'R', but the array TRIG and the current value
+               of N are inconsistent.
+
+          7. Accuracy
+
+          Some indication of accuracy can be obtained by performing a
+          subsequent inverse transform and comparing the results with the
+          original sequence (in exact arithmetic they would be identical).
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          nm*logn, but also depends on the factors of n. The routine is
+          fastest if the only prime factors of n are 2, 3 and 5, and is
+          particularly slow if n is a large prime, or has large prime
+          factors.
+
+          9. Example
+
+          This program reads in sequences of real data values and prints
+          their discrete Fourier transforms (as computed by C06FPF). The
+          Fourier transforms are expanded into full complex form using
+          C06GSF and printed. Inverse transforms are then calculated by
+          calling C06GQF followed by C06FQF showing that the original
+          sequences are restored.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc06fqf}{NAG On-line Documentation: c06fqf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C06FQF(3NAG)      Foundation Library (12/10/92)      C06FQF(3NAG)
+
+
+
+          C06 -- Summation of Series                                 C06FQF
+                  C06FQF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C06FQF computes the discrete Fourier transforms of m Hermitian
+          sequences, each containing n complex data values. This routine is
+          designed to be particularly efficient on vector processors.
+
+          2. Specification
+
+                 SUBROUTINE C06FQF (M, N, X, INIT, TRIG, WORK, IFAIL)
+                 INTEGER          M, N, IFAIL
+                 DOUBLE PRECISION X(M*N), TRIG(2*N), WORK(M*N)
+                 CHARACTER*1      INIT
+
+          3. Description
+
+                                                                p
+          Given m Hermitian sequences of n complex data values z , for
+                                                                j
+          j=0,1,...,n-1; p=1,2,...,m, this routine simultaneously
+          calculates the Fourier transforms of all the sequences defined
+          by:
+
+                     n-1
+             ^p   1  --  p    (   2(pi)jk)
+             x = --- >  z *exp(-i -------), k=0,1,...,n-1; p=1,2,...,m.
+              k      --  j    (      n   )
+                 \/n j=0
+
+                                   1
+          (Note the scale factor  --- in this definition.)
+                                    
+
+                                  \/n
+
+          The transformed values are purely real (see also the Chapter
+          Introduction).
+
+          The discrete Fourier transform is sometimes defined using a
+          positive sign in the exponential term
+
+                                   n-1
+                           ^p   1  --  p    (   2(pi)jk)
+                           x = --- >  z *exp(+i -------).
+                            k      --  j    (      n   )
+                               \/n j=0
+
+          To compute this form, this routine should be preceded by a call
+                                                          ^p
+          to C06GQF to form the complex conjugates of the z .
+                                                           j
+
+          The routine uses a variant of the fast Fourier transform (FFT)
+          algorithm (Brigham [1]) known as the Stockham self-sorting
+          algorithm, which is described in Temperton [2]. Special code is
+          included for the factors 2, 3, 4, 5 and 6. This routine is
+          designed to be particularly efficient on vector processors, and
+          it becomes especially fast as m, the number of transforms to be
+          computed in parallel, increases.
+
+          4. References
+
+          [1]   Brigham E O (1973) The Fast Fourier Transform. Prentice-
+                Hall.
+
+          [2]   Temperton C (1983) Fast Mixed-Radix Real Fourier Transforms.
+                J. Comput. Phys. 52 340--350.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number of sequences to be transformed, m.
+               Constraint: M >= 1.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of data values in each sequence, n.
+               Constraint: N >= 1.
+
+           3:  X(M,N) -- DOUBLE PRECISION array                Input/Output
+               On entry: the data must be stored in X as if in a two-
+               dimensional array of dimension (1:M,0:N-1); each of the m
+               sequences is stored in a row of the array in Hermitian form.
+                                     p                 p   p
+               If the n data values z  are written as x +iy , then for
+                                     j                 j   j
+                           p
+               0<=j<=n/2, x  is contained in X(p,j), and for 1<=j<=(n-1)/2,
+                           j
+                p
+               y  is contained in X(p,n-j). (See also Section 2.1.2 of the
+                j
+               Chapter Introduction.) On exit: the components of the m
+               discrete Fourier transforms, stored as if in a two-
+               dimensional array of dimension (1:M,0:N-1). Each of the m
+               transforms is stored as a row of the array, overwriting the
+               corresponding original sequence. If the n components of the
+                                                         ^p
+               discrete Fourier transform are denoted by x , for
+                                                          k
+               k=0,1,...,n-1, then the mn elements of the array X contain
+               the values
+                   ^1 ^2     ^m  ^1 ^2      ^m     ^1   ^2       ^m
+                   x ,x ,...,x , x ,x ,..., x ,...,x   ,x   ,...,x   .
+                    0  0      0   1  1       1      n-1  n-1      n-1
+
+           4:  INIT -- CHARACTER*1                                    Input
+               On entry: if the trigonometric coefficients required to
+               compute the transforms are to be calculated by the routine
+               and stored in the array TRIG, then INIT must be set equal to
+               'I' (Initial call).
+
+               If INIT contains 'S' (Subsequent call), then the routine
+               assumes that trigonometric coefficients for the specified
+               value of n are supplied in the array TRIG, having been
+               calculated in a previous call to one of C06FPF, C06FQF or
+               C06FRF.
+
+               If INIT contains 'R' (Restart), then the routine assumes
+               that trigonometric coefficients for the particular value of
+               N are supplied in the array TRIG, but does not check that
+               C06FPF, C06FQF or C06FRF have previously been called. This
+               option allows the TRIG array to be stored in an external
+               file, read in and re-used without the need for a call with
+               INIT equal to 'I'. The routine carries out a simple test to
+               check that the current value of n is compatible with the
+               array TRIG. Constraint: INIT = 'I', 'S' or 'R'.
+
+           5:  TRIG(2*N) -- DOUBLE PRECISION array             Input/Output
+               On entry: if INIT = 'S' or 'R', TRIG must contain the
+               required coefficients calculated in a previous call of the
+               routine. Otherwise TRIG need not be set. On exit: TRIG
+               contains the required coefficients (computed by the routine
+               if INIT = 'I').
+
+           6:  WORK(M*N) -- DOUBLE PRECISION array                Workspace
+
+           7:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M < 1.
+
+          IFAIL= 2
+               On entry N < 1.
+
+          IFAIL= 3
+               On entry INIT is not one of 'I', 'S' or 'R'.
+
+          IFAIL= 4
+               On entry INIT = 'S', but none of C06FPF, C06FQF and C06FRF
+               has previously been called.
+
+          IFAIL= 5
+               On entry INIT = 'S' or 'R', but the array TRIG and the
+               current value of n are inconsistent.
+
+          7. Accuracy
+
+          Some indication of accuracy can be obtained by performing a
+          subsequent inverse transform and comparing the results with the
+          original sequence (in exact arithmetic they would be identical).
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          nm*logn, but also depends on the factors of n. The routine is
+          fastest if the only prime factors of n are 2, 3 and 5, and is
+          particularly slow if n is a large prime, or has large prime
+          factors.
+
+          9. Example
+
+          This program reads in sequences of real data values which are
+          assumed to be Hermitian sequences of complex data stored in
+          Hermitian form. The sequences are expanded into full complex form
+          using C06GSF and printed. The discrete Fourier transforms are
+          then computed (using C06FQF) and printed out. Inverse transforms
+          are then calculated by calling C06FPF followed by C06GQF showing
+          that the original sequences are restored.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc06frf}{NAG On-line Documentation: c06frf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C06FRF(3NAG)      Foundation Library (12/10/92)      C06FRF(3NAG)
+
+
+
+          C06 -- Summation of Series                                 C06FRF
+                  C06FRF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C06FRF computes the discrete Fourier transforms of m sequences,
+          each containing n complex data values. This routine is designed
+          to be particularly efficient on vector processors.
+
+          2. Specification
+
+                 SUBROUTINE C06FRF (M, N, X, Y, INIT, TRIG, WORK, IFAIL)
+                 INTEGER          M, N, IFAIL
+                 DOUBLE PRECISION X(M*N), Y(M*N), TRIG(2*N), WORK(2*M*N)
+                 CHARACTER*1      INIT
+
+          3. Description
+
+                                                      p
+          Given m sequences of n complex data values z , for j=0,1,...,n-1;
+                                                      j
+          p=1,2,...,m, this routine simultaneously calculates the Fourier
+          transforms of all the sequences defined by:
+
+                     n-1
+             ^p   1  --  p    (   2(pi)jk)
+             z = --- >  z *exp(-i -------), k=0,1,...,n-1; p=1,2,...,m.
+              k      --  j    (      n   )
+                 \/n j=0
+
+                                   1
+          (Note the scale factor  --- in this definition.)
+                                    
+
+                                  \/n
+
+          The discrete Fourier transform is sometimes defined using a
+          positive sign in the exponential term
+
+                                   n-1
+                           ^p   1  --  p    (   2(pi)jk)
+                           z = --- >  z *exp(+i -------).
+                            k      --  j    (      n   )
+                               \/n j=0
+
+          To compute this form, this routine should be preceded and
+          followed by a call of C06GCF to form the complex conjugates of
+               p         ^p
+          the z  and the z .
+               j          k
+
+          The routine uses a variant of the fast Fourier transform (FFT)
+          algorithm (Brigham [1]) known as the Stockham self-sorting
+          algorithm, which is described in Temperton [2]. Special code is
+          provided for the factors 2, 3, 4, 5 and 6. This routine is
+          designed to be particularly efficient on vector processors, and
+          it becomes especially fast as m, the number of transforms to be
+          computed in parallel, increases.
+
+          4. References
+
+          [1]   Brigham E O (1973) The Fast Fourier Transform. Prentice-
+                Hall.
+
+          [2]   Temperton C (1983) Self-sorting Mixed-radix Fast Fourier
+                Transforms. J. Comput. Phys. 52 1--23.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number of sequences to be transformed, m.
+               Constraint: M >= 1.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of complex values in each sequence, n.
+               Constraint: N >= 1.
+
+           3:  X(M,N) -- DOUBLE PRECISION array                Input/Output
+
+           4:  Y(M,N) -- DOUBLE PRECISION array                Input/Output
+               On entry: the real and imaginary parts of the complex data
+               must be stored in X and Y respectively as if in a two-
+               dimensional array of dimension (1:M,0:N-1); each of the m
+               sequences is stored in a row of each array. In other words,
+               if the real parts of the pth sequence to be transformed are
+                           p
+               denoted by x , for j=0,1,...,n-1, then the mn elements of
+                           j
+               the array X must contain the values
+                    1  2      m   1  2      m       1    2        m
+                   x ,x ,...,x , x ,x ,...,x ,..., x   ,x   ,...,x   .
+                    0  0      0   1  1      1       n-1  n-1      n-1
+               On exit: X and Y are overwritten by the real and imaginary
+               parts of the complex transforms.
+
+           5:  INIT -- CHARACTER*1                                    Input
+               On entry: if the trigonometric coefficients required to
+               compute the transforms are to be calculated by the routine
+               and stored in the array TRIG, then INIT must be set equal to
+               'I' (Initial call).
+
+               If INIT contains 'S' (Subsequent call), then the routine
+               assumes that trigonometric coefficients for the specified
+               value of n are supplied in the array TRIG, having been
+               calculated in a previous call to one of C06FPF, C06FQF or
+               C06FRF.
+
+               If INIT contains 'R' (Restart) then the routine assumes that
+               trigonometric coefficients for the particular value of n are
+               supplied in the array TRIG, but does not check that C06FPF,
+               C06FQF or C06FRF have previously been called. This option
+               allows the TRIG array to be stored in an external file, read
+               in and re-used without the need for a call with INIT equal
+               to 'I'. The routine carries out a simple test to check that
+               the current value of n is compatible with the array TRIG.
+               Constraint: INIT = 'I', 'S' or 'R'.
+
+           6:  TRIG(2*N) -- DOUBLE PRECISION array             Input/Output
+               On entry: if INIT = 'S' or 'R', TRIG must contain the
+               required coefficients calculated in a previous call of the
+               routine. Otherwise TRIG need not be set. On exit: TRIG
+               contains the required coefficients (computed by the routine
+               if INIT = 'I').
+
+           7:  WORK(2*M*N) -- DOUBLE PRECISION array              Workspace
+
+           8:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M < 1.
+
+          IFAIL= 2
+               On entry N < 1.
+
+          IFAIL= 3
+               On entry INIT is not one of 'I', 'S' or 'R'.
+
+          IFAIL= 4
+               On entry INIT = 'S', but none of C06FPF, C06FQF and C06FRF
+               has previously been called.
+
+          IFAIL= 5
+               On entry INIT = 'S' or 'R', but the array TRIG and the
+               current value of n are inconsistent.
+
+          7. Accuracy
+
+          Some indication of accuracy can be obtained by performing a
+          subsequent inverse transform and comparing the results with the
+          original sequence (in exact arithmetic they would be identical).
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          nm*logn, but also depends on the factors of n. The routine is
+          fastest if the only prime factors of n are 2, 3 and 5, and is
+          particularly slow if n is a large prime, or has large prime
+          factors.
+
+          9. Example
+
+          This program reads in sequences of complex data values and prints
+          their discrete Fourier transforms (as computed by C06FRF).
+          Inverse transforms are then calculated using C06GCF and C06FRF
+          and printed out, showing that the original sequences are
+          restored.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc06fuf}{NAG On-line Documentation: c06fuf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C06FUF(3NAG)      Foundation Library (12/10/92)      C06FUF(3NAG)
+
+
+
+          C06 -- Summation of Series                                 C06FUF
+                  C06FUF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C06FUF computes the two-dimensional discrete Fourier transform of
+          a bivariate sequence of complex data values. This routine is
+          designed to be particularly efficient on vector processors.
+
+          2. Specification
+
+                 SUBROUTINE C06FUF (M, N, X, Y, INIT, TRIGM, TRIGN, WORK,
+                1                   IFAIL)
+                 INTEGER          M, N, IFAIL
+                 DOUBLE PRECISION X(M*N), Y(M*N), TRIGM(2*M), TRIGN(2*N),
+                1                 WORK(2*M*N)
+                 CHARACTER*1      INIT
+
+          3. Description
+
+          This routine computes the two-dimensional discrete Fourier
+          transform of a bivariate sequence of complex data values z    ,
+                                                                    j j
+                                                                     1 2
+          where j =0,1,...,m-1, j =0,1,...,n-1.
+                 1               2
+
+          The discrete Fourier transform is here defined by:
+
+                            m-1  n-1          (       ( j k   j k ))
+                ^       1   --   --           (       (  1 1   2 2))
+                z    = ---- >    >   z    *exp(-2(pi)i( ----+ ----)),
+                            --   --   j j     (       (  m     n  ))
+                 k k   \/mn j =0 j =0  1 2
+                  1 2        1    2
+
+          where k =0,1,...,m-1, k =0,1,...,n-1.
+                 1               2
+
+                                      1
+          (Note the scale factor of  ---- in this definition.)
+                                       
+
+                                     \/mn
+
+          To compute the inverse discrete Fourier transform, defined with
+          exp(+2(pi)i(...)) in the above formula instead of exp(-
+          2(pi)i(...)), this routine should be preceded and followed by
+          calls of C06GCF to form the complex conjugates of the data values
+          and the transform.
+
+          This routine calls C06FRF to perform multiple one-dimensional
+          discrete Fourier transforms by the fast Fourier transform (FFT)
+          algorithm in Brigham [1]. It is designed to be particularly
+          efficient on vector processors.
+
+          4. References
+
+          [1]   Brigham E O (1973) The Fast Fourier Transform. Prentice-
+                Hall.
+
+          [2]   Temperton C (1983) Self-sorting Mixed-radix Fast Fourier
+                Transforms. J. Comput. Phys. 52 1--23.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number of rows, m, of the arrays X and Y.
+               Constraint: M >= 1.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of columns, n, of the arrays X and Y.
+               Constraint: N >= 1.
+
+           3:  X(M,N) -- DOUBLE PRECISION array                Input/Output
+
+           4:  Y(M,N) -- DOUBLE PRECISION array                Input/Output
+               On entry: the real and imaginary parts of the complex data
+               values must be stored in arrays X and Y respectively. If X
+               and Y are regarded as two-dimensional arrays of dimension
+               (0:M-1,0:N-1), then X(j ,j ) and Y(j ,j ) must contain the
+                                      1  2         1  2
+               real and imaginary parts of z    . On exit: the real and
+                                            j j
+                                             1 2
+               imaginary parts respectively of the corresponding elements
+               of the computed transform.
+
+           5:  INIT -- CHARACTER*1                                    Input
+               On entry: if the trigonometric coefficients required to
+               compute the transforms are to be calculated by the routine
+               and stored in the arrays TRIGM and TRIGN, then INIT must be
+               set equal to 'I', (Initial call).
+
+               If INIT contains 'S', (Subsequent call), then the routine
+               assumes that trigonometric coefficients for the specified
+               values of m and n are supplied in the arrays TRIGM and
+               TRIGN, having been calculated in a previous call to the
+               routine.
+
+               If INIT contains 'R', (Restart), then the routine assumes
+               that trigonometric coefficients for the particular values of
+               m and n are supplied in the arrays TRIGM and TRIGN, but does
+               not check that the routine has previously been called. This
+               option allows the TRIGM and TRIGN arrays to be stored in an
+               external file, read in and re-used without the need for a
+               call with INIT equal to 'I'. The routine carries out a
+               simple test to check that the current values of m and n are
+               compatible with the arrays TRIGM and TRIGN. Constraint: INIT
+               = 'I', 'S' or 'R'.
+
+           6:  TRIGM(2*M) -- DOUBLE PRECISION array            Input/Output
+
+           7:  TRIGN(2*N) -- DOUBLE PRECISION array            Input/Output
+               On entry: if INIT = 'S' or 'R',TRIGM and TRIGN must contain
+               the required coefficients calculated in a previous call of
+               the routine. Otherwise TRIGM and TRIGN need not be set.
+
+               If m=n the same array may be supplied for TRIGM and TRIGN.
+               On exit: TRIGM and TRIGN contain the required coefficients
+               (computed by the routine if INIT = 'I').
+
+           8:  WORK(2*M*N) -- DOUBLE PRECISION array              Workspace
+
+           9:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M < 1.
+
+          IFAIL= 2
+               On entry N < 1.
+
+          IFAIL= 3
+               On entry INIT is not one of 'I', 'S' or 'R'.
+
+          IFAIL= 4
+               On entry INIT = 'S', but C06FUF has not previously been
+               called.
+
+          IFAIL= 5
+               On entry INIT = 'S' or 'R', but at least one of the arrays
+               TRIGM and TRIGN is inconsistent with the current value of M
+               or N.
+
+          7. Accuracy
+
+          Some indication of accuracy can be obtained by performing a
+          subsequent inverse transform and comparing the results with the
+          original sequence (in exact arithmetic they would be identical).
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          mn*log(mn), but also depends on the factorization of the
+          individual dimensions m and n. The routine is somewhat faster
+          than average if their only prime factors are 2, 3 or 5; and
+          fastest of all if they are powers of 2.
+
+          9. Example
+
+          This program reads in a bivariate sequence of complex data values
+          and prints the two-dimensional Fourier transform. It then
+          performs an inverse transform and prints the sequence so
+          obtained, which may be compared to the original data values.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc06gbf}{NAG On-line Documentation: c06gbf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C06GBF(3NAG)      Foundation Library (12/10/92)      C06GBF(3NAG)
+
+
+
+          C06 -- Summation of Series                                 C06GBF
+                  C06GBF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C06GBF forms the complex conjugate of a Hermitian sequence of n
+          data values.
+
+          2. Specification
+
+                 SUBROUTINE C06GBF (X, N, IFAIL)
+                 INTEGER          N, IFAIL
+                 DOUBLE PRECISION X(N)
+
+          3. Description
+
+          This is a utility routine for use in conjunction with C06EAF,
+          C06EBF, C06FAF(*) or C06FBF(*) to calculate inverse discrete
+          Fourier transforms (see the Chapter Introduction).
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  X(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: if the data values z  are written as x +iy  and
+                                             j                 j   j
+               if X is declared with bounds (0:N-1) in the (sub)program
+               from which C06GBF is called, then for 0<=j<=n/2, X(j) must
+               contain x (=x   ), while for n/2<j<=n-1, X(j) must contain
+                        j   n-j
+               -y (=y   ). In other words, X must contain the Hermitian
+                 j   n-j
+               sequence in Hermitian form. (See also Section 2.1.2 of the
+               Chapter Introduction). On exit: the imaginary parts y  are
+                                                                    j
+               negated. The real parts x  are not referenced.
+                                        j
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of data values, n. Constraint: N >= 1.
+
+           3:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               N < 1.
+
+          7. Accuracy
+
+          Exact.
+
+          8. Further Comments
+
+          The time taken by the routine is negligible.
+
+          9. Example
+
+          This program reads in a sequence of real data values, calls
+          C06EAF followed by C06GBF to compute their inverse discrete
+          Fourier transform, and prints this after expanding it from
+          Hermitian form into a full complex sequence.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc06gcf}{NAG On-line Documentation: c06gcf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C06GCF(3NAG)      Foundation Library (12/10/92)      C06GCF(3NAG)
+
+
+
+          C06 -- Summation of Series                                 C06GCF
+                  C06GCF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C06GCF forms the complex conjugate of a sequence of n data
+          values.
+
+          2. Specification
+
+                 SUBROUTINE C06GCF (Y, N, IFAIL)
+                 INTEGER          N, IFAIL
+                 DOUBLE PRECISION Y(N)
+
+          3. Description
+
+          This is a utility routine for use in conjunction with C06ECF or
+          C06FCF(*) to calculate inverse discrete Fourier transforms (see
+          the Chapter Introduction).
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  Y(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: if Y is declared with bounds (0:N-1) in the (sub)
+               program which C06GCF is called, then Y(j) must contain the
+               imaginary part of the jth data value, for 0<=j<=n-1. On
+               exit: these values are negated.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of data values, n. Constraint: N >= 1.
+
+           3:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               N < 1.
+
+          7. Accuracy
+
+          Exact.
+
+          8. Further Comments
+
+          The time taken by the routine is negligible.
+
+          9. Example
+
+          This program reads in a sequence of complex data values and
+          prints their inverse discrete Fourier transform as computed by
+          calling C06GCF, followed by C06ECF and C06GCF again.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc06gqf}{NAG On-line Documentation: c06gqf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C06GQF(3NAG)      Foundation Library (12/10/92)      C06GQF(3NAG)
+
+
+
+          C06 -- Summation of Series                                 C06GQF
+                  C06GQF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C06GQF forms the complex conjugates of m Hermitian sequences,
+          each containing n data values.
+
+          2. Specification
+
+                 SUBROUTINE C06GQF (M, N, X, IFAIL)
+                 INTEGER          M, N, IFAIL
+                 DOUBLE PRECISION X(M*N)
+
+          3. Description
+
+          This is a utility routine for use in conjunction with C06FPF and
+          C06FQF to calculate inverse discrete Fourier transforms (see the
+          Chapter Introduction).
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number of Hermitian sequences to be
+               conjugated, m. Constraint: M >= 1.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of data values in each Hermitian
+               sequence, n. Constraint: N >= 1.
+
+           3:  X(M,N) -- DOUBLE PRECISION array                Input/Output
+               On entry: the data must be stored in array X as if in a
+               two-dimensional array of dimension (1:M,0:N-1); each of the
+               m sequences is stored in a row of the array in Hermitian
+                                           p                 p   p
+               form. If the n data values z  are written as x +iy , then
+                                           j                 j   j
+                               p
+               for 0<=j<=n/2, x  is contained in X(p,j), and for 1<=j<=(n-
+                               j
+                      p
+               1)/2, y  is contained in X(p,n-j). (See also Section 2.1.2
+                      j
+               of the Chapter Introduction.) On exit: the imaginary parts
+                p                              p
+               y  are negated. The real parts x  are not referenced.
+                j                              j
+
+           4:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M < 1.
+
+          IFAIL= 2
+               On entry N < 1.
+
+          7. Accuracy
+
+          Exact.
+
+          8. Further Comments
+
+          None.
+
+          9. Example
+
+          This program reads in sequences of real data values which are
+          assumed to be Hermitian sequences of complex data stored in
+          Hermitian form. The sequences are expanded into full complex form
+          using C06GSF and printed. The sequences are then conjugated
+          (using C06GQF) and the conjugated sequences are expanded into
+          complex form using C06GSF and printed out.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXc06gsf}{NAG On-line Documentation: c06gsf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     C06GSF(3NAG)      Foundation Library (12/10/92)      C06GSF(3NAG)
+
+
+
+          C06 -- Summation of Series                                 C06GSF
+                  C06GSF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          C06GSF takes m Hermitian sequences, each containing n data
+          values, and forms the real and imaginary parts of the m
+          corresponding complex sequences.
+
+          2. Specification
+
+                 SUBROUTINE C06GSF (M, N, X, U, V, IFAIL)
+                 INTEGER          M, N, IFAIL
+                 DOUBLE PRECISION X(M*N), U(M*N), V(M*N)
+
+          3. Description
+
+          This is a utility routine for use in conjunction with C06FPF and
+          C06FQF (see the Chapter Introduction).
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number of Hermitian sequences, m, to be
+               converted into complex form. Constraint: M >= 1.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of data values, n, in each sequence.
+               Constraint: N >= 1.
+
+           3:  X(M,N) -- DOUBLE PRECISION array                       Input
+               On entry: the data must be stored in X as if in a two-
+               dimensional array of dimension (1:M,0:N-1); each of the m
+               sequences is stored in a row of the array in Hermitian form.
+                                     p                 p   p
+               If the n data values z  are written as x +iy , then for
+                                     j                 j   j
+                           p
+               0<=j<=n/2, x  is contained in X(p,j), and for 1<=j<=(n-1)/2,
+                           j
+                p
+               y  is contained in X(p,n-j). (See also Section 2.1.2 of the
+                j
+               Chapter Introduction.)
+
+           4:  U(M,N) -- DOUBLE PRECISION array                      Output
+
+           5:  V(M,N) -- DOUBLE PRECISION array                      Output
+               On exit: the real and imaginary parts of the m sequences of
+               length n, are stored in U and V respectively, as if in two-
+               dimensional arrays of dimension (1:M,0:N-1); each of the m
+               sequences is stored as if in a row of each array. In other
+               words, if the real parts of the pth sequence are denoted by
+                p
+               x , for j=0,1,...,n-1 then the mn elements of the array U
+                j
+               contain the values
+                     1  2      m   1  2      m       1    2        m
+                    x ,x ,...,x , x ,x ,...,x ,..., x   ,x   ,...,x
+                     0  0      0   1  1      1       n-1  n-1      n-1
+
+           6:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M < 1.
+
+          IFAIL= 2
+               On entry N < 1.
+
+          7. Accuracy
+
+          Exact.
+
+          8. Further Comments
+
+          None.
+
+          9. Example
+
+          This program reads in sequences of real data values which are
+          assumed to be Hermitian sequences of complex data stored in
+          Hermitian form. The sequences are then expanded into full complex
+          form using C06GSF and printed.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
diff --git a/src/hyper/pages/nagd.ht b/src/hyper/pages/nagd.ht
new file mode 100644
index 00000000..e93cb2ff
--- /dev/null
+++ b/src/hyper/pages/nagd.ht
@@ -0,0 +1,10328 @@
+\begin{page}{manpageXXd01}{NAG On-line Documentation: d01}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D01(3NAG)         Foundation Library (12/10/92)         D01(3NAG)
+
+
+
+          D01 -- Quadrature                             Introduction -- D01
+                                    Chapter D01
+                                    Quadrature
+
+          1. Scope of the Chapter
+
+          This chapter provides routines for the numerical evaluation of
+          definite integrals in one or more dimensions and for evaluating
+          weights and abscissae of integration rules.
+
+          2. Background to the Problems
+
+          The routines in this chapter are designed to estimate:
+
+          (a)   the value of a one-dimensional definite integral of the
+                form:
+                                        b
+                                        /
+                                        |f(x)dx                         (1)
+                                        /
+                                        a
+                where f(x) is defined by the user, either at a set of
+                points (x ,f(x )), for i=1,2,...,n where a=x <x <...<x =b,
+                         i    i                             1  2      n
+                or in the form of a function; and the limits of integration
+                a,b may be finite or infinite.
+
+                Some methods are specially designed for integrands of the
+                form
+                                    f(x)=w(x)g(x)                       (2)
+                which contain a factor w(x), called the weight-function, of
+                a specific form. These methods take full account of any
+                peculiar behaviour attributable to the w(x) factor.
+
+          (b)   the value of a multi-dimensional definite integral of the
+                form:
+                             /
+                             | f(x ,x ,...,x )dx ...dx dx               (3)
+                             /    1  2      n   n     2  1
+                             R
+                              n
+                where f(x ,x ,...,x ) is a function defined by the user and
+                         1  2      n
+                R  is some region of n-dimensional space.
+                 n
+
+                The simplest form of R  is the n-rectangle defined by:
+                                      n
+                               a <=x <=b ,  i=1,2,...,n                 (4)
+                                i   i   i
+                where a  and b  are constants. When a  and b  are functions
+                       i      i                      i      i
+                of x  (j<i), the region can easily be transformed to the
+                    j
+                rectangular form (see Davis and Rabinowitz [1] page 266).
+                Some of the methods described incorporate the
+                transformation procedure.
+
+          2.1. One-dimensional Integrals
+
+          To estimate the value of a one-dimensional integral, a quadrature
+          rule uses an approximation in the form of a weighted sum of
+          integrand values, i.e.,
+
+                              b         N
+                              /         --
+                              |f(x)dx~= >  w f(x ).                     (5)
+                              /         --  i   i
+                              a         i=1
+
+          The points x  within the interval [a,b] are known as the
+                      i
+          abscissae, and the w  are known as the weights.
+                              i
+
+          More generally, if the integrand has the form (2), the
+          corresponding formula is
+
+                            b             N
+                            /             --
+                            |w(x)g(x)dx~= >  w g(x ).                   (6)
+                            /             --  i   i
+                            a             i=1
+
+          If the integrand is known only at a fixed set of points, these
+          points must be used as the abscissae, and the weighted sum is
+          calculated using finite-difference methods. However, if the
+          functional form of the integrand is known, so that its value at
+          any abscissa is easily obtained, then a wide variety of
+          quadrature rules are available, each characterised by its choice
+          of abscissae and the corresponding weights.
+
+          The appropriate rule to use will depend on the interval [a,b] -
+          whether finite or otherwise - and on the form of any w(x) factor
+          in the integrand. A suitable value of N depends on the general
+          behaviour of f(x); or of g(x), if there is a w(x) factor present.
+
+          Among possible rules, we mention particularly the Gaussian
+          formulae, which employ a distribution of abscissae which is
+          optimal for f(x) or g(x) of polynomial form.
+
+          The choice of basic rules constitutes one of the principles on
+          which methods for one-dimensional integrals may be classified.
+          The other major basis of classification is the implementation
+          strategy, of which some types are now presented.
+
+          (a)   Single rule evaluation procedures
+
+                A fixed number of abscissae, N, is used. This number and
+                the particular rule chosen uniquely determine the weights
+                and abscissae. No estimate is made of the accuracy of the
+                result.
+
+          (b)   Automatic procedures
+
+                The number of abscissae, N, within [a,b] is gradually
+                increased until consistency is achieved to within a level
+                of accuracy (absolute or relative) requested by the user.
+                There are essentially two ways of doing this; hybrid forms
+                of these two methods are also possible:
+                (i)   whole interval procedures (non-adaptive)
+
+                      A series of rules using increasing values of N are
+                      successively applied over the whole interval [a,b].
+                      It is clearly more economical if abscissae already
+                      used for a lower value of N can be used again as part
+                      of a higher-order formula. This principle is known as
+                       optimal extension. There is no overlap between the
+                      abscissae used in Gaussian formulae of different
+                      orders. However, the Kronrod formulae are designed to
+                      give an optimal (2N+1)-point formula by adding (N+1)
+                      points to an N-point Gauss formula. Further
+                      extensions have been developed by Patterson.
+
+                (ii)  adaptive procedures
+
+                      The interval [a,b] is repeatedly divided into a
+                      number of sub-intervals, and integration rules are
+                      applied separately to each sub-interval. Typically,
+                      the subdivision process will be carried further in
+                      the neighbourhood of a sharp peak in the integrand,
+                      than where the curve is smooth. Thus, the
+                      distribution of abscissae is adapted to the shape of
+                      the integrand.
+
+                      Subdivision raises the problem of what constitutes an
+                      acceptable accuracy in each sub-interval. The usual
+                      global acceptability criterion  demands that the sum
+                      of the absolute values of the error estimates in the
+                      sub-intervals should meet the conditions required of
+                      the error over the whole interval. Automatic
+                      extrapolation over several levels of subdivision may
+                      eliminate the effects of some types of singularities.
+
+          An ideal general-purpose method would be an automatic method
+          which could be used for a wide variety of integrands, was
+          efficient (i.e., required the use of as few abscissae as
+          possible), and was reliable (i.e., always gave results within the
+          requested accuracy). Complete reliability is unobtainable, and
+          generally higher reliability is obtained at the expense of
+          efficiency, and vice versa.  It must therefore be emphasised that
+          the automatic routines in this chapter cannot be assumed to be
+          100% reliable. In general, however, the reliability is very high.
+
+          2.2. Multi-dimensional Integrals
+
+          A distinction must be made between cases of moderately low
+          dimensionality (say, up to 4 or 5 dimensions), and those of
+          higher dimensionality. Where the number of dimensions is limited,
+          a one-dimensional method may be applied to each dimension,
+          according to some suitable strategy, and high accuracy may be
+          obtainable (using product rules). However, the number of
+          integrand evaluations rises very rapidly with the number of
+          dimensions, so that the accuracy obtainable with an acceptable
+          amount of computational labour is limited; for example a product
+                                                                      9
+          of 3-point rules in 20 dimensions would require more than 10
+          integrand evaluations. Special techniques such as the Monte Carlo
+          methods can be used to deal with high dimensions.
+
+          (a)   Products of one-dimensional rules
+
+                Using a two-dimensional integral as an example, we have
+                         b  b                   [ b          ]
+                          1  2             N    [  2         ]
+                         /  /              --   [ /          ]
+                         |  | f(x,y)dydx~= >  w [ | f(x ,y)dy]          (7)
+                         /  /              --  i[ /    i     ]
+                         a  a              i=1  [ a          ]
+                          1  2                  [  2         ]
+
+                         b  b
+                          1  2             N   N
+                         /  /              --  --
+                         |  | f(x,y)dydx~= >   >  w v f(x ,y )          (8)
+                         /  /              --  --  i j   i  j
+                         a  a              i=1 j=1
+                          1  2
+                where (w ,x ) and (v ,y ) are the weights and abscissae of
+                        i  i        i  i
+                the rules used in the respective dimensions.
+
+                A different one-dimensional rule may be used for each
+                dimension, as appropriate to the range and any weight
+                function present, and a different strategy may be used, as
+                appropriate to the integrand behaviour as a function of
+                each independent variable.
+
+                For a rule-evaluation strategy in all dimensions, the
+                formula (8) is applied in a straightforward manner. For
+                automatic strategies (i.e., attempting to attain a
+                requested accuracy), there is a problem in deciding what
+                accuracy must be requested in the inner integral(s).
+                Reference to formula (7) shows that the presence of a
+                limited but random error in the y-integration for different
+                values of x  can produce a 'jagged' function of x, which
+                           i
+                may be difficult to integrate to the desired accuracy and
+                for this reason products of automatic one-dimensional
+                routines should be used with caution (see also Lyness [3]).
+
+          (b)   Monte Carlo methods
+
+                These are based on estimating the mean value of the
+                integrand sampled at points chosen from an appropriate
+                statistical distribution function. Usually a variance
+                reducing procedure is incorporated to combat the
+                fundamentally slow rate of convergence of the rudimentary
+                form of the technique. These methods can be effective by
+                comparison with alternative methods when the integrand
+                contains singularities or is erratic in some way, but they
+                are of quite limited accuracy.
+
+          (c)   Number theoretic methods
+
+                These are based on the work of Korobov and Conroy and
+                operate by exploiting implicitly the properties of the
+                Fourier expansion of the integrand. Special rules,
+                constructed from so-called optimal coefficients, give a
+                particularly uniform distribution of the points throughout
+                n-dimensional space and from their number theoretic
+                properties minimize the error on a prescribed class of
+                integrals. The method can be combined with the Monte Carlo
+                procedure.
+
+          (d)   Sag-Szekeres method
+
+                By transformation this method seeks to induce properties
+                into the integrand which make it accurately integrable by
+                the trapezoidal rule. The transformation also allows
+                effective control over the number of integrand evaluations.
+
+          (e)   Automatic adaptive procedures
+
+                An automatic adaptive strategy in several dimensions
+                normally involves division of the region into subregions,
+                concentrating the divisions in those parts of the region
+                where the integrand is worst behaved. It is difficult to
+                arrange with any generality for variable limits in the
+                inner integral(s). For this reason, some methods use a
+                region where all the limits are constants; this is called a
+                hyper-rectangle. Integrals over regions defined by variable
+                or infinite limits may be handled by transformation to a
+                hyper-rectangle. Integrals over regions so irregular that
+                such a transformation is not feasible may be handled by
+                surrounding the region by an appropriate hyper-rectangle
+                and defining the integrand to be zero outside the desired
+                region. Such a technique should always be followed by a
+                Monte Carlo method for integration.
+
+                The method used locally in each subregion produced by the
+                adaptive subdivision process is usually one of three types:
+                Monte Carlo, number theoretic or deterministic.
+                Deterministic methods are usually the most rapidly
+                convergent but are often expensive to use for high
+                dimensionality and not as robust as the other techniques.
+
+          2.3. References
+
+          Comprehensive reference:
+
+          [1]   Davis P J and Rabinowitz P (1975) Methods of Numerical
+                Integration. Academic Press.
+
+          Special topics:
+
+          [2]   Gladwell I (1986) Vectorisation of one dimensional
+                quadrature codes. Techincal Report. TR7/86  NAG.
+
+          [3]   Lyness J N (1983) When not to use an automatic quadrature
+                routine. SIAM Review. 25 63--87.
+
+          [4]   Piessens R, De Doncker-Kapenga E, Uberhuber C and Kahaner D
+                (1983) QUADPACK, A Subroutine Package for Automatic
+                Integration. Springer-Verlag.
+
+          [5]   Sobol I M (1974) The Monte Carlo Method. The University of
+                Chicago Press.
+
+          [6]   Stroud A H (1971) Approximate Calculation of Multiple
+                Integrals. Prentice-Hall.
+
+          3. Recommendations on Choice and Use of Routines
+
+          The following three sub-sections consider in turn routines for:
+          one-dimensional integrals over a finite interval, and over a
+          semi-infinite or an infinite interval; and multi-dimensional
+          integrals. Within each sub-section, routines are classified by
+          the type of method, which ranges from simple rule evaluation to
+          automatic adaptive algorithms. The recommendations apply
+          particularly when the primary objective is simply to compute the
+          value of one or more integrals, and in these cases the automatic
+          adaptive routines are generally the most convenient and reliable,
+          although also the most expensive in computing time.
+
+          Note however that in some circumstances it may be counter-
+          productive to use an automatic routine. If the results of the
+          quadrature are to be used in turn as input to a further
+          computation (e.g. an 'outer' quadrature or an optimization
+          problem), then this further computation may be adversely affected
+          by the 'jagged performance profile' of an automatic routine; a
+          simple rule-evaluation routine may provide much better overall
+          performance. For further guidance, the article Lyness [3] is
+          recommended.
+
+          3.1. One-dimensional Integrals over a Finite Interval
+
+          (a)   Integrand defined as a set of points
+
+                If f(x) is defined numerically at four or more points, then
+                the Gill-Miller finite difference method (D01GAF) should be
+                used. The interval of integration is taken to coincide with
+                the range of x-values of the points supplied. It is in the
+                nature of this problem that any routine may be unreliable.
+                In order to check results independently and so as to
+                provide an alternative technique the user may fit the
+                integrand by Chebyshev series using E02ADF and then use
+                routines E02AJF and E02AKF to evaluate its integral (which
+                need not be restricted to the range of the integration
+                points, as is the case for D01GAF). A further alternative
+                is to fit a cubic spline to the data using E02BAF and then
+                to evaluate its integral using E02BDF.
+
+          (b)   Integrand defined as a function
+
+                If the functional form of f(x) is known, then one of the
+                following approaches should be taken. They are arranged in
+                the order from most specific to most general, hence the
+                first applicable procedure in the list will be the most
+                efficient.  However, if the user does not wish to make any
+                assumptions about the integrand, the most reliable routine
+                to use will be D01AJF, although this will in general be
+                less efficient for simple integrals.
+                (i)   Rule-evaluation routines
+
+                      If f(x) is known to be sufficiently well-behaved
+                      (more precisely, can be closely approximated by a
+                      polynomial of moderate degree), a Gaussian routine
+                      with a suitable number of abscissae may be used.
+
+                      D01BBF may be used if it is required to examine the
+                      weights and abscissae. In this case, the user should
+                      write the code for the evaluation of quadrature
+                      summation (6).
+
+                (ii)  Automatic adaptive routines
+
+                      Firstly, several routines are available for
+                      integrands of the form w(x)g(x) where g(x) is a '
+                      smooth' function (i.e., has no singularities, sharp
+                      peaks or violent oscillations in the interval of
+                      integration) and w(x) is a weight function of one of
+                      the following forms:
+
+                                   (alpha)     (beta)          k          l
+                      if w(x)=(b-x)       (x-a)      (log(b-x)) (log(x-a))
+
+                      where k,l=0 or 1, (alpha),(beta)>-1: use D01APF;
+
+                      if w(x)=1/(x-c): use D01AQF (this integral is called
+                      the Hilbert transform of g);
+
+                      if w(x)=cos((omega)x) or sin((omega)x): use D01ANF
+                      (this routine can also handle certain types of
+                      singularities in g(x)).
+
+                      Secondly, there are some routines for general f(x).
+                      If f(x) is known to be free of singularities, though
+                      it may be oscillatory, D01AKF may be used.
+
+                      The most powerful of the finite interval integration
+                      routine is D01AJF (which can cope with singularities
+                      of several types). It may be used if none of the more
+                      specific situations described above applies. D01AJF
+                      is very reliable, particularly where the integrand
+                      has singularities other than at an end-point, or has
+                      discontinuities or cusps, and is therefore
+                      recommended where the integrand is known to be badly-
+                      behaved, or where its nature is completely unknown.
+
+                      Most of the routines in this chapter require the user
+                      to supply a function or subroutine to evaluate the
+                      integrand at a single point.
+
+                      If f(x) has singularities of certain types,
+                      discontinuities or sharp peaks  occurring at known
+                      points, the integral should be evaluated separately
+                      over each of the subranges or D01ALF may be used.
+
+          3.2. One-dimensional Integrals over a Semi-infinite or Infinite
+               Interval
+
+          (a)   Integrand defined as a set of points
+
+                If f(x) is defined numerically at four or more points, and
+                the portion of the integral lying outside the range of the
+                points supplied may be neglected, then the Gill-Miller
+                finite difference method, D01GAF, should be used.
+
+          (b)   Integrand defined as a function
+                (i)   Rule evaluation routines
+
+                      If f(x) behaves approximately like a polynomial in x,
+                      apart from a weight function of the form
+                       -(beta)x
+                      e                    (beta)>0 (semi-infinite interval,
+                                                     lower limit finite);
+
+                       -(beta)x
+                 or   e                    (beta)<0 (semi-infinite interval,
+                                                     upper limit finite);
+
+                                          2
+                        -(beta)(x-(alpha))
+                 or    e                   (beta)>0 (infinite interval);
+
+                               or if f(x) behaves approximately like a
+                                                  -1
+                               polynomial in (x+B)   (semi-infinite range);
+                               then the Gaussian routines may be used.
+
+                               D01BBF may be used if it is required to
+                               examine the weights and abscissae. In this
+                               case, the user should write the code for the
+                               evaluation of quadrature summation (6).
+
+                (ii)  Automatic adaptive routines
+
+                      D01AMF may be used, except for integrands which decay
+                      slowly towards an infinite end-point, and oscillate
+                      in sign over the entire range. For this class, it may
+                      be possible to calculate the integral by integrating
+                      between the zeros and invoking some extrapolation
+                      process.
+
+                      D01ASF may be used for integrals involving weight
+                      functions of the form cos((omega)x) and sin((omega)x)
+                      over a semi-infinite interval (lower limit finite).
+
+                      The following alternative procedures are mentioned
+                      for completeness, though their use will rarely be
+                      necessary:
+                      (1)   If the integrand decays rapidly towards an
+                            infinite end-point, a finite cut-off may be
+                            chosen, and the finite range methods applied.
+
+                      (2)   If the only irregularities occur in the finite
+                            part (apart from a singularity at the finite
+                            limit, with which D01AMF can cope), the range
+                            may be divided, with D01AMF used on the
+                            infinite part.
+
+                      (3)   A transformation to finite range may be
+                            employed, e.g.
+                                             1-t
+                                          x= ---  or x=-log t
+                                              t            e
+                            will transform (0,infty) to (1,0) while for
+                            infinite ranges we have
+                                   +infty        infty
+                                   /             /
+                                   |     f(x)dx= |    [f(x)+f(-x)]dx.
+                                   /             /
+                                   -infty        0
+                            If the integrand behaves badly on (-infty,0)
+                            and well on (0,infty) or vice versa it is
+                            better to compute it as
+                                       0             infty
+                                       /             /
+                                       |     f(x)dx+ |    f(x)dx.
+                                       /             /
+                                       -infty        0
+                            This saves computing unnecessary function
+                            values in the semi-infinite range where the
+                            function is well behaved.
+
+          3.3. Multi-dimensional Integrals
+
+          A number of techniques are available in this area and the choice
+          depends to a large extent on the dimension and the required
+          accuracy. It can be advantageous to use more than one technique
+          as a confirmation of accuracy particularly for high dimensional
+          integrations. Many of the routines incorporate the transformation
+          procedure REGION which allows general product regions to be
+          easily dealt with in terms of conversion to the standard n-cube
+          region.
+
+          (a)   Products of one-dimensional rules (suitable for up to about
+                5 dimensions)
+
+                If f(x ,x ,...,x ) is known to be a sufficiently well-
+                      1  2      n
+                behaved function of each variable x , apart possibly from
+                                                   i
+                weight functions of the types provided, a product of
+                Gaussian rules may be used. These are provided by D01BBF.
+                In this case, the user should write the code for the
+                evaluation of quadrature summation (6). Rules for finite,
+                semi-infinite and infinite ranges are included.
+
+                The one-dimensional routines may also be used recursively.
+                For example, the two-dimensional integral
+                                       b  b
+                                        1  2
+                                       /  /
+                                    I= |  | f(x,y)dydx
+                                       /  /
+                                       a  a
+                                        1  2
+                may be expressed as
+                                          b
+                                           1
+                                          /
+                                       I= | F(x)dx,
+                                          /
+                                          a
+                                           1
+                where
+                                           b
+                                            2
+                                           /
+                                     F(x)= | f(x,y)dy.
+                                           /
+                                           a
+                                            2
+                The user segment to evaluate F(x) will call the integration
+                routine for the y-integration, which will call another user
+                segment for f(x,y) as a function of y (x being effectively
+                a constant). Note that, as Fortran is not a recursive
+                language,  a different library integration routine must be
+                used for each dimension. Apart from this restriction, the
+                full range of one-dimensional routines are available, for
+                finite/infinite intervals, constant/variable limits, rule
+                evaluation/automatic strategies etc.
+
+          (b)   Automatic routines (D01GBF and D01FCF)
+
+                Both routines are for integrals of the form
+                         b  b     b
+                          1  2     n
+                         /  /     /
+                         |  | ... | f(x ,x ,...,x )dx dx   ...dx .
+                         /  /     /    1  2      n   n  n-1     1
+                         a  a     a
+                          1  2     n
+                D01GBF is an adaptive Monte Carlo routine. This routine is
+                       usually slow and not recommended for high accuracy
+                       work. It is a robust routine that can often be used
+                       for low accuracy results with highly irregular
+                       integrands or when n is large.
+
+                D01FCF is an adaptive deterministic routine. Convergence is
+                       fast for well-behaved integrands. Highly accurate
+                       results can often be obtained for n between 2 and 5,
+                       using significantly fewer integrand evaluations than
+                       would be required by D01GBF. The routine will
+                       usually work when the integrand is mildly singular
+                       and for n<=10 should be used before D01GBF. If it is
+                       known in advance that the integrand is highly
+                       irregular, it is best to compare results from at
+                       least two different routines.
+
+          There are many problems for which one or both of the routines
+          will require large amounts of computing time to obtain even
+          moderately accurate results. The amount of computing time is
+          controlled by the number of integrand evaluations allowed by the
+          user, and users should set this parameter carefully, with
+          reference to the time available and the accuracy desired.
+
+
+          3.4. Decision Trees
+
+          (i) One-dimensional integrals over a finite interval. (If in
+          doubt, follow the downward branch.)
+
+
+
+                   Please see figure in printed Reference Manual
+
+
+          (ii) One-dimensional integrals over a semi-infinite or infinite
+          interval. (If in doubt, follow the downward branch.)
+
+
+
+                   Please see figure in printed Reference Manual
+
+
+          D01 -- Quadrature                                 Contents -- D01
+          Chapter D01
+
+          Quadrature
+
+          D01AJF  1-D quadrature, adaptive, finite interval, strategy due
+                  to Piessens and de Doncker, allowing for badly-behaved
+                  integrands
+
+          D01AKF  1-D quadrature, adaptive, finite interval, method
+                  suitable for oscillating functions
+
+          D01ALF  1-D quadrature, adaptive, finite interval, allowing for
+                  singularities at user-specified break-points
+
+          D01AMF  1-D quadrature, adaptive, infinite or semi-infinite
+                  interval
+
+          D01ANF  1-D quadrature, adaptive, finite interval, weight
+                  function cos((omega)x) or sin((omega)x)
+
+          D01APF  1-D quadrature, adaptive, finite interval, weight
+                  function with end-point singularities of algebraico-
+                  logarithmic type
+
+          D01AQF  1-D quadrature, adaptive, finite interval, weight
+                  function 1/(x-c), Cauchy principal value (Hilbert
+                  transform)
+
+          D01ASF  1-D quadrature, adaptive, semi-infinite interval, weight
+                  function cos((omega)x) or sin((omega)x)
+
+          D01BBF  Weights and abscissae for Gaussian quadrature rules
+
+          D01FCF  Multi-dimensional adaptive quadrature over hyper-
+                  rectangle
+
+          D01GAF  1-D quadrature, integration of function defined by data
+                  values, Gill-Miller method
+
+          D01GBF  Multi-dimensional quadrature over hyper-rectangle, Monte
+                  Carlo method
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd01ajf}{NAG On-line Documentation: d01ajf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D01AJF(3NAG)      Foundation Library (12/10/92)      D01AJF(3NAG)
+
+
+
+          D01 -- Quadrature                                          D01AJF
+                  D01AJF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D01AJF is a general-purpose integrator which calculates an
+          approximation to the integral of a function f(x) over a finite
+          interval [a,b]:
+
+                                        b
+                                        /
+                                     I= |f(x)dx.
+                                        /
+                                        a
+
+          2. Specification
+
+                 SUBROUTINE D01AJF (F, A, B, EPSABS, EPSREL, RESULT,
+                1                   ABSERR, W, LW, IW, LIW, IFAIL)
+                 INTEGER          LW, IW(LIW), LIW, IFAIL
+                 DOUBLE PRECISION F, A, B, EPSABS, EPSREL, RESULT, ABSERR, W
+                1                 (LW)
+                 EXTERNAL         F
+
+          3. Description
+
+          D01AJF is based upon the QUADPACK routine QAGS (Piessens et al
+          [3]). It is an adaptive routine, using the Gauss 10-point and
+          Kronrod 21-point rules. The algorithm, described by de Doncker
+          [1], incorporates a global acceptance criterion (as defined by
+          Malcolm and Simpson [2]) together with the (epsilon)-algorithm
+          (Wynn [4]) to perform extrapolation. The local error estimation
+          is described by Piessens et al [3].
+
+          The routine is suitable as a general purpose integrator, and can
+          be used when the integrand has singularities, especially when
+          these are of algebraic or logarithmic type.
+
+          D01AJF requires the user to supply a function to evaluate the
+          integrand at a single point.
+
+          The routine D01ATF(*) uses an identical algorithm but requires
+          the user to supply a subroutine to evaluate the integrand at an
+          array of points. Therefore D01ATF(*) will be more efficient if
+          the evaluation can be performed in vector mode on a vector-
+          processing machine.
+
+          4. References
+
+          [1]   De Doncker E (1978) An Adaptive Extrapolation Algorithm for
+                Automatic Integration. Signum Newsletter. 13 (2) 12--18.
+
+          [2]   Malcolm M A and Simpson R B (1976) Local Versus Global
+                Strategies for Adaptive Quadrature. ACM Trans. Math. Softw.
+                1 129--146.
+
+          [3]   Piessens R, De Doncker-Kapenga E, Uberhuber C and Kahaner D
+                (1983) QUADPACK, A Subroutine Package for Automatic
+                Integration. Springer-Verlag.
+
+          [4]   Wynn P (1956) On a Device for Computing the e (S )
+                                                             m  n
+                Transformation. Math. Tables Aids Comput. 10 91--96.
+
+          5. Parameters
+
+           1:  F -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                                         External Procedure
+               F must return the value of the integrand f at a given point.
+
+               Its specification is:
+
+                      DOUBLE PRECISION FUNCTION F (X)
+                      DOUBLE PRECISION X
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the point at which the integrand f must be
+                    evaluated.
+               F must be declared as EXTERNAL in the (sub)program from
+               which D01AJF is called. Parameters denoted as Input
+               must not be changed by this procedure.
+
+           2:  A -- DOUBLE PRECISION                                  Input
+               On entry: the lower limit of integration, a.
+
+           3:  B -- DOUBLE PRECISION                                  Input
+               On entry: the upper limit of integration, b. It is not
+               necessary that a<b.
+
+           4:  EPSABS -- DOUBLE PRECISION                             Input
+               On entry: the absolute accuracy required. If EPSABS is
+               negative, the absolute value is used. See Section 7.
+
+           5:  EPSREL -- DOUBLE PRECISION                             Input
+               On entry: the relative accuracy required. If EPSREL is
+               negative, the absolute value is used. See Section 7.
+
+           6:  RESULT -- DOUBLE PRECISION                            Output
+               On exit: the approximation to the integral I.
+
+           7:  ABSERR -- DOUBLE PRECISION                            Output
+               On exit: an estimate of the modulus of the absolute error,
+               which should be an upper bound for |I-RESULT|.
+
+           8:  W(LW) -- DOUBLE PRECISION array                       Output
+               On exit: details of the computation, as described in
+               Section 8.
+
+           9:  LW -- INTEGER                                          Input
+               On entry:
+               the dimension of the array W as declared in the (sub)program
+               from which D01AJF is called.
+               The value of LW (together with that of LIW below) imposes a
+               bound on the number of sub-intervals into which the interval
+               of integration may be divided by the routine. The number of
+               sub-intervals cannot exceed LW/4. The more difficult the
+               integrand, the larger LW should be. Suggested value: a value
+               in the range 800 to 2000 is adequate for most problems.
+               Constraint: LW >= 4.
+
+          10:  IW(LIW) -- INTEGER array                              Output
+               On exit: IW(1) contains the actual number of sub-intervals
+               used. The rest of the array is used as workspace.
+
+          11:  LIW -- INTEGER                                         Input
+               On entry:
+               the dimension of the array IW as declared in the
+               (sub)program from which D01AJF is called.
+               The number of sub-intervals into which the interval of
+               integration may be divided cannot exceed LIW. Suggested
+               value: LIW = LW/4. Constraint: LIW >= 1.
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               The maximum number of subdivisions allowed with the given
+               workspace has been reached without the accuracy requirements
+               being achieved. Look at the integrand in order to determine
+               the integration difficulties. If the position of a local
+               difficulty within the interval can be determined (e.g. a
+               singularity of the integrand or its derivative, a peak, a
+               discontinuity, etc) you will probably gain from splitting up
+               the interval at this point and calling the integrator on the
+               subranges. If necessary, another integrator, which is
+               designed for handling the type of difficulty involved, must
+               be used. Alternatively, consider relaxing the accuracy
+               requirements specified by EPSABS and EPSREL, or increasing
+               the amount of workspace.
+
+          IFAIL= 2
+               Round-off error prevents the requested tolerance from being
+               achieved. The error may be under-estimated. Consider
+               requesting less accuracy.
+
+          IFAIL= 3
+               Extremely bad local integrand behaviour causes a very strong
+               subdivision around one (or more) points of the interval.
+
+          IFAIL= 4
+               The requested tolerance cannot be achieved, because the
+               extrapolation does not increase the accuracy satisfactorily;
+               the returned result is the best which can be obtained. The
+               same advice applies as in the case of IFAIL = 1.
+
+          IFAIL= 5
+               The integral is probably divergent, or slowly convergent.
+               Please note that divergence can occur with any non-zero
+               value of IFAIL.
+
+          IFAIL= 6
+               On entry LW < 4,
+
+               or       LIW < 1.
+
+          7. Accuracy
+
+          The routine cannot guarantee, but in practice usually achieves,
+          the following accuracy:
+
+                                  |I-RESULT|<=tol,
+
+          where
+
+                           tol=max{|EPSABS|,|EPSREL|*|I|},
+
+          and EPSABS and EPSREL are user-specified absolute and relative
+          error tolerance. Moreover it returns the quantity ABSERR which,
+          in normal circumstances, satisfies
+
+                              |I-RESULT|<=ABSERR<=tol.
+
+          8. Further Comments
+
+          The time taken by the routine depends on the integrand and the
+          accuracy required.
+
+          If IFAIL /= 0 on exit, then the user may wish to examine the
+          contents of the array W, which contains the end-points of the
+          sub-intervals used by D01AJF along with the integral
+          contributions and error estimates over the sub-intervals.
+
+          Specifically, for i=1,2,...,n, let r  denote the approximation to
+                                              i
+          the value of the integral over the sub-interval [a ,b ] in the
+                                                            i  i
+          partition of [a,b] and e  be the corresponding absolute error
+                                  i
+          estimate.
+
+                 b
+                  i                       n
+                 /                        --
+          Then,  | f(x)dx~=r  and RESULT= >  r , unless D01AJF terminates
+                 /          i             --  i
+                 a                        i=1
+                  i
+          while testing for divergence of the integral (see Piessens et al
+          [3], Section 3.4.3). In this case, RESULT (and ABSERR) are taken
+          to be the values returned from the extrapolation process. The
+          value of n is returned in IW(1), and the values a , b , e  and r
+                                                           i   i   i      i
+          are stored consecutively in the array W, that is:
+
+               a =W(i),
+                i
+
+               b =W(n+i),
+                i
+
+               e =W(2n+i) and
+                i
+
+               r =W(3n+i).
+                i
+
+          9. Example
+
+          To compute
+                              2(pi)
+                              /         x sin(30x)
+                              |     ----------------dx.
+                              /         
+                              0        /(  (  x  )2)
+                                      / (1-(-----) )
+                                    \/  (  (2(pi)) )
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd01akf}{NAG On-line Documentation: d01akf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D01AKF(3NAG)      Foundation Library (12/10/92)      D01AKF(3NAG)
+
+
+
+          D01 -- Quadrature                                          D01AKF
+                  D01AKF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D01AKF is an adaptive integrator, especially suited to
+          oscillating, non-singular integrands, which calculates an
+          approximation to the integral of a function f(x) over a finite
+          interval [a,b]:
+
+                                        b
+                                        /
+                                     I= |f(x)dx.
+                                        /
+                                        a
+
+          2. Specification
+
+                 SUBROUTINE D01AKF (F, A, B, EPSABS, EPSREL, RESULT,
+                1                   ABSERR, W, LW, IW, LIW, IFAIL)
+                 INTEGER          LW, IW(LIW), LIW, IFAIL
+                 DOUBLE PRECISION F, A, B, EPSABS, EPSREL, RESULT, ABSERR, W
+                1                 (LW)
+                 EXTERNAL         F
+
+          3. Description
+
+          D01AKF is based upon the QUADPACK routine QAG (Piessens et al [3]
+          ). It is an adaptive routine, using the Gauss 30-point and
+          Kronrod 61-point rules. A 'global' acceptance criterion (as
+          defined by Malcolm and Simpson [1]) is used. The local error
+          estimation is described in Piessens et al [3].
+
+          Because this routine is based on integration rules of high order,
+          it is especially suitable for non-singular oscillating
+          integrands.
+
+          D01AKF requires the user to supply a function to evaluate the
+          integrand at a single point.
+
+          The routine D01AUF(*) uses an identical algorithm but requires
+          the user to supply a subroutine to evaluate the integrand at an
+          array of points. Therefore D01AUF(*) will be more efficient if
+          the evaluation can be performed in vector mode on a vector-
+          processing machine.
+
+          D01AUF(*) also has an additional parameter KEY which allows the
+          user to select from six different Gauss-Kronrod rules.
+
+          4. References
+
+          [1]   Malcolm M A and Simpson R B (1976) Local Versus Global
+                Strategies for Adaptive Quadrature. ACM Trans. Math. Softw.
+                1 129--146.
+
+          [2]   Piessens R (1973) An Algorithm for Automatic Integration.
+                Angewandte Informatik. 15 399--401.
+
+          [3]   Piessens R, De Doncker-Kapenga E, Uberhuber C and Kahaner D
+                (1983) QUADPACK, A Subroutine Package for Automatic
+                Integration. Springer-Verlag.
+
+          5. Parameters
+
+           1:  F -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                                         External Procedure
+               F must return the value of the integrand f at a given point.
+
+               Its specification is:
+
+                      DOUBLE PRECISION FUNCTION F (X)
+                      DOUBLE PRECISION X
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the point at which the integrand f must be
+                    evaluated.
+               F must be declared as EXTERNAL in the (sub)program from
+               which D01AKF is called. Parameters denoted as Input
+               must not be changed by this procedure.
+
+           2:  A -- DOUBLE PRECISION                                  Input
+               On entry: the lower limit of integration, a.
+
+           3:  B -- DOUBLE PRECISION                                  Input
+               On entry: the upper limit of integration, b. It is not
+               necessary that a<b.
+
+           4:  EPSABS -- DOUBLE PRECISION                             Input
+               On entry: the absolute accuracy required. If EPSABS is
+               negative, the absolute value is used. See Section 7.
+
+           5:  EPSREL -- DOUBLE PRECISION                             Input
+               On entry: the relative accuracy required. If EPSREL is
+               negative, the absolute value is used. See Section 7.
+
+           6:  RESULT -- DOUBLE PRECISION                            Output
+               On exit: the approximation to the integral I.
+
+           7:  ABSERR -- DOUBLE PRECISION                            Output
+               On exit: an estimate of the modulus of the absolute error,
+               which should be an upper bound |I-RESULT|.
+
+           8:  W(LW) -- DOUBLE PRECISION array                       Output
+               On exit: details of the computation, as described in
+               Section 8.
+
+           9:  LW -- INTEGER                                          Input
+               On entry: the dimension of W, as declared in the (sub)
+               program from which D01AKF is called. The value of LW
+               (together with that of LIW below) imposes a bound on the
+               number of sub-intervals into which the interval of
+               integration may be divided by the routine. The number of
+               sub-intervals cannot exceed LW/4. The more difficult the
+               integrand, the larger LW should be. Suggested value: a value
+               in the range 800 to 2000 is adequate for most problems.
+               Constraint: LW >= 4. See IW below.
+
+          10:  IW(LIW) -- INTEGER array                              Output
+               On exit: IW(1) contains the actual number of sub-intervals
+               used. The rest of the array is used as workspace.
+
+          11:  LIW -- INTEGER                                         Input
+               On entry:
+               the dimension of the array IW as declared in the
+               (sub)program from which D01AKF is called.
+               The number of sub-intervals into which the interval of
+               integration may be divided cannot exceed LIW. Suggested
+               value: LIW = LW/4. Constraint: LIW >= 1.
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               The maximum number of subdivisions allowed with the given
+               workspace has been reached without the accuracy requirements
+               being achieved. Look at the integrand in order to determine
+               the integration difficulties. Probably another integrator
+               which is designed for handling the type of difficulty
+               involved must be used. Alternatively, consider relaxing the
+               accuracy requirements specified by EPSABS and EPSREL, or
+               increasing the amount of workspace.
+
+          IFAIL= 2
+               Round-off error prevents the requested tolerance from being
+               achieved. Consider  requesting less accuracy.
+
+          IFAIL= 3
+               Extremely bad local integrand behaviour causes a very strong
+               subdivision around one (or more) points of the interval.
+               The same advice applies as in the case of IFAIL = 1.
+
+          IFAIL= 4
+               On entry LW < 4,
+
+               or       LIW < 1.
+
+          7. Accuracy
+
+          The routine cannot guarantee, but in practice usually achieves,
+          the following accuracy:
+
+                                  |I-RESULT|<=tol,
+
+          where
+
+                           tol=max{|EPSABS|,|EPSREL|*|I|},
+
+          and EPSABS and EPSREL are user-specified absolute and relative
+          error tolerances. Moreover it returns the quantity ABSERR which,
+          in normal circumstances satisfies
+
+                              |I-RESULT|<=ABSERR<=tol.
+
+          8. Further Comments
+
+          The time taken by the routine depends on the integrand and the
+          accuracy required.
+
+          If IFAIL /= 0 on exit, then the user may wish to examine the
+          contents of the array W, which contains the end-points of the
+          sub-intervals used by D01AKF along with the integral
+          contributions and error estimates over these sub-intervals.
+
+          Specifically, for i=1,2,...,n, let r  denote the approximation to
+                                              i
+          the value of the integral over the sub-interval [a ,b ] in the
+                                                            i  i
+          partition of [a,b] and e  be the corresponding absolute error
+                                  i
+                           b
+                            i                       n
+                           /                        --
+          estimate. Then,  | f(x)dx~=r  and RESULT= >  r . The value of n
+                           /          i             --  i
+                           a                        i=1
+                            i
+          is returned in IW(1), and the values a , b , e  and r  are stored
+                                                i   i   i      i
+          consecutively in the array W, that is:
+
+               a =W(i),
+                i
+
+               b =W(n+i),
+                i
+
+               e =W(2n+i) and
+                i
+
+               r =W(3n+i).
+                i
+
+          9. Example
+
+          To compute
+
+                                2(pi)
+                                /
+                                |    x sin(30x) cosx dx.
+                                /
+                                0
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd01alf}{NAG On-line Documentation: d01alf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D01ALF(3NAG)      Foundation Library (12/10/92)      D01ALF(3NAG)
+
+
+
+          D01 -- Quadrature                                          D01ALF
+                  D01ALF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D01ALF is a general purpose integrator which calculates an
+          approximation to the integral of a function f(x) over a finite
+          interval [a,b]:
+
+                                        b
+                                        /
+                                     I= |f(x)dx
+                                        /
+                                        a
+
+          where the integrand may have local singular behaviour at a finite
+          number of points within the integration interval.
+
+          2. Specification
+
+                 SUBROUTINE D01ALF (F, A, B, NPTS, POINTS, EPSABS, EPSREL,
+                1                   RESULT, ABSERR, W, LW, IW, LIW, IFAIL)
+                 INTEGER          NPTS, LW, IW(LIW), LIW, IFAIL
+                 DOUBLE PRECISION F, A, B, POINTS(*), EPSABS, EPSREL,
+                1                 RESULT, ABSERR, W(LW)
+                 EXTERNAL         F
+
+          3. Description
+
+          D01ALF is based upon the QUADPACK routine QAGP (Piessens et al
+          [3]). It is very similar to D01AJF, but allows the user to supply
+          difficult. It is an adaptive routine, using the Gauss 10-point
+          and Kronrod 21-point rules. The algorithm described by de Doncker
+          [1], incorporates a global acceptance criterion (as defined by
+          Malcolm and Simpson [2]) together with the (epsilon)-algorithm
+          (Wynn [4]) to perform extrapolation. The user-supplied 'break-
+          points'always occur as the end-points of some sub-interval during
+          the adaptive process. The local error estimation is described by
+          Piessens et al [3].
+
+          4. References
+
+          [1]   De Doncker E (1978) An Adaptive Extrapolation Algorithm for
+                Automatic Integration. Signum Newsletter. 13 (2) 12--18.
+
+          [2]   Malcolm M A and Simpson R B (1976) Local Versus Global
+                Strategies for Adaptive Quadrature. ACM Trans. Math. Softw.
+                1 129--146.
+
+          [3]   Piessens R, De Doncker-Kapenga E, Uberhuber C and Kahaner D
+                (1983) QUADPACK, A Subroutine Package for Automatic
+                Integration. Springer-Verlag.
+
+          [4]   Wynn P (1956) On a Device for Computing the e (S )
+                                                             m  n
+                Transformation. Math. Tables Aids Comput. 10 91--96.
+
+          5. Parameters
+
+           1:  F -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                                    External Procedure
+               F must return the value of the integrand f at a given point.
+
+               Its specification is:
+
+                      DOUBLE PRECISION FUNCTION F (X)
+                      DOUBLE PRECISION X
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the point at which the integrand f must be
+                    evaluated.
+               F must be declared as EXTERNAL in the (sub)program from
+               which D01ALF is called. Parameters denoted as Input
+               must not be changed by this procedure.
+
+           2:  A -- DOUBLE PRECISION                                  Input
+               On entry: the lower limit of integration, a.
+
+           3:  B -- DOUBLE PRECISION                                  Input
+               On entry: the upper limit of integration, b. It is not
+               necessary that a<b.
+
+           4:  NPTS -- INTEGER                                        Input
+               On entry: the number of user-supplied break-points within
+               the integration interval. Constraint: NPTS >= 0.
+
+           5:  POINTS(NPTS) -- DOUBLE PRECISION array                 Input
+               On entry: the user-specified break-points. Constraint: the
+               break-points must all lie within the interval of integration
+               (but may be supplied in any order).
+
+           6:  EPSABS -- DOUBLE PRECISION                             Input
+               On entry: the absolute accuracy required. If EPSABS is
+               negative, the absolute value is used. See Section 7.
+
+           7:  EPSREL -- DOUBLE PRECISION                             Input
+               On entry: the relative accuracy required. If EPSREL is
+               negative, the absolute value is used. See Section 7.
+
+           8:  RESULT -- DOUBLE PRECISION                             Input
+               On entry: the approximation to the integral I.
+
+           9:  ABSERR -- DOUBLE PRECISION                            Output
+               On exit: an estimate of the modulus of the absolute error,
+               which should be an upper bound for |I-RESULT|.
+
+          10:  W(LW) -- DOUBLE PRECISION array                       Output
+               On exit: details of the computation, as described in
+               Section 8.
+
+          11:  LW -- INTEGER                                          Input
+               On entry:
+               the dimension of the array W as declared in the (sub)program
+               from which D01ALF is called.
+               The value of LW (together with that of LIW below) imposes a
+               bound on the number of sub-intervals into which the interval
+               of integration may be divided by the routine. The number of
+               sub-intervals cannot exceed (LW-2*NPTS-4)/4. The more
+               difficult the integrand, the larger LW should be. Suggested
+               value: a value in the range 800 to 2000 is adequate for most
+               problems. Constraint: LW>=2*NPTS+8.
+
+          12:  IW(LIW) -- INTEGER array                              Output
+               On exit: IW(1) contains the actual number of sub-intervals
+               used. The rest of the array is used as workspace.
+
+          13:  LIW -- INTEGER                                         Input
+               On entry:
+               the dimension of the array IW as declared in the
+               (sub)program from which D01ALF is called.
+               The number of sub-intervals into which the interval of
+               integration may be divided cannot exceed (LIW-NPTS-2)/2.
+               Suggested value: LIW = LW/2. Constraint: LIW >= NPTS + 4.
+
+          14:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               The maximum number of subdivisions allowed with the given
+               workspace has been reached, without the accuracy
+               requirements being achieved. Look at the integrand in order
+               to determine the integration difficulties. If the position
+               of a local difficulty within the interval can be determined
+               (e.g. a singularity of the integrand or its derivative, a
+               peak, a discontinuity, etc) it should be supplied to the
+               routine as an element of the vector POINTS. If necessary,
+               another integrator should be used, which is designed for
+               handling the type of difficulty involved. Alternatively,
+               consider relaxing the accuracy requirements specified by
+               EPSABS and EPSREL, or increasing the amount of workspace.
+
+          IFAIL= 2
+               Round-off error prevents the requested tolerance from being
+               achieved. The error may be under-estimated. Consider
+               requesting less accuracy.
+
+          IFAIL= 3
+               Extremely bad local integrand behaviour causes a very strong
+               subdivision around one (or more) points of the interval.
+               The same advice applies as in the case of IFAIL = 1.
+
+          IFAIL= 4
+               The requested tolerance cannot be achieved, because the
+               extrapolation does not increase the accuracy satisfactorily;
+               the result returned is the best which can be obtained.  The
+               same advice applies as in the case IFAIL = 1.
+
+          IFAIL= 5
+               The integral is probably divergent, or slowly convergent.
+               Please note that divergence can also occur with any other
+               non-zero value of IFAIL.
+
+          IFAIL= 6
+               The input is invalid: break-points are specified outside the
+               integration range, NPTS > LIMIT or NPTS < 0. RESULT and
+               ABSERR are set to zero.
+
+          IFAIL= 7
+               On entry LW<2*NPTS+8,
+
+               or       LIW<NPTS+4.
+
+          7. Accuracy
+
+          The routine cannot guarantee, but in practice usually achieves,
+          the following accuracy:
+                                  |I-RESULT|<=tol,
+
+          where
+
+                           tol=max{|EPSABS|,|EPSREL|*|I|},
+
+          and EPSABS and EPSREL are user-specified absolute and relative
+          error tolerances. Moreover it returns the quantity ABSERR which,
+          in normal circumstances, satisfies
+
+                              |I-RESULT|<=ABSERR<=tol.
+
+          8. Further Comments
+
+          The time taken by the routine depends on the integrand and on the
+          accuracy required.
+
+          If IFAIL /= 0 on exit, then the user may wish to examine the
+          contents of the array W, which contains the end-points of the
+          sub-intervals used by D01ALF along with the integral
+          contributions and error estimates over these sub-intervals.
+
+          Specifically, for i=1,2,...,n, let r  denote the approximation to
+                                              i
+          the value of the integral over the sub-interval [a ,b ] in the
+                                                            i  i
+          partition of [a,b] and e  be the corresponding absolute error
+                                  i
+                           b
+                            i                       n
+                           /                        --
+          estimate. Then,  | f(x)dx~=r  and RESULT= >  r  unless D01ALF
+                           /          i             --  i
+                           a                        i=1
+                            i
+          terminates while testing for divergence of the integral (see
+          Piessens et al [3] Section 3.4.3). In this case, RESULT (and
+          ABSERR) are taken to be the values returned from the
+          extrapolation process. The value of n is returned in IW(1), and
+          the values a , b , e  and r  are stored consecutively in the
+                      i   i   i      i
+          array W, that is:
+
+               a =W(i),
+                i
+
+               b =W(n+i),
+                i
+
+               e =W(2n+i) and
+                i
+
+               r =W(3n+i).
+                i
+
+          9. Example
+
+          To compute
+
+                                    1
+                                    /     1
+                                    | ---------dx.
+                                    /   
+                                    0 \/|x-1/7|
+
+          A break-point is specified at x=1/7, at which point the integrand
+          is infinite. (For definiteness the function FST returns the value
+          0.0 at this point.)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd01amf}{NAG On-line Documentation: d01amf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D01AMF(3NAG)      Foundation Library (12/10/92)      D01AMF(3NAG)
+
+
+
+          D01 -- Quadrature                                          D01AMF
+                  D01AMF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D01AMF calculates an approximation to the integral of a function
+          f(x) over an infinite or semi-infinite interval [a,b]:
+
+                                        b
+                                        /
+                                     I= |f(x)dx
+                                        /
+                                        a
+
+          2. Specification
+
+                 SUBROUTINE D01AMF (F, BOUND, INF, EPSABS, EPSREL, RESULT,
+                1                   ABSERR, W, LW, IW, LIW, IFAIL)
+                 INTEGER          INF, LW, IW(LIW), LIW, IFAIL
+                 DOUBLE PRECISION F, BOUND, EPSABS, EPSREL, RESULT, ABSERR,
+                1                 W(LW)
+                 EXTERNAL         F
+
+          3. Description
+
+          D01AMF is based on the QUADPACK routine QAGI (Piessens et al [3])
+          [0,1] using one of the identities:
+
+                            a             1
+                            /             / (   1-t) 1
+                            |     f(x)dx= |f(a- ---) --dt
+                            /             / (    t )  2
+                            -infty        0          t
+
+                             infty        1
+                             /            / (   1-t) 1
+                             |    f(x)dx= |f(a+ ---) --dt
+                             /            / (    t )  2
+                             a            0          t
+
+             infty         infty                1
+             /             /                    /[ ( 1-t)  ( -1+t)] 1
+             |     f(x)dx= |    (f(x)+f(-x))dx= |[f( ---)+f( ----)] --dt
+             /             /                    /[ (  t )  (  t  )]  2
+             -infty        0                    0                   t
+
+          where a represents a finite integration limit. An adaptive
+          procedure, based on the Gauss seven-point and Kronrod 15-point
+          rules, is then employed on the transformed integral. The
+          algorithm, described by de Doncker [1], incorporates a global
+          acceptance criterion (as defined by Malcolm and Simpson [2])
+          together with the (epsilon)-algorithm (Wynn [4]) to perform
+          extrapolation. The local error estimation is described by
+          Piessens et al [3].
+
+          4. References
+
+          [1]   De Doncker E (1978) An Adaptive Extrapolation Algorithm for
+                Automatic Integration. Signum Newsletter. 13 (2) 12--18.
+
+          [2]   Malcolm M A and Simpson R B (1976) Local Versus Global
+                Strategies for Adaptive Quadrature. ACM Trans. Math. Softw.
+                1 129--146.
+
+          [3]   Piessens R, De Doncker-Kapenga E, Uberhuber C and Kahaner D
+                (1983) QUADPACK, A Subroutine Package for Automatic
+                Integration. Springer-Verlag.
+
+          [4]   Wynn P (1956) On a Device for Computing the e (S )
+                                                             m  n
+                Transformation. Math. Tables Aids Comput. 10 91--96.
+
+          5. Parameters
+
+           1:  F -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                                         External Procedure
+               On entry: the point at which the integrand f must be
+               evaluated.
+
+               Its specification is:
+
+                      DOUBLE PRECISION FUNCTION F (X)
+                      DOUBLE PRECISION X
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the point at which the integrand f must be
+                    evaluated.
+               F must be declared as EXTERNAL in the (sub)program from
+               which D01AMF is called. Parameters denoted as Input
+               must not be changed by this procedure.
+
+           2:  BOUND -- DOUBLE PRECISION                              Input
+               On entry: the finite limit of the integration range (if
+               present). BOUND is not used if the interval is doubly
+               infinite.
+
+           3:  INF -- INTEGER                                         Input
+               On entry: indicates the kind of integration range:
+                    if INF =1, the range is [BOUND, +infty)
+
+                    if INF =-1, the range is (-infty, BOUND]
+
+                    if INF =+2, the range is (-infty, +infty).
+                Constraint: INF =-1, 1 or 2.
+
+           4:  EPSABS -- DOUBLE PRECISION                             Input
+               On entry: the absolute accuracy required. If EPSABS is
+               negative, the absolute value is used. See Section 7.
+
+           5:  EPSREL -- DOUBLE PRECISION                             Input
+               On entry: the relative accuracy required. If EPSREL is
+               negative, the absolute value is used. See Section 7.
+
+           6:  RESULT -- DOUBLE PRECISION                            Output
+               On exit: the approximation to the integral I.
+
+           7:  ABSERR -- DOUBLE PRECISION                            Output
+               On exit: an estimate of the modulus of the absolute error,
+               which should be an upper bound for |I-RESULT|.
+
+           8:  W(LW) -- DOUBLE PRECISION array                       Output
+               On exit: details of the computation, as described in
+               Section 8.
+
+           9:  LW -- INTEGER                                          Input
+               On entry:
+               the dimension of the array W as declared in the (sub)program
+               from which D01AMF is called.
+               The value of LW (together with that of LIW below) imposes a
+               bound on the number of sub-intervals into which the interval
+               of integration may be divided by the routine. The number of
+               sub-intervals cannot exceed LW/4. The more difficult the
+               integrand, the larger LW should be. Suggested value: a value
+               in the range 800 to 2000 is adequate for most problems.
+               Constraint: LW >= 4.
+
+          10:  IW(LIW) -- INTEGER array                              Output
+               On exit: IW(1) contains the actual number of sub-intervals
+               used. The rest of the array is used as workspace.
+
+          11:  LIW -- INTEGER                                         Input
+               On entry:
+               the dimension of the array IW as declared in the
+               (sub)program from which D01AMF is called.
+               The number of sub-intervals into which the interval of
+               integration may be divided cannot exceed LIW. Suggested
+               value: LIW = LW/4. Constraint: LIW >= 1.
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               The maximum number of subdivisions allowed with the given
+               workspace has been reached without the requested accuracy
+               requirements being achieved. Look at the integrand in order
+               to determine the integration difficulties. If the position
+               of a local difficulty within the interval can be determined
+               (e.g. a singularity of the integrand or its derivative, a
+               peak, a discontinuity, etc) you will probably gain from
+               splitting up the interval at this point and calling D01AMF
+               on the infinite subrange and an appropriate integrator on
+               the finite subrange. Alternatively, consider relaxing the
+               accuracy requirements specified by EPSABS and EPSREL, or
+               increasing the amount of workspace.
+
+          IFAIL= 2
+               Round-off error prevents the requested tolerance from being
+               achieved. The error may be underestimated. Consider
+               requesting less accuracy.
+
+          IFAIL= 3
+               Extremely bad local integrand behaviour causes a very strong
+               subdivision around one (or more) points of the interval.
+               The same advice applies as in the case of IFAIL = 1.
+
+          IFAIL= 4
+               The requested tolerance cannot be achieved, because the
+               extrapolation does not increase the accuracy satisfactorily;
+               the returned result is the best which can be obtained.  The
+               same advice applies as in the case of IFAIL = 1.
+
+          IFAIL= 5
+               The integral is probably divergent, or slowly convergent. It
+               must be noted that divergence can also occur with any other
+               non-zero value of IFAIL.
+
+          IFAIL= 6
+               On entry LW < 4,
+
+               or       LIW < 1,
+
+               or       INF /= -1, 1 or 2.
+               Please note that divergence can occur with any non-zero
+               value of IFAIL.
+
+          7. Accuracy
+
+          The routine cannot guarantee, but in practice usually achieves,
+          the following accuracy:
+
+                                  |I-RESULT|<=tol,
+
+          where
+
+                           tol=max{|EPSABS|,|EPSREL|*|I|},
+
+          and EPSABS and EPSREL are user-specified absolute and relative
+          error tolerances. Moreover it returns the quantity ABSERR, which,
+          in normal circumstances, satisfies
+
+                              |I-RESULT|<=ABSERR<=tol.
+
+          8. Further Comments
+
+          The time taken by the routine depends on the integrand and the
+          accuracy required.
+
+          If IFAIL /= 0 on exit, then the user may wish to examine the
+          contents of the array W, which contains the end-points of the
+          sub-intervals used by D01AMF along with the integral
+          contributions and error estimates over these sub-intervals.
+
+          Specifically, for i=1,2,...,n, let r  denote the approximation to
+                                              i
+          the value of the integral over the sub-interval [a ,b ] in the
+                                                            i  i
+          partition of [a,b] and e  be the corresponding absolute error
+                                  i
+                           b
+                            i                       n
+                           /                        --
+          estimate. Then,  | f(x)dx~=r  and RESULT= >  r  unless D01AMF
+                           /          i             --  i
+                           a                        i=1
+                            i
+          terminates while testing for divergence of the integral (see
+          Piessens et al [3] Section 3.4.3). In this case, RESULT (and
+          ABSERR) are taken to be the values returned from the
+          extrapolation process. The value of n is returned in IW(1), and
+          the values a , b , e  and r  are stored consecutively in the
+                      i   i   i      i
+          array W, that is:
+
+                    a =W(i),b =W(n+i),e =W(2n+i) and r =W(3n+i).
+                     i       i         i              i
+
+          Note: that this information applies to the integral transformed
+          to (0,1) as described in Section 3, not to the original integral.
+
+          9. Example
+
+          To compute
+
+                                  infty
+                                  /        1
+                                  |     --------dx.
+                                  /            
+                                  0     (x+1)\/x
+
+          The exact answer is (pi).
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd01anf}{NAG On-line Documentation: d01anf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D01ANF(3NAG)      Foundation Library (12/10/92)      D01ANF(3NAG)
+
+
+
+          D01 -- Quadrature                                          D01ANF
+                  D01ANF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D01ANF calculates an approximation to the sine or the cosine
+          transform of a function g over [a,b]:
+
+             b                                  b
+             /                                  /
+          I= |g(x)||,sin((omega)x)||,dx  or  I= |g(x)||,cos((omega)x)||,dx
+             /                                  /
+             a                                  a
+
+          (for a user-specified value of (omega)).
+
+          2. Specification
+
+                 SUBROUTINE D01ANF (G, A, B, OMEGA, KEY, EPSABS, EPSREL,
+                1                   RESULT, ABSERR, W, LW, IW, LIW, IFAIL)
+                 INTEGER          KEY, LW, IW(LIW), LIW, IFAIL
+                 DOUBLE PRECISION G, A, B, OMEGA, EPSABS, EPSREL, RESULT,
+                1                 ABSERR, W(LW)
+                 EXTERNAL         G
+
+          3. Description
+
+          D01ANF is based upon the QUADPACK routine QFOUR (Piessens et al
+          [3]). It is an adaptive routine, designed to integrate a function
+          of the form g(x)w(x), where w(x) is either sin((omega)x) or
+          cos((omega)x). If a sub-interval has length
+
+                                             -l
+                                     L=|b-a|2
+
+          then the integration over this sub-interval is performed by means
+          of a modified Clenshaw-Curtis procedure (Piessens and Branders
+          [2]) if L(omega)>4 and l<=20. In this case a Chebyshev-series
+          approximation of degree 24 is used to approximate g(x), while an
+          error estimate is computed from this approximation together with
+          that obtained using Chebyshev-series of degree 12. If the above
+          conditions do not hold then Gauss 7-point and Kronrod 15-point
+          rules are used. The algorithm, described in [3], incorporates a
+          global acceptance criterion (as defined in Malcolm and Simpson
+          [1]) together with the (epsilon)-algorithm Wynn [4] to perform
+          extrapolation. The local error estimation is described in [3].
+
+          4. References
+
+          [1]   Malcolm M A and Simpson R B (1976) Local Versus Global
+                Strategies for Adaptive Quadrature. ACM Trans. Math. Softw.
+                1 129--146.
+
+          [2]   Piessens R and Branders M (1975) Algorithm 002. Computation
+                of Oscillating Integrals. J. Comput. Appl. Math. 1 153--164.
+
+          [3]   Piessens R, De Doncker-Kapenga E, Uberhuber C and Kahaner D
+                (1983) QUADPACK, A Subroutine Package for Automatic
+                Integration. Springer-Verlag.
+
+          [4]   Wynn P (1956) On a Device for Computing the e (S )
+                                                             m  n
+                Transformation. Math. Tables Aids Comput. 10 91--96.
+
+          5. Parameters
+
+           1:  G -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                                         External Procedure
+               G must return the value of the function g at a given point.
+
+               Its specification is:
+
+                      DOUBLE PRECISION FUNCTION G (X)
+                      DOUBLE PRECISION X
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the point at which the function g must be
+                    evaluated.
+               G must be declared as EXTERNAL in the (sub)program from
+               which D01ANF is called. Parameters denoted as Input
+               must not be changed by this procedure.
+
+           2:  A -- DOUBLE PRECISION                                  Input
+               On entry: the lower limit of integration, a.
+
+           3:  B -- DOUBLE PRECISION                                  Input
+               On entry: the upper limit of integration, b. It is not
+               necessary that a<b.
+
+           4:  OMEGA -- DOUBLE PRECISION                              Input
+               On entry: the parameter (omega) in the weight function of
+               the transform.
+
+           5:  KEY -- INTEGER                                         Input
+               On entry: indicates which integral is to be computed:
+                    if KEY = 1, w(x)=cos((omega)x);
+
+                    if KEY = 2, w(x)=sin((omega)x).
+                Constraint: KEY = 1 or 2.
+
+           6:  EPSABS -- DOUBLE PRECISION                             Input
+               On entry: the absolute accuracy required. If EPSABS is
+               negative, the absolute value is used. See Section 7.
+
+           7:  EPSREL -- DOUBLE PRECISION                             Input
+               On entry: the relative accuracy required. If EPSREL is
+               negative, the absolute value is used. See Section 7.
+
+           8:  RESULT -- DOUBLE PRECISION                            Output
+               On exit: the approximation to the integral I.
+
+           9:  ABSERR -- DOUBLE PRECISION                            Output
+               On exit: an estimate of the modulus of the absolute error,
+               which should be an upper bound for |I-RESULT|.
+
+          10:  W(LW) -- DOUBLE PRECISION array                       Output
+               On exit: details of the computation, as described in
+               Section 8.
+
+          11:  LW -- INTEGER                                          Input
+               On entry:
+               the dimension of the array W as declared in the (sub)program
+               from which D01ANF is called.
+               The value of LW (together with that of LIW below) imposes a
+               bound on the number of sub-intervals into which the interval
+               of integration may be divided by the routine. The number of
+               sub-intervals cannot exceed LW/4. The more difficult the
+               integrand, the larger LW should be. Suggested value: a value
+               in the range 800 to 2000 is adequate for most problems.
+               Constraint: LW >= 4.
+
+          12:  IW(LIW) -- INTEGER array                              Output
+               On exit: IW(1) contains the actual number of sub-intervals
+               used. The rest of the array is used as workspace.
+
+          13:  LIW -- INTEGER                                         Input
+               On entry:
+               the dimension of the array IW as declared in the
+               (sub)program from which D01ANF is called.
+               The number of sub-intervals into which the interval of
+               integration may be divided cannot exceed LIW/2. Suggested
+               value: LIW = LW/2. Constraint: LIW >= 2.
+
+          14:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               The maximum number of subdivisions allowed with the given
+               workspace has been reached without the accuracy requested
+               being achieved. Look at the integrand in order to determine
+               the integration difficulties. If the position of a local
+               difficulty within the interval can be determined (e.g. a
+               singularity of the integrand or its derivative, a peak, a
+               discontinuity, etc) you will probably gain from splitting up
+               the interval at this point and calling the integrator on the
+               subranges. If necessary, another integrator, which is
+               designed for handling the type of difficulty involved, must
+               be used. Alternatively consider relaxing the accuracy
+               requirements specified by EPSABS and EPSREL, or increasing
+               amount of workspace.
+
+          IFAIL= 2
+               Round-off error prevents the requested tolerance from being
+               achieved. The error may be underestimated. Consider
+               requesting less accuracy.
+
+          IFAIL= 3
+               Extremely bad local behaviour of g(x) causes a very strong
+               subdivision around one (or more) points of the interval.
+               The same advice applies as in the case of IFAIL = 1.
+
+          IFAIL= 4
+               The requested tolerance cannot be achieved because the
+               extrapolation does not increase the accuracy satisfactorily;
+               the returned result is the best which can be obtained.  The
+               same advice applies as in the case of IFAIL = 1.
+
+          IFAIL= 5
+               The integral is probably divergent, or slowly convergent. It
+               must be noted that divergence can occur with any non-zero
+               value of IFAIL.
+
+          IFAIL= 6
+               On entry KEY < 1,
+
+               or       KEY > 2.
+
+          IFAIL= 7
+               On entry LW < 4,
+
+               or       LIW < 2.
+
+          7. Accuracy
+
+          The routine cannot guarantee, but in practice usually achieves,
+          the following accuracy:
+
+                                  |I-RESULT|<=tol,
+
+          where
+
+                           tol=max{|EPSABS|,|EPSREL|*|I|},
+
+          and EPSABS and EPSREL are user-specified absolute and relative
+          tolerances. Moreover it returns the quantity ABSERR, which, in
+          normal circumstances, satisfies
+
+                              |I-RESULT|<=ABSERR<=tol.
+
+          8. Further Comments
+
+          The time taken by the routine depends on the integrand and on the
+          accuracy required.
+
+          If IFAIL /= 0 on exit, then the user may wish to examine the
+          contents of the array W, which contains the end-points of the
+          sub-intervals used by D01ANF along with the integral
+          contributions and error estimates over these sub-intervals.
+
+          Specifically, for i=1,2,...,n, let r  denote the approximation to
+                                              i
+          the value of the integral over the sub-interval [a ,b ] in the
+                                                            i  i
+          partition of [a,b] and e  be the corresponding absolute error
+                                  i
+                           b
+                            i                           n
+                           /                            --
+          estimate. Then,  | g(x)w(x)dx~=r  and RESULT= >  r  unless D01ANF
+                           /              i             --  i
+                           a                            i=1
+                            i
+          terminates while testing for divergence of the integral (see
+          Piessens et al [3] Section 3.4.3). In this case, RESULT (and
+          ABSERR) are taken to be the values returned from the
+          extrapolation process. The value of n is returned in IW(1), and
+          the values a , b , e  and r  are stored consecutively in the
+                      i   i   i      i
+          array W, that is:
+
+               a =W(i),
+                i
+
+               b =W(n+i),
+                i
+
+               e =W(2n+i) and
+                i
+
+               r =W(3n+i).
+                i
+
+          9. Example
+
+          To compute
+
+                                 1
+                                 /
+                                 |ln(x)sin(10(pi)x)dx.
+                                 /
+                                 0
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd01apf}{NAG On-line Documentation: d01apf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D01APF(3NAG)      Foundation Library (12/10/92)      D01APF(3NAG)
+
+
+
+          D01 -- Quadrature                                          D01APF
+                  D01APF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D01APF is an adaptive integrator which calculates an
+          approximation to the integral of a function g(x)w(x) over a
+          finite interval [a,b]:
+
+                                      b
+                                      /
+                                   I= |g(x)w(x)dx
+                                      /
+                                      a
+
+          where the weight function w has end-point singularities of
+          algebraico-logarithmic type.
+
+          2. Specification
+
+                 SUBROUTINE D01APF (G, A, B, ALFA, BETA, KEY, EPSABS,
+                1                   EPSREL, RESULT, ABSERR, W, LW, IW, LIW,
+                2                   IFAIL)
+                 INTEGER          KEY, LW, IW(LIW), LIW, IFAIL
+                 DOUBLE PRECISION G, A, B, ALFA, BETA, EPSABS, EPSREL,
+                1                 RESULT, ABSERR, W(LW)
+                 EXTERNAL         G
+
+          3. Description
+
+          D01APF is based upon the QUADPACK routine QAWSE (Piessens et al
+          [3]) and integrates a function of the form g(x)w(x), where the
+          weight function w(x) may have algebraico-logarithmic
+          singularities at the end-points a and/or b. The strategy is a
+          modification of that in D01AKF. We start by bisecting the
+          original interval and applying modified Clenshaw-Curtis
+          integration of orders 12 and 24 to both halves. Clenshaw-Curtis
+          integration is then used on all sub-intervals which have a or b
+          as one of their end-points (Piessens et al [2]). On the other
+          sub-intervals Gauss-Kronrod (7-15 point) integration is carried
+          out.
+
+          A 'global' acceptance criterion (as defined by Malcolm and
+          Simpson [1]) is used. The local error estimation control is
+          described by Piessens et al [3].
+
+          4. References
+
+          [1]   Malcolm M A and Simpson R B (1976) Local Versus Global
+                Strategies for Adaptive Quadrature. ACM Trans. Math. Softw.
+                1 129--146.
+
+          [2]   Piessens R, Mertens I and Branders M (1974) Integration of
+                Functions having End-point Singularities. Angewandte
+                Informatik. 16 65--68.
+
+          [3]   Piessens R, De Doncker-Kapenga E, Uberhuber C and Kahaner D
+                (1983) QUADPACK, A Subroutine Package for Automatic
+                Integration. Springer-Verlag.
+
+          5. Parameters
+
+           1:  G -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                                        External Procedure
+               G must return the value of the function g at a given point
+               X.
+
+               Its specification is:
+
+                      DOUBLE PRECISION FUNCTION G (X)
+                      DOUBLE PRECISION X
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the point at which the function g must be
+                    evaluated.
+               G must be declared as EXTERNAL in the (sub)program from
+               which D01APF is called. Parameters denoted as Input
+               must not be changed by this procedure.
+
+           2:  A -- DOUBLE PRECISION                                  Input
+               On entry: the lower limit of integration, a.
+
+           3:  B -- DOUBLE PRECISION                                  Input
+               On entry: the upper limit of integration, b. Constraint: B
+               > A.
+
+           4:  ALFA -- DOUBLE PRECISION                               Input
+               On entry: the parameter (alpha) in the weight function.
+               Constraint: ALFA>-1.
+
+           5:  BETA -- DOUBLE PRECISION                               Input
+               On entry: the parameter (beta) in the weight function.
+               Constraint: BETA>-1.
+
+           6:  KEY -- INTEGER                                         Input
+               On entry: indicates which weight function is to be used:
+                                          (alpha)     (beta)
+                    if KEY = 1, w(x)=(x-a)       (b-x)
+
+                                          (alpha)     (beta)
+                    if KEY = 2, w(x)=(x-a)       (b-x)      ln(x-a)
+
+                                          (alpha)     (beta)
+                    if KEY = 3, w(x)=(x-a)       (b-x)      ln(b-x)
+
+                                          (alpha)     (beta)
+                    if KEY = 4, w(x)=(x-a)       (b-x)     ln(x-a)ln(b-x)
+
+               Constraint: KEY = 1, 2, 3 or 4
+
+           7:  EPSABS -- DOUBLE PRECISION                             Input
+               On entry: the absolute accuracy required. If EPSABS is
+               negative, the absolute value is used. See Section 7.
+
+           8:  EPSREL -- DOUBLE PRECISION                             Input
+               On entry: the relative accuracy required. If EPSREL is
+               negative, the absolute value is used. See Section 7.
+
+           9:  RESULT -- DOUBLE PRECISION                            Output
+               On exit: the approximation to the integral I.
+
+          10:  ABSERR -- DOUBLE PRECISION                            Output
+               On exit: an estimate of the modulus of the absolute error,
+               which should be an upper bound for |I-RESULT|.
+
+          11:  W(LW) -- DOUBLE PRECISION array                       Output
+               On exit: details of the computation, as described in
+               Section 8.
+
+          12:  LW -- INTEGER                                          Input
+               On entry:
+               the dimension of the array W as declared in the (sub)program
+               from which D01APF is called.
+               The value of LW (together with that of LIW below) imposes a
+               bound on the number of sub-intervals into which the interval
+               of integration may be divided by the routine. The number of
+               sub-intervals cannot exceed LW/4. The more difficult the
+               integrand, the larger LW should be. Suggested value: LW =
+               800 to 2000 is adequate for most problems. Constraint: LW >=
+               8.
+
+          13:  IW(LIW) -- INTEGER array                              Output
+               On exit: IW(1) contains the actual number of sub-intervals
+               used. The rest of the array is used as workspace.
+
+          14:  LIW -- INTEGER                                         Input
+               On entry:
+               the dimension of the array IW as declared in the
+               (sub)program from which D01APF is called.
+               The number of sub-intervals into which the interval of
+               integration may be divided cannot exceed LIW. Suggested
+               value: LIW = LW/4. Constraint: LIW >= 2.
+
+          15:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               The maximum number of subdivisions allowed with the given
+               workspace has been reached without the accuracy requirements
+               being achieved. Look at the integrand in order to determine
+               the integration difficulties. If the position of a
+               discontinuity or a singularity of algebraico-logarithmic
+               type within the interval can be determined, the interval
+               must be split up at this point and the integrator called on
+               the subranges. If necessary, another integrator, which is
+               designed for handling the difficulty involved, must be used.
+               Alternatively consider relaxing the accuracy requirements
+               specified by EPSABS and EPSREL, or increasing the amount of
+               workspace.
+
+          IFAIL= 2
+               Round-off error prevents the requested tolerance from being
+               achieved. Consider  requesting less accuracy.
+
+          IFAIL= 3
+               Extremely bad local integrand behaviour causes a very strong
+               subdivision around one (or more) points of the interval.
+               The same advice applies as in the case of IFAIL = 1.
+
+          IFAIL= 4
+               On entry B <= A,
+
+               or       ALFA <= -1,
+
+               or       BETA <= -1,
+
+               or       KEY < 1,
+
+               or       KEY > 4.
+
+          IFAIL= 5
+               On entry LW < 8,
+
+               or       LIW < 2.
+
+          7. Accuracy
+
+          The routine cannot guarantee, but in practice usually achieves,
+          the following accuracy:
+
+                                  |I-RESULT|<=tol,
+
+          where
+
+                           tol=max{|EPSABS|,|EPSREL|*|I|},
+
+          and EPSABS and EPSREL are user-specified absolute and relative
+          error tolerances.
+
+          Moreover it returns the quantity ABSERR which, in normal
+          circumstances, satisfies:
+
+                              |I-RESULT|<=ABSERR<=tol.
+
+          8. Further Comments
+
+          The time taken by the routine depends on the integrand and on the
+          accuracy required.
+
+          If IFAIL /= 0 on exit, then the user may wish to examine the
+          contents of the array W, which contains the end-points of the
+          sub-intervals used by D01APF along with the integral
+          contributions and error estimates over these sub-intervals.
+
+          Specifically, for i=1,2,...,n, let r  denote the approximation to
+                                              i
+          the value of the integral over the sub-interval [a ,b ] in the
+                                                            i  i
+          partition of [a,b] and e  be the corresponding absolute error
+                                  i
+                           b
+                            i                           n
+                           /                            --
+          estimate. Then,  | f(x)w(x)dx~=r  and RESULT= >  r . The value of
+                           /              i             --  i
+                           a                            i=1
+                            i
+          n is returned in IW(1), and the values a , b , e  and r  are
+                                                  i   i   i      i
+          stored consecutively in the array W, that is:
+
+               a =W(i),
+                i
+
+               b =W(n+i),
+                i
+
+               e =W(2n+i),
+                i
+
+               r =W(3n+i).
+                i
+
+          9. Example
+
+          To compute:
+
+                               1
+                               /
+                               |ln(x)cos(10(pi)x)dx  and
+                               /
+                               0
+
+                                    1
+                                    / sin(10x)
+                                    | --------dx.
+                                    /   
+                                    0 \/x(1-x)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd01aqf}{NAG On-line Documentation: d01aqf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D01AQF(3NAG)      Foundation Library (12/10/92)      D01AQF(3NAG)
+
+
+
+          D01 -- Quadrature                                          D01AQF
+                  D01AQF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D01AQF calculates an approximation to the Hilbert transform of a
+          function g(x) over [a,b]:
+
+                                        b
+                                        / g(x)
+                                     I= | ----dx
+                                        / x-c
+                                        a
+
+          for user-specified values of a, b and c.
+
+          2. Specification
+
+                 SUBROUTINE D01AQF (G, A, B, C, EPSABS, EPSREL, RESULT,
+                1                   ABSERR, W, LW, IW, LIW, IFAIL)
+                 INTEGER          LW, IW(LIW), LIW, IFAIL
+                 DOUBLE PRECISION G, A, B, C, EPSABS, EPSREL, RESULT,
+                1                 ABSERR, W(LW)
+                 EXTERNAL         G
+
+          3. Description
+
+          D01AQF is based upon the QUADPACK routine QAWC (Piessens et al
+          [3]) and integrates a function of the form g(x)w(x), where the
+          weight function
+
+                                             1
+                                      w(x)= ---
+                                            x-c
+
+          is that of the Hilbert transform. (If a<c<b the integral has to
+          be interpreted in the sense of a Cauchy principal value.) It is
+          an adaptive routine which employs a 'global' acceptance criterion
+          (as defined by Malcolm and Simpson [1]). Special care is taken to
+          ensure that c is never the end-point of a sub-interval (Piessens
+          et al[2]). On each sub-interval (c ,c ) modified Clenshaw-Curtis
+                                            1  2
+          integration of orders 12 and 24 is performed if c -d<=c<=c +d
+                                                           1        2
+          where d=(c -c )/20. Otherwise the Gauss 7-point and Kronrod 15-
+                    2  1
+          point rules are used. The local error estimation is described by
+          Piessens et al [3].
+
+          4. References
+
+          [1]   Malcolm M A and Simpson R B (1976) Local Versus Global
+                Strategies for Adaptive Quadrature. ACM Trans. Math. Softw.
+                1 129--146.
+
+          [2]   Piessens R, Van Roy-Branders M and Mertens I (1976) The
+                Automatic Evaluation of Cauchy Principal Value Integrals.
+                Angewandte Informatik. 18 31--35.
+
+          [3]   Piessens R, De Doncker-Kapenga E, Uberhuber C and Kahaner D
+                (1983) QUADPACK, A Subroutine Package for Automatic
+                Integration. Springer-Verlag.
+
+          5. Parameters
+
+           1:  G -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                                         External Procedure
+               G must return the value of the function g at a given point.
+
+               Its specification is:
+
+                      DOUBLE PRECISION FUNCTION G (X)
+                      DOUBLE PRECISION X
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the point at which the function g must be
+                    evaluated.
+               G must be declared as EXTERNAL in the (sub)program from
+               which D01AQF is called. Parameters denoted as Input
+               must not be changed by this procedure.
+
+           2:  A -- DOUBLE PRECISION                                  Input
+               On entry: the lower limit of integration, a.
+
+           3:  B -- DOUBLE PRECISION                                  Input
+               On entry: the upper limit of integration, b. It is not
+               necessary that a<b.
+
+           4:  C -- DOUBLE PRECISION                                  Input
+               On entry: the parameter c in the weight function.
+               Constraint: C must not equal A or B.
+
+           5:  EPSABS -- DOUBLE PRECISION                             Input
+               On entry: the absolute accuracy required. If EPSABS is
+               negative, the absolute value is used. See Section 7.
+
+           6:  EPSREL -- DOUBLE PRECISION                             Input
+               On entry: the relative accuracy required. If EPSREL is
+               negative, the absolute value is used. See Section 7.
+
+           7:  RESULT -- DOUBLE PRECISION                            Output
+               On exit: the approximation to the integral I.
+
+           8:  ABSERR -- DOUBLE PRECISION                            Output
+               On exit: an estimate of the modulus of the absolute error,
+               which should be an upper bound for |I-RESULT|.
+
+           9:  W(LW) -- DOUBLE PRECISION array                       Output
+               On exit: details of the computation, as described in
+               Section 8.
+
+          10:  LW -- INTEGER                                          Input
+               On entry:
+               the dimension of the array W as declared in the (sub)program
+               from which D01AQF is called.
+               The value of LW (together with that of LIW below) imposes a
+               bound on the number of sub-intervals into which the interval
+               of integration may be divided by the routine. The number of
+               sub-intervals cannot exceed LW/4. The more difficult the
+               integrand, the larger LW should be. Suggested value: LW =
+               800 to 2000 is adequate for most problems. Constraint: LW >=
+               4.
+
+          11:  IW(LIW) -- INTEGER array                              Output
+               On exit: IW(1) contains the actual number of sub-intervals
+               used. The rest of the array is used as workspace.
+
+          12:  LIW -- INTEGER                                         Input
+               On entry:
+               the dimension of the array IW as declared in the
+               (sub)program from which D01AQF is called.
+               The number of sub-intervals into which the interval of
+               integration may be divided cannot exceed LIW. Suggested
+               value: LIW = LW/4. Constraint: LIW >= 1.
+
+          13:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               The maximum number of subdivisions allowed with the given
+               workspace has been reached without the accuracy requirements
+               being achieved. Look at the integrand in order to determine
+               the integration difficulties. Another integrator which is
+               designed for handling the type of difficulty involved, must
+               be used. Alternatively consider relaxing the accuracy
+               requirements specified by EPSABS and EPSREL, or increasing
+               the workspace.
+
+          IFAIL= 2
+               Round-off error prevents the requested tolerance from being
+               achieved. Consider  requesting less accuracy.
+
+          IFAIL= 3
+               Extremely bad local behaviour of g(x) causes a very strong
+               subdivision around one (or more) points of the interval. The
+               same advice applies as in the case of IFAIL = 1.
+
+          IFAIL= 4
+               On entry C = A or C = B.
+
+          IFAIL= 5
+               On entry LW < 4,
+
+               or       LIW < 1.
+
+          7. Accuracy
+
+          The routine cannot guarantee, but in practice usually achieves,
+          the following accuracy:
+
+                                  |I-RESULT|<=tol,
+
+          where
+
+                           tol=max{|EPSABS|,|EPSREL|*|I|},
+
+          and EPSABS and EPSREL are user-specified absolute and relative
+          error tolerances. Moreover it returns the quantity ABSERR which,
+          in normal circumstances satisfies:
+
+                              |I-RESULT|<=ABSERR<=tol.
+
+          8. Further Comments
+
+          The time taken by the routine depends on the integrand and on the
+          accuracy required.
+
+          If IFAIL /= 0 on exit, then the user may wish to examine the
+          contents of the array W, which contains the end-points of the
+          sub-intervals used by D01AQF along with the integral
+          contributions and error estimates over these sub-intervals.
+
+          Specifically, for i=1,2,...,n, let r  denote the approximation to
+                                              i
+          the value of the integral over the sub-interval [a ,b ] in the
+                                                            i  i
+          partition of [a,b] and e  be the corresponding absolute error
+                                  i
+                           b
+                            i                           n
+                           /                            --
+          estimate. Then,  | g(x)w(x)dx~=r  and RESULT= >  r . The value of
+                           /              i             --  i
+                           a                            i=1
+                            i
+          n is returned in IW(1), and the values a , b , e  and r  are
+                                                  i   i   i      i
+          stored consecutively in the array W, that is:
+
+               a =W(i),
+                i
+
+               b =W(n+i),
+                i
+
+               e =W(2n+i) and
+                i
+
+               r =W(3n+i).
+                i
+
+          9. Example
+
+          To compute the Cauchy principal value of
+
+                                 1
+                                 /         dx
+                                 |  ----------------.
+                                 /    2     2     1
+                                 -1 (x +0.01 )(x- -)
+                                                  2
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd01asf}{NAG On-line Documentation: d01asf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D01ASF(3NAG)      Foundation Library (12/10/92)      D01ASF(3NAG)
+
+
+
+          D01 -- Quadrature                                          D01ASF
+                  D01ASF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D01ASF calculates an approximation to the sine or the cosine
+          transform of a function g over [a,infty):
+
+
+             infty
+             /
+          I= |    g(x)||,sin((omega)x)dx  or  I=
+             /
+             a
+           infty
+           /
+           |    g(x)||,cos((omega)x)dx
+           /
+           a
+
+          (for a user-specified value of (omega)).
+
+          2. Specification
+
+                 SUBROUTINE D01ASF (G, A, OMEGA, KEY, EPSABS, RESULT,
+                1                   ABSERR, LIMLST, LST, ERLST, RSLST,
+                2                   IERLST, W, LW, IW, LIW, IFAIL)
+                 INTEGER          KEY, LIMLST, LST, IERLST(LIMLST), LW, IW
+                1                 (LIW), LIW, IFAIL
+                 DOUBLE PRECISION G, A, OMEGA, EPSABS, RESULT, ABSERR, ERLST
+                1                 (LIMLST), RSLST(LIMLST), W(LW)
+                 EXTERNAL         G
+
+          3. Description
+
+          D01ASF is based upon the QUADPACK routine QAWFE (Piessens et al
+          [2]). It is an adaptive routine, designed to integrate a function
+          of the form g(x)w(x) over a semi-infinite interval, where w(x) is
+          either sin((omega)x) or cos((omega)x). Over successive intervals
+
+                          C =[a+(k-1)c,a+kc],   k=1,2,...,LST
+                           k
+
+          integration is performed by the same algorithm as is used by
+          D01ANF. The intervals C  are of constant length
+                                 k
+
+                    c={2[|(omega)|]+1}(pi)/|(omega)|,  (omega)/=0
+
+          where [|(omega)|] represents the largest integer less than or
+          equal to |(omega)|. Since c equals an odd number of half periods,
+          the integral contributions over succeeding intervals will
+          alternate in sign when the function g is positive and
+          monotonically decreasing over [a,infty). The algorithm, described
+          by [2], incorporates a global acceptance criterion (as defined by
+          Malcolm and Simpson [1]) together with the (epsilon)-algorithm
+          (Wynn [3]) to perform extrapolation. The local error estimation
+          is described by Piessens et al [2].
+
+          If (omega)=0 and KEY = 1, the routine uses the same algorithm as
+          D01AMF (with EPSREL = 0.0).
+
+          In contrast to the other routines in Chapter D01, D01ASF works
+          only with a user-specified absolute error tolerance (EPSABS).
+          Over the interval C  it attempts to satisfy the absolute accuracy
+                             k
+          requirement
+                                   EPSA =U *EPSABS
+                                       k  k
+                         k-1
+          where U =(1-p)p   , for k=1,2,... and p=0.9.
+                 k
+
+          However, when difficulties occur during the integration over the
+          kth sub-interval C  such that the error flag IERLST(k) is non-
+                            k
+          zero, the accuracy requirement over subsequent intervals is
+          relaxed. See Piessens et al [2] for more details.
+
+          4. References
+
+          [1]   Malcolm M A and Simpson R B (1976) Local Versus Global
+                Strategies for Adaptive Quadrature. ACM Trans. Math. Softw.
+                1 129--146.
+
+          [2]   Piessens R, De Doncker-Kapenga E, Uberhuber C and Kahaner D
+                (1983) QUADPACK, A Subroutine Package for Automatic
+                Integration. Springer-Verlag.
+
+          [3]   Wynn P (1956) On a Device for Computing the e (S )
+                                                             m  n
+                Transformation. Math. Tables Aids Comput. 10 91--96.
+
+          5. Parameters
+
+           1:  G -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                                         External Procedure
+               G must return the value of the function g at a given point.
+
+               Its specification is:
+
+                      DOUBLE PRECISION FUNCTION G (X)
+                      DOUBLE PRECISION X
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the point at which the function g must be
+                    evaluated.
+               G must be declared as EXTERNAL in the (sub)program from
+               which D01ASF is called. Parameters denoted as Input
+               must not be changed by this procedure.
+
+           2:  A -- DOUBLE PRECISION                                  Input
+               On entry: the lower limit of integration, a.
+
+           3:  OMEGA -- DOUBLE PRECISION                              Input
+               On entry: the parameter (omega) in the weight function of
+               the transform.
+
+           4:  KEY -- INTEGER                                         Input
+               On entry: indicates which integral is to be computed:
+                    if KEY = 1, w(x)=cos((omega)x);
+
+                    if KEY = 2, w(x)=sin((omega)x).
+                Constraint: KEY = 1 or 2.
+
+           5:  EPSABS -- DOUBLE PRECISION                             Input
+               On entry: the absolute accuracy requirement. If EPSABS is
+               negative, the absolute value is used. See Section 7.
+
+           6:  RESULT -- DOUBLE PRECISION                            Output
+               On exit: the approximation to the integral I.
+
+           7:  ABSERR -- DOUBLE PRECISION                            Output
+               On exit: an estimate of the modulus of the absolute error,
+               which should be an upper bound for |I-RESULT|.
+
+           8:  LIMLST -- INTEGER                                      Input
+               On entry: an upper bound on the number of intervals C
+                                                                     k
+               needed for the integration. Suggested value: LIMLST = 50 is
+               adequate for most problems. Constraint: LIMLST >= 3.
+
+           9:  LST -- INTEGER                                        Output
+               On exit: the number of intervals C  actually used for the
+                                                  k
+               integration.
+
+          10:  ERLST(LIMLST) -- DOUBLE PRECISION array               Output
+               On exit: ERLST(k) contains the error estimate corresponding
+               to the integral contribution over the interval C , for
+                                                               k
+               k=1,2,...,LST.
+
+          11:  RSLST(LIMLST) -- DOUBLE PRECISION array               Output
+               On exit: RSLST(k) contains the integral contribution over
+               the interval C  for k=1,2,...,LST.
+                             k
+
+          12:  IERLST(LIMLST) -- INTEGER array                       Output
+               On exit: IERLST(k) contains the error flag corresponding to
+               RSLST(k), for k=1,2,...,LST. See Section 6.
+
+          13:  W(LW) -- DOUBLE PRECISION array                    Workspace
+
+          14:  LW -- INTEGER                                          Input
+               On entry:
+               the dimension of the array W as declared in the (sub)program
+               from which D01ASF is called.
+               The value of LW (together with that of LIW below) imposes a
+               bound on the number of sub-intervals into which each
+               interval C  may be divided by the routine. The number of
+                         k
+               sub-intervals cannot exceed LW/4. The more difficult the
+               integrand, the larger LW should be. Suggested value: a value
+               in the range 800 to 2000 is adequate for most problems.
+               Constraint: LW >= 4.
+
+          15:  IW(LIW) -- INTEGER array                              Output
+               On exit: IW(1) contains the maximum number of sub-intervals
+               actually used for integrating over any of the intervals C .
+                                                                        k
+               The rest of the array is used as workspace.
+
+          16:  LIW -- INTEGER                                         Input
+               On entry:
+               the dimension of the array IW as declared in the
+               (sub)program from which D01ASF is called.
+               The number of sub-intervals into which each interval C  may
+                                                                     k
+               be divided cannot exceed LIW/2. Suggested value: LIW = LW/2.
+               Constraint: LIW >= 2.
+
+          17:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               The maximum number of subdivisions allowed with the given
+               workspace has been reached without the accuracy requirements
+               being achieved. Look at the integrand in order to determine
+               the integration difficulties. If the position of a local
+               difficulty within the interval can be determined (e.g. a
+               singularity of the integrand or its derivative, a peak, a
+               discontinuity, etc) you will probably gain from splitting up
+               the interval at this point and calling D01ASF on the
+               infinite subrange and an appropriate integrator on the
+               finite subrange. Alternatively, consider relaxing the
+               accuracy requirements specified by EPSABS or increasing the
+               amount of workspace.
+
+          IFAIL= 2
+               Round-off error prevents the requested tolerance from being
+               achieved. The error may be underestimated. Consider
+               requesting less accuracy.
+
+          IFAIL= 3
+               Extremely bad local integrand behaviour causes a very strong
+               subdivision around one (or more) points of the interval.
+               The same advice applies as in the case of IFAIL = 1.
+
+          IFAIL= 4
+               The requested tolerance cannot be achieved, because the
+               extrapolation does not increase the accuracy satisfactorily;
+               the returned result is the best which can be obtained.  The
+               same advice applies as in the case of IFAIL = 1.
+
+          IFAIL= 5
+               The integral is probably divergent, or slowly convergent.
+               Please note that divergence can occur with any non-zero
+               value of IFAIL.
+
+          IFAIL= 6
+               On entry KEY < 1,
+
+               or       KEY > 2,
+
+               or       LIMLST < 3.
+
+          IFAIL= 7
+               Bad integration behaviour occurs within one or more of the
+               intervals C . Location and type of the difficulty involved
+                          k
+               can be determined from the vector IERLST (see below).
+
+          IFAIL= 8
+               Maximum number of intervals C  (= LIMLST) allowed has been
+                                            k
+               achieved. Increase the value of LIMLST to allow more cycles.
+
+          IFAIL= 9
+               The extrapolation table constructed for convergence
+               acceleration of the series formed by the integral
+               contribution over the intervals C , does not converge to the
+                                                k
+               required accuracy.
+
+          IFAIL= 10
+               On entry LW < 4,
+
+               or       LIW < 2.
+
+          In the cases IFAIL = 7, 8 or 9, additional information about the
+          cause of the error can be obtained from the array IERLST, as
+          follows:
+
+          IERLST(k)=1
+                The maximum number of subdivisions = min(LW/4,LIW/2) has
+                been achieved on the kth interval.
+
+          IERLST(k)=2
+                Occurrence of round-off error is detected and prevents the
+                tolerance imposed on the kth interval from being achieved.
+
+          IERLST(k)=3
+                Extremely bad integrand behaviour occurs at some points of
+                the kth interval.
+
+          IERLST(k)=4
+                The integration procedure over the kth interval does not
+                converge (to within the required accuracy) due to round-off
+                in the extrapolation procedure invoked on this interval. It
+                is assumed that the result on this interval is the best
+                which can be obtained.
+
+          IERLST(k)=5
+                The integral over the kth interval is probably divergent or
+                slowly convergent. It must be noted that divergence can
+                occur with any other value of IERLST(k).
+
+          7. Accuracy
+
+          The routine cannot guarantee, but in practice usually achieves,
+          the following accuracy:
+
+                                |I-RESULT|<=|EPSABS|,
+
+          where EPSABS is the user-specified absolute error tolerance.
+          Moreover, it returns the quantity ABSERR, which, in normal
+          circumstances, satisfies
+
+                            |I-RESULT|<=ABSERR<=|EPSABS|.
+
+          8. Further Comments
+
+          None.
+
+          9. Example
+
+          To compute
+
+                               infty
+                               /      1
+                               |     ---cos((pi)x/2)dx.
+                               /       
+                               0     \/x
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd01bbf}{NAG On-line Documentation: d01bbf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D01BBF(3NAG)                 D01BBF                  D01BBF(3NAG)
+
+
+
+     D01 -- Quadrature                                          D01BBF
+             D01BBF -- NAG Foundation Library Routine Document
+
+     Note: Before using this routine, please read the Users' Note for
+     your implementation to check implementation-dependent details.
+     The symbol (*) after a NAG routine name denotes a routine that is
+     not included in the Foundation Library.
+
+     Note for users via the AXIOM system: the interface to this routine
+     has been enhanced for use with AXIOM and is slightly different to
+     that offered in the standard version of the Foundation Library.
+
+     1. Purpose
+
+     D01BBF returns the weights and abscissae appropriate to a
+     Gaussian quadrature formula with a specified number of abscissae.
+     The formulae provided are Gauss-Legendre, Gauss-Rational, Gauss-
+     Laguerre and Gauss-Hermite.
+
+     2. Specification
+
+            SUBROUTINE D01BBF (A, B, ITYPE, N, WEIGHT, ABSCIS, GTYPE,
+           1                   IFAIL)
+            INTEGER          ITYPE, N, GTYPE, IFAIL
+            DOUBLE PRECISION A, B, WEIGHT(N), ABSCIS(N)
+
+     3. Description
+
+     This routine returns the weights and abscissae for use in the
+     Gaussian quadrature of a function f(x). The quadrature takes the
+     form
+
+                                  n
+                                  --
+                               S= >  w f(x )
+                                  --  i   i
+                                  i=1
+
+     where w  are the weights and x  are the abscissae (see Davis and
+            i                      i
+     Rabinowitz [1], Froberg [2], Ralston [3] or Stroud and Secrest
+     [4]).
+
+     Weights and abscissae are available for Gauss-Legendre, Gauss-
+     Rational, Gauss-Laguerre and Gauss-Hermite quadrature, and for a
+     selection of values of n (see Section 5).
+
+     (a)   Gauss-Legendre Quadrature:
+                                       b
+                                       /
+                                   S~= |f(x)dx
+                                       /
+
+                                       a
+           where a and b are finite and it will be exact for any
+           function of the form
+                                       2n-1
+                                       --     i
+                                 f(x)= >   c x
+                                       --   i
+                                       i=0
+
+     (b)   Gauss-Rational quadrature:
+                infty                           a
+                /                               /
+            S~= |    f(x)dx  (a+b>0)   or   S~= |     f(x)dx  (a+b<0)
+                /                               /
+                a                               -infty
+           and will be exact for any function of the form
+                                          2n-1
+                                          --              i
+                                          >   c      (x+b)
+                            2n+1   c      --   2n+1-i
+                            --      i     i=0
+                      f(x)= >    ------= ------------------
+                            --        i           2n+1
+                            i=2  (x+b)       (x+b)
+
+     (c)   Gauss-Laguerre quadrature, adjusted weights option:
+                 infty                          a
+                 /                              /
+             S~= |    f(x)dx  (b>0<)   or   S~= |     f(x)dx  (b<0)
+                 /                              /
+                 a                              -infty
+           and will be exact for any function of the form
+                                         2n-1
+                                     -bx --     i
+                               f(x)=e    >   c x
+                                         --   i
+                                         i=0
+
+     (d)   Gauss-Hermite quadrature, adjusted weights option:
+                                    +infty
+                                    /
+                                S~= |     f(x)dx
+                                    /
+                                    -infty
+           and will be exact for any function of the form
+
+                                      2 2n-1
+                               -b(x-a)  --     i
+                         f(x)=e         >   c x   (b>0)
+                                        --   i
+                                        i=0
+
+     (e)   Gauss-Laguerre quadrature, normal weights option:
+
+               infty                             a
+               /     -bx                         /      -bx
+           S~= |    e   f(x)dx  (b>0)   or   S~= |     e   f(x)dx
+               /                                 /
+               a                                 -infty
+
+
+            (b<0)
+
+
+           and will be exact for any function of the form
+                                       2n-1
+                                       --     i
+                                 f(x)= >   c x
+                                       --   i
+                                       i=0
+
+     (f)   Gauss-Hermite quadrature, normal weights option:
+
+                                +infty        2
+                                /      -b(x-a)
+                            S~= |     e        f(x)dx
+                                /
+                                -infty
+           and will be exact for any function of the form:
+                                       2n-1
+                                       --     i
+                                 f(x)= >   c x
+                                       --   i
+                                       i=0
+
+     Note: that the Gauss-Legendre abscissae, with a=-1, b=+1, are the
+     zeros of the Legendre polynomials; the Gauss-Laguerre abscissae,
+     with a=0, b=1, are the zeros of the Laguerre polynomials; and the
+     Gauss-Hermite abscissae, with a=0, b=1, are the zeros of the
+     Hermite polynomials.
+
+     4. References
+
+     [1]   Davis P J and Rabinowitz P (1967) Numerical Integration.
+           Blaisdell Publishing Company. 33--52.
+
+     [2]   Froberg C E (1965) Introduction to Numerical Analysis.
+           Addison-Wesley. 181--187.
+
+     [3]   Ralston A (1965) A First Course in Numerical Analysis.
+           McGraw-Hill. 87--90.
+
+     [4]   Stroud A H and Secrest D (1966) Gaussian Quadrature
+           Formulas. Prentice-Hall.
+
+     5. Parameters
+
+      1:  A -- DOUBLE PRECISION                                  Input
+
+      2:  B -- DOUBLE PRECISION                                  Input
+          On entry: the quantities a and b as described in the
+          appropriate subsection of Section 3.
+
+      3:  ITYPE -- INTEGER                                       Input
+          On entry: indicates the type of weights for Gauss-Laguerre
+          or Gauss-Hermite quadrature (see Section 3):
+
+          if ITYPE = 1, adjusted weights will be returned;
+
+          if ITYPE = 0, normal weights will be returned.
+
+          Constraint: ITYPE = 0 or 1.
+
+          For Gauss-Legendre or Gauss-Rational quadrature, this
+          parameter is not used.
+
+      4:  N -- INTEGER                                           Input
+          On entry: the number of weights and abscissae to be
+          returned, n. Constraint: N =
+          1,2,3,4,5,6,8,10,12,14,16,20,24,32,48 or 64.
+
+      5:  WEIGHT(N) -- DOUBLE PRECISION array                   Output
+          On exit: the N weights. For Gauss-Laguerre and Gauss-
+          Hermite quadrature, these will be the adjusted weights if
+          ITYPE = 1, and the normal weights if ITYPE = 0.
+
+      6:  ABSCIS(N) -- DOUBLE PRECISION array                   Output
+          On exit: the N abscissae.
+
+      7:  GTYPE -- INTEGER                                       Input
+          On entry: The value of GTYPE indicates which quadrature
+          formula are to be used:
+               GTYPE = 0 for Gauss-Legendre weights and abscissae;
+
+               GTYPE = 1 for Gauss-Rational weights and abscissae;
+
+               GTYPE = 2 for Gauss-Laguerre weights and abscissae;
+
+               GTYPE = 3 for Gauss-Hermite weights and abscissae.
+
+      8:  IFAIL -- INTEGER                                Input/Output
+          On entry: IFAIL must be set to 0, -1 or 1. For users not
+          familiar with this parameter (described in the Essential
+          Introduction) the recommended value is 0.
+
+          On exit: IFAIL = 0 unless the routine detects an error (see
+          Section 6).
+
+     6. Error Indicators and Warnings
+
+     Errors detected by the routine:
+
+     IFAIL= 1
+          The N-point rule is not among those stored. If the soft fail
+          option is used, the weights and abscissae returned will be
+          those for the largest valid value of N less than the
+          requested value, and the excess elements of WEIGHT and
+          ABSCIS (i.e., up to the requested N) will be filled with
+          zeros.
+
+     IFAIL= 2
+          The value of A and/or B is invalid.
+               Gauss-Rational: A + B = 0
+
+               Gauss-Laguerre: B = 0
+
+               Gauss-Hermite: B <= 0
+          If the soft fail option is used the weights and abscissae
+          are returned as zero.
+
+     IFAIL= 3
+          Laguerre and Hermite normal weights only: underflow is
+          occurring in evaluating one or more of the normal weights.
+          If the soft fail option is used, the underflowing weights
+          are returned as zero. A smaller value of N must be used; or
+          adjusted weights should be used (ITYPE = 1). In the latter
+          case, take care that underflow does not occur when
+          evaluating the integrand appropriate for adjusted weights.
+
+     IFAIL=4
+         GTYPE < 0 or GTYPE > 3
+
+     7. Accuracy
+
+     The weights and abscissae are stored for standard values of A and
+     B to full machine accuracy.
+
+     8. Further Comments
+
+     Timing is negligible.
+
+     9. Example
+
+     This example program returns the abscissae and (adjusted) weights
+     for the six-point Gauss-Laguerre formula.
+
+     The example program is not reproduced here. The source code for
+     all example programs is distributed with the NAG Foundation
+     Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd01fcf}{NAG On-line Documentation: d01fcf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D01FCF(3NAG)      Foundation Library (12/10/92)      D01FCF(3NAG)
+
+
+
+          D01 -- Quadrature                                          D01FCF
+                  D01FCF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D01FCF attempts to evaluate a multi-dimensional integral (up to
+          15 dimensions), with constant and finite limits, to a specified
+          relative accuracy, using an adaptive subdivision strategy.
+
+          2. Specification
+
+                 SUBROUTINE D01FCF (NDIM, A, B, MINPTS, MAXPTS, FUNCTN,
+                1                   EPS, ACC, LENWRK, WRKSTR, FINVAL, IFAIL)
+                 INTEGER          NDIM, MINPTS, MAXPTS, LENWRK, IFAIL
+                 DOUBLE PRECISION A(NDIM), B(NDIM), FUNCTN, EPS, ACC, WRKSTR
+                1                 (LENWRK), FINVAL
+                 EXTERNAL         FUNCTN
+
+          3. Description
+
+          The routine returns an estimate of a multi-dimensional integral
+          over a hyper-rectangle (i.e., with constant limits), and also an
+          estimate of the relative error. The user sets the relative
+          accuracy required, supplies the integrand as a function
+          subprogram (FUNCTN), and also sets the minimum and maximum
+          acceptable number of calls to FUNCTN (in MINPTS and MAXPTS).
+
+          The routine operates by repeated subdivision of the hyper-
+          rectangular region into smaller hyper-rectangles. In each
+          subregion, the integral is estimated using a seventh-degree rule,
+          and an error estimate is obtained by comparison with a fifth-
+          degree rule which uses a subset of the same points. The fourth
+          differences of the integrand along each co-ordinate axis are
+          evaluated, and the subregion is marked for possible future
+          subdivision in half along that co-ordinate axis which has the
+          largest absolute fourth difference.
+
+          If the estimated errors, totalled over the subregions, exceed the
+          requested relative error (or if fewer than MINPTS calls to FUNCTN
+          have been made), further subdivision is necessary, and is
+          performed on the subregion with the largest estimated error, that
+          subregion being halved along the appropriate co-ordinate axis.
+
+          The routine will fail if the requested relative error level has
+          not been attained by the time MAXPTS calls to FUNCTN have been
+          made; or, if the amount LENWRK of working storage is
+          insufficient. A formula for the recommended value of LENWRK is
+          given in Section 5. If a smaller value is used, and is exhausted
+          in the course of execution, the routine switches to a less
+          efficient mode of operation; only if this mode also breaks down
+          is insufficient storage reported.
+
+          D01FCF is based on the HALF subroutine developed by van Dooren
+          and de Ridder [1]. It uses a different basic rule, described by
+          Genz and Malik [2].
+
+          4. References
+
+          [1]   Van Dooren P and De Ridder L (1976) An Adaptive Algorithm
+                for Numerical Integration over an N-dimensional Cube. J.
+                Comput. Appl. Math. 2 207--217.
+
+          [2]   Genz A C and Malik A A (1980) An Adaptive Algorithm for
+                Numerical Integration over an N-dimensional Rectangular
+                Region. J. Comput. Appl. Math. 6 295--302.
+
+          5. Parameters
+
+           1:  NDIM -- INTEGER                                        Input
+               On entry: the number of dimensions of the integral, n.
+               Constraint: 2 <= NDIM <= 15.
+
+           2:  A(NDIM) -- DOUBLE PRECISION array                      Input
+               On entry: the lower limits of integration, a , for
+                                                            i
+               i=1,2,...,n.
+
+           3:  B(NDIM) -- DOUBLE PRECISION array                      Input
+               On entry: the upper limits of integration, b , for
+                                                            i
+               i=1,2,...,n.
+
+           4:  MINPTS -- INTEGER                               Input/Output
+               On entry: MINPTS must be set to the minimum number of
+               integrand evaluations to be allowed. On exit: MINPTS
+               contains the actual number of integrand evaluations used by
+               D01FCF.
+
+           5:  MAXPTS -- INTEGER                                      Input
+               On entry: the maximum number of integrand evaluations to be
+               allowed.
+               Constraints:
+                        MAXPTS >= MINPTS
+
+                        MAXPTS >= (alpha),
+
+                                       NDIM       2
+                        where (alpha)=2    +2*NDIM +2*NDIM+1.
+
+           6:  FUNCTN -- DOUBLE PRECISION FUNCTION, supplied by the
+          user.
+                                                         External Procedure
+               FUNCTN must return the value of the integrand f at a given
+               point.
+
+               Its specification is:
+
+                      DOUBLE PRECISION FUNCTION FUNCTN (NDIM,Z)
+                      INTEGER          NDIM
+                      DOUBLE PRECISION Z(NDIM)
+
+                1:  NDIM -- INTEGER                                   Input
+                    On entry: the number of dimensions of the integral, n.
+
+                2:  Z(NDIM) -- DOUBLE PRECISION array                 Input
+                    On entry: the co-ordinates of the point at which the
+                    integrand must be evaluated.
+               FUNCTN must be declared as EXTERNAL in the (sub)program
+               from which D01FCF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           7:  EPS -- DOUBLE PRECISION                                Input
+               On entry: the relative error acceptable to the user. When
+               the solution is zero or very small relative accuracy may not
+               be achievable but the user may still set EPS to a reasonable
+               value and check for the error exit IFAIL = 2. Constraint:
+               EPS > 0.0.
+
+           8:  ACC -- DOUBLE PRECISION                               Output
+               On exit: the estimated relative error in FINVAL.
+
+           9:  LENWRK -- INTEGER                                      Input
+               On entry:
+               the dimension of the array WRKSTR as declared in the
+               (sub)program from which D01FCF is called.
+               Suggested value: for maximum efficiency, LENWRK >=
+               (NDIM+2)*(1+MAXPTS/(alpha)) (see parameter MAXPTS for
+               (alpha)).
+
+               If LENWRK is less than this, the routine will usually run
+               less efficiently and may fail. Constraint: LENWRK>=2*NDIM+4.
+
+          10:  WRKSTR(LENWRK) -- DOUBLE PRECISION array           Workspace
+
+          11:  FINVAL -- DOUBLE PRECISION                            Output
+               On exit: the best estimate obtained for the integral.
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit. To suppress
+               the output of an error message when soft failure occurs, set
+               IFAIL to 1.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          IFAIL= 1
+               On entry NDIM < 2,
+
+               or       NDIM > 15,
+
+               or       MAXPTS is too small,
+
+               or       LENWRK<2*NDIM+4,
+
+               or       EPS <= 0.0.
+
+          IFAIL= 2
+               MAXPTS was too small to obtain the required relative
+               accuracy EPS. On soft failure, FINVAL and ACC contain
+               estimates of the integral and the relative error, but ACC
+               will be greater than EPS.
+
+          IFAIL= 3
+               LENWRK was too small. On soft failure, FINVAL and ACC
+               contain estimates of the integral and the relative error,
+               but ACC will be greater than EPS.
+
+          7. Accuracy
+
+          A relative error estimate is output through the parameter ACC.
+
+          8. Further Comments
+
+          Execution time will usually be dominated by the time taken to
+          evaluate the integrand FUNCTN, and hence the maximum time that
+          could be taken will be proportional to MAXPTS.
+
+          9. Example
+
+          This example program estimates the integral
+
+                                  2
+                  1  1  1  1  4z z exp(2z z )
+                  /  /  /  /    1 3      1 3
+                  |  |  |  |  ---------------dz dz dz dz =0.575364.
+                  /  /  /  /             2     4  3  2  1
+                  0  0  0  0    (1+z +z )
+                                    2  4
+
+          The accuracy requested is one part in 10,000.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd01gaf}{NAG On-line Documentation: d01gaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D01GAF(3NAG)      Foundation Library (12/10/92)      D01GAF(3NAG)
+
+
+
+          D01 -- Quadrature                                          D01GAF
+                  D01GAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D01GAF integrates a function which is specified numerically at
+          four or more points, over the whole of its specified range, using
+          third-order finite-difference formulae with error estimates,
+          according to a method due to Gill and Miller.
+
+          2. Specification
+
+                 SUBROUTINE D01GAF (X, Y, N, ANS, ER, IFAIL)
+                 INTEGER          N, IFAIL
+                 DOUBLE PRECISION X(N), Y(N), ANS, ER
+
+          3. Description
+
+          This routine evaluates the definite integral
+
+                                       x
+                                        n
+                                       /
+                                    I= | y(x)dx,
+                                       /
+                                       x
+                                        1
+
+          where the function y is specified at the n-points x ,x ,...,x ,
+                                                             1  2      n
+          which should be all distinct, and in either ascending or
+          descending order. The integral between successive points is
+          calculated by a four-point finite-difference formula centred on
+          the interval concerned, except in the case of the first and last
+          intervals, where four-point forward and backward difference
+          formulae respectively are employed. If n is less than 4, the
+          routine fails. An approximation to the truncation error is
+          integrated and added to the result. It is also returned
+          separately to give an estimate of the uncertainty in the result.
+          The method is due to Gill and Miller.
+
+          4. References
+
+          [1]   Gill P E and Miller G F (1972) An Algorithm for the
+                Integration of Unequally Spaced Data. Comput. J. 15 80--83.
+
+          5. Parameters
+
+           1:  X(N) -- DOUBLE PRECISION array                         Input
+               On entry: the values of the independent variable, i.e., the
+               x ,x ,...,x . Constraint: either X(1) < X(2) <... < X(N) or
+                1  2      n
+               X(1) > X(2) >... > X(N).
+
+           2:  Y(N) -- DOUBLE PRECISION array                         Input
+               On entry: the values of the dependent variable y  at the
+                                                                i
+               points x , for i=1,2,...,n.
+                       i
+
+           3:  N -- INTEGER                                           Input
+               On entry: the number of points, n. Constraint: N >= 4.
+
+           4:  ANS -- DOUBLE PRECISION                               Output
+               On exit: the estimate of the integral.
+
+           5:  ER -- DOUBLE PRECISION                                Output
+               On exit: an estimate of the uncertainty in ANS.
+
+           6:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               Indicates that fewer than four-points have been supplied to
+               the routine.
+
+          IFAIL= 2
+               Values of X are neither strictly increasing nor strictly
+               decreasing.
+
+          IFAIL= 3
+               Two points have the same X-value.
+
+          No error is reported arising from the relative magnitudes of ANS
+          and ER on return, due to the difficulty when the true answer is
+          zero.
+
+          7. Accuracy
+
+          No accuracy level is specified by the user before calling the
+          routine but on return ABS(ER) is an approximation to, but not
+          necessarily a bound for, |I-ANS|. If on exit IFAIL > 0, both ANS
+          and ER are returned as zero.
+
+          8. Further Comments
+
+          The time taken by the routine depends on the number of points
+          supplied, n.
+
+          In their paper, Gill and Miller [1] do not add the quantity ER to
+          ANS before return. However, extensive tests have shown that a
+          dramatic reduction in the error often results from such addition.
+          In other cases, it does not make an improvement, but these tend
+          to be cases of low accuracy in which the modified answer is not
+          significantly inferior to the unmodified one. The user has the
+          option of recovering the Gill-Miller answer by subtracting ER
+          from ANS on return from the routine.
+
+          9. Example
+
+          The example program evaluates the integral
+
+                                    1
+                                    /  4
+                                    | ----dx=(pi)
+                                    /    2
+                                    0 1+x
+
+          reading in the function values at 21 unequally-spaced points.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd01gbf}{NAG On-line Documentation: d01gbf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D01GBF(3NAG)      Foundation Library (12/10/92)      D01GBF(3NAG)
+
+
+
+          D01 -- Quadrature                                          D01GBF
+                  D01GBF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D01GBF returns an approximation to the integral of a function
+          over a hyper-rectangular region, using a Monte Carlo method. An
+          approximate relative error estimate is also returned. This
+          routine is suitable for low accuracy work.
+
+          2. Specification
+
+                 SUBROUTINE D01GBF (NDIM, A, B, MINCLS, MAXCLS, FUNCTN,
+                1                   EPS, ACC, LENWRK, WRKSTR, FINEST, IFAIL)
+                 INTEGER          NDIM, MINCLS, MAXCLS, LENWRK, IFAIL
+                 DOUBLE PRECISION A(NDIM), B(NDIM), FUNCTN, EPS, ACC, WRKSTR
+                1                 (LENWRK), FINEST
+                 EXTERNAL         FUNCTN
+
+          3. Description
+
+          D01GBF uses an adaptive Monte Carlo method based on the algorithm
+          described by Lautrup [1]. It is implemented for integrals of the
+          form:
+
+                       b  b     b
+                        1  2     n
+                       /  /     /
+                       |  | ... | f(x ,x ,...,x )dx ...dx dx .
+                       /  /     /    1  2      n   n     2  1
+                       a  a     a
+                        1  2     n
+
+          Upon entry, unless LENWRK has been set to the minimum value
+          10*NDIM, the routine subdivides the integration region into a
+          number of equal volume subregions. Inside each subregion the
+          integral and the variance are estimated by means of pseudo-random
+          sampling. All contributions are added together to produce an
+          estimate for the whole integral and total variance. The variance
+          along each co-ordinate axis is determined and the routine uses
+          this information to increase the density and change the widths of
+          the sub-intervals along each axis, so as to reduce the total
+          variance. The total number of subregions is then increased by a
+          factor of two and the program recycles for another iteration. The
+          program stops when a desired accuracy has been reached or too
+          many integral evaluations are needed for the next cycle.
+
+          4. References
+
+          [1]   Lautrup B (1971) An Adaptive Multi-dimensional Integration
+                Procedure. Proc. 2nd Coll. on Advanced Methods in
+                Theoretical Physics, Marseille.
+
+          5. Parameters
+
+           1:  NDIM -- INTEGER                                        Input
+               On entry: the number of dimensions of the integral, n.
+               Constraint: NDIM >= 1.
+
+           2:  A(NDIM) -- DOUBLE PRECISION array                      Input
+               On entry: the lower limits of integration, a , for
+                                                           i
+               i=1,2,...,n.
+
+           3:  B(NDIM) -- DOUBLE PRECISION array                      Input
+               On entry: the upper limits of integration, b , for
+                                                           i
+               i=1,2,...,n.
+
+           4:  MINCLS -- INTEGER                               Input/Output
+               On entry: MINCLS must be set:
+
+               either to the minimum number of integrand evaluations to be
+               allowed, in which case MINCLS >= 0;
+
+               or to a negative value. In this case the routine assumes
+               that a previous call had been made with the same parameters
+               NDIM, A and B and with either the same integrand (in which
+               case D01GBF continues calculation) or a similar integrand
+               (in which case D01GBF begins the calculation with the
+               subdivision used in the last iteration of the previous call)
+               . See also WRKSTR. On exit: MINCLS contains the number of
+               integrand evaluations actually used by D01GBF.
+
+           5:  MAXCLS -- INTEGER                                      Input
+               On entry: the maximum number of integrand evaluations to be
+               allowed. In the continuation case this is the number of new
+               integrand evaluations to be allowed. These counts do not
+               include zero integrand values.
+               Constraints:
+                        MAXCLS > MINCLS,
+
+                        MAXCLS >= 4*(NDIM+1).
+
+           6:  FUNCTN -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                                         External Procedure
+               FUNCTN must return the value of the integrand f at a given
+               point.
+
+               Its specification is:
+
+                      DOUBLE PRECISION FUNCTION FUNCTN (NDIM, X)
+                      INTEGER          NDIM
+                      DOUBLE PRECISION X(NDIM)
+
+                1:  NDIM -- INTEGER                                   Input
+                    On entry: the number of dimensions of the integral, n.
+
+                2:  X(NDIM) -- DOUBLE PRECISION array                 Input
+                    On entry: the co-ordinates of the point at which the
+                    integrand must be evaluated.
+               FUNCTN must be declared as EXTERNAL in the (sub)program
+               from which D01GBF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           7:  EPS -- DOUBLE PRECISION                                Input
+               On entry: the relative accuracy required. Constraint: EPS
+               >= 0.0.
+
+           8:  ACC -- DOUBLE PRECISION                               Output
+               On exit: the estimated relative accuracy of FINEST.
+
+           9:  LENWRK -- INTEGER                                      Input
+               On entry:
+               the dimension of the array WRKSTR as declared in the
+               (sub)program from which D01GBF is called.
+               For maximum efficiency, LENWRK should be about
+                                              1/NDIM
+                             3*NDIM*(MAXCLS/4)      +7*NDIM.
+               If LENWRK is given the value 10*NDIM then the subroutine
+               uses only one iteration of a crude Monte Carlo method with
+               MAXCLS sample points. Constraint: LENWRK >= 10*NDIM.
+
+          10:  WRKSTR(LENWRK) -- DOUBLE PRECISION array        Input/Output
+               On entry: if MINCLS<0, WRKSTR must be unchanged from the
+               previous call of D01GBF - except that for a new integrand
+               WRKSTR(LENWRK) must be set to 0.0. See also MINCLS. On
+               exit: WRKSTR contains information about the current sub-
+               interval structure which could be used in later calls of
+               D01GBF. In particular, WRKSTR(j) gives the number of sub-
+               intervals used along the jth co-ordinate axis.
+
+          11:  FINEST -- DOUBLE PRECISION                            Output
+               On exit: the best estimate obtained for the integral.
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit. To suppress
+               the output of an error message when soft failure occurs, set
+               IFAIL to 1.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          IFAIL= 1
+               On entry NDIM < 1,
+
+               or       MINCLS >= MAXCLS,
+
+               or       LENWRK < 10*NDIM,
+
+               or       MAXCLS < 4*(NDIM+1),
+
+               or       EPS < 0.0.
+
+          IFAIL= 2
+               MAXCLS was too small for D01GBF to obtain the required
+               relative accuracy EPS. In this case D01GBF returns a value
+               of FINEST with estimated relative error ACC, but ACC will be
+               greater than EPS. This error exit may be taken before MAXCLS
+               non-zero integrand evaluations have actually occurred, if
+               the routine calculates that the current estimates could not
+               be improved before MAXCLS was exceeded.
+
+          7. Accuracy
+
+          A relative error estimate is output through the parameter ACC.
+          The confidence factor is set so that the actual error should be
+          less than ACC 90% of the time. If a user desires a higher
+          confidence level then a smaller value of EPS should be used.
+
+          8. Further Comments
+
+          The running time for D01GBF will usually be dominated by the time
+          used to evaluate the integrand FUNCTN, so the maximum time that
+          could be used is approximately proportional to MAXCLS.
+
+          For some integrands, particularly those that are poorly behaved
+          in a small part of the integration region, D01GBF may terminate
+          with a value of ACC which is significantly smaller than the
+          actual relative error. This should be suspected if the returned
+          value of MINCLS is small relative to the expected difficulty of
+          the integral. Where this occurs, D01GBF should be called again,
+          but with a higher entry value of MINCLS (e.g. twice the returned
+          value) and the results compared with those from the previous
+          call.
+
+          The exact values of FINEST and ACC on return will depend (within
+          statistical limits) on the sequence of random numbers generated
+          within D01GBF by calls to G05CAF. Separate runs will produce
+          identical answers unless the part of the program executed prior
+          to calling D01GBF also calls (directly or indirectly) routines
+          from Chapter G05, and the series of such calls differs between
+          runs. If desired, the user may ensure the identity or difference
+          between runs of the results returned by D01GBF, by calling G05CBF
+          or G05CCF respectively, immediately before calling D01GBF.
+
+          9. Example
+
+          This example program calculates the integral
+
+                  1  1  1  1  4x x exp(2x x )
+                  /  /  /  /    1 3      1 3
+                  |  |  |  |  ---------------dx dx dx dx =0.575364.
+                  /  /  /  /             2     1  2  3  4
+                  0  0  0  0    (1+x +x )
+                                    2  4
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd02}{NAG On-line Documentation: d02}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D02(3NAG)         Foundation Library (12/10/92)         D02(3NAG)
+
+
+
+          D02 -- Ordinary Differential Equations        Introduction -- D02
+                                    Chapter D02
+                          Ordinary Differential Equations
+
+          1. Scope of the Chapter
+
+          This chapter is concerned with the numerical solution of ordinary
+          differential equations. There are two main types of problem,
+          those in which all boundary conditions are specified at one point
+          (initial-value problems), and those in which the boundary
+          conditions are distributed between two or more points (boundary-
+          value problems and eigenvalue problems). Routines are available
+          for initial-value problems, two-point boundary-value problems and
+          Sturm-Liouville eigenvalue problems.
+
+          2. Background to the Problems
+
+          For most of the routines in this chapter a system of ordinary
+          differential equations must be written in the form
+
+                              y'  = f (x,y ,y ,...,y ),
+                                1    1    1  2      n
+
+                              y'  = f (x,y ,y ,...,y ),
+                                2    2    1  2      n
+
+                                .. ...
+
+                              y'  = f (x,y ,y ,...y ),
+                                n    n    1  2     n
+
+          that is the system must be given in first-order form. The n
+          dependent variables (also, the solution) y ,y ,...,y  are
+                                                    1  2      n
+          functions of the independent variable x, and the differential
+          equations give expressions for the first derivatives y' =dy /dx
+                                                                 i   i
+          in terms of x and y ,y ,...,y . For a system of n first-order
+                             1  2      n
+          equations, n associated boundary conditions are usually required
+          to define the solution.
+
+          A more general system may contain derivatives of higher order,
+          but such systems can almost always be reduced to the first-order
+          form by introducing new variables. For example, suppose we have
+          the third-order equation
+
+
+                                                2
+                                z'''+zz''+k(l-z' )=0.
+
+          We write y =z, y =z', y =z'', and the third order equation may
+                    1     2      3
+          then be written as the system of first-order equations
+
+                                y'  = y
+                                  1    2
+
+                                y'  = y
+                                  2    3
+                                                 2
+                                y'  = -y y -k(l-y ).
+                                  3     1 3      2
+
+          For this system n = 3 and we require 3 boundary conditions in
+          order to define the solution. These conditions must specify
+          values of the dependent variables at certain points. For example,
+          we have an initial-value problem if the conditions are:
+
+              y   =  0        at x=0
+               1
+
+              y   =  0        at x=0
+               2
+
+              y   =  0.1      at x=0.
+               3
+
+          These conditions would enable us to integrate the equations
+          numerically from the point x=0 to some specified end-point. We
+          have a boundary-value problem if the conditions are:
+
+              y   =  0        at x=0
+               1
+
+              y   =  0        at x=0
+               2
+
+              y   =  1        at x=10.
+               2
+
+          These conditions would be sufficient to define a solution in the
+          range 0<=x<=10, but the problem could not be solved by direct
+          integration (see Section 2.2 ). More general boundary conditions
+          are permitted in the boundary-value case.
+
+          2.1. Initial-value Problems
+
+          To solve first-order systems, initial values of the dependent
+          variables y , for i=1,2,...,n must be supplied at a given point,
+                     i
+          a. Also a point, b, at which the values of the dependent
+          variables are required, must be specified. The numerical solution
+          is then obtained by a step-by-step calculation which approximates
+          values of the variables y , for i=1,2,...,n at finite intervals
+                                   i
+          over the required range [a,b]. The routines in this chapter
+          adjust the step length automatically to meet specified accuracy
+          tolerances. Although the accuracy tests used are reliable over
+          each step individually, in general an accuracy requirement cannot
+          be guaranteed over a long range. For many problems there may be
+          no serious accumulation of error, but for unstable systems small
+          perturbations of the solution will often lead to rapid divergence
+          of the calculated values from the true values. A simple check for
+          stability is to carry out trial calculations with different
+          tolerances; if the results differ appreciably the system is
+          probably unstable. Over a short range, the difficulty may
+          possibly be overcome by taking sufficiently small tolerances, but
+          over a long range it may be better to try to reformulate the
+          problem.
+
+          A special class of initial-value problems are those for which the
+          solutions contain rapidly decaying transient terms. Such problems
+          are called stiff; an alternative way of describing them is to say
+          that certain eigenvalues of the Jacobian matrix (ddf /ddy ) have
+                                                              i    j
+          large negative real parts when compared to others. These problems
+          require special methods for efficient numerical solution; the
+          methods designed for non-stiff problems when applied to stiff
+          problems tend to be very slow, because they need small step
+          lengths to avoid numerical instability. A full discussion is
+          given in Hall and Watt [6] and a discussion of the methods for
+          stiff problems is given in Berzins, Brankin and Gladwell [1].
+
+          2.2. Boundary-value Problems
+
+          A full discussion of the design of the methods and codes for
+          boundary-value problems is given in Gladwell [4]. In general, a
+          system of nonlinear differential equations with boundary
+          conditions given at two or more points cannot be guaranteed to
+          have a solution. The solution has to be determined iteratively
+          (if it exists). Finite-difference equations are set up on a mesh
+          of points and estimated values for the solution at the grid
+          points are chosen. Using these estimated values as starting
+          values a Newton iteration is used to solve the finite-difference
+          equations. The accuracy of the solution is then improved by
+          deferred corrections or the addition of points to the mesh or a
+          combination of both. Good initial estimates of the solution may
+          be required in some cases but results may be difficult to compute
+          when the solution varies very rapidly over short ranges. A
+          discussion is given in Chapters 9 and 11 of Gladwell and Sayers
+          [5] and Chapter 4 of Childs et al [2].
+
+          2.3. Eigenvalue Problems
+
+          Sturm-Liouville problems of the form
+
+                             (p(x)y')'+q(x,(lambda))y=0
+
+          with appropriate boundary conditions given at two points, can be
+          solved by a Scaled Pruefer method. In this method the
+          differential equation is transformed to another which can be
+          solved for a specified eigenvalue by a shooting method. A
+          discussion is given in Chapter 11 of Gladwell and Sayers [5] and
+          a complete description is given in Pryce [7].
+
+          2.6. References
+
+          [1]   Berzins M, Brankin R W and Gladwell I (1987) Design of the
+                Stiff Integrators in the NAG Library. Technical Report.
+                TR14/87 NAG.
+
+          [2]   (1979) Codes for Boundary-value Problems in Ordinary
+                Differential Equations. Lecture Notes in Computer Science.
+                (ed Childs B, Scott M, Daniel J W, Denman E and Nelson P) 76
+                 Springer-Verlag.
+
+          [3]   Gladwell I (1979) Initial Value Routines in the NAG Library.
+                ACM Trans Math Softw. 5 386--400.
+
+          [4]   Gladwell I (1987) The NAG Library Boundary Value Codes.
+                Numerical Analysis Report. 134 Manchester University.
+
+          [5]   (1980) Computational Techniques for Ordinary Differential
+                Equations. (Gladwell I and Sayers D K) Academic Press.
+
+          [6]   Hall G and Watt J M (eds) (1976) Modern Numerical Methods
+                for Ordinary Differential Equations. Clarendon Press.
+
+          [7]   Pryce J D (1986) Error Estimation for Phase-function
+                Shooting Methods for Sturm-Liouville Problems. J. Num. Anal.
+                6 103--123.
+
+          3. Recommendations on Choice and Use of Routines
+
+          There are no routines which deal directly with COMPLEX equations.
+          These may however be transformed to larger systems of real
+          equations of the required form. Split each equation into its real
+          and imaginary parts and solve for the real and imaginary parts of
+          each component of the solution. Whilst this process doubles the
+          size of the system and may not always be appropriate it does make
+          available for use the full range of routines provided presently.
+
+          3.1. Initial-value Problems
+
+          For simple first-order problems with low accuracy requirements,
+          that is problems on a short range of integration, with derivative
+          functions f  which are inexpensive to calculate and where only a
+                     i
+          few correct figures are required, the best routines to use are
+          likely to be the Runge-Kutta-Merson (RK) routines, D02BBF and
+          D02BHF. For larger problems, over long ranges or with high
+          accuracy requirements the variable-order, variable-step Adams
+          routine D02CJF should usually be preferred. For stiff equations,
+          that is those with rapidly decaying transient solutions, the
+          Backward Differentiation Formula (BDF) variable-order, variable-
+          step routine D02EJF should be used.
+
+          There are four routines for initial-value problems, two of which
+          use the Runge-Kutta-Merson method:
+
+          D02BBF     integrates a system of first order ordinary
+                     differential equations over a range with intermediate
+                     output and a choice of error control
+
+          D02BHF     integrates a system of first order ordinary
+                     differential equations with a choice of error control
+                     until a position is determined where a function of the
+                     solution is zero.
+
+          one uses an Adams method:
+
+          D02CJF     combines the functionality of D02BBF and D02BHF
+
+          and one uses a BDF method:
+
+          D02EJF     combines the functionality of D02BBF and D02BHF.
+
+          3.2. Boundary-value Problems
+
+          D02GAF may be used for simple boundary-value problems with
+          assigned boundary values. The user may find that convergence is
+          difficult to achieve using D02GAF since only specifying the
+          unknown boundary values and the position of the finite-difference
+          mesh is permitted. In such cases the user may use D02RAF which
+          permits specification of an initial estimate for the solution at
+          all mesh points and allows the calculation to be influenced in
+          other ways too. D02RAF is designed to solve a general nonlinear
+          two-point boundary value problem with nonlinear boundary
+          conditions.
+
+          A routine, D02GBF, is also supplied specifically for the general
+          linear two-point boundary-value problem written in a standard
+
+          The user is advised to use interpolation routines from the E01
+          Chapter to obtain solution values at points not on the final
+          mesh.
+
+          3.3. Eigenvalue Problems
+
+          There is one general purpose routine for eigenvalue problems,
+          D02KEF. It may be used to solve regular or singular second-order
+          Sturm-Liouville problems on a finite or infinite range.
+          Discontinous coefficient functions can be treated and
+          eigenfunctions can be computed.
+
+
+          D02 -- Ordinary Differential Equations            Contents -- D02
+          Chapter D02
+
+          Ordinary Differential Equations
+
+          D02BBF  ODEs, IVP, Runge-Kutta-Merson method, over a range,
+                  intermediate output
+
+          D02BHF  ODEs, IVP, Runge-Kutta-Merson method, until function of
+                  solution is zero
+
+          D02CJF  ODEs, IVP, Adams method, until function of solution is
+                  zero, intermediate output
+
+          D02EJF  ODEs, stiff IVP, BDF method, until function of solution
+                  is zero, intermediate output
+
+          D02GAF  ODEs, boundary value problem, finite difference technique
+                  with deferred correction, simple nonlinear problem
+
+          D02GBF  ODEs, boundary value problem, finite difference technique
+                  with deferred correction, general linear problem
+
+          D02KEF  2nd order Sturm-Liouville problem, regular/singular
+                  system, finite/infinite range, eigenvalue and
+                  eigenfunction, user-specified break-points
+
+          D02RAF  ODEs, general nonlinear boundary value problem, finite
+                  difference technique with deferred correction,
+                  continuation facility
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd02bbf}{NAG On-line Documentation: d02bbf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D02BBF(3NAG)                 D02BBF                  D02BBF(3NAG)
+
+
+
+     D02 -- Ordinary Differential Equations                     D02BBF
+             D02BBF -- NAG Foundation Library Routine Document
+
+     Note: Before using this routine, please read the Users' Note for
+     your implementation to check implementation-dependent details.
+     The symbol (*) after a NAG routine name denotes a routine that is
+     not included in the Foundation Library.
+
+     Note for users via the AXIOM system: the interface to this routine
+     has been enhanced for use with AXIOM and is slightly different to
+     that offered in the standard version of the Foundation Library.
+
+     1. Purpose
+
+     D02BBF integrates a system of first-order ordinary differential
+     equations over an interval with suitable initial conditions,
+     using a Runge-Kutta-Merson method, and returns the solution at
+     points specified by the user.
+
+     2. Specification
+
+            SUBROUTINE D02BBF (X, XEND, M, N, Y, TOL, IRELAB, RESULT,
+           1                   FCN, OUTPUT, W, IFAIL)
+            INTEGER          M, N, IRELAB, IFAIL
+            DOUBLE PRECISION X, XEND, Y(N), TOL, W(N,7), RESULT(M,N)
+            EXTERNAL         FCN, OUTPUT
+
+     3. Description
+
+     The routine integrates a system of ordinary differential
+     equations
+
+                    y' =f (x,y ,y ,...,y )  i=1,2,...,n
+                      i  i    1  2      n
+
+     from x = X to x = XEND using a Merson form of the Runge-Kutta
+     method. The system is defined by a subroutine FCN supplied by the
+     user, which evaluates f  in terms of x and y ,y ,...,y , and the
+                            i                    1  2      n
+     values of y ,y ,...,y  must be given at x = X.
+                1  2      n
+
+     The solution is returned via the user-supplied routine OUTPUT at
+     a set of points specified by the user. This solution is obtained
+     by quintic Hermite interpolation on solution values produced by
+     the Runge-Kutta method.
+
+     The accuracy of the integration and, indirectly, the
+     interpolation is controlled by the parameters TOL and IRELAB.
+
+     For a description of Runge-Kutta methods and their practical
+     implementation see Hall and Watt [1].
+
+     4. References
+
+     [1]   Hall G and Watt J M (eds) (1976) Modern Numerical Methods
+           for Ordinary Differential Equations. Clarendon Press.
+
+     5. Parameters
+
+      1:  X -- DOUBLE PRECISION                           Input/Output
+          On entry: X must be set to the initial value of the
+          independent variable x. On exit: XEND, unless an error has
+          occurred, when it contains the value of x at the error.
+
+      2:  XEND -- DOUBLE PRECISION                               Input
+          On entry: the final value of the independent variable. If
+          XEND < X on entry, integration will proceed in a negative
+          direction.
+
+      3:  M -- INTEGER                                           Input
+          On entry: the first dimension of the array RESULT.  This
+          will usually be equal to the number of points at which the
+          solution is required.
+          Constraint: M > 0.
+
+      4:  N -- INTEGER                                           Input
+          On entry: the number of differential equations.
+          Constraint: N > 0.
+
+      5:  Y(N) -- DOUBLE PRECISION array                  Input/Output
+          On entry: the initial values of the solution y ,y ,...,y .
+                                                         1  2      n
+          On exit: the computed values of the solution at the final
+          value of X.
+
+      6:  TOL -- DOUBLE PRECISION                         Input/Output
+          On entry: TOL must be set to a positive tolerance for
+          controlling the error in the integration.
+
+          D02BBF has been designed so that, for most problems, a
+          reduction in TOL leads to an approximately proportional
+          reduction in the error in the solution at XEND. The relation
+          between changes in TOL and the error at intermediate output
+          points is less clear, but for TOL small enough the error at
+          intermediate output points should also be approximately
+          proportional to TOL. However, the actual relation between
+          TOL and the accuracy achieved cannot be guaranteed. The user
+          is strongly recommended to call D02BBF with more than one
+          value for TOL and to compare the results obtained to
+          estimate their accuracy. In the absence of any prior
+          knowledge, the user might compare the results obtained by
+                                      -p             -p-1
+          calling D02BBF with TOL=10.0   and TOL=10.0     if p correct
+          decimal digits in the solution are required. Constraint: TOL
+          > 0.0. On exit: normally unchanged. However if the range X
+          to XEND is so short that a small change in TOL is unlikely
+          to make any change in the computed solution then, on return,
+          TOL has its sign changed. This should be treated as a
+          warning that the computed solution is likely to be more
+          accurate than would be produced by using the same value of
+          TOL on a longer range of integration.
+
+      7:  IRELAB -- INTEGER                                      Input
+          On entry: IRELAB determines the type of error control. At
+          each step in the numerical solution an estimate of the local
+          error, EST, is made. For the current step to be accepted the
+          following condition must be satisfied:
+          IRELAB = 0
+                EST=10.0<=TOL*max{1.0,|y |,|y |,...,|y |};
+                                        1    2        n
+
+          IRELAB = 1
+                EST <= TOL;
+
+          IRELAB = 2
+                EST<=TOL*max{(epsilon),|y |,|y |,...,|y |}, where
+                                         1    2        n
+                (epsilon) is machine precision.
+          If the appropriate condition is not satisfied, the step size
+          is reduced and the solution is recomputed on the current
+          step.
+
+          If the user wishes to measure the error in the computed
+          solution in terms of the number of correct decimal places,
+          then IRELAB should be given the value 1 on entry, whereas if
+          the error requirement is in terms of the number of correct
+          significant digits, then IRELAB should be given the value 2.
+          Where there is no preference in the choice of error test
+          IRELAB = 0 will result in a mixed error test. Constraint: 0
+          <= IRELAB <= 2.
+
+      8:  RESULT(M,N) -- DOUBLE PRECISION array                 Output
+          On exit: the computed values of the solution at the points
+          given by OUTPUT.
+
+      9:  FCN -- SUBROUTINE, supplied by the user.
+                                                    External Procedure
+          FCN must evaluate the functions f  (i.e., the derivatives
+                                           i
+          y' ) for given values of its arguments x,y ,...,y .
+            i                                       1      n
+
+          Its specification is:
+
+                 SUBROUTINE FCN (X, Y, F)
+                 DOUBLE PRECISION X, Y(n), F(n)
+
+          where n is the actual value of N in the call of D02BBF.
+
+           1:  X -- DOUBLE PRECISION                             Input
+               On entry: the value of the argument x.
+
+           2:  Y(*) -- DOUBLE PRECISION array                    Input
+               On entry: the value of the argument y , for
+                                                     i
+               i=1,2,...,n.
+
+           3:  F(*) -- DOUBLE PRECISION array                   Output
+               On exit: the value of f , for i=1,2,...,n.
+                                       i
+          FCN must be declared as EXTERNAL in the (sub)program
+          from which D02BBF is called. Parameters denoted as
+          Input must not be changed by this procedure.
+
+     10:  OUTPUT -- SUBROUTINE, supplied by the user.
+                                                    External Procedure
+          OUTPUT allows the user to have access to intermediate values
+          of the computed solution at successive points specified by the
+          user. These solution values may be returned to the user via
+          the array RESULT if desired (this is a non-standard feature
+          added for use with the AXIOM system).  OUTPUT is initially
+          called by D02BBF with XSOL = X (the initial value of x). The
+          user must reset XSOL to the next point where OUTPUT is to be
+          called, and so on at each call to OUTPUT. If, after a call
+          to OUTPUT, the reset point XSOL is beyond XEND, D02BBF will
+          integrate to XEND with no further calls to OUTPUT; if a call
+          to OUTPUT is required at the point XSOL = XEND, then XSOL
+          must be given precisely the value XEND.
+
+          Its specification is:
+
+               SUBROUTINE OUTPUT(XSOL,Y,COUNT,M,N,RESULT)
+               DOUBLE PRECISION Y(N),RESULT(M,N),XSOL
+               INTEGER M,N,COUNT
+
+           1:  XSOL -- DOUBLE PRECISION                   Input/Output
+               On entry: the current value of the independent
+               variable x. On exit: the next value of x at which
+               OUTPUT is to be called.
+
+           2:  Y(N) -- DOUBLE PRECISION array                    Input
+               On entry: the computed solution at the point XSOL.
+
+           3:  COUNT -- INTEGER                           Input/Output
+               On entry: Zero if OUTPUT has not been called before, or
+               the previous value of COUNT.
+               On exit: A new value of COUNT: this can be used to keep
+               track of the number of times OUTPUT has been called.
+
+           4:  M -- INTEGER                                      Input
+               On entry: The first dimension of RESULT.
+
+           5:  N -- INTEGER                                      Input
+               On entry: The dimension of Y.
+
+           6:  RESULT(M,N) -- DOUBLE PRECISION array      Input/Output
+               On entry: the previous contents of RESULT.
+               On exit: RESULT may be used to return the values of the
+               intermediate solutions to the user.
+
+          OUTPUT must be declared as EXTERNAL in the (sub)program
+          from which D02BBF is called. Parameters denoted as
+          Input must not be changed by this procedure.
+
+     11:  W(N,7) -- DOUBLE PRECISION array                   Workspace
+
+     12:  IFAIL -- INTEGER                                Input/Output
+          On entry: IFAIL must be set to 0, -1 or 1. For users not
+          familiar with this parameter (described in the Essential
+          Introduction) the recommended value is 0.
+
+          On exit: IFAIL = 0 unless the routine detects an error (see
+          Section 6).
+
+     6. Error Indicators and Warnings
+
+     Errors detected by the routine:
+
+     IFAIL= 1
+          On entry TOL <= 0,
+
+          or       N <= 0,
+
+          or       IRELAB /= 0, 1 or 2.
+
+     IFAIL= 2
+          With the given value of TOL, no further progress can be made
+          across the integration range from the current point x = X,
+          or the dependence of the error on TOL would be lost if
+          further progress across the integration range were attempted
+          (see Section 8 for a discussion of this error exit). The
+          components Y(1),Y(2),...,Y(n) contain the computed values of
+          the solution at the current point x = X.
+
+     IFAIL= 3
+          TOL is too small for the routine to take an initial step
+          (see Section 8). X and Y(1),Y(2),...,Y(n) retain their
+          initial values.
+
+     IFAIL= 4
+          X = XEND and XSOL /= X after the initial call to OUTPUT.
+
+     IFAIL= 5
+          A value of XSOL returned by OUTPUT lies behind the previous
+          value of XSOL in the direction of integration.
+
+     IFAIL= 6
+          A serious error has occurred in an internal call to
+          D02PAF(*). Check all subroutine calls and array dimensions.
+          Seek expert help.
+
+     IFAIL= 7
+          A serious error has occurred in an internal call to
+          D02XAF(*). Check all subroutine calls and array dimensions.
+          Seek expert help.
+
+     7. Accuracy
+
+     The accuracy depends on TOL, on the mathematical properties of
+     the differential system, on the length of the range of
+     integration and on the method. It can be controlled by varying
+     TOL but the approximate proportionality of the error to TOL holds
+     only for a restricted range of values of TOL. For TOL too large,
+     the underlying theory may break down and the result of varying
+     TOL may be unpredictable. For TOL too small, rounding errors may
+     affect the solution significantly and an error exit with IFAIL =
+     2 or IFAIL = 3 is possible.
+
+     At the intermediate output points the same remarks apply. For
+     large values of TOL it is possible that the errors at some
+     intermediate output points may be much larger than at XEND. In
+     any case, it must not be expected that the error will have the
+     same size at all output points. At any point, it is a combination
+     of the errors arising from the integration of the differential
+     equation and the interpolation. The effect of combining these
+     errors will vary, though in most cases the integration error will
+     dominate.
+
+     The user who requires a more reliable estimate of the accuracy
+     achieved than can be obtained by varying TOL, is recommended to
+     call D02BDF(*) where both the solution and a global error
+     estimate are computed.
+
+     8. Further Comments
+
+     The time taken by the routine depends on the complexity and
+     mathematical properties of the system of differential equations
+     defined by FCN, on the range, the tolerance and the number of
+     calls to OUTPUT. There is also an overhead of the form a+b*n
+     where a and b are machine-dependent computing times.
+
+     If the routine fails with IFAIL = 3, then it can be called again
+     with a larger value of TOL (if this has not already been tried).
+     If the accuracy requested is really needed and cannot be obtained
+     with this routine, the system may be very stiff (see below) or so
+     badly scaled that it cannot be solved to the required accuracy.
+
+     If the routine fails with IFAIL = 2, it is probable that it has
+     been called with a value of TOL which is so small that the
+     solution cannot be obtained on the range X to XEND. This can
+     happen for well-behaved systems and very small values of TOL. The
+     user should, however, consider whether there is a more
+     fundamental difficulty. For example:
+
+     (a)   in the region of a singularity (infinite value) of the
+           solution, the routine will usually stop with IFAIL = 2,
+           unless overflow occurs first. If overflow occurs using
+           D02BBF, D02PAF(*) can be used instead to trap the
+           increasing solution before overflow occurs. In any case,
+           numerical integration cannot be continued through a
+           singularity, and analytic treatment should be considered;
+
+     (b)   for 'stiff' equations, where the solution contains rapidly
+           decaying components, the routine will use very small steps
+           in x (internally to D02BBF) to preserve stability. This
+           will usually exhibit itself by making the computing time
+           excessively long, or occasionally by an exit with IFAIL =
+           2. Merson's method is not efficient in such cases, and the
+           user should try using D02EBF(*) (Backward Differentiation
+           Formula). To determine whether a problem is stiff,
+           D02BDF(*) may be used.
+
+     For well-behaved systems with no difficulties such as stiffness
+     or singularities, the Merson method should work well for low
+     accuracy calculations (three or four figures). For higher
+     accuracy calculations or where FCN is costly to evaluate,
+     Merson's method may not be appropriate and a computationally less
+     expensive method may be D02CBF(*) which uses an Adams method.
+
+     Users with problems for which D02BBF is not sufficiently general
+     should consider using D02PAF(*) with D02XAF(*). D02PAF(*) is a
+     more general Merson routine with many facilities including more
+     general error control options and several criteria for
+     interrupting the calculations. D02XAF(*) interpolates on values
+     produced by D02PAF(*).
+
+     9. Example
+
+     To integrate the following equations (for a projectile)
+
+                                y'=tan(phi)
+
+                           -0.032tan(phi)   0.02v
+                       v'= --------------- --------
+                                 v         cos(phi)
+
+
+                                      -0.032
+                              (phi)'= ------
+                                         2
+                                        v
+
+     over an interval X = 0.0 to XEND = 8.0, starting with values
+     y=0.0, v=0.5 and (phi)=(pi)/5 and printing the solution at steps
+     of 1.0. We write y=Y(1), v=Y(2) and (phi)=Y(3), and we set
+     TOL=1.0E-4 and TOL=1.0E-5 in turn so that we may compare the
+     solutions. The value of (pi) is obtained by using X01AAF(*).
+
+     Note the careful construction of routine OUT to ensure that the
+     value of XEND is printed.
+
+     The example program is not reproduced here. The source code for
+     all example programs is distributed with the NAG Foundation
+     Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd02bhf}{NAG On-line Documentation: d02bhf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D02BHF(3NAG)      Foundation Library (12/10/92)      D02BHF(3NAG)
+
+
+
+          D02 -- Ordinary Differential Equations                     D02BHF
+                  D02BHF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D02BHF integrates a system of first-order ordinary differential
+          equations over an interval with suitable initial conditions,
+          using a Runge-Kutta-Merson method, until a user-specified
+          function of the solution is zero.
+
+          2. Specification
+
+                 SUBROUTINE D02BHF (X, XEND, N, Y, TOL, IRELAB, HMAX, FCN,
+                1                   G, W, IFAIL)
+                 INTEGER          N, IRELAB, IFAIL
+                 DOUBLE PRECISION X, XEND, Y(N), TOL, HMAX, G, W(N,7)
+                 EXTERNAL         FCN, G
+
+          3. Description
+
+          The routine advances the solution of a system of ordinary
+          differential equations
+
+                        y' =f (x,y ,y ,...,y ),  i=1,2,...,n,
+                          i  i    1  2      n
+
+          from x = X towards x = XEND using a Merson form of the Runge-
+          Kutta method. The system is defined by a subroutine FCN supplied
+          by the user, which evaluates f  in terms of x and y ,y ,...,y
+                                        i                    1  2      n
+          (see Section 5), and the values of y ,y ,...,y  must be given at
+                                              1  2      n
+          x = X.
+
+          As the integration proceeds, a check is made on the function
+          g(x,y) specified by the user, to determine an interval where it
+          changes sign. The position of this sign change is then determined
+          accurately by interpolating for the solution and its derivative.
+          It is assumed that g(x,y) is a continuous function of the
+          variables, so that a solution of g(x,y) = 0 can be determined by
+          searching for a change in sign in g(x,y).
+
+          The accuracy of the integration and, indirectly, of the
+          determination of the position where g(x,y) = 0, is controlled by
+          the parameter TOL.
+
+          For a description of Runge-Kutta methods and their practical
+          implementation see Hall and Watt [1].
+
+          4. References
+
+          [1]   Hall G and Watt J M (eds) (1976) Modern Numerical Methods
+                for Ordinary Differential Equations. Clarendon Press.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                           Input/Output
+               On entry: X must be set to the initial value of the
+               independent variable x. On exit: the point where g(x,y) = 0.
+               0 unless an error has occurred, when it contains the value
+               of x at the error. In particular, if g(x,y)/=0.0 anywhere on
+               the range X to XEND, it will contain XEND on exit.
+
+           2:  XEND -- DOUBLE PRECISION                               Input
+               On entry: the final value of the independent variable x.
+
+               If XEND < X on entry, integration proceeds in a negative
+               direction.
+
+           3:  N -- INTEGER                                           Input
+               On entry: the number of differential equations, n.
+               Constraint: N > 0.
+
+           4:  Y(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: the initial values of the solution y ,y ,...,y .
+                                                             1  2      n
+               On exit: the computed values of the solution at the final
+               point x = X.
+
+           5:  TOL -- DOUBLE PRECISION                         Input/Output
+               On entry: TOL must be set to a positive tolerance for
+               controlling the error in the integration and in the
+               determination of the position where g(x,y) = 0.0.
+
+               D02BHF has been designed so that, for most problems, a
+               reduction in TOL leads to an approximately proportional
+               reduction in the error in the solution obtained in the
+               integration. The relation between changes in TOL and the
+               error in the determination of the position where g(x,y) = 0.
+               0 is less clear, but for TOL small enough the error should
+               be approximately proportional to TOL. However, the actual
+               relation between TOL and the accuracy cannot be guaranteed.
+               The user is strongly recommended to call D02BHF with more
+               than one value for TOL and to compare the results obtained
+               to estimate their accuracy. In the absence of any prior
+               knowledge the user might compare results obtained by calling
+                                   -p             -p-1
+               D02BHF with TOL=10.0   and TOL=10.0     if p correct decimal
+               digits in the solution are required. Constraint: TOL > 0.0.
+
+               On exit: normally unchanged. However if the range from x = X
+               to the position where g(x,y) = 0.0 (or to the final value of
+               x if an error occurs) is so short that a small change in TOL
+               is unlikely to make any change in the computed solution,
+               then TOL is returned with its sign changed. To check results
+               returned with TOL < 0.0, D02BHF should be called again with
+               a positive value of TOL whose magnitude is considerably
+               smaller than that of the previous call.
+
+           6:  IRELAB -- INTEGER                                      Input
+               On entry: IRELAB determines the type of error control. At
+               each step in the numerical solution an estimate of the local
+               error, EST, is made. For the current step to be accepted the
+               following condition must be satisfied:
+               IRELAB = 0
+                     EST<=TOL*max{1.0,|y |,|y |,...,|y |};
+                                        1    2        n
+
+               IRELAB = 1
+                     EST <= TOL;
+
+               IRELAB = 2
+                     EST<=TOL*max{(epsilon),|y |,|y |,...,|y |},
+                                              1    2        n
+
+                     where (epsilon) is machine precision.
+               If the appropriate condition is not satisfied, the step size
+               is reduced and the solution recomputed on the current step.
+
+               If the user wishes to measure the error in the computed
+               solution in terms of the number of correct decimal places,
+               then IRELAB should be given the value 1 on entry, whereas if
+               the error requirement is in terms of the number of correct
+               significant digits, then IRELAB should be given the value 2.
+               Where there is no preference in the choice of error test,
+               IRELAB = 0 will result in a mixed error test. It should be
+               borne in mind that the computed solution will be used in
+               evaluating g(x,y). Constraint: 0 <= IRELAB <= 2.
+
+           7:  HMAX -- DOUBLE PRECISION                               Input
+               On entry: if HMAX = 0.0, no special action is taken.
+
+               If HMAX /= 0.0, a check is made for a change in sign of
+               g(x,y) at steps not greater than |HMAX|. This facility
+               should be used if there is any chance of 'missing' the
+               change in sign by checking too infrequently. For example, if
+               two changes of sign of g(x,y) are expected within a distance
+               h, say, of each other, then a suitable value for HMAX might
+               be HMAX = h/2. If only one change of sign in g(x,y) is
+               expected on the range X to XEND, then the choice HMAX = 0.0
+               is most appropriate.
+
+           8:  FCN -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               FCN must evaluate the functions f  (i.e., the derivatives
+                                                i
+               y' ) for given values of its arguments x,y ,...,y .
+                 i                                       1      n
+
+               Its specification is:
+
+                      SUBROUTINE FCN (X, Y, F)
+                      DOUBLE PRECISION X, Y(n), F(n)
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the value of the argument x.
+
+                2:  Y(*) -- DOUBLE PRECISION array                    Input
+                    On entry: the value of the argument y , for
+                                                         i
+                    i=1,2,...,n.
+
+                3:  F(*) -- DOUBLE PRECISION array                   Output
+                    On exit: the value of f , for i=1,2,...,n.
+                                           i
+               FCN must be declared as EXTERNAL in the (sub)program
+               from which D02BHF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           9:  G -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                                    External Procedure
+               G must evaluate the function g(x,y) at a specified point.
+
+               Its specification is:
+
+                      DOUBLE PRECISION FUNCTION G (X, Y)
+                      DOUBLE PRECISION X, Y(n)
+               where n is the actual value of N in the call of D02BHF.
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the value of the independent variable x.
+
+                2:  Y(*) -- DOUBLE PRECISION array                    Input
+                    On entry: the value of y , for i=1,2,...,n.
+                                             i
+               G must be declared as EXTERNAL in the (sub)program from
+               which D02BHF is called. Parameters denoted as Input
+               must not be changed by this procedure.
+
+          10:  W(N,7) -- DOUBLE PRECISION array                   Workspace
+
+          11:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry TOL <= 0.0,
+
+               or       N <= 0,
+
+               or       IRELAB /= 0, 1 or 2.
+
+          IFAIL= 2
+               With the given value of TOL, no further progress can be made
+               across the integration range from the current point x = X,
+               or dependence of the error on TOL would be lost if further
+               progress across the integration range were attempted (see
+               Section 8 for a discussion of this error exit). The
+               components Y(1),Y(2),...,Y(n) contain the computed values of
+               the solution at the current point x = X. No point at which
+               g(x,y) changes sign has been located up to the point x = X.
+
+          IFAIL= 3
+               TOL is too small for the routine to take an initial step
+               (see Section 8). X and Y(1),Y(2),...,Y(n) retain their
+               initial values.
+
+          IFAIL= 4
+               At no point in the range X to XEND did the function g(x,y)
+               change sign. It is assumed that g(x,y) = 0.0 has no
+               solution.
+
+          IFAIL= 5
+               A serious error has occurred in an internal call to
+               C05AZF(*). Check all subroutine calls and array dimensions.
+               Seek expert help.
+
+          IFAIL= 6
+               A serious error has occurred in an internal call to
+               D02PAF(*). Check all subroutine calls and array dimensions.
+               Seek expert help.
+
+          IFAIL= 7
+               A serious error has occurred in an internal call to
+               D02XAF(*). Check all subroutine calls and array dimensions.
+               Seek expert help.
+
+          7. Accuracy
+
+          The accuracy depends on TOL, on the mathematical properties of
+          the differential system, on the position where g(x,y) = 0.0 and
+          on the method. It can be controlled by varying TOL but the
+          approximate proportionality of the error to TOL holds only for a
+          restricted range of values of TOL. For TOL too large, the
+          underlying theory may break down and the result of varying TOL
+          may be unpredictable. For TOL too small, rounding error may
+          affect the solution significantly and an error exit with IFAIL =
+          2 or IFAIL = 3 is possible.
+
+          The accuracy may also be restricted by the properties of g(x,y).
+          The user should try to code G without introducing any unnecessary
+          cancellation errors.
+
+          8. Further Comments
+
+          The time taken by the routine depends on the complexity and
+          mathematical properties of the system of differential equations
+          defined by FCN, the complexity of G, on the range, the position
+          of the solution and the tolerance. There is also an overhead of
+          the form a+b*n where a and b are machine-dependent computing
+          times.
+
+          For some problems it is possible that D02BHF will return IFAIL =
+          4 because of inaccuracy of the computed values Y, leading to
+          inaccuracy in the computed values of g(x,y) used in the search
+          for the solution of g(x,y) = 0.0. This difficulty can be overcome
+          by reducing TOL sufficiently, and if necessary, by choosing HMAX
+          sufficiently small. If possible, the user should choose XEND well
+          beyond the expected point where g(x,y) = 0.0; for example make
+          |XEND-X| about 50 larger than the expected range. As a simple
+          check, if, with XEND fixed, a change in TOL does not lead to a
+          significant change in Y at XEND, then inaccuracy is not a likely
+          source of error.
+
+          If the routine fails with IFAIL = 3, then it could be called
+          again with a larger value of TOL if this has not already been
+          tried. If the accuracy requested is really needed and cannot be
+          obtained with this routine, the system may be very stiff (see
+          below) or so badly scaled that it cannot be solved to the
+          required accuracy.
+
+          If the routine fails with IFAIL = 2, it is likely that it has
+          been called with a value of TOL which is so small that a solution
+          cannot be obtained on the range X to XEND. This can happen for
+          well-behaved systems and very small values of TOL. The user
+          should, however, consider whether there is a more fundamental
+          difficulty. For example:
+
+          (a)   in the region of a singularity (infinite value) of the
+                solution, the routine will usually stop with IFAIL = 2,
+                unless overflow occurs first. If overflow occurs using
+                D02BHF, D02PAF(*) can be used instead to trap the
+                increasing solution, before overflow occurs. In any case,
+                numerical integration cannot be continued through a
+                singularity, and analytical treatment should be considered;
+
+          (b)   for 'stiff' equations, where the solution contains rapidly
+                decaying components, the routine will compute in very small
+                steps in x (internally to D02BHF) to preserve stability.
+                This will usually exhibit itself by making the computing
+                time excessively long, or occasionally by an exit with
+                IFAIL = 2. Merson's method is not efficient in such cases,
+                and the user should try D02EHF(*) which uses a Backward
+                Differentiation Formula method. To determine whether a
+                problem is stiff, D02BDF(*) may be used.
+
+          For well-behaved systems with no difficulties such as stiffness
+          or singularities, the Merson method should work well for low
+          accuracy calculations (three or four figures). For high accuracy
+          calculations or where FCN is costly to evaluate, Merson's method
+          may not be appropriate and a computationally less expensive
+          method may be D02CHF(*) which uses an Adams method.
+
+          For problems for which D02BHF is not sufficiently general, the
+          user should consider D02PAF(*). D02PAF(*) is a more general
+          Merson routine with many facilities including more general error
+          control options and several criteria for interrupting the
+          calculations. D02PAF(*) can be combined with the rootfinder
+          C05AZF(*) and the interpolation routine D02XAF(*) to solve
+          equations involving y ,y ,...,y  and their derivatives.
+                               1  2      n
+
+          D02BHF can also be used to solve an equation involving x,
+          y ,y ,...,y  and the derivatives of y ,y ,...,y . For example in
+           1  2      n                         1  2      n
+          Section 9, D02BHF is used to find a value of X > 0.0 where Y(1) =
+          0.0. It could instead be used to find a turning-point of y  by
+                                                                    1
+          replacing the function g(x,y) in the program by:
+
+                DOUBLE PRECISION FUNCTION G(X,Y)
+                DOUBLE PRECISION X,Y(3),F(3)
+                CALL FCN(X,Y,F)
+                G = F(1)
+                RETURN
+                END
+
+          This routine is only intended to locate the first zero of g(x,y).
+          If later zeros are required, users are strongly advised to
+          construct their own more general root finding routines as
+          discussed above.
+
+          9. Example
+
+          To find the value X > 0.0 at which y=0.0, where y, v, (phi) are
+          defined by
+
+                                     y'=tan(phi)
+
+                                -0.032tan(phi)   0.02v
+                            v'= --------------- --------
+                                      v         cos(phi)
+
+                                            0.032
+                                   (phi)'=- -----
+                                              2
+                                             v
+
+          and where at X = 0.0 we are given y=0.5, v=0.5 and (phi)=(pi)/5.
+          We write y=Y(1), v=Y(2) and (phi)=Y(3) and we set TOL=1.0E-4 and
+          TOL=1.0E-5 in turn so that we can compare the solutions. We
+          expect the solution X ~= 7.3 and so we set XEND = 10.0 to avoid
+          determining the solution of y=0.0 too near the end of the range
+          of integration. The value of (pi) is obtained by using X01AAF(*).
+
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd02cjf}{NAG On-line Documentation: d02cjf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D02CJF(3NAG)                 D02CJF                  D02CJF(3NAG)
+
+
+
+     D02 -- Ordinary Differential Equations                     D02CJF
+             D02CJF -- NAG Foundation Library Routine Document
+
+     Note: Before using this routine, please read the Users' Note for
+     your implementation to check implementation-dependent details.
+     The symbol (*) after a NAG routine name denotes a routine that is
+     not included in the Foundation Library.
+
+     Note for users via the AXIOM system: the interface to this routine
+     has been enhanced for use with AXIOM and is slightly different to
+     that offered in the standard version of the Foundation Library.
+
+     1. Purpose
+
+     D02CJF integrates a system of first-order ordinary differential
+     equations over a range with suitable initial conditions, using a
+     variable-order, variable-step Adams method until a user-specified
+     function, if supplied, of the solution is zero, and returns the
+     solution at points specified by the user, if desired.
+
+     2. Specification
+
+            SUBROUTINE D02CJF (X, XEND, M, N, Y, FCN, TOL, RELABS,
+           1                   RESULT, OUTPUT, G, W, IFAIL)
+            INTEGER          M, N, IFAIL
+            DOUBLE PRECISION X, XEND, Y(N), TOL, G, W(28+21*N), RESULT(M,N)
+            CHARACTER*1      RELABS
+            EXTERNAL         FCN, OUTPUT, G
+
+     3. Description
+
+     The routine advances the solution of a system of ordinary
+     differential equations
+
+                   y' =f (x,y ,y ,...,y ),  i=1,2,...,n,
+                     i  i    1  2      n
+
+     from x = X to x = XEND using a variable-order, variable-step
+     Adams method. The system is defined by a subroutine FCN supplied
+     by the user, which evaluates f  in terms of x and y ,y ,...,y .
+                                   i                    1  2      n
+     The initial values of y ,y ,...,y  must be given at x = X.
+                            1  2      n
+
+     The solution is returned via the user-supplied routine OUTPUT at
+     points specified by the user, if desired: this solution is
+                  1
+     obtained by C  interpolation on solution values produced by the
+     method. As the integration proceeds a check can be made on the
+     user-specified function g(x,y) to determine an interval where it
+     changes sign. The position of this sign change is then determined
+                    1
+     accurately by C  interpolation to the solution. It is assumed
+     that g(x,y) is a continuous function of the variables, so that a
+     solution of g(x,y)=0.0 can be determined by searching for a
+     change in sign in g(x,y). The accuracy of the integration, the
+     interpolation and, indirectly, of the determination of the
+     position where g(x,y)=0.0, is controlled by the parameters TOL
+     and RELABS.
+
+     For a description of Adams methods and their practical
+     implementation see Hall and Watt [1].
+
+     4. References
+
+     [1]   Hall G and Watt J M (eds) (1976) Modern Numerical Methods
+           for Ordinary Differential Equations. Clarendon Press.
+
+     5. Parameters
+
+      1:  X -- DOUBLE PRECISION                           Input/Output
+          On entry: the initial value of the independent variable x.
+          Constraint: X /= XEND. On exit: if g is supplied by the
+          user, it contains the point where g(x,y)=0.0, unless
+          g(x,y)/=0.0 anywhere on the range X to XEND, in which case,
+          X will contain XEND. If g is not supplied by the user it
+          contains XEND, unless an error has occurred, when it
+          contains the value of x at the error.
+
+      2:  XEND -- DOUBLE PRECISION                               Input
+          On entry: the final value of the independent variable. If
+          XEND < X, integration proceeds in the negative direction.
+          Constraint: XEND /= X.
+
+      3:  M -- INTEGER                                           Input
+          On entry: the first dimension of the array RESULT.  This
+          will usually be equal to the number of points at which the
+          solution is required.
+          Constraint: M > 0.
+
+      4:  N -- INTEGER                                           Input
+          On entry: the number of differential equations.
+          Constraint: N >= 1.
+
+      5:  Y(N) -- DOUBLE PRECISION array                  Input/Output
+          On entry: the initial values of the solution y ,y ,...,y
+                                                         1  2      n
+          at x = X. On exit: the computed values of the solution at
+          the final point x = X.
+
+      6:  FCN -- SUBROUTINE, supplied by the user.
+                                                    External Procedure
+          FCN must evaluate the functions f  (i.e., the derivatives
+                                           i
+          y' ) for given values of their arguments x,y ,y ,...,y .
+            i                                         1  2      n
+
+          Its specification is:
+
+                 SUBROUTINE FCN (X, Y, F)
+                 DOUBLE PRECISION X, Y(n), F(n)
+          where n is the actual value of N in the call of D02CJF.
+
+           1:  X -- DOUBLE PRECISION                             Input
+               On entry: the value of the independent variable x.
+
+           2:  Y(*) -- DOUBLE PRECISION array                    Input
+               On entry: the value of the variable y , for
+                                                     i
+               i=1,2,...,n.
+
+           3:  F(*) -- DOUBLE PRECISION array                   Output
+               On exit: the value of f , for i=1,2,...,n.
+                                       i
+          FCN must be declared as EXTERNAL in the (sub)program
+          from which D02CJF is called. Parameters denoted as
+          Input must not be changed by this procedure.
+
+      7:  TOL -- DOUBLE PRECISION                                Input
+          On entry: a positive tolerance for controlling the error in
+          the integration. Hence TOL affects the determination of the
+          position where g(x,y)=0.0, if g is supplied.
+
+          D02CJF has been designed so that, for most problems, a
+          reduction in TOL leads to an approximately proportional
+          reduction in the error in the solution. However, the actual
+          relation between TOL and the accuracy achieved cannot be
+          guaranteed. The user is strongly recommended to call D02CJF
+          with more than one value for TOL and to compare the results
+          obtained to estimate their accuracy. In the absence of any
+          prior knowledge, the user might compare the results obtained
+                                         -p             -p-1
+          by calling D02CJF with TOL=10.0   and TOL=10.0     where p
+          correct decimal digits are required in the solution.
+          Constraint: TOL > 0.0.
+
+      8:  RELABS -- CHARACTER*1                                  Input
+          On entry: the type of error control. At each step in the
+          numerical solution an estimate of the local error, EST, is
+          made. For the current step to be accepted the following
+          condition must be satisfied:
+                           ______________________________
+                          / n
+                         /  --                          2
+                 EST=   /   >  (e /((tau) *|y |+(tau) )) <=1.0
+                       /    --   i       r   i       a
+                     \/     i=1
+          where (tau)  and (tau)  are defined by
+                     r          a
+               RELABS   (tau)    (tau)
+                             r        a
+
+               'M'      TOL      TOL
+
+               'A'      0.0      TOL
+
+               'R'      TOL      (epsilon)
+
+               'D'      TOL      TOL
+          where (epsilon) is a small machine-dependent number and e
+                                                                   i
+          is an estimate of the local error at y , computed
+                                                i
+          internally. If the appropriate condition is not satisfied,
+          the step size is reduced and the solution is recomputed on
+          the current step. If the user wishes to measure the error in
+          the computed solution in terms of the number of correct
+          decimal places, then RELABS should be set to 'A' on entry,
+          whereas if the error requirement is in terms of the number
+          of correct significant digits, then RELABS should be set to
+          'R'. If the user prefers a mixed error test, then RELABS
+          should be set to 'M', otherwise if the user has no
+          preference, RELABS should be set to the default 'D'. Note
+          that in this case 'D' is taken to be 'M'. Constraint: RELABS
+          = 'M', 'A', 'R', 'D'.
+
+      9:  RESULT(M,N) -- DOUBLE PRECISION array                 Output
+          On exit: the computed values of the solution at the points
+          given by OUTPUT.
+
+     10:  OUTPUT -- SUBROUTINE, supplied by the user.
+                                                    External Procedure
+          OUTPUT allows the user to have access to intermediate values
+          of the computed solution at successive points specified by the
+          user. These solution values may be returned to the user via
+          the array RESULT if desired (this is a non-standard feature
+          added for use with the AXIOM system).  OUTPUT is initially
+          called by D02CJF with XSOL = X (the initial value of x). The
+          user must reset XSOL to the next point where OUTPUT is to be
+          called, and so on at each call to OUTPUT. If, after a call
+          to OUTPUT, the reset point XSOL is beyond XEND, D02CJF will
+          integrate to XEND with no further calls to OUTPUT; if a call
+          to OUTPUT is required at the point XSOL = XEND, then XSOL
+          must be given precisely the value XEND.
+
+          Its specification is:
+
+               SUBROUTINE OUTPUT(XSOL,Y,COUNT,M,N,RESULT)
+               DOUBLE PRECISION Y(N),RESULT(M,N),XSOL
+               INTEGER M,N,COUNT
+
+           1:  XSOL -- DOUBLE PRECISION                   Input/Output
+               On entry: the current value of the independent
+               variable x. On exit: the next value of x at which
+               OUTPUT is to be called.
+
+           2:  Y(N) -- DOUBLE PRECISION array                    Input
+               On entry: the computed solution at the point XSOL.
+
+           3:  COUNT -- INTEGER                           Input/Output
+               On entry: Zero if OUTPUT has not been called before, or
+               the previous value of COUNT.
+               On exit: A new value of COUNT: this can be used to keep
+               track of the number of times OUTPUT has been called.
+
+           4:  M -- INTEGER                                      Input
+               On entry: The first dimension of RESULT.
+
+           5:  N -- INTEGER                                      Input
+               On entry: The dimension of Y.
+
+           6:  RESULT(M,N) -- DOUBLE PRECISION array      Input/Output
+               On entry: the previous contents of RESULT.
+               On exit: RESULT may be used to return the values of the
+               intermediate solutions to the user.
+
+          OUTPUT must be declared as EXTERNAL in the (sub)program
+          from which D02CJF is called. Parameters denoted as
+          Input must not be changed by this procedure.
+
+     11:  G -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                               External Procedure
+          G must evaluate the function g(x,y) for specified values x,y
+          . It specifies the function g for which the first position x
+          where g(x,y)=0 is to be found.
+
+          If the user does not require the root finding option, the
+          actual argument G  mustbe the dummy routine D02CJW. (D02CJW
+          is included in the NAG Foundation Library and so need not be
+          supplied by the user).
+
+          Its specification is:
+
+                 DOUBLE PRECISION FUNCTION G (X, Y)
+                 DOUBLE PRECISION X, Y(n)
+          where n is the actual value of N in the call of D02CJF.
+
+           1:  X -- DOUBLE PRECISION                             Input
+               On entry: the value of the independent variable x.
+
+           2:  Y(*) -- DOUBLE PRECISION array                    Input
+               On entry: the value of the variable y , for
+                                                    i
+               i=1,2,...,n.
+          G must be declared as EXTERNAL in the (sub)program from
+          which D02CJF is called. Parameters denoted as Input
+          must not be changed by this procedure.
+
+     12:  W(28+21*N) -- DOUBLE PRECISION array               Workspace
+
+     13:  IFAIL -- INTEGER                                Input/Output
+          On entry: IFAIL must be set to 0, -1 or 1. For users not
+          familiar with this parameter (described in the Essential
+          Introduction) the recommended value is 0.
+
+          On exit: IFAIL = 0 unless the routine detects an error (see
+          Section 6).
+
+     6. Error Indicators and Warnings
+
+     Errors detected by the routine:
+
+     If on entry IFAIL = 0 or -1, explanatory error messages are
+     output on the current error message unit (as defined by X04AAF).
+
+     IFAIL= 1
+          On entry TOL <= 0.0,
+
+          or       N <= 0,
+
+          or       RELABS /= 'M', 'A', 'R' or 'D'.
+
+          or       X = XEND.
+
+     IFAIL= 2
+          With the given value of TOL, no further progress can be made
+          across the integration range from the current point x = X.
+          (See Section 8 for a discussion of this error exit.) The
+          components Y(1),Y(2),...,Y(N) contain the computed values of
+          the solution at the current point x = X. If the user has
+          supplied g, then no point at which g(x,y) changes sign has
+          been located up to the point x = X.
+
+     IFAIL= 3
+          TOL is too small for D02CJF to take an initial step. X and Y
+          (1),Y(2),...,Y(N) retain their initial values.
+
+     IFAIL= 4
+          XSOL has not been reset or XSOL lies behind X in the
+          direction of integration, after the initial call to OUTPUT,
+          if the OUTPUT option was selected.
+
+     IFAIL= 5
+          A value of XSOL returned by OUTPUT has not been reset or
+          lies behind the last value of XSOL in the direction of
+          integration, if the OUTPUT option was selected.
+
+     IFAIL= 6
+          At no point in the range X to XEND did the function g(x,y)
+          change sign, if g was supplied. It is assumed that g(x,y)=0
+          has no solution.
+
+     IFAIL= 7
+          A serious error has occurred in an internal call. Check all
+          subroutine calls and array sizes. Seek expert help.
+
+     7. Accuracy
+
+     The accuracy of the computation of the solution vector Y may be
+     controlled by varying the local error tolerance TOL. In general,
+     a decrease in local error tolerance should lead to an increase in
+     accuracy. Users are advised to choose RELABS = 'M' unless they
+     have a good reason for a different choice.
+
+     If the problem is a root-finding one, then the accuracy of the
+     root determined will depend on the properties of g(x,y). The user
+     should try to code G without introducing any unnecessary
+     cancellation errors.
+
+     8. Further Comments
+
+     If more than one root is required then D02QFF(*) should be used.
+
+     If the routine fails with IFAIL = 3, then it can be called again
+     with a larger value of TOL if this has not already been tried. If
+     the accuracy requested is really needed and cannot be obtained
+     with this routine, the system may be very stiff (see below) or so
+     badly scaled that it cannot be solved to the required accuracy.
+
+     If the routine fails with IFAIL = 2, it is probable that it has
+     been called with a value of TOL which is so small that a solution
+     cannot be obtained on the range X to XEND. This can happen for
+     well-behaved systems and very small values of TOL. The user
+     should, however, consider whether there is a more fundamental
+     difficulty. For example:
+
+     (a)   in the region of a singularity (infinite value) of the
+           solution, the routine will usually stop with IFAIL = 2,
+           unless overflow occurs first. Numerical integration cannot
+           be continued through a singularity, and analytic treatment
+           should be considered;
+
+     (b)   for 'stiff' equations where the solution contains rapidly
+           decaying components, the routine will use very small steps
+           in x (internally to D02CJF) to preserve stability. This
+           will exhibit itself by making the computing time
+           excessively long, or occasionally by an exit with IFAIL =
+           2. Adams methods are not efficient in such cases, and the
+           user should try D02EJF.
+
+     9. Example
+
+     We illustrate the solution of four different problems. In each
+     case the differential system (for a projectile) is
+
+                                y'=tan(phi)
+
+                           -0.032tan(phi)   0.02v
+                       v'= --------------- --------
+                                 v         cos(phi)
+
+                                      -0.032
+                              (phi)'= ------
+                                         2
+                                        v
+
+     over an interval X = 0.0 to XEND = 10.0 starting with values
+     y=0.5, v=0.5 and (phi)=(pi)/5. We solve each of the following
+     problems with local error tolerances 1.0E-4 and 1.0E-5.
+
+     (i)   To integrate to x=10.0 producing output at intervals of 2.0
+           until a point is encountered where y=0.0.
+
+     (ii)  As (i) but with no intermediate output.
+
+     (iii) As (i) but with no termination on a root-finding condition.
+
+     (iv)  As (i) but with no intermediate output and no root-finding
+           termination condition.
+
+     The example program is not reproduced here. The source code for
+     all example programs is distributed with the NAG Foundation
+     Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd02ejf}{NAG On-line Documentation: d02ejf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D02EJF(3NAG)                 D02EJF                  D02EJF(3NAG)
+
+
+
+     D02 -- Ordinary Differential Equations                     D02EJF
+             D02EJF -- NAG Foundation Library Routine Document
+
+     Note: Before using this routine, please read the Users' Note for
+     your implementation to check implementation-dependent details.
+     The symbol (*) after a NAG routine name denotes a routine that is
+     not included in the Foundation Library.
+
+     Note for users via the AXIOM system: the interface to this routine
+     has been enhanced for use with AXIOM and is slightly different to
+     that offered in the standard version of the Foundation Library.
+
+     1. Purpose
+
+     D02EJF integrates a stiff system of first-order ordinary
+     differential equations over an interval with suitable initial
+     conditions, using a variable-order, variable-step method
+     implementing the Backward Differentiation Formulae (BDF), until a
+     user-specified function, if supplied, of the solution is zero,
+     and returns the solution at points specified by the user, if
+     desired.
+
+     2. Specification
+
+            SUBROUTINE D02EJF (X, XEND, M, N, Y, FCN, PEDERV, TOL,
+           1                   RELABS, OUTPUT, G, W, IW, RESULT, IFAIL)
+            INTEGER          M, N, IW, IFAIL
+            DOUBLE PRECISION X, XEND, Y(N), TOL, G, W(IW), RESULT(M,N)
+            CHARACTER*1      RELABS
+            EXTERNAL         FCN, PEDERV, OUTPUT, G
+
+     3. Description
+
+     The routine advances the solution of a system of ordinary
+     differential equations
+
+                   y' =f (x,y ,y ,...,y ),  i=1,2,...,n,
+                     i  i    1  2      n
+
+     from x = X to x = XEND using a variable-order, variable-step
+     method implementing the BDF. The system is defined by a
+     subroutine FCN supplied by the user, which evaluates f  in terms
+                                                           i
+     of x and y ,y ,...,y  (see Section 5). The initial values of
+               1  2      n
+     y ,y ,...,y  must be given at x = X.
+      1  2      n
+
+     The solution is returned via the user-supplied routine OUTPUT at
+     points specified by the user, if desired: this solution is
+                  1
+     obtained by C  interpolation on solution values produced by the
+     method. As the integration proceeds a check can be made on the
+     user-specified function g(x,y) to determine an interval where it
+     changes sign. The position of this sign change is then determined
+                    1
+     accurately by C  interpolation to the solution. It is assumed
+     that g(x,y) is a continuous function of the variables, so that a
+     solution of g(x,y) = 0.0 can be determined by searching for a
+     change in sign in g(x,y). The accuracy of the integration, the
+     interpolation and, indirectly, of the determination of the
+     position where g(x,y) = 0.0, is controlled by the parameters TOL
+     and RELABS. The Jacobian of the system y'=f(x,y) may be supplied
+     in routine PEDERV, if it is available.
+
+     For a description of BDF and their practical implementation see
+     Hall and Watt [1].
+
+     4. References
+
+     [1]   Hall G and Watt J M (eds) (1976) Modern Numerical Methods
+           for Ordinary Differential Equations. Clarendon Press.
+
+     5. Parameters
+
+      1:  X -- DOUBLE PRECISION                           Input/Output
+          On entry: the initial value of the independent variable x.
+          Constraint: X /= XEND On exit: if G is supplied by the user,
+          X contains the point where g(x,y) = 0.0, unless g(x,y) /= 0.
+          0 anywhere on the range X to XEND, in which case, X will
+          contain XEND. If G is not supplied X contains XEND, unless
+          an error has occured, when it contains the value of x at the
+          error.
+
+      2:  XEND -- DOUBLE PRECISION                               Input
+          On entry: the final value of the independent variable. If
+          XEND < X, integration proceeds in the negative direction.
+          Constraint: XEND /= X.
+
+      3:  M -- INTEGER                                           Input
+          On entry: the first dimension of the array RESULT.  This
+          will usually be equal to the number of points at which the
+          solution is required.
+          Constraint: M > 0.
+
+      4:  N -- INTEGER                                           Input
+          On entry: the number of differential equations, n.
+          Constraint: N >= 1.
+
+      5:  Y(N) -- DOUBLE PRECISION array                  Input/Output
+          On entry: the initial values of the solution y ,y ,...,y
+                                                        1  2      n
+          at x = X. On exit: the computed values of the solution at
+          the final point x = X.
+
+      6:  FCN -- SUBROUTINE, supplied by the user.
+                                                    External Procedure
+          FCN must evaluate the functions f  (i.e., the derivatives
+                                           i
+          y' ) for given values of their arguments x,y ,y ,...,y .
+            i                                         1  2      n
+
+          Its specification is:
+
+                 SUBROUTINE FCN (X, Y, F)
+                 DOUBLE PRECISION X, Y(n), F(n)
+          where n is the actual value of N in the call of D02EJF.
+
+           1:  X -- DOUBLE PRECISION                             Input
+               On entry: the value of the independent variable x.
+
+           2:  Y(*) -- DOUBLE PRECISION array                    Input
+               On entry: the value of the variable y , for
+                                                    i
+               i=1,2,...,n.
+
+           3:  F(*) -- DOUBLE PRECISION array                   Output
+               On exit: the value of f , for i=1,2,...,n.
+                                      i
+          FCN must be declared as EXTERNAL in the (sub)program
+          from which D02EJF is called. Parameters denoted as
+          Input must not be changed by this procedure.
+
+      7:  PEDERV -- SUBROUTINE, supplied by the user.
+                                                    External Procedure
+          PEDERV must evaluate the Jacobian of the system (that is,
+                                   ddf
+                                      i
+          the partial derivatives  ----) for given values of the
+                                   ddy
+                                      j
+          variables x,y ,y ,...,y .
+                       1  2      n
+
+          Its specification is:
+
+                 SUBROUTINE PEDERV (X, Y, PW)
+                 DOUBLE PRECISION X, Y(n), PW(n,n)
+          where n is the actual value of N in the call of D02EJF.
+
+           1:  X -- DOUBLE PRECISION                             Input
+               On entry: the value of the independent variable x.
+
+           2:  Y(*) -- DOUBLE PRECISION array                    Input
+               On entry: the value of the variable y , for
+                                                    i
+               i=1,2,...,n.
+
+           3:  PW(n,*) -- DOUBLE PRECISION array                Output
+                                       ddf
+                                          i
+               On exit: the value of  ----, for i,j=1,2,...,n.
+                                       ddy
+                                          j
+          If the user does not wish to supply the Jacobian, the
+          actual argument PEDERV must be the dummy routine D02EJY
+          . (D02EJY is included in the NAG Foundation Library and
+          so need not be supplied by the user. The name may be
+          implementation dependent: see the User's Note for your
+          implementation for details).
+          PEDERV must be declared as EXTERNAL in the (sub)program
+          from which D02EJF is called. Parameters denoted as
+          Input must not be changed by this procedure.
+
+      8:  TOL -- DOUBLE PRECISION                         Input/Output
+          On entry: TOL must be set to a positive tolerance for
+          controlling the error in the integration. Hence TOL affects
+          the determination of the position where g(x,y) = 0.0, if G
+          is supplied.
+
+          D02EJF has been designed so that, for most problems, a
+          reduction in TOL leads to an approximately proportional
+          reduction in the error in the solution. However, the actual
+          relation between TOL and the accuracy achieved cannot be
+          guaranteed. The user is strongly recommended to call D02EJF
+          with more than one value for TOL and to compare the results
+          obtained to estimate their accuracy. In the absence of any
+          prior knowledge, the user might compare the results obtained
+                                       -p           -p-1
+          by calling D02EJF with TOL=10   and TOL=10     if p correct
+          decimal digits are required in the solution. Constraint: TOL
+          > 0.0. On exit: normally unchanged. However if the range X
+          to XEND is so short that a small change in TOL is unlikely
+          to make any change in the computed solution, then, on
+          return, TOL has its sign changed.
+
+      9:  RELABS -- CHARACTER*1                                  Input
+          On entry: the type of error control. At each step in the
+          numerical solution an estimate of the local error, EST, is
+          made. For the current step to be accepted the following
+          condition must be satisfied:
+                          ________________________________
+                         /   n
+                        /  1 --                          2
+                EST=   /   - >  (e /((tau) *|y |+(tau) )) <=1.0
+                      /    n --   i       r   i       a
+                    \/       i=1
+          where (tau)  and (tau)  are defined by
+                     r          a
+               RELABS   (tau)    (tau)
+                             r        a
+
+               'M'      TOL      TOL
+
+               'A'      0.0      TOL
+
+               'R'      TOL      (epsilon)
+
+               'D'      TOL      (epsilon)
+          where (epsilon) is a small machine-dependent number and e
+                                                                   i
+          is an estimate of the local error at y , computed
+                                                i
+          internally. If the appropriate condition is not satisfied,
+          the step size is reduced and the solution is recomputed on
+          the current step. If the user wishes to measure the error in
+          the computed solution in terms of the number of correct
+          decimal places, then RELABS should be set to 'A' on entry,
+          whereas if the error requirement is in terms of the number
+          of correct significant digits, then RELABS should be set to
+          'R'. If the user prefers a mixed error test, then RELABS
+          should be set to 'M', otherwise if the user has no
+          preference, RELABS should be set to the default 'D'. Note
+          that in this case 'D' is taken to be 'R'. Constraint: RELABS
+          = 'A', 'M', 'R' or 'D'.
+
+     10:  OUTPUT -- SUBROUTINE, supplied by the user.
+                                                    External Procedure
+          OUTPUT allows the user to have access to intermediate values
+          of the computed solution at successive points specified by the
+          user. These solution values may be returned to the user via
+          the array RESULT if desired (this is a non-standard feature
+          added for use with the AXIOM system).  OUTPUT is initially
+          called by D02EJF with XSOL = X (the initial value of x). The
+          user must reset XSOL to the next point where OUTPUT is to be
+          called, and so on at each call to OUTPUT. If, after a call
+          to OUTPUT, the reset point XSOL is beyond XEND, D02EJF will
+          integrate to XEND with no further calls to OUTPUT; if a call
+          to OUTPUT is required at the point XSOL = XEND, then XSOL
+          must be given precisely the value XEND.
+
+          Its specification is:
+
+               SUBROUTINE OUTPUT(XSOL,Y,COUNT,M,N,RESULT)
+               DOUBLE PRECISION Y(N),RESULT(M,N),XSOL
+               INTEGER M,N,COUNT
+
+           1:  XSOL -- DOUBLE PRECISION                   Input/Output
+               On entry: the current value of the independent
+               variable x. On exit: the next value of x at which
+               OUTPUT is to be called.
+
+           2:  Y(N) -- DOUBLE PRECISION array                    Input
+               On entry: the computed solution at the point XSOL.
+
+           3:  COUNT -- INTEGER                           Input/Output
+               On entry: Zero if OUTPUT has not been called before, or
+               the previous value of COUNT.
+               On exit: A new value of COUNT: this can be used to keep
+               track of the number of times OUTPUT has been called.
+
+           4:  M -- INTEGER                                      Input
+               On entry: The first dimension of RESULT.
+
+           5:  N -- INTEGER                                      Input
+               On entry: The dimension of Y.
+
+           6:  RESULT(M,N) -- DOUBLE PRECISION array      Input/Output
+               On entry: the previous contents of RESULT.
+               On exit: RESULT may be used to return the values of the
+               intermediate solutions to the user.
+
+          OUTPUT must be declared as EXTERNAL in the (sub)program
+          from which D02EJF is called. Parameters denoted as
+          Input must not be changed by this procedure.
+
+     11:  G -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                                    External Procedure
+          G must evaluate the function g(x,y) for specified values x,y
+          . It specifies the function g for which the first position x
+          where g(x,y) = 0 is to be found.
+
+          Its specification is:
+
+                 DOUBLE PRECISION FUNCTION G (X, Y)
+                 DOUBLE PRECISION X, Y(n)
+          where n is the actual value of N in the call of D02EJF.
+
+           1:  X -- DOUBLE PRECISION                             Input
+               On entry: the value of the independent variable x.
+
+           2:  Y(*) -- DOUBLE PRECISION array                    Input
+               On entry: the value of the variable y , for
+                                                    i
+               i=1,2,...,n.
+          If the user does not require the root finding option,
+          the actual argument G must be the dummy routine D02EJW.
+          (D02EJW is included in the NAG Foundation Library and
+          so need not be supplied by the user).
+          G must be declared as EXTERNAL in the (sub)program from
+          which D02EJF is called. Parameters denoted as Input
+          must not be changed by this procedure.
+
+     12:  W(IW) -- DOUBLE PRECISION array                    Workspace
+
+     13:  IW -- INTEGER                                          Input
+          On entry:
+          the dimension of the array W as declared in the (sub)program
+          from which D02EJF is called.
+          Constraint: IW>=(12+N)*N+50.
+
+     14:  RESULT(M,N) -- DOUBLE PRECISION array                 Output
+          On exit: the computed values of the solution at the points
+          given by OUTPUT.
+
+     15:  IFAIL -- INTEGER                                Input/Output
+          On entry: IFAIL must be set to 0, -1 or 1. For users not
+          familiar with this parameter (described in the Essential
+          Introduction) the recommended value is 0.
+
+          On exit: IFAIL = 0 unless the routine detects an error (see
+          Section 6).
+
+     6. Error Indicators and Warnings
+
+     Errors detected by the routine:
+
+     If on entry IFAIL = 0 or -1, explanatory error messages are
+     output on the current error message unit (as defined by X04AAF).
+
+     IFAIL= 1
+          On entry TOL <= 0.0,
+
+          or       X = XEND,
+
+          or       N <= 0,
+
+          or       RELABS /= 'M', 'A', 'R', 'D'.
+
+          or       IW<(12+N)*N+50.
+
+     IFAIL= 2
+          With the given value of TOL, no further progress can be made
+          across the integration range from the current point x = X.
+          (See Section 5 for a discussion of this error test.) The
+          components Y(1),Y(2),...,Y(n) contain the computed values of
+          the solution at the current point x = X. If the user has
+          supplied G, then no point at which g(x,y) changes sign has
+          been located up to the point x = X.
+
+     IFAIL= 3
+          TOL is too small for D02EJF to take an initial step. X and Y
+          (1),Y(2),...,Y(n) retain their initial values.
+
+     IFAIL= 4
+          XSOL lies behind X in the direction of integration, after
+          the initial call to OUTPUT, if the OUTPUT option was
+          selected.
+
+     IFAIL= 5
+          A value of XSOL returned by OUTPUT lies behind the last
+          value of XSOL in the direction of integration, if the OUTPUT
+          option was selected.
+
+     IFAIL= 6
+          At no point in the range X to XEND did the function g(x,y)
+          change sign, if G was supplied. It is assumed that g(x,y) =
+          0 has no solution.
+
+     IFAIL= 7
+          A serious error has occurred in an internal call to
+          C05AZF(*). Check all subroutine calls and array dimensions.
+          Seek expert help.
+
+     IFAIL= 8
+          A serious error has occurred in an internal call to
+          D02XKF(*). Check all subroutine calls and array dimensions.
+          Seek expert help.
+
+     IFAIL= 9
+          A serious error has occurred in an internal call to
+          D02NMF(*). Check all subroutine calls and array dimensions.
+          Seek expert help.
+
+     7. Accuracy
+
+     The accuracy of the computation of the solution vector Y may be
+     controlled by varying the local error tolerance TOL. In general,
+     a decrease in local error tolerance should lead to an increase in
+     accuracy. Users are advised to choose RELABS = 'R' unless they
+     have a good reason for a different choice. It is particularly
+     appropriate if the solution decays.
+
+     If the problem is a root-finding one, then the accuracy of the
+                                              ddg      ddg
+     root determined will depend strongly on  --- and  ----, for
+                                              ddx      ddy
+                                                          i
+     i=1,2,...,n. Large values for these quantities may imply large
+     errors in the root.
+
+     8. Further Comments
+
+     If more than one root is required, then to determine the second
+     and later roots D02EJF may be called again starting a short
+     distance past the previously determined roots. Alternatively the
+     user may construct his own root finding code using D02QDF(*) (or
+     the routines of the subchapter D02M-D02N), D02XKF(*) and
+     C05AZF(*).
+
+     If it is easy to code, the user should supply the routine PEDERV.
+     However, it is important to be aware that if PEDERV is coded
+     incorrectly, a very inefficient integration may result and
+     possibly even a failure to complete the integration (IFAIL = 2).
+
+     9. Example
+
+     We illustrate the solution of five different problems. In each
+     case the differential system is the well-known stiff Robertson
+     problem.
+
+                                      4
+                         a'= -0.04a-10 bc
+                                      4        7 2
+                         b'=  0.04a-10 bc -3*10 b
+                                               7 2
+                         c'=               3*10 b
+
+     with initial conditions a=1.0, b=c=0.0 at x=0.0. We solve each of
+     the following problems with local error tolerances 1.0E-3 and 1.0
+     E-4.
+
+     (i)   To integrate to x=10.0 producing output at intervals of 2.0
+           until a point is encountered where a=0.9. The Jacobian is
+           calculated numerically.
+
+     (ii)  As (i) but with the Jacobian calculated analytically.
+
+     (iii) As (i) but with no intermediate output.
+
+     (iv)  As (i) but with no termination on a root-finding condition.
+
+     (v)   Integrating the equations as in (i) but with no
+           intermediate output and no root-finding termination
+           condition.
+
+     The example program is not reproduced here. The source code for
+     all example programs is distributed with the NAG Foundation
+     Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd02gaf}{NAG On-line Documentation: d02gaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D02GAF(3NAG)      Foundation Library (12/10/92)      D02GAF(3NAG)
+
+
+
+          D02 -- Ordinary Differential Equations                     D02GAF
+                  D02GAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D02GAF solves the two-point boundary-value problem with assigned
+          boundary values for a system of ordinary differential equations,
+          using a deferred correction technique and a Newton iteration.
+
+          2. Specification
+
+                 SUBROUTINE D02GAF (U, V, N, A, B, TOL, FCN, MNP, X, Y, NP,
+                1                   W, LW, IW, LIW, IFAIL)
+                 INTEGER          N, MNP, NP, LW, IW(LIW), LIW, IFAIL
+                 DOUBLE PRECISION U(N,2), V(N,2), A, B, TOL, X(MNP), Y
+                1                 (N,MNP), W(LW)
+                 EXTERNAL         FCN
+
+          3. Description
+
+          D02GAF solves a two-point boundary-value problem for a system of
+          n differential equations in the interval [a,b]. The system is
+          written in the form
+
+                      y' =f (x,y ,y ,...,y ) , i=1,2,...,n              (1)
+                        i  i    1  2      n
+
+          and the derivatives are evaluated by a subroutine FCN supplied by
+          the user. Initially, n boundary values of the variables y  must
+                                                                   i
+          be specified (assigned), some at a and some at b. The user also
+          supplies estimates of the remaining n boundary values and all the
+          boundary values are used in constructing an initial approximation
+          to the solution. This approximate solution is corrected by a
+          finite-difference technique with deferred correction allied with
+          a Newton iteration to solve the finite-difference equations. The
+          technique used is described fully in Pereyra [1]. The Newton
+                                                ddf
+                                                   i
+          iteration requires a Jacobian matrix  ---- and this is calculated
+                                                ddy
+                                                   j
+          by numerical differentiation using an algorithm described in
+          Curtis et al [2].
+
+          The user supplies an absolute error tolerance and may also supply
+          an initial mesh for the construction of the finite-difference
+          equations (alternatively a default mesh is used). The algorithm
+          constructs a solution on a mesh defined by adding points to the
+          initial mesh. This solution is chosen so that the error is
+          everywhere less than the user's tolerance and so that the error
+          is approximately equidistributed on the final mesh. The solution
+          is returned on this final mesh.
+
+          If the solution is required at a few specific points then these
+          should be included in the initial mesh. If on the other hand the
+          solution is required at several specific points then the user
+          should use the interpolation routines provided in Chapter E01 if
+          these points do not themselves form a convenient mesh.
+
+          4. References
+
+          [1]   Pereyra V (1979) PASVA3: An Adaptive Finite-Difference
+                Fortran Program for First Order Nonlinear, Ordinary Boundary
+                Problems. Codes for Boundary Value Problems in Ordinary
+                Differential Equations. Lecture Notes in Computer Science.
+                (ed B Childs, M Scott, J W Daniel, E Denman and P Nelson) 76
+                Springer-Verlag.
+
+          [2]   Curtis A R, Powell M J D and Reid J K (1974) On the
+                Estimation of Sparse Jacobian Matrices. J. Inst. Maths
+                Applics. 13 117--119.
+
+          5. Parameters
+
+           1:  U(N,2) -- DOUBLE PRECISION array                       Input
+               On entry: U(i,1) must be set to the known (assigned) or
+               estimated values of y  at a and U(i,2) must be set to the
+                                    i
+               known or estimated values of y  at b, for i=1,2,...,n.
+                                             i
+
+           2:  V(N,2) -- DOUBLE PRECISION array                       Input
+               On entry: V(i,j) must be set to 0.0 if U(i,j) is a known
+               (assigned) value and to 1.0 if U(i,j) is an estimated value,
+               i=1,2,...,n; j=1,2. Constraint: precisely N of the V(i,j)
+               must be set to 0.0, i.e., precisely N of the U(i,j) must be
+               known values, and these must not be all at a or all at b.
+
+           3:  N -- INTEGER                                           Input
+               On entry: the number of equations. Constraint: N >= 2.
+
+           4:  A -- DOUBLE PRECISION                                  Input
+               On entry: the left-hand boundary point, a.
+
+           5:  B -- DOUBLE PRECISION                                  Input
+               On entry: the right-hand boundary point, b. Constraint: B >
+               A.
+
+           6:  TOL -- DOUBLE PRECISION                                Input
+               On entry: a positive absolute error tolerance. If
+                                    a=x <x <...<x  =b
+                                       1  2      NP
+               is the final mesh, z (x ) is the jth component of the
+                                   j  i
+               approximate solution at x , and y (x) is the jth component
+                                        i       j
+               of the true solution of equation (1) (see Section 3) and the
+               boundary conditions, then, except in extreme cases, it is
+               expected that
+                   |z (x )-y (x )|<=TOL , i=1,2,...,NP;j=1,2,...,n      (2)
+                     j  i   j  i
+                Constraint: TOL > 0.0.
+
+           7:  FCN -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               FCN must evaluate the functions f  (i.e., the derivatives
+                                                i
+               y' ) at the general point x.
+                 i
+
+               Its specification is:
+
+                      SUBROUTINE FCN (X, Y, F)
+                      DOUBLE PRECISION X, Y(n), F(n)
+               where n is the actual value of N in the call of D02GAF.
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the value of the argument x.
+
+                2:  Y(*) -- DOUBLE PRECISION array                    Input
+                    On entry: the value of the argument y , for
+                                                         i
+                    i=1,2,...,n.
+
+                3:  F(*) -- DOUBLE PRECISION array                   Output
+                    On exit: the values of f , for i=1,2,...,n.
+                                            i
+               FCN must be declared as EXTERNAL in the (sub)program
+               from which D02GAF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           8:  MNP -- INTEGER                                         Input
+               On entry: the maximum permitted number of mesh points.
+               Constraint: MNP >= 32.
+
+           9:  X(MNP) -- DOUBLE PRECISION array                Input/Output
+               On entry: if NP >= 4 (see NP below), the first NP elements
+               must define an initial mesh. Otherwise the elements of X
+               need not be set. Constraint:
+                          A=X(1)<X(2)<...<X(NP)=B for NP>=4             (3)
+               On exit: X(1),X(2),...,X(NP) define the final mesh (with
+               the returned value of NP) satisfying the relation (3).
+
+          10:  Y(N,MNP) -- DOUBLE PRECISION array                    Output
+               On exit: the approximate solution z (x ) satisfying (2), on
+                                                  j  i
+               the final mesh, that is
+                        Y(j,i)=z (x ) , i=1,2,...,NP;j=1,2,...,n,
+                                j  i
+               where NP is the number of points in the final mesh.
+
+               The remaining columns of Y are not used.
+
+          11:  NP -- INTEGER                                   Input/Output
+               On entry: determines whether a default or user-supplied
+               mesh is used. If NP = 0, a default value of 4 for NP and a
+               corresponding equispaced mesh X(1),X(2),...,X(NP) are used.
+               If NP >= 4, then the user must define an initial mesh using
+               the array X as described. Constraint: NP = 0 or 4 <= NP <=
+               MNP. On exit: the number of points in the final (returned)
+               mesh.
+
+          12:  W(LW) -- DOUBLE PRECISION array                    Workspace
+
+          13:  LW -- INTEGER                                          Input
+               On entry: the length of the array W as declared in the
+                                                            2         2
+               calling (sub)program. Constraint: LW>=MNP*(3N +6N+2)+4N +4N
+
+          14:  IW(LIW) -- INTEGER array                           Workspace
+
+          15:  LIW -- INTEGER                                         Input
+               On entry: the length of the array IW as declared in the
+                                                                  2
+               calling (sub)program. Constraint: LIW>=MNP*(2N+1)+N +4N+2.
+
+          16:  IFAIL -- INTEGER                                Input/Output
+               For this routine, the normal use of IFAIL is extended to
+               control the printing of error and warning messages as well
+               as specifying hard or soft failure (see the Essential
+               Introduction).
+
+               Before entry, IFAIL must be set to a value with the decimal
+               expansion cba, where each of the decimal digits c, b and a
+               must have a value of 0 or 1.
+               a=0   specifies hard failure, otherwise soft failure;
+
+               b=0   suppresses error messages, otherwise error messages
+                     will be printed (see Section 6);
+
+               c=0   suppresses warning messages, otherwise warning
+                     messages will be printed (see Section 6).
+               The recommended value for inexperienced users is 110 (i.e.,
+               hard failure with all messages printed).
+
+               Unless the routine detects an error (see Section 6), IFAIL
+               contains 0 on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               One or more of the parameters N, TOL, NP, MNP, LW or LIW has
+               been incorrectly set, or B <= A, or the condition (3) on X
+               is not satisfied, or the number of known boundary values
+               (specified by V) is not N.
+
+          IFAIL= 2
+               The Newton iteration has failed to converge. This could be
+               due to there being too few points in the initial mesh or to
+               the initial approximate solution being too inaccurate. If
+               this latter reason is suspected the user should use
+               subroutine D02RAF instead. If the warning 'Jacobian matrix
+               is singular' is printed this could be due to specifying zero
+               estimated boundary values and these should be varied. This
+               warning could also be printed in the unlikely event of the
+               Jacobian matrix being calculated inaccurately. If the user
+               cannot make changes to prevent the warning then subroutine
+               D02RAF should be used.
+
+          IFAIL= 3
+               The Newton iteration has reached round-off level. It could
+               be, however, that the answer returned is satisfactory. This
+               error might occur if too much accuracy is requested.
+
+          IFAIL= 4
+               A finer mesh is required for the accuracy requested; that is
+               MNP is not large enough.
+
+          IFAIL= 5
+               A serious error has occurred in a call to D02GAF. Check all
+               array subscripts and subroutine parameter lists in calls to
+               D02GAF. Seek expert help.
+
+          7. Accuracy
+
+          The solution returned by the routine will be accurate to the
+          user's tolerance as defined by the relation (2) except in extreme
+          circumstances. If too many points are specified in the initial
+          mesh, the solution may be more accurate than requested and the
+          error may not be approximately equidistributed.
+
+          8. Further Comments
+
+          The time taken by the routine depends on the difficulty of the
+          problem, the number of mesh points used (and the number of
+          different meshes used), the number of Newton iterations and the
+          number of deferred corrections.
+
+          The user is strongly recommended to set IFAIL to obtain self-
+          explanatory error messages, and also monitoring information about
+          the course of the computation. The user may select the channel
+          numbers on which this output is to appear by calls of X04AAF (for
+          error messages) or X04ABF (for monitoring information) - see
+          Section 9 for an example. Otherwise the default channel numbers
+          will be used, as specified in the implementation document.
+
+          A common cause of convergence problems in the Newton iteration is
+          the user specifying too few points in the initial mesh. Although
+          the routine adds points to the mesh to improve accuracy it is
+          unable to do so until the solution on the initial mesh has been
+          calculated in the Newton iteration.
+
+          If the user specifies zero known and estimated boundary values,
+          the routine constructs a zero initial approximation and in many
+          cases the Jacobian is singular when evaluated for this
+          approximation, leading to the breakdown of the Newton iteration.
+
+          The user may be unable to provide a sufficiently good choice of
+          initial mesh and estimated boundary values, and hence the Newton
+          iteration may never converge. In this case the continuation
+          facility provided in D02RAF is recommended.
+
+          In the case where the user wishes to solve a sequence of similar
+          problems, the final mesh from solving one case is strongly
+          recommended as the initial mesh for the next.
+
+          9. Example
+
+          We solve the differential equation
+
+                                                    2
+                              y'''=-yy''-(beta)(1-y' )
+
+          with boundary conditions
+
+                                    y(0)=y'(0)=0,
+
+                                      y'(10)=1
+
+          for (beta)=0.0 and (beta)=0.2 to an accuracy specified by TOL=1.0
+          E-3. We solve first the simpler problem with (beta)=0.0 using an
+          equispaced mesh of 26 points and then we solve the problem with
+          (beta)=0.2 using the final mesh from the first problem.
+
+          Note the call to X04ABF prior to the call to D02GAF.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd02gbf}{NAG On-line Documentation: d02gbf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D02GBF(3NAG)      Foundation Library (12/10/92)      D02GBF(3NAG)
+
+
+
+          D02 -- Ordinary Differential Equations                     D02GBF
+                  D02GBF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D02GBF solves a general linear two-point boundary value problem
+          for a system of ordinary differential equations using a deferred
+          correction technique.
+
+          2. Specification
+
+                 SUBROUTINE D02GBF (A, B, N, TOL, FCNF, FCNG, C, D, GAM,
+                1                   MNP, X, Y, NP, W, LW, IW, LIW, IFAIL)
+                 INTEGER          N, MNP, NP, LW, IW(LIW), LIW, IFAIL
+                 DOUBLE PRECISION A, B, TOL, C(N,N), D(N,N), GAM(N), X(MNP),
+                1                 Y(N,MNP), W(LW)
+                 EXTERNAL         FCNF, FCNG
+
+          3. Description
+
+          D02GBF solves the linear two-point boundary value problem for a
+          system of n ordinary differential equations in the interval
+
+          [a,b]. The system is written in the form
+
+                                 y'=F(x)y+g(x)                          (1)
+
+          and the boundary conditions are written in the form
+
+                              Cy(a)+Dy(b)=(gamma)                       (2)
+
+          Here F(x), C and D are n by n matrices, and g(x) and (gamma) are
+          n-component vectors. The approximate solution to (1) and (2) is
+          found using a finite-difference method with deferred correction.
+          The algorithm is a specialisation of that used in subroutine
+          D02RAF which solves a nonlinear version of (1) and (2). The
+          nonlinear version of the algorithm is described fully in Pereyra
+          [1].
+
+          The user supplies an absolute error tolerance and may also supply
+          an initial mesh for the construction of the finite-difference
+          equations (alternatively a default mesh is used). The algorithm
+          constructs a solution on a mesh defined by adding points to the
+          initial mesh. This solution is chosen so that the error is
+          everywhere less than the user's tolerance and so that the error
+          is approximately equidistributed on the final mesh. The solution
+          is returned on this final mesh.
+
+          If the solution is required at a few specific points then these
+          should be included in the initial mesh. If, on the other hand,
+          the solution is required at several specific points, then the
+          user should use the interpolation routines provided in Chapter
+          E01 if these points do not themselves form a convenient mesh.
+
+          4. References
+
+          [1]   Pereyra V (1979) PASVA3: An Adaptive Finite-Difference
+                Fortran Program for First Order Nonlinear, Ordinary Boundary
+                Problems. Codes for Boundary Value Problems in Ordinary
+                Differential Equations. Lecture Notes in Computer Science.
+                (ed B Childs, M Scott, J W Daniel, E Denman and P Nelson) 76
+                Springer-Verlag.
+
+          5. Parameters
+
+           1:  A -- DOUBLE PRECISION                                  Input
+               On entry: the left-hand boundary point, a.
+
+           2:  B -- DOUBLE PRECISION                                  Input
+               On entry: the right-hand boundary point, b. Constraint: B >
+               A.
+
+           3:  N -- INTEGER                                           Input
+               On entry: the number of equations; that is n is the order of
+               system (1). Constraint: N >= 2.
+
+           4:  TOL -- DOUBLE PRECISION                                Input
+               On entry: a positive absolute error tolerance. If
+                                    a=x <x <...<x  =b
+                                       1  2      NP
+               is the final mesh, z(x) is the approximate solution from
+               D02GBF and y(x) is the true solution of equations (1) and
+               (2) then, except in extreme cases, it is expected that
+                                    ||z-y||<=TOL                        (3)
+               where
+                           ||u||=max       max        |u (x )|.
+                                    1<=i<=N   1<=j<=NP  i  j
+               Constraint: TOL > 0.0.
+
+           5:  FCNF -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               FCNF must evaluate the matrix F(x) in (1) at a general point
+               x.
+
+               Its specification is:
+
+                      SUBROUTINE FCN (X, F)
+                      DOUBLE PRECISION X, F(n,n)
+               where n is the actual value of N in the call of D02GBF.
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the value of the independent variable x.
+
+                2:  F(n,n) -- DOUBLE PRECISION array                 Output
+                    On exit: the (i,j)th element of the matrix F(x), for
+                    i,j=1,2,...,n. (See Section 9 for an example.)
+               FCN must be declared as EXTERNAL in the (sub)program
+               from which D02GBF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           6:  FCNG -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               FCNG must evaluate the vector g(x) in (1) at a general point
+               x.
+
+               Its specification is:
+
+                      SUBROUTINE FCNG (X, G)
+                      DOUBLE PRECISION X, G(n)
+               where n is the actual value of N in the call of D02GBF.
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the value of the independent variable x.
+
+                2:  G(*) -- DOUBLE PRECISION array                   Output
+                    On exit: the ith element of the vector g(x), for
+                    i=1,2,...,n. (See Section Section 9 for an example.)
+               FCNG must be declared as EXTERNAL in the (sub)program
+               from which D02GBF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           7:  C(N,N) -- DOUBLE PRECISION array                Input/Output
+
+           8:  D(N,N) -- DOUBLE PRECISION array                Input/Output
+
+           9:  GAM(N) -- DOUBLE PRECISION array                Input/Output
+               On entry: the arrays C and D must be set to the matrices C
+               and D in (2). GAM must be set to the vector (gamma) in (2).
+               On exit: the rows of C and D and the components of GAM are
+               re-ordered so that the boundary conditions are in the order:
+                 (i)   conditions on y(a) only;
+
+                 (ii)  condition involving y(a) and y(b); and
+
+                 (iii) conditions on y(b) only.
+
+               The routine will be slightly more efficient if the arrays C,
+               D and GAM are ordered in this way before entry, and in this
+               event they will be unchanged on exit.
+
+               Note that the problems (1) and (2) must be of boundary value
+               type, that is neither C nor D may be identically zero. Note
+               also that the rank of the matrix [C,D] must be n for the
+               problem to be properly posed. Any violation of these
+               conditions will lead to an error exit.
+
+          10:  MNP -- INTEGER                                         Input
+               On entry: the maximum permitted number of mesh points.
+               Constraint: MNP >= 32.
+
+          11:  X(MNP) -- DOUBLE PRECISION array                Input/Output
+               On entry: if NP >= 4 (see NP below), the first NP elements
+               must define an initial mesh. Otherwise the elements of x
+               need not be set. Constraint:
+                         A=X(1)<X(2)<...<X(NP)=B, for NP>=4.            (4)
+               On exit: X(1),X(2),...,X(NP) define the final mesh (with the
+               returned value of NP) satisfying the relation (4).
+
+          12:  Y(N,MNP) -- DOUBLE PRECISION array                    Output
+               On exit: the approximate solution z(x) satisfying (3), on
+               the final mesh, that is
+                         Y(j,i)=z (x ) , i=1,2,...,NP;j=1,2,...,n
+                                 j  i
+               where NP is the number of points in the final mesh.
+
+               The remaining columns of Y are not used.
+
+          13:  NP -- INTEGER                                   Input/Output
+               On entry: determines whether a default mesh or user-supplied
+               mesh is used. If NP = 0, a default value of 4 for NP and a
+               corresponding equispaced mesh X(1),X(2),...,X(NP) are used.
+               If NP >= 4, then the user must define an initial mesh X as
+               in (4) above. On exit: the number of points in the final
+               (returned) mesh.
+
+          14:  W(LW) -- DOUBLE PRECISION array                    Workspace
+
+          15:  LW -- INTEGER                                          Input
+               On entry: the length of the array W, Constraint:
+                          2         2
+               LW>=MNP*(3N +5N+2)+3N +5N.
+
+          16:  IW(LIW) -- INTEGER array                           Workspace
+
+          17:  LIW -- INTEGER                                         Input
+               On entry: the length of the array IW. Constraint:
+               LIW>=MNP*(2N+1)+N.
+
+          18:  IFAIL -- INTEGER                                Input/Output
+               For this routine, the normal use of IFAIL is extended to
+               control the printing of error and warning messages as well
+               as specifying hard or soft failure (see the Essential
+               Introduction).
+
+               Before entry, IFAIL must be set to a value with the decimal
+               expansion cba, where each of the decimal digits c, b and a
+               must have a value of 0 or 1.
+               a=0   specifies hard failure, otherwise soft failure;
+
+               b=0   suppresses error messages, otherwise error messages
+                     will be printed (see Section 6);
+
+               c=0   suppresses warning messages, otherwise warning
+                     messages will be printed (see Section 6).
+               The recommended value for inexperienced users is 110 (i.e.,
+               hard failure with all messages printed).
+
+               Unless the routine detects an error (see Section 6), IFAIL
+               contains 0 on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          For each error, an explanatory error message is output on the
+          current error message unit (as defined by X04AAF), unless
+          suppressed by the value of IFAIL on entry.
+
+          IFAIL= 1
+               One or more of the parameters N, TOL, NP, MNP, LW or LIW is
+               incorrectly set, B <= A or the condition (4) on X is not
+               satisfied.
+
+          IFAIL= 2
+               There are three possible reasons for this error exit to be
+               taken:
+                 (i)  one of the matrices C or D is identically zero (that
+                      is the problem is of initial value and not boundary
+                      value type). In this case, IW(1) = 0 on exit;
+
+                 (ii) a row of C and the corresponding row of D are
+                      identically zero (that is the boundary conditions are
+                      rank deficient). In this case, on exit IW(1) contains
+                      the index of the first such row encountered; and
+
+                 (iii more than n of the columns of the n by 2n matrix [C,D
+                 )    ] are identically zero (that is the boundary
+                      conditions are rank deficient). In this case, on exit
+                      IW(1) contains minus the number of non-identically
+                      zero columns.
+
+          IFAIL= 3
+               The routine has failed to find a solution to the specified
+               accuracy. There are a variety of possible reasons including:
+                 (i)   the boundary conditions are rank deficient, which may
+                       be indicated by the message that the Jacobian is
+                       singular. However this is an unlikely explanation for
+                       the error exit as all rank deficient boundary
+                       conditions should lead instead to error exits with
+                       either IFAIL = 2 or IFAIL = 5; see also (iv) below;
+
+                 (ii)  not enough mesh points are permitted in order to
+                       attain the required accuracy. This is indicated by NP
+                       = MNP on return from a call to D02GBF. This
+                       difficulty may be aggravated by a poor initial choice
+                       of mesh points;
+
+                 (iii) the accuracy requested cannot be attained on the
+                       computer being used; and
+
+                 (iv)  an unlikely combination of values of F(x) has led to
+                       a singular Jacobian. The error should not persist if
+                       more mesh points are allowed.
+
+          IFAIL= 4
+               A serious error has occurred in a call to D02GBF. Check all
+               array subscripts and subroutine parameter lists in calls to
+               D02GBF. Seek expert help.
+
+          IFAIL= 5
+               There are two possible reasons for this error exit which
+               occurs when checking the rank of the boundary conditions by
+               reduction to a row echelon form:
+                    (i) at least one row of the n by 2n matrix [C,D] is a
+                    linear combination of the other rows and hence the
+                    boundary conditions are rank deficient. The index of
+                    the first such row encountered is given by IW(1) on
+                    exit; and
+
+                    (ii) as (i) but the rank deficiency implied by this
+                    error exit has only been determined up to a numerical
+                    tolerance. Minus the index of the first such row
+                    encountered is given by IW(1) on exit.
+               In case (ii) above there is some doubt as to the rank
+               deficiency of the boundary conditions. However even if the
+               boundary conditions are not rank deficient they are not
+               posed in a suitable form for use with this routine.
+
+               For example, if
+                                                          ((gamma) )
+                        (1 0        )     (1 0)           (       1)
+                      C=(1 (epsilon)) , D=(1 0) , (gamma)=((gamma) )
+                                                          (       2)
+               and (epsilon) is small enough, this error exit is likely to
+               be taken. A better form for the boundary conditions in this
+               case would be
+                                           ((gamma)                       )
+                                           (       1                      )
+                 (1 0)     (1 0)           (         -1                   )
+               C=(0 1) , D=(0 0) , (gamma)=((epsilon)  ((gamma) -(gamma) ))
+                                           (                   2        1 )
+
+          7. Accuracy
+
+          The solution returned by the routine will be accurate to the
+          user's tolerance as defined by the relation (3) except in extreme
+          circumstances. If too many points are specified in the initial
+          mesh, the solution may be more accurate than requested and the
+          error may not be approximately equidistributed.
+
+          8. Further Comments
+
+          The time taken by the routine depends on the difficulty of the
+          problem, the number of mesh points (and meshes) used and the
+          number of deferred corrections.
+
+          The user is strongly recommended to set IFAIL to obtain self-
+          explanatory error messages, and also monitoring information about
+          the course of the computation. The user may select the channel
+          numbers on which this output is to appear by calls of X04AAF (for
+          error messages) or X04ABF (for monitoring information) - see
+          Section 9 for an example. Otherwise the default channel numbers
+          will be used, as specified in the implementation document.
+
+          In the case where the user wishes to solve a sequence of similar
+          problems, the use of the final mesh from one case is strongly
+          recommended as the initial mesh for the next.
+
+          9. Example
+
+          We solve the problem (written as a first order system)
+
+                                  (epsilon)y''+y'=0
+
+          with boundary conditions
+
+                                   y(0)=0,  y(1)=1
+
+                                    -1                 -2
+          for the cases (epsilon)=10   and (epsilon)=10   using the default
+          initial mesh in the first case, and the final mesh of the first
+          case as initial mesh for the second (more difficult) case. We
+          give the solution and the error at each mesh point to illustrate
+          the accuracy of the method given the accuracy request TOL=1.0E-3.
+
+          Note the call to X04ABF prior to the call to D02GBF.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd02kef}{NAG On-line Documentation: d02kef}
+\beginscroll
+\begin{verbatim}
+
+
+
+D02KEF(3NAG)                 D02KEF                  D02KEF(3NAG)
+
+
+
+     D02 -- Ordinary Differential Equations                     D02KEF
+             D02KEF -- NAG Foundation Library Routine Document
+
+     Note: Before using this routine, please read the Users' Note for
+     your implementation to check implementation-dependent details.
+     The symbol (*) after a NAG routine name denotes a routine that is
+     not included in the Foundation Library.
+
+     1. Purpose
+
+     D02KEF finds a specified eigenvalue of a regular singular second-
+     order Sturm-Liouville system on a finite or infinite range, using
+     a Pruefer transformation and a shooting method. It also reports
+     values of the eigenfunction and its derivatives. Provision is
+     made for discontinuities in the coefficient functions or their
+     derivatives.
+
+     2. Specification
+
+            SUBROUTINE D02KEF (XPOINT, M, MATCH, COEFFN, BDYVAL, K,
+           1                   TOL, ELAM, DELAM, HMAX, MAXIT, MAXFUN,
+           2                   MONIT, REPORT, IFAIL)
+            INTEGER          M, MATCH, K, MAXIT, MAXFUN, IFAIL
+            DOUBLE PRECISION XPOINT(M), TOL, ELAM, DELAM, HMAX(2,M)
+            EXTERNAL         COEFFN, BDYVAL, MONIT, REPORT
+
+     3. Description
+
+     D02KEF has essentially the same purpose as D02KDF(*) with minor
+     modifications to enable values of the eigenfunction to be
+     obtained after convergence to the eigenvalue has been achieved.
+
+                                           ~~~~~~~~
+     It first finds a specified eigenvalue (lambda) of a Sturm-
+     Liouville system defined by a self-adjoint differential equation
+     of the second-order
+
+                     (p(x)y')'+q(x;(lambda))y=0, a<x<b
+
+     together with the appropriate boundary conditions at the two
+     (finite or infinite) end-points a and b. The functions p and q,
+     which are real-valued, must be defined by a subroutine COEFFN.
+     The boundary conditions must be defined by a subroutine BDYVAL,
+     and, in the case of a singularity at a or b, take the form of an
+     asymptotic formula for the solution near the relevant end-point.
+
+                                      ~~~~~~~~
+     When the final estimate (lambda)=(lambda) of the eigenvalue has
+     been found, the routine integrates the differential equation once
+     more with that value of (lambda), and with initial conditions
+     chosen so that the integral
+
+                       b
+                       /    2    ddq
+                    S= |y(x)  ----------(x;(lambda))dx
+                       /      dd(lambda)
+                       a
+
+     is approximately one. When q(x;(lambda)) is of the form
+     (lambda)w(x)+q(x), which is the most common case, S represents
+     the square of the norm of y induced by the inner product
+
+                                 b
+                                 /
+                          <f,g>= |f(x)g(x)w(x)dx,
+                                 /
+                                 a
+
+     with respect to which the eigenfunctions are mutually orthogonal.
+     This normalisation of y is only approximate, but experience shows
+     that S generally differs from unity by only one or two per cent.
+
+     During this final integration the REPORT routine supplied by the
+     user is called at each integration mesh point x. Sufficient
+     information is returned to permit the user to compute y(x) and
+     y'(x) for printing or plotting. For reasons described in Section
+     8.2, D02KEF passes across to REPORT, not y and y', but the Pru
+     efer variables (beta), (phi) and (rho) on which the numerical
+     method is based. Their relationship to y and y' is given by the
+     equations
+
+
+              ______   ( (rho))   ( (phi))
+     p(x)y'=\/(beta)exp( -----)cos( -----);
+                       (   2  )   (   2  )
+
+                1       ( (rho))   ( (phi))
+          y= --------exp( -----)sin( -----).
+               ______   (   2  )   (   2  )
+             \/(beta)
+
+     For the theoretical basis of the numerical method to be valid,
+     the following conditions should hold on the coefficient
+     functions:
+
+     (a)   p(x) must be non-zero and of one sign throughout the
+           interval (a,b); and,
+
+               ddq
+     (b)    ---------- must be of one sign throughout (a,b) for all
+            dd(lambda)
+           relevant values of (lambda), and must not be identically
+           zero as x varies, for any (lambda).
+
+     Points of discontinuity in the functions p and q or their
+     derivatives are allowed, and should be included as 'break-points'
+     in the array XPOINT.
+
+     A good account of the theory of Sturm-Liouville systems, with
+     some description of Pruefer transformations, is given in Birkhoff
+     and Rota [4], Chapter X. An introduction for the user of Pruefer
+     transformations for the numerical solution of eigenvalue problems
+     arising from physics and chemistry is Bailey [2].
+
+     The scaled Pruefer method is fairly recent, and is described in a
+     short note by Pryce [6] and in some detail in the technical
+     report [5].
+
+     4. References
+
+     [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+           Functions. Dover Publications.
+
+     [2]   Bailey P B (1966) Sturm-Liouville Eigenvalues via a Phase
+           Function. SIAM J. Appl. Math . 14 242--249.
+
+     [3]   Banks D O and Kurowski I (1968) Computation of Eigenvalues
+           of Singular Sturm-Liouville Systems. Math. Computing. 22
+           304--310.
+
+     [4]   Birkhoff G and Rota G C (1962) Ordinary Differential
+           Equations. Ginn & Co., Boston and New York.
+
+     [5]   Pryce J D (1981) Two codes for Sturm-Liouville problems.
+           Technical Report CS-81-01. Dept of Computer Science, Bristol
+           University .
+
+     [6]   Pryce J D and Hargrave B A (1977) The Scale Pruefer Method
+           for one-parameter and multi-parameter eigenvalue problems in
+           ODEs. Inst. Math. Appl., Numerical Analysis Newsletter. 1(3)
+
+     5. Parameters
+
+      1:  XPOINT(M) -- DOUBLE PRECISION array                    Input
+          On entry: the points where the boundary conditions computed
+          by BDYVAL are to be imposed, and also any break-points,
+          i.e., XPOINT(1) to XPOINT(m) must contain values x ,...,x
+                                                            1      m
+          such that
+                             x <=x <x <...<x   <=x
+                              1   2  3      m-1   m
+          with the following meanings:
+          (a)   x  and x  are the left and right end-points, a and b,
+                 1      m
+                of the domain of definition of the Sturm-Liouville
+                system if these are finite. If either a or b is
+                infinite, the corresponding value x  or x  may be a
+                                                   1     m
+                more-or-less arbitrarily 'large' number of appropriate
+                sign.
+
+          (b)   x  and x    are the Boundary Matching Points (BMP's),
+                 2      m-1
+                that is the points at which the left and right
+                boundary conditions computed in BDYVAL are imposed.
+
+                If the left-hand end-point is a regular point then the
+                user should set x =x (=a), while if it is a singular
+                                 2  1
+                point the user must set x >x . Similarly x   =x (=b)
+                                         2  1             m-1  m
+                if the right-hand end-point is regular, and x   <x  if
+                                                             m-1  m
+                it is singular.
+
+          (c)   The remaining m-4 points x ,...,x   , if any, define
+                                          3      m-2
+                `break-points' which divide the interval [x ,x   ]
+                                                           2  m-1
+                into m-3 sub-intervals
+                           i =[x ,x ],...,i   =[x   ,x   ]
+                            1   2  3       m-3   m-2  m-1
+                Numerical integration of the differential equation is
+                stopped and restarted at each break-point. In simple
+                cases no break-points are needed. However if p(x) or
+                q(x;(lambda)) are given by different formulae in
+                different parts of the range, then integration is more
+                efficient if the range is broken up by break-points in
+                the appropriate way. Similarly points where any jumps
+                occur in p(x) or q(x;(lambda)), or in their
+                derivatives up to the fifth order, should appear as
+                break-points.
+          Constraint: X(1) <= X(2) < ... < X(M-1) <= X(M).
+
+      2:  M -- INTEGER                                           Input
+          On entry: the number of points in the array XPOINT.
+          Constraint: M >= 4.
+
+      3:  MATCH -- INTEGER                                Input/Output
+          On entry: MATCH must be set to the index of the 'break-point
+          ' to be used as the matching point (see Section 8.3). If
+          MATCH is set to a value outside the range [2,m-1] then a
+          default value is taken, corresponding to the break-point
+          nearest the centre of the interval [XPOINT(2),XPOINT(m-1)].
+          On exit: the index of the break-point actually used as the
+          matching point.
+
+      4:  COEFFN -- SUBROUTINE, supplied by the user.
+                                               External Procedure
+          COEFFN must compute the values of the coefficient functions
+          p(x) and q(x;(lambda)) for given values of x and (lambda).
+          Section 3 states conditions which p and q must satisfy.
+
+          Its specification is:
+
+                 SUBROUTINE COEFFN (P, Q, DQDL, X, ELAM, JINT)
+                 DOUBLE PRECISION P, Q, DQDL, X, ELAM
+                 INTEGER          JINT
+
+           1:  P -- DOUBLE PRECISION                            Output
+               On exit: the value of p(x) for the current value of x.
+
+           2:  Q -- DOUBLE PRECISION                            Output
+               On exit: the value of q(x;(lambda)) for the current
+               value of x and the current trial value of (lambda).
+
+           3:  DQDL -- DOUBLE PRECISION                         Output
+                                         ddq
+               On exit: the value of  ----------(x;(lambda)) for the
+                                      dd(lambda)
+               current value of x and the current trial value of
+               (lambda). However DQDL is only used in error estimation
+               and an approximation (say to within 20%) will suffice.
+
+           4:  X -- DOUBLE PRECISION                             Input
+               On entry: the current value of x.
+
+           5:  ELAM -- DOUBLE PRECISION                          Input
+               On entry: the current trial value of the eigenvalue
+               parameter (lambda).
+
+           6:  JINT -- INTEGER                                   Input
+               On entry: the index j of the sub-interval i  (see
+                                                          j
+               specification of XPOINT) in which x lies.
+          See Section 8.4 and Section 9 for examples.
+
+          COEFFN must be declared as EXTERNAL in the (sub)program
+          from which D02KEF is called. Parameters denoted as
+          Input must not be changed by this procedure.
+
+      5:  BDYVAL -- SUBROUTINE, supplied by the user.
+                                               External Procedure
+          BDYVAL must define the boundary conditions. For each end-
+          point, BDYVAL must return (in YL or YR) values of y(x) and
+          p(x)y'(x) which are consistent with the boundary conditions
+          at the end-points; only the ratio of the values matters.
+          Here x is a given point (XL or XR) equal to, or close to,
+          the end-point.
+
+          For a regular end-point (a, say), x=a; and a boundary
+          condition of the form
+                                c y(a)+c y'(a)=0
+                                 1      2
+          can be handled by returning constant values in YL, e.g.
+          YL(1)=c  and YL(2)=-c p(a).
+                 2             1
+
+          For a singular end-point however, YL(1) and YL(2) will in
+          general be functions of XL and ELAM, and YR(1) and YR(2)
+          functions of XR and ELAM, usually derived analytically from
+          a power-series or asymptotic expansion. Examples are given
+          in Section 8.5 Section 9.
+
+          Its specification is:
+
+                 SUBROUTINE BDYVAL (XL, XR, ELAM, YL, YR))
+                 DOUBLE PRECISION XL, XR, ELAM, YL(3), YR(3)
+
+           1:  XL -- DOUBLE PRECISION                            Input
+               On entry: if a is a regular end-point of the system (so
+               that a=x =x ), then XL contains a. If a is a singular
+                       1  2
+               point (so that a<=x <x ), then XL contains a point x
+                                  1  2
+               such that x <x<=x ).
+                          1     2
+
+           2:  XR -- DOUBLE PRECISION                            Input
+               On entry: if b is a regular end-point of the system (so
+               that x   =x =b), then XR contains b. If b is a singular
+                     m-1  m
+               point (so that x   <x <=b), then XR contains a point x
+                               m-1  m
+               such that x   <=x<x .
+                          m-1     m
+
+           3:  ELAM -- DOUBLE PRECISION                          Input
+               On entry: the current trial value of (lambda).
+
+           4:  YL(3) -- DOUBLE PRECISION array                  Output
+               On exit: YL(1) and YL(2) should contain values of y(x)
+               and p(x)y'(x) respectively (not both zero) which are
+               consistent with the boundary condition at the left-hand
+               end-point, given by x = XL. YL(3) should not be set.
+
+           5:  YR(3) -- DOUBLE PRECISION array                  Output
+               On exit: YR(1) and YR(2) should contain values of y(x)
+               and p(x)y'(x) respectively (not both zero) which are
+               consistent with the boundary condition at the right-
+               hand end-point, given by x = XR. YR(3) should not be
+               set.
+
+          BDYVAL must be declared as EXTERNAL in the (sub)program
+          from which D02KEF is called. Parameters denoted as
+          Input must not be changed by this procedure.
+
+      6:  K -- INTEGER                                           Input
+          On entry: the index k of the required eigenvalue when the
+          eigenvalues are ordered
+          (lambda) <(lambda) <(lambda) <...<(lambda) <....
+                  0         1         2             k
+          Constraint: K >= 0.
+
+      7:  TOL -- DOUBLE PRECISION                                Input
+          On entry: the tolerance parameter which determines the
+          accuracy of the computed eigenvalue. The error estimate held
+          in DELAM on exit satisfies the mixed absolute/relative error
+          test
+                        DELAM<=TOL*max(1.0,|ELAM|)                 (*)
+          where ELAM is the final estimate of the eigenvalue. DELAM is
+          usually somewhat smaller than the right-hand side of (*) but
+          not several orders of magnitude smaller. Constraint: TOL >
+          0.0.
+
+      8:  ELAM -- DOUBLE PRECISION                        Input/Output
+                                                          ~~~~~~~~
+          On entry: an initial estimate of the eigenvalue (lambda).
+          On exit: the final computed estimate, whether or not an
+          error occurred.
+
+      9:  DELAM -- DOUBLE PRECISION                       Input/Output
+          On entry: an indication of the scale of the problem in the
+          (lambda)-direction. DELAM holds the initial 'search step'
+          (positive or negative). Its value is not critical but the
+          first two trial evaluations are made at ELAM and ELAM +
+          DELAM, so the routine will work most efficiently if the
+          eigenvalue lies between these values. A reasonable choice
+          (if a closer bound is not known) is half the distance
+          between adjacent eigenvalues in the neighbourhood of the one
+          sought. In practice, there will often be a problem, similar
+          to the one in hand but with known eigenvalues, which will
+          help one to choose initial values for ELAM and DELAM.
+
+          If DELAM = 0.0 on entry, it is given the default value of
+          0.25*max(1.0,|ELAM|). On exit: with IFAIL = 0, DELAM holds
+          an estimate of the absolute error in the computed
+                               ~~~~~~~~
+          eigenvalue, that is |(lambda)-ELAM|~=DELAM. (In Section 8.2
+          we discuss the assumptions under which this is true.) The
+          true error is rarely more than twice, or less than a tenth,
+          of the estimated error.
+
+          With IFAIL /= 0, DELAM may hold an estimate of the error, or
+          its initial value, depending on the value of IFAIL. See
+          Section 6 for further details.
+
+     10:  HMAX(2,M) -- DOUBLE PRECISION array             Input/Output
+          On entry: HMAX(1,j) a maximum step size to be used by the
+          differential equation code in the jth sub-interval i  (as
+                                                              j
+          described in the specification of parameter XPOINT), for
+          j=1,2,...,m-3. If it is zero the routine generates a maximum
+          step size internally.
+
+          It is recommended that HMAX(1,j) be set to zero unless the
+          coefficient functions p and q have features (such as a
+          narrow peak) within the jth sub-interval that could be '
+          missed' if a long step were taken. In such a case HMAX(1,j)
+          should be set to about half the distance over which the
+          feature should be observed. Too small a value will increase
+          the computing time for the routine. See Section 8 for
+          further suggestions.
+
+          The rest of the array is used as workspace. On exit: HMAX(
+          1,m-1) and HMAX(1,m) contain the sensitivity coefficients
+          (sigma) ,(sigma) , described in Section 8.6. Other entries
+                 l        r
+          contain diagnostic output in case of an error (see Section 6
+          ).
+
+     11:  MAXIT -- INTEGER                                Input/Output
+          On entry: a bound on n , the number of root-finding
+                                r
+          iterations allowed, that is the number of trial values of
+          (lambda) that are used. If MAXIT <= 0, no such bound is
+          assumed. (See also under MAXFUN.) Suggested value: MAXIT =
+          0. On exit: MAXIT will have been decreased by the number of
+          iterations actually performed, whether or not it was
+          positive on entry.
+
+     12:  MAXFUN -- INTEGER                                      Input
+          On entry: a bound on n , the number of calls to COEFFN made
+                                f
+          in any one root-finding iteration. If MAXFUN <= 0, no such
+          bound is assumed. Suggested value: MAXFUN = 0.
+
+          MAXFUN and MAXIT may be used to limit the computational cost
+          of a call to D02KEF, which is roughly proportional to n *n .
+                                                                 r  f
+
+     13:  MONIT -- SUBROUTINE, supplied by the user.
+                                               External Procedure
+          MONIT is called by D02KEF at the end of each root-finding
+          iteration and allows the user to monitor the course of the
+          computation by printing out the parameters (see Section 8
+          for an example).
+          If no monitoring is required, the dummy subroutine D02KAY
+          may be used. (D02KAY is included in the NAG Foundation
+          Library).
+
+          Its specification is:
+
+                 SUBROUTINE MONIT (MAXIT, IFLAG, ELAM, FINFO)
+                 INTEGER          MAXIT, IFLAG
+                 DOUBLE PRECISION ELAM, FINFO(15)
+
+           1:  MAXIT -- INTEGER                                  Input
+               On entry: the current value of the parameter MAXIT of
+               D02KEF; this is decreased by one at each iteration.
+
+           2:  IFLAG -- INTEGER                                  Input
+               On entry: IFLAG describes what phase the computation is
+               in, as follows:
+               IFLAG < 0
+                     an error occurred in the computation of the '
+                     miss-distance' at this iteration;
+
+                     an error exit from D02KEF with IFAIL =-IFLAG will
+                     follow.
+
+               IFLAG = 1
+                     the routine is trying to bracket the eigenvalue
+                     ~~~~~~~~
+                     (lambda).
+
+               IFLAG = 2
+                     the routine is converging to the eigenvalue
+                     ~~~~~~~~
+                     (lambda) (having already bracketed it).
+
+           3:  ELAM -- DOUBLE PRECISION                          Input
+               On entry: the current trial value of (lambda).
+
+           4:  FINFO(15) -- DOUBLE PRECISION array               Input
+               On entry: information about the behaviour of the
+               shooting method, and diagnostic information in the case
+               of errors. It should not normally be printed in full if
+               no error has occurred (that is, if IFLAG > 0), though
+               the first few components may be of interest to the
+               user. In case of an error (IFLAG < 0) all the
+               components of FINFO should be printed. The contents of
+               FINFO are as follows:
+
+               FINFO(1): the current value of the 'miss-distance' or '
+               residual' function f((lambda)) on which the shooting
+               method is based. FINFO(1) is set to zero if IFLAG < 0.
+
+               FINFO(2): an estimate of the quantity dd(lambda)
+               defined as follows. Consider the perturbation in the
+               miss-distance f((lambda)) that would result if the
+               local error, in the solution of the differential
+               equation, were always positive and equal to its maximum
+               permitted value. Then dd(lambda) is the perturbation in
+               (lambda) that would have the same effect on f((lambda))
+               . Thus, at the zero of f((lambda)),|dd(lambda)| is an
+               approximate bound on the perturbation of the zero (that
+               is the eigenvalue) caused by errors in numerical
+               solution. If dd(lambda) is very large then it is
+               possible that there has been a programming error in
+               COEFFN such that q is independent of (lambda). If this
+               is the case, an error exit with IFAIL = 5 should
+               follow. FINFO(2) is set to zero if IFLAG < 0.
+
+               FINFO(3): the number of internal iterations, using the
+               same value of (lambda) and tighter accuracy tolerances,
+               needed to bring the accuracy (that is the value of
+               dd(lambda)) to an acceptable value. Its value should
+               normally be 1.0, and should almost never exceed 2.0.
+
+               FINFO(4): the number of calls to COEFFN at this
+               iteration.
+
+               FINFO(5): the number of successful steps taken by the
+               internal differential equation solver at this
+               iteration. A step is successful if it is used to
+               advance the integration (cf. COUT(8) in specification
+               of D02PAF(*)).
+
+               FINFO(6): the number of unsuccessful steps used by the
+               internal integrator at this iteration (cf. COUT(9) in
+               specification of D02PAF(*)).
+
+               FINFO(7): the number of successful steps at the maximum
+               step size taken by the internal integrator at this
+               iteration (cf. COUT(3) in specification of D02PAF(*)).
+
+               FINFO(8): is not used.
+
+               FINFO(9) to FINFO(15): set to zero, unless IFLAG < 0 in
+               which case they hold the following values describing
+               the point of failure:
+
+               FINFO(9): contains the index of the sub-interval where
+               failure occurred, in the range 1 to m-3. In case of an
+               error in BDYVAL, it is set to 0 or m-2 depending on
+               whether the left or right boundary condition caused the
+               error.
+
+               FINFO(10): the value f the independent variable x, the
+               point at which error occurred. In case of an error in
+               BDYVAL, it is set to the value of XL or XR as
+               appropriate (see the specification of BDYVAL).
+
+               FINFO(11), FINFO(12), FINFO(13): the current values of
+               the Pruefer dependent variables (beta), (phi) and (rho)
+               respectively. These are set to zero in case of an error
+               in BDYVAL.
+
+               FINFO(14): the local-error tolerance being used by the
+               internal integrator at the point of failure. This is
+               set to zero in the case of an error in BDYVAL.
+
+               FINFO(15): the last integration mesh point. This is set
+               to zero in the case of an error in BDYVAL.
+          MONIT must be declared as EXTERNAL in the (sub)program
+          from which D02KEF is called. Parameters denoted as
+          Input must not be changed by this procedure.
+
+     14:  REPORT -- SUBROUTINE, supplied by the user.
+                                               External Procedure
+          This routine provides the means by which the user may
+          compute the eigenfunction y(x) and its derivative at each
+          integration mesh point x. (See Section 8 for an example).
+
+          Its specification is:
+
+                 SUBROUTINE REPORT (X, V, JINT)
+                 INTEGER          JINT
+                 DOUBLE PRECISION X, V(3)
+
+           1:  X -- DOUBLE PRECISION                             Input
+               On entry: the current value of the independent variable
+               x. See Section 8.3 for the order in which values of x
+               are supplied.
+
+           2:  V(3) -- DOUBLE PRECISION array                    Input
+               On entry: V(1), V(2), V(3) hold the current values of
+               the Pruefer variables (beta), (phi), (rho)
+               respectively.
+
+           3:  JINT -- INTEGER                                   Input
+               On entry: JINT indicates the sub-interval between
+               break-points in which X lies exactly as for the routine
+               COEFFN, except that at the extreme left end-point (when
+               x = XPOINT(2)) JINT is set to 0 and at the extreme
+               right end-point (when x=x =XPOINT(m-1)) JINT is set to
+                                        r
+               m-2.
+          REPORT must be declared as EXTERNAL in the (sub)program
+          from which D02KEF is called. Parameters denoted as
+          Input must not be changed by this procedure.
+
+     15:  IFAIL -- INTEGER                                Input/Output
+          On entry: IFAIL must be set to 0, -1 or 1. For users not
+          familiar with this parameter (described in the Essential
+          Introduction) the recommended value is 0.
+
+          On exit: IFAIL = 0 unless the routine detects an error (see
+          Section 6).
+
+     6. Error Indicators and Warnings
+
+     Errors detected by the routine:
+
+     IFAIL= 1
+          A parameter error. All parameters (except IFAIL) are left
+          unchanged. The reason for the error is shown by the value of
+          HMAX(2,1) as follows:
+          HMAX(2,1) = 1: M < 4;
+
+          HMAX(2,1) = 2: K < 0;
+
+          HMAX(2,1) = 3: TOL <= 0.0;
+
+          HMAX(2,1) = 4: XPOINT(1) to XPOINT(m) are not in ascending
+                         order.
+
+                         HMAX(2,2) gives the position i in XPOINT
+                         where this was detected.
+
+     IFAIL= 2
+          At some call to BDYVAL, invalid values were returned, that
+          is, either YL(1) = YL(2) = 0.0, or YR(1) = YR(2) = 0.0 (a
+          programming error in BDYVAL). See the last call of MONIT for
+          details.
+
+          This error exit will also occur if p(x) is zero at the point
+          where the boundary condition is imposed. Probably BDYVAL was
+          called with XL equal to a singular end-point a or with XR
+          equal to a singular end-point b.
+
+     IFAIL= 3
+          At some point between XL and XR the value of p(x) computed
+          by COEFFN became zero or changed sign. See the last call of
+          MONIT for details.
+
+     IFAIL= 4
+          MAXIT > 0 on entry, and after MAXIT iterations the
+          eigenvalue had not been found to the required accuracy.
+
+     IFAIL= 5
+          The 'bracketing' phase (with parameter IFLAG of MONIT equal
+          to 1) failed to bracket the eigenvalue within ten
+          iterations. This is caused by an error in formulating the
+          problem (for example, q is independent of (lambda)), or by
+          very poor initial estimates of ELAM, DELAM.
+
+          On exit ELAM and ELAM + DELAM give the end-points of the
+          interval within which no eigenvalue was located by the
+          routine.
+
+     IFAIL= 6
+          MAXFUN > 0 on entry, and the last iteration was terminated
+          because more than MAXFUN calls to COEFFN were used. See the
+          last call of MONIT for details.
+
+     IFAIL= 7
+          To obtain the desired accuracy the local error tolerance was
+          set so small at the start of some sub-interval that the
+          differential equation solver could not choose an initial
+          step size large enough to make significant progress. See the
+          last call of MONIT for diagnostics.
+
+     IFAIL= 8
+          At some point inside a sub-interval the step size in the
+          differenital equation solver was reduced to a value too
+          small to make significant progress (for the same reasons as
+          with IFAIL = 7). This could be due to pathological behaviour
+          of p(x) and q(x;(lambda)) or to an unreasonable accuracy
+          requirement or to the current value of (lambda) making the
+          equations 'stiff'. See the last call of MONIT for details.
+
+     IFAIL= 9
+          TOL is too small for the problem being solved and the
+          machine precision is being used. The final value of ELAM
+          should be a very good approximation to the eigenvalue.
+
+     IFAIL= 10
+          C05AZF(*), called by D02KEF, has terminated with the error
+          exit corresponding to a pole of the residual function
+          f((lambda)). This error exit should not occur, but if it
+          does, try solving the problem again with a smaller value for
+          TOL.
+
+     IFAIL= 11
+          A serious error has occurred in an internal call to D02KDY.
+          Check all subroutine calls and array dimensions. Seek expert
+          help.
+
+     IFAIL= 12
+          A serious error has occurred in an internal call to
+          C05AZF(*). Check all subroutine calls and array dimensions.
+          Seek expert help.
+
+          HMAX(2,1) holds the failure exit number from the routine
+          where the failure occurred. In the case of a failure in
+          C05AZF(*), HMAX(2,2) holds the value of parameter IND of
+          C05AZF(*).
+
+     Note: error exits with IFAIL = 2, 3, 6, 7, 8, 11 are caused by
+     being unable to set up or solve the differential equation at some
+     iteration, and will be immediately preceded by a call of MONIT
+     giving diagnostic information. For other errors, diagnostic
+     information is contained in HMAX(2,j), for j=1,2,...,m, where
+     appropriate.
+
+     7. Accuracy
+
+     See the discussion in Section 8.2.
+
+     8. Further Comments
+
+     8.1. Timing
+
+     The time taken by the routine depends on the complexity of the
+     coefficient functions, whether they or their derivatives are
+     rapidly changing, the tolerance demanded, and how many iterations
+     are needed to obtain convergence. The amount of work per
+     iteration is roughly doubled when TOL is divided by 16. To make
+     the most economical use of the routine, one should try to obtain
+     good initial values for ELAM and DELAM, and, where appropriate,
+     good asymptotic formulae. The boundary matching points should not
+     be set unnecessarily close to singular points. The extra time
+     needed to compute the eigenfunction is principally the cost of
+     one additional integration once the eigenvalue has been found.
+
+     8.2. General Description of the Algorithm
+
+     A shooting method, for differential equation problems containing
+     unknown parameters, relies on the construction of a 'miss-
+     distance function', which for given trial values of the
+     parameters measures how far the conditions of the problem are
+     from being met. The problem is then reduced to one of finding the
+     values of the parameters for which the miss-distance function is
+     zero, that is to a root-finding process. Shooting methods differ
+     mainly in how the miss-distance is defined.
+
+     This routine defines a miss-distance f((lambda)) based on the
+     rotation around the origin of the point P(x)=(p(x)y'(x),y(x)) in
+     the Phase Plane as the solution proceeds from a to b. The
+     boundary-conditions define the ray (i.e., two-sided line through
+     the origin) on which p(x) should start, and the ray on which it
+     should finish. The eigenvalue index k defines the total number of
+     half-turns it should make. Numerical solution is actually done by
+     matching point x=c. Then f((lambda)) is taken as the angle
+     between the rays to the two resulting points P (c) and P (c). A
+                                                   a         b
+     relative scaling of the py' and y axes, based on the behaviour of
+     the coefficient functions p and q, is used to improve the
+     numerical behaviour.
+
+
+
+              Please see figure in printed Reference Manual
+
+     The resulting function f((lambda)) is monotonic over -
+                                             ddq
+     infty<(lambda)<infty, increasing if  ---------->0 and decreasing
+                                          dd(lambda)
+            ddq
+     if  ----------<0, with a unique zero at the desired eigenvalue
+         dd(lambda)
+     ~~~~~~~~
+     (lambda). The routine measures f((lambda)) in units of a half-
+     turn. This means that as (lambda) increases, f((lambda)) varies
+     by about 1 as each eigenvalue is passed. (This feature implies
+     that the values of f((lambda)) at successive iterations -
+     especially in the early stages of the iterative process - can be
+     used with suitable extrapolation or interpolation to help the
+     choice of initial estimates for eigenvalues near to the one
+     currently being found.)
+
+     The routine actually computes a value for f((lambda)) with
+     errors, arising from the local errors of the differential
+     equation code and from the asymptotic formulae provided by the
+     user if singular points are involved. However, the error estimate
+     output in DELAM is usually fairly realistic, in that the actual
+            ~~~~~~~~
+     error |(lambda)-ELAM| is within an order of magnitude of DELAM.
+
+     We pass the values of (beta), (phi), (rho) across through REPORT
+     rather than converting them to values of y, y' inside D02KEF, for
+     the following reasons. First, there may be cases where auxiliary
+     quantities can be more accurately computed from the Pruefer
+     variables than from y and y'. Second, in singular problems on an
+     infinite interval y and y' may underflow towards the end of the
+     range, whereas the Pruefer variables remain well-behaved. Third,
+     with high-order eigenvalues (and therefore highly oscillatory
+     eigenfunctions) the eigenfunction may have a complete oscillation
+     (or more than one oscillation) between two mesh points, so that
+     values of y and y' at mesh points give a very poor representation
+     of the curve. The probable behaviour of the Pruefer variables in
+     this case is that (beta) and (rho) vary slowly whilst (phi)
+     increases quickly: for all three Pruefer variables linear
+     interpolation between the values at adjacent mesh points is
+     probably sufficiently accurate to yield acceptable intermediate
+     values of (beta), (phi), (rho) (and hence of y,y') for graphical
+     purposes.
+
+     Similar considerations apply to the exponentially decaying 'tails
+     Here (phi) has approximately constant value whilst (rho)
+     increases rapidly in the direction of integration, though the
+     step length is generally fairly small over such a range.
+
+     If the solution is output through REPORT at x-values which are
+     too widely spaced, the step length can be controlled by choosing
+     HMAX suitably, or, preferably, by reducing TOL. Both these
+     choices will lead to more accurate eigenvalues and eigenfunctions
+     but at some computational cost.
+
+     8.3. The Position of the Shooting Matching Point c
+
+     This point is always one of the values x  in array XPOINT. It may
+                                             i
+     be specified using the parameter MATCH. The default value is
+     chosen to be the value of that x ,2<=i<=m-1, that lies closest to
+                                     i
+     the middle of the interval [x ,x   ]. If there is a tie, the
+                                  2  m-1
+     rightmost candidate is chosen. In particular if there are no
+     break-points then c=x   (=x ) - that is the shooting is from left
+                          m-1   3
+     to right in this case. A break-point may be inserted purely to
+     move c to an interior point of the interval, even though the form
+     of the equations does not require it. This often speeds up
+     convergence especially with singular problems.
+
+     Note that the shooting method used by the code integrates first
+     from the left-hand end x , then from the right-hand end x , to
+                             l                                r
+     meet at the matching point c in the middle. This will of course
+     be reflected in printed or graphical output. The diagram shows a
+     possible sequence of nine mesh points (tau)  through (tau)  in
+                                                1              9
+     the order in which they appear, assuming there are just two sub-
+     intervals (so m=5).
+
+
+                                Figure 1
+              Please see figure in printed Reference Manual
+
+     Since the shooting method usually fails to match up the two 'legs
+     p(x)y' or both, at the matching point c. The code in fact 'shares
+     large jump does not imply an inaccurate eigenvalue, but implies
+     either
+
+     (a)   a badly chosen matching point: if q(x;(lambda)) has a '
+           humped' shape, c should be chosen near the maximum value of
+           q, especially if q is negative at the ends of the interval.
+
+     (b)   An inherently ill-conditioned problem, typically one where
+           another eigenvalue is pathologically close to the one being
+           sought. In this case it is extremely difficult to obtain an
+           accurate eigenfunction.
+
+     In Section 9 below, we find the 11th eigenvalue and corresponding
+     eigenfunction of the equation
+
+                                     2
+                  y''+((lambda)-x-2/x )y=0  on  0<x<infty
+
+     the boundary conditions being that y should remain bounded as x
+     tends to 0 and x tends to infty. The coding of this problem is
+     discussed in detail in Section 8.5.
+
+     The choice of matching point c is open. If we choose c=30.0 as in
+     the D02KDF(*) example program we find that the exponentially
+     increasing component of the solution dominates and we get
+     extremely inaccurate values for the eigenfunction (though the
+     eigenvalue is determined accurately). The values of the
+     eigenfunction calculated with c=30.0 are given schematically in
+     Figure 2.
+
+
+                                Figure 2
+              Please see figure in printed Reference Manual
+
+     If we choose c as the maximum of the hump in q(x;(lambda)) (see
+     (a) above) we instead obtain the accurate results given in
+     Figure 3.
+
+
+                                Figure 3
+              Please see figure in printed Reference Manual
+
+     8.4. Examples of Coding the COEFFN Routine
+
+     Coding COEFFN is straightforward except when break-points are
+     needed. The examples below show:
+
+     (a)   a simple case,
+
+     (b)   a case in which discontinuities in the coefficient
+           functions or their derivatives necessitate break-points,
+           and
+
+     (c)   a case where break-points together with the HMAX parameter
+           are an efficient way to deal with a coefficient function
+           that is well-behaved except over one short interval.
+
+     Example A
+
+     The modified Bessel equation
+
+                                         2     2
+                       x(xy')'+((lambda)x -(nu) )y=0
+
+     Assuming the interval of solution does not contain the origin,
+     dividing through by x, we have p(x)=x,
+                                 2
+     q(x;(lambda))=(lambda)x-(nu) /x. The code could be
+
+
+           SUBROUTINE COEFFN(P,Q,DQDL,X,ELAM,JINT)
+           P = X
+           Q = ELAM*X + NU*NU/X
+           DQDL = X
+           RETURN
+           END
+
+
+     where NU (standing for (nu)) is a real variable that might be
+     defined in a DATA statement, or might be in user-declared COMMON
+     so that its value could be set in the main program.
+
+     Example B
+
+     The Schroedinger equation
+
+                          y''+((lambda)+q(x))y=0
+
+                { 2
+                {x -10 (|x|<=4),
+     where q(x)={6/|x| (|x|>4),
+
+     over some interval 'approximating to (-infty,infty)', say [-20,
+     20]. Here we need break-points at +- 4, forming three sub-
+     intervals i =[-20,-4], i =[-4,4],i =[4,20]. The code could be
+                1            2         3
+
+
+           SUBROUTINE COEFFN(P,Q,DQDL,X,ELAM,JINT)
+           IF (JINT.EQ.2) THEN
+             Q = ELAM + X*X - 10.0E0
+           ELSE
+             Q = ELAM + 6.0E0/ABS(X)
+           ENDIF
+           P = 1.0E0
+           DQDL = 1.0E0
+           RETURN
+           END
+
+
+     The array XPOINT would contain the values x , -20.0, -4.0, +4.0,
+                                                1
+     +20.0, x  and m would be 6. The choice of appropriate values for
+             6
+     x  and x  depends on the form of the asymptotic formula computed
+      1      6
+     by BDYVAL and the technique is discussed in the next subsection.
+
+     Example C
+
+                                     2
+                                -100x
+
+               y''+(lambda)(1-2e      )y=0, over -10<=x<=10
+
+     Here q(x;(lambda)) is nearly constant over the range except for a
+     sharp inverted spike over approximately -0.1<=x<=0.1. There is a
+     danger that the routine will build up to a large step size and '
+     step over' the spike without noticing it. By using break-points -
+     say at +- 0.5 - one can restrict the step size near the spike
+     without impairing the efficiency elsewhere.
+
+     The code for COEFFN could be
+
+
+           SUBROUTINE COEFFN(P,Q,DQDL,X,ELAM,JINT)
+           P = 1.0E0
+           DQDL = 1.0E0 - 2.0E0*EXP(-100.0E0*X*X)
+           Q = ELAM*DQDL
+           RETURN
+           END
+
+
+     XPOINT might contain -0.0, -10.0, -0.5, 0.5, 10.0, 10.0 (assuming
+     +- 10 are regular points) and m would be 6. HMAX(1,j), j=1,2,3
+     might contain 0.0, 0.1 and 0.0.
+
+     8.5. Examples of Boundary Conditions at Singular Points
+
+     Quoting from Bailey [2] page 243: 'Usually... the differential
+     equation has two essentially different types of solution near a
+     singular point, and the boundary condition there merely serves to
+     distinguish one kind from the other. This is the case in all the
+     standard examples of mathematical physics.'
+
+     In most cases the behaviour of the ratio p(x)y'/y near the point
+     is quite different for the two types of solution. Essentially
+     what the user provides through his BDYVAL routine is an
+     approximation to this ratio, valid as x tends to the singular
+     point (SP).
+
+     The user must decide (a) how accurate to make this approximation
+     or asymptotic formula, for example how many terms of a series to
+     use, and (b) where to place the boundary matching point (BMP) at
+     which the numerical solution of the differential equation takes
+     over from the asymptotic formula. Taking the BMP closer to the SP
+     will generally improve the accuracy of the asymptotic formula,
+     but will make the computation more expensive as the Pruefer
+     differential equations generally become progressively more ill-
+     behaved as the SP is approached. The user is strongly recommended
+     to experiment with placing the BMPs. In many singular problems
+     quite crude asymptotic formulae will do. To help the user avoid
+     needlessly accurate formulae, D02KEF outputs two 'sensitivity
+     coefficients' (sigma) ,(sigma)  which estimate how much the
+                          l        r
+     errors at the BMP's affect the computed eigenvalue. They are
+     described in detail below, see Section 8.6.
+
+     Example of coding BDYVAL:
+
+     The example below illustrates typical situations:
+
+                      (            2 )
+                  y''+((lambda)-x- --)y=0, for 0<x<infty
+                      (             2)
+                      (            x )
+
+     the boundary conditions being that y should remain bounded as x
+     tends to 0 and x tends to infty.
+
+                                                             2
+     At the end x=0 there is one solution that behaves like x  and
+                                -1
+     another that behaves like x  . For the first of these solutions
+     p(x)y'/y is asymptotically 2/x while for the second it is
+     asymptotically -1/x. Thus the desired ratio is specified by
+     setting
+
+                          YL(1)=x and YL(2)=2.0.
+
+     At the end x=infty the equation behaves like Airy's equation
+     shifted through (lambda), i.e., like y''-ty=0 where t=x-(lambda),
+     so again there are two types of solution. The solution we require
+     behaves as
+
+                                (  2 3/2) 4 _
+                             exp(- -t   )/\/t
+                                (  3    )
+
+     and the other as
+
+                                (  2 3/2) 4 _
+                             exp(+ -t   )/\/t
+                                (  3    )
+
+                                                _
+     once, the desired solution has p(x)y'/y~-\/t so that we could set
+                              __________
+     YR(1) = 1.0 and YR(2)=-\/x-(lambda). The complete subroutine
+     might thus be
+
+
+           SUBROUTINE BDYVAL(XL,XR,ELAM,YL,YR)
+           real XL, XR, ELAM, YL(3), YR(3)
+           YL(1) = XL
+           YL(2) = 2.0E0
+           YR(1) = 1.0E0
+           YR(2) = -SQRT(XR - ELAM)
+           RETURN
+           END
+
+
+     Clearly for this problem it is essential that any value given by
+     D02KEF to XR is well to the right of the value of ELAM, so that
+     the user must vary the right-hand BMP with the eigenvalue index k
+                                k
+     function Ai(x), so there is no problem in estimating ELAM.
+
+     More accurate asymptotic formulae are easily found - near x=0 by
+     the standard Frobenius method, and near x=infty by using standard
+     asymptotics for Ai(x), Ai'(x) (see [1], p. 448). For example, by
+     the Frobenius method the solution near x=0 has the expansion
+
+                              2           2
+                           y=x (c +c x+c x +...)
+                                 0  1   2
+
+     with
+
+                                                c   -(lambda)c
+                        (lambda)     1           n-3          n-2
+         c =1,c =0,c =- --------,c = --,...,c = -----------------
+          0    1    2      10     3  18      n       n(n+3)
+
+     This yields
+
+                                   2         2
+                                2- -(lambda)x +...
+                       p(x)y'      5
+                       ------= --------------------.
+                         y      (   (lambda) 2    )
+                               x(1- --------x +...)
+                                (      10         )
+
+     8.6. The Sensitivity Parameters (sigma)  and (sigma)
+                                            l            r
+
+     The sensitivity parameters (sigma) ,(sigma)  (held in HMAX(1,m-1)
+                                       l        r
+     and HMAX(1,m) on output) estimate the effect of errors in the
+     boundary conditions. For sufficiently small errors (Delta)y,
+     (Delta)py' in y and py' respectively, the relations
+
+           (Delta)(lambda)~=(y.(Delta)py'-py'.(Delta)y) (sigma)
+                                                       l       l
+
+           (Delta)(lambda)~=(y.(Delta)py'-py'.(Delta)y) (sigma)
+                                                       r       r
+
+     are satisfied where the subscripts l, r denote errors committed
+     at left- and right-hand BMP's respectively, and (Delta)(lambda)
+     denotes the consequent error in the computed eigenvalue.
+
+     8.7. Missed Zeros
+
+     This is a pitfall to beware of at a singular point. If the BMP is
+     chosen so far from the SP that a zero of the desired
+     eigenfunction lies in between them, then the routine will fail to
+     number of zeros of its eigenfunction, the result will be that:
+
+     (a)   The wrong eigenvalue will be computed for the given index k
+           - in fact some (lambda)     will be found where k'>=1.
+                                  k+k'
+
+     (b)   The same index k can cause convergence to any of several
+           eigenvalues depending on the initial values of ELAM and
+           DELAM.
+
+     It is up to the user to take suitable precautions - for instance
+     by varying the position of the BMP's in the light of his
+     knowledge of the asymptotic behaviour of the eigenfunction at
+     different eigenvalues.
+
+     9. Example
+
+     To find the 11th eigenvalue and eigenfunction of the example of
+     Section 8.5, using the simple asymptotic formulae for the
+     boundary conditions.
+
+     Comparison of the results from this example program with the
+     corresponding results from D02KDF(*) example program shows that
+     similar output is produced from the routine MONIT, followed by
+     the eigenfunction values from REPORT, and then a further line of
+     information from MONIT (corresponding to the integration to find
+     the eigenfunction). Final information is printed within the
+     example program exactly as with D02KDF(*).
+
+                                                   3 _
+     Note the discrepancy at the matching point c(=\/4 , the maximum
+     of q(x;(lambda)), in this case) between the solutions obtained by
+     integrations from left and right end-points.
+
+     The example program is not reproduced here. The source code for
+     all example programs is distributed with the NAG Foundation
+     Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd02raf}{NAG On-line Documentation: d02raf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D02RAF(3NAG)      Foundation Library (12/10/92)      D02RAF(3NAG)
+
+
+
+          D02 -- Ordinary Differential Equations                     D02RAF
+                  D02RAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D02RAF solves the two-point boundary-value problem with general
+          boundary conditions for a system of ordinary differential
+          equations, using a deferred correction technique and Newton
+          iteration.
+
+          2. Specification
+
+                 SUBROUTINE D02RAF (N, MNP, NP, NUMBEG, NUMMIX, TOL, INIT,
+                1                   X, Y, IY, ABT, FCN, G, IJAC, JACOBF,
+                2                   JACOBG, DELEPS, JACEPS, JACGEP, WORK,
+                3                   LWORK, IWORK, LIWORK, IFAIL)
+                 INTEGER          N, MNP, NP, NUMBEG, NUMMIX, INIT, IY,
+                1                 IJAC, LWORK, IWORK(LIWORK), LIWORK, IFAIL
+                 DOUBLE PRECISION TOL, X(MNP), Y(IY,MNP), ABT(N), DELEPS,
+                1                 WORK(LWORK)
+                 EXTERNAL         FCN, G, JACOBF, JACOBG, JACEPS, JACGEP
+
+          3. Description
+
+          D02RAF solves a two-point boundary-value problem for a system of
+          n ordinary differential equations in the interval (a,b) with b>a.
+          The system is written in the form
+
+                      y '=f (x,y ,y ,...,y ) , i=1,2,...,n              (1)
+                       i   i    1  2      n
+
+          and the derivatives f  are evaluated by a subroutine FCN supplied
+                               i
+          by the user. With the differential equations (1) must be given a
+          system of n (nonlinear) boundary conditions
+
+                            g (y(a),y(b))=0 , i=1,2,...,n
+                             i
+
+          where
+
+                                                     T
+                         y(x)=[y (x),y (x),...,y (x)] .                 (2)
+                                1     2         n
+
+          The functions g  are evaluated by a subroutine G supplied by the
+                         i
+          user. The solution is computed using a finite-difference
+          technique with deferred correction allied to a Newton iteration
+          to solve the finite-difference equations. The technique used is
+          described fully in Pereyra [1].
+
+          The user must supply an absolute error tolerance and may also
+          supply an initial mesh for the finite-difference equations and an
+          initial approximate solution (alternatively a default mesh and
+          approximation are used). The approximate solution is corrected
+          using Newton iteration and deferred correction. Then, additional
+          points are added to the mesh and the solution is recomputed with
+          the aim of making the error everywhere less than the user's
+          tolerance and of approximately equidistributing the error on the
+          final mesh. The solution is returned on this final mesh.
+
+          If the solution is required at a few specific points then these
+          should be included in the initial mesh. If, on the other hand,
+          the solution is required at several specific points then the user
+          should use the interpolation routines provided in Chapter E01 if
+          these points do not themselves form a convenient mesh.
+
+          The Newton iteration requires Jacobian matrices
+
+                          ( ddf )  (  ddg   )     (  ddg   )
+                          (    i)  (     i  )     (     i  )
+                          ( ----), ( -------) and ( -------).
+                          ( ddy )  ( ddy (a))     ( ddy (b))
+                          (    j)  (    j   )     (    j   )
+
+          These may be supplied by the user through subroutines JACOBF for
+          ( ddf )
+          (    i)
+          ( ----) and JACOBG for the others. Alternatively the Jacobians
+          ( ddy )
+          (    j)
+          may be calculated by numerical differentiation using the
+          algorithm described in Curtis et al [2].
+
+          For problems of the type (1) and (2) for which it is difficult to
+          determine an initial approximation from which the Newton
+          iteration will converge, a continuation facility is provided. The
+          user must set up a family of problems
+
+                y'=f(x,y,(epsilon)),   g(y(a),y(b),(epsilon))=0,        (3)
+
+                                T
+          where f=[f ,f ,...,f ]  etc, and where (epsilon) is a
+                    1  2      n
+          continuation parameter. The choice (epsilon)=0 must give a
+          problem (3) which is easy to solve and (epsilon)=1 must define
+          the problem whose solution is actually required. The routine
+          solves a sequence of problems with (epsilon) values
+
+                   0=(epsilon) <(epsilon) <...<(epsilon) =1.            (4)
+                              1          2              p
+
+          The number p and the values (epsilon)  are chosen by the routine
+                                               i
+          so that each problem can be solved using the solution of its
+                                                                  ddf
+          predecessor as a starting approximation. Jacobians  -----------
+                                                              dd(epsilon)
+                   ddg
+          and  ----------- are required and they may be supplied by the
+               dd(epsilon)
+          user via routines JACEPS and JACGEP respectively or may be
+          computed by numerical differentiation.
+
+          4. References
+
+          [1]   Pereyra V (1979) PASVA3: An Adaptive Finite-Difference
+                Fortran Program for First Order Nonlinear, Ordinary Boundary
+                Problems. Codes for Boundary Value Problems in Ordinary
+                Differential Equations. Lecture Notes in Computer Science.
+                (ed B Childs, M Scott, J W Daniel, E Denman and P Nelson) 76
+                Springer-Verlag.
+
+          [2]   Curtis A R, Powell M J D and Reid J K (1974) On the
+                Estimation of Sparse Jacobian Matrices. J. Inst. Maths
+                Applics. 13 117--119.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+               On entry: the number of differential equations, n.
+               Constraint: N > 0.
+
+           2:  MNP -- INTEGER                                         Input
+               On entry: MNP must be set to the maximum permitted number
+               of points in the finite-difference mesh. If LWORK or LIWORK
+               (see below) is too small then internally MNP will be
+               replaced by the maximum permitted by these values. (A
+               warning message will be output if on entry IFAIL is set to
+               obtain monitoring information.) Constraint: MNP >= 32.
+
+           3:  NP -- INTEGER                                   Input/Output
+               On entry: NP must be set to the number of points to be used
+               in the initial mesh. Constraint: 4 <= NP <= MNP. On exit:
+               the number of points in the final mesh.
+
+           4:  NUMBEG -- INTEGER                                      Input
+               On entry: the number of left-hand boundary conditions (that
+               is the number involving y(a) only). Constraint: 0 <= NUMBEG
+               < N.
+
+           5:  NUMMIX -- INTEGER                                      Input
+               On entry: the number of coupled boundary conditions (that
+               is the number involving both y(a) and y(b)). Constraint: 0
+               <= NUMMIX <= N - NUMBEG.
+
+           6:  TOL -- DOUBLE PRECISION                                Input
+               On entry: a positive absolute error tolerance. If
+                                    a=x <x <...<x  =b
+                                       1  2      NP
+               is the final mesh, z (x ) is the jth component of the
+                                   j  i
+               approximate solution at x , and y (x) is the jth component
+                                        i       j
+               of the true solution of (1) and (2), then, except in extreme
+               circumstances, it is expected that
+                 |z (x )-y (x )|<=TOL , i=1,2,...,NP ; j=1,2,...,n.     (5)
+                   j  i   j  i
+                Constraint: TOL > 0.0.
+
+           7:  INIT -- INTEGER                                        Input
+               On entry: indicates whether the user wishes to supply an
+               initial mesh and approximate solution (INIT /= 0) or whether
+               default values are to be used, (INIT = 0).
+
+           8:  X(MNP) -- DOUBLE PRECISION array                Input/Output
+               On entry: the user must set X(1) = a and X(NP) = b. If INIT
+               = 0 on entry a default equispaced mesh will be used,
+               otherwise the user must specify a mesh by setting X(i)=x ,
+                                                                       i
+               for i = 2,3,...NP-1. Constraints:
+                      X(1) < X(NP), if INIT = 0,
+
+                      X(1) < X(2) <... < X(NP), if INIT /= 0.
+               On exit: X(1),X(2),...,X(NP) define the final mesh (with
+               the returned value of NP) and X(1) = a and X(NP) = b.
+
+           9:  Y(IY,MNP) -- DOUBLE PRECISION array             Input/Output
+               On entry: if INIT = 0, then Y need not be set.
+
+               If INIT /= 0, then the array Y must contain an initial
+               approximation to the solution such that Y(j,i) contains an
+               approximation to
+                           y (x ) , i=1,2,...,NP ; j=1,2,...,n.
+                            j  i
+               On exit: the approximate solution z (x ) satisfying (5) on
+                                                  j  i
+               the final mesh, that is
+                       Y(j,i)=z (x ) , i=1,2,...,NP ; j=1,2,...,n,
+                               j  i
+               where NP is the number of points in the final mesh. If an
+               error has occurred then Y contains the latest approximation
+               to the solution. The remaining columns of Y are not used.
+
+          10:  IY -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array Y as declared in the
+               (sub)program from which D02RAF is called.
+               Constraint: IY >= N.
+
+          11:  ABT(N) -- DOUBLE PRECISION array                      Output
+               On exit: ABT(i), for i=1,2,...,n, holds the largest
+               estimated error (in magnitude) of the ith component of the
+               solution over all mesh points.
+
+          12:  FCN -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               FCN must evaluate the functions f  (i.e., the derivatives
+                                                i
+               y' ) at a general point x for a given value of (epsilon),
+                 i
+               the continuation parameter (see Section 3).
+
+               Its specification is:
+
+                      SUBROUTINE FCN (X, EPS, Y, F, N)
+                      INTEGER          N
+                      DOUBLE PRECISION X, EPS, Y(N), F(N)
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the value of the argument x.
+
+                2:  EPS -- DOUBLE PRECISION                           Input
+                    On entry: the value of the continuation parameter,
+                    (epsilon). This is 1 if continuation is not being used.
+
+                3:  Y(N) -- DOUBLE PRECISION array                    Input
+                    On entry: the value of the argument y , for
+                                                         i
+                    i=1,2,...,n.
+
+                4:  F(N) -- DOUBLE PRECISION array                   Output
+                    On exit: the values of f , for i=1,2,...,n.
+                                            i
+
+                5:  N -- INTEGER                                      Input
+                    On entry: the number of equations.
+               FCN must be declared as EXTERNAL in the (sub)program
+               from which D02RAF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+          13:  G -- SUBROUTINE, supplied by the user.
+                                                    External Procedure
+               G must evaluate the boundary conditions in equation (3) and
+               place them in the array BC.
+
+               Its specification is:
+
+                      SUBROUTINE G (EPS, YA, YB, BC, N)
+                      INTEGER          N
+                      DOUBLE PRECISION EPS, YA(N), YB(N), BC(N)
+
+                1:  EPS -- DOUBLE PRECISION                           Input
+                    On entry: the value of the continuation parameter,
+                    (epsilon). This is 1 if continuation is not being used.
+
+                2:  YA(N) -- DOUBLE PRECISION array                   Input
+                    On entry: the value y (a), for i=1,2,...,n.
+                                         i
+
+                3:  YB(N) -- DOUBLE PRECISION array                   Input
+                    On entry: the value y (b), for i=1,2,...,n.
+                                         i
+
+                4:  BC(N) -- DOUBLE PRECISION array                  Output
+                    On exit: the values g (y(a),y(b),(epsilon)), for
+                                         i
+                    i=1,2,...,n. These must be ordered as follows:
+                      (i)   first, the conditions involving only y(a) (see
+                            NUMBEG description above);
+
+                      (ii)  next, the NUMMIX coupled conditions involving
+                            both y(a) and y(b) (see NUMMIX description
+                            above); and,
+
+                      (iii) finally, the conditions involving only y(b) (N-
+                            NUMBEG-NUMMIX).
+
+                5:  N -- INTEGER                                      Input
+                    On entry: the number of equations, n.
+               G must be declared as EXTERNAL in the (sub)program from
+               which D02RAF is called. Parameters denoted as Input
+               must not be changed by this procedure.
+
+          14:  IJAC -- INTEGER                                   Input
+               On entry: indicates whether or not the user is supplying
+               Jacobian evaluation routines. If IJAC /= 0 then the user
+               must supply routines JACOBF and JACOBG and also, when
+               continuation is used, routines JACEPS and JACGEP. If IJAC =
+               0 numerical differentiation is used to calculate the
+               Jacobian and the routines D02GAZ, D02GAY, D02GAZ and D02GAX
+                                                 ( ddf )
+                                                 (    i)
+               JACOBF must evaluate the Jacobian ( ----) for i,j=1,2,...,n,
+                                                 ( ddy )
+                                                 (    j)
+               given x and y , for j=1,2,...,n.
+                            j
+
+               Its specification is:
+
+                      SUBROUTINE JACOBF (X, EPS, Y, F, N)
+                      INTEGER          N
+                      DOUBLE PRECISION X, EPS, Y(N), F(N,N)
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the value of the argument x.
+
+                2:  EPS -- DOUBLE PRECISION                           Input
+                    On entry: the value of the continuation parameter
+                    (epsilon). This is 1 if continuation is not being used.
+
+                3:  Y(N) -- DOUBLE PRECISION array                    Input
+                    On entry: the value of the argument y , for
+                                                         i
+                    i=1,2,...,n.
+
+                4:  F(N,N) -- DOUBLE PRECISION array                 Output
+                                                                  ddf
+                                                                     i
+                    On exit: F(i,j) must be set to the value of   ----,
+                                                                  ddy
+                                                                     j
+                    evaluated at the point (x,y), for i,j=1,2,...,n.
+
+                5:  N -- INTEGER                                      Input
+                    On entry: the number of equations, n.
+               JACOBF must be declared as EXTERNAL in the (sub)program
+               from which D02RAF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+          16:  JACOBG -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+                                                  (  ddg   )     (  ddg   )
+                                                  (     i  )     (     i  )
+               JACOBG must evaluate the Jacobians ( -------) and ( -------)
+                                                  ( ddy (a))     ( ddy (b))
+                                                  (    j   )     (    j   )
+               The ordering of the rows of AJ and BJ must correspond to
+               the ordering of the boundary conditions described in the
+               specification of subroutine G above.
+
+               Its specification is:
+
+                      SUBROUTINE JACOBG (EPS, YA, YB, AJ, BJ, N)
+                      INTEGER          N
+                      DOUBLE PRECISION EPS, YA(N), YB(N), AJ(N,N), BJ
+                     1                 (N,N)
+
+                1:  EPS -- DOUBLE PRECISION                           Input
+                    On entry: the value of the continuation parameter,
+                    (epsilon). This is 1 if continuation is not being used.
+
+                2:  YA(N) -- DOUBLE PRECISION array                   Input
+                    On entry: the value y (a), for i=1,2,...,n.
+                                          i
+
+                3:  YB(N) -- DOUBLE PRECISION array                   Input
+                    On entry: the value y (b), for i=1,2,...,n.
+                                         i
+
+                4:  AJ(N,N) -- DOUBLE PRECISION array                Output
+                                                                ddg
+                                                                   i
+                    On exit: AJ(i,j) must be set to the value  -------,
+                                                               ddy (a)
+                                                                  j
+                    for i,j=1,2,...,n.
+
+                5:  BJ(N,N) -- DOUBLE PRECISION array                Output
+                                                                ddg
+                                                                   i
+                    On exit: BJ(i,j) must be set to the value  -------,
+                                                               ddy (b)
+                                                                  j
+                    for i,j=1,2...,n.
+
+                6:  N -- INTEGER                                      Input
+                    On entry: the number of equations, n.
+               JACOBG must be declared as EXTERNAL in the (sub)program
+               from which D02RAF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+          17:  DELEPS -- DOUBLE PRECISION                      Input/Output
+               On entry: DELEPS must be given a value which specifies
+               whether continuation is required. If DELEPS <= 0.0 or DELEPS
+               >= 1.0 then it is assumed that continuation is not required.
+               If 0.0 < DELEPS < 1.0 then it is assumed that continuation
+               is required unless DELEPS < square root of machine precision
+               when an error exit is taken. DELEPS is used as the increment
+               (epsilon) -(epsilon)  (see (4)) and the choice DELEPS = 0.1
+                        2          1
+               is recommended. On exit: an overestimate of the increment
+               (epsilon) -(epsilon)    (in fact the value of the increment
+                        p          p-1
+               which would have been tried if the restriction (epsilon) =1
+                                                                       p
+               had not been imposed). If continuation was not requested
+               then DELEPS = 0.0.
+
+               If continuation is not requested then the parameters JACEPS
+               and JACGEP may be replaced by dummy actual parameters in the
+               call to D02RAF. (D02GAZ and D02GAX respectively may be used
+               as the dummy parameters.)
+
+          18:  JACEPS -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+                                                       ddf
+                                                          i
+               JACEPS must evaluate the derivative  ----------- given x and
+                                                    dd(epsilon)
+               y if continuation is being used.
+
+               Its specification is:
+
+                      SUBROUTINE JACEPS (X, EPS, Y, F, N)
+                      INTEGER          N
+                      DOUBLE PRECISION X, EPS, Y(N), F(N)
+
+                1:  X -- DOUBLE PRECISION                             Input
+                    On entry: the value of the argument x.
+
+                2:  EPS -- DOUBLE PRECISION                           Input
+                    On entry: the value of the continuation parameter,
+                    (epsilon).
+
+                3:  Y(N) -- DOUBLE PRECISION array                    Input
+                    On entry: the solution values y  at the point x, for
+                                                   i
+                    i=1,2,...,n.
+
+                4:  F(N) -- DOUBLE PRECISION array                   Output
+                                                             ddf
+                                                                i
+                    On exit: F(i) must contain the value  ----------- at
+                                                          dd(epsilon)
+                    the point (x,y), for i=1,2,...,n.
+
+                5:  N -- INTEGER                                      Input
+                    On entry: the number of equations, n.
+               JACEPS must be declared as EXTERNAL in the (sub)program
+               from which D02RAF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+          19:  JACGEP -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+                                                        ddg
+                                                           i
+               JACGEP must evaluate the derivatives  ----------- if
+                                                     dd(epsilon)
+               continuation is being used.
+
+               Its specification is:
+
+                      SUBROUTINE JACGEP (EPS, YA, YB, BCEP, N)
+                      INTEGER          N
+                      DOUBLE PRECISION EPS, YA(N), YB(N), BCEP(N)
+
+                1:  EPS -- DOUBLE PRECISION                           Input
+                    On entry: the value of the continuation parameter,
+                    (epsilon).
+
+                2:  YA(N) -- DOUBLE PRECISION array                   Input
+                    On entry: the value of y (a), for i=1,2,...,n.
+                                            i
+
+                3:  YB(N) -- DOUBLE PRECISION array                   Input
+                    On entry: the value of y (b), for i=1,2,...,n.
+                                            i
+
+                4:  BCEP(N) -- DOUBLE PRECISION array                Output
+                    On exit: BCEP(i) must contain the value of
+                        ddg
+                           i
+                     -----------, for i=1,2,...,n.
+                     dd(epsilon)
+
+                5:  N -- INTEGER                                      Input
+                    On entry: the number of equations, n.
+               JACGEP must be declared as EXTERNAL in the (sub)program
+               from which D02RAF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+          20:  WORK(LWORK) -- DOUBLE PRECISION array              Workspace
+
+          21:  LWORK -- INTEGER                                       Input
+               On entry:
+               the dimension of the array WORK as declared in the
+               (sub)program from which D02RAF is called.
+                                         2         2
+               Constraint: LWORK>=MNP*(3N +6N+2)+4N +3N.
+
+          22:  IWORK(LIWORK) -- INTEGER array                     Workspace
+
+          23:  LIWORK -- INTEGER                                      Input
+               On entry:
+               the dimension of the array IWORK as declared in the
+               (sub)program from which D02RAF is called.
+               Constraints:
+                    LIWORK>=MNP*(2*N+1)+N, if IJAC /= 0,
+
+                                         2
+                    LIWORK>=MNP*(2*N+1)+N +4*N+2, if IJAC = 0.
+
+          24:  IFAIL -- INTEGER                                Input/Output
+               For this routine, the normal use of IFAIL is extended to
+               control the printing of error and warning messages as well
+               as specifying hard or soft failure (see the Essential
+               Introduction).
+
+               Before entry, IFAIL must be set to a value with the decimal
+               expansion cba, where each of the decimal digits c, b and a
+               must have a value of 0 or 1.
+               a=0   specifies hard failure, otherwise soft failure;
+
+               b=0   suppresses error messages, otherwise error messages
+                     will be printed (see Section 6);
+
+               c=0   suppresses warning messages, otherwise warning
+                     messages will be printed (see Section 6).
+               The recommended value for inexperienced users is 110 (i.e.,
+               hard failure with all messages printed).
+
+               Unless the routine detects an error (see Section 6), IFAIL
+               contains 0 on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          For each error, an explanatory error message is output on the
+          current error message unit (as defined by X04AAF), unless
+          suppressed by the value of IFAIL on entry.
+
+          IFAIL= 1
+               One or more of the parameters N, MNP, NP, NUMBEG, NUMMIX,
+               TOL, DELEPS, LWORK or LIWORK has been incorrectly set, or X
+               (1) >= X(NP) or the mesh points X(i) are not in strictly
+               ascending order.
+
+          IFAIL= 2
+               A finer mesh is required for the accuracy requested; that is
+               MNP is not large enough. This error exit normally occurs
+               when the problem being solved is difficult (for example,
+               there is a boundary layer) and high accuracy is requested. A
+               poor initial choice of mesh points will make this error exit
+               more likely.
+
+          IFAIL= 3
+               The Newton iteration has failed to converge. There are
+               several possible causes for this error:
+               (i)   faulty coding in one of the Jacobian calculation
+                     routines;
+
+               (ii)  if IJAC = 0 then inaccurate Jacobians may have been
+                     calculated numerically (this is a very unlikely
+                     cause); or,
+
+               (iii) a poor initial mesh or initial approximate solution
+                     has been selected either by the user or by default or
+                     there are not enough points in the initial mesh.
+                     Possibly, the user should try the continuation
+                     facility.
+
+          IFAIL= 4
+               The Newton iteration has reached round-off error level. It
+               could be however that the answer returned is satisfactory.
+               The error is likely to occur if too high an accuracy is
+               requested.
+
+          IFAIL= 5
+               The Jacobian calculated by JACOBG (or the equivalent matrix
+               calculated by numerical differentiation) is singular. This
+               may occur due to faulty coding of JACOBG or, in some
+               circumstances, to a zero initial choice of approximate
+               solution (such as is chosen when INIT = 0).
+
+          IFAIL= 6
+               There is no dependence on (epsilon) when continuation is
+               being used. This can be due to faulty coding of JACEPS or
+               JACGEP or, in some circumstances, to a zero initial choice
+               of approximate solution (such as is chosen when INIT = 0).
+
+          IFAIL= 7
+               DELEPS is required to be less than machine precision for
+               continuation to proceed. It is likely that either the
+               problem (3) has no solution for some value near the current
+               value of (epsilon) (see the advisory print out from D02RAF)
+               or that the problem is so difficult that even with
+               continuation it is unlikely to be solved using this routine.
+               If the latter cause is suspected then using more mesh points
+               initially may help.
+
+          IFAIL= 8
+               Indicates that a serious error has occurred in a call to
+               D02RAF. Check all array subscripts and subroutine parameter
+               lists in calls to D02RAF. Seek expert help.
+
+          IFAIL= 9
+               Indicates that a serious error has occurred in a call to
+               D02RAR. Check all array subscripts and subroutine parameter
+               lists in calls to D02RAF. Seek expert help.
+
+          7. Accuracy
+
+          The solution returned by the routine will be accurate to the
+          user's tolerance as defined by the relation (5) except in extreme
+          circumstances. The final error estimate over the whole mesh for
+          each component is given in the array ABT. If too many points are
+          specified in the initial mesh, the solution may be more accurate
+          than requested and the error may not be approximately
+          equidistributed.
+
+          8. Further Comments
+
+          There are too many factors present to quantify the timing. The
+          time taken by the routine is negligible only on very simple
+          problems.
+
+          The user is strongly recommended to set IFAIL to obtain self-
+          explanatory error messages, and also monitoring information about
+          the course of the computation.
+
+          In the case where the user wishes to solve a sequence of similar
+          problems, the use of the final mesh and solution from one case as
+          the initial mesh is strongly recommended for the next.
+
+          9. Example
+
+          We solve the differential equation
+
+                                                      2
+                            y'''=-yy''-2(epsilon)(1-y' )
+
+          with (epsilon)=1 and boundary conditions
+
+                               y(0)=y'(0)=0,  y'(10)=1
+
+          to an accuracy specified by TOL=1.0E-4. The continuation facility
+          is used with the continuation parameter (epsilon) introduced as
+          in the differential equation above and with DELEPS = 0.1
+          initially. (The continuation facility is not needed for this
+          problem and is used here for illustration.)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd03}{NAG On-line Documentation: d03}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D03(3NAG)         Foundation Library (12/10/92)         D03(3NAG)
+
+
+
+          D03 -- Partial Differential Equations         Introduction -- D03
+                                    Chapter D03
+                          Partial Differential Equations
+
+          1. Scope of the Chapter
+
+          This chapter is concerned with the solution of partial
+          differential equations.
+
+          2. Background to the Problems
+
+          The definition of a partial differential equation problem
+          includes not only the equation itself but also the domain of
+          interest and appropriate subsidiary conditions. Indeed, partial
+          differential equations are usually classified as elliptic,
+          hyperbolic or parabolic according to the form of the equation and
+          the form of the subsidiary conditions which must be assigned to
+          produce a well-posed problem. Ultimately it is hoped that this
+          chapter will contain routines for the solution of equations of
+          each of these types together with automatic mesh generation
+          routines and other utility routines particular to the solution of
+          partial differential equations. The routines in this chapter will
+          often call upon routines from the Linear Algebra Chapter F04 --
+          Simultaneous Linear Equations.
+
+          The classification of partial differential equations is easily
+          described in the case of linear equations of the second order in
+          two independent variables, i.e., equations of the form
+
+                        au  +2bu  +cu  +du +eu +fu+g=0,                 (1)
+                          xx    xy   yy   x   y
+
+          where a, b, c, d, e, f and g are functions of x and y only.
+          Equation (1) is called elliptic, hyperbolic or parabolic
+                           2
+          according as ac-b  is positive, negative or zero. Useful
+          definitions of the concepts of elliptic, hyperbolic and parabolic
+          character can also be given for differential equations in more
+          than two independent variables, for systems and for nonlinear
+          differential equations.
+
+          For elliptic equations, of which Laplace's equation
+
+                                   u  +u  =0                            (2)
+                                    xx  yy
+
+          is the simplest example of second order, the subsidiary
+          conditions take the form of boundary conditions, i.e., conditions
+          which provide information about the solution at all points of a
+          closed boundary. For example, if equation (2) holds in a plane
+          domain D bounded by a contour C, a solution u may be sought
+          subject to the condition
+
+                                   u=f  on C,                           (3)
+
+          where f is a given function. The condition (3) is known as a
+          Dirichlet boundary condition. Equally common is the Neumann
+          boundary condition
+
+                                  u'=g  on C,                           (4)
+
+          which is one form of a more general condition
+
+                                 u'+fu=g  on C,                         (5)
+
+          where u' denotes the derivative of u normal to the contour C and
+          f and g are given functions. Provided that f and g satisfy
+          certain restrictions, condition (5) yields a well-posed  boundary
+          value problem  for Laplace's equation. In the case of the Neumann
+          problem, one further piece of information, e.g. the value of u at
+          a particular point, is necessary for uniqueness of the solution.
+          Boundary conditions similar to the above are applicable to more
+          general second order elliptic equations, whilst two such
+          conditions are required for equations of fourth order.
+
+          For hyperbolic equations, the wave equation
+
+                                   u  -u  =0                            (6)
+                                    tt  xx
+
+          is the simplest example of second order. It is equivalent to a
+          first order system
+
+                               u -v =0,  v -u =0.                       (7)
+                                t  x      t  x
+
+          The subsidiary conditions may take the form of initial
+          conditions, i.e., conditions which provide information about the
+          solution at points on a suitable open boundary. For example, if
+          equation (6) is satisfied for t>0, a solution u may be sought
+          such that
+
+                          u(x,0)=f(x),  u (x,0)=g(x),                   (8)
+                                         t
+
+          where f and g are given functions. This is an example of an
+          initial value problem, sometimes known as Cauchy's problem.
+
+          For parabolic equations, of which the heat conduction equation
+
+                                    u -u  =0                            (9)
+                                     t  xx
+
+          is the simplest example, the subsidiary conditions always include
+          some of initial type and may also include some of boundary type.
+          For example, if equation (9) is satisfied for t>0 and 0<x<1, a
+          solution u may be sought such that
+
+                             u(x,0)=f(x),  0<x<1,                      (10)
+
+          and
+
+                          u(0,t)=0,  u(1,t)=1,  t>0.                   (11)
+
+          This is an example of a mixed  initial/boundary value problem.
+
+          For all types of partial differential equations, finite
+          difference methods (Mitchell and Griffiths [5]) and finite
+          element methods (Wait and Mitchell [9]) are the most common means
+          of solution and such methods obviously feature prominently either
+          in this chapter or in the companion NAG Finite Element Library.
+          Many of the utility routines in this chapter are concerned with
+          the solution of the large sparse systems of equations which arise
+          from the finite difference and finite element methods.
+
+          Alternative methods of solution are often suitable for special
+          classes of problems. For example, the method of characteristics
+          is the most common for hyperbolic equations involving time and
+          one space dimension (Smith [7]). The method of lines (see Mikhlin
+          and Smolitsky [4]) may be used to reduce a parabolic equation to
+          a (stiff) system of ordinary differential equations, which may be
+          solved by means of routines from Chapter D02 -- Ordinary
+          Differential Equations. Similarly, integral equation or boundary
+          element methods (Jaswon and Symm [3]) are frequently used for
+          elliptic equations. Typically, in the latter case, the solution
+          of a boundary value problem is represented in terms of certain
+          boundary functions by an integral expression which satisfies the
+          differential equation throughout the relevant domain. The
+          boundary functions are obtained by applying the given boundary
+          conditions to this representation. Implementation of this method
+          necessitates discretisation of only the boundary of the domain,
+          the dimensionality of the problem thus being effectively reduced
+          by one. The boundary conditions yield a full system of
+          simultaneous equations, as opposed to the sparse systems yielded
+          by the finite difference and finite element methods, but the full
+          system is usually of much lower order. Solution of this system
+          yields the boundary functions, from which the solution of the
+          problem may be obtained, by quadrature, as and where required.
+
+          2.1. References
+
+          [1]   Ames W F (1977) Nonlinear Partial Differential Equations in
+                Engineering. Academic Press (2nd Edition).
+
+          [2]   Berzins M (1990) Developments in the NAG Library Software
+                for Parabolic Equations. Scientific Software Systems. (ed J
+                C Mason and M G Cox) Chapman and Hall. 59--72.
+
+          [3]   Jaswon M A and Symm G T (1977) Integral Equation Methods in
+                Potential Theory and Elastostatics. Academic Press.
+
+          [4]   Mikhlin S G and Smolitsky K L (1967) Approximate Methods for
+                the Solution of Differential and Integral Equations.
+                Elsevier.
+
+          [5]   Mitchell A R and Griffiths D F (1980) The Finite Difference
+                Method in Partial Differential Equations. Wiley.
+
+          [6]   Richtmyer R D and Morton K W (1967) Difference Methods for
+                Initial-value Problems. Interscience (2nd Edition).
+
+          [7]   Smith G D (1985) Numerical Solution of Partial Differential
+                Equations: Finite Difference Methods. Oxford University
+                Press (3rd Edition).
+
+          [8]   Swarztrauber P N and Sweet R A (1979) Efficient Fortran
+                Subprograms for the Solution of Separable Elliptic Partial
+                Differential Equations. ACM Trans. Math. Softw. 5 352--364.
+
+          [9]   Wait R and Mitchell A R (1985) Finite Element Analysis and
+                Application. Wiley.
+
+          3. Recommendations on Choice and Use of Routines
+
+          The choice of routine will depend first of all upon the type of
+          partial differential equation to be solved. At present no special
+          allowances are made for problems with boundary singularities such
+          as may arise at corners of domains or at points where boundary
+          conditions change. For such problems results should be treated
+          with caution.
+
+          Users may wish to construct their own partial differential
+          equation solution software for problems not solvable by the
+          routines described in Sections 3.1 to 3.4 below. In such cases
+          users can employ appropriate routines from the Linear Algebra
+          Chapters to solve the resulting linear systems; see Section 3.5
+          for further details.
+
+          3.1. Elliptic Equations
+
+          The routine D03EDF solves a system of seven-point difference
+          equations in a rectangular grid (in two dimensions), using the
+          multigrid iterative method. The equations are supplied by the
+          user, and the seven-point form allows cross-derivative terms to
+          be represented (see Mitchell and Griffiths [5]). The method is
+          particularly efficient for large systems of equations with
+          diagonal dominance.
+
+          The routine D03EEF discretises a second-order equation on a two-
+          dimensional rectangular region using finite differences and a
+          seven-point molecule. The routine allows for cross-derivative
+          terms, Dirichlet, Neumann or mixed boundary conditions, and
+          either central or upwind differences. The resulting seven-
+          diagonal difference equations are in a form suitable for passing
+          directly to the multigrid routine D03EDF, although other solution
+          methods could easily be used.
+
+          The routine D03FAF, based on the routine HW3CRT from FISHPACK
+          (Swarztrauber and Sweet [8]), solves the Helmholtz equation in a
+          three-dimensional cuboidal region, with any combination of
+          Dirichlet, Neumann or periodic boundary conditions. The method
+          used is based on the fast Fourier transform algorithm, and is
+          likely to be particularly efficient on vector-processing
+          machines.
+
+          3.2. Hyperbolic Equations
+
+          There are no routines available yet for the solution of these
+          equations.
+
+          3.3. Parabolic Equations
+
+          There are no routines available yet for the solution of these
+          equations.
+
+          But problems in two space dimensions plus time may be treated as
+          a succession of elliptic equations [1], [6] using appropriate
+          D03E- routines or one may use codes from the NAG Finite Element
+          Library.
+
+          3.4. Utility Routines
+
+          There are no utility routines available yet, but routines are
+          available in the Linear Algebra Chapters for the direct and
+          iterative solution of linear equations. Here we point to some of
+          the routines that may be of use in solving the linear systems
+          that arise from finite difference or finite element
+          approximations to partial differential equation solutions.
+          Chapters F01 and F04 should be consulted for further information
+          and for the routine documents. Decision trees for the solution of
+          linear systems are given in Section 3.5 of the F04 Chapter
+          Introduction.
+
+          The following routines allow the direct solution of symmetric
+          positive-definite systems:
+
+              Band               F04ACF
+
+              Variable band      F01MCF and F04MCF
+              (skyline)
+
+              Tridiagonal        F04FAF
+
+              Sparse             F01MAF* and F04MAF
+
+          (* the parameter DROPTL should be set to zero for F01MAF to give
+          a direct solution)
+
+          and the following routines allow the iterative solution of
+          symmetric positive-definite systems:
+
+              Sparse (incomplete F01MAF and F04MBF
+              Cholesky)
+
+              Sparse (conjugate  F04MBF
+              gradient)
+
+          The latter routine above allows the user to supply a pre-
+          conditioner and also allows the solution of indefinite symmetric
+          systems.
+
+          The following routines allow the direct solution of unsymmetric
+          systems:
+
+              Band               F01LBF and F04LDF
+
+              Almost block-      F01LHF and F04LHF
+              diagonal
+
+              Tridiagonal        F01LEF and F04LEF or F04EAF
+
+              Sparse             F01BRF (and F01BSF) and F04AXF
+
+          and the following routine allows the iterative solution of
+          unsymmetric systems:
+
+              Sparse             F04QAF
+
+          The above routine allows the user to supply a pre-conditioner and
+          also allows the solution of least-squares systems.
+
+          3.5. Index
+
+          Elliptic equations
+               equations on rectangular grid (seven-point 2-D        D03EDF
+               molecule)
+               discretisation on rectangular grid (seven-point 2-D   D03EEF
+               molecule)
+               Helmholtz's equation in three dimensions              D03FAF
+
+
+
+          D03 -- Partial Differential Equations             Contents -- D03
+          Chapter D03
+
+          Partial Differential Equations
+
+          D03EDF  Elliptic PDE, solution of finite difference equations by
+                  a multigrid technique
+
+          D03EEF  Discretize a 2nd order elliptic PDE on a rectangle
+
+          D03FAF  Elliptic PDE, Helmholtz equation, 3-D Cartesian co-
+                  ordinates
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd03edf}{NAG On-line Documentation: d03edf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D03EDF(3NAG)      Foundation Library (12/10/92)      D03EDF(3NAG)
+
+
+
+          D03 -- Partial Differential Equations                      D03EDF
+                  D03EDF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D03EDF solves seven-diagonal systems of linear equations which
+          arise from the discretization of an elliptic partial differential
+          equation on a rectangular region. This routine uses a multigrid
+          technique.
+
+          2. Specification
+
+                 SUBROUTINE D03EDF (NGX, NGY, LDA, A, RHS, UB, MAXIT, ACC,
+                1                   US, U, IOUT, NUMIT, IFAIL)
+                 INTEGER          NGX, NGY, LDA, MAXIT, IOUT, NUMIT, IFAIL
+                 DOUBLE PRECISION A(LDA,7), RHS(LDA), UB(NGX*NGY), ACC, US
+                1                 (LDA), U(LDA)
+
+          3. Description
+
+          D03EDF solves, by multigrid iteration, the seven-point scheme
+
+           6            7
+          A   u       +A   u
+           i,j i-1,j+1  i,j i,j+1
+
+            3           4        5
+          +A   u      +A   u   +A   u
+            i,j i-1,j   i,j ij   i,j i+1,j
+
+                        1          2
+                      +A   u     +A   u        =f  ,
+                        i,j i,j-1  i,j i+1,j-1  ij
+
+                                               i=1,2,...,n ; j=1,2,...,n ,
+                                                          x             y
+
+          which arises from the discretization of an elliptic partial
+          differential equation of the form
+
+
+          (alpha)(x,y)U  +(beta)(x,y)U  +(gamma)(x,y)U  +(delta)(x,y)U
+                       xx             xy              yy              x
+
+          +(epsilon)(x,y)U +(phi)(x,y)U=(psi)(x,y)
+                          y
+
+          and its boundary conditions, defined on a rectangular region.
+          This we write in matrix form as
+
+                                        Au=f
+
+          The algorithm is described in separate reports by Wesseling [2],
+          [3] and McCarthy [1].
+
+          Systems of linear equations, matching the seven-point stencil
+          defined above, are solved by a multigrid iteration. An initial
+          estimate of the solution must be provided by the user. A zero
+          guess may be supplied if no better approximation is available.
+
+          A 'smoother' based on incomplete Crout decomposition is used to
+          eliminate the high frequency components of the error. A
+          restriction operator is then used to map the system on to a
+          sequence of coarser grids. The errors are then smoothed and
+          prolongated (mapped onto successively finer grids). When the
+          finest cycle is reached, the approximation to the solution is
+          corrected. The cycle is repeated for MAXIT iterations or until
+          the required accuracy, ACC, is reached.
+
+          D03EDF will automatically determine the number l of possible
+          coarse grids, 'levels' of the multigrid scheme, for a particular
+          problem. In other words, D03EDF determines the maximum integer l
+          so that n  and n  can be expressed in the form
+                   x      y
+
+                        l-1          l-1
+                   n =m2   +1,  n =n2   +1,  with  m>=2 and n>=2.
+                    x            y
+
+          It should be noted that the rate of convergence improves
+          significantly with the number of levels used (see McCarthy [1]),
+          so that n  and n  should be carefully chosen so that n -1 and
+                   x      y                                     x
+                                         l
+          n -1 have factors of the form 2 , with l as large as possible.
+           y
+          For good convergence the integer l should be at least 2.
+
+          D03EDF has been found to be robust in application, but being an
+          iterative method the problem of divergence can arise. For a
+          strictly diagonally dominant matrix A
+
+                                   4           k
+                                    ij   --     ij
+                                 |A   |> >   |A   |
+                                         --
+                                         k/=4
+
+          no such problem is foreseen. The diagonal dominance of A is not a
+          necessary condition, but should this condition be strongly
+          violated then divergence may occur. The quickest test is to try
+          the routine.
+
+          4. References
+
+          [1]   McCarthy G J (1983) Investigation into the Multigrid Code
+                MGD1. Report AERE-R 10889. Harwell.
+
+          [2]   Wesseling P (1982) MGD1 - A Robust and Efficient Multigrid
+                Method. Multigrid Methods. Lecture Notes in Mathematics. 960
+                Springer-Verlag. 614--630.
+
+          [3]   Wesseling P (1982) Theoretical Aspects of a Multigrid
+                Method. SIAM J. Sci. Statist. Comput. 3 387--407.
+
+          5. Parameters
+
+           1:  NGX -- INTEGER                                         Input
+               On entry: the number of interior grid points in the x-
+               direction, n . NGX-1 should preferably be divisible by as
+                           x
+               high a power of 2 as possible. Constraint: NGX >= 3.
+
+           2:  NGY -- INTEGER                                         Input
+               On entry: the number of interior grid points in the y-
+               direction, n . NGY-1 should preferably be divisible by as
+                           y
+               high a power of 2 as possible. Constraint: NGY >= 3.
+
+           3:  LDA -- INTEGER                                         Input
+               On entry: the first dimension of the array A as declared in
+               the (sub)program from which D03EDF is called, which must
+               also be a lower bound for the dimensions of the arrays RHS,
+               US and U. It is always sufficient to set
+               LDA>=(4*(NGX+1)*(NGY+1))/3, but slightly smaller values may
+               be permitted, depending on the values of NGX and NGY. If on
+               entry, LDA is too small, an error message gives the minimum
+               permitted value. (LDA must be large enough to allow space
+               for the coarse-grid approximations).
+
+           4:  A(LDA,7) -- DOUBLE PRECISION array              Input/Output
+                                                           k
+               On entry: A(i+(j-1)*NGX,k) must be set to  A  ,   for i =
+                                                           ij
+               1,2,...,NGX; j = 1,2,...,NGY and k = 1,2,...,7. On exit: A
+               is overwritten.
+
+           5:  RHS(LDA) -- DOUBLE PRECISION array              Input/Output
+               On entry: RHS(i+(j-1)*NGX) must be set to f  , for i =
+                                                           ij
+               1,2,...,NGX; j = 1,2,...,NGY. On exit: the first NGX*NGY
+               elements are unchanged and the rest of the array is used as
+               workspace.
+
+           6:  UB(NGX*NGY) -- DOUBLE PRECISION array           Input/Output
+               On entry: UB(i+(j-1)*NGX) must be set to the initial
+               estimate for the solution u  . On exit: the corresponding
+                                          ij
+               component of the residual r=f-Au.
+
+           7:  MAXIT -- INTEGER                                       Input
+               On entry: the maximum permitted number of multigrid
+               iterations. If MAXIT = 0, no multigrid iterations are
+               performed, but the coarse-grid approximations and incomplete
+               Crout decompositions are computed, and may be output if IOUT
+               is set accordingly. Constraint: MAXIT >= 0.
+
+           8:  ACC -- DOUBLE PRECISION                                Input
+               On entry: the required tolerance for convergence of the
+               residual 2-norm:
+                                              
+
+                                             / NGX*NGY
+                                            /  --         2
+                                 ||r|| =   /   >      (r )
+                                      2   /    --       k
+                                        \/     k=1
+               where r=f-Au and u is the computed solution. Note that the
+               norm is not scaled by the number of equations. The routine
+               will stop after fewer than MAXIT iterations if the residual
+               2-norm is less than the specified tolerance. (If MAXIT > 0,
+               at least one iteration is always performed.)
+
+               If on entry ACC = 0.0, then the machine precision is used as
+               a default value for the tolerance; if ACC > 0.0, but ACC is
+               less than the machine precision, then the routine will stop
+               when the residual 2-norm is less than the machine precision
+               and IFAIL will be set to 4. Constraint: ACC >= 0.0.
+
+           9:  US(LDA) -- DOUBLE PRECISION array                     Output
+               On exit: the residual 2-norm, stored in element US(1).
+
+          10:  U(LDA) -- DOUBLE PRECISION array                      Output
+               On exit: the computed solution u   is returned in U(i+(j-
+                                                ij
+               1)*NGX), for i = 1,2,...,NGX; j = 1,2,...,NGY.
+
+          11:  IOUT -- INTEGER                                        Input
+               On entry: controls the output of printed information to the
+               advisory message unit as returned by X04ABF:
+               IOUT = 0
+                     No output.
+
+               IOUT = 1
+                     The solution u  , for i = 1,2,...,NGX; j = 1,2,...
+                                   ij
+                     ,NGY.
+
+               IOUT = 2
+                     The residual 2-norm after each iteration, with the
+                     reduction factor over the previous iteration.
+
+               IOUT = 3
+                     As for IOUT = 1 and IOUT = 2.
+
+               IOUT = 4
+                     As for IOUT = 3, plus the final residual (as returned
+                     in UB).
+
+               IOUT = 5
+                     As for IOUT = 4, plus the initial elements of A and
+                     RHS.
+
+               IOUT = 6
+                     As for IOUT = 5, plus the Galerkin coarse grid
+                     approximations.
+
+               IOUT = 7
+                     As for IOUT = 6, plus the incomplete Crout
+                     decompositions.
+
+               IOUT = 8
+                     As for IOUT = 7, plus the residual after each
+                     iteration.
+               The elements A(p,k), the Galerkin coarse grid approximations
+               and the incomplete Crout decompositions are output in the
+               format:
+                    Y-index = j
+
+                    X-index = i A(p,1) A(p,2) A(p,3) A(p,4) A(p,5) A(p,6)
+                    A(p,7)
+
+                    where p=i+(j-1)*NGX, i = 1,2,...,NGX and j = 1,2,...
+                    ,NGY.
+               The vectors U(p), UB(p), RHS(p) are output in matrix form
+               with NGY rows and NGX columns. Where NGX > 10, the NGX
+               values for a given j-value are produced in rows of 10.
+               Values of IOUT > 4 may yield considerable amounts of output.
+               Constraint: 0 <= IOUT <= 8.
+
+          12:  NUMIT -- INTEGER                                      Output
+               On exit: the number of iterations performed.
+
+          13:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry NGX < 3,
+
+               or       NGY < 3,
+
+               or       LDA is too small,
+
+               or       ACC < 0.0,
+
+               or       MAXIT < 0,
+
+               or       IOUT < 0,
+
+               or       IOUT > 8.
+
+          IFAIL= 2
+               MAXIT iterations have been performed with the residual 2-
+               norm decreasing at each iteration but the residual 2-norm
+               has not been reduced to less than the specified tolerance
+               (see ACC). Examine the progress of the iteration by setting
+               IOUT >= 2.
+
+          IFAIL= 3
+               As for IFAIL = 2, except that at one or more iterations the
+               residual 2-norm did not decrease. It is likely that the
+               method fails to converge for the given matrix A.
+
+          IFAIL= 4
+               On entry ACC is less than the machine precision. The routine
+               terminated because the residual norm is less than the
+               machine precision.
+
+          7. Accuracy
+
+          See ACC (Section 5).
+
+          8. Further Comments
+
+          The rate of convergence of this routine is strongly dependent
+          upon the number of levels, l, in the multigrid scheme, and thus
+          the choice of NGX and NGY is very important. The user is advised
+          to experiment with different values of NGX and NGY to see the
+          effect they have on the rate of convergence; for example, using a
+                                    6
+          value such as NGX = 65 (=2 +1) followed by NGX = 64 (for which l
+          = 1).
+
+          9. Example
+
+          The program solves the elliptic partial differential equation
+
+                        U  -(alpha)U  +U  =-4,   (alpha)=1.7
+                         xx         xy  yy
+
+          on the unit square 0<=x,y<=1, with boundary conditions
+
+                           {x=0, (0<=y<=1)
+                    U=0 on {y=0, (0<=x<=1) U=1 on x=1,  0<=y<=1.
+                           {y=1, (0<=x<=1)
+
+          For the equation to be elliptic, (alpha) must be less than 2.
+
+          The equation is discretized on a square grid with mesh spacing h
+          in both directions using the following approximations:
+
+
+
+                   Please see figure in printed Reference Manual
+
+                                       1
+                                 U  ~= --(U -2U +U )
+                                  xx    2  E   O  W
+                                       h
+
+                                       1
+                                 U  ~= --(U -2U +U )
+                                  yy    2  N   O  S
+                                       h
+
+                                1
+                         U  ~= ---(U -U  +U -2U +U -U  +U ).
+                          xy     2  N  NW  E   O  W  SE  S
+                               2h
+
+          Thus the following equations are solved:
+
+
+              1                      1
+              -(alpha)u        + (1- -(alpha))u
+              2        i-1,j+1       2         i,j+1
+
+               1                                           1
+
+          +(1- -(alpha))u      + (-4+(alpha))u       + (1- -(alpha))u
+               2         i-1,j                ij           2         i+1,j
+
+                                     1                  1
+                               + (1- -(alpha))u      +  -(alpha)u
+                                     2         i,j-1    2        i+1,j-1
+
+                    2
+                =-4h
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd03eef}{NAG On-line Documentation: d03eef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D03EEF(3NAG)      Foundation Library (12/10/92)      D03EEF(3NAG)
+
+
+
+          D03 -- Partial Differential Equations                      D03EEF
+                  D03EEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D03EEF discretizes a second order elliptic partial differential
+          equation (PDE) on a rectangular region.
+
+          2. Specification
+
+                 SUBROUTINE D03EEF (XMIN, XMAX, YMIN, YMAX, PDEF, BNDY,
+                1                   NGX, NGY, LDA, A, RHS, SCHEME, IFAIL)
+                 INTEGER          NGX, NGY, LDA, IFAIL
+                 DOUBLE PRECISION XMIN, XMAX, YMIN, YMAX, A(LDA,7), RHS(LDA)
+                 CHARACTER*1      SCHEME
+                 EXTERNAL         PDEF, BNDY
+
+          3. Description
+
+          D03EEF discretizes a second order linear elliptic partial
+          differential equation of the form
+
+
+                         2                 2                  2
+                       dd U              dd U               dd U
+          (alpha)(x,y) ----+(beta)(x,y) ------+(gamma)(x,y) ----
+                         2              ddxddy                2
+                       dd x                                 dd y
+
+                        ddU                ddU
+          +(delta)(x,y) ---+(epsilon)(x,y) ---+(phi)(x,y)U=(psi)(x,y)   (1)
+                        ddx                ddy
+
+
+          on a rectangular region
+
+                                      x <=x<=x
+                                       A      B
+
+                                      y <=y<=y
+                                       A      B
+
+          subject to boundary conditions of the form
+
+                                             ddU
+                              a(x,y)U+b(x,y) ---=c(x,y)
+                                             ddn
+
+                 ddU
+          where  --- denotes the outward pointing normal derivative on the
+                 ddn
+          boundary. Equation (1) is said to be elliptic if
+
+                                                              2
+                      4(alpha)(x,y)(gamma)(x,y)>=((beta)(x,y))
+
+          for all points in the rectangular region. The linear equations
+          produced are in a form suitable for passing directly to the
+          multigrid routine D03EDF.
+
+          The equation is discretized on a rectangular grid, with n  grid
+                                                                   x
+          points in the x-direction and n  grid points in the y-direction.
+                                         y
+          The grid spacing used is therefore
+
+                                  h =(x -x )/(n -1)
+                                   x   B  A    x
+
+                                  h =(y -y )/(n -1)
+                                   y   B  A    y
+
+          and the co-ordinates of the grid points (x ,y ) are
+                                                    i  j
+
+                            x =x +(i-1)h ,  i=1,2,...,n ,
+                             i  A       x              x
+
+                            y =y +(j-1)h ,  j=1,2,...,n .
+                             j  A       y              y
+
+          At each grid point (x ,y ) six neighbouring grid points are used
+                               i  j
+          to approximate the partial differential equation, so that the
+          equation is discretized on the following seven-point stencil:
+
+
+
+                   Please see figure in printed Reference Manual
+
+          For convenience the approximation u   to the exact solution
+                                             ij
+          U(x ,y ) is denoted by u , and the neighbouring approximations
+             i  j                 O
+          are labelled according to points of the compass as shown. Where
+          numerical labels for the seven points are required, these are
+          also shown above.
+
+          The following approximations are used for the second derivatives:
+
+                 2
+               dd U   1
+               ----~= --(u -2u +u )
+                 2     2  E   O  W
+               dd x   h
+                       x
+
+                 2
+               dd U   1
+               ----~= --(u -2u +u )
+                 2     2  N   O  S
+               dd y   h
+                       y
+
+                  2
+                dd U    1
+               ------~= -----(u -u  +u -2u +u -u  +u ).
+               ddxddy 2h h   N  NW  E   O  W  SE  S
+                        x y
+
+          Two possible schemes may be used to approximate the first
+          derivatives:
+
+          Central Differences
+
+               ddU     1
+               ---~=  ---(u -u )
+               ddx    2h   W  E
+                        x
+
+               ddU     1
+               ---~=  ---(u -u )
+               ddy    2h   N  S
+                        y
+
+          Upwind Differences
+
+               ddU    1
+               ---~=  --(u -u )if (delta)(x,y)>0
+               ddx    h   E  O
+                       x
+
+               ddU    1
+               ---~=  --(u -u )if (delta)(x,y)<0
+               ddx    h   O  W
+                       x
+
+               ddU    1
+               ---~=  --(u -u )if (epsilon)(x,y)>0
+               ddy    h   N  O
+                       y
+
+               ddU    1
+               ---~=  --(u -u )if (epsilon)(x,y)<0.
+               ddy    h   O  S
+                       y
+
+          Central differences are more accurate than upwind differences,
+          but upwind differences may lead to a more diagonally dominant
+          matrix for those problems where the coefficients of the first
+          derivatives are significantly larger than the coefficients of the
+          second derivatives.
+
+          The approximations used for the first derivatives may be written
+          in a more compact form as follows:
+
+               ddU     1
+               ---~=  ---((k -1)u -2k u +(k +1)u )
+               ddx    2h (  x    W   x O   x    E)
+                        x
+
+               ddU     1
+               ---~=  ---((k -1)u -2k u +(k +1)u )
+               ddy    2h (  y    S   y O   y    N)
+                        y
+
+          where k =sign (delta) and k =sign (epsilon) for upwind
+                 x                   y
+          differences, and k =k =0 for central differences.
+                            x  y
+
+          At all points in the rectangular domain, including the boundary,
+          the coefficients in the partial differential equation are
+          evaluated by calling the user-supplied subroutine PDEF, and
+          applying the approximations. This leads to a seven-diagonal
+          system of linear equations of the form:
+
+              6            7
+             A  u        +A  u
+              ij i-1,j+1   ij i,j+1
+               3          4       5
+             +A  u      +A  u   +A  u
+               ij i-1,j   ij ij   ij i+1,j
+                 1          2
+               +A  u      +A  u        =f  , i=1,2,...,n ;j=1,2,...,n ,
+                 ij i,j-1   ij i+1,j-1   ij             x            y
+
+          where the coefficients are given by
+
+           1                 1                   1                   1
+          A  =(beta)(x ,y )----- +(gamma)(x ,y )-- +(epsilon)(x ,y )---(k -1)
+           ij         i  j 2h h            i  j  2             i  j 2h   y
+                             x y                h                     y
+                                                 y
+           2                   1
+          A  =-(beta)(x ,y ) -----
+           ij          i  j  2h h
+                               x y
+
+           3                 1                  1                   1
+          A  =(alpha)(x ,y ) --+(beta)(x ,y ) -----+(delta)(x ,y ) ---(k -1)
+           ij          i  j   2         i  j  2h h           i  j  2h   x
+                             h                  x y                  x
+                              x
+
+           4                  2                  1                   2
+          A  =-(alpha)(x ,y ) -- -(beta)(x ,y ) ---- -(gamma)(x ,y ) --
+           ij           i  j   2          i  j  h h            i  j   2
+                              h                  x y                 h
+                               x                                      y
+
+                              k                    k
+                               y                    y
+              -(delta)(x ,y ) -- -(epsilon)(x ,y ) -- -(phi)(x ,y )
+                        i  j  h              i  j  h          i  j
+                               x                    y
+
+           5                 1                  1                   1
+          A  =(alpha)(x ,y ) --+(beta)(x ,y ) -----+(delta)(x ,y ) ---(k +1)
+           ij          i  j   2         i  j  2h h           i  j  2h   x
+                             h                  x y                  x
+                              x
+
+           6                   1
+          A  =-(beta)(x ,y ) -----
+           ij          i  j  2h h
+                               x y
+
+           7                 1                  1                   1
+          A  =(beta)(x ,y )-----+(gamma)(x ,y )--+(epsilon)(x ,y )---(k +1)
+           ij         i  j 2h h           i  j  2             i j 2h   y
+                             x y               h                    y
+                                                y
+
+          f  =(psi)(x ,y )
+           ij        i  j
+
+          These equations then have to be modified to take account of the
+          boundary conditions. These may be Dirichlet (where the solution
+          is given), Neumann (where the derivative of the solution is
+          given), or mixed (where a linear combination of solution and
+          derivative is given).
+
+          If the boundary conditions are Dirichlet, there are an infinity
+          of possible equations which may be applied:
+
+                           (mu)u  =(mu)f   , (mu)/=0.                   (2)
+                                ij      ij
+
+          If D03EDF is used to solve the discretized equations, it turns
+          out that the choice of (mu) can have a dramatic effect on the
+          rate of convergence, and the obvious choice (mu)=1 is not the
+          best. Some choices may even cause the multigrid method to fail
+          altogether. In practice it has been found that a value of the
+          same order as the other diagonal elements of the matrix is best,
+          and the following value has been found to work well in practice:
+
+                                       ( { 2   2 }  4 )
+                             (mu)=min  (-{ --+ --},A  ).
+                                     ij( {  2   2}  ij)
+                                       ( { h   h }    )
+                                       ( {  x   y}    )
+
+          If the boundary conditions are either mixed or Neumann (i.e., B
+          /= 0 on return from the user-supplied subroutine BNDY), then one
+          of the points in the seven-point stencil lies outside the domain.
+          In this case the normal derivative in the boundary conditions is
+          used to eliminate the 'fictitious' point, u       :
+                                                     outside
+
+                           ddU   1
+                           ---~= --(u       -u      ).                  (3)
+                           ddn   2h  outside  inside
+
+          It should be noted that if the boundary conditions are Neumann
+          and (phi)(x,y)==0, then there is no unique solution. The routine
+          returns with IFAIL = 5 in this case, and the seven-diagonal
+          matrix is singular.
+
+          The four corners are treated separately. The user-supplied
+          subroutine BNDY is called twice, once along each of the edges
+          meeting at the corner. If both boundary conditions at this point
+          are Dirichlet and the prescribed solution values agree, then this
+          value is used in an equation of the form (2). If the prescribed
+          solution is discontinuous at the corner, then the average of the
+          two values is used. If one boundary condition is Dirichlet and
+          the other is mixed, then the value prescribed by the Dirichlet
+          condition is used in an equation of the form given above.
+          Finally, if both conditions are mixed or Neumann, then two '
+          fictitious' points are eliminated using two equations of the form
+          (3).
+
+          It is possible that equations for which the solution is known at
+          all points on the boundary, have coefficients which are not
+          defined on the boundary. Since this routine calls the user-
+          supplied subroutine PDEF at all points in the domain, including
+          boundary points, arithmetic errors may occur in the user's
+          routine PDEF which this routine cannot trap. If the user has an
+          equation with Dirichlet boundary conditions (i.e., B = 0 at all
+          points on the boundary), but with PDE coefficients which are
+          singular on the boundary, then D03EDF could be called directly
+          only using interior grid points with the user's own
+          discretization.
+
+          After the equations have been set up as described above, they are
+          checked for diagonal dominance. That is to say,
+
+                     4     --    k
+                   |A  | > >   |A  |,  i=1,2,...,n ; j=1,2,...,n .
+                     ij    --    ij               x             y
+                           k/=4
+
+          If this condition is not satisfied then the routine returns with
+          IFAIL = 6. The multigrid routine D03EDF may still converge in
+          this case, but if the coefficients of the first derivatives in
+          the partial differential equation are large compared with the
+          coefficients of the second derivative, the user should consider
+          using upwind differences (SCHEME = 'U').
+
+          Since this routine is designed primarily for use with D03EDF,
+          this document should be read in conjunction with the document for
+          that routine.
+
+          4. References
+
+          [1]   Wesseling P (1982) MGD1 - A Robust and Efficient Multigrid
+                Method. Multigrid Methods. Lecture Notes in Mathematics. 960
+                Springer-Verlag. 614--630.
+
+          5. Parameters
+
+           1:  XMIN -- DOUBLE PRECISION                               Input
+
+           2:  XMAX -- DOUBLE PRECISION                               Input
+               On entry: the lower and upper x co-ordinates of the
+               rectangular region respectively, x  and x . Constraint: XMIN
+                                                 A      B
+               < XMAX.
+
+           3:  YMIN -- DOUBLE PRECISION                               Input
+
+           4:  YMAX -- DOUBLE PRECISION                               Input
+               On entry: the lower and upper y co-ordinates of the
+               rectangular region respectively, y  and y . Constraint: YMIN
+                                                 A      B
+               < YMAX.
+
+           5:  PDEF -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               PDEF must evaluate the functions (alpha)(x,y), (beta)(x,y),
+               (gamma)(x,y), (delta)(x,y), (epsilon)(x,y), (phi)(x,y) and
+               (psi)(x,y) which define the equation at a general point
+               (x,y).
+
+               Its specification is:
+
+                      SUBROUTINE PDEF (X, Y, ALPHA, BETA, GAMMA,
+                     1                 DELTA, EPSLON, PHI, PSI)
+                      DOUBLE PRECISION X, Y, ALPHA, BETA, GAMMA, DELTA,
+                     1                 EPSLON, PHI, PSI
+
+                1:  X -- DOUBLE PRECISION                             Input
+
+                2:  Y -- DOUBLE PRECISION                             Input
+                    On entry: the x and y co-ordinates of the point at
+                    which the coefficients of the partial differential
+                    equation are to be evaluated. 8
+
+                3:  ALPHA -- DOUBLE PRECISION                        Output
+
+                4:  BETA -- DOUBLE PRECISION                         Output
+
+                5:  GAMMA -- DOUBLE PRECISION                        Output
+
+                6:  DELTA -- DOUBLE PRECISION                        Output
+
+                7:  EPSLON -- DOUBLE PRECISION                       Output
+
+                8:  PHI -- DOUBLE PRECISION                          Output
+
+                9:  PSI -- DOUBLE PRECISION                          Output
+                    On exit: ALPHA, BETA, GAMMA, DELTA, EPSLON, PHI and PSI
+                    must be set to the values of (alpha)(x,y), (beta)(x,y),
+                    (gamma)(x,y), (delta)(x,y), (epsilon)(x,y), (phi)(x,y)
+                    and (psi)(x,y) respectively at the point specified by X
+                    and Y.
+               PDEF must be declared as EXTERNAL in the (sub)program
+               from which D03EEF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           6:  BNDY -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               BNDY must evaluate the functions a(x,y), b(x,y), and c(x,y)
+               involved in the boundary conditions.
+
+               Its specification is:
+
+                      SUBROUTINE BNDY (X, Y, A, B, C, IBND)
+                      INTEGER          IBND
+                      DOUBLE PRECISION X, Y, A, B, C
+
+                1:  X -- DOUBLE PRECISION                             Input
+
+                2:  Y -- DOUBLE PRECISION                             Input
+                    On entry: the x and y co-ordinates of the point at
+                    which the boundary conditions are to be evaluated.
+
+                3:  A -- DOUBLE PRECISION                            Output
+
+                4:  B -- DOUBLE PRECISION                            Output
+
+                5:  C -- DOUBLE PRECISION                            Output
+                    On exit: A, B and C must be set to the values of the
+                    functions appearing in the boundary conditions.
+
+                6:  IBND -- INTEGER                                   Input
+                    On entry: specifies on which boundary the point (X,Y)
+                    lies. IBND = 0, 1, 2 or 3 according as the point lies
+                    on the bottom, right, top or left boundary.
+               BNDY must be declared as EXTERNAL in the (sub)program
+               from which D03EEF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           7:  NGX -- INTEGER                                         Input
+
+           8:  NGY -- INTEGER                                         Input
+               On entry: the number of interior grid points in the x- and y
+               -directions respectively, n  and n . If the seven-diagonal
+                                          x      y
+               equations are to be solved by D03EDF, then NGX-1 and NGY-1
+               should preferably be divisible by as high a power of 2 as
+               possible. Constraint: NGX >= 3, NGY >= 3.
+
+           9:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which D03EEF is called.
+               Constraint: if only the seven-diagonal equations are
+               required, then LDA >= NGX*NGY. If a call to this routine is
+               to be followed by a call to D03EDF to solve the seven-
+               diagonal linear equations, LDA >= (4*(NGX+1)*(NGY+1))/3.
+
+               Note: this routine only checks the former condition. D03EDF,
+               if called, will check the latter condition.
+
+          10:  A(LDA,7) -- DOUBLE PRECISION array                    Output
+               On exit: A(i,j), for i=1,2,...,NGX*NGY; j = 1,2,...,7,
+               contains the seven-diagonal linear equations produced by the
+               discretization described above. If LDA > NGX*NGY, the
+               remaining elements are not referenced by the routine, but if
+               LDA >= (4*(NGX+1)*(NGY+1))/3 then the array A can be passed
+               directly to D03EDF, where these elements are used as
+               workspace.
+
+          11:  RHS(LDA) -- DOUBLE PRECISION array                    Output
+               On exit: the first NGX*NGY elements contain the right-hand
+               sides of the seven-diagonal linear equations produced by the
+               discretization described above. If LDA > NGX*NGY, the
+               remaining elements are not referenced by the routine, but if
+               LDA >= (4*(NGY+1)*(NGY+1))/3 then the array RHS can be
+               passed directly to D03EDF, where these elements are used as
+               workspace.
+
+          12:  SCHEME -- CHARACTER*1                                  Input
+               On entry: the type of approximation to be used for the first
+               derivatives which occur in the partial differential
+               equation.
+
+               If SCHEME = 'C', then central differences are used.
+
+               If SCHEME = 'U', then upwind differences are used.
+               Constraint: SCHEME = 'C' or 'U'.
+
+               Note: generally speaking, if at least one of the
+               coefficients multiplying the first derivatives (DELTA or
+               EPSLON as returned by PDEF) are large compared with the
+               coefficients multiplying the second derivatives, then upwind
+               differences may be more appropriate. Upwind differences are
+               less accurate than central differences, but may result in
+               more rapid convergence for strongly convective equations.
+               The easiest test is to try both schemes.
+
+          13:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry XMIN >= XMAX,
+
+               or       YMIN >= YMAX,
+
+               or       NGX < 3,
+
+               or       NGY < 3,
+
+               or       LDA < NGX*NGY,
+
+               or       SCHEME is not one of 'C' or 'U'.
+
+          IFAIL= 2
+               At some point on the boundary there is a derivative in the
+               boundary conditions (B /= 0 on return from a BNDY) and there
+                                                                     2
+                                                                   dd U
+               is a non-zero coefficient of the mixed derivative  ------
+                                                                  ddxddy
+               (BETA /= 0 on return from PDEF).
+
+          IFAIL= 3
+               A null boundary has been specified, i.e., at some point both
+               A and B are zero on return from a call to BNDY.
+
+          IFAIL= 4
+                                                                    2
+               The equation is not elliptic, i.e., 4*ALPHA*GAMMA<BETA  after
+               a call to PDEF. The discretization has been completed, but
+               the convergence of D03EDF cannot be guaranteed.
+
+          IFAIL= 5
+               The boundary conditions are purely Neumann (only the
+               derivative is specified) and there is, in general, no unique
+               solution.
+
+          IFAIL= 6
+               The equations were not diagonally dominant. (See Section 3).
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          If this routine is used as a pre-processor to the multigrid
+          routine D03EDF it should be noted that the rate of convergence of
+          that routine is strongly dependent upon the number of levels in
+          the multigrid scheme, and thus the choice of NGX and NGY is very
+          important.
+
+          9. Example
+
+          The program solves the elliptic partial differential equation
+
+                             2     2
+                           dd U  dd U   { ddU  ddU}
+                           ----+ ----+50{ ---+ ---}=f(x,y)
+                              2     2   { ddx  ddy}
+                           ddx   ddy
+
+          on the unit square 0<=x, y<=1, with boundary conditions
+
+                ddU
+                --- given on x=0 and y=0,
+                ddn
+
+               U given on x=1 and y=1.
+
+          The function f(x,y) and the exact form of the boundary conditions
+          are derived from the exact solution U(x,y)=sinxsiny.
+
+          The equation is first solved using central differences. Since the
+          coefficients of the first derivatives are large, the linear
+          equations are not diagonally dominated, and convergence is slow.
+          The equation is solved a second time with upwind differences,
+          showing that convergence is more rapid, but the solution is less
+          accurate.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXd03faf}{NAG On-line Documentation: d03faf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     D03FAF(3NAG)      Foundation Library (12/10/92)      D03FAF(3NAG)
+
+
+
+          D03 -- Partial Differential Equations                      D03FAF
+                  D03FAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          D03FAF solves the Helmholtz equation in Cartesian co-ordinates in
+          three dimensions using the standard seven-point finite difference
+          approximation. This routine is designed to be particularly
+          efficient on vector processors.
+
+          2. Specification
+
+                 SUBROUTINE D03FAF (XS, XF, L, LBDCND, BDXS, BDXF, YS, YF,
+                1                   M, MBDCND, BDYS, BDYF, ZS, ZF, N,
+                2                   NBDCND, BDZS, BDZF, LAMBDA, LDIMF,
+                3                   MDIMF, F, PERTRB, W, LWRK, IFAIL)
+                 INTEGER          L, LBDCND, M, MBDCND, N, NBDCND, LDIMF,
+                1                 MDIMF, LWRK, IFAIL
+                 DOUBLE PRECISION XS, XF, BDXS(MDIMF,N+1), BDXF(MDIMF,N+1),
+                1                 YS, YF, BDYS(LDIMF,N+1), BDYF(LDIMF,N+1),
+                2                 ZS, ZF, BDZS(LDIMF,M+1), BDZF(LDIMF,M+1),
+                3                 LAMBDA, F(LDIMF,MDIMF,N+1), PERTRB, W
+                4                 (LWRK)
+
+          3. Description
+
+          D03FAF solves the three-dimensional Helmholtz equation in
+          cartesian co-ordinates:
+
+                           2     2     2
+                         dd u  dd u  dd u
+                         ----+ ----+ ----+(lambda)u=f(x,y,z)
+                           2     2     2
+                         dd x  dd y  dd z
+
+          This subroutine forms the system of linear equations resulting
+          from the standard seven-point finite difference equations, and
+          then solves the system using a method based on the fast Fourier
+          transform (FFT) described by Swarztrauber [1]. This subroutine is
+          based on the routine HW3CRT from FISHPACK (see Swarztrauber and
+          Sweet [2]).
+
+          More precisely, the routine replaces all the second derivatives
+          by second-order central difference approximations, resulting in a
+          block tridiagonal system of linear equations. The equations are
+          modified to allow for the prescribed boundary conditions. Either
+          the solution or the derivative of the solution may be specified
+          on any of the boundaries, or the solution may be specified to be
+          periodic in any of the three dimensions. By taking the discrete
+          Fourier transform in the x- and y-directions, the equations are
+          reduced to sets of tridiagonal systems of equations. The Fourier
+          transforms required are computed using the multiple FFT routines
+          found in Chapter C06 of the NAG Fortran Library.
+
+          4. References
+
+          [1]   Swarztrauber P N (1984) Fast Poisson Solvers. Studies in
+                Numerical Analysis. (ed G H Golub) Mathematical Association
+                of America.
+
+          [2]   Swarztrauber P N and Sweet R A (1979) Efficient Fortran
+                Subprograms for the Solution of Separable Elliptic Partial
+                Differential Equations. ACM Trans. Math. Softw. 5 352--364.
+
+          5. Parameters
+
+           1:  XS -- DOUBLE PRECISION                                 Input
+               On entry: the lower bound of the range of x, i.e., XS <= x
+               <= XF. Constraint: XS < XF.
+
+           2:  XF -- DOUBLE PRECISION                                 Input
+               On entry: the upper bound of the range of x, i.e., XS <= x
+               <= XF. Constraint: XS < XF.
+
+           3:  L -- INTEGER                                           Input
+               On entry: the number of panels into which the interval
+               (XS,XF) is subdivided. Hence, there will be L+1 grid points
+               in the x-direction given by x =XS+(i-1)*(delta)x, for
+                                            i
+               i=1,2,...,L+1, where (delta)x=(XF-XS)/L is the panel width.
+               Constraint: L >= 5.
+
+           4:  LBDCND -- INTEGER                                      Input
+               On entry: indicates the type of boundary conditions at x =
+               XS and x = XF.
+               LBDCND = 0
+                     if the solution is periodic in x, i.e.,
+                     u(XS,y,z)=u(XF,y,z).
+
+               LBDCND = 1
+                     if the solution is specified at x = XS and x = XF.
+
+               LBDCND = 2
+                     if the solution is specified at x = XS and the
+                     derivative of the solution with respect to x is
+                     specified at x = XF.
+
+               LBDCND = 3
+                     if the derivative of the solution with respect to x is
+                     specified at x = XS and x = XF.
+
+               LBDCND = 4
+                     if the derivative of the solution with respect to x is
+                     specified at x = XS and the solution is specified at x
+                     = XF.
+                Constraint: 0 <= LBDCND <= 4.
+
+           5:  BDXS(MDIMF,N+1) -- DOUBLE PRECISION array              Input
+               On entry: the values of the derivative of the solution with
+               respect to x at x = XS. When LBDCND = 3 or 4, BDXS
+               (j,k)=u (XS,y ,z ), for j=1,2,...,M+1; k=1,2,...,N+1.
+                      x     j  k
+
+               When LBDCND has any other value, BDXS is not referenced.
+
+           6:  BDXF(MDIMF,N+1) -- DOUBLE PRECISION array              Input
+               On entry: the values of the derivative of the solution with
+               respect to x at x = XF. When LBDCND = 2 or 3, BDXF(j,k) =
+               u (XF,y ,z ), for j=1,2,...,M+1; k=1,2,...,N+1.
+                x     i  k
+
+               When LBDCND has any other value, BDXF is not referenced.
+
+           7:  YS -- DOUBLE PRECISION                                 Input
+               On entry: the lower bound of the range of y, i.e., YS <= y
+               <= YF. Constraint: YS < YF.
+
+           8:  YF -- DOUBLE PRECISION                                 Input
+               On entry: the upper bound of the range of y, i.e., YS <= y
+               <= YF. Constraint: YS < YF.
+
+           9:  M -- INTEGER                                           Input
+               On entry: the number of panels into which the interval
+               (YS,YF) is subdivided. Hence, there will be M+1 grid points
+               in the y-direction given by y  = YS + (j-1)*(delta)y for
+                                            j
+               j=1,2,...,M+1, where (delta)y=(YF-YS)/M is the panel width.
+               Constraint: M >= 5.
+
+          10:  MBDCND -- INTEGER                                      Input
+               On entry: indicates the type of boundary conditions at y =
+               YS and y = YF.
+               MBDCND = 0
+                     if the solution is periodic in y, i.e.,
+                     u(x,YF,z)=u(x,YS,z).
+
+               MBDCND = 1
+                     if the solution is specified at y = YS and y = YF.
+
+               MBDCND = 2
+                     if the solution is specified at y = YS and the
+                     derivative of the solution with respect to y is
+                     specified at y = YF.
+
+               MBDCND = 3
+                     if the derivative of the solution with respect to y is
+                     specified at y = YS and y = YF.
+
+               MBDCND = 4
+                     if the derivative of the solution with respect to y is
+                     specified at y = YS and the solution is specified at y
+                     = YF.
+                Constraint: 0 <= MBDCND <= 4.
+
+          11:  BDYS(LDIMF,N+1) -- DOUBLE PRECISION array              Input
+               On entry: the values of the derivative of the solution with
+               respect to y at y = YS. When MBDCND = 3 or 4, BDYS
+               (i,k)=u (x ,y ,z ), for i=1,2,...,L+1; k=1,2,...,N+1.
+                      y  i  s  k
+
+               When MBDCND has any other value, BDYS is not referenced.
+
+          12:  BDYF(LDIMF,N+1) -- DOUBLE PRECISION array              Input
+               On entry: the values of the derivative of the solution with
+               respect to y at y = YF. When MBDCND = 2 or 3, BDYF
+               (i,k)=u (x ,YF,z ), for i=1,2,...,L+1; k=1,2,...,N+1.
+                      y  i     k
+
+               When MBDCND has any other value, BDYF is not referenced.
+
+          13:  ZS -- DOUBLE PRECISION                                 Input
+               On entry: the lower bound of the range of z, i.e., ZS <= z
+               <= ZF. Constraint: ZS < ZF.
+
+          14:  ZF -- DOUBLE PRECISION                                 Input
+               On entry: the upper bound of the range of z, i.e., ZS <= z
+               <= ZF. Constraint: ZS < ZF.
+
+          15:  N -- INTEGER                                           Input
+               On entry: the number of panels into which the interval
+               (ZS,ZF) is subdivided. Hence, there will be N+1 grid points
+               in the z-direction given by z =ZS+(k-1)*(delta)z, for
+                                            k
+               k=1,2,...,N+1, where (delta)z=(ZF-ZS)/N is the panel width.
+               Constraint: N >= 5.
+
+          16:  NBDCND -- INTEGER                                      Input
+               On entry: specifies the type of boundary conditions at z =
+               ZS and z = ZF.
+               NBDCND = 0
+                     if the solution is periodic in z, i.e.,
+                     u(x,y,ZF)=u(x,y,ZS).
+
+               NBDCND = 1
+                     if the solution is specified at z = ZS and z = ZF.
+
+               NBDCND = 2
+                     if the solution is specified at z = ZS and the
+                     derivative of the solution with respect to z is
+                     specified at z = ZF.
+
+               NBDCND = 3
+                     if the derivative of the solution with respect to z is
+                     specified at z = ZS and z = ZF.
+
+               NBDCND = 4
+                     if the derivative of the solution with respect to z is
+                     specified at z = ZS and the solution is specified at z
+                     = ZF.
+                Constraint: 0 <= NBDCND <= 4.
+
+          17:  BDZS(LDIMF,M+1) -- DOUBLE PRECISION array              Input
+               On entry: the values of the derivative of the solution with
+               respect to z at z = ZS. When NBDCND = 3 or 4, BDZS
+               (i,j)=u (x ,y ,ZS)=u(x,y,z), for i=1,2,...,L+1;
+                      z  i  j
+               j=1,2,...,M+1.
+
+               When NBDCND has any other value, BDZS is not referenced.
+
+          18:  BDZF(LDIMF,M+1) -- DOUBLE PRECISION array              Input
+               On entry: the values of the derivative of the solution with
+               respect to z at z = ZF. When NBDCND = 2 or 3, BDZF
+               (i,j)=u (x ,y ,ZF)=u(x,y,z), for i=1,2,...,L+1;
+                      z  i  j
+               j=1,2,...,M+1.
+
+               When NBDCND has any other value, BDZF is not referenced.
+
+          19:  LAMBDA -- DOUBLE PRECISION                             Input
+               On entry: the constant (lambda) in the Helmholtz equation.
+               For certain positive values of (lambda) a solution to the
+               differential equation may not exist, and close to these
+               values the solution of the discretized problem will be
+               extremely ill-conditioned. If (lambda)>0, then D03FAF will
+               set IFAIL to 3, but will still attempt to find a solution.
+               However, since in general the values of (lambda) for which
+               no solution exists cannot be predicted a priori, the user is
+               advised to treat any results computed with (lambda)>0 with
+               great caution.
+
+          20:  LDIMF -- INTEGER                                       Input
+               On entry:
+               the first dimension of the arrays F, BDYS, BDYF, BDZS and
+               BDZF as declared in the (sub)program from which D03FAF is
+               called.
+               Constraint: LDIMF >= L + 1.
+
+          21:  MDIMF -- INTEGER                                       Input
+               On entry: the second dimension of the array F and
+               the first dimension of the arrays BDXS and BDXF as declared
+               in the (sub)program from which D03FAF is called.
+               Constraint: MDIMF >= M + 1.
+
+          22:  F(LDIMF,MDIMF,N+1) -- DOUBLE PRECISION array    Input/Output
+               On entry: the values of the right-side of the Helmholtz
+               equation and boundary values (if any).
+
+               F(i,j,k)=f(x ,y ,z ) i=2,3,...,L, j=2,3,...,M and k
+                           i  j  k
+               =2,3,...,N.
+
+               On the boundaries F is defined by
+               LBDCND    F(1,j,k)  F(L+1,j,k)
+
+               0         f(XS,y ,z )f(XS,y ,z )
+                               j  k      j  k
+
+               1         u(XS,y ,z )u(XF,y ,z )
+                               j  k      j  k
+
+               2         u(XS,y ,z )f(XF,y ,z )  j=1,2,...,M+1
+                               j  k      j  k
+
+               3         f(XS,y ,z )f(XF,y ,z )  k=1,2,...,N+1
+                               j  k      j  k
+
+               4         f(XS,y ,z )u(XF,y ,z )
+                               j  k      j  k
+
+
+
+               MBDCND    F(i,1,k)  F(
+                                   i,M+1,k)
+
+               0         f(x ,YS,z )f(x ,YS,z )
+                            i     k   i     k
+
+               1         u(x ,YS,z )u(x ,YF,z )
+                            i     k   i     k
+
+               2         u(x ,YS,z )f(x ,YF,z )  i=1,2,...,L+1
+                            i     k   i     k
+
+               3         f(x ,YS,z )f(x ,YF,z )  k=1,2,...,N+1
+                            i     k   i     k
+
+               4         f(x ,YS,z )u(x ,YF,z )
+                            i     k   i     k
+
+
+
+               NBDCND    F(i,j,1)  F(i,j,N+1
+                                   )
+
+               0         f(x ,y ,ZS)f(x ,y ,ZS)
+                            i  j      i  j
+
+               1         u(x ,y ,ZS)u(x ,y ,ZF)
+                            i  j      i  j
+
+               2         u(x ,y ,ZS)f(x ,y ,ZF)  i=1,2,...,L+1
+                            i  j      i  j
+
+               3         f(x ,y ,ZS)f(x ,y ,ZF)  j=1,2,...,M+1
+                            i  j      i  j
+
+               4         f(x ,y ,ZS)u(x ,y ,ZF)
+                            i  j      i  j
+               Note: if the table calls for both the solution u and the
+               right-hand side f on a boundary, then the solution must be
+               specified. On exit: F contains the solution u(i,j,k) of the
+               finite difference approximation for the grid point (
+               x ,y ,z ) for i=1,2,...,L+1, j=1,2,...,M+1 and k=1,2,...,N+1.
+                i  j  k
+
+          23:  PERTRB -- DOUBLE PRECISION                            Output
+               On exit: PERTRB = 0, unless a solution to Poisson's
+               equation ((lambda)=0) is required with a combination of
+               periodic or derivative boundary conditions (LBDCND, MBDCND
+               and NBDCND = 0 or 3). In this case a solution may not exist.
+               PERTRB is a constant, calculated and subtracted from the
+               array F, which ensures that a solution exists. D03FAF then
+               computes this solution, which is a least-squares solution to
+               the original approximation. This solution is not unique and
+               is unnormalised. The value of PERTRB should be small
+               compared to the right-hand side F, otherwise a solution has
+               been obtained to an essentially different problem. This
+               comparison should always be made to insure that a meaningful
+               solution has been obtained.
+
+          24:  W(LWRK) -- DOUBLE PRECISION array                  Workspace
+
+          25:  LWRK -- INTEGER                                        Input
+               On entry:
+               the dimension of the array W as declared in the (sub)program
+               from which D03FAF is called.
+               LRWK>=2*(N+1)*max(L,M)+3*L+3*M+4*N+6 is an upper bound on
+               the required size of W. If LWRK is too small, the routine
+               exits with IFAIL = 2, and if on entry IFAIL = 0 or IFAIL = -
+               1, a message is output giving the exact value of LWRK
+               required to solve the current problem.
+
+          26:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry XS >= XF,
+
+               or       L < 5,
+
+               or       LBDCND < 0,
+
+               or       LBDCND > 4,
+
+               or       YS >= YF,
+
+               or       M < 5,
+
+               or       MBDCND < 0,
+
+               or       MBDCND > 4,
+
+               or       ZS >= ZF,
+
+               or       N < 5,
+
+               or       NBDCND < 0,
+
+               or       NBDCND > 4,
+
+               or       LDIMF < L + 1 > 0,
+
+               or       MDIMF < M + 1.
+
+          IFAIL= 2
+               On entry LWRK is too small.
+
+          IFAIL= 3
+               On entry (lambda) > 0.
+
+          7. Accuracy
+
+          None.
+
+          8. Further Comments
+
+          The execution time is roughly proportional to
+          L*M*N*(log L+log M+5), but also depends on input parameters
+                    2     2
+          LBDCND and MBDCND.
+
+          9. Example
+
+          The example solves the Helmholz equation
+
+                           2     2     2
+                         dd u  dd u  dd u
+                         ----+ ----+ ----+(lambda)u=f(x,y,z)
+                            2     2     2
+                         ddx   ddy   ddz
+
+                                               (pi)
+          for (x,y,z)is in [0,1]*[0,2(pi)]*[0, ----] where (lambda)=-2, and
+                                                2
+          f(x,y,z) is derived from the exact solution
+
+                                          4
+                                u(x,y,z)=x sin(y)cos(z).
+
+          The equation is subject to the following boundary conditions,
+          again derived from the exact solution given above.
+
+               u(0,y,z) and u(1,y,z) are prescribed (i.e., LBDCND = 1).
+
+               u(x,0,z)=u(x,2(pi),z) (i.e., MBDCND = 0).
+
+                                    (pi)
+               u(x,y,0) and u (x,y, ----) are prescribed (i.e. NBDCND = 2).
+                             x       2
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
diff --git a/src/hyper/pages/nage.ht b/src/hyper/pages/nage.ht
new file mode 100644
index 00000000..39807e03
--- /dev/null
+++ b/src/hyper/pages/nage.ht
@@ -0,0 +1,16396 @@
+\begin{page}{manpageXXe01}{NAG On-line Documentation: e01}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E01(3NAG)         Foundation Library (12/10/92)         E01(3NAG)
+
+
+
+          E01 -- Interpolation                          Introduction -- E01
+                                    Chapter E01
+                                   Interpolation
+
+          1. Scope of the Chapter
+
+          This chapter is concerned with the interpolation of a function of
+          one or two variables. When provided with the value of the
+          function (and possibly one or more of its lowest-order
+          derivatives) at each of a number of values of the variable(s),
+          the routines provide either an interpolating function or an
+          interpolated value. For some of the interpolating functions,
+          there are supporting routines to evaluate, differentiate or
+          integrate them.
+
+          2. Background to the Problems
+
+          In motivation and in some of its numerical processes, this
+          chapter has much in common with Chapter E02 (Curve and Surface
+          Fitting). For this reason, we shall adopt the same terminology
+          and refer to dependent variable and independent variable(s)
+          instead of function and variable(s). Where there is only one
+          independent variable, we shall denote it by x and the dependent
+          variable by y. Thus, in the basic problem considered in this
+          chapter, we are given a set of distinct values x ,x ,...,x  of x
+                                                          1  2      m
+          and a corresponding set of values y ,y ,...,y  of y, and we shall
+                                             1  2      m
+          describe the problem as being one of interpolating the data
+          points (x ,y ), rather than interpolating a function. In modern
+                   r  r
+          usage, however, interpolation can have either of two rather
+          different meanings, both relevant to routines in this chapter.
+          They are
+
+          (a) the determination of a function of x which takes the value y
+                                                                          r
+              at x=x , for r=1,2,...,m (an interpolating function or
+                    r
+              interpolant),
+
+          (b) the determination of the value (interpolated value or
+              interpolate) of an interpolating function at any given value,
+                  ^
+              say x, of x within the range of the x  (so as to estimate the
+                                                   r
+                       ^
+              value at x of the function underlying the data).
+
+          The latter is the older meaning, associated particularly with the
+          use of mathematical tables. The term 'function underlying the
+          data', like the other terminology described above, is used so as
+          to cover situations additional to those in which the data points
+          have been computed from a known function, as with a mathematical
+          table. In some contexts, the function may be unknown, perhaps
+          representing the dependency of one physical variable on another,
+          say temperature upon time.
+
+          Whether the underlying function is known or unknown, the object
+          of interpolation will usually be to approximate it to acceptable
+          accuracy by a function which is easy to evaluate anywhere in some
+          range of interest. Piecewise polynomials such as cubic splines
+          (see Section 2.2 of the E02 Chapter Introduction for definitions
+          of terms in this case), being easy to evaluate and also capable
+          of approximating a wide variety of functions, are the types of
+          function mostly used in this chapter as interpolating functions.
+
+          Piecewise polynomials also, to a large extent, avoid the well-
+          known problem of large unwanted fluctuations which can arise when
+          interpolating a data set with a simple polynomial. Fluctuations
+          can still arise but much less frequently and much less severely
+          than with simple polynomials. Unwanted fluctuations are avoided
+          altogether by a routine using piecewise cubic polynomials having
+          only first derivative continuity. It is designed especially for
+          monotonic data, but for other data still provides an interpolant
+          which increases, or decreases, over the same intervals as the
+          data.
+
+          The concept of interpolation can be generalised in a number of
+          ways. For example, we may be required to estimate the value of
+                                             ^
+          the underlying function at a value x outside the range of the
+          data. This is the process of extrapolation. In general, it is a
+          good deal less accurate than interpolation and is to be avoided
+          whenever possible.
+
+          Interpolation can also be extended to the case of two independent
+          variables. We shall denote these by x and y, and the dependent
+          variable by f. Methods used depend markedly on whether or not the
+          data values of f are given at the intersections of a rectangular
+          mesh in the (x,y)-plane. If they are, bicubic splines (see
+          Section 2.3.2 of the E02 Chapter Introduction) are very suitable
+          and usually very effective for the problem. For other cases,
+          perhaps where the f values are quite arbitrarily scattered in the
+          (x,y)-plane, polynomials and splines are not at all appropriate
+          and special forms of interpolating function have to be employed.
+          Many such forms have been devised and two of the most successful
+          are in routines in this chapter. They both have continuity in
+          first, but not higher, derivatives.
+
+          2.1. References
+
+          [1]   Froberg C E (1970) Introduction to Numerical Analysis.
+                Addison-Wesley (2nd Edition).
+
+          [2]   Dahlquist G and Bjork A (1974) Numerical Methods. Prentice-
+                Hall.
+
+          3. Recommendations on Choice and Use of Routines
+
+          3.1. General
+
+          Before undertaking interpolation, in other than the simplest
+          cases, the user should seriously consider the alternative of
+          using a routine from Chapter E02 to approximate the data by a
+          polynomial or spline containing significantly fewer coefficients
+          than the corresponding interpolating function. This approach is
+          much less liable to produce unwanted fluctuations and so can
+          often provide a better approximation to the function underlying
+          the data.
+
+          When interpolation is employed to approximate either an
+          underlying function or its values, the user will need to be
+          satisfied that the accuracy of approximation achieved is
+          adequate. There may be a means for doing this which is particular
+          to the application, or the routine used may itself provide a
+          means. In other cases, one possibility is to repeat the
+          interpolation using one or more extra data points, if they are
+          available, or otherwise one or more fewer, and to compare the
+          results. Other possibilities, if it is an interpolating function
+          which is determined, are to examine the function graphically, if
+          that gives sufficient accuracy, or to observe the behaviour of
+          the differences in a finite-difference table, formed from
+          evaluations of the interpolating function at equally-spaced
+          values of x over the range of interest. The spacing should be
+          small enough to cause the typical size of the differences to
+          decrease as the order of difference increases.
+
+          3.2. One Independent Variable
+
+          E01BAF computes an interpolating cubic spline, using a particular
+          choice for the set of knots which has proved generally
+          satisfactory in practice. If the user wishes to choose a
+          different set, a cubic spline routine from Chapter E02, namely
+          E02BAF, may be used in its interpolating mode, setting NCAP7 = M+
+          4 and all elements of the parameter W to unity. These routines
+          provide the interpolating function in B-spline form (see Section
+          2.2.2 in the E02 Chapter Introduction). Routines for evaluating,
+          differentiating and integrating this form are discussed in
+          Section 3.7 of the E02 Chapter Introduction.
+
+          The cubic spline does not always avoid unwanted fluctuations,
+          especially when the data show a steep slope close to a region of
+          small slope, or when the data inadequately represent the
+          underlying curve. In such cases, E01BEF can be very useful. It
+          derives a piecewise cubic polynomial (with first derivative
+          continuity) which, between any adjacent pair of data points,
+          either increases all the way, or decreases all the way (or stays
+          constant). It is especially suited to data which are monotonic
+          over their whole range.
+
+          In this routine, the interpolating function is represented simply
+          by its value and first derivative at the data points. Supporting
+          routines compute its value and first derivative elsewhere, as
+          well as its definite integral over an arbitary interval.
+
+          3.3. Two Independent Variables
+
+          3.3.1.  Data on a rectangular mesh
+
+          Given the value f   of the dependent variable f at the point
+                           qr
+          (x ,y ) in the plane of the independent variables x and y, for
+            q  r
+          each q=1,2,...,m and r=1,2,...,n (so that the points (x ,y ) lie
+                                                                 q  r
+          at the m*n intersections of a rectangular mesh), E01DAF computes
+          an interpolating bicubic spline, using a particular choice for
+          each of the spline's knot-set. This choice, the same as in E01BAF
+          , has proved generally satisfactory in practice. If, instead, the
+          user wishes to specify his own knots, a routine from Chapter E02,
+          namely E02DAF, may be adapted (it is more cumbersome for the
+          purpose, however, and much slower for larger problems). Using m
+          and n in the above sense, the parameter M must be set to m*n, PX
+          and PY must be set to m+4 and n+4 respectively and all elements
+          of W should be set to unity. The recommended value for EPS is
+          zero.
+
+          3.3.2.  Arbitrary data
+
+          As remarked at the end of Section 2, special types of
+          interpolating are required for this problem, which can often be
+          difficult to solve satisfactorily. Two of the most successful are
+          employed in E01SAF and E01SEF, the two routines which (with their
+          respective evaluation routines E01SBF and E01SFF) are provided
+          for the problem. Definitions can be found in the routine
+          documents. Both interpolants have first derivative continuity and
+          are 'local', in that their value at any point depends only on
+          data in the immediate neighbourhood of the point. This latter
+          feature is necessary for large sets of data to avoid prohibitive
+          computing time.
+
+          The relative merits of the two methods vary with the data and it
+          is not possible to predict which will be the better in any
+          particular case.
+
+          3.4. Index
+
+          Derivative, of interpolant from E01BEF                     E01BGF
+
+          Evaluation, of interpolant
+               from E01BEF                                           E01BFF
+               from E01SAF                                           E01SBF
+               from E01SEF                                           E01SFF
+          Extrapolation, one variable                                E01BEF
+          Integration (definite) of interpolant from E01BEF          E01BHF
+          Interpolated values, one variable, from interpolant from   E01BFF
+          E01BEF
+                                                                     E01BGF
+          Interpolated values, two variables,
+               from interpolant from E01SAF                          E01SBF
+               from interpolant from E01SEF                          E01SFF
+          Interpolating function, one variable,
+               cubic spline                                          E01BAF
+               other piecewise polynomial                            E01BEF
+          Interpolating function, two variables
+               bicubic spline                                        E01DAF
+               other piecewise polynomial                            E01SAF
+               modified Shepard method                               E01SEF
+
+
+          E01 -- Interpolation                              Contents -- E01
+          Chapter E01
+
+          Interpolation
+
+          E01BAF  Interpolating functions, cubic spline interpolant, one
+                  variable
+
+          E01BEF  Interpolating functions, monotonicity-preserving,
+                  piecewise cubic Hermite, one variable
+
+          E01BFF  Interpolated values, interpolant computed by E01BEF,
+                  function only, one variable,
+
+          E01BGF  Interpolated values, interpolant computed by E01BEF,
+                  function and 1st derivative, one variable
+
+          E01BHF  Interpolated values, interpolant computed by E01BEF,
+                  definite integral, one variable
+
+          E01DAF  Interpolating functions, fitting bicubic spline, data on
+                  rectangular grid
+
+          E01SAF  Interpolating functions, method of Renka and Cline, two
+                  variables
+
+          E01SBF  Interpolated values, evaluate interpolant computed by
+                  E01SAF, two variables
+
+          E01SEF  Interpolating functions, modified Shepard's method, two
+                  variables
+
+          E01SFF  Interpolated values, evaluate interpolant computed by
+                  E01SEF, two variables
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe01baf}{NAG On-line Documentation: e01baf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E01BAF(3NAG)      Foundation Library (12/10/92)      E01BAF(3NAG)
+
+
+
+          E01 -- Interpolation                                       E01BAF
+                  E01BAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E01BAF determines a cubic spline interpolant to a given set of
+          data.
+
+          2. Specification
+
+                 SUBROUTINE E01BAF (M, X, Y, LAMDA, C, LCK, WRK, LWRK,
+                1                   IFAIL)
+                 INTEGER          M, LCK, LWRK, IFAIL
+                 DOUBLE PRECISION X(M), Y(M), LAMDA(LCK), C(LCK), WRK(LWRK)
+
+          3. Description
+
+          This routine determines a cubic spline s(x), defined in the range
+          x <=x<=x , which interpolates (passes exactly through) the set of
+           1      m
+          data points (x ,y ), for i=1,2,...,m, where m>=4 and x <x <...<x
+                        i  i                                    1  2      m
+          end conditions are not imposed. The spline interpolant chosen has
+          m-4 interior knots (lambda) ,(lambda) ,...,(lambda) , which are
+                                     5         6             m
+          set to the values of x ,x ,...,x    respectively. This spline is
+                                3  4      m-2
+          represented in its B-spline form (see Cox [1]):
+
+                                        m
+                                        --
+                                  s(x)= >  c N (x),
+                                        --  i i
+                                        i=1
+
+          where N (x) denotes the normalised B-Spline of degree 3, defined
+                 i
+          upon the knots (lambda) ,(lambda)   ,...,(lambda)   , and c
+                                 i         i+1             i+4       i
+          denotes its coefficient, whose value is to be determined by the
+          routine.
+
+          The use of B-splines requires eight additional knots (lambda) ,
+                                                                       1
+          (lambda) , (lambda) , (lambda) , (lambda)   , (lambda)   ,
+                  2          3          4          m+1          m+2
+          (lambda)    and (lambda)    to be specified; the routine sets the
+                  m+3             m+4
+          first four of these to x  and the last four to x .
+                                  1                       m
+
+          The algorithm for determining the coefficients is as described in
+          [1] except that QR factorization is used instead of LU
+          decomposition. The implementation of the algorithm involves
+          setting up appropriate information for the related routine E02BAF
+          followed by a call of that routine. (For further details of
+          E02BAF, see the routine document.)
+
+          Values of the spline interpolant, or of its derivatives or
+          definite integral, can subsequently be computed as detailed in
+          Section 8.
+
+          4. References
+
+          [1]   Cox M G (1975) An Algorithm for Spline Interpolation. J.
+                Inst. Math. Appl. 15 95--108.
+
+          [2]   Cox M G (1977) A Survey of Numerical Methods for Data and
+                Function Approximation. The State of the Art in Numerical
+                Analysis. (ed D A H Jacobs) Academic Press. 627--668.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: m, the number of data points. Constraint: M >= 4.
+
+           2:  X(M) -- DOUBLE PRECISION array                         Input
+               On entry: X(i) must be set to x , the ith data value of the
+                                              i
+               independent variable x, for i=1,2,...,m. Constraint: X(i) <
+               X(i+1), for i=1,2,...,M-1.
+
+           3:  Y(M) -- DOUBLE PRECISION array                         Input
+               On entry: Y(i) must be set to y , the ith data value of the
+                                              i
+               dependent variable y, for i=1,2,...,m.
+
+           4:  LAMDA(LCK) -- DOUBLE PRECISION array                  Output
+               On exit: the value of (lambda) , the ith knot, for
+                                             i
+               i=1,2,...,m+4.
+
+           5:  C(LCK) -- DOUBLE PRECISION array                      Output
+               On exit: the coefficient c  of the B-spline N (x), for
+                                         i                  i
+               i=1,2,...,m. The remaining elements of the array are not
+               used.
+
+           6:  LCK -- INTEGER                                         Input
+               On entry:
+               the dimension of the arrays LAMDA and C as declared in the
+               (sub)program from which E01BAF is called.
+               Constraint: LCK >= M + 4.
+
+           7:  WRK(LWRK) -- DOUBLE PRECISION array                Workspace
+
+           8:  LWRK -- INTEGER                                        Input
+               On entry:
+               the dimension of the array WRK as declared in the
+               (sub)program from which E01BAF is called.
+               Constraint: LWRK>=6*M+16.
+
+           9:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry M < 4,
+
+               or       LCK<M+4,
+
+               or       LWRK<6*M+16.
+
+          IFAIL= 2
+               The X-values fail to satisfy the condition
+
+               X(1) < X(2) < X(3) <... < X(M).
+
+          7. Accuracy
+
+          The rounding errors incurred are such that the computed spline is
+          an exact interpolant for a slightly perturbed set of ordinates
+          y +(delta)y . The ratio of the root-mean-square value of the
+           i         i
+          (delta)y  to that of the y  is no greater than a small multiple
+                  i                 i
+          of the relative machine precision.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to m.
+
+          All the x  are used as knot positions except x  and x   . This
+                   i                                    2      m-1
+          choice of knots (see Cox [2]) means that s(x) is composed of m-3
+          cubic arcs as follows. If m=4, there is just a single arc space
+          spanning the whole interval x  to x . If m>=5, the first and last
+                                       1     4
+          arcs span the intervals x  to x  and x    to x  respectively.
+                                   1     3      m-2     m
+          Additionally if m>=6, the ith arc, for i=2,3,...,m-4 spans the
+          interval x    to x   .
+                    i+1     i+2
+
+          After the call
+
+                 CALL E01BAF (M, X, Y, LAMDA, C, LCK, WRK, LWRK, IFAIL)
+
+          the following operations may be carried out on the interpolant
+          s(x).
+
+          The value of s(x) at x = XVAL can be provided in the real
+          variable SVAL by the call
+
+                 CALL E02BBF (M+4, LAMDA, C, XVAL, SVAL, IFAIL)
+
+          The values of s(x) and its first three derivatives at x = XVAL
+          can be provided in the real array SDIF of dimension 4, by the
+          call
+
+                 CALL E02BCF (M+4, LAMDA, C, XVAL, LEFT, SDIF, IFAIL)
+
+          Here LEFT must specify whether the left- or right-hand value of
+          the third derivative is required (see E02BCF for details).
+
+          The value of the integral of s(x) over the range x  to x  can be
+                                                            1     m
+          provided in the real variable SINT by
+
+                 CALL E02BDF (M+4, LAMDA, C, SINT, IFAIL)
+
+          9. Example
+
+          The example program sets up data from 7 values of the exponential
+          function in the interval 0 to 1. E01BAF is then called to compute
+          a spline interpolant to these data.
+
+          The spline is evaluated by E02BBF, at the data points and at
+          points halfway between each adjacent pair of data points, and the
+                                           x
+          spline values and the values of e  are printed out.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe01bef}{NAG On-line Documentation: e01bef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E01BEF(3NAG)      Foundation Library (12/10/92)      E01BEF(3NAG)
+
+
+
+          E01 -- Interpolation                                       E01BEF
+                  E01BEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E01BEF computes a monotonicity-preserving piecewise cubic Hermite
+          interpolant to a set of data points.
+
+          2. Specification
+
+                 SUBROUTINE E01BEF (N, X, F, D, IFAIL)
+                 INTEGER          N, IFAIL
+                 DOUBLE PRECISION X(N), F(N), D(N)
+
+          3. Description
+
+          This routine estimates first derivatives at the set of data
+          points (x ,f ), for r=1,2,...,n, which determine a piecewise
+                   r  r
+          cubic Hermite interpolant to the data, that preserves
+          monotonicity over ranges where the data points are monotonic. If
+          the data points are only piecewise monotonic, the interpolant
+          will have an extremum at each point where monotonicity switches
+          direction. The estimates of the derivatives are computed by a
+          formula due to Brodlie, which is described in Fritsch and Butland
+          [1], with suitable changes at the boundary points.
+
+          The routine is derived from routine PCHIM in Fritsch [2].
+
+          Values of the computed interpolant, and of its first derivative
+          and definite integral, can subsequently be computed by calling
+          E01BFF, E01BGF and E01BHF, as described in Section 8
+
+          4. References
+
+          [1]   Fritsch F N and Butland J (1984) A Method for Constructing
+                Local Monotone Piecewise Cubic Interpolants. SIAM J. Sci.
+                Statist. Comput. 5 300--304.
+
+          [2]   Fritsch F N (1982) PCHIP Final Specifications. Report UCID-
+                30194. Lawrence Livermore National Laboratory.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+               On entry: n, the number of data points. Constraint: N >= 2.
+
+           2:  X(N) -- DOUBLE PRECISION array                         Input
+               On entry: X(r) must be set to x , the rth value of the
+                                              r
+               independent variable (abscissa), for r=1,2,...,n.
+               Constraint: X(r) < X(r+1).
+
+           3:  F(N) -- DOUBLE PRECISION array                         Input
+               On entry: F(r) must be set to f , the rth value of the
+                                              r
+               dependent variable (ordinate), for r=1,2,...,n.
+
+           4:  D(N) -- DOUBLE PRECISION array                        Output
+               On exit: estimates of derivatives at the data points. D(r)
+               contains the derivative at X(r).
+
+           5:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry N < 2.
+
+          IFAIL= 2
+               The values of X(r), for r=1,2,...,N, are not in strictly
+               increasing order.
+
+          7. Accuracy
+
+          The computational errors in the array D should be negligible in
+          most practical situations.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to n.
+
+          The values of the computed interpolant at the points PX(i), for
+          i=1,2,...,M, may be obtained in the real array PF, of length at
+          least M, by the call:
+
+
+                  CALL E01BFF(N,X,F,D,M,PX,PF,IFAIL)
+
+          where N, X and F are the input parameters to E01BEF and D is the
+          output parameter from E01BEF.
+
+          The values of the computed interpolant at the points PX(i), for i
+          = 1,2,...,M, together with its first derivatives, may be obtained
+          in the real arrays PF and PD, both of length at least M, by the
+          call:
+
+                  CALL E01BGF(N,X,F,D,M,PX,PF,PD,IFAIL)
+
+          where N, X, F and D are as described above.
+
+          The value of the definite integral of the interpolant over the
+          interval A to B can be obtained in the real variable PINT by the
+          call:
+
+                  CALL E01BHF(N,X,F,D,A,B,PINT,IFAIL)
+
+          where N, X, F and D are as described above.
+
+          9. Example
+
+          This example program reads in a set of data points, calls E01BEF
+          to compute a piecewise monotonic interpolant, and then calls
+          E01BFF to evaluate the interpolant at equally spaced points.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe01bff}{NAG On-line Documentation: e01bff}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E01BFF(3NAG)      Foundation Library (12/10/92)      E01BFF(3NAG)
+
+
+
+          E01 -- Interpolation                                       E01BFF
+                  E01BFF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E01BFF evaluates a piecewise cubic Hermite interpolant at a set
+          of points.
+
+          2. Specification
+
+                 SUBROUTINE E01BFF (N, X, F, D, M, PX, PF, IFAIL)
+                 INTEGER          N, M, IFAIL
+                 DOUBLE PRECISION X(N), F(N), D(N), PX(M), PF(M)
+
+          3. Description
+
+          This routine evaluates a piecewise cubic Hermite interpolant, as
+          computed by E01BEF, at the points PX(i), for i=1,2,...,m. If any
+          point lies outside the interval from X(1) to X(N), a value is
+          extrapolated from the nearest extreme cubic, and a warning is
+          returned.
+
+          The routine is derived from routine PCHFE in Fritsch [1].
+
+          4. References
+
+          [1]   Fritsch F N (1982) PCHIP Final Specifications. Report UCID-
+                30194. Lawrence Livermore National Laboratory.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+
+           2:  X(N) -- DOUBLE PRECISION array                         Input
+
+           3:  F(N) -- DOUBLE PRECISION array                         Input
+
+           4:  D(N) -- DOUBLE PRECISION array                         Input
+               On entry: N, X, F and D must be unchanged from the previous
+               call of E01BEF.
+
+           5:  M -- INTEGER                                           Input
+               On entry: m, the number of points at which the interpolant
+               is to be evaluated. Constraint: M >= 1.
+
+           6:  PX(M) -- DOUBLE PRECISION array                        Input
+               On entry: the m values of x at which the interpolant is to
+               be evaluated.
+
+           7:  PF(M) -- DOUBLE PRECISION array                       Output
+               On exit: PF(i) contains the value of the interpolant
+               evaluated at the point PX(i), for i=1,2,...,m.
+
+           8:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry N < 2.
+
+          IFAIL= 2
+               The values of X(r), for r = 1,2,...,N, are not in strictly
+               increasing order.
+
+          IFAIL= 3
+               On entry M < 1.
+
+          IFAIL= 4
+               At least one of the points PX(i), for i = 1,2,...,M, lies
+               outside the interval [X(1),X(N)], and extrapolation was
+               performed at all such points. Values computed at such points
+               may be very unreliable.
+
+          7. Accuracy
+
+          The computational errors in the array PF should be negligible in
+          most practical situations.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          the number of evaluation points, m. The evaluation will be most
+          efficient if the elements of PX are in non-decreasing order (or,
+          more generally, if they are grouped in increasing order of the
+          intervals [X(r-1),X(r)]). A single call of E01BFF with m>1 is
+          more efficient than several calls with m=1.
+
+          9. Example
+
+          This example program reads in values of N, X, F and D, and then
+          calls E01BFF to evaluate the interpolant at equally spaced
+          points.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe01bgf}{NAG On-line Documentation: e01bgf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E01BGF(3NAG)      Foundation Library (12/10/92)      E01BGF(3NAG)
+
+
+
+          E01 -- Interpolation                                       E01BGF
+                  E01BGF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E01BGF evaluates a piecewise cubic Hermite interpolant and its
+          first derivative at a set of points.
+
+          2. Specification
+
+                 SUBROUTINE E01BGF (N, X, F, D, M, PX, PF, PD, IFAIL)
+                 INTEGER          N, M, IFAIL
+                 DOUBLE PRECISION X(N), F(N), D(N), PX(M), PF(M), PD(M)
+
+          3. Description
+
+          This routine evaluates a piecewise cubic Hermite interpolant, as
+          computed by E01BEF, at the points PX(i), for i=1,2,...,m. The
+          first derivatives at the points are also computed. If any point
+          lies outside the interval from X(1) to X(N), values of the
+          interpolant and its derivative are extrapolated from the nearest
+          extreme cubic, and a warning is returned.
+
+          If values of the interpolant only, and not of its derivative, are
+          required, E01BFF should be used.
+
+          The routine is derived from routine PCHFD in Fritsch [1].
+
+          4. References
+
+          [1]   Fritsch F N (1982) PCHIP Final Specifications. Report UCID-
+                30194. Lawrence Livermore National Laboratory.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+
+           2:  X(N) -- DOUBLE PRECISION array                         Input
+
+           3:  F(N) -- DOUBLE PRECISION array                         Input
+
+           4:  D(N) -- DOUBLE PRECISION array                         Input
+               On entry: N, X, F and D must be unchanged from the previous
+               call of E01BEF.
+
+           5:  M -- INTEGER                                           Input
+               On entry: m, the number of points at which the interpolant
+               is to be evaluated. Constraint: M >= 1.
+
+           6:  PX(M) -- DOUBLE PRECISION array                        Input
+               On entry: the m values of x at which the interpolant is to
+               be evaluated.
+
+           7:  PF(M) -- DOUBLE PRECISION array                       Output
+               On exit: PF(i) contains the value of the interpolant
+               evaluated at the point PX(i), for i=1,2,...,m.
+
+           8:  PD(M) -- DOUBLE PRECISION array                       Output
+               On exit: PD(i) contains the first derivative of the
+               interpolant evaluated at the point PX(i), for i=1,2,...,m.
+
+           9:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry N < 2.
+
+          IFAIL= 2
+               The values of X(r), for r = 1,2,...,N, are not in strictly
+               increasing order.
+
+          IFAIL= 3
+               On entry M < 1.
+
+          IFAIL= 4
+               At least one of the points PX(i), for i = 1,2,...,M, lies
+               outside the interval [X(1),X(N)], and extrapolation was
+               performed at all such points. Values computed at these
+               points may be very unreliable.
+
+          7. Accuracy
+
+          The computational errors in the arrays PF and PD should be
+          negligible in most practical situations.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          the number of evaluation points, m. The evaluation will be most
+          efficient if the elements of PX are in non-decreasing order (or,
+          more generally, if they are grouped in increasing order of the
+          intervals [X(r-1),X(r)]). A single call of E01BGF with m>1 is
+          more efficient than several calls with m=1.
+
+          9. Example
+
+          This example program reads in values of N, X, F and D, and calls
+          E01BGF to compute the values of the interpolant and its
+          derivative at equally spaced points.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe01bhf}{NAG On-line Documentation: e01bhf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E01BHF(3NAG)      Foundation Library (12/10/92)      E01BHF(3NAG)
+
+
+
+          E01 -- Interpolation                                       E01BHF
+                  E01BHF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E01BHF evaluates the definite integral of a piecewise cubic
+          Hermite interpolant over the interval [a,b].
+
+          2. Specification
+
+                 SUBROUTINE E01BHF (N, X, F, D, A, B, PINT, IFAIL)
+                 INTEGER          N, IFAIL
+                 DOUBLE PRECISION X(N), F(N), D(N), A, B, PINT
+
+          3. Description
+
+          This routine evaluates the definite integral of a piecewise cubic
+          Hermite interpolant, as computed by E01BEF, over the interval
+          [a,b].
+
+          If either a or b lies outside the interval from X(1) to X(N)
+          computation of the integral involves extrapolation and a warning
+          is returned.
+
+          The routine is derived from routine PCHIA in Fritsch [1].
+
+          4. References
+
+          [1]   Fritsch F N (1982) PCHIP Final Specifications. Report UCID-
+                30194. Lawrence Livermore National Laboratory .
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+
+           2:  X(N) -- DOUBLE PRECISION array                         Input
+
+           3:  F(N) -- DOUBLE PRECISION array                         Input
+
+           4:  D(N) -- DOUBLE PRECISION array                         Input
+               On entry: N, X, F and D must be unchanged from the previous
+               call of E01BEF.
+
+           5:  A -- DOUBLE PRECISION                                  Input
+
+           6:  B -- DOUBLE PRECISION                                  Input
+               On entry: the interval [a,b] over which integration is to
+               be performed.
+
+           7:  PINT -- DOUBLE PRECISION                              Output
+               On exit: the value of the definite integral of the
+               interpolant over the interval [a,b].
+
+           8:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry N < 2.
+
+          IFAIL= 2
+               The values of X(r), for r = 1,2,...,N, are not in strictly
+               increasing order.
+
+          IFAIL= 3
+               On entry at least one of A or B lies outside the interval [X
+               (1),X(N)], and extrapolation was performed to compute the
+               integral. The value returned is therefore unreliable.
+
+          7. Accuracy
+
+          The computational error in the value returned for PINT should be
+          negligible in most practical situations.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          the number of data points included within the interval [a,b].
+
+          9. Example
+
+          This example program reads in values of N, X, F and D. It then
+          reads in pairs of values for A and B, and evaluates the definite
+          integral of the interpolant over the interval [A,B] until end-of-
+          file is reached.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe01daf}{NAG On-line Documentation: e01daf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E01DAF(3NAG)      Foundation Library (12/10/92)      E01DAF(3NAG)
+
+
+
+          E01 -- Interpolation                                       E01DAF
+                  E01DAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E01DAF computes a bicubic spline interpolating surface through a
+          set of data values, given on a rectangular grid in the x-y plane.
+
+          2. Specification
+
+                 SUBROUTINE E01DAF (MX, MY, X, Y, F, PX, PY, LAMDA, MU, C,
+                1                   WRK, IFAIL)
+                 INTEGER          MX, MY, PX, PY, IFAIL
+                 DOUBLE PRECISION X(MX), Y(MY), F(MX*MY), LAMDA(MX+4), MU(MX
+                1                 +4), C(MX*MY), WRK((MX+6)*(MY+6))
+
+          3. Description
+
+          This routine determines a bicubic spline interpolant to the set
+          of data points (x ,y ,f   ), for q=1,2,...,m ; r=1,2,...,m . The
+                           q  r  q,r                  x             y
+          spline is given in the B-spline representation
+
+                                    m   m
+                                     x   y
+                                    --  --
+                            s(x,y)= >   >  c  M (x)N (y),
+                                    --  --  ij i    j
+                                    i=1 j=1
+
+          such that
+
+                                   s(x ,y )=f   ,
+                                      q  r   q,r
+
+          where M (x) and N (y) denote normalised cubic B-splines, the
+                 i         j
+          former defined on the knots (lambda)  to (lambda)    and the
+                                              i            i+4
+          latter on the knots (mu)  to (mu)   , and the c   are the spline
+                                  j        j+4           ij
+          coefficients. These knots, as well as the coefficients, are
+          determined by the routine, which is derived from the routine
+          B2IRE in Anthony et al[1]. The method used is described in
+          Section 8.2.
+
+          For further information on splines, see Hayes and Halliday [4]
+          for bicubic splines and de Boor [3] for normalised B-splines.
+
+          Values of the computed spline can subsequently be obtained by
+          calling E02DEF or E02DFF as described in Section 8.3.
+
+          4. References
+
+          [1]   Anthony G T, Cox M G and Hayes J G (1982) DASL - Data
+                Approximation Subroutine Library. National Physical
+                Laboratory.
+
+          [2]   Cox M G (1975) An Algorithm for Spline Interpolation. J.
+                Inst. Math. Appl. 15 95--108.
+
+          [3]   De Boor C (1972) On Calculating with B-splines. J. Approx.
+                Theory. 6 50--62.
+
+          [4]   Hayes J G and Halliday J (1974) The Least-squares Fitting of
+                Cubic Spline Surfaces to General Data Sets. J. Inst. Math.
+                Appl. 14 89--103.
+
+          5. Parameters
+
+           1:  MX -- INTEGER                                          Input
+
+           2:  MY -- INTEGER                                          Input
+               On entry: MX and MY must specify m  and m  respectively,
+                                                 x      y
+               the number of points along the x and y axis that define the
+               rectangular grid. Constraint: MX >= 4 and MY >= 4.
+
+           3:  X(MX) -- DOUBLE PRECISION array                        Input
+
+           4:  Y(MY) -- DOUBLE PRECISION array                        Input
+               On entry: X(q) and Y(r) must contain x , for q=1,2,...,m ,
+                                                     q                 x
+               and y , for r=1,2,...,m , respectively. Constraints:
+                    r                 y
+                    X(q) < X(q+1), for q=1,2,...,m -1,
+                                                  x
+
+                    Y(r) < Y(r+1), for r=1,2,...,m -1.
+                                                  y
+
+           5:  F(MX*MY) -- DOUBLE PRECISION array                     Input
+               On entry: F(m *(q-1)+r) must contain f   , for q=1,2,...,m ;
+                            y                        q,r                 x
+               r=1,2,...,m .
+                          y
+
+           6:  PX -- INTEGER                                         Output
+
+           7:  PY -- INTEGER                                         Output
+               On exit: PX and PY contain m +4 and m +4, the total number
+                                           x        y
+               of knots of the computed spline with respect to the x and y
+               variables, respectively.
+
+           8:  LAMDA(MX+4) -- DOUBLE PRECISION array                 Output
+
+           9:  MU(MY+4) -- DOUBLE PRECISION array                    Output
+               On exit: LAMDA contains the complete set of knots (lambda)
+                                                                         i
+               associated with the x variable, i.e., the interior knots
+               LAMDA(5), LAMDA(6), ..., LAMDA(PX-4), as well as the
+               additional knots LAMDA(1) = LAMDA(2) = LAMDA(3) = LAMDA(4) =
+               X(1) and LAMDA(PX-3) = LAMDA(PX-2) = LAMDA(PX-1) = LAMDA(PX)
+               = X(MX) needed for the B-spline representation. MU contains
+               the corresponding complete set of knots (mu)  associated
+                                                           i
+               with the y variable.
+
+          10:  C(MX*MY) -- DOUBLE PRECISION array                    Output
+               On exit: the coefficients of the spline interpolant. C(
+               m *(i-1)+j) contains the coefficient c   described in
+                y                                    ij
+               Section 3.
+
+          11:  WRK((MX+6)*(MY+6)) -- DOUBLE PRECISION array       Workspace
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry MX < 4,
+
+               or       MY < 4.
+
+          IFAIL= 2
+               On entry either the values in the X array or the values in
+               the Y array are not in increasing order.
+
+          IFAIL= 3
+               A system of linear equations defining the B-spline
+               coefficients was singular; the problem is too ill-
+               conditioned to permit solution.
+
+          7. Accuracy
+
+          The main sources of rounding errors are in steps (2), (3), (6)
+          and (7) of the algorithm described in Section 8.2. It can be
+          shown (Cox [2]) that the matrix A  formed in step (2) has
+                                           x
+          elements differing relatively from their true values by at most a
+          small multiple of 3(epsilon), where (epsilon) is the machine
+          precision. A  is 'totally positive', and a linear system with
+                      x
+          such a coefficient matrix can be solved quite safely by
+          elimination without pivoting. Similar comments apply to steps (6)
+          and (7). Thus the complete process is numerically stable.
+
+          8. Further Comments
+
+          8.1. Timing
+
+          The time taken by this routine is approximately proportional to
+          m m .
+           x y
+
+          8.2. Outline of method used
+
+          The process of computing the spline consists of the following
+          steps:
+
+          (1)   choice of the interior x-knots (lambda) , (lambda) ,...,
+                                                       5          6
+                (lambda)   as (lambda) =x   , for i=5,6,...,m ,
+                        m             i  i-2                 x
+                         x
+
+          (2)   formation of the system
+                                          A E=F,
+                                           x
+                where A  is a band matrix of order m  and bandwidth 4,
+                       x                            x
+                containing in its qth row the values at x  of the B-splines
+                                                         q
+                in x, F is the m  by m  rectangular matrix of values f   ,
+                                x     y                               q,r
+                and E denotes an m  by m  rectangular matrix of
+                                  x     y
+                intermediate coefficients,
+
+          (3)   use of Gaussian elimination to reduce this system to band
+                triangular form,
+
+          (4)   solution of this triangular system for E,
+
+          (5)   choice of the interior y knots (mu) , (mu) ,...,(mu)   as
+                                                   5      6         m
+                                                                     y
+                (mu) =y   , for i=5,6,...,m ,
+                    i  i-2                 y
+
+          (6)   formation of the system
+                                            T  T
+                                         A C =E ,
+                                          y
+                where A  is the counterpart of A  for the y variable, and C
+                       y                        x
+                denotes the m  by m  rectangular matrix of values of c  ,
+                             x     y                                  ij
+
+          (7)   use of Gaussian elimination to reduce this system to band
+                triangular form,
+
+                                                        T
+          (8)   solution of this triangular system for C  and hence C.
+
+          For computational convenience, steps (2) and (3), and likewise
+          steps (6) and (7), are combined so that the formation of A  and
+                                                                    x
+          A  and the reductions to triangular form are carried out one row
+           y
+          at a time.
+
+          8.3. Evaluation of Computed Spline
+
+          The values of the computed spline at the points (TX(r),TY(r)),
+          for r = 1,2,...,N, may be obtained in the double precision array
+          FF, of length at least N, by the following call:
+
+
+                IFAIL = 0
+                CALL E02DEF(N,PX,PY,TX,TY,LAMDA,MU,C,FF,WRK,IWRK,IFAIL)
+
+          where PX, PY, LAMDA, MU and C are the output parameters of E01DAF
+          , WRK is a double precision workspace array of length at least
+          PY-4, and IWRK is an integer workspace array of length at least
+          PY-4.
+
+          To evaluate the computed spline on an NX by NY rectangular grid
+          of points in the x-y plane, which is defined by the x co-
+          ordinates stored in TX(q), for q = 1,2,...,NX, and the y co-
+          ordinates stored in TY(r), for r = 1,2,...,NY, returning the
+          results in the double precision array FG which is of length at
+          least NX*NY, the following call may be used:
+
+
+                 IFAIL = 0
+                 CALL E02DFF(NX,NY,PX,PY,TX,TY,LAMDA,MU,C,FG,WRK,LWRK,
+                *            IWRK,LIWRK,IFAIL)
+
+          where PX, PY, LAMDA, MU and C are the output parameters of E01DAF
+          , WRK is a double precision workspace array of length at least
+          LWRK = min(NWRK1,NWRK2), NWRK1 = NX*4+PX, NWRK2 = NY*4+PY, and
+          IWRK is an integer workspace array of length at least LIWRK = NY
+          + PY - 4 if NWRK1 > NWRK2, or NX + PX - 4 otherwise. The result
+          of the spline evaluated at grid point (q,r) is returned in
+          element (NY*(q-1)+r) of the array FG.
+
+          9. Example
+
+          This program reads in values of m , x  for q=1,2,...,m , m  and
+                                           x   q                x   y
+          y  for r=1,2,...,m , followed by values of the ordinates f
+           r                y                                       q,r
+          defined at the grid points (x ,y ). It then calls E01DAF to
+                                       q  r
+          compute a bicubic spline interpolant of the data values, and
+          prints the values of the knots and B-spline coefficients. Finally
+          it evaluates the spline at a small sample of points on a
+          rectangular grid.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe01saf}{NAG On-line Documentation: e01saf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E01SAF(3NAG)      Foundation Library (12/10/92)      E01SAF(3NAG)
+
+
+
+          E01 -- Interpolation                                       E01SAF
+                  E01SAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E01SAF generates a two-dimensional surface interpolating a set of
+          scattered data points, using the method of Renka and Cline.
+
+          2. Specification
+
+                 SUBROUTINE E01SAF (M, X, Y, F, TRIANG, GRADS, IFAIL)
+                 INTEGER          M, TRIANG(7*M), IFAIL
+                 DOUBLE PRECISION X(M), Y(M), F(M), GRADS(2,M)
+
+          3. Description
+
+          This routine constructs an interpolating surface F(x,y) through a
+          set of m scattered data points (x ,y ,f ), for r=1,2,...,m, using
+                                           r  r  r
+          a method due to Renka and Cline. In the (x,y) plane, the data
+          points must be distinct. The constructed surface is continuous
+          and has continuous first derivatives.
+
+          The method involves firstly creating a triangulation with all the
+          (x,y) data points as nodes, the triangulation being as nearly
+          equiangular as possible (see Cline and Renka [1]). Then gradients
+          in the x- and y-directions are estimated at node r, for
+          r=1,2,...,m, as the partial derivatives of a quadratic function
+          of x and y which interpolates the data value f , and which fits
+                                                        r
+          the data values at nearby nodes (those within a certain distance
+          chosen by the algorithm) in a weighted least-squares sense. The
+          weights are chosen such that closer nodes have more influence
+          than more distant nodes on derivative estimates at node r. The
+          computed partial derivatives, with the f  values, at the three
+                                                  r
+          nodes of each triangle define a piecewise polynomial surface of a
+          certain form which is the interpolant on that triangle. See Renka
+          and Cline [4] for more detailed information on the algorithm, a
+          development of that by Lawson [2]. The code is derived from Renka
+          [3].
+
+          The interpolant F(x,y) can subsequently be evaluated at any point
+          (x,y) inside or outside the domain of the data by a call to
+          E01SBF. Points outside the domain are evaluated by extrapolation.
+
+          4. References
+
+          [1]   Cline A K and Renka R L (1984) A Storage-efficient Method
+                for Construction of a Thiessen Triangulation. Rocky Mountain
+                J. Math. 14 119--139.
+
+                                                1
+          [2]   Lawson C L (1977) Software for C  Surface Interpolation.
+                Mathematical Software III. (ed J R Rice) Academic Press.
+                161--194.
+
+          [3]   Renka R L (1984) Algorithm 624: Triangulation and
+                Interpolation of Arbitrarily Distributed Points in the
+                Plane. ACM Trans. Math. Softw. 10 440--442.
+
+                                                                 1
+          [4]   Renka R L and Cline A K (1984) A Triangle-based C
+                Interpolation Method. Rocky Mountain J. Math. 14 223--237.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: m, the number of data points. Constraint: M >= 3.
+
+           2:  X(M) -- DOUBLE PRECISION array                         Input
+
+           3:  Y(M) -- DOUBLE PRECISION array                         Input
+
+           4:  F(M) -- DOUBLE PRECISION array                         Input
+               On entry: the co-ordinates of the rth data point, for
+               r=1,2,...,m. The data points are accepted in any order, but
+               see Section 8. Constraint: The (x,y) nodes must not all be
+               collinear, and each node must be unique.
+
+           5:  TRIANG(7*M) -- INTEGER array                          Output
+               On exit: a data structure defining the computed
+               triangulation, in a form suitable for passing to E01SBF.
+
+           6:  GRADS(2,M) -- DOUBLE PRECISION array                  Output
+               On exit: the estimated partial derivatives at the nodes, in
+               a form suitable for passing to E01SBF. The derivatives at
+               node r with respect to x and y are contained in GRADS(1,r)
+               and GRADS(2,r) respectively, for r=1,2,...,m.
+
+           7:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M < 3.
+
+          IFAIL= 2
+               On entry all the (X,Y) pairs are collinear.
+
+          IFAIL= 3
+               On entry (X(i),Y(i)) = (X(j),Y(j)) for some i/=j.
+
+          7. Accuracy
+
+          On successful exit, the computational errors should be negligible
+          in most situations but the user should always check the computed
+          surface for acceptability, by drawing contours for instance. The
+          surface always interpolates the input data exactly.
+
+          8. Further Comments
+
+          The time taken for a call of E01SAF is approximately proportional
+          to the number of data points, m. The routine is more efficient
+          if, before entry, the values in X, Y, F are arranged so that the
+          X array is in ascending order.
+
+          9. Example
+
+          This program reads in a set of 30 data points and calls E01SAF to
+          construct an interpolating surface. It then calls E01SBF to
+          evaluate the interpolant at a sample of points on a rectangular
+          grid.
+
+          Note that this example is not typical of a realistic problem: the
+          number of data points would normally be larger, and the
+          interpolant would need to be evaluated on a finer grid to obtain
+          an accurate plot, say.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe01sbf}{NAG On-line Documentation: e01sbf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E01SBF(3NAG)      Foundation Library (12/10/92)      E01SBF(3NAG)
+
+
+
+          E01 -- Interpolation                                       E01SBF
+                  E01SBF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E01SBF evaluates at a given point the two-dimensional interpolant
+          function computed by E01SAF.
+
+          2. Specification
+
+                 SUBROUTINE E01SBF (M, X, Y, F, TRIANG, GRADS, PX, PY, PF,
+                1                   IFAIL)
+                 INTEGER          M, TRIANG(7*M), IFAIL
+                 DOUBLE PRECISION X(M), Y(M), F(M), GRADS(2,M), PX, PY, PF
+
+          3. Description
+
+          This routine takes as input the parameters defining the
+          interpolant F(x,y) of a set of scattered data points (x ,y ,f ),
+                                                                 r  r  r
+          for r=1,2,...,m, as computed by E01SAF, and evaluates the
+          interpolant at the point (px,py).
+
+          If (px,py) is equal to (x ,y ) for some value of r, the returned
+                                   r  r
+          value will be equal to f .
+                                  r
+
+          If (px,py) is not equal to (x ,y ) for any r, the derivatives in
+                                       r  r
+          GRADS will be used to compute the interpolant. A triangle is
+          sought which contains the point (px,py), and the vertices of the
+          triangle along with the partial derivatives and f  values at the
+                                                           r
+          vertices are used to compute the value F(px,py). If the point
+          (px,py) lies outside the triangulation defined by the input
+          parameters, the returned value is obtained by extrapolation. In
+          this case, the interpolating function F is extended linearly
+          beyond the triangulation boundary. The method is described in
+          more detail in Renka and Cline [2] and the code is derived from
+          Renka [1].
+
+          E01SBF must only be called after a call to E01SAF.
+
+          4. References
+
+          [1]   Renka R L (1984) Algorithm 624: Triangulation and
+                Interpolation of Arbitrarily Distributed Points in the
+                Plane. ACM Trans. Math. Softw. 10 440--442.
+
+                                                                 1
+          [2]   Renka R L and Cline A K (1984) A Triangle-based C
+                Interpolation Method. Rocky Mountain J. Math. 14 223--237.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+
+           2:  X(M) -- DOUBLE PRECISION array                         Input
+
+           3:  Y(M) -- DOUBLE PRECISION array                         Input
+
+           4:  F(M) -- DOUBLE PRECISION array                         Input
+
+           5:  TRIANG(7*M) -- INTEGER array                           Input
+
+           6:  GRADS(2,M) -- DOUBLE PRECISION array                   Input
+               On entry: M, X, Y, F, TRIANG and GRADS must be unchanged
+               from the previous call of E01SAF.
+
+           7:  PX -- DOUBLE PRECISION                                 Input
+
+           8:  PY -- DOUBLE PRECISION                                 Input
+               On entry: the point (px,py) at which the interpolant is to
+               be evaluated.
+
+           9:  PF -- DOUBLE PRECISION                                Output
+               On exit: the value of the interpolant evaluated at the
+               point (px,py).
+
+          10:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M < 3.
+
+          IFAIL= 2
+               On entry the triangulation information held in the array
+               TRIANG does not specify a valid triangulation of the data
+               points. TRIANG may have been corrupted since the call to
+               E01SAF.
+
+          IFAIL= 3
+               The evaluation point (PX,PY) lies outside the nodal
+               triangulation, and the value returned in PF is computed by
+               extrapolation.
+
+          7. Accuracy
+
+          Computational errors should be negligible in most practical
+          situations.
+
+          8. Further Comments
+
+          The time taken for a call of E01SBF is approximately proportional
+          to the number of data points, m.
+
+          The results returned by this routine are particularly suitable
+          for applications such as graph plotting, producing a smooth
+          surface from a number of scattered points.
+
+          9. Example
+
+          See the example for E01SAF.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe01sef}{NAG On-line Documentation: e01sef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E01SEF(3NAG)      Foundation Library (12/10/92)      E01SEF(3NAG)
+
+
+
+          E01 -- Interpolation                                       E01SEF
+                  E01SEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E01SEF generates a two-dimensional surface interpolating a set of
+          scattered data points, using a modified Shepard method.
+
+          2. Specification
+
+                 SUBROUTINE E01SEF (M, X, Y, F, RNW, RNQ, NW, NQ, FNODES,
+                1                   MINNQ, WRK, IFAIL)
+                 INTEGER          M, NW, NQ, MINNQ, IFAIL
+                 DOUBLE PRECISION X(M), Y(M), F(M), RNW, RNQ, FNODES(5*M),
+                1                 WRK(6*M)
+
+          3. Description
+
+          This routine constructs an interpolating surface F(x,y) through a
+          set of m scattered data points (x ,y ,f ), for r=1,2,...,m, using
+                                           r  r  r
+          a modification of Shepard's method. The surface is continuous and
+          has continuous first derivatives.
+
+          The basic Shepard method, described in [2], interpolates the
+          input data with the weighted mean
+
+                                        m
+                                        --
+                                        >  w (x,y)f
+                                        --  r      r
+                                        r=1
+                               F(x,y)= -------------,
+                                         m
+                                         --
+                                         >  w (x,y)
+                                         --  r
+                                         r=1
+
+                                    1         2       2       2
+                     where w (x,y)= --  and  d =(x-x ) +(y-y ) .
+                            r        2        r     r       r
+                                    d
+                                     r
+
+          The basic method is global in that the interpolated value at any
+          point depends on all the data, but this routine uses a
+          modification due to Franke and Nielson described in [1], whereby
+          the method becomes local by adjusting each w (x,y) to be zero
+                                                      r
+          outside a circle with centre (x ,y ) and some radius R . Also, to
+                                         r  r                   w
+          improve the performance of the basic method, each f  above is
+                                                             r
+          replaced by a function f (x,y), which is a quadratic fitted by
+                                  r
+          weighted least-squares to data local to (x ,y ) and forced to
+                                                    r  r
+          interpolate (x ,y ,f ). In this context, a point (x,y) is defined
+                        r  r  r
+          to be local to another point if it lies within some distance R
+                                                                        q
+          of it. Computation of these quadratics constitutes the main work
+          done by this routine. If there are less than 5 other points
+          within distance R  from (x ,y ), the quadratic is replaced by a
+                           q        r  r
+          linear function. In cases of rank-deficiency, the minimum norm
+          solution is computed.
+
+          The user may specify values for R  and R , but it is usually
+                                           w      q
+          easier to choose instead two integers N  and N , from which the
+                                                 w      q
+          routine will compute R  and R . These integers can be thought of
+                                w      q
+          as the average numbers of data points lying within distances R
+                                                                        w
+          and R  respectively from each node. Default values are provided,
+               q
+          and advice on alternatives is given in Section 8.2.
+
+          The interpolant F(x,y) generated by this routine can subsequently
+          be evaluated for any point (x,y) in the domain of the data by a
+          call to E01SFF.
+
+          4. References
+
+          [1]   Franke R and Nielson G (1980) Smooth Interpolation of Large
+                Sets of Scattered Data. Internat. J. Num. Methods Engrg. 15
+                1691--1704.
+
+          [2]   Shepard D (1968) A Two-dimensional Interpolation Function
+                for Irregularly Spaced Data. Proc. 23rd Nat. Conf. ACM.
+                Brandon/Systems Press Inc., Princeton. 517--523.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: m, the number of data points. Constraint: M >= 3.
+
+           2:  X(M) -- DOUBLE PRECISION array                         Input
+
+           3:  Y(M) -- DOUBLE PRECISION array                         Input
+
+           4:  F(M) -- DOUBLE PRECISION array                         Input
+               On entry: the co-ordinates of the rth data point, for
+               r=1,2,...,m. The order of the data points is immaterial.
+               Constraint: each of the (X(r),Y(r)) pairs must be unique.
+
+           5:  RNW -- DOUBLE PRECISION                         Input/Output
+
+           6:  RNQ -- DOUBLE PRECISION                         Input/Output
+               On entry: suitable values for the radii R  and R ,
+                                                         w      q
+               described in Section 3. Constraint: RNQ <= 0 or 0 < RNW <=
+               RNQ. On exit: if RNQ is set less than or equal to zero on
+               entry, then default values for both of them will be computed
+               from the parameters NW and NQ, and RNW and RNQ will contain
+               these values on exit.
+
+           7:  NW -- INTEGER                                          Input
+
+           8:  NQ -- INTEGER                                          Input
+               On entry: if RNQ > 0.0 and RNW > 0.0 then NW and NQ are not
+               referenced by the routine. Otherwise, NW and NQ must specify
+               suitable values for the integers N  and N  described in
+                                                 w      q
+               Section 3.
+
+               If NQ is less than or equal to zero on entry, then default
+               values for both of them, namely NW = 9 and NQ = 18, will be
+               used. Constraint: NQ <= 0 or 0 < NW <= NQ.
+
+           9:  FNODES(5*M) -- DOUBLE PRECISION array                 Output
+               On exit: the coefficients of the constructed quadratic
+               nodal functions. These are in a form suitable for passing to
+               E01SFF.
+
+          10:  MINNQ -- INTEGER                                      Output
+               On exit: the minimum number of data points that lie within
+               radius RNQ of any node, and thus define a nodal function. If
+               MINNQ is very small (say, less than 5), then the interpolant
+               may be unsatisfactory in regions where the data points are
+               sparse.
+
+          11:  WRK(6*M) -- DOUBLE PRECISION array                 Workspace
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M < 3.
+
+          IFAIL= 2
+               On entry RNQ > 0 and either RNW > RNQ or RNW <= 0.
+
+          IFAIL= 3
+               On entry NQ > 0 and either NW > NQ or NW <= 0.
+
+          IFAIL= 4
+               On entry (X(i),Y(i)) is equal to (X(j),Y(j)) for some i/=j.
+
+          7. Accuracy
+
+          On successful exit, the computational errors should be negligible
+          in most situations but the user should always check the computed
+          surface for acceptability, by drawing contours for instance. The
+          surface always interpolates the input data exactly.
+
+          8. Further Comments
+
+          8.1. Timing
+
+          The time taken for a call of E01SEF is approximately proportional
+          to the number of data points, m, provided that N  is of the same
+                                                          q
+          order as its default value (18). However if N  is increased so
+                                                       q
+          that the method becomes more global, the time taken becomes
+                                         2
+          approximately proportional to m .
+                            m N} {{\rm w}}$ and ${\rm N} {{\rm q}}$
+          8.2. Choice of ${\
+          Note first that the radii R  and R , described in Section 3, are
+                                     w      q
+                                            
+
+                            / N            / N
+                       D   /   w      D   /   q
+          computed as  -  /   -- and  -  /   -- respectively, where D is
+                       2\/    m       2/    m
+          the maximum distance between any pair of data points.
+          Default values N =9 and N =18 work quite well when the data
+                          w        q
+          points are fairly uniformly distributed. However, for data having
+          some regions with relatively few points or for small data sets
+          (m<25), a larger value of N  may be needed. This is to ensure a
+                                    w
+          reasonable number of data points within a distance R  of each
+                                                              w
+          node, and to avoid some regions in the data area being left
+          outside all the discs of radius R  on which the weights w (x,y)
+                                           w                       r
+          are non-zero. Maintaining N  approximately equal to 2N  is
+                                     q                          w
+          usually an advantage.
+
+          Note however that increasing N  and N  does not improve the
+                                        w      q
+          quality of the interpolant in all cases. It does increase the
+          computational cost and makes the method less local.
+
+          9. Example
+
+          This program reads in a set of 30 data points and calls E01SEF to
+          construct an interpolating surface. It then calls E01SFF to
+          evaluate the interpolant at a sample of points on a rectangular
+          grid.
+
+          Note that this example is not typical of a realistic problem: the
+          number of data points would normally be larger, and the
+          interpolant would need to be evaluated on a finer grid to obtain
+          an accurate plot, say.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe01sff}{NAG On-line Documentation: e01sff}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E01SFF(3NAG)      Foundation Library (12/10/92)      E01SFF(3NAG)
+
+
+
+          E01 -- Interpolation                                       E01SFF
+                  E01SFF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E01SFF evaluates at a given point the two-dimensional
+          interpolating function computed by E01SEF.
+
+          2. Specification
+
+                 SUBROUTINE E01SFF (M, X, Y, F, RNW, FNODES, PX, PY, PF,
+                1                   IFAIL)
+                 INTEGER          M, IFAIL
+                 DOUBLE PRECISION X(M), Y(M), F(M), RNW, FNODES(5*M), PX,
+                1                 PY, PF
+
+          3. Description
+
+          This routine takes as input the interpolant F(x,y) of a set of
+          scattered data points (x ,y ,f ), for r=1,2,...,m, as computed by
+                                  r  r  r
+          E01SEF, and evaluates the interpolant at the point (px,py).
+
+          If (px,py) is equal to (x ,y ) for some value of r, the returned
+                                   r  r
+          value will be equal to f .
+                                  r
+
+          If (px,py) is not equal to (x ,y ) for any r, all points that are
+                                       r  r
+          within distance RNW of (px,py), along with the corresponding
+          nodal functions given by FNODES, will be used to compute a value
+          of the interpolant.
+
+          E01SFF must only be called after a call to E01SEF.
+
+          4. References
+
+          [1]   Franke R and Nielson G (1980) Smooth Interpolation of Large
+                Sets of Scattered Data. Internat. J. Num. Methods Engrg. 15
+                1691--1704.
+
+          [2]   Shepard D (1968) A Two-dimensional Interpolation Function
+                for Irregularly Spaced Data. Proc. 23rd Nat. Conf. ACM.
+                Brandon/Systems Press Inc., Princeton. 517--523.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+
+           2:  X(M) -- DOUBLE PRECISION array                         Input
+
+           3:  Y(M) -- DOUBLE PRECISION array                         Input
+
+           4:  F(M) -- DOUBLE PRECISION array                         Input
+
+           5:  RNW -- DOUBLE PRECISION                                Input
+
+           6:  FNODES(5*M) -- DOUBLE PRECISION array                  Input
+               On entry: M, X, Y, F, RNW and FNODES must be unchanged from
+               the previous call of E01SEF.
+
+           7:  PX -- DOUBLE PRECISION                                 Input
+
+           8:  PY -- DOUBLE PRECISION                                 Input
+               On entry: the point (px,py) at which the interpolant is to
+               be evaluated.
+
+           9:  PF -- DOUBLE PRECISION                                Output
+               On exit: the value of the interpolant evaluated at the
+               point (px,py).
+
+          10:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M < 3.
+
+          IFAIL= 2
+               The interpolant cannot be evaluated because the evaluation
+               point (PX,PY) lies outside the support region of the data
+               supplied in X, Y and F. This error exit will occur if
+               (PX,PY) lies at a distance greater than or equal to RNW from
+               every point given by arrays X and Y.
+
+               The value 0.0 is returned in PF. This value will not provide
+               continuity with values obtained at other points (PX,PY),
+               i.e., values obtained when IFAIL = 0 on exit.
+
+          7. Accuracy
+
+          Computational errors should be negligible in most practical
+          situations.
+
+          8. Further Comments
+
+          The time taken for a call of E01SFF is approximately proportional
+          to the number of data points, m.
+
+          The results returned by this routine are particularly suitable
+          for applications such as graph plotting, producing a smooth
+          surface from a number of scattered points.
+
+          9. Example
+
+          See the example for E01SEF.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02}{NAG On-line Documentation: e02}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02(3NAG)         Foundation Library (12/10/92)         E02(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting              Introduction -- E02
+                                    Chapter E02
+                             Curve and Surface Fitting
+
+          Contents of this Introduction:
+
+          1.     Scope of the Chapter
+
+          2.     Background to the Problems
+
+          2.1.   Preliminary Considerations
+
+          2.1.1. Fitting criteria: norms
+
+          2.1.2. Weighting of data points
+
+          2.2.   Curve Fitting
+
+          2.2.1. Representation of polynomials
+
+          2.2.2. Representation of cubic splines
+
+          2.3.   Surface Fitting
+
+          2.3.1. Bicubic splines: definition and representation
+
+          2.4.   General Linear and Nonlinear Fitting Functions
+
+          2.5.   Constrained Problems
+
+          2.6.   References
+
+          3.     Recommendations on Choice and Use of Routines
+
+          3.1.   General
+
+          3.1.1. Data considerations
+
+          3.1.2. Transformation of variables
+
+          3.2.   Polynomial Curves
+
+          3.2.1. Least-squares polynomials: arbitrary data points
+
+          3.2.2. Least-squares polynomials: selected data points
+
+          3.3.   Cubic Spline Curves
+
+          3.3.1. Least-squares cubic splines
+
+          3.3.2. Automatic fitting with cubic splines
+
+          3.4.   Spline Surfaces
+
+          3.4.1. Least-squares bicubic splines
+
+          3.4.2. Automatic fitting with bicubic splines
+
+          3.5.   General Linear and Nonlinear Fitting Functions
+
+          3.5.1. General linear functions
+
+          3.5.2. Nonlinear functions
+
+          3.6.   Constraints
+
+          3.7.   Evaluation, Differentiation and Integration
+
+          3.8.   Index
+
+
+
+          1. Scope of the Chapter
+
+          The main aim of this chapter is to assist the user in finding a
+          function which approximates a set of data points. Typically the
+          data contain random errors, as of experimental measurement, which
+          need to be smoothed out. To seek an approximation to the data, it
+          is first necessary to specify for the approximating function a
+          mathematical form (a polynomial, for example) which contains a
+          number of unspecified coefficients: the appropriate fitting
+          routine then derives for the coefficients the values which
+          provide the best fit of that particular form. The chapter deals
+          mainly with curve and surface fitting (i.e., fitting with
+          functions of one and of two variables) when a polynomial or a
+          cubic spline is used as the fitting function, since these cover
+          the most common needs. However, fitting with other functions
+          and/or more variables can be undertaken by means of general
+          linear or nonlinear routines (some of which are contained in
+          other chapters) depending on whether the coefficients in the
+          function occur linearly or nonlinearly. Cases where a graph
+          rather than a set of data points is given can be treated simply
+          by first reading a suitable set of points from the graph.
+
+          The chapter also contains routines for evaluating,
+          differentiating and integrating polynomial and spline curves and
+          surfaces, once the numerical values of their coefficients have
+          been determined.
+
+          2. Background to the Problems
+
+          2.1. Preliminary Considerations
+
+          In the curve-fitting problems considered in this chapter, we have
+          a dependent variable y and an independent variable x, and we are
+          given a set of data points (x ,y ), for r=1,2,...,m. The
+                                       r  r
+          preliminary matters to be considered in this section will, for
+          simplicity, be discussed in this context of curve-fitting
+          problems. In fact, however, these considerations apply equally
+          well to surface and higher-dimensional problems. Indeed, the
+          discussion presented carries over essentially as it stands if,
+          for these cases, we interpret x as a vector of several
+          independent variables and correspondingly each x  as a vector
+                                                          r
+          containing the rth data value of each independent variable.
+
+          We wish, then, to approximate the set of data points as closely
+          as possible with a specified function, f(x) say, which is as
+          smooth as possible -- f(x) may, for example, be a polynomial. The
+          requirements of smoothness and closeness conflict, however, and a
+          balance has to be struck between them. Most often, the smoothness
+          requirement is met simply by limiting the number of coefficients
+          allowed in the fitting function -- for example, by restricting
+          the degree in the case of a polynomial. Given a particular number
+          of coefficients in the function in question, the fitting routines
+          of this chapter determine the values of the coefficients such
+          that the 'distance' of the function from the data points is as
+          small as possible. The necessary balance is struck by the user
+          comparing a selection of such fits having different numbers of
+          coefficients. If the number of coefficients is too low, the
+          approximation to the data will be poor. If the number is too
+          high, the fit will be too close to the data, essentially
+          following the random errors and tending to have unwanted
+          fluctuations between the data points. Between these extremes,
+          there is often a group of fits all similarly close to the data
+          points and then, particularly when least-squares polynomials are
+          used, the choice is clear: it is the fit from this group having
+          the smallest number of coefficients.
+
+          The above process can be seen as the user minimizing the
+          smoothness measure (i.e., the number of coefficients) subject to
+          the distance from the data points being acceptably small. Some of
+          the routines, however, do this task themselves. They use a
+          different measure of smoothness (in each case one that is
+          continuous) and minimize it subject to the distance being less
+          than a threshold specified by the user. This is a much more
+          automatic process, requiring only some experimentation with the
+          threshold.
+
+          2.1.1.  Fitting criteria: norms
+
+          A measure of the above 'distance' between the set of data points
+          and the function f(x) is needed. The distance from a single data
+          point (x ,y ) to the function can simply be taken as
+                  r  r
+
+                              (epsilon) =y -f(x ),                      (1)
+                                       r  r    r
+
+          and is called the residual of the point. (With this definition,
+          the residual is regarded as a function of the coefficients
+          contained in f(x); however, the term is also used to mean the
+          particular value of (epsilon)  which corresponds to the fitted
+                                       r
+          values of the coefficients.) However, we need a measure of
+          distance for the set of data points as a whole. Three different
+          measures are used in the different routines (which measure to
+          select, according to circumstances, is discussed later in this
+          sub-section). With (epsilon)  defined in (1), these measures, or
+                                      r
+          norms, are
+
+                                m
+                                --
+                                >  |(epsilon) |,                        (2)
+                                --           r
+                                r=1
+
+                                 
+
+                                / m
+                               /  --          2
+                              /   >  (epsilon) ,  and                   (3)
+                             /    --          r
+                           \/     r=1
+
+                               max |(epsilon) |,                        (4)
+                                  r          r
+
+          respectively the l  norm, the l  norm and the l      norm.
+                            1            2               infty
+
+          Minimization of one or other of these norms usually provides the
+          fitting criterion, the minimization being carried out with
+          respect to the coefficients in the mathematical form used for
+          f(x): with respect to the b  for example if the mathematical form
+                                     i
+          is the power series in (8) below. The fit which results from
+          minimizing (2) is known as the l  fit, or the fit in the l  norm:
+                                          1                         1
+          that which results from minimizing (3) is the l  fit, the well-
+                                                         2
+          known least-squares fit (minimizing (3) is equivalent to
+          minimizing the square of (3), i.e., the sum of squares of
+          residuals, and it is the latter which is used in practice), and
+          that from minimizing (4) is the l     , or minimax, fit.
+                                           infty
+
+          Strictly speaking, implicit in the use of the above norms are the
+          statistical assumptions that the random errors in the y  are
+                                                                 r
+          independent of one another and that any errors in the x  are
+                                                                 r
+          negligible by comparison. From this point of view, the use of the
+          l  norm is appropriate when the random errors in the y  have a
+           2                                                    r
+          normal distribution, and the l      norm is appropriate when they
+                                        infty
+          have a rectangular distribution, as when fitting a table of
+          values rounded to a fixed number of decimal places. The l  norm
+                                                                   1
+          is appropriate when the error distribution has its frequency
+          function proportional to the negative exponential of the modulus
+          of the normalised error -- not a common situation.
+
+          However, the user is often indifferent to these statistical
+          considerations, and simply seeks a fit which he can assess by
+          inspection, perhaps visually from a graph of the results. In this
+          event, the l  norm is particularly appropriate when the data are
+                      1
+          thought to contain some 'wild' points (since fitting in this norm
+          tends to be unaffected by the presence of a small number of such
+          points), though of course in simple situations the user may
+          prefer to identify and reject these points. The l      norm
+                                                           infty
+          should be used only when the maximum residual is of particular
+          concern, as may be the case for example when the data values have
+          been obtained by accurate computation, as of a mathematical
+          function. Generally, however, a routine based on least-squares
+          should be preferred, as being computationally faster and usually
+          providing more information on which to assess the results. In
+          many problems the three fits will not differ significantly for
+          practical purposes.
+
+          Some of the routines based on the l  norm do not minimize the
+                                             2
+          norm itself but instead minimize some (intuitively acceptable)
+          measure of smoothness subject to the norm being less than a user-
+          specified threshold. These routines fit with cubic or bicubic
+          splines (see (10) and (14) below) and the smoothing measures
+          relate to the size of the discontinuities in their third
+          derivatives. A much more automatic fitting procedure follows from
+          this approach.
+
+          2.1.2.  Weighting of data points
+
+          The use of the above norms also assumes that the data values y
+                                                                        r
+          are of equal (absolute) accuracy. Some of the routines enable an
+          allowance to be made to take account of differing accuracies. The
+          allowance takes the form of 'weights' applied to the y-values so
+          that those values known to be more accurate have a greater
+          influence on the fit than others. These weights, to be supplied
+          by the user, should be calculated from estimates of the absolute
+          accuracies of the y-values, these estimates being expressed as
+          standard deviations, probable errors or some other measure which
+          has the same dimensions as y. Specifically, for each y  the
+                                                                r
+          corresponding weight w  should be inversely proportional to the
+                                r
+          accuracy estimate of y . For example, if the percentage accuracy
+                                r
+          is the same for all y , then the absolute accuracy of y  is
+                               r                                 r
+          proportional to y  (assuming y  to be positive, as it usually is
+                           r            r
+          in such cases) and so w =K/y , for r=1,2,...,m, for an arbitrary
+                                 r    r
+          positive constant K. (This definition of weight is stressed
+          because often weight is defined as the square of that used here.)
+          The norms (2), (3) and (4) above are then replaced respectively
+          by
+
+                               m
+                               --
+                               >  |w (epsilon) |,                       (5)
+                               --   r         r
+                               r=1
+
+                                
+
+                               / m
+                              /  --  2         2
+                             /   >  w (epsilon) ,  and                  (6)
+                            /    --  r         r
+                          \/     r=1
+
+                              max |w (epsilon) |.                       (7)
+                                 r  r         r
+
+          Again it is the square of (6) which is used in practice rather
+          than (6) itself.
+
+          2.2. Curve Fitting
+
+          When, as is commonly the case, the mathematical form of the
+          fitting function is immaterial to the problem, polynomials and
+          cubic splines are to be preferred because their simplicity and
+          ease of handling confer substantial benefits. The cubic spline is
+          the more versatile of the two. It consists of a number of cubic
+          polynomial segments joined end to end with continuity in first
+          and second derivatives at the joins. The third derivative at the
+          joins is in general discontinuous. The x-values of the joins are
+          called knots, or, more precisely, interior knots. Their number
+          determines the number of coefficients in the spline, just as the
+          degree determines the number of coefficients in a polynomial.
+
+          2.2.1.  Representation of polynomials
+
+          Rather than using the power-series form
+
+                                           2        k
+                           f(x)==b +b x+b x +...+b x                    (8)
+                                  0  1   2        k
+
+          to represent a polynomial, the routines in this chapter use the
+          Chebyshev series form
+
+                         1
+                  f(x)== -a T (x)+a T (x)+a T (x)+...+a T (x),          (9)
+                         2 0 0     1 1     2 2         k k
+
+          where T (x) is the Chebyshev polynomial of the first kind of
+                 i
+          degree i (see Cox and Hayes [1], page 9), and where the range of
+          x has been normalised to run from -1 to +1. The use of either
+          form leads theoretically to the same fitted polynomial, but in
+          practice results may differ substantially because of the effects
+          of rounding error. The Chebyshev form is to be preferred, since
+          it leads to much better accuracy in general, both in the
+          computation of the coefficients and in the subsequent evaluation
+          of the fitted polynomial at specified points. This form also has
+          other advantages: for example, since the later terms in (9)
+          generally decrease much more rapidly from left to right than do
+          those in (8), the situation is more often encountered where the
+          last terms are negligible and it is obvious that the degree of
+          the polynomial can be reduced (note that on the interval -1<=x<=1
+          for all i, T (x) attains the value unity but never exceeds it, so
+                      i
+          that the coefficient a  gives directly the maximum value of the
+                                i
+          term containing it).
+
+          2.2.2.  Representation of cubic splines
+
+          A cubic spline is represented in the form
+
+                      f(x)==c N (x)+c N (x)+...+c N (x),               (10)
+                             1 1     2 2         p p
+
+          where N (x), for i=1,2,...,p, is a normalised cubic B-spline (see
+                 i
+          Hayes [2]). This form, also, has advantages of computational
+          speed and accuracy over alternative representations.
+
+          2.3. Surface Fitting
+
+          There are now two independent variables, and we shall denote
+          these by x and y. The dependent variable, which was denoted by y
+          in the curve-fitting case, will now be denoted by f. (This is a
+          rather different notation from that indicated for the general-
+          dimensional problem in the first paragraph of Section 2.1 , but
+          it has some advantages in presentation.)
+
+          Again, in the absence of contrary indications in the particular
+          application being considered, polynomials and splines are the
+          approximating functions most commonly used. Only splines are used
+          by the surface-fitting routines in this chapter.
+
+          2.3.1.  Bicubic splines: definition and representation
+
+          The bicubic spline is defined over a rectangle R in the (x,y)
+          plane, the sides of R being parallel to the x- and y-axes. R is
+          divided into rectangular panels, again by lines parallel to the
+          axes. Over each panel the bicubic spline is a bicubic polynomial,
+          that is it takes the form
+
+                                3   3
+                                --  --     i j
+                                >   >  a  x y .                        (13)
+                                --  --  ij
+                                i=0 j=0
+
+          Each of these polynomials joins the polynomials in adjacent
+          panels with continuity up to the second derivative. The constant
+          x-values of the dividing lines parallel to the y-axis form the
+          set of interior knots for the variable x, corresponding precisely
+          to the set of interior knots of a cubic spline. Similarly, the
+          constant y-values of dividing lines parallel to the x-axis form
+          the set of interior knots for the variable y. Instead of
+          representing the bicubic spline in terms of the above set of
+          bicubic polynomials, however, it is represented, for the sake of
+          computational speed and accuracy, in the form
+
+                                 p   q
+                                 --  --
+                         f(x,y)= >   >  c  M (x)N (y),                 (14)
+                                 --  --  ij i    j
+                                 i=1 j=1
+
+          where M (x), for i=1,2,...,p, and N (y), for j=1,2,...,q, are
+                 i                           j
+          normalised B-splines (see Hayes and Halliday [4] for further
+          details of bicubic splines and Hayes [2] for normalised B-
+          splines).
+
+          2.4. General Linear and Nonlinear Fitting Functions
+
+          We have indicated earlier that, unless the data-fitting
+          application under consideration specifically requires some other
+          type of fitting function, a polynomial or a spline is usually to
+          be preferred. Special routines for these functions, in one and in
+          two variables, are provided in this chapter. When the application
+          does specify some other fitting function, however, it may be
+          treated by a routine which deals with a general linear function,
+          or by one for a general nonlinear function, depending on whether
+          the coefficients in the given function occur linearly or
+          nonlinearly.
+
+          The general linear fitting function can be written in the form
+
+                f(x)==c (phi) (x)+c (phi) (x)+...+c (phi) (x),         (15)
+                       1     1     2     2         p     p
+
+          where x is a vector of one or more independent variables, and the
+          (phi)  are any given functions of these variables (though they
+               i
+          must be linearly independent of one another if there is to be the
+          possibility of a unique solution to the fitting problem). This is
+          not intended to imply that each (phi)  is necessarily a function
+                                               i
+          of all the variables: we may have, for example, that each (phi)
+                                                                         i
+          is a function of a different single variable, and even that one
+          of the (phi)  is a constant. All that is required is that a value
+                      i
+          of each (phi) (x) can be computed when a value of each
+                       i
+          independent variable is given.
+
+          When the fitting function f(x) is not linear in its coefficients,
+          no more specific representation is available in general than f(x)
+          itself. However, we shall find it helpful later on to indicate
+          the fact that f(x) contains a number of coefficients (to be
+          determined by the fitting process) by using instead the notation
+          f(x;c), where c denotes the vector of coefficients. An example of
+          a nonlinear fitting function is
+
+
+                      f(x;c)==c +c exp(-c x)+c exp(-c x),              (16)
+                               1  2      4    3      5
+
+          which is in one variable and contains five coefficients. Note
+          that here, as elsewhere in this Chapter Introduction, we use the
+          term 'coefficients' to include all the quantities whose values
+          are to be determined by the fitting process, not just those which
+          occur linearly. We may observe that it is only the presence of
+          the coefficients c  and c  which makes the form (16) nonlinear.
+                            4      5
+          If the values of these two coefficients were known beforehand,
+          (16) would instead be a linear function which, in terms of the
+          general linear form (15), has p=3 and
+
+           (phi) (x)==1,  (phi) (x)==exp(-c x), and (phi) (x)==exp(-c x).
+                1              2           4             3           5
+
+          We may note also that polynomials and splines, such as (9) and
+          (14), are themselves linear in their coefficients. Thus if, when
+          fitting with these functions, a suitable special routine is not
+          available (as when more than two independent variables are
+          involved or when fitting in the l  norm), it is appropriate to
+                                           1
+          use a routine designed for a general linear function.
+
+          2.5. Constrained Problems
+
+          So far, we have considered only fitting processes in which the
+          values of the coefficients in the fitting function are determined
+          by an unconstrained minimization of a particular norm. Some
+          fitting problems, however, require that further restrictions be
+          placed on the determination of the coefficient values. Sometimes
+          these restrictions are contained explicitly in the formulation of
+          the problem in the form of equalities or inequalities which the
+          coefficients, or some function of them, must satisfy. For
+          example, if the fitting function contains a term Aexp(-kx), it
+          may be required that k>=0. Often, however, the equality or
+          inequality constraints relate to the value of the fitting
+          function or its derivatives at specified values of the
+          independent variable(s), but these too can be expressed in terms
+          of the coefficients of the fitting function, and it is
+          appropriate to do this if a general linear or nonlinear routine
+          is being used. For example, if the fitting function is that given
+          in (10), the requirement that the first derivative of the
+          function at x=x  be non-negative can be expressed as
+                         0
+
+                     c N '(x )+c N '(x )+...+c N '(x )>=0,             (17)
+                      1 1   0   2 2   0       p p   0
+
+          where the prime denotes differentiation with respect to x and
+          each derivative is evaluated at x=x . On the other hand, if the
+                                             0
+          requirement had been that the derivative at x=x  be exactly zero,
+                                                         0
+          the inequality sign in (17) would be replaced by an equality.
+
+          Routines which provide a facility for minimizing the appropriate
+          norm subject to such constraints are discussed in Section 3.6.
+
+          2.6. References
+
+          [1]   Cox M G and Hayes J G (1973) Curve fitting: a guide and
+                suite of algorithms for the non-specialist user. Report
+                NAC26. National Physical Laboratory.
+
+          [2]   Hayes J G (1974 ) Numerical Methods for Curve and Surface
+                Fitting. Bull Inst Math Appl. 10 144--152.
+
+                (For definition of normalised B-splines and details of
+                numerical methods.)
+
+          [3]   Hayes J G (1970) Curve Fitting by Polynomials in One
+                Variable. Numerical Approximation to Functions and Data. (ed
+                J G Hayes) Athlone Press, London.
+
+          [4]   Hayes J G and Halliday J (1974) The Least-squares Fitting of
+                Cubic Spline Surfaces to General Data Sets. J. Inst. Math.
+                Appl. 14 89--103.
+
+          3. Recommendations on Choice and Use of Routines
+
+          3.1. General
+
+          The choice of a routine to treat a particular fitting problem
+          will depend first of all on the fitting function and the norm to
+          be used. Unless there is good reason to the contrary, the fitting
+          function should be a polynomial or a cubic spline (in the
+          appropriate number of variables) and the norm should be the l
+                                                                       2
+          norm (leading to the least-squares fit). If some other function
+          is to be used, the choice of routine will depend on whether the
+          function is nonlinear (in which case see Section 3.5.2) or linear
+          in its coefficients (see Section 3.5.1), and, in the latter case,
+          on whether the l  or l  norm is to be used. The latter section is
+                          1     2
+          appropriate for polynomials and splines, too, if the l  norm is
+                                                                1
+          preferred.
+
+          In the case of a polynomial or cubic spline, if there is only one
+          independent variable, the user should choose a spline (Section
+          3.3) when the curve represented by the data is of complicated
+          form, perhaps with several peaks and troughs. When the curve is
+          of simple form, first try a polynomial (see Section 3.2) of low
+          degree, say up to degree 5 or 6, and then a spline if the
+          polynomial fails to provide a satisfactory fit. (Of course, if
+          third-derivative discontinuities are unacceptable to the user, a
+          polynomial is the only choice.) If the problem is one of surface
+          fitting, one of the spline routines should be used (Section 3.4).
+          If the problem has more than two independent variables, it may be
+          treated by the general linear routine in Section 3.5.1, again
+          using a polynomial in the first instance.
+
+          Another factor which affects the choice of routine is the
+          presence of constraints, as previously discussed in Section 2.5.
+          Indeed this factor is likely to be overriding at present, because
+          of the limited number of routines which have the necessary
+          facility. See Section 3.6.
+
+          3.1.1.  Data considerations
+
+          A satisfactory fit cannot be expected by any means if the number
+          and arrangement of the data points do not adequately represent
+          the character of the underlying relationship: sharp changes in
+          behaviour, in particular, such as sharp peaks, should be well
+          covered. Data points should extend over the whole range of
+          interest of the independent variable(s): extrapolation outside
+          the data ranges is most unwise. Then, with polynomials, it is
+          advantageous to have additional points near the ends of the
+          ranges, to counteract the tendency of polynomials to develop
+          fluctuations in these regions. When, with polynomial curves, the
+          user can precisely choose the x-values of the data, the special
+          points defined in Section 3.2.2 should be selected. With splines
+          the choice is less critical as long as the character of the
+          relationship is adequately represented. All fits should be tested
+          graphically before accepting them as satisfactory.
+
+          For this purpose it should be noted that it is not sufficient to
+          plot the values of the fitted function only at the data values of
+          the independent variable(s); at the least, its values at a
+          similar number of intermediate points should also be plotted, as
+          unwanted fluctuations may otherwise go undetected. Such
+          fluctuations are the less likely to occur the lower the number of
+          coefficients chosen in the fitting function. No firm guide can be
+          given, but as a rough rule, at least initially, the number of
+          coefficients should not exceed half the number of data points
+          (points with equal or nearly equal values of the independent
+          variable, or both independent variables in surface fitting,
+          counting as a single point for this purpose). However, the
+          situation may be such, particularly with a small number of data
+          points, that a satisfactorily close fit to the data cannot be
+          achieved without unwanted fluctuations occurring. In such cases,
+          it is often possible to improve the situation by a transformation
+          of one or more of the variables, as discussed in the next
+          paragraph: otherwise it will be necessary to provide extra data
+          points. Further advice on curve fitting is given in Cox and Hayes
+          [1] and, for polynomials only, in Hayes [3] of Section 2.7. Much
+          of the advice applies also to surface fitting; see also the
+          Routine Documents.
+
+          3.1.2.  Transformation of variables
+
+          Before starting the fitting, consideration should be given to the
+          choice of a good form in which to deal with each of the
+          variables: often it will be satisfactory to use the variables as
+          they stand, but sometimes the use of the logarithm, square root,
+          or some other function of a variable will lead to a better-
+          behaved relationship. This question is customarily taken into
+          account in preparing graphs and tables of a relationship and the
+          same considerations apply when curve or surface fitting. The
+          practical context will often give a guide. In general, it is best
+          to avoid having to deal with a relationship whose behaviour in
+          one region is radically different from that in another. A steep
+          rise at the left-hand end of a curve, for example, can often best
+          be treated by curve fitting in terms of log(x+c) with some
+          suitable value of the constant c. A case when such a
+          transformation gave substantial benefit is discussed in Hayes [3]
+          page 60. According to the features exhibited in any particular
+          case, transformation of either dependent variable or independent
+          variable(s) or both may be beneficial. When there is a choice it
+          is usually better to transform the independent variable(s): if
+          the dependent variable is transformed, the weights attached to
+          the data points must be adjusted. Thus (denoting the dependent
+          variable by y, as in the notation for curves) if the y  to be
+                                                                r
+          fitted have been obtained by a transformation y=g(Y) from
+          original data values Y , with weights W , for r=1,2,...,m, we
+                                r                r
+          must take
+
+                                w =W /(dy/dY),                         (18)
+                                 r  r
+
+          where the derivative is evaluated at Y . Strictly, the
+                                                r
+          transformation of Y and the adjustment of weights are valid only
+          when the data errors in the Y  are small compared with the range
+                                       r
+          spanned by the Y , but this is usually the case.
+                          r
+
+          3.2. Polynomial Curves
+
+          3.2.1.  Least-squares polynomials: arbitrary data points
+
+          E02ADF fits to arbitrary data points, with arbitrary weights,
+          polynomials of all degrees up to a maximum degree k, which is at
+          choice. If the user is seeking only a low degree polynomial, up
+          to degree 5 or 6 say, k=10 is an appropriate value, providing
+          there are about 20 data points or more. To assist in deciding the
+          degree of polynomial which satisfactorily fits the data, the
+          routine provides the root-mean-square-residual s  for all degrees
+                                                          i
+          i=1,2,...,k. In a satisfactory case, these s  will decrease
+                                                      i
+          steadily as i increases and then settle down to a fairly constant
+          value, as shown in the example
+
+                  i    s
+                        i
+
+                  0    3.5215
+
+                  1    0.7708
+
+                  2    0.1861
+
+                  3    0.0820
+
+                  4    0.0554
+
+                  5    0.0251
+
+                  6    0.0264
+
+                  7    0.0280
+
+                  8    0.0277
+
+                  9    0.0297
+
+                  10   0.0271
+
+          If the s  values settle down in this way, it indicates that the
+                  i
+          closest polynomial approximation justified by the data has been
+          achieved. The degree which first gives the approximately constant
+          value of s  (degree 5 in the example) is the appropriate degree
+                    i
+          to select. (Users who are prepared to accept a fit higher than
+          sixth degree, should simply find a high enough value of k to
+          enable the type of behaviour indicated by the example to be
+          detected: thus they should seek values of k for which at least 4
+          or 5 consecutive values of s  are approximately the same.) If the
+                                      i
+          degree were allowed to go high enough, s  would, in most cases,
+                                                  i
+          eventually start to decrease again, indicating that the data
+          points are being fitted too closely and that undesirable
+          fluctuations are developing between the points. In some cases,
+          particularly with a small number of data points, this final
+          decrease is not distinguishable from the initial decrease in s .
+                                                                        i
+          In such cases, users may seek an acceptable fit by examining the
+          graphs of several of the polynomials obtained. Failing this, they
+          may (a) seek a transformation of variables which improves the
+          behaviour, (b) try fitting a spline, or (c) provide more data
+          points. If data can be provided simply by drawing an
+          approximating curve by hand and reading points from it, use the
+          points discussed in Section 3.2.2.
+
+          3.2.2.  Least-squares polynomials: selected data points
+
+          When users are at liberty to choose the x-values of data points,
+          such as when the points are taken from a graph, it is most
+          advantageous when fitting with polynomials to use the values
+          x =cos((pi)r/n), for r=0,1,...,n for some value of n, a suitable
+           r
+          value for which is discussed at the end of this section. Note
+          that these x  relate to the variable x after it has been
+                      r
+          normalised so that its range of interest is -1 to +1. E02ADF may
+          then be used as in Section 3.2.1 to seek a satisfactory fit.
+
+          3.3. Cubic Spline Curves
+
+          3.3.1.  Least-squares cubic splines
+
+          E02BAF fits to arbitrary data points, with arbitrary weights, a
+          cubic spline with interior knots specified by the user. The
+          choice of these knots so as to give an acceptable fit must
+          largely be a matter of trial and error, though with a little
+          experience a satisfactory choice can often be made after one or
+          two trials. It is usually best to start with a small number of
+          knots (too many will result in unwanted fluctuations in the fit,
+          or even in there being no unique solution) and, examining the fit
+          graphically at each stage, to add a few knots at a time at places
+          where the fit is particularly poor. Moving the existing knots
+          towards these places will also often improve the fit. In regions
+          where the behaviour of the curve underlying the data is changing
+          rapidly, closer knots will be needed than elsewhere. Otherwise,
+          positioning is not usually very critical and equally-spaced knots
+          are often satisfactory. See also the next section, however.
+
+          A useful feature of the routine is that it can be used in
+          applications which require the continuity to be less than the
+          normal continuity of the cubic spline. For example, the fit may
+          be required to have a discontinuous slope at some point in the
+          range. This can be achieved by placing three coincident knots at
+          the given point. Similarly a discontinuity in the second
+          derivative at a point can be achieved by placing two knots there.
+          Analogy with these discontinuous cases can provide guidance in
+          more usual cases: for example, just as three coincident knots can
+          produce a discontinuity in slope, so three close knots can
+          produce a rapid change in slope. The closer the knots are, the
+          more rapid can the change be.
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          An example set of data is given in Figure 1. It is a rather
+          tricky set, because of the scarcity of data on the right, but it
+          will serve to illustrate some of the above points and to show
+          some of the dangers to be avoided. Three interior knots
+          (indicated by the vertical lines at the top of the diagram) are
+          chosen as a start. We see that the resulting curve is not steep
+          enough in the middle and fluctuates at both ends, severely on the
+          right. The spline is unable to cope with the shape and more knots
+          are needed.
+
+          In Figure 2, three knots have been added in the centre, where the
+          data shows a rapid change in behaviour, and one further out at
+          each end, where the fit is poor. The fit is still poor, so a
+          further knot is added in this region and, in Figure 3, disaster
+          ensues in rather spectacular fashion.
+
+                                     Figure 2
+                   Please see figure in printed Reference Manual
+
+                                     Figure 3
+                   Please see figure in printed Reference Manual
+
+          The reason is that, at the right-hand end, the fits in Figure 1
+          and Figure 2 have been interpreted as poor simply because of the
+          fluctuations about the curve underlying the data (or what it is
+          naturally assumed to be). But the fitting process knows only
+          about the data and nothing else about the underlying curve, so it
+          is important to consider only closeness to the data when deciding
+          goodness of fit.
+
+          Thus, in Figure 1, the curve fits the last two data points quite
+          well compared with the fit elsewhere, so no knot should have been
+          added in this region. In Figure 2, the curve goes exactly through
+          the last two points, so a further knot is certainly not needed
+          here.
+
+
+                                     Figure 4
+                   Please see figure in printed Reference Manual
+
+          Figure 4 shows what can be achieved without the extra knot on
+          each of the flat regions. Remembering that within each knot
+          interval the spline is a cubic polynomial, there is really no
+          need to have more than one knot interval covering each flat
+          region.
+
+          What we have, in fact, in Figure 2 and Figure 3 is a case of too
+          many knots (so too many coefficients in the spline equation) for
+          the number of data points. The warning in the second paragraph of
+          Section 2.1 was that the fit will then be too close to the data,
+          tending to have unwanted fluctuations between the data points.
+          The warning applies locally for splines, in the sense that, in
+          localities where there are plenty of data points, there can be a
+          lot of knots, as long as there are few knots where there are few
+          points, especially near the ends of the interval. In the present
+          example, with so few data points on the right, just the one extra
+          knot in Figure 2 is too many! The signs are clearly present, with
+          the last two points fitted exactly (at least to the graphical
+          accuracy and actually much closer than that) and fluctuations
+          within the last two knot-intervals (cf. Figure 1, where only the
+          final point is fitted exactly and one of the wobbles spans
+          several data points).
+
+          The situation in Figure 3 is different. The fit, if computed
+          exactly, would still pass through the last two data points, with
+          even more violent fluctuations. However, the problem has become
+          so ill-conditioned that all accuracy has been lost. Indeed, if
+          the last interior knot were moved a tiny amount to the right,
+          there would be no unique solution and an error message would have
+          been caused. Near-singularity is, sadly, not picked up by the
+          routine, but can be spotted readily in a graph, as Figure 3. B-
+          spline coefficients becoming large, with alternating signs, is
+          another indication. However, it is better to avoid such
+          situations, firstly by providing, whenever possible, data
+          adequately covering the range of interest, and secondly by
+          placing knots only where there is a reasonable amount of data.
+
+          The example here could, in fact, have utilised from the start the
+          observation made in the second paragraph of this section, that
+          three close knots can produce a rapid change in slope. The
+          example has two such rapid changes and so requires two sets of
+          three close knots (in fact, the two sets can be so close that one
+          knot can serve in both sets, so only five knots prove sufficient
+          in Figure 4). It should be noted, however, that the rapid turn
+          occurs within the range spanned by the three knots. This is the
+          reason that the six knots in Figure 2 are not satisfactory as
+          they do not quite span the two turns.
+
+          Some more examples to illustrate the choice of knots are given in
+          Cox and Hayes [1].
+
+          3.3.2.  Automatic fitting with cubic splines
+
+          E02BEF also fits cubic splines to arbitrary data points with
+          arbitrary weights but itself chooses the number and positions of
+          the knots. The user has to supply only a threshold for the sum of
+          squares of residuals. The routine first builds up a knot set by a
+          series of trial fits in the l  norm. Then, with the knot set
+                                       2
+          decided, the final spline is computed to minimize a certain
+          smoothing measure subject to satisfaction of the chosen
+          threshold. Thus it is easier to use than E02BAF (see previous
+          section), requiring only some experimentation with this
+          threshold. It should therefore be first choice unless the user
+          has a preference for the ordinary least-squares fit or, for
+          example, wishes to experiment with knot positions, trying to keep
+          their number down (E02BEF aims only to be reasonably frugal with
+          knots).
+
+          3.4. Spline Surfaces
+
+          3.4.1.  Least-squares bicubic splines
+
+          E02DAF fits to arbitrary data points, with arbitrary weights, a
+          bicubic spline with its two sets of interior knots specified by
+          the user. For choosing these knots, the advice given for cubic
+          splines, in Section 3.3.1 above, applies here too. (See also the
+          next section, however.) If changes in the behaviour of the
+          surface underlying the data are more marked in the direction of
+          one variable than of the other, more knots will be needed for the
+          former variable than the latter. Note also that, in the surface
+          case, the reduction in continuity caused by coincident knots will
+          extend across the whole spline surface: for example, if three
+          knots associated with the variable x are chosen to coincide at a
+          value L, the spline surface will have a discontinuous slope
+          across the whole extent of the line x=L.
+
+          With some sets of data and some choices of knots, the least-
+          squares bicubic spline will not be unique. This will not occur,
+          with a reasonable choice of knots, if the rectangle R is well
+          covered with data points: here R is defined as the smallest
+          rectangle in the (x,y) plane, with sides parallel to the axes,
+          which contains all the data points. Where the least-squares
+          solution is not unique, the minimal least-squares solution is
+          computed, namely that least-squares solution which has the
+          smallest value of the sum of squares of the B-spline coefficients
+          c   (see the end of Section 2.3.2 above). This choice of least-
+           ij
+          squares solution tends to minimize the risk of unwanted
+          fluctuations in the fit. The fit will not be reliable, however,
+          in regions where there are few or no data points.
+
+          3.4.2.  Automatic fitting with bicubic splines
+
+          E02DDF also fits bicubic splines to arbitrary data points with
+          arbitrary weights but chooses the knot sets itself. The user has
+          to supply only a threshold for the sum of squares of residuals.
+          Just like the automatic curve E02BEF (Section 3.3.2), E02DDF then
+          builds up the knot sets and finally fits a spline minimizing a
+          smoothing measure subject to satisfaction of the threshold.
+          Again, this easier to use routine is normally to be preferred, at
+          least in the first instance.
+
+          E02DCF is a very similar routine to E02DDF but deals with data
+          points of equal weight which lie on a rectangular mesh in the
+          (x,y) plane. This kind of data allows a very much faster
+          computation and so is to be preferred when applicable.
+          Substantial departures from equal weighting can be ignored if the
+          user is not concerned with statistical questions, though the
+          quality of the fit will suffer if this is taken too far. In such
+          cases, the user should revert to E02DDF.
+
+          3.5. General Linear and Nonlinear Fitting Functions
+
+          3.5.1.  General linear functions
+
+          For the general linear function (15), routines are available for
+          fitting in the l  and l  norms. The least-squares routines (which
+                          1      2
+          are to be preferred unless there is good reason to use another
+          norm -- see Section 2.1.1) are in Chapter F04. The l  routine is
+                                                              1
+          E02GAF.
+
+          All the above routines are essentially linear algebra routines,
+          and in considering their use we need to view the fitting process
+          in a slightly different way from hitherto. Taking y to be the
+          dependent variable and x the vector of independent variables, we
+          have, as for equation (1) but with each x  now a vector,
+                                                   r
+
+                          (epsilon) =y -f(x )  r=1,2,...,m.
+                                   r  r    r
+
+          Substituting for f(x) the general linear form (15), we can write
+          this as
+
+          c (phi) (x )+c (phi) (x )+...+c (phi) (x )=y -(epsilon) ,
+           1     1  r   2     2  r       p     p  r   r          r
+           r=1,2,...,m                                                 (19)
+
+
+          Thus we have a system of linear equations in the coefficients c .
+                                                                         j
+          Usually, in writing these equations, the (epsilon)  are omitted
+                                                            r
+          and simply taken as implied. The system of equations is then
+          described as an overdetermined system (since we must have m>=p if
+          there is to be the possibility of a unique solution to our
+          fitting problem), and the fitting process of computing the c  to
+                                                                      j
+          minimize one or other of the norms (2), (3) and (4) can be
+          described, in relation to the system of equations, as solving the
+          overdetermined system in that particular norm. In matrix
+          notation, the system can be written as
+
+                                   (Phi)c=y,                           (20)
+
+          where (Phi) is the m by p matrix whose element in row r and
+          column j is (phi) (x ), for r=1,2,...,m; j=1,2,...,p. The vectors
+                           j  r
+          c and y respectively contain the coefficients c  and the data
+                                                         j
+          values y .
+                  r
+
+          The routines, however, use the standard notation of linear
+          algebra, the overdetermined system of equations being denoted by
+
+                                     Ax=b                              (21)
+
+          The correspondence between this notation and that which we have
+          used for the data-fitting problem (equation (20)) is therefore
+          given by
+
+                            A==(Phi),   x==c   b==y                    (22)
+
+          Note that the norms used by these routines are the unweighted
+          norms (2) and (3). If the user wishes to apply weights to the
+          data points, that is to use the norms (5) or (6), the
+          equivalences (22) should be replaced by
+
+                              A==D(Phi),   x==c   b==Dy
+
+          where D is a diagonal matrix with w  as the rth diagonal element.
+                                             r
+          Here w , for r=1,2,...,m, is the weight of the rth data point as
+                r
+          defined in Section 2.1.2.
+
+          3.5.2.  Nonlinear functions
+
+          Routines for fitting with a nonlinear function in the l  norm are
+                                                                 2
+          provided in Chapter E04, and that chapter's Introduction should
+          be consulted for the appropriate choice of routine. Again,
+          however, the notation adopted is different from that we have used
+          for data fitting. In the latter, we denote the fitting function
+          by f(x;c), where x is the vector of independent variables and c
+          is the vector of coefficients, whose values are to be determined.
+          The squared l  norm, to be minimized with respect to the elements
+                       2
+          of c, is then
+
+                               m
+                               --  2            2
+                               >  w [y -f(x ;c)]                       (23)
+                               --  r  r    r
+                               r=1
+
+          where y  is the rth data value of the dependent variable, x  is
+                 r                                                   r
+          the vector containing the rth values of the independent
+          variables, and w  is the corresponding weight as defined in
+                          r
+          Section 2.1.2.
+
+          On the other hand, in the nonlinear least-squares routines of
+          Chapter E04, the function to be minimized is denoted by
+
+                                   m
+                                   --  2
+                                   >  f (x),                           (24)
+                                   --  i
+                                   i=1
+
+          the minimization being carried out with respect to the elements
+          of the vector x. The correspondence between the two notations is
+          given by
+
+          x==c and
+
+          f (x)==w [y -f(x ;c)],  i=r=1,2,...,m.
+           i      r  r    r
+
+          Note especially that the vector x of variables of the nonlinear
+          least-squares routines is the vector c of coefficients of the
+          data-fitting problem, and in particular that, if the selected
+          routine requires derivatives of the f (x) to be provided, these
+                                               i
+          are derivatives of w [y -f(x ;c)] with respect to the
+                              r  r    r
+          coefficients of the data-fitting problem.
+
+          3.6. Constraints
+
+          At present, there are only a limited number of routines which fit
+          subject to constraints. Chapter E04 contains a routine, E04UCF,
+          which can be used for fitting with a nonlinear function in the l
+                                                                          2
+          norm subject to equality or inequality constraints. This routine,
+          unlike those in that chapter suited to the unconstrained case, is
+          not designed specifically for minimizing functions which are sums
+          of squares, and so the function (23) has to be treated as a
+          general nonlinear function. The E04 Chapter Introduction should
+          be consulted.
+
+          The remaining constraint routine relates to fitting with
+          polynomials in the l  norm. E02AGF deals with polynomial curves
+                              2
+          and allows precise values of the fitting function and (if
+          required) all its derivatives up to a given order to be
+          prescribed at one or more values of the independent variable.
+
+          3.7. Evaluation, Differentiation and Integration
+
+          Routines are available to evaluate, differentiate and integrate
+          polynomials in Chebyshev-series form and cubic or bicubic splines
+          in B-spline form. These polynomials and splines may have been
+          produced by the various fitting routines or, in the case of
+          polynomials, from prior calls of the differentiation and
+          integration routines themselves.
+
+          E02AEF and E02AKF evaluate polynomial curves: the latter has a
+          longer parameter list but does not require the user to normalise
+          the values of the independent variable and can accept
+          coefficients which are not stored in contiguous locations. E02BBF
+          evaluates cubic spline curves, and E02DEF and E02DFF bicubic
+          spline surfaces.
+
+          Differentiation and integration of polynomial curves are carried
+          out by E02AHF and E02AJF respectively. The results are provided
+          in Chebyshev-series form and so repeated differentiation and
+          integration are catered for. Values of the derivative or integral
+          can then be computed using the appropriate evaluation routine.
+
+          For splines the differentiation and integration routines provided
+          are of a different nature from those for polynomials. E02BCF
+          provides values of a cubic spline curve and its first three
+          derivatives (the rest, of course, are zero) at a given value of x
+          spline over its whole range. These routines can also be applied
+          to surfaces of the form (14). For example, if, for each value of
+          j in turn, the coefficients c  , for i=1,2,...,p are supplied to
+                                       ij
+          E02BCF with x=x  and on each occasion we select from the output
+                         0
+          the value of the second derivative, d  say, and if the whole set
+                                               j
+          of d  are then supplied to the same routine with x=y , the output
+              j                                               0
+          will contain all the values at (x ,y ) of
+                                           0  0
+
+                             2             r+2
+                           dd f          dd   f
+                           -----  and   --------,  r=1,2,3.
+                             2             2   r
+                           dd fx        ddx ddy
+
+          Equally, if after each of the first p calls of E02BCF we had
+          selected the function value (E02BBF would also provide this)
+          instead of the second derivative and we had supplied these values
+          to E02BDF, the result obtained would have been the value of
+
+                                     B
+                                     /
+                                     |f(x ,y)dy,
+                                     /   0
+                                     A
+
+          where A and B are the end-points of the y interval over which the
+          spline was defined.
+
+          3.8. Index
+
+          Automatic fitting,
+               with bicubic splines                                  E02DCF
+                                                                     E02DDF
+               with cubic splines                                    E02BEF
+          Data on rectangular mesh                                   E02DCF
+          Differentiation,
+               of cubic splines                                      E02BCF
+               of polynomials                                        E02AHF
+          Evaluation,
+               of bicubic splines                                    E02DEF
+                                                                     E02DFF
+               of cubic splines                                      E02BBF
+               of cubic splines and derivatives                      E02BCF
+               of definite integral of cubic splines                 E02BDF
+               of polynomials                                        E02AEF
+                                                                     E02AKF
+          Integration,
+               of cubic splines (definite integral)                  E02BDF
+               of polynomials                                        E02AJF
+          Least-squares curve fit,
+               with cubic splines                                    E02BAF
+               with polynomials,
+                    arbitrary data points                            E02ADF
+                    with constraints                                 E02AGF
+          Least-squares surface fit with bicubic splines             E02DAF
+          l  fit with general linear function,                       E02GAF
+           1
+          Sorting,
+               2-D data into panels                                  E02ZAF
+
+
+          E02 -- Curve and Surface Fitting                  Contents -- E02
+          Chapter E02
+
+          Curve and Surface Fitting
+
+          E02ADF  Least-squares curve fit, by polynomials, arbitrary data
+                  points
+
+          E02AEF  Evaluation of fitted polynomial in one variable from
+                  Chebyshev series form (simplified parameter list)
+
+          E02AGF  Least-squares polynomial fit, values and derivatives may
+                  be constrained, arbitrary data points,
+
+          E02AHF  Derivative of fitted polynomial in Chebyshev series form
+
+          E02AJF  Integral of fitted polynomial in Chebyshev series form
+
+          E02AKF  Evaluation of fitted polynomial in one variable, from
+                  Chebyshev series form
+
+          E02BAF  Least-squares curve cubic spline fit (including
+                  interpolation)
+
+          E02BBF  Evaluation of fitted cubic spline, function only
+
+          E02BCF  Evaluation of fitted cubic spline, function and
+                  derivatives
+
+          E02BDF  Evaluation of fitted cubic spline, definite integral
+
+          E02BEF  Least-squares cubic spline curve fit, automatic knot
+                  placement
+
+          E02DAF  Least-squares surface fit, bicubic splines
+
+          E02DCF  Least-squares surface fit by bicubic splines with
+                  automatic knot placement, data on rectangular grid
+
+          E02DDF  Least-squares surface fit by bicubic splines with
+                  automatic knot placement, scattered data
+
+          E02DEF  Evaluation of a fitted bicubic spline at a vector of
+                  points
+
+          E02DFF  Evaluation of a fitted bicubic spline at a mesh of points
+
+          E02GAF  L -approximation by general linear function
+                   1
+
+          E02ZAF  Sort 2-D data into panels for fitting bicubic splines
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02adf}{NAG On-line Documentation: e02adf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02ADF(3NAG)      Foundation Library (12/10/92)      E02ADF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02ADF
+                  E02ADF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02ADF computes weighted least-squares polynomial approximations
+          to an arbitrary set of data points.
+
+          2. Specification
+
+                 SUBROUTINE E02ADF (M, KPLUS1, NROWS, X, Y, W, WORK1,
+                1                   WORK2, A, S, IFAIL)
+                 INTEGER          M, KPLUS1, NROWS, IFAIL
+                 DOUBLE PRECISION X(M), Y(M), W(M), WORK1(3*M), WORK2
+                1                 (2*KPLUS1), A(NROWS,KPLUS1), S(KPLUS1)
+
+          3. Description
+
+          This routine determines least-squares polynomial approximations
+          of degrees 0,1,...,k to the set of data points (x ,y ) with
+                                                           r  r
+          weights w , for r=1,2,...,m.
+                   r
+
+          The approximation of degree i has the property that it minimizes
+          (sigma)  the sum of squares of the weighted residuals (epsilon) ,
+                 i                                                       r
+          where
+
+                                (epsilon) =w (y -f )
+                                         r  r  r  r
+
+          and f  is the value of the polynomial of degree i at the rth data
+               r
+          point.
+
+          Each polynomial is represented in Chebyshev-series form with
+                              
+
+          normalised argument x. This argument lies in the range -1 to +1
+          and is related to the original variable x by the linear
+          transformation
+
+                                    (2x-x   -x   )
+                                         max  min
+                                 x= --------------.
+                                     (x   -x   )
+                                       max  min
+
+          Here x    and x    are respectively the largest and smallest
+                max      min
+          values of x . The polynomial approximation of degree i is
+                     r
+          represented as
+
+               1                                                   
+               -a     T (x)+a     T (x)+a     T (x)+...+a       T (x),
+               2 i+1,1 0     i+1,2 1     i+1,3 2         i+1,i+1 i
+
+                   
+
+          where T (x) is the Chebyshev polynomial of the first kind of
+                 j
+                                  
+
+          degree j with argument (x).
+
+          For i=0,1,...,k, the routine produces the values of a       , for
+                                                               i+1,j+1
+          j=0,1,...,i, together with the value of the root mean square
+                           
+
+                          / (sigma)
+                         /         i
+          residual s =  /   --------. In the case m=i+1 the routine sets
+                    i \/     m-i-1
+          the value of s  to zero.
+                        i
+
+          The method employed is due to Forsythe [4] and is based upon the
+          generation of a set of polynomials orthogonal with respect to
+          summation over the normalised data set. The extensions due to
+          Clenshaw [1] to represent these polynomials as well as the
+          approximating polynomials in their Chebyshev-series forms are
+          incorporated. The modifications suggested by Reinsch and
+          Gentleman (see [5]) to the method originally employed by Clenshaw
+          for evaluating the orthogonal polynomials from their Chebyshev-
+          series representations are used to give greater numerical
+          stability.
+
+          For further details of the algorithm and its use see Cox [2] and
+          [3].
+
+          Subsequent evaluation of the Chebyshev-series representations of
+          the polynomial approximations should be carried out using E02AEF.
+
+          4. References
+
+          [1]   Clenshaw C W (1960) Curve Fitting with a Digital Computer.
+                Comput. J. 2 170--173.
+
+          [2]   Cox M G (1974) A Data-fitting Package for the Non-specialist
+                User. Software for Numerical Mathematics. (ed D J Evans)
+                Academic Press.
+
+          [3]   Cox M G and Hayes J G (1973) Curve fitting: a guide and
+                suite of algorithms for the non-specialist user. Report
+                NAC26. National Physical Laboratory.
+
+          [4]   Forsythe G E (1957) Generation and use of orthogonal
+                polynomials for data fitting with a digital computer. J.
+                Soc. Indust. Appl. Math. 5 74--88.
+
+          [5]   Gentlemen W M (1969) An Error Analysis of Goertzel's
+                (Watt's) Method for Computing Fourier Coefficients. Comput.
+                J. 12 160--165.
+
+          [6]   Hayes J G (1970) Curve Fitting by Polynomials in One
+                Variable. Numerical Approximation to Functions and Data. (ed
+                J G Hayes) Athlone Press, London.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number m of data points. Constraint: M >=
+               MDIST >= 2, where MDIST is the number of distinct x values
+               in the data.
+
+           2:  KPLUS1 -- INTEGER                                      Input
+               On entry: k+1, where k is the maximum degree required.
+               Constraint: 0 < KPLUS1 <= MDIST, where MDIST is the number
+               of distinct x values in the data.
+
+           3:  NROWS -- INTEGER                                       Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which E02ADF is called.
+               Constraint: NROWS >= KPLUS1.
+
+           4:  X(M) -- DOUBLE PRECISION array                         Input
+               On entry: the values x  of the independent variable, for
+                                     r
+               r=1,2,...,m. Constraint: the values must be supplied in non-
+               decreasing order with X(M) > X(1).
+
+           5:  Y(M) -- DOUBLE PRECISION array                         Input
+               On entry: the values y  of the dependent variable, for
+                                     r
+               r=1,2,...,m.
+
+           6:  W(M) -- DOUBLE PRECISION array                         Input
+               On entry: the set of weights, w , for r=1,2,...,m. For
+                                              r
+               advice on the choice of weights, see Section 2.1.2 of the
+               Chapter Introduction. Constraint: W(r) > 0.0, for r=1,2,...m.
+
+           7:  WORK1(3*M) -- DOUBLE PRECISION array               Workspace
+
+           8:  WORK2(2*KPLUS1) -- DOUBLE PRECISION array          Workspace
+
+           9:  A(NROWS,KPLUS1) -- DOUBLE PRECISION array             Output
+                                                
+
+               On exit: the coefficients of T (x) in the approximating
+                                             j
+               polynomial of degree i. A(i+1,j+1) contains the coefficient
+               a       , for i=0,1,...,k; j=0,1,...,i.
+                i+1,j+1
+
+          10:  S(KPLUS1) -- DOUBLE PRECISION array                   Output
+               On exit: S(i+1) contains the root mean square residual s ,
+                                                                       i
+               for i=0,1,...,k, as described in Section 3. For the
+               interpretation of the values of the s  and their use in
+                                                    i
+               selecting an appropriate degree, see Section 3.1 of the
+               Chapter Introduction.
+
+          11:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               The weights are not all strictly positive.
+
+          IFAIL= 2
+               The values of X(r), for r=1,2,...,M are not in non-
+               decreasing order.
+
+          IFAIL= 3
+               All X(r) have the same value: thus the normalisation of X is
+               not possible.
+
+          IFAIL= 4
+               On entry KPLUS1 < 1 (so the maximum degree required is
+                        negative)
+
+               or       KPLUS1 > MDIST, where MDIST is the number of
+                        distinct x values in the data (so there cannot be a
+                        unique solution for degree k=KPLUS1-1).
+
+          IFAIL= 5
+               NROWS < KPLUS1.
+
+          7. Accuracy
+
+          No error analysis for the method has been published. Practical
+          experience with the method, however, is generally extremely
+          satisfactory.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          m(k+1)(k+11).
+
+          The approximating polynomials may exhibit undesirable
+          oscillations (particularly near the ends of the range) if the
+          maximum degree k exceeds a critical value which depends on the
+          number of data points m and their relative positions. As a rough
+          guide, for equally-spaced data, this critical value is about
+              
+
+          2*\/m. For further details see Hayes [6] page 60.
+
+          9. Example
+
+          Determine weighted least-squares polynomial approximations of
+          degrees 0, 1, 2 and 3 to a set of 11 prescribed data points. For
+          the approximation of degree 3, tabulate the data and the
+          corresponding values of the approximating polynomial, together
+          with the residual errors, and also the values of the
+          approximating polynomial at points half-way between each pair of
+          adjacent data points.
+
+          The example program supplied is written in a general form that
+          will enable polynomial approximations of degrees 0,1,...,k to be
+          obtained to m data points, with arbitrary positive weights, and
+          the approximation of degree k to be tabulated. E02AEF is used to
+          evaluate the approximating polynomial. The program is self-
+          starting in that any number of data sets can be supplied.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02aef}{NAG On-line Documentation: e02aef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02AEF(3NAG)      Foundation Library (12/10/92)      E02AEF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02AEF
+                  E02AEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02AEF evaluates a polynomial from its Chebyshev-series
+          representation.
+
+          2. Specification
+
+                 SUBROUTINE E02AEF (NPLUS1, A, XCAP, P, IFAIL)
+                 INTEGER          NPLUS1, IFAIL
+                 DOUBLE PRECISION A(NPLUS1), XCAP, P
+
+          3. Description
+
+          This routine evaluates the polynomial
+
+                        1                                   
+                        -a T (x)+a T (x)+a T (x)+...+a   T (x)
+                        2 1 0     2 1     3 2         n+1 n
+
+                                                          
+
+          for any value of x satisfying -1<=x<=1. Here T (x) denotes the
+                                                        j
+          Chebyshev polynomial of the first kind of degree j with argument
+          
+
+          x. The value of n is prescribed by the user.
+
+                                    
+
+          In practice, the variable x will usually have been obtained from
+          an original variable x, where x   <=x<=x    and
+                                         min      max
+
+                                  ((x-x   )-(x   -x))
+                                       min    max
+                               x= -------------------
+                                      (x   -x   )
+                                        max  min
+
+          Note that this form of the transformation should be used
+          computationally rather than the mathematical equivalent
+
+                                     (2x-x   -x   )
+                                          min  max
+                                  x= --------------
+                                      (x   -x   )
+                                        max  min
+
+                                                                 
+
+          since the former guarantees that the computed value of x differs
+          from its true value by at most 4(epsilon), where (epsilon) is the
+          machine precision, whereas the latter has no such guarantee.
+
+          The method employed is based upon the three-term recurrence
+          relation due to Clenshaw [1], with modifications to give greater
+          numerical stability due to Reinsch and Gentleman (see [4]).
+
+          For further details of the algorithm and its use see Cox [2] and
+          [3].
+
+          4. References
+
+          [1]   Clenshaw C W (1955) A Note on the Summation of Chebyshev
+                Series. Math. Tables Aids Comput. 9 118--120.
+
+          [2]   Cox M G (1974) A Data-fitting Package for the Non-specialist
+                User. Software for Numerical Mathematics. (ed D J Evans)
+                Academic Press.
+
+          [3]   Cox M G and Hayes J G (1973) Curve fitting: a guide and
+                suite of algorithms for the non-specialist user. Report
+                NAC26. National Physical Laboratory.
+
+          [4]   Gentlemen W M (1969) An Error Analysis of Goertzel's
+                (Watt's) Method for Computing Fourier Coefficients. Comput.
+                J. 12 160--165.
+
+          5. Parameters
+
+           1:  NPLUS1 -- INTEGER                                      Input
+               On entry: the number n+1 of terms in the series (i.e., one
+               greater than the degree of the polynomial). Constraint:
+               NPLUS1 >= 1.
+
+           2:  A(NPLUS1) -- DOUBLE PRECISION array                    Input
+               On entry: A(i) must be set to the value of the ith
+               coefficient in the series, for i=1,2,...,n+1.
+
+           3:  XCAP -- DOUBLE PRECISION                               Input
+                         
+
+               On entry: x, the argument at which the polynomial is to be
+               evaluated. It should lie in the range -1 to +1, but a value
+               just outside this range is permitted (see Section 6) to
+               allow for possible rounding errors committed in the
+                                        
+
+               transformation from x to x discussed in Section 3. Provided
+               the recommended form of the transformation is used, a
+               successful exit is thus assured whenever the value of x lies
+               in the range x    to x   .
+                             min     max
+
+           4:  P -- DOUBLE PRECISION                                 Output
+               On exit: the value of the polynomial.
+
+           5:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               ABS(XCAP) > 1.0 + 4(epsilon), where (epsilon) is the
+               machine precision. In this case the value of P is set
+               arbitrarily to zero.
+
+          IFAIL= 2
+               On entry NPLUS1 < 1.
+
+          7. Accuracy
+
+          The rounding errors committed are such that the computed value of
+          the polynomial is exact for a slightly perturbed set of
+          coefficients a +(delta)a . The ratio of the sum of the absolute
+                        i         i
+          values of the (delta)a  to the sum of the absolute values of the
+                                i
+          a  is less than a small multiple of (n+1) times machine
+           i
+          precision.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          n+1.
+
+          It is expected that a common use of E02AEF will be the evaluation
+          of the polynomial approximations produced by E02ADF and E02AFF(*)
+
+          9. Example
+
+                                                                   
+
+          Evaluate at 11 equally-spaced points in the interval -1<=x<=1 the
+          polynomial of degree 4 with Chebyshev coefficients, 2.0, 0.5, 0.
+          25, 0.125, 0.0625.
+
+          The example program is written in a general form that will enable
+          a polynomial of degree n in its Chebyshev-series form to be
+                                                                   
+
+          evaluated at m equally-spaced points in the interval -1<=x<=1.
+          The program is self-starting in that any number of data sets can
+          be supplied.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02agf}{NAG On-line Documentation: e02agf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02AGF(3NAG)      Foundation Library (12/10/92)      E02AGF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02AGF
+                  E02AGF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02AGF computes constrained weighted least-squares polynomial
+          approximations in Chebyshev-series form to an arbitrary set of
+          data points. The values of the approximations and any number of
+          their derivatives can be specified at selected points.
+
+          2. Specification
+
+                 SUBROUTINE E02AGF (M, KPLUS1, NROWS, XMIN, XMAX, X, Y, W,
+                1                   MF, XF, YF, LYF, IP, A, S, NP1, WRK,
+                2                   LWRK, IWRK, LIWRK, IFAIL)
+                 INTEGER          M, KPLUS1, NROWS, MF, LYF, IP(MF), NP1,
+                1                 LWRK, IWRK(LIWRK), LIWRK, IFAIL
+                 DOUBLE PRECISION XMIN, XMAX, X(M), Y(M), W(M), XF(MF), YF
+                1                 (LYF), A(NROWS,KPLUS1), S(KPLUS1), WRK
+                2                 (LWRK)
+
+          3. Description
+
+          This routine determines least-squares polynomial approximations
+          of degrees up to k to the set of data points (x ,y ) with weights
+                                                         r  r
+          w , for r=1,2,...,m. The value of k, the maximum degree required,
+           r
+          is prescribed by the user. At each of the values XF , for r =
+                                                             r
+          1,2,...,MF, of the independent variable x, the approximations and
+          their derivatives up to order p  are constrained to have one of
+                                         r
+                                                                      MF
+                                                                      --
+          the user-specified values YF , for s=1,2,...,n, where n=MF+ >  p
+                                      s                               --  r
+                                                                      r=1
+
+          The approximation of degree i has the property that, subject to
+          the imposed contraints, it minimizes (Sigma) , the sum of the
+                                                      i
+          squares of the weighted residuals (epsilon)  for r=1,2,...,m
+                                                     r
+          where
+
+                              (epsilon) =w (y -f (x ))
+                                       r  r  r  i  r
+
+          and f (x ) is the value of the polynomial approximation of degree
+               i  r
+          i at the rth data point.
+
+          Each polynomial is represented in Chebyshev-series form with
+                              
+
+          normalised argument x. This argument lies in the range -1 to +1
+          and is related to the original variable x by the linear
+          transformation
+
+                                     2x-(x   +x   )
+                                          max  min
+                                  x= --------------
+                                      (x   -x   )
+                                        max  min
+
+          where x    and x   , specified by the user, are respectively the
+                 min      max
+          lower and upper end-points of the interval of x over which the
+          polynomials are to be defined.
+
+          The polynomial approximation of degree i can be written as
+
+                      1                                      
+                      -a   +a   T (x)+...+a  T (x)+...+a  T (x)
+                      2 i,0  i,1 1         ij j         ii i
+
+                   
+
+          where T (x) is the Chebyshev polynomial of the first kind of
+                 j
+                                 
+
+          degree j with argument x. For i=n,n+1,...,k, the routine produces
+          the values of the coefficients a  , for j=0,1,...,i, together
+                                          ij
+          with the value of the root mean square residual, S , defined as
+                                                            i
+                 
+
+                /     --
+               /      >
+              /       --
+             /        i
+            /     ----------, where m' is the number of data points with
+          \/      (m'+n-i-1)
+          non-zero weight.
+
+          Values of the approximations may subsequently be computed using
+          E02AEF or E02AKF.
+
+                                                    
+
+          First E02AGF determines a polynomial (mu)(x), of degree n-1,
+          which satisfies the given constraints, and a polynomial (nu)(x),
+          of degree n, which has value (or derivative) zero wherever a
+          constrained value (or derivative) is specified. It then fits
+          y -(mu)(x ), for r=1,2,...,m with polynomials of the required
+           r       r
+                                            
+
+          degree in x each with factor (nu)(x). Finally the coefficients of
+               
+
+          (mu)(x) are added to the coefficients of these fits to give the
+          coefficients of the constrained polynomial approximations to the
+          data points (x ,y ), for r=1,2,...,m. The method employed is
+                        r  r
+          given in Hayes [3]: it is an extension of Forsythe's orthogonal
+          polynomials method [2] as modified by Clenshaw [1].
+
+          4. References
+
+          [1]   Clenshaw C W (1960) Curve Fitting with a Digital Computer.
+                Comput. J. 2 170--173.
+
+          [2]   Forsythe G E (1957) Generation and use of orthogonal
+                polynomials for data fitting with a digital computer. J.
+                Soc. Indust. Appl. Math. 5 74--88.
+
+          [3]   Hayes J G (1970) Curve Fitting by Polynomials in One
+                Variable. Numerical Approximation to Functions and Data. (ed
+                J G Hayes) Athlone Press, London.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number m of data points to be fitted.
+               Constraint: M >= 1.
+
+           2:  KPLUS1 -- INTEGER                                      Input
+               On entry: k+1, where k is the maximum degree required.
+               Constraint: n+1<=KPLUS1<=m''+n, where n is the total number
+               of constraints and m'' is the number of data points with
+               non-zero weights and distinct abscissae which do not
+               coincide with any of the XF(r).
+
+           3:  NROWS -- INTEGER                                       Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which E02AGF is called.
+               Constraint: NROWS >= KPLUS1.
+
+           4:  XMIN -- DOUBLE PRECISION                               Input
+
+           5:  XMAX -- DOUBLE PRECISION                               Input
+               On entry: the lower and upper end-points, respectively, of
+               the interval [x   ,x   ]. Unless there are specific reasons
+                              min  max
+               to the contrary, it is recommended that XMIN and XMAX be set
+               respectively to the lowest and highest value among the x
+                                                                       r
+               and XF(r). This avoids the danger of extrapolation provided
+               there is a constraint point or data point with non-zero
+               weight at each end-point. Constraint: XMAX > XMIN.
+
+           6:  X(M) -- DOUBLE PRECISION array                         Input
+               On entry: the value x  of the independent variable at the r
+                                    r
+               th data point, for r=1,2,...,m. Constraint: the X(r) must be
+               in non-decreasing order and satisfy XMIN <= X(r) <= XMAX.
+
+           7:  Y(M) -- DOUBLE PRECISION array                         Input
+               On entry: Y(r) must contain y , the value of the dependent
+                                            r
+               variable at the rth data point, for r=1,2,...,m.
+
+           8:  W(M) -- DOUBLE PRECISION array                         Input
+               On entry: the weights w  to be applied to the data points
+                                      r
+               x , for r=1,2...,m. For advice on the choice of weights see
+                r
+               the Chapter Introduction. Negative weights are treated as
+               positive. A zero weight causes the corresponding data point
+               to be ignored. Zero weight should be given to any data point
+               whose x and y values both coincide with those of a
+               constraint (otherwise the denominators involved in the root-
+               mean-square residuals s  will be slightly in error).
+                                      i
+
+           9:  MF -- INTEGER                                          Input
+               On entry: the number of values of the independent variable
+               at which a constraint is specified. Constraint: MF >= 1.
+
+          10:  XF(MF) -- DOUBLE PRECISION array                       Input
+               On entry: the rth value of the independent variable at
+               which a constraint is specified, for r = 1,2,...,MF.
+               Constraint: these values need not be ordered but must be
+               distinct and satisfy XMIN <= XF(r) <= XMAX.
+
+          11:  YF(LYF) -- DOUBLE PRECISION array                      Input
+               On entry: the values which the approximating polynomials
+               and their derivatives are required to take at the points
+               specified in XF. For each value of XF(r), YF contains in
+               successive elements the required value of the approximation,
+               its first derivative, second derivative,..., p th
+                                                             r
+               derivative, for r = 1,2,...,MF. Thus the value which the kth
+               derivative of each approximation (k=0 referring to the
+               approximation itself) is required to take at the point XF(r)
+               must be contained in YF(s), where
+                                  s=r+k+p +p +...+p   ,
+                                         1  2      r-1
+               for k=0,1,...,p  and r = 1,2,...,MF. The derivatives are
+                              r
+               with respect to the user's variable x.
+
+          12:  LYF -- INTEGER                                         Input
+               On entry:
+               the dimension of the array YF as declared in the
+               (sub)program from which E02AGF is called.
+               Constraint: LYF>=n, where n=MF+p +p +...+p  .
+                                               1  2      MF
+
+          13:  IP(MF) -- INTEGER array                                Input
+               On entry: IP(r) must contain p , the order of the highest-
+                                             r
+               order derivative specified at XF(r), for r = 1,2,...,MF.
+               p =0 implies that the value of the approximation at XF(r) is
+                r
+               specified, but not that of any derivative. Constraint: IP(r)
+               >= 0, for r=1,2,...,MF.
+
+          14:  A(NROWS,KPLUS1) -- DOUBLE PRECISION array             Output
+               On exit: A(i+1,j+1) contains the coefficient a   in the
+                                                             ij
+               approximating polynomial of degree i, for i=n,n+1,...,k;
+               j=0,1,...,i.
+
+          15:  S(KPLUS1) -- DOUBLE PRECISION array                   Output
+               On exit: S(i+1) contains s , for i=n,n+1,...,k, the root-
+                                          i
+               mean-square residual corresponding to the approximating
+               polynomial of degree i. In the case where the number of data
+               points with non-zero weight is equal to k+1-n, s  is
+                                                               i
+               indeterminate: the routine sets it to zero. For the
+               interpretation of the values of s  and their use in
+                                                i
+               selecting an appropriate degree, see Section 3.1 of the
+               Chapter Introduction.
+
+          16:  NP1 -- INTEGER                                        Output
+               On exit: n+1, where n is the total number of constraint
+               conditions imposed: n=MF+p +p +...+p  .
+                                         1  2      MF
+
+          17:  WRK(LWRK) -- DOUBLE PRECISION array                   Output
+               On exit: WRK contains weighted residuals of the highest
+               degree of fit determined (k). The residual at x  is in
+               element 2(n+1)+3(m+k+1)+r, for r=1,2,...,m. The rest of the
+               array is used as workspace.
+
+          18:  LWRK -- INTEGER                                        Input
+               On entry:
+               the dimension of the array WRK as declared in the
+               (sub)program from which E02AGF is called.
+               Constraint: LWRK>=max(4*M+3*KPLUS1, 8*n+5*IPMAX+MF+10)+2*n+2
+               , where IPMAX = max(IP(R)).
+
+          19:  IWRK(LIWRK) -- INTEGER array                       Workspace
+
+          20:  LIWRK -- INTEGER                                       Input
+               On entry:
+               the dimension of the array IWRK as declared in the
+               (sub)program from which E02AGF is called.
+               Constraint: LIWRK>=2*MF+2.
+
+          21:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry M < 1,
+
+               or       KPLUS1 < n + 1,
+
+               or       NROWS < KPLUS1,
+
+               or       MF < 1,
+
+               or       LYF < n,
+
+               or       LWRK is too small (see Section 5),
+
+               or       LIWRK<2*MF+2.
+               (Here n is the total number of constraint conditions.)
+
+          IFAIL= 2
+               IP(r) < 0 for some r = 1,2,...,MF.
+
+          IFAIL= 3
+               XMIN >= XMAX, or XF(r) is not in the interval XMIN to XMAX
+               for some r = 1,2,...,MF, or the XF(r) are not distinct.
+
+          IFAIL= 4
+               X(r) is not in the interval XMIN to XMAX for some
+               r=1,2,...,M.
+
+          IFAIL= 5
+               X(r) < X(r-1) for some r=2,3,...,M.
+
+          IFAIL= 6
+               KPLUS1>m''+n, where m'' is the number of data points with
+               non-zero weight and distinct abscissae which do not coincide
+               with any XF(r). Thus there is no unique solution.
+
+          IFAIL= 7
+               The polynomials (mu)(x) and/or (nu)(x) cannot be determined.
+               The problem supplied is too ill-conditioned. This may occur
+               when the constraint points are very close together, or large
+               in number, or when an attempt is made to constrain high-
+               order derivatives.
+
+          7. Accuracy
+
+          No complete error analysis exists for either the interpolating
+          algorithm or the approximating algorithm. However, considerable
+          experience with the approximating algorithm shows that it is
+          generally extremely satisfactory. Also the moderate number of
+          constraints, of low order, which are typical of data fitting
+          applications, are unlikely to cause difficulty with the
+          interpolating routine.
+
+          8. Further Comments
+
+          The time taken by the routine to form the interpolating
+                                                       3
+          polynomial is approximately proportional to n , and that to form
+          the approximating polynomials is very approximately proportional
+          to m(k+1)(k+1-n).
+
+          To carry out a least-squares polynomial fit without constraints,
+          use E02ADF. To carry out polynomial interpolation only, use
+          E01AEF(*).
+
+          9. Example
+
+          The example program reads data in the following order, using the
+          notation of the parameter list above:
+
+               MF
+
+               IP(i), XF(i), Y-value and derivative values (if any) at
+               XF(i), for i= 1,2,...,MF
+
+               M
+
+               X(i), Y(i), W(i), for i=1,2,...,M
+
+               k, XMIN, XMAX
+
+          The output is:
+
+               the root-mean-square residual for each degree from n to k;
+
+               the Chebyshev coefficients for the fit of degree k;
+
+               the data points, and the fitted values and residuals for
+               the fit of degree k.
+
+          The program is written in a generalized form which will read any
+          number of data sets.
+
+          The data set supplied specifies 5 data points in the interval [0.
+          0,4.0] with unit weights, to which are to be fitted polynomials,
+          p, of degrees up to 4, subject to the 3 constraints:
+
+               p(0.0)=1.0,  p'(0.0)=-2.0,  p(4.0)=9.0.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02ahf}{NAG On-line Documentation: e02ahf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02AHF(3NAG)      Foundation Library (12/10/92)      E02AHF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02AHF
+                  E02AHF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02AHF determines the coefficients in the Chebyshev-series
+          representation of the derivative of a polynomial given in
+          Chebyshev-series form.
+
+          2. Specification
+
+                 SUBROUTINE E02AHF (NP1, XMIN, XMAX, A, IA1, LA, PATM1,
+                1                   ADIF, IADIF1, LADIF, IFAIL)
+                 INTEGER          NP1, IA1, LA, IADIF1, LADIF, IFAIL
+                 DOUBLE PRECISION XMIN, XMAX, A(LA), PATM1, ADIF(LADIF)
+
+          3. Description
+
+          This routine forms the polynomial which is the derivative of a
+          given polynomial. Both the original polynomial and its derivative
+          are represented in Chebyshev-series form. Given the coefficients
+          a , for i=0,1,...,n, of a polynomial p(x) of degree n, where
+           i
+
+                                  1                    
+                            p(x)= -a +a T (x)+...+a T (x)
+                                  2 0  1 1         n n
+
+                                               
+
+          the routine returns the coefficients a , for i=0,1,...,n-1, of
+                                                i
+          the polynomial q(x) of degree n-1, where
+
+                            dp(x)  1                        
+                      q(x)= -----= -a +a T (x)+...+a   T   (x).
+                             dx    2 0  1 1         n-1 n-1
+
+                  
+
+          Here T (x) denotes the Chebyshev polynomial of the first kind of
+                j
+                                 
+
+          degree j with argument x. It is assumed that the normalised
+                   
+
+          variable x in the interval [-1,+1] was obtained from the user's
+          original variable x in the interval [x   ,x   ] by the linear
+                                                min  max
+          transformation
+
+                                     2x-(x   +x   )
+                                          max  min
+                                  x= --------------
+                                       x   -x
+                                        max  min
+
+          and that the user requires the derivative to be with respect to
+                                                            
+
+          the variable x. If the derivative with respect to x is required,
+          set x   =1 and x   =-1.
+               max        min
+
+          Values of the derivative can subsequently be computed, from the
+          coefficients obtained, by using E02AKF.
+
+          The method employed is that of [1] modified to obtain the
+                                                               
+
+          derivative with respect to x. Initially setting a   =a =0, the
+                                                           n+1  n
+          routine forms successively
+
+                                    2
+                     a   =a   + ---------2ia ,   i=n,n-1,...,1.
+                      i-1  i+1  x   -x      i
+                                 max  min
+
+          4. References
+
+          [1]   Unknown (1961) Chebyshev-series. Modern Computing Methods,
+                Chapter 8. NPL Notes on Applied Science (2nd Edition). 16
+                HMSO.
+
+          5. Parameters
+
+           1:  NP1 -- INTEGER                                         Input
+               On entry: n+1, where n is the degree of the given
+               polynomial p(x). Thus NP1 is the number of coefficients in
+               this polynomial. Constraint: NP1 >= 1.
+
+           2:  XMIN -- DOUBLE PRECISION                               Input
+
+           3:  XMAX -- DOUBLE PRECISION                               Input
+               On entry: the lower and upper end-points respectively of
+               the interval [x   ,x   ]. The Chebyshev-series
+                              min  max
+                                                                     
+
+               representation is in terms of the normalised variable x,
+               where
+                                       2x-(x   +x   )
+                                            max  min
+                                    x= --------------.
+                                         x   -x
+                                          max  min
+               Constraint: XMAX > XMIN.
+
+           4:  A(LA) -- DOUBLE PRECISION array                        Input
+               On entry: the Chebyshev coefficients of the polynomial p(x).
+               Specifically, element 1 + i*IA1 of A must contain the
+               coefficient a , for i=0,1,...,n. Only these n+1 elements
+                            i
+               will be accessed.
+
+               Unchanged on exit, but see ADIF, below.
+
+           5:  IA1 -- INTEGER                                         Input
+               On entry: the index increment of A. Most frequently the
+               Chebyshev coefficients are stored in adjacent elements of A,
+               and IA1 must be set to 1. However, if, for example, they are
+               stored in A(1),A(4),A(7),..., then the value of IA1 must be
+               3. See also Section 8. Constraint: IA1 >= 1.
+
+           6:  LA -- INTEGER                                          Input
+               On entry:
+               the dimension of the array A as declared in the (sub)program
+               from which E02AHF is called.
+               Constraint: LA>=1+(NP1-1)*IA1.
+
+           7:  PATM1 -- DOUBLE PRECISION                             Output
+               On exit: the value of p(x   ). If this value is passed to
+                                         min
+               the integration routine E02AJF with the coefficients of q(x)
+               , then the original polynomial p(x) is recovered, including
+               its constant coefficient.
+
+           8:  ADIF(LADIF) -- DOUBLE PRECISION array                 Output
+               On exit: the Chebyshev coefficients of the derived
+               polynomial q(x). (The differentiation is with respect to the
+               variable x). Specifically, element 1+i*IADIF1 of ADIF
+                                        
+
+               contains the coefficient a , i=0,1,...n-1. Additionally
+                                         i
+               element 1+n*IADIF1 is set to zero. A call of the routine may
+               have the array name ADIF the same as A, provided that note
+               is taken of the order in which elements are overwritten,
+               when choosing the starting elements and increments IA1 and
+               IADIF1: i.e., the coefficients a ,a ,...,a    must be intact
+                                               0  1      i-1
+                                 
+
+               after coefficient a  is stored. In particular, it is
+                                  i
+               possible to overwrite the a  completely by having IA1 =
+                                          i
+               IADIF1, and the actual arrays for A and ADIF identical.
+
+           9:  IADIF1 -- INTEGER                                      Input
+               On entry: the index increment of ADIF. Most frequently the
+               Chebyshev coefficients are required in adjacent elements of
+               ADIF, and IADIF1 must be set to 1. However, if, for example,
+               they are to be stored in ADIF(1),ADIF(4),ADIF(7),..., then
+               the value of IADIF1 must be 3. See Section 8. Constraint:
+               IADIF1 >= 1.
+
+          10:  LADIF -- INTEGER                                       Input
+               On entry:
+               the dimension of the array ADIF as declared in the
+               (sub)program from which E02AHF is called.
+               Constraint: LADIF>=1+(NP1-1)*IADIF1.
+
+          11:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry NP1 < 1,
+
+               or       XMAX <= XMIN,
+
+               or       IA1 < 1,
+
+               or       LA<=(NP1-1)*IA1,
+
+               or       IADIF1 < 1,
+
+               or       LADIF<=(NP1-1)*IADIF1.
+
+          7. Accuracy
+
+          There is always a loss of precision in numerical differentiation,
+          in this case associated with the multiplication by 2i in the
+          formula quoted in Section 3.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          n+1.
+
+          The increments IA1, IADIF1 are included as parameters to give a
+          degree of flexibility which, for example, allows a polynomial in
+          two variables to be differentiated with respect to either
+          variable without rearranging the coefficients.
+
+          9. Example
+
+          Suppose a polynomial has been computed in Chebyshev-series form
+          to fit data over the interval [-0.5,2.5]. The example program
+          evaluates the 1st and 2nd derivatives of this polynomial at 4
+          equally spaced points over the interval. (For the purposes of
+          this example, XMIN, XMAX and the Chebyshev coefficients are
+          simply supplied in DATA statements. Normally a program would
+          first read in or generate data and compute the fitted
+          polynomial.)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02ajf}{NAG On-line Documentation: e02ajf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02AJF(3NAG)      Foundation Library (12/10/92)      E02AJF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02AJF
+                  E02AJF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02AJF determines the coefficients in the Chebyshev-series
+          representation of the indefinite integral of a polynomial given
+          in Chebyshev-series form.
+
+          2. Specification
+
+                 SUBROUTINE E02AJF (NP1, XMIN, XMAX, A, IA1, LA, QATM1,
+                1                   AINT, IAINT1, LAINT, IFAIL)
+                 INTEGER          NP1, IA1, LA, IAINT1, LAINT, IFAIL
+                 DOUBLE PRECISION XMIN, XMAX, A(LA), QATM1, AINT(LAINT)
+
+          3. Description
+
+          This routine forms the polynomial which is the indefinite
+          integral of a given polynomial. Both the original polynomial and
+          its integral are represented in Chebyshev-series form. If
+          supplied with the coefficients a , for i=0,1,...,n, of a
+                                          i
+          polynomial p(x) of degree n, where
+
+                                 1                    
+                           p(x)= -a +a T (x)+...+a T (x),
+                                 2 0  1 1         n n
+
+          the routine returns the coefficients a' , for i=0,1,...,n+1, of
+                                                 i
+          the polynomial q(x) of degree n+1, where
+
+                              1                           
+                        q(x)= -a' +a' T (x)+...+a'   T   (x),
+                              2  0   1 1          n+1 n+1
+
+          and
+
+                                         /
+                                   q(x)= |p(x)dx.
+                                         /
+
+                  
+
+          Here T (x) denotes the Chebyshev polynomial of the first kind of
+                j
+          degree j with argument x. It is assumed that the normalised
+                   
+
+          variable x in the interval [-1,+1] was obtained from the user's
+          original variable x in the interval [x   ,x   ] by the linear
+                                                min  max
+          transformation
+
+                                     2x-(x   +x   )
+                                          max  min
+                                  x= --------------
+                                       x   -x
+                                        max  min
+
+          and that the user requires the integral to be with respect to the
+                                                      
+
+          variable x. If the integral with respect to x is required, set
+          x   =1 and x   =-1.
+           max        min
+
+          Values of the integral can subsequently be computed, from the
+          coefficients obtained, by using E02AKF.
+
+          The method employed is that of Chebyshev-series [1] modified for
+          integrating with respect to x. Initially taking a   =a   =0, the
+                                                           n+1  n+2
+          routine forms successively
+
+                          a   -a     x   -x
+                           i-1  i+1   max  min
+                     a' = ---------* ---------,   i=n+1,n,...,1.
+                       i     2i          2
+
+          The constant coefficient a'  is chosen so that q(x) is equal to a
+                                     0
+          specified value, QATM1, at the lower end-point of the interval on
+                                     
+
+          which it is defined, i.e., x=-1, which corresponds to x=x   .
+                                                                   min
+
+          4. References
+
+          [1]   Unknown (1961) Chebyshev-series. Modern Computing Methods,
+                Chapter 8. NPL Notes on Applied Science (2nd Edition). 16
+                HMSO.
+
+          5. Parameters
+
+           1:  NP1 -- INTEGER                                         Input
+               On entry: n+1, where n is the degree of the given
+               polynomial p(x). Thus NP1 is the number of coefficients in
+               this polynomial. Constraint: NP1 >= 1.
+
+           2:  XMIN -- DOUBLE PRECISION                               Input
+
+           3:  XMAX -- DOUBLE PRECISION                               Input
+               On entry: the lower and upper end-points respectively of
+               the interval [x   ,x   ]. The Chebyshev-series
+                              min  max
+                                                                     
+
+               representation is in terms of the normalised variable x,
+               where
+                                       2x-(x   +x   )
+                                            max  min
+                                    x= --------------.
+                                         x   -x
+                                          max  min
+               Constraint: XMAX > XMIN.
+
+           4:  A(LA) -- DOUBLE PRECISION array                        Input
+               On entry: the Chebyshev coefficients of the polynomial p(x)
+               . Specifically, element 1+i*IA1 of A must contain the
+               coefficient a , for i=0,1,...,n. Only these n+1 elements
+                            i
+               will be accessed.
+
+               Unchanged on exit, but see AINT, below.
+
+           5:  IA1 -- INTEGER                                         Input
+               On entry: the index increment of A. Most frequently the
+               Chebyshev coefficients are stored in adjacent elements of A,
+               and IA1 must be set to 1. However, if for example, they are
+               stored in A(1),A(4),A(7),..., then the value of IA1 must be
+               3. See also Section 8. Constraint: IA1 >= 1.
+
+           6:  LA -- INTEGER                                          Input
+               On entry:
+               the dimension of the array A as declared in the (sub)program
+               from which E02AJF is called.
+               Constraint: LA>=1+(NP1-1)*IA1.
+
+           7:  QATM1 -- DOUBLE PRECISION                              Input
+               On entry: the value that the integrated polynomial is
+               required to have at the lower end-point of its interval of
+                                    
+
+               definition, i.e., at x=-1 which corresponds to x=x   . Thus,
+                                                                 min
+               QATM1 is a constant of integration and will normally be set
+               to zero by the user.
+
+           8:  AINT(LAINT) -- DOUBLE PRECISION array                 Output
+               On exit: the Chebyshev coefficients of the integral q(x).
+               (The integration is with respect to the variable x, and the
+               constant coefficient is chosen so that q(x   ) equals QATM1)
+                                                         min
+               Specifically, element 1+i*IAINT1 of AINT contains the
+               coefficient a' , for i=0,1,...,n+1. A call of the routine
+                             i
+               may have the array name AINT the same as A, provided that
+               note is taken of the order in which elements are overwritten
+               when choosing starting elements and increments IA1 and
+               IAINT1: i.e., the coefficients, a ,a ,...,a    must be
+                                                0  1      i-2
+               intact after coefficient a'  is stored. In particular it is
+                                          i
+               possible to overwrite the a  entirely by having IA1 =
+                                          i
+               IAINT1, and the actual array for A and AINT identical.
+
+           9:  IAINT1 -- INTEGER                                      Input
+               On entry: the index increment of AINT. Most frequently the
+               Chebyshev coefficients are required in adjacent elements of
+               AINT, and IAINT1 must be set to 1. However, if, for example,
+               they are to be stored in AINT(1),AINT(4),AINT(7),..., then
+               the value of IAINT1 must be 3. See also Section 8.
+               Constraint: IAINT1 >= 1.
+
+          10:  LAINT -- INTEGER                                       Input
+               On entry:
+               the dimension of the array AINT as declared in the
+               (sub)program from which E02AJF is called.
+               Constraint: LAINT>=1+NP1*IAINT1.
+
+          11:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry NP1 < 1,
+
+               or       XMAX <= XMIN,
+
+               or       IA1 < 1,
+
+               or       LA<=(NP1-1)*IA1,
+
+               or       IAINT1 < 1,
+
+               or       LAINT<=NP1*IAINT1.
+
+          7. Accuracy
+
+          In general there is a gain in precision in numerical integration,
+          in this case associated with the division by 2i in the formula
+          quoted in Section 3.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          n+1.
+
+          The increments IA1, IAINT1 are included as parameters to give a
+          degree of flexibility which, for example, allows a polynomial in
+          two variables to be integrated with respect to either variable
+          without rearranging the coefficients.
+
+          9. Example
+
+          Suppose a polynomial has been computed in Chebyshev-series form
+          to fit data over the interval [-0.5,2.5]. The example program
+          evaluates the integral of the polynomial from 0.0 to 2.0. (For
+          the purpose of this example, XMIN, XMAX and the Chebyshev
+          coefficients are simply supplied in DATA statements. Normally a
+          program would read in or generate data and compute the fitted
+          polynomial).
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02akf}{NAG On-line Documentation: e02akf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02AKF(3NAG)      Foundation Library (12/10/92)      E02AKF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02AKF
+                  E02AKF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02AKF evaluates a polynomial from its Chebyshev-series
+          representation, allowing an arbitrary index increment for
+          accessing the array of coefficients.
+
+          2. Specification
+
+                 SUBROUTINE E02AKF (NP1, XMIN, XMAX, A, IA1, LA, X, RESULT,
+                1                   IFAIL)
+                 INTEGER          NP1, IA1, LA, IFAIL
+                 DOUBLE PRECISION XMIN, XMAX, A(LA), X, RESULT
+
+          3. Description
+
+          If supplied with the coefficients a , for i=0,1,...,n, of a
+                                             i
+                       
+
+          polynomial p(x) of degree n, where
+
+                                 1                    
+                           p(x)= -a +a T (x)+...+a T (x),
+                                 2 0  1 1         n n
+
+                                              
+
+          this routine returns the value of p(x) at a user-specified value
+                                     
+
+          of the variable x. Here T (x) denotes the Chebyshev polynomial of
+                                   j
+                                                   
+
+          the first kind of degree j with argument x. It is assumed that
+                                   
+
+          the independent variable x in the interval [-1,+1] was obtained
+          from the user's original variable x in the interval [x   ,x   ]
+                                                                min  max
+          by the linear transformation
+
+                                    2x-(x   +x   )
+                                         max  min
+                                 x= --------------.
+                                      x   -x
+                                       max  min
+
+          The coefficients a  may be supplied in the array A, with any
+                            i
+          increment between the indices of array elements which contain
+          successive coefficients. This enables the routine to be used in
+          surface fitting and other applications, in which the array might
+          have two or more dimensions.
+
+          The method employed is based upon the three-term recurrence
+          relation due to Clenshaw [1], with modifications due to Reinsch
+          and Gentleman (see [4]). For further details of the algorithm and
+          its use see Cox [2] and Cox and Hayes [3].
+
+          4. References
+
+          [1]   Clenshaw C W (1955) A Note on the Summation of Chebyshev-
+                series. Math. Tables Aids Comput. 9 118--120.
+
+          [2]   Cox M G (1973) A data-fitting package for the non-specialist
+                user. Report NAC40. National Physical Laboratory.
+
+          [3]   Cox M G and Hayes J G (1973) Curve fitting: a guide and
+                suite of algorithms for the non-specialist user. Report
+                NAC26. National Physical Laboratory.
+
+          [4]   Gentlemen W M (1969) An Error Analysis of Goertzel's
+                (Watt's) Method for Computing Fourier Coefficients. Comput.
+                J. 12 160--165.
+
+          5. Parameters
+
+           1:  NP1 -- INTEGER                                         Input
+               On entry: n+1, where n is the degree of the given
+                            
+
+               polynomial p(x). Constraint: NP1 >= 1.
+
+           2:  XMIN -- DOUBLE PRECISION                               Input
+
+           3:  XMAX -- DOUBLE PRECISION                               Input
+               On entry: the lower and upper end-points respectively of
+               the interval [x   ,x   ]. The Chebyshev-series
+                              min  max
+                                                                     
+
+               representation is in terms of the normalised variable x,
+               where
+                                       2x-(x   +x   )
+                                            max  min
+                                    x= --------------.
+                                         x   -x
+                                          max  min
+               Constraint: XMIN < XMAX.
+
+           4:  A(LA) -- DOUBLE PRECISION array                        Input
+               On entry: the Chebyshev coefficients of the polynomial p(x).
+               Specifically, element 1+i*IA1 must contain the coefficient
+               a , for i=0,1,...,n. Only these n+1 elements will be
+                i
+               accessed.
+
+           5:  IA1 -- INTEGER                                         Input
+               On entry: the index increment of A. Most frequently, the
+               Chebyshev coefficients are stored in adjacent elements of A,
+               and IA1 must be set to 1. However, if, for example, they are
+               stored in A(1),A(4),A(7),..., then the value of IA1 must be
+               3. Constraint: IA1 >= 1.
+
+           6:  LA -- INTEGER                                          Input
+               On entry:
+               the dimension of the array A as declared in the (sub)program
+               from which E02AKF is called.
+               Constraint: LA>=(NP1-1)*IA1+1.
+
+           7:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x at which the polynomial is to be
+               evaluated. Constraint: XMIN <= X <= XMAX.
+
+           8:  RESULT -- DOUBLE PRECISION                            Output
+                                                      
+
+               On exit: the value of the polynomial p(x).
+
+           9:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry NP1 < 1,
+
+               or       IA1 < 1,
+
+               or       LA<=(NP1-1)*IA1,
+
+               or       XMIN >= XMAX.
+
+          IFAIL= 2
+               X does not satisfy the restriction XMIN <= X <= XMAX.
+
+          7. Accuracy
+
+          The rounding errors are such that the computed value of the
+          polynomial is exact for a slightly perturbed set of coefficients
+          a +(delta)a . The ratio of the sum of the absolute values of the
+           i         i
+          (delta)a  to the sum of the absolute values of the a  is less
+                  i                                           i
+          than a small multiple of (n+1)*machine precision.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          n+1.
+
+          9. Example
+
+          Suppose a polynomial has been computed in Chebyshev-series form
+          to fit data over the interval [-0.5,2.5]. The example program
+          evaluates the polynomial at 4 equally spaced points over the
+          interval. (For the purposes of this example, XMIN, XMAX and the
+          Chebyshev coefficients are supplied in DATA statements. Normally
+          a program would first read in or generate data and compute the
+          fitted polynomial.)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02baf}{NAG On-line Documentation: e02baf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02BAF(3NAG)      Foundation Library (12/10/92)      E02BAF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02BAF
+                  E02BAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02BAF computes a weighted least-squares approximation to an
+          arbitrary set of data points by a cubic spline with knots
+          prescribed by the user. Cubic spline interpolation can also be
+          carried out.
+
+          2. Specification
+
+                 SUBROUTINE E02BAF (M, NCAP7, X, Y, W, LAMDA, WORK1, WORK2,
+                1                   C, SS, IFAIL)
+                 INTEGER          M, NCAP7, IFAIL
+                 DOUBLE PRECISION X(M), Y(M), W(M), LAMDA(NCAP7), WORK1(M),
+                1                 WORK2(4*NCAP7), C(NCAP7), SS
+
+          3. Description
+
+          This routine determines a least-squares cubic spline
+          approximation s(x) to the set of data points (x ,y ) with weights
+                                                         r  r
+                                                               
+
+          w , for r=1,2,...,m. The value of NCAP7 = n+7, where n is the
+           r
+          number of intervals of the spline (one greater than the number of
+          interior knots), and the values of the knots
+          (lambda) ,(lambda) ,...,(lambda)   , interior to the data
+                  5         6             n+3
+          interval, are prescribed by the user.
+
+          s(x) has the property that it minimizes (theta), the sum of
+          squares of the weighted residuals (epsilon) , for r=1,2,...,m,
+                                                     r
+          where
+
+                              (epsilon) =w (y -s(x )).
+                                       r  r  r    r
+
+          The routine produces this minimizing value of (theta) and the
+                                             
+
+          coefficients c ,c ,...,c , where q=n+3, in the B-spline
+                        1  2      q
+          representation
+
+                                        q
+                                        --
+                                  s(x)= >  c N (x).
+                                        --  i i
+                                        i=1
+
+          Here N (x) denotes the normalised B-spline of degree 3 defined
+                i
+          upon the knots (lambda) ,(lambda)   ,...,(lambda)   .
+                                 i         i+1             i+4
+
+          In order to define the full set of B-splines required, eight
+          additional knots (lambda) ,(lambda) ,(lambda) ,(lambda)  and
+                                   1         2         3         4
+          (lambda)   ,(lambda)-  ,(lambda)   ,(lambda)    are inserted
+                  n+4         n+5         n+6         n+7
+          automatically by the routine. The first four of these are set
+          equal to the smallest x  and the last four to the largest x .
+                                 r                                   r
+
+          The representation of s(x) in terms of B-splines is the most
+
+          compact form possible in that only n+3 coefficients, in addition
+                 
+
+          to the n+7 knots, fully define s(x).
+
+          The method employed involves forming and then computing the
+          least-squares solution of a set of m linear equations in the
+                                     
+
+          coefficients c  (i=1,2,...,n+3). The equations are formed using a
+                        i
+          recurrence relation for B-splines that is unconditionally stable
+          (Cox [1], de Boor [5]), even for multiple (coincident) knots. The
+          least-squares solution is also obtained in a stable manner by
+          using orthogonal transformations, viz. a variant of Givens
+          rotations (Gentleman [6] and [7]). This requires only one
+          equation to be stored at a time. Full advantage is taken of the
+          structure of the equations, there being at most four non-zero
+          values of N (x) for any value of x and hence at most four
+                     i
+          coefficients in each equation.
+
+          For further details of the algorithm and its use see Cox [2], [3]
+          and [4].
+
+          Subsequent evaluation of s(x) from its B-spline representation
+          may be carried out using E02BBF. If derivatives of s(x) are also
+          required, E02BCF may be used. E02BDF can be used to compute the
+          definite integral of s(x).
+
+          4. References
+
+          [1]   Cox M G (1972) The Numerical Evaluation of B-splines. J.
+                Inst. Math. Appl. 10 134--149.
+
+          [2]   Cox M G (1974) A Data-fitting Package for the Non-specialist
+                User. Software for Numerical Mathematics. (ed D J Evans)
+                Academic Press.
+
+          [3]   Cox M G (1975) Numerical methods for the interpolation and
+                approximation of data by spline functions. PhD Thesis. City
+                University, London.
+
+          [4]   Cox M G and Hayes J G (1973) Curve fitting: a guide and
+                suite of algorithms for the non-specialist user. Report
+                NAC26. National Physical Laboratory.
+
+          [5]   De Boor C (1972) On Calculating with B-splines. J. Approx.
+                Theory. 6 50--62.
+
+          [6]   Gentleman W M (1974) Algorithm AS 75. Basic Procedures for
+                Large Sparse or Weighted Linear Least-squares Problems.
+                Appl. Statist. 23 448--454.
+
+          [7]   Gentleman W M (1973) Least-squares Computations by Givens
+                Transformations without Square Roots. J. Inst. Math. Applic.
+                12 329--336.
+
+          [8]   Schoenberg I J and Whitney A (1953) On Polya Frequency
+                Functions III. Trans. Amer. Math. Soc. 74  246--259.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number m of data points. Constraint: M >=
+               MDIST >= 4, where MDIST is the number of distinct x values
+               in the data.
+
+           2:  NCAP7 -- INTEGER                                       Input
+                                    
+
+               On entry: n+7, where n is the number of intervals of the
+               spline (which is one greater than the number of interior
+               knots, i.e., the knots strictly within the range x  to x )
+                                                                 1     m
+               over which the spline is defined. Constraint: 8 <= NCAP7 <=
+               MDIST + 4, where MDIST is the number of distinct x values in
+               the data.
+
+           3:  X(M) -- DOUBLE PRECISION array                         Input
+               On entry: the values x  of the independent variable
+                                     r
+               (abscissa), for r=1,2,...,m. Constraint: x <=x <=...<=x .
+                                                         1   2        m
+
+           4:  Y(M) -- DOUBLE PRECISION array                         Input
+               On entry: the values y  of the dependent variable
+                                     r
+               (ordinate), for r=1,2,...,m.
+
+           5:  W(M) -- DOUBLE PRECISION array                         Input
+               On entry: the values w  of the weights, for r=1,2,...,m.
+                                     r
+               For advice on the choice of weights, see the Chapter
+               Introduction. Constraint: W(r) > 0, for r=1,2,...,m.
+
+           6:  LAMDA(NCAP7) -- DOUBLE PRECISION array          Input/Output
+               On entry: LAMDA(i) must be set to the (i-4)th (interior)
+
+               knot, (lambda) , for i=5,6,...,n+3. Constraint: X(1) < LAMDA
+                             i
+               (5) <= LAMDA(6) <=... <= LAMDA(NCAP7-4) < X(M). On exit: the
+               input values are unchanged, and LAMDA(i), for i = 1, 2, 3,
+               4, NCAP7-3, NCAP7-2, NCAP7-1, NCAP7 contains the additional
+               (exterior) knots introduced by the routine. For advice on
+               the choice of knots, see Section 3.3 of the Chapter
+               Introduction.
+
+           7:  WORK1(M) -- DOUBLE PRECISION array                 Workspace
+
+           8:  WORK2(4*NCAP7) -- DOUBLE PRECISION array           Workspace
+
+           9:  C(NCAP7) -- DOUBLE PRECISION array                    Output
+               On exit: the coefficient c  of the B-spline N (x), for
+                                         i                  i
+                         
+
+               i=1,2,...,n+3. The remaining elements of the array are not
+               used.
+
+          10:  SS -- DOUBLE PRECISION                                Output
+               On exit: the residual sum of squares, (theta).
+
+          11:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               The knots fail to satisfy the condition
+
+               X(1) < LAMDA(5) <= LAMDA(6) <=... <= LAMDA(NCAP7-4) < X(M).
+               Thus the knots are not in correct order or are not interior
+               to the data interval.
+
+          IFAIL= 2
+               The weights are not all strictly positive.
+
+          IFAIL= 3
+               The values of X(r), for r = 1,2,...,M are not in non-
+               decreasing order.
+
+          IFAIL= 4
+               NCAP7 < 8 (so the number of interior knots is negative) or
+               NCAP7 > MDIST + 4, where MDIST is the number of distinct x
+               values in the data (so there cannot be a unique solution).
+
+          IFAIL= 5
+               The conditions specified by Schoenberg and Whitney [8] fail
+               to hold for at least one subset of the distinct data
+               abscissae. That is, there is no subset of NCAP7-4 strictly
+               increasing values, X(R(1)),X(R(2)),...,X(R(NCAP7-4)), among
+               the abscissae such that
+                    X(R(1)) < LAMDA(1) < X(R(5)),
+
+                    X(R(2)) < LAMDA(2) < X(R(6)),
+
+                    ...
+
+                    X(R(NCAP7-8)) < LAMDA(NCAP7-8) < X(R(NCAP7-4)).
+               This means that there is no unique solution: there are
+               regions containing too many knots compared with the number
+               of data points.
+
+          7. Accuracy
+
+          The rounding errors committed are such that the computed
+          coefficients are exact for a slightly perturbed set of ordinates
+          y +(delta)y . The ratio of the root-mean-square value for the
+           r         r
+          (delta)y  to the root-mean-square value of the y  can be expected
+                  r                                       r
+          to be less than a small multiple of (kappa)*m*machine precision,
+          where (kappa) is a condition number for the problem. Values of
+          (kappa) for 20-30 practical data sets all proved to lie between
+          4.5 and 7.8 (see Cox [3]). (Note that for these data sets,
+          replacing the coincident end knots at the end-points x  and x
+                                                                1      m
+          used in the routine by various choices of non-coincident exterior
+          knots gave values of (kappa) between 16 and 180. Again see Cox
+          [3] for further details.) In general we would not expect (kappa)
+          to be large unless the choice of knots results in near-violation
+          of the Schoenberg-Whitney conditions.
+
+          A cubic spline which adequately fits the data and is free from
+          spurious oscillations is more likely to be obtained if the knots
+          are chosen to be grouped more closely in regions where the
+          function (underlying the data) or its derivatives change more
+          rapidly than elsewhere.
+
+          8. Further Comments
+
+                                                               
+
+          The time taken by the routine is approximately C*(2m+n+7)
+          seconds, where C is a machine-dependent constant.
+
+          Multiple knots are permitted as long as their multiplicity does
+          not exceed 4, i.e., the complete set of knots must satisfy
+                                               
+
+          (lambda) <(lambda)   , for i=1,2,...,n+3, (cf. Section 6). At a
+                  i         i+4
+          knot of multiplicity one (the usual case), s(x) and its first two
+          derivatives are continuous. At a knot of multiplicity two, s(x)
+          and its first derivative are continuous. At a knot of
+          multiplicity three, s(x) is continuous, and at a knot of
+          multiplicity four, s(x) is generally discontinous.
+
+          The routine can be used efficiently for cubic spline
+                                   
+
+          interpolation, i.e.,if m=n+3. The abscissae must then of course
+          satisfy x <x <...<x . Recommended values for the knots in this
+                   1  2      m
+                                                 
+
+          case are (lambda) =x   , for i=5,6,...,n+3.
+                           i  i-2
+
+          9. Example
+
+          Determine a weighted least-squares cubic spline approximation
+          with five intervals (four interior knots) to a set of 14 given
+          data points. Tabulate the data and the corresponding values of
+          the approximating spline, together with the residual errors, and
+          also the values of the approximating spline at points half-way
+          between each pair of adjacent data points.
+
+          The example program is written in a general form that will enable
+                                                         
+
+          a cubic spline approximation with n intervals (n-1 interior
+          knots) to be obtained to m data points, with arbitrary positive
+          weights, and the approximation to be tabulated. Note that E02BBF
+          is used to evaluate the approximating spline. The program is
+          self-starting in that any number of data sets can be supplied.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02bbf}{NAG On-line Documentation: e02bbf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02BBF(3NAG)      Foundation Library (12/10/92)      E02BBF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02BBF
+                  E02BBF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02BBF evaluates a cubic spline from its B-spline representation.
+
+          2. Specification
+
+                 SUBROUTINE E02BBF (NCAP7, LAMDA, C, X, S, IFAIL)
+                 INTEGER          NCAP7, IFAIL
+                 DOUBLE PRECISION LAMDA(NCAP7), C(NCAP7), X, S
+
+          3. Description
+
+          This routine evaluates the cubic spline s(x) at a prescribed
+          argument x from its augmented knot set (lambda) , for
+                                                         i
+                    
+
+          i=1,2,...,n+7, (see E02BAF) and from the coefficients c , for
+                                                                 i
+          i=1,2,...,q in its B-spline representation
+
+                                        q
+                                        --
+                                  s(x)= >  c N (x)
+                                        --  i i
+                                        i=1
+
+                            
+
+          Here q=n+3, where n is the number of intervals of the spline, and
+          N (x) denotes the normalised B-spline of degree 3 defined upon
+           i
+          the knots (lambda) ,(lambda)   ,...,(lambda)   . The prescribed
+                            i         i+1             i+4
+          argument x must satisfy (lambda) <=x<=(lambda)   .
+                                          4             n+4
+
+                                                                   
+
+          It is assumed that (lambda) >=(lambda)   , for j=2,3,...,n+7, and
+                                     j          j-1
+          (lambda)   >(lambda) .
+                              4
+                  n+4
+
+          The method employed is that of evaluation by taking convex
+          combinations due to de Boor [4]. For further details of the
+          algorithm and its use see Cox [1] and [3].
+
+          It is expected that a common use of E02BBF will be the evaluation
+          of the cubic spline approximations produced by E02BAF. A
+          generalization of E02BBF which also forms the derivative of s(x)
+          is E02BCF. E02BCF takes about 50% longer than E02BBF.
+
+          4. References
+
+          [1]   Cox M G (1972) The Numerical Evaluation of B-splines. J.
+                Inst. Math. Appl. 10 134--149.
+
+          [2]   Cox M G (1978) The Numerical Evaluation of a Spline from its
+                B-spline Representation. J. Inst. Math. Appl. 21 135--143.
+
+          [3]   Cox M G and Hayes J G (1973) Curve fitting: a guide and
+                suite of algorithms for the non-specialist user. Report
+                NAC26. National Physical Laboratory.
+
+          [4]   De Boor C (1972) On Calculating with B-splines. J. Approx.
+                Theory. 6 50--62.
+
+          5. Parameters
+
+           1:  NCAP7 -- INTEGER                                       Input
+                                    
+
+               On entry: n+7, where n is the number of intervals (one
+               greater than the number of interior knots, i.e., the knots
+               strictly within the range (lambda)  to (lambda)   ) over
+                                                 4            
+                                                              n+4
+               which the spline is defined. Constraint: NCAP7 >= 8.
+
+           2:  LAMDA(NCAP7) -- DOUBLE PRECISION array                 Input
+               On entry: LAMDA(j) must be set to the value of the jth
+               member of the complete set of knots, (lambda)  for
+                                                            j
+                         
+
+               j=1,2,...,n+7. Constraint: the LAMDA(j) must be in non-
+               decreasing order with LAMDA(NCAP7-3) > LAMDA(4).
+
+           3:  C(NCAP7) -- DOUBLE PRECISION array                     Input
+               On entry: the coefficient c  of the B-spline N (x), for
+                                          i                  i
+                         
+
+               i=1,2,...,n+3. The remaining elements of the array are not
+               used.
+
+           4:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x at which the cubic spline is to be
+               evaluated. Constraint: LAMDA(4) <= X <= LAMDA(NCAP7-3).
+
+           5:  S -- DOUBLE PRECISION                                 Output
+               On exit: the value of the spline, s(x).
+
+           6:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               The argument X does not satisfy LAMDA(4) <= X <= LAMDA(
+               NCAP7-3).
+
+               In this case the value of S is set arbitrarily to zero.
+
+          IFAIL= 2
+               NCAP7 < 8, i.e., the number of interior knots is negative.
+
+          7. Accuracy
+
+          The computed value of s(x) has negligible error in most practical
+          situations. Specifically, this value has an absolute error
+          bounded in modulus by 18*c   * machine precision, where c    is
+                                    max                            max
+          the largest in modulus of c ,c   ,c    and c   , and j is an
+                                     j  j+1  j+2      j+3
+          integer such that (lambda)   <=x<=(lambda)   . If c ,c   ,c
+                                    j+3             j+4      j  j+1  j+2
+          and c    are all of the same sign, then the computed value of
+               j+3
+          s(x) has a relative error not exceeding 20*machine precision in
+          modulus. For further details see Cox [2].
+
+          8. Further Comments
+
+                                                                      
+
+          The time taken by the routine is approximately C*(1+0.1*log(n+7))
+          seconds, where C is a machine-dependent constant.
+
+          Note: the routine does not test all the conditions on the knots
+          given in the description of LAMDA in Section 5, since to do this
+          would result in a computation time approximately linear in n+7
+          instead of log(n+7). All the conditions are tested in E02BAF,
+          however.
+
+          9. Example
+
+          Evaluate at 9 equally-spaced points in the interval 1.0<=x<=9.0
+          the cubic spline with (augmented) knots 1.0, 1.0, 1.0, 1.0, 3.0,
+          6.0, 8.0, 9.0, 9.0, 9.0, 9.0 and normalised cubic B-spline
+          coefficients 1.0, 2.0, 4.0, 7.0, 6.0, 4.0, 3.0.
+
+          The example program is written in a general form that will enable
+                              
+
+          a cubic spline with n intervals, in its normalised cubic B-spline
+          form, to be evaluated at m equally-spaced points in the interval
+                                 
+
+          LAMDA(4) <= x <= LAMDA(n+4). The program is self-starting in that
+          any number of data sets may be supplied.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02bcf}{NAG On-line Documentation: e02bcf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02BCF(3NAG)      Foundation Library (12/10/92)      E02BCF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02BCF
+                  E02BCF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02BCF evaluates a cubic spline and its first three derivatives
+          from its B-spline representation.
+
+          2. Specification
+
+                 SUBROUTINE E02BCF (NCAP7, LAMDA, C, X, LEFT, S, IFAIL)
+                 INTEGER          NCAP7, LEFT, IFAIL
+                 DOUBLE PRECISION LAMDA(NCAP7), C(NCAP7), X, S(4)
+
+          3. Description
+
+          This routine evaluates the cubic spline s(x) and its first three
+          derivatives at a prescribed argument x. It is assumed that s(x)
+          is represented in terms of its B-spline coefficients c , for
+                                                                i
+                    
+
+          i=1,2,...,n+3 and (augmented) ordered knot set (lambda) , for
+                                                                 i
+                    
+
+          i=1,2,...,n+7, (see E02BAF), i.e.,
+
+                                        q
+                                        --
+                                  s(x)= >  c N (x)
+                                        --  i i
+                                        i=1
+
+                      
+
+          Here q=n+3, n is the number of intervals of the spline and N (x)
+                                                                      i
+          denotes the normalised B-spline of degree 3 (order 4) defined
+          upon the knots (lambda) ,(lambda)   ,...,(lambda)   . The
+                                 i         i+1             i+4
+          prescribed argument x must satisfy
+
+                              (lambda) <=x<=(lambda)
+                                      4             n+4
+
+          At a simple knot (lambda)  (i.e., one satisfying
+                                   i
+          (lambda)   <(lambda) <(lambda)   ), the third derivative of the
+                  i-1         i         i+1
+          spline is in general discontinuous. At a multiple knot (i.e., two
+          or more knots with the same value), lower derivatives, and even
+          the spline itself, may be discontinuous. Specifically, at a point
+          x=u where (exactly) r knots coincide (such a point is termed a
+          knot of multiplicity r), the values of the derivatives of order
+          4-j, for j=1,2,...,r, are in general discontinuous. (Here
+          1<=r<=4;r>4 is not meaningful.) The user must specify whether the
+          value at such a point is required to be the left- or right-hand
+          derivative.
+
+          The method employed is based upon:
+
+               (i) carrying out a binary search for the knot interval
+               containing the argument x (see Cox [3]),
+
+               (ii) evaluating the non-zero B-splines of orders 1,2,3 and
+               4 by recurrence (see Cox [2] and [3]),
+
+               (iii) computing all derivatives of the B-splines of order 4
+               by applying a second recurrence to these computed B-spline
+               values (see de Boor [1]),
+
+               (iv) multiplying the 4th-order B-spline values and their
+               derivative by the appropriate B-spline coefficients, and
+               summing, to yield the values of s(x) and its derivatives.
+
+          E02BCF can be used to compute the values and derivatives of cubic
+          spline fits and interpolants produced by E02BAF.
+
+          If only values and not derivatives are required, E02BBF may be
+          used instead of E02BCF, which takes about 50% longer than E02BBF.
+
+          4. References
+
+          [1]   De Boor C (1972) On Calculating with B-splines. J. Approx.
+                Theory. 6 50--62.
+
+          [2]   Cox M G (1972) The Numerical Evaluation of B-splines. J.
+                Inst. Math. Appl. 10 134--149.
+
+          [3]   Cox M G (1978) The Numerical Evaluation of a Spline from its
+                B-spline Representation. J. Inst. Math. Appl. 21 135--143.
+
+          5. Parameters
+
+           1:  NCAP7 -- INTEGER                                       Input
+                                    
+
+               On entry: n+7, where n is the number of intervals of the
+               spline (which is one greater than the number of interior
+               knots, i.e., the knots strictly within the range (lambda)
+                                                                        4
+               to (lambda)    over which the spline is defined).
+                          n+4
+               Constraint: NCAP7 >= 8.
+
+           2:  LAMDA(NCAP7) -- DOUBLE PRECISION array                 Input
+               On entry: LAMDA(j) must be set to the value of the jth
+               member of the complete set of knots, (lambda) , for
+                                                            j
+                         
+
+               j=1,2,...,n+7. Constraint: the LAMDA(j) must be in non-
+               decreasing order with
+
+               LAMDA(NCAP7-3) > LAMDA(4).
+
+           3:  C(NCAP7) -- DOUBLE PRECISION array                     Input
+               On entry: the coefficient c  of the B-spline N (x), for
+                                          i                  i
+
+               i=1,2,...,n+3. The remaining elements of the array are not
+               used.
+
+           4:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x at which the cubic spline and its
+               derivatives are to be evaluated. Constraint: LAMDA(4) <= X
+               <= LAMDA(NCAP7-3).
+
+           5:  LEFT -- INTEGER                                        Input
+               On entry: specifies whether left- or right-hand values of
+               the spline and its derivatives are to be computed (see
+               Section 3). Left- or right-hand values are formed according
+               to whether LEFT is equal or not equal to 1. If x does not
+               coincide with a knot, the value of LEFT is immaterial. If x
+               = LAMDA(4), right-hand values are computed, and if x = LAMDA
+               (NCAP7-3), left-hand values are formed, regardless of the
+               value of LEFT.
+
+           6:  S(4) -- DOUBLE PRECISION array                        Output
+               On exit: S(j) contains the value of the (j-1)th derivative
+               of the spline at the argument x, for j = 1,2,3,4. Note that
+               S(1) contains the value of the spline.
+
+           7:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               NCAP7 < 8, i.e., the number of intervals is not positive.
+
+          IFAIL= 2
+               Either LAMDA(4) >= LAMDA(NCAP7-3), i.e., the range over
+               which s(x) is defined is null or negative in length, or X is
+               an invalid argument, i.e., X < LAMDA(4) or X >
+               LAMDA(NCAP7-3).
+
+          7. Accuracy
+
+          The computed value of s(x) has negligible error in most practical
+          situations. Specifically, this value has an absolute error
+          bounded in modulus by 18*c   * machine precision, where c    is
+                                    max                            max
+          the largest in modulus of c ,c   ,c    and c   , and j is an
+                                     j  j+1  j+2      j+3
+          integer such that (lambda)   <=x<=(lambda)   . If c ,c   ,c
+                                    j+3             j+4      j  j+1  j+2
+          and c    are all of the same sign, then the computed value of
+               j+3
+          s(x) has relative error bounded by 18*machine precision. For full
+          details see Cox [3].
+
+          No complete error analysis is available for the computation of
+          the derivatives of s(x). However, for most practical purposes the
+          absolute errors in the computed derivatives should be small.
+
+          8. Further Comments
+
+          The time taken by this routine is approximately linear in
+              
+
+          log(n+7).
+
+          Note: the routine does not test all the conditions on the knots
+          given in the description of LAMDA in Section 5, since to do this
+                                                                     
+
+          would result in a computation time approximately linear in n+7
+                         
+
+          instead of log(n+7). All the conditions are tested in E02BAF,
+          however.
+
+          9. Example
+
+          Compute, at the 7 arguments x = 0, 1, 2, 3, 4, 5, 6, the left-
+          and right-hand values and first 3 derivatives of the cubic spline
+          defined over the interval 0<=x<=6 having the 6 interior knots x =
+          1, 3, 3, 3, 4, 4, the 8 additional knots 0, 0, 0, 0, 6, 6, 6, 6,
+          and the 10 B-spline coefficients 10, 12, 13, 15, 22, 26, 24, 18,
+          14, 12.
+
+          The input data items (using the notation of Section 5) comprise
+          the following values in the order indicated:
+
+              
+
+              n           m
+
+              LAMDA(j),   for j= 1,2,...,NCAP7
+
+              C(j),       for j= 1,2,...,NCAP7-4
+
+              x(i),       for i=1,2,...,m
+
+          The example program is written in a general form that will enable
+          the values and derivatives of a cubic spline having an arbitrary
+          number of knots to be evaluated at a set of arbitrary points. Any
+          number of data sets may be supplied. The only changes required to
+          the program relate to the dimensions of the arrays LAMDA and C.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02bdf}{NAG On-line Documentation: e02bdf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02BDF(3NAG)      Foundation Library (12/10/92)      E02BDF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02BDF
+                  E02BDF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02BDF computes the definite integral of a cubic spline from its
+          B-spline representation.
+
+          2. Specification
+
+                 SUBROUTINE E02BDF (NCAP7, LAMDA, C, DEFINT, IFAIL)
+                 INTEGER          NCAP7, IFAIL
+                 DOUBLE PRECISION LAMDA(NCAP7), C(NCAP7), DEFINT
+
+          3. Description
+
+          This routine computes the definite integral of the cubic spline
+          s(x) between the limits x=a and x=b, where a and b are
+          respectively the lower and upper limits of the range over which
+          s(x) is defined. It is assumed that s(x) is represented in terms
+                                                         
+
+          of its B-spline coefficients c , for i=1,2,...,n+3 and
+                                        i
+                                                                
+
+          (augmented) ordered knot set (lambda) , for i=1,2,...,n+7, with
+                                               i
+          (lambda) =a, for i = 1,2,3,4 and (lambda) =b, for
+                  i                                i
+                        
+
+          i=n+4,n+5,n+6,n+7, (see E02BAF), i.e.,
+
+                                        q
+                                        --
+                                  s(x)= >  c N (x).
+                                        --  i i
+                                        i=1
+
+                      
+
+          Here q=n+3, n is the number of intervals of the spline and N (x)
+                                                                      i
+          denotes the normalised B-spline of degree 3 (order 4) defined
+          upon the knots (lambda) ,(lambda)   ,...,(lambda)   .
+                                 i         i+1             i+4
+
+          The method employed uses the formula given in Section 3 of Cox
+          [1].
+
+          E02BDF can be used to determine the definite integrals of cubic
+          spline fits and interpolants produced by E02BAF.
+
+          4. References
+
+          [1]   Cox M G (1975) An Algorithm for Spline Interpolation. J.
+                Inst. Math. Appl. 15 95--108.
+
+          5. Parameters
+
+           1:  NCAP7 -- INTEGER                                       Input
+                                    
+
+               On entry: n+7, where n is the number of intervals of the
+               spline (which is one greater than the number of interior
+               knots, i.e., the knots strictly within the range a to b)
+               over which the spline is defined. Constraint: NCAP7 >= 8.
+
+           2:  LAMDA(NCAP7) -- DOUBLE PRECISION array                 Input
+               On entry: LAMDA(j) must be set to the value of the jth
+               member of the complete set of knots, (lambda)  for
+                                                            j
+                         
+
+               j=1,2,...,n+7. Constraint: the LAMDA(j) must be in non-
+               decreasing order with LAMDA(NCAP7-3) > LAMDA(4) and satisfy
+                           LAMDA(1)=LAMDA(2)=LAMDA(3)=LAMDA(4)
+               and
+
+               LAMDA(NCAP7-3)=LAMDA(NCAP7-2)=LAMDA(NCAP7-1)=LAMDA(NCAP7).
+
+           3:  C(NCAP7) -- DOUBLE PRECISION array                     Input
+               On entry: the coefficient c  of the B-spline N (x), for
+                                          i                  i
+                         
+
+               i=1,2,...,n+3. The remaining elements of the array are not
+               used.
+
+           4:  DEFINT -- DOUBLE PRECISION                            Output
+               On exit: the value of the definite integral of s(x) between
+               the limits x=a and x=b, where a=(lambda)  and b=(lambda)   .
+                                                       4               n+4
+
+           5:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               NCAP7 < 8, i.e., the number of intervals is not positive.
+
+          IFAIL= 2
+               At least one of the following restrictions on the knots is
+               violated:
+                    LAMDA(NCAP7-3) > LAMDA(4),
+
+                    LAMDA(j) >= LAMDA(j-1),
+               for j = 2,3,...,NCAP7, with equality in the cases
+               j=2,3,4,NCAP7-2,NCAP7-1, and NCAP7.
+
+          7. Accuracy
+
+          The rounding errors are such that the computed value of the
+          integral is exact for a slightly perturbed set of B-spline
+          coefficients c  differing in a relative sense from those supplied
+                        i      
+          by no more than 2.2*(n+3)*machine precision.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          
+
+          n+7.
+
+          9. Example
+
+          Determine the definite integral over the interval 0<=x<=6 of a
+          cubic spline having 6 interior knots at the positions (lambda)=1,
+          3, 3, 3, 4, 4, the 8 additional knots 0, 0, 0, 0, 6, 6, 6, 6, and
+          the 10 B-spline coefficients 10, 12, 13, 15, 22, 26, 24, 18, 14,
+          12.
+
+          The input data items (using the notation of Section 5) comprise
+          the following values in the order indicated:
+
+              
+
+              n
+
+              LAMDA(j) for j = 1,2,...,NCAP7
+              ,
+
+              C(j),    for j = 1,2,...,NCAP7-3
+
+          The example program is written in a general form that will enable
+          the definite integral of a cubic spline having an arbitrary
+          number of knots to be computed. Any number of data sets may be
+          supplied. The only changes required to the program relate to the
+          dimensions of the arrays LAMDA and C.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02bef}{NAG On-line Documentation: e02bef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02BEF(3NAG)      Foundation Library (12/10/92)      E02BEF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02BEF
+                  E02BEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02BEF computes a cubic spline approximation to an arbitrary set
+          of data points. The knots of the spline are located
+          automatically, but a single parameter must be specified to
+          control the trade-off between closeness of fit and smoothness of
+          fit.
+
+          2. Specification
+
+                 SUBROUTINE E02BEF (START, M, X, Y, W, S, NEST, N, LAMDA,
+                1                   C, FP, WRK, LWRK, IWRK, IFAIL)
+                 INTEGER          M, NEST, N, LWRK, IWRK(NEST), IFAIL
+                 DOUBLE PRECISION X(M), Y(M), W(M), S, LAMDA(NEST), C(NEST),
+                1                 FP, WRK(LWRK)
+                 CHARACTER*1      START
+
+          3. Description
+
+          This routine determines a smooth cubic spline approximation s(x)
+          to the set of data points (x ,y ), with weights w , for
+                                      r  r                 r
+          r=1,2,...,m.
+
+          The spline is given in the B-spline representation
+
+                                      n-4
+                                      --
+                                s(x)= >  c N (x)                        (1)
+                                      --  i i
+                                      i=1
+
+          where N (x) denotes the normalised cubic B-spline defined upon
+                 i
+          the knots (lambda) ,(lambda)   ,...,(lambda)   .
+                            i         i+1             i+4
+
+          The total number n of these knots and their values
+          (lambda) ,...,(lambda)  are chosen automatically by the routine.
+                  1             n
+          The knots (lambda) ,...,(lambda)    are the interior knots; they
+                            5             n-4
+          divide the approximation interval [x ,x ] into n-7 sub-intervals.
+                                              1  m
+          The coefficients c ,c ,...,c    are then determined as the
+                            1  2      n-4
+          solution of the following constrained minimization problem:
+
+          minimize
+
+                                      n-4
+                                      --        2
+                               (eta)= >  (delta)                        (2)
+                                      --        i
+                                      i=5
+
+          subject to the constraint
+
+                                    m
+                                    --          2
+                           (theta)= >  (epsilon) <=S                    (3)
+                                    --          r
+                                    r=1
+
+          where: (delta)    stands for the discontinuity jump in the third
+                        i   order derivative of s(x) at the interior knot
+                      (lambda) ,
+                                    i
+
+                 (epsilon)  denotes the weighted residual w (y -s(x )),
+                          r                                r  r    r
+
+          and    S          is a non-negative number to be specified by
+                            the user.
+
+          The quantity (eta) can be seen as a measure of the (lack of)
+          smoothness of s(x), while closeness of fit is measured through
+          (theta). By means of the parameter S, 'the smoothing factor', the
+          user will then control the balance between these two (usually
+          conflicting) properties. If S is too large, the spline will be
+          too smooth and signal will be lost (underfit); if S is too small,
+          the spline will pick up too much noise (overfit). In the extreme
+          cases the routine will return an interpolating spline ((theta)=0)
+          if S is set to zero, and the weighted least-squares cubic
+          polynomial ((eta)=0) if S is set very large. Experimenting with S
+          values between these two extremes should result in a good
+          compromise. (See Section 8.2 for advice on choice of S.)
+
+          The method employed is outlined in Section 8.3 and fully
+          described in Dierckx [1], [2] and [3]. It involves an adaptive
+          strategy for locating the knots of the cubic spline (depending on
+          the function underlying the data and on the value of S), and an
+          iterative method for solving the constrained minimization problem
+          once the knots have been determined.
+
+          Values of the computed spline, or of its derivatives or definite
+          integral, can subsequently be computed by calling E02BBF, E02BCF
+          or E02BDF, as described in Section 8.4.
+
+          4. References
+
+          [1]   Dierckx P (1975) An Algorithm for Smoothing, Differentiating
+                and Integration of Experimental Data Using Spline Functions.
+                J. Comput. Appl. Math. 1 165--184.
+
+          [2]   Dierckx P (1982) A Fast Algorithm for Smoothing Data on a
+                Rectangular Grid while using Spline Functions. SIAM J.
+                Numer. Anal. 19 1286--1304.
+
+          [3]   Dierckx P (1981) An Improved Algorithm for Curve Fitting
+                with Spline Functions. Report TW54. Department of Computer
+                Science, Katholieke Universiteit Leuven.
+
+          [4]   Reinsch C H (1967) Smoothing by Spline Functions. Num. Math.
+                10 177--183.
+
+          5. Parameters
+
+           1:  START -- CHARACTER*1                                   Input
+               On entry: START must be set to 'C' or 'W'.
+
+               If START = 'C' (Cold start), the routine will build up the
+               knot set starting with no interior knots. No values need be
+               assigned to the parameters N, LAMDA, WRK or IWRK.
+
+               If START = 'W' (Warm start), the routine will restart the
+               knot-placing strategy using the knots found in a previous
+               call of the routine. In this case, the parameters N, LAMDA,
+               WRK, and IWRK must be unchanged from that previous call.
+               This warm start can save much time in searching for a
+               satisfactory value of S. Constraint: START = 'C' or 'W'.
+
+           2:  M -- INTEGER                                           Input
+               On entry: m, the number of data points. Constraint: M >= 4.
+
+           3:  X(M) -- DOUBLE PRECISION array                         Input
+               On entry: the values x  of the independent variable
+                                     r
+               (abscissa) x, for r=1,2,...,m. Constraint: x <x <...<x
+                                                           1  2      m
+
+           4:  Y(M) -- DOUBLE PRECISION array                         Input
+               On entry: the values y  of the dependent variable
+                                     r
+               (ordinate) y, for r=1,2,...,m.
+
+           5:  W(M) -- DOUBLE PRECISION array                         Input
+               On entry: the values w  of the weights, for r=1,2,...,m.
+                                     r
+               For advice on the choice of weights, see the Chapter
+               Introduction, Section 2.1.2. Constraint: W(r) > 0, for
+               r=1,2,...,m.
+
+           6:  S -- DOUBLE PRECISION                                  Input
+               On entry: the smoothing factor, S.
+
+               If S=0.0, the routine returns an interpolating spline.
+
+               If S is smaller than machine precision, it is assumed equal
+               to zero.
+
+               For advice on the choice of S, see Section 3 and Section 8.2
+               Constraint: S >= 0.0.
+
+           7:  NEST -- INTEGER                                        Input
+               On entry: an over-estimate for the number, n, of knots
+               required. Constraint: NEST >= 8. In most practical
+               situations, NEST = M/2 is sufficient. NEST never needs to be
+               larger than M + 4, the number of knots needed for
+               interpolation (S = 0.0).
+
+           8:  N -- INTEGER                                    Input/Output
+               On entry: if the warm start option is used, the value of N
+               must be left unchanged from the previous call. On exit: the
+               total number, n, of knots of the computed spline.
+
+           9:  LAMDA(NEST) -- DOUBLE PRECISION array           Input/Output
+               On entry: if the warm start option is used, the values
+               LAMDA(1), LAMDA(2),...,LAMDA(N) must be left unchanged from
+               the previous call. On exit: the knots of the spline i.e.,
+               the positions of the interior knots LAMDA(5), LAMDA(6),...
+               ,LAMDA(N-4) as well as the positions of the additional knots
+               LAMDA(1) = LAMDA(2) = LAMDA(3) = LAMDA(4) = x  and
+                                                            1
+
+               LAMDA(N-3) = LAMDA(N-2) = LAMDA(N-1) = LAMDA(N) = x  needed
+                                                                  m
+               for the B-spline representation.
+
+          10:  C(NEST) -- DOUBLE PRECISION array                     Output
+               On exit: the coefficient c  of the B-spline N (x) in the
+                                          i                  i
+               spline approximation s(x), for i=1,2,...,n-4.
+
+          11:  FP -- DOUBLE PRECISION                                Output
+               On exit: the sum of the squared weighted residuals, (theta),
+               of the computed spline approximation. If FP = 0.0, this is
+               an interpolating spline. FP should equal S within a relative
+               tolerance of 0.001 unless n=8 when the spline has no
+               interior knots and so is simply a cubic polynomial. For
+               knots to be inserted, S must be set to a value below the
+               value of FP produced in this case.
+
+          12:  WRK(LWRK) -- DOUBLE PRECISION array                Workspace
+               On entry: if the warm start option is used, the values WRK
+               (1),...,WRK(n) must be left unchanged from the previous
+               call.
+
+          13:  LWRK -- INTEGER                                        Input
+               On entry:
+               the dimension of the array WRK as declared in the
+               (sub)program from which E02BEF is called.
+               Constraint: LWRK>=4*M+16*NEST+41.
+
+          14:  IWRK(NEST) -- INTEGER array                        Workspace
+               On entry: if the warm start option is used, the values IWRK
+               (1), ..., IWRK(n) must be left unchanged from the previous
+               call.
+
+               This array is used as workspace.
+
+          15:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry START /= 'C' or 'W',
+
+               or       M < 4,
+
+               or       S < 0.0,
+
+               or       S = 0.0 and NEST < M + 4,
+
+               or       NEST < 8,
+
+               or       LWRK<4*M+16*NEST+41.
+
+          IFAIL= 2
+               The weights are not all strictly positive.
+
+          IFAIL= 3
+               The values of X(r), for r=1,2,...,M, are not in strictly
+               increasing order.
+
+          IFAIL= 4
+               The number of knots required is greater than NEST. Try
+               increasing NEST and, if necessary, supplying larger arrays
+               for the parameters LAMDA, C, WRK and IWRK. However, if NEST
+               is already large, say NEST > M/2, then this error exit may
+               indicate that S is too small.
+
+          IFAIL= 5
+               The iterative process used to compute the coefficients of
+               the approximating spline has failed to converge. This error
+               exit may occur if S has been set very small. If the error
+               persists with increased S, consult NAG.
+
+          If IFAIL = 4 or 5, a spline approximation is returned, but it
+          fails to satisfy the fitting criterion (see (2) and (3) in
+          Section 3) - perhaps by only a small amount, however.
+
+          7. Accuracy
+
+          On successful exit, the approximation returned is such that its
+          weighted sum of squared residuals FP is equal to the smoothing
+          factor S, up to a specified relative tolerance of 0.001 - except
+          that if n=8, FP may be significantly less than S: in this case
+          the computed spline is simply a weighted least-squares polynomial
+          approximation of degree 3, i.e., a spline with no interior knots.
+
+          8. Further Comments
+
+          8.1. Timing
+
+          The time taken for a call of E02BEF depends on the complexity of
+          the shape of the data, the value of the smoothing factor S, and
+          the number of data points. If E02BEF is to be called for
+          different values of S, much time can be saved by setting START =
+
+          8.2. Choice of S
+
+          If the weights have been correctly chosen (see Section 2.1.2 of
+          the Chapter Introduction), the standard deviation of w y  would
+                                                                r r
+          be the same for all r, equal to (sigma), say. In this case,
+                                                              2      
+          choosing the smoothing factor S in the range (sigma) (m+-\/2m),
+          as suggested by Reinsch [4], is likely to give a good start in
+          the search for a satisfactory value. Otherwise, experimenting
+          with different values of S will be required from the start,
+          taking account of the remarks in Section 3.
+
+          In that case, in view of computation time and memory
+          requirements, it is recommended to start with a very large value
+          for S and so determine the least-squares cubic polynomial; the
+          value returned for FP, call it FP , gives an upper bound for S.
+                                           0
+          Then progressively decrease the value of S to obtain closer fits
+          - say by a factor of 10 in the beginning, i.e., S=FP /10, S=FP
+                                                              0         0
+          /100, and so on, and more carefully as the approximation shows
+          more details.
+
+          The number of knots of the spline returned, and their location,
+          generally depend on the value of S and on the behaviour of the
+          function underlying the data. However, if E02BEF is called with
+          START = 'W', the knots returned may also depend on the smoothing
+          factors of the previous calls. Therefore if, after a number of
+          trials with different values of S and START = 'W', a fit can
+          finally be accepted as satisfactory, it may be worthwhile to call
+          E02BEF once more with the selected value for S but now using
+          START = 'C'. Often, E02BEF then returns an approximation with the
+          same quality of fit but with fewer knots, which is therefore
+          better if data reduction is also important.
+
+          8.3. Outline of Method Used
+
+          If S=0, the requisite number of knots is known in advance, i.e.,
+          n=m+4; the interior knots are located immediately as (lambda)  =
+                                                                       i
+          x   , for i=5,6,...,n-4. The corresponding least-squares spline
+           i-2
+          (see E02BAF) is then an interpolating spline and therefore a
+          solution of the problem.
+
+          If S>0, a suitable knot set is built up in stages (starting with
+          no interior knots in the case of a cold start but with the knot
+          set found in a previous call if a warm start is chosen). At each
+          stage, a spline is fitted to the data by least-squares (see
+          E02BAF) and (theta), the weighted sum of squares of residuals, is
+          computed. If (theta)>S, new knots are added to the knot set to
+          reduce (theta) at the next stage. The new knots are located in
+          intervals where the fit is particularly poor, their number
+          depending on the value of S and on the progress made so far in
+          reducing (theta). Sooner or later, we find that (theta)<=S and at
+          that point the knot set is accepted. The routine then goes on to
+          compute the (unique) spline which has this knot set and which
+          satisfies the full fitting criterion specified by (2) and (3).
+          The theoretical solution has (theta)=S. The routine computes the
+          spline by an iterative scheme which is ended when (theta)=S
+          within a relative tolerance of 0.001. The main part of each
+          iteration consists of a linear least-squares computation of
+          special form, done in a similarly stable and efficient manner as
+          in E02BAF.
+
+          An exception occurs when the routine finds at the start that,
+          even with no interior knots (n=8), the least-squares spline
+          already has its weighted sum of squares of residuals <=S. In this
+          case, since this spline (which is simply a cubic polynomial) also
+          has an optimal value for the smoothness measure (eta), namely
+          zero, it is returned at once as the (trivial) solution. It will
+          usually mean that S has been chosen too large.
+
+          For further details of the algorithm and its use, see Dierckx [3]
+
+          8.4. Evaluation of Computed Spline
+
+          The value of the computed spline at a given value X may be
+          obtained in the double precision variable S by the call:
+
+
+                 CALL E02BBF(N,LAMDA,C,X,S,IFAIL)
+
+          where N, LAMDA and C are the output parameters of E02BEF.
+
+          The values of the spline and its first three derivatives at a
+          given value X may be obtained in the double precision array SDIF
+          of dimension at least 4 by the call:
+
+                 CALL E02BCF(N,LAMDA,C,X,LEFT,SDIF,IFAIL)
+
+          where if LEFT = 1, left-hand derivatives are computed and if LEFT
+          /= 1, right-hand derivatives are calculated. The value of LEFT is
+          only relevant if X is an interior knot.
+
+          The value of the definite integral of the spline over the
+          interval X(1) to X(M) can be obtained in the double precision
+          variable SINT by the call:
+
+                 CALL E02BDF(N,LAMDA,C,SINT,IFAIL)
+
+          9. Example
+
+          This example program reads in a set of data values, followed by a
+          set of values of S. For each value of S it calls E02BEF to
+          compute a spline approximation, and prints the values of the
+          knots and the B-spline coefficients c .
+                                               i
+
+          The program includes code to evaluate the computed splines, by
+          calls to E02BBF, at the points x  and at points mid-way between
+                                          r
+          them. These values are not printed out, however; instead the
+          results are illustrated by plots of the computed splines,
+          together with the data points (indicated by *) and the positions
+          of the knots (indicated by vertical lines): the effect of
+          decreasing S can be clearly seen. (The plots were obtained by
+          calling NAG Graphical Supplement routine J06FAF(*).)
+
+
+                   Please see figures in printed Reference Manual
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02daf}{NAG On-line Documentation: e02daf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02DAF(3NAG)      Foundation Library (12/10/92)      E02DAF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02DAF
+                  E02DAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02DAF forms a minimal, weighted least-squares bicubic spline
+          surface fit with prescribed knots to a given set of data points.
+
+          2. Specification
+
+                 SUBROUTINE E02DAF (M, PX, PY, X, Y, F, W, LAMDA, MU,
+                1                   POINT, NPOINT, DL, C, NC, WS, NWS, EPS,
+                2                   SIGMA, RANK, IFAIL)
+                 INTEGER          M, PX, PY, POINT(NPOINT), NPOINT, NC, NWS,
+                1                 RANK, IFAIL
+                 DOUBLE PRECISION X(M), Y(M), F(M), W(M), LAMDA(PX), MU(PY),
+                1                 DL(NC), C(NC), WS(NWS), EPS, SIGMA
+
+          3. Description
+
+          This routine determines a bicubic spline fit s(x,y) to the set of
+          data points (x ,y ,f ) with weights w , for r=1,2,...,m. The two
+                        r  r  r                r
+          sets of internal knots of the spline, {(lambda)} and {(mu)},
+          associated with the variables x and y respectively, are
+          prescribed by the user. These knots can be thought of as dividing
+          the data region of the (x,y) plane into panels (see diagram in
+          Section 5). A bicubic spline consists of a separate bicubic
+          polynomial in each panel, the polynomials joining together with
+          continuity up to the second derivative across the panel
+          boundaries.
+
+          s(x,y) has the property that (Sigma), the sum of squares of its
+          weighted residuals (rho) , for r=1,2,...,m, where
+                                  r
+
+                            (rho) =w (s(x ,y )-f ),                     (1)
+                                 r  r    r  r   r
+
+          is as small as possible for a bicubic spline with the given knot
+          sets. The routine produces this minimized value of (Sigma) and
+          the coefficients c   in the B-spline representation of s(x,y) -
+                            ij
+          see Section 8. E02DEF and E02DFF are available to compute values
+          of the fitted spline from the coefficients c  .
+                                                      ij
+
+          The least-squares criterion is not always sufficient to determine
+          the bicubic spline uniquely: there may be a whole family of
+          splines which have the same minimum sum of squares. In these
+          cases, the routine selects from this family the spline for which
+          the sum of squares of the coefficients c   is smallest: in other
+                                                  ij
+          words, the minimal least-squares solution. This choice, although
+          arbitrary, reduces the risk of unwanted fluctuations in the
+          spline fit. The method employed involves forming a system of m
+          linear equations in the coefficients c   and then computing its
+                                                ij
+          least-squares solution, which will be the minimal least-squares
+          solution when appropriate. The basis of the method is described
+          in Hayes and Halliday [4]. The matrix of the equation is formed
+          using a recurrence relation for B-splines which is numerically
+          stable (see Cox [1] and de Boor [2] - the former contains the
+          more elementary derivation but, unlike [2], does not cover the
+          case of coincident knots). The least-squares solution is also
+          obtained in a stable manner by using orthogonal transformations,
+          viz. a variant of Givens rotation (see Gentleman [3]). This
+          requires only one row of the matrix to be stored at a time.
+          Advantage is taken of the stepped-band structure which the matrix
+          possesses when the data points are suitably ordered, there being
+          at most sixteen non-zero elements in any row because of the
+          definition of B-splines. First the matrix is reduced to upper
+          triangular form and then the diagonal elements of this triangle
+          are examined in turn. When an element is encountered whose
+          square, divided by the mean squared weight, is less than a
+          threshold (epsilon), it is replaced by zero and the rest of the
+          elements in its row are reduced to zero by rotations with the
+          remaining rows. The rank of the system is taken to be the number
+          of non-zero diagonal elements in the final triangle, and the non-
+          zero rows of this triangle are used to compute the minimal least-
+          squares solution. If all the diagonal elements are non-zero, the
+          rank is equal to the number of coefficients c   and the solution
+                                                       ij
+          obtained is the ordinary least-squares solution, which is unique
+          in this case.
+
+          4. References
+
+          [1]   Cox M G (1972) The Numerical Evaluation of B-splines. J.
+                Inst. Math. Appl. 10 134--149.
+
+          [2]   De Boor C (1972) On Calculating with B-splines. J. Approx.
+                Theory. 6 50--62.
+
+          [3]   Gentleman W M (1973) Least-squares Computations by Givens
+                Transformations without Square Roots. J. Inst. Math. Applic.
+                12 329--336.
+
+          [4]   Hayes J G and Halliday J (1974) The Least-squares Fitting of
+                Cubic Spline Surfaces to General Data Sets. J. Inst. Math.
+                Appl. 14 89--103.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number of data points, m. Constraint: M > 1.
+
+           2:  PX -- INTEGER                                          Input
+
+           3:  PY -- INTEGER                                          Input
+               On entry: the total number of knots (lambda) and (mu)
+               associated with the variables x and y, respectively.
+               Constraint: PX >= 8 and PY >= 8.
+
+               (They are such that PX-8 and PY-8 are the corresponding
+               numbers of interior knots.) The running time and storage
+               required by the routine are both minimized if the axes are
+               labelled so that PY is the smaller of PX and PY.
+
+           4:  X(M) -- DOUBLE PRECISION array                         Input
+
+           5:  Y(M) -- DOUBLE PRECISION array                         Input
+
+           6:  F(M) -- DOUBLE PRECISION array                         Input
+               On entry: the co-ordinates of the data point (x ,y ,f ), for
+                                                              r  r  r
+               r=1,2,...,m. The order of the data points is immaterial, but
+               see the array POINT, below.
+
+           7:  W(M) -- DOUBLE PRECISION array                         Input
+               On entry: the weight w  of the rth data point. It is
+                                     r
+               important to note the definition of weight implied by the
+               equation (1) in Section 3, since it is also common usage to
+               define weight as the square of this weight. In this routine,
+               each w  should be chosen inversely proportional to the
+                     r
+               (absolute) accuracy of the corresponding f , as expressed,
+                                                         r
+               for example, by the standard deviation or probable error of
+               the f . When the f  are all of the same accuracy, all the w
+                    r            r                                        r
+               may be set equal to 1.0.
+
+           8:  LAMDA(PX) -- DOUBLE PRECISION array             Input/Output
+               On entry: LAMDA(i+4) must contain the ith interior knot
+               (lambda)     associated with the variable x, for
+                       i+4
+               i=1,2,...,PX-8. The knots must be in non-decreasing order
+               and lie strictly within the range covered by the data values
+               of x. A knot is a value of x at which the spline is allowed
+               to be discontinuous in the third derivative with respect to
+               x, though continuous up to the second derivative. This
+               degree of continuity can be reduced, if the user requires,
+               by the use of coincident knots, provided that no more than
+               four knots are chosen to coincide at any point. Two, or
+               three, coincident knots allow loss of continuity in,
+               respectively, the second and first derivative with respect
+               to x at the value of x at which they coincide. Four
+               coincident knots split the spline surface into two
+               independent parts. For choice of knots see Section 8. On
+               exit: the interior knots LAMDA(5) to LAMDA(PX-4) are
+               unchanged, and the segments LAMDA(1:4) and LAMDA(PX-3:PX)
+               contain additional (exterior) knots introduced by the
+               routine in order to define the full set of B-splines
+               required. The four knots in the first segment are all set
+               equal to the lowest data value of x and the other four
+               additional knots are all set equal to the highest value:
+               there is experimental evidence that coincident end-knots are
+               best for numerical accuracy. The complete array must be left
+               undisturbed if E02DEF or E02DFF is to be used subsequently.
+
+           9:  MU(PY) -- DOUBLE PRECISION array                       Input
+               On entry: MU(i+4) must contain the ith interior knot (mu)
+                                                                        i+4
+               associated with the variable y, i=1,2,...,PY-8. The same
+               remarks apply to MU as to LAMDA above, with Y replacing X,
+               and y replacing x.
+
+          10:  POINT(NPOINT) -- INTEGER array                         Input
+               On entry: indexing information usually provided by E02ZAF
+               which enables the data points to be accessed in the order
+               which produces the advantageous matrix structure mentioned
+               in Section 3. This order is such that, if the (x,y) plane is
+               thought of as being divided into rectangular panels by the
+               two sets of knots, all data in a panel occur before data in
+               succeeding panels, where the panels are numbered from bottom
+               to top and then left to right with the usual arrangement of
+               axes, as indicated in the diagram.
+
+                      Please see figure in printed Reference Manual
+
+               A data point lying exactly on one or more panel sides is
+               considered to be in the highest numbered panel adjacent to
+               the point. E02ZAF should be called to obtain the array
+               POINT, unless it is provided by other means.
+
+          11:  NPOINT -- INTEGER                                      Input
+               On entry:
+               the dimension of the array POINT as declared in the
+               (sub)program from which E02DAF is called.
+               Constraint: NPOINT >= M + (PX-7)*(PY-7).
+
+          12:  DL(NC) -- DOUBLE PRECISION array                      Output
+               On exit: DL gives the squares of the diagonal elements of
+               the reduced triangular matrix, divided by the mean squared
+               weight. It includes those elements, less than (epsilon),
+               which are treated as zero (see Section 3).
+
+          13:  C(NC) -- DOUBLE PRECISION array                       Output
+               On exit: C gives the coefficients of the fit. C((PY-4)*(i-
+               1)+j) is the coefficient c   of Section 3 and Section 8 for
+                                         ij
+               i=1,2,...,PX-4 and j=1,2,...,PY-4. These coefficients are
+               used by E02DEF or E02DFF to calculate values of the fitted
+               function.
+
+          14:  NC -- INTEGER                                          Input
+               On entry: the value (PX-4)*(PY-4).
+
+          15:  WS(NWS) -- DOUBLE PRECISION array                  Workspace
+
+          16:  NWS -- INTEGER                                         Input
+               On entry:
+               the dimension of the array WS as declared in the
+               (sub)program from which E02DAF is called.
+               Constraint: NWS>=(2*NC+1)*(3*PY-6)-2.
+
+          17:  EPS -- DOUBLE PRECISION                                Input
+               On entry: a threshold (epsilon) for determining the
+               effective rank of the system of linear equations. The rank
+               is determined as the number of elements of the array DL (see
+               below) which are non-zero. An element of DL is regarded as
+               zero if it is less than (epsilon). Machine precision is a
+               suitable value for (epsilon) in most practical applications
+               which have only 2 or 3 decimals accurate in data. If some
+               coefficients of the fit prove to be very large compared with
+               the data ordinates, this suggests that (epsilon) should be
+               increased so as to decrease the rank. The array DL will give
+               a guide to appropriate values of (epsilon) to achieve this,
+               as well as to the choice of (epsilon) in other cases where
+               some experimentation may be needed to determine a value
+               which leads to a satisfactory fit.
+
+          18:  SIGMA -- DOUBLE PRECISION                             Output
+               On exit: (Sigma), the weighted sum of squares of residuals.
+               This is not computed from the individual residuals but from
+               the right-hand sides of the orthogonally-transformed linear
+               equations. For further details see Hayes and Halliday [4]
+               page 97. The two methods of computation are theoretically
+               equivalent, but the results may differ because of rounding
+               error.
+
+          19:  RANK -- INTEGER                                       Output
+               On exit: the rank of the system as determined by the value
+               of the threshold (epsilon). When RANK = NC, the least-
+               squares solution is unique: in other cases the minimal
+               least-squares solution is computed.
+
+          20:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               At least one set of knots is not in non-decreasing order, or
+               an interior knot is outside the range of the data values.
+
+          IFAIL= 2
+               More than four knots coincide at a single point, possibly
+               because all data points have the same value of x (or y) or
+               because an interior knot coincides with an extreme data
+               value.
+
+          IFAIL= 3
+               Array POINT does not indicate the data points in panel
+               order. Call E02ZAF to obtain a correct array.
+
+          IFAIL= 4
+               On entry M <= 1,
+
+               or       PX < 8,
+
+               or       PY < 8,
+
+               or       NC /= (PX-4)*(PY-4),
+
+               or       NWS is too small,
+
+               or       NPOINT is too small.
+
+          IFAIL= 5
+               All the weights w  are zero or rank determined as zero.
+                                r
+
+          7. Accuracy
+
+          The computation of the B-splines and reduction of the observation
+          matrix to triangular form are both numerically stable.
+
+          8. Further Comments
+
+          The time taken by this routine is approximately proportional to
+                                                           2
+          the number of data points, m, and to (3*(PY-4)+4) .
+
+          The B-spline representation of the bicubic spline is
+
+                                       --
+                               s(x,y)= > c  M (x)N (y)
+                                       -- ij i    j
+                                       ij
+
+          summed over i=1,2,...,PX-4 and over j=1,2,...,PY-4. Here M (x)
+                                                                    i
+          and N (y) denote normalised cubic B-splines,the former defined on
+               j
+          the knots (lambda) ,(lambda)   ,...,(lambda)    and the latter on
+                            i         i+1             i+4
+          the knots (mu) ,(mu)   ,...,(mu)   . For further details, see
+                        j     j+1         j+4
+          Hayes and Halliday [4] for bicubic splines and de Boor [2] for
+          normalised B-splines.
+
+          The choice of the interior knots, which help to determine the
+          spline's shape, must largely be a matter of trial and error. It
+          is usually best to start with a small number of knots and,
+          examining the fit at each stage, add a few knots at a time at
+          places where the fit is particularly poor. In intervals of x or y
+          where the surface represented by the data changes rapidly, in
+          function value or derivatives, more knots will be needed than
+          elsewhere. In some cases guidance can be obtained by analogy with
+          the case of coincident knots: for example, just as three
+          coincident knots can produce a discontinuity in slope, three
+          close knots can produce rapid change in slope. Of course, such
+          rapid changes in behaviour must be adequately represented by the
+          data points, as indeed must the behaviour of the surface
+          generally, if a satisfactory fit is to be achieved. When there is
+          no rapid change in behaviour, equally-spaced knots will often
+          suffice.
+
+          In all cases the fit should be examined graphically before it is
+          accepted as satisfactory.
+
+          The fit obtained is not defined outside the rectangle
+
+                  (lambda) <=x<=(lambda)    ,   (mu) <=y<=(mu)
+                          4             PX-3        4         PY-3
+
+          The reason for taking the extreme data values of x and y for
+          these four knots is that, as is usual in data fitting, the fit
+          cannot be expected to give satisfactory values outside the data
+          region. If, nevertheless, the user requires values over a larger
+          rectangle, this can be achieved by augmenting the data with two
+          artificial data points (a,c,0) and (b,d,0) with zero weight,
+          where a<=x<=b, c<=y<=d defines the enlarged rectangle. In the
+          case when the data are adequate to make the least-squares
+          solution unique (RANK = NC), this enlargement will not affect the
+          fit over the original rectangle, except for possibly enlarged
+          rounding errors, and will simply continue the bicubic polynomials
+          in the panels bordering the rectangle out to the new boundaries:
+          in other cases the fit will be affected. Even using the original
+          rectangle there may be regions within it, particularly at its
+          corners, which lie outside the data region and where, therefore,
+          the fit will be unreliable. For example, if there is no data
+          point in panel 1 of the diagram in Section 5, the least-squares
+          criterion leaves the spline indeterminate in this panel: the
+          minimal spline determined by the subroutine in this case passes
+          through the value zero at the point ((lambda) ,(mu) ).
+                                                       4     4
+
+          9. Example
+
+          This example program reads a value for (epsilon), and a set of
+          data points, weights and knot positions. If there are more y
+          knots than x knots, it interchanges the x and y axes. It calls
+          E02ZAF to sort the data points into panel order, E02DAF to fit a
+          bicubic spline to them, and E02DEF to evaluate the spline at the
+          data points.
+
+          Finally it prints:
+
+               the weighted sum of squares of residuals computed from the
+               linear equations;
+
+               the rank determined by E02DAF;
+
+               data points, fitted values and residuals in panel order;
+
+               the weighted sum of squares of the residuals;
+
+               the coefficients of the spline fit.
+
+          The program is written to handle any number of data sets.
+
+          Note: the data supplied in this example is not typical of a
+          realistic problem: the number of data points would normally be
+          much larger (in which case the array dimensions and the value of
+          NWS in the program would have to be increased); and the value of
+          (epsilon) would normally be much smaller on most machines (see
+                                                     -6
+          Section 5; the relatively large value of 10   has been chosen in
+          order to illustrate a minimal least-squares solution when RANK <
+          NC; in this example NC = 24).
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02dcf}{NAG On-line Documentation: e02dcf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02DCF(3NAG)      Foundation Library (12/10/92)      E02DCF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02DCF
+                  E02DCF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02DCF computes a bicubic spline approximation to a set of data
+          values, given on a rectangular grid in the x-y plane. The knots
+          of the spline are located automatically, but a single parameter
+          must be specified to control the trade-off between closeness of
+          fit and smoothness of fit.
+
+          2. Specification
+
+                 SUBROUTINE E02DCF (START, MX, X, MY, Y, F, S, NXEST,
+                1                   NYEST, NX, LAMDA, NY, MU, C, FP, WRK,
+                2                   LWRK, IWRK, LIWRK, IFAIL)
+                 INTEGER          MX, MY, NXEST, NYEST, NX, NY, LWRK, IWRK
+                1                 (LIWRK), LIWRK, IFAIL
+                 DOUBLE PRECISION X(MX), Y(MY), F(MX*MY), S, LAMDA(NXEST),
+                1                 MU(NYEST), C((NXEST-4)*(NYEST-4)), FP, WRK
+                2                 (LWRK)
+                 CHARACTER*1      START
+
+          3. Description
+
+          This routine determines a smooth bicubic spline approximation
+          s(x,y) to the set of data points (x ,y ,f   ), for q=1,2,...,m
+                                             q  r  q,r                  x
+          and r=1,2,...,m .
+                         y
+
+          The spline is given in the B-spline representation
+
+                                n -4 n -4
+                                 x    y
+                                --   --
+                        s(x,y)= >    >   c  M (x)N (y),                 (1)
+                                --   --   ij i    j
+                                i=1  j=1
+
+          where M (x) and N (y) denote normalised cubic B-splines, the
+                 i         j
+          former defined on the knots (lambda)  to (lambda)    and the
+                                              i            i+4
+          latter on the knots (mu)  to (mu)   . For further details, see
+                                  j        j+4
+          Hayes and Halliday [4] for bicubic splines and de Boor [1] for
+          normalised B-splines.
+
+          The total numbers n  and n  of these knots and their values
+                             x      y
+          (lambda) ,...,(lambda)   and (mu) ,...,(mu)   are chosen
+                  1             n          1         n
+                                 x                    y
+          automatically by the routine. The knots (lambda) ,...,
+                                                          5
+          (lambda)     and (mu) ,...,(mu)     are the interior knots; they
+                  n -4         5         n -4
+                   x                      y
+          divide the approximation domain [x ,x  ]*[y ,y  ] into (
+                                            1  m     1  m
+                                               m        m
+          n -7)*(n -7) subpanels [(lambda) ,(lambda)   ]*[(mu) ,(mu)   ],
+           x      y                       i         i+1       j     j+1
+          for i=4,5,...,n -4, j=4,5,...,n -4. Then, much as in the curve
+                         x               y
+          case (see E02BEF), the coefficients c   are determined as the
+                                               ij
+          solution of the following constrained minimization problem:
+
+          minimize
+
+                                     (eta),                             (2)
+
+          subject to the constraint
+
+                                 m   m
+                                  x   y
+                                 --  --          2
+                        (theta)= >   >  (epsilon)   <=S,                (3)
+                                 --  --          q,r
+                                 q=1 r=1
+
+          where  (eta) is a measure of the (lack of) smoothness of s(x,y).
+                       Its value depends on the discontinuity jumps in
+                       s(x,y) across the boundaries of the subpanels. It is
+                       zero only when there are no discontinuities and is
+                       positive otherwise, increasing with the size of the
+                       jumps (see Dierckx [2] for details).
+
+                 (epsilon)    denotes the residual f   -s(x ,y ),
+                          q,r                       q,r    q  r
+
+          and    S    is a non-negative number to be specified by the user.
+
+          By means of the parameter S, 'the smoothing factor', the user
+          will then control the balance between smoothness and closeness of
+          fit, as measured by the sum of squares of residuals in (3). If S
+          is too large, the spline will be too smooth and signal will be
+          lost (underfit); if S is too small, the spline will pick up too
+          much noise (overfit). In the extreme cases the routine will
+          return an interpolating spline ((theta)=0) if S is set to zero,
+          and the least-squares bicubic polynomial ((eta)=0) if S is set
+          very large. Experimenting with S-values between these two
+          extremes should result in a good compromise. (See Section 8.3 for
+          advice on choice of S.)
+
+          The method employed is outlined in Section 8.5 and fully
+          described in Dierckx [2] and [3]. It involves an adaptive
+          strategy for locating the knots of the bicubic spline (depending
+          on the function underlying the data and on the value of S), and
+          an iterative method for solving the constrained minimization
+          problem once the knots have been determined.
+
+          Values of the computed spline can subsequently be computed by
+          calling E02DEF or E02DFF as described in Section 8.6.
+
+          4. References
+
+          [1]   De Boor C (1972) On Calculating with B-splines. J. Approx.
+                Theory. 6 50--62.
+
+          [2]   Dierckx P (1982) A Fast Algorithm for Smoothing Data on a
+                Rectangular Grid while using Spline Functions. SIAM J.
+                Numer. Anal. 19 1286--1304.
+
+          [3]   Dierckx P (1981) An Improved Algorithm for Curve Fitting
+                with Spline Functions. Report TW54. Department of Computer
+                Science, Katholieke Universiteit Leuven.
+
+          [4]   Hayes J G and Halliday J (1974) The Least-squares Fitting of
+                Cubic Spline Surfaces to General Data Sets. J. Inst. Math.
+                Appl. 14 89--103.
+
+          [5]   Reinsch C H (1967) Smoothing by Spline Functions. Num. Math.
+                10 177--183.
+
+          5. Parameters
+
+           1:  START -- CHARACTER*1                                   Input
+               On entry: START must be set to 'C' or 'W'.
+
+               If START = 'C' (Cold start), the routine will build up the
+               knot set starting with no interior knots. No values need be
+               assigned to the parameters NX, NY, LAMDA, MU, WRK or IWRK.
+
+               If START = 'W' (Warm start), the routine will restart the
+               knot-placing strategy using the knots found in a previous
+               call of the routine. In this case, the parameters NX, NY,
+               LAMDA, MU, WRK and IWRK must be unchanged from that previous
+               call. This warm start can save much time in searching for a
+               satisfactory value of S. Constraint: START = 'C' or 'W'.
+
+           2:  MX -- INTEGER                                          Input
+               On entry: m , the number of grid points along the x axis.
+                          x
+               Constraint: MX >= 4.
+
+           3:  X(MX) -- DOUBLE PRECISION array                        Input
+               On entry: X(q) must be set to x , the x co-ordinate of the
+                                              q
+               qth grid point along the x axis, for q=1,2,...,m .
+                                                               x
+               Constraint: x <x <...<x  .
+                            1  2      m
+                                       x
+
+           4:  MY -- INTEGER                                          Input
+               On entry: m , the number of grid points along the y axis.
+                          y
+               Constraint: MY >= 4.
+
+           5:  Y(MY) -- DOUBLE PRECISION array                        Input
+               On entry: Y(r) must be set to y , the y co-ordinate of the
+                                              r
+               rth grid point along the y axis, for r=1,2,...,m .
+                                                               y
+               Constraint: y <y <...<y  .
+                            1  2      m
+                                       y
+
+           6:  F(MX*MY) -- DOUBLE PRECISION array                     Input
+               On entry: F(m *(q-1)+r) must contain the data value f   ,
+                            y                                       q,r
+               for q=1,2,...,m  and r=1,2,...,m .
+                              x                y
+
+           7:  S -- DOUBLE PRECISION                                  Input
+               On entry: the smoothing factor, S.
+
+               If S=0.0, the routine returns an interpolating spline.
+
+               If S is smaller than machine precision, it is assumed equal
+               to zero.
+
+               For advice on the choice of S, see Section 3 and Section 8.3
+               Constraint: S >= 0.0.
+
+           8:  NXEST -- INTEGER                                       Input
+
+           9:  NYEST -- INTEGER                                       Input
+               On entry: an upper bound for the number of knots n  and n
+                                                                  x      y
+               required in the x- and y-directions respectively.
+
+               In most practical situations, NXEST =m /2 and NYEST m /2 is
+                                                     x              y
+               sufficient. NXEST and NYEST never need to be larger than
+               m +4 and m +4 respectively, the numbers of knots needed for
+                x        y
+               interpolation (S=0.0). See also Section 8.4. Constraint:
+               NXEST >= 8 and NYEST >= 8.
+
+          10:  NX -- INTEGER                                   Input/Output
+               On entry: if the warm start option is used, the value of NX
+               must be left unchanged from the previous call. On exit: the
+               total number of knots, n , of the computed spline with
+                                       x
+               respect to the x variable.
+
+          11:  LAMDA(NXEST) -- DOUBLE PRECISION array          Input/Output
+               On entry: if the warm start option is used, the values
+               LAMDA(1), LAMDA(2),...,LAMDA(NX) must be left unchanged from
+               the previous call. On exit: LAMDA contains the complete set
+               of knots (lambda)  associated with the x variable, i.e., the
+                                i
+               interior knots LAMDA(5), LAMDA(6), ..., LAMDA(NX-4) as well
+               as the additional knots LAMDA(1) = LAMDA(2) = LAMDA(3) =
+               LAMDA(4) = X(1) and LAMDA(NX-3) = LAMDA(NX-2) = LAMDA(NX-1)
+               = LAMDA(NX) = X(MX) needed for the B-spline representation.
+
+          12:  NY -- INTEGER                                   Input/Output
+               On entry: if the warm start option is used, the value of NY
+               must be left unchanged from the previous call. On exit: the
+               total number of knots, n , of the computed spline with
+                                       y
+               respect to the y variable.
+
+          13:  MU(NYEST) -- DOUBLE PRECISION array             Input/Output
+               On entry: if the warm start option is used, the values MU
+               (1), MU(2),...,MU(NY) must be left unchanged from the
+               previous call. On exit: MU contains the complete set of
+               knots (mu)  associated with the y variable, i.e., the
+                         i
+               interior knots MU(5), MU(6),...,MU(NY-4) as well as the
+               additional knots MU(1) = MU(2) = MU(3) = MU(4) = Y(1) and MU
+               (NY-3) = MU(NY-2) = MU(NY-1) = MU(NY) = Y(MY) needed for the
+               B-spline representation.
+
+          14:  C((NXEST-4)*(NYEST-4)) -- DOUBLE PRECISION array      Output
+               On exit: the coefficients of the spline approximation. C(
+               (n -4)*(i-1)+j) is the coefficient c   defined in Section 3.
+                 y                                 ij
+
+          15:  FP -- DOUBLE PRECISION                                Output
+               On exit: the sum of squared residuals, (theta), of the
+               computed spline approximation. If FP = 0.0, this is an
+               interpolating spline. FP should equal S within a relative
+               tolerance of 0.001 unless NX = NY = 8, when the spline has
+               no interior knots and so is simply a bicubic polynomial. For
+               knots to be inserted, S must be set to a value below the
+               value of FP produced in this case.
+
+          16:  WRK(LWRK) -- DOUBLE PRECISION array                Workspace
+               On entry: if the warm start option is used, the values WRK
+               (1),...,WRK(4) must be left unchanged from the previous
+               call.
+
+               This array is used as workspace.
+
+          17:  LWRK -- INTEGER                                        Input
+               On entry:
+               the dimension of the array WRK as declared in the
+               (sub)program from which E02DCF is called.
+               Constraint:
+                    LWRK>=4*(MX+MY)+11*(NXEST+NYEST)+NXEST*MY
+
+                          +max(MY,NXEST)+54.
+
+          18:  IWRK(LIWRK) -- INTEGER array                       Workspace
+               On entry: if the warm start option is used, the values IWRK
+               (1), ..., IWRK(3) must be left unchanged from the previous
+               call.
+
+               This array is used as workspace.
+
+          19:  LIWRK -- INTEGER                                       Input
+               On entry:
+               the dimension of the array IWRK as declared in the
+               (sub)program from which E02DCF is called.
+               Constraint: LIWRK >= 3 + MX + MY + NXEST + NYEST.
+
+          20:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry START /= 'C' or 'W',
+
+               or       MX < 4,
+
+               or       MY < 4,
+
+               or       S < 0.0,
+
+               or       S = 0.0 and NXEST < MX + 4,
+
+               or       S = 0.0 and NYEST < MY + 4,
+
+               or       NXEST < 8,
+
+               or       NYEST < 8,
+
+               or       LWRK < 4*(MX+MY)+11*(NXEST+NYEST)+NXEST*MY+
+                               +max(MY,NXEST)+54
+
+               or       LIWRK < 3 + MX + MY + NXEST + NYEST.
+
+          IFAIL= 2
+               The values of X(q), for q = 1,2,...,MX, are not in strictly
+               increasing order.
+
+          IFAIL= 3
+               The values of Y(r), for r = 1,2,...,MY, are not in strictly
+               increasing order.
+
+          IFAIL= 4
+               The number of knots required is greater than allowed by
+               NXEST and NYEST. Try increasing NXEST and/or NYEST and, if
+               necessary, supplying larger arrays for the parameters LAMDA,
+               MU, C, WRK and IWRK. However, if NXEST and NYEST are already
+               large, say NXEST > MX/2 and NYEST > MY/2, then this error
+               exit may indicate that S is too small.
+
+          IFAIL= 5
+               The iterative process used to compute the coefficients of
+               the approximating spline has failed to converge. This error
+               exit may occur if S has been set very small. If the error
+               persists with increased S, consult NAG.
+
+          If IFAIL = 4 or 5, a spline approximation is returned, but it
+          fails to satisfy the fitting criterion (see (2) and (3) in
+          Section 3) -- perhaps by only a small amount, however.
+
+          7. Accuracy
+
+          On successful exit, the approximation returned is such that its
+          sum of squared residuals FP is equal to the smoothing factor S,
+          up to a specified relative tolerance of 0.001 - except that if
+          n =8 and n =8, FP may be significantly less than S: in this case
+           x        y
+          the computed spline is simply the least-squares bicubic
+          polynomial approximation of degree 3, i.e., a spline with no
+          interior knots.
+
+          8. Further Comments
+
+          8.1. Timing
+
+          The time taken for a call of E02DCF depends on the complexity of
+          the shape of the data, the value of the smoothing factor S, and
+          the number of data points. If E02DCF is to be called for
+          different values of S, much time can be saved by setting START =
+
+          8.2. Weighting of Data Points
+
+          E02DCF does not allow individual weighting of the data values. If
+          these were determined to widely differing accuracies, it may be
+          better to use E02DDF. The computation time would be very much
+          longer, however.
+
+          8.3. Choice of S
+
+          If the standard deviation of f    is the same for all q and r
+                                        q,r
+          (the case for which this routine is designed - see Section 8.2.)
+          and known to be equal, at least approximately, to (sigma), say,
+          then following Reinsch [5] and choosing the smoothing factor S in
+                           2      
+          the range (sigma) (m+-\/2m), where m=m m , is likely to give a
+                                                x y
+          good start in the search for a satisfactory value. If the
+          standard deviations vary, the sum of their squares over all the
+          data points could be used. Otherwise experimenting with different
+          values of S will be required from the start, taking account of
+          the remarks in Section 3.
+
+          In that case, in view of computation time and memory
+          requirements, it is recommended to start with a very large value
+          for S and so determine the least-squares bicubic polynomial; the
+          value returned for FP, call it FP , gives an upper bound for S.
+                                           0
+          Then progressively decrease the value of S to obtain closer fits
+          - say by a factor of 10 in the beginning, i.e., S=FP /10,
+                                                              0
+          S=FP /100, and so on, and more carefully as the approximation
+              0
+          shows more details.
+
+          The number of knots of the spline returned, and their location,
+          generally depend on the value of S and on the behaviour of the
+          function underlying the data. However, if E02DCF is called with
+          START = 'W', the knots returned may also depend on the smoothing
+          factors of the previous calls. Therefore if, after a number of
+          trials with different values of S and START = 'W', a fit can
+          finally be accepted as satisfactory, it may be worthwhile to call
+          E02DCF once more with the selected value for S but now using
+          START = 'C'. Often, E02DCF then returns an approximation with the
+          same quality of fit but with fewer knots, which is therefore
+          better if data reduction is also important.
+
+          8.4. Choice of NXEST and NYEST
+
+          The number of knots may also depend on the upper bounds NXEST and
+          NYEST. Indeed, if at a certain stage in E02DCF the number of
+          knots in one direction (say n ) has reached the value of its
+                                       x
+          upper bound (NXEST), then from that moment on all subsequent
+          knots are added in the other (y) direction. Therefore the user
+          has the option of limiting the number of knots the routine
+          locates in any direction. For example, by setting NXEST = 8 (the
+          lowest allowable value for NXEST), the user can indicate that he
+          wants an approximation which is a simple cubic polynomial in the
+          variable x.
+
+          8.5. Outline of Method Used
+
+          If S=0, the requisite number of knots is known in advance, i.e.,
+          n =m +4 and n =m +4; the interior knots are located immediately
+           x  x        y  y
+          as (lambda)  = x    and (mu)  = y   , for i=5,6,...,n -4 and
+                     i    i-2         j    j-2                 x
+          j=5,6,...,n -4. The corresponding least-squares spline is then an
+                     y
+          interpolating spline and therefore a solution of the problem.
+
+          If S>0, suitable knot sets are built up in stages (starting with
+          no interior knots in the case of a cold start but with the knot
+          set found in a previous call if a warm start is chosen). At each
+          stage, a bicubic spline is fitted to the data by least-squares,
+          and (theta), the sum of squares of residuals, is computed. If
+          (theta)>S, new knots are added to one knot set or the other so as
+          to reduce (theta) at the next stage. The new knots are located in
+          intervals where the fit is particularly poor, their number
+          depending on the value of S and on the progress made so far in
+          reducing (theta). Sooner or later, we find that (theta)<=S and at
+          that point the knot sets are accepted. The routine then goes on
+          to compute the (unique) spline which has these knot sets and
+          which satisfies the full fitting criterion specified by (2) and
+          (3). The theoretical solution has (theta)=S. The routine computes
+          the spline by an iterative scheme which is ended when (theta)=S
+          within a relative tolerance of 0.001. The main part of each
+          iteration consists of a linear least-squares computation of
+          special form, done in a similarly stable and efficient manner as
+          in E02BAF for least-squares curve fitting.
+
+          An exception occurs when the routine finds at the start that,
+          even with no interior knots (n =n =8), the least-squares spline
+                                        x  y
+          already has its sum of residuals <=S. In this case, since this
+          spline (which is simply a bicubic polynomial) also has an optimal
+          value for the smoothness measure (eta), namely zero, it is
+          returned at once as the (trivial) solution. It will usually mean
+          that S has been chosen too large.
+
+          For further details of the algorithm and its use see Dierckx [2].
+
+          8.6. Evaluation of Computed Spline
+
+          The values of the computed spline at the points (TX(r),TY(r)),
+          for r = 1,2,...,N, may be obtained in the double precision array
+          FF, of length at least N, by the following code:
+
+                 IFAIL = 0
+                 CALL E02DEF(N,NX,NY,TX,TY,LAMDA,MU,C,FF,WRK,IWRK,IFAIL)
+
+          where NX, NY, LAMDA, MU and C are the output parameters of E02DCF
+          , WRK is a double precision workspace array of length at least
+          NY-4, and IWRK is an integer workspace array of length at least
+          NY-4.
+
+          To evaluate the computed spline on a KX by KY rectangular grid of
+          points in the x-y plane, which is defined by the x co-ordinates
+          stored in TX(q), for q=1,2,...,KX, and the y co-ordinates stored
+          in TY(r), for r=1,2,...,KY, returning the results in the double
+          precision array FG which is of length at least KX*KY, the
+          following call may be used:
+
+                IFAIL = 0
+                CALL E02DFF(KX,KY,NX,NY,TX,TY,LAMDA,MU,C,FG,WRK,LWRK,
+               *            IWRK,LIWRK,IFAIL)
+
+          where NX, NY, LAMDA, MU and C are the output parameters of E02DCF
+          , WRK is a double precision workspace array of length at least
+          LWRK = min(NWRK1,NWRK2), NWRK1 = KX*4+NX, NWRK2 = KY*4+NY, and
+          IWRK is an integer workspace array of length at least LIWRK = KY
+          + NY - 4 if NWRK1 >= NWRK2, or KX + NX - 4 otherwise. The result
+          of the spline evaluated at grid point (q,r) is returned in
+          element (KY*(q-1)+r) of the array FG.
+
+          9. Example
+
+          This example program reads in values of MX, MY, x , for q = 1,2,.
+                                                           q
+                      r
+          ordinates f    defined at the grid points (x ,y ). It then calls
+                     q,r                              q  r
+          E02DCF to compute a bicubic spline approximation for one
+          specified value of S, and prints the values of the computed knots
+          and B-spline coefficients. Finally it evaluates the spline at a
+          small sample of points on a rectangular grid.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02ddf}{NAG On-line Documentation: e02ddf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02DDF(3NAG)      Foundation Library (12/10/92)      E02DDF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02DDF
+                  E02DDF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02DDF computes a bicubic spline approximation to a set of
+          scattered data. The knots of the spline are located
+          automatically, but a single parameter must be specified to
+          control the trade-off between closeness of fit and smoothness of
+          fit.
+
+          2. Specification
+
+                 SUBROUTINE E02DDF (START, M, X, Y, F, W, S, NXEST, NYEST,
+                1                   NX, LAMDA, NY, MU, C, FP, RANK, WRK,
+                2                   LWRK, IWRK, LIWRK, IFAIL)
+                 INTEGER          M, NXEST, NYEST, NX, NY, RANK, LWRK, IWRK
+                1                 (LIWRK), LIWRK, IFAIL
+                 DOUBLE PRECISION X(M), Y(M), F(M), W(M), S, LAMDA(NXEST),
+                1                 MU(NYEST), C((NXEST-4)*(NYEST-4)), FP, WRK
+                2                 (LWRK)
+                 CHARACTER*1      START
+
+          3. Description
+
+          This routine determines a smooth bicubic spline approximation
+          s(x,y) to the set of data points (x ,y ,f ) with weights w , for
+                                             r  r  r                r
+          r=1,2,...,m.
+
+          The approximation domain is considered to be the rectangle
+          [x   ,x   ]*[y   ,y   ], where x    (y   ) and x    (y   ) denote
+            min  max    min  max          min   min       max   max
+          the lowest and highest data values of x (y).
+
+          The spline is given in the B-spline representation
+
+                                n -4 n -4
+                                 x    y
+                                --   --
+                        s(x,y)= >    >   c  M (x)N (y),                 (1)
+                                --   --   ij i    j
+                                i=1  j=1
+
+          where M (x) and N (y) denote normalised cubic B-splines, the
+                 i         j
+          former defined on the knots (lambda)  to (lambda)    and the
+                                              i            i+4
+          latter on the knots (mu)  to (mu)   . For further details, see
+                                  j        j+4
+          Hayes and Halliday [4] for bicubic splines and de Boor [1] for
+          normalised B-splines.
+
+          The total numbers n  and n  of these knots and their values
+                             x      y
+          (lambda) ,...,(lambda)   and (mu) ,...,(mu)   are chosen
+                  1             n          1         n
+                                 x                    y
+          automatically by the routine. The knots (lambda) ,...,
+                                                          5
+          (lambda)     and (mu) ,..., (mu)     are the interior knots; they
+                  n -4         5          n -4
+                   x                       y
+          divide the approximation domain [x   ,x   ]*[y   ,y   ] into (
+                                            min  max    min  max
+          n -7)*(n -7) subpanels [(lambda) ,(lambda)   ]*[(mu) ,(mu)   ],
+           x      y                       i         i+1       j     j+1
+          for i=4,5,...,n -4; j=4,5,...,n -4. Then, much as in the curve
+                         x               y
+          case (see E02BEF), the coefficients c   are determined as the
+                                               ij
+          solution of the following constrained minimization problem:
+
+          minimize
+
+                                     (eta),                             (2)
+
+          subject to the constraint
+
+                                    m
+                                    --          2
+                           (theta)= >  (epsilon) <=S                    (3)
+                                    --          r
+                                    r=1
+
+          where: (eta) is a measure of the (lack of) smoothness of s(x,y).
+                       Its value depends on the discontinuity jumps in
+                       s(x,y) across the boundaries of the subpanels. It is
+                       zero only when there are no discontinuities and is
+                       positive otherwise, increasing with the size of the
+                       jumps (see Dierckx [2] for details).
+
+                 (epsilon)  denotes the weighted residual w (f -s(x ,y )),
+                          r                                r  r    r  r
+
+          and    S     is a non-negative number to be specified by the user.
+
+          By means of the parameter S, 'the smoothing factor', the user
+          will then control the balance between smoothness and closeness of
+          fit, as measured by the sum of squares of residuals in (3). If S
+          is too large, the spline will be too smooth and signal will be
+          lost (underfit); if S is too small, the spline will pick up too
+          much noise (overfit). In the extreme cases the method would
+          return an interpolating spline ((theta)=0) if S were set to zero,
+          and returns the least-squares bicubic polynomial ((eta)=0) if S
+          is set very large. Experimenting with S-values between these two
+          extremes should result in a good compromise. (See Section 8.2 for
+          advice on choice of S.) Note however, that this routine, unlike
+          E02BEF and E02DCF, does not allow S to be set exactly to zero: to
+          compute an interpolant to scattered data, E01SAF or E01SEF should
+          be used.
+
+          The method employed is outlined in Section 8.5 and fully
+          described in Dierckx [2] and [3]. It involves an adaptive
+          strategy for locating the knots of the bicubic spline (depending
+          on the function underlying the data and on the value of S), and
+          an iterative method for solving the constrained minimization
+          problem once the knots have been determined.
+
+          Values of the computed spline can subsequently be computed by
+          calling E02DEF or E02DFF as described in Section 8.6.
+
+          4. References
+
+          [1]   De Boor C (1972) On Calculating with B-splines. J. Approx.
+                Theory. 6 50--62.
+
+          [2]   Dierckx P (1981) An Algorithm for Surface Fitting with
+                Spline Functions. IMA J. Num. Anal. 1 267--283.
+
+          [3]   Dierckx P (1981) An Improved Algorithm for Curve Fitting
+                with Spline Functions. Report TW54. Department of Computer
+                Science, Katholieke Universiteit Leuven.
+
+          [4]   Hayes J G and Halliday J (1974) The Least-squares Fitting of
+                Cubic Spline Surfaces to General Data Sets. J. Inst. Math.
+                Appl. 14 89--103.
+
+          [5]   Peters G and Wilkinson J H (1970) The Least-squares Problem
+                and Pseudo-inverses. Comput. J. 13 309--316.
+
+          [6]   Reinsch C H (1967) Smoothing by Spline Functions. Num. Math.
+                10 177--183.
+
+          5. Parameters
+
+           1:  START -- CHARACTER*1                                   Input
+               On entry: START must be set to 'C' or 'W'.
+
+               If START = 'C' (Cold start), the routine will build up the
+               knot set starting with no interior knots. No values need be
+               assigned to the parameters NX, NY, LAMDA, MU or WRK.
+
+               If START = 'W' (Warm start), the routine will restart the
+               knot-placing strategy using the knots found in a previous
+               call of the routine. In this case, the parameters NX, NY,
+               LAMDA, MU and WRK must be unchanged from that previous call.
+               This warm start can save much time in searching for a
+               satisfactory value of S. Constraint: START = 'C' or 'W'.
+
+           2:  M -- INTEGER                                           Input
+               On entry: m, the number of data points.
+
+               The number of data points with non-zero weight (see W below)
+               must be at least 16.
+
+           3:  X(M) -- DOUBLE PRECISION array                         Input
+
+           4:  Y(M) -- DOUBLE PRECISION array                         Input
+
+           5:  F(M) -- DOUBLE PRECISION array                         Input
+               On entry: X(r), Y(r), F(r) must be set to the co-ordinates
+               of (x ,y ,f ), the rth data point, for r=1,2,...,m. The
+                    r  r  r
+               order of the data points is immaterial.
+
+           6:  W(M) -- DOUBLE PRECISION array                         Input
+               On entry: W(r) must be set to w , the rth value in the set
+                                              r
+               of weights, for r=1,2,...,m. Zero weights are permitted and
+               the corresponding points are ignored, except when
+               determining x   , x   , y    and y    (see Section 8.4). For
+                            min   max   min      max
+               advice on the choice of weights, see Section 2.1.2 of the
+               Chapter Introduction. Constraint: the number of data points
+               with non-zero weight must be at least 16.
+
+           7:  S -- DOUBLE PRECISION                                  Input
+               On entry: the smoothing factor, S.
+
+               For advice on the choice of S, see Section 3 and Section 8.2
+               . Constraint: S > 0.0.
+
+           8:  NXEST -- INTEGER                                       Input
+
+
+           9:  NYEST -- INTEGER                                       Input
+               On entry: an upper bound for the number of knots n  and n
+                                                                 x      y
+               required in the x- and y-directions respectively.
+                                                                 ___
+               In most practical situations, NXEST = NYEST = 4+\/m/2 is
+               sufficient. See also Section 8.3. Constraint: NXEST >= 8 and
+               NYEST >= 8.
+
+          10:  NX -- INTEGER                                   Input/Output
+               On entry: if the warm start option is used, the value of NX
+               must be left unchanged from the previous call. On exit: the
+               total number of knots, n , of the computed spline with
+                                       x
+               respect to the x variable.
+
+          11:  LAMDA(NXEST) -- DOUBLE PRECISION array          Input/Output
+               On entry: if the warm start option is used, the values LAMDA
+               (1), LAMDA(2),...,LAMDA(NX) must be left unchanged from the
+               previous call. On exit: LAMDA contains the complete set of
+               knots (lambda)  associated with the x variable, i.e., the
+                             i
+               interior knots LAMDA(5), LAMDA(6),...,LAMDA(NX-4) as well as
+               the additional knots LAMDA(1) = LAMDA(2) = LAMDA(3) = LAMDA
+               (4) = x    and LAMDA(NX-3) = LAMDA(NX-2) = LAMDA(NX-1) =
+                      min
+               LAMDA(NX) = x    needed for the B-spline representation
+                            max
+               (where x    and x    are as described in Section 3).
+                       min      max
+
+          12:  NY -- INTEGER                                   Input/Output
+               On entry: if the warm start option is used, the value of NY
+               must be left unchanged from the previous call. On exit: the
+               total number of knots, n , of the computed spline with
+                                       y
+               respect to the y variable.
+
+          13:  MU(NYEST) -- DOUBLE PRECISION array             Input/Output
+               On entry: if the warm start option is used, the values MU(1)
+               MU(2),...,MU(NY) must be left unchanged from the previous
+               call. On exit: MU contains the complete set of knots (mu)
+                                                                        i
+               associated with the y variable, i.e., the interior knots MU
+               (5), MU(6),...,MU(NY-4) as well as the additional knots MU
+               (1) = MU(2) = MU(3) = MU(4) = y    and MU(NY-3) = MU(NY-2) =
+                                              min
+               MU(NY-1) = MU(NY) = y    needed for the B-spline
+                                    max
+               representation (where y    and y    are as described in
+                                      min      max
+               Section 3).
+
+          14:  C((NXEST-4)*(NYEST-4)) -- DOUBLE PRECISION array      Output
+               On exit: the coefficients of the spline approximation. C(
+               (n -4)*(i-1)+j) is the coefficient c   defined in Section 3.
+                 y                                 ij
+
+          15:  FP -- DOUBLE PRECISION                                Output
+               On exit: the weighted sum of squared residuals, (theta), of
+               the computed spline approximation. FP should equal S within
+               a relative tolerance of 0.001 unless NX = NY = 8, when the
+               spline has no interior knots and so is simply a bicubic
+               polynomial. For knots to be inserted, S must be set to a
+               value below the value of FP produced in this case.
+
+          16:  RANK -- INTEGER                                       Output
+               On exit: RANK gives the rank of the system of equations used
+               to compute the final spline (as determined by a suitable
+               machine-dependent threshold). When RANK = (NX-4)*(NY-4), the
+               solution is unique; otherwise the system is rank-deficient
+               and the minimum-norm solution is computed. The latter case
+               may be caused by too small a value of S.
+
+          17:  WRK(LWRK) -- DOUBLE PRECISION array                Workspace
+               On entry: if the warm start option is used, the value of WRK
+               (1) must be left unchanged from the previous call.
+
+               This array is used as workspace.
+
+          18:  LWRK -- INTEGER                                        Input
+               On entry:
+               the dimension of the array WRK as declared in the
+               (sub)program from which E02DDF is called.
+               Constraint: LWRK >= (7*u*v+25*w)*(w+1)+2*(u+v+4*M)+23*w+56,
+
+               where
+
+               u=NXEST-4, v=NYEST-4, and w=max(u,v).
+
+               For some problems, the routine may need to compute the
+               minimal least-squares solution of a rank-deficient system of
+               linear equations (see Section 3). The amount of workspace
+               required to solve such problems will be larger than
+               specified by the value given above, which must be increased
+               by an amount, LWRK2 say. An upper bound for LWRK2 is given
+               by 4*u*v*w+2*u*v+4*w, where u, v and w are as above.
+               However, if there are enough data points, scattered
+               uniformly over the approximation domain, and if the
+               smoothing factor S is not too small, there is a good chance
+               that this extra workspace is not needed. A lot of memory
+               might therefore be saved by assuming LWRK2 = 0.
+
+          19:  IWRK(LIWRK) -- INTEGER array                       Workspace
+
+
+          20:  LIWRK -- INTEGER                                       Input
+               On entry:
+               the dimension of the array IWRK as declared in the
+               (sub)program from which E02DDF is called.
+               Constraint: LIWRK>=M+2*(NXEST-7)*(NYEST-7).
+
+          21:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry START /= 'C' or 'W',
+
+               or       the number of data points with non-zero weight <
+                        16,
+
+               or       S <= 0.0,
+
+               or       NXEST < 8,
+
+               or       NYEST < 8,
+
+               or       LWRK < (7*u*v+25*w)*(w+1)+2*(u+v+4*M)+23*w+56,
+                        where u = NXEST - 4, v = NYEST - 4 and w=max(u,v),
+
+               or       LIWRK <M+2*(NXEST-7)*(NYEST-7).
+
+          IFAIL= 2
+               On entry either all the X(r), for r = 1,2,...,M, are equal,
+               or all the Y(r), for r = 1,2,...,M, are equal.
+
+          IFAIL= 3
+               The number of knots required is greater than allowed by
+               NXEST and NYEST. Try increasing NXEST and/or NYEST and, if
+               necessary, supplying larger arrays for the parameters LAMDA,
+               MU, C, WRK and IWRK. However, if NXEST and NYEST are already
+                                               
+
+               large, say NXEST, NYEST > 4 + \/M/2, then this error exit
+               may indicate that S is too small.
+
+          IFAIL= 4
+               No more knots can be added because the number of B-spline
+               coefficients (NX-4)*(NY-4) already exceeds the number of
+               data points M. This error exit may occur if either of S or M
+               is too small.
+
+          IFAIL= 5
+               No more knots can be added because the additional knot would
+               (quasi) coincide with an old one. This error exit may occur
+               if too large a weight has been given to an inaccurate data
+               point, or if S is too small.
+
+          IFAIL= 6
+               The iterative process used to compute the coefficients of
+               the approximating spline has failed to converge. This error
+               exit may occur if S has been set very small. If the error
+               persists with increased S, consult NAG.
+
+          IFAIL= 7
+               LWRK is too small; the routine needs to compute the minimal
+               least-squares solution of a rank-deficient system of linear
+               equations, but there is not enough workspace. There is no
+               approximation returned but, having saved the information
+               contained in NX, LAMDA, NY, MU and WRK, and having adjusted
+               the value of LWRK and the dimension of array WRK
+               accordingly, the user can continue at the point the program
+               was left by calling E02DDF with START = 'W'. Note that the
+               requested value for LWRK is only large enough for the
+               current phase of the algorithm. If the routine is restarted
+               with LWRK set to the minimum value requested, a larger
+               request may be made at a later stage of the computation. See
+               Section 5 for the upper bound on LWRK. On soft failure, the
+               minimum requested value for LWRK is returned in IWRK(1) and
+               the safe value for LWRK is returned in IWRK(2).
+
+          If IFAIL = 3,4,5 or 6, a spline approximation is returned, but it
+          fails to satisfy the fitting criterion (see (2) and (3) in
+          Section 3 -- perhaps only by a small amount, however.
+
+          7. Accuracy
+
+          On successful exit, the approximation returned is such that its
+          weighted sum of squared residuals FP is equal to the smoothing
+          factor S, up to a specified relative tolerance of 0.001 - except
+          that if n =8 and n =8, FP may be significantly less than S: in
+                   x        y
+          this case the computed spline is simply the least-squares bicubic
+          polynomial approximation of degree 3, i.e., a spline with no
+          interior knots.
+
+          8. Further Comments
+
+          8.1. Timing
+
+          The time taken for a call of E02DDF depends on the complexity of
+          the shape of the data, the value of the smoothing factor S, and
+          the number of data points. If E02DDF is to be called for
+          different values of S, much time can be saved by setting START =
+          It should be noted that choosing S very small considerably
+          increases computation time.
+
+          8.2. Choice of S
+
+          If the weights have been correctly chosen (see Section 2.1.2 of
+          the Chapter Introduction), the standard deviation of w f  would
+                                                                r r
+          be the same for all r, equal to (sigma), say. In this case,
+                                                              2      
+          choosing the smoothing factor S in the range (sigma) (m+-\/2m),
+          as suggested by Reinsch [6], is likely to give a good start in
+          the search for a satisfactory value. Otherwise, experimenting
+          with different values of S will be required from the start.
+
+          In that case, in view of computation time and memory
+          requirements, it is recommended to start with a very large value
+          for S and so determine the least-squares bicubic polynomial; the
+          value returned for FP, call it FP , gives an upper bound for S.
+                                           0
+          Then progressively decrease the value of S to obtain closer fits
+          - say by a factor of 10 in the beginning, i.e., S=FP /10,
+                                                              0
+          S=FP /100, and so on, and more carefully as the approximation
+              0
+          shows more details.
+
+          To choose S very small is strongly discouraged. This considerably
+          increases computation time and memory requirements. It may also
+          cause rank-deficiency (as indicated by the parameter RANK) and
+          endanger numerical stability.
+
+          The number of knots of the spline returned, and their location,
+          generally depend on the value of S and on the behaviour of the
+          function underlying the data. However, if E02DDF is called with
+          START = 'W', the knots returned may also depend on the smoothing
+          factors of the previous calls. Therefore if, after a number of
+          trials with different values of S and START = 'W', a fit can
+          finally be accepted as satisfactory, it may be worthwhile to call
+          E02DDF once more with the selected value for S but now using
+          START = 'C'. Often, E02DDF then returns an approximation with the
+          same quality of fit but with fewer knots, which is therefore
+          better if data reduction is also important.
+
+          8.3. Choice of NXEST and NYEST
+
+          The number of knots may also depend on the upper bounds NXEST and
+          NYEST. Indeed, if at a certain stage in E02DDF the number of
+          knots in one direction (say n ) has reached the value of its
+                                       x
+          upper bound (NXEST), then from that moment on all subsequent
+          knots are added in the other (y) direction. This may indicate
+          that the value of NXEST is too small. On the other hand, it gives
+          the user the option of limiting the number of knots the routine
+          locates in any direction. For example, by setting NXEST = 8 (the
+          lowest allowable value for NXEST), the user can indicate that he
+          wants an approximation which is a simple cubic polynomial in the
+          variable x.
+
+          8.4. Restriction of the approximation domain
+
+          The fit obtained is not defined outside the rectangle
+          [(lambda) ,(lambda)    ]*[(mu) ,(mu)    ]. The reason for taking
+                   4         n -3       4     n -3
+                              x                y
+          the extreme data values of x and y for these four knots is that,
+          as is usual in data fitting, the fit cannot be expected to give
+          satisfactory values outside the data region. If, nevertheless,
+          the user requires values over a larger rectangle, this can be
+          achieved by augmenting the data with two artificial data points
+          (a,c,0) and (b,d,0) with zero weight, where [a,b]*[c,d] denotes
+          the enlarged rectangle.
+
+          8.5. Outline of method used
+
+          First suitable knot sets are built up in stages (starting with no
+          interior knots in the case of a cold start but with the knot set
+          found in a previous call if a warm start is chosen). At each
+          stage, a bicubic spline is fitted to the data by least-squares
+          and (theta), the sum of squares of residuals, is computed. If
+          (theta)>S, a new knot is added to one knot set or the other so as
+          to reduce (theta) at the next stage. The new knot is located in
+          an interval where the fit is particularly poor. Sooner or later,
+          we find that (theta)<=S and at that point the knot sets are
+          accepted. The routine then goes on to compute a spline which has
+          these knot sets and which satisfies the full fitting criterion
+          specified by (2) and (3). The theoretical solution has (theta)=S.
+          The routine computes the spline by an iterative scheme which is
+          ended when (theta)=S within a relative tolerance of 0.001. The
+          main part of each iteration consists of a linear least-squares
+          computation of special form, done in a similarly stable and
+          efficient manner as in E02DAF. As there also, the minimal least-
+          squares solution is computed wherever the linear system is found
+          to be rank-deficient.
+
+          An exception occurs when the routine finds at the start that,
+          even with no interior knots (N = 8), the least-squares spline
+          already has its sum of squares of residuals <=S. In this case,
+          since this spline (which is simply a bicubic polynomial) also has
+          an optimal value for the smoothness measure (eta), namely zero,
+          it is returned at once as the (trivial) solution. It will usually
+          mean that S has been chosen too large.
+
+          For further details of the algorithm and its use see Dierckx [2].
+
+          8.6. Evaluation of computed spline
+
+          The values of the computed spline at the points (TX(r),TY(r)),
+          for r = 1,2,...,N, may be obtained in the double precision array
+          FF, of length at least N, by the following code:
+
+
+                IFAIL = 0
+                CALL E02DEF(N,NX,NY,TX,TY,LAMDA,MU,C,FF,WRK,IWRK,IFAIL)
+
+
+          where NX, NY, LAMDA, MU and C are the output parameters of E02DDF
+          , WRK is a double precision workspace array of length at least
+          NY-4, and IWRK is an integer workspace array of length at least
+          NY-4.
+
+          To evaluate the computed spline on a KX by KY rectangular grid of
+          points in the x-y plane, which is defined by the x co-ordinates
+          stored in TX(q), for q=1,2,...,KX, and the y co-ordinates stored
+          in TY(r), for r=1,2,...,KY, returning the results in the double
+          precision array FG which is of length at least KX*KY, the
+          following call may be used:
+
+
+                IFAIL = 0
+                CALL E02DFF(KX,KY,NX,NY,TX,TY,LAMDA,MU,C,FG,WRK,LWRK,
+               *            IWRK,LIWRK,IFAIL)
+
+
+          where NX, NY, LAMDA, MU and C are the output parameters of E02DDF
+          , WRK is a double precision workspace array of length at least
+          LWRK = min(NWRK1,NWRK2), NWRK1 = KX*4+NX, NWRK2 = KY*4+NY, and
+          IWRK is an integer workspace array of length at least LIWRK = KY
+          + NY - 4 if NWRK1 >= NWRK2, or KX + NX - 4 otherwise. The result
+          of the spline evaluated at grid point (q,r) is returned in
+          element (KY*(q-1)+r) of the array FG.
+
+          9. Example
+
+          This example program reads in a value of M, followed by a set of
+          M data points (x ,y ,f ) and their weights w . It then calls
+                          r  r  r                     r
+          E02DDF to compute a bicubic spline approximation for one
+          specified value of S, and prints the values of the computed knots
+          and B-spline coefficients. Finally it evaluates the spline at a
+          small sample of points on a rectangular grid.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02def}{NAG On-line Documentation: e02def}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02DEF(3NAG)      Foundation Library (12/10/92)      E02DEF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02DEF
+                  E02DEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02DEF calculates values of a bicubic spline from its B-spline
+          representation.
+
+          2. Specification
+
+                 SUBROUTINE E02DEF (M, PX, PY, X, Y, LAMDA, MU, C, FF, WRK,
+                1                   IWRK, IFAIL)
+                 INTEGER          M, PX, PY, IWRK(PY-4), IFAIL
+                 DOUBLE PRECISION X(M), Y(M), LAMDA(PX), MU(PY), C((PX-4)*
+                1                 (PY-4)), FF(M), WRK(PY-4)
+
+          3. Description
+
+          This routine calculates values of the bicubic spline s(x,y) at
+          prescribed points (x ,y ), for r=1,2,...,m, from its augmented
+                              r  r
+          knot sets {(lambda)} and {(mu)} and from the coefficients c  ,
+                                                                     ij
+          for i=1,2,...,PX-4; j=1,2,...,PY-4, in its B-spline
+          representation
+
+                                      --
+                              s(x,y)= > c  M (x)N (y).
+                                      -- ij i    j
+                                      ij
+
+          Here M (x) and N (y) denote normalised cubic B-splines, the
+                i         j
+          former defined on the knots (lambda)  to (lambda)    and the
+                                              i            i+4
+          latter on the knots (mu)  to (mu)   .
+                                  j        j+4
+
+          This routine may be used to calculate values of a bicubic spline
+          given in the form produced by E01DAF, E02DAF, E02DCF and E02DDF.
+          It is derived from the routine B2VRE in Anthony et al [1].
+
+          4. References
+
+          [1]   Anthony G T, Cox M G and Hayes J G (1982) DASL - Data
+                Approximation Subroutine Library. National Physical
+                Laboratory.
+
+          [2]   Cox M G (1978) The Numerical Evaluation of a Spline from its
+                B-spline Representation. J. Inst. Math. Appl. 21 135--143.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: m, the number of points at which values of the
+               spline are required. Constraint: M >= 1.
+
+           2:  PX -- INTEGER                                          Input
+
+           3:  PY -- INTEGER                                          Input
+               On entry: PX and PY must specify the total number of knots
+               associated with the variables x and y respectively. They are
+               such that PX-8 and PY-8 are the corresponding numbers of
+               interior knots. Constraint: PX >= 8 and PY >= 8.
+
+           4:  X(M) -- DOUBLE PRECISION array                         Input
+
+           5:  Y(M) -- DOUBLE PRECISION array                         Input
+               On entry: X and Y must contain x  and y , for r=1,2,...,m,
+                                               r      r
+               respectively. These are the co-ordinates of the points at
+               which values of the spline are required. The order of the
+               points is immaterial. Constraint: X and Y must satisfy
+
+               LAMDA(4) <= X(r) <= LAMDA(PX-3)
+
+               and
+
+               MU(4) <= Y(r) <= MU(PY-3), for r=1,2,...,m.
+
+               The spline representation is not valid outside these
+               intervals.
+
+           6:  LAMDA(PX) -- DOUBLE PRECISION array                    Input
+
+           7:  MU(PY) -- DOUBLE PRECISION array                       Input
+               On entry: LAMDA and MU must contain the complete sets of
+               knots {(lambda)} and {(mu)} associated with the x and y
+               variables respectively. Constraint: the knots in each set
+               must be in non-decreasing order, with LAMDA(PX-3) > LAMDA(4)
+               and MU(PY-3) > MU(4).
+
+           8:  C((PX-4)*(PY-4)) -- DOUBLE PRECISION array             Input
+               On entry: C((PY-4)*(i-1)+j) must contain the coefficient
+               c   described in Section 3, for i=1,2,...,PX-4;
+                ij
+               j=1,2,...,PY-4.
+
+           9:  FF(M) -- DOUBLE PRECISION array                       Output
+               On exit: FF(r) contains the value of the spline at the
+               point (x ,y ), for r=1,2,...,m.
+                       r  r
+
+          10:  WRK(PY-4) -- DOUBLE PRECISION array                Workspace
+
+          11:  IWRK(PY-4) -- INTEGER array                        Workspace
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M < 1,
+
+               or       PY < 8,
+
+               or       PX < 8.
+
+          IFAIL= 2
+               On entry the knots in array LAMDA, or those in array MU, are
+               not in non-decreasing order, or LAMDA(PX-3) <= LAMDA(4), or
+               MU(PY-3) <= MU(4).
+
+          IFAIL= 3
+               On entry at least one of the prescribed points (x ,y ) lies
+                                                                r  r
+               outside the rectangle defined by LAMDA(4), LAMDA(PX-3) and
+               MU(4), MU(PY-3).
+
+          7. Accuracy
+
+          The method used to evaluate the B-splines is numerically stable,
+          in the sense that each computed value of s(x ,y ) can be regarded
+                                                      r  r
+          as the value that would have been obtained in exact arithmetic
+          from slightly perturbed B-spline coefficients. See Cox [2] for
+          details.
+
+          8. Further Comments
+
+          Computation time is approximately proportional to the number of
+          points, m, at which the evaluation is required.
+
+          9. Example
+
+          This program reads in knot sets LAMDA(1),..., LAMDA(PX) and MU(1)
+          ,..., MU(PY), and a set of bicubic spline coefficients c  .
+                                                                  ij
+          Following these are a value for m and the co-ordinates (x ,y ),
+                                                                   r  r
+          for r=1,2,...,m, at which the spline is to be evaluated.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02dff}{NAG On-line Documentation: e02dff}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02DFF(3NAG)      Foundation Library (12/10/92)      E02DFF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02DFF
+                  E02DFF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02DFF calculates values of a bicubic spline from its B-spline
+          representation. The spline is evaluated at all points on a
+          rectangular grid.
+
+          2. Specification
+
+                 SUBROUTINE E02DFF (MX, MY, PX, PY, X, Y, LAMDA, MU, C, FF,
+                1                   WRK, LWRK, IWRK, LIWRK, IFAIL)
+                 INTEGER          MX, MY, PX, PY, LWRK, IWRK(LIWRK), LIWRK,
+                1                 IFAIL
+                 DOUBLE PRECISION X(MX), Y(MY), LAMDA(PX), MU(PY), C((PX-4)*
+                1                 (PY-4)), FF(MX*MY), WRK(LWRK)
+
+          3. Description
+
+          This routine calculates values of the bicubic spline s(x,y) on a
+          rectangular grid of points in the x-y plane, from its augmented
+          knot sets {(lambda)} and {(mu)} and from the coefficients c  ,
+                                                                     ij
+          for i=1,2,...,PX-4; j=1,2,...,PY-4, in its B-spline
+          representation
+
+                                      --
+                              s(x,y)= > c  M (x)N (y).
+                                      -- ij i    j
+                                      ij
+
+          Here M (x) and N (y) denote normalised cubic B-splines, the
+                i         j
+          former defined on the knots (lambda)  to (lambda)    and the
+                                              i            i+4
+          latter on the knots (mu)  to (mu)   .
+                                  j        j+4
+
+          The points in the grid are defined by co-ordinates x , for
+                                                              q
+          q=1,2,...,m , along the x axis, and co-ordinates y , for
+                     x                                      r
+          r=1,2,...,m  along the y axis.
+                     y
+
+          This routine may be used to calculate values of a bicubic spline
+          given in the form produced by E01DAF, E02DAF, E02DCF and E02DDF.
+          It is derived from the routine B2VRE in Anthony et al [1].
+
+          4. References
+
+          [1]   Anthony G T, Cox M G and Hayes J G (1982) DASL - Data
+                Approximation Subroutine Library. National Physical
+                Laboratory.
+
+          [2]   Cox M G (1978) The Numerical Evaluation of a Spline from its
+                B-spline Representation. J. Inst. Math. Appl. 21 135--143.
+
+          5. Parameters
+
+           1:  MX -- INTEGER                                          Input
+
+           2:  MY -- INTEGER                                          Input
+               On entry: MX and MY must specify m  and m  respectively,
+                                                 x      y
+               the number of points along the x and y axis that define the
+               rectangular grid. Constraint: MX >= 1 and MY >= 1.
+
+           3:  PX -- INTEGER                                          Input
+
+           4:  PY -- INTEGER                                          Input
+               On entry: PX and PY must specify the total number of knots
+               associated with the variables x and y respectively. They are
+               such that PX-8 and PY-8 are the corresponding numbers of
+               interior knots. Constraint: PX >= 8 and PY >= 8.
+
+           5:  X(MX) -- DOUBLE PRECISION array                        Input
+
+           6:  Y(MY) -- DOUBLE PRECISION array                        Input
+               On entry: X and Y must contain x , for q=1,2,...,m , and y ,
+                                               q                 x       r
+               for r=1,2,...,m , respectively. These are the x and y co-
+                              y
+               ordinates that define the rectangular grid of points at
+               which values of the spline are required. Constraint: X and Y
+               must satisfy
+
+               LAMDA(4) <= X(q) < X(q+1) <= LAMDA(PX-3), for q=1,2,...,m -1
+                                                                        x
+               and
+
+               MU(4) <= Y(r) < Y(r+1) <= MU(PY-3), for r=1,2,...,m -1.
+                                                                  y
+
+               The spline representation is not valid outside these
+               intervals.
+
+           7:  LAMDA(PX) -- DOUBLE PRECISION array                    Input
+
+           8:  MU(PY) -- DOUBLE PRECISION array                       Input
+               On entry: LAMDA and MU must contain the complete sets of
+               knots {(lambda)} and {(mu)} associated with the x and y
+               variables respectively. Constraint: the knots in each set
+               must be in non-decreasing order, with LAMDA(PX-3) > LAMDA(4)
+               and MU(PY-3) > MU(4).
+
+           9:  C((PX-4)*(PY-4)) -- DOUBLE PRECISION array             Input
+               On entry: C((PY-4)*(i-1)+j) must contain the coefficient
+               c   described in Section 3, for i=1,2,...,PX-4;
+                ij
+               j=1,2,...,PY-4.
+
+          10:  FF(MX*MY) -- DOUBLE PRECISION array                   Output
+               On exit: FF(MY*(q-1)+r) contains the value of the spline at
+               the point (x ,y ), for q=1,2,...,m ; r=1,2,...,m .
+                           q  r                  x             y
+
+          11:  WRK(LWRK) -- DOUBLE PRECISION array                Workspace
+
+          12:  LWRK -- INTEGER                                        Input
+               On entry:
+               the dimension of the array WRK as declared in the
+               (sub)program from which E02DFF is called.
+               Constraint: LWRK >= min(NWRK1,NWRK2), where NWRK1=4*MX+PX,
+               NWRK2=4*MY+PY.
+
+          13:  IWRK(LIWRK) -- INTEGER array                       Workspace
+
+          14:  LIWRK -- INTEGER                                       Input
+               On entry:
+               the dimension of the array IWRK as declared in the
+               (sub)program from which E02DFF is called.
+               Constraint: LIWRK >= MY + PY - 4 if NWRK1 > NWRK2, or MX +
+               PX - 4 otherwise, where NWRK1 and NWRK2 are as defined in
+               the description of argument LWRK.
+
+          15:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry MX < 1,
+
+               or       MY < 1,
+
+               or       PY < 8,
+
+               or       PX < 8.
+
+          IFAIL= 2
+               On entry LWRK is too small,
+
+               or       LIWRK is too small.
+
+          IFAIL= 3
+               On entry the knots in array LAMDA, or those in array MU, are
+               not in non-decreasing order, or LAMDA(PX-3) <= LAMDA(4), or
+               MU(PY-3) <= MU(4).
+
+          IFAIL= 4
+               On entry the restriction LAMDA(4) <= X(1) <... < X(MX) <=
+               LAMDA(PX-3), or the restriction MU(4) <= Y(1) <... < Y(MY)
+               <= MU(PY-3), is violated.
+
+          7. Accuracy
+
+          The method used to evaluate the B-splines is numerically stable,
+          in the sense that each computed value of s(x ,y ) can be regarded
+                                                      r  r
+          as the value that would have been obtained in exact arithmetic
+          from slightly perturbed B-spline coefficients. See Cox [2] for
+          details.
+
+          8. Further Comments
+
+          Computation time is approximately proportional to m m +4(m +m ).
+                                                             x y    x  y
+
+          9. Example
+
+          This program reads in knot sets LAMDA(1),..., LAMDA(PX) and MU(1)
+          ,..., MU(PY), and a set of bicubic spline coefficients c  .
+                                                                  ij
+          Following these are values for m  and the x co-ordinates x , for
+                                          x                         q
+          q=1,2,...,m , and values for m  and the y co-ordinates y , for
+                     x                  y                         r
+          r=1,2,...,m , defining the grid of points on which the spline is
+                     y
+          to be evaluated.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02gaf}{NAG On-line Documentation: e02gaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02GAF(3NAG)      Foundation Library (12/10/92)      E02GAF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02GAF
+                  E02GAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02GAF calculates an l  solution to an over-determined system of
+                                1
+          linear equations.
+
+          2. Specification
+
+                 SUBROUTINE E02GAF (M, A, LA, B, NPLUS2, TOLER, X, RESID,
+                1                   IRANK, ITER, IWORK, IFAIL)
+                 INTEGER          M, LA, NPLUS2, IRANK, ITER, IWORK(M),
+                1                 IFAIL
+                 DOUBLE PRECISION A(LA,NPLUS2), B(M), TOLER, X(NPLUS2),
+                1                 RESID
+
+          3. Description
+
+          Given a matrix A with m rows and n columns (m>=n) and a vector b
+          with m elements, the routine calculates an l  solution to the
+                                                      1
+          over-determined system of equations
+
+                                        Ax=b.
+
+          That is to say, it calculates a vector x, with n elements, which
+          minimizes the l -norm (the sum of the absolute values) of the
+                         1
+          residuals
+
+                                         m
+                                         --
+                                   r(x)= >  |r |,
+                                         --   i
+                                         i=1
+
+          where the residuals r  are given by
+                               i
+
+                                  n
+                                  --
+                           r =b - >  a  x ,   i=1,2,...,m.
+                            i  i  --  ij j
+                                  j=1
+
+          Here a   is the element in row i and column j of A, b  is the ith
+                ij                                             i
+          element of b and x  the jth element of x. The matrix A need not
+                            j
+          be of full rank.
+
+          Typically in applications to data fitting, data consisting of m
+          points with co-ordinates (t ,y ) are to be approximated in the l
+                                     i  i                                 1
+          -norm by a linear combination of known functions (phi) (t),
+                                                                j
+
+             (alpha) (phi) (t)+(alpha) (phi) (t)+...+(alpha) (phi) (t).
+                    1     1           2     2               n     n
+
+          This is equivalent to fitting an l  solution to the over-
+                                            1
+          determined system of equations
+
+                       n
+                       --
+                       >  (phi) (t )(alpha) =y ,   i=1,2,...,m.
+                       --      j  i        j  i
+                       j=1
+
+          Thus if, for each value of i and j, the element a   of the matrix
+                                                           ij
+          A in the previous paragraph is set equal to the value of
+          (phi) (t ) and b  is set equal to y , the solution vector x will
+               j  i       i                  i
+          contain the required values of the (alpha) . Note that the
+                                                    j
+          independent variable t above can, instead, be a vector of several
+          independent variables (this includes the case where each (phi)
+                                                                        i
+          is a function of a different variable, or set of variables).
+
+          The algorithm is a modification of the simplex method of linear
+          programming applied to the primal formulation of the l  problem
+                                                                1
+          (see Barrodale and Roberts [1] and [2]). The modification allows
+          several neighbouring simplex vertices to be passed through in a
+          single iteration, providing a substantial improvement in
+          efficiency.
+
+          4. References
+
+          [1]   Barrodale I and Roberts F D K (1973) An Improved Algorithm
+                for Discrete \\ll  Linear Approximation. SIAM J. Numer.
+                                 1
+                Anal. 10 839--848.
+
+          [2]   Barrodale I and Roberts F D K (1974) Solution of an
+                Overdetermined System of Equations in the \\ll -norm. Comm.
+                                                              1
+                ACM. 17, 6 319--320.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number of equations, m (the number of rows of
+               the matrix A). Constraint: M >= n >= 1.
+
+           2:  A(LA,NPLUS2) -- DOUBLE PRECISION array          Input/Output
+               On entry: A(i,j) must contain a  , the element in the ith
+                                              ij
+               row and jth column of the matrix A, for i=1,2,...,m and
+               j=1,2,...,n. The remaining elements need not be set. On
+               exit: A contains the last simplex tableau generated by the
+               simplex method.
+
+           3:  LA -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which E02GAF is called.
+               Constraint: LA >= M + 2.
+
+           4:  B(M) -- DOUBLE PRECISION array                  Input/Output
+               On entry: b , the ith element of the vector b, for
+                          i
+               i=1,2,...,m. On exit: the ith residual r  corresponding to
+                                                       i
+               the solution vector x, for i=1,2,...,m.
+
+           5:  NPLUS2 -- INTEGER                                      Input
+               On entry: n+2, where n is the number of unknowns (the
+               number of columns of the matrix A). Constraint: 3 <= NPLUS2
+               <= M + 2.
+
+           6:  TOLER -- DOUBLE PRECISION                              Input
+               On entry: a non-negative value. In general TOLER specifies
+               a threshold below which numbers are regarded as zero. The
+                                                       2/3
+               recommended threshold value is (epsilon)    where (epsilon)
+               is the machine precision. The recommended value can be
+               computed within the routine by setting TOLER to zero. If
+               premature termination occurs a larger value for TOLER may
+               result in a valid solution. Suggested value: 0.0.
+
+           7:  X(NPLUS2) -- DOUBLE PRECISION array                   Output
+               On exit: X(j) contains the jth element of the solution
+               vector x, for j=1,2,...,n. The elements X(n+1) and X(n+2)
+               are unused.
+
+           8:  RESID -- DOUBLE PRECISION                             Output
+               On exit: the sum of the absolute values of the residuals
+               for the solution vector x.
+
+           9:  IRANK -- INTEGER                                      Output
+               On exit: the computed rank of the matrix A.
+
+          10:  ITER -- INTEGER                                       Output
+               On exit: the number of iterations taken by the simplex
+               method.
+
+          11:  IWORK(M) -- INTEGER array                          Workspace
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               An optimal solution has been obtained but this may not be
+               unique.
+
+          IFAIL= 2
+               The calculations have terminated prematurely due to rounding
+               errors. Experiment with larger values of TOLER or try
+               scaling the columns of the matrix (see Section 8).
+
+          IFAIL= 3
+               On entry NPLUS2 < 3,
+
+               or       NPLUS2 > M + 2,
+
+               or       LA < M + 2.
+
+          7. Accuracy
+
+          Experience suggests that the computational accuracy of the
+          solution x is comparable with the accuracy that could be obtained
+          by applying Gaussian elimination with partial pivoting to the n
+          equations satisfied by this algorithm (i.e., those equations with
+          zero residuals). The accuracy therefore varies with the
+          conditioning of the problem, but has been found generally very
+          satisfactory in practice.
+
+          8. Further Comments
+
+          The effects of m and n on the time and on the number of
+          iterations in the Simplex Method vary from problem to problem,
+          but typically the number of iterations is a small multiple of n
+          and the total time taken by the routine is approximately
+                            2
+          proportional to mn .
+
+          It is recommended that, before the routine is entered, the
+          columns of the matrix A are scaled so that the largest element in
+          each column is of the order of unity. This should improve the
+          conditioning of the matrix, and also enable the parameter TOLER
+          to perform its correct function. The solution x obtained will
+          then, of course, relate to the scaled form of the matrix. Thus if
+          the scaling is such that, for each j=1,2,...,n, the elements of
+          the jth column are multiplied by the constant k , the element x
+                                                         j               j
+          of the solution vector x must be multiplied by k  if it is
+                                                          j
+          desired to recover the solution corresponding to the original
+          matrix A.
+
+          9. Example
+
+          Suppose we wish to approximate a set of data by a curve of the
+          form
+
+                   t   -t
+               y=Ke +Le  +M
+
+          where K, L and M are unknown. Given values y  at 5 points t  we
+                                                      i              i
+          may form the over-determined set of equations for K, L and M
+
+                            x    -x
+                             i     i
+                           e  K+e   L+M=y ,  i=1,2,...,5.
+                                         i
+
+          E02GAF is used to solve these in the l  sense.
+                                                1
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe02zaf}{NAG On-line Documentation: e02zaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E02ZAF(3NAG)      Foundation Library (12/10/92)      E02ZAF(3NAG)
+
+
+
+          E02 -- Curve and Surface Fitting                           E02ZAF
+                  E02ZAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E02ZAF sorts two-dimensional data into rectangular panels.
+
+          2. Specification
+
+                 SUBROUTINE E02ZAF (PX, PY, LAMDA, MU, M, X, Y, POINT,
+                1                   NPOINT, ADRES, NADRES, IFAIL)
+                 INTEGER          PX, PY, M, POINT(NPOINT), NPOINT, ADRES
+                1                 (NADRES), NADRES, IFAIL
+                 DOUBLE PRECISION LAMDA(PX), MU(PY), X(M), Y(M)
+
+          3. Description
+
+          A set of m data points with rectangular Cartesian co-ordinates
+          x ,y  are sorted into panels defined by lines parallel to the y
+           r  r
+          and x axes. The intercepts of these lines on the x and y axes are
+          given in LAMDA(i), for i=5,6,...,PX-4 and MU(j), for
+          j=5,6,...,PY-4, respectively. The subroutine orders the data so
+          that all points in a panel occur before data in succeeding
+          panels, where the panels are numbered from bottom to top and then
+          left to right, with the usual arrangement of axes, as shown in
+          the diagram. Within a panel the points maintain their original
+          order.
+
+                   Please see figure in printed Reference Manual
+
+          A data point lying exactly on one or more panel sides is taken to
+          be in the highest-numbered panel adjacent to the point. The
+          subroutine does not physically rearrange the data, but provides
+          the array POINT which contains a linked list for each panel,
+          pointing to the data in that panel. The total number of panels is
+          (PX-7)*(PY-7).
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  PX -- INTEGER                                          Input
+
+           2:  PY -- INTEGER                                          Input
+
+               On entry: PX and PY must specify eight more than the number
+               of intercepts on the x axis and y axis, respectively.
+               Constraint: PX >= 8 and PY >= 8.
+
+           3:  LAMDA(PX) -- DOUBLE PRECISION array                    Input
+               On entry: LAMDA(5) to LAMDA(PX-4) must contain, in non-
+               decreasing order, the intercepts on the x axis of the sides
+               of the panels parallel to the y axis.
+
+           4:  MU(PY) -- DOUBLE PRECISION array                       Input
+               On entry: MU(5) to MU(PY-4) must contain, in non-decreasing
+               order, the intercepts on the y axis of the sides of the
+               panels parallel to the x axis.
+
+           5:  M -- INTEGER                                           Input
+               On entry: the number m of data points.
+
+           6:  X(M) -- DOUBLE PRECISION array                         Input
+
+           7:  Y(M) -- DOUBLE PRECISION array                         Input
+               On entry: the co-ordinates of the rth data point (x ,y ),
+                                                                  r  r
+               for r=1,2,...,m.
+
+           8:  POINT(NPOINT) -- INTEGER array                        Output
+               On exit: for i = 1,2,...,NADRES, POINT(m+i) = I1 is the
+               index of the first point in panel i, POINT(I1) = I2 is the
+               index of the second point in panel i and so on.
+
+               POINT(IN) = 0 indicates that X(IN),Y(IN) was the last point
+               in the panel.
+
+               The co-ordinates of points in panel i can be accessed in
+               turn by means of the following instructions:
+                  IN = M + I
+               10 IN = POINT(IN)
+                  IF (IN.EQ. 0) GOTO 20
+                  XI = X(IN)
+                  YI = Y(IN)
+                  .
+                  .
+                  .
+                  GOTO 10
+               20...
+
+           9:  NPOINT -- INTEGER                                      Input
+               On entry:
+               the dimension of the array POINT as declared in the
+               (sub)program from which E02ZAF is called.
+               Constraint: NPOINT >= M + (PX-7)*(PY-7).
+
+          10:  ADRES(NADRES) -- INTEGER array                     Workspace
+
+          11:  NADRES -- INTEGER                                      Input
+               On entry: the value (PX-7)*(PY-7), the number of panels
+               into which the (x,y) plane is divided.
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               The intercepts in the array LAMDA, or in the array MU, are
+               not in non-decreasing order.
+
+          IFAIL= 2
+               On entry PX < 8,
+
+               or       PY < 8,
+
+               or       M <= 0,
+
+               or       NADRES /= (PX-7)*(PY-7),
+
+               or       NPOINT < M + (PX-7)*(PY-7).
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          The time taken by this routine is approximately proportional to
+          m*log(NADRES).
+
+          This subroutine was written to sort two dimensional data in the
+          manner required by routines E02DAF and E02DBF(*). The first 9
+          parameters of E02ZAF are the same as the parameters in E02DAF and
+          E02DBF(*) which have the same name.
+
+          9. Example
+
+          This example program reads in data points and the intercepts of
+          the panel sides on the x and y axes; it calls E02ZAF to set up
+          the index array POINT; and finally it prints the data points in
+          panel order.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe04}{NAG On-line Documentation: e04}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E04(3NAG)         Foundation Library (12/10/92)         E04(3NAG)
+
+
+
+          E04 -- Minimizing or Maximizing a Function    Introduction -- E04
+                                    Chapter E04
+                        Minimizing or Maximizing a Function
+
+          Contents of this Introduction:
+
+          1.     Scope of the Chapter
+
+          2.     Background to the Problems
+
+          2.1.   Types of Optimization Problems
+
+          2.1.1. Unconstrained minimization
+
+          2.1.2. Nonlinear least-squares problems
+
+          2.1.3. Minimization subject to bounds on the variables
+
+          2.1.4. Minimization subject to linear constraints
+
+          2.1.5. Minimization subject to nonlinear constraints
+          
+          2.2.   Geometric Representation and Terminology
+
+          2.2.1. Gradient vector
+
+          2.2.2. Hessian matrix
+
+          2.2.3. Jacobian matrix; matrix of constraint normals
+
+          2.3.   Sufficient Conditions for a Solution
+
+          2.3.1. Unconstrained minimization
+
+          2.3.2. Minimization subject to bounds on the variables
+
+          2.3.3. Linearly-constrained minimization
+
+          2.3.4. Nonlinearly-constrained minimization
+
+          2.4.   Background to Optimization Methods
+
+          2.4.1. Methods for unconstrained optimization
+
+          2.4.2. Methods for nonlinear least-squares problems
+
+          2.4.3. Methods for handling constraints
+
+          2.5.   Scaling
+
+          2.5.1. Transformation of variables
+
+          2.5.2. Scaling the objective function
+
+          2.5.3. Scaling the constraints
+
+          2.6.   Analysis of Computed Results
+
+          2.6.1. Convergence criteria
+
+          2.6.2. Checking results
+
+          2.6.3. Monitoring progress
+
+          2.6.4. Confidence intervals for least-squares solutions
+
+          2.7.   References
+
+          3.     Recommendations on Choice and Use of Routines
+
+          3.1.   Choice of Routine
+
+          3.2.   Service Routines
+
+          3.3.   Function Evaluations at Infeasible Points
+
+          3.4.   Related Problems
+
+
+
+          1. Scope of the Chapter
+
+          An optimization problem involves minimizing a function (called
+          the objective function) of several variables, possibly subject to
+          restrictions on the values of the variables defined by a set of
+          constraint functions. The routines in the NAG Foundation Library
+          are concerned with function minimization only, since the problem
+          of maximizing a given function can be transformed into a
+          minimization problem simply by multiplying the function by -1.
+
+          This introduction is only a brief guide to the subject of
+          optimization designed for the casual user. Anyone with a
+          difficult or protracted problem to solve will find it beneficial
+          to consult a more detailed text, such as Gill et al [5] or
+          Fletcher [3].
+
+          Readers who are unfamiliar with the mathematics of the subject
+          may find some sections difficult at first reading; if so, they
+          should concentrate on Sections 2.1, 2.2, 2.5, 2.6 and 3.
+
+          2. Background to the Problems
+
+          2.1. Types of Optimization Problems
+
+          Solution of optimization problems by a single, all-purpose,
+          method is cumbersome and inefficient. Optimization problems are
+          therefore classified into particular categories, where each
+          category is defined by the properties of the objective and
+          constraint functions, as illustrated by some examples below.
+
+          Properties of Objective   Properties of Constraints
+          Function
+
+          Nonlinear                 Nonlinear
+
+          Sums of squares of        Sparse linear
+          nonlinear functions
+
+          Quadratic                 Linear
+
+          Sums of squares of linear Bounds
+          functions
+
+          Linear                    None
+
+          For instance, a specific problem category involves the
+          minimization of a nonlinear objective function subject to bounds
+          on the variables. In the following sections we define the
+          particular categories of problems that can be solved by routines
+          contained in this Chapter.
+
+          2.1.1.  Unconstrained minimization
+
+          In unconstrained minimization problems there are no constraints
+          on the variables. The problem can be stated mathematically as
+          follows:
+
+                                    minimize F(x)
+                                     x
+
+                         n                           T
+          where x is in R , that is, x=(x ,x ,...,x ) .
+                                        1  2      n
+
+          2.1.2.  Nonlinear least-squares problems
+
+          Special consideration is given to the problem for which the
+          function to be minimized can be expressed as a sum of squared
+          functions. The least-squares problem can be stated mathematically
+          as follows:
+
+                                 {     m       }
+                                 { T   --  2   }            n
+                        minimize {f f= >  f (x)},  x is in R
+                         x       {     --  i   }
+                                 {     i=1     }
+
+          where the ith element of the m-vector f is the function f (x).
+                                                                   i
+
+          2.1.3.  Minimization subject to bounds on the variables
+
+          These problems differ from the unconstrained problem in that at
+          least one of the variables is subject to a simple restriction on
+          its value, e.g.x <=10, but no constraints of a more general form
+                          5
+          are present.
+
+          The problem can be stated mathematically as follows:
+
+                                                       n
+                              minimize F(x),  x is in R
+                               x
+
+          subject to l <=x <=u , i=1,2,...,n.
+                      i   i   i
+
+          This format assumes that upper and lower bounds exist on all the
+          variables. By conceptually allowing u =infty and l =-infty all
+                                               i            i
+          the variables need not be restricted.
+
+          2.1.4.  Minimization subject to linear constraints
+
+          A general linear constraint is defined as a constraint function
+          that is linear in more than one of the variables, e.g. 3x +2x >=4
+                                                                   1   2
+          The various types of linear constraint are reflected in the
+          following mathematical statement of the problem:
+
+
+                                                       n
+                              minimize F(x),  x is in R
+                               x
+
+          subject to the
+
+                             T
+             equality       a x=b        i=1,2,...,m ;
+             constraints:    i   i                1
+
+                             T
+             inequality     a x>=b       i=m +1,m +2,...,m ;
+             constraints:    i    i         1    1        2
+
+                             T
+                            a x<=b       i=m +1,m +2,...,m ;
+                             i    i         2    2        3
+
+                                 T
+             range          s <=a x<=t   i=m +1,m +2,...,m ;
+             constraints:    j   i    j     3    3        4
+                                         j=1,2,...,m -m ;
+                                                    4  3
+
+             bounds         l <=x <=u    i=1,2,...,n
+             constraints:    i   i   i
+
+          where each a  is a vector of length n; b , s  and t  are constant
+                      i                           i   j      j
+          scalars; and any of the categories may be empty.
+
+          Although the bounds on x  could be included in the definition of
+                                  i
+          general linear constraints, we prefer to distinguish between them
+          for reasons of computational efficiency.
+
+          If F(x) is a linear function, the linearly-constrained problem is
+          termed a linear programming problem (LP problem); if F(x) is a
+          quadratic function, the problem is termed a quadratic programming
+          problem (QP problem). For further discussion of LP and QP
+          problems, including the dual formulation of such problems, see
+          Dantzig [2].
+
+          2.1.5.  Minimization subject to nonlinear constraints
+
+          A problem is included in this category if at least one constraint
+                                       2
+          function is nonlinear, e.g. x +x +x -2>=0. The mathematical
+                                       1  3  4
+          statement of the problem is identical to that for the linearly-
+          constrained case, except for the addition of the following
+          constraints:
+
+             equality       c (x)=0        i=1,2,...,m ;
+             constraints:    i                        5
+
+             inequality     c (x)>=0       i=m +1,m +2,...,m ;
+             constraints:    i                5    5        6
+
+             range          v <=c (x)<=w   i=m +1,m +2,...,m ,
+             constraints:    j   i      j     6    6        7
+                                           j=1,2,...,m -m
+                                                      7  6
+
+          where each c  is a nonlinear function; v  and w  are constant
+                      i                           j      j
+          scalars; and any category may be empty. Note that we do not
+          include a separate category for constraints of the form c (x)<=0,
+                                                                   i
+          since this is equivalent to -c (x)>=0.
+                                        i
+
+          2.2. Geometric Representation and Terminology
+
+          To illustrate the nature of optimization problems it is useful to
+          consider the following example in two dimensions
+
+                                 x
+                                  1   2   2
+                           F(x)=e  (4x +2x +4x x +2x +1).
+                                      1   2   1 2   2
+
+          (This function is used as the example function in the
+          documentation for the unconstrained routines.)
+
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          Figure 1 is a contour diagram of F(x). The contours labelled
+          F ,F ,...,F  are isovalue contours, or lines along which the
+           0  1      4
+                                                                   *
+          function F(x) takes specific constant values. The point x  is a
+                                                                *
+          local unconstrained minimum, that is, the value of F(x ) is less
+          than at all the neighbouring points. A function may have several
+          such minima. The lowest of the local minima is termed a global
+                                                            *
+          minimum. In the problem illustrated in Figure 1, x  is the only
+                                   
+
+          local minimum. The point x is said to be a saddle point because
+          it is a minimum along the line AB, but a maximum along CD.
+
+          If we add the constraint x >=0 to the problem of minimizing F(x),
+                                    1
+          the solution remains unaltered. In Figure 1 this constraint is
+          represented by the straight line passing through x =0, and the
+                                                            1
+          shading on the line indicates the unacceptable region. The region
+              n
+          in R  satisfying the constraints of an optimization problem is
+          termed the feasible region. A point satisfying the constraints is
+          defined as a feasible point.
+
+          If we add the nonlinear constraint x +x -x x -1.5>=0, represented
+                                              1  2  1 2
+                                                       *
+          by the curved shaded line in Figure 1, then x  is not a feasible
+                                                                ^
+          point. The solution of the new constrained problem is x, the
+          feasible point with the smallest function value.
+
+          2.2.1.  Gradient vector
+
+          The vector of first partial derivatives of F(x) is called the
+          gradient vector, and is denoted by g(x), i.e.,
+
+                             [ ddF(x)  ddF(x)      ddF(x)]T
+                        g(x)=[ ------, ------,..., ------] .
+                             [  ddx     ddx         ddx  ]
+                             [     1       2           n ]
+
+          For the function illustrated in Figure 1,
+
+                                   [      x          ]
+                                   [       1         ]
+                                   [F(x)+e  (8x +4x )]
+                                   [           1   2 ]
+                                   [ x               ]
+                                   [  1              ]
+                              g(x)=[e  (4x +4x +2)   ].
+                                   [      2   1      ]
+
+          The gradient vector is of importance in optimization because it
+          must be zero at an unconstrained minimum of any function with
+          continuous first derivatives.
+
+          2.2.2.  Hessian matrix
+
+          The matrix of second partial derivatives of a function is termed
+          its Hessian matrix. The Hessian matrix of F(x) is denoted by G(x)
+                                                  2
+          and its (i,j)th element is given by dd F(x)/ddx ddx . If F(x)
+                                                           i   j
+          has continuous second derivatives, then G(x) must be positive
+          semi-definite at any unconstrained minimum of F.
+
+          2.2.3.  Jacobian matrix; matrix of constraint normals
+
+          In nonlinear least-squares problems, the matrix of first partial
+          derivatives of the vector-valued function f(x) is termed the
+          Jacobian matrix of f(x) and its (i,j)th component is ddf /ddx .
+                                                                  i    j
+
+          The vector of first partial derivatives of the constraint c (x)
+                                                                     i
+          is denoted by
+
+                                 [ ddc (x)      ddc (x)]T
+                                 [    i            i   ]
+                           a (x)=[ -------,..., -------] .
+                            i    [  ddx          ddx   ]
+                                 [     1            n  ]
+
+                      ^                ^
+          At a point, x, the vector a (x) is orthogonal (normal) to the
+                                     i
+                                                    ^
+          isovalue contour of c (x) passing through x; this relationship is
+                               i
+          illustrated for a two-dimensional function in Figure 2.
+
+
+                                     Figure 2
+                   Please see figure in printed Reference Manual
+
+          The matrix whose columns are the vectors {a } is termed the
+                                                     i
+          matrix of constraint normals. Note that if c (x) is a linear
+                                                      i
+                                T
+          constraint involving a x, then its vector of first partial
+                                i
+          derivatives is simply the vector a .
+                                            i
+
+          2.3. Sufficient Conditions for a Solution
+
+          All nonlinear functions will be assumed to have continuous second
+          derivatives in the neighbourhood of the solution.
+
+          2.3.1.  Unconstrained minimization
+
+                                                                 *
+          The following conditions are sufficient for the point x  to be an
+          unconstrained local minimum of F(x):
+
+                      *
+          (i)   |||g(x )|||=0; and
+
+                   *
+          (ii)  G(x ) is positive-definite,
+
+          where |||g||| denotes the Euclidean length of g.
+
+          2.3.2.  Minimization subject to bounds on the variables
+
+          At the solution of a bounds-constrained problem, variables which
+          are not on their bounds are termed free variables. If it is known
+          in advance which variables are on their bounds at the solution,
+          the problem can be solved as an unconstrained problem in just the
+          free variables; thus, the sufficient conditions for a solution
+          are similar to those for the unconstrained case, applied only to
+          the free variables.
+
+                                                      *
+          Sufficient conditions for a feasible point x  to be the solution
+          of a bound-constrained problem are as follows:
+
+                      *
+          (i)   |||g(x )|||=0; and
+
+                   *
+          (ii)  G(x ) is positive-definite; and
+
+                    *               *
+          (iii) g (x )<0,x =u ; g (x )>0,x =l ,
+                 j        j  j   j        j  j
+
+                
+
+          where g(x) is the gradient of F(x) with respect to the free
+                         
+
+          variables, and G(x) is the Hessian matrix of F(x) with respect to
+          the free variables. The extra condition (iii) ensures that F(x)
+          cannot be reduced by moving off one or more of the bounds.
+
+          2.3.3.  Linearly-constrained minimization
+
+          For the sake of simplicity, the following description does not
+          include a specific treatment of bounds or range constraints,
+          since the results for general linear inequality constraints can
+          be applied directly to these cases.
+
+                         *
+          At a solution x , of a linearly-constrained problem, the
+          constraints which hold as equalities are called the active or
+          binding constraints. Assume that there are t active constraints
+                           *          ^
+          at the solution x , and let A denote the matrix whose columns are
+                                                                         ^
+          the columns of A corresponding to the active constraints, with b
+          the vector similarly obtained from b; then
+
+                                       ^T * ^
+                                       A x =b.
+
+          The matrix Z is defined as an n by (n-t) matrix satisfying:
+
+                                   ^T        T
+                                   A Z=0;   Z Z=I.
+
+          The columns of Z form an orthogonal basis for the set of vectors
+                                       ^
+          orthogonal to the columns of A.
+
+          Define
+
+                      T
+               g (x)=Z g(x), the projected gradient vector of F(x);
+                z
+
+                      T
+               G (x)=Z G(x)Z, the projected Hessian matrix of F(x).
+                z
+
+          At the solution of a linearly-constrained problem, the projected
+          gradient vector must be zero, which implies that the gradient
+                    *
+          vector g(x ) can be written as a linear combination of the
+                                     t
+                     ^           *   --          ^  ^
+          columns of A, i.e., g(x )= >  (lambda) a =A(lambda). The scalar
+                                     --         i i
+                                     i=1
+          (lambda)  is defined as the Lagrange multiplier corresponding to
+                  i
+          the ith active constraint. A simple interpretation of the ith
+          Lagrange multiplier is that it gives the gradient of F(x) along
+          the ith active constraint normal; a convenient definition of the
+          Lagrange multiplier vector (although not a recommended method for
+          computation) is:
+
+
+                                        ^T^ -1^T   *
+                              (lambda)=(A A)  A g(x ).
+
+                                     *
+          Sufficient conditions for x  to be the solution of a linearly-
+          constrained problem are:
+
+                 *                  ^T * ^
+          (i)   x  is feasible, and A x =b; and
+
+                       *                            *  ^
+          (ii)  |||g (x )|||=0, or equivalently, g(x )=A(lambda); and
+                    z
+
+                    *
+          (iii) G (x ) is positive-definite; and
+                 z
+
+          (iv)  (lambda) >0 if (lambda)  corresponds to a constraint
+                        i              i
+                ^T *  ^
+                a x >=b ;
+                 i     i
+
+               (lambda) <0 if (lambda)  corresponds to a constraint
+                       i              i
+               ^T *  ^
+               a x <=b .
+                i     i
+
+               The sign of (lambda)  is immaterial for equality
+                                   i
+               constraints, which by definition are always active.
+
+          2.3.4.  Nonlinearly-constrained minimization
+
+          For nonlinearly-constrained problems, much of the terminology is
+          defined exactly as in the linearly-constrained case. The set of
+          active constraints at x again means the set of constraints that
+                                                                     ^
+          hold as equalities at x, with corresponding definitions of c and
+          ^             ^
+          A: the vector c(x) contains the active constraint functions, and
+                         ^
+          the columns of A(x) are the gradient vectors of the active
+                                                           ^
+          constraints. As before, Z is defined in terms of A(x) as a matrix
+          such that:
+
+
+                                   ^T        T
+                                   A Z=0;   Z Z=I
+
+          where the dependence on x has been suppressed for compactness.
+
+                                                             T
+          The projected gradient vector g (x) is the vector Z g(x). At the
+                                         z
+                    *
+          solution x  of a nonlinearly-constrained problem, the projected
+          gradient must be zero, which implies the existence of Lagrange
+          multipliers corresponding to the active constraints, i.e.,
+             *  ^  *
+          g(x )=A(x )(lambda).
+
+          The Lagrangian function is given by:
+
+
+                                                     T^
+                          L(x,(lambda))=F(x)-(lambda) c(x).
+
+          We define g (x) as the gradient of the Lagrangian function; G (x)
+                     L                                                 L
+                                     ^
+          as its Hessian matrix, and G (x) as its projected Hessian matrix,
+                                      L
+                ^   T
+          i.e., G =Z G Z.
+                 L    L
+                                     *
+          Sufficient conditions for x  to be a solution of nonlinearly-
+          constrained problem are:
+
+                 *                  ^  *
+          (i)   x  is feasible, and c(x )=0; and
+
+                       *                             *  ^  *
+          (ii)  |||g (x )|||=0, or, equivalently, g(x )=A(x )(lambda); and
+                    z
+
+                ^   *
+          (iii) G (x ) is positive-definite; and
+                 L
+
+          (iv)  (lambda) >0 if (lambda)  corresponds to a constraint of the
+                        i              i
+                     ^
+                form c >=0; the sign of (lambda)  is immaterial for an
+                      i                         i
+                equality constraint.
+
+          Note that condition (ii) implies that the projected gradient of
+                                                        *
+          the Lagrangian function must also be zero at x , since the
+                          T                        ^  *
+          application of Z  annihilates the matrix A(x ).
+
+          2.4. Background to Optimization Methods
+
+          All the algorithms contained in this Chapter generate an
+                               (k)                                  *
+          iterative sequence {x   } that converges to the solution x  in
+          the limit, except for some special problem categories (i.e.,
+          linear and quadratic programming). To terminate computation of
+          the sequence, a convergence test is performed to determine
+          whether the current estimate of the solution is an adequate
+          approximation. The convergence tests are discussed in Section 2.6
+
+                                                     (k)
+          Most of the methods construct a sequence {x   } satisfying:
+
+                              (k+1)  (k)        (k) (k)
+                             x     =x   +(alpha)   p   ,
+
+                            (k)
+          where the vector p    is termed the direction of search, and
+                 (k)                                          (k)
+          (alpha)    is the steplength. The steplength (alpha)    is chosen
+                     (k+1)     (k)
+          so that F(x     )<F(x   ).
+
+
+          2.4.1.  Methods for unconstrained optimization
+
+          The distinctions among methods arise primarily from the need to
+          use varying levels of information about derivatives of F(x) in
+          defining the search direction. We describe three basic approaches
+          to unconstrained problems, which may be extended to other problem
+          categories. Since a full description of the methods would fill
+          several volumes, the discussion here can do little more than
+          allude to the processes involved, and direct the reader to other
+          sources for a full explanation.
+
+          (a)   Newton-type Methods (Modified Newton Methods)
+
+                                                              (k)
+                Newton-type methods use the Hessian matrix G(x   ), or a
+                                                      (k)
+                finite difference approximation to G(x   ), to define the
+                search direction. The routines in the Library either
+                                                                      (k)
+                require a subroutine that computes the elements of G(x   ),
+                                       (k)
+                or they approximate G(x   ) by finite differences.
+
+                Newton-type methods are the most powerful methods available
+                for general problems and will find the minimum of a
+                quadratic function in one iteration. See Sections 4.4. and
+                4.5.1. of Gill et al [5].
+
+          (b)   Quasi-Newton Methods
+
+                                                                (k)
+                Quasi-Newton methods approximate the Hessian G(x   ) by a
+                        (k)
+                matrix B    which is modified at each iteration to include
+                information obtained about the curvature of F along the
+                latest search direction. Although not as robust as Newton-
+                type methods, quasi-Newton methods can be more efficient
+                           (k)
+                because G(x   ) is not computed, or approximated by finite-
+                differences. Quasi-Newton methods minimize a quadratic
+                function in n iterations. See Section 4.5.2 of Gill et al
+                [5].
+
+          (c)   Conjugate-Gradient Methods
+
+                Unlike Newton-type and quasi-Newton methods, conjugate
+                gradient methods do not require the storage of an n by n
+                matrix and so are ideally suited to solve large problems.
+                Conjugate-gradient type methods are not usually as reliable
+                or efficient as Newton-type, or quasi-Newton methods. See
+                Section 4.8.3 of Gill et al [5].
+
+
+          2.4.2.  Methods for nonlinear least-squares problems
+
+          These methods are similar to those for unconstrained
+          optimization, but exploit the special structure of the Hessian
+          matrix to give improved computational efficiency.
+
+          Since
+
+                                         m
+                                         --  2
+                                   F(x)= >  f (x)
+                                         --  i
+                                         i=1
+
+          the Hessian matrix G(x) is of the form
+
+                                            m
+                                     T      --
+                          G(x)=2[J(x) J(x)+ >  f (x)G (x)],
+                                            --  i    i
+                                            i=1
+
+          where J(x) is the Jacobian matrix of f(x), and G (x) is the
+                                                          i
+          Hessian matrix of f (x).
+                             i
+
+          In the neighbourhood of the solution, |||f(x)||| is often small
+                             T
+          compared to |||J(x) J(x)||| (for example, when f(x) represents
+          the goodness of fit of a nonlinear model to observed data). In
+                           T
+          such cases, 2J(x) J(x) may be an adequate approximation to G(x),
+          thereby avoiding the need to compute or approximate second
+          derivatives of {f (x)}. See Section 4.7 of Gill et al [5].
+                           i
+
+          2.4.3.  Methods for handling constraints
+
+          Bounds on the variables are dealt with by fixing some of the
+          variables on their bounds and adjusting the remaining free
+          variables to minimize the function. By examining estimates of the
+          Lagrange multipliers it is possible to adjust the set of
+          variables fixed on their bounds so that eventually the bounds
+          active at the solution should be correctly identified. This type
+          of method is called an active set method. One feature of such
+          methods is that, given an initial feasible point, all
+                          (k)
+          approximations x    are feasible. This approach can be extended
+          to general linear constraints. At a point, x, the set of
+          constraints which hold as equalities being used to predict, or
+          approximate, the set of active constraints is called the working
+          set.
+
+          Nonlinear constraints are more difficult to handle. If at all
+          possible, it is usually beneficial to avoid including nonlinear
+          constraints during the formulation of the problem. The methods
+          currently implemented in the Library handle nonlinearly
+          constrained problems either by transforming them into a sequence
+          of bound constraint problems, or by transforming them into a
+          sequence of quadratic programming problems. A feature of almost
+                                                         (k)
+          all methods for nonlinear constraints is that x    is not
+          guaranteed to be feasible except in the limit, and this is
+          certainly true of the routines currently in the Library. See
+          Chapter 6, particularly Section 6.4 and Section 6.5 of Gill et al
+          [5].
+
+          Anyone interested in a detailed description of methods for
+          optimization should consult the references.
+
+          2.5. Scaling
+
+          Scaling (in a broadly defined sense) often has a significant
+          influence on the performance of optimization methods. Since
+          convergence tolerances and other criteria are necessarily based
+          on an implicit definition of 'small' and 'large', problems with
+          unusual or unbalanced scaling may cause difficulties for some
+          algorithms. Nonetheless, there are currently no scaling routines
+          in the Library, although the position is under constant review.
+          In light of the present state of the art, it is considered that
+          sensible scaling by the user is likely to be more effective than
+          any automatic routine. The following sections present some
+          general comments on problem scaling.
+
+          2.5.1.  Transformation of variables
+
+          One method of scaling is to transform the variables from their
+          original representation, which may reflect the physical nature of
+          the problem, to variables that have certain desirable properties
+          in terms of optimization. It is generally helpful for the
+          following conditions to be satisfied:
+
+          (i)   the variables are all of similar magnitude in the region of
+                interest;
+
+          (ii)  a fixed change in any of the variables results in similar
+                changes in F(x). Ideally, a unit change in any variable
+                produces a unit change in F(x);
+
+          (iii) the variables are transformed so as to avoid cancellation
+                error in the evaluation of F(x).
+
+          Normally, users should restrict themselves to linear
+          transformations of variables, although occasionally nonlinear
+          transformations are possible. The most common such transformation
+          (and often the most appropriate) is of the form
+
+                                     x   =Dx   ,
+                                      new   old
+
+          where D is a diagonal matrix with constant coefficients. Our
+          experience suggests that more use should be made of the
+          transformation
+
+                                    x   =Dx   +v,
+                                     new   old
+
+          where v is a constant vector.
+
+          Consider, for example, a problem in which the variable x
+                                                                  3
+          represents the position of the peak of a Gaussian curve to be
+          fitted to data for which the extreme values are 150 and 170;
+          therefore x  is known to lie in the range 150--170. One possible
+                     3
+                                                    
+
+          scaling would be to define a new variable x , given by
+                                                     3
+
+
+                                          x
+                                           3
+                                      x = ---.
+                                       3  170
+
+                                                                 
+
+          A better transformation, however, is given by defining x  as
+                                                                  3
+
+                                         x -160
+                                          3
+                                     x = ------.
+                                      3    10
+
+          Frequently, an improvement in the accuracy of evaluation of F(x)
+          can result if the variables are scaled before the routines to
+          evaluate F(x) are coded. For instance, in the above problem just
+          mentioned of Gaussian curve fitting, x  may always occur in terms
+                                                3
+          of the form (x -x ), where x  is a constant representing the mean
+                        3  m          m
+          peak position.
+
+          2.5.2.  Scaling the objective function
+
+          The objective function has already been mentioned in the
+          discussion of scaling the variables. The solution of a given
+          problem is unaltered if F(x) is multiplied by a positive
+          constant, or if a constant value is added to F(x). It is
+          generally preferable for the objective function to be of the
+          order of unity in the region of interest; thus, if in the
+                                                                +5
+          original formulation F(x) is always of the order of 10   (say),
+                                                           -5
+          then the value of F(x) should be multiplied by 10   when
+          evaluating the function within the optimization routines. If a
+          constant is added or subtracted in the computation of F(x),
+          usually it should be omitted - i.e., it is better to formulate
+                   2  2                 2  2               2  2
+          F(x) as x +x  rather than as x +x +1000 or even x +x +1. The
+                   1  2                 1  2               1  2
+          inclusion of such a constant in the calculation of F(x) can
+          result in a loss of significant figures.
+
+          2.5.3.  Scaling the constraints
+
+          The solution of a nonlinearly-constrained problem is unaltered if
+          the ith constraint is multiplied by a positive weight w . At the
+                                                                 i
+          approximation of the solution determined by a Library routine,
+          the active constraints will not be satisfied exactly, but will
+                                                 -8       -6
+          have 'small' values (for example, c =10  , c =10  , etc.). In
+                                             1        2
+          general, this discrepancy will be minimized if the constraints
+          are weighted so that a unit change in x produces a similar change
+          in each constraint.
+
+          A second reason for introducing weights is related to the effect
+          of the size of the constraints on the Lagrange multiplier
+          estimates and, consequently, on the active set strategy.
+          Additional discussion is given in Gill et al [5].
+
+          2.6. Analysis of Computed Results
+
+          2.6.1.  Convergence criteria
+
+          The convergence criteria inevitably vary from routine to routine,
+          since in some cases more information is available to be checked
+          (for example, is the Hessian matrix positive-definite?), and
+          different checks need to be made for different problem categories
+          (for example, in constrained minimization it is necessary to
+          verify whether a trial solution is feasible). Nonetheless, the
+          underlying principles of the various criteria are the same; in
+          non-mathematical terms, they are:
+
+                                  (k)
+          (i)   is the sequence {x   } converging?
+
+                                  (k)
+          (ii)  is the sequence {F   } converging?
+
+          (iii) are the necessary and sufficient conditions for the
+                solution satisfied?
+
+          The decision as to whether a sequence is converging is
+          necessarily speculative. The criterion used in the present
+          routines is to assume convergence if the relative change
+          occurring between two successive iterations is less than some
+          prescribed quantity. Criterion (iii) is the most reliable but
+          often the conditions cannot be checked fully because not all the
+          required information may be available.
+
+          2.6.2.  Checking results
+
+          Little a priori guidance can be given as to the quality of the
+          solution found by a nonlinear optimization algorithm, since no
+          guarantees can be given that the methods will always work.
+          Therefore, it is necessary for the user to check the computed
+          solution even if the routine reports success. Frequently a '
+          solution' may have been found even when the routine does not
+          report a success. The reason for this apparent contradiction is
+          that the routine needs to assess the accuracy of the solution.
+          This assessment is not an exact process and consequently may be
+          unduly pessimistic. Any 'solution' is in general only an
+          approximation to the exact solution, and it is possible that the
+          accuracy specified by the user is too stringent.
+
+          Further confirmation can be sought by trying to check whether or
+          not convergence tests are almost satisfied, or whether or not
+          some of the sufficient conditions are nearly satisfied. When it
+          is thought that a routine has returned a non-zero value of IFAIL
+          only because the requirements for 'success' were too stringent it
+          may be worth restarting with increased convergence tolerances.
+
+          For nonlinearly-constrained problems, check whether the solution
+          returned is feasible, or nearly feasible; if not, the solution
+          returned is not an adequate solution.
+
+          Confidence in a solution may be increased by resolving the
+          problem with a different initial approximation to the solution.
+          See Section 8.3 of Gill et al [5] for further information.
+
+          2.6.3.  Monitoring progress
+
+          Many of the routines in the Chapter have facilities to allow the
+          user to monitor the progress of the minimization process, and
+          users are encouraged to make use of these facilities. Monitoring
+          information can be a great aid in assessing whether or not a
+          satisfactory solution has been obtained, and in indicating
+          difficulties in the minimization problem or in the routine's
+          ability to cope with the problem.
+
+          The behaviour of the function, the estimated solution and first
+          derivatives can help in deciding whether a solution is acceptable
+          and what to do in the event of a return with a non-zero value of
+          IFAIL.
+
+          2.6.4.  Confidence intervals for least-squares solutions
+
+          When estimates of the parameters in a nonlinear least-squares
+          problem have been found, it may be necessary to estimate the
+          variances of the parameters and the fitted function. These can be
+          calculated from the Hessian of F(x) at the solution.
+
+          In many least-squares problems, the Hessian is adequately
+                                              T
+          approximated at the solution by G=2J J (see Section 2.4.3). The
+          Jacobian, J, or a factorization of J is returned by all the
+          comprehensive least-squares routines and, in addition, a routine
+          is supplied in the Library to estimate variances of the
+          parameters following the use of most of the nonlinear least-
+                                                 T
+          squares routines, in the case that G=2J J is an adequate
+          approximation.
+
+          Let H be the inverse of G, and S be the sum of squares, both
+                                     
+
+          calculated at the solution x; an unbiased estimate of the
+          variance of the ith parameter x  is
+                                         i
+
+
+                                           2S
+                                   var x = ---H
+                                        i  m-n ii
+
+                                                               
+
+          and an unbiased estimate of the covariance of x  and x  is
+                                                         i      j
+
+                                              2S
+                                covar(x ,x )= ---H  .
+                                       i  j   m-n ij
+
+              *
+          If x  is the true solution, then the 100(1-(beta)) confidence
+                      
+
+          interval on x is
+
+                 
+
+                /                          * 
+          x -  / var x .t                <x <x
+           i \/       i  (1-(beta)/2,m-n)  i  i
+               
+
+              /    
+          +  / var x .t                ,   i=1,2,...,n
+           \/       i  (1-(beta)/2,m-n)
+
+          where t                 is the 100(1-(beta))/2 percentage point
+                 (1-(beta)/2,m-n)
+          of the t-distribution with m-n degrees of freedom.
+
+          In the majority of problems, the residuals f , for i=1,2,...,m,
+                                                      i
+          contain the difference between the values of a model function
+          (phi)(z,x) calculated for m different values of the independent
+          variable z, and the corresponding observed values at these
+          points. The minimization process determines the parameters, or
+                                                                         
+
+          constants x, of the fitted function (phi)(z,x). For any value, z,
+          of the independent variable z, an unbiased estimate of the
+          variance of (phi) is
+
+
+                                 n   n
+                             2S  --  -- [ dd(phi)] [ dd(phi)]
+                  var (phi)= --- >   >  [ -------] [ -------] H  .
+                             m-n --  -- [  ddx   ] [  ddx   ]  ij
+                                 i=1 j=1[     i  ]z[     j  ]z
+
+                                                                  
+
+          The 100(1-(beta)) confidence interval on F at the point z is
+
+
+                                                          *
+          (phi)(z,x)-\/var (phi).t               < (phi)(z,x )
+                                  ((beta)/2,m-n)
+                                             
+
+                             < (phi)(z,x) +\/var (phi).t              .
+                                                       ((beta)/2,m-n)
+
+          For further details on the analysis of least-squares solutions
+          see Bard [1] and Wolberg [7].
+
+          2.7. References
+
+          [1]   Bard Y (1974) Nonlinear Parameter Estimation. Academic
+                Press.
+
+          [2]   Dantzig G B (1963) Linear Programming and Extensions.
+                Princeton University Press.
+
+          [3]   Fletcher R (1987) Practical Methods of Optimization. Wiley
+                (2nd Edition).
+
+          [4]   Gill P E and Murray W (eds) (1974) Numerical Methods for
+                Constrained Optimization. Academic Press.
+
+          [5]   Gill P E, Murray W and Wright M H (1981) Practical
+                Optimization. Academic Press.
+
+          [6]   Murray W (ed) (1972) Numerical Methods for Unconstrained
+                Optimization. Academic Press.
+
+          [7]   Wolberg J R (1967) Prediction Analysis. Van Nostrand.
+
+          3. Recommendations on Choice and Use of Routines
+
+          The choice of routine depends on several factors: the type of
+          problem (unconstrained, etc.); the level of derivative
+          information available (function values only, etc.); the
+          experience of the user (there are easy-to-use versions of some
+          routines); whether or not storage is a problem; and whether
+          computational time has a high priority.
+
+          3.1. Choice of Routine
+
+          Routines are provided to solve the following types of problem:
+
+
+          Nonlinear Programming                                      E04UCF
+          Quadratic Programming                                      E04NAF
+          Linear Programming                                         E04MBF
+          Nonlinear Function                                         E04DGF
+          (using 1st derivatives)
+          Nonlinear Function, unconstrained or simple bounds         E04JAF
+          (using function values only)
+          Nonlinear least-squares                                    E04FDF
+          (using function values only)
+          Nonlinear least-squares                                    E04GCF
+          (using function values and 1st derivatives)
+
+          E04UCF can be used to solve unconstrained, bound-constrained and
+          linearly-constrained problems.
+
+          E04NAF can be used as a comprehensive linear programming solver;
+          however, in most cases the easy-to-use routine E04MBFwill be
+          adequate.
+
+          E04MBF can be used to obtain a feasible point for a set of linear
+          constraints.
+
+          E04DGF can be used to solve large scale unconstrained problems.
+
+          The routines can be used to solve problems in a single variable.
+
+          3.2. Service Routines
+
+          One of the most common errors in use of optimization routines is
+          that the user's subroutines incorrectly evaluate the relevant
+          partial derivatives. Because exact gradient information normally
+          enhances efficiency in all areas of optimization, the user should
+          be encouraged to provide analytical derivatives whenever
+          possible. However, mistakes in the computation of derivatives can
+          result in serious and obscure run-time errors, as well as
+          complaints that the Library routines are incorrect.
+
+          E04UCF incorporates a check on the gradients being supplied and
+          users are encouraged to utilize this option; E04GCF also
+          incorporates a call to a derivative checker.
+
+          E04YCF estimates selected elements of the variance-covariance
+          matrix for the computed regression parameters following the use
+          of a nonlinear least-squares routine.
+
+          3.3. Function Evaluations at Infeasible Points
+
+          Users must not assume that the routines for constrained problems
+          will require the objective function to be evaluated only at
+          points which satisfy the constraints, i.e., feasible points. In
+          the first place some of the easy-to-use routines call a service
+          routine which will evaluate the objective function at the user-
+          supplied initial point, and at neighbouring points (to check
+          user-supplied derivatives or to estimate intervals for finite
+          differencing). Apart from this, all routines will ensure that any
+          evaluations of the objective function occur at points which
+          approximately satisfy any simple bounds or linear constraints.
+          Satisfaction of such constraints is only approximate because:
+
+          (a)   routines which have a parameter FEATOL may allow such
+                constraints to be violated by a margin specified by FEATOL;
+
+          (b)   routines which estimate derivatives by finite differences
+                may require function evaluations at points which just
+                violate such constraints even though the current iteration
+                just satisfies them.
+
+          There is no attempt to ensure that the current iteration
+          satisfies any nonlinear constraints. Users who wish to prevent
+          their objective function being evaluated outside some known
+          region (where it may be undefined or not practically computable),
+          may try to confine the iteration within this region by imposing
+          suitable simple bounds or linear constraints (but beware as this
+          may create new local minima where these constraints are active).
+
+          Note also that some routines allow the user-supplied routine to
+          return a parameter (MODE) with a negative value to force an
+          immediate clean exit from the minimization when the objective
+          function cannot be evaluated.
+
+          3.4. Related Problems
+
+          Apart from the standard types of optimization problem, there are
+          other related problems which can be solved by routines in this or
+          other chapters of the Library.
+
+          E04MBF can be used to find a feasible point for a set of linear
+          constraints and simple bounds.
+
+          Two routines in Chapter F04 solve linear least-squares problems,
+                          m                         n
+                          --      2                 --
+          i.e., minimize  >  r (x)  where r (x)=b - >  a  x .
+                          --  i            i     i  --  ij j
+                          i=1                       j=1
+
+          E02GAF solves an overdetermined system of linear equations in the
+                                    m
+                                    --
+          l  norm, i.e., minimizes  >  |r (x)|, with r  as above.
+           1                        --   i            i
+                                    i=1
+
+
+          E04 -- Minimizing or Maximizing a Function        Contents -- E04
+          Chapter E04
+
+          Minimizing or Maximizing a Function
+
+          E04DGF  Unconstrained minimum, pre-conditioned conjugate gradient
+                  algorithm, function of several variables using 1st
+                  derivatives
+
+          E04DJF  Read optional parameter values for E04DGF from external
+                  file
+
+          E04DKF  Supply optional parameter values to E04DGF
+
+          E04FDF  Unconstrained minimum of a sum of squares, combined
+                  Gauss-Newton and modified Newton algorithm using function
+                  values only
+
+          E04GCF  Unconstrained minimum of a sum of squares, combined
+                  Gauss-Newton and quasi-Newton algorithm, using 1st
+                  derivatives
+
+          E04JAF  Minimum, function of several variables, quasi-Newton
+                  algorithm, simple bounds, using function values only
+
+          E04MBF  Linear programming problem
+
+          E04NAF  Quadratic programming problem
+
+          E04UCF  Minimum, function of several variables, sequential QP
+                  method, nonlinear constraints, using function values and
+                  optionally 1st derivatives
+
+          E04UDF  Read optional parameter values for E04UCF from external
+                  file
+
+          E04UEF  Supply optional parameter values to E04UCF
+
+          E04YCF  Covariance matrix for nonlinear least-squares problem
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe04dgf}{NAG On-line Documentation: e04dgf}
+\beginscroll
+\begin{verbatim}
+
+   
+   
+   
+   E04DGF(3NAG)                 E04DGF                  E04DGF(3NAG)
+   
+   
+   
+        E04 -- Minimizing or Maximizing a Function                 E04DGF
+                E04DGF -- NAG Foundation Library Routine Document
+   
+        Note: Before using this routine, please read the Users' Note for
+        your implementation to check implementation-dependent details.
+        The symbol (*) after a NAG routine name denotes a routine that is
+        not included in the Foundation Library.
+   
+        Note for users via the AXIOM system: the interface to this routine
+        has been enhanced for use with AXIOM and is slightly different to
+        that offered in the standard version of the Foundation Library.  In
+        particular, the optional parameters of the NAG routine are now
+        included in the parameter list.  These are described in section
+        5.1.2, below.
+   
+        1. Purpose
+   
+        E04DGF minimizes an unconstrained nonlinear function of several
+        variables using a pre-conditioned, limited memory quasi-Newton
+        conjugate gradient method. First derivatives are required. The
+        routine is intended for use on large scale problems.
+   
+        2. Specification
+   
+               SUBROUTINE E04DGF(N,OBJFUN,ITER,OBJF,OBJGRD,X,IWORK,WORK,IUSER,
+              1                  USER,ES,FU,IT,LIN,LIST,MA,OP,PR,STA,STO,
+              2                  VE,IFAIL)
+               INTEGER           N, ITER, IWORK(N+1), IUSER(*),
+              1                  IT, PR, STA, STO, VE, IFAIL
+               DOUBLE PRECISION OBJF, OBJGRD(N), X(N), WORK(13*N), USER(*)
+              1                  ES, FU, LIN, OP, MA
+               LOGICAL           LIST
+               EXTERNAL          OBJFUN
+   
+        3. Description
+   
+        E04DGF uses a pre-conditioned conjugate gradient method and is
+        based upon algorithm PLMA as described in Gill and Murray [1] and
+        Gill et al [2] Section 4.8.3.
+   
+        The algorithm proceeds as follows:
+   
+        Let x  be a given starting point and let k denote the current
+             0
+        iteration, starting with k=0. The iteration requires g , the
+                                                              k
+        gradient vector evaluated at x , the kth estimate of the minimum.
+                                      k
+        At each iteration a vector p  (known as the direction of search)
+                                    k
+        is computed and the new estimate x    is given by x +(alpha) p
+                                          k+1              k        k k
+        where (alpha)  (the step length) minimizes the function
+                     k
+        F(x +(alpha) p ) with respect to the scalar (alpha) . A choice of
+           k        k k                                    k
+        initial step (alpha)  is taken as
+                            0
+   
+                                                    T
+                         (alpha) =min{1,2|F -F   |/g g }
+                                0          k  est   k k
+   
+        where F    is a user-supplied estimate of the function value at
+               est
+        the solution. If F    is not specified, the software always
+                          est
+        chooses the unit step length for (alpha) . Subsequent step length
+                                                0
+        estimates are computed using cubic interpolation with safeguards.
+   
+        A quasi-Newton method can be used to compute the search direction
+        p  by updating the inverse of the approximate Hessian (H ) and
+         k                                                      k
+        computing
+   
+                               p   =-H   g                            (1)
+                                k+1   k+1 k+1
+   
+        The updating formula for the approximate inverse is given by
+   
+                                               (    T    )
+                                               (   y H y )
+                       1  (     T    T  )   1  (    k k k)   T
+             H   =H - ----(H y s +s y H )+ ----(1+ ------)s s         (2)
+              k+1  k   T  ( k k k  k k k)   T  (     T   ) k k
+                      y s                  y s (    y s  )
+                       k k                  k k(     k k )
+   
+        where y =g   -g  and s =x   -x =(alpha) p .
+               k  k-1  k      k  k+1  k        k k
+   
+        The method used by E04DGF to obtain the search direction is based
+        upon computing p    as -H   g    where H    is a matrix obtained
+                        k+1      k+1 k+1        k+1
+        by updating the identity matrix with a limited number of quasi-
+        Newton corrections. The storage of an n by n matrix is avoided by
+        storing only the vectors that define the rank two corrections -
+        hence the term limited-memory quasi-Newton method. The precise
+        method depends upon the number of updating vectors stored. For
+        example, the direction obtained with the 'one-step' limited
+        memory update is given by (1) using (2) with H  equal to the
+                                                      k
+        identity matrix, viz.
+                                                   T    (    T  )
+                                                  s g   (   y y )
+                          1  ( T        T      )   k k+1(    k k)
+             p   =-g   + ----(s g   y +y g   s )- ------(1+ ----)s
+              k+1   k+1   T  ( k k+1 k  k k+1 k)    T   (    T  ) k
+                         y s                       y s  (   y s )
+                          k k                       k k (    k k)
+   
+        E04DGF uses a two-step method described in detail in Gill and
+        Murray [1] in which restarts and pre-conditioning are
+        incorporated. Using a limited-memory quasi-Newton formula, such
+        as the one above, guarantees p    to be a descent direction if
+                                      k+1
+                                T
+        all the inner products y  are positive for all vectors y  and s
+                                k                               k      k
+        used in the updating formula.
+   
+        The termination criterion of E04DGF is as follows:
+   
+        Let (tau)  specify a parameter that indicates the number of
+                 F
+        correct figures desired in F  ((tau)  is equivalent to Optimality
+                                    k       F
+        Tolerance in the optional parameter list, see Section 5.1). If
+        the following three conditions are satisfied
+   
+             (i) F   -F <(tau) (1+|F |)
+                  k-1  k      F     k
+   
+                                 ______
+             (ii) ||x   -x ||<  /(tau)  (1+||x ||)
+                     k-1  k   \/      F       k
+   
+                               ______
+             (iii) ||g ||<= 3 /(tau)  (1+|F |) or ||g ||<(epsilon) ,
+                      k     \/      F      k         k            A
+             where (epsilon)  is the absolute error associated with
+                            A
+             computing the objective function
+   
+        then the algorithm is considered to have converged. For a full
+        discussion on termination criteria see Gill et al [2] Chapter 8.
+   
+        4. References
+   
+        [1]   Gill P E and Murray W (1979) Conjugate-gradient Methods for
+              Large-scale Nonlinear Optimization. Technical Report SOL 79-
+              15. Department of Operations Research, Stanford University.
+   
+        [2]   Gill P E, Murray W and Wright M H (1981) Practical
+              Optimization. Academic Press.
+   
+        5. Parameters
+   
+         1:  N -- INTEGER                                           Input
+             On entry: the number n of variables. Constraint: N >= 1.
+   
+         2:  OBJFUN -- SUBROUTINE, supplied by the user.
+                                                       External Procedure
+             OBJFUN must calculate the objective function F(x) and its
+             gradient for a specified n element vector x.
+   
+             Its specification is:
+   
+                    SUBROUTINE OBJFUN (MODE, N, X, OBJF, OBJGRD,
+                   1                   NSTATE, IUSER, USER)
+                    INTEGER          MODE, N, NSTATE, IUSER(*)
+                    DOUBLE PRECISION X(N), OBJF, OBJGRD(N), USER(*)
+   
+              1:  MODE -- INTEGER                            Input/Output
+                  MODE is a flag that the user may set within OBJFUN to
+                  indicate a failure in the evaluation of the objective
+                  function. On entry: MODE is always non-negative. On
+                  exit: if MODE is negative the execution of E04DGF is
+                  terminated with IFAIL set to MODE.
+   
+              2:  N -- INTEGER                                      Input
+                  On entry: the number n of variables.
+   
+              3:  X(N) -- DOUBLE PRECISION array                    Input
+                  On entry: the point x at which the objective function
+                  is required.
+   
+              4:  OBJF -- DOUBLE PRECISION                         Output
+                  On exit: the value of the objective function F at the
+                  current point x.
+   
+              5:  OBJGRD(N) -- DOUBLE PRECISION array              Output
+                                                                ddF
+                  On exit: OBJGRD(i) must contain the value of  ---- at
+                                                                ddx
+                                                                   i
+                  the point x, for i=1,2,...,n.
+   
+              6:  NSTATE -- INTEGER                                 Input
+                  On entry: NSTATE will be 1 on the first call of OBJFUN
+                  by E04DGF, and is 0 for all subsequent calls. Thus, if
+                  the user wishes, NSTATE may be tested within OBJFUN in
+                  order to perform certain calculations once only. For
+                  example the user may read data or initialise COMMON
+                  blocks when NSTATE = 1.
+   
+              7:  IUSER(*) -- INTEGER array                User Workspace
+   
+              8:  USER(*) -- DOUBLE PRECISION array        User Workspace
+                  OBJFUN is called from E04DGF with the parameters IUSER
+                  and USER as supplied to E04DGF. The user is free to use
+                  arrays IUSER and USER to supply information to OBJFUN
+                  as an alternative to using COMMON.
+             OBJFUN must be declared as EXTERNAL in the (sub)program
+             from which E04DGF is called. Parameters denoted as
+             Input must not be changed by this procedure.
+   
+         3:  ITER -- INTEGER                                       Output
+             On exit: the number of iterations performed.
+   
+         4:  OBJF -- DOUBLE PRECISION                              Output
+             On exit: the value of the objective function F(x) at the
+             final iterate.
+   
+         5:  OBJGRD(N) -- DOUBLE PRECISION array                   Output
+             On exit: the objective gradient at the final iterate.
+   
+         6:  X(N) -- DOUBLE PRECISION array                  Input/Output
+             On entry: an initial estimate of the solution. On exit: the
+             final estimate of the solution.
+   
+         7:  IWORK(N+1) -- INTEGER array                        Workspace
+   
+         8:  WORK(13*N) -- DOUBLE PRECISION array               Workspace
+   
+         9:  IUSER(*) -- INTEGER array                     User Workspace
+             Note: the dimension of the array IUSER must be at least 1.
+             This array is not used by E04DGF, but is passed directly to
+             routine OBJFUN and may be used to supply information to
+             OBJFUN.
+   
+        10:  USER(*) -- DOUBLE PRECISION array             User Workspace
+             Note: the dimension of the array USER must be at least 1.
+             This array is not used by E04DGF, but is passed directly to
+             routine OBJFUN and may be used to supply information to
+             OBJFUN.
+   
+        11:  IFAIL -- INTEGER                                Input/Output
+             On entry: IFAIL must be set to 0, -1 or 1. Users who are
+             unfamiliar with this parameter should refer to the Essential
+             Introduction for details.
+   
+             On exit: IFAIL = 0 unless the routine detects an error or
+             gives a warning (see Section 6).
+   
+             For this routine, because the values of output parameters
+             may be useful even if IFAIL /=0 on exit, users are
+             recommended to set IFAIL to -1 before entry. It is then
+             essential to test the value of IFAIL on exit.
+   
+   
+        5.1. Optional Input Parameters
+   
+        Several optional parameters in E04DGF define choices in the
+        behaviour of the routine. In order to reduce the number of formal
+        parameters of E04DGF these optional parameters have associated
+        default values (see Section 5.1.3) that are appropriate for most
+        problems. Therefore the user need only specify those optional
+        parameters whose values are to be different from their default
+        values.
+   
+        The remainder of this section can be skipped by users who wish to
+        use the default values for all optional parameters. A complete
+        list of optional parameters and their default values is given in
+        Section 5.1.3.
+   
+        5.1.1. Specification of the Optional Parameters
+   
+        Optional parameters may be specified by calling one, or both, of
+        E04DJF and E04DKF prior to a call to E04DGF.
+   
+        E04DJF reads options from an external options file, with Begin
+        and End as the first and last lines respectively and each
+        intermediate line defining a single optional parameter. For
+        example,
+   
+              Begin
+                Print Level = 1
+              End
+   
+        The call
+   
+              CALL E04DJF(IOPTNS, INFORM)
+   
+        can then be used to read the file on unit IOPTNS. INFORM will be
+        zero on successful exit. E04DJF should be consulted for a full
+        description of this method of supplying optional parameters.
+   
+        E04DKF can be called to supply options directly, one call being
+        necessary for each optional parameter.
+   
+        For example,
+   
+   
+              CALL E04DKF(`Print level = 1')
+   
+   
+        E04DKF should be consulted for a full description of this method
+        of supplying optional parameters.
+   
+        All optional parameters not specified by the user are set to
+        their default values. Optional parameters specified by the user
+        are unaltered by E04DGF (unless they define invalid values) and
+        so remain in effect for subsequent calls to E04DGF, unless
+        altered by the user.
+   
+        5.1.2. Description of the Optional Parameters
+   
+        The following list (in alphabetical order) gives the valid
+        options. For each option, we give the keyword, any essential
+        optional qualifiers, the default value, and the definition. The
+        minimum valid abbreviation of each keyword is underlined. If no
+        characters of an optional qualifier are underlined, the qualifier
+        may be omitted. The letter a denotes a phrase (character string)
+        that qualifies an option. The letters i and r denote INTEGER and
+        real values required with certain options. The number (epsilon)
+        is a generic notation for machine precision, and (epsilon)
+                                                                  R
+        denotes the relative precision of the objective function (the
+        optional parameter Function Precision; see below).
+   
+        Defaults
+   
+        This special keyword may be used to reset the default values
+        following a call to E04DGF.
+   
+        Estimated Optimal Function Value r
+   
+           (Axiom parameter ES)
+   
+        This value of r specifies the user-supplied guess of the optimum
+        objective function value. This value is used by E04DGF to
+        calculate an initial step length (see Section 3). If the value of
+        r is not specified by the user (the default), then this has the
+        effect of setting the initial step length to unity. It should be
+        noted that for badly scaled functions a unit step along the
+        steepest descent direction will often compute the function at
+        very large values of x.
+   
+                                                 0.9
+        Function Precision  r Default = (epsilon)
+   
+           (Axiom parameter FU)
+   
+        The parameter defines (epsilon) , which is intended to be a
+                                       R
+        measure of the accuracy with which the problem function F can be
+        computed. The value of (epsilon)  should reflect the relative
+                                        R
+        precision of 1+|F(x)|; i.e. (epsilon)  acts as a relative
+                                             R
+        precision when |F| is large, and as an absolute precision when
+        |F| is small. For example, if F(x) is typically of order 1000 and
+        the first six significant digits are known to be correct, an
+        appropriate value for (epsilon)  would be 1.0E-6. In contrast, if
+                                       R
+                                     -4
+        F(x) is typically of order 10   and the first six significant
+        digits are known to be correct, an appropriate value for
+        (epsilon)  would be 1.0E-10. The choice of (epsilon)  can be
+                 R                                          R
+        quite complicated for badly scaled problems; see Chapter 8 of
+        Gill and Murray [2], for a discussion of scaling techniques. The
+        default value is appropriate for most simple functions that are
+        computed with full accuracy. However when the accuracy of the
+        computed function values is known to be significantly worse than
+        full precision, the value of (epsilon)  should be large enough so
+                                              R
+        that E04DGF will not attempt to distinguish between function
+        values that differ by less than the error inherent in the
+        calculation. If 0<=r<(epsilon), where (epsilon) is the machine
+        precision then the default value is used.
+   
+        Iteration Limit  i Default = max(50,5n)
+   
+        Iters
+   
+        Itns
+   
+           (Axiom parameter IT)
+   
+        The value i (i>=0) specifies the maximum number of iterations
+        allowed before termination. If i<0 the default value is used. See
+        Section 8 for further information.
+   
+        Linesearch Tolerance  r Default = 0.9
+   
+           (Axiom parameter LIN)
+   
+        The value r (0<=r<1) controls the accuracy with which the step
+        (alpha) taken during each iteration approximates a minimum of the
+        function along the search direction (the smaller the value of r,
+        the more accurate the linesearch). The default value r=0.9
+        requests an inaccurate search, and is appropriate for most
+        problems. A more accurate search may be appropriate when it is
+        desirable to reduce the number of iterations - for example, if
+        the objective function is cheap to evaluate.
+   
+        List     Default =  List
+        Nolist
+   
+           (Axiom parameter LIST)
+   
+        Normally each optional parameter specification is printed as it
+        is supplied. Nolist may be used to suppress the printing and List
+        may be used to restore printing.
+                                           10
+        Maximum Step Length  r Default = 10
+   
+           (Axiom parameter MA)
+   
+        The value r (r>0) defines the maximum allowable step length for
+        the line search. If r<=0 the default value is used.
+   
+                                                   0.8
+        Optimality Tolerance  r Default = (epsilon)
+   
+           (Axiom parameter OP)
+                                                   R
+   
+        The parameter r ((epsilon) <=r<1) specifies the accuracy to which
+                                  R
+        the user wishes the final iterate to approximate a solution of
+        the problem. Broadly speaking, r indicates the number of correct
+        figures desired in the objective function at the solution. For
+                           - 6
+        example, if r is 10    and E04DGF terminates successfully, the
+        final value of F should have approximately six correct figures.
+        E04DGF will terminate successfully if the iterative sequence of x
+        -values is judged to have converged and the final point satisfies
+        the termination criteria (see Section 3, where (tau)  represents
+                                                            F
+        Optimality Tolerance).
+   
+        Print Level  i Default = 10
+   
+           (Axiom parameter PR)
+   
+        The value i controls the amount of printout produced by E04DGF.
+        The following levels of printing are available.
+   
+        i     Output.
+   
+        0     No output.
+   
+        1     The final solution.
+   
+        5     One line of output for each iteration.
+   
+        10    The final solution and one line of output for each
+              iteration.
+   
+        Start Objective Check at Variable  i Default = 1
+   
+           (Axiom parameter STA)
+   
+        Stop Objective Check at Variable  i Default = n
+   
+           (Axiom parameter STO)
+   
+        These keywords take effect only if Verify Level > 0 (see below).
+        They may be used to control the verification of gradient elements
+        computed by subroutine OBJFUN. For example if the first 30
+        variables appear linearly in the objective, so that the
+        corresponding gradient elements are constant, then it is
+        reasonable to specify Start Objective Check at Variable  31.
+   
+        Verify Level  i Default = 0
+   
+        Verify                           No
+   
+        Verify Level                     -1
+   
+        Verify Level                     0
+   
+        Verify
+   
+        Verify                           Yes
+   
+        Verify Objective Gradients
+   
+        Verify Gradients
+   
+        Verify Level                     1
+   
+           (Axiom parameter VE)
+   
+        These keywords refer to finite-difference checks on the gradient
+        elements computed by the user-provided subroutine OBJFUN. It is
+        possible to set Verify Level in several ways, as indicated above.
+        For example, the gradients will be verified if Verify, Verify
+        Yes, Verify Gradients, Verify Objective Gradients or Verify Level
+        = 1 is specified.
+   
+        If i<0 then no checking will be performed. If i>0 then the
+        gradients will be verified at the user-supplied point. If i=0
+        only a 'cheap' test will be performed, requiring one call to
+        OBJFUN. If i=1, a more reliable (but more expensive) check will
+        be made on individual gradient components, within the ranges
+        specified by the Start and Stop keywords as described above. A
+        result of the form OK or BAD? is printed by E04DGF to indicate
+        whether or not each component appears to be correct.
+   
+        5.1.3. Optional parameter checklist and default values
+   
+        For easy reference, the following sample list shows all valid
+        keywords and their default values. The default options Function
+        Precision and Optimality Tolerance depend upon (epsilon), the
+        machine precision.
+   
+        Optional Parameters          Default Values
+   
+        Estimated Optimal Function
+        Value
+   
+                                              0.9
+        Function precision           (epsilon)
+   
+        Iterations                   max(50,5n)
+   
+        Linesearch Tolerance         0.9
+   
+                                       10
+        Maximum Step Length          10
+   
+        List/Nolist                  List
+   
+                                              0.8
+        Optimality Tolerance         (epsilon)
+   
+        Print Level                  10
+   
+        Start Objective Check at     1
+        Variable
+   
+        Stop Objective Check at      n
+        Variable
+   
+        Verify Level                 0
+   
+        5.2. Description of Printed Output
+   
+        The level of printed output from E04DGF is controlled by the user
+        (see the description of Print Level in Section 5.1).
+   
+        When Print Level >= 5, the following line of output is produced
+        at each iteration.
+   
+        Itn            is the iteration count.
+   
+        Step           is the step (alpha) taken along the computed
+                       search direction. On reasonably well-behaved
+                       problems, the unit step will be taken as the
+                       solution is approached.
+   
+        Nfun           is the cumulated number of evaluations of the
+                       objective function needed for the linesearch.
+                       Evaluations needed for the estimation of the
+                       gradients by finite differences are not included.
+                       Nfun is printed as a guide to the amount of work
+                       required for the linesearch. E04DGF will perform
+                       at most 16 function evaluations per iteration.
+   
+        Objective      is the value of the objective function.
+   
+        Norm G         is the Euclidean norm of the gradient of the
+                       objective function.
+   
+        Norm X         is the Euclidean norm of x.
+   
+        Norm (X(k-1)-X(k)) is the Euclidean norm of x   -x .
+                                                     k-1  k
+   
+        When Print Level = 1 or Print Level >= 10 then the solution at
+        the end of execution of E04DGF is printed out.
+   
+        The following describes the printout for each variable:
+   
+        Variable       gives the name (VARBL) and index j (j = 1 to n) of
+                       the variable
+   
+        Value          is the value of the variable at the final iterate
+   
+        Gradient Value is the value of the gradient of the objective
+                       function with respect to the jth variable at the
+                       final iterate
+   
+        6. Error Indicators and Warnings
+   
+        Errors or warnings specified by the routine:
+   
+        If on entry IFAIL = 0 or -1, explanatory error messages are
+        output on the current error message unit (as defined by X04AAF).
+   
+        On exit from E04DGF, IFAIL should be tested. If Print Level > 0
+        then a short description of IFAIL is printed.
+   
+        Errors and diagnostics indicated by IFAIL from E04DGF are as
+        follows:
+   
+        IFAIL< 0
+             A negative value of IFAIL indicates an exit from E04DGF
+             because the user set MODE negative in routine OBJFUN. The
+             value of IFAIL will be the same as the user's setting of
+             MODE.
+   
+        IFAIL= 1
+             Not used by this routine.
+   
+        IFAIL= 2
+             Not used by this routine.
+   
+        IFAIL= 3
+             The maximum number of iterations has been performed. If the
+             algorithm appears to be making progress the iterations value
+             may be too small (see Section 5.1.2) so the user should
+             increase iterations and rerun E04DGF. If the algorithm seems
+             to be 'bogged down',the user should check for incorrect
+             gradients or ill-conditioning as described below under IFAIL
+             = 6.
+   
+        IFAIL= 4
+             The computed upper bound on the step length taken during the
+             linesearch was too small. A rerun with an increased value of
+             the Maximum Step Length ((rho) say) may be successful unless
+                      10
+             (rho)>=10   (the default value), in which case the current
+             point cannot be improved upon.
+   
+        IFAIL= 5
+             Not used by this routine.
+   
+        IFAIL= 6
+             A sufficient decrease in the function value could not be
+             attained during the final linesearch. If the subroutine
+             OBJFUN computes the function and gradients correctly, then
+             this may occur because an overly stringent accuracy has been
+             requested, i.e., Optimality Tolerance is too small or if the
+             minimum lies close to a step length of zero. In this case
+             the user should apply the four tests described in Section 3
+             to determine whether or not the final solution is acceptable
+             (the user will need to set Print Level >= 5). For a
+             discussion of attainable accuracy see Gill and Murray [2].
+   
+             If many iterations have occurred in which essentially no
+             progress has been made or E04DGF has failed to move from the
+             initial point, subroutine OBJFUN may be incorrect. The user
+             should refer to the comments below under IFAIL = 7 and check
+             the gradients using the Verify parameter. Unfortunately,
+             there may be small errors in the objective gradients that
+             cannot be detected by the verification process. Finite-
+             difference approximations to first derivatives are
+             catastrophically affected by even small inaccuracies.
+   
+        IFAIL= 7
+             Large errors were found in the derivatives of the objective
+             function. This value of IFAIL will occur if the verification
+             process indicated that at least one gradient component had
+             no correct figures. The user should refer to the printed
+             output to determine which elements are suspected to be in
+             error.
+   
+             As a first step, the user should check that the code for the
+             objective values is correct - for example, by computing the
+             function at a point where the correct value is known.
+             However, care should be taken that the chosen point fully
+             tests the evaluation of the function. It is remarkable how
+             often the values x=0 or x=1 are used to test function
+             evaluation procedures, and how often the special properties
+             of these numbers make the test meaningless.
+   
+             Special care should be used in this test if computation of
+             the objective function involves subsidiary data communicated
+             in COMMON storage. Although the first evaluation of the
+             function may be correct, subsequent calculations may be in
+             error because some of the subsidiary data has accidentally
+             been overwritten.
+   
+             Errors in programming the function may be quite subtle in
+             that the function value is 'almost' correct. For example,
+             the function may not be accurate to full precision because
+             of the inaccurate calculation of a subsidiary quantity, or
+             the limited accuracy of data upon which the function
+             depends. A common error on machines where numerical
+             calculations are usually performed in double precision is to
+             include even one single-precision constant in the
+             calculation of the function; since some compilers do not
+             convert such constants to double precision, half the correct
+             figures may be lost by such a seemingly trivial error.
+   
+        IFAIL= 8
+             The gradient (g) at the starting point is too small. The
+                    T
+             value g g is less than (epsilon) |F(x )|, where (epsilon)
+                                             m    o                   m
+             is the machine precision.
+   
+             The problem should be rerun at a different starting point.
+   
+        IFAIL= 9
+             On entry N < 1.
+   
+        7. Accuracy
+   
+        On successful exit the accuracy of the solution will be as
+        defined by the optional parameter Optimality Tolerance.
+   
+        8. Further Comments
+   
+        Problems whose Hessian matrices at the solution contain sets of
+        clustered eigenvalues are likely to be minimized in significantly
+        fewer than n iterations. Problems without this property may
+        require anything between n and 5n iterations, with approximately
+        2n iterations being a common figure for moderately difficult
+        problems.
+   
+        9. Example
+   
+        To find a minimum of the function
+   
+                              x
+                               1   2   2
+                           F=e  (4x +2x +4x x +2x +1).
+                                   1   2   1 2   2
+   
+        The example program is not reproduced here. The source code for
+        all example programs is distributed with the NAG Foundation
+        Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe04djf}{NAG On-line Documentation: e04djf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E04DJF(3NAG)      Foundation Library (12/10/92)      E04DJF(3NAG)
+
+
+
+          E04 -- Minimizing or Maximizing a Function                 E04DJF
+                  E04DJF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          To supply optional parameters to E04DGF from an external file.
+
+          2. Specification
+
+                 SUBROUTINE E04DJF (IOPTNS, INFORM)
+                 INTEGER          IOPTNS, INFORM
+
+          3. Description
+
+          E04DJF may be used to supply values for optional parameters to
+          E04DGF. E04DJF reads an external file and each line of the file
+          defines a single optional parameter. It is only necessary to
+          supply values for those parameters whose values are to be
+          different from their default values.
+
+          Each optional parameter is defined by a single character string
+          of up to 72 characters, consisting of one or more items. The
+          items associated with a given option must be separated by spaces,
+          or equal signs (=). Alphabetic characters may be upper or lower
+          case. The string
+
+                Print level = 1
+
+          is an example of a string used to set an optional parameter. For
+          each option the string contains one or more of the following
+          items:
+
+          (a)   A mandatory keyword.
+
+          (b)   A phrase that qualifies the keyword.
+
+          (c)   A number that specifies an INTEGER or real value. Such
+                numbers may be up to 16 contiguous characters in Fortran
+                77's I, F, E or D formats, terminated by a space if this is
+                not the last item on the line.
+
+          Blank strings and comments are ignored. A comment begins with an
+          asterisk (*) and all subsequent characters in the string are
+          regarded as part of the comment.
+
+          The file containing the options must start with begin and must
+          finish with end An example of a valid options file is:
+
+                Begin * Example options file
+                  Print level = 10
+                End
+
+
+          Normally each line of the file is printed as it is read, on the
+          current advisory message unit (see X04ABF), but printing may be
+          suppressed using the keyword nolist. To suppress printing of
+          begin, nolist must be the first option supplied as in the file:
+
+                Begin
+                  Nolist
+                  Print level = 10
+                End
+
+          Printing will automatically be turned on again after a call to
+          E04DGF and may be turned on again at any time by the user by
+          using the keyword list.
+
+          Optional parameter settings are preserved following a call to
+          E04DGF, and so the keyword defaults is provided to allow the user
+          to reset all the optional parameters to their default values
+          prior to a subsequent call to E04DGF.
+
+          A complete list of optional parameters, their abbreviations,
+          synonyms and default values is given in Section 5.1 of the
+          routine document for E04DGF.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  IOPTNS -- INTEGER                                      Input
+               On entry: IOPTNS must be the unit number of the options
+               file. Constraint: 0 <= IOPTNS <= 99.
+
+           2:  INFORM -- INTEGER                                     Output
+               On exit: INFORM will be zero if an options file with the
+               correct structure has been read. Otherwise INFORM will be
+               positive. Positive values of INFORM indicate that an options
+               file may not have been successfully read as follows:
+               INFORM = 1
+                     IOPTNS is not in the range [0,99].
+
+               INFORM = 2
+                     begin was found, but end-of-file was found before end
+                     was found.
+
+               INFORM = 3
+                     end-of-file was found before begin was found.
+
+          6. Error Indicators and Warnings
+
+          If a line is not recognised as a valid option, then a warning
+          message is output on the current advisory message unit (see
+          X04ABF).
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          E04DKF may also be used to supply optional parameters to E04DGF.
+
+          9. Example
+
+          See the example for E04DGF.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe04dkf}{NAG On-line Documentation: e04dkf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E04DKF(3NAG)      Foundation Library (12/10/92)      E04DKF(3NAG)
+
+
+
+          E04 -- Minimizing or Maximizing a Function                 E04DKF
+                  E04DKF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          To supply individual optional parameters to E04DGF.
+
+          2. Specification
+
+                 SUBROUTINE E04DKF (STRING)
+                 CHARACTER*(*)    STRING
+
+          3. Description
+
+          E04DKF may be used to supply values for optional parameters to
+          E04DGF. It is only necessary to call E04DKF for those parameters
+          whose values are to be different from their default values. One
+          call to E04DKF sets one parameter value.
+
+          Each optional parameter is defined by a single character string
+          of up to 72 characters, consisting of one or more items. The
+          items associated with a given option must be separated by spaces,
+          or equal signs (=). Alphabetic characters may be upper or lower
+          case. The string
+
+                Print Level = 1
+
+          is an example of a string used to set an optional parameter. For
+          each option the string contains one or more of the following
+          items:
+
+          (a)   A mandatory keyword.
+
+          (b)   A phrase that qualifies the keyword.
+
+          (c)   A number that specifies an INTEGER or real value. Such
+                numbers may be up to 16 contiguous characters in Fortran
+                77's I, F, E or D formats, terminated by a space if this is
+                not the last item on the line.
+
+          Blank strings and comments are ignored. A comment begins with an
+          asterisk (*) and all subsequent characters in the string are
+          regarded as part of the comment.
+
+          Normally, each user-specified option is printed as it is defined,
+          on the current advisory message unit (see X04ABF), but this
+          printing may be suppressed using the keyword nolist Thus the
+          statement
+
+               CALL E04DKF (`Nolist')
+
+          suppresses printing of this and subsequent options. Printing will
+          automatically be turned on again after a call to E04DGF, and may
+          be turned on again at any time by the user, by using the keyword
+          list.
+
+          Optional parameter settings are preserved following a call to
+          E04DGF, and so the keyword defaults is provided to allow the user
+          to reset all the optional parameters to their default values by
+          the statement,
+
+               CALL E04DKF (`Defaults')
+
+          prior to a subsequent call to E04DGF.
+
+          A complete list of optional parameters, their abbreviations,
+          synonyms and default values is given in Section 5.1 of the
+          routine document for E04DGF.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  STRING -- CHARACTER*(*)                                Input
+               On entry: STRING must be a single valid option string. See
+               Section 3 above, and Section 5.1 of the routine document for
+               E04DGF.
+
+          6. Error Indicators and Warnings
+
+          If the parameter STRING is not recognised as a valid option
+          string, then a warning message is output on the current advisory
+          message unit (see X04ABF).
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          E04DJF may also be used to supply optional parameters to E04DGF.
+
+          9. Example
+
+          See the example for E04DGF.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe04fdf}{NAG On-line Documentation: e04fdf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E04FDF(3NAG)      Foundation Library (12/10/92)      E04FDF(3NAG)
+
+
+
+          E04 -- Minimizing or Maximizing a Function                 E04FDF
+                  E04FDF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E04FDF is an easy-to-use algorithm for finding an unconstrained
+          minimum of a sum of squares of m nonlinear functions in n
+          variables (m>=n). No derivatives are required.
+
+          It is intended for functions which are continuous and which have
+          continuous first and second derivatives (although it will usually
+          work even if the derivatives have occasional discontinuities).
+
+          2. Specification
+
+                 SUBROUTINE E04FDF (M, N, X, FSUMSQ, IW, LIW, W, LW, IFAIL)
+                 INTEGER          M, N, IW(LIW), LIW, LW, IFAIL
+                 DOUBLE PRECISION X(N), FSUMSQ, W(LW)
+
+          3. Description
+
+          This routine is essentially identical to the subroutine LSNDN1 in
+          the National Physical Laboratory Algorithms Library. It is
+          applicable to problems of the form
+
+                                            m
+                                            --        2
+                             Minimize F(x)= >  [f (x)]
+                                            --   i
+                                            i=1
+
+                                T
+          where x=(x ,x ,...,x )  and m>=n. (The functions f (x) are often
+                    1  2      n                             i
+          referred to as 'residuals'.) The user must supply a subroutine
+          LSFUN1 to evaluate functions f (x) at any point x.
+                                        i
+
+          From a starting point supplied by the user, a sequence of points
+          is generated which is intended to converge to a local minimum of
+          the sum of squares. These points are generated using estimates of
+          the curvature of F(x).
+
+          4. References
+
+          [1]   Gill P E and Murray W (1978) Algorithms for the Solution of
+                the Nonlinear Least-squares Problem. SIAM J. Numer. Anal. 15
+                977--992.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number m of residuals f (x), and the number n
+                                                    i
+               of variables, x . Constraint: 1 <= N <= M.
+                              j
+
+           3:  X(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: X(j) must be set to a guess at the jth component
+               of the position of the minimum, for j=1,2,...,n. On exit:
+               the lowest point found during the calculations. Thus, if
+               IFAIL = 0 on exit, X(j) is the jth component of the position
+               of the minimum.
+
+           4:  FSUMSQ -- DOUBLE PRECISION                            Output
+               On exit: the value of the sum of squares, F(x),
+               corresponding to the final point stored in X.
+
+           5:  IW(LIW) -- INTEGER array                           Workspace
+
+           6:  LIW -- INTEGER                                         Input
+               On entry: the length of IW as declared in the (sub)program
+               from which E04FDF has been called. Constraint: LIW >= 1.
+
+           7:  W(LW) -- DOUBLE PRECISION array                    Workspace
+
+           8:  LW -- INTEGER                                          Input
+               On entry: the length of W as declared in the (sub)program
+               from which E04FDF is called. Constraints:
+                    LW >= N*(7 + N + 2*M + (N-1)/2) + 3*M, if N > 1,
+
+                    LW >= 9 + 5*>M, if N = 1.
+
+           9:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit.
+
+          5.1. Optional Parameters
+
+               LSFUN1 -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               This routine must be supplied by the user to calculate the
+               vector of values f (x) at any point x. Since the routine is
+                                 i
+               not a parameter to E04FDF, it must be called LSFUN1. It
+               should be tested separately before being used in conjunction
+               with E04FDF (see the Chapter Introduction).
+
+               Its specification is:
+
+                      SUBROUTINE LSFUN1 (M, N, XC, FVECC)
+                      INTEGER          M, N
+                      DOUBLE PRECISION XC(N), FVECC(M)
+
+                1:  M -- INTEGER                                      Input
+
+                2:  N -- INTEGER                                      Input
+                    On entry: the numbers m and n of residuals and
+                    variables, respectively.
+
+                3:  XC(N) -- DOUBLE PRECISION array                   Input
+                    On entry: the point x at which the values of the f
+                                                                       i
+                    are required.
+
+                4:  FVECC(M) -- DOUBLE PRECISION array               Output
+                    On exit: FVECC(i) must contain the value of f  at the
+                                                                 i
+                    point x, for i=1,2,...,m.
+               LSFUN1 must be declared as EXTERNAL in the (sub)program
+               from which E04FDF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry N < 1,
+
+               or       M < N,
+
+               or       LIW < 1,
+
+               or       LW < N*(7 + N + 2*M + (N-1)/2) + 3*M, when N > 1,
+
+               or       LW < 9 + 5*>M, when N = 1.
+
+          IFAIL= 2
+               There have been 400*n calls of LSFUN1, yet the algorithm
+               does not seem to have converged. This may be due to an
+               awkward function or to a poor starting point, so it is worth
+               restarting E04FDF from the final point held in X.
+
+          IFAIL= 3
+               The final point does not satisfy the conditions for
+               acceptance as a minimum, but no lower point could be found.
+
+          IFAIL= 4
+               An auxiliary routine has been unable to complete a singular
+               value decomposition in a reasonable number of sub-
+               iterations.
+
+          IFAIL= 5
+
+          IFAIL= 6
+
+          IFAIL= 7
+
+          IFAIL= 8
+               There is some doubt about whether the point x found by
+               E04FDF is a minimum of F(x). The degree of confidence in the
+               result decreases as IFAIL increases. Thus when IFAIL = 5, it
+               is probable that the final x gives a good estimate of the
+               position of a minimum, but when IFAIL = 8 it is very
+               unlikely that the routine has found a minimum.
+
+          If the user is not satisfied with the result (e.g. because IFAIL
+          lies between 3 and 8), it is worth restarting the calculations
+          from a different starting point (not the point at which the
+          failure occurred) in order to avoid the region which caused the
+          failure. Repeated failure may indicate some defect in the
+          formulation of the problem.
+
+          7. Accuracy
+
+          If the problem is reasonably well scaled and a successful exit is
+          made, then, for a computer with a mantissa of t decimals, one
+          would expect to get about t/2-1 decimals accuracy in the
+          components of x and between t-1 (if F(x) is of order 1 at the
+          minimum) and 2t-2 (if F(x) is close to zero at the minimum)
+          decimals accuracy in F(x).
+
+          8. Further Comments
+
+          The number of iterations required depends on the number of
+          variables, the number of residuals and their behaviour, and the
+          distance of the starting point from the solution. The number of
+          multiplications performed per iteration of E04FDF varies, but for
+                                   2    3
+          m>>n is approximately n*m +O(n ). In addition, each iteration
+          makes at least n+1 calls of LSFUN1. So, unless the residuals can
+          be evaluated very quickly, the run time will be dominated by the
+          time spent in LSFUN1.
+
+          Ideally, the problem should be scaled so that the minimum value
+          of the sum of squares is in the range (0,1), and so that at
+          points a unit distance away from the solution the sum of squares
+          is approximately a unit value greater than at the minimum. It is
+          unlikely that the user will be able to follow these
+          recommendations very closely, but it is worth trying (by
+          guesswork), as sensible scaling will reduce the difficulty of the
+          minimization problem, so that E04FDF will take less computer
+          time.
+
+          When the sum of squares represents the goodness of fit of a
+          nonlinear model to observed data, elements of the variance-
+          covariance matrix of the estimated regression coefficients can be
+          computed by a subsequent call to E04YCF, using information
+          returned in segments of the workspace array W. See E04YCF for
+          further details.
+
+          9. Example
+
+          To find least-squares estimates of x , x  and x  in the model
+                                              1   2      3
+
+                                            t
+                                             1
+                                   y=x + ---------
+                                      1  x t +x t
+                                          2 2  3 3
+
+          using the 15 sets of data given in the following table.
+
+                                  y    t   t    t
+                                        1   2    3
+                                 0.14  1.0 15.0 1.0
+                                 0.18  2.0 14.0 2.0
+                                 0.22  3.0 13.0 3.0
+                                 0.25  4.0 12.0 4.0
+                                 0.29  5.0 11.0 5.0
+                                 0.32  6.0 10.0 6.0
+                                 0.35  7.0  9.0 7.0
+                                 0.39  8.0  8.0 8.0
+                                 0.37  9.0  7.0 7.0
+                                 0.58 10.0  6.0 6.0
+                                 0.73 11.0  5.0 5.0
+                                 0.96 12.0  4.0 4.0
+                                 1.34 13.0  3.0 3.0
+                                 2.10 14.0  2.0 2.0
+                                 4.39 15.0  1.0 1.0
+
+          The program uses (0.5, 1.0, 1.5) as the initial guess at the
+          position of the minimum.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe04gcf}{NAG On-line Documentation: e04gcf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E04GCF(3NAG)      Foundation Library (12/10/92)      E04GCF(3NAG)
+
+
+
+          E04 -- Minimizing or Maximizing a Function                 E04GCF
+                  E04GCF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E04GCF is an easy-to-use quasi-Newton algorithm for finding an
+          unconstrained minimum of a sum of squares of m nonlinear
+          functions in n variables (m>=n). First derivatives are required.
+
+          It is intended for functions which are continuous and which have
+          continuous first and second derivatives (although it will usually
+          work even if the derivatives have occasional discontinuities).
+
+          2. Specification
+
+                 SUBROUTINE E04GCF (M, N, X, FSUMSQ, IW, LIW, W, LW, IFAIL)
+                 INTEGER          M, N, IW(LIW), LIW, LW, IFAIL
+                 DOUBLE PRECISION X(N), FSUMSQ, W(LW)
+
+          3. Description
+
+          This routine is essentially identical to the subroutine LSFDQ2 in
+          the National Physical Laboratory Algorithms Library. It is
+          applicable to problems of the form
+
+                                             m
+                                             --        2
+                             Minimize  F(x)= >  [f (x)]
+                                             --   i
+                                             i=1
+
+                                T
+          where x=(x ,x ,...,x )  and m>=n. (The functions f (x) are often
+                    1  2      n                             i
+          referred to as 'residuals'.) The user must supply a subroutine
+          LSFUN2 to evaluate the residuals and their first derivatives at
+          any point x.
+
+          Before attempting to minimize the sum of squares, the algorithm
+          checks LSFUN2 for consistency. Then, from a starting point
+          supplied by the user, a sequence of points is generated which is
+          intended to converge to a local minimum of the sum of squares.
+          These points are generated using estimates of the curvature of
+          F(x).
+
+          4. References
+
+          [1]   Gill P E and Murray W (1978) Algorithms for the Solution of
+                the Nonlinear Least-squares Problem. SIAM J. Numer. Anal. 15
+                977--992.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number m of residuals f (x), and the number n
+                                                    i
+               of variables, x . Constraint: 1 <= N <= M.
+                              j
+
+           3:  X(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: X(j) must be set to a guess at the jth component
+               of the position of the minimum, for j=1,2,...,n. The routine
+               checks the first derivatives calculated by LSFUN2 at the
+               starting point, and so is more likely to detect an error in
+               the user's routine if the initial X(j) are non-zero and
+               mutually distinct. On exit: the lowest point found during
+               the calculations. Thus, if IFAIL = 0 on exit, X(j) is the j
+               th component of the position of the minimum.
+
+           4:  FSUMSQ -- DOUBLE PRECISION                            Output
+               On exit: the value of the sum of squares, F(x),
+               corresponding to the final point stored in X.
+
+           5:  IW(LIW) -- INTEGER array                           Workspace
+
+           6:  LIW -- INTEGER                                         Input
+               On entry: the length of IW as declared in the (sub)program
+               from which E04GCF is called. Constraint: LIW >= 1.
+
+           7:  W(LW) -- DOUBLE PRECISION array                    Workspace
+
+           8:  LW -- INTEGER                                          Input
+               On entry: the length of W as declared in the (sub)program
+               from which E04GCF is called. Constraints:
+                    LW >= 2*N*(4 + N + M) + 3*M, if N > 1,
+
+                    LW >= 11 + 5*M, if N = 1.
+
+           9:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit.
+
+          5.1. Optional Parameters
+
+               LSFUN2 -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               This routine must be supplied by the user to calculate the
+               vector of values f (x) and the Jacobian matrix of first
+                                 i
+                            ddf
+                               i
+               derivatives  ---- at any point x. Since the routine is not a
+                            ddx
+                               j
+               parameter to E04GCF, it must be called LSFUN2. It should be
+               tested separately before being used in conjunction with
+               E04GCF (see the Chapter Introduction).
+
+               Its specification is:
+
+                      SUBROUTINE LSFUN2 (M, N, XC, FVECC, FJACC, LJC)
+                      INTEGER          M, N, LJC
+                      DOUBLE PRECISION XC(N), FVECC(M), FJACC(LJC,N)
+               Important: The dimension declaration for FJACC must
+               contain the variable LJC, not an integer constant.
+
+                1:  M -- INTEGER                                      Input
+
+                2:  N -- INTEGER                                      Input
+                    On entry: the numbers m and n of residuals and
+                    variables, respectively.
+
+                3:  XC(N) -- DOUBLE PRECISION array                   Input
+                    On entry: the point x at which the values of the f
+                                                                       i
+                             ddf
+                                i
+                    and the  ---- are required.
+                             ddx
+                                j
+
+                4:  FVECC(M) -- DOUBLE PRECISION array               Output
+                    On exit: FVECC(i) must contain the value of f  at the
+                                                                  i
+                    point x, for i=1,2,...,m.
+
+                5:  FJACC(LJC,N) -- DOUBLE PRECISION array           Output
+                                                                    ddf
+                                                                       i
+                    On exit: FJACC(i,j) must contain the value of  ---- at
+                                                                    ddx
+                                                                       j
+                    the point x, for i=1,2,...,m; j=1,2,...,n.
+
+                6:  LJC -- INTEGER                                    Input
+                    On entry: the first dimension of the array FJACC.
+               LSFUN2 must be declared as EXTERNAL in the (sub)program
+               from which E04GCF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry N < 1,
+
+               or       M < N,
+
+               or       LIW < 1,
+
+               or       LW < 2*N*(4 + N + M) + 3*M, when N > 1,
+
+               or       LW < 9 + 5*>M, when N = 1.
+
+          IFAIL= 2
+               There have been 50*n calls of LSFUN2, yet the algorithm does
+               not seem to have converged. This may be due to an awkward
+               function or to a poor starting point, so it is worth
+               restarting E04GCF from the final point held in X.
+
+          IFAIL= 3
+               The final point does not satisfy the conditions for
+               acceptance as a minimum, but no lower point could be found.
+
+          IFAIL= 4
+               An auxiliary routine has been unable to complete a singular
+               value decomposition in a reasonable number of sub-
+               iterations.
+
+          IFAIL= 5
+
+          IFAIL= 6
+
+          IFAIL= 7
+
+          IFAIL= 8
+               There is some doubt about whether the point X found by
+               E04GCF is a minimum of F(x). The degree of confidence in the
+               result decreases as IFAIL increases. Thus, when IFAIL = 5,
+               it is probable that the final x gives a good estimate of the
+               position of a minimum, but when IFAIL = 8 it is very
+               unlikely that the routine has found a minimum.
+
+          IFAIL= 9
+               It is very likely that the user has made an error in forming
+                                ddf
+                                   i
+               the derivatives  ---- in LSFUN2.
+                                ddx
+                                   j
+
+          If the user is not satisfied with the result (e.g. because IFAIL
+          lies between 3 and 8), it is worth restarting the calculations
+          from a different starting point (not the point at which the
+          failure occurred) in order to avoid the region which caused the
+          failure. Repeated failure may indicate some defect in the
+          formulation of the problem.
+
+          7. Accuracy
+
+          If the problem is reasonably well scaled and a successful exit is
+          made then, for a computer with a mantissa of t decimals, one
+          would expect to get t/2-1 decimals accuracy in the components of
+          x and between t-1 (if F(x) is of order 1 at the minimum) and 2t-2
+          (if F(x) is close to zero at the minimum) decimals accuracy in
+          F(x).
+
+          8. Further Comments
+
+          The number of iterations required depends on the number of
+          variables, the number of residuals and their behaviour, and the
+          distance of the starting point from the solution. The number of
+          multiplications performed per iteration of E04GCF varies, but for
+                                   2    3
+          m>>n is approximately n*m +O(n ). In addition, each iteration
+          makes at least one call of LSFUN2. So, unless the residuals and
+          their derivatives can be evaluated very quickly, the run time
+          will be dominated by the time spent in LSFUN2.
+
+          Ideally the problem should be scaled so that the minimum value of
+          the sum of squares is in the range (0,1) and so that at points a
+          unit distance away from the solution the sum of squares is
+          approximately a unit value greater than at the minimum. It is
+          unlikely that the user will be able to follow these
+          recommendations very closely, but it is worth trying (by
+          guesswork), as sensible scaling will reduce the difficulty of the
+          minimization problem, so that E04GCF will take less computer
+          time.
+
+          When the sum of squares represents the goodness of fit of a
+          nonlinear model to observed data, elements of the variance-
+          covariance matrix of the estimated regression coefficients can be
+          computed by a subsequent call to E04YCF, using information
+          returned in segments of the workspace array W. See E04YCF for
+          further details.
+
+          9. Example
+
+          To find the least-squares estimates of x , x  and x  in the model
+                                                  1   2      3
+
+                                            t
+                                             1
+                                   y=x + ---------
+                                      1  x t +x t
+                                          2 2  3 3
+
+          using the 15 sets of data given in the following table.
+
+                                  y    t   t    t
+                                        1   2    3
+                                 0.14  1.0 15.0 1.0
+                                 0.18  2.0 14.0 2.0
+                                 0.22  3.0 13.0 3.0
+                                 0.25  4.0 12.0 4.0
+                                 0.29  5.0 11.0 5.0
+                                 0.32  6.0 10.0 6.0
+                                 0.35  7.0  9.0 7.0
+                                 0.39  8.0  8.0 8.0
+                                 0.37  9.0  7.0 7.0
+                                 0.58 10.0  6.0 6.0
+                                 0.73 11.0  5.0 5.0
+                                 0.96 12.0  4.0 4.0
+                                 1.34 13.0  3.0 3.0
+                                 2.10 14.0  2.0 2.0
+                                 4.39 15.0  1.0 1.0
+
+          The program uses (0.5, 1.0, 1.5) as the initial guess at the
+          position of the minimum.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe04jaf}{NAG On-line Documentation: e04jaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E04JAF(3NAG)      Foundation Library (12/10/92)      E04JAF(3NAG)
+
+
+
+          E04 -- Minimizing or Maximizing a Function                 E04JAF
+                  E04JAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E04JAF is an easy-to-use quasi-Newton algorithm for finding a
+          minimum of a function F(x ,x ,...,x ), subject to fixed upper and
+                                   1  2      n
+          lower bounds of the independent variables x ,x ,...,x , using
+                                                     1  2      n
+          function values only.
+
+          It is intended for functions which are continuous and which have
+          continuous first and second derivatives (although it will usually
+          work even if the derivatives have occasional discontinuities).
+
+          2. Specification
+
+                 SUBROUTINE E04JAF (N, IBOUND, BL, BU, X, F, IW, LIW, W,
+                1                   LW, IFAIL)
+                 INTEGER          N, IBOUND, IW(LIW), LIW, LW, IFAIL
+                 DOUBLE PRECISION BL(N), BU(N), X(N), F, W(LW)
+
+          3. Description
+
+          This routine is applicable to problems of the form:
+
+            Minimize  F(x ,x ,...,x ) subject to l <=x <=u  , j=1,2,...,n
+                         1  2      n              j   j   j
+
+          when derivatives of F(x) are unavailable.
+
+          Special provision is made for problems which actually have no
+          bounds on the x , problems which have only non-negativity bounds
+                         j
+          and problems in which l =l =...=l  and u =u =...=u . The user
+                                 1  2      n      1  2      n
+          must supply a subroutine FUNCT1 to calculate the value of F(x) at
+          any point x.
+
+          From a starting point supplied by the user there is generated, on
+          the basis of estimates of the gradient and the curvature of F(x),
+          a sequence of feasible points which is intended to converge to a
+          local minimum of the constrained function. An attempt is made to
+          verify that the final point is a minimum.
+
+          4. References
+
+          [1]   Gill P E and Murray W (1976) Minimization subject to bounds
+                on the variables. Report NAC 72. National Physical
+                Laboratory.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+               On entry: the number n of independent variables.
+               Constraint: N >= 1.
+
+           2:  IBOUND -- INTEGER                                      Input
+               On entry: indicates whether the facility for dealing with
+               bounds of special forms is to be used.
+
+               It must be set to one of the following values:
+               IBOUND = 0
+                     if the user will be supplying all the l  and u
+                                                            j      j
+                     individually.
+
+               IBOUND = 1
+                     if there are no bounds on any x .
+                                                    j
+
+               IBOUND = 2
+                     if all the bounds are of the form 0<=x .
+                                                           j
+
+               IBOUND = 3
+                     if l =l =...=l  and u =u =...=u .
+                         1  2      n      1  2      n
+
+           3:  BL(N) -- DOUBLE PRECISION array                 Input/Output
+               On entry: the lower bounds l .
+                                           j
+
+               If IBOUND is set to 0, the user must set BL(j) to l , for
+                                                                  j
+               j=1,2,...,n. (If a lower bound is not specified for a
+                                                                          6
+               particular x , the corresponding BL(j) should be set to -10.)
+                           j
+
+               If IBOUND is set to 3, the user must set BL(1) to l ; E04JAF
+                                                                  1
+               will then set the remaining elements of BL equal to BL(1).
+               On exit: the lower bounds actually used by E04JAF.
+
+           4:  BU(N) -- DOUBLE PRECISION array                 Input/Output
+               On entry: the upper bounds u .
+                                           j
+
+               If IBOUND is set to 0, the user must set BU(j) to u , for
+                                                                  j
+               j=1,2,...,n. (If an upper bound is not specified for a
+                                                                         6
+               particular x , the corresponding BU(j) should be set to 10.)
+                           j
+
+               If IBOUND is set to 3, the user must set BU(1) to u ; E04JAF
+                                                                  1
+               will then set the remaining elements of BU equal to BU(1).
+               On exit: the upper bounds actually used by E04JAF.
+
+           5:  X(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: X(j) must be set to an estimate of the jth
+               component of the position of the minimum, for j=1,2,...,n.
+               On exit: the lowest point found during the calculations.
+               Thus, if IFAIL = 0 on exit, X(j) is the jth component of the
+               position of the minimum.
+
+           6:  F -- DOUBLE PRECISION                                 Output
+               On exit: the value of F(x) corresponding to the final point
+               stored in X.
+
+           7:  IW(LIW) -- INTEGER array                           Workspace
+
+           8:  LIW -- INTEGER                                         Input
+               On entry: the length of IW as declared in the (sub)program
+               from which E04JAF is called. Constraint: LIW >= N + 2.
+
+           9:  W(LW) -- DOUBLE PRECISION array                    Workspace
+
+          10:  LW -- INTEGER                                          Input
+               On entry: the length of W as declared in the (sub)program
+               from which E04JAF is called. Constraint: LW>=max(N*(N-
+               1)/2+12*N,13).
+
+          11:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit. To suppress
+               the output of an error message when soft failure occurs, set
+               IFAIL to 1.
+
+          5.1. Optional Parameters
+
+               FUNCT1 -- SUBROUTINE, supplied by the user.
+                                                    External Procedure
+               This routine must be supplied by the user to calculate the
+               value of the function F(x) at any point x. Since this
+               routine is not a parameter to E04JAF, it must be called
+               FUNCT1. It should be tested separately before being used in
+               conjunction with E04JAF (see the Chapter Introduction).
+
+               Its specification is:
+
+                      SUBROUTINE FUNCT1 (N, XC, FC)
+                      INTEGER          N
+                      DOUBLE PRECISION XC(N), FC
+
+                1:  N -- INTEGER                                      Input
+                    On entry: the number n of variables.
+
+                2:  XC(N) -- DOUBLE PRECISION array                   Input
+                    On entry: the point x at which the function value is
+                    required.
+
+                3:  FC -- DOUBLE PRECISION                           Output
+                    On exit: the value of the function F at the current
+                    point x.
+               FUNCT1 must be declared as EXTERNAL in the (sub)program
+               from which E04JAF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          IFAIL= 1
+               On entry N < 1,
+
+               or       IBOUND < 0,
+
+               or       IBOUND > 3,
+
+               or       IBOUND = 0 and BL(j) > BU(j) for some j,
+
+               or       IBOUND = 3 and BL(1) > BU(1),
+
+               or       LIW < N + 2,
+
+               or       LW<max(13,12*N+N*(N-1)/2).
+
+          IFAIL= 2
+               There have been 400*n function evaluations, yet the
+               algorithm does not seem to be converging. The calculations
+               can be restarted from the final point held in X. The error
+               may also indicate that F(x) has no minimum.
+
+          IFAIL= 3
+               The conditions for a minimum have not all been met but a
+               lower point could not be found and the algorithm has failed.
+
+          IFAIL= 4
+               An overflow has occurred during the computation. This is an
+               unlikely failure, but if it occurs the user should restart
+               at the latest point given in X.
+
+          IFAIL= 5
+
+          IFAIL= 6
+
+          IFAIL= 7
+
+          IFAIL= 8
+               There is some doubt about whether the point x found by
+               E04JAF is a minimum. The degree of confidence in the result
+               decreases as IFAIL increases. Thus, when IFAIL = 5 it is
+               probable that the final x gives a good estimate of the
+               position of a minimum, but when IFAIL = 8 it is very
+               unlikely that the routine has found a minimum.
+
+          IFAIL= 9
+               In the search for a minimum, the modulus of one of the
+                                                    6
+               variables has become very large  (~10 ).  This indicates
+               that there is a mistake in FUNCT1, that the user's problem
+               has no finite solution, or that the problem needs rescaling
+               (see Section 8).
+
+          If the user is dissatisfied with the result (e.g. because IFAIL =
+          5, 6, 7 or 8), it is worth restarting the calculations from a
+          different starting point (not the point at which the failure
+          occurred) in order to avoid the region which caused the failure.
+          If persistent trouble occurs and the gradient can be calculated,
+          it may be advisable to change to a routine which uses gradients
+          (see the Chapter Introduction).
+
+          7. Accuracy
+
+          When a successful exit is made then, for a computer with a
+          mantissa of t decimals, one would expect to get about t/2-1
+          decimals accuracy in x and about t-1 decimals accuracy in F,
+          provided the problem is reasonably well scaled.
+
+          8. Further Comments
+
+          The number of iterations required depends on the number of
+          variables, the behaviour of F(x) and the distance of the starting
+          point from the solution. The number of operations performed in an
+                                                          2
+          iteration of E04JAF is roughly proportional to n . In addition,
+          each iteration makes at least m+1 calls of FUNCT1, where m is the
+          number of variables not fixed on bounds. So, unless F(x) can be
+          evaluated very quickly, the run time will be dominated by the
+          time spent in FUNCT1.
+
+          Ideally the problem should be scaled so that at the solution the
+          value of F(x) and the corresponding values of x ,x ,...,x  are
+                                                         1  2      n
+          each in the range (-1,+1), and so that at points a unit distance
+          away from the solution, F is approximately a unit value greater
+          than at the minimum. It is unlikely that the user will be able to
+          follow these recommendations very closely, but it is worth trying
+          (by guesswork), as sensible scaling will reduce the difficulty of
+          the minimization problem, so that E04JAF will take less computer
+          time.
+
+          9. Example
+
+          To minimize
+
+                                2         2         4          4
+                     F=(x +10x ) +5(x -x ) +(x -2x ) +10(x -x )
+                         1    2      3  4     2   3       1  4
+
+          subject to
+
+                                      1<=x <=3
+                                          1
+
+                                      -2<=x <=0
+                                           2
+
+                                      1<=x <=3,
+                                          4
+
+          starting from the initial guess (3, - 1, 0, 1).
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe04mbf}{NAG On-line Documentation: e04mbf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E04MBF(3NAG)      Foundation Library (12/10/92)      E04MBF(3NAG)
+
+
+
+          E04 -- Minimizing or Maximizing a Function                 E04MBF
+                  E04MBF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E04MBF is an easy-to-use routine for solving linear programming
+          problems, or for finding a feasible point for such problems. It
+          is not intended for large sparse problems.
+
+          2. Specification
+
+                 SUBROUTINE E04MBF (ITMAX, MSGLVL, N, NCLIN, NCTOTL, NROWA,
+                1                   A, BL, BU, CVEC, LINOBJ, X, ISTATE,
+                2                   OBJLP, CLAMDA, IWORK, LIWORK, WORK,
+                3                   LWORK, IFAIL)
+                 INTEGER          ITMAX, MSGLVL, N, NCLIN, NCTOTL, NROWA,
+                1                 ISTATE(NCTOTL), IWORK(LIWORK), LIWORK,
+                2                 LWORK, IFAIL
+                 DOUBLE PRECISION A(NROWA,N), BL(NCTOTL), BU(NCTOTL), CVEC
+                1                 (N), X(N), OBJLP, CLAMDA(NCTOTL), WORK
+                2                 (LWORK)
+                 LOGICAL          LINOBJ
+
+          3. Description
+
+          E04MBF solves linear programming (LP) problems of the form
+
+                                T                   (x )
+                    Minimize   c x   subject to  l<=(Ax)<=u            (LP)
+                             n
+                    x is in R
+
+          where c is an n element vector and A is an m by n matrix i.e.,
+          there are n variables and m general linear constraints. m may be
+          zero in which case the LP problem is subject only to bounds on
+          the variables. Notice that upper and lower bounds are specified
+          for all the variables and constraints. This form allows full
+          generality in specifying other types of constraints. For example
+          the ith constraint may be specified as equality by setting l =u .
+                                                                      i  i
+          If certain bounds are not present the associated elements of l or
+          u can be set to special values that will be treated as -infty or
+          +infty.
+
+          The routine allows the linear objective function to be omitted in
+          which case a feasible point for the set of constraints is sought.
+
+          The user must supply an initial estimate of the solution.
+
+          Users who wish to exercise additional control and users with
+          problems whose solution would benefit from additional flexibility
+          should consider using the comprehensive routine E04NAF.
+
+          4. References
+
+          [1]   Gill P E, Murray W and Wright M H (1981) Practical
+                Optimization. Academic Press.
+
+          [2]   Gill P E, Murray W, Saunders M A and Wright M H (1983)
+                User's Guide for SOL/QPSOL. Report SOL 83-7. Department of
+                Operations Research, Stanford University.
+
+          5. Parameters
+
+           1:  ITMAX -- INTEGER                                       Input
+               On entry: an upper bound on the number of iterations to be
+               taken. If ITMAX is not positive, then the value 50 is used
+               in place of ITMAX.
+
+           2:  MSGLVL -- INTEGER                                      Input
+               On entry: indicates whether or not printout is required at
+               the final solution. When printing occurs the output is on
+               the advisory message channel (see X04ABF). A description of
+               the printed output is given in Section 5.1. The level of
+               printing is determined as follows:
+               MSGLVL < 0
+                     No printing.
+
+               MSGLVL = 0
+                     Printing only if an input parameter is incorrect, or
+                     if the problem is so ill-conditioned that subsequent
+                     overflow is likely. This setting is strongly
+                     recommended in preference to MSGLVL < 0.
+
+               MSGLVL = 1
+                     Printing at the solution.
+
+               MSGLVL > 1
+                     Values greater than 1 should normally be used only at
+                     the direction of NAG; such values may generate large
+                     amounts of printed output.
+
+           3:  N -- INTEGER                                           Input
+               On entry: the number n of variables. Constraint: N >= 1.
+
+           4:  NCLIN -- INTEGER                                       Input
+               On entry: the number of general linear constraints in the
+               problem. Constraint: NCLIN >= 0.
+
+           5:  NCTOTL -- INTEGER                                      Input
+               On entry: the value (N+NCLIN).
+
+           6:  NROWA -- INTEGER                                       Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which E04MBF is called.
+               Constraint: NROWA >= max(1,NCLIN).
+
+           7:  A(NROWA,N) -- DOUBLE PRECISION array                   Input
+               On entry: the leading NCLIN by n part of A must contain the
+               NCLIN general constraints, with the coefficients of the ith
+               constraint in the ith row of A. If NCLIN = 0, then A is not
+               referenced.
+
+           8:  BL(NCTOTL) -- DOUBLE PRECISION array                   Input
+               On entry: the first n elements of BL must contain the lower
+               bounds on the n variables, and when NCLIN > 0, the next
+               NCLIN elements of BL must contain the lower bounds on the
+               NCLIN general linear constraints. To specify a non-existent
+               lower bound (l =-infty), set BL(j)<=-1.0E+20.
+                             j
+
+           9:  BU(NCTOTL) -- DOUBLE PRECISION array                   Input
+               On entry: the first n elements of BU must contain the upper
+               bounds on the n variables, and when NCLIN > 0, the next
+               NCLIN elements of BU must contain the upper bounds on the
+               NCLIN general linear constraints. To specify a non-existent
+               upper bound (u =+infty), set BU(j)>=1.0E+20. Constraint:
+                             j
+               BL(j)<=BU(j), for j=1,2,...,NCTOTL.
+
+          10:  CVEC(N) -- DOUBLE PRECISION array                      Input
+               On entry: with LINOBJ = .TRUE., CVEC must contain the
+               coefficients of the objective function. If LINOBJ = .FALSE.,
+               then CVEC is not referenced.
+
+          11:  LINOBJ -- LOGICAL                                      Input
+               On entry: indicates whether or not a linear objective
+               function is present. If LINOBJ = .TRUE., then the full LP
+               problem is solved, but if LINOBJ = .FALSE., only a feasible
+               point is found and the array CVEC is not referenced.
+
+          12:  X(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: an estimate of the solution, or of a feasible
+               point. Even when LINOBJ = .TRUE. it is not necessary for the
+               point supplied in X to be feasible. In the absence of better
+               information all elements of X may be set to zero. On exit:
+               the solution to the LP problem when LINOBJ = .TRUE., or a
+               feasible point when LINOBJ = .FALSE..
+
+               When no feasible point exists (see IFAIL = 1 in Section 6)
+               then X contains the point for which the sum of the
+               infeasibilities is a minimum. On return with IFAIL = 2, 3 or
+               4, X contains the point at which E04MBF terminated.
+
+          13:  ISTATE(NCTOTL) -- INTEGER array                       Output
+               On exit: with IFAIL < 5, ISTATE indicates the status of
+               every constraint at the final point. The first n elements of
+               ISTATE refer to the upper and lower bounds on the variables
+               and when NCLIN > 0 the next NCLIN elements refer to the
+               general constraints.
+
+               Their meaning is:
+               ISTATE(j) Meaning
+
+               -2        The constraint violates its lower bound. This
+                         value cannot occur for any element of ISTATE when
+                         a feasible point has been found.
+
+               -1        The constraint violates its upper bound. This
+                         value cannot occur for any element of ISTATE when
+                         a feasible point has been found.
+
+               0         The constraint is not in the working set (is not
+                         active) at the final point. Usually this means
+                         that the constraint lies strictly between its
+                         bounds.
+
+               1         This inequality constraint is in the working set
+                         (is active) at its lower bound.
+
+               2         This inequality constraint is in the working set
+                         (is active) at its upper bound.
+
+               3         This constraint is included in the working set (is
+                         active) as an equality. This value can only occur
+                         when BL(j) = BU(j).
+
+          14:  OBJLP -- DOUBLE PRECISION                             Output
+               On exit: when LINOBJ = .TRUE., then on successful exit,
+               OBJLP contains the value of the objective function at the
+               solution, and on exit with IFAIL = 2, 3 or 4, OBJLP contains
+               the value of the objective function at the point returned in
+               X.
+
+               When LINOBJ = .FALSE., then on successful exit OBJLP will be
+               zero and on return with IFAIL = 1, OBJLP contains the
+               minimum sum of the infeasibilities corresponding to the
+               point returned in X.
+
+          15:  CLAMDA(NCTOTL) -- DOUBLE PRECISION array              Output
+               On exit: when LINOBJ = .TRUE., then on successful exit, or
+               on exit with IFAIL = 2, 3, or 4, CLAMDA contains the
+               Lagrange multipliers (reduced costs) for each constraint
+               with respect to the working set. The first n components of
+               CLAMDA contain the multipliers for the bound constraints on
+               the variables and the remaining NCLIN components contain the
+               multipliers for the general linear constraints.
+
+               If ISTATE(j) = 0 so that the jth constraint is not in the
+               working set then CLAMDA(j) is zero. If X is optimal and
+               ISTATE(j) = 1, then CLAMDA(j) should be non-negative, and if
+               ISTATE(j) = 2, then CLAMDA(j) should be non-positive.
+
+               When LINOBJ = .FALSE., all NCTOTL elements of CLAMDA are
+               returned as zero.
+
+          16:  IWORK(LIWORK) -- INTEGER array                     Workspace
+
+          17:  LIWORK -- INTEGER                                      Input
+               On entry: the length of the array IWORK as declared in the
+               (sub)program from which E04MBF is called. Constraint:
+               LIWORK>=2*N.
+
+          18:  WORK(LWORK) -- DOUBLE PRECISION array              Workspace
+
+          19:  LWORK -- INTEGER                                       Input
+               On entry: the length of the array WORK as declared in the
+               (sub)program from which E04MBF is called. Constraints:
+               when N <= NCLIN then
+                               2
+                     LWORK>=2*N +6*N+4*NCLIN+NROWA;
+
+               when 0 <= NCLIN < N then
+                                       2
+                     LWORK>=2*(NCLIN+1) +4*NCLIN+6*N+NROWA.
+
+          20:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit. To suppress
+               the output of an error message when soft failure occurs, set
+               IFAIL to 1.
+
+          5.1. Description of the Printed Output
+
+          When MSGLVL = 1, then E04MBF will produce output on the advisory
+          message channel (see X04ABF ), giving information on the final
+          point. The following describes the printout associated with each
+          variable.
+
+          Output         Meaning
+
+          VARBL          The name (V) and index j, for j=1,2,...,n, of the
+                         variable.
+
+          STATE          The state of the variable. (FR if neither bound is
+                         in the working set, EQ for a fixed variable, LL if
+                         on its lower bound, UL if on its upper bound and
+                         TB if held on a temporary bound.) If the value of
+                         the variable lies outside the upper or lower bound
+                         then STATE will be ++ or -- respectively.
+
+          VALUE          The value of the variable at the final iteration.
+
+          LOWER BOUND    The lower bound specified for the variable.
+
+          UPPER BOUND    The upper bound specified for the variable.
+
+          LAGR MULT      The value of the Lagrange multiplier for the
+                         associated bound.
+
+          RESIDUAL       The difference between the value of the variable
+                         and the nearer of its bounds.
+
+          For each of the general constraints the printout is as above with
+          refers to the jth element of Ax, except that VARBL is replaced
+          by:
+
+          LNCON          The name (L) and index j, for j = 1,2,...,NCLIN of
+                         the constraint.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          Note: when MSGLVL=1 a short description of the error is printed.
+
+          IFAIL= 1
+               No feasible point could be found. Moving violated
+               constraints so that they are satisfied at the point returned
+               in X gives the minimum moves necessary to make the LP
+               problem feasible.
+
+          IFAIL= 2
+               The solution to the LP problem is unbounded.
+
+          IFAIL= 3
+               A total of 50 changes were made to the working set without
+               altering x. Cycling is probably occurring. The user should
+               consider using E04NAF with MSGLVL >= 5 to monitor constraint
+               additions and deletions in order to determine whether or not
+               cycling is taking place.
+
+          IFAIL= 4
+               The limit on the number of iterations has been reached.
+               Increase ITMAX or consider using E04NAF to monitor progress.
+
+          IFAIL= 5
+               An input parameter is invalid. Unless MSGLVL < 0 a message
+               will be printed.
+
+          IFAILOverflow
+               If the printed output before the overflow occurred contains
+               a warning about serious ill-conditioning in the working set
+               when adding the jth constraint, then either the user should
+               try using E04NAF and experiment with the magnitude of FEATOL
+               (j) in that routine, or the offending linearly dependent
+               constraint (with index j) should be removed from the
+               problem.
+
+          7. Accuracy
+
+          The routine implements a numerically stable active set strategy
+          and returns solutions that are as accurate as the condition of
+          the LP problem warrants on the machine.
+
+          8. Further Comments
+
+          The time taken by each iteration is approximately proportional to
+               2      2
+          min(n ,NCLIN ).
+
+          Sensible scaling of the problem is likely to reduce the number of
+          iterations required and make the problem less sensitive to
+          perturbations in the data, thus improving the condition of the LP
+          problem. In the absence of better information it is usually
+          sensible to make the Euclidean lengths of each constraint of
+          comparable magnitude. See Gill et al [1] for further information
+          and advice.
+
+          Note that the routine allows constraints to be violated by an
+          absolute tolerance equal to the machine precision (see X02AJF(*))
+
+          9. Example
+
+          To minimize the function
+
+                    -0.02x -0.2x -0.2x -0.2x -0.2x +0.04x +0.04x
+                          1     2     3     4     5      6      7
+
+          subject to the bounds
+
+                                -0.01 <= x  <= 0.01
+                                          1
+                                -0.1 <= x  <= 0.15,
+                                         2
+                                -0.01 <= x  <= 0.03,
+                                          3
+                                -0.04 <= x  <= 0.02,
+                                          4
+                                -0.1 <= x  <= 0.05,
+                                         5
+                                -0.01 <= x
+                                          6
+                                -0.01 <= x
+                                          7
+
+          and the general constraints
+
+                             x +x +x +x +x +x +x =-0.13
+                              1  2  3  4  5  6  7
+
+              0.15x +0.04x +0.02x +0.04x +0.02x +0.01x +0.03x <=-0.0049
+                   1      2      3      4      5      6      7
+
+                 0.03x +0.05x +0.08x +0.02x +0.06x +0.01x <=-0.0064
+                      1      2      3      4      5      6
+
+                     0.02x +0.04x +0.01x +0.02x +0.02x <=-0.0037
+                          1      2      3      4      5
+
+                            0.02x +0.03x +0.01x <=-0.0012
+                                 1      2      5
+
+                 -0.0992<=0.70x +0.75x +0.80x +0.75x +0.80x +0.97x
+                               1      2      3      4      5      6
+
+           -0.003<=0.02x +0.06x +0.08x +0.12x +0.02x +0.01x +0.97x <=0.002
+                        1      2      3      4      5      6      7
+
+          The initial point, which is infeasible, is
+
+                                                                 T
+                  x =(-0.01, -0.03, 0.0, -0.01, -0.1, 0.02, 0.01) .
+                   0
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe04naf}{NAG On-line Documentation: e04naf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E04NAF(3NAG)      Foundation Library (12/10/92)      E04NAF(3NAG)
+
+
+
+          E04 -- Minimizing or Maximizing a Function                 E04NAF
+                  E04NAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E04NAF is a comprehensive routine for solving quadratic
+          programming (QP) or linear programming (LP) problems. It is not
+          intended for large sparse problems.
+
+          2. Specification
+
+                 SUBROUTINE E04NAF (ITMAX, MSGLVL, N, NCLIN, NCTOTL, NROWA,
+                1                   NROWH, NCOLH, BIGBND, A, BL, BU, CVEC,
+                2                   FEATOL, HESS, QPHESS, COLD, LP, ORTHOG,
+                3                   X, ISTATE, ITER, OBJ, CLAMDA, IWORK,
+                4                   LIWORK, WORK, LWORK, IFAIL)
+                 INTEGER          ITMAX, MSGLVL, N, NCLIN, NCTOTL, NROWA,
+                1                 NROWH, NCOLH, ISTATE(NCTOTL), ITER, IWORK
+                2                 (LIWORK), LIWORK, LWORK, IFAIL
+                 DOUBLE PRECISION BIGBND, A(NROWA,N), BL(NCTOTL),
+                1                 BU(NCTOTL), CVEC(N), FEATOL(NCTOTL), HESS
+                2                 (NROWH,NCOLH), X(N), OBJ, CLAMDA(NCTOTL),
+                3                 WORK(LWORK)
+                 LOGICAL          COLD, LP, ORTHOG
+                 EXTERNAL         QPHESS
+
+          3. Description
+
+          E04NAF is essentially identical to the subroutine SOL/QPSOL
+          described in Gill et al [1].
+
+          E04NAF is designed to solve the quadratic programming (QP)
+          problem - the minimization of a quadratic function subject to a
+          set of linear constraints on the variables. The problem is
+          assumed to be stated in the following form:
+
+                           T   1 T                    (x )
+                Minimize  c x+ -x Hx   subject to  l<=(Ax)<=u ,         (1)
+                               2
+
+          where c is a constant n-vector and H is a constant n by n
+          symmetric matrix; note that H is the Hessian matrix (matrix of
+          second partial derivatives) of the quadratic objective function.
+          The matrix A is m by n, where m may be zero; A is treated as a
+          dense matrix.
+
+          The constraints involving A will be called the general
+          constraints. Note that upper and lower bounds are specified for
+          all the variables and for all the general constraints. The form
+          of (1) allows full generality in specifying other types of
+          constraints. In particular, an equality constraint is specified
+          by setting l =u . If certain bounds are not present, the
+                      i  i
+          associated elements of l or u can be set to special values that
+          will be treated as -infty or +infty.
+
+          The user must supply an initial estimate of the solution to (1),
+          and a subroutine that computes the product Hx for any given
+          vector x. If H is positive-definite or positive semi-definite,
+          E04NAF will obtain a global minimum; otherwise, the solution
+          obtained will be a local minimum (which may or may not be a
+          global minimum). If H is defined as the zero matrix, E04NAF will
+          solve the resulting linear programming (LP) problem; however,
+          this can be accomplished more efficiently by setting a logical
+          variable in the call of the routine (see the parameter LP in
+          Section 5).
+
+          E04NAF allows the user to provide the indices of the constraints
+          that are believed to be exactly satisfied at the solution. This
+          facility, known as a warm start, can lead to significant savings
+          in computational effort when solving a sequence of related
+          problems.
+
+          The method has two distinct phases. In the first (the LP phase),
+          an iterative procedure is carried out to determine a feasible
+          point. In this context, feasibility is defined by a user-provided
+          array FEATOL; the jth constraint is considered satisfied if its
+          violation does not exceed FEATOL(j). The second phase (the QP
+          phase) generates a sequence of feasible iterates in order to
+          minimize the quadratic objective function. In both phases, a
+          subset of the constraints - called the working set - is used to
+          define the search direction at each iteration; typically, the
+          working set includes constraints that are satisfied to within the
+          corresponding tolerances in the FEATOL array.
+
+          We now briefly describe a typical iteration in the QP phase. Let
+          x  denote the estimate of the solution at the kth iteration; the
+           k
+          next iterate is defined by
+
+                                 x   =x +(alpha) p
+                                  k+1  k        k k
+
+          where p  is an n-dimensional search direction and (alpha)  is a
+                 k                                                 k
+          scalar step length. Assume that the working (active) set contains
+          t  linearly independent constraints, and let C  denote the matrix
+           k                                            k
+          of coefficients of the bounds and general constraints in the
+          current working set.
+
+          Let Z  denote a matrix whose columns form a basis for the null
+               k
+          space of C , so that C Z =0. (Note that Z  has n  columns, where
+                    k           k k                k      z
+                                T
+          n =n-t .) The vector Z (c+Hx ) is called the projected gradient
+           z    k               k     k
+          at x . If the projected gradient is zero at x  (i.e., x  is a
+              k                                        k         k
+          constrained stationary point in the subspace defined by Z ),
+                                                                   k
+          Lagrange multipliers (lambda)  are defined as the solution of the
+                                       k
+          compatible overdetermined system
+
+                                T
+                               C (lambda) =c+Hx                         (2)
+                                k        k     k
+
+          The Lagrange multiplier (lambda) corresponding to an inequality
+          constraint in the working set is said to be optimal if
+          (lambda)<=0 when the associated constraint is at its upper bound,
+          or if (lambda)>=0 when the associated constraint is at its lower
+          bound. If a multiplier is non-optimal, the objective function can
+          be reduced by deleting the corresponding constraint (with index
+          JDEL, see Section 5.1) from the working set.
+
+          If the projected gradient at x  is non-zero, the search direction
+                                        k
+          p  is defined as
+           k
+
+                                    p =Z p                              (3)
+                                     k  k z
+
+          where p  is an n -vector. In effect, the constraints in the
+                 z        z
+          working set are treated as equalities, by constraining p  to lie
+                                                                  k
+          within the subspace of vectors orthogonal to the rows of C . This
+                                                                    k
+          definition ensures that C p =0, and hence the values of the
+                                   k k
+          constraints in the working set are not altered by any move along
+          p .
+           k
+
+          The vector p  is obtained by solving the equations
+                      z
+
+                                T        T
+                               Z HZ p =-Z (c+Hx )                       (4)
+                                k  k z   k     k
+
+                       T
+          (The matrix Z HZ  is called the projected Hessian matrix.) If the
+                       k  k
+          projected Hessian is positive-definite, the vector defined by (3)
+          and (4) is the step to the minimum of the quadratic function in
+          the subspace defined by Z .
+                                   k
+
+          If the projected Hessian is positive-definite and x +p  is
+                                                             k  k
+          feasible, (alpha)  will be taken as unity. In this case, the
+                           k
+          projected gradient at x    will be zero (see NORM ZTG in Section
+                                 k+1
+          5.1), and Lagrange multipliers can be computed (see Gill et al
+          [2]). Otherwise, (alpha)  is set to the step to the 'nearest'
+                                  k
+          constraint (with index JADD, see Section 5.1), which is added to
+          the working set at the next iteration.
+
+          The matrix Z  is obtained from the TQ factorization of C , in
+                      k                                           k
+          which C  is represented as
+                 k
+
+                                   C Q=(0 T )                           (5)
+                                    k      k
+
+          where T  is reverse-triangular. It follows from (5) that Z  may
+                 k                                                  k
+          be taken as the first n  columns of Q. If the projected Hessian
+                                 z
+          is positive-definite, (3) is solved using the Cholesky
+          factorization
+
+                                      T     T
+                                     Z HZ =R R
+                                      k  k  k k
+
+          where R  is upper triangular. These factorizations are updated as
+                 k
+          constraints enter or leave the working set (see Gill et al [2]
+          for further details).
+
+          An important feature of E04NAF is the treatment of indefiniteness
+          in the projected Hessian. If the projected Hessian is positive-
+          definite, it may become indefinite only when a constraint is
+          deleted from the working set. In this case, a temporary
+          modification (of magnitude HESS MOD, see Section 5.1) is added to
+          the last diagonal element of the Cholesky factor. Once a
+          modification has occurred, no further constraints are deleted
+          from the working set until enough constraints have been added so
+          that the projected Hessian is again positive-definite. If
+          equation (1) has a finite solution, a move along the direction
+          obtained by solving (4) with the modified Cholesky factor must
+          encounter a constraint that is not already in the working set.
+
+          In order to resolve indefiniteness in this way, we must ensure
+          that the projected Hessian is positive-definite at the first
+          iterate in the QP phase. Given the n  by n  projected Hessian, a
+                                              z     z
+          step-wise Cholesky factorization is performed with symmetric
+          interchanges (and corresponding rearrangement of the columns of Z
+          ), terminating if the next step would cause the matrix to become
+          indefinite. This determines the largest possible positive-
+          definite principal sub-matrix of the (permuted) projected
+          Hessian. If n  steps of the Cholesky factorization have been
+                       R
+          successfully completed, the relevant projected Hessian is an n
+                                                                        R
+                                          T
+          by n  positive-definite matrix Z HZ , where Z  comprises the
+              R                           R  R         R
+          first n  columns of Z. The quadratic function will subsequently
+                 R
+          be minimized within subspaces of reduced dimension until the full
+          projected Hessian is positive-definite.
+
+          If a linear program is being solved and there are fewer general
+          constraints than variables, the method moves from one vertex to
+          another while minimizing the objective function. When necessary,
+          an initial vertex is defined by temporarily fixing some of the
+          variables at their initial values.
+
+          Several strategies are used to control ill-conditioning in the
+          working set. One such strategy is associated with the FEATOL
+          array. Allowing the jth constraint to be violated by as much as
+          FEATOL(j) often provides a choice of constraints that could be
+          added to the working set. When a choice exists, the decision is
+          based on the conditioning of the working set. Negative steps are
+          occasionally permitted, since x  may violate the constraint to be
+                                         k
+          added.
+
+          4. References
+
+          [1]   Gill P E, Murray W, Saunders M A and Wright M H (1983)
+                User's Guide for SOL/QPSOL. Report SOL 83-7. Department of
+                Operations Research, Stanford University.
+
+          [2]   Gill P E, Murray W, Saunders M A and Wright M H (1982) The
+                design and implementation of a quadratic programming
+                algorithm. Report SOL 82-7. Department of Operations
+                Research, Stanford University.
+
+          [3]   Gill P E, Murray W and Wright M H (1981) Practical
+                Optimization. Academic Press.
+
+          5. Parameters
+
+           1:  ITMAX -- INTEGER                                       Input
+               On entry: an upper bound on the number of iterations to be
+               taken during the LP phase or the QP phase. If ITMAX is not
+               positive, then the value 50 is used in place of ITMAX.
+
+           2:  MSGLVL -- INTEGER                                      Input
+               On entry: MSGLVL must indicate the amount of intermediate
+               output desired (see Section 5.1 for a description of the
+               printed output). All output is written to the current
+               advisory message unit (see X04ABF). For MSGLVL >= 10, each
+               level includes the printout for all lower levels.
+               Value   Definition
+
+               <0      No printing.
+
+               0       Printing only if an input parameter is incorrect, or
+                       if the working set is so ill-conditioned that
+                       subsequent overflow is likely. This setting is
+                       strongly recommended in preference to MSGLVL < 0.
+
+               1       The final solution only.
+
+               5       One brief line of output for each constraint
+                       addition or deletion (no printout of the final
+                       solution).
+
+               >=10    The final solution and one brief line of output for
+                       each constraint addition or deletion.
+
+               >=15    At each iteration, X, ISTATE, and the indices of the
+                       free variables (i.e.,the variables not currently
+                       held on a bound).
+
+               >=20    At each iteration, the Lagrange multiplier estimates
+                       and the general constraint values.
+
+               >=30    At each iteration, the diagonal elements of the
+                       matrix T associated with the TQ factorization of the
+                       working set, and the diagonal elements of the
+                       Cholesky factor R of the projected Hessian.
+
+               >=80    Debug printout.
+
+               99      The arrays CVEC and HESS.
+
+           3:  N -- INTEGER                                           Input
+               On entry: the number, n, of variables. Constraint: N >= 1.
+
+           4:  NCLIN -- INTEGER                                       Input
+               On entry: the number of general linear constraints in the
+               problem. Constraint: NCLIN >= 0.
+
+           5:  NCTOTL -- INTEGER                                      Input
+               On entry: the value (N+NCLIN).
+
+           6:  NROWA -- INTEGER                                       Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which E04NAF is called.
+               Constraint: NROWA >= max(1,NCLIN).
+
+           7:  NROWH -- INTEGER                                       Input
+               On entry: the first dimension of the array HESS as declared
+               in the (sub)program from which E04NAF is called.
+               Constraint: NROWH >= 1.
+
+           8:  NCOLH -- INTEGER                                       Input
+               On entry: the column dimension of the array HESS as declared
+               in the (sub)program from which E04NAF is called.
+               Constraint: NCOLH >= 1.
+
+           9:  BIGBND -- DOUBLE PRECISION                             Input
+               On entry: BIGBND must denote an 'infinite' component of l
+               and u. Any upper bound greater than or equal to BIGBND will
+               be regarded as plus infinity, and a lower bound less than or
+               equal to -BIGBND will be regarded as minus infinity.
+               Constraint: BIGBND > 0.0.
+
+          10:  A(NROWA,N) -- DOUBLE PRECISION array                   Input
+               On entry: the leading NCLIN by n part of A must contain the
+               NCLIN general constraints, with the ith constraint in the i
+               th row of A. If NCLIN = 0, then A is not referenced.
+
+          11:  BL(NCTOTL) -- DOUBLE PRECISION array                   Input
+               On entry: the lower bounds for all the constraints, in the
+               following order. The first n elements of BL must contain the
+               lower bounds on the variables. If NCLIN > 0, the next NCLIN
+               elements of BL must contain the lower bounds for the general
+               linear constraints. To specify a non-existent lower bound
+               (i.e., l =-infty), the value used must satisfy BL(j)<=-
+                       j
+               BIGBND To specify the jth constraint as an equality, the
+               user must set BL(j) = BU(j). Constraint: BL(j) <= BU(j),
+               j=1,2,...,NCTOTL.
+
+          12:  BU(NCTOTL) -- DOUBLE PRECISION array                   Input
+               On entry: the upper bounds for all the constraints, in the
+               following order. The first n elements of BU must contain the
+               upper bounds on the variables. If NCLIN > 0, the next NCLIN
+               elements of BU must contain the upper bounds for the general
+               linear constraints. To specify a non-existent upper bound
+               (i.e., u =+infty), the value used must satisfy BU(j) >=
+                       j
+               BIGBND. To specify the jth constraint as an equality, the
+               user must set BU(j) = BL(j). Constraint: BU(j) >= BL(j),
+               j=1,2,...,NCTOTL.
+
+          13:  CVEC(N) -- DOUBLE PRECISION array                      Input
+               On entry: the coefficients of the linear term of the
+               objective function (the vector c in equation  (1)).
+
+          14:  FEATOL(NCTOTL) -- DOUBLE PRECISION array               Input
+               On entry: a set of positive tolerances that define the
+               maximum permissible absolute violation in each constraint in
+               order for a point to be considered feasible, i.e., if the
+               violation in constraint j is less than FEATOL(j), the point
+               is considered to be feasible with respect to the jth
+               constraint. The ordering of the elements of FEATOL is the
+               same as that described above for BL.
+
+               The elements of FEATOL should not be too small and a warning
+               message will be printed on the current advisory message
+               channel if any element of FEATOL is less than the machine
+               precision (see X02AJF(*)). As the elements of FEATOL
+               increase, the algorithm is less likely to encounter
+               difficulties with ill-conditioning and degeneracy. However,
+               larger values of FEATOL(j) mean that constraint j could be
+               violated by a significant amount. It is recommended that
+               FEATOL(j) be set to a value equal to the largest acceptable
+               violation for constraint j. For example, if the data
+               defining the constraints are of order unity and are correct
+               to about 6 decimal digits, it would be appropriate to choose
+                              -6
+               FEATOL(j) as 10   for all relevant j. Often the square root
+               of the machine precision is a reasonable choice if the
+               constraint is well scaled.
+
+          15:  HESS(NROWH,NCOLH) -- DOUBLE PRECISION array            Input
+               On entry: HESS may be used to store the Hessian matrix H of
+               equation (1) if desired. HESS is accessed only by the
+               subroutine QPHESS and is not accessed if LP = .TRUE.. Refer
+               to the specification of QPHESS (below) for further details
+               of how HESS may be used to pass data to QPHESS.
+
+          16:  QPHESS -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               QPHESS must define the product of the Hessian matrix H and a
+               vector x. The elements of H need not be defined explicitly.
+               QPHESS is not accessed if LP is set to .TRUE. and in this
+               case QPHESS may be the dummy routine E04NAN. (E04NAN is
+               included in the NAG Foundation Library and so need not be
+               supplied by the user. Its name may be implementation-
+               dependent: see the Users' Note for your implementation for
+               details.)
+
+               Its specification is:
+
+                      SUBROUTINE QPHESS (N, NROWH, NCOLH, JTHCOL,
+                     1                   HESS, X, HX)
+                      INTEGER          N, NROWH, NCOLH, JTHCOL
+                      DOUBLE PRECISION HESS(NROWH,NCOLH), X(N), HX(N)
+
+                1:  N -- INTEGER                                      Input
+                    On entry: the number n of variables.
+
+                2:  NROWH -- INTEGER                                  Input
+                    On entry: the row dimension of the array HESS.
+
+                3:  NCOLH -- INTEGER                                  Input
+                    On entry: the column dimension of the array HESS.
+
+                4:  JTHCOL -- INTEGER                                 Input
+                    The input parameter JTHCOL is included to allow
+                    flexibility for the user in the special situation when
+                    x is the jth co-ordinate vector (i.e.,the jth column of
+                    the identity matrix). This may be of interest because
+                    the product Hx is then the jth column of H, which can
+                    sometimes be computed very efficiently. The user may
+                    code QPHESS to take advantage of this case. On entry:
+                    if JTHCOL = j, where j>0, HX must contain column JTHCOL
+                    of H, and hence special code may be included in QPHESS
+                    to test JTHCOL if desired. However, special code is not
+                    necessary, since the vector x always contains column
+                    JTHCOL of the identity matrix whenever QPHESS is called
+                    with JTHCOL > 0.
+
+                5:  HESS(NROWH,NCOLH) -- DOUBLE PRECISION array       Input
+                    On entry: the Hessian matrix H.
+
+                    In some cases, it may be desirable to use a one-
+                    dimensional array to transmit data or workspace to
+                    QPHESS; HESS should then be declared with dimension
+                    (NROWH) in the (sub)program from which E04NAF is called
+                    and the parameter NCOLH must be 1.
+
+                    In other situations, it may be desirable to compute Hx
+                    without accessing HESS - for example, if H is sparse or
+                    has special structure. (This is illustrated in the
+                    subroutine QPHES1 in the example program in Section 9.)
+                    The parameters HESS, NROWH and NCOLH may then refer to
+                    any convenient array.
+
+                    When MSGLVL = 99, the (possibly undefined) contents of
+                    HESS will be printed, except if NROWH and NCOLH are
+                    both 1. Also printed are the results of calling QPHESS
+                    with JTHCOL = 1,2,...,n.
+
+                6:  X(N) -- DOUBLE PRECISION array                    Input
+                    On entry: the vector x.
+
+                7:  HX(N) -- DOUBLE PRECISION array                  Output
+                    On exit: HX must contain the product Hx.
+               QPHESS must be declared as EXTERNAL in the (sub)program
+               from which E04NAF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+          17:  COLD -- LOGICAL                                        Input
+               On entry: COLD must indicate whether the user has specified
+               an initial estimate of the active set of constraints. If
+               COLD is set to .TRUE., the initial working set is determined
+               by E04NAF. If COLD is set to .FALSE. (a 'warm start'), the
+               user must define the ISTATE array which gives the status of
+               each constraint with respect to the working set. E04NAF will
+               override the user's specification of ISTATE if necessary, so
+               that a poor choice of working set will not cause a fatal
+               error.
+
+               The warm start option is particularly useful when E04NAF is
+               called repeatedly to solve related problems.
+
+          18:  LP -- LOGICAL                                          Input
+               On entry: if LP = .FALSE., E04NAF will solve the specified
+               quadratic programming problem. If LP = .TRUE., E04NAF will
+               treat H as zero and solve the resulting linear programming
+               problem; in this case, the parameters HESS and QPHESS will
+               not be referenced.
+
+          19:  ORTHOG -- LOGICAL                                      Input
+               On entry: ORTHOG must indicate whether orthogonal
+               transformations are to be used in computing and updating the
+               TQ factorization of the working set
+                                        A Q=(0 T),
+                                         s
+               where A  is a sub-matrix of A and T is reverse-triangular.
+                      s
+               If ORTHOG = .TRUE., the TQ factorization is computed using
+               Householder reflections and plane rotations, and the matrix
+               Q is orthogonal. If ORTHOG = .FALSE., stabilized elementary
+               transformations are used to maintain the factorization, and
+               Q is not orthogonal. A rule of thumb in making the choice is
+               that orthogonal transformations require more work, but
+               provide greater numerical stability. Thus, we recommend
+               setting ORTHOG to .TRUE. if the problem is reasonably small
+               or the active set is ill-conditioned. Otherwise, setting
+               ORTHOG to .FALSE. will often lead to a reduction in solution
+               time with negligible loss of reliability.
+
+          20:  X(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: an estimate of the solution. In the absence of
+               better information all elements of X may be set to zero. On
+               exit: from E04NAF, X contains the best estimate of the
+               solution.
+
+          21:  ISTATE(NCTOTL) -- INTEGER array                 Input/Output
+               On entry: with COLD as .FALSE., ISTATE must indicate the
+               status of every constraint with respect to the working set.
+               The ordering of ISTATE is as follows; the first n elements
+               of ISTATE refer to the upper and lower bounds on the
+               variables and elements n+1 through n + NCLIN refer to the
+               upper and lower bounds on Ax. The significance of each
+               possible value of ISTATE(j) is as follows:
+               ISTATE(j) Meaning
+
+               -2        The constraint violates its lower bound by more
+                         than FEATOL(j). This value of ISTATE cannot occur
+                         after a feasible point has been found.
+
+               -1        The constraint violates its upper bound by more
+                         than FEATOL(j). This value of ISTATE cannot occur
+                         after a feasible point has been found.
+
+               0         The constraint is not in the working set. Usually,
+                         this means that the constraint lies strictly
+                         between its bounds.
+
+               1         This inequality constraint is included in the
+                         working set at its lower bound. The value of the
+                         constraint is within FEATOL(j) of its lower bound.
+
+               2         This inequality constraint is included in the
+                         working set at its upper bound. The value of the
+                         constraint is within FEATOL(j) of its upper bound.
+
+               3         The constraint is included in the working set as
+                         an equality. This value of ISTATE can occur only
+                         when BL(j) = BU(j). The corresponding constraint
+                         is within FEATOL(j) of its required value.
+               If COLD = .TRUE., ISTATE need not be set by the user.
+               However, when COLD = .FALSE., every element of ISTATE must
+               be set to one of the values given above to define a
+               suggested initial working set (which will be changed by
+               E04NAF if necessary). The most likely values are:
+               ISTATE(j) Meaning
+
+               0         The corresponding constraint should not be in the
+                         initial working set.
+
+               1         The constraint should be in the initial working
+                         set at its lower bound.
+
+               2         The constraint should be in the initial working
+                         set at its upper bound.
+
+               3         The constraint should be in the initial working
+                         set as an equality. This value must not be
+                         specified unless BL(j) = BU(j). The values 1, 2 or
+                         3 all have the same effect when BL(j) = BU(j).
+               Note that if E04NAF has been called previously with the same
+               values of N and NCLIN, ISTATE already contains satisfactory
+               values. On exit: when E04NAF exits with IFAIL set to 0, 1 or
+               3, the values in the array ISTATE indicate the status of the
+               constraints in the active set at the solution. Otherwise,
+               ISTATE indicates the composition of the working set at the
+               final iterate.
+
+          22:  ITER -- INTEGER                                       Output
+               On exit: the number of iterations performed in either the LP
+               phase or the QP phase, whichever was last entered.
+
+               Note that ITER is reset to zero after the LP phase.
+
+          23:  OBJ -- DOUBLE PRECISION                               Output
+               On exit: the value of the quadratic objective function at x
+               if x is feasible (IFAIL <= 5), or the sum of infeasibilities
+               at x otherwise (6 <= IFAIL <= 8).
+
+          24:  CLAMDA(NCTOTL) -- DOUBLE PRECISION array              Output
+               On exit: the values of the Lagrange multiplier for each
+               constraint with respect to the current working set. The
+               ordering of CLAMDA is as follows; the first n components
+               contain the multipliers for the bound constraints on the
+               variables, and the remaining components contain the
+               multipliers for the general linear constraints. If ISTATE(j)
+               = 0 (i.e.,constraint j is not in the working set), CLAMDA(j)
+               is zero. If x is optimal and ISTATE(j) = 1, CLAMDA(j) should
+               be non-negative; if ISTATE(j) = 2, CLAMDA(j) should be non-
+               positive.
+
+          25:  IWORK(LIWORK) -- INTEGER array                     Workspace
+
+          26:  LIWORK -- INTEGER                                      Input
+               On entry:
+               the dimension of the array IWORK as declared in the
+               (sub)program from which E04NAF is called.
+               Constraint: LIWORK>=2*N.
+
+          27:  WORK(LWORK) -- DOUBLE PRECISION array              Workspace
+
+          28:  LWORK -- INTEGER                                       Input
+               On entry:
+               the dimension of the array WORK as declared in the
+               (sub)program from which E04NAF is called.
+               Constrai if LP = .FALSE. or NCLIN >= N then
+               nts:               2
+                        LWORK>=2*N +4*N*NCLIN+NROWA.
+
+                        if LP = .TRUE. and NCLIN < N then
+                                          2
+                        LWORK>=2*(NCLIN+1) +4*N+2*NCLIN+NROWA.
+               If MSGLVL > 0, the amount of workspace provided and the
+               amount of workspace required are output on the current
+               advisory message unit (as defined by X04ABF). As an
+               alternative to computing LWORK from the formula given above,
+               the user may prefer to obtain an appropriate value from the
+               output of a preliminary run with a positive value of MSGLVL
+               and LWORK set to 1 (E04NAF will then terminate with IFAIL =
+               9).
+
+          29:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit. To suppress
+               the output of an error message when soft failure occurs, set
+               IFAIL to 1.
+
+               IFAIL contains zero on exit if x is a strong local minimum.
+               i.e., the projected gradient is neglible, the Lagrange
+               multipliers are optimal, and the projected Hessian is
+               positive-definite. In some cases, a zero value of IFAIL
+               means that x is a global minimum (e.g. when the Hessian
+               matrix is positive-definite).
+
+          5.1. Description of the Printed Output
+
+          When MSGLVL >= 5, a line of output is produced for every change
+          in the working set (thus, several lines may be printed during a
+          single iteration).
+
+          To aid interpretation of the printed results, we mention the
+          convention for numbering the constraints: indices 1 through to n
+          refer to the bounds on the variables, and when NCLIN > 0 indices
+          n+1 through to n + NCLIN refer to the general constraints. When
+          the status of a constraint changes, the index of the constraint
+          is printed, along with the designation L (lower bound), U (upper
+          bound) or E (equality).
+
+          In the LP phase, the printout includes the following:
+
+          ITN            is the iteration count.
+
+          JDEL           is the index of the constraint deleted from the
+                         working set. If JDEL is zero, no constraint was
+                         deleted.
+
+          JADD           is the index of the constraint added to the
+                         working set. If JADD is zero, no constraint was
+                         added.
+
+          STEP           is the step taken along the computed search
+                         direction.
+
+          COND T         is a lower bound on the condition number of the
+                         matrix of predicted active constraints.
+
+          NUMINF         is the number of violated constraints
+                         (infeasibilities).
+
+          SUMINF         is a weighted sum of the magnitudes of the
+                         constraint violations.
+
+                                                                        T
+          LPOBJ          is the value of the linear objective function c x.
+                         It is printed only if LP = .TRUE..
+
+          During the QP phase, the printout includes the following:
+
+          ITN            is the iteration count (reset to zero after the LP
+                         phase).
+
+          JDEL           is the index of the constraint deleted from the
+                         working set. If JDEL is zero, no constraint was
+                         deleted.
+
+          JADD           is the index of the constraint added to the
+                         working set. If JADD is zero, no constraint was
+                         added.
+
+          STEP           is the step (alpha)  taken along the direction of
+                                            k
+                         search (if STEP is 1.0, the current point is a
+                         minimum in the subspace defined by the current
+                         working set).
+
+          NHESS          is the number of calls to subroutine QPHESS.
+
+          OBJECTIVE      is the value of the quadratic objective function.
+
+          NCOLZ          is the number of columns of Z (see Section 3). In
+                         general, it is the dimension of the subspace in
+                         which the quadratic is currently being minimized.
+
+          NORM GFREE     is the Euclidean norm of the gradient of the
+                         objective function with respect to the free
+                         variables, i.e. variables not currently held at a
+                         bound (NORM GFREE is not printed if ORTHOG = .
+                         FALSE.). In some cases, the objective function and
+                         gradient are updated rather than recomputed. If
+                         so, this entry will be -- to indicate that the
+                         gradient with respect to the free variables has
+                         not been computed.
+
+          NORM QTG       is a weighted norm of the gradient of the
+                         objective function with respect to the free
+                         variables (NORM QTG is not printed if ORTHOG = .
+                         TRUE.). In some cases, the objective function and
+                         gradient are updated rather than recomputed. If
+                         so, this entry will be -- to indicate that the
+                         gradient with respect to the free variables has
+                         not been computed.
+
+          NORM ZTG       is the Euclidean norm of the projected gradient
+                         (see Section 3).
+
+          COND T         is a lower bound on the condition number of the
+                         matrix of constraints in the working set.
+
+          COND ZHZ       is a lower bound on the condition number of the
+                         projected Hessian matrix.
+
+          HESS MOD       is the correction added to the diagonal of the
+                         projected Hessian to ensure that a satisfactory
+                         Cholesky factorization exists (see Section 3).
+                         When the projected Hessian is sufficiently
+                         positive-definite, HESS MOD will be zero.
+
+          When MSGLVL = 1 or MSGLVL >= 10, the summary printout at the end
+          of execution of E04NAF includes a listing of the status of every
+          constraint. Note that default names are assigned to all variables
+          and constraints.
+
+          The following describes the printout for each variable.
+
+          VARBL          is the name (V) and index j, j=1,2,...,n, of the
+                         variable.
+
+          STATE          gives the state of the variable (FR if neither
+                         bound is in the working set, EQ if a fixed
+                         variable, LL if on its lower bound, UL if on its
+                         upper bound, TB if held on a temporary bound). If
+                         VALUE lies outside the upper or lower bounds by
+                         more than FEATOL(j), STATE will be ++ or --
+                         respectively.
+
+          VALUE          is the value of the variable at the final
+                         iteration.
+
+          LOWER BOUND    is the lower bound specified for the variable.
+
+          UPPER BOUND    is the upper bound specified for the variable.
+
+          LAGR MULT      is the value of the Lagrange multiplier for the
+                         associated bound constraint. This will be zero if
+                         STATE is FR. If x is optimal and STATE is LL, the
+                         multiplier should be non-negative; if STATE is UL,
+                         the multiplier should be non-positive.
+
+          RESIDUAL       is the difference between the variable and the
+                         nearer of its bounds BL(j) and BU(j).
+
+          For each of the general constraints the printout is as above with
+          refers to the jth element of Ax, except that VARBL is replaced by
+
+          LNCON          The name (L) and index j, j=1,2,...,NCLIN, of the
+                         constraint.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          IFAIL= 1
+               x is a weak local minimum (the projected gradient is
+               negligible, the Lagrange multipliers are optimal, but the
+               projected Hessian is only semi-definite). This means that
+               the solution is not unique.
+
+          IFAIL= 2
+               The solution appears to be unbounded, i.e., the quadratic
+               function is unbounded below in the feasible region. This
+               value of IFAIL occurs when a step of infinity would have to
+               be taken in order to continue the algorithm.
+
+          IFAIL= 3
+               x appears to be a local minimum, but optimality cannot be
+               verified because some of the Lagrange multipliers are very
+               small in magnitude.
+
+               E04NAF has probably found a solution. However, the presence
+               of very small Lagrange multipliers means that the predicted
+               active set may be incorrect, or that x may be only a
+               constrained stationary point rather than a local minimum.
+               The method in E04NAF is not guaranteed to find the correct
+               active set when there are very small multipliers. E04NAF
+               attempts to delete constraints with zero multipliers, but
+               this does not necessarily resolve the issue. The
+               determination of the correct active set is a combinatorial
+               problem that may require an extremely large amount of time.
+               The occurrence of small multipliers often (but not always)
+               indicates that there are redundant constraints.
+
+          IFAIL= 4
+               The iterates of the QP phase could be cycling, since a total
+               of 50 changes were made to the working set without altering
+               x.
+
+               This value will occur if 50 iterations are performed in the
+               QP phase without changing x. The user should check the
+               printed output for a repeated pattern of constraint
+               deletions and additions. If a sequence of constraint changes
+               is being repeated, the iterates are probably cycling.
+               (E04NAF  does not contain a method that is guaranteed to
+               avoid cycling, which would be combinatorial in nature.)
+               Cycling may occur in two circumstances: at a constrained
+               stationary point where there are some small or zero Lagrange
+               multipliers (see the discussion of IFAIL = 3); or at a point
+               (usually a vertex) where the constraints that are satisfied
+               exactly are nearly linearly dependent. In the latter case,
+               the user has the option of identifying the offending
+               dependent constraints and removing them from the problem, or
+               restarting the run with larger values of FEATOL for nearly
+               dependent constraints. If E04NAF terminates with IFAIL = 4,
+               but no suspicious pattern of constraint changes can be
+               observed, it may be worthwhile to restart with the final x
+               (with or without the warm start option).
+
+          IFAIL= 5
+               The limit of ITMAX iterations was reached in the QP phase
+               before normal termination occurred.
+
+               The value of ITMAX may be too small. If the method appears
+               to be making progress (e.g. the objective function is being
+               satisfactorily reduced), increase ITMAX and rerun E04NAF
+               (possibly using the warm start facility to specify the
+               initial working set). If ITMAX is already large, but some of
+               the constraints could be nearly linearly dependent, check
+               the output for a repeated pattern of constraints entering
+               and leaving the working set. (Near-dependencies are often
+               indicated by wide variations in size in the diagonal
+               elements of the T matrix, which will be printed if MSGLVL >=
+               30.) In this case, the algorithm could be cycling (see the
+               comments for IFAIL = 4).
+
+          IFAIL= 6
+               The LP phase terminated without finding a feasible point,
+               and hence it is not possible to satisfy all the constraints
+               to within the tolerances specified by the FEATOL array. In
+               this case, the final iterate will reveal values for which
+               there will be a feasible point (e.g. a feasible point will
+               exist if the feasibility tolerance for each violated
+               constraint exceeds its RESIDUAL at the final point). The
+               modified problem (with altered values in FEATOL) may then be
+               solved using a warm start.
+
+               The user should check that there are no constraint
+               redundancies. If the data for the jth constraint are
+               accurate only to the absolute precision (delta), the user
+               should ensure that the value of FEATOL(j) is greater than
+               (delta). For example, if all elements of A are of order
+               unity and are accurate only to three decimal places, every
+                                                        -3
+               component of FEATOL should be at least 10  .
+
+          IFAIL= 7
+               The iterates may be cycling during the LP phase; see the
+               comments above under IFAIL = 4.
+
+          IFAIL= 8
+               The limit of ITMAX iterations was reached during the LP
+               phase. See comments above under IFAIL = 5.
+
+          IFAIL= 9
+               An input parameter is invalid.
+
+          Overflow
+               If the printed output before the overflow error contains a
+               warning about serious ill-conditioning in the working set
+               when adding the jth constraint, it may be possible to avoid
+               the difficulty by increasing the magnitude of FEATOL(j) and
+               rerunning the program. If the message recurs even after this
+               change, the offending linearly dependent constraint (with
+               index j) must be removed from the problem. If a warning
+               message did not precede the fatal overflow, the user should
+               contact NAG.
+
+          7. Accuracy
+
+          The routine implements a numerically stable active set strategy
+          and returns solutions that are as accurate as the condition of
+          the QP problem warrants on the machine.
+
+          8. Further Comments
+
+          The number of iterations depends upon factors such as the number
+          of variables and the distances of the starting point from the
+          solution. The number of operations performed per iteration is
+                                         2
+          roughly proportional to (NFREE) , where NFREE (NFREE<=n) is the
+          number of variables fixed on their upper or lower bounds.
+
+          Sensible scaling of the problem is likely to reduce the number of
+          iterations required and make the problem less sensitive to
+          perturbations in the data, thus improving the condition of the QP
+          problem. See the Chapter Introduction and Gill et al [1] for
+          further information and advice.
+
+          9. Example
+
+                                    T   1 T
+          To minimize the function c x+ -x Hx, where
+                                        2
+
+                                                             T
+                      c=[-0.02,-0.2,-0.2,-0.2,-0.2,0.04,0.04]
+
+                                   [2 0 0 0 0  0  0]
+                                   [0 2 0 0 0  0  0]
+                                   [0 0 2 2 0  0  0]
+                                 H=[0 0 2 2 0  0  0]
+                                   [0 0 0 0 2  0  0]
+                                   [0 0 0 0 0 -2 -2]
+                                   [0 0 0 0 0 -2 -2]
+
+          subject to the bounds
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe04ucf}{NAG On-line Documentation: e04ucf}
+\beginscroll
+\begin{verbatim}
+
+   
+   
+   
+   E04UCF(3NAG)                 E04UCF                  E04UCF(3NAG)
+   
+   
+   
+        E04 -- Minimizing or Maximizing a Function                 E04UCF
+                E04UCF -- NAG Foundation Library Routine Document
+   
+        Note: Before using this routine, please read the Users' Note for
+        your implementation to check implementation-dependent details.
+        The symbol (*) after a NAG routine name denotes a routine that is
+        not included in the Foundation Library.
+   
+        Note for users via the AXIOM system: the interface to this routine
+        has been enhanced for use with AXIOM and is slightly different to
+        that offered in the standard version of the Foundation Library.  In
+        particular, the optional parameters of the NAG routine are now
+        included in the parameter list.  These are described in section
+        5.1.2, below.
+   
+        1. Purpose
+   
+        E04UCF is designed to minimize an arbitrary smooth function
+        subject to constraints, which may include simple bounds on the
+        variables, linear constraints and smooth nonlinear constraints.
+        (E04UCF  may be used for unconstrained, bound-constrained and
+        linearly constrained optimization.) The user must provide
+        subroutines that define the objective and constraint functions
+        and as many of their first partial derivatives as possible.
+        Unspecified derivatives are approximated by finite differences.
+        All matrices are treated as dense, and hence E04UCF is not
+        intended for large sparse problems.
+   
+        E04UCF uses a sequential quadratic programming (SQP) algorithm in
+        which the search direction is the solution of a quadratic
+        programming (QP) problem. The algorithm treats bounds, linear
+        constraints and nonlinear constraints separately.
+   
+        2. Specification
+   
+               SUBROUTINE E04UCF (N, NCLIN, NCNLN, NROWA, NROWJ, NROWR,
+              1                   A, BL, BU, CONFUN, OBJFUN, ITER,
+              2                   ISTATE, C, CJAC, CLAMDA, OBJF, OBJGRD,
+              3                   R, X, IWORK, LIWORK, WORK, LWORK,
+              4                   IUSER, USER, STA, CRA, DER, FEA, FUN,
+              5                   HES, INFB, INFS, LINF, LINT, LIST,
+              6                   MAJI, MAJP, MINI, MINP, MON, NONF,
+              7                   OPT, STE, STAO, STAC, STOO, STOC, VE,
+              8                   IFAIL)
+               INTEGER          N, NCLIN, NCNLN, NROWA, NROWJ, NROWR,
+              1                 ITER, ISTATE(N+NCLIN+NCNLN), IWORK(LIWORK)
+              2                 , LIWORK, LWORK, IUSER(*), DER, MAJI,
+              3                 MAJP, MINI, MINP, MON, STAO, STAC, STOO,
+              4                 STOC, VE, IFAIL
+               DOUBLE PRECISION A(NROWA,*), BL(N+NCLIN+NCNLN), BU
+              1                 (N+NCLIN+NCNLN), C(*), CJAC(NROWJ,*),
+              2                 CLAMDA(N+NCLIN+NCNLN), OBJF, OBJGRD(N), R
+              3                 (NROWR,N), X(N), WORK(LWORK), USER(*),
+              4                 CRA, FEA, FUN,  INFB, INFS, LINF, LINT,
+              5                 NONF, OPT, STE
+               LOGICAL          LIST, STA, HES
+               EXTERNAL         CONFUN, OBJFUN
+   
+        3. Description
+   
+        E04UCF is designed to solve the nonlinear programming problem --
+        the minimization of a smooth nonlinear function subject to a set
+        of constraints on the variables. The problem is assumed to be
+        stated in the following form:
+   
+                                                   { x  }
+               Minimize      F(x)   subject to  l<={A x }<=u,         (1)
+                        n                          { L  }
+               x is in R                           {c(x)}
+   
+        where F(x), the objective function, is a nonlinear function, A
+                                                                      L
+        is an n  by n constant matrix, and c(x) is an n  element vector
+               L                                       N
+        of nonlinear constraint functions. (The matrix A  and the vector
+                                                        L
+        c(x) may be empty.) The objective function and the constraint
+        functions are assumed to be smooth, i.e., at least twice-
+        continuously differentiable. (The method of E04UCF will usually
+        solve (1) if there are only isolated discontinuities away from
+        the solution.)
+   
+        This routine is essentially identical to the subroutine SOL/NPSOL
+        described in Gill et al [8].
+   
+        Note that upper and lower bounds are specified for all the
+        variables and for all the constraints.
+   
+        An equality constraint can be specified by setting l =u . If
+                                                            i  i
+        certain bounds are not present, the associated elements of l or u
+        can be set to special values that will be treated as -infty or
+        +infty.
+   
+        If there are no nonlinear constraints in (1) and F is linear or
+        quadratic then one of E04MBF, E04NAF or E04NCF(*) will generally
+        be more efficient. If the problem is large and sparse the MINOS
+        package (see Murtagh and Saunders [13]) should be used, since
+        E04UCF treats all matrices as dense.
+   
+        The user must supply an initial estimate of the solution to (1),
+        together with subroutines that define F(x), c(x) and as many
+        first partial derivatives as possible; unspecified derivatives
+        are approximated by finite differences.
+   
+        The objective function is defined by subroutine OBJFUN, and the
+        nonlinear constraints are defined by subroutine CONFUN. On every
+        call, these subroutines must return appropriate values of the
+        objective and nonlinear constraints. The user should also provide
+        the available partial derivatives. Any unspecified derivatives
+        are approximated by finite differences; see Section 5.1 for a
+        discussion of the optional parameter Derivative Level. Just
+        before either OBJFUN or CONFUN is called, each element of the
+        current gradient array OBJGRD or CJAC is initialised to a special
+        value. On exit, any element that retains the value is estimated
+        by finite differences. Note that if there are nonlinear
+        costraints, then the first call to CONFUN will precede the first
+        call to OBJFUN.
+   
+        For maximum reliability, it is preferable for the user to provide
+        all partial derivatives (see Chapter 8 of Gill et al [10], for a
+        detailed discussion). If all gradients cannot be provided, it is
+        similarly advisable to provide as many as possible. While
+        developing the subroutines OBJFUN and CONFUN, the optional
+        parameter Verify (see Section 5.1) should be used to check the
+        calculation of any known gradients.
+   
+        E04UCF implements a sequential quadratic programming (SQP)
+        method. The document for E04NCF(*) should be consulted in
+        conjunction with this document.
+   
+        In the rest of this section we briefly summarize the main
+        features of the method of E04UCF. Where possible, explicit
+        reference is made to the names of variables that are parameters
+        of subroutines E04UCF or appear in the printed output (see
+        Section 5.2).
+   
+        At a solution of (1), some of the constraints will be active,
+        i.e., satisfied exactly. An active simple bound constraint
+        implies that the corresponding variable is fixed at its bound,
+        and hence the variables are partitioned into fixed and free
+        variables. Let C denote the m by n matrix of gradients of the
+        active general linear and nonlinear constraints. The number of
+        fixed variables will be denoted by n  , with n   (n  =n-n  ) the
+                                            FX        FR   FR    FX
+        number of free variables. The subscripts 'FX' and 'FR' on a
+        vector or matrix will denote the vector or matrix composed of the
+        components corresponding to fixed or free variables.
+   
+        A point x is a first-order Kuhn-Tucker point for (1) (see, e.g.,
+        Powell [14]) if the following conditions hold:
+   
+             (i) x is feasible;
+   
+             (ii) there exist vectors (xi) and (lambda) (the Lagrange
+             multiplier vectors for the bound and general constraints)
+             such that
+                                  T
+                               g=C (lambda)+(xi),                    (2)
+             where g is the gradient of F evaluated at x, and (xi) =0 if
+                                                                  j
+             the jth variable is free.
+   
+             (iii) The Lagrange multiplier corresponding to an
+             inequality constraint active at its lower bound must be
+             non-negative, and non-positive for an inequality constraint
+             active at its upper bound.
+   
+        Let Z denote a matrix whose columns form a basis for the set of
+        vectors orthogonal to the rows of C  ; i.e., C  Z=0. An
+                                           FR         FR
+        equivalent statement of the condition (2) in terms of Z is
+   
+                                     T
+                                    Z g  =0.
+                                       FR
+   
+                    T
+        The vector Z g   is termed the projected gradient of F at x.
+                      FR
+        Certain additional conditions must be satisfied in order for a
+        first-order Kuhn-Tucker point to be a solution of (1) (see, e.g.,
+        Powell [14]).
+   
+        The method of E04UCF is a sequential quadratic programming (SQP)
+        method. For an overview of SQP methods, see, for example,
+        Fletcher [5], Gill et al [10] and Powell [15].
+   
+        The basic structure of E04UCF involves major and minor
+        iterations. The major iterations generate a sequence of iterates
+                               *
+        {x } that converge to x , a first-order Kuhn-Tucker point of (1).
+          k
+                                                      _
+        At a typical major iteration, the new iterate x is defined by
+   
+                                _
+                                x=x+(alpha)p                          (3)
+   
+        where x is the current iterate, the non-negative scalar (alpha)
+        is the step length, and p is the search direction. (For
+        simplicity, we shall always consider a typical iteration and
+        avoid reference to the index of the iteration.) Also associated
+        with each major iteration are estimates of the Lagrange
+        multipliers and a prediction of the active set.
+   
+        The search direction p in (3) is the solution of a quadratic
+        programming subproblem of the form
+   
+                         T   1 T                  _  { p }  _
+              Minimize  g p+ -p Hp,   subject to  l<={A p}<=u,        (4)
+               p             2                       { L }
+                                                     {A p}
+                                                     { N }
+   
+        where g is the gradient of F at x, the matrix H is a positive-
+        definite quasi-Newton approximation to the Hessian of the
+        Lagrangian function (see Section 8.3), and A  is the Jacobian
+                                                    N
+        matrix of c evaluated at x. (Finite-difference estimates may be
+        used for g and A ; see the optional parameter Derivative Level in
+                        N
+        Section 5.1.) Let l in (1) be partitioned into three sections:
+        l , l  and l , corresponding to the bound, linear and nonlinear
+         B   L      N
+                                _
+        constraints. The vector l in (4) is similarly partitioned, and is
+        defined as
+   
+                       _         _               _
+                       l =l -x,  l =l -A x,  and l =l -c,
+                        B  B      L  L  L         N  N
+   
+        where c is the vector of nonlinear constraints evaluated at x.
+                   _
+        The vector u is defined in an analogous fashion.
+   
+        The estimated Lagrange multipliers at each major iteration are
+        the Lagrange multipliers from the subproblem (4) (and similarly
+        for the predicted active set). (The numbers of bounds, general
+        linear and nonlinear constraints in the QP active set are the
+        quantities Bnd, Lin and Nln in the printed output of E04UCF.) In
+        E04UCF, (4) is solved using E04NCF(*). Since solving a quadratic
+        program as an iterative procedure, the minor iterations of E04UCF
+        are the iterations of E04NCF(*). (More details about solving the
+        subproblem are given in Section 8.1.)
+   
+        Certain matrices associated with the QP subproblem are relevant
+        in the major iterations. Let the subscripts 'FX' and 'FR' refer
+        to the predicted fixed and free variables, and let C denote the m
+        by n matrix of gradients of the general linear and nonlinear
+        constraints in the predicted active set. First, we have available
+        the TQ factorization of C  :
+                                 FR
+   
+                               C  Q  =(0 T),                          (5)
+                                FR FR
+   
+        where T is a nonsingular m by m reverse-triangular matrix (i.e.,
+        t  =0 if i+j<m), and the non-singular n   by n   matrix Q   is
+         ij                                    FR     FR         FR
+        the product of orthogonal transformations (see Gill et al [6]).
+        Second, we have the upper-triangular Cholesky factor R of the
+        transformed and re-ordered Hessian matrix
+   
+                                T       T~
+                               R R=H ==Q HQ,                          (6)
+                                    Q
+   
+              ~
+        where H is the Hessian H with rows and columns permuted so that
+        the free variables are first, and Q is the n by n matrix
+   
+                                   (Q    )
+                                   ( FR  )
+                                 Q=(  I  ),                           (7)
+                                   (   FX)
+   
+        with I   the identity matrix of order n  . If the columns of Q
+              FX                               FX                     FR
+        are partitioned so that
+   
+                                   Q  =(Z Y),
+                                    FR
+   
+        the n (n ==n  -m) columns of Z form a basis for the null space of
+             z  z   FR
+                                                                     T
+        C  . The matrix Z is used to compute the projected gradient Z g
+         FR                                                            FR
+        at the current iterate. (The values Nz, Norm Gf and Norm Gz
+                                                            T
+        printed by E04UCF give n  and the norms of g   and Z g  .)
+                                z                   FR        FR
+   
+        A theoretical characteristic of SQP methods is that the predicted
+        active set from the QP subproblem (4) is identical to the correct
+                                          *
+        active set in a neighbourhood of x . In E04UCF, this feature is
+        exploited by using the QP active set from the previous iteration
+        as a prediction of the active set for the next QP subproblem,
+        which leads in practice to optimality of the subproblems in only
+        one iteration as the solution is approached. Separate treatment
+        of bound and linear constraints in E04UCF also saves computation
+        in factorizing C   and H .
+                        FR      Q
+   
+        Once p has been computed, the major iteration proceeds by
+        determining a step length (alpha) that produces a 'sufficient
+        decrease' in an augmented Lagrangian merit function (see Section
+        8.2). Finally, the approximation to the transformed Hessian
+        matrix H  is updated using a modified BFGS quasi-Newton update
+                Q
+        (see Section 8.3) to incorporate new curvature information
+                                       _
+        obtained in the move from x to x.
+   
+        On entry to E04UCF, an iterative procedure from E04NCF(*) is
+        executed, starting with the user-provided initial point, to find
+        a point that is feasible with respect to the bounds and linear
+        constraints (using the tolerance specified by Linear Feasibility
+        Tolerance see Section 5.1). If no feasible point exists for the
+        bound and linear constraints, (1) has no solution and E04UCF
+        terminates. Otherwise, the problem functions will thereafter be
+        evaluated only at points that are feasible with respect to the
+        bounds and linear constraints. The only exception involves
+        variables whose bounds differ by an amount comparable to the
+        finite-difference interval (see the discussion of Difference
+        Interval in Section 5.1). In contrast to the bounds and linear
+        constraints, it must be emphasised that the nonlinear constraints
+        will not generally be satisfied until an optimal point is
+        reached.
+   
+        Facilities are provided to check whether the user-provided
+        gradients appear to be correct (see the optional parameter Verify
+        in Section 5.1). In general, the check is provided at the first
+        point that is feasible with respect to the linear constraints and
+        bounds. However, the user may request that the check be performed
+        at the initial point.
+   
+        In summary, the method of E04UCF first determines a point that
+        satisfies the bound and linear constraints. Thereafter, each
+        iteration includes:
+   
+        (a)   the solution of a quadratic programming subproblem;
+   
+        (b)   a linesearch with an augmented Lagrangian merit function;
+              and
+   
+        (c)   a quasi-Newton update of the approximate Hessian of the
+              Lagrangian function.
+   
+        These three procedures are described in more detail in Section 8.
+   
+        4. References
+   
+        [1]   Dennis J E Jr and More J J (1977) Quasi-Newton Methods,
+              Motivation and Theory. SIAM Review. 19 46--89.
+   
+        [2]   Dennis J E Jr and Schnabel R B (1981) A New Derivation of
+              Symmetric Positive-Definite Secant Updates. Nonlinear
+              Programming 4. (ed O L Mangasarian, R R Meyer and S M.
+              Robinson) Academic Press. 167--199.
+   
+        [3]   Dennis J E Jr and Schnabel R B (1983) Numerical Methods for
+              Unconstrained Optimixation and Nonlinear Equations.
+              Prentice-Hall.
+   
+        [4]   Dongarra J J, Du Croz J J, Hammarling S and Hanson R J
+              (1985) A Proposal for an Extended set of Fortran Basic
+              Linear Algebra Subprograms. SIGNUM Newsletter. 20 (1) 2--18.
+   
+        [5]   Fletcher R (1981) Practical Methods of Optimization, Vol 2.
+              Constrained Optimization. Wiley.
+   
+        [6]   Gill P E, Murray W, Saunders M A and Wright M H (1984)
+              User's Guide for SOL/QPSOL Version 3.2. Report SOL 84-5.
+              Department of Operations Research, Stanford University.
+   
+        [7]   Gill P E, Murray W, Saunders M A and Wright M H (1984)
+              Procedures for Optimization Problems with a Mixture of
+              Bounds and General Linear Constraints. ACM Trans. Math.
+              Softw. 10 282--298.
+   
+        [8]   Gill P E, Hammarling S, Murray W, Saunders M A and Wright M
+              H (1986) User's Guide for LSSOL (Version 1.0). Report SOL
+              86-1. Department of Operations Research, Stanford
+              University.
+   
+        [9]   Gill P E, Murray W, Saunders M A and Wright M H (1986) Some
+              Theoretical Properties of an Augmented Lagrangian Merit
+              Function. Report SOL 86-6R. Department of Operations
+              Research, Stanford University.
+   
+        [10]  Gill P E, Murray W and Wright M H (1981) Practical
+              Optimization. Academic Press.
+   
+        [11]  Hock W and Schittkowski K (1981) Test Examples for Nonlinear
+              Programming Codes. Lecture Notes in Economics and
+              Mathematical Systems. 187 Springer-Verlag.
+   
+        [12]  Lawson C L, Hanson R J, Kincaid D R and Krogh F T (1979)
+              Basic Linear Algebra Subprograms for Fortran Usage. ACM
+              Trans. Math. Softw. 5 308--325.
+   
+        [13]  Murtagh B A and Saunders M A (1983) MINOS 5.0 User's Guide.
+              Report SOL 83-20. Department of Operations Research,
+              Stanford University.
+   
+        [14]  Powell M J D (1974) Introduction to Constrained
+              Optimization. Numerical Methods for Constrained
+              Optimization. (ed P E Gill and W Murray) Academic Press. 1--
+              28.
+   
+        [15]  Powell M J D (1983) Variable Metric Methods in Constrained
+              Optimization. Mathematical Programming: The State of the
+              Art. (ed A Bachem, M Groetschel and B Korte) Springer-
+              Verlag. 288--311.
+   
+        5. Parameters
+   
+         1:  N -- INTEGER                                           Input
+             On entry: the number, n, of variables in the problem.
+             Constraint: N > 0.
+   
+         2:  NCLIN -- INTEGER                                       Input
+             On entry: the number, n , of general linear constraints in
+                                    L
+             the problem. Constraint: NCLIN >= 0.
+   
+         3:  NCNLN -- INTEGER                                       Input
+             On entry: the number, n , of nonlinear constraints in the
+                                    N
+             problem. Constraint: NCNLN >= 0.
+   
+         4:  NROWA -- INTEGER                                       Input
+             On entry:
+             the first dimension of the array A as declared in the
+             (sub)program from which E04UCF is called.
+             Constraint: NROWA >= max(1,NCLIN).
+   
+         5:  NROWJ -- INTEGER                                       Input
+             On entry:
+             the first dimension of the array CJAC as declared in the
+             (sub)program from which E04UCF is called.
+             Constraint: NROWJ >= max(1,NCNLN).
+   
+         6:  NROWR -- INTEGER                                       Input
+             On entry:
+             the first dimension of the array R as declared in the
+             (sub)program from which E04UCF is called.
+             Constraint: NROWR >= N.
+   
+         7:  A(NROWA,*) -- DOUBLE PRECISION array                   Input
+             The second dimension of the array A must be >= N for NCLIN >
+             0. On entry: the ith row of the array A must contain the ith
+             row of the matrix A  of general linear constraints in (1).
+                                L
+             That is, the ith row contains the coefficients of the ith
+             general linear constraint, for i = 1,2,...,NCLIN.
+   
+             If NCLIN = 0 then the array A is not referenced.
+   
+         8:  BL(N+NCLIN+NCNLN) -- DOUBLE PRECISION array            Input
+             On entry: the lower bounds for all the constraints, in the
+             following order. The first n elements of BL must contain the
+             lower bounds on the variables. If NCLIN > 0, the next n
+                                                                    L
+             elements of BL must contain the lower bounds on the general
+             linear constraints. If NCNLN > 0, the next n  elements of BL
+                                                         N
+             must contain the lower bounds for the nonlinear constraints.
+             To specify a non-existent lower bound (i.e., l =-infty), the
+                                                           j
+             value used must satisfy BL(j)<=-BIGBND, where BIGBND is the
+             value of the optional parameter Infinite Bound Size whose
+                                10
+             default value is 10   (see Section 5.1). To specify the jth
+             constraint as an equality, the user must set BL(j) = BU(j) =
+             (beta), say, where |(beta)|<BIGBND. Constraint: BL(j) <= BU(
+             j), for j=1,2,...,N+NCLIN+NCNLN.
+   
+         9:  BU(N+NCLIN+NCNLN) -- DOUBLE PRECISION array            Input
+             On entry: the upper bounds for all the constraints in the
+             following order. The first n elements of BU must contain the
+             upper bounds on the variables. If NCLIN > 0, the next n
+                                                                    L
+             elements of BU must contain the upper bounds on the general
+             linear constraints. If NCNLN > 0, the next n  elements of BU
+                                                         N
+             must contain the upper bounds for the nonlinear constraints.
+             To specify a non-existent upper bound (i.e., u =+infty), the
+                                                           j
+             value used must satisfy BU(j) >= BIGBND, where BIGBND is the
+             value of the optional parameter Infinite Bound Size, whose
+                                10
+             default value is 10   (see Section 5.1). To specify the jth
+             constraint as an equality, the user must set BU(j) = BL(j) =
+             (beta), say, where |(beta)| < BIGBND. Constraint: BU(j) >=
+             BL(j), for j=1,2,...,N+NCLIN+NCNLN.
+   
+        10:  CONFUN -- SUBROUTINE, supplied by the user.
+                                                       External Procedure
+             CONFUN must calculate the vector c(x) of nonlinear
+             constraint functions and (optionally) its Jacobian for a
+             specified n element vector x. If there are no nonlinear
+             constraints (NCNLN=0), CONFUN will never be called by E04UCF
+             and CONFUN may be the dummy routine E04UDM. (E04UDM is
+             included in the NAG Foundation Library and so need not be
+             supplied by the user. Its name may be implementation-
+             dependent: see the Users' Note for your implementation for
+             details.) If there are nonlinear constraints, the first call
+             to CONFUN will occur before the first call to OBJFUN.
+   
+             Its specification is:
+   
+                    SUBROUTINE CONFUN (MODE, NCNLN, N, NROWJ, NEEDC,
+                   1                   X, C, CJAC, NSTATE, IUSER,
+                   2                   USER)
+                    INTEGER          MODE, NCNLN, N, NROWJ, NEEDC
+                   1                 (NCNLN), NSTATE, IUSER(*)
+                    DOUBLE PRECISION X(N), C(NCNLN), CJAC(NROWJ,N),
+                   1                 USER(*)
+   
+              1:  MODE -- INTEGER                            Input/Output
+                  On entry: MODE indicates the values that must be
+                  assigned during each call of CONFUN. MODE will always
+                  have the value 2 if all elements of the Jacobian are
+                  available, i.e., if Derivative Level is either 2 or 3
+                  (see Section 5.1). If some elements of CJAC are
+                  unspecified, E04UCF will call CONFUN with MODE = 0, 1,
+                  or 2:
+   
+                  If MODE = 2, only the elements of C corresponding to
+                  positive values of NEEDC must be set (and similarly for
+                  the available components of the rows of CJAC).
+   
+                  If MODE = 1, the available components of the rows of
+                  CJAC corresponding to positive values in NEEDC must be
+                  set. Other rows of CJAC and the array C will be
+                  ignored.
+   
+                  If MODE = 0, the components of C corresponding to
+                  positive values in NEEDC must be set. Other components
+                  and the array CJAC are ignored. On exit: MODE may be
+                  set to a negative value if the user wishes to terminate
+                  the solution to the current problem. If MODE is
+                  negative on exit from CONFUN then E04UCF will terminate
+                  with IFAIL set to MODE.
+   
+              2:  NCNLN -- INTEGER                                  Input
+                  On entry: the number, n , of nonlinear constraints.
+                                         N
+   
+              3:  N -- INTEGER                                      Input
+                  On entry: the number, n, of variables.
+   
+              4:  NROWJ -- INTEGER                                  Input
+                  On entry: the first dimension of the array CJAC.
+   
+              5:  NEEDC(NCNLN) -- INTEGER array                     Input
+                  On entry: the indices of the elements of C or CJAC that
+                  must be evaluated by CONFUN. If NEEDC(i)>0 then the ith
+                  element of C and/or the ith row of CJAC (see parameter
+                  MODE above) must be evaluated at x.
+   
+              6:  X(N) -- DOUBLE PRECISION array                    Input
+                  On entry: the vector x of variables at which the
+                  constraint functions are to be evaluated.
+   
+              7:  C(NCNLN) -- DOUBLE PRECISION array               Output
+                  On exit: if NEEDC(i)>0 and MODE = 0 or 2, C(i) must
+                  contain the value of the ith constraint at x. The
+                  remaining components of C, corresponding to the non-
+                  positive elements of NEEDC, are ignored.
+   
+              8:  CJAC(NROWJ,N) -- DOUBLE PRECISION array          Output
+                  On exit: if NEEDC(i)>0 and MODE = 1 or 2, the ith row
+                  of CJAC must contain the available components of the
+                  vector (nabla)c  given by
+                                 i
+                                      ( ddc   ddc       ddc )
+                                      (    i     i         i)T
+                            (nabla)c =( ----, ----,..., ----) ,
+                                    i ( ddx   ddx       ddx )
+                                      (    1     2         n)
+                         ddc
+                            i
+                  where  ---- is the partial derivative of the ith
+                         ddx
+                            j
+                  constraint with respect to the jth variable, evaluated
+                  at the point x. See also the parameter NSTATE below.
+                  The remaining rows of CJAC, corresponding to non-
+                  positive elements of NEEDC, are ignored.
+   
+                  If all constraint gradients (Jacobian elements) are
+                  known (i.e., Derivative Level = 2 or 3; see Section 5.1)
+                  any constant elements may be assigned to CJAC one
+                  time only at the start of the optimization. An element
+                  of CJAC that is not subsequently assigned in CONFUN
+                  will retain its initial value throughout. Constant
+                  elements may be loaded into CJAC either before the call
+                  to E04UCF or during the first call to CONFUN (signalled
+                  by the value NSTATE = 1). The ability to preload
+                  constants is useful when many Jacobian elements are
+                  identically zero, in which case CJAC may be initialised
+                  to zero and non-zero elements may be reset by CONFUN.
+   
+                  Note that constant non-zero elements do affect the
+                  values of the constraints. Thus, if CJAC(i,j) is set to
+                  a constant value, it need not be reset in subsequent
+                  calls to CONFUN, but the value CJAC(i,j)*X(j) must
+                  nonetheless be added to C(i).
+   
+                  It must be emphasized that, if Derivative Level < 2,
+                  unassigned elements of CJAC are not treated as
+                  constant; they are estimated by finite differences, at
+                  non-trivial expense. If the user does not supply a
+                  value for Difference Interval (see Section 5.1), an
+                  interval for each component of x is computed
+                  automatically at the start of the optimization. The
+                  automatic procedure can usually identify constant
+                  elements of CJAC, which are then computed once only by
+                  finite differences.
+   
+              9:  NSTATE -- INTEGER                                 Input
+                  On entry: if NSTATE = 1 then E04UCF is calling CONFUN
+                  for the first time. This parameter setting allows the
+                  user to save computation time if certain data must be
+                  read or calculated only once.
+   
+             10:  IUSER(*) -- INTEGER array                User Workspace
+   
+             11:  USER(*) -- DOUBLE PRECISION array        User Workspace
+                  CONFUN is called from E04UCF with the parameters IUSER
+                  and USER as supplied to E04UCF. The user is free to use
+                  the arrays IUSER and USER to supply information to
+                  CONFUN as an alternative to using COMMON.
+             CONFUN must be declared as EXTERNAL in the (sub)program
+             from which E04UCF is called. Parameters denoted as
+             Input must not be changed by this procedure.
+   
+        11:  OBJFUN -- SUBROUTINE, supplied by the user.
+                                                       External Procedure
+             OBJFUN must calculate the objective function F(x) and
+             (optionally) the gradient g(x) for a specified n element
+             vector x.
+   
+             Its specification is:
+   
+                    SUBROUTINE OBJFUN (MODE, N, X, OBJF, OBJGRD,
+                   1                   NSTATE, IUSER, USER)
+                    INTEGER          MODE, N, NSTATE, IUSER(*)
+                    DOUBLE PRECISION X(N), OBJF, OBJGRD(N), USER(*)
+   
+              1:  MODE -- INTEGER                            Input/Output
+                  On entry: MODE indicates the values that must be
+                  assigned during each call of OBJFUN.
+   
+                  MODE will always have the value 2 if all components of
+                  the objective gradient are specified by the user, i.e.,
+                  if Derivative Level is either 1 or 3. If some gradient
+                  elements are unspecified, E04UCF will call OBJFUN with
+                  MODE = 0, 1 or 2.
+                       If MODE = 2, compute OBJF and the available
+                       components of OBJGRD.
+   
+                       If MODE = 1, compute all available components of
+                       OBJGRD; OBJF is not required.
+   
+                       If MODE = 0, only OBJF needs to be computed;
+                       OBJGRD is ignored.
+                  On exit: MODE may be set to a negative value if the
+                  user wishes to terminate the solution to the current
+                  problem. If MODE is negative on exit from OBJFUN, then
+                  E04UCF will terminate with IFAIL set to MODE.
+   
+              2:  N -- INTEGER                                      Input
+                  On entry: the number, n, of variables.
+   
+              3:  X(N) -- DOUBLE PRECISION array                    Input
+                  On entry: the vector x of variables at which the
+                  objective function is to be evaluated.
+   
+              4:  OBJF -- DOUBLE PRECISION                         Output
+                  On exit: if MODE = 0 or 2, OBJF must be set to the
+                  value of the objective function at x.
+   
+              5:  OBJGRD(N) -- DOUBLE PRECISION array              Output
+                  On exit: if MODE = 1 or 2, OBJGRD must return the
+                  available components of the gradient evaluated at x.
+   
+              6:  NSTATE -- INTEGER                                 Input
+                  On entry: if NSTATE = 1 then E04UCF is calling OBJFUN
+                  for the first time. This parameter setting allows the
+                  user to save computation time if certain data must be
+                  read or calculated only once.
+   
+              7:  IUSER(*) -- INTEGER array                User Workspace
+   
+              8:  USER(*) -- DOUBLE PRECISION array        User Workspace
+                  OBJFUN is called from E04UCF with the parameters IUSER
+                  and USER as supplied to E04UCF. The user is free to use
+                  the arrays IUSER and USER to supply information to
+                  OBJFUN as an alternative to using COMMON.
+             OBJFUN must be declared as EXTERNAL in the (sub)program
+             from which E04UCF is called. Parameters denoted as
+             Input must not be changed by this procedure.
+   
+        12:  ITER -- INTEGER                                       Output
+             On exit: the number of iterations performed.
+   
+        13:  ISTATE(N+NCLIN+NCNLN) -- INTEGER array          Input/Output
+             On entry: ISTATE need not be initialised if E04UCF is called
+             with (the default) Cold Start option. The ordering of ISTATE
+             is as follows. The first n elements of ISTATE refer to the
+             upper and lower bounds on the variables, elements n+1
+             through n+n  refer to the upper and lower bounds on A x, and
+                        L                                         L
+             elements n+n +1 through n+n +n  refer to the upper and lower
+                         L              L  N
+             bounds on c(x). When a Warm Start option is chosen, the
+             elements of ISTATE corresponding to the bounds and linear
+             constraints define the initial working set for the procedure
+             that finds a feasible point for the linear constraints and
+             bounds. The active set at the conclusion of this procedure
+             and the elements of ISTATE corresponding to nonlinear
+             constraints then define the initial working set for the
+             first QP subproblem. Possible values for ISTATE(j) are:
+   
+             ISTATE(j) Meaning
+   
+             0         The corresponding constraint is not in the initial
+                       QP working set.
+   
+             1         This inequality constraint should be in the
+                       working set at its lower bound.
+   
+             2         This inequality constraint should be in the
+                       working set at its upper bound.
+   
+             3         This equality constraint should be in the initial
+                       working set. This value must not be specified
+                       unless BL(j) = BU(j). The values 1,2 or 3 all have
+                       the same effect when BL(j) = BU(j).
+             Note that if E04UCF has been called previously with the same
+             values of N, NCLIN and NCNLN, ISTATE already contains
+             satisfactory values. If necessary, E04UCF will override the
+             user's specification of ISTATE so that a poor choice will
+             not cause the algorithm to fail. On exit: with IFAIL = 0 or
+             1, the values in the array ISTATE correspond to the active
+             set of the final QP subproblem, and are a prediction of the
+             status of the constraints at the solution of the problem.
+             Otherwise, ISTATE indicates the composition of the QP
+             working set at the final iterate. The significance of each
+             possible value of ISTATE(j) is as follows:
+             -2        This constraint violates its lower bound by more
+                       than the appropriate feasibility tolerance (see
+                       the optional parameters LinearFeasibility
+                       Tolerance and Nonlinear Feasibility Tolerance in
+                       Section 5.1). This value can occur only when no
+                       feasible point can be found for a QP subproblem.
+   
+             -1        This constraint violates its upper bound by more
+                       than the appropriate feasibility tolerance (see
+                       the optional parameters Linearear Feasibility
+                       Tolerance and Nonlinear Feasibility Tolerance in
+                       Section 5.1). This value can occur only when no
+                       feasible point can be found for a QP subproblem.
+   
+             0         The constraint is satisfied to within the
+                       feasibility tolerance, but is not in the working
+                       set.
+   
+             1         This inequality constraint is included in the QP
+                       working set at its upper bound.
+   
+             2         This inequality constraint is included in the QP
+                       working set at its upper bound.
+   
+             3         This constraint is included in the QP working set
+                       as an equality. This value of ISTATE can occur
+                       only when BL(j) = BU(j).
+   
+        14:  C(*) -- DOUBLE PRECISION array                        Output
+             Note: the dimension of the array C must be at least
+             max(1,NCNLN).
+             On exit: if NCNLN > 0, C(i) contains the value of the ith
+             nonlinear constraint function c  at the final iterate, for
+                                            i
+             i=1,2,...,NCNLN. If NCNLN = 0, then the array C is not
+             referenced.
+   
+        15:  CJAC(NROWJ,*) -- DOUBLE PRECISION array         Input/Output
+             Note: the second dimension of the array CJAC must be at
+             least N for NCNLN >0 and 1 otherwise On entry: in general,
+             CJAC need not be initialised before the call to E04UCF.
+             However, if Derivative Level = 3, the user may optionally
+             set the constant elements of CJAC (see parameter NSTATE in
+             the description of CONFUN). Such constant elements need not
+             be re-assigned on subsequent calls to CONFUN. If NCNLN = 0,
+             then the array CJAC is not referenced. On exit: if NCNLN >
+             0, CJAC contains the Jacobian matrix of the nonlinear
+             constraint functions at the final iterate, i.e., CJAC(i,j)
+             contains the partial derivative of the ith constraint
+             function with respect to the jth variable, for i=1,2,...,
+             NCNLN; j = 1,2,...,N. (See the discussion of parameter CJAC
+             under CONFUN.)
+   
+        16:  CLAMDA(N+NCLIN+NCNLN) -- DOUBLE PRECISION array Input/Output
+             On entry: CLAMDA need not be initialised if E04UCF is called
+             with the (default) Cold Start option. With the Warm Start
+             option, CLAMDA must contain a multiplier estimate for each
+             nonlinear constraint with a sign that matches the status of
+             the constraint specified by the ISTATE array (as above). The
+             ordering of CLAMDA is as follows; the first n elements
+             contain the multipliers for the bound constraints on the
+             variables, elements n+1 through n+n  contain the multipliers
+                                                L
+             for the general linear constraints, and elements n+n +1
+                                                                 L
+             through n+n +n  contain the multipliers for the nonlinear
+                        L  N
+             constraints. If the jth constraint is defined as 'inactive'
+             by the initial value of the ISTATE array, CLAMDA(j) should
+             be zero; if the jth constraint is an inequality active at
+             its lower bound, CLAMDA(j) should be non-negative; if the j
+             th constraint is an inequality active at its upper bound,
+             CLAMDA(j) should be non-positive. On exit: the values of the
+             QP multipliers from the last QP subproblem. CLAMDA(j) should
+             be non-negative if ISTATE(j) = 1 and non-positive if ISTATE(
+             j) = 2.
+   
+        17:  OBJF -- DOUBLE PRECISION                              Output
+             On exit: the value of the objective function, F(x), at the
+             final iterate.
+   
+        18:  OBJGRD(N) -- DOUBLE PRECISION array                   Output
+             On exit: the gradient (or its finite-difference
+             approximation) of the objective function at the final
+             iterate.
+   
+        19:  R(NROWR,N) -- DOUBLE PRECISION array            Input/Output
+             On entry: R need not be initialised if E04UCF is called with
+             a Cold Start option (the default), and will be taken as the
+             identity. With a Warm Start R must contain the upper-
+             triangular Cholesky factor R of the initial approximation of
+             the Hessian of the Lagrangian function, with the variables
+             in the natural order. Elements not in the upper-triangular
+             part of R are assumed to be zero and need not be assigned.
+             On exit: if Hessian = No, (the default; see Section 5.1), R
+                                                                 T~
+             contains the upper-triangular Cholesky factor R of Q HQ, an
+             estimate of the transformed and re-ordered Hessian of the
+             Lagrangian at x (see (6) in Section 3). If Hessian = Yes, R
+             contains the upper-triangular Cholesky factor R of H, the
+             approximate (untransformed) Hessian of the Lagrangian, with
+             the variables in the natural order.
+   
+        20:  X(N) -- DOUBLE PRECISION array                  Input/Output
+             On entry: an initial estimate of the solution. On exit: the
+             final estimate of the solution.
+   
+        21:  IWORK(LIWORK) -- INTEGER array                     Workspace
+   
+        22:  LIWORK -- INTEGER                                      Input
+             On entry:
+             the dimension of the array IWORK as declared in the
+             (sub)program from which E04UCF is called.
+             Constraint: LIWORK>=3*N+NCLIN+2*NCNLN.
+   
+        23:  WORK(LWORK) -- DOUBLE PRECISION array              Workspace
+   
+        24:  LWORK -- INTEGER                                       Input
+             On entry:
+             the dimension of the array WORK as declared in the
+             (sub)program from which E04UCF is called.
+             Constraints:
+             if NCLIN = NCNLN = 0 then
+                   LWORK >=20*N
+   
+             if NCNLN = 0 and NCLIN > 0 then
+                              2
+                   LWORK >=2*N +20*N+11*NCLIN
+   
+             if NCNLN > 0 and NCLIN >= 0 then
+                             2
+                   LWORK>=2*N +N*NCLIN+20*N*NCNLN+20*N+ 11*NCLIN+21*NCNLN
+   
+             If Major Print Level > 0, the required amounts of workspace
+             are output on the current advisory message channel (see
+             X04ABF). As an alternative to computing LIWORK and LWORK
+             from the formulas given above, the user may prefer to obtain
+             appropriate values from the output of a preliminary run with
+             a positive value of Major Print Level and LIWORK and LWORK
+             set to 1. (E04UCF will then terminate with IFAIL = 9.)
+   
+        25:  IUSER(*) -- INTEGER array                     User Workspace
+             Note: the dimension of the array IUSER must be at least 1.
+             IUSER is not used by E04UCF, but is passed directly to
+             routines CONFUN and OBJFUN and may be used to pass
+             information to those routines.
+   
+        26:  USER(*) -- DOUBLE PRECISION array             User Workspace
+             Note: the dimension of the array USER must be at least 1.
+             USER is not used by E04UCF, but is passed directly to
+             routines CONFUN and OBJFUN and may be used to pass
+             information to those routines.
+   
+        27:  IFAIL -- INTEGER                                Input/Output
+             On entry: IFAIL must be set to 0, -1 or 1. Users who are
+             unfamiliar with this parameter should refer to the Essential
+             Introduction for details.
+   
+             On exit: IFAIL = 0 unless the routine detects an error or
+             gives a warning (see Section 6).
+   
+             For this routine, because the values of output parameters
+             may be useful even if IFAIL /=0 on exit, users are
+             recommended to set IFAIL to -1 before entry. It is then
+             essential to test the value of IFAIL on exit.
+   
+             E04UCF returns with IFAIL = 0 if the iterates have
+             converged to a point x that satisfies the first-order Kuhn-
+             Tucker conditions to the accuracy requested by the optional
+             parameter Optimality Tolerance (see Section 5.1), i.e., the
+             projected gradient and active constraint residuals are
+             negligible at x.
+   
+             The user should check whether the following four conditions
+             are satisfied:
+             (i)   the final value of Norm Gz is significantly less than
+                   that at the starting point;
+   
+             (ii)  during the final major iterations, the values of Step
+                   and ItQP are both one;
+   
+             (iii) the last few values of both Norm Gz and Norm C become
+                   small at a fast linear rate;
+   
+             (iv)  Cond Hz is small.
+             If all these conditions hold, x is almost certainly a local
+             minimum of (1). (See Section 9 for a specific example.)
+   
+        5.1. Optional Input Parameters
+   
+        Several optional parameters in E04UCF define choices in the
+        behaviour of the routine. In order to reduce the number of formal
+        parameters of E04UCF these optional parameters have associated
+        default values (see Section 5.1.3) that are appropriate for most
+        problems. Therefore the user need only specify those optional
+        parameters whose values are to be different from their default
+        values.
+   
+        The remainder of this section can be skipped by users who wish to
+        use the default values for all optional prameters. A complete
+        list of optional parameters and their default values is given in
+        Section 5.1.3
+   
+        5.1.1. Specification of the optional parameters
+   
+        Optional parameters may be specified by calling one, or both, of
+        E04UDF and E04UEF prior to a call to E04UCF.
+   
+        E04UDF reads options from an external options file, with Begin
+        and End as the first and last lines respectively and each
+        intermediate line defining a single optional parameter. For
+        example,
+   
+              Begin
+                Print Level = 1
+              End
+   
+        The call
+   
+              CALL E04UDF (IOPTNS, INFORM)
+   
+        can then be used to read the file on unit IOPTNS. INFORM will be
+        zero on successful exit. E04UDF should be consulted for a full
+        description of this method of supplying optional parameters.
+   
+        E04UEF can be called directly to supply options, one call being
+        necessary for each optional parameter. For example,
+   
+              CALL E04UEF (`Print level = 1')
+   
+        E04UEF should be consulted for a full description of this method
+        of supplying optional parameters.
+   
+        All optional parameters not specified by the user are set to
+        their default values. Optional parameters specified by the user
+        are unaltered by E04UCF (unless they define invalid values) and
+        so remain in effect for subsequent calls to E04UCF, unless
+        altered by the user.
+   
+        5.1.2. Description of the optional parameters
+   
+        The following list (in alphabetical order) gives the valid
+        options. For each option, we give the keyword, any essential
+        optional qualifiers, the default value, and the definition. The
+        minimum valid abbreviation of each keyword is underlined. If no
+        characters of an optional qualifier are underlined, the qualifier
+        may be omitted. The letter a denotes a phrase (character string)
+        that qualifies an option. The letters i and r denote INTEGER and
+        DOUBLE PRECISION values required with certain options. The number
+        (epsilon) is a generic notation for machine precision (see
+        X02AJF(*) ), and (epsilon)  denotes the relative precision of the
+                                  R
+        objective function (the optional parameter Function Precision see
+        below).
+   
+        Central Difference Interval r Default values are computed
+   
+        If the algorithm switches to central differences because the
+        forward-difference approximation is not sufficiently accurate,
+        the value of r is used as the difference interval for every
+        component of x. The use of finite-differences is discussed
+        further below under the optional parameter Difference Interval.
+   
+        Cold Start Default = Cold Start
+   
+        Warm Start
+   
+           (AXIOM parameter STA, warm start when .TRUE.)
+   
+        This option controls the specification of the initial working set
+        in both the procedure for finding a feasible point for the linear
+        constraints and bounds, and in the first QP subproblem
+        thereafter. With a Cold Start, the first working set is chosen by
+        E04UCF based on the values of the variables and constraints at
+        the initial point. Broadly speaking, the initial working set will
+        include equality constraints and bounds or inequality constraints
+        that violate or 'nearly' satisfy their bounds (within Crash
+        Tolerance; see below). With a Warm Start, the user must set the
+        ISTATE array and define CLAMDA and R as discussed in Section 5.
+        ISTATE values associated with bounds and linear constraints
+        determine the initial working set of the procedure to find a
+        feasible point with respect to the bounds and linear constraints.
+        ISTATE values associated with nonlinear constraints determine the
+        initial working set of the first QP subproblem after such a
+        feasible point has been found. E04UCF will override the user's
+        specification of ISTATE if necessary, so that a poor choice of
+        the working set will not cause a fatal error. A warm start will
+        be advantageous if a good estimate of the initial working set is
+        available - for example, when E04UCF is called repeatedly to
+        solve related problems.
+   
+        Crash Tolerance r Default = 0.01
+   
+           (AXIOM parameter CRA)
+   
+        This value is used in conjunction with the optional parameter
+        Cold Start (the default value). When making a cold start, the QP
+        algorithm in E04UCF must select an initial working set. When r>=0
+        , the initial working set will include (if possible) bounds or
+        general inequality constraints that lie within r of their bounds.
+                                                 T
+        In particular, a constraint of the form a x>=l will be included
+                                                 j
+                                        T
+        in the initial working set if |a x-l|<=r(1+|l|). If r<0 or r>1,
+                                        j
+        the default value is used.
+   
+        Defaults
+   
+        This special keyword may be used to reset the default values
+        following a call to E04UCF.
+   
+        Derivative Level i Default = 3
+   
+           (AXIOM parameter DER)
+   
+        This parameter indicates which derivatives are provided by the
+        user in subroutines OBJFUN and CONFUN. The possible choices for i
+        are the following.
+   
+            i    Meaning
+   
+            3    All objective and constraint gradients are provided by
+                 the user.
+   
+            2    All of the Jacobian is provided, but some components of
+                 the objective gradient are not specified by the user.
+   
+            1    All elements of the objective gradient are known, but
+                 some elements of the Jacobian matrix are not specified
+                 by the user.
+   
+            0    Some elements of both the objective gradient and the
+                 Jacobian matrix are not specified by the user.
+   
+        The value i=3 should be used whenever possible, since E04UCF is
+        more reliable and will usually be more efficient when all
+        derivatives are exact.
+   
+        If i=0 or 2, E04UCF will estimate the unspecified components of
+        the objective gradient, using finite differences. The computation
+        of finite-difference approximations usually increases the total
+        run-time, since a call to OBJFUN is required for each unspecified
+        element. Furthermore, less accuracy can be attained in the
+        solution (see Chapter 8 of Gill et al [10], for a discussion of
+        limiting accuracy).
+   
+        If i=0 or 1, E04UCF will approximate unspecified elements of the
+        Jacobian. One call to CONFUN is needed for each variable for
+        which partial derivatives are not available. For example, if the
+        Jacobian has the form
+   
+                                    (* * * *)
+                                    (* ? ? *)
+                                    (* * ? *)
+                                    (* * * *)
+   
+        where '*' indicates an element provided by the user and '?'
+        indicates an unspecified element, E04UCF will call CONFUN twice:
+        once to estimate the missing element in column 2, and again to
+        estimate the two missing elements in column 3. (Since columns 1
+        and 4 are known, they require no calls to CONFUN.)
+   
+        At times, central differences are used rather than forward
+        differences, in which case twice as many calls to OBJFUN and
+        CONFUN are needed. (The switch to central differences is not
+        under the user's control.)
+   
+        Difference Interval r Default values are computed
+   
+           (AXIOM parameter DIF)
+   
+        This option defines an interval used to estimate gradients by
+        finite differences in the following circumstances:
+   
+        (a)   For verifying the objective and/or constraint gradients
+              (see the description of Verify, below).
+   
+        (b)   For estimating unspecified elements of the objective
+              gradient of the Jacobian matrix.
+   
+        In general, a derivative with respect to the jth variable is
+                                                                      ^
+        approximated using the interval (delta) , where (delta) =r(1+|x |)
+                                               j               j       j
+                ^
+        with x the first point feasible with respect to the bounds and
+        linear constraints. If the functions are well scaled, the
+        resulting derivative approximation should be accurate to O(r).
+   
+        See Gill et al [10] for a discussion of the accuracy in finite-
+        difference approximations.
+   
+        If a difference interval is not specified by the user, a finite-
+        difference interval will be computed automatically for each
+        variable by a procedure that requires up to six calls of CONFUN
+        and OBJFUN for each component. This option is recommended if the
+        function is badly scaled or the user wishes to have E04UCF
+        determine constant elements in the objective and constraint
+        gradients (see the descriptions of CONFUN and OBJFUN in
+        Section 5).
+   
+                                            _________
+        Feasibility Tolerance r Default = \/(epsilon)
+   
+           (AXIOM parameter FEA)
+   
+        The scalar r defines the maximum acceptable absolute violations
+        in linear and nonlinear constraints at a 'feasible' point; i.e.,
+        a constraint is considered satisfied if its violation does not
+        exceed r. If r<(epsilon) or r>=1, the default value is used.
+        Using this keyword sets both optional parameters Linear
+        Feasibility Tolerance and Nonlinear Feasibility Tolerance to r,
+        if (epsilon)<=r<1. (Additional details are given below under the
+        descriptions of these parameters.)
+   
+                                                0.9
+        Function Precision r Default = (epsilon)
+   
+           (AXIOM parameter FUN)
+   
+        This parameter defines (epsilon) , which is intended to be a
+                                        R
+        measure of the accuracy with which the problem functions f and c
+        can be computed. If r<(epsilon) or r>=1, the default value is
+        used. The value of (epsilon)  should reflect the relative
+                                    R
+        precision of 1+|F(x)|; i.e., (epsilon)  acts as a relative
+                                              R
+        precision when |F| is large, and as an absolute precision when
+        |F| is small. For example, if F(x) is typically of order 1000 and
+        the first six significant digits are known to be correct, an
+        appropriate value for (epsilon)  would be 1.0E-6. In contrast, if
+                                       R
+                                     -4
+        F(x) is typically of order 10   and the first six significant
+        digits are known to be correct, an appropriate value for
+        (epsilon)  would be 1.0E-10. The choice of (epsilon)  can be
+                 R                                          R
+        quite complicated for badly scaled problems; see Chapter 8 of
+        Gill et al [10] for a discussion of scaling techniques. The
+        default value is appropriate for most simple functions that are
+        computed with full accuracy. However, when the accuracy of the
+        computed function values is known to be significantly worse than
+        full precision, the value of (epsilon)  should be large enough so
+                                              R
+        that E04UCF will not attempt to distinguish between function
+        values that differ by less than the error inherent in the
+        calculation.
+   
+        Hessian No Default = No
+   
+        Hessian Yes
+   
+           (No AXIOM parameter - fixed as Yes)
+   
+        This option controls the contents of the upper-triangular matrix
+        R (see Section 5). E04UCF works exclusively with the transformed
+        and re-ordered Hessian H  (6), and hence extra computation is
+                                Q
+        required to form the Hessian itself. If Hessian = No, R contains
+        the Cholesky factor of the transformed and re-ordered Hessian. If
+        Hessian = Yes the Cholesky factor of the approximate Hessian
+        itself is formed and stored in R. The user should select Hessian
+        = Yes if a warm start will be used for the next call to E04UCF.
+   
+                                          10
+        Infinite Bound Size r Default = 10
+   
+           (AXIOM parameter INFB)
+   
+        If r>0, r defines the 'infinite' bound BIGBND in the definition
+        of the problem constraints. Any upper bound greater than or equal
+        to BIGBND will be regarded as plus infinity (and similarly for a
+        lower bound less than or equal to -BIGBND). If r<=0, the default
+        value is used.
+   
+                                                    10
+        Infinite Step Size r Default = max(BIGBND,10  )
+   
+           (AXIOM parameter INFS)
+   
+        If r>0, r specifies the magnitude of the change in variables that
+        is treated as a step to an unbounded solution. If the change in x
+        during an iteration would exceed the value of Infinite Step Size,
+        the objective function is considered to be unbounded below in the
+        feasible region. If r<=0, the default value is used.
+   
+        Iteration limit i Default = max(50,3(n+n )+10n )
+                                                L     N
+   
+        See Major Iteration Limit below.
+   
+                                                    _________
+        Linear Feasibility Tolerance r  Default = \/(epsilon)
+                                      1
+   
+           (AXIOM parameter LINF)
+   
+                                                       _________
+        Nonlinear Feasibility Tolerance r  Default = \/(epsilon) if
+                                         2
+   
+           (AXIOM parameter NONF)
+                                           0.33
+        Derivative Level >= 2 and (epsilon)     otherwise
+   
+        The scalars r  and r  define the maximum acceptable absolute
+                     1      2
+        violations in linear and nonlinear constraints at a 'feasible'
+        point; i.e., a linear constraint is considered satisfied if its
+        violation does not exceed r , and similarly for a nonlinear
+                                   1
+        constraint and r . If r <(epsilon) or r >=1, the default value is
+                        2      i               i
+        used, for i=1,2.
+   
+        On entry to E04UCF, an iterative procedure is executed in order
+        to find a point that satisfies the linear constraint and bounds
+        on the variables to within the tolerance r . All subsequent
+                                                  1
+        iterates will satisfy the linear constraints to within the same
+        tolerance (unless r  is comparable to the finite-difference
+                           1
+        interval).
+   
+        For nonlinear constraints, the feasibility tolerance r  defines
+                                                              2
+        the largest constraint violation that is acceptable at an optimal
+        point. Since nonlinear constraints are generally not satisfied
+        until the final iterate, the value of Nonlinear Feasibility
+        Tolerance acts as a partial termination criterion for the
+        iterative sequence generated by E04UCF (see the discussion of
+        Optimality Tolerance).
+   
+        These tolerances should reflect the precision of the
+        corresponding constraints. For example, if the variables and the
+        coefficients in the linear constraints are of order unity, and
+        the latter are correct to about 6 decimal digits, it would be
+                                       - 6
+        appropriate to specify r  as 10   .
+                                1
+   
+        Linesearch Tolerance r Default = 0.9
+   
+           (AXIOM parameter LINT)
+   
+   
+        The value r (0 <= r < 1) controls the accuracy with which the
+        step (alpha) taken during each iteration approximates a minimum
+        of the merit function along the search direction (the smaller the
+        value of r, the more accurate the linesearch). The default value
+        r=0.9 requests an inaccurate search, and is appropriate for most
+        problems, particularly those with any nonlinear constraints.
+   
+        If there are no nonlinear constraints, a more accurate search may
+        be appropriate when it is desirable to reduce the number of major
+        iterations - for example, if the objective function is cheap to
+        evaluate, or if a substantial number of gradients are
+        unspecified.
+   
+        List Default = List
+   
+        Nolist
+   
+           (AXIOM parameter LIST)
+   
+        Normally each optional parameter specification is printed as it
+        is supplied. Nolist may be used to suppress the printing and List
+        may be used to restore printing.
+   
+        Major Iteration Limit i Default = max(50,3(n+n )+10n )
+                                                      L     N
+   
+        Iteration Limit
+   
+        Iters
+   
+        Itns
+   
+           (AXIOM parameter MAJI)
+   
+        The value of i specifies the maximum number of major iterations
+        allowed before termination. Setting i=0 and Major Print Level> 0
+        means that the workspace needed will be computed and printed, but
+        no iterations will be performed.
+   
+        Major Print level i Default = 10
+   
+        Print Level
+   
+           (AXIOM parameter MAJP)
+   
+        The value of i controls the amount of printout produced by the
+        major iterations of E04UCF. (See also Minor Print level below.)
+        The levels of printing are indicated below.
+   
+        i        Output
+   
+        0        No output.
+   
+        1        The final solution only.
+   
+        5        One line for each major iteration (no printout of the
+                 final solution).
+   
+        >=10     The final solution and one line of output for each
+                 iteration.
+   
+        >=20     At each major iteration, the objective function, the
+                 Euclidean norm of the nonlinear constraint violations,
+                 the values of the nonlinear constraints (the vector c),
+                 the values of the linear constraints (the vector A x),
+                                                                   L
+                 and the current values of the variables (the vector x).
+   
+        >=30     At each major iteration, the diagonal elements of the
+                 matrix T associated with the TQ factorization (5) of the
+                 QP working set, and the diagonal elements of R, the
+                 triangular factor of the transformed and re-ordered
+                 Hessian (6).
+   
+        Minor Iteration Limit i Default = max(50,3(n+n +n ))
+                                                      L  N
+   
+           (AXIOM parameter MINI)
+   
+        The value of i specifies the maximum number of iterations for the
+        optimality phase of each QP subproblem.
+   
+        Minor Print Level i Default = 0
+   
+           (AXIOM parameter MINP)
+   
+        The value of i controls the amount of printout produced by the
+        minor iterations of E04UCF, i.e., the iterations of the quadratic
+        programming algorithm. (See also Major Print Level, above.) The
+        following levels of printing are available.
+   
+        i        Output
+   
+        0        No output.
+   
+        1        The final QP solution.
+   
+        5        One line of output for each minor iteration (no printout
+                 of the final QP solution).
+   
+        >=10     The final QP solution and one brief line of output for
+                 each minor iteration.
+   
+        >=20     At each minor iteration, the current estimates of the QP
+                 multipliers, the current estimate of the QP search
+                 direction, the QP constraint values, and the status of
+                 each QP constraint.
+   
+        >=30     At each minor iteration, the diagonal elements of the
+                 matrix T associated with the TQ factorization (5) of the
+                 QP working set, and the diagonal elements of the
+                 Cholesky factor R of the transformed Hessian (6).
+   
+                                                      _________
+        Nonlinear Feasibility Tolerance r Default = \/(epsilon)
+   
+        See Linear Feasibility Tolerance, above.
+   
+                                                  0.8
+        Optimality Tolerance r Default = (epsilon)
+   
+           (AXIOM parameter OPT)
+   
+        The parameter r ((epsilon) <=r<1) specifies the accuracy to which
+                                  R
+        the user wishes the final iterate to approximate a solution of
+        the problem. Broadly speaking, r indicates the number of correct
+        figures desired in the objective function at the solution. For
+                           - 6
+        example, if r is 10    and E04UCF terminates successfully, the
+        final value of F should have approximately six correct figures.
+        If r<(epsilon)  or r>=1 the default value is used.
+                      R
+   
+        E04UCF will terminate successfully if the iterative sequence of x
+        -values is judged to have converged and the final point satisfies
+        the first-order Kuhn-Tucker conditions (see Section 3). The
+        sequence of iterates is considered to have converged at x if
+   
+                                        _
+                       (alpha) ||p||<=\/r(1+||x||),                  (8a)
+   
+        where p is the search direction and (alpha) the step length from
+        (3). An iterate is considered to satisfy the first-order
+        conditions for a minimum if
+   
+                     T         _
+                  ||Z g  ||<=\/r(1+max(1+|F(x)|,||g  ||))            (8b)
+                       FR                          FR
+   
+        and
+   
+                          |res |<=ftol for all j,                    (8c)
+                              j
+   
+               T
+        where Z g   is the projected gradient (see Section 3), g   is the
+                 FR                                             FR
+        gradient of F(x) with respect to the free variables, res  is the
+                                                                j
+        violation of the jth active nonlinear constraint, and ftol is the
+        Nonlinear Feasibility Tolerance.
+   
+        Step Limit r Default = 2.0
+   
+           (AXIOM parameter STE)
+   
+        If r>0, r specifies the maximum change in variables at the first
+                                                              bx
+        step of the linesearch. In some cases, such as F(x)=ae   or
+               b
+        F(x)=ax , even a moderate change in the components of x can lead
+        to floating-point overflow. The parameter r is therefore used to
+        encourage evaluation of the problem functions at meaningful
+                                                           ~
+        points. Given any major iterate x, the first point x at which F
+        and c are evaluated during the linesearch is restricted so that
+   
+                               ~
+                             ||x-x|| <=r(1+||x|| ).
+                                    2           2
+   
+        The linesearch may go on and evaluate F and c at points further
+        from x if this will result in a lower value of the merit
+        function. In this case, the character L is printed at the end of
+        the optional line of printed output, (see Section 5.2). If L is
+        printed for most of the iterations, r should be set to a larger
+        value.
+   
+        Wherever possible, upper and lower bounds on x should be used to
+        prevent evaluation of nonlinear functions at wild values. The
+        default value Step Limit = 2.0 should not affect progress on
+        well-behaved functions, but values 0.1 or 0.01 may be helpful
+        when rapidly varying functions are present. If a small value of
+        Step Limit is selected, a good starting point may be required. An
+        important application is to the class of nonlinear least-squares
+        problems. If r<=0, the default value is used.
+   
+        Start Objective Check At Variable k Default = 1
+   
+           (AXIOM parameter STAO)
+   
+        Start Constraint Check At Variable k Default = 1
+   
+           (AXIOM parameter STAC)
+   
+        Stop Objective Check At Variable l Default = n
+   
+           (AXIOM parameter STOO)
+   
+        Stop Constraint Check At Variable l Default = n
+   
+           (AXIOM parameter STOC)
+   
+        These keywords take effect only if Verify Level > 0 (see below).
+        They may be used to control the verification of gradient elements
+        computed by subroutines OBJFUN and CONFUN. For example, if the
+        first 30 components of the objective gradient appeared to be
+        correct in an earlier run, so that only component 31 remains
+        questionable, it is reasonable to specify Start Objective Check
+        At Variable 31. If the first 30 variables appear linearly in the
+        objective, so that the corresponding gradient elements are
+        constant, the above choice would also be appropriate.
+   
+        Verify Level i Default = 0
+   
+        Verify No
+   
+        Verify Level - 1
+   
+        Verify Level 0
+   
+        Verify Objective Gradients
+   
+        Verify Level 1
+   
+        Verify Constraint Gradients
+   
+        Verify Level 2
+   
+        Verify
+   
+        Verify Yes
+   
+        Verify Gradients
+   
+        Verify Level 3
+   
+           (AXIOM parameter VE)
+   
+        These keywords refer to finite-difference checks on the gradient
+        elements computed by the user-provided subroutines OBJFUN and
+        CONFUN. (Unspecified gradient components are not checked.) It is
+        possible to specify Verify Levels 0-3 in several ways, as
+        indicated above. For example, the nonlinear objective gradient
+        (if any) will be verified if either Verify Objective Gradients or
+        Verify Level 1 is specified. Similarly, the objective and the
+        constraint gradients will be verified if Verify Yes or Verify
+        Level 3 or Verify is specified.
+   
+        If 0<=i<=3, gradients will be verified at the first point that
+        satisfies the linear constraints and bounds. If i=0, only a '
+        cheap' test will be performed, requiring one call to OBJFUN and
+        one call to CONFUN. If 1<=i<=3, a more reliable (but more
+        expensive) check will be made on individual gradient components,
+        within the ranges specified by the Start and Stop keywords
+        described above. A result of the form OK or BAD? is printed by
+        E04UCF to indicate whether or not each component appears to be
+        correct.
+   
+        If 10<=i<=13, the action is the same as for i - 10, except that
+        it will take place at the user-specified initial value of x.
+   
+        We suggest that Verify Level 3 be specified whenever a new
+        function routine is being developed.
+   
+        5.1.3. Optional parameter checklist and default values
+   
+        For easy reference, the following list shows all the valid
+        keywords and their default values. The symbol (epsilon)
+        represents the machine precision (see X02AJF(*) ).
+   
+        Optional Parameters      Default Values
+   
+   
+   
+        Central difference       Computed automatically
+        interval
+   
+        Cold/Warm start          Cold start
+   
+        Crash tolerance          0.01
+   
+        Defaults
+   
+        Derivative level         3
+   
+        Difference interval      Computed automatically
+   
+                                   _________
+        Feasibility tolerance    \/(epsilon)
+   
+                                          0.9
+        Function precision       (epsilon)
+   
+        Hessian                  No
+   
+                                   10
+        Infinite bound size      10
+   
+                                   10
+        Infinite step size       10
+   
+                                   _________
+        Linear feasibility       \/(epsilon)
+        tolerance
+   
+        Linesearch tolerance     0.9
+   
+        List/Nolist              List
+   
+        Major iteration limit    max(50,3(n+n )+10n )
+                                             L     N
+   
+        Major print level        10
+   
+        Minor iteration limit    max(50,3(n+n +n ))
+                                             L  N
+   
+        Minor print level        0
+   
+                                   _________
+        Nonlinear feasibility    \/(epsilon) if Derivative Level >= 2
+        tolerance                                   0.33
+                                 otherwise (epsilon)
+   
+                                          0.8
+        Optimality tolerance     (epsilon)
+                                          R
+   
+        Step limit               2.0
+   
+        Start objective check    1
+   
+        Start constraint check   1
+   
+        Stop objective check     n
+   
+        Stop constraint check    n
+   
+        Verify level             0
+   
+        5.2. Description of Printed Output
+   
+        The level of printed output from E04UCF is controlled by the user
+        (see the description of Major Print Level and Minor Print Level
+        in Section 5.1). If Minor Print Level > 0, output is obtained
+        from the subroutines that solve the QP subproblem. For a detailed
+        description of this information the reader should refer to
+        E04NCF(*).
+   
+        When Major Print Level >= 5, the following line of output is
+        produced at every major iteration of E04UCF. In all cases, the
+        values of the quantities printed are those in effect on
+        completion of the given iteration.
+   
+        Itn            is the iteration count.
+   
+        ItQP           is the sum of the iterations required by the
+                       feasibility and optimality phases of the QP
+                       subproblem. Generally, ItQP will be 1 in the later
+                       iterations, since theoretical analysis predicts
+                       that the correct active set will be identified
+                       near the solution (see Section 3).
+   
+                       Note that ItQP may be greater than the Minor
+                       Iteration Limit if some iterations are required
+                       for the feasibility phase.
+   
+        Step           is the step (alpha) taken along the computed
+                       search direction. On reasonably well-behaved
+                       problems, the unit step will be taken as the
+                       solution is approached.
+   
+        Nfun           is the cumulative number of evaluations of the
+                       objective function needed for the linesearch.
+                       Evaluations needed for the estimation of the
+                       gradients by finite differences are not included.
+                       Nfun is printed as a guide to the amount of work
+                       required for the linesearch.
+   
+        Merit          is the value of the augmented Lagrangian merit
+                       function (12) at the current iterate. This
+                       function will decrease at each iteration unless it
+                       was necessary to increase the penalty parameters
+                       (see Section 8.2). As the solution is approached,
+                       Merit will converge to the value of the objective
+                       function at the solution.
+   
+                       If the QP subproblem does not have a feasible
+                       point (signified by I at the end of the current
+                       output line), the merit function is a large
+                       multiple of the constraint violations, weighted by
+                       the penalty parameters. During a sequence of major
+                       iterations with infeasible subproblems, the
+                       sequence of Merit values will decrease
+                       monotonically until either a feasible subproblem
+                       is obtained or E04UCF terminates with IFAIL = 3
+                       (no feasible point could be found for the
+                       nonlinear constraints).
+   
+                       If no nonlinear constraints are present (i.e.,
+                       NCNLN = 0), this entry contains Objective, the
+                       value of the objective function F(x). The
+                       objective function will decrease monotonically to
+                       its optimal value when there are no nonlinear
+                       constraints.
+   
+        Bnd            is the number of simple bound constraints in the
+                       predicted active set.
+   
+        Lin            is the number of general linear constraints in the
+                       predicted active set.
+   
+        Nln            is the number of nonlinear constraints in the
+                       predicted active set (not printed if NCNLN is
+                       zero).
+   
+        Nz             is the number of columns of Z (see Section 8.1).
+                       The value of Nz is the number of variables minus
+                       the number of constraints in the predicted active
+                       set; i.e., Nz = n-(Bnd + Lin + Nln).
+   
+        Norm Gf        is the Euclidean norm of g  , the gradient of the
+                                                 FR
+                       objective function with respect to the free
+                       variables, i.e.,variables not currently held at a
+                       bound.
+   
+                             T
+        Norm Gz        is ||Z g  ||, the Euclidean norm of the projected
+                               FR
+                       gradient (see Section 8.1). Norm  Gz will be
+                       approximately zero in the neighbourhood of a
+                       solution.
+   
+        Cond H         is a lower bound on the condition number of the
+                       Hessian approximation H.
+   
+        Cond Hz        is a lower bound on the condition number of the
+                       projected Hessian approximation H  (
+                                                        z
+                            T      T
+                       (H =Z H  Z=R R ; see (6) and (12) in Sections 3
+                         z    FR   z z
+                       and 8.1). The larger this number, the more
+                       difficult the problem.
+   
+        Cond T         is a lower bound on the condition number of the
+                       matrix of predicted active constraints.
+   
+        Norm C         is the Euclidean norm of the residuals of
+                       constraints that are violated or in the predicted
+                       active set (not printed if NCNLN is zero). Norm C
+                       will be approximately zero in the neighbourhood of
+                       a solution.
+   
+        Penalty        is the Euclidean norm of the vector of penalty
+                       parameters used in the augumented Lagrangian merit
+                       function (not printed if NCNLN is zero).
+   
+        Conv           is a three-letter indication of the status of the
+                       three convergence tests (8a)-(8c) defined in the
+                       description of the optional parameter Optimality
+                       Tolerance in Section 5.1 Each letter is T if the
+                       test is satisfied, and F otherwise. The three
+                       tests indicate whether:
+                       (a)   the sequence of iterates has converged;
+   
+                       (b)   the projected gradient (Norm  Gz) is
+                             sufficiently small; and
+   
+                       (c)   the norm of the residuals of constraints in
+                             the predicted active set (Norm  C) is small
+                             enough.
+                       If any of these indicators is F when E04UCF
+                       terminates with IFAIL = 0, the user should check
+                       the solution carefully.
+   
+        M              is printed if the Quasi-Newton update was modified
+                       to ensure that the Hessian approximation is
+                       positive-definite (see Section 8.3).
+   
+        I              is printed if the QP subproblem has no feasible
+                       point.
+   
+        C              is printed if central differences were used to
+                       compute the unspecified objective and constraint
+                       gradients. If the value of Step is zero, the
+                       switch to central differences was made because no
+                       lower point could be found in the linesearch. (In
+                       this case, the QP subproblem is resolved with the
+                       central-difference gradient and Jacobian.) If the
+                       value of Step is non-zero, central differences
+                       were computed because Norm  Gz and Norm  C imply
+                       that x is close to a Kuhn-Tucker point.
+   
+        L              is printed if the linesearch has produced a
+                       relative change in x greater than the value
+                       defined by the optional parameter Step Limit. If
+                       this output occurs frequently during later
+                       iterations of the run, Step Limit should be set to
+                       a larger value.
+   
+        R              is printed if the approximate Hessian has been
+                       refactorized. If the diagonal condition estimator
+                       of R indicates that the approximate Hessian is
+                       badly conditioned, the approximate Hessian is
+                       refactorized using column interchanges. If
+                       necessary, R is modified so that its diagonal
+                       condition estimator is bounded.
+   
+        When Major Print Level = 1 or Major Print Level >= 10, the
+        summary printout at the end of execution of E04UCF includes a
+        listing of the status of every variable and constraint. Note that
+        default names are assigned to all variables and constraints.
+   
+        The following describes the printout for each variable.
+   
+        Varbl          gives the name (V) and index j=1,2,...,n of the
+                       variable.
+   
+        State          gives the state of the variable in the predicted
+                       active set (FR if neither bound is in the active
+                       set, EQ if a fixed variable, LL if on its lower
+                       bound, UL if on its upper bound). If the variable
+                       is predicted to lie outside its upper or lower
+                       bound by more than the feasibility tolerance,
+                       State will be ++ or -- respectively. (The latter
+                       situation can occur only when there is no feasible
+                       point for the bounds and linear constraints.)
+   
+        Value          is the value of the variable at the final
+                       iteration.
+   
+        Lower bound    is the lower bound specified for the variable.
+                       (None indicates that BL(j)<=- BIGBND.)
+   
+        Upper bound    is the upper bound specified for the variable.
+                       (None indicates that BL(j)>=BIGBND.)
+   
+        Lagr Mult      is the value of the Lagrange-multiplier for the
+                       associated bound constraint. This will be zero if
+                       State is FR. If x is optimal, the multiplier
+                       should be non-negative if State is LL, and non-
+                       positive if State is UL.
+   
+        Residual       is the difference between the variable Value and
+                       the nearer of its bounds BL(j) and BU(j).
+   
+        The printout for general constraints is the same as for
+        variables, except for the following:
+   
+        L Con      is the name (L) and index i, for i = 1,2,...,NCLIN of
+                   a linear constraint.
+   
+        N Con      is the name (N) and index i, for i = 1,2,...,NCNLN of
+                   a nonlinear constraint.
+   
+        6. Error Indicators and Warnings
+   
+        Errors or warnings specified by the routine:
+   
+        If on entry IFAIL = 0 or -1, explanatory error messages are
+        output on the current error message unit (as defined by X04AAF).
+   
+        The input data for E04UCF should always be checked (even if
+        E04UCF terminates with IFAIL=0).
+   
+        Note that when Print Level>0, a short description of IFAIL is
+        printed.
+   
+        Errors and diagnostics indicated by IFAIL, together with some
+        recommendations for recovery are indicated below.
+   
+        IFAIL= 1
+             The final iterate x satisfies the first-order Kuhn-Tucker
+             conditions to the accuracy requested, but the sequence of
+             iterates has not yet converged. E04UCF was terminated
+             because no further improvement could be made in the merit
+             function.
+   
+             This value of IFAIL may occur in several circumstances. The
+             most common situation is that the user asks for a solution
+             with accuracy that is not attainable with the given
+             precision of the problem (as specified by Function Precision
+             see Section 5). This condition will also occur if, by
+             chance, an iterate is an 'exact' Kuhn-Tucker point, but the
+             change in the variables was significant at the previous
+             iteration. (This situation often happens when minimizing
+             very simple functions, such as quadratics.)
+   
+             If the four conditions listed in Section 5 for IFAIL = 0 are
+             satisfied, x is likely to be a solution of (1) even if IFAIL
+             = 1.
+   
+        IFAIL= 2
+             E04UCF has terminated without finding a feasible point for
+             the linear constraints and bounds, which means that no
+             feasible point exists for the given value of Linear
+             Feasibility Tolerance (see Section 5.1). The user should
+             check that there are no constraint redundancies. If the data
+             for the constraints are accurate only to an absolute
+             precision (sigma), the user should ensure that the value of
+             the optional parameter Linear Feasibility Tolerance is
+             greater than (sigma). For example, if all elements of A are
+             of order unity and are accurate to only three decimal
+                                                                       -3
+             places, Linear Feasibility Tolerance should be at least 10  .
+   
+        IFAIL= 3
+             No feasible point could be found for the nonlinear
+             constraints. The problem may have no feasible solution. This
+             means that there has been a sequence of QP subproblems for
+             which no feasible point could be found (indicated by I at
+             the end of each terse line of output). This behaviour will
+             occur if there is no feasible point for the nonlinear
+             constraints. (However, there is no general test that can
+             determine whether a feasible point exists for a set of
+             nonlinear constraints.) If the infeasible subproblems occur
+             from the very first major iteration, it is highly likely
+             that no feasible point exists. If infeasibilities occur when
+             earlier subproblems have been feasible, small constraint
+             inconsistencies may be present. The user should check the
+             validity of constraints with negative values of ISTATE. If
+             the user is convinced that a feasible point does exist,
+             E04UCF should be restarted at a different starting point.
+   
+        IFAIL= 4
+             The limiting number of iterations (determined by the
+             optional parameter Major Iteration Limit see Section 5.1)
+             has been reached.
+   
+             If the algorithm appears to be making progress, Major
+             Iteration Limit may be too small. If so, increase its value
+             and rerun E04UCF (possibly using the Warm Start option). If
+             the algorithm seems to be 'bogged down', the user should
+             check for incorrect gradients or ill-conditioning as
+             described below under IFAIL = 6.
+   
+             Note that ill-conditioning in the working set is sometimes
+             resolved automatically by the algorithm, in which case
+             performing additional iterations may be helpful. However,
+             ill-conditioning in the Hessian approximation tends to
+             persist once it has begun, so that allowing additional
+             iterations without altering R is usually inadvisable. If the
+             quasi-Newton update of the Hessian approximation was
+             modified during the latter iterations (i.e., an M occurs at
+             the end of each terse line), it may be worthwhile to try a
+             warm start at the final point as suggested above.
+   
+        IFAIL= 6
+             x does not satisfy the first-order Kuhn-Tucker conditions,
+             and no improved point for the merit function could be found
+             during the final line search.
+   
+             A sufficient decrease in the merit function could not be
+             attained during the final line search. This sometimes occurs
+             because an overly stringent accuracy has been requested,
+             i.e., Optimality Tolerance is too small. In this case the
+             user should apply the four tests described under IFAIL = 0
+             above to determine whether or not the final solution is
+             acceptable (see Gill et al [10], for a discussion of the
+             attainable accuracy).
+   
+             If many iterations have occurred in which essentially no
+             progress has been made and E04UCF has failed completely to
+             move from the initial point then subroutines OBJFUN or
+             CONFUN may be incorrect. The user should refer to comments
+             below under IFAIL = 7 and check the gradients using the
+             Verify parameter. Unfortunately, there may be small errors
+             in the objective and constraint gradients that cannot be
+             detected by the verification process. Finite-difference
+             approximations to first derivatives are catastrophically
+             affected by even small inaccuracies. An indication of this
+             situation is a dramatic alteration in the iterates if the
+             finite-difference interval is altered. One might also
+             suspect this type of error if a switch is made to central
+             differences even when Norm Gz and Norm C are large.
+   
+             Another possibility is that the search direction has become
+             inaccurate because of ill-conditioning in the Hessian
+             approximation or the matrix of constraints in the working
+             set; either form of ill-conditioning tends to be reflected
+             in large values of ItQP (the number of iterations required
+             to solve each QP subproblem).
+   
+             If the condition estimate of the projected Hessian (Cond Hz)
+             is extremely large, it may be worthwhile to rerun E04UCF
+             from the final point with the Warm Start option. In this
+             situation, ISTATE should be left unaltered and R should be
+             reset to the identity matrix.
+   
+             If the matrix of constraints in the working set is ill-
+             conditioned (i.e., Cond T is extremely large), it may be
+             helpful to run E04UCF with a relaxed value of the
+             Feasibility Tolerance (Constraint dependencies are often
+             indicated by wide variations in size in the diagonal
+             elements of the matrix T, whose diagonals will be printed
+             for Major Print Level >= 30).
+   
+        IFAIL= 7
+             The user-provided derivatives of the objective function
+             and/or nonlinear constraints appear to be incorrect.
+   
+             Large errors were found in the derivatives of the objective
+             function and/or nonlinear constraints. This value of IFAIL
+             will occur if the verification process indicated that at
+             least one gradient or Jacobian component had no correct
+             figures. The user should refer to the printed output to
+             determine which elements are suspected to be in error.
+   
+             As a first-step, the user should check that the code for the
+             objective and constraint values is correct - for example, by
+             computing the function at a point where the correct value is
+             known. However, care should be taken that the chosen point
+             fully tests the evaluation of the function. It is remarkable
+             how often the values x=0 or x=1 are used to test function
+             evaluation procedures, and how often the special properties
+             of these numbers make the test meaningless.
+   
+             Special care should be used in this test if computation of
+             the objective function involves subsidiary data communicated
+             in COMMON storage. Although the first evaluation of the
+             function may be correct, subsequent calculations may be in
+             error because some of the subsidiary data has accidently
+             been overwritten.
+   
+             Errors in programming the function may be quite subtle in
+             that the function value is 'almost' correct. For example,
+             the function may not be accurate to full precision because
+             of the inaccurate calculation of a subsidiary quantity, or
+             the limited accuracy of data upon which the function
+             depends. A common error on machines where numerical
+             calculations are usually performed in double precision is to
+             include even one single-precision constant in the
+             calculation of the function; since some compilers do not
+             convert such constants to double precision, half the correct
+             figures may be lost by such a seemingly trivial error.
+   
+        IFAIL= 9
+             An input parameter is invalid. The user should refer to the
+             printed output to determine which parameter must be
+             redefined.
+   
+        IFAILOverflow
+             If the printed output before the overflow error contains a
+             warning about serious ill-conditioning in the working set
+             when adding the jth constraint, it may be possible to avoid
+             the difficulty by increasing the magnitude of the optional
+             parameter Linear Feasiblity Tolerance or Nonlinear
+             Feasiblity Tolerance, and rerunning the program. If the
+             message recurs even after this change, the offending
+             linearly dependent constraint (with index 'j') must be
+             removed from the problem. If overflow occurs in one of the
+             user-supplied routines (e.g. if the nonlinear functions
+             involve exponentials or singularities), it may help to
+             specify tighter bounds for some of the variables (i.e.,
+             reduce the gap between appropriate l  and u ).
+                                                 j      j
+   
+        7. Accuracy
+   
+        If IFAIL = 0 on exit then the vector returned in the array X is
+        an estimate of the solution to an accuracy of approximately
+        Feasiblity Tolerance (see Section 5.1), whose default value is
+                 0.8
+        (epsilon)   , where (epsilon) is the machine precision (see
+        X02AJF(*)).
+   
+        8. Further Comments
+   
+        In this section we give some further details of the method used
+        by E04UCF.
+   
+        8.1. Solution of the Quadratic Programming Subproblem
+   
+        The search direction p is obtained by solving (4) using the
+        method of E04NCF(*) (Gill et al [8]), which was specifically
+        designed to be used within an SQP algorithm for nonlinear
+        programming.
+   
+        The method of E04UCF is a two-phase (primal) quadratic
+        programming method. The two phases of the method are: finding an
+        initial feasible point by minimizing the sum of infeasibilities
+        (the feasibility phase), and minimizing the quadratic objective
+        function within the feasible region (the optimality phase). The
+        computations in both phases are perfomed by the same subroutines.
+        The two-phase nature of the algorithm is reflected by changing
+        the function being minimized from the sum of infeasibilities to
+        the quadratic objective function.
+   
+        In general, a quadratic program must be solved by iteration. Let
+        p denote the current estimate of the solution of (4); the new
+                _
+        iterate p is defined by
+   
+                               _
+                               p=p+(sigma)d,                          (9)
+   
+        where, as in (3), (sigma) is a non-negative step length and d is
+        a search direction.
+   
+        At the beginning of each iteration of E04UCF, a working set is
+        defined of constraints (general and bound) that are satisfied
+        exactly. The vector d is then constructed so that the values of
+        constraints in the working set remain unaltered for any move
+        along d. For a bound constraint in the working set, this property
+        is achieved by setting the corresponding component of d to zero,
+        i.e., by fixing the variable at its bound. As before, the
+        subscripts 'FX' and 'FR' denote selection of the components
+        associated with the fixed and free variables.
+   
+        Let C denote the sub-matrix of rows of
+   
+                                      (A )
+                                      ( L)
+                                      (A )
+                                      ( N)
+   
+        corresponding to general constraints in the working set. The
+        general constraints in the working set will remain unaltered if
+   
+                                 C  d  =0,                           (10)
+                                  FR FR
+   
+        which is equivalent to defining d   as
+                                         FR
+   
+                                  d  =Zd                             (11)
+                                   FR   z
+   
+        for some vector d , where Z is the matrix associated with the TQ
+                         z
+        factorization (5) of C  .
+                              FR
+   
+        The definition of d  in (11) depends on whether the current p is
+                           z
+        feasible. If not, d  is zero except for a component (gamma) in
+                           z
+        the jth position, where j and (gamma) are chosen so that the sum
+        of infeasibilities is decreasing along d. (For further details,
+        see Gill et al [8].) In the feasible case, d  satisfies the
+                                                    z
+        equations
+   
+                               T       T
+                              R R d =-Z q  ,                         (12)
+                               z z z     FR
+   
+                                            T
+        where R  is the Cholesky factor of Z H  Z and q is the gradient
+               z                              FR
+                                                                   T
+        of the quadratic objective function (q=g+Hp). (The vector Z q
+                                                                     FR
+        is the projected gradient of the QP.) With (12), P+d is the
+        minimizer of the quadratic objective function subject to treating
+        the constraints in the working set as equalities.
+   
+        If the QP projected gradient is zero, the current point is a
+        constrained stationary point in the subspace defined by the
+        working set. During the feasiblity phase, the projected gradient
+        will usually be zero only at a vertex (although it may vanish at
+        non-vertices in the presence of constraint dependencies). During
+        the optimality phase, a zero projected gradient implies that p
+        minimizes the quadratic objective function when the constraints
+        in the working set are treated as equalities. In either case,
+        Lagrange multipliers are computed. Given a positive constant
+        (delta) of the order of the machine precision, the Lagrange
+        multiplier (mu)  corresponding to an inequality constraint in the
+                       j
+        working set at its upper bound is said to be optimal if
+        (mu) <=(delta) when the jth constraint is at its upper bound, or
+            j
+        if (mu) >=-(delta) when the associated constraint is at its lower
+               j
+        bound. If any multiplier is non-optimal, the current objective
+        function (either the true objective or the sum of
+        infeasibilities) can be reduced by deleting the corresponding
+        constraint from the working set.
+   
+        If optimal multipliers occur during the feasibility phase and the
+        sum of infeasibilities is non-zero, no feasible point exists. The
+        QP algorithm will then continue iterating to determine the
+        minimum sum of infeasibilities. At this point, the Lagrange
+        multiplier (mu)  will satisfy -(1+(delta))<=(mu) <=(delta) for an
+                       j                                j
+        inequality constraint at its upper bound, and
+        -(delta)<=(mu) <=1+(delta) for an inequality at its lower bound.
+                      j
+        The Lagrange multiplier for an equality constraint will satisfy
+        |(mu) |<=1+(delta).
+             j
+   
+        The choice of step length (sigma) in the QP iteration (9) is
+        based on remaining feasible with respect to the satisfied
+        constraints. During the optimality phase, if p+d is feasible,
+        (sigma) will be taken as unity. (In this case, the projected
+                    _
+        gradient at p will be zero.) Otherwise, (sigma) is set to
+        (sigma) , the step to the 'nearest'constraint, which is added to
+               M
+        the working set at the next iteration.
+   
+        Each change in the working set leads to a simple change to C  :
+                                                                    FR
+        if the status of a general constraint changes, a row of C   is
+                                                                 FR
+        altered; if a bound constraint enters or leaves the working set,
+        a column of C   changes. Explicit representations are recurred of
+                     FR
+                                                       T       T
+        the matrices T, Q   and R, and of the vectors Q q and Q g.
+                         FR
+   
+        8.2. The Merit Function
+   
+        After computing the search direction as described in Section 3,
+        each major iteration proceeds by determining a step length
+        (alpha) in (3) that produces a 'sufficient decrease' in the
+        augmented Lagrangian merit function
+   
+   
+                              --
+        L(x,(lambda),s)=F(x)- > (lambda) (c (x)-s )
+                              --        i  i     i
+                              i
+   
+                              1 --                2
+                            + - > (rho) (c (x)-s ) ,                 (13)
+                              2 --     i  i     i
+                                i
+   
+        where x, (lambda) and s vary during the linsearch. The summation
+        terms in (13) involve only the nonlinear constraints. The vector
+        (lambda) is an estimate of the Lagrange multipliers for the
+        nonlinear constraints of (1). The non-negative slack variables
+        {s } allow nonlinear inequality constraints to be treated without
+          i
+        introducing discontinuities. The solution of the QP subproblem
+        (4) provides a vector triple that serves as a direction of search
+        for the three sets of variables. The non-negative vector (rho) of
+        penalty parameters is initialised to zero at the beginning of the
+        first major iteration. Thereafter, selected components are
+        increased whenever necessary to ensure descent for the merit
+        function. Thus, the sequence of norms of (rho) (the printed
+        quantity Penalty, see Section 5.2) is generally non-decreasing,
+        although each (rho)  may be reduced a limited number of times.
+                           i
+   
+        The merit function (13) and its global convergence properties are
+        described in Gill et al [9].
+   
+        8.3. The Quasi-Newton Update
+   
+        The matrix H in (4) is a positive-definite quasi-Newton
+        approximation to the Hessian of the Lagrangian function. (For a
+        review of quasi-Newton methods, see Dennis and Schnabel [3].) At
+                                                                     _
+        the end of each major iteration, a new Hessian approximation H is
+        defined as a rank-two modification of H. In E04UCF, the BFGS
+        quasi-Newton update is used:
+   
+                          _     1     T    1   T
+                          H=H- ----Hss H+ ---yy ,                    (14)
+                                T          T
+                               s Hs       y s
+   
+                _
+        where s=x-x (the change in x).
+   
+        In E04UCF, H is required to be positive-definite. If H is
+                           _
+        positive-definite, H defined by (14) will be positive-definite if
+                     T
+        and only if y s is positive (see, e.g. Dennis and More [1]).
+        Ideally, y in (14) would be taken as y , the change in gradient
+                                              L
+        of the Lagrangian function
+   
+                             _ _T         T
+                          y =g-A (mu) -g+A (mu) ,                    (15)
+                           L    N    N    N    N
+   
+        where (mu)  denotes the QP multipliers associated with the
+                  N
+                                                           T
+        nonlinear constraints of the original problem. If y s is not
+                                                           L
+        sufficiently positive, an attempt is made to perform the update
+        with a vector y of the form
+   
+                          m
+                           N
+                          --             _    _
+                    y=y + >  (omega) (a (x)c (x)-a (x)c (x)),
+                       L  --        i  i    i     i    i
+                          i=1
+   
+        where (omega) >=0. If no such vector can be found, the update is
+                     i
+        perfomed with a scaled y ; in this case, M is printed to indicate
+                                L
+        that the update is modified.
+   
+        Rather than modifying H itself, the Cholesky factor of the
+        transformed Hessian H  (6) is updated, where Q is the matrix from
+                             Q
+        (5) associated with the active set of the QP subproblem. The
+        update (13) is equivalent to the following update to H :
+                                                              Q
+   
+                     _        1        T     1     T
+                     H =H - ------H s s H + ----y y ,                (16)
+                      Q  Q   T     Q Q Q Q   T   Q Q
+                            s H s           y s
+                             Q Q Q           Q Q
+   
+                  T           T
+        where y =Q y, and s =Q s. This update may be expressed as a rank-
+               Q           Q
+        one update to R (see Dennis and Schnabel [2]).
+   
+        9. Example
+   
+        This section describes one version of the so-called 'hexagon'
+        problem (a different formulation is given as Problem 108 in Hock
+        and Schittkowski [11]). The problem is to determine the hexagon
+        of maximum area such that no two of its vertices are more than
+        one unit apart (the solution is not a regular hexagon).
+   
+        All constraint types are included (bounds, linear, nonlinear),
+        and the Hessian of the Lagrangian function is not positive-
+        definite at the solution. The problem has nine variables, non-
+        infinite bounds on seven of the variables, four general linear
+        constraints, and fourteen nonlinear constraints.
+   
+        The objective function is
+   
+                      F(x)=-x x +x x -x x -x x +x x +x x .
+                             2 6  1 7  3 7  5 8  4 9  3 8
+   
+        The bounds on the variables are
+   
+             x >=0, -1<=x <=1, x >=0,x >=0, x >=0,x <=0, and x <=0.
+              1          3      5     6      7     8          9
+   
+        Thus,
+   
+                                                             T
+                  l =(0,-infty,-1,-infty,0,0,0,-infty,-infty)
+                   B
+   
+                                                               T
+                 u =(infty,infty,1,infty,infty,infty,infty,0,0)
+                  B
+   
+        The general linear constraints are
+   
+             x -x >=0,x -x >=0, x -x >=0,and x -x >=0.
+              2  1     3  2      3  4         4  5
+   
+        Hence,
+   
+                  (0)     (-1  1 0  0  0 0 0 0 0)        (infty)
+                  (0)     ( 0 -1 1  0  0 0 0 0 0)        (infty)
+               l =(0), A =( 0  0 1 -1  0 0 0 0 0) and u =(infty).
+                L (0)   L ( 0  0 0  1 -1 0 0 0 0)      L (infty)
+   
+        The nonlinear constraints are all of the form c (x)<=1, for
+                                                       i
+        i=1,2,...,14; hence, all components of l  are -infty, and all
+                                                N
+        components of u  are 1. The fourteen functions {c (x)} are
+                       N                                 i
+   
+                    2  2
+             c (x)=x +x ,
+              1     1  6
+   
+                          2        2
+             c (x)=(x -x ) +(x -x ) ,
+              2      2  1     7  6
+   
+                          2  2
+             c (x)=(x -x ) +x ,
+              3      3  1    6
+   
+                          2        2
+             c (x)=(x -x ) +(x -x ) ,
+              4      1  4     6  8
+   
+                          2        2
+             c (x)=(x -x ) +(x -x ) ,
+              5      1  5     6  9
+   
+                    2  2
+             c (x)=x +x ,
+              6     2  7
+   
+                          2  2
+             c (x)=(x -x ) +x ,
+              7      3  2    7
+   
+                          2        2
+             c (x)=(x -x ) +(x -x ) ,
+              8      4  2     8  7
+   
+                          2        2
+             c (x)=(x -x ) +(x -x ) ,
+              9      2  5     7  9
+   
+                           2  2
+             c  (x)=(x -x ) +x ,
+              10      4  3    8
+   
+                           2  2
+             c  (x)=(x -x ) +x ,
+              11      5  3    9
+   
+                     2  2
+             c  (x)=x +x ,
+              12     4  8
+   
+                           2        2
+             c  (x)=(x -x ) +(x -x ) ,
+              13      4  5     9  8
+   
+                     2  2
+             c  (x)=x +x .
+              14     5  9
+   
+        An optimal solution (to five figures) is
+   
+   
+         *
+        x =(0.060947,0.59765,1.0,0.59765,0.060947,0.34377,0.5,
+                         T
+            -0.5,0.34377) ,
+   
+               *
+        and F(x )=-1.34996. (The optimal objective function is unique,
+        but is achieved for other values of x.) Five nonlinear
+                                                        *
+        constraints and one simple bound are active at x . The sample
+        solution output is given later in this section, following the
+        sample main program and problem definition.
+   
+        Two calls are made to E04UCF in order to demonstrate some of its
+        features. For the first call, the starting point is:
+   
+                                                                      T
+         x =(0.1,0.125,0.666666,0.142857,0.111111,0.2,0.25,-0.2,-0.25) .
+          0
+   
+        All objective and constraint derivatives are specified in the
+        user-provided subroutines OBJFN1 and CONFN1, i.e., the default
+        option Derivative Level =3 is used.
+   
+        On completion of the first call to E04UCF, the optimal variables
+        are perturbed to produce the initial point for a second run in
+        which the problem functions are defined by the subroutines OBJFN2
+        and CONFN2. To illustrate one of the finite-difference options in
+        E04UCF, these routines are programmed so that the first six
+        components of the objective gradient and the constant elements of
+        the Jacobian matrix are not specified; hence, the option
+        Derivative Level =0 is chosen. During computation of the finite-
+        difference intervals, the constant Jacobian elements are
+        identified and set, and E04UCF automatically increases the
+        derivative level to 2.
+   
+        The second call to E04UCF illustrates the use of the Warm Start
+        Level option to utilize the final active set, nonlinear
+        multipliers and approximate Hessian from the first run. Note that
+        Hessian = Yes was specified for the first run so that the array R
+        would contain the Cholesky factor of the approximate Hessian of
+        the Lagrangian.
+   
+        The two calls to E04UCF illustrate the alternative methods of
+        assigning default parameters. (There is no special significance
+        in the order of these assignments; an options file may just as
+        easily be used to modify parameters set by E04UEF.)
+   
+        The results are typical of those obtained from E04UCF when
+        solving well behaved (non-trivial) nonlinear problems. The
+        approximate Hessian and working set remain relatively well-
+        conditioned. Similarly the penalty parameters remain small and
+        approximately constant. The numerical results illustrate much of
+        the theoretically predicted behaviour of a quasi-Newton SQP
+        method. As x approaches the solution, only one minor iteration is
+        perfomed per major iteration, and the Norm Gz and Norm C columns
+        exhibit the fast linear convergence rate mentioned in Sections 5
+        and 6. Note that the constraint violations converge earlier than
+        the projected gradient. The final values of the project gradient
+        norm and constraint norm reflect the limiting accuracy of the two
+        quantities. It is possible to achieve almost full precision in
+        the constraint norm but only half precision in the projected
+        gradient norm. Note that the final accuracy in the nonlinear
+        constraints is considerably better than the feasibility
+        tolerance, because the constraint violations are being refined
+        during the last few iterations while the algorithm is working to
+        reduce the projected gradient norm. In this problem, the
+        constraint values and Lagrange multipliers at the solution are '
+        well balanced', i.e., all the multipliers are approximately the
+        same order of magnitude. The behaviour is typical of a well-
+        scaled problem.
+   
+        The example program is not reproduced here. The source code for
+        all example programs is distributed with the NAG Foundation
+        Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe04udf}{NAG On-line Documentation: e04udf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E04UDF(3NAG)      Foundation Library (12/10/92)      E04UDF(3NAG)
+
+
+
+          E04 -- Minimizing or Maximizing a Function                 E04UDF
+                  E04UDF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          To supply optional parameters to E04UCF from an external file.
+
+          2. Specification
+
+                 SUBROUTINE E04UDF (IOPTNS, INFORM)
+                 INTEGER          IOPTNS, INFORM
+
+          3. Description
+
+          E04UDF may be used to supply values for optional parameters to
+          E04UCF. E04UDF reads an external file and each line of the file
+          defines a single optional parameter. It is only necessary to
+          supply values for those parameters whose values are to be
+          different from their default values.
+
+          Each optional parameter is defined by a single character string
+          of up to 72 characters, consisting of one or more items. The
+          items associated with a given option must be separated by spaces,
+          or equal signs (=). Alphabetic characters may be upper or lower
+          case. The string
+
+                 Print level = 1
+
+          is an example of a string used to set an optional parameter. For
+          each option the string contains one or more of the following
+          items:
+
+          (a)   A mandatory keyword.
+
+          (b)   A phrase that qualifies the keyword.
+
+          (c)   A number that specifies an INTEGER or real value. Such
+                numbers may be up to 16 contiguous characters in Fortran
+                77's I, F, E or D formats, terminated by a space if this is
+                not the last item on the line.
+
+          Blank strings and comments are ignored. A comment begins with an
+          asterisk (*) and all subsequent characters in the string are
+          regarded as part of the comment.
+
+          The file containing the options must start with begin and must
+          finish with end An example of a valid options file is:
+
+                Begin * Example options file
+                 Print level =10
+                End
+
+          Normally each line of the file is printed as it is read, on the
+          current advisory message unit (see X04ABF), but printing may be
+          suppressed using the keyword nolist To suppress printing of begin,
+          nolist must be the first option supplied as in the file:
+
+                Begin
+                  Nolist
+                  Print level = 10
+                End
+
+          Printing will automatically be turned on again after a call to
+          E04UCF and may be turned on again at any time by the user by
+          using the keyword list.
+
+          Optional parameter settings are preserved following a call to
+          E04UCF, and so the keyword defaults is provided to allow the user
+          to reset all the optional parameters to their default values
+          prior to a subsequent call to E04UCF.
+
+          A complete list of optional parameters, their abbreviations,
+          synonyms and default values is given in Section 5.1 of the
+          document for E04UCF.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  IOPTNS -- INTEGER                                      Input
+               On entry: IOPTNS must be the unit number of the options
+               file. Constraint: 0 <= IOPTNS <= 99.
+
+           2:  INFORM -- INTEGER                                     Output
+               On exit: INFORM will be zero, if an options file with the
+               current structure has been read. Otherwise INFORM will be
+               positive. Positive values of INFORM indicate that an options
+               file may not have been successfully read as follows:
+               INFORM = 1
+                     IOPTNS is not in the range [0,99].
+
+               INFORM = 2
+                     begin was found, but end-of-file was found before end
+                     was found.
+
+               INFORM = 3
+                     end-of-file was found before begin was found.
+
+          6. Error Indicators and Warnings
+
+          If a line is not recognised as a valid option, then a warning
+          message is output on the current advisory message unit (X04ABF).
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          E04UEF may also be used to supply optional parameters to E04UCF.
+
+          9. Example
+
+          See the example for E04UCF.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe04uef}{NAG On-line Documentation: e04uef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E04UEF(3NAG)      Foundation Library (12/10/92)      E04UEF(3NAG)
+
+
+
+          E04 -- Minimizing or Maximizing a Function                 E04UEF
+                  E04UEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          To supply individual optional parameters to E04UCF.
+
+          2. Specification
+
+                 SUBROUTINE E04UEF (STRING)
+                 CHARACTER*(*)    STRING
+
+          3. Description
+
+          E04UEF may be used to supply values for optional parameters to
+          E04UCF. It is only necessary to call E04UEF for those parameters
+          whose values are to be different from their default values. One
+          call to E04UEF sets one parameter value.
+
+          Each optional parameter is defined by a single character string
+          of up to 72 characters, consisting of one or more items. The
+          items associated with a given option must be separated by spaces,
+          or equal signs (=). Alphabetic characters may be upper or lower
+          case. The string
+
+                Print level = 1
+
+          is an example of a string used to set an optional parameter. For
+          each option the string contains one or more of the following
+          items:
+
+          (a)   A mandatory keyword.
+
+          (b)   A phrase that qualifies the keyword.
+
+          (c)   A number that specifies an INTEGER or real value. Such
+                numbers may be up to 16 contiguous characters in Fortran
+                77's I, F, E or D formats, terminated by a space if this is
+                not the last item on the line.
+
+          Blank strings and comments are ignored. A comment begins with an
+          asterisk (*) and all subsequent characters in the string are
+          regarded as part of the comment.
+
+          Normally, each user-specified option is printed as it is defined,
+          on the current advisory message unit (see X04ABF), but this
+          printing may be suppressed using the keyword nolist Thus the
+          statement
+
+                CALL E04UEF (`Nolist')
+
+          suppresses printing of this and subsequent options. Printing will
+          automatically be turned on again after a call to E04UCF, and may
+          be turned on again at any time by the user, by using the keyword
+          list.
+
+          Optional parameter settings are preserved following a call to
+          E04UCF, and so the keyword defaults is provided to allow the user
+          to reset all the optional parameters to their default values by
+          the statement,
+
+                CALL E04UEF (`Defaults')
+
+          prior to a subsequent call to E04UCF.
+
+          A complete list of optional parameters, their abbreviations,
+          synonyms and default values is given in Section 5.1 of the
+          document for E04UCF.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  STRING -- CHARACTER*(*)                                Input
+               On entry: STRING must be a single valid option string. See
+               Section 3 above and Section 5.1 of the routine document for
+               E04UCF. On entry: STRING must be a single valid option
+               string. See Section 3 above and Section 5.1 of the routine
+               document for E04UCF.
+
+          6. Error Indicators and Warnings
+
+          If the parameter STRING is not recognised as a valid option
+          string, then a warning message is output on the current advisory
+          message unit (X04ABF).
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          E04UDF may also be used to supply optional parameters to E04UCF.
+
+          9. Example
+
+          See the example for E04UCF.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXe04ycf}{NAG On-line Documentation: e04ycf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     E04YCF(3NAG)      Foundation Library (12/10/92)      E04YCF(3NAG)
+
+
+
+          E04 -- Minimizing or Maximizing a Function                 E04YCF
+                  E04YCF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          E04YCF returns estimates of elements of the variance-covariance
+          matrix of the estimated regression coefficients for a nonlinear
+          least squares problem. The estimates are derived from the
+          Jacobian of the function f(x) at the solution.
+
+          This routine may be used following any one of the nonlinear
+          least-squares routines E04FCF(*), E04FDF, E04GBF(*), E04GCF,
+          E04GDF(*), E04GEF(*), E04HEF(*), E04HFF(*).
+
+          2. Specification
+
+                 SUBROUTINE E04YCF (JOB, M, N, FSUMSQ, S, V, LV, CJ, WORK,
+                1                   IFAIL)
+                 INTEGER          JOB, M, N, LV, IFAIL
+                 DOUBLE PRECISION FSUMSQ, S(N), V(LV,N), CJ(N), WORK(N)
+
+          3. Description
+
+          E04YCF is intended for use when the nonlinear least-squares
+                          T
+          function, F(x)=f (x)f(x), represents the goodness of fit of a
+          nonlinear model to observed data. The routine assumes that the
+          Hessian of F(x), at the solution, can be adequately approximated
+               T
+          by 2J J, where J is the Jacobian of f(x) at the solution. The
+          estimated variance-covariance matrix C is then given by
+
+                                 2  T  -1    T
+                        C=(sigma) (J J)     J J non-singular,
+
+                       2
+          where (sigma)  is the estimated variance of the residual at the
+                    
+
+          solution, x, given by
+
+                                               
+
+                                          2  F(x)
+                                   (sigma) = ----,
+                                             m-n
+
+          m being the number of observations and n the number of variables.
+
+          The diagonal elements of C are estimates of the variances of the
+          estimated regression coefficients. See the Chapter Introduction
+          E04 and Bard [1] and Wolberg [2] for further information on the
+          use of C.
+
+                T
+          When J J is singular then C is taken to be
+
+                                           2  T  *
+                                  C=(sigma) (J J) ,
+
+                  T  *                           T
+          where (J J)  is the pseudo-inverse of J J, but in this case the
+          parameter IFAIL is returned as non-zero as a warning to the user
+          that J has linear dependencies in its columns. The assumed rank
+          of J can be obtained from IFAIL.
+
+          The routine can be used to find either the diagonal elements of
+          C, or the elements of the jth column of C, or the whole of C.
+
+          E04YCF must be preceded by one of the nonlinear least-squares
+          routines mentioned in Section 1, and requires the parameters
+          FSUMSQ, S and V to be supplied by those routines. FSUMSQ is the
+                                    
+
+          residual sum of squares F(x), and S and V contain the singular
+          values and right singular vectors respectively in the singular
+          value decomposition of J. S and V are returned directly by the
+          comprehensive routines E04FCF(*), E04GBF(*), E04GDF(*) and
+          E04HEF(*), but are returned as part of the workspace parameter W
+          from the easy-to-use routines E04FDF, E04GCF, E04GEF(*) and
+          E04HFF(*). In the case of E04FDF, S starts at W(NS), where
+
+                          NS=6*N+2*M+M*N+1+max(1,N*(N-1)/2)
+
+          and in the cases of the remaining easy-to-use routines, S starts
+          at W(NS), where
+
+                     NS=7*N+2*M+M*N+N*(N+1)/2+1+max(1,N*(N-1)/2)
+
+          The parameter V starts immediately following the elements of S,
+          so that V starts at W(NV), where
+
+                                      NV=NS+N.
+
+          For all the easy-to-use routines the parameter LV must be
+          supplied as N. Thus a call to E04YCF following E04FDF can be
+          illustrated as
+
+
+                 CALL E04FDF (M, N, X, FSUMSQ, IW, LIW, W, LW, IFAIL)
+                 NS = 6*N + 2*M + M*N + 1 + MAX((1,(N*(N-1))/2)
+                 NV = NS + N
+                 CALL E04YCF (JOB, M, N, FSUMSQ, W(NS), W(NV),
+                *             N, CJ, WORK, IFAIL)
+
+                                                        2
+          where the parameters M, N, FSUMSQ and the (n+n ) elements W(NS),
+          WS(NS+1),..., W(NV+N*N-1) must not be altered between the calls
+          to E04FDF and E04YCF. The above illustration also holds for a
+          call to E04YCF following a call to one of E04GCF, E04GEF(*),
+          E04HFF(*) except that NS must be computed as
+
+                NS = 7*N + 2*M + M*N + (N*(N+1))/2 + 1 + MAX((1,N*(N-1))/2)
+
+          4. References
+
+          [1]   Bard Y (1974) Nonlinear Parameter Estimation. Academic
+                Press.
+
+          [2]   Wolberg J R (1967) Prediction Analysis. Van Nostrand.
+
+          5. Parameters
+
+           1:  JOB -- INTEGER                                         Input
+               On entry: which elements of C are returned as follows:
+               JOB = -1
+                     The n by n symmetric matrix C is returned.
+
+               JOB = 0
+                     The diagonal elements of C are returned.
+
+               JOB > 0
+                     The elements of column JOB of C are returned.
+                Constraint: -1 <= JOB <= N.
+
+           2:  M -- INTEGER                                           Input
+               On entry: the number m of observations (residuals f (x)).
+                                                                  i
+               Constraint: M >= N.
+
+           3:  N -- INTEGER                                           Input
+               On entry: the number n of variables (x ). Constraint: 1 <=
+                                                     j
+               N <= M.
+
+           4:  FSUMSQ -- DOUBLE PRECISION                             Input
+                                                                
+
+               On entry: the sum of squares of the residuals, F(x), at the
+                        
+
+               solution x, as returned by the nonlinear least-squares
+               routine. Constraint: FSUMSQ >= 0.0.
+
+           5:  S(N) -- DOUBLE PRECISION array                         Input
+               On entry: the n singular values of the Jacobian as returned
+               by the nonlinear least-squares routine. See Section 3 for
+               information on supplying S following one of the easy-to-use
+               routines.
+
+           6:  V(LV,N) -- DOUBLE PRECISION array               Input/Output
+               On entry: the n by n right-hand orthogonal matrix (the
+               right singular vectors) of J as returned by the nonlinear
+               least-squares routine. See Section 3 for information on
+               supplying V following one of the easy-to-use routines. On
+               exit: when JOB >= 0 then V is unchanged.
+
+               When JOB = -1 then the leading n by n part of V is
+               overwritten by the n by n matrix C. When E04YCF is called
+               with JOB = -1 following an easy-to-use routine this means
+                                                             2
+               that C is returned, column by column, in the n  elements of
+                                                  2
+               W given by W(NV),W(NV+1),...,W(NV+N -1). (See Section 3 for
+               the definition of NV).
+
+           7:  LV -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array V as declared in the
+               (sub)program from which E04YCF is called.
+               When V is passed in the workspace parameter W following one
+               of the easy-to-use least-square routines, LV must be the
+               value N.
+
+           8:  CJ(N) -- DOUBLE PRECISION array                       Output
+               On exit: with JOB = 0, CJ returns the n diagonal elements
+               of C.
+
+               With JOB = j>0, CJ returns the n elements of the jth column
+               of C.
+
+               When JOB = -1, CJ is not referenced.
+
+           9:  WORK(N) -- DOUBLE PRECISION array                  Workspace
+               When JOB = -1 or 0 then WORK is used as internal workspace.
+
+               When JOB > 0, WORK is not referenced.
+
+          10:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit. To suppress
+               the output of an error message when soft failure occurs, set
+               IFAIL to 1.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+          IFAIL= 1
+               On entry JOB < -1,
+
+               or       JOB > N,
+
+               or       N < 1,
+
+               or       M < N,
+
+               or       FSUMSQ < 0.0.
+
+          IFAIL= 2
+               The singular values are all zero, so that at the solution
+               the Jacobian matrix J has rank 0.
+
+          IFAIL> 2
+               At the solution the Jacobian matrix contains linear, or near
+               linear, dependencies amongst its columns. In this case the
+               required elements of C have still been computed based upon J
+               having an assumed rank given by (IFAIL-2). The rank is
+               computed by regarding singular values SV(j) that are not
+               larger than 10*(epsilon)*SV(1) as zero, where (epsilon) is
+               the machine precision (see X02AJF(*)). Users who expect near
+               linear dependencies at the solution and are happy with this
+               tolerance in determining rank should call E04YCF with IFAIL
+               = 1 in order to prevent termination by P01ABF(*). It is then
+               essential to test the value of IFAIL on exit from E04YCF.
+
+          IFAILOverflow
+               If overflow occurs then either an element of C is very
+               large, or the singular values or singular vectors have been
+               incorrectly supplied.
+
+          7. Accuracy
+
+          The computed elements of C will be the exact covariances
+          corresponding to a closely neighbouring Jacobian matrix J.
+
+          8. Further Comments
+
+          When JOB = -1 the time taken by the routine is approximately
+                           3
+          proportional to n . When JOB >= 0 the time taken by the routine
+                                            2
+          is approximately proportional to n .
+
+          9. Example
+
+          To estimate the variance-covariance matrix C for the least-
+          squares estimates of x , x  and x  in the model
+                                1   2      3
+
+                                            t
+                                             1
+                                   y=x + ---------
+                                      1  x t +x t
+                                          2 2  3 3
+
+          using the 15 sets of data given in the following table:
+
+                                  y    t   t    t
+                                        1   2    3
+                                 0.14  1.0 15.0 1.0
+                                 0.18  2.0 14.0 2.0
+                                 0.22  3.0 13.0 3.0
+                                 0.25  4.0 12.0 4.0
+                                 0.29  5.0 11.0 5.0
+                                 0.32  6.0 10.0 6.0
+                                 0.35  7.0  9.0 7.0
+                                 0.39  8.0  8.0 8.0
+                                 0.37  9.0  7.0 7.0
+                                 0.58 10.0  6.0 6.0
+                                 0.73 11.0  5.0 5.0
+                                 0.96 12.0  4.0 4.0
+                                 1.34 13.0  3.0 3.0
+                                 2.10 14.0  2.0 2.0
+                                 4.39 15.0  1.0 1.0
+
+          The program uses (0.5,1.0,1.5) as the initial guess at the
+          position of the minimum and computes the least-squares solution
+          using E04FDF. See the routine document E04FDF for further
+          information.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
diff --git a/src/hyper/pages/nagf.ht b/src/hyper/pages/nagf.ht
new file mode 100644
index 00000000..7a3b37d2
--- /dev/null
+++ b/src/hyper/pages/nagf.ht
@@ -0,0 +1,12302 @@
+\begin{page}{manpageXXf}{NAG On-line Documentation: f}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F(3NAG)           Foundation Library (12/10/92)           F(3NAG)
+
+
+
+          F -- Linear Algebra                             Introduction -- F
+                                     Chapter F
+                                  Linear Algebra
+
+          1. Introduction
+
+          The F Chapters of the Library are concerned with linear algebra
+          and cover a large area. This general introduction is intended to
+          help users decide which particular F Chapter is relevant to their
+          problem. There are F Chapters with the following titles:
+
+               F01 -- Matrix Factorizations
+
+               F02 -- Eigenvalues and Eigenvectors
+
+               F04 -- Simultaneous Linear Equations
+
+               F06 -- Linear Algebra Support Routines
+
+               F07 -- Linear Equations (LAPACK)
+
+          The principal problem areas addressed by the above Chapters are:
+
+               Systems of linear equations
+
+               Linear least-squares problems
+
+               Eigenvalue and singular value problems
+
+          The solution of these problems usually involves several matrix
+          operations, such as a matrix factorization followed by the
+          solution of the factorized form, and the routines for these
+          operations themselves utilize lower level support routines;
+          typically routines from Chapter F06. Most users will not normally
+          need to be concerned with these support routines.
+
+          NAG has been involved in a project, called LAPACK [1], to develop
+          a linear algebra package for modern high-performance computers
+          and some of the routines developed within that project are
+          incorporated into the Library as Chapter F07. It should be
+          emphasised that, while the LAPACK project has been concerned with
+          high-performance computers, the routines do not compromise
+          efficiency on conventional machines.
+
+          For background information on numerical algorithms for the
+          solution of linear algebra problems see Golub and Van Loan [4].
+          For some problem areas the user has the choice of selecting a
+          single routine to solve the problem, a so-called Black Box
+          routine, or selecting more than one routine to solve the problem,
+          such as a factorization routine followed by a solve routine, so-
+          called General Purpose routines. The following sections indicate
+          which chapters are relevant to particular problem areas.
+
+          2. Linear Equations
+
+          The Black Box routines for solving linear equations of the form
+
+                                  Ax=b  and  AX=B,
+
+          where A is an n by n real or complex, non-singular matrix, are to
+          be found in Chapter F04. Such equations can also be solved by
+          selecting a General Purpose factorization routine from Chapter
+          F01 and combining it with a solve routine in Chapter F04, or by
+          selecting a factorization and a solve routine from Chapter F07.
+
+          There are routines to cater for a variety of types of matrix,
+          including general, symmetric or Hermitian, symmetric or Hermitian
+          positive definite, tridiagonal, skyline and sparse matrices.
+
+          In order to select the appropriate routine, users are recommended
+          to consult the F04 Chapter Introduction in the first instance.
+
+          3. Linear Least-squares
+
+          Routines for solving linear least-squares problems of the form
+
+
+                                      T
+                            minimize r r,  where  r=b-Ax,
+                             x
+
+          where A is an m by n, possibly rank deficient, matrix are to be
+          found in Chapter F04. Linear least-squares problems can also be
+          solved by routines in the statistical Chapter G02.
+
+          In order to select the appropriate routine, users are recommended
+          to consult the F04 Chapter Introduction in the first instance,
+          but users with additional statistical requirements may prefer to
+          consult the G02 Chapter Introduction.
+
+          4. Eigenvalue Problems and Singular Value Problems
+
+          Routines for solving standard matrix eigenvalue problems of the
+          form
+
+
+                                    Ax=(lambda)x,
+
+          where A is an n by n real or complex matrix, and generalized
+          matrix eigenvalue problems of the form
+
+
+                                    Ax=(lambda)Bx
+
+          where B is also an n by n matrix are to be found in Chapter F02.
+
+          There are routines to cater for various types of matrices,
+          including general, symmetric or Hermitian and sparse matrices.
+
+          Similarly, the routines for finding singular values and/or
+          singular vectors of an m by n real or complex matrix A are to be
+          found in Chapter F02.
+
+          In order to select the appropriate routine, users are recommended
+          to consult the F02 Chapter Introduction in the first instance.
+
+          5. Matrix Factorizations
+
+          Routines for various sorts of matrix factorization are to be
+          found in Chapters F01 and F07 together with associated
+          transformation routines. In order to select the appropriate
+          routine users are recommended to consult the F01 Chapter
+          Introduction in the first instance.
+
+          6. Support Routines
+
+          Chapter F06 contains a variety of routines to perform elementary
+          algebraic operations involving scalars, vectors and matrices,
+          such as setting up a plane rotation, performing a dot product and
+          computing a matrix-vector product. Chapter F06 contains routines
+          that meet the specification of the BLAS (Basic Linear Algebra
+          Subprograms) [5, 3, 2]. The routines in this chapter will not
+          normally be required by the general user, but are intended for
+          use by those who require to build specialist linear algebra
+          modules. The BLAS are extensively used by other NAG Foundation
+          Library routines.
+
+          References
+
+          [1]   Anderson E, Bai Z, Bischof C, Demmel J, Dongarra J J, Du
+                Croz J, Greenbaum A, Hammarling S, McKenney A, Ostrouchov S
+                and Sorensen D (1992) LAPACK Users' Guide. SIAM
+                Philadelphia.
+
+          [2]   Dongarra J J, Du Croz J J, Duff I S and Hammarling S (1990)
+                A Set of Level 3 Basic Linear Algebra Subprograms. ACM
+                Trans. Math. Softw. 16 1--28.
+
+          [3]   Dongarra J J, Du Croz J J, Hammarling S and Hanson R J
+                (1988) An Extended Set of FORTRAN Basic Linear Algebra
+                Subprograms. ACM Trans. Math. Softw. 14 1--32.
+
+          [4]   Golub G H and Van Loan C F (1989) Matrix Computations (2nd
+                Edition). Johns Hopkins University Press, Baltimore,
+                Maryland.
+
+          [5]   Lawson C L, Hanson R J, Kincaid D R and Krogh F T (1979)
+                Basic Linear Algebra Subprograms for Fortran Usage. ACM
+                Trans. Math. Softw. 5 308--325.
+
+          [6]   Parlett B N (1980) The Symmetric Eigenvalue Problem.
+                Prentice-Hall.
+
+          [7]   Wilkinson J H (1965) The Algebraic Eigenvalue Problem.
+                Clarendon Press.
+
+          [8]   Wilkinson J H (1977) Some Recent Advances in Numerical
+                Linear Algebra. The State of the Art in Numerical Analysis.
+                (ed D A H Jacobs) Academic Press.
+
+          [9]   Wilkinson J H (1978) Singular Value Decomposition -- Basic
+                Aspects. Numerical Software -- Needs and Availability. (ed D
+                A H Jacobs) Academic Press.
+
+          [10]  Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf01}{NAG On-line Documentation: f01}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F01(3NAG)         Foundation Library (12/10/92)         F01(3NAG)
+
+
+
+          F01 -- Matrix Factorization                   Introduction -- F01
+                                    Chapter F01
+                               Matrix Factorization
+
+          1. Scope of the Chapter
+
+          This chapter provides facilities for matrix factorizations and
+          associated transformations.
+
+          2. Background to the Problems
+
+          An n by n matrix may be factorized as
+
+                                            T
+                                      A=PLUQ ,
+
+          where L and U are respectively lower and uper triangular
+          matrices, and P and Q are permutation matrices. This is called an
+          LU factorization. For general dense matrices it is usual to
+          choose Q=I and to then choose P to ensure that the factorization
+          is numerically stable. For sparse matrices, judicious choice of P
+          and Q ensures numerical stability as well as maintaining as much
+          sparsity as possible in the factors L and U. The LU factorization
+          is normally used in connection with the solution of the linear
+          equations
+
+
+                                        Ax=b,
+
+          whose solution, x, may then be obtained by solving in succession
+          the simpler equations
+
+                                    T
+                                Ly=P b,  Uz=y,  x=Qz
+
+          the first by forward substitution and the second by backward
+          substitution. Routines to perform this solution are to be found
+          in Chapter F04.
+
+                                                                      T
+          When A is symmetric positive-definite then we can choose U=L  and
+          Q=P, to give the Cholesky factorization. This factorization is
+          numerically stable without permutations, but in the sparse case
+          the permutations can again be used to try to maintain sparsity.
+          The Cholesky factorization is sometimes expressed as
+
+                                           T T
+                                     A=PLDL P ,
+
+          where D is a diagonal matrix with positive diagonal elements and
+          L is unit lower triangular.
+
+          The LU factorization can also be performed on rectangular
+          matrices, but in this case it is more usual to perform a QR
+          factorization. When A is an m by n (m>=n) matrix this is given by
+
+
+                                          (R)
+                                       A=Q(0),
+
+          where R is an n by n upper triangular matrix and Q is an
+          orthogonal (unitary in the complex case) matrix.
+
+          3. Recommendations on Choice and Use of Routines
+
+          Routine F07ADF performs the LU factorization of a real m by n
+          dense matrix.
+
+          The LU factorization of a sparse matrix is performed by routine
+          F01BRF. Following the use of F01BRF, matrices with the same
+          sparsity pattern may be factorized by routine F01BSF.
+
+          The Cholesky factorization of a real symmetric positive-definite
+          dense matrix is performed by routine F07FDF.
+
+          Routine F01MCF performs the Cholesky factorization of a real
+          symmetric positive-definite variable band (skyline) matrix, and a
+          general sparse symmetric positive-definite matrix may be
+          factorized using routine F01MAF.
+
+          The QR factorization of an m by n (m>=n) matrix is performed by
+          routine F01QCF in the real case, and F01RCF in the complex case.
+          Following the use of F01QCF, operations with Q may be performed
+          using routine F01QDF and some, or all, of the columns of Q may be
+          formed using routine F01QEF. Routines F01RDF and F01REF perform
+          the same tasks following the use of F01RCF.
+
+
+          F01 -- Matrix Factorizations                      Contents -- F01
+          Chapter F01
+
+          Matrix Factorizations
+
+          F01BRF  LU factorization of real sparse matrix
+
+          F01BSF  LU factorization of real sparse matrix with known
+                  sparsity pattern
+
+                    T
+          F01MAF  LL  factorization of real sparse symmetric positive-
+                  definite matrix
+
+                     T
+          F01MCF  LDL  factorization of real symmetric positive-definite
+                  variable-bandwidth matrix
+
+          F01QCF  QR factorization of real m by n matrix (m>=n)
+
+                                                                      T
+          F01QDF  Operations with orthogonal matrices, compute QB or Q B
+                  after factorization by F01QCF
+
+          F01QEF  Operations with orthogonal matrices, form columns of Q
+                  after factorization by F01QCF
+
+          F01RCF  QR factorization of complex m by n matrix (m>=n)
+
+                                                                   H
+          F01RDF  Operations with unitary matrices, compute QB or Q B after
+                  factorization by F01RCF
+
+          F01REF  Operations with unitary matrices, form columns of Q after
+                  factorization by F01RCF
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf01brf}{NAG On-line Documentation: f01brf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F01BRF(3NAG)      Foundation Library (12/10/92)      F01BRF(3NAG)
+
+
+
+          F01 -- Matrix Factorizations                               F01BRF
+                  F01BRF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F01BRF factorizes a real sparse matrix. The routine either forms
+          the LU factorization of a permutation of the entire matrix, or,
+          optionally, first permutes the matrix to block lower triangular
+          form and then only factorizes the diagonal blocks.
+
+          2. Specification
+
+                 SUBROUTINE F01BRF (N, NZ, A, LICN, IRN, LIRN, ICN, PIVOT,
+                1                   IKEEP, IW, W, LBLOCK, GROW, ABORT,
+                2                   IDISP, IFAIL)
+                 INTEGER          N, NZ, LICN, IRN(LIRN), LIRN, ICN(LICN),
+                1                 IKEEP(5*N), IW(8*N), IDISP(10), IFAIL
+                 DOUBLE PRECISION A(LICN), PIVOT, W(N)
+                 LOGICAL          LBLOCK, GROW, ABORT(4)
+
+          3. Description
+
+          Given a real sparse matrix A, this routine may be used to obtain
+          the LU factorization of a permutation of A,
+
+                                       PAQ=LU
+
+          where P and Q are permutation matrices, L is unit lower
+          triangular and U is upper triangular. The routine uses a sparse
+          variant of Gaussian elimination, and the pivotal strategy is
+          designed to compromise between maintaining sparsity and
+          controlling loss of accuracy through round-off.
+
+          Optionally the routine first permutes the matrix into block lower
+          triangular form and then only factorizes the diagonal blocks. For
+          some matrices this gives a considerable saving in storage and
+          execution time.
+
+          Extensive data checks are made; duplicated non-zeros can be
+          accumulated.
+
+          The factorization is intended to be used by F04AXF to solve
+                                                      T
+          sparse systems of linear equations Ax=b or A x=b. If several
+          matrices of the same sparsity pattern are to be factorized,
+          F01BSF should be used for the second and subsequent matrices.
+
+          The method is fully described by Duff [1].
+
+          4. References
+
+          [1]   Duff I S (1977) MA28 -- a set of Fortran subroutines for
+                sparse unsymmetric linear equations. A.E.R.E. Report R.8730.
+                HMSO.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A. Constraint: N > 0.
+
+           2:  NZ -- INTEGER                                          Input
+               On entry: the number of non-zero elements in the matrix A.
+               Constraint: NZ > 0.
+
+           3:  A(LICN) -- DOUBLE PRECISION array               Input/Output
+               On entry: A(i), for i = 1,2,...,NZ must contain the non-
+               zero elements of the sparse matrix A. They can be in any
+               order since the routine will reorder them. On exit: the non-
+               zero elements in the LU factorization. The array must not be
+               changed by the user between a call of this routine and a
+               call of F04AXF.
+
+           4:  LICN -- INTEGER                                        Input
+               On entry:
+               the dimension of the arrays A and ICN as declared in the
+               (sub)program from which F01BRF is called.
+               Since the factorization is returned in A and ICN, LICN
+               should be large enough to accommodate this and should
+               ordinarily be 2 to 4 times as large as NZ. Constraint: LICN
+               >= NZ.
+
+           5:  IRN(LIRN) -- INTEGER array                      Input/Output
+               On entry: IRN(i), for i = 1,2,...,NZ must contain the row
+               index of the non-zero element stored in A(i). On exit: the
+               array is overwritten and is not needed for subsequent calls
+               of F01BSF or F04AXF.
+
+           6:  LIRN -- INTEGER                                        Input
+               On entry:
+               the dimension of the array IRN as declared in the
+               (sub)program from which F01BRF is called.
+               It need not be as large as LICN; normally it will not need
+               to be very much greater than NZ. Constraint: LIRN >= NZ.
+
+           7:  ICN(LICN) -- INTEGER array                      Input/Output
+               On entry: ICN(i), for i = 1,2,...,NZ must contain the
+               column index of the non-zero element stored in A(i). On
+               exit: the column indices of the non-zero elements in the
+               factorization. The array must not be changed by the user
+               between a call of this routine and subsequent calls of
+               F01BSF or F04AXF.
+
+           8:  PIVOT -- DOUBLE PRECISION                              Input
+               On entry: PIVOT should have a value in the range 0.0 <=
+               PIVOT <= 0.9999 and is used to control the choice of pivots.
+               If PIVOT < 0.0, the value 0.0 is assumed, and if PIVOT > 0.
+               9999, the value 0.9999 is assumed. When searching a row for
+               a pivot, any element is excluded which is less than PIVOT
+               times the largest of those elements in the row available as
+               pivots. Thus decreasing PIVOT biases the algorithm to
+               maintaining sparsity at the expense of stability. Suggested
+               value: PIVOT = 0.1 has been found to work well on test
+               examples.
+
+           9:  IKEEP(5*N) -- INTEGER array                           Output
+               On exit: indexing information about the factorization. The
+               array must not be changed by the user between a call of this
+               routine and calls of F01BSF or F04AXF.
+
+          10:  IW(8*N) -- INTEGER array                           Workspace
+
+          11:  W(N) -- DOUBLE PRECISION array                        Output
+               On exit: if GROW = .TRUE., W(1) contains an estimate (an
+               upper bound) of the increase in size of elements encountered
+               during the factorization (see GROW); the rest of the array
+               is used as workspace.
+
+               If GROW = .FALSE., the array is not used.
+
+          12:  LBLOCK -- LOGICAL                                      Input
+               On entry: if LBLOCK = .TRUE., the matrix is pre-ordered
+               into block lower triangular form before the LU factorization
+               is performed; otherwise the entire matrix is factorized.
+               Suggested value: LBLOCK = .TRUE. unless the matrix is known
+               to be irreducible.
+
+          13:  GROW -- LOGICAL                                        Input
+               On entry: if GROW = .TRUE., then on exit W(1) contains an
+               estimate (an upper bound) of the increase in size of
+               elements encountered during the factorization. If the matrix
+               is well-scaled (see Section 8.2), then a high value for W(1)
+               indicates that the LU factorization may be inaccurate and
+               the user should be wary of the results and perhaps increase
+               the parameter PIVOT for subsequent runs (see Section 7).
+               Suggested value: GROW = .TRUE..
+
+          14:  ABORT(4) -- LOGICAL array                              Input
+               On entry:
+                    if ABORT(1) = .TRUE., the routine will exit
+                    immediately on detecting a structural singularity (one
+                    that depends on the pattern of non-zeros) and return
+
+                    IFAIL = 1; otherwise it will complete the
+                    factorization (see Section 8.3).
+
+                    If ABORT(2) = .TRUE., the routine will exit
+                    immediately on detecting a numerical singularity (one
+                    that depends on the numerical values) and return IFAIL
+                    = 2; otherwise it will complete the factorization (see
+                    Section 8.3).
+
+                    If ABORT(3) = .TRUE., the routine will exit
+                    immediately (with IFAIL = 5) when the arrays A and ICN
+                    are filled up by the previously factorized, active and
+                    unfactorized parts of the matrix; otherwise it
+                    continues so that better guidance on necessary array
+                    sizes can be given in IDISP(6) and IDISP(7), and will
+                    exit with IFAIL in the range 4 to 6. Note that there
+                    is always an immediate error exit if the array IRN is
+                    too small.
+
+                    If ABORT(4) = .TRUE., the routine exits immediately
+                    (with IFAIL = 13) if it finds duplicate elements in
+                    the input matrix. If ABORT(4) = .FALSE., the routine
+                    proceeds using a value equal to the sum of the
+                    duplicate elements. In either case details of each
+                    duplicate element are output on the current advisory
+                    message unit (see X04ABF), unless suppressed by the
+                    value of IFAIL on entry.
+                Suggested values:
+                    ABORT(1) = .TRUE.
+
+                    ABORT(2) = .TRUE.
+
+                    ABORT(3) = .FALSE.
+
+                    ABORT(4) = .TRUE..
+
+          15:  IDISP(10) -- INTEGER array                            Output
+               On exit: IDISP is used to communicate information about the
+               factorization to the user and also between a call of F01BRF
+               and subsequent calls to F01BSF or F04AXF.
+                    IDISP(1) and IDISP(2), indicate the position in arrays
+                    A and ICN of the first and last elements in the LU
+                    factorization of the diagonal blocks. (IDISP(2) gives
+                    the number of non-zeros in the factorization.)
+
+                    IDISP(3) and IDISP(4), monitor the adequacy of 'elbow
+                    room' in the arrays IRN and A/ICN respectively, by
+                    giving the number of times that the data in these
+                    arrays has been compressed during the factorization to
+                    release more storage. If either IDISP(3) or IDISP(4)
+                    is quite large (say greater than 10), it will probably
+                    pay the user to increase the size of the corresponding
+                    array(s) for subsequent runs. If either is very low or
+                    zero, then the user can perhaps save storage by
+                    reducing the size of the corresponding array(s).
+
+                    IDISP(5), gives an upper bound on the rank of the
+                    matrix.
+
+                    IDISP(6) and IDISP(7), give the minimum size of arrays
+                    IRN and A/ICN respectively which would enable a
+                    successful run on an identical matrix (but some '
+                    elbow-room' should be allowed - see Section 8).
+
+                    IDISP(8) to (10), are only used if LBLOCK = .TRUE..
+
+                    IDISP(8), gives the structural rank of the matrix.
+
+                    IDISP(9), gives the number of diagonal blocks.
+
+                    IDISP(10), gives the size of the largest diagonal
+                    block.
+
+                    IDISP(1) and IDISP(2), must not be changed by the user
+                    between a call of F01BRF and subsequent calls to
+                    F01BSF or F04AXF.
+
+          16:  IFAIL -- INTEGER                                Input/Output
+               For this routine, the normal use of IFAIL is extended to
+               control the printing of error and warning messages as well
+               as specifying hard or soft failure (see the Essential
+               Introduction).
+
+               Before entry, IFAIL must be set to a value with the decimal
+               expansion cba, where each of the decimal digits c, b and a
+               must have a value of 0 or 1.
+               a=0   specifies hard failure, otherwise soft failure;
+
+               b=0   suppresses error messages, otherwise error messages
+                     will be printed (see Section 6);
+
+               c=0   suppresses warning messages, otherwise warning
+                     messages will be printed (see Section 6).
+               The recommended value for inexperienced users is 110 (i.e.,
+               hard failure with all messages printed).
+
+               Unless the routine detects an error (see Section 6), IFAIL
+               contains 0 on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          For each error, an explanatory error message is output on the
+          current error message unit (as defined by X04AAF), unless
+          suppressed by the value of IFAIL on entry.
+
+          IFAIL=-2
+               Successful factorization of a numerically singular matrix
+               (which may also be structurally singular) (see Section 8.3).
+
+          IFAIL=-1
+               Successful factorization of a structurally singular matrix
+               (see Section 8.3).
+
+          IFAIL= 1
+               The matrix is structurally singular and the factorization
+               has been abandoned (ABORT(1) was .TRUE. on entry).
+
+          IFAIL= 2
+               The matrix is numerically singular and the factorization has
+               been abandoned (ABORT(2) was .TRUE. on entry).
+
+          IFAIL= 3
+               LIRN is too small: there is not enough space in the array
+               IRN to continue the factorization. The user is recommended
+               to try again with LIRN (and the length of IRN) equal to at
+               least IDISP(6) + N/2.
+
+          IFAIL= 4
+               LICN is much too small: there is much too little space in
+               the arrays A and ICN to continue the factorization.
+
+          IFAIL= 5
+               LICN is too small: there is not enough space in the arrays A
+               and ICN to store the factorization. If ABORT(3) was .FALSE.
+               on entry, the factorization has been completed but some of
+               the LU factors have been discarded to create space, IDISP(7)
+               then gives the minimum value of LICN (i.e., the minimum
+               length of A and ICN) required for a successful factorization
+               of the same matrix.
+
+          IFAIL= 6
+               LICN and LIRN are both too small: effectively this is a
+               combination of IFAIL = 3 and IFAIL = 5 (with ABORT(3) = .
+               FALSE.).
+
+          IFAIL= 7
+               LICN is too small: there is not enough space in the arrays A
+               and ICN for the permutation to block triangular form.
+
+          IFAIL= 8
+               On entry N <= 0.
+
+          IFAIL= 9
+               On entry NZ <= 0.
+
+          IFAIL= 10
+               On entry LICN < NZ.
+
+          IFAIL= 11
+               On entry LIRN < NZ.
+
+          IFAIL= 12
+               On entry an element of the input matrix has a row or column
+               index (i.e., an element of IRN or ICN) outside the range 1
+               to N.
+
+          IFAIL= 13
+               Duplicate elements have been found in the input matrix and
+               the factorization has been abandoned (ABORT(4) = .TRUE. on
+               entry).
+
+          7. Accuracy
+
+          The factorization obtained is exact for a perturbed matrix whose
+          (i,j)th element differs from a   by less than 3(epsilon)(rho)m
+                                        ij                              ij
+          where (epsilon) is the machine precision, (rho) is the growth
+          value returned in W(1) if GROW = .TRUE., and m   the number of
+                                                        ij
+          Gaussian elimination operations applied to element (i,j). The
+          value of m   is not greater than n and is usually much less.
+                    ij
+          Small (rho) values therefore guarantee accurate results, but
+          unfortunately large (rho) values may give a very pessimistic
+          indication of accuracy.
+
+          8. Further Comments
+
+          8.1. Timing
+
+          The time required may be estimated very roughly from the number
+          (tau) of non-zeros in the factorized form (output as IDISP(2))
+          and for this routine and its associates is
+
+                             2
+               F01BRF: 5(tau) /n units
+
+                            2
+               F01BSF: (tau) /n units
+
+               F04AXF: 2(tau) units
+
+          where our unit is the time for the inner loop of a full matrix
+                                                                  1 3
+          code (e.g. solving a full set of equations takes about  -n
+                                                                  3
+          units). Note that the faster F01BSF time makes it well worthwhile
+          to use this for a sequence of problems with the same pattern.
+
+          It should be appreciated that (tau) varies widely from problem to
+          problem. For network problems it may be little greater than NZ,
+          the number of non-zeros in A; for discretisation of 2-dimensional
+          and 3-dimensional partial differential equations it may be about
+                       1 5/3
+          3nlog n and  -n   , respectively.
+               2       2
+
+          The time taken to find the block lower triangular form (LBLOCK =
+          it is not found (LBLOCK = .FALSE.). If the matrix is irreducible
+          (IDISP(9) = 1 after a call with LBLOCK = .TRUE.) then this time
+          is wasted. Otherwise, particularly if the largest block is small
+          (IDISP(10)<<n), the consequent savings are likely to be greater.
+
+          The time taken to estimate growth (GROW = .TRUE.) is typically
+          under 2% of the overall time.
+
+          The overall time may be substantially increased if there is
+          inadequate 'elbow-room' in the arrays A, IRN and ICN. When the
+          sizes of the arrays are minimal (IDISP(6) and IDISP(7)) it can
+          execute as much as three times slower. Values of IDISP(3) and
+          IDISP(4) greater than about 10 indicate that it may be worthwhile
+          to increase array sizes.
+
+          8.2. Scaling
+
+          The use of a relative pivot tolerance PIVOT essentially
+          presupposes that the matrix is well-scaled, i.e., that the matrix
+          elements are broadly comparable in size. Practical problems are
+          often naturally well-scaled but particular care is needed for
+          problems containing mixed types of variables (for example
+          millimetres and neutron fluxes).
+
+          8.3. Singular and Rectangular Systems
+
+          It is envisaged that this routine will almost always be called
+          for square non-singular matrices and that singularity indicates
+          an error condition. However, even if the matrix is singular it is
+          possible to complete the factorization. It is even possible for
+          F04AXF to solve a set of equations whose matrix is singular
+          provided the set is consistent.
+
+          Two forms of singularity are possible. If the matrix would be
+          singular for any values of the non-zeros (e.g. if it has a whole
+          row of zeros), then we say it is structurally singular, and
+          continue only if ABORT(1) = .FALSE.. If the matrix is non-
+          singular by virtue of the particular values of the non-zeros,
+          then we say that it is numerically singular and continue only if
+          ABORT(2) = .FALSE..
+
+          Rectangular matrices may be treated by setting N to the larger of
+          the number of rows and numbers of columns and setting ABORT(1) =
+
+          Note: the soft failure option should be used (last digit of IFAIL
+          = 1) if the user wishes to factorize singular matrices with ABORT
+          (1) or ABORT(2) set to .FALSE..
+
+          8.4. Duplicated Non-zeros
+
+          The matrix A may consist of a sum of contributions from different
+          sub-systems (for example finite elements). In such cases the user
+          may rely on this routine to perform assembly, since duplicated
+          elements are summed.
+
+          9. Example
+
+          To factorize the real sparse matrix:
+
+                                 ( 5  0  0  0 0  0)
+                                 ( 0  2 -1  2 0  0)
+                                 ( 0  0  3  0 0  0)
+                                 (-2  0  0  1 1  0).
+                                 (-1  0  0 -1 2 -3)
+                                 (-1 -1  0  0 0  6)
+
+          This example program simply prints out some information about the
+          factorization as returned by F01BRF in W(1) and IDISP. Normally
+          the call of F01BRF would be followed by a call of F04AXF (see
+          Example for F04AXF).
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf01bsf}{NAG On-line Documentation: f01bsf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F01BSF(3NAG)      Foundation Library (12/10/92)      F01BSF(3NAG)
+
+
+
+          F01 -- Matrix Factorizations                               F01BSF
+                  F01BSF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F01BSF factorizes a real sparse matrix using the pivotal sequence
+          previously obtained by F01BRF when a matrix of the same sparsity
+          pattern was factorized.
+
+          2. Specification
+
+                 SUBROUTINE F01BSF (N, NZ, A, LICN, IVECT, JVECT, ICN,
+                1                   IKEEP, IW, W, GROW, ETA, RPMIN, ABORT,
+                2                   IDISP, IFAIL)
+                 INTEGER          N, NZ, LICN, IVECT(NZ), JVECT(NZ), ICN
+                1                 (LICN), IKEEP(5*N), IW(8*N), IDISP(2),
+                2                 IFAIL
+                 DOUBLE PRECISION A(LICN), W(N), ETA, RPMIN
+                 LOGICAL          GROW, ABORT
+
+          3. Description
+
+          This routine accepts as input a real sparse matrix of the same
+          sparsity pattern as a matrix previously factorized by a call of
+          F01BRF. It first applies to the matrix the same permutations as
+          were used by F01BRF, both for permutation to block triangular
+          form and for pivoting, and then performs Gaussian elimination to
+          obtain the LU factorization of the diagonal blocks.
+
+          Extensive data checks are made; duplicated non-zeros can be
+          accumulated.
+
+          The factorization is intended to be used by F04AXF to solve
+                                                      T
+          sparse systems of linear equations Ax=b or A x=b.
+
+          F01BSF is much faster than F01BRF and in some applications it is
+          expected that there will be many calls of F01BSF for each call of
+          F01BRF.
+
+          The method is fully described in Duff [1].
+
+          4. References
+
+          [1]   Duff I S (1977) MA28 -- a set of Fortran subroutines for
+                sparse unsymmetric linear equations. A.E.R.E. Report R.8730.
+                HMSO.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A. Constraint: N > 0.
+
+           2:  NZ -- INTEGER                                          Input
+               On entry: the number of non-zeros in the matrix A.
+               Constraint: NZ > 0.
+
+           3:  A(LICN) -- DOUBLE PRECISION array               Input/Output
+               On entry: A(i), for i = 1,2,...,NZ must contain the non-
+               zero elements of the sparse matrix A. They can be in any
+               order since the routine will reorder them. On exit: the non-
+               zero elements in the factorization. The array must not be
+               changed by the user between a call of this routine and a
+               call of F04AXF.
+
+           4:  LICN -- INTEGER                                        Input
+               On entry:
+               the dimension of the arrays A and ICN as declared in the
+               (sub)program from which F01BSF is called.
+               It should have the same value as it had for F01BRF.
+               Constraint: LICN >= NZ.
+
+           5:  IVECT(NZ) -- INTEGER array                             Input
+
+           6:  JVECT(NZ) -- INTEGER array                             Input
+               On entry: IVECT(i) and JVECT(i), for i = 1,2,...,NZ must
+               contain the row index and the column index respectively of
+               the non-zero element stored in A(i).
+
+           7:  ICN(LICN) -- INTEGER array                             Input
+               On entry: the same information as output by F01BRF. It must
+               not be changed by the user between a call of this routine
+               and a call of F04AXF.
+
+           8:  IKEEP(5*N) -- INTEGER array                            Input
+               On entry: the same indexing information about the
+               factorization as output from F01BRF. It must not be changed
+               between a call of this routine and a call of F04AXF.
+
+           9:  IW(8*N) -- INTEGER array                           Workspace
+
+          10:  W(N) -- DOUBLE PRECISION array                        Output
+               On exit: if GROW = .TRUE., W(1) contains an estimate (an
+               upper bound) of the increase in size of elements encountered
+               during the factorization (see GROW); the rest of the array
+               is used as workspace.
+
+               If GROW = .FALSE., the array is not used.
+
+          11:  GROW -- LOGICAL                                        Input
+               On entry: if GROW = .TRUE., then on exit W(1) contains an
+               estimate (an upper bound) of the increase in size of
+               elements encountered during the factorization. If the matrix
+               is well-scaled (see Section 8.2), then a high value for W(1)
+               indicates that the LU factorization may be inaccurate and
+               the user should be wary of the results and perhaps increase
+               the parameter PIVOT for subsequent runs (see Section 7).
+
+          12:  ETA -- DOUBLE PRECISION                                Input
+               On entry: the relative pivot threshold below which an error
+               diagnostic is provoked and IFAIL is set to 7. If ETA is
+               greater than 1.0, then no check on pivot size is made.
+                                        -4
+               Suggested value: ETA = 10  .
+
+          13:  RPMIN -- DOUBLE PRECISION                             Output
+               On exit: if ETA is less than 1.0, then RPMIN gives the
+               smallest ratio of the pivot to the largest element in the
+               row of the corresponding upper triangular factor thus
+               monitoring the stability of the factorization. If RPMIN is
+               very small it may be advisable to perform a new
+               factorization using F01BRF.
+
+          14:  ABORT -- LOGICAL                                       Input
+               On entry: if ABORT = .TRUE., the routine exits immediately
+               (with IFAIL = 8) if it finds duplicate elements in the input
+               matrix. If ABORT = .FALSE., the routine proceeds using a
+               value equal to the sum of the duplicate elements. In either
+               case details of each duplicate element are output on the
+               current advisory message unit (see X04ABF), unless
+               suppressed by the value of IFAIL on entry. Suggested value:
+               ABORT = .TRUE..
+
+          15:  IDISP(2) -- INTEGER array                              Input
+               On entry: IDISP(1) and IDISP(2) must be unchanged since the
+               previous call of F01BRF.
+
+          16:  IFAIL -- INTEGER                                Input/Output
+               For this routine, the normal use of IFAIL is extended to
+               control the printing of error and warning messages as well
+               as specifying hard or soft failure (see the Essential
+               Introduction).
+
+               Before entry, IFAIL must be set to a value with the decimal
+               expansion cba, where each of the decimal digits c, b and a
+               must have a value of 0 or 1.
+               a=0   specifies hard failure, otherwise soft failure;
+
+               b=0   suppresses error messages, otherwise error messages
+                     will be printed (see Section 6);
+
+               c=0   suppresses warning messages, otherwise warning
+                     messages will be printed (see Section 6).
+               The recommended value for inexperienced users is 110 (i.e.,
+               hard failure with all messages printed).
+
+               Unless the routine detects an error (see Section 6), IFAIL
+               contains 0 on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          For each error, an explanatory error message is output on the
+          current error message unit (as defined by X04AAF), unless
+          suppressed by the value of IFAIL on entry.
+
+          IFAIL= 1
+               On entry N <= 0.
+
+          IFAIL= 2
+               On entry NZ <= 0.
+
+          IFAIL= 3
+               On entry LICN < NZ.
+
+          IFAIL= 4
+               On entry an element of the input matrix has a row or column
+               index (i.e., an element of IVECT or JVECT) outside the range
+               1 to N.
+
+          IFAIL= 5
+               The input matrix is incompatible with the matrix factorized
+               by the previous call of F01BRF (see Section 8).
+
+          IFAIL= 6
+               The input matrix is numerically singular.
+
+          IFAIL= 7
+               A very small pivot has been detected (see Section 5, ETA).
+               The factorization has been completed but is potentially
+               unstable.
+
+          IFAIL= 8
+               Duplicate elements have been found in the input matrix and
+               the factorization has been abandoned (ABORT = .TRUE. on
+               entry).
+
+          7. Accuracy
+
+          The factorization obtained is exact for a perturbed matrix whose
+          (i,j)th element differs from a   by less than 3(epsilon)(rho)m
+                                        ij                              ij
+          where (epsilon) is the machine precision, (rho) is the growth
+          value returned in W(1) if GROW = .TRUE., and m   the number of
+                                                        ij
+          Gaussian elimination operations applied to element (i,j).
+
+          If (rho) = W(1) is very large or RPMIN is very small, then a
+          fresh call of F01BRF is recommended.
+
+          8. Further Comments
+
+          If the user has a sequence of problems with the same sparsity
+          pattern then this routine is recommended after F01BRF has been
+          called for one such problem. It is typically 4 to 7 times faster
+          but is potentially unstable since the previous pivotal sequence
+          is used. Further details on timing are given in document F01BRF.
+
+          If growth estimation is performed (GROW = .TRUE.), then the time
+          increases by between 5% and 10%. Pivot size monitoring (ETA <= 1.
+          0) involves a similar overhead.
+
+          We normally expect this routine to be entered with a matrix
+          having the same pattern of non-zeros as was earlier presented to
+          F01BRF. However there is no record of this pattern, but rather a
+          record of the pattern including all fill-ins. Therefore we permit
+          additional non-zeros in positions corresponding to fill-ins.
+
+          If singular matrices are being treated then it is also required
+          that the present matrix be sufficiently like the previous one for
+          the same permutations to be suitable for factorization with the
+          same set of zero pivots.
+
+          9. Example
+
+          To factorize the real sparse matrices
+
+                                 ( 5  0  0  0 0  0)
+                                 ( 0  2 -1  2 0  0)
+                                 ( 0  0  3  0 0  0)
+                                 (-2  0  0  1 1  0)
+                                 (-1  0  0 -1 2 -3)
+                                 (-1 -1  0  0 0  6)
+
+          and
+
+                                (10  0  0  0  0  0)
+                                ( 0 12 -3 -1  0  0)
+                                ( 0  0 15  0  0  0)
+                                (-2  0  0 10 -1  0).
+                                (-1  0  0 -5  1 -1)
+                                (-1 -2  0  0  0  6)
+
+          This example program simply prints the values of W(1) and RPMIN
+          returned by F01BSF. Normally the calls of F01BRF and F01BSF would
+          be followed by calls of F04AXF.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf01maf}{NAG On-line Documentation: f01maf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F01MAF(3NAG)      Foundation Library (12/10/92)      F01MAF(3NAG)
+
+
+
+          F01 -- Matrix Factorizations                               F01MAF
+                  F01MAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F01MAF computes an incomplete Cholesky factorization of a real
+          sparse symmetric positive-definite matrix A.
+
+          2. Specification
+
+                 SUBROUTINE F01MAF (N, NZ, A, LICN, IRN, LIRN, ICN, DROPTL,
+                1                   DENSW, WKEEP, IKEEP, IWORK, ABORT,
+                2                   INFORM, IFAIL)
+                 INTEGER          N, NZ, LICN, IRN(LIRN), LIRN, ICN(LICN),
+                1                 IKEEP(2*N), IWORK(6*N), INFORM(4), IFAIL
+                 DOUBLE PRECISION A(LICN), DROPTL, DENSW, WKEEP(3*N)
+                 LOGICAL          ABORT(3)
+
+          3. Description
+
+          F01MAF computes an incomplete Cholesky factorization
+
+                                      T T
+                                C=PLDL P ,   WAW=C+E
+
+          for the sparse symmetric positive-definite matrix A, where P is a
+          permutation matrix, L is a unit lower triangular matrix, D is a
+          diagonal matrix with positive diagonal elements, E is an error
+          matrix representing elements dropped during the factorization and
+          diagonal elements that have been modified to ensure that C is
+          positive-definite, and W is a diagonal matrix, chosen to make the
+          diagonal elements of WAW unity.
+
+           -1  -1
+          W  CW   is a pre-conditioning matrix for A, and the factorization
+          of C is intended to be used by F04MAF to solve systems of linear
+          equations Ax=b.
+
+          The permutation matrix P is chosen to reduce the amount of fill-
+          in that occurs in L and the user-supplied parameter DROPTL can
+          also be used to control the amount of fill-in that occurs.
+
+          Full details on the factorization can be found in Munksgaard [1].
+
+          F01MAF is based on the Harwell Library routine MA31A.
+
+          4. References
+
+          [1]   Munksgaard N (1980) Solving Sparse Symmetric Sets of Linear
+                Equations by Pre-conditioned Conjugate Gradients. ACM Trans.
+                Math. Softw. 6 206--219.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A. Constraint: N >= 1.
+
+           2:  NZ -- INTEGER                                          Input
+               On entry: the number of non-zero elements in the upper
+               triangular part of the matrix A, including the number of
+               elements on the leading diagonal. Constraint: NZ >= N.
+
+           3:  A(LICN) -- DOUBLE PRECISION array               Input/Output
+               On entry: the first NZ elements of the array A must contain
+               the non-zero elements of the upper triangular part of the
+               sparse positive-definite symmetric matrix A, including the
+               elements on the leading diagonal. On exit: the first (NZ-N)
+               elements of A contain the elements above the diagonal of the
+               matrix WAW, where W is a diagonal matrix whose ith diagonal
+                              -1/2
+               element is w =a    . These elements are returned in order by
+                           i  ii
+               rows and the value returned in ICN(k) gives the column index
+               of the element returned in A(k). The value w  is returned in
+                                                           i
+               the ith element of the array WKEEP. The remaining LROW-NZ+N
+               elements of A, where LROW is the value returned in INFORM(1),
+               return details of the factorization for use by F04MAF.
+
+           4:  LICN -- INTEGER                                        Input
+               On entry:
+               the dimension of the array A as declared in the (sub)program
+               from which F01MAF is called.
+               If fill-in is expected during the factorization, then a
+               larger value of LICN will allow fewer elements to be dropped
+               during the factorization, thus giving a more accurate
+               factorization, which in turn will almost certainly mean that
+               fewer iterations will be required by F04MAF. Constraint:
+               LICN>=2*NZ.
+
+           5:  IRN(LIRN) -- INTEGER array                      Input/Output
+               On entry: IRN(k), for k = 1,2,...,NZ must contain the row
+               index of the non-zero element of the matrix A supplied in A
+               (k). On exit: the first LCOL elements of IRN, where LCOL is
+               the value returned in INFORM(2), return details of the
+               factorization for use by F04MAF.
+
+           6:  LIRN -- INTEGER                                        Input
+               On entry:
+               the dimension of the array IRN as declared in the
+               (sub)program from which F01MAF is called.
+               LIRN must be at least NZ, but, as with LICN, if fill-in is
+               expected then a larger value of LIRN will allow a more
+               accurate factorization. For this purpose LIRN should exceed
+               NZ by the same amount that LICN exceeds 2*NZ. Constraint:
+               LIRN >= NZ.
+
+           7:  ICN(LICN) -- INTEGER array                      Input/Output
+               On entry: ICN(k), for k = 1,2,...,NZ must contain the column
+               index of the non-zero element of the matrix A supplied in A
+               (k). Thus a  =A(k), where i = IRN(k) and j = ICN(k). On
+                          ij
+               exit: the first (NZ-N) elements of ICN give the column
+               indices of the first (NZ-N) elements returned in A. The
+               remaining LROW - NZ + N elements of ICN return details of
+               the factorization for use by F04MAF.
+
+           8:  DROPTL -- DOUBLE PRECISION                      Input/Output
+               On entry: a value in the range [-1.0,1.0] to be used as a
+               tolerance in deciding whether or not to drop elements during
+                                                                     (k+1)
+               the factorization. At the kth pivot step the element a
+                                                                     ij
+               is dropped if it would cause fill-in and if
+                                                   
+
+                               (k+1)              / (k) (k)
+                             |a     |<|DROPTL|*  / a   a   .
+                               ij              \/   ii  jj
+               If DROPTL is supplied as negative, then it is not altered
+               during the factorization and so is unchanged on exit, but if
+               DROPTL is supplied as positive then it may be altered by the
+               routine with the aim of obtaining an accurate factorization
+               in the space available. If DROPTL is supplied as -1.0, then
+               no fill-in will occur during the factorization; and if
+               DROPTL is supplied as 0.0 then a complete factorization is
+               performed. On exit: may be overwritten with the value used
+               by the routine in order to obtain an accurate factorization
+               in the space available, if DROPTL > 0.0 on entry.
+
+           9:  DENSW -- DOUBLE PRECISION                       Input/Output
+               On entry: a value in the range [0.0,1.0] to be used in
+               deciding whether or not to regard the active part of the
+               matrix at the kth pivot step as being full. If the ratio of
+               non-zero elements to the total number of elements is greater
+               than or equal to DENSW, then the active part is regarded as
+               full. If DENSW < 1.0, then the storage used is likely to
+               increase compared to the case where DENSW = 0, but the
+               execution time is likely to decrease. Suggested value: DENSW
+               = 0.8. On exit: if on entry DENSW is not in the range
+               [0.0,1.0], then it is set to 0.8. Otherwise it is unchanged.
+
+          10:  WKEEP(3*N) -- DOUBLE PRECISION array                  Output
+               On exit: information which must be passed unchanged to
+               F04MAF. The first N elements contain the values w , for
+                                                                i
+               i=1,2,...,n, and the next N elements contain the diagonal
+               elements of D.
+
+          11:  IKEEP(2*N) -- INTEGER array                           Output
+               On exit: information which must be passed unchanged to
+               F04MAF.
+
+          12:  IWORK(6*N) -- INTEGER array                        Workspace
+
+          13:  ABORT(3) -- LOGICAL array                              Input
+               On entry:
+                    if ABORT(1) = .TRUE., the routine will exit
+                    immediately on detecting duplicate elements and return
+                    IFAIL = 5. Otherwise when ABORT(1) = .FALSE., the
+                    calculations will continue using the sum of the
+                    duplicate entries. In either case details of the
+                    duplicate elements are output on the current advisory
+                    message unit (see X04ABF), unless suppressed by the
+                    value of IFAIL on entry.
+
+                    If ABORT(2) = .TRUE., the routine will exit
+                    immediately on detecting a zero or negative pivot
+                    element and return IFAIL = 6. Otherwise when ABORT(2)
+                    = .FALSE., the zero or negative pivot element will be
+                    modified to ensure positive-definiteness and a message
+                    will be printed on the current advisory message unit,
+                    unless suppressed by the value of IFAIL on entry.
+
+                    If ABORT(3) = .TRUE., the routine will exit
+                    immediately if the arrays A and ICN have been filled
+                    up and return IFAIL = 7. Otherwise when ABORT(3) = .
+                    FALSE., the data in the arrays is compressed to
+                    release more storage and a message will be printed on
+                    the current advisory message unit, unless suppressed
+                    by the value of IFAIL on entry. If DROPTL is positive
+                    on entry, it may be modified in order to allow a
+                    factorization to be completed in the available space.
+               Suggested values:
+                    ABORT(1) = .TRUE.,
+
+                    ABORT(2) = .TRUE.,
+
+                    ABORT(3) = .TRUE..
+
+          14:  INFORM(4) -- INTEGER array                            Output
+               On exit:
+                    INFORM(1) returns the number of elements of A and ICN
+                    that have been used by the routine. Thus at least the
+                    first INFORM(1) elements of A and of ICN must be
+                    supplied to F04MAF.
+
+                    Similarly, INFORM(2) returns the number of elements of
+                    IRN that have been used by the routine and so at least
+                    the first INFORM(2) elements must be supplied to
+                    F04MAF.
+
+                    INFORM(3) returns the number of entries supplied in A
+                    that corresponded to diagonal and duplicate elements.
+                    If no duplicate entries were found, then INFORM(3)
+                    will return the value of N.
+
+                    INFORM(4) returns the value k of the pivot step from
+                    which the active matrix was regarded as full.
+               INFORM must be passed unchanged to F04MAF.
+
+          15:  IFAIL -- INTEGER                                Input/Output
+               For this routine, the normal use of IFAIL is extended to
+               control the printing of error and warning messages as well
+               as specifying hard or soft failure (see the Essential
+               Introduction).
+
+               Before entry, IFAIL must be set to a value with the decimal
+               expansion cba, where each of the decimal digits c, b and a
+               must have a value of 0 or 1.
+               a=0   specifies hard failure, otherwise soft failure;
+
+               b=0   suppresses error messages, otherwise error messages
+                     will be printed (see Section 6);
+
+               c=0   suppresses warning messages, otherwise warning
+                     messages will be printed (see Section 6).
+               The recommended value for inexperienced users is 110 (i.e.,
+               hard failure with all messages printed).
+
+               Unless the routine detects an error (see Section 6), IFAIL
+               contains 0 on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          For each error, an explanatory error message is output on the
+          current error message unit (as defined by X04AAF), unless
+          suppressed by the value of IFAIL on entry.
+
+          IFAIL= 1
+               On entry N < 1,
+
+               or       NZ < N,
+
+               or       LIRN < NZ,
+
+               or       LICN<2*NZ.
+
+          IFAIL= 2
+               One of the conditions 0 < IRN(k) <= ICN(k) <= N is not
+               satisfied so that A(k) is not in the upper triangle of the
+               matrix. No further computation is attempted.
+
+          IFAIL= 3
+               One of the diagonal elements of the matrix A is zero or
+               negative so that A is not positive-definite. No further
+               computation is attempted.
+
+          IFAIL= 4
+               The available space has been used and no further
+               compressions are possible. The user should either increase
+               DROPTL, or allocate more space to A, IRN and ICN.
+
+               For all the remaining values of IFAIL the computations will
+               continue in the case of soft failure, so that more than one
+               advisory message may be printed.
+
+          IFAIL= 5
+               Duplicate elements have been detected and ABORT(1) = .TRUE..
+
+          IFAIL= 6
+               A zero or negative pivot element has been detected during
+               the factorization and ABORT(2) = .TRUE..
+
+               This should not happen if A is an M-matrix (see Munksgaard
+               [1]), but may occur for other types of positive-definite
+               matrix.
+
+          IFAIL= 7
+               The available space has been used and ABORT(3) = .TRUE..
+
+          7. Accuracy
+
+          The accuracy of the factorization will be determined by the size
+          of the elements that are dropped and the size of the
+          modifications made to the diagonal elements. If these sizes are
+          small then the computed factors will correspond to a matrix close
+          to A and the number of iterations required by F04MAF will be
+          small. The more incomplete the factorization, the higher the
+          number of iterations required by F04MAF.
+
+          8. Further Comments
+
+          The time taken by the routine will depend upon the sparsity
+          pattern of the matrix and the number of fill-ins that occur
+          during the factorization. At the very least the time taken can be
+          expected to be roughly proportional to n(tau), where (tau) is the
+          number of non-zeros.
+          The routine is intended for use with positive-definite matrices,
+          but the user is warned that it will not necessarily detect non-
+          positive-definiteness. Indeed the routine may return a
+          factorization that can satisfactorily be used by F04MAF even when
+          A is not positive-definite, but this should not be relied upon as
+          F04MAF may not converge.
+
+          9. Example
+
+          The example program illustrates the use of F01MAF in conjunction
+          with F04MAF to solve the 16 linear equations Ax=b, where
+
+                          (1 z     z                      )
+                          (z 1 z     z                    )
+                          (  z 1 z     z                  )
+                          (    z 1 0     z                )
+                          (z     0 1 z     z              )
+                          (  z     z 1 z     z            )
+                          (    z     z 1 z     z          )
+                          (      z     z 1 0     z        )
+                        A=(        z     0 1 z     z      ).
+                          (          z     z 1 z     z    )
+                          (            z     z 1 z     z  )
+                          (              z     z 1 0     z)
+                          (                z     0 1 z    )
+                          (                  z     z 1 z  )
+                          (                    z     z 1 z)
+                          (                      z     z 1)
+
+           T ( 1  1  1  1  1      1  1      1  1  1  1  1)
+          b =( -  -  -  -  - 0 0  -  - 0 0  -  -  -  -  -),
+             ( 2  4  4  2  4      4  4      4  2  4  4  2)
+
+                    1
+          where z=- -.
+                    4
+
+          The n by n matrix A arises in the solution of Laplace's equation
+          in a unit square, using a 5-point formula with a 6 by 6
+          discretisation, with unity on the boundaries.
+
+          The drop tolerance, DROPTL, is taken as 0.1, and the density
+          factor, DENSW, is taken as 0.8. The value IFAIL = 111 is used so
+          that advisory and error messages will be printed, but soft
+          failure would occur if IFAIL were returned as non-zero.
+
+          A relative accuracy of about 0.0001 is requested in the solution
+          from F04MAF, with a maximum of 50 iterations.
+
+          The example program for F02FJF illustrates the use of F01MAF and
+          F04MAF in solving an eigenvalue problem.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf01mcf}{NAG On-line Documentation: f01mcf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F01MCF(3NAG)      Foundation Library (12/10/92)      F01MCF(3NAG)
+
+
+
+          F01 -- Matrix Factorizations                               F01MCF
+                  F01MCF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F01MCF computes the Cholesky factorization of a real symmetric
+          positive-definite variable-bandwidth matrix.
+
+          2. Specification
+
+                 SUBROUTINE F01MCF (N, A, LAL, NROW, AL, D, IFAIL)
+                 INTEGER          N, LAL, NROW(N), IFAIL
+                 DOUBLE PRECISION A(LAL), AL(LAL), D(N)
+
+          3. Description
+
+          This routine determines the unit lower triangular matrix L and
+                                                                   T
+          the diagonal matrix D in the Cholesky factorization A=LDL  of a
+          symmetric positive-definite variable-bandwidth matrix A of order
+          n. (Such a matrix is sometimes called a 'sky-line' matrix.)
+
+          The matrix A is represented by the elements lying within the
+          envelope of its lower triangular part, that is, between the first
+          non-zero of each row and the diagonal (see Section 9 for an
+          example). The width NROW(i) of the ith row is the number of
+          elements between the first non-zero element and the element on
+          the diagonal, inclusive. Although, of course, any matrix
+          possesses an envelope as defined, this routine is primarily
+          intended for the factorization of symmetric positive-definite
+          matrices with an average bandwidth which is small compared with n
+          (also see Section 8).
+
+          The method is based on the property that during Cholesky
+          factorization there is no fill-in outside the envelope.
+
+          The determination of L and D is normally the first of two steps
+          in the solution of the system of equations Ax=b. The remaining
+                                        T
+          step, viz. the solution of LDL x=b may be carried out using
+          F04MCF.
+
+          4. References
+
+          [1]   Jennings A (1966) A Compact Storage Scheme for the Solution
+                of Symmetric Linear Simultaneous Equations. Comput. J. 9
+                281--285.
+
+          [2]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A. Constraint: N >= 1.
+
+           2:  A(LAL) -- DOUBLE PRECISION array                       Input
+               On entry: the elements within the envelope of the lower
+               triangle of the positive-definite symmetric matrix A, taken
+               in row by row order. The following code assigns the matrix
+               elements within the envelope to the correct elements of the
+               array:
+
+                     K = 0
+                     DO 20 I = 1, N
+                     DO 10 J = I-NROW(I)+1, I
+                     K = K + 1
+                     A(K) = matrix (I,J)
+                     10    CONTINUE
+                     20 CONTINUE
+
+               See also Section 8.
+
+           3:  LAL -- INTEGER                                         Input
+               On entry: the smaller of the dimensions of the arrays A and
+               AL as declared in the calling (sub)program from which F01MCF
+               is called. Constraint: LAL >= NROW(1) + NROW(2) +... + NROW(
+               n).
+
+           4:  NROW(N) -- INTEGER array                               Input
+               On entry: NROW(i) must contain the width of row i of the
+               matrix A, i.e., the number of elements between the first
+               (leftmost) non-zero element and the element on the diagonal,
+               inclusive. Constraint: 1 <= NROW(i) <= i, for i=1,2,...,n.
+
+           5:  AL(LAL) -- DOUBLE PRECISION array                     Output
+               On exit: the elements within the envelope of the lower
+               triangular matrix L, taken in row by row order. The envelope
+               of L is identical to that of the lower triangle of A. The
+               unit diagonal elements of L are stored explicitly. See also
+               Section 8.
+
+           6:  D(N) -- DOUBLE PRECISION array                        Output
+               On exit: the diagonal elements of the the diagonal matrix D
+               . Note that the determinant of A is equal to the product of
+               these diagonal elements. If the value of the determinant is
+               required it should not be determined by forming the product
+               explicitly, because of the possibility of overflow or
+               underflow. The logarithm of the determinant may safely be
+               formed from the sum of the logarithms of the diagonal
+               elements.
+
+           7:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry N < 1,
+
+               or       for some i, NROW(i) < 1 or NROW(i) > i,
+
+               or       LAL < NROW(1) + NROW(2) +... + NROW(N).
+
+          IFAIL= 2
+               A is not positive-definite, or this property has been
+               destroyed by rounding errors. The factorization has not been
+               completed.
+
+          IFAIL= 3
+               A is not positive-definite, or this property has been
+               destroyed by rounding errors. The factorization has been
+               completed but may be very inaccurate (see Section 7).
+
+          7. Accuracy
+
+          If IFAIL = 0 on exit, then the computed L and D satisfy the
+                      T
+          relation LDL =A+F, where
+
+                                        2
+                              ||F|| <=km (epsilon)max a
+                                   2                 i ii
+
+          and
+
+                                       2
+                             ||F|| <=km (epsilon)||A|| ,
+                                  2                   2
+
+          where k is a constant of order unity, m is the largest value of
+          NROW(i), and (epsilon) is the machine precision. See Wilkinson
+          and Reinsch [2], pp 25--27, 54--55. If IFAIL = 3 on exit, then
+          the factorization has been completed although the matrix was not
+          positive-definite. However the factorization may be very
+          inaccurate and should be used only with great caution. For
+          instance, if it is used to solve a set of equations Ax=b using
+          F04MCF, the residual vector b-Ax should be checked.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          the sum of squares of the values of NROW(i).
+
+          The distribution of row widths may be very non-uniform without
+          undue loss of efficiency. Moreover, the routine has been designed
+          to be as competitive as possible in speed with routines designed
+          for full or uniformly banded matrices, when applied to such
+          matrices.
+
+          Unless otherwise stated in the Users' Note for your
+          implementation, the routine may be called with the same actual
+          array supplied for parameters A and AL, in which case L
+          overwrites the lower triangle of A. However this is not standard
+          Fortran 77 and may not work in all implementations.
+
+          9. Example
+
+          To obtain the Cholesky factorization of the symmetric matrix,
+          whose lower triangle is:
+
+                                 (1               )
+                                 (2  5            )
+                                 (0  3 13         )
+                                 (0  0  0 16      ).
+                                 (5 14 18  8 55   )
+                                 (0  0  0 24 17 77)
+
+          For this matrix, the elements of NROW must be set to 1, 2, 2, 1,
+          5, 3, and the elements within the envelope must be supplied in
+          row order as:
+
+          1, 2, 5, 3, 13, 16, 5, 14, 18, 8, 55, 24, 17, 77.
+
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf01qcf}{NAG On-line Documentation: f01qcf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F01QCF(3NAG)      Foundation Library (12/10/92)      F01QCF(3NAG)
+
+
+
+          F01 -- Matrix Factorizations                               F01QCF
+                  F01QCF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F01QCF finds the QR factorization of the real m by n matrix A,
+          where m>=n.
+
+          2. Specification
+
+                 SUBROUTINE F01QCF (M, N, A, LDA, ZETA, IFAIL)
+                 INTEGER          M, N, LDA, IFAIL
+                 DOUBLE PRECISION A(LDA,*), ZETA(*)
+
+          3. Description
+
+          The m by n matrix A is factorized as
+
+                    (R)
+                 A=Q(0)       when m>n,
+
+                 A=QR         when m=n,
+
+          where Q is an m by m orthogonal matrix and R is an n by n upper
+          triangular matrix. The factorization is obtained by Householder's
+          method. The kth transformation matrix, Q , which is used to
+                                                  k
+          introduce zeros into the kth column of A is given in the form
+
+                                         (I 0 )
+                                      Q =(0 T )
+                                       k (   k)
+
+          where
+
+                                             T
+                                     T =I-u u ,
+                                      k    k k
+
+                                       ((zeta) )
+                                       (      k)
+                                    u =(z      ),
+                                     k ( k     )
+
+          (zeta)  is a scalar and z  is an (m-k) element vector. (zeta)
+                k                  k                                   k
+          and z  are chosen to annihilate the elements below the triangular
+               k
+          part of A.
+
+          The vector u  is returned in the kth element of the array ZETA
+                      k
+          and in the kth column of A, such that (zeta)  is in ZETA(k) and
+                                                      k
+          the elements of z  are in A(k+1,k),...,A(m,k). The elements of R
+                           k
+          are returned in the upper triangular part of A.
+
+          Q is given by
+
+                                                 T
+                                  Q=(Q Q   ...Q ) .
+                                      n n-1    1
+
+          Good background descriptions to the QR factorization are given in
+          Dongarra et al [1] and Golub and Van Loan [2], but note that this
+          routine is not based upon LINPACK routine DQRDC.
+
+          4. References
+
+          [1]   Dongarra J J, Moler C B, Bunch J R and Stewart G W (1979)
+                LINPACK Users' Guide. SIAM, Philadelphia.
+
+          [2]   Golub G H and Van Loan C F (1989) Matrix Computations (2nd
+                Edition). Johns Hopkins University Press, Baltimore,
+                Maryland.
+
+          [3]   Wilkinson J H (1965) The Algebraic Eigenvalue Problem.
+                Oxford University Press.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: m, the number of rows of A. Constraint: M >= N.
+
+           2:  N -- INTEGER                                           Input
+               On entry: n, the number of columns of A.
+
+               When N = 0 then an immediate return is effected.
+               Constraint: N >= 0.
+
+           3:  A(LDA,*) -- DOUBLE PRECISION array              Input/Output
+               Note: the second dimension of the array A must be at least
+               max(1,n).
+               On entry: the leading m by n part of the array A must
+               contain the matrix to be factorized. On exit: the n by n
+               upper triangular part of A will contain the upper triangular
+               matrix R and the m by n strictly lower triangular part of A
+               will contain details of the factorization as described in
+               Section 3.
+
+           4:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F01QCF is called.
+               Constraint: LDA >= max(1,M).
+
+           5:  ZETA(*) -- DOUBLE PRECISION array                     Output
+               Note: the dimension of the array ZETA must be at least max
+               (1,n) On exit: ZETA(k) contains the scalar (zeta)  for the k
+                                                                k
+               th transformation. If T =I then ZETA(k)=0.0, otherwise ZETA(
+                                      k
+               k) contains (zeta)  as described in Section 3 and (zeta)  is
+                                 k                                     k
+
+               always in the range (1.0, \/2.0).
+
+           6:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL=-1
+               On entry M < N,
+
+               or       N < 0,
+
+               or       LDA < M.
+
+          7. Accuracy
+
+          The computed factors Q and R satisfy the relation
+
+                                       (R)
+                                      Q(0)=A+E,
+
+          where
+
+                             |||E|||<=c(epsilon)|||A|||,
+
+          and (epsilon) is the machine precision (see X02AJF(*)), c is a
+          modest function of m and n and |||.||| denotes the spectral (two)
+          norm.
+
+          8. Further Comments
+
+          The approximate number of floating-point operations is given by
+            2
+          2n (3m-n)/3.
+
+          Following the use of this routine the operations
+
+                                                 T
+                                 B:=QB  and  B:=Q B,
+
+          where B is an m by k matrix, can be performed by calls to F01QDF.
+          The operation B:=QB can be obtained by the call:
+
+
+                IFAIL = 0
+                CALL F01QDF('No transpose', 'Separate', M, N, A, LDA, ZETA,
+               *            K, B, LDB, WORK, IFAIL)
+
+                  T
+          and B:=Q B can be obtained by the call:
+
+
+                IFAIL = 0
+                CALL F01QDF('Transpose', 'Separate', M, N, A, LDA, ZETA,
+               *            K, B, LDB, WORK, IFAIL)
+
+          In both cases WORK must be a k element array that is used as
+          workspace. If B is a one-dimensional array (single column) then
+          the parameter LDB can be replaced by M. See F01QDF for further
+          details.
+
+          The first k columns of the orthogonal matrix Q can either be
+          obtained by setting B to the first k columns of the unit matrix
+          and using the first of the above two calls, or by calling F01QEF,
+          which overwrites the k columns of Q on the first k columns of the
+          array A. Q is obtained by the call:
+
+
+               CALL F01QEF('Separate', M, N, K, A, LDA, ZETA, WORK, IFAIL)
+
+          As above WORK must be a k element array. If k is larger than N,
+          then A must have been declared to have at least k columns.
+
+          Operations involving the matrix R can readily be performed by the
+          Level 2 BLAS routines DTRSV and DTRMV (see Chapter F06), but note
+          that no test for near singularity of R is incorporated in DTRSV.
+          If R is singular, or nearly singular then F02WUF(*) can be used
+          to determine the singular value decomposition of R.
+
+          9. Example
+
+          To obtain the QR factorization of the 5 by 3 matrix
+
+                                   (2.0  2.5  2.5)
+                                   (2.0  2.5  2.5)
+                                 A=(1.6 -0.4  2.8).
+                                   (2.0 -0.5  0.5)
+                                   (1.2 -0.3 -2.9)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf01qdf}{NAG On-line Documentation: f01qdf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F01QDF(3NAG)      Foundation Library (12/10/92)      F01QDF(3NAG)
+
+
+
+          F01 -- Matrix Factorizations                               F01QDF
+                  F01QDF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F01QDF performs one of the transformations
+
+                                               T
+                                  B:=QB or B:=Q B,
+
+          where B is an m by ncolb real matrix and Q is an m by m
+          orthogonal matrix, given as the product of Householder
+          transformation matrices.
+
+          This routine is intended for use following F01QCF or F01QFF(*).
+
+          2. Specification
+
+                 SUBROUTINE F01QDF (TRANS, WHERET, M, N, A, LDA, ZETA,
+                1                   NCOLB, B, LDB, WORK, IFAIL)
+                 INTEGER          M, N, LDA, NCOLB, LDB, IFAIL
+                 DOUBLE PRECISION A(LDA,*), ZETA(*), B(LDB,*), WORK(*)
+                 CHARACTER*1      TRANS, WHERET
+
+          3. Description
+
+          Q is assumed to be given by
+
+                                                 T
+                                  Q=(Q Q   ...Q ) ,
+                                      n n-1    1
+
+          Q  being given in the form
+           k
+
+                                         (I 0 )
+                                      Q =(0 T )
+                                       k (   k)
+
+          where
+
+                                             T
+                                     T =I-u u ,
+                                      k    k k
+
+                                       ((zeta) )
+                                       (      k)
+                                    u =(z      ),
+                                     k ( k     )
+
+          (zeta)  is a scalar and z  is an (m-k) element vector. z  must be
+                k                  k                              k
+          supplied in the kth column of A in elements A(k+1,k),...,A(m,k)
+          and (zeta)  must be supplied either in A(k,k) or in ZETA(k),
+                    k
+          depending upon the parameter WHERET.
+
+          To obtain Q explicitly B may be set to I and pre-multiplied by Q.
+                                                 T
+          This is more efficient than obtaining Q .
+
+          4. References
+
+          [1]   Golub G H and Van Loan C F (1989) Matrix Computations (2nd
+                Edition). Johns Hopkins University Press, Baltimore,
+                Maryland.
+
+          [2]   Wilkinson J H (1965) The Algebraic Eigenvalue Problem.
+                Oxford University Press.
+
+          5. Parameters
+
+           1:  TRANS -- CHARACTER*1                                   Input
+               On entry: the operation to be performed as follows:
+               TRANS = 'N' (No transpose)
+                     Perform the operation B:=QB.
+
+               TRANS = 'T' or 'C' (Transpose)
+                                               T
+                     Perform the operation B:=Q B.
+                Constraint: TRANS must be one of 'N', 'T' or 'C'.
+
+           2:  WHERET -- CHARACTER*1                                  Input
+               On entry: indicates where the elements of (zeta) are to be
+               found as follows:
+               WHERET = 'I' (In A)
+                     The elements of (zeta) are in A.
+
+               WHERET = 'S' (Separate)
+                     The elements of (zeta) are separate from A, in ZETA.
+               Constraint: WHERET must be one of 'I' or 'S'.
+
+           3:  M -- INTEGER                                           Input
+               On entry: m, the number of rows of A. Constraint: M >= N.
+
+           4:  N -- INTEGER                                           Input
+               On entry: n, the number of columns of A.
+
+               When N = 0 then an immediate return is effected.
+               Constraint: N >= 0.
+
+           5:  A(LDA,*) -- DOUBLE PRECISION array                     Input
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the leading m by n strictly lower triangular part
+               of the array A must contain details of the matrix Q. In
+               addition, when WHERET = 'I', then the diagonal elements of A
+               must contain the elements of (zeta) as described under the
+               argument ZETA below.
+
+               When WHERET = 'S', the diagonal elements of the array A are
+               referenced, since they are used temporarily to store the
+               (zeta) , but they contain their original values on return.
+                     k
+
+           6:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F01QDF is called.
+               Constraint: LDA >= max(1,M).
+
+           7:  ZETA(*) -- DOUBLE PRECISION array                      Input
+               Note: when WHERET = 'S', the dimension of the array ZETA
+               must be greater than or equal to max(1,N). On entry: if
+               WHERET = 'S', the array ZETA must contain the elements of
+               (zeta). If ZETA(k) = 0.0 then T  is assumed to be I
+                                              k
+               otherwise ZETA(k) is assumed to contain (zeta) .
+                                                             k
+
+               When WHERET = 'I', ZETA is not referenced.
+
+           8:  NCOLB -- INTEGER                                       Input
+               On entry: ncolb, number of columns of B.
+
+               When NCOLB = 0 then an immediate return is effected.
+               Constraint: NCOLB >= 0.
+
+           9:  B(LDB,*) -- DOUBLE PRECISION array              Input/Output
+               Note: the second dimension of the array B must be at least
+               max(1,NCOLB).
+               On entry: the leading m by ncolb part of the array B must
+               contain the matrix to be transformed. On exit: B is
+               overwritten by the transformed matrix.
+
+          10:  LDB -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array B as declared in the
+               (sub)program from which F01QDF is called.
+               Constraint: LDB >= max(1,M).
+
+          11:  WORK(*) -- DOUBLE PRECISION array                  Workspace
+               Note: the dimension of the array WORK must be at least
+               max(1,NCOLB).
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL=-1
+               On entry TRANS /= 'N', 'T' or 'C',
+
+               or       WHERET /= 'I' or 'S',
+
+               or       M < N,
+
+               or       N < 0,
+
+               or       LDA < M,
+
+               or       NCOLB < 0,
+
+               or       LDB < M.
+
+          7. Accuracy
+
+                                                T
+          Letting C denote the computed matrix Q B, C satisfies the
+          relation
+
+                                       QC=B+E,
+
+          where
+
+                               ||E||<=c(epsilon)||B||,
+
+          and (epsilon) the machine precision (see X02AJF(*)), c is a
+          modest function of m and |||.||| denotes the spectral (two) norm.
+          An equivalent result holds for the computed matrix QB. See also
+          Section 7 of F01QCF.
+
+          8. Further Comments
+
+          The approximate number of floating-point operations is given by
+          2n(2m-n)ncolb.
+
+          9. Example
+
+                                T
+          To obtain the matrix Q B for the matrix B given by
+
+                                     ( 1.1  0.00)
+                                     ( 0.9  0.00)
+                                   B=( 0.6  1.32)
+                                     ( 0.0  1.10)
+                                     (-0.8 -0.26)
+
+          following the QR factorization of the 5 by 3 matrix A given by
+
+                                   (2.0  2.5  2.5)
+                                   (2.0  2.5  2.5)
+                                 A=(1.6 -0.4  2.8).
+                                   (2.0 -0.5  0.5)
+                                   (1.2 -0.3 -2.9)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf01qef}{NAG On-line Documentation: f01qef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F01QEF(3NAG)      Foundation Library (12/10/92)      F01QEF(3NAG)
+
+
+
+          F01 -- Matrix Factorizations                               F01QEF
+                  F01QEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F01QEF returns the first ncolq columns of the real m by m
+          orthogonal matrix Q, where Q is given as the product of
+          Householder transformation matrices.
+
+          This routine is intended for use following F01QCF or F01QFF(*).
+
+          2. Specification
+
+                 SUBROUTINE F01QEF (WHERET, M, N, NCOLQ, A, LDA, ZETA,
+                1                   WORK, IFAIL)
+                 INTEGER          M, N, NCOLQ, LDA, IFAIL
+                 DOUBLE PRECISION A(LDA,*), ZETA(*), WORK(*)
+                 CHARACTER*1      WHERET
+
+          3. Description
+
+          Q is assumed to be given by
+
+                                                 T
+                                  Q=(Q Q   ...Q ) ,
+                                      n n-1    1
+
+          Q  being given in the form
+           k
+
+                                         (I 0 )
+                                      Q =(0 T )
+                                       k (   k)
+
+          where
+
+                                             T
+                                     T =I-u u ,
+                                      k    k k
+
+                                       ((zeta) )
+                                       (      k)
+                                    u =(z      ),
+                                     k ( k     )
+
+          (zeta)  is a scalar and z  is an (m-k) element vector. z  must be
+                k                  k                              k
+          supplied in the kth column of A in elements A(k+1,k),...,A(m,k)
+          and (zeta)  must be supplied either in A(k,k) or in ZETA(k),
+                    k
+          depending upon the parameter WHERET.
+
+          4. References
+
+          [1]   Golub G H and Van Loan C F (1989) Matrix Computations (2nd
+                Edition). Johns Hopkins University Press, Baltimore,
+                Maryland.
+
+          [2]   Wilkinson J H (1965) The Algebraic Eigenvalue Problem.
+                Oxford University Press.
+
+          5. Parameters
+
+           1:  WHERET -- CHARACTER*1                                  Input
+               On entry: indicates where the elements of (zeta) are to be
+               found as follows:
+               WHERET = 'I' (In A)
+                     The elements of (zeta) are in A.
+
+               WHERET = 'S' (Separate)
+                     The elements of (zeta) are separate from A, in ZETA.
+               Constraint: WHERET must be one of 'I' or 'S'.
+
+           2:  M -- INTEGER                                           Input
+               On entry: m, the number of rows of A. Constraint: M >= N.
+
+           3:  N -- INTEGER                                           Input
+               On entry: n, the number of columns of A. Constraint: N >=
+               0.
+
+           4:  NCOLQ -- INTEGER                                       Input
+               On entry: ncolq, the required number of columns of Q.
+               Constraint: 0 <= NCOLQ <= M.
+
+               When NCOLQ = 0 then an immediate return is effected.
+
+           5:  A(LDA,*) -- DOUBLE PRECISION array              Input/Output
+               Note: the second dimension of the array A must be at least
+               max(1,N,NCOLQ).
+               On entry: the leading m by n strictly lower triangular part
+               of the array A must contain details of the matrix Q. In
+               addition, when WHERET = 'I', then the diagonal elements of A
+               must contain the elements of (zeta) as described under the
+               argument ZETA below. On exit: the first NCOLQ columns of the
+               array A are overwritten by the first NCOLQ columns of the m
+               by m orthogonal matrix Q. When N = 0 then the first NCOLQ
+               columns of A are overwritten by the first NCOLQ columns of
+               the identity matrix.
+
+           6:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F01QEF is called.
+               Constraint: LDA >= max(1,M).
+
+           7:  ZETA(*) -- DOUBLE PRECISION array                      Input
+               Note: the dimension of the array ZETA must be at least
+               max(1,N).
+               On entry: with WHERET = 'S', the array ZETA must contain the
+               elements of (zeta). If ZETA(k) = 0.0 then T  is assumed to
+                                                          k
+               be I, otherwise ZETA(k) is assumed to contain (zeta) .
+                                                                   k
+
+               When WHERET = 'I', the array ZETA is not referenced.
+
+           8:  WORK(*) -- DOUBLE PRECISION array                  Workspace
+               Note: the dimension of the array WORK must be at least
+               max(1,NCOLQ).
+
+           9:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL=-1
+               On entry WHERET /= 'I' or 'S',
+
+               or       M < N,
+
+               or       N < 0,
+
+               or       NCOLQ < 0 or NCOLQ > M,
+
+               or       LDA < M.
+
+          7. Accuracy
+
+          The computed matrix Q satisfies the relation
+
+                                       Q=P+E,
+
+          where P is an exactly orthogonal matrix and
+
+                                 ||E||<=c(epsilon)
+
+          (epsilon) is the machine precision (see X02AJF(*)), c is a modest
+          function of m and |||.||| denotes the spectral (two) norm. See
+          also Section 7 of F01QCF.
+
+          8. Further Comments
+
+          The approximate number of floating-point operations required is
+          given by
+
+               2
+               -n{(3m-n)(2ncolq-n)-n(ncolq-n)},    ncolq>n,
+               3
+
+               2     2
+               -ncolq (3m-ncolq),                  ncolq<=n.
+               3
+
+          9. Example
+
+          To obtain the 5 by 5 orthogonal matrix Q following the QR
+          factorization of the 5 by 3 matrix A given by
+
+                                   (2.0  2.5  2.5)
+                                   (2.0  2.5  2.5)
+                                 A=(1.6 -0.4  2.8).
+                                   (2.0 -0.5  0.5)
+                                   (1.2 -0.3 -2.9)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf01rcf}{NAG On-line Documentation: f01rcf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F01RCF(3NAG)      Foundation Library (12/10/92)      F01RCF(3NAG)
+
+
+
+          F01 -- Matrix Factorizations                               F01RCF
+                  F01RCF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F01RCF finds the QR factorization of the complex m by n matrix A,
+          where m>=n.
+
+          2. Specification
+
+                 SUBROUTINE   F01RCF (M, N, A, LDA, THETA, IFAIL)
+                 INTEGER              M, N, LDA, IFAIL
+                 COMPLEX(KIND(1.0D0)) A(LDA,*), THETA(*)
+
+          3. Description
+
+          The m by n matrix A is factorized as
+
+                    (R)
+                 A=Q(0)       when m>n,
+
+                 A=QR         when m=n,
+
+          where Q is an m by m unitary matrix and R is an n by n upper
+          triangular matrix with real diagonal elements.
+
+          The factorization is obtained by Householder's method. The kth
+          transformation matrix, Q , which is used to introduce zeros into
+                                  k
+          the kth column of A is given in the form
+
+                                        (I 0 )
+                                     Q =(0 T ),
+                                      k (   k)
+
+          where
+
+                                                 H
+                                 T =I-(gamma) u u ,
+                                  k          k k k
+
+                                       ((zeta) )
+                                       (      k)
+                                    u =(z      ),
+                                     k ( k     )
+
+          (gamma)  is a scalar for which Re (gamma) =1.0, (zeta)  is a real
+                 k                                 k            k
+          scalar and z  is an (m-k) element vector. (gamma) , (zeta)  and
+                      k                                    k        k
+          z  are chosen to annihilate the elements below the triangular
+           k
+          part of A and to make the diagonal elements real.
+
+          The scalar (gamma)  and the vector u  are returned in the kth
+                            k                 k
+          element of the array THETA and in the kth column of A, such that
+          (theta) , given by
+                 k
+
+                           (theta) =((zeta) ,Im(gamma) ),
+                                  k        k          k
+
+          is in THETA(k) and the elements of z  are in a     ,...,a   . The
+                                              k         k+1,k      m,k
+          elements of R are returned in the upper triangular part of A.
+
+          Q is given by
+
+                                                 H
+                                  Q=(Q Q   ...Q ) .
+                                      n n-1    1
+
+          A good background description to the QR factorization is given in
+          Dongarra et al [1], but note that this routine is not based upon
+          LINPACK routine ZQRDC.
+
+          4. References
+
+          [1]   Dongarra J J, Moler C B, Bunch J R and Stewart G W (1979)
+                LINPACK Users' Guide. SIAM, Philadelphia.
+
+          [2]   Wilkinson J H (1965) The Algebraic Eigenvalue Problem.
+                Oxford University Press.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: m, the number of rows of A. Constraint: M >= N.
+
+           2:  N -- INTEGER                                           Input
+               On entry: n, the number of columns of A. Constraint: N >=
+               0.
+
+               When N = 0 then an immediate return is effected.
+
+           3:  A(LDA,*) -- COMPLEX(KIND(1.0D0)) array          Input/Output
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the leading m by n part of the array A must
+               contain the matrix to be factorized. On exit: the n by n
+               upper triangular part of A will contain the upper triangular
+               matrix R, with the imaginary parts of the diagonal elements
+               set to zero, and the m by n strictly lower triangular part
+               of A will contain details of the factorization as described
+               above.
+
+           4:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F01RCF is called.
+               Constraint: LDA >= max(1,M).
+
+           5:  THETA(*) -- COMPLEX(KIND(1.0D)) array                 Output
+               Note: the dimension of the array THETA must be at least
+               max(1,N).
+               On exit: the scalar (theta)  for the kth transformation. If
+                                          k
+               T =I then THETA(k) = 0.0; if
+                k
+                                ((alpha) 0)
+                             T =(      0 I)   Re(alpha)<0.0,
+                              k
+               then THETA(k)=(alpha); otherwise THETA(k) contains THETA(k)
+               as described in Section 3 and Re(THETA(k)) is always in the
+                             
+
+               range (1.0, \/2.0).
+
+           6:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL=-1
+               On entry M < N,
+
+               or       N < 0,
+
+               or       LDA < M.
+
+          7. Accuracy
+          The computed factors Q and R satisfy the relation
+
+                                       (R)
+                                      Q(0)=A+E,
+
+          where
+
+                             ||E||<=c(epsilon)||A||,
+
+          (epsilon) being the machine precision, c is a modest function of
+          m and n and ||.|| denotes the spectral (two) norm.
+
+          8. Further Comments
+
+          The approximate number of real floating-point operations is given
+               2
+          by 8n (3m-n)/3.
+
+          Following the use of this routine the operations
+
+                                                 H
+                                 B:=QB  and  B:=Q B,
+
+          where B is an m by k matrix, can be performed by calls to F01RDF.
+          The operation B:=QB can be obtained by the call:
+
+
+                IFAIL = 0
+                CALL F01RDF(`No conjugate', 'Separate', M, N, A, LDA, THETA,
+               *            K, B, LDB, WORK, IFAIL)
+
+                  H
+          and B:=Q B can be obtained by the call:
+
+
+                IFAIL = 0
+                CALL F01RDF(`Conjugate', 'Separate', M, N, A, LDA, THETA,
+               *            K, B, LDB, WORK, IFAIL)
+
+          In both cases WORK must be a k element array that is used as
+          workspace. If B is a one-dimensional array (single column) then
+          the parameter LDB can be replaced by M. See F01RDF for further
+          details.
+
+          The first k columns of the unitary matrix Q can either be
+          obtained by setting B to the first k columns of the unit matrix
+          and using the first of the above two calls, or by calling F01REF,
+          which overwrites the k columns of Q on the first k columns of the
+          array A. Q is obtained by the call:
+
+
+                CALL F01REF(`Separate', M, N, K, A, LDA, THETA, WORK, IFAIL)
+
+          As above, WORK must be a k element array. If k is larger than n,
+          then A must have been declared to have at least k columns.
+
+          Operations involving the matrix R can readily be performed by the
+          Level 2 BLAS routines ZTRSV and ZTRMV (see Chapter F06), but note
+          that no test for near singularity of R is incorporated in ZTRSV.
+          If R is singular, or nearly singular, then F02XUF(*) can be used
+          to determine the singular value decomposition of R.
+
+          9. Example
+
+          To obtain the QR factorization of the 5 by 3 matrix
+
+                             (  0.5i -0.5+1.5i -1.0+1.0i)
+                             (0.4+0.3i 0.9+1.3i 0.2+1.4i)
+                           A=(  0.4 -0.4+0.4i    1.8    ).
+                             (0.3-0.4i 0.1+0.7i    0.0  )
+                             ( -0.3i 0.3+0.3i   2.4i    )
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf01rdf}{NAG On-line Documentation: f01rdf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F01RDF(3NAG)      Foundation Library (12/10/92)      F01RDF(3NAG)
+
+
+
+          F01 -- Matrix Factorizations                               F01RDF
+                  F01RDF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F01RDF performs one of the transformations
+
+                                               H
+                                  B:=QB or B:=Q B,
+
+          where B is an m by ncolb complex matrix and Q is an m by m
+          unitary matrix, given as the product of Householder
+          transformation matrices.
+
+          This routine is intended for use following F01RCF or F01RFF(*).
+
+          2. Specification
+
+                 SUBROUTINE F01RDF (TRANS, WHERET, M, N, A, LDA, THETA,
+                1                   NCOLB, B, LDB, WORK, IFAIL)
+                 INTEGER              M, N, LDA, NCOLB, LDB, IFAIL
+                 COMPLEX(KIND(1.0D0)) A(LDA,*), THETA(*), B(LDB,*), WORK(*)
+                 CHARACTER*1          TRANS, WHERET
+
+          3. Description
+
+          The unitary matrix Q is assumed to be given by
+
+                                                 H
+                                  Q=(Q Q   ...Q ) ,
+                                      n n-1    1
+
+          Q  being given in the form
+           k
+
+                                        (I 0 )
+                                     Q =(0 T ),
+                                      k (   k)
+
+          where
+
+                                                 H
+                                 T =I-(gamma) u u ,
+                                  k          k k k
+
+                                       ((zeta) )
+                                       (      k)
+                                    u =(z      ),
+                                     k ( k     )
+
+          (gamma)  is a scalar for which Re (gamma) =1.0, (zeta)  is a real
+                 k                                 k            k
+          scalar and z  is an (m-k) element vector.
+                      k
+
+          z  must be supplied in the kth column of A in elements
+           k
+          a     ,...,a    and (theta) , given by
+           k+1,k      m,k            k
+
+                           (theta) =((zeta) ,Im (gamma) ),
+                                  k        k           k
+
+          must be supplied either in a    or in THETA(k), depending upon
+                                      k,k
+          the parameter WHERET.
+
+          To obtain Q explicitly B may be set to I and pre-multiplied by Q.
+                                                 H
+          This is more efficient than obtaining Q . Alternatively, F01REF
+          may be used to obtain Q overwritten on A.
+
+          4. References
+
+          [1]   Wilkinson J H (1965) The Algebraic Eigenvalue Problem.
+                Oxford University Press.
+
+          5. Parameters
+
+           1:  TRANS -- CHARACTER*1                                   Input
+               On entry: the operation to be performed as follows:
+               TRANS = 'N' (No transpose)
+                     Perform the operation B:=QB.
+
+               TRANS = 'C' (Conjugate transpose)
+                                               H
+                     Perform the operation B:=Q B.
+               Constraint: TRANS must be one of 'N' or 'C'.
+
+           2:  WHERET -- CHARACTER*1                                  Input
+               On entry: the elements of (theta) are to be found as
+               follows:
+               WHERET = 'I' (In A)
+                     The elements of (theta) are in A.
+
+               WHERET = 'S' (Separate)
+                     The elements of (theta) are separate from A, in THETA.
+               Constraint: WHERET must be one of 'I' or 'S'.
+
+           3:  M -- INTEGER                                           Input
+               On entry: m, the number of rows of A. Constraint: M >= N.
+
+           4:  N -- INTEGER                                           Input
+               On entry: n, the number of columns of A. Constraint: N >=
+               0.
+
+               When N = 0 then an immediate return is effected.
+
+           5:  A(LDA,*) -- COMPLEX(KIND(1.0D)) array                  Input
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the leading m by n strictly lower triangular part
+               of the array A must contain details of the matrix Q. In
+               addition, when WHERET = 'I', then the diagonal elements of A
+               must contain the elements of (theta) as described under the
+               argument THETA below.
+
+               When WHERET = 'S', then the diagonal elements of the array A
+               are referenced, since they are used temporarily to store the
+               (zeta) , but they contain their original values on return.
+                     k
+
+           6:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F01RDF is called.
+               Constraint: LDA >= max(1,M).
+
+           7:  THETA(*) -- COMPLEX(KIND(1.0D)) array                  Input
+               Note: the dimension of the array THETA must be at least
+               max(1,N).
+               On entry: with WHERET = 'S', the array THETA must contain
+               the elements of (theta). If THETA(k)=0.0 then T  is assumed
+                                                              k
+               to be I; if THETA(k)=(alpha), with Re(alpha)<0.0, then T  is
+                                                                       k
+               assumed to be of the form
+                                        ((alpha) 0)
+                                     T =(0       I);
+                                      k
+               otherwise THETA(k) is assumed to contain (theta)  given by
+                                                               k
+               (theta) =((zeta) ,Im(gamma) ).
+                      k        k          k
+
+               When WHERET = 'I', the array THETA is not referenced, and
+               may be dimensioned of length 1.
+
+           8:  NCOLB -- INTEGER                                       Input
+               On entry: ncolb, the number of columns of B. Constraint:
+               NCOLB >= 0.
+               When NCOLB = 0 then an immediate return is effected.
+
+           9:  B(LDB,*) -- COMPLEX(KIND(1.0D)) array           Input/Output
+               Note: the second dimension of the array B must be at least
+               max(1,NCOLB).
+               On entry: the leading m by ncolb part of the array B must
+               contain the matrix to be transformed. On exit: B is
+               overwritten by the transformed matrix.
+
+          10:  LDB -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array B as declared in the
+               (sub)program from which F01RDF is called.
+               Constraint: LDB >= max(1,M).
+
+          11:  WORK(*) -- COMPLEX(KIND(1.0D)) array               Workspace
+               Note: the dimension of the array WORK must be at least
+               max(1,NCOLB).
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL=-1
+               On entry TRANS /= 'N' or 'C',
+
+               or       WHERET /= 'I' or 'S',
+
+               or       M < N,
+
+               or       N < 0,
+
+               or       LDA < M,
+
+               or       NCOLB < 0,
+
+               or       LDB < M.
+
+          7. Accuracy
+
+          Letting C denote the computed matrix Q B, C satisfies the
+          relation
+
+                                       QC=B+E,
+
+          where
+
+                             ||E||<=c(epsilon)||B||,
+
+          (epsilon) being the machine precision, c is a modest function of
+          m and |||.||| denotes the spectral (two) norm. An equivalent
+          result holds for the computed matrix QB. See also Section 7 of
+          F01RCF.
+
+          8. Further Comments
+
+          The approximate number of real floating-point operations is given
+          by 8n(2m-n)ncolb.
+
+          9. Example
+
+                                H
+          To obtain the matrix Q B for the matrix B given by
+
+                               (-0.55+1.05i  0.45+1.05i)
+                               ( 0.49+0.93i  1.09+0.13i)
+                             B=( 0.56-0.16i  0.64+0.16i)
+                               ( 0.39+0.23i -0.39-0.23i)
+                               ( 1.13+0.83i -1.13+0.77i)
+
+          following the QR factorization of the 5 by 3 matrix A given by
+
+                             (  0.5i   -0.5+1.5i -1.0+1.0i)
+                             (0.4+0.3i  0.9+1.3i  0.2+1.4i)
+                           A=(  0.4    -0.4+0.4i  1.8     ).
+                             (0.3-0.4i  0.1+0.7i  0.0     )
+                             ( -0.3i    0.3+0.3i      2.4i)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf01ref}{NAG On-line Documentation: f01ref}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F01REF(3NAG)      Foundation Library (12/10/92)      F01REF(3NAG)
+
+
+
+          F01 -- Matrix Factorizations                               F01REF
+                  F01REF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F01REF returns the first ncolq columns of the complex m by m
+          unitary matrix Q, where Q is given as the product of Householder
+          transformation matrices.
+
+          This routine is intended for use following F01RCF.
+
+          2. Specification
+
+                 SUBROUTINE F01REF (WHERET, M, N, NCOLQ, A, LDA, THETA,
+                1                   WORK, IFAIL)
+                 INTEGER              M, N, NCOLQ, LDA, IFAIL
+                 COMPLEX(KIND(1.0D0)) A(LDA,*), THETA(*), WORK(*)
+                 CHARACTER*1          WHERET
+
+          3. Description
+
+          The unitary matrix Q is assumed to be given by
+
+                                                 H
+                                  Q=(Q Q   ...Q ) ,
+                                      n n-1    1
+
+          Q  being given in the form
+           k
+
+                                        (I 0 )
+                                     Q =(0 T ),
+                                      k (   k)
+
+          where
+
+                                                 H
+                                 T =I-(gamma) u u ,
+                                  k          k k k
+
+                                       ((zeta) )
+                                       (      k)
+                                    u =(z      ),
+                                     k ( k     )
+
+          (gamma)  is a scalar for which Re (gamma) =1.0, (zeta)  is a real
+                 k                                 k            k
+
+          scalar and z  is an (m-k) element vector.
+                      k
+
+          z  must be supplied in the kth column of A in elements
+           k
+          a     ,...,a    and (theta) , given by
+           k+1,k      m,k            k
+
+                           (theta) =((zeta) ,Im (gamma) ),
+                                  k        k           k
+
+          must be supplied either in a    or in THETA(k) depending upon the
+                                      k,k
+          parameter WHERET.
+
+          4. References
+
+          [1]   Wilkinson J H (1965) The Algebraic Eigenvalue Problem.
+                Oxford University Press.
+
+          5. Parameters
+
+           1:  WHERET -- CHARACTER*1                                  Input
+               On entry: the elements of (theta) are to be found as
+               follows:
+               WHERET = 'I' (In A)
+                     The elements of (theta) are in A.
+
+               WHERET = 'S' (Separate)
+                     The elements of (theta) are separate from A, in THETA.
+               Constraint: WHERET must be one of 'I' or 'S'.
+
+           2:  M -- INTEGER                                           Input
+               On entry: m, the number of rows of A. Constraint: M >= N.
+
+           3:  N -- INTEGER                                           Input
+               On entry: n, the number of columns of A. Constraint: N >=
+               0.
+
+           4:  NCOLQ -- INTEGER                                       Input
+               On entry: ncolq, the required number of columns of Q.
+               Constraint: 0 <= NCOLQ <= M.
+
+               When NCOLQ = 0 then an immediate return is effected.
+
+           5:  A(LDA,*) -- COMPLEX(KIND(1.0D)) array           Input/Output
+               Note: the second dimension of the array A must be at least
+               max(1,N,NCOLQ).
+               On entry: the leading m by n strictly lower triangular part
+               of the array A must contain details of the matrix Q. In
+               addition, when WHERET = 'I', then the diagonal elements of A
+               must contain the elements of (theta) as described under the
+               argument THETA below. On exit: the first NCOLQ columns of
+               the array A are overwritten by the first NCOLQ columns of
+               the m by m unitary matrix Q. When N = 0 then the first NCOLQ
+               columns of A are overwritten by the first NCOLQ columns of
+               the unit matrix.
+
+           6:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F01REF is called.
+               Constraint: LDA >= max(1,M).
+
+           7:  THETA(*) -- COMPLEX(KIND(1.0D)) array                  Input
+               Note: the dimension of the array THETA must be at least
+               max(1,N).
+               On entry: if WHERET = 'S', the array THETA must contain the
+               elements of (theta). If THETA(k)=0.0 then T  is assumed to
+                                                          k
+               be I; if THETA(k)=(alpha), with Re(alpha)<0.0, then T  is
+                                                                    k
+               assumed to be of the form
+                                        ((alpha) 0)
+                                     T =(   0    I);
+                                      k
+               otherwise THETA(k) is assumed to contain (theta)  given by
+                                                               k
+               (theta) =((zeta) ,Im(gamma) ).
+                      k        k          k
+
+               When WHERET = 'I', the array THETA is not referenced.
+
+           8:  WORK(*) -- COMPLEX(KIND(1.0D)) array               Workspace
+               Note: the dimension of the array WORK must be at least
+               max(1,NCOLQ).
+
+           9:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL=-1
+               On entry WHERET /= 'I' or 'S',
+
+               or       M < N,
+
+               or       N < 0,
+
+               or       NCOLQ < 0 or NCOLQ > M,
+
+               or       LDA < M.
+
+          7. Accuracy
+
+          The computed matrix Q satisfies the relation
+
+                                       Q=P+E,
+
+          where P is an exactly unitary matrix and
+
+                                ||E||<=c(epsilon),
+
+          (epsilon) being the machine precision, c is a modest function of
+          m and |||.||| denotes the spectral (two) norm. See also Section 7
+          of F01RCF.
+
+          8. Further Comments
+
+          The approximate number of real floating-point operations required
+          is given by
+
+               8
+               -n{(3m-n)(2ncolq-n)-n(ncolq-n)},  ncolq>n
+               3
+
+               8     2
+               -ncolq (3m-ncolq),                ncolq<=n
+               3
+
+          9. Example
+
+          To obtain the 5 by 5 unitary matrix Q following the QR
+          factorization of the 5 by 3 matrix A given by
+
+                             (    0.5i -0.5+1.5i -1.0+1.4i)
+                             (0.4+0.3i  0.9+1.3i  0.2+1.4i)
+                           A=(0.4      -0.4+0.4i  1.8     ).
+                             (0.3-0.4i  0.1+0.7i  0.0     )
+                             (   -0.3i  0.3+0.3i      2.4i)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02}{NAG On-line Documentation: f02}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02(3NAG)         Foundation Library (12/10/92)         F02(3NAG)
+
+
+
+          F02 -- Eigenvalues and Eigenvectors           Introduction -- F02
+                                    Chapter F02
+                           Eigenvalues and Eigenvectors
+
+          1. Scope of the Chapter
+
+          This chapter is concerned with computing
+
+          --    eigenvalues and eigenvectors of a matrix
+
+          --    eigenvalues and eigenvectors of generalized matrix
+                eigenvalue problems
+
+          --    singular values and singular vectors of a matrix.
+
+          2. Background to the Problems
+
+          2.1. Eigenvalue Problems
+
+          In the most usual form of eigenvalue problem we are given a
+          square n by n matrix A and wish to compute (lambda) (an
+          eigenvalue) and x/=0 (an eigenvector) which satisfy the equation
+
+                                    Ax=(lambda)x
+
+          Such problems are called 'standard' eigenvalue problems in
+          contrast to 'generalized' eigenvalue problems where we wish to
+          satisfy the equation
+
+                                    Ax=(lambda)Bx
+
+          B also being a square n by n matrix.
+
+          Section 2.1.1 and Section 2.1.2 discuss, respectively, standard
+          and generalized eigenvalue problems where the matrices involved
+          are dense; Section 2.1.3 discusses both types of problem in the
+          case where A and B are sparse (and symmetric).
+
+          2.1.1.  Standard eigenvalue problems
+
+          Some of the routines in this chapter find all the n eigenvalues,
+          some find all the n eigensolutions (eigenvalues and corresponding
+          eigenvectors), and some find a selected group of eigenvalues
+          and/or eigenvectors. The matrix A may be:
+
+          (i)   general (real or complex)
+
+          (ii)  real symmetric, or
+
+          (iii) complex Hermitian (so that if a  =(alpha)+i(beta) then
+                                               ij
+                a  =(alpha)-i(beta)).
+                 ji
+
+          In all cases the computation starts with a similarity
+                          -1
+          transformation S  AS=T, where S is non-singular and is the
+          product of fairly simple matrices, and T has an 'easier form'
+          than A so that its eigensolutions are easily determined. The
+          matrices A and T, of course, have the same eigenvalues, and if y
+          is an eigenvector of T then Sy is the corresponding eigenvector
+          of A.
+
+          In case (i) (general real or complex A), the selected form of T
+          is an upper Hessenberg matrix (t  =0 if i-j>1) and S is the
+                                          ij
+          product of n-2 stabilised elementary transformation matrices.
+          There is no easy method of computing selected eigenvalues of a
+          Hessenberg matrix, so that all eigenvalues are always calculated.
+          In the real case this computation is performed via the Francis QR
+          algorithm with double shifts, and in the complex case by means of
+          the LR algorithm. If the eigenvectors are required they are
+          computed by back-substitution following the QR and LR algorithm.
+
+          In case (ii) (real and symmetric A) the selected simple form of T
+          is a tridiagonal matrix (t  =0 if |i-j|>1), and S is the product
+                                    ij
+          of n-2 orthogonal Householder transformation matrices. If only
+          selected eigenvalues are required, they are obtained by the
+          method of bisection using the Sturm sequence property, and the
+          corresponding eigenvectors of T are computed by inverse
+          iteration. If all eigenvalues are required, they are computed
+          from T via the QL algorithm (an adaptation of the QR algorithm),
+          and the corresponding eigenvectors of T are the product of the
+          transformations for the QL reduction. In all cases the
+          corresponding eigenvectors of A are recovered from the
+          computation of x=Sy.
+
+          In case (iii) (complex Hermitian A) analogous transformations as
+          in case (ii) are used. T has complex elements in off-diagonal
+          positions, but a simple diagonal similarity transformation is
+          then used to produce a real tridiagonal form, after which the QL
+          algorithm and succeeding methods described in the previous
+          paragraph are used to complete the solution.
+
+          2.1.2.  Generalized eigenvalue problems
+
+          Here we distinguish as a special case those problems in which
+          both A and B are symmetric and B is positive-definite and well-
+          conditioned with respect to inversion (i.e., all the eigenvalues
+          of B are significantly greater than zero). Such problems can be
+          satisfactorily treated by first reducing them to case (ii) of
+          Section 2.1.1 and then using the methods described there to
+                                                              T
+          compute the eigensolutions. If B is factorized as LL  (L lower
+          triangular), then Ax=(lambda)Bx is equivalent to the standard
+                                                   -1   T -1        T
+          symmetric problem Ry=(lambda)y, where R=L  A(L )   and y=L x.
+          After finding an eigenvector y of R, the required x is computed
+                                     T
+          by back-substitution in y=L x.
+
+          For generalized problems of the form Ax=(lambda)Bx which do not
+          fall into the special case, the QZ algorithm is provided.
+
+          In order to appreciate the domain in which this algorithm is
+          appropriate we remark first that when B is non-singular the
+          problem Ax=(lambda)Bx is fully equivalent to the problem
+            -1
+          (B  A)x=(lambda)x; both the eigenvalues and eigenvectors being
+          the same. When A is non-singular Ax=(lambda)Bx is equivalent to
+                        -1
+          the problem (A  B)x=(mu)x; the eigenvalues (mu) being the
+          reciprocals of the required eigenvalues and the eigenvectors
+          remaining the same. In theory then, provided at least one of the
+          matrices A and B is non-singular, the generalized problem
+          Ax=(lambda)Bx could be solved via the standard problem
+          Cx=(lambda)x with an appropriate matrix C, and as far as economy
+          of effort is concerned this is quite satisfactory. However, in
+          practice, for this reduction to be satisfactory from the
+          standpoint of numerical stability, one requires more than the
+                                                                -1
+          mere non-singularity of A or B. It is necessary that B  A (or
+           -1
+          A  B) should not only exist but that B (or A) should be well-
+          conditioned with respect to inversion. The nearer B (or A) is to
+                                               -1       -1
+          singularity the more unsatisfactory B  A (or A  B) will be as a
+          vehicle for determining the required eigenvalues. Unfortunately
+                                                                        -1
+          one cannot counter ill-conditioning in B (or A) by computing B  A
+               -1
+          (or A  B) accurately to single precision using iterative
+          refinement. Well-determined eigenvalues of the original
+          Ax=(lambda)Bx may be poorly determined even by the correctly
+                              -1       -1
+          rounded version of B  A (or A  B). The situation may in some
+          instances be saved by the observation that if Ax=(lambda)Bx then
+          (A-kB)x=((lambda)-k)Bx. Hence if A-kB is non-singular we may
+                                            -1
+          solve the standard problem [(A-kB)  B]x=(mu)x and for numerical
+          stability we require only that (A-kB) be well-conditioned with
+          respect to inversion.
+
+          In practice one may well be in a situation where no k is known
+          for which (A-kB) is well-conditioned with respect to inversion
+          and indeed (A-kB) may be singular for all k. The QZ algorithm is
+          designed to deal directly with the problem Ax=(lambda)Bx itself
+          and its performance is unaffected by singularity or near-
+          singularity of A, B or A-kB.
+
+          2.1.3.  Sparse symmetric problems
+
+          If the matrices A and B are large and sparse (i.e., only a small
+          proportion of the elements are non-zero), then the methods
+          described in the previous Section are unsuitable, because in
+          reducing the problem to a simpler form, much of the sparsity of
+          the problem would be lost; hence the computing time and the
+          storage required would be very large. Instead, for symmetric
+          problems, the method of simultaneous iteration may be used to
+          determine selected eigenvalues and the corresponding
+          eigenvectors. The routine provided has been designed to handle
+          both symmetric and generalized symmetric problems.
+
+          2.2. Singular Value Problems
+
+          The singular value decomposition of an m by n real matrix A is
+          given by
+
+                                            T
+                                       A=QDP ,
+
+          where Q is an m by m orthogonal matrix, P is an n by n orthogonal
+          matrix and D is an m by n diagonal matrix with non-negative
+          diagonal elements. The first k==min(m,n) columns of Q and P are
+          the left- and right-hand singular vectors of A and the k diagonal
+          elements of D are the singular values.
+
+          When A is complex then the singular value decomposition is given
+          by
+
+                                            H
+                                       A=QDP ,
+
+                                      H                                   T
+          where Q and P are unitary, P  denotes the complex conjugate of P
+          and D is as above for the real case.
+
+          If the matrix A has column means of zero, then AP is the matrix
+          of principal components of A and the singular values are the
+          square roots of the sample variances of the observations with
+          respect to the principal components. (See also Chapter G03.)
+
+          Routines are provided to return the singular values and vectors
+          of a general real or complex matrix.
+
+          3. Recommendations on Choice and Use of Routines
+
+          3.1. General Discussion
+
+          There is one routine, F02FJF, which is designed for sparse
+          symmetric eigenvalue problems, either standard or generalized.
+          The remainder of the routines are designed for dense matrices.
+
+          3.2. Eigenvalue and Eigenvector Routines
+
+          These reduce the matrix A to a simpler form by a similarity
+                          -1
+          transformation S  AS=T where T is an upper Hessenberg or
+          tridiagonal matrix, compute the eigensolutions of T, and then
+          recover the eigenvectors of A via the matrix S. The eigenvectors
+          are normalised so that
+
+                                      n
+                                      --     2
+                                      >  |x | =1
+                                      --   r
+                                      r=1
+
+          x  being the rth component of the eigenvector x, and so that the
+           r
+          element of largest modulus is real if x is complex. For problems
+          of the type Ax=(lambda)Bx with A and B symmetric and B positive-
+                                                             T
+          definite, the eigenvectors are normalised so that x Bx=1, x
+          always being real for such problems.
+
+          3.3. Singular Value and Singular Vector Routines
+
+          These reduce the matrix A to real bidiagonal form, B say, by
+                                      T
+          orthogonal transformations Q AP=B in the real case, and by
+                                   H
+          unitary transformations Q AP=B in the complex case, and the
+          singular values and vectors are computed via this bidiagonal
+          form. The singular values are returned in descending order.
+
+          3.4. Decision Trees
+
+          (i)  Eigenvalues and Eigenvectors
+
+
+                   Please see figure in printed Reference Manual
+
+
+          (ii)  Singular Values and Singular Vectors
+
+
+                   Please see figure in printed Reference Manual
+
+          F02 -- Eigenvalues and Eigenvectors               Contents -- F02
+          Chapter F02
+
+          Eigenvalues and Eigenvectors
+
+          F02AAF  All eigenvalues of real symmetric matrix
+
+          F02ABF  All eigenvalues and eigenvectors of real symmetric matrix
+
+          F02ADF  All eigenvalues of generalized real symmetric-definite
+                  eigenproblem
+
+          F02AEF  All eigenvalues and eigenvectors of generalized real
+                  symmetric-definite eigenproblem
+
+          F02AFF  All eigenvalues of real matrix
+
+          F02AGF  All eigenvalues and eigenvectors of real matrix
+
+          F02AJF  All eigenvalues of complex matrix
+
+          F02AKF  All eigenvalues and eigenvectors of complex matrix
+
+          F02AWF  All eigenvalues of complex Hermitian matrix
+
+          F02AXF  All eigenvalues and eigenvectors of complex Hermitian
+                  matrix
+
+          F02BBF  Selected eigenvalues and eigenvectors of real symmetric
+                  matrix
+
+          F02BJF  All eigenvalues and optionally eigenvectors of
+                  generalized eigenproblem by QZ algorithm, real matrices
+
+          F02FJF  Selected eigenvalues and eigenvectors of sparse symmetric
+                  eigenproblem
+
+          F02WEF  SVD of real matrix
+
+          F02XEF  SVD of complex matrix
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02aaf}{NAG On-line Documentation: f02aaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02AAF(3NAG)      Foundation Library (12/10/92)      F02AAF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02AAF
+                  F02AAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02AAF calculates all the eigenvalues of a real symmetric matrix.
+
+          2. Specification
+
+                 SUBROUTINE F02AAF (A, IA, N, R, E, IFAIL)
+                 INTEGER          IA, N, IFAIL
+                 DOUBLE PRECISION A(IA,N), R(N), E(N)
+
+          3. Description
+
+          This routine reduces the real symmetric matrix A to a real
+          symmetric tridiagonal matrix using Householder's method. The
+          eigenvalues of the tridiagonal matrix are then determined using
+          the QL algorithm.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  A(IA,N) -- DOUBLE PRECISION array               Input/Output
+               On entry: the lower triangle of the n by n symmetric matrix
+               A. The elements of the array above the diagonal need not be
+               set. On exit: the elements of A below the diagonal are
+               overwritten, and the rest of the array is unchanged.
+
+           2:  IA -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F02AAF is called.
+               Constraint: IA >= N.
+
+           3:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A.
+
+           4:  R(N) -- DOUBLE PRECISION array                        Output
+               On exit: the eigenvalues in ascending order.
+
+           5:  E(N) -- DOUBLE PRECISION array                     Workspace
+
+           6:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               Failure in F02AVF(*) indicating that more than 30*N
+               iterations are required to isolate all the eigenvalues.
+
+          7. Accuracy
+
+          The accuracy of the eigenvalues depends on the sensitivity of the
+          matrix to rounding errors produced in tridiagonalisation. For a
+          detailed error analysis see Wilkinson and Reinsch [1] pp 222 and
+          235.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          9. Example
+
+          To calculate all the eigenvalues of the real symmetric matrix:
+
+                               ( 0.5  0.0  2.3 -2.6)
+                               ( 0.0  0.5 -1.4 -0.7)
+                               ( 2.3 -1.4  0.5  0.0).
+                               (-2.6 -0.7  0.0  0.5)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02abf}{NAG On-line Documentation: f02abf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02ABF(3NAG)      Foundation Library (12/10/92)      F02ABF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02ABF
+                  F02ABF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02ABF calculates all the eigenvalues and eigenvectors of a real
+          symmetric matrix.
+
+          2. Specification
+
+                 SUBROUTINE F02ABF (A, IA, N, R, V, IV, E, IFAIL)
+                 INTEGER          IA, N, IV, IFAIL
+                 DOUBLE PRECISION A(IA,N), R(N), V(IV,N), E(N)
+
+          3. Description
+
+          This routine reduces the real symmetric matrix A to a real
+          symmetric tridiagonal matrix by Householder's method. The
+          eigenvalues and eigenvectors are calculated using the QL
+          algorithm.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  A(IA,N) -- DOUBLE PRECISION array                      Input
+               On entry: the lower triangle of the n by n symmetric matrix
+               A. The elements of the array above the diagonal need not be
+               set. See also Section 8.
+
+           2:  IA -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F02ABF is called.
+               Constraint: IA >= N.
+
+           3:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A.
+
+           4:  R(N) -- DOUBLE PRECISION array                        Output
+               On exit: the eigenvalues in ascending order.
+
+           5:  V(IV,N) -- DOUBLE PRECISION array                     Output
+               On exit: the normalised eigenvectors, stored by columns;
+               the ith column corresponds to the ith eigenvalue. The
+               eigenvectors are normalised so that the sum of squares of
+               the elements is equal to 1.
+
+           6:  IV -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array V as declared in the
+               (sub)program from which F02ABF is called.
+               Constraint: IV >= N.
+
+           7:  E(N) -- DOUBLE PRECISION array                     Workspace
+
+           8:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               Failure in F02AMF(*) indicating that more than 30*N
+               iterations are required to isolate all the eigenvalues.
+
+          7. Accuracy
+
+          The eigenvectors are always accurately orthogonal but the
+          accuracy of the individual eigenvectors is dependent on their
+          inherent sensitivity to changes in the original matrix. For a
+          detailed error analysis see Wilkinson and Reinsch [1] pp 222 and
+          235.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          Unless otherwise stated in the Users' Note for your
+          implementation, the routine may be called with the same actual
+          array supplied for parameters A and V, in which case the
+          eigenvectors will overwrite the original matrix. However this is
+          not standard Fortran 77, and may not work on all systems.
+
+          9. Example
+
+          To calculate all the eigenvalues and eigenvectors of the real
+          symmetric matrix:
+
+                               ( 0.5  0.0  2.3 -2.6)
+                               ( 0.0  0.5 -1.4 -0.7)
+                               ( 2.3 -1.4  0.5  0.0).
+                               (-2.6 -0.7  0.0  0.5)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02adf}{NAG On-line Documentation: f02adf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02ADF(3NAG)      Foundation Library (12/10/92)      F02ADF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02ADF
+                  F02ADF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02ADF calculates all the eigenvalues of  Ax=(lambda)Bx,  where A
+          is a real symmetric matrix and B is a real symmetric positive-
+          definite matrix.
+
+          2. Specification
+
+                 SUBROUTINE F02ADF (A, IA, B, IB, N, R, DE, IFAIL)
+                 INTEGER          IA, IB, N, IFAIL
+                 DOUBLE PRECISION A(IA,N), B(IB,N), R(N), DE(N)
+
+          3. Description
+
+          The problem is reduced to the standard symmetric eigenproblem
+          using Cholesky's method to decompose B into triangular matrices,
+              T
+          B=LL , where L is lower triangular. Then Ax=(lambda)Bx implies
+            -1  -T   T             T
+          (L  AL  )(L x)=(lambda)(L x); hence the eigenvalues of
+          Ax=(lambda)Bx are those of Py=(lambda)y where P is the symmetric
+                  -1  -T
+          matrix L  AL  . Householder's method is used to tridiagonalise
+          the matrix P and the eigenvalues are then found using the QL
+          algorithm.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  A(IA,N) -- DOUBLE PRECISION array               Input/Output
+               On entry: the upper triangle of the n by n symmetric matrix
+               A. The elements of the array below the diagonal need not be
+               set. On exit: the lower triangle of the array is
+               overwritten. The rest of the array is unchanged.
+
+           2:  IA -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F02ADF is called.
+               Constraint: IA >= N.
+
+           3:  B(IB,N) -- DOUBLE PRECISION array               Input/Output
+               On entry: the upper triangle of the n by n symmetric
+               positive-definite matrix B. The elements of the array below
+               the diagonal need not be set. On exit: the elements below
+               the diagonal are overwritten. The rest of the array is
+               unchanged.
+
+           4:  IB -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array B as declared in the
+               (sub)program from which F02ADF is called.
+               Constraint: IB >= N.
+
+           5:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrices A and B.
+
+           6:  R(N) -- DOUBLE PRECISION array                        Output
+               On exit: the eigenvalues in ascending order.
+
+           7:  DE(N) -- DOUBLE PRECISION array                    Workspace
+
+           8:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               Failure in F01AEF(*); the matrix B is not positive-definite
+               possibly due to rounding errors.
+
+          IFAIL= 2
+               Failure in F02AVF(*), more than 30*N iterations are required
+               to isolate all the eigenvalues.
+
+          7. Accuracy
+
+          In general this routine is very accurate. However, if B is ill-
+          conditioned with respect to inversion, the eigenvalues could be
+          inaccurately determined. For a detailed error analysis see
+          Wilkinson and Reinsch [1] pp 310, 222 and 235.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          9. Example
+
+          To calculate all the eigenvalues of the general symmetric
+          eigenproblem Ax=(lambda) Bx where A is the symmetric matrix:
+
+                                (0.5  1.5  6.6   4.8)
+                                (1.5  6.5 16.2   8.6)
+                                (6.6 16.2 37.6   9.8)
+                                (4.8  8.6  9.8 -17.1)
+
+          and B is the symmetric positive-definite matrix:
+
+                                    (1  3  4  1)
+                                    (3 13 16 11)
+                                    (4 16 24 18).
+                                    (1 11 18 27)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02aef}{NAG On-line Documentation: f02aef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02AEF(3NAG)      Foundation Library (12/10/92)      F02AEF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02AEF
+                  F02AEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02AEF calculates all the eigenvalues and eigenvectors of
+          Ax=(lambda)Bx,  where A is a real symmetric matrix and B is a
+          real symmetric positive-definite matrix.
+
+          2. Specification
+
+                 SUBROUTINE F02AEF (A, IA, B, IB, N, R, V, IV, DL, E, IFAIL)
+                 INTEGER          IA, IB, N, IV, IFAIL
+                 DOUBLE PRECISION A(IA,N), B(IB,N), R(N), V(IV,N), DL(N), E
+                1                 (N)
+
+          3. Description
+
+          The problem is reduced to the standard symmetric eigenproblem
+          using Cholesky's method to decompose B into triangular matrices
+              T
+          B=LL , where L is lower triangular. Then Ax=(lambda)Bx implies
+            -1  -T   T             T
+          (L  AL  )(L x)=(lambda)(L x); hence the eigenvalues of
+          Ax=(lambda)Bx are those of Py=(lambda)y, where P is the symmetric
+                  -1  -T
+          matrix L  AL  . Householder's method is used to tridiagonalise
+          the matrix P and the eigenvalues are found using the QL
+          algorithm. An eigenvector z of the derived problem is related to
+                                                         T
+          an eigenvector x of the original problem by z=L x. The
+          eigenvectors z are determined using the QL algorithm and are
+                              T
+          normalised so that z z=1; the eigenvectors of the original
+                                                  T
+          problem are then determined by solving L x=z, and are normalised
+                   T
+          so that x Bx=1.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  A(IA,N) -- DOUBLE PRECISION array               Input/Output
+
+               On entry: the upper triangle of the n by n symmetric matrix
+               A. The elements of the array below the diagonal need not be
+               set. On exit: the lower triangle of the array is
+               overwritten. The rest of the array is unchanged. See also
+               Section 8.
+
+           2:  IA -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F02AEF is called.
+               Constraint: IA >= N.
+
+           3:  B(IB,N) -- DOUBLE PRECISION array               Input/Output
+               On entry: the upper triangle of the n by n symmetric
+               positive-definite matrix B. The elements of the array below
+               the diagonal need not be set. On exit: the elements below
+               the diagonal are overwritten. The rest of the array is
+               unchanged.
+
+           4:  IB -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array B as declared in the
+               (sub)program from which F02AEF is called.
+               Constraint: IB >= N.
+
+           5:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrices A and B.
+
+           6:  R(N) -- DOUBLE PRECISION array                        Output
+               On exit: the eigenvalues in ascending order.
+
+           7:  V(IV,N) -- DOUBLE PRECISION array                     Output
+               On exit: the normalised eigenvectors, stored by columns;
+               the ith column corresponds to the ith eigenvalue. The
+                                                      T
+               eigenvectors x are normalised so that x Bx=1. See also
+               Section 8.
+
+           8:  IV -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array V as declared in the
+               (sub)program from which F02AEF is called.
+               Constraint: IV >= N.
+
+           9:  DL(N) -- DOUBLE PRECISION array                    Workspace
+
+          10:  E(N) -- DOUBLE PRECISION array                     Workspace
+
+          11:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               Failure in F01AEF(*); the matrix B is not positive-definite,
+               possibly due to rounding errors.
+
+          IFAIL= 2
+               Failure in F02AMF(*); more than 30*N iterations are required
+               to isolate all the eigenvalues.
+
+          7. Accuracy
+
+          In general this routine is very accurate. However, if B is ill-
+          conditioned with respect to inversion, the eigenvectors could be
+          inaccurately determined. For a detailed error analysis see
+          Wilkinson and Reinsch [1] pp 310, 222 and 235.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          Unless otherwise stated in the Users' Note for your
+          implementation, the routine may be called with the same actual
+          array supplied for parameters A and V, in which case the
+          eigenvectors will overwrite the original matrix A. However this
+          is not standard Fortran 77, and may not work on all systems.
+
+          9. Example
+
+          To calculate all the eigenvalues and eigenvectors of the general
+          symmetric eigenproblem Ax=(lambda) Bx where A is the symmetric
+          matrix:
+
+                                (0.5  1.5  6.6   4.8)
+                                (1.5  6.5 16.2   8.6)
+                                (6.6 16.2 37.6   9.8)
+                                (4.8  8.6  9.8 -17.1)
+
+          and B is the symmetric positive-definite matrix:
+
+                                    (1  3  4  1)
+                                    (3 13 16 11)
+                                    (4 16 24 18).
+                                    (1 11 18 27)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02aff}{NAG On-line Documentation: f02aff}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02AFF(3NAG)      Foundation Library (12/10/92)      F02AFF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02AFF
+                  F02AFF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02AFF calculates all the eigenvalues of a real unsymmetric
+          matrix.
+
+          2. Specification
+
+                 SUBROUTINE F02AFF (A, IA, N, RR, RI, INTGER, IFAIL)
+                 INTEGER          IA, N, INTGER(N), IFAIL
+                 DOUBLE PRECISION A(IA,N), RR(N), RI(N)
+
+          3. Description
+
+          The matrix A is first balanced and then reduced to upper
+          Hessenberg form using stabilised elementary similarity
+          transformations. The eigenvalues are then found using the QR
+          algorithm for real Hessenberg matrices.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  A(IA,N) -- DOUBLE PRECISION array               Input/Output
+               On entry: the n by n matrix A. On exit: the array is
+               overwritten.
+
+           2:  IA -- INTEGER                                          Input
+               On entry:
+               the dimension of the array A as declared in the (sub)program
+               from which F02AFF is called.
+               Constraint: IA >= N.
+
+           3:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A.
+
+           4:  RR(N) -- DOUBLE PRECISION array                       Output
+               On exit: the real parts of the eigenvalues.
+
+           5:  RI(N) -- DOUBLE PRECISION array                       Output
+               On exit: the imaginary parts of the eigenvalues.
+
+           6:  INTGER(N) -- INTEGER array                            Output
+               On exit: INTGER(i) contains the number of iterations used
+               to find the ith eigenvalue. If INTGER(i) is negative, the i
+               th eigenvalue is the second of a pair found simultaneously.
+
+               Note that the eigenvalues are found in reverse order,
+               starting with the nth.
+
+           7:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               More than 30*N iterations are required to isolate all the
+               eigenvalues.
+
+          7. Accuracy
+
+          The accuracy of the results depends on the original matrix and
+          the multiplicity of the roots. For a detailed error analysis see
+          Wilkinson and Reinsch [1] pp 352 and 367.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          9. Example
+
+          To calculate all the eigenvalues of the real matrix:
+
+                               (  1.5 0.1  4.5 -1.5)
+                               (-22.5 3.5 12.5 -2.5)
+                               ( -2.5 0.3  4.5 -2.5).
+                               ( -2.5 0.1  4.5  2.5)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02agf}{NAG On-line Documentation: f02agf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02AGF(3NAG)      Foundation Library (12/10/92)      F02AGF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02AGF
+                  F02AGF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02AGF calculates all the eigenvalues and eigenvectors of a real
+          unsymmetric matrix.
+
+          2. Specification
+
+                 SUBROUTINE F02AGF (A, IA, N, RR, RI, VR, IVR, VI, IVI,
+                1                   INTGER, IFAIL)
+                 INTEGER          IA, N, IVR, IVI, INTGER(N), IFAIL
+                 DOUBLE PRECISION A(IA,N), RR(N), RI(N), VR(IVR,N), VI
+                1                 (IVI,N)
+
+          3. Description
+
+          The matrix A is first balanced and then reduced to upper
+          Hessenberg form using real stabilised elementary similarity
+          transformations. The eigenvalues and eigenvectors of the
+          Hessenberg matrix are calculated using the QR algorithm. The
+          eigenvectors of the Hessenberg matrix are back-transformed to
+          give the eigenvectors of the original matrix A.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  A(IA,N) -- DOUBLE PRECISION array               Input/Output
+               On entry: the n by n matrix A. On exit: the array is
+               overwritten.
+
+           2:  IA -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F02AGF is called.
+               Constraint: IA >= N.
+
+           3:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A.
+
+           4:  RR(N) -- DOUBLE PRECISION array                       Output
+               On exit: the real parts of the eigenvalues.
+
+           5:  RI(N) -- DOUBLE PRECISION array                       Output
+               On exit: the imaginary parts of the eigenvalues.
+
+           6:  VR(IVR,N) -- DOUBLE PRECISION array                   Output
+               On exit: the real parts of the eigenvectors, stored by
+               columns. The ith column corresponds to the ith eigenvalue.
+               The eigenvectors are normalised so that the sum of the
+               squares of the moduli of the elements is equal to 1 and the
+               element of largest modulus is real. This ensures that real
+               eigenvalues have real eigenvectors.
+
+           7:  IVR -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array VR as declared in the
+               (sub)program from which F02AGF is called.
+               Constraint: IVR >= N.
+
+           8:  VI(IVI,N) -- DOUBLE PRECISION array                   Output
+               On exit: the imaginary parts of the eigenvectors, stored by
+               columns. The ith column corresponds to the ith eigenvalue.
+
+           9:  IVI -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array VI as declared in the
+               (sub)program from which F02AGF is called.
+               Constraint: IVI >= N.
+
+          10:  INTGER(N) -- INTEGER array                            Output
+               On exit: INTGER(i) contains the number of iterations used
+               to find the ith eigenvalue. If INTGER(i) is negative, the i
+               th eigenvalue is the second of a pair found simultaneously.
+
+               Note that the eigenvalues are found in reverse order,
+               starting with the nth.
+
+          11:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               More than 30*N iterations are required to isolate all the
+               eigenvalues.
+
+          7. Accuracy
+
+          The accuracy of the results depends on the original matrix and
+          the multiplicity of the roots. For a detailed error analysis see
+          Wilkinson and Reinsch [1] pp 352 and 390.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          9. Example
+
+          To calculate all the eigenvalues and eigenvectors of the real
+          matrix:
+
+                               (  1.5 0.1  4.5 -1.5)
+                               (-22.5 3.5 12.5 -2.5)
+                               ( -2.5 0.3  4.5 -2.5).
+                               ( -2.5 0.1  4.5  2.5)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02ajf}{NAG On-line Documentation: f02ajf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02AJF(3NAG)      Foundation Library (12/10/92)      F02AJF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02AJF
+                  F02AJF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02AJF calculates all the eigenvalues of a complex matrix.
+
+          2. Specification
+
+                 SUBROUTINE F02AJF (AR, IAR, AI, IAI, N, RR, RI, INTGER,
+                1                   IFAIL)
+                 INTEGER          IAR, IAI, N, INTGER(N), IFAIL
+                 DOUBLE PRECISION AR(IAR,N), AI(IAI,N), RR(N), RI(N)
+
+          3. Description
+
+          The complex matrix A is first balanced and then reduced to upper
+          Hessenberg form using stabilised elementary similarity
+          transformations. The eigenvalues are then found using the
+          modified LR algorithm for complex Hessenberg matrices.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  AR(IAR,N) -- DOUBLE PRECISION array             Input/Output
+               On entry: the real parts of the elements of the n by n
+               complex matrix A. On exit: the array is overwritten.
+
+           2:  IAR -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array AR as declared in the
+               (sub)program from which F02AJF is called.
+               Constraint: IAR >= N.
+
+           3:  AI(IAI,N) -- DOUBLE PRECISION array             Input/Output
+               On entry: the imaginary parts of the elements of the n by n
+               complex matrix A. On exit: the array is overwritten.
+
+           4:  IAI -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array AI as declared in the
+               (sub)program from which F02AJF is called.
+               Constraint: IAI >= N.
+
+           5:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A.
+
+           6:  RR(N) -- DOUBLE PRECISION array                       Output
+               On exit: the real parts of the eigenvalues.
+
+           7:  RI(N) -- DOUBLE PRECISION array                       Output
+               On exit: the imaginary parts of the eigenvalues.
+
+           8:  INTGER(N) -- INTEGER array                         Workspace
+
+           9:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               More than 30*N iterations are required to isolate all the
+               eigenvalues.
+
+          7. Accuracy
+
+          The accuracy of the results depends on the original matrix and
+          the multiplicity of the roots. For a detailed error analysis see
+          Wilkinson and Reinsch [1] pp 352 and 401.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          9. Example
+
+          To calculate all the eigenvalues of the complex matrix:
+
+                     (-21.0-5.0i      24.60i 13.6+10.2i     4.0i)
+                     (     22.5i 26.00-5.00i  7.5-10.0i 2.5     )
+                     ( -2.0+1.5i  1.68+2.24i  4.5-5.0i  1.5+2.0i).
+                     (     -2.5i -2.60       -2.7+3.6i  2.5-5.0i)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02akf}{NAG On-line Documentation: f02akf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02AKF(3NAG)      Foundation Library (12/10/92)      F02AKF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02AKF
+                  F02AKF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02AKF calculates all the eigenvalues and eigenvectors of a
+          complex matrix.
+
+          2. Specification
+
+                 SUBROUTINE F02AKF (AR, IAR, AI, IAI, N, RR, RI, VR, IVR,
+                1                   VI, IVI, INTGER, IFAIL)
+                 INTEGER          IAR, IAI, N, IVR, IVI, INTGER(N), IFAIL
+                 DOUBLE PRECISION AR(IAR,N), AI(IAI,N), RR(N), RI(N), VR
+                1                 (IVR,N), VI(IVI,N)
+
+          3. Description
+
+          The complex matrix A is first balanced and then reduced to upper
+          Hessenberg form by stabilised elementary similarity
+          transformations. The eigenvalues and eigenvectors of the
+          Hessenberg matrix are calculated using the LR algorithm. The
+          eigenvectors of the Hessenberg matrix are back-transformed to
+          give the eigenvectors of the original matrix.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  AR(IAR,N) -- DOUBLE PRECISION array             Input/Output
+               On entry: the real parts of the elements of the n by n
+               complex matrix A. On exit: the array is overwritten.
+
+           2:  IAR -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array AR as declared in the
+               (sub)program from which F02AKF is called.
+               Constraint: IAR >= N.
+
+           3:  AI(IAI,N) -- DOUBLE PRECISION array             Input/Output
+               On entry: the imaginary parts of the elements of the n by n
+               complex matrix A. On exit: the array is overwritten.
+
+           4:  IAI -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array AI as declared in the
+               (sub)program from which F02AKF is called.
+               Constraint: IAI >= N.
+
+           5:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A.
+
+           6:  RR(N) -- DOUBLE PRECISION array                       Output
+               On exit: the real parts of the eigenvalues.
+
+           7:  RI(N) -- DOUBLE PRECISION array                       Output
+               On exit: the imaginary parts of the eigenvalues.
+
+           8:  VR(IVR,N) -- DOUBLE PRECISION array                   Output
+               On exit: the real parts of the eigenvectors, stored by
+               columns. The ith column corresponds to the ith eigenvalue.
+               The eigenvectors are normalised so that the sum of squares
+               of the moduli of the elements is equal to 1 and the element
+               of largest modulus is real.
+
+           9:  IVR -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array VR as declared in the
+               (sub)program from which F02AKF is called.
+               Constraint: IVR >= N.
+
+          10:  VI(IVI,N) -- DOUBLE PRECISION array                   Output
+               On exit: the imaginary parts of the eigenvectors, stored by
+               columns. The ith column corresponds to the ith eigenvalue.
+
+          11:  IVI -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array VI as declared in the
+               (sub)program from which F02AKF is called.
+               Constraint: IVI >= N.
+
+          12:  INTGER(N) -- INTEGER array                         Workspace
+
+          13:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               More than 30*N iterations are required to isolate all the
+               eigenvalues.
+
+          7. Accuracy
+
+          The accuracy of the results depends on the conditioning of the
+          original matrix and the multiplicity of the roots. For a detailed
+          error analysis see Wilkinson and Reinsch [1] pp 352 and 390.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          9. Example
+
+          To calculate all the eigenvalues and eigenvectors of the complex
+          matrix:
+
+                     (-21.0-5.0i      24.60i 13.6+10.2i     4.0i)
+                     (     22.5i 26.00-5.00i  7.5-10.0i 2.5     )
+                     ( -2.0+1.5i  1.68+2.24i  4.5-5.0i  1.5+2.0i).
+                     (     -2.5i -2.60       -2.7+3.6i  2.5-5.0i)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02awf}{NAG On-line Documentation: f02awf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02AWF(3NAG)      Foundation Library (12/10/92)      F02AWF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02AWF
+                  F02AWF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02AWF calculates all the eigenvalues of a complex Hermitian
+          matrix.
+
+          2. Specification
+
+                 SUBROUTINE F02AWF (AR, IAR, AI, IAI, N, R, WK1, WK2, WK3,
+                1                   IFAIL)
+                 INTEGER          IAR, IAI, N, IFAIL
+                 DOUBLE PRECISION AR(IAR,N), AI(IAI,N), R(N), WK1(N),
+                1                 WK2(N), WK3(N)
+
+          3. Description
+
+          The complex Hermitian matrix A is first reduced to a real
+          tridiagonal matrix by n-2 unitary transformations, and a
+          subsequent diagonal transformation. The eigenvalues are then
+          derived using the QL algorithm, an adaptation of the QR
+          algorithm.
+
+          4. References
+
+          [1]   Peters G (1967) NPL Algorithms Library. Document No.
+                F1/04/A.
+
+          [2]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  AR(IAR,N) -- DOUBLE PRECISION array             Input/Output
+               On entry: the real parts of the elements of the lower
+               triangle of the n by n complex Hermitian matrix A. Elements
+               of the array above the diagonal need not be set. On exit:
+               the array is overwritten.
+
+           2:  IAR -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array AR as declared in the
+               (sub)program from which F02AWF is called.
+               Constraint: IAR >= N.
+
+           3:  AI(IAI,N) -- DOUBLE PRECISION array             Input/Output
+               On entry: the imaginary parts of the elements of the lower
+               triangle of the n by n complex Hermitian matrix A. Elements
+               of the array above the diagonal need not be set. On exit:
+               the array is overwritten.
+
+           4:  IAI -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array AI as declared in the
+               (sub)program from which F02AWF is called.
+               Constraint: IAI >= N.
+
+           5:  N -- INTEGER                                           Input
+               On entry: n, the order of the complex Hermitian matrix, A.
+
+           6:  R(N) -- DOUBLE PRECISION array                        Output
+               On exit: the eigenvalues in ascending order.
+
+           7:  WK1(N) -- DOUBLE PRECISION array                   Workspace
+
+           8:  WK2(N) -- DOUBLE PRECISION array                   Workspace
+
+           9:  WK3(N) -- DOUBLE PRECISION array                   Workspace
+
+          10:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               More than 30*N iterations are required to isolate all the
+               eigenvalues.
+
+          7. Accuracy
+
+          For a detailed error analysis see Peters [1] page 3 and Wilkinson
+          and Reinsch [2] page 235.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          9. Example
+
+          To calculate all the eigenvalues of the complex Hermitian matrix:
+
+                      (0.50        0.00       1.84+1.38i  2.08-1.56i)
+                      (0.00        0.50       1.12+0.84i -0.56+0.42i)
+                      (1.84-1.38i  1.12-0.84i 0.50        0.00      ).
+                      (2.08+1.56i -0.56-0.42i 0.00        0.50      )
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02axf}{NAG On-line Documentation: f02axf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02AXF(3NAG)      Foundation Library (12/10/92)      F02AXF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02AXF
+                  F02AXF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02AXF calculates all the eigenvalues and eigenvectors of a
+          complex Hermitian matrix.
+
+          2. Specification
+
+                 SUBROUTINE F02AXF (AR, IAR, AI, IAI, N, R, VR, IVR, VI,
+                1                   IVI, WK1, WK2, WK3, IFAIL)
+                 INTEGER          IAR, IAI, N, IVR, IVI, IFAIL
+                 DOUBLE PRECISION AR(IAR,N), AI(IAI,N), R(N), VR(IVR,N), VI
+                1                 (IVI,N), WK1(N), WK2(N), WK3(N)
+
+          3. Description
+
+          The complex Hermitian matrix is first reduced to a real
+          tridiagonal matrix by n-2 unitary transformations and a
+          subsequent diagonal transformation. The eigenvalues and
+          eigenvectors are then derived using the QL algorithm, an
+          adaptation of the QR algorithm.
+
+          4. References
+
+          [1]   Peters G (1967) NPL Algorithms Library. Document No.
+                F2/03/A.
+
+          [2]   Peters G (1967) NPL Algorithms Library. Document No.
+                F1/04/A.
+
+          5. Parameters
+
+           1:  AR(IAR,N) -- DOUBLE PRECISION array                    Input
+               On entry: the real parts of the elements of the lower
+               triangle of the n by n complex Hermitian matrix A. Elements
+               of the array above the diagonal need not be set. See also
+               Section 8.
+
+           2:  IAR -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array AR as declared in the
+               (sub)program from which F02AXF is called.
+               Constraint: IAR >= N.
+
+           3:  AI(IAI,N) -- DOUBLE PRECISION array                    Input
+               On entry: the imaginary parts of the elements of the lower
+               triangle of the n by n complex Hermitian matrix A. Elements
+               of the array above the diagonal need not be set. See also
+               Section 8.
+
+           4:  IAI -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array AI as declared in the
+               (sub)program from which F02AXF is called.
+               Constraint: IAI >= N.
+
+           5:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix, A.
+
+           6:  R(N) -- DOUBLE PRECISION array                        Output
+               On exit: the eigenvalues in ascending order.
+
+           7:  VR(IVR,N) -- DOUBLE PRECISION array                   Output
+               On exit: the real parts of the eigenvectors, stored by
+               columns. The ith column corresponds to the ith eigenvector.
+               The eigenvectors are normalised so that the sum of the
+               squares of the moduli of the elements is equal to 1 and the
+               element of largest modulus is real. See also Section 8.
+
+           8:  IVR -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array VR as declared in the
+               (sub)program from which F02AXF is called.
+               Constraint: IVR >= N.
+
+           9:  VI(IVI,N) -- DOUBLE PRECISION array                   Output
+               On exit: the imaginary parts of the eigenvectors, stored by
+               columns. The ith column corresponds to the ith eigenvector.
+               See also Section 8.
+
+          10:  IVI -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array VI as declared in the
+               (sub)program from which F02AXF is called.
+               Constraint: IVI >= N.
+
+          11:  WK1(N) -- DOUBLE PRECISION array                   Workspace
+
+          12:  WK2(N) -- DOUBLE PRECISION array                   Workspace
+
+          13:  WK3(N) -- DOUBLE PRECISION array                   Workspace
+
+          14:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               More than 30*N iterations are required to isolate all the
+               eigenvalues.
+
+          IFAIL= 2
+               The diagonal elements of AI are not all zero, i.e., the
+               complex matrix is not Hermitian.
+
+          7. Accuracy
+
+          The eigenvectors are always accurately orthogonal but the
+          accuracy of the individual eigenvalues and eigenvectors is
+          dependent on their inherent sensitivity to small changes in the
+          original matrix. For a detailed error analysis see Peters [1]
+          page 3 and [2] page 3.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          Unless otherwise stated in the implementation document, the
+          routine may be called with the same actual array supplied for
+          parameters AR and VR, and for AI and VI, in which case the
+          eigenvectors will overwrite the original matrix A. However this
+          is not standard Fortran 77, and may not work on all systems.
+
+          9. Example
+
+          To calculate the eigenvalues and eigenvectors of the complex
+          Hermitian matrix:
+
+                      (0.50        0.00       1.84+1.38i  2.08-1.56i)
+                      (0.00        0.50       1.12+0.84i -0.56+0.42i)
+                      (1.84-1.38i  1.12-0.84i 0.50        0.00      ).
+                      (2.08+1.56i -0.56-0.42i 0.00        0.50      )
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02bbf}{NAG On-line Documentation: f02bbf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02BBF(3NAG)      Foundation Library (12/10/92)      F02BBF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02BBF
+                  F02BBF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02BBF calculates selected eigenvalues and eigenvectors of a real
+          symmetric matrix by reduction to tridiagonal form, bisection and
+          inverse iteration, where the selected eigenvalues lie within a
+          given interval.
+
+          2. Specification
+
+                 SUBROUTINE F02BBF (A, IA, N, ALB, UB, M, MM, R, V, IV, D,
+                1                   E, E2, X, G, C, ICOUNT, IFAIL)
+                 INTEGER          IA, N, M, MM, IV, ICOUNT(M), IFAIL
+                 DOUBLE PRECISION A(IA,N), ALB, UB, R(M), V(IV,M), D(N), E
+                1                 (N), E2(N), X(N,7), G(N)
+                 LOGICAL          C(N)
+
+          3. Description
+
+          The real symmetric matrix A is reduced to a symmetric tridiagonal
+          matrix T by Householder's method. The eigenvalues which lie
+          within a given interval [l,u], are calculated by the method of
+          bisection. The corresponding eigenvectors of T are calculated by
+          inverse iteration. A back-transformation is then performed to
+          obtain the eigenvectors of the original matrix A.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  A(IA,N) -- DOUBLE PRECISION array               Input/Output
+               On entry: the lower triangle of the n by n symmetric matrix
+               A. The elements of the array above the diagonal need not be
+               set. On exit: the elements of A below the diagonal are
+               overwritten, and the rest of the array is unchanged.
+
+           2:  IA -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F02BBF is called.
+               Constraint: IA >= N.
+
+           3:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A.
+
+           4:  ALB -- DOUBLE PRECISION                                Input
+
+           5:  UB -- DOUBLE PRECISION                                 Input
+               On entry: l and u, the lower and upper end-points of the
+               interval within which eigenvalues are to be calculated.
+
+           6:  M -- INTEGER                                           Input
+               On entry: an upper bound for the number of eigenvalues
+               within the interval.
+
+           7:  MM -- INTEGER                                         Output
+               On exit: the actual number of eigenvalues within the
+               interval.
+
+           8:  R(M) -- DOUBLE PRECISION array                        Output
+               On exit: the eigenvalues, not necessarily in ascending
+               order.
+
+           9:  V(IV,M) -- DOUBLE PRECISION array                     Output
+               On exit: the eigenvectors, stored by columns. The ith
+               column corresponds to the ith eigenvalue. The eigenvectors
+               are normalised so that the sum of the squares of the
+               elements are equal to 1.
+
+          10:  IV -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array V as declared in the
+               (sub)program from which F02BBF is called.
+               Constraint: IV >= N.
+
+          11:  D(N) -- DOUBLE PRECISION array                     Workspace
+
+          12:  E(N) -- DOUBLE PRECISION array                     Workspace
+
+          13:  E2(N) -- DOUBLE PRECISION array                    Workspace
+
+          14:  X(N,7) -- DOUBLE PRECISION array                   Workspace
+
+          15:  G(N) -- DOUBLE PRECISION array                     Workspace
+
+          16:  C(N) -- LOGICAL array                              Workspace
+
+          17:  ICOUNT(M) -- INTEGER array                            Output
+               On exit: ICOUNT(i) contains the number of iterations for
+               the ith eigenvalue.
+
+          18:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               M is less than the number of eigenvalues in the given
+               interval. On exit MM contains the number of eigenvalues in
+               the interval. Rerun with this value for M.
+
+          IFAIL= 2
+               More than 5 iterations are required to determine any one
+               eigenvector.
+
+          7. Accuracy
+
+          There is no guarantee of the accuracy of the eigenvectors as the
+          results depend on the original matrix and the multiplicity of the
+          roots. For a detailed error analysis see Wilkinson and Reinsch
+          [1] pp 222 and 436.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          This subroutine should only be used when less than 25% of the
+          eigenvalues and the corresponding eigenvectors are required. Also
+          this subroutine is less efficient with matrices which have
+          multiple eigenvalues.
+
+          9. Example
+
+          To calculate the eigenvalues lying between -2.0 and 3.0, and the
+          corresponding eigenvectors of the real symmetric matrix:
+
+                               ( 0.5  0.0  2.3 -2.6)
+                               ( 0.0  0.5 -1.4 -0.7)
+                               ( 2.3 -1.4  0.5  0.0).
+                               (-2.6 -0.7  0.0  0.5)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02bjf}{NAG On-line Documentation: f02bjf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02BJF(3NAG)      Foundation Library (12/10/92)      F02BJF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02BJF
+                  F02BJF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02BJF calculates all the eigenvalues and, if required, all the
+          eigenvectors of the generalized eigenproblem  Ax=(lambda)Bx
+          where A and B are real, square matrices, using the QZ algorithm.
+
+          2. Specification
+
+                 SUBROUTINE F02BJF (N, A, IA, B, IB, EPS1, ALFR, ALFI,
+                1                   BETA, MATV, V, IV, ITER, IFAIL)
+                 INTEGER          N, IA, IB, IV, ITER(N), IFAIL
+                 DOUBLE PRECISION A(IA,N), B(IB,N), EPS1, ALFR(N), ALFI(N),
+                1                 BETA(N), V(IV,N)
+                 LOGICAL          MATV
+
+          3. Description
+
+          All the eigenvalues and, if required, all the eigenvectors of the
+          generalized eigenproblem Ax=(lambda)Bx where A and B are real,
+          square matrices, are determined using the QZ algorithm. The QZ
+          algorithm consists of 4 stages:
+
+          (a)   A is reduced to upper Hessenberg form and at the same time
+                B is reduced to upper triangular form.
+
+          (b)   A is further reduced to quasi-triangular form while the
+                triangular form of B is maintained.
+
+          (c)   The quasi-triangular form of A is reduced to triangular
+                form and the eigenvalues extracted. This routine does not
+                actually produce the eigenvalues (lambda) , but instead
+                                                         j
+                returns (alpha)  and (beta)  such that
+
+                               j           j
+                        (lambda) =(alpha) /(beta) ,   j=1,2,...,.n
+                                j        j       j
+                The division by (beta)  becomes the responsibility of the
+                                      j
+                user's program, since (beta)  may be zero indicating an
+                                            j
+                infinite eigenvalue. Pairs of complex eigenvalues occur
+                with (alpha) /(beta)  and (alpha)   /(beta)    complex
+                            j       j            j+1       j+1
+
+                conjugates, even though (alpha)  and (alpha)    are not
+                                               j            j+1
+                conjugate.
+
+          (d)   If the eigenvectors are required (MATV = .TRUE.), they are
+                obtained from the triangular matrices and then transformed
+                back into the original co-ordinate system.
+
+          4. References
+
+          [1]   Moler C B and Stewart G W (1973) An Algorithm for
+                Generalized Matrix Eigenproblems. SIAM J. Numer. Anal. 10
+                241--256.
+
+          [2]   Ward R C (1975) The Combination Shift QZ Algorithm. SIAM J.
+                Numer. Anal. 12 835--853.
+
+          [3]   Wilkinson J H (1979) Kronecker's Canonical Form and the QZ
+                Algorithm. Linear Algebra and Appl. 28 285--303.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrices A and B.
+
+           2:  A(IA,N) -- DOUBLE PRECISION array               Input/Output
+               On entry: the n by n matrix A. On exit: the array is
+               overwritten.
+
+           3:  IA -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F02BJF is called.
+               Constraint: IA >= N.
+
+           4:  B(IB,N) -- DOUBLE PRECISION array               Input/Output
+               On entry: the n by n matrix B. On exit: the array is
+               overwritten.
+
+           5:  IB -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array B as declared in the
+               (sub)program from which F02BJF is called.
+               Constraint: IB >= N.
+
+           6:  EPS1 -- DOUBLE PRECISION                               Input
+               On entry: the tolerance used to determine negligible
+               elements. If EPS1 > 0.0, an element will be considered
+               negligible if it is less than EPS1 times the norm of its
+               matrix. If EPS1 <= 0.0, machine precision is used in place
+               of EPS1. A positive value of EPS1 may result in faster
+               execution but less accurate results.
+
+           7:  ALFR(N) -- DOUBLE PRECISION array                     Output
+
+           8:  ALFI(N) -- DOUBLE PRECISION array                     Output
+               On exit: the real and imaginary parts of (alpha) , for
+                                                               j
+               j=1,2,...,n.
+
+           9:  BETA(N) -- DOUBLE PRECISION array                     Output
+               On exit: (beta) , for j=1,2,...,n.
+                               j
+
+          10:  MATV -- LOGICAL                                        Input
+               On entry: MATV must be set .TRUE. if the eigenvectors are
+               required, otherwise .FALSE..
+
+          11:  V(IV,N) -- DOUBLE PRECISION array                     Output
+               On exit: if MATV = .TRUE., then
+                    (i)if the jth eigenvalue is real, the jth column of V
+                    contains its eigenvector;
+
+                    (ii) if the jth and (j+1)th eigenvalues form a complex
+                    pair, the jth and (j+1)th columns of V contain the
+                    real and imaginary parts of the eigenvector associated
+                    with the first eigenvalue of the pair. The conjugate
+                    of this vector is the eigenvector for the conjugate
+                    eigenvalue.
+               Each eigenvector is normalised so that the component of
+               largest modulus is real and the sum of squares of the moduli
+               equal one.
+
+               If MATV = .FALSE., V is not used.
+
+          12:  IV -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array V as declared in the
+               (sub)program from which F02BJF is called.
+               Constraint: IV >= N.
+
+          13:  ITER(N) -- INTEGER array                              Output
+               On exit: ITER(j) contains the number of iterations needed
+               to obtain the jth eigenvalue. Note that the eigenvalues are
+               obtained in reverse order, starting with the nth.
+
+          14:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= i
+               More than 30*N iterations are required to determine all the
+               diagonal 1 by 1 or 2 by 2 blocks of the quasi-triangular
+               form in the second step of the QZ algorithm. IFAIL is set to
+               the index i of the eigenvalue at which this failure occurs.
+               If the soft failure option is used, (alpha)  and (beta)  are
+                                                          j           j
+               correct for j=i+1,i+2,...,n, but V does not contain any
+               correct eigenvectors.
+
+          7. Accuracy
+
+          The computed eigenvalues are always exact for a problem
+          (A+E)x=(lambda)(B+F)x where ||E||/||A|| and ||F||/||B||
+          are both of the order of max(EPS1,(epsilon)), EPS1 being defined
+          as in Section 5 and (epsilon) being the machine precision.
+
+          Note: interpretation of results obtained with the QZ algorithm
+          often requires a clear understanding of the effects of small
+          changes in the original data. These effects are reviewed in
+          Wilkinson [3], in relation to the significance of small values of
+          (alpha)  and (beta) . It should be noted that if (alpha)  and
+                 j           j                                    j
+          (beta)  are both small for any j, it may be that no reliance can
+                j
+          be placed on any of the computed eigenvalues
+          (lambda) =(alpha) /(beta) . The user is recommended to study [3]
+                  i        i       i
+          and, if in difficulty, to seek expert advice on determining the
+          sensitivity of the eigenvalues to perturbations in the data.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+          and also depends on the value chosen for parameter EPS1.
+
+          9. Example
+
+          To find all the eigenvalues and eigenvectors of Ax=(lambda) Bx
+          where
+
+                                 (3.9 12.5 -34.5 -0.5)
+                                 (4.3 21.5 -47.5  7.5)
+                               A=(4.3 21.5 -43.5  3.5)
+                                 (4.4 26.0 -46.0  6.0)
+
+          and
+
+                                      (1 2 -3 1)
+                                      (1 3 -5 4)
+                                    B=(1 3 -4 3).
+                                      (1 3 -4 4)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02fjf}{NAG On-line Documentation: f02fjf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02FJF(3NAG)      Foundation Library (12/10/92)      F02FJF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02FJF
+                  F02FJF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          To find eigenvalues and eigenvectors of a real sparse symmetric
+          or generalized symmetric eigenvalue problem.
+
+          2. Specification
+
+                 SUBROUTINE F02FJF (N, M, K, NOITS, TOL, DOT, IMAGE, MONIT,
+                1                   NOVECS, X, NRX, D, WORK, LWORK, RWORK,
+                2                   LRWORK, IWORK, LIWORK, IFAIL)
+                 INTEGER          N, M, K, NOITS, NOVECS, NRX, LWORK,
+                1                 LRWORK, IWORK(LIWORK), LIWORK, IFAIL
+                 DOUBLE PRECISION TOL, DOT, X(NRX,K), D(K), WORK(LWORK),
+                1                 RWORK(LRWORK)
+                 EXTERNAL         DOT, IMAGE, MONIT
+
+          3. Description
+
+          F02FJF finds the m eigenvalues of largest absolute value and the
+          corresponding eigenvectors for the real eigenvalue problem
+
+                                  Cx=(lambda)x                          (1)
+
+          where C is an n by n matrix such that
+
+                                         T
+                                     BC=C B                             (2)
+
+          for a given positive-definite matrix B. C is said to be B-
+          symmetric. Different specifications of C allow for the solution
+          of a variety of eigenvalue problems. For example, when
+
+                                                     T
+                              C=A and B=I where   A=A
+
+          the routine finds the m eigenvalues of largest absolute magnitude
+          for the standard symmetric eigenvalue problem
+
+                                 Ax=(lambda)x.                          (3)
+
+          The routine is intended for the case where A is sparse.
+
+          As a second example, when
+
+                                          -1
+                                       C=B  A
+
+          where
+
+                                           T
+                                        A=A
+
+          the routine finds the m eigenvalues of largest absolute magnitude
+          for the generalized symmetric eigenvalue problem
+
+                                 Ax=(lambda)Bx.                         (4)
+
+          The routine is intended for the case where A and B are sparse.
+
+          The routine does not require C explicitly, but C is specified via
+          a user-supplied routine IMAGE which, given an n element vector z,
+          computes the image w given by
+
+                                        w=Cz.
+
+                                                       -1
+          For instance, in the above example, where C=B  A, routine IMAGE
+          will need to solve the positive-definite system of equations
+          Bw=Az for w.
+
+          To find the m eigenvalues of smallest absolute magnitude of (3)
+                           -1
+          we can choose C=A   and hence find the reciprocals of the
+          required eigenvalues, so that IMAGE will need to solve Aw=z for
+                                                         -1
+          w, and correspondingly for (4) we can choose C=A  B and solve
+          Aw=Bz for w.
+
+          A table of examples of choice of IMAGE is given in Table 3.1. It
+          should be remembered that the routine also returns the
+          corresponding eigenvectors and that B is positive-definite.
+          Throughout A is assumed to be symmetric and, where necessary,
+          non-singularity is also assumed.
+
+          Eigenvalues             Problem
+          Required
+
+                      Ax=(lambda)x     (B=I)Ax=(lambda)Bx ABx=(lambda)x
+
+          Largest     Compute          Solve              Compute
+                      w=Az             Bw=Az              w=ABz
+
+          Smallest    Solve            Solve              Solve
+          (Find       Aw=z             Aw=Bz              Av=z,  Bw=(nu)
+          1/(lambda))
+
+          Furthest    Compute          Solve              Compute
+          from        w=(A-(sigma)I)z  Bw=(A-(sigma)B)z   w=(AB-(sigma)I)z
+          (sigma)
+          (Find
+          (lambda)-
+          (sigma))
+
+          Closest to Solve             Solve              Solve
+          (sigma)    (A-(sigma)I)w=z   (A-(sigma)B)w=Bz   (AB-(sigma)I)w=z
+          (Find 1/((
+          lambda)-
+          (sigma)))
+
+
+                                     Table 3.1
+                   The Requirement of IMAGE for Various Problems
+
+          The matrix B also need not be supplied explicitly, but is
+          specified via a user-supplied routine DOT which, given n element
+                                                                 T
+          vectors z and w, computes the generalized dot product w Bz.
+
+          F02FJF is based upon routine SIMITZ (see Nikolai [1]), which is
+          itself a derivative of the Algol procedure ritzit (see
+          Rutishauser [4]), and uses the method of simultaneous (subspace)
+          iteration. (See Parlett [2] for description, analysis and advice
+          on the use of the method.)
+
+          The routine performs simultaneous iteration on k>m vectors.
+          Initial estimates to p<=k eigenvectors, corresponding to the p
+          eigenvalues of C of largest absolute value, may be supplied by
+          the user to F02FJF. When possible k should be chosen so that the
+          kth eigenvalue is not too close to the m required eigenvalues,
+          but if k is initially chosen too small then F02FJF may be re-
+          entered, supplying approximations to the k eigenvectors found so
+          far and with k then increased.
+
+          At each major iteration F02FJF solves an r by r (r<=k) eigenvalue
+          sub-problem in order to obtain an approximation to the
+          eigenvalues for which convergence has not yet occurred. This
+          approximation is refined by Chebyshev acceleration.
+
+          4. References
+
+          [1]   Nikolai P J (1979) Algorithm 538: Eigenvectors and
+                eigenvalues of real generalized symmetric matrices by
+                simultaneous iteration. ACM Trans. Math. Softw. 5 118--125.
+
+          [2]   Parlett B N (1980) The Symmetric Eigenvalue Problem.
+                Prentice-Hall.
+
+          [3]   Rutishauser H (1969) Computational aspects of F L Bauer's
+                simultaneous iteration method. Num. Math. 13 4--13.
+
+          [4]   Rutishauser H (1970) Simultaneous iteration method for
+                symmetric matrices. Num. Math. 16 205--223.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix C. Constraint: N >= 1.
+
+           2:  M -- INTEGER                                    Input/Output
+               On entry: m, the number of eigenvalues required.
+                                            '
+               Constraint: M >= 1. On exit: m, the number of eigenvalues
+               actually found. It is equal to m if IFAIL = 0 on exit, and
+               is less than m if IFAIL = 2, 3 or 4. See Section 6 and
+               Section 8 for further information.
+
+           3:  K -- INTEGER                                           Input
+               On entry: the number of simultaneous iteration vectors to be
+               used. Too small a value of K may inhibit convergence, while
+               a larger value of K incurs additional storage and additional
+               work per iteration. Suggested value: K = M + 4 will often be
+               a reasonable choice in the absence of better information.
+               Constraint: M < K <= N.
+
+           4:  NOITS -- INTEGER                                Input/Output
+               On entry: the maximum number of major iterations (eigenvalue
+               sub-problems) to be performed. If NOITS <= 0, then the value
+               100 is used in place of NOITS. On exit: the number of
+               iterations actually performed.
+
+           5:  TOL -- DOUBLE PRECISION                                Input
+               On entry: a relative tolerance to be used in accepting
+               eigenvalues and eigenvectors. If the eigenvalues are
+               required to about t significant figures, then TOL should be
+                              -t
+               set to about 10  . d  is accepted as an eigenvalue as soon
+                                   i
+               as two successive approximations to d  differ by less than
+                                                    i
+                 ~                   ~
+               (|d |*TOL)/10,  where d  is the latest aproximation to d .
+                  i                   i                                i
+               Once an eigenvalue has been accepted, then an eigenvector is
+               accepted as soon as  (d f )/(d -d )<TOL, where f  is the
+                                      i i    i  k              i
+               normalised residual of the current approximation to the
+               eigenvector (see Section 8 for further information). The
+               values of the f  and d  can be printed from routine MONIT.
+                              i      i
+               If TOL is supplied outside the range ((epsilon), 1.0), where
+               (epsilon) is the machine precision, then the value (epsilon)
+               is used in place of TOL.
+
+           6:  DOT -- DOUBLE PRECISION FUNCTION, supplied by the user.
+                                                         External Procedure
+                                          T
+               DOT must return the value w Bz for given vectors w and z.
+               For the standard eigenvalue problem, where B=I, DOT must
+                                       T
+               return the dot product w z.
+
+               Its specification is:
+
+                      DOUBLE PRECISION FUNCTION DOT (IFLAG, N, Z, W,
+                     1                               RWORK, LRWORK,
+                     2                               IWORK, LIWORK)
+                      INTEGER          IFLAG, N, LRWORK, IWORK(LIWORK),
+                     1                 LIWORK
+                      DOUBLE PRECISION Z(N), W(N), RWORK(LRWORK)
+
+                1:  IFLAG -- INTEGER                           Input/Output
+                    On entry: IFLAG is always non-negative. On exit: IFLAG
+                    may be used as a flag to indicate a failure in the
+                                    T
+                    computation of w Bz. If IFLAG is negative on exit from
+                    DOT, then F02FJF will exit immediately with IFAIL set
+                    to IFLAG. Note that in this case DOT must still be
+                    assigned a value.
+
+                2:  N -- INTEGER                                      Input
+                    On entry: the number of elements in the vectors z and w
+                    and the order of the matrix B.
+
+                3:  Z(N) -- DOUBLE PRECISION array                    Input
+                                                      T
+                    On entry: the vector z for which w Bz is required.
+
+                4:  W(N) -- DOUBLE PRECISION array                    Input
+                                                      T
+                    On entry: the vector w for which w Bz is required.
+
+                5:  RWORK(LRWORK) -- DOUBLE PRECISION array  User Workspace
+
+
+                6:  LRWORK -- INTEGER                                 Input
+
+
+                7:  IWORK(LIWORK) -- INTEGER array           User Workspace
+
+
+                8:  LIWORK -- INTEGER                                 Input
+                    DOT is called from F02FJF with the parameters RWORK,
+                    LRWORK, IWORK and LIWORK as supplied to F02FJF. The
+                    user is free to use the arrays RWORK and IWORK to
+                    supply information to DOT and to IMAGE as an
+                    alternative to using COMMON.
+               DOT must be declared as EXTERNAL in the (sub)program
+               from which F02FJF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           7:  IMAGE -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               IMAGE must return the vector w=Cz for a given vector z.
+
+               Its specification is:
+
+                      SUBROUTINE IMAGE (IFLAG, N, Z, W, RWORK, LRWORK,
+                     1                  IWORK, LIWORK)
+                      INTEGER          IFLAG, N, LRWORK, IWORK(LIWORK),
+                     1                 LIWORK
+                      DOUBLE PRECISION Z(N), W(N), RWORK(LRWORK)
+
+                1:  IFLAG -- INTEGER                           Input/Output
+                    On entry: IFLAG is always non-negative. On exit: IFLAG
+                    may be used as a flag to indicate a failure in the
+                    computation of w. If IFLAG is negative on exit from
+                    IMAGE, then F02FJF will exit immediately with IFAIL set
+                    to IFLAG.
+
+                2:  N -- INTEGER                                      Input
+                    On entry: n, the number of elements in the vectors w
+                    and z, and the order of the matrix C.
+
+                3:  Z(N) -- DOUBLE PRECISION array                    Input
+                    On entry: the vector z for which Cz is required.
+
+                4:  W(N) -- DOUBLE PRECISION array                   Output
+                    On exit: the vector w=Cz.
+
+                5:  RWORK(LRWORK) -- DOUBLE PRECISION array  User Workspace
+
+                6:  LRWORK -- INTEGER                                 Input
+
+                7:  IWORK(LIWORK) -- INTEGER array           User Workspace
+
+                8:  LIWORK -- INTEGER                                 Input
+                    IMAGE is called from F02FJF with the parameters RWORK,
+                    LRWORK, IWORK and LIWORK as supplied to F02FJF. The
+                    user is free to use the arrays RWORK and IWORK to
+                    supply information to IMAGE and DOT as an alternative
+                    to using COMMON.
+               IMAGE must be declared as EXTERNAL in the (sub)program
+               from which F02FJF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           8:  MONIT -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               MONIT is used to monitor the progress of F02FJF. MONIT may
+               be the dummy subroutine F02FJZ if no monitoring is actually
+               required. (F02FJZ is included in the NAG Foundation Library
+               and so need not be supplied by the user. The routine name
+               F02FJZ may be implementation dependent: see the Users' Note
+               for your implementation for details.) MONIT is called after
+               the solution of each eigenvalue sub-problem and also just
+               prior to return from F02FJF. The parameters ISTATE and
+               NEXTIT allow selective printing by MONIT.
+
+               Its specification is:
+
+                      SUBROUTINE MONIT (ISTATE, NEXTIT, NEVALS,
+                     1                  NEVECS, K, F, D)
+                      INTEGER          ISTATE, NEXTIT, NEVALS, NEVECS,
+                     1                 K
+                      DOUBLE PRECISION F(K), D(K)
+
+                1:  ISTATE -- INTEGER                                 Input
+                    On entry: ISTATE specifies the state of F02FJF and will
+                    have values as follows:
+                    ISTATE = 0
+                          No eigenvalue or eigenvector has just been
+                          accepted.
+
+                    ISTATE = 1
+                          One or more eigenvalues have been accepted since
+                          the last call to MONIT.
+
+                    ISTATE = 2
+                          One or more eigenvectors have been accepted since
+                          the last call to MONIT.
+
+                    ISTATE = 3
+                          One or more eigenvalues and eigenvectors have
+                          been accepted since the last call to MONIT.
+
+                    ISTATE = 4
+                          Return from F02FJF is about to occur.
+
+                2:  NEXTIT -- INTEGER                                 Input
+                    On entry: the number of the next iteration.
+
+                3:  NEVALS -- INTEGER                                 Input
+                    On entry: the number of eigenvalues accepted so far.
+
+                4:  NEVECS -- INTEGER                                 Input
+                    On entry: the number of eigenvectors accepted so far.
+
+                5:  K -- INTEGER                                      Input
+                    On entry: k, the number of simultaneous iteration
+                    vectors.
+
+                6:  F(K) -- DOUBLE PRECISION array                    Input
+                    On entry: a vector of error quantities measuring the
+                    state of convergence of the simultaneous iteration
+                    vectors. See the parameter TOL of F02FJF above and
+                    Section 8 for further details. Each element of F is
+                    initially set to the value 4.0 and an element remains
+                    at 4.0 until the corresponding vector is tested.
+
+                7:  D(K) -- DOUBLE PRECISION array                    Input
+                    On entry: D(i) contains the latest approximation to the
+                    absolute value of the ith eigenvalue of C.
+               MONIT must be declared as EXTERNAL in the (sub)program
+               from which F02FJF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           9:  NOVECS -- INTEGER                                      Input
+               On entry: the number of approximate vectors that are being
+               supplied in X. If NOVECS is outside the range (0,K), then
+               the value 0 is used in place of NOVECS.
+
+          10:  X(NRX,K) -- DOUBLE PRECISION array              Input/Output
+               On entry: if 0 < NOVECS <= K, the first NOVECS columns of X
+               must contain approximations to the eigenvectors
+               corresponding to the NOVECS eigenvalues of largest absolute
+               value of C. Supplying approximate eigenvectors can be useful
+               when reasonable approximations are known, or when the
+               routine is being restarted with a larger value of K.
+               Otherwise it is not necessary to supply approximate vectors,
+               as simultaneous iteration vectors will be generated randomly
+               by the routine. On exit: if IFAIL = 0, 2, 3 or 4, the first
+               m' columns contain the eigenvectors corresponding to the
+               eigenvalues returned in the first m' elements of D (see
+               below); and the next k-m'-1 columns contain approximations
+               to the eigenvectors corresponding to the approximate
+               eigenvalues returned in the next k-m'-1 elements of D. Here
+               m' is the value returned in M (see above), the number of
+               eigenvalues actually found. The kth column is used as
+               workspace.
+
+          11:  NRX -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array X as declared in the
+               (sub)program from which F02FJF is called.
+               Constraint: NRX >= N.
+
+          12:  D(K) -- DOUBLE PRECISION array                        Output
+               On exit: if IFAIL = 0, 2, 3 or 4, the first m' elements
+               contain the first m' eigenvalues in decreasing order of
+               magnitude; and the next k-m'-1 elements contain
+               approximations to the next k-m'-1 eigenvalues. Here m' is
+               the value returned in M (see above), the number of
+               eigenvalues actually found. D(k) contains the value e where
+               (-e,e) is the latest interval over which Chebyshev
+               acceleration is performed.
+
+          13:  WORK(LWORK) -- DOUBLE PRECISION array              Workspace
+
+          14:  LWORK -- INTEGER                                       Input
+               On entry: the length of the array WORK, as declared in the
+               (sub)program from which F02FJF is called. Constraint:
+               LWORK>=3*K+max(K*K,2*N).
+
+          15:  RWORK(LRWORK) -- DOUBLE PRECISION array       User Workspace
+               RWORK is not used by F02FJF, but is passed directly to
+               routines DOT and IMAGE and may be used to supply information
+               to these routines.
+
+          16:  LRWORK -- INTEGER                                      Input
+               On entry: the length of the array RWORK, as declared in the
+               (sub)program from which F02FJF is called. Constraint: LRWORK
+               >= 1.
+
+          17:  IWORK(LIWORK) -- INTEGER array                User Workspace
+               IWORK is not used by F02FJF, but is passed directly to
+               routines DOT and IMAGE and may be used to supply information
+               to these routines.
+
+          18:  LIWORK -- INTEGER                                      Input
+               On entry: the length of the array IWORK, as declared in the
+               (sub)program from which F02FJF is called. Constraint: LIWORK
+               >= 1.
+
+          19:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. Users who are
+               unfamiliar with this parameter should refer to the Essential
+               Introduction for details.
+
+               On exit: IFAIL = 0 unless the routine detects an error or
+               gives a warning (see Section 6).
+
+               For this routine, because the values of output parameters
+               may be useful even if IFAIL /=0 on exit, users are
+               recommended to set IFAIL to -1 before entry. It is then
+               essential to test the value of IFAIL on exit. To suppress
+               the output of an error message when soft failure occurs, set
+               IFAIL to 1.
+
+          6. Error Indicators and Warnings
+
+          Errors or warnings specified by the routine:
+
+
+          IFAIL< 0
+               A negative value of IFAIL indicates an exit from F02FJF
+               because the user has set IFLAG negative in DOT or IMAGE. The
+               value of IFAIL will be the same as the user's setting of
+               IFLAG.
+
+          IFAIL= 1
+               On entry N < 1,
+
+               or       M < 1,
+
+               or       M >= K,
+
+               or       K > N,
+
+               or       NRX < N,
+
+               or       LWORK <3*K+max(K*K*N),
+
+               or       LRWORK < 1,
+
+               or       LIWORK < 1.
+
+          IFAIL= 2
+               Not all the requested eigenvalues and vectors have been
+               obtained. Approximations to the rth eigenvalue are
+               oscillating rapidly indicating that severe cancellation is
+               occurring in the rth eigenvector and so M is returned as (r-
+               1). A restart with a larger value of K may permit
+               convergence.
+
+          IFAIL= 3
+               Not all the requested eigenvalues and vectors have been
+               obtained. The rate of convergence of the remaining
+               eigenvectors suggests that more than NOITS iterations would
+               be required and so the input value of M has been reduced. A
+               restart with a larger value of K may permit convergence.
+
+          IFAIL= 4
+               Not all the requested eigenvalues and vectors have been
+               obtained. NOITS iterations have been performed. A restart,
+               possibly with a larger value of K, may permit convergence.
+
+          IFAIL= 5
+               This error is very unlikely to occur, but indicates that
+               convergence of the eigenvalue sub-problem has not taken
+               place. Restarting with a different set of approximate
+               vectors may allow convergence. If this error occurs the user
+               should check carefully that F02FJF is being called
+               correctly.
+
+          7. Accuracy
+
+          Eigenvalues and eigenvectors will normally be computed to the
+          accuracy requested by the parameter TOL, but eigenvectors
+          corresponding to small or to close eigenvalues may not always be
+          computed to the accuracy requested by the parameter TOL. Use of
+          the routine MONIT to monitor acceptance of eigenvalues and
+          eigenvectors is recommended.
+
+          8. Further Comments
+
+          The time taken by the routine will be principally determined by
+          the time taken to solve the eigenvalue sub-problem and the time
+          taken by the routines DOT and IMAGE. The time taken to solve an
+                                                                    2
+          eigenvalue sub-problem is approximately proportional to nk . It
+          is important to be aware that several calls to DOT and IMAGE may
+          occur on each major iteration.
+
+          As can be seen from Table 3.1, many applications of F02FJF will
+          require routine IMAGE to solve a system of linear equations. For
+          example, to find the smallest eigenvalues of Ax=(lambda)Bx, IMAGE
+          needs to solve equations of the form Aw=Bz for w and routines
+          from Chapters F01 and F04 of the NAG Foundation Library will
+          frequently be useful in this context. In particular, if A is a
+          positive-definite variable band matrix, F04MCF may be used after
+          A has been factorized by F01MCF. Thus factorization need be
+          performed only once prior to calling F02FJF. An illustration of
+          this type of use is given in the example program in Section 9.
+
+                           ~
+          An approximation d , to the ith eigenvalue, is accepted as soon
+                            h
+             ~
+          as d  and the previous approximation differ by less than
+              h
+           ~
+          |d |*TOL/10. Eigenvectors are accepted in groups corresponding to
+            h
+          clusters of eigenvalues that are equal, or nearly equal, in
+          absolute value and that have already been accepted. If d  is the
+                                                                  r
+          last eigenvalue in such a group and we define the residual r  as
+                                                                      j
+
+                                      r =Cx -y
+                                       j   j  r
+
+          where y  is the projection of Cx , with respect to B, onto the
+                 r                        j
+          space spanned by x ,x ,...,x  and x  is the current approximation
+                            1  2      r      j
+          to the jth eigenvector, then the value f  returned in MONIT is
+                                                  i
+          given by
+
+                                                       2  T
+                        f =max||r || /||Cx ||     ||x|| =x Bx
+                         i       j  B     j  B         B
+
+          and each vector in the group is accepted as an eigenvector if
+
+                                (|d |f )/(|d |-e)<TOL
+                                   r  r     r
+
+                                                   ~
+          where e is the current approximation to |d |. The values of the
+                                                    k
+          f  are systematically increased if the convergence criteria
+           i
+          appear to be too strict. See Rutishauser [4] for further details.
+
+          The algorithm implemented by F02FJF differs slightly from SIMITZ
+          (Nikolai [1]) in that the eigenvalue sub-problem is solved using
+          the singular value decomposition of the upper triangular matrix R
+                                                                         T
+          of the Gram-Schmidt factorization of Cx , rather than forming R R
+                                                 r
+
+          9. Example
+
+          To find the four eigenvalues of smallest absolute value and
+          corresponding eigenvectors for the generalized symmetric
+          eigenvalue problem Ax=(lambda)Bx, where A and B are the 16 by 16
+          matrices
+
+                           (1 a     a                      )
+                           (a 1 a     a                    )
+                           (  a 1 a     a                  )
+                           (    a 1 a     a                )
+                           (a     a 1 a     a              )
+                           (  a     a 1 a     a            )
+                           (    a     a 1 a     a          )
+                           (      a     a 1 a     a        )
+                         A=(        a     a 1 a     a      )
+                           (          a     a 1 a     a    )
+                           (            a     a 1 a     a  )
+                           (              a     a 1 a     a)
+                           (                a     a 1 a    )
+                           (                  a     a 1 a  )
+                           (                    a     a 1 a)
+                           (                      a     a 1)
+
+                    1
+          where a=- -
+                    4
+
+                           (1 b                            )
+                           (b 1 b                          )
+                           (  b 1 b                        )
+                           (    b 1 b                      )
+                           (      b 1 b                    )
+                           (        b 1 b                  )
+                           (          b 1 b                )
+                         B=(            b 1 b              )
+                           (              b 1 b            )
+                           (                b 1 b          )
+                           (                  b 1 b        )
+                           (                    b 1 b      )
+                           (                      b 1 b    )
+                           (                        b 1 b  )
+                           (                          b 1 b)
+
+                    1
+          where b=- -
+                    2
+
+          TOL is taken as 0.0001 and 6 iteration vectors are used. F01MAF
+          is used to factorize the matrix A, prior to calling F02FJF, and
+          F04MAF is used within IMAGE to solve the equations Aw=Bz for w.
+          Details of the factorization of A are passed from F01MAF to
+          F04MAF by means of the COMMON block BLOCK1.
+
+          Output from MONIT occurs each time ISTATE is non-zero. Note that
+          the required eigenvalues are the reciprocals of the eigenvalues
+          returned by F02FJF.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02wef}{NAG On-line Documentation: f02wef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02WEF(3NAG)      Foundation Library (12/10/92)      F02WEF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02WEF
+                  F02WEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02WEF returns all, or part, of the singular value decomposition
+          of a general real matrix.
+
+          2. Specification
+
+                 SUBROUTINE F02WEF (M, N, A, LDA, NCOLB, B, LDB, WANTQ, Q,
+                1                   LDQ, SV, WANTP, PT, LDPT, WORK, IFAIL)
+                 INTEGER          M, N, LDA, NCOLB, LDB, LDQ, LDPT, IFAIL
+                 DOUBLE PRECISION A(LDA,*), B(LDB,*), Q(LDQ,*), SV(*), PT
+                1                 (LDPT,*), WORK(*)
+                 LOGICAL          WANTQ, WANTP
+
+          3. Description
+
+          The m by n matrix A is factorized as
+
+                                            T
+                                       A=QDP ,
+
+          where
+
+                (S)
+              D=(0),          m>n,
+
+              D=S,            m=n,
+
+              D=(S 0),        m<n,
+
+          Q is an m by m orthogonal matrix, P is an n by n orthogonal
+          matrix and S is a min(m,n) by min(m,n) diagonal matrix with non-
+          negative diagonal elements, sv ,sv ,...,sv        , ordered such
+                                        1   2       min(m,n)
+          that
+
+                            sv >=sv >=...>=sv        >=0.
+                              1    2         min(m,n)
+
+          The first min(m,n) columns of Q are the left-hand singular
+          vectors of A, the diagonal elements of S are the singular values
+          of A and the first min(m,n) columns of P are the right-hand
+          singular vectors of A.
+
+          Either or both of the left-hand and right-hand singular vectors
+          of A may be requested and the matrix C given by
+
+                                          T
+                                       C=Q B,
+
+          where B is an m by ncolb given matrix, may also be requested.
+
+          The routine obtains the singular value decomposition by first
+          reducing A to upper triangular form by means of Householder
+          transformations, from the left when m>=n and from the right when
+          m<n. The upper triangular form is then reduced to bidiagonal form
+          by Givens plane rotations and finally the QR algorithm is used to
+          obtain the singular value decomposition of the bidiagonal form.
+
+          Good background descriptions to the singular value decomposition
+          are given in Dongarra et al [1], Hammarling [2] and Wilkinson [3]
+          DSVDC.
+
+          Note that if K is any orthogonal diagonal matrix so that
+
+                                         T
+                                       KK =I,
+
+          (so that K has elements +1 or -1 on the diagonal)
+
+          then
+
+                                               T
+                                    A=(QK)D(PK)
+
+          is also a singular value decomposition of A.
+
+          4. References
+
+          [1]   Dongarra J J, Moler C B, Bunch J R and Stewart G W (1979)
+                LINPACK Users' Guide. SIAM, Philadelphia.
+
+          [2]   Hammarling S (1985) The Singular Value Decomposition in
+                Multivariate Statistics. ACM Signum Newsletter. 20, 3 2--25.
+
+          [3]   Wilkinson J H (1978) Singular Value Decomposition -- Basic
+                Aspects. Numerical Software -- Needs and Availability. (ed D
+                A H Jacobs) Academic Press.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number of rows, m, of the matrix A.
+               Constraint: M >= 0.
+
+               When M = 0 then an immediate return is effected.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of columns, n, of the matrix A.
+               Constraint: N >= 0.
+
+               When N = 0 then an immediate return is effected.
+
+           3:  A(LDA,*) -- DOUBLE PRECISION array              Input/Output
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the leading m by n part of the array A must
+               contain the matrix A whose singular value decomposition is
+               required. On exit: if M >= N and WANTQ = .TRUE., then the
+               leading m by n part of A will contain the first n columns of
+               the orthogonal matrix Q.
+
+               If M < N and WANTP = .TRUE., then the leading m by n part of
+                                                                         T
+               A will contain the first m rows of the orthogonal matrix P .
+
+               If M >= N and WANTQ = .FALSE. and WANTP = .TRUE., then the
+               leading n by n part of A will contain the first n rows of
+                                      T
+               the orthogonal matrix P .
+
+               Otherwise the leading m by n part of A is used as internal
+               workspace.
+
+           4:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F02WEF is called.
+               Constraint: LDA >= max(1,M).
+
+           5:  NCOLB -- INTEGER                                       Input
+               On entry: ncolb, the number of columns of the matrix B.
+               When NCOLB = 0 the array B is not referenced. Constraint:
+               NCOLB >= 0.
+
+           6:  B(LDB,*) -- DOUBLE PRECISION array              Input/Output
+               Note: the second dimension of the array B must be at least
+               max(1,ncolb) On entry: if NCOLB > 0, the leading m by ncolb
+               part of the array B must contain the matrix to be
+               transformed. On exit: B is overwritten by the m by ncolb
+                        T
+               matrix  Q B.
+
+           7:  LDB -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array B as declared in the
+               (sub)program from which F02WEF is called.
+               Constraint: if NCOLB > 0 then LDB >= max(1,M).
+
+           8:  WANTQ -- LOGICAL                                       Input
+               On entry: WANTQ must be .TRUE., if the left-hand singular
+               vectors are required. If WANTQ = .FALSE., then the array Q
+               is not referenced.
+
+           9:  Q(LDQ,*) -- DOUBLE PRECISION array                    Output
+               Note: the second dimension of the array Q must be at least
+               max(1,M).
+               On exit: if M < N and WANTQ = .TRUE., the leading m by m
+               part of the array Q will contain the orthogonal matrix Q.
+               Otherwise the array Q is not referenced.
+
+          10:  LDQ -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array Q as declared in the
+               (sub)program from which F02WEF is called.
+               Constraint: if M < N and WANTQ = .TRUE., LDQ >= max(1,M).
+
+          11:  SV(*) -- DOUBLE PRECISION array                       Output
+               Note: the length of SV must be at least min(M,N). On exit:
+               the min(M,N) diagonal elements of the matrix S.
+
+          12:  WANTP -- LOGICAL                                       Input
+               On entry: WANTP must be .TRUE. if the right-hand singular
+               vectors are required. If WANTP = .FALSE., then the array PT
+               is not referenced.
+
+          13:  PT(LDPT,*) -- DOUBLE PRECISION array                  Output
+               Note: the second dimension of the array PT must be at least
+               max(1,N).
+               On exit: if M >= N and WANTQ and WANTP are .TRUE., the
+               leading n by n part of the array PT will contain the
+                                  T
+               orthogonal matrix P . Otherwise the array PT is not
+               referenced.
+
+          14:  LDPT -- INTEGER                                        Input
+               On entry:
+               the first dimension of the array PT as declared in the
+               (sub)program from which F02WEF is called.
+               Constraint: if M >= N and WANTQ and WANTP are .TRUE., LDPT
+               >= max(1,N).
+
+          15:  WORK(*) -- DOUBLE PRECISION array                     Output
+               Note: the length of WORK must be at least max(1,lwork),
+               where lwork must be as given in the following table:
+
+               M >= N
+               WANTQ is .TRUE. and WANTP = .TRUE.
+                                2
+                     lwork=max(N +5*(N-1),N+NCOLB,4)
+
+               WANTQ = .TRUE. and WANTP = .FALSE.
+                                2
+                     lwork=max(N +4*(N-1),N+NCOLB,4)
+
+               WANTQ = .FALSE. and WANTP = .TRUE.
+                     lwork=max(3*(N-1),2) when NCOLB = 0
+
+                     lwork=max(5*(N-1),2) when NCOLB > 0
+
+               WANTQ = .FALSE. and WANTP = .FALSE.
+                     lwork=max(2*(N-1),2) when NCOLB = 0
+
+                     lwork=max(3*(N-1),2) when NCOLB > 0
+
+               M < N
+               WANTQ = .TRUE. and WANTP = .TRUE.
+                                2
+                     lwork=max(M +5*(M-1),2)
+
+               WANTQ = .TRUE. and WANTP = .FALSE.
+                     lwork=max(3*(M-1),1)
+
+               WANTQ = .FALSE. and WANTP = .TRUE.
+                                2
+                     lwork=max(M +3*(M-1),2) when NCOLB = 0
+
+                                2
+                     lwork=max(M +5*(M-1),2) when NCOLB > 0
+
+               WANTQ = .FALSE. and WANTP = .FALSE.
+                     lwork=max(2*(M-1),1) when NCOLB = 0
+
+                     lwork=max(3*(M-1),1) when NCOLB > 0
+               On exit: WORK(min(M,N)) contains the total number of
+               iterations taken by the R algorithm.
+
+               The rest of the array is used as workspace.
+
+          16:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL=-1
+               One or more of the following conditions holds:
+                    M < 0,
+
+                    N < 0,
+
+                    LDA < M,
+
+                    NCOLB < 0,
+
+                    LDB < M and NCOLB > 0,
+
+                    LDQ < M and M < N and WANTQ = .TRUE.,
+
+                    LDPT < N and M >= N and WANTQ = .TRUE., and WANTP = .
+                    TRUE..
+
+          IFAIL> 0
+               The QR algorithm has failed to converge in 50*min(m,n)
+               iterations. In this case SV(1), SV(2),..., SV(IFAIL) may not
+               have been found correctly and the remaining singular values
+               may not be the smallest. The matrix A will nevertheless have
+                                        T
+               been factorized as  A=QEP ,  where the leading min(m,n) by
+               min(m,n) part of E is a bidiagonal matrix with SV(1), SV(2),
+               ..., SV(min(m,n)) as the diagonal elements and WORK(1), WORK
+               (2),..., WORK(min(m,n)-1) as the super-diagonal elements.
+
+               This failure is not likely to occur.
+
+          7. Accuracy
+
+          The computed factors Q, D and P satisfy the relation
+
+                                         T
+                                      QDP =A+E,
+
+          where
+
+                             ||E||<=c(epsilon)||A||,
+
+          (epsilon) being the machine precision, c is a modest function of
+          m and n and ||.|| denotes the spectral (two) norm. Note that
+          ||A||=sv .
+                  1
+
+          8. Further Comments
+
+          Following the use of this routine the rank of A may be estimated
+          by a call to the INTEGER FUNCTION F06KLF(*). The statement:
+
+                 IRANK = F06KLF(MIN(M, N), SV, 1, TOL)
+
+          returns the value (k-1) in IRANK, where k is the smallest integer
+          for which SV(k)<tol*SV(1), where tol is the tolerance supplied in
+          TOL, so that IRANK is an estimate of the rank of S and thus also
+          of A. If TOL is supplied as negative then the machine precision
+          is used in place of TOL.
+
+          9. Example
+
+          9.1. Example 1
+
+          To find the singular value decomposition of the 5 by 3 matrix
+
+                                    (2.0  2.5  2.5)
+                                    (2.0  2.5  2.5)
+                                  A=(1.6 -0.4  2.8)
+                                    (2.0 -0.5  0.5)
+                                    (1.2 -0.3 -2.9)
+
+                                    T
+          together with the vector Q b for the vector
+
+                                        ( 1.1)
+                                        ( 0.9)
+                                      b=( 0.6)
+                                        ( 0.0)
+                                        (-0.8)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+          9.2. Example 2
+
+          To find the singular value decomposition of the 3 by 5 matrix
+
+                               (2.0 2.0  1.6  2.0  1.2)
+                             A=(2.5 2.5 -0.4 -0.5 -0.3)
+                               (2.5 2.5  2.8  0.5 -2.9)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf02xef}{NAG On-line Documentation: f02xef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F02XEF(3NAG)      Foundation Library (12/10/92)      F02XEF(3NAG)
+
+
+
+          F02 -- Eigenvalue and Eigenvectors                         F02XEF
+                  F02XEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F02XEF returns all, or part, of the singular value decomposition
+          of a general complex matrix.
+
+          2. Specification
+
+                 SUBROUTINE F02XEF (M, N, A, LDA, NCOLB, B, LDB, WANTQ, Q,
+                1                   LDQ, SV, WANTP, PH, LDPH, RWORK, CWORK,
+                2                   IFAIL)
+                 INTEGER                   M, N, LDA, NCOLB, LDB, LDQ, LDPH,
+                1                          IFAIL
+                 DOUBLE PRECISION          SV(*), RWORK(*)
+                 COMPLEX(KIND=KIND(1.0D0)) A(LDA,*), B(LDB,*), Q(LDQ,*),
+                1                          PH(LDPH,*), CWORK(*)
+                 LOGICAL                   WANTQ, WANTP
+
+          3. Description
+
+          The m by n matrix A is factorized as
+
+                                            H
+                                       A=QDP ,
+
+          where
+
+                (S)
+              D=(0)           m>n,
+
+              D=S,            m=n,
+
+              D=(S 0),        m<n,
+
+          Q is an m by m unitary matrix, P is an n by n unitary matrix and
+          S is a min(m,n) by min(m,n) diagonal matrix with real non-
+          negative diagonal elements, sv ,sv ,...,sv        , ordered such
+                                        1   2       min(m,n)
+          that
+
+                            sv >=sv >=...>=sv        >=0.
+                              1    2         min(m,n)
+
+          The first min(m,n) columns of Q are the left-hand singular
+          vectors of A, the diagonal elements of S are the singular values
+          of A and the first min(m,n) columns of P are the right-hand
+          singular vectors of A.
+
+          Either or both of the left-hand and right-hand singular vectors
+          of A may be requested and the matrix C given by
+
+                                          H
+                                       C=Q B,
+
+          where B is an m by ncolb given matrix, may also be requested.
+
+          The routine obtains the singular value decomposition by first
+          reducing A to upper triangular form by means of Householder
+          transformations, from the left when m>=n and from the right when
+          m<n. The upper triangular form is then reduced to bidiagonal form
+          by Givens plane rotations and finally the QR algorithm is used to
+          obtain the singular value decomposition of the bidiagonal form.
+
+          Good background descriptions to the singular value decomposition
+          are given in Dongarra et al [1], Hammarling [2] and Wilkinson [3]
+          ZSVDC.
+
+          Note that if K is any unitary diagonal matrix so that
+
+                                         H
+                                       KK =I,
+
+          then
+
+                                               H
+                                    A=(QK)D(PK)
+
+          is also a singular value decomposition of A.
+
+          4. References
+
+          [1]   Dongarra J J, Moler C B, Bunch J R and Stewart G W (1979)
+                LINPACK Users' Guide. SIAM, Philadelphia.
+
+          [2]   Hammarling S (1985) The Singular Value Decomposition in
+                Multivariate Statistics. ACM Signum Newsletter. 20, 3 2--25.
+
+          [3]   Wilkinson J H (1978) Singular Value Decomposition -- Basic
+                Aspects. Numerical Software -- Needs and Availability. (ed D
+                A H Jacobs) Academic Press.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the number of rows, m, of the matrix A.
+               Constraint: M >= 0.
+
+               When M = 0 then an immediate return is effected.
+
+           2:  N -- INTEGER                                           Input
+               On entry: the number of columns, n, of the matrix A.
+               Constraint: N >= 0.
+
+               When N = 0 then an immediate return is effected.
+
+           3:  A(LDA,*) -- COMPLEX(KIND(1.0D)) array           Input/Output
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the leading m by n part of the array A must
+               contain the matrix A whose singular value decomposition is
+               required. On exit: if M >= N and WANTQ = .TRUE., then the
+               leading m by n part of A will contain the first n columns of
+               the unitary matrix Q.
+               If M < N and WANTP = .TRUE., then the leading m by n part of
+                                                                      H
+               A will contain the first m rows of the unitary matrix P .
+               will contain the first m rows of the unitary matrix P If M
+               >= N and WANTQ = .FALSE. and WANTP = .TRUE., then the
+               leading n by n part of A will contain the first n
+                                           H
+               rows of the unitary matrix P . Otherwise the leading m by n
+               part of A is used as internal workspace.
+
+           4:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F02XEF is called.
+               Constraint: LDA >= max(1,M).
+
+           5:  NCOLB -- INTEGER                                       Input
+               On entry: ncolb, the number of columns of the matrix B.
+               When NCOLB = 0 the array B is not referenced. Constraint:
+               NCOLB >= 0.
+
+           6:  B(LDB,*) -- COMPLEX(KIND(1.0D)) array           Input/Output
+               Note: the second dimension of the array B must be at least
+               max(1,NCOLB).
+               On entry: if NCOLB > 0, the leading m by ncolb part of the
+               array B must contain the matrix to be transformed. On exit:
+                                                          H
+               B is overwritten by the m by ncolb matrix Q B.
+
+           7:  LDB -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array B as declared in the
+               (sub)program from which F02XEF is called.
+               Constraint: if NCOLB > 0, then LDB >= max(1,M).
+
+           8:  WANTQ -- LOGICAL                                       Input
+               On entry: WANTQ must be .TRUE. if the left-hand singular
+               vectors are required. If WANTQ = .FALSE. then the array Q is
+               not referenced.
+
+           9:  Q(LDQ,*) -- COMPLEX(KIND(1.0D)) array                 Output
+               Note: the second dimension of the array Q must be at least
+               max(1,M).
+               On exit: if M < N and WANTQ = .TRUE., the leading m by m
+               part of the array Q will contain the unitary matrix Q.
+               Otherwise the array Q is not referenced.
+
+          10:  LDQ -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array Q as declared in the
+               (sub)program from which F02XEF is called.
+               Constraint: if M < N and WANTQ = .TRUE., LDQ >= max(1,M).
+
+          11:  SV(*) -- DOUBLE PRECISION array                       Output
+               Note: the length of SV must be at least min(M,N). On exit:
+               the min(m,n) diagonal elements of the matrix S.
+
+          12:  WANTP -- LOGICAL                                       Input
+               On entry: WANTP must be .TRUE. if the right-hand singular
+               vectors are required. If WANTP = .FALSE. then the array PH
+               is not referenced.
+
+          13:  PH(LDPH,*) -- DOUBLE PRECISION array                  Output
+               Note: the second dimension of the array PH must be at least
+               max(1,N).
+               On exit: if M >= N and WANTQ and WANTP are .TRUE., the
+               leading n by n part of the array PH will contain the unitary
+                       H
+               matrix P . Otherwise the array PH is not referenced.
+
+          14:  LDPH -- INTEGER                                        Input
+               On entry:
+               the first dimension of the array PH as declared in the
+               (sub)program from which F02XEF is called.
+               Constraint: if M >= N and WANTQ and WANTP are .TRUE.,  LDPH
+               >= max(1,N).
+
+          15:  RWORK(*) -- DOUBLE PRECISION array                    Output
+               Note: the length of RWORK must be at least max(1,lrwork),
+               where lrwork must satisfy:
+               lrwork=2*(min(M,N)-1) when
+                     NCOLB = 0 and WANTQ and WANTP are .FALSE.,
+
+               lrwork=3*(min(M,N)-1) when
+                     either NCOLB = 0 and WANTQ = .FALSE. and WANTP = .
+                     TRUE., or WANTP = .FALSE. and one or both of NCOLB > 0
+                     and WANTQ = .TRUE.
+
+               lrwork=5*(min(M,N)-1)
+                     otherwise.
+               On exit: RWORK(min(M,N)) contains the total number of
+               iterations taken by the QR algorithm.
+
+               The rest of the array is used as workspace.
+
+          16:  CWORK(*) -- COMPLEX(KIND(1.0D)) array              Workspace
+               Note: the length of CWORK must be at least max(1,lcwork),
+               where lcwork must satisfy:
+                             2
+               lcwork=N+max(N ,NCOLB) when
+                     M >= N and WANTQ and WANTP are both .TRUE.
+
+                             2
+               lcwork=N+max(N +N,NCOLB) when
+                     M >= N and WANTQ = .TRUE., but WANTP = .FALSE.
+
+               lcwork=N+max(N,NCOLB) when
+                     M >= N and WANTQ = .FALSE.
+
+                       2
+               lcwork=M +M when
+                     M < N and WANTP = .TRUE.
+
+               lcwork = M when
+                     M < N and WANTP = .FALSE.
+
+          17:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL=-1
+               One or more of the following conditions holds:
+                    M < 0,
+
+                    N < 0,
+
+                    LDA < M,
+
+                    NCOLB < 0,
+
+                    LDB < M and NCOLB > 0,
+
+                    LDQ < M and M < N and WANTQ = .TRUE.,
+
+                    LDPH < N and M >= N and WANTQ = .TRUE. and WANTP = .
+                    TRUE..
+
+          IFAIL> 0
+               The QR algorithm has failed to converge in 50*min(m,n)
+               iterations. In this case SV(1), SV(2),..., SV(IFAIL) may not
+               have been found correctly and the remaining singular values
+               may not be the smallest. The matrix A will nevertheless have
+                                        H
+               been factorized as  A=QEP   where the leading min(m,n) by
+               min(m,n) part of E is a bidiagonal matrix with SV(1), SV(2),
+               ..., SV(min(m,n)) as the diagonal elements and RWORK(1),
+               RWORK(2),..., RWORK(min(m,n)-1) as the super-diagonal
+               elements.
+
+               This failure is not likely to occur.
+
+          7. Accuracy
+
+          The computed factors Q, D and P satisfy the relation
+
+                                         H
+                                      QDP =A+E,
+
+          where
+
+                             ||E||<=c(epsilon)||A||,
+
+          (epsilon) being the machine precision, c is a modest function of
+          m and n and ||.|| denotes the spectral (two) norm. Note that
+          ||A||=sv .
+                  1
+
+          8. Further Comments
+
+          Following the use of this routine the rank of A may be estimated
+          by a call to the INTEGER FUNCTION F06KLF(*). The statement:
+
+
+                  IRANK = F06KLF(MIN(M, N), SV, 1, TOL)
+
+          returns the value (k-1) in IRANK, where k is the smallest integer
+          for which SV(k)<tol*SV(1), where tol is the tolerance supplied in
+          TOL, so that IRANK is an estimate of the rank of S and thus also
+          of A. If TOL is supplied as negative then the machine precision
+          is used in place of TOL.
+
+          9. Example
+
+          9.1. Example 1
+
+          To find the singular value decomposition of the 5 by 3 matrix
+
+                             (    0.5i -0.5+1.5i -1.0+1.0i)
+                             (0.4+0.3i  0.9+1.3i  0.2+1.4i)
+                           A=(0.4      -0.4+0.4i  1.8     )
+                             (0.3-0.4i  0.1+0.7i  0.0     )
+                             (   -0.3i  0.3+0.3i      2.4i)
+
+                                    H
+          together with the vector Q b for the vector
+
+                                     (-0.55+1.05i)
+                                     ( 0.49+0.93i)
+                                   b=( 0.56-0.16i)
+                                     ( 0.39+0.23i)
+                                     ( 1.13+0.83i)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+          9.2. Example 2
+
+          To find the singular value decompostition of the 3 by 5 matrix
+
+                   (     0.5i 0.4-0.3i  0.4      0.3+0.4i     0.3i)
+                 A=(-0.5-1.5i 0.9-1.3i -0.4-0.4i 0.1-0.7i 0.3-0.3i)
+                   (-1.0-1.0i 0.2-1.4i  1.8      0.0         -2.4i)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf04}{NAG On-line Documentation: f04}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F04(3NAG)         Foundation Library (12/10/92)         F04(3NAG)
+
+
+
+          F04 -- Simultaneous Linear Equations          Introduction -- F04
+                                    Chapter F04
+                           Simultaneous Linear Equations
+
+          1. Scope of the Chapter
+
+          This chapter, together with two routines in Chapter F07, is
+          concerned with the solution of the matrix equation AX=B, where B
+          may be a single vector or a matrix of multiple right-hand sides.
+          The matrix A may be real, complex, symmetric, Hermitian positive-
+          definite, or sparse. It may also be rectangular, in which case a
+          least-squares solution is obtained.
+
+          2. Background to the Problems
+
+          A set of linear equations may be written in the form
+
+                                        Ax=b
+
+          where the known matrix A, with real or complex coefficients, is
+          of size m by n, (m rows and n columns), the known right-hand
+          vector b has m components (m rows and one column), and the
+          required solution vector x has n components (n rows and one
+          column). There may sometimes be p vectors b , i=1,2,...,p on the
+                                                     i
+          right-hand side and the equations may then be written as
+
+                                        AX=B
+
+          the required matrix X having as its p columns the solutions of
+          Ax =b , i=1,2,...,p. Some routines deal with the latter case, but
+            i  i
+          for clarity only the case p=1 is discussed here.
+
+          The most common problem, the determination of the unique solution
+          of Ax=b, occurs when m=n and A is non-singular, that is rank(A)=n
+          problem, discussed in Section 2.2 below, is the determination of
+          the least-squares solution of Ax~=b, i.e., the determination of a
+          vector x which minimizes the Euclidean length (two norm) of the
+          residual vector r=b-Ax. The usual case has m>n and rank(A)=n, in
+          which case x is unique.
+
+          2.1. Unique Solution of Ax=b
+
+          Most of the routines in this chapter, as well as two routines in
+          Chapter F07, solve this particular problem. The solution is
+          obtained by performing either an LU factorization, or a Cholesky
+          factorization, as discussed in Section 2 of the F01 Chapter
+          Introduction.
+
+          Two of the routines in this chapter use a process called
+          iterative refinement to improve the initial solution in order to
+          obtain a solution that is correct to working accuracy. It should
+          be emphasised that if A and b are not known exactly then not all
+          the figures in this solution may be meaningful. To be more
+          precise, if x is the exact solution of the equations
+
+
+                                        Ax=b
+
+              
+
+          and x is the solution of the perturbed equations
+
+                                          
+
+                                     (A+E)x=b+e,
+
+                                         ||E||
+          then, provided that (kappa)(A) -------<=1,
+                                         ||A||
+
+                   
+
+                 ||x-x||        (kappa)(A)      ( ||E||   ||e|| )
+                 ------- <= --------------------( ----- + ----- ),
+                  ||x||                  ||E||  ( ||A||   ||b|| )
+                            1-(kappa)(A) -----
+                                         ||A||
+
+                                   -1
+          where (kappa)(A)=||A||||A  || is the condition number of A
+          with respect to inversion. Thus, if A is ill-conditioned (
+                                
+
+          (kappa)(A) is large), x may differ significantly from x. Often
+                     ||E||
+          (kappa)(A) -----<<1 in which case the above bound effectively
+                     ||A||
+          reduces to
+
+
+                        
+
+                      ||x-x||              ( ||E||   ||e|| )
+                      ------- <= (kappa)(A)( ----- + ----- ).
+                       ||x||               ( ||A||   ||b|| )
+
+          2.2. The Least-squares Solution of Ax~=b
+
+          The least-squares problem is to find a vector x to minimize
+
+
+                                  T
+                                 r r,  where r=b-Ax.
+
+          When m>=n and rank(A)=n then the solution vector x is unique. For
+          the cases where x is not unique the routines in this chapter
+          obtain the minimal length solution, that is the vector x for
+                 T
+          which x x is a minimum.
+
+          2.3. Calculating the Inverse of a Matrix
+
+          The routines in this chapter can also be used to calculate the
+          inverse of a square matrix A by solving the equation
+
+                                        AX=I,
+
+          where I is the identity matrix.
+
+          3. Recommendations on Choice and Use of Routines
+
+          3.1. General Purpose Routines
+
+          Many of the routines in this chapter perform the complete
+          solution of the required equations, but some of the routines, as
+          well as the routines in Chapter F07, assume that a prior
+          factorization has been performed, using the appropriate
+          factorization routine from Chapter F01 or Chapter F07. These, so-
+          called, general purpose routines can be useful when explicit
+          information on the factorization is required, as well as the
+          solution of the equations, or when the solution is required for
+          multiple right-hand sides, or for a sequence of right-hand sides.
+
+          Note that some of the routines that perform a complete solution
+          also allow multiple right-hand sides.
+
+          3.2. Iterative Refinement
+
+          The routines that perform iterative refinement are more costly
+          than those that do not perform iterative refinement, both in
+          terms of time and storage, and should only be used if the problem
+          really warrants the additional accuracy provided by these
+          routines. The storage requirements are approximately doubled,
+          while the additional time is not usually prohibitive since the
+          initial factorization is used at each iteration.
+
+          3.3. Sparse Matrix Routines
+
+          The routines for sparse matrices should usually be used only when
+          the number of non-zero elements is very small, less than 10% of
+          the total number of elements of A. Additionally, when the matrix
+          is symmetric positive-definite the sparse routines should
+          generally be used only when A does not have a (variable) band
+          structure.
+
+          There are four routines for solving sparse linear equations, two
+          for solving general real systems (F04AXF and F04QAF), one for
+          solving symmetric positive-definite systems (F04MAF) and one for
+          solving symmetric systems that may, or may not, be positive-
+          definite (F04MBF). F04AXF and F04MAF utilise factorizations of
+          the matrix A obtained by routines in Chapter F01, while the other
+          two routines use iterative techniques and require a user-supplied
+                                                             T
+          function to compute matrix-vector products Ac and A c for any
+          given vector c. The routines requiring factorizations will
+          usually be faster and the factorization can be utilised to solve
+          for several right-hand sides, but the original matrix has to be
+          explicitly supplied and is overwritten by the factorization, and
+          the storage requirements will usually be substantially more than
+          those of the iterative routines.
+
+          Routines F04MBF and F04QAF both allow the user to supply a pre-
+          conditioner.
+
+          F04MBF can be used to solve systems of the form (A-(lambda)I)x=b,
+          which can be useful in applications such as Rayleigh quotient
+          iteration.
+
+          F04QAF also solves sparse least-squares problems and allows the
+          solution of damped (regularized) least-squares problems.
+
+          3.4. Decision Trees
+
+          If at any stage the answer to a question is 'Don't know' this
+          should be read as 'No'.
+
+          For those routines that need to be preceded by a factorization
+          routine, the appropriate routine name is given in brackets after
+          the name of the routine for solving the equations. Note also that
+          you may be directed to a routine in Chapter F07.
+
+
+          3.4.1. Routines for unique solution of Ax=b
+
+
+                   Please see figure in printed Reference Manual
+
+
+
+          3.4.2.  Routines for Least-squares problems
+
+
+                   Please see figure in printed Reference Manual
+
+
+
+
+          F04 -- Simultaneous Linear Equations              Contents -- F04
+          Chapter F04
+
+          Eigenvalues and Eigenvectors
+
+          F04ADF  Approximate solution of complex simultaneous linear
+                  equations with multiple right-hand sides
+
+          F04ARF  Approximate solution of real simultaneous linear
+                  equations, one right-hand side
+
+          F04ASF  Accurate solution of real symmetric positive-definite
+                  simultaneous linear equations, one right-hand side
+
+          F04ATF  Accurate solution of real simultaneous linear equations,
+                  one right-hand side
+
+          F04AXF  Approximate solution of real sparse simultaneous linear
+                  equations (coefficient matrix already factorized by
+                  F01BRF or F01BSF)
+
+          F04FAF  Approximate solution of real symmetric positive-definite
+                  tridiagonal simultaneous linear equations, one right-hand
+                  side
+
+          F04JGF  Least-squares (if rank = n) or minimal least-squares (if
+                  rank <n) solution of m real equations in n unknowns, rank
+                  <=n, m>=n
+
+          F04MAF  Real sparse symmetric positive-definite simultaneous
+                  linear equations (coefficient matrix already factorized)
+
+          F04MBF  Real sparse symmetric simultaneous linear equations
+
+          F04MCF  Approximate solution of real symmetric positive-definite
+                  variable-bandwidth simultaneous linear equations
+                  (coefficient matrix already factorized)
+
+          F04QAF  Sparse linear least-squares problem, m real equations in
+                  n unknowns
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf04adf}{NAG On-line Documentation: f04adf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F04ADF(3NAG)      Foundation Library (12/10/92)      F04ADF(3NAG)
+
+
+
+          F04 -- Simultaneous Linear Equations                       F04ADF
+                  F04ADF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F04ADF calculates the approximate solution of a set of complex
+          linear equations with multiple right-hand sides, using an LU
+          factorization with partial pivoting.
+
+          2. Specification
+
+                 SUBROUTINE F04ADF (A, IA, B, IB, N, M, C, IC, WKSPCE,
+                1                   IFAIL)
+                 INTEGER              IA, IB, N, M, IC, IFAIL
+                 DOUBLE PRECISION     WKSPCE(*)
+                 COMPLEX(KIND(1.0D0)) A(IA,*), B(IB,*), C(IC,*)
+
+          3. Description
+
+          Given a set of complex linear equations AX=B, the routine first
+          computes an LU factorization of A with partial pivoting, PA=LU,
+          where P is a permutation matrix, L is lower triangular and U is
+          unit upper triangular. The columns x of the solution X are found
+          by forward and backward substitution in Ly=Pb and Ux=y, where b
+          is a column of the right-hand side matrix B.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  A(IA,*) -- COMPLEX(KIND(1.0D)) array            Input/Output
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the n by n matrix A. On exit: A is overwritten by
+               the lower triangular matrix L and the off-diagonal elements
+               of the upper triangular matrix U. The unit diagonal elements
+               of U are not stored.
+
+           2:  IA -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F04ADF is called.
+               Constraint: IA >= max(1,N).
+
+           3:  B(IB,*) -- COMPLEX(KIND(1.0D)) array                   Input
+               Note: the second dimension of the array B must be at least
+               max(1,M).
+               On entry: the n by m right-hand side matrix B. See also
+               Section 8.
+
+           4:  IB -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array B as declared in the
+               (sub)program from which F04ADF is called.
+               Constraint: IB >= max(1,N).
+
+           5:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A. Constraint: N >= 0.
+
+           6:  M -- INTEGER                                           Input
+               On entry: m, the number of right-hand sides. Constraint: M
+               >= 0.
+
+           7:  C(IC,*) -- COMPLEX(KIND(1.0D)) array                  Output
+               Note: the second dimension of the array C must be at least
+               max(1,M).
+               On exit: the n by m solution matrix X. See also Section 8.
+
+           8:  IC -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array C as declared in the
+               (sub)program from which F04ADF is called.
+               Constraint: IC >= max(1,N).
+
+           9:  WKSPCE(*) -- DOUBLE PRECISION array                Workspace
+               Note: the dimension of the array WKSPCE must be at least
+               max(1,N).
+
+          10:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               The matrix A is singular, possibly due to rounding errors.
+
+          IFAIL= 2
+               On entry N < 0,
+
+               or       M < 0,
+
+               or       IA < max(1,N),
+
+               or       IB < max(1,N),
+
+               or       IC < max(1,N).
+
+          7. Accuracy
+
+          The accuracy of the computed solution depends on the conditioning
+          of the original matrix. For a detailed error analysis see
+          Wilkinson and Reinsch [1] page 106.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          Unless otherwise stated in the Users' Note for your
+          implementation, the routine may be called with the same actual
+          array supplied for parameters B and C, in which case the solution
+          vectors will overwrite the right-hand sides. However this is not
+          standard Fortran 77, and may not work on all systems.
+
+          9. Example
+
+          To solve the set of linear equations AX=B where
+
+                                   (1  1+2i  2+10i)
+                                 A=(1+i  3i -5+14i)
+                                   (1+i  5i -8+20i)
+
+          and
+
+
+                                         (1)
+                                       B=(0).
+                                         (0)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf04arf}{NAG On-line Documentation: f04arf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F04ARF(3NAG)      Foundation Library (12/10/92)      F04ARF(3NAG)
+
+
+
+          F04 -- Simultaneous Linear Equations                       F04ARF
+                  F04ARF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F04ARF calculates the approximate solution of a set of real
+          linear equations with a single right-hand side, using an LU
+          factorization with partial pivoting.
+
+          2. Specification
+
+                 SUBROUTINE F04ARF (A, IA, B, N, C, WKSPCE, IFAIL)
+                 INTEGER          IA, N, IFAIL
+                 DOUBLE PRECISION A(IA,*), B(*), C(*), WKSPCE(*)
+
+          3. Description
+
+          Given a set of linear equations, Ax=b, the routine first computes
+          an LU factorization of A with partial pivoting, PA=LU, where P is
+          a permutation matrix, L is lower triangular and U is unit upper
+          triangular. The approximate solution x is found by forward and
+          backward substitution in Ly=Pb and Ux=y, where b is the right-
+          hand side.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  A(IA,*) -- DOUBLE PRECISION array               Input/Output
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the n by n matrix A. On exit: A is overwritten by
+               the lower triangular matrix L and the off-diagonal elements
+               of the upper triangular matrix U. The unit diagonal elements
+               of U are not stored.
+
+           2:  IA -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F04ARF is called.
+               Constraint: IA >= max(1,N).
+
+           3:  B(*) -- DOUBLE PRECISION array                         Input
+               Note: the dimension of the array B must be at least
+               max(1,N).
+               On entry: the right-hand side vector b.
+
+           4:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A. Constraint: N >= 0.
+
+           5:  C(*) -- DOUBLE PRECISION array                        Output
+               Note: the dimension of the array C must be at least
+               max(1,N).
+               On exit: the solution vector x.
+
+           6:  WKSPCE(*) -- DOUBLE PRECISION array                Workspace
+               Note: the dimension of the array WKSPCE must be at least
+               max(1,N).
+
+           7:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               The matrix A is singular, possibly due to rounding errors.
+
+          IFAIL= 2
+               On entry N < 0,
+
+               or       IA < max(1,N).
+
+          7. Accuracy
+
+          The accuracy of the computed solution depends on the conditioning
+          of the original matrix. For a detailed error analysis see
+          Wilkinson and Reinsch [1] page 107.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          Unless otherwise stated in the Users' Note for your
+          implementation, the routine may be called with the same actual
+          array supplied for parameters B and C, in which case the solution
+          vector will overwrite the right-hand side. However this is not
+          standard Fortran 77, and may not work on all systems.
+
+          9. Example
+
+          To solve the set of linear equations Ax=b where
+
+                                     ( 33  16  72)
+                                   A=(-24 -10 -57)
+                                     ( -8  -4 -17)
+
+          and
+
+
+                                        (-359)
+                                      b=( 281).
+                                        (  85)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf04asf}{NAG On-line Documentation: f04asf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F04ASF(3NAG)      Foundation Library (12/10/92)      F04ASF(3NAG)
+
+
+
+          F04 -- Simultaneous Linear Equations                       F04ASF
+                  F04ASF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F04ASF calculates the accurate solution of a set of real
+          symmetric positive-definite linear equations with a single right-
+          hand side,  Ax=b, using a Cholesky factorization and iterative
+          refinement.
+
+          2. Specification
+
+                 SUBROUTINE F04ASF (A, IA, B, N, C, WK1, WK2, IFAIL)
+                 INTEGER          IA, N, IFAIL
+                 DOUBLE PRECISION A(IA,*), B(*), C(*), WK1(*), WK2(*)
+
+          3. Description
+
+          Given a set of real linear equations Ax=b, where A is a symmetric
+          positive-definite matrix, the routine first computes a Cholesky
+                                    T
+          factorization of A as A=LL  where L is lower triangular. An
+          approximation to x is found by forward and backward substitution.
+          The residual vector r=b-Ax is then calculated using additional
+                                                                  T
+          precision and a correction d to x is found by solving LL d=r. x
+          is then replaced by x+d, and this iterative refinement of the
+          solution is repeated until machine accuracy is obtained.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  A(IA,*) -- DOUBLE PRECISION array               Input/Output
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the upper triangle of the n by n positive-definite
+               symmetric matrix A. The elements of the array below the
+               diagonal need not be set. On exit: the elements of the array
+               below the diagonal are overwritten; the upper triangle of A
+               is unchanged.
+
+           2:  IA -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F04ASF is called.
+               Constraint: IA >= max(1,N).
+
+           3:  B(*) -- DOUBLE PRECISION array                         Input
+               Note: the dimension of the array B must be at least
+               max(1,N).
+               On entry: the right-hand side vector b.
+
+           4:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A. Constraint: N >= 0.
+
+           5:  C(*) -- DOUBLE PRECISION array                        Output
+               Note: the dimension of the array C must be at least
+               max(1,N).
+               On exit: the solution vector x.
+
+           6:  WK1(*) -- DOUBLE PRECISION array                   Workspace
+               Note: the dimension of the array WK1 must be at least
+               max(1,N).
+
+           7:  WK2(*) -- DOUBLE PRECISION array                   Workspace
+               Note: the dimension of the array WK2 must be at least
+               max(1,N).
+
+           8:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               The matrix A is not positive-definite, possibly due to
+               rounding errors.
+
+          IFAIL= 2
+               Iterative refinement fails to improve the solution, i.e.,
+               the matrix A is too ill-conditioned.
+
+          IFAIL= 3
+               On entry N < 0,
+
+               or       IA < max(1,N).
+
+          7. Accuracy
+
+          The computed solutions should be correct to full machine
+          accuracy. For a detailed error analysis see Wilkinson and Reinsch
+          [1] page 39.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          The routine must not be called with the same name for parameters
+          B and C.
+
+          9. Example
+
+          To solve the set of linear equations Ax=b where
+
+                                     (5  7  6  5)
+                                     (7 10  8  7)
+                                   A=(6  8 10  9)
+                                     (5  7  9 10)
+
+          and
+
+
+                                         (23)
+                                         (32)
+                                       b=(33).
+                                         (31)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf04atf}{NAG On-line Documentation: f04atf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F04ATF(3NAG)      Foundation Library (12/10/92)      F04ATF(3NAG)
+
+
+
+          F04 -- Simultaneous Linear Equations                       F04ATF
+                  F04ATF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F04ATF calculates the accurate solution of a set of real linear
+          equations with a single right-hand side, using an LU
+          factorization with partial pivoting, and iterative refinement.
+
+          2. Specification
+
+                 SUBROUTINE F04ATF (A, IA, B, N, C, AA, IAA, WKS1, WKS2,
+                1                   IFAIL)
+                 INTEGER          IA, N, IAA, IFAIL
+                 DOUBLE PRECISION A(IA,*), B(*), C(*), AA(IAA,*), WKS1(*),
+                1                 WKS2(*)
+
+          3. Description
+
+          Given a set of real linear equations, Ax=b, the routine first
+          computes an LU factorization of A with partial pivoting, PA=LU,
+          where P is a permutation matrix, L is lower triangular and U is
+          unit upper triangular. An approximation to x is found by forward
+          and backward substitution in Ly=Pb and Ux=y. The residual vector
+          r=b-Ax is then calculated using additional precision, and a
+          correction d to x is found by solving LUd=r. x is replaced by x+d
+          , and this iterative refinement of the solution is repeated until
+          full machine accuracy is obtained.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  A(IA,*) -- DOUBLE PRECISION array                      Input
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the n by n matrix A.
+
+           2:  IA -- INTEGER                                          Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F04ATF is called.
+               Constraint: IA >= max(1,N).
+
+           3:  B(*) -- DOUBLE PRECISION array                         Input
+               Note: the dimension of the array B must be at least
+               max(1,N).
+               On entry: the right-hand side vector b.
+
+           4:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A. Constraint: N >= 0.
+
+           5:  C(*) -- DOUBLE PRECISION array                        Output
+               Note: the dimension of the array C must be at least
+               max(1,N).
+               On exit: the solution vector x.
+
+           6:  AA(IAA,*) -- DOUBLE PRECISION array                   Output
+               Note: the second dimension of the array AA must be at least
+               max(1,N).
+               On exit: the triangular factors L and U, except that the
+               unit diagonal elements of U are not stored.
+
+           7:  IAA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array AA as declared in the
+               (sub)program from which F04ATF is called.
+               Constraint: IAA >= max(1,N).
+
+           8:  WKS1(*) -- DOUBLE PRECISION array                  Workspace
+               Note: the dimension of the array WKS1 must be at least
+               max(1,N).
+
+           9:  WKS2(*) -- DOUBLE PRECISION array                  Workspace
+               Note: the dimension of the array WKS2 must be at least
+               max(1,N).
+
+          10:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               The matrix A is singular, possibly due to rounding errors.
+
+          IFAIL= 2
+
+               Iterative refinement fails to improve the solution, i.e.,
+               the matrix A is too ill-conditioned.
+
+          IFAIL= 3
+               On entry N < 0,
+
+               or       IA < max(1,N),
+
+               or       IAA < max(1,N).
+
+          7. Accuracy
+
+          The computed solutions should be correct to full machine
+          accuracy. For a detailed error analysis see Wilkinson and Reinsch
+          [1] page 107.
+
+          8. Further Comments
+
+                                                                          3
+          The time taken by the routine is approximately proportional to n
+
+          The routine must not be called with the same name for parameters
+          B and C.
+
+          9. Example
+
+          To solve the set of linear equations Ax=b where
+
+                                     ( 33  16  72)
+                                   A=(-24 -10 -57)
+                                     ( -8  -4 -17)
+
+          and
+
+
+                                        (-359)
+                                      b=( 281).
+                                        (  85)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf04axf}{NAG On-line Documentation: f04axf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F04AXF(3NAG)      Foundation Library (12/10/92)      F04AXF(3NAG)
+
+
+
+          F04 -- Simultaneous Linear Equations                       F04AXF
+                  F04AXF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F04AXF calculates the approximate solution of a set of real
+          sparse linear equations with a single right-hand side, Ax=b or
+           T
+          A x=b, where A has been factorized by F01BRF or F01BSF.
+
+          2. Specification
+
+                 SUBROUTINE F04AXF (N, A, LICN, ICN, IKEEP, RHS, W, MTYPE,
+                1                   IDISP, RESID)
+                 INTEGER          N, LICN, ICN(LICN), IKEEP(5*N), MTYPE,
+                1                 IDISP(2)
+                 DOUBLE PRECISION A(LICN), RHS(N), W(N), RESID
+
+          3. Description
+
+                                                              T
+          To solve a system of real linear equations Ax=b or A x=b, where A
+          is a general sparse matrix, A must first be factorized by F01BRF
+          or F01BSF. F04AXF then computes x by block forward or backward
+          substitution using simple forward and backward substitution
+          within each diagonal block.
+
+          The method is fully described in Duff [1].
+
+          4. References
+
+          [1]   Duff I S (1977) MA28 -- a set of Fortran subroutines for
+                sparse unsymmetric linear equations. A.E.R.E. Report R.8730.
+                HMSO.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A.
+
+           2:  A(LICN) -- DOUBLE PRECISION array                      Input
+               On entry: the non-zero elements in the factorization of the
+               matrix A, as returned by F01BRF or F01BSF.
+
+           3:  LICN -- INTEGER                                        Input
+               On entry:
+               the dimension of the arrays A and ICN as declared in the
+               (sub)program from which F04AXF is called.
+
+           4:  ICN(LICN) -- INTEGER array                             Input
+               On entry: the column indices of the non-zero elements of
+               the factorization, as returned by F01BRF or F01BSF.
+
+           5:  IKEEP(5*N) -- INTEGER array                            Input
+               On entry: the indexing information about the factorization,
+               as returned by F01BRF or F01BSF.
+
+           6:  RHS(N) -- DOUBLE PRECISION array                Input/Output
+               On entry: the right-hand side vector b. On exit: RHS is
+               overwritten by the solution vector x.
+
+           7:  W(N) -- DOUBLE PRECISION array                     Workspace
+
+           8:  MTYPE -- INTEGER                                       Input
+               On entry: MTYPE specifies the task to be performed:
+                    if MTYPE = 1, solve Ax=b,
+
+                                          T
+                    if MTYPE /= 1, solve A x=b.
+
+           9:  IDISP(2) -- INTEGER array                              Input
+               On entry: the values returned in IDISP by F01BRF.
+
+          10:  RESID -- DOUBLE PRECISION                             Output
+               On exit: the value of the maximum residual,
+                        --
+               max(|b - > a  x |), over all the unsatisfied equations, in
+                     i  -- ij j
+                        j
+               case F01BRF or F01BSF has been used to factorize a singular
+               or rectangular matrix.
+
+          6. Error Indicators and Warnings
+
+          None.
+
+          7. Accuracy
+
+          The accuracy of the computed solution depends on the conditioning
+          of the original matrix. Since F04AXF is always used with either
+          F01BRF or F01BSF, the user is recommended to set GROW = .TRUE. on
+          entry to these routines and to examine the value of W(1) on exit
+          (see the routine documents for F01BRF and F01BSF). For a detailed
+          error analysis see Duff [1] page 17.
+
+          If storage for the original matrix is available then the error
+          can be estimated by calculating the residual
+
+                                                T
+                                 r=b-Ax  (or b-A x)
+
+          and calling F04AXF again to find a correction (delta) for x by
+          solving
+
+                                             T
+                            A(delta)=r  (or A (delta)=r).
+
+          8. Further Comments
+
+          If the factorized form contains (tau) non-zeros (IDISP(2) = (tau)
+          ) then the time taken is very approximately 2(tau) times longer
+          than the inner loop of full matrix code. Some advantage is taken
+                                                        T
+          of zeros in the right-hand side when solving A x=b (MTYPE /= 1).
+
+          9. Example
+
+          To solve the set of linear equations Ax=b where
+
+                                  ( 5  0  0  0 0  0)
+                                  ( 0  2 -1  2 0  0)
+                                  ( 0  0  3  0 0  0)
+                                A=(-2  0  0  1 1  0)
+                                  (-1  0  0 -1 2 -3)
+                                  (-1 -1  0  0 0  6)
+
+          and
+
+
+                                         (15)
+                                         (12)
+                                         (18)
+                                       b=( 3).
+                                         (-6)
+                                         ( 0)
+
+          The non-zero elements of A and indexing information are read in
+          by the program, as described in the document for F01BRF.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf04faf}{NAG On-line Documentation: f04faf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F04FAF(3NAG)      Foundation Library (12/10/92)      F04FAF(3NAG)
+
+
+
+          F04 -- Simultaneous Linear Equations                       F04FAF
+                  F04FAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F04FAF calculates the approximate solution of a set of real
+          symmetric positive-definite tridiagonal linear equations.
+
+          2. Specification
+
+                 SUBROUTINE F04FAF (JOB, N, D, E, B, IFAIL)
+                 INTEGER          JOB, N, IFAIL
+                 DOUBLE PRECISION D(N), E(N), B(N)
+
+          3. Description
+
+          F04FAF is based upon the Linpack routine DPTSL (see Dongarra et
+          al [1]) and solves the equations
+
+                                        Tx=b,
+
+          where T is a real n by n symmetric positive-definite tridiagonal
+          matrix, using a modified symmetric Gaussian elimination algorithm
+                                 T
+          to factorize T as T=MKM , where K is diagonal and M is a matrix
+          of multipliers as described in Section 8.
+
+          When the input parameter JOB is supplied as 1, then the routine
+          assumes that a previous call to F04FAF has already factorized T;
+          otherwise JOB must be supplied as 0.
+
+          4. References
+
+          [1]   Dongarra J J, Moler C B, Bunch J R and Stewart G W (1979)
+                LINPACK Users' Guide. SIAM, Philadelphia.
+
+          5. Parameters
+
+           1:  JOB -- INTEGER                                         Input
+               On entry: specifies the job to be performed by F04FAF as
+               follows:
+               JOB = 0
+                     The matrix T is factorized and the equations Tx=b are
+                     solved for x.
+
+               JOB = 1
+                     The matrix T is assumed to have already been
+                     factorized by a previous call to F04FAF with JOB = 0;
+                     the equations Tx=b are solved for x.
+
+           2:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix T. Constraint: N >= 1.
+
+           3:  D(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: if JOB = 0, D must contain the diagonal elements
+               of T. If JOB = 1, D must contain the diagonal matrix K, as
+               returned by a previous call of F04FAF with JOB = 0. On
+               exit: if JOB = 0, D is overwritten by the diagonal matrix K
+               of the factorization. If JOB = 1, D is unchanged.
+
+           4:  E(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: if JOB = 0, E must contain the super-diagonal
+               elements of T, stored in E(2) to E(n). If JOB = 1, E must
+               contain the off-diagonal elements of the matrix M, as
+               returned by a previous call of F04FAF with JOB = 0. E(1) is
+               not used. On exit: if JOB = 0, E(2) to E(n) are overwritten
+               by the off-diagonal elements of the matrix M of the
+               factorization. If JOB = 1, E is unchanged.
+
+           5:  B(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: the right-hand side vector b. On exit: B is
+               overwritten by the solution vector x.
+
+           6:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry N < 1,
+
+               or       JOB /= 0 or 1.
+
+          IFAIL= 2
+               The matrix T is either not positive-definite or is nearly
+               singular. This failure can only occur when JOB = 0 and
+               inspection of the elements of D will give an indication of
+               why failure has occurred. If an element of D is close to
+               zero, then T is probably nearly singular; if an element of D
+               is negative but not close to zero, then T is not positive-
+               definite.
+
+          IFAILOverflow
+               If overflow occurs during the execution of this routine,
+               then either T is very nearly singular or an element of the
+               right-hand side vector b is very large. In this latter case
+               the equations should be scaled so that no element of b is
+               very large. Note that to preserve symmetry it is necessary
+                                                            T
+               to scale by a transformation of the form (PTP )b=Px, where P
+               is a diagonal matrix.
+
+          IFAILUnderflow
+               Any underflows that occur during the execution of this
+               routine are harmless.
+
+          7. Accuracy
+
+          The computed factorization (see Section 8) will satisfy the
+          equation
+
+                                         T
+                                      MKM =T+E
+
+          where ||E|| <=2(epsilon)||T|| , p=1,F,infty,
+                     p                 p
+
+          (epsilon) being the machine precision. The computed solution of
+                                  
+
+          the equations Tx=b, say x, will satisfy an equation of the form
+
+                                           
+
+                                      (T+F)x=b,
+
+          where F can be expected to satisfy a bound of the form
+
+                          ||F||<=(alpha)(epsilon)||T||,
+
+          (alpha) being a modest constant. This implies that the relative
+                   
+
+          error in x satisfies
+
+                              
+
+                           ||x-x||
+                           -------<=c(T)(alpha)(epsilon),
+                            ||x||
+
+          where c(T) is the condition number of T with respect to
+                                                   
+
+          inversion. Thus if T is nearly singular, x can be expected to
+          have a large relative error.
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to n.
+
+          The routine eliminates the off-diagonal elements of T by
+          simultaneously performing symmetric Gaussian elimination from the
+          top and the bottom of T. The result is that T is factorized as
+
+                                            T
+                                       T=MKM ,
+
+          where K is a diagonal matrix and M is a matrix of the form
+
+                           (1  0  0 ..  0    0 0    .. 0 0    0 )
+                           (m  1  0 ..  0    0 0    .. 0 0    0 )
+                           ( 2                                  )
+                           (0  m  1 ..  0    0 0    .. 0 0    0 )
+                           (    3                               )
+                           (.  .  . ..  .    . .    .. . .    . )
+                           (.  .  . ..  .    . .    .. . .    . )
+                           (0  0  0 ..  1    0 0    .. 0 0    0 )
+                         M=(0  0  0 ..  m    1 m    .. 0 0    0 )
+                           (             j+1    j+2             )
+                           (0  0  0 ..  0    0 ..   1 0  0    0 )
+                           (.  .  . ..  .    . .    .. . .    . )
+                           (.  .  . ..  .    . .    .. . .    . )
+                           (0  0  0 ..  0    0 0    .. 1 m    0 )
+                           (                              n-1   )
+                           (0  0  0 ..  0    0 0    .. 0 1    m )
+                           (                                   n)
+                           (0  0  0 ..  0    0 0 .   . 0 0    1 )
+
+          j being the integer part of n/2. (For example when n=5,j=2.) The
+          diagonal elements of K are returned in D with k  in the ith
+                                                         i
+          element of D and m  is returned in the ith element of E.
+                            i
+
+          The routine fails with IFAIL = 2 if any diagonal element of K is
+          non-positive. It should be noted that T may be nearly singular
+          even if all the diagonal elements of K are positive, but in this
+          case at least one element of K is almost certain to be small
+          relative to |||T|||. If there is any doubt as to whether or not T
+          is nearly singular, then the user should consider examining the
+          diagonal elements of K.
+
+          9. Example
+
+          To solve the symmetric positive-definite equations
+
+                                       Tx =b
+                                         1  1
+
+          and
+
+                                       Tx =b
+                                         2  2
+
+          where
+
+                         ( 4 -2  0  0 0)       ( 6)      (10)
+                         (-2 10 -6  0 0)       ( 9)      ( 4)
+                       T=( 0 -6 29 15 0),   b =( 2),  b =( 9).
+                         ( 0  0 15 25 8)     1 (14)    2 (65)
+                         ( 0  0  0  8 5)       ( 7)      (23)
+
+          The equations are solved by two calls to F04FAF, the first with
+          JOB = 0 and the second, using the factorization from the first
+          call, with JOB = 1.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf04jgf}{NAG On-line Documentation: f04jgf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F04JGF(3NAG)      Foundation Library (12/10/92)      F04JGF(3NAG)
+
+
+
+          F04 -- Simultaneous Linear Equations                       F04JGF
+                  F04JGF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F04JGF finds the solution of a linear least-squares problem, Ax=b
+          , where A is a real m by n (m>=n) matrix and b is an m element
+          vector. If the matrix of observations is not of full rank, then
+          the minimal least-squares solution is returned.
+
+          2. Specification
+
+                 SUBROUTINE F04JGF (M, N, A, NRA, B, TOL, SVD, SIGMA,
+                1                   IRANK, WORK, LWORK, IFAIL)
+                 INTEGER          M, N, NRA, IRANK, LWORK, IFAIL
+                 DOUBLE PRECISION A(NRA,N), B(M), TOL, SIGMA, WORK(LWORK)
+                 LOGICAL          SVD
+
+          3. Description
+
+          The minimal least-squares solution of the problem Ax=b is the
+          vector x of minimum (Euclidean) length which minimizes the length
+          of the residual vector r=b-Ax.
+
+          The real m by n (m>=n) matrix A is factorized as
+
+                                          (U)
+                                       A=Q(0)
+
+          where Q is an m by m orthogonal matrix and U is an n by n upper
+          triangular matrix. If U is of full rank, then the least-squares
+          solution is given by
+
+                                        -1    T
+                                    x=(U   0)Q b.
+
+          If U is not of full rank, then the singular value decomposition
+          of U is obtained so that U is factorized as
+
+                                            T
+                                       U=RDP ,
+
+          where R and P are n by n orthogonal matrices and D is the n by n
+          diagonal matrix
+
+                       D=diag((sigma) ,(sigma) ,...,(sigma) ),
+                                     1        2            n
+
+          with (sigma) >=(sigma) >=...(sigma) >=0, these being the singular
+                      1         2            n
+          values of A. If the singular values (sigma)   ,...,(sigma)  are
+                                                     k+1            n
+          negligible, but (sigma)  is not negligible, relative to the data
+                                 k
+          errors in A, then the rank of A is taken to be k and the minimal
+          least-squares solution is given by
+
+                                   ( -1  )( T  )
+                                   (S   0)(R  0) T
+                                x=P(0 0  )(0 I )Q b,
+
+          where S=diag((sigma) ,(sigma) ,...,(sigma) ).
+                              1        2            k
+
+          This routine obtains the factorizations by a call to F02WDF(*).
+
+          The routine also returns the value of the standard error
+
+
+                      /  T
+                     /  r r
+          (sigma)=  /   --- ,     if m>k,
+                  \/    m-k
+
+                                           T
+                 =  0,            if m=k, r r being the residual sum of
+                                  squares and k the rank of A.
+
+          4. References
+
+          [1]   Lawson C L and Hanson R J (1974) Solving Least-squares
+                Problems. Prentice-Hall.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: m, the number of rows of A. Constraint: M >= N.
+
+           2:  N -- INTEGER                                           Input
+               On entry: n, the number of columns of A. Constraint: 1 <= N
+               <= M.
+
+           3:  A(NRA,N) -- DOUBLE PRECISION array              Input/Output
+               On entry: the m by n matrix A. On exit: if SVD is returned
+               as .FALSE., A} is overwritten by details of the QU
+               factorization of A (see F02WDF(*) for further details). If
+               SVD is returned as .TRUE., the first n rows of A are
+               overwritten by the right-hand singular vectors, stored by
+               rows; and the remaining rows of the array are used as
+               workspace.
+
+           4:  NRA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F04JGF is called.
+               Constraint: NRA >= M.
+
+           5:  B(M) -- DOUBLE PRECISION array                  Input/Output
+               On entry: the right-hand side vector b. On exit: the first
+               n elements of B contain the minimal least-squares solution
+               vector x. The remaining m-n elements are used for workspace.
+
+           6:  TOL -- DOUBLE PRECISION                                Input
+               On entry: a relative tolerance to be used to determine the
+               rank of A. TOL should be chosen as approximately the largest
+               relative error in the elements of A. For example, if the
+               elements of A are correct to about 4 significant figures
+                                                   -4
+               then TOL should be set to about 5*10  . See Section 8 for a
+               description of how TOL is used to determine rank. If TOL is
+               outside the range ((epsilon),1.0), where (epsilon) is the
+               machine precision, then the value (epsilon) is used in place
+               of TOL. For most problems this is unreasonably small.
+
+           7:  SVD -- LOGICAL                                        Output
+               On exit: SVD is returned as .FALSE. if the least-squares
+               solution has been obtained from the QU factorization of A.
+               In this case A is of full rank. SVD is returned as .TRUE. if
+               the least-squares solution has been obtained from the
+               singular value decomposition of A.
+
+           8:  SIGMA -- DOUBLE PRECISION                             Output
+                                                               
+
+                                                              / T
+               On exit: the standard error, i.e., the value \/ r r/(m-k)
+               when m>k, and the value zero when m=k. Here r is the
+               residual vector b-Ax and k is the rank of A.
+
+           9:  IRANK -- INTEGER                                      Output
+               On exit: k, the rank of the matrix A. It should be noted
+               that it is possible for IRANK to be returned as n and SVD to
+               be returned as .TRUE.. This means that the matrix U only
+               just failed the test for non-singularity.
+
+          10:  WORK(LWORK) -- DOUBLE PRECISION array                 Output
+               On exit: if SVD is returned as .FALSE., then the first n
+               elements of WORK contain information on the QU factorization
+               of A (see parameter A above and F02WDF(*)), and WORK(n+1)
+                                                      -1
+               contains the condition number ||U|| ||U  ||  of the
+                                                  E       E
+               upper triangular matrix U.
+
+               If SVD is returned as .TRUE., then the first n elements of
+               WORK contain the singular values of A arranged in descending
+               order and WORK(n+1) contains the total number of iterations
+               taken by the QR algorithm. The rest of WORK is used as
+               workspace.
+
+          11:  LWORK -- INTEGER                                       Input
+               On entry:
+               the dimension of the array WORK as declared in the
+               (sub)program from which F04JGF is called.
+               Constraint: LWORK >= 4*N.
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry N < 1,
+
+               or       M < N,
+
+               or       NRA < M,
+
+               or       LWORK < 4*N.
+
+          IFAIL= 2
+               The QR algorithm has failed to converge to the singular
+               values in 50*N iterations. This failure can only happen when
+               the singular value decomposition is employed, but even then
+               it is not likely to occur.
+
+          7. Accuracy
+
+                                               T
+          The computed factors Q, U, R, D and P  satisfy the relations
+
+                              (U)        (R 0)(D) T
+                             Q(0)=A+E , Q(0 I)(0)P =A+F,
+
+          where
+
+                           ||E|| <=c (epsilon)||A|| ,
+                                2   1              2
+
+
+                           ||F|| <=c (epsilon)||A|| ,
+                                2   2              2
+
+          (epsilon) being the machine precision, and c  and c  being modest
+                                                      1      2
+          functions of m and n. Note that ||A|| =(sigma) .
+                                               2        1
+
+          For a fuller discussion, covering the accuracy of the solution x
+          see Lawson and Hanson [1], especially pp 50 and 95.
+
+          8. Further Comments
+
+          If the least-squares solution is obtained from the QU
+          factorization then the time taken by the routine is approximately
+                           2
+          proportional to n (3m-n). If the least-squares solution is
+          obtained from the singular value decomposition then the time
+                                                  2
+          taken is approximately proportional to n (3m+19n). The
+          approximate proportionality factor is the same in each case.
+
+          This routine is column biased and so is suitable for use in paged
+          environments.
+
+          Following the QU factorization of A the condition number
+
+                                             -1
+                               c(U)=||U|| ||U  ||
+                                         E       E
+
+          is determined and if c(U) is such that
+
+                                    c(U)*TOL>1.0
+
+          then U is regarded as singular and the singular values of A are
+          computed. If this test is not satisfied, U is regarded as non-
+          singular and the rank of A is set to n. When the singular values
+          are computed the rank of A, say k, is returned as the largest
+          integer such that
+
+                               (sigma) >TOL*(sigma) ,
+                                      k            1
+
+          unless (sigma) =0 in which case k is returned as zero. That is,
+                        1
+          singular values which satisfy (sigma) <=TOL*(sigma)  are regarded
+                                               i             1
+          as negligible because relative perturbations of order TOL can
+          make such singular values zero.
+
+          9. Example
+
+          To obtain a least-squares solution for r=b-Ax, where
+
+                           (0.05  0.05 0.25 -0.25)      (1)
+                           (0.25  0.25 0.05 -0.05)      (2)
+                           (0.35  0.35 1.75 -1.75)      (3)
+                         A=(1.75  1.75 0.35 -0.35),   B=(4)
+                           (0.30 -0.30 0.30  0.30)      (5)
+                           (0.40 -0.40 0.40  0.40)      (6)
+
+                                                  -4
+          and the value TOL is to be taken as 5*10  .
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf04maf}{NAG On-line Documentation: f04maf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F04MAF(3NAG)      Foundation Library (12/10/92)      F04MAF(3NAG)
+
+
+
+          F04 -- Simultaneous Linear Equations                       F04MAF
+                  F04MAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          To solve a sparse symmetric positive-definite system of linear
+          equations, Ax=b, using a pre-conditioned conjugate gradient
+          method, where A has been factorized by F01MAF.
+
+          2. Specification
+
+                 SUBROUTINE F04MAF (N, NZ, A, LICN, IRN, LIRN, ICN, B, ACC,
+                1                   NOITS, WKEEP, WORK, IKEEP, INFORM,
+                2                   IFAIL)
+                 INTEGER          N, NZ, LICN, IRN(LIRN), LIRN, ICN(LICN),
+                1                 NOITS(2), IKEEP(2*N), INFORM(4), IFAIL
+                 DOUBLE PRECISION A(LICN), B(N), ACC(2), WKEEP(3*N), WORK
+                1                 (3*N)
+
+          3. Description
+
+          F04MAF solves the n linear equations
+
+                                     Ax=b,                              (1)
+
+          where A is a sparse symmetric positive-definite matrix, following
+          the incomplete Cholesky factorization by F01MAF, given by
+
+                                      T T
+                                C=PLDL P  , WAW=C+E,
+
+          where P is a permutation matrix, L is a unit lower triangular
+          matrix, D is a diagonal matrix with positive diagonal elements, E
+          is an error matrix representing elements dropped during the
+          factorization and diagonal elements that have been modified to
+          ensure that C is positive-definite, and W is a diagonal matrix,
+          chosen to make the diagonal elements of WAW unity.
+
+          Equation (1) is solved by applying a pre-conditioned conjugate
+          gradient method to the equations
+
+                                       -1
+                                (WAW)(W  x)=Wb,                         (2)
+
+          using C as the pre-conditioning matrix. Details of the conjugate
+          gradient method are given in Munksgaard [1].
+
+          The iterative procedure is terminated if
+
+                                ||Wr|| <=(eta),                         (3)
+                                      2
+
+          where r is the residual vector r=b-Ax, ||r||  denotes the
+                                                      2
+          Euclidean length of r, (eta) is a user-supplied tolerance and x
+          is the current approximation to the solution. Notice that
+
+                                               -1
+                                  Wr=Wb-(WAW)(W  x)
+
+          so that Wr is the residual of the normalised equations (2).
+
+          F04MAF is based on the Harwell Library routine MA31B.
+
+          4. References
+
+          [1]   Munksgaard N (1980) Solving Sparse Symmetric Sets of Linear
+                Equations by Pre-conditioned Conjugate Gradients. ACM Trans.
+                Math. Softw. 6 206--219.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A. Constraint: N >= 1.
+
+           2:  NZ -- INTEGER                                          Input
+               On entry: the number of non-zero elements in the upper
+               triangular part of the matrix A, including the number of
+               elements on the leading diagonal. Constraint: NZ >= N.
+
+           3:  A(LICN) -- DOUBLE PRECISION array                      Input
+               On entry: the first LROW elements, where LROW is the value
+               supplied in INFORM(1), must contain details of the
+               factorization, as returned by F01MAF.
+
+           4:  LICN -- INTEGER                                        Input
+               On entry: the length of the array A, as declared in the
+               (sub)program from which F04MAF is called. It need never be
+               larger than the value of LICN supplied to F01MAF.
+               Constraint: LICN >= INFORM(1).
+
+           5:  IRN(LIRN) -- INTEGER array                             Input
+               On entry: the first LCOL elements, where LCOL is the value
+               supplied in INFORM(2), must contain details of the
+               factorization, as returned by F01MAF.
+
+           6:  LIRN -- INTEGER                                        Input
+               On entry: the length of the array IRN, as declared in the
+               (sub)program from which F04MAF is called. It need never be
+               larger than the value of LIRN supplied to F01MAF.
+               Constraint: LIRN >= INFORM(2).
+
+           7:  ICN(LICN) -- INTEGER array                             Input
+               On entry: the first LROW elements, where LROW is the value
+               supplied in INFORM(1), must contain details of the
+               factorization, as returned by F01MAF.
+
+           8:  B(N) -- DOUBLE PRECISION array                  Input/Output
+               On entry: the right-hand side vector b. On exit: B is
+               overwritten by the solution vector x.
+
+           9:  ACC(2) -- DOUBLE PRECISION array                Input/Output
+               On entry: ACC(1) specifies the tolerance for convergence,
+               (eta), in equation (3) of Section 3. If ACC(1) is outside
+               the range [(epsilon),1], where (epsilon) is the machine
+               precision, then the value (epsilon) is used in place of ACC
+               (1). ACC(2) need not be set. On exit: ACC(2) contains the
+               actual value of ||Wr||  at the final point. ACC(1) is
+                                     2
+               unchanged.
+
+          10:  NOITS(2) -- INTEGER array                       Input/Output
+               On entry: NOITS(1) specifies the maximum permitted number of
+               iterations. If NOITS(1) < 1, then the value 100 is used in
+               its place. NOITS(2) need not be set. On exit: NOITS(2)
+               contains the number of iterations taken to converge. NOITS
+               (1) is unchanged.
+
+          11:  WKEEP(3*N) -- DOUBLE PRECISION array                   Input
+               On entry: WKEEP must be unchanged from the previous call of
+               F01MAF.
+
+          12:  WORK(3*N) -- DOUBLE PRECISION array                   Output
+               On exit: WORK(1) contains a lower bound for the condition
+               number of A. The rest of the array is used for workspace.
+
+          13:  IKEEP(2*N) -- INTEGER array                            Input
+               On entry: IKEEP must be unchanged from the previous call of
+               F01MAF.
+
+          14:  INFORM(4) -- INTEGER array                             Input
+               On entry: INFORM must be unchanged from the previous call of
+               F01MAF.
+
+          15:  IFAIL -- INTEGER                                Input/Output
+               For this routine, the normal use of IFAIL is extended to
+               control the printing of error and warning messages as well
+               as specifying hard or soft failure (see the Essential
+               Introduction).
+
+               Before entry, IFAIL must be set to a value with the decimal
+               expansion cba, where each of the decimal digits c, b and a
+               must have a value of 0 or 1.
+               a=0   specifies hard failure, otherwise soft failure;
+
+               b=0   suppresses error messages, otherwise error messages
+                     will be printed (see Section 6);
+
+               c=0   suppresses warning messages, otherwise warning
+                     messages will be printed (see Section 6).
+               The recommended value for inexperienced users is 110 (i.e.,
+               hard failure with all messages printed).
+
+               Unless the routine detects an error (see Section 6), IFAIL
+               contains 0 on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          For each error, an explanatory error message is output on the
+          current error message unit (as defined by X04AAF), unless
+          suppressed by the value of IFAIL on entry.
+
+          IFAIL= 1
+               On entry N < 1,
+
+               or       NZ < N,
+
+               or       LICN < INFORM(1),
+
+               or       LIRN < INFORM(2).
+
+          IFAIL= 2
+               Convergence has not taken place within the requested NOITS
+               (1) number of iterations. ACC(2) gives the value ||Wr|| ,
+                                                                      2
+               for the final point. Either too few iterations have been
+               allowed, or the requested convergence criterion cannot be
+               met.
+
+          IFAIL= 3
+               The matrix A is singular, or nearly singular. Singularity
+               has been detected during the conjugate gradient iterations,
+               so that the computations are not complete.
+
+          IFAIL= 4
+               The matrix A is singular, or nearly singular. The message
+               output on the current error message channel will include an
+               estimate of the condition number of A. In the case of soft
+               failure an approximate solution is returned such that the
+               value ||Wr||  is given by ACC(2) and the estimate (a lower
+                           2
+               bound) of the condition number is returned in WORK(1).
+
+          7. Accuracy
+
+          On successful return, or on return with IFAIL = 2 or IFAIL = 4
+          the computed solution will satisfy equation (3) of Section 3,
+          with (eta) = ACC(2).
+
+          8. Further Comments
+
+          The time taken by the routine will depend upon the sparsity of
+          the factorization and the number of iterations required. The
+          number of iterations will be affected by the nature of the
+          factorization supplied by F01MAF. The more incomplete the
+          factorization, the higher the number of iterations required by
+          F04MAF.
+
+          When the solution of several systems of equations, all with the
+          same matrix of coefficients, A, is required, then F01MAF need be
+          called only once to factorize A. This is illustrated in the
+          context of an eigenvalue problem in the example program for
+          F02FJF.
+
+          9. Example
+
+          The example program illustrates the use of F01MAF in conjunction
+          with F04MAF to solve the 16 linear equations Ax=b, where
+
+                           (1 a     a                      )
+                           (a 1 a     a                    )
+                           (  a 1 a     a                  )
+                           (    a 1 0     a                )
+                           (a     0 1 a     a              )
+                           (  a     a 1 a     a            )
+                           (    a     a 1 a     a          )
+                           (      a     a 1 0     a        )
+                         A=(        a     0 1 a     a      )
+                           (          a     a 1 a     a    )
+                           (            a     a 1 a     a  )
+                           (              a     a 1 0     a)
+                           (                a     0 1 a    )
+                           (                  a     a 1 a  )
+                           (                    a     a 1 a)
+                           (                      a     a 1)
+
+                    1
+          where a=- -.
+                    4
+
+                   T (  1  1  1  1  1      1  1      1  1  1  1  1)
+                  b =(  -  -  -  -  - 0 0  -  - 0 0  -  -  -  -  -)
+                     (  2  4  4  2  4      4  4      4  2  4  4  2)
+
+          The n by n matrix A arises in the solution of Laplace's equation
+          in a unit-square, using a five-point formula with a 6 by 6
+          discretisation, with unity on the boundaries.
+
+          The drop tolerance, DROPTL, is taken as 0.1 and the density
+          factor, DENSW, is taken as 0.8. The value IFAIL = 111 is used so
+          that advisory and error messages will be printed, but soft
+          failure would occur if IFAIL were returned as non-zero.
+
+          A relative accuracy of about 0.0001 is requested in the solution
+          from F04MAF, with a maximum of 50 iterations.
+
+          The example program for F02FJF illustrates the use of routines
+          F01MAF and F04MAF in solving an eigenvalue problem.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf04mbf}{NAG On-line Documentation: f04mbf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F04MBF(3NAG)      Foundation Library (12/10/92)      F04MBF(3NAG)
+
+
+
+          F04 -- Simultaneous Linear Equations                       F04MBF
+                  F04MBF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F04MBF solves a system of real sparse symmetric linear equations
+          using a Lanczos algorithm.
+
+          2. Specification
+
+                 SUBROUTINE F04MBF (N, B, X, APROD, MSOLVE, PRECON, SHIFT,
+                1                   RTOL, ITNLIM, MSGLVL, ITN, ANORM,
+                2                   ACOND, RNORM, XNORM, WORK, RWORK,
+                3                   LRWORK, IWORK, LIWORK, INFORM, IFAIL)
+                 INTEGER          N, ITNLIM, MSGLVL, ITN, LRWORK, IWORK
+                1                 (LIWORK), LIWORK, INFORM, IFAIL
+                 DOUBLE PRECISION B(N), X(N), SHIFT, RTOL, ANORM, ACOND,
+                1                 RNORM, XNORM, WORK(N,5), RWORK(LRWORK)
+                 LOGICAL          PRECON
+                 EXTERNAL         APROD, MSOLVE
+
+          3. Description
+
+          F04MBF solves the system of linear equations
+
+                               (A-(lambda)I)x=b                       (3.1)
+
+          where A is an n by n sparse symmetric matrix and (lambda) is a
+          scalar, which is of course zero if the solution of the equations
+
+                                        Ax=b
+
+          is required. It should be noted that neither A nor (A-(lambda)I)
+          need be positive-definite.
+
+          (lambda) is supplied as the parameter SHIFT, and allows F04MBF to
+          be used for finding eigenvectors of A in methods such as Rayleigh
+          quotient iteration (see for example Lewis [1]), in which case
+          (lambda) will be an approximation to an eigenvalue of A and b an
+          approximation to an eigenvector of A.
+
+          The routine also provides an option to allow pre-conditioning and
+          this will often reduce the number of iterations required by
+          F04MBF.
+
+          F04MBF is based upon algorithm SYMMLQ (see Paige and Saunders
+          [2]) and solves the equations by an algorithm based upon the
+
+          Lanczos process. Details of the method are given in Paige and
+          Saunders [2]. The routine does not require A explicitly, but A is
+          specified via a user-supplied routine APROD which, given an n
+          element vector c, must return the vector z given by
+
+                                        z=Ac.
+
+          The pre-conditioning option is based on the following reasoning.
+          If A can be expressed in the form
+
+                                        A=I+B
+
+          where B is of rank (rho), then the Lanczos process converges (in
+          exact arithmetic) in at most (rho) iterations. If more generally
+          A can be expressed in the form
+
+                                        A=M+C
+
+          where M is symmetric positive-definite and C has rank (rho), then
+
+                           -(1/2)  -(1/2)    -(1/2)  -(1/2)
+                          M      AM      =I+M      CM
+
+               -(1/2)  -(1/2)
+          and M      AM       also has rank (rho), and the Lanczos process
+                      -(1/2)  -(1/2)
+          applied to M      AM       would again converge in at most (rho)
+          iterations. On a computer, the number of iterations may be
+          greater than (rho), but the Lanczos process may still be expected
+                                                        -(1/2)  -(1/2)
+          to converge rapidly. F04MBF does not require M      AM       to
+          be formed explicitly, but implicitly solves the equations
+
+                -(1/2)              -(1/2)   -(1/2)       1/2
+               M      (A-(lambda)I)M      y=M      b , y=M   x        (3.2)
+
+          with the user being required to supply a routine MSOLVE to solve
+          the equations
+
+                                    Mz=c.                             (3.3)
+
+          For the pre-conditioning option to be effective, it is desirable
+          that equations (3.3) can be solved efficiently. The example
+          program in Section 9 illustrates the use of this option.
+
+          If we let r denote the residual vector
+
+                                 r=b-(A-(lambda)I)x
+
+          corresponding to an iterate x, then, when pre-conditioning has
+          not been requested, the iterative procedure is terminated if it
+          is estimated that
+
+                      ||r||<=tol.||A-(lambda)I||.||x||,               (3.4)
+
+          where tol is a user-supplied tolerance, ||r|| denotes the
+          Euclidean length of the vector r and ||A|| denotes the Frobenius
+          (Euclidean) norm of the matrix A. When pre-conditioning has been
+          requested, the iterative procedure is terminated if it is
+          estimated that
+
+             -(1/2)            -(1/2)             -(1/2)      1/2
+          ||M      r||<=tol.||M      (A-(lambda)I)M     ||.||M   x||. (3.5)
+
+          Note that
+
+                -(1/2)    -(1/2)    -(1/2)              -(1/2)  1/2
+               M      r=(M      b)-M      (A-(lambda)I)M      (M   x)
+
+                   -(1/2)
+          so that M      r is the residual vector corresponding to equation
+          (3.2). The routine will also terminate if it is estimated that
+
+                   ||A-(lambda)I||.||x||>=||b||/(epsilon),            (3.6)
+
+          where (epsilon) is the machine precision, when pre-conditioning
+          has not been requested; or if it is estimated that
+
+
+             -(1/2)              -(1/2)      1/2        -(1/2)
+          ||M      (A-(lambda)I)M      ||.||M   x||>=||M      b||/(epsilon)
+
+                                                                      (3.7)
+
+          when pre-conditioning has been requested. If (3.6) is satisfied
+          then x is almost certainly an eigenvector of A corresponding to
+          the eigenvalue (lambda). If (lambda) was set to 0 (for the
+          solution of Ax=b), then this condition simply means that A is
+          effectively singular.
+
+          4. References
+
+          [1]   Lewis J G (1977) Algorithms for sparse matrix eigenvalue
+                problems. Technical Report STAN-CS-77-595. Computer Science
+                Department, Stanford University.
+
+          [2]   Paige C C and Saunders M A (1975) Solution of Sparse
+                Indefinite Systems of Linear Equations. SIAM J. Numer. Anal.
+                12 617--629.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A. Constraint: N >= 1.
+
+           2:  B(N) -- DOUBLE PRECISION array                         Input
+               On entry: the right-hand side vector b.
+
+           3:  X(N) -- DOUBLE PRECISION array                        Output
+               On exit: the solution vector x.
+
+           4:  APROD -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               APROD must return the vector y=Ax for a given vector x.
+
+               Its specification is:
+
+                      SUBROUTINE APROD (IFLAG, N, X, Y, RWORK, LRWORK,
+                     1                  IWORK,LIWORK)
+                      INTEGER          IFLAG, N, LRWORK, LIWORK, IWORK
+                     1                 (LIWORK)
+                      DOUBLE PRECISION X(N), Y(N), RWORK(LRWORK)
+
+                1:  IFLAG -- INTEGER                           Input/Output
+                    On entry: IFLAG is always non-negative. On exit: IFLAG
+                    may be used as a flag to indicate a failure in the
+                    computation of  Ax.   If IFLAG is negative on exit from
+                    APROD, F04MBF will exit immediately with IFAIL set to
+                    IFLAG.
+
+                2:  N -- INTEGER                                      Input
+                    On entry: n, the order of the matrix A.
+
+                3:  X(N) -- DOUBLE PRECISION array                    Input
+                    On entry: the vector x for which Ax is required.
+
+                4:  Y(N) -- DOUBLE PRECISION array                   Output
+                    On exit: the vector y=Ax.
+
+                5:  RWORK(LRWORK) -- DOUBLE PRECISION array  User Workspace
+
+                6:  LRWORK -- INTEGER                                 Input
+
+                7:  IWORK(LIWORK) -- INTEGER array           User Workspace
+
+                8:  LIWORK -- INTEGER                                 Input
+                    APROD is called from F04MBF with the parameters RWORK,
+                    LRWORK, IWORK and LIWORK as supplied to F04MBF. The
+                    user is free to use the arrays RWORK and IWORK to
+                    supply information to APROD and MSOLVE as an
+                    alternative to using COMMON.
+               APROD must be declared as EXTERNAL in the (sub)program
+               from which F04MBF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           5:  MSOLVE -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+               MSOLVE is only referenced when PRECON is supplied as .TRUE..
+               When PRECON is supplied as .FALSE., then F04MBF may be
+               called with APROD as the actual argument for MSOLVE. When
+               PRECON is supplied as .TRUE., then MSOLVE must return the
+               solution y of the equations My=x for a given vector x, where
+               M must be symmetric positive-definite.
+
+               Its specification is:
+
+                      SUBROUTINE MSOLVE (IFLAG, N, X, Y, RWORK,
+                     1                   LRWORK, IWORK,LIWORK)
+                      INTEGER          IFLAG, N, LRWORK, LIWORK, IWORK
+                     1                 (LIWORK)
+                      DOUBLE PRECISION X(N), Y(N), RWORK(LRWORK)
+
+                1:  IFLAG -- INTEGER                           Input/Output
+                    On entry: IFLAG is always non-negative. On exit: IFLAG
+                    may be used as a flag to indicate a failure in the
+                    solution of My=x.
+
+                    If IFLAG is negative on exit from MSOLVE, F04MBF will
+                    exit immediately with IFAIL set to IFLAG.
+
+                2:  N -- INTEGER                                      Input
+                    On entry: n, the order of the matrix M.
+
+                3:  X(N) -- DOUBLE PRECISION array                    Input
+                    On entry: the vector x for which the equations My=x are
+                    to be solved.
+
+                4:  Y(N) -- DOUBLE PRECISION array                   Output
+                    On exit: the solution to the equations My=x.
+
+                5:  RWORK(LRWORK) -- DOUBLE PRECISION array  User Workspace
+
+                6:  LRWORK -- INTEGER                                 Input
+
+                7:  IWORK(LIWORK) -- INTEGER array           User Workspace
+
+                8:  LIWORK -- INTEGER                                 Input
+                    MSOLVE is called from F04MBF with the parameters RWORK,
+                    LRWORK, IWORK and LIWORK as supplied to F04MBF. The
+                    user is free to use the arrays RWORK and IWORK to
+                    supply information to APROD and MSOLVE as an
+                    alternative to using COMMON.
+               MSOLVE must be declared as EXTERNAL in the (sub)program
+               from which F04MBF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           6:  PRECON -- LOGICAL                                      Input
+               On entry: PRECON specifies whether or not pre-conditioning
+               is required. If PRECON = .TRUE., then pre-conditioning will
+               be invoked and MSOLVE will be referenced by F04MBF; if
+               PRECON = .FALSE., then MSOLVE is not referenced.
+
+           7:  SHIFT -- DOUBLE PRECISION                              Input
+               On entry: the value of (lambda). If the equations Ax=b are
+               to be solved, then SHIFT must be supplied as zero.
+
+           8:  RTOL -- DOUBLE PRECISION                               Input
+               On entry: the tolerance for convergence, tol, of equation
+               (3.4). RTOL should not normally be less than (epsilon),
+               where (epsilon) is the machine precision.
+
+           9:  ITNLIM -- INTEGER                                      Input
+               On entry: an upper limit on the number of iterations. If
+               ITNLIM <= 0, then the value N is used in place of ITNLIM.
+
+          10:  MSGLVL -- INTEGER                                      Input
+               On entry: the level of printing from F04MBF. If MSGLVL <= 0,
+               then no printing occurs, but otherwise messages will be
+               output on the advisory message channel (see X04ABF). A
+               description of the printed output is given in Section 5.1
+               below. The level of printing is determined as follows:
+               MSGLVL <= 0
+                     No printing.
+
+               MSGLVL = 1
+                     A brief summary is printed just prior to return from
+                     F04MBF.
+
+               MSGLVL >= 2
+                     A summary line is printed periodically to monitor the
+                     progress of F04MBF, together with a brief summary just
+                     prior to return from F04MBF.
+
+          11:  ITN -- INTEGER                                        Output
+               On exit: the number of iterations performed.
+
+          12:  ANORM -- DOUBLE PRECISION                             Output
+               On exit: an estimate of ||A-(lambda)I|| when PRECON =
+                              -(1/2)              -(1/2)
+               .FALSE., and ||M      (A-(lambda)I)M      || when PRECON =
+               .TRUE..
+
+          13:  ACOND -- DOUBLE PRECISION                             Output
+               On exit: an estimate of the condition number of (A-
+               (lambda)I) when PRECON = .FALSE., and of
+                -(1/2)              -(1/2)
+               M      (A-(lambda)I)M       when PRECON = .TRUE.. This will
+               usually be a substantial under-estimate.
+
+          14:  RNORM -- DOUBLE PRECISION                             Output
+               On exit: ||r||, where r=b-(A-(lambda)I)x and x is the
+               solution returned in X.
+
+          15:  XNORM -- DOUBLE PRECISION                             Output
+               On exit: ||x||, where x is the solution returned in X.
+
+          16:  WORK(5*N) -- DOUBLE PRECISION array                Workspace
+
+          17:  RWORK(LRWORK) -- DOUBLE PRECISION array       User Workspace
+               RWORK is not used by F04MBF, but is passed directly to
+               routines APROD and MSOLVE and may be used to pass
+               information to these routines.
+
+          18:  LRWORK -- INTEGER                                      Input
+               On entry: the length of the array RWORK as declared in the
+               (sub)program from which F04MBF is called. Constraint: LRWORK
+               >= 1.
+
+          19:  IWORK(LIWORK) -- INTEGER array                User Workspace
+               IWORK is not used by F04MBF, but is passed directly to
+               routines APROD and MSOLVE and may be used to pass
+               information to these routines.
+
+          20:  LIWORK -- INTEGER                                      Input
+               On entry: the length of the array IWORK as declared in the
+               (sub)program from which F04MBF is called. Constraint: LIWORK
+               >= 1.
+
+          21:  INFORM -- INTEGER                                     Output
+               On exit: the reason for termination of F04MBF as follows:
+               INFORM = 0
+                     The right-hand side vector b=0 so that the exact
+                     solution is x=0. No iterations are performed in this
+                     case.
+
+               INFORM = 1
+                     The termination criterion of equation (3.4) has been
+                     satisfied with tol as the value supplied in RTOL.
+
+               INFORM = 2
+                     The termination criterion of equation (3.4) has been
+                     satisfied with tol equal to (epsilon), where (epsilon)
+                     is the machine precision. The value supplied in RTOL
+                     must have been less than (epsilon) and was too small
+                     for the machine.
+
+               INFORM = 3
+                     The termination criterion of equation (3.5) has been
+                     satisfied so that X is almost certainly an eigenvector
+                     of A corresponding to the eigenvalue SHIFT.
+               The values INFORM = 4 and INFORM = 5 correspond to failure
+               with IFAIL = 3 or IFAIL = 2 respectively (see Section 6) and
+               when IFAIL is negative, INFORM will be set to the same
+               negative value.
+
+          22:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          5.1. Description of the Printed Output
+
+          When MSGLVL > 0, then F04MBF will produce output (except in the
+          case where the routine fails with IFAIL = 1) on the advisory
+          message channel (see X04ABF ).
+
+          The following notation is used in the output.
+
+             Output     Meaning
+
+                         -(1/2)                   
+             RBAR       M      (b-(A-(lambda)I)x)=r
+
+                         -(1/2)              -(1/2) 
+             ABAR       M      (A-(lambda)I)M      =A
+
+                         1/2
+             Y          M   x
+
+             R          b-(A-(lambda)I)x
+
+             NORM(A)    ||A||
+
+          Of course, when pre-conditioning has not been requested then the
+          first three reduce to (b-(A-(lambda)I)x), (A-(lambda)I) and x
+          respectively. When MSGLVL >= 2 then some initial information is
+          printed and the following notation is used.
+
+             Output     Meaning
+
+                          T -1  1/2
+             BETA1      (b M  b)   ==(beta)
+                                           1
+
+                                   2  -(1/2)  T  -(1/2)  -(1/2)   -(1/2)
+             ALFA1      (1/(beta) ) (M      b) (M      AM      )(M      b)
+                                 1
+                        ==(alpha)
+                                 1
+          and a summary line is printed periodically giving the following
+          information:
+
+             Output     Meaning
+
+             ITN        Iteration number, k.
+
+                                                         L         L
+             X1(LQ)     The first element of the vector x , where x  is the
+                                                         k         k
+                        current iterate. See Paige and Saunders [2] for
+                        details.
+
+                                                         C         C
+             X1(CG)     The first element of the vector x , where x  is the
+                                                         k         k
+                        vector that would be obtained by conjugate
+                        gradients. See Paige and Saunders [2] for details.
+
+                                     
+
+             NORM(RBAR) ||r||, where r is as defined above and x is either
+                         L     C
+                        x  or x  depending upon which is the best current
+                         k     k
+                        approximation to the solution. (See LQ/CG below).
+
+             NORM(T)    The value ||T ||, where T  is the tridiagonal
+                                     k           k
+                        matrix of the Lanczos process. This increases
+                                                                
+
+                        monotonically and is a lower bound on ||A||.
+
+             COND(L)    A monotonically increasing lower bound on the
+                                                         -1
+                        condition number of A, ||A||||(A)  ||.
+
+                                         L
+             LQ/CG      L is printed if x  is the best current
+                                         k
+                        approximation to the solution and C is printed
+                        otherwise.
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL< 0
+               A negative value of IFAIL indicates an exit from F04MBF
+               because the user has set IFLAG negative in APROD or MSOLVE.
+               The value of IFAIL will be the same as the user's setting of
+               IFLAG.
+
+          IFAIL= 1
+               On entry N < 1,
+
+               or       LRWORK < 1,
+
+               or       LIWORK < 1.
+
+          IFAIL= 2
+               The pre-conditioning matrix M does not appear to be
+               positive-definite. The user should check that MSOLVE is
+               working correctly.
+
+          IFAIL= 3
+               The limit on the number of iterations has been reached. If
+               IFAIL = 1 on entry then the latest approximation to the
+               solution is returned in X and the values ANORM, ACOND, RNORM
+               and XNORM are also returned.
+
+          The value of INFORM contains additional information about the
+          termination of the routine and users must examine INFORM to judge
+          whether the routine has performed successfully for the problem in
+          hand. In particular INFORM = 3 denotes that the matrix A-
+          (lambda)I is effectively singular: if the purpose of calling
+          F04MBF is to solve a system of equations Ax=b, then this
+          condition must be regarded as a failure, but if the purpose is to
+          compute an eigenvector, this result would be very satisfactory.
+
+          7. Accuracy
+
+          The computed solution x will satisfy the equation
+
+                                 r=b-(A-(lambda)I)x
+
+          where the value ||r|| is as returned in the parameter RNORM.
+
+          8. Further Comments
+
+          The time taken by the routine is likely to be principally
+          determined by the time taken in APROD and, when pre-conditioning
+          has been requested, in MSOLVE. Each of these routines is called
+          once every iteration.
+
+          The time taken by the remaining operations in F04MBF is
+          approximately proportional to n.
+
+          9. Example
+
+          To solve the 10 equations Ax=b given by
+
+                            (2 1 0 0 0 0 0 0 0 3)      (6)
+                            (1 2 1 0 0 0 0 0 0 0)      (4)
+                            (0 1 2 1 0 0 0 0 0 0)      (4)
+                            (0 0 1 2 1 0 0 0 0 0)      (4)
+                            (0 0 0 1 2 1 0 0 0 0)      (4)
+                          A=(0 0 0 0 1 2 1 0 0 0),   b=(4).
+                            (0 0 0 0 0 1 2 1 0 0)      (4)
+                            (0 0 0 0 0 0 1 2 1 0)      (4)
+                            (0 0 0 0 0 0 0 1 2 1)      (4)
+                            (3 0 0 0 0 0 0 0 1 2)      (6)
+
+          The tridiagonal part of A is positive-definite and such
+          tridiagonal equations can be solved efficiently by F04FAF. The
+          form of A suggests that this tridiagonal part is a good candidate
+          for the pre-conditioning matrix M and so we illustrate the use of
+          F04MBF by pre-conditioning with the 10 by 10 matrix
+
+                                    (2 1 0 ... 0)
+                                    (1 2 1 ... 0)
+                                    (0 1 2 ... 0)
+                                  M=(. . .   .  ).
+                                    (. . .   .  )
+                                    (. . .   .  )
+                                    (0 0 0 ... 2)
+
+          Since A-M has only 2 non-zero elements and is obviously of rank
+          2, we can expect F04MBF to converge very quickly in this example.
+          Of course, in practical problems we shall not usually be able to
+          make such a good choice of M.
+
+                                                  -5
+          The example sets the tolerance RTOL = 10  .
+
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf04mcf}{NAG On-line Documentation: f04mcf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F04MCF(3NAG)      Foundation Library (12/10/92)      F04MCF(3NAG)
+
+
+
+          F04 -- Simultaneous Linear Equations                       F04MCF
+                  F04MCF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F04MCF computes the approximate solution of a system of real
+          linear equations with multiple right-hand sides,  AX=B,  where A
+          is a symmetric positive-definite variable-bandwidth matrix, which
+          has previously been factorized by F01MCF. Related systems may
+          also be solved.
+
+          2. Specification
+
+                 SUBROUTINE F04MCF (N, AL, LAL, D, NROW, IR, B, NRB,
+                1                   ISELCT, X, NRX, IFAIL)
+                 INTEGER          N, LAL, NROW(N), IR, NRB, ISELCT, NRX,
+                1                 IFAIL
+                 DOUBLE PRECISION AL(LAL), D(N), B(NRB,IR), X(NRX,IR)
+
+          3. Description
+
+          The normal use of this routine is the solution of the systems
+          AX=B, following a call of F01MCF to determine the Cholesky
+                             T
+          factorization A=LDL  of the symmetric positive-definite variable-
+          bandwidth matrix A.
+
+          However, the routine may be used to solve any one of the
+          following systems of linear algebraic equations:
+
+                  T
+          (1)  LDL X = B (usual system),
+
+          (2)  LDX   = B (lower triangular system),
+
+                 T
+          (3)  DL X  = B (upper triangular system),
+
+                 T
+          (4)  LL X  = B
+
+          (5)  LX    = B (unit lower triangular system),
+
+                T
+          (6)  L X   = B (unit upper triangular system).
+
+          L denotes a unit lower triangular variable-bandwidth matrix of
+          order n, D a diagonal matrix of order n, and B a set of right-
+          hand sides.
+
+          The matrix L is represented by the elements lying within its
+          envelope i.e., between the first non-zero of each row and the
+          diagonal (see Section 9 for an example). The width NROW(i) of the
+          ith row is the number of elements between the first non-zero
+          element and the element on the diagonal inclusive.
+
+          4. References
+
+          [1]   Wilkinson J H and Reinsch C (1971) Handbook for Automatic
+                Computation II, Linear Algebra. Springer-Verlag.
+
+          5. Parameters
+
+           1:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix L. Constraint: N >= 1.
+
+           2:  AL(LAL) -- DOUBLE PRECISION array                      Input
+               On entry: the elements within the envelope of the lower
+               triangular matrix L, taken in row by row order, as returned
+               by F01MCF. The unit diagonal elements of L must be stored
+               explicitly.
+
+           3:  LAL -- INTEGER                                         Input
+               On entry:
+               the dimension of the array AL as declared in the
+               (sub)program from which F04MCF is called.
+               Constraint: LAL >= NROW(1) + NROW(2) +... + NROW(n).
+
+           4:  D(N) -- DOUBLE PRECISION array                         Input
+               On entry: the diagonal elements of the diagonal matrix D. D
+               is not referenced if ISELCT >= 4.
+
+           5:  NROW(N) -- INTEGER array                               Input
+               On entry: NROW(i) must contain the width of row i of L,
+               i.e.,the number of elements between the first (leftmost)
+               non-zero element and the element on the diagonal, inclusive.
+               Constraint: 1 <= NROW(i)<=i.
+
+           6:  IR -- INTEGER                                          Input
+               On entry: r, the number of right-hand sides. Constraint: IR
+               >= 1.
+
+           7:  B(NRB,IR) -- DOUBLE PRECISION array                    Input
+               On entry: the n by r right-hand side matrix B. See also
+               Section 8.
+
+           8:  NRB -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array B as declared in the
+               (sub)program from which F04MCF is called.
+               Constraint: NRB >= N.
+
+           9:  ISELCT -- INTEGER                                      Input
+               On entry: ISELCT must specify the type of system to be
+               solved, as follows:
+
+                                     T
+               ISELCT = 1:  solve LDL X = B,
+
+               ISELCT = 2:  solve LDX = B,
+
+                                    T
+               ISELCT = 3:  solve DL X = B,
+
+                                    T
+               ISELCT = 4:  solve LL X = B,
+
+               ISELCT = 5:  solve LX = B,
+
+                                   T
+               ISELCT = 6:  solve L X = B.
+
+          10:  X(NRX,IR) -- DOUBLE PRECISION array                   Output
+               On exit: the n by r solution matrix X. See also Section 8.
+
+          11:  NRX -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array X as declared in the
+               (sub)program from which F04MCF is called.
+               Constraint: NRX >= N.
+
+          12:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry N < 1,
+
+               or       for some i, NROW(i)<1 or NROW(i) > i,
+
+               or       LAL < NROW(1) + NROW(2) +... + NROW(N).
+
+          IFAIL= 2
+
+               On entry IR < 1,
+
+               or       NRB < N,
+
+               or       NRX < N.
+
+          IFAIL= 3
+               On entry ISELCT < 1,
+
+               or       ISELCT > 6.
+
+          IFAIL= 4
+               The diagonal matrix D is singular, i.e., at least one of the
+               elements of D is zero. This can only occur if ISELCT <= 3.
+
+          IFAIL= 5
+               At least one of the diagonal elements of L is not equal to
+               unity.
+
+          7. Accuracy
+
+          The usual backward error analysis of the solution of triangular
+          system applies: each computed solution vector is exact for
+          slightly perturbed matrices L and D, as appropriate (cf.
+          Wilkinson and Reinsch [1] pp 25--27, 54--55).
+
+          8. Further Comments
+
+          The time taken by the routine is approximately proportional to
+          pr, where
+
+                           p=NROW(1)+NROW(2)+...+NROW(n).
+
+          Unless otherwise stated in the Users' Note for your
+          implementation, the routine may be called with the same actual
+          array supplied for the parameters B and X, in which case the
+          solution matrix will overwrite the right-hand side matrix.
+          However this is not standard Fortran 77 and may not work in all
+          implementations.
+
+          9. Example
+
+          To solve the system of equations AX=B, where
+
+                                  (1  2  0  0  5  0)
+                                  (2  5  3  0 14  0)
+                                  (0  3 13  0 18  0)
+                                A=(0  0  0 16  8 24)
+                                  (5 14 18  8 55 17)
+                                  (0  0  0 24 17 77)
+
+          and
+
+                                       ( 6 -10)
+                                       (15 -21)
+                                       (11  -3)
+                                     B=( 0  24)
+                                       (51 -39)
+                                       (46  67)
+
+          Here A is symmetric and positive-definite and must first be
+          factorized by F01MCF.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf04qaf}{NAG On-line Documentation: f04qaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F04QAF(3NAG)      Foundation Library (12/10/92)      F04QAF(3NAG)
+
+
+
+          F04 -- Simultaneous Linear Equations                       F04QAF
+                  F04QAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F04QAF solves sparse unsymmetric equations, sparse linear least-
+          squares problems and sparse damped linear least-squares problems,
+          using a Lanczos algorithm.
+
+          2. Specification
+
+                 SUBROUTINE F04QAF (M, N, B, X, SE, APROD, DAMP, ATOL,
+                1                   BTOL, CONLIM, ITNLIM, MSGLVL, ITN,
+                2                   ANORM, ACOND, RNORM, ARNORM, XNORM,
+                3                   WORK, RWORK, LRWORK, IWORK, LIWORK,
+                4                   INFORM, IFAIL)
+                 INTEGER          M, N, ITNLIM, MSGLVL, ITN, LRWORK, IWORK
+                1                 (LIWORK), LIWORK, INFORM, IFAIL
+                 DOUBLE PRECISION B(M), X(N), SE(N), DAMP, ATOL, BTOL,
+                1                 CONLIM, ANORM, ACOND, RNORM, ARNORM,
+                2                 XNORM, WORK(N,2), RWORK(LRWORK)
+                 EXTERNAL         APROD
+
+          3. Description
+
+          F04QAF can be used to solve a system of linear equations
+
+                                     Ax=b                             (3.1)
+
+          where A is an n by n sparse unsymmetric matrix, or can be used to
+          solve linear least-squares problems, so that F04QAF minimizes the
+          value (rho) given by
+
+                             (rho)=||r||,  r=b-Ax                     (3.2)
+
+          where A is an m by n sparse matrix and ||r|| denotes the
+                                             2  T
+          Euclidean length of r so that ||r|| =r r. A damping parameter,
+          (lambda), may be included in the least-squares problem in which
+          case F04QAF minimizes the value (rho) given by
+
+                             2      2         2     2
+                        (rho) =||r|| +(lambda) ||x||                  (3.3)
+
+          (lambda) is supplied as the parameter DAMP and should of course
+          be zero if the solution to problems (3.1) or (3.2) is required.
+          Minimizing (rho) in (3.3) is often called ridge regression.
+
+          F04QAF is based upon algorithm LSQR (see Paige and Saunders [1]
+          and [2]) and solves the problems by an algorithm based upon the
+          Lanczos process. Details of the method are given in [1]. The
+          routine does not require A explicitly, but A is specified via a
+          user-supplied routine APROD which must perform the operations (
+                        T
+          y+Ax) and (x+A y) for a given n element vector x and m element
+          vector y. A parameter to APROD specifies which of the two
+          operations is required on a given entry.
+
+          The routine also returns estimates of the standard errors of the
+          sample regression coefficients (x , for i=1,2,...,n) given by the
+                                           i
+          diagonal elements of the estimated variance-covariance matrix V.
+          When problem (3.2) is being solved and A is of full rank, then V
+          is given by
+
+                            2  T  -1    2      2
+                         V=s (A A)  ,  s =(rho) /(m-n),  m>n
+
+          and when problem (3.3) is being solved then V is given by
+
+                    2  T          2  -1    2      2
+                 V=s (A A+(lambda) I)  ,  s =(rho) /m,  (lambda)/=0.
+
+              
+
+          Let A denote the matrix
+
+                                    (A        )
+               A=A,  (lambda)=0 ; A=((lambda)I),  (lambda)/=0,        (3.4)
+
+              
+
+          let r denote the residual vector
+
+                                       (b) 
+                  r=r,  (lambda)=0 ; r=(0)-Ax,  (lambda)/=0           (3.5)
+
+                                                         
+
+          corresponding to an iterate x, so that (rho)=||r|| is the
+          function being minimized, and let ||A|| denote the Frobenius
+          (Euclidean) norm of A. Then the routine accepts x as a solution
+          if it is estimated that one of the following two conditions is
+          satisfied:
+
+                                    
+
+                       (rho)<=tol ||A||.||x||+tol ||b||               (3.6)
+                                 1               2
+
+                              T           
+                           ||A r||<=tol ||A||(rho)                    (3.7)
+                                       1
+
+          where tol  and tol  are user-supplied tolerances which estimate
+                   1        2
+          the relative errors in A and b respectively. Condition (3.6) is
+          appropriate for compatible problems where, in theory, we expect
+          the residual to be zero and will be satisfied by an acceptable
+          solution x to a compatible problem. Condition (3.7) is
+          appropriate for incompatible systems where we do not expect the
+          residual to be zero and is based upon the observation that, in
+          theory,
+
+                                         T
+                                        A r=0
+
+          when x is a solution to the least-squares problem, and so (3.7)
+          will be satisfied by an acceptable solution x to a linear least-
+          squares problem.
+
+          The routine also includes a test to prevent convergence to
+          solutions, x, with unacceptably large elements. This can happen
+          if A is nearly singular or is nearly rank deficient. If we let
+                                 
+
+          the singular values of A be
+
+                        (sigma) >=(sigma) >=...>=(sigma) >=0
+                               1         2              n
+
+                                       
+
+          then the condition number of A is defined as
+
+                                   
+
+                              cond(A)=(sigma) /(sigma)
+                                             1        k
+
+                                                                    
+
+          where (sigma)  is the smallest non-zero singular value of A and
+                       k
+                                                   
+
+          hence k is the rank of A. When k<n, then A is rank deficient, the
+          least-squares solution is not unique and F04QAF will normally
+                                                               
+
+          converge to the minimal length solution. In practice A will not
+          have exactly zero singular values, but may instead have small
+          singular values that we wish to regard as zero.
+
+          The routine provides for this possibility by terminating if
+
+                                     
+
+                                cond(A)>=c                            (3.8)
+                                          lim
+
+                                                                         
+
+          where c    is a user-supplied limit on the condition number of A.
+                 lim
+          For problem (3.1) termination with this condition indicates that
+          A is nearly singular and for problem (3.2) indicates that A is
+          nearly rank deficient and so has near linear dependencies in its
+                                                        T
+          columns. In this case inspection of ||r||, ||A r|| and ||x||,
+          which are all returned by the routine, will indicate whether or
+          not an acceptable solution has been found. Condition (3.8),
+          perhaps in conjunction with (lambda)/=0, can be used to try and '
+          regularise' least-squares solutions. A full discussion of the
+          stopping criteria is given in Section 6 of reference Paige and
+          Saunders [1].
+
+          Introduction of a non-zero damping parameter (lambda) tends to
+          reduce the size of the computed solution and to make its
+          components less sensitive to changes in the data, and F04QAF is
+          applicable when a value of (lambda) is known a priori. To have an
+                                                         
+
+          effect, (lambda) should normally be at least \/(epsilon)||A||
+          where (epsilon) is the machine precision. For further discussion
+          see Paige and Saunders [2] and the references given there.
+
+          Whenever possible the matrix A should be scaled so that the
+          relative errors in the elements of A are all of comparable size.
+          Such a scaling helps to prevent the least-squares problem from
+          being unnecessarily sensitive to data errors and will normally
+          reduce the number of iterations required. At the very least, in
+          the absence of better information, the columns of A should be
+          scaled to have roughly equal column length.
+
+          4. References
+
+          [1]   Paige C C and Saunders M A (1982) LSQR: An Algorithm for
+                Sparse Linear Equations and Sparse Least-squares. ACM Trans.
+                Math. Softw. 8 43--71.
+
+          [2]   Paige C C and Saunders M A (1982) ALGORITHM 583 LSQR: Sparse
+                Linear Equations and Least-squares Problems. ACM Trans.
+                Math. Softw. 8 195--209.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: m, the number of rows of the matrix A.
+               Constraint: M >= 1.
+
+           2:  N -- INTEGER                                           Input
+               On entry: n, the number of columns of the matrix A.
+               Constraint: N >= 1.
+
+           3:  B(M) -- DOUBLE PRECISION array                  Input/Output
+               On entry: the right-hand side vector b. On exit: the array
+               is overwritten.
+
+           4:  X(N) -- DOUBLE PRECISION array                        Output
+               On exit: the solution vector x.
+
+           5:  SE(N) -- DOUBLE PRECISION array                       Output
+               On exit: the estimates of the standard errors of the
+               components of x. Thus SE(i) contains an estimate of the
+               element v   of the estimated variance-covariance matrix V.
+                        ii
+               The estimates returned in SE will be the lower bounds on the
+               actual estimated standard errors, but will usually have at
+               least one correct figure.
+
+           6:  APROD -- SUBROUTINE, supplied by the user.
+                                                         External Procedure
+                                                                   T
+               APROD must perform the operations y:=y+Ax and x:=x+A y for
+               given vectors x and y.
+
+               Its specification is:
+
+                      SUBROUTINE APROD (MODE, M, N, X, Y, RWORK,
+                     1                  LRWORK, IWORK, LIWORK)
+                      INTEGER          MODE, M, N, LRWORK, LIWORK,
+                     1                 IWORK(LIWORK)
+                      DOUBLE PRECISION X(N), Y(M), RWORK(LRWORK)
+
+                1:  MODE -- INTEGER                            Input/Output
+                    On entry: MODE specifies which operation is to be
+                    performed:
+                         If MODE = 1, then APROD must compute y+Ax.
+
+                                                                 T
+                         If MODE = 2, then APROD must compute x+A y.
+                    On exit: MODE may be used as a flag to indicate a
+                                                             T
+                    failure in the computation of y+Ax or x+A y. If MODE is
+                    negative on exit from APROD, F04QAF will exit
+                    immediately with IFAIL set to MODE.
+
+                2:  M -- INTEGER                                      Input
+                    On entry: m, the number of rows of A.
+
+                3:  N -- INTEGER                                      Input
+                    On entry: n, the number of columns of A.
+
+                4:  X(N) -- DOUBLE PRECISION array             Input/Output
+                    On entry: the vector x. On exit: if MODE = 1, X must be
+                    unchanged;
+
+                                                   T
+                    If MODE = 2, X must contain x+A y.
+
+                5:  Y(M) -- DOUBLE PRECISION array             Input/Output
+                    On entry: the vector y. On exit: if MODE = 1, Y must
+                    contain y+Ax;
+
+                    If MODE = 2, Y must be unchanged.
+
+                6:  RWORK(LRWORK) -- DOUBLE PRECISION array  User Workspace
+
+                7:  LRWORK -- INTEGER                                 Input
+
+                8:  IWORK(LIWORK) -- INTEGER array           User Workspace
+
+                9:  LIWORK -- INTEGER                                 Input
+                    APROD is called from F04QAF with the parameters RWORK,
+                    LRWORK, IWORK and LIWORK as supplied to F04QAF. The
+                    user is free to use the arrays RWORK and IWORK to
+                    supply information to APROD as an alternative to using
+                    COMMON.
+               APROD must be declared as EXTERNAL in the (sub)program
+               from which F04QAF is called. Parameters denoted as
+               Input must not be changed by this procedure.
+
+           7:  DAMP -- DOUBLE PRECISION                               Input
+               On entry: the value (lambda). If either problem (3.1) or
+               problem (3.2) is to be solved, then DAMP must be supplied as
+               zero.
+
+           8:  ATOL -- DOUBLE PRECISION                               Input
+               On entry: the tolerance, tol , of the convergence criteria
+                                           1
+               (3.6) and (3.7); it should be an estimate of the largest
+               relative error in the elements of A. For example, if the
+               elements of A are correct to about 4 significant figures,
+                                                    -4
+               then ATOL should be set to about 5*10  . If ATOL is supplied
+               as less than (epsilon), where (epsilon) is the machine
+               precision, then the value (epsilon) is used in place of
+               ATOL.
+
+           9:  BTOL -- DOUBLE PRECISION                               Input
+               On entry: the tolerance, tol , of the convergence criterion
+                                           2
+               (3.6);  it should be an estimate of the largest relative
+               error in the elements of B. For example, if the elements of
+               B are correct to about 4 significant figures, then BTOL
+                                          -4
+               should be set to about 5*10  . If BTOL is supplied as less
+               than (epsilon), where (epsilon) is the machine precision,
+               then the value (epsilon) is used in place of BTOL.
+
+          10:  CONLIM -- DOUBLE PRECISION                             Input
+               On entry: the value c    of equation  (3.8); it should be an
+                                    lim
+                                                      
+
+               upper limit on the condition number of A. CONLIM should not
+               normally be chosen much larger than 1.0/ATOL. If CONLIM is
+               supplied as zero then the value 1.0/(epsilon), where
+               (epsilon) is the machine precision, is used in place of
+               CONLIM.
+
+          11:  ITNLIM -- INTEGER                                      Input
+               On entry: an upper limit on the number of iterations. If
+               ITNLIM <= 0, then the value N is used in place of ITNLIM,
+               but for ill-conditioned problems a higher value of ITNLIM is
+               likely to be necessary.
+
+          12:  MSGLVL -- INTEGER                                      Input
+               On entry: the level of printing from F04QAF. If MSGLVL <= 0,
+               then no printing occurs, but otherwise messages will be
+               output on the advisory message channel (see X04ABF). A
+               description of the printed output is given in Section 5.2
+               below. The level of printing is determined as follows:
+               MSGLVL <= 0
+                     No printing.
+
+               MSGLVL = 1
+                     A brief summary is printed just prior to return from
+                     F04QAF.
+
+               MSGLVL >= 2
+                     A summary line is printed periodically to monitor the
+                     progress of F04QAF, together with a brief summary just
+                     prior to return from F04QAF.
+
+          13:  ITN -- INTEGER                                        Output
+               On exit: the number of iterations performed.
+
+          14:  ANORM -- DOUBLE PRECISION                             Output
+                                                            
+
+               On exit: an estimate of ||A|| for the matrix A of equation
+               (3.4).
+
+          15:  ACOND -- DOUBLE PRECISION                             Output
+                                            
+
+               On exit: an estimate of cond(A) which is a lower bound.
+
+          16:  RNORM -- DOUBLE PRECISION                             Output
+                                                                
+
+               On exit: an estimate of ||r|| for the residual,  r, of
+               equation (3.5) corresponding to the solution x returned in
+                              
+
+               X. Note that ||r|| is the function being minimized.
+
+          17:  ARNORM -- DOUBLE PRECISION                            Output
+                                              T
+               On exit: an estimate of the ||A r|| corresponding to the
+               solution x returned in X.
+
+          18:  XNORM -- DOUBLE PRECISION                             Output
+               On exit: an estimate of ||x|| for the solution x returned in
+               X.
+
+          19:  WORK(2*N) -- DOUBLE PRECISION array                Workspace
+
+          20:  RWORK(LRWORK) -- DOUBLE PRECISION array       User Workspace
+               RWORK is not used by F04QAF, but is passed directly to
+               routine APROD and may be used to pass information to that
+               routine.
+
+          21:  LRWORK -- INTEGER                                      Input
+               On entry: the length of the array RWORK as declared in the
+               (sub)program from which F04QAF is called. Constraint: LRWORK
+               >= 1.
+
+          22:  IWORK(LIWORK) -- INTEGER array                User Workspace
+               IWORK is not used by F04QAF, but is passed directly to
+               routine APROD and may be used to pass information to that
+               routine.
+
+          23:  LIWORK -- INTEGER                                      Input
+               On entry: the length of the array IWORK as declared in the
+               (sub)program from which F04QAF is called. Constraint: LIWORK
+               >= 1.
+
+          24:  INFORM -- INTEGER                                     Output
+               On exit: the reason for termination of F04QAF as follows:
+               INFORM = 0
+                     The exact solution is x=0. No iterations are performed
+                     in this case.
+
+               INFORM = 1
+                     The termination criterion of equation (3.6) has been
+                     satisfied with tol  and tol  as the values supplied in
+                                       1        2
+                     ATOL and BTOL respectively.
+
+               INFORM = 2
+                     The termination criterion of equation (3.7) has been
+                     satisfied with tol  as the value supplied in ATOL.
+                                       1
+
+               INFORM = 3
+                     The termination criterion of equation (3.6) has been
+                     satisfied with tol  and/or tol  as the value (epsilon)
+                                       1           2
+                     , where (epsilon) is the machine precision. One or
+                     both of the values supplied in ATOL and BTOL must have
+                     been less than (epsilon) and was too small for this
+                     machine.
+
+               INFORM = 4
+                     The termination criterion of equation (3.7) has been
+                     satisfied with tol  as the value (epsilon), where
+                                       1
+                     (epsilon) is the machine precision. The value supplied
+                     in ATOL must have been less than (epsilon) and was too
+                     small for this machine.
+               The values INFORM = 5, INFORM = 6 and INFORM = 7 correspond
+               to failure with IFAIL = 2, IFAIL = 3 and IFAIL = 4
+               respectively (see Section 6) and when IFAIL is negative
+               INFORM will be set to the same negative value.
+
+          25:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          5.1. Description of the printed output
+
+          When MSGLVL > 0, then F04QAF will produce output (except in the
+          case where the routine fails with IFAIL = 1) on the advisory
+          message channel (see X04ABF ).
+
+          When MSGLVL >= 2 then a summary line is printed periodically
+          giving the following information:
+
+             Output       Meaning
+
+             ITN          Iteration number, k.
+
+             X(1)         The first element of the current iterate x .
+                                                                    k
+
+             FUNCTION     The current value of the function, (rho), being
+                          minimized.
+
+                                                             
+
+             COMPAT       An estimate of ||r ||/||b||, where r  is the
+                                            k                 k
+                          residual corresponding to x . This value should
+                                                     k
+                          converge to zero (in theory) if and only if the
+                          problem is compatible. COMPAT decreases
+                          monotonically.
+
+                                            T             
+             INCOMPAT     An estimate of ||A r ||/(||A||||r ||) which
+                                              k            k
+                          should converge to zero if and only if at the
+                          solution (rho) is non-zero. INCOMPAT is not
+                          usually monotonic.
+
+                                                                   
+
+             NRM(ABAR)    A monotonically increasing estimate of ||A||.
+
+             COND(ABAR)   A monotonically increasing estimate of the
+                                                
+
+                          condition number cond(A).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL< 0
+               A negative value of IFAIL indicates an exit from F04QAF
+               because the user has set MODE negative in APROD. The value
+               of IFAIL will be the same as the user's setting of MODE.
+
+          IFAIL= 1
+               On entry M < 1,
+
+               or       N < 1,
+
+               or       LRWORK < 1,
+
+               or       LIWORK < 1.
+
+          IFAIL= 2
+               The condition of equation (3.8) has been satisfied for the
+               value of c    supplied in CONLIM. If this failure is
+                         lim
+               unexpected the user should check that APROD is working
+               correctly. Although conditions (3.6) or (3.7) have not been
+               satisfied, the values returned in RNORM, ARNORM and XNORM
+               may nevertheless indicate that an acceptable solution has
+               been reached.
+
+          IFAIL= 3
+               The conditions of equation (3.8) has been satisified for the
+               value c   =1.0/(epsilon), where (epsilon) is the machine
+                      lim
+                                     
+
+               precision. The matrix A is nearly singular or rank deficient
+               and the problem is too ill-conditioned for this machine. If
+               this failure is unexpected, the user should check that APROD
+               is working correctly.
+
+          IFAIL= 4
+               The limit on the number of iterations has been reached. The
+               number of iterations required by F04QAF and the condition of
+                          
+
+               the matrix A can depend strongly on the scaling of the
+               problem. Poor scaling of the rows and columns of A should be
+               avoided whenever possible.
+
+          7. Accuracy
+
+          When the problem is compatible, the computed solution x will
+          satisfy the equation
+
+                                       r=b-Ax,
+
+          where an estimate of ||r|| is returned in the parameter RNORM.
+          When the problem is incompatible, the computed solution x will
+          satisfy the equation
+
+                                        T
+                                       A r=e,
+
+          where an estimate of ||e|| is returned in the parameter ARNORM.
+          See also Section 6.2 of Paige and Saunders [1].
+
+          8. Further Comments
+
+          The time taken by the routine is likely to be principally
+          determined by the time taken in APROD, which is called twice on
+          each iteration, once with MODE = 1 and once with MODE = 2. The
+          time taken per iteration by the remaining operations in F04QAF is
+          approximately proportional to max(m,n).
+
+          The Lanczos process will usually converge more quickly if A is
+          pre-conditioned by a non-singular matrix M that approximates A in
+          some sense and is also chosen so that equations of the form My=c
+          can efficiently be solved for y. Some discussion of pre-
+          conditioning in the context of symmetric matrices is given in
+          Section 3 of the document for F04MBF. In the context of F04QAF,
+          problem (3.1) is equivalent to
+
+                                     -1
+                                  (AM  )y=b,  Mx=y
+
+          and problem (3.2) is equivalent to minimizing
+
+                                               -1
+                          (rho)=||r||,  r=b-(AM  )y,  Mx=y.
+
+                                         -1 T   -1   -T  T   -1
+          Note that the normal matrix (AM  ) (AM  )=M  (A A)M   so that the
+                             -1
+          pre-conditioning AM   is equivalent to the pre-conditioning
+           -T  T   -1                       T
+          M  (A A)M   of the normal matrix A A.
+
+          Pre-conditioning can be incorporated into F04QAF simply by coding
+                                           -1         -T T
+          the routine APROD to compute y+AM  x and x+M  A y in place of
+                      T
+          y+Ax and x+A y respectively, and then solving the equations Mx=y
+                                           -1
+          for x on return from F04QAF. y+AM  x should be computed by
+                                                             -T T
+          solving Mz=x for z and then computing y+Az, and x+M  A y should
+                                  T   T
+          be computed by solving M z=A y for z and then forming x+z.
+
+          9. Example
+
+          To solve the linear least-squares problem
+
+                            minimize (rho)=||r||,  r=b-Ax
+
+          where A is the 13 by 12 matrix and b is the 13 element vector
+          given by
+
+                        ( 1  0  0 -1  0  0  0  0  0  0  0  0)
+                        ( 0  1  0  0 -1  0  0  0  0  0  0  0)
+                        ( 0  0  1 -1  0  0  0  0  0  0  0  0)
+                        (-1  0 -1  4 -1  0  0 -1  0  0  0  0)
+                        ( 0 -1  0 -1  4 -1  0  0 -1  0  0  0)
+                        ( 0  0  0  0 -1  1  0  0  0  0  0  0)
+                      A=( 0  0  0  0  0  0  1 -1  0  0  0  0),
+                        ( 0  0  0 -1  0  0 -1  4 -1  0 -1  0)
+                        ( 0  0  0  0 -1  0  0 -1  4 -1  0 -1)
+                        ( 0  0  0  0  0  0  0  0 -1  1  0  0)
+                        ( 0  0  0  0  0  0  0 -1  0  0  1  0)
+                        ( 0  0  0  0  0  0  0  0 -1  0  0  1)
+                        ( 1  1  1  0  0  1  1  0  0  1  1  1)
+
+                                          ( 0  )
+                                          ( 0  )
+                                          ( 0  )
+                                          ( 1  )
+                                          ( 1  )
+                                         2( 0  )
+                                     b=-h ( 0  )
+                                          ( 1  )
+                                          ( 1  )
+                                          ( 0  )
+                                          ( 0  )
+                                          ( 0  )
+                                          (  -3)
+                                          (-h  )
+
+          with h=0.1.
+
+          Such a problem can arise by considering the Neumann problem on a
+          rectangle
+
+                              (delta)u
+                              --------=0
+                              (delta)n
+
+               (delta)u            2           (delta)u      /
+               --------=0   (nabla) u=g(x,y)   --------=0    |u=1
+               (delta)n                        (delta)n      /
+                                                             c
+
+                              (delta)u
+                              --------=0
+                              (delta)n
+
+          where C is the boundary of the rectangle, and discretising as
+          illustrated below with the square mesh
+
+
+
+                   Please see figure in printed Reference Manual
+
+          The 12 by 12 symmetric part of A represents the difference
+          equations and the final row comes from the normalising condition.
+          The example program has g(x,y)=1 at all the internal mesh points,
+          but apart from this is written in a general manner so that the
+          number of rows (NROWS) and columns (NCOLS) in the grid can
+          readily be altered.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf06}{NAG On-line Documentation: f06}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F06(3NAG)         Foundation Library (12/10/92)         F06(3NAG)
+
+
+
+          F06 -- Linear Algebra Support Routines        Introduction -- F06
+                                    Chapter F06
+                          Linear Algebra Support Routines
+
+          Contents of this Introduction:
+
+          1.     Scope of the Chapter
+
+          2.     Background to the Problems
+
+          2.1.   The Use of BLAS Names
+
+          2.2.   Background Information
+
+          2.2.1. Real plane rotations
+
+          2.3.   References
+
+          3.     Recommendations on Choice and Use of Routines
+
+          3.1.   The Level-0 Scalar Routines
+
+          3.2.   The Level-1 Vector Routines
+
+          3.3.   The Level-2 Matrix-vector Routines
+
+          3.4.   The Level-3 Matrix-matrix Routines
+
+          4.     Description of the F06 Routines
+
+          4.1.   The Level-0 Scalar Routines
+
+          4.2.   The Level-1 Vector Routines
+
+          4.3.   The Level-2 Matrix-vector Routines
+
+          4.4.   The Level-3 Matrix-matrix Routines
+
+
+
+          1. Scope of the Chapter
+
+          This Chapter is concerned with basic linear algebra routines
+          which perform elementary algebraic operations involving scalars,
+          vectors and matrices.
+
+          2. Background to the Problems
+
+          All the routines in this Chapter meet the specification of the
+          Basic Linear Algebra Subprograms (BLAS) as described in Lawson
+          et al [6], Dongarra et al [3] and [4]. The first reference
+          describes a set of routines concerned with operations on scalars
+          and vectors: these will be referred to here as the Level-0 and
+          the Level-1 BLAS; the second reference describes a set of
+          routines concerned with matrix-vector operations: these will be
+          referred to here as the Level-2 BLAS; and the third reference
+          describes a set of routines concerned with matrix-matrix
+          operations: these will be referred to here as the Level-3 BLAS.
+          The terminology reflects the number of operations involved. For
+                                                 2
+          example, a Level-2 routine involves 0(n ) operations for an n by
+          n matrix.
+
+          Table 1.1 indicates the naming scheme for the routines in this
+          Chapter. The heading 'mixed type' is for routines where a mixture
+          of data types is involved, such as a routine that returns the
+          real Euclidean length of a complex vector.
+
+
+                                       Level-0  Level-1  Level-2  Level-3
+                'real'   BLAS routine  F06A F   F06E F   F06P F   F06Y F
+             'complex'   BLAS routine  -        F06G F   F06S F   F06Z F
+
+                                     Table 1.1
+
+
+
+          The routines in this chapter do not have full routine documents,
+          but instead are covered by some relevant background material, in
+          Section 2.2, together with general descriptions, in Section 4,
+          sufficient to enable their use. As this chapter is concerned only
+          with basic linear algebra operations, the routines will not
+          normally be required by the general user. The functionality of
+          each routine is indicated in Section 3 so that those users
+          requiring these routines to build specialist linear algebra
+          modules can determine which routines are of interest.
+
+          2.1. The Use of BLAS Names
+
+          Many of the routines in other chapters of the Library call the
+          BLAS in this chapter. These routines are usually called by the
+          BLAS name and so, for correct operation of the Library, it is
+          essential that users do not attempt to link their own versions of
+          these routines. If users are in any doubt about how to avoid
+          this, please consult your local support staff or the NAG Response
+          Centre.
+
+          The BLAS names are used in order to make use of efficient
+          implementations of the routines when these exist. Such
+          implementations are stringently tested before being used, to
+          ensure that they correctly meet the specification of the BLAS,
+          and that they return the desired accuracy (see, for example,
+          Dongarra et al. [3] and [4]).
+
+          2.2. Background Information
+
+          Most of the routines in this chapter implement straightforward
+          scalar, vector and matrix operations that need no further
+          explanation beyond a statement of the purpose of the routine. In
+          this section we give some additional background information for
+          those few cases where additional explanation may be necessary.
+
+          2.2.1.  Real plane rotations
+
+          Two routines in the chapter are concerned with setting up and
+          applying plane rotations. For further background information see
+          Golub and Van Loan [5].
+
+          A plane rotation matrix for the (i,j) plane, R  , is an
+                                                        ij
+          orthogonal matrix that is different from the unit matrix only in
+          the elements r  , r  , r   and r  . If we put
+                        ii   jj   ij      ji
+
+
+                                      (r   r  )
+                                      ( ii  ij)
+                                    R=(r   r  ),
+                                      ( ji  jj)
+
+          then, in the real case, it is usual to choose R   so that
+                                                         ij
+
+                     ( c s)
+                   R=(-s c),  c=cos(theta),  s=sin(theta).            (2.1)
+
+          The application of plane rotations is straightforward and needs
+          no further elaboration, so further comment is made only on the
+          construction of plane rotations.
+
+          The most common use of plane rotations is to choose c and s so
+          that for given a and b,
+
+
+                                ( c s)(a) (d)
+                                (-s c)(b)=(0)                         (2.2)
+
+          In such an application the matrix R is often termed a Givens
+          rotation matrix.
+
+          The BLAS routine F06AAF(*) (DROTG), see Lawson et al [6] and
+          Dodson and Grimes [1, 2], computes c, s and d as
+
+
+                                           2  2 1/2
+                                d=(sigma)(a +b )   ,
+                          {a/d, d/=0      {b/d, d/=0
+                        c={1,   d=0 ,   s={0,   d=0                   (2.3)
+
+                        {sign a,|a|>|b|
+          where (sigma)={sign b,|a|<=|b|.
+
+          The value z defined as
+
+                               {s,   |s|<c or c=0
+                             z={1/c, 0<|c|<=s                         (2.4)
+
+          is also computed and this enables c and s to be reconstructed
+          from the single value z as
+
+
+                         {0,          z=1
+                         {    2 1/2             {1,       z=1
+                       c={(1-z )   , |z|<1    s={z,      |z|<1
+                         {1/z,       |z|>1,     {    2
+                                                {(1-c ), |z|>1.
+
+          2.3. References
+
+          [1]   Dodson D S and Grimes R G (1982) Remark on Algorithm 539.
+                ACM Trans Math Softw. 8 403--404.
+
+          [2]   Dodson D S and Grimes R G (1982) Remark on Algorithm 539.
+                ACM Trans. Math. Softw. 9 140.
+
+          [3]   Dongarra J J, Du Croz J J, Hammarling S and Hanson R J
+                (1988) An Extended Set of FORTRAN Basic Linear Algebra
+                Subprograms. ACM Trans. Math. Softw. 14 1--32.
+
+          [4]   Dongarra J J, Du Croz J J, Duff I S and Hammarling S (1990)
+                A Set of Level 3 Basic Linear Algebra Subprograms. ACM
+                Trans. Math. Softw. 16 1--28.
+
+          [5]   Golub G H and Van Loan C F (1989) Matrix Computations (2nd
+                Edition). Johns Hopkins University Press, Baltimore,
+                Maryland.
+
+          [6]   Lawson C L, Hanson R J, Kincaid D R and Krogh F T (1979)
+                Basic Linear Algebra Subprograms for Fortran Usage. ACM
+                Trans. Math. Softw. 5 308--325.
+
+          3. Recommendations on Choice and Use of Routines
+
+          This section lists the routines in each of the categories Level-0
+          (scalar), Level-1 (vector), Level-2 (matrix-vector and matrix)
+          and Level-3 (matrix-matrix). The corresponding double precision
+          BLAS name is indicated in brackets.
+
+          Within each section routines are listed in alphabetic order of
+          the fifth character in the routine name, so that corresponding
+          real and complex routines may have adjacent entries.
+
+          3.1. The Level-0 Scalar Routine
+
+          The Level-0 routine performs the scalar operation of generating a
+          plane rotation.
+
+          F06AAF (DROTG)  generates a real plane rotation.
+
+          3.2. The Level-1 Vector Routines
+
+          The Level-1 routines perform operations on or between vectors,
+          such as computing dot products and Euclidean lengths.
+
+          F06EAF (DDOT)   computes the dot product of two real vectors
+
+          F06GAF (ZDOTU)  computes the dot product of two complex vectors
+                          (unconjugated)
+
+          F06GBF (ZDOTC)  computes the dot product of two complex vectors
+                          (conjugated)
+
+          F06ECF (DAXPY)  adds a scalar times a vector to another real
+                          vector
+
+          F06GCF (ZAXPY)  adds a scalar times a vector to another complex
+                          vector
+
+          F06EDF (DSCAL)  multiplies a real vector by a scalar
+
+          F06GDF (ZSCAL)  multiplies a complex vector by a scalar
+
+          F06JDF (ZDSCAL) multiplies a complex vector by a real scalar
+
+          F06EFF (DCOPY)  copies a real vector
+
+          F06GFF (ZCOPY)  copies a complex vector
+
+          F06EGF (DSWAP)  swaps two real vectors
+
+          F06GGF (ZSWAP)  swaps two complex vectors
+
+          F06EJF (DNRM2)  computes the Euclidean length of a real vector
+
+          F06JJF (DZNRM2) computes the Euclidean length of a complex vector
+
+          F06EKF (DASUM)  sums the absolute values of the elements of a
+                          real vector
+
+          F06JKF (DZASUM) sums the absolute values of the elements of a
+                          complex vector
+
+          F06JLF (IDAMAX) finds the index of the element of largest
+                          absolute value of a real vector
+
+          F06JMF (IZAMAX) finds the index of the element of largest
+                          absolute value of a complex vector
+
+          F06EPF (DROT)   applies a real plane rotation
+
+          3.3. The Level-2 Matrix-vector Routines
+
+          The Level-2 routines perform matrix-vector operations, such as
+          forming the product between a matrix and a vector.
+
+          F06PAF (DGEMV)  computes a matrix-vector product;
+                          real general matrix
+
+          F06SAF (ZGEMV)  computes a matrix-vector product;
+                          complex general matrix
+
+          F06PBF (DGBMV)  computes a matrix-vector product;
+                          real general band matrix
+
+          F06SBF (ZGBMV)  computes a matrix-vector product;
+                          complex general band matrix
+
+          F06PCF (DSYMV)  computes a matrix-vector product;
+                          real symmetric matrix
+
+          F06SCF (ZHEMV)  computes a matrix-vector product;
+                          complex Hermitian matrix
+
+          F06PDF (DSBMV)  computes a matrix-vector product;
+                          real symmetric band matrix
+
+          F06SDF (ZHBMV)  computes a matrix-vector product;
+                          complex Hermitian band matrix
+
+          F06PEF (DSPMV)  computes a matrix-vector product;
+                          real symmetric packed matrix
+
+          F06SEF (ZHPMV)  computes a matrix-vector product;
+                          complex Hermitian packed matrix
+
+          F06PFF (DTRMV)  computes a matrix-vector product;
+                          real triangular matrix
+
+          F06SFF (ZTRMV)  computes a matrix-vector product;
+                          complex triangular matrix
+
+          F06PGF (DTBMV)  computes a matrix-vector product;
+                          real triangular band matrix
+
+          F06SGF (ZTBMV)  computes a matrix-vector product;
+                          complex triangular band matrix
+
+          F06PHF (DTPMV)  computes a matrix-vector product;
+                          real triangular packed matrix
+
+          F06SHF (ZTPMV)  computes a matrix-vector product;
+                          complex triangular packed matrix
+
+          F06PJF (DTRSV)  solves a system of equations;
+                          real triangular coefficient matrix
+
+          F06SJF (ZTRSV)  solves a system of equations;
+                          complex triangular coefficient matrix
+
+          F06PKF (DTBSV)  solves a system of equations;
+                          real triangular band coefficient matrix
+
+          F06SKF (ZTBSV)  solves a system of equations;
+                          complex triangular band coefficient matrix
+
+          F06PLF (DTPSV)  solves a system of equations;
+                          real triangular packed coefficient matrix
+
+          F06SLF (ZTPSV)  solves a system of equations;
+                          complex triangular packed coefficient matrix
+
+          F06PMF (DGER)   performs a rank-one update;
+                          real general matrix
+
+          F06SMF (ZGERU)  performs a rank-one update;
+                          complex general matrix (unconjugated vector)
+
+          F06SNF (ZGERC)  performs a rank-one update;
+                          complex general matrix (conjugated vector)
+
+          F06PPF (DSYR)   performs a rank-one update;
+                          real symmetric matrix
+
+          F06SPF (ZHER)   performs a rank-one update;
+                          complex Hermitian matrix
+
+          F06PQF (DSPR)   performs a rank-one update;
+                          real symmetric packed matrix
+
+          F06SQF (ZHPR)   performs a rank-one update;
+                          complex Hermitian packed matrix
+
+          F06PRF (DSYR2)  performs a rank-two update;
+                          real symmetric matrix
+
+          F06SRF (ZHER2)  performs a rank-two update;
+                          complex Hermitian matrix
+
+          F06PSF (DSPR2)  performs a rank-two update;
+                          real symmetric packed matrix
+
+          F06SSF (ZHPR2)  performs a rank-two update;
+                          complex Hermitian packed matrix
+
+          3.4. The Level-3 Matrix-matrix Routines
+
+          The Level-3 routines perform matrix-matrix operations, such as
+          forming the product of two matrices.
+
+          F06YAF (DGEMM)  computes a matrix-matrix product; two real
+                          rectangular matrices
+
+          F06ZAF (ZGEMM)  computes a matrix-matrix product; two complex
+                          rectangular matrices
+
+          F06YCF (DSYMM)  computes a matrix-matrix product; one real
+                          symmetric matrix, one real rectangular matrix
+
+          F06ZCF (ZHEMM)  computes a matrix-matrix product; one complex
+                          Hermitian matrix, one complex rectangular matrix
+
+          F06YFF (DTRMM)  computes a matrix-matrix product; one real
+                          triangular matrix, one real rectangular matrix
+
+          F06ZFF (ZTRMM)  computes a matrix-matrix product; one complex
+                          triangular matrix, one complex rectangular matrix
+
+          F06YJF (DTRSM)  solves a system of equations with multiple right-
+                          hand sides, real triangular coefficient matrix
+
+          F06ZJF (ZTRSM)  solves a system of equations with multiple right-
+                          hand sides, complex triangular coefficient matrix
+
+          F06YPF (DSYRK)  performs a rank-k update of a real symmetric
+                          matrix
+
+          F06ZPF (ZHERK)  performs a rank-k update of a complex hermitian
+                          matrix
+
+          F06YRF (DSYR2K) performs a rank-2k update of a real symmetric
+                          matrix
+
+          F06ZRF (ZHER2K) performs a rank-2k update of a complex Hermitian
+                          matrix
+
+          F06ZTF (ZSYMM)  computes a matrix-matrix product: one complex
+                          symmetric matrix, one complex rectangular matrix
+
+          F06ZUF (ZSYRK)  performs a rank-k update of a complex symmetric
+                          matrix
+
+          F06ZWF (ZSYR2K) performs a rank-2k update of a complex symmetric
+                          matrix
+
+          4. Description of the F06 Routines
+
+          In this section we describe the purpose of each routine and give
+          information on the parameter lists, where appropriate indicating
+          their general nature. Usually the association between the routine
+          arguments and the mathematical variables is obvious and in such
+          cases a description of the argument is omitted.
+
+          Within each section, the parameter lists for all routines are
+          presented, followed by the purpose of the routines and
+          information on the parameter lists. The double precision BLAS
+          names are given in ENTRY statements.
+
+          Within each section routines are listed in alphabetic order of
+          the fifth character in the routine name, so that corresponding
+          real and complex routines may have adjacent entries.
+
+          4.1. The Level-0 Scalar Routines
+
+          The scalar routines have no array arguments.
+
+
+          SUBROUTINE   F06AAF( A,B,C,S )
+          ENTRY        DROTG ( A,B,C,S )
+          DOUBLE PRECISION     A,B,C,S
+
+          F06AAF(*) generates the parameters c and s of a Givens rotation
+          as defined by equations (2.3) and (2.4), from given a and b. On
+          exit, A is overwritten by d and B is overwritten by z.
+
+          4.2. The Level-1 Vector Routines
+
+          The vector routines all have one or more one-dimensional arrays
+          as arguments, each representing a vector.
+
+          The length of each vector, n, is represented by the argument N,
+          and the routines may be called with non-positive values of N, in
+          which case the routine returns immediately except for the
+          functions, which set the function value to zero before returning.
+
+          In addition to the argument N, each array argument is also
+          associated with an increment argument that immediately follows
+          the array argument, and whose name consists of the three
+          characters INC, followed by the name of the array. For example, a
+          vector x will be represented by the two arguments X, INCX. The
+          increment argument is the spacing (stride) in the array for which
+          the elements of the vector occur. For instance, if INCX = 2, then
+          the elements of x are in locations X(1),X(3),...,X(2*N-1) of the
+          array X and the intermediate locations X(2),X(4),...,X(2*N-2) are
+          not referenced.
+
+          Thus when INCX > 0, the vector element x  is in the array element
+                                                  i
+          X(1+(i-1)*INCX). When INCX <= 0 the elements are stored in the
+          reverse order so that the vector element x  is in the array
+                                                    i
+          element X(1-(n-i)*INCX) and hence, in particular, the element x
+                                                                         n
+          is in X(1). The declared length of the array X in the calling
+          (sub)program must be at least (1+(N-1)*|INCX|).
+
+          Non-positive increments are permitted only for those routines
+          that have more than one array argument. While zero increments are
+          formally permitted for such routines, their use in Chapter F06 is
+          strongly discouraged since the effect may be implementation
+          dependent.
+
+
+
+          DOUBLE PRECISION FUNCTION F06EAF ( N,      X,INCX,Y,INCY )
+          DOUBLE PRECISION          DDOT
+          ENTRY                     DDOT   ( N,      X,INCX,Y,INCY )
+          INTEGER                            N,        INCX,  INCY
+          DOUBLE PRECISION                           X(*),  Y(*)
+
+          COMPLEX(KIND(1.0D0)) FUNCTION F06GAF ( N,      X,INCX,Y,INCY )
+          COMPLEX(KIND(1.0D0))          ZDOTU
+          ENTRY                         ZDOTU  ( N,      X,INCX,Y,INCY )
+          INTEGER                                N,        INCX,  INCY
+          COMPLEX(KIND(1.0D0))                           X(*),  Y(*)
+
+          COMPLEX(KIND(1.0D0)) FUNCTION F06GBF ( N,      X,INCX,Y,INCY )
+          COMPLEX(KIND(1.0D0))          ZDOTC
+          ENTRY                         ZDOTC  ( N,      X,INCX,Y,INCY )
+          INTEGER                                N,        INCX,  INCY
+          COMPLEX(KIND(1.0D0))                           X(*),  Y(*)
+
+          SUBROUTINE                F06ECF ( N,ALPHA,X,INCX,Y,INCY )
+          ENTRY                     DAXPY  ( N,ALPHA,X,INCX,Y,INCY )
+          INTEGER                            N,        INCX,  INCY
+          DOUBLE PRECISION                     ALPHA,X(*),  Y(*)
+
+          SUBROUTINE                F06GCF ( N,ALPHA,X,INCX,Y,INCY )
+          ENTRY                     ZAXPY  ( N,ALPHA,X,INCX,Y,INCY )
+          INTEGER                            N,        INCX,  INCY
+          COMPLEX(KIND(1.0D0))                 ALPHA,X(*),  Y(*)
+
+          SUBROUTINE                F06EDF ( N,ALPHA,X,INCX )
+          ENTRY                     DSCAL  ( N,ALPHA,X,INCX )
+          INTEGER                            N,        INCX
+          DOUBLE PRECISION                     ALPHA,X(*)
+
+          SUBROUTINE                F06GDF ( N,ALPHA,X,INCX )
+          ENTRY                     ZSCAL  ( N,ALPHA,X,INCX )
+          INTEGER                            N,        INCX
+          COMPLEX(KIND(1.0D0))                 ALPHA,X(*)
+
+          SUBROUTINE                F06JDF ( N,ALPHA,X,INCX )
+          ENTRY                     ZDSCAL ( N,ALPHA,X,INCX )
+          INTEGER                            N,        INCX
+          DOUBLE PRECISION          ALPHA
+          COMPLEX(KIND(1.0D0))                       X(*)
+
+          SUBROUTINE                F06EFF ( N,      X,INCX,Y,INCY )
+          ENTRY                     DCOPY  ( N,      X,INCX,Y,INCY )
+          INTEGER                            N,        INCX,  INCY
+          DOUBLE PRECISION                           X(*),  Y(*)
+
+          SUBROUTINE                F06GFF ( N,      X,INCX,Y,INCY )
+          ENTRY                     ZCOPY  ( N,      X,INCX,Y,INCY )
+          INTEGER                            N,        INCX,  INCY
+          COMPLEX(KIND(1.0D0))                       X(*),  Y(*)
+
+          SUBROUTINE                F06EGF ( N,      X,INCX,Y,INCY )
+          ENTRY                     DSWAP  ( N,      X,INCX,Y,INCY )
+          INTEGER                            N,        INCX,  INCY
+          DOUBLE PRECISION                           X(*),  Y(*)
+
+          SUBROUTINE                F06GGF ( N,      X,INCX,Y,INCY )
+          ENTRY                     ZSWAP  ( N,      X,INCX,Y,INCY )
+          INTEGER                            N,        INCX,  INCY
+          COMPLEX(KIND(1.0D0))                       X(*),  Y(*)
+
+          DOUBLE PRECISION FUNCTION F06EJF ( N,      X,INCX )
+          DOUBLE PRECISION          DNRM2
+          ENTRY                     DNRM2  ( N,      X,INCX )
+          INTEGER                            N,        INCX
+          DOUBLE PRECISION                           X(*)
+
+          DOUBLE PRECISION FUNCTION F06JJF ( N,      X,INCX )
+          DOUBLE PRECISION          DZNRM2
+          ENTRY                     DZNRM2 ( N,      X,INCX )
+          INTEGER                            N,        INCX
+          COMPLEX(KIND(1.0D0))                       X(*)
+
+          DOUBLE PRECISION FUNCTION F06EKF ( N,      X,INCX )
+          DOUBLE PRECISION          DASUM
+          ENTRY                     DASUM  ( N,      X,INCX )
+          INTEGER                            N,        INCX
+          DOUBLE PRECISION                           X(*)
+
+          DOUBLE PRECISION FUNCTION F06JKF ( N,      X,INCX )
+          DOUBLE PRECISION          DZASUM
+          ENTRY                     DZASUM ( N,      X,INCX )
+          INTEGER                            N,        INCX
+          COMPLEX(KIND(1.0D0))                       X(*)
+
+          INTEGER FUNCTION          F06JLF ( N,      X,INCX )
+          INTEGER                   IDAMAX
+          ENTRY                     IDAMAX ( N,      X,INCX )
+          INTEGER                            N,        INCX
+          DOUBLE PRECISION                            X(*)
+
+          INTEGER FUNCTION          F06JMF ( N,      X,INCX )
+          INTEGER                   IZAMAX
+          ENTRY                     IZAMAX ( N,      X,INCX )
+          INTEGER                            N,        INCX
+          COMPLEX(KIND(1.0D0))                       X(*)
+
+          SUBROUTINE                F06EPF ( N,      X,INCX,Y,INCY,C,S )
+          ENTRY                     DROT   ( N,      X,INCX,Y,INCY,C,S )
+          INTEGER                            N,        INCX,  INCY
+          DOUBLE PRECISION                           X(*),  Y(*),  C,S
+
+          F06EAF(*) and F06GAF(*)
+
+                                  T
+          return the dot product x y.
+
+          F06GBF(*)
+
+                                   H          H
+          returns the dot product x y, where x  denotes the complex
+                        T
+          conjugate of x .
+
+          F06ECF(*) and F06GCF(*)
+
+          perform the operation y<-(alpha)x+y, often called an axpy
+          operation.
+
+          F06EDF(*), F06GDF(*) and F06JDF(*)
+
+          perform the operation x<-(alpha)x.
+
+          F06EFF(*) and F06GFF(*)
+
+          perform the operation y<-x.
+
+          F06EGF(*) and F06GGF(*)
+
+          perform the operation x<=>y, that is x and y are swapped.
+
+          F06EJF(*) and F06JJF(*)
+
+                                                     ( n       )1/2
+                                                     ( --     2)
+          return the value ||x||  defined by ||x|| = ( >  |x | )   .
+                                2                 2  ( --   i  )
+                                                     ( i=1     )
+
+          F06EKF(*)
+
+                                                      n
+                                                      --
+          returns the value ||x||  defined by ||x|| = >  |x |.
+                                 1                 1  --   i
+                                                      i=1
+
+          F06JKF(*)
+
+                                                  n
+                                                  --
+          returns the value asum defined by asum= >  (|(Re(x )|+|Im(x )|).
+                                                  --        i        i
+                                                  i=1
+
+          F06JLF(*)
+
+          returns the first index j such that |x |=max |x |.
+                                                j     i  i
+
+          F06JMF(*)
+
+          returns the first index j such that
+          |Re(x )|+|Im(x )|=max (|(Re(x )|+|Im(x )|).
+               j        j      i       i        i
+
+          F06EPF(*)
+
+                                      ( T)          ( T)
+                                      (x )          (x )
+                                      ( T)    ( c s)( T)
+          performs the plane rotation (y ) <- (-s c)(y ).
+
+          4.3. The Level-2 Matrix-vector Routines
+
+          The matrix-vector routines all have one array argument
+          representing a matrix; usually this is a two-dimensional array
+          but in some cases the matrix is represented by a one-dimensional
+          array.
+
+          The size of the matrix is determined by the arguments M and N for
+          an m by n rectangular matrix; and by the argument N for an n by n
+          symmetric, Hermitian, or triangular matrix. Note that it is
+          permissible to call the routines with M or N = 0, in which case
+          the routines exit immediately without referencing their array
+          arguments. For band matrices, the bandwidth is determined by the
+          arguments KL and KU for a rectangular matrix with kl sub-
+          diagonals and ku super-diagonals; and by the argument K for a
+          symmetric, Hermitian, or triangular matrix with k sub-diagonals
+          and/or super-diagonals.
+
+          The description of the matrix consists either of the array name
+          (A) followed by the first dimension of the array as declared in
+          the calling (sub)program (LDA), when the matrix is being stored
+          in a two-dimensional array; or the array name (AP) alone when the
+          matrix is being stored as a (packed) vector. In the former case
+          the actual array must contain at least ((n-1)d+l) elements, where
+          d is the first dimension of the array, d>=l , and l=m for arrays
+          representing general matrices, l=n for arrays representing
+          symmetric, Hermitian and triangular matrices, l=kl+ku+1 for
+          arrays representing general band matrices and l=k+1 for
+          symmetric, Hermitian and triangular band matrices. For one-
+          dimensional arrays representing matrices (packed storage) the
+                                             1
+          actual array must contain at least -n(n+1) elements.
+                                             2
+
+          As with the vector routines, vectors are represented by one-
+          dimensional arrays together with a corresponding increment
+          argument (see Section 4.2). The only difference is that for these
+          routines a zero increment is not permitted.
+
+          When the vector x consists of k elements then the declared length
+          of the array X in the calling (sub)program must be at least
+          (1+(k-1)|INCX|).
+
+          The arguments that specify options are character arguments with
+          the names TRANS, UPLO and DIAG. TRANS is used by the matrix-
+          vector product routines as follows:
+
+              Value      Meaning
+
+              'N' or 'n' Operate with the matrix
+
+              'T' or 't' Operate with the transpose of the matrix
+
+              'C' or 'c' Operate with the conjugate transpose of the matrix
+
+          In the real case the values 'T', 't', 'C' and 'c' have the same
+          meaning.
+
+          UPLO is used by the Hermitian, symmetric, and triangular matrix
+          routines to specify whether the upper or lower triangle is being
+          referenced as follows:
+
+
+              Value      Meaning
+
+              'U' or 'u' Upper triangle
+
+              'L' or 'l' Lower triangle
+
+          DIAG is used by the triangular matrix routines to specify whether
+          or not the matrix is unit triangular, as follows:
+
+              Value      Meaning
+
+              'U' or 'u' Unit triangular
+
+              'N' or 'n' Non-unit triangular
+
+          When DIAG is supplied as 'U' or 'u' the diagonal elements are not
+          referenced.
+
+          It is worth noting that actual character arguments in Fortran may
+          be longer than the corresponding dummy arguments. So that, for
+          example, the value 'T' for TRANS may be passed as 'TRANSPOSE'.
+
+          The routines for real symmetric and complex Hermitian matrices
+          allow for the matrix to be stored in either the upper (UPLO = 'U
+          to be packed in a one-dimensional array. In the latter case the
+          upper triangle may be packed sequentially column by column (UPLO
+          = 'U'), or the lower triangle may be packed sequentially column
+          by column (UPLO = 'L'). Note that for real symmetric matrices
+          packing the upper triangle by column is equivalent to packing the
+          lower triangle by rows, and packing the lower triangle by columns
+          is equivalent to packing the upper triangle by rows. (For complex
+          Hermitian matrices the only difference is that the off-diagonal
+          elements are conjugated.)
+
+          For triangular matrices the argument UPLO serves to define
+          whether the matrix is upper (UPLO = 'U') or lower (UPLO = 'L')
+          triangular. In packed storage the triangle has to be packed by
+          column.
+
+          The band matrix routines allow storage so that the jth column of
+          the matrix is stored in the jth column of the Fortran array. For
+          a general band matrix the diagonal of the matrix is stored in the
+          (ku+1)th row of the array. For a Hermitian or symmetric matrix
+          either the upper triangle (UPLO = 'U') may be stored in which
+          case the leading diagonal is in the (k+1)th row of the array, or
+          the lower triangle (UPLO = 'L') may be stored in which case the
+          leading diagonal is in the first row of the array. For an upper
+          triangular band matrix (UPLO = 'U') the leading diagonal is in
+          the (k+1)th row of the array and for a lower triangular band
+          matrix (UPLO = 'L') the leading diagonal is in the first row.
+
+          For a Hermitian matrix the imaginary parts of the diagonal
+          elements are of course zero and thus the imaginary parts of the
+          corresponding Fortran array elements need not be set, but are
+          assumed to be zero.
+
+          For packed triangular matrices the same storage layout is used
+          whether or not DIAG = 'U', i.e., space is left for the diagonal
+          elements even if those array elements are not referenced.
+
+                                             H
+          Throughout the following sections A  denotes the complex
+                        T
+          conjugate of A .
+
+
+          SUBROUTINE F06PAF( TRANS,M,N,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          ENTRY DGEMV      ( TRANS,M,N,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          CHARACTER*1        TRANS
+          INTEGER            M,N,LDA,INCX,INCY
+          DOUBLE PRECISION   ALPHA,A(LDA,*),X(*),BETA,Y(*)
+
+          SUBROUTINE F06SAF( TRANS,M,N,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          ENTRY ZGEMV      ( TRANS,M,N,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          CHARACTER*1          TRANS
+          INTEGER              M,N,LDA,INCX,INCY
+          COMPLEX(KIND(1.0D0)) ALPHA,A(LDA,*),X(*),BETA,Y(*)
+
+          SUBROUTINE F06PBF( TRANS,M,N,KL,KU,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          ENTRY DGBMV      ( TRANS,M,N,KL,KU,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          CHARACTER*1        TRANS
+          INTEGER            M,N,KL,KU,LDA,INCX,INCY
+          DOUBLE PRECISION   ALPHA,A(LDA,*),X(*),BETA,Y(*)
+
+          SUBROUTINE F06SBF( TRANS,M,N,KL,KU,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          ENTRY ZGBMV      ( TRANS,M,N,KL,KU,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          CHARACTER*1          TRANS
+          INTEGER              M,N,KL,KU,LDA,INCX,INCY
+          COMPLEX(KIND(1.0D0)) ALPHA,A(LDA,*),X(*),BETA,Y(*)
+
+          SUBROUTINE F06PCF( UPLO,N,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          ENTRY DSYMV      ( UPLO,N,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          CHARACTER*1        UPLO
+          INTEGER            N,LDA,INCX,INCY
+          DOUBLE PRECISION   ALPHA,A(LDA,*),X(*),BETA,Y(*)
+
+          SUBROUTINE F06SCF( UPLO,N,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          ENTRY ZHEMV      ( UPLO,N,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          CHARACTER*1          UPLO
+          INTEGER              N,LDA,INCX,INCY
+          COMPLEX(KIND(1.0D0)) ALPHA,A(LDA,*),X(*),BETA,Y(*)
+
+          SUBROUTINE F06PDF( UPLO,N,K,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          ENTRY DSBMV      ( UPLO,N,K,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          CHARACTER*1        UPLO
+          INTEGER            N,K,LDA,INCX,INCY
+          DOUBLE PRECISION   ALPHA,A(LDA,*),X(*),BETA,Y(*)
+
+          SUBROUTINE F06SDF( UPLO,N,K,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          ENTRY ZHBMV      ( UPLO,N,K,ALPHA,A,LDA,X,INCX,BETA,Y,INCY )
+          CHARACTER*1          UPLO
+          INTEGER              N,K,LDA,INCX,INCY
+          COMPLEX(KIND(1.0D0)) ALPHA,A(LDA,*),X(*),BETA,Y(*)
+
+          SUBROUTINE F06PEF( UPLO,N,ALPHA,AP,X,INCX,BETA,Y,INCY )
+          ENTRY DSPMV      ( UPLO,N,ALPHA,AP,X,INCX,BETA,Y,INCY )
+          CHARACTER*1        UPLO
+          INTEGER            N,INCX,INCY
+          DOUBLE PRECISION   ALPHA,AP(*),X(*),BETA,Y(*)
+
+          SUBROUTINE F06SEF( UPLO,N,ALPHA,AP,X,INCX,BETA,Y,INCY )
+          ENTRY ZHPMV      ( UPLO,N,ALPHA,AP,X,INCX,BETA,Y,INCY )
+          CHARACTER*1          UPLO
+          INTEGER              N,INCX,INCY
+          COMPLEX(KIND(1.0D0)) ALPHA,AP(*),X(*),BETA,Y(*)
+
+          SUBROUTINE F06PFF( UPLO,TRANS,DIAG,N,A,LDA,X,INCX )
+          ENTRY DTRMV      ( UPLO,TRANS,DIAG,N,A,LDA,X,INCX )
+          CHARACTER*1        UPLO,TRANS,DIAG
+          INTEGER            N,LDA,INCX
+          DOUBLE PRECISION   A(LDA,*),X(*)
+
+          SUBROUTINE F06SFF( UPLO,TRANS,DIAG,N,A,LDA,X,INCX )
+          ENTRY ZTRMV      ( UPLO,TRANS,DIAG,N,A,LDA,X,INCX )
+          CHARACTER*1          UPLO,TRANS,DIAG
+          INTEGER              N,LDA,INCX
+          COMPLEX(KIND(1.0D0)) A(LDA,*),X(*)
+
+          SUBROUTINE F06PGF( UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX )
+          ENTRY DTBMV      ( UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX )
+          CHARACTER*1        UPLO,TRANS,DIAG
+          INTEGER            N,K,LDA,INCX
+          DOUBLE PRECISION   A(LDA,*),X(*)
+
+          SUBROUTINE F06SGF( UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX )
+          ENTRY ZTBMV      ( UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX )
+          CHARACTER*1          UPLO,TRANS,DIAG
+          INTEGER              N,K,LDA,INCX
+          COMPLEX(KIND(1.0D0)) A(LDA,*),X(*)
+
+          SUBROUTINE F06PHF( UPLO,TRANS,DIAG,N,AP,X,INCX )
+          ENTRY DTPMV      ( UPLO,TRANS,DIAG,N,AP,X,INCX )
+          CHARACTER*1        UPLO,TRANS,DIAG
+          INTEGER            N,INCX
+          DOUBLE PRECISION   AP(*),X(*)
+
+          SUBROUTINE F06SHF( UPLO,TRANS,DIAG,N,AP,X,INCX )
+          ENTRY ZTPMV      ( UPLO,TRANS,DIAG,N,AP,X,INCX )
+          CHARACTER*1          UPLO,TRANS,DIAG
+          INTEGER              N,INCX
+          COMPLEX(KIND(1.0D0)) AP(*),X(*)
+
+          SUBROUTINE F06PJF( UPLO,TRANS,DIAG,N,A,LDA,X,INCX )
+          ENTRY DTRSV      ( UPLO,TRANS,DIAG,N,A,LDA,X,INCX )
+          CHARACTER*1        UPLO,TRANS,DIAG
+          INTEGER            N,LDA,INCX
+          DOUBLE PRECISION   A(LDA,*),X(*)
+
+          SUBROUTINE F06SJF( UPLO,TRANS,DIAG,N,A,LDA,X,INCX )
+          ENTRY ZTRSV      ( UPLO,TRANS,DIAG,N,A,LDA,X,INCX )
+          CHARACTER*1          UPLO,TRANS,DIAG
+          INTEGER              N,LDA,INCX
+          COMPLEX(KIND(1.0D0)) A(LDA,*),X(*)
+
+          SUBROUTINE F06PKF( UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX )
+          ENTRY DTBSV      ( UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX )
+          CHARACTER*1        UPLO,TRANS,DIAG
+          INTEGER            N,K,LDA,INCX
+          DOUBLE PRECISION   A(LDA,*),X(*)
+
+          SUBROUTINE F06SKF( UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX )
+          ENTRY ZTBSV      ( UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX )
+          CHARACTER*1          UPLO,TRANS,DIAG
+          INTEGER              N,K,LDA,INCX
+          COMPLEX(KIND(1.0D0)) A(LDA,*),X(*)
+
+          SUBROUTINE F06PLF( UPLO,TRANS,DIAG,N,AP,X,INCX )
+          ENTRY DTPSV      ( UPLO,TRANS,DIAG,N,AP,X,INCX )
+          CHARACTER*1        UPLO,TRANS,DIAG
+          INTEGER            N,INCX
+          DOUBLE PRECISION   AP(*),X(*)
+
+          SUBROUTINE F06SLF( UPLO,TRANS,DIAG,N,AP,X,INCX )
+          ENTRY ZTPSV      ( UPLO,TRANS,DIAG,N,AP,X,INCX )
+          CHARACTER*1          UPLO,TRANS,DIAG
+          INTEGER              N,INCX
+          COMPLEX(KIND(1.0D0)) AP(*),X(*)
+
+          SUBROUTINE F06PMF( M,N,ALPHA,X,INCX,Y,INCY,A,LDA )
+          ENTRY DGER       ( M,N,ALPHA,X,INCX,Y,INCY,A,LDA )
+          INTEGER            M,N,INCX,INCY,LDA
+          DOUBLE PRECISION   ALPHA,X(*),Y(*),A(LDA,*)
+
+          SUBROUTINE F06SMF( M,N,ALPHA,X,INCX,Y,INCY,A,LDA )
+          ENTRY ZGERU      ( M,N,ALPHA,X,INCX,Y,INCY,A,LDA )
+          INTEGER              M,N,INCX,INCY,LDA
+          COMPLEX(KIND(1.0D0)) ALPHA,X(*),Y(*),A(LDA,*)
+
+          SUBROUTINE F06SNF( M,N,ALPHA,X,INCX,Y,INCY,A,LDA )
+          ENTRY ZGERC      ( M,N,ALPHA,X,INCX,Y,INCY,A,LDA )
+          INTEGER              M,N,INCX,INCY,LDA
+          COMPLEX(KIND(1.0D0)) ALPHA,X(*),Y(*),A(LDA,*)
+
+          SUBROUTINE F06PPF( UPLO,N,ALPHA,X,INCX,A,LDA )
+          ENTRY DSYR       ( UPLO,N,ALPHA,X,INCX,A,LDA )
+          CHARACTER*1        UPLO
+          INTEGER            N,INCX,LDA
+          DOUBLE PRECISION   ALPHA,X(*),A(LDA,*)
+
+          SUBROUTINE F06SPF( UPLO,N,ALPHA,X,INCX,A,LDA )
+          ENTRY ZHER       ( UPLO,N,ALPHA,X,INCX,A,LDA )
+          CHARACTER*1          UPLO
+          INTEGER              N,INCX,LDA
+          DOUBLE PRECISION     ALPHA
+          COMPLEX(KIND(1.0D0)) X(*),A(LDA,*)
+
+          SUBROUTINE F06PQF( UPLO,N,ALPHA,X,INCX,AP )
+          ENTRY DSPR       ( UPLO,N,ALPHA,X,INCX,AP )
+          CHARACTER*1        UPLO
+          INTEGER                   N,INCX
+          DOUBLE PRECISION            ALPHA,X(*),AP(*)
+
+          SUBROUTINE F06SQF( UPLO,N,ALPHA,X,INCX,AP )
+          ENTRY ZHPR       ( UPLO,N,ALPHA,X,INCX,AP )
+          CHARACTER*1          UPLO
+          INTEGER              N,INCX
+          DOUBLE PRECISION     ALPHA
+          COMPLEX(KIND(1.0D0)) X(*),AP(*)
+
+          SUBROUTINE F06PRF( UPLO,N,ALPHA,X,INCX,Y,INCY,A,LDA )
+          ENTRY DSYR2      ( UPLO,N,ALPHA,X,INCX,Y,INCY,A,LDA )
+          CHARACTER*1        UPLO
+          INTEGER            N,INCX,INCY,LDA
+          DOUBLE PRECISION   ALPHA,X(*),Y(*),A(LDA,*)
+
+          SUBROUTINE F06SRF( UPLO,N,ALPHA,X,INCX,Y,INCY,A,LDA )
+          ENTRY ZHER2      ( UPLO,N,ALPHA,X,INCX,Y,INCY,A,LDA )
+          CHARACTER*1          UPLO
+          INTEGER              N,INCX,INCY,LDA
+          COMPLEX(KIND(1.0D0)) ALPHA,X(*),Y(*),A(LDA,*)
+
+          SUBROUTINE F06PSF( UPLO,N,ALPHA,X,INCX,Y,INCY,AP )
+          ENTRY DSPR2      ( UPLO,N,ALPHA,X,INCX,Y,INCY,AP )
+          CHARACTER*1        UPLO
+          INTEGER            N,INCX,INCY
+          DOUBLE PRECISION  ALPHA,X(*),Y(*),AP(*)
+
+          SUBROUTINE F06SSF( UPLO,N,ALPHA,X,INCX,Y,INCY,AP )
+          ENTRY ZHPR2      ( UPLO,N,ALPHA,X,INCX,Y,INCY,AP )
+          CHARACTER*1          UPLO
+          INTEGER              N,INCX,INCY
+          COMPLEX(KIND(1.0D0)) ALPHA,X(*),Y(*),AP(*)
+
+          F06PAF(*), F06SAF(*), F06PBF(*) and F06SBF(*)
+
+          perform the operation
+
+              y<-(alpha)Ax+(beta)y, when TRANS = 'N',
+
+                         T
+              y<-(alpha)A x+(beta)y, when TRANS = 'T',
+
+                         H
+              y<-(alpha)A x+(beta)y, when TRANS = 'C',
+
+          where A is a general matrix for F06PAF(*) and F06SAF(*), and is a
+          general band matrix for F06PBF(*) and F06SBF(*).
+
+          F06PCF(*), F06SCF(*), F06PEF(*), F06SEF(*), F06PDF(*) and
+          F06SDF(*)
+
+          perform the operation
+
+          y<-(alpha)Ax+(beta)y
+
+          where A is symmetric and Hermitian for F06PCF(*) and F06SCF(*)
+          respectively, is symmetric and Hermitian stored in packed form
+          for F06PEF(*) and F06SEF(*) respectively, and is symmetric and
+          Hermitian band for F06PDF(*) and F06SDF(*).
+
+          F06PFF(*), F06SFF(*), F06PHF(*), F06SHF(*), F06PGF(*) and
+          F06SGF(*)
+
+          perform the operation
+
+              x<-Ax,       when TRANS = 'N',
+
+                  T
+              x<-A x,      when TRANS = 'T',
+
+                  H
+              x<-A x,      when TRANS = 'C',
+
+          where A is a triangular matrix for F06PFF(*) and F06SFF(*), is a
+          triangular matrix stored in packed form for F06PHF(*) and
+          F06SHF(*), and is a triangular band matrix for F06PGF(*) and
+          F06SGF(*).
+
+          F06PJF(*), F06SJF(*), F06PLF(*), F06SLF(*), F06PKF(*) and
+          F06SKF(*)
+
+          solve the equations
+
+              Ax=b,        when TRANS = 'N',
+
+               T
+              A x=b,       when TRANS = 'T',
+
+               H
+              A x=b,       when TRANS = 'C',
+
+          where A is a triangular matrix for F06PJF(*) and F06SJF(*), is a
+          triangular matrix stored in packed form for F06PLF(*) and
+          F06SLF(*), and is a triangular band matrix for F06PKF(*) and
+          F06SKF(*). The vector b must be supplied in the array X and is
+          overwritten by the solution. It is important to note that no test
+          for singularity is included in these routines.
+
+          F06PMF(*) and F06SMF(*)
+
+                                            T
+          perform the operation A<-(alpha)xy +A, where A is a general
+          matrix.
+
+          F06SNF(*)
+
+                                             H
+          performs the operation A<-(alpha)xy +A, where A is a general
+          complex matrix.
+
+          F06PPF(*) and F06PQF(*)
+
+                                            T
+          perform the operation A<-(alpha)xx +A, where A is a symmetric
+          matrix for F06PPF(*) and is a symmetric matrix stored in packed
+          form for F06PQF(*).
+
+          F06SPF(*) and F06SQF(*)
+
+                                            H
+          perform the operation A<-(alpha)xx +A, where A is an Hermitian
+          matrix for F06SPF(*) and is an Hermitian matrix stored in packed
+          form for F06SQF(*).
+
+          F06PRF(*) and F06PSF(*)
+
+                                            T          T
+          perform the operation A<-(alpha)xy +(alpha)yx +A, where A is a
+          symmetric matrix for F06PRF(*) and is a symmetric matrix stored
+          in packed form for F06PSF(*).
+
+          F06SRF(*) and F06SSF(*)
+
+                                            H          H
+          perform the operation A<-(alpha)xy +(alpha)yx +A, where A is an
+          Hermitian matrix for F06SRF(*) and is an Hermitian matrix stored
+          in packed form for F06SSF(*).
+
+          The following argument values are invalid:
+
+               Any value of the character arguments DIAG, TRANS, or UPLO
+               whose meaning is not specified.
+
+               M < 0
+
+               N < 0
+
+               KL < 0
+
+               KU < 0
+
+               K < 0
+
+               LDA < M
+
+               LDA < KL + KU + 1
+
+               LDA < N for the routines involving full Hermitian,
+               symmetric or triangular matrices
+
+               LDA < K + 1 for the routines involving band Hermitian,
+               symmetric or triangular matrices
+
+               INCX = 0
+
+               INCY = 0
+
+          If a routine is called with an invalid value then an error
+          message is output, on the error message unit (see X04AAF), giving
+          the name of the routine and the number of the first invalid
+          argument, and execution is terminated.
+
+          4.4. The Level-3 Matrix-matrix Routines
+
+          The matrix-matrix routines all have either two or three arguments
+          representing a matrix, one of which is an input-output argument,
+          and in each case the arguments are two-dimensional arrays.
+
+          The sizes of the matrices are determined by one or more of the
+          arguments M, N and K. The size of the input-output array is
+          always determined by the arguments M and N for a rectangular m by
+          n matrix, and by the argument N for a square n by n matrix. It is
+          permissible to call the routines with M or N = 0, in which case
+          the routines exit immediately without referencing their array
+          arguments.
+
+          Many of the routines perform an operation of the form
+
+                                    C<-P+(beta)C,
+
+          where P is the product of two matrices, or the sum of two such
+          products. When the inner dimension of the matrix product is
+          different from m or n it is denoted by K. Again it is permissible
+          to call the routines with K = 0 and if M > 0, but K = 0, then the
+          routines perform the operation
+
+
+                                     C<-(beta)C.
+
+          As with the Level-2 routines (see Section 4.3) the description of
+          the matrix consists of the array name (A or B or C) followed by
+          the first dimension (LDA or LDB or LDC).
+
+          The arguments that specify options are character arguments with
+          the names SIDE, TRANSA, TRANSB, TRANS, UPLO and DIAG. UPLO and
+          DIAG have the same values and meanings as for the Level-2
+          routines (see Section 4.3); TRANSA, TRANSB and TRANS have the
+          same values and meanings as TRANS in the Level-2 routines, where
+          TRANSA and TRANSB apply to the matrices A and B respectively.
+          SIDE is used by the routines as follows:
+
+              Value   Meaning
+
+              'L'     Multiply general matrix by symmetric, Hermitian or
+                      triangular matrix on the left
+
+              'R'     Multiply general matrix by symmetric, Hermitian or
+                      triangular matrix on the right
+
+          The storage conventions for matrices are as for the Level-2
+          routines (see Section 4.3).
+
+          SUBROUTINE F06YAF( TRANSA,TRANSB,M,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          ENTRY      DGEMM ( TRANSA,TRANSB,M,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          CHARACTER          TRANSA,TRANSB
+          INTEGER            M,N,K,LDA,LDB,LDC
+          DOUBLE PRECISION   ALPHA,A(LDA,*),B(LDB,*),BETA,C(LDC,*)
+
+          SUBROUTINE F06ZAF( TRANSA,TRANSB,M,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          ENTRY      ZGEMM ( TRANSA,TRANSB,M,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          CHARACTER            TRANSA,TRANSB
+          INTEGER              M,N,K,LDA,LDB,LDC
+          COMPLEX(KIND(1.0D0)) ALPHA,A(LDA,*),B(LDB,*),BETA,C(LDC,*)
+
+          SUBROUTINE F06YCF( SIDE,UPLO,M,N,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          ENTRY      DSYMM ( SIDE,UPLO,M,N,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          CHARACTER          SIDE,UPLO
+          INTEGER            M,N,LDA,LDB,LDC
+          DOUBLE PRECISION   ALPHA,A(LDA,*),B(LDB,*),BETA,C(LDC,*)
+
+          SUBROUTINE F06ZCF( SIDE,UPLO,M,N,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          ENTRY      ZHEMM ( SIDE,UPLO,M,N,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          CHARACTER            SIDE,UPLO
+          INTEGER              M,N,LDA,LDB,LDC
+          COMPLEX(KIND(1.0D0)) ALPHA,A(LDA,*),B(LDB,*),BETA,C(LDC,*)
+
+          SUBROUTINE F06ZTF( SIDE,UPLO,M,N,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          ENTRY      ZSYMM ( SIDE,UPLO,M,N,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          CHARACTER            SIDE,UPLO
+          INTEGER              M,N,LDA,LDB,LDC
+          COMPLEX(KIND(1.0D0)) ALPHA,A(LDA,*),B(LDB,*),BETA,C(LDC,*)
+
+          SUBROUTINE F06YFF( SIDE,UPLO,TRANSA,DIAG,M,N,ALPHA,A,LDA,B,LDB )
+          ENTRY      DTRMM ( SIDE,UPLO,TRANSA,DIAG,M,N,ALPHA,A,LDA,B,LDB )
+          CHARACTER          SIDE,UPLO,TRANSA,DIAG
+          INTEGER            M,N,LDA,LDB
+          DOUBLE PRECISION   ALPHA,A(LDA,*),B(LDB,*)
+
+          SUBROUTINE F06ZFF( SIDE,UPLO,TRANSA,DIAG,M,N,ALPHA,A,LDA,B,LDB )
+          ENTRY      ZTRMM ( SIDE,UPLO,TRANSA,DIAG,M,N,ALPHA,A,LDA,B,LDB )
+          CHARACTER            SIDE,UPLO,TRANSA,DIAG
+          INTEGER              M,N,LDA,LDB
+          COMPLEX(KIND(1.0D0)) ALPHA,A(LDA,*),B(LDB,*)
+
+          SUBROUTINE F06YJF( SIDE,UPLO,TRANSA,DIAG,M,N,ALPHA,A,LDA,B,LDB )
+          ENTRY      DTRSM ( SIDE,UPLO,TRANSA,DIAG,M,N,ALPHA,A,LDA,B,LDB )
+          CHARACTER          SIDE,UPLO,TRANSA,DIAG
+          INTEGER            M,N,LDA,LDB
+          DOUBLE PRECISION   ALPHA,A(LDA,*),B(LDB,*)
+
+          SUBROUTINE F06ZJF( SIDE,UPLO,TRANSA,DIAG,M,N,ALPHA,A,LDA,B,LDB )
+          ENTRY      ZTRSM ( SIDE,UPLO,TRANSA,DIAG,M,N,ALPHA,A,LDA,B,LDB )
+          CHARACTER            SIDE,UPLO,TRANSA,DIAG
+          INTEGER              M,N,LDA,LDB
+          COMPLEX(KIND(1.0D0)) ALPHA,A(LDA,*),B(LDB,*)
+
+          SUBROUTINE F06YPF( UPLO,TRANS,N,K,ALPHA,A,LDA,BETA,C,LDC )
+          ENTRY      DSYRK ( UPLO,TRANS,N,K,ALPHA,A,LDA,BETA,C,LDC )
+          CHARACTER          UPLO,TRANS
+          INTEGER            N,K,LDA,LDC
+          DOUBLE PRECISION   ALPHA,A(LDA,*),BETA,C(LDC,*)
+
+          SUBROUTINE F06ZPF( UPLO,TRANS,N,K,ALPHA,A,LDA,BETA,C,LDC )
+          ENTRY      ZHERK ( UPLO,TRANS,N,K,ALPHA,A,LDA,BETA,C,LDC )
+          CHARACTER            UPLO,TRANS
+          INTEGER              N,K,LDA,LDC
+          DOUBLE PRECISION     ALPHA,BETA
+          COMPLEX(KIND(1.0D0)) A(LDA,*),C(LDC,*)
+
+          SUBROUTINE F06ZUF( UPLO,TRANS,N,K,ALPHA,A,LDA,BETA,C,LDC )
+          ENTRY      ZSYRK ( UPLO,TRANS,N,K,ALPHA,A,LDA,BETA,C,LDC )
+          CHARACTER            UPLO,TRANS
+          INTEGER              N,K,LDA,LDC
+          COMPLEX(KIND(1.0D0)) ALPHA,A(LDA,*),BETA,C(LDC,*)
+
+          SUBROUTINE F06YRF( UPLO,TRANS,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          ENTRY      DSYR2K( UPLO,TRANS,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          CHARACTER          UPLO,TRANS
+          INTEGER            N,K,LDA,LDB,LDC
+          DOUBLE PRECISION   ALPHA,A(LDA,*),B(LDB,*),BETA,C(LDC,*)
+
+          SUBROUTINE F06ZRF( UPLO,TRANS,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          ENTRY      ZHER2K( UPLO,TRANS,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          CHARACTER            UPLO,TRANS
+          INTEGER              N,K,LDA,LDB,LDC
+          DOUBLE PRECISION     BETA
+          COMPLEX(KIND(1.0D0)) ALPHA,A(LDA,*),B(LDB,*),C(LDC,*)
+
+          SUBROUTINE F06ZWF( UPLO,TRANS,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          ENTRY      ZSYR2K( UPLO,TRANS,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC )
+          CHARACTER            UPLO,TRANS
+          INTEGER              N,K,LDA,LDB,LDC
+          COMPLEX(KIND(1.0D0)) ALPHA,A(LDA,*),B(LDB,*),BETA,C(LDC,*)
+
+
+          F06YAF(*) and F06ZAF(*)
+
+          perform the operation indicated in the following table:
+
+                        TRANSA = 'N'    TRANSA = 'T'      TRANSA = 'C'
+
+                                                     T               H
+          TRANSB='N'   C <- (alpha)AB   C <- (alpha)A B   C <- (alpha)A B
+                            +(beta)C        +(beta)C          +(beta)C
+                       A is m*k,        A is k*m,         A is k*m,
+                       B is k*n         B is k*n          B is k*n
+
+                                     T               T T               H T
+          TRANSB='T'   C <- (alpha)AB   C <- (alpha)A B   C <- (alpha)A B
+                           +(beta)C         +(beta)C          +(beta)C
+                       A is m*k,        A is k*m,         A is k*m,
+                       B is n*k         B is n*k          B is n*k
+
+                                     H               T H               H H
+          TRANSB='C'   C <- (alpha)AB   C <- (alpha)A B   C <- (alpha)A B
+                           +(beta)C         +(beta)C          +(beta)C
+                       A is m*k,        A is k*m,         A is k*m,
+                       B is n*k         B is n*k          B is n*k
+
+
+          where A and B are general matrices and C is a general m by n
+          matrix.
+
+          F06YCF(*), F06ZCF(*) and F06ZTF(*) perform the operation
+
+          indicated in the following table:
+
+              SIDE = 'L'               SIDE = 'R'
+
+              C<-(alpha)AB+(beta)C     C<-(alpha)BA+(beta)C
+
+              A is m*m                 B is m*n
+
+              B is m*n                 A is n*n
+
+          where A is symmetric for F06YCF(*) and F06ZTF(*) and is Hermitian
+          for F06ZCF(*), B is a general matrix and C is a general m by n
+          matrix.
+
+          F06YFF(*) and F06ZFF(*) perform the operation indicated in the
+          following table:
+
+                    TRANSA = 'N'       TRANSA = 'T'        TRANSA = 'C'
+
+                                                    T                   H
+          SIDE='L'  B <- (alpha)AB     B <- (alpha)A B     B <- (alpha)A B
+
+                    A is               A is                A is
+                    triangular         triangular          triangular
+                    m*m                m*m                 m*m
+
+                                                     T                   H
+          SIDE='R'  B <- (alpha)BA     B <- (alpha)BA      B <- (alpha)BA
+
+                    A is               A is                A is
+                    triangular         triangular          triangular
+                    n*n                n*n                 n*n
+
+          where B is a general m by n matrix.
+
+          F06YJF(*) and F06ZJF(*) solve the equations, indicated in the
+          following table, for X:
+
+                    TRANSA = 'N'       TRANSA = 'T'       TRANSA = 'C'
+
+                                        T                  H
+          SIDE='L'  AX=(alpha)B        A X=(alpha)B       A X=(alpha)B
+
+                    A is               A is               A is
+                    triangular         triangular         triangular
+                    m*m                m*m                m*m
+
+                                         T                  H
+          SIDE='R'  XA=(alpha)B        XA =(alpha)B       XA =(alpha)B
+
+                    A is               A is               A is
+                    triangular         triangular         triangular
+                    n*n                n*n                n*n
+
+          where B is a general m by n matrix. The m by n solution matrix X
+          is overwritten on the array B. It is important to note that no
+          test for singularity is included in these routines.
+
+          F06YPF(*), F06ZPF(*) and F06ZUF(*) perform the operation
+          indicated in the following table:
+
+                    TRANS = 'N'        TRANS = 'T'        TRANS = 'C'
+
+                                  T                 T                  T
+          F06YPF    C <- (alpha)AA     C <- (alpha)A A    C <- (alpha)A A
+                        +(beta)C           +(beta)C           +(beta)C
+                                  T                 T
+          F06ZUF    C <- (alpha)AA     C <- (alpha)A A         --
+                        +(beta)C           +(beta)C
+                                  H                                    H
+          F06ZPF    C <- (alpha)AA          --            C <- (alpha)A A
+                        +(beta)C                              +(beta)C
+
+                    A is n*k           A is k*n           A is k*n
+
+          where A is a general matrix and C is an n by n symmetric matrix
+          for F06YPF(*) and F06ZUF(*), and is an n by n Hermitian matrix
+          for F06ZPF(*).
+
+          F06YRF(*), F06ZRF(*) and F06ZWF(*) perform the operation
+          indicated in the following table:
+
+
+                    TRANS = 'N'        TRANS = 'T'        TRANS = 'C'
+
+                                  T                 T                  T
+          F06YRF    C <- (alpha)AB     C <- (alpha)A B    C <- (alpha)A B
+                                  T                 T                  T
+                        +(alpha)BA         +(alpha)B A        +(alpha)B A
+                        +(beta)C           +(beta)C           +(beta)C
+                                  T                 T
+          F06ZWF    C <- (alpha)AB     C <- (alpha)A B           --
+                                  T                 T
+                        +(alpha)BA         +(alpha)B A
+                        +(beta)C           +(beta)C
+                                  H                                    H
+          F06ZRF    C <- (alpha)AB           --           C <- (alpha)A B
+                                  H                                    H
+                        +(alpha)BA                            +(alpha)B A
+                        +(beta)C                              +(beta)C
+
+                    A and B are n*k    A and B are k*n    A and B are k*n
+
+          where A and B are general matrices and C is an n by n symmetric
+          matrix for F06YRF(*) and F06ZWF(*), and is an n by n Hermitian
+          matrix for F06ZPF(*).
+
+          The following values of arguments are invalid:
+
+               Any value of the character arguments SIDE, TRANSA, TRANSB,
+               TRANS, UPLO or DIAG, whose meaning is not specified.
+
+               M < 0
+
+               N < 0
+
+               K < 0
+
+               LDA < the number of rows in the matrix A.
+
+               LDB < the number of rows in the matrix B.
+
+               LDC < the number of rows in the matrix C.
+
+          If a routine is called with an invalid value then an error
+          message is output, on the error message unit (see X04AAF), giving
+          the name of the routine and the number of the first invalid
+          argument, and execution is terminated.
+
+
+          F06 -- Linear Algebra Support Routines            Contents -- F06
+          Chapter F06
+
+          Linear Algebra Support Routines
+
+          F06AAF  (DROTG) Generate real plane rotation
+
+          F06EAF  (DDOT) Dot product of two real vectors
+
+          F06ECF  (DAXPY) Add scalar times real vector to real vector
+
+          F06EDF  (DSCAL) Multiply real vector by scalar
+
+          F06EFF  (DCOPY) Copy real vector
+
+          F06EGF  (DSWAP) Swap two real vectors
+
+          F06EJF  (DNRM2) Compute Euclidean norm of real vector
+
+          F06EKF  (DASUM) Sum the absolute values of real vector elements
+
+          F06EPF  (DROT) Apply real plane rotation
+
+          F06GAF  (ZDOTU) Dot product of two complex vectors, unconjugated
+
+          F06GBF  (ZDOTC) Dot product of two complex vectors, conjugated
+
+          F06GCF  (ZAXPY) Add scalar times complex vector to complex vector
+
+          F06GDF  (ZSCAL) Multiply complex vector by complex scalar
+
+          F06GFF  (ZCOPY) Copy complex vector
+
+          F06GGF  (ZSWAP) Swap two complex vectors
+
+          F06JDF  (ZDSCAL) Multiply complex vector by real scalar
+
+          F06JJF  (DZNRM2) Compute Euclidean norm of complex vector
+
+          F06JKF  (DZASUM) Sum the absolute values of complex vector
+                  elements
+
+          F06JLF  (IDAMAX) Index, real vector element with largest absolute
+                  value
+
+          F06JMF  (IZAMAX) Index, complex vector element with largest
+                  absolute value
+
+          F06PAF  (DGEMV) Matrix-vector product, real rectangular matrix
+
+          F06PBF  (DGBMV) Matrix-vector product, real rectangular band
+                  matrix
+
+          F06PCF  (DSYMV) Matrix-vector product, real symmetric matrix
+
+          F06PDF  (DSBMV) Matrix-vector product, real symmetric band matrix
+
+          F06PEF  (DSPMV) Matrix-vector product, real symmetric packed
+                  matrix
+
+          F06PFF  (DTRMV) Matrix-vector product, real triangular matrix
+
+          F06PGF  (DTBMV) Matrix-vector product, real triangular band
+                  matrix
+
+          F06PHF  (DTPMV) Matrix-vector product, real triangular packed
+                  matrix
+
+          F06PJF  (DTRSV) System of equations, real triangular matrix
+
+          F06PKF  (DTBSV) System of equations, real triangular band matrix
+
+          F06PLF  (DTPSV) System of equations, real triangular packed
+                  matrix
+
+          F06PMF  (DGER) Rank-1 update, real rectangular matrix
+
+          F06PPF  (DSYR) Rank-1 update, real symmetric matrix
+
+          F06PQF  (DSPR) Rank-1 update, real symmetric packed matrix
+
+          F06PRF  (DSYR2) Rank-2 update, real symmetric matrix
+
+          F06PSF  (DSPR2) Rank-2 update, real symmetric packed matrix
+
+          F06SAF  (ZGEMV) Matrix-vector product, complex rectangular matrix
+
+          F06SBF  (ZGBMV) Matrix-vector product, complex rectangular band
+                  matrix
+
+          F06SCF  (ZHEMV) Matrix-vector product, complex Hermitian matrix
+
+          F06SDF  (ZHBMV) Matrix-vector product, complex Hermitian band
+                  matrix
+
+          F06SEF  (ZHPMV) Matrix-vector product, complex Hermitian packed
+                  matrix
+
+          F06SFF  (ZTRMV) Matrix-vector product, complex triangular matrix
+
+          F06SGF  (ZTBMV) Matrix-vector product, complex triangular band
+                  matrix
+
+          F06SHF  (ZTPMV) Matrix-vector product, complex triangular packed
+                  matrix
+
+          F06SJF  (ZTRSV) System of equations, complex triangular matrix
+
+          F06SKF  (ZTBSV) System of equations, complex triangular band
+                  matrix
+
+          F06SLF  (ZTPSV) System of equations, complex triangular packed
+                  matrix
+
+          F06SMF  (ZGERU) Rank-1 update, complex rectangular matrix,
+                  unconjugated vector
+
+          F06SNF  (ZGERC) Rank-1 update, complex rectangular matrix,
+                  conjugated vector
+
+          F06SPF  (ZHER) Rank-1 update, complex Hermitian matrix
+
+          F06SQF  (ZHPR) Rank-1 update, complex Hermitian packed matrix
+
+          F06SRF  (ZHER2) Rank-2 update, complex Hermitian matrix
+
+          F06SSF  (ZHPR2) Rank-2 update, complex Hermitian packed matrix
+
+          F06YAF  (DGEMM) Matrix-matrix product, two real rectangular
+                  matrices
+
+          F06YCF  (DSYMM) Matrix-matrix product, one real symmetric matrix,
+                  one real rectangular matrix
+
+          F06YFF  (DTRMM) Matrix-matrix product, one real triangular
+                  matrix, one real rectangular matrix
+
+          F06YJF  (DTRSM) Solves a system of equations with multiple right-
+                  hand sides, real triangular coefficient matrix
+
+          F06YPF  (DSYRK) Rank-k update of a real symmetric matrix
+
+          F06YRF  (DSYR2K) Rank-2k update of a real symmetric matrix
+
+          F06ZAF  (ZGEMM) Matrix-matrix product, two complex rectangular
+                  matrices
+
+          F06ZCF  (ZHEMM) Matrix-matrix product, one complex Hermitian
+                  matrix, one complex rectangular matrix
+
+          F06ZFF  (ZTRMM) Matrix-matrix product, one complex triangular
+                  matrix, one complex rectangular matrix
+
+          F06ZJF  (ZTRSM) Solves system of equations with multiple right-
+                  hand sides, complex triangular coefficient matrix
+
+          F06ZPF  (ZHERK) Rank-k update of a complex Hermitian matrix
+
+          F06ZRF  (ZHER2K) Rank-2k update of a complex Hermitian matrix
+
+          F06ZTF  (ZSYMM) Matrix-matrix product, one complex symmetric
+                  matrix, one complex rectangular matrix
+
+          F06ZUF  (ZSYRK) Rank-k update of a complex symmetric matrix
+
+          F06ZWF  (ZSYR2K) Rank-2k update of a complex symmetric matrix
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf07}{NAG On-line Documentation: f07}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F07(3NAG)         Foundation Library (12/10/92)         F07(3NAG)
+
+
+
+          F07 -- Linear Equations (LAPACK)              Introduction -- F07
+                                    Chapter F07
+                             Linear Equations (LAPACK)
+
+          1. Scope of the Chapter
+
+          This chapter provides four routines concerned with matrix
+          factorization, and the solution of systems of linear equations
+          following the matrix factorizations.
+
+          2. Background to the Problems
+
+          Background material, together with pointers to the routines in
+          this chapter, are to be found in the F01 and F04 Chapter
+          Introductions.
+
+          3. Recommendations on Choice and Use of Routines
+
+          The routines in this chapter are derived from the LAPACK project
+          and may also be called using the LAPACK name, which is given in
+          brackets following the F07 name in the following descriptions.
+
+          Routine F07ADF (DGETRF) performs an LU factorization of a real m
+          by n matrix A. Following the use of this routine, F07AEF (DGETRS)
+          may be used to solve a system of n non-singular linear equations,
+          with one or more right-hand sides.
+
+          Routine F07FDF (DPOTRF) performs the Cholesky factorization of a
+          real symmetric positive-definite matrix A. Following the use of
+          this routine, F07FEF (DPOTRS) may be used to solve a system of
+          symmetric positive-definite linear equations, with one or more
+          right-hand sides.
+
+
+          F07 -- Linear Equations (LAPACK)                  Contents -- F07
+          Chapter F07
+
+          Linear Equations (LAPACK)
+
+          F07ADF  (DGETRF) LU factorization of real m by n matrix
+
+          F07AEF  (DGETRS) Solution of real system of linear equations,
+                  multiple right-hand sides, matrix already factorized by
+                  F07ADF
+
+          F07FDF  (DPOTRF) Cholesky factorization of real symmetric
+                  positive-definite matrix
+
+          F07FEF  (DPOTRS) Solution of real symmetric positive-definite
+                  system of linear equations, multiple right-hand sides,
+                  matrix already factorized by F07FDF
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf07adf}{NAG On-line Documentation: f07adf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F07ADF(3NAG)      Foundation Library (12/10/92)      F07ADF(3NAG)
+
+
+
+          F07 -- Linear Equations (LAPACK)                           F07ADF
+                  F07ADF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F07ADF (DGETRF) computes the LU factorization of a real m by n
+          matrix.
+
+          2. Specification
+
+                 SUBROUTINE F07ADF (M, N, A, LDA, IPIV, INFO)
+                 ENTRY            M, N, A, LDA, IPIV, INFO
+                 INTEGER          M, N, LDA, IPIV(*), INFO
+                 DOUBLE PRECISION A(LDA,*)
+
+          The ENTRY statement enables the routine to be called by its
+          LAPACK name.
+
+          3. Description
+
+          This routine forms the LU factorization of a real m by n matrix A
+          as A=PLU, where P is a permutation matrix, L is lower triangular
+          with unit diagonal elements (lower trapezoidal if m>n) and U is
+          upper triangular (upper trapezoidal if m<n). Usually A is square
+          (m=n), and both L and U are triangular. The routine uses partial
+          pivoting, with row interchanges.
+
+          4. References
+
+          [1]   Golub G H and Van Loan C F (1989) Matrix Computations (2nd
+                Edition). Johns Hopkins University Press, Baltimore,
+                Maryland.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: m, the number of rows of the matrix A.
+               Constraint: M >= 0.
+
+           2:  N -- INTEGER                                           Input
+               On entry: n, the number of columns of the matrix A.
+               Constraint: N >= 0.
+
+           3:  A(LDA,*) -- DOUBLE PRECISION array              Input/Output
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the m by n matrix A. On exit: A is overwritten by
+               the factors L and U; the unit diagonal elements of L are not
+               stored.
+
+           4:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F07ADF is called.
+               Constraint: LDA >= max(1,M).
+
+           5:  IPIV(*) -- INTEGER array                              Output
+               Note: the dimension of the array IPIV must be at least
+               max(1,min(M,N)).
+               On exit: the pivot indices. Row i of the matrix A was
+               interchanged with row IPIV(i) for i=1,2,...,min(m,n).
+
+           6:  INFO -- INTEGER                                       Output
+               On exit: INFO = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          INFO < 0
+               If INFO = -i, the ith parameter has an illegal value. An
+               explanatory message is output, and execution of the program
+               is terminated.
+
+          INFO > 0
+               If INFO = i, u   is exactly zero. The factorization has been
+                             ii
+               completed but the factor U is exactly singular, and division
+               by zero will occur if it is subsequently used to solve a
+                                                         -1
+               system of linear equations or to compute A  .
+
+          7. Accuracy
+
+          The computed factors L and U are the exact factors of a perturbed
+          matrix A+E, where
+
+                          |E|<=c(min(m,n))(epsilon)P|L||U|,
+
+          c(n) is a modest linear function of n, and (epsilon) is the
+          machine precision.
+
+          8. Further Comments
+
+          The total number of floating-point operations is approximately
+           2 3                           1 2                   1 2
+           -n  if m=n (the usual case),  -n (3m-n) if m>n and  -m (3n-m) if
+           3                             3                     3
+          m<n.
+
+          A call to this routine with m=n may be followed by calls to the
+          routines:
+
+                                                 T
+               F07AEF (DGETRS) to solve AX=B or A X=B;
+
+               F07AGF (DGECON)(*) to estimate the condition number of A;
+
+               F07AJF (DGETRI)(*) to compute the inverse of A.
+
+          The complex analogue of this routine is F07ARF (ZGETRF)(*).
+
+          9. Example
+
+          To compute the LU factorization of the matrix A, where
+
+                              ( 1.80  2.88  2.05 -0.89)
+                              ( 5.25 -2.95 -0.95 -3.80)
+                            A=( 1.58 -2.69 -2.90 -1.04).
+                              (-1.11 -0.66 -0.59  0.80)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf07aef}{NAG On-line Documentation: f07aef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F07AEF(3NAG)      Foundation Library (12/10/92)      F07AEF(3NAG)
+
+
+
+          F07 -- Linear Equations (LAPACK)                           F07AEF
+                  F07AEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F07AEF (DGETRS) solves a real system of linear equations with
+                                              T
+          multiple right-hand sides, AX=B or A X=B, where A has been
+          factorized by F07ADF (DGETRF).
+
+          2. Specification
+
+                 SUBROUTINE F07AEF (TRANS, N, NRHS, A, LDA, IPIV, B, LDB,
+                1                   INFO)
+                 ENTRY            TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO
+                 INTEGER          N, NRHS, LDA, IPIV(*), LDB, INFO
+                 DOUBLE PRECISION A(LDA,*), B(LDB,*)
+                 CHARACTER*1      TRANS
+
+          The ENTRY statement enables the routine to be called by its
+          LAPACK name.
+
+          3. Description
+
+                                                              T
+          To solve a real system of linear equations AX=B or A X=B, this
+          routine must be preceded by a call to F07ADF (DGETRF)which
+          computes the LU factorization of A as A=PLU. The solution is
+          computed by forward and backward substitution.
+
+          If TRANS = 'N', the solution is computed by solving PLY=B and
+          then UX=Y.
+
+                                                                      T
+          If TRANS = 'T' or 'C', the solution is computed by solving U Y=B
+                    T T
+          and then L P X=Y.
+
+          4. References
+
+          [1]   Golub G H and Van Loan C F (1989) Matrix Computations (2nd
+                Edition). Johns Hopkins University Press, Baltimore,
+                Maryland.
+
+          5. Parameters
+
+           1:  TRANS -- CHARACTER*1                                   Input
+               On entry: indicates the form of the equations as follows:
+                    if TRANS = 'N', then AX=B is solved for X;
+
+                                                 T
+                    if TRANS = 'T' or 'C', then A X=B is solved for X.
+                Constraint: TRANS = 'N', 'T' or 'C'.
+
+           2:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A. Constraint: N >= 0.
+
+           3:  NRHS -- INTEGER                                        Input
+               On entry: r, the number of right-hand sides. Constraint:
+               NRHS >= 0.
+
+           4:  A(LDA,*) -- DOUBLE PRECISION array                     Input
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the LU factorization of A, as returned by F07ADF
+               (DGETRF).
+
+           5:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F07AEF is called.
+               Constraint: LDA >= max(1,N).
+
+           6:  IPIV(*) -- INTEGER array                               Input
+               Note: the dimension of the array IPIV must be at least
+               max(1,N).
+               On entry: the pivot indices, as returned by F07ADF (DGETRF).
+
+           7:  B(LDB,*) -- DOUBLE PRECISION array              Input/Output
+               Note: the second dimension of the array B must be at least
+               max(1,NRHS).
+               On entry: the n by r right-hand side matrix B. On exit: the
+               n by r solution matrix X.
+
+           8:  LDB -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array B as declared in the
+               (sub)program from which F07AEF is called.
+               Constraint: LDB >= max(1,N).
+
+           9:  INFO -- INTEGER                                       Output
+               On exit: INFO = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          INFO < 0
+               If INFO = -i, the ith parameter has an illegal value. An
+               explanatory message is output, and execution of the program
+               is terminated.
+
+          7. Accuracy
+
+          For each right-hand side vector b, the computed solution x is the
+          exact solution of a perturbed system of equations (A+E)x=b, where
+
+                             |E|<=c(n)(epsilon)P|L||U|,
+
+          c(n) is a modest linear function of n, and (epsilon) is the
+          machine precision.
+
+             ^
+          If x is the true solution, then the computed solution x satisfies
+          a forward error bound of the form
+
+                             ^
+                        ||x-x||
+                               infty
+                        ------------ <= c(n)cond(A,x)(epsilon)
+                         ||x||
+                              infty
+
+                              -1
+          where cond(A,x)=|||A  ||A||x|||     /||x||      <=
+                                         infty      infty
+                      -1
+          cond(A)=|||A  ||A|||     <=(kappa)     (A). Note that cond(A,x)
+                              infty         infty
+                                                      T
+          can be much smaller than cond(A), and cond(A ) can be much larger
+          (or smaller) than cond(A).
+
+          Forward and backward error bounds can be computed by calling
+          F07AHF (DGERFS)(*), and an estimate for (kappa)     (A) can be
+                                                         infty
+          obtained by calling F07AGF (DGECON)(*) with NORM ='I'.
+
+          8. Further Comments
+
+          The total number of floating-point operations is approximately
+            2
+          2n r.
+
+          This routine may be followed by a call to F07AHF (DGERFS)(*) to
+          refine the solution and return an error estimate.
+
+          The complex analogue of this routine is F07ASF (ZGETRS)(*).
+
+          9. Example
+
+          To solve the system of equations AX=B, where
+
+                               ( 1.80  2.88  2.05 -0.89)
+                               ( 5.25 -2.95 -0.95 -3.80)
+                             A=( 1.58 -2.69 -2.90 -1.04)
+                               (-1.11 -0.66 -0.59  0.80)
+
+          and
+
+
+                                    ( 9.52  18.47)
+                                    (24.35   2.25)
+                                  B=( 0.77 -13.28).
+                                    (-6.22  -6.21)
+
+          Here A is unsymmetric and must first be factorized by F07ADF
+          (DGETRF)).
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf07fdf}{NAG On-line Documentation: f07fdf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F07FDF(3NAG)      Foundation Library (12/10/92)      F07FDF(3NAG)
+
+
+
+          F07 -- Linear Equations (LAPACK)                           F07FDF
+                  F07FDF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F07FDF (DPOTRF) computes the Cholesky factorization of a real
+          symmetric positive-definite matrix.
+
+          2. Specification
+
+                 SUBROUTINE F07FDF (UPLO, N, A, LDA, INFO)
+                 ENTRY            UPLO, N, A, LDA, INFO
+                 INTEGER          N, LDA, INFO
+                 DOUBLE PRECISION A(LDA,*)
+                 CHARACTER*1      UPLO
+
+          The ENTRY statement enables the routine to be called by its
+          LAPACK name.
+
+          3. Description
+
+          This routine forms the Cholesky factorization of a real symmetric
+                                                  T                       T
+          positive-definite matrix A either as A=U U if UPLO = 'U' or A=LL
+          if UPLO = 'L', where U is an upper triangular matrix and L is
+          lower triangular.
+
+          4. References
+
+          [1]   Demmel J W (1989) On Floating-point Errors in Cholesky.
+                LAPACK Working Note No. 14. University of Tennessee,
+                Knoxville.
+
+          [2]   Golub G H and Van Loan C F (1989) Matrix Computations (2nd
+                Edition). Johns Hopkins University Press, Baltimore,
+                Maryland.
+
+          5. Parameters
+
+           1:  UPLO -- CHARACTER*1                                    Input
+               On entry: indicates whether the upper or lower triangular
+               part of A is stored and how A is factorized, as follows:
+                    if UPLO = 'U', then the upper triangular part of A is
+                                                   T
+                    stored and A is factorized as U U, where U is upper
+                    triangular;
+
+                    if UPLO = 'L', then the lower triangular part of A is
+                                                    T
+                    stored and A is factorized as LL , where L is lower
+                    triangular.
+                Constraint: UPLO = 'U' or 'L'.
+
+           2:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A. Constraint: N >= 0.
+
+           3:  A(LDA,*) -- DOUBLE PRECISION array              Input/Output
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the n by n symmetric positive-definite matrix A.
+               If UPLO = 'U', the upper triangle of A must be stored and
+               the elements of the array below the diagonal are not
+               referenced; if UPLO = 'L', the lower triangle of A must be
+               stored and the elements of the array above the diagonal are
+               not referenced. On exit: the upper or lower triangle of A is
+               overwritten by the Cholesky factor U or L as specified by
+               UPLO.
+
+           4:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F07FDF is called.
+               Constraint: LDA >= max(1,N).
+
+           5:  INFO -- INTEGER                                       Output
+               On exit: INFO = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          INFO < 0
+               If INFO = -i, the ith parameter has an illegal value. An
+               explanatory message is output, and execution of the program
+               is terminated.
+
+          INFO > 0
+               If INFO = i, the leading minor of order i is not positive-
+               definite and the factorization could not be completed. Hence
+               A itself is not positive-definite. This may indicate an
+               error in forming the matrix A. To factorize a symmetric
+               matrix which is not positive-definite, call F07MDF
+               (DSYTRF)(*) instead.
+
+          7. Accuracy
+
+          If UPLO = 'U', the computed factor U is the exact factor of a
+          perturbed matrix A+E, where
+
+                                                 T
+                             |E|<=c(n)(epsilon)|U ||U|,
+
+          c(n) is a modest linear function of n, and (epsilon) is the
+          machine precision. If UPLO = 'L', a similar statement holds for
+          the computed factor L. It follows that
+                                 
+
+          |e  |<=c(n)(epsilon)  /a  a  .
+            ij                \/  ii jj
+
+          8. Further Comments
+
+          The total number of floating-point operations is approximately
+           1 3
+           -n .
+           3
+
+          A call to this routine may be followed by calls to the routines:
+
+               F07FEF (DPOTRS) to solve AX=B;
+
+               F07FGF (DPOCON)(*) to estimate the condition number of A;
+
+               F07FJF (DPOTRI)(*) to compute the inverse of A.
+
+          The complex analogue of this routine is F07FRF (ZPOTRF)(*).
+
+          9. Example
+
+          To compute the Cholesky factorization of the matrix A, where
+
+                              ( 4.16 -3.12  0.56 -0.10)
+                              (-3.12  5.03 -0.83  1.18)
+                            A=( 0.56 -0.83  0.76  0.34).
+                              (-0.10  1.18  0.34  1.18)
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXf07fef}{NAG On-line Documentation: f07fef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     F07FEF(3NAG)      Foundation Library (12/10/92)      F07FEF(3NAG)
+
+
+
+          F07 -- Linear Equations (LAPACK)                           F07FEF
+                  F07FEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          F07FEF (DPOTRS) solves a real symmetric positive-definite system
+          of linear equations with multiple right-hand sides, AX=B, where A
+          has been factorized by F07FDF (DPOTRF).
+
+          2. Specification
+
+                 SUBROUTINE F07FEF (UPLO, N, NRHS, A, LDA, B, LDB, INFO)
+                 ENTRY            UPLO, N, NRHS, A, LDA, B, LDB, INFO
+                 INTEGER          N, NRHS, LDA, LDB, INFO
+                 DOUBLE PRECISION A(LDA,*), B(LDB,*)
+                 CHARACTER*1      UPLO
+
+          The ENTRY statement enables the routine to be called by its
+          LAPACK name.
+
+          3. Description
+
+          To solve a real symmetric positive-definite system of linear
+          equations AX=B, this routine must be preceded by a call to
+          F07FDF (DPOTRF) which computes the Cholesky factorization of A.
+          The solution X is computed by forward and backward substitution.
+
+                            T
+          If UPLO = 'U', A=U U, where U is upper triangular; the solution X
+                                  T
+          is computed by solving U Y=B and then UX=Y.
+
+                             T
+          If UPLO = 'L', A=LL , where L is lower triangular; the solution X
+                                                T
+          is computed by solving LY=B and then L X=Y.
+
+          4. References
+
+          [1]   Golub G H and Van Loan C F (1989) Matrix Computations (2nd
+                Edition). Johns Hopkins University Press, Baltimore,
+                Maryland.
+
+          5. Parameters
+
+           1:  UPLO -- CHARACTER*1                                    Input
+               On entry: indicates whether the upper or lower triangular
+               part of A is stored and how A is factorized, as follows:
+                                           T
+                    if UPLO = 'U', then A=U U where U is upper triangular;
+
+                                            T
+                    if UPLO = 'L', then A=LL  where L is lower triangular.
+                Constraint: UPLO = 'U' or 'L'.
+
+           2:  N -- INTEGER                                           Input
+               On entry: n, the order of the matrix A. Constraint: N >= 0.
+
+           3:  NRHS -- INTEGER                                        Input
+               On entry: r, the number of right-hand sides. Constraint:
+               NRHS >= 0.
+
+           4:  A(LDA,*) -- DOUBLE PRECISION array                     Input
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the Cholesky factor of A, as returned by F07FDF
+               (DPOTRF).
+
+           5:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which F07FEF is called.
+               Constraint: LDA >=max(1,N).
+
+           6:  B(LDB,*) -- DOUBLE PRECISION array              Input/Output
+               Note: the second dimension of the array B must be at least
+               max(1,NRHS).
+               On entry: the n by r right-hand side matrix B.
+
+           7:  LDB -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array B as declared in the
+               (sub)program from which F07FEF is called.
+               Constraint: LDB >=max(1,N).
+
+           8:  INFO -- INTEGER                                       Output
+               On exit: INFO = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          INFO < 0
+               If INFO = -i, the ith parameter has an illegal value. An
+               explanatory message is output, and execution of the program
+               is terminated.
+
+          7. Accuracy
+
+          For each right-hand side vector b, the computed solution x is the
+          exact solution of a perturbed system of equations (A+E)x=b, where
+
+                                   T
+               |E|<=c(n)(epsilon)|U ||U| if UPLO = 'U',
+
+                                      T
+               |E|<=c(n)(epsilon)|L||L | if UPLO = 'L',
+
+          c(n) is a modest linear function of n, and (epsilon) is the
+          machine precision.
+
+             ^
+          If x is the true solution, then the computed solution x satisfies
+          a forward bound of the form
+
+                             ^
+                        ||x-x||
+                               infty
+                        ------------<=c(n)cond(A,x)(epsilon)
+                         ||x||
+                              infty
+
+                              -1
+          where cond(A,x)=|||A  ||A||x|||     /||x||      <=
+                                         infty      infty
+                      -1
+          cond(A)=|||A  ||A|||     <=(kappa)     (A). Note that cond(A,x)
+                              infty         infty
+          can be much smaller than cond(A).
+
+          Forward and backward error bounds can be computed by calling
+          F07FHF (DPORFS)(*), and an estimate for (kappa)     (A) (
+                                                         infty
+          =(kappa) (A)) can be obtained by calling F07FGF (DPOCON)(*).
+                  1
+
+          8. Further Comments
+
+          The total number of floating-point operations is approximately
+            2
+          2n r.
+
+          This routine may be followed by a call to F07FHF (DPORFS)(*) to
+          refine the solution and return an error estimate.
+
+          The complex analogue of this routine is F07FSF (ZPOTRS)(*).
+
+          9. Example
+
+          To compute the Cholesky factorization of the matrix A, where
+
+                              ( 4.16 -3.12  0.56 -0.10)
+                              (-3.12  5.03 -0.83  1.18)
+                            A=( 0.56 -0.83  0.76  0.34).
+                              (-0.10  1.18  0.34  1.18)
+
+          and
+
+                                    (  8.70 8.30)
+                                    (-13.35 2.13)
+                                  B=(  1.89 1.61).
+                                    ( -4.14 5.00)
+
+          Here A is symmetric positive-definite and must first be
+          factorized by F07FDF (DPOTRF).
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
diff --git a/src/hyper/pages/nagm.ht b/src/hyper/pages/nagm.ht
new file mode 100644
index 00000000..497b91f1
--- /dev/null
+++ b/src/hyper/pages/nagm.ht
@@ -0,0 +1,948 @@
+\begin{page}{manpageXXm01}{NAG On-line Documentation: m01}
+\beginscroll
+\begin{verbatim}
+
+
+
+     M01(3NAG)         Foundation Library (12/10/92)         M01(3NAG)
+
+
+
+          M01 -- Sorting                                Introduction -- M01
+                                    Chapter M01
+                                      Sorting
+
+          1. Scope of the Chapter
+
+          This chapter is concerned with sorting numeric data. It handles
+          only the simplest types of data structure and it is concerned
+          only with internal sorting -- that is, sorting a set of data
+          which can all be stored within the program.
+
+          Users with large files of data or complex data structures to be
+          sorted should use a comprehensive sorting program or package.
+
+          2. Background to the Problems
+
+          The usefulness of sorting is obvious (perhaps a little too
+          obvious, since sorting can be expensive and is sometimes done
+          when not strictly necessary). Sorting may traditionally be
+          associated with data-processing and non-numerical programming,
+          but it has many uses within the realm of numerical analysis, for
+          example, to arrange eigenvalues in ascending order of absolute
+          value and in the ranking of observations for nonparametric
+          statistics.
+
+          The general problem may be defined as follows. We are given N
+          items of data
+
+
+                                    R ,R ,...,R .
+                                     1  2      N
+
+          Each item R  contains a key K  which can be ordered relative to
+                     i                 i
+          any other key according to some specified criterion (for example,
+          ascending numeric value). The problem is to determine a
+          permutation
+
+
+                                 p(1),p(2),...,p(N)
+
+          which puts the keys in order:
+
+                              K    <=K    <=...<=K
+                               p(1)   p(2)        p(N)
+
+          Sometimes we may wish actually to rearrange the items so that
+          their keys are in order; for other purposes we may simply require
+          a table of indices so that the items can be referred to in sorted
+          order; or yet again we may require a table of ranks, that is, the
+          positions of each item in the sorted order.
+
+          For example, given the single-character items, to be sorted into
+          alphabetic order:
+
+               E    B    A    D    C
+
+          the indices of the items in sorted order are
+
+               3    2    5    4    1
+
+          and the ranks of the items are
+
+               5    2    1    4    3
+
+          Indices may be converted to ranks, and vice versa, by simply
+          computing the inverse permutation.
+
+          The items may consist solely of the key (each item may simply be
+          a number). On the other hand, the items may contain additional
+          information (for example, each item may be an eigenvalue of a
+          matrix and its associated eigenvector, the eigenvalue being the
+          key). In the latter case there may be many distinct items with
+          equal keys, and it may be important to preserve the original
+          order among them (if this is achieved, the sorting is called '
+          stable').
+
+          There are a number of ingenious algorithms for sorting. For a
+          fascinating discussion of them, and of the whole subject, see
+          Knuth [1].
+
+          2.1. References
+
+          [1]   Knuth D E (1973) The Art of Computer Programming, Vol 3.
+                Addison-Wesley.
+
+          3. Recommendations on Choice and Use of Routines
+
+          Four categories of routines are provided:
+
+          - routines which rearrange the data into sorted order (M01C-);
+
+          - routines which determine the ranks of the data, leaving the
+          data unchanged (M01D-);
+
+          - routines which rearrange the data according to pre-determined
+          ranks (M01E-);
+
+          - service routines (M01Z-).
+
+          Routines are provided for sorting double precision data only.
+
+          If the task is simply to rearrange a one-dimensional array of
+          data into sorted order, then M01CAF should be used, since this
+          requires no extra workspace and is faster than any other method.
+
+          For many applications it is in fact preferable to separate the
+          task of determining the sorted order (ranking) from the task of
+          rearranging data into a pre-determined order; the latter task may
+          not need to be performed at all. Frequently it may be sufficient
+          to refer to the data in sorted order via an index vector, without
+          rearranging it. Frequently also one set of data (e.g. a column of
+          a matrix) is used for determining a set of ranks, which are then
+          applied to other data (e.g. the remaining columns of the matrix).
+
+          To determine the ranks of a set of data, use an M01D- routine.
+          Routines are provided for ranking one-dimensional arrays, and for
+          ranking rows or columns of two-dimensional arrays.
+
+          To create an index vector so that data can be referred to in
+          sorted order, first call an M01D- routine to determine the ranks,
+          and then call M01ZAF to convert the vector of ranks into an index
+          vector.
+
+          To rearrange data according to pre-determined ranks: use M01EAF
+          if the data is stored in a one-dimensional array; or if the data
+          is stored in a more complicated structure, use an index vector to
+          generate a new copy of the data in the desired order.
+
+          Examples of these operations can be found in the routine
+          documents of the relevant routines.
+
+          4. Index
+
+           Ranking:
+               columns of a matrix, double precision numbers         M01DJF
+               rows of a matrix, double precision numbers            M01DEF
+               vector, double precision numbers                      M01DAF
+          Rearranging (according to pre-determined ranks):
+               vector, double precision numbers                      M01EAF
+          Service routines:
+               invert a permutation (ranks to indices or vice        M01ZAF
+               versa)
+          Sorting (i.e., rearranging into sorted order):
+               vector, double precision numbers                      M01CAF
+
+
+          M01 -- Sorting                                    Contents -- M01
+          Chapter M01
+
+          Sorting
+
+          M01CAF  Sort a vector, double precision numbers
+
+          M01DAF  Rank a vector, double precision numbers
+
+          M01DEF  Rank rows of a matrix, double precision numbers
+
+          M01DJF  Rank columns of a matrix, double precision numbers
+
+          M01EAF  Rearrange a vector according to given ranks, double
+                  precision numbers
+
+          M01ZAF  Invert a permutation
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXm01caf}{NAG On-line Documentation: m01caf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     M01CAF(3NAG)      Foundation Library (12/10/92)      M01CAF(3NAG)
+
+
+
+          M01 -- Sorting                                             M01CAF
+                  M01CAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          M01CAF rearranges a vector of double precision numbers into
+          ascending or descending order.
+
+          2. Specification
+
+                 SUBROUTINE M01CAF (RV, M1, M2, ORDER, IFAIL)
+                 INTEGER          M1, M2, IFAIL
+                 DOUBLE PRECISION RV(M2)
+                 CHARACTER*1      ORDER
+
+          3. Description
+
+          M01CAF is based on Singleton's implementation of the 'median-of-
+          three' Quicksort algorithm [2], but with two additional
+          modifications. First, small subfiles are sorted by an insertion
+          sort on a separate final pass (Sedgewick [1]). Second, if a
+          subfile is partitioned into two very unbalanced subfiles, the
+          larger of them is flagged for special treatment: before it is
+          partitioned, its end-points are swapped with two random points
+          within it; this makes the worst case behaviour extremely
+          unlikely.
+
+          4. References
+
+          [1]   Sedgewick R (1978) Implementing Quicksort Programs. Comm.
+                ACM. 21 847--857.
+
+          [2]   Singleton R C (1969) An Efficient Algorithm for Sorting with
+                Minimal Storage: Algorithm 347. Comm. ACM. 12 185--187.
+
+          5. Parameters
+
+           1:  RV(M2) -- DOUBLE PRECISION array                Input/Output
+               On entry: elements M1 to M2 of RV must contain double
+               precision values to be sorted. On exit: these values are
+               rearranged into sorted order.
+
+           2:  M1 -- INTEGER                                          Input
+               On entry: the index of the first element of RV to be sorted.
+               Constraint: M1 > 0.
+
+           3:  M2 -- INTEGER                                          Input
+
+               On entry: the index of the last element of RV to be sorted.
+               Constraint: M2 >= M1.
+
+           4:  ORDER -- CHARACTER*1                                   Input
+               On entry: if ORDER is 'A', the values will be sorted into
+               ascending (i.e., non-decreasing) order; if ORDER is 'D',
+               into descending order. Constraint: ORDER = 'A' or 'D'.
+
+           5:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M2 < 1,
+
+               or       M1 < 1,
+
+               or       M1 > M2.
+
+          IFAIL= 2
+               On entry ORDER is not 'A' or 'D'.
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          The average time taken by the routine is approximately
+          proportional to n*logn, where n = M2 - M1 + 1. The worst case
+                                   2
+          time is proportional to n  but this is extremely unlikely to
+          occur.
+
+          9. Example
+
+          The example program reads a list of double precision numbers and
+          sorts them into ascending order.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXm01daf}{NAG On-line Documentation: m01daf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     M01DAF(3NAG)      Foundation Library (12/10/92)      M01DAF(3NAG)
+
+
+
+          M01 -- Sorting                                             M01DAF
+                  M01DAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          M01DAF ranks a vector of double precision numbers in ascending or
+          descending order.
+
+          2. Specification
+
+                 SUBROUTINE M01DAF (RV, M1, M2, ORDER, IRANK, IFAIL)
+                 INTEGER          M1, M2, IRANK(M2), IFAIL
+                 DOUBLE PRECISION RV(M2)
+                 CHARACTER*1      ORDER
+
+          3. Description
+
+          M01DAF uses a variant of list-merging, as described by Knuth [1]
+          pp 165-166. The routine takes advantage of natural ordering in
+          the data, and uses a simple list insertion in a preparatory pass
+          to generate ordered lists of length at least 10. The ranking is
+          stable: equal elements preserve their ordering in the input data.
+
+          4. References
+
+          [1]   Knuth D E (1973) The Art of Computer Programming, Vol 3.
+                Addison-Wesley.
+
+          5. Parameters
+
+           1:  RV(M2) -- DOUBLE PRECISION array                       Input
+               On entry: elements M1 to M2 of RV must contain double
+               precision values to be ranked.
+
+           2:  M1 -- INTEGER                                          Input
+               On entry: the index of the first element of RV to be ranked.
+               Constraint: M1 > 0.
+
+           3:  M2 -- INTEGER                                          Input
+               On entry: the index of the last element of RV to be ranked.
+               Constraint: M2 >= M1.
+
+           4:  ORDER -- CHARACTER*1                                   Input
+               On entry: if ORDER is 'A', the values will be ranked in
+               ascending (i.e., non-decreasing) order; if ORDER is 'D',
+               into descending order. Constraint: ORDER = 'A' or 'D'.
+
+           5:  IRANK(M2) -- INTEGER array                            Output
+               On exit: elements M1 to M2 of IRANK contain the ranks of the
+               corresponding elements of RV. Note that the ranks are in the
+               range M1 to M2: thus, if RV(i) is the first element in the
+               rank order, IRANK(i) is set to M1.
+
+           6:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M2 < 1,
+
+               or       M1 < 1,
+
+               or       M1 > M2.
+
+          IFAIL= 2
+               On entry ORDER is not 'A' or 'D'.
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          The average time taken by the routine is approximately
+          proportional to n*logn, where n = M2 - M1 + 1.
+
+          9. Example
+
+          The example program reads a list of double precision numbers and
+          ranks them in ascending order.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXm01def}{NAG On-line Documentation: m01def}
+\beginscroll
+\begin{verbatim}
+
+
+
+     M01DEF(3NAG)      Foundation Library (12/10/92)      M01DEF(3NAG)
+
+
+
+          M01 -- Sorting                                             M01DEF
+                  M01DEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          M01DEF ranks the rows of a matrix of double precision numbers in
+          ascending or descending order.
+
+          2. Specification
+
+                 SUBROUTINE M01DEF (RM, LDM, M1, M2, N1, N2, ORDER, IRANK,
+                1                   IFAIL)
+                 INTEGER          LDM, M1, M2, N1, N2, IRANK(M2), IFAIL
+                 DOUBLE PRECISION RM(LDM,N2)
+                 CHARACTER*1      ORDER
+
+          3. Description
+
+          M01DEF ranks rows M1 to M2 of a matrix, using the data in columns
+          N1 to N2 of those rows. The ordering is determined by first
+          ranking the data in column N1, then ranking any tied rows
+          according to the data in column N1 + 1, and so on up to column
+          N2.
+
+          M01DEF uses a variant of list-merging, as described by Knuth [1]
+          pp 165-166. The routine takes advantage of natural ordering in
+          the data, and uses a simple list insertion in a preparatory pass
+          to generate ordered lists of length at least 10. The ranking is
+          stable: equal rows preserve their ordering in the input data.
+
+          4. References
+
+          [1]   Knuth D E (1973) The Art of Computer Programming, Vol 3.
+                Addison-Wesley.
+
+          5. Parameters
+
+           1:  RM(LDM,N2) -- DOUBLE PRECISION array                   Input
+               On entry: columns N1 to N2 of rows M1 to M2 of RM must
+               contain double precision data to be ranked.
+
+           2:  LDM -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array RM as declared in the
+               (sub)program from which M01DEF is called.
+               Constraint: LDM >= M2.
+
+           3:  M1 -- INTEGER                                          Input
+               On entry: the index of the first row of RM to be ranked.
+               Constraint: M1 > 0.
+
+           4:  M2 -- INTEGER                                          Input
+               On entry: the index of the last row of RM to be ranked.
+               Constraint: M2 >= M1.
+
+           5:  N1 -- INTEGER                                          Input
+               On entry: the index of the first column of RM to be used.
+               Constraint: N1 > 0.
+
+           6:  N2 -- INTEGER                                          Input
+               On entry: the index of the last column of RM to be used.
+               Constraint: N2 >= N1.
+
+           7:  ORDER -- CHARACTER*1                                   Input
+               On entry: if ORDER is 'A', the rows will be ranked in
+               ascending (i.e., non-decreasing) order; if ORDER is 'D',
+               into descending order. Constraint: ORDER = 'A' or 'D'.
+
+           8:  IRANK(M2) -- INTEGER array                            Output
+               On exit: elements M1 to M2 of IRANK contain the ranks of the
+               corresponding rows of RM. Note that the ranks are in the
+               range M1 to M2: thus, if the ith row of RM is the first in
+               the rank order, IRANK(i) is set to M1.
+
+           9:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M2 < 1,
+
+               or       N2 < 1,
+
+               or       M1 < 1,
+
+               or       M1 > M2,
+
+               or       N1 < 1,
+
+               or       N1 > N2,
+
+               or       LDM < M2.
+
+          IFAIL= 2
+               On entry ORDER is not 'A' or 'D'.
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          The average time taken by the routine is approximately
+          proportional to n*logn, where n = M2 - M1 + 1.
+
+          9. Example
+
+          The example program reads a matrix of double precision numbers
+          and ranks the rows in ascending order.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXm01djf}{NAG On-line Documentation: m01djf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     M01DJF(3NAG)      Foundation Library (12/10/92)      M01DJF(3NAG)
+
+
+
+          M01 -- Sorting                                             M01DJF
+                  M01DJF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          M01DJF ranks the columns of a matrix of double precision numbers
+          in ascending or descending order.
+
+          2. Specification
+
+                 SUBROUTINE M01DJF (RM, LDM, M1, M2, N1, N2, ORDER, IRANK,
+                1                   IFAIL)
+                 INTEGER          LDM, M1, M2, N1, N2, IRANK(N2), IFAIL
+                 DOUBLE PRECISION RM(LDM,N2)
+                 CHARACTER*1      ORDER
+
+          3. Description
+
+          M01DJF ranks columns N1 to N2 of a matrix, using the data in rows
+          M1 to M2 of those columns. The ordering is determined by first
+          ranking the data in row M1, then ranking any tied columns
+          according to the data in row M1 + 1, and so on up to row M2.
+
+          M01DJF uses a variant of list-merging, as described by Knuth [1]
+          pp 165-166. The routine takes advantage of natural ordering in
+          the data, and uses a simple list insertion in a preparatory pass
+          to generate ordered lists of length at least 10. The ranking is
+          stable: equal columns preserve their ordering in the input data.
+
+          4. References
+
+          [1]   Knuth D E (1973) The Art of Computer Programming, Vol 3.
+                Addison-Wesley.
+
+          5. Parameters
+
+           1:  RM(LDM,N2) -- DOUBLE PRECISION array                   Input
+               On entry: rows M1 to M2 of columns N1 to N2 of RM must
+               contain double precision data to be ranked.
+
+           2:  LDM -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array RM as declared in the
+               (sub)program from which M01DJF is called.
+               Constraint: LDM >= M2.
+
+           3:  M1 -- INTEGER                                          Input
+               On entry: the index of the first row of RM to be used.
+               Constraint: M1 > 0.
+
+           4:  M2 -- INTEGER                                          Input
+               On entry: the index of the last row of RM to be used.
+               Constraint: M2 >= M1.
+
+           5:  N1 -- INTEGER                                          Input
+               On entry: the index of the first column of RM to be ranked.
+               Constraint: N1 > 0.
+
+           6:  N2 -- INTEGER                                          Input
+               On entry: the index of the last column of RM to be ranked.
+               Constraint: N2 >= N1.
+
+           7:  ORDER -- CHARACTER*1                                   Input
+               On entry: if ORDER is 'A', the columns will be ranked in
+               ascending (i.e., non-decreasing) order; if ORDER is 'D',
+               into descending order. Constraint: ORDER = 'A' or 'D'.
+
+           8:  IRANK(N2) -- INTEGER array                            Output
+               On exit: elements N1 to N2 of IRANK contain the ranks of the
+               corresponding columns of RM. Note that the ranks are in the
+               range N1 to N2: thus, if the ith column of RM is the first
+               in the rank order, IRANK(i) is set to N1.
+
+           9:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M2 < 1,
+
+               or       N2 < 1,
+
+               or       M1 < 1,
+
+               or       M1 > M2,
+
+               or       N1 < 1,
+
+               or       N1 > N2,
+
+               or       LDM < M2.
+
+          IFAIL= 2
+               On entry ORDER is not 'A' or 'D'.
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          The average time taken by the routine is approximately
+          proportional to n*logn, where n = N2 - N1 + 1.
+
+          9. Example
+
+          The example program reads a matrix of double precision numbers
+          and ranks the columns in ascending order.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXm01eaf}{NAG On-line Documentation: m01eaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     M01EAF(3NAG)      Foundation Library (12/10/92)      M01EAF(3NAG)
+
+
+
+          M01 -- Sorting                                             M01EAF
+                  M01EAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          M01EAF rearranges a vector of double precision numbers into the
+          order specified by a vector of ranks.
+
+          2. Specification
+
+                 SUBROUTINE M01EAF (RV, M1, M2, IRANK, IFAIL)
+                 INTEGER          M1, M2, IRANK(M2), IFAIL
+                 DOUBLE PRECISION RV(M2)
+
+          3. Description
+
+          M01EAF is designed to be used typically in conjunction with the
+          M01D- ranking routines. After one of the M01D- routines has been
+          called to determine a vector of ranks, M01EAF can be called to
+          rearrange a vector of real numbers into the rank order. If the
+          vector of ranks has been generated in some other way, then
+          M01ZBF(*) should be called to check its validity before M01EAF is
+          called.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  RV(M2) -- DOUBLE PRECISION array                Input/Output
+               On entry: elements M1 to M2 of RV must contain double
+               precision values to be rearranged. On exit: these values are
+               rearranged into rank order. For example, if IRANK(i) = M1,
+               then the initial value of RV(i) is moved to RV(M1).
+
+           2:  M1 -- INTEGER                                          Input
+
+           3:  M2 -- INTEGER                                          Input
+               On entry: M1 and M2 must specify the range of the ranks
+               supplied in IRANK and the elements of RV to be rearranged.
+               Constraint: 0 < M1 <= M2.
+
+           4:  IRANK(M2) -- INTEGER array                             Input
+               On entry: elements M1 to M2 of IRANK must contain a
+               permutation of the integers M1 to M2, which are interpreted
+               as a vector of ranks.
+
+           5:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M2 < 1,
+
+               or       M1 < 1,
+
+               or       M1 > M2.
+
+          IFAIL= 2
+               Elements M1 to M2 of IRANK contain a value outside the range
+               M1 to M2.
+
+          IFAIL= 3
+               Elements M1 to M2 of IRANK contain a repeated value.
+
+          If IFAIL = 2 or 3, elements M1 to M2 of IRANK do not contain a
+          permutation of the integers M1 to M2. On exit, the contents of RV
+          may be corrupted. To check the validity of IRANK without the risk
+          of corrupting RV, use M01ZBF(*).
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          The average time taken by the routine is approximately
+          proportional to n, where n = M2 - M1 + 1.
+
+          9. Example
+
+          The example program reads a matrix of double precision numbers
+          and rearranges its rows so that the elements of the kth column
+          are in ascending order. To do this, the program first calls
+          M01DAF to rank the elements of the kth column, and then calls
+          M01EAF to rearrange each column into the order specified by the
+          ranks. The value of k is read from the data-file.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXm01zaf}{NAG On-line Documentation: m01zaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     M01ZAF(3NAG)      Foundation Library (12/10/92)      M01ZAF(3NAG)
+
+
+
+          M01 -- Sorting                                             M01ZAF
+                  M01ZAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          M01ZAF inverts a permutation, and hence converts a rank vector to
+          an index vector, or vice versa.
+
+          2. Specification
+
+                 SUBROUTINE M01ZAF (IPERM, M1, M2, IFAIL)
+                 INTEGER          IPERM(M2), M1, M2, IFAIL
+
+          3. Description
+
+          There are two common ways of describing a permutation using an
+          integer vector IPERM. The first uses ranks: IPERM(i) holds the
+          position to which the ith data element should be moved in order
+          to sort the data; in other words its rank in the sorted order.
+          The second uses indices: IPERM(i) holds the current position of
+          the data element which would occur in ith position in sorted
+          order. For example, given the values
+
+                 3.5    5.9    2.9    0.5
+
+          to be sorted in ascending order, the ranks would be
+
+                 3      4      2      1
+
+          and the indices would be
+
+                 4      3      1      2
+
+          The M01D- routines generate ranks, and the M01E- routines require
+          ranks to be supplied to specify the re-ordering. However if it is
+          desired simply to refer to the data in sorted order without
+          actually re-ordering them, indices are more convenient than ranks
+          (see the example in Section 9).
+
+          M01ZAF can be used to convert ranks to indices, or indices to
+          ranks, as the two permutations are inverses of one another.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  IPERM(M2) -- INTEGER array                      Input/Output
+               On entry: elements M1 to M2 of IPERM must contain a
+               permutation of the integers M1 to M2. On exit: these
+               elements contain the inverse permutation of the integers M1
+               to M2.
+
+           2:  M1 -- INTEGER                                          Input
+
+           3:  M2 -- INTEGER                                          Input
+               On entry: M1 and M2 must specify the range of elements used
+               in the array IPERM and the range of values in the
+               permutation, as specified under IPERM. Constraint: 0 < M1 <=
+               M2.
+
+           4:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M2 < 1,
+
+               or       M1 < 1,
+               or       M1 > M2.
+          IFAIL= 2
+               Elements M1 to M2 of IPERM contain a value outside the range
+               M1 to M2.
+
+          IFAIL= 3
+               Elements M1 to M2 of IPERM contain a repeated value.
+
+          If IFAIL = 2 or 3, elements M1 to M2 of IPERM do not contain a
+          permutation of the integers M1 to M2; on exit these elements are
+          usually corrupted. To check the validity of a permutation without
+          the risk of corrupting it, use M01ZBF(*).
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          None.
+
+          9. Example
+
+          The example program reads a matrix of double precision numbers
+          and prints its rows in ascending order as ranked by M01DEF. The
+          program first calls M01DEF to rank the rows, and then calls
+          M01ZAF to convert the rank vector to an index vector, which is
+          used to refer to the rows in sorted order.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
diff --git a/src/hyper/pages/nags.ht b/src/hyper/pages/nags.ht
new file mode 100644
index 00000000..c6ee4a2c
--- /dev/null
+++ b/src/hyper/pages/nags.ht
@@ -0,0 +1,7689 @@
+\begin{page}{manpageXXs}{NAG On-line Documentation: s}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S(3NAG)           Foundation Library (12/10/92)           S(3NAG)
+
+
+
+          S -- Approximations of Special Functions        Introduction -- S
+                                     Chapter S
+                        Approximations of Special Functions
+
+          1. Scope of the Chapter
+
+          This chapter is concerned with the provision of some commonly
+          occurring physical and mathematical functions.
+
+          2. Background to the Problems
+
+          The majority of the routines in this chapter approximate real-
+          valued functions of a single real argument, and the techniques
+          involved are described in Section 2.1. In addition the chapter
+          contains routines for elliptic integrals (see Section 2.2),
+          Bessel and Airy functions of a complex argument (see Section 2.3)
+          , and exponential of a complex argument.
+
+          2.1. Functions of a Single Real Argument
+
+          Most of the routines for functions of a single real argument have
+          been based on truncated Chebyshev expansions. This method of
+          approximation was adopted as a compromise between the conflicting
+          requirements of efficiency and ease of implementation on many
+          different machine ranges. For details of the reasons behind this
+          choice and the production and testing procedures followed in
+          constructing this chapter see Schonfelder [7].
+
+          Basically if the function to be approximated is f(x), then for
+          xis in  [a,b] an approximation of the form
+
+
+                                          --'
+                                f(x)=g(x) >  C T (t)
+                                          --  r r
+                                          r=0
+
+                     --'
+          is used, ( >  denotes, according to the usual convention, a
+                     --
+          summation in which the first term is halved), where g(x) is some
+          suitable auxiliary function which extracts any singularities,
+          asymptotes and, if possible, zeros of the function in the range
+          in question and t=t(x) is a mapping of the general range [a,b] to
+          the specific range [-1,+1] required by the Chebyshev polynomials,
+          T (t). For a detailed description of the properties of the
+           r
+          Chebyshev polynomials see Clenshaw [5] and Fox and Parker [6].
+
+          The essential property of these polynomials for the purposes of
+          function approximation is that T (t) oscillates between +-1 and
+                                          n
+          it takes its extreme values n+1 times in the interval [-1,+1].
+          Therefore, provided the coefficients C  decrease in magnitude
+                                                r
+          sufficiently rapidly the error made by truncating the Chebyshev
+          expansion after n terms is approximately given by
+
+
+                                    E(t)~=C T (t)
+                                           n n
+
+          That is the error oscillates between +-C  and takes its extreme
+                                                  n
+          value n+1 times in the interval in question. Now this is just the
+          condition that the approximation be a mini-max representation,
+          one which minimizes the maximum error. By suitable choice of the
+          interval, [a,b], the auxiliary function, g(x), and the mapping of
+          the independent variable, t(x), it is almost always possible to
+          obtain a Chebyshev expansion with rapid convergence and hence
+          truncations that provide near mini-max polynomial approximations
+          to the required function. The difference between the true mini-
+          max polynomial and the truncated Chebyshev expansion is seldom
+          sufficiently great to be of significance.
+
+          The evaluation of the Chebyshev expansions follows one of two
+          methods. The first and most efficient, and hence most commonly
+          used, works with the equivalent simple polynomial. The second
+          method, which is used on the few occasions when the first method
+          proves to be unstable, is based directly on the truncated
+          Chebyshev series and uses backward recursion to evaluate the sum.
+          For the first method, a suitably truncated Chebyshev expansion
+          (truncation is chosen so that the error is less than the machine
+          precision) is converted to the equivalent simple polynomial. That
+          is we evaluate the set of coefficients b  such that
+                                                  r
+
+                                   n-1      n-1
+                                   --    r  --'
+                             y(t)= >  b t = >  C T (t).
+                                   --  r    --  r r
+                                   r=0      r=0
+
+          The polynomial can then be evaluated by the efficient Horner's
+          method of nested multiplications,
+
+                     y(t)=(b +t(b +t(b +...t(b   +tb   )))...).
+                            0    1    2       n-2   n-1
+
+          This method of evaluation results in efficient routines but for
+          some expansions there is considerable loss of accuracy due to
+          cancellation effects. In these cases the second method is used.
+          It is well known that if
+
+                        b    =C
+                         n-1   n-1
+
+                        b    =2tb   +C
+                         n-2     n-1  n-2
+
+                        b  =2tb   -b   +C ,  j=n-3,n-4,...,0
+                         j     j+1  j+2  j
+
+          then
+
+                                --'         1
+                                >  C T (t)= -(b -b )
+                                --  r r     2  0  2
+                                r=0
+
+          and this is always stable. This method is most efficiently
+          implemented by using three variables cyclically and explicitly
+          constructing the recursion.
+
+          That is,
+
+                        (alpha) = C
+                                   n-1
+                        (beta) = 2t(alpha)+C
+                                            n-2
+                        (gamma) = 2t(beta)-(alpha)+C
+                                                    n-3
+                        (alpha) = 2t(gamma)-(beta)+C
+                                                    n-4
+                        (beta) = 2t(alpha)-(gamma)+C
+                                                    n-5
+                          ...
+                          ...
+                        (alpha) = 2t(gamma)-(beta)+C   (say)
+                                                    2
+                        (beta) = 2t(alpha)-(gamma)+C
+                                                    1
+                                                1
+                        y(t) = t(beta)-(alpha)+ -C
+                                                2 0
+
+          The auxiliary functions used are normally functions compounded of
+          simple polynomial (usually linear) factors extracting zeros, and
+          the primary compiler-provided functions, sin, cos, ln, exp, sqrt,
+          which extract singularities and/or asymptotes or in some cases
+          basic oscillatory behaviour, leaving a smooth well-behaved
+          function to be approximated by the Chebyshev expansion which can
+          therefore be rapidly convergent.
+
+          The mappings of [a,b] to [-1,+1] used, range from simple linear
+          mappings to the case when b is infinite and considerable
+          improvement in convergence can be obtained by use of a bilinear
+          form of mapping. Another common form of mapping is used when the
+          function is even, that is it involves only even powers in its
+          expansion. In this case an approximation over the whole interval
+                                                    (x)2
+          [-a,a] can be provided using a mapping t=2(-) -1. This embodies
+                                                    (a)
+          the evenness property but the expansion in t involves all powers
+          and hence removes the necessity of working with an expansion with
+          half its coefficients zero.
+
+          For many of the routines an analysis of the error in principle is
+          given, viz, if E and (nabla) are the absolute errors in function
+          and argument and (epsilon) and (delta) are the corresponding
+          relative errors, then
+
+                                  E~=|f'(x)|(nabla)
+
+                                 E~=|xf'(x)|(delta)
+
+                                        | xf'(x)|
+                             (epsilon)~=| ------|(delta)
+                                        |  f(x) |
+
+          If we ignore errors that arise in the argument of the function by
+          propagation of data errors etc and consider only those errors
+          that result from the fact that a real number is being represented
+          in the computer in floating-point form with finite precision,
+          then (delta) is bounded and this bound is independent of the
+          magnitude of x; e.g. on an 11-digit machine
+
+                                               -11
+                                  |(delta)|<=10   .
+
+          (This of course implies that the absolute error (nabla)=x(delta)
+          is also bounded but the bound is now dependent on x). However
+          because of this the last two relations above are probably of more
+          interest. If possible the relative error propagation is
+          discussed; that is the behaviour of the error amplification
+          factor |xf'(x)/f(x)| is described, but in some cases, such as
+          near zeros of the function which cannot be extracted explicitly,
+          absolute error in the result is the quantity of significance and
+          here the factor |xf'(x)| is described. In general, testing of the
+          functions has shown that their error behaviour follows fairly
+          well these theoretical error behaviours. In regions, where the
+          error amplification factors are less than or of the order of one,
+          the errors are slightly larger than the above predictions. The
+          errors are here limited largely by the finite precision of
+          arithmetic in the machine but (epsilon) is normally no more than
+          a few times greater than the bound on (delta). In regions where
+          the amplification factors are large, order of ten or greater, the
+          theoretical analysis gives a good measure of the accuracy
+          obtainable.
+
+          It should be noted that the definitions and notations used for
+          the functions in this chapter are all taken from Abramowitz and
+          Stegun [1]. Users are strongly recommended to consult this book
+          for details before using the routines in this chapter.
+
+          2.2. Approximations to Elliptic Integrals
+
+          The functions provided here are symmetrised variants of the
+          classic elliptic integrals. These alternative definitions have
+          been suggested by Carlson (see [2], [3] and [4]) and he also
+          developed the basic algorithms used in this chapter.
+
+          The standard integral of the first kind is represented by
+
+
+                                     infty
+                                   1 /            dt
+                        R (x,y,z)= - |     -----------------
+                         F         2 /       
+                                     0     \/(t+x)(t+y)(t+z)
+
+          where x,y,z>=0 and at most one may be equal to zero.
+
+                                     1
+          The normalisation factor,  -, is chosen so as to make
+                                     2
+
+                                                
+
+                                  R (x,x,x)=1/\/x.
+                                   F
+
+          If any two of the variables are equal, R  degenerates into the
+                                                  F
+          second function
+
+
+                                             infty
+                                           1 /         dt
+                        R (x,y)=R (x,y,y)= - |     ----------
+                         C       F         2 /       
+                                             0     \/t+x(t+y)
+
+          where the argument restrictions are now x>=0 and y/=0.
+
+          This function is related to the logarithm or inverse hyperbolic
+          functions if 0<y<x, and to the inverse circular functions if
+          0<=x<=y.
+
+          The integrals of the second kind are defined by
+
+                                    infty
+                                  3 /             dt
+                       R (x,y,z)= - |     -------------------
+                        D         2 /        
+                                    0       /               3
+                                          \/ (t+x)(t+y)(t+z)
+
+          with z>0, x>=0 and y>=0 but only one of x or y may be zero.
+
+          The function is a degenerate special case of the integral of the
+          third kind
+
+                                   infty
+                                 3 /                  dt
+                R (x,y,z,(rho))= - |     ----------------------------
+                 J               2 /     (                 )
+                                   0     (\/(t+x)(t+y)(t+z))(t+(rho))
+
+          with (rho)/=0, x,y,z>=0 with at most one equality holding. Thus
+          R (x,y,z)=R (x,y,z,z). The normalisation of both these functions
+           D         J
+          is chosen so that
+
+                                                       
+
+                           R (x,x,x)=R (x,x,x,x)=1/(x\/x)
+                            D         J
+
+          The algorithms used for all these functions are based on
+          duplication theorems. These allow a recursion system to be
+          established which constructs a new set of arguments from the old
+          using a combination of arithmetic and geometric means. The value
+          of the function at the original arguments can then be simply
+          related to the value at the new arguments. These recursive
+          reductions are used until the arguments differ from the mean by
+          an amount small enough for a Taylor series about the mean to give
+          sufficient accuracy when retaining terms of order less than six.
+          Each step of the recurrences reduces the difference from the mean
+          by a factor of four, and as the truncation error is of order six,
+                                               -n
+          the truncation error goes like (4096)  , where n is the number of
+          iterations.
+
+          The above forms can be related to the more traditional canonical
+          forms (see Abramowitz and Stegun [1], 17.2).
+
+                           2               2               2
+          If we write q=cos (phi),r=1-m.sin (phi),s=1+n.sin (phi), where
+                    1
+          0<(phi)<= -(pi), we have: the elliptic integral of the first
+                    2
+          kind:
+
+                        sin(phi)      -1/2      1-/2
+                        /           2         2
+            F((phi)|m)= |       (1-t )   (1-mt )   dt=sin(phi).R (q,r,1);
+                        /                                       F
+                        0
+
+          the elliptic integral of the second kind:
+
+                                 sin(phi)      -1/2      -1/2
+                                 /           2         2
+                   E((phi)|m) =  |       (1-t )   (1-mt )  dt
+                                 /
+                                 0
+                                                    1     3
+                              = sin(phi).R (q,r,1)- -m.sin (phi).R (q,r,1)
+                                          F         3             D
+
+          the elliptic integral of the third kind:
+
+                                 sin(phi)      -1/2      -1/2
+                                 /           2         2         2 -1
+              (Pi)(n;(phi)|m) =  |       (1-t )   (1-mt )   (1+nt )  dt
+                                 /
+                                 0
+                                                    1     3
+                              =sin(phi).R (q,r,1)- -n.sin (phi).R (q,r,1,s)
+                                         F         3             J
+
+          Also the complete elliptic integral of the first kind:
+
+                      (pi)/2
+                      /             2       -1/2
+                K(m)= |     (1-m.sin (theta))   d(theta)=R (0,1-m,1);
+                      /                                   F
+                      0
+
+          the complete elliptic integral of the second kind:
+
+
+                (pi)/2
+                /         2        1/2                     1
+          E(m)= | (1-m.sin (theta))  d(theta)=R (0,1-m,1)- -mR (0,1-m,1).
+                /                              F           3  D
+                0
+
+
+          2.3. Bessel and Airy Functions of a Complex Argument
+
+          The routines for Bessel and Airy functions of a real argument are
+          based on Chebyshev expansions, as described in Section 2.1. The
+          routines for functions of a complex argument, however, use
+          different methods. These routines relate all functions to the
+          modified Bessel functions I    (z) and K    (z) computed in the
+                                     (nu)         (nu)
+          right-half complex plane, including their analytic continuations.
+          I     and K     are computed by different methods according to
+           (nu)      (nu)
+          the values of zand (nu). The methods include power series,
+          asymptotic expansions and Wronskian evaluations. The relations
+          between functions are based on well known formulae (see
+          Abramowitz and Stegun [1]).
+
+          2.4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Carlson B C (1977) Special Functions of Applied Mathematics.
+                Academic Press.
+
+          [3]   Carlson B C (1965) On Computing Elliptic Integrals and
+                Functions. J Math Phys. 44 36--51.
+
+          [4]   Carlson B C (1977) Elliptic Integrals of the First Kind.
+                SIAM J Math Anal. 8 231--242.
+
+          [5]   Clenshaw C W (1962) Mathematical Tables. Chebyshev Series
+                for Mathematical Functions. HMSO.
+
+          [6]   Fox L and Parker I B (1968) Chebyshev Polynomials in
+                Numerical Analysis. Oxford University Press.
+
+          [7]   Schonfelder J L (1976 ) The Production of Special Function
+                Routines for a Multi-Machine Library. Software Practice and
+                Experience. 6(1)
+
+          3. Recommendations on Choice and Use of Routines
+
+          3.1. Elliptic Integrals
+
+          IMPORTANT ADVICE: users who encounter elliptic integrals in the
+          course of their work are strongly recommended to look at
+          transforming their analysis directly to one of the Carlson forms,
+          rather than the traditional canonical Legendre forms. In general,
+          the extra symmetry of the Carlson forms is likely to simplify the
+          analysis, and these symmetric forms are much more stable to
+          calculate.
+
+          The routine S21BAF for R  is largely included as an auxiliary to
+                                  C
+          the other routines for elliptic integrals. This integral
+          essentially calculates elementary functions, e.g.
+
+
+                                        (( 1+x)2  )
+                            lnx=(x-1).R (( ---) ,x),x>0;
+                                       C((  2 )   )
+
+                                            2
+                            arcsinx=x.R (1-x ,1),|x|<=1;
+                                       C
+
+                                              2
+                             arcsinhx=x.R (1+x ,1),  etc
+                                         C
+
+          In general this method of calculating these elementary functions
+          is not recommended as there are usually much more efficient
+          specific routines available in the Library. However, S21BAF may
+          be used, for example, to compute lnx/(x-1) when x is close to 1,
+          without the loss of significant figures that occurs when lnx and
+          x-1 are computed separately.
+
+          3.2. Bessel and Airy Functions
+
+          For computing the Bessel functions J    (x), Y    (x), I    (x)
+                                              (nu)      (nu)      (nu)
+          and K    (x) where x is real and (nu)=0 or 1, special routines
+               (nu)
+          are provided, which are much faster than the more general
+          routines that allow a complex argument and arbitrary real (nu)>=0
+          functions and their derivatives Ai(x), Bi(x), Ai'(x), Bi'(x) for
+          a real argument which are much faster than the routines for
+          complex arguments.
+
+          3.3. Index
+
+          Airy function, Ai, real argument                           S17AGF
+          Airy function, Ai', real argument                          S17AJF
+          Airy function, Ai or Ai', complex argument, optionally     S17DGF
+          scaled
+          Airy function, Bi, real argument                           S17AHF
+          Airy function, Bi', real argument                          S17AKF
+          Airy function, Bi or Bi', complex argument, optionally     S17DHF
+          scaled
+          Bessel function, J , real argument                         S17AEF
+                            0
+          Bessel function, J , real argument                         S17AFF
+                            1
+          Bessel function, J    , complex argument, optionally       S17DEF
+                            (nu)
+          scaled
+          Bessel function, Y , real argument                         S17ACF
+                            0
+          Bessel function, Y , real argument                         S17ADF
+                            1
+          Bessel function, Y    , complex argument, optionally       S17DCF
+                            (nu)
+          scaled
+          Complement of the Error function                           S15ADF
+          Cosine Integral                                            S13ACF
+          Elliptic integral, symmetrised, degenerate of 1st kind,    S21BAF
+          R
+           C
+          Elliptic integral, symmetrised, of 1st kind, R             S21BBF
+                                                        F
+          Elliptic integral, symmetrised, of 2nd kind, R             S21BCF
+                                                        D
+          Elliptic integral, symmetrised, of 3rd kind, R             S21BDF
+                                                        J
+          Erf, real argument                                         S15AEF
+          Erfc, real argument                                        S15ADF
+          Error function                                             S15AEF
+          Exponential, complex                                       S01EAF
+          Exponential Integral                                       S13AAF
+          Fresnel Integral, C                                        S20ADF
+          Fresnel Integral, S                                        S20ACF
+          Gamma function                                             S14AAF
+          Gamma function, incomplete                                 S14BAF
+          Generalized Factorial function                             S14AAF
+                           (1)      (2)
+          Hankel function H     or H    , complex argument,          S17DLF
+                           (nu)     (nu)
+          optionally scaled
+          Incomplete Gamma function                                  S14BAF
+          Jacobian elliptic functions, sn, cn, dn                    S21CAF
+          Kelvin function, bei x                                     S19ABF
+          Kelvin function, ber x                                     S19AAF
+          Kelvin function, kei x                                     S19ADF
+          Kelvin function, ker x                                     S19ACF
+          Logarithm of Gamma function                                S14ABF
+          Modified Bessel function, I , real argument                S18AEF
+                                     0
+          Modified Bessel function, I , real argument                S18AFF
+                                     1
+          Modified Bessel function, I    , complex argument,         S18DEF
+                                     (nu)
+          optionally scaled
+          Modified Bessel function, K , real argument                S18ACF
+                                     0
+          Modified Bessel function, K , real argument                S18ADF
+                                     1
+          Modified Bessel function, K    , complex argument,         S18DCF
+                                     (nu)
+          optionally scaled
+          Sine integral                                              S13ADF
+
+
+          S -- Approximations of Special Functions            Contents -- S
+          Chapter S
+
+          Approximations of Special Functions
+
+                                        z
+          S01EAF  Complex exponential, e
+
+          S13AAF  Exponential integral E (x)
+                                        1
+
+          S13ACF  Cosine integral Ci(x)
+
+          S13ADF  Sine integral Si(x)
+
+          S14AAF  Gamma function
+
+          S14ABF  Log Gamma function
+
+          S14BAF  Incomplete gamma functions P(a,x) and Q(a,x)
+
+          S15ADF  Complement of error function erfc x
+
+          S15AEF  Error function erf x
+
+          S17ACF  Bessel function Y (x)
+                                   0
+
+          S17ADF  Bessel function Y (x)
+                                   1
+
+          S17AEF  Bessel function J (x)
+                                   0
+
+          S17AFF  Bessel function J (x)
+                                   1
+
+          S17AGF  Airy function Ai(x)
+
+          S17AHF  Airy function Bi(x)
+
+          S17AJF  Airy function Ai'(x)
+
+          S17AKF  Airy function Bi'(x)
+
+          S17DCF  Bessel functions Y      (z), real a>=0, complex z,
+                                    (nu)+a
+                  (nu)=0,1,2,...
+
+          S17DEF  Bessel functions J      (z), real a>=0, complex z,
+                                    (nu)+a
+                  (nu)=0,1,2,...
+
+          S17DGF  Airy functions Ai(z) and Ai'(z), complex z
+
+          S17DHF  Airy functions Bi(z) and Bi'(z), complex z
+
+                                    (j)
+          S17DLF  Hankel functions H      (z), j=1,2, real a>=0, complex z,
+                                    (nu)+a
+                  (nu)=0,1,2,...
+
+          S18ACF  Modified Bessel function K (x)
+                                            0
+
+          S18ADF  Modified Bessel function K (x)
+                                            1
+
+          S18AEF  Modified Bessel function I (x)
+                                            0
+
+          S18AFF  Modified Bessel function I (x)
+                                            1
+
+          S18DCF  Modified Bessel functions K      (z), real a>=0, complex
+                                             (nu)+a
+                  z, (nu)=0,1,2,...
+
+          S18DEF  Modified Bessel functions I      (z), real a>=0, complex
+                                             (nu)+a
+                  z, (nu)=0,1,2,...
+
+          S19AAF  Kelvin function ber x
+
+          S19ABF  Kelvin function bei x
+
+          S19ACF  Kelvin function ker x
+
+          S19ADF  Kelvin function kei x
+
+          S20ACF  Fresnel integral S(x)
+
+          S20ADF  Fresnel integral C(x)
+
+          S21BAF  Degenerate symmetrised elliptic integral of 1st kind
+                  R (x,y)
+                   C
+
+          S21BBF  Symmetrised elliptic integral of 1st kind R (x,y,z)
+                                                             F
+
+          S21BCF  Symmetrised elliptic integral of 2nd kind R (x,y,z)
+                                                             D
+
+          S21BDF  Symmetrised elliptic integral of 3rd kind R (x,y,z,r)
+                                                             J
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs01eaf}{NAG On-line Documentation: s01eaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S01EAF(3NAG)      Foundation Library (12/10/92)      S01EAF(3NAG)
+
+
+
+          S01 -- Approximations of Special Functions                 S01EAF
+                  S01EAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+                                                     z
+          S01EAF evaluates the exponential function e , for complex z.
+
+          2. Specification
+
+                 COMPLEX(KIND(1.0D0)) FUNCTION S01EAF (Z, IFAIL)
+                 INTEGER                               IFAIL
+                 COMPLEX(KIND(1.0D0))                  Z
+
+          3. Description
+
+                                                           z
+          This routine evaluates the exponential function e , taking care
+          to avoid machine overflow, and giving a warning if the result
+          cannot be computed to more than half precision. The function is
+                        z  x
+          evaluated as e =e (cosy+isiny), where x and y are the real and
+          imaginary parts respectively of z.
+
+          Since cosy and siny are less than or equal to 1 in magnitude, it
+                            x                        x         x
+          is possible that e  may overflow although e cosy or e siny does
+                                                               x+ln|cosy|
+          not. In this case the alternative formula sign(cosy)e
+          is used for the real part of the result, and
+                     x+ln|siny|
+          sign(siny)e           for the imaginary part. If either part of
+          the result still overflows, a warning is returned through
+          parameter IFAIL.
+
+          If Im z is too large, precision may be lost in the evaluation of
+          siny and cosy. Again, a warning is returned through IFAIL.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  Z -- COMPLEX(KIND(1.0D0))                              Input
+               On entry: the argument z of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               The real part of the result overflows, and is set to the
+               largest safe number with the correct sign. The imaginary
+               part of the result is meaningful.
+
+          IFAIL= 2
+               The imaginary part of the result overflows, and is set to
+               the largest safe number with the correct sign. The real part
+               of the result is meaningful.
+
+          IFAIL= 3
+               Both real and imaginary parts of the result overflow, and
+               are set to the largest safe number with the correct signs.
+
+          IFAIL= 4
+               The computed result is accurate to less than half precision,
+               due to the size of Im z.
+
+          IFAIL= 5
+               The computed result has no precision, due to the size of Im
+               z, and is set to zero.
+
+          7. Accuracy
+
+          Accuracy is limited in general only by the accuracy of the
+          Fortran intrinsic functions in the computation of siny, cosy and
+           x
+          e , where x=Re z, y=Im z. As y gets larger, precision will
+          probably be lost due to argument reduction in the evaluation of
+          the sine and cosine functions, until the warning error IFAIL = 4
+                                           
+
+          occurs when y gets larger than \/1/(epsilon), where (epsilon) is
+          the machine precision. Note that on some machines, the intrinsic
+          functions SIN and COS will not operate on arguments larger than
+                  
+
+          about \/1/(epsilon), and so IFAIL can never return as 4.
+
+          In the comparatively rare event that the result is computed by
+                                  x+ln|cosy|                x+ln|siny|
+          the formulae sign(cosy)e           and sign(siny)e          , a
+          further small loss of accuracy may be expected due to rounding
+          errors in the logarithmic function.
+
+          8. Further Comments
+
+          None.
+
+          9. Example
+
+          The example program reads values of the argument z from a file,
+          evaluates the function at each value of z and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs13aaf}{NAG On-line Documentation: s13aaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S13AAF(3NAG)      Foundation Library (12/10/92)      S13AAF(3NAG)
+
+
+
+          S13 -- Approximations of Special Functions                 S13AAF
+                  S13AAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S13AAF returns the value of the exponential integral E (x), via
+                                                                1
+          the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S13AAF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          The routine calculates an approximate value for
+
+                                     infty
+                                     /     e-u
+                              E (x)= |     -  du , x>0.
+                               1     /     u
+                                     x
+
+          For 0<x<=4, the approximation is based on the Chebyshev expansion
+
+                                   --                        1
+                   E (x)=y(t)-lnx= > 'a T (t)-lnx , where t= -x-1.
+                    1              --  r r                   2
+                                   r
+
+          For x>4,
+
+                                   -x       -x
+                                  e        e   --'
+                           E (x)= ---y(t)= --- >  a T (t),
+                            1      x        x  --  r r
+                                               r
+
+                                                     11.25-x
+                        where  t=-1.0+14.5/(x+3.25)= -------.
+                                                     3.25+x
+
+          In both cases, -1<=t<=+1.
+
+          To guard against producing underflows, if x>x   the result is set
+                                                       hi
+          directly to zero. For the value x   see the Users' Note for your
+                                           hi
+          implementation.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1965) Handbook of Mathematical
+                Functions. Dover Publications. Ch. 26.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function. Constraint: X > 0.
+               0.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               Before entry, IFAIL must be assigned a value. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               Unless the routine detects an error (see Section 6), IFAIL
+               contains 0 on exit.
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               The routine has been called with an argument less than or
+               equal to zero for which the function is not defined. The
+               result returned is zero.
+
+          7. Accuracy
+
+          If (delta) and (epsilon) are the relative errors in argument and
+          result respectively, then in principle,
+
+                                         |   -x         |
+                                         |  e           |
+                            |(epsilon)|~=| -----*(delta)|
+                                         | E (x)        |
+                                         |  1           |
+
+          so the relative error in the argument is amplified in the result
+                                -x
+          by at least a factor e  /E (x). The equality should hold if
+                                    1
+          (delta) is greater than the machine precision ((delta) due to
+          data errors etc) but if (delta) is simply a result of round-off
+          in the machine representation, it is possible that an extra
+          figure may be lost in internal calculation and round-off.
+
+          The behaviour of this amplification factor is shown in Figure 1.
+
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          It should be noted that, for small x, the amplification factor
+          tends to zero and eventually the error in the result will be
+          limited by machine precision.
+
+          For large x,
+
+                             (epsilon)~x(delta)=(Delta),
+
+          the absolute error in the argument.
+
+          8. Further Comments
+
+          None.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs13acf}{NAG On-line Documentation: s13acf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S13ACF(3NAG)      Foundation Library (12/10/92)      S13ACF(3NAG)
+
+
+
+          S13 -- Approximations of Special Functions                 S13ACF
+                  S13ACF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S13ACF returns the value of the cosine integral
+
+                                            x
+                                            / cosu-1
+                         Ci(x)=(gamma)+lnx+ | ------du,  x>0
+                                            /   u
+                                            0
+
+          via the routine name, where (gamma) denotes Euler's constant.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S13ACF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          The routine calculates an approximate value for Ci(x).
+
+          For 0<x<=16 it is based on the Chebyshev expansion
+
+                                   --'              ( x )2
+                        Ci(x)=lnx+ >   a T (t) , t=2( --) -1.
+                                   --   r r         ( 16)
+                                   r=0
+
+          For 16<x<x   where the value of x   is given in the Users' Note
+                    hi                     hi
+          for your implementation,
+
+                                     f(x)sinx  g(x)cosx
+                              Ci(x)= --------- --------
+                                        x          2
+                                                  x
+
+                      --'                   --'             ( 16)2
+          where f(x)= >   f T (t) and g(x)= >   g T (t), t=2( --) -1.
+                      --   r r              --   r r        ( x )
+                      r=0                   r=0
+
+          For x>=x  , Ci(x)=0 to within the accuracy possible (see Section
+                  hi
+          7).
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function. Constraint: X > 0.
+               0.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               The routine has been called with an argument less than or
+               equal to zero for which the function is not defined. The
+               result returned is zero.
+
+          7. Accuracy
+
+          If E and (epsilon) are the absolute and relative errors in the
+          result and (delta) is the relative error in the argument then in
+          principle these are related by
+
+                                                     | (delta)cosx|
+                 |E|~=|(delta)cosx| and |(epsilon)|~=| -----------|.
+                                                     |    Ci(x)   |
+
+          That is accuracy will be limited by machine precision near the
+          origin and near the zeros of cosx, but near the zeros of Ci(x)
+          only absolute accuracy can be maintained.
+
+          The behaviour of this amplification is shown in Figure 1.
+
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+                                        sinx
+          For large values of x, Ci(x)~ ---- therefore
+                                         x
+          (epsilon)~(delta)xcotx and since (delta) is limited by the finite
+          precision of the machine it becomes impossible to return results
+          which have any relative accuracy. That is, when x>=1/(delta) we
+          have that |Ci(x)|<=1/x~E and hence is not significantly different
+          from zero.
+
+          Hence x   is chosen such that for values of x>=x  , Ci(x) in
+                 hi                                       hi
+          principle would have values less than the machine precision and
+          so is essentially zero.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs13adf}{NAG On-line Documentation: s13adf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S13ADF(3NAG)      Foundation Library (12/10/92)      S13ADF(3NAG)
+
+
+
+          S13 -- Approximations of Special Functions                 S13ADF
+                  S13ADF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S13ADF returns the value of the sine integral
+
+                                         x
+                                         / sinu
+                                  Si(x)= | ----du,
+                                         /  u
+                                         0
+
+          via the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S13ADF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          The routine calculates an approximate value for Si(x).
+
+          For |x|<=16.0 it is based on the Chebyshev expansion
+
+                                 --'              ( x )2
+                         Si(x)=x >   a T (t) , t=2( --) -1.
+                                 --   r r         ( 16)
+                                 r=0
+
+          For 16<|x|<x  , where x   is an implementation dependent number,
+                      hi         hi
+
+                                   { (pi)  f(x)cosx  g(x)sinx}
+                      Si(x)=sign(x){ ----- --------- --------}
+                                   {  2       x          2   }
+                                   {                    x    }
+
+                      --'                   --'             ( 16)2
+          where f(x)= >   f T (t) and g(x)= >   g T (t), t=2( --) -1.
+                      --   r r              --   r r        ( x )
+                      r=0                   r=0
+
+                               1
+          For |x|>=x  , Si(x)= -(pi)signx to within machine precision.
+                    hi         2
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          There are no failure exits from this routine. The parameter IFAIL
+          has been included for consistency with other routines in this
+          chapter.
+
+          7. Accuracy
+
+          If (delta) and (epsilon) are the relative errors in the argument
+          and result, respectively, then in principle
+
+                                         | (delta)sinx|
+                            |(epsilon)|~=| -----------|.
+                                         |    Si(x)   |
+
+          The equality may hold if (delta) is greater than the machine
+          precision ((delta) due to data errors etc) but if (delta) is
+          simply due to round-off in the machine representation, then since
+          the factor relating (delta) to (epsilon) is always less than one,
+          the accuracy will be limited by machine precision.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs14aaf}{NAG On-line Documentation: s14aaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S14AAF(3NAG)      Foundation Library (12/10/92)      S14AAF(3NAG)
+
+
+
+          S14 -- Approximations of Special Functions                 S14AAF
+                  S14AAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S14AAF returns the value of the Gamma function (Gamma)(x), via
+          the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S14AAF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the Gamma function
+          (Gamma)(x). The routine is based on the Chebyshev expansion:
+
+                               --'
+                 (Gamma)(1+u)= >   a T (t) , where 0<=u<1 , t=2u-1,
+                               --   r r
+                               r=0
+
+          and uses the property (Gamma)(1+x)=x(Gamma)(x). If x=N+1+u where
+          N is integral and 0<=u<1 then it follows that:
+
+               for N>0  (Gamma)(x)=(x-1)(x-2)...(x-N)(Gamma)(1+u),
+
+               for N=0  (Gamma)(x)=(Gamma)(1+u),
+
+                                        (Gamma)(1+u)
+               for N<0  (Gamma)(x)= ---------------------.
+                                    x(x+1)(x+2)...(x-N-1)
+
+          There are four possible failures for this routine:
+
+               (i) if x is too large, there is a danger of overflow since
+               (Gamma)(x) could become too large to be represented in the
+               machine;
+
+               (ii) if x is too large and negative, there is a danger of
+               underflow;
+
+               (iii) if x is equal to a negative integer, (Gamma)(x) would
+               overflow since it has poles at such points;
+
+               (iv) if x is too near zero, there is again the danger of
+                                                                    1
+               overflow on some machines. For small x, (Gamma)(x)~= -, and
+                                                                    x
+               on some machines there exists a range of non-zero but small
+               values of x for which 1/x is larger than the greatest
+               representable value.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function. Constraint: X must
+               not be a negative integer.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               The argument is too large. On soft failure the routine
+               returns the approximate value of (Gamma)(x) at the nearest
+               valid argument.
+
+          IFAIL= 2
+               The argument is too large and negative. On soft failure the
+               routine returns zero.
+
+          IFAIL= 3
+               The argument is too close to zero. On soft failure the
+               routine returns the approximate value of (Gamma)(x) at the
+               nearest valid argument.
+
+          IFAIL= 4
+               The argument is a negative integer, at which value
+               (Gamma)(x) is infinite. On soft failure the routine returns
+               a large positive value.
+
+          7. Accuracy
+
+          Let (delta) and (epsilon) be the relative errors in the argument
+          and the result respectively. If (delta) is somewhat larger than
+          the machine precision (i.e., is due to data errors etc), then
+          (epsilon) and (delta) are approximately related by:
+
+                            (epsilon)~=|x(Psi)(x)|(delta)
+
+          (provided (epsilon) is also greater than the representation
+                                                         (Gamma)'(x)
+          error). Here (Psi)(x) is the digamma function  -----------.
+                                                         (Gamma)(x)
+          Figure 1 shows the behaviour of the error amplification factor
+          |x(Psi)(x)|.
+
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          If (delta) is of the same order as machine precision, then
+          rounding errors could make (epsilon) slightly larger than the
+          above relation predicts.
+
+          There is clearly a severe, but unavoidable, loss of accuracy for
+          arguments close to the poles of (Gamma)(x) at negative integers.
+          However relative accuracy is preserved near the pole at x=0 right
+          up to the point of failure arising from the danger of overflow.
+
+          Also accuracy will necessarily be lost as x becomes large since
+          in this region
+
+                               (epsilon)~=(delta)xlnx.
+
+          However since (Gamma)(x) increases rapidly with x, the routine
+          must fail due to the danger of overflow before this loss of
+          accuracy is too great. (e.g. for x=20, the amplification factor
+          ~= 60.)
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs14abf}{NAG On-line Documentation: s14abf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S14ABF(3NAG)      Foundation Library (12/10/92)      S14ABF(3NAG)
+
+
+
+          S14 -- Approximations of Special Functions                 S14ABF
+                  S14ABF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S14ABF returns a value for the logarithm of the Gamma function,
+          ln(Gamma)(x), via the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S14ABF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to ln(Gamma)(x). It is
+          based on two Chebyshev expansions.
+
+          For 0<x<=x     , ln(Gamma)(x)=-lnx to within machine accuracy.
+                    small
+
+          For x     <x<=15.0, the recursive relation
+               small
+          (Gamma)(1+x)=x(Gamma)(x) is used to reduce the calculation to one
+          involving (Gamma)(1+u), 0<=u<1 which is evaluated as:
+
+                                       --'
+                         (Gamma)(1+u)= >   a T (t),  t=2u-1.
+                                       --   r r
+                                       r=0
+
+          Once (Gamma)(x) has been calculated, the required result is
+          produced by taking the logarithm.
+
+          For 15.0<x<=x   ,
+                       big
+
+                                       1        1
+                      ln(Gamma)(x)=(x- -)lnx-x+ -ln2(pi)+y(x)/x
+                                       2        2
+                                           2
+                      --'             ( 15)
+          where y(x)= >   b T (t), t=2( --) -1.
+                      --   r r        ( x )
+                      r=0
+
+          For x   <x<=x     the term y(x)/x is negligible and so its
+               big     vbig
+          calculation is omitted.
+
+          For x>x     there is a danger of setting overflow so the routine
+                 vbig
+          must fail.
+
+          For x<=0.0 the function is not defined and the routine fails.
+
+          Note: x      is calculated so that if x<x     , (Gamma)(x)=1/x to
+                 small                             small
+          within machine accuracy. x    is calculated so that if x>x   ,
+                                    big                             big
+
+                                          1        1
+                         ln(Gamma)(x)=(x- -)lnx-x+ -ln2(pi)
+                                          2        2
+
+          to within machine accuracy. x     is calculated so that
+                                       vbig
+          ln(Gamma)(x    ) is close to the value returned by X02ALF(*).
+                     vbig
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function. Constraint: X > 0.
+               0.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               X <= 0.0, the function is undefined. On soft failure, the
+               routine returns zero.
+
+          IFAIL= 2
+               X is too large, the function would overflow. On soft
+               failure, the routine returns the value of the function at
+               the largest permissible argument.
+
+          7. Accuracy
+
+          Let (delta) and (epsilon) be the relative errors in the argument
+          and result respectively, and E be the absolute error in the
+          result.
+
+          If (delta) is somewhat larger than the relative machine
+          precision, then
+
+                                                  |  x*(Psi)(x) |
+            E~=|x*(Psi)(x)|(delta) and (epsilon)~=| ------------|(delta)
+                                                  | ln(Gamma)(x)|
+
+                                                  (Gamma)'(x)
+          where (Psi)(x) is the digamma function  -----------. Figure 1 and
+                                                  (Gamma)(x)
+          Figure 2 show the behaviour of these error amplification factors.
+
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+                                     Figure 2
+                   Please see figure in printed Reference Manual
+
+          These show that relative error can be controlled, since except
+          near x=1 or 2 relative error is attenuated by the function or at
+          least is not greatly amplified.
+
+                                  (    1 )
+          For large x, (epsilon)~=(1+ ---)(delta) and for small x,
+                                  (   lnx)
+                       1
+          (epsilon)~= ---(delta).
+                      lnx
+
+          The function ln(Gamma)(x) has zeros at x=1 and 2 and hence
+          relative accuracy is not maintainable near those points. However
+          absolute accuracy can still be provided near those zeros as is
+          shown above.
+
+          If however, (delta) is of the order of the machine precision,
+          then rounding errors in the routine's internal arithmetic may
+          result in errors which are slightly larger than those predicted
+          by the equalities. It should be noted that even in areas where
+          strong attenuation of errors is predicted the relative precision
+          is bounded by the effective machine precision.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs14baf}{NAG On-line Documentation: s14baf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S14BAF(3NAG)      Foundation Library (12/10/92)      S14BAF(3NAG)
+
+
+
+          S14 -- Approximations of Special Functions                 S14BAF
+                  S14BAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S14BAF computes values for the incomplete gamma functions P(a,x)
+          and Q(a,x).
+
+          2. Specification
+
+                 SUBROUTINE S14BAF (A, X, TOL, P, Q, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION A, X, TOL, P, Q
+
+          3. Description
+
+          This subroutine evaluates the incomplete gamma functions in the
+          normalised form
+
+                                              x
+                                       1      / a-1 -t
+                           P(a,x)= ---------- |t   e  dt,
+                                   (Gamma)(a) /
+                                              0
+
+                                            infty
+                                     1      /     a-1 -t
+                         Q(a,x)= ---------- |    t   e  dt,
+                                 (Gamma)(a) /
+                                            x
+
+          with x>=0 and a>0, to a user-specified accuracy. With this
+          normalisation, P(a,x)+Q(a,x)=1.
+
+          Several methods are used to evaluate the functions depending on
+          the arguments a and x, the methods including Taylor expansion for
+          P(a,x), Legendre's continued fraction for Q(a,x), and power
+          series for Q(a,x). When both a and x are large, and a~=x, the
+          uniform asymptotic expansion of Temme [3] is employed for greater
+          efficiency - specifically, this expansion is used when a>=20 and
+          0.7a<=x<=1.4a.
+
+          Once either of P or Q is computed, the other is obtained by
+          subtraction from 1. In order to avoid loss of relative precision
+          in this subtraction, the smaller of P and Q is computed first.
+
+          This routine is derived from subroutine GAM in Gautschi [2].
+
+          4. References
+
+          [1]   Gautschi W (1979) A Computational Procedure for Incomplete
+                Gamma Functions. ACM Trans. Math. Softw. 5 466--481.
+
+          [2]   Gautschi W (1979) Algorithm 542: Incomplete Gamma Functions.
+                ACM Trans. Math. Softw. 5 482--489.
+
+          [3]   Temme N M (1987) On the Computation of the Incomplete Gamma
+                Functions for Large Values of the Parameters. Algorithms for
+                Approximation. (ed J C Mason and M G Cox) Oxford University
+                Press.
+
+          5. Parameters
+
+           1:  A -- DOUBLE PRECISION                                  Input
+               On entry: the argument a of the functions. Constraint: A >
+               0.0.
+
+           2:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the functions. Constraint: X >=
+               0.0.
+
+           3:  TOL -- DOUBLE PRECISION                                Input
+               On entry: the relative accuracy required by the user in the
+               results. If S14BAF is entered with TOL greater than 1.0 or
+               less than machine precision, then the value of machine
+               precision is used instead.
+
+           4:  P -- DOUBLE PRECISION                                 Output
+
+           5:  Q -- DOUBLE PRECISION                                 Output
+               On exit: the values of the functions P(a,x) and Q(a,x)
+               respectively.
+
+           6:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry A <= 0.0.
+
+          IFAIL= 2
+               On entry X < 0.0.
+
+          IFAIL= 3
+               Convergence of the Taylor series or Legendre continued
+               fraction fails within 600 iterations. This error is
+               extremely unlikely to occur; if it does, contact NAG.
+
+          7. Accuracy
+
+          There are rare occasions when the relative accuracy attained is
+          somewhat less than that specified by parameter TOL. However, the
+          error should never exceed more than one or two decimal places.
+          Note also that there is a limit of 18 decimal places on the
+          achievable accuracy, because constants in the routine are given
+          to this precision.
+
+          8. Further Comments
+
+          The time taken for a call of S14BAF depends on the precision
+          requested through TOL, and also varies slightly with the input
+          arguments a and x.
+
+          9. Example
+
+          The example program reads values of the argument a and x from a
+          file, evaluates the function and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs15adf}{NAG On-line Documentation: s15adf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S15ADF(3NAG)      Foundation Library (12/10/92)      S15ADF(3NAG)
+
+
+
+          S15 -- Approximations of Special Functions                 S15ADF
+                  S15ADF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S15ADF returns the value of the complementary error function,
+          erfcx, via the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S15ADF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          The routine calculates an approximate value for the complement of
+          the error function
+
+                                                2
+                                        infty -u
+                                   2    /
+                         erfc x= ------ |    e   du=1-erf x.
+                                        /
+                                 \/(pi) x
+
+          For x>=0, it is based on the Chebyshev expansion
+
+                                            2
+                                          -x
+                                  erfc x=e   y(x),
+
+                      --'
+          where y(x)= >   a T (t) and t=(x-3.75)/(x+3.75), -1<=t<=+1.
+                      --   r r
+                      r=0
+
+          For x>=x  , where there is a danger of setting underflow, the
+                  hi
+          result is returned as zero.
+
+                               2
+                             -x
+          For x<0, erfc x=2-e   y(|x|).
+
+          For x<x   <0, the result is returned as 2.0 which is correct to
+                 low
+          within machine precision. The values of x   and x    are given in
+                                                   hi      low
+          the Users' Note for your implementation.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          There are no failure exits from this routine. The parameter IFAIL
+          has been included for consistency with other routines in this
+          chapter.
+
+          7. Accuracy
+
+          If (delta) and (epsilon) are relative errors in the argument and
+          result, respectively, then in principle
+
+                                     |         2          |
+                                     |       -x           |
+                                     |    2xe             |
+                        |(epsilon)|~=| ------------(delta)|.
+                                     |                    |
+
+                                     | \/(pi)erfc x       |
+
+          That is, the relative error in the argument, x, is amplified by a
+                          2
+                        -x
+                     2xe
+          factor  ------------ in the result.
+                    
+
+                  \/(pi)erfc x
+
+          The behaviour of this factor is shown in Figure 1.
+
+
+                                     Figure 1
+
+                   Please see figure in printed Reference Manual
+
+                                                                     2x
+          It should be noted that near x=0 this factor behaves as  ------
+                                                                     
+
+                                                                   \/(pi)
+          and hence the accuracy is largely determined by the machine
+          precision. Also for large negative x, where the factor is
+                2
+              -x
+            xe
+          ~ ------, accuracy is mainly limited by machine precision.
+              
+
+            \/(pi)
+                                                               2
+          However, for large positive x, the factor becomes ~2x  and to an
+          extent relative accuracy is necessarily lost. The absolute
+          accuracy E is given by
+
+                                           2
+                                         -x
+                                      2xe
+                                  E~= ------(delta)
+                                        
+
+                                      \/(pi)
+
+          so absolute accuracy is guaranteed for all x.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs15aef}{NAG On-line Documentation: s15aef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S15AEF(3NAG)      Foundation Library (12/10/92)      S15AEF(3NAG)
+
+
+
+          S15 -- Approximations of Special Functions                 S15AEF
+                  S15AEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S15AEF returns the value of the error function erfx, via the
+          routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S15AEF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          Evaluates the error function,
+
+                                   2
+                               x -t
+                          2    /
+               erf x =  ------ |e   dt.
+                               /
+                        \/(pi) 0
+
+          For |x|<=2,
+                          --'                   1 2
+                erf x = x >   a T (t), where t= -x -1.
+                          --   r r              2
+                          r=0
+
+          For 2<|x|<x  ,
+                     hi
+                             {        2               }
+                             {      -x                }
+                             {     e       --         }           x-7
+                erf x = signx{1- --------- >  'b T (t)}, where t= ---.
+                             {             --   r r   }           x+3
+                             {   |x|\/(pi) r=0        }
+
+          For |x|>=x  ,
+                    hi
+                erf x =  signx.
+
+          x   is the value above which erf x=+-1 within machine precision.
+           hi
+          Its value is given in the Users' Note for your implementation.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          There are no failure exits from this routine. The parameter IFAIL
+          has been included for consistency with other routines in this
+          chapter.
+
+          7. Accuracy
+
+          On a machine with approximately 11 significant figures the
+          routine agrees with available tables to 10 figures and
+          consistency checking with S15ADF of the relation
+
+                                  erf x+erfc x=1.0
+
+          shows errors in at worst the 11th figure.
+
+          8. Further Comments
+
+          None.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs17acf}{NAG On-line Documentation: s17acf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S17ACF(3NAG)      Foundation Library (12/10/92)      S17ACF(3NAG)
+
+
+
+          S17 -- Approximations of Special Functions                 S17ACF
+                  S17ACF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S17ACF returns the value of the Bessel Function Y (x), via the
+                                                           0
+          routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S17ACF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the Bessel Function of
+          the second kind Y (x).
+                           0
+
+          Note: Y (x) is undefined for x<=0 and the routine will fail for
+                 0
+          such arguments.
+
+          The routine is based on four Chebyshev expansions:
+
+          For 0<x<=8,
+                                                                   2
+                      2      --'          --'                  ( x)
+              Y (x)= ----lnx >   a T (t)+ >   b T (t), with t=2( -) -1,
+               0     (pi)    --   r r     --   r r             ( 8)
+                             r=0          r=0
+
+          For x>8,
+
+                          
+
+                         /   2  {        (   (pi))         (   (pi))}
+                Y (x)=  /  -----{P (x)sin(x- ----)+Q (x)cos(x- ----)}
+                 0    \/   (pi)x{ 0      (    4  )  0      (    4  )}
+
+                       --'
+          where P (x)= >   c T (t),
+                 0     --   r r
+                       r=0
+                                                2
+                     8 --'                  ( 8)
+          and Q (x)= - >   d T (t), with t=2( -) -1.
+               0     x --   r r             ( x)
+                       r=0
+
+                                    2  (  ( x)        )
+          For x near zero, Y (x)~= ----(ln( -)+(gamma)), where (gamma)
+                            0      (pi)(  ( 2)        )
+          denotes Euler's constant. This approximation is used when x is
+          sufficiently small for the result to be correct to machine
+          precision.
+
+          For very large x, it becomes impossible to provide results with
+          any reasonable accuracy (see Section 7), hence the routine fails.
+          Such arguments contain insufficient information to determine the
+                                                                 
+
+                                                                / 2
+          phase of oscillation of Y (x); only the amplitude,   /  -, can be
+                                   0                         \/   x
+          determined and this is returned on soft failure. The range for
+          which this occurs is roughly related to the machine precision:
+          the routine will fail if x>~1/machine precision (see the Users'
+          Note for your implementation for details).
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Clenshaw C W (1962) Mathematical Tables. Chebyshev Series
+                for Mathematical Functions. HMSO.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function. Constraint: X > 0.
+               0.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               X is too large. On soft failure the routine returns the
+               amplitude of the Y  oscillation, \/2/(pi)x.
+                                 0
+
+          IFAIL= 2
+               X <= 0.0, Y  is undefined. On soft failure the routine
+                          0
+               returns zero.
+
+          7. Accuracy
+
+          Let (delta) be the relative error in the argument and E be the
+          absolute error in the result. (Since Y (x) oscillates about zero,
+                                                0
+          absolute error and not relative error is significant, except for
+          very small x.)
+
+          If (delta) is somewhat larger than the machine representation
+          error (e.g. if (delta) is due to data errors etc), then E and
+          (delta) are approximately related by
+
+                                 E~=|xY (x)|(delta)
+                                       1
+
+          (provided E is also within machine bounds). Figure 1 displays the
+          behaviour of the amplification factor |xY (x)|.
+                                                   1
+
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          However, if (delta) is of the same order as the machine
+          representation errors, then rounding errors could make E slightly
+          larger than the above relation predicts.
+
+          For very small x, the errors are essentially independent of
+          (delta) and the routine should provide relative accuracy bounded
+          by the machine precision.
+
+          For very large x, the above relation ceases to apply. In this
+                                                                   
+
+                            /   2     (   (pi))                   /   2
+          region, Y (x)~=  /  -----sin(x- ----). The amplitude   /  -----
+                   0     \/   (pi)x   (    4  )                /   (pi)x
+          can be calculated with reasonable accuracy for all x, but
+             (   (pi))               (pi)
+          sin(x- ----) cannot. If x- ---- is written as 2N(pi)+(theta)
+             (    4  )                4
+                                                              (   (pi))
+          where N is an integer and 0<=(theta)<2(pi), then sin(x- ----) is
+                                                              (    4  )
+                                                   -1
+          determined by (theta) only. If x>~(delta)  , (theta) cannot be
+          determined with any accuracy at all. Thus if x is greater than,
+          or of the order of the inverse of machine precision, it is
+          impossible to calculate the phase of Y (x) and the routine must
+                                                0
+          fail.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs17adf}{NAG On-line Documentation: s17adf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S17ADF(3NAG)      Foundation Library (12/10/92)      S17ADF(3NAG)
+
+
+
+          S17 -- Approximations of Special Functions                 S17ADF
+                  S17ADF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S17ADF returns the value of the Bessel Function Y (x), via the
+                                                           1
+          routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S17ADF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the Bessel Function of
+          the second kind Y (x).
+                           1
+
+          Note: Y (x) is undefined for x<=0 and the routine will fail for
+                 1
+          such arguments.
+
+          The routine is based on four Chebyshev expansions:
+
+          For 0<x<=8,
+
+                        2      x --'           2     x --'
+                Y (x)= ----lnx - >   a T (t)- ----x+ - >   b T (t) , with t
+                 1     (pi)    8 --   r r     (pi)   8 --   r r
+                                 r=0                   r=0
+                           2
+                       ( x)
+                     =2( -) -1;
+                       ( 8)
+
+
+          For x>8,
+                            
+
+                           /   2  {        (    (pi))         (    (pi))}
+                  Y (x)=  /  -----{P (x)sin(x-3 ----)+Q (x)cos(x-3 ----)}
+                   1    \/   (pi)x{ 1      (     4  )  1      (     4  )}
+
+                             --'
+                where P (x)= >   c T (t),
+                       1     --   r r
+                             r=0
+                                                      2
+                           8 --'                  ( 8)
+                and Q (x)= - >   d T (t), with t=2( -) -1.
+                     1     x --   r r             ( x)
+                             r=0
+
+                                      2
+          For x near zero, Y (x)~=- -----. This approximation is used when
+                            1       (pi)x
+          x is sufficiently small for the result to be correct to machine
+          precision. For extremely small x, there is a danger of overflow
+                             2
+          in calculating - ----- and for such arguments the routine will
+                           (pi)x
+          fail.
+
+          For very large x, it becomes impossible to provide results with
+          any reasonable accuracy (see Section 7), hence the routine fails.
+          Such arguments contain insufficient information to determine the
+                                                                 
+
+                                                                /   2
+          phase of oscillation of Y (x), only the amplitude,   /  -----,
+                                   1                         \/   (pi)x
+          can be determined and this is returned on soft failure. The range
+          for which this occurs is roughly related to machine precision;
+          the routine will fail if x>~1/machine precision (see the Users'
+          Note for your implementation for details).
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Clenshaw C W (1962) Mathematical Tables. Chebyshev Series
+                for Mathematical Functions. HMSO.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function. Constraint: X > 0.
+               0.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               X is too large. On soft failure the routine returns the
+                                                   
+
+                                                  /   2
+               amplitude of the Y  oscillation,  /  -----.
+                                 1             \/   (pi)x
+
+          IFAIL= 2
+               X <= 0.0, Y  is undefined. On soft failure the routine
+                          1
+               returns zero.
+
+          IFAIL= 3
+               X is too close to zero, there is a danger of overflow. On
+               soft failure, the routine returns the value of Y (x) at the
+                                                               1
+               smallest valid argument.
+
+          7. Accuracy
+
+          Let (delta) be the relative error in the argument and E be the
+          absolute error in the result. (Since Y (x) oscillates about zero,
+                                                1
+          absolute error and not relative error is significant, except for
+          very small x.)
+
+          If (delta) is somewhat larger than the machine precision (e.g. if
+          (delta) is due to data errors etc), then E and (delta) are
+          approximately related by:
+
+                              E~=|xY (x)-Y (x)|(delta)
+                                    0     1
+
+          (provided E is also within machine bounds). Figure 1 displays the
+          behaviour of the amplification factor |xY (x)-Y (x)|.
+                                                   0     1
+
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          However, if (delta) is of the same order as machine precision,
+          then rounding errors could make E slightly larger than the above
+          relation predicts.
+
+          For very small x, absolute error becomes large, but the relative
+          error in the result is of the same order as (delta).
+
+          For very large x, the above relation ceases to apply. In this
+                          2(pi)   (   3(pi))                 2(pi)
+          region, Y (x)~= -----sin(x- -----). The amplitude  ----- can be
+                   1        x     (     4  )                   x
+                                                                (   3(pi))
+          calculated with reasonable accuracy for all x, but sin(x- -----)
+                                                                (     4  )
+                        3(pi)
+          cannot. If x- ----- is written as 2N(pi)+(theta) where N is an
+                          4
+                                                (   3(pi))
+          integer and 0<=(theta)<2(pi), then sin(x- -----) is determined by
+                                                (     4  )
+                                    -1
+          (theta) only. If x>(delta)  , (theta) cannot be determined with
+          any accuracy at all. Thus if x is greater than, or of the order
+          of, the inverse of the machine precision, it is impossible to
+          calculate the phase of Y (x) and the routine must fail.
+                                  1
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs17aef}{NAG On-line Documentation: s17aef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S17AEF(3NAG)      Foundation Library (12/10/92)      S17AEF(3NAG)
+
+
+
+          S17 -- Approximations of Special Functions                 S17AEF
+                  S17AEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S17AEF returns the value of the Bessel Function J (x), via the
+                                                           0
+          routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S17AEF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the Bessel Function of
+          the first kind J (x).
+                          0
+
+          Note: J (-x)=J (x), so the approximation need only consider x>=0.
+                 0      0
+
+          The routine is based on three Chebyshev expansions:
+
+          For 0<x<=8,                           2
+                       --'                  ( x)
+                J (x)= >   a T (t), with t=2( -) -1.
+                 0     --   r r             ( 8)
+                       r=0
+
+          For x>8,
+                          
+
+                         /   2  {        (   (pi))         (   (pi))}
+                J (x)=  /  -----{P (x)cos(x- ----)-Q (x)sin(x- ----)}
+                 0    \/   (pi)x{ 0      (    4  )  0      (    4  )}
+
+                               --'
+                where P (x) =  >   b T (t),
+                       0       --   r r
+                               r=0
+                                                        2
+                             8 --'                  ( 8)
+                and Q (x) =  - >   c T (t), with t=2( -) -1.
+                     0       x --   r r             ( x)
+                               r=0
+
+          For x near zero, J (x)~=1. This approximation is used when x is
+                            0
+          sufficiently small for the result to be correct to machine
+          precision.
+
+          For very large x, it becomes impossible to provide results with
+          any reasonable accuracy (see Section 7), hence the routine fails.
+          Such arguments contain insufficient information to determine the
+                                                                 
+
+                                                                /    2
+          phase of oscillation of J (x); only the amplitude,   /  -------,
+                                   0                         \/   (pi)|x|
+          can be determined and this is returned on soft failure. The range
+          for which this occurs is roughly related to the machine
+          precision; the routine will fail if |x|>~1/machine precision (see
+          the Users' Note for your implementation).
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Clenshaw C W (1962) Mathematical Tables. Chebyshev Series
+                for Mathematical Functions. HMSO.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               X is too large. On soft failure the routine returns the
+                                                     
+
+                                                    /    2
+               amplitude of the J  oscillation,    /  -------.
+                                 0               \/   (pi)|x|
+
+          7. Accuracy
+
+          Let (delta) be the relative error in the argument and E be the
+          absolute error in the result. (Since J (x) oscillates about zero,
+                                                0
+          absolute error and not relative error is significant.)
+
+          If (delta) is somewhat larger than the machine precision (e.g. if
+          (delta) is due to data errors etc), then E and (delta) are
+          approximately related by:
+
+                                 E~=|xJ (x)|(delta)
+                                       1
+
+          (provided E is also within machine bounds). Figure 1 displays the
+          behaviour of the amplification factor |xJ (x)|.
+                                                   1
+
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          However, if (delta) is of the same order as machine precision,
+          then rounding errors could make E slightly larger than the above
+          relation predicts.
+
+          For very large x, the above relation ceases to apply. In this
+                             
+
+                            /    2      (   (pi))
+          region, J (x)~=  /  -------cos(x- ----). The amplitude
+                   0     \/   (pi)|x|   (    4  )
+              
+
+             /    2
+            /  ------- can be calculated with reasonable accuracy for all x
+          \/   (pi)|x|
+                 (   (pi))               (pi)
+          but cos(x- ----) cannot. If x- ---- is written as
+                 (    4  )                4
+          2N(pi)+(theta) where N is an integer and 0 <= (theta) < 2(pi),
+                  (   (pi))                                             -1
+          then cos(x- ----) is determined by (theta) only. If x>~(delta)  ,
+                  (    4  )
+          (theta) cannot be determined with any accuracy at all. Thus if x
+          is greater than, or of the order of, the inverse of the machine
+          precision, it is impossible to calculate the phase of J (x) and
+                                                                 0
+          the routine must fail.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs17aff}{NAG On-line Documentation: s17aff}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S17AFF(3NAG)      Foundation Library (12/10/92)      S17AFF(3NAG)
+
+
+
+          S17 -- Approximations of Special Functions                 S17AFF
+                  S17AFF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S17AFF returns the value of the Bessel Function J (x), via the
+                                                           1
+          routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S17AFF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the Bessel Function of
+          the first kind J (x).
+                          1
+
+          Note: J (-x)=-J (x), so the approximation need only consider x>=0
+                 1       1
+
+          The routine is based on three Chebyshev expansions:
+
+          For 0<x<=8,                              2
+                       x --'                   ( x)
+                J (x)= - >   a T (t) , with t=2( -) -1.
+                 1     8 --   r r              ( 8)
+                         r=0
+
+          For x>8,
+                          
+
+                         /  2   {        (   3(pi))         (   3(pi))}
+                J (x)=  /  ----x{P (x)cos(x- -----)-Q (x)sin(x- -----)}
+                 1    \/   (pi) { 1      (     4  )  1      (     4  )}
+
+                            --'
+                whereP (x)= >   b T (t),
+                      1     --   r r
+                            r=0
+                                                      2
+                           8 --'                  ( 8)
+                and Q (x)= - >   c T (t), with t=2( -) -1.
+                     1     x --   r r             ( x)
+                             r=0
+                                   x
+          For x near zero, J (x)~= -. This approximation is used when x is
+                            1      2
+          sufficiently small for the result to be correct to machine
+          precision.
+
+          For very large x, it becomes impossible to provide results with
+          any reasonable accuracy (see Section 7), hence the routine fails.
+          Such arguments contain insufficient information to determine the
+                                                                 
+
+                                                                /    2
+          phase of oscillation of J (x); only the amplitude,   /  -------,
+                                   1                         \/   (pi)|x|
+          can be determined and this is returned on soft failure. The range
+          for which this occurs is roughly related to the machine
+          precision; the routine will fail if |x|>~1/machine precision (see
+          the Users' Note for your implementation for details).
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Clenshaw C W (1962) Mathematical Tables. Chebyshev Series
+                for Mathematical Functions. HMSO.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               X is too large. On soft failure the routine returns the
+                                                     
+
+                                                    /    2
+               amplitude of the J  oscillation,    /  -------.
+                                 1               \/   (pi)|x|
+
+          7. Accuracy
+
+
+          Let (delta) be the relative error in the argument and E be the
+          absolute error in the result. (Since J (x) oscillates about zero,
+                                                1
+          absolute error and not relative error is significant.)
+
+          If (delta) is somewhat larger than machine precision (e.g. if
+          (delta) is due to data errors etc), then E and (delta) are
+          approximately related by:
+
+                              E~=|xJ (x)-J (x)|(delta)
+                                    0     1
+
+          (provided E is also within machine bounds). Figure 1 displays the
+          behaviour of the amplification factor |xJ (x)-J (x)|.
+                                                   0     1
+
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          However, if (delta) is of the same order as machine precision,
+          then rounding errors could make E slightly larger than the above
+          relation predicts.
+
+          For very large x, the above relation ceases to apply. In this
+                             
+
+                            /    2      (   3(pi))
+          region, J (x)~=  /  -------cos(x- -----). The amplitude
+                   1     \/   (pi)|x|   (     4  )
+              
+
+             /    2
+            /  ------- can be calculated with reasonable accuracy for all x
+          \/   (pi)|x|
+                 (   3(pi))               3(pi)
+          but cos(x- -----) cannot. If x- ----- is written as
+                 (     4  )                 4
+          2N(pi)+(theta) where N is an integer and 0<=(theta)<2(pi), then
+             (   3(pi))                                             -1
+          cos(x- -----) is determined by (theta) only. If x>~(delta)  ,
+             (     4  )
+          (theta) cannot be determined with any accuracy at all. Thus if x
+          is greater than, or of the order of, machine precision, it is
+          impossible to calculate the phase of J (x) and the routine must
+                                                1
+          fail.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs17agf}{NAG On-line Documentation: s17agf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S17AGF(3NAG)      Foundation Library (12/10/92)      S17AGF(3NAG)
+
+
+
+          S17 -- Approximations of Special Functions                 S17AGF
+                  S17AGF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S17AGF returns a value for the Airy function, Ai(x), via the
+          routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S17AGF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the Airy function,
+          Ai(x). It is based on a number of Chebyshev expansions:
+
+          For x<-5,
+
+                                     a(t)sinz-b(t)cosz
+                              Ai(x)= -----------------
+                                              1/4
+                                          (-x)
+
+                             
+
+                   (pi)  2  /  3
+          where z= ----+ -\/ -x , and a(t) and b(t) are expansions in the
+                    4    3
+                           3
+                       ( 5)
+          variable t=-2( -) -1.
+                       ( x)
+
+          For -5<=x<=0,
+
+                                  Ai(x)=f(t)-xg(t),
+                                                  3
+                                              ( x)
+          where f and g are expansions in t=-2( -) -1.
+                                              ( 5)
+
+          For 0<x<4.5,
+
+                                         -3x/2
+                                  Ai(x)=e     y(t),
+
+          where y is an expansion in t=4x/9-1.
+
+          For 4.5<=x<9,
+
+                                         -5x/2
+                                  Ai(x)=e     u(t),
+
+          where u is an expansion in t=4x/9-3.
+
+          For x>=9,
+
+                                           -z
+                                          e  v(t)
+                                   Ai(x)= -------,
+                                            1/4
+                                           x
+
+                       
+
+                   2  / 3                             ( 18)
+          where z= -\/ x  and v is an expansion in t=2( --)-1.
+                   3                                  ( z )
+
+          For |x|< the machine precision, the result is set directly to
+          Ai(0). This both saves time and guards against underflow in
+          intermediate calculations.
+
+          For large negative arguments, it becomes impossible to calculate
+          the phase of the oscillatory function with any precision and so
+                                                                2/3
+                                                   (     3     )
+          the routine must fail. This occurs if x<-( ----------)   , where
+                                                   ( 2(epsilon))
+          (epsilon) is the machine precision.
+
+          For large positive arguments, where Ai decays in an essentially
+          exponential manner, there is a danger of underflow so the routine
+          must fail.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               X is too large and positive. On soft failure, the routine
+               returns zero.
+
+          IFAIL= 2
+               X is too large and negative. On soft failure, the routine
+               returns zero.
+
+          7. Accuracy
+
+          For negative arguments the function is oscillatory and hence
+          absolute error is the appropriate measure. In the positive region
+          the function is essentially exponential-like and here relative
+          error is appropriate. The absolute error, E, and the relative
+          error, (epsilon), are related in principle to the relative error
+          in the argument, (delta), by
+
+                                                 | xAi'(x)|
+                 E~=|xAi'(x)|(delta), (epsilon)~=| -------|(delta).
+                                                 |  Ai(x) |
+
+          In practice, approximate equality is the best that can be
+          expected. When (delta), (epsilon) or E is of the order of the
+          machine precision, the errors in the result will be somewhat
+          larger.
+
+          For small x, errors are strongly damped by the function and hence
+          will be bounded by the machine precision.
+
+          For moderate negative x, the error behaviour is oscillatory but
+          the amplitude of the error grows like
+
+                                                     5/4
+                                      (    E   )  |x|
+                            amplitude ( -------)~ ------.
+                                      ( (delta))    
+                                                  \/(pi)
+
+                                                                    
+
+                                                                2  /   3
+          However the phase error will be growing roughly like  -\/ |x|
+                                                                3
+          and hence all accuracy will be lost for large negative arguments
+          due to the impossibility of calculating sin and cos to any
+                           
+
+                       2  /   3     1
+          accuracy if  -\/ |x| > -------.
+                       3         (delta)
+
+          For large positive arguments, the relative error amplification is
+          considerable:
+
+                                                
+
+                                   (epsilon)   / 3
+                                   ---------~\/ x .
+                                    (delta)
+
+          This means a loss of roughly two decimal places accuracy for
+          arguments in the region of 20. However very large arguments are
+          not possible due to the danger of setting underflow and so the
+          errors are limited in practice.
+
+          8. Further Comments
+
+          None.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs17ahf}{NAG On-line Documentation: s17ahf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S17AHF(3NAG)      Foundation Library (12/10/92)      S17AHF(3NAG)
+
+
+
+          S17 -- Approximations of Special Functions                 S17AHF
+                  S17AHF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S17AHF returns a value of the Airy function, Bi(x), via the
+          routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S17AHF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the Airy function
+          Bi(x). It is based on a number of Chebyshev expansions.
+
+          For x<-5,
+
+                                     a(t)cosz+b(t)sinz
+                              Bi(x)= -----------------,
+                                              1/4
+                                          (-x)
+
+                             
+
+                   (pi)  2  /  3
+          where z= ----+ -\/ -x  and a(t) and b(t) are expansions in the
+                    4    3
+                           3
+                       ( 5)
+          variable t=-2( -) -1.
+                       ( x)
+
+          For -5<=x<=0,
+
+                                       
+
+                               Bi(x)=\/3(f(t)+xg(t)),
+                                                  3
+                                              ( x)
+          where f and g are expansions in t=-2( -) -1.
+                                              ( 5)
+
+          For 0<x<4.5,
+
+                                         11x/8
+                                  Bi(x)=e     y(t),
+
+          where y is an expansion in t=4x/9-1.
+
+          For 4.5<=x<=9,
+
+                                         5x/2
+                                  Bi(x)=e    v(t),
+
+          where v is an expansion in t=4x/9-3.
+
+          For x>=9,
+
+                                           z
+                                          e u(t)
+                                   Bi(x)= ------,
+                                            1/4
+                                           x
+
+                       
+
+                   2  / 3                             ( 18)
+          where z= -\/ x  and u is an expansion in t=2( --)-1.
+                   3                                  ( z )
+
+          For |x|< the machine precision, the result is set directly to
+          Bi(0). This both saves time and avoids possible intermediate
+          underflows.
+
+          For large negative arguments, it becomes impossible to calculate
+          the phase of the oscillating function with any accuracy so the
+                                                            2/3
+                                               (     3     )
+          routine must fail. This occurs if x<-( ----------)   , where
+                                               ( 2(epsilon))
+          (epsilon) is the machine precision.
+
+          For large positive arguments, there is a danger of causing
+          overflow since Bi grows in an essentially exponential manner, so
+          the routine must fail.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               X is too large and positive. On soft failure, the routine
+               returns zero.
+
+          IFAIL= 2
+               X is too large and negative. On soft failure, the routine
+               returns zero.
+
+          7. Accuracy
+
+          For negative arguments the function is oscillatory and hence
+          absolute error is the appropriate measure. In the positive region
+          the function is essentially exponential-like and here relative
+          error is appropriate. The absolute error, E, and the relative
+          error, (epsilon), are related in principle to the relative error
+          in the argument, (delta), by
+
+                                                 | xBi'(x)|
+                 E~=|xBi'(x)|(delta), (epsilon)~=| -------|(delta).
+                                                 |  Bi(x) |
+
+          In practice, approximate equality is the best that can be
+          expected. When (delta), (epsilon) or E is of the order of the
+          machine precision, the errors in the result will be somewhat
+          larger.
+
+          For small x, errors are strongly damped and hence will be bounded
+          essentially by the machine precision.
+
+          For moderate to large negative x, the error behaviour is clearly
+          oscillatory but the amplitude of the error grows like amplitude
+                         5/4
+          (    E   )  |x|
+          ( -------)~ ------.
+          ( (delta))    
+                      \/(pi)
+
+                                                                  
+
+                                                              2  /   3
+          However the phase error will be growing roughly as  -\/ |x|  and
+                                                              3
+          hence all accuracy will be lost for large negative arguments.
+
+          This is due to the impossibility of calculating sin and cos to
+                               
+
+                           2  /   3     1
+          any accuracy if  -\/ |x| > -------.
+                           3         (delta)
+
+          For large positive arguments, the relative error amplification is
+          considerable:
+
+                                                
+
+                                   (epsilon)   / 3
+                                   ---------~\/ x .
+                                    (delta)
+
+          This means a loss of roughly two decimal places accuracy for
+          arguments in the region of 20. However very large arguments are
+          not possible due to the danger of causing overflow and errors are
+          therefore limited in practice.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs17ajf}{NAG On-line Documentation: s17ajf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S17AJF(3NAG)      Foundation Library (12/10/92)      S17AJF(3NAG)
+
+
+
+          S17 -- Approximations of Special Functions                 S17AJF
+                  S17AJF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S17AJF returns a value of the derivative of the Airy function
+          Ai(x), via the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S17AJF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the derivative of the
+          Airy function Ai(x). It is based on a number of Chebyshev
+          expansions.
+
+          For x<-5,
+
+                                4   [           b(t)     ]
+                         Ai'(x)=\/-x[a(t)cosz+ ------sinz],
+                                    [          (zeta)    ]
+
+                                            
+
+                   (pi)                 2  /  3
+          where z= ----+(zeta), (zeta)= -\/ -x  and a(t) and b(t) are
+                    4                   3
+                                         3
+                                     ( 5)
+          expansions in variable t=-2( -) -1.
+                                     ( x)
+
+          For -5<=x<=0,
+
+                                         2
+                                 Ai'(x)=x f(t)-g(t),
+                                                  3
+                                              ( x)
+          where f and g are expansions in t=-2( -) -1.
+                                              ( 5)
+
+          For 0<x<4.5,
+
+                                         -11x/8
+                                 Ai'(x)=e      y(t),
+
+                                           ( x)
+          where y(t) is an expansion in t=4( -)-1.
+                                           ( 9)
+
+          For 4.5<=x<9,
+
+                                         -5x/2
+                                 Ai'(x)=e     v(t),
+
+                                           ( x)
+          where v(t) is an expansion in t=4( -)-3.
+                                           ( 9)
+
+          For x>=9,
+
+                                        4    -z
+                                 Ai'(x)=\/-xe  u(t),
+
+
+                   2  / 3                                ( 18)
+          where z= -\/ x  and u(t) is an expansion in t=2( --)-1.
+                   3                                     ( z )
+
+          For |x|< the square of the machine precision, the result is set
+          directly to Ai'(0). This both saves time and avoids possible
+          intermediate underflows.
+
+          For large negative arguments, it becomes impossible to calculate
+          a result for the oscillating function with any accuracy and so
+                                                                4/7
+                                                    (          )
+                                                    (  \/(pi)  )
+          the routine must fail. This occurs for x<-( ---------)   , where
+                                                    ( (epsilon))
+          (epsilon) is the machine precision.
+
+          For large positive arguments, where Ai' decays in an essentially
+          exponential manner, there is a danger of underflow so the routine
+          must fail.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               X is too large and positive. On soft failure, the routine
+               returns zero.
+
+          IFAIL= 2
+               X is too large and negative. On soft failure, the routine
+               returns zero.
+
+          7. Accuracy
+
+          For negative arguments the function is oscillatory and hence
+          absolute error is the appropriate measure. In the positive region
+          the function is essentially exponential in character and here
+          relative error is needed. The absolute error, E, and the relative
+          error, (epsilon), are related in principle to the relative error
+          in the argument, (delta), by
+
+                                                  |  2     |
+                     2                            | x Ai(x)|
+                E~=|x Ai(x)|(delta)    (epsilon)~=| -------|(delta).
+                                                  | Ai'(x) |
+
+          In practice, approximate equality is the best that can be
+          expected. When (delta), (epsilon) or E is of the order of the
+          machine precision, the errors in the result will be somewhat
+          larger.
+
+          For small x, positive or negative, errors are strongly attenuated
+          by the function and hence will be roughly bounded by the machine
+          precision.
+
+          For moderate to large negative x, the error, like the function,
+          is oscillatory; however the amplitude of the error grows like
+
+                                          7/4
+                                       |x|
+                                       ------.
+
+                                       \/(pi)
+
+
+          Therefore it becomes impossible to calculate the function with
+                                    
+
+                             7/4  \/(pi)
+          any accuracy if |x|   > -------.
+                                  (delta)
+
+          For large positive x, the relative error amplification is
+          considerable:
+
+                                                
+
+                                  (epsilon)    / 3
+                                  ---------~=\/ x .
+                                   (delta)
+
+          However, very large arguments are not possible due to the danger
+          of underflow. Thus in practice error amplification is limited.
+
+          8. Further Comments
+
+          None.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs17akf}{NAG On-line Documentation: s17akf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S17AKF(3NAG)      Foundation Library (12/10/92)      S17AKF(3NAG)
+
+
+
+          S17 -- Approximations of Special Functions                 S17AKF
+                  S17AKF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S17AKF returns a value for the derivative of the Airy function
+          Bi(x), via the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S17AKF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine calculates an approximate value for the derivative
+          of the Airy function Bi(x). It is based on a number of Chebyshev
+          expansions.
+
+          For x<-5,
+
+                                4   [            b(t)     ]
+                         Bi'(x)=\/-x[-a(t)sinz+ ------cosz],
+                                    [           (zeta)    ]
+
+                                           
+
+                   (pi)                2  /  3
+          where z= ----+(zeta),(zeta)= -\/ -x  and a(t) and b(t) are
+                    4                  3
+                                             3
+                                         ( 5)
+          expansions in the variable t=-2( -) -1.
+                                         ( x)
+
+          For -5<=x<=0,
+
+                                          2
+                              Bi'(x)=\/3(x f(t)+g(t)),
+                                                  3
+                                              ( x)
+          where f and g are expansions in t=-2( -) -1.
+                                              ( 5)
+
+          For 0<x<4.5,
+
+                                          3x/2
+                                  Bi'(x)=e    y(t),
+
+          where y(t) is an expansion in t=4x/9-1.
+
+          For 4.5<=x<9,
+
+                                         21x/8
+                                 Bi'(x)=e     u(t),
+
+          where u(t) is an expansion in t=4x/9-3.
+
+          For x>=9,
+
+                                         4   z
+                                  Bi'(x)=\/xe v(t),
+
+                       
+
+                   2  / 3                                ( 18)
+          where z= -\/ x  and v(t) is an expansion in t=2( --)-1.
+                   3                                     ( z )
+
+          For |x|< the square of the machine precision, the result is set
+          directly to Bi'(0). This saves time and avoids possible
+          underflows in calculation.
+
+          For large negative arguments, it becomes impossible to calculate
+          a result for the oscillating function with any accuracy so the
+                                                            4/7
+                                                (          )
+                                                (  \/(pi)  )
+          routine must fail. This occurs for x<-( ---------)   , where
+                                                ( (epsilon))
+          (epsilon) is the machine precision.
+
+          For large positive arguments, where Bi' grows in an essentially
+          exponential manner, there is a danger of overflow so the routine
+          must fail.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               X is too large and positive. On soft failure the routine
+               returns zero.
+
+          IFAIL= 2
+               X is too large and negative. On soft failure the routine
+               returns zero.
+
+          7. Accuracy
+
+          For negative arguments the function is oscillatory and hence
+          absolute error is appropriate. In the positive region the
+          function has essentially exponential behaviour and hence relative
+          error is needed. The absolute error, E, and the relative error
+          (epsilon), are related in principle to the relative error in the
+          argument (delta), by
+
+                                                  |  2     |
+                     2                            | x Bi(x)|
+                E~=|x Bi(x)|(delta)    (epsilon)~=| -------|(delta).
+                                                  | Bi'(x) |
+
+          In practice, approximate equality is the best that can be
+          expected. When (delta), (epsilon) or E is of the order of the
+          machine precision, the errors in the result will be somewhat
+          larger.
+
+          For small x, positive or negative, errors are strongly attenuated
+          by the function and hence will effectively be bounded by the
+          machine precision.
+
+          For moderate to large negative x, the error is, like the
+          function, oscillatory. However, the amplitude of the absolute
+                               7/4
+                            |x|
+          error grows like  ------. Therefore it becomes impossible to
+                              
+
+                            \/(pi)
+                                                                
+
+                                                         7/4  \/(pi)
+          calculate the function with any accuracy if |x|   > -------.
+                                                              (delta)
+
+          For large positive x, the relative error amplification is
+                                      
+
+                         (epsilon)   / 3
+          considerable:  ---------~\/ x . However, very large arguments are
+                          (delta)
+          not possible due to the danger of overflow. Thus in practice the
+          actual amplification that occurs is limited.
+
+          8. Further Comments
+
+          None.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs17dcf}{NAG On-line Documentation: s17dcf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S17DCF(3NAG)      Foundation Library (12/10/92)      S17DCF(3NAG)
+
+
+
+          S17 -- Approximations of Special Functions                 S17DCF
+                  S17DCF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S17DCF returns a sequence of values for the Bessel functions
+          Y      (z) for complex z, non-negative (nu) and n=0,1,...,N-1,
+           (nu)+n
+          with an option for exponential scaling.
+
+          2. Specification
+
+                 SUBROUTINE S17DCF (FNU, Z, N, SCALE, CY, NZ, CWRK, IFAIL)
+                 INTEGER              N, NZ, IFAIL
+                 DOUBLE PRECISION     FNU
+                 COMPLEX(KIND(1.0D0)) Z, CY(N), CWRK(N)
+                 CHARACTER*1          SCALE
+
+          3. Description
+
+          This subroutine evaluates a sequence of values for the Bessel
+          function Y    (z), where z is complex, -(pi) < arg z <= (pi), and
+                    (nu)
+          (nu) is the real, non-negative order. The N-member sequence is
+          generated for orders (nu), (nu)+1,...,(nu)+N-1. Optionally, the
+                                            -|Im z|
+          sequence is scaled by the factor e       .
+
+          Note: although the routine may not be called with (nu) less than
+          zero, for negative orders the formula
+          Y     (z)=Y    (z)cos((pi)(nu))+J    (z)sin((pi)(nu)) may be used
+           -(nu)     (nu)                  (nu)
+          (for the Bessel function J    (z), see S17DEF).
+                                    (nu)
+
+          The routine is derived from the routine CBESY in Amos [2]. It is
+                                           (1)      (2)
+                                          H    (z)-H    (z)
+                                           (nu)     (nu)            (1)
+          based on the relation Y    (z)= -----------------, where H    (z)
+                                 (nu)            2i                 (nu)
+               (2)
+          and H    (z) are the Hankel functions of the first and second
+               (nu)
+          kinds respectively (see S17DLF).
+
+          When N is greater than 1, extra values of Y    (z) are computed
+                                                     (nu)
+          using recurrence relations.
+
+          For very large |z| or ((nu)+N-1), argument reduction will cause
+          total loss of accuracy, and so no computation is performed. For
+          slightly smaller |z| or ((nu)+N-1), the computation is performed
+          but results are accurate to less than half of machine precision.
+          If |z| is very small, near the machine underflow threshold, or
+          ((nu)+N-1) is too large, there is a risk of overflow and so no
+          computation is performed. In all the above cases, a warning is
+          given by the routine.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Amos D E (1986) Algorithm 644: A Portable Package for Bessel
+                Functions of a Complex Argument and Nonnegative Order. ACM
+                Trans. Math. Softw. 12 265--273.
+
+          5. Parameters
+
+           1:  FNU -- DOUBLE PRECISION                                Input
+               On entry: the order, (nu), of the first member of the
+               sequence of functions. Constraint: FNU >= 0.0.
+
+           2:  Z -- COMPLEX(KIND(1.0D0))                              Input
+               On entry: the argument, z, of the functions. Constraint: Z
+               /= (0.0, 0.0).
+
+           3:  N -- INTEGER                                           Input
+               On entry: the number, N, of members required in the sequence
+               Y    (z), Y      (z),...,Y        (z). Constraint: N >= 1.
+                (nu)      (nu)+1         (nu)+N-1
+
+           4:  SCALE -- CHARACTER*1                                   Input
+               On entry: the scaling option.
+
+               If SCALE = 'U', the results are returned unscaled.
+
+               If SCALE = 'S', the results are returned scaled by the
+                       -|Imz|
+               factor e      . Constraint: SCALE = 'U' or 'S'.
+
+           5:  CY(N) -- COMPLEX(KIND(1.0D)) array                    Output
+               On exit: the N required function values: CY(i) contains
+               Y        (z), for i=1,2,...,N.
+                (nu)+i-1
+
+           6:  NZ -- INTEGER                                         Output
+               On exit: the number of components of CY that are set to zero
+               due to underflow. The positions of such components in the
+               array CY are arbitrary.
+
+           7:  CWRK(N) -- COMPLEX(KIND(1.0D)) array               Workspace
+
+           8:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry FNU < 0.0,
+
+               or       Z = (0.0, 0.0),
+
+               or       N < 1,
+
+               or       SCALE /= 'U' or 'S'.
+
+          IFAIL= 2
+               No computation has been performed due to the likelihood of
+               overflow, because ABS(Z) is less than a machine-dependent
+               threshold value (given in the Users' Note for your
+               implementation).
+
+          IFAIL= 3
+               No computation has been performed due to the likelihood of
+               overflow, because FNU + N - 1 is too large - how large
+               depends on Z as well as the overflow threshold of the
+               machine.
+
+          IFAIL= 4
+               The computation has been performed, but the errors due to
+               argument reduction in elementary functions make it likely
+               that the results returned by S17DCF are accurate to less
+               than half of machine precision. This error exit may occur if
+               either ABS(Z) or FNU + N - 1 is greater than a machine-
+               dependent threshold value (given in the Users' Note for
+               your implementation).
+
+          IFAIL= 5
+               No computation has been performed because the errors due to
+               argument reduction in elementary functions mean that all
+               precision in results returned by S17DCF would be lost. This
+               error exit may occur if either ABS(Z) or FNU + N - 1 is
+               greater than a machine-dependent threshold value (given in
+               the Users' Note for your implementation).
+
+          IFAIL= 6
+               No results are returned because the algorithm termination
+               condition has not been met. This may occur because the
+               parameters supplied to S17DCF would have caused overflow or
+               underflow.
+
+          7. Accuracy
+
+          All constants in subroutine S17DCF are given to approximately 18
+          digits of precision. Calling the number of digits of precision in
+          the floating-point arithmetic being used t, then clearly the
+          maximum number of correct digits in the results obtained is
+          limited by p=min(t,18). Because of errors in argument reduction
+          when computing elementary functions inside S17DCF, the actual
+          number of correct digits is limited, in general, by p-s, where
+          s~max(1,|log  |z||,|log  (nu)|) represents the number of digits
+                      10         10
+          lost due to the argument reduction. Thus the larger the values of
+          |z| and (nu), the less the precision in the result. If S17DCF is
+          called with N>1, then computation of function values via
+          recurrence may lead to some further small loss of accuracy.
+
+          If function values which should nominally be identical are
+          computed by calls to S17DCF with different base values of (nu)
+          and different N, the computed values may not agree exactly.
+          Empirical tests with modest values of (nu) and z have shown that
+          the discrepancy is limited to the least significant 3-4 digits of
+          precision.
+
+          8. Further Comments
+
+          The time taken by the routine for a call of S17DCF is
+          approximately proportional to the value of N, plus a constant. In
+          general it is much cheaper to call S17DCF with N greater than 1,
+          rather than to make N separate calls to S17DCF.
+
+          Paradoxically, for some values of z and (nu), it is cheaper to
+          call S17DCF with a larger value of N than is required, and then
+          discard the extra function values returned. However, it is not
+          possible to state the precise circumstances in which this is
+          likely to occur. It is due to the fact that the base value used
+          to start recurrence may be calculated in different regions for
+          different N, and the costs in each region may differ greatly.
+
+          Note that if the function required is Y (x) or Y (x), i.e., (nu)
+                                                 0        1
+          = 0.0 or 1.0, where x is real and positive, and only a single
+          unscaled function value is required, then it may be much cheaper
+          to call S17ACF or S17ADF respectively.
+
+          9. Example
+
+          The example program prints a caption and then proceeds to read
+          sets of data from the input data stream. The first datum is a
+          value for the order FNU, the second is a complex value for the
+          argument, Z, and the third is a value for the parameter SCALE.
+          The program calls the routine with N = 2 to evaluate the function
+          for orders FNU and FNU + 1, and it prints the results. The
+          process is repeated until the end of the input data stream is
+          encountered.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs17def}{NAG On-line Documentation: s17def}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S17DEF(3NAG)      Foundation Library (12/10/92)      S17DEF(3NAG)
+
+
+
+          S17 -- Approximations of Special Functions                 S17DEF
+                  S17DEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S17DEF returns a sequence of values for the Bessel functions
+          J      (z) for complex z, non-negative (nu) and n=0,1,...,N-1,
+           (nu)+n
+          with an option for exponential scaling.
+
+          2. Specification
+
+                 SUBROUTINE S17DEF (FNU, Z, N, SCALE, CY, NZ, IFAIL)
+                 INTEGER              N, NZ, IFAIL
+                 DOUBLE PRECISION     FNU
+                 COMPLEX(KIND(1.0D0)) Z, CY(N)
+                 CHARACTER*1          SCALE
+
+          3. Description
+
+          This subroutine evaluates a sequence of values for the Bessel
+          function J    (z), where z is complex, -(pi) < arg z <= (pi), and
+                    (nu)
+          (nu) is the real, non-negative order. The N-member sequence is
+          generated for orders (nu), (nu)+1,...,(nu)+N-1. Optionally, the
+                                            -|Im z|
+          sequence is scaled by the factor e       .
+
+          Note: although the routine may not be called with (nu) less than
+          zero, for negative orders the formula
+          J     (z)=J    (z)cos((pi)(nu))-Y    (z)sin((pi)(nu)) may be used
+           -(nu)     (nu)                  (nu)
+          (for the Bessel function Y    (z), see S17DCF).
+                                    (nu)
+
+          The routine is derived from the routine CBESJ in Amos [2]. It is
+                                           (nu)(pi)i/2
+          based on the relations J    (z)=e           I    (-iz), Im z>=0.0
+                                  (nu)                 (nu)
+                        -(nu)(pi)i/2
+          and J    (z)=e            I    (iz), Im z<0.0.
+               (nu)                  (nu)
+
+          The Bessel function I    (z) is computed using a variety of
+                               (nu)
+          techniques depending on the region under consideration.
+
+          When N is greater than 1, extra values of J    (z) are computed
+                                                     (nu)
+          using recurrence relations.
+
+          For very large |z| or ((nu)+N-1), argument reduction will cause
+          total loss of accuracy, and so no computation is performed. For
+          slightly smaller |z| or ((nu)+N-1), the computation is performed
+          but results are accurate to less than half of machine precision.
+          If Im z is large, there is a risk of overflow and so no
+          computation is performed. In all the above cases, a warning is
+          given by the routine.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Amos D E (1986) Algorithm 644: A Portable Package for Bessel
+                Functions of a Complex Argument and Nonnegative Order. ACM
+                Trans. Math. Softw. 12 265--273.
+
+          5. Parameters
+
+           1:  FNU -- DOUBLE PRECISION                                Input
+               On entry: the order, (nu), of the first member of the
+               sequence of functions. Constraint: FNU >= 0.0.
+
+           2:  Z -- COMPLEX(KIND(1.0D0))                              Input
+               On entry: the argument z of the functions.
+
+           3:  N -- INTEGER                                           Input
+               On entry: the number, N, of members required in the sequence
+               J    (z),J      (z),...,J        (z). Constraint: N >= 1.
+                (nu)     (nu)+1         (nu)+N-1
+
+           4:  SCALE -- CHARACTER*1                                   Input
+               On entry: the scaling option.
+                    If SCALE = 'U', the results are returned unscaled.
+
+                    If SCALE = 'S', the results are returned scaled by the
+                            -|Imz|
+                    factor e      .
+               Constraint: SCALE = 'U' or 'S'.
+
+           5:  CY(N) -- COMPLEX(KIND(1.0D)) array                    Output
+               On exit: the N required function values: CY(i) contains
+               J        (z), for i=1,2,...,N.
+                (nu)+i-1
+
+           6:  NZ -- INTEGER                                         Output
+               On exit: the number of components of CY that are set to zero
+               due to underflow. If NZ > 0, then elements CY(N-NZ+1), CY(N-
+               NZ+2),...,CY(N) are set to zero.
+
+           7:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry FNU < 0.0,
+
+               or       N < 1,
+
+               or       SCALE /= 'U' or 'S'.
+
+          IFAIL= 2
+               No computation has been performed due to the likelihood of
+               overflow, because Im Z is larger than a machine-dependent
+               threshold value (given in the Users' Note for your
+               implementation). This error exit can only occur when SCALE =
+               'U'.
+
+          IFAIL= 3
+               The computation has been performed, but the errors due to
+               argument reduction in elementary functions make it likely
+               that the results returned by S17DEF are accurate to less
+               than half of machine precision. This error exit may occur if
+               either ABS(Z) or FNU + N - 1 is greater than a machine-
+               dependent threshold value (given in the Users' Note for
+               your implementation).
+
+          IFAIL= 4
+               No computation has been performed because the errors due to
+               argument reduction in elementary functions mean that all
+               precision in results returned by S17DEF would be lost. This
+               error exit may occur when either ABS(Z) or FNU + N - 1 is
+               greater than a machine-dependent threshold value (given in
+               the Users' Note for your implementation).
+
+          IFAIL= 5
+               No results are returned because the algorithm termination
+               condition has not been met. This may occur because the
+               parameters supplied to S17DEF would have caused overflow or
+               underflow.
+
+          7. Accuracy
+
+          All constants in subroutine S17DEF are given to approximately 18
+          digits of precision. Calling the number of digits of precision in
+          the floating-point arithmetic being used t, then clearly the
+          maximum number of correct digits in the results obtained is
+          limited by p=min(t,18). Because of errors in argument reduction
+          when computing elementary functions inside S17DEF, the actual
+          number of correct digits is limited, in general, by p-s, where
+          s~max(1,|log  |(|z|),|log  (nu)|) represents the number of digits
+                      10           10
+          lost due to the argument reduction. Thus the larger the values of
+          |z| and (nu), the less the precision in the result. If S17DEF is
+          called with N>1, then computation of function values via
+          recurrence may lead to some further small loss of accuracy.
+
+          If function values which should nominally be identical are
+          computed by calls to S17DEF with different base values of (nu)
+          and different N, the computed values may not agree exactly.
+          Empirical tests with modest values of (nu) and z have shown that
+          the discrepancy is limited to the least significant 3-4 digits of
+          precision.
+
+          8. Further Comments
+
+          The time taken by the routine for a call of S17DEF is
+          approximately proportional to the value of N, plus a constant. In
+          general it is much cheaper to call S17DEF with N greater than 1,
+          rather than to make N separate calls to S17DEF.
+
+          Paradoxically, for some values of z and (nu), it is cheaper to
+          call S17DEF with a larger value of N than is required, and then
+          discard the extra function values returned. However, it is not
+          possible to state the precise circumstances in which this is
+          likely to occur. It is due to the fact that the base value used
+          to start recurrence may be calculated in different regions for
+          different N, and the costs in each region may differ greatly.
+
+          Note that if the function required is J (x) or J (x), i.e., (nu)
+                                                 0        1
+          = 0.0 or 1.0, where x is real and positive, and only a single
+          unscaled function value is required, then it may be much cheaper
+          to call S17AEF or S17AFF respectively.
+
+          9. Example
+
+          The example program prints a caption and then proceeds to read
+          sets of data from the input data stream. The first datum is a
+          value for the order FNU, the second is a complex value for the
+          argument, Z, and the third is a value for the parameter SCALE.
+
+          The program calls the routine with N = 2 to evaluate the function
+          for orders FNU and FNU + 1, and it prints the results. The
+          process is repeated until the end of the input data stream is
+          encountered.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs17dgf}{NAG On-line Documentation: s17dgf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S17DGF(3NAG)      Foundation Library (12/10/92)      S17DGF(3NAG)
+
+
+
+          S17 -- Approximations of Special Functions                 S17DGF
+                  S17DGF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S17DGF returns the value of the Airy function Ai(z) or its
+          derivative Ai'(z) for complex z, with an option for exponential
+          scaling.
+
+          2. Specification
+
+                 SUBROUTINE S17DGF (DERIV, Z, SCALE, AI, NZ, IFAIL)
+                 INTEGER              NZ, IFAIL
+                 COMPLEX(KIND(1.0D0)) Z, AI
+                 CHARACTER*1          DERIV, SCALE
+
+          3. Description
+
+          This subroutine returns a value for the Airy function Ai(z) or
+          its derivative Ai'(z), where z is complex, -(pi) < arg z <= (pi).
+                                                             
+
+                                                         2z\/z/3
+          Optionally, the value is scaled by the factor e       .
+
+          The routine is derived from the routine CAIRY in Amos [2]. It is
+                                          
+
+                                        \/zK   (w)              -zK   (w)
+                                            1/3                    2/3
+          based on the relations Ai(z)= ----------, and Ai'(z)= ---------,
+                                                                       
+
+                                         (pi)\/3                 (pi)/3
+                                                                
+
+          where K     is the modified Bessel function and w=2z\/z/3.
+                 (nu)
+
+          For very large |z|, argument reduction will cause total loss of
+          accuracy, and so no computation is performed. For slightly
+          smaller |z|, the computation is performed but results are
+          accurate to less than half of machine precision. If Re w is too
+          large, and the unscaled function is required, there is a risk of
+          overflow and so no computation is performed. In all the above
+          cases, a warning is given by the routine.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Amos D E (1986) Algorithm 644: A Portable Package for Bessel
+                Functions of a Complex Argument and Nonnegative Order. ACM
+                Trans. Math. Softw. 12 265--273.
+
+          5. Parameters
+
+           1:  DERIV -- CHARACTER*1                                   Input
+               On entry: specifies whether the function or its derivative
+               is required.
+                    If DERIV = 'F', Ai(z) is returned.
+
+                    If DERIV = 'D', Ai'(z) is returned.
+               Constraint: DERIV = 'F' or 'D'.
+
+           2:  Z -- COMPLEX(KIND(1.0D0))                              Input
+               On entry: the argument z of the function.
+
+           3:  SCALE -- CHARACTER*1                                   Input
+               On entry: the scaling option.
+
+               If SCALE = 'U', the result is returned unscaled.
+
+               If SCALE = 'S', the result is returned scaled by the factor
+                    
+
+                2z\/z/3
+               e       . Constraint: SCALE = 'U' or 'S'.
+
+           4:  AI -- COMPLEX(KIND(1.0D0))                            Output
+               On exit: the required function or derivative value.
+
+           5:  NZ -- INTEGER                                         Output
+               On exit: NZ indicates whether or not AI is set to zero due
+               to underflow. This can only occur when SCALE = 'U'.
+
+               If NZ = 0, AI is not set to zero.
+
+               If NZ = 1, AI is set to zero.
+
+           6:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry DERIV /= 'F' or 'D'.
+
+               or       SCALE /= 'U' or 'S'.
+
+          IFAIL= 2
+               No computation has been performed due to the likelihood of
+                                                                
+
+               overflow, because Re w is too large, where w=2Z\/Z/3 -- how
+               large depends on Z and the overflow threshold of the
+               machine. This error exit can only occur when SCALE = 'U'.
+
+          IFAIL= 3
+               The computation has been performed, but the errors due to
+               argument reduction in elementary functions make it likely
+               that the result returned by S17DGF is accurate to less than
+               half of machine precision. This error exit may occur if ABS
+               (Z) is greater than a machine-dependent threshold value
+               (given in the Users' Note for your implementation).
+
+          IFAIL= 4
+               No computation has been performed because the errors due to
+               argument reduction in elementary functions mean that all
+               precision in the result returned by S17DGF would be lost.
+               This error exit may occur if ABS(Z) is greater than a
+               machine-dependent threshold value (given in the Users' Note
+               for your implementation).
+
+          IFAIL= 5
+               No result is returned because the algorithm termination
+               condition has not been met. This may occur because the
+               parameters supplied to S17DGF would have caused overflow or
+               underflow.
+
+          7. Accuracy
+
+          All constants in subroutine S17DGF are given to approximately 18
+          digits of precision. Calling the number of digits of precision in
+          the floating-point arithmetic being used t, then clearly the
+          maximum number of correct digits in the results obtained is
+          limited by p=min(t,18). Because of errors in argument reduction
+          when computing elementary functions inside S17DGF, the actual
+          number of correct digits is limited, in general, by p-s, where
+          s~max(1,|log  |z||) represents the number of digits lost due to
+                      10
+          the argument reduction. Thus the larger the value of |z|, the
+          less the precision in the result.
+
+          Empirical tests with modest values of z, checking relations
+          between Airy functions Ai(z), Ai'(z), Bi(z) and Bi'(z), have
+          shown errors limited to the least significant 3-4 digits of
+          precision.
+
+          8. Further Comments
+
+          Note that if the function is required to operate on a real
+          argument only, then it may be much cheaper to call S17AGF or
+          S17AJF.
+
+          9. Example
+
+          The example program prints a caption and then proceeds to read
+          sets of data from the input data stream. The first datum is a
+          value for the parameter DERIV, the second is a complex value for
+          the argument, Z, and the third is a value for the parameter
+          SCALE. The program calls the routine and prints the results. The
+          process is repeated until the end of the input data stream is
+          encountered.
+
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs17dhf}{NAG On-line Documentation: s17dhf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S17DHF(3NAG)      Foundation Library (12/10/92)      S17DHF(3NAG)
+
+
+
+          S17 -- Approximations of Special Functions                 S17DHF
+                  S17DHF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S17DHF returns the value of the Airy function Bi(z) or its
+          derivative Bi'(z) for complex z, with an option for exponential
+          scaling.
+
+          2. Specification
+
+                 SUBROUTINE S17DHF (DERIV, Z, SCALE, BI, IFAIL)
+                 INTEGER              IFAIL
+                 COMPLEX(KIND(1.0D0)) Z, BI
+                 CHARACTER*1          DERIV, SCALE
+
+          3. Description
+
+          This subroutine returns a value for the Airy function Bi(z) or
+          its derivative Bi'(z), where z is complex, -(pi) < argz <= (pi).
+                                                                  
+
+                                                         |Re (2z\/z/3)|
+          Optionally, the value is scaled by the factor e              .
+
+          The routine is derived from the routine CBIRY in Amos [2]. It is
+                                          
+
+                                        \/z
+          based on the relations Bi(z)= ---(I    (w)+I   (w)), and
+                                             -1/3     1/3
+                                        \/3
+                   z
+          Bi'(z)= ---(I    (w)+I   (w)), where I     is the modified Bessel
+                       -2/3     2/3             (nu)
+                  \/3
+                             
+
+          function and w=2z\/z/3.
+
+          For very large |z|, argument reduction will cause total loss of
+          accuracy, and so no computation is performed. For slightly
+          smaller |z|, the computation is performed but results are
+          accurate to less than half of machine precision. If Re  z is too
+          large, and the unscaled function is required, there is a risk of
+          overflow and so no computation is performed. In all the above
+          cases, a warning is given by the routine.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Hammersley J M and Handscomb D C (1967) Monte-Carlo Methods.
+                Methuen.
+
+          5. Parameters
+
+           1:  DERIV -- CHARACTER*1                                   Input
+               On entry: specifies whether the function or its derivative
+               is required.
+                    If DERIV = 'F', Bi(z) is returned.
+
+                    If DERIV = 'D', Bi'(z) is returned.
+               Constraint: DERIV = 'F' or 'D'.
+
+           2:  Z -- COMPLEX(KIND(1.0D0))                              Input
+               On entry: the argument z of the function.
+
+           3:  SCALE -- CHARACTER*1                                   Input
+               On entry: the scaling option.
+                    If SCALE = 'U', the result is returned unscaled.
+
+                    If SCALE = 'S', the result is returned scaled by the
+                                    
+
+                            |Re(2z\/z/3)|
+                    factor e             .
+               Constraint: SCALE = 'U' or 'S'.
+
+           4:  BI -- COMPLEX(KIND(1.0D0))                            Output
+               On exit: the required function or derivative value.
+
+           5:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry DERIV /= 'F' or 'D'.
+
+               or       SCALE /= 'U' or 'S'.
+
+          IFAIL= 2
+               No computation has been performed due to the likelihood of
+               overflow, because real(Z) is too large - how large depends
+               on the overflow threshold of the machine. This error exit
+               can only occur when SCALE = 'U'.
+
+          IFAIL= 3
+               The computation has been performed, but the errors due to
+               argument reduction in elementary functions make it likely
+               that the result returned by S17DHF is accurate to less than
+               half of machine precision. This error exit may occur if ABS
+               (Z) is greater than a machine-dependent threshold value
+               (given in the Users' Note for your implementation).
+
+          IFAIL= 4
+               No computation has been performed because the errors due to
+               argument reduction in elementary functions mean that all
+               precision in the result returned by S17DHF would be lost.
+               This error exit may occur if ABS(Z) is greater than a
+               machine-dependent threshold value (given in the Users' Note
+               for your implementation).
+
+          IFAIL= 5
+               No result is returned because the algorithm termination
+               condition has not been met. This may occur because the
+               parameters supplied to S17DHF would have caused overflow or
+               underflow.
+
+          7. Accuracy
+
+          All constants in subroutine S17DHF are given to approximately 18
+          digits of precision. Calling the number of digits of precision in
+          the floating-point arithmetic being used t, then clearly the
+          maximum number of correct digits in the results obtained is
+          limited by p=min(t,18). Because of errors in argument reduction
+          when computing elementary functions inside S17DHF, the actual
+          number of correct digits is limited, in general, by p-s, where
+          s~max(1,|log  |z||) represents the number of digits lost due to
+                      10
+          the argument reduction. Thus the larger the value of |z|, the
+          less the precision in the result.
+
+          Empirical tests with modest values of z, checking relations
+          between Airy functions Ai(z), Ai'(z), Bi(z) and Bi'(z), have
+          shown errors limited to the least significant 3-4 digits of
+          precision.
+
+          8. Further Comments
+
+          Note that if the function is required to operate on a real
+          argument only, then it may be much cheaper to call S17AHF or
+          S17AKF.
+
+          9. Example
+
+          The example program prints a caption and then proceeds to read
+          sets of data from the input data stream. The first datum is a
+          value for the parameter DERIV, the second is a complex value for
+          the argument, Z, and the third is a value for the parameter
+          SCALE. The program calls the routine and prints the results. The
+          process is repeated until the end of the input data stream is
+          encountered.
+
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs17dlf}{NAG On-line Documentation: s17dlf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S17DLF(3NAG)      Foundation Library (12/10/92)      S17DLF(3NAG)
+
+
+
+          S17 -- Approximations of Special Functions                 S17DLF
+                  S17DLF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S17DLF returns a sequence of values for the Hankel functions
+           (1)           (2)
+          H      (z) or H      (z) for complex z, non-negative (nu) and
+           (nu)+n        (nu)+n
+          n=0,1,...,N-1, with an option for exponential scaling.
+
+          2. Specification
+
+                 SUBROUTINE S17DLF (M, FNU, Z, N, SCALE, CY, NZ, IFAIL)
+                 INTEGER              M, N, NZ, IFAIL
+                 DOUBLE PRECISION     FNU
+                 COMPLEX(KIND(1.0D0)) Z, CY(N)
+                 CHARACTER*1          SCALE
+
+          3. Description
+
+          This subroutine evaluates a sequence of values for the Hankel
+                    (1)         (2)
+          function H    (z) or H    (z), where z is complex, -(pi) < argz
+                    (nu)        (nu)
+          <= (pi), and (nu) is the real, non-negative order. The N-member
+          sequence is generated for orders (nu), (nu)+1,...,(nu)+N-1.
+                                                            -iz
+          Optionally, the sequence is scaled by the factor e    if the
+                       (1)                       iz
+          function is H    (z) or by the factor e   if the function is
+                       (nu)
+           (2)
+          H    (z).
+           (nu)
+
+          Note: although the routine may not be called with (nu) less than
+          zero, for negative orders the formulae
+           (1)       (nu)(pi)i (1)           (2)       -(nu)(pi)i (2)
+          H     (z)=e         H    (z), and H     (z)=e          H    (z)
+           -(nu)               (nu)          -(nu)                (nu)
+          may be used.
+
+          The routine is derived from the routine CBESH in Amos [2]. It is
+          based on the relation
+
+                            (m)      1 -p(nu)        -p
+                           H    (z)= -e      K    (ze  ),
+                            (nu)     p        (nu)
+
+                    (pi)                 (pi)
+          where p=i ---- if m=1 and p=-i ---- if m=2, and the Bessel
+                     2                    2
+          function K    (z) is computed in the right half-plane only.
+                    (nu)
+          Continuation of K    (z) to the left half-plane is computed in
+                           (nu)
+          terms of the Bessel function I    (z). These functions are
+                                        (nu)
+          evaluated using a variety of different techniques, depending on
+          the region under consideration.
+
+                                                     (m)
+          When N is greater than 1, extra values of H    (z) are computed
+                                                     (nu)
+          using recurrence relations.
+
+          For very large |z| or ((nu)+N-1), argument reduction will cause
+          total loss of accuracy, and so no computation is performed. For
+          slightly smaller |z| or ((nu)+N-1), the computation is performed
+          but results are accurate to less than half of machine precision.
+          If |z| is very small, near the machine underflow threshold, or
+          ((nu)+N-1) is too large, there is a risk of overflow and so no
+          computation is performed. In all the above cases, a warning is
+          given by the routine.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Amos D E (1986) Algorithm 644: A Portable Package for Bessel
+                Functions of a Complex Argument and Nonnegative Order. ACM
+                Trans. Math. Softw. 12 265--273.
+
+          5. Parameters
+
+           1:  M -- INTEGER                                           Input
+               On entry: the kind of functions required.
+                                                 (1)
+                    If M = 1, the functions are H    (z).
+                                                 (nu)
+
+                                                 (2)
+                    If M = 2, the functions are H    (z).
+                                                 (nu)
+               Constraint: M = 1 or 2.
+
+           2:  FNU -- DOUBLE PRECISION                                Input
+               On entry: the order, (nu), of the first member of the
+               sequence of functions. Constraint: FNU >= 0.0.
+
+           3:  Z -- COMPLEX(KIND(1.0D0))                              Input
+               On entry: the argument z of the functions. Constraint: Z /=
+               (0.0, 0.0).
+
+           4:  N -- INTEGER                                           Input
+               On entry: the number, N, of members required in the sequence
+                (M)    (M)          (M)
+               H    , H      ,..., H        . Constraint: N >= 1.
+                (nu)   (nu)+1       (nu)+N-1
+
+           5:  SCALE -- CHARACTER*1                                   Input
+               On entry: the scaling option.
+                    If SCALE = 'U', the results are returned unscaled.
+
+                    If SCALE = 'S', the results are returned scaled by the
+                            -iz                               iz
+                    factor e    when M = 1, or by the factor e   when M =
+                    2.
+               Constraint: SCALE = 'U' or 'S'.
+
+           6:  CY(N) -- COMPLEX(KIND(1.0D)) array                    Output
+               On exit: the N required function values: CY(i) contains
+                (M)
+               H        , for i=1,2,...,N.
+                (nu)+i-1
+
+           7:  NZ -- INTEGER                                         Output
+               On exit: the number of components of CY that are set to zero
+               due to underflow. If NZ > 0, then if Imz>0.0 and M = 1, or
+               Imz<0.0 and M = 2, elements CY(1), CY(2),...,CY(NZ) are set
+               to zero. In the complementary half-planes, NZ simply states
+               the number of underflows, and not which elements they are.
+
+           8:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry M /= 1 and M /= 2,
+
+               or       FNU < 0.0,
+
+               or       Z = (0.0, 0.0),
+
+               or       N < 1,
+
+               or       SCALE /= 'U' or 'S'.
+
+          IFAIL= 2
+               No computation has been performed due to the likelihood of
+               overflow, because ABS(Z) is less than a machine-dependent
+               threshold value (given in the Users' Note for your
+               implementation).
+
+          IFAIL= 3
+               No computation has been performed due to the likelihood of
+               overflow, because FNU + N - 1 is too large - how large
+               depends on Z and the overflow threshold of the machine.
+
+          IFAIL= 4
+               The computation has been performed, but the errors due to
+               argument reduction in elementary functions make it likely
+               that the results returned by S17DLF are accurate to less
+               than half of machine precision. This error exit may occur if
+               either ABS(Z) or FNU + N - 1 is greater than a machine-
+               dependent threshold value (given in the Users' Note for
+               your implementation).
+
+          IFAIL= 5
+               No computation has been performed because the errors due to
+               argument reduction in elementary functions mean that all
+               precision in results returned by S17DLF would be lost. This
+               error exit may occur when either of ABS(Z) or FNU + N - 1 is
+               greater than a machine-dependent threshold value (given in
+               the Users' Note for your implementation).
+
+          IFAIL= 6
+               No results are returned because the algorithm termination
+               condition has not been met. This may occur because the
+               parameters supplied to S17DLF would have caused overflow or
+               underflow.
+
+          7. Accuracy
+
+          All constants in subroutine S17DLF are given to approximately 18
+          digits of precision. Calling the number of digits of precision in
+          the floating-point arithmetic being used t, then clearly the
+          maximum number of correct digits in the results obtained is
+          limited by p=min(t,18). Because of errors in argument reduction
+          when computing elementary functions inside S17DLF, the actual
+          number of correct digits is limited, in general, by p-s, where
+          s~max(1,|log  |z||,|log  (nu)|) represents the number of digits
+                      10         10
+          lost due to the argument reduction. Thus the larger the values of
+          |z| and (nu), the less the precision in the result. If S17DLF is
+          called with N>1, then computation of function values via
+          recurrence may lead to some further small loss of accuracy.
+
+          If function values which should nominally be identical are
+          computed by calls to S17DLF with different base values of (nu)
+          and different N, the computed values may not agree exactly.
+          Empirical tests with modest values of (nu) and z have shown that
+          the discrepancy is limited to the least significant 3-4 digits of
+          precision.
+
+          8. Further Comments
+
+          The time taken by the routine for a call of S17DLF is
+          approximately proportional to the value of N, plus a constant. In
+          general it is much cheaper to call S17DLF with N greater than 1,
+          rather than to make N separate calls to S17DLF.
+
+          Paradoxically, for some values of z and (nu), it is cheaper to
+          call S17DLF with a larger value of N than is required, and then
+          discard the extra function values returned. However, it is not
+          possible to state the precise circumstances in which this is
+          likely to occur. It is due to the fact that the base value used
+          to start recurrence may be calculated in different regions for
+          different N, and the costs in each region may differ greatly.
+
+          9. Example
+
+          The example program prints a caption and then proceeds to read
+          sets of data from the input data stream. The first datum is a
+          value for the kind of function, M, the second is a value for the
+          order FNU, the third is a complex value for the argument, Z, and
+          the fourth is a value for the parameter SCALE. The program calls
+          the routine with N = 2 to evaluate the function for orders FNU
+          and FNU + 1, and it prints the results. The process is repeated
+          until the end of the input data stream is encountered.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs18acf}{NAG On-line Documentation: s18acf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S18ACF(3NAG)      Foundation Library (12/10/92)      S18ACF(3NAG)
+
+
+
+          S18 -- Approximations of Special Functions                 S18ACF
+                  S18ACF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S18ACF returns the value of the modified Bessel Function K (x),
+                                                                    0
+          via the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S18ACF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the modified Bessel
+          Function of the second kind K (x).
+                                       0
+
+          Note: K (x) is undefined for x<=0 and the routine will fail for
+                 0
+          such arguments.
+
+          The routine is based on five Chebyshev expansions:
+
+          For 0<x<=1,
+
+                           --'          --'                     2
+                K (x)=-lnx >   a T (t)+ >   b T (t) , where t=2x -1;
+                 0         --   r r     --   r r
+                           r=0          r=0
+
+          For 1<x<=2,
+
+                               -x --'
+                        K (x)=e   >   c T (t) , where t=2x-3;
+                         0        --   r r
+                                  r=0
+
+          For 2<x<=4,
+
+                               -x --'
+                        K (x)=e   >   d T (t) , where t=x-3;
+                         0        --   r r
+                                  r=0
+          For x>4,
+
+                               -x
+                              e   --'                    9-x
+                       K (x)= --- >   e T (t) , where t= ---.
+                        0         --   r r               1+x
+                              \/x r=0
+
+                                             ( x)
+          For x near zero, K (x)~=-(gamma)-ln( -), where (gamma) denotes
+                            0                ( 2)
+          Euler's constant. This approximation is used when x is
+          sufficiently small for the result to be correct to machine
+          precision.
+
+          For large x, where there is a danger of underflow due to the
+          smallness of K , the result is set exactly to zero.
+                        0
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function. Constraint: X > 0.
+               0.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               X <= 0.0, K  is undefined. On soft failure the routine
+                          0
+               returns zero.
+
+          7. Accuracy
+
+          Let (delta) and (epsilon) be the relative errors in the argument
+          and result respectively.
+
+          If (delta) is somewhat larger than the machine precision (i.e.,
+          if (delta) is due to data errors etc), then (epsilon) and (delta)
+          are approximately related by:
+
+                                       | xK (x)|
+                                       |   1   |
+                            (epsilon)~=| ------|(delta).
+                                       | K (x) |
+                                       |  0    |
+
+          Figure 1 shows the behaviour of the error amplification factor
+
+                                     | xK (x)|
+                                     |   1   |
+                                     | ------|.
+                                     | K (x) |
+                                     |  0    |
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          However, if (delta) is of the same order as machine precision,
+          then rounding errors could make (epsilon) slightly larger than
+          the above relation predicts.
+
+                                                                 |  1 |
+          For small x, the amplification factor is approximately | ---|,
+                                                                 | lnx|
+          which implies strong attenuation of the error, but in general
+          (epsilon) can never be less than the machine precision.
+
+          For large x, (epsilon)~=x(delta) and we have strong amplification
+          of the relative error. Eventually K , which is asymptotically
+                                             0
+                     -x
+                    e
+          given by  ---, becomes so small that it cannot be calculated
+                      
+
+                    \/x
+          without underflow and hence the routine will return zero. Note
+          that for large x the errors will be dominated by those of the
+          Fortran intrinsic function EXP.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the appropriate
+          the Users' Note for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs18adf}{NAG On-line Documentation: s18adf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S18ADF(3NAG)      Foundation Library (12/10/92)      S18ADF(3NAG)
+
+
+
+          S18 -- Approximations of Special Functions                 S18ADF
+                  S18ADF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S18ADF returns the value of the modified Bessel Function K (x),
+                                                                    1
+          via the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S18ADF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the modified Bessel
+          Function of the second kind K (x).
+                                       1
+
+          Note: K (x) is undefined for x<=0 and the routine will fail for
+                 1
+          such arguments.
+
+          The routine is based on five Chebyshev expansions:
+
+          For 0<x<=1,
+
+                     1      --'           --'                     2
+              K (x)= -+xlnx >   a T (t)-x >   b T (t) , where t=2x -1;
+               1     x      --   r r      --   r r
+                            r=0           r=0
+
+          For 1<x<=2,
+
+                               -x --'
+                        K (x)=e   >   c T (t) , where t=2x-3;
+                         1        --   r r
+                                  r=0
+
+          For 2<x<=4,
+
+                               -x --'
+                        K (x)=e   >   d T (t) , where t=x-3;
+                         1        --   r r
+                                  r=0
+
+          For x>4,
+
+                               -x
+                              e   --'                    9-x
+                       K (x)= --- >   e T (t) , where t= ---.
+                        1         --   r r               1+x
+                              \/x r=0
+
+                                   1
+          For x near zero, K (x)~= -. This approximation is used when x is
+                            1      x
+          sufficiently small for the result to be correct to machine
+          precision. For very small x on some machines, it is impossible to
+                     1
+          calculate  - without overflow and the routine must fail.
+                     x
+
+          For large x, where there is a danger of underflow due to the
+          smallness of K , the result is set exactly to zero.
+                        1
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function. Constraint: X > 0.
+               0.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               X <= 0.0, K  is undefined. On soft failure the routine
+                          1
+               returns zero.
+
+          IFAIL= 2
+               X is too small, there is a danger of overflow. On soft
+               failure the routine returns approximately the largest
+               representable value.
+
+          7. Accuracy
+
+          Let (delta) and (epsilon) be the relative errors in the argument
+          and result respectively.
+
+          If (delta) is somewhat larger than the machine precision (i.e.,
+          if (delta) is due to data errors etc), then (epsilon) and (delta)
+          are approximately related by:
+
+                                    | xK (x)-K (x)|
+                                    |   0     1   |
+                         (epsilon)~=| ------------|(delta).
+                                    |    K (x)    |
+                                    |     1       |
+
+          Figure 1 shows the behaviour of the error amplification factor
+
+                                  | xK (x)-K (x)|
+                                  |   0     1   |
+                                  | ------------|.
+                                  |    K (x)    |
+                                  |     1       |
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          However if (delta) is of the same order as the machine precision,
+          then rounding errors could make (epsilon) slightly larger than
+          the above relation predicts.
+
+          For small x, (epsilon)~=(delta) and there is no amplification of
+          errors.
+
+          For large x, (epsilon)~=x(delta) and we have strong amplification
+          of the relative error. Eventually K , which is asymptotically
+                                             1
+                     -x
+                    e
+          given by  ---, becomes so small that it cannot be calculated
+                      
+
+                    \/x
+          without underflow and hence the routine will return zero. Note
+          that for large x the errors will be dominated by those of the
+          Fortran intrinsic function EXP.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs18aef}{NAG On-line Documentation: s18aef}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S18AEF(3NAG)      Foundation Library (12/10/92)      S18AEF(3NAG)
+
+
+
+          S18 -- Approximations of Special Functions                 S18AEF
+                  S18AEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S18AEF returns the value of the modified Bessel Function I (x),
+                                                                    0
+          via the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S18AEF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the modified Bessel
+          Function of the first kind I (x).
+                                      0
+
+          Note: I (-x)=I (x), so the approximation need only consider x>=0.
+                 0      0
+
+          The routine is based on three Chebyshev expansions:
+
+          For 0<x<=4,
+
+                              x --'                    ( x)
+                       I (x)=e  >   a T (t) , where t=2( -)-1;
+                        0       --   r r               ( 4)
+                                r=0
+
+          For 4<x<=12,
+
+                               x --'                    x-8
+                        I (x)=e  >   b T (t) , where t= ---;
+                         0       --   r r                4
+                                 r=0
+
+          For x>12,
+
+                             x
+                            e   --'                    ( 12)
+                     I (x)= --- >   c T (t) , where t=2( --)-1.
+                      0         --   r r               ( x )
+                            \/x r=0
+
+          For small x, I (x)~=1. This approximation is used when x is
+                        0
+          sufficiently small for the result to be correct to machine
+          precision.
+
+          For large x, the routine must fail because of the danger of
+                                   x
+          overflow in calculating e .
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               X is too large. On soft failure the routine returns the
+               approximate value of I (x) at the nearest valid argument.
+                                     0
+
+          7. Accuracy
+
+          Let (delta) and (epsilon) be the relative errors in the argument
+          and result respectively.
+
+          If (delta) is somewhat larger than the machine precision (i.e.,
+          if (delta) is due to data errors etc), then (epsilon) and (delta)
+          are approximately related by:
+
+                                       | xI (x)|
+                                       |   1   |
+                            (epsilon)~=| ------|(delta).
+                                       | I (x) |
+                                       |  0    |
+
+          Figure 1 shows the behaviour of the error amplification factor
+
+                                     | xI (x)|
+                                     |   1   |
+                                     | ------|.
+                                     | I (x) |
+                                     |  0    |
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          However if (delta) is of the same order as machine precision,
+          then rounding errors could make (epsilon) slightly larger than
+          the above relation predicts.
+
+                                                                  2
+                                                                 x
+          For small x the amplification factor is approximately  --, which
+                                                                 2
+          implies strong attenuation of the error, but in general (epsilon)
+          can never be less than the machine precision.
+
+          For large x, (epsilon)~=x(delta) and we have strong amplification
+          of errors. However the routine must fail for quite moderate
+          values of x, because I (x) would overflow; hence in practice the
+                                0
+          loss of accuracy for large x is not excessive. Note that for
+          large x the errors will be dominated by those of the Fortran
+          intrinsic function EXP.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs18aff}{NAG On-line Documentation: s18aff}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S18AFF(3NAG)      Foundation Library (12/10/92)      S18AFF(3NAG)
+
+
+
+          S18 -- Approximations of Special Functions                 S18AFF
+                  S18AFF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S18AFF returns a value for the modified Bessel Function I (x),
+                                                                   1
+          via the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S18AFF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the modified Bessel
+          Function of the first kind I (x).
+                                      1
+
+          Note: I (-x)=-I (x), so the approximation need only consider x>=0
+                 1       1
+
+          The routine is based on three Chebyshev expansions:
+
+          For 0<x<=4,
+
+                               --'                    ( x)2
+                       I (x)=x >   a T (t) , where t=2( -) -1;
+                        1      --   r r               ( 4)
+                               r=0
+
+          For 4<x<=12,
+
+                               x --'                    x-8
+                        I (x)=e  >   b T (t) , where t= ---;
+                         1       --   r r                4
+                                 r=0
+
+          For x>12,
+
+                             x
+                            e   --'                    ( 12)
+                     I (x)= --- >   c T (t) , where t=2( --)-1.
+                      1         --   r r               ( x )
+                            \/x r=0
+
+          For small x, I (x)~=x. This approximation is used when x is
+                        1
+          sufficiently small for the result to be correct to machine
+          precision.
+
+          For large x, the routine must fail because I (x) cannot be
+                                                      1
+          represented without overflow.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               X is too large. On soft failure the routine returns the
+               approximate value of I (x) at the nearest valid argument.
+                                     1
+
+          7. Accuracy
+
+          Let (delta) and (epsilon) be the relative errors in the argument
+          and result respectively.
+
+          If (delta) is somewhat larger than the machine precision (i.e.,
+          if (delta) is due to data errors etc), then (epsilon) and (delta)
+          are approximately related by:
+
+                                    | xI (x)-I (x)|
+                                    |   0     1   |
+                         (epsilon)~=| ------------|(delta).
+                                    |    I (x)    |
+                                    |     1       |
+
+          Figure 1 shows the behaviour of the error amplification factor
+
+                                  | xI (x)-I (x)|
+                                  |   0     1   |
+                                  | ------------|,
+                                  |    I (x)    |
+                                  |     1       |
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          However if (delta) is of the same order as machine precision,
+          then rounding errors could make (epsilon) slightly larger than
+          the above relation predicts.
+
+          For small x, (epsilon)~=(delta) and there is no amplification of
+          errors.
+
+          For large x, (epsilon)~=x(delta) and we have strong amplification
+          of errors. However the routine must fail for quite moderate
+          values of x because I (x) would overflow; hence in practice the
+                               1
+          loss of accuracy for large x is not excessive. Note that for
+          large x, the errors will be dominated by those of the Fortran
+          intrinsic function EXP.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs18dcf}{NAG On-line Documentation: s18dcf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S18DCF(3NAG)      Foundation Library (12/10/92)      S18DCF(3NAG)
+
+
+
+          S18 -- Approximations of Special Functions                 S18DCF
+                  S18DCF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S18DCF returns a sequence of values for the modified Bessel
+          functions K      (z) for complex z, non-negative (nu) and
+                     (nu)+n
+          n=0,1,...,N-1, with an option for exponential scaling.
+
+          2. Specification
+
+                 SUBROUTINE S18DCF (FNU, Z, N, SCALE, CY, NZ, IFAIL)
+                 INTEGER              N, NZ, IFAIL
+                 DOUBLE PRECISION     FNU
+                 COMPLEX(KIND(1.0D0)) Z, CY(N)
+                 CHARACTER*1          SCALE
+
+          3. Description
+
+          This subroutine evaluates a sequence of values for the modified
+          Bessel function K    (z), where z is complex, -(pi) < arg z <=
+                           (nu)
+          (pi), and (nu) is the real, non-negative order. The N-member
+          sequence is generated for orders (nu), (nu)+1,...,(nu)+N-1.
+                                                            z
+          Optionally, the sequence is scaled by the factor e .
+
+          The routine is derived from the routine CBESK in Amos [2].
+
+          Note: although the routine may not be called with (nu) less than
+          zero, for negative orders the formula K     (z)=K    (z) may be
+                                                 -(nu)     (nu)
+          used.
+
+          When N is greater than 1, extra values of K    (z) are computed
+                                                     (nu)
+          using recurrence relations.
+
+          For very large |z| or ((nu)+N-1), argument reduction will cause
+          total loss of accuracy, and so no computation is performed. For
+          slightly smaller |z| or ((nu)+N-1), the computation is performed
+          but results are accurate to less than half of machine precision.
+          If |z| is very small, near the machine underflow threshold, or
+          ((nu)+N-1) is too large, there is a risk of overflow and so no
+          computation is performed. In all the above cases, a warning is
+          given by the routine.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Amos D E (1986) Algorithm 644: A Portable Package for Bessel
+                Functions of a Complex Argument and Nonnegative Order. ACM
+                Trans. Math. Softw. 12 265--273.
+
+          5. Parameters
+
+           1:  FNU -- DOUBLE PRECISION                                Input
+               On entry: the order, (nu), of the first member of the
+               sequence of functions. Constraint: FNU >= 0.0.
+
+           2:  Z -- COMPLEX(KIND(1.0D0))                              Input
+               On entry: the argument z of the functions. Constraint: Z /=
+               (0.0, 0.0).
+
+           3:  N -- INTEGER                                           Input
+               On entry: the number, N, of members required in the sequence
+               K    (z), K      (z),..., K        (z). Constraint: N >= 1.
+                (nu)      (nu)+1          (nu)+N-1
+
+           4:  SCALE -- CHARACTER*1                                   Input
+               On entry: the scaling option.
+
+               If SCALE = 'U', the results are returned unscaled.
+
+               If SCALE = 'S', the results are returned scaled by the
+                       z
+               factor e . Constraint: SCALE = 'U' or 'S'.
+
+           5:  CY(N) -- COMPLEX(KIND(1.0D)) array                    Output
+               On exit: the N required function values: CY(i) contains
+               K        (z), for i=1,2,...,N.
+                (nu)+i-1
+
+           6:  NZ -- INTEGER                                         Output
+               On exit: the number of components of CY that are set to zero
+               due to underflow. If NZ > 0 and Rez>=0.0, elements CY(1),CY
+               (2),...,CY(NZ) are set to zero. If Rez<0.0, NZ simply states
+               the number of underflows, and not which elements they are.
+
+           7:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry FNU < 0.0,
+
+               or       Z = (0.0, 0.0),
+
+               or       N < 1,
+
+               or       SCALE /= 'U' or 'S'.
+
+          IFAIL= 2
+               No computation has been performed due to the likelihood of
+               overflow, because ABS(Z) is less than a machine-dependent
+               threshold value (given in the Users' Note for your
+               implementation).
+
+          IFAIL= 3
+               No computation has been performed due to the likelihood of
+               overflow, because FNU + N - 1 is too large - how large
+               depends on Z and the overflow threshold of the machine.
+
+          IFAIL= 4
+               The computation has been performed, but the errors due to
+               argument reduction in elementary functions make it likely
+               that the results returned by S18DCF are accurate to less
+               than half of machine precision. This error exit may occur if
+               either ABS(Z) or FNU + N - 1 is greater than a machine-
+               dependent threshold value (given in the Users' Note for
+               your implementation).
+
+          IFAIL= 5
+               No computation has been performed because the errors due to
+               argument reduction in elementary functions mean that all
+               precision in results returned by S18DCF would be lost. This
+               error exit may occur when either ABS(Z) or FNU + N - 1 is
+               greater than a machine-dependent threshold value (given in
+               the Users' Note for your implementation).
+
+          IFAIL= 6
+               No results are returned because the algorithm termination
+               condition has not been met. This may occur because the
+               parameters supplied to S18DCF would have caused overflow or
+               underflow.
+
+          7. Accuracy
+
+
+          All constants in subroutine S18DCF are given to approximately 18
+          digits of precision. Calling the number of digits of precision in
+          the floating-point arithmetic being used t, then clearly the
+          maximum number of correct digits in the results obtained is
+          limited by p=min(t,18). Because of errors in argument reduction
+          when computing elementary functions inside S18DCF, the actual
+          number of correct digits is limited, in general, by p-s, where
+          s~max(1,|log  |z||,|log  (nu)|) represents the number of digits
+                      10         10
+          lost due to the argument reduction. Thus the larger the values of
+          |z| and (nu), the less the precision in the result. If S18DCF is
+          called with N>1, then computation of function values via
+          recurrence may lead to some further small loss of accuracy.
+
+          If function values which should nominally be identical are
+          computed by calls to S18DCF with different base values of (nu)
+          and different N, the computed values may not agree exactly.
+          Empirical tests with modest values of (nu) and z have shown that
+          the discrepancy is limited to the least significant 3-4 digits of
+          precision.
+
+          8. Further Comments
+
+          The time taken by the routine for a call of S18DCF is
+          approximately proportional to the value of N, plus a constant. In
+          general it is much cheaper to call S18DCF with N greater than 1,
+          rather than to make N separate calls to S18DCF.
+
+          Paradoxically, for some values of z and (nu), it is cheaper to
+          call S18DCF with a larger value of N than is required, and then
+          discard the extra function values returned. However, it is not
+          possible to state the precise circumstances in which this is
+          likely to occur. It is due to the fact that the base value used
+          to start recurrence may be calculated in different regions for
+          different N, and the costs in each region may differ greatly.
+
+          Note that if the function required is K (x) or K (x), i.e.,
+                                                 0        1
+          (nu)=0.0 or 1.0, where x is real and positive, and only a single
+          function value is required, then it may be much cheaper to call
+          S18ACF, S18ADF, S18CCF(*) or S18CDF(*), depending on whether a
+          scaled result is required or not.
+
+          9. Example
+
+          The example program prints a caption and then proceeds to read
+          sets of data from the input data stream. The first datum is a
+          value for the order FNU, the second is a complex value for the
+          argument, Z, and the third is a value for the parameter SCALE.
+          The program calls the routine with N = 2 to evaluate the function
+          for orders FNU and FNU + 1, and it prints the results. The
+          process is repeated until the end of the input data stream is
+          encountered.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs18def}{NAG On-line Documentation: s18def}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S18DEF(3NAG)      Foundation Library (12/10/92)      S18DEF(3NAG)
+
+
+
+          S18 -- Approximations of Special Functions                 S18DEF
+                  S18DEF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S18DEF returns a sequence of values for the modified Bessel
+          functions I      (z) for complex z, non-negative (nu) and
+                     (nu)+n
+          n=0,1,...,N-1, with an option for exponential scaling.
+
+          2. Specification
+
+                 SUBROUTINE S18DEF (FNU, Z, N, SCALE, CY, NZ, IFAIL)
+                 INTEGER              N, NZ, IFAIL
+                 DOUBLE PRECISION     FNU
+                 COMPLEX(KIND(1.0D0)) Z, CY(N)
+                 CHARACTER*1          SCALE
+
+          3. Description
+
+          This subroutine evaluates a sequence of values for the modified
+          Bessel function I    (z), where z is complex, -(pi) < argz <=
+                           (nu)
+          (pi), and (nu) is the real, non-negative order. The N-member
+          sequence is generated for orders (nu), (nu)+1,...,(nu)+N-1.
+                                                            -|Rez|
+          Optionally, the sequence is scaled by the factor e      .
+
+          The routine is derived from the routine CBESI in Amos [2].
+
+          Note: although the routine may not be called with (nu) less than
+          zero, for negative orders the formula
+                               2
+          I     (z)=I    (z)+ ----sin((pi)(nu))K    (z) may be used (for
+           -(nu)     (nu)     (pi)              (nu)
+          the Bessel function K    (z), see S18DCF).
+                               (nu)
+
+          When N is greater than 1, extra values of I    (z) are computed
+                                                     (nu)
+          using recurrence relations.
+
+          For very large |z| or ((nu)+N-1), argument reduction will cause
+          total loss of accuracy, and so no computation is performed. For
+          slightly smaller |z| or ((nu)+N-1), the computation is performed
+          but results are accurate to less than half of machine precision.
+          If Re(z) is too large and the unscaled function is required,
+          there is a risk of overflow and so no computation is performed.
+          In all the above cases, a warning is given by the routine.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Amos D E (1986) Algorithm 644: A Portable Package for Bessel
+                Functions of a Complex Argument and Nonnegative Order. ACM
+                Trans. Math. Softw. 12 265--273.
+
+          5. Parameters
+
+           1:  FNU -- DOUBLE PRECISION                                Input
+               On entry: the order, (nu), of the first member of the
+               sequence of functions. Constraint: FNU >= 0.0.
+
+           2:  Z -- COMPLEX(KIND(1.0D0))                              Input
+               On entry: the argument z of the functions.
+
+           3:  N -- INTEGER                                           Input
+               On entry: the number, N, of members required in the sequence
+               I    (z),I      (z),...,I        (z). Constraint: N >= 1.
+                (nu)     (nu)+1         (nu)+N-1
+
+           4:  SCALE -- CHARACTER*1                                   Input
+               On entry: the scaling option.
+                    If SCALE = 'U', the results are returned unscaled.
+
+                    If SCALE = 'S', the results are returned scaled by the
+                            -|Rez|
+                    factor e      .
+               Constraint: SCALE = 'U' or 'S'.
+
+           5:  CY(N) -- COMPLEX(KIND(1.0D)) array                    Output
+               On exit: the N required function values: CY(i) contains
+               I        (z), for i=1,2,...,N.
+                (nu)+i-1
+
+           6:  NZ -- INTEGER                                         Output
+               On exit: the number of components of CY that are set to zero
+               due to underflow.
+
+               If NZ > 0, then elements CY(N-NZ+1),CY(N-NZ+2),...,CY(N) are
+               set to zero.
+
+           7:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry FNU < 0.0,
+
+               or       N < 1,
+
+               or       SCALE /= 'U' or 'S'.
+
+          IFAIL= 2
+               No computation has been performed due to the likelihood of
+               overflow, because real(Z) is greater than a machine-
+               dependent threshold value (given in the Users' Note for
+               your implementation). This error exit can only occur when
+               SCALE = 'U'.
+
+          IFAIL= 3
+               The computation has been performed, but the errors due to
+               argument reduction in elementary functions make it likely
+               that the results returned by S18DEF are accurate to less
+               than half of machine precision. This error exit may occur
+               when either ABS(Z) or FNU + N - 1 is greater than a machine-
+               dependent threshold value (given in the Users' Note for
+               your implementation).
+
+          IFAIL= 4
+               No computation has been performed because the errors due to
+               argument reduction in elementary functions mean that all
+               precision in results returned by S18DEF would be lost. This
+               error exit may occur when either ABS(Z) or FNU + N - 1 is
+               greater than a machine-dependent threshold value (given in
+               the Users' Note for your implementation).
+
+          IFAIL= 5
+               No results are returned because the algorithm termination
+               condition has not been met. This may occur because the
+               parameters supplied to S18DEF would have caused overflow or
+               underflow.
+
+          7. Accuracy
+
+          All constants in subroutine S18DEF are given to approximately 18
+          digits of precision. Calling the number of digits of precision in
+          the floating-point arithmetic being used t, then clearly the
+          maximum number of correct digits in the results obtained is
+          limited by p=min(t,18). Because of errors in argument reduction
+          when computing elementary functions inside S18DEF, the actual
+          number of correct digits is limited, in general, by p-s, where
+          s~max(1,|log  |z||,|log  (nu)|) represents the number of digits
+                      10         10
+          lost due to the argument reduction. Thus the larger the values of
+          |z| and (nu), the less the precision in the result. If S18DEF is
+          called with N>1, then computation of function values via
+          recurrence may lead to some further small loss of accuracy.
+
+          If function values which should nominally be identical are
+          computed by calls to S18DEF with different base values of (nu)
+          and different N, the computed values may not agree exactly.
+          Empirical tests with modest values of (nu) and z have shown that
+          the discrepancy is limited to the least significant 3-4 digits of
+          precision.
+
+          8. Further Comments
+
+          The time taken by the routine for a call of S18DEF is
+          approximately proportional to the value of N, plus a constant. In
+          general it is much cheaper to call S18DEF with N greater than 1,
+          rather than to make N separate calls to S18DEF.
+
+          Paradoxically, for some values of z and (nu), it is cheaper to
+          call S18DEF with a larger value of N than is required, and then
+          discard the extra function values returned. However, it is not
+          possible to state the precise circumstances in which this is
+          likely to occur. It is due to the fact that the base value used
+          to start recurrence may be calculated in different regions for
+          different N, and the costs in each region may differ greatly.
+
+          Note that if the function required is I (x) or I (x), i.e.,
+                                                 0        1
+          (nu)=0.0 or 1.0, where x is real and positive, and only a single
+          function value is required, then it may be much cheaper to call
+          S18AEF, S18AFF, S18CEF(*) or S18CFF(*), depending on whether a
+          scaled result is required or not.
+
+          9. Example
+
+          The example program prints a caption and then proceeds to read
+          sets of data from the input data stream. The first datum is a
+          value for the order FNU, the second is a complex value for the
+          argument, Z, and the third is a value for the parameter SCALE.
+          The program calls the routine with N = 2 to evaluate the function
+          for orders FNU and FNU + 1, and it prints the results. The
+          process is repeated until the end of the input data stream is
+          encountered.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs19aaf}{NAG On-line Documentation: s19aaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S19AAF(3NAG)      Foundation Library (12/10/92)      S19AAF(3NAG)
+
+
+
+          S19 -- Approximations of Special Functions                 S19AAF
+                  S19AAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S19AAF returns a value for the Kelvin function ber x via the
+          routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S19AAF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the Kelvin function
+          berx.
+
+          Note: ber(-x)=berx, so the approximation need only consider
+          x>=0.0.
+
+          The routine is based on several Chebyshev expansions:
+
+          For 0<=x<=5,
+
+                               --'                 ( x)4
+                         berx= >   a T (t) with t=2( -) -1;
+                               --   r r            ( 5)
+                               r=0
+
+          For x>5,
+
+                            
+
+                        x/\/2
+                       e      [(   1    )            1              ]
+                berx= --------[(1+ -a(t))cos(alpha)+ -b(t)sin(alpha)]
+                              [(   x    )            x              ]
+                      \/2(pi)x
+
+                            
+
+                       -x/\/2
+                      e      [(   1    )           1             ]
+                   + --------[(1+ -c(t))sin(beta)+ -d(t)cos(beta)]
+                             [(   x    )           x             ]
+                     \/2(pi)x
+
+                          x   (pi)           x   (pi)
+          where (alpha)= ---- ----, (beta)= ---+ ----,
+                               8                  8
+                         \/2                /2
+
+          and a(t), b(t), c(t), and d(t) are expansions in the variable
+             10
+          t= ---1.
+             x
+
+          When x is sufficiently close to zero, the result is set directly
+          to ber 0=1.0.
+
+          For large x, there is a danger of the result being totally
+          inaccurate, as the error amplification factor grows in an
+          essentially exponential manner; therefore the routine must fail.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry ABS(X) is too large for an accurate result to be
+               returned. On soft failure, the routine returns zero.
+
+          7. Accuracy
+
+          Since the function is oscillatory, the absolute error rather than
+          the relative error is important. Let E be the absolute error in
+          the result and (delta) be the relative error in the argument. If
+          (delta) is somewhat larger than the machine precision, then we
+          have:
+
+                               |  x              |
+                            E~=| ---(ber x+bei x)|(delta)
+                               |        1     1  |
+                               | \/2             |
+
+          (provided E is within machine bounds).
+
+          For small x the error amplification is insignificant and thus the
+          absolute error is effectively bounded by the machine precision.
+
+          For medium and large x, the error behaviour is oscillatory and
+                                                  
+
+                                      /   x   x/\/2
+          its amplitude grows like   /  -----e    . Therefore it is not
+                                   \/   2(pi)
+
+          possible to calculate the function with any accuracy when
+                       
+
+              x/\/2  /2(pi)
+          \/xe     > -------. Note that this value of x is much smaller than
+                     (delta)
+
+          the minimum value of x for which the function overflows.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs19abf}{NAG On-line Documentation: s19abf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S19ABF(3NAG)      Foundation Library (12/10/92)      S19ABF(3NAG)
+
+
+
+          S19 -- Approximations of Special Functions                 S19ABF
+                  S19ABF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S19ABF returns a value for the Kelvin function bei x via the
+          routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S19ABF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the Kelvin function
+          beix.
+
+          Note: bei(-x)=beix, so the approximation need only consider
+          x>=0.0.
+
+          The routine is based on several Chebyshev expansions:
+
+          For 0<=x<=5,
+
+                              2
+                             x  --'                    ( x)4
+                      bei x= -- >   a T (t),  with  t=2( -) -1;
+                             4  --   r r               ( 5)
+                                r=0
+
+          For x>5,
+
+                            
+
+                        x/\/2
+                       e      [(   1    )            1              ]
+               bei x= --------[(1+ -a(t))sin(alpha)- -b(t)cos(alpha)]
+                              [(   x    )            x              ]
+                      \/2(pi)x
+
+                            
+
+                        x/\/2
+                       e     [(   1    )           1             ]
+                   + --------[(1+ -c(t))cos(beta)- -d(t)sin(beta)]
+                             [(   x    )           x             ]
+                     \/2(pi)x
+
+                         x   (pi)          x   (pi)
+          where(alpha)= ---- ----,(beta)= ---+ ----,
+                              8                 8
+                        \/2               /2
+
+          and a(t), b(t), c(t), and d(t) are expansions in the variable
+             10
+          t= ---1.
+             x
+
+          When x is sufficiently close to zero, the result is computed as
+                  2
+                 x
+          bei x= --. If this result would underflow, the result returned is
+                 4
+          beix=0.0.
+
+          For large x, there is a danger of the result being totally
+          inaccurate, as the error amplification factor grows in an
+          essentially exponential manner; therefore the routine must fail.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry ABS(X) is too large for an accurate result to be
+               returned. On soft failure, the routine returns zero.
+
+          7. Accuracy
+
+          Since the function is oscillatory, the absolute error rather than
+          the relative error is important. Let E be the absolute error in
+          the function, and (delta) be the relative error in the argument.
+          If (delta) is somewhat larger than the machine precision, then we
+          have:
+
+                              |  x               |
+                           E~=| ---(-ber x+bei x)|(delta)
+                              |         1     1  |
+                              | \/2              |
+
+          (provided E is within machine bounds).
+
+          For small x the error amplification is insignificant and thus the
+          absolute error is effectively bounded by the machine precision.
+
+          For medium and large x, the error behaviour is oscillatory and
+                                                 
+
+                                      /   x   x\/2
+          its amplitude grows like   /  -----e    . Therefore it is
+                                   \/   2(pi)
+
+          impossible to calculate the functions with any accuracy when
+                  
+
+              x/\/2 /2(pi)
+          \/xe     > -------. Note that this value of x is much smaller than
+                     (delta)
+
+          the minimum value of x for which the function overflows.
+
+          8. Further Comments
+
+          For details of the time taken by the routine see the Users' Note
+          for your implementation.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs19acf}{NAG On-line Documentation: s19acf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S19ACF(3NAG)      Foundation Library (12/10/92)      S19ACF(3NAG)
+
+
+
+          S19 -- Approximations of Special Functions                 S19ACF
+                  S19ACF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S19ACF returns a value for the Kelvin function ker x, via the
+          routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S19ACF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the Kelvin function
+          ker x.
+
+          Note: for x<0 the function is undefined and at x=0 it is infinite
+          so we need only consider x>0.
+
+          The routine is based on several Chebyshev expansions:
+
+          For 0<x<=1,
+
+                                           (pi) 2
+                          ker x=-f(t)logx+ ----x g(t)+y(t)
+                                            16
+
+                                                                       4
+          where f(t), g(t) and y(t) are expansions in the variable t=2x -1;
+
+          For 1<x<=3,
+
+                                         (  11 )
+                                ker x=exp(- --x)q(t)
+                                         (  16 )
+
+          where q(t) is an expansion in the variable t=x-2;
+
+          For x>3,
+
+                                  
+
+                             -x/\/2
+                      / (pi)       [(   1    )           1             ]
+             ker x=  /  ----e      [(1+ -c(t))cos(beta)- -d(t)sin(beta)]
+                   \/    2x        [(   x    )           x             ]
+
+
+                         x   (pi)
+          where (beta)= ---+ ----, and c(t) and d(t) are expansions in the
+                              8
+                        \/2
+                      6
+          variable t= --1.
+                      x
+
+          When x is sufficiently close to zero, the result is computed as
+
+                                                            2
+                                          ( x) (      3 2) x
+                        ker x=-(gamma)-log( -)+((pi)- -x ) --
+                                          ( 2) (      8  ) 16
+
+          and when x is even closer to zero, simply as
+                            ( x)
+          ker x=-(gamma)-log( -).
+                            ( 2)
+
+                                                                       
+
+                                                           / (pi) -x/\/2
+          For large x, ker x is asymptotically given by   /  ----e      and
+                                                        \/    2x
+
+          this becomes so small that it cannot be computed without
+          underflow and the routine fails.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function. Constraint: X > 0.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry X is too large, the result underflows. On soft
+               failure, the routine returns zero.
+
+          IFAIL= 2
+               On entry X <= 0, the function is undefined. On soft failure
+               the routine returns zero.
+
+          7. Accuracy
+
+          Let E be the absolute error in the result, (epsilon) be the
+          relative error in the result and (delta) be the relative error in
+          the argument. If (delta) is somewhat larger than the machine
+          precision, then we have:
+
+                              |  x               |
+                           E~=| ---(ker  x+kei x)|(delta),
+                              |         1     1  |
+                              | \/2              |
+
+                                  |     ker  x+kei x|
+                                  |  x      1     1 |
+                       (epsilon)~=| --- ------------|(delta).
+                                  |        ker x    |
+                                  | \/2             |
+
+          For very small x, the relative error amplification factor is
+                                    1
+          approximately given by  ------, which implies a strong
+                                  |logx|
+          attenuation of relative error. However, (epsilon) in general
+          cannot be less than the machine precision.
+
+          For small x, errors are damped by the function and hence are
+          limited by the machine precision.
+
+          For medium and large x, the error behaviour, like the function
+          itself, is oscillatory, and hence only the absolute accuracy for
+          the function can be maintained. For this range of x, the
+                                                                      
+
+                                                         / (pi)x -x/\/2
+          amplitude of the absolute error decays like   /  -----e
+                                                      \/     2
+
+          which implies a strong attenuation of error. Eventually, ker x,
+                                                           
+
+                                               / (pi) -x/\/2
+          which asymptotically behaves like   /  ----e     , becomes so
+                                            \/    2x
+
+          small that it cannot be calculated without causing underflow, and
+          the routine returns zero. Note that for large x the errors are
+          dominated by those of the Fortran intrinsic function EXP.
+
+          8. Further Comments
+
+          Underflow may occur for a few values of x close to the zeros of
+          ker x, below the limit which causes a failure with IFAIL = 1.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs19adf}{NAG On-line Documentation: s19adf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S19ADF(3NAG)      Foundation Library (12/10/92)      S19ADF(3NAG)
+
+
+
+          S19 -- Approximations of Special Functions                 S19ADF
+                  S19ADF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S19ADF returns a value for the Kelvin function keix via the
+          routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S19ADF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the Kelvin function
+          keix.
+
+          Note: for x<0 the function is undefined, so we need only consider
+          x>=0.
+
+          The routine is based on several Chebyshev expansions:
+
+          For 0<=x<=1,
+
+                                           2
+                                (pi)      x
+                         keix=- ----f(t)+ --[-g(t)logx+v(t)]
+                                 4        4
+
+                                                                       4
+          where f(t), g(t) and v(t) are expansions in the variable t=2x -1;
+
+          For 1<x<=3,
+
+                                         (  9 )
+                                 keix=exp(- -x)u(t)
+                                         (  8 )
+
+          where u(t) is an expansion in the variable t=x-2;
+
+          For x>3,
+
+                                  
+
+                      / (pi) -x/\/2[(   1)               1             ]
+              keix=  /  ----e      [(1+ -)c(t)sin(beta)+ -d(t)cos(beta)]
+                   \/    2x        [(   x)               x             ]
+
+
+                         x   (pi)
+          where (beta)= ---+ ----, and c(t) and d(t) are expansions in the
+                              8
+                        \/2
+                      6
+          variable t= --1.
+                      x
+
+          For x<0, the function is undefined, and hence the routine fails
+          and returns zero.
+
+          When x is sufficiently close to zero, the result is computed as
+
+                                                          2
+                                (pi) (             ( x)) x
+                         keix=- ----+(1-(gamma)-log( -)) --
+                                 4   (             ( 2)) 4
+
+          and when x is even closer to zero simply as
+
+                                           (pi)
+                                    keix=- ----.
+                                            4
+
+                                                                      
+
+                                                          / (pi) -x/\/2
+          For large x, keix is asymptotically given by   /  ----e       and
+                                                       \/    2x
+
+          this becomes so small that it cannot be computed without
+          underflow and the routine fails.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function. Constraint: X >=
+               0.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          IFAIL= 1
+               On entry X is too large, the result underflows. On soft
+               failure, the routine returns zero.
+
+          IFAIL= 2
+               On entry X < 0, the function is undefined. On soft failure
+               the routine returns zero.
+
+          7. Accuracy
+
+          Let E be the absolute error in the result, and (delta) be the
+          relative error in the argument. If (delta) is somewhat larger
+          than the machine representation error, then we have:
+
+                             |  x                |
+                          E~=| ---(-ker  x+kei x)|(delta).
+                             |          1     1  |
+                             | \/2               |
+
+          For small x, errors are attenuated by the function and hence are
+          limited by the machine precision.
+
+          For medium and large x, the error behaviour, like the function
+          itself, is oscillatory and hence only absolute accuracy of the
+          function can be maintained. For this range of x, the amplitude of
+                                                         
+
+                                            / (pi)x -x/\/2
+          the absolute error decays like   /  -----e      , which implies a
+                                         \/     2
+
+          strong attenuation of error. Eventually, keix, which is
+                                                 
+
+                                     / (pi) -x/\/2
+          asymptotically given by   /  ----e      , becomes so small that it
+                                  \/    2x
+
+          cannot be calculated without causing underflow and therefore the
+          routine returns zero. Note that for large x, the errors are
+          dominated by those of the Fortran intrinsic function EXP.
+
+          8. Further Comments
+
+          Underflow may occur for a few values of x close to the zeros of
+          keix, below the limit which causes a failure with IFAIL = 1.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs20acf}{NAG On-line Documentation: s20acf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S20ACF(3NAG)      Foundation Library (12/10/92)      S20ACF(3NAG)
+
+
+
+          S20 -- Approximations of Special Functions                 S20ACF
+                  S20ACF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S20ACF returns a value for the Fresnel Integral S(x), via the
+          routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S20ACF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the Fresnel Integral
+
+                                     x
+                                     /   ( (pi) 2)
+                               S(x)= |sin( ----t )dt.
+                                     /   (  2    )
+                                     0
+
+          Note: S(x)=-S(-x), so the approximation need only consider x>=0.0
+
+          The routine is based on three Chebyshev expansions:
+
+          For 0<x<=3,
+
+                             3 --'                   ( x)4
+                       S(x)=x  >   a T (t) , with t=2( -) -1;
+                               --   r r              ( 3)
+                               r=0
+
+          For x>3,
+
+                          1  f(x)   ( (pi) 2)  g(x)   ( (pi) 2)
+                    S(x)= -- ----cos( ----x )- ----sin( ----x ),
+                          2   x     (  2    )    3    (  2    )
+                                                x
+
+                        --'
+          where f(x) =  >   b T (t),
+                        --   r r
+                        r=0
+
+                      --'                  ( 3)4
+          and g(x) =  >   c T (t), with t=2( -) -1.
+                      --   r r             ( x)
+                      r=0
+
+                              (pi) 3
+          For small x, S(x)~= ----x . This approximation is used when x is
+                               6
+          sufficiently small for the result to be correct to machine
+          precision. For very small x, this approximation would underflow;
+          the result is then set exactly to zero.
+
+                               1                1
+          For large x, f(x)~= ---- and g(x)~= -----. Therefore for
+                              (pi)                2
+                                              (pi)
+                                       1                                 1
+          moderately large x, when  ------- is negligible compared with  -,
+                                        2 3                              2
+                                    (pi) x
+          the second term in the approximation for x>3 may be dropped. For
+                                1                              1
+          very large x, when  ----- becomes negligible, S(x)~= -. However
+                              (pi)x                            2
+          there will be considerable difficulties in calculating
+             ( (pi) 2)
+          cos( ----x ) accurately before this final limiting value can be
+             (  2    )
+                         ( (pi) 2)
+          used. Since cos( ----x ) is periodic, its value is essentially
+                         (  2    )
+                                                2      2
+          determined by the fractional part of x . If x =N+(theta) where N
+                                                  ( (pi) 2)
+          is an integer and 0<=(theta)<1, then cos( ----x ) depends on
+                                                  (  2    )
+          (theta) and on N modulo 4. By exploiting this fact, it is
+          possible to retain significance in the calculation of
+             ( (pi) 2)
+          cos( ----x ) either all the way to the very large x limit, or at
+             (  2    )
+                                           x
+          least until the integer part of  - is equal to the maximum
+                                           2
+          integer allowed on the machine.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          There are no failure exits from this routine. The parameter IFAIL
+          has been included for consistency with other routines in this
+          chapter.
+
+          7. Accuracy
+
+          Let (delta) and (epsilon) be the relative errors in the argument
+          and result respectively.
+
+          If (delta) is somewhat larger than the machine precision (i.e.,
+          if (delta) is due to data errors etc), then (epsilon) and (delta)
+          are approximately related by:
+
+                                    |     ( (pi) 2)|
+                                    | xsin( ----x )|
+                                    |     (  2    )|
+                         (epsilon)~=| -------------|(delta).
+                                    |     S(x)     |
+
+          Figure 1 shows the behaviour of the error amplification factor
+
+          |     ( (pi) 2)|
+          | xsin( ----x )|
+          |     (  2    )|
+          | -------------|.
+          |     S(x)     |
+
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          However if (delta) is of the same order as the machine precision,
+          then rounding errors could make (epsilon) slightly larger than
+          the above relation predicts.
+
+          For small x, (epsilon)~=3(delta) and hence there is only moderate
+          amplification of relative error. Of course for very small x where
+          the correct result would underflow and exact zero is returned,
+          relative error-control is lost.
+
+          For moderately large values of x,
+
+                                    |     ( (pi) 2)|
+                       |(epsilon)|~=|2xsin( ----x )||(delta)|
+                                    |     (  2    )|
+
+          and the result will be subject to increasingly large
+          amplification of errors. However the above relation breaks down
+                                             1
+          for large values of x (i.e., when  -- is of the order of the
+                                              2
+                                             x
+          machine precision in this region the relative error in the result
+                                       2
+          is essentially bounded by  -----).
+                                     (pi)x
+
+          Hence the effects of error amplification are limited and at worst
+          the relative error loss should not exceed half the possible
+          number of significant figures.
+
+          8. Further Comments
+
+          None.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs20adf}{NAG On-line Documentation: s20adf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S20ADF(3NAG)      Foundation Library (12/10/92)      S20ADF(3NAG)
+
+
+
+          S20 -- Approximations of Special Functions                 S20ADF
+                  S20ADF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S20ADF returns a value for the Fresnel Integral C(x), via the
+          routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S20ADF (X, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X
+
+          3. Description
+
+          This routine evaluates an approximation to the Fresnel Integral
+
+                                     x
+                                     /   ( (pi) 2)
+                               C(x)= |cos( ----t )dt.
+                                     /   (  2    )
+                                     0
+
+          Note: C(x)=-C(-x), so the approximation need only consider x>=0.0
+
+          The routine is based on three Chebyshev expansions:
+
+          For 0<x<=3,
+
+                               --'                   ( x)4
+                        C(x)=x >   a T (t) , with t=2( -) -1;
+                               --   r r              ( 3)
+                               r=0
+
+          For x>3,
+
+                          1  f(x)   ( (pi) 2)  g(x)   ( (pi) 2)
+                    C(x)= -+ ----sin( ----x )- ----cos( ----x ),
+                          2   x     (  2    )    3    (  2    )
+                                                x
+
+                        --'
+          where f(x) =  >   b T (t),
+                        --   r r
+                        r=0
+
+                      --'                  ( 3)4
+          and g(x) =  >   c T (t), with t=2( -) -1.
+                      --   r r             ( x)
+                      r=0
+
+          For small x, C(x)~=x. This approximation is used when x is
+          sufficiently small for the result to be correct to machine
+          precision.
+
+                               1                1
+          For large x, f(x)~= ---- and g(x)~= -----. Therefore for
+                              (pi)                2
+                                              (pi)
+                                       1                                 1
+          moderately large x, when  ------- is negligible compared with  -,
+                                        2 3                              2
+                                    (pi) x
+          the second term in the approximation for x>3 may be dropped. For
+                                1                              1
+          very large x, when  ----- becomes negligible, C(x)~= -. However
+                              (pi)x                            2
+          there will be considerable difficulties in calculating
+             ( (pi) 2)
+          sin( ----x ) accurately before this final limiting value can be
+             (  2    )
+                         ( (pi) 2)
+          used. Since sin( ----x ) is periodic, its value is essentially
+                         (  2    )
+                                                2      2
+          determined by the fractional part of x . If x =N+(theta), where N
+                                                  ( (pi) 2)
+          is an integer and 0<=(theta)<1, then sin( ----x ) depends on
+                                                  (  2    )
+          (theta) and on N modulo 4. By exploiting this fact, it is
+          possible to retain some significance in the calculation of
+             ( (pi) 2)
+          sin( ----x ) either all the way to the very large x limit, or at
+             (  2    )
+                                           x
+          least until the integer part of  - is equal to the maximum
+                                           2
+          integer allowed on the machine.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+               On entry: the argument x of the function.
+
+           2:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          There are no failure exits from this routine. The parameter IFAIL
+          has been included for consistency with other routines in this
+          chapter.
+
+          7. Accuracy
+
+          Let (delta) and (epsilon) be the relative errors in the argument
+          and result respectively.
+
+          If (delta) is somewhat larger than the machine precision (i.e if
+          (delta) is due to data errors etc), then (epsilon) and (delta)
+          are approximately related by:
+
+                                    |     ( (pi) 2)|
+                                    | xcos( ----x )|
+                                    |     (  2    )|
+                         (epsilon)~=| -------------|(delta).
+                                    |     C(x)     |
+
+          Figure 1 shows the behaviour of the error amplification factor
+
+          |     ( (pi) 2)|
+          | xcos( ----x )|
+          |     (  2    )|
+          | -------------|.
+          |     C(x)     |
+
+
+                                     Figure 1
+                   Please see figure in printed Reference Manual
+
+          However if (delta) is of the same order as the machine precision,
+          then rounding errors could make (epsilon) slightly larger than
+          the above relation predicts.
+
+          For small x, (epsilon)~=(delta) and there is no amplification of
+          relative error.
+
+          For moderately large values of x,
+
+                                    |     ( (pi) 2)|
+                       |(epsilon)|~=|2xcos( ----x )||(delta)|
+                                    |     (  2    )|
+
+          and the result will be subject to increasingly large
+          amplification of errors. However the above relation breaks down
+                                             1
+          for large values of x (i.e., when  -- is of the order of the
+                                              2
+                                             x
+          machine precision); in this region the relative error in the
+                                              2
+          result is essentially bounded by  -----).
+                                            (pi)x
+
+          Hence the effects of error amplification are limited and at worst
+          the relative error loss should not exceed half the possible
+          number of significant figures.
+
+          8. Further Comments
+
+          None.
+
+          9. Example
+
+          The example program reads values of the argument x from a file,
+          evaluates the function at each value of x and prints the results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs21baf}{NAG On-line Documentation: s21baf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S21BAF(3NAG)      Foundation Library (12/10/92)      S21BAF(3NAG)
+
+
+
+          S21 -- Approximations of Special Functions                 S21BAF
+                  S21BAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S21BAF returns a value of an elementary integral, which occurs as
+          a degenerate case of an elliptic integral of the first kind, via
+          the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S21BAF (X, Y, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X, Y
+
+          3. Description
+
+          This routine calculates an approximate value for the integral
+
+                                        infty
+                                      1 /         dt
+                             R (x,y)= - |     ----------
+                              C       2 /       
+                                        0     \/t+x(t+y)
+
+          where x>=0 and y/=0.
+
+          This function, which is related to the logarithm or inverse
+          hyperbolic functions for y<x and to inverse circular functions if
+          x<y, arises as a degenerate form of the elliptic integral of the
+          first kind. If y<0, the result computed is the Cauchy principal
+          value of the integral.
+
+          The basic algorithm, which is due to Carlson [2] and [3], is to
+          reduce the arguments recursively towards their mean by the
+          system:
+
+              x =x,
+               0
+
+              y =y
+               0
+
+              (mu) =(x +2y )/3,
+                  n   n   n
+
+              S =(y -x )/3(mu)
+               n   n  n       n
+
+                               
+
+              (lambda) =y +2  /x y
+                      n  n  \/  n n
+
+              x   =(x +(lambda) )/4,
+               n+1   n         n
+
+              y   =(y +(lambda) )/4.
+               n+1   n         n
+
+          The quantity |S | for n=0,1,2,3,... decreases with increasing n,
+                         n
+                             n
+          eventually |S |~1/4 . For small enough S  the required function
+                       n                          n
+          value can be approximated by the first few terms of the Taylor
+          series about the mean. That is
+
+                              (     2   3    4    5)
+                              (   3S   S   3S   9S )
+                              (     n   n    n    n)    
+                      R (x,y)=(1+ ---+ --+ ---+ ---)/  /(mu) .
+                       C      (   10   7    8   22 ) \/     n
+
+          The truncation error involved in using this approximation is
+                           6
+          bounded by 16|S | /(1-2|S |) and the recursive process is stopped
+                         n         n
+          when S  is small enough for this truncation error to be
+                n
+          negligible compared to the machine precision.
+
+          Within the domain of definition, the function value is itself
+          representable for all representable values of its arguments.
+          However, for values of the arguments near the extremes the above
+          algorithm must be modified so as to avoid causing underflows or
+          overflows in intermediate steps. In extreme regions arguments are
+          pre-scaled away from the extremes and compensating scaling of the
+          result is done before returning to the calling program.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Carlson B C (1978) Computing Elliptic Integrals by
+                Duplication. (Preprint) Department of Physics, Iowa State
+                University.
+
+          [3]   Carlson B C (1988) A Table of Elliptic Integrals of the
+                Third Kind. Math. Comput. 51 267--280.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+
+           2:  Y -- DOUBLE PRECISION                                  Input
+               On entry: the arguments x and y of the function,
+               respectively. Constraint: X >= 0.0 and Y /= 0.0.
+
+           3:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry X < 0.0; the function is undefined.
+
+          IFAIL= 2
+               On entryY = 0.0; the function is undefined.
+
+               On soft failure the routine returns zero.
+
+          7. Accuracy
+
+          In principle the routine is capable of producing full machine
+          precision. However round-off errors in internal arithmetic will
+          result in slight loss of accuracy. This loss should never be
+          excessive as the algorithm does not involve any significant
+          amplification of round-off error. It is reasonable to assume that
+          the result is accurate to within a small multiple of the machine
+          precision.
+
+          8. Further Comments
+
+          Users should consult the Chapter Introduction which shows the
+          relationship of this function to the classical definitions of the
+          elliptic integrals.
+
+          9. Example
+
+          This example program simply generates a small set of non-extreme
+          arguments which are used with the routine to produce the table of
+          low accuracy results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs21bbf}{NAG On-line Documentation: s21bbf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S21BBF(3NAG)      Foundation Library (12/10/92)      S21BBF(3NAG)
+
+
+
+          S21 -- Approximations of Special Functions                 S21BBF
+                  S21BBF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S21BBF returns a value of the symmetrised elliptic integral of
+          the first kind, via the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S21BBF (X, Y, Z, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X, Y, Z
+
+          3. Description
+
+          This routine calculates an approximation to the integral
+
+                                     infty
+                                   1 /            dt
+                        R (x,y,z)= - |     -----------------
+                         F         2 /       
+                                     0     \/(t+x)(t+y)(t+z)
+
+          where x, y, z>=0 and at most one is zero.
+
+          The basic algorithm, which is due to Carlson [2] and [3], is to
+          reduce the arguments recursively towards their mean by the rule:
+
+               x =min(x,y,z),z =max(x,y,z),
+                0             0
+
+               y = remaining third intermediate value argument.
+                0
+
+          (This ordering, which is possible because of the symmetry of the
+          function, is done for technical reasons related to the avoidance
+          of overflow and underflow.)
+
+              (mu) =   (x +y +3z )/3
+                  n      n  n   n
+
+              X    =   (1-x )/(mu)
+               n           n      n
+
+              Y    =   (1-y )/(mu)
+               n           n      n
+
+              Z    =   (1-z )/(mu)
+               n           n      n
+
+                                           
+
+              (lambda) =  /x y +  /y z +  /z x
+                      n \/  n n /  n n /  n n
+
+              x    =   (x +(lambda) )/4
+               n+1       n         n
+
+              y    =   (y +(lambda) )/4
+               n+1       n         n
+
+              z    =   (z +(lambda) )/4
+               n+1       n         n
+
+          (epsilon) =max(|X |,|Y |,|Z |) and the function may be
+                   n       n    n    n
+          approximated adequately by a 5th order power series:
+
+                                         (        2           )
+                                         (   E   E   3E E   E )
+                                    1    (    2   2    2 3   3)
+                      R (x,y,z)= --------(1- --+ --- -----+ --)
+                       F                 (   10  24   44    14)
+                                   /(mu)
+                                 \/     n
+
+          where E =X Y +Y Z +Z X ,E =X Y Z .
+                 2  n n  n n  n n  3  n n n
+
+          The truncation error involved in using this approximation is
+                              6
+          bounded by (epsilon) /4(1-(epsilon) ) and the recursive process
+                              n              n
+          is stopped when this truncation error is negligible compared with
+          the machine precision.
+
+          Within the domain of definition, the function value is itself
+          representable for all representable values of its arguments.
+          However, for values of the arguments near the extremes the above
+          algorithm must be modified so as to avoid causing underflows or
+          overflows in intermediate steps. In extreme regions arguments are
+          pre-scaled away from the extremes and compensating scaling of the
+          result is done before returning to the calling program.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Carlson B C (1978) Computing Elliptic Integrals by
+                Duplication. (Preprint) Department of Physics, Iowa State
+                University.
+
+          [3]   Carlson B C (1988) A Table of Elliptic Integrals of the
+                Third Kind. Math. Comput. 51 267--280.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+
+           2:  Y -- DOUBLE PRECISION                                  Input
+
+           3:  Z -- DOUBLE PRECISION                                  Input
+               On entry: the arguments x, y and z of the function.
+               Constraint: X, Y, Z >= 0.0 and only one of X, Y and Z may be
+               zero.
+
+           4:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry one or more of X, Y and Z is negative; the function
+               is undefined.
+
+          IFAIL= 2
+               On entry two or more of X, Y and Z are zero; the function is
+               undefined.
+
+               On soft failure, the routine returns zero.
+
+          7. Accuracy
+
+          In principle the routine is capable of producing full machine
+          precision. However round-off errors in internal arithmetic will
+          result in slight loss of accuracy. This loss should never be
+          excessive as the algorithm does not involve any significant
+          amplification of round-off error. It is reasonable to assume that
+          the result is accurate to within a small multiple of the machine
+          precision.
+
+          8. Further Comments
+
+          Users should consult the Chapter Introduction which shows the
+          relationship of this function to the classical definitions of the
+          elliptic integrals.
+
+          If two arguments are equal, the function reduces to the
+          elementary integral R , computed by S21BAF.
+                               C
+
+          9. Example
+
+          This example program simply generates a small set of non-extreme
+          arguments which are used with the routine to produce the table of
+          low accuracy results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs21bcf}{NAG On-line Documentation: s21bcf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S21BCF(3NAG)      Foundation Library (12/10/92)      S21BCF(3NAG)
+
+
+
+          S21 -- Approximations of Special Functions                 S21BCF
+                  S21BCF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S21BCF returns a value of the symmetrised elliptic integral of
+          the second kind, via the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S21BCF (X, Y, Z, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X, Y, Z
+
+          3. Description
+
+          This routine calculates an approximate value for the integral
+
+                                    infty
+                                  3 /             dt
+                       R (x,y,z)= - |     -------------------
+                        D         2 /        
+                                    0       /               3
+                                          \/ (t+x)(t+y)(t+z)
+
+          where x, y>=0, at most one of x and y is zero, and z>0.
+
+          The basic algorithm, which is due to Carlson [2] and [3], is to
+          reduce the arguments recursively towards their mean by the rule:
+
+              x    =   x
+               0
+
+              y    =   y
+               0
+
+              z    =   z
+               0
+
+              (mu) =   (x +y +3z )/5
+                  n      n  n   n
+
+              X    =   (1-x )/(mu)
+               n           n      n
+
+              Y    =   (1-y )/(mu)
+               n           n      n
+
+              Z    =   (1-z )/(mu)
+               n           n      n
+
+                                           
+
+              (lambda) =  /x y +  /y z +  /z x
+                       n\/  n n /  n n /  n n
+
+              x    =   (x +(lambda) )/4
+               n+1       n         n
+
+              y    =   (y +(lambda) )/4
+               n+1       n         n
+
+              z    =   (z +(lambda) )/4
+               n+1       n         n
+
+          For n sufficiently large,
+
+                                                       ( 1)n
+                        (epsilon) =max(|X |,|Y |,|Z |)~( -)
+                                 n       n    n    n   ( 4)
+
+          and the function may be approximated adequately by a 5th order
+                                   n-1          -m
+                                   --          4
+          power series R (x,y,z)=3 >   -------------------+
+                        D          --                   
+                                   m=0 (z +(lambda) )  /z
+                                         m         n \/  m
+
+
+               -n
+              4     [   3 (2)  1 (3)  3   (2) 2  3  (4)  3  (2) (3)  3  (5)]
+           ---------[1+ -S   + -S   + --(S   ) + --S   + --S   S   + --S   ]
+                    [   7 n    3 n    22  n      11 n    13 n   n    13 n  ]
+              /    3
+             / (mu)
+           \/      n
+
+          where
+
+                                 (m) ( m  m   m)
+                                S   =(X +Y +3Z )/2m.
+                                 n   ( n  n   n)
+
+          The truncation error in this expansion is bounded by
+                         6
+               3(epsilon)
+                         n
+           ------------------- and the recursive process is terminated when
+               
+
+              /              3
+             / (1-(epsilon) )
+           \/              n
+          this quantity is negligible compared with the machine precision.
+
+          The routine may fail either because it has been called with
+          arguments outside the domain of definition, or with arguments so
+          extreme that there is an unavoidable danger of setting underflow
+          or overflow.
+
+                           -3/2
+          Note: R (x,x,x)=x    , so there exists a region of extreme
+                 D
+          arguments for which the function value is not representable.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Carlson B C (1978) Computing Elliptic Integrals by
+                Duplication. (Preprint) Department of Physics, Iowa State
+                University.
+
+          [3]   Carlson B C (1988) A Table of Elliptic Integrals of the
+                Third Kind. Math. Comput. 51 267--280.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+
+           2:  Y -- DOUBLE PRECISION                                  Input
+
+           3:  Z -- DOUBLE PRECISION                                  Input
+               On entry: the arguments x, y and z of the function.
+               Constraint: X, Y >= 0.0, Z > 0.0 and only one of X and Y may
+               be zero.
+
+           4:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry either X or Y is negative, or both X and Y are
+               zero; the function is undefined.
+
+          IFAIL= 2
+               On entry Z <= 0.0; the function is undefined.
+
+          IFAIL= 3
+               On entry either Z is too close to zero or both X and Y are
+               too close to zero: there is a danger of setting overflow.
+
+          IFAIL= 4
+               On entry at least one of X, Y and Z is too large: there is a
+               danger of setting underflow.
+
+               On soft failure the routine returns zero.
+
+          7. Accuracy
+
+          In principle the routine is capable of producing full machine
+          precision. However round-off errors in internal arithmetic will
+          result in slight loss of accuracy. This loss should never be
+          excessive as the algorithm does not involve any significant
+          amplification of round-off error. It is reasonable to assume that
+          the result is accurate to within a small multiple of the machine
+          precision.
+
+          8. Further Comments
+
+          Users should consult the Chapter Introduction which shows the
+          relationship of this function to the classical definitions of the
+          elliptic integrals.
+
+          9. Example
+
+          This example program simply generates a small set of non-extreme
+          arguments which are used with the routine to produce the table of
+          low accuracy results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXs21bdf}{NAG On-line Documentation: s21bdf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     S21BDF(3NAG)      Foundation Library (12/10/92)      S21BDF(3NAG)
+
+
+
+          S21 -- Approximations of Special Functions                 S21BDF
+                  S21BDF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          S21BDF returns a value of the symmetrised elliptic integral of
+          the third kind, via the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION S21BDF (X, Y, Z, R, IFAIL)
+                 INTEGER          IFAIL
+                 DOUBLE PRECISION X, Y, Z, R
+
+          3. Description
+
+          This routine calculates an approximation to the integral
+
+                                    infty
+                                  3 /                 dt
+                 R (x,y,z,(rho))= - |     --------------------------
+                  J               2 /                
+                                    0     (t+(rho))\/(t+x)(t+y)(t+z)
+
+          where x, y, z>=0, (rho)/=0 and at most one of x, y and z is zero.
+
+          If p<0, the result computed is the Cauchy principal value of the
+          integral.
+
+          The basic algorithm, which is due to Carlson [2] and [3], is to
+          reduce the arguments recursively towards their mean by the rule:
+
+              x    =   x
+               0
+
+              y    =   y
+               0
+
+              z    =   z
+               0
+
+              (rho) =  (rho)
+                   0
+
+              (mu) =   (x +y +z +2(rho) )/5
+                  n      n  n  n       n
+
+              X    =   (1-x )/(mu)
+               n           n      n
+
+              Y    =   (1-y )/(mu)
+               n           n      n
+
+              Z    =   (1-z )/(mu)
+               n           n      n
+
+              P    =   (1-(rho) )/(mu)
+               n               n      n
+
+                                            
+
+              (lambda) =   /x y +  /y z +  /z x
+                        n\/  n n /  n n /  n n
+
+              x    =   (x +(lambda) )/4
+               n+1       n         n
+
+              y    =   (y +(lambda) )/4
+               n+1       n         n
+
+              z    =   (z +(lambda) )/4
+               n+1       n         n
+
+              (rho) =  ((rho) +(lambda) )/4
+                   n+1       n         n
+                                                            2
+                       [                                   ]
+              (alpha) =[(rho) (  /x +  /y +  /z )+  /x y z ]
+                     n [     n \/  n /  n /  n  /  n n n]
+
+                                               2
+              (beta) = (rho) ((rho) +(lambda) )
+                    n       n      n         n
+
+          For n sufficiently large,
+
+                                                            1
+                       (epsilon) =max(|X |,|Y |,|Z |,|P |)~ --
+                                n       n    n    n    n     n
+                                                            4
+
+          and the function may be approximated by a 5th order power series
+                            n-1
+                            --  -m
+          R (x,y,z,(rho))=3 >  4  R ((alpha) ,(beta) )
+           J                --     C        m       m
+                            m=0
+
+
+                -n
+               4     [   3 (2)  1 (3)  3   (2) 2  3  (4)  3  (2) (3)  3  (5)]
+          + ---------[1+ -S   + -S   + --(S   ) + --S   + --S   S   + --S   ]
+                     [   7 n    3 n    22  n      11 n    13 n   n    13 n  ]
+               /    3
+              / (mu)
+            \/      n
+
+                 (m)     m  m  m   m
+          where S    = (X +Y +Z +2P )/2m.
+                 n       n  n  n   n
+
+          The truncation error in this expansion is bounded by
+                          
+
+                    6    /              3
+          3(epsilon) /  / (1-(epsilon) )  and the recursion process is
+                    n \/              n
+          terminated when this quantity is negligible compared with the
+          machine precision. The routine may fail either because it has
+          been called with arguments outside the domain of definition or
+          with arguments so extreme that there is an unavoidable danger of
+          setting underflow or overflow.
+
+                               3
+                             - -
+                               2
+
+          Note: R (x,x,x,x)=x   , so there exists a region of extreme
+                 J
+          arguments for which the function value is not representable.
+
+          4. References
+
+          [1]   Abramowitz M and Stegun I A (1968) Handbook of Mathematical
+                Functions. Dover Publications.
+
+          [2]   Carlson B C (1978) Computing Elliptic Integrals by
+                Duplication. (Preprint) Department of Physics, Iowa State
+                University.
+
+          [3]   Carlson B C (1988) A Table of Elliptic Integrals of the
+                Third Kind. Math. Comput. 51 267--280.
+
+          5. Parameters
+
+           1:  X -- DOUBLE PRECISION                                  Input
+
+           2:  Y -- DOUBLE PRECISION                                  Input
+
+           3:  Z -- DOUBLE PRECISION                                  Input
+
+           4:  R -- DOUBLE PRECISION                                  Input
+               On entry: the arguments x, y, z and (rho) of the function.
+               Constraint: X, Y, Z >= 0.0, R /= 0.0 and at most one of X, Y
+               and Z may be zero.
+
+           5:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry at least one of X, Y and Z is negative, or at least
+               two of them are zero; the function is undefined.
+
+          IFAIL= 2
+               On entry R = 0.0; the function is undefined.
+
+          IFAIL= 3
+               On entry either R is too close to zero, or any two of X, Y
+               and Z are too close to zero; there is a danger of setting
+               overflow.
+
+          IFAIL= 4
+               On entry at least one of X, Y, Z and R is too large; there
+               is a danger of setting underflow.
+
+          IFAIL= 5
+               An error has occurred in a call to S21BAF. Any such
+               occurrence should be reported to NAG.
+
+               On soft failure, the routine returns zero.
+
+          7. Accuracy
+
+          In principle the routine is capable of producing full machine
+          precision. However round-off errors in internal arithmetic will
+          result in slight loss of accuracy. This loss should never be
+          excessive as the algorithm does not involve any significant
+          amplification of round-off error. It is reasonable to assume that
+          the result is accurate to within a small multiple of the machine
+          precision.
+
+          8. Further Comments
+
+          Users should consult the Chapter Introduction which shows the
+          relationship of this function to the classical definitions of the
+          elliptic integrals.
+
+          If the argument R is equal to any of the other arguments, the
+          function reduces to the integral R , computed by S21BCF.
+                                            D
+
+          9. Example
+
+          This example program simply generates a small set of non-extreme
+          arguments which are used with the routine to produce the table of
+          low accuracy results.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
diff --git a/src/hyper/pages/nagx.ht b/src/hyper/pages/nagx.ht
new file mode 100644
index 00000000..0676bfe3
--- /dev/null
+++ b/src/hyper/pages/nagx.ht
@@ -0,0 +1,1510 @@
+\begin{page}{manpageXXx01}{NAG On-line Documentation: x01}
+\beginscroll
+\begin{verbatim}
+
+
+
+     X01(3NAG)         Foundation Library (12/10/92)         X01(3NAG)
+
+
+
+          X01 -- Mathematical Constants                 Introduction -- X01
+                                    Chapter X01
+                              Mathematical Constants
+
+          1. Scope of the Chapter
+
+          This chapter is concerned with the provision of mathematical
+          constants required by other routines within the Library.
+
+          It should be noted that because of the trivial nature of the
+          routines individual routine documents are not provided.
+
+          2. Background to the Problems
+
+          Some Library routines require mathematical constants to maximum
+          machine precision. These routines call Chapter X01 and thus
+          lessen the number of changes that have to be made between
+          different implementations of the Library.
+
+          3. Recommendations on Choice and Use of Routines
+
+          Although these routines are primarily intended for use by other
+          routines they may be accessed directly by the user:
+
+          Constant           Fortran Specification
+
+          (pi)               DOUBLE PRECISION FUNCTION X01AAF(X)
+                             DOUBLE PRECISION X
+
+          (gamma)            DOUBLE PRECISION FUNCTION X01ABF(X)
+          (Euler constant)   DOUBLE PRECISION X
+
+          The argument X of these routines is a dummy argument.
+
+
+          X01 -- Mathematical Constants                     Contents -- X01
+          Chapter X01
+
+          Mathematical Constants
+
+          X01AAF  (pi)
+
+          X01ABF  Euler's constant, (gamma)
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXx02}{NAG On-line Documentation: x02}
+\beginscroll
+\begin{verbatim}
+
+
+
+     X02(3NAG)         Foundation Library (12/10/92)         X02(3NAG)
+
+
+
+          X02 -- Machine Constants                      Introduction -- X02
+                                    Chapter X02
+                                 Machine Constants
+
+          1. Scope of the Chapter
+
+          This chapter is concerned with parameters which characterise
+          certain aspects of the computing environment in which the NAG
+          Foundation Library is implemented. They relate primarily to
+          floating-point arithmetic, but also to integer arithmetic and the
+          elementary functions. The values of the parameters vary from one
+          implementation of the Library to another, but within the context
+          of a single implementation they are constants.
+
+          The parameters are intended for use primarily by other routines
+          in the Library, but users of the Library may sometimes need to
+          refer to them directly.
+
+          Each parameter-value is returned by a separate Fortran function.
+          Because of the trivial nature of the functions, individual
+          routine documents are not provided; the necessary details are
+          given in Section 3 of this Introduction.
+
+          2. Background to the Problems
+
+          2.1. Floating-Point Arithmetic
+
+          2.1.1.  A model of floating-point arithmetic
+
+          In order to characterise the important properties of floating-
+          point arithmetic by means of a small number of parameters, NAG
+          uses a simplified model of floating-point arithmetic. The
+          parameters of the model can be chosen to provide a sufficiently
+          close description of the behaviour of actual implementations of
+          floating-point arithmetic, but not, in general, an exact
+          description; actual implementations vary too much in the details
+          of how numbers are represented or arithmetic operations are
+          performed.
+
+          The model is based on that developed by Brown [1], but differs in
+          some respects. The essential features are summarised here.
+
+          The model is characterised by four integer parameters and one
+          logical parameter. The four integer parameters are:
+
+                b  :  the base
+
+                p  :  the precision (i.e. the number of significant base-B
+                      digits)
+
+              e    :  the minimum exponent
+               min
+
+              e    :  the maximum exponent
+               max
+
+          These parameters define a set of numerical values of the form:
+
+                                           e
+                                        f*b
+
+          where the exponent e must lie in the range [e   ,e   ], and the
+                                                       min  max
+          fraction f (also called the mantissa or significand) lies in the
+          range [1/b,1), and may be written:
+
+                                    f=0.f f ...f
+                                         1 2    p
+
+          Thus f is a p-digit fraction to the base b; the f  are the base-b
+                                                           i
+          digits of the fraction: they are integers in the range 0 to b-1,
+          and the leading digit f  must not be zero.
+                                 1
+
+          The set of values so defined (together with zero) are called
+          model numbers. For example, if b=10, p=5, e   =-99 and e   =+99,
+                                                     min          max
+                                                   67
+          then a typical model number is 0.12345*10  .
+
+          The model numbers must obey certain rules for the computed
+          results of the following basic arithmetic operations: addition,
+          subtraction, multiplication, negation, absolute value, and
+          comparisons. The rules depend on the value of the logical
+          parameter ROUNDS.
+
+          If ROUNDS is true, then the computed result must be the nearest
+          model number to the exact result (assuming that overflow or
+          underflow does not occur); if the exact result is midway between
+          two model numbers, then it may be rounded either way.
+
+          If ROUNDS is false, then: if the exact result is a model number,
+          the computed result must be equal to the exact result; otherwise,
+          the computed result may be either of the adjacent model numbers
+          on either side of the exact result.
+
+          For division and square root, this latter rule is further relaxed
+          (regardless of the value of ROUNDS): the computed result may also
+          be one of the next adjacent model numbers on either side of the
+          permitted values just stated.
+
+          On some machines, the full set of representable floating-point
+          numbers conforms to the rules of the model with appropriate
+          values of b, p, e   , e    and ROUNDS. For example, for machines
+                           min   max
+          with IEEE arithmetic, in double precision:
+
+              b    =  2
+
+              p    =  53
+
+              e    =-1021
+               min
+
+              e    =1024 and ROUNDS is true.
+               max
+
+          For other machines, values of the model parameters must be chosen
+          which define a large subset of the representable numbers;
+          typically it may be necessary to decrease p by 1 (in which case
+          ROUNDS is always set to false), or to increase e    or decrease
+                                                          min
+          e    by a little bit. There are additional rules to ensure that
+           max
+          arithmetic operations on those representable numbers which are
+          not model numbers, are consistent with arithmetic on model
+          numbers.
+
+          (Note: the model used here differs from that described in Brown
+          [1] in the following respects: square-root is treated, like
+          division, as a weakly supported operator; and the logical
+          parameter ROUNDS has been introduced to take account of machines
+          with good rounding.)
+
+          2.1.2.  Derived parameters of floating-point arithmetic
+
+          Most numerical algorithms require access, not to the basic
+          parameters of the model, but to certain derived values, of which
+          the most important are:
+
+                                   (1)  1-p
+          the machine precision =  (-)*b    if ROUNDS is true,
+          (epsilon):               (2)
+
+                                    1-p
+                                =  b    otherwise (but see Note below).
+
+                                    e   -1
+                                     min
+          the smallest positive =  b
+          model number:
+
+                                            e
+                                       -p    max
+          the largest positive  =  (1-b  )*b
+          model number:
+
+          Note: this value is increased very slightly in some
+          implementations to ensure that the computed result of 1+(epsilon)
+          or 1-(epsilon) differs from 1. For example in IEEE binary single
+                                                    -24  -47
+          precision arithmetic the value is set to 2   +2   .
+
+          Two additional derived values are used in the NAG Foundation
+          Library. Their definitions depend not only on the properties of
+          the basic arithmetic operations just considered, but also on
+          properties of some of the elementary functions. We define the
+          safe range parameter to be the smallest positive model number z
+          such that for any x in the range [z,1/z] the following can be
+          computed without undue loss of accuracy, overflow, underflow or
+          other error:
+
+              -x
+
+              1/x
+
+              -1/x
+
+              SQRT(x)
+
+              LOG(x)
+
+              EXP(LOG(x))
+
+              y**(LOG(x)/LOG(y)) for any y
+
+          In a similar fashion we define the safe range parameter for
+          complex arithmetic as the smallest positive model number z such
+          that for any x in the range [z,1/z] the following can be computed
+          without any undue loss of accuracy, overflow, underflow or other
+          error:
+
+              -w
+
+              1/w
+
+              -1/w
+
+              SQRT(w)
+
+              LOG(w)
+
+              EXP(LOG(w))
+
+              y**(LOG(w)/LOG(y)) for any y
+
+              ABS(w)
+
+          where w is any of x, ix, x+ix, 1/x, i/x, 1/x+i/x, and i is the
+          square root of -1.
+
+          This parameter was introduced to take account of the quality of
+          complex arithmetic on the machine. On machines with well
+          implemented complex arithmetic, its value will differ from that
+          of the real safe range parameter by a small multiplying factor
+          less than 10. For poorly implemented complex arithmetic this
+          factor may be larger by many orders of magnitude.
+
+          2.2. Other Aspects of the Computing Environment
+
+          No attempt has been made to characterise comprehensively any
+          other aspects of the computing environment. The other functions
+          in this chapter provide specific information that is occasionally
+          required by routines in the Library.
+
+          2.3. References
+
+          [1]   Brown W S (1981) A Simple but Realistic Model of Floating-
+                point Computation. ACM Trans. Math. Softw. 7 445--480.
+
+          3. Recommendations on Choice and Use of Routines
+
+          3.1. Parameters of Floating-point Arithmetic
+
+          DOUBLE PRECISION FUNCTION returns the machine precision i.e.
+          X02AJF()                  (1)  1-p                       1-p
+                                    (-)*b    if ROUNDS is true or b
+                                    (2)
+                                    otherwise (or a value very slightly
+                                    larger than this, see Section 2.1.2)
+
+          DOUBLE PRECISION FUNCTION returns the smallest positive model
+          X02AKF()                               e   -1
+                                                  min
+                                    number i.e. b
+
+          DOUBLE PRECISION FUNCTION returns the largest positive model
+          X02ALF()                                       e
+                                                    -p    max
+                                    number i.e. (1-b  )*b
+
+
+          DOUBLE PRECISION FUNCTION returns the safe range parameter as
+          X02AMF()                  defined in Section 2.1.2
+
+          DOUBLE PRECISION FUNCTION returns the safe range parameter for
+          X02ANF()                  complex arithmetic as defined in
+                                    Section 2.1.2
+
+          INTEGER FUNCTION X02BHF() returns the model parameter b
+
+          INTEGER FUNCTION X02BJF() returns the model parameter p
+
+          INTEGER FUNCTION X02BKF() returns the model parameter e
+                                                                 min
+
+          INTEGER FUNCTION X02BLF() returns the model parameter e
+                                                                 max
+
+          LOGICAL FUNCTION X02DJF() returns the model parameter ROUNDS
+
+          3.2. Parameters of Other Aspects of the Computing Environment
+
+          DOUBLE PRECISION FUNCTION returns the largest positive real
+          X02AHF(X)                 argument for which the SIN and COS
+          DOUBLE PRECISION X        routines return a result with some
+                                    meaningful accuracy
+
+          INTEGER FUNCTION X02BBF   returns the largest positive integer
+          (X)                       value
+          DOUBLE PRECISION X
+
+          INTEGER FUNCTION X02BEF   returns the maximum number of decimal
+          (X)                       digits which can be accurately
+          DOUBLE PRECISION X        represented over the whole range of
+                                    floating-point numbers
+
+          The argument X of these routines is a dummy argument.
+
+          4. Example Program Text
+
+          The example program simply prints the values of all the functions
+          in Chapter X02. Obviously the results will vary from one
+          implementation of the Library to another.
+
+
+          X02 -- Machine Constants                          Contents -- X02
+          Chapter X02
+
+          Machine Constants
+
+          X02AHF  Largest permissible argument for SIN and COS
+
+          X02AJF  Machine precision
+
+          X02AKF  Smallest positive model number
+
+          X02ALF  Largest positive model number
+
+          X02AMF  Safe range of floating-point arithmetic
+
+          X02ANF  Safe range of complex floating-point arithmetic
+
+          X02BBF  Largest representable integer
+
+          X02BEF  Maximum number of decimal digits that can be represented
+
+          X02BHF  Parameter of floating-point arithmetic model, b
+
+          X02BJF  Parameter of floating-point arithmetic model, p
+
+          X02BKF  Parameter of floating-point arithmetic model, e
+                                                                 min
+
+          X02BLF  Parameter of floating-point arithmetic model, e
+                                                                 max
+
+          X02DJF  Parameter of floating-point arithmetic model, ROUNDS
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXx04}{NAG On-line Documentation: x04}
+\beginscroll
+\begin{verbatim}
+
+
+
+     X04(3NAG)         Foundation Library (12/10/92)         X04(3NAG)
+
+
+
+          X04 -- Input/Output Utilities                 Introduction -- X04
+                                    Chapter X04
+                              Input/Output Utilities
+
+          1. Scope of the Chapter
+
+          This chapter contains utility routines concerned with input and
+          output to or from an external file.
+
+          2. Background to the Problems
+
+          2.1. Output from NAG Foundation Library Routines
+
+          Output from NAG Foundation Library routines to an external file
+          falls into two categories:
+
+          (a)   Error messages which are always associated with an error
+                exit from a routine, that is, with a non-zero value of
+                IFAIL as specified in Section 6 of the routine document.
+
+          (b)   Advisory messages which include output of final results,
+                output of intermediate results to monitor the course of a
+                computation, and various warning or informative messages.
+
+          Each category of output is written to its own Fortran output unit
+          - the  error message unit  or the  advisory message unit. In
+          practice these may be the same unit number. Default unit numbers
+          are provided for each implementation of the Library (see the
+          Users' Note); they may be changed by users. Output of error
+          messages may be controlled by the setting of IFAIL (see the
+          Essential Introduction). Output of advisory messages may usually
+          be controlled by the setting of some other parameter (e.g.
+          MSGLVL) (or in some routines also by IFAIL). An alternative
+          mechanism for completely suppressing output is to set the
+          relevant unit number < 0.
+
+          For further information about error and advisory messages, see
+          Chapter P01.
+
+          2.2. Matrix Printing Routines
+
+          Routines are provided to allow formatted output of general
+          rectangular or triangular matrices stored in a two-dimensional
+          array (real and complex data types).
+
+          All output is directed to the unit number for output of advisory
+          messages, which may be altered by a call to X04ABF.
+
+          3. Recommendations on Choice and Use of Routines
+
+          Apart from the obvious utility of the matrix printing routines,
+          users of the Library may need to call routines in Chapter X04 for
+          the following purposes:
+
+               if the default unit number for error messages (given in the
+               Users' Note for your implementation) is not satisfactory,
+               it may be changed to a new value NERR by the statement
+
+                     CALL X04AAF(1,NERR)
+
+               Similarly the unit number for advisory messages may be
+               changed to a new value NADV by the statement
+
+                     CALL X04ABF(1,NADV)
+
+          4. Index
+
+          Accessing unit number:
+               of advisory message unit                              X04ABF
+               of error message unit                                 X04AAF
+          Printing matrices:
+               general complex matrix                                X04DAF
+               general real matrix                                   X04CAF
+
+
+
+          X04 -- Input/Output Utilities                     Contents -- X04
+          Chapter X04
+
+          Input/Output Utilities
+
+          X04AAF  Return or set unit number for error messages
+
+          X04ABF  Return or set unit number for advisory messages
+
+          X04CAF  Print a real general matrix
+
+          X04DAF  Print a complex general matrix
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXx04aaf}{NAG On-line Documentation: x04aaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     X04AAF(3NAG)      Foundation Library (12/10/92)      X04AAF(3NAG)
+
+
+
+          X04 -- Input/Output Utilities                              X04AAF
+                  X04AAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          X04AAF returns the value of the current error message unit
+          number, or sets the current error message unit number to a new
+          value.
+
+          2. Specification
+
+                 SUBROUTINE X04AAF (IFLAG, NERR)
+                 INTEGER          IFLAG, NERR
+
+          3. Description
+
+          This routine enables those library routines which output error
+          messages, to determine the number of the output unit to which the
+          error messages are to be sent; in this case X04AAF is called with
+          IFLAG = 0. X04AAF may also be called with IFLAG = 1 to set the
+          unit number to a specified value. Otherwise a default value
+          (stated in the Users' Note for your implementation) is returned.
+
+          Records written to this output unit by other library routines are
+          at most 80 characters long (including a line-printer carriage
+          control character).
+
+          Note that if the unit number is set < 0, no messages will be
+          output.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  IFLAG -- INTEGER                                       Input
+               On entry: the action to be taken (see NERR). Constraint:
+               IFLAG = 0 or 1.
+
+           2:  NERR -- INTEGER                                 Input/Output
+               On entry:
+                    if IFLAG = 0, NERR need not be set;
+
+                    if IFLAG = 1, NERR must specify the new error message
+                    unit number.
+               On exit:
+                    if IFLAG = 0, NERR is set to the current error message
+                    unit number,
+
+                    if IFLAG = 1, NERR is unchanged.
+               Note that Fortran unit numbers must be positive or zero. If
+               NERR is set < 0, output of error messages is totally
+               suppressed.
+
+          6. Error Indicators and Warnings
+
+          None.
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          The time taken by the routine is negligible.
+
+          9. Example
+
+          In this example X04AAF is called by the user's main program to
+          make the error message from the routine DUMMY appear on the same
+          unit as the rest of the output (unit 6). Normally a NAG
+          Foundation Library routine with an IFAIL parameter (see Essential
+          Introduction) would take the place of DUMMY.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXx04abf}{NAG On-line Documentation: x04abf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     X04ABF(3NAG)      Foundation Library (12/10/92)      X04ABF(3NAG)
+
+
+
+          X04 -- Input/Output Utilities                              X04ABF
+                  X04ABF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          X04ABF returns the value of the current advisory message unit
+          number, or sets the current advisory message unit number to a new
+          value.
+
+          2. Specification
+
+                 SUBROUTINE X04ABF (IFLAG, NADV)
+                 INTEGER          IFLAG, NADV
+
+          3. Description
+
+          This routine enables those library routines which output advisory
+          messages, to determine the number of the output unit to which the
+          advisory messages are to be sent; in this case X04ABF is called
+          with IFLAG = 0. X04ABF may also be called with IFLAG = 1 to set
+          the unit number to a specified value. Otherwise a default value
+          (stated in the User's Note for your implementation) is returned.
+
+          Records written to this output unit by other library routines are
+          at most 120 characters long (including a line-printer carriage
+          control character), unless those library routines allow users to
+          specify longer records.
+
+          Note that if the unit number is set < 0, no messages will be
+          output.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  IFLAG -- INTEGER                                       Input
+               On entry: the action to be taken (see NADV). Constraint:
+               IFLAG = 0 or 1.
+
+           2:  NADV -- INTEGER                                 Input/Output
+               On entry:
+                    if IFLAG = 0, NADV need not be set;
+
+                    if IFLAG = 1, NADV must specify the new advisory
+                    message unit number.
+               On exit:
+                    if IFLAG = 0, NADV is set to the current advisory
+                    message unit number;
+
+                    if IFLAG = 1, NADV is unchanged.
+               Note that Fortran unit numbers must be positive or zero. If
+               NADV is set < 0, output of advisory messages is totally
+               suppressed.
+
+          6. Error Indicators and Warnings
+
+          None.
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          The time taken by this routine is negligible.
+
+          9. Example
+
+          In this example X04ABF is called by the user's main program to
+          make the advisory message from the routine DUMMY appear on the
+          same unit as the rest of the output (unit 6). Normally a NAG
+          Foundation Library routine with an IFAIL parameter (see Essential
+          Introduction) would take the place of DUMMY.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXx04caf}{NAG On-line Documentation: x04caf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     X04CAF(3NAG)      Foundation Library (12/10/92)      X04CAF(3NAG)
+
+
+
+          X04 -- Input/Output Utilities                              X04CAF
+                  X04CAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          X04CAF is an easy-to-use routine to print a real matrix stored in
+          a two-dimensional array.
+
+          2. Specification
+
+                 SUBROUTINE X04CAF (MATRIX, DIAG, M, N, A, LDA, TITLE,
+                1                   IFAIL)
+                 INTEGER          M, N, LDA, IFAIL
+                 DOUBLE PRECISION A(LDA,*)
+                 CHARACTER*1      MATRIX, DIAG
+                 CHARACTER*(*)    TITLE
+
+          3. Description
+
+          X04CAF prints a real matrix. It is an easy-to-use driver for
+          X04CBF(*). The routine uses default values for the format in
+          which numbers are printed, for labelling the rows and columns,
+          and for output record length.
+
+          X04CAF will choose a format code such that numbers will be
+          printed with either an F8.4, F11.4 or a 1PE13.4 format. The F8.4
+          code is chosen if the sizes of all the matrix elements to be
+          printed lie between 0.001 and 1.0. The F11.4 code is chosen if
+          the sizes of all the matrix elements to be printed lie between 0.
+          001 and 9999.9999. Otherwise the 1PE13.4 code is chosen.
+
+          The matrix is printed with integer row and column labels, and
+          with a maximum record length of 80.
+
+          The matrix is output to the unit defined by X04ABF.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  MATRIX -- CHARACTER*1                                  Input
+               On entry: indicates the part of the matrix to be printed, as
+               follows:
+
+               MATRIX = 'G' (General), the whole of the rectangular matrix.
+
+               MATRIX = 'L' (Lower), the lower triangle of the matrix, or
+               the lower trapezium if the matrix has more rows than
+               columns.
+
+               MATRIX = 'U' (Upper), the upper triangle of the matrix, or
+               the upper trapezium if the matrix has more columns than
+               rows. Constraint: MATRIX must be one of 'G', 'L' or 'U'.
+
+           2:  DIAG -- CHARACTER*1                                    Input
+               On entry: unless MATRIX = 'G', DIAG must specify whether the
+               diagonal elements of the matrix are to be printed, as
+               follows:
+
+               DIAG = 'B' (Blank), the diagonal elements of the matrix are
+               not referenced and not printed.
+
+               DIAG = 'U' (Unit diagonal), the diagonal elements of the
+               matrix are not referenced, but are assumed all to be unity,
+               and are printed as such.
+
+               DIAG = 'N' (Non-unit diagonal), the diagonal elements of the
+               matrix are referenced and printed.
+
+               If MATRIX = 'G', then DIAG need not be set. Constraint: If
+               MATRIX /= 'G', then DIAG must be one of 'B', 'U' or 'N'.
+
+           3:  M -- INTEGER                                           Input
+
+           4:  N -- INTEGER                                           Input
+               On entry: the number of rows and columns of the matrix,
+               respectively, to be printed.
+
+               If either of M or N is less than 1, X04CAF will exit
+               immediately after printing TITLE; no row or column labels
+               are printed.
+
+           5:  A(LDA,*) -- DOUBLE PRECISION array                     Input
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the matrix to be printed. Only the elements that
+               will be referred to, as specified by parameters MATRIX and
+               DIAG, need be set.
+
+           6:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which X04CAF is called.
+               Constraint: LDA >= M.
+
+           7:  TITLE -- CHARACTER*(*)                                 Input
+               On entry: a title to be printed above the matrix. If TITLE =
+               ' ', no title (and no blank line) will be printed.
+
+               If TITLE contains more than 80 characters, the contents of
+               TITLE will be wrapped onto more than one line, with the
+               break after 80 characters.
+
+               Any trailing blank characters in TITLE are ignored.
+
+           8:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry MATRIX /= 'G', 'L' or 'U'.
+
+          IFAIL= 2
+               On entry MATRIX = 'L' or 'U', but DIAG /= 'N', 'U' or 'B'.
+
+          IFAIL= 3
+               On entry LDA < M.
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          A call to X04CAF is equivalent to a call to X04CBF(*) with the
+          following argument values:
+
+
+                NCOLS = 80
+                INDENT = 0
+                LABROW = 'I'
+                LABCOL = 'I'
+                FORMAT = ' '
+
+
+          9. Example
+
+          This example program calls X04CAF twice, first to print a 3 by 5
+          rectangular matrix, and then to print a 5 by 5 lower triangular
+          matrix.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXx04daf}{NAG On-line Documentation: x04daf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     X04DAF(3NAG)      Foundation Library (12/10/92)      X04DAF(3NAG)
+
+
+
+          X04 -- Input/Output Utilities                              X04DAF
+                  X04DAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          X04DAF is an easy-to-use routine to print a complex matrix stored
+          in a two-dimensional array.
+
+          2. Specification
+
+                 SUBROUTINE X04DAF (MATRIX, DIAG, M, N, A, LDA, TITLE,
+                1                   IFAIL)
+                 INTEGER              M, N, LDA, IFAIL
+                 COMPLEX(KIND(1.0D0)) A(LDA,*)
+                 CHARACTER*1          MATRIX, DIAG
+                 CHARACTER*(*)        TITLE
+
+          3. Description
+
+          X04DAF prints a complex matrix. It is an easy-to-use driver for
+          X04DBF(*). The routine uses default values for the format in
+          which numbers are printed, for labelling the rows and columns,
+          and for output record length.
+
+          X04DAF will choose a format code such that numbers will be
+          printed with either an F8.4, F11.4 or a 1PE13.4 format. The F8.4
+          code is chosen if the sizes of all the matrix elements to be
+          printed lie between 0.001 and 1.0. The F11.4 code is chosen if
+          the sizes of all the matrix elements to be printed lie between 0.
+          001 and 9999.9999. Otherwise the 1PE13.4 code is chosen. The
+          chosen code is used to print each complex element of the matrix
+          with the real part above the imaginary part.
+
+          The matrix is printed with integer row and column labels, and
+          with a maximum record length of 80.
+
+          The matrix is output to the unit defined by X04ABF.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  MATRIX -- CHARACTER*1                                  Input
+               On entry: indicates the part of the matrix to be printed, as
+               follows:
+
+               MATRIX = 'G' (General), the whole of the rectangular matrix.
+
+               MATRIX = 'L' (Lower), the lower triangle of the matrix, or
+               the lower trapezium if the matrix has more rows than
+               columns.
+
+               MATRIX = 'U' (Upper), the upper triangle of the matrix, or
+               the upper trapezium if the matrix has more columns than
+               rows. Constraint: MATRIX must be one of 'G', 'L' or 'U'.
+
+           2:  DIAG -- CHARACTER*1                                    Input
+               On entry: unless MATRIX = 'G', DIAG must specify whether the
+               diagonal elements of the matrix are to be printed, as
+               follows:
+
+               DIAG = 'B' (Blank), the diagonal elements of the matrix are
+               not referenced and not printed.
+
+               DIAG = 'U' (Unit diagonal), the diagonal elements of the
+               matrix are not referenced, but are assumed all to be unity,
+               and are printed as such.
+
+               DIAG = 'N' (Non-unit diagonal), the diagonal elements of the
+               matrix are referenced and printed.
+
+               If MATRIX = 'G', then DIAG need not be set. Constraint: If
+               MATRIX /= 'G', then DIAG must be one of 'B', 'U' or 'N'.
+
+           3:  M -- INTEGER                                           Input
+
+           4:  N -- INTEGER                                           Input
+               On entry: the number of rows and columns of the matrix,
+               respectively, to be printed.
+
+               If either of M or N is less than 1, X04DAF will exit
+               immediately after printing TITLE; no row or column labels
+               are printed.
+
+           5:  A(LDA,*) -- COMPLEX(KIND(1.0D)) array                  Input
+               Note: the second dimension of the array A must be at least
+               max(1,N).
+               On entry: the matrix to be printed. Only the elements that
+               will be referred to, as specified by parameters MATRIX and
+               DIAG, need be set.
+
+           6:  LDA -- INTEGER                                         Input
+               On entry:
+               the first dimension of the array A as declared in the
+               (sub)program from which X04DAF is called.
+               Constraint: LDA >= M.
+
+           7:  TITLE -- CHARACTER*(*)                                 Input
+
+               On entry: a title to be printed above the matrix. If TITLE =
+               ' ', no title (and no blank line) will be printed.
+
+               If TITLE contains more than 80 characters, the contents of
+               TITLE will be wrapped onto more than one line, with the
+               break after 80 characters.
+
+               Any trailing blank characters in TITLE are ignored.
+
+           8:  IFAIL -- INTEGER                                Input/Output
+               On entry: IFAIL must be set to 0, -1 or 1. For users not
+               familiar with this parameter (described in the Essential
+               Introduction) the recommended value is 0.
+
+               On exit: IFAIL = 0 unless the routine detects an error (see
+               Section 6).
+
+          6. Error Indicators and Warnings
+
+          Errors detected by the routine:
+
+          If on entry IFAIL = 0 or -1, explanatory error messages are
+          output on the current error message unit (as defined by X04AAF).
+
+          IFAIL= 1
+               On entry MATRIX /= 'G', 'L' or 'U'.
+
+          IFAIL= 2
+               On entry MATRIX = 'L' or 'U', but DIAG /= 'N', 'U' or 'B'.
+
+          IFAIL= 3
+               On entry LDA < M.
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          A call to X04DAF is equivalent to a call to X04DBF(*) with the
+          following argument values:
+
+
+                NCOLS = 80
+                INDENT = 0
+                LABROW = 'I'
+                LABCOL = 'I'
+                FORMAT = ' '
+                USEFRM = 'A'
+
+
+          9. Example
+
+          This example program calls X04DAF twice, first to print a 4 by 3
+          rectangular matrix, and then to print a 4 by 4 lower triangular
+          matrix.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXx05}{NAG On-line Documentation: x05}
+\beginscroll
+\begin{verbatim}
+
+
+
+     X05(3NAG)         Foundation Library (12/10/92)         X05(3NAG)
+
+
+
+          X05 -- Date and Time Utilities                Introduction -- X05
+                                    Chapter X05
+                              Date and Time Utilities
+
+          1. Scope of the Chapter
+
+          This chapter provides routines to obtain the current real time,
+          and the amount of processor time used.
+
+          2. Background to the Problems
+
+          2.1. Real Time
+
+          Routines are provided to obtain the current time in two different
+          formats, and to compare two such times.
+
+          2.2. Processor Time
+
+          A routine is provided to return the current amount of processor
+          time used. This allows the timing of a particular routine or
+          section of code.
+
+          3. Recommendations on Choice and Use of Routines
+
+          X05AAF     returns the current date/time in integer format.
+
+          X05ABF     converts from integer to character string date/time.
+
+          X05ACF     compares two date/time character strings.
+
+          X05BAF     returns the amount of processor time used.
+
+
+          X05 -- Date and Time Utilities                    Contents -- X05
+          Chapter X05
+
+          Date and Time Utilities
+
+          X05AAF  Return date and time as an array of integers
+
+          X05ABF  Convert array of integers representing date and time to
+                  character string
+
+          X05ACF  Compare two character strings representing date and time
+
+          X05BAF  Return the CPU time
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXx05aaf}{NAG On-line Documentation: x05aaf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     X05AAF(3NAG)      Foundation Library (12/10/92)      X05AAF(3NAG)
+
+
+
+          X05 -- Date and Time Utilities                             X05AAF
+                  X05AAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          X05AAF returns the current date and time.
+
+          2. Specification
+
+                 SUBROUTINE X05AAF (ITIME)
+                 INTEGER          ITIME(7)
+
+          3. Description
+
+          X05AAF returns the current date and time as a set of seven
+          integers.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  ITIME(7) -- INTEGER array                             Output
+               On exit: the current date and time, as follows:
+
+               ITIME(1) contains the current year.
+
+               ITIME(2) contains the current month, in the range 1--12.
+
+               ITIME(3) contains the current day, in the range 1--31.
+
+               ITIME(4) contains the current hour, in the range 0--23.
+
+               ITIME(5) contains the current minute, in the range 0--59.
+
+               ITIME(6) contains the current second, in the range 0--59.
+
+               ITIME(7) contains the current millisecond, in the range 0--
+               999.
+
+          6. Error Indicators and Warnings
+
+          None.
+
+          7. Accuracy
+
+          The accuracy of this routine depends on the accuracy of the host
+          machine. In particular, on some machines it may not be possible
+          to return a value for the current millisecond, for example. In
+          this case, the value returned will be zero.
+
+          8. Further Comments
+
+          None.
+
+          9. Example
+
+          This program prints out the vector ITIME after a call to X05AAF.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXx05abf}{NAG On-line Documentation: x05abf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     X05ABF(3NAG)      Foundation Library (12/10/92)      X05ABF(3NAG)
+
+
+
+          X05 -- Date and Time Utilities                             X05ABF
+                  X05ABF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          X05ABF converts from a seven-integer format time and date, as
+          returned by X05AAF, into a character string, returned via the
+          routine name.
+
+          2. Specification
+
+                 CHARACTER*30 FUNCTION X05ABF (ITIME)
+                 INTEGER          ITIME(7)
+
+          3. Description
+
+          X05ABF returns a character string of length 30 which contains the
+          date and time as supplied in argument ITIME. On exit, the
+          character string has the following format:
+
+
+                `DAY XXTH MTH YEAR HR:MN:SC.MIL',
+
+
+            where  DAY   is one of 'Sun', 'Mon', 'Tue', 'Wed', 'Thu',
+                         'Fri', 'Sat',
+
+                   XX    is an integer denoting the day of the month,
+
+                   TH    is one of 'st', 'nd', 'rd', 'th',
+
+                   MTH   is one of 'Jan', 'Feb', 'Mar', 'Apr', 'May',
+                         'Jun', 'Jul', 'Aug', 'Sep', 'Oct', 'Nov', 'Dec',
+
+                   YEAR  is the year as a four digit integer,
+
+                   HR    is the hour,
+
+                   MN    is the minute,
+
+                   SC    is the second,
+
+                   MIL   is the millisecond.
+
+          If on entry the date in ITIME is invalid, the string returned is
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  ITIME(7) -- INTEGER array                              Input
+               On entry: a date and time in the format returned by X05AAF,
+               as follows:
+                   ITIME   must contain the year as a positive integer.
+                   (1)
+
+                   ITIME   must contain the month, in the range 1-12.
+                   (2)
+
+                   ITIME   must contain the day, in the range 1 to p, where
+                   (3)     p = 28, 29, 30 or 31, depending on the month and
+                           year.
+
+                   ITIME   must contain the hour, in the range 0-23.
+                   (4)
+
+                   ITIME   must contain the minute, in the range 0-59.
+                   (5)
+
+                   ITIME   must contain the second, in the range 0-59.
+                   (6)
+
+                   ITIME   must contain the millisecond, in the range 0-
+                   (7)     999.
+
+          6. Error Indicators and Warnings
+
+          None.
+
+          7. Accuracy
+
+          The day name included as part of the character string returned by
+          this routine is calculated assuming that the date is part of the
+          Gregorian calendar. This calendar has been in operation in Europe
+          since October the 15th 1582, and in Great Britain since September
+          the 14th 1752. Entry to this routine with a date earlier than
+          these will therefore not return a day name that is historically
+          accurate.
+
+          8. Further Comments
+
+          Two dates stored in character string format, as returned by this
+          routine, may be compared by X05ACF.
+
+          9. Example
+
+          This program initialises a time in ITIME, and converts it to
+          character format by a call to X05ABF.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXx05acf}{NAG On-line Documentation: x05acf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     X05ACF(3NAG)      Foundation Library (12/10/92)      X05ACF(3NAG)
+
+
+
+          X05 -- Date and Time Utilities                             X05ACF
+                  X05ACF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          X05ACF compares two date/time character strings, each stored in
+          the format returned by X05ABF.
+
+          2. Specification
+
+                 INTEGER FUNCTION X05ACF (CTIME1, CTIME2)
+                 CHARACTER*(*)    CTIME1, CTIME2
+
+          3. Description
+
+          X05ACF compares two date/time character strings, and returns an
+          integer that specifies which one is the earliest. The result is
+          an integer returned through the routine name, with meaning as
+          follows:
+
+               X05ACF = -1: the first date/time string is earlier than the
+               second.
+
+               X05ACF = 0: the two date/time strings are equivalent.
+
+               X05ACF = 1: the first date/time string is later than the
+               second.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+           1:  CTIME1 -- CHARACTER*(*)                                Input
+
+           2:  CTIME2 -- CHARACTER*(*)                                Input
+               On entry: the date/time strings to be compared. These are
+               expected be in the format returned by X05ABF, although
+               X05ACF will still attempt to interpret the strings if they
+               vary slightly from this format. See Section 8 for further
+               details.
+
+          6. Error Indicators and Warnings
+
+          None.
+
+          7. Accuracy
+
+          Not applicable.
+
+          8. Further Comments
+
+          For flexibility, X05ACF will accept various formats for the two
+          date/time strings CTIME1 and CTIME2.
+
+          The strings do not have to be the same length. It is permissible,
+          for example, to enter with one or both of the strings truncated
+          to a smaller length, in which case missing fields are treated as
+          zero.
+
+          Each character string may be of any length, but everything after
+          character 80 is ignored.
+
+          Each string may or may not include an alphabetic day name, such
+          as 'Wednesday', at its start. These day names are ignored, and no
+          check is made that the day name corresponds correctly to the rest
+          of the date.
+
+          The month name may contain any number of characters provided it
+          uniquely identifies the month, however all characters that are
+          supplied are significant.
+
+          Each field in the character string must be separated by one or
+          more spaces.
+
+          The case of all alphabetic characters is insignificant.
+
+          Any field in a date time string that is indecipherable according
+          to the above rules will be converted to a zero value internally.
+          Thus two strings that are completely indecipherable will compare
+          equal.
+
+          According to these rules, all the following date/time strings are
+          equivalent:
+
+              'Thursday 10th July 1958 12:43:17.320'
+
+              'THU 10th JULY 1958 12:43:17.320'
+
+              '10th Jul 1958 12:43:17.320'
+
+          9. Example
+
+          This program initialises two date/time strings, and compares them
+          by a call to X05ACF.
+
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+
+\end{verbatim}
+\endscroll
+\end{page}
+\begin{page}{manpageXXx05baf}{NAG On-line Documentation: x05baf}
+\beginscroll
+\begin{verbatim}
+
+
+
+     X05BAF(3NAG)      Foundation Library (12/10/92)      X05BAF(3NAG)
+
+
+
+          X05 -- Date and Time Utilities                             X05BAF
+                  X05BAF -- NAG Foundation Library Routine Document
+
+          Note: Before using this routine, please read the Users' Note for
+          your implementation to check implementation-dependent details.
+          The symbol (*) after a NAG routine name denotes a routine that is
+          not included in the Foundation Library.
+
+          1. Purpose
+
+          X05BAF returns the amount of processor time used since an
+          unspecified previous time, via the routine name.
+
+          2. Specification
+
+                 DOUBLE PRECISION FUNCTION X05BAF ()
+
+          3. Description
+
+          X05BAF returns the number of seconds of processor time used since
+          some previous time. The previous time is system dependent, but
+          may be, for example, the time the current job or the current
+          program started running.
+
+          If the system clock of the host machine is inaccessible for any
+          reason, X05BAF returns the value zero.
+
+          4. References
+
+          None.
+
+          5. Parameters
+
+          None.
+
+          6. Error Indicators and Warnings
+
+          None.
+
+          7. Accuracy
+
+          The accuracy of the value returned depends on the accuracy of the
+          system clock on the host machine.
+
+          8. Further Comments
+
+          Since the value returned by X05BAF is the amount of processor
+          time since some unspecified earlier time, no significance should
+          be placed on the value other than as a marker to be compared with
+          some later figure returned by X05BAF. The amount of processor
+          time that has elapsed between two calls of X05BAF can be simply
+          calculated as the earlier value subtracted from the later value.
+
+          9. Example
+
+          This program makes a call to X05BAF, performs some computations,
+          makes another call to X05BAF, and gives the time used by the
+          computations as the difference between the two returned values.
+
+          The example program is not reproduced here. The source code for
+          all example programs is distributed with the NAG Foundation
+          Library software and should be available on-line.
+\end{verbatim}
+\endscroll
+\end{page}
diff --git a/src/hyper/pages/newuser.ht b/src/hyper/pages/newuser.ht
new file mode 100644
index 00000000..0ff95d6c
--- /dev/null
+++ b/src/hyper/pages/newuser.ht
@@ -0,0 +1,287 @@
+% Copyright The Numerical Algorithms Group Limited 1991, 1992, 1993.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% --------------------------------------------------------------------
+\begin{page}{NoMoreHelpPage}{No More Help :-(}
+% --------------------------------------------------------------------
+\beginscroll\vspace{2}
+\centerline{No additional or specific help information is available.}
+\centerline{Click on \ \ExitButton{QuitPage} \  to get back.}
+\endscroll
+\end{page}
+
+% ----------------------------------------------------------------------
+\begin{page}{YouTriedIt}{You Tried It!}
+% ----------------------------------------------------------------------
+\beginscroll
+\upbutton{Click here}{UpPage} to get back.
+\endscroll
+\end{page}
+
+% Getting Started
+
+%% % Now using text from book
+
+%%  % --------------------------------------------------------------------
+%%  \begin{page}{GettingStarted}{Getting Started}
+%%  % --------------------------------------------------------------------
+%%  \beginscroll
+%%  \par
+%%  \HyperName{} is the gateway to \Language{}.
+%%  It's both an on-line tutorial and an on-line reference.  It also enables you
+%%  to use \Language{} simply by using the mouse and filling in templates.
+%%  \HyperName{} is available to you if you are running \Language{} under the
+%%  X Window System.
+%%  \par
+%%  Pages usually have active areas, marked in \downlink{this
+%%  font.}{YouTriedIt}
+%%  As you move the mouse pointer to an active area, the pointer changes from a
+%%  filled dot to an open circle.
+%%  The active areas are usually linked to other pages.
+%%  When you click on an active area, you move to the linked page.
+%%  Try clicking \downlink{here}{YouTriedIt} now.
+%%  \par
+%%  Now we suggest that you learn more about other features of
+%%  \HyperName{} by clicking on an active area in the menu below:
+%%  %
+%%  \beginmenu
+%%  \menumemolink{Headings}{ugHyperHeadingsPage}\tab{15}How to use the headings at the top of the page
+%%  \menulink{Scroll Bars}{ugHyperScrollPage}\tab{15}All about {\it scroll bars} on \HyperName{} pages
+%%  \menulink{Input Areas}{ugHyperInputPage}\tab{15}All about {\it input areas} in \HyperName{}
+%%  \menulink{Buttons}{ugHyperButtonsPage}\tab{15}Learn about {\it radio buttons} and {\it toggles}
+%%  \menulink{Search Strings}{SearchStrings}\tab{15}Learn about {\it search strings} in \HyperName{}
+%%  \menulink{Example Pages}{ugHyperExamplePage}\tab{15}How to view or run {\it examples} on \HyperName{} pages
+%%  \menulink{Settings}{ugHyperResourcesPage}\tab{15}X Window Resources for \HyperName{}
+%%  \endmenu
+%%  \endscroll
+%%  \end{page}
+
+%%  % --------------------------------------------------------------------
+%%  \begin{page}{ExamplesIntro}{\Language{} Examples}
+%%  % --------------------------------------------------------------------
+%%  \pp
+%%  \beginscroll
+%%  Many pages have \Language{} examples.
+%%  Here are two:
+%%  \spadpaste{a:= x**2 + 1 \bound{a}} \newline
+%%  \spadpaste{(a - 2)**2 \free{a}} \newline
+%%  Each example has an active ``button'' along the left margin.
+%%  When you click on this button the output for the
+%%  command is ``pasted-in.''
+%%  Try it!
+%%  Click again on the button and you'll see
+%%  that the pasted-in output disappears. Got the idea?
+%%  \par
+%%  Maybe you would like to run an example?
+%%  To do so, just click on any part of it!
+%%  When you do, the example line is copied into a new interactive \Language{}
+%%  buffer for this \HyperName{} page.
+%%  \par
+%%  Sometimes one example line cannot be run before you run an earlier one.
+%%  Don't worry---this is all automatic!
+%%  For instance, the second example line above refers to \spad{a} which is
+%%  assigned in the first example line.
+%%  What happens if you first click on the second example line?
+%%  \Language{} first issues the first line (to assign \spad{a}), then the
+%%  second (to do the computation using \spad{a}).
+%%  \par
+%%  The new interactive \Language{} buffer will disappear when you leave
+%%  \HyperName{}.
+%%  If you want to get rid of it beforehand,
+%%  use the ``Cancel'' button of the X window manager.
+%%  \endscroll
+%%  %\autobutt{HelpHelp}
+%%  \end{page}
+
+%%  % --------------------------------------------------------------------
+%%  \begin{page}{RadioButtons}{Radio Buttons and Toggles}
+%%  % --------------------------------------------------------------------
+%%  \beginscroll
+%%  \radioboxes{sample}{\htbmfile{pick}}{\htbmfile{unpick}}
+%%  \par
+%%  Radio buttons are a group of round buttons like those on car radios: you can
+%%  select only one.
+%%  Here are three radio buttons:
+%%  \centerline{
+%%  {\em\radiobox[1]{rone}{sample}\space{}The first one}\space{3}
+%%  {\em\radiobox[0]{rtwo}{sample}\space{}The second one}\space{3}
+%%  {\em\radiobox[0]{rthree}{sample}\space{}The third one}
+%%  }
+%%  \newline
+%%  The selected button has an {\it X} in the box.
+%%  The others which are not selected are open, i.e. they have no {\it X}.
+%%  To change the selection, move the cursor with the mouse to an
+%%  alternate radio button and click. Try it now.
+%%  %\vspace{1}\centerline{To see another kind of button click on:}
+%%  %\centerline{\box{\downlink{Next Page}{ToggleButton}}}
+%%  %\endscroll
+%%  %\autobuttons\end{page}
+%%  %\begin{page}{ToggleButton}{Toggles}
+%%  %\beginscroll
+%%  \vspace{1}
+%%  \par
+%%  A toggle is a square button you can either select (it has an {\it X}) or
+%%  not (it has no {\it X}).
+%%  Unlike radio buttons, you can set a group of them any way you like.
+%%  Here are three:
+%%  \centerline{
+%%  {\em\inputbox[1]{one}{\htbmfile{pick}}{\htbmfile{unpick}}\space{}The first one}
+%%  \space{3}
+%%  {\em\inputbox[0]{two}{\htbmfile{pick}}{\htbmfile{unpick}}\space{}The second one}
+%%  \space{3}
+%%  {\em\inputbox[1]{three}{\htbmfile{pick}}{\htbmfile{unpick}}\space{}The third one}
+%%  }
+%%  \newline
+%%  To change the selections, move the cursor with the mouse
+%%  to a toggle and click.
+%%  \endscroll
+%%  \autobuttons\end{page}
+
+%%  % --------------------------------------------------------------------
+%%  \begin{page}{InputAreas}{Input Areas}
+%%  % --------------------------------------------------------------------
+%%  \beginscroll
+%%  \par Input areas are boxes that you can fill in.
+%%  Here is one:
+%%  \centerline{\inputstring{one}{40}{some text}}
+%%  \newline As you can see, the input area has some initial text {\it some text}
+%%  followed by an underscore (the character {\it _}).
+%%  First, make sure that the mouse cursor is
+%%  on this page. Now type some
+%%  characters at the keyboard.
+%%  The characters that you type are now inserted in front of the underscore.
+%%  You may type as many characters as you like.
+%%  The input area will grow to accomodate as many characters as you type.
+%%  Use the {\it Backspace} key to erase
+%%  characters to the left.
+%%  Use the keys {\it Insert}, {\it Delete}, {\it Home} and {\it End}
+%%  to modify what you type.
+%%  Also try right- and left-arrow keys immediately to the right of the
+%%  standard keyboard.
+%%  \vspace{1}\newline\centerline{\box{\downlink{Next Page}{MultipleugHyperInputPage}}}
+%%  \endscroll
+%%  \end{page}
+
+
+%%  % --------------------------------------------------------------------
+%%  \begin{page}{MultipleInputAreas}{Multiple Input Areas}
+%%  % --------------------------------------------------------------------
+%%  \beginscroll
+%%  Here is a large input area like the one on the last page:
+%%  \centerline{\inputstring{one}{40}{one}}
+%%  \newline
+%%  and here are two smaller ones:
+%%  \centerline{\inputstring{two}{15}{two}\space{8}\inputstring{three}{7}{three}}
+%%  Move your mouse cursor to somewhere within this page.
+%%  Note that only the first input area has an underscore cursor.
+%%  This means that when you type characters at your keyboard, they
+%%  will go into this first input area. Try it!
+%%  \par
+%%  To type information into another input area,
+%%  use the {\it Enter} or {\it Tab} key to move from one input area to another.
+%%  To move in the reverse order, use {\it Shift + Tab}.
+%%  \par
+%%  You can also move from one input area to another using your mouse.
+%%  Notice that each input area is active. Click on one of the areas.
+%%  As you can see, the underscore cursor now moves to that window.
+%%  \endscroll
+%%  \end{page}
+
+%%  % Now using text from the book
+%%
+%%  % --------------------------------------------------------------------
+%%  \begin{page}{ScrollBars}{Using Scroll Bars}
+%%  % --------------------------------------------------------------------
+%%  When all of the text does not fit within a window, part of the window
+%%  is like a ``looking glass'' you can
+%%  move up and down over the length of the text.
+%%  The text seen by the looking glass has a {\it scroll bar}
+%%  down its right side.
+%%  The {\it scroll bar} allows you to move this looking glass.
+%%  It also tells you the position of the looking glass
+%%  relative to the whole text.
+%%  \beginscroll
+%%  \par
+%%  The part of this \HyperName{} window beginning with this line has a
+%%  {\it scroll bar} along its right side.
+%%  Move the cursor with the mouse to the scroll bar.
+%%  Now move the cursor to the `down-arrow' at the
+%%  bottom of the scroll bar and click. See that the looking glass moves
+%%  down one line. Do it again and again. Each time you click, the
+%%  looking glass moves down one line.
+%%  \par
+%%  Now move the mouse to the 'up-arrow' at the top of the scroll
+%%  bar and click. The looking glass moves up one line each time you click.
+%%  \par
+%%  Next move the mouse to any position along the middle of the
+%%  scroll bar and click.
+%%  This will attempt to move the top of the looking glass to the point where you
+%%  click.
+%%  However, you cannot make the looking glass to go off the bottom edge.
+%%  For this example page, the looking glass region is approximately
+%%  half of the whole region. So the lowest point you can
+%%  set top of the looking glass is halfway down.
+%%  Get the idea?
+%%  \par
+%%  Want to use the keyboard instead of the mouse?
+%%  Then use the {\it Page Up} and {\it Page Down} keys on your
+%%  keyboard. They move the visible part of the region up and down
+%%  one page each time you type them. Try them!
+%%  \par
+%%  If a page does not have an input area, you can also use the {\it Home}
+%%  and up and down arrow keys to move the visible part of the region.
+%%  The {\it Home} key moves the region to the very top of the page.
+%%  The up and down arrow keys move the region up and down one line,
+%%  respectively.
+%%  (If a page does have an input area, these three keys operate on the
+%%  input area.)
+%%  \endscroll
+%%  \autobuttons\end{page}
+
+
+%%  % --------------------------------------------------------------------
+%%  \begin{page}{StartingButtonHelp}{Know These Buttons}
+%%  % --------------------------------------------------------------------
+%%  \beginscroll
+%%  Most pages have a standard set of buttons at the top of the page.
+%%  \newline
+%%  This is what they mean:
+%%  \par \ExitBitmap   \space{} {\it Exit} from \HyperName{}
+%%  \par \HelpBitmap   \space{} Get {\it Help}
+%%  \par \ReturnBitmap \space{} {\it Jump back} to main page
+%%  \par \UpBitmap     \space{} {\it Go back} one page
+%%  \newline
+%%  \pp
+%%  The {\it Help} button shows you pages that can give you additional
+%%  information.  You can always
+%%  click on {\it Help} while you're
+%%  exploring.
+%%  You can always make forays into new topics.
+%%  \HyperName{} remembers where you came from.
+%%  Don't worry about how to get back. Just click on either the ``up arrow''
+%%  or the ``HOME'' button. Now click on the ``up arrow'' to go back one page.
+%%  \endscroll
+%%  \autobutt{DummyHelp}\end{page}
+
+%%  % --------------------------------------------------------------------
+%%  \begin{page}{StartingMenuHelp}{Menus}
+%%  % --------------------------------------------------------------------
+%%  \pp
+%%  \beginscroll
+%%  A `menu' is a list of topics. Each topic has at least one active area.
+%%  Clicking on the active area marked {\it Menus} is how you got here.
+%%  \horizontalline
+%%  Here is another menu to practice on:
+%%  \newline
+%%  \beginmenu
+%%  %\menulink{Riddle}{RiddlePage}
+%%  %    A classic riddle.
+%%  %\menulink{\HyperName{}}{WhatIsHyperName}
+%%  %    What is \HyperName{}
+%%  \menulink{Buttons}{ButtonHelp}
+%%      Buttons in \HyperName{}.
+%%  \endmenu
+%%  \endscroll
+%%  \autobuttons\end{page}
+
diff --git a/src/hyper/pages/numbers.ht b/src/hyper/pages/numbers.ht
new file mode 100644
index 00000000..ea0a8b55
--- /dev/null
+++ b/src/hyper/pages/numbers.ht
@@ -0,0 +1,321 @@
+% Copyright The Numerical Algorithms Group Limited 1991, 1992, 1993
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Number page
+
+%-------------------------------------------------------------------------
+\begin{page}{NumberPage}{\Language{} Number Types}
+%-------------------------------------------------------------------------
+\beginscroll
+The following types of numbers are among those available in \Language{}.
+\beginmenu
+
+\menulink{Integers}{IntegerPage}\tab{16}
+Arithmetic with arbitrarily large integers.
+
+\menulink{Fractions}{FractionPage} \tab{16}
+Rational numbers and general fractions.
+
+\menulink{Machine Floats}{DoubleFloatXmpPage} \tab{16}
+Fixed precision machine floating-point.
+
+\menulink{Real Numbers}{FloatXmpPage} \tab{16}
+Arbitrary precision decimal arithmetic.
+
+\menulink{Complex Numbers}{ComplexXmpPage} \tab{16}
+Complex numbers in general.
+
+\menulink{Finite Fields}{ugProblemFinitePage} \tab{16}
+Arithmetic in characteristic \spad{p}.
+\endmenu
+\horizontalline\newline
+Additional Topics
+\beginmenu
+
+\menulink{Numeric Functions}{ugProblemNumericPage}
+\menulink{Cardinal Numbers}{CardinalNumberXmpPage}
+\menulink{Machine-sized Integers}{SingleIntegerXmpPage}
+\menulink{Roman Numerals}{RomanNumeralXmpPage}
+\menulink{Continued Fractions}{ContinuedFractionXmpPage}
+\menulink{Partial Fractions}{PartialFractionXmpPage}
+\menulink{Quaternions}{QuaternionXmpPage}
+\menulink{Octonions}{OctonionXmpPage}
+\menulink{Repeating Decimals}{DecimalExpansionXmpPage}
+\menulink{Repeating Binary Expansions}{BinaryExpansionXmpPage}
+\menulink{Repeating Hexadecimal Expansions}{HexadecimalExpansionXmpPage}
+\menulink{Expansions in other Bases}{RadixExpansionXmpPage}
+
+%\menulink{p-adic Numbers}{PAdicPage}
+%\menulink{Algebraic Numbers}{AlgebraicNumberPage}
+\endmenu
+
+\endscroll
+\autobuttons
+\end{page}
+
+% Fraction Page
+
+%-------------------------------------------------------------------------
+\begin{page}{FractionPage}{Fractions}
+%-------------------------------------------------------------------------
+
+\beginscroll
+\Language{} handles fractions in many different contexts
+and will automatically simplify fractions whenever possible.
+Here are some examples:
+\spadpaste{1/4 - 1/5}
+\spadpaste{f := (x**2 + 1)/(x - 1) \bound{f}}
+\spadpaste{g := (x**2 - 3*x + 2)/(x + 2) \bound{g}}
+\spadpaste{f * g \free{f g}}
+\endscroll
+Additional Topics:
+\beginmenu
+
+\menulink{Rational Numbers}{RationalNumberPage} \tab{18}
+Quotients of integers
+
+\menulink{Quotient Fields}{FractionXmpPage}  \tab{18}
+Quotients over an arbitrary integral domain
+
+%\menulink{Localizations}{LocalizationPage}  \tab{18}
+%Fractions in the most general setting
+\endmenu
+\autobuttons
+\end{page}
+
+%-------------------------------------------------------------------------
+\begin{page}{RationalNumberPage}{Rational Numbers}
+%-------------------------------------------------------------------------
+\beginscroll
+Like integers, rational numbers can be arbitrarily large.
+For example:
+\spadpaste{61657 ** 10 / 999983 ** 12}
+Rational numbers will not be converted to decimals unless you explicitly
+ask \Language{} to do so.
+To convert a rational number to a decimal, use the function
+\spadfun{numeric}.
+Here's an example:
+\spadpaste{x := 104348/33215 \bound{x}}
+\spadpaste{numeric x \free{x}}
+You can find the numerator and denominator of rational numbers using
+the functions \spadfun{numer} and \spadfun{denom}, respectively.
+\spadpaste{numer(x) \free{x}}
+\spadpaste{denom(x) \free{x}}
+To factor the numerator and denominator of a fraction, use the following
+command:
+\spadpaste{factor(numer x) / factor(denom x) \free{x}}
+\endscroll
+\autobuttons
+\end{page}
+
+%\begin{page}{LocalizationPage}{Localizations}
+%\beginscroll
+%\endscroll
+%\autobuttons
+
+% Algebraic Number Page
+
+%\begin{page}{AlgebraicNumberPage}{Algebraic Numbers}
+%\beginscroll
+%\endscroll
+%\autobuttons
+
+% p-adic Number Page
+
+%\begin{page}{PAdicPage}{p-adic Numbers}
+%\beginscroll
+%\endscroll
+%\autobuttons
+
+% Integer Page
+
+%-------------------------------------------------------------------------
+\begin{page}{IntegerPage}{Integers}
+%-------------------------------------------------------------------------
+
+\beginscroll
+In \Language{}, integers can be as large as you like.
+Try the following examples:
+\spadpaste{x := factorial(200) \bound{x}}
+\spadpaste{y := 2**90 - 1 \bound{y}}
+Of course, you can now do arithmetic as usual on these (very)
+large integers:
+\spadpaste{x + y \free{x y}}
+\spadpaste{x - y \free{x y}}
+\spadpaste{x * y \free{x y}}
+\Language{} can factor integers, but numbers with small prime factors
+\spadpaste{factor(x) \free{x}}
+will factor more rapidly than numbers with large prime factors.
+\spadpaste{factor(y) \free{y}}
+\horizontalline
+Additional Topics
+\beginmenu
+
+\menulink{General Info}{IntegerXmpPage} \tab{16}
+General information and examples of integers.
+
+\menulink{Factorization}{ugxIntegerPrimesPage} \tab{16}
+Primes and factorization.
+
+\menulink{Functions}{IntegerNumberTheoryFunctionsXmpPage} \tab{16}
+Number theoretic functions.
+
+\menulink{Examples}{IntegerExamplePage} \tab{16}
+Examples from number theory.
+
+\menulink{Problems}{IntegerProblemPage} \tab{16}
+Problems from number theory.
+
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+
+%\begin{page}{IntegerFactorizationPage}{Factorization of Integers}
+%\beginscroll
+%\endscroll
+%\autobuttons
+
+%-------------------------------------------------------------------------
+\begin{page}{IntegerExamplePage}{Examples}
+%-------------------------------------------------------------------------
+\beginscroll
+One can show that if an integer of the form 2**k + 1 is prime, then
+k must be a power of 2.
+\downlink{Proof}{IntegerExampleProofPage}
+\par
+Pierre Fermat conjectured that every integer of the forn 2**(2**n) + 1
+is prime.
+Let's look for a counterexample.
+First define a function:
+\spadpaste{f: NNI -> INT \bound{f1}}
+\spadpaste{f(n) == 2**(2**n) + 1 \bound{f} \free{f1}}
+Now try commands like:
+\spadpaste{factor f(1) \free{f}}
+\spadpaste{factor f(2) \free{f}}
+until you find an integer of this form which is composite.
+You can also try the following command:
+\spadpaste{for n in 1..6 repeat output factor f(n) \free{f}}
+Obviously, Fermat didn't have access to \Language{}!
+\endscroll
+\autobuttons
+\end{page}
+
+%-------------------------------------------------------------------------
+\begin{page}{IntegerExampleProofPage}{Proof}
+%-------------------------------------------------------------------------
+
+\beginscroll
+Proposition.  If 2**k + 1 is prime, then k is a power of 2.
+\newline
+Proof.  Suppose that k = m * n with m > 1 odd.  Then
+%
+\centerline{2**n = -1 (mod (2**n + 1))}
+\centerline{2**(n * m) = (-1)**m = -1 (mod (2**n + 1))}
+\centerline{2**k + 1 = 0 (mod (2**n + 1))}
+%
+Therefore, 2**k + 1 is divisible by 2**n + 1.
+Now 1 < 2**n + 1 and since m > 1, 2**n + 1 < 2**k + 1.
+Hence, 2**k + 1 has a non-trivial factor.
+\newline
+QED
+\endscroll
+\autobuttons
+\end{page}
+
+%-------------------------------------------------------------------------
+\begin{page}{IntegerProblemPage}{Problems}
+%-------------------------------------------------------------------------
+
+\beginscroll
+One can show that if an integer of the form 2**k - 1 is prime, then
+k must be prime.
+\downlink{Proof}{IntegerProblemProofPage}
+\newline
+Problem \#1:  Find the smallest prime p such that \spad{2**p - 1} is not prime.
+\downlink{Answer}{IntegerProblemAnswerPage1}
+\newline
+Problem \#2:  Find the smallest positive integer \spad{n} such that
+\spad{n**2 - n + 41} isn't prime.
+\downlink{Answer}{IntegerProblemAnswerPage2}
+\endscroll
+\autobuttons
+\end{page}
+
+
+%-------------------------------------------------------------------------
+\begin{page}{IntegerProblemProofPage}{Proof}
+%-------------------------------------------------------------------------
+\beginscroll
+Proposition.  If \spad{2**k - 1} is prime, then \spad{k} is prime.
+\newline
+Proof.  Suppose that k = m * n is a non-trivial factorization.
+Then
+%
+\centerline{2**m = 1 (mod (2**m - 1))}
+\centerline{2**(m * n) = 1 (mod (2**m - 1))}
+\newline
+and 2**m - 1 is a non-trivial factor of 2**k - 1.
+\newline
+QED
+\endscroll
+\autobuttons
+\end{page}
+
+%-------------------------------------------------------------------------
+\begin{page}{IntegerProblemAnswerPage1}{Solution to Problem \#1}
+%-------------------------------------------------------------------------
+\beginscroll
+Problem \#1:  Find the smallest prime p such that \spad{2**p - 1}
+is not prime.
+\newline
+First, define a function:
+\spadpaste{f: NNI -> INT \bound{f1}}
+\spadpaste{f(n) == 2**n - 1 \bound{f} \free{f1}}
+You can try factoring f(p) as p ranges through the set of primes.
+For example,
+\spadpaste{factor f(7) \free{f}}
+This gets tedious after a while, so let's use \Language{}'s stream
+facility.  (A stream is essentially an infinite sequence.)
+\newline
+First, we create a stream consisting of the positive integers:
+\spadpaste{ints := [n for n in 1..] \bound{ints}}
+Now, we create a stream consisting of the primes:
+\spadpaste{primes := [x for x in ints | prime? x] \bound{primes} \free{ints}}
+Here's the 25th prime:
+\spadpaste{primes.25 \free{primes}}
+Next, create the stream of numbers of the form 2**p - 1 with p prime:
+\spadpaste{numbers := [f(n) for n in primes] \bound{numbers} \free{primes f}}
+Finally, form
+the stream of factorizations of the elements of \spad{numbers}:
+\spadpaste{factors := [factor n for n in numbers] \bound{factors} \free{numbers}}
+You can see that the fifth number in the stream (2047 = 23*89)
+is the first one that has a non-trivial factorization.
+Since 2**11 = 2048, the solution to the problem is 11.
+\newline
+Here's another way to see that 2047 is the first number in the stream that
+is composite:
+\spadpaste{nums := [x for x in numbers | not prime? x] \bound{nums} \free{numbers}}
+\endscroll
+\autobuttons
+\end{page}
+
+%-------------------------------------------------------------------------
+\begin{page}{IntegerProblemAnswerPage2}{Solution to Problem \#2}
+%-------------------------------------------------------------------------
+\beginscroll
+Problem \#2:  Find the smallest positive integer n such that
+\spad{n**2 - n + 41} is not prime.
+\newline
+When n = 41, n**2 - n + 41 = 41**2, which certainly isn't prime.
+Let's see if any smaller integer works.
+Here are the first 40 values:
+\spadpaste{numbers := [n**2 - n + 41 for n in 0..40] \bound{numbers}}
+Now have \Language{} factor the numbers on this list:
+\spadpaste{[factor n for n in numbers] \free{numbers}}
+You can see that 41 is the smallest positive integer n such that
+n**n - n + 41 is not prime.
+\endscroll
+\autobuttons
+\end{page}
diff --git a/src/hyper/pages/numbers.pht b/src/hyper/pages/numbers.pht
new file mode 100644
index 00000000..52f963f8
--- /dev/null
+++ b/src/hyper/pages/numbers.pht
@@ -0,0 +1,607 @@
+\begin{patch}{IntegerPagePatch1}
+\begin{paste}{IntegerPageFull1}{IntegerPageEmpty1}
+\pastebutton{IntegerPageFull1}{\hidepaste}
+\tab{5}\spadcommand{x := factorial(200)\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (1)
+  788657867364790503552363213932185062295135977687173263_
+   29474253324435944996340334292030428401198462390417721_
+   21389196388302576427902426371050619266249528299311134_
+   62857270763317237396988943922445621451664240254033291_
+   86413122742829485327752424240757390324032125740557956_
+   86602260319041703240623517008587961789222227896237038_
+   97374720000000000000000000000000000000000000000000000_
+   000
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerPageEmpty1}
+\begin{paste}{IntegerPageEmpty1}{IntegerPagePatch1}
+\pastebutton{IntegerPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{x := factorial(200)\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerPagePatch2}
+\begin{paste}{IntegerPageFull2}{IntegerPageEmpty2}
+\pastebutton{IntegerPageFull2}{\hidepaste}
+\tab{5}\spadcommand{y := 2**90 - 1\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (2)  1237940039285380274899124223
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerPageEmpty2}
+\begin{paste}{IntegerPageEmpty2}{IntegerPagePatch2}
+\pastebutton{IntegerPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{y := 2**90 - 1\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerPagePatch3}
+\begin{paste}{IntegerPageFull3}{IntegerPageEmpty3}
+\pastebutton{IntegerPageFull3}{\hidepaste}
+\tab{5}\spadcommand{x + y\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (3)
+  788657867364790503552363213932185062295135977687173263_
+   29474253324435944996340334292030428401198462390417721_
+   21389196388302576427902426371050619266249528299311134_
+   62857270763317237396988943922445621451664240254033291_
+   86413122742829485327752424240757390324032125740557956_
+   86602260319041703240623517008587961789222227896237038_
+   97374720000000000000000000001237940039285380274899124_
+   223
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerPageEmpty3}
+\begin{paste}{IntegerPageEmpty3}{IntegerPagePatch3}
+\pastebutton{IntegerPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{x + y\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerPagePatch4}
+\begin{paste}{IntegerPageFull4}{IntegerPageEmpty4}
+\pastebutton{IntegerPageFull4}{\hidepaste}
+\tab{5}\spadcommand{x - y\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (4)
+  788657867364790503552363213932185062295135977687173263_
+   29474253324435944996340334292030428401198462390417721_
+   21389196388302576427902426371050619266249528299311134_
+   62857270763317237396988943922445621451664240254033291_
+   86413122742829485327752424240757390324032125740557956_
+   86602260319041703240623517008587961789222227896237038_
+   97374719999999999999999999998762059960714619725100875_
+   777
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerPageEmpty4}
+\begin{paste}{IntegerPageEmpty4}{IntegerPagePatch4}
+\pastebutton{IntegerPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{x - y\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerPagePatch5}
+\begin{paste}{IntegerPageFull5}{IntegerPageEmpty5}
+\pastebutton{IntegerPageFull5}{\hidepaste}
+\tab{5}\spadcommand{x * y\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (5)
+  976311151308292982184363119660950225776642966765414042_
+   37079496488133389834070329188092355479782812765687260_
+   17975734913119466356078732929100728088106228471338396_
+   75509315106953260921744797014165125163884859138819053_
+   52475858689630194698878995048210905618067437176553811_
+   33973032509524956986554360537566475497856969235918273_
+   09521182392695050703382396859842560000000000000000000_
+   000000000000000000000000000000
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerPageEmpty5}
+\begin{paste}{IntegerPageEmpty5}{IntegerPagePatch5}
+\pastebutton{IntegerPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{x * y\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerPagePatch6}
+\begin{paste}{IntegerPageFull6}{IntegerPageEmpty6}
+\pastebutton{IntegerPageFull6}{\hidepaste}
+\tab{5}\spadcommand{factor(x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (6)
+      197 97 49 32  19  16  11  10  8  6  6  5  4  4  4  3
+     2   3  5  7  11  13  17  19  23 29 31 37 41 43 47 53
+  *
+       3  3  2  2  2  2  2  2  2
+     59 61 67 71 73 79 83 89 97 101 103 107 109 113 127
+  *
+     131 137 139 149 151 157 163 167 173 179 181 191 193
+  *
+     197 199
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerPageEmpty6}
+\begin{paste}{IntegerPageEmpty6}{IntegerPagePatch6}
+\pastebutton{IntegerPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{factor(x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerPagePatch7}
+\begin{paste}{IntegerPageFull7}{IntegerPageEmpty7}
+\pastebutton{IntegerPageFull7}{\hidepaste}
+\tab{5}\spadcommand{factor(y)\free{y }}
+\indentrel{3}\begin{verbatim}
+         3
+   (7)  3 7 11 19 31 73 151 331 631 23311 18837001
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerPageEmpty7}
+\begin{paste}{IntegerPageEmpty7}{IntegerPagePatch7}
+\pastebutton{IntegerPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{factor(y)\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionPagePatch1}
+\begin{paste}{FractionPageFull1}{FractionPageEmpty1}
+\pastebutton{FractionPageFull1}{\hidepaste}
+\tab{5}\spadcommand{1/4 - 1/5}
+\indentrel{3}\begin{verbatim}
+         1
+   (1)  ÄÄ
+        20
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionPageEmpty1}
+\begin{paste}{FractionPageEmpty1}{FractionPagePatch1}
+\pastebutton{FractionPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{1/4 - 1/5}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionPagePatch2}
+\begin{paste}{FractionPageFull2}{FractionPageEmpty2}
+\pastebutton{FractionPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f := (x**2 + 1)/(x - 1)\bound{f }}
+\indentrel{3}\begin{verbatim}
+         2
+        x  + 1
+   (2)  ÄÄÄÄÄÄ
+         x - 1
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionPageEmpty2}
+\begin{paste}{FractionPageEmpty2}{FractionPagePatch2}
+\pastebutton{FractionPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f := (x**2 + 1)/(x - 1)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionPagePatch3}
+\begin{paste}{FractionPageFull3}{FractionPageEmpty3}
+\pastebutton{FractionPageFull3}{\hidepaste}
+\tab{5}\spadcommand{g := (x**2 - 3*x + 2)/(x + 2)\bound{g }}
+\indentrel{3}\begin{verbatim}
+         2
+        x  - 3x + 2
+   (3)  ÄÄÄÄÄÄÄÄÄÄÄ
+           x + 2
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionPageEmpty3}
+\begin{paste}{FractionPageEmpty3}{FractionPagePatch3}
+\pastebutton{FractionPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{g := (x**2 - 3*x + 2)/(x + 2)\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{FractionPagePatch4}
+\begin{paste}{FractionPageFull4}{FractionPageEmpty4}
+\pastebutton{FractionPageFull4}{\hidepaste}
+\tab{5}\spadcommand{f * g\free{f g }}
+\indentrel{3}\begin{verbatim}
+         3     2
+        x  - 2x  + x - 2
+   (4)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+              x + 2
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{FractionPageEmpty4}
+\begin{paste}{FractionPageEmpty4}{FractionPagePatch4}
+\pastebutton{FractionPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{f * g\free{f g }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Patch1}
+\begin{paste}{IntegerProblemAnswerPage1Full1}{IntegerProblemAnswerPage1Empty1}
+\pastebutton{IntegerProblemAnswerPage1Full1}{\hidepaste}
+\tab{5}\spadcommand{f: NNI -> INT\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Empty1}
+\begin{paste}{IntegerProblemAnswerPage1Empty1}{IntegerProblemAnswerPage1Patch1}
+\pastebutton{IntegerProblemAnswerPage1Empty1}{\showpaste}
+\tab{5}\spadcommand{f: NNI -> INT\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Patch2}
+\begin{paste}{IntegerProblemAnswerPage1Full2}{IntegerProblemAnswerPage1Empty2}
+\pastebutton{IntegerProblemAnswerPage1Full2}{\hidepaste}
+\tab{5}\spadcommand{f(n) == 2**n - 1\bound{f }\free{f1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Empty2}
+\begin{paste}{IntegerProblemAnswerPage1Empty2}{IntegerProblemAnswerPage1Patch2}
+\pastebutton{IntegerProblemAnswerPage1Empty2}{\showpaste}
+\tab{5}\spadcommand{f(n) == 2**n - 1\bound{f }\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Patch3}
+\begin{paste}{IntegerProblemAnswerPage1Full3}{IntegerProblemAnswerPage1Empty3}
+\pastebutton{IntegerProblemAnswerPage1Full3}{\hidepaste}
+\tab{5}\spadcommand{factor f(7)\free{f }}
+\indentrel{3}\begin{verbatim}
+   (3)  127
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Empty3}
+\begin{paste}{IntegerProblemAnswerPage1Empty3}{IntegerProblemAnswerPage1Patch3}
+\pastebutton{IntegerProblemAnswerPage1Empty3}{\showpaste}
+\tab{5}\spadcommand{factor f(7)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Patch4}
+\begin{paste}{IntegerProblemAnswerPage1Full4}{IntegerProblemAnswerPage1Empty4}
+\pastebutton{IntegerProblemAnswerPage1Full4}{\hidepaste}
+\tab{5}\spadcommand{ints := [n for n in 1..]\bound{ints }}
+\indentrel{3}\begin{verbatim}
+   (4)  [1,2,3,4,5,6,7,8,9,10,...]
+                           Type: Stream PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Empty4}
+\begin{paste}{IntegerProblemAnswerPage1Empty4}{IntegerProblemAnswerPage1Patch4}
+\pastebutton{IntegerProblemAnswerPage1Empty4}{\showpaste}
+\tab{5}\spadcommand{ints := [n for n in 1..]\bound{ints }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Patch5}
+\begin{paste}{IntegerProblemAnswerPage1Full5}{IntegerProblemAnswerPage1Empty5}
+\pastebutton{IntegerProblemAnswerPage1Full5}{\hidepaste}
+\tab{5}\spadcommand{primes := [x for x in ints | prime? x]\bound{primes }\free{ints }}
+\indentrel{3}\begin{verbatim}
+   (5)  [2,3,5,7,11,13,17,19,23,29,...]
+                           Type: Stream PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Empty5}
+\begin{paste}{IntegerProblemAnswerPage1Empty5}{IntegerProblemAnswerPage1Patch5}
+\pastebutton{IntegerProblemAnswerPage1Empty5}{\showpaste}
+\tab{5}\spadcommand{primes := [x for x in ints | prime? x]\bound{primes }\free{ints }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Patch6}
+\begin{paste}{IntegerProblemAnswerPage1Full6}{IntegerProblemAnswerPage1Empty6}
+\pastebutton{IntegerProblemAnswerPage1Full6}{\hidepaste}
+\tab{5}\spadcommand{primes.25\free{primes }}
+\indentrel{3}\begin{verbatim}
+   (6)  97
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Empty6}
+\begin{paste}{IntegerProblemAnswerPage1Empty6}{IntegerProblemAnswerPage1Patch6}
+\pastebutton{IntegerProblemAnswerPage1Empty6}{\showpaste}
+\tab{5}\spadcommand{primes.25\free{primes }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Patch7}
+\begin{paste}{IntegerProblemAnswerPage1Full7}{IntegerProblemAnswerPage1Empty7}
+\pastebutton{IntegerProblemAnswerPage1Full7}{\hidepaste}
+\tab{5}\spadcommand{numbers := [f(n) for n in primes]\bound{numbers }\free{primes f }}
+\indentrel{3}\begin{verbatim}
+   (7)
+   [3, 7, 31, 127, 2047, 8191, 131071, 524287, 8388607,
+    536870911, ...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Empty7}
+\begin{paste}{IntegerProblemAnswerPage1Empty7}{IntegerProblemAnswerPage1Patch7}
+\pastebutton{IntegerProblemAnswerPage1Empty7}{\showpaste}
+\tab{5}\spadcommand{numbers := [f(n) for n in primes]\bound{numbers }\free{primes f }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Patch8}
+\begin{paste}{IntegerProblemAnswerPage1Full8}{IntegerProblemAnswerPage1Empty8}
+\pastebutton{IntegerProblemAnswerPage1Full8}{\hidepaste}
+\tab{5}\spadcommand{factors := [factor n for n in numbers]\bound{factors }\free{numbers }}
+\indentrel{3}\begin{verbatim}
+   (8)
+   [3, 7, 31, 127, 23 89, 8191, 131071, 524287,
+    47 178481, 233 1103 2089, ...]
+                          Type: Stream Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Empty8}
+\begin{paste}{IntegerProblemAnswerPage1Empty8}{IntegerProblemAnswerPage1Patch8}
+\pastebutton{IntegerProblemAnswerPage1Empty8}{\showpaste}
+\tab{5}\spadcommand{factors := [factor n for n in numbers]\bound{factors }\free{numbers }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Patch9}
+\begin{paste}{IntegerProblemAnswerPage1Full9}{IntegerProblemAnswerPage1Empty9}
+\pastebutton{IntegerProblemAnswerPage1Full9}{\hidepaste}
+\tab{5}\spadcommand{nums := [x for x in numbers | not prime? x]\bound{nums }\free{numbers }}
+\indentrel{3}\begin{verbatim}
+   (9)
+   [2047, 8388607, 536870911, 137438953471,
+    2199023255551, 8796093022207, 140737488355327,
+    9007199254740991, 576460752303423487,
+    147573952589676412927, ...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage1Empty9}
+\begin{paste}{IntegerProblemAnswerPage1Empty9}{IntegerProblemAnswerPage1Patch9}
+\pastebutton{IntegerProblemAnswerPage1Empty9}{\showpaste}
+\tab{5}\spadcommand{nums := [x for x in numbers | not prime? x]\bound{nums }\free{numbers }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage2Patch1}
+\begin{paste}{IntegerProblemAnswerPage2Full1}{IntegerProblemAnswerPage2Empty1}
+\pastebutton{IntegerProblemAnswerPage2Full1}{\hidepaste}
+\tab{5}\spadcommand{numbers := [n**2 - n + 41 for n in 0..40]\bound{numbers }}
+\indentrel{3}\begin{verbatim}
+   (1)
+   [41, 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151,
+    173, 197, 223, 251, 281, 313, 347, 383, 421, 461,
+    503, 547, 593, 641, 691, 743, 797, 853, 911, 971,
+    1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage2Empty1}
+\begin{paste}{IntegerProblemAnswerPage2Empty1}{IntegerProblemAnswerPage2Patch1}
+\pastebutton{IntegerProblemAnswerPage2Empty1}{\showpaste}
+\tab{5}\spadcommand{numbers := [n**2 - n + 41 for n in 0..40]\bound{numbers }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage2Patch2}
+\begin{paste}{IntegerProblemAnswerPage2Full2}{IntegerProblemAnswerPage2Empty2}
+\pastebutton{IntegerProblemAnswerPage2Full2}{\hidepaste}
+\tab{5}\spadcommand{[factor n for n in numbers]\free{numbers }}
+\indentrel{3}\begin{verbatim}
+   (2)
+   [41, 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151,
+    173, 197, 223, 251, 281, 313, 347, 383, 421, 461,
+    503, 547, 593, 641, 691, 743, 797, 853, 911, 971,
+    1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601]
+                            Type: List Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerProblemAnswerPage2Empty2}
+\begin{paste}{IntegerProblemAnswerPage2Empty2}{IntegerProblemAnswerPage2Patch2}
+\pastebutton{IntegerProblemAnswerPage2Empty2}{\showpaste}
+\tab{5}\spadcommand{[factor n for n in numbers]\free{numbers }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerExamplePagePatch1}
+\begin{paste}{IntegerExamplePageFull1}{IntegerExamplePageEmpty1}
+\pastebutton{IntegerExamplePageFull1}{\hidepaste}
+\tab{5}\spadcommand{f: NNI -> INT\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerExamplePageEmpty1}
+\begin{paste}{IntegerExamplePageEmpty1}{IntegerExamplePagePatch1}
+\pastebutton{IntegerExamplePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{f: NNI -> INT\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerExamplePagePatch2}
+\begin{paste}{IntegerExamplePageFull2}{IntegerExamplePageEmpty2}
+\pastebutton{IntegerExamplePageFull2}{\hidepaste}
+\tab{5}\spadcommand{f(n) == 2**(2**n) + 1\bound{f }\free{f1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerExamplePageEmpty2}
+\begin{paste}{IntegerExamplePageEmpty2}{IntegerExamplePagePatch2}
+\pastebutton{IntegerExamplePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f(n) == 2**(2**n) + 1\bound{f }\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerExamplePagePatch3}
+\begin{paste}{IntegerExamplePageFull3}{IntegerExamplePageEmpty3}
+\pastebutton{IntegerExamplePageFull3}{\hidepaste}
+\tab{5}\spadcommand{factor f(1)\free{f }}
+\indentrel{3}\begin{verbatim}
+   (3)  5
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerExamplePageEmpty3}
+\begin{paste}{IntegerExamplePageEmpty3}{IntegerExamplePagePatch3}
+\pastebutton{IntegerExamplePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{factor f(1)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerExamplePagePatch4}
+\begin{paste}{IntegerExamplePageFull4}{IntegerExamplePageEmpty4}
+\pastebutton{IntegerExamplePageFull4}{\hidepaste}
+\tab{5}\spadcommand{factor f(2)\free{f }}
+\indentrel{3}\begin{verbatim}
+   (4)  17
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerExamplePageEmpty4}
+\begin{paste}{IntegerExamplePageEmpty4}{IntegerExamplePagePatch4}
+\pastebutton{IntegerExamplePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{factor f(2)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{IntegerExamplePagePatch5}
+\begin{paste}{IntegerExamplePageFull5}{IntegerExamplePageEmpty5}
+\pastebutton{IntegerExamplePageFull5}{\hidepaste}
+\tab{5}\spadcommand{for n in 1..6 repeat output factor f(n)\free{f }}
+\indentrel{3}\begin{verbatim}
+   5
+   17
+   257
+   65537
+   641 6700417
+   274177 67280421310721
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{IntegerExamplePageEmpty5}
+\begin{paste}{IntegerExamplePageEmpty5}{IntegerExamplePagePatch5}
+\pastebutton{IntegerExamplePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{for n in 1..6 repeat output factor f(n)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{RationalNumberPagePatch1}
+\begin{paste}{RationalNumberPageFull1}{RationalNumberPageEmpty1}
+\pastebutton{RationalNumberPageFull1}{\hidepaste}
+\tab{5}\spadcommand{61657 ** 10 / 999983 ** 12}
+\indentrel{3}\begin{verbatim}
+   (1)
+     794006207119672937688869745365148806136551203249
+  /
+    9997960190729191813417704955587887712239578440952258_
+     46167460930641229761
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RationalNumberPageEmpty1}
+\begin{paste}{RationalNumberPageEmpty1}{RationalNumberPagePatch1}
+\pastebutton{RationalNumberPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{61657 ** 10 / 999983 ** 12}
+\end{paste}\end{patch}
+
+\begin{patch}{RationalNumberPagePatch2}
+\begin{paste}{RationalNumberPageFull2}{RationalNumberPageEmpty2}
+\pastebutton{RationalNumberPageFull2}{\hidepaste}
+\tab{5}\spadcommand{x := 104348/33215\bound{x }}
+\indentrel{3}\begin{verbatim}
+        104348
+   (2)  ÄÄÄÄÄÄ
+         33215
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RationalNumberPageEmpty2}
+\begin{paste}{RationalNumberPageEmpty2}{RationalNumberPagePatch2}
+\pastebutton{RationalNumberPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{x := 104348/33215\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{RationalNumberPagePatch3}
+\begin{paste}{RationalNumberPageFull3}{RationalNumberPageEmpty3}
+\pastebutton{RationalNumberPageFull3}{\hidepaste}
+\tab{5}\spadcommand{numeric x\free{x }}
+\indentrel{3}\begin{verbatim}
+   (3)  3.1415926539 214210447
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RationalNumberPageEmpty3}
+\begin{paste}{RationalNumberPageEmpty3}{RationalNumberPagePatch3}
+\pastebutton{RationalNumberPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{numeric x\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{RationalNumberPagePatch4}
+\begin{paste}{RationalNumberPageFull4}{RationalNumberPageEmpty4}
+\pastebutton{RationalNumberPageFull4}{\hidepaste}
+\tab{5}\spadcommand{numer(x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (4)  104348
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RationalNumberPageEmpty4}
+\begin{paste}{RationalNumberPageEmpty4}{RationalNumberPagePatch4}
+\pastebutton{RationalNumberPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{numer(x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{RationalNumberPagePatch5}
+\begin{paste}{RationalNumberPageFull5}{RationalNumberPageEmpty5}
+\pastebutton{RationalNumberPageFull5}{\hidepaste}
+\tab{5}\spadcommand{denom(x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (5)  33215
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RationalNumberPageEmpty5}
+\begin{paste}{RationalNumberPageEmpty5}{RationalNumberPagePatch5}
+\pastebutton{RationalNumberPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{denom(x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{RationalNumberPagePatch6}
+\begin{paste}{RationalNumberPageFull6}{RationalNumberPageEmpty6}
+\pastebutton{RationalNumberPageFull6}{\hidepaste}
+\tab{5}\spadcommand{factor(numer x) / factor(denom x)\free{x }}
+\indentrel{3}\begin{verbatim}
+         2
+        2 19 1373
+   (6)  ÄÄÄÄÄÄÄÄÄ
+        5 7 13 73
+                        Type: Fraction Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{RationalNumberPageEmpty6}
+\begin{paste}{RationalNumberPageEmpty6}{RationalNumberPagePatch6}
+\pastebutton{RationalNumberPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{factor(numer x) / factor(denom x)\free{x }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/patch.ht b/src/hyper/pages/patch.ht
new file mode 100644
index 00000000..6e74ff36
--- /dev/null
+++ b/src/hyper/pages/patch.ht
@@ -0,0 +1,66 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+\begin{patch}{patch0}
+Hi There Mom
+\end{patch}
+
+\begin{patch}{SmallPatch}
+Hi
+\end{patch}
+
+\begin{patch}{dooutI}
+\begin{paste}{FileI}{doinI}
+\begin{items}
+\item {\it i)} Here is the first item in the popped up sublist.
+I hope it looks okay
+\item {\it ii)} Here is the second item in the popped up sublist.
+I hope it looks okay
+\item {\it iii)} Here is the third item in the popped up sublist.
+I hope it looks okay
+\end{items}
+\end{paste}
+\end{patch}
+
+\begin{patch}{dooutII}
+\begin{paste}{FileII}{doinII}
+\begin{items}
+\item {\it i)} Here is the first item in the popped up sublist.
+I hope it looks okay
+\item {\it ii)} Here is the second item in the popped up sublist.
+I hope it looks okay
+\item {\it iii)} Here is the third item in the popped up sublist.
+I hope it looks okay
+\end{items}
+\end{paste}
+\end{patch}
+
+\begin{patch}{dooutIII}
+\begin{paste}{FileIII}{doinIII}
+\begin{items}
+\item {\it i)} Here is the first item in the popped up sublist.
+I hope it looks okay
+\item {\it ii)} Here is the second item in the popped up sublist.
+I hope it looks okay
+\item {\it iii)} Here is the third item in the popped up sublist.
+I hope it looks okay
+\end{items}
+\end{paste}
+\end{patch}
+
+
+\begin{patch}{doinI}
+\begin{paste}{FileI}{dooutI} \end{paste}
+\end{patch}
+
+\begin{patch}{doinII}
+\begin{paste}{FileII}{dooutII} \end{paste}
+\end{patch}
+
+\begin{patch}{doinIII}
+\begin{paste}{FileIII}{dooutIII} \end{paste}
+\end{patch}
+
+
+
diff --git a/src/hyper/pages/polys.ht b/src/hyper/pages/polys.ht
new file mode 100644
index 00000000..5b22ed9e
--- /dev/null
+++ b/src/hyper/pages/polys.ht
@@ -0,0 +1,170 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% Polynomial page
+% @(#)poly.ht   1.1 90/07/12 18:32:24
+% added PolynomialNumberFieldFactorizationPage: 90/12/05 J. Grabmeier
+
+\begin{page}{PolynomialPage}{Polynomials}
+\beginscroll
+\beginmenu
+\menulink{Basic Functions}{PolynomialBasicPage} \tab{18}
+Create and manipulate polynomials.
+\menulink{Substitutions}{PolynomialSubstitutionPage} \tab{18}
+Evaluate polynomials.
+\menulink{Factorization}{ugProblemFactorPage} \tab{18}
+Factor in different contexts.
+\menulink{GCDs and Friends}{PolynomialGCDPage} \tab{18}
+Greatest common divisors etc..
+\menulink{Roots}{PolynomialRootPage}\tab{18}
+Work with and solve for roots.
+\menulink{Specific Types}{PolynomialTypesPage}\tab{18}
+More specific information.
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+
+\begin{page}{PolynomialTypesPage}{The Specific Polynomial Types}
+\beginscroll
+\beginmenu
+\menulink{Polynomial}{PolynomialXmpPage} \newline
+The general type.
+\menulink{UnivariatePolynomial}{UnivariatePolynomialXmpPage} \newline
+One variable polynomials.
+\menulink{MultivariatePolynomial}{MultivariatePolynomialXmpPage} \newline
+Multiple variable polynomials, recursive structure.
+\menulink{DistributedMultivariatePolynomial}{DistributedMultivariatePolynomialXmpPage}
+\newline
+Multiple variable polynomials, non-recursive structure.
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+
+\begin{page}{PolynomialBasicPage}{Basic Operations On Polynomials}
+
+\beginscroll
+You create
+polynomials using the usual operations of \spadopFrom{+}{Polynomial},
+\spadopFrom{-}{Polynomial}, \spadopFrom{*}{Polynomial}
+(for multiplication), and \spadopFrom{**}{Polynomial} (for exponentiation).
+Here are two examples: \newline
+\spadpaste{p := a*x**2 + b*x*y + c*y**2 \bound{p}} 
+\spadpaste{q := 13*x**2 + 3*z \bound{q}} 
+These operations can also be used to combine polynomials.
+Try the following:
+\spadpaste{p + q \free{p q}} 
+\spadpaste{p - 3*q \free{p q}} 
+\spadpaste{p**2 + p*q \free{p q}} 
+\spadpaste{r := (p + q)**2 \bound{r} \free{p q}}
+As you can see from the above examples, the variables are ordered
+by defaults \spad{z > y > x > c > b > a},
+that is, \spad{z} is the main variable, then
+\spad{y} and so on in reverse alphabetical order.
+You can redefine this
+ordering (for display purposes only) with the \spadfun{setVariableOrder}
+command.
+For example, the following
+makes \spad{a} the main variable, then \spad{b}, and so on:
+\spadpaste{setVariableOrder [a,b,c,x,y,z] \bound{vord}}
+Now compare the way polynomials are displayed:
+\spadpaste{p \free{p vord}}
+\spadpaste{q \free{q vord}}
+\spadpaste{r \free{r vord}}
+To return to the system's default ordering,
+use \spadfun{resetVariableOrder}.
+\spadpaste{resetVariableOrder() \bound{rvord}}
+\spadpaste{p \free{p rvord}}
+Polynomial coefficients can be pulled out
+using the function \spadfun{coefficient}. \newline
+For example:
+\spadpaste{coefficient(q,x,2) \free{q}} 
+will give you the coefficient of \spad{x**2} in the polynomial \spad{q}.
+\newline
+Try these commands:
+\spadpaste{coefficient(r,x,3) \free{r}} 
+\spadpaste{c := coefficient(r,z,1) \free{r} \bound{c}} 
+\spadpaste{coefficient(c,x,2) \free{c}}
+Coefficients of monomials can be obtained as follows:
+\spadpaste{coefficient(q**2, [x,z], [2,1]) \free{q}}
+This will return the coefficient of x**2 * z in the polynomial q**2.
+Also,
+\spadpaste{coefficient(r, [x,y], [2,2]) \free{r}}
+will return the coefficient of \spad{x**2 * y**2}
+in the polynomial \spad{r(x,y)}.
+\endscroll
+\autobuttons
+\end{page}
+
+\begin{page}{PolynomialSubstitutionPage}{Polynomial Evaluation and Substitution}
+\beginscroll
+The function \spadfun{eval} is used to substitute values into polynomials.
+Here's an example of how to use it:
+\spadpaste{p := x**2 + y**2 \bound{p}}
+\spadpaste{eval(p,x=5) \free{p}}
+\newline
+This example would give you the value of the polynomial \spad{p} at 5.
+You can also substitute into polynomials with
+several variables.  First, specify the polynomial, then give
+a list of bindings of the form \spad{variable = value}. For example:
+\spadpaste{eval(p,[x = a + b,y = c + d]) \free{p}}
+Here \spad{x} was replaced by \spad{a + b},
+and \spad{y} was replaced by \spad{c + d}.
+Here's another example:
+\spadpaste{q := x**3 + 5*x - y**4 \bound{q}}
+\spadpaste{eval(q,[x=y,y=x]) \free{q}}
+Substitution is done ``in parallel.''
+That is, \Language{} takes
+\spad{q(x,y)} and returns \spad{q(y,x)}.
+\newline
+You can also substitute numerical values for some or all of the
+variables:
+\spadpaste{px := eval(p, y = sin(2.0)) \bound{px} \free{p}}
+\spadpaste{eval(px, x = cos(2.0)) \free{px}}
+\endscroll
+\autobuttons
+\end{page}
+
+\begin{page}{PolynomialGCDPage}{Greatest Common Divisors, Resultants, and Discriminants}
+\beginscroll
+You can compute the greatest common divisor of two polynomials using the
+function \spadfun{gcd}.
+Here's an example:
+\spadpaste{p := 3*x**8 + 2*x**7 + 6*x**2 + 7*x + 2 \bound{p}}
+\spadpaste{q := 2*x**13 + 9*x**7 + 2*x**6 + 10*x + 5 \bound{q}}
+\spadpaste{gcd(p,q) \free{p q}}
+You could
+also see that \spad{p} and \spad{q} have a factor in common by using the
+function \spadfun{resultant}:
+\spadpaste{resultant(p,q,x) \free{p q}}
+The resultant of two polynomials vanishes precisely when they have a
+factor in common.
+(In the example above
+we specified the variable with
+which we wanted to compute the resultant because the
+polynomials could have involved variables other than x.)
+\endscroll
+\autobuttons
+\end{page}
+
+\begin{page}{PolynomialRootPage}{Roots of Polynomials}
+\beginscroll
+\beginmenu
+\menulink{Using a Single Root of a Polynomial}{ugxProblemSymRootOnePage}
+\newline
+Working with a single root of a polynomial.
+\menulink{Using All Roots of a Polynomial}{ugxProblemSymRootAllPage}
+\newline
+Working with all the roots of a polynomial.
+\menulink{Solution of a Single Polynomial Equation}{ugxProblemOnePolPage}
+\newline
+Finding the roots of one polynomial.
+\menulink{Solution of Systems of Polynomial Equations}{ugxProblemPolSysPage}
+\newline
+Finding the roots of a system of polynomials.
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
diff --git a/src/hyper/pages/polys.pht b/src/hyper/pages/polys.pht
new file mode 100644
index 00000000..8a94c238
--- /dev/null
+++ b/src/hyper/pages/polys.pht
@@ -0,0 +1,501 @@
+\begin{patch}{PolynomialGCDPagePatch1}
+\begin{paste}{PolynomialGCDPageFull1}{PolynomialGCDPageEmpty1}
+\pastebutton{PolynomialGCDPageFull1}{\hidepaste}
+\tab{5}\spadcommand{p := 3*x**8 + 2*x**7 + 6*x**2 + 7*x + 2\bound{p }}
+\indentrel{3}\begin{verbatim}
+          8     7     2
+   (1)  3x  + 2x  + 6x  + 7x + 2
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialGCDPageEmpty1}
+\begin{paste}{PolynomialGCDPageEmpty1}{PolynomialGCDPagePatch1}
+\pastebutton{PolynomialGCDPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{p := 3*x**8 + 2*x**7 + 6*x**2 + 7*x + 2\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialGCDPagePatch2}
+\begin{paste}{PolynomialGCDPageFull2}{PolynomialGCDPageEmpty2}
+\pastebutton{PolynomialGCDPageFull2}{\hidepaste}
+\tab{5}\spadcommand{q := 2*x**13 + 9*x**7 + 2*x**6 + 10*x + 5\bound{q }}
+\indentrel{3}\begin{verbatim}
+          13     7     6
+   (2)  2x   + 9x  + 2x  + 10x + 5
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialGCDPageEmpty2}
+\begin{paste}{PolynomialGCDPageEmpty2}{PolynomialGCDPagePatch2}
+\pastebutton{PolynomialGCDPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{q := 2*x**13 + 9*x**7 + 2*x**6 + 10*x + 5\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialGCDPagePatch3}
+\begin{paste}{PolynomialGCDPageFull3}{PolynomialGCDPageEmpty3}
+\pastebutton{PolynomialGCDPageFull3}{\hidepaste}
+\tab{5}\spadcommand{gcd(p,q)\free{p q }}
+\indentrel{3}\begin{verbatim}
+         7
+   (3)  x  + 2x + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialGCDPageEmpty3}
+\begin{paste}{PolynomialGCDPageEmpty3}{PolynomialGCDPagePatch3}
+\pastebutton{PolynomialGCDPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{gcd(p,q)\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialGCDPagePatch4}
+\begin{paste}{PolynomialGCDPageFull4}{PolynomialGCDPageEmpty4}
+\pastebutton{PolynomialGCDPageFull4}{\hidepaste}
+\tab{5}\spadcommand{resultant(p,q,x)\free{p q }}
+\indentrel{3}\begin{verbatim}
+   (4)  0
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialGCDPageEmpty4}
+\begin{paste}{PolynomialGCDPageEmpty4}{PolynomialGCDPagePatch4}
+\pastebutton{PolynomialGCDPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{resultant(p,q,x)\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPagePatch1}
+\begin{paste}{PolynomialSubstitutionPageFull1}{PolynomialSubstitutionPageEmpty1}
+\pastebutton{PolynomialSubstitutionPageFull1}{\hidepaste}
+\tab{5}\spadcommand{p := x**2 + y**2\bound{p }}
+\indentrel{3}\begin{verbatim}
+         2    2
+   (1)  y  + x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPageEmpty1}
+\begin{paste}{PolynomialSubstitutionPageEmpty1}{PolynomialSubstitutionPagePatch1}
+\pastebutton{PolynomialSubstitutionPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{p := x**2 + y**2\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPagePatch2}
+\begin{paste}{PolynomialSubstitutionPageFull2}{PolynomialSubstitutionPageEmpty2}
+\pastebutton{PolynomialSubstitutionPageFull2}{\hidepaste}
+\tab{5}\spadcommand{eval(p,x=5)\free{p }}
+\indentrel{3}\begin{verbatim}
+         2
+   (2)  y  + 25
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPageEmpty2}
+\begin{paste}{PolynomialSubstitutionPageEmpty2}{PolynomialSubstitutionPagePatch2}
+\pastebutton{PolynomialSubstitutionPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{eval(p,x=5)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPagePatch3}
+\begin{paste}{PolynomialSubstitutionPageFull3}{PolynomialSubstitutionPageEmpty3}
+\pastebutton{PolynomialSubstitutionPageFull3}{\hidepaste}
+\tab{5}\spadcommand{eval(p,[x = a + b,y = c + d])\free{p }}
+\indentrel{3}\begin{verbatim}
+         2           2    2           2
+   (3)  d  + 2c d + c  + b  + 2a b + a
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPageEmpty3}
+\begin{paste}{PolynomialSubstitutionPageEmpty3}{PolynomialSubstitutionPagePatch3}
+\pastebutton{PolynomialSubstitutionPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{eval(p,[x = a + b,y = c + d])\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPagePatch4}
+\begin{paste}{PolynomialSubstitutionPageFull4}{PolynomialSubstitutionPageEmpty4}
+\pastebutton{PolynomialSubstitutionPageFull4}{\hidepaste}
+\tab{5}\spadcommand{q := x**3 + 5*x - y**4\bound{q }}
+\indentrel{3}\begin{verbatim}
+           4    3
+   (4)  - y  + x  + 5x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPageEmpty4}
+\begin{paste}{PolynomialSubstitutionPageEmpty4}{PolynomialSubstitutionPagePatch4}
+\pastebutton{PolynomialSubstitutionPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{q := x**3 + 5*x - y**4\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPagePatch5}
+\begin{paste}{PolynomialSubstitutionPageFull5}{PolynomialSubstitutionPageEmpty5}
+\pastebutton{PolynomialSubstitutionPageFull5}{\hidepaste}
+\tab{5}\spadcommand{eval(q,[x=y,y=x])\free{q }}
+\indentrel{3}\begin{verbatim}
+         3         4
+   (5)  y  + 5y - x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPageEmpty5}
+\begin{paste}{PolynomialSubstitutionPageEmpty5}{PolynomialSubstitutionPagePatch5}
+\pastebutton{PolynomialSubstitutionPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{eval(q,[x=y,y=x])\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPagePatch6}
+\begin{paste}{PolynomialSubstitutionPageFull6}{PolynomialSubstitutionPageEmpty6}
+\pastebutton{PolynomialSubstitutionPageFull6}{\hidepaste}
+\tab{5}\spadcommand{px := eval(p, y = sin(2.0))\bound{px }\free{p }}
+\indentrel{3}\begin{verbatim}
+         2
+   (6)  x  + 0.8268218104 3180595732
+                                 Type: Polynomial Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPageEmpty6}
+\begin{paste}{PolynomialSubstitutionPageEmpty6}{PolynomialSubstitutionPagePatch6}
+\pastebutton{PolynomialSubstitutionPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{px := eval(p, y = sin(2.0))\bound{px }\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPagePatch7}
+\begin{paste}{PolynomialSubstitutionPageFull7}{PolynomialSubstitutionPageEmpty7}
+\pastebutton{PolynomialSubstitutionPageFull7}{\hidepaste}
+\tab{5}\spadcommand{eval(px, x = cos(2.0))\free{px }}
+\indentrel{3}\begin{verbatim}
+   (7)  1.0
+                                 Type: Polynomial Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialSubstitutionPageEmpty7}
+\begin{paste}{PolynomialSubstitutionPageEmpty7}{PolynomialSubstitutionPagePatch7}
+\pastebutton{PolynomialSubstitutionPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{eval(px, x = cos(2.0))\free{px }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch1}
+\begin{paste}{PolynomialBasicPageFull1}{PolynomialBasicPageEmpty1}
+\pastebutton{PolynomialBasicPageFull1}{\hidepaste}
+\tab{5}\spadcommand{p := a*x**2 + b*x*y + c*y**2\bound{p }}
+\indentrel{3}\begin{verbatim}
+           2              2
+   (1)  c y  + b x y + a x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty1}
+\begin{paste}{PolynomialBasicPageEmpty1}{PolynomialBasicPagePatch1}
+\pastebutton{PolynomialBasicPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{p := a*x**2 + b*x*y + c*y**2\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch2}
+\begin{paste}{PolynomialBasicPageFull2}{PolynomialBasicPageEmpty2}
+\pastebutton{PolynomialBasicPageFull2}{\hidepaste}
+\tab{5}\spadcommand{q := 13*x**2 + 3*z\bound{q }}
+\indentrel{3}\begin{verbatim}
+                2
+   (2)  3z + 13x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty2}
+\begin{paste}{PolynomialBasicPageEmpty2}{PolynomialBasicPagePatch2}
+\pastebutton{PolynomialBasicPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{q := 13*x**2 + 3*z\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch3}
+\begin{paste}{PolynomialBasicPageFull3}{PolynomialBasicPageEmpty3}
+\pastebutton{PolynomialBasicPageFull3}{\hidepaste}
+\tab{5}\spadcommand{p + q\free{p q }}
+\indentrel{3}\begin{verbatim}
+                2                    2
+   (3)  3z + c y  + b x y + (a + 13)x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty3}
+\begin{paste}{PolynomialBasicPageEmpty3}{PolynomialBasicPagePatch3}
+\pastebutton{PolynomialBasicPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{p + q\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch4}
+\begin{paste}{PolynomialBasicPageFull4}{PolynomialBasicPageEmpty4}
+\pastebutton{PolynomialBasicPageFull4}{\hidepaste}
+\tab{5}\spadcommand{p - 3*q\free{p q }}
+\indentrel{3}\begin{verbatim}
+                  2                    2
+   (4)  - 9z + c y  + b x y + (a - 39)x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty4}
+\begin{paste}{PolynomialBasicPageEmpty4}{PolynomialBasicPagePatch4}
+\pastebutton{PolynomialBasicPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{p - 3*q\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch5}
+\begin{paste}{PolynomialBasicPageFull5}{PolynomialBasicPageEmpty5}
+\pastebutton{PolynomialBasicPageFull5}{\hidepaste}
+\tab{5}\spadcommand{p**2 + p*q\free{p q }}
+\indentrel{3}\begin{verbatim}
+   (5)
+          2                2      2 4           3
+     (3c y  + 3b x y + 3a x )z + c y  + 2b c x y
+   + 
+                  2  2 2               3      2        4
+   ((2a + 13)c + b )x y  + (2a + 13)b x y + (a  + 13a)x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty5}
+\begin{paste}{PolynomialBasicPageEmpty5}{PolynomialBasicPagePatch5}
+\pastebutton{PolynomialBasicPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{p**2 + p*q\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch6}
+\begin{paste}{PolynomialBasicPageFull6}{PolynomialBasicPageEmpty6}
+\pastebutton{PolynomialBasicPageFull6}{\hidepaste}
+\tab{5}\spadcommand{r := (p + q)**2\bound{r }\free{p q }}
+\indentrel{3}\begin{verbatim}
+   (6)
+       2        2                      2      2 4
+     9z  + (6c y  + 6b x y + (6a + 78)x )z + c y
+   + 
+             3                  2  2 2               3
+     2b c x y  + ((2a + 26)c + b )x y  + (2a + 26)b x y
+   + 
+       2              4
+     (a  + 26a + 169)x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty6}
+\begin{paste}{PolynomialBasicPageEmpty6}{PolynomialBasicPagePatch6}
+\pastebutton{PolynomialBasicPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{r := (p + q)**2\bound{r }\free{p q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch7}
+\begin{paste}{PolynomialBasicPageFull7}{PolynomialBasicPageEmpty7}
+\pastebutton{PolynomialBasicPageFull7}{\hidepaste}
+\tab{5}\spadcommand{setVariableOrder [a,b,c,x,y,z]\bound{vord }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty7}
+\begin{paste}{PolynomialBasicPageEmpty7}{PolynomialBasicPagePatch7}
+\pastebutton{PolynomialBasicPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{setVariableOrder [a,b,c,x,y,z]\bound{vord }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch8}
+\begin{paste}{PolynomialBasicPageFull8}{PolynomialBasicPageEmpty8}
+\pastebutton{PolynomialBasicPageFull8}{\hidepaste}
+\tab{5}\spadcommand{p\free{p vord }}
+\indentrel{3}\begin{verbatim}
+         2             2
+   (8)  x a + y x b + y c
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty8}
+\begin{paste}{PolynomialBasicPageEmpty8}{PolynomialBasicPagePatch8}
+\pastebutton{PolynomialBasicPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{p\free{p vord }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch9}
+\begin{paste}{PolynomialBasicPageFull9}{PolynomialBasicPageEmpty9}
+\pastebutton{PolynomialBasicPageFull9}{\hidepaste}
+\tab{5}\spadcommand{q\free{q vord }}
+\indentrel{3}\begin{verbatim}
+           2
+   (9)  13x  + 3z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty9}
+\begin{paste}{PolynomialBasicPageEmpty9}{PolynomialBasicPagePatch9}
+\pastebutton{PolynomialBasicPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{q\free{q vord }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch10}
+\begin{paste}{PolynomialBasicPageFull10}{PolynomialBasicPageEmpty10}
+\pastebutton{PolynomialBasicPageFull10}{\hidepaste}
+\tab{5}\spadcommand{r\free{r vord }}
+\indentrel{3}\begin{verbatim}
+   (10)
+      4 2        3      2 2       4       2      2 2 2
+     x a  + (2y x b + 2y x c + 26x  + 6z x )a + y x b
+   + 
+        3           3               4 2
+     (2y x c + 26y x  + 6z y x)b + y c
+   + 
+         2 2       2         4        2     2
+     (26y x  + 6z y )c + 169x  + 78z x  + 9z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty10}
+\begin{paste}{PolynomialBasicPageEmpty10}{PolynomialBasicPagePatch10}
+\pastebutton{PolynomialBasicPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{r\free{r vord }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch11}
+\begin{paste}{PolynomialBasicPageFull11}{PolynomialBasicPageEmpty11}
+\pastebutton{PolynomialBasicPageFull11}{\hidepaste}
+\tab{5}\spadcommand{resetVariableOrder()\bound{rvord }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty11}
+\begin{paste}{PolynomialBasicPageEmpty11}{PolynomialBasicPagePatch11}
+\pastebutton{PolynomialBasicPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{resetVariableOrder()\bound{rvord }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch12}
+\begin{paste}{PolynomialBasicPageFull12}{PolynomialBasicPageEmpty12}
+\pastebutton{PolynomialBasicPageFull12}{\hidepaste}
+\tab{5}\spadcommand{p\free{p rvord }}
+\indentrel{3}\begin{verbatim}
+            2              2
+   (12)  c y  + b x y + a x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty12}
+\begin{paste}{PolynomialBasicPageEmpty12}{PolynomialBasicPagePatch12}
+\pastebutton{PolynomialBasicPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{p\free{p rvord }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch13}
+\begin{paste}{PolynomialBasicPageFull13}{PolynomialBasicPageEmpty13}
+\pastebutton{PolynomialBasicPageFull13}{\hidepaste}
+\tab{5}\spadcommand{coefficient(q,x,2)\free{q }}
+\indentrel{3}\begin{verbatim}
+   (13)  13
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty13}
+\begin{paste}{PolynomialBasicPageEmpty13}{PolynomialBasicPagePatch13}
+\pastebutton{PolynomialBasicPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{coefficient(q,x,2)\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch14}
+\begin{paste}{PolynomialBasicPageFull14}{PolynomialBasicPageEmpty14}
+\pastebutton{PolynomialBasicPageFull14}{\hidepaste}
+\tab{5}\spadcommand{coefficient(r,x,3)\free{r }}
+\indentrel{3}\begin{verbatim}
+   (14)  (2a + 26)b y
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty14}
+\begin{paste}{PolynomialBasicPageEmpty14}{PolynomialBasicPagePatch14}
+\pastebutton{PolynomialBasicPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{coefficient(r,x,3)\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch15}
+\begin{paste}{PolynomialBasicPageFull15}{PolynomialBasicPageEmpty15}
+\pastebutton{PolynomialBasicPageFull15}{\hidepaste}
+\tab{5}\spadcommand{c := coefficient(r,z,1)\free{r }\bound{c }}
+\indentrel{3}\begin{verbatim}
+             2                      2
+   (15)  6c y  + 6b x y + (6a + 78)x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty15}
+\begin{paste}{PolynomialBasicPageEmpty15}{PolynomialBasicPagePatch15}
+\pastebutton{PolynomialBasicPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{c := coefficient(r,z,1)\free{r }\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch16}
+\begin{paste}{PolynomialBasicPageFull16}{PolynomialBasicPageEmpty16}
+\pastebutton{PolynomialBasicPageFull16}{\hidepaste}
+\tab{5}\spadcommand{coefficient(c,x,2)\free{c }}
+\indentrel{3}\begin{verbatim}
+   (16)  6a + 78
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty16}
+\begin{paste}{PolynomialBasicPageEmpty16}{PolynomialBasicPagePatch16}
+\pastebutton{PolynomialBasicPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{coefficient(c,x,2)\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch17}
+\begin{paste}{PolynomialBasicPageFull17}{PolynomialBasicPageEmpty17}
+\pastebutton{PolynomialBasicPageFull17}{\hidepaste}
+\tab{5}\spadcommand{coefficient(q**2, [x,z], [2,1])\free{q }}
+\indentrel{3}\begin{verbatim}
+   (17)  78
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty17}
+\begin{paste}{PolynomialBasicPageEmpty17}{PolynomialBasicPagePatch17}
+\pastebutton{PolynomialBasicPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{coefficient(q**2, [x,z], [2,1])\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPagePatch18}
+\begin{paste}{PolynomialBasicPageFull18}{PolynomialBasicPageEmpty18}
+\pastebutton{PolynomialBasicPageFull18}{\hidepaste}
+\tab{5}\spadcommand{coefficient(r, [x,y], [2,2])\free{r }}
+\indentrel{3}\begin{verbatim}
+                       2
+   (18)  (2a + 26)c + b
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{PolynomialBasicPageEmpty18}
+\begin{paste}{PolynomialBasicPageEmpty18}{PolynomialBasicPagePatch18}
+\pastebutton{PolynomialBasicPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{coefficient(r, [x,y], [2,2])\free{r }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/record.ht b/src/hyper/pages/record.ht
new file mode 100644
index 00000000..42900612
--- /dev/null
+++ b/src/hyper/pages/record.ht
@@ -0,0 +1,54 @@
+%---------------------------------------------------------------------------
+%                   Record Constructor Page
+%---------------------------------------------------------------------------
+
+\begin{page}{DomainRecord}{Domain {\em Record(a:A,...,b:B)}}
+\beginscroll
+{\em Record} takes any number of selector-domain pairs as arguments:
+\indentrel{2}
+\newline \spad{a}, a selector, an element of domain \spadtype{Symbol}
+\newline \spad{A}, a domain of category \spadtype{SetCategory}
+\newline\tab{10}...
+\newline \spad{b}, a selector, an element of domain \spadtype{Symbol}
+\newline \spad{B}, a domain of category \spadtype{SetCategory}
+\indentrel{-2}\newline
+This constructor is a primitive in \Language{}. 
+\newline
+\beginmenu
+\item\menulispdownlink{Description}{(|dbSpecialDescription| '|Record|)}\tab{15}General description
+\item\menulispdownlink{Operations}{(|dbSpecialOperations| '|Record|)}\tab{15}All exported operations of {\em Record(a:A,b:B)}
+%\item\menudownlink{Examples}   {RecordExamples}   \tab{15}Examples illustrating use
+%\item\menudownlink{Exports}    {RecordExports}    \tab{15}Explicit categories and operations
+\endmenu
+\vspace{1}\newline
+The selectors {\em a,...,b} of a \spad{Record} type must be distinct.
+\endscroll\end{page}
+ 
+\begin{page}{RecordDescription}{Domain Constructor {\em Record}}
+\beginscroll
+\newline\menuitemstyle{}\tab{2}Record({\em a:A},{\em b:B})
+\newline\tab{2}{\em Arguments:}\indent{17}\tab{-2}
+{\em a}, a selector, an element of domain \spadtype{Symbol}
+\newline\tab{-2}
+{\em A}, a domain of category \spadtype{SetCategory}
+\newline\tab{10}...
+\newline\tab{-2}
+{\em b}, a selector, an element of domain \spadtype{Symbol}
+\newline\tab{-2}
+{\em B}, a domain of category \spadtype{SetCategory}
+\indent{0}\newline\tab{2}{\em Returns:}\indent{15}\tab{0}
+a record object with component objects of type {\em A},...,{\em B} with 
+correponding selectors {\em a},...,{\em b}
+as described below.
+\indent{0}\newline\tab{2}{\em Description:}\indent{15}\tab{0}
+{\em Record(a:A,b:B)} is used to create the class of pairs of objects made
+up of a value of type {\em A} selected by the symbol {\em a} and
+a value of type {\em B} selected by the symbol {\em b}.
+In general, the {\em Record} constructor can take any number of arguments and thus can 
+be used to create aggregates of 
+heterogeneous components of arbitrary size selectable by name.
+{\em Record} is a primitive domain of \Language{} which cannot be
+defined in the \Language{} language. 
+\endscroll
+\end{page}
+
diff --git a/src/hyper/pages/releaseNotes.ht b/src/hyper/pages/releaseNotes.ht
new file mode 100644
index 00000000..3a3884b6
--- /dev/null
+++ b/src/hyper/pages/releaseNotes.ht
@@ -0,0 +1,41 @@
+% =====================================================================
+\begin{page}{releaseNotes}{0. What's New in \Language{}}
+% =====================================================================
+\beginscroll
+\beginmenu
+    \menudownlink{Online information}{onlineInformation}
+    \menudownlink{February 2005}{feb2005}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+
+% =====================================================================
+\begin{page}{onlineInformation}{Online information}
+% =====================================================================
+\beginscroll
+\Language{} information can be found online at
+{http://page.axiom-developer.org/zope/mathaction/FrontPage}
+{http://savannah.nongnu.org/projects/axiom}
+{http://sourceforge.net/projects/axiom}
+{http://www.caissny.org}
+\endscroll
+\autobuttons
+\end{page}
+
+% =====================================================================
+\begin{page}{feb2005}{Feature Complete release}
+% =====================================================================
+\beginscroll
+The February 2005 release is the first complete release of the 
+\Language{} system since it was first made available as open source.
+
+This release includes the full complement of algebra, the graphics
+subsystem, and the hyperdoc system. 
+
+This full release runs on Linux and Solaris 9. 
+The algebra runs on Windows.
+\endscroll
+\autobuttons
+\end{page}
+
diff --git a/src/hyper/pages/rootpage.ht b/src/hyper/pages/rootpage.ht
new file mode 100644
index 00000000..3556aedc
--- /dev/null
+++ b/src/hyper/pages/rootpage.ht
@@ -0,0 +1,419 @@
+% Copyright The Numerical Algorithms Group Limited 1991-1994.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+
+%------------------------------------------------------------------------
+\begin{page}{RootPageLogo}{The Scientific Computation System}
+%------------------------------------------------------------------------
+%\downlink{\inputbitmap{\htbmdir{}/axiom.xbm}}{RootPage1}
+\beginscroll
+\centerline{\inputbitmap{\htbmdir{}/axiom1.bitmap}}
+Copyright 1991, 1992, 1993, 1994, 1995, 1996, 1997 by
+the Numerical Algorithms Group Limited, which is also the proprietor of the
+trademarks AXIOM and the AXIOM logo and of the registered
+trademarks NAG and the NAG logo.
+AXIOM was originally developed by the Research Division of the International
+Business Machines Corporation, Yorktown Heights, New York, USA.
+\centerline{\inputbitmap{\htbmdir{}/naglogo.bitmap}}
+Incorporating ANNA : the AXIOM/NAG Numerical Analyst Expert System
+\centerline{\inputbitmap{\htbmdir{}/anna_logo.xbm}}
+\endscroll
+\end{page}
+
+
+%------------------------------------------------------------------------
+\begin{page}{RootPage}{AXIOM HyperDoc Top Level}
+%------------------------------------------------------------------------
+\beginscroll
+\centerline{\inputbitmap{\htbmdir{}/axiom1.bitmap}}
+%This is the top level of \HyperName{} for \Language{}.
+%The mouse cursor becomes a hand when the cursor is over a
+%\downlink{selectable}{YouTriedIt} item.
+%For an introduction to \HyperName, click on the
+%\windowlink{\fbox{HELP}}{ugHyperPage} button.
+
+\horizontalline
+\newline\tab{0}
+What would you like to do?
+
+\beginmenu
+
+\menuwindowlink{Basic Commands}{BasicCommand}
+\tab{16}Solve problems by filling in templates.
+
+\menuwindowlink{Reference}{TopReferencePage}
+\tab{16}Scan on-line documentation for \Language{}.
+
+\menuwindowlink{Topics}{TopicPage}
+\tab{16}Learn how to use \Language{}, by topic.
+
+\menuwindowlink{Browse}{Man0Page}
+\tab{16}Browse through the \Language{} library.
+
+\menuwindowlink{Examples}{TopExamplePage}
+\tab{16}See examples of use of the library.
+
+\menuwindowlink{Settings}{TopSettingsPage}
+\tab{16}Display and change the system environment.
+
+\menuwindowlink{NAG Link}{htxl}
+\tab{16} Link to NAG Numerical Library.
+
+%\menuwindowlink{\inputbitmap{\htbmdir{}/anna.xbm.tiny}}{UXANNA}
+%\tab{16} The AXIOM/NAG Numerical Analyst Expert System
+
+\menuwindowlink{About AXIOM}{RootPageLogo}
+\tab{16}See some basic information about \Language{}.
+
+%\menuwindowlink{What's New}{ugWhatsNewTwoTwoPage}
+\menuwindowlink{What's New}{releaseNotes}
+\tab{16}Enhancements in this version of \Language{}.
+
+%menulispwindowlink is unsafe:TTT
+%\menulispwindowlink{Settings}{(|htSystemVariables|)}
+%\tab{16}Change an \Language{} system variable.
+%\menulispwindowlink{User Variables}{(|htUserVariables|)}
+%\tab{16}Review user variables and \lispwindowlink{history}{(|htHistory|)}
+
+
+\localinfo
+\endmenu
+\endscroll
+\autobutt{ugHyperPage}
+\end{page}
+
+%------------------------------------------------------------------------
+\begin{page}{TopSettingsPage}{System Commands}
+%------------------------------------------------------------------------
+\beginscroll
+System commands are used to perform \Language{} environment
+management and change \Language{} system variables.
+\beginmenu
+\menumemolink{Commands}{ugSysCmdPage}
+\tab{12}System commands that control your environment.
+
+\menulispmemolink{Settings}{(|htSystemVariables|)}
+\tab{12}Change an \Language{} variable.
+
+\endmenu
+\endscroll
+\end{page}
+
+
+%------------------------------------------------------------------------
+\begin{page}{TopExamplePage}{\Language{} Examples}
+%------------------------------------------------------------------------
+\beginscroll
+What would you like to see?
+\beginmenu
+\item\menudownlink{Graphics}{GraphicsExamplePage}
+\tab{12}Examples of \Language{} Graphics
+\item\menudownlink{Domains}{ExamplesExposedPage}
+\tab{12}Examples of use of \Language{} domains and packages
+\item\menudownlink{Operations}{ExampleCoverPage}
+\tab{12}Examples of \Language{} Operations, by topic
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+
+\beginmenu
+\item\menudownlink{Graphics}{GraphicsExamplePage}
+\tab{12}Examples of \Language{} Graphics
+\item\menudownlink{Domains}{ExamplesExposedPage}
+\tab{12}Examples of use of \Language{} domains and packages
+\item\menudownlink{Operations}{ExampleCoverPage}
+\tab{12}Examples of \Language{} Operations, by topic
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+
+%------------------------------------------------------------------------
+\begin{page}{TopReferencePage}{\Language{} Reference}
+%------------------------------------------------------------------------
+\newline
+\pp
+To select an item, move the cursor with the mouse to
+any word in \downlink{this font}{YouTriedIt} and click the left mouse button.
+\beginscroll
+\beginmenu
+\menumemolink{What's New}{ugWhatsNewTwoTwoPage}
+\tab{12}A synopsis of what's new in this release.
+
+\menumemolink{AXIOM Book}{UsersGuidePage}
+\tab{12}The on-line version of the Jenks/Sutor book.
+
+%\menumemolink{\asharp{} Guide}{AsUsersGuidePage}
+%\tab{12}The on-line \asharp{} Users Guide.
+
+\menumemolink{NAG Library}{FoundationLibraryDocPage}
+\tab{12}The on-line \naglib{} documentation.
+
+\menumemolink{Topics}{TopicPage}
+\tab{12}Learn how to use \Language{}, by topic.
+
+\menumemolink{Language}{ugLangPage}
+\tab{12}Introduction to the \Language{} language.
+
+\menumemolink{Examples}{ExamplesExposedPage}
+\tab{12}Examples for exposed domains and packages
+
+\menumemolink{Commands}{ugSysCmdPage}
+\tab{12}System commands that control your workspace.
+
+%\menumemolink{Operations}{NoPageYet} \tab{12}
+%A guide to useful operations
+
+%\menulispcommand{System Variables}{(|htsv|)}\tab{12}
+%\tab{16}View and change a system-defined variable
+
+\menumemolink{Glossary}{GlossaryPage}\tab{12}
+A glossary of \Language{} terms.
+
+\menumemolink{\HyperName{}}{HTXTopPage}
+\tab{12} How to write your own \HyperName{} pages.
+
+\menumemolink{Search}{RefSearchPage}
+\tab{12} Reference pages for occurrences of a string.
+
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+
+
+\begin{page}{FoundationLibraryDocPage}{\naglib{} Documentation}
+\begin{scroll}
+
+\beginitems
+\item\downlink{\menuitemstyle{Essential Introduction to the NAG Foundation Library}}{manpageXXintro}
+\item Foundation Library Chapter Manual Pages
+\beginitems
+\item\downlink{\menuitemstyle{C02}}{manpageXXc02}\tab{8} Zeros of Polynomials
+\indentrel{8}\newline
+\table{
+{\downlink{c02aff}{manpageXXc02aff}}
+{\downlink{c02agf}{manpageXXc02agf}}
+}\indentrel{-8}
+\item\downlink{\menuitemstyle{C05}}{manpageXXc05}\tab{8} Roots of One or More Transcendental Equations
+\indentrel{8}\newline
+\table{
+{\downlink{c05adf}{manpageXXc05adf}}
+{\downlink{c05nbf}{manpageXXc05nbf}}
+{\downlink{c05pbf}{manpageXXc05pbf}}
+{\downlink{c05zaf}{manpageXXc05zaf}}
+}\indentrel{-8}
+\item\downlink{\menuitemstyle{C06}}{manpageXXc06}\tab{8} Summation of Series
+\indentrel{8}\newline
+\table{
+{\downlink{c06eaf}{manpageXXc06eaf}}
+{\downlink{c06ebf}{manpageXXc06ebf}}
+{\downlink{c06ecf}{manpageXXc06ecf}}
+{\downlink{c06ekf}{manpageXXc06ekf}}
+{\downlink{c06fpf}{manpageXXc06fpf}}
+{\downlink{c06fqf}{manpageXXc06fqf}}
+{\downlink{c06frf}{manpageXXc06frf}}
+{\downlink{c06fuf}{manpageXXc06fuf}}
+{\downlink{c06gbf}{manpageXXc06gbf}}
+{\downlink{c06gcf}{manpageXXc06gcf}}
+{\downlink{c06gqf}{manpageXXc06gqf}}
+{\downlink{c06gsf}{manpageXXc06gsf}}
+}\indentrel{-8}
+\item\downlink{\menuitemstyle{D01}}{manpageXXd01}\tab{8} Quadrature
+\indentrel{8}\newline
+\table{
+{\downlink{d01ajf}{manpageXXd01ajf}}
+{\downlink{d01akf}{manpageXXd01akf}}
+{\downlink{d01alf}{manpageXXd01alf}}
+{\downlink{d01amf}{manpageXXd01amf}}
+{\downlink{d01anf}{manpageXXd01anf}}
+{\downlink{d01apf}{manpageXXd01apf}}
+{\downlink{d01aqf}{manpageXXd01aqf}}
+{\downlink{d01asf}{manpageXXd01asf}}
+{\downlink{d01bbf}{manpageXXd01bbf}}
+{\downlink{d01fcf}{manpageXXd01fcf}}
+{\downlink{d01gaf}{manpageXXd01gaf}}
+{\downlink{d01gbf}{manpageXXd01gbf}}
+}\indentrel{-8}
+\item\downlink{\menuitemstyle{D02}}{manpageXXd02}\tab{8} Ordinary Differential Equations
+\indentrel{8}\newline
+\table{
+{\downlink{d02bbf}{manpageXXd02bbf}}
+{\downlink{d02bhf}{manpageXXd02bhf}}
+{\downlink{d02cjf}{manpageXXd02cjf}}
+{\downlink{d02ejf}{manpageXXd02ejf}}
+{\downlink{d02gaf}{manpageXXd02gaf}}
+{\downlink{d02gbf}{manpageXXd02gbf}}
+{\downlink{d02kef}{manpageXXd02kef}}
+{\downlink{d02raf}{manpageXXd02raf}}
+}\indentrel{-8}
+\item\downlink{\menuitemstyle{D03}}{manpageXXd03}\tab{8} Partial Differential Equations
+\indentrel{8}\newline
+\table{
+{\downlink{d03edf}{manpageXXd03edf}}
+{\downlink{d03eef}{manpageXXd03eef}}
+{\downlink{d03faf}{manpageXXd03faf}}
+}\indentrel{-8}
+\item\downlink{\menuitemstyle{E01}}{manpageXXe01}\tab{8} Interpolation
+\indentrel{8}\newline
+\table{
+{\downlink{e01baf}{manpageXXe01baf}}
+{\downlink{e01bef}{manpageXXe01bef}}
+{\downlink{e01bff}{manpageXXe01bff}}
+{\downlink{e01bgf}{manpageXXe01bgf}}
+{\downlink{e01bhf}{manpageXXe01bhf}}
+{\downlink{e01daf}{manpageXXe01daf}}
+{\downlink{e01saf}{manpageXXe01saf}}
+{\downlink{e01sbf}{manpageXXe01sbf}}
+{\downlink{e01sef}{manpageXXe01sef}}
+{\downlink{e01sff}{manpageXXe01sff}}
+}\indentrel{-8}
+\item\downlink{\menuitemstyle{E02}}{manpageXXe02}\tab{8} Curve and Surface Fitting
+\indentrel{8}\newline
+\table{
+{\downlink{e02adf}{manpageXXe02adf}}
+{\downlink{e02aef}{manpageXXe02aef}}
+{\downlink{e02agf}{manpageXXe02agf}}
+{\downlink{e02ahf}{manpageXXe02ahf}}
+{\downlink{e02ajf}{manpageXXe02ajf}}
+{\downlink{e02akf}{manpageXXe02akf}}
+{\downlink{e02baf}{manpageXXe02baf}}
+{\downlink{e02bbf}{manpageXXe02bbf}}
+{\downlink{e02bcf}{manpageXXe02bcf}}
+{\downlink{e02bdf}{manpageXXe02bdf}}
+{\downlink{e02bef}{manpageXXe02bef}}
+{\downlink{e02daf}{manpageXXe02daf}}
+{\downlink{e02dcf}{manpageXXe02dcf}}
+{\downlink{e02ddf}{manpageXXe02ddf}}
+{\downlink{e02def}{manpageXXe02def}}
+{\downlink{e02dff}{manpageXXe02dff}}
+{\downlink{e02gaf}{manpageXXe02gaf}}
+{\downlink{e02zaf}{manpageXXe02zaf}}
+}\indentrel{-8}
+\item\downlink{\menuitemstyle{E04}}{manpageXXe04}\tab{8} Minimizing or Maximizing a Function
+\indentrel{8}\newline
+\table{
+{\downlink{e04dgf}{manpageXXe04dgf}}
+{\downlink{e04djf}{manpageXXe04djf}}
+{\downlink{e04dkf}{manpageXXe04dkf}}
+{\downlink{e04fdf}{manpageXXe04fdf}}
+{\downlink{e04gcf}{manpageXXe04gcf}}
+{\downlink{e04jaf}{manpageXXe04jaf}}
+{\downlink{e04mbf}{manpageXXe04mbf}}
+{\downlink{e04naf}{manpageXXe04naf}}
+{\downlink{e04ucf}{manpageXXe04ucf}}
+{\downlink{e04udf}{manpageXXe04udf}}
+{\downlink{e04uef}{manpageXXe04uef}}
+{\downlink{e04ycf}{manpageXXe04ycf}}
+}\indentrel{-8}
+\item\downlink{\menuitemstyle{F}}{manpageXXf}\tab{8} Linear Algebra
+\item\downlink{\menuitemstyle{F01}}{manpageXXf01}\tab{8} Matrix Operations, Including Inversion
+\indentrel{8}\newline
+\table{
+{\downlink{f01brf}{manpageXXf01brf}}
+{\downlink{f01bsf}{manpageXXf01bsf}}
+{\downlink{f01maf}{manpageXXf01maf}}
+{\downlink{f01mcf}{manpageXXf01mcf}}
+{\downlink{f01qcf}{manpageXXf01qcf}}
+{\downlink{f01qdf}{manpageXXf01qdf}}
+{\downlink{f01qef}{manpageXXf01qef}}
+{\downlink{f01rcf}{manpageXXf01rcf}}
+{\downlink{f01rdf}{manpageXXf01rdf}}
+{\downlink{f01ref}{manpageXXf01ref}}
+}\indentrel{-8}
+\item\downlink{\menuitemstyle{F02}}{manpageXXf02}\tab{8} Eigenvalues and Eigenvectors
+\indentrel{8}\newline
+\table{
+{\downlink{f02aaf}{manpageXXf02aaf}}
+{\downlink{f02abf}{manpageXXf02abf}}
+{\downlink{f02adf}{manpageXXf02adf}}
+{\downlink{f02aef}{manpageXXf02aef}}
+{\downlink{f02aff}{manpageXXf02aff}}
+{\downlink{f02agf}{manpageXXf02agf}}
+{\downlink{f02ajf}{manpageXXf02ajf}}
+{\downlink{f02akf}{manpageXXf02akf}}
+{\downlink{f02awf}{manpageXXf02awf}}
+{\downlink{f02axf}{manpageXXf02axf}}
+{\downlink{f02bbf}{manpageXXf02bbf}}
+{\downlink{f02bjf}{manpageXXf02bjf}}
+{\downlink{f02fjf}{manpageXXf02fjf}}
+{\downlink{f02wef}{manpageXXf02wef}}
+{\downlink{f02xef}{manpageXXf02xef}}
+}\indentrel{-8}
+\item\downlink{\menuitemstyle{F04}}{manpageXXf04}\tab{8} Simultaneous Linear Equations
+\indentrel{8}\newline
+\table{
+{\downlink{f04adf}{manpageXXf04adf}}
+{\downlink{f04arf}{manpageXXf04arf}}
+{\downlink{f04asf}{manpageXXf04asf}}
+{\downlink{f04atf}{manpageXXf04atf}}
+{\downlink{f04axf}{manpageXXf04axf}}
+{\downlink{f04faf}{manpageXXf04faf}}
+{\downlink{f04jgf}{manpageXXf04jgf}}
+{\downlink{f04maf}{manpageXXf04maf}}
+{\downlink{f04mbf}{manpageXXf04mbf}}
+{\downlink{f04mcf}{manpageXXf04mcf}}
+{\downlink{f04qaf}{manpageXXf04qaf}}
+}\indentrel{-8}
+\item\downlink{\menuitemstyle{F07}}{manpageXXf07}\tab{8} Linear Equations (LAPACK)
+\indentrel{8}\newline
+\table{
+{\downlink{f07adf}{manpageXXf07adf}}
+{\downlink{f07aef}{manpageXXf07aef}}
+{\downlink{f07fdf}{manpageXXf07fdf}}
+{\downlink{f07fef}{manpageXXf07fef}}
+}\indentrel{-8}
+\item\downlink{\menuitemstyle{S}}{manpageXXs}\tab{8} Approximations of Special Functions
+\indentrel{8}\newline
+\table{
+{\downlink{s01eaf}{manpageXXs01eaf}}
+{\downlink{s13aaf}{manpageXXs13aaf}}
+{\downlink{s13acf}{manpageXXs13acf}}
+{\downlink{s13adf}{manpageXXs13adf}}
+{\downlink{s14aaf}{manpageXXs14aaf}}
+{\downlink{s14abf}{manpageXXs14abf}}
+{\downlink{s14baf}{manpageXXs14baf}}
+{\downlink{s15adf}{manpageXXs15adf}}
+{\downlink{s15aef}{manpageXXs15aef}}
+{\downlink{s17acf}{manpageXXs17acf}}
+{\downlink{s17adf}{manpageXXs17adf}}
+{\downlink{s17aef}{manpageXXs17aef}}
+{\downlink{s17aff}{manpageXXs17aff}}
+{\downlink{s17agf}{manpageXXs17agf}}
+{\downlink{s17ahf}{manpageXXs17ahf}}
+{\downlink{s17ajf}{manpageXXs17ajf}}
+{\downlink{s17akf}{manpageXXs17akf}}
+{\downlink{s17dcf}{manpageXXs17dcf}}
+{\downlink{s17def}{manpageXXs17def}}
+{\downlink{s17dgf}{manpageXXs17dgf}}
+{\downlink{s17dhf}{manpageXXs17dhf}}
+{\downlink{s17dlf}{manpageXXs17dlf}}
+{\downlink{s18acf}{manpageXXs18acf}}
+{\downlink{s18adf}{manpageXXs18adf}}
+{\downlink{s18aef}{manpageXXs18aef}}
+{\downlink{s18aff}{manpageXXs18aff}}
+{\downlink{s18dcf}{manpageXXs18dcf}}
+{\downlink{s18def}{manpageXXs18def}}
+{\downlink{s19aaf}{manpageXXs19aaf}}
+{\downlink{s19abf}{manpageXXs19abf}}
+{\downlink{s19acf}{manpageXXs19acf}}
+{\downlink{s19adf}{manpageXXs19adf}}
+{\downlink{s20acf}{manpageXXs20acf}}
+{\downlink{s20adf}{manpageXXs20adf}}
+{\downlink{s21baf}{manpageXXs21baf}}
+{\downlink{s21bbf}{manpageXXs21bbf}}
+{\downlink{s21bcf}{manpageXXs21bcf}}
+{\downlink{s21bdf}{manpageXXs21bdf}}
+}\indentrel{-8}
+\enditems
+\item\downlink{\menuitemstyle{Introduction to NAG On-Line Documentation}}{manpageXXonline}
+\item\downlink{\menuitemstyle{Keywords in Context}}{manpageXXkwic}
+\item\downlink{\menuitemstyle{List of all \naglib{} Routines}}{manpageXXsummary}
+\item\downlink{\menuitemstyle{Converting from the Workstation Library}}{manpageXXconvert}
+\enditems
+\end{scroll}
+\end{page}
+
+
diff --git a/src/hyper/pages/srchkey.ht b/src/hyper/pages/srchkey.ht
new file mode 100644
index 00000000..7d62a3ad
--- /dev/null
+++ b/src/hyper/pages/srchkey.ht
@@ -0,0 +1,142 @@
+% Copyright The Numerical Algorithms Group Limited 1991.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+%%  Using text from book now.
+
+%%  \begin{page}{SearchStrings}{All about {\em Search Strings}}
+%%  \newline Here is an \downlink{input area}{ugHyperInputPage}:
+%%  \inputstring{filter}{35}{}\newline
+%%  and here are some names of greek letters we will
+%%  \lispdownlink{search}{(|htGreekSearch| '|\stringvalue{filter}|)}
+%%  for:{\em
+%%  \table{{alpha}{beta}{gamma}{delta}{epsilon}{zeta}{eta}{theta}{iota}{kappa}{lambda}
+%%  {mu}{nu}{pi}}}
+%%  \beginscroll
+%%  A {\em search string} can be used to select entries from a list.
+%%  For example, if you enter a search string into the input area above and
+%%  then click on {\em search} above, \HyperName{} will select those greek letters
+%%  indicated by the search string.
+%%  \par
+%%  Let's try it.
+%%  The simplest search string is a word, e.g. {\em beta}. A word will only
+%%  match an entry having exactly that spelling. Enter the word {\em beta} into the
+%%  input area above then click on {\em search}. As you can see, {\em beta} matches
+%%  only the single entry {\em beta}.
+%%  \par
+%%  Normally matching is insensitive to whether the alphabetic characters of your
+%%  search string are in upper- or lower- case.
+%%  Thus {\em Beta} and {\em BETA} will both match {\em beta}.
+%%  \par
+%%  You will very often want to use the "wildcard" {\em *} in your search string
+%%  so as to match multiple entries in the list. The search key {\em *} will
+%%  match every entry in the list. Try it!
+%%  Enter {\em *} as your search string above then click on {\em search}.
+%%  As you can see, the {\em *} matches every greek letter.
+%%  \par
+%%  You can also use {\em *} anywhere within a search string to match
+%%  an arbitrary substring. Try {\em *eta} for example.
+%%  It will match {\em beta}, {\em zeta}, {\em eta}, {\em theta}.
+%%  In general, the wildcard matches any substring of zero or more arbitrary characters.
+%%  \par
+%%  You use any number of wildcards in a search string as long as they are not
+%%  adjacent. Thus {\em *e*} will match {\em beta}, {\em delta}, {\em epsilon}, and others, that is, any word containing the letter {\em e}.
+%%  Also {\em z*a} will match only {\em zeta}, that is, only words
+%%  beginning with {\em z} and ending with {\em a}.
+%%  Finally, {\em *a*a*} will match {\em alpha}, {\em gamma}, in general, all
+%%  words which have at least two {\em a}'s their name.
+%%  \horizontalline
+%%  For more information, see:\newline
+%%  \downlink{Logical Searches}{LogicalSearches}\tab{17}Using {\em and}, {\em or}, and {\em not} in search strings
+%%  \newline
+%%  %\downlink{Searching Text}{SearchingText}\tab{17}Using search keys to search text
+%%  %\newline
+%%  %\downlink{Some Details}{SearchOptions}\tab{17}Conventions and options you should know
+%%  \endscroll
+%%  \end{page}
+%%
+%%  \begin{page}{LogicalSearches}{Logical Searches}
+%%  \newline Here is an \downlink{input area}{ugHyperInputPage}:
+%%  \inputstring{filter}{35}{}\newline
+%%  and here are some names of greek letters we will
+%%  \lispdownlink{search}{(|htGreekSearch| '|\stringvalue{filter}|)}
+%%  for:{\em
+%%  \table{{alpha}{beta}{gamma}{delta}{epsilon}{zeta}{eta}{theta}{iota}{kappa}{lambda}
+%%  {mu}{nu}{pi}}}
+%%  \beginscroll
+%%  For more complicated searches, you can use
+%%  {\em and}, {\em or}, and {\em not} with basic search strings.
+%%  Just write logical expressions using these three operators just like
+%%  in the \Language{} language.
+%%  For example, {\em alpha or beta} will match the first two greek letters,
+%%  {\em *a* and *p*} will match all greek letters which have both an
+%%  {\em a} and an {\em p}, and
+%%  {\em not alpha} will match all the greek letters but {\em alpha}.
+%%  \par
+%%  Use parentheses for grouping.
+%%  For example, {\em *a* and (not *p*)} will match the greek letters that
+%%  contain the letter {\em a} but no letter {\em p}. Get the idea?
+%%  \par
+%%  There is no limit to how complex your logical expression can be.
+%%  For example {\em a* or b* or c* or d* or e* and (not *a*)} is ok!
+%%  \endscroll
+%%  \end{page}
+%%
+%%  \begin{page}{SearchingText}{SearchingText}
+%%  Here is an \downlink{input area}{ugHyperInputPage}:
+%%  \inputstring{filter}{35}{}\newline
+%%  and here are two lines of text we will
+%%  \lispdownlink{search}{(|htTextSearch| '|\stringvalue{filter}|)}:\indent{2}
+%%  \table{
+%%  {{\em Fruit flies} *like* a {\em banana with cauliflower ears.}}
+%%  {{\em Sneak Sears Silas with Savings Snatch}}
+%%  }\indent{0}
+%%
+%%  \beginscroll
+%%  Up until now, you have only learned how search strings match single
+%%  words.
+%%  Search strings can also be used to search text, e.g. the documentation
+%%  for the library and \HyperName pages.
+%%  Here the conventions are a bit different.
+%%  \par
+%%  If the search string is a word containing no wildcard {\em *},
+%%  it will only match text containing that identical word.
+%%  For example, the search string {\em ears} will match the
+%%  first line (with {\em ears})
+%%  but not the second line (with {\em Sears}).
+%%  \par
+%%  If the search string is a word containing {\em *} at its beginning
+%%  or end, it will match a string with arbitrary characters at the beginning
+%%  or end, respectively.
+%%  For example, the search string {\em *ears} will match both the
+%%  first line with {\em ears} and the second with {\em Sears}.
+%%  \par
+%%  As usual, you can use \downlink{and, or, not}{LogicalSearches} in your
+%%  search string. For example, {\em *ears and sa*} will only match the
+%%  second line with {\em Sears} and {\em Savings}.
+%%  \par
+%%  If your search string needs to contain a {\em *}, the words {\em and},
+%%  {\em or}, or {\em not}, precede the first character of these special strings
+%%  with  an underscore
+%%  ({\em _}) (the escape character).
+%%  Thus {\em _*like_*} and {\em _and} will match the first line.
+%%  The search string {\em _**} (meaning any word beginning with {\em *}) will also
+%%  match the first line.
+%%  \par
+%%  Your search string can also have two or more words.
+%%  For example, {\em califlower ears} will match the first line.
+%%  If the phrase contains an {\em and}, {\em or} or {\em not},
+%%  be sure to place an underscore before the first character.
+%%  For example {\em bananas and califlower} and {\em bananas _and califlower} have
+%%  quite different meanings!
+%%  \par
+%%  Search strings however will not match adjacent words formatted differently.
+%%  For example, {\em a banana} will not match the first line since "banana"
+%%  is in italics and "a" is not.
+%%  \horizontalline
+%%  %See Also:
+%%  %\newline
+%%  %\downlink{Some Details}{SearchOptions}\tab{17}Conventions and options you should know
+%%  \endscroll
+%%  \end{page}
+
diff --git a/src/hyper/pages/topics.ht b/src/hyper/pages/topics.ht
new file mode 100644
index 00000000..1ffcdb26
--- /dev/null
+++ b/src/hyper/pages/topics.ht
@@ -0,0 +1,124 @@
+% Copyright The Numerical Algorithms Group Limited 1992-1994.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+%------------------------------------------------------------------------
+\begin{page}{TopicPage}{\Language{} Topics}
+%------------------------------------------------------------------------
+
+\beginscroll
+Select a topic below: % or
+%\lispmemolink{search}{(|htTutorialSearch| '|\stringvalue{pattern}|))}
+%for string (use {\em *} for wild card):
+%\newline\inputstring{pattern}{58}{}
+\beginmenu
+\menumemolink{Numbers}{NumberPage}\tab{18}
+A look at different types of numbers
+
+\menumemolink{Polynomials}{PolynomialPage}\tab{18}
+Polynomials in \Language{}
+%
+\menumemolink{Functions}{FunctionPage}\tab{18}
+Built-in and user-defined functions
+%
+\menumemolink{Solving Equations}{EquationPage}\tab{18}
+Facilities for solving equations
+%
+\menumemolink{Calculus}{CalculusPage}\tab{18}
+Using \Language{} to do calculus
+%
+\menumemolink{Linear Algebra}{LinAlgPage}\tab{18}
+\Language{}'s linear algebra facilities
+%
+\menumemolink{Graphics}{GraphicsPage}\tab{18}
+\Language{}'s graphics facilities
+%
+\menumemolink{Algebra}{AlgebraPage}\tab{18}
+\Language{}'s abstract algebra facilities
+%
+\endmenu
+\endscroll
+\end{page}
+
+%------------------------------------------------------------------------
+\begin{page}{EquationPage}{Solving Equations}
+%------------------------------------------------------------------------
+\beginscroll
+\Language{} lets you solve equations of various types:
+\beginmenu
+    \menulink{Solution of Systems of Linear Equations}{ugxProblemLinSysPage}
+      \newline Solve systems of linear equations.
+    \menulink{Solution of a Single Polynomial Equation}{ugxProblemOnePolPage}
+      \newline Find roots of polynomials.
+    \menulink{Solution of Systems of Polynomial Equations}{ugxProblemPolSysPage}
+    \newline Solve systems of polynomial equations.
+    \menulink{Solution of Differential Equations}{ugProblemDEQPage}
+    \newline Closed form and series solutions of differential equations.
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+
+%------------------------------------------------------------------------
+\begin{page}{LinAlgPage}{Linear Algebra}
+%------------------------------------------------------------------------
+\beginscroll
+\beginmenu
+
+\menulink{Introduction}{ugIntroTwoDimPage}\newline
+
+Create and manipulate matrices.
+Work with the entries of a matrix.
+Perform matrix arithmetic.
+
+\menulink{Creating Matrices}{ugxMatrixCreatePage} \newline
+
+Create matrices from scratch and from other matrices.
+
+\menulink{Operations on Matrices}{ugxMatrixOpsPage} \newline
+
+Algebraic manipulations with matrices.
+Compute the inverse, determinant and trace of a matrix.
+Find the rank, nullspace and row echelon form of a matrix.
+
+\menulink{Eigenvalues and Eigenvectors}{ugProblemEigenPage} \newline
+
+How to compute eigenvalues and eigenvectors.
+\endmenu
+\horizontalline\newline
+Additional Topics:
+\beginmenu
+\menulink{Example: Determinant of a Hilbert Matrix}{ugxFloatHilbertPage}
+\menulink{Computing the Permanent}{PermanentXmpPage}
+\menulink{Working with Vectors}{VectorXmpPage}
+\menulink{Working with Square Matrices}{SquareMatrixXmpPage}
+\menulink{Working with One-Dimensional Arrays}{OneDimensionalArrayXmpPage}
+\menulink{Working with Two-Dimensional Arrays}{TwoDimensionalArrayXmpPage}
+\menulink{Conversion (Polynomials of Matrices)}{ugTypesConvertPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+
+%------------------------------------------------------------------------
+\begin{page}{CalculusPage}{Calculus}
+%------------------------------------------------------------------------
+\beginscroll
+\beginmenu
+\menulink{Limits}{ugProblemLimitsPage} \tab{17}
+Compute limits of functional expressions.
+\menulink{Derivatives}{ugIntroCalcDerivPage}\tab{17}
+Compute derivatives and partial derivatives.
+\menulink{Integrals}{ugIntroIntegratePage}\tab{17}
+Introduction to \Language{}'s symbolic integrator.
+\menulink{More Integrals}{ugProblemIntegrationPage}\tab{17}
+More information about symbolic integration.
+\menulink{Laplace}{ugProblemLaplacePage}\tab{17}
+Computing Laplace transforms.
+\menulink{Series}{ugProblemSeriesPage}\tab{17}
+Compute series expansions of expressions.
+\menulink{Differential Eqns}{ugProblemDEQPage}\tab{17}
+Solve differential equations.
+\endmenu
+\endscroll
+\autobuttons \end{page}
diff --git a/src/hyper/pages/type.ht b/src/hyper/pages/type.ht
new file mode 100644
index 00000000..63a9bf0b
--- /dev/null
+++ b/src/hyper/pages/type.ht
@@ -0,0 +1,16 @@
+%---------------------------------------------------------------------------
+%                   Category Type Constructor Page
+%---------------------------------------------------------------------------
+ 
+\begin{page}{CategoryType}{Category {\em Type}}
+\beginscroll
+{\em Type} is a primitive category in AXIOM,
+one which is an ancestor of all AXIOM categories.
+
+{\em Type} is the root of AXIOM's category hierarchy,
+a category with no properties (exported operations 
+and attributes) of which all other categories are descendants.
+Two important children of {\em Type} are
+\spadtype{SetCategory}, the category of all algebraic domains,
+and \spadtype{Aggregate}, the category of all data structures.
+\endscroll
diff --git a/src/hyper/pages/ug.ht b/src/hyper/pages/ug.ht
new file mode 100644
index 00000000..3bc873dd
--- /dev/null
+++ b/src/hyper/pages/ug.ht
@@ -0,0 +1,54 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by chapmenu.awk.
+% =====================================================================
+\begin{page}{UsersGuidePage}{Users Guide}
+% =====================================================================
+This is the table of contents for the Users Guide.
+Click on any item below to see that section.
+\beginscroll
+\indent{3}
+\beginmenu
+\menudownlink{{0. What's New in \Language{} Version 2.2}}{ugWhatsNewTwoTwoPage}
+\endmenu
+\indent{0}
+Part I.   Basic Features of \Language{}
+\indent{3}
+\beginmenu
+\menudownlink{{1. An Overview of \Language{}}}{ugIntroPage}
+\menudownlink{{2. Using Types and Modes}}{ugTypesPage}
+\menudownlink{{3. Using \HyperName{}}}{ugHyperPage}
+\menudownlink{{4. Input Files and Output Styles}}{ugInOutPage}
+\menudownlink{{5. Introduction to the \Language{} Interactive Language}}{ugLangPage}
+\menudownlink{{6. User-Defined Functions, Macros and Rules}}{ugUserPage}
+\menudownlink{{7. Graphics}}{ugGraphPage}
+\endmenu
+\indent{0}
+Part II.  Advanced Problem Solving and Examples
+\indent{3}
+\beginmenu
+\menudownlink{{8. Advanced Problem Solving}}{ugProblemPage}
+\menudownlink{{9. Some Examples for Domains and Packages}}{ExamplesExposedPage}
+\endmenu
+\indent{0}
+Part III. Advanced Programming in \Language{}
+\indent{3}
+\beginmenu
+\menudownlink{{10. Interactive Programming}}{ugIntProgPage}
+\menudownlink{{11. Packages}}{ugPackagesPage}
+\menudownlink{{12. Categories}}{ugCategoriesPage}
+\menudownlink{{13. Domains}}{ugDomainsPage}
+\menudownlink{{14. Browse}}{ugBrowsePage}
+\menudownlink{{15. What's New in \Language{} Version 2.0}}{ugWhatsNewPage}
+\endmenu
+\indent{0}
+Appendices.
+\indent{3}
+\beginmenu
+\menudownlink{{A. \Language{} System Commands}}{ugSysCmdPage}
+\menudownlink{{F. Programs for AXIOM Images}}{ugAppGraphicsPage}
+\menudownlink{{G. Glossary}}{GlossaryPage}
+\endmenu
+\indent{0}
+\endscroll
+\autobuttons
+\end{page}
diff --git a/src/hyper/pages/ug00.ht b/src/hyper/pages/ug00.ht
new file mode 100644
index 00000000..b8ec3213
--- /dev/null
+++ b/src/hyper/pages/ug00.ht
@@ -0,0 +1,462 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+
+\texht{\setcounter{chapter}{-1}}{} % 
+
+%
+\newcommand{\ugWhatsNewTwoTwoTitle}{What's New in \Language{} Version 2.2}
+\newcommand{\ugWhatsNewTwoTwoNumber}{0.}
+%
+% =====================================================================
+\begin{page}{ugWhatsNewTwoTwoPage}{0. What's New in \Language{} Version 2.2}
+% =====================================================================
+\beginscroll
+
+\beginmenu
+    \menudownlink{{0.1. \axiomxl{} compiler - Enhancements and Additions}}{ugTwoTwoAldorPage}
+    \menudownlink{{0.2. New polynomial domains and algorithms}}{ugTwoTwoPolynomialsPage}
+    \menudownlink{{0.3. Enhancements to HyperDoc and Graphics}}{ugTwoTwoHyperdocPage}
+    \menudownlink{{0.4. Enhancements to NAGLink}}{ugTwoTwoNAGLinkPage}
+    \menudownlink{{0.5. Enhancements to the Lisp system}}{ugTwoTwoCCLPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTwoTwoAldorTitle}{\axiomxl{} compiler - Enhancements and Additions}
+\newcommand{\ugTwoTwoAldorNumber}{0.1.}
+%
+% =====================================================================
+\begin{page}{ugTwoTwoAldorPage}{0.1. \axiomxl{} compiler - Enhancements and Additions}
+% =====================================================================
+\beginscroll
+
+Numerous bug fixes and improvements have been implemented in version 1.1.12
+of \axiomxl{} which is included in this release. You may also notice the name
+{\it Aldor} being used when referring to \axiomxl{}.
+
+The format of {\bf .ao} files has changed somewhat so you need
+to recompile your {\bf .as} source files.
+
+An updated User's Guide is included for on-line use. We provide
+TeX {\bf .dvi} format (including hyper-references), Acrobat {\bf .pdf}
+format, PostScript {\bf .ps} format and plain ASCII {bf .txt}. 
+An up-to-date {\bf .dvi}
+viewer should be able to allow you to follow the cross-references.
+These files can be found in 
+{\bf \env{AXIOM}/compiler/doc/axlugII.*}.
+
+An exception mechanism has been implemented. You can now throw and 
+catch exceptions which are fully-fledged domains that can carry information.
+See Chapter 31 in the on-line guide for details.
+
+Support for linking with Fortran-77 code has been added. You can call
+Fortran-77 code from {\it Aldor}  and vice-versa. See Chapter 32
+in the on-line guide for details.
+
+The \spad{ref} keyword has been added. You must use \spad{_ref} to 
+enable use as an identifier.
+
+A generic configurable C compiler and linker driver i{\bf unicl} has been 
+implemented. You can now select C compiler and linker options with
+greater ease and you can give names to specific configurations. 
+See Chapter 28 in the on-line guide for details.
+
+There is now support for accessing IEEE floating-point rounding modes
+and exception flags. You must ensure however that your C compiler 
+and linker will respect the IEEE standard by enabling the appropriate
+options through {\bf unicl}. 
+See BasicIeeeControlPackage (26.7) in the on-line guide.
+
+Special runtime libraries ({\bf libfoam-gmp.a}) have been prepared to enable
+a drop-in replacement of Aldor's big-integer arithmetic with the GNU gmp
+big-integer library. The executable script {\bf axiomxl.gmp} (in the same place
+as {\bf axiomxl}) expects the {\bf GmpDir} environment variable to be set 
+to the directory where a compiled GNU gmp library resides ({\bf libgmp.a}) and 
+arranges for the appropriate libraries to be linked in when generating 
+a stand-alone executable. 
+
+The language-defined type TrailingArray has been introduced. See 
+TrailingArray (13.7) in the on-line guide.
+
+The supplied library {\bf libaxllib.al}, supporting stand-alone programs, 
+has been updated. Automatically
+generated documentation can be found in Chapters 25 and 26 of the on-line
+guide.
+
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTwoTwoPolynomialsTitle}{New polynomial domains and algorithms}
+\newcommand{\ugTwoTwoPolynomialsNumber}{0.2.}
+%
+% =====================================================================
+\begin{page}{ugTwoTwoPolynomialsPage}{0.2. New polynomial domains and algorithms}
+% =====================================================================
+\beginscroll
+
+Univariate polynomial factorisation over the integers has been 
+enhanced by updates to the \spadtype{GaloisGroupFactorizer} type
+and friends from Frederic Lehobey (Frederic.Lehobey@lifl.fr, University of 
+Lille I, France).
+
+The package constructor \spadtype{PseudoRemainderSequence}
+provides efficient algorithms by Lionel Ducos 
+(Lionel.Ducos@mathlabo.univ-poitiers.fr, University of Poitiers, France) 
+for computing sub-resultants.
+This leads to a speed up in many places in \Language{} where 
+sub-resultants are computed (polynomial system solving, 
+algebraic factorization, integration).
+
+Based on this package, the domain constructor 
+\spadtype{NewSparseUnivariatePolynomial}
+extends the constructor \spadtype{SparseUnivariatePolynomial}.
+In a similar way, the \spadtype{NewSparseMultivariatePolynomial} extends
+the constructor \spadtype{SparseUnivariatePolynomial};
+it also provides some additional operations related 
+to polynomial system solving by means of triangular sets.
+
+Several domain constructors implement
+regular triangular sets (or regular chains).
+Among them \spadtype{RegularTriangularSet}
+and \spadtype{SquareFreeRegularTriangularSet}.
+They also implement an algorithm by Marc Moreno Maza (marc@nag.co.uk, NAG)
+for computing triangular decompositions of polynomial systems.
+This method is refined in the package \spadtype{LazardSetSolvingPackage}
+in order to produce decompositions by means of Lazard triangular sets.
+For the case of polynomial systems with finitely many solutions,
+these decompositions can also be computed by
+the package \spadtype{LexTriangularPackage}.
+
+The domain constructor \spadtype{RealClosure} by Renaud Rioboo
+(Renaud.Rioboo@lip6.fr, University of Paris 6, France)
+provides the real closure of an ordered field.
+The implementation is based on interval arithmetic.
+Moreover, the design of this constructor and its related 
+packages allows an easy use of other codings for real algebraic numbers.
+
+Based on triangular decompositions and the \spadtype{RealClosure} constructor,
+the package \spadtype{ZeroDimensionalSolvePackage}
+provides operations for computing symbolically the real or complex roots
+of polynomial systems with finitely many solutions.
+
+Polynomial arithmetic with non-commutative variables 
+has been improved too by a contribution of Michel Petitot
+(Michel.Petitot@lifl.fr, University of Lille I, France).
+The domain constructors \spadtype{XRecursivePolynomial}
+and \spadtype{XDistributedPolynomial} provide
+recursive and distributed representations for these polynomials.
+They are the non-commutative equivalents for
+the \spadtype{SparseMultivariatePolynomial} 
+and \spadtype{DistributedMultivariatePolynomial} constructors.
+The constructor \spadtype{LiePolynomial} implement Lie
+polynomials in the Lyndon basis.
+The constructor \spadtype{XPBWPolynomial} manage polynomials
+with non-commutative variables in
+the \texht{Poincar\'e}{Poincare\space{-1}'}-Birkhoff-Witt basis from the Lyndon basis.
+This allows to compute in the Lie Group associated with a
+free nilpotent Lie algebra by using the \spadtype{LieExponentials}
+domain constructor.
+
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTwoTwoHyperdocTitle}{Enhancements to HyperDoc and Graphics}
+\newcommand{\ugTwoTwoHyperdocNumber}{0.3.}
+%
+% =====================================================================
+\begin{page}{ugTwoTwoHyperdocPage}{0.3. Enhancements to HyperDoc and Graphics}
+% =====================================================================
+\beginscroll
+
+From this version of AXIOM onwards, the pixmap format used to save graphics
+images in color and to display them in HyperDoc has been changed to the 
+industry-standard XPM format. See {\tt ftp://koala.inria.fr/pub/xpm}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTwoTwoNAGLinkTitle}{Enhancements to NAGLink}
+\newcommand{\ugTwoTwoNAGLinkNumber}{0.4.}
+%
+% =====================================================================
+\begin{page}{ugTwoTwoNAGLinkPage}{0.4. Enhancements to NAGLink}
+% =====================================================================
+\beginscroll
+
+This version of AXIOM incorporates the AXIOM/NAG Numerical Analyst (ANNA)
+software developed in the University of Bath. ANNA is an expert system which 
+uses AXIOM algebra functionality in choosing the most appropriate NAG Fortran Library
+routine for a particular problem. Documentation is provided on-line on the main HyperDoc page.
+
+
+ 
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTwoTwoCCLTitle}{Enhancements to the Lisp system}
+\newcommand{\ugTwoTwoCCLNumber}{0.5.}
+%
+% =====================================================================
+\begin{page}{ugTwoTwoCCLPage}{0.5. Enhancements to the Lisp system}
+% =====================================================================
+\beginscroll
+
+In this version of AXIOM, certain Library operations have been accelerated by converting 
+their Lisp implementations into kernel operations implemented directly in C. Here is a list
+of the accelerated operations. The given names encode the abbreviated type the operation 
+comes from, the name of the operation and, in the case of exported functions, an encoded
+signature and numerical index. 
+
+\begin{verbatim}
+|A1AGG-;=;2AB;28|
+|A1AGG-;copy;2A;19|
+|A1AGG-;sort!;M2A;2|
+|ABELGRP-;*;Nni2S;3|
+|ABELGRP-;-;3S;1|
+|ABELMON-;*;Pi2S;2|
+|ABELMON-;zero?;SB;1|
+|AGG-;empty?;SB;3|
+|AGG-;size?;SNniB;6|
+|ALIST;keys;$L;6|
+|ALIST;search;Key$U;15|
+|ARR2CAT-;copy;2S;10|
+|BOP;<;2$B;29|
+|BOP;=;2$B;27|
+|BOP;has?;$SB;9|
+|BOP;is?;$SB;1|
+|BOP;name;$S;2|
+|BOP;properties;$Al;3|
+|BOP;property;$SU;7|
+|BOP;weight;$Nni;28|
+|COMPCAT-;*;R2S;18|
+|COMPCAT-;-;2S;17|
+|COMPCAT-;=;2SB;15|
+|COMPCAT-;exquo;SRU;46|
+|COMPCAT-;recip;SU;48|
+|COMPCAT-;rem;3S;53|
+|COMPCAT-;unitNormal;SR;49|
+|COMPLEX;*;3$;10|
+|COMPLEX;+;3$;9|
+|COMPLEX;coerce;R$;5|
+|COMPLEX;exquo;2$U;11|
+|COMPLEX;one?;$B;4|
+|COMPLEX;zero?;$B;3|
+|DIRPROD;<;2$B;18|
+|DIRPROD;subtractIfCan;2$U;14|
+|ELAGG-;removeDuplicates;2A;12|
+|ELTAGG-;qelt;SDomIm;1|
+|ELTAGG-;qsetelt!;SDom2Im;2|
+|EUCDOM-;gcd;3S;5|
+|EUCDOM-;unitNormalizeIdealElt|
+|EXPR;*;3$;11|
+|EXPR;+;3$;12|
+|EXPR;-;2$;8|
+|EXPR;/;3$;14|
+|EXPR;=;2$B;21|
+|EXPR;algkernels!0|
+|EXPR;algkernels|
+|EXPR;coerce;I$;10|
+|EXPR;coerce;Smp$;24|
+|EXPR;commonk0|
+|EXPR;commonk|
+|EXPR;denom;$Smp;23|
+|EXPR;numer;$Smp;22|
+|EXPR;reduc|
+|EXPR;zero?;$B;7|
+|FACUTIL;lowerPolynomial;SupSup;1|
+|FAMR-;coefficients;SL;4|
+|FAMR-;ground;SR;2|
+|FFP;*;3$;21|
+|FFP;+;3$;22|
+|FFP;-;3$;23|
+|FFP;=;2$B;24|
+|FIELD-;/;3S;11|
+|FIELD-;inv;2S;4|
+|FLAGG-;sort!;2A;8|
+|FLAGG-;sort;M2A;6|
+|FLASORT;QuickSort|
+|FLASORT;partition|
+|FLASORT;quickSort;M2V;1|
+|FM;*;R2$;1|
+|FRAC;*;3$;18|
+|FRAC;*;I2$;19|
+|FRAC;+;3$;16|
+|FRAC;=;2$B;22|
+|FRAC;cancelGcd|
+|FRAC;coerce;S$;1|
+|FRAC;normalize|
+|FRAC;one?;$B;23|
+|FRAC;recip;$U;13|
+|FRAC;zero?;$B;2|
+|FSAGG-;brace;LA;3|
+|GDMP;univariate;$OvlSup;21|
+|HDP;<;2$B;1|
+|IARRAY1;#;$Nni;1|
+|IARRAY1;elt;$IS;16|
+|IARRAY1;fill!;$S$;2|
+|IARRAY1;map;M3$;8|
+|IARRAY1;maxIndex;$I;13|
+|IARRAY1;minIndex;$I;3|
+|IARRAY1;new;NniS$;5|
+|IARRAY1;qelt;$IS;14|
+|IARRAY1;qsetelt!;$I2S;15|
+|IARRAY1;setelt;$I2S;17|
+|IDPAM;+;3$;4|
+|IDPAM;map;M2$;7|
+|IDPAM;monomial;AS$;6|
+|IDPO;=;2$B;1|
+|IFARRAY;#;$Nni;4|
+|IFARRAY;concat!;$S$;21|
+|IFARRAY;elt;$IS;17|
+|IFARRAY;empty;$;3|
+|IFARRAY;growAndFill|
+|IFARRAY;growWith|
+|IFARRAY;maxIndex;$I;6|
+|IFARRAY;minIndex;$I;7|
+|IFARRAY;new;NniS$;8|
+|IFARRAY;removeDuplicates!;2$;30|
+|IFARRAY;setelt;$I2S;18|
+|IIARRAY2;elt;$2IR;11|
+|IIARRAY2;empty?;$B;1|
+|IIARRAY2;maxColIndex;$I;7|
+|IIARRAY2;maxRowIndex;$I;6|
+|IIARRAY2;minColIndex;$I;5|
+|IIARRAY2;minRowIndex;$I;4|
+|IIARRAY2;ncols;$Nni;9|
+|IIARRAY2;nrows;$Nni;8|
+|IIARRAY2;qelt;$2IR;10|
+|IIARRAY2;qsetelt!;$2I2R;12|
+|ILIST;concat!;3$;25|
+|ILIST;copy;2$;20|
+|ILIST;empty;$;6|
+|ILIST;first;$S;4|
+|ILIST;member?;S$B;24|
+|ILIST;mergeSort|
+|ILIST;minIndex;$I;18|
+|ILIST;removeDuplicates!;2$;26|
+|ILIST;rest;$Nni$;19|
+|ILIST;sort!;M2$;27|
+|ILIST;split!;$I$;29|
+|IMATLIN;rowEchelon;2M;3|
+|INS-;symmetricRemainder;3S;27|
+|INT;exquo;2$U;44|
+|INT;one?;$B;2|
+|INT;positiveRemainder;3$;23|
+|INT;unitNormal;$R;47|
+|INTDOM-;unitCanonical;2S;2|
+|ISTRING;<;2$B;6|
+|KDAGG-;key?;KeySB;1|
+|KERNEL;<;2$B;14|
+|KERNEL;=;2$B;13|
+|KERNEL;B2Z|
+|KERNEL;argument;$L;6|
+|KERNEL;operator;$Bo;5|
+|KERNEL;position;$Nni;7|
+|KERNEL;triage|
+|LO;-;2$;3|
+|LO;=;2$B;4|
+|LSAGG-;<;2AB;25|
+|LSAGG-;reduce;MA2S;16|
+|LSAGG-;select!;M2A;5|
+|MATCAT-;*;3S;29|
+|MATCAT-;*;S2Col;32|
+|MDDFACT;reduction!0|
+|MDDFACT;reduction|
+|MODRING;reduce;RMod$;6|
+|MODRING;zero?;$B;10|
+|MONOID-;one?;SB;2|
+|NNI;subtractIfCan;2$U;3|
+|NSMP;mvar;$VarSet;5|
+|OVAR;<;2$B;10|
+|PERMGRP;inv|
+|PERMGRP;orbitWithSvc|
+|PERMGRP;testIdentity|
+|PERMGRP;times|
+|PGCD;better|
+|PGCD;gcd;3P;15|
+|PGCD;gcdTermList|
+|POLYCATQ;variables;FL;5|
+|PR;*;3$;20|
+|PR;addm!|
+|PR;coerce;R$;12|
+|PR;degree;$E;4|
+|PR;leadingCoefficient;$R;6|
+|PR;reductum;2$;8|
+|PRIMARR;#;$Nni;1|
+|PRIMARR;fill!;$S$;9|
+|PRIMARR;new;NniS$;4|
+|PRTITION;<;2$B;5|
+|REPSQ;expt;SPiS;1|
+|RING-;coerce;IS;1|
+|SAE;*;3$;15|
+|SAE;+;3$;13|
+|SAE;-;2$;14|
+|SAE;=;2$B;12|
+|SAE;reduce;UP$;11|
+|SCACHE;enterInCache;SMS;5|
+|SET;construct;L$;19|
+|SET;empty;$;4|
+|SGROUP-;**;SPiS;1|
+|SINT;zero?;$B;33|
+|SMP;*;3$;29|
+|SMP;*;R2$;25|
+|SMP;+;3$;26|
+|SMP;-;2$;23|
+|SMP;=;2$B;28|
+|SMP;coerce;R$;20|
+|SMP;coerce;VarSet$;7|
+|SMP;exquo;2$U;35|
+|SMP;exquo;2$U;36|
+|SMP;gcd;3$;41|
+|SMP;gcd;3$;44|
+|SMP;gcd;3$;48|
+|SMP;gcd;3$;51|
+|SMP;ground?;$B;15|
+|SMP;leadingCoefficient;$R;73|
+|SMP;mainVariable;$U;59|
+|SMP;one?;$B;4|
+|SMP;retract;$R;55|
+|SMP;unitNormal;$R;31|
+|SMP;variables;$L;58|
+|SMP;zero?;$B;3|
+|STAGG-;c2|
+|STAGG-;concat;3A;7|
+|STAGG-;elt;AIS;5|
+|STAGG-;first;ANniA;3|
+|STREAM2;map;MSS;2|
+|STREAM2;mapp!0|
+|STREAM2;mapp|
+|STREAM;empty;$;33|
+|STREAM;lazyEval|
+|STREAM;setfrst!|
+|STREAM;setrst!|
+|STTAYLOR;+;3S;2|
+|SUP;exquo;2$U;19|
+|SUP;exquo;2$U;20|
+|SUP;fmecg;$NniR2$;21|
+|SUP;ground?;$B;3|
+|SUP;monicDivide;2$R;28|
+|URAGG-;tail;2A;16|
+|ZMOD;*;3$;30|
+|ZMOD;+;3$;32|
+|ZMOD;-;2$;36|
+|ZMOD;-;3$;33|
+\end{verbatim}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug01.ht b/src/hyper/pages/ug01.ht
new file mode 100644
index 00000000..b193daeb
--- /dev/null
+++ b/src/hyper/pages/ug01.ht
@@ -0,0 +1,2591 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+
+\texht{\setcounter{chapter}{0}}{} % Chapter 1
+
+
+%
+\newcommand{\ugIntroTitle}{An Overview of \Language{}}
+\newcommand{\ugIntroNumber}{1.}
+%
+% =====================================================================
+\begin{page}{ugIntroPage}{1. An Overview of \Language{}}
+% =====================================================================
+\beginscroll
+
+Welcome to the \Language{} environment for interactive computation
+and problem solving.
+Consider this chapter a brief, whirlwind tour of the \Language{}
+world.
+We introduce you to \Language{}'s graphics and the \Language{}
+language.
+Then we give a sampling of the large variety of facilities
+in the \Language{} system, ranging from the various kinds of
+numbers, to data types (like lists, arrays, and sets) and
+mathematical objects (like matrices, integrals, and differential
+equations).
+We conclude with the discussion of system commands and an
+interactive ``undo.''
+
+Before embarking on the tour, we need to brief those readers
+working interactively with \Language{} on some details.
+Others can skip right immediately to
+\downlink{``\ugIntroTypoTitle''}{ugIntroTypoPage} in Section \ugIntroTypoNumber\ignore{ugIntroTypo}.
+
+\beginmenu
+    \menudownlink{{1.1. Starting Up and Winding Down}}{ugIntroStartPage}
+    \menudownlink{{1.2. Typographic Conventions}}{ugIntroTypoPage}
+    \menudownlink{{1.3. The \Language{} Language}}{ugIntroExpressionsPage}
+    \menudownlink{{1.4. Graphics}}{ugIntroGraphicsPage}
+    \menudownlink{{1.5. Numbers}}{ugIntroNumbersPage}
+    \menudownlink{{1.6. Data Structures}}{ugIntroCollectPage}
+    \menudownlink{{1.7. Expanding to Higher Dimensions}}{ugIntroTwoDimPage}
+    \menudownlink{{1.8. Writing Your Own Functions}}{ugIntroYouPage}
+    \menudownlink{{1.9. Polynomials}}{ugIntroVariablesPage}
+    \menudownlink{{1.10. Limits}}{ugIntroCalcLimitsPage}
+    \menudownlink{{1.11. Series}}{ugIntroSeriesPage}
+    \menudownlink{{1.12. Derivatives}}{ugIntroCalcDerivPage}
+    \menudownlink{{1.13. Integration}}{ugIntroIntegratePage}
+    \menudownlink{{1.14. Differential Equations}}{ugIntroDiffEqnsPage}
+    \menudownlink{{1.15. Solution of Equations}}{ugIntroSolutionPage}
+    \menudownlink{{1.16. System Commands}}{ugIntroSysCmmandsPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroStartTitle}{Starting Up and Winding Down}
+\newcommand{\ugIntroStartNumber}{1.1.}
+%
+% =====================================================================
+\begin{page}{ugIntroStartPage}{1.1. Starting Up and Winding Down}
+% =====================================================================
+\beginscroll
+%
+
+You need to know how to start the \Language{} system and how to stop it.
+We assume that \Language{} has been correctly installed on your
+machine (as described in another \Language{} document).
+
+To begin using \Language{}, issue the command {\bf axiom} to the
+%-% \HDindex{starting @{starting \Language{}}}{ugIntroStartPage}{1.1.}{Starting Up and Winding Down}
+operating system shell.
+%-% \HDindex{axiom @{\bf axiom}}{ugIntroStartPage}{1.1.}{Starting Up and Winding Down}
+There is a brief pause, some start-up messages, and then one
+or more windows appear.
+
+If you are not running \Language{} under the X Window System, there is
+only one window (the console).
+At the lower left of the screen there is a prompt that
+%-% \HDindex{prompt}{ugIntroStartPage}{1.1.}{Starting Up and Winding Down}
+looks like
+\begin{verbatim}
+(1) ->
+\end{verbatim}
+%%--> do you want to talk about equation numbers on the right, etc.
+When you want to enter input to \Language{}, you do so on the same line
+after the prompt.
+The ``1'' in ``(1)'' is the computation step number and is incremented
+%-% \HDindex{step number}{ugIntroStartPage}{1.1.}{Starting Up and Winding Down}
+after you enter \Language{} statements.
+Note, however, that a system command
+such as \spadsys{)clear all}
+may change the step number in other ways.
+We talk about step numbers more when we discuss system commands
+and the workspace history facility.
+
+If you are running \Language{} under the X Window System, there may be two
+%-% \HDindex{X Window System}{ugIntroStartPage}{1.1.}{Starting Up and Winding Down}
+windows: the console window (as just described) and the \HyperName{}
+main menu.
+%-% \HDindex{Hyper @{\HyperName{}}}{ugIntroStartPage}{1.1.}{Starting Up and Winding Down}
+\HyperName{} is a multiple-window hypertext system that lets you
+%-% \HDindex{window}{ugIntroStartPage}{1.1.}{Starting Up and Winding Down}
+view \Language{} documentation and examples on-line,
+execute \Language{} expressions, and generate graphics.
+If you are in a graphical windowing environment,
+it is usually started automatically when \Language{} begins.
+If it is not running, issue \spadsys{)hd} to start it.
+We discuss the basics of \HyperName{} in \downlink{``\ugHyperTitle''}{ugHyperPage} in Chapter \ugHyperNumber\ignore{ugHyper}.
+%-% \HDsyscmdindex{hd}{ugIntroStartPage}{1.1.}{Starting Up and Winding Down}
+
+To interrupt an \Language{} computation, hold down the
+%-% \HDindex{interrupt}{ugIntroStartPage}{1.1.}{Starting Up and Winding Down}
+\texht{\fbox{\bf Ctrl}}{{\bf Ctrl}} (control) key and press
+\texht{\fbox{\bf c}}{{\bf c}}.
+This  brings you back to the \Language{} prompt.
+
+\beginImportant
+To exit from \Language{},  move to the console window,
+%-% \HDindex{stopping @{stopping \Language{}}}{ugIntroStartPage}{1.1.}{Starting Up and Winding Down}
+type \spadsys{)quit}
+%-% \HDindex{exiting @{exiting \Language{}}}{ugIntroStartPage}{1.1.}{Starting Up and Winding Down}
+at the input prompt and press the \texht{\fbox{\bf Enter}}{{\bf Enter}} key.
+%-% \HDsyscmdindex{quit}{ugIntroStartPage}{1.1.}{Starting Up and Winding Down}
+You will probably be prompted with the following message:
+\centerline{{Please enter {\bf y} or {\bf yes} if you really want to leave the }}
+\centerline{{interactive environment and return to the operating system}}
+You should respond {\bf yes}, for example, to exit \Language{}.
+\endImportant
+
+We are purposely vague in describing exactly what your screen
+looks like or what messages \Language{} displays.
+\Language{} runs on a number of different machines, operating
+systems and window environments, and these differences all affect
+the physical look of the system.
+You can also change the way that \Language{} behaves via
+\spadgloss{system commands} described later in this chapter and in
+\downlink{``\ugSysCmdTitle''}{ugSysCmdPage} in Appendix \ugSysCmdNumber\ignore{ugSysCmd}.
+System commands are special commands, like \spadcmd{)set}, that begin
+with a closing parenthesis and are used to change your
+environment.
+For example, you can set a system variable so that you are not
+prompted for confirmation when you want to leave \Language{}.
+
+\beginmenu
+    \menudownlink{{1.1.1. \Clef{}}}{ugAvailCLEFPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugAvailCLEFTitle}{\Clef{}}
+\newcommand{\ugAvailCLEFNumber}{1.1.1.}
+%
+% =====================================================================
+\begin{page}{ugAvailCLEFPage}{1.1.1. \Clef{}}
+% =====================================================================
+\beginscroll
+%
+If you are using \Language{} under the X Window System, the
+%-% \HDindex{Clef@{\Clef{}}}{ugAvailCLEFPage}{1.1.1.}{\Clef{}}
+%-% \HDindex{command line editor}{ugAvailCLEFPage}{1.1.1.}{\Clef{}}
+\Clef{} command line editor is probably available and installed.
+With this editor you can recall previous lines with the up and
+down arrow keys\texht{ (\fbox{$\uparrow$} and
+\fbox{$\downarrow$})}{}.
+To move forward and backward on a line, use the right and
+left arrows\texht{ (\fbox{$\rightarrow$} and
+\fbox{$\leftarrow$})}{}.
+You can use the
+\texht{\fbox{\bf Insert}}{{\bf Insert}}
+key to toggle insert mode on or off.
+When you are in insert mode,
+the cursor appears as a large block and if you type
+anything, the characters are inserted into the line without
+deleting the previous ones.
+
+If you press the
+\texht{\fbox{\bf Home}}{{\bf Home}}
+key, the cursor moves to the beginning of the line and if you press the
+\texht{\fbox{\bf End}}{{\bf End}}
+key, the cursor moves to the end of the line.
+Pressing
+\texht{\fbox{\bf Ctrl}--\fbox{\bf End}}{{\bf Ctrl-End}}
+deletes all the text from the cursor to the end of the line.
+
+\Clef{} also provides \Language{} operation name completion for
+%-% \HDindex{operation name completion}{ugAvailCLEFPage}{1.1.1.}{\Clef{}}
+a limited set of operations.
+If you enter a few letters and then press the
+\texht{\fbox{\bf Tab}}{{\bf Tab}} key,
+\Clef{} tries to use those letters as the prefix of an \Language{}
+operation name.
+If a name appears and it is not what you want, press
+\texht{\fbox{\bf Tab}}{{\bf Tab}} again
+to see another name.
+
+You are ready to begin your journey into the world of \Language{}.
+Proceed to the first stop.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroTypoTitle}{Typographic Conventions}
+\newcommand{\ugIntroTypoNumber}{1.2.}
+%
+% =====================================================================
+\begin{page}{ugIntroTypoPage}{1.2. Typographic Conventions}
+% =====================================================================
+\beginscroll
+
+In this book we have followed these typographical conventions:
+\indent{4}
+\beginitems
+%
+\item[-] Categories, domains and packages are displayed in
+\texht{a sans-serif typeface:}{this font:}
+\axiomType{Ring}, \axiomType{Integer}, \axiomType{DiophantineSolutionPackage}.
+%
+\item[-] Prefix operators, infix operators, and punctuation symbols in the \Language{}
+language are displayed in the text like this:
+\axiomOp{+}, \axiomSyntax{\$}, \axiomSyntax{+->}.
+%
+\item[-] \Language{} expressions or expression fragments are displayed in
+\texht{a mon\-o\-space typeface:}{this font:}
+\axiom{inc(x) == x + 1}.
+%
+\item[-] For clarity of presentation, \TeX{} is often
+used to format expressions\texht{: $g(x)=x^2+1.$}{.}
+%
+\item[-] Function names and \HyperName{} button names
+are displayed in the text in
+\texht{a bold typeface:}{this font:}
+\axiomFun{factor}, \axiomFun{integrate},  {\bf Lighting}.
+%
+\item[-] Italics are used for emphasis and for words defined in the
+glossary: \spadgloss{category}.
+\enditems
+\indent{0}
+
+This book contains over 2500 examples of \Language{} input and output.
+All examples were run though \Language{} and their output was
+created in \texht{\TeX{}}{TeX} form for this book by the \Language{}
+\axiomType{TexFormat} package.
+%-% \HDexptypeindex{TexFormat}{ugIntroTypoPage}{1.2.}{Typographic Conventions}
+We have deleted system messages from the example output if those
+messages are not important for the discussions in which the examples
+appear.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroExpressionsTitle}{The \Language{} Language}
+\newcommand{\ugIntroExpressionsNumber}{1.3.}
+%
+% =====================================================================
+\begin{page}{ugIntroExpressionsPage}{1.3. The \Language{} Language}
+% =====================================================================
+\beginscroll
+%
+
+The \Language{} language is a rich language for performing
+interactive computations and for building components of the
+\Language{} library.
+Here we present only some basic aspects of the language that you
+need to know for the rest of this chapter.
+Our discussion here is intentionally informal, with details
+unveiled on an ``as needed'' basis.
+For more information on a particular construct, we suggest you
+consult the index at the back of the book.
+
+\beginmenu
+    \menudownlink{{1.3.1. Arithmetic Expressions}}{ugIntroArithmeticPage}
+    \menudownlink{{1.3.2. Previous Results}}{ugIntroPreviousPage}
+    \menudownlink{{1.3.3. Some Types}}{ugIntroTypesPage}
+    \menudownlink{{1.3.4. Symbols, Variables, Assignments, and Declarations}}{ugIntroAssignPage}
+    \menudownlink{{1.3.5. Conversion}}{ugIntroConversionPage}
+    \menudownlink{{1.3.6. Calling Functions}}{ugIntroCallFunPage}
+    \menudownlink{{1.3.7. Some Predefined Macros}}{ugIntroMacrosPage}
+    \menudownlink{{1.3.8. Long Lines}}{ugIntroLongPage}
+    \menudownlink{{1.3.9. Comments}}{ugIntroCommentsPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroArithmeticTitle}{Arithmetic Expressions}
+\newcommand{\ugIntroArithmeticNumber}{1.3.1.}
+%
+% =====================================================================
+\begin{page}{ugIntroArithmeticPage}{1.3.1. Arithmetic Expressions}
+% =====================================================================
+\beginscroll
+
+For arithmetic expressions, use the \spadop{+} and \spadop{-}
+\spadglossSee{operators}{operator} as in mathematics.
+Use \spadop{*} for multiplication, and \spadop{**} for
+exponentiation.
+To create a fraction, use \spadop{/}.
+When an expression contains several operators, those of highest
+\spadgloss{precedence} are evaluated first.
+For arithmetic operators, \spadop{**} has highest precedence,
+\spadop{*} and \spadop{/} have the next highest
+precedence, and \spadop{+} and \spadop{-} have the lowest
+precedence.
+
+\xtc{
+\Language{} puts implicit parentheses around operations of higher
+precedence, and groups those of equal precedence from left to right.
+}{
+\spadpaste{1 + 2 - 3 / 4 * 3 ** 2 - 1}
+}
+\xtc{
+The above expression is equivalent to this.
+}{
+\spadpaste{((1 + 2) - ((3 / 4) * (3 ** 2))) - 1}
+}
+\xtc{
+If an expression contains subexpressions enclosed in parentheses,
+the parenthesized subexpressions are evaluated first (from left to
+right, from inside out).
+}{
+\spadpaste{1 + 2 - 3/ (4 * 3 ** (2 - 1))}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroPreviousTitle}{Previous Results}
+\newcommand{\ugIntroPreviousNumber}{1.3.2.}
+%
+% =====================================================================
+\begin{page}{ugIntroPreviousPage}{1.3.2. Previous Results}
+% =====================================================================
+\beginscroll
+
+Use the percent sign (\axiomSyntax{\%}) to refer to the last
+result.
+%-% \HDindex{result!previous}{ugIntroPreviousPage}{1.3.2.}{Previous Results}
+Also, use \axiomSyntax{\%\%} to refer to previous results.
+%-% \HDindex{percentpercent@{\%\%}}{ugIntroPreviousPage}{1.3.2.}{Previous Results}
+\axiom{\%\%(-1)} is equivalent to \axiomSyntax{\%},
+\axiom{\%\%(-2)} returns the next to the last result, and so on.
+\axiom{\%\%(1)} returns the result from step number 1,
+\axiom{\%\%(2)} returns the result from step number 2, and so on.
+\axiom{\%\%(0)} is not defined.
+
+\xtc{
+This is ten to the tenth power.
+}{
+\spadpaste{10 ** 10 \bound{prev}}
+}
+\xtc{
+This is the last result minus one.
+}{
+\spadpaste{\% - 1 \free{prev}\bound{prev1}}
+}
+\xtc{
+This is the last result.
+}{
+\spadpaste{\%\%(-1) \free{prev1}\bound{prev2}}
+}
+\xtc{
+This is the result from step number 1.
+}{
+\spadpaste{\%\%(1) \free{prev2}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroTypesTitle}{Some Types}
+\newcommand{\ugIntroTypesNumber}{1.3.3.}
+%
+% =====================================================================
+\begin{page}{ugIntroTypesPage}{1.3.3. Some Types}
+% =====================================================================
+\beginscroll
+
+Everything in \Language{} has a type.
+The type determines what operations you can perform on an object and
+how the object can be used.
+An entire chapter of this book (\downlink{``\ugTypesTitle''}{ugTypesPage} in Chapter \ugTypesNumber\ignore{ugTypes}) is dedicated to
+the interactive use of types.
+Several of the final chapters discuss how types are built and how
+they are organized in the \Language{} library.
+
+\xtc{
+Positive integers are given type \spadtype{PositiveInteger}.
+}{
+\spadpaste{8}
+}
+\xtc{
+Negative ones are given type \spadtype{Integer}.
+This fine distinction is helpful to the
+\Language{} interpreter.
+}{
+\spadpaste{-8}
+}
+\xtc{
+Here a positive integer exponent gives a polynomial result.
+}{
+\spadpaste{x**8}
+}
+\xtc{
+Here a negative integer exponent produces a fraction.
+}{
+\spadpaste{x**(-8)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroAssignTitle}{Symbols, Variables, Assignments, and Declarations}
+\newcommand{\ugIntroAssignNumber}{1.3.4.}
+%
+% =====================================================================
+\begin{page}{ugIntroAssignPage}{1.3.4. Symbols, Variables, Assignments, and Declarations}
+% =====================================================================
+\beginscroll
+
+A \spadgloss{symbol} is a literal used for the input of things like
+the ``variables'' in polynomials and power series.
+
+\labelSpace{2pc}
+\xtc{
+We use the three symbols \axiom{x}, \axiom{y}, and \axiom{z} in
+entering this polynomial.
+}{
+\spadpaste{(x - y*z)**2}
+}
+A symbol has a name beginning with an uppercase or lowercase alphabetic
+%-% \HDindex{symbol!naming}{ugIntroAssignPage}{1.3.4.}{Symbols, Variables, Assignments, and Declarations}
+character, \axiomSyntax{\%}, or \axiomSyntax{!}.
+Successive characters (if any) can be any of the above, digits, or
+\axiomSyntax{?}.
+Case is distinguished: the symbol \axiom{points} is different
+from the symbol \axiom{Points}.
+
+A symbol can also be used in \Language{} as a \spadgloss{variable}.
+A variable refers to a value.
+To \spadglossSee{assign}{assignment}
+a value to a variable,
+%-% \HDindex{variable!naming}{ugIntroAssignPage}{1.3.4.}{Symbols, Variables, Assignments, and Declarations}
+the operator \axiomSyntax{:=}
+%-% \HDindex{assignment}{ugIntroAssignPage}{1.3.4.}{Symbols, Variables, Assignments, and Declarations}
+is used.\footnote{\Language{} actually has two forms of assignment:
+{\it immediate} assignment, as discussed here,
+and {\it delayed assignment}. See \downlink{``\ugLangAssignTitle''}{ugLangAssignPage} in Section \ugLangAssignNumber\ignore{ugLangAssign} for details.}
+A variable initially has no restrictions on the kinds of
+%-% \HDindex{declaration}{ugIntroAssignPage}{1.3.4.}{Symbols, Variables, Assignments, and Declarations}
+values to which it can refer.
+
+\xtc{
+This assignment gives the value \axiom{4} (an integer) to
+a variable named \axiom{x}.
+}{
+\spadpaste{x := 4}
+}
+\xtc{
+This gives the value \axiom{z + 3/5} (a polynomial)  to \axiom{x}.
+}{
+\spadpaste{x := z + 3/5}
+}
+\xtc{
+To restrict the types of objects that can be assigned to a variable,
+use a \spadgloss{declaration}
+}{
+\spadpaste{y : Integer \bound{y}}
+}
+\xtc{
+After a variable is declared to be of some type, only values
+of that type can be assigned to that variable.
+}{
+\spadpaste{y := 89\bound{y1}\free{y}}
+}
+\xtc{
+The declaration for \axiom{y} forces values assigned to \axiom{y} to
+be converted to integer values.
+}{
+\spadpaste{y := sin \%pi}
+}
+\xtc{
+If no such conversion is possible,
+\Language{} refuses to assign a value to \axiom{y}.
+}{
+\spadpaste{y := 2/3}
+}
+\xtc{
+A type declaration can also be given together with an assignment.
+The declaration can assist \Language{} in choosing the correct
+operations to apply.
+}{
+\spadpaste{f : Float := 2/3}
+}
+
+Any number of expressions can be given on input line.
+Just separate them by semicolons.
+Only the result of evaluating the last expression is displayed.
+
+\xtc{
+These two expressions have the same effect as
+the previous single expression.
+}{
+\spadpaste{f : Float; f := 2/3 \bound{fff}}
+}
+
+The type of a symbol is either \axiomType{Symbol}
+%-% \HDexptypeindex{Symbol}{ugIntroAssignPage}{1.3.4.}{Symbols, Variables, Assignments, and Declarations}
+or \axiomType{Variable({\it name})} where {\it name} is the name
+of the symbol.
+
+\xtc{
+By default, the interpreter
+%-% \HDexptypeindex{Variable}{ugIntroAssignPage}{1.3.4.}{Symbols, Variables, Assignments, and Declarations}
+gives this symbol the type \axiomType{Variable(q)}.
+}{
+\spadpaste{q}
+}
+\xtc{
+When multiple symbols are involved, \axiomType{Symbol} is used.
+}{
+\spadpaste{[q, r]}
+}
+
+\xtc{
+What happens when you try to use a symbol that is the name of a variable?
+}{
+\spadpaste{f \free{fff}}
+}
+\xtc{
+Use a single quote (\axiomSyntax{'}) before
+%-% \HDindex{quote}{ugIntroAssignPage}{1.3.4.}{Symbols, Variables, Assignments, and Declarations}
+the name to get the symbol.
+}{
+\spadpaste{'f}
+}
+
+Quoting a name creates a symbol by
+preventing evaluation of the name as a variable.
+Experience will teach you when you are most likely going to need to use
+a quote.
+We try to point out the location of such trouble spots.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroConversionTitle}{Conversion}
+\newcommand{\ugIntroConversionNumber}{1.3.5.}
+%
+% =====================================================================
+\begin{page}{ugIntroConversionPage}{1.3.5. Conversion}
+% =====================================================================
+\beginscroll
+
+Objects of one type can usually be ``converted'' to objects of several
+other types.
+To \spadglossSee{convert}{conversion}
+an object to a new type, use the \axiomSyntax{::} infix
+operator.\footnote{Conversion is discussed in detail in \downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert}.}
+For example, to display an object, it is necessary to
+convert the object to type \spadtype{OutputForm}.
+
+\xtc{
+This produces a polynomial with rational number coefficients.
+}{
+\spadpaste{p := r**2 + 2/3 \bound{p}}
+}
+\xtc{
+Create a quotient of polynomials with integer coefficients
+by using \axiomSyntax{::}.
+}{
+\spadpaste{p :: Fraction Polynomial Integer \free{p}}
+}
+
+Some conversions can be performed automatically when
+\Language{} tries to evaluate your input.
+Others conversions must be explicitly requested.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroCallFunTitle}{Calling Functions}
+\newcommand{\ugIntroCallFunNumber}{1.3.6.}
+%
+% =====================================================================
+\begin{page}{ugIntroCallFunPage}{1.3.6. Calling Functions}
+% =====================================================================
+\beginscroll
+
+As we saw earlier, when you want to add or subtract two values,
+you place the arithmetic operator \spadop{+}
+or \spadop{-} between the two
+\spadglossSee{arguments}{argument} denoting the values.
+To use most other \Language{} operations, however, you use another syntax:
+%-% \HDindex{function!calling}{ugIntroCallFunPage}{1.3.6.}{Calling Functions}
+write the name
+of the operation first, then an open parenthesis, then each of the
+arguments separated by commas, and, finally, a closing parenthesis.
+If the operation takes only one argument and the argument is a number
+or a symbol, you can omit the parentheses.
+
+\xtc{
+This calls the operation \axiomFun{factor} with the single
+integer argument \axiom{120}.
+}{
+\spadpaste{factor(120)}
+}
+\xtc{
+This is a call to \axiomFun{divide} with the two integer arguments
+\axiom{125} and \axiom{7}.
+}{
+\spadpaste{divide(125,7)}
+}
+\xtc{
+This calls \axiomFun{quatern} with four floating-point arguments.
+}{
+\spadpaste{quatern(3.4,5.6,2.9,0.1)}
+}
+\xtc{
+This is the same as \axiom{factorial(10)}.
+}{
+\spadpaste{factorial 10}
+}
+
+An operations that returns a \spadtype{Boolean} value (that is,
+\spad{true} or \spad{false}) frequently has a name suffixed with
+a question mark (``?'').  For example, the \spadfun{even?}
+operation returns \spad{true} if its integer argument is an even
+number, \spad{false} otherwise.
+
+An operation that can be destructive on one or more arguments
+usually has a name ending in a exclamation point (``!'').
+This actually means that it is {\it allowed} to update its
+arguments but it is not {\it required} to do so. For example,
+the underlying representation of a collection type may not allow
+the very last element to removed and so an empty object may be
+returned instead. Therefore, it is important that you use the
+object returned by the operation and not rely on a physical
+change having occurred within the object. Usually, destructive
+operations are provided for efficiency reasons.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroMacrosTitle}{Some Predefined Macros}
+\newcommand{\ugIntroMacrosNumber}{1.3.7.}
+%
+% =====================================================================
+\begin{page}{ugIntroMacrosPage}{1.3.7. Some Predefined Macros}
+% =====================================================================
+\beginscroll
+
+\Language{} provides several \spadglossSee{macros}{macro}
+for your convenience.\footnote{See \downlink{``\ugUserMacrosTitle''}{ugUserMacrosPage} in Section \ugUserMacrosNumber\ignore{ugUserMacros}
+for a discussion on how to write your own macros.}
+Macros are names
+%-% \HDindex{macro!predefined}{ugIntroMacrosPage}{1.3.7.}{Some Predefined Macros}
+(or forms) that expand to larger expressions for commonly used values.
+
+\texht{
+\centerline{{\begin{tabular}{ll}}}
+\centerline{{\spadgloss{\%i}             &  The square root of -1. }}
+\centerline{{\spadgloss{\%e}             &  The base of the natural logarithm. }}
+\centerline{{\spadgloss{\%pi}            &  $\pi$. }}
+\centerline{{\spadgloss{\%infinity}      &  $\infty$. }}
+\centerline{{\spadgloss{\%plusInfinity}  &  $+\infty$. }}
+\centerline{{\spadgloss{\%minusInfinity} &  $-\infty$.}}
+\centerline{{\end{tabular}}}
+%-% \HDindex{\%i}{ugIntroMacrosPage}{1.3.7.}{Some Predefined Macros}
+%-% \HDindex{\%e}{ugIntroMacrosPage}{1.3.7.}{Some Predefined Macros}
+%-% \HDindex{\%pi}{ugIntroMacrosPage}{1.3.7.}{Some Predefined Macros}
+%-% \HDindex{pi@{$\pi$ (= \%pi)}}{ugIntroMacrosPage}{1.3.7.}{Some Predefined Macros}
+%-% \HDindex{\%infinity}{ugIntroMacrosPage}{1.3.7.}{Some Predefined Macros}
+%-% \HDindex{infinity@{$\infty$ (= \%infinity)}}{ugIntroMacrosPage}{1.3.7.}{Some Predefined Macros}
+%-% \HDindex{\%plusInfinity}{ugIntroMacrosPage}{1.3.7.}{Some Predefined Macros}
+%-% \HDindex{\%minusInfinity}{ugIntroMacrosPage}{1.3.7.}{Some Predefined Macros}
+}{
+\indent{0}
+\beginitems
+\item[\axiomSyntax{\%i}]             \tab{17} The square root of -1.
+\item[\axiomSyntax{\%e}]             \tab{17} The base of the natural logarithm.
+\item[\axiomSyntax{\%pi}]            \tab{17} Pi.
+\item[\axiomSyntax{\%infinity}]      \tab{17} Infinity.
+\item[\axiomSyntax{\%plusInfinity}]  \tab{17} Plus infinity.
+\item[\axiomSyntax{\%minusInfinity}] \tab{17} Minus infinity.
+\enditems
+\indent{0}
+}
+
+%To display all the macros (along with anything you have
+%defined in the workspace), issue the system command \spadsys{)display all}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroLongTitle}{Long Lines}
+\newcommand{\ugIntroLongNumber}{1.3.8.}
+%
+% =====================================================================
+\begin{page}{ugIntroLongPage}{1.3.8. Long Lines}
+% =====================================================================
+\beginscroll
+
+When you enter \Language{} expressions from your keyboard, there
+will be times when they are too long to fit on one line.
+\Language{} does not care how long your lines are, so you can let
+them continue from the right margin to the left side of the
+next line.
+
+Alternatively, you may want to enter several shorter lines and
+have \Language{} glue them together.
+To get this glue, put an underscore (\_) at the end of
+each line you wish to continue.
+\begin{verbatim}
+2_
++_
+3
+\end{verbatim}
+is the same as if you had entered
+\begin{verbatim}
+2+3
+\end{verbatim}
+
+If you are putting your \Language{} statements in an input file
+(see \downlink{``\ugInOutInTitle''}{ugInOutInPage} in Section \ugInOutInNumber\ignore{ugInOutIn}),
+you can use indentation to indicate the structure of your program.
+(see \downlink{``\ugLangBlocksTitle''}{ugLangBlocksPage} in Section \ugLangBlocksNumber\ignore{ugLangBlocks}).
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroCommentsTitle}{Comments}
+\newcommand{\ugIntroCommentsNumber}{1.3.9.}
+%
+% =====================================================================
+\begin{page}{ugIntroCommentsPage}{1.3.9. Comments}
+% =====================================================================
+\beginscroll
+
+Comment statements begin with two consecutive hyphens or two
+consecutive plus signs and continue until the end of the line.
+
+\xtc{
+The comment beginning with {\tt --} is ignored by \Language{}.
+}{
+\spadpaste{2 + 3   -- this is rather simple, no?}
+}
+
+There is no way to write long multi-line comments
+other than starting each line with \axiomSyntax{--} or
+\axiomSyntax{++}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroGraphicsTitle}{Graphics}
+\newcommand{\ugIntroGraphicsNumber}{1.4.}
+%
+% =====================================================================
+\begin{page}{ugIntroGraphicsPage}{1.4. Graphics}
+% =====================================================================
+\beginscroll
+%
+
+\Language{} has a two- and three-dimensional drawing and rendering
+%-% \HDindex{graphics}{ugIntroGraphicsPage}{1.4.}{Graphics}
+package that allows you to draw, shade, color, rotate, translate, map,
+clip, scale and combine graphic output of \Language{} computations.
+The graphics interface is capable of plotting functions of one or more
+variables and plotting parametric surfaces.
+Once the graphics figure appears in a window,
+move your mouse to the window and click.
+A control panel appears immediately  and allows you to
+interactively transform the object.
+
+\psXtc{
+This is an example of \Language{}'s two-dimensional plotting.
+From the 2D Control Panel you can rescale the plot, turn axes and units
+on and off and save the image, among other things.
+This PostScript image was produced by clicking on the
+\texht{\fbox{\bf PS}}{{\bf PS}} 2D Control Panel button.
+}{
+\graphpaste{draw(cos(5*t/8), t=0..16*\%pi, coordinates==polar)}
+}{
+\epsffile[72 72 300 300]{../ps/rose-1.ps}
+}
+
+\psXtc{
+This is an example of \Language{}'s three-dimensional plotting.
+It is a monochrome graph of the complex arctangent
+function.
+The image displayed was rotated and had the ``shade'' and ``outline''
+display options set from the 3D Control Panel.
+The PostScript output was produced by clicking on the
+\texht{\fbox{\bf save}}{{\bf save}} 3D Control Panel button and then
+clicking on the \texht{\fbox{\bf PS}}{{\bf PS}} button.
+See \downlink{``\ugProblemNumericTitle''}{ugProblemNumericPage} in Section \ugProblemNumericNumber\ignore{ugProblemNumeric} for more details and examples
+of \Language{}'s numeric and graphics capabilities.
+}{
+\graphpaste{draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction == (x,y) +-> argument atan complex(x,y))}
+}{
+\epsffile[72 72 285 285]{../ps/atan-1.ps}
+}
+
+An exhibit of \Gallery{} is given in the
+center section of this book.
+For a description of the commands and programs that
+produced these figures, see \downlink{``\ugAppGraphicsTitle''}{ugAppGraphicsPage} in Appendix \ugAppGraphicsNumber\ignore{ugAppGraphics}.
+PostScript
+%-% \HDindex{PostScript}{ugIntroGraphicsPage}{1.4.}{Graphics}
+output is available so that \Language{} images can be
+printed.\footnote{PostScript is a trademark of Adobe Systems Incorporated,
+registered in the United States.}
+See \downlink{``\ugGraphTitle''}{ugGraphPage} in Chapter \ugGraphNumber\ignore{ugGraph} for more examples and details about using
+\Language{}'s graphics facilities.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroNumbersTitle}{Numbers}
+\newcommand{\ugIntroNumbersNumber}{1.5.}
+%
+% =====================================================================
+\begin{page}{ugIntroNumbersPage}{1.5. Numbers}
+% =====================================================================
+\beginscroll
+%
+
+\Language{} distinguishes very carefully between different kinds
+of numbers, how they are represented and what their properties
+are.
+Here are a sampling of some of these kinds of numbers and some
+things you can do with them.
+
+\xtc{
+Integer arithmetic is always exact.
+}{
+\spadpaste{11**13 * 13**11 * 17**7 - 19**5 * 23**3}
+}
+\xtc{
+Integers can be represented in factored form.
+}{
+\spadpaste{factor 643238070748569023720594412551704344145570763243 \bound{ex1}}
+}
+\xtc{
+Results stay factored when you do arithmetic.
+Note that the \axiom{12} is automatically factored for you.
+}{
+\spadpaste{\% * 12 \free{ex1}}
+}
+%-% \HDindex{radix}{ugIntroNumbersPage}{1.5.}{Numbers}
+\xtc{
+Integers can also be displayed to bases other than 10.
+This is an integer in base 11.
+}{
+\spadpaste{radix(25937424601,11)}
+}
+\xtc{
+Roman numerals are also available for those special occasions.
+%-% \HDindex{Roman numerals}{ugIntroNumbersPage}{1.5.}{Numbers}
+}{
+\spadpaste{roman(1992)}
+}
+\xtc{
+Rational number arithmetic is also exact.
+}{
+\spadpaste{r := 10 + 9/2 + 8/3 + 7/4 + 6/5 + 5/6 + 4/7 + 3/8 + 2/9\bound{r}}
+}
+\xtc{
+To factor fractions, you have to
+map \axiomFun{factor} onto the numerator and denominator.
+}{
+\spadpaste{map(factor,r) \free{r}}
+}
+\xtc{
+Type \spadtype{SingleInteger} refers to machine word-length
+integers.
+%-% \HDexptypeindex{SingleInteger}{ugIntroNumbersPage}{1.5.}{Numbers}
+In English, this expression means ``\axiom{11} as a small
+integer''.
+}{
+\spadpaste{11@SingleInteger}
+}
+\xtc{
+Machine double-precision floating-point numbers are also
+available for numeric and graphical
+applications.
+%-% \HDexptypeindex{DoubleFloat}{ugIntroNumbersPage}{1.5.}{Numbers}
+}{
+\spadpaste{123.21@DoubleFloat}
+}
+
+The normal floating-point type in \Language{}, \spadtype{Float},
+is a software implementation of floating-point numbers in which
+the exponent and the mantissa may have any number of
+digits.\footnote{See \downlink{`Float'}{FloatXmpPage}\ignore{Float} and \downlink{`DoubleFloat'}{DoubleFloatXmpPage}\ignore{DoubleFloat} for
+additional information on floating-point types.}
+The types \spadtype{Complex(Float)} and
+\spadtype{Complex(DoubleFloat)} are the corresponding software
+implementations of complex floating-point numbers.
+
+\xtc{
+This is a floating-point approximation to about twenty digits.
+%-% \HDindex{floating point}{ugIntroNumbersPage}{1.5.}{Numbers}
+The \axiomSyntax{::}
+is used here to change from one kind of object
+(here, a rational number) to another (a floating-point number).
+}{
+\spadpaste{r :: Float \free{r}}
+}
+\xtc{
+Use \spadfunFrom{digits}{Float} to change the number of digits in
+the representation.
+This operation returns the previous value so you can reset it
+later.
+}{
+\spadpaste{digits(22) \bound{fewerdigits}}
+}
+\xtc{
+To \axiom{22} digits of precision, the number
+\texht{$e^{\pi {\sqrt {163.0}}}$}{\axiom{exp(\%pi * sqrt 163.0)}}
+appears to be an integer.
+}{
+\spadpaste{exp(\%pi * sqrt 163.0) \free{fewerdigits}}
+}
+\xtc{
+Increase the precision to forty digits and try again.
+}{
+\spadpaste{digits(40);  exp(\%pi * sqrt 163.0) \free{moredigits}}
+}
+\xtc{
+Here are complex numbers with rational numbers as real and
+%-% \HDindex{complex numbers}{ugIntroNumbersPage}{1.5.}{Numbers}
+imaginary parts.
+}{
+\spadpaste{(2/3 + \%i)**3 \bound{gaussint}}
+}
+\xtc{
+The standard operations on complex numbers are available.
+}{
+\spadpaste{conjugate \% \free{gaussint}}
+}
+\xtc{
+You can factor complex integers.
+}{
+\spadpaste{factor(89 - 23 * \%i)}
+}
+\xtc{
+Complex numbers with floating point parts are also available.
+}{
+\spadpaste{exp(\%pi/4.0 * \%i)}
+}
+%%--> These are not numbers:
+%\xtc{
+%The real and imaginary parts can be symbolic.
+%}{
+%\spadcommand{complex(u,v) \bound{cuv}}
+%}
+%\xtc{
+%Of course, you can do complex arithmetic with these also.
+%See \downlink{`Complex'}{ComplexXmpPage}\ignore{Complex} for more information.
+%}{
+%\spadcommand{\% ** 2 \free{cuv}}
+%}
+\xtc{
+Every rational number has an exact representation as a
+repeating decimal expansion
+(see \downlink{`DecimalExpansion'}{DecimalExpansionXmpPage}\ignore{DecimalExpansion}).
+}{
+\spadpaste{decimal(1/352)}
+}
+\xtc{
+A rational number can also be expressed as a continued fraction (see
+%-% \HDindex{continued fraction}{ugIntroNumbersPage}{1.5.}{Numbers}
+\downlink{`ContinuedFraction'}{ContinuedFractionXmpPage}\ignore{ContinuedFraction}).
+%-% \HDindex{fraction!continued}{ugIntroNumbersPage}{1.5.}{Numbers}
+}{
+\spadpaste{continuedFraction(6543/210)}
+}
+\xtc{
+Also, partial fractions can be used and can be displayed in a
+%-% \HDindex{partial fraction}{ugIntroNumbersPage}{1.5.}{Numbers}
+compact \ldots
+%-% \HDindex{fraction!partial}{ugIntroNumbersPage}{1.5.}{Numbers}
+}{
+\spadpaste{partialFraction(1,factorial(10)) \bound{partfrac}}
+}
+\xtc{
+or expanded format (see \downlink{`PartialFraction'}{PartialFractionXmpPage}\ignore{PartialFraction}).
+}{
+\spadpaste{padicFraction(\%) \free{partfrac}}
+}
+\xtc{
+Like integers, bases (radices) other than ten can be used for rational
+numbers (see \downlink{`RadixExpansion'}{RadixExpansionXmpPage}\ignore{RadixExpansion}).
+Here we use base eight.
+}{
+\spadpaste{radix(4/7, 8)\bound{rad}}
+}
+\xtc{
+Of course, there are complex versions of these as well.
+\Language{} decides to make the result a complex rational number.
+}{
+\spadpaste{\% + 2/3*\%i\free{rad}}
+}
+\xtc{
+You can also use \Language{} to manipulate fractional powers.
+%-% \HDindex{radical}{ugIntroNumbersPage}{1.5.}{Numbers}
+}{
+\spadpaste{(5 + sqrt 63 + sqrt 847)**(1/3)}
+}
+\xtc{
+You can also compute with integers modulo a prime.
+}{
+\spadpaste{x : PrimeField 7 := 5 \bound{x}}
+}
+\xtc{
+Arithmetic is then done modulo \mathOrSpad{7}.
+}{
+\spadpaste{x**3 \free{x}}
+}
+\xtc{
+Since \mathOrSpad{7} is prime, you can invert nonzero values.
+}{
+\spadpaste{1/x \free{x}}
+}
+\xtc{
+You can also compute modulo an integer that is not a prime.
+}{
+\spadpaste{y : IntegerMod 6 := 5 \bound{y}}
+}
+\xtc{
+All of the usual arithmetic operations are available.
+}{
+\spadpaste{y**3 \free{y}}
+}
+\xtc{
+Inversion is not available if the modulus is not a prime
+number.
+Modular arithmetic and prime fields are discussed in
+\downlink{``\ugxProblemFinitePrimeTitle''}{ugxProblemFinitePrimePage} in Section \ugxProblemFinitePrimeNumber\ignore{ugxProblemFinitePrime}.
+}{
+\spadpaste{1/y \free{y}}
+}
+\xtc{
+This defines \axiom{a} to be an algebraic number, that is,
+a root of a polynomial equation.
+}{
+\spadpaste{a := rootOf(a**5 + a**3 + a**2 + 3,a) \bound{a}}
+}
+\xtc{
+Computations with \axiom{a} are reduced according
+to the polynomial equation.
+}{
+\spadpaste{(a + 1)**10\free{a}}
+}
+\xtc{
+Define \axiom{b} to be an algebraic number involving \axiom{a}.
+}{
+\spadpaste{b := rootOf(b**4 + a,b) \bound{b}\free{a}}
+}
+\xtc{
+Do some arithmetic.
+}{
+\spadpaste{2/(b - 1) \free{b}\bound{check}}
+}
+\xtc{
+To expand and simplify this, call \axiomFun{ratDenom}
+to rationalize the denominator.
+}{
+\spadpaste{ratDenom(\%) \free{check}\bound{check1}}
+}
+\xtc{
+If we do this, we should get \axiom{b}.
+}{
+\spadpaste{2/\%+1 \free{check1}\bound{check2}}
+}
+\xtc{
+But we need to rationalize the denominator again.
+}{
+\spadpaste{ratDenom(\%) \free{check2}}
+}
+\xtc{
+Types \spadtype{Quaternion} and \spadtype{Octonion} are also available.
+Multiplication of quaternions is non-commutative, as expected.
+}{
+\spadpaste{q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroCollectTitle}{Data Structures}
+\newcommand{\ugIntroCollectNumber}{1.6.}
+%
+% =====================================================================
+\begin{page}{ugIntroCollectPage}{1.6. Data Structures}
+% =====================================================================
+\beginscroll
+%
+
+\Language{} has a large variety of data structures available.
+Many data structures are particularly useful for interactive
+computation and others are useful for building applications.
+The data structures of \Language{} are organized into
+\spadglossSee{category hierarchies}{hierarchy} as shown on
+the inside back cover.
+
+A \spadgloss{list} is the most commonly used data structure in
+\Language{} for holding objects all of the same
+type.\footnote{Lists are discussed in \downlink{`List'}{ListXmpPage}\ignore{List} and in
+\downlink{``\ugLangItsTitle''}{ugLangItsPage} in Section \ugLangItsNumber\ignore{ugLangIts}.}
+The name {\it list} is short for ``linked-list of nodes.'' Each
+node consists of a value (\spadfunFrom{first}{List}) and a link
+(\spadfunFrom{rest}{List}) that
+\spadglossSee{points}{pointer} to the next node, or to a
+distinguished value denoting the empty list.
+To get to, say, the third element, \Language{} starts at the front
+of the list, then traverses across two links to the third node.
+
+\xtc{
+Write a list of elements using
+square brackets with commas separating the elements.
+}{
+\spadpaste{u := [1,-7,11] \bound{u}}
+}
+\xtc{
+This is the value at the third node.
+Alternatively, you can say \axiom{u.3}.
+}{
+\spadpaste{first rest rest u\free{u}}
+}
+
+Many operations are defined on lists, such as:
+\axiomFun{empty?}, to test that a list has no elements;
+\axiomFun{cons}\axiom{(x,l)}, to create a new list with
+\axiomFun{first} element \axiom{x} and \axiomFun{rest} \axiom{l};
+\axiomFun{reverse}, to create a new list with elements in reverse
+order; and \axiomFun{sort}, to arrange elements in order.
+
+An important point about lists is that they are ``mutable'': their
+constituent elements and links can be changed ``in place.''
+To do this, use any of the operations whose names end with the
+character \axiomSyntax{!}.
+
+\xtc{
+The operation \spadfunFromX{concat}{List}\axiom{(u,v)}
+replaces the last link of the list
+\axiom{u} to point to some other list \axiom{v}.
+Since \axiom{u} refers to the original list,
+this change is seen by \axiom{u}.
+}{
+\spadpaste{concat!(u,[9,1,3,-4]); u\free{u}\bound{u1}}
+}
+\xtc{
+A {\it cyclic list} is a list with a ``cycle'':
+%-% \HDindex{list!cyclic}{ugIntroCollectPage}{1.6.}{Data Structures}
+a link pointing back to an earlier node of the list.
+%-% \HDindex{cyclic list}{ugIntroCollectPage}{1.6.}{Data Structures}
+To create a cycle, first get a node somewhere down
+the list.
+}{
+\spadpaste{lastnode := rest(u,3)\free{u1}\bound{u2}}
+}
+\xtc{
+Use \spadfunFromX{setrest}{List} to
+change the link emanating from that node to point back to an
+earlier part of the list.
+}{
+\spadpaste{setrest!(lastnode,rest(u,2)); u\free{u2}}
+}
+
+A \spadgloss{stream}
+is a structure that (potentially) has an infinite number of
+distinct elements.\footnote{Streams are discussed in
+\downlink{`Stream'}{StreamXmpPage}\ignore{Stream} and in \downlink{``\ugLangItsTitle''}{ugLangItsPage} in Section \ugLangItsNumber\ignore{ugLangIts}.}
+Think of a stream as an ``infinite list'' where elements are
+computed successively.
+
+\xtc{
+Create an infinite stream of factored integers.
+Only a certain number of initial elements are computed
+and displayed.
+}{
+\spadpaste{[factor(i) for i in 2.. by 2] \bound{stream1}}
+}
+\xtc{
+\Language{} represents streams by a collection of already-computed
+elements together with a function to compute the next element
+``on demand.''
+Asking for the \eth{\axiom{n}} element causes elements \axiom{1} through
+\axiom{n} to be evaluated.
+}{
+\spadpaste{\%.36 \free{stream1}}
+}
+
+Streams can also be finite or cyclic.
+They are implemented by a linked list structure similar to lists
+and have many of the same operations.
+For example, \axiomFun{first} and \axiomFun{rest} are used to access
+elements and successive nodes of a stream.
+%%> reverse and sort do not exist for streams
+%%Don't try to reverse or sort a stream: the
+%%operation will generally run forever!
+
+A \spadgloss{one-dimensional array} is another data structure used
+to hold objects of the same type.\footnote{See \downlink{`OneDimensionalArray'}{OneDimensionalArrayXmpPage}\ignore{OneDimensionalArray} for
+details.}
+Unlike lists, one-dimensional arrays are inflexible---they are
+%-% \HDindex{array!one-dimensional}{ugIntroCollectPage}{1.6.}{Data Structures}
+implemented using a fixed block of storage.
+Their advantage is that they give quick and equal access time to
+any element.
+
+\xtc{
+A simple way to create a one-dimensional array is to apply the
+operation \axiomFun{oneDimensionalArray} to a list of elements.
+}{
+\spadpaste{a := oneDimensionalArray [1, -7, 3, 3/2]\bound{a}}
+}
+\xtc{
+One-dimensional arrays are also mutable:
+you can change their constituent elements ``in place.''
+}{
+\spadpaste{a.3 := 11; a\bound{a1}\free{a}}
+}
+\xtc{
+However, one-dimensional arrays are not flexible structures.
+You cannot destructively \spadfunX{concat} them together.
+}{
+\spadpaste{concat!(a,oneDimensionalArray [1,-2])\free{a1}}
+}
+
+Examples of datatypes similar to \spadtype{OneDimensionalArray}
+are: \spadtype{Vector} (vectors are mathematical structures
+implemented by one-dimensional arrays), \spadtype{String} (arrays
+of ``characters,'' represented by byte vectors), and
+\spadtype{Bits} (represented by ``bit vectors'').
+
+\xtc{
+A vector of 32 bits, each representing the \spadtype{Boolean} value \axiom{true}.
+}{
+\spadpaste{bits(32,true)}
+}
+
+A \spadgloss{flexible array} is a cross between a list
+%-% \HDindex{array!flexible}{ugIntroCollectPage}{1.6.}{Data Structures}
+and a one-dimensional array.\footnote{See \downlink{`FlexibleArray'}{FlexibleArrayXmpPage}\ignore{FlexibleArray} for
+details.}
+Like a one-dimensional array, a flexible array occupies a fixed
+block of storage.
+Its block of storage, however, has room to expand!
+When it gets full, it grows (a new, larger block of storage is
+allocated); when it has too much room, it contracts.
+
+\xtc{
+Create a flexible array of three elements.
+}{
+\spadpaste{f := flexibleArray [2, 7, -5]\bound{f}}
+}
+\xtc{
+Insert some elements between the second and third elements.
+}{
+\spadpaste{insert!(flexibleArray [11, -3],f,2)\free{f}}
+}
+
+Flexible arrays are used to implement ``heaps.'' A
+\spadgloss{heap} is an example of a data structure called a
+\spadgloss{priority queue}, where elements are ordered with
+respect to one another.\footnote{See \downlink{`Heap'}{HeapXmpPage}\ignore{Heap} for more details.
+Heaps are also examples of data structures called
+\spadglossSee{bags}{bag}.
+Other bag data structures are \spadtype{Stack}, \spadtype{Queue},
+and \spadtype{Dequeue}.}
+A heap is organized
+so as to optimize insertion and extraction of maximum elements.
+The \spadfunX{extract} operation
+returns the maximum element of the heap, after destructively
+removing that element and
+reorganizing the heap
+so that the next maximum element is ready to be delivered.
+
+\xtc{
+An easy way to create a heap is to apply the
+operation \spadfun{heap} to a list of values.
+}{
+\spadpaste{h := heap [-4,7,11,3,4,-7]\bound{h}}
+}
+\xtc{
+This loop extracts elements one-at-a-time from \spad{h}
+until the heap is exhausted, returning the elements
+as a list in the order they were extracted.
+}{
+\spadpaste{[extract!(h) while not empty?(h)]\free{h}}
+}
+
+A \spadgloss{binary tree} is a ``tree'' with at most two branches
+%-% \HDindex{tree}{ugIntroCollectPage}{1.6.}{Data Structures}
+per node: it is either empty, or else is a node consisting of a
+value, and a left and right subtree (again, binary trees).\footnote{Example of binary tree types are
+\spadtype{BinarySearchTree} (see \downlink{`BinarySearchTree'}{BinarySearchTreeXmpPage}\ignore{BinarySearchTree},
+\spadtype{PendantTree}, \spadtype{TournamentTree},
+and \spadtype{BalancedBinaryTree} (see \downlink{`BalancedBinaryTree'}{BalancedBinaryTreeXmpPage}\ignore{BalancedBinaryTree}).}
+
+\xtc{
+A {\it binary search tree} is a binary tree such that,
+%-% \HDindex{tree!binary search}{ugIntroCollectPage}{1.6.}{Data Structures}
+for each node, the value of the node is
+%-% \HDindex{binary search tree}{ugIntroCollectPage}{1.6.}{Data Structures}
+greater than all values (if any) in the left subtree,
+and less than or equal all values (if any) in the right subtree.
+}{
+\spadpaste{binarySearchTree [5,3,2,9,4,7,11]}
+}
+
+\xtc{
+A {\it balanced binary tree} is useful for doing modular computations.
+%-% \HDindex{balanced binary tree}{ugIntroCollectPage}{1.6.}{Data Structures}
+Given a list \axiom{lm} of moduli,
+%-% \HDindex{tree!balanced binary}{ugIntroCollectPage}{1.6.}{Data Structures}
+\axiomFun{modTree}\axiom{(a,lm)} produces a balanced binary
+tree with the values \texht{$a \bmod m$}{a {\tt mod} m}
+at its leaves.
+}{
+\spadpaste{modTree(8,[2,3,5,7])}
+}
+
+A \spadgloss{set} is a collection of elements where duplication
+and order is irrelevant.\footnote{See \downlink{`Set'}{SetXmpPage}\ignore{Set} for more
+details.}
+Sets are always finite and have no corresponding
+structure like streams for infinite collections.
+
+\xtc{
+%Create sets using braces (\axiomSyntax{\{} and \axiomSyntax{\}})
+%rather than brackets.
+}{
+\spadpaste{fs := set[1/3,4/5,-1/3,4/5] \bound{fs}}
+}
+
+A \spadgloss{multiset}
+is a set that keeps track of the number
+of duplicate values.\footnote{See \downlink{`MultiSet'}{MultiSetXmpPage}\ignore{MultiSet} for details.}
+\xtc{
+For all the primes \axiom{p} between 2 and 1000, find the
+distribution of \texht{$p \bmod 5$}{p mod 5}.
+}{
+\spadpaste{multiset [x rem 5 for x in primes(2,1000)]}
+}
+
+A \spadgloss{table}
+is conceptually a set of ``key--value'' pairs and
+is a generalization of a multiset.\footnote{For examples of tables, see
+\spadtype{AssociationList} (\downlink{`AssociationList'}{AssociationListXmpPage}\ignore{AssociationList}),
+\spadtype{HashTable},
+\spadtype{KeyedAccessFile} (\downlink{`KeyedAccessFile'}{KeyedAccessFileXmpPage}\ignore{KeyedAccessFile}),
+\spadtype{Library} (\downlink{`Library'}{LibraryXmpPage}\ignore{Library}),
+\spadtype{SparseTable} (\downlink{`SparseTable'}{SparseTableXmpPage}\ignore{SparseTable}),
+\spadtype{StringTable} (\downlink{`StringTable'}{StringTableXmpPage}\ignore{StringTable}),
+and \spadtype{Table} (\downlink{`Table'}{TableXmpPage}\ignore{Table}).}
+The domain \spadtype{Table(Key, Entry)} provides a general-purpose
+type for tables with {\it values} of type \axiom{Entry} indexed
+by {\it keys} of type \axiom{Key}.
+
+\xtc{
+Compute the above distribution of primes using tables.
+First, let \axiom{t} denote an empty table of keys and values,
+each of type \spadtype{Integer}.
+}{
+\spadpaste{t : Table(Integer,Integer) := empty()\bound{t}}
+}
+
+We define a function \userfun{howMany} to return the number
+of values of a given modulus \axiom{k} seen so far.
+It calls \axiomFun{search}\axiom{(k,t)} which returns the number of
+values stored under the key \axiom{k} in table \axiom{t}, or
+\axiom{"failed"} if no such value is yet stored in \axiom{t} under
+\axiom{k}.
+
+\xtc{
+In English, this says ``Define \axiom{howMany(k)} as follows.
+First, let \smath{n} be the value of \axiomFun{search}\smath{(k,t)}.
+Then, if \smath{n} has the value \smath{"failed"}, return the value
+\smath{1}; otherwise return \smath{n + 1}.''
+}{
+\spadpaste{howMany(k) == (n:=search(k,t); n case "failed" => 1; n+1)\bound{how}}
+}
+\xtc{
+Run through the primes to create the table, then print the table.
+The expression \axiom{t.m := howMany(m)} updates the value in table \axiom{t}
+stored under key \axiom{m}.
+}{
+\spadpaste{for p in primes(2,1000) repeat (m:= p rem 5; t.m:= howMany(m)); t\free{how t}}
+}
+
+A {\it record}
+is an example of an inhomogeneous collection
+of objects.\footnote{See \downlink{``\ugTypesRecordsTitle''}{ugTypesRecordsPage} in Section \ugTypesRecordsNumber\ignore{ugTypesRecords} for details.}
+A record consists of a set of named {\it selectors} that
+can be used to access its components.
+%-% \HDindex{Record@{\sf Record}}{ugIntroCollectPage}{1.6.}{Data Structures}
+
+\xtc{
+Declare that \axiom{daniel} can only be
+assigned a record with two prescribed fields.
+}{
+\spadpaste{daniel : Record(age : Integer, salary : Float) \bound{danieldec}}
+}
+\xtc{
+Give \axiom{daniel} a value, using square brackets to enclose the values of
+the fields.
+}{
+\spadpaste{daniel := [28, 32005.12] \free{danieldec}\bound{daniel}}
+}
+\xtc{
+Give \axiom{daniel} a raise.
+}{
+\spadpaste{daniel.salary := 35000; daniel \free{daniel}}
+}
+
+A {\it union}
+is a data structure used when objects
+have multiple types.\footnote{See \downlink{``\ugTypesUnionsTitle''}{ugTypesUnionsPage} in Section \ugTypesUnionsNumber\ignore{ugTypesUnions} for details.}
+%-% \HDindex{Union@{\sf Union}}{ugIntroCollectPage}{1.6.}{Data Structures}
+
+\xtc{
+Let \axiom{dog} be either an integer or a string value.
+}{
+\spadpaste{dog: Union(licenseNumber: Integer, name: String)\bound{xint}}
+}
+\xtc{
+Give \axiom{dog} a name.
+}{
+\spadpaste{dog := "Whisper"\free{xint}}
+}
+
+All told, there are over forty different data structures in
+\Language{}.
+Using the domain constructors described in \downlink{``\ugDomainsTitle''}{ugDomainsPage} in Chapter \ugDomainsNumber\ignore{ugDomains}, you
+can add your own data structure or extend an existing one.
+Choosing the right data structure for your application may be the key
+to obtaining good performance.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroTwoDimTitle}{Expanding to Higher Dimensions}
+\newcommand{\ugIntroTwoDimNumber}{1.7.}
+%
+% =====================================================================
+\begin{page}{ugIntroTwoDimPage}{1.7. Expanding to Higher Dimensions}
+% =====================================================================
+\beginscroll
+%
+
+To get higher dimensional aggregates, you can create one-dimensional
+aggregates with elements that are themselves
+aggregates, for example, lists of lists, one-dimensional arrays of
+lists of multisets, and so on.
+For applications requiring two-dimensional homogeneous aggregates,
+you will likely find {\it two-dimensional arrays}
+%-% \HDindex{matrix}{ugIntroTwoDimPage}{1.7.}{Expanding to Higher Dimensions}
+and {\it matrices} most useful.
+%-% \HDindex{array!two-dimensional}{ugIntroTwoDimPage}{1.7.}{Expanding to Higher Dimensions}
+
+The entries in \spadtype{TwoDimensionalArray} and
+\spadtype{Matrix} objects
+are all the same type, except that those for
+\spadtype{Matrix} must belong to a \spadtype{Ring}.
+You create and access elements in roughly the same way.
+Since matrices have an understood algebraic structure, certain algebraic
+operations are available for matrices but not for arrays.
+Because of this, we limit our discussion here to
+\spadtype{Matrix}, that can be regarded as an extension of
+\spadtype{TwoDimensionalArray}.\footnote{See
+\downlink{`TwoDimensionalArray'}{TwoDimensionalArrayXmpPage}\ignore{TwoDimensionalArray} for more information about arrays.
+For more information about \Language{}'s linear algebra
+facilities, see \downlink{`Matrix'}{MatrixXmpPage}\ignore{Matrix}, \downlink{`Permanent'}{PermanentXmpPage}\ignore{Permanent},
+\downlink{`SquareMatrix'}{SquareMatrixXmpPage}\ignore{SquareMatrix}, \downlink{`Vector'}{VectorXmpPage}\ignore{Vector},
+\downlink{``\ugProblemEigenTitle''}{ugProblemEigenPage} in Section \ugProblemEigenNumber\ignore{ugProblemEigen}\texht{(computation of eigenvalues and
+eigenvectors)}{}, and
+\downlink{``\ugProblemLinPolEqnTitle''}{ugProblemLinPolEqnPage} in Section \ugProblemLinPolEqnNumber\ignore{ugProblemLinPolEqn}\texht{(solution of linear and
+polynomial equations)}{}.}
+
+\xtc{
+You can create a matrix from a list of lists,
+%-% \HDindex{matrix!creating}{ugIntroTwoDimPage}{1.7.}{Expanding to Higher Dimensions}
+where each of the inner lists represents a row of the matrix.
+}{
+\spadpaste{m := matrix([[1,2], [3,4]]) \bound{m}}
+}
+\xtc{
+The ``collections'' construct (see \downlink{``\ugLangItsTitle''}{ugLangItsPage} in Section \ugLangItsNumber\ignore{ugLangIts}) is
+useful for creating matrices whose entries are given by formulas.
+%-% \HDindex{matrix!Hilbert}{ugIntroTwoDimPage}{1.7.}{Expanding to Higher Dimensions}
+}{
+\spadpaste{matrix([[1/(i + j - x) for i in 1..4] for j in 1..4]) \bound{hilb}}
+}
+\xtc{
+Let \axiom{vm} denote the three by three Vandermonde matrix.
+}{
+\spadpaste{vm := matrix [[1,1,1], [x,y,z], [x*x,y*y,z*z]] \bound{vm}}
+}
+\xtc{
+Use this syntax to extract an entry in the matrix.
+}{
+\spadpaste{vm(3,3) \free{vm}}
+}
+\xtc{
+You can also pull out a \axiomFun{row} or a \axiom{column}.
+}{
+\spadpaste{column(vm,2) \free{vm}}
+}
+\xtc{
+You can do arithmetic.
+}{
+\spadpaste{vm * vm \free{vm}}
+}
+\xtc{
+You can perform operations such as
+\axiomFun{transpose}, \axiomFun{trace}, and \axiomFun{determinant}.
+}{
+\spadpaste{factor determinant vm \free{vm}\bound{d}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroYouTitle}{Writing Your Own Functions}
+\newcommand{\ugIntroYouNumber}{1.8.}
+%
+% =====================================================================
+\begin{page}{ugIntroYouPage}{1.8. Writing Your Own Functions}
+% =====================================================================
+\beginscroll
+%
+
+\Language{} provides you with a very large library of predefined
+operations and objects to compute with.
+You can use the \Language{} library of constructors to create new
+objects dynamically of quite arbitrary complexity.
+For example, you can make lists of matrices of fractions of
+polynomials with complex floating point numbers as coefficients.
+Moreover, the library provides a wealth of operations that allow
+you to create and manipulate these objects.
+
+For many applications,
+you need to interact with the interpreter and write some
+\Language{} programs to tackle your application.
+\Language{} allows you to write functions interactively,
+%-% \HDindex{function}{ugIntroYouPage}{1.8.}{Writing Your Own Functions}
+thereby effectively extending the system library.
+Here we give a few simple examples, leaving the details to \downlink{``\ugUserTitle''}{ugUserPage} in Chapter \ugUserNumber\ignore{ugUser}.
+
+We begin by looking at several ways that you can define the
+``factorial'' function in \Language{}.
+The first way is to give a
+%-% \HDindex{function!piece-wise definition}{ugIntroYouPage}{1.8.}{Writing Your Own Functions}
+piece-wise definition of the function.
+%-% \HDindex{piece-wise function definition}{ugIntroYouPage}{1.8.}{Writing Your Own Functions}
+This method is best for a general recurrence
+relation since the pieces are gathered together and compiled into
+an efficient iterative function.
+Furthermore, enough previously computed values are automatically
+saved so that a subsequent call to the function can pick up from
+where it left off.
+
+\xtc{
+Define the value of \userfun{fact} at \axiom{0}.
+}{
+\spadpaste{fact(0) == 1 \bound{fact}}
+}
+\xtc{
+Define the value of \axiom{fact(n)} for general \axiom{n}.
+}{
+\spadpaste{fact(n) == n*fact(n-1)\bound{facta}\free{fact}}
+}
+\xtc{
+Ask for the value at \axiom{50}.
+The resulting function created by \Language{}
+computes the value by iteration.
+}{
+\spadpaste{fact(50) \free{facta}}
+}
+\xtc{
+A second definition uses an \axiom{if-then-else} and recursion.
+}{
+\spadpaste{fac(n) == if n < 3 then n else n * fac(n - 1) \bound{fac}}
+}
+\xtc{
+This function is less efficient than the previous version since
+each iteration involves a recursive function call.
+}{
+\spadpaste{fac(50) \free{fac}}
+}
+\xtc{
+A third version directly uses iteration.
+}{
+\spadpaste{fa(n) == (a := 1; for i in 2..n repeat a := a*i; a) \bound{fa}}
+}
+\xtc{
+This is the least space-consumptive version.
+}{
+\spadpaste{fa(50) \free{fa}}
+}
+\xtc{
+A final version appears to construct a large list and then reduces over
+it with multiplication.
+}{
+\spadpaste{f(n) == reduce(*,[i for i in 2..n]) \bound{f}}
+}
+\xtc{
+In fact, the resulting computation is optimized into an efficient
+iteration loop equivalent to that of the third version.
+}{
+\spadpaste{f(50) \free{f}}
+}
+\xtc{
+The library version uses an algorithm that is different from the four
+above because it highly optimizes the recurrence relation definition of
+\axiomFun{factorial}.
+}{
+\spadpaste{factorial(50)}
+}
+
+You are not limited to one-line functions in \Language{}.
+If you place your function definitions in {\bf .input} files
+%-% \HDindex{file!input}{ugIntroYouPage}{1.8.}{Writing Your Own Functions}
+(see \downlink{``\ugInOutInTitle''}{ugInOutInPage} in Section \ugInOutInNumber\ignore{ugInOutIn}), you can have
+multi-line functions that use indentation for grouping.
+
+Given \axiom{n} elements, \axiomFun{diagonalMatrix} creates an
+\axiom{n} by \axiom{n} matrix with those elements down the diagonal.
+This function uses a permutation matrix
+that interchanges the \axiom{i}th and \axiom{j}th rows of a matrix
+by which it is right-multiplied.
+
+\xtc{
+This function definition shows a style of definition that can be used
+in {\bf .input} files.
+Indentation is used to create \spadglossSee{blocks}{block}\texht{\/}{}:
+sequences of expressions that are evaluated in sequence except as
+modified by control statements such as \axiom{if-then-else} and \axiom{return}.
+}{
+\begin{spadsrc}[\bound{permMat}]
+permMat(n, i, j) ==
+  m := diagonalMatrix
+    [(if i = k or j = k then 0 else 1)
+      for k in 1..n]
+  m(i,j) := 1
+  m(j,i) := 1
+  m
+\end{spadsrc}
+}
+\xtc{
+This creates a four by four matrix that interchanges the second and third
+rows.
+}{
+\spadpaste{p := permMat(4,2,3) \free{permMat}\bound{p}}
+}
+\xtc{
+Create an example matrix to permute.
+}{
+\spadpaste{m := matrix [[4*i + j for j in 1..4] for i in 0..3]\bound{m}}
+}
+\xtc{
+Interchange the second and third rows of m.
+}{
+\spadpaste{permMat(4,2,3) * m \free{p m}}
+}
+
+A function can also be passed as an argument to another function,
+which then applies the function or passes it off to some other
+function that does.
+You often have to declare the type of a function that has
+functional arguments.
+
+\xtc{
+This declares \userfun{t} to be a two-argument function that
+returns a \spadtype{Float}.
+The first argument is a function that takes one \spadtype{Float}
+argument and returns a \spadtype{Float}.
+}{
+\spadpaste{t : (Float -> Float, Float) -> Float \bound{tdecl}}
+}
+\xtc{
+This is the definition of \userfun{t}.
+}{
+\spadpaste{t(fun, x) == fun(x)**2 + sin(x)**2 \free{tdecl}\bound{t}}
+}
+\xtc{
+We have not defined a \axiomFun{cos} in the workspace. The one from the
+\Language{} library will do.
+}{
+\spadpaste{t(cos, 5.2058) \free{t}}
+}
+\xtc{
+Here we define our own (user-defined) function.
+}{
+\spadpaste{cosinv(y) == cos(1/y) \bound{cosinv}}
+}
+\xtc{
+Pass this function as an argument to \userfun{t}.
+}{
+\spadpaste{t(cosinv, 5.2058) \free{t}\free{cosinv}}
+}
+
+\Language{} also has pattern matching capabilities for
+%-% \HDindex{simplification}{ugIntroYouPage}{1.8.}{Writing Your Own Functions}
+simplification
+%-% \HDindex{pattern matching}{ugIntroYouPage}{1.8.}{Writing Your Own Functions}
+of expressions and for defining new functions by rules.
+For example, suppose that you want to apply regularly a transformation
+that groups together products of radicals:
+\texht{$$\sqrt{a}\:\sqrt{b} \mapsto \sqrt{ab}, \quad
+(\forall a)(\forall b)$$}{\axiom{sqrt(a) * sqrt(b) by sqrt(a*b)} for any \axiom{a} and \axiom{b}}
+Note that such a transformation is not generally correct.
+\Language{} never uses it automatically.
+
+\xtc{
+Give this rule the name \userfun{groupSqrt}.
+}{
+\spadpaste{groupSqrt := rule(sqrt(a) * sqrt(b) == sqrt(a*b)) \bound{g}}
+}
+\xtc{
+Here is a test expression.
+}{
+\spadpaste{a := (sqrt(x) + sqrt(y) + sqrt(z))**4 \bound{sxy}}
+}
+\xtc{
+The rule
+\userfun{groupSqrt} successfully simplifies the expression.
+}{
+\spadpaste{groupSqrt a \free{sxy} \free{g}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroVariablesTitle}{Polynomials}
+\newcommand{\ugIntroVariablesNumber}{1.9.}
+%
+% =====================================================================
+\begin{page}{ugIntroVariablesPage}{1.9. Polynomials}
+% =====================================================================
+\beginscroll
+%
+
+Polynomials are the commonly used algebraic types in symbolic
+computation.
+%-% \HDindex{polynomial}{ugIntroVariablesPage}{1.9.}{Polynomials}
+Interactive users of \Language{} generally only see one type
+of polynomial: \spadtype{Polynomial(R)}.
+This type represents polynomials in any number of unspecified
+variables over a particular coefficient domain \axiom{R}.
+This type represents its coefficients
+\spadglossSee{sparsely}{sparse}: only terms with non-zero
+coefficients are represented.
+%-% \HDexptypeindex{Polynomial}{ugIntroVariablesPage}{1.9.}{Polynomials}
+
+In building applications, many other kinds of polynomial
+representations are useful.
+Polynomials may have one variable or multiple variables, the
+variables can be named or unnamed, the coefficients can be stored
+sparsely or densely.
+So-called ``distributed multivariate polynomials'' store
+polynomials as coefficients paired with vectors of exponents.
+This type is particularly efficient for use in algorithms for
+solving systems of non-linear polynomial equations.
+
+\xtc{
+The polynomial constructor most familiar to the interactive user
+is \spadtype{Polynomial}.
+}{
+\spadpaste{(x**2 - x*y**3 +3*y)**2}
+}
+\xtc{
+If you wish to restrict the variables used,
+\spadtype{UnivariatePolynomial}
+provides polynomials in one variable.
+%-% \HDexptypeindex{UnivariatePolynomial}{ugIntroVariablesPage}{1.9.}{Polynomials}
+}{
+\spadpaste{p: UP(x,INT) := (3*x-1)**2 * (2*x + 8)}
+}
+\xtc{
+The constructor
+\spadtype{MultivariatePolynomial} provides polynomials in one or more
+specified variables.
+%-% \HDexptypeindex{MultivariatePolynomial}{ugIntroVariablesPage}{1.9.}{Polynomials}
+}{
+\spadpaste{m: MPOLY([x,y],INT) := (x**2-x*y**3+3*y)**2 \bound{m}}
+}
+\xtc{
+You can change the way the polynomial appears by modifying the variable
+ordering in the explicit list.
+}{
+\spadpaste{m :: MPOLY([y,x],INT) \free{m}}
+}
+\xtc{
+The constructor
+\spadtype{DistributedMultivariatePolynomial} provides
+polynomials in one or more specified variables with the monomials
+ordered lexicographically.
+%-% \HDexptypeindex{DistributedMultivariatePolynomial}{ugIntroVariablesPage}{1.9.}{Polynomials}
+}{
+\spadpaste{m :: DMP([y,x],INT) \free{m}}
+}
+\xtc{
+The constructor
+\spadtype{HomogeneousDistributedMultivariatePolynomial} is similar except that
+the monomials are ordered by total order refined by reverse
+lexicographic order.
+%-% \HDexptypeindex{HomogeneousDistributedMultivariatePolynomial}{ugIntroVariablesPage}{1.9.}{Polynomials}
+}{
+\spadpaste{m :: HDMP([y,x],INT) \free{m}}
+}
+
+More generally, the domain constructor
+\spadtype{GeneralDistributedMultivariatePolynomial} allows the
+user to provide an arbitrary predicate to define his own term ordering.
+%-% \HDexptypeindex{GeneralDistributedMultivariatePolynomial}{ugIntroVariablesPage}{1.9.}{Polynomials}
+These last three constructors are typically used in
+\texht{Gr\"{o}bner}{Groebner} basis
+%-% \HDindex{Groebner basis@{Gr\protect\"{o}bner basis}}{ugIntroVariablesPage}{1.9.}{Polynomials}
+applications and when a flat (that is, non-recursive) display is
+wanted and the term ordering is critical for controlling the computation.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroCalcLimitsTitle}{Limits}
+\newcommand{\ugIntroCalcLimitsNumber}{1.10.}
+%
+% =====================================================================
+\begin{page}{ugIntroCalcLimitsPage}{1.10. Limits}
+% =====================================================================
+\beginscroll
+%
+
+\Language{}'s \axiomFun{limit} function is usually used to
+evaluate limits of quotients where the numerator and denominator
+%-% \HDindex{limit}{ugIntroCalcLimitsPage}{1.10.}{Limits}
+both tend to zero or both tend to infinity.
+To find the limit of an expression \axiom{f} as a real variable
+\axiom{x} tends to a limit value \axiom{a}, enter \axiom{limit(f, x=a)}.
+Use \axiomFun{complexLimit} if the variable is complex.
+Additional information and examples of limits are in
+\downlink{``\ugProblemLimitsTitle''}{ugProblemLimitsPage} in Section \ugProblemLimitsNumber\ignore{ugProblemLimits}.
+
+\xtc{
+You can take limits of functions with parameters.
+%-% \HDindex{limit!of function with parameters}{ugIntroCalcLimitsPage}{1.10.}{Limits}
+}{
+\spadpaste{g := csc(a*x) / csch(b*x) \bound{g}}
+}
+\xtc{
+As you can see, the limit is expressed in terms of the parameters.
+}{
+\spadpaste{limit(g,x=0) \free{g}}
+}
+%
+\xtc{
+A variable may also approach plus or minus infinity:
+}{
+\spadpaste{h := (1 + k/x)**x \bound{h}}
+}
+\xtc{
+\texht{Use \axiom{\%plusInfinity} and \axiom{\%minusInfinity} to
+denote $\infty$ and $-\infty$.}{}
+}{
+\spadpaste{limit(h,x=\%plusInfinity) \free{h}}
+}
+\xtc{
+A function can be defined on both sides of a particular value, but
+may tend to different limits as its variable approaches that value from the
+left and from the right.
+}{
+\spadpaste{limit(sqrt(y**2)/y,y = 0)}
+}
+\xtc{
+As \axiom{x} approaches \axiom{0} along the real axis, \axiom{exp(-1/x**2)}
+tends to \axiom{0}.
+}{
+\spadpaste{limit(exp(-1/x**2),x = 0)}
+}
+\xtc{
+However, if \axiom{x} is allowed to approach \axiom{0} along any path in the
+complex plane, the limiting value of \axiom{exp(-1/x**2)} depends on the
+path taken because the function has an essential singularity at \axiom{x=0}.
+This is reflected in the error message returned by the function.
+}{
+\spadpaste{complexLimit(exp(-1/x**2),x = 0)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroSeriesTitle}{Series}
+\newcommand{\ugIntroSeriesNumber}{1.11.}
+%
+% =====================================================================
+\begin{page}{ugIntroSeriesPage}{1.11. Series}
+% =====================================================================
+\beginscroll
+%
+
+\Language{} also provides power series.
+%-% \HDindex{series!power}{ugIntroSeriesPage}{1.11.}{Series}
+By default, \Language{} tries to compute and display the first ten elements
+of a series.
+Use \spadsys{)set streams calculate} to change the default value
+to something else.
+%-% \HDsyscmdindex{set streams calculate}{ugIntroSeriesPage}{1.11.}{Series}
+For the purposes of this book, we have used this system command to display
+fewer than ten terms.
+For more information about working with series, see
+\downlink{``\ugProblemSeriesTitle''}{ugProblemSeriesPage} in Section \ugProblemSeriesNumber\ignore{ugProblemSeries}.
+
+\xtc{
+You can convert a functional expression to a power series by using the
+operation \axiomFun{series}.
+In this example,
+\axiom{sin(a*x)} is expanded in powers of \axiom{(x - 0)},
+that is, in powers of \axiom{x}.
+}{
+\spadpaste{series(sin(a*x),x = 0)}
+}
+\xtc{
+This expression expands
+\axiom{sin(a*x)} in powers of \axiom{(x - \%pi/4)}.
+}{
+\spadpaste{series(sin(a*x),x = \%pi/4)}
+}
+\xtc{
+\Language{} provides
+%-% \HDindex{series!Puiseux}{ugIntroSeriesPage}{1.11.}{Series}
+{\it Puiseux series:}
+%-% \HDindex{Puiseux series}{ugIntroSeriesPage}{1.11.}{Series}
+series with rational number exponents.
+The first argument to \axiomFun{series} is an in-place function that
+computes the \eth{\axiom{n}} coefficient.
+(Recall that
+the \axiomSyntax{+->} is an infix operator meaning ``maps to.'')
+}{
+\spadpaste{series(n +-> (-1)**((3*n - 4)/6)/factorial(n - 1/3),x = 0,4/3..,2)}
+}
+\xtc{
+Once you have created a power series, you can perform arithmetic operations
+on that series.
+We compute the Taylor expansion of \axiom{1/(1-x)}.
+%-% \HDindex{series!Taylor}{ugIntroSeriesPage}{1.11.}{Series}
+}{
+\spadpaste{f := series(1/(1-x),x = 0) \bound{f}}
+}
+\xtc{
+Compute the square of the series.
+}{
+\spadpaste{f ** 2 \free{f}}
+}
+\xtc{
+The usual elementary functions
+(\axiomFun{log}, \axiomFun{exp}, trigonometric functions, and so on)
+are defined for power series.
+}{
+\spadpaste{f := series(1/(1-x),x = 0) \bound{f1}}
+}
+\xtc{
+}{
+\spadpaste{g := log(f) \free{f1}\bound{g}}
+}
+\xtc{
+}{
+\spadpaste{exp(g) \free{g}}
+}
+\xtc{
+Here is a way to obtain numerical approximations of
+\axiom{e} from the Taylor series expansion of \axiom{exp(x)}.
+First create the desired Taylor expansion.
+}{
+\spadpaste{f := taylor(exp(x)) \bound{f2}}
+}
+\xtc{
+Evaluate the series at the value \axiom{1.0}.
+As you see, you get a sequence of partial sums.
+}{
+\spadpaste{eval(f,1.0) \free{f2}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroCalcDerivTitle}{Derivatives}
+\newcommand{\ugIntroCalcDerivNumber}{1.12.}
+%
+% =====================================================================
+\begin{page}{ugIntroCalcDerivPage}{1.12. Derivatives}
+% =====================================================================
+\beginscroll
+%
+Use the \Language{} function \axiomFun{D} to differentiate an
+%-% \HDindex{derivative}{ugIntroCalcDerivPage}{1.12.}{Derivatives}
+expression.
+%-% \HDindex{differentiation}{ugIntroCalcDerivPage}{1.12.}{Derivatives}
+
+\texht{\vskip 2pc}{}
+\xtc{
+To find the derivative of an expression \axiom{f} with respect to a
+variable \axiom{x}, enter \axiom{D(f, x)}.
+}{
+\spadpaste{f := exp exp x \bound{f}}
+}
+\xtc{
+}{
+\spadpaste{D(f, x) \free{f}}
+}
+\xtc{
+An optional third argument \axiom{n} in \axiomFun{D} asks
+\Language{} for the \eth{\axiom{n}} derivative of \axiom{f}.
+This finds the fourth derivative of \axiom{f} with respect to \axiom{x}.
+}{
+\spadpaste{D(f, x, 4) \free{f}}
+}
+\xtc{
+You can also compute partial derivatives by specifying the order of
+%-% \HDindex{differentiation!partial}{ugIntroCalcDerivPage}{1.12.}{Derivatives}
+differentiation.
+}{
+\spadpaste{g := sin(x**2 + y) \bound{g}}
+}
+\xtc{
+}{
+\spadpaste{D(g, y) \free{g}}
+}
+\xtc{
+}{
+\spadpaste{D(g, [y, y, x, x]) \free{g}}
+}
+
+\Language{} can manipulate the derivatives (partial and iterated) of
+%-% \HDindex{differentiation!formal}{ugIntroCalcDerivPage}{1.12.}{Derivatives}
+expressions involving formal operators.
+All the dependencies must be explicit.
+\xtc{
+This returns \axiom{0} since \axiom{F} (so far)
+does not explicitly depend on \axiom{x}.
+}{
+\spadpaste{D(F,x)}
+}
+Suppose that we have \axiom{F} a function of \axiom{x},
+\axiom{y}, and \axiom{z}, where \axiom{x} and \axiom{y} are themselves
+functions of \axiom{z}.
+\xtc{
+Start by declaring that \axiom{F}, \axiom{x}, and \axiom{y}
+are operators.
+%-% \HDindex{operator}{ugIntroCalcDerivPage}{1.12.}{Derivatives}
+}{
+\spadpaste{F := operator 'F; x := operator 'x; y := operator 'y\bound{F x y}}
+}
+\xtc{
+You can use \axiom{F}, \axiom{x}, and \axiom{y} in expressions.
+}{
+\spadpaste{a := F(x z, y z, z**2) + x y(z+1) \bound{a}\free{F}\free{x}\free{y}}
+}
+\xtc{
+Differentiate formally with respect to \axiom{z}.
+The formal derivatives appearing in \axiom{dadz} are not just formal symbols,
+but do represent the derivatives of \axiom{x}, \axiom{y}, and \axiom{F}.
+}{
+\spadpaste{dadz := D(a, z)\bound{da}\free{a}}
+}
+\xtc{
+You can evaluate the above for particular functional
+values of \axiom{F}, \axiom{x}, and \axiom{y}.
+If \axiom{x(z)} is \axiom{exp(z)} and \axiom{y(z)} is \axiom{log(z+1)}, then
+this evaluates \axiom{dadz}.
+}{
+\spadpaste{eval(eval(dadz, 'x, z +-> exp z), 'y, z +-> log(z+1))\free{da}}
+}
+\xtc{
+You obtain the same result by first evaluating \axiom{a} and
+then differentiating.
+}{
+\spadpaste{eval(eval(a, 'x, z +-> exp z), 'y, z +-> log(z+1)) \free{a}\bound{eva}}
+}
+\xtc{
+}{
+\spadpaste{D(\%, z)\free{eva}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroIntegrateTitle}{Integration}
+\newcommand{\ugIntroIntegrateNumber}{1.13.}
+%
+% =====================================================================
+\begin{page}{ugIntroIntegratePage}{1.13. Integration}
+% =====================================================================
+\beginscroll
+%
+
+\Language{} has extensive library facilities for integration.
+%-% \HDindex{integration}{ugIntroIntegratePage}{1.13.}{Integration}
+
+The first example is the integration of a fraction with
+denominator that factors into a quadratic and a quartic
+irreducible polynomial.
+The usual partial fraction approach used by most other computer
+algebra systems either fails or introduces expensive unneeded
+algebraic numbers.
+
+\xtc{
+We use a factorization-free algorithm.
+}{
+\spadpaste{integrate((x**2+2*x+1)/((x+1)**6+1),x)}
+}
+
+When real parameters are present, the form of the integral can depend on
+the signs of some expressions.
+
+\xtc{
+Rather than query the user or make sign assumptions, \Language{} returns
+all possible answers.
+}{
+\spadpaste{integrate(1/(x**2 + a),x)}
+}
+
+The \axiomFun{integrate} operation generally assumes that all
+parameters are real.
+The only exception is when the integrand has complex valued
+quantities.
+
+\xtc{
+If the parameter is complex instead of real, then the notion of sign is
+undefined and there is a unique answer.
+You can request this answer by ``prepending'' the word ``complex'' to the
+command name:
+}{
+\spadpaste{complexIntegrate(1/(x**2 + a),x)}
+}
+
+The following two examples illustrate the limitations of
+table-based approaches.
+The two integrands are very similar, but the answer to one of them
+requires the addition of two new algebraic numbers.
+
+\xtc{
+This one is the easy one.
+The next one looks very similar
+but the answer is much more complicated.
+}{
+\spadpaste{integrate(x**3 / (a+b*x)**(1/3),x)}
+}
+\xtc{
+Only an algorithmic approach
+is guaranteed to find what new constants must be added in order to
+find a solution.
+}{
+\spadpaste{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)}
+}
+
+Some computer algebra systems use heuristics or table-driven
+approaches to integration.
+When these systems cannot determine the answer to an integration
+problem, they reply ``I don't know.'' \Language{} uses a
+algorithm for integration.
+that conclusively proves that an integral cannot be expressed in
+terms of elementary functions.
+
+\xtc{
+When \Language{} returns an integral sign, it has proved
+that no answer exists as an elementary function.
+}{
+\spadpaste{integrate(log(1 + sqrt(a*x + b)) / x,x)}
+}
+\Language{} can handle complicated mixed functions much beyond what you
+can find in tables.
+\xtc{
+Whenever possible, \Language{} tries to express the answer using the functions
+present in the integrand.
+}{
+\spadpaste{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b)) / (sqrt(x+b) * (x + cosh(1+sqrt(x + b)))), x)}
+}
+\xtc{
+A strong structure-checking algorithm in \Language{} finds hidden algebraic
+relationships between functions.
+}{
+\spadpaste{integrate(tan(atan(x)/3),x)}
+}
+\noindent
+%%--> Bob---> please make these formulas in this section smaller.
+The discovery of this algebraic relationship is necessary for correct
+integration of this function.
+Here are the details:
+\indent{4}
+\beginitems
+\item[1. ] 
+If \texht{$x=\tan t$}{\axiom{x=tan(t)}} and
+\texht{$g=\tan (t/3)$}{\axiom{g=tan(t/3)}} then the following
+algebraic relation is true:
+\texht{$${g^3-3xg^2-3g+x=0}$$}{\centerline{\axiom{g**3 - 3*x*g**2 - 3*g + x = 0}}}
+\item[2. ] 
+Integrate \axiom{g} using this algebraic relation; this produces:
+\texht{$${%
+{(24g^2 - 8)\log(3g^2 - 1) + (81x^2 + 24)g^2 + 72xg - 27x^2 - 16}
+\over{54g^2 - 18}}$$}{\centerline{\axiom{(24g**2 - 8)log(3g**2 - 1) +
+(81x**2 + 24)g**2 + 72xg - 27x**2 - 16/ (54g**2 - 18)}}}
+\item[3. ] 
+Rationalize the denominator, producing:
+\texht{\narrowDisplay{{8\log(3g^2-1) - 3g^2 + 18xg + 16} \over
+{18}}}{\centerline{\axiom{(8*log(3*g**2-1) - 3*g**2 + 18*x*g + 16)/18}}}
+Replace \axiom{g} by the initial definition
+\texht{$g = \tan(\arctan(x)/3)$}{\axiom{g = tan(arctan(x)/3)}}
+to produce the final result.
+\enditems
+\indent{0}
+
+\xtc{
+This is an example of a mixed function where
+the algebraic layer is over the transcendental one.
+}{
+\spadpaste{integrate((x + 1) / (x*(x + log x) ** (3/2)), x)}
+}
+\xtc{
+While incomplete for non-elementary functions, \Language{} can
+handle some of them.
+}{
+\spadpaste{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)}
+}
+
+More examples of \Language{}'s integration capabilities are discussed in
+\downlink{``\ugProblemIntegrationTitle''}{ugProblemIntegrationPage} in Section \ugProblemIntegrationNumber\ignore{ugProblemIntegration}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroDiffEqnsTitle}{Differential Equations}
+\newcommand{\ugIntroDiffEqnsNumber}{1.14.}
+%
+% =====================================================================
+\begin{page}{ugIntroDiffEqnsPage}{1.14. Differential Equations}
+% =====================================================================
+\beginscroll
+%
+The general approach used in integration also carries over to the
+solution of linear differential equations.
+
+\labelSpace{2pc}
+\xtc{
+Let's solve some differential equations.
+Let \axiom{y} be the unknown function in terms of \axiom{x}.
+}{
+\spadpaste{y := operator 'y \bound{y}}
+}
+\xtc{
+Here we solve a third order equation with polynomial coefficients.
+}{
+\spadpaste{deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4 \bound{e3}\free{y}}
+}
+\xtc{
+}{
+\spadpaste{solve(deq, y, x) \free{e3}\free{y}}
+}
+\xtc{
+Here we find all the algebraic function solutions of the equation.
+}{
+\spadpaste{deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0 \bound{e5}\free{y}}
+}
+\xtc{
+}{
+\spadpaste{solve(deq, y, x) \free{e5}\free{y}}
+}
+
+Coefficients of differential equations can come from arbitrary
+constant fields.
+For example, coefficients can contain algebraic numbers.
+
+\xtc{
+This example has solutions
+whose logarithmic derivative is an algebraic function of
+degree two.
+}{
+\spadpaste{eq := 2*x**3 * D(y x,x,2) + 3*x**2 * D(y x,x) - 2 * y x\bound{eq}\free{y}}
+}
+\xtc{
+}{
+\spadpaste{solve(eq,y,x).basis\free{eq}}
+}
+
+\xtc{
+Here's another differential equation to solve.
+}{
+\spadpaste{deq := D(y x, x) = y(x) / (x + y(x) * log y x) \bound{deqi}\free{y}}
+}
+\xtc{
+}{
+\spadpaste{solve(deq, y, x) \free{deqi y}}
+}
+
+Rather than attempting to get a closed form solution of
+a differential equation, you instead might want to find an
+approximate solution in the form of a series.
+
+\xtc{
+Let's solve a system of nonlinear first order equations and get a
+solution in power series.
+Tell \Language{} that \axiom{x} is also an operator.
+}{
+\spadpaste{x := operator 'x\bound{x}}
+}
+\xtc{
+Here are the two equations forming our system.
+}{
+\spadpaste{eq1 := D(x(t), t) = 1 + x(t)**2\free{x}\free{y}\bound{eq1}}
+}
+\xtc{
+}{
+\spadpaste{eq2 := D(y(t), t) = x(t) * y(t)\free{x}\free{y}\bound{eq2}}
+}
+\xtc{
+We can solve the system around \axiom{t = 0} with the initial conditions
+\axiom{x(0) = 0} and \axiom{y(0) = 1}.
+Notice that since we give the unknowns in the
+order \axiom{[x, y]}, the answer is a list of two series in the order
+\axiom{[series for x(t), series for y(t)]}.
+}{
+\spadpaste{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x}\free{y}\free{eq1}\free{eq2}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroSolutionTitle}{Solution of Equations}
+\newcommand{\ugIntroSolutionNumber}{1.15.}
+%
+% =====================================================================
+\begin{page}{ugIntroSolutionPage}{1.15. Solution of Equations}
+% =====================================================================
+\beginscroll
+%
+
+\Language{} also has state-of-the-art algorithms for the solution
+of systems of polynomial equations.
+When the number of equations and unknowns is the same, and you
+have no symbolic coefficients, you can use \spadfun{solve} for
+real roots and \spadfun{complexSolve} for complex roots.
+In each case, you tell \Language{} how accurate you want your
+result to be.
+All operations in the \spadfun{solve} family return answers in
+the form of a list of solution sets, where each solution set is a
+list of equations.
+
+\xtc{
+A system of two equations involving a symbolic
+parameter \axiom{t}.
+}{
+\spadpaste{S(t) == [x**2-2*y**2 - t,x*y-y-5*x + 5]\bound{S1}}
+}
+\xtc{
+Find the real roots of \spad{S(19)} with
+rational arithmetic, correct to within \smath{1/10^{20}}.
+}{
+\spadpaste{solve(S(19),1/10**20)\free{S1}}
+}
+\xtc{
+Find the complex roots of \spad{S(19)} with floating
+point coefficients to \spad{20} digits accuracy in the mantissa.
+}{
+\spadpaste{complexSolve(S(19),10.e-20)\free{S1}}
+}
+\xtc{
+If a system of equations has symbolic coefficients and you want
+a solution in radicals, try \spadfun{radicalSolve}.
+}{
+\spadpaste{radicalSolve(S(a),[x,y])\free{S1}}
+}
+For systems of equations with symbolic coefficients, you can
+apply \spadfun{solve}, listing the variables that you want
+\Language{} to solve for.
+For polynomial equations, a solution cannot usually be expressed
+solely in terms of the other variables.
+Instead, the solution is presented as a ``triangular'' system of
+equations, where each polynomial has coefficients involving
+only the succeeding variables. This is analogous to converting  a linear system
+of equations to ``triangular form''.
+\xtc{
+A system of three equations in five variables.
+}{
+\spadpaste{eqns := [x**2 - y + z,x**2*z + x**4 - b*y, y**2 *z - a - b*x]\bound{e}}
+}
+\xtc{
+Solve the system for unknowns \smath{[x,y,z]},
+reducing the solution to triangular form.
+}{
+\spadpaste{solve(eqns,[x,y,z])\free{e}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntroSysCmmandsTitle}{System Commands}
+\newcommand{\ugIntroSysCmmandsNumber}{1.16.}
+%
+% =====================================================================
+\begin{page}{ugIntroSysCmmandsPage}{1.16. System Commands}
+% =====================================================================
+\beginscroll
+%
+
+We conclude our tour of \Language{} with a brief discussion
+of \spadgloss{system commands}.
+System commands are special statements that start with a
+closing parenthesis (\axiomSyntax{)}). They are used to control or
+display your \Language{} environment, start the \HyperName{}
+system, issue operating system commands and leave \Language{}.
+For example, \spadsys{)system} is used
+to issue commands to the operating system from \Language{}.
+%-% \HDsyscmdindex{system}{ugIntroSysCmmandsPage}{1.16.}{System Commands}
+Here is a brief description of some of these commands.
+For more information on specific commands, see
+\downlink{``\ugSysCmdTitle''}{ugSysCmdPage} in Appendix \ugSysCmdNumber\ignore{ugSysCmd}.
+
+Perhaps the most important user command is the \spadsys{)clear all}
+command that initializes your environment.
+Every section and subsection in this book has an invisible
+\spadsys{)clear all} that is read prior to the examples given in
+the section.
+\spadsys{)clear all} gives you a fresh, empty environment with no
+user variables defined and the step number reset to \axiom{1}.
+The \spadsys{)clear} command can also be used to selectively clear
+values and properties of system variables.
+
+Another useful system command is \spadsys{)read}.
+A preferred way to develop an application in \Language{} is to put
+your interactive commands into a file, say {\bf my.input} file.
+To get \Language{} to read this file, you use the system command
+\spadsys{)read my.input}.
+If you need to make changes to your approach or definitions, go
+into your favorite editor, change {\bf my.input}, then
+\spadsys{)read my.input} again.
+
+Other system commands include: \spadsys{)history}, to display
+previous input and/or output lines; \spadsys{)display}, to display
+properties and values of workspace variables; and \spadsys{)what}.
+
+\xtc{
+Issue \spadsys{)what} to get a list of \Language{} objects that
+contain a given substring in their name.
+}{
+\spadpaste{)what operations integrate}
+}
+
+%\head{subsection}{Undo}{ugIntroUndo}
+
+A useful system command is \spadcmd{)undo}.
+Sometimes while computing interactively with \Language{}, you make
+a mistake and enter an incorrect definition or assignment.
+Or perhaps you
+need to try one of several alternative approaches, one after
+another, to find the best way to approach an application.
+For this, you will find the \spadgloss{undo} facility of
+\Language{} helpful.
+
+System command \spadsys{)undo n} means ``undo back to step \axiom{n}''; it
+restores the values of user variables to those that existed
+immediately after input expression \axiom{n} was evaluated.
+Similarly, \spadsys{)undo -n} undoes changes caused by the last
+\axiom{n} input expressions.
+Once you have done an \spadsys{)undo},
+you can continue on from there, or make a change and
+{\bf redo} all your input expressions from the point
+of the \spadsys{)undo} forward.
+The \spadsys{)undo} is completely general: it changes the environment
+like any user expression.
+Thus you can \spadsys{)undo} any previous undo.
+
+Here is a sample dialogue between user and \Language{}.
+\xtc{
+``Let me define
+two mutually dependent functions \axiom{f} and \axiom{g} piece-wise.''
+}{
+\spadpaste{f(0) == 1; g(0) == 1\bound{u1}}
+}
+\xtc{
+``Here is the general term for \axiom{f}.''
+}{
+\spadpaste{f(n) == e/2*f(n-1) - x*g(n-1)\bound{u2}\free{u1}}
+}
+\xtc{
+``And here is the general term for \axiom{g}.''
+}{
+\spadpaste{g(n) == -x*f(n-1) + d/3*g(n-1)\bound{u3}\free{u2}}
+}
+\xtc{
+``What is value of \axiom{f(3)}?''
+}{
+\spadpaste{f(3)\bound{u4}\free{u3}}
+}
+\noOutputXtc{
+``Hmm, I think I want to define \axiom{f} differently.
+Undo to the environment right after I defined \axiom{f}.''
+}{
+\spadpaste{)undo 2\bound{u5}\free{u4}}
+}
+\xtc{
+``Here is how I think I want \axiom{f} to be defined instead.''
+}{
+\spadpaste{f(n) == d/3*f(n-1) - x*g(n-1)\bound{u6}\free{u5}}
+}
+\noOutputXtc{
+Redo the computation from expression \axiom{3} forward.
+}{
+\spadpaste{)undo )redo\bound{u7}\free{u6}}
+}
+\noOutputXtc{
+``I want my old definition of
+\axiom{f} after all. Undo the undo and restore
+the environment to that immediately after \axiom{(4)}.''
+}{
+\spadpaste{)undo 4\bound{u8}\free{u7}}
+}
+\xtc{
+``Check that the value of \axiom{f(3)} is restored.''
+}{
+\spadpaste{f(3)\bound{u9}\free{u8}}
+}
+
+After you have gone off on several tangents, then backtracked to
+previous points in your conversation using \spadsys{)undo}, you
+might want to save all the ``correct'' input commands you issued,
+disregarding those undone.
+The system command \spadsys{)history )write mynew.input} writes a
+clean straight-line program onto the file {\bf mynew.input} on
+your disk.
+
+This concludes your tour of \Language{}.
+To disembark, issue the system command \spadsys{)quit} to leave \Language{}
+and return to the operating system.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug01.pht b/src/hyper/pages/ug01.pht
new file mode 100644
index 00000000..0ef9c533
--- /dev/null
+++ b/src/hyper/pages/ug01.pht
@@ -0,0 +1,3673 @@
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+            x
+          %e
+   (1)  %e
+                               Type: Expression Integer
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+   (3)  ((%e )  + 6(%e )  + 7(%e )  + %e )%e
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+                                   2
+   (9)  x(y(z + 1)) + F(x(z),y(z),z )
+                               Type: Expression Integer
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+        ,3                       ,2
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+                               Type: Expression Integer
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+   (11)
+          2            z             2
+       (2z  + 2z)F  (%e ,log(z + 1),z )
+                  ,3
+     + 
+             z             2
+       F  (%e ,log(z + 1),z )
+        ,2
+     + 
+                z      z             2
+       (z + 1)%e F  (%e ,log(z + 1),z ) + z + 1
+                  ,1
+  /
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+                               Type: Expression Integer
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+                               Type: Expression Integer
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+   (13)
+          2            z             2
+       (2z  + 2z)F  (%e ,log(z + 1),z )
+                  ,3
+     + 
+             z             2
+       F  (%e ,log(z + 1),z )
+        ,2
+     + 
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+       (z + 1)%e F  (%e ,log(z + 1),z ) + z + 1
+                  ,1
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+                               Type: Expression Integer
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+   (1)  x y  - 6x y  - 2x y  + 9y  + 6x y + x
+                               Type: Polynomial Integer
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+                  Type: UnivariatePolynomial(x,Integer)
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+\begin{paste}{ugIntroVariablesPageFull3}{ugIntroVariablesPageEmpty3}
+\pastebutton{ugIntroVariablesPageFull3}{\hidepaste}
+\tab{5}\spadcommand{m: MPOLY([x,y],INT) := (x**2-x*y**3+3*y)**2\bound{m }}
+\indentrel{3}\begin{verbatim}
+         4     3 3     6       2     4      2
+   (3)  x  - 2y x  + (y  + 6y)x  - 6y x + 9y
+            Type: MultivariatePolynomial([x,y],Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroVariablesPageEmpty3}
+\begin{paste}{ugIntroVariablesPageEmpty3}{ugIntroVariablesPagePatch3}
+\pastebutton{ugIntroVariablesPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{m: MPOLY([x,y],INT) := (x**2-x*y**3+3*y)**2\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroVariablesPagePatch4}
+\begin{paste}{ugIntroVariablesPageFull4}{ugIntroVariablesPageEmpty4}
+\pastebutton{ugIntroVariablesPageFull4}{\hidepaste}
+\tab{5}\spadcommand{m :: MPOLY([y,x],INT)\free{m }}
+\indentrel{3}\begin{verbatim}
+         2 6       4     3 3     2     2     4
+   (4)  x y  - 6x y  - 2x y  + 9y  + 6x y + x
+            Type: MultivariatePolynomial([y,x],Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroVariablesPageEmpty4}
+\begin{paste}{ugIntroVariablesPageEmpty4}{ugIntroVariablesPagePatch4}
+\pastebutton{ugIntroVariablesPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{m :: MPOLY([y,x],INT)\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroVariablesPagePatch5}
+\begin{paste}{ugIntroVariablesPageFull5}{ugIntroVariablesPageEmpty5}
+\pastebutton{ugIntroVariablesPageFull5}{\hidepaste}
+\tab{5}\spadcommand{m :: DMP([y,x],INT)\free{m }}
+\indentrel{3}\begin{verbatim}
+         6 2     4      3 3     2       2    4
+   (5)  y x  - 6y x - 2y x  + 9y  + 6y x  + x
+ Type: DistributedMultivariatePolynomial([y,x],Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroVariablesPageEmpty5}
+\begin{paste}{ugIntroVariablesPageEmpty5}{ugIntroVariablesPagePatch5}
+\pastebutton{ugIntroVariablesPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{m :: DMP([y,x],INT)\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroVariablesPagePatch6}
+\begin{paste}{ugIntroVariablesPageFull6}{ugIntroVariablesPageEmpty6}
+\pastebutton{ugIntroVariablesPageFull6}{\hidepaste}
+\tab{5}\spadcommand{m :: HDMP([y,x],INT)\free{m }}
+\indentrel{3}\begin{verbatim}
+         6 2     3 3     4     4       2     2
+   (6)  y x  - 2y x  - 6y x + x  + 6y x  + 9y
+Type: HomogeneousDistributedMultivariatePolynomial([y,x],Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroVariablesPageEmpty6}
+\begin{paste}{ugIntroVariablesPageEmpty6}{ugIntroVariablesPagePatch6}
+\pastebutton{ugIntroVariablesPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{m :: HDMP([y,x],INT)\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePagePatch1}
+\begin{paste}{ugIntroIntegratePageFull1}{ugIntroIntegratePageEmpty1}
+\pastebutton{ugIntroIntegratePageFull1}{\hidepaste}
+\tab{5}\spadcommand{integrate((x**2+2*x+1)/((x+1)**6+1),x)}
+\indentrel{3}\begin{verbatim}
+              3     2
+        atan(x  + 3x  + 3x + 1)
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                   3
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePageEmpty1}
+\begin{paste}{ugIntroIntegratePageEmpty1}{ugIntroIntegratePagePatch1}
+\pastebutton{ugIntroIntegratePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{integrate((x**2+2*x+1)/((x+1)**6+1),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePagePatch2}
+\begin{paste}{ugIntroIntegratePageFull2}{ugIntroIntegratePageEmpty2}
+\pastebutton{ugIntroIntegratePageFull2}{\hidepaste}
+\tab{5}\spadcommand{integrate(1/(x**2 + a),x)}
+\indentrel{3}\begin{verbatim}
+               2      ÚÄÄÄ¿
+             (x  - a)\³- a  + 2a x         ÚÄ¿
+         log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)      x\³a
+                      2             atan(ÄÄÄÄÄ)
+                     x  + a                a
+   (2)  [ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄ]
+                     ÚÄÄÄ¿               ÚÄ¿
+                   2\³- a               \³a
+               Type: Union(List Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePageEmpty2}
+\begin{paste}{ugIntroIntegratePageEmpty2}{ugIntroIntegratePagePatch2}
+\pastebutton{ugIntroIntegratePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{integrate(1/(x**2 + a),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePagePatch3}
+\begin{paste}{ugIntroIntegratePageFull3}{ugIntroIntegratePageEmpty3}
+\pastebutton{ugIntroIntegratePageFull3}{\hidepaste}
+\tab{5}\spadcommand{complexIntegrate(1/(x**2 + a),x)}
+\indentrel{3}\begin{verbatim}
+              ÚÄÄÄ¿              ÚÄÄÄ¿
+            x\³- a  + a        x\³- a  - a
+        log(ÄÄÄÄÄÄÄÄÄÄÄ) - log(ÄÄÄÄÄÄÄÄÄÄÄ)
+                ÚÄÄÄ¿              ÚÄÄÄ¿
+               \³- a              \³- a
+   (3)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                        ÚÄÄÄ¿
+                      2\³- a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePageEmpty3}
+\begin{paste}{ugIntroIntegratePageEmpty3}{ugIntroIntegratePagePatch3}
+\pastebutton{ugIntroIntegratePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{complexIntegrate(1/(x**2 + a),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePagePatch4}
+\begin{paste}{ugIntroIntegratePageFull4}{ugIntroIntegratePageEmpty4}
+\pastebutton{ugIntroIntegratePageFull4}{\hidepaste}
+\tab{5}\spadcommand{integrate(x**3 / (a+b*x)**(1/3),x)}
+\indentrel{3}\begin{verbatim}
+   (4)
+        3 3         2 2       2          3 3ÚÄÄÄÄÄÄÄ¿2
+   (120b x  - 135a b x  + 162a b x - 243a )\³b x + a
+   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                              4
+                          440b
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePageEmpty4}
+\begin{paste}{ugIntroIntegratePageEmpty4}{ugIntroIntegratePagePatch4}
+\pastebutton{ugIntroIntegratePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{integrate(x**3 / (a+b*x)**(1/3),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePagePatch5}
+\begin{paste}{ugIntroIntegratePageFull5}{ugIntroIntegratePageEmpty5}
+\pastebutton{ugIntroIntegratePageFull5}{\hidepaste}
+\tab{5}\spadcommand{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)}
+\indentrel{3}\begin{verbatim}
+   (5)
+       -
+              2 2 ÚÄ¿
+            2b x \³3
+         *
+                3ÚÄ¿3ÚÄÄÄÄÄÄÄ¿2   3ÚÄ¿2 3ÚÄÄÄÄÄÄÄ¿
+            log(\³a \³b x + a   + \³a   \³b x + a + a)
+     + 
+         2 2 ÚÄ¿    3ÚÄ¿2 3ÚÄÄÄÄÄÄÄ¿
+       4b x \³3 log(\³a   \³b x + a - a)
+     + 
+                    ÚÄ¿3ÚÄ¿2 3ÚÄÄÄÄÄÄÄ¿    ÚÄ¿
+          2 2     2\³3 \³a   \³b x + a + a\³3
+       12b x atan(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
+                               3a
+     + 
+                    ÚÄ¿3ÚÄ¿3ÚÄÄÄÄÄÄÄ¿2
+       (12b x - 9a)\³3 \³a \³b x + a
+  /
+        2 2 ÚÄ¿3ÚÄ¿
+     18a x \³3 \³a
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePageEmpty5}
+\begin{paste}{ugIntroIntegratePageEmpty5}{ugIntroIntegratePagePatch5}
+\pastebutton{ugIntroIntegratePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePagePatch6}
+\begin{paste}{ugIntroIntegratePageFull6}{ugIntroIntegratePageEmpty6}
+\pastebutton{ugIntroIntegratePageFull6}{\hidepaste}
+\tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)}
+\indentrel{3}\begin{verbatim}
+           x      ÚÄÄÄÄÄÄÄÄ¿
+         Ú¿  log(\³b + %S a  + 1)
+   (6)   ³   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ d%S
+        ÀÙ            %S
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePageEmpty6}
+\begin{paste}{ugIntroIntegratePageEmpty6}{ugIntroIntegratePagePatch6}
+\pastebutton{ugIntroIntegratePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePagePatch7}
+\begin{paste}{ugIntroIntegratePageFull7}{ugIntroIntegratePageEmpty7}
+\pastebutton{ugIntroIntegratePageFull7}{\hidepaste}
+\tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b)) / (sqrt(x+b) * (x + cosh(1+sqrt(x + b)))), x)}
+\indentrel{3}\begin{verbatim}
+   (7)
+                          ÚÄÄÄÄÄ¿
+                 - 2cosh(\³x + b  + 1) - 2x
+     2log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
+                ÚÄÄÄÄÄ¿              ÚÄÄÄÄÄ¿
+          sinh(\³x + b  + 1) - cosh(\³x + b  + 1)
+   + 
+         ÚÄÄÄÄÄ¿
+     - 2\³x + b
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePageEmpty7}
+\begin{paste}{ugIntroIntegratePageEmpty7}{ugIntroIntegratePagePatch7}
+\pastebutton{ugIntroIntegratePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b)) / (sqrt(x+b) * (x + cosh(1+sqrt(x + b)))), x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePagePatch8}
+\begin{paste}{ugIntroIntegratePageFull8}{ugIntroIntegratePageEmpty8}
+\pastebutton{ugIntroIntegratePageFull8}{\hidepaste}
+\tab{5}\spadcommand{integrate(tan(atan(x)/3),x)}
+\indentrel{3}\begin{verbatim}
+   (8)
+                 atan(x) 2             atan(x) 2
+       8log(3tan(ÄÄÄÄÄÄÄ)  - 1) - 3tan(ÄÄÄÄÄÄÄ)
+                    3                     3
+     + 
+               atan(x)
+       18x tan(ÄÄÄÄÄÄÄ)
+                  3
+  /
+     18
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePageEmpty8}
+\begin{paste}{ugIntroIntegratePageEmpty8}{ugIntroIntegratePagePatch8}
+\pastebutton{ugIntroIntegratePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{integrate(tan(atan(x)/3),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePagePatch9}
+\begin{paste}{ugIntroIntegratePageFull9}{ugIntroIntegratePageEmpty9}
+\pastebutton{ugIntroIntegratePageFull9}{\hidepaste}
+\tab{5}\spadcommand{integrate((x + 1) / (x*(x + log x) ** (3/2)), x)}
+\indentrel{3}\begin{verbatim}
+            ÚÄÄÄÄÄÄÄÄÄÄ¿
+          2\³log(x) + x
+   (9)  - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+            log(x) + x
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePageEmpty9}
+\begin{paste}{ugIntroIntegratePageEmpty9}{ugIntroIntegratePagePatch9}
+\pastebutton{ugIntroIntegratePageEmpty9}{\showpaste}
+\tab{5}\spadcommand{integrate((x + 1) / (x*(x + log x) ** (3/2)), x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePagePatch10}
+\begin{paste}{ugIntroIntegratePageFull10}{ugIntroIntegratePageEmpty10}
+\pastebutton{ugIntroIntegratePageFull10}{\hidepaste}
+\tab{5}\spadcommand{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)}
+\indentrel{3}\begin{verbatim}
+                      ÚÄÄÄ¿    erf(x) - 1      ÚÄÄÄ¿
+         (erf(x) - 1)\³%pi log(ÄÄÄÄÄÄÄÄÄÄ) - 2\³%pi
+                               erf(x) + 1
+   (10)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                         8erf(x) - 8
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroIntegratePageEmpty10}
+\begin{paste}{ugIntroIntegratePageEmpty10}{ugIntroIntegratePagePatch10}
+\pastebutton{ugIntroIntegratePageEmpty10}{\showpaste}
+\tab{5}\spadcommand{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPagePatch1}
+\begin{paste}{ugIntroCalcLimitsPageFull1}{ugIntroCalcLimitsPageEmpty1}
+\pastebutton{ugIntroCalcLimitsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{g := csc(a*x) / csch(b*x)\bound{g }}
+\indentrel{3}\begin{verbatim}
+         csc(a x)
+   (1)  ÄÄÄÄÄÄÄÄÄ
+        csch(b x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPageEmpty1}
+\begin{paste}{ugIntroCalcLimitsPageEmpty1}{ugIntroCalcLimitsPagePatch1}
+\pastebutton{ugIntroCalcLimitsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{g := csc(a*x) / csch(b*x)\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPagePatch2}
+\begin{paste}{ugIntroCalcLimitsPageFull2}{ugIntroCalcLimitsPageEmpty2}
+\pastebutton{ugIntroCalcLimitsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{limit(g,x=0)\free{g }}
+\indentrel{3}\begin{verbatim}
+        b
+   (2)  Ä
+        a
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPageEmpty2}
+\begin{paste}{ugIntroCalcLimitsPageEmpty2}{ugIntroCalcLimitsPagePatch2}
+\pastebutton{ugIntroCalcLimitsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{limit(g,x=0)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPagePatch3}
+\begin{paste}{ugIntroCalcLimitsPageFull3}{ugIntroCalcLimitsPageEmpty3}
+\pastebutton{ugIntroCalcLimitsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{h := (1 + k/x)**x\bound{h }}
+\indentrel{3}\begin{verbatim}
+         x + k x
+   (3)  (ÄÄÄÄÄ)
+           x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPageEmpty3}
+\begin{paste}{ugIntroCalcLimitsPageEmpty3}{ugIntroCalcLimitsPagePatch3}
+\pastebutton{ugIntroCalcLimitsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{h := (1 + k/x)**x\bound{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPagePatch4}
+\begin{paste}{ugIntroCalcLimitsPageFull4}{ugIntroCalcLimitsPageEmpty4}
+\pastebutton{ugIntroCalcLimitsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{limit(h,x=\%plusInfinity)\free{h }}
+\indentrel{3}\begin{verbatim}
+          k
+   (4)  %e
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPageEmpty4}
+\begin{paste}{ugIntroCalcLimitsPageEmpty4}{ugIntroCalcLimitsPagePatch4}
+\pastebutton{ugIntroCalcLimitsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{limit(h,x=\%plusInfinity)\free{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPagePatch5}
+\begin{paste}{ugIntroCalcLimitsPageFull5}{ugIntroCalcLimitsPageEmpty5}
+\pastebutton{ugIntroCalcLimitsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)}
+\indentrel{3}\begin{verbatim}
+   (5)  [leftHandLimit= - 1,rightHandLimit= 1]
+Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPageEmpty5}
+\begin{paste}{ugIntroCalcLimitsPageEmpty5}{ugIntroCalcLimitsPagePatch5}
+\pastebutton{ugIntroCalcLimitsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPagePatch6}
+\begin{paste}{ugIntroCalcLimitsPageFull6}{ugIntroCalcLimitsPageEmpty6}
+\pastebutton{ugIntroCalcLimitsPageFull6}{\hidepaste}
+\tab{5}\spadcommand{limit(exp(-1/x**2),x = 0)}
+\indentrel{3}\begin{verbatim}
+   (6)  0
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPageEmpty6}
+\begin{paste}{ugIntroCalcLimitsPageEmpty6}{ugIntroCalcLimitsPagePatch6}
+\pastebutton{ugIntroCalcLimitsPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{limit(exp(-1/x**2),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPagePatch7}
+\begin{paste}{ugIntroCalcLimitsPageFull7}{ugIntroCalcLimitsPageEmpty7}
+\pastebutton{ugIntroCalcLimitsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{complexLimit(exp(-1/x**2),x = 0)}
+\indentrel{3}\begin{verbatim}
+   (7)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCalcLimitsPageEmpty7}
+\begin{paste}{ugIntroCalcLimitsPageEmpty7}{ugIntroCalcLimitsPagePatch7}
+\pastebutton{ugIntroCalcLimitsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{complexLimit(exp(-1/x**2),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPagePatch1}
+\begin{paste}{ugIntroSysCmmandsPageFull1}{ugIntroSysCmmandsPageEmpty1}
+\pastebutton{ugIntroSysCmmandsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)what operations integrate}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPageEmpty1}
+\begin{paste}{ugIntroSysCmmandsPageEmpty1}{ugIntroSysCmmandsPagePatch1}
+\pastebutton{ugIntroSysCmmandsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)what operations integrate}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPagePatch2}
+\begin{paste}{ugIntroSysCmmandsPageFull2}{ugIntroSysCmmandsPageEmpty2}
+\pastebutton{ugIntroSysCmmandsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f(0) == 1; g(0) == 1\bound{u1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPageEmpty2}
+\begin{paste}{ugIntroSysCmmandsPageEmpty2}{ugIntroSysCmmandsPagePatch2}
+\pastebutton{ugIntroSysCmmandsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f(0) == 1; g(0) == 1\bound{u1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPagePatch3}
+\begin{paste}{ugIntroSysCmmandsPageFull3}{ugIntroSysCmmandsPageEmpty3}
+\pastebutton{ugIntroSysCmmandsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{f(n) == e/2*f(n-1) - x*g(n-1)\bound{u2 }\free{u1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPageEmpty3}
+\begin{paste}{ugIntroSysCmmandsPageEmpty3}{ugIntroSysCmmandsPagePatch3}
+\pastebutton{ugIntroSysCmmandsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f(n) == e/2*f(n-1) - x*g(n-1)\bound{u2 }\free{u1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPagePatch4}
+\begin{paste}{ugIntroSysCmmandsPageFull4}{ugIntroSysCmmandsPageEmpty4}
+\pastebutton{ugIntroSysCmmandsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{g(n) == -x*f(n-1) + d/3*g(n-1)\bound{u3 }\free{u2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPageEmpty4}
+\begin{paste}{ugIntroSysCmmandsPageEmpty4}{ugIntroSysCmmandsPagePatch4}
+\pastebutton{ugIntroSysCmmandsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{g(n) == -x*f(n-1) + d/3*g(n-1)\bound{u3 }\free{u2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPagePatch5}
+\begin{paste}{ugIntroSysCmmandsPageFull5}{ugIntroSysCmmandsPageEmpty5}
+\pastebutton{ugIntroSysCmmandsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{f(3)\bound{u4 }\free{u3 }}
+\indentrel{3}\begin{verbatim}
+   (4)
+      3        1    2      1  2   1       1  2     1  3
+   - x  + (e + Ä d)x  + (- Ä e  - Ä d e - Ä d )x + Ä e
+               3           4      6       9        8
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPageEmpty5}
+\begin{paste}{ugIntroSysCmmandsPageEmpty5}{ugIntroSysCmmandsPagePatch5}
+\pastebutton{ugIntroSysCmmandsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{f(3)\bound{u4 }\free{u3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPagePatch6}
+\begin{paste}{ugIntroSysCmmandsPageFull6}{ugIntroSysCmmandsPageEmpty6}
+\pastebutton{ugIntroSysCmmandsPageFull6}{\hidepaste}
+\tab{5}\spadcommand{)undo 2\bound{u5 }\free{u4 }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPageEmpty6}
+\begin{paste}{ugIntroSysCmmandsPageEmpty6}{ugIntroSysCmmandsPagePatch6}
+\pastebutton{ugIntroSysCmmandsPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{)undo 2\bound{u5 }\free{u4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPagePatch7}
+\begin{paste}{ugIntroSysCmmandsPageFull7}{ugIntroSysCmmandsPageEmpty7}
+\pastebutton{ugIntroSysCmmandsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{f(n) == d/3*f(n-1) - x*g(n-1)\bound{u6 }\free{u5 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPageEmpty7}
+\begin{paste}{ugIntroSysCmmandsPageEmpty7}{ugIntroSysCmmandsPagePatch7}
+\pastebutton{ugIntroSysCmmandsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{f(n) == d/3*f(n-1) - x*g(n-1)\bound{u6 }\free{u5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPagePatch8}
+\begin{paste}{ugIntroSysCmmandsPageFull8}{ugIntroSysCmmandsPageEmpty8}
+\pastebutton{ugIntroSysCmmandsPageFull8}{\hidepaste}
+\tab{5}\spadcommand{)undo )redo\bound{u7 }\free{u6 }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPageEmpty8}
+\begin{paste}{ugIntroSysCmmandsPageEmpty8}{ugIntroSysCmmandsPagePatch8}
+\pastebutton{ugIntroSysCmmandsPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{)undo )redo\bound{u7 }\free{u6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPagePatch9}
+\begin{paste}{ugIntroSysCmmandsPageFull9}{ugIntroSysCmmandsPageEmpty9}
+\pastebutton{ugIntroSysCmmandsPageFull9}{\hidepaste}
+\tab{5}\spadcommand{)undo 4\bound{u8 }\free{u7 }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPageEmpty9}
+\begin{paste}{ugIntroSysCmmandsPageEmpty9}{ugIntroSysCmmandsPagePatch9}
+\pastebutton{ugIntroSysCmmandsPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{)undo 4\bound{u8 }\free{u7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPagePatch10}
+\begin{paste}{ugIntroSysCmmandsPageFull10}{ugIntroSysCmmandsPageEmpty10}
+\pastebutton{ugIntroSysCmmandsPageFull10}{\hidepaste}
+\tab{5}\spadcommand{f(3)\bound{u9 }\free{u8 }}
+\indentrel{3}\begin{verbatim}
+   (6)
+      3        1    2      1  2   1       1  2     1  3
+   - x  + (e + Ä d)x  + (- Ä e  - Ä d e - Ä d )x + Ä e
+               3           4      6       9        8
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSysCmmandsPageEmpty10}
+\begin{paste}{ugIntroSysCmmandsPageEmpty10}{ugIntroSysCmmandsPagePatch10}
+\pastebutton{ugIntroSysCmmandsPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{f(3)\bound{u9 }\free{u8 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroArithmeticPagePatch1}
+\begin{paste}{ugIntroArithmeticPageFull1}{ugIntroArithmeticPageEmpty1}
+\pastebutton{ugIntroArithmeticPageFull1}{\hidepaste}
+\tab{5}\spadcommand{1 + 2 - 3 / 4 * 3 ** 2 - 1}
+\indentrel{3}\begin{verbatim}
+          19
+   (1)  - ÄÄ
+           4
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroArithmeticPageEmpty1}
+\begin{paste}{ugIntroArithmeticPageEmpty1}{ugIntroArithmeticPagePatch1}
+\pastebutton{ugIntroArithmeticPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{1 + 2 - 3 / 4 * 3 ** 2 - 1}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroArithmeticPagePatch2}
+\begin{paste}{ugIntroArithmeticPageFull2}{ugIntroArithmeticPageEmpty2}
+\pastebutton{ugIntroArithmeticPageFull2}{\hidepaste}
+\tab{5}\spadcommand{((1 + 2) - ((3 / 4) * (3 ** 2))) - 1}
+\indentrel{3}\begin{verbatim}
+          19
+   (2)  - ÄÄ
+           4
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroArithmeticPageEmpty2}
+\begin{paste}{ugIntroArithmeticPageEmpty2}{ugIntroArithmeticPagePatch2}
+\pastebutton{ugIntroArithmeticPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{((1 + 2) - ((3 / 4) * (3 ** 2))) - 1}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroArithmeticPagePatch3}
+\begin{paste}{ugIntroArithmeticPageFull3}{ugIntroArithmeticPageEmpty3}
+\pastebutton{ugIntroArithmeticPageFull3}{\hidepaste}
+\tab{5}\spadcommand{1 + 2 - 3/ (4 * 3 ** (2 - 1))}
+\indentrel{3}\begin{verbatim}
+        11
+   (3)  ÄÄ
+         4
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroArithmeticPageEmpty3}
+\begin{paste}{ugIntroArithmeticPageEmpty3}{ugIntroArithmeticPagePatch3}
+\pastebutton{ugIntroArithmeticPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{1 + 2 - 3/ (4 * 3 ** (2 - 1))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroConversionPagePatch1}
+\begin{paste}{ugIntroConversionPageFull1}{ugIntroConversionPageEmpty1}
+\pastebutton{ugIntroConversionPageFull1}{\hidepaste}
+\tab{5}\spadcommand{p := r**2 + 2/3\bound{p }}
+\indentrel{3}\begin{verbatim}
+         2   2
+   (1)  r  + Ä
+             3
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroConversionPageEmpty1}
+\begin{paste}{ugIntroConversionPageEmpty1}{ugIntroConversionPagePatch1}
+\pastebutton{ugIntroConversionPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{p := r**2 + 2/3\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroConversionPagePatch2}
+\begin{paste}{ugIntroConversionPageFull2}{ugIntroConversionPageEmpty2}
+\pastebutton{ugIntroConversionPageFull2}{\hidepaste}
+\tab{5}\spadcommand{p :: Fraction Polynomial Integer\free{p }}
+\indentrel{3}\begin{verbatim}
+          2
+        3r  + 2
+   (2)  ÄÄÄÄÄÄÄ
+           3
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroConversionPageEmpty2}
+\begin{paste}{ugIntroConversionPageEmpty2}{ugIntroConversionPagePatch2}
+\pastebutton{ugIntroConversionPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{p :: Fraction Polynomial Integer\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch1}
+\begin{paste}{ugIntroYouPageFull1}{ugIntroYouPageEmpty1}
+\pastebutton{ugIntroYouPageFull1}{\hidepaste}
+\tab{5}\spadcommand{fact(0) == 1\bound{fact }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty1}
+\begin{paste}{ugIntroYouPageEmpty1}{ugIntroYouPagePatch1}
+\pastebutton{ugIntroYouPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{fact(0) == 1\bound{fact }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch2}
+\begin{paste}{ugIntroYouPageFull2}{ugIntroYouPageEmpty2}
+\pastebutton{ugIntroYouPageFull2}{\hidepaste}
+\tab{5}\spadcommand{fact(n) == n*fact(n-1)\bound{facta }\free{fact }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty2}
+\begin{paste}{ugIntroYouPageEmpty2}{ugIntroYouPagePatch2}
+\pastebutton{ugIntroYouPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{fact(n) == n*fact(n-1)\bound{facta }\free{fact }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch3}
+\begin{paste}{ugIntroYouPageFull3}{ugIntroYouPageEmpty3}
+\pastebutton{ugIntroYouPageFull3}{\hidepaste}
+\tab{5}\spadcommand{fact(50)\free{facta }}
+\indentrel{3}\begin{verbatim}
+   (3)
+  304140932017133780436126081660647688443776415689605120_
+   00000000000
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty3}
+\begin{paste}{ugIntroYouPageEmpty3}{ugIntroYouPagePatch3}
+\pastebutton{ugIntroYouPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{fact(50)\free{facta }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch4}
+\begin{paste}{ugIntroYouPageFull4}{ugIntroYouPageEmpty4}
+\pastebutton{ugIntroYouPageFull4}{\hidepaste}
+\tab{5}\spadcommand{fac(n) == if n < 3 then n else n * fac(n - 1)\bound{fac }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty4}
+\begin{paste}{ugIntroYouPageEmpty4}{ugIntroYouPagePatch4}
+\pastebutton{ugIntroYouPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{fac(n) == if n < 3 then n else n * fac(n - 1)\bound{fac }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch5}
+\begin{paste}{ugIntroYouPageFull5}{ugIntroYouPageEmpty5}
+\pastebutton{ugIntroYouPageFull5}{\hidepaste}
+\tab{5}\spadcommand{fac(50)\free{fac }}
+\indentrel{3}\begin{verbatim}
+   (5)
+  304140932017133780436126081660647688443776415689605120_
+   00000000000
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty5}
+\begin{paste}{ugIntroYouPageEmpty5}{ugIntroYouPagePatch5}
+\pastebutton{ugIntroYouPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{fac(50)\free{fac }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch6}
+\begin{paste}{ugIntroYouPageFull6}{ugIntroYouPageEmpty6}
+\pastebutton{ugIntroYouPageFull6}{\hidepaste}
+\tab{5}\spadcommand{fa(n) == (a := 1; for i in 2..n repeat a := a*i; a)\bound{fa }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty6}
+\begin{paste}{ugIntroYouPageEmpty6}{ugIntroYouPagePatch6}
+\pastebutton{ugIntroYouPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{fa(n) == (a := 1; for i in 2..n repeat a := a*i; a)\bound{fa }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch7}
+\begin{paste}{ugIntroYouPageFull7}{ugIntroYouPageEmpty7}
+\pastebutton{ugIntroYouPageFull7}{\hidepaste}
+\tab{5}\spadcommand{fa(50)\free{fa }}
+\indentrel{3}\begin{verbatim}
+   (7)
+  304140932017133780436126081660647688443776415689605120_
+   00000000000
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty7}
+\begin{paste}{ugIntroYouPageEmpty7}{ugIntroYouPagePatch7}
+\pastebutton{ugIntroYouPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{fa(50)\free{fa }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch8}
+\begin{paste}{ugIntroYouPageFull8}{ugIntroYouPageEmpty8}
+\pastebutton{ugIntroYouPageFull8}{\hidepaste}
+\tab{5}\spadcommand{f(n) == reduce(*,[i for i in 2..n])\bound{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty8}
+\begin{paste}{ugIntroYouPageEmpty8}{ugIntroYouPagePatch8}
+\pastebutton{ugIntroYouPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{f(n) == reduce(*,[i for i in 2..n])\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch9}
+\begin{paste}{ugIntroYouPageFull9}{ugIntroYouPageEmpty9}
+\pastebutton{ugIntroYouPageFull9}{\hidepaste}
+\tab{5}\spadcommand{f(50)\free{f }}
+\indentrel{3}\begin{verbatim}
+   (9)
+  304140932017133780436126081660647688443776415689605120_
+   00000000000
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty9}
+\begin{paste}{ugIntroYouPageEmpty9}{ugIntroYouPagePatch9}
+\pastebutton{ugIntroYouPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{f(50)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch10}
+\begin{paste}{ugIntroYouPageFull10}{ugIntroYouPageEmpty10}
+\pastebutton{ugIntroYouPageFull10}{\hidepaste}
+\tab{5}\spadcommand{factorial(50)}
+\indentrel{3}\begin{verbatim}
+   (10)
+  304140932017133780436126081660647688443776415689605120_
+   00000000000
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty10}
+\begin{paste}{ugIntroYouPageEmpty10}{ugIntroYouPagePatch10}
+\pastebutton{ugIntroYouPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{factorial(50)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch11}
+\begin{paste}{ugIntroYouPageFull11}{ugIntroYouPageEmpty11}
+\pastebutton{ugIntroYouPageFull11}{\hidepaste}
+\tab{5}\spadcommand{permMat(n, i, j) ==
+  m := diagonalMatrix
+    [(if i = k or j = k then 0 else 1)
+      for k in 1..n]
+  m(i,j) := 1
+  m(j,i) := 1
+  m
+\bound{permMat }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty11}
+\begin{paste}{ugIntroYouPageEmpty11}{ugIntroYouPagePatch11}
+\pastebutton{ugIntroYouPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{permMat(n, i, j) ==
+  m := diagonalMatrix
+    [(if i = k or j = k then 0 else 1)
+      for k in 1..n]
+  m(i,j) := 1
+  m(j,i) := 1
+  m
+\bound{permMat }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch12}
+\begin{paste}{ugIntroYouPageFull12}{ugIntroYouPageEmpty12}
+\pastebutton{ugIntroYouPageFull12}{\hidepaste}
+\tab{5}\spadcommand{p := permMat(4,2,3)\free{permMat }\bound{p }}
+\indentrel{3}\begin{verbatim}
+         Ú1  0  0  0¿
+         ³          ³
+         ³0  0  1  0³
+   (12)  ³          ³
+         ³0  1  0  0³
+         ³          ³
+         À0  0  0  1Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty12}
+\begin{paste}{ugIntroYouPageEmpty12}{ugIntroYouPagePatch12}
+\pastebutton{ugIntroYouPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{p := permMat(4,2,3)\free{permMat }\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch13}
+\begin{paste}{ugIntroYouPageFull13}{ugIntroYouPageEmpty13}
+\pastebutton{ugIntroYouPageFull13}{\hidepaste}
+\tab{5}\spadcommand{m := matrix [[4*i + j for j in 1..4] for i in 0..3]\bound{m }}
+\indentrel{3}\begin{verbatim}
+         Ú1   2   3   4 ¿
+         ³              ³
+         ³5   6   7   8 ³
+   (13)  ³              ³
+         ³9   10  11  12³
+         ³              ³
+         À13  14  15  16Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty13}
+\begin{paste}{ugIntroYouPageEmpty13}{ugIntroYouPagePatch13}
+\pastebutton{ugIntroYouPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{m := matrix [[4*i + j for j in 1..4] for i in 0..3]\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch14}
+\begin{paste}{ugIntroYouPageFull14}{ugIntroYouPageEmpty14}
+\pastebutton{ugIntroYouPageFull14}{\hidepaste}
+\tab{5}\spadcommand{permMat(4,2,3) * m\free{p m }}
+\indentrel{3}\begin{verbatim}
+         Ú1   2   3   4 ¿
+         ³              ³
+         ³9   10  11  12³
+   (14)  ³              ³
+         ³5   6   7   8 ³
+         ³              ³
+         À13  14  15  16Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty14}
+\begin{paste}{ugIntroYouPageEmpty14}{ugIntroYouPagePatch14}
+\pastebutton{ugIntroYouPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{permMat(4,2,3) * m\free{p m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch15}
+\begin{paste}{ugIntroYouPageFull15}{ugIntroYouPageEmpty15}
+\pastebutton{ugIntroYouPageFull15}{\hidepaste}
+\tab{5}\spadcommand{t : (Float -> Float, Float) -> Float\bound{tdecl }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty15}
+\begin{paste}{ugIntroYouPageEmpty15}{ugIntroYouPagePatch15}
+\pastebutton{ugIntroYouPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{t : (Float -> Float, Float) -> Float\bound{tdecl }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch16}
+\begin{paste}{ugIntroYouPageFull16}{ugIntroYouPageEmpty16}
+\pastebutton{ugIntroYouPageFull16}{\hidepaste}
+\tab{5}\spadcommand{t(fun, x) == fun(x)**2 + sin(x)**2\free{tdecl }\bound{t }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty16}
+\begin{paste}{ugIntroYouPageEmpty16}{ugIntroYouPagePatch16}
+\pastebutton{ugIntroYouPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{t(fun, x) == fun(x)**2 + sin(x)**2\free{tdecl }\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch17}
+\begin{paste}{ugIntroYouPageFull17}{ugIntroYouPageEmpty17}
+\pastebutton{ugIntroYouPageFull17}{\hidepaste}
+\tab{5}\spadcommand{t(cos, 5.2058)\free{t }}
+\indentrel{3}\begin{verbatim}
+   (17)  1.0
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty17}
+\begin{paste}{ugIntroYouPageEmpty17}{ugIntroYouPagePatch17}
+\pastebutton{ugIntroYouPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{t(cos, 5.2058)\free{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch18}
+\begin{paste}{ugIntroYouPageFull18}{ugIntroYouPageEmpty18}
+\pastebutton{ugIntroYouPageFull18}{\hidepaste}
+\tab{5}\spadcommand{cosinv(y) == cos(1/y)\bound{cosinv }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty18}
+\begin{paste}{ugIntroYouPageEmpty18}{ugIntroYouPagePatch18}
+\pastebutton{ugIntroYouPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{cosinv(y) == cos(1/y)\bound{cosinv }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch19}
+\begin{paste}{ugIntroYouPageFull19}{ugIntroYouPageEmpty19}
+\pastebutton{ugIntroYouPageFull19}{\hidepaste}
+\tab{5}\spadcommand{t(cosinv, 5.2058)\free{t }\free{cosinv }}
+\indentrel{3}\begin{verbatim}
+   (19)  1.7392237241 800516493
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty19}
+\begin{paste}{ugIntroYouPageEmpty19}{ugIntroYouPagePatch19}
+\pastebutton{ugIntroYouPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{t(cosinv, 5.2058)\free{t }\free{cosinv }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch20}
+\begin{paste}{ugIntroYouPageFull20}{ugIntroYouPageEmpty20}
+\pastebutton{ugIntroYouPageFull20}{\hidepaste}
+\tab{5}\spadcommand{groupSqrt := rule(sqrt(a) * sqrt(b) == sqrt(a*b))\bound{g }}
+\indentrel{3}\begin{verbatim}
+             ÚÄ¿ ÚÄ¿        ÚÄÄÄ¿
+   (20)  %BD\³a \³b  == %BD\³a b
+  Type: RewriteRule(Integer,Integer,Expression Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty20}
+\begin{paste}{ugIntroYouPageEmpty20}{ugIntroYouPagePatch20}
+\pastebutton{ugIntroYouPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{groupSqrt := rule(sqrt(a) * sqrt(b) == sqrt(a*b))\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch21}
+\begin{paste}{ugIntroYouPageFull21}{ugIntroYouPageEmpty21}
+\pastebutton{ugIntroYouPageFull21}{\hidepaste}
+\tab{5}\spadcommand{a := (sqrt(x) + sqrt(y) + sqrt(z))**4\bound{sxy }}
+\indentrel{3}\begin{verbatim}
+   (21)
+                      ÚÄ¿                   ÚÄ¿  ÚÄ¿
+     ((4z + 4y + 12x)\³y  + (4z + 12y + 4x)\³x )\³z
+   + 
+                     ÚÄ¿ ÚÄ¿    2                 2
+     (12z + 4y + 4x)\³x \³y  + z  + (6y + 6x)z + y  + 6x y
+   + 
+      2
+     x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty21}
+\begin{paste}{ugIntroYouPageEmpty21}{ugIntroYouPagePatch21}
+\pastebutton{ugIntroYouPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{a := (sqrt(x) + sqrt(y) + sqrt(z))**4\bound{sxy }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPagePatch22}
+\begin{paste}{ugIntroYouPageFull22}{ugIntroYouPageEmpty22}
+\pastebutton{ugIntroYouPageFull22}{\hidepaste}
+\tab{5}\spadcommand{groupSqrt a\free{sxy }\free{g }}
+\indentrel{3}\begin{verbatim}
+   (22)
+                     ÚÄÄÄ¿                   ÚÄÄÄ¿
+     (4z + 4y + 12x)\³y z  + (4z + 12y + 4x)\³x z
+   + 
+                     ÚÄÄÄ¿    2                 2
+     (12z + 4y + 4x)\³x y  + z  + (6y + 6x)z + y  + 6x y
+   + 
+      2
+     x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroYouPageEmpty22}
+\begin{paste}{ugIntroYouPageEmpty22}{ugIntroYouPagePatch22}
+\pastebutton{ugIntroYouPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{groupSqrt a\free{sxy }\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTypesPagePatch1}
+\begin{paste}{ugIntroTypesPageFull1}{ugIntroTypesPageEmpty1}
+\pastebutton{ugIntroTypesPageFull1}{\hidepaste}
+\tab{5}\spadcommand{8}
+\indentrel{3}\begin{verbatim}
+   (1)  8
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTypesPageEmpty1}
+\begin{paste}{ugIntroTypesPageEmpty1}{ugIntroTypesPagePatch1}
+\pastebutton{ugIntroTypesPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{8}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTypesPagePatch2}
+\begin{paste}{ugIntroTypesPageFull2}{ugIntroTypesPageEmpty2}
+\pastebutton{ugIntroTypesPageFull2}{\hidepaste}
+\tab{5}\spadcommand{-8}
+\indentrel{3}\begin{verbatim}
+   (2)  - 8
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTypesPageEmpty2}
+\begin{paste}{ugIntroTypesPageEmpty2}{ugIntroTypesPagePatch2}
+\pastebutton{ugIntroTypesPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{-8}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTypesPagePatch3}
+\begin{paste}{ugIntroTypesPageFull3}{ugIntroTypesPageEmpty3}
+\pastebutton{ugIntroTypesPageFull3}{\hidepaste}
+\tab{5}\spadcommand{x**8}
+\indentrel{3}\begin{verbatim}
+         8
+   (3)  x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTypesPageEmpty3}
+\begin{paste}{ugIntroTypesPageEmpty3}{ugIntroTypesPagePatch3}
+\pastebutton{ugIntroTypesPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{x**8}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTypesPagePatch4}
+\begin{paste}{ugIntroTypesPageFull4}{ugIntroTypesPageEmpty4}
+\pastebutton{ugIntroTypesPageFull4}{\hidepaste}
+\tab{5}\spadcommand{x**(-8)}
+\indentrel{3}\begin{verbatim}
+         1
+   (4)  ÄÄ
+         8
+        x
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTypesPageEmpty4}
+\begin{paste}{ugIntroTypesPageEmpty4}{ugIntroTypesPagePatch4}
+\pastebutton{ugIntroTypesPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{x**(-8)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPagePatch1}
+\begin{paste}{ugIntroTwoDimPageFull1}{ugIntroTwoDimPageEmpty1}
+\pastebutton{ugIntroTwoDimPageFull1}{\hidepaste}
+\tab{5}\spadcommand{m := matrix([[1,2], [3,4]])\bound{m }}
+\indentrel{3}\begin{verbatim}
+        Ú1  2¿
+   (1)  ³    ³
+        À3  4Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPageEmpty1}
+\begin{paste}{ugIntroTwoDimPageEmpty1}{ugIntroTwoDimPagePatch1}
+\pastebutton{ugIntroTwoDimPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{m := matrix([[1,2], [3,4]])\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPagePatch2}
+\begin{paste}{ugIntroTwoDimPageFull2}{ugIntroTwoDimPageEmpty2}
+\pastebutton{ugIntroTwoDimPageFull2}{\hidepaste}
+\tab{5}\spadcommand{matrix([[1/(i + j - x) for i in 1..4] for j in 1..4])\bound{hilb }}
+\indentrel{3}\begin{verbatim}
+        Ú    1        1        1        1  ¿
+        ³- ÄÄÄÄÄ  - ÄÄÄÄÄ  - ÄÄÄÄÄ  - ÄÄÄÄij
+        ³  x - 2    x - 3    x - 4    x - 5³
+        ³                                  ³
+        ³    1        1        1        1  ³
+        ³- ÄÄÄÄÄ  - ÄÄÄÄÄ  - ÄÄÄÄÄ  - ÄÄÄÄij
+        ³  x - 3    x - 4    x - 5    x - 6³
+   (2)  ³                                  ³
+        ³    1        1        1        1  ³
+        ³- ÄÄÄÄÄ  - ÄÄÄÄÄ  - ÄÄÄÄÄ  - ÄÄÄÄij
+        ³  x - 4    x - 5    x - 6    x - 7³
+        ³                                  ³
+        ³    1        1        1        1  ³
+        ³- ÄÄÄÄÄ  - ÄÄÄÄÄ  - ÄÄÄÄÄ  - ÄÄÄÄij
+        À  x - 5    x - 6    x - 7    x - 8Ù
+               Type: Matrix Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPageEmpty2}
+\begin{paste}{ugIntroTwoDimPageEmpty2}{ugIntroTwoDimPagePatch2}
+\pastebutton{ugIntroTwoDimPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{matrix([[1/(i + j - x) for i in 1..4] for j in 1..4])\bound{hilb }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPagePatch3}
+\begin{paste}{ugIntroTwoDimPageFull3}{ugIntroTwoDimPageEmpty3}
+\pastebutton{ugIntroTwoDimPageFull3}{\hidepaste}
+\tab{5}\spadcommand{vm := matrix [[1,1,1], [x,y,z], [x*x,y*y,z*z]]\bound{vm }}
+\indentrel{3}\begin{verbatim}
+        Ú1   1   1 ¿
+        ³          ³
+   (3)  ³x   y   z ³
+        ³          ³
+        ³ 2   2   2³
+        Àx   y   z Ù
+                        Type: Matrix Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPageEmpty3}
+\begin{paste}{ugIntroTwoDimPageEmpty3}{ugIntroTwoDimPagePatch3}
+\pastebutton{ugIntroTwoDimPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{vm := matrix [[1,1,1], [x,y,z], [x*x,y*y,z*z]]\bound{vm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPagePatch4}
+\begin{paste}{ugIntroTwoDimPageFull4}{ugIntroTwoDimPageEmpty4}
+\pastebutton{ugIntroTwoDimPageFull4}{\hidepaste}
+\tab{5}\spadcommand{vm(3,3)\free{vm }}
+\indentrel{3}\begin{verbatim}
+         2
+   (4)  z
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPageEmpty4}
+\begin{paste}{ugIntroTwoDimPageEmpty4}{ugIntroTwoDimPagePatch4}
+\pastebutton{ugIntroTwoDimPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{vm(3,3)\free{vm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPagePatch5}
+\begin{paste}{ugIntroTwoDimPageFull5}{ugIntroTwoDimPageEmpty5}
+\pastebutton{ugIntroTwoDimPageFull5}{\hidepaste}
+\tab{5}\spadcommand{column(vm,2)\free{vm }}
+\indentrel{3}\begin{verbatim}
+              2
+   (5)  [1,y,y ]
+                        Type: Vector Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPageEmpty5}
+\begin{paste}{ugIntroTwoDimPageEmpty5}{ugIntroTwoDimPagePatch5}
+\pastebutton{ugIntroTwoDimPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{column(vm,2)\free{vm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPagePatch6}
+\begin{paste}{ugIntroTwoDimPageFull6}{ugIntroTwoDimPageEmpty6}
+\pastebutton{ugIntroTwoDimPageFull6}{\hidepaste}
+\tab{5}\spadcommand{vm * vm\free{vm }}
+\indentrel{3}\begin{verbatim}
+   (6)
+   Ú    2                2              2          ¿
+   ³   x  + x + 1       y  + y + 1     z  + z + 1  ³
+   ³                                               ³
+   ³  2                 2     2        3           ³
+   ³ x z + x y + x     y z + y  + x   z  + y z + x ³
+   ³                                               ³
+   ³ 2 2      2    2   2 2    3    2   4    2     2³
+   Àx z  + x y  + x   y z  + y  + x   z  + y z + x Ù
+                        Type: Matrix Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPageEmpty6}
+\begin{paste}{ugIntroTwoDimPageEmpty6}{ugIntroTwoDimPagePatch6}
+\pastebutton{ugIntroTwoDimPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{vm * vm\free{vm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPagePatch7}
+\begin{paste}{ugIntroTwoDimPageFull7}{ugIntroTwoDimPageEmpty7}
+\pastebutton{ugIntroTwoDimPageFull7}{\hidepaste}
+\tab{5}\spadcommand{factor determinant vm\free{vm }\bound{d }}
+\indentrel{3}\begin{verbatim}
+   (7)  (y - x)(z - y)(z - x)
+                      Type: Factored Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroTwoDimPageEmpty7}
+\begin{paste}{ugIntroTwoDimPageEmpty7}{ugIntroTwoDimPagePatch7}
+\pastebutton{ugIntroTwoDimPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{factor determinant vm\free{vm }\bound{d }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPagePatch1}
+\begin{paste}{ugIntroAssignPageFull1}{ugIntroAssignPageEmpty1}
+\pastebutton{ugIntroAssignPageFull1}{\hidepaste}
+\tab{5}\spadcommand{(x - y*z)**2}
+\indentrel{3}\begin{verbatim}
+         2 2             2
+   (1)  y z  - 2x y z + x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPageEmpty1}
+\begin{paste}{ugIntroAssignPageEmpty1}{ugIntroAssignPagePatch1}
+\pastebutton{ugIntroAssignPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{(x - y*z)**2}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPagePatch2}
+\begin{paste}{ugIntroAssignPageFull2}{ugIntroAssignPageEmpty2}
+\pastebutton{ugIntroAssignPageFull2}{\hidepaste}
+\tab{5}\spadcommand{x := 4}
+\indentrel{3}\begin{verbatim}
+   (2)  4
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPageEmpty2}
+\begin{paste}{ugIntroAssignPageEmpty2}{ugIntroAssignPagePatch2}
+\pastebutton{ugIntroAssignPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{x := 4}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPagePatch3}
+\begin{paste}{ugIntroAssignPageFull3}{ugIntroAssignPageEmpty3}
+\pastebutton{ugIntroAssignPageFull3}{\hidepaste}
+\tab{5}\spadcommand{x := z + 3/5}
+\indentrel{3}\begin{verbatim}
+            3
+   (3)  z + Ä
+            5
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPageEmpty3}
+\begin{paste}{ugIntroAssignPageEmpty3}{ugIntroAssignPagePatch3}
+\pastebutton{ugIntroAssignPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{x := z + 3/5}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPagePatch4}
+\begin{paste}{ugIntroAssignPageFull4}{ugIntroAssignPageEmpty4}
+\pastebutton{ugIntroAssignPageFull4}{\hidepaste}
+\tab{5}\spadcommand{y : Integer\bound{y }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPageEmpty4}
+\begin{paste}{ugIntroAssignPageEmpty4}{ugIntroAssignPagePatch4}
+\pastebutton{ugIntroAssignPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{y : Integer\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPagePatch5}
+\begin{paste}{ugIntroAssignPageFull5}{ugIntroAssignPageEmpty5}
+\pastebutton{ugIntroAssignPageFull5}{\hidepaste}
+\tab{5}\spadcommand{y := 89\bound{y1 }\free{y }}
+\indentrel{3}\begin{verbatim}
+   (5)  89
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPageEmpty5}
+\begin{paste}{ugIntroAssignPageEmpty5}{ugIntroAssignPagePatch5}
+\pastebutton{ugIntroAssignPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{y := 89\bound{y1 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPagePatch6}
+\begin{paste}{ugIntroAssignPageFull6}{ugIntroAssignPageEmpty6}
+\pastebutton{ugIntroAssignPageFull6}{\hidepaste}
+\tab{5}\spadcommand{y := sin \%pi}
+\indentrel{3}\begin{verbatim}
+   (6)  0
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPageEmpty6}
+\begin{paste}{ugIntroAssignPageEmpty6}{ugIntroAssignPagePatch6}
+\pastebutton{ugIntroAssignPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{y := sin \%pi}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPagePatch7}
+\begin{paste}{ugIntroAssignPageFull7}{ugIntroAssignPageEmpty7}
+\pastebutton{ugIntroAssignPageFull7}{\hidepaste}
+\tab{5}\spadcommand{y := 2/3}
+\indentrel{3}\begin{verbatim}
+   2
+   Ä
+   3
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPageEmpty7}
+\begin{paste}{ugIntroAssignPageEmpty7}{ugIntroAssignPagePatch7}
+\pastebutton{ugIntroAssignPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{y := 2/3}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPagePatch8}
+\begin{paste}{ugIntroAssignPageFull8}{ugIntroAssignPageEmpty8}
+\pastebutton{ugIntroAssignPageFull8}{\hidepaste}
+\tab{5}\spadcommand{f : Float := 2/3}
+\indentrel{3}\begin{verbatim}
+   (7)  0.6666666666 6666666667
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPageEmpty8}
+\begin{paste}{ugIntroAssignPageEmpty8}{ugIntroAssignPagePatch8}
+\pastebutton{ugIntroAssignPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{f : Float := 2/3}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPagePatch9}
+\begin{paste}{ugIntroAssignPageFull9}{ugIntroAssignPageEmpty9}
+\pastebutton{ugIntroAssignPageFull9}{\hidepaste}
+\tab{5}\spadcommand{f : Float; f := 2/3\bound{fff }}
+\indentrel{3}\begin{verbatim}
+   (8)  0.6666666666 6666666667
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPageEmpty9}
+\begin{paste}{ugIntroAssignPageEmpty9}{ugIntroAssignPagePatch9}
+\pastebutton{ugIntroAssignPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{f : Float; f := 2/3\bound{fff }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPagePatch10}
+\begin{paste}{ugIntroAssignPageFull10}{ugIntroAssignPageEmpty10}
+\pastebutton{ugIntroAssignPageFull10}{\hidepaste}
+\tab{5}\spadcommand{q}
+\indentrel{3}\begin{verbatim}
+   (9)  q
+                                       Type: Variable q
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPageEmpty10}
+\begin{paste}{ugIntroAssignPageEmpty10}{ugIntroAssignPagePatch10}
+\pastebutton{ugIntroAssignPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{q}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPagePatch11}
+\begin{paste}{ugIntroAssignPageFull11}{ugIntroAssignPageEmpty11}
+\pastebutton{ugIntroAssignPageFull11}{\hidepaste}
+\tab{5}\spadcommand{[q, r]}
+\indentrel{3}\begin{verbatim}
+   (10)  [q,r]
+                   Type: List OrderedVariableList [q,r]
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPageEmpty11}
+\begin{paste}{ugIntroAssignPageEmpty11}{ugIntroAssignPagePatch11}
+\pastebutton{ugIntroAssignPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{[q, r]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPagePatch12}
+\begin{paste}{ugIntroAssignPageFull12}{ugIntroAssignPageEmpty12}
+\pastebutton{ugIntroAssignPageFull12}{\hidepaste}
+\tab{5}\spadcommand{f\free{fff }}
+\indentrel{3}\begin{verbatim}
+   (11)  0.6666666666 6666666667
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPageEmpty12}
+\begin{paste}{ugIntroAssignPageEmpty12}{ugIntroAssignPagePatch12}
+\pastebutton{ugIntroAssignPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{f\free{fff }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPagePatch13}
+\begin{paste}{ugIntroAssignPageFull13}{ugIntroAssignPageEmpty13}
+\pastebutton{ugIntroAssignPageFull13}{\hidepaste}
+\tab{5}\spadcommand{'f}
+\indentrel{3}\begin{verbatim}
+   (12)  f
+                                       Type: Variable f
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroAssignPageEmpty13}
+\begin{paste}{ugIntroAssignPageEmpty13}{ugIntroAssignPagePatch13}
+\pastebutton{ugIntroAssignPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{'f}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPagePatch1}
+\begin{paste}{ugIntroSeriesPageFull1}{ugIntroSeriesPageEmpty1}
+\pastebutton{ugIntroSeriesPageFull1}{\hidepaste}
+\tab{5}\spadcommand{series(sin(a*x),x = 0)}
+\indentrel{3}\begin{verbatim}
+   (1)
+            3        5        7          9
+           a   3    a   5    a    7     a     9
+     a x - ÄÄ x  + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x
+            6      120      5040      362880
+   + 
+           11
+          a      11      12
+     - ÄÄÄÄÄÄÄÄ x   + O(x  )
+       39916800
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPageEmpty1}
+\begin{paste}{ugIntroSeriesPageEmpty1}{ugIntroSeriesPagePatch1}
+\pastebutton{ugIntroSeriesPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{series(sin(a*x),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPagePatch2}
+\begin{paste}{ugIntroSeriesPageFull2}{ugIntroSeriesPageEmpty2}
+\pastebutton{ugIntroSeriesPageFull2}{\hidepaste}
+\tab{5}\spadcommand{series(sin(a*x),x = \%pi/4)}
+\indentrel{3}\begin{verbatim}
+   (2)
+         a %pi          a %pi      %pi
+     sin(ÄÄÄÄÄ) + a cos(ÄÄÄÄÄ)(x - ÄÄÄ)
+           4              4         4
+   + 
+        2    a %pi                3    a %pi
+       a sin(ÄÄÄÄÄ)              a cos(ÄÄÄÄÄ)
+               4         %pi 2           4         %pi 3
+     - ÄÄÄÄÄÄÄÄÄÄÄÄ (x - ÄÄÄ)  - ÄÄÄÄÄÄÄÄÄÄÄÄ (x - ÄÄÄ)
+             2            4            6            4
+   + 
+      4    a %pi                5    a %pi
+     a sin(ÄÄÄÄÄ)              a cos(ÄÄÄÄÄ)
+             4         %pi 4           4         %pi 5
+     ÄÄÄÄÄÄÄÄÄÄÄÄ (x - ÄÄÄ)  + ÄÄÄÄÄÄÄÄÄÄÄÄ (x - ÄÄÄ)
+          24            4           120           4
+   + 
+        6    a %pi                7    a %pi
+       a sin(ÄÄÄÄÄ)              a cos(ÄÄÄÄÄ)
+               4         %pi 6           4         %pi 7
+     - ÄÄÄÄÄÄÄÄÄÄÄÄ (x - ÄÄÄ)  - ÄÄÄÄÄÄÄÄÄÄÄÄ (x - ÄÄÄ)
+            720           4          5040           4
+   + 
+      8    a %pi                9    a %pi
+     a sin(ÄÄÄÄÄ)              a cos(ÄÄÄÄÄ)
+             4         %pi 8           4         %pi 9
+     ÄÄÄÄÄÄÄÄÄÄÄÄ (x - ÄÄÄ)  + ÄÄÄÄÄÄÄÄÄÄÄÄ (x - ÄÄÄ)
+         40320          4         362880          4
+   + 
+        10    a %pi
+       a  sin(ÄÄÄÄÄ)
+                4         %pi 10          %pi 11
+     - ÄÄÄÄÄÄÄÄÄÄÄÄÄ (x - ÄÄÄ)   + O((x - ÄÄÄ)  )
+          3628800          4               4
+Type: UnivariatePuiseuxSeries(Expression Integer,x,pi/4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPageEmpty2}
+\begin{paste}{ugIntroSeriesPageEmpty2}{ugIntroSeriesPagePatch2}
+\pastebutton{ugIntroSeriesPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{series(sin(a*x),x = \%pi/4)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPagePatch3}
+\begin{paste}{ugIntroSeriesPageFull3}{ugIntroSeriesPageEmpty3}
+\pastebutton{ugIntroSeriesPageFull3}{\hidepaste}
+\tab{5}\spadcommand{series(n +-> (-1)**((3*n - 4)/6)/factorial(n - 1/3),x = 0,4/3..,2)}
+\indentrel{3}\begin{verbatim}
+         4      10
+         Ä      ÄÄ
+         3   1   3      5
+   (3)  x  - Ä x   + O(x )
+             6
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPageEmpty3}
+\begin{paste}{ugIntroSeriesPageEmpty3}{ugIntroSeriesPagePatch3}
+\pastebutton{ugIntroSeriesPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{series(n +-> (-1)**((3*n - 4)/6)/factorial(n - 1/3),x = 0,4/3..,2)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPagePatch4}
+\begin{paste}{ugIntroSeriesPageFull4}{ugIntroSeriesPageEmpty4}
+\pastebutton{ugIntroSeriesPageFull4}{\hidepaste}
+\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }}
+\indentrel{3}\begin{verbatim}
+   (4)
+              2    3    4    5    6    7    8    9    10
+     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
+   + 
+        11
+     O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPageEmpty4}
+\begin{paste}{ugIntroSeriesPageEmpty4}{ugIntroSeriesPagePatch4}
+\pastebutton{ugIntroSeriesPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPagePatch5}
+\begin{paste}{ugIntroSeriesPageFull5}{ugIntroSeriesPageEmpty5}
+\pastebutton{ugIntroSeriesPageFull5}{\hidepaste}
+\tab{5}\spadcommand{f ** 2\free{f }}
+\indentrel{3}\begin{verbatim}
+   (5)
+                2     3     4     5     6     7     8
+     1 + 2x + 3x  + 4x  + 5x  + 6x  + 7x  + 8x  + 9x
+   + 
+        9      10      11
+     10x  + 11x   + O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPageEmpty5}
+\begin{paste}{ugIntroSeriesPageEmpty5}{ugIntroSeriesPagePatch5}
+\pastebutton{ugIntroSeriesPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{f ** 2\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPagePatch6}
+\begin{paste}{ugIntroSeriesPageFull6}{ugIntroSeriesPageEmpty6}
+\pastebutton{ugIntroSeriesPageFull6}{\hidepaste}
+\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+   (6)
+              2    3    4    5    6    7    8    9    10
+     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
+   + 
+        11
+     O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPageEmpty6}
+\begin{paste}{ugIntroSeriesPageEmpty6}{ugIntroSeriesPagePatch6}
+\pastebutton{ugIntroSeriesPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPagePatch7}
+\begin{paste}{ugIntroSeriesPageFull7}{ugIntroSeriesPageEmpty7}
+\pastebutton{ugIntroSeriesPageFull7}{\hidepaste}
+\tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }}
+\indentrel{3}\begin{verbatim}
+   (7)
+         1  2   1  3   1  4   1  5   1  6   1  7   1  8
+     x + Ä x  + Ä x  + Ä x  + Ä x  + Ä x  + Ä x  + Ä x
+         2      3      4      5      6      7      8
+   + 
+     1  9    1  10    1  11      12
+     Ä x  + ÄÄ x   + ÄÄ x   + O(x  )
+     9      10       11
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPageEmpty7}
+\begin{paste}{ugIntroSeriesPageEmpty7}{ugIntroSeriesPagePatch7}
+\pastebutton{ugIntroSeriesPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPagePatch8}
+\begin{paste}{ugIntroSeriesPageFull8}{ugIntroSeriesPageEmpty8}
+\pastebutton{ugIntroSeriesPageFull8}{\hidepaste}
+\tab{5}\spadcommand{exp(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (8)
+              2    3    4    5    6    7    8    9    10
+     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
+   + 
+        11
+     O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPageEmpty8}
+\begin{paste}{ugIntroSeriesPageEmpty8}{ugIntroSeriesPagePatch8}
+\pastebutton{ugIntroSeriesPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{exp(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPagePatch9}
+\begin{paste}{ugIntroSeriesPageFull9}{ugIntroSeriesPageEmpty9}
+\pastebutton{ugIntroSeriesPageFull9}{\hidepaste}
+\tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }}
+\indentrel{3}\begin{verbatim}
+   (9)
+             1  2   1  3    1  4    1   5    1   6
+     1 + x + Ä x  + Ä x  + ÄÄ x  + ÄÄÄ x  + ÄÄÄ x
+             2      6      24      120      720
+   + 
+     1   7     1    8      1    9      1     10      11
+   ÄÄÄÄ x  + ÄÄÄÄÄ x  + ÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x   + O(x  )
+   5040      40320      362880      3628800
+   Type: UnivariateTaylorSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPageEmpty9}
+\begin{paste}{ugIntroSeriesPageEmpty9}{ugIntroSeriesPagePatch9}
+\pastebutton{ugIntroSeriesPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPagePatch10}
+\begin{paste}{ugIntroSeriesPageFull10}{ugIntroSeriesPageEmpty10}
+\pastebutton{ugIntroSeriesPageFull10}{\hidepaste}
+\tab{5}\spadcommand{eval(f,1.0)\free{f2 }}
+\indentrel{3}\begin{verbatim}
+   (10)
+   [1.0, 2.0, 2.5, 2.6666666666 666666667,
+    2.7083333333 333333333, 2.7166666666 666666667,
+    2.7180555555 555555556, 2.7182539682 53968254,
+    2.7182787698 412698413, 2.7182815255 731922399, ...]
+                          Type: Stream Expression Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSeriesPageEmpty10}
+\begin{paste}{ugIntroSeriesPageEmpty10}{ugIntroSeriesPagePatch10}
+\pastebutton{ugIntroSeriesPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{eval(f,1.0)\free{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCallFunPagePatch1}
+\begin{paste}{ugIntroCallFunPageFull1}{ugIntroCallFunPageEmpty1}
+\pastebutton{ugIntroCallFunPageFull1}{\hidepaste}
+\tab{5}\spadcommand{factor(120)}
+\indentrel{3}\begin{verbatim}
+         3
+   (1)  2 3 5
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCallFunPageEmpty1}
+\begin{paste}{ugIntroCallFunPageEmpty1}{ugIntroCallFunPagePatch1}
+\pastebutton{ugIntroCallFunPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{factor(120)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCallFunPagePatch2}
+\begin{paste}{ugIntroCallFunPageFull2}{ugIntroCallFunPageEmpty2}
+\pastebutton{ugIntroCallFunPageFull2}{\hidepaste}
+\tab{5}\spadcommand{divide(125,7)}
+\indentrel{3}\begin{verbatim}
+   (2)  [quotient= 17,remainder= 6]
+     Type: Record(quotient: Integer,remainder: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCallFunPageEmpty2}
+\begin{paste}{ugIntroCallFunPageEmpty2}{ugIntroCallFunPagePatch2}
+\pastebutton{ugIntroCallFunPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{divide(125,7)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCallFunPagePatch3}
+\begin{paste}{ugIntroCallFunPageFull3}{ugIntroCallFunPageEmpty3}
+\pastebutton{ugIntroCallFunPageFull3}{\hidepaste}
+\tab{5}\spadcommand{quatern(3.4,5.6,2.9,0.1)}
+\indentrel{3}\begin{verbatim}
+   (3)  3.4 + 5.6 i + 2.9 j + 0.1 k
+                                 Type: Quaternion Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCallFunPageEmpty3}
+\begin{paste}{ugIntroCallFunPageEmpty3}{ugIntroCallFunPagePatch3}
+\pastebutton{ugIntroCallFunPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{quatern(3.4,5.6,2.9,0.1)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCallFunPagePatch4}
+\begin{paste}{ugIntroCallFunPageFull4}{ugIntroCallFunPageEmpty4}
+\pastebutton{ugIntroCallFunPageFull4}{\hidepaste}
+\tab{5}\spadcommand{factorial 10}
+\indentrel{3}\begin{verbatim}
+   (4)  3628800
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCallFunPageEmpty4}
+\begin{paste}{ugIntroCallFunPageEmpty4}{ugIntroCallFunPagePatch4}
+\pastebutton{ugIntroCallFunPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{factorial 10}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch1}
+\begin{paste}{ugIntroCollectPageFull1}{ugIntroCollectPageEmpty1}
+\pastebutton{ugIntroCollectPageFull1}{\hidepaste}
+\tab{5}\spadcommand{u := [1,-7,11]\bound{u }}
+\indentrel{3}\begin{verbatim}
+   (1)  [1,- 7,11]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty1}
+\begin{paste}{ugIntroCollectPageEmpty1}{ugIntroCollectPagePatch1}
+\pastebutton{ugIntroCollectPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{u := [1,-7,11]\bound{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch2}
+\begin{paste}{ugIntroCollectPageFull2}{ugIntroCollectPageEmpty2}
+\pastebutton{ugIntroCollectPageFull2}{\hidepaste}
+\tab{5}\spadcommand{first rest rest u\free{u }}
+\indentrel{3}\begin{verbatim}
+   (2)  11
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty2}
+\begin{paste}{ugIntroCollectPageEmpty2}{ugIntroCollectPagePatch2}
+\pastebutton{ugIntroCollectPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{first rest rest u\free{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch3}
+\begin{paste}{ugIntroCollectPageFull3}{ugIntroCollectPageEmpty3}
+\pastebutton{ugIntroCollectPageFull3}{\hidepaste}
+\tab{5}\spadcommand{concat!(u,[9,1,3,-4]); u\free{u }\bound{u1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  [1,- 7,11,9,1,3,- 4]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty3}
+\begin{paste}{ugIntroCollectPageEmpty3}{ugIntroCollectPagePatch3}
+\pastebutton{ugIntroCollectPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{concat!(u,[9,1,3,-4]); u\free{u }\bound{u1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch4}
+\begin{paste}{ugIntroCollectPageFull4}{ugIntroCollectPageEmpty4}
+\pastebutton{ugIntroCollectPageFull4}{\hidepaste}
+\tab{5}\spadcommand{lastnode := rest(u,3)\free{u1 }\bound{u2 }}
+\indentrel{3}\begin{verbatim}
+   (4)  [9,1,3,- 4]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty4}
+\begin{paste}{ugIntroCollectPageEmpty4}{ugIntroCollectPagePatch4}
+\pastebutton{ugIntroCollectPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{lastnode := rest(u,3)\free{u1 }\bound{u2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch5}
+\begin{paste}{ugIntroCollectPageFull5}{ugIntroCollectPageEmpty5}
+\pastebutton{ugIntroCollectPageFull5}{\hidepaste}
+\tab{5}\spadcommand{setrest!(lastnode,rest(u,2)); u\free{u2 }}
+\indentrel{3}\begin{verbatim}
+               ____
+   (5)  [1,- 7,11,9]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty5}
+\begin{paste}{ugIntroCollectPageEmpty5}{ugIntroCollectPagePatch5}
+\pastebutton{ugIntroCollectPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{setrest!(lastnode,rest(u,2)); u\free{u2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch6}
+\begin{paste}{ugIntroCollectPageFull6}{ugIntroCollectPageEmpty6}
+\pastebutton{ugIntroCollectPageFull6}{\hidepaste}
+\tab{5}\spadcommand{[factor(i) for i in 2.. by 2]\bound{stream1 }}
+\indentrel{3}\begin{verbatim}
+            2      3      2       4    2  2
+   (6)  [2,2 ,2 3,2 ,2 5,2 3,2 7,2 ,2 3 ,2 5,...]
+                          Type: Stream Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty6}
+\begin{paste}{ugIntroCollectPageEmpty6}{ugIntroCollectPagePatch6}
+\pastebutton{ugIntroCollectPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{[factor(i) for i in 2.. by 2]\bound{stream1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch7}
+\begin{paste}{ugIntroCollectPageFull7}{ugIntroCollectPageEmpty7}
+\pastebutton{ugIntroCollectPageFull7}{\hidepaste}
+\tab{5}\spadcommand{\%.36\free{stream1 }}
+\indentrel{3}\begin{verbatim}
+         3 2
+   (7)  2 3
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty7}
+\begin{paste}{ugIntroCollectPageEmpty7}{ugIntroCollectPagePatch7}
+\pastebutton{ugIntroCollectPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{\%.36\free{stream1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch8}
+\begin{paste}{ugIntroCollectPageFull8}{ugIntroCollectPageEmpty8}
+\pastebutton{ugIntroCollectPageFull8}{\hidepaste}
+\tab{5}\spadcommand{a := oneDimensionalArray [1, -7, 3, 3/2]\bound{a }}
+\indentrel{3}\begin{verbatim}
+                 3
+   (8)  [1,- 7,3,Ä]
+                 2
+             Type: OneDimensionalArray Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty8}
+\begin{paste}{ugIntroCollectPageEmpty8}{ugIntroCollectPagePatch8}
+\pastebutton{ugIntroCollectPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{a := oneDimensionalArray [1, -7, 3, 3/2]\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch9}
+\begin{paste}{ugIntroCollectPageFull9}{ugIntroCollectPageEmpty9}
+\pastebutton{ugIntroCollectPageFull9}{\hidepaste}
+\tab{5}\spadcommand{a.3 := 11; a\bound{a1 }\free{a }}
+\indentrel{3}\begin{verbatim}
+                  3
+   (9)  [1,- 7,11,Ä]
+                  2
+             Type: OneDimensionalArray Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty9}
+\begin{paste}{ugIntroCollectPageEmpty9}{ugIntroCollectPagePatch9}
+\pastebutton{ugIntroCollectPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{a.3 := 11; a\bound{a1 }\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch10}
+\begin{paste}{ugIntroCollectPageFull10}{ugIntroCollectPageEmpty10}
+\pastebutton{ugIntroCollectPageFull10}{\hidepaste}
+\tab{5}\spadcommand{concat!(a,oneDimensionalArray [1,-2])\free{a1 }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty10}
+\begin{paste}{ugIntroCollectPageEmpty10}{ugIntroCollectPagePatch10}
+\pastebutton{ugIntroCollectPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{concat!(a,oneDimensionalArray [1,-2])\free{a1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch11}
+\begin{paste}{ugIntroCollectPageFull11}{ugIntroCollectPageEmpty11}
+\pastebutton{ugIntroCollectPageFull11}{\hidepaste}
+\tab{5}\spadcommand{bits(32,true)}
+\indentrel{3}\begin{verbatim}
+   (10)  "11111111111111111111111111111111"
+                                             Type: Bits
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty11}
+\begin{paste}{ugIntroCollectPageEmpty11}{ugIntroCollectPagePatch11}
+\pastebutton{ugIntroCollectPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{bits(32,true)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch12}
+\begin{paste}{ugIntroCollectPageFull12}{ugIntroCollectPageEmpty12}
+\pastebutton{ugIntroCollectPageFull12}{\hidepaste}
+\tab{5}\spadcommand{f := flexibleArray [2, 7, -5]\bound{f }}
+\indentrel{3}\begin{verbatim}
+   (11)  [2,7,- 5]
+                            Type: FlexibleArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty12}
+\begin{paste}{ugIntroCollectPageEmpty12}{ugIntroCollectPagePatch12}
+\pastebutton{ugIntroCollectPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{f := flexibleArray [2, 7, -5]\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch13}
+\begin{paste}{ugIntroCollectPageFull13}{ugIntroCollectPageEmpty13}
+\pastebutton{ugIntroCollectPageFull13}{\hidepaste}
+\tab{5}\spadcommand{insert!(flexibleArray [11, -3],f,2)\free{f }}
+\indentrel{3}\begin{verbatim}
+   (12)  [2,11,- 3,7,- 5]
+                            Type: FlexibleArray Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty13}
+\begin{paste}{ugIntroCollectPageEmpty13}{ugIntroCollectPagePatch13}
+\pastebutton{ugIntroCollectPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{insert!(flexibleArray [11, -3],f,2)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch14}
+\begin{paste}{ugIntroCollectPageFull14}{ugIntroCollectPageEmpty14}
+\pastebutton{ugIntroCollectPageFull14}{\hidepaste}
+\tab{5}\spadcommand{h := heap [-4,7,11,3,4,-7]\bound{h }}
+\indentrel{3}\begin{verbatim}
+   (13)  [11,4,7,- 4,3,- 7]
+                                     Type: Heap Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty14}
+\begin{paste}{ugIntroCollectPageEmpty14}{ugIntroCollectPagePatch14}
+\pastebutton{ugIntroCollectPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{h := heap [-4,7,11,3,4,-7]\bound{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch15}
+\begin{paste}{ugIntroCollectPageFull15}{ugIntroCollectPageEmpty15}
+\pastebutton{ugIntroCollectPageFull15}{\hidepaste}
+\tab{5}\spadcommand{[extract!(h) while not empty?(h)]\free{h }}
+\indentrel{3}\begin{verbatim}
+   (14)  [11,7,4,3,- 4,- 7]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty15}
+\begin{paste}{ugIntroCollectPageEmpty15}{ugIntroCollectPagePatch15}
+\pastebutton{ugIntroCollectPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{[extract!(h) while not empty?(h)]\free{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch16}
+\begin{paste}{ugIntroCollectPageFull16}{ugIntroCollectPageEmpty16}
+\pastebutton{ugIntroCollectPageFull16}{\hidepaste}
+\tab{5}\spadcommand{binarySearchTree [5,3,2,9,4,7,11]}
+\indentrel{3}\begin{verbatim}
+   (15)  [[2,3,4],5,[7,9,11]]
+                 Type: BinarySearchTree PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty16}
+\begin{paste}{ugIntroCollectPageEmpty16}{ugIntroCollectPagePatch16}
+\pastebutton{ugIntroCollectPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{binarySearchTree [5,3,2,9,4,7,11]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch17}
+\begin{paste}{ugIntroCollectPageFull17}{ugIntroCollectPageEmpty17}
+\pastebutton{ugIntroCollectPageFull17}{\hidepaste}
+\tab{5}\spadcommand{modTree(8,[2,3,5,7])}
+\indentrel{3}\begin{verbatim}
+   (16)  [0,2,3,1]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty17}
+\begin{paste}{ugIntroCollectPageEmpty17}{ugIntroCollectPagePatch17}
+\pastebutton{ugIntroCollectPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{modTree(8,[2,3,5,7])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch18}
+\begin{paste}{ugIntroCollectPageFull18}{ugIntroCollectPageEmpty18}
+\pastebutton{ugIntroCollectPageFull18}{\hidepaste}
+\tab{5}\spadcommand{fs := set[1/3,4/5,-1/3,4/5]\bound{fs }}
+\indentrel{3}\begin{verbatim}
+            1 1 4
+   (17)  {- Ä,Ä,Ä}
+            3 3 5
+                             Type: Set Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty18}
+\begin{paste}{ugIntroCollectPageEmpty18}{ugIntroCollectPagePatch18}
+\pastebutton{ugIntroCollectPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{fs := set[1/3,4/5,-1/3,4/5]\bound{fs }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch19}
+\begin{paste}{ugIntroCollectPageFull19}{ugIntroCollectPageEmpty19}
+\pastebutton{ugIntroCollectPageFull19}{\hidepaste}
+\tab{5}\spadcommand{multiset [x rem 5 for x in primes(2,1000)]}
+\indentrel{3}\begin{verbatim}
+   (18)  {0,42: 3,40: 1,38: 4,47: 2}
+                                 Type: Multiset Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty19}
+\begin{paste}{ugIntroCollectPageEmpty19}{ugIntroCollectPagePatch19}
+\pastebutton{ugIntroCollectPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{multiset [x rem 5 for x in primes(2,1000)]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch20}
+\begin{paste}{ugIntroCollectPageFull20}{ugIntroCollectPageEmpty20}
+\pastebutton{ugIntroCollectPageFull20}{\hidepaste}
+\tab{5}\spadcommand{t : Table(Integer,Integer) := empty()\bound{t }}
+\indentrel{3}\begin{verbatim}
+   (19)  table()
+                           Type: Table(Integer,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty20}
+\begin{paste}{ugIntroCollectPageEmpty20}{ugIntroCollectPagePatch20}
+\pastebutton{ugIntroCollectPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{t : Table(Integer,Integer) := empty()\bound{t }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch21}
+\begin{paste}{ugIntroCollectPageFull21}{ugIntroCollectPageEmpty21}
+\pastebutton{ugIntroCollectPageFull21}{\hidepaste}
+\tab{5}\spadcommand{howMany(k) == (n:=search(k,t); n case "failed" => 1; n+1)\bound{how }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty21}
+\begin{paste}{ugIntroCollectPageEmpty21}{ugIntroCollectPagePatch21}
+\pastebutton{ugIntroCollectPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{howMany(k) == (n:=search(k,t); n case "failed" => 1; n+1)\bound{how }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch22}
+\begin{paste}{ugIntroCollectPageFull22}{ugIntroCollectPageEmpty22}
+\pastebutton{ugIntroCollectPageFull22}{\hidepaste}
+\tab{5}\spadcommand{for p in primes(2,1000) repeat (m:= p rem 5; t.m:= howMany(m)); t\free{how t }}
+\indentrel{3}\begin{verbatim}
+   (21)  table(2= 47,4= 38,1= 40,3= 42,0= 1)
+                           Type: Table(Integer,Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty22}
+\begin{paste}{ugIntroCollectPageEmpty22}{ugIntroCollectPagePatch22}
+\pastebutton{ugIntroCollectPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{for p in primes(2,1000) repeat (m:= p rem 5; t.m:= howMany(m)); t\free{how t }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch23}
+\begin{paste}{ugIntroCollectPageFull23}{ugIntroCollectPageEmpty23}
+\pastebutton{ugIntroCollectPageFull23}{\hidepaste}
+\tab{5}\spadcommand{daniel : Record(age : Integer, salary : Float)\bound{danieldec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty23}
+\begin{paste}{ugIntroCollectPageEmpty23}{ugIntroCollectPagePatch23}
+\pastebutton{ugIntroCollectPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{daniel : Record(age : Integer, salary : Float)\bound{danieldec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch24}
+\begin{paste}{ugIntroCollectPageFull24}{ugIntroCollectPageEmpty24}
+\pastebutton{ugIntroCollectPageFull24}{\hidepaste}
+\tab{5}\spadcommand{daniel := [28, 32005.12]\free{danieldec }\bound{daniel }}
+\indentrel{3}\begin{verbatim}
+   (23)  [age= 28,salary= 32005.12]
+               Type: Record(age: Integer,salary: Float)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty24}
+\begin{paste}{ugIntroCollectPageEmpty24}{ugIntroCollectPagePatch24}
+\pastebutton{ugIntroCollectPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{daniel := [28, 32005.12]\free{danieldec }\bound{daniel }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch25}
+\begin{paste}{ugIntroCollectPageFull25}{ugIntroCollectPageEmpty25}
+\pastebutton{ugIntroCollectPageFull25}{\hidepaste}
+\tab{5}\spadcommand{daniel.salary := 35000; daniel\free{daniel }}
+\indentrel{3}\begin{verbatim}
+   (24)  [age= 28,salary= 35000.0]
+               Type: Record(age: Integer,salary: Float)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty25}
+\begin{paste}{ugIntroCollectPageEmpty25}{ugIntroCollectPagePatch25}
+\pastebutton{ugIntroCollectPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{daniel.salary := 35000; daniel\free{daniel }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch26}
+\begin{paste}{ugIntroCollectPageFull26}{ugIntroCollectPageEmpty26}
+\pastebutton{ugIntroCollectPageFull26}{\hidepaste}
+\tab{5}\spadcommand{dog: Union(licenseNumber: Integer, name: String)\bound{xint }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty26}
+\begin{paste}{ugIntroCollectPageEmpty26}{ugIntroCollectPagePatch26}
+\pastebutton{ugIntroCollectPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{dog: Union(licenseNumber: Integer, name: String)\bound{xint }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPagePatch27}
+\begin{paste}{ugIntroCollectPageFull27}{ugIntroCollectPageEmpty27}
+\pastebutton{ugIntroCollectPageFull27}{\hidepaste}
+\tab{5}\spadcommand{dog := "Whisper"\free{xint }}
+\indentrel{3}\begin{verbatim}
+   (26)  "Whisper"
+                          Type: Union(name: String,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCollectPageEmpty27}
+\begin{paste}{ugIntroCollectPageEmpty27}{ugIntroCollectPagePatch27}
+\pastebutton{ugIntroCollectPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{dog := "Whisper"\free{xint }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch1}
+\begin{paste}{ugIntroNumbersPageFull1}{ugIntroNumbersPageEmpty1}
+\pastebutton{ugIntroNumbersPageFull1}{\hidepaste}
+\tab{5}\spadcommand{11**13 * 13**11 * 17**7 - 19**5 * 23**3}
+\indentrel{3}\begin{verbatim}
+   (1)  25387751112538918594666224484237298
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty1}
+\begin{paste}{ugIntroNumbersPageEmpty1}{ugIntroNumbersPagePatch1}
+\pastebutton{ugIntroNumbersPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{11**13 * 13**11 * 17**7 - 19**5 * 23**3}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch2}
+\begin{paste}{ugIntroNumbersPageFull2}{ugIntroNumbersPageEmpty2}
+\pastebutton{ugIntroNumbersPageFull2}{\hidepaste}
+\tab{5}\spadcommand{factor 643238070748569023720594412551704344145570763243\bound{ex1 }}
+\indentrel{3}\begin{verbatim}
+          13  11  7  5  3  2
+   (2)  11  13  17 19 23 29
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty2}
+\begin{paste}{ugIntroNumbersPageEmpty2}{ugIntroNumbersPagePatch2}
+\pastebutton{ugIntroNumbersPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{factor 643238070748569023720594412551704344145570763243\bound{ex1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch3}
+\begin{paste}{ugIntroNumbersPageFull3}{ugIntroNumbersPageEmpty3}
+\pastebutton{ugIntroNumbersPageFull3}{\hidepaste}
+\tab{5}\spadcommand{\% * 12\free{ex1 }}
+\indentrel{3}\begin{verbatim}
+         2    13  11  7  5  3  2
+   (3)  2 3 11  13  17 19 23 29
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty3}
+\begin{paste}{ugIntroNumbersPageEmpty3}{ugIntroNumbersPagePatch3}
+\pastebutton{ugIntroNumbersPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{\% * 12\free{ex1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch4}
+\begin{paste}{ugIntroNumbersPageFull4}{ugIntroNumbersPageEmpty4}
+\pastebutton{ugIntroNumbersPageFull4}{\hidepaste}
+\tab{5}\spadcommand{radix(25937424601,11)}
+\indentrel{3}\begin{verbatim}
+   (4)  10000000000
+                                Type: RadixExpansion 11
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty4}
+\begin{paste}{ugIntroNumbersPageEmpty4}{ugIntroNumbersPagePatch4}
+\pastebutton{ugIntroNumbersPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{radix(25937424601,11)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch5}
+\begin{paste}{ugIntroNumbersPageFull5}{ugIntroNumbersPageEmpty5}
+\pastebutton{ugIntroNumbersPageFull5}{\hidepaste}
+\tab{5}\spadcommand{roman(1992)}
+\indentrel{3}\begin{verbatim}
+   (5)  MCMXCII
+                                     Type: RomanNumeral
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty5}
+\begin{paste}{ugIntroNumbersPageEmpty5}{ugIntroNumbersPagePatch5}
+\pastebutton{ugIntroNumbersPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{roman(1992)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch6}
+\begin{paste}{ugIntroNumbersPageFull6}{ugIntroNumbersPageEmpty6}
+\pastebutton{ugIntroNumbersPageFull6}{\hidepaste}
+\tab{5}\spadcommand{r := 10 + 9/2 + 8/3 + 7/4 + 6/5 + 5/6 + 4/7 + 3/8 + 2/9\bound{r }}
+\indentrel{3}\begin{verbatim}
+        55739
+   (6)  ÄÄÄÄÄ
+         2520
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty6}
+\begin{paste}{ugIntroNumbersPageEmpty6}{ugIntroNumbersPagePatch6}
+\pastebutton{ugIntroNumbersPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{r := 10 + 9/2 + 8/3 + 7/4 + 6/5 + 5/6 + 4/7 + 3/8 + 2/9\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch7}
+\begin{paste}{ugIntroNumbersPageFull7}{ugIntroNumbersPageEmpty7}
+\pastebutton{ugIntroNumbersPageFull7}{\hidepaste}
+\tab{5}\spadcommand{map(factor,r)\free{r }}
+\indentrel{3}\begin{verbatim}
+        139 401
+   (7)  ÄÄÄÄÄÄÄ
+         3 2
+        2 3 5 7
+                        Type: Fraction Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty7}
+\begin{paste}{ugIntroNumbersPageEmpty7}{ugIntroNumbersPagePatch7}
+\pastebutton{ugIntroNumbersPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{map(factor,r)\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch8}
+\begin{paste}{ugIntroNumbersPageFull8}{ugIntroNumbersPageEmpty8}
+\pastebutton{ugIntroNumbersPageFull8}{\hidepaste}
+\tab{5}\spadcommand{11@SingleInteger}
+\indentrel{3}\begin{verbatim}
+   (8)  11
+                                    Type: SingleInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty8}
+\begin{paste}{ugIntroNumbersPageEmpty8}{ugIntroNumbersPagePatch8}
+\pastebutton{ugIntroNumbersPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{11@SingleInteger}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch9}
+\begin{paste}{ugIntroNumbersPageFull9}{ugIntroNumbersPageEmpty9}
+\pastebutton{ugIntroNumbersPageFull9}{\hidepaste}
+\tab{5}\spadcommand{123.21@DoubleFloat}
+\indentrel{3}\begin{verbatim}
+   (9)  123.21000000000001
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty9}
+\begin{paste}{ugIntroNumbersPageEmpty9}{ugIntroNumbersPagePatch9}
+\pastebutton{ugIntroNumbersPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{123.21@DoubleFloat}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch10}
+\begin{paste}{ugIntroNumbersPageFull10}{ugIntroNumbersPageEmpty10}
+\pastebutton{ugIntroNumbersPageFull10}{\hidepaste}
+\tab{5}\spadcommand{r :: Float\free{r }}
+\indentrel{3}\begin{verbatim}
+   (10)  22.1186507936 50793651
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty10}
+\begin{paste}{ugIntroNumbersPageEmpty10}{ugIntroNumbersPagePatch10}
+\pastebutton{ugIntroNumbersPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{r :: Float\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch11}
+\begin{paste}{ugIntroNumbersPageFull11}{ugIntroNumbersPageEmpty11}
+\pastebutton{ugIntroNumbersPageFull11}{\hidepaste}
+\tab{5}\spadcommand{digits(22)\bound{fewerdigits }}
+\indentrel{3}\begin{verbatim}
+   (11)  20
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty11}
+\begin{paste}{ugIntroNumbersPageEmpty11}{ugIntroNumbersPagePatch11}
+\pastebutton{ugIntroNumbersPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{digits(22)\bound{fewerdigits }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch12}
+\begin{paste}{ugIntroNumbersPageFull12}{ugIntroNumbersPageEmpty12}
+\pastebutton{ugIntroNumbersPageFull12}{\hidepaste}
+\tab{5}\spadcommand{exp(\%pi * sqrt 163.0)\free{fewerdigits }}
+\indentrel{3}\begin{verbatim}
+   (12)  26253741 2640768744.0
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty12}
+\begin{paste}{ugIntroNumbersPageEmpty12}{ugIntroNumbersPagePatch12}
+\pastebutton{ugIntroNumbersPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{exp(\%pi * sqrt 163.0)\free{fewerdigits }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch13}
+\begin{paste}{ugIntroNumbersPageFull13}{ugIntroNumbersPageEmpty13}
+\pastebutton{ugIntroNumbersPageFull13}{\hidepaste}
+\tab{5}\spadcommand{digits(40); exp(\%pi * sqrt 163.0)\free{moredigits }}
+\indentrel{3}\begin{verbatim}
+   (13)  26253741 2640768743.9999999999 9925007259 76
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty13}
+\begin{paste}{ugIntroNumbersPageEmpty13}{ugIntroNumbersPagePatch13}
+\pastebutton{ugIntroNumbersPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{digits(40); exp(\%pi * sqrt 163.0)\free{moredigits }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch14}
+\begin{paste}{ugIntroNumbersPageFull14}{ugIntroNumbersPageEmpty14}
+\pastebutton{ugIntroNumbersPageFull14}{\hidepaste}
+\tab{5}\spadcommand{(2/3 + \%i)**3\bound{gaussint }}
+\indentrel{3}\begin{verbatim}
+           46   1
+   (14)  - ÄÄ + Ä %i
+           27   3
+                         Type: Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty14}
+\begin{paste}{ugIntroNumbersPageEmpty14}{ugIntroNumbersPagePatch14}
+\pastebutton{ugIntroNumbersPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{(2/3 + \%i)**3\bound{gaussint }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch15}
+\begin{paste}{ugIntroNumbersPageFull15}{ugIntroNumbersPageEmpty15}
+\pastebutton{ugIntroNumbersPageFull15}{\hidepaste}
+\tab{5}\spadcommand{conjugate \%\free{gaussint }}
+\indentrel{3}\begin{verbatim}
+           46   1
+   (15)  - ÄÄ - Ä %i
+           27   3
+                         Type: Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty15}
+\begin{paste}{ugIntroNumbersPageEmpty15}{ugIntroNumbersPagePatch15}
+\pastebutton{ugIntroNumbersPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{conjugate \%\free{gaussint }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch16}
+\begin{paste}{ugIntroNumbersPageFull16}{ugIntroNumbersPageEmpty16}
+\pastebutton{ugIntroNumbersPageFull16}{\hidepaste}
+\tab{5}\spadcommand{factor(89 - 23 * \%i)}
+\indentrel{3}\begin{verbatim}
+                           2         2
+   (16)  - (1 + %i)(2 + %i) (3 + 2%i)
+                         Type: Factored Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty16}
+\begin{paste}{ugIntroNumbersPageEmpty16}{ugIntroNumbersPagePatch16}
+\pastebutton{ugIntroNumbersPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{factor(89 - 23 * \%i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch17}
+\begin{paste}{ugIntroNumbersPageFull17}{ugIntroNumbersPageEmpty17}
+\pastebutton{ugIntroNumbersPageFull17}{\hidepaste}
+\tab{5}\spadcommand{exp(\%pi/4.0 * \%i)}
+\indentrel{3}\begin{verbatim}
+   (17)
+     0.7071067811 8654752440 0844362104 8490392849
+   + 
+     0.7071067811 8654752440 0844362104 8490392848 %i
+                                    Type: Complex Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty17}
+\begin{paste}{ugIntroNumbersPageEmpty17}{ugIntroNumbersPagePatch17}
+\pastebutton{ugIntroNumbersPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{exp(\%pi/4.0 * \%i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch18}
+\begin{paste}{ugIntroNumbersPageFull18}{ugIntroNumbersPageEmpty18}
+\pastebutton{ugIntroNumbersPageFull18}{\hidepaste}
+\tab{5}\spadcommand{decimal(1/352)}
+\indentrel{3}\begin{verbatim}
+                __
+   (18)  0.0028409
+                                 Type: DecimalExpansion
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty18}
+\begin{paste}{ugIntroNumbersPageEmpty18}{ugIntroNumbersPagePatch18}
+\pastebutton{ugIntroNumbersPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{decimal(1/352)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch19}
+\begin{paste}{ugIntroNumbersPageFull19}{ugIntroNumbersPageEmpty19}
+\pastebutton{ugIntroNumbersPageFull19}{\hidepaste}
+\tab{5}\spadcommand{continuedFraction(6543/210)}
+\indentrel{3}\begin{verbatim}
+                1 ³     1 ³     1 ³     1 ³
+   (19)  31 + ÚÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÙ + ÚÄÄÄÙ
+              ³ 6     ³ 2     ³ 1     ³ 3
+                        Type: ContinuedFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty19}
+\begin{paste}{ugIntroNumbersPageEmpty19}{ugIntroNumbersPagePatch19}
+\pastebutton{ugIntroNumbersPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{continuedFraction(6543/210)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch20}
+\begin{paste}{ugIntroNumbersPageFull20}{ugIntroNumbersPageEmpty20}
+\pastebutton{ugIntroNumbersPageFull20}{\hidepaste}
+\tab{5}\spadcommand{partialFraction(1,factorial(10))\bound{partfrac }}
+\indentrel{3}\begin{verbatim}
+         159   23   12   1
+   (20)  ÄÄÄ - ÄÄ - ÄÄ + Ä
+           8    4    2   7
+          2    3    5
+                          Type: PartialFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty20}
+\begin{paste}{ugIntroNumbersPageEmpty20}{ugIntroNumbersPagePatch20}
+\pastebutton{ugIntroNumbersPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{partialFraction(1,factorial(10))\bound{partfrac }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch21}
+\begin{paste}{ugIntroNumbersPageFull21}{ugIntroNumbersPageEmpty21}
+\pastebutton{ugIntroNumbersPageFull21}{\hidepaste}
+\tab{5}\spadcommand{padicFraction(\%)\free{partfrac }}
+\indentrel{3}\begin{verbatim}
+   (21)
+   1    1    1    1    1    1    2    1    2   2    2   1
+   Ä + ÄÄ + ÄÄ + ÄÄ + ÄÄ + ÄÄ - ÄÄ - ÄÄ - ÄÄ - Ä - ÄÄ + Ä
+   2    4    5    6    7    8    2    3    4   5    2   7
+       2    2    2    2    2    3    3    3        5
+                          Type: PartialFraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty21}
+\begin{paste}{ugIntroNumbersPageEmpty21}{ugIntroNumbersPagePatch21}
+\pastebutton{ugIntroNumbersPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{padicFraction(\%)\free{partfrac }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch22}
+\begin{paste}{ugIntroNumbersPageFull22}{ugIntroNumbersPageEmpty22}
+\pastebutton{ugIntroNumbersPageFull22}{\hidepaste}
+\tab{5}\spadcommand{radix(4/7, 8)\bound{rad }}
+\indentrel{3}\begin{verbatim}
+           _
+   (22)  0.4
+                                 Type: RadixExpansion 8
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty22}
+\begin{paste}{ugIntroNumbersPageEmpty22}{ugIntroNumbersPagePatch22}
+\pastebutton{ugIntroNumbersPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{radix(4/7, 8)\bound{rad }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch23}
+\begin{paste}{ugIntroNumbersPageFull23}{ugIntroNumbersPageEmpty23}
+\pastebutton{ugIntroNumbersPageFull23}{\hidepaste}
+\tab{5}\spadcommand{\% + 2/3*\%i\free{rad }}
+\indentrel{3}\begin{verbatim}
+         4   2
+   (23)  Ä + Ä %i
+         7   3
+                         Type: Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty23}
+\begin{paste}{ugIntroNumbersPageEmpty23}{ugIntroNumbersPagePatch23}
+\pastebutton{ugIntroNumbersPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{\% + 2/3*\%i\free{rad }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch24}
+\begin{paste}{ugIntroNumbersPageFull24}{ugIntroNumbersPageEmpty24}
+\pastebutton{ugIntroNumbersPageFull24}{\hidepaste}
+\tab{5}\spadcommand{(5 + sqrt 63 + sqrt 847)**(1/3)}
+\indentrel{3}\begin{verbatim}
+          ÚÄÄÄÄÄÄÄÄÄÄ¿
+         3³   ÚÄ¿
+   (24)  \³14\³7  + 5
+                                  Type: AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty24}
+\begin{paste}{ugIntroNumbersPageEmpty24}{ugIntroNumbersPagePatch24}
+\pastebutton{ugIntroNumbersPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{(5 + sqrt 63 + sqrt 847)**(1/3)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch25}
+\begin{paste}{ugIntroNumbersPageFull25}{ugIntroNumbersPageEmpty25}
+\pastebutton{ugIntroNumbersPageFull25}{\hidepaste}
+\tab{5}\spadcommand{x : PrimeField 7 := 5\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (25)  5
+                                     Type: PrimeField 7
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty25}
+\begin{paste}{ugIntroNumbersPageEmpty25}{ugIntroNumbersPagePatch25}
+\pastebutton{ugIntroNumbersPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{x : PrimeField 7 := 5\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch26}
+\begin{paste}{ugIntroNumbersPageFull26}{ugIntroNumbersPageEmpty26}
+\pastebutton{ugIntroNumbersPageFull26}{\hidepaste}
+\tab{5}\spadcommand{x**3\free{x }}
+\indentrel{3}\begin{verbatim}
+   (26)  6
+                                     Type: PrimeField 7
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty26}
+\begin{paste}{ugIntroNumbersPageEmpty26}{ugIntroNumbersPagePatch26}
+\pastebutton{ugIntroNumbersPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{x**3\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch27}
+\begin{paste}{ugIntroNumbersPageFull27}{ugIntroNumbersPageEmpty27}
+\pastebutton{ugIntroNumbersPageFull27}{\hidepaste}
+\tab{5}\spadcommand{1/x\free{x }}
+\indentrel{3}\begin{verbatim}
+   (27)  3
+                                     Type: PrimeField 7
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty27}
+\begin{paste}{ugIntroNumbersPageEmpty27}{ugIntroNumbersPagePatch27}
+\pastebutton{ugIntroNumbersPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{1/x\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch28}
+\begin{paste}{ugIntroNumbersPageFull28}{ugIntroNumbersPageEmpty28}
+\pastebutton{ugIntroNumbersPageFull28}{\hidepaste}
+\tab{5}\spadcommand{y : IntegerMod 6 := 5\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (28)  5
+                                     Type: IntegerMod 6
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty28}
+\begin{paste}{ugIntroNumbersPageEmpty28}{ugIntroNumbersPagePatch28}
+\pastebutton{ugIntroNumbersPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{y : IntegerMod 6 := 5\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch29}
+\begin{paste}{ugIntroNumbersPageFull29}{ugIntroNumbersPageEmpty29}
+\pastebutton{ugIntroNumbersPageFull29}{\hidepaste}
+\tab{5}\spadcommand{y**3\free{y }}
+\indentrel{3}\begin{verbatim}
+   (29)  5
+                                     Type: IntegerMod 6
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty29}
+\begin{paste}{ugIntroNumbersPageEmpty29}{ugIntroNumbersPagePatch29}
+\pastebutton{ugIntroNumbersPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{y**3\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch30}
+\begin{paste}{ugIntroNumbersPageFull30}{ugIntroNumbersPageEmpty30}
+\pastebutton{ugIntroNumbersPageFull30}{\hidepaste}
+\tab{5}\spadcommand{1/y\free{y }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty30}
+\begin{paste}{ugIntroNumbersPageEmpty30}{ugIntroNumbersPagePatch30}
+\pastebutton{ugIntroNumbersPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{1/y\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch31}
+\begin{paste}{ugIntroNumbersPageFull31}{ugIntroNumbersPageEmpty31}
+\pastebutton{ugIntroNumbersPageFull31}{\hidepaste}
+\tab{5}\spadcommand{a := rootOf(a**5 + a**3 + a**2 + 3,a)\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (30)  a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty31}
+\begin{paste}{ugIntroNumbersPageEmpty31}{ugIntroNumbersPagePatch31}
+\pastebutton{ugIntroNumbersPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{a := rootOf(a**5 + a**3 + a**2 + 3,a)\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch32}
+\begin{paste}{ugIntroNumbersPageFull32}{ugIntroNumbersPageEmpty32}
+\pastebutton{ugIntroNumbersPageFull32}{\hidepaste}
+\tab{5}\spadcommand{(a + 1)**10\free{a }}
+\indentrel{3}\begin{verbatim}
+              4       3       2
+   (31)  - 85a  - 264a  - 378a  - 458a - 287
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty32}
+\begin{paste}{ugIntroNumbersPageEmpty32}{ugIntroNumbersPagePatch32}
+\pastebutton{ugIntroNumbersPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{(a + 1)**10\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch33}
+\begin{paste}{ugIntroNumbersPageFull33}{ugIntroNumbersPageEmpty33}
+\pastebutton{ugIntroNumbersPageFull33}{\hidepaste}
+\tab{5}\spadcommand{b := rootOf(b**4 + a,b)\bound{b }\free{a }}
+\indentrel{3}\begin{verbatim}
+   (32)  b
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty33}
+\begin{paste}{ugIntroNumbersPageEmpty33}{ugIntroNumbersPagePatch33}
+\pastebutton{ugIntroNumbersPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{b := rootOf(b**4 + a,b)\bound{b }\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch34}
+\begin{paste}{ugIntroNumbersPageFull34}{ugIntroNumbersPageEmpty34}
+\pastebutton{ugIntroNumbersPageFull34}{\hidepaste}
+\tab{5}\spadcommand{2/(b - 1)\free{b }\bound{check }}
+\indentrel{3}\begin{verbatim}
+           2
+   (33)  ÄÄÄÄÄ
+         b - 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty34}
+\begin{paste}{ugIntroNumbersPageEmpty34}{ugIntroNumbersPagePatch34}
+\pastebutton{ugIntroNumbersPageEmpty34}{\showpaste}
+\tab{5}\spadcommand{2/(b - 1)\free{b }\bound{check }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch35}
+\begin{paste}{ugIntroNumbersPageFull35}{ugIntroNumbersPageEmpty35}
+\pastebutton{ugIntroNumbersPageFull35}{\hidepaste}
+\tab{5}\spadcommand{ratDenom(\%)\free{check }\bound{check1 }}
+\indentrel{3}\begin{verbatim}
+   (34)
+       4    3     2          3     4    3     2          2
+     (a  - a  + 2a  - a + 1)b  + (a  - a  + 2a  - a + 1)b
+   + 
+       4    3     2              4    3     2
+     (a  - a  + 2a  - a + 1)b + a  - a  + 2a  - a + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty35}
+\begin{paste}{ugIntroNumbersPageEmpty35}{ugIntroNumbersPagePatch35}
+\pastebutton{ugIntroNumbersPageEmpty35}{\showpaste}
+\tab{5}\spadcommand{ratDenom(\%)\free{check }\bound{check1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch36}
+\begin{paste}{ugIntroNumbersPageFull36}{ugIntroNumbersPageEmpty36}
+\pastebutton{ugIntroNumbersPageFull36}{\hidepaste}
+\tab{5}\spadcommand{2/\%+1\free{check1 }\bound{check2 }}
+\indentrel{3}\begin{verbatim}
+   (35)
+         4    3     2          3
+       (a  - a  + 2a  - a + 1)b
+     + 
+         4    3     2          2
+       (a  - a  + 2a  - a + 1)b
+     + 
+         4    3     2              4    3     2
+       (a  - a  + 2a  - a + 1)b + a  - a  + 2a  - a + 3
+  /
+         4    3     2          3
+       (a  - a  + 2a  - a + 1)b
+     + 
+         4    3     2          2
+       (a  - a  + 2a  - a + 1)b
+     + 
+         4    3     2              4    3     2
+       (a  - a  + 2a  - a + 1)b + a  - a  + 2a  - a + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty36}
+\begin{paste}{ugIntroNumbersPageEmpty36}{ugIntroNumbersPagePatch36}
+\pastebutton{ugIntroNumbersPageEmpty36}{\showpaste}
+\tab{5}\spadcommand{2/\%+1\free{check1 }\bound{check2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch37}
+\begin{paste}{ugIntroNumbersPageFull37}{ugIntroNumbersPageEmpty37}
+\pastebutton{ugIntroNumbersPageFull37}{\hidepaste}
+\tab{5}\spadcommand{ratDenom(\%)\free{check2 }}
+\indentrel{3}\begin{verbatim}
+   (36)  b
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty37}
+\begin{paste}{ugIntroNumbersPageEmpty37}{ugIntroNumbersPagePatch37}
+\pastebutton{ugIntroNumbersPageEmpty37}{\showpaste}
+\tab{5}\spadcommand{ratDenom(\%)\free{check2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPagePatch38}
+\begin{paste}{ugIntroNumbersPageFull38}{ugIntroNumbersPageEmpty38}
+\pastebutton{ugIntroNumbersPageFull38}{\hidepaste}
+\tab{5}\spadcommand{q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)}
+\indentrel{3}\begin{verbatim}
+   (37)  - 8i + 16j - 8k
+                               Type: Quaternion Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroNumbersPageEmpty38}
+\begin{paste}{ugIntroNumbersPageEmpty38}{ugIntroNumbersPagePatch38}
+\pastebutton{ugIntroNumbersPageEmpty38}{\showpaste}
+\tab{5}\spadcommand{q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPagePatch1}
+\begin{paste}{ugIntroDiffEqnsPageFull1}{ugIntroDiffEqnsPageEmpty1}
+\pastebutton{ugIntroDiffEqnsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{y := operator 'y\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (1)  y
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPageEmpty1}
+\begin{paste}{ugIntroDiffEqnsPageEmpty1}{ugIntroDiffEqnsPagePatch1}
+\pastebutton{ugIntroDiffEqnsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{y := operator 'y\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPagePatch2}
+\begin{paste}{ugIntroDiffEqnsPageFull2}{ugIntroDiffEqnsPageEmpty2}
+\pastebutton{ugIntroDiffEqnsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4\bound{e3 }\free{y }}
+\indentrel{3}\begin{verbatim}
+         3 ,,,       2 ,,         ,               4
+   (2)  x y   (x) + x y  (x) - 2xy (x) + 2y(x)= 2x
+
+                      Type: Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPageEmpty2}
+\begin{paste}{ugIntroDiffEqnsPageEmpty2}{ugIntroDiffEqnsPagePatch2}
+\pastebutton{ugIntroDiffEqnsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4\bound{e3 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPagePatch3}
+\begin{paste}{ugIntroDiffEqnsPageFull3}{ugIntroDiffEqnsPageEmpty3}
+\pastebutton{ugIntroDiffEqnsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }}
+\indentrel{3}\begin{verbatim}
+   (3)
+                 5      3      2
+                x  - 10x  + 20x  + 4
+   [particular= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,
+                         15x
+              3     2      3      3     2
+            2x  - 3x  + 1 x  - 1 x  - 3x  - 1
+    basis= [ÄÄÄÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄ]]
+                  x          x         x
+Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPageEmpty3}
+\begin{paste}{ugIntroDiffEqnsPageEmpty3}{ugIntroDiffEqnsPagePatch3}
+\pastebutton{ugIntroDiffEqnsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPagePatch4}
+\begin{paste}{ugIntroDiffEqnsPageFull4}{ugIntroDiffEqnsPageEmpty4}
+\pastebutton{ugIntroDiffEqnsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0\bound{e5 }\free{y }}
+\indentrel{3}\begin{verbatim}
+          2      ,,         ,
+   (4)  (x  + 1)y  (x) + 3xy (x) + y(x)= 0
+
+                      Type: Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPageEmpty4}
+\begin{paste}{ugIntroDiffEqnsPageEmpty4}{ugIntroDiffEqnsPagePatch4}
+\pastebutton{ugIntroDiffEqnsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0\bound{e5 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPagePatch5}
+\begin{paste}{ugIntroDiffEqnsPageFull5}{ugIntroDiffEqnsPageEmpty5}
+\pastebutton{ugIntroDiffEqnsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }}
+\indentrel{3}\begin{verbatim}
+   (5)
+                                         ÚÄÄÄÄÄÄ¿
+                                         ³ 2
+                              1     log(\³x  + 1  - x)
+   [particular= 0,basis= [ÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ]]
+                           ÚÄÄÄÄÄÄ¿       ÚÄÄÄÄÄÄ¿
+                           ³ 2            ³ 2
+                          \³x  + 1       \³x  + 1
+Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPageEmpty5}
+\begin{paste}{ugIntroDiffEqnsPageEmpty5}{ugIntroDiffEqnsPagePatch5}
+\pastebutton{ugIntroDiffEqnsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPagePatch6}
+\begin{paste}{ugIntroDiffEqnsPageFull6}{ugIntroDiffEqnsPageEmpty6}
+\pastebutton{ugIntroDiffEqnsPageFull6}{\hidepaste}
+\tab{5}\spadcommand{eq := 2*x**3 * D(y x,x,2) + 3*x**2 * D(y x,x) - 2 * y x\bound{eq }\free{y }}
+\indentrel{3}\begin{verbatim}
+          3 ,,        2 ,
+   (6)  2x y  (x) + 3x y (x) - 2y(x)
+
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPageEmpty6}
+\begin{paste}{ugIntroDiffEqnsPageEmpty6}{ugIntroDiffEqnsPagePatch6}
+\pastebutton{ugIntroDiffEqnsPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{eq := 2*x**3 * D(y x,x,2) + 3*x**2 * D(y x,x) - 2 * y x\bound{eq }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPagePatch7}
+\begin{paste}{ugIntroDiffEqnsPageFull7}{ugIntroDiffEqnsPageEmpty7}
+\pastebutton{ugIntroDiffEqnsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{solve(eq,y,x).basis\free{eq }}
+\indentrel{3}\begin{verbatim}
+               2      2
+           - ÄÄÄÄ   ÄÄÄÄ
+              ÚÄ¿    ÚÄ¿
+             \³x    \³x
+   (7)  [%e      ,%e    ]
+                          Type: List Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPageEmpty7}
+\begin{paste}{ugIntroDiffEqnsPageEmpty7}{ugIntroDiffEqnsPagePatch7}
+\pastebutton{ugIntroDiffEqnsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{solve(eq,y,x).basis\free{eq }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPagePatch8}
+\begin{paste}{ugIntroDiffEqnsPageFull8}{ugIntroDiffEqnsPageEmpty8}
+\pastebutton{ugIntroDiffEqnsPageFull8}{\hidepaste}
+\tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }}
+\indentrel{3}\begin{verbatim}
+         ,            y(x)
+   (8)  y (x)= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+               y(x)log(y(x)) + x
+                      Type: Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPageEmpty8}
+\begin{paste}{ugIntroDiffEqnsPageEmpty8}{ugIntroDiffEqnsPagePatch8}
+\pastebutton{ugIntroDiffEqnsPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPagePatch9}
+\begin{paste}{ugIntroDiffEqnsPageFull9}{ugIntroDiffEqnsPageEmpty9}
+\pastebutton{ugIntroDiffEqnsPageFull9}{\hidepaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }}
+\indentrel{3}\begin{verbatim}
+                     2
+        y(x)log(y(x))  - 2x
+   (9)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+               2y(x)
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPageEmpty9}
+\begin{paste}{ugIntroDiffEqnsPageEmpty9}{ugIntroDiffEqnsPagePatch9}
+\pastebutton{ugIntroDiffEqnsPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPagePatch10}
+\begin{paste}{ugIntroDiffEqnsPageFull10}{ugIntroDiffEqnsPageEmpty10}
+\pastebutton{ugIntroDiffEqnsPageFull10}{\hidepaste}
+\tab{5}\spadcommand{x := operator 'x\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (10)  x
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPageEmpty10}
+\begin{paste}{ugIntroDiffEqnsPageEmpty10}{ugIntroDiffEqnsPagePatch10}
+\pastebutton{ugIntroDiffEqnsPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{x := operator 'x\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPagePatch11}
+\begin{paste}{ugIntroDiffEqnsPageFull11}{ugIntroDiffEqnsPageEmpty11}
+\pastebutton{ugIntroDiffEqnsPageFull11}{\hidepaste}
+\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2\free{x }\free{y }\bound{eq1 }}
+\indentrel{3}\begin{verbatim}
+          ,         2
+   (11)  x (t)= x(t)  + 1
+
+                      Type: Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPageEmpty11}
+\begin{paste}{ugIntroDiffEqnsPageEmpty11}{ugIntroDiffEqnsPagePatch11}
+\pastebutton{ugIntroDiffEqnsPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2\free{x }\free{y }\bound{eq1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPagePatch12}
+\begin{paste}{ugIntroDiffEqnsPageFull12}{ugIntroDiffEqnsPageEmpty12}
+\pastebutton{ugIntroDiffEqnsPageFull12}{\hidepaste}
+\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }}
+\indentrel{3}\begin{verbatim}
+          ,
+   (12)  y (t)= x(t)y(t)
+
+                      Type: Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPageEmpty12}
+\begin{paste}{ugIntroDiffEqnsPageEmpty12}{ugIntroDiffEqnsPagePatch12}
+\pastebutton{ugIntroDiffEqnsPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPagePatch13}
+\begin{paste}{ugIntroDiffEqnsPageFull13}{ugIntroDiffEqnsPageEmpty13}
+\pastebutton{ugIntroDiffEqnsPageFull13}{\hidepaste}
+\tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }}
+\indentrel{3}\begin{verbatim}
+   (13)
+        1  3    2  5    17  7    62   9      11
+   [t + Ä t  + ÄÄ t  + ÄÄÄ t  + ÄÄÄÄ t  + O(t  ),
+        3      15      315      2835
+
+           1  2    5  4    61  6    277  8    50521   10
+       1 + Ä t  + ÄÄ t  + ÄÄÄ t  + ÄÄÄÄ t  + ÄÄÄÄÄÄÄ t
+           2      24      720      8064      3628800
+     + 
+          11
+       O(t  )
+     ]
+Type: List UnivariateTaylorSeries(Expression Integer,t,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroDiffEqnsPageEmpty13}
+\begin{paste}{ugIntroDiffEqnsPageEmpty13}{ugIntroDiffEqnsPagePatch13}
+\pastebutton{ugIntroDiffEqnsPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroGraphicsPagePatch1}
+\begin{paste}{ugIntroGraphicsPageFull1}{ugIntroGraphicsPageEmpty1}
+\pastebutton{ugIntroGraphicsPageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(cos(5*t/8), t=0..16*\%pi, coordinates==polar)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugIntroGraphicsPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugIntroGraphicsPage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroGraphicsPageEmpty1}
+\begin{paste}{ugIntroGraphicsPageEmpty1}{ugIntroGraphicsPagePatch1}
+\pastebutton{ugIntroGraphicsPageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(cos(5*t/8), t=0..16*\%pi, coordinates==polar)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroGraphicsPagePatch2}
+\begin{paste}{ugIntroGraphicsPageFull2}{ugIntroGraphicsPageEmpty2}
+\pastebutton{ugIntroGraphicsPageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction == (x,y) +-> argument atan complex(x,y))}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugIntroGraphicsPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugIntroGraphicsPage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroGraphicsPageEmpty2}
+\begin{paste}{ugIntroGraphicsPageEmpty2}{ugIntroGraphicsPagePatch2}
+\pastebutton{ugIntroGraphicsPageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction == (x,y) +-> argument atan complex(x,y))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCommentsPagePatch1}
+\begin{paste}{ugIntroCommentsPageFull1}{ugIntroCommentsPageEmpty1}
+\pastebutton{ugIntroCommentsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{2 + 3 -- this is rather simple, no?}
+\indentrel{3}\begin{verbatim}
+   (1)  5
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroCommentsPageEmpty1}
+\begin{paste}{ugIntroCommentsPageEmpty1}{ugIntroCommentsPagePatch1}
+\pastebutton{ugIntroCommentsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{2 + 3 -- this is rather simple, no?}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSolutionPagePatch1}
+\begin{paste}{ugIntroSolutionPageFull1}{ugIntroSolutionPageEmpty1}
+\pastebutton{ugIntroSolutionPageFull1}{\hidepaste}
+\tab{5}\spadcommand{S(t) == [x**2-2*y**2 - t,x*y-y-5*x + 5]\bound{S1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSolutionPageEmpty1}
+\begin{paste}{ugIntroSolutionPageEmpty1}{ugIntroSolutionPagePatch1}
+\pastebutton{ugIntroSolutionPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{S(t) == [x**2-2*y**2 - t,x*y-y-5*x + 5]\bound{S1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSolutionPagePatch2}
+\begin{paste}{ugIntroSolutionPageFull2}{ugIntroSolutionPageEmpty2}
+\pastebutton{ugIntroSolutionPageFull2}{\hidepaste}
+\tab{5}\spadcommand{solve(S(19),1/10**20)\free{S1 }}
+\indentrel{3}\begin{verbatim}
+   (2)
+               2451682632253093442511
+   [[y= 5,x= - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ],
+                295147905179352825856
+             2451682632253093442511
+    [y= 5,x= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ]]
+              295147905179352825856
+   Type: List List Equation Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSolutionPageEmpty2}
+\begin{paste}{ugIntroSolutionPageEmpty2}{ugIntroSolutionPagePatch2}
+\pastebutton{ugIntroSolutionPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{solve(S(19),1/10**20)\free{S1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSolutionPagePatch3}
+\begin{paste}{ugIntroSolutionPageFull3}{ugIntroSolutionPageEmpty3}
+\pastebutton{ugIntroSolutionPageFull3}{\hidepaste}
+\tab{5}\spadcommand{complexSolve(S(19),10.e-20)\free{S1 }}
+\indentrel{3}\begin{verbatim}
+   (3)
+   [
+     [y= 5.0,
+      x= 8.3066238629 1807485256 1669055295 290320373]
+     ,
+
+     [y= 5.0,
+      x= - 8.3066238629 1807485256 1669055295 290320373]
+     ,
+    [y= - 3.0 %i,x= 1.0], [y= 3.0 %i,x= 1.0]]
+      Type: List List Equation Polynomial Complex Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSolutionPageEmpty3}
+\begin{paste}{ugIntroSolutionPageEmpty3}{ugIntroSolutionPagePatch3}
+\pastebutton{ugIntroSolutionPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{complexSolve(S(19),10.e-20)\free{S1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSolutionPagePatch4}
+\begin{paste}{ugIntroSolutionPageFull4}{ugIntroSolutionPageEmpty4}
+\pastebutton{ugIntroSolutionPageFull4}{\hidepaste}
+\tab{5}\spadcommand{radicalSolve(S(a),[x,y])\free{S1 }}
+\indentrel{3}\begin{verbatim}
+   (4)
+           ÚÄÄÄÄÄÄ¿             ÚÄÄÄÄÄÄ¿
+   [[x= - \³a + 50 ,y= 5], [x= \³a + 50 ,y= 5],
+              ÚÄÄÄÄÄÄÄ¿               ÚÄÄÄÄÄÄÄ¿
+              ³- a + 1                ³- a + 1
+    [x= 1,y=  ³ÄÄÄÄÄÄÄ ], [x= 1,y= -  ³ÄÄÄÄÄÄÄ ]]
+             \³   2                  \³   2
+            Type: List List Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSolutionPageEmpty4}
+\begin{paste}{ugIntroSolutionPageEmpty4}{ugIntroSolutionPagePatch4}
+\pastebutton{ugIntroSolutionPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{radicalSolve(S(a),[x,y])\free{S1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSolutionPagePatch5}
+\begin{paste}{ugIntroSolutionPageFull5}{ugIntroSolutionPageEmpty5}
+\pastebutton{ugIntroSolutionPageFull5}{\hidepaste}
+\tab{5}\spadcommand{eqns := [x**2 - y + z,x**2*z + x**4 - b*y, y**2 *z - a - b*x]\bound{e }}
+\indentrel{3}\begin{verbatim}
+                  2  2           4  2
+   (5)  [z - y + x ,x z - b y + x ,y z - b x - a]
+                          Type: List Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSolutionPageEmpty5}
+\begin{paste}{ugIntroSolutionPageEmpty5}{ugIntroSolutionPagePatch5}
+\pastebutton{ugIntroSolutionPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{eqns := [x**2 - y + z,x**2*z + x**4 - b*y, y**2 *z - a - b*x]\bound{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSolutionPagePatch6}
+\begin{paste}{ugIntroSolutionPageFull6}{ugIntroSolutionPageEmpty6}
+\pastebutton{ugIntroSolutionPageFull6}{\hidepaste}
+\tab{5}\spadcommand{solve(eqns,[x,y,z])\free{e }}
+\indentrel{3}\begin{verbatim}
+   (6)
+                       2
+          a           a
+   [[x= - Ä,y= 0,z= - ÄÄ],
+          b            2
+                      b
+
+          3       2    2
+         z  + 2b z  + b z - a
+     [x= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ, y= z + b,
+                   b
+
+          6       5     2 4      3       3     4         2
+         z  + 4b z  + 6b z  + (4b  - 2a)z  + (b  - 4a b)z
+       + 
+               2     3    2
+         - 2a b z - b  + a
+         =
+         0
+       ]
+     ]
+   Type: List List Equation Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroSolutionPageEmpty6}
+\begin{paste}{ugIntroSolutionPageEmpty6}{ugIntroSolutionPagePatch6}
+\pastebutton{ugIntroSolutionPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{solve(eqns,[x,y,z])\free{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroPreviousPagePatch1}
+\begin{paste}{ugIntroPreviousPageFull1}{ugIntroPreviousPageEmpty1}
+\pastebutton{ugIntroPreviousPageFull1}{\hidepaste}
+\tab{5}\spadcommand{10 ** 10\bound{prev }}
+\indentrel{3}\begin{verbatim}
+   (1)  10000000000
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroPreviousPageEmpty1}
+\begin{paste}{ugIntroPreviousPageEmpty1}{ugIntroPreviousPagePatch1}
+\pastebutton{ugIntroPreviousPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{10 ** 10\bound{prev }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroPreviousPagePatch2}
+\begin{paste}{ugIntroPreviousPageFull2}{ugIntroPreviousPageEmpty2}
+\pastebutton{ugIntroPreviousPageFull2}{\hidepaste}
+\tab{5}\spadcommand{\% - 1\free{prev }\bound{prev1 }}
+\indentrel{3}\begin{verbatim}
+   (2)  9999999999
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroPreviousPageEmpty2}
+\begin{paste}{ugIntroPreviousPageEmpty2}{ugIntroPreviousPagePatch2}
+\pastebutton{ugIntroPreviousPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{\% - 1\free{prev }\bound{prev1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroPreviousPagePatch3}
+\begin{paste}{ugIntroPreviousPageFull3}{ugIntroPreviousPageEmpty3}
+\pastebutton{ugIntroPreviousPageFull3}{\hidepaste}
+\tab{5}\spadcommand{\%\%(-1)\free{prev1 }\bound{prev2 }}
+\indentrel{3}\begin{verbatim}
+   (3)  9999999999
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroPreviousPageEmpty3}
+\begin{paste}{ugIntroPreviousPageEmpty3}{ugIntroPreviousPagePatch3}
+\pastebutton{ugIntroPreviousPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{\%\%(-1)\free{prev1 }\bound{prev2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntroPreviousPagePatch4}
+\begin{paste}{ugIntroPreviousPageFull4}{ugIntroPreviousPageEmpty4}
+\pastebutton{ugIntroPreviousPageFull4}{\hidepaste}
+\tab{5}\spadcommand{\%\%(1)\free{prev2 }}
+\indentrel{3}\begin{verbatim}
+   (4)  10000000000
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntroPreviousPageEmpty4}
+\begin{paste}{ugIntroPreviousPageEmpty4}{ugIntroPreviousPagePatch4}
+\pastebutton{ugIntroPreviousPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{\%\%(1)\free{prev2 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ug02.ht b/src/hyper/pages/ug02.ht
new file mode 100644
index 00000000..1ac5cdcf
--- /dev/null
+++ b/src/hyper/pages/ug02.ht
@@ -0,0 +1,2225 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\texht{\setcounter{chapter}{1}}{} % Chapter 2
+
+%
+\newcommand{\ugTypesTitle}{Using Types and Modes}
+\newcommand{\ugTypesNumber}{2.}
+%
+% =====================================================================
+\begin{page}{ugTypesPage}{2. Using Types and Modes}
+% =====================================================================
+\beginscroll
+
+In this chapter we look at the key notion of \spadgloss{type} and its
+generalization \spadgloss{mode}.
+We show that every \Language{} object has a type that
+determines what you can do with the object.
+In particular, we explain how to use types to call specific functions
+from particular parts of the library and how types and modes can be used
+to create new objects from old.
+We also look at \pspadtype{Record} and \pspadtype{Union} types and
+the special type \axiomType{Any}.
+Finally, we give you an idea of how \Language{} manipulates types and
+modes internally to resolve ambiguities.
+
+\beginmenu
+    \menudownlink{{2.1. The Basic Idea}}{ugTypesBasicPage}
+    \menudownlink{{2.2. Writing Types and Modes}}{ugTypesWritingPage}
+    \menudownlink{{2.3. Declarations}}{ugTypesDeclarePage}
+    \menudownlink{{2.4. Records}}{ugTypesRecordsPage}
+    \menudownlink{{2.5. Unions}}{ugTypesUnionsPage}
+    \menudownlink{{2.6. The ``Any'' Domain}}{ugTypesAnyNonePage}
+    \menudownlink{{2.7. Conversion}}{ugTypesConvertPage}
+    \menudownlink{{2.8. Subdomains Again}}{ugTypesSubdomainsPage}
+    \menudownlink{{2.9. Package Calling and Target Types}}{ugTypesPkgCallPage}
+    \menudownlink{{2.10. Resolving Types}}{ugTypesResolvePage}
+    \menudownlink{{2.11. Exposing Domains and Packages}}{ugTypesExposePage}
+    \menudownlink{{2.12. Commands for Snooping}}{ugAvailSnoopPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesBasicTitle}{The Basic Idea}
+\newcommand{\ugTypesBasicNumber}{2.1.}
+%
+% =====================================================================
+\begin{page}{ugTypesBasicPage}{2.1. The Basic Idea}
+% =====================================================================
+\beginscroll
+
+The \Language{} world deals with many kinds of objects.
+There are mathematical objects such as numbers and polynomials,
+data structure objects such as lists and arrays, and graphics
+objects such as points and graphic images.
+Functions are objects too.
+
+\Language{} organizes objects using the notion of \spadglossSee{domain of
+computation}{domain}, or simply \spadgloss{domain}.
+Each domain denotes a class of objects.
+The class of objects it denotes is usually given by the name of the
+domain: \axiomType{Integer} for the integers, \axiomType{Float} for
+floating-point numbers, and so on.
+The convention is that the first letter of a domain name is capitalized.
+Similarly, the domain \axiomType{Polynomial(Integer)} denotes ``polynomials
+with integer coefficients.''
+Also, \axiomType{Matrix(Float)} denotes ``matrices with floating-point
+entries.''
+
+Every basic \Language{} object belongs to a unique domain.
+The integer \axiom{3} belongs to the domain \axiomType{Integer} and
+the polynomial \axiom{x + 3} belongs to the domain
+\axiomType{Polynomial(Integer)}.
+The domain of an object is also called its \spadgloss{type}.
+Thus we speak of ``the type \axiomType{Integer}''
+and ``the type \axiomType{Polynomial(Integer)}.''
+
+\xtc{
+After an \Language{} computation, the type is displayed toward the
+right-hand side of the page (or screen).
+}{
+\spadpaste{-3}
+}
+\xtc{
+Here we create a rational number but it looks like the last result.
+The type however tells you it is different.
+You cannot identify the type of an object by how \Language{}
+displays the object.
+}{
+\spadpaste{-3/1}
+}
+\xtc{
+When a computation produces a result of a simpler type, \Language{} leaves
+the type unsimplified.
+Thus no information is lost.
+}{
+\spadpaste{x + 3 - x \bound{three}}
+}
+\xtc{
+This seldom matters since \Language{} retracts the answer to the
+simpler type if it is necessary.
+}{
+\spadpaste{factorial(\%) \free{three}}
+}
+\xtc{
+When you issue a positive number, the type \axiomType{PositiveInteger} is
+printed.
+Surely, \axiom{3} also has type \axiomType{Integer}!
+The curious reader may now have two questions.
+First, is the type of an object not unique?
+Second, how is \axiomType{PositiveInteger} related to \axiomType{Integer}?
+Read on!
+}{
+\spadpaste{3}
+}
+
+Any domain can be refined to a \spadgloss{subdomain} by a membership
+\spadgloss{predicate}.\footnote{A predicate is a function that,
+when applied to an object of the domain, returns either
+\axiom{true} or \axiom{false}.}
+For example, the domain \axiomType{Integer} can be refined to the
+subdomain \axiomType{PositiveInteger}, the set of integers
+\axiom{x} such that \axiom{x > 0}, by giving the \Language{}
+predicate \axiom{x +-> x > 0}.
+Similarly, \Language{} can define subdomains such as ``the
+subdomain of diagonal matrices,'' ``the subdomain of lists of length
+two,'' ``the subdomain of monic irreducible polynomials in
+\axiom{x},'' and so on.
+Trivially, any domain is a subdomain of itself.
+
+While an object belongs to a unique domain, it can belong to any
+number of subdomains.
+Any subdomain of the domain of an object can be used as the
+{\it type} of that object.
+The type of \axiom{3} is indeed both \axiomType{Integer} and
+\axiomType{PositiveInteger} as well as any other subdomain of
+integer whose predicate is satisfied, such as ``the prime
+integers,'' ``the odd positive integers between 3 and 17,'' and so
+on.
+
+\beginmenu
+    \menudownlink{{2.1.1. Domain Constructors}}{ugTypesBasicDomainConsPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesBasicDomainConsTitle}{Domain Constructors}
+\newcommand{\ugTypesBasicDomainConsNumber}{2.1.1.}
+%
+% =====================================================================
+\begin{page}{ugTypesBasicDomainConsPage}{2.1.1. Domain Constructors}
+% =====================================================================
+\beginscroll
+
+In \Language{}, domains are objects.
+You can create them, pass them to functions, and, as we'll see later, test
+them for certain properties.
+
+In \Language{}, you ask for a value of a function by applying its name
+to a set of arguments.
+
+\xtc{
+To ask for ``the factorial of 7'' you enter this expression to
+\Language{}.
+This applies the function \axiom{factorial} to the value \axiom{7}
+to compute the result.
+}{
+\spadpaste{factorial(7)}
+}
+\xtc{
+Enter the type \axiomType{Polynomial (Integer)} as an expression to
+\Language{}.
+This looks much like a function call as well.
+It is!
+The result is appropriately stated to be of type
+\axiomType{Domain}, which
+according to our usual convention, denotes the class of all domains.
+}{
+\spadpaste{Polynomial(Integer)}
+}
+
+The most basic operation involving domains is that of building a new
+domain from a given one.
+To create the domain of ``polynomials over the integers,'' \Language{}
+applies the function \axiomType{Polynomial} to the domain
+\axiomType{Integer}.
+A function like \axiomType{Polynomial} is called a \spadgloss{domain
+constructor} or,
+%-% \HDindex{constructor!domain}{ugTypesBasicDomainConsPage}{2.1.1.}{Domain Constructors}
+more simply, a
+\spadgloss{constructor}.
+A domain constructor is a function that creates a domain.
+An argument to a domain constructor can be another domain or, in general,
+an arbitrary kind of object.
+\axiomType{Polynomial} takes a single domain argument while
+\axiomType{SquareMatrix} takes a positive integer as an argument
+to give its dimension and
+a domain argument to give the type of its components.
+
+What kinds of domains can you use as the argument to
+\axiomType{Polynomial} or \axiomType{SquareMatrix} or
+\axiomType{List}?
+Well, the first two are mathematical in nature.
+You want to be able to perform algebraic operations like
+\axiomOp{+} and \axiomOp{*} on polynomials and square matrices,
+and operations such as \axiomFun{determinant} on square matrices.
+So you want to allow polynomials of integers {\it and} polynomials
+of square matrices with complex number coefficients and, in
+general, anything that ``makes sense.'' At the same time, you
+don't want \Language{} to be able to build nonsense domains such
+as ``polynomials of strings!''
+
+In contrast to algebraic structures, data structures can hold any
+kind of object.
+Operations on lists such as \axiomFunFrom{insert}{List},
+\axiomFunFrom{delete}{List}, and \axiomFunFrom{concat}{List} just
+manipulate the list itself without changing or operating on its
+elements.
+Thus you can build \axiomType{List} over almost any datatype,
+including itself.
+\xtc{
+Create a complicated algebraic domain.
+}{
+\spadpaste{List (List (Matrix (Polynomial (Complex (Fraction (Integer))))))}
+}
+\xtc{
+Try to create a meaningless domain.
+}{
+\spadpaste{Polynomial(String)}
+}
+Evidently from our last example, \Language{} has some mechanism
+that tells what a constructor can use as an argument.
+This brings us to the notion of \spadgloss{category}.
+As domains are objects, they too have a domain.
+The domain of a domain is a category.
+A category is simply a type whose members are domains.
+
+A common algebraic category is \axiomType{Ring}, the class of all domains
+that are ``rings.''
+A ring is an algebraic structure with constants \axiom{0} and \axiom{1} and
+operations \axiomOpFrom{+}{Ring}, \axiomOpFrom{-}{Ring}, and
+\axiomOpFrom{*}{Ring}.
+These operations are assumed ``closed'' with respect to the domain,
+meaning that they take two objects of the domain and produce a result
+object also in the domain.
+The operations are understood to satisfy certain ``axioms,'' certain
+mathematical principles providing the algebraic foundation for rings.
+For example, the {\it additive inverse axiom} for rings states:
+\centerline{{Every element \axiom{x} has an additive inverse \axiom{y} such}}
+\centerline{{that \axiom{x + y = 0}.}}
+The prototypical example of a domain that is a ring is the integers.
+Keep them in mind whenever we mention \axiomType{Ring}.
+
+Many algebraic domain constructors such as \axiomType{Complex},
+\axiomType{Polynomial}, \axiomType{Fraction}, take rings as
+arguments and return rings as values.
+You can use the infix operator ``\axiom{has}''
+\spadkey{has}
+to ask a domain if it belongs to a particular category.
+
+\xtc{
+All numerical types are rings.
+Domain constructor \axiomType{Polynomial} builds ``the ring of polynomials
+over any other ring.''
+}{
+\spadpaste{Polynomial(Integer) has Ring}
+}
+\xtc{
+Constructor \axiomType{List} never produces a ring.
+}{
+\spadpaste{List(Integer) has Ring}
+}
+\xtc{
+The constructor \axiomType{Matrix(R)} builds ``the domain of all matrices
+over the ring \axiom{R}.'' This domain is never a ring since the operations
+\axiomSyntax{+}, \axiomSyntax{-}, and \axiomSyntax{*} on matrices of arbitrary
+shapes are undefined.
+}{
+\spadpaste{Matrix(Integer) has Ring}
+}
+\xtc{
+Thus you can never build polynomials over matrices.
+}{
+\spadpaste{Polynomial(Matrix(Integer))}
+}
+\xtc{
+Use \axiomType{SquareMatrix(n,R)} instead.
+For any positive integer \axiom{n}, it builds ``the ring of \axiom{n} by
+\axiom{n} matrices over \axiom{R}.''
+}{
+\spadpaste{Polynomial(SquareMatrix(7,Complex(Integer)))}
+}
+
+Another common category is \axiomType{Field}, the class of all fields.
+%-% \HDindex{field}{ugTypesBasicDomainConsPage}{2.1.1.}{Domain Constructors}
+A field is a ring with additional operations.
+For example, a field has commutative multiplication and
+a closed operation \axiomOpFrom{/}{Field} for the
+division of two elements.
+\axiomType{Integer} is not a field since, for example, \axiom{3/2} does not
+have an integer result.
+The prototypical example of a field is the rational numbers, that is, the
+domain \axiomType{Fraction(Integer)}.
+In general, the constructor \axiomType{Fraction} takes a ring as an
+argument and returns a field.\footnote{Actually,
+the argument domain must have some additional
+properties so as to belong to category \axiomType{IntegralDomain}.}
+Other domain constructors, such as \axiomType{Complex}, build fields only if
+their argument domain is a field.
+
+\xtc{
+The complex integers (often called the ``Gaussian integers'') do not form
+a field.
+}{
+\spadpaste{Complex(Integer) has Field}
+}
+\xtc{
+But fractions of complex integers do.
+}{
+\spadpaste{Fraction(Complex(Integer)) has Field}
+}
+\xtc{
+The algebraically equivalent domain of complex rational numbers is a field
+since domain constructor \axiomType{Complex} produces a field whenever its
+argument is a field.
+}{
+\spadpaste{Complex(Fraction(Integer)) has Field}
+}
+
+The most basic category is \axiomType{Type}.
+%-% \HDexptypeindex{Type}{ugTypesBasicDomainConsPage}{2.1.1.}{Domain Constructors}
+It denotes the class of all domains and
+subdomains.\footnote{\axiomType{Type} does not denote the class of
+all types.
+The type of all categories is \axiomType{Category}.
+The type of \axiomType{Type} itself is undefined.}
+Domain constructor \axiomType{List} is able to build ``lists of
+elements from domain \axiom{D}'' for arbitrary \axiom{D} simply by
+requiring that \axiom{D} belong to category \axiomType{Type}.
+
+Now, you may ask, what exactly is a category?
+%-% \HDindex{category}{ugTypesBasicDomainConsPage}{2.1.1.}{Domain Constructors}
+Like domains, categories can be defined in the \Language{} language.
+A category is defined by three components:
+%
+\indent{4}
+\beginitems
+\item[1. ] a name (for example, \axiomType{Ring}),
+used to refer to the class of domains that the category represents;
+\item[2. ] a set of operations, used to refer to the operations that
+the domains of this class support
+(for example, \axiomOp{+}, \axiomOp{-}, and \axiomOp{*} for rings); and
+\item[3. ] an optional list of other categories that this category extends.
+\enditems
+\indent{0}
+%
+This last component is a new idea.
+And it is key to the design of \Language{}!
+Because categories can extend one another, they form hierarchies.
+\texht{Detailed charts showing the category hierarchies in \Language{} are
+displayed in the endpages of this book.
+There you see that all categories are extensions of \axiomType{Type} and that
+\axiomType{Field} is an extension of \axiomType{Ring}.}{}
+
+The operations supported by the domains of a category are called the
+\spadglossSee{exports}{export} of that category because these are the
+operations made available for system-wide use.
+The exports of a domain of a given category are not only the ones
+explicitly mentioned by the category.
+Since a category extends other categories, the operations of these other
+categories---and all categories these other categories extend---are also
+exported by the domains.
+
+For example, polynomial domains belong to \axiomType{PolynomialCategory}.
+This category explicitly mentions some twenty-nine
+operations on polynomials, but it
+extends eleven other categories (including \axiomType{Ring}).
+As a result, the current system has over one hundred operations on polynomials.
+
+If a domain belongs to a category that extends, say, \axiomType{Ring}, it
+is convenient to say that the domain exports \axiomType{Ring}.
+The name of the category thus provides a convenient shorthand for the list
+of operations exported by the category.
+Rather than listing operations such as \axiomOpFrom{+}{Ring} and
+\axiomOpFrom{*}{Ring} of \axiomType{Ring} each time they are needed, the
+definition of a type simply asserts that it exports category
+\axiomType{Ring}.
+
+The category name, however, is more than a shorthand.
+The name \axiomType{Ring}, in fact, implies that the operations exported by
+rings are required to satisfy a set of ``axioms'' associated with the name
+\axiomType{Ring}.\footnote{This subtle
+but important feature distinguishes \Language{} from
+other abstract datatype designs.}
+
+Why is it not correct to assume that some type is a ring if it exports all
+of the operations of \axiomType{Ring}?
+Here is why.
+Some languages such as {\bf APL}
+%-% \HDindex{APL}{ugTypesBasicDomainConsPage}{2.1.1.}{Domain Constructors}
+denote the \axiomType{Boolean} constants \axiom{true} and
+\axiom{false} by the integers \axiom{1} and \axiom{0}
+respectively, then use \axiomOp{+} and \axiomOp{*} to denote the
+logical operators \axiomFun{or} and \axiomFun{and}.
+But with these definitions
+\axiomType{Boolean} is not a ring since the additive inverse
+axiom is violated.\footnote{There is no inverse element \axiom{a}
+such that \axiom{1 + a = 0}, or, in the usual terms: \axiom{true
+or a = false}.}
+This alternative definition of \axiomType{Boolean} can be easily
+and correctly implemented in \Language{}, since
+\axiomType{Boolean} simply does not assert that it is of category
+\axiomType{Ring}.
+This prevents the system from building meaningless domains such as
+\axiomType{Polynomial(Boolean)} and then wrongfully applying
+algorithms that presume that the ring axioms hold.
+
+
+Enough on categories. To learn more about them, see
+\downlink{``\ugCategoriesTitle''}{ugCategoriesPage} in Chapter \ugCategoriesNumber\ignore{ugCategories}.
+We now return to our discussion of domains.
+
+Domains \spadgloss{export} a set of operations to make them
+available for system-wide use.
+\axiomType{Integer}, for example, exports the operations
+\axiomOpFrom{+}{Integer} and \axiomOpFrom{=}{Integer} given by
+the \spadglossSee{signatures}{signature}
+\axiomOpFrom{+}{Integer}: \spadsig{(Integer,Integer)}{Integer} and
+\axiomOpFrom{=}{Integer}: \spadsig{(Integer,Integer)}{Boolean},
+respectively.
+Each of these operations takes two \axiomType{Integer} arguments.
+The \axiomOpFrom{+}{Integer} operation also returns an \axiomType{Integer} but
+\axiomOpFrom{=}{Integer} returns a \axiomType{Boolean}: \axiom{true} or
+\axiom{false}.
+The operations exported by a domain usually manipulate objects of
+the domain---but not always.
+
+The operations of a domain may actually take as arguments, and return as
+values, objects from any domain.
+For example, \axiomType{Fraction (Integer)} exports the operations
+\axiomOpFrom{/}{Fraction}: \spadsig{(Integer,Integer)}{Fraction(Integer)}
+and \axiomFunFrom{characteristic}{Fraction}:
+\spadsig{}{NonNegativeInteger}.
+
+Suppose all operations of a domain take as arguments and return
+as values, only objects from {\it other} domains.
+%-% \HDindex{package}{ugTypesBasicDomainConsPage}{2.1.1.}{Domain Constructors}
+This kind of domain
+%-% \HDindex{constructor!package}{ugTypesBasicDomainConsPage}{2.1.1.}{Domain Constructors}
+is what \Language{} calls a \spadgloss{package}.
+
+A package does not designate a class of objects at all.
+Rather, a package is just a collection of operations.
+Actually the bulk of the \Language{} library of algorithms consists
+of packages.
+The facilities for factorization; integration; solution of linear,
+polynomial, and differential equations; computation of limits; and so
+on, are all defined in packages.
+Domains needed by algorithms can be passed to a package as arguments or
+used by name if they are not ``variable.''
+Packages are useful for defining operations that convert objects of one
+type to another, particularly when these types have different
+parameterizations.
+As an example, the package \axiomType{PolynomialFunction2(R,S)} defines
+operations that convert polynomials over a domain \axiom{R} to polynomials
+over \axiom{S}.
+To convert an object from \axiomType{Polynomial(Integer)} to
+\axiomType{Polynomial(Float)}, \Language{} builds the package
+\axiomType{PolynomialFunctions2(Integer,Float)} in order to create the
+required conversion function.
+(This happens ``behind the scenes'' for you: see \downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert}
+for details on how to convert objects.)
+
+\Language{} categories, domains and packages and all their contained
+functions are written in the \Language{} programming language and have
+been compiled into machine code.
+This is what comprises the \Language{} \spadgloss{library}.
+In the rest of this book we show you how to use these domains and
+their functions and how to write your own functions.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesWritingTitle}{Writing Types and Modes}
+\newcommand{\ugTypesWritingNumber}{2.2.}
+%
+% =====================================================================
+\begin{page}{ugTypesWritingPage}{2.2. Writing Types and Modes}
+% =====================================================================
+\beginscroll
+%
+
+We have already seen in \texht{the last
+section}{\downlink{``\ugTypesBasicTitle''}{ugTypesBasicPage} in Section \ugTypesBasicNumber\ignore{ugTypesBasic}} several examples of types.
+Most of these examples had either no arguments (for example,
+\axiomType{Integer}) or one argument (for example,
+\axiomType{Polynomial (Integer)}).
+In this section we give details about writing arbitrary types.
+We then define \spadglossSee{modes}{mode} and discuss how to write
+them.
+We conclude the section with a discussion on constructor
+abbreviations.
+
+\xtc{
+When might you need to write a type or mode?
+You need to do so when you declare variables.
+}{
+\spadpaste{a : PositiveInteger}
+}
+\xtc{
+You need to do so when you declare functions
+(\downlink{``\ugTypesDeclareTitle''}{ugTypesDeclarePage} in Section \ugTypesDeclareNumber\ignore{ugTypesDeclare}),
+}{
+\spadpaste{f : Integer -> String}
+}
+\xtc{
+You need to do so when you convert an object from one type to another
+(\downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert}).
+}{
+\spadpaste{factor(2 :: Complex(Integer))}
+}
+\xtc{
+}{
+\spadpaste{(2 = 3)\$Integer}
+}
+\xtc{
+You need to do so when you give computation target type information
+(\downlink{``\ugTypesPkgCallTitle''}{ugTypesPkgCallPage} in Section \ugTypesPkgCallNumber\ignore{ugTypesPkgCall}).
+}{
+\spadpaste{(2 = 3)@Boolean}
+}
+
+\beginmenu
+    \menudownlink{{2.2.1. Types with No Arguments}}{ugTypesWritingZeroPage}
+    \menudownlink{{2.2.2. Types with One Argument}}{ugTypesWritingOnePage}
+    \menudownlink{{2.2.3. Types with More Than One Argument}}{ugTypesWritingMorePage}
+    \menudownlink{{2.2.4. Modes}}{ugTypesWritingModesPage}
+    \menudownlink{{2.2.5. Abbreviations}}{ugTypesWritingAbbrPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesWritingZeroTitle}{Types with No Arguments}
+\newcommand{\ugTypesWritingZeroNumber}{2.2.1.}
+%
+% =====================================================================
+\begin{page}{ugTypesWritingZeroPage}{2.2.1. Types with No Arguments}
+% =====================================================================
+\beginscroll
+
+A constructor with no arguments can be written either
+%-% \HDindex{type!using parentheses}{ugTypesWritingZeroPage}{2.2.1.}{Types with No Arguments}
+with or without
+%-% \HDindex{parentheses!using with types}{ugTypesWritingZeroPage}{2.2.1.}{Types with No Arguments}
+trailing opening and closing parentheses (\axiomSyntax{()}).
+\texht{
+\centerline{{\begin{tabular}{ccc}}}
+\centerline{{\axiomType{Boolean()} is the same as \axiomType{Boolean} & \quad &}}
+\centerline{{\axiomType{Integer()} is the same as \axiomType{Integer} }}
+\centerline{{\axiomType{String()} is the same as \axiomType{String} & \quad &}}
+\centerline{{\axiomType{Void()} is the same as \axiomType{Void} }}
+\centerline{{\end{tabular}}}
+}{
+\centerline{{\axiomType{Boolean()} is the same as \axiomType{Boolean} }}
+\centerline{{\axiomType{Integer()} is the same as \axiomType{Integer} }}
+\centerline{{\axiomType{String()} is the same as \axiomType{String} }}
+\centerline{{\axiomType{Void()} is the same as \axiomType{Void} }}
+and so on.
+}
+It is customary to omit the parentheses.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesWritingOneTitle}{Types with One Argument}
+\newcommand{\ugTypesWritingOneNumber}{2.2.2.}
+%
+% =====================================================================
+\begin{page}{ugTypesWritingOnePage}{2.2.2. Types with One Argument}
+% =====================================================================
+\beginscroll
+
+A constructor with one argument can frequently be
+%-% \HDindex{type!using parentheses}{ugTypesWritingOnePage}{2.2.2.}{Types with One Argument}
+written with no
+%-% \HDindex{parentheses!using with types}{ugTypesWritingOnePage}{2.2.2.}{Types with One Argument}
+parentheses.
+Types nest from right to left so that \axiomType{Complex Fraction
+Polynomial Integer} is the same as
+\axiomType{Complex (Fraction (Polynomial (Integer)))}.
+You need to use parentheses to force the application of a constructor
+to the correct argument, but you need not use any more than is
+necessary to remove ambiguities.
+
+Here are some guidelines for using parentheses (they are possibly slightly
+more restrictive than they need to be).
+\xtc{
+If the argument is an expression like \axiom{2 + 3}
+then you must enclose the argument in parentheses.
+}{
+\spadpaste{e : PrimeField(2 + 3)}
+}
+%
+\xtc{
+If the type is to be used with package calling
+then you must enclose the argument in parentheses.
+}{
+\spadpaste{content(2)\$Polynomial(Integer)}
+}
+\xtc{
+Alternatively, you can write the type without parentheses
+then enclose the whole type expression with parentheses.
+}{
+\spadpaste{content(2)\$(Polynomial Complex Fraction Integer)}
+}
+\xtc{
+If you supply computation target type information
+(\downlink{``\ugTypesPkgCallTitle''}{ugTypesPkgCallPage} in Section \ugTypesPkgCallNumber\ignore{ugTypesPkgCall})
+then you should enclose the argument in parentheses.
+}{
+\spadpaste{(2/3)@Fraction(Polynomial(Integer))}
+}
+%
+\xtc{
+If the type itself has parentheses around it and we are
+not in the case of the first example above,
+then the parentheses can usually be omitted.
+}{
+\spadpaste{(2/3)@Fraction(Polynomial Integer)}
+}
+%
+\xtc{
+If the type is used in a declaration and the argument is a single-word
+type, integer or symbol,
+then the parentheses can usually be omitted.
+}{
+\spadpaste{(d,f,g) : Complex Polynomial Integer}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesWritingMoreTitle}{Types with More Than One Argument}
+\newcommand{\ugTypesWritingMoreNumber}{2.2.3.}
+%
+% =====================================================================
+\begin{page}{ugTypesWritingMorePage}{2.2.3. Types with More Than One Argument}
+% =====================================================================
+\beginscroll
+
+If a constructor
+%-% \HDindex{type!using parentheses}{ugTypesWritingMorePage}{2.2.3.}{Types with More Than One Argument}
+has more than
+%-% \HDindex{parentheses!using with types}{ugTypesWritingMorePage}{2.2.3.}{Types with More Than One Argument}
+one argument, you must use parentheses.
+Some examples are
+\centerline{{\axiomType{UnivariatePolynomial(x, Float)} }}
+\centerline{{\axiomType{MultivariatePolynomial([z,w,r], Complex Float)} }}
+\centerline{{\axiomType{SquareMatrix(3, Integer)} }}
+\centerline{{\axiomType{FactoredFunctions2(Integer,Fraction Integer)}}}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesWritingModesTitle}{Modes}
+\newcommand{\ugTypesWritingModesNumber}{2.2.4.}
+%
+% =====================================================================
+\begin{page}{ugTypesWritingModesPage}{2.2.4. Modes}
+% =====================================================================
+\beginscroll
+
+A \spadgloss{mode} is a type that possibly is a
+question mark (\axiomSyntax{?}) or contains one in an argument
+position.
+For example, the following are all modes.
+\texht{
+\centerline{{\begin{tabular}{ccc}}}
+\centerline{{\axiomType{?} & \quad &}}
+\centerline{{\axiomType{Polynomial ?} }}
+\centerline{{\axiomType{Matrix Polynomial ?} & \quad &}}
+\centerline{{\axiomType{SquareMatrix(3,?)} }}
+\centerline{{\axiomType{Integer} & \quad &}}
+\centerline{{\axiomType{OneDimensionalArray(Float)}}}
+\centerline{{\end{tabular}}}
+}{
+\centerline{{\axiomType{?} }}
+\centerline{{\axiomType{Polynomial ?} }}
+\centerline{{\axiomType{Matrix Polynomial ?} }}
+\centerline{{\axiomType{SquareMatrix(3,?)} }}
+\centerline{{\axiomType{Integer} }}
+\centerline{{\axiomType{OneDimensionalArray(Float)}}}
+}
+As is evident from these examples, a mode is a type with a
+part that is not specified (indicated by a question mark).
+Only one \axiomSyntax{?} is allowed per mode and it must appear in the
+most deeply nested argument that is a type. Thus
+\nonLibAxiomType{?(Integer)},
+\nonLibAxiomType{Matrix(? (Polynomial))},
+\nonLibAxiomType{SquareMatrix(?, Integer)} and
+\nonLibAxiomType{SquareMatrix(?, ?)} are all invalid.
+The question mark must take the place of a domain, not data (for example,
+the integer that is the dimension of a square matrix).
+This rules out, for example, the two \axiomType{SquareMatrix}
+expressions.
+
+Modes can be used for declarations
+(\downlink{``\ugTypesDeclareTitle''}{ugTypesDeclarePage} in Section \ugTypesDeclareNumber\ignore{ugTypesDeclare})
+and conversions
+(\downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert}).
+However, you cannot use a mode for package calling or giving target
+type information.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesWritingAbbrTitle}{Abbreviations}
+\newcommand{\ugTypesWritingAbbrNumber}{2.2.5.}
+%
+% =====================================================================
+\begin{page}{ugTypesWritingAbbrPage}{2.2.5. Abbreviations}
+% =====================================================================
+\beginscroll
+
+Every constructor has an abbreviation that
+%-% \HDindex{abbreviation!constructor}{ugTypesWritingAbbrPage}{2.2.5.}{Abbreviations}
+you can freely
+%-% \HDindex{constructor!abbreviation}{ugTypesWritingAbbrPage}{2.2.5.}{Abbreviations}
+substitute for the constructor name.
+In some cases, the abbreviation is nothing more than the
+capitalized version of the constructor name.
+
+\beginImportant
+Aside from allowing types to be written more concisely,
+abbreviations are used by \Language{} to name various system
+files for constructors (such as library filenames, test input
+files and example files).
+Here are some common abbreviations.
+\texht{}{\table{
+{\axiomType{COMPLEX}    abbreviates \axiomType{Complex}              }
+{\axiomType{DFLOAT}     abbreviates \axiomType{DoubleFloat}          }
+{\axiomType{EXPR}       abbreviates \axiomType{Expression}           }
+{\axiomType{FLOAT}      abbreviates \axiomType{Float}                }
+{\axiomType{FRAC}       abbreviates \axiomType{Fraction}             }
+{\axiomType{INT}        abbreviates \axiomType{Integer}              }
+{\axiomType{MATRIX}     abbreviates \axiomType{Matrix}               }
+{\axiomType{NNI}        abbreviates \axiomType{NonNegativeInteger}   }
+{\axiomType{PI}         abbreviates \axiomType{PositiveInteger}      }
+{\axiomType{POLY}       abbreviates \axiomType{Polynomial}           }
+{\axiomType{STRING}     abbreviates \axiomType{String}               }
+{\axiomType{UP}         abbreviates \axiomType{UnivariatePolynomial} }
+}}
+\endImportant
+
+You can combine both full constructor names and abbreviations
+in a type expression.
+Here are some types using abbreviations.
+\centerline{{\axiomType{POLY INT} is the same as \axiomType{Polynomial(INT)} }}
+\centerline{{\axiomType{POLY(Integer)} is the same as \axiomType{Polynomial(Integer)} }}
+\centerline{{\axiomType{POLY(Integer)} is the same as \axiomType{Polynomial(INT)} }}
+\centerline{{\axiomType{FRAC(COMPLEX(INT))} is the same as \axiomType{Fraction Complex Integer} }}
+\centerline{{\axiomType{FRAC(COMPLEX(INT))} is the same as \axiomType{FRAC(Complex Integer)} }}
+
+There are several ways of finding the names of constructors and
+their abbreviations.
+For a specific constructor, use \spadcmd{)abbreviation query}.
+%-% \HDsyscmdindex{abbreviation}{ugTypesWritingAbbrPage}{2.2.5.}{Abbreviations}
+You can also use the \spadcmd{)what} system command to see the names
+and abbreviations of constructors.
+%-% \HDsyscmdindex{what}{ugTypesWritingAbbrPage}{2.2.5.}{Abbreviations}
+For more information about \spadcmd{)what}, see
+\downlink{``\ugSysCmdwhatTitle''}{ugSysCmdwhatPage} in Section \ugSysCmdwhatNumber\ignore{ugSysCmdwhat}.
+\xtc{
+\spadcmd{)abbreviation query} can be
+abbreviated (no pun intended) to \spadcmd{)abb q}.
+}{
+\spadpaste{)abb q Integer}
+}
+\xtc{
+The \spadcmd{)abbreviation query} command lists
+the constructor name if you give the abbreviation.
+Issue \spadcmd{)abb q} if you want to see the names and abbreviations
+of all \Language{} constructors.
+}{
+\spadpaste{)abb q DMP}
+}
+\xtc{
+Issue this to see all packages whose names contain the string ``ode''.
+%-% \HDsyscmdindex{what packages}{ugTypesWritingAbbrPage}{2.2.5.}{Abbreviations}
+}{
+\spadpaste{)what packages ode}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesDeclareTitle}{Declarations}
+\newcommand{\ugTypesDeclareNumber}{2.3.}
+%
+% =====================================================================
+\begin{page}{ugTypesDeclarePage}{2.3. Declarations}
+% =====================================================================
+\beginscroll
+%
+A \spadgloss{declaration} is an expression used
+to restrict the type of values that can be assigned to variables.
+A colon (\axiomSyntax{:}) is always used after a variable or
+list of variables to be declared.
+
+\beginImportant
+For a single variable, the syntax for declaration is
+\centerline{{{\it variableName \axiom{:} typeOrMode}}}
+For multiple variables, the syntax is
+\centerline{{{\tt (\subscriptIt{variableName}{1}, \subscriptIt{variableName}{2}, \ldots \subscriptIt{variableName}{N}): {\it typeOrMode}}}}
+\endImportant
+
+You can always combine a declaration with an assignment.
+When you do, it is equivalent to first giving a declaration statement,
+then giving an assignment.
+For more information on assignment, see
+\downlink{``\ugIntroAssignTitle''}{ugIntroAssignPage} in Section \ugIntroAssignNumber\ignore{ugIntroAssign} and
+\downlink{``\ugLangAssignTitle''}{ugLangAssignPage} in Section \ugLangAssignNumber\ignore{ugLangAssign}.
+To see how to declare your own functions, see
+\downlink{``\ugUserDeclareTitle''}{ugUserDeclarePage} in Section \ugUserDeclareNumber\ignore{ugUserDeclare}.
+
+\xtc{
+This declares one variable to have a type.
+}{
+\spadpaste{a : Integer \bound{a}}
+}
+\xtc{
+This declares several variables to have a type.
+}{
+\spadpaste{(b,c) : Integer \bound{b c}}
+}
+\xtc{
+\axiom{a, b} and \axiom{c} can only hold integer values.
+}{
+\spadpaste{a := 45 \free{a}}
+}
+\xtc{
+If a value cannot be converted to a declared type,
+an error message is displayed.
+}{
+\spadpaste{b := 4/5 \free{b}}
+}
+\xtc{
+This declares a variable with a mode.
+}{
+\spadpaste{n : Complex ? \bound{n}}
+}
+\xtc{
+This declares several variables with a mode.
+}{
+\spadpaste{(p,q,r) : Matrix Polynomial ? \bound{p q r}}
+}
+\xtc{
+This complex object has integer real and imaginary parts.
+}{
+\spadpaste{n := -36 + 9 * \%i \free{n}}
+}
+\xtc{
+This complex object has fractional symbolic real and imaginary parts.
+}{
+\spadpaste{n := complex(4/(x + y),y/x) \free{n}}
+}
+\xtc{
+This matrix has entries that are polynomials with integer
+coefficients.
+}{
+\spadpaste{p := [[1,2],[3,4],[5,6]] \free{p}}
+}
+\xtc{
+This matrix has a single entry that is a polynomial with
+rational number coefficients.
+}{
+\spadpaste{q := [[x - 2/3]] \free{q}}
+}
+\xtc{
+This matrix has entries that are polynomials with complex integer
+coefficients.
+}{
+\spadpaste{r := [[1-\%i*x,7*y+4*\%i]] \free{r}}
+}
+%
+\xtc{
+Note the difference between this and the next example.
+This is a complex object with polynomial real and imaginary parts.
+}{
+\spadpaste{f : COMPLEX POLY ? := (x + y*\%i)**2}
+}
+\xtc{
+This is a polynomial with complex integer coefficients.
+The objects are convertible from one to the other.
+See \downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert} for more information.
+}{
+\spadpaste{g : POLY COMPLEX ? := (x + y*\%i)**2}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesRecordsTitle}{Records}
+\newcommand{\ugTypesRecordsNumber}{2.4.}
+%
+% =====================================================================
+\begin{page}{ugTypesRecordsPage}{2.4. Records}
+% =====================================================================
+\beginscroll
+%
+A \pspadtype{Record} is an object composed of one or more other objects,
+%-% \HDindex{Record@\protect\nonLibAxiomType{Record}}{ugTypesRecordsPage}{2.4.}{Records}
+each of which is referenced
+%-% \HDindex{selector!record}{ugTypesRecordsPage}{2.4.}{Records}
+with
+%-% \HDindex{record!selector}{ugTypesRecordsPage}{2.4.}{Records}
+a \spadgloss{selector}.
+Components can all belong to the same type or each can have a different type.
+
+\beginImportant
+The syntax for writing a \pspadtype{Record} type is
+\centerline{{{\tt Record(\subscriptIt{selector}{1}:\subscriptIt{type}{1}, \subscriptIt{selector}{2}:\subscriptIt{type}{2}, \ldots, \subscriptIt{selector}{N}:\subscriptIt{type}{N})}}}
+You must be careful if a selector has the same name as a variable in the
+workspace.
+If this occurs, precede the selector name by a single
+%-% \HDindex{quote}{ugTypesRecordsPage}{2.4.}{Records}
+quote.
+\endImportant
+
+Record components are implicitly ordered.
+All the components of a record can
+be set at once by assigning the record a
+bracketed \spadgloss{tuple} of values of the proper length
+(for example, \axiom{r : Record(a: Integer, b: String) := [1, "two"]}).
+To access a component of a record \axiom{r},
+write the name \axiom{r}, followed by a period, followed by a selector.
+
+%
+\xtc{
+The object returned by this computation is a record with two components: a
+\axiom{quotient} part and a \axiom{remainder} part.
+}{
+\spadpaste{u := divide(5,2) \bound{u}}
+}
+%
+\xtc{
+This is the quotient part.
+}{
+\spadpaste{u.quotient \free{u}}
+}
+\xtc{
+This is the remainder part.
+}{
+\spadpaste{u.remainder \free{u}}
+}
+%
+\xtc{
+You can use selector expressions on the left-hand side of an assignment
+to change destructively the components of a record.
+}{
+\spadpaste{u.quotient := 8978 \free{u}\bound{u1}}
+}
+\xtc{
+The selected component \axiom{quotient} has the value \axiom{8978},
+which is what is returned by the assignment.
+Check that the value of \axiom{u} was modified.
+}{
+\spadpaste{u \free{u}\free{u1}}
+}
+\xtc{
+Selectors are evaluated.
+Thus you can use variables that evaluate to selectors instead of the
+selectors themselves.
+}{
+\spadpaste{s := 'quotient \bound{s}}
+}
+\xtc{
+Be careful!
+A selector could have the same name as a variable in the workspace.
+If this occurs, precede the selector name by a single quote, as in
+\axiom{u.'quotient}.
+%-% \HDindex{selector!quoting}{ugTypesRecordsPage}{2.4.}{Records}
+}{
+\spadpaste{divide(5,2).s \free{s}}
+}
+\xtc{
+Here we declare that the value of \axiom{bd}
+has two components: a string,
+to be accessed via \axiom{name}, and an integer,
+to be accessed via \axiom{birthdayMonth}.
+}{
+\spadpaste{bd : Record(name : String, birthdayMonth : Integer) \bound{bddec}}
+}
+\xtc{
+You must initially set the value of the entire \pspadtype{Record}
+at once.
+}{
+\spadpaste{bd := ["Judith", 3] \free{bddec}\bound{bd}}
+}
+\xtc{
+Once set, you can change any of the individual components.
+}{
+\spadpaste{bd.name := "Katie" \free{bd}}
+}
+\xtc{
+Records may be nested and the selector names can be shared at
+different levels.
+}{
+\spadpaste{r : Record(a : Record(b: Integer, c: Integer), b: Integer) \bound{rdec}}
+}
+\xtc{
+The record \axiom{r} has a \axiom{b} selector at two different levels.
+Here is an initial value for \axiom{r}.
+}{
+\spadpaste{r := [[1,2],3] \bound{r}\free{rdec}}
+}
+\xtc{
+This extracts the \axiom{b} component from the \axiom{a} component of \axiom{r}.
+}{
+\spadpaste{r.a.b \free{r}}
+}
+\xtc{
+This extracts the \axiom{b} component from \axiom{r}.
+}{
+\spadpaste{r.b \free{r}}
+}
+%
+\xtc{
+You can also use spaces or parentheses to refer to \pspadtype{Record}
+components.
+This is the same as \axiom{r.a}.
+}{
+\spadpaste{r(a) \free{r}}
+}
+\xtc{
+This is the same as \axiom{r.b}.
+}{
+\spadpaste{r b \free{r}}
+}
+\xtc{
+This is the same as \axiom{r.b := 10}.
+}{
+\spadpaste{r(b) := 10 \free{r}\bound{r1}}
+}
+\xtc{
+Look at \axiom{r} to make sure it was modified.
+}{
+\spadpaste{r \free{r1}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesUnionsTitle}{Unions}
+\newcommand{\ugTypesUnionsNumber}{2.5.}
+%
+% =====================================================================
+\begin{page}{ugTypesUnionsPage}{2.5. Unions}
+% =====================================================================
+\beginscroll
+%
+Type \pspadtype{Union} is used for objects that
+can be of any of a specific finite set of types.
+%-% \HDindex{Union@\protect\nonLibAxiomType{Union}}{ugTypesUnionsPage}{2.5.}{Unions}
+Two versions of unions are available,
+one with selectors (like records) and one without.
+%-% \HDindex{union}{ugTypesUnionsPage}{2.5.}{Unions}
+
+\beginmenu
+    \menudownlink{{2.5.1. Unions Without Selectors}}{ugTypesUnionsWOSelPage}
+    \menudownlink{{2.5.2. Unions With Selectors}}{ugTypesUnionsWSelPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesUnionsWOSelTitle}{Unions Without Selectors}
+\newcommand{\ugTypesUnionsWOSelNumber}{2.5.1.}
+%
+% =====================================================================
+\begin{page}{ugTypesUnionsWOSelPage}{2.5.1. Unions Without Selectors}
+% =====================================================================
+\beginscroll
+
+The declaration \axiom{x : Union(Integer, String, Float)}
+states that \axiom{x} can have values that are integers,
+strings or ``big'' floats.
+If, for example, the \pspadtype{Union} object is an integer, the object is
+said to belong to the \axiomType{Integer} {\it branch}
+of the \pspadtype{Union}.\footnote{
+Note that we are being a bit careless with the language here.
+Technically, the type of \axiom{x} is always
+\pspadtype{Union(Integer, String, Float)}.
+If it belongs to the \axiomType{Integer} branch, \axiom{x}
+may be converted to an object of type \axiomType{Integer}.}
+
+\beginImportant
+The syntax for writing a \pspadtype{Union} type without selectors is
+\centerline{{{\tt Union(\subscriptIt{type}{1}, \subscriptIt{type}{2}, \ldots, \subscriptIt{type}{N})}}}
+The types in a union without selectors must be distinct.
+\endImportant
+
+It is possible to create unions like
+\pspadtype{Union(Integer, PositiveInteger)} but they are
+difficult to work with because of the overlap in the branch
+types.
+See below for the rules \Language{} uses for converting something
+into a union object.
+
+The \axiom{case} infix
+\spadkey{case}
+operator returns a \axiomType{Boolean}
+and can be used to determine the branch in which an object lies.
+
+\xtc{
+This function displays a message stating in which
+branch of the \pspadtype{Union} the object (defined as \axiom{x}
+above) lies.
+}{
+\begin{spadsrc}[\bound{sayBranch}]
+sayBranch(x : Union(Integer,String,Float)) : Void  ==
+  output
+    x case Integer => "Integer branch"
+    x case String  => "String branch"
+    "Float branch"
+\end{spadsrc}
+}
+%
+\xtc{
+This tries \userfun{sayBranch} with an integer.
+}{
+\spadpaste{sayBranch 1 \free{sayBranch}}
+}
+\xtc{
+This tries \userfun{sayBranch} with a string.
+}{
+\spadpaste{sayBranch "hello" \free{sayBranch}}
+}
+\xtc{
+This tries \userfun{sayBranch} with a floating-point number.
+}{
+\spadpaste{sayBranch 2.718281828 \free{sayBranch}}
+}
+%
+
+There are two things of interest about this particular
+example to which we would like to draw your attention.
+\indent{4}
+\beginitems
+%
+\item[1. ] \Language{} normally converts a result to the target value
+before passing it to the function.
+If we left the declaration information out of this function definition
+then the \axiom{sayBranch} call would have been attempted with an
+\axiomType{Integer} rather than a \pspadtype{Union}, and an error would have
+resulted.
+%
+\item[2. ] The types in a \pspadtype{Union} are searched in the order given.
+So if the type were given as
+
+\noindent
+{\small\axiom{sayBranch(x: Union(String,Integer,Float,Any)): Void}}
+
+\noindent
+then the result would have been ``String branch'' because there
+is a conversion from \axiomType{Integer} to \axiomType{String}.
+\enditems
+\indent{0}
+
+Sometimes \pspadtype{Union} types can have extremely
+long names.
+\Language{} therefore abbreviates the names of unions by printing
+the type of the branch first within the \pspadtype{Union} and then
+eliding the remaining types with an ellipsis (\axiomSyntax{...}).
+
+\xtc{
+Here the \axiomType{Integer} branch is displayed first.
+Use \axiomSyntax{::} to create a \pspadtype{Union} object from an object.
+}{
+\spadpaste{78 :: Union(Integer,String)}
+}
+\xtc{
+Here the \axiomType{String} branch is displayed first.
+}{
+\spadpaste{s := "string" :: Union(Integer,String) \bound{s}}
+}
+\xtc{
+Use \axiom{typeOf} to see the full and actual \pspadtype{Union} type.
+\spadkey{typeOf}
+}{
+\spadpaste{typeOf s}
+}
+\xtc{
+A common operation that returns a union is \axiomFunFrom{exquo}{Integer}
+which returns the ``exact quotient'' if the quotient is exact,...
+}{
+\spadpaste{three := exquo(6,2) \bound{three}}
+}
+\xtc{
+and \axiom{"failed"} if the quotient is not exact.
+}{
+\spadpaste{exquo(5,2)}
+}
+\xtc{
+A union with a \axiom{"failed"} is frequently used to indicate the failure
+or lack of applicability of an object.
+As another example, assign an integer a variable \axiom{r} declared to be a
+rational number.
+}{
+\spadpaste{r: FRAC INT := 3 \bound{r}\bound{rdec}}
+}
+\xtc{
+The operation \axiomFunFrom{retractIfCan}{Fraction} tries to retract the
+fraction to the underlying domain \axiomType{Integer}.
+It produces a union object.
+Here it succeeds.
+}{
+\spadpaste{retractIfCan(r) \free{r}}
+}
+\xtc{
+Assign it a rational number.
+}{
+\spadpaste{r := 3/2 \bound{r1}\free{rdec}}
+}
+\xtc{
+Here the retraction fails.
+}{
+\spadpaste{retractIfCan(r) \free{r1}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesUnionsWSelTitle}{Unions With Selectors}
+\newcommand{\ugTypesUnionsWSelNumber}{2.5.2.}
+%
+% =====================================================================
+\begin{page}{ugTypesUnionsWSelPage}{2.5.2. Unions With Selectors}
+% =====================================================================
+\beginscroll
+
+Like records (\downlink{``\ugTypesRecordsTitle''}{ugTypesRecordsPage} in Section \ugTypesRecordsNumber\ignore{ugTypesRecords}),
+you can write \pspadtype{Union} types
+%-% \HDindex{selector!union}{ugTypesUnionsWSelPage}{2.5.2.}{Unions With Selectors}
+with selectors.
+%-% \HDindex{union!selector}{ugTypesUnionsWSelPage}{2.5.2.}{Unions With Selectors}
+
+\beginImportant
+The syntax for writing a \pspadtype{Union} type with selectors is
+\centerline{{{\tt Union(\subscriptIt{selector}{1}:\subscriptIt{type}{1}, \subscriptIt{selector}{2}:\subscriptIt{type}{2}, \ldots, \subscriptIt{selector}{N}:\subscriptIt{type}{N})}}}
+You must be careful if a selector has the same name as a variable in the
+workspace.
+If this occurs, precede the selector name by a single
+%-% \HDindex{quote}{ugTypesUnionsWSelPage}{2.5.2.}{Unions With Selectors}
+quote.
+%-% \HDindex{selector!quoting}{ugTypesUnionsWSelPage}{2.5.2.}{Unions With Selectors}
+It is an error to use a selector that does not correspond to the branch of
+the \pspadtype{Union} in which the element actually lies.
+\endImportant
+
+Be sure to understand the difference between records and unions
+with selectors.
+%-% \HDindex{union!difference from record}{ugTypesUnionsWSelPage}{2.5.2.}{Unions With Selectors}
+Records can have more than one component and the selectors are
+used to refer to the components.
+%-% \HDindex{record!difference from union}{ugTypesUnionsWSelPage}{2.5.2.}{Unions With Selectors}
+Unions always have one component but the type of that one
+component can vary.
+An object of type \pspadtype{Record(a: Integer, b: Float, c:
+String)} contains an integer {\it and} a float  {\it and} a
+string.
+An object of type \pspadtype{Union(a: Integer, b: Float, c:
+String)} contains an integer {\it or} a float  {\it or} a
+string.
+
+Here is a version of the \userfun{sayBranch} function (cf.
+\downlink{``\ugTypesUnionsWOSelTitle''}{ugTypesUnionsWOSelPage} in Section \ugTypesUnionsWOSelNumber\ignore{ugTypesUnionsWOSel}) that works with a union with selectors.
+It displays a message stating in which branch of the \pspadtype{Union} the
+object lies.
+\begin{verbatim}
+sayBranch(x:Union(i:Integer,s:String,f:Float)):Void==
+  output
+    x case i => "Integer branch"
+    x case s  => "String branch"
+    "Float branch"
+\end{verbatim}
+Note that \axiom{case} uses the selector name as its right-hand argument.
+\spadkey{case}
+If you accidentally use the branch type on the right-hand side of
+\axiom{case}, \axiom{false} will be returned.
+
+\xtc{
+Declare variable \axiom{u} to have a union type with selectors.
+}{
+\spadpaste{u : Union(i : Integer, s : String) \bound{undec}}
+}
+\xtc{
+Give an initial value to \axiom{u}.
+}{
+\spadpaste{u := "good morning" \bound{u}\free{undec}}
+}
+\xtc{
+Use \axiom{case} to determine in which
+branch of a \pspadtype{Union} an object lies.
+}{
+\spadpaste{u case i \free{u}}
+}
+\xtc{
+}{
+\spadpaste{u case s \free{u}}
+}
+\xtc{
+To access the element in a particular branch, use the selector.
+}{
+\spadpaste{u.s \free{u}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesAnyNoneTitle}{The ``Any'' Domain}
+\newcommand{\ugTypesAnyNoneNumber}{2.6.}
+%
+% =====================================================================
+\begin{page}{ugTypesAnyNonePage}{2.6. The ``Any'' Domain}
+% =====================================================================
+\beginscroll
+
+With the exception of objects of type \pspadtype{Record}, all \Language{}
+data structures are homogenous, that is, they hold objects all of the same
+type.
+%-% \HDexptypeindex{Any}{ugTypesAnyNonePage}{2.6.}{The ``Any'' Domain}
+If you need to get around this, you can use type \axiomType{Any}.
+Using \axiomType{Any}, for example, you can create lists whose
+elements are integers, rational numbers, strings, and even other lists.
+
+\xtc{
+Declare \axiom{u} to have type \axiomType{Any}.
+}{
+\spadpaste{u: Any\bound{uany}}
+}
+\xtc{
+Assign a list of mixed type values to \axiom{u}
+}{
+\spadpaste{u := [1, 7.2, 3/2, x**2, "wally"]\free{uany}\bound{u}}
+}
+\xtc{
+When we ask for the elements, \Language{} displays these types.
+}{
+\spadpaste{u.1 \free{u}}
+}
+\xtc{
+Actually, these objects belong to \axiomType{Any} but \Language{}
+automatically converts them to their natural types for you.
+}{
+\spadpaste{u.3 \free{u}}
+}
+\xtc{
+Since type \axiomType{Any} can be anything,
+it can only belong to type \axiomType{Type}.
+Therefore it cannot be used in algebraic domains.
+}{
+\spadpaste{v : Matrix(Any)}
+}
+
+Perhaps you are wondering how \Language{} internally represents
+objects of type \axiomType{Any}.
+An object of type \axiomType{Any} consists not only a data part
+representing its normal value, but also a type part (a {\it badge}) giving
+%-% \HDindex{badge}{ugTypesAnyNonePage}{2.6.}{The ``Any'' Domain}
+its type.
+For example, the value \axiom{1} of type \axiomType{PositiveInteger} as an
+object of type \axiomType{Any} internally looks like
+\axiom{[1,\axiomType{PositiveInteger()}]}.
+
+%When should you use \axiomType{Any} instead of a \pspadtype{Union} type?
+%Can you plan ahead?
+%For a \pspadtype{Union}, you must know in advance exactly which types you
+%are
+%\index{union!vs. Any@{vs. \protect\nonLibAxiomType{Any}}}
+%going to allow.
+%For \axiomType{Any}, anything that comes along can be accommodated.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesConvertTitle}{Conversion}
+\newcommand{\ugTypesConvertNumber}{2.7.}
+%
+% =====================================================================
+\begin{page}{ugTypesConvertPage}{2.7. Conversion}
+% =====================================================================
+\beginscroll
+%
+\beginImportant
+\spadglossSee{Conversion}{conversion}
+is the process of changing an object of one type
+into an object of another type.
+The syntax for conversion is:
+\centerline{{{\it object} {\tt ::} {\it newType}}}
+\endImportant
+
+\xtc{
+By default, \axiom{3} has the type \axiomType{PositiveInteger}.
+}{
+\spadpaste{3}
+}
+\xtc{
+We can change this into an object of type \axiomType{Fraction Integer}
+by using \axiomSyntax{::}.
+}{
+\spadpaste{3 :: Fraction Integer}
+}
+
+A \spadgloss{coercion} is a special kind of conversion that \Language{} is
+allowed to do automatically when you enter an expression.
+Coercions are usually somewhat safer than more general conversions.
+The \Language{} library contains operations called
+\axiomFun{coerce} and \axiomFun{convert}.
+Only the \axiomFun{coerce} operations can be used by the
+interpreter to change an object into an object of another type unless
+you explicitly use a \axiomSyntax{::}.
+
+By now you will be quite familiar with what types and modes look like.
+It is useful to think of a type or mode as a pattern
+for what you want the result to be.
+\xtc{
+Let's start with a square matrix of polynomials with complex rational number
+coefficients.
+%-% \HDexptypeindex{SquareMatrix}{ugTypesConvertPage}{2.7.}{Conversion}
+}{
+\spadpaste{m : SquareMatrix(2,POLY COMPLEX FRAC INT) \bound{mdec}}
+}
+\xtc{
+}{
+\spadpaste{m := matrix [[x-3/4*\%i,z*y**2+1/2],[3/7*\%i*y**4 - x,12-\%i*9/5]] \bound{m}\free{mdec}}
+}
+\xtc{
+We first want to interchange the \axiomType{Complex} and
+\axiomType{Fraction} layers.
+We do the conversion by doing the interchange in the type expression.
+}{
+\spadpaste{m1 := m :: SquareMatrix(2,POLY FRAC COMPLEX INT) \free{m}\bound{m1}}
+}
+\xtc{
+Interchange the \axiomType{Polynomial} and the
+\axiomType{Fraction} levels.
+}{
+\spadpaste{m2 := m1 :: SquareMatrix(2,FRAC POLY COMPLEX INT) \free{m1}\bound{m2}}
+}
+\xtc{
+Interchange the \axiomType{Polynomial} and the
+\axiomType{Complex} levels.
+}{
+\spadpaste{m3 := m2 :: SquareMatrix(2,FRAC COMPLEX POLY INT) \free{m2}\bound{m3}}
+}
+
+All the entries have changed types, although in comparing the
+last two results only the entry in the lower left corner looks different.
+We did all the intermediate steps to show you what \Language{} can do.
+
+\xtc{
+In fact, we could have combined all these into one conversion.
+}{
+\spadpaste{m :: SquareMatrix(2,FRAC COMPLEX POLY INT) \free{m}}
+}
+
+There are times when \Language{} is not be able to do the conversion
+in one step.
+You may need to break up the transformation into several conversions
+in order to get an object of the desired type.
+
+We cannot move either \axiomType{Fraction} or \axiomType{Complex}
+above (or to the left of, depending on how you look at it)
+\axiomType{SquareMatrix} because each of these levels requires that its
+argument type have commutative multiplication, whereas
+\axiomType{SquareMatrix} does not.\footnote{\axiomType{Fraction} requires
+that its argument belong to the category \axiomType{IntegralDomain} and
+%-% \HDindex{category}{ugTypesConvertPage}{2.7.}{Conversion}
+\axiomType{Complex} requires that its argument belong to
+\axiomType{CommutativeRing}. See
+\downlink{``\ugTypesBasicTitle''}{ugTypesBasicPage} in Section \ugTypesBasicNumber\ignore{ugTypesBasic}
+for a brief discussion of categories.}
+The \axiomType{Integer} level did not move anywhere
+because it does not allow any arguments.
+We also did not move the \axiomType{SquareMatrix} part anywhere, but
+we could have.
+\xtc{
+Recall that \axiom{m} looks like this.
+}{
+\spadpaste{m \free{m}}
+}
+\xtc{
+If we want a polynomial with matrix coefficients rather than a matrix
+with polynomial entries, we can just do the conversion.
+}{
+\spadpaste{m :: POLY SquareMatrix(2,COMPLEX FRAC INT) \free{m}}
+}
+\xtc{
+We have not yet used modes for any conversions.
+Modes are a great shorthand for indicating the type of the
+object you want.
+Instead of using the long type expression in the
+last example, we could have simply said this.
+}{
+\spadpaste{m :: POLY ? \free{m}}
+}
+\xtc{
+We can also indicate more structure if we want the entries
+of the matrices to be fractions.
+}{
+\spadpaste{m :: POLY SquareMatrix(2,FRAC ?) \free{m}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesSubdomainsTitle}{Subdomains Again}
+\newcommand{\ugTypesSubdomainsNumber}{2.8.}
+%
+% =====================================================================
+\begin{page}{ugTypesSubdomainsPage}{2.8. Subdomains Again}
+% =====================================================================
+\beginscroll
+
+A \spadgloss{subdomain} \axiom{S} of a domain \axiom{D} is a domain
+consisting of
+\indent{4}
+\beginitems
+\item[1. ] those elements of \axiom{D} that satisfy some
+\spadgloss{predicate} (that is, a test that returns \axiom{true} or
+\axiom{false}) and
+\item[2. ] a subset of the operations of \axiom{D}.
+\enditems
+\indent{0}
+Every domain is a subdomain of itself, trivially satisfying the
+membership test: \axiom{true}.
+
+Currently, there are only two system-defined subdomains in \Language{} that receive
+substantial use.
+\axiomType{PositiveInteger} and
+\axiomType{NonNegativeInteger} are subdomains of \axiomType{Integer}.
+An element \axiom{x} of \axiomType{NonNegativeInteger} is an integer
+that is greater than or equal to zero, that is, satisfies
+\axiom{x >= 0.}
+An element \axiom{x} of \axiomType{PositiveInteger} is a nonnegative integer
+that is, in fact, greater than zero, that is, satisfies \axiom{x > 0.}
+Not all operations from \axiomType{Integer} are available for these
+subdomains.
+For example, negation and subtraction are not provided since the subdomains
+are not closed under those operations.
+When you use an integer in an expression, \Language{} assigns to it the
+type that is the most specific subdomain whose predicate is satisfied.
+\xtc{
+This is a positive integer.
+}{
+\spadpaste{5}
+}
+\xtc{
+This is a nonnegative integer.
+}{
+\spadpaste{0}
+}
+\xtc{
+This is neither of the above.
+}{
+\spadpaste{-5}
+}
+\xtc{
+Furthermore, unless you are assigning an integer to a declared variable
+or using a conversion, any integer result has as type the most
+specific subdomain.
+}{
+\spadpaste{(-2) - (-3)}
+}
+\xtc{
+}{
+\spadpaste{0 :: Integer}
+}
+\xtc{
+}{
+\spadpaste{x : NonNegativeInteger := 5}
+}
+
+When necessary, \Language{} converts an integer object into one belonging
+to a less specific subdomain.
+For example, in \axiom{3-2}, the arguments to \axiomOpFrom{-}{Integer} are both
+elements of \axiomType{PositiveInteger}, but this type does not provide
+a subtraction operation.
+Neither does \axiomType{NonNegativeInteger}, so \axiom{3} and \axiom{2}
+are viewed as elements of \axiomType{Integer}, where their difference
+can be calculated.
+The result is \axiom{1}, which \Language{} then automatically assigns
+the type \axiomType{PositiveInteger}.
+
+\xtc{
+Certain operations are very sensitive to the subdomains to which their
+arguments belong.
+This is an element of \axiomType{PositiveInteger}.
+}{
+\spadpaste{2 ** 2}
+}
+\xtc{
+This is an element of \axiomType{Fraction Integer}.
+}{
+\spadpaste{2 ** (-2)}
+}
+\xtc{
+It makes sense then that this
+is a list of elements of \axiomType{PositiveInteger}.
+}{
+\spadpaste{[10**i for i in 2..5]}
+}
+What should the type of \axiom{[10**(i-1) for i in 2..5]} be?
+On one hand, \axiom{i-1} is always an integer greater than zero
+as \axiom{i} ranges from \axiom{2} to \axiom{5} and so \axiom{10**i}
+is also always a positive integer.
+On the other, \axiom{i-1} is a very simple function of \axiom{i}.
+\Language{} does not try to analyze every such function over the
+index's range of values to determine whether it is always positive
+or nowhere negative.
+For an arbitrary \Language{} function, this analysis is not possible.
+
+\xtc{
+So, to be consistent no such analysis is done and we get this.
+}{
+\spadpaste{[10**(i-1) for i in 2..5]}
+}
+\xtc{
+To get a list of elements of \axiomType{PositiveInteger} instead, you
+have two choices.
+You can use a conversion.
+}{
+\spadpaste{[10**((i-1) :: PI) for i in 2..5]}
+}
+\xtc{
+Or you can use \axiom{pretend}.
+\spadkey{pretend}
+}{
+\spadpaste{[10**((i-1) pretend PI) for i in 2..5]}
+}
+
+The operation \axiom{pretend} is used to defeat the \Language{}
+type system.
+The expression \axiom{object pretend D} means ``make a new object
+(without copying) of type \axiom{D} from \axiom{object}.''
+If \axiom{object} were an integer and you told \Language{}
+to pretend it was a list, you would probably see a message about a
+fatal error being caught and memory possibly being damaged.
+Lists do not have the same internal representation as integers!
+
+You use \axiom{pretend} at your peril.
+%-% \HDindex{peril}{ugTypesSubdomainsPage}{2.8.}{Subdomains Again}
+
+\xtc{
+Use \axiom{pretend} with great care!
+\Language{} trusts you that the value is of the specified type.
+}{
+\spadpaste{(2/3) pretend Complex Integer}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesPkgCallTitle}{Package Calling and Target Types}
+\newcommand{\ugTypesPkgCallNumber}{2.9.}
+%
+% =====================================================================
+\begin{page}{ugTypesPkgCallPage}{2.9. Package Calling and Target Types}
+% =====================================================================
+\beginscroll
+
+\Language{} works hard to figure out what you mean by an
+expression without your having to qualify it with type
+information.
+Nevertheless, there are times when you need to help it along by
+providing hints (or even orders!) to get \Language{} to do what
+you want.
+
+We saw in \downlink{``\ugTypesDeclareTitle''}{ugTypesDeclarePage} in Section \ugTypesDeclareNumber\ignore{ugTypesDeclare} that declarations using types
+and modes control the type of the results produced.
+For example, we can either produce a complex object with
+polynomial real and imaginary parts or a polynomial with complex
+integer coefficients, depending on the declaration.
+
+\spadglossSee{Package calling}{package call} is how you tell
+\Language{} to use a particular function from a particular part of
+the library.
+
+\xtc{
+Use the \axiomOpFrom{/}{Fraction} from \axiomType{Fraction Integer}
+to create a fraction of two integers.
+}{
+\spadpaste{2/3}
+}
+\xtc{
+If we wanted a floating point number, we can say ``use the
+\axiomOpFrom{/}{Float} in \axiomType{Float}.''
+}{
+\spadpaste{(2/3)\$Float}
+}
+\xtc{
+Perhaps we actually wanted a fraction of complex integers.
+}{
+\spadpaste{(2/3)\$Fraction(Complex Integer)}
+}
+
+In each case, \Language{} used the indicated operations, sometimes
+first needing to convert the two integers into objects of an
+appropriate type.
+In these examples, \axiomOpFrom{/}{Fraction} is written as an
+infix operator.
+
+\beginImportant
+To use package calling with an infix operator, use the
+following syntax:
+\centerline{{{\tt ( \subscriptIt{arg}{1} {\it op} \subscriptIt{arg}{1} )\${\it type} }}}
+\endImportant
+
+We used, for example, \axiom{(2/3)\$Float}.
+The expression \axiom{2 + 3 + 4} is equivalent to \axiom{(2+3) + 4.}
+Therefore in the expression
+\axiom{(2 + 3 + 4)\$Float} the second
+\axiomOp{+} comes from the \axiomType{Float} domain.
+Can you guess whether the first \axiomOp{+} comes from
+\axiomType{Integer} or \axiomType{Float}?\footnote{\axiomType{Float},
+because the package call causes \Language{} to convert
+\axiom{(2 + 3)} and \axiom{4} to type \axiomType{Float}.
+Before the sum is converted, it is given a target type (see below) of
+\axiomType{Float} by \Language{} and then evaluated.
+The target type causes the \axiomOp{+} from \axiomType{Float} to be used.}
+
+\beginImportant
+For an operator written before its arguments, you must use
+parentheses around the arguments (even if there is only one),
+and follow the closing parenthesis by a \axiomSyntax{\$}
+and then the type.
+\centerline{{{\tt {\it fun} ( \subscriptIt{arg}{1}, \subscriptIt{arg}{1}, \ldots, \subscriptIt{arg}{N} )\${\it type}}}}
+\endImportant
+
+For example, to call the ``minimum'' function from \axiomType{DoubleFloat}
+on two integers, you could write \axiom{min(4,89)\$DoubleFloat}.
+Another use of package calling is to tell \Language{} to use a library
+function rather than a function you defined.
+We discuss this in \downlink{``\ugUserUseTitle''}{ugUserUsePage} in Section \ugUserUseNumber\ignore{ugUserUse}.
+
+Sometimes rather than specifying where an operation comes from, you just
+want to say what type the result should be.
+We say that you provide
+%-% \HDindex{type!target}{ugTypesPkgCallPage}{2.9.}{Package Calling and Target Types}
+a
+\spadglossSee{target type}{target} for the expression.
+%-% \HDindex{target type}{ugTypesPkgCallPage}{2.9.}{Package Calling and Target Types}
+Instead of using a \axiomSyntax{\$}, use a \axiomSyntax{@} to specify
+the requested target type.
+Otherwise, the syntax is the same.
+Note that giving a target type is not the same as explicitly doing a
+conversion.
+The first says ``try to pick operations so that the result has
+such-and-such a type.''
+The second says ``compute the result and then convert to an object of
+such-and-such a type.''
+
+\xtc{
+Sometimes it makes sense, as in this expression,
+to say ``choose the operations in this expression so that
+the final result is a \axiomType{Float}.''
+}{
+\spadpaste{(2/3)@Float}
+}
+
+Here we used \axiomSyntax{@} to say that the target type of the
+left-hand side was \axiomType{Float}.
+In this simple case, there was no real difference
+between using \axiomSyntax{\$} and \axiomSyntax{@}.
+You can see the difference if you try the following.
+\xtc{
+This says to try to choose \axiomOp{+} so that the result is
+a string.
+\Language{} cannot do this.
+}{
+\spadpaste{(2 + 3)@String}
+}
+\xtc{
+This says to get the \axiomOp{+} from \axiomType{String} and apply
+it to the two integers.
+\Language{} also cannot do this because there is no \axiomOp{+}
+exported by \axiomType{String}.
+}{
+\spadpaste{(2 + 3)\$String}
+}
+(By the way, the operation \axiomFunFrom{concat}{String} or juxtaposition
+is used to concatenate two strings.)
+%-% \HDexptypeindex{String}{ugTypesPkgCallPage}{2.9.}{Package Calling and Target Types}
+
+When we have more than one operation in an expression, the
+difference is even more evident.
+The following two expressions show that \Language{} uses the
+target type to create different objects.
+The \axiomOp{+}, \axiomOp{*} and \axiomOp{**} operations are all
+chosen so that an object of the correct final type is created.
+
+\xtc{
+This says that the operations should be chosen so
+that the result is a \axiomType{Complex} object.
+}{
+\spadpaste{((x + y * \%i)**2)@(Complex Polynomial Integer)}
+}
+\xtc{
+This says that the operations should be chosen so
+that the result is a \axiomType{Polynomial} object.
+}{
+\spadpaste{((x + y * \%i)**2)@(Polynomial Complex Integer)}
+}
+\xtc{
+What do you think might happen if we left off all
+target type and package call information in this last example?
+}{
+\spadpaste{(x + y * \%i)**2 \bound{prevC}}
+}
+\xtc{
+We can convert it to \axiomType{Complex} as an afterthought.
+But this is more work than just saying making what we want in the first
+place.
+}{
+\spadpaste{\% :: Complex ? \free{prevC}}
+}
+
+Finally, another use of package calling is to qualify fully an
+operation that is passed as an argument to a function.
+
+\xtc{
+Start with a small matrix of integers.
+}{
+\spadpaste{h := matrix [[8,6],[-4,9]] \bound{h}}
+}
+%
+\xtc{
+We want to produce a new matrix that has for entries the multiplicative
+inverses of the entries of \axiom{h}.
+One way to do this is by calling
+\axiomFunFrom{map}{MatrixCategoryFunctions2} with the
+\axiomFunFrom{inv}{Fraction} function from \axiomType{Fraction (Integer)}.
+}{
+\spadpaste{map(inv\$Fraction(Integer),h) \free{h}}
+}
+\xtc{
+We could have been a bit less verbose and used abbreviations.
+}{
+\spadpaste{map(inv\$FRAC(INT),h) \free{h}\bound{h1}}
+}
+%
+\xtc{
+As it turns out, \Language{} is smart enough to know what we mean
+anyway.
+We can just say this.
+}{
+\spadpaste{map(inv,h) \free{h}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesResolveTitle}{Resolving Types}
+\newcommand{\ugTypesResolveNumber}{2.10.}
+%
+% =====================================================================
+\begin{page}{ugTypesResolvePage}{2.10. Resolving Types}
+% =====================================================================
+\beginscroll
+
+In this section we briefly describe an internal process by which
+%-% \HDindex{resolve}{ugTypesResolvePage}{2.10.}{Resolving Types}
+\Language{} determines a type to which two objects of possibly
+different types can be converted.
+We do this to give you further insight into how \Language{} takes
+your input, analyzes it, and produces a result.
+
+What happens when you enter \axiom{x + 1} to \Language{}?
+Let's look at what you get from the two terms of this expression.
+
+\xtc{
+This is a symbolic object whose type indicates the name.
+}{
+\spadpaste{x}
+}
+\xtc{
+This is a positive integer.
+}{
+\spadpaste{1}
+}
+
+There are no operations in \axiomType{PositiveInteger} that add
+positive integers to objects of type \axiomType{Variable(x)} nor
+are there any in \axiomType{Variable(x)}.
+Before it can add the two parts, \Language{} must come up with
+a common type to which both \axiom{x} and \axiom{1} can be
+converted.
+We say that \Language{} must {\it resolve} the two types
+into a common type.
+In this example, the common type is \axiomType{Polynomial(Integer)}.
+
+\xtc{
+Once this is determined, both parts are converted into polynomials,
+and the addition operation from \axiomType{Polynomial(Integer)} is used to
+get the answer.
+}{
+\spadpaste{x + 1}
+}
+\xtc{
+\Language{} can always resolve two types: if nothing resembling
+the original types can be found, then \axiomType{Any} is be used.
+%-% \HDexptypeindex{Any}{ugTypesResolvePage}{2.10.}{Resolving Types}
+This is fine and useful in some cases.
+}{
+\spadpaste{["string",3.14159]}
+}
+\xtc{
+In other cases objects of type \axiomType{Any} can't be used
+by the operations you specified.
+}{
+\spadpaste{"string" + 3.14159}
+}
+Although this example was contrived, your expressions may need
+to be qualified slightly to help \Language{} resolve the
+types involved.
+You may need to declare a few variables, do some package calling,
+provide some target type information or do some explicit
+conversions.
+
+We suggest that you just enter the expression you want evaluated and
+see what \Language{} does.
+We think you will be impressed with its ability to ``do what I
+mean.''
+If \Language{} is still being obtuse, give it some hints.
+As you work with \Language{}, you will learn where it needs a
+little help to analyze quickly and perform your computations.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugTypesExposeTitle}{Exposing Domains and Packages}
+\newcommand{\ugTypesExposeNumber}{2.11.}
+%
+% =====================================================================
+\begin{page}{ugTypesExposePage}{2.11. Exposing Domains and Packages}
+% =====================================================================
+\beginscroll
+
+In this section we discuss how \Language{} makes some operations
+available to you while hiding others that are meant to be used by
+developers or only in rare cases.
+If you are a new user of \Language{}, it is likely that everything
+you need is available by default and you may want
+to skip over this section on first reading.
+
+Every
+%-% \HDindex{constructor!exposed}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+domain and package in the \Language{} library
+%-% \HDindex{constructor!hidden}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+is
+%-% \HDindex{exposed!constructor}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+either
+\spadglossSee{exposed}{expose} (meaning that you can use its operations without doing
+anything special) or it is {\it hidden} (meaning you have to either
+package call
+(see \downlink{``\ugTypesPkgCallTitle''}{ugTypesPkgCallPage} in Section \ugTypesPkgCallNumber\ignore{ugTypesPkgCall})
+the operations it contains or explicitly expose it to use the
+operations).
+The initial exposure status for a constructor is set in the
+file {\bf exposed.lsp} (see the {\it Installer's Note}
+%-% \HDindex{exposed.lsp @{\bf exposed.lsp}}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+for \Language{}
+%-% \HDindex{file!exposed.lsp @{\bf exposed.lsp}}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+if you need to know the location of this file).
+Constructors are collected together in
+%-% \HDindex{group!exposure}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+{\it exposure groups}.
+%-% \HDindex{exposure!group}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+Categories are all in the exposure group ``categories'' and the
+bulk of the basic set of packages and domains that are exposed
+are in the exposure group ``basic.''
+Here is an abbreviated sample of the file (without the Lisp parentheses):
+\begin{verbatim}
+basic
+        AlgebraicNumber                          AN
+        AlgebraGivenByStructuralConstants        ALGSC
+        Any                                      ANY
+        AnyFunctions1                            ANY1
+        BinaryExpansion                          BINARY
+        Boolean                                  BOOLEAN
+        CardinalNumber                           CARD
+        CartesianTensor                          CARTEN
+        Character                                CHAR
+        CharacterClass                           CCLASS
+        CliffordAlgebra                          CLIF
+        Color                                    COLOR
+        Complex                                  COMPLEX
+        ContinuedFraction                        CONTFRAC
+        DecimalExpansion                         DECIMAL
+        ...
+\end{verbatim}
+\begin{verbatim}
+categories
+        AbelianGroup                             ABELGRP
+        AbelianMonoid                            ABELMON
+        AbelianMonoidRing                        AMR
+        AbelianSemiGroup                         ABELSG
+        Aggregate                                AGG
+        Algebra                                  ALGEBRA
+        AlgebraicallyClosedField                 ACF
+        AlgebraicallyClosedFunctionSpace         ACFS
+        ArcHyperbolicFunctionCategory            AHYP
+        ...
+\end{verbatim}
+For each constructor in a group, the full name and the abbreviation
+is given.
+There are other groups in {\bf exposed.lsp} but initially only the
+constructors in exposure groups ``basic'' ``categories'' ``naglink'' and ``anna'' are exposed.
+
+As an interactive user of \Language{}, you do not need to modify
+this file.
+Instead, use \spadcmd{)set expose} to expose, hide or query the exposure
+status of an individual constructor or exposure group.
+%-% \HDsyscmdindex{set expose}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+The reason for having exposure groups is to be able to expose or hide
+multiple constructors with a single command.
+For example, you might group together into exposure group ``quantum'' a
+number of domains and packages useful for quantum mechanical computations.
+These probably should not be available to every user, but you want an easy
+way to make the whole collection visible to \Language{} when it is looking
+for operations to apply.
+
+If you wanted to hide all the basic constructors available by default, you
+would issue \spadcmd{)set expose drop group basic}.
+%-% \HDsyscmdindex{set expose drop group} We do not recommend that you do this.{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+If, however, you discover that you have hidden all the basic constructors,
+you should issue \spadcmd{)set expose add group basic} to restore your
+default environment.
+%-% \HDsyscmdindex{set expose add group}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+
+It is more likely that you would want to expose or hide individual
+constructors.
+In \downlink{``\ugUserTriangleTitle''}{ugUserTrianglePage} in Section \ugUserTriangleNumber\ignore{ugUserTriangle} we use several operations from
+\axiomType{OutputForm}, a domain usually hidden.
+To avoid package calling every operation from \axiomType{OutputForm}, we
+expose the domain and let \Language{} conclude that those operations should
+be used.
+Use \spadcmd{)set expose add constructor} and \spadcmd{)set expose drop
+constructor} to expose and hide a constructor, respectively.
+%-% \HDsyscmdindex{set expose drop constructor}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+You should use the constructor name, not the abbreviation.
+The \spadcmd{)set expose} command guides you through these options.
+%-% \HDsyscmdindex{set expose add constructor}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+
+If you expose a previously hidden constructor, \Language{}
+exhibits new behavior (that was your intention) though you might not
+expect the results that you get.
+\axiomType{OutputForm} is, in fact, one of the worst offenders in this
+regard.
+%-% \HDexptypeindex{OutputForm}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+This domain is meant to be used by other domains for creating a
+structure that \Language{} knows how to display.
+It has functions like \axiomOpFrom{+}{OutputForm} that form output
+representations rather than do mathematical calculations.
+Because of the order in which \Language{} looks at constructors
+when it is deciding what operation to apply, \axiomType{OutputForm}
+might be used instead of what you expect.
+\xtc{
+This is a polynomial.
+}{
+\spadpaste{x + x}
+}
+\xtc{
+Expose \axiomType{OutputForm}.
+}{
+\spadpaste{)set expose add constructor OutputForm \bound{setexposeadd}}
+}
+\xtc{
+This is what we get when \axiomType{OutputForm} is automatically
+available.
+}{
+\spadpaste{x + x \free{setexposeadd}}
+}
+\xtc{
+Hide \axiomType{OutputForm} so we don't run into problems
+with any later examples!
+}{
+\spadpaste{)set expose drop constructor OutputForm \bound{setexposedrop}}
+}
+
+Finally, exposure is done on a frame-by-frame basis.
+A \spadgloss{frame} (see \downlink{``\ugSysCmdframeTitle''}{ugSysCmdframePage} in Section \ugSysCmdframeNumber\ignore{ugSysCmdframe})
+%-% \HDindex{frame!exposure and}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+is one of possibly several
+logical \Language{} workspaces within a physical one, each having
+its own environment (for example, variables and function definitions).
+If you have several \Language{} workspace windows on your screen, they
+are all different frames, automatically created for you by \HyperName{}.
+Frames can be manually created, made active and destroyed by the
+\spadcmd{)frame} system command.
+%-% \HDsyscmdindex{frame}{ugTypesExposePage}{2.11.}{Exposing Domains and Packages}
+They do not share exposure information, so you need to use
+\spadcmd{)set expose} in each one to add or drop constructors from view.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugAvailSnoopTitle}{Commands for Snooping}
+\newcommand{\ugAvailSnoopNumber}{2.12.}
+%
+% =====================================================================
+\begin{page}{ugAvailSnoopPage}{2.12. Commands for Snooping}
+% =====================================================================
+\beginscroll
+
+To conclude this chapter, we introduce you to some system commands
+that you can use for getting more information about domains,
+packages, categories, and operations.
+The most powerful \Language{} facility for getting information about
+constructors and operations is the \Browse{} component of \HyperName{}.
+This is discussed in \downlink{``\ugBrowseTitle''}{ugBrowsePage} in Chapter \ugBrowseNumber\ignore{ugBrowse}.
+
+Use the \spadsys{)what} system command to see lists of system objects
+whose name contain a particular substring (uppercase or lowercase is
+not significant).
+%-% \HDsyscmdindex{what}{ugAvailSnoopPage}{2.12.}{Commands for Snooping}
+
+\xtc{
+Issue this to see a list of all operations with
+``{\tt complex}'' in their names.
+%-% \HDsyscmdindex{what operation}{ugAvailSnoopPage}{2.12.}{Commands for Snooping}
+}{
+\spadpaste{)what operation complex}
+}
+\xtc{
+If you want to see all domains with ``{\tt matrix}'' in their names, issue
+this.
+%-% \HDsyscmdindex{what domain}{ugAvailSnoopPage}{2.12.}{Commands for Snooping}
+}{
+\spadpaste{)what domain matrix}
+}
+\xtc{
+Similarly, if you wish to see all packages whose names contain
+``{\tt gauss}'', enter this.
+%-% \HDsyscmdindex{what packages}{ugAvailSnoopPage}{2.12.}{Commands for Snooping}
+}{
+\spadpaste{)what package gauss}
+}
+\xtc{
+This command shows all
+the operations that \axiomType{Any} provides.
+Wherever \axiomSyntax{\$} appears, it means ``\axiomType{Any}''.
+%-% \HDsyscmdindex{show}{ugAvailSnoopPage}{2.12.}{Commands for Snooping}
+}{
+\spadpaste{)show Any}
+}
+\xtc{
+This displays all operations with the name \axiomFun{complex}.
+%-% \HDsyscmdindex{display operation}{ugAvailSnoopPage}{2.12.}{Commands for Snooping}
+}{
+\spadpaste{)display operation complex}
+}
+Let's analyze this output.
+\xtc{
+First we find out what some of the abbreviations mean.
+}{
+\spadpaste{)abbreviation query COMPCAT}
+}
+\xtc{
+}{
+\spadpaste{)abbreviation query COMRING}
+}
+
+So if \axiom{D1} is a commutative ring (such as the integers or
+floats) and \axiom{D} belongs to \axiomType{ComplexCategory
+D1}, then there is an operation called \axiomFun{complex} that
+takes two elements of \axiom{D1} and creates an element of
+\axiom{D}.
+The primary example of a constructor implementing domains
+belonging to \axiomType{ComplexCategory} is \axiomType{Complex}.
+See \downlink{`Complex'}{ComplexXmpPage}\ignore{Complex} for more information on that and see
+\downlink{``\ugUserDeclareTitle''}{ugUserDeclarePage} in Section \ugUserDeclareNumber\ignore{ugUserDeclare} for more information on function types.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug02.pht b/src/hyper/pages/ug02.pht
new file mode 100644
index 00000000..b96c56b0
--- /dev/null
+++ b/src/hyper/pages/ug02.pht
@@ -0,0 +1,2328 @@
+\begin{patch}{ugTypesUnionsWSelPagePatch1}
+\begin{paste}{ugTypesUnionsWSelPageFull1}{ugTypesUnionsWSelPageEmpty1}
+\pastebutton{ugTypesUnionsWSelPageFull1}{\hidepaste}
+\tab{5}\spadcommand{u : Union(i : Integer, s : String)\bound{undec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWSelPageEmpty1}
+\begin{paste}{ugTypesUnionsWSelPageEmpty1}{ugTypesUnionsWSelPagePatch1}
+\pastebutton{ugTypesUnionsWSelPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{u : Union(i : Integer, s : String)\bound{undec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWSelPagePatch2}
+\begin{paste}{ugTypesUnionsWSelPageFull2}{ugTypesUnionsWSelPageEmpty2}
+\pastebutton{ugTypesUnionsWSelPageFull2}{\hidepaste}
+\tab{5}\spadcommand{u := "good morning"\bound{u }\free{undec }}
+\indentrel{3}\begin{verbatim}
+   (2)  "good morning"
+                             Type: Union(s: String,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWSelPageEmpty2}
+\begin{paste}{ugTypesUnionsWSelPageEmpty2}{ugTypesUnionsWSelPagePatch2}
+\pastebutton{ugTypesUnionsWSelPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{u := "good morning"\bound{u }\free{undec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWSelPagePatch3}
+\begin{paste}{ugTypesUnionsWSelPageFull3}{ugTypesUnionsWSelPageEmpty3}
+\pastebutton{ugTypesUnionsWSelPageFull3}{\hidepaste}
+\tab{5}\spadcommand{u case i\free{u }}
+\indentrel{3}\begin{verbatim}
+   (3)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWSelPageEmpty3}
+\begin{paste}{ugTypesUnionsWSelPageEmpty3}{ugTypesUnionsWSelPagePatch3}
+\pastebutton{ugTypesUnionsWSelPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{u case i\free{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWSelPagePatch4}
+\begin{paste}{ugTypesUnionsWSelPageFull4}{ugTypesUnionsWSelPageEmpty4}
+\pastebutton{ugTypesUnionsWSelPageFull4}{\hidepaste}
+\tab{5}\spadcommand{u case s\free{u }}
+\indentrel{3}\begin{verbatim}
+   (4)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWSelPageEmpty4}
+\begin{paste}{ugTypesUnionsWSelPageEmpty4}{ugTypesUnionsWSelPagePatch4}
+\pastebutton{ugTypesUnionsWSelPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{u case s\free{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWSelPagePatch5}
+\begin{paste}{ugTypesUnionsWSelPageFull5}{ugTypesUnionsWSelPageEmpty5}
+\pastebutton{ugTypesUnionsWSelPageFull5}{\hidepaste}
+\tab{5}\spadcommand{u.s\free{u }}
+\indentrel{3}\begin{verbatim}
+   (5)  "good morning"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWSelPageEmpty5}
+\begin{paste}{ugTypesUnionsWSelPageEmpty5}{ugTypesUnionsWSelPagePatch5}
+\pastebutton{ugTypesUnionsWSelPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{u.s\free{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingOnePagePatch1}
+\begin{paste}{ugTypesWritingOnePageFull1}{ugTypesWritingOnePageEmpty1}
+\pastebutton{ugTypesWritingOnePageFull1}{\hidepaste}
+\tab{5}\spadcommand{e : PrimeField(2 + 3)}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingOnePageEmpty1}
+\begin{paste}{ugTypesWritingOnePageEmpty1}{ugTypesWritingOnePagePatch1}
+\pastebutton{ugTypesWritingOnePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{e : PrimeField(2 + 3)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingOnePagePatch2}
+\begin{paste}{ugTypesWritingOnePageFull2}{ugTypesWritingOnePageEmpty2}
+\pastebutton{ugTypesWritingOnePageFull2}{\hidepaste}
+\tab{5}\spadcommand{content(2)$Polynomial(Integer)}
+\indentrel{3}\begin{verbatim}
+   (2)  2
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingOnePageEmpty2}
+\begin{paste}{ugTypesWritingOnePageEmpty2}{ugTypesWritingOnePagePatch2}
+\pastebutton{ugTypesWritingOnePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{content(2)$Polynomial(Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingOnePagePatch3}
+\begin{paste}{ugTypesWritingOnePageFull3}{ugTypesWritingOnePageEmpty3}
+\pastebutton{ugTypesWritingOnePageFull3}{\hidepaste}
+\tab{5}\spadcommand{content(2)$(Polynomial Complex Fraction Integer)}
+\indentrel{3}\begin{verbatim}
+   (3)  2
+                         Type: Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingOnePageEmpty3}
+\begin{paste}{ugTypesWritingOnePageEmpty3}{ugTypesWritingOnePagePatch3}
+\pastebutton{ugTypesWritingOnePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{content(2)$(Polynomial Complex Fraction Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingOnePagePatch4}
+\begin{paste}{ugTypesWritingOnePageFull4}{ugTypesWritingOnePageEmpty4}
+\pastebutton{ugTypesWritingOnePageFull4}{\hidepaste}
+\tab{5}\spadcommand{(2/3)@Fraction(Polynomial(Integer))}
+\indentrel{3}\begin{verbatim}
+        2
+   (4)  Ä
+        3
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingOnePageEmpty4}
+\begin{paste}{ugTypesWritingOnePageEmpty4}{ugTypesWritingOnePagePatch4}
+\pastebutton{ugTypesWritingOnePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{(2/3)@Fraction(Polynomial(Integer))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingOnePagePatch5}
+\begin{paste}{ugTypesWritingOnePageFull5}{ugTypesWritingOnePageEmpty5}
+\pastebutton{ugTypesWritingOnePageFull5}{\hidepaste}
+\tab{5}\spadcommand{(2/3)@Fraction(Polynomial Integer)}
+\indentrel{3}\begin{verbatim}
+        2
+   (5)  Ä
+        3
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingOnePageEmpty5}
+\begin{paste}{ugTypesWritingOnePageEmpty5}{ugTypesWritingOnePagePatch5}
+\pastebutton{ugTypesWritingOnePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{(2/3)@Fraction(Polynomial Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingOnePagePatch6}
+\begin{paste}{ugTypesWritingOnePageFull6}{ugTypesWritingOnePageEmpty6}
+\pastebutton{ugTypesWritingOnePageFull6}{\hidepaste}
+\tab{5}\spadcommand{(d,f,g) : Complex Polynomial Integer}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingOnePageEmpty6}
+\begin{paste}{ugTypesWritingOnePageEmpty6}{ugTypesWritingOnePagePatch6}
+\pastebutton{ugTypesWritingOnePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{(d,f,g) : Complex Polynomial Integer}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPagePatch1}
+\begin{paste}{ugTypesSubdomainsPageFull1}{ugTypesSubdomainsPageEmpty1}
+\pastebutton{ugTypesSubdomainsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{5}
+\indentrel{3}\begin{verbatim}
+   (1)  5
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPageEmpty1}
+\begin{paste}{ugTypesSubdomainsPageEmpty1}{ugTypesSubdomainsPagePatch1}
+\pastebutton{ugTypesSubdomainsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{5}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPagePatch2}
+\begin{paste}{ugTypesSubdomainsPageFull2}{ugTypesSubdomainsPageEmpty2}
+\pastebutton{ugTypesSubdomainsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{0}
+\indentrel{3}\begin{verbatim}
+   (2)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPageEmpty2}
+\begin{paste}{ugTypesSubdomainsPageEmpty2}{ugTypesSubdomainsPagePatch2}
+\pastebutton{ugTypesSubdomainsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{0}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPagePatch3}
+\begin{paste}{ugTypesSubdomainsPageFull3}{ugTypesSubdomainsPageEmpty3}
+\pastebutton{ugTypesSubdomainsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{-5}
+\indentrel{3}\begin{verbatim}
+   (3)  - 5
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPageEmpty3}
+\begin{paste}{ugTypesSubdomainsPageEmpty3}{ugTypesSubdomainsPagePatch3}
+\pastebutton{ugTypesSubdomainsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{-5}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPagePatch4}
+\begin{paste}{ugTypesSubdomainsPageFull4}{ugTypesSubdomainsPageEmpty4}
+\pastebutton{ugTypesSubdomainsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{(-2) - (-3)}
+\indentrel{3}\begin{verbatim}
+   (4)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPageEmpty4}
+\begin{paste}{ugTypesSubdomainsPageEmpty4}{ugTypesSubdomainsPagePatch4}
+\pastebutton{ugTypesSubdomainsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{(-2) - (-3)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPagePatch5}
+\begin{paste}{ugTypesSubdomainsPageFull5}{ugTypesSubdomainsPageEmpty5}
+\pastebutton{ugTypesSubdomainsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{0 :: Integer}
+\indentrel{3}\begin{verbatim}
+   (5)  0
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPageEmpty5}
+\begin{paste}{ugTypesSubdomainsPageEmpty5}{ugTypesSubdomainsPagePatch5}
+\pastebutton{ugTypesSubdomainsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{0 :: Integer}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPagePatch6}
+\begin{paste}{ugTypesSubdomainsPageFull6}{ugTypesSubdomainsPageEmpty6}
+\pastebutton{ugTypesSubdomainsPageFull6}{\hidepaste}
+\tab{5}\spadcommand{x : NonNegativeInteger := 5}
+\indentrel{3}\begin{verbatim}
+   (6)  5
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPageEmpty6}
+\begin{paste}{ugTypesSubdomainsPageEmpty6}{ugTypesSubdomainsPagePatch6}
+\pastebutton{ugTypesSubdomainsPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{x : NonNegativeInteger := 5}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPagePatch7}
+\begin{paste}{ugTypesSubdomainsPageFull7}{ugTypesSubdomainsPageEmpty7}
+\pastebutton{ugTypesSubdomainsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{2 ** 2}
+\indentrel{3}\begin{verbatim}
+   (7)  4
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPageEmpty7}
+\begin{paste}{ugTypesSubdomainsPageEmpty7}{ugTypesSubdomainsPagePatch7}
+\pastebutton{ugTypesSubdomainsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{2 ** 2}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPagePatch8}
+\begin{paste}{ugTypesSubdomainsPageFull8}{ugTypesSubdomainsPageEmpty8}
+\pastebutton{ugTypesSubdomainsPageFull8}{\hidepaste}
+\tab{5}\spadcommand{2 ** (-2)}
+\indentrel{3}\begin{verbatim}
+        1
+   (8)  Ä
+        4
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPageEmpty8}
+\begin{paste}{ugTypesSubdomainsPageEmpty8}{ugTypesSubdomainsPagePatch8}
+\pastebutton{ugTypesSubdomainsPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{2 ** (-2)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPagePatch9}
+\begin{paste}{ugTypesSubdomainsPageFull9}{ugTypesSubdomainsPageEmpty9}
+\pastebutton{ugTypesSubdomainsPageFull9}{\hidepaste}
+\tab{5}\spadcommand{[10**i for i in 2..5]}
+\indentrel{3}\begin{verbatim}
+   (9)  [100,1000,10000,100000]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPageEmpty9}
+\begin{paste}{ugTypesSubdomainsPageEmpty9}{ugTypesSubdomainsPagePatch9}
+\pastebutton{ugTypesSubdomainsPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{[10**i for i in 2..5]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPagePatch10}
+\begin{paste}{ugTypesSubdomainsPageFull10}{ugTypesSubdomainsPageEmpty10}
+\pastebutton{ugTypesSubdomainsPageFull10}{\hidepaste}
+\tab{5}\spadcommand{[10**(i-1) for i in 2..5]}
+\indentrel{3}\begin{verbatim}
+   (10)  [10,100,1000,10000]
+                            Type: List Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPageEmpty10}
+\begin{paste}{ugTypesSubdomainsPageEmpty10}{ugTypesSubdomainsPagePatch10}
+\pastebutton{ugTypesSubdomainsPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{[10**(i-1) for i in 2..5]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPagePatch11}
+\begin{paste}{ugTypesSubdomainsPageFull11}{ugTypesSubdomainsPageEmpty11}
+\pastebutton{ugTypesSubdomainsPageFull11}{\hidepaste}
+\tab{5}\spadcommand{[10**((i-1) :: PI) for i in 2..5]}
+\indentrel{3}\begin{verbatim}
+   (11)  [10,100,1000,10000]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPageEmpty11}
+\begin{paste}{ugTypesSubdomainsPageEmpty11}{ugTypesSubdomainsPagePatch11}
+\pastebutton{ugTypesSubdomainsPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{[10**((i-1) :: PI) for i in 2..5]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPagePatch12}
+\begin{paste}{ugTypesSubdomainsPageFull12}{ugTypesSubdomainsPageEmpty12}
+\pastebutton{ugTypesSubdomainsPageFull12}{\hidepaste}
+\tab{5}\spadcommand{[10**((i-1) pretend PI) for i in 2..5]}
+\indentrel{3}\begin{verbatim}
+   (12)  [10,100,1000,10000]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPageEmpty12}
+\begin{paste}{ugTypesSubdomainsPageEmpty12}{ugTypesSubdomainsPagePatch12}
+\pastebutton{ugTypesSubdomainsPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{[10**((i-1) pretend PI) for i in 2..5]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPagePatch13}
+\begin{paste}{ugTypesSubdomainsPageFull13}{ugTypesSubdomainsPageEmpty13}
+\pastebutton{ugTypesSubdomainsPageFull13}{\hidepaste}
+\tab{5}\spadcommand{(2/3) pretend Complex Integer}
+\indentrel{3}\begin{verbatim}
+   (13)  2 + 3%i
+                                  Type: Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesSubdomainsPageEmpty13}
+\begin{paste}{ugTypesSubdomainsPageEmpty13}{ugTypesSubdomainsPagePatch13}
+\pastebutton{ugTypesSubdomainsPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{(2/3) pretend Complex Integer}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPagePatch1}
+\begin{paste}{ugTypesUnionsWOSelPageFull1}{ugTypesUnionsWOSelPageEmpty1}
+\pastebutton{ugTypesUnionsWOSelPageFull1}{\hidepaste}
+\tab{5}\spadcommand{sayBranch(x : Union(Integer,String,Float)) : Void  ==
+  output
+    x case Integer => "Integer branch"
+    x case String  => "String branch"
+    "Float branch"
+\bound{sayBranch }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPageEmpty1}
+\begin{paste}{ugTypesUnionsWOSelPageEmpty1}{ugTypesUnionsWOSelPagePatch1}
+\pastebutton{ugTypesUnionsWOSelPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{sayBranch(x : Union(Integer,String,Float)) : Void  ==
+  output
+    x case Integer => "Integer branch"
+    x case String  => "String branch"
+    "Float branch"
+\bound{sayBranch }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPagePatch2}
+\begin{paste}{ugTypesUnionsWOSelPageFull2}{ugTypesUnionsWOSelPageEmpty2}
+\pastebutton{ugTypesUnionsWOSelPageFull2}{\hidepaste}
+\tab{5}\spadcommand{sayBranch 1\free{sayBranch }}
+\indentrel{3}\begin{verbatim}
+   Integer branch
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPageEmpty2}
+\begin{paste}{ugTypesUnionsWOSelPageEmpty2}{ugTypesUnionsWOSelPagePatch2}
+\pastebutton{ugTypesUnionsWOSelPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{sayBranch 1\free{sayBranch }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPagePatch3}
+\begin{paste}{ugTypesUnionsWOSelPageFull3}{ugTypesUnionsWOSelPageEmpty3}
+\pastebutton{ugTypesUnionsWOSelPageFull3}{\hidepaste}
+\tab{5}\spadcommand{sayBranch "hello"\free{sayBranch }}
+\indentrel{3}\begin{verbatim}
+   String branch
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPageEmpty3}
+\begin{paste}{ugTypesUnionsWOSelPageEmpty3}{ugTypesUnionsWOSelPagePatch3}
+\pastebutton{ugTypesUnionsWOSelPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{sayBranch "hello"\free{sayBranch }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPagePatch4}
+\begin{paste}{ugTypesUnionsWOSelPageFull4}{ugTypesUnionsWOSelPageEmpty4}
+\pastebutton{ugTypesUnionsWOSelPageFull4}{\hidepaste}
+\tab{5}\spadcommand{sayBranch 2.718281828\free{sayBranch }}
+\indentrel{3}\begin{verbatim}
+   Float branch
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPageEmpty4}
+\begin{paste}{ugTypesUnionsWOSelPageEmpty4}{ugTypesUnionsWOSelPagePatch4}
+\pastebutton{ugTypesUnionsWOSelPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{sayBranch 2.718281828\free{sayBranch }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPagePatch5}
+\begin{paste}{ugTypesUnionsWOSelPageFull5}{ugTypesUnionsWOSelPageEmpty5}
+\pastebutton{ugTypesUnionsWOSelPageFull5}{\hidepaste}
+\tab{5}\spadcommand{78 :: Union(Integer,String)}
+\indentrel{3}\begin{verbatim}
+   (5)  78
+                               Type: Union(Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPageEmpty5}
+\begin{paste}{ugTypesUnionsWOSelPageEmpty5}{ugTypesUnionsWOSelPagePatch5}
+\pastebutton{ugTypesUnionsWOSelPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{78 :: Union(Integer,String)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPagePatch6}
+\begin{paste}{ugTypesUnionsWOSelPageFull6}{ugTypesUnionsWOSelPageEmpty6}
+\pastebutton{ugTypesUnionsWOSelPageFull6}{\hidepaste}
+\tab{5}\spadcommand{s := "string" :: Union(Integer,String)\bound{s }}
+\indentrel{3}\begin{verbatim}
+   (6)  "string"
+                                Type: Union(String,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPageEmpty6}
+\begin{paste}{ugTypesUnionsWOSelPageEmpty6}{ugTypesUnionsWOSelPagePatch6}
+\pastebutton{ugTypesUnionsWOSelPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{s := "string" :: Union(Integer,String)\bound{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPagePatch7}
+\begin{paste}{ugTypesUnionsWOSelPageFull7}{ugTypesUnionsWOSelPageEmpty7}
+\pastebutton{ugTypesUnionsWOSelPageFull7}{\hidepaste}
+\tab{5}\spadcommand{typeOf s}
+\indentrel{3}\begin{verbatim}
+   (7)  Union(Integer,String)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPageEmpty7}
+\begin{paste}{ugTypesUnionsWOSelPageEmpty7}{ugTypesUnionsWOSelPagePatch7}
+\pastebutton{ugTypesUnionsWOSelPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{typeOf s}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPagePatch8}
+\begin{paste}{ugTypesUnionsWOSelPageFull8}{ugTypesUnionsWOSelPageEmpty8}
+\pastebutton{ugTypesUnionsWOSelPageFull8}{\hidepaste}
+\tab{5}\spadcommand{three := exquo(6,2)\bound{three }}
+\indentrel{3}\begin{verbatim}
+   (8)  3
+                               Type: Union(Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPageEmpty8}
+\begin{paste}{ugTypesUnionsWOSelPageEmpty8}{ugTypesUnionsWOSelPagePatch8}
+\pastebutton{ugTypesUnionsWOSelPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{three := exquo(6,2)\bound{three }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPagePatch9}
+\begin{paste}{ugTypesUnionsWOSelPageFull9}{ugTypesUnionsWOSelPageEmpty9}
+\pastebutton{ugTypesUnionsWOSelPageFull9}{\hidepaste}
+\tab{5}\spadcommand{exquo(5,2)}
+\indentrel{3}\begin{verbatim}
+   (9)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPageEmpty9}
+\begin{paste}{ugTypesUnionsWOSelPageEmpty9}{ugTypesUnionsWOSelPagePatch9}
+\pastebutton{ugTypesUnionsWOSelPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{exquo(5,2)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPagePatch10}
+\begin{paste}{ugTypesUnionsWOSelPageFull10}{ugTypesUnionsWOSelPageEmpty10}
+\pastebutton{ugTypesUnionsWOSelPageFull10}{\hidepaste}
+\tab{5}\spadcommand{r: FRAC INT := 3\bound{r }\bound{rdec }}
+\indentrel{3}\begin{verbatim}
+   (10)  3
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPageEmpty10}
+\begin{paste}{ugTypesUnionsWOSelPageEmpty10}{ugTypesUnionsWOSelPagePatch10}
+\pastebutton{ugTypesUnionsWOSelPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{r: FRAC INT := 3\bound{r }\bound{rdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPagePatch11}
+\begin{paste}{ugTypesUnionsWOSelPageFull11}{ugTypesUnionsWOSelPageEmpty11}
+\pastebutton{ugTypesUnionsWOSelPageFull11}{\hidepaste}
+\tab{5}\spadcommand{retractIfCan(r)\free{r }}
+\indentrel{3}\begin{verbatim}
+   (11)  3
+                               Type: Union(Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPageEmpty11}
+\begin{paste}{ugTypesUnionsWOSelPageEmpty11}{ugTypesUnionsWOSelPagePatch11}
+\pastebutton{ugTypesUnionsWOSelPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{retractIfCan(r)\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPagePatch12}
+\begin{paste}{ugTypesUnionsWOSelPageFull12}{ugTypesUnionsWOSelPageEmpty12}
+\pastebutton{ugTypesUnionsWOSelPageFull12}{\hidepaste}
+\tab{5}\spadcommand{r := 3/2\bound{r1 }\free{rdec }}
+\indentrel{3}\begin{verbatim}
+         3
+   (12)  Ä
+         2
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPageEmpty12}
+\begin{paste}{ugTypesUnionsWOSelPageEmpty12}{ugTypesUnionsWOSelPagePatch12}
+\pastebutton{ugTypesUnionsWOSelPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{r := 3/2\bound{r1 }\free{rdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPagePatch13}
+\begin{paste}{ugTypesUnionsWOSelPageFull13}{ugTypesUnionsWOSelPageEmpty13}
+\pastebutton{ugTypesUnionsWOSelPageFull13}{\hidepaste}
+\tab{5}\spadcommand{retractIfCan(r)\free{r1 }}
+\indentrel{3}\begin{verbatim}
+   (13)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesUnionsWOSelPageEmpty13}
+\begin{paste}{ugTypesUnionsWOSelPageEmpty13}{ugTypesUnionsWOSelPagePatch13}
+\pastebutton{ugTypesUnionsWOSelPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{retractIfCan(r)\free{r1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingAbbrPagePatch1}
+\begin{paste}{ugTypesWritingAbbrPageFull1}{ugTypesWritingAbbrPageEmpty1}
+\pastebutton{ugTypesWritingAbbrPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)abb q Integer}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingAbbrPageEmpty1}
+\begin{paste}{ugTypesWritingAbbrPageEmpty1}{ugTypesWritingAbbrPagePatch1}
+\pastebutton{ugTypesWritingAbbrPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)abb q Integer}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingAbbrPagePatch2}
+\begin{paste}{ugTypesWritingAbbrPageFull2}{ugTypesWritingAbbrPageEmpty2}
+\pastebutton{ugTypesWritingAbbrPageFull2}{\hidepaste}
+\tab{5}\spadcommand{)abb q DMP}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingAbbrPageEmpty2}
+\begin{paste}{ugTypesWritingAbbrPageEmpty2}{ugTypesWritingAbbrPagePatch2}
+\pastebutton{ugTypesWritingAbbrPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{)abb q DMP}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingAbbrPagePatch3}
+\begin{paste}{ugTypesWritingAbbrPageFull3}{ugTypesWritingAbbrPageEmpty3}
+\pastebutton{ugTypesWritingAbbrPageFull3}{\hidepaste}
+\tab{5}\spadcommand{)what packages ode}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingAbbrPageEmpty3}
+\begin{paste}{ugTypesWritingAbbrPageEmpty3}{ugTypesWritingAbbrPagePatch3}
+\pastebutton{ugTypesWritingAbbrPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{)what packages ode}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicPagePatch1}
+\begin{paste}{ugTypesBasicPageFull1}{ugTypesBasicPageEmpty1}
+\pastebutton{ugTypesBasicPageFull1}{\hidepaste}
+\tab{5}\spadcommand{-3}
+\indentrel{3}\begin{verbatim}
+   (1)  - 3
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicPageEmpty1}
+\begin{paste}{ugTypesBasicPageEmpty1}{ugTypesBasicPagePatch1}
+\pastebutton{ugTypesBasicPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{-3}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicPagePatch2}
+\begin{paste}{ugTypesBasicPageFull2}{ugTypesBasicPageEmpty2}
+\pastebutton{ugTypesBasicPageFull2}{\hidepaste}
+\tab{5}\spadcommand{-3/1}
+\indentrel{3}\begin{verbatim}
+   (2)  - 3
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicPageEmpty2}
+\begin{paste}{ugTypesBasicPageEmpty2}{ugTypesBasicPagePatch2}
+\pastebutton{ugTypesBasicPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{-3/1}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicPagePatch3}
+\begin{paste}{ugTypesBasicPageFull3}{ugTypesBasicPageEmpty3}
+\pastebutton{ugTypesBasicPageFull3}{\hidepaste}
+\tab{5}\spadcommand{x + 3 - x\bound{three }}
+\indentrel{3}\begin{verbatim}
+   (3)  3
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicPageEmpty3}
+\begin{paste}{ugTypesBasicPageEmpty3}{ugTypesBasicPagePatch3}
+\pastebutton{ugTypesBasicPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{x + 3 - x\bound{three }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicPagePatch4}
+\begin{paste}{ugTypesBasicPageFull4}{ugTypesBasicPageEmpty4}
+\pastebutton{ugTypesBasicPageFull4}{\hidepaste}
+\tab{5}\spadcommand{factorial(\%)\free{three }}
+\indentrel{3}\begin{verbatim}
+   (4)  6
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicPageEmpty4}
+\begin{paste}{ugTypesBasicPageEmpty4}{ugTypesBasicPagePatch4}
+\pastebutton{ugTypesBasicPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{factorial(\%)\free{three }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicPagePatch5}
+\begin{paste}{ugTypesBasicPageFull5}{ugTypesBasicPageEmpty5}
+\pastebutton{ugTypesBasicPageFull5}{\hidepaste}
+\tab{5}\spadcommand{3}
+\indentrel{3}\begin{verbatim}
+   (5)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicPageEmpty5}
+\begin{paste}{ugTypesBasicPageEmpty5}{ugTypesBasicPagePatch5}
+\pastebutton{ugTypesBasicPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{3}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPagePatch1}
+\begin{paste}{ugTypesBasicDomainConsPageFull1}{ugTypesBasicDomainConsPageEmpty1}
+\pastebutton{ugTypesBasicDomainConsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{factorial(7)}
+\indentrel{3}\begin{verbatim}
+   (1)  5040
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPageEmpty1}
+\begin{paste}{ugTypesBasicDomainConsPageEmpty1}{ugTypesBasicDomainConsPagePatch1}
+\pastebutton{ugTypesBasicDomainConsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{factorial(7)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPagePatch2}
+\begin{paste}{ugTypesBasicDomainConsPageFull2}{ugTypesBasicDomainConsPageEmpty2}
+\pastebutton{ugTypesBasicDomainConsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{Polynomial(Integer)}
+\indentrel{3}\begin{verbatim}
+   (2)  Polynomial Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPageEmpty2}
+\begin{paste}{ugTypesBasicDomainConsPageEmpty2}{ugTypesBasicDomainConsPagePatch2}
+\pastebutton{ugTypesBasicDomainConsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{Polynomial(Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPagePatch3}
+\begin{paste}{ugTypesBasicDomainConsPageFull3}{ugTypesBasicDomainConsPageEmpty3}
+\pastebutton{ugTypesBasicDomainConsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{List (List (Matrix (Polynomial (Complex (Fraction (Integer))))))}
+\indentrel{3}\begin{verbatim}
+   (3)
+   List List Matrix Polynomial Complex Fraction Integer
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPageEmpty3}
+\begin{paste}{ugTypesBasicDomainConsPageEmpty3}{ugTypesBasicDomainConsPagePatch3}
+\pastebutton{ugTypesBasicDomainConsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{List (List (Matrix (Polynomial (Complex (Fraction (Integer))))))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPagePatch4}
+\begin{paste}{ugTypesBasicDomainConsPageFull4}{ugTypesBasicDomainConsPageEmpty4}
+\pastebutton{ugTypesBasicDomainConsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{Polynomial(String)}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPageEmpty4}
+\begin{paste}{ugTypesBasicDomainConsPageEmpty4}{ugTypesBasicDomainConsPagePatch4}
+\pastebutton{ugTypesBasicDomainConsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{Polynomial(String)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPagePatch5}
+\begin{paste}{ugTypesBasicDomainConsPageFull5}{ugTypesBasicDomainConsPageEmpty5}
+\pastebutton{ugTypesBasicDomainConsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{Polynomial(Integer) has Ring}
+\indentrel{3}\begin{verbatim}
+   (4)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPageEmpty5}
+\begin{paste}{ugTypesBasicDomainConsPageEmpty5}{ugTypesBasicDomainConsPagePatch5}
+\pastebutton{ugTypesBasicDomainConsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{Polynomial(Integer) has Ring}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPagePatch6}
+\begin{paste}{ugTypesBasicDomainConsPageFull6}{ugTypesBasicDomainConsPageEmpty6}
+\pastebutton{ugTypesBasicDomainConsPageFull6}{\hidepaste}
+\tab{5}\spadcommand{List(Integer) has Ring}
+\indentrel{3}\begin{verbatim}
+   (5)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPageEmpty6}
+\begin{paste}{ugTypesBasicDomainConsPageEmpty6}{ugTypesBasicDomainConsPagePatch6}
+\pastebutton{ugTypesBasicDomainConsPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{List(Integer) has Ring}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPagePatch7}
+\begin{paste}{ugTypesBasicDomainConsPageFull7}{ugTypesBasicDomainConsPageEmpty7}
+\pastebutton{ugTypesBasicDomainConsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{Matrix(Integer) has Ring}
+\indentrel{3}\begin{verbatim}
+   (6)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPageEmpty7}
+\begin{paste}{ugTypesBasicDomainConsPageEmpty7}{ugTypesBasicDomainConsPagePatch7}
+\pastebutton{ugTypesBasicDomainConsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{Matrix(Integer) has Ring}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPagePatch8}
+\begin{paste}{ugTypesBasicDomainConsPageFull8}{ugTypesBasicDomainConsPageEmpty8}
+\pastebutton{ugTypesBasicDomainConsPageFull8}{\hidepaste}
+\tab{5}\spadcommand{Polynomial(Matrix(Integer))}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPageEmpty8}
+\begin{paste}{ugTypesBasicDomainConsPageEmpty8}{ugTypesBasicDomainConsPagePatch8}
+\pastebutton{ugTypesBasicDomainConsPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{Polynomial(Matrix(Integer))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPagePatch9}
+\begin{paste}{ugTypesBasicDomainConsPageFull9}{ugTypesBasicDomainConsPageEmpty9}
+\pastebutton{ugTypesBasicDomainConsPageFull9}{\hidepaste}
+\tab{5}\spadcommand{Polynomial(SquareMatrix(7,Complex(Integer)))}
+\indentrel{3}\begin{verbatim}
+   (7)  Polynomial SquareMatrix(7,Complex Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPageEmpty9}
+\begin{paste}{ugTypesBasicDomainConsPageEmpty9}{ugTypesBasicDomainConsPagePatch9}
+\pastebutton{ugTypesBasicDomainConsPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{Polynomial(SquareMatrix(7,Complex(Integer)))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPagePatch10}
+\begin{paste}{ugTypesBasicDomainConsPageFull10}{ugTypesBasicDomainConsPageEmpty10}
+\pastebutton{ugTypesBasicDomainConsPageFull10}{\hidepaste}
+\tab{5}\spadcommand{Complex(Integer) has Field}
+\indentrel{3}\begin{verbatim}
+   (8)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPageEmpty10}
+\begin{paste}{ugTypesBasicDomainConsPageEmpty10}{ugTypesBasicDomainConsPagePatch10}
+\pastebutton{ugTypesBasicDomainConsPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{Complex(Integer) has Field}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPagePatch11}
+\begin{paste}{ugTypesBasicDomainConsPageFull11}{ugTypesBasicDomainConsPageEmpty11}
+\pastebutton{ugTypesBasicDomainConsPageFull11}{\hidepaste}
+\tab{5}\spadcommand{Fraction(Complex(Integer)) has Field}
+\indentrel{3}\begin{verbatim}
+   (9)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPageEmpty11}
+\begin{paste}{ugTypesBasicDomainConsPageEmpty11}{ugTypesBasicDomainConsPagePatch11}
+\pastebutton{ugTypesBasicDomainConsPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{Fraction(Complex(Integer)) has Field}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPagePatch12}
+\begin{paste}{ugTypesBasicDomainConsPageFull12}{ugTypesBasicDomainConsPageEmpty12}
+\pastebutton{ugTypesBasicDomainConsPageFull12}{\hidepaste}
+\tab{5}\spadcommand{Complex(Fraction(Integer)) has Field}
+\indentrel{3}\begin{verbatim}
+   (10)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesBasicDomainConsPageEmpty12}
+\begin{paste}{ugTypesBasicDomainConsPageEmpty12}{ugTypesBasicDomainConsPagePatch12}
+\pastebutton{ugTypesBasicDomainConsPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{Complex(Fraction(Integer)) has Field}
+\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPagePatch1}
+\begin{paste}{ugAvailSnoopPageFull1}{ugAvailSnoopPageEmpty1}
+\pastebutton{ugAvailSnoopPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)what operation complex}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPageEmpty1}
+\begin{paste}{ugAvailSnoopPageEmpty1}{ugAvailSnoopPagePatch1}
+\pastebutton{ugAvailSnoopPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)what operation complex}
+\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPagePatch2}
+\begin{paste}{ugAvailSnoopPageFull2}{ugAvailSnoopPageEmpty2}
+\pastebutton{ugAvailSnoopPageFull2}{\hidepaste}
+\tab{5}\spadcommand{)what domain matrix}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPageEmpty2}
+\begin{paste}{ugAvailSnoopPageEmpty2}{ugAvailSnoopPagePatch2}
+\pastebutton{ugAvailSnoopPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{)what domain matrix}
+\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPagePatch3}
+\begin{paste}{ugAvailSnoopPageFull3}{ugAvailSnoopPageEmpty3}
+\pastebutton{ugAvailSnoopPageFull3}{\hidepaste}
+\tab{5}\spadcommand{)what package gauss}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPageEmpty3}
+\begin{paste}{ugAvailSnoopPageEmpty3}{ugAvailSnoopPagePatch3}
+\pastebutton{ugAvailSnoopPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{)what package gauss}
+\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPagePatch4}
+\begin{paste}{ugAvailSnoopPageFull4}{ugAvailSnoopPageEmpty4}
+\pastebutton{ugAvailSnoopPageFull4}{\hidepaste}
+\tab{5}\spadcommand{)show Any}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPageEmpty4}
+\begin{paste}{ugAvailSnoopPageEmpty4}{ugAvailSnoopPagePatch4}
+\pastebutton{ugAvailSnoopPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{)show Any}
+\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPagePatch5}
+\begin{paste}{ugAvailSnoopPageFull5}{ugAvailSnoopPageEmpty5}
+\pastebutton{ugAvailSnoopPageFull5}{\hidepaste}
+\tab{5}\spadcommand{)display operation complex}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPageEmpty5}
+\begin{paste}{ugAvailSnoopPageEmpty5}{ugAvailSnoopPagePatch5}
+\pastebutton{ugAvailSnoopPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{)display operation complex}
+\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPagePatch6}
+\begin{paste}{ugAvailSnoopPageFull6}{ugAvailSnoopPageEmpty6}
+\pastebutton{ugAvailSnoopPageFull6}{\hidepaste}
+\tab{5}\spadcommand{)abbreviation query COMPCAT}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPageEmpty6}
+\begin{paste}{ugAvailSnoopPageEmpty6}{ugAvailSnoopPagePatch6}
+\pastebutton{ugAvailSnoopPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{)abbreviation query COMPCAT}
+\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPagePatch7}
+\begin{paste}{ugAvailSnoopPageFull7}{ugAvailSnoopPageEmpty7}
+\pastebutton{ugAvailSnoopPageFull7}{\hidepaste}
+\tab{5}\spadcommand{)abbreviation query COMRING}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugAvailSnoopPageEmpty7}
+\begin{paste}{ugAvailSnoopPageEmpty7}{ugAvailSnoopPagePatch7}
+\pastebutton{ugAvailSnoopPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{)abbreviation query COMRING}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesExposePagePatch1}
+\begin{paste}{ugTypesExposePageFull1}{ugTypesExposePageEmpty1}
+\pastebutton{ugTypesExposePageFull1}{\hidepaste}
+\tab{5}\spadcommand{x + x}
+\indentrel{3}\begin{verbatim}
+   (1)  2x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesExposePageEmpty1}
+\begin{paste}{ugTypesExposePageEmpty1}{ugTypesExposePagePatch1}
+\pastebutton{ugTypesExposePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{x + x}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesExposePagePatch2}
+\begin{paste}{ugTypesExposePageFull2}{ugTypesExposePageEmpty2}
+\pastebutton{ugTypesExposePageFull2}{\hidepaste}
+\tab{5}\spadcommand{)set expose add constructor OutputForm\bound{setexposeadd }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesExposePageEmpty2}
+\begin{paste}{ugTypesExposePageEmpty2}{ugTypesExposePagePatch2}
+\pastebutton{ugTypesExposePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{)set expose add constructor OutputForm\bound{setexposeadd }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesExposePagePatch3}
+\begin{paste}{ugTypesExposePageFull3}{ugTypesExposePageEmpty3}
+\pastebutton{ugTypesExposePageFull3}{\hidepaste}
+\tab{5}\spadcommand{x + x\free{setexposeadd }}
+\indentrel{3}\begin{verbatim}
+   (2)  x + x
+                                       Type: OutputForm
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesExposePageEmpty3}
+\begin{paste}{ugTypesExposePageEmpty3}{ugTypesExposePagePatch3}
+\pastebutton{ugTypesExposePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{x + x\free{setexposeadd }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesExposePagePatch4}
+\begin{paste}{ugTypesExposePageFull4}{ugTypesExposePageEmpty4}
+\pastebutton{ugTypesExposePageFull4}{\hidepaste}
+\tab{5}\spadcommand{)set expose drop constructor OutputForm\bound{setexposedrop }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesExposePageEmpty4}
+\begin{paste}{ugTypesExposePageEmpty4}{ugTypesExposePagePatch4}
+\pastebutton{ugTypesExposePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{)set expose drop constructor OutputForm\bound{setexposedrop }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch1}
+\begin{paste}{ugTypesPkgCallPageFull1}{ugTypesPkgCallPageEmpty1}
+\pastebutton{ugTypesPkgCallPageFull1}{\hidepaste}
+\tab{5}\spadcommand{2/3}
+\indentrel{3}\begin{verbatim}
+        2
+   (1)  Ä
+        3
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty1}
+\begin{paste}{ugTypesPkgCallPageEmpty1}{ugTypesPkgCallPagePatch1}
+\pastebutton{ugTypesPkgCallPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{2/3}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch2}
+\begin{paste}{ugTypesPkgCallPageFull2}{ugTypesPkgCallPageEmpty2}
+\pastebutton{ugTypesPkgCallPageFull2}{\hidepaste}
+\tab{5}\spadcommand{(2/3)$Float}
+\indentrel{3}\begin{verbatim}
+   (2)  0.6666666666 6666666667
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty2}
+\begin{paste}{ugTypesPkgCallPageEmpty2}{ugTypesPkgCallPagePatch2}
+\pastebutton{ugTypesPkgCallPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{(2/3)$Float}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch3}
+\begin{paste}{ugTypesPkgCallPageFull3}{ugTypesPkgCallPageEmpty3}
+\pastebutton{ugTypesPkgCallPageFull3}{\hidepaste}
+\tab{5}\spadcommand{(2/3)$Fraction(Complex Integer)}
+\indentrel{3}\begin{verbatim}
+        2
+   (3)  Ä
+        3
+                         Type: Fraction Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty3}
+\begin{paste}{ugTypesPkgCallPageEmpty3}{ugTypesPkgCallPagePatch3}
+\pastebutton{ugTypesPkgCallPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{(2/3)$Fraction(Complex Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch4}
+\begin{paste}{ugTypesPkgCallPageFull4}{ugTypesPkgCallPageEmpty4}
+\pastebutton{ugTypesPkgCallPageFull4}{\hidepaste}
+\tab{5}\spadcommand{(2/3)@Float}
+\indentrel{3}\begin{verbatim}
+   (4)  0.6666666666 6666666667
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty4}
+\begin{paste}{ugTypesPkgCallPageEmpty4}{ugTypesPkgCallPagePatch4}
+\pastebutton{ugTypesPkgCallPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{(2/3)@Float}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch5}
+\begin{paste}{ugTypesPkgCallPageFull5}{ugTypesPkgCallPageEmpty5}
+\pastebutton{ugTypesPkgCallPageFull5}{\hidepaste}
+\tab{5}\spadcommand{(2 + 3)@String}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty5}
+\begin{paste}{ugTypesPkgCallPageEmpty5}{ugTypesPkgCallPagePatch5}
+\pastebutton{ugTypesPkgCallPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{(2 + 3)@String}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch6}
+\begin{paste}{ugTypesPkgCallPageFull6}{ugTypesPkgCallPageEmpty6}
+\pastebutton{ugTypesPkgCallPageFull6}{\hidepaste}
+\tab{5}\spadcommand{(2 + 3)$String}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty6}
+\begin{paste}{ugTypesPkgCallPageEmpty6}{ugTypesPkgCallPagePatch6}
+\pastebutton{ugTypesPkgCallPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{(2 + 3)$String}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch7}
+\begin{paste}{ugTypesPkgCallPageFull7}{ugTypesPkgCallPageEmpty7}
+\pastebutton{ugTypesPkgCallPageFull7}{\hidepaste}
+\tab{5}\spadcommand{((x + y * \%i)**2)@(Complex Polynomial Integer)}
+\indentrel{3}\begin{verbatim}
+           2    2
+   (5)  - y  + x  + 2x y %i
+                       Type: Complex Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty7}
+\begin{paste}{ugTypesPkgCallPageEmpty7}{ugTypesPkgCallPagePatch7}
+\pastebutton{ugTypesPkgCallPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{((x + y * \%i)**2)@(Complex Polynomial Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch8}
+\begin{paste}{ugTypesPkgCallPageFull8}{ugTypesPkgCallPageEmpty8}
+\pastebutton{ugTypesPkgCallPageFull8}{\hidepaste}
+\tab{5}\spadcommand{((x + y * \%i)**2)@(Polynomial Complex Integer)}
+\indentrel{3}\begin{verbatim}
+           2              2
+   (6)  - y  + 2%i x y + x
+                       Type: Polynomial Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty8}
+\begin{paste}{ugTypesPkgCallPageEmpty8}{ugTypesPkgCallPagePatch8}
+\pastebutton{ugTypesPkgCallPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{((x + y * \%i)**2)@(Polynomial Complex Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch9}
+\begin{paste}{ugTypesPkgCallPageFull9}{ugTypesPkgCallPageEmpty9}
+\pastebutton{ugTypesPkgCallPageFull9}{\hidepaste}
+\tab{5}\spadcommand{(x + y * \%i)**2\bound{prevC }}
+\indentrel{3}\begin{verbatim}
+           2              2
+   (7)  - y  + 2%i x y + x
+                       Type: Polynomial Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty9}
+\begin{paste}{ugTypesPkgCallPageEmpty9}{ugTypesPkgCallPagePatch9}
+\pastebutton{ugTypesPkgCallPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{(x + y * \%i)**2\bound{prevC }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch10}
+\begin{paste}{ugTypesPkgCallPageFull10}{ugTypesPkgCallPageEmpty10}
+\pastebutton{ugTypesPkgCallPageFull10}{\hidepaste}
+\tab{5}\spadcommand{\% :: Complex ?\free{prevC }}
+\indentrel{3}\begin{verbatim}
+           2    2
+   (8)  - y  + x  + 2x y %i
+                       Type: Complex Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty10}
+\begin{paste}{ugTypesPkgCallPageEmpty10}{ugTypesPkgCallPagePatch10}
+\pastebutton{ugTypesPkgCallPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{\% :: Complex ?\free{prevC }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch11}
+\begin{paste}{ugTypesPkgCallPageFull11}{ugTypesPkgCallPageEmpty11}
+\pastebutton{ugTypesPkgCallPageFull11}{\hidepaste}
+\tab{5}\spadcommand{h := matrix [[8,6],[-4,9]]\bound{h }}
+\indentrel{3}\begin{verbatim}
+        Ú 8   6¿
+   (9)  ³      ³
+        À- 4  9Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty11}
+\begin{paste}{ugTypesPkgCallPageEmpty11}{ugTypesPkgCallPagePatch11}
+\pastebutton{ugTypesPkgCallPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{h := matrix [[8,6],[-4,9]]\bound{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch12}
+\begin{paste}{ugTypesPkgCallPageFull12}{ugTypesPkgCallPageEmpty12}
+\pastebutton{ugTypesPkgCallPageFull12}{\hidepaste}
+\tab{5}\spadcommand{map(inv$Fraction(Integer),h)\free{h }}
+\indentrel{3}\begin{verbatim}
+         Ú 1   1¿
+         ³ Ä   ij
+         ³ 8   6³
+   (10)  ³      ³
+         ³  1  1³
+         ³- Ä  ij
+         À  4  9Ù
+                          Type: Matrix Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty12}
+\begin{paste}{ugTypesPkgCallPageEmpty12}{ugTypesPkgCallPagePatch12}
+\pastebutton{ugTypesPkgCallPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{map(inv$Fraction(Integer),h)\free{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch13}
+\begin{paste}{ugTypesPkgCallPageFull13}{ugTypesPkgCallPageEmpty13}
+\pastebutton{ugTypesPkgCallPageFull13}{\hidepaste}
+\tab{5}\spadcommand{map(inv$FRAC(INT),h)\free{h }\bound{h1 }}
+\indentrel{3}\begin{verbatim}
+         Ú 1   1¿
+         ³ Ä   ij
+         ³ 8   6³
+   (11)  ³      ³
+         ³  1  1³
+         ³- Ä  ij
+         À  4  9Ù
+                          Type: Matrix Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty13}
+\begin{paste}{ugTypesPkgCallPageEmpty13}{ugTypesPkgCallPagePatch13}
+\pastebutton{ugTypesPkgCallPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{map(inv$FRAC(INT),h)\free{h }\bound{h1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPagePatch14}
+\begin{paste}{ugTypesPkgCallPageFull14}{ugTypesPkgCallPageEmpty14}
+\pastebutton{ugTypesPkgCallPageFull14}{\hidepaste}
+\tab{5}\spadcommand{map(inv,h)\free{h }}
+\indentrel{3}\begin{verbatim}
+         Ú 1   1¿
+         ³ Ä   ij
+         ³ 8   6³
+   (12)  ³      ³
+         ³  1  1³
+         ³- Ä  ij
+         À  4  9Ù
+                          Type: Matrix Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesPkgCallPageEmpty14}
+\begin{paste}{ugTypesPkgCallPageEmpty14}{ugTypesPkgCallPagePatch14}
+\pastebutton{ugTypesPkgCallPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{map(inv,h)\free{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePagePatch1}
+\begin{paste}{ugTypesDeclarePageFull1}{ugTypesDeclarePageEmpty1}
+\pastebutton{ugTypesDeclarePageFull1}{\hidepaste}
+\tab{5}\spadcommand{a : Integer\bound{a }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePageEmpty1}
+\begin{paste}{ugTypesDeclarePageEmpty1}{ugTypesDeclarePagePatch1}
+\pastebutton{ugTypesDeclarePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{a : Integer\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePagePatch2}
+\begin{paste}{ugTypesDeclarePageFull2}{ugTypesDeclarePageEmpty2}
+\pastebutton{ugTypesDeclarePageFull2}{\hidepaste}
+\tab{5}\spadcommand{(b,c) : Integer\bound{b c }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePageEmpty2}
+\begin{paste}{ugTypesDeclarePageEmpty2}{ugTypesDeclarePagePatch2}
+\pastebutton{ugTypesDeclarePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{(b,c) : Integer\bound{b c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePagePatch3}
+\begin{paste}{ugTypesDeclarePageFull3}{ugTypesDeclarePageEmpty3}
+\pastebutton{ugTypesDeclarePageFull3}{\hidepaste}
+\tab{5}\spadcommand{a := 45\free{a }}
+\indentrel{3}\begin{verbatim}
+   (3)  45
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePageEmpty3}
+\begin{paste}{ugTypesDeclarePageEmpty3}{ugTypesDeclarePagePatch3}
+\pastebutton{ugTypesDeclarePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{a := 45\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePagePatch4}
+\begin{paste}{ugTypesDeclarePageFull4}{ugTypesDeclarePageEmpty4}
+\pastebutton{ugTypesDeclarePageFull4}{\hidepaste}
+\tab{5}\spadcommand{b := 4/5\free{b }}
+\indentrel{3}\begin{verbatim}
+   4
+   Ä
+   5
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePageEmpty4}
+\begin{paste}{ugTypesDeclarePageEmpty4}{ugTypesDeclarePagePatch4}
+\pastebutton{ugTypesDeclarePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{b := 4/5\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePagePatch5}
+\begin{paste}{ugTypesDeclarePageFull5}{ugTypesDeclarePageEmpty5}
+\pastebutton{ugTypesDeclarePageFull5}{\hidepaste}
+\tab{5}\spadcommand{n : Complex ?\bound{n }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePageEmpty5}
+\begin{paste}{ugTypesDeclarePageEmpty5}{ugTypesDeclarePagePatch5}
+\pastebutton{ugTypesDeclarePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{n : Complex ?\bound{n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePagePatch6}
+\begin{paste}{ugTypesDeclarePageFull6}{ugTypesDeclarePageEmpty6}
+\pastebutton{ugTypesDeclarePageFull6}{\hidepaste}
+\tab{5}\spadcommand{(p,q,r) : Matrix Polynomial ?\bound{p q r }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePageEmpty6}
+\begin{paste}{ugTypesDeclarePageEmpty6}{ugTypesDeclarePagePatch6}
+\pastebutton{ugTypesDeclarePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{(p,q,r) : Matrix Polynomial ?\bound{p q r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePagePatch7}
+\begin{paste}{ugTypesDeclarePageFull7}{ugTypesDeclarePageEmpty7}
+\pastebutton{ugTypesDeclarePageFull7}{\hidepaste}
+\tab{5}\spadcommand{n := -36 + 9 * \%i\free{n }}
+\indentrel{3}\begin{verbatim}
+   (6)  - 36 + 9%i
+                                  Type: Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePageEmpty7}
+\begin{paste}{ugTypesDeclarePageEmpty7}{ugTypesDeclarePagePatch7}
+\pastebutton{ugTypesDeclarePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{n := -36 + 9 * \%i\free{n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePagePatch8}
+\begin{paste}{ugTypesDeclarePageFull8}{ugTypesDeclarePageEmpty8}
+\pastebutton{ugTypesDeclarePageFull8}{\hidepaste}
+\tab{5}\spadcommand{n := complex(4/(x + y),y/x)\free{n }}
+\indentrel{3}\begin{verbatim}
+          4     y
+   (7)  ÄÄÄÄÄ + Ä %i
+        y + x   x
+              Type: Complex Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePageEmpty8}
+\begin{paste}{ugTypesDeclarePageEmpty8}{ugTypesDeclarePagePatch8}
+\pastebutton{ugTypesDeclarePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{n := complex(4/(x + y),y/x)\free{n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePagePatch9}
+\begin{paste}{ugTypesDeclarePageFull9}{ugTypesDeclarePageEmpty9}
+\pastebutton{ugTypesDeclarePageFull9}{\hidepaste}
+\tab{5}\spadcommand{p := [[1,2],[3,4],[5,6]]\free{p }}
+\indentrel{3}\begin{verbatim}
+        Ú1  2¿
+        ³    ³
+   (8)  ³3  4³
+        ³    ³
+        À5  6Ù
+                        Type: Matrix Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePageEmpty9}
+\begin{paste}{ugTypesDeclarePageEmpty9}{ugTypesDeclarePagePatch9}
+\pastebutton{ugTypesDeclarePageEmpty9}{\showpaste}
+\tab{5}\spadcommand{p := [[1,2],[3,4],[5,6]]\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePagePatch10}
+\begin{paste}{ugTypesDeclarePageFull10}{ugTypesDeclarePageEmpty10}
+\pastebutton{ugTypesDeclarePageFull10}{\hidepaste}
+\tab{5}\spadcommand{q := [[x - 2/3]]\free{q }}
+\indentrel{3}\begin{verbatim}
+        Ú    2¿
+   (9)  ³x - ij
+        À    3Ù
+               Type: Matrix Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePageEmpty10}
+\begin{paste}{ugTypesDeclarePageEmpty10}{ugTypesDeclarePagePatch10}
+\pastebutton{ugTypesDeclarePageEmpty10}{\showpaste}
+\tab{5}\spadcommand{q := [[x - 2/3]]\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePagePatch11}
+\begin{paste}{ugTypesDeclarePageFull11}{ugTypesDeclarePageEmpty11}
+\pastebutton{ugTypesDeclarePageFull11}{\hidepaste}
+\tab{5}\spadcommand{r := [[1-\%i*x,7*y+4*\%i]]\free{r }}
+\indentrel{3}\begin{verbatim}
+   (10)  [- %i x + 1  7y + 4%i]
+                Type: Matrix Polynomial Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePageEmpty11}
+\begin{paste}{ugTypesDeclarePageEmpty11}{ugTypesDeclarePagePatch11}
+\pastebutton{ugTypesDeclarePageEmpty11}{\showpaste}
+\tab{5}\spadcommand{r := [[1-\%i*x,7*y+4*\%i]]\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePagePatch12}
+\begin{paste}{ugTypesDeclarePageFull12}{ugTypesDeclarePageEmpty12}
+\pastebutton{ugTypesDeclarePageFull12}{\hidepaste}
+\tab{5}\spadcommand{f : COMPLEX POLY ? := (x + y*\%i)**2}
+\indentrel{3}\begin{verbatim}
+            2    2
+   (11)  - y  + x  + 2x y %i
+                       Type: Complex Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePageEmpty12}
+\begin{paste}{ugTypesDeclarePageEmpty12}{ugTypesDeclarePagePatch12}
+\pastebutton{ugTypesDeclarePageEmpty12}{\showpaste}
+\tab{5}\spadcommand{f : COMPLEX POLY ? := (x + y*\%i)**2}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePagePatch13}
+\begin{paste}{ugTypesDeclarePageFull13}{ugTypesDeclarePageEmpty13}
+\pastebutton{ugTypesDeclarePageFull13}{\hidepaste}
+\tab{5}\spadcommand{g : POLY COMPLEX ? := (x + y*\%i)**2}
+\indentrel{3}\begin{verbatim}
+            2              2
+   (12)  - y  + 2%i x y + x
+                       Type: Polynomial Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesDeclarePageEmpty13}
+\begin{paste}{ugTypesDeclarePageEmpty13}{ugTypesDeclarePagePatch13}
+\pastebutton{ugTypesDeclarePageEmpty13}{\showpaste}
+\tab{5}\spadcommand{g : POLY COMPLEX ? := (x + y*\%i)**2}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesAnyNonePagePatch1}
+\begin{paste}{ugTypesAnyNonePageFull1}{ugTypesAnyNonePageEmpty1}
+\pastebutton{ugTypesAnyNonePageFull1}{\hidepaste}
+\tab{5}\spadcommand{u: Any\bound{uany }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesAnyNonePageEmpty1}
+\begin{paste}{ugTypesAnyNonePageEmpty1}{ugTypesAnyNonePagePatch1}
+\pastebutton{ugTypesAnyNonePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{u: Any\bound{uany }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesAnyNonePagePatch2}
+\begin{paste}{ugTypesAnyNonePageFull2}{ugTypesAnyNonePageEmpty2}
+\pastebutton{ugTypesAnyNonePageFull2}{\hidepaste}
+\tab{5}\spadcommand{u := [1, 7.2, 3/2, x**2, "wally"]\free{uany }\bound{u }}
+\indentrel{3}\begin{verbatim}
+               3  2
+   (2)  [1,7.2,Ä,x ,"wally"]
+               2
+                                         Type: List Any
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesAnyNonePageEmpty2}
+\begin{paste}{ugTypesAnyNonePageEmpty2}{ugTypesAnyNonePagePatch2}
+\pastebutton{ugTypesAnyNonePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{u := [1, 7.2, 3/2, x**2, "wally"]\free{uany }\bound{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesAnyNonePagePatch3}
+\begin{paste}{ugTypesAnyNonePageFull3}{ugTypesAnyNonePageEmpty3}
+\pastebutton{ugTypesAnyNonePageFull3}{\hidepaste}
+\tab{5}\spadcommand{u.1\free{u }}
+\indentrel{3}\begin{verbatim}
+   (3)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesAnyNonePageEmpty3}
+\begin{paste}{ugTypesAnyNonePageEmpty3}{ugTypesAnyNonePagePatch3}
+\pastebutton{ugTypesAnyNonePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{u.1\free{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesAnyNonePagePatch4}
+\begin{paste}{ugTypesAnyNonePageFull4}{ugTypesAnyNonePageEmpty4}
+\pastebutton{ugTypesAnyNonePageFull4}{\hidepaste}
+\tab{5}\spadcommand{u.3\free{u }}
+\indentrel{3}\begin{verbatim}
+        3
+   (4)  Ä
+        2
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesAnyNonePageEmpty4}
+\begin{paste}{ugTypesAnyNonePageEmpty4}{ugTypesAnyNonePagePatch4}
+\pastebutton{ugTypesAnyNonePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{u.3\free{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesAnyNonePagePatch5}
+\begin{paste}{ugTypesAnyNonePageFull5}{ugTypesAnyNonePageEmpty5}
+\pastebutton{ugTypesAnyNonePageFull5}{\hidepaste}
+\tab{5}\spadcommand{v : Matrix(Any)}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesAnyNonePageEmpty5}
+\begin{paste}{ugTypesAnyNonePageEmpty5}{ugTypesAnyNonePagePatch5}
+\pastebutton{ugTypesAnyNonePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{v : Matrix(Any)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch1}
+\begin{paste}{ugTypesRecordsPageFull1}{ugTypesRecordsPageEmpty1}
+\pastebutton{ugTypesRecordsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{u := divide(5,2)\bound{u }}
+\indentrel{3}\begin{verbatim}
+   (1)  [quotient= 2,remainder= 1]
+     Type: Record(quotient: Integer,remainder: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty1}
+\begin{paste}{ugTypesRecordsPageEmpty1}{ugTypesRecordsPagePatch1}
+\pastebutton{ugTypesRecordsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{u := divide(5,2)\bound{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch2}
+\begin{paste}{ugTypesRecordsPageFull2}{ugTypesRecordsPageEmpty2}
+\pastebutton{ugTypesRecordsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{u.quotient\free{u }}
+\indentrel{3}\begin{verbatim}
+   (2)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty2}
+\begin{paste}{ugTypesRecordsPageEmpty2}{ugTypesRecordsPagePatch2}
+\pastebutton{ugTypesRecordsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{u.quotient\free{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch3}
+\begin{paste}{ugTypesRecordsPageFull3}{ugTypesRecordsPageEmpty3}
+\pastebutton{ugTypesRecordsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{u.remainder\free{u }}
+\indentrel{3}\begin{verbatim}
+   (3)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty3}
+\begin{paste}{ugTypesRecordsPageEmpty3}{ugTypesRecordsPagePatch3}
+\pastebutton{ugTypesRecordsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{u.remainder\free{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch4}
+\begin{paste}{ugTypesRecordsPageFull4}{ugTypesRecordsPageEmpty4}
+\pastebutton{ugTypesRecordsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{u.quotient := 8978\free{u }\bound{u1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  8978
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty4}
+\begin{paste}{ugTypesRecordsPageEmpty4}{ugTypesRecordsPagePatch4}
+\pastebutton{ugTypesRecordsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{u.quotient := 8978\free{u }\bound{u1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch5}
+\begin{paste}{ugTypesRecordsPageFull5}{ugTypesRecordsPageEmpty5}
+\pastebutton{ugTypesRecordsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{u\free{u }\free{u1 }}
+\indentrel{3}\begin{verbatim}
+   (5)  [quotient= 8978,remainder= 1]
+     Type: Record(quotient: Integer,remainder: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty5}
+\begin{paste}{ugTypesRecordsPageEmpty5}{ugTypesRecordsPagePatch5}
+\pastebutton{ugTypesRecordsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{u\free{u }\free{u1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch6}
+\begin{paste}{ugTypesRecordsPageFull6}{ugTypesRecordsPageEmpty6}
+\pastebutton{ugTypesRecordsPageFull6}{\hidepaste}
+\tab{5}\spadcommand{s := 'quotient\bound{s }}
+\indentrel{3}\begin{verbatim}
+   (6)  quotient
+                                Type: Variable quotient
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty6}
+\begin{paste}{ugTypesRecordsPageEmpty6}{ugTypesRecordsPagePatch6}
+\pastebutton{ugTypesRecordsPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{s := 'quotient\bound{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch7}
+\begin{paste}{ugTypesRecordsPageFull7}{ugTypesRecordsPageEmpty7}
+\pastebutton{ugTypesRecordsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{divide(5,2).s\free{s }}
+\indentrel{3}\begin{verbatim}
+   (7)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty7}
+\begin{paste}{ugTypesRecordsPageEmpty7}{ugTypesRecordsPagePatch7}
+\pastebutton{ugTypesRecordsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{divide(5,2).s\free{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch8}
+\begin{paste}{ugTypesRecordsPageFull8}{ugTypesRecordsPageEmpty8}
+\pastebutton{ugTypesRecordsPageFull8}{\hidepaste}
+\tab{5}\spadcommand{bd : Record(name : String, birthdayMonth : Integer)\bound{bddec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty8}
+\begin{paste}{ugTypesRecordsPageEmpty8}{ugTypesRecordsPagePatch8}
+\pastebutton{ugTypesRecordsPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{bd : Record(name : String, birthdayMonth : Integer)\bound{bddec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch9}
+\begin{paste}{ugTypesRecordsPageFull9}{ugTypesRecordsPageEmpty9}
+\pastebutton{ugTypesRecordsPageFull9}{\hidepaste}
+\tab{5}\spadcommand{bd := ["Judith", 3]\free{bddec }\bound{bd }}
+\indentrel{3}\begin{verbatim}
+   (9)  [name= "Judith",birthdayMonth= 3]
+      Type: Record(name: String,birthdayMonth: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty9}
+\begin{paste}{ugTypesRecordsPageEmpty9}{ugTypesRecordsPagePatch9}
+\pastebutton{ugTypesRecordsPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{bd := ["Judith", 3]\free{bddec }\bound{bd }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch10}
+\begin{paste}{ugTypesRecordsPageFull10}{ugTypesRecordsPageEmpty10}
+\pastebutton{ugTypesRecordsPageFull10}{\hidepaste}
+\tab{5}\spadcommand{bd.name := "Katie"\free{bd }}
+\indentrel{3}\begin{verbatim}
+   (10)  "Katie"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty10}
+\begin{paste}{ugTypesRecordsPageEmpty10}{ugTypesRecordsPagePatch10}
+\pastebutton{ugTypesRecordsPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{bd.name := "Katie"\free{bd }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch11}
+\begin{paste}{ugTypesRecordsPageFull11}{ugTypesRecordsPageEmpty11}
+\pastebutton{ugTypesRecordsPageFull11}{\hidepaste}
+\tab{5}\spadcommand{r : Record(a : Record(b: Integer, c: Integer), b: Integer)\bound{rdec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty11}
+\begin{paste}{ugTypesRecordsPageEmpty11}{ugTypesRecordsPagePatch11}
+\pastebutton{ugTypesRecordsPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{r : Record(a : Record(b: Integer, c: Integer), b: Integer)\bound{rdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch12}
+\begin{paste}{ugTypesRecordsPageFull12}{ugTypesRecordsPageEmpty12}
+\pastebutton{ugTypesRecordsPageFull12}{\hidepaste}
+\tab{5}\spadcommand{r := [[1,2],3]\bound{r }\free{rdec }}
+\indentrel{3}\begin{verbatim}
+   (12)  [a= [b= 1,c= 2],b= 3]
+Type: Record(a: Record(b: Integer,c: Integer),b: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty12}
+\begin{paste}{ugTypesRecordsPageEmpty12}{ugTypesRecordsPagePatch12}
+\pastebutton{ugTypesRecordsPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{r := [[1,2],3]\bound{r }\free{rdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch13}
+\begin{paste}{ugTypesRecordsPageFull13}{ugTypesRecordsPageEmpty13}
+\pastebutton{ugTypesRecordsPageFull13}{\hidepaste}
+\tab{5}\spadcommand{r.a.b\free{r }}
+\indentrel{3}\begin{verbatim}
+   (13)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty13}
+\begin{paste}{ugTypesRecordsPageEmpty13}{ugTypesRecordsPagePatch13}
+\pastebutton{ugTypesRecordsPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{r.a.b\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch14}
+\begin{paste}{ugTypesRecordsPageFull14}{ugTypesRecordsPageEmpty14}
+\pastebutton{ugTypesRecordsPageFull14}{\hidepaste}
+\tab{5}\spadcommand{r.b\free{r }}
+\indentrel{3}\begin{verbatim}
+   (14)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty14}
+\begin{paste}{ugTypesRecordsPageEmpty14}{ugTypesRecordsPagePatch14}
+\pastebutton{ugTypesRecordsPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{r.b\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch15}
+\begin{paste}{ugTypesRecordsPageFull15}{ugTypesRecordsPageEmpty15}
+\pastebutton{ugTypesRecordsPageFull15}{\hidepaste}
+\tab{5}\spadcommand{r(a)\free{r }}
+\indentrel{3}\begin{verbatim}
+   (15)  [b= 1,c= 2]
+                    Type: Record(b: Integer,c: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty15}
+\begin{paste}{ugTypesRecordsPageEmpty15}{ugTypesRecordsPagePatch15}
+\pastebutton{ugTypesRecordsPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{r(a)\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch16}
+\begin{paste}{ugTypesRecordsPageFull16}{ugTypesRecordsPageEmpty16}
+\pastebutton{ugTypesRecordsPageFull16}{\hidepaste}
+\tab{5}\spadcommand{r b\free{r }}
+\indentrel{3}\begin{verbatim}
+   (16)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty16}
+\begin{paste}{ugTypesRecordsPageEmpty16}{ugTypesRecordsPagePatch16}
+\pastebutton{ugTypesRecordsPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{r b\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch17}
+\begin{paste}{ugTypesRecordsPageFull17}{ugTypesRecordsPageEmpty17}
+\pastebutton{ugTypesRecordsPageFull17}{\hidepaste}
+\tab{5}\spadcommand{r(b) := 10\free{r }\bound{r1 }}
+\indentrel{3}\begin{verbatim}
+   (17)  10
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty17}
+\begin{paste}{ugTypesRecordsPageEmpty17}{ugTypesRecordsPagePatch17}
+\pastebutton{ugTypesRecordsPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{r(b) := 10\free{r }\bound{r1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPagePatch18}
+\begin{paste}{ugTypesRecordsPageFull18}{ugTypesRecordsPageEmpty18}
+\pastebutton{ugTypesRecordsPageFull18}{\hidepaste}
+\tab{5}\spadcommand{r\free{r1 }}
+\indentrel{3}\begin{verbatim}
+   (18)  [a= [b= 1,c= 2],b= 10]
+Type: Record(a: Record(b: Integer,c: Integer),b: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesRecordsPageEmpty18}
+\begin{paste}{ugTypesRecordsPageEmpty18}{ugTypesRecordsPagePatch18}
+\pastebutton{ugTypesRecordsPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{r\free{r1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesResolvePagePatch1}
+\begin{paste}{ugTypesResolvePageFull1}{ugTypesResolvePageEmpty1}
+\pastebutton{ugTypesResolvePageFull1}{\hidepaste}
+\tab{5}\spadcommand{x}
+\indentrel{3}\begin{verbatim}
+   (1)  x
+                                       Type: Variable x
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesResolvePageEmpty1}
+\begin{paste}{ugTypesResolvePageEmpty1}{ugTypesResolvePagePatch1}
+\pastebutton{ugTypesResolvePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{x}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesResolvePagePatch2}
+\begin{paste}{ugTypesResolvePageFull2}{ugTypesResolvePageEmpty2}
+\pastebutton{ugTypesResolvePageFull2}{\hidepaste}
+\tab{5}\spadcommand{1}
+\indentrel{3}\begin{verbatim}
+   (2)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesResolvePageEmpty2}
+\begin{paste}{ugTypesResolvePageEmpty2}{ugTypesResolvePagePatch2}
+\pastebutton{ugTypesResolvePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{1}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesResolvePagePatch3}
+\begin{paste}{ugTypesResolvePageFull3}{ugTypesResolvePageEmpty3}
+\pastebutton{ugTypesResolvePageFull3}{\hidepaste}
+\tab{5}\spadcommand{x + 1}
+\indentrel{3}\begin{verbatim}
+   (3)  x + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesResolvePageEmpty3}
+\begin{paste}{ugTypesResolvePageEmpty3}{ugTypesResolvePagePatch3}
+\pastebutton{ugTypesResolvePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{x + 1}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesResolvePagePatch4}
+\begin{paste}{ugTypesResolvePageFull4}{ugTypesResolvePageEmpty4}
+\pastebutton{ugTypesResolvePageFull4}{\hidepaste}
+\tab{5}\spadcommand{["string",3.14159]}
+\indentrel{3}\begin{verbatim}
+   (4)  ["string",3.14159]
+                                         Type: List Any
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesResolvePageEmpty4}
+\begin{paste}{ugTypesResolvePageEmpty4}{ugTypesResolvePagePatch4}
+\pastebutton{ugTypesResolvePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{["string",3.14159]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesResolvePagePatch5}
+\begin{paste}{ugTypesResolvePageFull5}{ugTypesResolvePageEmpty5}
+\pastebutton{ugTypesResolvePageFull5}{\hidepaste}
+\tab{5}\spadcommand{"string" + 3.14159}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesResolvePageEmpty5}
+\begin{paste}{ugTypesResolvePageEmpty5}{ugTypesResolvePagePatch5}
+\pastebutton{ugTypesResolvePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{"string" + 3.14159}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPagePatch1}
+\begin{paste}{ugTypesConvertPageFull1}{ugTypesConvertPageEmpty1}
+\pastebutton{ugTypesConvertPageFull1}{\hidepaste}
+\tab{5}\spadcommand{3}
+\indentrel{3}\begin{verbatim}
+   (1)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPageEmpty1}
+\begin{paste}{ugTypesConvertPageEmpty1}{ugTypesConvertPagePatch1}
+\pastebutton{ugTypesConvertPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{3}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPagePatch2}
+\begin{paste}{ugTypesConvertPageFull2}{ugTypesConvertPageEmpty2}
+\pastebutton{ugTypesConvertPageFull2}{\hidepaste}
+\tab{5}\spadcommand{3 :: Fraction Integer}
+\indentrel{3}\begin{verbatim}
+   (2)  3
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPageEmpty2}
+\begin{paste}{ugTypesConvertPageEmpty2}{ugTypesConvertPagePatch2}
+\pastebutton{ugTypesConvertPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{3 :: Fraction Integer}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPagePatch3}
+\begin{paste}{ugTypesConvertPageFull3}{ugTypesConvertPageEmpty3}
+\pastebutton{ugTypesConvertPageFull3}{\hidepaste}
+\tab{5}\spadcommand{m : SquareMatrix(2,POLY COMPLEX FRAC INT)\bound{mdec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPageEmpty3}
+\begin{paste}{ugTypesConvertPageEmpty3}{ugTypesConvertPagePatch3}
+\pastebutton{ugTypesConvertPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{m : SquareMatrix(2,POLY COMPLEX FRAC INT)\bound{mdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPagePatch4}
+\begin{paste}{ugTypesConvertPageFull4}{ugTypesConvertPageEmpty4}
+\pastebutton{ugTypesConvertPageFull4}{\hidepaste}
+\tab{5}\spadcommand{m := matrix [[x-3/4*\%i,z*y**2+1/2],[3/7*\%i*y**4 - x,12-\%i*9/5]]\bound{m }\free{mdec }}
+\indentrel{3}\begin{verbatim}
+        Ú     3         2    1 ¿
+        ³ x - Ä %i     y z + Ä ³
+        ³     4              2 ³
+   (4)  ³                      ³
+        ³3     4           9   ³
+        ³Ä %i y  - x  12 - Ä %i³
+        À7                 5   Ù
+Type: SquareMatrix(2,Polynomial Complex Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPageEmpty4}
+\begin{paste}{ugTypesConvertPageEmpty4}{ugTypesConvertPagePatch4}
+\pastebutton{ugTypesConvertPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{m := matrix [[x-3/4*\%i,z*y**2+1/2],[3/7*\%i*y**4 - x,12-\%i*9/5]]\bound{m }\free{mdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPagePatch5}
+\begin{paste}{ugTypesConvertPageFull5}{ugTypesConvertPageEmpty5}
+\pastebutton{ugTypesConvertPageFull5}{\hidepaste}
+\tab{5}\spadcommand{m1 := m :: SquareMatrix(2,POLY FRAC COMPLEX INT)\free{m }\bound{m1 }}
+\indentrel{3}\begin{verbatim}
+        Ú     3%i     2    1 ¿
+        ³ x - ÄÄÄ    y z + Ä ³
+        ³      4           2 ³
+   (5)  ³                    ³
+        ³3%i  4      60 - 9%i³
+        ³ÄÄÄ y  - x  ÄÄÄÄÄÄÄij
+        À 7              5   Ù
+Type: SquareMatrix(2,Polynomial Fraction Complex Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPageEmpty5}
+\begin{paste}{ugTypesConvertPageEmpty5}{ugTypesConvertPagePatch5}
+\pastebutton{ugTypesConvertPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{m1 := m :: SquareMatrix(2,POLY FRAC COMPLEX INT)\free{m }\bound{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPagePatch6}
+\begin{paste}{ugTypesConvertPageFull6}{ugTypesConvertPageEmpty6}
+\pastebutton{ugTypesConvertPageFull6}{\hidepaste}
+\tab{5}\spadcommand{m2 := m1 :: SquareMatrix(2,FRAC POLY COMPLEX INT)\free{m1 }\bound{m2 }}
+\indentrel{3}\begin{verbatim}
+        Ú               2     ¿
+        ³ 4x - 3%i    2y z + 1³
+        ³ ÄÄÄÄÄÄÄÄ    ÄÄÄÄÄÄÄij
+        ³     4           2   ³
+   (6)  ³                     ³
+        ³     4               ³
+        ³3%i y  - 7x  60 - 9%i³
+        ³ÄÄÄÄÄÄÄÄÄÄÄ  ÄÄÄÄÄÄÄij
+        À     7           5   Ù
+Type: SquareMatrix(2,Fraction Polynomial Complex Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPageEmpty6}
+\begin{paste}{ugTypesConvertPageEmpty6}{ugTypesConvertPagePatch6}
+\pastebutton{ugTypesConvertPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{m2 := m1 :: SquareMatrix(2,FRAC POLY COMPLEX INT)\free{m1 }\bound{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPagePatch7}
+\begin{paste}{ugTypesConvertPageFull7}{ugTypesConvertPageEmpty7}
+\pastebutton{ugTypesConvertPageFull7}{\hidepaste}
+\tab{5}\spadcommand{m3 := m2 :: SquareMatrix(2,FRAC COMPLEX POLY INT)\free{m2 }\bound{m3 }}
+\indentrel{3}\begin{verbatim}
+        Ú                2     ¿
+        ³  4x - 3%i    2y z + 1³
+        ³  ÄÄÄÄÄÄÄÄ    ÄÄÄÄÄÄÄij
+        ³      4           2   ³
+   (7)  ³                      ³
+        ³         4            ³
+        ³- 7x + 3y %i  60 - 9%i³
+        ³ÄÄÄÄÄÄÄÄÄÄÄÄ  ÄÄÄÄÄÄÄij
+        À      7           5   Ù
+Type: SquareMatrix(2,Fraction Complex Polynomial Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPageEmpty7}
+\begin{paste}{ugTypesConvertPageEmpty7}{ugTypesConvertPagePatch7}
+\pastebutton{ugTypesConvertPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{m3 := m2 :: SquareMatrix(2,FRAC COMPLEX POLY INT)\free{m2 }\bound{m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPagePatch8}
+\begin{paste}{ugTypesConvertPageFull8}{ugTypesConvertPageEmpty8}
+\pastebutton{ugTypesConvertPageFull8}{\hidepaste}
+\tab{5}\spadcommand{m :: SquareMatrix(2,FRAC COMPLEX POLY INT)\free{m }}
+\indentrel{3}\begin{verbatim}
+        Ú                2     ¿
+        ³  4x - 3%i    2y z + 1³
+        ³  ÄÄÄÄÄÄÄÄ    ÄÄÄÄÄÄÄij
+        ³      4           2   ³
+   (8)  ³                      ³
+        ³         4            ³
+        ³- 7x + 3y %i  60 - 9%i³
+        ³ÄÄÄÄÄÄÄÄÄÄÄÄ  ÄÄÄÄÄÄÄij
+        À      7           5   Ù
+Type: SquareMatrix(2,Fraction Complex Polynomial Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPageEmpty8}
+\begin{paste}{ugTypesConvertPageEmpty8}{ugTypesConvertPagePatch8}
+\pastebutton{ugTypesConvertPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{m :: SquareMatrix(2,FRAC COMPLEX POLY INT)\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPagePatch9}
+\begin{paste}{ugTypesConvertPageFull9}{ugTypesConvertPageEmpty9}
+\pastebutton{ugTypesConvertPageFull9}{\hidepaste}
+\tab{5}\spadcommand{m\free{m }}
+\indentrel{3}\begin{verbatim}
+        Ú     3         2    1 ¿
+        ³ x - Ä %i     y z + Ä ³
+        ³     4              2 ³
+   (9)  ³                      ³
+        ³3     4           9   ³
+        ³Ä %i y  - x  12 - Ä %i³
+        À7                 5   Ù
+Type: SquareMatrix(2,Polynomial Complex Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPageEmpty9}
+\begin{paste}{ugTypesConvertPageEmpty9}{ugTypesConvertPagePatch9}
+\pastebutton{ugTypesConvertPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{m\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPagePatch10}
+\begin{paste}{ugTypesConvertPageFull10}{ugTypesConvertPageEmpty10}
+\pastebutton{ugTypesConvertPageFull10}{\hidepaste}
+\tab{5}\spadcommand{m :: POLY SquareMatrix(2,COMPLEX FRAC INT)\free{m }}
+\indentrel{3}\begin{verbatim}
+   (10)
+                 Ú 0    0¿
+     Ú0  1¿ 2    ³       ³ 4   Ú 1   0¿
+     ³    ³y z + ³3      ³y  + ³      ³x
+     À0  0Ù      ³Ä %i  0³     À- 1  0Ù
+                 À7      Ù
+   + 
+     Ú  3         1    ¿
+     ³- Ä %i      Ä    ³
+     ³  4         2    ³
+     ³                 ³
+     ³             9   ³
+     ³  0     12 - Ä %i³
+     À             5   Ù
+Type: Polynomial SquareMatrix(2,Complex Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPageEmpty10}
+\begin{paste}{ugTypesConvertPageEmpty10}{ugTypesConvertPagePatch10}
+\pastebutton{ugTypesConvertPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{m :: POLY SquareMatrix(2,COMPLEX FRAC INT)\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPagePatch11}
+\begin{paste}{ugTypesConvertPageFull11}{ugTypesConvertPageEmpty11}
+\pastebutton{ugTypesConvertPageFull11}{\hidepaste}
+\tab{5}\spadcommand{m :: POLY ?\free{m }}
+\indentrel{3}\begin{verbatim}
+   (11)
+                 Ú 0    0¿
+     Ú0  1¿ 2    ³       ³ 4   Ú 1   0¿
+     ³    ³y z + ³3      ³y  + ³      ³x
+     À0  0Ù      ³Ä %i  0³     À- 1  0Ù
+                 À7      Ù
+   + 
+     Ú  3         1    ¿
+     ³- Ä %i      Ä    ³
+     ³  4         2    ³
+     ³                 ³
+     ³             9   ³
+     ³  0     12 - Ä %i³
+     À             5   Ù
+Type: Polynomial SquareMatrix(2,Complex Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPageEmpty11}
+\begin{paste}{ugTypesConvertPageEmpty11}{ugTypesConvertPagePatch11}
+\pastebutton{ugTypesConvertPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{m :: POLY ?\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPagePatch12}
+\begin{paste}{ugTypesConvertPageFull12}{ugTypesConvertPageEmpty12}
+\pastebutton{ugTypesConvertPageFull12}{\hidepaste}
+\tab{5}\spadcommand{m :: POLY SquareMatrix(2,FRAC ?)\free{m }}
+\indentrel{3}\begin{verbatim}
+   (12)
+                                        Ú  3%i     1    ¿
+               Ú 0   0¿                 ³- ÄÄÄ     Ä    ³
+   Ú0  1¿ 2    ³      ³ 4   Ú 1   0¿    ³   4      2    ³
+   ³    ³y z + ³3%i   ³y  + ³      ³x + ³               ³
+   À0  0Ù      ³ÄÄÄ  0³     À- 1  0Ù    ³       60 - 9%i³
+               À 7    Ù                 ³  0    ÄÄÄÄÄÄÄij
+                                        À           5   Ù
+Type: Polynomial SquareMatrix(2,Fraction Complex Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesConvertPageEmpty12}
+\begin{paste}{ugTypesConvertPageEmpty12}{ugTypesConvertPagePatch12}
+\pastebutton{ugTypesConvertPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{m :: POLY SquareMatrix(2,FRAC ?)\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingPagePatch1}
+\begin{paste}{ugTypesWritingPageFull1}{ugTypesWritingPageEmpty1}
+\pastebutton{ugTypesWritingPageFull1}{\hidepaste}
+\tab{5}\spadcommand{a : PositiveInteger}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingPageEmpty1}
+\begin{paste}{ugTypesWritingPageEmpty1}{ugTypesWritingPagePatch1}
+\pastebutton{ugTypesWritingPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{a : PositiveInteger}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingPagePatch2}
+\begin{paste}{ugTypesWritingPageFull2}{ugTypesWritingPageEmpty2}
+\pastebutton{ugTypesWritingPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f : Integer -> String}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingPageEmpty2}
+\begin{paste}{ugTypesWritingPageEmpty2}{ugTypesWritingPagePatch2}
+\pastebutton{ugTypesWritingPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f : Integer -> String}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingPagePatch3}
+\begin{paste}{ugTypesWritingPageFull3}{ugTypesWritingPageEmpty3}
+\pastebutton{ugTypesWritingPageFull3}{\hidepaste}
+\tab{5}\spadcommand{factor(2 :: Complex(Integer))}
+\indentrel{3}\begin{verbatim}
+                     2
+   (3)  - %i (1 + %i)
+                         Type: Factored Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingPageEmpty3}
+\begin{paste}{ugTypesWritingPageEmpty3}{ugTypesWritingPagePatch3}
+\pastebutton{ugTypesWritingPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{factor(2 :: Complex(Integer))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingPagePatch4}
+\begin{paste}{ugTypesWritingPageFull4}{ugTypesWritingPageEmpty4}
+\pastebutton{ugTypesWritingPageFull4}{\hidepaste}
+\tab{5}\spadcommand{(2 = 3)$Integer}
+\indentrel{3}\begin{verbatim}
+   (4)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingPageEmpty4}
+\begin{paste}{ugTypesWritingPageEmpty4}{ugTypesWritingPagePatch4}
+\pastebutton{ugTypesWritingPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{(2 = 3)$Integer}
+\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingPagePatch5}
+\begin{paste}{ugTypesWritingPageFull5}{ugTypesWritingPageEmpty5}
+\pastebutton{ugTypesWritingPageFull5}{\hidepaste}
+\tab{5}\spadcommand{(2 = 3)@Boolean}
+\indentrel{3}\begin{verbatim}
+   (5)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugTypesWritingPageEmpty5}
+\begin{paste}{ugTypesWritingPageEmpty5}{ugTypesWritingPagePatch5}
+\pastebutton{ugTypesWritingPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{(2 = 3)@Boolean}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ug03.ht b/src/hyper/pages/ug03.ht
new file mode 100644
index 00000000..542041e6
--- /dev/null
+++ b/src/hyper/pages/ug03.ht
@@ -0,0 +1,517 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\texht{\setcounter{chapter}{2}}{} % Chapter 3
+
+%
+\newcommand{\ugHyperTitle}{Using \HyperName{}}
+\newcommand{\ugHyperNumber}{3.}
+%
+% =====================================================================
+\begin{page}{ugHyperPage}{3. Using \HyperName{}}
+% =====================================================================
+\beginscroll
+
+
+\HyperName{} is the gateway to \Language{}.
+%-% \HDindex{HyperDoc @{\protect\HyperName{}}}{ugHyperPage}{3.}{Using \HyperName{}}
+It's both an on-line tutorial and an on-line reference manual.
+It also enables you to use \Language{} simply by using the mouse and
+filling in templates.
+\HyperName{} is available to you if you are running \Language{} under the
+X Window System.
+
+Pages usually have active areas, marked in
+\texht{{\bf this font} (bold face).}{\downlink{this font.}{YouTriedIt}}
+As you move the mouse pointer to an active area, the pointer changes from
+a filled dot to an open circle.
+The active areas are usually linked to other pages.
+When you click on an active area, you move to the linked page.
+\texht{}{Try clicking \downlink{here}{YouTriedIt} now.}
+
+We suggest that you learn more about other features of
+\HyperName{} by clicking on an active area in the menu below.
+
+\beginmenu
+    \menudownlink{{3.1. Headings}}{ugHyperHeadingsPage}
+    \menudownlink{{3.2. Key Definitions}}{ugHyperKeysPage}
+    \menudownlink{{3.3. Scroll Bars}}{ugHyperScrollPage}
+    \menudownlink{{3.4. Input Areas}}{ugHyperInputPage}
+    \menudownlink{{3.5. Radio Buttons and Toggles}}{ugHyperButtonsPage}
+    \menudownlink{{3.6. Search Strings}}{ugHyperSearchPage}
+    \menudownlink{{3.7. Example Pages}}{ugHyperExamplePage}
+    \menudownlink{{3.8. X Window Resources for \HyperName{}}}{ugHyperResourcesPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugHyperHeadingsTitle}{Headings}
+\newcommand{\ugHyperHeadingsNumber}{3.1.}
+%
+% =====================================================================
+\begin{page}{ugHyperHeadingsPage}{3.1. Headings}
+% =====================================================================
+\beginscroll
+%
+Most pages have a standard set of buttons at the top of the page.
+This is what they mean:
+
+\indent{0}
+\beginitems
+\item[\StdHelpButton{}] Click on this to get help.
+The button only appears if there is specific help for the page you are
+viewing.
+You can get {\it general} help for \HyperName{} by clicking the help
+button on the home page.
+
+\item[\UpButton{}] Click here to go back one page.
+By clicking on this button repeatedly, you can go back several pages and
+then take off in a new direction.
+
+\item[\ReturnButton{}] Go back to the home page, that is,
+the page on which you started.
+Use \HyperName{} to explore, to make forays into new topics.
+Don't worry about how to get back.
+\HyperName{} remembers where you came from.
+Just click on this button to return.
+
+\item[\StdExitButton{}] From the root window (the one that is displayed when
+you start the system) this button leaves the \HyperName{} program, and it
+must be restarted if you want to use it again.
+From any other \HyperName{} window, it just makes that one window go away.
+You {\it must} use this button to get rid of a window.
+If you use the window manager ``Close'' button, then all of \HyperName{}
+goes away.
+\enditems
+\indent{0}
+%
+The buttons are not displayed if they are not applicable to the page
+you are viewing.
+For example, there is no \ReturnButton{} button on the top-level menu.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugHyperKeysTitle}{Key Definitions}
+\newcommand{\ugHyperKeysNumber}{3.2.}
+%
+% =====================================================================
+\begin{page}{ugHyperKeysPage}{3.2. Key Definitions}
+% =====================================================================
+\beginscroll
+
+The following keyboard definitions are in effect throughout
+\HyperName{}.
+See \downlink{``\ugHyperScrollTitle''}{ugHyperScrollPage} in Section \ugHyperScrollNumber\ignore{ugHyperScroll} and
+\downlink{``\ugHyperInputTitle''}{ugHyperInputPage} in Section \ugHyperInputNumber\ignore{ugHyperInput}
+for some contextual key definitions.
+%
+\indent{0}
+\beginitems
+\item[F1] Display the main help page.
+\item[F3] Same as \StdExitButton{}, makes the window go away if you are not at the top-level window or quits the \HyperName{} facility if you are at the top-level.
+\item[F5] Rereads the \HyperName{} database, if necessary (for system developers).
+\item[F9] Displays this information about key definitions.
+\item[F12] Same as {\bf F3}.
+\item[Up Arrow] Scroll up one line.
+\item[Down Arrow] Scroll down one line.
+\item[Page Up] Scroll up one page.
+\item[Page Down] Scroll down one page.
+\enditems
+\indent{0}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugHyperScrollTitle}{Scroll Bars}
+\newcommand{\ugHyperScrollNumber}{3.3.}
+%
+% =====================================================================
+\begin{page}{ugHyperScrollPage}{3.3. Scroll Bars}
+% =====================================================================
+\beginscroll
+%
+
+Whenever there is too much text to fit on a page, a {\it scroll
+%-% \HDindex{scroll bar}{ugHyperScrollPage}{3.3.}{Scroll Bars}
+bar} automatically appears along the right side.
+
+With a scroll bar, your page becomes an aperture, that is, a
+window into a larger amount of text than can be displayed at one
+time.
+The scroll bar lets you move up and down in the text to see
+different parts.
+It also shows where the aperture is relative to the whole text.
+The aperture is indicated by a strip on the scroll bar.
+
+Move the cursor with the mouse to the ``down-arrow'' at the bottom
+of the scroll bar and click.
+See that the aperture moves down one line.
+Do it several times.
+Each time you click, the aperture moves down one line.
+Move the mouse to the ``up-arrow'' at the top of the scroll bar and
+click.
+The aperture moves up one line each time you click.
+
+Next move the mouse to any position along the middle of the scroll bar and
+click.
+\HyperName{} attempts to move the top of the aperture to this point in
+the text.
+
+You cannot make the aperture go off the bottom edge.
+When the aperture is about half the size of text, the lowest you can move
+the aperture is halfway down.
+
+To move up or down one screen at a time, use the
+\texht{\fbox{\bf PageUp}}{{\bf PageUp}} and \texht{\fbox{\bf PageDown}}{{\bf
+PageDown}} keys on your keyboard.
+They move the visible part of the region up and down one page each time you
+press them.
+
+If the \HyperName{} page does not contain an input area
+(see \downlink{``\ugHyperInputTitle''}{ugHyperInputPage} in Section \ugHyperInputNumber\ignore{ugHyperInput}), you can also use the
+\texht{\fbox{\bf Home}}{{\bf Home}} and
+\texht{\fbox{$\uparrow$}}{up} and
+\texht{\fbox{$\downarrow$}}{down} arrow keys to navigate.
+When you press the \texht{\fbox{\bf Home}}{{\bf Home}} key,
+the screen is positioned at the very top of the page.
+Use the \texht{\fbox{$\uparrow$}}{up} and
+\texht{\fbox{$\downarrow$}}{down} arrow keys to move the screen up
+and down one line at a time, respectively.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugHyperInputTitle}{Input Areas}
+\newcommand{\ugHyperInputNumber}{3.4.}
+%
+% =====================================================================
+\begin{page}{ugHyperInputPage}{3.4. Input Areas}
+% =====================================================================
+\beginscroll
+%
+Input areas are boxes where you can put data.
+Here is one:
+\centerline{\inputstring{one}{40}{some text}}
+\newline As you can see, the input area has some initial text {\it some text}
+followed by an underscore cursor (the character {\it _}).
+
+To enter characters, first
+move your mouse cursor to somewhere within the \HyperName{} page.
+Characters that you type are inserted in front of the underscore.
+This means that when you type characters at your keyboard, they
+go into this first input area.
+
+The input area grows to accommodate as many characters as you type.
+Use the \texht{\fbox{\bf Backspace}}{{\bf Backspace}}
+key to erase characters to the left.
+To modify what you type, use the right-arrow \texht{\fbox{$\rightarrow$}}{}
+and left-arrow keys \texht{\fbox{$\leftarrow$}}{} and the
+keys \texht{\fbox{\bf Insert}}{{\bf Insert}},
+\texht{\fbox{\bf Delete}}{{\bf Delete}},
+\texht{\fbox{\bf Home}}{{\bf Home}} and
+\texht{\fbox{\bf End}}{{\bf End}}.
+These keys are found immediately on the right of the standard IBM keyboard.
+
+If you press the
+\texht{\fbox{\bf Home}}{{\bf Home}}
+key, the cursor moves to the beginning of the line and if you press the
+\texht{\fbox{\bf End}}{{\bf End}}
+key, the cursor moves to the end of the line.
+Pressing
+\texht{\fbox{\bf Ctrl}--\fbox{\bf End}}{{\bf Ctrl-End}}
+deletes all the text from the cursor to the end of the line.
+
+A page may have more than one input area.
+Only one input area has an underscore cursor.
+When you first see apage, the top-most input area contains the
+cursor.
+To type information into another input area,
+use the \texht{\fbox{\bf Enter}}{{\bf Enter}} or
+\texht{\fbox{\bf Tab}}{{\bf Tab}} key to move
+from one input area to another.
+To move in the reverse order, use
+\texht{\fbox{\bf Shift}--\fbox{\bf Tab}}{{\bf Shift-Tab}}.
+
+You can also move from one input area to another using your mouse.
+Notice that each input area is active. Click on one of the areas.
+As you can see, the underscore cursor moves to that window.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugHyperButtonsTitle}{Radio Buttons and Toggles}
+\newcommand{\ugHyperButtonsNumber}{3.5.}
+%
+% =====================================================================
+\begin{page}{ugHyperButtonsPage}{3.5. Radio Buttons and Toggles}
+% =====================================================================
+\beginscroll
+%
+Some pages have {\it radio buttons} and {\it toggles}.
+Radio buttons are a group of buttons like those on car radios: you can
+select only one at a time.
+\radioboxes{sample}{\htbmfile{pick}}{\htbmfile{unpick}}
+Here are three radio buttons:
+\centerline{
+{\em\radiobox[1]{rone}{sample}\space{}\ First one}\space{3}
+{\em\radiobox[0]{rtwo}{sample}\space{}\ Second one}\space{3}
+{\em\radiobox[0]{rthree}{sample}\space{}\ Third one}
+}
+\newline
+Once you have selected a button, it appears to be inverted and
+contains a checkmark.
+To change the selection, move the cursor with the mouse to a different radio
+button and click.
+\texht{}{Try it now.}
+
+A toggle is an independent button that displays some on/off
+state.
+When ``on'', the button appears to be inverted and
+contains a checkmark.
+When ``off'', the button is raised.
+%
+Unlike radio buttons, you can set a group of them any way you like.
+Here are three:
+\centerline{
+{\em\inputbox[1]{one}{\htbmfile{pick}}{\htbmfile{unpick}}\space{}\ First one}
+\space{3}
+{\em\inputbox[0]{two}{\htbmfile{pick}}{\htbmfile{unpick}}\space{}\ Second one}
+\space{3}
+{\em\inputbox[1]{three}{\htbmfile{pick}}{\htbmfile{unpick}}\space{}\ Third one}
+}
+\newline
+To change toggle the selection, move the cursor with the mouse
+to the button and click.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugHyperSearchTitle}{Search Strings}
+\newcommand{\ugHyperSearchNumber}{3.6.}
+%
+% =====================================================================
+\begin{page}{ugHyperSearchPage}{3.6. Search Strings}
+% =====================================================================
+\beginscroll
+%
+A {\it search string} is used for searching some database.
+To learn about search strings, we suggest that
+you bring up the \HyperName{} glossary.
+To do this from the top-level page of \HyperName{}:
+\indent{4}
+\beginitems
+\item[1. ] Click on \windowlink{Reference}{TopReferencePage},
+bringing up the \Language{} Reference page.
+\item[2. ] Click on \windowlink{Glossary}{GlossaryPage}, bringing up the glossary.
+\texht{}{(You can also just click on the word ``Glossary'' in the
+last sentence.)}
+\enditems
+\indent{0}
+Once you get the window containing the glossary, move it so that
+it and this window are both visible.
+
+The glossary has an input area at its bottom.
+We review the various kinds of search strings
+you can enter to search the glossary.
+
+The simplest search string is a word, for example, {\tt operation}.
+A word only matches an entry having exactly that spelling.
+Enter the word {\tt operation} into the input area above then click on
+{\bf Search}.
+As you can see, {\tt operation} matches only one entry, namely with {\tt
+operation} itself.
+
+Normally matching is insensitive to whether the alphabetic characters of your
+search string are in uppercase or lowercase.
+Thus {\tt operation} and {\tt OperAtion} both have the same effect.
+%If you prefer that matching be case-sensitive, issue the command
+%\spadsys{set HHyperName mixedCase} command to the interpreter.
+
+You will very often want to use the wildcard \spadSyntax{*} in your search
+string so as to match multiple entries in the list.
+The search key \spadSyntax{*}  matches every entry in the list.
+You can also use \spadSyntax{*} anywhere within a search string to match an
+arbitrary substring.
+Try {\tt cat*} for example:
+enter {\tt cat*} into the input area and click on {\bf Search}.
+This matches several entries.
+
+You use any number of wildcards in a search string as long as they are
+not adjacent.
+Try search strings such as {\tt *dom*}.
+As you see, this search string  matches {\tt domain}, {\tt domain
+constructor}, {\tt subdomain}, and so on.
+
+\beginmenu
+    \menudownlink{{3.6.1. Logical Searches}}{ugLogicalSearchesPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLogicalSearchesTitle}{Logical Searches}
+\newcommand{\ugLogicalSearchesNumber}{3.6.1.}
+%
+% =====================================================================
+\begin{page}{ugLogicalSearchesPage}{3.6.1. Logical Searches}
+% =====================================================================
+\beginscroll
+
+For more complicated searches, you can use
+\spadSyntax{and}, \spadSyntax{or}, and \spadSyntax{not}
+with basic search strings;
+write logical expressions using these three operators just as
+in the \Language{} language.
+For example, {\tt domain or package}  matches the two
+entries {\tt domain} and {\tt package}.
+Similarly, {\tt dom* and *con*} matches {\tt domain constructor}
+and others.
+Also {\tt not *a*} matches every entry that does not contain
+the letter {\tt a} somewhere.
+
+Use parentheses for grouping.
+For example, {\tt dom* and (not *con*)}
+matches {\tt domain} but not {\tt domain constructor}.
+
+There is no limit to how complex your logical expression can be.
+For example,
+\centerline{{{\tt a* or b* or c* or d* or e* and (not *a*)}}}
+is a valid expression.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugHyperExampleTitle}{Example Pages}
+\newcommand{\ugHyperExampleNumber}{3.7.}
+%
+% =====================================================================
+\begin{page}{ugHyperExamplePage}{3.7. Example Pages}
+% =====================================================================
+\beginscroll
+%
+Many pages have \Language{} example commands.
+%
+Here are two:
+\spadpaste{a:= x**2 + 1 \bound{a}} 
+\spadpaste{(a - 2)**2 \free{a}} 
+%
+Each command has an active ``button'' along the left margin.
+When you click on this button, the output for the command is
+``pasted-in.''
+Click again on the button and you see that the pasted-in output
+disappears.
+
+Maybe you would like to run an example?
+To do so, just click on any part of its text!
+When you do, the example line is copied into a new interactive
+\Language{} buffer for this \HyperName{} page.
+
+Sometimes one example line cannot be run before you run an earlier one.
+Don't worry---\HyperName{} automatically runs all the necessary
+lines in the right order!
+For instance, the second example line above refers to \spad{a} which is
+assigned in the first example line.
+What happens if you first click on the second example line?
+\Language{} first issues the first line (to assign \spad{a}), then the
+second (to do the computation using \spad{a}).
+
+The new interactive \Language{} buffer disappears when you leave
+\HyperName{}.
+If you want to get rid of it beforehand,
+use the {\bf Cancel} button of the X Window manager
+or issue the \Language{} system command \spadsys{)close.}
+%-% \HDsyscmdindex{close}{ugHyperExamplePage}{3.7.}{Example Pages}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugHyperResourcesTitle}{X Window Resources for \HyperName{}}
+\newcommand{\ugHyperResourcesNumber}{3.8.}
+%
+% =====================================================================
+\begin{page}{ugHyperResourcesPage}{3.8. X Window Resources for \HyperName{}}
+% =====================================================================
+\beginscroll
+%
+You can control the appearance of \HyperName{} while running under Version 11
+%-% \HDindex{HyperDoc @{\protect\HyperName{}}!X Window System defaults}{ugHyperResourcesPage}{3.8.}{X Window Resources for \HyperName{}}
+of the X Window System by placing the following resources
+%-% \HDindex{X Window System}{ugHyperResourcesPage}{3.8.}{X Window Resources for \HyperName{}}
+in the file {\bf .Xdefaults} in your home directory.
+%-% \HDindex{file!.Xdefaults @{\bf .Xdefaults}}{ugHyperResourcesPage}{3.8.}{X Window Resources for \HyperName{}}
+In what follows, {\it font} is any valid X11 font name
+%-% \HDindex{font}{ugHyperResourcesPage}{3.8.}{X Window Resources for \HyperName{}}
+(for example, {\tt Rom14}) and {\it color} is any valid X11 color
+%-% \HDindex{color}{ugHyperResourcesPage}{3.8.}{X Window Resources for \HyperName{}}
+specification (for example, {\tt NavyBlue}).
+For more information about fonts and colors, refer to the
+X Window documentation for your system.
+\indent{0}
+\beginitems
+\item[{\tt Axiom.hyperdoc.RmFont:} {\it font}] \ \newline
+This is the standard text font.  \xdefault{Rom14}
+\item[{\tt Axiom.hyperdoc.RmColor:} {\it color}] \ \newline
+This is the standard text color.  \xdefault{black}
+\item[{\tt Axiom.hyperdoc.ActiveFont:} {\it font}] \ \newline
+This is the font used for \HyperName{} link buttons.  \xdefault{Bld14}
+\item[{\tt Axiom.hyperdoc.ActiveColor:} {\it color}] \ \newline
+This is the color used for \HyperName{} link buttons.  \xdefault{black}
+\item[{\tt Axiom.hyperdoc.AxiomFont:} {\it font}] \ \newline
+This is the font used for active \Language{} commands.\footnote{
+This was called {\tt Axiom.hyperdoc.SpadFont} in early versions
+of \Language{}.}
+\xdefault{Bld14}
+\item[{\tt Axiom.hyperdoc.AxiomColor:} {\it color}] \ \newline
+This is the color used for active \Language{} commands.\footnote{
+This was called {\tt Axiom.hyperdoc.SpadColor} in early versions
+of \Language{}.}
+\xdefault{black}
+\item[{\tt Axiom.hyperdoc.BoldFont:} {\it font}] \ \newline
+This is the font used for bold face.  \xdefault{Bld14}
+\item[{\tt Axiom.hyperdoc.BoldColor:} {\it color}] \ \newline
+This is the color used for bold face.  \xdefault{black}
+\item[{\tt Axiom.hyperdoc.TtFont:} {\it font}] \ \newline
+This is the font used for \Language{} output in \HyperName{}.
+This font must be fixed-width.  \xdefault{Rom14}
+\item[{\tt Axiom.hyperdoc.TtColor:} {\it color}] \ \newline
+This is the color used for \Language{} output in \HyperName{}.
+\xdefault{black}
+\item[{\tt Axiom.hyperdoc.EmphasizeFont:} {\it font}] \ \newline
+This is the font used for italics.  \xdefault{Itl14}
+\item[{\tt Axiom.hyperdoc.EmphasizeColor:} {\it color}] \ \newline
+This is the color used for italics.  \xdefault{black}
+\item[{\tt Axiom.hyperdoc.InputBackground:} {\it color}] \ \newline
+This is the color used as the background for input areas.
+\xdefault{black}
+\item[{\tt Axiom.hyperdoc.InputForeground:} {\it color}] \ \newline
+This is the color used as the foreground for input areas.
+\xdefault{white}
+\item[{\tt Axiom.hyperdoc.BorderColor:} {\it color}] \ \newline
+This is the color used for drawing border lines.
+\xdefault{black}
+\item[{\tt Axiom.hyperdoc.Background:} {\it color}] \ \newline
+This is the color used for the background of all windows.
+\xdefault{white}
+\enditems
+\indent{0}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug03.pht b/src/hyper/pages/ug03.pht
new file mode 100644
index 00000000..5c0fdc7f
--- /dev/null
+++ b/src/hyper/pages/ug03.pht
@@ -0,0 +1,34 @@
+\begin{patch}{ugHyperExamplePagePatch1}
+\begin{paste}{ugHyperExamplePageFull1}{ugHyperExamplePageEmpty1}
+\pastebutton{ugHyperExamplePageFull1}{\hidepaste}
+\tab{5}\spadcommand{a:= x**2 + 1\bound{a }}
+\indentrel{3}\begin{verbatim}
+         2
+   (1)  x  + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugHyperExamplePageEmpty1}
+\begin{paste}{ugHyperExamplePageEmpty1}{ugHyperExamplePagePatch1}
+\pastebutton{ugHyperExamplePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{a:= x**2 + 1\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugHyperExamplePagePatch2}
+\begin{paste}{ugHyperExamplePageFull2}{ugHyperExamplePageEmpty2}
+\pastebutton{ugHyperExamplePageFull2}{\hidepaste}
+\tab{5}\spadcommand{(a - 2)**2\free{a }}
+\indentrel{3}\begin{verbatim}
+         4     2
+   (2)  x  - 2x  + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugHyperExamplePageEmpty2}
+\begin{paste}{ugHyperExamplePageEmpty2}{ugHyperExamplePagePatch2}
+\pastebutton{ugHyperExamplePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{(a - 2)**2\free{a }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ug04.ht b/src/hyper/pages/ug04.ht
new file mode 100644
index 00000000..d0b59bfa
--- /dev/null
+++ b/src/hyper/pages/ug04.ht
@@ -0,0 +1,637 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\texht{\setcounter{chapter}{3}}{} % Chapter 4
+
+%
+\newcommand{\ugInOutTitle}{Input Files and Output Styles}
+\newcommand{\ugInOutNumber}{4.}
+%
+% =====================================================================
+\begin{page}{ugInOutPage}{4. Input Files and Output Styles}
+% =====================================================================
+\beginscroll
+
+In this chapter we discuss how to collect \Language{} statements
+and commands into files and then read the contents into the
+workspace.
+We also show how to display the results of your computations in
+several different styles including \texht{\TeX}{TeX}, FORTRAN and
+monospace two-dimensional format.\footnote{\texht{\TeX}{TeX} is a
+trademark of the American Mathematical Society.}
+
+The printed version of this book uses the \Language{}
+\texht{\TeX}{TeX} output formatter.
+When we demonstrate a particular output style, we will need to
+turn \texht{\TeX}{TeX} formatting off and the output style on so
+that the correct output is shown in the text.
+
+\beginmenu
+    \menudownlink{{4.1. Input Files}}{ugInOutInPage}
+    \menudownlink{{4.2. The axiom.input File}}{ugInOutSpadprofPage}
+    \menudownlink{{4.3. Common Features of Using Output Formats}}{ugInOutOutPage}
+    \menudownlink{{4.4. Monospace Two-Dimensional Mathematical Format}}{ugInOutAlgebraPage}
+    \menudownlink{{4.5. TeX Format}}{ugInOutTeXPage}
+    \menudownlink{{4.6. IBM Script Formula Format}}{ugInOutScriptPage}
+    \menudownlink{{4.7. FORTRAN Format}}{ugInOutFortranPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugInOutInTitle}{Input Files}
+\newcommand{\ugInOutInNumber}{4.1.}
+%
+% =====================================================================
+\begin{page}{ugInOutInPage}{4.1. Input Files}
+% =====================================================================
+\beginscroll
+%
+In this section we explain what an {\it input file} is and
+%-% \HDindex{file!input}{ugInOutInPage}{4.1.}{Input Files}
+why you would want to know about it.
+We discuss where \Language{} looks for input files and how you can
+direct it to look elsewhere.
+We also show how to read the contents of an input file into the
+\spadgloss{workspace} and how to use the \spadgloss{history}
+facility to generate an input file from the statements you have
+entered directly into the workspace.
+
+An {\it input} file contains \Language{} expressions and system
+commands.
+Anything that you can enter directly to \Language{} can be put
+into an input file.
+This is how you save input functions and expressions that you wish
+to read into \Language{} more than one time.
+
+To read an input file into \Language{}, use the \spadcmd{)read}
+system command.
+%-% \HDsyscmdindex{read}{ugInOutInPage}{4.1.}{Input Files}
+For example, you can read a file in a particular directory by issuing
+\begin{verbatim}
+)read /spad/src/input/matrix.input
+\end{verbatim}
+The ``{\bf .input}'' is optional; this also works:
+\begin{verbatim}
+)read /spad/src/input/matrix
+\end{verbatim}
+What happens if you just enter
+\spadcmd{)read matrix.input} or even \spadcmd{)read matrix}?
+\Language{} looks in your current working directory for input files
+that are not qualified by a directory name.
+Typically, this directory is the directory from which you invoked
+\Language{}.
+To change the current working directory, use the \spadcmd{)cd} system command.
+The command \spadsys{)cd} by itself shows the current
+working
+%-% \HDindex{directory!default for searching}{ugInOutInPage}{4.1.}{Input Files}
+directory.
+%-% \HDsyscmdindex{cd}{ugInOutInPage}{4.1.}{Input Files}
+To change it to
+%-% \HDindex{file!input!where found}{ugInOutInPage}{4.1.}{Input Files}
+the \spadsys{src/input} subdirectory for user ``babar'',
+issue
+\begin{verbatim}
+)cd /u/babar/src/input
+\end{verbatim}
+\Language{} looks first in this directory for an input file.
+If it is not found, it looks in the system's directories, assuming
+you meant some input file that was provided with \Language{}.
+
+\beginImportant
+If you have the \Language{} history facility turned on (which it is
+by default), you can save all the lines you have entered into the
+workspace by entering
+\begin{verbatim}
+)history )write
+\end{verbatim}
+%-% \HDsyscmdindex{history )write}{ugInOutInPage}{4.1.}{Input Files}
+
+\Language{} tells you what input file to edit to see your
+statements.
+The file is in your home directory or in the directory you
+specified with \spadsys{)cd}.
+%-% \HDsyscmdindex{cd}{ugInOutInPage}{4.1.}{Input Files}
+\endImportant
+
+In \downlink{``\ugLangBlocksTitle''}{ugLangBlocksPage} in Section \ugLangBlocksNumber\ignore{ugLangBlocks}
+we discuss using indentation in input files to group statements
+into {\it blocks.}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugInOutSpadprofTitle}{The axiom.input File}
+\newcommand{\ugInOutSpadprofNumber}{4.2.}
+%
+% =====================================================================
+\begin{page}{ugInOutSpadprofPage}{4.2. The axiom.input File}
+% =====================================================================
+\beginscroll
+
+When \Language{} starts up, it tries to read the input file
+{\bf axiom.input} from your home
+%-% \HDindex{start-up profile file}{ugInOutSpadprofPage}{4.2.}{The axiom.input File}
+directory.
+%-% \HDindex{file!start-up profile}{ugInOutSpadprofPage}{4.2.}{The axiom.input File}
+It there is no {\bf axiom.input} in your home directory, it reads the copy
+located in its own {\bf src/input} directory.
+%-% \HDindex{file!axiom.input @{\bf axiom.input}}{ugInOutSpadprofPage}{4.2.}{The axiom.input File}
+The file usually contains
+system commands to personalize your \Language{} environment.
+In the remainder of this section we mention a few things
+that users frequently place in their
+{\bf axiom.input} files.
+
+In order to have FORTRAN output always produced from your
+computations, place the system command
+\spadcmd{)set output fortran on}
+in {\bf axiom.input}.
+%-% \HDsyscmdindex{quit}{ugInOutSpadprofPage}{4.2.}{The axiom.input File}
+If you do not want to be prompted for confirmation when you issue
+the \spadcmd{)quit} system command, place
+\spadcmd{)set quit unprotected}
+in {\bf axiom.input}.
+%-% \HDsyscmdindex{set quit unprotected}{ugInOutSpadprofPage}{4.2.}{The axiom.input File}
+If you then decide that you do want to be prompted, issue
+\spadcmd{)set quit protected}.
+%-% \HDsyscmdindex{set quit protected}{ugInOutSpadprofPage}{4.2.}{The axiom.input File}
+This is the default setting
+so that new users do not leave \Language{}
+inadvertently.\footnote{The
+system command \spadsys{)pquit} always prompts you for
+confirmation.}
+
+To see the other system variables you can set, issue \spadsys{)set}
+or use the \HyperName{} {\bf Settings} facility to view and change
+\Language{} system variables.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugInOutOutTitle}{Common Features of Using Output Formats}
+\newcommand{\ugInOutOutNumber}{4.3.}
+%
+% =====================================================================
+\begin{page}{ugInOutOutPage}{4.3. Common Features of Using Output Formats}
+% =====================================================================
+\beginscroll
+
+In this section we discuss how to start and stop the display
+%-% \HDindex{output formats!common features}{ugInOutOutPage}{4.3.}{Common Features of Using Output Formats}
+of the different output formats and how to send the output to the
+screen or to a file.
+%-% \HDindex{file!sending output to}{ugInOutOutPage}{4.3.}{Common Features of Using Output Formats}
+To fix ideas, we use FORTRAN output format for most of the
+examples.
+
+You can use the \spadcmd{)set output}
+system
+%-% \HDindex{output formats!starting}{ugInOutOutPage}{4.3.}{Common Features of Using Output Formats}
+command to
+%-% \HDindex{output formats!stopping}{ugInOutOutPage}{4.3.}{Common Features of Using Output Formats}
+toggle or redirect the different kinds of output.
+%-% \HDsyscmdindex{set output}{ugInOutOutPage}{4.3.}{Common Features of Using Output Formats}
+The name of the kind of output follows ``output'' in the command.
+The names are
+
+\indent{0}
+\beginitems
+\item[fortran]  for FORTRAN output.
+\item[algebra]  for monospace two-dimensional mathematical output.
+\item[tex]      for \texht{\TeX}{TeX} output.
+\item[script]   for IBM Script Formula Format output.
+\enditems
+\indent{0}
+
+For example, issue \spadsys{)set output fortran on} to turn on
+FORTRAN format and
+issue \spadsys{)set output fortran off} to turn it off.
+By default, {\tt algebra} is {\tt on} and all others are {\tt off}.
+%-% \HDsyscmdindex{set output fortran}{ugInOutOutPage}{4.3.}{Common Features of Using Output Formats}
+When output is started, it is sent to the screen.
+To send the output to a file, give the file name without
+%-% \HDindex{output formats!sending to file}{ugInOutOutPage}{4.3.}{Common Features of Using Output Formats}
+directory or extension.
+\Language{} appends a file extension depending on the kind of
+output being produced.
+\xtc{
+Issue this to redirect FORTRAN output to, for example, the file
+{\bf linalg.sfort}.
+}{
+\spadpaste{)set output fortran linalg}
+}
+\noOutputXtc{
+You must {\it also} turn on the creation of FORTRAN output.
+The above just says where it goes if it is created.
+}{
+\spadpaste{)set output fortran on}
+}
+In what directory is this output placed?
+It goes into the directory from which you started \Language{},
+or if you have used the \spadsys{)cd} system command, the one
+that you specified with \spadsys{)cd}.
+%-% \HDsyscmdindex{cd}{ugInOutOutPage}{4.3.}{Common Features of Using Output Formats}
+You should use \spadcmd{)cd} before you send the output to the file.
+
+\noOutputXtc{
+You can always direct output back to the screen by issuing this.
+%-% \HDindex{output formats!sending to screen}{ugInOutOutPage}{4.3.}{Common Features of Using Output Formats}
+}{
+\spadpaste{)set output fortran console}
+}
+\noOutputXtc{
+Let's make sure FORTRAN formatting is off so that nothing we
+do from now on produces FORTRAN output.
+}{
+\spadpaste{)set output fortran off}
+}
+\noOutputXtc{
+We also delete the demonstrated output file we created.
+}{
+\spadpaste{)system rm linalg.sfort}
+}
+
+You can abbreviate the words ``\spad{on},'' ``\spad{off}'' and
+``\spad{console}'' to the minimal number
+of characters needed to distinguish them.
+Because of this, you cannot send output to files called
+{\bf on.sfort, off.sfort, of.sfort,
+console.sfort, consol.sfort} and so on.
+
+The width of the output on the page is set by
+%-% \HDindex{output formats!line length}{ugInOutOutPage}{4.3.}{Common Features of Using Output Formats}
+\spadcmd{)set output length}
+for all formats except FORTRAN.
+%-% \HDsyscmdindex{set output length}{ugInOutOutPage}{4.3.}{Common Features of Using Output Formats}
+Use \spadcmd{)set fortran fortlength} to
+change the FORTRAN line length from its default value of \spad{72}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugInOutAlgebraTitle}{Monospace Two-Dimensional Mathematical Format}
+\newcommand{\ugInOutAlgebraNumber}{4.4.}
+%
+% =====================================================================
+\begin{page}{ugInOutAlgebraPage}{4.4. Monospace Two-Dimensional Mathematical Format}
+% =====================================================================
+\beginscroll
+
+This is the default output format for \Language{}.
+%-% \HDsyscmdindex{set output algebra}{ugInOutAlgebraPage}{4.4.}{Monospace Two-Dimensional Mathematical Format}
+It is usually on when you start the system.
+%-% \HDindex{output formats!monospace 2D}{ugInOutAlgebraPage}{4.4.}{Monospace Two-Dimensional Mathematical Format}
+%-% \HDindex{monospace 2D output format}{ugInOutAlgebraPage}{4.4.}{Monospace Two-Dimensional Mathematical Format}
+
+\texht{\vskip 4pc}{}
+\noOutputXtc{
+If it is not, issue this.
+}{
+\spadpaste{)set output algebra on \bound{algon}}
+}
+\noOutputXtc{
+Since the printed version of this book
+(as opposed to the \HyperName{} version)
+shows output produced by the
+\texht{\TeX}{TeX}{} output formatter,
+let us temporarily turn off
+\texht{\TeX}{TeX}{} output.
+}{
+\spadpaste{)set output tex off \bound{texoff}}
+}
+\xtc{
+Here is an example of what it looks like.
+}{
+\spadpaste{matrix [[i*x**i + j*\%i*y**j for i in 1..2] for j in 3..4] \free{algon texoff}}
+}
+\noOutputXtc{
+Issue this to turn off this kind of formatting.
+}{
+\spadpaste{)set output algebra off}
+}
+\noOutputXtc{
+Turn \texht{\TeX}{TeX}{} output on again.
+}{
+\spadpaste{)set output tex on}
+}
+
+The characters used for the matrix brackets above are rather ugly.
+You get this character set when you issue
+%-% \HDindex{character set}{ugInOutAlgebraPage}{4.4.}{Monospace Two-Dimensional Mathematical Format}
+\spadcmd{)set output characters plain}.
+%-% \HDsyscmdindex{set output characters}{ugInOutAlgebraPage}{4.4.}{Monospace Two-Dimensional Mathematical Format}
+This character set should be used when you are running on a machine
+that does not support the IBM extended ASCII character set.
+If you are running on an IBM workstation, for example, issue
+\spadcmd{)set output characters default}
+to get better looking output.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugInOutTeXTitle}{TeX Format}
+\newcommand{\ugInOutTeXNumber}{4.5.}
+%
+% =====================================================================
+\begin{page}{ugInOutTeXPage}{4.5. TeX Format}
+% =====================================================================
+\beginscroll
+
+\Language{} can produce \texht{\TeX}{TeX}{} output for your
+%-% \HDindex{output formats!TeX @{\TeX}}{ugInOutTeXPage}{4.5.}{TeX Format}
+expressions.
+%-% \HDindex{TeX output format @{\TeX} output format}{ugInOutTeXPage}{4.5.}{TeX Format}
+The output is produced using macros from the
+\texht{\LaTeX}{LaTeX} document preparation system by
+Leslie Lamport.\footnote{See Leslie Lamport, {\it LaTeX: A Document
+Preparation System,} Reading, Massachusetts: Addison-Wesley
+Publishing Company, Inc., 1986.}
+The printed version of this book was produced using this formatter.
+
+\noOutputXtc{
+To turn on \texht{\TeX}{TeX}{} output formatting, issue this.
+%-% \HDsyscmdindex{set output tex}{ugInOutTeXPage}{4.5.}{TeX Format}
+}{
+\spadpaste{)set output tex on \bound{texon}}
+}
+Here is an example of its output.
+\begin{verbatim}
+matrix [[i*x**i + j*\%i*y**j for i in 1..2] for j in 3..4]
+
+\[
+\left[
+\begin{array}{cc}
+\displaystyle {{3 \  \%i \  {y \sp 3}}+x}&
+\displaystyle {{3 \  \%i \  {y \sp 3}}+{2 \  {x \sp 2}}} \\
+\displaystyle {{4 \  \%i \  {y \sp 4}}+x}&
+\displaystyle {{4 \  \%i \  {y \sp 4}}+{2 \  {x \sp 2}}}
+\end{array}
+\right]
+\leqno(3)
+\]
+%AXIOM STEP NUMBER: 3
+\end{verbatim}
+To turn \texht{\TeX}{TeX}{} output formatting off, issue \spadsys{)set
+output tex off}.
+The \texht{\LaTeX}{LaTeX} macros in the output generated by \Language{}
+are all standard except for the following definitions:
+\begin{verbatim}
+\def\csch{\mathop{\rm csch}\nolimits}
+
+\def\erf{\mathop{\rm erf}\nolimits}
+
+\def\zag#1#2{
+  {{\hfill \left. {#1} \right|}
+   \over
+   {\left| {#2} \right. \hfill}
+  }
+}
+\end{verbatim}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugInOutScriptTitle}{IBM Script Formula Format}
+\newcommand{\ugInOutScriptNumber}{4.6.}
+%
+% =====================================================================
+\begin{page}{ugInOutScriptPage}{4.6. IBM Script Formula Format}
+% =====================================================================
+\beginscroll
+
+\Language{} can
+%-% \HDindex{output formats!IBM Script Formula Format}{ugInOutScriptPage}{4.6.}{IBM Script Formula Format}
+produce IBM Script Formula Format output for your
+%-% \HDindex{IBM Script Formula Format}{ugInOutScriptPage}{4.6.}{IBM Script Formula Format}
+expressions.
+
+\texht{\vskip 2pc}{}
+\noOutputXtc{
+To turn IBM Script Formula Format on, issue this.
+%-% \HDsyscmdindex{set output script}{ugInOutScriptPage}{4.6.}{IBM Script Formula Format}
+}{
+\spadpaste{)set output script on}
+}
+Here is an example of its output.
+\begin{verbatim}
+matrix [[i*x**i + j*%i*y**j for i in 1..2] for j in 3..4]
+
+.eq set blank @
+:df.
+<left lb <<<<3 @@ %i @@ <y sup 3>>+x> here <<3 @@ %i @@
+<y sup 3>>+<2 @@ <x sup 2>>>> habove <<<4 @@ %i @@
+<y sup 4>>+x> here <<4 @@ %i @@ <y sup 4>>+<2 @@
+<x up 2>>>>> right rb>
+:edf.
+\end{verbatim}
+\noOutputXtc{
+To turn IBM Script Formula Format output formatting off, issue this.
+}{
+\spadpaste{)set output script off}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugInOutFortranTitle}{FORTRAN Format}
+\newcommand{\ugInOutFortranNumber}{4.7.}
+%
+% =====================================================================
+\begin{page}{ugInOutFortranPage}{4.7. FORTRAN Format}
+% =====================================================================
+\beginscroll
+
+In addition to turning FORTRAN output on and off and stating where the
+%-% \HDindex{output formats!FORTRAN}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+output should be placed, there are many options that control the
+%-% \HDindex{FORTRAN output format}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+appearance of the generated code.
+In this section we describe some of the basic options.
+Issue \spadcmd{)set fortran} to see a full list with their current
+settings.
+
+The output FORTRAN expression usually begins in column 7.
+If the expression needs more than one line, the ampersand character
+\spadSyntax{\&} is used in column 6.
+Since some versions of FORTRAN have restrictions on the number of lines
+per statement, \Language{} breaks long expressions into segments with
+a maximum of 1320 characters (20 lines of 66 characters) per segment.
+%-% \HDsyscmdindex{set fortran}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+If you want to change this, say, to 660 characters,
+issue the system command
+%-% \HDsyscmdindex{set fortran explength}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+\spadcmd{)set fortran explength 660}.
+%-% \HDindex{FORTRAN output format!breaking into multiple statements}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+You can turn off the line breaking by issuing
+\spadcmd{)set fortran segment off}.
+%-% \HDsyscmdindex{set fortran segment}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+Various code optimization levels are available.
+%
+\noOutputXtc{
+FORTRAN output is produced after you issue this.
+%-% \HDsyscmdindex{set output fortran}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+}{
+\spadpaste{)set output fortran on \bound{forton}}
+}
+\noOutputXtc{
+For the initial examples, we set the optimization level to 0, which is the
+lowest level.
+%-% \HDsyscmdindex{set fortran optlevel}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+}{
+\spadpaste{)set fortran optlevel 0 \bound{opt0}\free{forton}}
+}
+\noOutputXtc{
+The output is usually in columns 7 through 72, although fewer columns
+are used in the following examples so that the output
+%-% \HDindex{FORTRAN output format!line length}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+fits nicely on the page.
+}{
+\spadpaste{)set fortran fortlength 60}
+}
+\xtc{
+By default, the output goes to the screen and is displayed
+before the standard \Language{} two-dimensional output.
+In this example, an
+assignment to the variable \spad{R1} was generated because this is
+the result of step 1.
+}{
+\spadpaste{(x+y)**3 \free{opt0}}
+}
+\xtc{
+Here is an example that illustrates the line breaking.
+}{
+\spadpaste{(x+y+z)**3 \free{opt0}}
+}
+
+Note in the above examples that integers are generally converted to
+%-% \HDindex{FORTRAN output format!integers vs. floats}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+floating point numbers, except in exponents.
+This is the default behavior but can be turned off by issuing
+\spadcmd{)set fortran ints2floats off}.
+%-% \HDsyscmdindex{set fortran ints2floats}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+The rules governing when the conversion is done are:
+\indent{4}
+\beginitems
+\item[1. ] If an integer is an exponent, convert it to a floating point
+number if it is greater than 32767 in absolute value, otherwise leave it
+as an integer.
+\item[2. ] Convert all other integers in an expression to floating
+point numbers.
+\enditems
+\indent{0}
+These rules only govern integers in expressions.
+Numbers generated by \Language{} for \spad{DIMENSION} statements are also
+integers.
+
+To set the type of generated FORTRAN data,
+%-% \HDindex{FORTRAN output format!data types}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+use one of the following:
+\begin{verbatim}
+)set fortran defaulttype REAL
+)set fortran defaulttype INTEGER
+)set fortran defaulttype COMPLEX
+)set fortran defaulttype LOGICAL
+)set fortran defaulttype CHARACTER
+\end{verbatim}
+
+\xtc{
+When temporaries are created, they are given a default type of
+{\tt REAL.}
+Also, the {\tt REAL} versions of functions are used by default.
+}{
+\spadpaste{sin(x) \free{opt1}}
+}
+\noOutputXtc{
+At optimization level 1, \Language{} removes common subexpressions.
+%-% \HDindex{FORTRAN output format!optimization level}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+%-% \HDsyscmdindex{set fortran optlevel}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+}{
+\spadpaste{)set fortran optlevel 1 \bound{opt1}\free{forton}}
+}
+\xtc{
+}{
+\spadpaste{(x+y+z)**3 \free{opt1}}
+}
+\noOutputXtc{
+This changes the precision to {\tt DOUBLE}.
+%-% \HDsyscmdindex{set fortran precision double}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+Substitute \spad{single} for \spad{double}
+%-% \HDindex{FORTRAN output format!precision}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+to return to single precision.
+%-% \HDsyscmdindex{set fortran precision single}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+}{
+\spadpaste{)set fortran precision double \free{opt1}\bound{double1}}
+}
+\xtc{
+Complex constants display the precision.
+}{
+\spadpaste{2.3 + 5.6*\%i  \free{double1}}
+}
+\xtc{
+The function names that \Language{} generates depend on the chosen
+precision.
+}{
+\spadpaste{sin \%e  \free{double1}}
+}
+\noOutputXtc{
+Reset the precision to \spad{single} and look at these two
+examples again.
+}{
+\spadpaste{)set fortran precision single \free{opt1}\bound{single1}}
+}
+\xtc{
+}{
+\spadpaste{2.3 + 5.6*\%i  \free{single1}}
+}
+\xtc{
+}{
+\spadpaste{sin \%e  \free{single1}}
+}
+\xtc{
+Expressions that look like lists, streams, sets or matrices cause
+array code to be generated.
+}{
+\spadpaste{[x+1,y+1,z+1] \free{opt1}}
+}
+\xtc{
+A temporary variable is generated to be the name of the array.
+%-% \HDindex{FORTRAN output format!arrays}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+This may have to be changed in your particular application.
+}{
+\spadpaste{set[2,3,4,3,5] \free{opt1}}
+}
+\xtc{
+By default, the starting index for generated FORTRAN arrays is \spad{0}.
+}{
+\spadpaste{matrix [[2.3,9.7],[0.0,18.778]] \free{opt1}}
+}
+\noOutputXtc{
+To change the starting index for generated FORTRAN arrays to be \spad{1},
+%-% \HDsyscmdindex{set fortran startindex}{ugInOutFortranPage}{4.7.}{FORTRAN Format}
+issue this.
+This value can only be \spad{0} or \spad{1}.
+}{
+\spadpaste{)set fortran startindex 1 \free{opt1}\bound{start1}}
+}
+\xtc{
+Look at the code generated for the matrix again.
+}{
+\spadpaste{matrix [[2.3,9.7],[0.0,18.778]] \free{start1}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug04.pht b/src/hyper/pages/ug04.pht
new file mode 100644
index 00000000..992f12ee
--- /dev/null
+++ b/src/hyper/pages/ug04.pht
@@ -0,0 +1,493 @@
+\begin{patch}{ugInOutTeXPagePatch1}
+\begin{paste}{ugInOutTeXPageFull1}{ugInOutTeXPageEmpty1}
+\pastebutton{ugInOutTeXPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)set output tex on\bound{texon }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutTeXPageEmpty1}
+\begin{paste}{ugInOutTeXPageEmpty1}{ugInOutTeXPagePatch1}
+\pastebutton{ugInOutTeXPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)set output tex on\bound{texon }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutOutPagePatch1}
+\begin{paste}{ugInOutOutPageFull1}{ugInOutOutPageEmpty1}
+\pastebutton{ugInOutOutPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)set output fortran linalg}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutOutPageEmpty1}
+\begin{paste}{ugInOutOutPageEmpty1}{ugInOutOutPagePatch1}
+\pastebutton{ugInOutOutPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)set output fortran linalg}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutOutPagePatch2}
+\begin{paste}{ugInOutOutPageFull2}{ugInOutOutPageEmpty2}
+\pastebutton{ugInOutOutPageFull2}{\hidepaste}
+\tab{5}\spadcommand{)set output fortran on}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutOutPageEmpty2}
+\begin{paste}{ugInOutOutPageEmpty2}{ugInOutOutPagePatch2}
+\pastebutton{ugInOutOutPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{)set output fortran on}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutOutPagePatch3}
+\begin{paste}{ugInOutOutPageFull3}{ugInOutOutPageEmpty3}
+\pastebutton{ugInOutOutPageFull3}{\hidepaste}
+\tab{5}\spadcommand{)set output fortran console}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutOutPageEmpty3}
+\begin{paste}{ugInOutOutPageEmpty3}{ugInOutOutPagePatch3}
+\pastebutton{ugInOutOutPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{)set output fortran console}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutOutPagePatch4}
+\begin{paste}{ugInOutOutPageFull4}{ugInOutOutPageEmpty4}
+\pastebutton{ugInOutOutPageFull4}{\hidepaste}
+\tab{5}\spadcommand{)set output fortran off}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutOutPageEmpty4}
+\begin{paste}{ugInOutOutPageEmpty4}{ugInOutOutPagePatch4}
+\pastebutton{ugInOutOutPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{)set output fortran off}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutOutPagePatch5}
+\begin{paste}{ugInOutOutPageFull5}{ugInOutOutPageEmpty5}
+\pastebutton{ugInOutOutPageFull5}{\hidepaste}
+\tab{5}\spadcommand{)system rm linalg.sfort}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutOutPageEmpty5}
+\begin{paste}{ugInOutOutPageEmpty5}{ugInOutOutPagePatch5}
+\pastebutton{ugInOutOutPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{)system rm linalg.sfort}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutScriptPagePatch1}
+\begin{paste}{ugInOutScriptPageFull1}{ugInOutScriptPageEmpty1}
+\pastebutton{ugInOutScriptPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)set output script on}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutScriptPageEmpty1}
+\begin{paste}{ugInOutScriptPageEmpty1}{ugInOutScriptPagePatch1}
+\pastebutton{ugInOutScriptPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)set output script on}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutScriptPagePatch2}
+\begin{paste}{ugInOutScriptPageFull2}{ugInOutScriptPageEmpty2}
+\pastebutton{ugInOutScriptPageFull2}{\hidepaste}
+\tab{5}\spadcommand{)set output script off}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutScriptPageEmpty2}
+\begin{paste}{ugInOutScriptPageEmpty2}{ugInOutScriptPagePatch2}
+\pastebutton{ugInOutScriptPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{)set output script off}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutAlgebraPagePatch1}
+\begin{paste}{ugInOutAlgebraPageFull1}{ugInOutAlgebraPageEmpty1}
+\pastebutton{ugInOutAlgebraPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)set output algebra on\bound{algon }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutAlgebraPageEmpty1}
+\begin{paste}{ugInOutAlgebraPageEmpty1}{ugInOutAlgebraPagePatch1}
+\pastebutton{ugInOutAlgebraPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)set output algebra on\bound{algon }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutAlgebraPagePatch2}
+\begin{paste}{ugInOutAlgebraPageFull2}{ugInOutAlgebraPageEmpty2}
+\pastebutton{ugInOutAlgebraPageFull2}{\hidepaste}
+\tab{5}\spadcommand{)set output tex off\bound{texoff }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutAlgebraPageEmpty2}
+\begin{paste}{ugInOutAlgebraPageEmpty2}{ugInOutAlgebraPagePatch2}
+\pastebutton{ugInOutAlgebraPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{)set output tex off\bound{texoff }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutAlgebraPagePatch3}
+\begin{paste}{ugInOutAlgebraPageFull3}{ugInOutAlgebraPageEmpty3}
+\pastebutton{ugInOutAlgebraPageFull3}{\hidepaste}
+\tab{5}\spadcommand{matrix [[i*x**i + j*\%i*y**j for i in 1..2] for j in 3..4]\free{algon texoff }}
+\indentrel{3}\begin{verbatim}
+        Ú     3           3     2¿
+        ³3%i y  + x  3%i y  + 2x ³
+   (1)  ³                        ³
+        ³     4           4     2³
+        À4%i y  + x  4%i y  + 2x Ù
+                Type: Matrix Polynomial Complex Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutAlgebraPageEmpty3}
+\begin{paste}{ugInOutAlgebraPageEmpty3}{ugInOutAlgebraPagePatch3}
+\pastebutton{ugInOutAlgebraPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{matrix [[i*x**i + j*\%i*y**j for i in 1..2] for j in 3..4]\free{algon texoff }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutAlgebraPagePatch4}
+\begin{paste}{ugInOutAlgebraPageFull4}{ugInOutAlgebraPageEmpty4}
+\pastebutton{ugInOutAlgebraPageFull4}{\hidepaste}
+\tab{5}\spadcommand{)set output algebra off}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutAlgebraPageEmpty4}
+\begin{paste}{ugInOutAlgebraPageEmpty4}{ugInOutAlgebraPagePatch4}
+\pastebutton{ugInOutAlgebraPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{)set output algebra off}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutAlgebraPagePatch5}
+\begin{paste}{ugInOutAlgebraPageFull5}{ugInOutAlgebraPageEmpty5}
+\pastebutton{ugInOutAlgebraPageFull5}{\hidepaste}
+\tab{5}\spadcommand{)set output tex on}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutAlgebraPageEmpty5}
+\begin{paste}{ugInOutAlgebraPageEmpty5}{ugInOutAlgebraPagePatch5}
+\pastebutton{ugInOutAlgebraPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{)set output tex on}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch1}
+\begin{paste}{ugInOutFortranPageFull1}{ugInOutFortranPageEmpty1}
+\pastebutton{ugInOutFortranPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)set output fortran on\bound{forton }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty1}
+\begin{paste}{ugInOutFortranPageEmpty1}{ugInOutFortranPagePatch1}
+\pastebutton{ugInOutFortranPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)set output fortran on\bound{forton }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch2}
+\begin{paste}{ugInOutFortranPageFull2}{ugInOutFortranPageEmpty2}
+\pastebutton{ugInOutFortranPageFull2}{\hidepaste}
+\tab{5}\spadcommand{)set fortran optlevel 0\bound{opt0 }\free{forton }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty2}
+\begin{paste}{ugInOutFortranPageEmpty2}{ugInOutFortranPagePatch2}
+\pastebutton{ugInOutFortranPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{)set fortran optlevel 0\bound{opt0 }\free{forton }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch3}
+\begin{paste}{ugInOutFortranPageFull3}{ugInOutFortranPageEmpty3}
+\pastebutton{ugInOutFortranPageFull3}{\hidepaste}
+\tab{5}\spadcommand{)set fortran fortlength 60}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty3}
+\begin{paste}{ugInOutFortranPageEmpty3}{ugInOutFortranPagePatch3}
+\pastebutton{ugInOutFortranPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{)set fortran fortlength 60}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch4}
+\begin{paste}{ugInOutFortranPageFull4}{ugInOutFortranPageEmpty4}
+\pastebutton{ugInOutFortranPageFull4}{\hidepaste}
+\tab{5}\spadcommand{(x+y)**3\free{opt0 }}
+\indentrel{3}\begin{verbatim}
+         3       2     2     3
+   (1)  y  + 3x y  + 3x y + x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty4}
+\begin{paste}{ugInOutFortranPageEmpty4}{ugInOutFortranPagePatch4}
+\pastebutton{ugInOutFortranPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{(x+y)**3\free{opt0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch5}
+\begin{paste}{ugInOutFortranPageFull5}{ugInOutFortranPageEmpty5}
+\pastebutton{ugInOutFortranPageFull5}{\hidepaste}
+\tab{5}\spadcommand{(x+y+z)**3\free{opt0 }}
+\indentrel{3}\begin{verbatim}
+   (2)
+      3             2      2            2      3       2
+     z  + (3y + 3x)z  + (3y  + 6x y + 3x )z + y  + 3x y
+   + 
+       2     3
+     3x y + x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty5}
+\begin{paste}{ugInOutFortranPageEmpty5}{ugInOutFortranPagePatch5}
+\pastebutton{ugInOutFortranPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{(x+y+z)**3\free{opt0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch6}
+\begin{paste}{ugInOutFortranPageFull6}{ugInOutFortranPageEmpty6}
+\pastebutton{ugInOutFortranPageFull6}{\hidepaste}
+\tab{5}\spadcommand{sin(x)\free{opt1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  sin(x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty6}
+\begin{paste}{ugInOutFortranPageEmpty6}{ugInOutFortranPagePatch6}
+\pastebutton{ugInOutFortranPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{sin(x)\free{opt1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch7}
+\begin{paste}{ugInOutFortranPageFull7}{ugInOutFortranPageEmpty7}
+\pastebutton{ugInOutFortranPageFull7}{\hidepaste}
+\tab{5}\spadcommand{)set fortran optlevel 1\bound{opt1 }\free{forton }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty7}
+\begin{paste}{ugInOutFortranPageEmpty7}{ugInOutFortranPagePatch7}
+\pastebutton{ugInOutFortranPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{)set fortran optlevel 1\bound{opt1 }\free{forton }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch8}
+\begin{paste}{ugInOutFortranPageFull8}{ugInOutFortranPageEmpty8}
+\pastebutton{ugInOutFortranPageFull8}{\hidepaste}
+\tab{5}\spadcommand{(x+y+z)**3\free{opt1 }}
+\indentrel{3}\begin{verbatim}
+   (4)
+      3             2      2            2      3       2
+     z  + (3y + 3x)z  + (3y  + 6x y + 3x )z + y  + 3x y
+   + 
+       2     3
+     3x y + x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty8}
+\begin{paste}{ugInOutFortranPageEmpty8}{ugInOutFortranPagePatch8}
+\pastebutton{ugInOutFortranPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{(x+y+z)**3\free{opt1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch9}
+\begin{paste}{ugInOutFortranPageFull9}{ugInOutFortranPageEmpty9}
+\pastebutton{ugInOutFortranPageFull9}{\hidepaste}
+\tab{5}\spadcommand{)set fortran precision double\free{opt1 }\bound{double1 }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty9}
+\begin{paste}{ugInOutFortranPageEmpty9}{ugInOutFortranPagePatch9}
+\pastebutton{ugInOutFortranPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{)set fortran precision double\free{opt1 }\bound{double1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch10}
+\begin{paste}{ugInOutFortranPageFull10}{ugInOutFortranPageEmpty10}
+\pastebutton{ugInOutFortranPageFull10}{\hidepaste}
+\tab{5}\spadcommand{2.3 + 5.6*\%i\free{double1 }}
+\indentrel{3}\begin{verbatim}
+   (5)  2.3 + 5.6 %i
+                                    Type: Complex Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty10}
+\begin{paste}{ugInOutFortranPageEmpty10}{ugInOutFortranPagePatch10}
+\pastebutton{ugInOutFortranPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{2.3 + 5.6*\%i\free{double1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch11}
+\begin{paste}{ugInOutFortranPageFull11}{ugInOutFortranPageEmpty11}
+\pastebutton{ugInOutFortranPageFull11}{\hidepaste}
+\tab{5}\spadcommand{sin \%e\free{double1 }}
+\indentrel{3}\begin{verbatim}
+   (6)  sin(%e)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty11}
+\begin{paste}{ugInOutFortranPageEmpty11}{ugInOutFortranPagePatch11}
+\pastebutton{ugInOutFortranPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{sin \%e\free{double1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch12}
+\begin{paste}{ugInOutFortranPageFull12}{ugInOutFortranPageEmpty12}
+\pastebutton{ugInOutFortranPageFull12}{\hidepaste}
+\tab{5}\spadcommand{)set fortran precision single\free{opt1 }\bound{single1 }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty12}
+\begin{paste}{ugInOutFortranPageEmpty12}{ugInOutFortranPagePatch12}
+\pastebutton{ugInOutFortranPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{)set fortran precision single\free{opt1 }\bound{single1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch13}
+\begin{paste}{ugInOutFortranPageFull13}{ugInOutFortranPageEmpty13}
+\pastebutton{ugInOutFortranPageFull13}{\hidepaste}
+\tab{5}\spadcommand{2.3 + 5.6*\%i\free{single1 }}
+\indentrel{3}\begin{verbatim}
+   (7)  2.3 + 5.6 %i
+                                    Type: Complex Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty13}
+\begin{paste}{ugInOutFortranPageEmpty13}{ugInOutFortranPagePatch13}
+\pastebutton{ugInOutFortranPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{2.3 + 5.6*\%i\free{single1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch14}
+\begin{paste}{ugInOutFortranPageFull14}{ugInOutFortranPageEmpty14}
+\pastebutton{ugInOutFortranPageFull14}{\hidepaste}
+\tab{5}\spadcommand{sin \%e\free{single1 }}
+\indentrel{3}\begin{verbatim}
+   (8)  sin(%e)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty14}
+\begin{paste}{ugInOutFortranPageEmpty14}{ugInOutFortranPagePatch14}
+\pastebutton{ugInOutFortranPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{sin \%e\free{single1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch15}
+\begin{paste}{ugInOutFortranPageFull15}{ugInOutFortranPageEmpty15}
+\pastebutton{ugInOutFortranPageFull15}{\hidepaste}
+\tab{5}\spadcommand{[x+1,y+1,z+1]\free{opt1 }}
+\indentrel{3}\begin{verbatim}
+   (9)  [x + 1,y + 1,z + 1]
+                          Type: List Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty15}
+\begin{paste}{ugInOutFortranPageEmpty15}{ugInOutFortranPagePatch15}
+\pastebutton{ugInOutFortranPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{[x+1,y+1,z+1]\free{opt1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch16}
+\begin{paste}{ugInOutFortranPageFull16}{ugInOutFortranPageEmpty16}
+\pastebutton{ugInOutFortranPageFull16}{\hidepaste}
+\tab{5}\spadcommand{set[2,3,4,3,5]\free{opt1 }}
+\indentrel{3}\begin{verbatim}
+   (10)  {2,3,4,5}
+                              Type: Set PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty16}
+\begin{paste}{ugInOutFortranPageEmpty16}{ugInOutFortranPagePatch16}
+\pastebutton{ugInOutFortranPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{set[2,3,4,3,5]\free{opt1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch17}
+\begin{paste}{ugInOutFortranPageFull17}{ugInOutFortranPageEmpty17}
+\pastebutton{ugInOutFortranPageFull17}{\hidepaste}
+\tab{5}\spadcommand{matrix [[2.3,9.7],[0.0,18.778]]\free{opt1 }}
+\indentrel{3}\begin{verbatim}
+         Ú2.3   9.7  ¿
+   (11)  ³           ³
+         À0.0  18.778Ù
+                                     Type: Matrix Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty17}
+\begin{paste}{ugInOutFortranPageEmpty17}{ugInOutFortranPagePatch17}
+\pastebutton{ugInOutFortranPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{matrix [[2.3,9.7],[0.0,18.778]]\free{opt1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch18}
+\begin{paste}{ugInOutFortranPageFull18}{ugInOutFortranPageEmpty18}
+\pastebutton{ugInOutFortranPageFull18}{\hidepaste}
+\tab{5}\spadcommand{)set fortran startindex 1\free{opt1 }\bound{start1 }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty18}
+\begin{paste}{ugInOutFortranPageEmpty18}{ugInOutFortranPagePatch18}
+\pastebutton{ugInOutFortranPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{)set fortran startindex 1\free{opt1 }\bound{start1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPagePatch19}
+\begin{paste}{ugInOutFortranPageFull19}{ugInOutFortranPageEmpty19}
+\pastebutton{ugInOutFortranPageFull19}{\hidepaste}
+\tab{5}\spadcommand{matrix [[2.3,9.7],[0.0,18.778]]\free{start1 }}
+\indentrel{3}\begin{verbatim}
+         Ú2.3   9.7  ¿
+   (12)  ³           ³
+         À0.0  18.778Ù
+                                     Type: Matrix Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugInOutFortranPageEmpty19}
+\begin{paste}{ugInOutFortranPageEmpty19}{ugInOutFortranPagePatch19}
+\pastebutton{ugInOutFortranPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{matrix [[2.3,9.7],[0.0,18.778]]\free{start1 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ug05.ht b/src/hyper/pages/ug05.ht
new file mode 100644
index 00000000..6d861686
--- /dev/null
+++ b/src/hyper/pages/ug05.ht
@@ -0,0 +1,1824 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\texht{\setcounter{chapter}{4}}{} % Chapter 5
+
+%
+\newcommand{\ugLangTitle}{Introduction to the \Language{} Interactive Language}
+\newcommand{\ugLangNumber}{5.}
+%
+% =====================================================================
+\begin{page}{ugLangPage}{5. Introduction to the \Language{} Interactive Language}
+% =====================================================================
+\beginscroll
+
+In this chapter we look at some of the basic components of the
+\Language{} language that you can use interactively.
+We show how to create a \spadgloss{block} of expressions,
+how to form loops and list iterations, how to modify the sequential
+evaluation of a block and how to use {\tt if-then-else} to
+evaluate parts of your program conditionally.
+We suggest you first read the boxed material in each section and then
+proceed to a more thorough reading of the chapter.
+
+\beginmenu
+    \menudownlink{{5.1. Immediate and Delayed Assignments}}{ugLangAssignPage}
+    \menudownlink{{5.2. Blocks}}{ugLangBlocksPage}
+    \menudownlink{{5.3. if-then-else}}{ugLangIfPage}
+    \menudownlink{{5.4. Loops}}{ugLangLoopsPage}
+    \menudownlink{{5.5. Creating Lists and Streams with Iterators}}{ugLangItsPage}
+    \menudownlink{{5.6. An Example: Streams of Primes}}{ugLangStreamsPrimesPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangAssignTitle}{Immediate and Delayed Assignments}
+\newcommand{\ugLangAssignNumber}{5.1.}
+%
+% =====================================================================
+\begin{page}{ugLangAssignPage}{5.1. Immediate and Delayed Assignments}
+% =====================================================================
+\beginscroll
+
+A \spadgloss{variable} in \Language{} refers to a value.
+A variable has a name beginning with an uppercase or lowercase alphabetic
+character, \axiomSyntax{\%}, or \axiomSyntax{!}.
+Successive characters (if any) can be any of the above, digits, or
+\axiomSyntax{?}.
+Case is distinguished.
+The following are all examples of valid, distinct variable names:
+\begin{verbatim}
+a             tooBig?    a1B2c3%!?
+A             %j         numberOfPoints
+beta6         %J         numberofpoints
+\end{verbatim}
+
+The \axiomSyntax{:=} operator is the immediate \spadgloss{assignment}
+operator.
+%-% \HDindex{assignment!immediate}{ugLangAssignPage}{5.1.}{Immediate and Delayed Assignments}
+Use it to associate a value with a variable.
+%-% \HDindex{immediate assignment}{ugLangAssignPage}{5.1.}{Immediate and Delayed Assignments}
+
+\beginImportant
+The syntax for immediate assignment for a single variable is
+\centerline{{{\it variable} \axiom{:=} {\it expression}}}
+The value returned by an immediate assignment is the value of {\it expression}.
+\endImportant
+
+\xtc{
+The right-hand side of the expression is evaluated,
+yielding \axiom{1}.  This value is then assigned to \axiom{a}.
+}{
+\spadpaste{a := 1 \bound{a}}
+}
+\xtc{
+The right-hand side of the expression is evaluated,
+yielding \axiom{1}.  This value is then assigned to \axiom{b}.
+Thus \axiom{a} and \axiom{b} both have the value \axiom{1} after the sequence
+of assignments.
+}{
+\spadpaste{b := a \free{a}\bound{b}}
+}
+\xtc{
+What is the value of \axiom{b} if \axiom{a} is
+assigned the value \axiom{2}?
+}{
+\spadpaste{a := 2 \bound{a2}}
+}
+\xtc{
+As you see, the value of \axiom{b} is left unchanged.
+}{
+\spadpaste{b \free{b}}
+}
+This is what we mean when we say this kind of assignment is
+{\it immediate};
+\axiom{b} has no dependency on \axiom{a} after the initial assignment.
+This is the usual notion of assignment found in programming
+languages such as C,
+%-% \HDindex{C language!assignment}{ugLangAssignPage}{5.1.}{Immediate and Delayed Assignments}
+PASCAL
+%-% \HDindex{PASCAL!assignment}{ugLangAssignPage}{5.1.}{Immediate and Delayed Assignments}
+and FORTRAN.
+%-% \HDindex{FORTRAN!assignment}{ugLangAssignPage}{5.1.}{Immediate and Delayed Assignments}
+
+\Language{} provides delayed assignment with \axiomSyntax{==}.
+%-% \HDindex{assignment!delayed}{ugLangAssignPage}{5.1.}{Immediate and Delayed Assignments}
+This implements a
+%-% \HDindex{delayed assignment}{ugLangAssignPage}{5.1.}{Immediate and Delayed Assignments}
+delayed evaluation of the right-hand side and dependency
+checking.
+
+\beginImportant
+The syntax for delayed assignment is
+\centerline{{{\it variable} \axiom{==} {\it expression}}}
+The value returned by a delayed assignment is \void{}.
+\endImportant
+
+\xtc{
+Using \axiom{a} and \axiom{b} as above, these are the corresponding delayed
+assignments.
+}{
+\spadpaste{a == 1 \bound{ad}}
+}
+\xtc{
+}{
+\spadpaste{b == a \free{ad}\bound{bd}}
+}
+\xtc{
+The right-hand side of each delayed assignment
+is left unevaluated until the
+variables on the left-hand sides are evaluated.
+Therefore this evaluation and \ldots
+}{
+\spadpaste{a \free{ad}}
+}
+\xtc{
+this evaluation seem the same as before.
+}{
+\spadpaste{b \free{bd}}
+}
+\xtc{
+If we change \axiom{a} to \axiom{2}
+}{
+\spadpaste{a == 2 \bound{ad2}}
+}
+\xtc{
+then
+\axiom{a} evaluates to \axiom{2}, as expected, but
+}{
+\spadpaste{a \free{ad2}}
+}
+\xtc{
+the value of \axiom{b} reflects the change to \axiom{a}.
+}{
+\spadpaste{b \free{bd ad2}}
+}
+
+It is possible to set several variables at the same time
+%-% \HDindex{assignment!multiple immediate}{ugLangAssignPage}{5.1.}{Immediate and Delayed Assignments}
+by using
+%-% \HDindex{multiple immediate assignment}{ugLangAssignPage}{5.1.}{Immediate and Delayed Assignments}
+a \spadgloss{tuple} of variables and a tuple of expressions.\footnote{A
+\spadgloss{tuple} is a collection of things separated by commas, often
+surrounded by parentheses.}
+
+\beginImportant
+The syntax for multiple immediate assignments is
+\centerline{{{\tt ( \subscriptIt{var}{1}, \subscriptIt{var}{2}, \ldots, \subscriptIt{var}{N} ) := ( \subscriptIt{expr}{1}, \subscriptIt{expr}{2}, \ldots, \subscriptIt{expr}{N} ) }}}
+The value returned by an immediate assignment is the value of
+\subscriptIt{expr}{N}.
+\endImportant
+
+\xtc{
+This sets \axiom{x} to \axiom{1} and \axiom{y} to \axiom{2}.
+}{
+\spadpaste{(x,y) := (1,2) \bound{x}\bound{y}}
+}
+Multiple immediate assigments are parallel in the sense that the
+expressions on the right are all evaluated before any assignments
+on the left are made.
+However, the order of evaluation of these expressions is undefined.
+\xtc{
+You can use multiple immediate assignment to swap the
+values held by variables.
+}{
+\spadpaste{(x,y) := (y,x) \free{x y}\bound{swap}}
+}
+\xtc{
+\axiom{x} has the previous value of \axiom{y}.
+}{
+\spadpaste{x \free{swap}}
+}
+\xtc{
+\axiom{y} has the previous value of \axiom{x}.
+}{
+\spadpaste{y \free{swap}}
+}
+
+There is no syntactic form for multiple delayed assignments.
+See the discussion in
+\downlink{``\ugUserDelayTitle''}{ugUserDelayPage} in Section \ugUserDelayNumber\ignore{ugUserDelay}
+about how \Language{} differentiates between delayed assignments and
+user functions of no arguments.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangBlocksTitle}{Blocks}
+\newcommand{\ugLangBlocksNumber}{5.2.}
+%
+% =====================================================================
+\begin{page}{ugLangBlocksPage}{5.2. Blocks}
+% =====================================================================
+\beginscroll
+
+%%
+%% We should handle tabs in pile correctly but so far we do not.
+%%
+
+A \spadgloss{block} is a sequence of expressions evaluated
+in the order that they appear, except as modified by control expressions
+such as \axiom{break},
+\spadkey{break}
+\axiom{return},
+\spadkey{return}
+\axiom{iterate} and
+\spadkey{iterate}
+\axiom{if-then-else} constructions.
+The value of a block is the value of the expression last evaluated
+in the block.
+
+To leave a block early, use \axiomSyntax{=>}.
+For example, \axiom{i < 0 => x}.
+The expression before the \axiomSyntax{=>} must evaluate to
+\axiom{true} or \axiom{false}.
+The expression following the \axiomSyntax{=>} is the return value
+for the block.
+
+A block can be constructed in two ways:
+\indent{4}
+\beginitems
+\item[1. ] the expressions can be separated by semicolons
+and the resulting expression surrounded by parentheses, and
+\item[2. ] the expressions can be written on succeeding lines with each line
+indented the same number of spaces (which must be greater than zero).
+%-% \HDindex{indentation}{ugLangBlocksPage}{5.2.}{Blocks}
+A block entered in this form is
+called a \spadgloss{pile}.
+\enditems
+\indent{0}
+Only the first form is available if you are entering expressions
+directly to \Language{}.
+Both forms are available in {\bf .input} files.
+
+\beginImportant
+The syntax for a simple block of expressions entered interactively is
+\centerline{{{\tt ( \subscriptIt{expression}{1}; \subscriptIt{expression}{2}; \ldots; \subscriptIt{expression}{N} )}}}
+The value returned by a block is the value of an
+\axiomSyntax{=>} expression, or \subscriptIt{expression}{N}
+if no \axiomSyntax{=>} is encountered.
+\endImportant
+
+In {\bf .input} files, blocks can also be written using
+\spadglossSee{piles}{pile}.
+The examples throughout this book are assumed to come from {\bf .input} files.
+
+\xtc{
+In this example, we assign a rational number to \axiom{a} using a block
+consisting of three expressions.
+This block is written as a pile.
+Each expression in the pile has the same indentation, in this case two
+spaces to the right of the first line.
+}{
+\begin{spadsrc}
+a :=
+  i := gcd(234,672)
+  i := 3*i**5 - i + 1
+  1 / i
+\end{spadsrc}
+}
+\xtc{
+Here is the same block written on one line.
+This is how you are required to enter it at the input prompt.
+}{
+\spadpaste{a := (i := gcd(234,672); i := 3*i**5 - i + 1; 1 / i)}
+}
+\xtc{
+Blocks can be used to put several expressions on one line.
+The value returned is that of the last expression.
+}{
+\spadpaste{(a := 1; b := 2; c := 3; [a,b,c]) \bound{a b c}}
+}
+
+\Language{} gives you two ways of writing a block and the
+preferred way in an {\bf .input} file is to use a pile.
+%-% \HDindex{file!input}{ugLangBlocksPage}{5.2.}{Blocks}
+Roughly speaking, a pile is
+a block whose constituent expressions are indented the same amount.
+You begin a pile by starting a new line for the first expression,
+indenting it to the right of the previous line.
+You then enter the second expression on a new line, vertically aligning
+it with the first line. And so on.
+If you need to enter an inner pile, further indent its lines to the right
+of the outer pile.
+\Language{} knows where a pile ends.
+It ends when a subsequent line is indented to the left of the pile or
+the end of the file.
+
+\xtc{
+Blocks can be used to perform several steps before an assignment
+(immediate or delayed) is made.
+}{
+\begin{spadsrc}[\free{a b}]
+d :=
+   c := a**2 + b**2
+   sqrt(c * 1.3)
+\end{spadsrc}
+}
+\xtc{
+Blocks can be used in the arguments to functions.
+(Here \axiom{h} is assigned \axiom{2.1 + 3.5}.)
+}{
+\begin{spadsrc}[\bound{h}]
+h := 2.1 +
+   1.0
+   3.5
+\end{spadsrc}
+}
+\xtc{
+Here the second argument to \axiomFun{eval} is \axiom{x = z}, where
+the value of \axiom{z} is computed in the first line of the block
+starting on the second line.
+}{
+\begin{spadsrc}
+eval(x**2 - x*y**2,
+     z := %pi/2.0 - exp(4.1)
+     x = z
+   )
+\end{spadsrc}
+}
+\xtc{
+Blocks can be used in the clauses of \axiom{if-then-else}
+expressions (see \downlink{``\ugLangIfTitle''}{ugLangIfPage} in Section \ugLangIfNumber\ignore{ugLangIf}).
+}{
+\spadpaste{if h > 3.1 then 1.0 else (z := cos(h); max(z,0.5)) \free{h}}
+}
+\xtc{
+This is the pile version of the last block.
+}{
+\begin{spadsrc}[\free{h}]
+if h > 3.1 then
+    1.0
+  else
+    z := cos(h)
+    max(z,0.5)
+\end{spadsrc}
+}
+\xtc{
+Blocks can be nested.
+}{
+\spadpaste{a := (b := factorial(12); c := (d := eulerPhi(22); factorial(d));b+c)}
+}
+\xtc{
+This is the pile version of the last block.
+}{
+\begin{spadsrc}
+a :=
+  b := factorial(12)
+  c :=
+    d := eulerPhi(22)
+    factorial(d)
+  b+c
+\end{spadsrc}
+}
+
+\xtc{
+Since \axiom{c + d} does equal \axiom{3628855}, \axiom{a} has the value
+of \axiom{c} and the last line is never evaluated.
+}{
+\begin{spadsrc}
+a :=
+  c := factorial 10
+  d := fibonacci 10
+  c + d = 3628855 => c
+  d
+\end{spadsrc}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangIfTitle}{if-then-else}
+\newcommand{\ugLangIfNumber}{5.3.}
+%
+% =====================================================================
+\begin{page}{ugLangIfPage}{5.3. if-then-else}
+% =====================================================================
+\beginscroll
+
+Like many other programming languages, \Language{} uses the three
+keywords \spadkey{if} \axiom{if, then} \spadkey{then} and \axiom{else}
+\spadkey{else} to form
+%-% \HDindex{conditional}{ugLangIfPage}{5.3.}{if-then-else}
+conditional expressions.
+The \axiom{else} part of the conditional is optional.
+The expression between the \axiom{if} and \axiom{then} keywords
+is a
+\spadgloss{predicate}: an expression that evaluates to or is convertible to
+either {\tt true} or {\tt false}, that is,
+a \axiomType{Boolean}.
+%-% \HDexptypeindex{Boolean}{ugLangIfPage}{5.3.}{if-then-else}
+
+\beginImportant
+The syntax for conditional expressions is
+\centerline{{{\tt if {\it predicate} then \subscriptIt{expression}{1} else \subscriptIt{expression}{2}}}}
+where the \axiom{else} \subscriptIt{\it expression}{2} part is optional.
+The value returned from a conditional expression is
+\subscriptIt{\it expression}{1} if the predicate evaluates to \axiom{true}
+and \subscriptIt{\it expression}{2} otherwise.
+If no \axiom{else} clause is given, the value is always \void{}.
+\endImportant
+
+An \axiom{if-then-else} expression always returns a value.
+If the
+\axiom{else} clause is missing then the entire expression returns
+\void{}.
+If both clauses are present, the type of the value returned by \axiom{if}
+is obtained by resolving the types of the values of the two clauses.
+See \downlink{``\ugTypesResolveTitle''}{ugTypesResolvePage} in Section \ugTypesResolveNumber\ignore{ugTypesResolve}
+for more information.
+
+The predicate must evaluate to, or be convertible to, an object of type
+\axiomType{Boolean}: {\tt true} or {\tt false}.
+By default, the equal sign \spadopFrom{=}{Equation} creates
+%-% \HDindex{equation}{ugLangIfPage}{5.3.}{if-then-else}
+an equation.
+\xtc{
+This is an equation.
+%-% \HDexptypeindex{Equation}{ugLangIfPage}{5.3.}{if-then-else}
+In particular, it is an object of type \axiomType{Equation Polynomial Integer}.
+}{
+\spadpaste{x + 1 = y}
+}
+However, for predicates in \axiom{if} expressions, \Language{}
+%-% \HDindex{equality testing}{ugLangIfPage}{5.3.}{if-then-else}
+places a default target type of \axiomType{Boolean} on the
+predicate and equality testing is performed.
+%-% \HDexptypeindex{Boolean}{ugLangIfPage}{5.3.}{if-then-else}
+Thus you need not qualify the \axiomSyntax{=} in any way.
+In other contexts you may need to tell \Language{} that you want
+to test for equality rather than create an equation.
+In those cases, use \axiomSyntax{@} and a target type of
+\axiomType{Boolean}.
+See \downlink{``\ugTypesPkgCallTitle''}{ugTypesPkgCallPage} in Section \ugTypesPkgCallNumber\ignore{ugTypesPkgCall} for more information.
+
+The compound symbol meaning ``not equal'' in \Language{} is
+%-% \HDindex{inequality testing}{ugLangIfPage}{5.3.}{if-then-else}
+``\texht{$\sim =$}{\axiom{~=}}''.
+%-% \HDindex{\_notequal@$\sim =$}{ugLangIfPage}{5.3.}{if-then-else}
+This can be used directly without a package call or a target specification.
+The expression
+\axiom{a} \texht{$\sim =$}{\axiom{~=}} \axiom{b} is directly translated into
+\axiom{not (a = b)}.
+
+Many other functions have return values of type \axiomType{Boolean}.
+These include \axiom{<}, \axiom{<=}, \axiom{>},
+\axiom{>=}, \texht{$\sim =$}{\axiom{~=}} and \axiom{member?}.
+By convention, operations with names ending in \axiomSyntax{?}
+return \axiomType{Boolean} values.
+
+The usual rules for piles are suspended for conditional expressions.
+In {\bf .input} files, the \axiom{then} and
+\axiom{else} keywords can begin in the same column as the corresponding
+\axiom{if} but may also appear to the right.
+Each of the following styles of writing \axiom{if-then-else}
+expressions is acceptable:
+\begin{verbatim}
+if i>0 then output("positive") else output("nonpositive")
+
+if i > 0 then output("positive")
+  else output("nonpositive")
+
+if i > 0 then output("positive")
+else output("nonpositive")
+
+if i > 0
+then output("positive")
+else output("nonpositive")
+
+if i > 0
+  then output("positive")
+  else output("nonpositive")
+\end{verbatim}
+
+A block can follow the \axiom{then} or \axiom{else} keywords.
+In the following two assignments to \axiom{a}, the \axiom{then} and \axiom{else}
+clauses each are followed by two-line piles.
+The value returned in each is the value of the second line.
+
+\begin{verbatim}
+a :=
+  if i > 0 then
+    j := sin(i * pi())
+    exp(j + 1/j)
+  else
+    j := cos(i * 0.5 * pi())
+    log(abs(j)**5 + 1)
+
+a :=
+  if i > 0
+    then
+      j := sin(i * pi())
+      exp(j + 1/j)
+    else
+      j := cos(i * 0.5 * pi())
+      log(abs(j)**5 + 1)
+\end{verbatim}
+These are both equivalent to the following:
+\begin{verbatim}
+a :=
+  if i > 0 then (j := sin(i * pi()); exp(j + 1/j))
+  else (j := cos(i * 0.5 * pi()); log(abs(j)**5 + 1))
+\end{verbatim}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsTitle}{Loops}
+\newcommand{\ugLangLoopsNumber}{5.4.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsPage}{5.4. Loops}
+% =====================================================================
+\beginscroll
+
+A \spadgloss{loop} is an expression that contains another expression,
+%-% \HDindex{loop}{ugLangLoopsPage}{5.4.}{Loops}
+called the {\it loop body}, which is to be evaluated zero or more
+%-% \HDindex{loop!body}{ugLangLoopsPage}{5.4.}{Loops}
+times.
+All loops contain the \axiom{repeat} keyword and return \void{}.
+Loops can contain inner loops to any depth.
+
+\beginImportant
+The most basic loop is of the form
+\centerline{{\axiom{repeat} {\it loopBody}}}
+Unless {\it loopBody} contains a \axiom{break} or \axiom{return} expression,
+the loop repeats forever.
+The value returned by the loop is \void{}.
+\endImportant
+
+\beginmenu
+    \menudownlink{{5.4.1. Compiling vs. Interpreting Loops}}{ugLangLoopsCompIntPage}
+    \menudownlink{{5.4.2. return in Loops}}{ugLangLoopsReturnPage}
+    \menudownlink{{5.4.3. break in Loops}}{ugLangLoopsBreakPage}
+    \menudownlink{{5.4.4. break vs. {\tt =>} in Loop Bodies}}{ugLangLoopsBreakVsPage}
+    \menudownlink{{5.4.5. More Examples of break}}{ugLangLoopsBreakMorePage}
+    \menudownlink{{5.4.6. iterate in Loops}}{ugLangLoopsIteratePage}
+    \menudownlink{{5.4.7. while Loops}}{ugLangLoopsWhilePage}
+    \menudownlink{{5.4.8. for Loops}}{ugLangLoopsForInPage}
+    \menudownlink{{5.4.9. for i in n..m repeat}}{ugLangLoopsForInNMPage}
+    \menudownlink{{5.4.10. for i in n..m by s repeat}}{ugLangLoopsForInNMSPage}
+    \menudownlink{{5.4.11. for i in n.. repeat}}{ugLangLoopsForInNPage}
+    \menudownlink{{5.4.12. for x in l repeat}}{ugLangLoopsForInXLPage}
+    \menudownlink{{5.4.13. ``Such that'' Predicates}}{ugLangLoopsForInPredPage}
+    \menudownlink{{5.4.14. Parallel Iteration}}{ugLangLoopsParPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsCompIntTitle}{Compiling vs. Interpreting Loops}
+\newcommand{\ugLangLoopsCompIntNumber}{5.4.1.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsCompIntPage}{5.4.1. Compiling vs. Interpreting Loops}
+% =====================================================================
+\beginscroll
+
+\Language{} tries to determine completely the type of every
+object in a loop and then to translate the loop body to LISP or even to
+machine code.
+This translation is called \spadglossSee{compilation}{compiler}.
+
+If \Language{} decides that it cannot compile the loop, it issues a
+%-% \HDindex{loop!compilation}{ugLangLoopsCompIntPage}{5.4.1.}{Compiling vs. Interpreting Loops}
+message stating the problem and then the following message:
+%
+\centerline{{{\bf We will attempt to step through and interpret the code.}}}
+%
+It is still possible that \Language{} can evaluate the loop but in
+\spadgloss{interpret-code mode}.
+See \downlink{``\ugUserCompIntTitle''}{ugUserCompIntPage} in Section \ugUserCompIntNumber\ignore{ugUserCompInt} where this is discussed in terms
+%-% \HDindex{panic!avoiding}{ugLangLoopsCompIntPage}{5.4.1.}{Compiling vs. Interpreting Loops}
+of compiling versus interpreting functions.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsReturnTitle}{return in Loops}
+\newcommand{\ugLangLoopsReturnNumber}{5.4.2.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsReturnPage}{5.4.2. return in Loops}
+% =====================================================================
+\beginscroll
+
+A \axiom{return} expression is used to exit a function with
+%-% \HDindex{loop!leaving via return}{ugLangLoopsReturnPage}{5.4.2.}{return in Loops}
+a particular value.
+In particular, if a \axiom{return} is in a loop within the
+\spadkey{return}
+function, the loop is terminated whenever the \axiom{return}
+is evaluated.
+%> This is a bug! The compiler should never accept allow
+%> Void to be the return type of a function when it has to use
+%> resolve to determine it.
+\xtc{
+Suppose we start with this.
+}{
+\begin{spadsrc}[\bound{f}]
+f() ==
+  i := 1
+  repeat
+    if factorial(i) > 1000 then return i
+    i := i + 1
+\end{spadsrc}
+}
+\xtc{
+When \axiom{factorial(i)} is big enough, control passes from
+inside the loop all the way outside the function, returning the
+value of \axiom{i} (or so we think).
+}{
+\spadpaste{f() \free{f}}
+}
+
+What went wrong?
+Isn't it obvious that this function should return an integer?
+Well, \Language{} makes no attempt to analyze the structure of a
+loop to determine if it always returns a value because, in
+general, this is impossible.
+So \Language{} has this simple rule: the type of the function is
+determined by the type of its body, in this case a block.
+The normal value of a block is the value of its last expression,
+in this case, a loop.
+And the value of every loop is \void{}!
+So the return type of \userfun{f} is \axiomType{Void}.
+
+There are two ways to fix this.
+The best way is for you to tell \Language{} what the return type
+of \axiom{f} is.
+You do this by giving \axiom{f} a declaration \axiom{f: () ->
+Integer} prior to calling for its value.
+This tells \Language{}: ``trust me---an integer is returned.''
+We'll explain more about this in the next chapter.
+Another clumsy way is to add a dummy expression as follows.
+
+\xtc{
+Since we want an integer, let's stick in a dummy final expression that is
+an integer and will never be evaluated.
+}{
+\begin{spadsrc}[\bound{f1}]
+f() ==
+  i := 1
+  repeat
+    if factorial(i) > 1000 then return i
+    i := i + 1
+  0
+\end{spadsrc}
+}
+\xtc{
+When we try \userfun{f} again we get what we wanted.
+See
+\downlink{``\ugUserBlocksTitle''}{ugUserBlocksPage} in Section \ugUserBlocksNumber\ignore{ugUserBlocks}
+for more information.
+}{
+\spadpaste{f() \free{f1}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsBreakTitle}{break in Loops}
+\newcommand{\ugLangLoopsBreakNumber}{5.4.3.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsBreakPage}{5.4.3. break in Loops}
+% =====================================================================
+\beginscroll
+
+The \axiom{break} keyword is often more useful
+\spadkey{break}
+in terminating
+%-% \HDindex{loop!leaving via break}{ugLangLoopsBreakPage}{5.4.3.}{break in Loops}
+a loop.
+%>  and more in keeping with the ideas of structured programming.
+A \axiom{break} causes control to transfer to the expression
+immediately following the loop.
+As loops always return \void{},
+you cannot return a value with \axiom{break}.
+That is, \axiom{break} takes no argument.
+
+\xtc{
+This example is a modification of the last example in
+\texht{the previous section}{\downlink{``\ugLangLoopsReturnTitle''}{ugLangLoopsReturnPage} in Section \ugLangLoopsReturnNumber\ignore{ugLangLoopsReturn}}.
+Instead of using \axiom{return}, we'll use \axiom{break}.
+}{
+\begin{spadsrc}[\bound{f1}]
+f() ==
+  i := 1
+  repeat
+    if factorial(i) > 1000 then break
+    i := i + 1
+  i
+\end{spadsrc}
+}
+\xtc{
+The loop terminates when \axiom{factorial(i)} gets big enough,
+the last line of the function evaluates to the corresponding ``good''
+value of \axiom{i}, and the function terminates, returning that value.
+}{
+\spadpaste{f() \free{f1}}
+}
+\xtc{
+You can only use \axiom{break} to terminate the evaluation of one loop.
+Let's consider a loop within a loop, that is, a loop with a nested loop.
+First, we initialize two counter variables.
+}{
+\spadpaste{(i,j) := (1, 1) \bound{i}\bound{j}}
+}
+\xtc{
+Nested loops must have multiple \axiom{break}
+%-% \HDindex{loop!nested}{ugLangLoopsBreakPage}{5.4.3.}{break in Loops}
+expressions at the appropriate nesting level.
+How would you rewrite this so \axiom{(i + j) > 10} is only evaluated once?
+}{
+\begin{spadsrc}[\free{i j}]
+repeat
+  repeat
+    if (i + j) > 10 then break
+    j := j + 1
+  if (i + j) > 10 then break
+  i := i + 1
+\end{spadsrc}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsBreakVsTitle}{break vs. {\tt =>} in Loop Bodies}
+\newcommand{\ugLangLoopsBreakVsNumber}{5.4.4.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsBreakVsPage}{5.4.4. break vs. {\tt =>} in Loop Bodies}
+% =====================================================================
+\beginscroll
+
+Compare the following two loops:
+
+\begin{verbatim}
+i := 1                            i := 1
+repeat                            repeat
+  i := i + 1                        i := i + 1
+  i > 3 => i                        if i > 3 then break
+  output(i)                         output(i)
+\end{verbatim}
+
+In the example on the left, the values
+\mathOrSpad{2} and \mathOrSpad{3} for \axiom{i} are displayed
+but then the \axiomSyntax{=>} does not allow control to reach the call to
+\axiomFunFrom{output}{OutputForm} again.
+The loop will not terminate
+until you run out of space or interrupt the execution.
+The variable \axiom{i} will continue to be incremented because
+the \axiomSyntax{=>} only means to leave the {\it block,} not the loop.
+
+In the example on the right,
+upon reaching \mathOrSpad{4}, the \axiom{break} will be
+executed, and both the block and the loop will terminate.
+This is one of the reasons why both \axiomSyntax{=>} and \axiom{break} are
+provided.
+Using a \axiom{while} clause (see below) with the \axiomSyntax{=>}
+\spadkey{while}
+lets you simulate the action of \axiom{break}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsBreakMoreTitle}{More Examples of break}
+\newcommand{\ugLangLoopsBreakMoreNumber}{5.4.5.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsBreakMorePage}{5.4.5. More Examples of break}
+% =====================================================================
+\beginscroll
+
+Here we give four examples of \axiom{repeat} loops that
+terminate when a value exceeds a given bound.
+
+\texht{\vskip 1pc}{}
+\xtc{
+First, initialize \axiom{i} as the loop counter.
+}{
+\spadpaste{i := 0 \bound{i}}
+}
+\xtc{
+Here is the first loop.
+When the square of \axiom{i} exceeds \axiom{100}, the loop terminates.
+}{
+\begin{spadsrc}[\free{i}\bound{i1}]
+repeat
+  i := i + 1
+  if i**2 > 100 then break
+\end{spadsrc}
+}
+\xtc{
+Upon completion, \axiom{i} should have the value \axiom{11}.
+}{
+\spadpaste{i \free{i1}}
+}
+%
+%
+\xtc{
+Do the same thing except use \axiomSyntax{=>} instead
+an \axiom{if-then} expression.
+}{
+\spadpaste{i := 0 \bound{i2}}
+}
+\xtc{
+}{
+\begin{spadsrc}[\free{i2}\bound{i3}]
+repeat
+  i := i + 1
+  i**2 > 100 => break
+\end{spadsrc}
+}
+\xtc{
+}{
+\spadpaste{i \free{i3}}
+}
+%
+%
+\xtc{
+As a third example, we use a simple loop to compute \axiom{n!}.
+}{
+\spadpaste{(n, i, f) := (100, 1, 1) \bound{n}\bound{i4}\bound{f}}
+}
+\xtc{
+Use \axiom{i} as the iteration variable and \axiom{f}
+to compute the factorial.
+}{
+\begin{spadsrc}[\bound{f1}\bound{i5}\free{f i4 n}]
+repeat
+  if i > n then break
+  f := f * i
+  i := i + 1
+\end{spadsrc}
+}
+\xtc{
+Look at the value of \axiom{f}.
+}{
+\spadpaste{f \free{f1}}
+}
+%
+%
+\xtc{
+Finally, we show an example of nested loops.
+First define a four by four matrix.
+}{
+\spadpaste{m := matrix [[21,37,53,14], [8,-24,22,-16], [2,10,15,14], [26,33,55,-13]] \bound{m2}}
+}
+\xtc{
+Next, set row counter \axiom{r} and column counter \axiom{c} to
+\mathOrSpad{1}.
+Note: if we were writing a function, these would all be local
+variables rather than global workspace variables.
+}{
+\spadpaste{(r, c) := (1, 1) \bound{r}\bound{c}}
+}
+\xtc{
+Also, let \axiom{lastrow} and
+\axiom{lastcol} be the final row and column index.
+}{
+\spadpaste{(lastrow, lastcol) := (nrows(m), ncols(m)) \bound{lastrow}\bound{lastcol}\free{m2}}
+}
+%
+\xtc{
+Scan the rows looking for the first negative element.
+We remark that you can reformulate this example in a better, more
+concise form by using a \axiom{for} clause with \axiom{repeat}.
+See
+\downlink{``\ugLangLoopsForInTitle''}{ugLangLoopsForInPage} in Section \ugLangLoopsForInNumber\ignore{ugLangLoopsForIn}
+for more information.
+}{
+\begin{spadsrc}[\free{m2 r c lastrow lastcol}]
+repeat
+  if r > lastrow then break
+  c := 1
+  repeat
+    if c > lastcol then break
+    if elt(m,r,c) < 0 then
+      output [r, c, elt(m,r,c)]
+      r := lastrow
+      break     -- don't look any further
+    c := c + 1
+  r := r + 1
+\end{spadsrc}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsIterateTitle}{iterate in Loops}
+\newcommand{\ugLangLoopsIterateNumber}{5.4.6.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsIteratePage}{5.4.6. iterate in Loops}
+% =====================================================================
+\beginscroll
+
+\Language{} provides an \axiom{iterate} expression that
+\spadkey{iterate}
+skips over the remainder of a loop body and starts the next loop iteration.
+\xtc{
+We first initialize a counter.
+}{
+\spadpaste{i := 0 \bound{i}}
+}
+\xtc{
+Display the even integers from \axiom{2} to \axiom{5}.
+}{
+\begin{spadsrc}[\free{i}]
+repeat
+  i := i + 1
+  if i > 5 then break
+  if odd?(i) then iterate
+  output(i)
+\end{spadsrc}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsWhileTitle}{while Loops}
+\newcommand{\ugLangLoopsWhileNumber}{5.4.7.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsWhilePage}{5.4.7. while Loops}
+% =====================================================================
+\beginscroll
+
+The \axiom{repeat} in a loop can be modified by adding one or
+more \axiom{while} clauses.
+\spadkey{while}
+Each clause contains a \spadgloss{predicate}
+immediately following the \axiom{while} keyword.
+The predicate is tested {\it before}
+the evaluation of the body of the loop.
+The loop body is evaluated whenever the predicates in a \axiom{while}
+clause are all \axiom{true}.
+
+\beginImportant
+The syntax for a simple loop using \axiom{while} is
+\centerline{{\axiom{while} {\it predicate} \axiom{repeat} {\it loopBody}}}
+The {\it predicate} is evaluated before {\it loopBody} is evaluated.
+A \axiom{while} loop terminates immediately when {\it predicate}
+evaluates to \axiom{false} or when a \axiom{break} or \axiom{return}
+expression is evaluated in {\it loopBody}.
+The value returned by the loop is \void{}.
+\endImportant
+
+\xtc{
+Here is a simple example of using \axiom{while} in a loop.
+We first initialize the counter.
+}{
+\spadpaste{i := 1 \bound{i}}
+}
+\xtc{
+The steps involved in computing this example are
+(1) set \axiom{i} to \axiom{1}, (2) test the condition \axiom{i < 1} and
+determine that it is not true, and (3) do not evaluate the
+loop body and therefore do not display \axiom{"hello"}.
+}{
+\begin{spadsrc}[\free{i}]
+while i < 1 repeat
+  output "hello"
+  i := i + 1
+\end{spadsrc}
+}
+\xtc{
+If you have multiple predicates to be tested use the
+logical \axiom{and} operation to separate them.
+\Language{} evaluates these predicates from left to right.
+}{
+\spadpaste{(x, y) := (1, 1) \bound{x}\bound{y}}
+}
+\xtc{
+}{
+\begin{spadsrc}[\free{x y}]
+while x < 4 and y < 10 repeat
+  output [x,y]
+  x := x + 1
+  y := y + 2
+\end{spadsrc}
+}
+\xtc{
+A \axiom{break} expression can be included in a loop body to terminate a
+loop even if the predicate in any \axiom{while} clauses are not \axiom{false}.
+}{
+\spadpaste{(x, y) := (1, 1) \bound{x1}\bound{y1}}
+}
+\xtc{
+This loop has multiple \axiom{while} clauses and the loop terminates
+before any one of their conditions evaluates to \axiom{false}.
+}{
+\begin{spadsrc}[\free{x1 y1}]
+while x < 4 while y < 10 repeat
+  if x + y > 7 then break
+  output [x,y]
+  x := x + 1
+  y := y + 2
+\end{spadsrc}
+}
+\xtc{
+Here's a different version of the nested loops that looked
+for the first negative element in a matrix.
+}{
+\spadpaste{m := matrix [[21,37,53,14], [8,-24,22,-16], [2,10,15,14], [26,33,55,-13]] \bound{m2}}
+}
+\xtc{
+Initialized the row index to \axiom{1} and
+get the number of rows and columns.
+If we were writing a function, these would all be
+local variables.
+}{
+\spadpaste{r := 1 \bound{r}}
+}
+\xtc{
+}{
+\spadpaste{(lastrow, lastcol) := (nrows(m), ncols(m)) \bound{lastrow}\bound{lastcol}\free{m2}}
+}
+%
+\xtc{
+Scan the rows looking for the first negative element.
+}{
+\begin{spadsrc}[\free{m2 r lastrow lastcol}]
+while r <= lastrow repeat
+  c := 1  -- index of first column
+  while c <= lastcol repeat
+    if elt(m,r,c) < 0 then
+      output [r, c, elt(m,r,c)]
+      r := lastrow
+      break     -- don't look any further
+    c := c + 1
+  r := r + 1
+\end{spadsrc}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsForInTitle}{for Loops}
+\newcommand{\ugLangLoopsForInNumber}{5.4.8.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsForInPage}{5.4.8. for Loops}
+% =====================================================================
+\beginscroll
+
+\Language{} provides the \axiom{for}
+\spadkey{for}
+and \axiom{in}
+\spadkey{in}
+keywords in \axiom{repeat} loops,
+allowing you to iterate across all
+%-% \HDindex{iteration}{ugLangLoopsForInPage}{5.4.8.}{for Loops}
+elements of a list, or to have a variable take on integral values
+from a lower bound to an upper bound.
+We shall refer to these modifying clauses of \axiom{repeat} loops as
+\axiom{for} clauses.
+These clauses can be present in addition to \axiom{while} clauses.
+As with all other types of \axiom{repeat} loops, \axiom{break} can
+\spadkey{break}
+be used to prematurely terminate the evaluation of the loop.
+
+\beginImportant
+The syntax for a simple loop using \axiom{for} is
+\centerline{{\axiom{for} {\it iterator} \axiom{repeat} {\it loopBody}}}
+The {\it iterator} has several forms.
+Each form has an end test which is evaluated
+before {\it loopBody} is evaluated.
+A \axiom{for} loop terminates immediately when the end test
+succeeds (evaluates to \axiom{true}) or when a \axiom{break} or \axiom{return}
+expression is evaluated in {\it loopBody}.
+The value returned by the loop is \void{}.
+\endImportant
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsForInNMTitle}{for i in n..m repeat}
+\newcommand{\ugLangLoopsForInNMNumber}{5.4.9.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsForInNMPage}{5.4.9. for i in n..m repeat}
+% =====================================================================
+\beginscroll
+
+If \axiom{for}
+\spadkey{for}
+is followed by a variable name, the \axiom{in}
+\spadkey{in}
+keyword and then an integer segment of the form \axiom{n..m},
+%-% \HDindex{segment}{ugLangLoopsForInNMPage}{5.4.9.}{for i in n..m repeat}
+the end test for this loop is the predicate \axiom{i > m}.
+The body of the loop is evaluated \axiom{m-n+1} times if this
+number is greater than 0.
+If this number is less than or equal to 0, the loop body is not evaluated
+at all.
+
+The variable \axiom{i} has the value
+\axiom{n, n+1, ..., m} for successive iterations
+of the loop body.
+The loop variable is a \spadgloss{local variable}
+within the loop body: its value is not available outside the loop body
+and its value and type within the loop body completely mask any outer
+definition of a variable with the same name.
+
+%
+\xtc{
+This loop prints the values of
+\texht{${10}^3$, ${11}^3$, and $12^3$}{\axiom{10**3, 11**3, and 12**3}}:
+}{
+\spadpaste{for i in 10..12 repeat output(i**3)}
+}
+%
+\xtc{
+Here is a sample list.
+}{
+\spadpaste{a := [1,2,3] \bound{a}}
+}
+\xtc{
+Iterate across this list, using \axiomSyntax{.} to access the elements of a list and
+the \axiomFun{\#} operation to count its elements.
+}{
+\spadpaste{for i in 1..\#a repeat output(a.i) \free{a}}
+}
+%
+This type of iteration is applicable to anything that uses \axiomSyntax{.}.
+You can also use it with functions that use indices to extract elements.
+%
+\xtc{
+Define \axiom{m} to be a matrix.
+}{
+\spadpaste{m := matrix [[1,2],[4,3],[9,0]] \bound{m}}
+}
+\xtc{
+Display the rows of \axiom{m}.
+}{
+\spadpaste{for i in 1..nrows(m) repeat output row(m,i) \free{m}}
+}
+%
+You can use \axiom{iterate} with \axiom{for}-loops.
+\spadkey{iterate}
+\xtc{
+Display the even integers in a segment.
+}{
+\begin{spadsrc}
+for i in 1..5 repeat
+  if odd?(i) then iterate
+  output(i)
+\end{spadsrc}
+}
+
+See \downlink{`Segment'}{SegmentXmpPage}\ignore{Segment} for more information about segments.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsForInNMSTitle}{for i in n..m by s repeat}
+\newcommand{\ugLangLoopsForInNMSNumber}{5.4.10.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsForInNMSPage}{5.4.10. for i in n..m by s repeat}
+% =====================================================================
+\beginscroll
+
+By default, the difference between values taken on by a variable in loops
+such as \axiom{for i in n..m repeat ...} is \mathOrSpad{1}.
+It is possible to supply another, possibly negative, step value by using
+the \axiom{by}
+\spadkey{by}
+keyword along with \axiom{for} and \axiom{in}.
+Like the upper and lower bounds, the step value following the
+\axiom{by} keyword must be an integer.
+Note that the loop
+\axiom{for i in 1..2 by 0 repeat output(i)}
+will not terminate by itself, as the step value does not change the index
+from its initial value of \mathOrSpad{1}.
+
+\xtc{
+This expression displays the odd integers between two bounds.
+}{
+\spadpaste{for i in 1..5 by 2 repeat output(i)}
+}
+\xtc{
+Use this to display the numbers in reverse order.
+}{
+\spadpaste{for i in 5..1 by -2 repeat output(i)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsForInNTitle}{for i in n.. repeat}
+\newcommand{\ugLangLoopsForInNNumber}{5.4.11.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsForInNPage}{5.4.11. for i in n.. repeat}
+% =====================================================================
+\beginscroll
+
+If the value after the \axiomSyntax{..}
+is omitted, the loop has no end test.
+A potentially infinite loop is thus created.
+The variable is given the successive values \axiom{n, n+1, n+2, ...}
+and the loop is terminated only if a \axiom{break} or \axiom{return}
+expression is evaluated in the loop body.
+However you may also add some other modifying clause on the
+\axiom{repeat} (for example, a \axiom{while} clause) to stop the loop.
+
+\xtc{
+This loop displays the integers greater than or equal to \axiom{15}
+and less than the first prime greater than \axiom{15}.
+}{
+\spadpaste{for i in 15.. while not prime?(i) repeat output(i)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsForInXLTitle}{for x in l repeat}
+\newcommand{\ugLangLoopsForInXLNumber}{5.4.12.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsForInXLPage}{5.4.12. for x in l repeat}
+% =====================================================================
+\beginscroll
+
+Another variant of the \axiom{for} loop has the form:
+\centerline{{{\it \axiom{for} x \axiom{in} list \axiom{repeat} loopBody}}}
+This form is used when you want to iterate directly over the
+elements of a list.
+In this form of the \axiom{for} loop, the variable
+\axiom{x} takes on the value of each successive element in \axiom{l}.
+The end test is most simply stated in English: ``are there no more
+\axiom{x} in \axiom{l}?''
+
+\xtc{
+If \axiom{l} is this list,
+}{
+\spadpaste{l := [0,-5,3] \bound{l}}
+}
+\xtc{
+display all elements of \axiom{l}, one per line.
+}{
+\spadpaste{for x in l repeat output(x) \free{l}}
+}
+
+Since the list constructing expression \axiom{expand [n..m]} creates the
+list \axiom{[n, n+1, ..., m]}\footnote{This list is empty if \axiom{n >
+m}.}, you might be tempted to think that the loops
+\begin{verbatim}
+for i in n..m repeat output(i)
+\end{verbatim}
+and
+\begin{verbatim}
+for x in expand [n..m] repeat output(x)
+\end{verbatim}
+are equivalent.
+The second form first creates the list
+\axiom{expand [n..m]} (no matter how large it might be) and
+then does the iteration.
+The first form potentially runs in much less space, as the index variable
+\axiom{i} is simply incremented once per loop and the list is not actually
+created.
+Using the first form is much more efficient.
+%
+\xtc{
+Of course, sometimes you really want to iterate across a specific list.
+This displays each of the factors of \axiom{2400000}.
+}{
+\spadpaste{for f in factors(factor(2400000)) repeat output(f)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsForInPredTitle}{``Such that'' Predicates}
+\newcommand{\ugLangLoopsForInPredNumber}{5.4.13.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsForInPredPage}{5.4.13. ``Such that'' Predicates}
+% =====================================================================
+\beginscroll
+
+A \axiom{for} loop can be followed by a \axiomSyntax{|} and then a
+predicate.
+The predicate qualifies the use of the values from the iterator following
+the \axiom{for}.
+Think of the vertical bar
+\axiomSyntax{|} as the phrase ``such that.''
+\xtc{
+This loop expression
+prints out the integers \axiom{n} in the given segment
+such that \axiom{n} is odd.
+}{
+\spadpaste{for n in 0..4 | odd? n repeat output n}
+}
+
+\beginImportant
+A \axiom{for} loop can also be written
+\centerline{{\axiom{for} {\it iterator} \axiom{|} {\it predicate}  \axiom{repeat} {\it loopBody}}}
+which is equivalent to:
+\centerline{{\axiom{for} {\it iterator} \axiom{repeat if}}}
+\centerline{{{\it predicate} \axiom{then} {\it loopBody} \axiom{else} \axiom{iterate}}}
+\endImportant
+
+The predicate need not refer only to the variable in the \axiom{for} clause:
+any variable in an outer scope can be part of the predicate.
+\xtc{
+In this example, the predicate on the inner \axiom{for} loop uses
+\axiom{i} from the outer loop and the \axiom{j} from the \axiom{for}
+%-% \HDindex{iteration!nested}{ugLangLoopsForInPredPage}{5.4.13.}{``Such that'' Predicates}
+clause that it directly modifies.
+}{
+\begin{spadsrc}
+for i in 1..50 repeat
+  for j in 1..50 | factorial(i+j) < 25 repeat
+    output [i,j]
+\end{spadsrc}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangLoopsParTitle}{Parallel Iteration}
+\newcommand{\ugLangLoopsParNumber}{5.4.14.}
+%
+% =====================================================================
+\begin{page}{ugLangLoopsParPage}{5.4.14. Parallel Iteration}
+% =====================================================================
+\beginscroll
+
+The last example of
+\texht{the previous section}{\downlink{``\ugLangLoopsForInPredTitle''}{ugLangLoopsForInPredPage} in Section \ugLangLoopsForInPredNumber\ignore{ugLangLoopsForInPred}}
+gives an example of
+\spadgloss{nested iteration}: a loop is contained
+%-% \HDindex{iteration!nested}{ugLangLoopsParPage}{5.4.14.}{Parallel Iteration}
+in another loop.
+%-% \HDindex{iteration!parallel}{ugLangLoopsParPage}{5.4.14.}{Parallel Iteration}
+Sometimes you want to iterate across two lists in parallel, or perhaps
+you want to traverse a list while incrementing a variable.
+
+\beginImportant
+The general syntax of a repeat loop is
+\centerline{{{\tt \subscriptIt{iterator}{1} \subscriptIt{iterator}{2} \ldots \subscriptIt{iterator}{N} repeat {\it loopBody}}}}
+where each {\it iterator} is either a \axiom{for} or a \axiom{while} clause.
+The loop terminates immediately when the end test of any {\it iterator}
+succeeds or when a \axiom{break} or \axiom{return} expression is evaluated
+in {\it loopBody}.
+The value returned by the loop is \void{}.
+\endImportant
+
+\xtc{
+Here we write a loop to iterate across
+two lists, computing the sum of the pairwise product
+of elements. Here is the first list.
+}{
+\spadpaste{l := [1,3,5,7] \bound{l}}
+}
+\xtc{
+And the second.
+}{
+\spadpaste{m := [100,200] \bound{m}}
+}
+\xtc{
+The initial value of the sum counter.
+}{
+\spadpaste{sum := 0 \bound{sum}}
+}
+\xtc{
+The last two elements of \axiom{l} are not used in the calculation
+because \axiom{m} has two fewer elements than \axiom{l}.
+}{
+\begin{spadsrc}[\bound{doit}\free{sum l m}]
+for x in l for y in m repeat
+    sum := sum + x*y
+\end{spadsrc}
+}
+\xtc{
+Display the ``dot product.''
+}{
+\spadpaste{sum \free{doit}}
+}
+
+\xtc{
+Next, we write a loop to compute the sum of the products of the loop elements with
+their positions in the loop.
+}{
+\spadpaste{l := [2,3,5,7,11,13,17,19,23,29,31,37] \bound{l1}}
+}
+\xtc{
+The initial sum.
+}{
+\spadpaste{sum := 0 \bound{sum1}}
+}
+\xtc{
+Here looping stops when the list \axiom{l} is exhausted, even though
+the \axiom{for i in 0..} specifies no terminating condition.
+}{
+\spadpaste{for i in 0.. for x in l repeat sum := i * x \bound{doit1}\free{sum1 l1}}
+}
+\xtc{
+Display this weighted sum.
+}{
+\spadpaste{sum \free{doit1}}
+}
+
+When \axiomSyntax{|} is used to qualify any of the \axiom{for} clauses in a
+parallel iteration, the variables in the predicates can be from an outer
+scope or from a \axiom{for} clause in or to the left of a modified clause.
+
+This is correct:
+\begin{verbatim}
+for i in 1..10 repeat
+  for j in 200..300 | odd? (i+j) repeat
+    output [i,j]
+\end{verbatim}
+This is not correct since the variable \axiom{j} has not been
+defined outside the inner loop.
+\begin{verbatim}
+for i in 1..10 | odd? (i+j) repeat  -- wrong, j not defined
+  for j in 200..300 repeat
+    output [i,j]
+\end{verbatim}
+
+%>% *********************************************************************
+%>\head{subsection}{Mixing Loop Modifiers}{ugLangLoopsMix}
+%>% *********************************************************************
+
+\xtc{
+This example shows that it is possible to mix several of the
+%-% \HDindex{loop!mixing modifiers}{ugLangLoopsParPage}{5.4.14.}{Parallel Iteration}
+forms of \axiom{repeat} modifying clauses on a loop.
+}{
+\begin{spadsrc}
+for i in 1..10
+    for j in 151..160 | odd? j
+      while i + j < 160 repeat
+        output [i,j]
+\end{spadsrc}
+}
+%
+Here are useful rules for composing loop expressions:
+\indent{4}
+\beginitems
+\item[1. ] \axiom{while} predicates can only refer to variables that
+are global (or in an outer scope)
+or that are defined in \axiom{for} clauses to the left of the
+predicate.
+\item[2. ] A ``such that'' predicate (something following \axiomSyntax{|})
+must directly follow a \axiom{for} clause and can only refer to
+variables that are global (or in an outer scope)
+or defined in the modified \axiom{for} clause
+or any \axiom{for} clause to the left.
+\enditems
+\indent{0}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangItsTitle}{Creating Lists and Streams with Iterators}
+\newcommand{\ugLangItsNumber}{5.5.}
+%
+% =====================================================================
+\begin{page}{ugLangItsPage}{5.5. Creating Lists and Streams with Iterators}
+% =====================================================================
+\beginscroll
+
+All of what we did for loops in \downlink{``\ugLangLoopsTitle''}{ugLangLoopsPage} in Section \ugLangLoopsNumber\ignore{ugLangLoops}
+%-% \HDindex{iteration}{ugLangItsPage}{5.5.}{Creating Lists and Streams with Iterators}
+can be transformed into expressions that create lists
+%-% \HDindex{list!created by iterator}{ugLangItsPage}{5.5.}{Creating Lists and Streams with Iterators}
+and streams.
+%-% \HDindex{stream!created by iterator}{ugLangItsPage}{5.5.}{Creating Lists and Streams with Iterators}
+The \axiom{repeat,} \axiom{break} or \axiom{iterate} words are not used but
+all the other ideas carry over.
+Before we give you the general rule, here are some examples which
+give you the idea.
+
+\xtc{
+This creates a simple list of the integers from \axiom{1} to \axiom{10}.
+}{
+\spadpaste{list := [i for i in 1..10] \bound{list}}
+}
+\xtc{
+Create a stream of the integers greater than or equal to \axiom{1}.
+}{
+\spadpaste{stream := [i for i in 1..] \bound{stream}}
+}
+\xtc{
+This is a list of the prime integers between \axiom{1} and \axiom{10},
+inclusive.
+}{
+\spadpaste{[i for i in 1..10 | prime? i]}
+}
+\xtc{
+This is a stream of the prime integers greater than or equal to \axiom{1}.
+}{
+\spadpaste{[i for i in 1..   | prime? i]}
+}
+\xtc{
+This is a list of the integers between \axiom{1} and \axiom{10},
+inclusive, whose squares are less than \axiom{700}.
+}{
+\spadpaste{[i for i in 1..10 while i*i < 700]}
+}
+\xtc{
+This is a stream of the integers greater than or equal to \axiom{1}
+whose squares are less than \axiom{700}.
+}{
+\spadpaste{[i for i in 1..   while i*i < 700]}
+}
+
+Got the idea?
+Here is the general rule.
+%-% \HDindex{collection}{ugLangItsPage}{5.5.}{Creating Lists and Streams with Iterators}
+
+\beginImportant
+The general syntax of a collection is
+\centerline{{{\tt [ {\it collectExpression} \subscriptIt{iterator}{1}  \subscriptIt{iterator}{2}  \ldots  \subscriptIt{iterator}{N} ]}}}
+where each \subscriptIt{iterator}{i} is either a \axiom{for} or a
+\axiom{while} clause.
+The loop terminates immediately when the end test of any
+\subscriptIt{iterator}{i} succeeds or when a \axiom{return} expression is
+evaluated in {\it collectExpression}.
+The value returned by the collection is either a list or a stream of
+elements, one for each iteration of the {\it collectExpression}.
+\endImportant
+
+Be careful when you use \axiom{while}
+%-% \HDindex{stream!using while @{using {\tt while}}}{ugLangItsPage}{5.5.}{Creating Lists and Streams with Iterators}
+to create a stream.
+By default, \Language{} tries to compute and display the first ten elements
+of a stream.
+If the \axiom{while} condition is not satisfied quickly, \Language{}
+can spend a long (possibly infinite) time trying to compute
+%-% \HDindex{stream!number of elements computed}{ugLangItsPage}{5.5.}{Creating Lists and Streams with Iterators}
+the elements.
+Use \spadcmd{)set streams calculate} to change the default
+to something else.
+%-% \HDsyscmdindex{set streams calculate}{ugLangItsPage}{5.5.}{Creating Lists and Streams with Iterators}
+This also affects the number of terms computed and displayed for power
+series.
+For the purposes of this book, we have used this system
+command to display fewer than ten terms.
+\xtc{
+Use nested iterators to create lists of
+%-% \HDindex{iteration!nested}{ugLangItsPage}{5.5.}{Creating Lists and Streams with Iterators}
+lists which can then be given as an argument to \axiomFun{matrix}.
+}{
+\spadpaste{matrix [[x**i+j for i in 1..3] for j in 10..12]}
+}
+\xtc{
+You can also create lists of streams, streams of lists and
+streams of streams.
+Here is a stream of streams.
+}{
+\spadpaste{[[i/j for i in j+1..] for j in 1..]}
+}
+\xtc{
+You can use parallel iteration across lists and streams to create
+%-% \HDindex{iteration!parallel}{ugLangItsPage}{5.5.}{Creating Lists and Streams with Iterators}
+new lists.
+}{
+\spadpaste{[i/j for i in 3.. by 10 for j in 2..]}
+}
+\xtc{
+Iteration stops if the end of a list or stream is reached.
+}{
+\spadpaste{[i**j for i in 1..7 for j in 2.. ]}
+}
+%\xtc{
+%or a while condition fails.
+%}{
+%\spadcommand{[i**j for i in 1..  for j in 2.. while i + j < 5 ]}
+%}
+\xtc{
+As with loops, you can combine these modifiers to make very
+complicated conditions.
+}{
+\spadpaste{[[[i,j] for i in 10..15 | prime? i] for j in 17..22 | j = squareFreePart j]}
+}
+
+See \downlink{`List'}{ListXmpPage}\ignore{List} and \downlink{`Stream'}{StreamXmpPage}\ignore{Stream} for more information on creating and
+manipulating lists and streams, respectively.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugLangStreamsPrimesTitle}{An Example: Streams of Primes}
+\newcommand{\ugLangStreamsPrimesNumber}{5.6.}
+%
+% =====================================================================
+\begin{page}{ugLangStreamsPrimesPage}{5.6. An Example: Streams of Primes}
+% =====================================================================
+\beginscroll
+
+We conclude this chapter with an example of the creation and manipulation
+of infinite streams of prime integers.
+This might be useful for experiments with numbers or other applications
+where you are using sequences of primes over and over again.
+As for all streams, the stream of primes is only computed as far out as you
+need.
+Once computed, however, all the primes up to that point are saved for
+future reference.
+
+Two useful operations provided by the \Language{} library are
+\axiomFunFrom{prime?}{IntegerPrimesPackage} and
+\axiomFunFrom{nextPrime}{IntegerPrimesPackage}.
+A straight-forward way to create a stream of
+prime numbers is to start with the stream of positive integers \axiom{[2,..]} and
+filter out those that are prime.
+\xtc{
+Create a stream of primes.
+}{
+\spadpaste{primes : Stream Integer := [i for i in 2.. | prime? i]}
+}
+A more elegant way, however, is to use the \axiomFunFrom{generate}{Stream}
+operation from \axiomType{Stream}.
+Given an initial value \axiom{a} and a function \axiom{f},
+\axiomFunFrom{generate}{Stream}
+constructs the stream \axiom{[a, f(a), f(f(a)), ...]}.
+This function gives you the quickest method of getting the stream of primes.
+\xtc{
+This is how you use
+\axiomFunFrom{generate}{Stream} to
+generate an infinite stream of primes.
+}{
+\spadpaste{primes := generate(nextPrime,2)}
+}
+\xtc{
+Once the stream is generated, you might only be interested in
+primes starting at a particular value.
+}{
+\spadpaste{smallPrimes := [p for p in primes | p > 1000] \bound{smallPrimes}}
+}
+\xtc{
+Here are the first 11 primes greater than 1000.
+}{
+\spadpaste{[p for p in smallPrimes for i in 1..11] \free{smallPrimes}}
+}
+\xtc{
+Here is a stream of primes between 1000 and 1200.
+}{
+\spadpaste{[p for p in smallPrimes while p < 1200] \free{smallPrimes}}
+}
+\xtc{
+To get these expanded into a finite stream,
+you call \axiomFunFrom{complete}{Stream} on the stream.
+}{
+\spadpaste{complete \%}
+}
+\xtc{
+Twin primes are consecutive odd number pairs which are prime.
+Here is the stream of twin primes.
+}{
+\spadpaste{twinPrimes := [[p,p+2] for p in primes | prime?(p + 2)]}
+}
+\xtc{
+Since we already have the primes computed we can
+avoid the call to \axiomFunFrom{prime?}{IntegerPrimesPackage}
+by using a double iteration.
+This time we'll just generate a stream of the first of the twin primes.
+}{
+\spadpaste{firstOfTwins:= [p for p in primes for q in rest primes | q=p+2]}
+}
+
+Let's try to compute the infinite stream of triplet primes,
+the set of primes \axiom{p} such that \axiom{[p,p+2,p+4]}
+are primes. For example, \axiom{[3,5,7]} is a triple prime.
+We could do this by a triple \axiom{for} iteration.
+A more economical way is to use \userfun{firstOfTwins}.
+This time however, put a semicolon at the end of the line.
+
+\xtc{Create the stream of firstTriplets.
+Put a semicolon at the end so that no
+elements are computed.
+}{
+\spadpaste{firstTriplets := [p for p in firstOfTwins for q in rest firstOfTwins | q = p+2];}
+}
+
+What happened?
+As you know, by default
+\Language{} displays the first ten
+elements of a stream when you first display it.
+And, therefore, it needs to compute them!
+If you want {\it no} elements computed, just terminate the expression by a
+semicolon (\axiomSyntax{;}).\footnote{
+Why does this happen? The semi-colon prevents the display of the
+result of evaluating the expression.
+Since no stream elements are needed for display (or anything else, so far),
+none are computed.
+}
+
+\xtc{
+Compute the first triplet prime.
+}{
+\spadpaste{firstTriplets.1}
+}
+
+If you want to compute another, just ask for it.
+But wait a second!
+Given three consecutive odd integers, one of them must be divisible
+by 3. Thus there is only one triplet prime.
+But suppose that you did not know this and wanted to know what was the
+tenth triplet prime.
+\begin{verbatim}
+firstTriples.10
+\end{verbatim}
+To compute the tenth triplet prime, \Language{} first must compute the second,
+the third, and so on.
+But since there isn't even a second triplet prime, \Language{} will
+compute forever.
+Nonetheless, this effort can produce a useful result.
+After waiting a bit, hit
+\texht{\fbox{\bf Ctrl}--\fbox{\bf c}}{{\bf Ctrl-c}}.
+The system responds as follows.
+\begin{verbatim}
+   >> System error:
+   Console interrupt.
+   You are being returned to the top level of
+   the interpreter.
+\end{verbatim}
+Let's say that you want to know how many primes have been computed.
+Issue
+\begin{verbatim}
+numberOfComputedEntries primes
+\end{verbatim}
+and, for this discussion, let's say that the result is \axiom{2045.}
+\xtc{
+How big is the \eth{\axiom{2045}} prime?
+}{
+\spadpaste{primes.2045}
+}
+
+What you have learned is that there are no triplet primes between 5 and
+17837.
+Although this result is well known (some might even say trivial), there
+are many experiments you could make where the result is not known.
+What you see here is a paradigm for testing of hypotheses.
+Here our hypothesis could have been: ``there is more than one triplet
+prime.''
+We have tested this hypothesis for 17837 cases.
+With streams, you can let your machine run, interrupt it to see how far
+it has progressed,
+then start it up and let it continue from where it left off.
+
+%> RDJ note to RSS:
+%> Expressions not statements or lines--
+%>   By an expression I mean any syntactically correct program fragment.
+%>   Everything in AXIOM is an expression since every fragment has a value and a type.
+%>   In most languages including LISP, a "statement" is different from an expression:
+%>   it is executed for side-effect only and an error is incurred if you assign it a value.
+%>   This "gimmick" takes care of incomplete expressions such as "if x > 0 then y" in blocks.
+%>   In LISP, "u := (if x > 0 then y)" is illegal but in AXIOM it is legal.
+%>   Also, in AXIOM the value of a repeat loop is void even though you might be
+%>   be able to prove that it always returns a valid value (you have an example of this)!
+%>   This will be considered a bug not a feature. But it is how things stand.
+%>   In any case---this point should be in a box somewhere since it is key
+%>   to a user's understanding to the language. I am not sure where. You only
+%>   gain an appreciation for it after are awhile in chapter 5.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug05.pht b/src/hyper/pages/ug05.pht
new file mode 100644
index 00000000..78695934
--- /dev/null
+++ b/src/hyper/pages/ug05.pht
@@ -0,0 +1,2002 @@
+\begin{patch}{ugLangLoopsWhilePagePatch1}
+\begin{paste}{ugLangLoopsWhilePageFull1}{ugLangLoopsWhilePageEmpty1}
+\pastebutton{ugLangLoopsWhilePageFull1}{\hidepaste}
+\tab{5}\spadcommand{i := 1\bound{i }}
+\indentrel{3}\begin{verbatim}
+   (1)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePageEmpty1}
+\begin{paste}{ugLangLoopsWhilePageEmpty1}{ugLangLoopsWhilePagePatch1}
+\pastebutton{ugLangLoopsWhilePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{i := 1\bound{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePagePatch2}
+\begin{paste}{ugLangLoopsWhilePageFull2}{ugLangLoopsWhilePageEmpty2}
+\pastebutton{ugLangLoopsWhilePageFull2}{\hidepaste}
+\tab{5}\spadcommand{while i < 1 repeat
+  output "hello"
+  i := i + 1
+\free{i }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePageEmpty2}
+\begin{paste}{ugLangLoopsWhilePageEmpty2}{ugLangLoopsWhilePagePatch2}
+\pastebutton{ugLangLoopsWhilePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{while i < 1 repeat
+  output "hello"
+  i := i + 1
+\free{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePagePatch3}
+\begin{paste}{ugLangLoopsWhilePageFull3}{ugLangLoopsWhilePageEmpty3}
+\pastebutton{ugLangLoopsWhilePageFull3}{\hidepaste}
+\tab{5}\spadcommand{(x, y) := (1, 1)\bound{x }\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (3)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePageEmpty3}
+\begin{paste}{ugLangLoopsWhilePageEmpty3}{ugLangLoopsWhilePagePatch3}
+\pastebutton{ugLangLoopsWhilePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{(x, y) := (1, 1)\bound{x }\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePagePatch4}
+\begin{paste}{ugLangLoopsWhilePageFull4}{ugLangLoopsWhilePageEmpty4}
+\pastebutton{ugLangLoopsWhilePageFull4}{\hidepaste}
+\tab{5}\spadcommand{while x < 4 and y < 10 repeat
+  output [x,y]
+  x := x + 1
+  y := y + 2
+\free{x y }}
+\indentrel{3}\begin{verbatim}
+   [1,1]
+   [2,3]
+   [3,5]
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePageEmpty4}
+\begin{paste}{ugLangLoopsWhilePageEmpty4}{ugLangLoopsWhilePagePatch4}
+\pastebutton{ugLangLoopsWhilePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{while x < 4 and y < 10 repeat
+  output [x,y]
+  x := x + 1
+  y := y + 2
+\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePagePatch5}
+\begin{paste}{ugLangLoopsWhilePageFull5}{ugLangLoopsWhilePageEmpty5}
+\pastebutton{ugLangLoopsWhilePageFull5}{\hidepaste}
+\tab{5}\spadcommand{(x, y) := (1, 1)\bound{x1 }\bound{y1 }}
+\indentrel{3}\begin{verbatim}
+   (5)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePageEmpty5}
+\begin{paste}{ugLangLoopsWhilePageEmpty5}{ugLangLoopsWhilePagePatch5}
+\pastebutton{ugLangLoopsWhilePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{(x, y) := (1, 1)\bound{x1 }\bound{y1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePagePatch6}
+\begin{paste}{ugLangLoopsWhilePageFull6}{ugLangLoopsWhilePageEmpty6}
+\pastebutton{ugLangLoopsWhilePageFull6}{\hidepaste}
+\tab{5}\spadcommand{while x < 4 while y < 10 repeat
+  if x + y > 7 then break
+  output [x,y]
+  x := x + 1
+  y := y + 2
+\free{x1 y1 }}
+\indentrel{3}\begin{verbatim}
+   [1,1]
+   [2,3]
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePageEmpty6}
+\begin{paste}{ugLangLoopsWhilePageEmpty6}{ugLangLoopsWhilePagePatch6}
+\pastebutton{ugLangLoopsWhilePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{while x < 4 while y < 10 repeat
+  if x + y > 7 then break
+  output [x,y]
+  x := x + 1
+  y := y + 2
+\free{x1 y1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePagePatch7}
+\begin{paste}{ugLangLoopsWhilePageFull7}{ugLangLoopsWhilePageEmpty7}
+\pastebutton{ugLangLoopsWhilePageFull7}{\hidepaste}
+\tab{5}\spadcommand{m := matrix [[21,37,53,14], [8,-24,22,-16], [2,10,15,14], [26,33,55,-13]]\bound{m2 }}
+\indentrel{3}\begin{verbatim}
+        Ú21   37   53   14 ¿
+        ³                  ³
+        ³8   - 24  22  - 16³
+   (7)  ³                  ³
+        ³2    10   15   14 ³
+        ³                  ³
+        À26   33   55  - 13Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePageEmpty7}
+\begin{paste}{ugLangLoopsWhilePageEmpty7}{ugLangLoopsWhilePagePatch7}
+\pastebutton{ugLangLoopsWhilePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{m := matrix [[21,37,53,14], [8,-24,22,-16], [2,10,15,14], [26,33,55,-13]]\bound{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePagePatch8}
+\begin{paste}{ugLangLoopsWhilePageFull8}{ugLangLoopsWhilePageEmpty8}
+\pastebutton{ugLangLoopsWhilePageFull8}{\hidepaste}
+\tab{5}\spadcommand{r := 1\bound{r }}
+\indentrel{3}\begin{verbatim}
+   (8)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePageEmpty8}
+\begin{paste}{ugLangLoopsWhilePageEmpty8}{ugLangLoopsWhilePagePatch8}
+\pastebutton{ugLangLoopsWhilePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{r := 1\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePagePatch9}
+\begin{paste}{ugLangLoopsWhilePageFull9}{ugLangLoopsWhilePageEmpty9}
+\pastebutton{ugLangLoopsWhilePageFull9}{\hidepaste}
+\tab{5}\spadcommand{(lastrow, lastcol) := (nrows(m), ncols(m))\bound{lastrow }\bound{lastcol }\free{m2 }}
+\indentrel{3}\begin{verbatim}
+   (9)  4
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePageEmpty9}
+\begin{paste}{ugLangLoopsWhilePageEmpty9}{ugLangLoopsWhilePagePatch9}
+\pastebutton{ugLangLoopsWhilePageEmpty9}{\showpaste}
+\tab{5}\spadcommand{(lastrow, lastcol) := (nrows(m), ncols(m))\bound{lastrow }\bound{lastcol }\free{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePagePatch10}
+\begin{paste}{ugLangLoopsWhilePageFull10}{ugLangLoopsWhilePageEmpty10}
+\pastebutton{ugLangLoopsWhilePageFull10}{\hidepaste}
+\tab{5}\spadcommand{while r <= lastrow repeat
+  c := 1  -- index of first column
+  while c <= lastcol repeat
+    if elt(m,r,c) < 0 then
+      output [r, c, elt(m,r,c)]
+      r := lastrow
+      break     -- don't look any further
+    c := c + 1
+  r := r + 1
+\free{m2 r lastrow lastcol }}
+\indentrel{3}\begin{verbatim}
+   [2,2,- 24]
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsWhilePageEmpty10}
+\begin{paste}{ugLangLoopsWhilePageEmpty10}{ugLangLoopsWhilePagePatch10}
+\pastebutton{ugLangLoopsWhilePageEmpty10}{\showpaste}
+\tab{5}\spadcommand{while r <= lastrow repeat
+  c := 1  -- index of first column
+  while c <= lastcol repeat
+    if elt(m,r,c) < 0 then
+      output [r, c, elt(m,r,c)]
+      r := lastrow
+      break     -- don't look any further
+    c := c + 1
+  r := r + 1
+\free{m2 r lastrow lastcol }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNPagePatch1}
+\begin{paste}{ugLangLoopsForInNPageFull1}{ugLangLoopsForInNPageEmpty1}
+\pastebutton{ugLangLoopsForInNPageFull1}{\hidepaste}
+\tab{5}\spadcommand{for i in 15.. while not prime?(i) repeat output(i)}
+\indentrel{3}\begin{verbatim}
+   15
+   16
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNPageEmpty1}
+\begin{paste}{ugLangLoopsForInNPageEmpty1}{ugLangLoopsForInNPagePatch1}
+\pastebutton{ugLangLoopsForInNPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{for i in 15.. while not prime?(i) repeat output(i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMPagePatch1}
+\begin{paste}{ugLangLoopsForInNMPageFull1}{ugLangLoopsForInNMPageEmpty1}
+\pastebutton{ugLangLoopsForInNMPageFull1}{\hidepaste}
+\tab{5}\spadcommand{for i in 10..12 repeat output(i**3)}
+\indentrel{3}\begin{verbatim}
+   1000
+   1331
+   1728
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMPageEmpty1}
+\begin{paste}{ugLangLoopsForInNMPageEmpty1}{ugLangLoopsForInNMPagePatch1}
+\pastebutton{ugLangLoopsForInNMPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{for i in 10..12 repeat output(i**3)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMPagePatch2}
+\begin{paste}{ugLangLoopsForInNMPageFull2}{ugLangLoopsForInNMPageEmpty2}
+\pastebutton{ugLangLoopsForInNMPageFull2}{\hidepaste}
+\tab{5}\spadcommand{a := [1,2,3]\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (2)  [1,2,3]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMPageEmpty2}
+\begin{paste}{ugLangLoopsForInNMPageEmpty2}{ugLangLoopsForInNMPagePatch2}
+\pastebutton{ugLangLoopsForInNMPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{a := [1,2,3]\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMPagePatch3}
+\begin{paste}{ugLangLoopsForInNMPageFull3}{ugLangLoopsForInNMPageEmpty3}
+\pastebutton{ugLangLoopsForInNMPageFull3}{\hidepaste}
+\tab{5}\spadcommand{for i in 1..\#a repeat output(a.i)\free{a }}
+\indentrel{3}\begin{verbatim}
+   1
+   2
+   3
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMPageEmpty3}
+\begin{paste}{ugLangLoopsForInNMPageEmpty3}{ugLangLoopsForInNMPagePatch3}
+\pastebutton{ugLangLoopsForInNMPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{for i in 1..\#a repeat output(a.i)\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMPagePatch4}
+\begin{paste}{ugLangLoopsForInNMPageFull4}{ugLangLoopsForInNMPageEmpty4}
+\pastebutton{ugLangLoopsForInNMPageFull4}{\hidepaste}
+\tab{5}\spadcommand{m := matrix [[1,2],[4,3],[9,0]]\bound{m }}
+\indentrel{3}\begin{verbatim}
+        Ú1  2¿
+        ³    ³
+   (4)  ³4  3³
+        ³    ³
+        À9  0Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMPageEmpty4}
+\begin{paste}{ugLangLoopsForInNMPageEmpty4}{ugLangLoopsForInNMPagePatch4}
+\pastebutton{ugLangLoopsForInNMPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{m := matrix [[1,2],[4,3],[9,0]]\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMPagePatch5}
+\begin{paste}{ugLangLoopsForInNMPageFull5}{ugLangLoopsForInNMPageEmpty5}
+\pastebutton{ugLangLoopsForInNMPageFull5}{\hidepaste}
+\tab{5}\spadcommand{for i in 1..nrows(m) repeat output row(m,i)\free{m }}
+\indentrel{3}\begin{verbatim}
+   [1,2]
+   [4,3]
+   [9,0]
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMPageEmpty5}
+\begin{paste}{ugLangLoopsForInNMPageEmpty5}{ugLangLoopsForInNMPagePatch5}
+\pastebutton{ugLangLoopsForInNMPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{for i in 1..nrows(m) repeat output row(m,i)\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMPagePatch6}
+\begin{paste}{ugLangLoopsForInNMPageFull6}{ugLangLoopsForInNMPageEmpty6}
+\pastebutton{ugLangLoopsForInNMPageFull6}{\hidepaste}
+\tab{5}\spadcommand{for i in 1..5 repeat
+  if odd?(i) then iterate
+  output(i)
+}
+\indentrel{3}\begin{verbatim}
+   2
+   4
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMPageEmpty6}
+\begin{paste}{ugLangLoopsForInNMPageEmpty6}{ugLangLoopsForInNMPagePatch6}
+\pastebutton{ugLangLoopsForInNMPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{for i in 1..5 repeat
+  if odd?(i) then iterate
+  output(i)
+}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsReturnPagePatch1}
+\begin{paste}{ugLangLoopsReturnPageFull1}{ugLangLoopsReturnPageEmpty1}
+\pastebutton{ugLangLoopsReturnPageFull1}{\hidepaste}
+\tab{5}\spadcommand{f() ==
+  i := 1
+  repeat
+    if factorial(i) > 1000 then return i
+    i := i + 1
+\bound{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsReturnPageEmpty1}
+\begin{paste}{ugLangLoopsReturnPageEmpty1}{ugLangLoopsReturnPagePatch1}
+\pastebutton{ugLangLoopsReturnPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{f() ==
+  i := 1
+  repeat
+    if factorial(i) > 1000 then return i
+    i := i + 1
+\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsReturnPagePatch2}
+\begin{paste}{ugLangLoopsReturnPageFull2}{ugLangLoopsReturnPageEmpty2}
+\pastebutton{ugLangLoopsReturnPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f()\free{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsReturnPageEmpty2}
+\begin{paste}{ugLangLoopsReturnPageEmpty2}{ugLangLoopsReturnPagePatch2}
+\pastebutton{ugLangLoopsReturnPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f()\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsReturnPagePatch3}
+\begin{paste}{ugLangLoopsReturnPageFull3}{ugLangLoopsReturnPageEmpty3}
+\pastebutton{ugLangLoopsReturnPageFull3}{\hidepaste}
+\tab{5}\spadcommand{f() ==
+  i := 1
+  repeat
+    if factorial(i) > 1000 then return i
+    i := i + 1
+  0
+\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsReturnPageEmpty3}
+\begin{paste}{ugLangLoopsReturnPageEmpty3}{ugLangLoopsReturnPagePatch3}
+\pastebutton{ugLangLoopsReturnPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f() ==
+  i := 1
+  repeat
+    if factorial(i) > 1000 then return i
+    i := i + 1
+  0
+\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsReturnPagePatch4}
+\begin{paste}{ugLangLoopsReturnPageFull4}{ugLangLoopsReturnPageEmpty4}
+\pastebutton{ugLangLoopsReturnPageFull4}{\hidepaste}
+\tab{5}\spadcommand{f()\free{f1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  7
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsReturnPageEmpty4}
+\begin{paste}{ugLangLoopsReturnPageEmpty4}{ugLangLoopsReturnPagePatch4}
+\pastebutton{ugLangLoopsReturnPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{f()\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInXLPagePatch1}
+\begin{paste}{ugLangLoopsForInXLPageFull1}{ugLangLoopsForInXLPageEmpty1}
+\pastebutton{ugLangLoopsForInXLPageFull1}{\hidepaste}
+\tab{5}\spadcommand{l := [0,-5,3]\bound{l }}
+\indentrel{3}\begin{verbatim}
+   (1)  [0,- 5,3]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInXLPageEmpty1}
+\begin{paste}{ugLangLoopsForInXLPageEmpty1}{ugLangLoopsForInXLPagePatch1}
+\pastebutton{ugLangLoopsForInXLPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{l := [0,-5,3]\bound{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInXLPagePatch2}
+\begin{paste}{ugLangLoopsForInXLPageFull2}{ugLangLoopsForInXLPageEmpty2}
+\pastebutton{ugLangLoopsForInXLPageFull2}{\hidepaste}
+\tab{5}\spadcommand{for x in l repeat output(x)\free{l }}
+\indentrel{3}\begin{verbatim}
+   0
+   - 5
+   3
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInXLPageEmpty2}
+\begin{paste}{ugLangLoopsForInXLPageEmpty2}{ugLangLoopsForInXLPagePatch2}
+\pastebutton{ugLangLoopsForInXLPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{for x in l repeat output(x)\free{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInXLPagePatch3}
+\begin{paste}{ugLangLoopsForInXLPageFull3}{ugLangLoopsForInXLPageEmpty3}
+\pastebutton{ugLangLoopsForInXLPageFull3}{\hidepaste}
+\tab{5}\spadcommand{for f in factors(factor(2400000)) repeat output(f)}
+\indentrel{3}\begin{verbatim}
+   [factor= 2,exponent= 8]
+   [factor= 3,exponent= 1]
+   [factor= 5,exponent= 5]
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInXLPageEmpty3}
+\begin{paste}{ugLangLoopsForInXLPageEmpty3}{ugLangLoopsForInXLPagePatch3}
+\pastebutton{ugLangLoopsForInXLPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{for f in factors(factor(2400000)) repeat output(f)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangIfPagePatch1}
+\begin{paste}{ugLangIfPageFull1}{ugLangIfPageEmpty1}
+\pastebutton{ugLangIfPageFull1}{\hidepaste}
+\tab{5}\spadcommand{x + 1 = y}
+\indentrel{3}\begin{verbatim}
+   (1)  x + 1= y
+                      Type: Equation Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangIfPageEmpty1}
+\begin{paste}{ugLangIfPageEmpty1}{ugLangIfPagePatch1}
+\pastebutton{ugLangIfPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{x + 1 = y}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMSPagePatch1}
+\begin{paste}{ugLangLoopsForInNMSPageFull1}{ugLangLoopsForInNMSPageEmpty1}
+\pastebutton{ugLangLoopsForInNMSPageFull1}{\hidepaste}
+\tab{5}\spadcommand{for i in 1..5 by 2 repeat output(i)}
+\indentrel{3}\begin{verbatim}
+   1
+   3
+   5
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMSPageEmpty1}
+\begin{paste}{ugLangLoopsForInNMSPageEmpty1}{ugLangLoopsForInNMSPagePatch1}
+\pastebutton{ugLangLoopsForInNMSPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{for i in 1..5 by 2 repeat output(i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMSPagePatch2}
+\begin{paste}{ugLangLoopsForInNMSPageFull2}{ugLangLoopsForInNMSPageEmpty2}
+\pastebutton{ugLangLoopsForInNMSPageFull2}{\hidepaste}
+\tab{5}\spadcommand{for i in 5..1 by -2 repeat output(i)}
+\indentrel{3}\begin{verbatim}
+   5
+   3
+   1
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInNMSPageEmpty2}
+\begin{paste}{ugLangLoopsForInNMSPageEmpty2}{ugLangLoopsForInNMSPagePatch2}
+\pastebutton{ugLangLoopsForInNMSPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{for i in 5..1 by -2 repeat output(i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsIteratePagePatch1}
+\begin{paste}{ugLangLoopsIteratePageFull1}{ugLangLoopsIteratePageEmpty1}
+\pastebutton{ugLangLoopsIteratePageFull1}{\hidepaste}
+\tab{5}\spadcommand{i := 0\bound{i }}
+\indentrel{3}\begin{verbatim}
+   (1)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsIteratePageEmpty1}
+\begin{paste}{ugLangLoopsIteratePageEmpty1}{ugLangLoopsIteratePagePatch1}
+\pastebutton{ugLangLoopsIteratePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{i := 0\bound{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsIteratePagePatch2}
+\begin{paste}{ugLangLoopsIteratePageFull2}{ugLangLoopsIteratePageEmpty2}
+\pastebutton{ugLangLoopsIteratePageFull2}{\hidepaste}
+\tab{5}\spadcommand{repeat
+  i := i + 1
+  if i > 5 then break
+  if odd?(i) then iterate
+  output(i)
+\free{i }}
+\indentrel{3}\begin{verbatim}
+   2
+   4
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsIteratePageEmpty2}
+\begin{paste}{ugLangLoopsIteratePageEmpty2}{ugLangLoopsIteratePagePatch2}
+\pastebutton{ugLangLoopsIteratePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{repeat
+  i := i + 1
+  if i > 5 then break
+  if odd?(i) then iterate
+  output(i)
+\free{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPagePatch1}
+\begin{paste}{ugLangItsPageFull1}{ugLangItsPageEmpty1}
+\pastebutton{ugLangItsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{list := [i for i in 1..10]\bound{list }}
+\indentrel{3}\begin{verbatim}
+   (1)  [1,2,3,4,5,6,7,8,9,10]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPageEmpty1}
+\begin{paste}{ugLangItsPageEmpty1}{ugLangItsPagePatch1}
+\pastebutton{ugLangItsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{list := [i for i in 1..10]\bound{list }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPagePatch2}
+\begin{paste}{ugLangItsPageFull2}{ugLangItsPageEmpty2}
+\pastebutton{ugLangItsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{stream := [i for i in 1..]\bound{stream }}
+\indentrel{3}\begin{verbatim}
+   (2)  [1,2,3,4,5,6,7,8,9,10,...]
+                           Type: Stream PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPageEmpty2}
+\begin{paste}{ugLangItsPageEmpty2}{ugLangItsPagePatch2}
+\pastebutton{ugLangItsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{stream := [i for i in 1..]\bound{stream }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPagePatch3}
+\begin{paste}{ugLangItsPageFull3}{ugLangItsPageEmpty3}
+\pastebutton{ugLangItsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{[i for i in 1..10 | prime? i]}
+\indentrel{3}\begin{verbatim}
+   (3)  [2,3,5,7]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPageEmpty3}
+\begin{paste}{ugLangItsPageEmpty3}{ugLangItsPagePatch3}
+\pastebutton{ugLangItsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{[i for i in 1..10 | prime? i]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPagePatch4}
+\begin{paste}{ugLangItsPageFull4}{ugLangItsPageEmpty4}
+\pastebutton{ugLangItsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{[i for i in 1.. | prime? i]}
+\indentrel{3}\begin{verbatim}
+   (4)  [2,3,5,7,11,13,17,19,23,29,...]
+                           Type: Stream PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPageEmpty4}
+\begin{paste}{ugLangItsPageEmpty4}{ugLangItsPagePatch4}
+\pastebutton{ugLangItsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{[i for i in 1.. | prime? i]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPagePatch5}
+\begin{paste}{ugLangItsPageFull5}{ugLangItsPageEmpty5}
+\pastebutton{ugLangItsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{[i for i in 1..10 while i*i < 700]}
+\indentrel{3}\begin{verbatim}
+   (5)  [1,2,3,4,5,6,7,8,9,10]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPageEmpty5}
+\begin{paste}{ugLangItsPageEmpty5}{ugLangItsPagePatch5}
+\pastebutton{ugLangItsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{[i for i in 1..10 while i*i < 700]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPagePatch6}
+\begin{paste}{ugLangItsPageFull6}{ugLangItsPageEmpty6}
+\pastebutton{ugLangItsPageFull6}{\hidepaste}
+\tab{5}\spadcommand{[i for i in 1.. while i*i < 700]}
+\indentrel{3}\begin{verbatim}
+   (6)  [1,2,3,4,5,6,7,8,9,10,...]
+                           Type: Stream PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPageEmpty6}
+\begin{paste}{ugLangItsPageEmpty6}{ugLangItsPagePatch6}
+\pastebutton{ugLangItsPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{[i for i in 1.. while i*i < 700]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPagePatch7}
+\begin{paste}{ugLangItsPageFull7}{ugLangItsPageEmpty7}
+\pastebutton{ugLangItsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{matrix [[x**i+j for i in 1..3] for j in 10..12]}
+\indentrel{3}\begin{verbatim}
+        Ú         2        3     ¿
+        ³x + 10  x  + 10  x  + 10³
+        ³                        ³
+   (7)  ³         2        3     ³
+        ³x + 11  x  + 11  x  + 11³
+        ³                        ³
+        ³         2        3     ³
+        Àx + 12  x  + 12  x  + 12Ù
+                        Type: Matrix Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPageEmpty7}
+\begin{paste}{ugLangItsPageEmpty7}{ugLangItsPagePatch7}
+\pastebutton{ugLangItsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{matrix [[x**i+j for i in 1..3] for j in 10..12]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPagePatch8}
+\begin{paste}{ugLangItsPageFull8}{ugLangItsPageEmpty8}
+\pastebutton{ugLangItsPageFull8}{\hidepaste}
+\tab{5}\spadcommand{[[i/j for i in j+1..] for j in 1..]}
+\indentrel{3}\begin{verbatim}
+   (8)
+   [[2,3,4,5,6,7,8,9,10,11,...],
+     3   5   7   9   11
+    [Ä,2,Ä,3,Ä,4,Ä,5,ÄÄ,6,...],
+     2   2   2   2    2
+     4 5   7 8   10 11   13
+    [Ä,Ä,2,Ä,Ä,3,ÄÄ,ÄÄ,4,ÄÄ,...],
+     3 3   3 3    3  3    3
+     5 3 7   9 5 11   13 7
+    [Ä,Ä,Ä,2,Ä,Ä,ÄÄ,3,ÄÄ,Ä,...],
+     4 2 4   4 2  4    4 2
+     6 7 8 9   11 12 13 14
+    [Ä,Ä,Ä,Ä,2,ÄÄ,ÄÄ,ÄÄ,ÄÄ,3,...],
+     5 5 5 5    5  5  5  5
+     7 4 3 5 11   13 7 5 8
+    [Ä,Ä,Ä,Ä,ÄÄ,2,ÄÄ,Ä,Ä,Ä,...],
+     6 3 2 3  6    6 3 2 3
+     8 9 10 11 12 13   15 16 17
+    [Ä,Ä,ÄÄ,ÄÄ,ÄÄ,ÄÄ,2,ÄÄ,ÄÄ,ÄÄ,...],
+     7 7  7  7  7  7    7  7  7
+     9 5 11 3 13 7 15   17 9
+    [Ä,Ä,ÄÄ,Ä,ÄÄ,Ä,ÄÄ,2,ÄÄ,Ä,...],
+     8 4  8 2  8 4  8    8 4
+     10 11 4 13 14 5 16 17   19
+    [ÄÄ,ÄÄ,Ä,ÄÄ,ÄÄ,Ä,ÄÄ,ÄÄ,2,ÄÄ,...],
+      9  9 3  9  9 3  9  9    9
+     11 6 13 7 3 8 17 9 19
+    [ÄÄ,Ä,ÄÄ,Ä,Ä,Ä,ÄÄ,Ä,ÄÄ,2,...], ...]
+     10 5 10 5 2 5 10 5 10
+                   Type: Stream Stream Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPageEmpty8}
+\begin{paste}{ugLangItsPageEmpty8}{ugLangItsPagePatch8}
+\pastebutton{ugLangItsPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{[[i/j for i in j+1..] for j in 1..]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPagePatch9}
+\begin{paste}{ugLangItsPageFull9}{ugLangItsPageEmpty9}
+\pastebutton{ugLangItsPageFull9}{\hidepaste}
+\tab{5}\spadcommand{[i/j for i in 3.. by 10 for j in 2..]}
+\indentrel{3}\begin{verbatim}
+         3 13 23 33 43 53 63 73 83 93
+   (9)  [Ä,ÄÄ,ÄÄ,ÄÄ,ÄÄ,ÄÄ,ÄÄ,ÄÄ,ÄÄ,ÄÄ,...]
+         2  3  4  5  6  7  8  9 10 11
+                          Type: Stream Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPageEmpty9}
+\begin{paste}{ugLangItsPageEmpty9}{ugLangItsPagePatch9}
+\pastebutton{ugLangItsPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{[i/j for i in 3.. by 10 for j in 2..]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPagePatch10}
+\begin{paste}{ugLangItsPageFull10}{ugLangItsPageEmpty10}
+\pastebutton{ugLangItsPageFull10}{\hidepaste}
+\tab{5}\spadcommand{[i**j for i in 1..7 for j in 2.. ]}
+\indentrel{3}\begin{verbatim}
+   (10)  [1,8,81,1024,15625,279936,5764801]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPageEmpty10}
+\begin{paste}{ugLangItsPageEmpty10}{ugLangItsPagePatch10}
+\pastebutton{ugLangItsPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{[i**j for i in 1..7 for j in 2.. ]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPagePatch11}
+\begin{paste}{ugLangItsPageFull11}{ugLangItsPageEmpty11}
+\pastebutton{ugLangItsPageFull11}{\hidepaste}
+\tab{5}\spadcommand{[[[i,j] for i in 10..15 | prime? i] for j in 17..22 | j = squareFreePart j]}
+\indentrel{3}\begin{verbatim}
+   (11)
+   [[[11,17],[13,17]], [[11,19],[13,19]],
+    [[11,21],[13,21]], [[11,22],[13,22]]]
+                   Type: List List List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangItsPageEmpty11}
+\begin{paste}{ugLangItsPageEmpty11}{ugLangItsPagePatch11}
+\pastebutton{ugLangItsPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{[[[i,j] for i in 10..15 | prime? i] for j in 17..22 | j = squareFreePart j]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPagePatch1}
+\begin{paste}{ugLangStreamsPrimesPageFull1}{ugLangStreamsPrimesPageEmpty1}
+\pastebutton{ugLangStreamsPrimesPageFull1}{\hidepaste}
+\tab{5}\spadcommand{primes : Stream Integer := [i for i in 2.. | prime? i]}
+\indentrel{3}\begin{verbatim}
+   (1)  [2,3,5,7,11,13,17,19,23,29,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPageEmpty1}
+\begin{paste}{ugLangStreamsPrimesPageEmpty1}{ugLangStreamsPrimesPagePatch1}
+\pastebutton{ugLangStreamsPrimesPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{primes : Stream Integer := [i for i in 2.. | prime? i]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPagePatch2}
+\begin{paste}{ugLangStreamsPrimesPageFull2}{ugLangStreamsPrimesPageEmpty2}
+\pastebutton{ugLangStreamsPrimesPageFull2}{\hidepaste}
+\tab{5}\spadcommand{primes := generate(nextPrime,2)}
+\indentrel{3}\begin{verbatim}
+   (2)  [2,3,5,7,11,13,17,19,23,29,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPageEmpty2}
+\begin{paste}{ugLangStreamsPrimesPageEmpty2}{ugLangStreamsPrimesPagePatch2}
+\pastebutton{ugLangStreamsPrimesPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{primes := generate(nextPrime,2)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPagePatch3}
+\begin{paste}{ugLangStreamsPrimesPageFull3}{ugLangStreamsPrimesPageEmpty3}
+\pastebutton{ugLangStreamsPrimesPageFull3}{\hidepaste}
+\tab{5}\spadcommand{smallPrimes := [p for p in primes | p > 1000]\bound{smallPrimes }}
+\indentrel{3}\begin{verbatim}
+   (3)
+   [1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPageEmpty3}
+\begin{paste}{ugLangStreamsPrimesPageEmpty3}{ugLangStreamsPrimesPagePatch3}
+\pastebutton{ugLangStreamsPrimesPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{smallPrimes := [p for p in primes | p > 1000]\bound{smallPrimes }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPagePatch4}
+\begin{paste}{ugLangStreamsPrimesPageFull4}{ugLangStreamsPrimesPageEmpty4}
+\pastebutton{ugLangStreamsPrimesPageFull4}{\hidepaste}
+\tab{5}\spadcommand{[p for p in smallPrimes for i in 1..11]\free{smallPrimes }}
+\indentrel{3}\begin{verbatim}
+   (4)
+   [1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPageEmpty4}
+\begin{paste}{ugLangStreamsPrimesPageEmpty4}{ugLangStreamsPrimesPagePatch4}
+\pastebutton{ugLangStreamsPrimesPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{[p for p in smallPrimes for i in 1..11]\free{smallPrimes }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPagePatch5}
+\begin{paste}{ugLangStreamsPrimesPageFull5}{ugLangStreamsPrimesPageEmpty5}
+\pastebutton{ugLangStreamsPrimesPageFull5}{\hidepaste}
+\tab{5}\spadcommand{[p for p in smallPrimes while p < 1200]\free{smallPrimes }}
+\indentrel{3}\begin{verbatim}
+   (5)
+   [1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPageEmpty5}
+\begin{paste}{ugLangStreamsPrimesPageEmpty5}{ugLangStreamsPrimesPagePatch5}
+\pastebutton{ugLangStreamsPrimesPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{[p for p in smallPrimes while p < 1200]\free{smallPrimes }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPagePatch6}
+\begin{paste}{ugLangStreamsPrimesPageFull6}{ugLangStreamsPrimesPageEmpty6}
+\pastebutton{ugLangStreamsPrimesPageFull6}{\hidepaste}
+\tab{5}\spadcommand{complete \%}
+\indentrel{3}\begin{verbatim}
+   (6)
+   [1009,1013,1019,1021,1031,1033,1039,1049,1051,1061,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPageEmpty6}
+\begin{paste}{ugLangStreamsPrimesPageEmpty6}{ugLangStreamsPrimesPagePatch6}
+\pastebutton{ugLangStreamsPrimesPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{complete \%}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPagePatch7}
+\begin{paste}{ugLangStreamsPrimesPageFull7}{ugLangStreamsPrimesPageEmpty7}
+\pastebutton{ugLangStreamsPrimesPageFull7}{\hidepaste}
+\tab{5}\spadcommand{twinPrimes := [[p,p+2] for p in primes | prime?(p + 2)]}
+\indentrel{3}\begin{verbatim}
+   (7)
+   [[3,5], [5,7], [11,13], [17,19], [29,31], [41,43],
+    [59,61], [71,73], [101,103], [107,109], ...]
+                              Type: Stream List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPageEmpty7}
+\begin{paste}{ugLangStreamsPrimesPageEmpty7}{ugLangStreamsPrimesPagePatch7}
+\pastebutton{ugLangStreamsPrimesPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{twinPrimes := [[p,p+2] for p in primes | prime?(p + 2)]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPagePatch8}
+\begin{paste}{ugLangStreamsPrimesPageFull8}{ugLangStreamsPrimesPageEmpty8}
+\pastebutton{ugLangStreamsPrimesPageFull8}{\hidepaste}
+\tab{5}\spadcommand{firstOfTwins:= [p for p in primes for q in rest primes | q=p+2]}
+\indentrel{3}\begin{verbatim}
+   (8)  [3,5,11,17,29,41,59,71,101,107,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPageEmpty8}
+\begin{paste}{ugLangStreamsPrimesPageEmpty8}{ugLangStreamsPrimesPagePatch8}
+\pastebutton{ugLangStreamsPrimesPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{firstOfTwins:= [p for p in primes for q in rest primes | q=p+2]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPagePatch9}
+\begin{paste}{ugLangStreamsPrimesPageFull9}{ugLangStreamsPrimesPageEmpty9}
+\pastebutton{ugLangStreamsPrimesPageFull9}{\hidepaste}
+\tab{5}\spadcommand{firstTriplets := [p for p in firstOfTwins for q in rest firstOfTwins | q = p+2];}
+\indentrel{3}\begin{verbatim}
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPageEmpty9}
+\begin{paste}{ugLangStreamsPrimesPageEmpty9}{ugLangStreamsPrimesPagePatch9}
+\pastebutton{ugLangStreamsPrimesPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{firstTriplets := [p for p in firstOfTwins for q in rest firstOfTwins | q = p+2];}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPagePatch10}
+\begin{paste}{ugLangStreamsPrimesPageFull10}{ugLangStreamsPrimesPageEmpty10}
+\pastebutton{ugLangStreamsPrimesPageFull10}{\hidepaste}
+\tab{5}\spadcommand{firstTriplets.1}
+\indentrel{3}\begin{verbatim}
+   (10)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPageEmpty10}
+\begin{paste}{ugLangStreamsPrimesPageEmpty10}{ugLangStreamsPrimesPagePatch10}
+\pastebutton{ugLangStreamsPrimesPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{firstTriplets.1}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPagePatch11}
+\begin{paste}{ugLangStreamsPrimesPageFull11}{ugLangStreamsPrimesPageEmpty11}
+\pastebutton{ugLangStreamsPrimesPageFull11}{\hidepaste}
+\tab{5}\spadcommand{primes.2045}
+\indentrel{3}\begin{verbatim}
+   (11)  17837
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangStreamsPrimesPageEmpty11}
+\begin{paste}{ugLangStreamsPrimesPageEmpty11}{ugLangStreamsPrimesPagePatch11}
+\pastebutton{ugLangStreamsPrimesPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{primes.2045}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInPredPagePatch1}
+\begin{paste}{ugLangLoopsForInPredPageFull1}{ugLangLoopsForInPredPageEmpty1}
+\pastebutton{ugLangLoopsForInPredPageFull1}{\hidepaste}
+\tab{5}\spadcommand{for n in 0..4 | odd? n repeat output n}
+\indentrel{3}\begin{verbatim}
+   1
+   3
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInPredPageEmpty1}
+\begin{paste}{ugLangLoopsForInPredPageEmpty1}{ugLangLoopsForInPredPagePatch1}
+\pastebutton{ugLangLoopsForInPredPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{for n in 0..4 | odd? n repeat output n}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInPredPagePatch2}
+\begin{paste}{ugLangLoopsForInPredPageFull2}{ugLangLoopsForInPredPageEmpty2}
+\pastebutton{ugLangLoopsForInPredPageFull2}{\hidepaste}
+\tab{5}\spadcommand{for i in 1..50 repeat
+  for j in 1..50 | factorial(i+j) < 25 repeat
+    output [i,j]
+}
+\indentrel{3}\begin{verbatim}
+   [1,1]
+   [1,2]
+   [1,3]
+   [2,1]
+   [2,2]
+   [3,1]
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsForInPredPageEmpty2}
+\begin{paste}{ugLangLoopsForInPredPageEmpty2}{ugLangLoopsForInPredPagePatch2}
+\pastebutton{ugLangLoopsForInPredPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{for i in 1..50 repeat
+  for j in 1..50 | factorial(i+j) < 25 repeat
+    output [i,j]
+}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePagePatch1}
+\begin{paste}{ugLangLoopsBreakMorePageFull1}{ugLangLoopsBreakMorePageEmpty1}
+\pastebutton{ugLangLoopsBreakMorePageFull1}{\hidepaste}
+\tab{5}\spadcommand{i := 0\bound{i }}
+\indentrel{3}\begin{verbatim}
+   (1)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePageEmpty1}
+\begin{paste}{ugLangLoopsBreakMorePageEmpty1}{ugLangLoopsBreakMorePagePatch1}
+\pastebutton{ugLangLoopsBreakMorePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{i := 0\bound{i }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePagePatch2}
+\begin{paste}{ugLangLoopsBreakMorePageFull2}{ugLangLoopsBreakMorePageEmpty2}
+\pastebutton{ugLangLoopsBreakMorePageFull2}{\hidepaste}
+\tab{5}\spadcommand{repeat
+  i := i + 1
+  if i**2 > 100 then break
+\free{i }\bound{i1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePageEmpty2}
+\begin{paste}{ugLangLoopsBreakMorePageEmpty2}{ugLangLoopsBreakMorePagePatch2}
+\pastebutton{ugLangLoopsBreakMorePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{repeat
+  i := i + 1
+  if i**2 > 100 then break
+\free{i }\bound{i1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePagePatch3}
+\begin{paste}{ugLangLoopsBreakMorePageFull3}{ugLangLoopsBreakMorePageEmpty3}
+\pastebutton{ugLangLoopsBreakMorePageFull3}{\hidepaste}
+\tab{5}\spadcommand{i\free{i1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  11
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePageEmpty3}
+\begin{paste}{ugLangLoopsBreakMorePageEmpty3}{ugLangLoopsBreakMorePagePatch3}
+\pastebutton{ugLangLoopsBreakMorePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{i\free{i1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePagePatch4}
+\begin{paste}{ugLangLoopsBreakMorePageFull4}{ugLangLoopsBreakMorePageEmpty4}
+\pastebutton{ugLangLoopsBreakMorePageFull4}{\hidepaste}
+\tab{5}\spadcommand{i := 0\bound{i2 }}
+\indentrel{3}\begin{verbatim}
+   (4)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePageEmpty4}
+\begin{paste}{ugLangLoopsBreakMorePageEmpty4}{ugLangLoopsBreakMorePagePatch4}
+\pastebutton{ugLangLoopsBreakMorePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{i := 0\bound{i2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePagePatch5}
+\begin{paste}{ugLangLoopsBreakMorePageFull5}{ugLangLoopsBreakMorePageEmpty5}
+\pastebutton{ugLangLoopsBreakMorePageFull5}{\hidepaste}
+\tab{5}\spadcommand{repeat
+  i := i + 1
+  i**2 > 100 => break
+\free{i2 }\bound{i3 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePageEmpty5}
+\begin{paste}{ugLangLoopsBreakMorePageEmpty5}{ugLangLoopsBreakMorePagePatch5}
+\pastebutton{ugLangLoopsBreakMorePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{repeat
+  i := i + 1
+  i**2 > 100 => break
+\free{i2 }\bound{i3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePagePatch6}
+\begin{paste}{ugLangLoopsBreakMorePageFull6}{ugLangLoopsBreakMorePageEmpty6}
+\pastebutton{ugLangLoopsBreakMorePageFull6}{\hidepaste}
+\tab{5}\spadcommand{i\free{i3 }}
+\indentrel{3}\begin{verbatim}
+   (6)  11
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePageEmpty6}
+\begin{paste}{ugLangLoopsBreakMorePageEmpty6}{ugLangLoopsBreakMorePagePatch6}
+\pastebutton{ugLangLoopsBreakMorePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{i\free{i3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePagePatch7}
+\begin{paste}{ugLangLoopsBreakMorePageFull7}{ugLangLoopsBreakMorePageEmpty7}
+\pastebutton{ugLangLoopsBreakMorePageFull7}{\hidepaste}
+\tab{5}\spadcommand{(n, i, f) := (100, 1, 1)\bound{n }\bound{i4 }\bound{f }}
+\indentrel{3}\begin{verbatim}
+   (7)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePageEmpty7}
+\begin{paste}{ugLangLoopsBreakMorePageEmpty7}{ugLangLoopsBreakMorePagePatch7}
+\pastebutton{ugLangLoopsBreakMorePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{(n, i, f) := (100, 1, 1)\bound{n }\bound{i4 }\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePagePatch8}
+\begin{paste}{ugLangLoopsBreakMorePageFull8}{ugLangLoopsBreakMorePageEmpty8}
+\pastebutton{ugLangLoopsBreakMorePageFull8}{\hidepaste}
+\tab{5}\spadcommand{repeat
+  if i > n then break
+  f := f * i
+  i := i + 1
+\bound{f1 }\bound{i5 }\free{f i4 n }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePageEmpty8}
+\begin{paste}{ugLangLoopsBreakMorePageEmpty8}{ugLangLoopsBreakMorePagePatch8}
+\pastebutton{ugLangLoopsBreakMorePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{repeat
+  if i > n then break
+  f := f * i
+  i := i + 1
+\bound{f1 }\bound{i5 }\free{f i4 n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePagePatch9}
+\begin{paste}{ugLangLoopsBreakMorePageFull9}{ugLangLoopsBreakMorePageEmpty9}
+\pastebutton{ugLangLoopsBreakMorePageFull9}{\hidepaste}
+\tab{5}\spadcommand{f\free{f1 }}
+\indentrel{3}\begin{verbatim}
+   (9)
+  933262154439441526816992388562667004907159682643816214_
+   68592963895217599993229915608941463976156518286253697_
+   920827223758251185210916864000000000000000000000000
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePageEmpty9}
+\begin{paste}{ugLangLoopsBreakMorePageEmpty9}{ugLangLoopsBreakMorePagePatch9}
+\pastebutton{ugLangLoopsBreakMorePageEmpty9}{\showpaste}
+\tab{5}\spadcommand{f\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePagePatch10}
+\begin{paste}{ugLangLoopsBreakMorePageFull10}{ugLangLoopsBreakMorePageEmpty10}
+\pastebutton{ugLangLoopsBreakMorePageFull10}{\hidepaste}
+\tab{5}\spadcommand{m := matrix [[21,37,53,14], [8,-24,22,-16], [2,10,15,14], [26,33,55,-13]]\bound{m2 }}
+\indentrel{3}\begin{verbatim}
+         Ú21   37   53   14 ¿
+         ³                  ³
+         ³8   - 24  22  - 16³
+   (10)  ³                  ³
+         ³2    10   15   14 ³
+         ³                  ³
+         À26   33   55  - 13Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePageEmpty10}
+\begin{paste}{ugLangLoopsBreakMorePageEmpty10}{ugLangLoopsBreakMorePagePatch10}
+\pastebutton{ugLangLoopsBreakMorePageEmpty10}{\showpaste}
+\tab{5}\spadcommand{m := matrix [[21,37,53,14], [8,-24,22,-16], [2,10,15,14], [26,33,55,-13]]\bound{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePagePatch11}
+\begin{paste}{ugLangLoopsBreakMorePageFull11}{ugLangLoopsBreakMorePageEmpty11}
+\pastebutton{ugLangLoopsBreakMorePageFull11}{\hidepaste}
+\tab{5}\spadcommand{(r, c) := (1, 1)\bound{r }\bound{c }}
+\indentrel{3}\begin{verbatim}
+   (11)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePageEmpty11}
+\begin{paste}{ugLangLoopsBreakMorePageEmpty11}{ugLangLoopsBreakMorePagePatch11}
+\pastebutton{ugLangLoopsBreakMorePageEmpty11}{\showpaste}
+\tab{5}\spadcommand{(r, c) := (1, 1)\bound{r }\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePagePatch12}
+\begin{paste}{ugLangLoopsBreakMorePageFull12}{ugLangLoopsBreakMorePageEmpty12}
+\pastebutton{ugLangLoopsBreakMorePageFull12}{\hidepaste}
+\tab{5}\spadcommand{(lastrow, lastcol) := (nrows(m), ncols(m))\bound{lastrow }\bound{lastcol }\free{m2 }}
+\indentrel{3}\begin{verbatim}
+   (12)  4
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePageEmpty12}
+\begin{paste}{ugLangLoopsBreakMorePageEmpty12}{ugLangLoopsBreakMorePagePatch12}
+\pastebutton{ugLangLoopsBreakMorePageEmpty12}{\showpaste}
+\tab{5}\spadcommand{(lastrow, lastcol) := (nrows(m), ncols(m))\bound{lastrow }\bound{lastcol }\free{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePagePatch13}
+\begin{paste}{ugLangLoopsBreakMorePageFull13}{ugLangLoopsBreakMorePageEmpty13}
+\pastebutton{ugLangLoopsBreakMorePageFull13}{\hidepaste}
+\tab{5}\spadcommand{repeat
+  if r > lastrow then break
+  c := 1
+  repeat
+    if c > lastcol then break
+    if elt(m,r,c) < 0 then
+      output [r, c, elt(m,r,c)]
+      r := lastrow
+      break     -- don't look any further
+    c := c + 1
+  r := r + 1
+\free{m2 r c lastrow lastcol }}
+\indentrel{3}\begin{verbatim}
+   [2,2,- 24]
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakMorePageEmpty13}
+\begin{paste}{ugLangLoopsBreakMorePageEmpty13}{ugLangLoopsBreakMorePagePatch13}
+\pastebutton{ugLangLoopsBreakMorePageEmpty13}{\showpaste}
+\tab{5}\spadcommand{repeat
+  if r > lastrow then break
+  c := 1
+  repeat
+    if c > lastcol then break
+    if elt(m,r,c) < 0 then
+      output [r, c, elt(m,r,c)]
+      r := lastrow
+      break     -- don't look any further
+    c := c + 1
+  r := r + 1
+\free{m2 r c lastrow lastcol }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPagePatch1}
+\begin{paste}{ugLangBlocksPageFull1}{ugLangBlocksPageEmpty1}
+\pastebutton{ugLangBlocksPageFull1}{\hidepaste}
+\tab{5}\spadcommand{a :=
+  i := gcd(234,672)
+  i := 3*i**5 - i + 1
+  1 / i
+}
+\indentrel{3}\begin{verbatim}
+          1
+   (1)  ÄÄÄÄÄ
+        23323
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPageEmpty1}
+\begin{paste}{ugLangBlocksPageEmpty1}{ugLangBlocksPagePatch1}
+\pastebutton{ugLangBlocksPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{a :=
+  i := gcd(234,672)
+  i := 3*i**5 - i + 1
+  1 / i
+}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPagePatch2}
+\begin{paste}{ugLangBlocksPageFull2}{ugLangBlocksPageEmpty2}
+\pastebutton{ugLangBlocksPageFull2}{\hidepaste}
+\tab{5}\spadcommand{a := (i := gcd(234,672); i := 3*i**5 - i + 1; 1 / i)}
+\indentrel{3}\begin{verbatim}
+          1
+   (2)  ÄÄÄÄÄ
+        23323
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPageEmpty2}
+\begin{paste}{ugLangBlocksPageEmpty2}{ugLangBlocksPagePatch2}
+\pastebutton{ugLangBlocksPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{a := (i := gcd(234,672); i := 3*i**5 - i + 1; 1 / i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPagePatch3}
+\begin{paste}{ugLangBlocksPageFull3}{ugLangBlocksPageEmpty3}
+\pastebutton{ugLangBlocksPageFull3}{\hidepaste}
+\tab{5}\spadcommand{(a := 1; b := 2; c := 3; [a,b,c])\bound{a b c }}
+\indentrel{3}\begin{verbatim}
+   (3)  [1,2,3]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPageEmpty3}
+\begin{paste}{ugLangBlocksPageEmpty3}{ugLangBlocksPagePatch3}
+\pastebutton{ugLangBlocksPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{(a := 1; b := 2; c := 3; [a,b,c])\bound{a b c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPagePatch4}
+\begin{paste}{ugLangBlocksPageFull4}{ugLangBlocksPageEmpty4}
+\pastebutton{ugLangBlocksPageFull4}{\hidepaste}
+\tab{5}\spadcommand{d :=
+   c := a**2 + b**2
+   sqrt(c * 1.3)
+\free{a b }}
+\indentrel{3}\begin{verbatim}
+   (4)  2.5495097567 96392415
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPageEmpty4}
+\begin{paste}{ugLangBlocksPageEmpty4}{ugLangBlocksPagePatch4}
+\pastebutton{ugLangBlocksPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{d :=
+   c := a**2 + b**2
+   sqrt(c * 1.3)
+\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPagePatch5}
+\begin{paste}{ugLangBlocksPageFull5}{ugLangBlocksPageEmpty5}
+\pastebutton{ugLangBlocksPageFull5}{\hidepaste}
+\tab{5}\spadcommand{h := 2.1 +
+   1.0
+   3.5
+\bound{h }}
+\indentrel{3}\begin{verbatim}
+   (5)  5.6
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPageEmpty5}
+\begin{paste}{ugLangBlocksPageEmpty5}{ugLangBlocksPagePatch5}
+\pastebutton{ugLangBlocksPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{h := 2.1 +
+   1.0
+   3.5
+\bound{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPagePatch6}
+\begin{paste}{ugLangBlocksPageFull6}{ugLangBlocksPageEmpty6}
+\pastebutton{ugLangBlocksPageFull6}{\hidepaste}
+\tab{5}\spadcommand{eval(x**2 - x*y**2,
+     z := \%pi/2.0 - exp(4.1)
+     x = z
+   )
+}
+\indentrel{3}\begin{verbatim}
+   (6)
+                           2
+   58.7694912705 67072878 y  + 3453.8531042012 59382
+                                 Type: Polynomial Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPageEmpty6}
+\begin{paste}{ugLangBlocksPageEmpty6}{ugLangBlocksPagePatch6}
+\pastebutton{ugLangBlocksPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{eval(x**2 - x*y**2,
+     z := \%pi/2.0 - exp(4.1)
+     x = z
+   )
+}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPagePatch7}
+\begin{paste}{ugLangBlocksPageFull7}{ugLangBlocksPageEmpty7}
+\pastebutton{ugLangBlocksPageFull7}{\hidepaste}
+\tab{5}\spadcommand{if h > 3.1 then 1.0 else (z := cos(h); max(z,0.5))\free{h }}
+\indentrel{3}\begin{verbatim}
+   (7)  1.0
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPageEmpty7}
+\begin{paste}{ugLangBlocksPageEmpty7}{ugLangBlocksPagePatch7}
+\pastebutton{ugLangBlocksPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{if h > 3.1 then 1.0 else (z := cos(h); max(z,0.5))\free{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPagePatch8}
+\begin{paste}{ugLangBlocksPageFull8}{ugLangBlocksPageEmpty8}
+\pastebutton{ugLangBlocksPageFull8}{\hidepaste}
+\tab{5}\spadcommand{if h > 3.1 then
+    1.0
+  else
+    z := cos(h)
+    max(z,0.5)
+\free{h }}
+\indentrel{3}\begin{verbatim}
+   (8)  1.0
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPageEmpty8}
+\begin{paste}{ugLangBlocksPageEmpty8}{ugLangBlocksPagePatch8}
+\pastebutton{ugLangBlocksPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{if h > 3.1 then
+    1.0
+  else
+    z := cos(h)
+    max(z,0.5)
+\free{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPagePatch9}
+\begin{paste}{ugLangBlocksPageFull9}{ugLangBlocksPageEmpty9}
+\pastebutton{ugLangBlocksPageFull9}{\hidepaste}
+\tab{5}\spadcommand{a := (b := factorial(12); c := (d := eulerPhi(22); factorial(d));b+c)}
+\indentrel{3}\begin{verbatim}
+   (9)  482630400
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPageEmpty9}
+\begin{paste}{ugLangBlocksPageEmpty9}{ugLangBlocksPagePatch9}
+\pastebutton{ugLangBlocksPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{a := (b := factorial(12); c := (d := eulerPhi(22); factorial(d));b+c)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPagePatch10}
+\begin{paste}{ugLangBlocksPageFull10}{ugLangBlocksPageEmpty10}
+\pastebutton{ugLangBlocksPageFull10}{\hidepaste}
+\tab{5}\spadcommand{a :=
+  b := factorial(12)
+  c :=
+    d := eulerPhi(22)
+    factorial(d)
+  b+c
+}
+\indentrel{3}\begin{verbatim}
+   (10)  482630400
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPageEmpty10}
+\begin{paste}{ugLangBlocksPageEmpty10}{ugLangBlocksPagePatch10}
+\pastebutton{ugLangBlocksPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{a :=
+  b := factorial(12)
+  c :=
+    d := eulerPhi(22)
+    factorial(d)
+  b+c
+}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPagePatch11}
+\begin{paste}{ugLangBlocksPageFull11}{ugLangBlocksPageEmpty11}
+\pastebutton{ugLangBlocksPageFull11}{\hidepaste}
+\tab{5}\spadcommand{a :=
+  c := factorial 10
+  d := fibonacci 10
+  c + d = 3628855 => c
+  d
+}
+\indentrel{3}\begin{verbatim}
+   (11)  3628800
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangBlocksPageEmpty11}
+\begin{paste}{ugLangBlocksPageEmpty11}{ugLangBlocksPagePatch11}
+\pastebutton{ugLangBlocksPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{a :=
+  c := factorial 10
+  d := fibonacci 10
+  c + d = 3628855 => c
+  d
+}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch1}
+\begin{paste}{ugLangAssignPageFull1}{ugLangAssignPageEmpty1}
+\pastebutton{ugLangAssignPageFull1}{\hidepaste}
+\tab{5}\spadcommand{a := 1\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (1)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty1}
+\begin{paste}{ugLangAssignPageEmpty1}{ugLangAssignPagePatch1}
+\pastebutton{ugLangAssignPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{a := 1\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch2}
+\begin{paste}{ugLangAssignPageFull2}{ugLangAssignPageEmpty2}
+\pastebutton{ugLangAssignPageFull2}{\hidepaste}
+\tab{5}\spadcommand{b := a\free{a }\bound{b }}
+\indentrel{3}\begin{verbatim}
+   (2)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty2}
+\begin{paste}{ugLangAssignPageEmpty2}{ugLangAssignPagePatch2}
+\pastebutton{ugLangAssignPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{b := a\free{a }\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch3}
+\begin{paste}{ugLangAssignPageFull3}{ugLangAssignPageEmpty3}
+\pastebutton{ugLangAssignPageFull3}{\hidepaste}
+\tab{5}\spadcommand{a := 2\bound{a2 }}
+\indentrel{3}\begin{verbatim}
+   (3)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty3}
+\begin{paste}{ugLangAssignPageEmpty3}{ugLangAssignPagePatch3}
+\pastebutton{ugLangAssignPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{a := 2\bound{a2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch4}
+\begin{paste}{ugLangAssignPageFull4}{ugLangAssignPageEmpty4}
+\pastebutton{ugLangAssignPageFull4}{\hidepaste}
+\tab{5}\spadcommand{b\free{b }}
+\indentrel{3}\begin{verbatim}
+   (4)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty4}
+\begin{paste}{ugLangAssignPageEmpty4}{ugLangAssignPagePatch4}
+\pastebutton{ugLangAssignPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{b\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch5}
+\begin{paste}{ugLangAssignPageFull5}{ugLangAssignPageEmpty5}
+\pastebutton{ugLangAssignPageFull5}{\hidepaste}
+\tab{5}\spadcommand{a == 1\bound{ad }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty5}
+\begin{paste}{ugLangAssignPageEmpty5}{ugLangAssignPagePatch5}
+\pastebutton{ugLangAssignPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{a == 1\bound{ad }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch6}
+\begin{paste}{ugLangAssignPageFull6}{ugLangAssignPageEmpty6}
+\pastebutton{ugLangAssignPageFull6}{\hidepaste}
+\tab{5}\spadcommand{b == a\free{ad }\bound{bd }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty6}
+\begin{paste}{ugLangAssignPageEmpty6}{ugLangAssignPagePatch6}
+\pastebutton{ugLangAssignPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{b == a\free{ad }\bound{bd }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch7}
+\begin{paste}{ugLangAssignPageFull7}{ugLangAssignPageEmpty7}
+\pastebutton{ugLangAssignPageFull7}{\hidepaste}
+\tab{5}\spadcommand{a\free{ad }}
+\indentrel{3}\begin{verbatim}
+   (7)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty7}
+\begin{paste}{ugLangAssignPageEmpty7}{ugLangAssignPagePatch7}
+\pastebutton{ugLangAssignPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{a\free{ad }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch8}
+\begin{paste}{ugLangAssignPageFull8}{ugLangAssignPageEmpty8}
+\pastebutton{ugLangAssignPageFull8}{\hidepaste}
+\tab{5}\spadcommand{b\free{bd }}
+\indentrel{3}\begin{verbatim}
+   (8)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty8}
+\begin{paste}{ugLangAssignPageEmpty8}{ugLangAssignPagePatch8}
+\pastebutton{ugLangAssignPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{b\free{bd }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch9}
+\begin{paste}{ugLangAssignPageFull9}{ugLangAssignPageEmpty9}
+\pastebutton{ugLangAssignPageFull9}{\hidepaste}
+\tab{5}\spadcommand{a == 2\bound{ad2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty9}
+\begin{paste}{ugLangAssignPageEmpty9}{ugLangAssignPagePatch9}
+\pastebutton{ugLangAssignPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{a == 2\bound{ad2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch10}
+\begin{paste}{ugLangAssignPageFull10}{ugLangAssignPageEmpty10}
+\pastebutton{ugLangAssignPageFull10}{\hidepaste}
+\tab{5}\spadcommand{a\free{ad2 }}
+\indentrel{3}\begin{verbatim}
+   (10)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty10}
+\begin{paste}{ugLangAssignPageEmpty10}{ugLangAssignPagePatch10}
+\pastebutton{ugLangAssignPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{a\free{ad2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch11}
+\begin{paste}{ugLangAssignPageFull11}{ugLangAssignPageEmpty11}
+\pastebutton{ugLangAssignPageFull11}{\hidepaste}
+\tab{5}\spadcommand{b\free{bd ad2 }}
+\indentrel{3}\begin{verbatim}
+   (11)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty11}
+\begin{paste}{ugLangAssignPageEmpty11}{ugLangAssignPagePatch11}
+\pastebutton{ugLangAssignPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{b\free{bd ad2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch12}
+\begin{paste}{ugLangAssignPageFull12}{ugLangAssignPageEmpty12}
+\pastebutton{ugLangAssignPageFull12}{\hidepaste}
+\tab{5}\spadcommand{(x,y) := (1,2)\bound{x }\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (12)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty12}
+\begin{paste}{ugLangAssignPageEmpty12}{ugLangAssignPagePatch12}
+\pastebutton{ugLangAssignPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{(x,y) := (1,2)\bound{x }\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch13}
+\begin{paste}{ugLangAssignPageFull13}{ugLangAssignPageEmpty13}
+\pastebutton{ugLangAssignPageFull13}{\hidepaste}
+\tab{5}\spadcommand{(x,y) := (y,x)\free{x y }\bound{swap }}
+\indentrel{3}\begin{verbatim}
+   (13)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty13}
+\begin{paste}{ugLangAssignPageEmpty13}{ugLangAssignPagePatch13}
+\pastebutton{ugLangAssignPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{(x,y) := (y,x)\free{x y }\bound{swap }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch14}
+\begin{paste}{ugLangAssignPageFull14}{ugLangAssignPageEmpty14}
+\pastebutton{ugLangAssignPageFull14}{\hidepaste}
+\tab{5}\spadcommand{x\free{swap }}
+\indentrel{3}\begin{verbatim}
+   (14)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty14}
+\begin{paste}{ugLangAssignPageEmpty14}{ugLangAssignPagePatch14}
+\pastebutton{ugLangAssignPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{x\free{swap }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPagePatch15}
+\begin{paste}{ugLangAssignPageFull15}{ugLangAssignPageEmpty15}
+\pastebutton{ugLangAssignPageFull15}{\hidepaste}
+\tab{5}\spadcommand{y\free{swap }}
+\indentrel{3}\begin{verbatim}
+   (15)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangAssignPageEmpty15}
+\begin{paste}{ugLangAssignPageEmpty15}{ugLangAssignPagePatch15}
+\pastebutton{ugLangAssignPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{y\free{swap }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPagePatch1}
+\begin{paste}{ugLangLoopsParPageFull1}{ugLangLoopsParPageEmpty1}
+\pastebutton{ugLangLoopsParPageFull1}{\hidepaste}
+\tab{5}\spadcommand{l := [1,3,5,7]\bound{l }}
+\indentrel{3}\begin{verbatim}
+   (1)  [1,3,5,7]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPageEmpty1}
+\begin{paste}{ugLangLoopsParPageEmpty1}{ugLangLoopsParPagePatch1}
+\pastebutton{ugLangLoopsParPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{l := [1,3,5,7]\bound{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPagePatch2}
+\begin{paste}{ugLangLoopsParPageFull2}{ugLangLoopsParPageEmpty2}
+\pastebutton{ugLangLoopsParPageFull2}{\hidepaste}
+\tab{5}\spadcommand{m := [100,200]\bound{m }}
+\indentrel{3}\begin{verbatim}
+   (2)  [100,200]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPageEmpty2}
+\begin{paste}{ugLangLoopsParPageEmpty2}{ugLangLoopsParPagePatch2}
+\pastebutton{ugLangLoopsParPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{m := [100,200]\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPagePatch3}
+\begin{paste}{ugLangLoopsParPageFull3}{ugLangLoopsParPageEmpty3}
+\pastebutton{ugLangLoopsParPageFull3}{\hidepaste}
+\tab{5}\spadcommand{sum := 0\bound{sum }}
+\indentrel{3}\begin{verbatim}
+   (3)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPageEmpty3}
+\begin{paste}{ugLangLoopsParPageEmpty3}{ugLangLoopsParPagePatch3}
+\pastebutton{ugLangLoopsParPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{sum := 0\bound{sum }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPagePatch4}
+\begin{paste}{ugLangLoopsParPageFull4}{ugLangLoopsParPageEmpty4}
+\pastebutton{ugLangLoopsParPageFull4}{\hidepaste}
+\tab{5}\spadcommand{for x in l for y in m repeat
+    sum := sum + x*y
+\bound{doit }\free{sum l m }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPageEmpty4}
+\begin{paste}{ugLangLoopsParPageEmpty4}{ugLangLoopsParPagePatch4}
+\pastebutton{ugLangLoopsParPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{for x in l for y in m repeat
+    sum := sum + x*y
+\bound{doit }\free{sum l m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPagePatch5}
+\begin{paste}{ugLangLoopsParPageFull5}{ugLangLoopsParPageEmpty5}
+\pastebutton{ugLangLoopsParPageFull5}{\hidepaste}
+\tab{5}\spadcommand{sum\free{doit }}
+\indentrel{3}\begin{verbatim}
+   (5)  700
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPageEmpty5}
+\begin{paste}{ugLangLoopsParPageEmpty5}{ugLangLoopsParPagePatch5}
+\pastebutton{ugLangLoopsParPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{sum\free{doit }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPagePatch6}
+\begin{paste}{ugLangLoopsParPageFull6}{ugLangLoopsParPageEmpty6}
+\pastebutton{ugLangLoopsParPageFull6}{\hidepaste}
+\tab{5}\spadcommand{l := [2,3,5,7,11,13,17,19,23,29,31,37]\bound{l1 }}
+\indentrel{3}\begin{verbatim}
+   (6)  [2,3,5,7,11,13,17,19,23,29,31,37]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPageEmpty6}
+\begin{paste}{ugLangLoopsParPageEmpty6}{ugLangLoopsParPagePatch6}
+\pastebutton{ugLangLoopsParPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{l := [2,3,5,7,11,13,17,19,23,29,31,37]\bound{l1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPagePatch7}
+\begin{paste}{ugLangLoopsParPageFull7}{ugLangLoopsParPageEmpty7}
+\pastebutton{ugLangLoopsParPageFull7}{\hidepaste}
+\tab{5}\spadcommand{sum := 0\bound{sum1 }}
+\indentrel{3}\begin{verbatim}
+   (7)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPageEmpty7}
+\begin{paste}{ugLangLoopsParPageEmpty7}{ugLangLoopsParPagePatch7}
+\pastebutton{ugLangLoopsParPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{sum := 0\bound{sum1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPagePatch8}
+\begin{paste}{ugLangLoopsParPageFull8}{ugLangLoopsParPageEmpty8}
+\pastebutton{ugLangLoopsParPageFull8}{\hidepaste}
+\tab{5}\spadcommand{for i in 0.. for x in l repeat sum := i * x\bound{doit1 }\free{sum1 l1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPageEmpty8}
+\begin{paste}{ugLangLoopsParPageEmpty8}{ugLangLoopsParPagePatch8}
+\pastebutton{ugLangLoopsParPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{for i in 0.. for x in l repeat sum := i * x\bound{doit1 }\free{sum1 l1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPagePatch9}
+\begin{paste}{ugLangLoopsParPageFull9}{ugLangLoopsParPageEmpty9}
+\pastebutton{ugLangLoopsParPageFull9}{\hidepaste}
+\tab{5}\spadcommand{sum\free{doit1 }}
+\indentrel{3}\begin{verbatim}
+   (9)  407
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPageEmpty9}
+\begin{paste}{ugLangLoopsParPageEmpty9}{ugLangLoopsParPagePatch9}
+\pastebutton{ugLangLoopsParPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{sum\free{doit1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPagePatch10}
+\begin{paste}{ugLangLoopsParPageFull10}{ugLangLoopsParPageEmpty10}
+\pastebutton{ugLangLoopsParPageFull10}{\hidepaste}
+\tab{5}\spadcommand{for i in 1..10
+    for j in 151..160 | odd? j
+      while i + j < 160 repeat
+        output [i,j]
+}
+\indentrel{3}\begin{verbatim}
+   [1,151]
+   [3,153]
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsParPageEmpty10}
+\begin{paste}{ugLangLoopsParPageEmpty10}{ugLangLoopsParPagePatch10}
+\pastebutton{ugLangLoopsParPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{for i in 1..10
+    for j in 151..160 | odd? j
+      while i + j < 160 repeat
+        output [i,j]
+}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakPagePatch1}
+\begin{paste}{ugLangLoopsBreakPageFull1}{ugLangLoopsBreakPageEmpty1}
+\pastebutton{ugLangLoopsBreakPageFull1}{\hidepaste}
+\tab{5}\spadcommand{f() ==
+  i := 1
+  repeat
+    if factorial(i) > 1000 then break
+    i := i + 1
+  i
+\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakPageEmpty1}
+\begin{paste}{ugLangLoopsBreakPageEmpty1}{ugLangLoopsBreakPagePatch1}
+\pastebutton{ugLangLoopsBreakPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{f() ==
+  i := 1
+  repeat
+    if factorial(i) > 1000 then break
+    i := i + 1
+  i
+\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakPagePatch2}
+\begin{paste}{ugLangLoopsBreakPageFull2}{ugLangLoopsBreakPageEmpty2}
+\pastebutton{ugLangLoopsBreakPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f()\free{f1 }}
+\indentrel{3}\begin{verbatim}
+   (2)  7
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakPageEmpty2}
+\begin{paste}{ugLangLoopsBreakPageEmpty2}{ugLangLoopsBreakPagePatch2}
+\pastebutton{ugLangLoopsBreakPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f()\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakPagePatch3}
+\begin{paste}{ugLangLoopsBreakPageFull3}{ugLangLoopsBreakPageEmpty3}
+\pastebutton{ugLangLoopsBreakPageFull3}{\hidepaste}
+\tab{5}\spadcommand{(i,j) := (1, 1)\bound{i }\bound{j }}
+\indentrel{3}\begin{verbatim}
+   (3)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakPageEmpty3}
+\begin{paste}{ugLangLoopsBreakPageEmpty3}{ugLangLoopsBreakPagePatch3}
+\pastebutton{ugLangLoopsBreakPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{(i,j) := (1, 1)\bound{i }\bound{j }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakPagePatch4}
+\begin{paste}{ugLangLoopsBreakPageFull4}{ugLangLoopsBreakPageEmpty4}
+\pastebutton{ugLangLoopsBreakPageFull4}{\hidepaste}
+\tab{5}\spadcommand{repeat
+  repeat
+    if (i + j) > 10 then break
+    j := j + 1
+  if (i + j) > 10 then break
+  i := i + 1
+\free{i j }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugLangLoopsBreakPageEmpty4}
+\begin{paste}{ugLangLoopsBreakPageEmpty4}{ugLangLoopsBreakPagePatch4}
+\pastebutton{ugLangLoopsBreakPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{repeat
+  repeat
+    if (i + j) > 10 then break
+    j := j + 1
+  if (i + j) > 10 then break
+  i := i + 1
+\free{i j }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ug06.ht b/src/hyper/pages/ug06.ht
new file mode 100644
index 00000000..92842e31
--- /dev/null
+++ b/src/hyper/pages/ug06.ht
@@ -0,0 +1,2970 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\texht{\setcounter{chapter}{5}}{} % Chapter 6
+
+\newcommand{\pred}[1]{\subscriptIt{pred}{#1}}
+\newcommand{\expr}[1]{\subscriptIt{expression}{#1}}
+
+%
+\newcommand{\ugUserTitle}{User-Defined Functions, Macros and Rules}
+\newcommand{\ugUserNumber}{6.}
+%
+% =====================================================================
+\begin{page}{ugUserPage}{6. User-Defined Functions, Macros and Rules}
+% =====================================================================
+\beginscroll
+
+In this chapter we show you how to write functions and macros,
+and we explain how \Language{} looks for and applies them.
+We show some simple one-line examples of functions, together
+with larger ones that are defined piece-by-piece or through the use of
+piles.
+
+\beginmenu
+    \menudownlink{{6.1. Functions vs. Macros}}{ugUserFunMacPage}
+    \menudownlink{{6.2. Macros}}{ugUserMacrosPage}
+    \menudownlink{{6.3. Introduction to Functions}}{ugUserIntroPage}
+    \menudownlink{{6.4. Declaring the Type of Functions}}{ugUserDeclarePage}
+    \menudownlink{{6.5. One-Line Functions}}{ugUserOnePage}
+    \menudownlink{{6.6. Declared vs. Undeclared Functions}}{ugUserDecUndecPage}
+    \menudownlink{{6.7. Functions vs. Operations}}{ugUserDecOpersPage}
+    \menudownlink{{6.8. Delayed Assignments vs. Functions with No Arguments}}{ugUserDelayPage}
+    \menudownlink{{6.9. How \Language{} Determines What Function to Use}}{ugUserUsePage}
+    \menudownlink{{6.10. Compiling vs. Interpreting}}{ugUserCompIntPage}
+    \menudownlink{{6.11. Piece-Wise Function Definitions}}{ugUserPiecePage}
+    \menudownlink{{6.12. Caching Previously Computed Results}}{ugUserCachePage}
+    \menudownlink{{6.13. Recurrence Relations}}{ugUserRecurPage}
+    \menudownlink{{6.14. Making Functions from Objects}}{ugUserMakePage}
+    \menudownlink{{6.15. Functions Defined with Blocks}}{ugUserBlocksPage}
+    \menudownlink{{6.16. Free and Local Variables}}{ugUserFreeLocalPage}
+    \menudownlink{{6.17. Anonymous Functions}}{ugUserAnonPage}
+    \menudownlink{{6.18. Example: A Database}}{ugUserDatabasePage}
+    \menudownlink{{6.19. Example: A Famous Triangle}}{ugUserTrianglePage}
+    \menudownlink{{6.20. Example: Testing for Palindromes}}{ugUserPalPage}
+    \menudownlink{{6.21. Rules and Pattern Matching}}{ugUserRulesPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserFunMacTitle}{Functions vs. Macros}
+\newcommand{\ugUserFunMacNumber}{6.1.}
+%
+% =====================================================================
+\begin{page}{ugUserFunMacPage}{6.1. Functions vs. Macros}
+% =====================================================================
+\beginscroll
+
+A function is a program to perform some
+%-% \HDindex{function!vs. macro}{ugUserFunMacPage}{6.1.}{Functions vs. Macros}
+computation.
+%-% \HDindex{macro!vs. function}{ugUserFunMacPage}{6.1.}{Functions vs. Macros}
+Most functions have names so that it is easy to refer to them.
+A simple example of a function is one named
+\axiomFunFrom{abs}{Integer} which
+computes the absolute value of an integer.
+%
+\xtc{
+This is a use of the ``absolute value'' library function for integers.
+}{
+\spadpaste{abs(-8)}
+}
+\xtc{
+This is an unnamed function that does the same thing, using the
+``maps-to'' syntax \axiomSyntax{+->} that we discuss in
+\downlink{``\ugUserAnonTitle''}{ugUserAnonPage} in Section \ugUserAnonNumber\ignore{ugUserAnon}.
+}{
+\spadpaste{(x +-> if x < 0 then -x else x)(-8)}
+}
+%
+Functions can be used alone or serve as the building blocks for larger
+programs.
+Usually they return a value that you might want to use in the next stage
+of a computation, but not always (for example, see
+\downlink{`Exit'}{ExitXmpPage}\ignore{Exit} and \downlink{`Void'}{VoidXmpPage}\ignore{Void}).
+They may also read data from your keyboard, move information from one
+place to another, or format and display results on your screen.
+
+In \Language{}, as in mathematics, functions
+%-% \HDindex{function!parameters}{ugUserFunMacPage}{6.1.}{Functions vs. Macros}
+are usually \spadglossSee{parameterized}{parameterized form}.
+Each time you {\it call} (some people say \spadgloss{apply} or
+\spadglossSee{invoke}{invocation}) a function, you give
+%-% \HDindex{parameters to a function}{ugUserFunMacPage}{6.1.}{Functions vs. Macros}
+values to the parameters (variables).
+Such a value is called an \spadgloss{argument} of
+%-% \HDindex{function!arguments}{ugUserFunMacPage}{6.1.}{Functions vs. Macros}
+the function.
+\Language{} uses the arguments for the computation.
+In this way you get different results depending on what you ``feed'' the
+function.
+
+Functions can have local variables or refer to global variables in the
+workspace.
+\Language{} can often \spadglossSee{compile}{compiler} functions so that
+they execute very efficiently.
+Functions can be passed as arguments to other functions.
+
+Macros are textual substitutions.
+They are used to clarify the meaning of constants or expressions and to be
+templates for frequently used expressions.
+Macros can be parameterized but they are not objects that can be passed as
+arguments to functions.
+In effect, macros are extensions to the \Language{} expression parser.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserMacrosTitle}{Macros}
+\newcommand{\ugUserMacrosNumber}{6.2.}
+%
+% =====================================================================
+\begin{page}{ugUserMacrosPage}{6.2. Macros}
+% =====================================================================
+\beginscroll
+
+A \spadgloss{macro} provides general textual substitution of
+%-% \HDindex{macro}{ugUserMacrosPage}{6.2.}{Macros}
+an \Language{} expression for a name.
+You can think of a macro as being a generalized abbreviation.
+You can only have one macro in your workspace with
+a given name, no matter how many arguments it has.
+
+\beginImportant
+The two general forms for macros are
+\centerline{{{\tt macro} {\it name} {\tt ==} {\it body} }}
+\centerline{{{\tt macro} {\it name(arg1,...)} {\tt ==} {\it body}}}
+where the body of the macro can be any \Language{} expression.
+\endImportant
+
+%
+\xtc{
+For example, suppose you decided that you
+like to use \axiom{df} for \axiomFun{D}.
+You define the macro \axiom{df} like this.
+}{
+\spadpaste{macro df == D \bound{df}}
+}
+\xtc{
+Whenever you type \axiom{df}, the system expands it to
+\axiomFun{D}.
+}{
+\spadpaste{df(x**2 + x + 1,x) \free{df}}
+}
+\xtc{
+Macros can be parameterized and so can be used for many different
+kinds of objects.
+}{
+\spadpaste{macro ff(x) == x**2 + 1 \bound{ff}}
+}
+\xtc{
+Apply it to a number, a symbol, or an expression.
+}{
+\spadpaste{ff z \free{ff}}
+}
+\xtc{
+Macros can also be nested, but you get an error message if you
+run out of space because of an infinite nesting loop.
+}{
+\spadpaste{macro gg(x) == ff(2*x - 2/3) \bound{gg}\free{ff}}
+}
+\xtc{
+This new macro is fine as it does not produce a loop.
+}{
+\spadpaste{gg(1/w) \free{gg}}
+}
+%
+\xtc{
+This, however, loops since \axiom{gg} is
+defined in terms of \axiom{ff}.
+}{
+\spadpaste{macro ff(x) == gg(-x) \free{gg}}
+}
+\xtc{
+The body of a macro can be a block.
+}{
+\spadpaste{macro next == (past := present; present := future; future := past + present) \bound{next}}
+}
+\xtc{
+Before entering \axiom{next}, we need
+values for \axiom{present} and \axiom{future}.
+}{
+\spadpaste{present : Integer := 0 \bound{present}}
+}
+\xtc{
+}{
+\spadpaste{future : Integer := 1 \bound{future}}
+}
+\xtc{
+Repeatedly evaluating \axiom{next} produces the next Fibonacci number.
+}{
+\spadpaste{next \free{future}\free{present}}
+}
+\xtc{
+And the next one.
+}{
+\spadpaste{next \free{future}\free{present}}
+}
+\xtc{
+Here is the infinite stream of the rest of the Fibonacci numbers.
+}{
+\spadpaste{[next for i in 1..] \free{future}\free{present}}
+}
+\xtc{
+Bundle all the above lines into a single macro.
+}{
+\begin{spadsrc}[\bound{fibstr}]
+macro fibStream ==
+  present : Integer := 1
+  future : Integer := 1
+  [next for i in 1..] where
+    macro next ==
+      past := present
+      present := future
+      future := past + present
+\end{spadsrc}
+}
+\xtc{
+Use \axiomFunFrom{concat}{Stream} to start with the first two
+%-% \HDindex{Fibonacci numbers}{ugUserMacrosPage}{6.2.}{Macros}
+Fibonacci numbers.
+}{
+\spadpaste{concat([0,1],fibStream) \free{fibstr}}
+}
+\xtc{
+An easier way to compute these numbers is to
+use the library operation \axiomFun{fibonacci}.
+}{
+\spadpaste{[fibonacci i for i in 1..]}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserIntroTitle}{Introduction to Functions}
+\newcommand{\ugUserIntroNumber}{6.3.}
+%
+% =====================================================================
+\begin{page}{ugUserIntroPage}{6.3. Introduction to Functions}
+% =====================================================================
+\beginscroll
+
+Each name in your workspace can refer to a single object.
+This may be any kind of object including a function.
+You can use interactively any function from the library or any that you
+define in the workspace.
+In the library the same name can have very many functions, but you
+can have only one function with a given name, although it can have any
+number of arguments that you choose.
+
+If you define a function in the workspace that has the same name and number
+of arguments as one in the library, then your definition takes precedence.
+In fact, to get the library function you must
+\spadglossSee{package-call}{package call} it (see \downlink{``\ugTypesPkgCallTitle''}{ugTypesPkgCallPage} in Section \ugTypesPkgCallNumber\ignore{ugTypesPkgCall}).
+
+To use a function in \Language{}, you apply it to its arguments.
+Most functions are applied by entering the name of the function followed by
+its argument or arguments.
+\xtc{
+}{
+\spadpaste{factor(12)}
+}
+%
+\xtc{
+Some functions like \axiomOp{+} have {\it infix} \spadgloss{operators} as names.
+}{
+\spadpaste{3 + 4}
+}
+\xtc{
+The function \axiomOp{+} has two arguments.
+When you give it more than two arguments,
+\Language{} groups the arguments to the left.
+This expression is equivalent to \axiom{(1 + 2) + 7}.
+}{
+\spadpaste{1 + 2 + 7}
+}
+
+All operations, including infix operators, can be written in prefix form,
+that is, with the operation name followed by the arguments
+in parentheses.
+For example, \axiom{2 + 3} can alternatively be written as \axiom{+(2,3)}.
+But \axiom{+(2,3,4)} is an error since \axiomOp{+}
+takes only two arguments.
+
+Prefix operations are generally applied before the infix operation.
+Thus \axiom{factorial 3 + 1} means \axiom{factorial(3) + 1} producing
+\axiom{7}, and
+\axiom{- 2 + 5} means \axiom{(-2) + 5} producing \axiom{3}.
+An example of a prefix operator is prefix \axiomOp{-}.
+For example, \axiom{- 2 + 5} converts to \axiom{(- 2) + 5} producing
+the value \axiom{3}.
+Any prefix function taking two arguments can be written in
+an infix manner by putting an
+ampersand (\axiomSyntax{\&}) before the name.
+Thus \axiom{D(2*x,x)} can be written as
+\axiom{2*x \&D x} returning \axiom{2}.
+
+Every function in \Language{} is identified by
+a \spadgloss{name} and \spadgloss{type}.\footnote{An exception is
+an ``anonymous function''
+discussed in
+\downlink{``\ugUserAnonTitle''}{ugUserAnonPage} in Section \ugUserAnonNumber\ignore{ugUserAnon}.}
+The type of a function is always a mapping of the form
+\spadsig{Source}{Target}
+where \axiom{Source} and \axiom{Target} are types.
+To enter a type from the keyboard, enter the arrow by using
+a hyphen \axiomSyntax{-} followed by a greater-than sign
+\axiomSyntax{>}, e.g. {\tt Integer -> Integer}.
+
+Let's go back to \axiomOp{+}.
+There are many \axiomOp{+} functions in the
+\Language{} library: one for integers, one for floats, another for
+rational numbers, and so on.
+These \axiomOp{+} functions have different types and thus are
+different functions.
+You've seen examples of this \spadgloss{overloading}
+before---using the same name for different functions.
+Overloading is the rule rather than the exception.
+You can add two integers, two polynomials, two matrices or
+two power series.
+These are all done with the same function name
+but with different functions.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserDeclareTitle}{Declaring the Type of Functions}
+\newcommand{\ugUserDeclareNumber}{6.4.}
+%
+% =====================================================================
+\begin{page}{ugUserDeclarePage}{6.4. Declaring the Type of Functions}
+% =====================================================================
+\beginscroll
+
+In \downlink{``\ugTypesDeclareTitle''}{ugTypesDeclarePage} in Section \ugTypesDeclareNumber\ignore{ugTypesDeclare} we discussed how to declare a variable
+to restrict the kind of values that can be assigned to it.
+In this section we show how to declare a variable that refers to
+function objects.
+
+\beginImportant
+A function is an object of type
+\centerline{{\spadsig{Source}{Type}}}
+where \axiom{Source} and \axiom{Target} can be any type.
+A common type for \axiom{Source} is
+\axiomType{Tuple}(\subscriptIt{T}{1}, \ldots, \subscriptIt{T}{n}),
+usually written
+(\subscriptIt{T}{1}, \ldots, \subscriptIt{T}{n}),
+to indicate a function of \axiom{n} arguments.
+\endImportant
+
+\xtc{
+If \axiom{g} takes an \axiomType{Integer}, a \axiomType{Float} and
+another \axiomType{Integer}, and returns a
+\axiomType{String}, the declaration is written this way.
+}{
+\spadpaste{g: (Integer,Float,Integer) -> String}
+}
+\xtc{
+The types need not be written fully; using abbreviations, the above
+declaration is:
+}{
+\spadpaste{g: (INT,FLOAT,INT) -> STRING}
+}
+\xtc{
+It is possible for a function to take no arguments.
+If \axiom{ h} takes no arguments
+but returns a \axiomType{Polynomial} \axiomType{Integer}, any
+of the following declarations is acceptable.
+}{
+\spadpaste{h: () -> POLY INT}
+}
+\xtc{
+}{
+\spadpaste{h: () -> Polynomial INT}
+}
+\xtc{
+}{
+\spadpaste{h: () -> POLY Integer}
+}
+
+
+\beginImportant
+Functions can also be declared when they are being defined.
+The syntax for combined declaration/definition is:
+\centerline{{\frenchspacing{\tt {\it functionName}(\subscriptIt{parm}{1}: \subscriptIt{parmType}{1}, \ldots, \subscriptIt{parm}{N}: \subscriptIt{parmType}{N}): {\it functionReturnType}}}}
+\endImportant
+
+The following definition fragments show how this can be done for
+the functions \axiom{g} and \axiom{h} above.
+\begin{verbatim}
+g(arg1: INT, arg2: FLOAT, arg3: INT): STRING == ...
+
+h(): POLY INT == ...
+\end{verbatim}
+
+A current restriction on function declarations is that they must
+involve fully specified types (that is, cannot include modes involving
+explicit or implicit \axiomSyntax{?}).
+For more information on declaring things in general, see
+\downlink{``\ugTypesDeclareTitle''}{ugTypesDeclarePage} in Section \ugTypesDeclareNumber\ignore{ugTypesDeclare}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserOneTitle}{One-Line Functions}
+\newcommand{\ugUserOneNumber}{6.5.}
+%
+% =====================================================================
+\begin{page}{ugUserOnePage}{6.5. One-Line Functions}
+% =====================================================================
+\beginscroll
+
+As you use \Language{}, you will find that you will write many short functions
+%-% \HDindex{function!one-line definition}{ugUserOnePage}{6.5.}{One-Line Functions}
+to codify sequences of operations that you often perform.
+In this section we write some simple one-line functions.
+
+\xtc{
+This is a simple recursive factorial function for positive integers.
+}{
+\spadpaste{fac n == if n < 3 then n else n * fac(n-1) \bound{fac}}
+}
+\xtc{
+}{
+\spadpaste{fac 10 \free{fac}}
+}
+%>> Thankfully, the $ is no longer needed in the next example.
+\xtc{
+This function computes \axiom{1 + 1/2 + 1/3 + ... + 1/n}.
+}{
+\spadpaste{s n == reduce(+,[1/i for i in 1..n]) \bound{s}}
+}
+\xtc{
+}{
+\spadpaste{s 50 \free{s}}
+}
+\xtc{
+This function computes a Mersenne number, several of which are prime.
+%-% \HDindex{Mersenne number}{ugUserOnePage}{6.5.}{One-Line Functions}
+}{
+\spadpaste{mersenne i == 2**i - 1 \bound{mersenne}}
+}
+\xtc{
+If you type \axiom{mersenne}, \Language{} shows you
+the function definition.
+}{
+\spadpaste{mersenne \free{mersenne}}
+}
+\xtc{
+Generate a stream of Mersenne numbers.
+}{
+\spadpaste{[mersenne i for i in 1..] \free{mersenne}}
+}
+\xtc{
+Create a stream of those values of \axiom{i} such that
+\axiom{mersenne(i)} is prime.
+}{
+\spadpaste{mersenneIndex := [n for n in 1.. | prime?(mersenne(n))] \bound{mersenneIndex}\free{mersenne}}
+}
+\xtc{
+Finally, write a function that returns the \eth{\axiom{n}} Mersenne
+prime.
+}{
+\spadpaste{mersennePrime n == mersenne mersenneIndex(n) \free{mersenne mersenneIndex}\bound{mersennePrime}}
+}
+\xtc{
+}{
+\spadpaste{mersennePrime 5 \free{mersennePrime}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserDecUndecTitle}{Declared vs. Undeclared Functions}
+\newcommand{\ugUserDecUndecNumber}{6.6.}
+%
+% =====================================================================
+\begin{page}{ugUserDecUndecPage}{6.6. Declared vs. Undeclared Functions}
+% =====================================================================
+\beginscroll
+
+If you declare the type of a function, you can apply
+it to any data that can be converted to the source type
+of the function.
+
+\labelSpace{2pc}
+\xtc{
+Define \userfun{f} with type \spadsig{Integer}{Integer}.
+}{
+\spadpaste{f(x: Integer): Integer == x + 1 \bound{f}}
+}
+\xtc{
+The function
+\userfun{f} can be applied to integers, \ldots
+}{
+\spadpaste{f 9 \free{f}}
+}
+\xtc{
+and to values that convert to integers, \ldots
+}{
+\spadpaste{f(-2.0) \free{f}}
+}
+\xtc{
+but not to values that cannot be converted to integers.
+}{
+\spadpaste{f(2/3) \free{f}}
+}
+
+To make the function over a wide range of types, do not
+declare its type.
+\xtc{
+Give the same definition with no declaration.
+}{
+\spadpaste{g x == x + 1 \bound{g}}
+}
+\xtc{
+If \axiom{x + 1} makes sense, you can apply \userfun{g} to \axiom{x}.
+}{
+\spadpaste{g 9 \free{g}}
+}
+\xtc{
+A version of \userfun{g} with different argument types
+get compiled for each new kind of argument used.
+}{
+\spadpaste{g(2/3)  \free{g}}
+}
+\xtc{
+Here \axiom{x+1} for \axiom{x = "axiom"} makes no sense.
+}{
+\spadpaste{g("axiom")\free{g}}
+}
+
+As you will see in \downlink{``\ugCategoriesTitle''}{ugCategoriesPage} in Chapter \ugCategoriesNumber\ignore{ugCategories},
+\Language{} has a formal idea of categories for what ``makes sense.''
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserDecOpersTitle}{Functions vs. Operations}
+\newcommand{\ugUserDecOpersNumber}{6.7.}
+%
+% =====================================================================
+\begin{page}{ugUserDecOpersPage}{6.7. Functions vs. Operations}
+% =====================================================================
+\beginscroll
+
+A function is an object that you can create, manipulate, pass to,
+and return from functions (for some interesting examples of
+library functions that manipulate functions, see
+\downlink{`MappingPackage1'}{MappingPackageOneXmpPage}\ignore{MappingPackage1}).
+Yet, we often seem to use the term \spadgloss{operation} and
+function interchangeably in \Language{}.
+What is the distinction?
+
+First consider values and types associated with some variable \axiom{n} in
+your workspace.
+You can make the declaration \axiom{n : Integer}, then assign \axiom{n} an
+integer value.
+You then speak of the integer \axiom{n}.
+However, note that the integer is not the name \axiom{n} itself, but
+the value that you assign to \axiom{n}.
+
+Similarly, you can declare a variable \axiom{f} in your workspace to have
+type \spadsig{Integer}{Integer}, then assign \axiom{f}, through a definition
+or an assignment of an anonymous function.
+You then speak of the function \axiom{f}.
+However, the function is not \axiom{f}, but the value that you
+assign to \axiom{f}.
+
+A function is a value, in fact, some machine code for doing something.
+Doing what?
+Well, performing some \spadgloss{operation}.
+Formally, an operation consists of the constituent parts of \axiom{f} in your
+workspace, excluding the value; thus an operation has a name and a type.
+An operation is what domains and packages export.
+Thus \axiomType{Ring} exports one operation \axiomOp{+}.
+Every ring also exports this operation.
+Also, the author of every ring in the system is obliged under contract
+(see \downlink{``\ugPackagesAbstractTitle''}{ugPackagesAbstractPage} in Section \ugPackagesAbstractNumber\ignore{ugPackagesAbstract})
+to provide an implementation for this operation.
+
+This chapter is all about functions---how you create them interactively and
+how you apply them to meet your needs.
+In \downlink{``\ugPackagesTitle''}{ugPackagesPage} in Chapter \ugPackagesNumber\ignore{ugPackages} you will learn how to create them for the
+\Language{} library.
+Then in \downlink{``\ugCategoriesTitle''}{ugCategoriesPage} in Chapter \ugCategoriesNumber\ignore{ugCategories}, you will learn about categories and
+exported operations.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserDelayTitle}{Delayed Assignments vs. Functions with No Arguments}
+\newcommand{\ugUserDelayNumber}{6.8.}
+%
+% =====================================================================
+\begin{page}{ugUserDelayPage}{6.8. Delayed Assignments vs. Functions with No Arguments}
+% =====================================================================
+\beginscroll
+
+In \downlink{``\ugLangAssignTitle''}{ugLangAssignPage} in Section \ugLangAssignNumber\ignore{ugLangAssign} we discussed the difference between immediate and
+%-% \HDindex{function!with no arguments}{ugUserDelayPage}{6.8.}{Delayed Assignments vs. Functions with No Arguments}
+delayed assignments.
+In this section we show the difference between delayed
+assignments and functions of no arguments.
+
+\labelSpace{2pc}
+\xtc{
+A function of no arguments is sometimes called a {\it nullary function.}
+}{
+\spadpaste{sin24() == sin(24.0) \bound{sin24}}
+}
+\xtc{
+You must use the parentheses (\axiomSyntax{()}) to evaluate it.
+Like a delayed assignment, the right-hand-side of a function evaluation
+is not evaluated until the left-hand-side is used.
+}{
+\spadpaste{sin24() \free{sin24}}
+}
+\xtc{
+If you omit the parentheses, you just get the function definition.
+%(Note how the explicit floating point number in the definition
+%has been translated into a function call involving a mantissa,
+%exponent and radix.)
+}{
+\spadpaste{sin24 \free{sin24}}
+}
+\xtc{
+You do not use the parentheses \axiomSyntax{()} in a delayed assignment\ldots
+}{
+\spadpaste{cos24 == cos(24.0) \bound{cos24}}
+}
+\xtc{
+nor in the evaluation.
+}{
+\spadpaste{cos24 \free{cos24}}
+}
+The only syntactic difference between delayed assignments
+and nullary functions is that you use \axiomSyntax{()} in the latter case.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserUseTitle}{How \Language{} Determines What Function to Use}
+\newcommand{\ugUserUseNumber}{6.9.}
+%
+% =====================================================================
+\begin{page}{ugUserUsePage}{6.9. How \Language{} Determines What Function to Use}
+% =====================================================================
+\beginscroll
+
+What happens if you define a function that has the same name as a library
+function?
+Well, if your function has the same name and number of arguments (we
+sometimes say \spadgloss{arity}) as another function
+in the library, then your function covers up the library function.
+If you want then to call the library function, you will have to package-call
+it.
+\Language{} can use both the functions you write and those that come
+from the library.
+Let's do a simple example to illustrate this.
+\xtc{
+Suppose you (wrongly!) define \userfun{sin} in this way.
+}{
+\spadpaste{sin x == 1.0 \bound{sin}}
+}
+\xtc{
+The value \axiom{1.0} is returned for any argument.
+}{
+\spadpaste{sin 4.3 \free{sin}}
+}
+\xtc{
+If you want the library operation, we have to package-call it
+(see \downlink{``\ugTypesPkgCallTitle''}{ugTypesPkgCallPage} in Section \ugTypesPkgCallNumber\ignore{ugTypesPkgCall}
+for more information).
+}{
+\spadpaste{sin(4.3)\$Float}
+}
+\xtc{
+}{
+\spadpaste{sin(34.6)\$Float}
+}
+\xtc{
+Even worse, say we accidentally used the same name as a library
+function in the function.
+}{
+\spadpaste{sin x == sin x \bound{sin1}}
+}
+\xtc{
+Then \Language{} definitely does not understand us.
+}{
+\spadpaste{sin 4.3 \free{sin1}}
+}
+\xtc{
+Again, we could package-call the inside function.
+}{
+\spadpaste{sin x == sin(x)\$Float \bound{sin2}}
+}
+\xtc{
+}{
+\spadpaste{sin 4.3 \free{sin2}}
+}
+Of course, you are unlikely to make such obvious errors.
+It is more probable that you would write a function and in the body use a
+function that you think is a library function.
+If you had also written a function by that same name, the library function
+would be invisible.
+
+How does \Language{} determine what library function to call?
+It very much depends on the particular example, but the simple case of
+creating the polynomial
+\axiom{x + 2/3} will give you an idea.
+\indent{4}
+\beginitems
+\item[1. ] The \axiom{x} is analyzed and its default type is
+\axiomType{Variable(x)}.
+\item[2. ] The \axiom{2} is analyzed and its default type is
+\axiomType{PositiveInteger}.
+\item[3. ] The \axiom{3} is analyzed and its default type is
+\axiomType{PositiveInteger}.
+\item[4. ] Because the arguments to \axiomOp{/} are integers, \Language{}
+gives the expression \axiom{2/3} a default target type of
+\axiomType{Fraction(Integer)}.
+\item[5. ] \Language{} looks in \axiomType{PositiveInteger} for \axiomOp{/}.
+It is not found.
+\item[6. ] \Language{} looks in \axiomType{Fraction(Integer)} for \axiomOp{/}.
+It is found for arguments of type \axiomType{Integer}.
+\item[7. ] The \axiom{2} and \axiom{3} are converted to objects of type
+\axiomType{Integer} (this is trivial) and \axiomOp{/} is applied,
+creating an object of type \axiomType{Fraction(Integer)}.
+\item[8. ] No \axiomOp{+} for arguments of types \axiomType{Variable(x)} and
+\axiomType{Fraction(Integer)} are found in either domain.
+\item[9. ] \Language{} resolves
+%-% \HDindex{resolve}{ugUserUsePage}{6.9.}{How \Language{} Determines What Function to Use}
+(see \downlink{``\ugTypesResolveTitle''}{ugTypesResolvePage} in Section \ugTypesResolveNumber\ignore{ugTypesResolve})
+the types and gets \axiomType{Polynomial (Fraction (Integer))}.
+\item[10. ] The \axiom{x} and the \axiom{2/3} are converted to objects of this
+type and \axiomOp{+} is applied, yielding the answer, an object of type
+\axiomType{Polynomial (Fraction (Integer))}.
+\enditems
+\indent{0}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserCompIntTitle}{Compiling vs. Interpreting}
+\newcommand{\ugUserCompIntNumber}{6.10.}
+%
+% =====================================================================
+\begin{page}{ugUserCompIntPage}{6.10. Compiling vs. Interpreting}
+% =====================================================================
+\beginscroll
+
+When possible, \Language{} completely determines the type of every object in
+a function, then translates the function definition to \Lisp{} or
+to machine code (see next section).
+This translation,
+%-% \HDindex{function!compiler}{ugUserCompIntPage}{6.10.}{Compiling vs. Interpreting}
+called \spadglossSee{compilation}{compiler}, happens the first time you call
+the function and results in a computational delay.
+Subsequent function calls with the same argument types use the compiled
+version of the code without delay.
+
+If \Language{} cannot determine the type of everything, the
+function may still be executed
+%-% \HDindex{function!interpretation}{ugUserCompIntPage}{6.10.}{Compiling vs. Interpreting}
+but
+%-% \HDindex{interpret-code mode}{ugUserCompIntPage}{6.10.}{Compiling vs. Interpreting}
+in \spadglossSee{interpret-code mode}{interpreter} :
+each statement in the function is analyzed and executed as the control
+flow indicates.
+This process is slower than executing a compiled function, but it
+allows the execution of code that may involve objects whose types
+change.
+
+\beginImportant
+If \Language{} decides that it cannot compile the code, it
+issues a message stating the problem and then the following
+message:
+%
+\centerline{{{\bf We will attempt to step through and interpret the code.}}}
+%
+This is not a time to panic.
+%-% \HDindex{panic!avoiding}{ugUserCompIntPage}{6.10.}{Compiling vs. Interpreting}
+Rather, it just means that what you gave to \Language{}
+is somehow ambiguous: either it is not specific enough to be analyzed
+completely, or it is beyond \Language{}'s present interactive
+compilation abilities.
+\endImportant
+
+\xtc{
+This function runs in interpret-code mode, but it does not compile.
+}{
+\begin{spadsrc}[\bound{varPolys}]
+varPolys(vars) ==
+  for var in vars repeat
+    output(1 :: UnivariatePolynomial(var,Integer))
+\end{spadsrc}
+}
+\xtc{
+For \axiom{vars} equal to \axiom{['x, 'y, 'z]}, this function displays
+\axiom{1} three times.
+}{
+\spadpaste{varPolys ['x,'y,'z] \free{varPolys}}
+}
+\xtc{
+The type of the argument to \axiomFun{output} changes in each iteration,
+so \Language{} cannot compile the function.
+In this case, even the inner loop by itself would have a problem:
+}{
+\begin{spadsrc}
+for var in ['x,'y,'z] repeat
+  output(1 :: UnivariatePolynomial(var,Integer))
+\end{spadsrc}
+}
+
+Sometimes you can help a function to compile by using an extra conversion
+or by using \axiom{pretend}.
+\spadkey{pretend}
+See \downlink{``\ugTypesSubdomainsTitle''}{ugTypesSubdomainsPage} in Section \ugTypesSubdomainsNumber\ignore{ugTypesSubdomains} for details.
+
+When a function is compilable, you have the choice of whether it is
+compiled to \Lisp{} and then interpreted by the \Lisp{}
+interpreter or then further compiled from \Lisp{} to machine code.
+%-% \HDindex{machine code}{ugUserCompIntPage}{6.10.}{Compiling vs. Interpreting}
+The option is controlled via \spadcmd{)set functions compile}.
+%-% \HDsyscmdindex{set function compile}{ugUserCompIntPage}{6.10.}{Compiling vs. Interpreting}
+Issue \spadcmd{)set functions compile on} to compile all the way to
+machine code.
+With
+the default setting \spadcmd{)set functions compile off},
+\Language{} has its \Lisp{} code interpreted
+because the overhead of further compilation is larger than the run-time
+of most of the functions our users have defined.
+You may find that selectively turning this option on and off will
+%-% \HDindex{performance}{ugUserCompIntPage}{6.10.}{Compiling vs. Interpreting}
+give you the best performance in your particular application.
+For example, if you are writing functions for graphics applications
+where hundreds of points are being computed, it is almost certainly true
+that you will get the best performance by issuing
+\spadcmd{)set functions compile on}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserPieceTitle}{Piece-Wise Function Definitions}
+\newcommand{\ugUserPieceNumber}{6.11.}
+%
+% =====================================================================
+\begin{page}{ugUserPiecePage}{6.11. Piece-Wise Function Definitions}
+% =====================================================================
+\beginscroll
+
+To move beyond functions defined in one line, we introduce in this section
+functions that are defined piece-by-piece.
+That is, we say ``use this definition when the argument is such-and-such and
+use this other definition when the argument is that-and-that.''
+
+\beginmenu
+    \menudownlink{{6.11.1. A Basic Example}}{ugUserPieceBasicPage}
+    \menudownlink{{6.11.2. Picking Up the Pieces}}{ugUserPiecePickingPage}
+    \menudownlink{{6.11.3. Predicates}}{ugUserPiecePredPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserPieceBasicTitle}{A Basic Example}
+\newcommand{\ugUserPieceBasicNumber}{6.11.1.}
+%
+% =====================================================================
+\begin{page}{ugUserPieceBasicPage}{6.11.1. A Basic Example}
+% =====================================================================
+\beginscroll
+
+There are many other ways to define a factorial function for nonnegative
+integers.
+You might
+%-% \HDindex{function!piece-wise definition}{ugUserPieceBasicPage}{6.11.1.}{A Basic Example}
+say
+%-% \HDindex{piece-wise function definition}{ugUserPieceBasicPage}{6.11.1.}{A Basic Example}
+factorial of \axiom{0} is \axiom{1,} otherwise factorial of \axiom{n} is
+\axiom{n} times factorial of \axiom{n-1}.
+Here is one way to do this in \Language{}.
+%
+\xtc{
+Here is the value for \axiom{n = 0}.
+}{
+\spadpaste{fact(0) == 1 \bound{fact0}}
+}
+\xtc{
+Here is the value for \axiom{n > 0}.
+The vertical bar \axiomSyntax{|} means
+``such that''.
+}{
+\spadpaste{fact(n | n > 0) == n * fact(n - 1) \free{fact0}\bound{factn}}
+}
+%-% \HDindex{such that}{ugUserPieceBasicPage}{6.11.1.}{A Basic Example}
+%>> am moving this back
+%The vertical bar \axiomSyntax{|} is read as ``such that'' and so
+%\index{such that}
+%the second line means that that part of the definition for \userfun{fact}
+%is for any \axiom{n} such that \axiom{n} is greater than 0.
+%In fact, the first line is really just a shorthand expression for
+%\axiom{fact(n | n = 0) == 1}.
+%>> prefer scratching next 4 lines
+%We are implicitly using a \spadgloss{predicate} with a \axiomSyntax{|} in
+%this line (see \downlink{``\ugUserPiecePredTitle''}{ugUserPiecePredPage} in Section \ugUserPiecePredNumber\ignore{ugUserPiecePred} for more on predicates).
+%So this piece of the function is applicable to all (the not so many!)
+%values of \axiom{n} that are equal to zero.
+\xtc{
+What is the value for \axiom{n = 3}?
+}{
+\spadpaste{fact(3) \free{factn}}
+}
+\xtc{
+What is the value for \axiom{n = -3}?
+}{
+\spadpaste{fact(-3) \free{factn}}
+}
+\xtc{
+Now for a second definition.
+Here is the value for \axiom{n = 0}.
+}{
+\spadpaste{facto(0) == 1 \bound{facto0}}
+}
+\xtc{
+Give an error message if \axiom{n < 0}.
+}{
+\spadpaste{facto(n | n < 0) == error "arguments to facto must be non-negative" \free{facto0}\bound{factop}}
+}
+\xtc{
+Here is the value otherwise.
+}{
+\spadpaste{facto(n) == n * facto(n - 1) \free{factop}\bound{facton}}
+}
+\xtc{
+What is the value for \axiom{n = 7}?
+}{
+\spadpaste{facto(3) \free{facton}}
+}
+\xtc{
+What is the value for \axiom{n = -7}?
+}{
+\spadpaste{facto(-7) \free{facton}}
+}
+\xtc{
+To see the current piece-wise definition of a function,
+use \spadsys{)display value}.
+}{
+\spadpaste{)display value facto \free{facton}}
+}
+
+In general a {\it piece-wise definition} of a function consists of two or
+more parts.
+Each part gives a ``piece'' of the entire definition.
+\Language{} collects the pieces of a function as you enter them.
+When you ask for a value of the function, it then ``glues''
+the pieces together to form a function.
+
+The two piece-wise definitions for the factorial function
+are examples of recursive functions, that is, functions that
+are defined in terms of themselves.
+Here is an interesting doubly-recursive function.
+This function returns the value \axiom{11} for all positive integer arguments.
+\xtc{
+Here is the first of two pieces.
+}{
+\spadpaste{eleven(n | n < 1) == n + 11\bound{ff0}}
+}
+\xtc{
+And the general case.
+}{
+\spadpaste{eleven(m) == eleven(eleven(m - 12))\bound{ff1}\free{ff0}}
+}
+\xtc{
+Compute \axiom{elevens}, the infinite stream
+of values of \axiom{eleven}.
+}{
+\spadpaste{elevens := [eleven(i) for i in 0..]\bound{ff2}\free{ff1}}
+}
+\xtc{
+What is the value at \axiom{n = 200}?
+}{
+\spadpaste{elevens 200\free{ff2}}
+}
+\xtc{
+What is the \Language{}'s definition of \axiom{eleven}?
+}{
+\spadpaste{)display value eleven\free{ff2}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserPiecePickingTitle}{Picking Up the Pieces}
+\newcommand{\ugUserPiecePickingNumber}{6.11.2.}
+%
+% =====================================================================
+\begin{page}{ugUserPiecePickingPage}{6.11.2. Picking Up the Pieces}
+% =====================================================================
+\beginscroll
+
+Here are the details about how \Language{} creates a function from its
+pieces.
+\Language{} converts the \eth{\axiom{i}} piece of a function definition into a
+conditional expression of the form: \axiom{if} \pred{i} \axiom{then}
+\expr{i}.
+If any new piece has a \pred{i} that is identical\footnote{after all
+variables are uniformly named} to an earlier \pred{j}, the earlier piece is
+removed.
+Otherwise, the new piece is always added at the end.
+
+\beginImportant
+If there are \axiom{n} pieces to a function definition for \axiom{f},
+the function defined \axiom{f} is: \newline
+%
+\texht{\hspace*{3pc}}{\tab{6}}
+{\tt if} \pred{1} {\tt then} \expr{1} {\tt else}\newline
+\texht{\hspace*{6pc}}{\tab{12}}. . . \newline
+\texht{\hspace*{3pc}}{\tab{6}}
+{\tt if} \pred{n} {\tt then} \expr{n} {\tt else}\newline
+\texht{\hspace*{3pc}}{\tab{6}}
+{\tt  error "You did not define f for argument <arg>."}
+%
+\endImportant
+
+You can give definitions of any number of mutually recursive function
+definitions, piece-wise or otherwise.
+No computation is done until you ask for a value.
+When you do ask for a value, all the relevant definitions are gathered,
+analyzed, and translated into separate functions and compiled.
+
+\xtc{
+Let's recall the definition of \userfun{eleven} from
+\texht{the previous section}{\downlink{``\ugUserPieceBasicTitle''}{ugUserPieceBasicPage} in Section \ugUserPieceBasicNumber\ignore{ugUserPieceBasic}}.
+}{
+\spadpaste{eleven(n | n < 1) == n + 11\bound{ff0}}
+}
+\xtc{
+}{
+\spadpaste{eleven(m) == eleven(eleven(m - 12))\bound{ff1}\free{ff0}}
+}
+
+A similar doubly-recursive function below produces \axiom{-11} for all
+negative positive integers.
+If you haven't worked out why or how \userfun{eleven} works,
+the structure of this definition gives a clue.
+\xtc{
+This definition we write as a block.
+}{
+\begin{spadsrc}[\bound{rf1}]
+minusEleven(n) ==
+  n >= 0 => n - 11
+  minusEleven (5 + minusEleven(n + 7))
+\end{spadsrc}
+}
+\xtc{
+Define \axiom{s(n)} to be the
+sum of plus and minus ``eleven'' functions divided by \axiom{n}.
+Since \axiom{11 - 11 = 0}, we define \axiom{s(0)} to be \axiom{1}.
+}{
+\spadpaste{s(0) == 1\bound{rf2}}
+}
+\xtc{
+And the general term.
+}{
+\spadpaste{s(n) == (eleven(n) + minusEleven(n))/n\bound{rf3}\free{rf2 rf1 ff1}}
+}
+\xtc{
+What are the first ten values of \axiom{s}?
+}{
+\spadpaste{[s(n) for n in 0..]\free{rf3}}
+}
+%% interpreter puts the rule at the end - should fix
+\Language{} can create infinite streams in the positive direction (for
+example, for index values \axiom{0,1, \ldots}) or negative direction (for
+example, for index values \axiom{0,-1,-2, \ldots}).
+Here we would like a stream of values of \axiom{s(n)} that is infinite in
+both directions.
+The function \axiom{t(n)} below returns the \eth{\axiom{n}} term of the infinite
+stream \axiom{[s(0), s(1), s(-1), s(2), s(-2), \ldots].}
+Its definition has three pieces.
+\xtc{
+Define the initial term.
+}{
+\spadpaste{t(1) == s(0)\bound{t1}\free{rf4}}
+}
+\xtc{
+The even numbered terms are the \axiom{s(i)} for positive \axiom{i}.
+We use \axiomOp{quo} rather than \axiomOp{/}
+since we want the result to be an integer.
+}{
+\spadpaste{t(n | even?(n)) == s(n quo 2)\free{t1}\bound{t2}}
+}
+\xtc{
+Finally, the odd numbered terms are the
+\axiom{s(i)} for negative \axiom{i}.
+In piece-wise definitions, you can use different variables
+to define different pieces. \Language{} will not get confused.
+}{
+\spadpaste{t(p) == s(- p quo 2)\free{t2}\bound{t3}}
+}
+\xtc{
+Look at the definition of \axiom{t}.
+In the first piece, the variable \axiom{n}
+was used; in the second piece, \axiom{p}.
+\Language{} always uses
+your last variable to display your definitions
+back to you.
+}{
+\spadpaste{)display value t\free{t2}}
+}
+\xtc{
+Create a series of values of \axiom{s} applied to
+alternating positive and negative arguments.
+}{
+\spadpaste{[t(i) for i in 1..]\free{t3}\bound{t4}}
+}
+\xtc{
+Evidently \axiom{t(n) = 1} for all \axiom{i.}
+Check it at \axiom{n= 100}.
+}{
+\spadpaste{t(100)\free{t4}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserPiecePredTitle}{Predicates}
+\newcommand{\ugUserPiecePredNumber}{6.11.3.}
+%
+% =====================================================================
+\begin{page}{ugUserPiecePredPage}{6.11.3. Predicates}
+% =====================================================================
+\beginscroll
+
+We have already seen some examples of
+%-% \HDindex{function!predicate}{ugUserPiecePredPage}{6.11.3.}{Predicates}
+predicates
+%-% \HDindex{predicate!in function definition}{ugUserPiecePredPage}{6.11.3.}{Predicates}
+(\downlink{``\ugUserPieceBasicTitle''}{ugUserPieceBasicPage} in Section \ugUserPieceBasicNumber\ignore{ugUserPieceBasic}).
+Predicates are \axiomType{Boolean}-valued expressions and \Language{} uses them
+for filtering collections
+(see \downlink{``\ugLangItsTitle''}{ugLangItsPage} in Section \ugLangItsNumber\ignore{ugLangIts})
+and for placing
+constraints on function arguments.
+In this section we discuss their latter usage.
+
+\xtc{
+The simplest use of a predicate is one you don't see at all.
+}{
+\spadpaste{opposite 'right == 'left}
+}
+\xtc{
+Here is a longer way to give the ``opposite definition.''
+}{
+\spadpaste{opposite (x | x = 'left) == 'right}
+}
+\xtc{
+Try it out.
+}{
+\spadpaste{for x in ['right,'left,'inbetween] repeat output opposite x}
+}
+
+Explicit predicates tell \Language{} that the given function definition
+piece is to be applied if the predicate evaluates to {\tt true} for the
+arguments to the function.
+You can use such ``constant'' arguments for integers,
+%-% \HDindex{function!constant argument}{ugUserPiecePredPage}{6.11.3.}{Predicates}
+strings, and quoted symbols.
+%-% \HDindex{constant function argument}{ugUserPiecePredPage}{6.11.3.}{Predicates}
+The \axiomType{Boolean} values \axiom{true} and \axiom{false} can also be used
+if qualified with ``\spad{@}'' or ``\spad{\$}'' and \axiomType{Boolean}.
+The following are all valid function definition fragments using
+constant arguments.
+\begin{verbatim}
+a(1) == ...
+b("unramified") == ...
+c('untested) == ...
+d(true@Boolean) == ...
+\end{verbatim}
+
+If a function has more than one argument,
+each argument can have its own predicate.
+However, if a predicate involves two or more arguments, it must be given
+{\it after} all the arguments mentioned in the predicate have been given.
+You are always safe to give
+a single predicate at the end of the argument list.
+\xtc{
+A function involving predicates on two arguments.
+}{
+\spadpaste{inFirstHalfQuadrant(x | x > 0,y | y < x) == true}
+}
+\xtc{
+This is incorrect as it gives a predicate on \axiom{y}
+before the argument \axiom{y} is given.
+}{
+\spadpaste{inFirstHalfQuadrant(x | x > 0 and y < x,y) == true}
+}
+\xtc{
+It is always correct to write the predicate at the end.
+}{
+\spadpaste{inFirstHalfQuadrant(x,y | x > 0 and y < x) == true \bound{ifq1a}}
+}
+\xtc{
+Here is the rest of the definition.
+}{
+\spadpaste{inFirstHalfQuadrant(x,y) == false \bound{ifq1b}}
+}
+\xtc{
+Try it out.
+}{
+\spadpaste{[inFirstHalfQuadrant(i,3) for i in 1..5]\bound{ifq1b}}
+}
+
+{\bf Remark:} Very old versions of \Language{} allowed predicates
+to be given after a {\tt when} keyword as in
+{\tt inFirstHalfQuadrant(x ,y) == true when x >0 and y < x}.
+This is no longer supported, is WRONG, and will cause a syntax
+error or strange behavior.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserCacheTitle}{Caching Previously Computed Results}
+\newcommand{\ugUserCacheNumber}{6.12.}
+%
+% =====================================================================
+\begin{page}{ugUserCachePage}{6.12. Caching Previously Computed Results}
+% =====================================================================
+\beginscroll
+
+By default, \Language{} does not save the values of any function.
+%-% \HDindex{function!caching values}{ugUserCachePage}{6.12.}{Caching Previously Computed Results}
+You can cause it to save values and not to recompute unnecessarily
+%-% \HDindex{remembering function values}{ugUserCachePage}{6.12.}{Caching Previously Computed Results}
+by using \spadcmd{)set functions cache}.
+%-% \HDsyscmdindex{set functions cache}{ugUserCachePage}{6.12.}{Caching Previously Computed Results}
+This should be used before the functions are defined or, at least, before
+they are executed.
+The word following ``cache'' should be \axiom{0} to turn off
+caching, a positive integer \axiom{n} to save the last \axiom{n}
+computed values or ``all'' to save all computed values.
+If you then give a list of names of functions, the caching
+only affects those functions.
+Use no list of names or ``all'' when you want to define the default
+behavior for functions not specifically mentioned in other
+\spadcmd{)set functions cache} statements.
+If you give no list of names, all functions will have the caching behavior.
+If you explicitly turn on caching for one or more names, you must
+explicitly turn off caching for those names when you want to stop
+saving their values.
+
+\xtc{
+This causes the functions \userfun{f} and \userfun{g} to have
+the last three computed values saved.
+}{
+\spadpaste{)set functions cache 3 f g \bound{cache}}
+}
+\xtc{
+This is a sample definition for \userfun{f}.
+}{
+\spadpaste{f x == factorial(2**x) \bound{fdef}\free{cache}}
+}
+\xtc{
+A message is displayed stating what \userfun{f} will cache.
+}{
+\spadpaste{f(4) \free{}\free{cache}}
+}
+\xtc{
+This causes all other functions to have all computed values saved by
+default.
+}{
+\spadpaste{)set functions cache all}
+}
+\xtc{
+This causes all functions that have not been specifically cached in some way
+to have no computed values saved.
+}{
+\spadpaste{)set functions cache 0}
+}
+\xtc{
+We also make \userfun{f} and \userfun{g} uncached.
+}{
+\spadpaste{)set functions cache 0 f g}
+}
+
+\beginImportant
+Be careful about caching functions that have
+\spadglossSee{side effects}{side effect}.
+Such a function might destructively modify the elements of an array or
+issue a \axiomFun{draw} command, for example.
+A function that you expect to execute every time it is called should
+not be cached.
+Also, it is highly unlikely that a function with no arguments should
+be cached.
+\endImportant
+
+You should also be careful about caching functions that depend on
+free variables.
+See \downlink{``\ugUserFreeLocalTitle''}{ugUserFreeLocalPage} in Section \ugUserFreeLocalNumber\ignore{ugUserFreeLocal}
+for an example.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserRecurTitle}{Recurrence Relations}
+\newcommand{\ugUserRecurNumber}{6.13.}
+%
+% =====================================================================
+\begin{page}{ugUserRecurPage}{6.13. Recurrence Relations}
+% =====================================================================
+\beginscroll
+
+One of the most useful classes of function are those defined via a
+``recurrence relation.''
+A {\it recurrence relation} makes each successive
+%-% \HDindex{recurrence relation}{ugUserRecurPage}{6.13.}{Recurrence Relations}
+value depend on some or all of the previous values.
+A simple example is the ordinary ``factorial'' function:
+\begin{verbatim}
+fact(0) == 1
+fact(n | n > 0) == n * fact(n-1)
+\end{verbatim}
+
+The value of
+\axiom{fact(10)} depends on the value of \axiom{fact(9)}, \axiom{fact(9)}
+on \axiom{fact(8)}, and so on.
+Because it depends on only one previous value, it is usually called a
+{\it first order recurrence relation.}
+You can easily imagine a function based on two, three or more previous
+values.
+The Fibonacci numbers are probably the most famous function defined by a
+%-% \HDindex{Fibonacci numbers}{ugUserRecurPage}{6.13.}{Recurrence Relations}
+second order recurrence relation.
+\xtc{
+The library function \axiomFun{fibonacci} computes Fibonacci numbers.
+It is obviously optimized for speed.
+}{
+\spadpaste{[fibonacci(i) for i in 0..]}
+}
+\xtc{
+Define the
+Fibonacci numbers ourselves using a piece-wise definition.
+}{
+\spadpaste{fib(1) == 1 \bound{fib0}}
+}
+\xtc{
+}{
+\spadpaste{fib(2) == 1 \bound{fib1}\free{fib0}}
+}
+\xtc{
+}{
+\spadpaste{fib(n) == fib(n-1) + fib(n-2) \bound{fibn}\free{fib1}}
+}
+
+As defined, this recurrence relation is obviously doubly-recursive.
+To compute \axiom{fib(10)}, we need to compute \axiom{fib(9)} and
+\axiom{fib(8)}.
+And to  \axiom{fib(9)}, we need to compute \axiom{fib(8)} and
+\axiom{fib(7)}.
+And so on.
+It seems that to compute \axiom{fib(10)} we need to compute
+\axiom{fib(9)} once, \axiom{fib(8)} twice, \axiom{fib(7)} three times.
+Look familiar?
+The number of function calls needed to compute {\it any} second order
+recurrence relation in the obvious way is exactly \axiom{fib(n)}.
+These numbers grow!
+For example, if \Language{} actually did this, then \axiom{fib(500)}
+requires more than \texht{$10^{104}$}{\axiom{10**104}} function calls.
+And, given all this, our definition of \userfun{fib} obviously could not be
+used to calculate the five-hundredth Fibonacci number.
+\xtc{
+Let's try it anyway.
+}{
+\spadpaste{fib(500) \free{fibn}}
+}
+
+Since this takes a short time to compute, it obviously didn't do
+as many as \texht{$10^{104}$}{\axiom{10**104}} operations!
+By default, \Language{} transforms any recurrence relation it recognizes
+into an iteration.
+Iterations are efficient.
+To compute the value of the \eth{\axiom{n}}
+term of a recurrence relation using an iteration requires only
+\axiom{n} function calls.\footnote{If
+you compare the speed of our \userfun{fib} function
+to the library function, our version is still slower.
+This is because the library
+\axiomFunFrom{fibonacci}{IntegerNumberTheoryFunctions}
+uses a ``powering algorithm'' with a computing time
+proportional to \texht{$\log^3(n)$}{\axiom{log(n)**3}} to compute
+\axiom{fibonacci(n).}}
+
+To turn off this special recurrence relation compilation, issue
+%-% \HDsyscmdindex{set function recurrence}{ugUserRecurPage}{6.13.}{Recurrence Relations}
+\begin{verbatim}
+)set functions recurrence off
+\end{verbatim}
+To turn it back on, substitute ``{\tt on}'' for ``{\tt off}''.
+
+The transformations that \Language{} uses for \userfun{fib} caches the
+last two values.\footnote{For a more general \eth{\axiom{k}} order recurrence
+relation, \Language{} caches the last \axiom{k} values.}
+If, after computing a value for \userfun{fib}, you ask
+for some larger value, \Language{} picks up the cached values
+and continues computing from there.
+See \downlink{``\ugUserFreeLocalTitle''}{ugUserFreeLocalPage} in Section \ugUserFreeLocalNumber\ignore{ugUserFreeLocal}
+for an example of a function definition that has this same behavior.
+Also see \downlink{``\ugUserCacheTitle''}{ugUserCachePage} in Section \ugUserCacheNumber\ignore{ugUserCache}
+for a more general discussion of how you can cache function values.
+
+Recurrence relations can be used for defining recurrence relations
+involving polynomials, rational functions, or anything you like.
+Here we compute the infinite stream of Legendre polynomials.
+\xtc{
+The Legendre polynomial of degree \axiom{0.}
+}{
+\spadpaste{p(0) == 1\bound{p0}}
+}
+\xtc{
+The Legendre polynomial of degree \axiom{1.}
+}{
+\spadpaste{p(1) == x\bound{p1}}
+}
+
+\xtc{
+The Legendre polynomial of degree \axiom{n}.
+}{
+\spadpaste{p(n) == ((2*n-1)*x*p(n-1) - (n-1)*p(n-2))/n\bound{pn}\free{p1}}
+}
+\xtc{
+Compute the Legendre polynomial of degree \axiom{6.}
+}{
+\spadpaste{p(6)\free{pn}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserMakeTitle}{Making Functions from Objects}
+\newcommand{\ugUserMakeNumber}{6.14.}
+%
+% =====================================================================
+\begin{page}{ugUserMakePage}{6.14. Making Functions from Objects}
+% =====================================================================
+\beginscroll
+
+There are many times when you compute a complicated expression
+and then wish to use that expression as the body of a function.
+\Language{} provides an operation called \axiomFun{function} to do
+%-% \HDindex{function!from an object}{ugUserMakePage}{6.14.}{Making Functions from Objects}
+this.
+%-% \HDindex{function!made by function @{made by {\bf function}}}{ugUserMakePage}{6.14.}{Making Functions from Objects}
+It creates a function object and places it into the workspace.
+There are several versions, depending on how many arguments the function
+has.
+The first argument to \axiomFun{function} is always the expression to be
+converted into the function body, and the second is always the name to be
+used for the function.
+For more information, see \downlink{`MakeFunction'}{MakeFunctionXmpPage}\ignore{MakeFunction}.
+
+\xtc{
+Start with a simple example of a polynomial in three variables.
+}{
+\spadpaste{p := -x + y**2 - z**3 \bound{p}}
+}
+\xtc{
+To make this into a function of no arguments that
+simply returns the polynomial, use the two argument form of
+\axiomFun{function}.
+}{
+\spadpaste{function(p,'f0) \free{p}\bound{f0}}
+}
+\xtc{
+To avoid possible conflicts (see below), it is a good idea to
+quote always this second argument.
+}{
+\spadpaste{f0 \free{f0}}
+}
+\xtc{
+This is what you get when you evaluate the function.
+}{
+\spadpaste{f0() \free{f0}}
+}
+\xtc{
+To make a function in \axiom{x}, use a version of
+\axiomFun{function} that takes three arguments.
+The last argument is the name of the variable to use as the parameter.
+Typically, this variable occurs in the expression and, like the function
+name, you should quote it to avoid possible confusion.
+}{
+\spadpaste{function(p,'f1,'x) \free{p}\bound{f1}}
+}
+\xtc{
+This is what the new function looks like.
+}{
+\spadpaste{f1 \free{f1}}
+}
+\xtc{
+This is the value of \userfun{f1} at \axiom{x = 3}.
+Notice that the return type of the function is
+\axiomType{Polynomial (Integer)}, the same as \axiom{p}.
+}{
+\spadpaste{f1(3) \free{f1}}
+}
+\xtc{
+To use \axiom{x} and \axiom{y} as parameters, use the
+four argument form of \axiomFun{function}.
+}{
+\spadpaste{function(p,'f2,'x,'y) \free{p}\bound{f2}}
+}
+\xtc{
+}{
+\spadpaste{f2 \free{f2}}
+}
+\xtc{
+Evaluate \axiom{f2} at \axiom{x = 3} and \axiom{y = 0}.
+The return type of \userfun{f2} is still
+\axiomType{Polynomial(Integer)} because the variable \axiom{z}
+is still present and not one of the parameters.
+}{
+\spadpaste{f2(3,0) \free{f2}}
+}
+\xtc{
+Finally, use all three variables as parameters.
+There is no five argument form of \axiomFun{function}, so use the one with
+three arguments, the third argument being a list of the parameters.
+}{
+\spadpaste{function(p,'f3,['x,'y,'z]) \free{p}\bound{f3}}
+}
+\xtc{
+Evaluate this using the same values for \axiom{x} and \axiom{y}
+as above, but let \axiom{z} be \axiom{-6}.
+The result type of \userfun{f3} is \axiomType{Integer}.
+}{
+\spadpaste{f3 \free{f3}}
+}
+\xtc{
+}{
+\spadpaste{f3(3,0,-6) \free{f3}}
+}
+
+The four functions we have defined via \axiom{p} have been undeclared.
+To declare a function whose body is to be generated by
+%-% \HDindex{function!declaring}{ugUserMakePage}{6.14.}{Making Functions from Objects}
+\axiomFun{function}, issue the declaration {\it before} the function is created.
+\xtc{
+}{
+\spadpaste{g: (Integer, Integer) -> Float \bound{g}}
+}
+\xtc{
+}{
+\spadpaste{D(sin(x-y)/cos(x+y),x) \bound{prev}}
+}
+\xtc{
+}{
+\spadpaste{function(\%,'g,'x,'y) \free{g}\free{prev}}
+}
+\xtc{
+}{
+\spadpaste{g \free{g}}
+}
+It is an error to use \axiom{g} without the quote in the
+penultimate expression since \axiom{g} had been declared but did not have
+a value.
+Similarly, since it is common to overuse variable names like \axiom{x},
+\axiom{y}, and so on,
+you avoid problems if you always quote the variable names
+for \axiomFun{function}.
+In general,
+if \axiom{x} has a value and you use \axiom{x} without a quote in a call to
+\axiomFun{function}, then
+\Language{} does not know what you are trying to do.
+
+What kind of object is allowable as the first argument to \axiomFun{function}?
+Let's use the \Browse{} facility of \HyperName{} to find out.
+%-% \HDindex{Browse@\Browse{}}{ugUserMakePage}{6.14.}{Making Functions from Objects}
+At the main \Browse{} menu, enter the string {\tt function} and then
+click on {\bf Operations.}
+The exposed operations called \axiomFun{function} all take an object
+whose type belongs to category \axiomType{ConvertibleTo InputForm}.
+What domains are those?
+Go back to the main \Browse{} menu, erase {\tt function},
+enter {\tt ConvertibleTo} in the
+input area, and click on {\bf categories} on the {\bf Constructors} line.
+At the bottom of the page, enter {\tt InputForm} in the input area
+following {\bf S =}.
+Click on {\bf Cross Reference} and then on {\bf Domains}.
+The list you see contains over forty domains that belong to the
+category \axiomType{ConvertibleTo InputForm}.
+Thus you can use \axiomFun{function} for \axiomType{Integer},
+\axiomType{Float},
+\axiomType{String},
+\axiomType{Complex},
+\axiomType{Expression}, and so on.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserBlocksTitle}{Functions Defined with Blocks}
+\newcommand{\ugUserBlocksNumber}{6.15.}
+%
+% =====================================================================
+\begin{page}{ugUserBlocksPage}{6.15. Functions Defined with Blocks}
+% =====================================================================
+\beginscroll
+
+You need not restrict yourself to functions that only fit on one line
+or are written in a piece-wise manner.
+The body of the function can be a block, as discussed in
+\downlink{``\ugLangBlocksTitle''}{ugLangBlocksPage} in Section \ugLangBlocksNumber\ignore{ugLangBlocks}.
+
+\labelSpace{1pc}
+\xtc{
+Here is a short function that swaps two elements of a list,
+array or vector.
+}{
+\begin{spadsrc}[\bound{swap}]
+swap(m,i,j) ==
+  temp := m.i
+  m.i := m.j
+  m.j := temp
+\end{spadsrc}
+}
+\xtc{
+The significance of \userfun{swap} is that it has a destructive
+effect on its first argument.
+}{
+\spadpaste{k := [1,2,3,4,5] \bound{k}}
+}
+\xtc{
+}{
+\spadpaste{swap(k,2,4) \free{l swap}\bound{swapk}}
+}
+\xtc{
+You see that the second and fourth elements are interchanged.
+}{
+\spadpaste{k \free{swapk}}
+}
+
+\xtc{
+Using this, we write a couple of different sort functions.
+First, a simple bubble sort.
+%-% \HDindex{sort!bubble}{ugUserBlocksPage}{6.15.}{Functions Defined with Blocks}
+The operation \axiomOpFrom{\#}{List} returns the number of elements in
+an aggregate.
+}{
+\begin{spadsrc}[\bound{bubbleSort}]
+bubbleSort(m) ==
+  n := #m
+  for i in 1..(n-1) repeat
+    for j in n..(i+1) by -1 repeat
+      if m.j < m.(j-1) then swap(m,j,j-1)
+  m
+\end{spadsrc}
+}
+\xtc{
+Let this be the list we want to sort.
+}{
+\spadpaste{m := [8,4,-3,9] \bound{m}}
+}
+\xtc{
+This is the result of sorting.
+}{
+\spadpaste{bubbleSort(m) \free{m swap bubbleSort}\bound{sortm}}
+}
+\xtc{
+Moreover, \axiom{m} is destructively changed to be the sorted version.
+}{
+\spadpaste{m \free{sortm}}
+}
+
+\xtc{
+This function implements an insertion sort.
+%-% \HDindex{sort!insertion}{ugUserBlocksPage}{6.15.}{Functions Defined with Blocks}
+The basic idea is to traverse the list and insert the \eth{\axiom{i}}
+element in its correct position among the \axiom{i-1} previous
+elements.
+Since we start at the beginning of the list, the list elements before the
+\eth{\axiom{i}} element have already been placed in ascending order.
+}{
+\begin{spadsrc}[\bound{insertionSort}]
+insertionSort(m) ==
+  for i in 2..#m repeat
+    j := i
+    while j > 1 and m.j < m.(j-1) repeat
+      swap(m,j,j-1)
+      j := j - 1
+  m
+\end{spadsrc}
+}
+\xtc{
+As with our bubble sort, this is a destructive function.
+}{
+\spadpaste{m := [8,4,-3,9] \bound{m1}}
+}
+\xtc{
+}{
+\spadpaste{insertionSort(m) \free{m1 swap insertionSort}\bound{sortm1}}
+}
+\xtc{
+}{
+\spadpaste{m \free{sortm1}}
+}
+
+Neither of the above functions is efficient for sorting large lists since
+they reference elements by asking for the \eth{\axiom{j}} element of the
+structure \axiom{m}.
+%For lists, compute \axiom{m.(j+1) = rest(m,j).first}, and thus, starting at
+%the first node of \axiom{m}, walk down to the \eth{\axiom{j}} node, then call
+%\axiomFun{first}.
+
+\xtc{
+Here is a more efficient bubble sort for lists.
+}{
+\begin{spadsrc}[\bound{bubbleSort2}]
+bubbleSort2(m: List Integer): List Integer ==
+  null m => m
+  l := m
+  while not null (r := l.rest) repeat
+     r := bubbleSort2 r
+     x := l.first
+     if x < r.first then
+       l.first := r.first
+       r.first := x
+     l.rest := r
+     l := l.rest
+  m
+\end{spadsrc}
+}
+\xtc{
+Try it out.
+}{
+\spadpaste{bubbleSort2 [3,7,2]\free{bubbleSort2}}
+}
+
+This definition is both recursive and iterative, and is tricky!
+Unless you are {\it really} curious about this definition,
+we suggest you skip immediately to the next section.
+
+Here are the key points in the definition.
+First notice that if you are sorting a list with less than two elements,
+there is nothing to do: just return the list.
+This definition returns immediately if there are zero elements, and skips
+the entire \axiom{while} loop if there is just one element.
+
+The second point to realize is that on each outer iteration, the bubble sort
+ensures that the minimum element is propagated leftmost.
+Each iteration of the \axiom{while} loop calls \userfun{bubbleSort2}
+recursively to sort all but the first element.
+When finished, the minimum element is either in the first or second position.
+The conditional expression ensures that it comes first.
+If it is in the second, then a swap occurs.
+In any case, the \axiomFun{rest} of the original list must be updated to hold
+the result of the recursive call.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserFreeLocalTitle}{Free and Local Variables}
+\newcommand{\ugUserFreeLocalNumber}{6.16.}
+%
+% =====================================================================
+\begin{page}{ugUserFreeLocalPage}{6.16. Free and Local Variables}
+% =====================================================================
+\beginscroll
+
+When you want to refer to a variable that is not local to your
+function, use a ``\axiom{free}'' declaration.
+\spadkey{free}
+Variables declared to be \axiom{free}
+%-% \HDindex{free variable}{ugUserFreeLocalPage}{6.16.}{Free and Local Variables}
+are assumed to be defined globally
+%-% \HDindex{variable!free}{ugUserFreeLocalPage}{6.16.}{Free and Local Variables}
+in the
+%-% \HDindex{variable!global}{ugUserFreeLocalPage}{6.16.}{Free and Local Variables}
+workspace.
+%-% \HDindex{global variable}{ugUserFreeLocalPage}{6.16.}{Free and Local Variables}
+
+\labelSpace{1pc}
+\xtc{
+This is a global workspace variable.
+}{
+\spadpaste{counter := 0 \bound{counter}}
+}
+\xtc{
+This function refers to the global \axiom{counter}.
+}{
+\begin{spadsrc}[\free{counter}\bound{f}]
+f() ==
+  free counter
+  counter := counter + 1
+\end{spadsrc}
+}
+\xtc{
+The global \axiom{counter} is incremented by \axiom{1}.
+}{
+\spadpaste{f() \free{f}\bound{f1}}
+}
+\xtc{
+}{
+\spadpaste{counter \free{f1}}
+}
+
+Usually \Language{} can tell that you mean to refer to a global
+variable and so \axiom{free} isn't always necessary.
+However, for clarity and the sake of self-documentation, we encourage
+you to use it.
+
+Declare a variable to be ``\axiom{local}'' when you do not want to refer to
+%-% \HDindex{variable!local}{ugUserFreeLocalPage}{6.16.}{Free and Local Variables}
+a global variable by the same name.
+%-% \HDindex{local variable}{ugUserFreeLocalPage}{6.16.}{Free and Local Variables}
+
+\xtc{
+This function uses \axiom{counter} as a local variable.
+}{
+\begin{spadsrc}[\bound{g}]
+g() ==
+  local counter
+  counter := 7
+\end{spadsrc}
+}
+\xtc{
+Apply the function.
+}{
+\spadpaste{g() \free{g}}
+}
+\xtc{
+Check that the global value of \axiom{counter} is unchanged.
+}{
+\spadpaste{counter\free{g f1}}
+}
+
+Parameters to a function are local variables in the function.
+Even if you issue a \axiom{free} declaration for a parameter, it is
+still local.
+
+What happens if you do not declare that a variable \axiom{x} in
+the body of your function is \axiom{local} or \axiom{free}?
+Well, \Language{} decides on this basis:
+
+\indent{4}
+\beginitems
+\item[1. ] \Language{} scans your function line-by-line, from top-to-bottom.
+The right-hand side of an assignment is looked at before the left-hand
+side.
+\item[2. ] If \axiom{x} is referenced before it is assigned a value, it is a
+\axiom{free} (global) variable.
+\item[3. ] If \axiom{x} is assigned a value before it is referenced, it is a
+\axiom{local} variable.
+\enditems
+\indent{0}
+
+\xtc{
+Set two global variables to 1.
+}{
+\spadpaste{a := b := 1\bound{ab1}}
+}
+\xtc{
+Refer to \axiom{a} before it is assigned a value, but
+assign a value to \axiom{b} before it is referenced.
+}{
+\begin{spadsrc}[\bound{hh}]
+h() ==
+  b := a + 1
+  a := b + a
+\end{spadsrc}
+}
+\xtc{
+Can you predict this result?
+}{
+\spadpaste{h() \free{ab1 hh}\bound{hhh}}
+}
+\xtc{
+How about this one?
+}{
+\spadpaste{[a, b] \free{hhh}}
+}
+
+What happened?
+In the first line of the function body for \axiom{h}, \axiom{a} is
+referenced on the right-hand side of the assignment.
+Thus \axiom{a} is a free variable.
+The variable \axiom{b} is not referenced in that line, but it is
+assigned a value.
+Thus \axiom{b} is a local variable and is given the value
+\axiom{a + 1 = 2}.
+In the second line, the free variable \axiom{a} is assigned the value
+\axiom{b + a}which equals \axiom{2 + 1 = 3.}
+This is the value returned by the function.
+Since \axiom{a} was free in \userfun{h}, the global variable \axiom{a}
+has value \axiom{3.}
+Since \axiom{b} was local in \userfun{h}, the global variable \axiom{b}
+is unchanged---it still has the value \axiom{1.}
+
+It is good programming practice always to declare global variables.
+However, by far the most common situation is to have local variables in
+your functions.
+No declaration is needed for this situation, but be sure to
+initialize their values.
+
+Be careful if you use free variables and you cache the value of
+your function (see \downlink{``\ugUserCacheTitle''}{ugUserCachePage} in Section \ugUserCacheNumber\ignore{ugUserCache}).
+Caching {\it only} checks if the values of the function arguments
+are the same as in a function call previously seen.
+It does not check if any of the free variables on which the
+function depends have changed between function calls.
+\xtc{
+Turn on caching for \userfun{p}.
+}{
+\spadpaste{)set fun cache all p \bound{pcache}}
+}
+\xtc{
+Define \userfun{p} to depend on the free variable \axiom{N}.
+}{
+\spadpaste{p(i,x) == ( free N; reduce( + , [ (x-i)**n for n in 1..N ] ) ) \free{pcache}\bound{pdef}}
+}
+\xtc{
+Set the value of \axiom{N}.
+}{
+\spadpaste{N := 1 \bound{Nass}}
+}
+\xtc{
+Evaluate \userfun{p} the first time.
+}{
+\spadpaste{p(0, x) \free{pdef Nass}\bound{pfirst}}
+}
+\xtc{
+Change the value of \axiom{N}.
+}{
+\spadpaste{N := 2 \bound{Nass2}}
+}
+\xtc{
+Evaluate \userfun{p} the second time.
+}{
+\spadpaste{p(0, x) \free{pfirst Nass2}}
+}
+If caching had been turned off, the second evaluation would have
+reflected the changed value of \axiom{N}.
+\xtc{
+Turn off caching for \userfun{p}.
+}{
+\spadpaste{)set fun cache 0 p}
+}
+
+\Language{} does not allow {\it fluid variables}, that is, variables
+%-% \HDindex{variable!fluid}{ugUserFreeLocalPage}{6.16.}{Free and Local Variables}
+\spadglossSee{bound}{binding} by a function \spad{f} that can be referenced by
+functions called by \spad{f}.
+%-% \HDindex{fluid variable}{ugUserFreeLocalPage}{6.16.}{Free and Local Variables}
+
+Values are passed to functions by \spadgloss{reference}: a pointer
+to the value is passed rather than a copy of the value or a pointer to
+a copy.
+
+\xtc{
+This is a global variable that is bound to a record object.
+}{
+\spadpaste{r : Record(i : Integer) := [1] \free{r}}
+}
+\xtc{
+This function first modifies the one component of its
+record argument and then rebinds the parameter to another
+record.
+}{
+\begin{spadsrc}[\bound{resetRecord}]
+resetRecord rr ==
+  rr.i := 2
+  rr := [10]
+\end{spadsrc}
+}
+\xtc{
+Pass \axiom{r} as an argument to \userfun{resetRecord}.
+}{
+\spadpaste{resetRecord r \free{r resetRecord}\bound{rr}}
+}
+\xtc{
+The value of \axiom{r} was changed by the expression
+\axiom{rr.i := 2} but not by \axiom{rr := [10]}.
+}{
+\spadpaste{r \free{rr}}
+}
+
+To conclude this section, we give an iterative definition of
+%-% \HDindex{Fibonacci numbers}{ugUserFreeLocalPage}{6.16.}{Free and Local Variables}
+a function that computes Fibonacci numbers.
+This definition approximates the definition into which \Language{}
+transforms the recurrence relation definition of \userfun{fib} in
+\downlink{``\ugUserRecurTitle''}{ugUserRecurPage} in Section \ugUserRecurNumber\ignore{ugUserRecur}.
+
+\xtc{
+Global variables
+\axiom{past} and \axiom{present} are used
+to hold the last computed Fibonacci numbers.
+}{
+\spadpaste{past := present := 1\bound{f0}}
+}
+\xtc{
+Global variable \axiom{index} gives the
+current index of \axiom{present}.
+}{
+\spadpaste{index := 2\bound{f1}\free{f0}}
+}
+\xtc{
+Here is a recurrence relation defined in terms
+of these three global variables.
+}{
+\begin{spadsrc}[\bound{f3}\free{f2}]
+fib(n) ==
+  free past, present, index
+  n < 3 => 1
+  n = index - 1 => past
+  if n < index-1 then
+    (past,present) := (1,1)
+    index := 2
+  while (index < n) repeat
+    (past,present) := (present, past+present)
+    index := index + 1
+  present
+\end{spadsrc}
+}
+\xtc{
+Compute the infinite stream of Fibonacci numbers.
+}{
+\spadpaste{fibs := [fib(n) for n in 1..] \bound{fibs}\free{f3}}
+}
+\xtc{
+What is the 1000th Fibonacci number?
+}{
+\spadpaste{fibs 1000 \free{fibs}}
+}
+
+As an exercise, we suggest you write a function in an iterative
+style that computes the value of the recurrence relation
+\texht{$p(n) = p(n-1) - 2 \, p(n-2) + 4 \, p(n-3)$}{\axiom{p(n) = p(n-1) - 2*p(n-2) + 4*p(n-3)}}
+having the initial values
+\texht{$p(1) = 1,\, p(2) = 3 \hbox{ and } p(3) = 9.$}{\axiom{p(1) = 1, p(2) = 3 {\rm and} p(3) = 9.}}
+How would you write the function using an element
+\axiomType{OneDimensionalArray} or \axiomType{Vector}
+to hold the previously computed values?
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserAnonTitle}{Anonymous Functions}
+\newcommand{\ugUserAnonNumber}{6.17.}
+%
+% =====================================================================
+\begin{page}{ugUserAnonPage}{6.17. Anonymous Functions}
+% =====================================================================
+\beginscroll
+
+\beginImportant
+An {\it anonymous function} is a function that is
+%-% \HDindex{function!anonymous}{ugUserAnonPage}{6.17.}{Anonymous Functions}
+defined
+%-% \HDindex{anonymous function}{ugUserAnonPage}{6.17.}{Anonymous Functions}
+by giving a list of parameters, the ``maps-to'' compound
+%-% \HDindex{+-> @{\tt +->}}{ugUserAnonPage}{6.17.}{Anonymous Functions}
+symbol \axiomSyntax{+->} \texht{(from the mathematical symbol
+$\mapsto$)}{},
+and by an expression involving the parameters, the evaluation of
+which determines the return value of the function.
+
+\centerline{{{\tt ( \subscriptIt{parm}{1}, \subscriptIt{parm}{2}, \ldots, \subscriptIt{parm}{N} ) +-> {\it expression}}}}
+\endImportant
+
+You can apply an anonymous function in several ways.
+\indent{4}
+\beginitems
+\item[1. ] Place the anonymous function definition in parentheses
+directly followed by a list of arguments.
+\item[2. ] Assign the anonymous function to a variable and then
+use the variable name when you would normally use a function name.
+\item[3. ] Use \axiomSyntax{==} to use the anonymous function definition as
+the arguments and body of a regular function definition.
+\item[4. ] Have a named function contain a declared anonymous function and
+use the result returned by the named function.
+\enditems
+\indent{0}
+
+\beginmenu
+    \menudownlink{{6.17.1. Some Examples}}{ugUserAnonExampPage}
+    \menudownlink{{6.17.2. Declaring Anonymous Functions}}{ugUserAnonDeclarePage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserAnonExampTitle}{Some Examples}
+\newcommand{\ugUserAnonExampNumber}{6.17.1.}
+%
+% =====================================================================
+\begin{page}{ugUserAnonExampPage}{6.17.1. Some Examples}
+% =====================================================================
+\beginscroll
+
+Anonymous functions are particularly useful for defining functions
+``on the fly.'' That is, they are handy for simple functions that
+are used only in one place.
+In the following examples, we show how to write some simple
+anonymous functions.
+
+\xtc{
+This is a simple absolute value function.
+}{
+\spadpaste{x +-> if x < 0 then -x else x \bound{anon0}}
+}
+\xtc{
+}{
+\spadpaste{abs1 := \% \free{anon0}\bound{abs1}}
+}
+\xtc{
+This function returns {\tt true} if the absolute value of
+the first argument is greater than the absolute value of the
+second, {\tt false} otherwise.
+}{
+\spadpaste{(x,y) +-> abs1(x) > abs1(y) \bound{anon1}\free{abs1}}
+}
+\xtc{
+We use the above function to ``sort'' a list of integers.
+}{
+\spadpaste{sort(\%,[3,9,-4,10,-3,-1,-9,5]) \free{anon1}}
+}
+
+\xtc{
+This function returns \axiom{1} if \axiom{i + j} is even, \axiom{-1} otherwise.
+}{
+\spadpaste{ev := ( (i,j) +-> if even?(i+j) then 1 else -1) \bound{ev}}
+}
+\xtc{
+We create a four-by-four matrix containing \axiom{1} or \axiom{-1}
+depending on whether the row plus the column index is even or not.
+}{
+\spadpaste{matrix([[ev(row,col) for row in 1..4] for col in 1..4]) \free{ev}}
+}
+
+\xtc{
+This function returns {\tt true} if a polynomial in \axiom{x} has multiple
+roots, {\tt false} otherwise.
+It is defined and applied in the same expression.
+}{
+\spadpaste{( p +-> not one?(gcd(p,D(p,x))) )(x**2+4*x+4)}
+}
+
+\xtc{
+This and the next expression are equivalent.
+}{
+\spadpaste{g(x,y,z) == cos(x + sin(y + tan(z)))}
+}
+\xtc{
+The one you use is a matter of taste.
+}{
+\spadpaste{g == (x,y,z) +-> cos(x + sin(y + tan(z)))}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserAnonDeclareTitle}{Declaring Anonymous Functions}
+\newcommand{\ugUserAnonDeclareNumber}{6.17.2.}
+%
+% =====================================================================
+\begin{page}{ugUserAnonDeclarePage}{6.17.2. Declaring Anonymous Functions}
+% =====================================================================
+\beginscroll
+
+If you declare any of the arguments you must declare all of them.
+Thus,
+\begin{verbatim}
+(x: INT,y): FRAC INT +-> (x + 2*y)/(y - 1)
+\end{verbatim}
+is not legal.
+
+\xtc{
+This is an example of a fully declared anonymous
+%-% \HDindex{function!declaring}{ugUserAnonDeclarePage}{6.17.2.}{Declaring Anonymous Functions}
+function.
+%-% \HDindex{function!anonymous!declaring}{ugUserAnonDeclarePage}{6.17.2.}{Declaring Anonymous Functions}
+The output shown just indicates that the object you created is a
+particular kind of map, that is, function.
+}{
+\spadpaste{(x: INT,y: INT): FRAC INT +-> (x + 2*y)/(y - 1)}
+}
+\xtc{
+\Language{} allows you to declare the arguments and not declare
+the return type.
+}{
+\spadpaste{(x: INT,y: INT) +-> (x + 2*y)/(y - 1)}
+}
+The return type is computed from the types of the arguments and the
+body of the function.
+You cannot declare the return type if you do not declare the arguments.
+Therefore,
+\begin{verbatim}
+(x,y): FRAC INT +-> (x + 2*y)/(y - 1)
+\end{verbatim}
+is not legal.
+
+\xtc{
+This and the next expression are equivalent.
+}{
+\spadpaste{h(x: INT,y: INT): FRAC INT == (x + 2*y)/(y - 1)}
+}
+\xtc{
+The one you use is a matter of taste.
+}{
+\spadpaste{h == (x: INT,y: INT): FRAC INT +-> (x + 2*y)/(y - 1)}
+}
+
+When should you declare an anonymous function?
+\indent{4}
+\beginitems
+\item[1. ] If you use an anonymous function and \Language{} can't figure
+out what you are trying to do, declare the function.
+\item[2. ] If the function has nontrivial argument types or a
+nontrivial return type that
+\Language{} may be able to determine eventually, but you are not
+willing to wait that long, declare the function.
+\item[3. ] If the function will only be used for arguments of specific
+types and it is not too much trouble to declare the function, do so.
+\item[4. ] If you are using the anonymous function as an argument to
+another function (such as \axiomFun{map} or \axiomFun{sort}),
+consider declaring the function.
+\item[5. ] If you define an anonymous function inside a named function,
+you {\it must} declare the anonymous function.
+\enditems
+\indent{0}
+
+\xtc{
+This is an example of a named function for integers that returns a
+function.
+}{
+\spadpaste{addx x == ((y: Integer): Integer +-> x + y) \bound{addx}}
+}
+\xtc{
+We define \userfun{g} to be a function that adds \axiom{10} to its
+argument.
+}{
+\spadpaste{g := addx 10 \free{addx}\bound{g}}
+}
+\xtc{
+Try it out.
+}{
+\spadpaste{g 3 \free{g}}
+}
+\xtc{
+}{
+\spadpaste{g(-4) \free{g}}
+}
+
+%-% \HDindex{function!anonymous!restrictions}{ugUserAnonDeclarePage}{6.17.2.}{Declaring Anonymous Functions}
+An anonymous function cannot be recursive: since it does not have a
+name, you cannot even call it within itself!
+If you place an anonymous function inside a named function, the
+anonymous function must be declared.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserDatabaseTitle}{Example: A Database}
+\newcommand{\ugUserDatabaseNumber}{6.18.}
+%
+% =====================================================================
+\begin{page}{ugUserDatabasePage}{6.18. Example: A Database}
+% =====================================================================
+\beginscroll
+
+This example shows how you can use \Language{} to organize a database of
+lineage data and then query the database for relationships.
+
+\labelSpace{1.5pc}
+\xtc{
+The database is entered as ``assertions'' that are really
+pieces of a function definition.
+}{
+\spadpaste{children("albert") == ["albertJr","richard","diane"]\bound{d1}}
+}
+\xtc{
+Each piece
+\axiom{children(x) == y} means
+``the children of \axiom{x} are \axiom{y}''.
+}{
+\spadpaste{children("richard") == ["douglas","daniel","susan"]\free{d1}\bound{d2}}
+}
+\xtc{
+This family tree thus spans four generations.
+}{
+\spadpaste{children("douglas") == ["dougie","valerie"]\free{d2}\bound{d3}}
+}
+\xtc{
+Say ``no one else has children.''
+}{
+\spadpaste{children(x) == []\free{d3}\bound{d4}}
+}
+
+\xtc{
+We need some functions for computing lineage.
+Start with \axiom{childOf}.
+}{
+\spadpaste{childOf(x,y) == member?(x,children(y))\bound{d9}\free{d10}}
+}
+\xtc{
+To find the \axiom{parentOf} someone,
+you have to scan the database of
+people applying \axiom{children}.
+}{
+\begin{spadsrc}[\bound{d8a}\free{d9}]
+parentOf(x) ==
+  for y in people repeat
+    (if childOf(x,y) then return y)
+  "unknown"
+\end{spadsrc}
+}
+\xtc{
+And a grandparent of \axiom{x} is just a parent of a parent of \axiom{x}.
+}{
+\spadpaste{grandParentOf(x) == parentOf parentOf x\bound{d8}\free{d8a}}
+}
+\xtc{
+The grandchildren of \axiom{x}
+are the people \axiom{y} such that
+\axiom{x} is a grandparent of \axiom{y}.
+}{
+\spadpaste{grandchildren(x) == [y for y in people | grandParentOf(y) = x]\free{d7}\bound{d8}}
+}
+\xtc{
+Suppose you want to make a list of all great-grandparents.
+Well, a great-grandparent is a grandparent of a person who has children.
+}{
+\begin{spadsrc}[\free{d6}\bound{d7}]
+greatGrandParents == [x for x in people |
+  reduce(_or,[not empty? children(y) for y in grandchildren(x)],false)]
+\end{spadsrc}
+}
+\xtc{
+Define \axiom{descendants} to include the parent as well.
+}{
+\begin{spadsrc}[\free{d5}\bound{d6}]
+descendants(x) ==
+  kids := children(x)
+  null kids => [x]
+  concat(x,reduce(concat,[descendants(y)
+    for y in kids],[]))
+\end{spadsrc}
+}
+\xtc{
+Finally, we need a list of people.
+Since all people are descendants of ``albert'', let's say so.
+}{
+\spadpaste{people == descendants "albert"\free{d4}\bound{d5}}
+}
+
+We have used \axiomSyntax{==} to define the database and some functions to
+query the database.
+But no computation is done until we ask for some information.
+Then, once and for all, the functions are analyzed and compiled to machine
+code for run-time efficiency.
+Notice that no types are given anywhere in this example.
+They are not needed.
+
+\xtc{
+Who are the grandchildren of ``richard''?
+}{
+\spadpaste{grandchildren "richard"\bound{d10}\free{d11}}
+}
+\xtc{
+Who are the great-grandparents?
+}{
+\spadpaste{greatGrandParents\bound{d11}\free{d12}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserTriangleTitle}{Example: A Famous Triangle}
+\newcommand{\ugUserTriangleNumber}{6.19.}
+%
+% =====================================================================
+\begin{page}{ugUserTrianglePage}{6.19. Example: A Famous Triangle}
+% =====================================================================
+\beginscroll
+
+In this example we write some functions that display
+Pascal's triangle.
+%-% \HDindex{Pascal's triangle}{ugUserTrianglePage}{6.19.}{Example: A Famous Triangle}
+It demonstrates the use of piece-wise definitions and some output
+operations you probably haven't seen before.
+
+\labelSpace{1pc}
+\xtc{
+To make these output operations
+available, we have to \spadgloss{expose} the domain
+\axiomType{OutputForm}.
+%-% \HDexptypeindex{OutputForm}{ugUserTrianglePage}{6.19.}{Example: A Famous Triangle}
+See \downlink{``\ugTypesExposeTitle''}{ugTypesExposePage} in Section \ugTypesExposeNumber\ignore{ugTypesExpose} for more information about exposing domains
+and packages.
+}{
+\spadpaste{)set expose add constructor OutputForm \bound{expose}}
+}
+\xtc{
+Define the values along the first
+row and any column \axiom{i}.
+}{
+\spadpaste{pascal(1,i) == 1 \bound{pas1}}
+}
+\xtc{
+Define the values for when the row
+and column index \axiom{i} are equal.
+Repeating the argument name indicates that
+the two index values are equal.
+}{
+\spadpaste{pascal(n,n) == 1 \bound{pas2}\free{pas1}}
+}
+\xtc{
+}{
+\begin{spadsrc}[\bound{pas3}\free{pas1 pas2}]
+pascal(i,j | 1 < i and i < j) ==
+   pascal(i-1,j-1)+pascal(i,j-1)
+\end{spadsrc}
+}
+Now that we have defined the coefficients in Pascal's triangle,
+let's write a couple of one-liners to display it.
+\xtc{
+First, define a function that gives the \eth{\axiom{n}} row.
+}{
+\spadpaste{pascalRow(n) == [pascal(i,n) for i in 1..n] \bound{pascalRow}\free{pas3}}
+}
+\xtc{
+Next, we write the function \userfun{displayRow}
+to display the row, separating entries by blanks and centering.
+}{
+\spadpaste{displayRow(n) == output center blankSeparate pascalRow(n) \free{pascalRow}\bound{displayRow}\free{expose}}
+}
+%
+Here we have used three output operations.
+Operation \axiomFunFrom{output}{OutputForm}
+displays the printable form of objects on the screen,
+\axiomFunFrom{center}{OutputForm} centers a printable form in the
+width of the screen, and \axiomFunFrom{blankSeparate}{OutputForm} takes a list of
+printable forms and inserts a blank between successive elements.
+\xtc{
+Look at the result.
+}{
+\spadpaste{for i in 1..7 repeat displayRow i \free{displayRow}}
+}
+Being purists, we find this less than satisfactory.
+Traditionally, elements of Pascal's triangle are centered between
+the left and right elements on the line above.
+%
+\xtc{
+To fix this misalignment, we go back and
+redefine \userfun{pascalRow} to right adjust the entries within the
+triangle within a width of four characters.
+}{
+\spadpaste{pascalRow(n) == [right(pascal(i,n),4) for i in 1..n] \bound{pascalRow2}}
+}
+%
+\xtc{
+Finally let's look at our purely reformatted triangle.
+}{
+\spadpaste{for i in 1..7 repeat displayRow i \free{pascalRow2}\free{displayRow}}
+}
+\xtc{
+Unexpose \axiomType{OutputForm} so we don't get unexpected
+results later.
+}{
+\spadpaste{)set expose drop constructor OutputForm}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserPalTitle}{Example: Testing for Palindromes}
+\newcommand{\ugUserPalNumber}{6.20.}
+%
+% =====================================================================
+\begin{page}{ugUserPalPage}{6.20. Example: Testing for Palindromes}
+% =====================================================================
+\beginscroll
+
+
+In this section we define a function \userfun{pal?} that tests whether its
+%-% \HDindex{palindrome}{ugUserPalPage}{6.20.}{Example: Testing for Palindromes}
+argument is a {\it palindrome}, that is, something that reads the same
+backwards and forwards.
+For example, the string ``Madam I'm Adam'' is a palindrome (excluding blanks
+and punctuation) and so is the number \axiom{123454321.}
+The definition works for any datatype that has \axiom{n} components that
+are accessed by the indices \axiom{1\ldots n}.
+
+\xtc{
+Here is the definition for \userfun{pal?}.
+It is simply a call to an auxiliary function called
+\userfun{palAux?}.
+We are following the convention of ending a function's name with
+\axiomSyntax{?} if the function returns a \axiomType{Boolean} value.
+}{
+\spadpaste{pal? s ==  palAux?(s,1,\#s) \bound{pal}}
+}
+\xtc{
+Here is \userfun{palAux?}.
+It works by comparing elements that are equidistant from the start and end
+of the object.
+}{
+\begin{spadsrc}[\bound{palAux}]
+palAux?(s,i,j) ==
+  j > i =>
+    (s.i = s.j) and palAux?(s,i+1,i-1)
+  true
+\end{spadsrc}
+}
+\xtc{
+Try \userfun{pal?} on some examples.
+First, a string.
+}{
+\spadpaste{pal? "Oxford"  \free{pal palAux}}
+}
+\xtc{
+A list of polynomials.
+}{
+\spadpaste{pal? [4,a,x-1,0,x-1,a,4]  \free{pal palAux}}
+}
+\xtc{
+A list of integers from the example in
+\texht{the last section.}{\downlink{``\ugUserTriangleTitle''}{ugUserTrianglePage} in Section \ugUserTriangleNumber\ignore{ugUserTriangle}.}
+}{
+\spadpaste{pal? [1,6,15,20,15,6,1] \free{pal palAux}}
+}
+\xtc{
+To use \userfun{pal?} on an integer, first convert it to a string.
+}{
+\spadpaste{pal?(1441::String)\free{pal palAux}}
+}
+\xtc{
+Compute an infinite stream of decimal numbers,
+each of which is an obvious palindrome.
+}{
+\spadpaste{ones := [reduce(+,[10**j for j in 0..i]) for i in 1..]\free{pal palAux}\bound{pal5}}
+}
+\xtc{
+How about their squares?
+}{
+\spadpaste{squares := [x**2 for x in ones]\free{pal5}\bound{pal6}}
+}
+\xtc{
+Well, let's test them all!
+}{
+\spadpaste{[pal?(x::String) for x in squares]\free{pal6}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugUserRulesTitle}{Rules and Pattern Matching}
+\newcommand{\ugUserRulesNumber}{6.21.}
+%
+% =====================================================================
+\begin{page}{ugUserRulesPage}{6.21. Rules and Pattern Matching}
+% =====================================================================
+\beginscroll
+
+A common mathematical formula is
+\texht{\narrowDisplay{%
+\log(x) + \log(y) = \log(x y) \quad\forall \, x \hbox{\ and\ } y.}}{
+\axiom{log(x) + log(y) == log(x * y)} for any \axiom{x} and \axiom{y}.}
+The presence of
+\texht{``$\forall$''}{the word ``any''}
+indicates that \axiom{x} and \axiom{y} can stand for arbitrary mathematical
+expressions in the above formula.
+You can use such mathematical formulas in \Language{} to specify ``rewrite
+rules''.
+Rewrite rules are objects in \Language{} that can be assigned to variables for
+later use, often for the purpose of simplification.
+Rewrite rules look like ordinary function definitions except that they are
+preceded by the reserved word \axiom{rule}.
+\spadkey{rule}
+For example, a rewrite rule for the above formula is:
+\begin{verbatim}
+rule log(x) + log(y) == log(x * y)
+\end{verbatim}
+Like function definitions, no action is taken when a rewrite rule is issued.
+Think of rewrite rules as functions that take one argument.
+When a rewrite rule \axiom{A = B} is applied to an argument \axiom{f}, its
+meaning is: ``rewrite every subexpression of \axiom{f} that {\it matches}
+\axiom{A} by \axiom{B.}''
+The left-hand side of a rewrite rule is called a \spadgloss{pattern}; its
+right-side side is called its \spadgloss{substitution}.
+
+\xtc{
+Create a rewrite rule named \userfun{logrule}.
+The generated symbol beginning with a \axiomSyntax{\%} is a place-holder
+for any other terms that might occur in the sum.
+}{
+\spadpaste{logrule := rule log(x) + log(y) == log(x * y) \bound{logrule}}
+}
+\xtc{
+Create an expression with logarithms.
+}{
+\spadpaste{f := log sin x + log x \bound{f}}
+}
+\xtc{
+Apply \userfun{logrule} to \axiom{f}.
+}{
+\spadpaste{logrule f \free{f}\free{logrule}}
+}
+
+The meaning of our example rewrite rule is:
+``for all expressions \axiom{x} and \axiom{y}, rewrite
+\axiom{log(x) + log(y)} by \axiom{log(x * y)}.''
+Patterns generally have both operation names
+(here, \axiomFun{log} and \axiomOp{+})
+and variables (here, \axiom{x} and \axiom{y}).
+By default, every operation name stands for itself.
+Thus \axiomFun{log}  matches only ``\axiom{log}'' and not any
+other operation such as \axiomFun{sin}.
+On the other hand, variables do not stand for themselves.
+Rather, a variable denotes a
+{\it pattern variable} that is free to match any expression whatsoever.
+%-% \HDindex{pattern!variables}{ugUserRulesPage}{6.21.}{Rules and Pattern Matching}
+
+When a rewrite rule is applied, a process called
+\spadgloss{pattern matching} goes to work by systematically
+scanning
+%-% \HDindex{pattern!matching}{ugUserRulesPage}{6.21.}{Rules and Pattern Matching}
+the subexpressions of the argument.
+When a subexpression is found that ``matches'' the pattern, the subexpression
+is replaced by the right-hand side of the rule.
+The details of what happens will be covered later.
+
+The customary \Language{} notation for patterns is actually a shorthand for a
+longer, more general notation.
+Pattern variables can be made explicit by using a percent
+(\axiomSyntax{\%}) as the first character of the variable name.
+To say that a name stands for itself, you can prefix that name with a quote
+operator (\axiomSyntax{'}).
+Although the current \Language{} parser does not let you quote an operation
+name, this more general notation gives you an alternate way of giving the same
+rewrite rule:
+\begin{verbatim}
+rule log(%x) + log(%y) == log(x * y)
+\end{verbatim}
+This longer notation gives you patterns that the
+standard notation won't handle.
+For example, the rule
+\texht{\typeout{check this example}}{}
+\begin{verbatim}
+rule %f(c * 'x) ==  c*%f(x)
+\end{verbatim}
+means ``for all \axiom{f} and \axiom{c}, replace \axiom{f(y)} by
+\axiom{c * f(x)} when \axiom{y} is the product of \axiom{c}
+and the explicit variable \axiom{x}.''
+
+Thus the pattern can have several adornments on the names that appear there.
+Normally, all these adornments are dropped in the substitution on the
+right-hand side.
+
+To summarize:
+
+\beginImportant
+To enter a single rule in \Language{}, use the following syntax:
+\spadkey{rule}
+\centerline{{{\tt rule {\it leftHandSide} == {\it rightHandSide}}}}
+The {\it leftHandSide} is a pattern to be matched and
+the {\it rightHandSide} is its substitution.
+The rule is an object of type \axiomType{RewriteRule} that can be
+assigned to a variable and applied to expressions to transform them.
+\endImportant
+
+Rewrite rules can be collected
+into rulesets so that a set of rules can be applied at once.
+Here is another simplification rule for logarithms.
+\texht{\narrowDisplay{y \log(x) = \log(x^y) \quad\forall \, x \hbox{\ and\ } y.}}{
+\axiom{y * log(x) == log(x ** y)} for any \axiom{x} and \axiom{y}.}
+If instead of giving a single rule following the reserved word \axiom{rule}
+you give a ``pile'' of rules, you create
+what is called a {\it ruleset.}
+%-% \HDindex{ruleset}{ugUserRulesPage}{6.21.}{Rules and Pattern Matching}
+Like rules, rulesets are objects in \Language{} and
+can be assigned to variables.
+You will find it useful to group commonly used rules into input files, and read
+them in as needed.
+\xtc{
+Create a ruleset named \axiom{logrules}.
+}{
+\begin{spadsrc}[\bound{logrules}]
+logrules := rule
+  log(x) + log(y) == log(x * y)
+  y * log x       == log(x ** y)
+\end{spadsrc}
+}
+\xtc{
+Again, create an expression \axiom{f} containing logarithms.
+}{
+\spadpaste{f := a * log(sin x) - 2 * log x \bound{f1}}
+}
+\xtc{
+Apply the ruleset \userfun{logrules} to \axiom{f}.
+}{
+\spadpaste{logrules f \free{f1}\free{logrules}}
+}
+
+We have allowed pattern variables to match arbitrary expressions in the
+above examples.
+Often you want a variable only to match expressions
+satisfying some predicate.
+For example, we may want to apply the transformation
+\texht{\narrowDisplay{y \log(x) = \log(x^y)}}{\axiom{y * log(x) == log(x ** y)}}
+only when \axiom{y} is an integer.
+%
+The way to restrict a pattern variable \axiom{y} by a predicate \axiom{f(y)}
+%-% \HDindex{pattern!variable!predicate}{ugUserRulesPage}{6.21.}{Rules and Pattern Matching}
+is by using a vertical bar \axiomSyntax{|}, which means ``such that,'' in
+%-% \HDindex{such that}{ugUserRulesPage}{6.21.}{Rules and Pattern Matching}
+much the same way it is used in function definitions.
+%-% \HDindex{predicate!on a pattern variable}{ugUserRulesPage}{6.21.}{Rules and Pattern Matching}
+You do this only once, but at the earliest
+(meaning deepest and leftmost) part of the pattern.
+\xtc{
+This restricts the logarithmic rule to create integer exponents only.
+}{
+\begin{spadsrc}[\bound{logrules2}]
+logrules2 := rule
+  log(x) + log(y)          == log(x * y)
+  (y | integer? y) * log x == log(x ** y)
+\end{spadsrc}
+}
+\xtc{
+Compare this with the result of applying the previous set of rules.
+}{
+\spadpaste{f \free{f1}}
+}
+\xtc{
+}{
+\spadpaste{logrules2 f \free{f1}\free{logrules2}}
+}
+You should be aware that you might need to apply a function like
+\spadfun{integer} within your predicate expression to actually apply
+the test function.
+\xtc{
+Here we use \spadfun{integer} because \spad{n} has
+type \spadtype{Expression Integer} but \spadfun{even?} is an operation
+defined on integers.
+}{
+\spadpaste{evenRule := rule cos(x)**(n | integer? n and even? integer n)==(1-sin(x)**2)**(n/2) \bound{evenRule}}
+}
+\xtc{
+Here is the application of the rule.
+}{
+\spadpaste{evenRule( cos(x)**2 ) \free{evenRule}}
+}
+\xtc{
+This is an example of some of the usual identities involving products of
+sines and cosines.
+}{
+\begin{spadsrc}[\bound{sinCosProducts}]
+sinCosProducts == rule
+  sin(x) * sin(y) == (cos(x-y) - cos(x + y))/2
+  cos(x) * cos(y) == (cos(x-y) + cos(x+y))/2
+  sin(x) * cos(y) == (sin(x-y) + sin(x + y))/2
+\end{spadsrc}
+}
+\xtc{
+}{
+\spadpaste{g := sin(a)*sin(b) + cos(b)*cos(a) + sin(2*a)*cos(2*a) \bound{g}}
+}
+\xtc{
+}{
+\spadpaste{sinCosProducts g \free{sinCosProducts g}}
+}
+
+Another qualification you will often want to use is to allow a pattern to
+match an identity element.
+Using the pattern \axiom{x + y}, for example, neither \axiom{x} nor \axiom{y}
+matches the expression \axiom{0}.
+Similarly, if a pattern contains a product \axiom{x*y} or an exponentiation
+\axiom{x**y}, then neither \axiom{x} or \axiom{y} matches \axiom{1}.
+%
+\xtc{
+If identical elements were matched, pattern matching would generally loop.
+Here is an expansion rule for exponentials.
+}{
+\spadpaste{exprule := rule exp(a + b) == exp(a) * exp(b)\bound{exprule}}
+}
+\xtc{
+This rule would cause infinite rewriting on this if either \axiom{a} or
+\axiom{b} were allowed to match \axiom{0}.
+}{
+\spadpaste{exprule exp x \free{exprule}}
+}
+%
+There are occasions when you do want a pattern variable in a sum or
+product to match \axiom{0} or \axiom{1}.
+If so, prefix its name
+with a \axiomSyntax{?} whenever it appears in a left-hand side of a rule.
+For example, consider the following rule for the exponential integral:
+\texht{\narrowDisplay{\int \left(\frac{y+e^x}{x}\right)\: dx = \int \frac{y}{x}\: dx + \hbox{\rm Ei}(x)
+\quad\forall \, x \hbox{\ and\ } y.}}{
+\axiom{integral((y + exp x)/x, x) == integral(y/x, x) + Ei x}
+for any \axiom{x} and \axiom{y}.}
+This rule is valid for \axiom{y = 0}.
+One solution is to create a \axiomType{Ruleset} with two
+rules, one with and one without \axiom{y}.
+A better solution is to use an ``optional'' pattern variable.
+%
+\xtc{
+Define rule \axiom{eirule} with
+a pattern variable \axiom{?y} to indicate
+that an expression may or may not occur.
+}{
+\spadpaste{eirule := rule integral((?y + exp x)/x,x) == integral(y/x,x) + Ei x \bound{eirule}}
+}
+\xtc{
+Apply rule \axiom{eirule} to an integral without this term.
+}{
+\spadpaste{eirule integral(exp u/u, u) \free{eirule}}
+}
+\xtc{
+Apply rule \axiom{eirule} to an integral with this term.
+}{
+\spadpaste{eirule integral(sin u + exp u/u, u) \free{eirule}}
+}
+
+Here is one final adornment you will find useful.
+When matching a pattern of the form \axiom{x + y} to an expression containing a
+long sum of the form \axiom{a +\ldots+ b}, there is no way to predict in
+advance which subset of the sum  matches \axiom{x} and which matches
+\axiom{y}.
+Aside from efficiency, this is generally unimportant since the rule holds for
+any possible combination of matches for \axiom{x} and \axiom{y}.
+In some situations, however, you may want to say which pattern variable is a sum
+(or product) of several terms, and which should match only a single term.
+To do this, put a prefix colon \axiomSyntax{:} before the pattern variable
+that you want to match multiple terms.
+%-% \HDindex{pattern!variable!matching several terms}{ugUserRulesPage}{6.21.}{Rules and Pattern Matching}
+%
+\xtc{
+The remaining rules involve operators \axiom{u} and \axiom{v}.
+%-% \HDindex{operator}{ugUserRulesPage}{6.21.}{Rules and Pattern Matching}
+}{
+\spadpaste{u := operator 'u \bound{u}}
+}
+\xtc{
+These definitions tell \Language{} that
+\axiom{u} and \axiom{v} are formal operators to be used in expressions.
+}{
+\spadpaste{v := operator 'v \bound{v}}
+}
+\xtc{
+First define \axiom{myRule}
+with no restrictions on the pattern variables
+\axiom{x} and \axiom{y}.
+}{
+\spadpaste{myRule := rule u(x + y) == u x + v y \free{u v}\bound{m}}
+}
+\xtc{
+Apply \axiom{myRule} to an expression.
+}{
+\spadpaste{myRule u(a + b + c + d) \free{m}}
+}
+\xtc{
+Define \axiom{myOtherRule} to match several terms
+so that the rule gets applied recursively.
+}{
+\spadpaste{myOtherRule := rule u(:x + y) == u x + v y \free{u v}\bound{m2}}
+}
+\xtc{
+Apply \axiom{myOtherRule} to the same expression.
+}{
+\spadpaste{myOtherRule u(a + b + c + d) \free{m2}}
+}
+
+
+Here are some final remarks on pattern matching.
+Pattern matching provides a very useful paradigm for solving
+certain classes of problems, namely, those that involve
+transformations of one form to another and back.
+However, it is important to recognize its limitations.
+%-% \HDindex{pattern!matching!caveats}{ugUserRulesPage}{6.21.}{Rules and Pattern Matching}
+
+First, pattern matching slows down as the number of rules you have to apply
+increases.
+Thus it is good practice to organize the sets of rules you use optimally so
+that irrelevant rules are never included.
+
+Second, careless use of pattern matching can lead to wrong answers.
+You should avoid using pattern matching to handle hidden algebraic
+relationships that can go undetected by other programs.
+As a simple example, a symbol such as ``J'' can easily be used to represent
+the square root of \axiom{-1} or some other important algebraic quantity.
+Many algorithms branch on whether an expression is zero or not, then divide by
+that expression if it is not.
+If you fail to simplify an expression involving powers of
+\axiom{J} to \axiom{-1,}
+algorithms may incorrectly assume an expression is non-zero, take a wrong
+branch, and produce a meaningless result.
+
+Pattern matching should also not be used as a substitute for a domain.
+In \Language{}, objects of one domain are transformed to objects of other
+domains using well-defined \axiomFun{coerce} operations.
+Pattern matching should be used on objects that are all the same type.
+Thus if your application can be handled by type \axiomType{Expression} in
+\Language{} and you think you need pattern matching, consider this choice
+carefully.
+%-% \HDexptypeindex{Expression}{ugUserRulesPage}{6.21.}{Rules and Pattern Matching}
+You may well be better served by extending an existing domain
+or by building a new domain of objects for your application.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug06.pht b/src/hyper/pages/ug06.pht
new file mode 100644
index 00000000..aa3e0a7f
--- /dev/null
+++ b/src/hyper/pages/ug06.pht
@@ -0,0 +1,4063 @@
+\begin{patch}{ugUserAnonDeclarePagePatch1}
+\begin{paste}{ugUserAnonDeclarePageFull1}{ugUserAnonDeclarePageEmpty1}
+\pastebutton{ugUserAnonDeclarePageFull1}{\hidepaste}
+\tab{5}\spadcommand{(x: INT,y: INT): FRAC INT +-> (x + 2*y)/(y - 1)}
+\indentrel{3}\begin{verbatim}
+   (1)  theMap(NIL)
+          Type: ((Integer,Integer) -> Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePageEmpty1}
+\begin{paste}{ugUserAnonDeclarePageEmpty1}{ugUserAnonDeclarePagePatch1}
+\pastebutton{ugUserAnonDeclarePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{(x: INT,y: INT): FRAC INT +-> (x + 2*y)/(y - 1)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePagePatch2}
+\begin{paste}{ugUserAnonDeclarePageFull2}{ugUserAnonDeclarePageEmpty2}
+\pastebutton{ugUserAnonDeclarePageFull2}{\hidepaste}
+\tab{5}\spadcommand{(x: INT,y: INT) +-> (x + 2*y)/(y - 1)}
+\indentrel{3}\begin{verbatim}
+   (2)  theMap(NIL)
+          Type: ((Integer,Integer) -> Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePageEmpty2}
+\begin{paste}{ugUserAnonDeclarePageEmpty2}{ugUserAnonDeclarePagePatch2}
+\pastebutton{ugUserAnonDeclarePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{(x: INT,y: INT) +-> (x + 2*y)/(y - 1)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePagePatch3}
+\begin{paste}{ugUserAnonDeclarePageFull3}{ugUserAnonDeclarePageEmpty3}
+\pastebutton{ugUserAnonDeclarePageFull3}{\hidepaste}
+\tab{5}\spadcommand{h(x: INT,y: INT): FRAC INT == (x + 2*y)/(y - 1)}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePageEmpty3}
+\begin{paste}{ugUserAnonDeclarePageEmpty3}{ugUserAnonDeclarePagePatch3}
+\pastebutton{ugUserAnonDeclarePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{h(x: INT,y: INT): FRAC INT == (x + 2*y)/(y - 1)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePagePatch4}
+\begin{paste}{ugUserAnonDeclarePageFull4}{ugUserAnonDeclarePageEmpty4}
+\pastebutton{ugUserAnonDeclarePageFull4}{\hidepaste}
+\tab{5}\spadcommand{h == (x: INT,y: INT): FRAC INT +-> (x + 2*y)/(y - 1)}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePageEmpty4}
+\begin{paste}{ugUserAnonDeclarePageEmpty4}{ugUserAnonDeclarePagePatch4}
+\pastebutton{ugUserAnonDeclarePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{h == (x: INT,y: INT): FRAC INT +-> (x + 2*y)/(y - 1)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePagePatch5}
+\begin{paste}{ugUserAnonDeclarePageFull5}{ugUserAnonDeclarePageEmpty5}
+\pastebutton{ugUserAnonDeclarePageFull5}{\hidepaste}
+\tab{5}\spadcommand{addx x == ((y: Integer): Integer +-> x + y)\bound{addx }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePageEmpty5}
+\begin{paste}{ugUserAnonDeclarePageEmpty5}{ugUserAnonDeclarePagePatch5}
+\pastebutton{ugUserAnonDeclarePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{addx x == ((y: Integer): Integer +-> x + y)\bound{addx }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePagePatch6}
+\begin{paste}{ugUserAnonDeclarePageFull6}{ugUserAnonDeclarePageEmpty6}
+\pastebutton{ugUserAnonDeclarePageFull6}{\hidepaste}
+\tab{5}\spadcommand{g := addx 10\free{addx }\bound{g }}
+\indentrel{3}\begin{verbatim}
+   (6)  theMap(LAMBDA_f647nv_704,826)
+                             Type: (Integer -> Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePageEmpty6}
+\begin{paste}{ugUserAnonDeclarePageEmpty6}{ugUserAnonDeclarePagePatch6}
+\pastebutton{ugUserAnonDeclarePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{g := addx 10\free{addx }\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePagePatch7}
+\begin{paste}{ugUserAnonDeclarePageFull7}{ugUserAnonDeclarePageEmpty7}
+\pastebutton{ugUserAnonDeclarePageFull7}{\hidepaste}
+\tab{5}\spadcommand{g 3\free{g }}
+\indentrel{3}\begin{verbatim}
+   (7)  13
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePageEmpty7}
+\begin{paste}{ugUserAnonDeclarePageEmpty7}{ugUserAnonDeclarePagePatch7}
+\pastebutton{ugUserAnonDeclarePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{g 3\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePagePatch8}
+\begin{paste}{ugUserAnonDeclarePageFull8}{ugUserAnonDeclarePageEmpty8}
+\pastebutton{ugUserAnonDeclarePageFull8}{\hidepaste}
+\tab{5}\spadcommand{g(-4)\free{g }}
+\indentrel{3}\begin{verbatim}
+   (8)  6
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonDeclarePageEmpty8}
+\begin{paste}{ugUserAnonDeclarePageEmpty8}{ugUserAnonDeclarePagePatch8}
+\pastebutton{ugUserAnonDeclarePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{g(-4)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPagePatch1}
+\begin{paste}{ugUserPiecePickingPageFull1}{ugUserPiecePickingPageEmpty1}
+\pastebutton{ugUserPiecePickingPageFull1}{\hidepaste}
+\tab{5}\spadcommand{eleven(n | n < 1) == n + 11\bound{ff0 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPageEmpty1}
+\begin{paste}{ugUserPiecePickingPageEmpty1}{ugUserPiecePickingPagePatch1}
+\pastebutton{ugUserPiecePickingPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{eleven(n | n < 1) == n + 11\bound{ff0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPagePatch2}
+\begin{paste}{ugUserPiecePickingPageFull2}{ugUserPiecePickingPageEmpty2}
+\pastebutton{ugUserPiecePickingPageFull2}{\hidepaste}
+\tab{5}\spadcommand{eleven(m) == eleven(eleven(m - 12))\bound{ff1 }\free{ff0 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPageEmpty2}
+\begin{paste}{ugUserPiecePickingPageEmpty2}{ugUserPiecePickingPagePatch2}
+\pastebutton{ugUserPiecePickingPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{eleven(m) == eleven(eleven(m - 12))\bound{ff1 }\free{ff0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPagePatch3}
+\begin{paste}{ugUserPiecePickingPageFull3}{ugUserPiecePickingPageEmpty3}
+\pastebutton{ugUserPiecePickingPageFull3}{\hidepaste}
+\tab{5}\spadcommand{minusEleven(n) ==
+  n >= 0 => n - 11
+  minusEleven (5 + minusEleven(n + 7))
+\bound{rf1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPageEmpty3}
+\begin{paste}{ugUserPiecePickingPageEmpty3}{ugUserPiecePickingPagePatch3}
+\pastebutton{ugUserPiecePickingPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{minusEleven(n) ==
+  n >= 0 => n - 11
+  minusEleven (5 + minusEleven(n + 7))
+\bound{rf1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPagePatch4}
+\begin{paste}{ugUserPiecePickingPageFull4}{ugUserPiecePickingPageEmpty4}
+\pastebutton{ugUserPiecePickingPageFull4}{\hidepaste}
+\tab{5}\spadcommand{s(0) == 1\bound{rf2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPageEmpty4}
+\begin{paste}{ugUserPiecePickingPageEmpty4}{ugUserPiecePickingPagePatch4}
+\pastebutton{ugUserPiecePickingPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{s(0) == 1\bound{rf2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPagePatch5}
+\begin{paste}{ugUserPiecePickingPageFull5}{ugUserPiecePickingPageEmpty5}
+\pastebutton{ugUserPiecePickingPageFull5}{\hidepaste}
+\tab{5}\spadcommand{s(n) == (eleven(n) + minusEleven(n))/n\bound{rf3 }\free{rf2 rf1 ff1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPageEmpty5}
+\begin{paste}{ugUserPiecePickingPageEmpty5}{ugUserPiecePickingPagePatch5}
+\pastebutton{ugUserPiecePickingPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{s(n) == (eleven(n) + minusEleven(n))/n\bound{rf3 }\free{rf2 rf1 ff1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPagePatch6}
+\begin{paste}{ugUserPiecePickingPageFull6}{ugUserPiecePickingPageEmpty6}
+\pastebutton{ugUserPiecePickingPageFull6}{\hidepaste}
+\tab{5}\spadcommand{[s(n) for n in 0..]\free{rf3 }}
+\indentrel{3}\begin{verbatim}
+   (6)  [1,1,1,1,1,1,1,1,1,1,...]
+                          Type: Stream Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPageEmpty6}
+\begin{paste}{ugUserPiecePickingPageEmpty6}{ugUserPiecePickingPagePatch6}
+\pastebutton{ugUserPiecePickingPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{[s(n) for n in 0..]\free{rf3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPagePatch7}
+\begin{paste}{ugUserPiecePickingPageFull7}{ugUserPiecePickingPageEmpty7}
+\pastebutton{ugUserPiecePickingPageFull7}{\hidepaste}
+\tab{5}\spadcommand{t(1) == s(0)\bound{t1 }\free{rf4 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPageEmpty7}
+\begin{paste}{ugUserPiecePickingPageEmpty7}{ugUserPiecePickingPagePatch7}
+\pastebutton{ugUserPiecePickingPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{t(1) == s(0)\bound{t1 }\free{rf4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPagePatch8}
+\begin{paste}{ugUserPiecePickingPageFull8}{ugUserPiecePickingPageEmpty8}
+\pastebutton{ugUserPiecePickingPageFull8}{\hidepaste}
+\tab{5}\spadcommand{t(n | even?(n)) == s(n quo 2)\free{t1 }\bound{t2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPageEmpty8}
+\begin{paste}{ugUserPiecePickingPageEmpty8}{ugUserPiecePickingPagePatch8}
+\pastebutton{ugUserPiecePickingPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{t(n | even?(n)) == s(n quo 2)\free{t1 }\bound{t2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPagePatch9}
+\begin{paste}{ugUserPiecePickingPageFull9}{ugUserPiecePickingPageEmpty9}
+\pastebutton{ugUserPiecePickingPageFull9}{\hidepaste}
+\tab{5}\spadcommand{t(p) == s(- p quo 2)\free{t2 }\bound{t3 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPageEmpty9}
+\begin{paste}{ugUserPiecePickingPageEmpty9}{ugUserPiecePickingPagePatch9}
+\pastebutton{ugUserPiecePickingPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{t(p) == s(- p quo 2)\free{t2 }\bound{t3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPagePatch10}
+\begin{paste}{ugUserPiecePickingPageFull10}{ugUserPiecePickingPageEmpty10}
+\pastebutton{ugUserPiecePickingPageFull10}{\hidepaste}
+\tab{5}\spadcommand{)display value t\free{t2 }}
+\indentrel{3}\begin{verbatim}
+   Definition:
+     t 1 == s(0)
+     t (p | even?(p)) == s(p quo 2)
+     t p == s(- p quo 2)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPageEmpty10}
+\begin{paste}{ugUserPiecePickingPageEmpty10}{ugUserPiecePickingPagePatch10}
+\pastebutton{ugUserPiecePickingPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{)display value t\free{t2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPagePatch11}
+\begin{paste}{ugUserPiecePickingPageFull11}{ugUserPiecePickingPageEmpty11}
+\pastebutton{ugUserPiecePickingPageFull11}{\hidepaste}
+\tab{5}\spadcommand{[t(i) for i in 1..]\free{t3 }\bound{t4 }}
+\indentrel{3}\begin{verbatim}
+   (10)  [1,1,1,1,1,1,1,1,1,1,...]
+                          Type: Stream Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPageEmpty11}
+\begin{paste}{ugUserPiecePickingPageEmpty11}{ugUserPiecePickingPagePatch11}
+\pastebutton{ugUserPiecePickingPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{[t(i) for i in 1..]\free{t3 }\bound{t4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPagePatch12}
+\begin{paste}{ugUserPiecePickingPageFull12}{ugUserPiecePickingPageEmpty12}
+\pastebutton{ugUserPiecePickingPageFull12}{\hidepaste}
+\tab{5}\spadcommand{t(100)\free{t4 }}
+\indentrel{3}\begin{verbatim}
+   (11)  1
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePickingPageEmpty12}
+\begin{paste}{ugUserPiecePickingPageEmpty12}{ugUserPiecePickingPagePatch12}
+\pastebutton{ugUserPiecePickingPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{t(100)\free{t4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPagePatch1}
+\begin{paste}{ugUserPalPageFull1}{ugUserPalPageEmpty1}
+\pastebutton{ugUserPalPageFull1}{\hidepaste}
+\tab{5}\spadcommand{pal? s == palAux?(s,1,\#s)\bound{pal }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPageEmpty1}
+\begin{paste}{ugUserPalPageEmpty1}{ugUserPalPagePatch1}
+\pastebutton{ugUserPalPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{pal? s == palAux?(s,1,\#s)\bound{pal }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPagePatch2}
+\begin{paste}{ugUserPalPageFull2}{ugUserPalPageEmpty2}
+\pastebutton{ugUserPalPageFull2}{\hidepaste}
+\tab{5}\spadcommand{palAux?(s,i,j) ==
+  j > i =>
+    (s.i = s.j) and palAux?(s,i+1,i-1)
+  true
+\bound{palAux }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPageEmpty2}
+\begin{paste}{ugUserPalPageEmpty2}{ugUserPalPagePatch2}
+\pastebutton{ugUserPalPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{palAux?(s,i,j) ==
+  j > i =>
+    (s.i = s.j) and palAux?(s,i+1,i-1)
+  true
+\bound{palAux }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPagePatch3}
+\begin{paste}{ugUserPalPageFull3}{ugUserPalPageEmpty3}
+\pastebutton{ugUserPalPageFull3}{\hidepaste}
+\tab{5}\spadcommand{pal? "Oxford"\free{pal palAux }}
+\indentrel{3}\begin{verbatim}
+   (3)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPageEmpty3}
+\begin{paste}{ugUserPalPageEmpty3}{ugUserPalPagePatch3}
+\pastebutton{ugUserPalPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{pal? "Oxford"\free{pal palAux }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPagePatch4}
+\begin{paste}{ugUserPalPageFull4}{ugUserPalPageEmpty4}
+\pastebutton{ugUserPalPageFull4}{\hidepaste}
+\tab{5}\spadcommand{pal? [4,a,x-1,0,x-1,a,4]\free{pal palAux }}
+\indentrel{3}\begin{verbatim}
+   (4)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPageEmpty4}
+\begin{paste}{ugUserPalPageEmpty4}{ugUserPalPagePatch4}
+\pastebutton{ugUserPalPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{pal? [4,a,x-1,0,x-1,a,4]\free{pal palAux }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPagePatch5}
+\begin{paste}{ugUserPalPageFull5}{ugUserPalPageEmpty5}
+\pastebutton{ugUserPalPageFull5}{\hidepaste}
+\tab{5}\spadcommand{pal? [1,6,15,20,15,6,1]\free{pal palAux }}
+\indentrel{3}\begin{verbatim}
+   (5)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPageEmpty5}
+\begin{paste}{ugUserPalPageEmpty5}{ugUserPalPagePatch5}
+\pastebutton{ugUserPalPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{pal? [1,6,15,20,15,6,1]\free{pal palAux }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPagePatch6}
+\begin{paste}{ugUserPalPageFull6}{ugUserPalPageEmpty6}
+\pastebutton{ugUserPalPageFull6}{\hidepaste}
+\tab{5}\spadcommand{pal?(1441::String)\free{pal palAux }}
+\indentrel{3}\begin{verbatim}
+   (6)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPageEmpty6}
+\begin{paste}{ugUserPalPageEmpty6}{ugUserPalPagePatch6}
+\pastebutton{ugUserPalPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{pal?(1441::String)\free{pal palAux }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPagePatch7}
+\begin{paste}{ugUserPalPageFull7}{ugUserPalPageEmpty7}
+\pastebutton{ugUserPalPageFull7}{\hidepaste}
+\tab{5}\spadcommand{ones := [reduce(+,[10**j for j in 0..i]) for i in 1..]\free{pal palAux }\bound{pal5 }}
+\indentrel{3}\begin{verbatim}
+   (7)
+   [11, 111, 1111, 11111, 111111, 1111111, 11111111,
+    111111111, 1111111111, 11111111111, ...]
+                           Type: Stream PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPageEmpty7}
+\begin{paste}{ugUserPalPageEmpty7}{ugUserPalPagePatch7}
+\pastebutton{ugUserPalPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{ones := [reduce(+,[10**j for j in 0..i]) for i in 1..]\free{pal palAux }\bound{pal5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPagePatch8}
+\begin{paste}{ugUserPalPageFull8}{ugUserPalPageEmpty8}
+\pastebutton{ugUserPalPageFull8}{\hidepaste}
+\tab{5}\spadcommand{squares := [x**2 for x in ones]\free{pal5 }\bound{pal6 }}
+\indentrel{3}\begin{verbatim}
+   (8)
+   [121, 12321, 1234321, 123454321, 12345654321,
+    1234567654321, 123456787654321, 12345678987654321,
+    1234567900987654321, 123456790120987654321, ...]
+                           Type: Stream PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPageEmpty8}
+\begin{paste}{ugUserPalPageEmpty8}{ugUserPalPagePatch8}
+\pastebutton{ugUserPalPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{squares := [x**2 for x in ones]\free{pal5 }\bound{pal6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPagePatch9}
+\begin{paste}{ugUserPalPageFull9}{ugUserPalPageEmpty9}
+\pastebutton{ugUserPalPageFull9}{\hidepaste}
+\tab{5}\spadcommand{[pal?(x::String) for x in squares]\free{pal6 }}
+\indentrel{3}\begin{verbatim}
+   (9)
+   [true,true,true,true,true,true,true,true,true,true,...]
+                                   Type: Stream Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPalPageEmpty9}
+\begin{paste}{ugUserPalPageEmpty9}{ugUserPalPagePatch9}
+\pastebutton{ugUserPalPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{[pal?(x::String) for x in squares]\free{pal6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePagePatch1}
+\begin{paste}{ugUserOnePageFull1}{ugUserOnePageEmpty1}
+\pastebutton{ugUserOnePageFull1}{\hidepaste}
+\tab{5}\spadcommand{fac n == if n < 3 then n else n * fac(n-1)\bound{fac }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePageEmpty1}
+\begin{paste}{ugUserOnePageEmpty1}{ugUserOnePagePatch1}
+\pastebutton{ugUserOnePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{fac n == if n < 3 then n else n * fac(n-1)\bound{fac }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePagePatch2}
+\begin{paste}{ugUserOnePageFull2}{ugUserOnePageEmpty2}
+\pastebutton{ugUserOnePageFull2}{\hidepaste}
+\tab{5}\spadcommand{fac 10\free{fac }}
+\indentrel{3}\begin{verbatim}
+   (2)  3628800
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePageEmpty2}
+\begin{paste}{ugUserOnePageEmpty2}{ugUserOnePagePatch2}
+\pastebutton{ugUserOnePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{fac 10\free{fac }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePagePatch3}
+\begin{paste}{ugUserOnePageFull3}{ugUserOnePageEmpty3}
+\pastebutton{ugUserOnePageFull3}{\hidepaste}
+\tab{5}\spadcommand{s n == reduce(+,[1/i for i in 1..n])\bound{s }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePageEmpty3}
+\begin{paste}{ugUserOnePageEmpty3}{ugUserOnePagePatch3}
+\pastebutton{ugUserOnePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{s n == reduce(+,[1/i for i in 1..n])\bound{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePagePatch4}
+\begin{paste}{ugUserOnePageFull4}{ugUserOnePageEmpty4}
+\pastebutton{ugUserOnePageFull4}{\hidepaste}
+\tab{5}\spadcommand{s 50\free{s }}
+\indentrel{3}\begin{verbatim}
+        13943237577224054960759
+   (4)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+         3099044504245996706400
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePageEmpty4}
+\begin{paste}{ugUserOnePageEmpty4}{ugUserOnePagePatch4}
+\pastebutton{ugUserOnePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{s 50\free{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePagePatch5}
+\begin{paste}{ugUserOnePageFull5}{ugUserOnePageEmpty5}
+\pastebutton{ugUserOnePageFull5}{\hidepaste}
+\tab{5}\spadcommand{mersenne i == 2**i - 1\bound{mersenne }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePageEmpty5}
+\begin{paste}{ugUserOnePageEmpty5}{ugUserOnePagePatch5}
+\pastebutton{ugUserOnePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{mersenne i == 2**i - 1\bound{mersenne }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePagePatch6}
+\begin{paste}{ugUserOnePageFull6}{ugUserOnePageEmpty6}
+\pastebutton{ugUserOnePageFull6}{\hidepaste}
+\tab{5}\spadcommand{mersenne\free{mersenne }}
+\indentrel{3}\begin{verbatim}
+                       i
+   (6)  mersenne i == 2  - 1
+                          Type: FunctionCalled mersenne
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePageEmpty6}
+\begin{paste}{ugUserOnePageEmpty6}{ugUserOnePagePatch6}
+\pastebutton{ugUserOnePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{mersenne\free{mersenne }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePagePatch7}
+\begin{paste}{ugUserOnePageFull7}{ugUserOnePageEmpty7}
+\pastebutton{ugUserOnePageFull7}{\hidepaste}
+\tab{5}\spadcommand{[mersenne i for i in 1..]\free{mersenne }}
+\indentrel{3}\begin{verbatim}
+   (7)  [1,3,7,15,31,63,127,255,511,1023,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePageEmpty7}
+\begin{paste}{ugUserOnePageEmpty7}{ugUserOnePagePatch7}
+\pastebutton{ugUserOnePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{[mersenne i for i in 1..]\free{mersenne }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePagePatch8}
+\begin{paste}{ugUserOnePageFull8}{ugUserOnePageEmpty8}
+\pastebutton{ugUserOnePageFull8}{\hidepaste}
+\tab{5}\spadcommand{mersenneIndex := [n for n in 1.. | prime?(mersenne(n))]\bound{mersenneIndex }\free{mersenne }}
+\indentrel{3}\begin{verbatim}
+   (8)  [2,3,5,7,13,17,19,31,61,89,...]
+                           Type: Stream PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePageEmpty8}
+\begin{paste}{ugUserOnePageEmpty8}{ugUserOnePagePatch8}
+\pastebutton{ugUserOnePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{mersenneIndex := [n for n in 1.. | prime?(mersenne(n))]\bound{mersenneIndex }\free{mersenne }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePagePatch9}
+\begin{paste}{ugUserOnePageFull9}{ugUserOnePageEmpty9}
+\pastebutton{ugUserOnePageFull9}{\hidepaste}
+\tab{5}\spadcommand{mersennePrime n == mersenne mersenneIndex(n)\free{mersenne mersenneIndex }\bound{mersennePrime }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePageEmpty9}
+\begin{paste}{ugUserOnePageEmpty9}{ugUserOnePagePatch9}
+\pastebutton{ugUserOnePageEmpty9}{\showpaste}
+\tab{5}\spadcommand{mersennePrime n == mersenne mersenneIndex(n)\free{mersenne mersenneIndex }\bound{mersennePrime }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePagePatch10}
+\begin{paste}{ugUserOnePageFull10}{ugUserOnePageEmpty10}
+\pastebutton{ugUserOnePageFull10}{\hidepaste}
+\tab{5}\spadcommand{mersennePrime 5\free{mersennePrime }}
+\indentrel{3}\begin{verbatim}
+   (10)  8191
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserOnePageEmpty10}
+\begin{paste}{ugUserOnePageEmpty10}{ugUserOnePagePatch10}
+\pastebutton{ugUserOnePageEmpty10}{\showpaste}
+\tab{5}\spadcommand{mersennePrime 5\free{mersennePrime }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePagePatch1}
+\begin{paste}{ugUserUsePageFull1}{ugUserUsePageEmpty1}
+\pastebutton{ugUserUsePageFull1}{\hidepaste}
+\tab{5}\spadcommand{sin x == 1.0\bound{sin }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePageEmpty1}
+\begin{paste}{ugUserUsePageEmpty1}{ugUserUsePagePatch1}
+\pastebutton{ugUserUsePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{sin x == 1.0\bound{sin }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePagePatch2}
+\begin{paste}{ugUserUsePageFull2}{ugUserUsePageEmpty2}
+\pastebutton{ugUserUsePageFull2}{\hidepaste}
+\tab{5}\spadcommand{sin 4.3\free{sin }}
+\indentrel{3}\begin{verbatim}
+   (2)  1.0
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePageEmpty2}
+\begin{paste}{ugUserUsePageEmpty2}{ugUserUsePagePatch2}
+\pastebutton{ugUserUsePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{sin 4.3\free{sin }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePagePatch3}
+\begin{paste}{ugUserUsePageFull3}{ugUserUsePageEmpty3}
+\pastebutton{ugUserUsePageFull3}{\hidepaste}
+\tab{5}\spadcommand{sin(4.3)$Float}
+\indentrel{3}\begin{verbatim}
+   (3)  - 0.9161659367 4945498404
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePageEmpty3}
+\begin{paste}{ugUserUsePageEmpty3}{ugUserUsePagePatch3}
+\pastebutton{ugUserUsePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{sin(4.3)$Float}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePagePatch4}
+\begin{paste}{ugUserUsePageFull4}{ugUserUsePageEmpty4}
+\pastebutton{ugUserUsePageFull4}{\hidepaste}
+\tab{5}\spadcommand{sin(34.6)$Float}
+\indentrel{3}\begin{verbatim}
+   (4)  - 0.0424680347 1695010154 3
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePageEmpty4}
+\begin{paste}{ugUserUsePageEmpty4}{ugUserUsePagePatch4}
+\pastebutton{ugUserUsePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{sin(34.6)$Float}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePagePatch5}
+\begin{paste}{ugUserUsePageFull5}{ugUserUsePageEmpty5}
+\pastebutton{ugUserUsePageFull5}{\hidepaste}
+\tab{5}\spadcommand{sin x == sin x\bound{sin1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePageEmpty5}
+\begin{paste}{ugUserUsePageEmpty5}{ugUserUsePagePatch5}
+\pastebutton{ugUserUsePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{sin x == sin x\bound{sin1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePagePatch6}
+\begin{paste}{ugUserUsePageFull6}{ugUserUsePageEmpty6}
+\pastebutton{ugUserUsePageFull6}{\hidepaste}
+\tab{5}\spadcommand{sin 4.3\free{sin1 }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePageEmpty6}
+\begin{paste}{ugUserUsePageEmpty6}{ugUserUsePagePatch6}
+\pastebutton{ugUserUsePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{sin 4.3\free{sin1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePagePatch7}
+\begin{paste}{ugUserUsePageFull7}{ugUserUsePageEmpty7}
+\pastebutton{ugUserUsePageFull7}{\hidepaste}
+\tab{5}\spadcommand{sin x == sin(x)$Float\bound{sin2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePageEmpty7}
+\begin{paste}{ugUserUsePageEmpty7}{ugUserUsePagePatch7}
+\pastebutton{ugUserUsePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{sin x == sin(x)$Float\bound{sin2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePagePatch8}
+\begin{paste}{ugUserUsePageFull8}{ugUserUsePageEmpty8}
+\pastebutton{ugUserUsePageFull8}{\hidepaste}
+\tab{5}\spadcommand{sin 4.3\free{sin2 }}
+\indentrel{3}\begin{verbatim}
+   (7)  - 0.9161659367 4945498404
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserUsePageEmpty8}
+\begin{paste}{ugUserUsePageEmpty8}{ugUserUsePagePatch8}
+\pastebutton{ugUserUsePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{sin 4.3\free{sin2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch1}
+\begin{paste}{ugUserMakePageFull1}{ugUserMakePageEmpty1}
+\pastebutton{ugUserMakePageFull1}{\hidepaste}
+\tab{5}\spadcommand{p := -x + y**2 - z**3\bound{p }}
+\indentrel{3}\begin{verbatim}
+           3    2
+   (1)  - z  + y  - x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty1}
+\begin{paste}{ugUserMakePageEmpty1}{ugUserMakePagePatch1}
+\pastebutton{ugUserMakePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{p := -x + y**2 - z**3\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch2}
+\begin{paste}{ugUserMakePageFull2}{ugUserMakePageEmpty2}
+\pastebutton{ugUserMakePageFull2}{\hidepaste}
+\tab{5}\spadcommand{function(p,'f0)\free{p }\bound{f0 }}
+\indentrel{3}\begin{verbatim}
+   (2)  f0
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty2}
+\begin{paste}{ugUserMakePageEmpty2}{ugUserMakePagePatch2}
+\pastebutton{ugUserMakePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{function(p,'f0)\free{p }\bound{f0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch3}
+\begin{paste}{ugUserMakePageFull3}{ugUserMakePageEmpty3}
+\pastebutton{ugUserMakePageFull3}{\hidepaste}
+\tab{5}\spadcommand{f0\free{f0 }}
+\indentrel{3}\begin{verbatim}
+                    3    2
+   (3)  f0 () == - z  + y  - x
+                                Type: FunctionCalled f0
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty3}
+\begin{paste}{ugUserMakePageEmpty3}{ugUserMakePagePatch3}
+\pastebutton{ugUserMakePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f0\free{f0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch4}
+\begin{paste}{ugUserMakePageFull4}{ugUserMakePageEmpty4}
+\pastebutton{ugUserMakePageFull4}{\hidepaste}
+\tab{5}\spadcommand{f0()\free{f0 }}
+\indentrel{3}\begin{verbatim}
+           3    2
+   (4)  - z  + y  - x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty4}
+\begin{paste}{ugUserMakePageEmpty4}{ugUserMakePagePatch4}
+\pastebutton{ugUserMakePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{f0()\free{f0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch5}
+\begin{paste}{ugUserMakePageFull5}{ugUserMakePageEmpty5}
+\pastebutton{ugUserMakePageFull5}{\hidepaste}
+\tab{5}\spadcommand{function(p,'f1,'x)\free{p }\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+   (5)  f1
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty5}
+\begin{paste}{ugUserMakePageEmpty5}{ugUserMakePagePatch5}
+\pastebutton{ugUserMakePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{function(p,'f1,'x)\free{p }\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch6}
+\begin{paste}{ugUserMakePageFull6}{ugUserMakePageEmpty6}
+\pastebutton{ugUserMakePageFull6}{\hidepaste}
+\tab{5}\spadcommand{f1\free{f1 }}
+\indentrel{3}\begin{verbatim}
+                   3    2
+   (6)  f1 x == - z  + y  - x
+                                Type: FunctionCalled f1
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty6}
+\begin{paste}{ugUserMakePageEmpty6}{ugUserMakePagePatch6}
+\pastebutton{ugUserMakePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{f1\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch7}
+\begin{paste}{ugUserMakePageFull7}{ugUserMakePageEmpty7}
+\pastebutton{ugUserMakePageFull7}{\hidepaste}
+\tab{5}\spadcommand{f1(3)\free{f1 }}
+\indentrel{3}\begin{verbatim}
+           3    2
+   (7)  - z  + y  - 3
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty7}
+\begin{paste}{ugUserMakePageEmpty7}{ugUserMakePagePatch7}
+\pastebutton{ugUserMakePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{f1(3)\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch8}
+\begin{paste}{ugUserMakePageFull8}{ugUserMakePageEmpty8}
+\pastebutton{ugUserMakePageFull8}{\hidepaste}
+\tab{5}\spadcommand{function(p,'f2,'x,'y)\free{p }\bound{f2 }}
+\indentrel{3}\begin{verbatim}
+   (8)  f2
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty8}
+\begin{paste}{ugUserMakePageEmpty8}{ugUserMakePagePatch8}
+\pastebutton{ugUserMakePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{function(p,'f2,'x,'y)\free{p }\bound{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch9}
+\begin{paste}{ugUserMakePageFull9}{ugUserMakePageEmpty9}
+\pastebutton{ugUserMakePageFull9}{\hidepaste}
+\tab{5}\spadcommand{f2\free{f2 }}
+\indentrel{3}\begin{verbatim}
+                       3    2
+   (9)  f2 (x,y) == - z  + y  - x
+                                Type: FunctionCalled f2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty9}
+\begin{paste}{ugUserMakePageEmpty9}{ugUserMakePagePatch9}
+\pastebutton{ugUserMakePageEmpty9}{\showpaste}
+\tab{5}\spadcommand{f2\free{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch10}
+\begin{paste}{ugUserMakePageFull10}{ugUserMakePageEmpty10}
+\pastebutton{ugUserMakePageFull10}{\hidepaste}
+\tab{5}\spadcommand{f2(3,0)\free{f2 }}
+\indentrel{3}\begin{verbatim}
+            3
+   (10)  - z  - 3
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty10}
+\begin{paste}{ugUserMakePageEmpty10}{ugUserMakePagePatch10}
+\pastebutton{ugUserMakePageEmpty10}{\showpaste}
+\tab{5}\spadcommand{f2(3,0)\free{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch11}
+\begin{paste}{ugUserMakePageFull11}{ugUserMakePageEmpty11}
+\pastebutton{ugUserMakePageFull11}{\hidepaste}
+\tab{5}\spadcommand{function(p,'f3,['x,'y,'z])\free{p }\bound{f3 }}
+\indentrel{3}\begin{verbatim}
+   (11)  f3
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty11}
+\begin{paste}{ugUserMakePageEmpty11}{ugUserMakePagePatch11}
+\pastebutton{ugUserMakePageEmpty11}{\showpaste}
+\tab{5}\spadcommand{function(p,'f3,['x,'y,'z])\free{p }\bound{f3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch12}
+\begin{paste}{ugUserMakePageFull12}{ugUserMakePageEmpty12}
+\pastebutton{ugUserMakePageFull12}{\hidepaste}
+\tab{5}\spadcommand{f3\free{f3 }}
+\indentrel{3}\begin{verbatim}
+                          3    2
+   (12)  f3 (x,y,z) == - z  + y  - x
+                                Type: FunctionCalled f3
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty12}
+\begin{paste}{ugUserMakePageEmpty12}{ugUserMakePagePatch12}
+\pastebutton{ugUserMakePageEmpty12}{\showpaste}
+\tab{5}\spadcommand{f3\free{f3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch13}
+\begin{paste}{ugUserMakePageFull13}{ugUserMakePageEmpty13}
+\pastebutton{ugUserMakePageFull13}{\hidepaste}
+\tab{5}\spadcommand{f3(3,0,-6)\free{f3 }}
+\indentrel{3}\begin{verbatim}
+   (13)  213
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty13}
+\begin{paste}{ugUserMakePageEmpty13}{ugUserMakePagePatch13}
+\pastebutton{ugUserMakePageEmpty13}{\showpaste}
+\tab{5}\spadcommand{f3(3,0,-6)\free{f3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch14}
+\begin{paste}{ugUserMakePageFull14}{ugUserMakePageEmpty14}
+\pastebutton{ugUserMakePageFull14}{\hidepaste}
+\tab{5}\spadcommand{g: (Integer, Integer) -> Float\bound{g }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty14}
+\begin{paste}{ugUserMakePageEmpty14}{ugUserMakePagePatch14}
+\pastebutton{ugUserMakePageEmpty14}{\showpaste}
+\tab{5}\spadcommand{g: (Integer, Integer) -> Float\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch15}
+\begin{paste}{ugUserMakePageFull15}{ugUserMakePageEmpty15}
+\pastebutton{ugUserMakePageFull15}{\hidepaste}
+\tab{5}\spadcommand{D(sin(x-y)/cos(x+y),x)\bound{prev }}
+\indentrel{3}\begin{verbatim}
+         - sin(y - x)sin(y + x) + cos(y - x)cos(y + x)
+   (15)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                                    2
+                          cos(y + x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty15}
+\begin{paste}{ugUserMakePageEmpty15}{ugUserMakePagePatch15}
+\pastebutton{ugUserMakePageEmpty15}{\showpaste}
+\tab{5}\spadcommand{D(sin(x-y)/cos(x+y),x)\bound{prev }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch16}
+\begin{paste}{ugUserMakePageFull16}{ugUserMakePageEmpty16}
+\pastebutton{ugUserMakePageFull16}{\hidepaste}
+\tab{5}\spadcommand{function(\%,'g,'x,'y)\free{g }\free{prev }}
+\indentrel{3}\begin{verbatim}
+   (16)  g
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty16}
+\begin{paste}{ugUserMakePageEmpty16}{ugUserMakePagePatch16}
+\pastebutton{ugUserMakePageEmpty16}{\showpaste}
+\tab{5}\spadcommand{function(\%,'g,'x,'y)\free{g }\free{prev }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePagePatch17}
+\begin{paste}{ugUserMakePageFull17}{ugUserMakePageEmpty17}
+\pastebutton{ugUserMakePageFull17}{\hidepaste}
+\tab{5}\spadcommand{g\free{g }}
+\indentrel{3}\begin{verbatim}
+   (17)
+   g (x,y) ==
+     - sin(y - x)sin(y + x) + cos(y - x)cos(y + x)
+     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                                2
+                      cos(y + x)
+                                 Type: FunctionCalled g
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMakePageEmpty17}
+\begin{paste}{ugUserMakePageEmpty17}{ugUserMakePagePatch17}
+\pastebutton{ugUserMakePageEmpty17}{\showpaste}
+\tab{5}\spadcommand{g\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserCachePagePatch1}
+\begin{paste}{ugUserCachePageFull1}{ugUserCachePageEmpty1}
+\pastebutton{ugUserCachePageFull1}{\hidepaste}
+\tab{5}\spadcommand{)set functions cache 3 f g\bound{cache }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserCachePageEmpty1}
+\begin{paste}{ugUserCachePageEmpty1}{ugUserCachePagePatch1}
+\pastebutton{ugUserCachePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)set functions cache 3 f g\bound{cache }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserCachePagePatch2}
+\begin{paste}{ugUserCachePageFull2}{ugUserCachePageEmpty2}
+\pastebutton{ugUserCachePageFull2}{\hidepaste}
+\tab{5}\spadcommand{f x == factorial(2**x)\bound{fdef }\free{cache }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserCachePageEmpty2}
+\begin{paste}{ugUserCachePageEmpty2}{ugUserCachePagePatch2}
+\pastebutton{ugUserCachePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f x == factorial(2**x)\bound{fdef }\free{cache }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserCachePagePatch3}
+\begin{paste}{ugUserCachePageFull3}{ugUserCachePageEmpty3}
+\pastebutton{ugUserCachePageFull3}{\hidepaste}
+\tab{5}\spadcommand{f(4)\free{}\free{cache }}
+\indentrel{3}\begin{verbatim}
+   (2)  20922789888000
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserCachePageEmpty3}
+\begin{paste}{ugUserCachePageEmpty3}{ugUserCachePagePatch3}
+\pastebutton{ugUserCachePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f(4)\free{}\free{cache }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserCachePagePatch4}
+\begin{paste}{ugUserCachePageFull4}{ugUserCachePageEmpty4}
+\pastebutton{ugUserCachePageFull4}{\hidepaste}
+\tab{5}\spadcommand{)set functions cache all}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserCachePageEmpty4}
+\begin{paste}{ugUserCachePageEmpty4}{ugUserCachePagePatch4}
+\pastebutton{ugUserCachePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{)set functions cache all}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserCachePagePatch5}
+\begin{paste}{ugUserCachePageFull5}{ugUserCachePageEmpty5}
+\pastebutton{ugUserCachePageFull5}{\hidepaste}
+\tab{5}\spadcommand{)set functions cache 0}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserCachePageEmpty5}
+\begin{paste}{ugUserCachePageEmpty5}{ugUserCachePagePatch5}
+\pastebutton{ugUserCachePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{)set functions cache 0}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserCachePagePatch6}
+\begin{paste}{ugUserCachePageFull6}{ugUserCachePageEmpty6}
+\pastebutton{ugUserCachePageFull6}{\hidepaste}
+\tab{5}\spadcommand{)set functions cache 0 f g}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserCachePageEmpty6}
+\begin{paste}{ugUserCachePageEmpty6}{ugUserCachePagePatch6}
+\pastebutton{ugUserCachePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{)set functions cache 0 f g}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDelayPagePatch1}
+\begin{paste}{ugUserDelayPageFull1}{ugUserDelayPageEmpty1}
+\pastebutton{ugUserDelayPageFull1}{\hidepaste}
+\tab{5}\spadcommand{sin24() == sin(24.0)\bound{sin24 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDelayPageEmpty1}
+\begin{paste}{ugUserDelayPageEmpty1}{ugUserDelayPagePatch1}
+\pastebutton{ugUserDelayPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{sin24() == sin(24.0)\bound{sin24 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDelayPagePatch2}
+\begin{paste}{ugUserDelayPageFull2}{ugUserDelayPageEmpty2}
+\pastebutton{ugUserDelayPageFull2}{\hidepaste}
+\tab{5}\spadcommand{sin24()\free{sin24 }}
+\indentrel{3}\begin{verbatim}
+   (2)  - 0.9055783620 0662384514
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDelayPageEmpty2}
+\begin{paste}{ugUserDelayPageEmpty2}{ugUserDelayPagePatch2}
+\pastebutton{ugUserDelayPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{sin24()\free{sin24 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDelayPagePatch3}
+\begin{paste}{ugUserDelayPageFull3}{ugUserDelayPageEmpty3}
+\pastebutton{ugUserDelayPageFull3}{\hidepaste}
+\tab{5}\spadcommand{sin24\free{sin24 }}
+\indentrel{3}\begin{verbatim}
+   (3)  sin24 () == sin(24.0)
+                             Type: FunctionCalled sin24
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDelayPageEmpty3}
+\begin{paste}{ugUserDelayPageEmpty3}{ugUserDelayPagePatch3}
+\pastebutton{ugUserDelayPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{sin24\free{sin24 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDelayPagePatch4}
+\begin{paste}{ugUserDelayPageFull4}{ugUserDelayPageEmpty4}
+\pastebutton{ugUserDelayPageFull4}{\hidepaste}
+\tab{5}\spadcommand{cos24 == cos(24.0)\bound{cos24 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDelayPageEmpty4}
+\begin{paste}{ugUserDelayPageEmpty4}{ugUserDelayPagePatch4}
+\pastebutton{ugUserDelayPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{cos24 == cos(24.0)\bound{cos24 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDelayPagePatch5}
+\begin{paste}{ugUserDelayPageFull5}{ugUserDelayPageEmpty5}
+\pastebutton{ugUserDelayPageFull5}{\hidepaste}
+\tab{5}\spadcommand{cos24\free{cos24 }}
+\indentrel{3}\begin{verbatim}
+   (5)  0.4241790073 3699697594
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDelayPageEmpty5}
+\begin{paste}{ugUserDelayPageEmpty5}{ugUserDelayPagePatch5}
+\pastebutton{ugUserDelayPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{cos24\free{cos24 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPagePatch1}
+\begin{paste}{ugUserRecurPageFull1}{ugUserRecurPageEmpty1}
+\pastebutton{ugUserRecurPageFull1}{\hidepaste}
+\tab{5}\spadcommand{[fibonacci(i) for i in 0..]}
+\indentrel{3}\begin{verbatim}
+   (1)  [0,1,1,2,3,5,8,13,21,34,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPageEmpty1}
+\begin{paste}{ugUserRecurPageEmpty1}{ugUserRecurPagePatch1}
+\pastebutton{ugUserRecurPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{[fibonacci(i) for i in 0..]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPagePatch2}
+\begin{paste}{ugUserRecurPageFull2}{ugUserRecurPageEmpty2}
+\pastebutton{ugUserRecurPageFull2}{\hidepaste}
+\tab{5}\spadcommand{fib(1) == 1\bound{fib0 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPageEmpty2}
+\begin{paste}{ugUserRecurPageEmpty2}{ugUserRecurPagePatch2}
+\pastebutton{ugUserRecurPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{fib(1) == 1\bound{fib0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPagePatch3}
+\begin{paste}{ugUserRecurPageFull3}{ugUserRecurPageEmpty3}
+\pastebutton{ugUserRecurPageFull3}{\hidepaste}
+\tab{5}\spadcommand{fib(2) == 1\bound{fib1 }\free{fib0 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPageEmpty3}
+\begin{paste}{ugUserRecurPageEmpty3}{ugUserRecurPagePatch3}
+\pastebutton{ugUserRecurPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{fib(2) == 1\bound{fib1 }\free{fib0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPagePatch4}
+\begin{paste}{ugUserRecurPageFull4}{ugUserRecurPageEmpty4}
+\pastebutton{ugUserRecurPageFull4}{\hidepaste}
+\tab{5}\spadcommand{fib(n) == fib(n-1) + fib(n-2)\bound{fibn }\free{fib1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPageEmpty4}
+\begin{paste}{ugUserRecurPageEmpty4}{ugUserRecurPagePatch4}
+\pastebutton{ugUserRecurPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{fib(n) == fib(n-1) + fib(n-2)\bound{fibn }\free{fib1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPagePatch5}
+\begin{paste}{ugUserRecurPageFull5}{ugUserRecurPageEmpty5}
+\pastebutton{ugUserRecurPageFull5}{\hidepaste}
+\tab{5}\spadcommand{fib(500)\free{fibn }}
+\indentrel{3}\begin{verbatim}
+   (5)
+  139423224561697880139724382870407283950070256587697307_
+   264108962948325571622863290691557658876222521294125
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPageEmpty5}
+\begin{paste}{ugUserRecurPageEmpty5}{ugUserRecurPagePatch5}
+\pastebutton{ugUserRecurPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{fib(500)\free{fibn }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPagePatch6}
+\begin{paste}{ugUserRecurPageFull6}{ugUserRecurPageEmpty6}
+\pastebutton{ugUserRecurPageFull6}{\hidepaste}
+\tab{5}\spadcommand{p(0) == 1\bound{p0 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPageEmpty6}
+\begin{paste}{ugUserRecurPageEmpty6}{ugUserRecurPagePatch6}
+\pastebutton{ugUserRecurPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{p(0) == 1\bound{p0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPagePatch7}
+\begin{paste}{ugUserRecurPageFull7}{ugUserRecurPageEmpty7}
+\pastebutton{ugUserRecurPageFull7}{\hidepaste}
+\tab{5}\spadcommand{p(1) == x\bound{p1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPageEmpty7}
+\begin{paste}{ugUserRecurPageEmpty7}{ugUserRecurPagePatch7}
+\pastebutton{ugUserRecurPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{p(1) == x\bound{p1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPagePatch8}
+\begin{paste}{ugUserRecurPageFull8}{ugUserRecurPageEmpty8}
+\pastebutton{ugUserRecurPageFull8}{\hidepaste}
+\tab{5}\spadcommand{p(n) == ((2*n-1)*x*p(n-1) - (n-1)*p(n-2))/n\bound{pn }\free{p1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPageEmpty8}
+\begin{paste}{ugUserRecurPageEmpty8}{ugUserRecurPagePatch8}
+\pastebutton{ugUserRecurPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{p(n) == ((2*n-1)*x*p(n-1) - (n-1)*p(n-2))/n\bound{pn }\free{p1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPagePatch9}
+\begin{paste}{ugUserRecurPageFull9}{ugUserRecurPageEmpty9}
+\pastebutton{ugUserRecurPageFull9}{\hidepaste}
+\tab{5}\spadcommand{p(6)\free{pn }}
+\indentrel{3}\begin{verbatim}
+        231  6   315  4   105  2    5
+   (9)  ÄÄÄ x  - ÄÄÄ x  + ÄÄÄ x  - ÄÄ
+         16       16       16      16
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRecurPageEmpty9}
+\begin{paste}{ugUserRecurPageEmpty9}{ugUserRecurPagePatch9}
+\pastebutton{ugUserRecurPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{p(6)\free{pn }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch1}
+\begin{paste}{ugUserRulesPageFull1}{ugUserRulesPageEmpty1}
+\pastebutton{ugUserRulesPageFull1}{\hidepaste}
+\tab{5}\spadcommand{logrule := rule log(x) + log(y) == log(x * y)\bound{logrule }}
+\indentrel{3}\begin{verbatim}
+   (1)  log(y) + log(x) + %B == log(x y) + %B
+  Type: RewriteRule(Integer,Integer,Expression Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty1}
+\begin{paste}{ugUserRulesPageEmpty1}{ugUserRulesPagePatch1}
+\pastebutton{ugUserRulesPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{logrule := rule log(x) + log(y) == log(x * y)\bound{logrule }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch2}
+\begin{paste}{ugUserRulesPageFull2}{ugUserRulesPageEmpty2}
+\pastebutton{ugUserRulesPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f := log sin x + log x\bound{f }}
+\indentrel{3}\begin{verbatim}
+   (2)  log(sin(x)) + log(x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty2}
+\begin{paste}{ugUserRulesPageEmpty2}{ugUserRulesPagePatch2}
+\pastebutton{ugUserRulesPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f := log sin x + log x\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch3}
+\begin{paste}{ugUserRulesPageFull3}{ugUserRulesPageEmpty3}
+\pastebutton{ugUserRulesPageFull3}{\hidepaste}
+\tab{5}\spadcommand{logrule f\free{f }\free{logrule }}
+\indentrel{3}\begin{verbatim}
+   (3)  log(x sin(x))
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty3}
+\begin{paste}{ugUserRulesPageEmpty3}{ugUserRulesPagePatch3}
+\pastebutton{ugUserRulesPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{logrule f\free{f }\free{logrule }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch4}
+\begin{paste}{ugUserRulesPageFull4}{ugUserRulesPageEmpty4}
+\pastebutton{ugUserRulesPageFull4}{\hidepaste}
+\tab{5}\spadcommand{logrules := rule
+  log(x) + log(y) == log(x * y)
+  y * log x       == log(x ** y)
+\bound{logrules }}
+\indentrel{3}\begin{verbatim}
+   (4)
+   {log(y) + log(x) + %C == log(x y) + %C,
+                     y
+    y log(x) == log(x )}
+      Type: Ruleset(Integer,Integer,Expression Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty4}
+\begin{paste}{ugUserRulesPageEmpty4}{ugUserRulesPagePatch4}
+\pastebutton{ugUserRulesPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{logrules := rule
+  log(x) + log(y) == log(x * y)
+  y * log x       == log(x ** y)
+\bound{logrules }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch5}
+\begin{paste}{ugUserRulesPageFull5}{ugUserRulesPageEmpty5}
+\pastebutton{ugUserRulesPageFull5}{\hidepaste}
+\tab{5}\spadcommand{f := a * log(sin x) - 2 * log x\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+   (5)  a log(sin(x)) - 2log(x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty5}
+\begin{paste}{ugUserRulesPageEmpty5}{ugUserRulesPagePatch5}
+\pastebutton{ugUserRulesPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{f := a * log(sin x) - 2 * log x\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch6}
+\begin{paste}{ugUserRulesPageFull6}{ugUserRulesPageEmpty6}
+\pastebutton{ugUserRulesPageFull6}{\hidepaste}
+\tab{5}\spadcommand{logrules f\free{f1 }\free{logrules }}
+\indentrel{3}\begin{verbatim}
+                  a
+            sin(x)
+   (6)  log(ÄÄÄÄÄÄÄ)
+                2
+               x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty6}
+\begin{paste}{ugUserRulesPageEmpty6}{ugUserRulesPagePatch6}
+\pastebutton{ugUserRulesPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{logrules f\free{f1 }\free{logrules }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch7}
+\begin{paste}{ugUserRulesPageFull7}{ugUserRulesPageEmpty7}
+\pastebutton{ugUserRulesPageFull7}{\hidepaste}
+\tab{5}\spadcommand{logrules2 := rule
+  log(x) + log(y)          == log(x * y)
+  (y | integer? y) * log x == log(x ** y)
+\bound{logrules2 }}
+\indentrel{3}\begin{verbatim}
+   (7)
+   {log(y) + log(x) + %E == log(x y) + %E,
+                     y
+    y log(x) == log(x )}
+      Type: Ruleset(Integer,Integer,Expression Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty7}
+\begin{paste}{ugUserRulesPageEmpty7}{ugUserRulesPagePatch7}
+\pastebutton{ugUserRulesPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{logrules2 := rule
+  log(x) + log(y)          == log(x * y)
+  (y | integer? y) * log x == log(x ** y)
+\bound{logrules2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch8}
+\begin{paste}{ugUserRulesPageFull8}{ugUserRulesPageEmpty8}
+\pastebutton{ugUserRulesPageFull8}{\hidepaste}
+\tab{5}\spadcommand{f\free{f1 }}
+\indentrel{3}\begin{verbatim}
+   (8)  a log(sin(x)) - 2log(x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty8}
+\begin{paste}{ugUserRulesPageEmpty8}{ugUserRulesPagePatch8}
+\pastebutton{ugUserRulesPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{f\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch9}
+\begin{paste}{ugUserRulesPageFull9}{ugUserRulesPageEmpty9}
+\pastebutton{ugUserRulesPageFull9}{\hidepaste}
+\tab{5}\spadcommand{logrules2 f\free{f1 }\free{logrules2 }}
+\indentrel{3}\begin{verbatim}
+                             1
+   (9)  a log(sin(x)) + log(ÄÄ)
+                             2
+                            x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty9}
+\begin{paste}{ugUserRulesPageEmpty9}{ugUserRulesPagePatch9}
+\pastebutton{ugUserRulesPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{logrules2 f\free{f1 }\free{logrules2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch10}
+\begin{paste}{ugUserRulesPageFull10}{ugUserRulesPageEmpty10}
+\pastebutton{ugUserRulesPageFull10}{\hidepaste}
+\tab{5}\spadcommand{evenRule := rule cos(x)**(n | integer? n and even? integer n)==(1-sin(x)**2)**(n/2)\bound{evenRule }}
+\indentrel{3}\begin{verbatim}
+                                   n
+                                   Ä
+               n             2     2
+   (10)  cos(x)  == (- sin(x)  + 1)
+  Type: RewriteRule(Integer,Integer,Expression Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty10}
+\begin{paste}{ugUserRulesPageEmpty10}{ugUserRulesPagePatch10}
+\pastebutton{ugUserRulesPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{evenRule := rule cos(x)**(n | integer? n and even? integer n)==(1-sin(x)**2)**(n/2)\bound{evenRule }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch11}
+\begin{paste}{ugUserRulesPageFull11}{ugUserRulesPageEmpty11}
+\pastebutton{ugUserRulesPageFull11}{\hidepaste}
+\tab{5}\spadcommand{evenRule( cos(x)**2 )\free{evenRule }}
+\indentrel{3}\begin{verbatim}
+                 2
+   (11)  - sin(x)  + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty11}
+\begin{paste}{ugUserRulesPageEmpty11}{ugUserRulesPagePatch11}
+\pastebutton{ugUserRulesPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{evenRule( cos(x)**2 )\free{evenRule }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch12}
+\begin{paste}{ugUserRulesPageFull12}{ugUserRulesPageEmpty12}
+\pastebutton{ugUserRulesPageFull12}{\hidepaste}
+\tab{5}\spadcommand{sinCosProducts == rule
+  sin(x) * sin(y) == (cos(x-y) - cos(x + y))/2
+  cos(x) * cos(y) == (cos(x-y) + cos(x+y))/2
+  sin(x) * cos(y) == (sin(x-y) + sin(x + y))/2
+\bound{sinCosProducts }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty12}
+\begin{paste}{ugUserRulesPageEmpty12}{ugUserRulesPagePatch12}
+\pastebutton{ugUserRulesPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{sinCosProducts == rule
+  sin(x) * sin(y) == (cos(x-y) - cos(x + y))/2
+  cos(x) * cos(y) == (cos(x-y) + cos(x+y))/2
+  sin(x) * cos(y) == (sin(x-y) + sin(x + y))/2
+\bound{sinCosProducts }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch13}
+\begin{paste}{ugUserRulesPageFull13}{ugUserRulesPageEmpty13}
+\pastebutton{ugUserRulesPageFull13}{\hidepaste}
+\tab{5}\spadcommand{g := sin(a)*sin(b) + cos(b)*cos(a) + sin(2*a)*cos(2*a)\bound{g }}
+\indentrel{3}\begin{verbatim}
+   (13)  sin(a)sin(b) + cos(2a)sin(2a) + cos(a)cos(b)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty13}
+\begin{paste}{ugUserRulesPageEmpty13}{ugUserRulesPagePatch13}
+\pastebutton{ugUserRulesPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{g := sin(a)*sin(b) + cos(b)*cos(a) + sin(2*a)*cos(2*a)\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch14}
+\begin{paste}{ugUserRulesPageFull14}{ugUserRulesPageEmpty14}
+\pastebutton{ugUserRulesPageFull14}{\hidepaste}
+\tab{5}\spadcommand{sinCosProducts g\free{sinCosProducts g }}
+\indentrel{3}\begin{verbatim}
+         sin(4a) + 2cos(b - a)
+   (14)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                   2
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty14}
+\begin{paste}{ugUserRulesPageEmpty14}{ugUserRulesPagePatch14}
+\pastebutton{ugUserRulesPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{sinCosProducts g\free{sinCosProducts g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch15}
+\begin{paste}{ugUserRulesPageFull15}{ugUserRulesPageEmpty15}
+\pastebutton{ugUserRulesPageFull15}{\hidepaste}
+\tab{5}\spadcommand{exprule := rule exp(a + b) == exp(a) * exp(b)\bound{exprule }}
+\indentrel{3}\begin{verbatim}
+           b + a      a  b
+   (15)  %e      == %e %e
+  Type: RewriteRule(Integer,Integer,Expression Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty15}
+\begin{paste}{ugUserRulesPageEmpty15}{ugUserRulesPagePatch15}
+\pastebutton{ugUserRulesPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{exprule := rule exp(a + b) == exp(a) * exp(b)\bound{exprule }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch16}
+\begin{paste}{ugUserRulesPageFull16}{ugUserRulesPageEmpty16}
+\pastebutton{ugUserRulesPageFull16}{\hidepaste}
+\tab{5}\spadcommand{exprule exp x\free{exprule }}
+\indentrel{3}\begin{verbatim}
+           x
+   (16)  %e
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty16}
+\begin{paste}{ugUserRulesPageEmpty16}{ugUserRulesPagePatch16}
+\pastebutton{ugUserRulesPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{exprule exp x\free{exprule }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch17}
+\begin{paste}{ugUserRulesPageFull17}{ugUserRulesPageEmpty17}
+\pastebutton{ugUserRulesPageFull17}{\hidepaste}
+\tab{5}\spadcommand{eirule := rule integral((?y + exp x)/x,x) == integral(y/x,x) + Ei x\bound{eirule }}
+\indentrel{3}\begin{verbatim}
+            x   %N
+          Ú¿  %e   + y                   y
+   (17)   ³   ÄÄÄÄÄÄÄÄ d%N  == 'integral(Ä,x) + 'Ei(x)
+         ÀÙ      %N                      x
+  Type: RewriteRule(Integer,Integer,Expression Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty17}
+\begin{paste}{ugUserRulesPageEmpty17}{ugUserRulesPagePatch17}
+\pastebutton{ugUserRulesPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{eirule := rule integral((?y + exp x)/x,x) == integral(y/x,x) + Ei x\bound{eirule }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch18}
+\begin{paste}{ugUserRulesPageFull18}{ugUserRulesPageEmpty18}
+\pastebutton{ugUserRulesPageFull18}{\hidepaste}
+\tab{5}\spadcommand{eirule integral(exp u/u, u)\free{eirule }}
+\indentrel{3}\begin{verbatim}
+   (18)  Ei(u)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty18}
+\begin{paste}{ugUserRulesPageEmpty18}{ugUserRulesPagePatch18}
+\pastebutton{ugUserRulesPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{eirule integral(exp u/u, u)\free{eirule }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch19}
+\begin{paste}{ugUserRulesPageFull19}{ugUserRulesPageEmpty19}
+\pastebutton{ugUserRulesPageFull19}{\hidepaste}
+\tab{5}\spadcommand{eirule integral(sin u + exp u/u, u)\free{eirule }}
+\indentrel{3}\begin{verbatim}
+            u
+          Ú¿
+   (19)   ³   sin(%N)d%N  + Ei(u)
+         ÀÙ
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty19}
+\begin{paste}{ugUserRulesPageEmpty19}{ugUserRulesPagePatch19}
+\pastebutton{ugUserRulesPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{eirule integral(sin u + exp u/u, u)\free{eirule }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch20}
+\begin{paste}{ugUserRulesPageFull20}{ugUserRulesPageEmpty20}
+\pastebutton{ugUserRulesPageFull20}{\hidepaste}
+\tab{5}\spadcommand{u := operator 'u\bound{u }}
+\indentrel{3}\begin{verbatim}
+   (20)  u
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty20}
+\begin{paste}{ugUserRulesPageEmpty20}{ugUserRulesPagePatch20}
+\pastebutton{ugUserRulesPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{u := operator 'u\bound{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch21}
+\begin{paste}{ugUserRulesPageFull21}{ugUserRulesPageEmpty21}
+\pastebutton{ugUserRulesPageFull21}{\hidepaste}
+\tab{5}\spadcommand{v := operator 'v\bound{v }}
+\indentrel{3}\begin{verbatim}
+   (21)  v
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty21}
+\begin{paste}{ugUserRulesPageEmpty21}{ugUserRulesPagePatch21}
+\pastebutton{ugUserRulesPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{v := operator 'v\bound{v }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch22}
+\begin{paste}{ugUserRulesPageFull22}{ugUserRulesPageEmpty22}
+\pastebutton{ugUserRulesPageFull22}{\hidepaste}
+\tab{5}\spadcommand{myRule := rule u(x + y) == u x + v y\free{u v }\bound{m }}
+\indentrel{3}\begin{verbatim}
+   (22)  u(y + x) == 'v(y) + 'u(x)
+  Type: RewriteRule(Integer,Integer,Expression Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty22}
+\begin{paste}{ugUserRulesPageEmpty22}{ugUserRulesPagePatch22}
+\pastebutton{ugUserRulesPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{myRule := rule u(x + y) == u x + v y\free{u v }\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch23}
+\begin{paste}{ugUserRulesPageFull23}{ugUserRulesPageEmpty23}
+\pastebutton{ugUserRulesPageFull23}{\hidepaste}
+\tab{5}\spadcommand{myRule u(a + b + c + d)\free{m }}
+\indentrel{3}\begin{verbatim}
+   (23)  v(d + c + b) + u(a)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty23}
+\begin{paste}{ugUserRulesPageEmpty23}{ugUserRulesPagePatch23}
+\pastebutton{ugUserRulesPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{myRule u(a + b + c + d)\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch24}
+\begin{paste}{ugUserRulesPageFull24}{ugUserRulesPageEmpty24}
+\pastebutton{ugUserRulesPageFull24}{\hidepaste}
+\tab{5}\spadcommand{myOtherRule := rule u(:x + y) == u x + v y\free{u v }\bound{m2 }}
+\indentrel{3}\begin{verbatim}
+   (24)  u(y + x) == 'v(y) + 'u(x)
+  Type: RewriteRule(Integer,Integer,Expression Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty24}
+\begin{paste}{ugUserRulesPageEmpty24}{ugUserRulesPagePatch24}
+\pastebutton{ugUserRulesPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{myOtherRule := rule u(:x + y) == u x + v y\free{u v }\bound{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPagePatch25}
+\begin{paste}{ugUserRulesPageFull25}{ugUserRulesPageEmpty25}
+\pastebutton{ugUserRulesPageFull25}{\hidepaste}
+\tab{5}\spadcommand{myOtherRule u(a + b + c + d)\free{m2 }}
+\indentrel{3}\begin{verbatim}
+   (25)  v(c) + v(b) + v(a) + u(d)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserRulesPageEmpty25}
+\begin{paste}{ugUserRulesPageEmpty25}{ugUserRulesPagePatch25}
+\pastebutton{ugUserRulesPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{myOtherRule u(a + b + c + d)\free{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserIntroPagePatch1}
+\begin{paste}{ugUserIntroPageFull1}{ugUserIntroPageEmpty1}
+\pastebutton{ugUserIntroPageFull1}{\hidepaste}
+\tab{5}\spadcommand{factor(12)}
+\indentrel{3}\begin{verbatim}
+         2
+   (1)  2 3
+                                 Type: Factored Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserIntroPageEmpty1}
+\begin{paste}{ugUserIntroPageEmpty1}{ugUserIntroPagePatch1}
+\pastebutton{ugUserIntroPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{factor(12)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserIntroPagePatch2}
+\begin{paste}{ugUserIntroPageFull2}{ugUserIntroPageEmpty2}
+\pastebutton{ugUserIntroPageFull2}{\hidepaste}
+\tab{5}\spadcommand{3 + 4}
+\indentrel{3}\begin{verbatim}
+   (2)  7
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserIntroPageEmpty2}
+\begin{paste}{ugUserIntroPageEmpty2}{ugUserIntroPagePatch2}
+\pastebutton{ugUserIntroPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{3 + 4}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserIntroPagePatch3}
+\begin{paste}{ugUserIntroPageFull3}{ugUserIntroPageEmpty3}
+\pastebutton{ugUserIntroPageFull3}{\hidepaste}
+\tab{5}\spadcommand{1 + 2 + 7}
+\indentrel{3}\begin{verbatim}
+   (3)  10
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserIntroPageEmpty3}
+\begin{paste}{ugUserIntroPageEmpty3}{ugUserIntroPagePatch3}
+\pastebutton{ugUserIntroPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{1 + 2 + 7}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFunMacPagePatch1}
+\begin{paste}{ugUserFunMacPageFull1}{ugUserFunMacPageEmpty1}
+\pastebutton{ugUserFunMacPageFull1}{\hidepaste}
+\tab{5}\spadcommand{abs(-8)}
+\indentrel{3}\begin{verbatim}
+   (1)  8
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFunMacPageEmpty1}
+\begin{paste}{ugUserFunMacPageEmpty1}{ugUserFunMacPagePatch1}
+\pastebutton{ugUserFunMacPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{abs(-8)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFunMacPagePatch2}
+\begin{paste}{ugUserFunMacPageFull2}{ugUserFunMacPageEmpty2}
+\pastebutton{ugUserFunMacPageFull2}{\hidepaste}
+\tab{5}\spadcommand{(x +-> if x < 0 then -x else x)(-8)}
+\indentrel{3}\begin{verbatim}
+   (2)  8
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFunMacPageEmpty2}
+\begin{paste}{ugUserFunMacPageEmpty2}{ugUserFunMacPagePatch2}
+\pastebutton{ugUserFunMacPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{(x +-> if x < 0 then -x else x)(-8)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch1}
+\begin{paste}{ugUserBlocksPageFull1}{ugUserBlocksPageEmpty1}
+\pastebutton{ugUserBlocksPageFull1}{\hidepaste}
+\tab{5}\spadcommand{swap(m,i,j) ==
+  temp := m.i
+  m.i := m.j
+  m.j := temp
+\bound{swap }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty1}
+\begin{paste}{ugUserBlocksPageEmpty1}{ugUserBlocksPagePatch1}
+\pastebutton{ugUserBlocksPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{swap(m,i,j) ==
+  temp := m.i
+  m.i := m.j
+  m.j := temp
+\bound{swap }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch2}
+\begin{paste}{ugUserBlocksPageFull2}{ugUserBlocksPageEmpty2}
+\pastebutton{ugUserBlocksPageFull2}{\hidepaste}
+\tab{5}\spadcommand{k := [1,2,3,4,5]\bound{k }}
+\indentrel{3}\begin{verbatim}
+   (2)  [1,2,3,4,5]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty2}
+\begin{paste}{ugUserBlocksPageEmpty2}{ugUserBlocksPagePatch2}
+\pastebutton{ugUserBlocksPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{k := [1,2,3,4,5]\bound{k }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch3}
+\begin{paste}{ugUserBlocksPageFull3}{ugUserBlocksPageEmpty3}
+\pastebutton{ugUserBlocksPageFull3}{\hidepaste}
+\tab{5}\spadcommand{swap(k,2,4)\free{l swap }\bound{swapk }}
+\indentrel{3}\begin{verbatim}
+   (3)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty3}
+\begin{paste}{ugUserBlocksPageEmpty3}{ugUserBlocksPagePatch3}
+\pastebutton{ugUserBlocksPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{swap(k,2,4)\free{l swap }\bound{swapk }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch4}
+\begin{paste}{ugUserBlocksPageFull4}{ugUserBlocksPageEmpty4}
+\pastebutton{ugUserBlocksPageFull4}{\hidepaste}
+\tab{5}\spadcommand{k\free{swapk }}
+\indentrel{3}\begin{verbatim}
+   (4)  [1,4,3,2,5]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty4}
+\begin{paste}{ugUserBlocksPageEmpty4}{ugUserBlocksPagePatch4}
+\pastebutton{ugUserBlocksPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{k\free{swapk }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch5}
+\begin{paste}{ugUserBlocksPageFull5}{ugUserBlocksPageEmpty5}
+\pastebutton{ugUserBlocksPageFull5}{\hidepaste}
+\tab{5}\spadcommand{bubbleSort(m) ==
+  n := \#m
+  for i in 1..(n-1) repeat
+    for j in n..(i+1) by -1 repeat
+      if m.j < m.(j-1) then swap(m,j,j-1)
+  m
+\bound{bubbleSort }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty5}
+\begin{paste}{ugUserBlocksPageEmpty5}{ugUserBlocksPagePatch5}
+\pastebutton{ugUserBlocksPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{bubbleSort(m) ==
+  n := \#m
+  for i in 1..(n-1) repeat
+    for j in n..(i+1) by -1 repeat
+      if m.j < m.(j-1) then swap(m,j,j-1)
+  m
+\bound{bubbleSort }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch6}
+\begin{paste}{ugUserBlocksPageFull6}{ugUserBlocksPageEmpty6}
+\pastebutton{ugUserBlocksPageFull6}{\hidepaste}
+\tab{5}\spadcommand{m := [8,4,-3,9]\bound{m }}
+\indentrel{3}\begin{verbatim}
+   (6)  [8,4,- 3,9]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty6}
+\begin{paste}{ugUserBlocksPageEmpty6}{ugUserBlocksPagePatch6}
+\pastebutton{ugUserBlocksPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{m := [8,4,-3,9]\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch7}
+\begin{paste}{ugUserBlocksPageFull7}{ugUserBlocksPageEmpty7}
+\pastebutton{ugUserBlocksPageFull7}{\hidepaste}
+\tab{5}\spadcommand{bubbleSort(m)\free{m swap bubbleSort }\bound{sortm }}
+\indentrel{3}\begin{verbatim}
+   (7)  [- 3,4,8,9]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty7}
+\begin{paste}{ugUserBlocksPageEmpty7}{ugUserBlocksPagePatch7}
+\pastebutton{ugUserBlocksPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{bubbleSort(m)\free{m swap bubbleSort }\bound{sortm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch8}
+\begin{paste}{ugUserBlocksPageFull8}{ugUserBlocksPageEmpty8}
+\pastebutton{ugUserBlocksPageFull8}{\hidepaste}
+\tab{5}\spadcommand{m\free{sortm }}
+\indentrel{3}\begin{verbatim}
+   (8)  [- 3,4,8,9]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty8}
+\begin{paste}{ugUserBlocksPageEmpty8}{ugUserBlocksPagePatch8}
+\pastebutton{ugUserBlocksPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{m\free{sortm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch9}
+\begin{paste}{ugUserBlocksPageFull9}{ugUserBlocksPageEmpty9}
+\pastebutton{ugUserBlocksPageFull9}{\hidepaste}
+\tab{5}\spadcommand{insertionSort(m) ==
+  for i in 2..\#m repeat
+    j := i
+    while j > 1 and m.j < m.(j-1) repeat
+      swap(m,j,j-1)
+      j := j - 1
+  m
+\bound{insertionSort }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty9}
+\begin{paste}{ugUserBlocksPageEmpty9}{ugUserBlocksPagePatch9}
+\pastebutton{ugUserBlocksPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{insertionSort(m) ==
+  for i in 2..\#m repeat
+    j := i
+    while j > 1 and m.j < m.(j-1) repeat
+      swap(m,j,j-1)
+      j := j - 1
+  m
+\bound{insertionSort }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch10}
+\begin{paste}{ugUserBlocksPageFull10}{ugUserBlocksPageEmpty10}
+\pastebutton{ugUserBlocksPageFull10}{\hidepaste}
+\tab{5}\spadcommand{m := [8,4,-3,9]\bound{m1 }}
+\indentrel{3}\begin{verbatim}
+   (10)  [8,4,- 3,9]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty10}
+\begin{paste}{ugUserBlocksPageEmpty10}{ugUserBlocksPagePatch10}
+\pastebutton{ugUserBlocksPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{m := [8,4,-3,9]\bound{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch11}
+\begin{paste}{ugUserBlocksPageFull11}{ugUserBlocksPageEmpty11}
+\pastebutton{ugUserBlocksPageFull11}{\hidepaste}
+\tab{5}\spadcommand{insertionSort(m)\free{m1 swap insertionSort }\bound{sortm1 }}
+\indentrel{3}\begin{verbatim}
+   (11)  [- 3,4,8,9]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty11}
+\begin{paste}{ugUserBlocksPageEmpty11}{ugUserBlocksPagePatch11}
+\pastebutton{ugUserBlocksPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{insertionSort(m)\free{m1 swap insertionSort }\bound{sortm1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch12}
+\begin{paste}{ugUserBlocksPageFull12}{ugUserBlocksPageEmpty12}
+\pastebutton{ugUserBlocksPageFull12}{\hidepaste}
+\tab{5}\spadcommand{m\free{sortm1 }}
+\indentrel{3}\begin{verbatim}
+   (12)  [- 3,4,8,9]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty12}
+\begin{paste}{ugUserBlocksPageEmpty12}{ugUserBlocksPagePatch12}
+\pastebutton{ugUserBlocksPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{m\free{sortm1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch13}
+\begin{paste}{ugUserBlocksPageFull13}{ugUserBlocksPageEmpty13}
+\pastebutton{ugUserBlocksPageFull13}{\hidepaste}
+\tab{5}\spadcommand{bubbleSort2(m: List Integer): List Integer ==
+  null m => m
+  l := m
+  while not null (r := l.rest) repeat
+     r := bubbleSort2 r
+     x := l.first
+     if x < r.first then
+       l.first := r.first
+       r.first := x
+     l.rest := r
+     l := l.rest
+  m
+\bound{bubbleSort2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty13}
+\begin{paste}{ugUserBlocksPageEmpty13}{ugUserBlocksPagePatch13}
+\pastebutton{ugUserBlocksPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{bubbleSort2(m: List Integer): List Integer ==
+  null m => m
+  l := m
+  while not null (r := l.rest) repeat
+     r := bubbleSort2 r
+     x := l.first
+     if x < r.first then
+       l.first := r.first
+       r.first := x
+     l.rest := r
+     l := l.rest
+  m
+\bound{bubbleSort2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPagePatch14}
+\begin{paste}{ugUserBlocksPageFull14}{ugUserBlocksPageEmpty14}
+\pastebutton{ugUserBlocksPageFull14}{\hidepaste}
+\tab{5}\spadcommand{bubbleSort2 [3,7,2]\free{bubbleSort2 }}
+\indentrel{3}\begin{verbatim}
+   (14)  [7,3,2]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserBlocksPageEmpty14}
+\begin{paste}{ugUserBlocksPageEmpty14}{ugUserBlocksPagePatch14}
+\pastebutton{ugUserBlocksPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{bubbleSort2 [3,7,2]\free{bubbleSort2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch1}
+\begin{paste}{ugUserMacrosPageFull1}{ugUserMacrosPageEmpty1}
+\pastebutton{ugUserMacrosPageFull1}{\hidepaste}
+\tab{5}\spadcommand{macro df == D\bound{df }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty1}
+\begin{paste}{ugUserMacrosPageEmpty1}{ugUserMacrosPagePatch1}
+\pastebutton{ugUserMacrosPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{macro df == D\bound{df }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch2}
+\begin{paste}{ugUserMacrosPageFull2}{ugUserMacrosPageEmpty2}
+\pastebutton{ugUserMacrosPageFull2}{\hidepaste}
+\tab{5}\spadcommand{df(x**2 + x + 1,x)\free{df }}
+\indentrel{3}\begin{verbatim}
+   (2)  2x + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty2}
+\begin{paste}{ugUserMacrosPageEmpty2}{ugUserMacrosPagePatch2}
+\pastebutton{ugUserMacrosPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{df(x**2 + x + 1,x)\free{df }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch3}
+\begin{paste}{ugUserMacrosPageFull3}{ugUserMacrosPageEmpty3}
+\pastebutton{ugUserMacrosPageFull3}{\hidepaste}
+\tab{5}\spadcommand{macro ff(x) == x**2 + 1\bound{ff }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty3}
+\begin{paste}{ugUserMacrosPageEmpty3}{ugUserMacrosPagePatch3}
+\pastebutton{ugUserMacrosPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{macro ff(x) == x**2 + 1\bound{ff }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch4}
+\begin{paste}{ugUserMacrosPageFull4}{ugUserMacrosPageEmpty4}
+\pastebutton{ugUserMacrosPageFull4}{\hidepaste}
+\tab{5}\spadcommand{ff z\free{ff }}
+\indentrel{3}\begin{verbatim}
+         2
+   (4)  z  + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty4}
+\begin{paste}{ugUserMacrosPageEmpty4}{ugUserMacrosPagePatch4}
+\pastebutton{ugUserMacrosPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{ff z\free{ff }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch5}
+\begin{paste}{ugUserMacrosPageFull5}{ugUserMacrosPageEmpty5}
+\pastebutton{ugUserMacrosPageFull5}{\hidepaste}
+\tab{5}\spadcommand{macro gg(x) == ff(2*x - 2/3)\bound{gg }\free{ff }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty5}
+\begin{paste}{ugUserMacrosPageEmpty5}{ugUserMacrosPagePatch5}
+\pastebutton{ugUserMacrosPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{macro gg(x) == ff(2*x - 2/3)\bound{gg }\free{ff }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch6}
+\begin{paste}{ugUserMacrosPageFull6}{ugUserMacrosPageEmpty6}
+\pastebutton{ugUserMacrosPageFull6}{\hidepaste}
+\tab{5}\spadcommand{gg(1/w)\free{gg }}
+\indentrel{3}\begin{verbatim}
+           2
+        13w  - 24w + 36
+   (6)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                2
+              9w
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty6}
+\begin{paste}{ugUserMacrosPageEmpty6}{ugUserMacrosPagePatch6}
+\pastebutton{ugUserMacrosPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{gg(1/w)\free{gg }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch7}
+\begin{paste}{ugUserMacrosPageFull7}{ugUserMacrosPageEmpty7}
+\pastebutton{ugUserMacrosPageFull7}{\hidepaste}
+\tab{5}\spadcommand{macro ff(x) == gg(-x)\free{gg }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty7}
+\begin{paste}{ugUserMacrosPageEmpty7}{ugUserMacrosPagePatch7}
+\pastebutton{ugUserMacrosPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{macro ff(x) == gg(-x)\free{gg }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch8}
+\begin{paste}{ugUserMacrosPageFull8}{ugUserMacrosPageEmpty8}
+\pastebutton{ugUserMacrosPageFull8}{\hidepaste}
+\tab{5}\spadcommand{macro next == (past := present; present := future; future := past + present)\bound{next }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty8}
+\begin{paste}{ugUserMacrosPageEmpty8}{ugUserMacrosPagePatch8}
+\pastebutton{ugUserMacrosPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{macro next == (past := present; present := future; future := past + present)\bound{next }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch9}
+\begin{paste}{ugUserMacrosPageFull9}{ugUserMacrosPageEmpty9}
+\pastebutton{ugUserMacrosPageFull9}{\hidepaste}
+\tab{5}\spadcommand{present : Integer := 0\bound{present }}
+\indentrel{3}\begin{verbatim}
+   (9)  0
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty9}
+\begin{paste}{ugUserMacrosPageEmpty9}{ugUserMacrosPagePatch9}
+\pastebutton{ugUserMacrosPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{present : Integer := 0\bound{present }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch10}
+\begin{paste}{ugUserMacrosPageFull10}{ugUserMacrosPageEmpty10}
+\pastebutton{ugUserMacrosPageFull10}{\hidepaste}
+\tab{5}\spadcommand{future : Integer := 1\bound{future }}
+\indentrel{3}\begin{verbatim}
+   (10)  1
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty10}
+\begin{paste}{ugUserMacrosPageEmpty10}{ugUserMacrosPagePatch10}
+\pastebutton{ugUserMacrosPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{future : Integer := 1\bound{future }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch11}
+\begin{paste}{ugUserMacrosPageFull11}{ugUserMacrosPageEmpty11}
+\pastebutton{ugUserMacrosPageFull11}{\hidepaste}
+\tab{5}\spadcommand{next\free{future }\free{present }}
+\indentrel{3}\begin{verbatim}
+   (11)  1
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty11}
+\begin{paste}{ugUserMacrosPageEmpty11}{ugUserMacrosPagePatch11}
+\pastebutton{ugUserMacrosPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{next\free{future }\free{present }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch12}
+\begin{paste}{ugUserMacrosPageFull12}{ugUserMacrosPageEmpty12}
+\pastebutton{ugUserMacrosPageFull12}{\hidepaste}
+\tab{5}\spadcommand{next\free{future }\free{present }}
+\indentrel{3}\begin{verbatim}
+   (12)  2
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty12}
+\begin{paste}{ugUserMacrosPageEmpty12}{ugUserMacrosPagePatch12}
+\pastebutton{ugUserMacrosPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{next\free{future }\free{present }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch13}
+\begin{paste}{ugUserMacrosPageFull13}{ugUserMacrosPageEmpty13}
+\pastebutton{ugUserMacrosPageFull13}{\hidepaste}
+\tab{5}\spadcommand{[next for i in 1..]\free{future }\free{present }}
+\indentrel{3}\begin{verbatim}
+   (13)  [3,5,8,13,21,34,55,89,144,233,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty13}
+\begin{paste}{ugUserMacrosPageEmpty13}{ugUserMacrosPagePatch13}
+\pastebutton{ugUserMacrosPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{[next for i in 1..]\free{future }\free{present }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch14}
+\begin{paste}{ugUserMacrosPageFull14}{ugUserMacrosPageEmpty14}
+\pastebutton{ugUserMacrosPageFull14}{\hidepaste}
+\tab{5}\spadcommand{macro fibStream ==
+  present : Integer := 1
+  future : Integer := 1
+  [next for i in 1..] where
+    macro next ==
+      past := present
+      present := future
+      future := past + present
+\bound{fibstr }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty14}
+\begin{paste}{ugUserMacrosPageEmpty14}{ugUserMacrosPagePatch14}
+\pastebutton{ugUserMacrosPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{macro fibStream ==
+  present : Integer := 1
+  future : Integer := 1
+  [next for i in 1..] where
+    macro next ==
+      past := present
+      present := future
+      future := past + present
+\bound{fibstr }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch15}
+\begin{paste}{ugUserMacrosPageFull15}{ugUserMacrosPageEmpty15}
+\pastebutton{ugUserMacrosPageFull15}{\hidepaste}
+\tab{5}\spadcommand{concat([0,1],fibStream)\free{fibstr }}
+\indentrel{3}\begin{verbatim}
+   (15)  [0,1,2,3,5,8,13,21,34,55,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty15}
+\begin{paste}{ugUserMacrosPageEmpty15}{ugUserMacrosPagePatch15}
+\pastebutton{ugUserMacrosPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{concat([0,1],fibStream)\free{fibstr }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPagePatch16}
+\begin{paste}{ugUserMacrosPageFull16}{ugUserMacrosPageEmpty16}
+\pastebutton{ugUserMacrosPageFull16}{\hidepaste}
+\tab{5}\spadcommand{[fibonacci i for i in 1..]}
+\indentrel{3}\begin{verbatim}
+   (16)  [1,1,2,3,5,8,13,21,34,55,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserMacrosPageEmpty16}
+\begin{paste}{ugUserMacrosPageEmpty16}{ugUserMacrosPagePatch16}
+\pastebutton{ugUserMacrosPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{[fibonacci i for i in 1..]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDeclarePagePatch1}
+\begin{paste}{ugUserDeclarePageFull1}{ugUserDeclarePageEmpty1}
+\pastebutton{ugUserDeclarePageFull1}{\hidepaste}
+\tab{5}\spadcommand{g: (Integer,Float,Integer) -> String}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDeclarePageEmpty1}
+\begin{paste}{ugUserDeclarePageEmpty1}{ugUserDeclarePagePatch1}
+\pastebutton{ugUserDeclarePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{g: (Integer,Float,Integer) -> String}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDeclarePagePatch2}
+\begin{paste}{ugUserDeclarePageFull2}{ugUserDeclarePageEmpty2}
+\pastebutton{ugUserDeclarePageFull2}{\hidepaste}
+\tab{5}\spadcommand{g: (INT,FLOAT,INT) -> STRING}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDeclarePageEmpty2}
+\begin{paste}{ugUserDeclarePageEmpty2}{ugUserDeclarePagePatch2}
+\pastebutton{ugUserDeclarePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{g: (INT,FLOAT,INT) -> STRING}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDeclarePagePatch3}
+\begin{paste}{ugUserDeclarePageFull3}{ugUserDeclarePageEmpty3}
+\pastebutton{ugUserDeclarePageFull3}{\hidepaste}
+\tab{5}\spadcommand{h: () -> POLY INT}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDeclarePageEmpty3}
+\begin{paste}{ugUserDeclarePageEmpty3}{ugUserDeclarePagePatch3}
+\pastebutton{ugUserDeclarePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{h: () -> POLY INT}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDeclarePagePatch4}
+\begin{paste}{ugUserDeclarePageFull4}{ugUserDeclarePageEmpty4}
+\pastebutton{ugUserDeclarePageFull4}{\hidepaste}
+\tab{5}\spadcommand{h: () -> Polynomial INT}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDeclarePageEmpty4}
+\begin{paste}{ugUserDeclarePageEmpty4}{ugUserDeclarePagePatch4}
+\pastebutton{ugUserDeclarePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{h: () -> Polynomial INT}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDeclarePagePatch5}
+\begin{paste}{ugUserDeclarePageFull5}{ugUserDeclarePageEmpty5}
+\pastebutton{ugUserDeclarePageFull5}{\hidepaste}
+\tab{5}\spadcommand{h: () -> POLY Integer}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDeclarePageEmpty5}
+\begin{paste}{ugUserDeclarePageEmpty5}{ugUserDeclarePagePatch5}
+\pastebutton{ugUserDeclarePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{h: () -> POLY Integer}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserCompIntPagePatch1}
+\begin{paste}{ugUserCompIntPageFull1}{ugUserCompIntPageEmpty1}
+\pastebutton{ugUserCompIntPageFull1}{\hidepaste}
+\tab{5}\spadcommand{varPolys(vars) ==
+  for var in vars repeat
+    output(1 :: UnivariatePolynomial(var,Integer))
+\bound{varPolys }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserCompIntPageEmpty1}
+\begin{paste}{ugUserCompIntPageEmpty1}{ugUserCompIntPagePatch1}
+\pastebutton{ugUserCompIntPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{varPolys(vars) ==
+  for var in vars repeat
+    output(1 :: UnivariatePolynomial(var,Integer))
+\bound{varPolys }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserCompIntPagePatch2}
+\begin{paste}{ugUserCompIntPageFull2}{ugUserCompIntPageEmpty2}
+\pastebutton{ugUserCompIntPageFull2}{\hidepaste}
+\tab{5}\spadcommand{varPolys ['x,'y,'z]\free{varPolys }}
+\indentrel{3}\begin{verbatim}
+   1
+   1
+   1
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserCompIntPageEmpty2}
+\begin{paste}{ugUserCompIntPageEmpty2}{ugUserCompIntPagePatch2}
+\pastebutton{ugUserCompIntPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{varPolys ['x,'y,'z]\free{varPolys }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserCompIntPagePatch3}
+\begin{paste}{ugUserCompIntPageFull3}{ugUserCompIntPageEmpty3}
+\pastebutton{ugUserCompIntPageFull3}{\hidepaste}
+\tab{5}\spadcommand{for var in ['x,'y,'z] repeat
+  output(1 :: UnivariatePolynomial(var,Integer))
+}
+\indentrel{3}\begin{verbatim}
+   1
+   1
+   1
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserCompIntPageEmpty3}
+\begin{paste}{ugUserCompIntPageEmpty3}{ugUserCompIntPagePatch3}
+\pastebutton{ugUserCompIntPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{for var in ['x,'y,'z] repeat
+  output(1 :: UnivariatePolynomial(var,Integer))
+}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPagePatch1}
+\begin{paste}{ugUserDecUndecPageFull1}{ugUserDecUndecPageEmpty1}
+\pastebutton{ugUserDecUndecPageFull1}{\hidepaste}
+\tab{5}\spadcommand{f(x: Integer): Integer == x + 1\bound{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPageEmpty1}
+\begin{paste}{ugUserDecUndecPageEmpty1}{ugUserDecUndecPagePatch1}
+\pastebutton{ugUserDecUndecPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{f(x: Integer): Integer == x + 1\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPagePatch2}
+\begin{paste}{ugUserDecUndecPageFull2}{ugUserDecUndecPageEmpty2}
+\pastebutton{ugUserDecUndecPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f 9\free{f }}
+\indentrel{3}\begin{verbatim}
+   (2)  10
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPageEmpty2}
+\begin{paste}{ugUserDecUndecPageEmpty2}{ugUserDecUndecPagePatch2}
+\pastebutton{ugUserDecUndecPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f 9\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPagePatch3}
+\begin{paste}{ugUserDecUndecPageFull3}{ugUserDecUndecPageEmpty3}
+\pastebutton{ugUserDecUndecPageFull3}{\hidepaste}
+\tab{5}\spadcommand{f(-2.0)\free{f }}
+\indentrel{3}\begin{verbatim}
+   (3)  - 1
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPageEmpty3}
+\begin{paste}{ugUserDecUndecPageEmpty3}{ugUserDecUndecPagePatch3}
+\pastebutton{ugUserDecUndecPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f(-2.0)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPagePatch4}
+\begin{paste}{ugUserDecUndecPageFull4}{ugUserDecUndecPageEmpty4}
+\pastebutton{ugUserDecUndecPageFull4}{\hidepaste}
+\tab{5}\spadcommand{f(2/3)\free{f }}
+\indentrel{3}\begin{verbatim}
+   2
+   Ä
+   3
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPageEmpty4}
+\begin{paste}{ugUserDecUndecPageEmpty4}{ugUserDecUndecPagePatch4}
+\pastebutton{ugUserDecUndecPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{f(2/3)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPagePatch5}
+\begin{paste}{ugUserDecUndecPageFull5}{ugUserDecUndecPageEmpty5}
+\pastebutton{ugUserDecUndecPageFull5}{\hidepaste}
+\tab{5}\spadcommand{g x == x + 1\bound{g }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPageEmpty5}
+\begin{paste}{ugUserDecUndecPageEmpty5}{ugUserDecUndecPagePatch5}
+\pastebutton{ugUserDecUndecPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{g x == x + 1\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPagePatch6}
+\begin{paste}{ugUserDecUndecPageFull6}{ugUserDecUndecPageEmpty6}
+\pastebutton{ugUserDecUndecPageFull6}{\hidepaste}
+\tab{5}\spadcommand{g 9\free{g }}
+\indentrel{3}\begin{verbatim}
+   (5)  10
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPageEmpty6}
+\begin{paste}{ugUserDecUndecPageEmpty6}{ugUserDecUndecPagePatch6}
+\pastebutton{ugUserDecUndecPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{g 9\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPagePatch7}
+\begin{paste}{ugUserDecUndecPageFull7}{ugUserDecUndecPageEmpty7}
+\pastebutton{ugUserDecUndecPageFull7}{\hidepaste}
+\tab{5}\spadcommand{g(2/3)\free{g }}
+\indentrel{3}\begin{verbatim}
+        5
+   (6)  Ä
+        3
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPageEmpty7}
+\begin{paste}{ugUserDecUndecPageEmpty7}{ugUserDecUndecPagePatch7}
+\pastebutton{ugUserDecUndecPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{g(2/3)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPagePatch8}
+\begin{paste}{ugUserDecUndecPageFull8}{ugUserDecUndecPageEmpty8}
+\pastebutton{ugUserDecUndecPageFull8}{\hidepaste}
+\tab{5}\spadcommand{g("axiom")\free{g }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDecUndecPageEmpty8}
+\begin{paste}{ugUserDecUndecPageEmpty8}{ugUserDecUndecPagePatch8}
+\pastebutton{ugUserDecUndecPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{g("axiom")\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePagePatch1}
+\begin{paste}{ugUserDatabasePageFull1}{ugUserDatabasePageEmpty1}
+\pastebutton{ugUserDatabasePageFull1}{\hidepaste}
+\tab{5}\spadcommand{children("albert") == ["albertJr","richard","diane"]\bound{d1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePageEmpty1}
+\begin{paste}{ugUserDatabasePageEmpty1}{ugUserDatabasePagePatch1}
+\pastebutton{ugUserDatabasePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{children("albert") == ["albertJr","richard","diane"]\bound{d1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePagePatch2}
+\begin{paste}{ugUserDatabasePageFull2}{ugUserDatabasePageEmpty2}
+\pastebutton{ugUserDatabasePageFull2}{\hidepaste}
+\tab{5}\spadcommand{children("richard") == ["douglas","daniel","susan"]\free{d1 }\bound{d2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePageEmpty2}
+\begin{paste}{ugUserDatabasePageEmpty2}{ugUserDatabasePagePatch2}
+\pastebutton{ugUserDatabasePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{children("richard") == ["douglas","daniel","susan"]\free{d1 }\bound{d2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePagePatch3}
+\begin{paste}{ugUserDatabasePageFull3}{ugUserDatabasePageEmpty3}
+\pastebutton{ugUserDatabasePageFull3}{\hidepaste}
+\tab{5}\spadcommand{children("douglas") == ["dougie","valerie"]\free{d2 }\bound{d3 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePageEmpty3}
+\begin{paste}{ugUserDatabasePageEmpty3}{ugUserDatabasePagePatch3}
+\pastebutton{ugUserDatabasePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{children("douglas") == ["dougie","valerie"]\free{d2 }\bound{d3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePagePatch4}
+\begin{paste}{ugUserDatabasePageFull4}{ugUserDatabasePageEmpty4}
+\pastebutton{ugUserDatabasePageFull4}{\hidepaste}
+\tab{5}\spadcommand{children(x) == []\free{d3 }\bound{d4 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePageEmpty4}
+\begin{paste}{ugUserDatabasePageEmpty4}{ugUserDatabasePagePatch4}
+\pastebutton{ugUserDatabasePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{children(x) == []\free{d3 }\bound{d4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePagePatch5}
+\begin{paste}{ugUserDatabasePageFull5}{ugUserDatabasePageEmpty5}
+\pastebutton{ugUserDatabasePageFull5}{\hidepaste}
+\tab{5}\spadcommand{childOf(x,y) == member?(x,children(y))\bound{d9 }\free{d10 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePageEmpty5}
+\begin{paste}{ugUserDatabasePageEmpty5}{ugUserDatabasePagePatch5}
+\pastebutton{ugUserDatabasePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{childOf(x,y) == member?(x,children(y))\bound{d9 }\free{d10 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePagePatch6}
+\begin{paste}{ugUserDatabasePageFull6}{ugUserDatabasePageEmpty6}
+\pastebutton{ugUserDatabasePageFull6}{\hidepaste}
+\tab{5}\spadcommand{parentOf(x) ==
+  for y in people repeat
+    (if childOf(x,y) then return y)
+  "unknown"
+\bound{d8a }\free{d9 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePageEmpty6}
+\begin{paste}{ugUserDatabasePageEmpty6}{ugUserDatabasePagePatch6}
+\pastebutton{ugUserDatabasePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{parentOf(x) ==
+  for y in people repeat
+    (if childOf(x,y) then return y)
+  "unknown"
+\bound{d8a }\free{d9 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePagePatch7}
+\begin{paste}{ugUserDatabasePageFull7}{ugUserDatabasePageEmpty7}
+\pastebutton{ugUserDatabasePageFull7}{\hidepaste}
+\tab{5}\spadcommand{grandParentOf(x) == parentOf parentOf x\bound{d8 }\free{d8a }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePageEmpty7}
+\begin{paste}{ugUserDatabasePageEmpty7}{ugUserDatabasePagePatch7}
+\pastebutton{ugUserDatabasePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{grandParentOf(x) == parentOf parentOf x\bound{d8 }\free{d8a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePagePatch8}
+\begin{paste}{ugUserDatabasePageFull8}{ugUserDatabasePageEmpty8}
+\pastebutton{ugUserDatabasePageFull8}{\hidepaste}
+\tab{5}\spadcommand{grandchildren(x) == [y for y in people | grandParentOf(y) = x]\free{d7 }\bound{d8 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePageEmpty8}
+\begin{paste}{ugUserDatabasePageEmpty8}{ugUserDatabasePagePatch8}
+\pastebutton{ugUserDatabasePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{grandchildren(x) == [y for y in people | grandParentOf(y) = x]\free{d7 }\bound{d8 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePagePatch9}
+\begin{paste}{ugUserDatabasePageFull9}{ugUserDatabasePageEmpty9}
+\pastebutton{ugUserDatabasePageFull9}{\hidepaste}
+\tab{5}\spadcommand{greatGrandParents == [x for x in people |
+  reduce(_or,[not empty? children(y) for y in grandchildren(x)],false)]
+\free{d6 }\bound{d7 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePageEmpty9}
+\begin{paste}{ugUserDatabasePageEmpty9}{ugUserDatabasePagePatch9}
+\pastebutton{ugUserDatabasePageEmpty9}{\showpaste}
+\tab{5}\spadcommand{greatGrandParents == [x for x in people |
+  reduce(_or,[not empty? children(y) for y in grandchildren(x)],false)]
+\free{d6 }\bound{d7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePagePatch10}
+\begin{paste}{ugUserDatabasePageFull10}{ugUserDatabasePageEmpty10}
+\pastebutton{ugUserDatabasePageFull10}{\hidepaste}
+\tab{5}\spadcommand{descendants(x) ==
+  kids := children(x)
+  null kids => [x]
+  concat(x,reduce(concat,[descendants(y)
+    for y in kids],[]))
+\free{d5 }\bound{d6 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePageEmpty10}
+\begin{paste}{ugUserDatabasePageEmpty10}{ugUserDatabasePagePatch10}
+\pastebutton{ugUserDatabasePageEmpty10}{\showpaste}
+\tab{5}\spadcommand{descendants(x) ==
+  kids := children(x)
+  null kids => [x]
+  concat(x,reduce(concat,[descendants(y)
+    for y in kids],[]))
+\free{d5 }\bound{d6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePagePatch11}
+\begin{paste}{ugUserDatabasePageFull11}{ugUserDatabasePageEmpty11}
+\pastebutton{ugUserDatabasePageFull11}{\hidepaste}
+\tab{5}\spadcommand{people == descendants "albert"\free{d4 }\bound{d5 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePageEmpty11}
+\begin{paste}{ugUserDatabasePageEmpty11}{ugUserDatabasePagePatch11}
+\pastebutton{ugUserDatabasePageEmpty11}{\showpaste}
+\tab{5}\spadcommand{people == descendants "albert"\free{d4 }\bound{d5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePagePatch12}
+\begin{paste}{ugUserDatabasePageFull12}{ugUserDatabasePageEmpty12}
+\pastebutton{ugUserDatabasePageFull12}{\hidepaste}
+\tab{5}\spadcommand{grandchildren "richard"\bound{d10 }\free{d11 }}
+\indentrel{3}\begin{verbatim}
+   (12)  ["dougie","valerie"]
+                                      Type: List String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePageEmpty12}
+\begin{paste}{ugUserDatabasePageEmpty12}{ugUserDatabasePagePatch12}
+\pastebutton{ugUserDatabasePageEmpty12}{\showpaste}
+\tab{5}\spadcommand{grandchildren "richard"\bound{d10 }\free{d11 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePagePatch13}
+\begin{paste}{ugUserDatabasePageFull13}{ugUserDatabasePageEmpty13}
+\pastebutton{ugUserDatabasePageFull13}{\hidepaste}
+\tab{5}\spadcommand{greatGrandParents\bound{d11 }\free{d12 }}
+\indentrel{3}\begin{verbatim}
+   (13)  ["albert"]
+                                      Type: List String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserDatabasePageEmpty13}
+\begin{paste}{ugUserDatabasePageEmpty13}{ugUserDatabasePagePatch13}
+\pastebutton{ugUserDatabasePageEmpty13}{\showpaste}
+\tab{5}\spadcommand{greatGrandParents\bound{d11 }\free{d12 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePagePatch1}
+\begin{paste}{ugUserTrianglePageFull1}{ugUserTrianglePageEmpty1}
+\pastebutton{ugUserTrianglePageFull1}{\hidepaste}
+\tab{5}\spadcommand{)set expose add constructor OutputForm\bound{expose }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePageEmpty1}
+\begin{paste}{ugUserTrianglePageEmpty1}{ugUserTrianglePagePatch1}
+\pastebutton{ugUserTrianglePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)set expose add constructor OutputForm\bound{expose }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePagePatch2}
+\begin{paste}{ugUserTrianglePageFull2}{ugUserTrianglePageEmpty2}
+\pastebutton{ugUserTrianglePageFull2}{\hidepaste}
+\tab{5}\spadcommand{pascal(1,i) == 1\bound{pas1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePageEmpty2}
+\begin{paste}{ugUserTrianglePageEmpty2}{ugUserTrianglePagePatch2}
+\pastebutton{ugUserTrianglePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{pascal(1,i) == 1\bound{pas1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePagePatch3}
+\begin{paste}{ugUserTrianglePageFull3}{ugUserTrianglePageEmpty3}
+\pastebutton{ugUserTrianglePageFull3}{\hidepaste}
+\tab{5}\spadcommand{pascal(n,n) == 1\bound{pas2 }\free{pas1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePageEmpty3}
+\begin{paste}{ugUserTrianglePageEmpty3}{ugUserTrianglePagePatch3}
+\pastebutton{ugUserTrianglePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{pascal(n,n) == 1\bound{pas2 }\free{pas1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePagePatch4}
+\begin{paste}{ugUserTrianglePageFull4}{ugUserTrianglePageEmpty4}
+\pastebutton{ugUserTrianglePageFull4}{\hidepaste}
+\tab{5}\spadcommand{pascal(i,j | 1 < i and i < j) ==
+   pascal(i-1,j-1)+pascal(i,j-1)
+\bound{pas3 }\free{pas1 pas2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePageEmpty4}
+\begin{paste}{ugUserTrianglePageEmpty4}{ugUserTrianglePagePatch4}
+\pastebutton{ugUserTrianglePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{pascal(i,j | 1 < i and i < j) ==
+   pascal(i-1,j-1)+pascal(i,j-1)
+\bound{pas3 }\free{pas1 pas2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePagePatch5}
+\begin{paste}{ugUserTrianglePageFull5}{ugUserTrianglePageEmpty5}
+\pastebutton{ugUserTrianglePageFull5}{\hidepaste}
+\tab{5}\spadcommand{pascalRow(n) == [pascal(i,n) for i in 1..n]\bound{pascalRow }\free{pas3 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePageEmpty5}
+\begin{paste}{ugUserTrianglePageEmpty5}{ugUserTrianglePagePatch5}
+\pastebutton{ugUserTrianglePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{pascalRow(n) == [pascal(i,n) for i in 1..n]\bound{pascalRow }\free{pas3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePagePatch6}
+\begin{paste}{ugUserTrianglePageFull6}{ugUserTrianglePageEmpty6}
+\pastebutton{ugUserTrianglePageFull6}{\hidepaste}
+\tab{5}\spadcommand{displayRow(n) == output center blankSeparate pascalRow(n)\free{pascalRow }\bound{displayRow }\free{expose }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePageEmpty6}
+\begin{paste}{ugUserTrianglePageEmpty6}{ugUserTrianglePagePatch6}
+\pastebutton{ugUserTrianglePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{displayRow(n) == output center blankSeparate pascalRow(n)\free{pascalRow }\bound{displayRow }\free{expose }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePagePatch7}
+\begin{paste}{ugUserTrianglePageFull7}{ugUserTrianglePageEmpty7}
+\pastebutton{ugUserTrianglePageFull7}{\hidepaste}
+\tab{5}\spadcommand{for i in 1..7 repeat displayRow i\free{displayRow }}
+\indentrel{3}\begin{verbatim}
+                                   1
+                                  1 1
+                                 1 2 1
+                                1 3 3 1
+                               1 4 6 4 1
+                             1 5 10 10 5 1
+                            1 6 15 20 15 6 1
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePageEmpty7}
+\begin{paste}{ugUserTrianglePageEmpty7}{ugUserTrianglePagePatch7}
+\pastebutton{ugUserTrianglePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{for i in 1..7 repeat displayRow i\free{displayRow }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePagePatch8}
+\begin{paste}{ugUserTrianglePageFull8}{ugUserTrianglePageEmpty8}
+\pastebutton{ugUserTrianglePageFull8}{\hidepaste}
+\tab{5}\spadcommand{pascalRow(n) == [right(pascal(i,n),4) for i in 1..n]\bound{pascalRow2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePageEmpty8}
+\begin{paste}{ugUserTrianglePageEmpty8}{ugUserTrianglePagePatch8}
+\pastebutton{ugUserTrianglePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{pascalRow(n) == [right(pascal(i,n),4) for i in 1..n]\bound{pascalRow2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePagePatch9}
+\begin{paste}{ugUserTrianglePageFull9}{ugUserTrianglePageEmpty9}
+\pastebutton{ugUserTrianglePageFull9}{\hidepaste}
+\tab{5}\spadcommand{for i in 1..7 repeat displayRow i\free{pascalRow2 }\free{displayRow }}
+\indentrel{3}\begin{verbatim}
+                                     1
+                                  1    1
+                                1    2    1
+                             1    3    3    1
+                           1    4    6    4    1
+                        1    5   10   10    5    1
+                      1    6   15   20   15    6    1
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePageEmpty9}
+\begin{paste}{ugUserTrianglePageEmpty9}{ugUserTrianglePagePatch9}
+\pastebutton{ugUserTrianglePageEmpty9}{\showpaste}
+\tab{5}\spadcommand{for i in 1..7 repeat displayRow i\free{pascalRow2 }\free{displayRow }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePagePatch10}
+\begin{paste}{ugUserTrianglePageFull10}{ugUserTrianglePageEmpty10}
+\pastebutton{ugUserTrianglePageFull10}{\hidepaste}
+\tab{5}\spadcommand{)set expose drop constructor OutputForm}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserTrianglePageEmpty10}
+\begin{paste}{ugUserTrianglePageEmpty10}{ugUserTrianglePagePatch10}
+\pastebutton{ugUserTrianglePageEmpty10}{\showpaste}
+\tab{5}\spadcommand{)set expose drop constructor OutputForm}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch1}
+\begin{paste}{ugUserFreeLocalPageFull1}{ugUserFreeLocalPageEmpty1}
+\pastebutton{ugUserFreeLocalPageFull1}{\hidepaste}
+\tab{5}\spadcommand{counter := 0\bound{counter }}
+\indentrel{3}\begin{verbatim}
+   (1)  0
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty1}
+\begin{paste}{ugUserFreeLocalPageEmpty1}{ugUserFreeLocalPagePatch1}
+\pastebutton{ugUserFreeLocalPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{counter := 0\bound{counter }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch2}
+\begin{paste}{ugUserFreeLocalPageFull2}{ugUserFreeLocalPageEmpty2}
+\pastebutton{ugUserFreeLocalPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f() ==
+  free counter
+  counter := counter + 1
+\free{counter }\bound{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty2}
+\begin{paste}{ugUserFreeLocalPageEmpty2}{ugUserFreeLocalPagePatch2}
+\pastebutton{ugUserFreeLocalPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f() ==
+  free counter
+  counter := counter + 1
+\free{counter }\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch3}
+\begin{paste}{ugUserFreeLocalPageFull3}{ugUserFreeLocalPageEmpty3}
+\pastebutton{ugUserFreeLocalPageFull3}{\hidepaste}
+\tab{5}\spadcommand{f()\free{f }\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty3}
+\begin{paste}{ugUserFreeLocalPageEmpty3}{ugUserFreeLocalPagePatch3}
+\pastebutton{ugUserFreeLocalPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f()\free{f }\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch4}
+\begin{paste}{ugUserFreeLocalPageFull4}{ugUserFreeLocalPageEmpty4}
+\pastebutton{ugUserFreeLocalPageFull4}{\hidepaste}
+\tab{5}\spadcommand{counter\free{f1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  1
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty4}
+\begin{paste}{ugUserFreeLocalPageEmpty4}{ugUserFreeLocalPagePatch4}
+\pastebutton{ugUserFreeLocalPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{counter\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch5}
+\begin{paste}{ugUserFreeLocalPageFull5}{ugUserFreeLocalPageEmpty5}
+\pastebutton{ugUserFreeLocalPageFull5}{\hidepaste}
+\tab{5}\spadcommand{g() ==
+  local counter
+  counter := 7
+\bound{g }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty5}
+\begin{paste}{ugUserFreeLocalPageEmpty5}{ugUserFreeLocalPagePatch5}
+\pastebutton{ugUserFreeLocalPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{g() ==
+  local counter
+  counter := 7
+\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch6}
+\begin{paste}{ugUserFreeLocalPageFull6}{ugUserFreeLocalPageEmpty6}
+\pastebutton{ugUserFreeLocalPageFull6}{\hidepaste}
+\tab{5}\spadcommand{g()\free{g }}
+\indentrel{3}\begin{verbatim}
+   (6)  7
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty6}
+\begin{paste}{ugUserFreeLocalPageEmpty6}{ugUserFreeLocalPagePatch6}
+\pastebutton{ugUserFreeLocalPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{g()\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch7}
+\begin{paste}{ugUserFreeLocalPageFull7}{ugUserFreeLocalPageEmpty7}
+\pastebutton{ugUserFreeLocalPageFull7}{\hidepaste}
+\tab{5}\spadcommand{counter\free{g f1 }}
+\indentrel{3}\begin{verbatim}
+   (7)  1
+                               Type: NonNegativeInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty7}
+\begin{paste}{ugUserFreeLocalPageEmpty7}{ugUserFreeLocalPagePatch7}
+\pastebutton{ugUserFreeLocalPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{counter\free{g f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch8}
+\begin{paste}{ugUserFreeLocalPageFull8}{ugUserFreeLocalPageEmpty8}
+\pastebutton{ugUserFreeLocalPageFull8}{\hidepaste}
+\tab{5}\spadcommand{a := b := 1\bound{ab1 }}
+\indentrel{3}\begin{verbatim}
+   (8)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty8}
+\begin{paste}{ugUserFreeLocalPageEmpty8}{ugUserFreeLocalPagePatch8}
+\pastebutton{ugUserFreeLocalPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{a := b := 1\bound{ab1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch9}
+\begin{paste}{ugUserFreeLocalPageFull9}{ugUserFreeLocalPageEmpty9}
+\pastebutton{ugUserFreeLocalPageFull9}{\hidepaste}
+\tab{5}\spadcommand{h() ==
+  b := a + 1
+  a := b + a
+\bound{hh }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty9}
+\begin{paste}{ugUserFreeLocalPageEmpty9}{ugUserFreeLocalPagePatch9}
+\pastebutton{ugUserFreeLocalPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{h() ==
+  b := a + 1
+  a := b + a
+\bound{hh }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch10}
+\begin{paste}{ugUserFreeLocalPageFull10}{ugUserFreeLocalPageEmpty10}
+\pastebutton{ugUserFreeLocalPageFull10}{\hidepaste}
+\tab{5}\spadcommand{h()\free{ab1 hh }\bound{hhh }}
+\indentrel{3}\begin{verbatim}
+   (10)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty10}
+\begin{paste}{ugUserFreeLocalPageEmpty10}{ugUserFreeLocalPagePatch10}
+\pastebutton{ugUserFreeLocalPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{h()\free{ab1 hh }\bound{hhh }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch11}
+\begin{paste}{ugUserFreeLocalPageFull11}{ugUserFreeLocalPageEmpty11}
+\pastebutton{ugUserFreeLocalPageFull11}{\hidepaste}
+\tab{5}\spadcommand{[a, b]\free{hhh }}
+\indentrel{3}\begin{verbatim}
+   (11)  [3,1]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty11}
+\begin{paste}{ugUserFreeLocalPageEmpty11}{ugUserFreeLocalPagePatch11}
+\pastebutton{ugUserFreeLocalPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{[a, b]\free{hhh }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch12}
+\begin{paste}{ugUserFreeLocalPageFull12}{ugUserFreeLocalPageEmpty12}
+\pastebutton{ugUserFreeLocalPageFull12}{\hidepaste}
+\tab{5}\spadcommand{)set fun cache all p\bound{pcache }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty12}
+\begin{paste}{ugUserFreeLocalPageEmpty12}{ugUserFreeLocalPagePatch12}
+\pastebutton{ugUserFreeLocalPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{)set fun cache all p\bound{pcache }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch13}
+\begin{paste}{ugUserFreeLocalPageFull13}{ugUserFreeLocalPageEmpty13}
+\pastebutton{ugUserFreeLocalPageFull13}{\hidepaste}
+\tab{5}\spadcommand{p(i,x) == ( free N; reduce( + , [ (x-i)**n for n in 1..N ] ) )\free{pcache }\bound{pdef }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty13}
+\begin{paste}{ugUserFreeLocalPageEmpty13}{ugUserFreeLocalPagePatch13}
+\pastebutton{ugUserFreeLocalPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{p(i,x) == ( free N; reduce( + , [ (x-i)**n for n in 1..N ] ) )\free{pcache }\bound{pdef }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch14}
+\begin{paste}{ugUserFreeLocalPageFull14}{ugUserFreeLocalPageEmpty14}
+\pastebutton{ugUserFreeLocalPageFull14}{\hidepaste}
+\tab{5}\spadcommand{N := 1\bound{Nass }}
+\indentrel{3}\begin{verbatim}
+   (13)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty14}
+\begin{paste}{ugUserFreeLocalPageEmpty14}{ugUserFreeLocalPagePatch14}
+\pastebutton{ugUserFreeLocalPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{N := 1\bound{Nass }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch15}
+\begin{paste}{ugUserFreeLocalPageFull15}{ugUserFreeLocalPageEmpty15}
+\pastebutton{ugUserFreeLocalPageFull15}{\hidepaste}
+\tab{5}\spadcommand{p(0, x)\free{pdef Nass }\bound{pfirst }}
+\indentrel{3}\begin{verbatim}
+   (14)  x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty15}
+\begin{paste}{ugUserFreeLocalPageEmpty15}{ugUserFreeLocalPagePatch15}
+\pastebutton{ugUserFreeLocalPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{p(0, x)\free{pdef Nass }\bound{pfirst }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch16}
+\begin{paste}{ugUserFreeLocalPageFull16}{ugUserFreeLocalPageEmpty16}
+\pastebutton{ugUserFreeLocalPageFull16}{\hidepaste}
+\tab{5}\spadcommand{N := 2\bound{Nass2 }}
+\indentrel{3}\begin{verbatim}
+   (15)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty16}
+\begin{paste}{ugUserFreeLocalPageEmpty16}{ugUserFreeLocalPagePatch16}
+\pastebutton{ugUserFreeLocalPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{N := 2\bound{Nass2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch17}
+\begin{paste}{ugUserFreeLocalPageFull17}{ugUserFreeLocalPageEmpty17}
+\pastebutton{ugUserFreeLocalPageFull17}{\hidepaste}
+\tab{5}\spadcommand{p(0, x)\free{pfirst Nass2 }}
+\indentrel{3}\begin{verbatim}
+   (16)  x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty17}
+\begin{paste}{ugUserFreeLocalPageEmpty17}{ugUserFreeLocalPagePatch17}
+\pastebutton{ugUserFreeLocalPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{p(0, x)\free{pfirst Nass2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch18}
+\begin{paste}{ugUserFreeLocalPageFull18}{ugUserFreeLocalPageEmpty18}
+\pastebutton{ugUserFreeLocalPageFull18}{\hidepaste}
+\tab{5}\spadcommand{)set fun cache 0 p}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty18}
+\begin{paste}{ugUserFreeLocalPageEmpty18}{ugUserFreeLocalPagePatch18}
+\pastebutton{ugUserFreeLocalPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{)set fun cache 0 p}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch19}
+\begin{paste}{ugUserFreeLocalPageFull19}{ugUserFreeLocalPageEmpty19}
+\pastebutton{ugUserFreeLocalPageFull19}{\hidepaste}
+\tab{5}\spadcommand{r : Record(i : Integer) := [1]\free{r }}
+\indentrel{3}\begin{verbatim}
+   (17)  [i= 1]
+                               Type: Record(i: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty19}
+\begin{paste}{ugUserFreeLocalPageEmpty19}{ugUserFreeLocalPagePatch19}
+\pastebutton{ugUserFreeLocalPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{r : Record(i : Integer) := [1]\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch20}
+\begin{paste}{ugUserFreeLocalPageFull20}{ugUserFreeLocalPageEmpty20}
+\pastebutton{ugUserFreeLocalPageFull20}{\hidepaste}
+\tab{5}\spadcommand{resetRecord rr ==
+  rr.i := 2
+  rr := [10]
+\bound{resetRecord }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty20}
+\begin{paste}{ugUserFreeLocalPageEmpty20}{ugUserFreeLocalPagePatch20}
+\pastebutton{ugUserFreeLocalPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{resetRecord rr ==
+  rr.i := 2
+  rr := [10]
+\bound{resetRecord }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch21}
+\begin{paste}{ugUserFreeLocalPageFull21}{ugUserFreeLocalPageEmpty21}
+\pastebutton{ugUserFreeLocalPageFull21}{\hidepaste}
+\tab{5}\spadcommand{resetRecord r\free{r resetRecord }\bound{rr }}
+\indentrel{3}\begin{verbatim}
+   (19)  [i= 10]
+                               Type: Record(i: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty21}
+\begin{paste}{ugUserFreeLocalPageEmpty21}{ugUserFreeLocalPagePatch21}
+\pastebutton{ugUserFreeLocalPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{resetRecord r\free{r resetRecord }\bound{rr }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch22}
+\begin{paste}{ugUserFreeLocalPageFull22}{ugUserFreeLocalPageEmpty22}
+\pastebutton{ugUserFreeLocalPageFull22}{\hidepaste}
+\tab{5}\spadcommand{r\free{rr }}
+\indentrel{3}\begin{verbatim}
+   (20)  [i= 2]
+                               Type: Record(i: Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty22}
+\begin{paste}{ugUserFreeLocalPageEmpty22}{ugUserFreeLocalPagePatch22}
+\pastebutton{ugUserFreeLocalPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{r\free{rr }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch23}
+\begin{paste}{ugUserFreeLocalPageFull23}{ugUserFreeLocalPageEmpty23}
+\pastebutton{ugUserFreeLocalPageFull23}{\hidepaste}
+\tab{5}\spadcommand{past := present := 1\bound{f0 }}
+\indentrel{3}\begin{verbatim}
+   (21)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty23}
+\begin{paste}{ugUserFreeLocalPageEmpty23}{ugUserFreeLocalPagePatch23}
+\pastebutton{ugUserFreeLocalPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{past := present := 1\bound{f0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch24}
+\begin{paste}{ugUserFreeLocalPageFull24}{ugUserFreeLocalPageEmpty24}
+\pastebutton{ugUserFreeLocalPageFull24}{\hidepaste}
+\tab{5}\spadcommand{index := 2\bound{f1 }\free{f0 }}
+\indentrel{3}\begin{verbatim}
+   (22)  2
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty24}
+\begin{paste}{ugUserFreeLocalPageEmpty24}{ugUserFreeLocalPagePatch24}
+\pastebutton{ugUserFreeLocalPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{index := 2\bound{f1 }\free{f0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch25}
+\begin{paste}{ugUserFreeLocalPageFull25}{ugUserFreeLocalPageEmpty25}
+\pastebutton{ugUserFreeLocalPageFull25}{\hidepaste}
+\tab{5}\spadcommand{fib(n) ==
+  free past, present, index
+  n < 3 => 1
+  n = index - 1 => past
+  if n < index-1 then
+    (past,present) := (1,1)
+    index := 2
+  while (index < n) repeat
+    (past,present) := (present, past+present)
+    index := index + 1
+  present
+\bound{f3 }\free{f2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty25}
+\begin{paste}{ugUserFreeLocalPageEmpty25}{ugUserFreeLocalPagePatch25}
+\pastebutton{ugUserFreeLocalPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{fib(n) ==
+  free past, present, index
+  n < 3 => 1
+  n = index - 1 => past
+  if n < index-1 then
+    (past,present) := (1,1)
+    index := 2
+  while (index < n) repeat
+    (past,present) := (present, past+present)
+    index := index + 1
+  present
+\bound{f3 }\free{f2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch26}
+\begin{paste}{ugUserFreeLocalPageFull26}{ugUserFreeLocalPageEmpty26}
+\pastebutton{ugUserFreeLocalPageFull26}{\hidepaste}
+\tab{5}\spadcommand{fibs := [fib(n) for n in 1..]\bound{fibs }\free{f3 }}
+\indentrel{3}\begin{verbatim}
+   (24)  [1,1,2,3,5,8,13,21,34,55,...]
+                           Type: Stream PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty26}
+\begin{paste}{ugUserFreeLocalPageEmpty26}{ugUserFreeLocalPagePatch26}
+\pastebutton{ugUserFreeLocalPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{fibs := [fib(n) for n in 1..]\bound{fibs }\free{f3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPagePatch27}
+\begin{paste}{ugUserFreeLocalPageFull27}{ugUserFreeLocalPageEmpty27}
+\pastebutton{ugUserFreeLocalPageFull27}{\hidepaste}
+\tab{5}\spadcommand{fibs 1000\free{fibs }}
+\indentrel{3}\begin{verbatim}
+   (25)
+  434665576869374564356885276750406258025646605173717804_
+   02481729089536555417949051890403879840079255169295922_
+   59308032263477520968962323987332247116164299644090653_
+   3187938298969649928516003704476137795166849228875
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserFreeLocalPageEmpty27}
+\begin{paste}{ugUserFreeLocalPageEmpty27}{ugUserFreeLocalPagePatch27}
+\pastebutton{ugUserFreeLocalPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{fibs 1000\free{fibs }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPagePatch1}
+\begin{paste}{ugUserPiecePredPageFull1}{ugUserPiecePredPageEmpty1}
+\pastebutton{ugUserPiecePredPageFull1}{\hidepaste}
+\tab{5}\spadcommand{opposite 'right == 'left}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPageEmpty1}
+\begin{paste}{ugUserPiecePredPageEmpty1}{ugUserPiecePredPagePatch1}
+\pastebutton{ugUserPiecePredPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{opposite 'right == 'left}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPagePatch2}
+\begin{paste}{ugUserPiecePredPageFull2}{ugUserPiecePredPageEmpty2}
+\pastebutton{ugUserPiecePredPageFull2}{\hidepaste}
+\tab{5}\spadcommand{opposite (x | x = 'left) == 'right}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPageEmpty2}
+\begin{paste}{ugUserPiecePredPageEmpty2}{ugUserPiecePredPagePatch2}
+\pastebutton{ugUserPiecePredPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{opposite (x | x = 'left) == 'right}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPagePatch3}
+\begin{paste}{ugUserPiecePredPageFull3}{ugUserPiecePredPageEmpty3}
+\pastebutton{ugUserPiecePredPageFull3}{\hidepaste}
+\tab{5}\spadcommand{for x in ['right,'left,'inbetween] repeat output opposite x}
+\indentrel{3}\begin{verbatim}
+   left
+   right
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPageEmpty3}
+\begin{paste}{ugUserPiecePredPageEmpty3}{ugUserPiecePredPagePatch3}
+\pastebutton{ugUserPiecePredPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{for x in ['right,'left,'inbetween] repeat output opposite x}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPagePatch4}
+\begin{paste}{ugUserPiecePredPageFull4}{ugUserPiecePredPageEmpty4}
+\pastebutton{ugUserPiecePredPageFull4}{\hidepaste}
+\tab{5}\spadcommand{inFirstHalfQuadrant(x | x > 0,y | y < x) == true}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPageEmpty4}
+\begin{paste}{ugUserPiecePredPageEmpty4}{ugUserPiecePredPagePatch4}
+\pastebutton{ugUserPiecePredPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{inFirstHalfQuadrant(x | x > 0,y | y < x) == true}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPagePatch5}
+\begin{paste}{ugUserPiecePredPageFull5}{ugUserPiecePredPageEmpty5}
+\pastebutton{ugUserPiecePredPageFull5}{\hidepaste}
+\tab{5}\spadcommand{inFirstHalfQuadrant(x | x > 0 and y < x,y) == true}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPageEmpty5}
+\begin{paste}{ugUserPiecePredPageEmpty5}{ugUserPiecePredPagePatch5}
+\pastebutton{ugUserPiecePredPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{inFirstHalfQuadrant(x | x > 0 and y < x,y) == true}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPagePatch6}
+\begin{paste}{ugUserPiecePredPageFull6}{ugUserPiecePredPageEmpty6}
+\pastebutton{ugUserPiecePredPageFull6}{\hidepaste}
+\tab{5}\spadcommand{inFirstHalfQuadrant(x,y | x > 0 and y < x) == true\bound{ifq1a }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPageEmpty6}
+\begin{paste}{ugUserPiecePredPageEmpty6}{ugUserPiecePredPagePatch6}
+\pastebutton{ugUserPiecePredPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{inFirstHalfQuadrant(x,y | x > 0 and y < x) == true\bound{ifq1a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPagePatch7}
+\begin{paste}{ugUserPiecePredPageFull7}{ugUserPiecePredPageEmpty7}
+\pastebutton{ugUserPiecePredPageFull7}{\hidepaste}
+\tab{5}\spadcommand{inFirstHalfQuadrant(x,y) == false\bound{ifq1b }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPageEmpty7}
+\begin{paste}{ugUserPiecePredPageEmpty7}{ugUserPiecePredPagePatch7}
+\pastebutton{ugUserPiecePredPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{inFirstHalfQuadrant(x,y) == false\bound{ifq1b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPagePatch8}
+\begin{paste}{ugUserPiecePredPageFull8}{ugUserPiecePredPageEmpty8}
+\pastebutton{ugUserPiecePredPageFull8}{\hidepaste}
+\tab{5}\spadcommand{[inFirstHalfQuadrant(i,3) for i in 1..5]\bound{ifq1b }}
+\indentrel{3}\begin{verbatim}
+   (7)  [false,false,false,true,true]
+                                     Type: List Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPiecePredPageEmpty8}
+\begin{paste}{ugUserPiecePredPageEmpty8}{ugUserPiecePredPagePatch8}
+\pastebutton{ugUserPiecePredPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{[inFirstHalfQuadrant(i,3) for i in 1..5]\bound{ifq1b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPagePatch1}
+\begin{paste}{ugUserAnonExampPageFull1}{ugUserAnonExampPageEmpty1}
+\pastebutton{ugUserAnonExampPageFull1}{\hidepaste}
+\tab{5}\spadcommand{x +-> if x < 0 then -x else x\bound{anon0 }}
+\indentrel{3}\begin{verbatim}
+   (1)
+     x
+   +-> 
+     if x < 0
+       then - x
+       else x
+                                Type: AnonymousFunction
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPageEmpty1}
+\begin{paste}{ugUserAnonExampPageEmpty1}{ugUserAnonExampPagePatch1}
+\pastebutton{ugUserAnonExampPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{x +-> if x < 0 then -x else x\bound{anon0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPagePatch2}
+\begin{paste}{ugUserAnonExampPageFull2}{ugUserAnonExampPageEmpty2}
+\pastebutton{ugUserAnonExampPageFull2}{\hidepaste}
+\tab{5}\spadcommand{abs1 := \%\free{anon0 }\bound{abs1 }}
+\indentrel{3}\begin{verbatim}
+   (2)
+     x
+   +-> 
+     if x < 0
+       then - x
+       else x
+                                Type: AnonymousFunction
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPageEmpty2}
+\begin{paste}{ugUserAnonExampPageEmpty2}{ugUserAnonExampPagePatch2}
+\pastebutton{ugUserAnonExampPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{abs1 := \%\free{anon0 }\bound{abs1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPagePatch3}
+\begin{paste}{ugUserAnonExampPageFull3}{ugUserAnonExampPageEmpty3}
+\pastebutton{ugUserAnonExampPageFull3}{\hidepaste}
+\tab{5}\spadcommand{(x,y) +-> abs1(x) > abs1(y)\bound{anon1 }\free{abs1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  (x,y) +-> abs1(y) < abs1(x)
+                                Type: AnonymousFunction
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPageEmpty3}
+\begin{paste}{ugUserAnonExampPageEmpty3}{ugUserAnonExampPagePatch3}
+\pastebutton{ugUserAnonExampPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{(x,y) +-> abs1(x) > abs1(y)\bound{anon1 }\free{abs1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPagePatch4}
+\begin{paste}{ugUserAnonExampPageFull4}{ugUserAnonExampPageEmpty4}
+\pastebutton{ugUserAnonExampPageFull4}{\hidepaste}
+\tab{5}\spadcommand{sort(\%,[3,9,-4,10,-3,-1,-9,5])\free{anon1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  [10,- 9,9,5,- 4,- 3,3,- 1]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPageEmpty4}
+\begin{paste}{ugUserAnonExampPageEmpty4}{ugUserAnonExampPagePatch4}
+\pastebutton{ugUserAnonExampPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{sort(\%,[3,9,-4,10,-3,-1,-9,5])\free{anon1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPagePatch5}
+\begin{paste}{ugUserAnonExampPageFull5}{ugUserAnonExampPageEmpty5}
+\pastebutton{ugUserAnonExampPageFull5}{\hidepaste}
+\tab{5}\spadcommand{ev := ( (i,j) +-> if even?(i+j) then 1 else -1)\bound{ev }}
+\indentrel{3}\begin{verbatim}
+   (5)
+     (i,j)
+   +-> 
+     if even?(i + j)
+       then 1
+       else - 1
+                                Type: AnonymousFunction
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPageEmpty5}
+\begin{paste}{ugUserAnonExampPageEmpty5}{ugUserAnonExampPagePatch5}
+\pastebutton{ugUserAnonExampPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{ev := ( (i,j) +-> if even?(i+j) then 1 else -1)\bound{ev }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPagePatch6}
+\begin{paste}{ugUserAnonExampPageFull6}{ugUserAnonExampPageEmpty6}
+\pastebutton{ugUserAnonExampPageFull6}{\hidepaste}
+\tab{5}\spadcommand{matrix([[ev(row,col) for row in 1..4] for col in 1..4])\free{ev }}
+\indentrel{3}\begin{verbatim}
+        Ú 1   - 1   1   - 1¿
+        ³                  ³
+        ³- 1   1   - 1   1 ³
+   (6)  ³                  ³
+        ³ 1   - 1   1   - 1³
+        ³                  ³
+        À- 1   1   - 1   1 Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPageEmpty6}
+\begin{paste}{ugUserAnonExampPageEmpty6}{ugUserAnonExampPagePatch6}
+\pastebutton{ugUserAnonExampPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{matrix([[ev(row,col) for row in 1..4] for col in 1..4])\free{ev }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPagePatch7}
+\begin{paste}{ugUserAnonExampPageFull7}{ugUserAnonExampPageEmpty7}
+\pastebutton{ugUserAnonExampPageFull7}{\hidepaste}
+\tab{5}\spadcommand{( p +-> not one?(gcd(p,D(p,x))) )(x**2+4*x+4)}
+\indentrel{3}\begin{verbatim}
+   (7)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPageEmpty7}
+\begin{paste}{ugUserAnonExampPageEmpty7}{ugUserAnonExampPagePatch7}
+\pastebutton{ugUserAnonExampPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{( p +-> not one?(gcd(p,D(p,x))) )(x**2+4*x+4)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPagePatch8}
+\begin{paste}{ugUserAnonExampPageFull8}{ugUserAnonExampPageEmpty8}
+\pastebutton{ugUserAnonExampPageFull8}{\hidepaste}
+\tab{5}\spadcommand{g(x,y,z) == cos(x + sin(y + tan(z)))}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPageEmpty8}
+\begin{paste}{ugUserAnonExampPageEmpty8}{ugUserAnonExampPagePatch8}
+\pastebutton{ugUserAnonExampPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{g(x,y,z) == cos(x + sin(y + tan(z)))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPagePatch9}
+\begin{paste}{ugUserAnonExampPageFull9}{ugUserAnonExampPageEmpty9}
+\pastebutton{ugUserAnonExampPageFull9}{\hidepaste}
+\tab{5}\spadcommand{g == (x,y,z) +-> cos(x + sin(y + tan(z)))}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserAnonExampPageEmpty9}
+\begin{paste}{ugUserAnonExampPageEmpty9}{ugUserAnonExampPagePatch9}
+\pastebutton{ugUserAnonExampPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{g == (x,y,z) +-> cos(x + sin(y + tan(z)))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch1}
+\begin{paste}{ugUserPieceBasicPageFull1}{ugUserPieceBasicPageEmpty1}
+\pastebutton{ugUserPieceBasicPageFull1}{\hidepaste}
+\tab{5}\spadcommand{fact(0) == 1\bound{fact0 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty1}
+\begin{paste}{ugUserPieceBasicPageEmpty1}{ugUserPieceBasicPagePatch1}
+\pastebutton{ugUserPieceBasicPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{fact(0) == 1\bound{fact0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch2}
+\begin{paste}{ugUserPieceBasicPageFull2}{ugUserPieceBasicPageEmpty2}
+\pastebutton{ugUserPieceBasicPageFull2}{\hidepaste}
+\tab{5}\spadcommand{fact(n | n > 0) == n * fact(n - 1)\free{fact0 }\bound{factn }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty2}
+\begin{paste}{ugUserPieceBasicPageEmpty2}{ugUserPieceBasicPagePatch2}
+\pastebutton{ugUserPieceBasicPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{fact(n | n > 0) == n * fact(n - 1)\free{fact0 }\bound{factn }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch3}
+\begin{paste}{ugUserPieceBasicPageFull3}{ugUserPieceBasicPageEmpty3}
+\pastebutton{ugUserPieceBasicPageFull3}{\hidepaste}
+\tab{5}\spadcommand{fact(3)\free{factn }}
+\indentrel{3}\begin{verbatim}
+   (3)  6
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty3}
+\begin{paste}{ugUserPieceBasicPageEmpty3}{ugUserPieceBasicPagePatch3}
+\pastebutton{ugUserPieceBasicPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{fact(3)\free{factn }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch4}
+\begin{paste}{ugUserPieceBasicPageFull4}{ugUserPieceBasicPageEmpty4}
+\pastebutton{ugUserPieceBasicPageFull4}{\hidepaste}
+\tab{5}\spadcommand{fact(-3)\free{factn }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty4}
+\begin{paste}{ugUserPieceBasicPageEmpty4}{ugUserPieceBasicPagePatch4}
+\pastebutton{ugUserPieceBasicPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{fact(-3)\free{factn }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch5}
+\begin{paste}{ugUserPieceBasicPageFull5}{ugUserPieceBasicPageEmpty5}
+\pastebutton{ugUserPieceBasicPageFull5}{\hidepaste}
+\tab{5}\spadcommand{facto(0) == 1\bound{facto0 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty5}
+\begin{paste}{ugUserPieceBasicPageEmpty5}{ugUserPieceBasicPagePatch5}
+\pastebutton{ugUserPieceBasicPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{facto(0) == 1\bound{facto0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch6}
+\begin{paste}{ugUserPieceBasicPageFull6}{ugUserPieceBasicPageEmpty6}
+\pastebutton{ugUserPieceBasicPageFull6}{\hidepaste}
+\tab{5}\spadcommand{facto(n | n < 0) == error "arguments to facto must be non-negative"\free{facto0 }\bound{factop }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty6}
+\begin{paste}{ugUserPieceBasicPageEmpty6}{ugUserPieceBasicPagePatch6}
+\pastebutton{ugUserPieceBasicPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{facto(n | n < 0) == error "arguments to facto must be non-negative"\free{facto0 }\bound{factop }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch7}
+\begin{paste}{ugUserPieceBasicPageFull7}{ugUserPieceBasicPageEmpty7}
+\pastebutton{ugUserPieceBasicPageFull7}{\hidepaste}
+\tab{5}\spadcommand{facto(n) == n * facto(n - 1)\free{factop }\bound{facton }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty7}
+\begin{paste}{ugUserPieceBasicPageEmpty7}{ugUserPieceBasicPagePatch7}
+\pastebutton{ugUserPieceBasicPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{facto(n) == n * facto(n - 1)\free{factop }\bound{facton }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch8}
+\begin{paste}{ugUserPieceBasicPageFull8}{ugUserPieceBasicPageEmpty8}
+\pastebutton{ugUserPieceBasicPageFull8}{\hidepaste}
+\tab{5}\spadcommand{facto(3)\free{facton }}
+\indentrel{3}\begin{verbatim}
+   (7)  6
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty8}
+\begin{paste}{ugUserPieceBasicPageEmpty8}{ugUserPieceBasicPagePatch8}
+\pastebutton{ugUserPieceBasicPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{facto(3)\free{facton }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch9}
+\begin{paste}{ugUserPieceBasicPageFull9}{ugUserPieceBasicPageEmpty9}
+\pastebutton{ugUserPieceBasicPageFull9}{\hidepaste}
+\tab{5}\spadcommand{facto(-7)\free{facton }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty9}
+\begin{paste}{ugUserPieceBasicPageEmpty9}{ugUserPieceBasicPagePatch9}
+\pastebutton{ugUserPieceBasicPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{facto(-7)\free{facton }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch10}
+\begin{paste}{ugUserPieceBasicPageFull10}{ugUserPieceBasicPageEmpty10}
+\pastebutton{ugUserPieceBasicPageFull10}{\hidepaste}
+\tab{5}\spadcommand{)display value facto\free{facton }}
+\indentrel{3}\begin{verbatim}
+   Definition:
+     facto 0 == 1
+     facto (n | n < 0) ==
+       error(arguments to facto must be non-negative)
+     facto n == n facto(n - 1)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty10}
+\begin{paste}{ugUserPieceBasicPageEmpty10}{ugUserPieceBasicPagePatch10}
+\pastebutton{ugUserPieceBasicPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{)display value facto\free{facton }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch11}
+\begin{paste}{ugUserPieceBasicPageFull11}{ugUserPieceBasicPageEmpty11}
+\pastebutton{ugUserPieceBasicPageFull11}{\hidepaste}
+\tab{5}\spadcommand{eleven(n | n < 1) == n + 11\bound{ff0 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty11}
+\begin{paste}{ugUserPieceBasicPageEmpty11}{ugUserPieceBasicPagePatch11}
+\pastebutton{ugUserPieceBasicPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{eleven(n | n < 1) == n + 11\bound{ff0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch12}
+\begin{paste}{ugUserPieceBasicPageFull12}{ugUserPieceBasicPageEmpty12}
+\pastebutton{ugUserPieceBasicPageFull12}{\hidepaste}
+\tab{5}\spadcommand{eleven(m) == eleven(eleven(m - 12))\bound{ff1 }\free{ff0 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty12}
+\begin{paste}{ugUserPieceBasicPageEmpty12}{ugUserPieceBasicPagePatch12}
+\pastebutton{ugUserPieceBasicPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{eleven(m) == eleven(eleven(m - 12))\bound{ff1 }\free{ff0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch13}
+\begin{paste}{ugUserPieceBasicPageFull13}{ugUserPieceBasicPageEmpty13}
+\pastebutton{ugUserPieceBasicPageFull13}{\hidepaste}
+\tab{5}\spadcommand{elevens := [eleven(i) for i in 0..]\bound{ff2 }\free{ff1 }}
+\indentrel{3}\begin{verbatim}
+   (10)  [11,11,11,11,11,11,11,11,11,11,...]
+                                   Type: Stream Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty13}
+\begin{paste}{ugUserPieceBasicPageEmpty13}{ugUserPieceBasicPagePatch13}
+\pastebutton{ugUserPieceBasicPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{elevens := [eleven(i) for i in 0..]\bound{ff2 }\free{ff1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch14}
+\begin{paste}{ugUserPieceBasicPageFull14}{ugUserPieceBasicPageEmpty14}
+\pastebutton{ugUserPieceBasicPageFull14}{\hidepaste}
+\tab{5}\spadcommand{elevens 200\free{ff2 }}
+\indentrel{3}\begin{verbatim}
+   (11)  11
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty14}
+\begin{paste}{ugUserPieceBasicPageEmpty14}{ugUserPieceBasicPagePatch14}
+\pastebutton{ugUserPieceBasicPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{elevens 200\free{ff2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPagePatch15}
+\begin{paste}{ugUserPieceBasicPageFull15}{ugUserPieceBasicPageEmpty15}
+\pastebutton{ugUserPieceBasicPageFull15}{\hidepaste}
+\tab{5}\spadcommand{)display value eleven\free{ff2 }}
+\indentrel{3}\begin{verbatim}
+   Definition:
+     eleven (m | m < 1) == m + 11
+     eleven m == eleven(eleven(m - 12))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugUserPieceBasicPageEmpty15}
+\begin{paste}{ugUserPieceBasicPageEmpty15}{ugUserPieceBasicPagePatch15}
+\pastebutton{ugUserPieceBasicPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{)display value eleven\free{ff2 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ug07.ht b/src/hyper/pages/ug07.ht
new file mode 100644
index 00000000..9a9cf528
--- /dev/null
+++ b/src/hyper/pages/ug07.ht
@@ -0,0 +1,3095 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+
+\texht{\setcounter{chapter}{6}}{} % Chapter 7
+
+\newcommand{\optArg}[1]{{{\tt [}{#1}{\tt ]}}}
+\newcommand{\argDef}[1]{{\tt ({#1})}}
+\newcommand{\funSyntax}[2]{\axiomFun{#1}{\tt ({\small\it{#2}})}}
+\newcommand{\funArgs}[1]{{\tt ({\small\it {#1}})}\newline}
+
+%
+\newcommand{\ugGraphTitle}{Graphics}
+\newcommand{\ugGraphNumber}{7.}
+%
+% =====================================================================
+\begin{page}{ugGraphPage}{7. Graphics}
+% =====================================================================
+\beginscroll
+
+%
+
+This chapter shows how to use the \Language{} graphics facilities
+%-% \HDindex{graphics}{ugGraphPage}{7.}{Graphics}
+under the X Window System.
+\Language{} has \twodim{} and \threedim{} drawing and rendering
+packages that allow the drawing, coloring, transforming, mapping,
+clipping, and combining of graphic output from \Language{}
+computations.
+This facility is particularly useful for investigating problems in
+areas such as topology.
+The graphics package is capable of plotting functions of one or
+more variables or plotting parametric surfaces and curves.
+Various coordinate systems are also available, such as polar and
+spherical.
+
+A graph is displayed in a viewport window and it has a
+%-% \HDindex{viewport}{ugGraphPage}{7.}{Graphics}
+control-panel that uses interactive mouse commands.
+PostScript and other output forms are available so that \Language{}
+%-% \HDindex{PostScript}{ugGraphPage}{7.}{Graphics}
+images can be printed or used by other programs.\footnote{PostScript
+is a trademark of Adobe Systems Incorporated, registered in the United
+States.}
+
+\beginmenu
+    \menudownlink{{7.1. Two-Dimensional Graphics}}{ugGraphTwoDPage}
+    \menudownlink{{7.2. Three-Dimensional Graphics}}{ugGraphThreeDPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphTwoDTitle}{Two-Dimensional Graphics}
+\newcommand{\ugGraphTwoDNumber}{7.1.}
+%
+% =====================================================================
+\begin{page}{ugGraphTwoDPage}{7.1. Two-Dimensional Graphics}
+% =====================================================================
+\beginscroll
+%
+The \Language{} \twodim{} graphics package provides the ability to
+%-% \HDindex{graphics!two-dimensional}{ugGraphTwoDPage}{7.1.}{Two-Dimensional Graphics}
+display
+%
+\indent{4}
+\beginitems
+%
+\item[-] curves defined by functions of a single real variable
+%
+\item[-] curves defined by parametric equations
+%
+\item[-] implicit non-singular curves defined by polynomial equations
+%
+\item[-] planar graphs generated from lists of point components.
+\enditems
+\indent{0}
+These graphs
+can be modified by specifying various options, such as
+calculating points in the polar
+coordinate system or changing the size of the graph viewport window.
+
+\beginmenu
+    \menudownlink{{7.1.1. Plotting Two-Dimensional Functions of One Variable}}{ugGraphTwoDPlotPage}
+    \menudownlink{{7.1.2. Plotting Two-Dimensional Parametric Plane Curves}}{ugGraphTwoDParPage}
+    \menudownlink{{7.1.3. Plotting Plane Algebraic Curves}}{ugGraphTwoDPlanePage}
+    \menudownlink{{7.1.4. Two-Dimensional Options}}{ugGraphTwoDOptionsPage}
+    \menudownlink{{7.1.5. Color}}{ugGraphColorPage}
+    \menudownlink{{7.1.6. Palette}}{ugGraphColorPalettePage}
+    \menudownlink{{7.1.7. Two-Dimensional Control-Panel}}{ugGraphTwoDControlPage}
+    \menudownlink{{7.1.8. Operations for Two-Dimensional Graphics}}{ugGraphTwoDopsPage}
+    \menudownlink{{7.1.9. Addendum: Building Two-Dimensional Graphs}}{ugGraphTwoDbuildPage}
+    \menudownlink{{7.1.10. Addendum: Appending a Graph to a Viewport Window Containing a Graph}}{ugGraphTwoDappendPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphTwoDPlotTitle}{Plotting Two-Dimensional Functions of One Variable}
+\newcommand{\ugGraphTwoDPlotNumber}{7.1.1.}
+%
+% =====================================================================
+\begin{page}{ugGraphTwoDPlotPage}{7.1.1. Plotting Two-Dimensional Functions of One Variable}
+% =====================================================================
+\beginscroll
+
+%-% \HDindex{curve!one variable function}{ugGraphTwoDPlotPage}{7.1.1.}{Plotting Two-Dimensional Functions of One Variable}
+The first kind of \twodim{} graph is that of a curve defined by a function
+\axiom{y = f(x)} over a finite interval of the \axiom{x} axis.
+
+%
+\beginImportant
+The general format for drawing a function defined by a formula
+\axiom{f(x)} is:
+%
+\centerline{{{\tt draw(f(x), x = a..b, {\it options})}}}
+where \axiom{a..b} defines the range of \axiom{x}, and where
+{\it options} prescribes zero or more options as described in
+\downlink{``\ugGraphTwoDOptionsTitle''}{ugGraphTwoDOptionsPage} in Section \ugGraphTwoDOptionsNumber\ignore{ugGraphTwoDOptions}.
+An example of an option is \axiom{curveColor == bright red().}
+An alternative format involving functions \axiom{f} and \axiom{g}
+is also available.
+\endImportant
+
+A simple way to plot a function is to use a formula.
+The first argument is the formula.
+For the second argument, write the name of the independent variable (here, \axiom{x}),
+followed by an \spadSyntax{=}, and the range of values.
+
+\psXtc{
+Display this formula over the range
+\texht{$0 \leq x \leq 6$}{0 <= x <= 6}.
+\Language{} converts your formula to a compiled
+function so that the results can be computed
+quickly and efficiently.
+}{
+\graphpaste{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)}
+}{
+\epsffile[0 0 295 295]{../ps/2D1VarA.ps}
+}
+
+Notice that \Language{} compiled the function before the graph was put
+on the screen.
+
+\psXtc{
+Here is the same graph on a different interval.
+This time we give the graph a title.
+}{
+\graphpaste{draw(sin(tan(x)) - tan(sin(x)),x = 10..16)}
+}{
+%window was 300 x 300
+\epsffile[0 0 295 295]{../ps/2D1VarB.ps}
+}
+%
+Once again the formula is converted to a compiled function before
+any points were computed.
+If you want to graph the same function on several intervals, it is
+a good idea to define the function first so that the function has
+to be compiled only once.
+\xtc{
+This time we first define the function.
+}{
+\spadpaste{f(x) == (x-1)*(x-2)*(x-3) \bound{f}}
+}
+\psXtc{
+To draw the function, the first argument is its name
+and the second is just the range with no independent variable.
+}{
+\graphpaste{draw(f, 0..4) \free{f}}
+}{
+\epsffile[0 0 295 295]{../ps/2D1VarD.ps}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphTwoDParTitle}{Plotting Two-Dimensional Parametric Plane Curves}
+\newcommand{\ugGraphTwoDParNumber}{7.1.2.}
+%
+% =====================================================================
+\begin{page}{ugGraphTwoDParPage}{7.1.2. Plotting Two-Dimensional Parametric Plane Curves}
+% =====================================================================
+\beginscroll
+
+The second kind of \twodim{} graph is that of
+%-% \HDindex{parametric plane curve}{ugGraphTwoDParPage}{7.1.2.}{Plotting Two-Dimensional Parametric Plane Curves}
+curves produced by parametric equations.
+%-% \HDindex{curve!parametric plane}{ugGraphTwoDParPage}{7.1.2.}{Plotting Two-Dimensional Parametric Plane Curves}
+Let \axiom{x = f(t)} and \axiom{y = g(t)} be formulas or two
+functions \axiom{f} and \axiom{g} as the parameter \axiom{t} ranges
+over an interval \axiom{[a,b]}.
+The function \axiomFun{curve} takes the two functions \axiom{f} and
+\axiom{g} as its parameters.
+
+\beginImportant
+The general format for drawing a \twodim{} plane curve defined by
+parametric formulas \axiom{x = f(t)} and \axiom{y = g(t)} is:
+%
+\centerline{{{\tt draw(curve(f(t), g(t)), t = a..b, {\it options})}}}
+where \axiom{a..b} defines the range of the independent variable \axiom{t},
+and where {\it options} prescribes zero or more options as
+described in \downlink{``\ugGraphThreeDOptionsTitle''}{ugGraphThreeDOptionsPage} in Section \ugGraphThreeDOptionsNumber\ignore{ugGraphThreeDOptions}.
+An example of an option is \axiom{curveColor == bright red().}
+\endImportant
+
+Here's an example:
+
+\psXtc{
+Define a parametric curve using a range involving
+\axiom{\%pi}, \Language{}'s way of saying \texht{$\pi$}{``pi''}.
+For parametric curves, \Language{} compiles two
+functions, one for each of the functions \axiom{f} and \axiom{g}.
+}{
+\graphpaste{draw(curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)), t = 0..2*\%pi)}
+}{
+\epsffile[0 0 295 295]{../ps/2DppcA.ps}
+}
+%
+%
+\psXtc{
+The title may be an arbitrary string and is an
+optional argument to the \axiomFun{draw} command.
+}{
+\graphpaste{draw(curve(cos(t), sin(t)), t = 0..2*\%pi)}
+}{
+\epsffile[0 0 295 295]{../ps/2DppcB.ps}
+}
+%
+If you plan on plotting \axiom{x = f(t)}, \axiom{y = g(t)} as \axiom{t} ranges over
+several intervals, you may want to define functions \axiom{f} and \axiom{g} first, so
+that they need not be recompiled every time you create a new graph.
+Here's an example:
+\xtc{
+As before, you can first define the functions you wish to draw.
+}{
+\spadpaste{f(t:DFLOAT):DFLOAT == sin(3*t/4) \bound{f}}
+}
+\xtc{
+\Language{} compiles them to map \axiomType{DoubleFloat}
+values to \axiomType{DoubleFloat} values.
+}{
+\spadpaste{g(t:DFLOAT):DFLOAT == sin(t) \bound{g}}
+}
+
+\psXtc{
+Give to {\tt curve} the names of the functions,
+then write the range without the name of the
+independent variable.
+}{
+\graphpaste{draw(curve(f,g),0..\%pi) \free{f g}}
+}{
+\epsffile[0 0 295 295]{../ps/2DppcC.ps}
+}
+%
+%
+\psXtc{
+Here is another look at the same curve but over a different
+range. Notice that \axiom{f} and \axiom{g} are not recompiled.
+Also note that \Language{} provides a default title based on
+the first function specified in \axiomFun{curve}.
+}{
+\graphpaste{draw(curve(f,g),-4*\%pi..4*\%pi) \free{f g}}
+}{
+\epsffile[0 0 295 295]{../ps/2DppcE.ps}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphTwoDPlaneTitle}{Plotting Plane Algebraic Curves}
+\newcommand{\ugGraphTwoDPlaneNumber}{7.1.3.}
+%
+% =====================================================================
+\begin{page}{ugGraphTwoDPlanePage}{7.1.3. Plotting Plane Algebraic Curves}
+% =====================================================================
+\beginscroll
+
+A third kind of \twodim{} graph is a non-singular ``solution curve''
+%-% \HDindex{curve!plane algebraic}{ugGraphTwoDPlanePage}{7.1.3.}{Plotting Plane Algebraic Curves}
+in a rectangular region of the plane.
+A solution curve is a curve defined by a polynomial equation
+\axiom{p(x,y) = 0}.
+%-% \HDindex{plane algebraic curve}{ugGraphTwoDPlanePage}{7.1.3.}{Plotting Plane Algebraic Curves}
+Non-singular means that the curve is ``smooth'' in that it does not
+cross itself or come to a point (cusp).
+Algebraically, this means that for any point \axiom{(x,y)} on the curve,
+that is, a point such that \axiom{p(x,y) = 0}, the partial derivatives
+\texht{${{\partial p}\over{\partial x}}(x,y)$ and
+${{\partial p}\over{\partial y}}(x,y)$}{\axiom{dp/dx(x,y)}
+and \axiom{dp/dy(a,b)}}
+are not both zero.
+%-% \HDindex{curve!smooth}{ugGraphTwoDPlanePage}{7.1.3.}{Plotting Plane Algebraic Curves}
+%-% \HDindex{curve!non-singular}{ugGraphTwoDPlanePage}{7.1.3.}{Plotting Plane Algebraic Curves}
+%-% \HDindex{smooth curve}{ugGraphTwoDPlanePage}{7.1.3.}{Plotting Plane Algebraic Curves}
+%-% \HDindex{non-singular curve}{ugGraphTwoDPlanePage}{7.1.3.}{Plotting Plane Algebraic Curves}
+
+%
+\beginImportant
+The general format for drawing a non-singular solution curve
+given by a polynomial of the form \axiom{p(x,y) = 0} is:
+%
+\centerline{{{\tt draw(p(x,y) = 0, x, y, range == [a..b, c..d], {\it options})}}}
+where the second and third arguments name the first and second
+independent variables of \axiom{p}.
+A {\tt range} option is always given to designate a bounding
+rectangular region of the plane \texht{$a \leq x \leq b, c \leq y
+\leq d$}{a <= x <= b, c <= y <= d}.
+Zero or more additional options as described in
+\downlink{``\ugGraphTwoDOptionsTitle''}{ugGraphTwoDOptionsPage} in Section \ugGraphTwoDOptionsNumber\ignore{ugGraphTwoDOptions} may be given.
+\endImportant
+
+\xtc{
+We require that the polynomial has rational or integral coefficients.
+Here is an algebraic curve example (``Cartesian ovals''):
+%-% \HDindex{Cartesian!ovals}{ugGraphTwoDPlanePage}{7.1.3.}{Plotting Plane Algebraic Curves}
+}{
+\spadpaste{p := ((x**2 + y**2 + 1) - 8*x)**2 - (8*(x**2 + y**2 + 1)-4*x-1) \bound{p}}
+}
+
+\psXtc{
+The first argument is always expressed as an equation of the form \axiom{p = 0}
+where \axiom{p} is a polynomial.
+}{
+\graphpaste{draw(p = 0, x, y, range == [-1..11, -7..7]) \free{p}}
+}{
+\epsffile[0 0 295 295]{../ps/2DpacA.ps}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphTwoDOptionsTitle}{Two-Dimensional Options}
+\newcommand{\ugGraphTwoDOptionsNumber}{7.1.4.}
+%
+% =====================================================================
+\begin{page}{ugGraphTwoDOptionsPage}{7.1.4. Two-Dimensional Options}
+% =====================================================================
+\beginscroll
+
+The \axiomFun{draw} commands take an optional list of options,
+such as {\tt title} shown above.
+Each option is given by the syntax: {\it name} {\tt ==} {\it value}.
+Here is a list of the available options in the order that they are
+described below.
+
+\table{ {adaptive} {clip} {unit} {clip} {curveColor} {range}
+{toScale} {pointColor} {coordinates}}
+
+The \axiom{adaptive} option turns adaptive plotting on or off.
+%-% \HDindex{adaptive plotting}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+Adaptive plotting uses an algorithm that traverses a graph and computes
+more points for those parts of the graph with high curvature.
+The higher the curvature of a region is, the more points the algorithm
+computes.
+%-% \HDindex{graphics!2D options!adaptive}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+%
+%
+\psXtc{
+The {\tt adaptive} option is normally on.
+Here we turn it off.
+}{
+\graphpaste{draw(sin(1/x),x=-2*\%pi..2*\%pi, adaptive == false)}
+}{
+\epsffile[0 0 295 295]{../ps/2DOptAd.ps}
+}
+%
+%
+\psXtc{
+The {\tt clip} option turns clipping on or off.
+%-% \HDindex{graphics!2D options!clipping}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+If on, large values are cut off according to
+\axiomFunFrom{clipPointsDefault}{GraphicsDefaults}.
+}{
+\graphpaste{draw(tan(x),x=-2*\%pi..2*\%pi, clip == true)}
+}{
+\epsffile[0 0 295 295]{../ps/2DOptCp.ps}
+}
+%
+%
+\psXtc{
+Option {\tt toScale} does plotting to scale if {\tt true} or uses
+the entire viewport if {\tt false}.
+The default can be determined using
+\axiomFunFrom{drawToScale}{GraphicsDefaults}.
+%-% \HDindex{graphics!2D options!to scale}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+}{
+\graphpaste{draw(sin(x),x=-\%pi..\%pi, toScale == true, unit == [1.0,1.0])}
+}{
+\epsffile[0 0 295 295]{../ps/2DOptSc.ps}
+}
+%
+%
+\psXtc{
+Option {\tt clip} with a range sets point clipping of a graph within the
+%-% \HDindex{graphics!2D options!clip in a range}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+ranges specified in the list \axiom{[x range,y range]}.
+%-% \HDindex{clipping}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+If only one range is specified, clipping applies to the y-axis.
+}{
+\graphpaste{draw(sec(x),x=-2*\%pi..2*\%pi, clip == [-2*\%pi..2*\%pi,-\%pi..\%pi], unit == [1.0,1.0])}
+}{
+\epsffile[0 0 295 295]{../ps/2DOptCpR.ps}
+}
+%
+\psXtc{
+Option {\tt curveColor} sets the color of the graph curves or lines to be the
+%-% \HDindex{graphics!2D options!curve color}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+indicated palette color
+%-% \HDindex{curve!color}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+(see \downlink{``\ugGraphColorTitle''}{ugGraphColorPage} in Section \ugGraphColorNumber\ignore{ugGraphColor} and \downlink{``\ugGraphColorPaletteTitle''}{ugGraphColorPalettePage} in Section \ugGraphColorPaletteNumber\ignore{ugGraphColorPalette}).
+%-% \HDindex{color!curve}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+}{
+\graphpaste{draw(sin(x),x=-\%pi..\%pi, curveColor == bright red())}
+}{
+\epsffile[0 0 295 295]{../ps/2DOptCvC.ps}
+}
+%
+\psXtc{
+Option {\tt pointColor}
+sets the color of the graph points to the indicated
+%-% \HDindex{graphics!2D options!point color}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+palette color
+(see \downlink{``\ugGraphColorTitle''}{ugGraphColorPage} in Section \ugGraphColorNumber\ignore{ugGraphColor} and \downlink{``\ugGraphColorPaletteTitle''}{ugGraphColorPalettePage} in Section \ugGraphColorPaletteNumber\ignore{ugGraphColorPalette}).
+%-% \HDindex{color!point}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+}{
+\graphpaste{draw(sin(x),x=-\%pi..\%pi, pointColor == pastel yellow())}
+}{
+\epsffile[0 0 295 295]{../ps/2DOptPtC.ps}
+}
+%
+\psXtc{
+Option {\tt unit} sets the intervals at which the axis units are plotted
+%-% \HDindex{graphics!2D options!set units}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+according to the indicated steps [\axiom{x} interval, \axiom{y} interval].
+}{
+\graphpaste{draw(curve(9*sin(3*t/4),8*sin(t)), t = -4*\%pi..4*\%pi, unit == [2.0,1.0])}
+}{
+\epsffile[0 0 295 295]{../ps/2DOptUt.ps}
+}
+%
+%
+\psXtc{
+Option {\tt range} sets the range of variables in a graph to be
+within the ranges
+%-% \HDindex{graphics!2D options!range}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+for solving plane algebraic curve plots.
+}{
+\graphpaste{draw(y**2 + y - (x**3 - x) = 0, x, y, range == [-2..2,-2..1], unit==[1.0,1.0])}
+}{
+\epsffile[0 0 295 295]{../ps/2DOptRgA.ps}
+}
+%
+%
+\psXtc{
+A second example of a solution plot.
+}{
+\graphpaste{draw(x**2 + y**2 = 1, x, y, range == [-3/2..3/2,-3/2..3/2], unit==[0.5,0.5])}
+}{
+\epsffile[0 0 295 295]{../ps/2DOptRgB.ps}
+}
+%
+%
+\psXtc{
+Option \axiom{coordinates} indicates the coordinate system
+in which the graph
+%-% \HDindex{graphics!2D options!coordinates}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+is plotted.
+The default is to use the Cartesian coordinate system.
+%-% \HDindex{Cartesian!coordinate system}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+For more details, see \downlink{``\ugGraphCoordTitle''}{ugGraphCoordPage} in Section \ugGraphCoordNumber\ignore{ugGraphCoord} \texht{.}{or
+\axiomType{CoordinateSystems}.}
+%-% \HDindex{coordinate system!Cartesian}{ugGraphTwoDOptionsPage}{7.1.4.}{Two-Dimensional Options}
+}{
+\graphpaste{draw(curve(sin(5*t),t),t=0..2*\%pi, coordinates == polar)}
+}{
+\epsffile[0 0 295 295]{../ps/2DOptPlr.ps}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphColorTitle}{Color}
+\newcommand{\ugGraphColorNumber}{7.1.5.}
+%
+% =====================================================================
+\begin{page}{ugGraphColorPage}{7.1.5. Color}
+% =====================================================================
+\beginscroll
+
+The domain \axiomType{Color}
+%-% \HDexptypeindex{Color}{ugGraphColorPage}{7.1.5.}{Color}
+provides operations for manipulating
+%-% \HDindex{graphics!color}{ugGraphColorPage}{7.1.5.}{Color}
+colors in \twodim{} graphs.
+%-% \HDindex{color}{ugGraphColorPage}{7.1.5.}{Color}
+Colors are objects of \axiomType{Color}.
+Each color has a {\it hue} and a {\it weight}.
+%-% \HDindex{hue}{ugGraphColorPage}{7.1.5.}{Color}
+Hues are represented by integers that range from \axiom{1} to the
+\axiomFunFrom{numberOfHues()}{Color}, normally
+%-% \HDindex{graphics!color!number of hues}{ugGraphColorPage}{7.1.5.}{Color}
+\axiom{27}.
+%\footnote{Use \axiomFun{colorDef} to
+%change these values to any range you want for a given \threedim{} viewport}
+%-% \HDindex{weight}{ugGraphColorPage}{7.1.5.}{Color}
+Weights are floats and  have the value \axiom{1.0} by default.
+%
+\indent{0}
+\beginitems
+%
+\item[\axiomFun{color}]\funArgs{integer}
+creates a color of hue {\it integer} and weight \axiom{1.0}.
+%
+\item[\axiomFun{hue}]\funArgs{color}
+returns the hue of {\it color} as an integer.
+%-% \HDindex{graphics!color!hue function}{ugGraphColorPage}{7.1.5.}{Color}
+%
+\item[\axiomFun{red}]\funArgs{},
+\funSyntax{blue}{},
+\funSyntax{green}{}, and \funSyntax{yellow}{}
+%-% \HDindex{graphics!color!primary color functions}{ugGraphColorPage}{7.1.5.}{Color}
+create colors of that hue with weight \axiom{1.0}.
+%
+\item[\subscriptIt{color}{1} {\tt +} \subscriptIt{color}{2}] returns the
+color that results from additively combining the indicated
+\subscriptIt{color}{1} and \subscriptIt{color}{2}.
+Color addition is not commutative: changing the order of the arguments
+produces different results.
+%
+\item[{\it integer} {\tt *} {\it color}]
+changes the weight of {\it color} by {\it integer}
+without affecting its hue.
+%-% \HDindex{graphics!color!multiply function}{ugGraphColorPage}{7.1.5.}{Color}
+For example,
+\axiom{red() + 3*yellow()} produces a color closer to yellow than to red.
+Color multiplication is not associative: changing the order of grouping
+%-% \HDindex{color!multiplication}{ugGraphColorPage}{7.1.5.}{Color}
+produces different results.
+\enditems
+\indent{0}
+%
+\psXtc{
+These functions can be used to change the point and curve colors
+for two- and \threedim{} graphs.
+Use the {\tt pointColor} option for points.
+}{
+\graphpaste{draw(x**2,x=-1..1,pointColor == green())}
+}{
+\epsffile[0 0 295 295]{../ps/23DColA.ps}
+}
+%
+\psXtc{
+Use the {\tt curveColor} option for curves.
+}{
+\graphpaste{draw(x**2,x=-1..1,curveColor == color(13) + 2*blue())}
+}{
+\epsffile[0 0 295 295]{../ps/23DColB.ps}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphColorPaletteTitle}{Palette}
+\newcommand{\ugGraphColorPaletteNumber}{7.1.6.}
+%
+% =====================================================================
+\begin{page}{ugGraphColorPalettePage}{7.1.6. Palette}
+% =====================================================================
+\beginscroll
+%-% \HDindex{graphics!palette}{ugGraphColorPalettePage}{7.1.6.}{Palette}
+
+Domain \axiomType{Palette} is the domain of shades of colors:
+\axiomFun{dark}, \axiomFun{dim}, \axiomFun{bright}, \axiomFun{pastel}, and \axiomFun{light},
+designated by the integers \axiom{1} through \axiom{5}, respectively.
+%-% \HDexptypeindex{Palette}{ugGraphColorPalettePage}{7.1.6.}{Palette}
+\xtc{
+Colors are normally ``bright.''
+}{
+\spadpaste{shade red()}
+}
+\xtc{
+To change the shade of a color, apply the name of a shade to it.
+%-% \HDindex{color!shade}{ugGraphColorPalettePage}{7.1.6.}{Palette}
+%-% \HDindex{shade}{ugGraphColorPalettePage}{7.1.6.}{Palette}
+}{
+\spadpaste{myFavoriteColor := dark blue() \bound{mfc}}
+}
+\xtc{
+The expression \axiom{shade(color)}
+returns the value of a shade of \axiom{color}.
+}{
+\spadpaste{shade myFavoriteColor \free{mfc}}
+}
+\xtc{
+The expression \axiom{hue(color)} returns its hue.
+}{
+\spadpaste{hue myFavoriteColor \free{mfc}}
+}
+\psXtc{
+Palettes can be used in specifying colors in \twodim{} graphs.
+}{
+\graphpaste{draw(x**2,x=-1..1,curveColor == dark blue())}
+}{
+\epsffile[0 0 295 295]{../ps/23DPal.ps}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphTwoDControlTitle}{Two-Dimensional Control-Panel}
+\newcommand{\ugGraphTwoDControlNumber}{7.1.7.}
+%
+% =====================================================================
+\begin{page}{ugGraphTwoDControlPage}{7.1.7. Two-Dimensional Control-Panel}
+% =====================================================================
+\beginscroll
+%-% \HDindex{graphics!2D control-panel}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+Once you have created a viewport, move your mouse to the viewport and click
+with your left mouse button to display a control-panel.
+The panel is displayed on the side of the viewport closest to
+where you clicked.  Each of the buttons which toggle on and off show the
+current state of the graph.
+
+
+\subsubsection{Transformations}
+%-% \HDindex{graphics!2D control-panel!transformations}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+
+Object transformations are executed from the control-panel by mouse-activated
+potentiometer windows.
+%
+\indent{0}
+\beginitems
+%
+\item[Scale:] To scale a graph, click on a mouse button
+%-% \HDindex{graphics!2D control-panel!scale}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+within the {\bf Scale} window in the upper left corner of the control-panel.
+The axes along which the scaling is to occur are indicated by setting the
+toggles above the arrow.
+With {\tt X On} and {\tt Y On} appearing, both axes are selected and scaling
+is uniform.
+If either is not selected, for example, if {\tt X Off} appears, scaling is
+non-uniform.
+%
+\item[Translate:] To translate a graph, click the mouse in the
+%-% \HDindex{graphics!2D control-panel!translate}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+{\bf Translate} window in the direction you wish the graph to move.
+This window is located in the upper right corner of the control-panel.
+Along the top of the {\bf Translate} window are two buttons for selecting
+the direction of translation.
+Translation along both coordinate axes results when {\tt X On} and {\tt Y
+On} appear or along one axis when one is on, for example, {\tt X On} and
+{\tt Y Off} appear.
+\enditems
+\indent{0}
+
+\subsubsection{Messages}
+%-% \HDindex{graphics!2D control-panel!messages}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+
+The window directly below the transformation potentiometer windows is
+used to display system messages relating to the viewport and the control-panel.
+The following format is displayed: \newline
+%
+\centerline{{[scaleX, scaleY] \axiom{>}graph\axiom{<} [translateX, translateY] \newline}}
+The two values to the left show the scale factor along the {\tt X} and
+{\tt Y} coordinate axes.  The two values to the right show the distance of
+translation from the center in the {\tt X} and {\tt Y} directions.  The number
+in the center shows which graph in the viewport this data pertains to.
+When multiple graphs exist in the same viewport,
+the graph must be selected (see ``Multiple Graphs,'' below) in
+order for its transformation data to be shown, otherwise the number
+is 1.
+
+\subsubsection{Multiple Graphs}
+
+%-% \HDindex{graphics!2D control-panel!multiple graphs}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+The {\bf Graphs} window contains buttons that allow the placement
+of \twodim{} graphs into one of nine available slots in any other
+\twodim{} viewport.
+In the center of the window are numeral buttons from one to nine
+that show whether a graph is displayed in the viewport.
+Below each number button is a button showing whether a graph
+that is present is selected for application of some
+transformation.
+When the caret symbol is displayed, then the graph in that slot
+will be manipulated.
+Initially, the graph for which the viewport is created occupies
+the first slot, is displayed, and is selected.
+%
+%
+\indent{0}
+\beginitems
+%
+\item[Clear:]  The {\bf Clear} button deselects every viewport graph slot.
+%-% \HDindex{graphics!2D control-panel!clear}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+A graph slot is reselected by selecting the button below its number.
+%
+\item[Query:]  The {\bf Query} button is used to display the scale and
+%-% \HDindex{graphics!2D control-panel!query}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+translate data for the indicated graph.  When this button is selected the
+message ``Click on the graph to query'' appears.  Select a slot
+number button from the {\bf Graphs} window. The scaling factor and translation
+offset of the graph are then displayed in the message window.
+%
+\item[Pick:]  The {\bf Pick} button is used to select a graph
+%-% \HDindex{graphics!2D control-panel!pick}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+to be placed or dropped into the indicated viewport.  When this button is
+selected, the message ``Click on the graph to pick'' appears.
+Click on the slot with the graph number of the desired
+graph.  The graph information is held waiting for
+you to execute a {\bf Drop} in some other graph.
+%
+\item[Drop:]  Once a graph has been picked up using the {\bf Pick} button,
+%-% \HDindex{graphics!2D control-panel!drop}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+the {\bf Drop} button places it into a new viewport slot.
+The message ``Click on the graph to drop'' appears in the message
+window when the {\bf Drop} button is selected.
+By selecting one of the slot number buttons in the {\bf Graphs}
+window, the graph currently being held is dropped into this slot
+and displayed.
+\enditems
+\indent{0}
+
+\subsubsection{Buttons}
+%-% \HDindex{graphics!2D control-panel!buttons}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+
+%
+\indent{0}
+\beginitems
+%
+\item[Axes] turns the coordinate axes on or off.
+%-% \HDindex{graphics!2D control-panel!axes}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+%
+\item[Units] turns the units along the {\tt x}
+and {\tt y} axis on or off.
+%-% \HDindex{graphics!2D control-panel!units}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+%
+\item[Box] encloses the area of the viewport graph
+in a bounding box, or removes the box if already enclosed.
+%-% \HDindex{graphics!2D control-panel!box}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+%
+\item[Pts] turns on or off the display of points.
+%-% \HDindex{graphics!2D control-panel!points}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+%
+\item[Lines] turns on or off the display
+of lines connecting points.
+%-% \HDindex{graphics!2D control-panel!lines}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+%
+\item[PS] writes the current viewport contents to
+%-% \HDindex{graphics!2D control-panel!ps}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+a file {\bf axiom2D.ps} or to a name specified in the user's {\bf
+%-% \HDindex{graphics!.Xdefaults!PostScript file name}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+.Xdefaults} file.
+%-% \HDindex{file!.Xdefaults @{\bf .Xdefaults}}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+The file is placed in the directory from which \Language{} or the {\bf
+viewAlone} program was invoked.
+%-% \HDindex{PostScript}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+%
+\item[Reset] resets the object transformation
+characteristics and attributes back to their initial states.
+%-% \HDindex{graphics!2D control-panel!reset}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+%
+\item[Hide] makes the control-panel disappear.
+%-% \HDindex{graphics!2D control-panel!hide}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+%
+\item[Quit] queries whether the current viewport
+%-% \HDindex{graphics!2D control-panel!quit}{ugGraphTwoDControlPage}{7.1.7.}{Two-Dimensional Control-Panel}
+session should be terminated.
+\enditems
+\indent{0}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphTwoDopsTitle}{Operations for Two-Dimensional Graphics}
+\newcommand{\ugGraphTwoDopsNumber}{7.1.8.}
+%
+% =====================================================================
+\begin{page}{ugGraphTwoDopsPage}{7.1.8. Operations for Two-Dimensional Graphics}
+% =====================================================================
+\beginscroll
+
+Here is a summary of useful \Language{} operations for \twodim{}
+graphics.
+Each operation name is followed by a list of arguments.
+Each argument is written as a variable informally named according
+to the type of the argument (for example, {\it integer}).
+If appropriate, a default value for an argument is given in
+parentheses immediately following the name.
+
+%
+\texht{\bgroup\hbadness = 10001\sloppy}{}
+\indent{0}
+\beginitems
+%
+\item[\axiomFun{adaptive}]\funArgs{\optArg{boolean\argDef{true}}}
+%-% \HDindex{adaptive plotting}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+sets or indicates whether graphs are plotted
+%-% \HDindex{graphics!set 2D defaults!adaptive}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+according to the adaptive refinement algorithm.
+%
+\item[\axiomFun{axesColorDefault}]\funArgs{\optArg{color\argDef{dark blue()}}}
+sets or indicates the default color of the
+%-% \HDindex{graphics!set 2D defaults!axes color}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+axes in a \twodim{} graph viewport.
+%
+\item[\axiomFun{clipPointsDefault}]\funArgs{\optArg{boolean\argDef{false}}}
+sets or
+indicates whether point clipping is
+%-% \HDindex{graphics!set 2D defaults!clip points}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+to be applied as the default for graph plots.
+%
+\item[\axiomFun{drawToScale}]\funArgs{\optArg{boolean\argDef{false}}}
+sets or
+indicates whether the plot of a graph
+%-% \HDindex{graphics!set 2D defaults!to scale}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+is ``to scale'' or uses the entire viewport space as the default.
+%
+\item[\axiomFun{lineColorDefault}]\funArgs{\optArg{color\argDef{pastel yellow()}}}
+sets or indicates the default color of the
+%-% \HDindex{graphics!set 2D defaults!line color}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+lines or curves in a \twodim{} graph viewport.
+%
+\item[\axiomFun{maxPoints}]\funArgs{\optArg{integer\argDef{500}}}
+sets or indicates
+the default maximum number of
+%-% \HDindex{graphics!set 2D defaults!max points}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+possible points to be used when constructing a \twodim{} graph.
+%
+\item[\axiomFun{minPoints}]\funArgs{\optArg{integer\argDef{21}}}
+sets or indicates the default minimum number of
+%-% \HDindex{graphics!set 2D defaults!min points}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+possible points to be used when constructing a \twodim{} graph.
+%
+\item[\axiomFun{pointColorDefault}]\funArgs{\optArg{color\argDef{bright red()}}}
+sets or indicates the default color of the
+%-% \HDindex{graphics!set 2D defaults!point color}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+points in a \twodim{} graph viewport.
+%
+\item[\axiomFun{pointSizeDefault}]\funArgs{\optArg{integer\argDef{5}}}
+sets or indicates the default size of the
+%-% \HDindex{graphics!set 2D defaults!point size}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+dot used to plot points in a \twodim{} graph.
+%
+\item[\axiomFun{screenResolution}]\funArgs{\optArg{integer\argDef{600}}}
+sets or indicates the default screen
+%-% \HDindex{graphics!set 2D defaults!screen resolution}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+resolution constant used in setting the computation limit of adaptively
+%-% \HDindex{adaptive plotting}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+generated curve plots.
+%
+\item[\axiomFun{unitsColorDefault}]\funArgs{\optArg{color\argDef{dim green()}}}
+sets or indicates the default color of the
+%-% \HDindex{graphics!set 2D defaults!units color}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+unit labels in a \twodim{} graph viewport.
+%
+\item[\axiomFun{viewDefaults}]\funArgs{}
+resets the default settings for the following
+%-% \HDindex{graphics!set 2D defaults!reset viewport}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+attributes:  point color, line color, axes color, units color, point size,
+viewport upper left-hand corner position, and the viewport size.
+%
+\item[\axiomFun{viewPosDefault}]\funArgs{\optArg{list\argDef{[100,100]}}}
+sets or indicates the default position of the
+%-% \HDindex{graphics!set 2D defaults!viewport position}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+upper left-hand corner of a \twodim{} viewport, relative to the
+display root window.
+The upper left-hand corner of the display is considered to be at the
+(0, 0) position.
+%
+\item[\axiomFun{viewSizeDefault}]\funArgs{\optArg{list\argDef{[200,200]}}}
+sets or
+indicates the default size in which two
+%-% \HDindex{graphics!set 2D defaults!viewport size}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+dimensional viewport windows are shown.
+It is defined by a width and then a height.
+%
+\item[\axiomFun{viewWriteAvailable}]\funArgs{\optArg{list\argDef{["pixmap",
+"bitmap", "postscript", \"image"}}}
+indicates the possible file types
+%-% \HDindex{graphics!2D defaults!available viewport writes}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+that can be created with the \axiomFunFrom{write}{TwoDimensionalViewport} function.
+%
+\item[\axiomFun{viewWriteDefault}]
+\funArgs{\optArg{list\argDef{[]}}}
+sets or indicates the default types of files, in
+%-% \HDindex{graphics!set 2D defaults!write viewport}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+addition to the {\bf data} file, that are created when a
+\axiomFun{write} function is executed on a viewport.
+%
+\item[\axiomFun{units}]\funArgs{viewport, integer\argDef{1}, string\argDef{"off"}}
+turns the units on or off for the graph with index {\it integer}.
+%
+\item[\axiomFun{axes}]\funArgs{viewport, integer\argDef{1}, string\argDef{"on"}}
+turns the axes on
+%-% \HDindex{graphics!2D commands!axes}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+or off for the graph with index {\it integer}.
+%
+\item[\axiomFun{close}]\funArgs{viewport}
+closes {\it viewport}.
+%-% \HDindex{graphics!2D commands!close}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+%
+\item[\axiomFun{connect}]\funArgs{viewport, integer\argDef{1}, string\argDef{"on"}}
+declares whether lines
+%-% \HDindex{graphics!2D commands!connect}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+connecting the points are displayed or not.
+%
+\item[\axiomFun{controlPanel}]\funArgs{viewport, string\argDef{"off"}}
+declares
+whether the \twodim{} control-panel is automatically displayed
+or not.
+%
+\item[\axiomFun{graphs}]\funArgs{viewport}
+returns a list
+%-% \HDindex{graphics!2D commands!graphs}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+describing the state of each graph.
+If the graph state is not being used this is shown by {\tt "undefined"},
+otherwise a description of the graph's contents is shown.
+%
+\item[\axiomFun{graphStates}]\funArgs{viewport}
+displays
+%-% \HDindex{graphics!2D commands!state of graphs}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+a list of all the graph states available for {\it viewport}, giving the
+values for every property.
+%
+\item[\axiomFun{key}]\funArgs{viewport}
+returns the process
+%-% \HDindex{graphics!2D commands!key}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+ID number for {\it viewport}.
+%
+\item[\axiomFun{move}]\funArgs{viewport,
+\subscriptText{integer}{x}(viewPosDefault),
+\subscriptText{integer}{y}(viewPosDefault)}
+moves {\it viewport} on the screen so that the
+%-% \HDindex{graphics!2D commands!move}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+upper left-hand corner of {\it viewport} is at the position {\it (x,y)}.
+%
+\item[\axiomFun{options}]\funArgs{\it viewport}
+returns a list
+%-% \HDindex{graphics!2D commands!options}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+of all the \axiomType{DrawOption}s used by {\it viewport}.
+%
+\item[\axiomFun{points}]\funArgs{viewport, integer\argDef{1}, string\argDef{"on"}}
+specifies whether the graph points for graph {\it integer} are
+%-% \HDindex{graphics!2D commands!points}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+to be displayed or not.
+%
+\item[\axiomFun{region}]\funArgs{viewport, integer\argDef{1}, string\argDef{"off"}}
+declares whether graph {\it integer} is or is not to be displayed
+with a bounding rectangle.
+%
+\item[\axiomFun{reset}]\funArgs{viewport}
+resets all the properties of {\it viewport}.
+%
+\item[\axiomFun{resize}]\funArgs{viewport,
+\subscriptText{integer}{width}, \subscriptText{integer}{height}}
+%-% \HDindex{graphics!2D commands!resize}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+resizes {\it viewport} with a new {\it width} and {\it height}.
+%
+\item[\axiomFun{scale}]\funArgs{viewport, \subscriptText{integer}{n}\argDef{1},
+\subscriptText{integer}{x}\argDef{0.9}, \subscriptText{integer}{y}\argDef{0.9}}
+scales values for the
+%-% \HDindex{graphics!2D commands!scale}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+{\it x} and {\it y} coordinates of graph {\it n}.
+%
+\item[\axiomFun{show}]\funArgs{viewport, \subscriptText{integer}{n}\argDef{1},
+string\argDef{"on"}}
+indicates if graph {\it n} is shown or not.
+%
+\item[\axiomFun{title}]\funArgs{viewport, string\argDef{"Axiom 2D"}}
+designates the title for {\it viewport}.
+%
+\item[\axiomFun{translate}]\funArgs{viewport,
+\subscriptText{integer}{n}\argDef{1},
+\subscriptText{float}{x}\argDef{0.0}, \subscriptText{float}{y}\argDef{0.0}}
+%-% \HDindex{graphics!2D commands!translate}{ugGraphTwoDopsPage}{7.1.8.}{Operations for Two-Dimensional Graphics}
+causes graph {\it n} to be moved {\it x} and {\it y} units in the respective directions.
+%
+\item[\axiomFun{write}]\funArgs{viewport, \subscriptText{string}{directory},
+\optArg{strings}}
+if no third argument is given, writes the {\bf data} file onto the directory
+with extension {\bf data}.
+The third argument can be a single string or a list of strings with some or
+all the entries {\tt "pixmap"}, {\tt "bitmap"}, {\tt "postscript"}, and
+{\tt "image"}.
+\enditems
+\indent{0}
+\texht{\egroup}{}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphTwoDbuildTitle}{Addendum: Building Two-Dimensional Graphs}
+\newcommand{\ugGraphTwoDbuildNumber}{7.1.9.}
+%
+% =====================================================================
+\begin{page}{ugGraphTwoDbuildPage}{7.1.9. Addendum: Building Two-Dimensional Graphs}
+% =====================================================================
+\beginscroll
+
+In this section we demonstrate how to create \twodim{} graphs from
+lists of points and give an example showing how to read the lists
+of points from a file.
+
+\subsubsection{Creating a Two-Dimensional Viewport from a List of Points}
+
+\Language{} creates lists of points in a \twodim{} viewport by utilizing
+the \axiomType{GraphImage} and \axiomType{TwoDimensionalViewport} domains.
+In this example, the \axiomFunFrom{makeGraphImage}{GraphImage}
+function takes a list of lists of points parameter, a list of colors for
+each point in the graph, a list of colors for each line in the graph, and
+a list of sizes for each point in the graph.
+%
+\xtc{
+The following expressions create a list of lists of points which will be read
+by \Language{} and made into a \twodim{} viewport.
+}{
+\spadpaste{p1 := point [1,1]\$(Point DFLOAT) \bound{p1}}
+}
+\xtc{
+}{
+\spadpaste{p2 := point [0,1]\$(Point DFLOAT) \bound{p2}}
+}
+\xtc{
+}{
+\spadpaste{p3 := point [0,0]\$(Point DFLOAT) \bound{p3}}
+}
+\xtc{
+}{
+\spadpaste{p4 := point [1,0]\$(Point DFLOAT) \bound{p4}}
+}
+\xtc{
+}{
+\spadpaste{p5 := point [1,.5]\$(Point DFLOAT) \bound{p5}}
+}
+\xtc{
+}{
+\spadpaste{p6 := point [.5,0]\$(Point DFLOAT) \bound{p6}}
+}
+\xtc{
+}{
+\spadpaste{p7 := point [0,0.5]\$(Point DFLOAT) \bound{p7}}
+}
+\xtc{
+}{
+\spadpaste{p8 := point [.5,1]\$(Point DFLOAT) \bound{p8}}
+}
+\xtc{
+}{
+\spadpaste{p9 := point [.25,.25]\$(Point DFLOAT) \bound{p9}}
+}
+\xtc{
+}{
+\spadpaste{p10 := point [.25,.75]\$(Point DFLOAT) \bound{p10}}
+}
+\xtc{
+}{
+\spadpaste{p11 := point [.75,.75]\$(Point DFLOAT) \bound{p11}}
+}
+\xtc{
+}{
+\spadpaste{p12 := point [.75,.25]\$(Point DFLOAT) \bound{p12}}
+}
+\xtc{
+Finally, here is the list.
+}{
+\spadpaste{llp := [[p1,p2], [p2,p3], [p3,p4], [p4,p1], [p5,p6], [p6,p7], [p7,p8], [p8,p5], [p9,p10], [p10,p11], [p11,p12], [p12,p9]] \free{p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12} \bound{llp}}
+}
+\xtc{
+Now we set the point sizes for all components of the graph.
+}{
+\spadpaste{size1 := 6::PositiveInteger \bound{size1}}
+}
+\xtc{
+}{
+}
+\xtc{
+}{
+\spadpaste{size2 := 8::PositiveInteger \bound{size2}}
+}
+\xtc{
+}{
+\spadpaste{size3 := 10::PositiveInteger \bound{size3}}
+}
+\xtc{
+}{
+\spadpaste{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3] \bound{lsize} \free{size1 size2 size3}}
+}
+\xtc{
+Here are the colors for the points.
+}{
+\spadpaste{pc1 := pastel red() \bound{pc1}}
+}
+\xtc{
+}{
+\spadpaste{pc2 := dim green() \bound{pc2}}
+}
+\xtc{
+}{
+\spadpaste{pc3 := pastel yellow() \bound{pc3}}
+}
+\xtc{
+}{
+\spadpaste{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3] \free{pc1 pc2 pc3} \bound{lpc}}
+}
+\xtc{
+Here are the colors for the lines.
+}{
+\spadpaste{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()] \bound{lc}}
+}
+\xtc{
+Now the \axiomType{GraphImage} is created according to the component
+specifications indicated above.
+}{
+\spadpaste{g := makeGraphImage(llp,lpc,lc,lsize)\$GRIMAGE \bound{g} \free{llp lpc lc lsize}}
+}
+\psXtc{
+The \axiomFunFrom{makeViewport2D}{TwoDimensionalViewport} function now
+creates a \axiomType{TwoDimensionalViewport} for this graph according to the
+list of options specified within the brackets.
+}{
+\graphpaste{makeViewport2D(g,[title("Lines")])\$VIEW2D \free{g}}
+}{
+%
+}
+%See Figure #.#.
+\xtc{
+This example demonstrates the use of the \axiomType{GraphImage} functions
+\axiomFunFrom{component}{GraphImage} and \axiomFunFrom{appendPoint}{GraphImage}
+in adding points to an empty \axiomType{GraphImage}.
+}{
+\spadpaste{)clear all \bound{clearAll}}
+}
+\xtc{
+}{
+\spadpaste{g := graphImage()\$GRIMAGE \bound{Sg}\free{clearAll}}
+}
+\xtc{
+}{
+\spadpaste{p1 := point [0,0]\$(Point DFLOAT) \bound{Sp1}}
+}
+\xtc{
+}{
+\spadpaste{p2 := point [.25,.25]\$(Point DFLOAT) \bound{Sp2}}
+}
+\xtc{
+}{
+\spadpaste{p3 := point [.5,.5]\$(Point DFLOAT) \bound{Sp3}}
+}
+\xtc{
+}{
+\spadpaste{p4 := point [.75,.75]\$(Point DFLOAT) \bound{Sp4}}
+}
+\xtc{
+}{
+\spadpaste{p5 := point [1,1]\$(Point DFLOAT) \bound{Sp5}}
+}
+\xtc{
+}{
+\spadpaste{component(g,p1)\$GRIMAGE\free{Sg Sp1}\bound{gp1}}
+}
+\xtc{
+}{
+\spadpaste{component(g,p2)\$GRIMAGE\free{Sg Sp2}\bound{gp2}}
+}
+\xtc{
+}{
+\spadpaste{appendPoint(g,p3)\$GRIMAGE\free{gp1 gp2 Sp3}\bound{gp3}}
+}
+\xtc{
+}{
+\spadpaste{appendPoint(g,p4)\$GRIMAGE\free{gp3 Sp4}\bound{gp4}}
+}
+\xtc{
+}{
+\spadpaste{appendPoint(g,p5)\$GRIMAGE\free{gp4 Sp5}\bound{gp5}}
+}
+\xtc{
+}{
+\spadpaste{g1 := makeGraphImage(g)\$GRIMAGE \bound{Sg1} \free{gp5}}
+}
+\psXtc{
+Here is the graph.
+}{
+\graphpaste{makeViewport2D(g1,[title("Graph Points")])\$VIEW2D \free{Sg1}}
+}{
+%
+}
+%
+%See Figure #.#.
+%
+\xtc{
+A list of points can also be made into a \axiomType{GraphImage} by using
+the operation \axiomFunFrom{coerce}{GraphImage}.  It is equivalent to adding
+each point to \axiom{g2} using \axiomFunFrom{component}{GraphImage}.
+}{
+\spadpaste{g2 := coerce([[p1],[p2],[p3],[p4],[p5]])\$GRIMAGE  \free{Sp1 Sp2 Sp3 Sp4 Sp5} \bound{Sg2}}
+}
+\xtc{
+Now, create an empty \axiomType{TwoDimensionalViewport}.
+}{
+\spadpaste{v := viewport2D()\$VIEW2D \bound{Sv}}
+}
+\xtc{
+}{
+\spadpaste{options(v,[title("Just Points")])\$VIEW2D \free{Sv}\bound{Svo}}
+}
+\xtc{
+Place the graph into the viewport.
+}{
+\spadpaste{putGraph(v,g2,1)\$VIEW2D \free{Sg2 Svo}\bound{Svog2}}
+}
+\psXtc{
+Take a look.
+}{
+\graphpaste{makeViewport2D(v)\$VIEW2D \free{Svog2}}
+}{
+%
+}
+
+%See Figure #.#.
+
+\subsubsection{Creating a Two-Dimensional Viewport of a List of Points from a File}
+
+The following three functions read a list of points from a
+file and then draw the points and the connecting lines. The
+points are stored in the file in readable form as floating point numbers
+(specifically, \axiomType{DoubleFloat} values) as an alternating
+stream of \axiom{x}- and \axiom{y}-values. For example,
+\begin{verbatim}
+0.0 0.0     1.0 1.0     2.0 4.0
+3.0 9.0     4.0 16.0    5.0 25.0
+\end{verbatim}
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ drawPoints(lp:List\ Point\ DoubleFloat):VIEW2D\ ==}\newline
+{\tt 2.\ \ \ \ \ g\ :=\ graphImage()\$GRIMAGE}\newline
+{\tt 3.\ \ \ \ \ for\ p\ in\ lp\ repeat}\newline
+{\tt 4.\ \ \ \ \ \ \ component(g,p,pointColorDefault(),lineColorDefault(),}\newline
+{\tt 5.\ \ \ \ \ \ \ \ \ pointSizeDefault())}\newline
+{\tt 6.\ \ \ \ \ gi\ :=\ makeGraphImage(g)\$GRIMAGE}\newline
+{\tt 7.\ \ \ \ \ makeViewport2D(gi,[title("Points")])\$VIEW2D}\newline
+{\tt 8.\ \ \ }\newline
+{\tt 9.\ \ \ drawLines(lp:List\ Point\ DoubleFloat):VIEW2D\ ==}\newline
+{\tt 10.\ \ \ \ g\ :=\ graphImage()\$GRIMAGE}\newline
+{\tt 11.\ \ \ \ component(g,\ lp,\ pointColorDefault(),\ lineColorDefault(),}\newline
+{\tt 12.\ \ \ \ \ \ pointSizeDefault())\$GRIMAGE}\newline
+{\tt 13.\ \ \ \ gi\ :=\ makeGraphImage(g)\$GRIMAGE}\newline
+{\tt 14.\ \ \ \ makeViewport2D(gi,[title("Points")])\$VIEW2D}\newline
+{\tt 15.\ \ }\newline
+{\tt 16.\ \ plotData2D(name,\ title)\ ==}\newline
+{\tt 17.\ \ \ \ f:File(DFLOAT)\ :=\ open(name,"input")}\newline
+{\tt 18.\ \ \ \ lp:LIST(Point\ DFLOAT)\ :=\ empty()}\newline
+{\tt 19.\ \ \ \ while\ ((x\ :=\ readIfCan!(f))\ case\ DFLOAT)\ repeat}\newline
+{\tt 20.\ \ \ \ \ \ y\ :\ DFLOAT\ :=\ read!(f)}\newline
+{\tt 21.\ \ \ \ \ \ lp\ :=\ cons(point\ [x,y]\$(Point\ DFLOAT),\ lp)}\newline
+{\tt 22.\ \ \ \ \ \ lp}\newline
+{\tt 23.\ \ \ \ close!(f)}\newline
+{\tt 24.\ \ \ \ drawPoints(lp)}\newline
+{\tt 25.\ \ \ \ drawLines(lp)}\newline
+\endImportant
+%
+This command will actually create the viewport and the graph if
+the point data is in the file \axiom{"file.data"}.
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ plotData2D("file.data",\ "2D\ Data\ Plot")}\newline
+\endImportant
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphTwoDappendTitle}{Addendum: Appending a Graph to a Viewport Window Containing a Graph}
+\newcommand{\ugGraphTwoDappendNumber}{7.1.10.}
+%
+% =====================================================================
+\begin{page}{ugGraphTwoDappendPage}{7.1.10. Addendum: Appending a Graph to a Viewport Window Containing a Graph}
+% =====================================================================
+\beginscroll
+
+This section demonstrates how to append a \twodim{} graph to a viewport
+already containing other graphs.
+The default \axiomFun{draw} command places a graph into the first
+\axiomType{GraphImage} slot position of the \axiomType{TwoDimensionalViewport}.
+
+\xtc{
+This graph is in the first slot in its viewport.
+}{
+\spadpaste{v1 := draw(sin(x),x=0..2*\%pi) \bound{v1}}
+}
+\xtc{
+So is this graph.
+}{
+\spadpaste{v2 := draw(cos(x),x=0..2*\%pi, curveColor==light red()) \bound{v2}}
+}
+\xtc{
+The operation \axiomFunFrom{getGraph}{TwoDimensionalViewport}
+retrieves the \axiomType{GraphImage} \axiom{g1} from the first slot position
+in the viewport \axiom{v1}.
+}{
+\spadpaste{g1 := getGraph(v1,1) \bound{g1}\free{v1}}
+}
+\xtc{
+Now \axiomFunFrom{putGraph}{TwoDimensionalViewport}
+places \axiom{g1} into the the second slot position of \axiom{v2}.
+}{
+\spadpaste{putGraph(v2,g1,2) \bound{v22}\free{g1 v2}}
+}
+\psXtc{
+Display the new \axiomType{TwoDimensionalViewport} containing both graphs.
+}{
+\graphpaste{makeViewport2D(v2) \free{v22}}
+}{
+%
+}
+%
+%See Figure #.#.
+%
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphThreeDTitle}{Three-Dimensional Graphics}
+\newcommand{\ugGraphThreeDNumber}{7.2.}
+%
+% =====================================================================
+\begin{page}{ugGraphThreeDPage}{7.2. Three-Dimensional Graphics}
+% =====================================================================
+\beginscroll
+%
+The \Language{} \threedim{} graphics package provides the ability to
+%-% \HDindex{graphics!three-dimensional}{ugGraphThreeDPage}{7.2.}{Three-Dimensional Graphics}
+%
+\indent{4}
+\beginitems
+%
+\item[-] generate surfaces defined by a function of two real variables
+%
+\item[-] generate space curves and tubes defined by parametric equations
+%
+\item[-] generate surfaces defined by parametric equations
+\enditems
+\indent{0}
+These graphs can be modified by using various options, such as calculating
+points in the spherical coordinate system or changing the polygon grid size
+of a surface.
+
+\beginmenu
+    \menudownlink{{7.2.1. Plotting Three-Dimensional Functions of Two Variables}}{ugGraphThreeDPlotPage}
+    \menudownlink{{7.2.2. Plotting Three-Dimensional Parametric Space Curves}}{ugGraphThreeDParmPage}
+    \menudownlink{{7.2.3. Plotting Three-Dimensional Parametric Surfaces}}{ugGraphThreeDParPage}
+    \menudownlink{{7.2.4. Three-Dimensional Options}}{ugGraphThreeDOptionsPage}
+    \menudownlink{{7.2.5. The makeObject Command}}{ugGraphMakeObjectPage}
+    \menudownlink{{7.2.6. Building Three-Dimensional Objects From Primitives}}{ugGraphThreeDBuildPage}
+    \menudownlink{{7.2.7. Coordinate System Transformations}}{ugGraphCoordPage}
+    \menudownlink{{7.2.8. Three-Dimensional Clipping}}{ugGraphClipPage}
+    \menudownlink{{7.2.9. Three-Dimensional Control-Panel}}{ugGraphThreeDControlPage}
+    \menudownlink{{7.2.10. Operations for Three-Dimensional Graphics}}{ugGraphThreeDopsPage}
+    \menudownlink{{7.2.11. Customization using .Xdefaults}}{ugXdefaultsPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphThreeDPlotTitle}{Plotting Three-Dimensional Functions of Two Variables}
+\newcommand{\ugGraphThreeDPlotNumber}{7.2.1.}
+%
+% =====================================================================
+\begin{page}{ugGraphThreeDPlotPage}{7.2.1. Plotting Three-Dimensional Functions of Two Variables}
+% =====================================================================
+\beginscroll
+
+%-% \HDindex{surface!two variable function}{ugGraphThreeDPlotPage}{7.2.1.}{Plotting Three-Dimensional Functions of Two Variables}
+The simplest \threedim{} graph is that of a surface defined by a function
+of two variables, \axiom{z = f(x,y)}.
+
+%
+\beginImportant
+The general format for drawing a surface defined by a formula \axiom{f(x,y)}
+of two variables \axiom{x} and \axiom{y} is:
+%
+\centerline{{{\tt draw(f(x,y), x = a..b, y = c..d, {\it options})}}}
+where \axiom{a..b} and \axiom{c..d} define the range of \axiom{x}
+and \axiom{y}, and where {\it options} prescribes zero or more
+options as described in \downlink{``\ugGraphThreeDOptionsTitle''}{ugGraphThreeDOptionsPage} in Section \ugGraphThreeDOptionsNumber\ignore{ugGraphThreeDOptions}.
+An example of an option is \axiom{title == "Title of Graph".}
+An alternative format involving a function \axiom{f} is also
+available.
+\endImportant
+
+%
+\psXtc{
+The simplest way to plot a function of two variables is to use a formula.
+With formulas you always precede the range specifications with
+the variable name and an \spadSyntax{=} sign.
+}{
+\graphpaste{draw(cos(x*y),x=-3..3,y=-3..3)}
+}{
+\epsffile[0 0 295 295]{../ps/3D2VarA.ps}
+}
+%
+\xtc{
+If you intend to use a function more than once,
+or it is long and complex, then first
+give its definition to \Language{}.
+}{
+\spadpaste{f(x,y) == sin(x)*cos(y) \bound{f}}
+}
+%
+%
+\psXtc{
+To draw the function, just give its name and drop the variables
+from the range specifications.
+\Language{} compiles your function for efficient computation
+of data for the graph.
+Notice that \Language{} uses the text of your function as a
+default title.
+}{
+\graphpaste{draw(f,-\%pi..\%pi,-\%pi..\%pi) \free{f}}
+}{
+\epsffile[0 0 295 295]{../ps/3D2VarB.ps}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphThreeDParmTitle}{Plotting Three-Dimensional Parametric Space Curves}
+\newcommand{\ugGraphThreeDParmNumber}{7.2.2.}
+%
+% =====================================================================
+\begin{page}{ugGraphThreeDParmPage}{7.2.2. Plotting Three-Dimensional Parametric Space Curves}
+% =====================================================================
+\beginscroll
+
+A second kind of \threedim{} graph is a \threedim{} space curve
+%-% \HDindex{curve!parametric space}{ugGraphThreeDParmPage}{7.2.2.}{Plotting Three-Dimensional Parametric Space Curves}
+defined by the parametric equations for \axiom{x(t)}, \axiom{y(t)},
+%-% \HDindex{parametric space curve}{ugGraphThreeDParmPage}{7.2.2.}{Plotting Three-Dimensional Parametric Space Curves}
+and \axiom{z(t)} as a function of an independent variable \axiom{t}.
+
+%
+\beginImportant
+The general format for drawing a \threedim{} space curve defined by
+parametric formulas \axiom{x = f(t)}, \axiom{y = g(t)}, and
+\axiom{z = h(t)} is:
+%
+\centerline{{{\tt draw(curve(f(t),g(t),h(t)), t = a..b, {\it options})}}}
+where \axiom{a..b} defines the range of the independent variable
+\axiom{t}, and where {\it options} prescribes zero or more options
+as described in \downlink{``\ugGraphThreeDOptionsTitle''}{ugGraphThreeDOptionsPage} in Section \ugGraphThreeDOptionsNumber\ignore{ugGraphThreeDOptions}.
+An example of an option is \axiom{title == "Title of Graph".}
+An alternative format involving functions \axiom{f}, \axiom{g} and
+\axiom{h} is also available.
+\endImportant
+
+%
+\psXtc{
+If you use explicit formulas to draw a space curve, always precede
+the range specification with the variable name and an
+\spadSyntax{=} sign.
+}{
+\graphpaste{draw(curve(5*cos(t), 5*sin(t),t), t=-12..12)}
+}{
+\epsffile[0 0 295 295]{../ps/3DpscA.ps}
+}
+%
+\xtc{
+Alternatively, you can draw space curves by referring to functions.
+}{
+\spadpaste{i1(t:DFLOAT):DFLOAT == sin(t)*cos(3*t/5) \bound{i1}}
+}
+\xtc{
+This is useful if the functions are to be used more than once \ldots
+}{
+\spadpaste{i2(t:DFLOAT):DFLOAT == cos(t)*cos(3*t/5) \bound{i2}}
+}
+\xtc{
+or if the functions are long and complex.
+}{
+\spadpaste{i3(t:DFLOAT):DFLOAT == cos(t)*sin(3*t/5) \bound{i3}}
+}
+%
+%
+\psXtc{
+Give the names of the functions and
+drop the variable name specification in the second argument.
+Again, \Language{} supplies a default title.
+}{
+\graphpaste{draw(curve(i1,i2,i3),0..15*\%pi) \free{i1 i2 i3}}
+}{
+\epsffile[0 0 295 295]{../ps/3DpscB.ps}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphThreeDParTitle}{Plotting Three-Dimensional Parametric Surfaces}
+\newcommand{\ugGraphThreeDParNumber}{7.2.3.}
+%
+% =====================================================================
+\begin{page}{ugGraphThreeDParPage}{7.2.3. Plotting Three-Dimensional Parametric Surfaces}
+% =====================================================================
+\beginscroll
+
+%-% \HDindex{surface!parametric}{ugGraphThreeDParPage}{7.2.3.}{Plotting Three-Dimensional Parametric Surfaces}
+A third kind of \threedim{} graph is a surface defined by
+%-% \HDindex{parametric surface}{ugGraphThreeDParPage}{7.2.3.}{Plotting Three-Dimensional Parametric Surfaces}
+parametric equations for \axiom{x(u,v)}, \axiom{y(u,v)}, and
+\axiom{z(u,v)} of two independent variables \axiom{u} and \axiom{v}.
+
+%
+\beginImportant
+The general format for drawing a \threedim{} graph defined by
+parametric formulas \axiom{x = f(u,v)}, \axiom{y = g(u,v)},
+and \axiom{z = h(u,v)} is:
+%
+\centerline{{{\tt draw(surface(f(u,v),g(u,v),h(u,v)), u = a..b, v = c..d, {\it options})}}}
+where \axiom{a..b} and \axiom{c..d} define the range of the
+independent variables \axiom{u} and \axiom{v}, and where
+{\it options} prescribes zero or more options as described in
+\downlink{``\ugGraphThreeDOptionsTitle''}{ugGraphThreeDOptionsPage} in Section \ugGraphThreeDOptionsNumber\ignore{ugGraphThreeDOptions}.
+An example of an option is \axiom{title == "Title of Graph".}
+An alternative format involving functions \axiom{f}, \axiom{g} and
+\axiom{h} is also available.
+\endImportant
+
+%
+\psXtc{
+This example draws a graph of a surface plotted using the
+parabolic cylindrical coordinate system option.
+%-% \HDindex{coordinate system!parabolic cylindrical}{ugGraphThreeDParPage}{7.2.3.}{Plotting Three-Dimensional Parametric Surfaces}
+The values of the functions supplied to \axiomFun{surface} are
+%-% \HDindex{parabolic cylindrical coordinate system}{ugGraphThreeDParPage}{7.2.3.}{Plotting Three-Dimensional Parametric Surfaces}
+interpreted in coordinates as given by a {\tt coordinates} option,
+here as parabolic cylindrical coordinates (see
+\downlink{``\ugGraphCoordTitle''}{ugGraphCoordPage} in Section \ugGraphCoordNumber\ignore{ugGraphCoord}).
+}{
+\graphpaste{draw(surface(u*cos(v), u*sin(v), v*cos(u)), u=-4..4, v=0..\%pi, coordinates== parabolicCylindrical)}
+}{
+\epsffile[0 0 295 295]{../ps/3DpsA.ps}
+}
+%
+Again, you can graph these parametric surfaces using functions,
+if the functions are long and complex.
+\xtc{
+Here we declare the types of arguments and values to be of type
+\axiomType{DoubleFloat}.
+}{
+\spadpaste{n1(u:DFLOAT,v:DFLOAT):DFLOAT == u*cos(v) \bound{n1}}
+}
+\xtc{
+As shown by previous examples, these declarations are necessary.
+}{
+\spadpaste{n2(u:DFLOAT,v:DFLOAT):DFLOAT == u*sin(v) \bound{n2}}
+}
+\xtc{
+In either case, \Language{} compiles the functions
+when needed to graph a result.
+}{
+\spadpaste{n3(u:DFLOAT,v:DFLOAT):DFLOAT == u \bound{n3}}
+}
+\xtc{
+Without these declarations, you have to suffix floats
+with \axiom{@DFLOAT} to get a \axiomType{DoubleFloat} result.
+However, a call here with an unadorned float produces a \axiomType{DoubleFloat}.
+}{
+\spadpaste{n3(0.5,1.0)\free{n3}}
+}
+%
+%
+\psXtc{
+Draw the surface by referencing the function names, this time
+choosing the toroidal coordinate system.
+%-% \HDindex{coordinate system!toroidal}{ugGraphThreeDParPage}{7.2.3.}{Plotting Three-Dimensional Parametric Surfaces}
+%-% \HDindex{toroidal coordinate system}{ugGraphThreeDParPage}{7.2.3.}{Plotting Three-Dimensional Parametric Surfaces}
+}{
+\graphpaste{draw(surface(n1,n2,n3), 1..4, 1..2*\%pi, coordinates == toroidal(1\$DFLOAT)) \free{n1 n2 n3}}
+}{
+\epsffile[0 0 295 295]{../ps/3DpsB.ps}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphThreeDOptionsTitle}{Three-Dimensional Options}
+\newcommand{\ugGraphThreeDOptionsNumber}{7.2.4.}
+%
+% =====================================================================
+\begin{page}{ugGraphThreeDOptionsPage}{7.2.4. Three-Dimensional Options}
+% =====================================================================
+\beginscroll
+
+%-% \HDindex{graphics!3D options}{ugGraphThreeDOptionsPage}{7.2.4.}{Three-Dimensional Options}
+The \axiomFun{draw} commands optionally take an optional list of options such
+as {\tt coordinates} as shown in the last example.
+Each option is given by the syntax: \axiom{name} {\tt ==} \axiom{value}.
+Here is a list of the available options in the order that they are
+described below:
+
+\table{ {title} {coordinates} {var1Steps} {style} {tubeRadius} {var2Steps}
+{colorFunction} {tubePoints} {space}}
+
+\psXtc{
+The option \axiom{title} gives your graph a title.
+%-% \HDindex{graphics!3D options!title}{ugGraphThreeDOptionsPage}{7.2.4.}{Three-Dimensional Options}
+}{
+\graphpaste{draw(cos(x*y),x=0..2*\%pi,y=0..\%pi,title == "Title of Graph") }
+}{
+\epsffile[0 0 295 295]{../ps/3DOptTtl.ps}
+}
+%
+\psXtc{
+The \axiom{style} determines which of four rendering algorithms is used for
+%-% \HDindex{rendering}{ugGraphThreeDOptionsPage}{7.2.4.}{Three-Dimensional Options}
+the graph.
+The choices are
+{\tt "wireMesh"}, {\tt "solid"}, {\tt "shade"}, and {\tt "smooth"}.
+}{
+\graphpaste{draw(cos(x*y),x=-3..3,y=-3..3, style=="smooth", title=="Smooth Option")}
+}{
+\epsffile[0 0 295 295]{../ps/3DOptSty.ps}
+}
+%
+
+In all but the wire-mesh style, polygons in a surface or tube plot
+are normally colored in a graph according to their
+\axiom{z}-coordinate value.  Space curves are colored according to their
+parametric variable value.
+%-% \HDindex{graphics!3D options!color function}{ugGraphThreeDOptionsPage}{7.2.4.}{Three-Dimensional Options}
+To change this, you can give a coloring function.
+%-% \HDindex{function!coloring}{ugGraphThreeDOptionsPage}{7.2.4.}{Three-Dimensional Options}
+The coloring function is sampled across the range of its arguments, then
+normalized onto the standard \Language{} colormap.
+
+\xtc{
+A function of one variable  makes the color depend on the
+value of the parametric variable specified for a tube plot.
+}{
+\spadpaste{color1(t) == t \bound{colorFxn1}}
+}
+\psXtc{
+}{
+\graphpaste{draw(curve(sin(t), cos(t),0), t=0..2*\%pi, tubeRadius == .3, colorFunction == color1) \free{colorFxn1}}
+}{
+\epsffile[0 0 295 295]{../ps/3DOptCf1.ps}
+}
+%
+\xtc{
+A function of two variables makes the color depend on the
+values of the independent variables.
+}{
+\spadpaste{color2(u,v) == u**2 - v**2 \bound{colorFxn2}}
+}
+\psXtc{
+Use the option {\tt colorFunction} for special coloring.
+}{
+\graphpaste{draw(cos(u*v), u=-3..3, v=-3..3, colorFunction == color2) \free{colorFxn2}}
+}{
+\epsffile[0 0 295 295]{../ps/3DOptCf2.ps}
+}
+%
+\xtc{
+With a three variable function, the
+color also depends on the value of the function.
+}{
+\spadpaste{color3(x,y,fxy) == sin(x*fxy) + cos(y*fxy) \bound{colorFxn3}}
+}
+\psXtc{
+}{
+\graphpaste{draw(cos(x*y), x=-3..3, y=-3..3, colorFunction == color3) \free{colorFxn3}}
+}{
+\epsffile[0 0 295 295]{../ps/3DOptCf3.ps}
+}
+%
+Normally the Cartesian coordinate system is used.
+%-% \HDindex{Cartesian!coordinate system}{ugGraphThreeDOptionsPage}{7.2.4.}{Three-Dimensional Options}
+To change this, use the {\tt coordinates} option.
+%-% \HDindex{coordinate system!Cartesian}{ugGraphThreeDOptionsPage}{7.2.4.}{Three-Dimensional Options}
+For details, see \downlink{``\ugGraphCoordTitle''}{ugGraphCoordPage} in Section \ugGraphCoordNumber\ignore{ugGraphCoord}.
+%
+%
+\xtc{
+}{
+\spadpaste{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 \bound{m}}
+}
+\psXtc{
+Use the spherical
+%-% \HDindex{spherical coordinate system}{ugGraphThreeDOptionsPage}{7.2.4.}{Three-Dimensional Options}
+coordinate system.
+%-% \HDindex{coordinate system!spherical}{ugGraphThreeDOptionsPage}{7.2.4.}{Three-Dimensional Options}
+}{
+\graphpaste{draw(m, 0..2*\%pi,0..\%pi, coordinates == spherical, style=="shade") \free{m}}
+}{
+\epsffile[0 0 295 295]{../ps/3DOptCrd.ps}
+}
+%
+Space curves may be displayed as tubes with polygonal cross sections.
+%-% \HDindex{tube}{ugGraphThreeDOptionsPage}{7.2.4.}{Three-Dimensional Options}
+Two options, {\tt tubeRadius} and {\tt tubePoints},  control the size and
+shape of this cross section.
+%
+\psXtc{
+The {\tt tubeRadius} option specifies the radius of the tube that
+%-% \HDindex{tube!radius}{ugGraphThreeDOptionsPage}{7.2.4.}{Three-Dimensional Options}
+encircles the specified space curve.
+}{
+\graphpaste{draw(curve(sin(t),cos(t),0),t=0..2*\%pi, style=="shade", tubeRadius == .3)}
+}{
+\epsffile[0 0 295 295]{../ps/3DOptRad.ps}
+}
+%
+%
+\psXtc{
+The {\tt tubePoints} option specifies the number of vertices
+%-% \HDindex{tube!points in polygon}{ugGraphThreeDOptionsPage}{7.2.4.}{Three-Dimensional Options}
+defining the polygon that is used to create a tube around the
+specified space curve.
+The larger this number is, the more cylindrical the tube becomes.
+}{
+\graphpaste{draw(curve(sin(t), cos(t), 0), t=0..2*\%pi, style=="shade", tubeRadius == .25, tubePoints == 3)}
+}{
+\epsffile[0 0 295 295]{../ps/3DOptPts.ps}
+}
+%
+%-% \HDindex{graphics!3D options!variable steps}{ugGraphThreeDOptionsPage}{7.2.4.}{Three-Dimensional Options}
+%
+%
+\psXtc{
+Options \axiomFunFrom{var1Steps}{DrawOption} and
+\axiomFunFrom{var2Steps}{DrawOption} specify the number of intervals into
+which the grid defining a surface plot is subdivided with respect to the
+first and second parameters of the surface function(s).
+}{
+\graphpaste{draw(cos(x*y),x=-3..3,y=-3..3, style=="shade", var1Steps == 30, var2Steps == 30)}
+}{
+\epsffile[0 0 295 295]{../ps/3DOptvB.ps}
+}
+%
+The {\tt space} option
+of a \axiomFun{draw} command lets you build multiple graphs in three space.
+To use this option, first create an empty three-space object,
+then use the {\tt space} option thereafter.
+There is no restriction as to the number or kinds
+of graphs that can be combined this way.
+\xtc{
+Create an empty three-space object.
+}{
+\spadpaste{s := create3Space()\$(ThreeSpace DFLOAT) \bound{s}}
+}
+%
+%
+\xtc{
+}{
+\spadpaste{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 \bound{m}}
+}
+\psXtc{
+Add a graph to this three-space object.
+The new graph destructively inserts the graph
+into \axiom{s}.
+}{
+\graphpaste{draw(m,0..\%pi,0..2*\%pi, coordinates == spherical, space == s) \free{s m}}
+}{
+\epsffile[0 0 295 295]{../ps/3Dmult1A.ps}
+}
+%
+%
+\psXtc{
+Add a second graph to \axiom{s}.
+}{
+\graphpaste{v := draw(curve(1.5*sin(t), 1.5*cos(t),0), t=0..2*\%pi, tubeRadius == .25, space == s) \free{s} \bound{v}}
+}{
+\epsffile[0 0 295 295]{../ps/3Dmult1B.ps}
+}
+%
+A three-space object can also be obtained from an existing \threedim{} viewport
+using the \axiomFunFrom{subspace}{ThreeSpace} command.
+You can then use \axiomFun{makeViewport3D} to create a viewport window.
+\xtc{
+Assign to \axiom{subsp} the three-space object in viewport \axiom{v}.
+}{
+\spadpaste{subsp := subspace v \free{v} \bound{su}}
+}
+\xtc{
+Reset the space component of \axiom{v} to the value of \axiom{subsp}.
+}{
+\spadpaste{subspace(v, subsp) \bound{sp} \free{su}}
+}
+\noOutputXtc{
+Create a viewport window from a three-space object.
+}{
+\graphpaste{makeViewport3D(subsp,"Graphs") \free{sp}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphMakeObjectTitle}{The makeObject Command}
+\newcommand{\ugGraphMakeObjectNumber}{7.2.5.}
+%
+% =====================================================================
+\begin{page}{ugGraphMakeObjectPage}{7.2.5. The makeObject Command}
+% =====================================================================
+\beginscroll
+
+An alternate way to create multiple graphs is to use
+\axiomFun{makeObject}.
+The \axiomFun{makeObject} command is similar to the \axiomFun{draw}
+command, except that it returns a three-space object rather than a
+\axiomType{ThreeDimensionalViewport}.
+In fact, \axiomFun{makeObject} is called by the \axiomFun{draw}
+command to create the \axiomType{ThreeSpace} then
+\axiomFunFrom{makeViewport3D}{ThreeDimensionalViewport} to create a
+viewport window.
+
+\xtc{
+}{
+\spadpaste{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 \bound{m}}
+}
+\noOutputXtc{
+Do the last example a new way.
+First use \axiomFun{makeObject} to
+create a three-space object \axiom{sph}.
+}{
+\spadpaste{sph := makeObject(m, 0..\%pi, 0..2*\%pi, coordinates==spherical)\bound{sph}\free{m}}
+}
+\noOutputXtc{
+Add a second object to \axiom{sph}.
+}{
+\spadpaste{makeObject(curve(1.5*sin(t), 1.5*cos(t), 0), t=0..2*\%pi, space == sph, tubeRadius == .25) \free{sph}\bound{v1}}
+}
+\noOutputXtc{
+Create and display a viewport
+containing \axiom{sph}.
+}{
+\graphpaste{makeViewport3D(sph,"Multiple Objects") \free{v1}}
+}
+
+Note that an undefined \axiomType{ThreeSpace} parameter declared in a
+\axiomFun{makeObject} or \axiomFun{draw} command results in an error.
+Use the \axiomFunFrom{create3Space}{ThreeSpace} function to define a
+\axiomType{ThreeSpace}, or obtain a \axiomType{ThreeSpace} that has been
+previously generated before including it in a command line.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphThreeDBuildTitle}{Building Three-Dimensional Objects From Primitives}
+\newcommand{\ugGraphThreeDBuildNumber}{7.2.6.}
+%
+% =====================================================================
+\begin{page}{ugGraphThreeDBuildPage}{7.2.6. Building Three-Dimensional Objects From Primitives}
+% =====================================================================
+\beginscroll
+
+Rather than using the \axiomFun{draw} and \axiomFun{makeObject} commands,
+%-% \HDindex{graphics!advanced!build 3D objects}{ugGraphThreeDBuildPage}{7.2.6.}{Building Three-Dimensional Objects From Primitives}
+you can create \threedim{} graphs from primitives.
+Operation \axiomFunFrom{create3Space}{ThreeSpace} creates a
+three-space object to which points, curves and polygons
+can be added using the operations from the \axiomType{ThreeSpace}
+domain.
+The resulting object can then be displayed in a viewport using
+\axiomFunFrom{makeViewport3D}{ThreeDimensionalViewport}.
+
+\xtc{
+Create the empty three-space object \axiom{space}.
+}{
+\spadpaste{space := create3Space()\$(ThreeSpace DFLOAT) \bound{space}}
+}
+
+Objects can be sent to this \axiom{space} using the operations
+exported by the \axiomType{ThreeSpace} domain.
+%-% \HDexptypeindex{ThreeSpace}{ugGraphThreeDBuildPage}{7.2.6.}{Building Three-Dimensional Objects From Primitives}
+The following examples place curves into \axiom{space}.
+
+\xtc{
+Add these eight curves to the space.
+}{
+\spadpaste{closedCurve(space,[[0,30,20], [0,30,30], [0,40,30], [0,40,100], [0,30,100],[0,30,110], [0,60,110], [0,60,100], [0,50,100], [0,50,30], [0,60,30], [0,60,20]]) \bound{curve1} \free{space}}
+}
+\xtc{
+}{
+\spadpaste{closedCurve(space,[[80,0,30], [80,0,100], [70,0,110], [40,0,110], [30,0,100], [30,0,90], [40,0,90], [40,0,95], [45,0,100], [65,0,100], [70,0,95], [70,0,35]]) \bound{curve2} \free{space}}
+}
+\xtc{
+}{
+\spadpaste{closedCurve(space,[[70,0,35], [65,0,30], [45,0,30], [40,0,35], [40,0,60], [50,0,60], [50,0,70], [30,0,70], [30,0,30], [40,0,20], [70,0,20], [80,0,30]]) \bound{curve3} \free{space}}
+}
+\xtc{
+}{
+\spadpaste{closedCurve(space,[[0,70,20], [0,70,110], [0,110,110], [0,120,100], [0,120,70], [0,115,65], [0,120,60], [0,120,30], [0,110,20], [0,80,20], [0,80,30], [0,80,20]]) \bound{curve4} \free{space}}
+}
+\xtc{
+}{
+\spadpaste{closedCurve(space,[[0,105,30], [0,110,35], [0,110,55], [0,105,60], [0,80,60], [0,80,70], [0,105,70], [0,110,75], [0,110,95], [0,105,100], [0,80,100], [0,80,20], [0,80,30]]) \bound{curve5} \free{space}}
+}
+\xtc{
+}{
+\spadpaste{closedCurve(space,[[140,0,20], [140,0,110], [130,0,110], [90,0,20], [101,0,20],[114,0,50], [130,0,50], [130,0,60], [119,0,60], [130,0,85], [130,0,20]]) \bound{curve6} \free{space}}
+}
+\xtc{
+}{
+\spadpaste{closedCurve(space,[[0,140,20], [0,140,110], [0,150,110], [0,170,50], [0,190,110], [0,200,110], [0,200,20], [0,190,20], [0,190,75], [0,175,35], [0,165,35],[0,150,75], [0,150,20]]) \bound{curve7} \free{space}}
+}
+\xtc{
+}{
+\spadpaste{closedCurve(space,[[200,0,20], [200,0,110], [189,0,110], [160,0,45], [160,0,110], [150,0,110], [150,0,20], [161,0,20], [190,0,85], [190,0,20]]) \bound{curve8} \free{space}}
+}
+\psXtc{
+Create and display the viewport using \axiomFun{makeViewport3D}.
+Options may also be given but here are displayed as a list with values
+enclosed in parentheses.
+}{
+\graphpaste{makeViewport3D(space, title == "Letters") \free{space curve1 curve2 curve3 curve4 curve5 curve6 curve7 curve8}}
+}{
+\epsffile[0 0 295 295]{../ps/3DBuildA.ps}
+}
+
+\subsubsection{Cube Example}
+
+As a second example of the use of primitives, we generate a cube using a
+polygon mesh.
+It is important to use a consistent orientation of the polygons for
+correct generation of \threedim{} objects.
+
+\xtc{
+Again start with an empty three-space object.
+}{
+\spadpaste{spaceC := create3Space()\$(ThreeSpace DFLOAT) \bound{spaceC}}
+}
+\xtc{
+For convenience,
+give \axiomType{DoubleFloat} values \axiom{+1} and \axiom{-1} names.
+}{
+\spadpaste{x: DFLOAT := 1 \bound{x}}
+}
+\xtc{
+}{
+\spadpaste{y: DFLOAT := -1 \bound{y}}
+}
+\xtc{
+Define the vertices of the cube.
+}{
+\spadpaste{a := point [x,x,y,1::DFLOAT]\$(Point DFLOAT) \bound{a} \free{x y}}
+}
+\xtc{
+}{
+\spadpaste{b := point [y,x,y,4::DFLOAT]\$(Point DFLOAT) \bound{b} \free{x y}}
+}
+\xtc{
+}{
+\spadpaste{c := point [y,x,x,8::DFLOAT]\$(Point DFLOAT) \bound{c} \free{x y}}
+}
+\xtc{
+}{
+\spadpaste{d := point [x,x,x,12::DFLOAT]\$(Point DFLOAT) \bound{d} \free{x y}}
+}
+\xtc{
+}{
+\spadpaste{e := point [x,y,y,16::DFLOAT]\$(Point DFLOAT) \bound{e} \free{x y}}
+}
+\xtc{
+}{
+\spadpaste{f := point [y,y,y,20::DFLOAT]\$(Point DFLOAT) \bound{f} \free{x y}}
+}
+\xtc{
+}{
+\spadpaste{g := point [y,y,x,24::DFLOAT]\$(Point DFLOAT) \bound{g} \free{x y}}
+}
+\xtc{
+}{
+\spadpaste{h := point [x,y,x,27::DFLOAT]\$(Point DFLOAT) \bound{h} \free{x y}}
+}
+\xtc{
+Add the faces of the cube as polygons to the space using a
+consistent orientation.
+}{
+\spadpaste{polygon(spaceC,[d,c,g,h]) \free{d c g h spaceC} \bound{pol1}}
+}
+\xtc{
+}{
+\spadpaste{polygon(spaceC,[d,h,e,a]) \free{d h e a spaceC} \bound{pol2}}
+}
+\xtc{
+}{
+\spadpaste{polygon(spaceC,[c,d,a,b]) \free{c d a b spaceC} \bound{pol3}}
+}
+\xtc{
+}{
+\spadpaste{polygon(spaceC,[g,c,b,f]) \free{g c b f spaceC} \bound{pol4}}
+}
+\xtc{
+}{
+\spadpaste{polygon(spaceC,[h,g,f,e]) \free{h g f e spaceC} \bound{pol5}}
+}
+\xtc{
+}{
+\spadpaste{polygon(spaceC,[e,f,b,a]) \free{e f b a spaceC} \bound{pol6}}
+}
+\psXtc{
+Create and display the viewport.
+}{
+\graphpaste{makeViewport3D(spaceC, title == "Cube") \free{pol1 pol2 pol3 pol4 pol5 pol6}}
+}{
+\epsffile[0 0 295 295]{../ps/3DBuildB.ps}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphCoordTitle}{Coordinate System Transformations}
+\newcommand{\ugGraphCoordNumber}{7.2.7.}
+%
+% =====================================================================
+\begin{page}{ugGraphCoordPage}{7.2.7. Coordinate System Transformations}
+% =====================================================================
+\beginscroll
+%-% \HDindex{graphics!advanced!coordinate systems}{ugGraphCoordPage}{7.2.7.}{Coordinate System Transformations}
+
+The \axiomType{CoordinateSystems} package provides coordinate transformation
+functions that map a given data point from the coordinate system specified
+into the Cartesian coordinate system.
+%-% \HDexptypeindex{CoordinateSystems}{ugGraphCoordPage}{7.2.7.}{Coordinate System Transformations}
+The default coordinate system, given a triplet \axiom{(f(u,v), u, v)}, assumes
+that \axiom{z = f(u, v)}, \axiom{x = u} and \axiom{y = v},
+that is, reads the coordinates in \axiom{(z, x, y)} order.
+
+\xtc{
+}{
+\spadpaste{m(u:DFLOAT,v:DFLOAT):DFLOAT == u**2 \bound{m}}
+}
+%
+\psXtc{
+Graph plotted in default coordinate system.
+}{
+\graphpaste{draw(m,0..3,0..5) \free{m}}
+}{
+\epsffile[0 0 295 295]{../ps/defcoord.ps}
+}
+
+The \axiom{z} coordinate comes first since the first argument of
+the \axiomFun{draw} command gives its values.
+In general, the coordinate systems \Language{} provides, or any
+that you make up, must provide a map to an \axiom{(x, y, z)} triplet in
+order to be compatible with the
+\axiomFunFrom{coordinates}{DrawOption} \axiomType{DrawOption}.
+%-% \HDexptypeindex{DrawOption}{ugGraphCoordPage}{7.2.7.}{Coordinate System Transformations}
+Here is an example.
+
+\xtc{
+Define the identity function.
+}{
+\spadpaste{cartesian(point:Point DFLOAT):Point DFLOAT == point \bound{cart}}
+}
+\psXtc{
+Pass \axiom{cartesian} as the \axiomFunFrom{coordinates}{DrawOption}
+parameter to the \axiomFun{draw} command.
+}{
+\graphpaste{draw(m,0..3,0..5,coordinates==cartesian) \free{m cart}}
+}{
+\epsffile[0 0 295 295]{../ps/cartcoord.ps}
+}
+%
+
+What happened?
+The option {\tt coordinates == cartesian} directs \Language{} to
+treat the dependent variable \axiom{m} defined by
+\texht{$m=u^2$}{m=u**2} as the \axiom{x} coordinate.
+Thus the triplet of values \axiom{(m, u, v)} is transformed to
+coordinates \axiom{(x, y, z)} and so we get the graph of
+\texht{$x=y^2$}{x=y**2}.
+
+Here is another example.
+The \axiomFunFrom{cylindrical}{CoordinateSystems} transform takes
+%-% \HDindex{coordinate system!cylindrical}{ugGraphCoordPage}{7.2.7.}{Coordinate System Transformations}
+input of the form \axiom{(w,u,v)}, interprets it in the order
+%-% \HDindex{cylindrical coordinate system}{ugGraphCoordPage}{7.2.7.}{Coordinate System Transformations}
+\texht{($r$,$\theta$,$z$)}{(\axiom{r}, \axiom{theta}, \axiom{z})}
+and maps it to the Cartesian coordinates
+\texht{$x=r\cos(\theta)$, $y=r\sin(\theta)$, $z=z$}
+{\axiom{x = r * cos(theta)}, \axiom{y = r * sin(theta)}, \axiom{z = z}}
+in which
+\texht{$r$}{\axiom{r}} is the radius,
+\texht{$\theta$}{\axiom{theta}} is the angle and
+\texht{$z$}{\axiom{z}} is the z-coordinate.
+\xtc{
+An example using the \axiomFunFrom{cylindrical}{CoordinateSystems}
+coordinates for the constant \axiom{r = 3}.
+}{
+\spadpaste{f(u:DFLOAT,v:DFLOAT):DFLOAT == 3 \bound{f}}
+}
+\psXtc{
+Graph plotted in cylindrical coordinates.
+}{
+\graphpaste{draw(f,0..\%pi,0..6,coordinates==cylindrical) \free{f}}
+}{
+\epsffile[0 0 295 295]{../ps/cylCoord.ps}
+}
+
+Suppose you would like to specify \smath{z} as a function of
+\smath{r} and \texht{$\theta$}{\axiom{theta}} instead of just
+\smath{r}?
+Well, you still can use the \axiomFun{cylindrical} \Language{}
+transformation but we have to reorder the triplet before
+passing it to the transformation.
+
+\xtc{
+First, let's create a point to
+work with and call it \axiom{pt} with some color \axiom{col}.
+}{
+\spadpaste{col := 5 \bound{c}}
+}
+\xtc{
+}{
+\spadpaste{pt := point[1,2,3,col]\$(Point DFLOAT) \free{c} \bound{pt}}
+}
+The reordering you want is
+\texht{$(z,r, \theta)$}{\axiom{(z,r,theta)}} to
+\texht{$(r, \theta,z)$}{\axiom{(r,theta,z)}}
+so that the first element is moved to the third element, while the second
+and third elements move forward and the color element does not change.
+\xtc{
+Define a function \userfun{reorder} to reorder the point elements.
+}{
+\spadpaste{reorder(p:Point DFLOAT):Point DFLOAT == point[p.2, p.3, p.1, p.4] \bound{freo}}
+}
+\xtc{
+The function moves the second and third elements
+forward but the color does not change.
+}{
+\spadpaste{reorder pt \free{pt freo}}
+}
+\xtc{
+The function \userfun{newmap} converts our reordered version of
+the cylindrical coordinate system to the standard
+\texht{$(x,y,z)$}{\axiom{(x,y,z)}} Cartesian system.
+}{
+\spadpaste{newmap(pt:Point DFLOAT):Point DFLOAT == cylindrical(reorder pt) \free{freo} \bound{fnewmap}}
+}
+\xtc{
+}{
+\spadpaste{newmap pt \free{fnewmap pt} \bound{new}}
+}
+%
+\psXtc{
+Graph the same function \axiom{f} using the coordinate mapping of the function
+\axiom{newmap}, so it is now interpreted as
+\texht{$z=3$}{\axiom{z = 3}}:
+}{
+\graphpaste{draw(f,0..3,0..2*\%pi,coordinates==newmap) \free{f new}}
+}{
+\epsffile[0 0 295 295]{../ps/newmap.ps}
+}
+
+{\texht{\sloppy}{}
+The \axiomType{CoordinateSystems} package exports the following
+%-% \HDindex{coordinate system}{ugGraphCoordPage}{7.2.7.}{Coordinate System Transformations}
+operations:
+\axiomFun{bipolar},
+\axiomFun{bipolarCylindrical},
+\axiomFun{cartesian},
+\axiomFun{conical},
+\axiomFun{cylindrical},
+\axiomFun{elliptic},
+\axiomFun{ellipticCylindrical},
+\axiomFun{oblateSpheroidal},
+\axiomFun{parabolic},
+\axiomFun{parabolicCylindrical},
+\axiomFun{paraboloidal},
+\axiomFun{polar},
+\axiomFun{prolateSpheroidal},
+\axiomFun{spherical}, and
+\axiomFun{toroidal}.
+Use \Browse{} or the \spadcmd{)show} system command
+%-% \HDsyscmdindex{show}{ugGraphCoordPage}{7.2.7.}{Coordinate System Transformations}
+to get more information.
+
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphClipTitle}{Three-Dimensional Clipping}
+\newcommand{\ugGraphClipNumber}{7.2.8.}
+%
+% =====================================================================
+\begin{page}{ugGraphClipPage}{7.2.8. Three-Dimensional Clipping}
+% =====================================================================
+\beginscroll
+
+A \threedim{} graph can be explicitly clipped within the \axiomFun{draw}
+%-% \HDindex{graphics!advanced!clip}{ugGraphClipPage}{7.2.8.}{Three-Dimensional Clipping}
+command by indicating a minimum and maximum threshold for the
+%-% \HDindex{clipping}{ugGraphClipPage}{7.2.8.}{Three-Dimensional Clipping}
+given function definition.
+These thresholds can be defined using the \Language{} \axiomFun{min}
+and \axiomFun{max} functions.
+\xtc{
+}{
+\begin{spadsrc}[\bound{g}]
+gamma(x,y) ==
+  g := Gamma complex(x,y)
+  point [x, y, max( min(real g, 4), -4), argument g]
+\end{spadsrc}
+}
+\psXtc{
+Here is an example that clips
+the gamma function in order to eliminate the extreme divergence it creates.
+}{
+\graphpaste{draw(gamma,-\%pi..\%pi,-\%pi..\%pi,var1Steps==50,var2Steps==50) \free{g}}
+}{
+\epsffile[0 0 295 295]{../ps/clipGamma.ps}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphThreeDControlTitle}{Three-Dimensional Control-Panel}
+\newcommand{\ugGraphThreeDControlNumber}{7.2.9.}
+%
+% =====================================================================
+\begin{page}{ugGraphThreeDControlPage}{7.2.9. Three-Dimensional Control-Panel}
+% =====================================================================
+\beginscroll
+%-% \HDindex{graphics!3D control-panel}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+Once you have created a viewport, move your mouse to the viewport
+and click with your left mouse button.
+This displays a control-panel on the side of the viewport
+that is closest to where you clicked.
+
+
+\subsubsection{Transformations}
+
+We recommend you first select the {\bf Bounds} button while
+%-% \HDindex{graphics!3D control-panel!transformations}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+executing transformations since the bounding box displayed
+indicates the object's position as it changes.
+%
+\indent{0}
+\beginitems
+%
+\item[Rotate:]  A rotation transformation occurs by clicking the mouse
+%-% \HDindex{graphics!3D control-panel!rotate}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+within the {\bf Rotate} window in the upper left corner of the
+control-panel.
+The rotation is computed in spherical coordinates, using the
+horizontal mouse position to increment or decrement the value of
+the longitudinal angle \texht{$\theta$}{\axiom{theta}} within the
+range of 0 to \texht{2$\pi$}{2*pi} and the vertical mouse position
+to increment or decrement the value of the latitudinal angle
+\texht{$\phi$}{\axiom{phi}} within the range of \texht{-$\pi$}{pi}
+to \texht{$\pi$}{pi}.
+The active mode of rotation is displayed in green on a color
+monitor or in clear text on a black and white monitor, while the
+inactive mode is displayed in red for color display or a mottled
+pattern for black and white.
+%
+\indent{0}
+\beginitems
+%
+\item[origin:]  The {\bf origin} button indicates that the
+rotation is to occur with respect to the origin of the viewing space, that is
+indicated by the axes.
+%
+\item[object:]  The {\bf object} button indicates that the
+rotation is to occur with respect to the center of volume of the object,
+independent of the axes' origin position.
+\enditems
+\indent{0}
+%
+\item[Scale:]  A scaling transformation occurs by clicking the mouse
+%-% \HDindex{graphics!3D control-panel!scale}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+within the {\bf Scale} window in the upper center of the
+control-panel, containing a zoom arrow.
+The axes along which the scaling is to occur are indicated by
+selecting the appropriate button above the zoom arrow window.
+The selected axes are displayed in green on a color monitor or in
+clear text on a black and white monitor, while the unselected axes
+are displayed in red for a color display or a mottled pattern for
+black and white.
+%
+\indent{0}
+\beginitems
+%
+\item[uniform:]  Uniform scaling along the {\tt x}, {\tt y}
+and {\tt z} axes occurs when all the axes buttons are selected.
+%
+\item[non-uniform:]  If any of the axes buttons are
+not selected, non-uniform scaling occurs, that is, scaling occurs only in the
+direction of the axes that are selected.
+\enditems
+\indent{0}
+%
+\item[Translate:]  Translation occurs by indicating with the mouse in the
+%-% \HDindex{graphics!3D control-panel!translate}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+{\bf Translate} window the direction you want the graph to move.
+This window is located in the upper right corner of the
+control-panel and contains a potentiometer with crossed arrows
+pointing up, down, left and right.
+Along the top of the {\bf Translate} window are three buttons
+({\bf XY},
+{\bf XZ}, and {\bf YZ}) indicating the three orthographic projection planes.
+Each orientates the group as a view into that plane.
+Any translation of the graph occurs only along this plane.
+\enditems
+\indent{0}
+
+\subsubsection{Messages}
+%-% \HDindex{graphics!3D control-panel!messages}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+
+The window directly below the potentiometer windows for transformations is
+used to display system messages relating to the viewport, the control-panel
+and the current graph displaying status.
+
+\subsubsection{Colormap}
+%-% \HDindex{graphics!3D control-panel!color map}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+
+Directly below the message window is the colormap range indicator
+window.
+%-% \HDindex{colormap}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+The \Language{} Colormap shows a sampling of the spectrum from
+which hues can be drawn to represent the colors of a surface.
+The Colormap is composed of five shades for each of the hues along
+this spectrum.
+By moving the markers above and below the Colormap, the range of
+hues that are used to color the existing surface are set.
+The bottom marker shows the hue for the low end of the color range
+and the top marker shows the hue for the upper end of the range.
+Setting the bottom and top markers at the same hue results in
+monochromatic smooth shading of the graph when {\bf Smooth} mode is selected.
+At each end of the Colormap are {\bf +} and {\bf -} buttons.
+When clicked on, these increment or decrement the top or bottom
+marker.
+
+\subsubsection{Buttons}
+%-% \HDindex{graphics!3D control-panel!buttons}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+
+Below the Colormap window and to the left are located various
+buttons that determine the characteristics of a graph.
+The buttons along the bottom and right hand side all have special
+meanings; the remaining buttons in the first row indicate the mode
+or style used to display the graph.
+The second row are toggles that turn on or off a property of the
+graph.
+On a color monitor, the property is on if green (clear text, on a
+monochrome monitor) and off if red (mottled pattern, on a
+monochrome monitor).
+Here is a list of their functions.
+%
+\indent{0}
+\beginitems
+%
+\item[Wire] displays surface and tube plots as a
+%-% \HDindex{graphics!3D control-panel!wire}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+wireframe image in a single color (blue) with no hidden surfaces removed,
+or displays space curve plots in colors based upon their parametric variables.
+This is the fastest mode for displaying a graph.
+This is very useful when you
+want to find a good orientation of your graph.
+%
+\item[Solid] displays the graph with hidden
+%-% \HDindex{graphics!3D control-panel!solid}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+surfaces removed, drawing each polygon beginning with the furthest
+from the viewer.
+The edges of the polygons are displayed in the hues specified by
+the range in the Colormap window.
+%
+\item[Shade] displays the graph with hidden
+%-% \HDindex{graphics!3D control-panel!shade}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+surfaces removed and with the polygons shaded, drawing each
+polygon beginning with the furthest from the viewer.
+Polygons are shaded in the hues specified by the range in the
+Colormap window using the Phong illumination model.
+%-% \HDindex{Phong!illumination model}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+%
+\item[Smooth] displays the graph using a
+%-% \HDindex{graphics!3D control-panel!smooth}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+renderer that computes the graph one line at a time.
+The location and color of the graph at each visible point on the
+screen are determined and displayed using the Phong illumination
+%-% \HDindex{Phong!illumination model}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+model.
+Smooth shading is done in one of two ways, depending on the range
+selected in the colormap window and the number of colors available
+from the hardware and/or window manager.
+When the top and bottom markers of the colormap range are set to
+different hues, the graph is rendered by dithering between the
+%-% \HDindex{dithering}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+transitions in color hue.
+When the top and bottom markers of the colormap range are set to
+the same hue, the graph is rendered using the Phong smooth shading
+model.
+%-% \HDindex{Phong!smooth shading model}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+However, if enough colors cannot be allocated for this purpose,
+the renderer reverts to the color dithering method until a
+sufficient color supply is available.
+For this reason, it may not be possible to render multiple Phong
+smooth shaded graphs at the same time on some systems.
+%
+\item[Bounds] encloses the entire volume of the
+viewgraph within a bounding box, or removes the box if previously selected.
+%-% \HDindex{graphics!3D control-panel!bounds}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+The region that encloses the entire volume of the viewport graph is displayed.
+%
+\item[Axes] displays Cartesian
+%-% \HDindex{graphics!3D control-panel!axes}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+coordinate axes of the space, or turns them off if previously selected.
+%
+\item[Outline] causes
+%-% \HDindex{graphics!3D control-panel!outline}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+quadrilateral polygons forming the graph surface to be outlined in black when
+the graph is displayed in {\bf Shade} mode.
+%
+\item[BW] converts a color viewport to black and white, or vice-versa.
+%-% \HDindex{graphics!3D control-panel!bw}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+When this button is selected the
+control-panel and viewport switch to an immutable colormap composed of a range
+of grey scale patterns or tiles that are used wherever shading is necessary.
+%
+\item[Light] takes you to a control-panel described below.
+%
+\item[ViewVolume] takes you to another control-panel as described below.
+%-% \HDindex{graphics!3D control-panel!save}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+%
+\item[Save] creates a menu of the possible file types that can
+be written using the control-panel.
+The {\bf Exit} button leaves the save menu.
+The {\bf Pixmap} button writes an \Language{} pixmap of
+%-% \HDindex{graphics!3D control-panel!pixmap}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+the current viewport contents.  The file is called {\bf axiom3D.pixmap} and is
+located in the directory from which \Language{} or {\bf viewAlone} was
+started.
+The {\bf PS} button writes the current viewport contents to
+%-% \HDindex{graphics!3D control-panel!ps}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+PostScript output rather than to the viewport window.
+By default the file is called {\bf axiom3D.ps}; however, if a file
+%-% \HDindex{file!.Xdefaults @{\bf .Xdefaults}}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+name is specified in the user's {\bf .Xdefaults} file it is
+%-% \HDindex{graphics!.Xdefaults!PostScript file name}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+used.
+The file is placed in the directory from which the \Language{} or
+{\bf viewAlone} session was begun.
+See also the \axiomFunFrom{write}{ThreeDimensionalViewport}
+function.
+%-% \HDindex{PostScript}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+%
+\item[Reset] returns the object transformation
+%-% \HDindex{graphics!3D control-panel!reset}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+characteristics back to their initial states.
+%
+\item[Hide] causes the control-panel for the
+%-% \HDindex{graphics!3D control-panel!hide}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+corresponding viewport to disappear from the screen.
+%
+\item[Quit]  queries whether the current viewport
+%-% \HDindex{graphics!3D control-panel!quit}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+session should be terminated.
+\enditems
+\indent{0}
+
+\subsubsection{Light}
+%-% \HDindex{graphics!3D control-panel!light}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+
+%
+%>>>\begin{texonly}
+%
+%>>>\begin{figure}[htbp]
+%>>>\begin{picture}(183,252)(-125,0)
+%>>>\special{psfile=../ps/3Dlight.ps}
+%>>>\end{picture}
+%>>>\caption{Three-Dimensional Lighting Panel.}
+%>>>\end{figure}
+%>>>\end{texonly}
+%
+
+The {\bf Light} button changes the control-panel into the
+{\bf Lighting Control-Panel}.  At the top of this panel, the three axes
+are shown with the same orientation as the object.  A light vector from
+the origin of the axes shows the current position of the light source
+relative to the object.  At the bottom of the panel is an {\bf Abort}
+button that cancels any changes to the lighting that were made, and a
+{\bf Return} button that carries out the current set of lighting changes
+on the graph.
+%
+\indent{0}
+\beginitems
+%
+\item[XY:]  The {\bf XY} lighting axes window is below the
+%-% \HDindex{graphics!3D control-panel!move xy}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+{\bf Lighting Control-Panel} title and to the left.
+This changes the light vector within the {\bf XY} view plane.
+%
+\item[Z:]  The {\bf Z} lighting axis window is below the
+%-% \HDindex{graphics!3D control-panel!move z}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+{\bf Lighting Control-Panel} title and in the center.  This
+changes the {\bf Z}
+location of the light vector.
+%
+\item[Intensity:]
+Below the {\bf Lighting Control-Panel} title
+%-% \HDindex{graphics!3D control-panel!intensity}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+and to the right is the light intensity meter.
+Moving the intensity indicator down decreases the amount of
+light emitted from the light source.
+When the indicator is at the top of the meter the light source is
+emitting at 100\% intensity.
+At the bottom of the meter the light source is emitting at a level
+slightly above ambient lighting.
+\enditems
+\indent{0}
+
+\subsubsection{View Volume}
+%-% \HDindex{graphics!3D control-panel!view volume}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+
+The {\bf View Volume} button changes the control-panel into
+the {\bf Viewing Volume Panel}.
+At the bottom of the viewing panel is an {\bf Abort} button that
+cancels any changes to the viewing volume that were made and a
+{\it Return} button that carries out the current set of
+viewing changes to the graph.
+%
+%>>>\begin{texonly}
+%
+%>>>\begin{figure}[htbp]
+%>>>\begin{picture}(183,252)(-125,0)
+%>>>\special{psfile=../ps/3Dvolume.ps}
+%>>>\end{picture}
+%>>>\caption{Three-Dimensional Volume Panel.}
+%>>>\end{figure}
+%>>>\end{texonly}
+%
+\indent{0}
+\beginitems
+%
+\item[Eye Reference:]  At the top of this panel is the
+%-% \HDindex{graphics!3D control-panel!eye reference}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+{\bf Eye Reference} window.
+It shows a planar projection of the viewing pyramid from the eye
+of the viewer relative to the location of the object.
+This has a bounding region represented by the rectangle on the
+left.
+Below the object rectangle is the {\bf Hither} window.
+By moving the slider in this window the hither clipping plane sets
+%-% \HDindex{hither clipping plane}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+the front of the view volume.
+As a result of this depth clipping all points of the object closer
+to the eye than this hither plane are not shown.
+The {\bf Eye Distance} slider to the right of the {\bf Hither}
+slider is used to change the degree of perspective in the image.
+%
+\item[Clip Volume:]  The {\bf Clip Volume} window is at the
+%-% \HDindex{graphics!3D control-panel!clip volume}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+bottom of the {\bf Viewing Volume Panel}.
+On the right is a {\bf Settings} menu.
+In this menu are buttons to select viewing attributes.
+Selecting the {\bf Perspective} button computes the image using
+perspective projection.
+%-% \HDindex{graphics!3D control-panel!perspective}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+The {\bf Show Region} button indicates whether the clipping region
+of the
+%-% \HDindex{graphics!3D control-panel!show clip region}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+volume is to be drawn in the viewport and the {\bf Clipping On}
+button shows whether the view volume clipping is to be in effect
+when the image
+%-% \HDindex{graphics!3D control-panel!clipping on}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+is drawn.
+The left side of the {\bf Clip Volume} window shows the clipping
+%-% \HDindex{graphics!3D control-panel!clip volume}{ugGraphThreeDControlPage}{7.2.9.}{Three-Dimensional Control-Panel}
+boundary of the graph.
+Moving the knobs along the {\bf X}, {\bf Y}, and {\bf Z} sliders
+adjusts the volume of the clipping region accordingly.
+\enditems
+\indent{0}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugGraphThreeDopsTitle}{Operations for Three-Dimensional Graphics}
+\newcommand{\ugGraphThreeDopsNumber}{7.2.10.}
+%
+% =====================================================================
+\begin{page}{ugGraphThreeDopsPage}{7.2.10. Operations for Three-Dimensional Graphics}
+% =====================================================================
+\beginscroll
+
+Here is a summary of useful \Language{} operations for \threedim{}
+graphics.
+Each operation name is followed by a list of arguments.
+Each argument is written as a variable informally named according
+to the type of the argument (for example, {\it integer}).
+If appropriate, a default value for an argument is given in
+parentheses immediately following the name.
+
+%
+\texht{\bgroup\hbadness = 10001\sloppy}{}
+\indent{0}
+\beginitems
+%
+\item[\axiomFun{adaptive3D?}]\funArgs{}
+tests whether space curves are to be plotted
+%-% \HDindex{graphics!plot3d defaults!adaptive}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+according to the
+%-% \HDindex{adaptive plotting}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+adaptive refinement algorithm.
+
+%
+\item[\axiomFun{axes}]\funArgs{viewport, string\argDef{"on"}}
+turns the axes on and off.
+%-% \HDindex{graphics!3D commands!axes}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+
+%
+\item[\axiomFun{close}]\funArgs{viewport}
+closes the viewport.
+%-% \HDindex{graphics!3D commands!close}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+
+%
+\item[\axiomFun{colorDef}]\funArgs{viewport,
+\subscriptIt{color}{1}\argDef{1}, \subscriptIt{color}{2}\argDef{27}}
+sets the colormap
+%-% \HDindex{graphics!3D commands!define color}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+range to be from
+\subscriptIt{color}{1} to \subscriptIt{color}{2}.
+
+%
+\item[\axiomFun{controlPanel}]\funArgs{viewport, string\argDef{"off"}}
+declares whether the
+%-% \HDindex{graphics!3D commands!control-panel}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+control-panel for the viewport is to be displayed or not.
+
+%
+\item[\axiomFun{diagonals}]\funArgs{viewport, string\argDef{"off"}}
+declares whether the
+%-% \HDindex{graphics!3D commands!diagonals}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+polygon outline includes the diagonals or not.
+
+%
+\item[\axiomFun{drawStyle}]\funArgs{viewport, style}
+selects which of four drawing styles
+%-% \HDindex{graphics!3D commands!drawing style}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+are used: {\tt "wireMesh", "solid", "shade",} or {\tt "smooth".}
+
+%
+\item[\axiomFun{eyeDistance}]\funArgs{viewport,float\argDef{500}}
+sets the distance of the eye from the origin of the object
+%-% \HDindex{graphics!3D commands!eye distance}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+for use in the \axiomFunFrom{perspective}{ThreeDimensionalViewport}.
+
+%
+\item[\axiomFun{key}]\funArgs{viewport}
+returns the operating
+%-% \HDindex{graphics!3D commands!key}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+system process ID number for the viewport.
+
+%
+\item[\axiomFun{lighting}]\funArgs{viewport,
+\subscriptText{float}{x}\argDef{-0.5},
+\subscriptText{float}{y}\argDef{0.5}, \subscriptText{float}{z}\argDef{0.5}}
+sets the Cartesian
+%-% \HDindex{graphics!3D commands!lighting}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+coordinates of the light source.
+
+%
+\item[\axiomFun{modifyPointData}]\funArgs{viewport,integer,point}
+replaces the coordinates of the point with
+%-% \HDindex{graphics!3D commands!modify point data}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+the index {\it integer} with {\it point}.
+
+%
+\item[\axiomFun{move}]\funArgs{viewport,
+\subscriptText{integer}{x}\argDef{viewPosDefault},
+\subscriptText{integer}{y}\argDef{viewPosDefault}}
+moves the upper
+%-% \HDindex{graphics!3D commands!move}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+left-hand corner of the viewport to screen position
+\allowbreak
+({\small \subscriptText{integer}{x}, \subscriptText{integer}{y}}).
+
+%
+\item[\axiomFun{options}]\funArgs{viewport}
+returns a list of all current draw options.
+
+%
+\item[\axiomFun{outlineRender}]\funArgs{viewport, string\argDef{"off"}}
+turns polygon outlining
+%-% \HDindex{graphics!3D commands!outline}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+off or on when drawing in {\tt "shade"} mode.
+
+%
+\item[\axiomFun{perspective}]\funArgs{viewport, string\argDef{"on"}}
+turns perspective
+%-% \HDindex{graphics!3D commands!perspective}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+viewing on and off.
+
+%
+\item[\axiomFun{reset}]\funArgs{viewport}
+resets the attributes of a viewport to their
+%-% \HDindex{graphics!3D commands!reset}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+initial settings.
+
+%
+\item[\axiomFun{resize}]\funArgs{viewport,
+\subscriptText{integer}{width} \argDef{viewSizeDefault},
+\subscriptText{integer}{height} \argDef{viewSizeDefault}}
+resets the width and height
+%-% \HDindex{graphics!3D commands!resize}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+values for a viewport.
+
+%
+\item[\axiomFun{rotate}]\funArgs{viewport,
+\subscriptText{number}{\texht{$\theta$}{\axiom{theta}}}\argDef{viewThetaDefault},
+\subscriptText{number}{\texht{$\phi$}{\axiom{phi}}}\argDef{viewPhiDefault}}
+rotates the viewport by rotation angles for longitude
+({\it \texht{$\theta$}{\axiom{theta}}}) and
+latitude ({\it \texht{$\phi$}{\axiom{phi}}}).
+Angles designate radians if given as floats, or degrees if given
+%-% \HDindex{graphics!3D commands!rotate}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+as integers.
+
+%
+\item[\axiomFun{setAdaptive3D}]\funArgs{boolean\argDef{true}}
+sets whether space curves are to be plotted
+%-% \HDindex{graphics!plot3d defaults!set adaptive}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+according to the adaptive
+%-% \HDindex{adaptive plotting}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+refinement algorithm.
+
+%
+\item[\axiomFun{setMaxPoints3D}]\funArgs{integer\argDef{1000}}
+ sets the default maximum number of possible
+%-% \HDindex{graphics!plot3d defaults!set max points}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+points to be used when constructing a \threedim{} space curve.
+
+%
+\item[\axiomFun{setMinPoints3D}]\funArgs{integer\argDef{49}}
+sets the default minimum number of possible
+%-% \HDindex{graphics!plot3d defaults!set min points}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+points to be used when constructing a \threedim{} space curve.
+
+%
+\item[\axiomFun{setScreenResolution3D}]\funArgs{integer\argDef{500}}
+sets the default screen resolution constant
+%-% \HDindex{graphics!plot3d defaults!set screen resolution}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+used in setting the computation limit of adaptively
+%-% \HDindex{adaptive plotting}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+generated \threedim{} space curve plots.
+
+%
+\item[\axiomFun{showRegion}]\funArgs{viewport, string\argDef{"off"}}
+declares whether the bounding
+%-% \HDindex{graphics!3D commands!showRegion}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+box of a graph is shown or not.
+%
+\item[\axiomFun{subspace}]\funArgs{viewport}
+returns the space component.
+%
+\item[\axiomFun{subspace}]\funArgs{viewport, subspace}
+resets the space component
+%-% \HDindex{graphics!3D commands!subspace}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+to {\it subspace}.
+
+%
+\item[\axiomFun{title}]\funArgs{viewport, string}
+gives the viewport the
+%-% \HDindex{graphics!3D commands!title}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+title {\it string}.
+
+%
+\item[\axiomFun{translate}]\funArgs{viewport,
+\subscriptText{float}{x}\argDef{viewDeltaXDefault},
+\subscriptText{float}{y}\argDef{viewDeltaYDefault}}
+translates
+%-% \HDindex{graphics!3D commands!translate}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+the object horizontally and vertically relative to the center of the viewport.
+
+%
+\item[\axiomFun{intensity}]\funArgs{viewport,float\argDef{1.0}}
+resets the intensity {\it I} of the light source,
+%-% \HDindex{graphics!3D commands!intensity}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+\texht{$0 \le I \le 1.$}{{\it 0 \le I \le 1}.}
+
+%
+\item[\axiomFun{tubePointsDefault}]\funArgs{\optArg{integer\argDef{6}}}
+sets or indicates the default number of
+%-% \HDindex{graphics!3D defaults!tube points}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+vertices defining the polygon that is used to create a tube around
+a space curve.
+
+%
+\item[\axiomFun{tubeRadiusDefault}]\funArgs{\optArg{float\argDef{0.5}}}
+sets or indicates the default radius of
+%-% \HDindex{graphics!3D defaults!tube radius}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+the tube that encircles a space curve.
+
+%
+\item[\axiomFun{var1StepsDefault}]\funArgs{\optArg{integer\argDef{27}}}
+sets or indicates the default number of
+%-% \HDindex{graphics!3D defaults!var1 steps}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+increments into which the grid defining a surface plot is subdivided with
+respect to the first parameter declared in the surface function.
+
+%
+\item[\axiomFun{var2StepsDefault}]\funArgs{\optArg{integer\argDef{27}}}
+sets or indicates the default number of
+%-% \HDindex{graphics!3D defaults!var2 steps}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+increments into which the grid defining a surface plot is subdivided with
+respect to the second parameter declared in the surface function.
+
+%
+\item[\axiomFun{viewDefaults}]\funArgs{{\tt [}\subscriptText{integer}{%
+point}, \subscriptText{integer}{line}, \subscriptText{integer}{axes},
+\subscriptText{integer}{units}, \subscriptText{float}{point},
+\allowbreak\subscriptText{list}{position},
+\subscriptText{list}{size}{\tt ]}}
+resets the default settings for the
+%-% \HDindex{graphics!3D defaults!reset viewport defaults}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+point color, line color, axes color, units color, point size,
+viewport upper left-hand corner position, and the viewport size.
+
+%
+\item[\axiomFun{viewDeltaXDefault}]\funArgs{\optArg{float\argDef{0}}}
+resets the default horizontal offset
+%-% \HDindex{graphics!3D commands!deltaX default}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+from the center of the viewport, or returns the current default offset if no argument is given.
+
+%
+\item[\axiomFun{viewDeltaYDefault}]\funArgs{\optArg{float\argDef{0}}}
+resets the default vertical offset
+%-% \HDindex{graphics!3D commands!deltaY default}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+from the center of the viewport, or returns the current default offset if no argument is given.
+
+%
+\item[\axiomFun{viewPhiDefault}]\funArgs{\optArg{float\argDef{-\texht{$\pi$}{{\it pi}}/4}}}
+resets the default latitudinal view angle,
+or returns the current default angle if no argument is given.
+%-% \HDindex{graphics!3D commands!phi default}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+\texht{$\phi$}{{\it phi}} is set to this value.
+
+%
+\item[\axiomFun{viewpoint}]\funArgs{viewport, \subscriptText{float}{x},
+\subscriptText{float}{y}, \subscriptText{float}{z}}
+sets the viewing position in Cartesian coordinates.
+
+%
+\item[\axiomFun{viewpoint}]\funArgs{viewport,
+\subscriptText{float}{\texht{$\theta$}{\axiom{theta}}},
+\subscriptText{Float}{\texht{$\phi$}{\axiom{phi}}}}
+sets the viewing position in spherical coordinates.
+
+%
+\item[\axiomFun{viewpoint}]\funArgs{viewport,
+\subscriptText{Float}{\texht{$\theta$}{\axiom{theta}}},
+\subscriptText{Float}{\texht{$\phi$}{\axiom{phi}}},
+\subscriptText{Float}{scaleFactor},
+\subscriptText{Float}{xOffset}, \subscriptText{Float}{yOffset}}
+sets the viewing position in spherical coordinates,
+the scale factor, and offsets.
+%-% \HDindex{graphics!3D commands!viewpoint}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+\texht{$\theta$}{{\it theta}} (longitude) and
+\texht{$\phi$}{{\it phi}} (latitude) are in radians.
+
+%
+\item[\axiomFun{viewPosDefault}]\funArgs{\optArg{list\argDef{[0,0]}}}
+sets or indicates the position of the upper
+%-% \HDindex{graphics!3D defaults!viewport position}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+left-hand corner of a \twodim{} viewport, relative to the display root
+window (the upper left-hand corner of the display is \axiom{[0, 0]}).
+
+%
+\item[\axiomFun{viewSizeDefault}]\funArgs{\optArg{list\argDef{[400,400]}}}
+sets or indicates the width and height dimensions
+%-% \HDindex{graphics!3D defaults!viewport size}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+of a viewport.
+
+%
+\item[\axiomFun{viewThetaDefault}]\funArgs{\optArg{float\argDef{\texht{$\pi$}{{\it pi}}/4}}}
+resets the default longitudinal view angle,
+or returns the current default angle if no argument is given.
+%-% \HDindex{graphics!3D commands!theta default}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+When a parameter is specified, the default longitudinal view angle
+\texht{$\theta$}{{\it theta}} is set to this value.
+
+%
+\item[\axiomFun{viewWriteAvailable}]\funArgs{\optArg{list\argDef{["pixmap",
+"bitmap", "postscript", "image"}}}
+indicates the possible file types
+%-% \HDindex{graphics!3D defaults!available viewport writes}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+that can be created with the \axiomFunFrom{write}{ThreeDimensionalViewport} function.
+
+%
+\item[\axiomFun{viewWriteDefault}]\funArgs{\optArg{list\argDef{[]}}}
+sets or indicates the default types of files
+that are created in addition to the {\bf data} file when a
+\axiomFunFrom{write}{ThreeDimensionalViewport} command
+%-% \HDindex{graphics!3D defaults!viewport writes}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+is executed on a viewport.
+
+%
+\item[\axiomFun{viewScaleDefault}]\funArgs{\optArg{float}}
+sets the default scaling factor, or returns
+%-% \HDindex{graphics!3D commands!scale default}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+the current factor if no argument is given.
+
+%
+\item[\axiomFun{write}]\funArgs{viewport, directory, \optArg{option}}
+writes the file {\bf data} for {\it viewport}
+in the directory {\it directory}.
+An optional third argument specifies a file type (one of {\tt
+pixmap}, {\tt bitmap}, {\tt postscript}, or {\tt image}), or a
+list of file types.
+An additional file is written for each file type listed.
+
+%
+\item[\axiomFun{scale}]\funArgs{viewport, float\argDef{2.5}}
+specifies the scaling factor.
+%-% \HDindex{graphics!3D commands!scale}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+%-% \HDindex{scaling graphs}{ugGraphThreeDopsPage}{7.2.10.}{Operations for Three-Dimensional Graphics}
+\enditems
+\indent{0}
+\texht{\egroup}{}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugXdefaultsTitle}{Customization using .Xdefaults}
+\newcommand{\ugXdefaultsNumber}{7.2.11.}
+%
+% =====================================================================
+\begin{page}{ugXdefaultsPage}{7.2.11. Customization using .Xdefaults}
+% =====================================================================
+\beginscroll
+%-% \HDindex{graphics!.Xdefaults}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+
+Both the \twodim{} and \threedim{} drawing facilities consult
+the {\bf .Xdefaults} file for various defaults.
+%-% \HDindex{file!.Xdefaults @{\bf .Xdefaults}}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+The list of defaults that are recognized by the graphing routines
+is discussed in this section.
+These defaults are preceded by {\tt Axiom.3D.}
+for \threedim{} viewport defaults, {\tt Axiom.2D.}
+for \twodim{} viewport defaults, or {\tt Axiom*} (no dot) for
+those defaults that are acceptable to either viewport type.
+
+%
+\indent{0}
+\beginitems
+%
+\item[{\tt Axiom*buttonFont:\ \it font}] \ \newline
+This indicates which
+%-% \HDindex{graphics!.Xdefaults!button font}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+font type is used for the button text on the control-panel.
+\xdefault{Rom11}
+%
+\item[{\tt Axiom.2D.graphFont:\ \it font}] \quad (2D only) \newline
+This indicates
+%-% \HDindex{graphics!.Xdefaults!graph number font}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+which font type is used for displaying the graph numbers and
+slots in the {\bf Graphs} section of the \twodim{} control-panel.
+\xdefault{Rom22}
+%
+\item[{\tt Axiom.3D.headerFont:\ \it font}] \ \newline
+This indicates which
+%-% \HDindex{graphics!.Xdefaults!graph label font}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+font type is used for the axes labels and potentiometer
+header names on \threedim{} viewport windows.
+This is also used for \twodim{} control-panels for indicating
+which font type is used for potentionmeter header names and
+multiple graph title headers.
+%for example, {\tt Axiom.2D.headerFont: 8x13}.
+\xdefault{Itl14}
+%
+\item[{\tt Axiom*inverse:\ \it switch}] \ \newline
+This indicates whether the
+%-% \HDindex{graphics!.Xdefaults!inverting background}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+background color is to be inverted from white to black.
+If {\tt on}, the graph viewports use black as the background
+color.
+If {\tt off} or no declaration is made, the graph viewports use a
+white background.
+\xdefault{off}
+%
+\item[{\tt Axiom.3D.lightingFont:\ \it font}] \quad (3D only) \newline
+This indicates which font type is used for the {\bf x},
+%-% \HDindex{graphics!.Xdefaults!lighting font}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+{\bf y}, and {\bf z} labels of the two lighting axes potentiometers, and for
+the {\bf Intensity} title on the lighting control-panel.
+\xdefault{Rom10}
+%
+\item[{\tt Axiom.2D.messageFont, Axiom.3D.messageFont:\ \it font}] \ \newline
+These indicate the font type
+%-% \HDindex{graphics!.Xdefaults!message font}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+to be used for the text in the control-panel message window.
+\xdefault{Rom14}
+%
+\item[{\tt Axiom*monochrome:\ \it switch}] \ \newline
+This indicates whether the
+%-% \HDindex{graphics!.Xdefaults!monochrome}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+graph viewports are to be displayed as if the monitor is black and
+white, that is, a 1 bit plane.
+If {\tt on} is specified, the viewport display is black and white.
+If {\tt off} is specified, or no declaration for this default is
+given, the viewports are displayed in the normal fashion for the
+monitor in use.
+\xdefault{off}
+%
+\item[{\tt Axiom.2D.postScript:\ \it filename}] \ \newline
+This specifies
+%-% \HDindex{graphics!.Xdefaults!PostScript file name}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+the name of the file that is generated when a 2D PostScript graph
+%-% \HDindex{PostScript}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+is saved.
+\xdefault{axiom2D.ps}
+%
+\item[{\tt Axiom.3D.postScript:\ \it filename}] \ \newline
+This specifies
+%-% \HDindex{graphics!.Xdefaults!PostScript file name}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+the name of the file that is generated when a 3D PostScript graph
+%-% \HDindex{PostScript}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+is saved.
+\xdefault{axiom3D.ps}
+%
+\item[{\tt Axiom*titleFont \it font}] \ \newline
+This
+%-% \HDindex{graphics!.Xdefaults!title font}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+indicates which font type is used
+for the title text and, for \threedim{} graphs,
+in the lighting and viewing-volume control-panel windows.
+%-% \HDindex{graphics!Xdefaults!2d}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+\xdefault{Rom14}
+%
+\item[{\tt Axiom.2D.unitFont:\ \it font}] \quad (2D only) \newline
+This indicates
+%-% \HDindex{graphics!.Xdefaults!unit label font}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+which font type is used for displaying the unit labels on
+\twodim{} viewport graphs.
+\xdefault{6x10}
+%
+\item[{\tt Axiom.3D.volumeFont:\ \it font}] \quad (3D only) \newline
+This indicates which font type is used for the {\bf x},
+%-% \HDindex{graphics!.Xdefaults!volume label font}{ugXdefaultsPage}{7.2.11.}{Customization using .Xdefaults}
+{\bf y}, and {\bf z} labels of the clipping region sliders; for the
+{\bf Perspective}, {\bf Show Region}, and {\bf Clipping On} buttons under
+{\bf Settings}, and above the windows for the {\bf Hither} and
+{\bf Eye Distance} sliders in the {\bf Viewing Volume Panel} of the
+\threedim{} control-panel.
+\xdefault{Rom8}
+\enditems
+\indent{0}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug07.pht b/src/hyper/pages/ug07.pht
new file mode 100644
index 00000000..91ad424e
--- /dev/null
+++ b/src/hyper/pages/ug07.pht
@@ -0,0 +1,2386 @@
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+\begin{paste}{ugGraphTwoDbuildPageEmpty16}{ugGraphTwoDbuildPagePatch16}
+\pastebutton{ugGraphTwoDbuildPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{size3 := 10::PositiveInteger\bound{size3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch17}
+\begin{paste}{ugGraphTwoDbuildPageFull17}{ugGraphTwoDbuildPageEmpty17}
+\pastebutton{ugGraphTwoDbuildPageFull17}{\hidepaste}
+\tab{5}\spadcommand{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3]\bound{lsize }\free{size1 size2 size3 }}
+\indentrel{3}\begin{verbatim}
+   (17)  [6,6,6,6,8,8,8,8,10,10,10,10]
+                             Type: List PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty17}
+\begin{paste}{ugGraphTwoDbuildPageEmpty17}{ugGraphTwoDbuildPagePatch17}
+\pastebutton{ugGraphTwoDbuildPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3]\bound{lsize }\free{size1 size2 size3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch18}
+\begin{paste}{ugGraphTwoDbuildPageFull18}{ugGraphTwoDbuildPageEmpty18}
+\pastebutton{ugGraphTwoDbuildPageFull18}{\hidepaste}
+\tab{5}\spadcommand{pc1 := pastel red()\bound{pc1 }}
+\indentrel{3}\begin{verbatim}
+   (18)  [Hue: 1  Weight: 1.0] from the Pastel palette
+                                          Type: Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty18}
+\begin{paste}{ugGraphTwoDbuildPageEmpty18}{ugGraphTwoDbuildPagePatch18}
+\pastebutton{ugGraphTwoDbuildPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{pc1 := pastel red()\bound{pc1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch19}
+\begin{paste}{ugGraphTwoDbuildPageFull19}{ugGraphTwoDbuildPageEmpty19}
+\pastebutton{ugGraphTwoDbuildPageFull19}{\hidepaste}
+\tab{5}\spadcommand{pc2 := dim green()\bound{pc2 }}
+\indentrel{3}\begin{verbatim}
+   (19)  [Hue: 14  Weight: 1.0] from the Dim palette
+                                          Type: Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty19}
+\begin{paste}{ugGraphTwoDbuildPageEmpty19}{ugGraphTwoDbuildPagePatch19}
+\pastebutton{ugGraphTwoDbuildPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{pc2 := dim green()\bound{pc2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch20}
+\begin{paste}{ugGraphTwoDbuildPageFull20}{ugGraphTwoDbuildPageEmpty20}
+\pastebutton{ugGraphTwoDbuildPageFull20}{\hidepaste}
+\tab{5}\spadcommand{pc3 := pastel yellow()\bound{pc3 }}
+\indentrel{3}\begin{verbatim}
+   (20)  [Hue: 11  Weight: 1.0] from the Pastel palette
+                                          Type: Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty20}
+\begin{paste}{ugGraphTwoDbuildPageEmpty20}{ugGraphTwoDbuildPagePatch20}
+\pastebutton{ugGraphTwoDbuildPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{pc3 := pastel yellow()\bound{pc3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch21}
+\begin{paste}{ugGraphTwoDbuildPageFull21}{ugGraphTwoDbuildPageEmpty21}
+\pastebutton{ugGraphTwoDbuildPageFull21}{\hidepaste}
+\tab{5}\spadcommand{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3]\free{pc1 pc2 pc3 }\bound{lpc }}
+\indentrel{3}\begin{verbatim}
+   (21)
+   [[Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 11  Weight: 1.0] from the Pastel palette,
+    [Hue: 11  Weight: 1.0] from the Pastel palette,
+    [Hue: 11  Weight: 1.0] from the Pastel palette,
+    [Hue: 11  Weight: 1.0] from the Pastel palette]
+                                     Type: List Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty21}
+\begin{paste}{ugGraphTwoDbuildPageEmpty21}{ugGraphTwoDbuildPagePatch21}
+\pastebutton{ugGraphTwoDbuildPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3]\free{pc1 pc2 pc3 }\bound{lpc }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch22}
+\begin{paste}{ugGraphTwoDbuildPageFull22}{ugGraphTwoDbuildPageEmpty22}
+\pastebutton{ugGraphTwoDbuildPageFull22}{\hidepaste}
+\tab{5}\spadcommand{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()]\bound{lc }}
+\indentrel{3}\begin{verbatim}
+   (22)
+   [[Hue: 22  Weight: 1.0] from the Pastel palette,
+    [Hue: 11  Weight: 1.0] from the Light palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 1  Weight: 1.0] from the Bright palette,
+    [Hue: 14  Weight: 1.0] from the Light palette,
+    [Hue: 11  Weight: 1.0] from the Dim palette,
+    [Hue: 22  Weight: 1.0] from the Bright palette,
+    [Hue: 1  Weight: 1.0] from the Dark palette,
+    [Hue: 1  Weight: 1.0] from the Pastel palette,
+    [Hue: 22  Weight: 1.0] from the Light palette,
+    [Hue: 14  Weight: 1.0] from the Dim palette,
+    [Hue: 11  Weight: 1.0] from the Light palette]
+                                     Type: List Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty22}
+\begin{paste}{ugGraphTwoDbuildPageEmpty22}{ugGraphTwoDbuildPagePatch22}
+\pastebutton{ugGraphTwoDbuildPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()]\bound{lc }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch23}
+\begin{paste}{ugGraphTwoDbuildPageFull23}{ugGraphTwoDbuildPageEmpty23}
+\pastebutton{ugGraphTwoDbuildPageFull23}{\hidepaste}
+\tab{5}\spadcommand{g := makeGraphImage(llp,lpc,lc,lsize)$GRIMAGE\bound{g }\free{llp lpc lc lsize }}
+\indentrel{3}\begin{verbatim}
+   (23)  Graph with 12 point lists
+                                       Type: GraphImage
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty23}
+\begin{paste}{ugGraphTwoDbuildPageEmpty23}{ugGraphTwoDbuildPagePatch23}
+\pastebutton{ugGraphTwoDbuildPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{g := makeGraphImage(llp,lpc,lc,lsize)$GRIMAGE\bound{g }\free{llp lpc lc lsize }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch24}
+\begin{paste}{ugGraphTwoDbuildPageFull24}{ugGraphTwoDbuildPageEmpty24}
+\pastebutton{ugGraphTwoDbuildPageFull24}{\hidepaste}
+\tab{5}\spadgraph{makeViewport2D(g,[title("Lines")])$VIEW2D\free{g }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDbuildPage24.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDbuildPage24}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty24}
+\begin{paste}{ugGraphTwoDbuildPageEmpty24}{ugGraphTwoDbuildPagePatch24}
+\pastebutton{ugGraphTwoDbuildPageEmpty24}{\showpaste}
+\tab{5}\spadgraph{makeViewport2D(g,[title("Lines")])$VIEW2D\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch25}
+\begin{paste}{ugGraphTwoDbuildPageFull25}{ugGraphTwoDbuildPageEmpty25}
+\pastebutton{ugGraphTwoDbuildPageFull25}{\hidepaste}
+\tab{5}\spadcommand{)clear all\bound{clearAll }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty25}
+\begin{paste}{ugGraphTwoDbuildPageEmpty25}{ugGraphTwoDbuildPagePatch25}
+\pastebutton{ugGraphTwoDbuildPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{)clear all\bound{clearAll }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch26}
+\begin{paste}{ugGraphTwoDbuildPageFull26}{ugGraphTwoDbuildPageEmpty26}
+\pastebutton{ugGraphTwoDbuildPageFull26}{\hidepaste}
+\tab{5}\spadcommand{g := graphImage()$GRIMAGE\bound{Sg }\free{clearAll }}
+\indentrel{3}\begin{verbatim}
+   (1)  Graph with 0 point lists
+                                       Type: GraphImage
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty26}
+\begin{paste}{ugGraphTwoDbuildPageEmpty26}{ugGraphTwoDbuildPagePatch26}
+\pastebutton{ugGraphTwoDbuildPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{g := graphImage()$GRIMAGE\bound{Sg }\free{clearAll }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch27}
+\begin{paste}{ugGraphTwoDbuildPageFull27}{ugGraphTwoDbuildPageEmpty27}
+\pastebutton{ugGraphTwoDbuildPageFull27}{\hidepaste}
+\tab{5}\spadcommand{p1 := point [0,0]$(Point DFLOAT)\bound{Sp1 }}
+\indentrel{3}\begin{verbatim}
+   (2)  [0.0,0.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty27}
+\begin{paste}{ugGraphTwoDbuildPageEmpty27}{ugGraphTwoDbuildPagePatch27}
+\pastebutton{ugGraphTwoDbuildPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{p1 := point [0,0]$(Point DFLOAT)\bound{Sp1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch28}
+\begin{paste}{ugGraphTwoDbuildPageFull28}{ugGraphTwoDbuildPageEmpty28}
+\pastebutton{ugGraphTwoDbuildPageFull28}{\hidepaste}
+\tab{5}\spadcommand{p2 := point [.25,.25]$(Point DFLOAT)\bound{Sp2 }}
+\indentrel{3}\begin{verbatim}
+   (3)  [0.25,0.25]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty28}
+\begin{paste}{ugGraphTwoDbuildPageEmpty28}{ugGraphTwoDbuildPagePatch28}
+\pastebutton{ugGraphTwoDbuildPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{p2 := point [.25,.25]$(Point DFLOAT)\bound{Sp2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch29}
+\begin{paste}{ugGraphTwoDbuildPageFull29}{ugGraphTwoDbuildPageEmpty29}
+\pastebutton{ugGraphTwoDbuildPageFull29}{\hidepaste}
+\tab{5}\spadcommand{p3 := point [.5,.5]$(Point DFLOAT)\bound{Sp3 }}
+\indentrel{3}\begin{verbatim}
+   (4)  [0.5,0.5]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty29}
+\begin{paste}{ugGraphTwoDbuildPageEmpty29}{ugGraphTwoDbuildPagePatch29}
+\pastebutton{ugGraphTwoDbuildPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{p3 := point [.5,.5]$(Point DFLOAT)\bound{Sp3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch30}
+\begin{paste}{ugGraphTwoDbuildPageFull30}{ugGraphTwoDbuildPageEmpty30}
+\pastebutton{ugGraphTwoDbuildPageFull30}{\hidepaste}
+\tab{5}\spadcommand{p4 := point [.75,.75]$(Point DFLOAT)\bound{Sp4 }}
+\indentrel{3}\begin{verbatim}
+   (5)  [0.75,0.75]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty30}
+\begin{paste}{ugGraphTwoDbuildPageEmpty30}{ugGraphTwoDbuildPagePatch30}
+\pastebutton{ugGraphTwoDbuildPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{p4 := point [.75,.75]$(Point DFLOAT)\bound{Sp4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch31}
+\begin{paste}{ugGraphTwoDbuildPageFull31}{ugGraphTwoDbuildPageEmpty31}
+\pastebutton{ugGraphTwoDbuildPageFull31}{\hidepaste}
+\tab{5}\spadcommand{p5 := point [1,1]$(Point DFLOAT)\bound{Sp5 }}
+\indentrel{3}\begin{verbatim}
+   (6)  [1.0,1.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty31}
+\begin{paste}{ugGraphTwoDbuildPageEmpty31}{ugGraphTwoDbuildPagePatch31}
+\pastebutton{ugGraphTwoDbuildPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{p5 := point [1,1]$(Point DFLOAT)\bound{Sp5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch32}
+\begin{paste}{ugGraphTwoDbuildPageFull32}{ugGraphTwoDbuildPageEmpty32}
+\pastebutton{ugGraphTwoDbuildPageFull32}{\hidepaste}
+\tab{5}\spadcommand{component(g,p1)$GRIMAGE\free{Sg Sp1 }\bound{gp1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty32}
+\begin{paste}{ugGraphTwoDbuildPageEmpty32}{ugGraphTwoDbuildPagePatch32}
+\pastebutton{ugGraphTwoDbuildPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{component(g,p1)$GRIMAGE\free{Sg Sp1 }\bound{gp1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch33}
+\begin{paste}{ugGraphTwoDbuildPageFull33}{ugGraphTwoDbuildPageEmpty33}
+\pastebutton{ugGraphTwoDbuildPageFull33}{\hidepaste}
+\tab{5}\spadcommand{component(g,p2)$GRIMAGE\free{Sg Sp2 }\bound{gp2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty33}
+\begin{paste}{ugGraphTwoDbuildPageEmpty33}{ugGraphTwoDbuildPagePatch33}
+\pastebutton{ugGraphTwoDbuildPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{component(g,p2)$GRIMAGE\free{Sg Sp2 }\bound{gp2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch34}
+\begin{paste}{ugGraphTwoDbuildPageFull34}{ugGraphTwoDbuildPageEmpty34}
+\pastebutton{ugGraphTwoDbuildPageFull34}{\hidepaste}
+\tab{5}\spadcommand{appendPoint(g,p3)$GRIMAGE\free{gp1 gp2 Sp3 }\bound{gp3 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty34}
+\begin{paste}{ugGraphTwoDbuildPageEmpty34}{ugGraphTwoDbuildPagePatch34}
+\pastebutton{ugGraphTwoDbuildPageEmpty34}{\showpaste}
+\tab{5}\spadcommand{appendPoint(g,p3)$GRIMAGE\free{gp1 gp2 Sp3 }\bound{gp3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch35}
+\begin{paste}{ugGraphTwoDbuildPageFull35}{ugGraphTwoDbuildPageEmpty35}
+\pastebutton{ugGraphTwoDbuildPageFull35}{\hidepaste}
+\tab{5}\spadcommand{appendPoint(g,p4)$GRIMAGE\free{gp3 Sp4 }\bound{gp4 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty35}
+\begin{paste}{ugGraphTwoDbuildPageEmpty35}{ugGraphTwoDbuildPagePatch35}
+\pastebutton{ugGraphTwoDbuildPageEmpty35}{\showpaste}
+\tab{5}\spadcommand{appendPoint(g,p4)$GRIMAGE\free{gp3 Sp4 }\bound{gp4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch36}
+\begin{paste}{ugGraphTwoDbuildPageFull36}{ugGraphTwoDbuildPageEmpty36}
+\pastebutton{ugGraphTwoDbuildPageFull36}{\hidepaste}
+\tab{5}\spadcommand{appendPoint(g,p5)$GRIMAGE\free{gp4 Sp5 }\bound{gp5 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty36}
+\begin{paste}{ugGraphTwoDbuildPageEmpty36}{ugGraphTwoDbuildPagePatch36}
+\pastebutton{ugGraphTwoDbuildPageEmpty36}{\showpaste}
+\tab{5}\spadcommand{appendPoint(g,p5)$GRIMAGE\free{gp4 Sp5 }\bound{gp5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch37}
+\begin{paste}{ugGraphTwoDbuildPageFull37}{ugGraphTwoDbuildPageEmpty37}
+\pastebutton{ugGraphTwoDbuildPageFull37}{\hidepaste}
+\tab{5}\spadcommand{g1 := makeGraphImage(g)$GRIMAGE\bound{Sg1 }\free{gp5 }}
+\indentrel{3}\begin{verbatim}
+   (12)  Graph with 2 point lists
+                                       Type: GraphImage
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty37}
+\begin{paste}{ugGraphTwoDbuildPageEmpty37}{ugGraphTwoDbuildPagePatch37}
+\pastebutton{ugGraphTwoDbuildPageEmpty37}{\showpaste}
+\tab{5}\spadcommand{g1 := makeGraphImage(g)$GRIMAGE\bound{Sg1 }\free{gp5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch38}
+\begin{paste}{ugGraphTwoDbuildPageFull38}{ugGraphTwoDbuildPageEmpty38}
+\pastebutton{ugGraphTwoDbuildPageFull38}{\hidepaste}
+\tab{5}\spadgraph{makeViewport2D(g1,[title("Graph Points")])$VIEW2D\free{Sg1 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDbuildPage38.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDbuildPage38}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty38}
+\begin{paste}{ugGraphTwoDbuildPageEmpty38}{ugGraphTwoDbuildPagePatch38}
+\pastebutton{ugGraphTwoDbuildPageEmpty38}{\showpaste}
+\tab{5}\spadgraph{makeViewport2D(g1,[title("Graph Points")])$VIEW2D\free{Sg1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch39}
+\begin{paste}{ugGraphTwoDbuildPageFull39}{ugGraphTwoDbuildPageEmpty39}
+\pastebutton{ugGraphTwoDbuildPageFull39}{\hidepaste}
+\tab{5}\spadcommand{g2 := coerce([[p1],[p2],[p3],[p4],[p5]])$GRIMAGE\free{Sp1 Sp2 Sp3 Sp4 Sp5 }\bound{Sg2 }}
+\indentrel{3}\begin{verbatim}
+   (14)  Graph with 5 point lists
+                                       Type: GraphImage
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty39}
+\begin{paste}{ugGraphTwoDbuildPageEmpty39}{ugGraphTwoDbuildPagePatch39}
+\pastebutton{ugGraphTwoDbuildPageEmpty39}{\showpaste}
+\tab{5}\spadcommand{g2 := coerce([[p1],[p2],[p3],[p4],[p5]])$GRIMAGE\free{Sp1 Sp2 Sp3 Sp4 Sp5 }\bound{Sg2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch40}
+\begin{paste}{ugGraphTwoDbuildPageFull40}{ugGraphTwoDbuildPageEmpty40}
+\pastebutton{ugGraphTwoDbuildPageFull40}{\hidepaste}
+\tab{5}\spadcommand{v := viewport2D()$VIEW2D\bound{Sv }}
+\indentrel{3}\begin{verbatim}
+   (15)
+   Closed or Undefined TwoDimensionalViewport: "AXIOM2D"
+                           Type: TwoDimensionalViewport
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty40}
+\begin{paste}{ugGraphTwoDbuildPageEmpty40}{ugGraphTwoDbuildPagePatch40}
+\pastebutton{ugGraphTwoDbuildPageEmpty40}{\showpaste}
+\tab{5}\spadcommand{v := viewport2D()$VIEW2D\bound{Sv }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch41}
+\begin{paste}{ugGraphTwoDbuildPageFull41}{ugGraphTwoDbuildPageEmpty41}
+\pastebutton{ugGraphTwoDbuildPageFull41}{\hidepaste}
+\tab{5}\spadcommand{options(v,[title("Just Points")])$VIEW2D\free{Sv }\bound{Svo }}
+\indentrel{3}\begin{verbatim}
+   (16)
+   Closed or Undefined TwoDimensionalViewport: "AXIOM2D"
+                           Type: TwoDimensionalViewport
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty41}
+\begin{paste}{ugGraphTwoDbuildPageEmpty41}{ugGraphTwoDbuildPagePatch41}
+\pastebutton{ugGraphTwoDbuildPageEmpty41}{\showpaste}
+\tab{5}\spadcommand{options(v,[title("Just Points")])$VIEW2D\free{Sv }\bound{Svo }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch42}
+\begin{paste}{ugGraphTwoDbuildPageFull42}{ugGraphTwoDbuildPageEmpty42}
+\pastebutton{ugGraphTwoDbuildPageFull42}{\hidepaste}
+\tab{5}\spadcommand{putGraph(v,g2,1)$VIEW2D\free{Sg2 Svo }\bound{Svog2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty42}
+\begin{paste}{ugGraphTwoDbuildPageEmpty42}{ugGraphTwoDbuildPagePatch42}
+\pastebutton{ugGraphTwoDbuildPageEmpty42}{\showpaste}
+\tab{5}\spadcommand{putGraph(v,g2,1)$VIEW2D\free{Sg2 Svo }\bound{Svog2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPagePatch43}
+\begin{paste}{ugGraphTwoDbuildPageFull43}{ugGraphTwoDbuildPageEmpty43}
+\pastebutton{ugGraphTwoDbuildPageFull43}{\hidepaste}
+\tab{5}\spadgraph{makeViewport2D(v)$VIEW2D\free{Svog2 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDbuildPage43.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDbuildPage43}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDbuildPageEmpty43}
+\begin{paste}{ugGraphTwoDbuildPageEmpty43}{ugGraphTwoDbuildPagePatch43}
+\pastebutton{ugGraphTwoDbuildPageEmpty43}{\showpaste}
+\tab{5}\spadgraph{makeViewport2D(v)$VIEW2D\free{Svog2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDParmPagePatch1}
+\begin{paste}{ugGraphThreeDParmPageFull1}{ugGraphThreeDParmPageEmpty1}
+\pastebutton{ugGraphThreeDParmPageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(5*cos(t), 5*sin(t),t), t=-12..12)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDParmPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDParmPage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDParmPageEmpty1}
+\begin{paste}{ugGraphThreeDParmPageEmpty1}{ugGraphThreeDParmPagePatch1}
+\pastebutton{ugGraphThreeDParmPageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(curve(5*cos(t), 5*sin(t),t), t=-12..12)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDParmPagePatch2}
+\begin{paste}{ugGraphThreeDParmPageFull2}{ugGraphThreeDParmPageEmpty2}
+\pastebutton{ugGraphThreeDParmPageFull2}{\hidepaste}
+\tab{5}\spadcommand{i1(t:DFLOAT):DFLOAT == sin(t)*cos(3*t/5)\bound{i1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDParmPageEmpty2}
+\begin{paste}{ugGraphThreeDParmPageEmpty2}{ugGraphThreeDParmPagePatch2}
+\pastebutton{ugGraphThreeDParmPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{i1(t:DFLOAT):DFLOAT == sin(t)*cos(3*t/5)\bound{i1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDParmPagePatch3}
+\begin{paste}{ugGraphThreeDParmPageFull3}{ugGraphThreeDParmPageEmpty3}
+\pastebutton{ugGraphThreeDParmPageFull3}{\hidepaste}
+\tab{5}\spadcommand{i2(t:DFLOAT):DFLOAT == cos(t)*cos(3*t/5)\bound{i2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDParmPageEmpty3}
+\begin{paste}{ugGraphThreeDParmPageEmpty3}{ugGraphThreeDParmPagePatch3}
+\pastebutton{ugGraphThreeDParmPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{i2(t:DFLOAT):DFLOAT == cos(t)*cos(3*t/5)\bound{i2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDParmPagePatch4}
+\begin{paste}{ugGraphThreeDParmPageFull4}{ugGraphThreeDParmPageEmpty4}
+\pastebutton{ugGraphThreeDParmPageFull4}{\hidepaste}
+\tab{5}\spadcommand{i3(t:DFLOAT):DFLOAT == cos(t)*sin(3*t/5)\bound{i3 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDParmPageEmpty4}
+\begin{paste}{ugGraphThreeDParmPageEmpty4}{ugGraphThreeDParmPagePatch4}
+\pastebutton{ugGraphThreeDParmPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{i3(t:DFLOAT):DFLOAT == cos(t)*sin(3*t/5)\bound{i3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDParmPagePatch5}
+\begin{paste}{ugGraphThreeDParmPageFull5}{ugGraphThreeDParmPageEmpty5}
+\pastebutton{ugGraphThreeDParmPageFull5}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(i1,i2,i3),0..15*\%pi)\free{i1 i2 i3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDParmPage5.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDParmPage5}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDParmPageEmpty5}
+\begin{paste}{ugGraphThreeDParmPageEmpty5}{ugGraphThreeDParmPagePatch5}
+\pastebutton{ugGraphThreeDParmPageEmpty5}{\showpaste}
+\tab{5}\spadgraph{draw(curve(i1,i2,i3),0..15*\%pi)\free{i1 i2 i3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphMakeObjectPagePatch1}
+\begin{paste}{ugGraphMakeObjectPageFull1}{ugGraphMakeObjectPageEmpty1}
+\pastebutton{ugGraphMakeObjectPageFull1}{\hidepaste}
+\tab{5}\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1\bound{m }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphMakeObjectPageEmpty1}
+\begin{paste}{ugGraphMakeObjectPageEmpty1}{ugGraphMakeObjectPagePatch1}
+\pastebutton{ugGraphMakeObjectPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphMakeObjectPagePatch2}
+\begin{paste}{ugGraphMakeObjectPageFull2}{ugGraphMakeObjectPageEmpty2}
+\pastebutton{ugGraphMakeObjectPageFull2}{\hidepaste}
+\tab{5}\spadcommand{sph := makeObject(m, 0..\%pi, 0..2*\%pi, coordinates==spherical)\bound{sph }\free{m }}
+\indentrel{3}\begin{verbatim}
+   (2)  3-Space with 1 component
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphMakeObjectPageEmpty2}
+\begin{paste}{ugGraphMakeObjectPageEmpty2}{ugGraphMakeObjectPagePatch2}
+\pastebutton{ugGraphMakeObjectPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{sph := makeObject(m, 0..\%pi, 0..2*\%pi, coordinates==spherical)\bound{sph }\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphMakeObjectPagePatch3}
+\begin{paste}{ugGraphMakeObjectPageFull3}{ugGraphMakeObjectPageEmpty3}
+\pastebutton{ugGraphMakeObjectPageFull3}{\hidepaste}
+\tab{5}\spadcommand{makeObject(curve(1.5*sin(t), 1.5*cos(t), 0), t=0..2*\%pi, space == sph, tubeRadius == .25)\free{sph }\bound{v1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  3-Space with 2 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphMakeObjectPageEmpty3}
+\begin{paste}{ugGraphMakeObjectPageEmpty3}{ugGraphMakeObjectPagePatch3}
+\pastebutton{ugGraphMakeObjectPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{makeObject(curve(1.5*sin(t), 1.5*cos(t), 0), t=0..2*\%pi, space == sph, tubeRadius == .25)\free{sph }\bound{v1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphMakeObjectPagePatch4}
+\begin{paste}{ugGraphMakeObjectPageFull4}{ugGraphMakeObjectPageEmpty4}
+\pastebutton{ugGraphMakeObjectPageFull4}{\hidepaste}
+\tab{5}\spadgraph{makeViewport3D(sph,"Multiple Objects")\free{v1 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphMakeObjectPage4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphMakeObjectPage4}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphMakeObjectPageEmpty4}
+\begin{paste}{ugGraphMakeObjectPageEmpty4}{ugGraphMakeObjectPagePatch4}
+\pastebutton{ugGraphMakeObjectPageEmpty4}{\showpaste}
+\tab{5}\spadgraph{makeViewport3D(sph,"Multiple Objects")\free{v1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDPlotPagePatch1}
+\begin{paste}{ugGraphThreeDPlotPageFull1}{ugGraphThreeDPlotPageEmpty1}
+\pastebutton{ugGraphThreeDPlotPageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDPlotPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDPlotPage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDPlotPageEmpty1}
+\begin{paste}{ugGraphThreeDPlotPageEmpty1}{ugGraphThreeDPlotPagePatch1}
+\pastebutton{ugGraphThreeDPlotPageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDPlotPagePatch2}
+\begin{paste}{ugGraphThreeDPlotPageFull2}{ugGraphThreeDPlotPageEmpty2}
+\pastebutton{ugGraphThreeDPlotPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f(x,y) == sin(x)*cos(y)\bound{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDPlotPageEmpty2}
+\begin{paste}{ugGraphThreeDPlotPageEmpty2}{ugGraphThreeDPlotPagePatch2}
+\pastebutton{ugGraphThreeDPlotPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f(x,y) == sin(x)*cos(y)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDPlotPagePatch3}
+\begin{paste}{ugGraphThreeDPlotPageFull3}{ugGraphThreeDPlotPageEmpty3}
+\pastebutton{ugGraphThreeDPlotPageFull3}{\hidepaste}
+\tab{5}\spadgraph{draw(f,-\%pi..\%pi,-\%pi..\%pi)\free{f }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDPlotPage3.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDPlotPage3}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDPlotPageEmpty3}
+\begin{paste}{ugGraphThreeDPlotPageEmpty3}{ugGraphThreeDPlotPagePatch3}
+\pastebutton{ugGraphThreeDPlotPageEmpty3}{\showpaste}
+\tab{5}\spadgraph{draw(f,-\%pi..\%pi,-\%pi..\%pi)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDappendPagePatch1}
+\begin{paste}{ugGraphTwoDappendPageFull1}{ugGraphTwoDappendPageEmpty1}
+\pastebutton{ugGraphTwoDappendPageFull1}{\hidepaste}
+\tab{5}\spadcommand{v1 := draw(sin(x),x=0..2*\%pi)\bound{v1 }}
+\indentrel{3}\begin{verbatim}
+   (1)  TwoDimensionalViewport: "sin x"
+                           Type: TwoDimensionalViewport
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDappendPageEmpty1}
+\begin{paste}{ugGraphTwoDappendPageEmpty1}{ugGraphTwoDappendPagePatch1}
+\pastebutton{ugGraphTwoDappendPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{v1 := draw(sin(x),x=0..2*\%pi)\bound{v1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDappendPagePatch2}
+\begin{paste}{ugGraphTwoDappendPageFull2}{ugGraphTwoDappendPageEmpty2}
+\pastebutton{ugGraphTwoDappendPageFull2}{\hidepaste}
+\tab{5}\spadcommand{v2 := draw(cos(x),x=0..2*\%pi, curveColor==light red())\bound{v2 }}
+\indentrel{3}\begin{verbatim}
+   (2)  TwoDimensionalViewport: "cos x"
+                           Type: TwoDimensionalViewport
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDappendPageEmpty2}
+\begin{paste}{ugGraphTwoDappendPageEmpty2}{ugGraphTwoDappendPagePatch2}
+\pastebutton{ugGraphTwoDappendPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{v2 := draw(cos(x),x=0..2*\%pi, curveColor==light red())\bound{v2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDappendPagePatch3}
+\begin{paste}{ugGraphTwoDappendPageFull3}{ugGraphTwoDappendPageEmpty3}
+\pastebutton{ugGraphTwoDappendPageFull3}{\hidepaste}
+\tab{5}\spadcommand{g1 := getGraph(v1,1)\bound{g1 }\free{v1 }}
+\indentrel{3}\begin{verbatim}
+   (3)  Graph with 1 point list
+                                       Type: GraphImage
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDappendPageEmpty3}
+\begin{paste}{ugGraphTwoDappendPageEmpty3}{ugGraphTwoDappendPagePatch3}
+\pastebutton{ugGraphTwoDappendPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{g1 := getGraph(v1,1)\bound{g1 }\free{v1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDappendPagePatch4}
+\begin{paste}{ugGraphTwoDappendPageFull4}{ugGraphTwoDappendPageEmpty4}
+\pastebutton{ugGraphTwoDappendPageFull4}{\hidepaste}
+\tab{5}\spadcommand{putGraph(v2,g1,2)\bound{v22 }\free{g1 v2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDappendPageEmpty4}
+\begin{paste}{ugGraphTwoDappendPageEmpty4}{ugGraphTwoDappendPagePatch4}
+\pastebutton{ugGraphTwoDappendPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{putGraph(v2,g1,2)\bound{v22 }\free{g1 v2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDappendPagePatch5}
+\begin{paste}{ugGraphTwoDappendPageFull5}{ugGraphTwoDappendPageEmpty5}
+\pastebutton{ugGraphTwoDappendPageFull5}{\hidepaste}
+\tab{5}\spadgraph{makeViewport2D(v2)\free{v22 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDappendPage5.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDappendPage5}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDappendPageEmpty5}
+\begin{paste}{ugGraphTwoDappendPageEmpty5}{ugGraphTwoDappendPagePatch5}
+\pastebutton{ugGraphTwoDappendPageEmpty5}{\showpaste}
+\tab{5}\spadgraph{makeViewport2D(v2)\free{v22 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch1}
+\begin{paste}{ugGraphThreeDBuildPageFull1}{ugGraphThreeDBuildPageEmpty1}
+\pastebutton{ugGraphThreeDBuildPageFull1}{\hidepaste}
+\tab{5}\spadcommand{space := create3Space()$(ThreeSpace DFLOAT)\bound{space }}
+\indentrel{3}\begin{verbatim}
+   (1)  3-Space with 0 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty1}
+\begin{paste}{ugGraphThreeDBuildPageEmpty1}{ugGraphThreeDBuildPagePatch1}
+\pastebutton{ugGraphThreeDBuildPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{space := create3Space()$(ThreeSpace DFLOAT)\bound{space }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch2}
+\begin{paste}{ugGraphThreeDBuildPageFull2}{ugGraphThreeDBuildPageEmpty2}
+\pastebutton{ugGraphThreeDBuildPageFull2}{\hidepaste}
+\tab{5}\spadcommand{closedCurve(space,[[0,30,20], [0,30,30], [0,40,30], [0,40,100], [0,30,100],[0,30,110], [0,60,110], [0,60,100], [0,50,100], [0,50,30], [0,60,30], [0,60,20]])\bound{curve1 }\free{space }}
+\indentrel{3}\begin{verbatim}
+   (2)  3-Space with 1 component
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty2}
+\begin{paste}{ugGraphThreeDBuildPageEmpty2}{ugGraphThreeDBuildPagePatch2}
+\pastebutton{ugGraphThreeDBuildPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{closedCurve(space,[[0,30,20], [0,30,30], [0,40,30], [0,40,100], [0,30,100],[0,30,110], [0,60,110], [0,60,100], [0,50,100], [0,50,30], [0,60,30], [0,60,20]])\bound{curve1 }\free{space }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch3}
+\begin{paste}{ugGraphThreeDBuildPageFull3}{ugGraphThreeDBuildPageEmpty3}
+\pastebutton{ugGraphThreeDBuildPageFull3}{\hidepaste}
+\tab{5}\spadcommand{closedCurve(space,[[80,0,30], [80,0,100], [70,0,110], [40,0,110], [30,0,100], [30,0,90], [40,0,90], [40,0,95], [45,0,100], [65,0,100], [70,0,95], [70,0,35]])\bound{curve2 }\free{space }}
+\indentrel{3}\begin{verbatim}
+   (3)  3-Space with 2 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty3}
+\begin{paste}{ugGraphThreeDBuildPageEmpty3}{ugGraphThreeDBuildPagePatch3}
+\pastebutton{ugGraphThreeDBuildPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{closedCurve(space,[[80,0,30], [80,0,100], [70,0,110], [40,0,110], [30,0,100], [30,0,90], [40,0,90], [40,0,95], [45,0,100], [65,0,100], [70,0,95], [70,0,35]])\bound{curve2 }\free{space }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch4}
+\begin{paste}{ugGraphThreeDBuildPageFull4}{ugGraphThreeDBuildPageEmpty4}
+\pastebutton{ugGraphThreeDBuildPageFull4}{\hidepaste}
+\tab{5}\spadcommand{closedCurve(space,[[70,0,35], [65,0,30], [45,0,30], [40,0,35], [40,0,60], [50,0,60], [50,0,70], [30,0,70], [30,0,30], [40,0,20], [70,0,20], [80,0,30]])\bound{curve3 }\free{space }}
+\indentrel{3}\begin{verbatim}
+   (4)  3-Space with 3 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty4}
+\begin{paste}{ugGraphThreeDBuildPageEmpty4}{ugGraphThreeDBuildPagePatch4}
+\pastebutton{ugGraphThreeDBuildPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{closedCurve(space,[[70,0,35], [65,0,30], [45,0,30], [40,0,35], [40,0,60], [50,0,60], [50,0,70], [30,0,70], [30,0,30], [40,0,20], [70,0,20], [80,0,30]])\bound{curve3 }\free{space }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch5}
+\begin{paste}{ugGraphThreeDBuildPageFull5}{ugGraphThreeDBuildPageEmpty5}
+\pastebutton{ugGraphThreeDBuildPageFull5}{\hidepaste}
+\tab{5}\spadcommand{closedCurve(space,[[0,70,20], [0,70,110], [0,110,110], [0,120,100], [0,120,70], [0,115,65], [0,120,60], [0,120,30], [0,110,20], [0,80,20], [0,80,30], [0,80,20]])\bound{curve4 }\free{space }}
+\indentrel{3}\begin{verbatim}
+   (5)  3-Space with 4 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty5}
+\begin{paste}{ugGraphThreeDBuildPageEmpty5}{ugGraphThreeDBuildPagePatch5}
+\pastebutton{ugGraphThreeDBuildPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{closedCurve(space,[[0,70,20], [0,70,110], [0,110,110], [0,120,100], [0,120,70], [0,115,65], [0,120,60], [0,120,30], [0,110,20], [0,80,20], [0,80,30], [0,80,20]])\bound{curve4 }\free{space }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch6}
+\begin{paste}{ugGraphThreeDBuildPageFull6}{ugGraphThreeDBuildPageEmpty6}
+\pastebutton{ugGraphThreeDBuildPageFull6}{\hidepaste}
+\tab{5}\spadcommand{closedCurve(space,[[0,105,30], [0,110,35], [0,110,55], [0,105,60], [0,80,60], [0,80,70], [0,105,70], [0,110,75], [0,110,95], [0,105,100], [0,80,100], [0,80,20], [0,80,30]])\bound{curve5 }\free{space }}
+\indentrel{3}\begin{verbatim}
+   (6)  3-Space with 5 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty6}
+\begin{paste}{ugGraphThreeDBuildPageEmpty6}{ugGraphThreeDBuildPagePatch6}
+\pastebutton{ugGraphThreeDBuildPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{closedCurve(space,[[0,105,30], [0,110,35], [0,110,55], [0,105,60], [0,80,60], [0,80,70], [0,105,70], [0,110,75], [0,110,95], [0,105,100], [0,80,100], [0,80,20], [0,80,30]])\bound{curve5 }\free{space }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch7}
+\begin{paste}{ugGraphThreeDBuildPageFull7}{ugGraphThreeDBuildPageEmpty7}
+\pastebutton{ugGraphThreeDBuildPageFull7}{\hidepaste}
+\tab{5}\spadcommand{closedCurve(space,[[140,0,20], [140,0,110], [130,0,110], [90,0,20], [101,0,20],[114,0,50], [130,0,50], [130,0,60], [119,0,60], [130,0,85], [130,0,20]])\bound{curve6 }\free{space }}
+\indentrel{3}\begin{verbatim}
+   (7)  3-Space with 6 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty7}
+\begin{paste}{ugGraphThreeDBuildPageEmpty7}{ugGraphThreeDBuildPagePatch7}
+\pastebutton{ugGraphThreeDBuildPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{closedCurve(space,[[140,0,20], [140,0,110], [130,0,110], [90,0,20], [101,0,20],[114,0,50], [130,0,50], [130,0,60], [119,0,60], [130,0,85], [130,0,20]])\bound{curve6 }\free{space }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch8}
+\begin{paste}{ugGraphThreeDBuildPageFull8}{ugGraphThreeDBuildPageEmpty8}
+\pastebutton{ugGraphThreeDBuildPageFull8}{\hidepaste}
+\tab{5}\spadcommand{closedCurve(space,[[0,140,20], [0,140,110], [0,150,110], [0,170,50], [0,190,110], [0,200,110], [0,200,20], [0,190,20], [0,190,75], [0,175,35], [0,165,35],[0,150,75], [0,150,20]])\bound{curve7 }\free{space }}
+\indentrel{3}\begin{verbatim}
+   (8)  3-Space with 7 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty8}
+\begin{paste}{ugGraphThreeDBuildPageEmpty8}{ugGraphThreeDBuildPagePatch8}
+\pastebutton{ugGraphThreeDBuildPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{closedCurve(space,[[0,140,20], [0,140,110], [0,150,110], [0,170,50], [0,190,110], [0,200,110], [0,200,20], [0,190,20], [0,190,75], [0,175,35], [0,165,35],[0,150,75], [0,150,20]])\bound{curve7 }\free{space }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch9}
+\begin{paste}{ugGraphThreeDBuildPageFull9}{ugGraphThreeDBuildPageEmpty9}
+\pastebutton{ugGraphThreeDBuildPageFull9}{\hidepaste}
+\tab{5}\spadcommand{closedCurve(space,[[200,0,20], [200,0,110], [189,0,110], [160,0,45], [160,0,110], [150,0,110], [150,0,20], [161,0,20], [190,0,85], [190,0,20]])\bound{curve8 }\free{space }}
+\indentrel{3}\begin{verbatim}
+   (9)  3-Space with 8 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty9}
+\begin{paste}{ugGraphThreeDBuildPageEmpty9}{ugGraphThreeDBuildPagePatch9}
+\pastebutton{ugGraphThreeDBuildPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{closedCurve(space,[[200,0,20], [200,0,110], [189,0,110], [160,0,45], [160,0,110], [150,0,110], [150,0,20], [161,0,20], [190,0,85], [190,0,20]])\bound{curve8 }\free{space }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch10}
+\begin{paste}{ugGraphThreeDBuildPageFull10}{ugGraphThreeDBuildPageEmpty10}
+\pastebutton{ugGraphThreeDBuildPageFull10}{\hidepaste}
+\tab{5}\spadgraph{makeViewport3D(space, title == "Letters")\free{space curve1 curve2 curve3 curve4 curve5 curve6 curve7 curve8 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDBuildPage10.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDBuildPage10}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty10}
+\begin{paste}{ugGraphThreeDBuildPageEmpty10}{ugGraphThreeDBuildPagePatch10}
+\pastebutton{ugGraphThreeDBuildPageEmpty10}{\showpaste}
+\tab{5}\spadgraph{makeViewport3D(space, title == "Letters")\free{space curve1 curve2 curve3 curve4 curve5 curve6 curve7 curve8 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch11}
+\begin{paste}{ugGraphThreeDBuildPageFull11}{ugGraphThreeDBuildPageEmpty11}
+\pastebutton{ugGraphThreeDBuildPageFull11}{\hidepaste}
+\tab{5}\spadcommand{spaceC := create3Space()$(ThreeSpace DFLOAT)\bound{spaceC }}
+\indentrel{3}\begin{verbatim}
+   (11)  3-Space with 0 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty11}
+\begin{paste}{ugGraphThreeDBuildPageEmpty11}{ugGraphThreeDBuildPagePatch11}
+\pastebutton{ugGraphThreeDBuildPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{spaceC := create3Space()$(ThreeSpace DFLOAT)\bound{spaceC }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch12}
+\begin{paste}{ugGraphThreeDBuildPageFull12}{ugGraphThreeDBuildPageEmpty12}
+\pastebutton{ugGraphThreeDBuildPageFull12}{\hidepaste}
+\tab{5}\spadcommand{x: DFLOAT := 1\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (12)  1.0
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty12}
+\begin{paste}{ugGraphThreeDBuildPageEmpty12}{ugGraphThreeDBuildPagePatch12}
+\pastebutton{ugGraphThreeDBuildPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{x: DFLOAT := 1\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch13}
+\begin{paste}{ugGraphThreeDBuildPageFull13}{ugGraphThreeDBuildPageEmpty13}
+\pastebutton{ugGraphThreeDBuildPageFull13}{\hidepaste}
+\tab{5}\spadcommand{y: DFLOAT := -1\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (13)  - 1.0
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty13}
+\begin{paste}{ugGraphThreeDBuildPageEmpty13}{ugGraphThreeDBuildPagePatch13}
+\pastebutton{ugGraphThreeDBuildPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{y: DFLOAT := -1\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch14}
+\begin{paste}{ugGraphThreeDBuildPageFull14}{ugGraphThreeDBuildPageEmpty14}
+\pastebutton{ugGraphThreeDBuildPageFull14}{\hidepaste}
+\tab{5}\spadcommand{a := point [x,x,y,1::DFLOAT]$(Point DFLOAT)\bound{a }\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (14)  [1.0,1.0,- 1.0,1.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty14}
+\begin{paste}{ugGraphThreeDBuildPageEmpty14}{ugGraphThreeDBuildPagePatch14}
+\pastebutton{ugGraphThreeDBuildPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{a := point [x,x,y,1::DFLOAT]$(Point DFLOAT)\bound{a }\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch15}
+\begin{paste}{ugGraphThreeDBuildPageFull15}{ugGraphThreeDBuildPageEmpty15}
+\pastebutton{ugGraphThreeDBuildPageFull15}{\hidepaste}
+\tab{5}\spadcommand{b := point [y,x,y,4::DFLOAT]$(Point DFLOAT)\bound{b }\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (15)  [- 1.0,1.0,- 1.0,4.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty15}
+\begin{paste}{ugGraphThreeDBuildPageEmpty15}{ugGraphThreeDBuildPagePatch15}
+\pastebutton{ugGraphThreeDBuildPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{b := point [y,x,y,4::DFLOAT]$(Point DFLOAT)\bound{b }\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch16}
+\begin{paste}{ugGraphThreeDBuildPageFull16}{ugGraphThreeDBuildPageEmpty16}
+\pastebutton{ugGraphThreeDBuildPageFull16}{\hidepaste}
+\tab{5}\spadcommand{c := point [y,x,x,8::DFLOAT]$(Point DFLOAT)\bound{c }\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (16)  [- 1.0,1.0,1.0,8.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty16}
+\begin{paste}{ugGraphThreeDBuildPageEmpty16}{ugGraphThreeDBuildPagePatch16}
+\pastebutton{ugGraphThreeDBuildPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{c := point [y,x,x,8::DFLOAT]$(Point DFLOAT)\bound{c }\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch17}
+\begin{paste}{ugGraphThreeDBuildPageFull17}{ugGraphThreeDBuildPageEmpty17}
+\pastebutton{ugGraphThreeDBuildPageFull17}{\hidepaste}
+\tab{5}\spadcommand{d := point [x,x,x,12::DFLOAT]$(Point DFLOAT)\bound{d }\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (17)  [1.0,1.0,1.0,12.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty17}
+\begin{paste}{ugGraphThreeDBuildPageEmpty17}{ugGraphThreeDBuildPagePatch17}
+\pastebutton{ugGraphThreeDBuildPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{d := point [x,x,x,12::DFLOAT]$(Point DFLOAT)\bound{d }\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch18}
+\begin{paste}{ugGraphThreeDBuildPageFull18}{ugGraphThreeDBuildPageEmpty18}
+\pastebutton{ugGraphThreeDBuildPageFull18}{\hidepaste}
+\tab{5}\spadcommand{e := point [x,y,y,16::DFLOAT]$(Point DFLOAT)\bound{e }\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (18)  [1.0,- 1.0,- 1.0,16.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty18}
+\begin{paste}{ugGraphThreeDBuildPageEmpty18}{ugGraphThreeDBuildPagePatch18}
+\pastebutton{ugGraphThreeDBuildPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{e := point [x,y,y,16::DFLOAT]$(Point DFLOAT)\bound{e }\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch19}
+\begin{paste}{ugGraphThreeDBuildPageFull19}{ugGraphThreeDBuildPageEmpty19}
+\pastebutton{ugGraphThreeDBuildPageFull19}{\hidepaste}
+\tab{5}\spadcommand{f := point [y,y,y,20::DFLOAT]$(Point DFLOAT)\bound{f }\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (19)  [- 1.0,- 1.0,- 1.0,20.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty19}
+\begin{paste}{ugGraphThreeDBuildPageEmpty19}{ugGraphThreeDBuildPagePatch19}
+\pastebutton{ugGraphThreeDBuildPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{f := point [y,y,y,20::DFLOAT]$(Point DFLOAT)\bound{f }\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch20}
+\begin{paste}{ugGraphThreeDBuildPageFull20}{ugGraphThreeDBuildPageEmpty20}
+\pastebutton{ugGraphThreeDBuildPageFull20}{\hidepaste}
+\tab{5}\spadcommand{g := point [y,y,x,24::DFLOAT]$(Point DFLOAT)\bound{g }\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (20)  [- 1.0,- 1.0,1.0,24.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty20}
+\begin{paste}{ugGraphThreeDBuildPageEmpty20}{ugGraphThreeDBuildPagePatch20}
+\pastebutton{ugGraphThreeDBuildPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{g := point [y,y,x,24::DFLOAT]$(Point DFLOAT)\bound{g }\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch21}
+\begin{paste}{ugGraphThreeDBuildPageFull21}{ugGraphThreeDBuildPageEmpty21}
+\pastebutton{ugGraphThreeDBuildPageFull21}{\hidepaste}
+\tab{5}\spadcommand{h := point [x,y,x,27::DFLOAT]$(Point DFLOAT)\bound{h }\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (21)  [1.0,- 1.0,1.0,27.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty21}
+\begin{paste}{ugGraphThreeDBuildPageEmpty21}{ugGraphThreeDBuildPagePatch21}
+\pastebutton{ugGraphThreeDBuildPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{h := point [x,y,x,27::DFLOAT]$(Point DFLOAT)\bound{h }\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch22}
+\begin{paste}{ugGraphThreeDBuildPageFull22}{ugGraphThreeDBuildPageEmpty22}
+\pastebutton{ugGraphThreeDBuildPageFull22}{\hidepaste}
+\tab{5}\spadcommand{polygon(spaceC,[d,c,g,h])\free{d c g h spaceC }\bound{pol1 }}
+\indentrel{3}\begin{verbatim}
+   (22)  3-Space with 1 component
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty22}
+\begin{paste}{ugGraphThreeDBuildPageEmpty22}{ugGraphThreeDBuildPagePatch22}
+\pastebutton{ugGraphThreeDBuildPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{polygon(spaceC,[d,c,g,h])\free{d c g h spaceC }\bound{pol1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch23}
+\begin{paste}{ugGraphThreeDBuildPageFull23}{ugGraphThreeDBuildPageEmpty23}
+\pastebutton{ugGraphThreeDBuildPageFull23}{\hidepaste}
+\tab{5}\spadcommand{polygon(spaceC,[d,h,e,a])\free{d h e a spaceC }\bound{pol2 }}
+\indentrel{3}\begin{verbatim}
+   (23)  3-Space with 2 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty23}
+\begin{paste}{ugGraphThreeDBuildPageEmpty23}{ugGraphThreeDBuildPagePatch23}
+\pastebutton{ugGraphThreeDBuildPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{polygon(spaceC,[d,h,e,a])\free{d h e a spaceC }\bound{pol2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch24}
+\begin{paste}{ugGraphThreeDBuildPageFull24}{ugGraphThreeDBuildPageEmpty24}
+\pastebutton{ugGraphThreeDBuildPageFull24}{\hidepaste}
+\tab{5}\spadcommand{polygon(spaceC,[c,d,a,b])\free{c d a b spaceC }\bound{pol3 }}
+\indentrel{3}\begin{verbatim}
+   (24)  3-Space with 3 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty24}
+\begin{paste}{ugGraphThreeDBuildPageEmpty24}{ugGraphThreeDBuildPagePatch24}
+\pastebutton{ugGraphThreeDBuildPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{polygon(spaceC,[c,d,a,b])\free{c d a b spaceC }\bound{pol3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch25}
+\begin{paste}{ugGraphThreeDBuildPageFull25}{ugGraphThreeDBuildPageEmpty25}
+\pastebutton{ugGraphThreeDBuildPageFull25}{\hidepaste}
+\tab{5}\spadcommand{polygon(spaceC,[g,c,b,f])\free{g c b f spaceC }\bound{pol4 }}
+\indentrel{3}\begin{verbatim}
+   (25)  3-Space with 4 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty25}
+\begin{paste}{ugGraphThreeDBuildPageEmpty25}{ugGraphThreeDBuildPagePatch25}
+\pastebutton{ugGraphThreeDBuildPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{polygon(spaceC,[g,c,b,f])\free{g c b f spaceC }\bound{pol4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch26}
+\begin{paste}{ugGraphThreeDBuildPageFull26}{ugGraphThreeDBuildPageEmpty26}
+\pastebutton{ugGraphThreeDBuildPageFull26}{\hidepaste}
+\tab{5}\spadcommand{polygon(spaceC,[h,g,f,e])\free{h g f e spaceC }\bound{pol5 }}
+\indentrel{3}\begin{verbatim}
+   (26)  3-Space with 5 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty26}
+\begin{paste}{ugGraphThreeDBuildPageEmpty26}{ugGraphThreeDBuildPagePatch26}
+\pastebutton{ugGraphThreeDBuildPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{polygon(spaceC,[h,g,f,e])\free{h g f e spaceC }\bound{pol5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch27}
+\begin{paste}{ugGraphThreeDBuildPageFull27}{ugGraphThreeDBuildPageEmpty27}
+\pastebutton{ugGraphThreeDBuildPageFull27}{\hidepaste}
+\tab{5}\spadcommand{polygon(spaceC,[e,f,b,a])\free{e f b a spaceC }\bound{pol6 }}
+\indentrel{3}\begin{verbatim}
+   (27)  3-Space with 6 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty27}
+\begin{paste}{ugGraphThreeDBuildPageEmpty27}{ugGraphThreeDBuildPagePatch27}
+\pastebutton{ugGraphThreeDBuildPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{polygon(spaceC,[e,f,b,a])\free{e f b a spaceC }\bound{pol6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPagePatch28}
+\begin{paste}{ugGraphThreeDBuildPageFull28}{ugGraphThreeDBuildPageEmpty28}
+\pastebutton{ugGraphThreeDBuildPageFull28}{\hidepaste}
+\tab{5}\spadgraph{makeViewport3D(spaceC, title == "Cube")\free{pol1 pol2 pol3 pol4 pol5 pol6 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDBuildPage28.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDBuildPage28}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDBuildPageEmpty28}
+\begin{paste}{ugGraphThreeDBuildPageEmpty28}{ugGraphThreeDBuildPagePatch28}
+\pastebutton{ugGraphThreeDBuildPageEmpty28}{\showpaste}
+\tab{5}\spadgraph{makeViewport3D(spaceC, title == "Cube")\free{pol1 pol2 pol3 pol4 pol5 pol6 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPagePatch1}
+\begin{paste}{ugGraphTwoDOptionsPageFull1}{ugGraphTwoDOptionsPageEmpty1}
+\pastebutton{ugGraphTwoDOptionsPageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(1/x),x=-2*\%pi..2*\%pi, adaptive == false)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPageEmpty1}
+\begin{paste}{ugGraphTwoDOptionsPageEmpty1}{ugGraphTwoDOptionsPagePatch1}
+\pastebutton{ugGraphTwoDOptionsPageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(sin(1/x),x=-2*\%pi..2*\%pi, adaptive == false)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPagePatch2}
+\begin{paste}{ugGraphTwoDOptionsPageFull2}{ugGraphTwoDOptionsPageEmpty2}
+\pastebutton{ugGraphTwoDOptionsPageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(tan(x),x=-2*\%pi..2*\%pi, clip == true)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPageEmpty2}
+\begin{paste}{ugGraphTwoDOptionsPageEmpty2}{ugGraphTwoDOptionsPagePatch2}
+\pastebutton{ugGraphTwoDOptionsPageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(tan(x),x=-2*\%pi..2*\%pi, clip == true)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPagePatch3}
+\begin{paste}{ugGraphTwoDOptionsPageFull3}{ugGraphTwoDOptionsPageEmpty3}
+\pastebutton{ugGraphTwoDOptionsPageFull3}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(x),x=-\%pi..\%pi, toScale == true, unit == [1.0,1.0])}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage3.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage3}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPageEmpty3}
+\begin{paste}{ugGraphTwoDOptionsPageEmpty3}{ugGraphTwoDOptionsPagePatch3}
+\pastebutton{ugGraphTwoDOptionsPageEmpty3}{\showpaste}
+\tab{5}\spadgraph{draw(sin(x),x=-\%pi..\%pi, toScale == true, unit == [1.0,1.0])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPagePatch4}
+\begin{paste}{ugGraphTwoDOptionsPageFull4}{ugGraphTwoDOptionsPageEmpty4}
+\pastebutton{ugGraphTwoDOptionsPageFull4}{\hidepaste}
+\tab{5}\spadgraph{draw(sec(x),x=-2*\%pi..2*\%pi, clip == [-2*\%pi..2*\%pi,-\%pi..\%pi], unit == [1.0,1.0])}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage4}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPageEmpty4}
+\begin{paste}{ugGraphTwoDOptionsPageEmpty4}{ugGraphTwoDOptionsPagePatch4}
+\pastebutton{ugGraphTwoDOptionsPageEmpty4}{\showpaste}
+\tab{5}\spadgraph{draw(sec(x),x=-2*\%pi..2*\%pi, clip == [-2*\%pi..2*\%pi,-\%pi..\%pi], unit == [1.0,1.0])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPagePatch5}
+\begin{paste}{ugGraphTwoDOptionsPageFull5}{ugGraphTwoDOptionsPageEmpty5}
+\pastebutton{ugGraphTwoDOptionsPageFull5}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(x),x=-\%pi..\%pi, curveColor == bright red())}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage5.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage5}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPageEmpty5}
+\begin{paste}{ugGraphTwoDOptionsPageEmpty5}{ugGraphTwoDOptionsPagePatch5}
+\pastebutton{ugGraphTwoDOptionsPageEmpty5}{\showpaste}
+\tab{5}\spadgraph{draw(sin(x),x=-\%pi..\%pi, curveColor == bright red())}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPagePatch6}
+\begin{paste}{ugGraphTwoDOptionsPageFull6}{ugGraphTwoDOptionsPageEmpty6}
+\pastebutton{ugGraphTwoDOptionsPageFull6}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(x),x=-\%pi..\%pi, pointColor == pastel yellow())}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage6.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage6}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPageEmpty6}
+\begin{paste}{ugGraphTwoDOptionsPageEmpty6}{ugGraphTwoDOptionsPagePatch6}
+\pastebutton{ugGraphTwoDOptionsPageEmpty6}{\showpaste}
+\tab{5}\spadgraph{draw(sin(x),x=-\%pi..\%pi, pointColor == pastel yellow())}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPagePatch7}
+\begin{paste}{ugGraphTwoDOptionsPageFull7}{ugGraphTwoDOptionsPageEmpty7}
+\pastebutton{ugGraphTwoDOptionsPageFull7}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(9*sin(3*t/4),8*sin(t)), t = -4*\%pi..4*\%pi, unit == [2.0,1.0])}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage7.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage7}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPageEmpty7}
+\begin{paste}{ugGraphTwoDOptionsPageEmpty7}{ugGraphTwoDOptionsPagePatch7}
+\pastebutton{ugGraphTwoDOptionsPageEmpty7}{\showpaste}
+\tab{5}\spadgraph{draw(curve(9*sin(3*t/4),8*sin(t)), t = -4*\%pi..4*\%pi, unit == [2.0,1.0])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPagePatch8}
+\begin{paste}{ugGraphTwoDOptionsPageFull8}{ugGraphTwoDOptionsPageEmpty8}
+\pastebutton{ugGraphTwoDOptionsPageFull8}{\hidepaste}
+\tab{5}\spadgraph{draw(y**2 + y - (x**3 - x) = 0, x, y, range == [-2..2,-2..1], unit==[1.0,1.0])}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage8.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage8}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPageEmpty8}
+\begin{paste}{ugGraphTwoDOptionsPageEmpty8}{ugGraphTwoDOptionsPagePatch8}
+\pastebutton{ugGraphTwoDOptionsPageEmpty8}{\showpaste}
+\tab{5}\spadgraph{draw(y**2 + y - (x**3 - x) = 0, x, y, range == [-2..2,-2..1], unit==[1.0,1.0])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPagePatch9}
+\begin{paste}{ugGraphTwoDOptionsPageFull9}{ugGraphTwoDOptionsPageEmpty9}
+\pastebutton{ugGraphTwoDOptionsPageFull9}{\hidepaste}
+\tab{5}\spadgraph{draw(x**2 + y**2 = 1, x, y, range == [-3/2..3/2,-3/2..3/2], unit==[0.5,0.5])}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage9.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage9}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPageEmpty9}
+\begin{paste}{ugGraphTwoDOptionsPageEmpty9}{ugGraphTwoDOptionsPagePatch9}
+\pastebutton{ugGraphTwoDOptionsPageEmpty9}{\showpaste}
+\tab{5}\spadgraph{draw(x**2 + y**2 = 1, x, y, range == [-3/2..3/2,-3/2..3/2], unit==[0.5,0.5])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPagePatch10}
+\begin{paste}{ugGraphTwoDOptionsPageFull10}{ugGraphTwoDOptionsPageEmpty10}
+\pastebutton{ugGraphTwoDOptionsPageFull10}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(sin(5*t),t),t=0..2*\%pi, coordinates == polar)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage10.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDOptionsPage10}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDOptionsPageEmpty10}
+\begin{paste}{ugGraphTwoDOptionsPageEmpty10}{ugGraphTwoDOptionsPagePatch10}
+\pastebutton{ugGraphTwoDOptionsPageEmpty10}{\showpaste}
+\tab{5}\spadgraph{draw(curve(sin(5*t),t),t=0..2*\%pi, coordinates == polar)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphColorPalettePagePatch1}
+\begin{paste}{ugGraphColorPalettePageFull1}{ugGraphColorPalettePageEmpty1}
+\pastebutton{ugGraphColorPalettePageFull1}{\hidepaste}
+\tab{5}\spadcommand{shade red()}
+\indentrel{3}\begin{verbatim}
+   (1)  3
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphColorPalettePageEmpty1}
+\begin{paste}{ugGraphColorPalettePageEmpty1}{ugGraphColorPalettePagePatch1}
+\pastebutton{ugGraphColorPalettePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{shade red()}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphColorPalettePagePatch2}
+\begin{paste}{ugGraphColorPalettePageFull2}{ugGraphColorPalettePageEmpty2}
+\pastebutton{ugGraphColorPalettePageFull2}{\hidepaste}
+\tab{5}\spadcommand{myFavoriteColor := dark blue()\bound{mfc }}
+\indentrel{3}\begin{verbatim}
+   (2)  [Hue: 22  Weight: 1.0] from the Dark palette
+                                          Type: Palette
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphColorPalettePageEmpty2}
+\begin{paste}{ugGraphColorPalettePageEmpty2}{ugGraphColorPalettePagePatch2}
+\pastebutton{ugGraphColorPalettePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{myFavoriteColor := dark blue()\bound{mfc }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphColorPalettePagePatch3}
+\begin{paste}{ugGraphColorPalettePageFull3}{ugGraphColorPalettePageEmpty3}
+\pastebutton{ugGraphColorPalettePageFull3}{\hidepaste}
+\tab{5}\spadcommand{shade myFavoriteColor\free{mfc }}
+\indentrel{3}\begin{verbatim}
+   (3)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphColorPalettePageEmpty3}
+\begin{paste}{ugGraphColorPalettePageEmpty3}{ugGraphColorPalettePagePatch3}
+\pastebutton{ugGraphColorPalettePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{shade myFavoriteColor\free{mfc }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphColorPalettePagePatch4}
+\begin{paste}{ugGraphColorPalettePageFull4}{ugGraphColorPalettePageEmpty4}
+\pastebutton{ugGraphColorPalettePageFull4}{\hidepaste}
+\tab{5}\spadcommand{hue myFavoriteColor\free{mfc }}
+\indentrel{3}\begin{verbatim}
+   (4)  Hue: 22  Weight: 1.0
+                                            Type: Color
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphColorPalettePageEmpty4}
+\begin{paste}{ugGraphColorPalettePageEmpty4}{ugGraphColorPalettePagePatch4}
+\pastebutton{ugGraphColorPalettePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{hue myFavoriteColor\free{mfc }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphColorPalettePagePatch5}
+\begin{paste}{ugGraphColorPalettePageFull5}{ugGraphColorPalettePageEmpty5}
+\pastebutton{ugGraphColorPalettePageFull5}{\hidepaste}
+\tab{5}\spadgraph{draw(x**2,x=-1..1,curveColor == dark blue())}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphColorPalettePage5.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphColorPalettePage5}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphColorPalettePageEmpty5}
+\begin{paste}{ugGraphColorPalettePageEmpty5}{ugGraphColorPalettePagePatch5}
+\pastebutton{ugGraphColorPalettePageEmpty5}{\showpaste}
+\tab{5}\spadgraph{draw(x**2,x=-1..1,curveColor == dark blue())}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch1}
+\begin{paste}{ugGraphThreeDOptionsPageFull1}{ugGraphThreeDOptionsPageEmpty1}
+\pastebutton{ugGraphThreeDOptionsPageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(cos(x*y),x=0..2*\%pi,y=0..\%pi,title == "Title of Graph")}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage1}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty1}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty1}{ugGraphThreeDOptionsPagePatch1}
+\pastebutton{ugGraphThreeDOptionsPageEmpty1}{\showpaste}
+\tab{5}\spadgraph{draw(cos(x*y),x=0..2*\%pi,y=0..\%pi,title == "Title of Graph")}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch2}
+\begin{paste}{ugGraphThreeDOptionsPageFull2}{ugGraphThreeDOptionsPageEmpty2}
+\pastebutton{ugGraphThreeDOptionsPageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3, style=="smooth", title=="Smooth Option")}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty2}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty2}{ugGraphThreeDOptionsPagePatch2}
+\pastebutton{ugGraphThreeDOptionsPageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3, style=="smooth", title=="Smooth Option")}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch3}
+\begin{paste}{ugGraphThreeDOptionsPageFull3}{ugGraphThreeDOptionsPageEmpty3}
+\pastebutton{ugGraphThreeDOptionsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{color1(t) == t\bound{colorFxn1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty3}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty3}{ugGraphThreeDOptionsPagePatch3}
+\pastebutton{ugGraphThreeDOptionsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{color1(t) == t\bound{colorFxn1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch4}
+\begin{paste}{ugGraphThreeDOptionsPageFull4}{ugGraphThreeDOptionsPageEmpty4}
+\pastebutton{ugGraphThreeDOptionsPageFull4}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(sin(t), cos(t),0), t=0..2*\%pi, tubeRadius == .3, colorFunction == color1)\free{colorFxn1 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage4}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty4}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty4}{ugGraphThreeDOptionsPagePatch4}
+\pastebutton{ugGraphThreeDOptionsPageEmpty4}{\showpaste}
+\tab{5}\spadgraph{draw(curve(sin(t), cos(t),0), t=0..2*\%pi, tubeRadius == .3, colorFunction == color1)\free{colorFxn1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch5}
+\begin{paste}{ugGraphThreeDOptionsPageFull5}{ugGraphThreeDOptionsPageEmpty5}
+\pastebutton{ugGraphThreeDOptionsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{color2(u,v) == u**2 - v**2\bound{colorFxn2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty5}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty5}{ugGraphThreeDOptionsPagePatch5}
+\pastebutton{ugGraphThreeDOptionsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{color2(u,v) == u**2 - v**2\bound{colorFxn2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch6}
+\begin{paste}{ugGraphThreeDOptionsPageFull6}{ugGraphThreeDOptionsPageEmpty6}
+\pastebutton{ugGraphThreeDOptionsPageFull6}{\hidepaste}
+\tab{5}\spadgraph{draw(cos(u*v), u=-3..3, v=-3..3, colorFunction == color2)\free{colorFxn2 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage6.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage6}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty6}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty6}{ugGraphThreeDOptionsPagePatch6}
+\pastebutton{ugGraphThreeDOptionsPageEmpty6}{\showpaste}
+\tab{5}\spadgraph{draw(cos(u*v), u=-3..3, v=-3..3, colorFunction == color2)\free{colorFxn2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch7}
+\begin{paste}{ugGraphThreeDOptionsPageFull7}{ugGraphThreeDOptionsPageEmpty7}
+\pastebutton{ugGraphThreeDOptionsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{color3(x,y,fxy) == sin(x*fxy) + cos(y*fxy)\bound{colorFxn3 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty7}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty7}{ugGraphThreeDOptionsPagePatch7}
+\pastebutton{ugGraphThreeDOptionsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{color3(x,y,fxy) == sin(x*fxy) + cos(y*fxy)\bound{colorFxn3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch8}
+\begin{paste}{ugGraphThreeDOptionsPageFull8}{ugGraphThreeDOptionsPageEmpty8}
+\pastebutton{ugGraphThreeDOptionsPageFull8}{\hidepaste}
+\tab{5}\spadgraph{draw(cos(x*y), x=-3..3, y=-3..3, colorFunction == color3)\free{colorFxn3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage8.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage8}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty8}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty8}{ugGraphThreeDOptionsPagePatch8}
+\pastebutton{ugGraphThreeDOptionsPageEmpty8}{\showpaste}
+\tab{5}\spadgraph{draw(cos(x*y), x=-3..3, y=-3..3, colorFunction == color3)\free{colorFxn3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch9}
+\begin{paste}{ugGraphThreeDOptionsPageFull9}{ugGraphThreeDOptionsPageEmpty9}
+\pastebutton{ugGraphThreeDOptionsPageFull9}{\hidepaste}
+\tab{5}\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1\bound{m }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty9}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty9}{ugGraphThreeDOptionsPagePatch9}
+\pastebutton{ugGraphThreeDOptionsPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch10}
+\begin{paste}{ugGraphThreeDOptionsPageFull10}{ugGraphThreeDOptionsPageEmpty10}
+\pastebutton{ugGraphThreeDOptionsPageFull10}{\hidepaste}
+\tab{5}\spadgraph{draw(m, 0..2*\%pi,0..\%pi, coordinates == spherical, style=="shade")\free{m }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage10.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage10}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty10}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty10}{ugGraphThreeDOptionsPagePatch10}
+\pastebutton{ugGraphThreeDOptionsPageEmpty10}{\showpaste}
+\tab{5}\spadgraph{draw(m, 0..2*\%pi,0..\%pi, coordinates == spherical, style=="shade")\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch11}
+\begin{paste}{ugGraphThreeDOptionsPageFull11}{ugGraphThreeDOptionsPageEmpty11}
+\pastebutton{ugGraphThreeDOptionsPageFull11}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(sin(t),cos(t),0),t=0..2*\%pi, style=="shade", tubeRadius == .3)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage11.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage11}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty11}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty11}{ugGraphThreeDOptionsPagePatch11}
+\pastebutton{ugGraphThreeDOptionsPageEmpty11}{\showpaste}
+\tab{5}\spadgraph{draw(curve(sin(t),cos(t),0),t=0..2*\%pi, style=="shade", tubeRadius == .3)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch12}
+\begin{paste}{ugGraphThreeDOptionsPageFull12}{ugGraphThreeDOptionsPageEmpty12}
+\pastebutton{ugGraphThreeDOptionsPageFull12}{\hidepaste}
+\tab{5}\spadgraph{draw(curve(sin(t), cos(t), 0), t=0..2*\%pi, style=="shade", tubeRadius == .25, tubePoints == 3)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage12.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage12}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty12}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty12}{ugGraphThreeDOptionsPagePatch12}
+\pastebutton{ugGraphThreeDOptionsPageEmpty12}{\showpaste}
+\tab{5}\spadgraph{draw(curve(sin(t), cos(t), 0), t=0..2*\%pi, style=="shade", tubeRadius == .25, tubePoints == 3)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch13}
+\begin{paste}{ugGraphThreeDOptionsPageFull13}{ugGraphThreeDOptionsPageEmpty13}
+\pastebutton{ugGraphThreeDOptionsPageFull13}{\hidepaste}
+\tab{5}\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3, style=="shade", var1Steps == 30, var2Steps == 30)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage13.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage13}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty13}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty13}{ugGraphThreeDOptionsPagePatch13}
+\pastebutton{ugGraphThreeDOptionsPageEmpty13}{\showpaste}
+\tab{5}\spadgraph{draw(cos(x*y),x=-3..3,y=-3..3, style=="shade", var1Steps == 30, var2Steps == 30)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch14}
+\begin{paste}{ugGraphThreeDOptionsPageFull14}{ugGraphThreeDOptionsPageEmpty14}
+\pastebutton{ugGraphThreeDOptionsPageFull14}{\hidepaste}
+\tab{5}\spadcommand{s := create3Space()$(ThreeSpace DFLOAT)\bound{s }}
+\indentrel{3}\begin{verbatim}
+   (14)  3-Space with 0 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty14}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty14}{ugGraphThreeDOptionsPagePatch14}
+\pastebutton{ugGraphThreeDOptionsPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{s := create3Space()$(ThreeSpace DFLOAT)\bound{s }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch15}
+\begin{paste}{ugGraphThreeDOptionsPageFull15}{ugGraphThreeDOptionsPageEmpty15}
+\pastebutton{ugGraphThreeDOptionsPageFull15}{\hidepaste}
+\tab{5}\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1\bound{m }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty15}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty15}{ugGraphThreeDOptionsPagePatch15}
+\pastebutton{ugGraphThreeDOptionsPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == 1\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch16}
+\begin{paste}{ugGraphThreeDOptionsPageFull16}{ugGraphThreeDOptionsPageEmpty16}
+\pastebutton{ugGraphThreeDOptionsPageFull16}{\hidepaste}
+\tab{5}\spadgraph{draw(m,0..\%pi,0..2*\%pi, coordinates == spherical, space == s)\free{s m }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage16.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage16}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty16}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty16}{ugGraphThreeDOptionsPagePatch16}
+\pastebutton{ugGraphThreeDOptionsPageEmpty16}{\showpaste}
+\tab{5}\spadgraph{draw(m,0..\%pi,0..2*\%pi, coordinates == spherical, space == s)\free{s m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch17}
+\begin{paste}{ugGraphThreeDOptionsPageFull17}{ugGraphThreeDOptionsPageEmpty17}
+\pastebutton{ugGraphThreeDOptionsPageFull17}{\hidepaste}
+\tab{5}\spadgraph{v := draw(curve(1.5*sin(t), 1.5*cos(t),0), t=0..2*\%pi, tubeRadius == .25, space == s)\free{s }\bound{v }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage17.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage17}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty17}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty17}{ugGraphThreeDOptionsPagePatch17}
+\pastebutton{ugGraphThreeDOptionsPageEmpty17}{\showpaste}
+\tab{5}\spadgraph{v := draw(curve(1.5*sin(t), 1.5*cos(t),0), t=0..2*\%pi, tubeRadius == .25, space == s)\free{s }\bound{v }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch18}
+\begin{paste}{ugGraphThreeDOptionsPageFull18}{ugGraphThreeDOptionsPageEmpty18}
+\pastebutton{ugGraphThreeDOptionsPageFull18}{\hidepaste}
+\tab{5}\spadcommand{subsp := subspace v\free{v }\bound{su }}
+\indentrel{3}\begin{verbatim}
+   (18)  3-Space with 2 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty18}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty18}{ugGraphThreeDOptionsPagePatch18}
+\pastebutton{ugGraphThreeDOptionsPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{subsp := subspace v\free{v }\bound{su }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch19}
+\begin{paste}{ugGraphThreeDOptionsPageFull19}{ugGraphThreeDOptionsPageEmpty19}
+\pastebutton{ugGraphThreeDOptionsPageFull19}{\hidepaste}
+\tab{5}\spadcommand{subspace(v, subsp)\bound{sp }\free{su }}
+\indentrel{3}\begin{verbatim}
+   (19)
+   Closed or Undefined ThreeDimensionalViewport: "AXIOM3D"
+                         Type: ThreeDimensionalViewport
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty19}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty19}{ugGraphThreeDOptionsPagePatch19}
+\pastebutton{ugGraphThreeDOptionsPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{subspace(v, subsp)\bound{sp }\free{su }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPagePatch20}
+\begin{paste}{ugGraphThreeDOptionsPageFull20}{ugGraphThreeDOptionsPageEmpty20}
+\pastebutton{ugGraphThreeDOptionsPageFull20}{\hidepaste}
+\tab{5}\spadgraph{makeViewport3D(subsp,"Graphs")\free{sp }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage20.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDOptionsPage20}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphThreeDOptionsPageEmpty20}
+\begin{paste}{ugGraphThreeDOptionsPageEmpty20}{ugGraphThreeDOptionsPagePatch20}
+\pastebutton{ugGraphThreeDOptionsPageEmpty20}{\showpaste}
+\tab{5}\spadgraph{makeViewport3D(subsp,"Graphs")\free{sp }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphClipPagePatch1}
+\begin{paste}{ugGraphClipPageFull1}{ugGraphClipPageEmpty1}
+\pastebutton{ugGraphClipPageFull1}{\hidepaste}
+\tab{5}\spadcommand{gamma(x,y) ==
+  g := Gamma complex(x,y)
+  point [x, y, max( min(real g, 4), -4), argument g]
+\bound{g }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphClipPageEmpty1}
+\begin{paste}{ugGraphClipPageEmpty1}{ugGraphClipPagePatch1}
+\pastebutton{ugGraphClipPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{gamma(x,y) ==
+  g := Gamma complex(x,y)
+  point [x, y, max( min(real g, 4), -4), argument g]
+\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphClipPagePatch2}
+\begin{paste}{ugGraphClipPageFull2}{ugGraphClipPageEmpty2}
+\pastebutton{ugGraphClipPageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(gamma,-\%pi..\%pi,-\%pi..\%pi,var1Steps==50,var2Steps==50)\free{g }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphClipPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphClipPage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphClipPageEmpty2}
+\begin{paste}{ugGraphClipPageEmpty2}{ugGraphClipPagePatch2}
+\pastebutton{ugGraphClipPageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(gamma,-\%pi..\%pi,-\%pi..\%pi,var1Steps==50,var2Steps==50)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphCoordPagePatch1}
+\begin{paste}{ugGraphCoordPageFull1}{ugGraphCoordPageEmpty1}
+\pastebutton{ugGraphCoordPageFull1}{\hidepaste}
+\tab{5}\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == u**2\bound{m }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphCoordPageEmpty1}
+\begin{paste}{ugGraphCoordPageEmpty1}{ugGraphCoordPagePatch1}
+\pastebutton{ugGraphCoordPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{m(u:DFLOAT,v:DFLOAT):DFLOAT == u**2\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphCoordPagePatch2}
+\begin{paste}{ugGraphCoordPageFull2}{ugGraphCoordPageEmpty2}
+\pastebutton{ugGraphCoordPageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(m,0..3,0..5)\free{m }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphCoordPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphCoordPage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphCoordPageEmpty2}
+\begin{paste}{ugGraphCoordPageEmpty2}{ugGraphCoordPagePatch2}
+\pastebutton{ugGraphCoordPageEmpty2}{\showpaste}
+\tab{5}\spadgraph{draw(m,0..3,0..5)\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphCoordPagePatch3}
+\begin{paste}{ugGraphCoordPageFull3}{ugGraphCoordPageEmpty3}
+\pastebutton{ugGraphCoordPageFull3}{\hidepaste}
+\tab{5}\spadcommand{cartesian(point:Point DFLOAT):Point DFLOAT == point\bound{cart }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphCoordPageEmpty3}
+\begin{paste}{ugGraphCoordPageEmpty3}{ugGraphCoordPagePatch3}
+\pastebutton{ugGraphCoordPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{cartesian(point:Point DFLOAT):Point DFLOAT == point\bound{cart }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphCoordPagePatch4}
+\begin{paste}{ugGraphCoordPageFull4}{ugGraphCoordPageEmpty4}
+\pastebutton{ugGraphCoordPageFull4}{\hidepaste}
+\tab{5}\spadgraph{draw(m,0..3,0..5,coordinates==cartesian)\free{m cart }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphCoordPage4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphCoordPage4}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphCoordPageEmpty4}
+\begin{paste}{ugGraphCoordPageEmpty4}{ugGraphCoordPagePatch4}
+\pastebutton{ugGraphCoordPageEmpty4}{\showpaste}
+\tab{5}\spadgraph{draw(m,0..3,0..5,coordinates==cartesian)\free{m cart }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphCoordPagePatch5}
+\begin{paste}{ugGraphCoordPageFull5}{ugGraphCoordPageEmpty5}
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+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
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+
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+\indentrel{3}\begin{verbatim}
+   (7)  5
+                                  Type: PositiveInteger
+\end{verbatim}
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+
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+\indentrel{3}\begin{verbatim}
+   (8)  [1.0,2.0,3.0,5.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
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+
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+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
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+
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+\indentrel{3}\begin{verbatim}
+   (10)  [2.0,3.0,1.0,5.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphCoordPageEmpty10}
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+\tab{5}\spadcommand{newmap(pt:Point DFLOAT):Point DFLOAT == cylindrical(reorder pt)\free{freo }\bound{fnewmap }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphCoordPageEmpty11}
+\begin{paste}{ugGraphCoordPageEmpty11}{ugGraphCoordPagePatch11}
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+\tab{5}\spadcommand{newmap pt\free{fnewmap pt }\bound{new }}
+\indentrel{3}\begin{verbatim}
+   (12)
+   [- 1.9799849932008908,0.28224001611973443,1.0,5.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
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+
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+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDParPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDParPage2}}
+\end{paste}\end{patch}
+
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+
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+\tab{5}\spadcommand{f(t:DFLOAT):DFLOAT == sin(3*t/4)\bound{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDParPageEmpty3}
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+\tab{5}\spadcommand{g(t:DFLOAT):DFLOAT == sin(t)\bound{g }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
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+
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+\tab{5}\spadcommand{g(t:DFLOAT):DFLOAT == sin(t)\bound{g }}
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+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDParPage5.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDParPage5}}
+\end{paste}\end{patch}
+
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+
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+\pastebutton{ugGraphTwoDParPageFull6}{\hidepaste}
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+
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+
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+\pastebutton{ugGraphTwoDPlotPageFull1}{\hidepaste}
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+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDPlotPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDPlotPage1}}
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+
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+\tab{5}\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)}
+\end{paste}\end{patch}
+
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+\pastebutton{ugGraphTwoDPlotPageFull2}{\hidepaste}
+\tab{5}\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 10..16)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDPlotPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDPlotPage2}}
+\end{paste}\end{patch}
+
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+\tab{5}\spadgraph{draw(sin(tan(x)) - tan(sin(x)),x = 10..16)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDPlotPagePatch3}
+\begin{paste}{ugGraphTwoDPlotPageFull3}{ugGraphTwoDPlotPageEmpty3}
+\pastebutton{ugGraphTwoDPlotPageFull3}{\hidepaste}
+\tab{5}\spadcommand{f(x) == (x-1)*(x-2)*(x-3)\bound{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDPlotPageEmpty3}
+\begin{paste}{ugGraphTwoDPlotPageEmpty3}{ugGraphTwoDPlotPagePatch3}
+\pastebutton{ugGraphTwoDPlotPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f(x) == (x-1)*(x-2)*(x-3)\bound{f }}
+\end{paste}\end{patch}
+
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+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphTwoDPlotPage4.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphTwoDPlotPage4}}
+\end{paste}\end{patch}
+
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+
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+\begin{paste}{ugGraphThreeDParPageFull1}{ugGraphThreeDParPageEmpty1}
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+\tab{5}\spadgraph{draw(surface(u*cos(v), u*sin(v), v*cos(u)), u=-4..4, v=0..\%pi, coordinates== parabolicCylindrical)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugGraphThreeDParPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugGraphThreeDParPage1}}
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+
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+\tab{5}\spadgraph{draw(surface(u*cos(v), u*sin(v), v*cos(u)), u=-4..4, v=0..\%pi, coordinates== parabolicCylindrical)}
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+
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+\tab{5}\spadcommand{n1(u:DFLOAT,v:DFLOAT):DFLOAT == u*cos(v)\bound{n1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
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+\tab{5}\spadcommand{n2(u:DFLOAT,v:DFLOAT):DFLOAT == u*sin(v)\bound{n2 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
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+
+\begin{patch}{ugGraphThreeDParPageEmpty3}
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+\tab{5}\spadcommand{n2(u:DFLOAT,v:DFLOAT):DFLOAT == u*sin(v)\bound{n2 }}
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+
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+\tab{5}\spadcommand{n3(u:DFLOAT,v:DFLOAT):DFLOAT == u\bound{n3 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
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+
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+\tab{5}\spadcommand{n3(0.5,1.0)\free{n3 }}
+\indentrel{3}\begin{verbatim}
+   (5)  0.5
+                                      Type: DoubleFloat
+\end{verbatim}
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+
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+\begin{paste}{ugGraphThreeDParPageEmpty5}{ugGraphThreeDParPagePatch5}
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+\tab{5}\spadcommand{n3(0.5,1.0)\free{n3 }}
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+
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+\tab{5}\spadcommand{p := ((x**2 + y**2 + 1) - 8*x)**2 - (8*(x**2 + y**2 + 1)-4*x-1)\bound{p }}
+\indentrel{3}\begin{verbatim}
+   (1)
+    4      2            2    4      3      2
+   y  + (2x  - 16x - 6)y  + x  - 16x  + 58x  - 12x - 6
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugGraphTwoDPlanePageEmpty1}
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+\end{paste}\end{patch}
+
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+
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+\tab{5}\spadgraph{draw(p = 0, x, y, range == [-1..11, -7..7])\free{p }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ug08.ht b/src/hyper/pages/ug08.ht
new file mode 100644
index 00000000..f14e0289
--- /dev/null
+++ b/src/hyper/pages/ug08.ht
@@ -0,0 +1,4986 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\texht{\setcounter{chapter}{7}}{} % Chapter 8
+
+%
+\newcommand{\ugProblemTitle}{Advanced Problem Solving}
+\newcommand{\ugProblemNumber}{8.}
+%
+% =====================================================================
+\begin{page}{ugProblemPage}{8. Advanced Problem Solving}
+% =====================================================================
+\beginscroll
+
+In this chapter we describe techniques useful in solving advanced problems
+with \Language{}.
+
+\beginmenu
+    \menudownlink{{8.1. Numeric Functions}}{ugProblemNumericPage}
+    \menudownlink{{8.2. Polynomial Factorization}}{ugProblemFactorPage}
+    \menudownlink{{8.3. Manipulating Symbolic Roots of a Polynomial}}{ugProblemSymRootPage}
+    \menudownlink{{8.4. Computation of Eigenvalues and Eigenvectors}}{ugProblemEigenPage}
+    \menudownlink{{8.5. Solution of Linear and Polynomial Equations}}{ugProblemLinPolEqnPage}
+    \menudownlink{{8.6. Limits}}{ugProblemLimitsPage}
+    \menudownlink{{8.7. Laplace Transforms}}{ugProblemLaplacePage}
+    \menudownlink{{8.8. Integration}}{ugProblemIntegrationPage}
+    \menudownlink{{8.9. Working with Power Series}}{ugProblemSeriesPage}
+    \menudownlink{{8.10. Solution of Differential Equations}}{ugProblemDEQPage}
+    \menudownlink{{8.11. Finite Fields}}{ugProblemFinitePage}
+    \menudownlink{{8.12. Primary Decomposition of Ideals}}{ugProblemIdealPage}
+    \menudownlink{{8.13. Computation of Galois Groups}}{ugProblemGaloisPage}
+    \menudownlink{{8.14. Non-Associative Algebras and Modelling Genetic Laws}}{ugProblemGeneticPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemNumericTitle}{Numeric Functions}
+\newcommand{\ugProblemNumericNumber}{8.1.}
+%
+% =====================================================================
+\begin{page}{ugProblemNumericPage}{8.1. Numeric Functions}
+% =====================================================================
+\beginscroll
+%
+\Language{} provides two basic floating-point types: \axiomType{Float} and
+\axiomType{DoubleFloat}.  This section describes how to use numerical
+%-% \HDindex{function!numeric}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+operations defined on these types and the related complex types.
+%-% \HDindex{numeric operations}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+%
+As we mentioned in \downlink{``\ugIntroTitle''}{ugIntroPage} in Chapter \ugIntroNumber\ignore{ugIntro}, the \axiomType{Float} type is a software
+implementation of floating-point numbers in which the exponent and the
+%-% \HDindex{floating-point number}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+significand may have any number of digits.
+%-% \HDindex{number!floating-point}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+See \downlink{`Float'}{FloatXmpPage}\ignore{Float} for detailed information about this domain.
+The \axiomType{DoubleFloat} (see \downlink{`DoubleFloat'}{DoubleFloatXmpPage}\ignore{DoubleFloat}) is usually a hardware
+implementation of floating point numbers, corresponding to machine double
+precision.
+The types \axiomType{Complex Float} and \axiomType{Complex DoubleFloat} are
+%-% \HDindex{floating-point number!complex}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+the corresponding software implementations of complex floating-point numbers.
+%-% \HDindex{complex!floating-point number}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+In this section the term {\it floating-point type}  means any of these
+%-% \HDindex{number!complex floating-point}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+four types.
+%
+The floating-point types implement the basic elementary functions.
+These include (where \axiomSyntax{\$} means
+\axiomType{DoubleFloat},
+\axiomType{Float},
+\axiomType{Complex DoubleFloat}, or
+\axiomType{Complex Float}):
+
+\noindent
+\axiomFun{exp},  \axiomFun{log}: \axiom{\$ -> \$} \newline
+\axiomFun{sin},  \axiomFun{cos}, \axiomFun{tan}, \axiomFun{cot}, \axiomFun{sec}, \axiomFun{csc}: \axiom{\$ -> \$} \newline
+\axiomFun{sin},  \axiomFun{cos}, \axiomFun{tan}, \axiomFun{cot}, \axiomFun{sec}, \axiomFun{csc}: \axiom{\$ -> \$}  \newline
+\axiomFun{asin}, \axiomFun{acos}, \axiomFun{atan}, \axiomFun{acot}, \axiomFun{asec}, \axiomFun{acsc}: \axiom{\$ -> \$}  \newline
+\axiomFun{sinh},  \axiomFun{cosh}, \axiomFun{tanh}, \axiomFun{coth}, \axiomFun{sech}, \axiomFun{csch}: \axiom{\$ -> \$}  \newline
+\axiomFun{asinh}, \axiomFun{acosh}, \axiomFun{atanh}, \axiomFun{acoth}, \axiomFun{asech}, \axiomFun{acsch}: \axiom{\$ -> \$}  \newline
+\axiomFun{pi}: \axiom{() -> \$}  \newline
+\axiomFun{sqrt}: \axiom{\$ -> \$} \newline
+\axiomFun{nthRoot}: \axiom{(\$, Integer) -> \$}  \newline
+\axiomFunFrom{**}{Float}: \axiom{(\$, Fraction Integer) -> \$} \newline
+\axiomFunFrom{**}{Float}: \axiom{(\$,\$) -> \$}  \newline
+
+The handling of roots depends on whether the floating-point type
+%-% \HDindex{root!numeric approximation}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+is real or complex: for the real floating-point types,
+\axiomType{DoubleFloat} and \axiomType{Float}, if a real root exists
+the one with the same sign as the radicand is returned; for the
+complex floating-point types, the principal value is returned.
+%-% \HDindex{principal value}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+Also, for real floating-point types the inverse functions
+produce errors if the results are not real.
+This includes cases such as \axiom{asin(1.2)}, \axiom{log(-3.2)},
+\axiom{sqrt(-1.1)}.
+%
+\xtc{
+The default floating-point type is \axiomType{Float} so to evaluate
+functions using \axiomType{Float} or \axiomType{Complex Float}, just use
+normal decimal notation.
+}{
+\spadpaste{exp(3.1)}
+}
+\xtc{
+}{
+\spadpaste{exp(3.1 + 4.5 * \%i)}
+}
+\xtc{
+To evaluate functions using \axiomType{DoubleFloat}
+or \axiomType{Complex DoubleFloat},
+a declaration or conversion is required.
+}{
+\spadpaste{r: DFLOAT := 3.1; t: DFLOAT := 4.5; exp(r + t*\%i)}
+}
+\xtc{
+}{
+\spadpaste{exp(3.1::DFLOAT + 4.5::DFLOAT * \%i)}
+}
+%
+A number of special functions are provided by the package
+\axiomType{DoubleFloatSpecialFunctions} for the machine-precision
+%-% \HDindex{special functions}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+floating-point types.
+%-% \HDexptypeindex{DoubleFloatSpecialFunctions}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+The special functions provided are listed below, where \axiom{F} stands for
+the types \axiomType{DoubleFloat} and \axiomType{Complex DoubleFloat}.
+The real versions of the functions yield an error if the result is not real.
+%-% \HDindex{function!special}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+
+\noindent
+\axiomFun{Gamma}: \axiom{F -> F}\hfill\newline
+\axiom{Gamma(z)} is the Euler gamma function,
+%-% \HDindex{function!Gamma}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{$\Gamma(z)$}{\axiom{Gamma(z)}},
+   defined by
+%-% \HDindex{Euler!gamma function}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{\narrowDisplay{\Gamma(z) = \int_{0}^{\infty} t^{z-1} e^{-t} dt.}%
+   }{
+\newline
+\centerline{{\axiom{Gamma(z) = integrate(t**(z-1)*exp(-t), t=0..\%infinity).}}}
+   }
+
+\noindent
+\axiomFun{Beta}: \axiom{F -> F}\hfill\newline
+   \axiom{Beta(u, v)} is the Euler Beta function,
+%-% \HDindex{function!Euler Beta}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{$B(u,v)$}{\axiom{B(u,v)}}, defined by
+%-% \HDindex{Euler!Beta function}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{\narrowDisplay{B(u,v) = \int_{0}^{1} t^{u-1} (1-t)^{v-1} dt.}%
+   }{
+\newline
+\centerline{{      \axiom{Beta(u,v) = integrate(t**(u-1)*(1-t)**(b-1), t=0..1).}}}
+   }
+   This is related to \texht{$\Gamma(z)$}{\axiom{Gamma(z)}} by
+   \texht{\narrowDisplay{B(u,v) = \frac{\Gamma(u) \Gamma(v)}{\Gamma(u + v)}.}%
+   }{
+\newline
+\centerline{{      \axiom{Beta(u,v) = Gamma(u)*Gamma(v)/Gamma(u + v).}}}
+   }%
+
+\noindent
+\axiomFun{logGamma}: \axiom{F -> F}\hfill\newline
+   \axiom{logGamma(z)} is the natural logarithm of
+\texht{$\Gamma(z)$}{\axiom{Gamma(z)}}.
+   This can often be computed even if \texht{$\Gamma(z)$}{\axiom{Gamma(z)}}
+cannot.
+%
+
+\noindent
+\axiomFun{digamma}: \axiom{F -> F}\hfill\newline
+   \axiom{digamma(z)}, also called \axiom{psi(z)},
+%-% \HDindex{psi @ $\psi$}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+is the function \texht{$\psi(z)$}{\axiom{psi(z)}},
+%-% \HDindex{function!digamma}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   defined by \texht{\narrowDisplay{\psi(z) = \Gamma'(z)/\Gamma(z).}%
+   }{
+\newline
+\centerline{{      \axiom{psi(z) = Gamma'(z)/Gamma(z).}}}
+  }%
+
+\noindent
+\axiomFun{polygamma}: \axiom{(NonNegativeInteger, F) -> F}\hfill\newline
+   \axiom{polygamma(n, z)} is the \eth{\axiom{n}} derivative of
+%-% \HDindex{function!polygamma}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{$\psi(z)$}{\axiom{digamma(z)}}\texht{, written $\psi^{(n)}(z)$}{}.
+
+\noindent
+\axiomFun{besselJ}: \axiom{(F,F) -> F}\hfill\newline
+   \axiom{besselJ(v,z)} is the Bessel function of the first kind,
+%-% \HDindex{function!Bessel}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{$J_\nu (z)$}{\axiom{J(v,z)}}.
+   This function satisfies the differential equation
+   \texht{\narrowDisplay{z^2 w''(z) + z w'(z) + (z^2-\nu^2)w(z) = 0.}%
+   }{
+\newline
+\centerline{{      \axiom{z**2*w''(z) + z*w'(z) + (z**2-v**2)*w(z) = 0.}}}
+  }%
+
+\noindent
+\axiomFun{besselY}: \axiom{(F,F) -> F}\hfill\newline
+   \axiom{besselY(v,z)} is the Bessel function of the second kind,
+%-% \HDindex{function!Bessel}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{$Y_\nu (z)$}{\axiom{Y(v,z)}}.
+   This function satisfies the same differential equation as
+   \axiomFun{besselJ}.
+   The implementation simply uses the relation
+   \texht{\narrowDisplay{Y_\nu (z) = \frac{J_\nu (z) \cos(\nu \pi) - J_{-\nu} (z)}{\sin(\nu \pi)}.}%
+   }{
+\newline
+\centerline{{      \axiom{Y(v,z) = (J(v,z)*cos(v*\%pi) - J(-v,z))/sin(v*\%pi).}}}
+   }%
+
+\noindent
+\axiomFun{besselI}: \axiom{(F,F) -> F}\hfill\newline
+   \axiom{besselI(v,z)} is the modified Bessel function of the first kind,
+%-% \HDindex{function!Bessel}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{$I_\nu (z)$}{\axiom{I(v,z)}}.
+   This function satisfies the differential equation
+   \texht{\narrowDisplay{z^2 w''(z) + z w'(z) - (z^2+\nu^2)w(z) = 0.}%
+   }{
+\newline
+\centerline{{      \axiom{z**2 * w''(z) + z * w'(z) - (z**2+v**2)*w(z) = 0.}}}
+   }%
+
+\noindent
+\axiomFun{besselK}: \axiom{(F,F) -> F}\hfill\newline
+   \axiom{besselK(v,z)} is the modified Bessel function of the second kind,
+%-% \HDindex{function!Bessel}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{$K_\nu (z)$}{\axiom{K(v,z)}}.
+   This function satisfies the same differential equation as \axiomFun{besselI}.
+%-% \HDindex{Bessel function}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   The implementation simply uses the relation
+   \texht{\narrowDisplay{K_\nu (z) = \pi \frac{I_{-\nu} (z) - I_{\nu} (z)}{2 \sin(\nu \pi)}.}%
+   }{
+\newline
+\centerline{{      \axiom{K(v,z) = \%pi*(I(v,z) - I(-v,z))/(2*sin(v*\%pi)).}}}
+   }
+
+\noindent
+\axiomFun{airyAi}: \axiom{F -> F}\hfill\newline
+   \axiom{airyAi(z)} is the Airy function \texht{$Ai(z)$}{\axiom{Ai(z)}}.
+%-% \HDindex{function!Airy Ai}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   This function satisfies the differential equation
+   \texht{$w''(z) - z w(z) = 0.$}{\axiom{w''(z) - z * w(z) = 0.}}
+   The implementation simply uses the relation
+   \texht{\narrowDisplay{Ai(-z) = \frac{1}{3}\sqrt{z} ( J_{-1/3} (\frac{2}{3}z^{3/2}) + J_{1/3} (\frac{2}{3}z^{3/2}) ).}%
+   }{
+\newline
+\centerline{{      \axiom{Ai(-z) = 1/3 * sqrt(z) * (J(-1/3, 2/3*z**(3/2)) + J(1/3, 2/3*z**(3/2)) ).}}}
+   }%
+
+\noindent
+\axiomFun{airyBi}: \axiom{F -> F}\hfill\newline
+   \axiom{airyBi(z)} is the Airy function \texht{$Bi(z)$}{\axiom{Bi(z)}}.
+%-% \HDindex{function!Airy Bi}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   This function satisfies the same differential equation as \axiomFun{airyAi}.
+%-% \HDindex{Airy function}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   The implementation simply uses the relation
+   \texht{\narrowDisplay{Bi(-z) = \frac{1}{3}\sqrt{3 z} ( J_{-1/3} (\frac{2}{3}z^{3/2}) - J_{1/3} (\frac{2}{3}z^{3/2}) ).}%
+   }{
+\newline
+\centerline{{      \axiom{Bi(-z) = 1/3 *sqrt(3*z) * (J(-1/3, 2/3*z**(3/2)) - J(1/3, 2/3*z**(3/2)) ).}}}
+   }
+
+\noindent
+\axiomFun{hypergeometric0F1}: \axiom{(F,F) -> F}\hfill\newline
+   \axiom{hypergeometric0F1(c,z)} is the hypergeometric function
+%-% \HDindex{function!hypergeometric}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{${}_0 F_1 ( ; c; z)$}{\axiom{0F1(; c; z)}}.%
+
+\xtc{
+The above special functions are defined only for small floating-point types.
+If you give \axiomType{Float} arguments, they are converted to
+\axiomType{DoubleFloat} by \Language{}.
+}{
+\spadpaste{Gamma(0.5)**2}
+}
+\xtc{
+}{
+\spadpaste{a := 2.1; b := 1.1; besselI(a + \%i*b, b*a + 1)}
+}
+%
+A number of additional operations may be used to compute numerical values.
+These are special polynomial functions that can be evaluated for values in
+any commutative ring \axiom{R}, and in particular for values in any
+floating-point type.
+The following operations are provided by the package
+\axiomType{OrthogonalPolynomialFunctions}:
+%-% \HDexptypeindex{OrthogonalPolynomialFunctions}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+
+\noindent
+\axiomFun{chebyshevT}: \axiom{(NonNegativeInteger, R) -> R}\hbox{}\hfill\newline
+   \axiom{chebyshevT(n,z)} is the \eth{\axiom{n}} Chebyshev polynomial of the first
+   kind, \texht{$T_n (z)$}{\axiom{T[n](z)}}.  These are defined by
+   \texht{\narrowDisplay{\frac{1-t z}{1-2 t z+t^2} = \sum_{n=0}^{\infty} T_n (z) t^n.}%
+   }{
+\newline
+\centerline{{   \axiom{(1-t*z)/(1-2*t*z+t**2) = sum(T[n](z) *t**n, n = 0..).}}}
+   }%
+
+\noindent
+\axiomFun{chebyshevU}: \axiom{(NonNegativeInteger, R) -> R}\hbox{}\hfill\newline
+   \axiom{chebyshevU(n,z)} is the \eth{\axiom{n}} Chebyshev polynomial of the second
+   kind, \texht{$U_n (z)$}{\axiom{U[n](z)}}. These are defined by
+   \texht{\narrowDisplay{\frac{1}{1-2 t z+t^2} = \sum_{n=0}^{\infty} U_n (z) t^n.}%
+   }{
+\newline
+\centerline{{   \axiom{1/(1-2*t*z+t**2) = sum(U[n](z) *t**n, n = 0..).}}}
+   }%
+
+\noindent
+\axiomFun{hermiteH}:   \axiom{(NonNegativeInteger, R) -> R}\hbox{}\hfill\newline
+   \axiom{hermiteH(n,z)} is the \eth{\axiom{n}} Hermite polynomial,
+   \texht{$H_n (z)$}{\axiom{H[n](z)}}.
+   These are defined by
+   \texht{\narrowDisplay{e^{2 t z - t^2} = \sum_{n=0}^{\infty} H_n (z) \frac{t^n}{n!}.}%
+   }{
+\newline
+\centerline{{   \axiom{exp(2*t*z-t**2) = sum(H[n](z)*t**n/n!, n = 0..).}}}
+   }%
+
+\noindent
+\axiomFun{laguerreL}:  \axiom{(NonNegativeInteger, R) -> R}\hbox{}\hfill\newline
+   \axiom{laguerreL(n,z)} is the \eth{\axiom{n}} Laguerre polynomial,
+   \texht{$L_n (z)$}{\axiom{L[n](z)}}.
+   These are defined by
+   \texht{\narrowDisplay{\frac{e^{-\frac{t z}{1-t}}}{1-t} = \sum_{n=0}^{\infty} L_n (z) \frac{t^n}{n!}.}%
+   }{
+\newline
+\centerline{{   \axiom{exp(-t*z/(1-t))/(1-t) = sum(L[n](z)*t**n/n!, n = 0..).}}}
+   }%
+
+\noindent
+\axiomFun{laguerreL}:  \axiom{(NonNegativeInteger, NonNegativeInteger, R) -> R}\hbox{}\hfill\newline
+   \axiom{laguerreL(m,n,z)} is the associated Laguerre polynomial,
+   \texht{$L^m_n (z)$}{\axiom{L<m>[n](z)}}.
+   This is the \eth{\axiom{m}} derivative of \texht{$L_n (z)$}{\axiom{L[n](z)}}.
+
+\noindent
+\axiomFun{legendreP}:  \axiom{(NonNegativeInteger, R) -> R}\hbox{}\hfill\newline
+   \axiom{legendreP(n,z)} is the \eth{\axiom{n}} Legendre polynomial,
+   \texht{$P_n (z)$}{\axiom{P[n](z)}}.  These are defined by
+   \texht{\narrowDisplay{\frac{1}{\sqrt{1-2 t z+t^2}} = \sum_{n=0}^{\infty} P_n (z) t^n.}%
+   }{
+\newline
+\centerline{{   \axiom{1/sqrt(1-2*z*t+t**2) = sum(P[n](z)*t**n, n = 0..).}}}
+   }%
+
+%
+\xtc{
+These operations require non-negative integers for the indices, but otherwise
+the argument can be given as desired.
+}{
+\spadpaste{[chebyshevT(i, z) for i in 0..5]}
+}
+\xtc{
+The expression \axiom{chebyshevT(n,z)} evaluates to the \eth{\axiom{n}} Chebyshev
+%-% \HDindex{polynomial!Chebyshev!of the first kind}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+polynomial of the first kind.
+}{
+\spadpaste{chebyshevT(3, 5.0 + 6.0*\%i)}
+}
+\xtc{
+}{
+\spadpaste{chebyshevT(3, 5.0::DoubleFloat)}
+}
+\xtc{
+The expression \axiom{chebyshevU(n,z)} evaluates to the \eth{\axiom{n}} Chebyshev
+%-% \HDindex{polynomial!Chebyshev!of the second kind}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+polynomial of the second kind.
+}{
+\spadpaste{[chebyshevU(i, z) for i in 0..5]}
+}
+\xtc{
+}{
+\spadpaste{chebyshevU(3, 0.2)}
+}
+\xtc{
+The expression \axiom{hermiteH(n,z)} evaluates to the \eth{\axiom{n}} Hermite
+%-% \HDindex{polynomial!Hermite}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+polynomial.
+}{
+\spadpaste{[hermiteH(i, z) for i in 0..5]}
+}
+\xtc{
+}{
+\spadpaste{hermiteH(100, 1.0)}
+}
+\xtc{
+The expression \axiom{laguerreL(n,z)} evaluates to the \eth{\axiom{n}} Laguerre
+%-% \HDindex{polynomial!Laguerre}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+polynomial.
+}{
+\spadpaste{[laguerreL(i, z) for i in 0..4]}
+}
+\xtc{
+}{
+\spadpaste{laguerreL(4, 1.2)}
+}
+\xtc{
+}{
+\spadpaste{[laguerreL(j, 3, z) for j in 0..4]}
+}
+\xtc{
+}{
+\spadpaste{laguerreL(1, 3, 2.1)}
+}
+\xtc{
+The expression
+%-% \HDindex{polynomial!Legendre}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+\axiom{legendreP(n,z)} evaluates to the \eth{\axiom{n}} Legendre polynomial,
+}{
+\spadpaste{[legendreP(i,z) for i in 0..5]}
+}
+\xtc{
+}{
+\spadpaste{legendreP(3, 3.0*\%i)}
+}
+%
+
+Finally, three number-theoretic polynomial operations may be evaluated.
+%-% \HDindex{number theory}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+The following operations are provided by the package
+\axiomType{NumberTheoreticPolynomialFunctions}.
+%-% \HDexptypeindex{NumberTheoreticPolynomialFunctions}.{ugProblemNumericPage}{8.1.}{Numeric Functions}
+
+\noindent
+\axiomFun{bernoulliB}: \axiom{(NonNegativeInteger, R) -> R} \hbox{}\hfill\newline
+   \axiom{bernoulliB(n,z)} is the \eth{\axiom{n}} Bernoulli polynomial,
+%-% \HDindex{polynomial!Bernoulli}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{$B_n (z)$}{\axiom{B[n](z)}}.  These are defined by
+   \texht{\narrowDisplay{\frac{t e^{z t}}{e^t - 1} = \sum_{n=0}^{\infty} B_n (z) \frac{t^n}{n!}.}
+   }{
+\newline
+\centerline{{   \axiom{t*exp(z*t)/(exp t - 1) = sum(B[n](z)*t**n/n! for n - 0..)}}}
+   }%
+
+\noindent
+\axiomFun{eulerE}: \axiom{(NonNegativeInteger, R) -> R} \hbox{}\hfill\newline
+   \axiom{eulerE(n,z)} is the \eth{\axiom{n}} Euler polynomial,
+%-% \HDindex{Euler!polynomial}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{$E_n (z)$}{\axiom{E[n](z)}}.  These are defined by
+%-% \HDindex{polynomial!Euler}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{\narrowDisplay{\frac{2 e^{z t}}{e^t + 1} = \sum_{n=0}^{\infty} E_n (z) \frac{t^n}{n!}.}%
+   }{
+\newline
+\centerline{{   \axiom{2*exp(z*t)/(exp t + 1) = sum(E[n](z)*t**n/n! for n - 0..)}}}
+   }%
+
+\noindent
+\axiomFun{cyclotomic}: \axiom{(NonNegativeInteger, R) -> R}\hbox{}\hfill\newline
+   \axiom{cyclotomic(n,z)} is the \eth{\axiom{n}} cyclotomic polynomial
+   \texht{$\Phi_n (z)$}{\axiom{phi(n,z)}}.  This is the polynomial whose
+   roots are precisely the primitive \eth{\axiom{n}} roots of unity.
+%-% \HDindex{Euler!totient function}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   This polynomial has degree given by the Euler totient function
+%-% \HDindex{function!totient}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+   \texht{$\phi(n)$}{\axiom{phi(n)}}.
+
+\xtc{
+The expression \axiom{bernoulliB(n,z)} evaluates to the \eth{\axiom{n}} Bernoulli
+%-% \HDindex{polynomial!Bernouilli}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+polynomial.
+}{
+\spadpaste{bernoulliB(3, z)}
+}
+\xtc{
+}{
+\spadpaste{bernoulliB(3, 0.7 + 0.4 * \%i)}
+}
+\xtc{
+The expression
+%-% \HDindex{polynomial!Euler}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+\axiom{eulerE(n,z)} evaluates to the \eth{\axiom{n}} Euler polynomial.
+}{
+\spadpaste{eulerE(3, z)}
+}
+\xtc{
+}{
+\spadpaste{eulerE(3, 0.7 + 0.4 * \%i)}
+}
+\xtc{
+The expression
+%-% \HDindex{polynomial!cyclotomic}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+\axiom{cyclotomic(n,z)} evaluates to the \eth{\axiom{n}} cyclotomic polynomial.
+%-% \HDindex{cyclotomic polynomial}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+}{
+\spadpaste{cyclotomic(3, z)}
+}
+\xtc{
+}{
+\spadpaste{cyclotomic(3, (-1.0 + 0.0 * \%i)**(2/3))}
+}
+
+Drawing complex functions in \Language{} is presently somewhat
+awkward compared to drawing real functions.
+It is necessary to use the \axiomFun{draw} operations that operate
+on functions rather than expressions.
+
+\psXtc{
+This is the complex exponential function\texht{ (rotated interactively).}{.}
+%-% \HDindex{function!complex exponential}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+When this is displayed in color, the height is the value of the real part of
+the function and the color is the imaginary part.
+Red indicates large negative imaginary values, green indicates imaginary
+values near zero and blue/violet indicates large positive imaginary values.
+}{
+\graphpaste{draw((x,y)+-> real exp complex(x,y), -2..2, -2*\%pi..2*\%pi, colorFunction == (x, y) +->  imag exp complex(x,y), title=="exp(x+\%i*y)", style=="smooth")}
+}{
+\epsffile[0 0 295 295]{../ps/compexp.ps}
+}
+
+\psXtc{
+This is the complex arctangent function.
+%-% \HDindex{function!complex arctangent}{ugProblemNumericPage}{8.1.}{Numeric Functions}
+Again, the height is the real part of the function value but here
+the color indicates the function value's phase.
+The position of the branch cuts are clearly visible and one can
+see that the function is real only for a real argument.
+}{
+\graphpaste{vp := draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction==(x,y) +->argument atan complex(x,y), title=="atan(x+\%i*y)", style=="shade"); rotate(vp,-160,-45); vp}
+}{
+\epsffile[0 0 295 295]{../ps/compatan.ps}
+}
+
+\psXtc{
+This is the complex Gamma function.
+}{
+\graphpaste{draw((x,y) +-> max(min(real Gamma complex(x,y),4),-4), -\%pi..\%pi, -\%pi..\%pi, style=="shade", colorFunction == (x,y) +-> argument Gamma complex(x,y), title == "Gamma(x+\%i*y)", var1Steps == 50, var2Steps== 50)}
+}{
+\epsffile[0 0 295 295]{../ps/compgamm.ps}
+}
+
+\psXtc{
+This shows the real Beta function near the origin.
+}{
+\graphpaste{draw(Beta(x,y)/100, x=-1.6..1.7, y = -1.6..1.7, style=="shade", title=="Beta(x,y)", var1Steps==40, var2Steps==40)}
+}{
+\epsffile[0 0 295 295]{../ps/realbeta.ps}
+}
+
+\psXtc{
+This is the Bessel function \texht{$J_\alpha (x)$}{\axiom{J(alpha,x)}}
+for index \texht{$\alpha$}{\axiom{alpha}} in the range \axiom{-6..4} and
+argument \texht{$x$}{\axiom{x}} in the range \axiom{2..14}.
+}{
+\graphpaste{draw((alpha,x) +-> min(max(besselJ(alpha, x+8), -6), 6), -6..4, -6..6, title=="besselJ(alpha,x)", style=="shade", var1Steps==40, var2Steps==40)}
+}{
+\epsffile[0 0 295 295]{../ps/bessel.ps}
+}
+
+\psXtc{
+This is the modified Bessel function
+\texht{$I_\alpha (x)$}{\axiom{I(alpha,x)}}
+evaluated for various real values of the index \texht{$\alpha$}{\axiom{alpha}}
+and fixed argument \texht{$x = 5$}{\axiom{x = 5}}.
+}{
+\graphpaste{draw(besselI(alpha, 5), alpha = -12..12, unit==[5,20])}
+}{
+\epsffile[0 0 295 295]{../ps/modbess.ps}
+}
+
+\psXtc{
+This is similar to the last example
+except the index \texht{$\alpha$}{\axiom{alpha}}
+takes on complex values in a \axiom{6 x 6} rectangle  centered on the origin.
+}{
+\graphpaste{draw((x,y) +-> real besselI(complex(x/20, y/20),5), -60..60, -60..60, colorFunction == (x,y)+-> argument besselI(complex(x/20,y/20),5), title=="besselI(x+i*y,5)", style=="shade")}
+}{
+\epsffile[0 0 295 295]{../ps/modbessc.ps}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemFactorTitle}{Polynomial Factorization}
+\newcommand{\ugProblemFactorNumber}{8.2.}
+%
+% =====================================================================
+\begin{page}{ugProblemFactorPage}{8.2. Polynomial Factorization}
+% =====================================================================
+\beginscroll
+%
+The \Language{} polynomial factorization
+%-% \HDindex{polynomial!factorization}{ugProblemFactorPage}{8.2.}{Polynomial Factorization}
+facilities are available for all polynomial types and a wide variety of
+coefficient domains.
+%-% \HDindex{factorization}{ugProblemFactorPage}{8.2.}{Polynomial Factorization}
+Here are some examples.
+
+\beginmenu
+    \menudownlink{{8.2.1. Integer and Rational Number Coefficients}}{ugProblemFactorIntRatPage}
+    \menudownlink{{8.2.2. Finite Field Coefficients}}{ugProblemFactorFFPage}
+    \menudownlink{{8.2.3. Simple Algebraic Extension Field Coefficients}}{ugProblemFactorAlgPage}
+    \menudownlink{{8.2.4. Factoring Rational Functions}}{ugProblemFactorRatFunPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemFactorIntRatTitle}{Integer and Rational Number Coefficients}
+\newcommand{\ugProblemFactorIntRatNumber}{8.2.1.}
+%
+% =====================================================================
+\begin{page}{ugProblemFactorIntRatPage}{8.2.1. Integer and Rational Number Coefficients}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+\xtc{
+Polynomials with integer
+%-% \HDindex{polynomial!factorization!integer coefficients}{ugProblemFactorIntRatPage}{8.2.1.}{Integer and Rational Number Coefficients}
+coefficients can be be factored.
+}{
+\spadpaste{v := (4*x**3+2*y**2+1)*(12*x**5-x**3*y+12) \bound{v}}
+}
+\xtc{
+}{
+\spadpaste{factor v \free{v}}
+}
+\xtc{
+Also, \Language{} can factor polynomials with
+%-% \HDindex{polynomial!factorization!rational number coefficients}{ugProblemFactorIntRatPage}{8.2.1.}{Integer and Rational Number Coefficients}
+rational number coefficients.
+}{
+\spadpaste{w := (4*x**3+(2/3)*x**2+1)*(12*x**5-(1/2)*x**3+12) \bound{w}}
+}
+\xtc{
+}{
+\spadpaste{factor w \free{w}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemFactorFFTitle}{Finite Field Coefficients}
+\newcommand{\ugProblemFactorFFNumber}{8.2.2.}
+%
+% =====================================================================
+\begin{page}{ugProblemFactorFFPage}{8.2.2. Finite Field Coefficients}
+% =====================================================================
+\beginscroll
+
+Polynomials with coefficients in a finite field
+%-% \HDindex{polynomial!factorization!finite field coefficients}{ugProblemFactorFFPage}{8.2.2.}{Finite Field Coefficients}
+can be also be factored.
+%-% \HDindex{finite field!factoring polynomial with coefficients in}{ugProblemFactorFFPage}{8.2.2.}{Finite Field Coefficients}
+
+\labelSpace{3pc}
+\xtc{
+}{
+\spadpaste{u : POLY(PF(19)) :=3*x**4+2*x**2+15*x+18 \bound{u}}
+}
+\xtc{
+These include the integers mod \axiom{p}, where \axiom{p} is prime, and
+extensions of these fields.
+}{
+\spadpaste{factor u \free{u}}
+}
+\xtc{
+Convert this to have coefficients in the finite
+field with \texht{$19^3$}{\axiom{19**3}} elements.
+See \downlink{``\ugProblemFiniteTitle''}{ugProblemFinitePage} in Section \ugProblemFiniteNumber\ignore{ugProblemFinite} for more information
+about finite fields.
+}{
+\spadpaste{factor(u :: POLY FFX(PF 19,3)) \free{u}}
+}
+%
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemFactorAlgTitle}{Simple Algebraic Extension Field Coefficients}
+\newcommand{\ugProblemFactorAlgNumber}{8.2.3.}
+%
+% =====================================================================
+\begin{page}{ugProblemFactorAlgPage}{8.2.3. Simple Algebraic Extension Field Coefficients}
+% =====================================================================
+\beginscroll
+
+Polynomials with coefficients in simple algebraic extensions
+%-% \HDindex{polynomial!factorization!algebraic extension field coefficients}{ugProblemFactorAlgPage}{8.2.3.}{Simple Algebraic Extension Field Coefficients}
+of the rational numbers can be factored.
+%-% \HDindex{algebraic number}{ugProblemFactorAlgPage}{8.2.3.}{Simple Algebraic Extension Field Coefficients}
+%-% \HDindex{number!algebraic}{ugProblemFactorAlgPage}{8.2.3.}{Simple Algebraic Extension Field Coefficients}
+
+\labelSpace{2pc}
+\xtc{
+Here, \axiom{aa} and \axiom{bb} are symbolic roots of polynomials.
+}{
+\spadpaste{aa := rootOf(aa**2+aa+1) \bound{aa}}
+}
+\xtc{
+}{
+\spadpaste{p:=(x**3+aa**2*x+y)*(aa*x**2+aa*x+aa*y**2)**2 \free{aa}\bound{p}}
+}
+\xtc{
+Note that the second argument to factor can be a list of
+algebraic extensions to factor over.
+}{
+\spadpaste{factor(p,[aa]) \free{p aa}}
+}
+\xtc{
+This factors \axiom{x**2+3} over the integers.
+}{
+\spadpaste{factor(x**2+3)}
+}
+\xtc{
+Factor the same polynomial over the field obtained by adjoining
+\axiom{aa} to the rational numbers.
+}{
+\spadpaste{factor(x**2+3,[aa]) \free{aa}}
+}
+\xtc{
+Factor \axiom{x**6+108} over the same field.
+}{
+\spadpaste{factor(x**6+108,[aa]) \free{aa}}
+}
+\xtc{
+}{
+\spadpaste{bb:=rootOf(bb**3-2) \bound{bb}}
+}
+\xtc{
+}{
+\spadpaste{factor(x**6+108,[bb]) \free{bb}}
+}
+\xtc{
+Factor again over the field obtained by adjoining both \axiom{aa}
+and \axiom{bb} to the rational numbers.
+}{
+\spadpaste{factor(x**6+108,[aa,bb]) \free{aa bb}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemFactorRatFunTitle}{Factoring Rational Functions}
+\newcommand{\ugProblemFactorRatFunNumber}{8.2.4.}
+%
+% =====================================================================
+\begin{page}{ugProblemFactorRatFunPage}{8.2.4. Factoring Rational Functions}
+% =====================================================================
+\beginscroll
+
+Since fractions of polynomials form a field, every element (other than zero)
+%-% \HDindex{rational function!factoring}{ugProblemFactorRatFunPage}{8.2.4.}{Factoring Rational Functions}
+divides any other, so there is no useful notion of irreducible factors.
+Thus the \axiomFun{factor} operation is not very useful for fractions
+of polynomials.
+
+\xtc{
+There is, instead, a specific operation \axiomFun{factorFraction}
+that separately factors the numerator and denominator and returns
+a fraction of the factored results.
+}{
+\spadpaste{factorFraction((x**2-4)/(y**2-4))}
+}
+\xtc{
+You can also use \axiomFun{map}. This expression
+applies the \axiomFun{factor} operation
+to the numerator and denominator.
+}{
+\spadpaste{map(factor,(x**2-4)/(y**2-4))}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemSymRootTitle}{Manipulating Symbolic Roots of a Polynomial}
+\newcommand{\ugProblemSymRootNumber}{8.3.}
+%
+% =====================================================================
+\begin{page}{ugProblemSymRootPage}{8.3. Manipulating Symbolic Roots of a Polynomial}
+% =====================================================================
+\beginscroll
+%
+In this section we show you how to work with one root or all roots
+%-% \HDindex{root!symbolic}{ugProblemSymRootPage}{8.3.}{Manipulating Symbolic Roots of a Polynomial}
+of a polynomial.
+These roots are represented symbolically (as opposed to being
+numeric approximations).
+See \downlink{``\ugxProblemOnePolTitle''}{ugxProblemOnePolPage} in Section \ugxProblemOnePolNumber\ignore{ugxProblemOnePol} and \downlink{``\ugxProblemPolSysTitle''}{ugxProblemPolSysPage} in Section \ugxProblemPolSysNumber\ignore{ugxProblemPolSys} for
+information about solving for the roots of one or more
+polynomials.
+
+\beginmenu
+    \menudownlink{{8.3.1. Using a Single Root of a Polynomial}}{ugxProblemSymRootOnePage}
+    \menudownlink{{8.3.2. Using All Roots of a Polynomial}}{ugxProblemSymRootAllPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemSymRootOneTitle}{Using a Single Root of a Polynomial}
+\newcommand{\ugxProblemSymRootOneNumber}{8.3.1.}
+%
+% =====================================================================
+\begin{page}{ugxProblemSymRootOnePage}{8.3.1. Using a Single Root of a Polynomial}
+% =====================================================================
+\beginscroll
+
+Use \axiomFun{rootOf} to get a symbolic root of a polynomial:
+\axiom{rootOf(p, x)} returns a root of \axiom{p(x)}.
+
+\labelSpace{2pc}
+\xtc{
+This creates an algebraic number \axiom{a}.
+%-% \HDindex{algebraic number}{ugxProblemSymRootOnePage}{8.3.1.}{Using a Single Root of a Polynomial}
+%-% \HDindex{number!algebraic}{ugxProblemSymRootOnePage}{8.3.1.}{Using a Single Root of a Polynomial}
+}{
+\spadpaste{a := rootOf(a**4+1,a) \bound{a}}
+}
+\xtc{
+To find the algebraic relation that defines \axiom{a},
+use \axiomFun{definingPolynomial}.
+}{
+\spadpaste{definingPolynomial a \free{a}}
+}
+\xtc{
+You can use \axiom{a} in any further expression,
+including a nested \axiomFun{rootOf}.
+}{
+\spadpaste{b := rootOf(b**2-a-1,b) \free{a}\bound{b}}
+}
+\xtc{
+Higher powers of the roots are automatically reduced during
+calculations.
+}{
+\spadpaste{a + b \free{a b}\bound{c}}
+}
+\xtc{
+}{
+\spadpaste{\% ** 5 \free{c}}
+}
+\xtc{
+The operation \axiomFun{zeroOf} is similar to \axiomFun{rootOf},
+except that it may express the root using radicals in some cases.
+%-% \HDindex{radical}{ugxProblemSymRootOnePage}{8.3.1.}{Using a Single Root of a Polynomial}
+}{
+\spadpaste{rootOf(c**2+c+1,c)}
+}
+\xtc{
+}{
+\spadpaste{zeroOf(d**2+d+1,d)}
+}
+\xtc{
+}{
+\spadpaste{rootOf(e**5-2,e)}
+}
+\xtc{
+}{
+\spadpaste{zeroOf(f**5-2,f)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemSymRootAllTitle}{Using All Roots of a Polynomial}
+\newcommand{\ugxProblemSymRootAllNumber}{8.3.2.}
+%
+% =====================================================================
+\begin{page}{ugxProblemSymRootAllPage}{8.3.2. Using All Roots of a Polynomial}
+% =====================================================================
+\beginscroll
+
+Use \axiomFun{rootsOf} to get all symbolic roots of a polynomial:
+\axiom{rootsOf(p, x)} returns a
+list of all the roots of \axiom{p(x)}.
+If \axiom{p(x)} has a multiple root of order \axiom{n}, then that root
+%-% \HDindex{root!multiple}{ugxProblemSymRootAllPage}{8.3.2.}{Using All Roots of a Polynomial}
+appears \axiom{n} times in the list.
+\texht{\typeout{Make sure these variables are x0 etc}}{}
+
+\xtc{
+Compute all the roots of \axiom{x**4 + 1}.
+}{
+\spadpaste{l := rootsOf(x**4+1,x) \bound{l}}
+}
+\xtc{
+As a side effect, the variables \axiom{\%x0, \%x1} and \axiom{\%x2} are bound
+to the first three roots of \axiom{x**4+1}.
+}{
+\spadpaste{\%x0**5 \free{l}}
+}
+\xtc{
+Although they all satisfy \axiom{x**4 + 1 = 0, \%x0, \%x1,}
+and \axiom{\%x2} are different algebraic numbers.
+To find the algebraic relation that defines each of them,
+use \axiomFun{definingPolynomial}.
+}{
+\spadpaste{definingPolynomial \%x0 \free{l}}
+}
+\xtc{
+}{
+\spadpaste{definingPolynomial \%x1 \free{l}}
+}
+\xtc{
+}{
+\spadpaste{definingPolynomial \%x2 \free{l}}
+}
+\xtc{
+We can check that the sum and product of the roots of \axiom{x**4+1} are
+its trace and norm.
+}{
+\spadpaste{x3 := last l \free{l} \bound{x3}}
+}
+\xtc{
+}{
+\spadpaste{\%x0 + \%x1 + \%x2 + x3 \free{x3}}
+}
+\xtc{
+}{
+\spadpaste{\%x0 * \%x1 * \%x2 * x3 \free{x3}}
+}
+\xtc{
+Corresponding to the pair of operations
+\axiomFun{rootOf}/\axiomFun{zeroOf} in
+\downlink{``\ugxProblemOnePolTitle''}{ugxProblemOnePolPage} in Section \ugxProblemOnePolNumber\ignore{ugxProblemOnePol}, there is
+an operation \axiomFun{zerosOf} that, like \axiomFun{rootsOf},
+computes all the roots
+of a given polynomial, but which expresses some of them in terms of
+radicals.
+}{
+\spadpaste{zerosOf(y**4+1,y) \bound{z}}
+}
+\xtc{
+As you see, only one implicit algebraic number was created
+(\axiom{\%y1}), and its defining equation is this.
+The other three roots are expressed in radicals.
+}{
+\spadpaste{definingPolynomial \%y1 \free{z}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemEigenTitle}{Computation of Eigenvalues and Eigenvectors}
+\newcommand{\ugProblemEigenNumber}{8.4.}
+%
+% =====================================================================
+\begin{page}{ugProblemEigenPage}{8.4. Computation of Eigenvalues and Eigenvectors}
+% =====================================================================
+\beginscroll
+%
+In this section we show you
+some of \Language{}'s facilities for computing and
+%-% \HDindex{eigenvalue}{ugProblemEigenPage}{8.4.}{Computation of Eigenvalues and Eigenvectors}
+manipulating eigenvalues and eigenvectors, also called
+%-% \HDindex{eigenvector}{ugProblemEigenPage}{8.4.}{Computation of Eigenvalues and Eigenvectors}
+characteristic values and characteristic vectors,
+%-% \HDindex{characteristic!value}{ugProblemEigenPage}{8.4.}{Computation of Eigenvalues and Eigenvectors}
+respectively.
+%-% \HDindex{characteristic!vector}{ugProblemEigenPage}{8.4.}{Computation of Eigenvalues and Eigenvectors}
+
+\texht{\vskip 4pc}{}
+
+\xtc{
+Let's first create a matrix with integer entries.
+}{
+\spadpaste{m1 := matrix [[1,2,1],[2,1,-2],[1,-2,4]] \bound{m1}}
+}
+\xtc{
+To get a list of the {\it rational} eigenvalues,
+use the operation \axiomFun{eigenvalues}.
+}{
+\spadpaste{leig := eigenvalues(m1) \free{m1} \bound{leig}}
+}
+\xtc{
+Given an explicit eigenvalue, \axiomFun{eigenvector} computes the eigenvectors
+corresponding to it.
+}{
+\spadpaste{eigenvector(first(leig),m1) \free{m1 leig}}
+}
+
+The operation \axiomFun{eigenvectors} returns a list of pairs of values and
+vectors. When an eigenvalue is rational, \Language{} gives you
+the value explicitly; otherwise, its minimal polynomial is given,
+(the polynomial of lowest degree with the eigenvalues as roots),
+together with a parametric representation of the eigenvector using the
+eigenvalue.
+This means that if you ask \Language{} to \axiomFun{solve}
+the minimal polynomial, then you can substitute these roots
+%-% \HDindex{polynomial!minimal}{ugProblemEigenPage}{8.4.}{Computation of Eigenvalues and Eigenvectors}
+into the parametric form of the corresponding eigenvectors.
+%-% \HDindex{minimal polynomial}{ugProblemEigenPage}{8.4.}{Computation of Eigenvalues and Eigenvectors}
+
+\xtc{
+You must be aware that unless an exact eigenvalue has been computed,
+the eigenvector may be badly in error.
+}{
+\spadpaste{eigenvectors(m1) \free{m1}}
+}
+\xtc{
+Another possibility is to use the operation
+\axiomFun{radicalEigenvectors}
+tries to compute explicitly the eigenvectors
+in terms of radicals.
+%-% \HDindex{radical}{ugProblemEigenPage}{8.4.}{Computation of Eigenvalues and Eigenvectors}
+}{
+\spadpaste{radicalEigenvectors(m1) \free{m1}}
+}
+
+Alternatively, \Language{} can compute real or complex approximations to the
+%-% \HDindex{approximation}{ugProblemEigenPage}{8.4.}{Computation of Eigenvalues and Eigenvectors}
+eigenvectors and eigenvalues using the operations \axiomFun{realEigenvectors}
+or \axiomFun{complexEigenvectors}.
+They each take an additional argument \texht{$\epsilon$}{\axiom{epsilon}}
+to specify the ``precision'' required.
+%-% \HDindex{precision}{ugProblemEigenPage}{8.4.}{Computation of Eigenvalues and Eigenvectors}
+In the real case, this means that each approximation will be within
+\texht{$\pm\epsilon$}{plus or minus \axiom{epsilon}} of the actual
+result.
+In the complex case, this means that each approximation will be within
+\texht{$\pm\epsilon$}{plus or minus \axiom{epsilon}} of the actual result
+in each of the real and imaginary parts.
+
+\xtc{
+The precision can be specified as a \axiomType{Float} if the results are
+desired in floating-point notation, or as \axiomType{Fraction Integer} if the
+results are to be expressed using rational (or complex rational) numbers.
+}{
+\spadpaste{realEigenvectors(m1,1/1000) \free{m1}}
+}
+\xtc{
+If an \axiom{n} by \axiom{n} matrix has \axiom{n} distinct eigenvalues (and
+therefore \axiom{n} eigenvectors) the operation \axiomFun{eigenMatrix}
+gives you a matrix of the eigenvectors.
+}{
+\spadpaste{eigenMatrix(m1) \free{m1}}
+}
+\xtc{
+}{
+\spadpaste{m2 := matrix [[-5,-2],[18,7]] \bound{m2}}
+}
+\xtc{
+}{
+\spadpaste{eigenMatrix(m2) \free{m2}}
+}
+%
+%
+\xtc{
+If a symmetric matrix
+%-% \HDindex{matrix!symmetric}{ugProblemEigenPage}{8.4.}{Computation of Eigenvalues and Eigenvectors}
+has a basis of orthonormal eigenvectors, then
+%-% \HDindex{basis!orthonormal}{ugProblemEigenPage}{8.4.}{Computation of Eigenvalues and Eigenvectors}
+\axiomFun{orthonormalBasis} computes a list of these vectors.
+%-% \HDindex{orthonormal basis}{ugProblemEigenPage}{8.4.}{Computation of Eigenvalues and Eigenvectors}
+}{
+\spadpaste{m3 := matrix [[1,2],[2,1]] \bound{m3}}
+}
+\xtc{
+}{
+\spadpaste{orthonormalBasis(m3) \free{m3}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemLinPolEqnTitle}{Solution of Linear and Polynomial Equations}
+\newcommand{\ugProblemLinPolEqnNumber}{8.5.}
+%
+% =====================================================================
+\begin{page}{ugProblemLinPolEqnPage}{8.5. Solution of Linear and Polynomial Equations}
+% =====================================================================
+\beginscroll
+%
+In this section we discuss the \Language{} facilities for solving
+systems of linear equations, finding the roots of polynomials and
+%-% \HDindex{linear equation}{ugProblemLinPolEqnPage}{8.5.}{Solution of Linear and Polynomial Equations}
+solving systems of polynomial equations.
+For a discussion of the solution of differential equations, see
+\downlink{``\ugProblemDEQTitle''}{ugProblemDEQPage} in Section \ugProblemDEQNumber\ignore{ugProblemDEQ}.
+
+\beginmenu
+    \menudownlink{{8.5.1. Solution of Systems of Linear Equations}}{ugxProblemLinSysPage}
+    \menudownlink{{8.5.2. Solution of a Single Polynomial Equation}}{ugxProblemOnePolPage}
+    \menudownlink{{8.5.3. Solution of Systems of Polynomial Equations}}{ugxProblemPolSysPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemLinSysTitle}{Solution of Systems of Linear Equations}
+\newcommand{\ugxProblemLinSysNumber}{8.5.1.}
+%
+% =====================================================================
+\begin{page}{ugxProblemLinSysPage}{8.5.1. Solution of Systems of Linear Equations}
+% =====================================================================
+\beginscroll
+
+You can use the operation \axiomFun{solve} to solve systems of linear equations.
+%-% \HDindex{equation!linear!solving}{ugxProblemLinSysPage}{8.5.1.}{Solution of Systems of Linear Equations}
+
+The operation \axiomFun{solve} takes two arguments, the list of equations and the
+list of the unknowns to be solved for.
+A system of linear equations need not have a unique solution.
+
+\xtc{
+To solve the linear system:
+\centerline{{\axiom{  x +   y +   z = 8} }}
+\centerline{{\axiom{3*x - 2*y +   z = 0} }}
+\centerline{{\axiom{  x + 2*y + 2*z = 17}}}
+evaluate this expression.
+}{
+\spadpaste{solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])}
+}
+
+Parameters are given as new variables starting with a percent sign and
+\spadSyntax{\%} and
+the variables are expressed in terms of the parameters.
+If the system has no solutions then the empty list is returned.
+
+\xtc{
+When you solve the linear system
+\centerline{{\axiom{  x + 2*y + 3*z = 2} }}
+\centerline{{\axiom{2*x + 3*y + 4*z = 2} }}
+\centerline{{\axiom{3*x + 4*y + 5*z = 2}}}
+with this expression
+you get a solution involving a parameter.
+}{
+\spadpaste{solve([x+2*y+3*z=2,2*x+3*y+4*z=2,3*x+4*y+5*z=2],[x,y,z])}
+}
+
+The system can also be presented as a matrix and a vector.
+The matrix contains the coefficients of the linear equations and
+the vector contains the numbers appearing on the right-hand sides
+of the equations.
+You may input the matrix as a list of rows and the vector as a
+list of its elements.
+
+\xtc{
+To solve the system:
+\centerline{{\axiom{  x +   y +   z = 8} }}
+\centerline{{\axiom{3*x - 2*y +   z = 0} }}
+\centerline{{\axiom{  x + 2*y + 2*z = 17}}}
+in matrix form you would evaluate this expression.
+}{
+\spadpaste{solve([[1,1,1],[3,-2,1],[1,2,2]],[8,0,17])}
+}
+
+The solutions are presented as a \pspadtype{Record} with two
+components: the component \texht{{\it particular}}{{\tt
+particular}} contains a particular solution of the given system or
+the item {\tt "failed"} if there are no solutions, the component
+\texht{{\it basis}}{{\tt basis}} contains a list of vectors that
+are a basis for the space of solutions of the corresponding
+homogeneous system.
+If the system of linear equations does not have a unique solution,
+then the \texht{{\it basis}}{{\tt basis}} component contains
+non-trivial vectors.
+
+\xtc{
+This happens when you solve the linear system
+\centerline{{\axiom{  x + 2*y + 3*z = 2} }}
+\centerline{{\axiom{2*x + 3*y + 4*z = 2} }}
+\centerline{{\axiom{3*x + 4*y + 5*z = 2}}}
+with this command.
+}{
+\spadpaste{solve([[1,2,3],[2,3,4],[3,4,5]],[2,2,2])}
+}
+
+All solutions of this system are obtained by adding the particular
+solution with a linear combination of the \texht{{\it basis}}{{\tt
+basis}} vectors.
+
+When no solution exists then {\tt "failed"} is returned as the
+\texht{{\it particular}}{{\tt particular}} component, as follows:
+
+\xtc{
+}{
+\spadpaste{solve([[1,2,3],[2,3,4],[3,4,5]],[2,3,2])}
+}
+
+When you want to solve a system of homogeneous equations (that is,
+a system where the numbers on the right-hand sides of the
+%-% \HDindex{nullspace}{ugxProblemLinSysPage}{8.5.1.}{Solution of Systems of Linear Equations}
+equations are all zero) in the matrix form you can omit the second
+argument and use the \axiomFun{nullSpace} operation.
+
+\xtc{
+This computes the solutions of the following system of equations:
+\centerline{{\axiom{  x + 2*y + 3*z = 0} }}
+\centerline{{\axiom{2*x + 3*y + 4*z = 0} }}
+\centerline{{\axiom{3*x + 4*y + 5*z = 0}}}
+The result is given as a list of vectors and
+these vectors form a basis for the solution space.
+}{
+\spadpaste{nullSpace([[1,2,3],[2,3,4],[3,4,5]])}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemOnePolTitle}{Solution of a Single Polynomial Equation}
+\newcommand{\ugxProblemOnePolNumber}{8.5.2.}
+%
+% =====================================================================
+\begin{page}{ugxProblemOnePolPage}{8.5.2. Solution of a Single Polynomial Equation}
+% =====================================================================
+\beginscroll
+
+\Language{} can solve polynomial equations producing either approximate
+%-% \HDindex{polynomial!root finding}{ugxProblemOnePolPage}{8.5.2.}{Solution of a Single Polynomial Equation}
+or exact solutions.
+%-% \HDindex{equation!polynomial!solving}{ugxProblemOnePolPage}{8.5.2.}{Solution of a Single Polynomial Equation}
+Exact solutions are either members of the ground
+field or can be presented symbolically as roots of irreducible polynomials.
+
+\xtc{
+This returns the one rational root along with an irreducible
+polynomial describing the other solutions.
+}{
+\spadpaste{solve(x**3  = 8,x)}
+}
+\xtc{
+If you want solutions expressed in terms of radicals you would use this
+instead.
+%-% \HDindex{radical}{ugxProblemOnePolPage}{8.5.2.}{Solution of a Single Polynomial Equation}
+}{
+\spadpaste{radicalSolve(x**3  = 8,x)}
+}
+
+The \axiomFun{solve} command always returns a value but
+\axiomFun{radicalSolve} returns only the solutions that it is
+able to express in terms of radicals.
+%-% \HDindex{radical}{ugxProblemOnePolPage}{8.5.2.}{Solution of a Single Polynomial Equation}
+
+If the polynomial equation has rational coefficients
+you can ask for approximations to its real roots by calling
+solve with a second argument that specifies the ``precision''
+%-% \HDindex{precision}{ugxProblemOnePolPage}{8.5.2.}{Solution of a Single Polynomial Equation}
+\texht{$\epsilon$}{\axiom{epsilon}}.
+This means that each approximation will be within
+\texht{$\pm\epsilon$}{plus or minus \axiom{epsilon}} of the actual
+result.
+
+\xtc{
+Notice that the type of second argument controls the type of the result.
+}{
+\spadpaste{solve(x**4 - 10*x**3 + 35*x**2 - 50*x + 25,.0001)}
+}
+\xtc{
+If you give a floating-point precision you get a floating-point result;
+if you give the precision as a rational number you get a rational result.
+}{
+\spadpaste{solve(x**3-2,1/1000)}
+}
+\xtc{
+If you want approximate complex results you should use the
+%-% \HDindex{approximation}{ugxProblemOnePolPage}{8.5.2.}{Solution of a Single Polynomial Equation}
+command \axiomFun{complexSolve} that takes the same precision argument
+\texht{$\epsilon$}{\axiom{epsilon}}.
+}{
+\spadpaste{complexSolve(x**3-2,.0001)}
+}
+\xtc{
+Each approximation will be within
+\texht{$\pm\epsilon$}{plus or minus \axiom{epsilon}} of the actual result
+in each of the real and imaginary parts.
+}{
+\spadpaste{complexSolve(x**2-2*\%i+1,1/100)}
+}
+
+Note that if you omit the \axiomOp{=} from the first argument
+\Language{} generates an equation by equating the first argument to zero.
+Also, when only one variable is present in the equation, you
+do not need to specify the variable to be solved for, that is,
+you can omit the second argument.
+
+\xtc{
+\Language{} can also solve equations involving rational functions.
+Solutions where the denominator vanishes are discarded.
+}{
+\spadpaste{radicalSolve(1/x**3 + 1/x**2 + 1/x = 0,x)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemPolSysTitle}{Solution of Systems of Polynomial Equations}
+\newcommand{\ugxProblemPolSysNumber}{8.5.3.}
+%
+% =====================================================================
+\begin{page}{ugxProblemPolSysPage}{8.5.3. Solution of Systems of Polynomial Equations}
+% =====================================================================
+\beginscroll
+
+Given a system of equations of rational functions with exact coefficients:
+%-% \HDindex{equation!polynomial!solving}{ugxProblemPolSysPage}{8.5.3.}{Solution of Systems of Polynomial Equations}
+\centerline{{\axiom{p1(x1,...,xn)} }}
+\centerline{{\axiom{.}             }}
+\centerline{{\axiom{.}             }}
+\centerline{{\axiom{.}             }}
+\centerline{{\axiom{pm(x1,...,xn)}}}
+\Language{} can find
+numeric or symbolic solutions.
+The system is first split into irreducible components, then for
+each component, a triangular system of equations is found that reduces
+the problem to sequential solution of univariate polynomials resulting
+from substitution of partial solutions from the previous stage.
+\centerline{{\axiom{q1(x1,...,xn)} }}
+\centerline{{\axiom{.} }}
+\centerline{{\axiom{.} }}
+\centerline{{\axiom{.} }}
+\centerline{{\axiom{qm(xn)}}}
+
+Symbolic solutions can be presented using ``implicit'' algebraic numbers
+defined as roots of irreducible polynomials or in terms of radicals.
+\Language{} can also find approximations to the real or complex roots
+of a system of polynomial equations to any user-specified accuracy.
+
+The operation \axiomFun{solve} for systems is used in a way similar
+to \axiomFun{solve} for single equations.
+Instead of a polynomial equation, one has to give a list of
+equations and instead of a single variable to solve for, a list of
+variables.
+For solutions of single equations see \downlink{``\ugxProblemOnePolTitle''}{ugxProblemOnePolPage} in Section \ugxProblemOnePolNumber\ignore{ugxProblemOnePol}.
+
+%
+\xtc{
+Use the operation \axiomFun{solve} if you want implicitly presented
+solutions.
+}{
+\spadpaste{solve([3*x**3 + y + 1,y**2 -4],[x,y])}
+}
+\xtc{
+}{
+\spadpaste{solve([x = y**2-19,y = z**2+x+3,z = 3*x],[x,y,z])}
+}
+\xtc{
+Use \axiomFun{radicalSolve} if you want your solutions expressed
+in terms of radicals.
+}{
+\spadpaste{radicalSolve([3*x**3 + y + 1,y**2 -4],[x,y])}
+}
+
+To get numeric solutions you only need to give the list of
+equations and the precision desired.
+The list of variables would be redundant information since there
+can be no parameters for the numerical solver.
+
+\xtc{
+If the precision is expressed as a floating-point number you get
+results expressed as floats.
+}{
+\spadpaste{solve([x**2*y - 1,x*y**2 - 2],.01)}
+}
+\xtc{
+To get complex numeric solutions, use the operation \axiomFun{complexSolve},
+which takes the same arguments as in the real case.
+}{
+\spadpaste{complexSolve([x**2*y - 1,x*y**2 - 2],1/1000)}
+}
+\xtc{
+It is also possible to solve systems of equations in rational functions
+over the rational numbers.
+Note that \axiom{[x = 0.0, a = 0.0]} is not returned as a solution since
+the denominator vanishes there.
+}{
+\spadpaste{solve([x**2/a = a,a = a*x],.001)}
+}
+\xtc{
+When solving equations with
+denominators, all solutions where the denominator vanishes are
+discarded.
+}{
+\spadpaste{radicalSolve([x**2/a + a + y**3 - 1,a*y + a + 1],[x,y])}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemLimitsTitle}{Limits}
+\newcommand{\ugProblemLimitsNumber}{8.6.}
+%
+% =====================================================================
+\begin{page}{ugProblemLimitsPage}{8.6. Limits}
+% =====================================================================
+\beginscroll
+%
+To compute a limit, you must specify a functional expression,
+%-% \HDindex{limit}{ugProblemLimitsPage}{8.6.}{Limits}
+a variable, and a limiting value for that variable.
+If you do not specify a direction, \Language{} attempts to
+compute a two-sided limit.
+
+\xtc{
+Issue this to compute the limit
+\texht{$$\lim_{x \rightarrow 1}{{\displaystyle x^2 - 3x +
+2}\over{\displaystyle x^2 - 1}}.$$}{
+of \axiom{(x**2 - 3*x + 2)/(x**2 - 1)} as \axiom{x} approaches \axiom{1}.}
+}{
+\spadpaste{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)}
+}
+
+Sometimes the limit when approached from the left is different from the
+limit from the right and, in this case, you may wish to ask for a
+one-sided limit.
+Also,
+if you have a function that is only defined on one side of a particular value,
+%-% \HDindex{limit!one-sided vs. two-sided}{ugProblemLimitsPage}{8.6.}{Limits}
+you can compute a one-sided limit.
+
+\xtc{
+The function \axiom{log(x)} is only defined to the right of zero,
+that is, for \axiom{x > 0}.
+Thus, when computing limits of functions involving \axiom{log(x)},
+you probably want a ``right-hand'' limit.
+}{
+\spadpaste{limit(x * log(x),x = 0,"right")}
+}
+\xtc{
+When you do not specify \axiom{"right"} or \axiom{"left"} as the
+optional fourth argument, \axiomFun{limit} tries to compute a
+two-sided limit.
+Here the limit from the left does not exist, as \Language{}
+indicates when you try to take a two-sided limit.
+}{
+\spadpaste{limit(x * log(x),x = 0)}
+}
+
+A function can be defined on both sides of a particular value, but
+tend to different limits as its variable approaches that value from the
+left and from the right.
+We can construct an example of this as follows:
+Since
+\texht{$\sqrt{y^2}$}{\axiom{sqrt(y**2)}}
+is simply the absolute value of \axiom{y},
+the function
+\texht{$\sqrt{y^2} / y$}{\axiom{sqrt(y**2) / y}}
+is simply the sign (\axiom{+1} or \axiom{-1}) of the nonzero
+real number \axiom{y}.
+Therefore,
+\texht{$\sqrt{y^2} / y = -1$}{\axiom{sqrt(y**2) / y = -1}}
+for \axiom{y < 0} and
+\texht{$\sqrt{y^2} / y = +1$}{\axiom{sqrt(y**2) / y = +1}}
+for \axiom{y > 0}.
+\xtc{
+This is what happens when we take the limit at \axiom{y = 0}.
+The answer returned by \Language{} gives both a
+``left-hand'' and a ``right-hand'' limit.
+}{
+\spadpaste{limit(sqrt(y**2)/y,y = 0)}
+}
+\xtc{
+Here is another example, this time using a more complicated function.
+}{
+\spadpaste{limit(sqrt(1 - cos(t))/t,t = 0)}
+}
+
+You can compute limits at infinity by passing either
+%-% \HDindex{limit!at infinity}{ugProblemLimitsPage}{8.6.}{Limits}
+\texht{$+\infty$ or $-\infty$}{``plus infinity'' or ``minus infinity''}
+as the third argument of \axiomFun{limit}.
+\xtc{
+To do this, use the constants \axiom{\%plusInfinity} and \axiom{\%minusInfinity}.
+}{
+\spadpaste{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)}
+}
+\xtc{
+}{
+\spadpaste{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)}
+}
+
+\xtc{
+You can take limits of functions with parameters.
+%-% \HDindex{limit!of function with parameters}{ugProblemLimitsPage}{8.6.}{Limits}
+As you can see, the limit is expressed in terms of the parameters.
+}{
+\spadpaste{limit(sinh(a*x)/tan(b*x),x = 0)}
+}
+
+When you use \axiomFun{limit}, you are taking the limit of a real
+function of a real variable.
+\xtc{
+When you compute this,
+\Language{} returns \axiom{0} because, as a function of a real variable,
+\axiom{sin(1/z)} is always between \axiom{-1} and \axiom{1}, so \axiom{z * sin(1/z)}
+tends to \axiom{0} as \axiom{z} tends to \axiom{0}.
+}{
+\spadpaste{limit(z * sin(1/z),z = 0)}
+}
+However, as a function of a {\it complex} variable, \axiom{sin(1/z)} is badly
+%-% \HDindex{limit!real vs. complex}{ugProblemLimitsPage}{8.6.}{Limits}
+behaved near \axiom{0} (one says that \axiom{sin(1/z)} has an
+%-% \HDindex{essential singularity}{ugProblemLimitsPage}{8.6.}{Limits}
+{\it essential singularity} at \axiom{z = 0}).
+%-% \HDindex{singularity!essential}{ugProblemLimitsPage}{8.6.}{Limits}
+\xtc{
+When viewed as a function of a complex variable, \axiom{z * sin(1/z)}
+does not approach any limit as \axiom{z} tends to \axiom{0} in the complex plane.
+\Language{} indicates this when we call \axiomFun{complexLimit}.
+}{
+\spadpaste{complexLimit(z * sin(1/z),z = 0)}
+}
+
+%% This is used in chapter 1
+
+You can also take complex limits at infinity, that is, limits of a function of
+\axiom{z} as \axiom{z} approaches infinity on the Riemann sphere.
+Use the symbol \axiom{\%infinity} to denote ``complex infinity.''
+\xtc{
+As above, to compute complex limits rather than real limits, use
+\axiomFun{complexLimit}.
+}{
+\spadpaste{complexLimit((2 + z)/(1 - z),z = \%infinity)}
+}
+\xtc{
+In many cases, a limit of a real function of a real variable
+exists when the corresponding complex limit does not.
+This limit exists.
+}{
+\spadpaste{limit(sin(x)/x,x = \%plusInfinity)}
+}
+\xtc{
+But this limit does not.
+}{
+\spadpaste{complexLimit(sin(x)/x,x = \%infinity)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemLaplaceTitle}{Laplace Transforms}
+\newcommand{\ugProblemLaplaceNumber}{8.7.}
+%
+% =====================================================================
+\begin{page}{ugProblemLaplacePage}{8.7. Laplace Transforms}
+% =====================================================================
+\beginscroll
+%
+\Language{} can compute some forward Laplace transforms, mostly
+%-% \HDindex{Laplace transform}{ugProblemLaplacePage}{8.7.}{Laplace Transforms}
+of elementary
+%-% \HDindex{function!elementary}{ugProblemLaplacePage}{8.7.}{Laplace Transforms}
+functions
+%-% \HDindex{transform!Laplace}{ugProblemLaplacePage}{8.7.}{Laplace Transforms}
+not involving logarithms, although some cases of
+special functions are handled.
+\xtc{
+To compute the forward Laplace transform of \axiom{F(t)} with respect to
+\axiom{t} and express the result as \axiom{f(s)}, issue the command
+\axiom{laplace(F(t), t, s)}.
+}{
+\spadpaste{laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t), t, s)}
+}
+\xtc{
+Here are some other non-trivial examples.
+}{
+\spadpaste{laplace((exp(a*t) - exp(b*t))/t, t, s)}
+}
+\xtc{
+}{
+\spadpaste{laplace(2/t * (1 - cos(a*t)), t, s)}
+}
+\xtc{
+}{
+\spadpaste{laplace(exp(-a*t) * sin(b*t) / b**2, t, s)}
+}
+\xtc{
+}{
+\spadpaste{laplace((cos(a*t) - cos(b*t))/t, t, s)}
+}
+\xtc{
+\Language{} also knows about a few special functions.
+}{
+\spadpaste{laplace(exp(a*t+b)*Ei(c*t), t, s)}
+}
+\xtc{
+}{
+\spadpaste{laplace(a*Ci(b*t) + c*Si(d*t), t, s)}
+}
+\xtc{
+When \Language{} does not know about a particular transform,
+it keeps it as a formal transform in the answer.
+}{
+\spadpaste{laplace(sin(a*t) - a*t*cos(a*t) + exp(t**2), t, s)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemIntegrationTitle}{Integration}
+\newcommand{\ugProblemIntegrationNumber}{8.8.}
+%
+% =====================================================================
+\begin{page}{ugProblemIntegrationPage}{8.8. Integration}
+% =====================================================================
+\beginscroll
+%
+Integration is the reverse process of differentiation, that is,
+%-% \HDindex{integration}{ugProblemIntegrationPage}{8.8.}{Integration}
+an {\it integral} of a function \axiom{f} with respect to a variable
+\axiom{x} is any function \axiom{g} such that \axiom{D(g,x)}
+is equal to \axiom{f}.
+\xtc{
+The package \axiomType{FunctionSpaceIntegration} provides the top-level
+integration operation, \axiomFunFrom{integrate}{FunctionSpaceIntegration},
+for integrating real-valued elementary functions.
+%-% \HDexptypeindex{FunctionSpaceIntegration}{ugProblemIntegrationPage}{8.8.}{Integration}
+}{
+\spadpaste{integrate(cosh(a*x)*sinh(a*x), x)}
+}
+\xtc{
+Unfortunately, antiderivatives of most functions cannot be expressed in
+terms of elementary functions.
+}{
+\spadpaste{integrate(log(1 + sqrt(a * x + b)) / x, x)}
+}
+Given an elementary function to integrate, \Language{} returns a formal
+integral as above only when it can prove that the integral is not
+elementary and not when it cannot determine the integral.
+In this rare case it prints a message that it cannot
+determine if an elementary integral exists.
+%
+\xtc{
+Similar functions may have antiderivatives
+%-% \HDindex{antiderivative}{ugProblemIntegrationPage}{8.8.}{Integration}
+that look quite different because the form of the antiderivative
+depends on the sign of a constant that appears in the function.
+}{
+\spadpaste{integrate(1/(x**2 - 2),x)}
+}
+\xtc{
+}{
+\spadpaste{integrate(1/(x**2 + 2),x)}
+}
+%
+If the integrand contains parameters, then there may be several possible
+antiderivatives, depending on the signs of expressions of the parameters.
+\xtc{
+In this case \Language{} returns a list of answers that cover all
+the possible cases.
+Here you
+use the answer involving the square root of \axiom{a} when \axiom{a > 0} and
+%-% \HDindex{integration!result as list of real functions}{ugProblemIntegrationPage}{8.8.}{Integration}
+the answer involving the square root of \axiom{-a} when \axiom{a < 0}.
+}{
+\spadpaste{integrate(x**2 / (x**4 - a**2), x)}
+}
+
+If the parameters and the variables of integration can be complex
+numbers rather than real, then the notion of sign is not defined.
+In this case all the possible answers can be expressed as one
+complex function.
+To get that function, rather than a list of real functions, use
+\axiomFunFrom{complexIntegrate}{FunctionSpaceComplexIntegration},
+which is provided by the package
+%-% \HDindex{integration!result as a complex functions}{ugProblemIntegrationPage}{8.8.}{Integration}
+\axiomType{FunctionSpaceComplexIntegration}.
+%-% \HDexptypeindex{FunctionSpaceComplexIntegration}{ugProblemIntegrationPage}{8.8.}{Integration}
+
+\xtc{
+This operation is used for
+integrating complex-valued elementary functions.
+}{
+\spadpaste{complexIntegrate(x**2 / (x**4 - a**2), x)}
+}
+\xtc{
+As with the real case,
+antiderivatives for most complex-valued functions cannot be expressed
+in terms of elementary functions.
+}{
+\spadpaste{complexIntegrate(log(1 + sqrt(a * x + b)) / x, x)}
+}
+
+Sometimes \axiomFun{integrate} can involve symbolic algebraic numbers
+such as those returned by \axiomFunFrom{rootOf}{Expression}.
+To see how to work with these strange generated symbols (such as
+\axiom{\%\%a0}), see \downlink{``\ugxProblemSymRootAllTitle''}{ugxProblemSymRootAllPage} in Section \ugxProblemSymRootAllNumber\ignore{ugxProblemSymRootAll}.
+
+Definite integration is the process of computing the area between
+%-% \HDindex{integration!definite}{ugProblemIntegrationPage}{8.8.}{Integration}
+the \axiom{x}-axis and the curve of a function \axiom{f(x)}.
+The fundamental theorem of calculus states that if \axiom{f} is
+continuous on an interval \axiom{a..b} and if there exists a function \axiom{g}
+that is differentiable on \axiom{a..b} and such that \axiom{D(g, x)}
+is equal to \axiom{f}, then the definite integral of \axiom{f}
+for \axiom{x} in the interval \axiom{a..b} is equal to \axiom{g(b) - g(a)}.
+
+\xtc{
+The package \axiomType{RationalFunctionDefiniteIntegration} provides
+the top-level definite integration operation,
+\axiomFunFrom{integrate}{RationalFunctionDefiniteIntegration},
+for integrating real-valued rational functions.
+}{
+\spadpaste{integrate((x**4 - 3*x**2 + 6)/(x**6-5*x**4+5*x**2+4), x = 1..2)}
+}
+\Language{} checks beforehand that the function you are integrating is
+defined on the interval \axiom{a..b}, and prints an error message if it
+finds that this is not case, as in the following example:
+\begin{verbatim}
+integrate(1/(x**2-2), x = 1..2)
+
+    >> Error detected within library code:
+       Pole in path of integration
+       You are being returned to the top level
+       of the interpreter.
+\end{verbatim}
+When parameters are present in the function, the function may or may not be
+defined on the interval of integration.
+
+\xtc{
+If this is the case, \Language{} issues a warning that a pole might
+lie in the path of integration, and does not compute the integral.
+}{
+\spadpaste{integrate(1/(x**2-a), x = 1..2)}
+}
+
+If you know that you are using values of the parameter for which
+the function has no pole in the interval of integration, use the
+string {\tt "noPole"} as a third argument to
+\axiomFunFrom{integrate}{RationalFunctionDefiniteIntegration}:
+
+%
+\xtc{
+The value here is, of course, incorrect if \axiom{sqrt(a)} is between
+\axiom{1} and \axiom{2.}
+}{
+\spadpaste{integrate(1/(x**2-a), x = 1..2, "noPole")}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemSeriesTitle}{Working with Power Series}
+\newcommand{\ugProblemSeriesNumber}{8.9.}
+%
+% =====================================================================
+\begin{page}{ugProblemSeriesPage}{8.9. Working with Power Series}
+% =====================================================================
+\beginscroll
+%
+\Language{} has very sophisticated facilities for working with power
+%-% \HDindex{series}{ugProblemSeriesPage}{8.9.}{Working with Power Series}
+series.
+%-% \HDindex{power series}{ugProblemSeriesPage}{8.9.}{Working with Power Series}
+Infinite series are represented by a list of the
+coefficients that have already been determined, together with a
+function for computing the additional coefficients if needed.
+%
+The system command that determines how many terms of a series is displayed
+is \spadcmd{)set streams calculate}.
+For the purposes of this book, we have used this system command to display
+fewer than ten terms.
+%-% \HDsyscmdindex{set streams calculate}{ugProblemSeriesPage}{8.9.}{Working with Power Series}
+Series can be created from expressions, from functions for the
+series coefficients, and from applications of operations on
+existing series.
+The most general function for creating a series is called
+\axiomFun{series}, although you can also use \axiomFun{taylor},
+\axiomFun{laurent} and \axiomFun{puiseux} in situations where you
+know what kind of exponents are involved.
+
+For information about solving differential equations in terms of
+power series, see \downlink{``\ugxProblemDEQSeriesTitle''}{ugxProblemDEQSeriesPage} in Section \ugxProblemDEQSeriesNumber\ignore{ugxProblemDEQSeries}.
+
+\beginmenu
+    \menudownlink{{8.9.1. Creation of Power Series}}{ugxProblemSeriesCreatePage}
+    \menudownlink{{8.9.2. Coefficients of Power Series}}{ugxProblemSeriesCoefficientsPage}
+    \menudownlink{{8.9.3. Power Series Arithmetic}}{ugxProblemSeriesArithmeticPage}
+    \menudownlink{{8.9.4. Functions on Power Series}}{ugxProblemSeriesFunctionsPage}
+    \menudownlink{{8.9.5. Converting to Power Series}}{ugxProblemSeriesConversionsPage}
+    \menudownlink{{8.9.6. Power Series from Formulas}}{ugxProblemSeriesFormulaPage}
+    \menudownlink{{8.9.7. Substituting Numerical Values in Power Series}}{ugxProblemSeriesSubstitutePage}
+    \menudownlink{{8.9.8. Example: Bernoulli Polynomials and Sums of Powers}}{ugxProblemSeriesBernoulliPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemSeriesCreateTitle}{Creation of Power Series}
+\newcommand{\ugxProblemSeriesCreateNumber}{8.9.1.}
+%
+% =====================================================================
+\begin{page}{ugxProblemSeriesCreatePage}{8.9.1. Creation of Power Series}
+% =====================================================================
+\beginscroll
+
+\labelSpace{4pc}
+\xtc{
+This is the easiest way to create a power series.
+This tells \Language{} that \axiom{x} is to be treated as a power series,
+%-% \HDindex{series!creating}{ugxProblemSeriesCreatePage}{8.9.1.}{Creation of Power Series}
+so functions of \axiom{x} are again power series.
+}{
+\spadpaste{x := series 'x \bound{x}}
+}
+%
+We didn't say anything about the coefficients of the power
+series, so the coefficients are general expressions over the integers.
+This allows us to introduce denominators, symbolic constants, and other
+variables as needed.
+\xtc{
+Here the coefficients are integers (note that the coefficients
+are the Fibonacci
+%-% \HDindex{Fibonacci numbers}{ugxProblemSeriesCreatePage}{8.9.1.}{Creation of Power Series}
+numbers).
+}{
+\spadpaste{1/(1 - x - x**2) \free{x}}
+}
+\xtc{
+This series has coefficients that are rational numbers.
+}{
+\spadpaste{sin(x) \free{x}}
+}
+\xtc{
+When you enter this expression
+you introduce the symbolic constants \axiom{sin(1)} and \axiom{cos(1).}
+}{
+\spadpaste{sin(1 + x) \free{x}}
+}
+\xtc{
+When you enter the expression
+the variable \axiom{a} appears in the resulting series expansion.
+}{
+\spadpaste{sin(a * x) \free{x}}
+}
+
+\xtc{
+You can also convert an expression into a series expansion.
+This expression creates the series expansion of \axiom{1/log(y)}
+about \axiom{y = 1}.
+For details and more examples, see
+\downlink{``\ugxProblemSeriesConversionsTitle''}{ugxProblemSeriesConversionsPage} in Section \ugxProblemSeriesConversionsNumber\ignore{ugxProblemSeriesConversions}.
+}{
+\spadpaste{series(1/log(y),y = 1)}
+}
+
+You can create power series with more general coefficients.
+You normally accomplish this via a type declaration (see
+\downlink{``\ugTypesDeclareTitle''}{ugTypesDeclarePage} in Section \ugTypesDeclareNumber\ignore{ugTypesDeclare}).
+See \downlink{``\ugxProblemSeriesFunctionsTitle''}{ugxProblemSeriesFunctionsPage} in Section \ugxProblemSeriesFunctionsNumber\ignore{ugxProblemSeriesFunctions} for some warnings about
+working with declared series.
+
+\xtc{
+We declare that \axiom{y} is a one-variable Taylor series
+%-% \HDindex{series!Taylor}{ugxProblemSeriesCreatePage}{8.9.1.}{Creation of Power Series}
+(\axiomType{UTS} is the abbreviation for \axiomType{UnivariateTaylorSeries})
+in the variable \axiom{z} with \axiomType{FLOAT}
+(that is, floating-point) coefficients, centered about \axiom{0.}
+Then, by assignment, we obtain the Taylor expansion of
+\axiom{exp(z)} with floating-point coefficients.
+%-% \HDexptypeindex{UnivariateTaylorSeries}{ugxProblemSeriesCreatePage}{8.9.1.}{Creation of Power Series}
+}{
+\spadpaste{y : UTS(FLOAT,'z,0) := exp(z) \bound{y}}
+}
+
+You can also create a power series by giving an explicit formula
+for its \eth{\axiom{n}} coefficient.
+For details and more examples, see
+\downlink{``\ugxProblemSeriesFormulaTitle''}{ugxProblemSeriesFormulaPage} in Section \ugxProblemSeriesFormulaNumber\ignore{ugxProblemSeriesFormula}.
+
+\xtc{
+To create a series about
+\axiom{w = 0} whose \eth{\axiom{n}} Taylor coefficient
+is \axiom{1/n!}, you can evaluate this expression.
+This is the Taylor expansion of \axiom{exp(w)} at \axiom{w = 0}.
+}{
+\spadpaste{series(1/factorial(n),n,w = 0)}
+}
+%
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemSeriesCoefficientsTitle}{Coefficients of Power Series}
+\newcommand{\ugxProblemSeriesCoefficientsNumber}{8.9.2.}
+%
+% =====================================================================
+\begin{page}{ugxProblemSeriesCoefficientsPage}{8.9.2. Coefficients of Power Series}
+% =====================================================================
+\beginscroll
+
+You can extract any coefficient from a power series---even one
+that hasn't been computed yet.
+This is possible because in \Language{}, infinite series are
+represented by a list of the coefficients that have already been
+determined, together with a function for computing the additional
+coefficients.
+(This is known as {\it lazy evaluation}.) When you ask for a
+%-% \HDindex{series!lazy evaluation}{ugxProblemSeriesCoefficientsPage}{8.9.2.}{Coefficients of Power Series}
+coefficient that hasn't yet been computed, \Language{} computes
+%-% \HDindex{lazy evaluation}{ugxProblemSeriesCoefficientsPage}{8.9.2.}{Coefficients of Power Series}
+whatever additional coefficients it needs and then stores them in
+the representation of the power series.
+
+\xtc{
+Here's an example of how to extract the coefficients of a power series.
+%-% \HDindex{series!extracting coefficients}{ugxProblemSeriesCoefficientsPage}{8.9.2.}{Coefficients of Power Series}
+}{
+\spadpaste{x := series(x) \bound{x}}
+}
+\xtc{
+}{
+\spadpaste{y := exp(x) * sin(x) \free{x} \bound{y}}
+}
+\xtc{
+This coefficient is readily available.
+}{
+\spadpaste{coefficient(y,6) \free{y}}
+}
+\xtc{
+But let's get the fifteenth coefficient of \axiom{y}.
+}{
+\spadpaste{coefficient(y,15) \free{y} \bound{y15}}
+}
+\xtc{
+If you look at \axiom{y}
+then you see that the coefficients up to order \axiom{15}
+have all been computed.
+}{
+\spadpaste{y \free{y15}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemSeriesArithmeticTitle}{Power Series Arithmetic}
+\newcommand{\ugxProblemSeriesArithmeticNumber}{8.9.3.}
+%
+% =====================================================================
+\begin{page}{ugxProblemSeriesArithmeticPage}{8.9.3. Power Series Arithmetic}
+% =====================================================================
+\beginscroll
+
+You can manipulate power series using the usual arithmetic operations
+%-% \HDindex{series!arithmetic}{ugxProblemSeriesArithmeticPage}{8.9.3.}{Power Series Arithmetic}
+\axiomOpFrom{+}{UnivariatePuiseuxSeries},
+\axiomOpFrom{-}{UnivariatePuiseuxSeries},
+\axiomOpFrom{*}{UnivariatePuiseuxSeries}, and
+\axiomOpFrom{/}{UnivariatePuiseuxSeries}.
+%
+
+\labelSpace{1pc}
+\xtc{
+The results of these operations are also power series.
+}{
+\spadpaste{x := series x \bound{x}}
+}
+\xtc{
+}{
+\spadpaste{(3 + x) / (1 + 7*x)}
+}
+\xtc{
+You can also compute \axiom{f(x) ** g(x)}, where \axiom{f(x)} and \axiom{g(x)}
+are two power series.
+}{
+\spadpaste{base := 1 / (1 - x) \free{x} \bound{base}}
+}
+%
+\xtc{
+}{
+\spadpaste{expon := x * base \free{x base} \bound{expon}}
+}
+%
+\xtc{
+}{
+\spadpaste{base ** expon \free{base expon}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemSeriesFunctionsTitle}{Functions on Power Series}
+\newcommand{\ugxProblemSeriesFunctionsNumber}{8.9.4.}
+%
+% =====================================================================
+\begin{page}{ugxProblemSeriesFunctionsPage}{8.9.4. Functions on Power Series}
+% =====================================================================
+\beginscroll
+
+Once you have created a power series, you can apply transcendental
+functions
+(for example, \axiomFun{exp}, \axiomFun{log}, \axiomFun{sin}, \axiomFun{tan},
+\axiomFun{cosh}, etc.) to it.
+
+\labelSpace{1pc}
+%
+\xtc{
+To demonstrate this, we first create the power series
+expansion of the rational function
+\texht{
+${\displaystyle x^2} \over {\displaystyle 1 - 6x + x^2}$
+}{
+\axiom{x**2/(1 - 6*x + x**2)}
+}
+about \axiom{x = 0}.
+}{
+\spadpaste{x := series 'x \bound{x}}
+}
+%
+\xtc{
+}{
+\spadpaste{rat := x**2 / (1 - 6*x + x**2) \free{x} \bound{rat}}
+}
+%
+%
+\xtc{
+If you want to compute the series expansion of
+\texht{
+$\sin\left({\displaystyle x^2} \over {\displaystyle 1 - 6x + x^2}\right)$
+}{
+\axiom{sin(x**2/(1 - 6*x + x**2))}
+}
+you simply compute the sine of \axiom{rat}.
+}{
+\spadpaste{sin(rat) \free{rat}}
+}
+%
+
+\beginImportant
+\noindent {\bf Warning:}
+the type of the coefficients of a power series may
+affect the kind of computations that you can do with that series.
+This can only happen when you have made a declaration to
+specify a series domain with a certain type of coefficient.
+\endImportant
+
+\xtc{
+If you evaluate
+then you have declared that \axiom{y} is a one variable Taylor series
+%-% \HDindex{series!Taylor}{ugxProblemSeriesFunctionsPage}{8.9.4.}{Functions on Power Series}
+(\axiomType{UTS} is the abbreviation for \axiomType{UnivariateTaylorSeries})
+in the variable \axiom{y} with \axiomType{FRAC INT}
+(that is, fractions of integer) coefficients, centered about \axiom{0}.
+}{
+\spadpaste{y : UTS(FRAC INT,y,0) := y \bound{y}}
+}
+%
+\xtc{
+You can now compute certain power series in \axiom{y},
+{\it provided} that these series have rational coefficients.
+}{
+\spadpaste{exp(y) \free{y}}
+}
+%
+\xtc{
+You can get examples of such series
+by applying transcendental functions to
+series in \axiom{y} that have no constant terms.
+}{
+\spadpaste{tan(y**2) \free{y}}
+}
+%
+\xtc{
+}{
+\spadpaste{cos(y + y**5) \free{y}}
+}
+%
+%
+\xtc{
+Similarly, you can compute the logarithm of a power series with rational
+coefficients if the constant coefficient is \axiom{1.}
+}{
+\spadpaste{log(1 + sin(y)) \free{y}}
+}
+%
+If you wanted to apply, say, the operation \axiomFun{exp} to a power
+series with a nonzero constant coefficient \texht{$a_0$}{\axiom{a0}},
+then the constant coefficient of the result would be
+\texht{$e^{a_0}$}{\axiom{exp(a0)}}, which is {\it not} a rational number.
+Therefore, evaluating \axiom{exp(2 + tan(y))} would generate an error
+message.
+
+If you want to compute the Taylor expansion of \axiom{exp(2 + tan(y))}, you must
+ensure that the coefficient domain has an operation \axiomFun{exp} defined
+for it.
+An example of such a domain is \axiomType{Expression Integer}, the type
+of formal functional expressions over the integers.
+%
+\xtc{
+When working with coefficients of this type,
+}{
+\spadpaste{z : UTS(EXPR INT,z,0) := z \bound{z}}
+}
+\xtc{
+this presents no problems.
+}{
+\spadpaste{exp(2 + tan(z)) \free{z}}
+}
+%
+Another way to create Taylor series whose coefficients are expressions over
+the integers is to use \axiomFun{taylor} which works similarly to
+%-% \HDindex{series!Taylor}{ugxProblemSeriesFunctionsPage}{8.9.4.}{Functions on Power Series}
+\axiomFun{series}.
+%
+\xtc{
+This is equivalent to the previous computation, except that now we
+are using the variable \axiom{w} instead of \axiom{z}.
+}{
+\spadpaste{w := taylor 'w \bound{w}}
+}
+\xtc{
+}{
+\spadpaste{exp(2 + tan(w)) \free{w}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemSeriesConversionsTitle}{Converting to Power Series}
+\newcommand{\ugxProblemSeriesConversionsNumber}{8.9.5.}
+%
+% =====================================================================
+\begin{page}{ugxProblemSeriesConversionsPage}{8.9.5. Converting to Power Series}
+% =====================================================================
+\beginscroll
+
+The \axiomType{ExpressionToUnivariatePowerSeries} package provides
+operations for computing series expansions of functions.
+%-% \HDexptypeindex{ExpressionToUnivariatePowerSeries}{ugxProblemSeriesConversionsPage}{8.9.5.}{Converting to Power Series}
+
+\labelSpace{1pc}
+\xtc{
+Evaluate this
+to compute the Taylor expansion of \axiom{sin x} about
+%-% \HDindex{series!Taylor}{ugxProblemSeriesConversionsPage}{8.9.5.}{Converting to Power Series}
+\axiom{x = 0}.
+The first argument, \axiom{sin(x)}, specifies the function whose series
+expansion is to be computed and the second argument, \axiom{x = 0},
+specifies that the series is to be expanded in power of \axiom{(x - 0)},
+that is, in power of \axiom{x}.
+}{
+\spadpaste{taylor(sin(x),x = 0)}
+}
+\xtc{
+Here is the Taylor expansion of \axiom{sin x} about
+\texht{$x = \frac{\pi}{6}$}{\axiom{x = \%pi/6}}:
+}{
+\spadpaste{taylor(sin(x),x = \%pi/6)}
+}
+
+The function to be expanded into a series may have variables other than
+%-% \HDindex{series!multiple variables}{ugxProblemSeriesConversionsPage}{8.9.5.}{Converting to Power Series}
+the series variable.
+%
+\xtc{
+For example, we may expand \axiom{tan(x*y)} as a Taylor series in
+\axiom{x}
+}{
+\spadpaste{taylor(tan(x*y),x = 0)}
+}
+%
+\xtc{
+or as a Taylor series in \axiom{y}.
+}{
+\spadpaste{taylor(tan(x*y),y = 0)}
+}
+\xtc{
+A more interesting function is
+\texht{${\displaystyle t e^{x t}} \over {\displaystyle e^t - 1}$}{(t *
+\%e**(x*t))/(\%e**t - 1)}.
+When we expand this function as a Taylor series in \axiom{t}
+the \eth{\axiom{n}} order coefficient is the \eth{\axiom{n}} Bernoulli
+%-% \HDindex{Bernoulli!polynomial}{ugxProblemSeriesConversionsPage}{8.9.5.}{Converting to Power Series}
+polynomial
+%-% \HDindex{polynomial!Bernoulli}{ugxProblemSeriesConversionsPage}{8.9.5.}{Converting to Power Series}
+divided by \axiom{n!}.
+}{
+\spadpaste{bern := taylor(t*exp(x*t)/(exp(t) - 1),t = 0) \bound{bern}}
+}
+\xtc{
+Therefore, this and the next expression
+produce the same result.
+}{
+\spadpaste{factorial(6) * coefficient(bern,6) \free{bern}}
+}
+\xtc{
+}{
+\spadpaste{bernoulliB(6,x)}
+}
+
+Technically, a series with terms of negative degree is not considered to
+be a Taylor series, but, rather, a
+%-% \HDindex{series!Laurent}{ugxProblemSeriesConversionsPage}{8.9.5.}{Converting to Power Series}
+{\it Laurent series}.
+%-% \HDindex{Laurent series}{ugxProblemSeriesConversionsPage}{8.9.5.}{Converting to Power Series}
+If you try to compute a Taylor series expansion of
+\texht{$\frac{x}{\log x}$}{x/log(x)}
+at \axiom{x = 1} via \axiom{taylor(x/log(x),x = 1)}
+you get an error message.
+The reason is that the function has a {\it pole} at \axiom{x =
+1}, meaning that
+its series expansion about this point has terms of negative degree.
+A series with finitely many terms of negative degree is called a Laurent
+series.
+\xtc{
+You get the desired series expansion by issuing this.
+}{
+\spadpaste{laurent(x/log(x),x = 1)}
+}
+
+Similarly, a series with terms of fractional degree is neither a Taylor
+series nor a Laurent series.
+Such a series is called a
+%-% \HDindex{series!Puiseux}{ugxProblemSeriesConversionsPage}{8.9.5.}{Converting to Power Series}
+{\it Puiseux series}.
+%-% \HDindex{Puiseux series}{ugxProblemSeriesConversionsPage}{8.9.5.}{Converting to Power Series}
+The expression \axiom{laurent(sqrt(sec(x)),x = 3 * \%pi/2)}
+results in an error message
+because the series expansion about this point has terms of fractional degree.
+\xtc{
+However, this command produces what you want.
+}{
+\spadpaste{puiseux(sqrt(sec(x)),x = 3 * \%pi/2)}
+}
+
+Finally, consider the case of functions that do not have Puiseux
+expansions about certain points.
+An example of this is \texht{$x^x$}{\axiom{x^x}} about \axiom{x = 0}.
+\axiom{puiseux(x**x,x=0)}
+produces an error message because of the
+type of singularity of the function at \axiom{x = 0}.
+\xtc{
+The general function \axiomFun{series} can be used in this case.
+Notice that the series returned is not, strictly speaking, a power series
+because of the \axiom{log(x)} in the expansion.
+}{
+\spadpaste{series(x**x,x=0)}
+}
+
+\beginImportant
+The operation \axiomFun{series} returns the most general type of
+infinite series.
+The user who is not interested in distinguishing
+between various types of infinite series may wish to use this operation
+exclusively.
+\endImportant
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemSeriesFormulaTitle}{Power Series from Formulas}
+\newcommand{\ugxProblemSeriesFormulaNumber}{8.9.6.}
+%
+% =====================================================================
+\begin{page}{ugxProblemSeriesFormulaPage}{8.9.6. Power Series from Formulas}
+% =====================================================================
+\beginscroll
+
+The \axiomType{GenerateUnivariatePowerSeries} package enables you to
+%-% \HDindex{series!giving formula for coefficients}{ugxProblemSeriesFormulaPage}{8.9.6.}{Power Series from Formulas}
+create power series from explicit formulas for their
+\eth{\axiom{n}} coefficients.
+In what follows, we construct series expansions for certain
+transcendental functions by giving formulas for their
+coefficients.
+You can also compute such series expansions directly simply by
+specifying the function and the point about which the series is to
+be expanded.
+%-% \HDexptypeindex{GenerateUnivariatePowerSeries}{ugxProblemSeriesFormulaPage}{8.9.6.}{Power Series from Formulas}
+See \downlink{``\ugxProblemSeriesConversionsTitle''}{ugxProblemSeriesConversionsPage} in Section \ugxProblemSeriesConversionsNumber\ignore{ugxProblemSeriesConversions} for more information.
+
+Consider the Taylor expansion of \texht{$e^x$}{\axiom{\%e**x}}
+%-% \HDindex{series!Taylor}{ugxProblemSeriesFormulaPage}{8.9.6.}{Power Series from Formulas}
+about \axiom{x = 0}:
+\texht{\narrowDisplay{%
+\begin{array}{ccl}
+e^x &=& \displaystyle 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots \\ \\
+    &=& \displaystyle\sum_{n=0}^\infty \frac{x^n}{n!}
+\end{array}}%
+}{
+\begin{verbatim}
+%e**x = 1 + x + x**2/2 + x**3/6 + ...
+      = sum from n=0 to n=%infty of x**n / n!
+\end{verbatim}
+}
+The \eth{\axiom{n}} Taylor coefficient is \axiom{1/n!}.
+%
+\xtc{
+This is how you create this series in \Language{}.
+}{
+\spadpaste{series(n +-> 1/factorial(n),x = 0)}
+}
+
+The first argument specifies a formula for the \eth{\axiom{n}}
+coefficient by giving a function that maps \axiom{n} to
+\axiom{1/n!}.
+The second argument specifies that the series is to be expanded in
+powers of \axiom{(x - 0)}, that is, in powers of \axiom{x}.
+Since we did not specify an initial degree, the first term in the
+series was the term of degree 0 (the constant term).
+Note that the formula was given as an anonymous function.
+These are discussed in \downlink{``\ugUserAnonTitle''}{ugUserAnonPage} in Section \ugUserAnonNumber\ignore{ugUserAnon}.
+
+Consider the Taylor expansion of \axiom{log x} about \axiom{x = 1}:
+\texht{\narrowDisplay{%
+\begin{array}{ccl}
+\log(x) &=& \displaystyle (x - 1) - \frac{(x - 1)^2}{2} + \frac{(x - 1)^3}{3} - \cdots \\ \\
+        &=& \displaystyle\sum_{n = 1}^\infty (-1)^{n-1} \frac{(x - 1)^n}{n}
+\end{array}}%
+}{
+\begin{verbatim}
+log x = (x - 1) - \(x - 1)^2/2 + (x - 1)^3/3 - ...
+      = sum from n=1 to n=%infty of (-1)**(n-1) (x - 1)**n / n
+\end{verbatim}
+}
+If you were to evaluate the expression
+\axiom{series(n +-> (-1)**(n-1) / n, x = 1)}
+you would get an error message because \Language{} would try to
+calculate a term of degree \axiom{0} and therefore divide by \axiom{0.}
+
+\xtc{
+Instead, evaluate this.
+The third argument, \axiom{1..}, indicates that only terms of degree
+\axiom{n = 1, ...} are to be computed.
+}{
+\spadpaste{series(n +-> (-1)**(n-1)/n,x = 1,1..)}
+}
+%
+
+Next consider the Taylor expansion of an odd function, say,
+\axiom{sin(x)}:
+\begin{verbatim}
+sin x = x - x**3/3! + x**5/5! - ...
+\end{verbatim}
+Here every other coefficient is zero and we would like to give an
+explicit formula only for the odd Taylor coefficients.
+%
+\xtc{
+This is one way to do it.
+The third argument, \axiom{1..}, specifies that the first term to be computed
+is the term of degree 1.
+The fourth argument, \axiom{2}, specifies that we
+increment by \axiom{2} to find the degrees of subsequent terms, that is, the next
+term
+is of degree \axiom{1 + 2}, the next of degree \axiom{1 + 2 + 2}, etc.
+}{
+\spadpaste{series(n +-> (-1)**((n-1)/2)/factorial(n),x = 0,1..,2)}
+}
+%
+
+\xtc{
+The initial degree and the increment do not have to be integers.
+For example, this expression produces a series expansion of
+\texht{$\sin(x^{\frac{1}{3}})$}{\axiom{sin(x**(1/3))}}.
+}{
+\spadpaste{series(n +-> (-1)**((3*n-1)/2)/factorial(3*n),x = 0,1/3..,2/3)}
+}
+\xtc{
+While the increment must be positive, the initial degree may be negative.
+This yields the Laurent expansion of \axiom{csc(x)} at
+\axiom{x = 0}.
+%
+}{
+\spadpaste{cscx := series(n +-> (-1)**((n-1)/2) * 2 * (2**n-1) * bernoulli(numer(n+1)) / factorial(n+1), x=0, -1..,2) \bound{cscx}}
+}
+\xtc{
+Of course, the reciprocal of this power series is the Taylor expansion
+of \axiom{sin(x)}.
+}{
+\spadpaste{1/cscx \free{cscx}}
+}
+%
+\xtc{
+As a final example,
+here is the Taylor expansion of \axiom{asin(x)} about \axiom{x = 0}.
+}{
+\spadpaste{asinx := series(n +-> binomial(n-1,(n-1)/2)/(n*2**(n-1)),x=0,1..,2) \bound{asinx}}
+}
+\xtc{
+When we compute the \axiom{sin} of this series, we get \axiom{x}
+(in the sense that all higher terms computed so far are zero).
+}{
+\spadpaste{sin(asinx) \free{asinx}}
+}
+
+As we discussed in \downlink{``\ugxProblemSeriesConversionsTitle''}{ugxProblemSeriesConversionsPage} in Section \ugxProblemSeriesConversionsNumber\ignore{ugxProblemSeriesConversions},
+you can also use the operations \axiomFun{taylor}, \axiomFun{laurent} and
+\axiomFun{puiseux} instead of \axiomFun{series} if you know ahead of time
+what kind of exponents a series has.
+You can't go wrong using \axiomFun{series}, though.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemSeriesSubstituteTitle}{Substituting Numerical Values in Power Series}
+\newcommand{\ugxProblemSeriesSubstituteNumber}{8.9.7.}
+%
+% =====================================================================
+\begin{page}{ugxProblemSeriesSubstitutePage}{8.9.7. Substituting Numerical Values in Power Series}
+% =====================================================================
+\beginscroll
+
+Use \axiomFunFrom{eval}{UnivariatePowerSeriesCategory}
+%-% \HDindex{approximation}{ugxProblemSeriesSubstitutePage}{8.9.7.}{Substituting Numerical Values in Power Series}
+to substitute a numerical value for a variable in
+%-% \HDindex{series!numerical approximation}{ugxProblemSeriesSubstitutePage}{8.9.7.}{Substituting Numerical Values in Power Series}
+a power series.
+For example, here's a way to obtain numerical approximations of
+\axiom{\%e} from the Taylor series
+expansion of \axiom{exp(x)}.
+
+\labelSpace{1pc}
+\xtc{
+First you create the desired Taylor expansion.
+}{
+\spadpaste{f := taylor(exp(x)) \bound{f}}
+}
+\xtc{
+Then you evaluate the series at the value \axiom{1.0}.
+The result is a sequence of the partial sums.
+}{
+\spadpaste{eval(f,1.0)}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemSeriesBernoulliTitle}{Example: Bernoulli Polynomials and Sums of Powers}
+\newcommand{\ugxProblemSeriesBernoulliNumber}{8.9.8.}
+%
+% =====================================================================
+\begin{page}{ugxProblemSeriesBernoulliPage}{8.9.8. Example: Bernoulli Polynomials and Sums of Powers}
+% =====================================================================
+\beginscroll
+
+\Language{} provides operations for computing definite and
+%-% \HDindex{summation!definite}{ugxProblemSeriesBernoulliPage}{8.9.8.}{Example: Bernoulli Polynomials and Sums of Powers}
+indefinite sums.
+%-% \HDindex{summation!indefinite}{ugxProblemSeriesBernoulliPage}{8.9.8.}{Example: Bernoulli Polynomials and Sums of Powers}
+
+\labelSpace{3pc}
+\xtc{
+You can compute the sum of the first
+ten fourth powers by evaluating this.
+This creates a list whose entries are
+\texht{$m^4$}{\axiom{m**4}} as \texht{$m$}{\axiom{m}} ranges from 1
+to 10, and then computes the sum of the entries of that list.
+}{
+\spadpaste{reduce(+,[m**4 for m in 1..10])}
+}
+\xtc{
+You can also compute a formula for the sum of the first
+\texht{$k$}{\axiom{k}} fourth powers, where \texht{$k$}{\axiom{k}} is an
+unspecified positive integer.
+}{
+\spadpaste{sum4 := sum(m**4, m = 1..k) \bound{sum4}}
+}
+\xtc{
+This formula is valid for any positive integer \texht{$k$}{\axiom{k}}.
+For instance, if we replace \texht{$k$}{\axiom{k}} by 10,
+%-% \HDindex{summation!definite}{ugxProblemSeriesBernoulliPage}{8.9.8.}{Example: Bernoulli Polynomials and Sums of Powers}
+we obtain the number we computed earlier.
+}{
+\spadpaste{eval(sum4, k = 10) \free{sum4}}
+}
+
+You can compute a formula for the sum of the first
+\texht{$k$}{\axiom{k}} \eth{\axiom{n}} powers in a similar fashion.
+Just replace the \axiom{4} in the definition of \userfun{sum4} by
+any expression not involving \texht{$k$}{\axiom{k}}.
+\Language{} computes these formulas using Bernoulli polynomials;
+%-% \HDindex{Bernoulli!polynomial}{ugxProblemSeriesBernoulliPage}{8.9.8.}{Example: Bernoulli Polynomials and Sums of Powers}
+we
+%-% \HDindex{polynomial!Bernoulli}{ugxProblemSeriesBernoulliPage}{8.9.8.}{Example: Bernoulli Polynomials and Sums of Powers}
+use the rest of this section to describe this method.
+
+%
+\xtc{
+First consider this function of \axiom{t} and \axiom{x}.
+}{
+\spadpaste{f := t*exp(x*t) / (exp(t) - 1) \bound{f}}
+}
+\noOutputXtc{
+Since the expressions involved get quite large, we tell
+\Language{} to show us only terms of degree up to \axiom{5.}
+}{
+\spadpaste{)set streams calculate 5 \bound{set}}
+}
+%-% \HDsyscmdindex{set streams calculate}{ugxProblemSeriesBernoulliPage}{8.9.8.}{Example: Bernoulli Polynomials and Sums of Powers}
+%
+%
+\xtc{
+If we look at the Taylor expansion of \axiom{f(x, t)} about \axiom{t = 0,}
+we see that the coefficients of the powers of \axiom{t} are polynomials
+in \axiom{x}.
+}{
+\spadpaste{ff := taylor(f,t = 0) \free{f set} \bound{ff}}
+}
+%
+In fact, the \eth{\axiom{n}} coefficient in this series is essentially
+the \eth{\axiom{n}} Bernoulli polynomial:
+the \eth{\axiom{n}} coefficient of the series is
+\texht{${1 \over {n!}} B_n(x)$}{\axiom{1/n! * Bn(x)}}, where
+\texht{$B_n(x)$}{\axiom{Bn(x)}}
+is the \eth{\axiom{n}} Bernoulli polynomial.
+Thus, to obtain the \eth{\axiom{n}} Bernoulli polynomial, we multiply
+the \eth{\axiom{n}} coefficient
+of the series \axiom{ff} by \axiom{n!}.
+%
+\xtc{
+For example, the sixth Bernoulli polynomial is this.
+}{
+\spadpaste{factorial(6) * coefficient(ff,6) \free{ff}}
+}
+%
+\xtc{
+We derive some properties of the function \axiom{f(x,t)}.
+First we compute \axiom{f(x + 1,t) - f(x,t)}.
+}{
+\spadpaste{g := eval(f, x = x + 1) - f \bound{g} \free{f}}
+}
+%
+\xtc{
+If we normalize \axiom{g}, we see that it has a particularly simple form.
+}{
+\spadpaste{normalize(g) \free{g}}
+}
+%
+From this it follows that the \eth{\axiom{n}}
+coefficient in the Taylor expansion of
+\axiom{g(x,t)} at \axiom{t = 0} is
+\texht{${1\over{(n-1)\:!}}\:x^{n-1}$}{\axiom{1/(n-1)! * x**(n-1)}}.
+\xtc{
+If you want to check this, evaluate the next expression.
+}{
+\spadpaste{taylor(g,t = 0) \free{g}}
+}
+%
+However, since \axiom{g(x,t) = f(x+1,t)-f(x,t)}, it follows that the
+\eth{\axiom{n}} coefficient is
+\texht{${1 \over {n!}}\:(B_n(x+1)-B_n(x))$}{\axiom{1/n! * (Bn(x + 1) -
+Bn(x))}}.
+Equating coefficients, we see that
+\texht{${1\over{(n-1)\:!}}\:x^{n-1} = {1\over{n!}}\:(B_n(x + 1) -
+B_n(x))$}{\axiom{1/(n-1)! * x**(n-1) = 1/n! * (Bn(x + 1) - Bn(x))}}
+and, therefore,
+\texht{$x^{n-1} = {1\over{n}}\:(B_n(x + 1) -
+B_n(x))$}{\axiom{x**(n-1) = 1/n * (Bn(x + 1) - Bn(x))}}.
+%
+Let's apply this formula repeatedly, letting \axiom{x} vary between two
+integers \axiom{a} and \axiom{b}, with \axiom{a < b}:
+%
+\begin{verbatim}
+  a**(n-1)       = 1/n * (Bn(a + 1) - Bn(a))
+  (a + 1)**(n-1) = 1/n * (Bn(a + 2) - Bn(a + 1))
+  (a + 2)**(n-1) = 1/n * (Bn(a + 3) - Bn(a + 2))
+     .
+     .
+     .
+  (b - 1)**(n-1) = 1/n * (Bn(b) - Bn(b - 1))
+  b**(n-1)       = 1/n * (Bn(b + 1) - Bn(b))
+\end{verbatim}
+
+When we add these equations we find that
+the sum of the left-hand sides is
+\texht{$\sum_{m=a}^{b} m^{n-1},$}{\axiom{sum(m = a..b,m ** (n-1))},}%
+the sum of the
+\texht{$(n-1)^{\hbox{\small\rm st}}$}{\axiom{(n-1)}-st}
+powers from \axiom{a} to \axiom{b}.
+The sum of the right-hand sides is a ``telescoping series.''
+After cancellation, the sum is simply
+\texht{${1\over{n}}\:(B_n(b + 1) -
+B_n(a))$}{\axiom{1/n * (Bn(b + 1) - Bn(a))}}.
+
+Replacing \axiom{n} by \axiom{n + 1}, we have shown that
+\centerline{{\axiom{sum(m = a..b,m ** n) = 1/(n + 1) * (B<n+1>(b + 1) - B<n+1>(a))}.}}
+
+Let's use this to obtain the formula for the sum of fourth powers.
+\xtc{
+First we obtain the Bernoulli polynomial \texht{$B_5$}{\axiom{B5}}.
+}{
+\spadpaste{B5 := factorial(5) * coefficient(ff,5) \free{ff} \bound{B5}}
+}
+%
+\xtc{
+To find the sum of the first \texht{$k$}{\axiom{k}} 4th powers,
+we multiply \axiom{1/5} by
+\texht{$B_5(k+1) - B_5(1)$}{\axiom{B5(k + 1) - B5(1)}}.
+}{
+\spadpaste{1/5 * (eval(B5, x = k + 1) - eval(B5, x = 1)) \free{B5}}
+}
+%
+\xtc{
+This is the same formula that we obtained via \axiom{sum(m**4, m = 1..k)}.
+}{
+\spadpaste{sum4 \free{sum4}}
+}
+
+At this point you may want to do the same computation, but with an
+exponent other than \axiom{4.}
+For example, you might try to find a formula for the sum of the
+first \texht{$k$}{\axiom{k}} 20th powers.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemDEQTitle}{Solution of Differential Equations}
+\newcommand{\ugProblemDEQNumber}{8.10.}
+%
+% =====================================================================
+\begin{page}{ugProblemDEQPage}{8.10. Solution of Differential Equations}
+% =====================================================================
+\beginscroll
+%
+In this section we discuss \Language{}'s facilities for
+%-% \HDindex{equation!differential!solving}{ugProblemDEQPage}{8.10.}{Solution of Differential Equations}
+solving
+%-% \HDindex{differential equation}{ugProblemDEQPage}{8.10.}{Solution of Differential Equations}
+differential equations in closed-form and in series.
+
+\Language{} provides facilities for closed-form solution of
+%-% \HDindex{equation!differential!solving in closed-form}{ugProblemDEQPage}{8.10.}{Solution of Differential Equations}
+single differential equations of the following kinds:
+\indent{4}
+\beginitems
+\item[-] linear ordinary differential equations, and
+\item[-] non-linear first order ordinary differential equations
+when integrating factors can be found just by integration.
+\enditems
+\indent{0}
+
+For a discussion of the solution of systems of linear and polynomial
+equations, see
+\downlink{``\ugProblemLinPolEqnTitle''}{ugProblemLinPolEqnPage} in Section \ugProblemLinPolEqnNumber\ignore{ugProblemLinPolEqn}.
+
+\beginmenu
+    \menudownlink{{8.10.1. Closed-Form Solutions of Linear Differential Equations}}{ugxProblemLDEQClosedPage}
+    \menudownlink{{8.10.2. Closed-Form Solutions of Non-Linear Differential Equations}}{ugxProblemNLDEQClosedPage}
+    \menudownlink{{8.10.3. Power Series Solutions of Differential Equations}}{ugxProblemDEQSeriesPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemLDEQClosedTitle}{Closed-Form Solutions of Linear Differential Equations}
+\newcommand{\ugxProblemLDEQClosedNumber}{8.10.1.}
+%
+% =====================================================================
+\begin{page}{ugxProblemLDEQClosedPage}{8.10.1. Closed-Form Solutions of Linear Differential Equations}
+% =====================================================================
+\beginscroll
+
+A {\it differential equation} is an equation involving an unknown {\it
+function} and one or more of its derivatives.
+%-% \HDindex{differential equation}{ugxProblemLDEQClosedPage}{8.10.1.}{Closed-Form Solutions of Linear Differential Equations}
+The equation is called {\it ordinary} if derivatives with respect to
+%-% \HDindex{equation!differential}{ugxProblemLDEQClosedPage}{8.10.1.}{Closed-Form Solutions of Linear Differential Equations}
+only one dependent variable appear in the equation (it is called {\it
+partial} otherwise).
+The package \axiomType{ElementaryFunctionODESolver} provides the
+top-level operation \spadfun {solve} for finding closed-form solutions
+of ordinary differential equations.
+%-% \HDexptypeindex{ElementaryFunctionODESolver}{ugxProblemLDEQClosedPage}{8.10.1.}{Closed-Form Solutions of Linear Differential Equations}
+
+To solve a differential equation, you must first create an operator for
+%-% \HDindex{operator}{ugxProblemLDEQClosedPage}{8.10.1.}{Closed-Form Solutions of Linear Differential Equations}
+the unknown function.
+%
+\xtc{
+We let \axiom{y} be the unknown function in terms of \axiom{x}.
+}{
+\spadpaste{y := operator 'y \bound{y}}
+}
+%
+You then type the equation using \axiomFun{D} to create the
+derivatives of the unknown function \axiom{y(x)} where \axiom{x} is any
+symbol you choose (the so-called {\it dependent variable}).
+%
+\xtc{
+This is how you enter
+the equation \axiom{y'' + y' + y = 0}.
+}{
+\spadpaste{deq := D(y x, x, 2) + D(y x, x) + y x = 0\bound{e1}\free{y}}
+}
+%
+The simplest way to invoke the \axiomFun{solve} command is with three
+arguments.
+\begin{items}
+\item the differential equation,
+\item the operator representing the unknown function,
+\item the dependent variable.
+\end{items}
+%
+\xtc{
+So, to solve the above equation, we enter this.
+}{
+\spadpaste{solve(deq, y, x) \free{e1}\free{y}}
+}
+%
+Since linear ordinary differential equations have infinitely many
+solutions, \axiomFun{solve} returns a {\it particular solution}
+\texht{$f_p$}{\axiom{f_p}}
+and a basis
+\texht{$f_1,\dots,f_n$}{\axiom{f1,...,fn}}
+for the solutions of the corresponding homogenuous equation.
+Any expression of the form
+\texht{$f_p + c_1 f_1 + \dots c_n f_n$}{\axiom{fp + c1 f1 + ... + cn fn}}
+where the \texht{$c_i$}{\axiom{ci}} do not involve
+the dependent variable is also a solution.
+This is similar to what you get when you solve systems of linear
+algebraic equations.
+
+A way to select a unique solution is to specify {\it initial
+conditions}: choose a value \axiom{a} for the dependent variable
+and specify the values of the unknown function and its derivatives
+at \axiom{a}.
+If the number of initial conditions is equal to the order of the
+equation, then the solution is unique (if it exists in closed
+form!) and \axiomFun{solve} tries to find it.
+To specify initial conditions to \axiomFun{solve}, use an
+\axiomType{Equation} of the form \axiom{x = a} for the third
+parameter instead of the dependent variable, and add a fourth
+parameter consisting of the list of values \axiom{y(a), y'(a), ...}.
+
+\xtc{
+To find the solution of \axiom{y'' + y = 0} satisfying \axiom{y(0) = y'(0) = 1},
+do this.
+}{
+\spadpaste{deq := D(y x, x, 2) + y x \bound{e2}\free{y}}
+}
+\xtc{
+You can omit the \axiom{= 0} when you enter the equation to be solved.
+}{
+\spadpaste{solve(deq, y, x = 0, [1, 1]) \free{e2}\free{y}}
+}
+%
+
+\Language{} is not limited to linear differential equations with
+constant coefficients.
+It can also find solutions when the coefficients are rational or
+algebraic functions of the dependent variable.
+Furthermore, \Language{} is not limited by the order of the equation.
+%
+\xtc{
+\Language{} can solve the following third order equations with
+polynomial coefficients.
+}{
+\spadpaste{deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4 \bound{e3}\free{y}}
+}
+\xtc{
+}{
+\spadpaste{solve(deq, y, x) \free{e3}\free{y}}
+}
+%
+%
+\xtc{
+Here we are solving a homogeneous equation.
+}{
+\spadpaste{deq := (x**9+x**3) * D(y x, x, 3) + 18 * x**8 * D(y x, x, 2) - 90 * x * D(y x, x) - 30 * (11 * x**6 - 3) * y x \bound{e4}\free{y}}
+}
+\xtc{
+}{
+\spadpaste{solve(deq, y, x) \free{e4}\free{y}}
+}
+%
+On the other hand, and in contrast with the operation
+\axiomFun{integrate}, it can happen that \Language{} finds no solution
+and that some closed-form solution still exists.
+While it is mathematically complicated to describe exactly when the
+solutions are guaranteed to be found, the following statements are
+correct and form good guidelines for linear ordinary differential
+equations:
+\begin{items}
+\item If the coefficients are constants, \Language{} finds a complete basis
+of solutions (i,e, all solutions).
+\item If the coefficients are rational functions in the dependent variable,
+\Language{} at least finds all solutions that do not involve algebraic
+functions.
+\end{items}
+%
+Note that this last statement does not mean that \Language{} does not find
+the solutions that are algebraic functions.
+It means that it is not
+guaranteed that the algebraic function solutions will be found.
+%
+\xtc{
+This is an example where all the algebraic solutions are found.
+}{
+\spadpaste{deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0 \bound{e5}\free{y}}
+}
+\xtc{
+}{
+\spadpaste{solve(deq, y, x) \free{e5}\free{y}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemNLDEQClosedTitle}{Closed-Form Solutions of Non-Linear Differential Equations}
+\newcommand{\ugxProblemNLDEQClosedNumber}{8.10.2.}
+%
+% =====================================================================
+\begin{page}{ugxProblemNLDEQClosedPage}{8.10.2. Closed-Form Solutions of Non-Linear Differential Equations}
+% =====================================================================
+\beginscroll
+
+This is an example that shows how to solve a non-linear
+first order ordinary differential equation manually when an integrating
+factor can be found just by integration.
+At the end, we show you how to solve it directly.
+
+Let's solve the differential equation \axiom{y' = y / (x + y log y)}.
+%
+\xtc{
+Using the notation
+\axiom{m(x, y) + n(x, y) y' = 0},
+we have \axiom{m = -y} and \axiom{n = x + y log y}.
+}{
+\spadpaste{m := -y \bound{m}}
+}
+\xtc{
+}{
+\spadpaste{n := x + y * log y \bound{n}}
+}
+%
+\xtc{
+We first check for exactness, that is, does \axiom{dm/dy = dn/dx}?
+}{
+\spadpaste{D(m, y) - D(n, x) \free{m n}}
+}
+%
+This is not zero, so the equation is not exact.
+Therefore we must look for
+an integrating factor: a function \axiom{mu(x,y)} such that
+\axiom{d(mu m)/dy = d(mu n)/dx}.
+Normally, we first search for \axiom{mu(x,y)} depending only on
+\axiom{x} or only on \axiom{y}.
+%
+\xtc{
+Let's search for such a \axiom{mu(x)} first.
+}{
+\spadpaste{mu := operator 'mu \bound{mu}}
+}
+\xtc{
+}{
+\spadpaste{a := D(mu(x) * m, y) - D(mu(x) * n, x) \bound{a}\free{m n mu}}
+}
+%
+%
+\xtc{
+If the above is zero for a function
+\axiom{mu} that does {\it not} depend on \axiom{y}, then
+\axiom{mu(x)} is an integrating factor.
+}{
+\spadpaste{solve(a = 0, mu, x) \free{mu a}}
+}
+%
+The solution depends on \axiom{y}, so there is no integrating
+factor that depends on \axiom{x} only.
+%
+\xtc{
+Let's look for one that depends on \axiom{y} only.
+}{
+\spadpaste{b := D(mu(y) * m, y) - D(mu(y) * n, x) \bound{b}\free{mu m}}
+}
+\xtc{
+}{
+\spadpaste{sb := solve(b = 0, mu, y) \free{mu b}\bound{sb}}
+}
+\noindent
+We've found one!
+%
+\xtc{
+The above \axiom{mu(y)} is an integrating factor.
+We must multiply our initial equation
+(that is, \axiom{m} and \axiom{n}) by the integrating factor.
+}{
+\spadpaste{intFactor := sb.basis.1 \bound{intFactor}\free{sb}}
+}
+\xtc{
+}{
+\spadpaste{m := intFactor * m \bound{m1}\free{m intFactor}}
+}
+\xtc{
+}{
+\spadpaste{n := intFactor * n \bound{n1}\free{n intFactor}}
+}
+%
+\xtc{
+Let's check for exactness.
+}{
+\spadpaste{D(m, y) - D(n, x) \free{m1 n1}}
+}
+%
+We must solve the exact equation, that is, find a function
+\axiom{s(x,y)} such that
+\axiom{ds/dx = m}  and \axiom{ds/dy = n}.
+%
+\xtc{
+We start by writing \axiom{s(x, y) = h(y) + integrate(m, x)}
+where \axiom{h(y)} is an unknown function of \axiom{y}.
+This guarantees that \axiom{ds/dx = m}.
+}{
+\spadpaste{h := operator 'h \bound{h}}
+}
+\xtc{
+}{
+\spadpaste{sol := h y + integrate(m, x) \bound{sol}\free{h m1}}
+}
+%
+%
+\xtc{
+All we want is to find \axiom{h(y)} such that
+\axiom{ds/dy = n}.
+}{
+\spadpaste{dsol := D(sol, y) \free{sol}\bound{dsol}}
+}
+\xtc{
+}{
+\spadpaste{nsol := solve(dsol = n, h, y) \free{dsol n1 h}\bound{nsol}}
+}
+%
+\xtc{
+The above particular solution is the \axiom{h(y)} we want, so we just replace
+\axiom{h(y)} by it in the implicit solution.
+}{
+\spadpaste{eval(sol, h y = nsol.particular) \free{sol h nsol}}
+}
+%
+A first integral of the initial equation is obtained by setting
+this result equal to an arbitrary constant.
+
+Now that we've seen how to solve the equation ``by hand,''
+we show you how to do it with the \axiomFun{solve} operation.
+\xtc{
+First define \axiom{y} to be an operator.
+}{
+\spadpaste{y := operator 'y \bound{y}}
+}
+\xtc{
+Next we create the differential equation.
+}{
+\spadpaste{deq := D(y x, x) = y(x) / (x + y(x) * log y x) \bound{deqi}\free{y}}
+}
+\xtc{
+Finally, we solve it.
+}{
+\spadpaste{solve(deq, y, x) \free{deqi y}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemDEQSeriesTitle}{Power Series Solutions of Differential Equations}
+\newcommand{\ugxProblemDEQSeriesNumber}{8.10.3.}
+%
+% =====================================================================
+\begin{page}{ugxProblemDEQSeriesPage}{8.10.3. Power Series Solutions of Differential Equations}
+% =====================================================================
+\beginscroll
+
+The command to solve differential equations in power
+%-% \HDindex{equation!differential!solving in power series}{ugxProblemDEQSeriesPage}{8.10.3.}{Power Series Solutions of Differential Equations}
+series
+%-% \HDindex{power series}{ugxProblemDEQSeriesPage}{8.10.3.}{Power Series Solutions of Differential Equations}
+around
+%-% \HDindex{series!power}{ugxProblemDEQSeriesPage}{8.10.3.}{Power Series Solutions of Differential Equations}
+a particular initial point with specific initial conditions is called
+\axiomFun{seriesSolve}.
+It can take a variety of parameters, so we illustrate
+its use with some examples.
+
+\labelSpace{1pc}
+\noOutputXtc{
+Since the coefficients of some solutions
+are quite large, we reset the default to compute only seven terms.
+}{
+\spadpaste{)set streams calculate 7 \bound{c7}}
+}
+
+You can solve a single nonlinear equation of any order. For example,
+we solve  \axiom{y''' = sin(y'') * exp(y) + cos(x)}
+subject to \axiom{y(0) = 1, y'(0) = 0, y''(0) = 0}.
+
+\xtc{
+We first tell \Language{}
+that the symbol \axiom{'y} denotes a new operator.
+}{
+\spadpaste{y := operator 'y \bound{y}}
+}
+\xtc{
+Enter the differential equation using \axiom{y} like any system
+function.
+}{
+\spadpaste{eq := D(y(x), x, 3) - sin(D(y(x), x, 2))*exp(y(x)) = cos(x)\bound{eq}\free{y}}
+}
+%
+\xtc{
+Solve it around \axiom{x = 0} with the initial conditions
+\axiom{y(0) = 1, y'(0) = y''(0) = 0}.
+}{
+\spadpaste{seriesSolve(eq, y, x = 0, [1, 0, 0])\free{y}\free{eq}\free{c7}}
+}
+
+You can also solve a system of nonlinear first order equations.
+For example, we solve a system that has \axiom{tan(t)} and
+\axiom{sec(t)} as solutions.
+
+\xtc{
+We tell \Language{} that \axiom{x} is also an operator.
+}{
+\spadpaste{x := operator 'x\bound{x}}
+}
+\xtc{
+Enter the two equations forming our system.
+}{
+\spadpaste{eq1 := D(x(t), t) = 1 + x(t)**2\free{x}\free{y}\bound{eq1}}
+}
+%
+\xtc{
+}{
+\spadpaste{eq2 := D(y(t), t) = x(t) * y(t)\free{x}\free{y}\bound{eq2}}
+}
+%
+\xtc{
+Solve the system around \axiom{t = 0} with the initial conditions
+\axiom{x(0) = 0} and \axiom{y(0) = 1}.
+Notice that since we give the unknowns in the
+order \axiom{[x, y]}, the answer is a list of two series in the order
+\axiom{[series for x(t), series for y(t)]}.
+}{
+\spadpaste{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x}\free{y}\free{eq1}\free{eq2}\free{c7}}
+}
+\noindent
+The order in which we give the
+equations and the initial conditions has no effect on the order of
+the solution.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemFiniteTitle}{Finite Fields}
+\newcommand{\ugProblemFiniteNumber}{8.11.}
+%
+% =====================================================================
+\begin{page}{ugProblemFinitePage}{8.11. Finite Fields}
+% =====================================================================
+\beginscroll
+%
+
+A {\it finite field} (also called a {\it Galois field}) is a
+finite algebraic structure where one can add, multiply and divide
+under the same laws (for example, commutativity, associativity or
+distributivity) as apply to the rational, real or complex numbers.
+Unlike those three fields, for any finite field there exists a
+positive prime integer \smath{p}, called the
+\axiomFun{characteristic}, such that \texht{$p \: x = 0$}{\axiom{p *
+x = 0}} for any element \smath{x} in the finite field.
+In fact, the number of elements in a finite field is a power of
+the characteristic and for each prime \smath{p} and positive
+integer \smath{n} there exists exactly one finite field with
+\texht{$p^n$}{\axiom{p**n}} elements, up to
+isomorphism.\footnote{For more information about the algebraic
+structure and properties of finite fields, see, for example, S.
+Lang, {\it Algebra}, Second Edition, New York: Addison-Wesley
+Publishing Company, Inc., 1984, ISBN 0 201 05487 6; or R.
+Lidl, H.
+Niederreiter, {\it Finite Fields}, Encyclopedia of Mathematics and
+Its Applications, Vol.
+20, Cambridge: Cambridge Univ.
+Press, 1983, ISBN 0 521 30240 4.}
+
+When \axiom{n = 1,} the field has \smath{p} elements and is
+called a {\it prime field}, discussed in \texht{the next
+section}{\downlink{``\ugxProblemFinitePrimeTitle''}{ugxProblemFinitePrimePage} in Section \ugxProblemFinitePrimeNumber\ignore{ugxProblemFinitePrime}}.
+There are several ways of implementing extensions of finite
+fields, and \Language{} provides quite a bit of freedom to allow
+you to choose the one that is best for your application.
+Moreover, we provide operations for converting among the different
+representations of extensions and different extensions of a single
+field.
+Finally, note that you usually need to package-call operations
+from finite fields if the operations do not take as an argument an
+object of the field.
+See \downlink{``\ugTypesPkgCallTitle''}{ugTypesPkgCallPage} in Section \ugTypesPkgCallNumber\ignore{ugTypesPkgCall} for more information on
+package-calling.
+
+\beginmenu
+    \menudownlink{{8.11.1. Modular Arithmetic and Prime Fields}}{ugxProblemFinitePrimePage}
+    \menudownlink{{8.11.2. Extensions of Finite Fields}}{ugxProblemFiniteExtensionFinitePage}
+    \menudownlink{{8.11.3. Irreducible Modulus Polynomial Representations}}{ugxProblemFiniteModulusPage}
+    \menudownlink{{8.11.4. Cyclic Group Representations}}{ugxProblemFiniteCyclicPage}
+    \menudownlink{{8.11.5. Normal Basis Representations}}{ugxProblemFiniteNormalPage}
+    \menudownlink{{8.11.6. Conversion Operations for Finite Fields}}{ugxProblemFiniteConversionPage}
+    \menudownlink{{8.11.7. Utility Operations for Finite Fields}}{ugxProblemFiniteUtilityPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemFinitePrimeTitle}{Modular Arithmetic and Prime Fields}
+\newcommand{\ugxProblemFinitePrimeNumber}{8.11.1.}
+%
+% =====================================================================
+\begin{page}{ugxProblemFinitePrimePage}{8.11.1. Modular Arithmetic and Prime Fields}
+% =====================================================================
+\beginscroll
+%-% \HDindex{finite field}{ugxProblemFinitePrimePage}{8.11.1.}{Modular Arithmetic and Prime Fields}
+%-% \HDindex{Galois!field}{ugxProblemFinitePrimePage}{8.11.1.}{Modular Arithmetic and Prime Fields}
+%-% \HDindex{field!finite!prime}{ugxProblemFinitePrimePage}{8.11.1.}{Modular Arithmetic and Prime Fields}
+%-% \HDindex{field!prime}{ugxProblemFinitePrimePage}{8.11.1.}{Modular Arithmetic and Prime Fields}
+%-% \HDindex{field!Galois}{ugxProblemFinitePrimePage}{8.11.1.}{Modular Arithmetic and Prime Fields}
+%-% \HDindex{prime field}{ugxProblemFinitePrimePage}{8.11.1.}{Modular Arithmetic and Prime Fields}
+%-% \HDindex{modular arithmetic}{ugxProblemFinitePrimePage}{8.11.1.}{Modular Arithmetic and Prime Fields}
+%-% \HDindex{arithmetic!modular}{ugxProblemFinitePrimePage}{8.11.1.}{Modular Arithmetic and Prime Fields}
+
+Let \smath{n} be a positive integer.
+It is well known that you can get the same result if you perform
+addition, subtraction or multiplication of integers and then take
+the remainder on dividing by \smath{n} as if
+you had first done such remaindering on the operands, performed the
+arithmetic and then (if necessary) done remaindering again.
+This allows us to speak of arithmetic
+{\it modulo} \smath{n} or, more simply
+{\it mod} \smath{n}.
+\xtc{
+In \Language{}, you use \axiomType{IntegerMod} to do such arithmetic.
+}{
+\spadpaste{(a,b) : IntegerMod 12 \bound{abdec}}
+}
+\xtc{
+}{
+\spadpaste{(a, b) := (16, 7) \free{abdec}\bound{a b}}
+}
+\xtc{
+}{
+\spadpaste{[a - b, a * b] \free{a b}}
+}
+\xtc{
+If \smath{n} is not prime, there is only a limited notion of
+reciprocals and division.
+}{
+\spadpaste{a / b \free{a b}}
+}
+\xtc{
+}{
+\spadpaste{recip a \free{a}}
+}
+\xtc{
+Here \axiom{7} and \axiom{12} are relatively prime, so \axiom{7}
+has a multiplicative inverse modulo \axiom{12}.
+}{
+\spadpaste{recip b \free{b}}
+}
+
+If we take \smath{n} to be a prime number \smath{p},
+then taking inverses and, therefore, division are generally defined.
+\xtc{
+Use \axiomType{PrimeField} instead of \axiomType{IntegerMod}
+for \smath{n} prime.
+}{
+\spadpaste{c : PrimeField 11 := 8 \bound{c}}
+}
+\xtc{
+}{
+\spadpaste{inv c \free{c}}
+}
+\xtc{
+You can also use \axiom{1/c} and \axiom{c**(-1)} for the inverse of
+\smath{c}.
+}{
+\spadpaste{9/c \free{c}}
+}
+
+\axiomType{PrimeField} (abbreviation \axiomType{PF}) checks if its
+argument is prime when you try to use an operation from it.
+If you know the argument is prime (particularly if it is large),
+\axiomType{InnerPrimeField} (abbreviation \axiomType{IPF}) assumes
+the argument has already been verified to be prime.
+If you do use a number that is not prime, you will eventually get
+an error message, most likely a division by zero message.
+For computer science applications, the most important finite fields
+are \axiomType{PrimeField 2} and its extensions.
+
+\xtc{
+In the following examples, we work with the finite field with
+\smath{p = 101} elements.
+}{
+\spadpaste{GF101 := PF 101 \bound{GF101} }
+}
+\xtc{
+Like many domains in \Language{}, finite fields provide an operation
+for returning a random element of the domain.
+}{
+\spadpaste{x := random()\$GF101 \bound{x}\free{GF101}}
+}
+\xtc{
+}{
+\spadpaste{y : GF101 := 37 \bound{y}\free{GF101}}
+}
+\xtc{
+}{
+\spadpaste{z := x/y \bound{z}\free{x y}}
+}
+\xtc{
+}{
+\spadpaste{z * y - x \free{x y z}}
+}
+%
+\xtc{
+The element \axiom{2} is a {\it primitive element} of this field,
+%-% \HDindex{primitive element}{ugxProblemFinitePrimePage}{8.11.1.}{Modular Arithmetic and Prime Fields}
+%-% \HDindex{element!primitive}{ugxProblemFinitePrimePage}{8.11.1.}{Modular Arithmetic and Prime Fields}
+}{
+\spadpaste{pe := primitiveElement()\$GF101 \bound{pe}\free{GF101}}
+}
+%
+\xtc{
+in the sense that its powers enumerate all nonzero elements.
+}{
+\spadpaste{[pe**i for i in 0..99] \free{pe}}
+}
+%
+%
+\xtc{
+If every nonzero element is a power of a primitive element, how do you
+determine what the exponent is?
+Use
+%-% \HDindex{discrete logarithm}{ugxProblemFinitePrimePage}{8.11.1.}{Modular Arithmetic and Prime Fields}
+\axiomFun{discreteLog}.
+%-% \HDindex{logarithm!discrete}{ugxProblemFinitePrimePage}{8.11.1.}{Modular Arithmetic and Prime Fields}
+}{
+\spadpaste{ex := discreteLog(y) \bound{ex}\free{y}}
+}
+\xtc{
+}{
+\spadpaste{pe ** ex \free{ex pe}}
+}
+%
+\xtc{
+The \axiomFun{order} of a nonzero element \smath{x} is the
+smallest positive integer \smath{t} such
+\texht{$x^t = 1$}{\axiom{x**t = 1}}.
+}{
+\spadpaste{order y \free{y}}
+}
+\xtc{
+The order of a primitive element is the defining \smath{p-1}.
+}{
+\spadpaste{order pe \free{pe}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemFiniteExtensionFiniteTitle}{Extensions of Finite Fields}
+\newcommand{\ugxProblemFiniteExtensionFiniteNumber}{8.11.2.}
+%
+% =====================================================================
+\begin{page}{ugxProblemFiniteExtensionFinitePage}{8.11.2. Extensions of Finite Fields}
+% =====================================================================
+\beginscroll
+%-% \HDindex{finite field}{ugxProblemFiniteExtensionFinitePage}{8.11.2.}{Extensions of Finite Fields}
+%-% \HDindex{field!finite!extension of}{ugxProblemFiniteExtensionFinitePage}{8.11.2.}{Extensions of Finite Fields}
+
+When you want to work with an extension of a finite field in \Language{},
+you have three choices to make:
+\indent{4}
+\beginitems
+\item[1. ] Do you want to generate an extension of the prime field
+(for example, \axiomType{PrimeField 2}) or an extension of a given field?
+%
+\item[2. ] Do you want to use a representation that is particularly
+efficient for multiplication, exponentiation and addition but
+uses a lot of computer memory (a representation that models the cyclic
+group structure of the multiplicative group of the field extension
+and uses a Zech logarithm table),  one that
+%-% \HDindex{Zech logarithm}{ugxProblemFiniteExtensionFinitePage}{8.11.2.}{Extensions of Finite Fields}
+uses a normal basis for the vector space structure of the field
+extension, or one that performs arithmetic modulo an irreducible
+polynomial?
+The cyclic group representation is only usable up to ``medium''
+(relative to your machine's performance)
+sized fields.
+If the field is large and the normal basis is relatively simple,
+the normal basis representation is more efficient for exponentiation than
+the irreducible polynomial representation.
+%
+\item[3. ] Do you want to provide a polynomial explicitly, a root of which
+``generates'' the extension in one of the three senses in (2),
+or do you wish to have the polynomial generated for you?
+\enditems
+\indent{0}
+
+This illustrates one of the most important features of \Language{}:
+you can choose exactly the right data-type and representation to
+suit your application best.
+
+We first tell you what domain constructors to use for each case
+above, and then give some examples.
+
+\texht{\hangafter=1\hangindent=2pc}{\noindent}
+Constructors that automatically generate extensions of the prime field:
+\newline
+\axiomType{FiniteField} \newline
+\axiomType{FiniteFieldCyclicGroup} \newline
+\axiomType{FiniteFieldNormalBasis}
+
+\texht{\hangafter=1\hangindent=2pc}{\noindent}
+Constructors that generate extensions of an arbitrary field:
+\newline
+\axiomType{FiniteFieldExtension} \newline
+\axiomType{FiniteFieldExtensionByPolynomial} \newline
+\axiomType{FiniteFieldCyclicGroupExtension} \newline
+\axiomType{FiniteFieldCyclicGroupExtensionByPolynomial} \newline
+\axiomType{FiniteFieldNormalBasisExtension} \newline
+\axiomType{FiniteFieldNormalBasisExtensionByPolynomial}
+
+\texht{\hangafter=1\hangindent=2pc}{\noindent}
+Constructors that use a cyclic group representation:
+\newline
+\axiomType{FiniteFieldCyclicGroup} \newline
+\axiomType{FiniteFieldCyclicGroupExtension} \newline
+\axiomType{FiniteFieldCyclicGroupExtensionByPolynomial}
+
+\texht{\hangafter=1\hangindent=2pc}{\noindent}
+Constructors that use a normal basis representation:
+\newline
+\axiomType{FiniteFieldNormalBasis} \newline
+\axiomType{FiniteFieldNormalBasisExtension} \newline
+\axiomType{FiniteFieldNormalBasisExtensionByPolynomial}
+
+\texht{\hangafter=1\hangindent=2pc}{\noindent}
+Constructors that use an irreducible modulus polynomial representation:
+\newline
+\axiomType{FiniteField} \newline
+\axiomType{FiniteFieldExtension} \newline
+\axiomType{FiniteFieldExtensionByPolynomial}
+
+\texht{\hangafter=1\hangindent=2pc}{\noindent}
+Constructors that generate a polynomial for you:
+\newline
+\axiomType{FiniteField} \newline
+\axiomType{FiniteFieldExtension} \newline
+\axiomType{FiniteFieldCyclicGroup} \newline
+\axiomType{FiniteFieldCyclicGroupExtension} \newline
+\axiomType{FiniteFieldNormalBasis} \newline
+\axiomType{FiniteFieldNormalBasisExtension}
+
+\texht{\hangafter=1\hangindent=2pc}{\noindent}
+Constructors for which you provide a polynomial:
+\newline
+\axiomType{FiniteFieldExtensionByPolynomial} \newline
+\axiomType{FiniteFieldCyclicGroupExtensionByPolynomial} \newline
+\axiomType{FiniteFieldNormalBasisExtensionByPolynomial}
+
+These constructors are discussed in the following sections where
+we collect together descriptions of extension fields that have the
+same underlying representation.\footnote{For
+more information on the implementation aspects of finite
+fields, see
+J. Grabmeier, A. Scheerhorn, {\it Finite Fields in AXIOM,}
+Technical Report, IBM Heidelberg Scientific Center, 1992.}
+
+If you don't really care about all this detail, just use
+\axiomType{FiniteField}.
+As your knowledge of your application and its \Language{} implementation
+grows, you can come back and choose an alternative constructor that
+may improve the efficiency of your code.
+Note that the exported operations are almost the same for all constructors
+of finite field extensions and include the operations exported by
+\axiomType{PrimeField}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemFiniteModulusTitle}{Irreducible Modulus Polynomial Representations}
+\newcommand{\ugxProblemFiniteModulusNumber}{8.11.3.}
+%
+% =====================================================================
+\begin{page}{ugxProblemFiniteModulusPage}{8.11.3. Irreducible Modulus Polynomial Representations}
+% =====================================================================
+\beginscroll
+
+All finite field extension constructors discussed in this
+%-% \HDindex{finite field}{ugxProblemFiniteModulusPage}{8.11.3.}{Irreducible Modulus Polynomial Representations}
+section
+%-% \HDindex{field!finite!extension of}{ugxProblemFiniteModulusPage}{8.11.3.}{Irreducible Modulus Polynomial Representations}
+use a representation that performs arithmetic with univariate
+(one-variable) polynomials modulo an irreducible polynomial.
+This polynomial may be given explicitly by you or automatically
+generated.
+The ground field may be the prime field or one you specify.
+See \downlink{``\ugxProblemFiniteExtensionFiniteTitle''}{ugxProblemFiniteExtensionFinitePage} in Section \ugxProblemFiniteExtensionFiniteNumber\ignore{ugxProblemFiniteExtensionFinite} for general
+information about finite field extensions.
+
+For \axiomType{FiniteField} (abbreviation \axiomType{FF}) you provide a
+prime number \smath{p} and an extension degree \smath{n}.
+This degree can be 1.
+%
+\xtc{
+\Language{} uses the prime field \axiomType{PrimeField(p)},
+here \axiomType{PrimeField 2},
+and it chooses an irreducible polynomial of degree \smath{n},
+here 12, over the ground field.
+}{
+\spadpaste{GF4096 := FF(2,12); \bound{GF4096}}
+}
+%
+
+The objects in the generated field extension are polynomials
+of degree at most \smath{n-1} with coefficients in the
+prime field.
+The polynomial indeterminate is automatically chosen by \Language{} and
+is typically something like \axiom{\%A} or \axiom{\%D}.
+These (strange) variables are {\it only} for output display;
+there are several ways to construct elements of this field.
+
+The operation \axiomFun{index} enumerates the elements of the field
+extension and accepts as argument the integers from 1 to
+\smath{p \texht{^}{**} n}.
+%
+\xtc{
+The expression
+\axiom{index(p)} always gives the indeterminate.
+}{
+\spadpaste{a := index(2)\$GF4096 \bound{a}\free{GF4096}}
+}
+%
+%
+\xtc{
+You can build polynomials in \smath{a} and calculate in
+\axiom{GF4096}.
+}{
+\spadpaste{b := a**12 - a**5 + a \bound{b}\free{a}}
+}
+\xtc{
+}{
+\spadpaste{b ** 1000 \free{b}}
+}
+\xtc{
+}{
+\spadpaste{c := a/b \free{a b}\bound{c}}
+}
+%
+\xtc{
+Among the available operations are \axiomFun{norm} and \axiomFun{trace}.
+}{
+\spadpaste{norm c \free{c}}
+}
+\xtc{
+}{
+\spadpaste{trace c \free{c}}
+}
+%
+%
+
+Since any nonzero element is a power of a primitive element, how
+do we discover what the exponent is?
+%
+\xtc{
+The operation \axiomFun{discreteLog} calculates
+%-% \HDindex{discrete logarithm}{ugxProblemFiniteModulusPage}{8.11.3.}{Irreducible Modulus Polynomial Representations}
+the exponent and,
+%-% \HDindex{logarithm!discrete}{ugxProblemFiniteModulusPage}{8.11.3.}{Irreducible Modulus Polynomial Representations}
+if it is called with only one argument, always refers to the primitive
+element returned by \axiomFun{primitiveElement}.
+}{
+\spadpaste{dL := discreteLog a \free{a}\bound{dL}}
+}
+\xtc{
+}{
+\spadpaste{g ** dL \free{dL g}}
+}
+
+\axiomType{FiniteFieldExtension} (abbreviation \axiomType{FFX}) is
+similar to \axiomType{FiniteField} except that the
+ground-field for \axiomType{FiniteFieldExtension} is arbitrary and
+chosen by you.
+%
+\xtc{
+In case you select the prime field as ground field, there is
+essentially no difference between the constructed two finite field
+extensions.
+}{
+\spadpaste{GF16 := FF(2,4); \bound{GF16}}
+}
+\xtc{
+}{
+\spadpaste{GF4096 := FFX(GF16,3); \bound{GF4096x}\free{GF16}}
+}
+\xtc{
+}{
+\spadpaste{r := (random()\$GF4096) ** 20 \free{GF4096x}\bound{r}}
+}
+\xtc{
+}{
+\spadpaste{norm(r) \free{r}}
+}
+%
+
+\axiomType{FiniteFieldExtensionByPolynomial} (abbreviation \axiomType{FFP})
+is similar to \axiomType{FiniteField} and \axiomType{FiniteFieldExtension}
+but is more general.
+%
+\xtc{
+}{
+\spadpaste{GF4 := FF(2,2); \bound{GF4}}
+}
+\xtc{
+}{
+\spadpaste{f := nextIrreduciblePoly(random(6)\$FFPOLY(GF4))\$FFPOLY(GF4) \free{GF4}\bound{f}}
+}
+\xtc{
+For \axiomType{FFP} you choose both the
+ground field and the irreducible polynomial used in the representation.
+The degree of the extension is the degree of the polynomial.
+}{
+\spadpaste{GF4096 := FFP(GF4,f); \bound{GF4096y}\free{f GF4}}
+}
+\xtc{
+}{
+\spadpaste{discreteLog random()\$GF4096 \free{GF4096y}}
+}
+%
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemFiniteCyclicTitle}{Cyclic Group Representations}
+\newcommand{\ugxProblemFiniteCyclicNumber}{8.11.4.}
+%
+% =====================================================================
+\begin{page}{ugxProblemFiniteCyclicPage}{8.11.4. Cyclic Group Representations}
+% =====================================================================
+\beginscroll
+%-% \HDindex{finite field}{ugxProblemFiniteCyclicPage}{8.11.4.}{Cyclic Group Representations}
+%-% \HDindex{field!finite!extension of}{ugxProblemFiniteCyclicPage}{8.11.4.}{Cyclic Group Representations}
+
+In every finite field there exist elements whose powers are all the
+nonzero elements of the field.
+Such an element is called a {\it primitive element}.
+
+In \axiomType{FiniteFieldCyclicGroup} (abbreviation \axiomType{FFCG})
+%-% \HDindex{group!cyclic}{ugxProblemFiniteCyclicPage}{8.11.4.}{Cyclic Group Representations}
+the nonzero elements are represented by the
+powers of a fixed primitive
+%-% \HDindex{element!primitive}{ugxProblemFiniteCyclicPage}{8.11.4.}{Cyclic Group Representations}
+element
+%-% \HDindex{primitive element}{ugxProblemFiniteCyclicPage}{8.11.4.}{Cyclic Group Representations}
+of the field (that is, a generator of its
+cyclic multiplicative group).
+Multiplication (and hence exponentiation) using this representation is easy.
+To do addition, we consider our primitive element as the root of a primitive
+polynomial (an irreducible polynomial whose
+roots are all primitive).
+See \downlink{``\ugxProblemFiniteUtilityTitle''}{ugxProblemFiniteUtilityPage} in Section \ugxProblemFiniteUtilityNumber\ignore{ugxProblemFiniteUtility} for examples of how to
+compute such a polynomial.
+
+%
+\xtc{
+To use \axiomType{FiniteFieldCyclicGroup} you provide a prime number and an
+extension degree.
+}{
+\spadpaste{GF81 := FFCG(3,4); \bound{GF81}}
+}
+%
+%
+\xtc{
+\Language{} uses the prime field, here \axiomType{PrimeField 3}, as the
+ground field and it chooses a primitive polynomial of degree
+\smath{n}, here 4, over the prime field.
+}{
+\spadpaste{a := primitiveElement()\$GF81 \bound{a}\free{GF81}}
+}
+%
+%
+\xtc{
+You can calculate in \axiom{GF81}.
+}{
+\spadpaste{b  := a**12 - a**5 + a \bound{b}\free{a}}
+}
+%
+\xtc{
+In this representation of finite fields the discrete logarithm
+of an element can be seen directly in its output form.
+}{
+\spadpaste{b \free{b}}
+}
+\xtc{
+}{
+\spadpaste{discreteLog b \free{b}}
+}
+%
+
+\axiomType{FiniteFieldCyclicGroupExtension} (abbreviation
+\axiomType{FFCGX}) is similar to \axiomType{FiniteFieldCyclicGroup}
+except that the ground field for
+\axiomType{FiniteFieldCyclicGroupExtension} is arbitrary and chosen
+by you.
+In case you select the prime field as ground field, there is
+essentially no difference between the constructed two finite field
+extensions.
+%
+\xtc{
+}{
+\spadpaste{GF9 := FF(3,2); \bound{GF9}}
+}
+\xtc{
+}{
+\spadpaste{GF729 := FFCGX(GF9,3); \bound{GF729}\free{GF9}}
+}
+\xtc{
+}{
+\spadpaste{r := (random()\$GF729) ** 20 \free{GF729}\bound{r}}
+}
+\xtc{
+}{
+\spadpaste{trace(r) \free{r}}
+}
+%
+
+\axiomType{FiniteFieldCyclicGroupExtensionByPolynomial}
+(abbreviation \axiomType{FFCGP})
+is similar to \axiomType{FiniteFieldCyclicGroup} and
+\axiomType{FiniteFieldCyclicGroupExtension}
+but is more general.
+For \axiomType{FiniteFieldCyclicGroupExtensionByPolynomial} you choose both the
+ground field and the irreducible polynomial used in the representation.
+The degree of the extension is the degree of the polynomial.
+%
+\xtc{
+}{
+\spadpaste{GF3  := PrimeField 3; \bound{GF3}}
+}
+\xtc{
+We use a utility operation to generate an irreducible primitive
+polynomial (see \downlink{``\ugxProblemFiniteUtilityTitle''}{ugxProblemFiniteUtilityPage} in Section \ugxProblemFiniteUtilityNumber\ignore{ugxProblemFiniteUtility}).
+The polynomial has one variable that is ``anonymous'': it displays
+as a question mark.
+}{
+\spadpaste{f := createPrimitivePoly(4)\$FFPOLY(GF3) \bound{f}\free{GF3}}
+}
+\xtc{
+}{
+\spadpaste{GF81 := FFCGP(GF3,f); \bound{GF81x}\free{f GF3}}
+}
+\xtc{
+Let's look at a random element from this field.
+}{
+\spadpaste{random()\$GF81 \free{GF81x}}
+}
+%
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemFiniteNormalTitle}{Normal Basis Representations}
+\newcommand{\ugxProblemFiniteNormalNumber}{8.11.5.}
+%
+% =====================================================================
+\begin{page}{ugxProblemFiniteNormalPage}{8.11.5. Normal Basis Representations}
+% =====================================================================
+\beginscroll
+%-% \HDindex{finite field}{ugxProblemFiniteNormalPage}{8.11.5.}{Normal Basis Representations}
+%-% \HDindex{field!finite!extension of}{ugxProblemFiniteNormalPage}{8.11.5.}{Normal Basis Representations}
+%-% \HDindex{basis!normal}{ugxProblemFiniteNormalPage}{8.11.5.}{Normal Basis Representations}
+%-% \HDindex{normal basis}{ugxProblemFiniteNormalPage}{8.11.5.}{Normal Basis Representations}
+
+Let \smath{K} be a finite extension of degree \smath{n} of the
+finite field \smath{F} and let \smath{F} have \smath{q}
+elements.
+An element \smath{x} of \smath{K} is said to be
+{\it normal} over \smath{F} if the elements
+\centerline{{\axiom{1, x**q, x**(q**2), ..., x**(q**(n-1))}}}
+form a basis of \smath{K} as a vector space over \smath{F}.
+Such a basis is called a {\it normal basis}.\footnote{This
+agrees with the general definition of a normal basis because the
+\smath{n} distinct powers of the automorphism
+\texht{$x \mapsto x^q$}{\axiom{x +-> x**q}}
+constitute the Galois group of \smath{K/F}.}
+
+If \smath{x} is normal over \smath{F}, its minimal
+%-% \HDindex{polynomial!minimal}{ugxProblemFiniteNormalPage}{8.11.5.}{Normal Basis Representations}
+polynomial is also said to be {\it normal} over \smath{F}.
+%-% \HDindex{minimal polynomial}{ugxProblemFiniteNormalPage}{8.11.5.}{Normal Basis Representations}
+There exist normal bases for all finite extensions of arbitrary
+finite fields.
+
+In \axiomType{FiniteFieldNormalBasis} (abbreviation
+\axiomType{FFNB}), the elements of the finite field are represented
+by coordinate vectors with respect to a normal basis.
+
+\xtc{
+You provide a prime \smath{p} and an extension degree
+\smath{n}.
+}{
+\spadpaste{K := FFNB(3,8) \bound{K}}
+}
+%
+\Language{} uses the prime field \axiomType{PrimeField(p)},
+here \axiomType{PrimeField 3},
+and it chooses a normal polynomial of degree
+\smath{n}, here 8, over the ground field.
+The remainder class of the indeterminate is used
+as the normal element.
+The polynomial indeterminate is automatically chosen by \Language{} and
+is typically something like \axiom{\%A} or \axiom{\%D}.
+These (strange) variables are only for output display;
+there are several ways to construct elements of this field.
+The output of the basis elements is something like
+\texht{$\%A^{q^i}.$}{
+\begin{verbatim}
+   i
+  q
+%A  .
+\end{verbatim}
+}
+%
+\xtc{
+}{
+\spadpaste{a := normalElement()\$K \bound{a}\free{K}}
+}
+%
+%
+\xtc{
+You can calculate in \smath{K} using \smath{a}.
+}{
+\spadpaste{b  := a**12 - a**5 + a \bound{b}\free{a}}
+}
+
+\axiomType{FiniteFieldNormalBasisExtension} (abbreviation
+\axiomType{FFNBX}) is
+similar to \axiomType{FiniteFieldNormalBasis} except that the
+groundfield for \axiomType{FiniteFieldNormalBasisExtension} is arbitrary and
+chosen by you.
+In case you select the prime field as ground field, there is
+essentially no difference between the constructed two finite field
+extensions.
+\xtc{
+}{
+\spadpaste{GF9 := FFNB(3,2); \bound{GF9}}
+}
+\xtc{
+}{
+\spadpaste{GF729 := FFNBX(GF9,3); \bound{GF729}\free{GF9}}
+}
+\xtc{
+}{
+\spadpaste{r := random()\$GF729 \bound{r}\free{GF729}}
+}
+\xtc{
+}{
+\spadpaste{r + r**3 + r**9 + r**27 \free{r}}
+}
+
+\axiomType{FiniteFieldNormalBasisExtensionByPolynomial}
+(abbreviation \axiomType{FFNBP}) is similar to
+\axiomType{FiniteFieldNormalBasis} and
+\axiomType{FiniteFieldNormalBasisExtension} but is more general.
+For \axiomType{FiniteFieldNormalBasisExtensionByPolynomial} you
+choose both the ground field and the irreducible polynomial used
+in the representation.
+The degree of the extension is the degree of the polynomial.
+
+%
+\xtc{
+}{
+\spadpaste{GF3 := PrimeField 3; \bound{GF3}}
+}
+\xtc{
+We use a utility operation to generate an irreducible normal
+polynomial (see \downlink{``\ugxProblemFiniteUtilityTitle''}{ugxProblemFiniteUtilityPage} in Section \ugxProblemFiniteUtilityNumber\ignore{ugxProblemFiniteUtility}).
+The polynomial has one variable that is ``anonymous'': it displays
+as a question mark.
+}{
+\spadpaste{f := createNormalPoly(4)\$FFPOLY(GF3) \free{GF3}\bound{f}}
+}
+\xtc{
+}{
+\spadpaste{GF81 := FFNBP(GF3,f); \bound{GF81}\free{f GF3}}
+}
+\xtc{
+Let's look at a random element from this field.
+}{
+\spadpaste{r := random()\$GF81 \free{GF81}\bound{r1}}
+}
+\xtc{
+}{
+\spadpaste{r * r**3 * r**9 * r**27 \free{r1}}
+}
+\xtc{
+}{
+\spadpaste{norm r \free{r1}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemFiniteConversionTitle}{Conversion Operations for Finite Fields}
+\newcommand{\ugxProblemFiniteConversionNumber}{8.11.6.}
+%
+% =====================================================================
+\begin{page}{ugxProblemFiniteConversionPage}{8.11.6. Conversion Operations for Finite Fields}
+% =====================================================================
+\beginscroll
+%-% \HDindex{field!finite!conversions}{ugxProblemFiniteConversionPage}{8.11.6.}{Conversion Operations for Finite Fields}
+
+\labelSpace{5pc}
+%
+\xtc{
+Let \texht{$K$}{\axiom{K}} be a finite field.
+}{
+\spadpaste{K := PrimeField 3 \bound{K}}
+}
+%
+An extension field \texht{$K_m$}{\axiom{Km}} of degree
+\smath{m} over \texht{$K$}{\axiom{K}} is a subfield of an
+extension field \texht{$K_n$}{\axiom{Kn}} of degree \smath{n}
+over \texht{$K$}{\axiom{K}} if and only if \smath{m} divides
+\smath{n}.
+\texht{
+\centerline{{\begin{tabular}{ccc}}}
+\centerline{{$K_n$ }}
+\centerline{{$|$ }}
+\centerline{{$K_m$ & $\Longleftrightarrow$ & $m | n$ }}
+\centerline{{$|$ }}
+\centerline{{K}}
+\centerline{{\end{tabular}}}
+}{
+\begin{verbatim}
+Kn
+|
+Km   <==>  m | n
+|
+K
+\end{verbatim}
+}
+\axiomType{FiniteFieldHomomorphisms} provides conversion operations
+between different extensions of one
+fixed finite ground field and between different representations of
+these finite fields.
+\xtc{
+Let's choose \smath{m} and \smath{n},
+}{
+\spadpaste{(m,n) := (4,8) \bound{m n}}
+}
+\xtc{
+build the field extensions,
+}{
+\spadpaste{Km := FiniteFieldExtension(K,m) \bound{Km}\free{K m}}
+}
+\xtc{
+and pick two random elements from the smaller field.
+}{
+\spadpaste{Kn := FiniteFieldExtension(K,n) \bound{Kn}\free{K n}}
+}
+\xtc{
+}{
+\spadpaste{a1 := random()\$Km \bound{a1}\free{Km}}
+}
+\xtc{
+}{
+\spadpaste{b1 := random()\$Km \bound{b1}\free{Km}}
+}
+%
+\xtc{
+Since \smath{m} divides \smath{n},
+\texht{$K_m$}{\axiom{Km}} is a subfield of \texht{$K_n$}{\axiom{Kn}}.
+}{
+\spadpaste{a2 := a1 :: Kn \bound{a2}\free{a1 Kn}}
+}
+\xtc{
+Therefore we can convert the elements of \texht{$K_m$}{\axiom{Km}}
+into elements of \texht{$K_n$}{\axiom{Kn}}.
+}{
+\spadpaste{b2 := b1 :: Kn \bound{b2}\free{b1 Kn}}
+}
+%
+%
+\xtc{
+To check this, let's do some arithmetic.
+}{
+\spadpaste{a1+b1 - ((a2+b2) :: Km) \free{a1 a2 b1 b2 Km Kn}}
+}
+\xtc{
+}{
+\spadpaste{a1*b1 - ((a2*b2) :: Km) \free{a1 a2 b1 b2 Km Kn}}
+}
+%
+There are also conversions available for the
+situation, when \texht{$K_m$}{\axiom{Km}} and \texht{$K_n$}{\axiom{Kn}}
+are represented in different ways (see
+\downlink{``\ugxProblemFiniteExtensionFiniteTitle''}{ugxProblemFiniteExtensionFinitePage} in Section \ugxProblemFiniteExtensionFiniteNumber\ignore{ugxProblemFiniteExtensionFinite}).
+For example let's choose \texht{$K_m$}{\axiom{Km}} where the
+representation is 0 plus the cyclic multiplicative group and
+\texht{$K_n$}{\axiom{Kn}} with a normal basis representation.
+\xtc{
+}{
+\spadpaste{Km := FFCGX(K,m) \bound{Km2}\free{K m}}
+}
+\xtc{
+}{
+\spadpaste{Kn := FFNBX(K,n) \bound{Kn2}\free{K n}}
+}
+\xtc{
+}{
+\spadpaste{(a1,b1) := (random()\$Km,random()\$Km) \bound{a12 b12}\free{Km2}}
+}
+\xtc{
+}{
+\spadpaste{a2 := a1 :: Kn \bound{a22}\free{a12 Kn2}}
+}
+\xtc{
+}{
+\spadpaste{b2 := b1 :: Kn \bound{b22}\free{b12 Kn2}}
+}
+%
+\xtc{
+Check the arithmetic again.
+}{
+\spadpaste{a1+b1 - ((a2+b2) :: Km) \free{a12 a22 b12 b22 Km2}}
+}
+\xtc{
+}{
+\spadpaste{a1*b1 - ((a2*b2) :: Km) \free{a12 a22 b12 b22 Km2}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugxProblemFiniteUtilityTitle}{Utility Operations for Finite Fields}
+\newcommand{\ugxProblemFiniteUtilityNumber}{8.11.7.}
+%
+% =====================================================================
+\begin{page}{ugxProblemFiniteUtilityPage}{8.11.7. Utility Operations for Finite Fields}
+% =====================================================================
+\beginscroll
+
+\axiomType{FiniteFieldPolynomialPackage} (abbreviation
+\axiomType{FFPOLY})
+provides operations for generating, counting and testing polynomials
+over finite fields. Let's start with a couple of definitions:
+\indent{4}
+\beginitems
+\item[-] A polynomial is {\it primitive} if its roots are primitive
+%-% \HDindex{polynomial!primitive}{ugxProblemFiniteUtilityPage}{8.11.7.}{Utility Operations for Finite Fields}
+elements in an extension of the coefficient field of degree equal
+to the degree of the polynomial.
+\item[-] A polynomial is {\it normal} over its coefficient field
+%-% \HDindex{polynomial!normal}{ugxProblemFiniteUtilityPage}{8.11.7.}{Utility Operations for Finite Fields}
+if its roots are linearly independent
+elements in an extension of the coefficient field of degree equal
+to the degree of the polynomial.
+\enditems
+\indent{0}
+In what follows, many of the generated polynomials have one
+``anonymous'' variable.
+This indeterminate is displayed as a question mark (\spadSyntax{?}).
+
+\xtc{
+To fix ideas, let's use the field with five elements for the first
+few examples.
+}{
+\spadpaste{GF5 := PF 5; \bound{GF5}}
+}
+%
+%
+\xtc{
+You can generate irreducible polynomials of any (positive) degree
+%-% \HDindex{polynomial!irreducible}{ugxProblemFiniteUtilityPage}{8.11.7.}{Utility Operations for Finite Fields}
+(within the storage capabilities of the computer and your ability
+to wait) by using
+\axiomFunFrom{createIrreduciblePoly}{FiniteFieldPolynomialPackage}.
+}{
+\spadpaste{f := createIrreduciblePoly(8)\$FFPOLY(GF5) \bound{f}\free{GF5}}
+}
+%
+\xtc{
+Does this polynomial have other important properties? Use
+\axiomFun{primitive?} to test whether it is a primitive polynomial.
+}{
+\spadpaste{primitive?(f)\$FFPOLY(GF5) \free{f}}
+}
+\xtc{
+Use \axiomFun{normal?} to test whether it is a normal polynomial.
+}{
+\spadpaste{normal?(f)\$FFPOLY(GF5) \free{f}}
+}
+\noindent
+Note that this is actually a trivial case,
+because a normal polynomial of degree \smath{n}
+must have a nonzero term of degree \smath{n-1}.
+We will refer back to this later.
+
+\xtc{
+To get a primitive polynomial of degree 8 just issue this.
+}{
+\spadpaste{p := createPrimitivePoly(8)\$FFPOLY(GF5) \bound{p}\free{GF5}}
+}
+\xtc{
+}{
+\spadpaste{primitive?(p)\$FFPOLY(GF5) \free{p}}
+}
+\xtc{
+This polynomial is not normal,
+}{
+\spadpaste{normal?(p)\$FFPOLY(GF5) \free{p}}
+}
+\xtc{
+but if you want a normal one simply write this.
+}{
+\spadpaste{n := createNormalPoly(8)\$FFPOLY(GF5) \bound{n} \free{GF5}}
+}
+\xtc{
+This polynomial is not primitive!
+}{
+\spadpaste{primitive?(n)\$FFPOLY(GF5) \free{n}}
+}
+This could have been seen directly, as
+the constant term is 1 here, which is not a primitive
+element up to the factor (\axiom{-1}) raised to the degree of the
+polynomial.\footnote{Cf. Lidl, R. \& Niederreiter,
+H., {\it Finite Fields,} Encycl. of Math. 20, (Addison-Wesley, 1983),
+p.90, Th. 3.18.}
+
+What about
+polynomials that are both primitive and normal?
+The existence of such a polynomial is by no means obvious.
+\footnote{The existence of such polynomials is proved in
+Lenstra, H. W. \& Schoof, R. J., {\it Primitive
+Normal Bases for Finite Fields,} Math. Comp. 48, 1987, pp. 217-231.}
+%
+\xtc{
+If you really need one use either
+\axiomFunFrom{createPrimitiveNormalPoly}{FiniteFieldPolynomialPackage} or
+\axiomFunFrom{createNormalPrimitivePoly}{FiniteFieldPolynomialPackage}.
+}{
+\spadpaste{createPrimitiveNormalPoly(8)\$FFPOLY(GF5) \free{GF5}}
+}
+%
+
+If you want to obtain additional polynomials of the various types above
+as given by the {\bf create...} operations above, you can use the {\bf
+next...} operations.
+For instance,
+\axiomFunFrom{nextIrreduciblePoly}{FiniteFieldPolynomialPackage} yields
+the next monic irreducible polynomial with the same degree as the input
+polynomial.
+By ``next'' we mean ``next in a natural order using the terms and
+coefficients.''
+This will become more clear in the following examples.
+
+\xtc{
+This is the field with five elements.
+}{
+\spadpaste{GF5 := PF 5; \bound{GF5}}
+}
+%
+\xtc{
+Our first example irreducible polynomial, say
+of degree 3, must be ``greater'' than this.
+}{
+\spadpaste{h := monomial(1,8)\$SUP(GF5) \bound{h}\free{GF5}}
+}
+\xtc{
+You can generate it by doing this.
+}{
+\spadpaste{nh := nextIrreduciblePoly(h)\$FFPOLY(GF5) \bound{nh}\free{h}}
+}
+%
+\xtc{
+Notice that this polynomial is not the same as the one
+\axiomFunFrom{createIrreduciblePoly}{FiniteFieldPolynomialPackage}.
+}{
+\spadpaste{createIrreduciblePoly(3)\$FFPOLY(GF5) \free{GF5}}
+}
+\xtc{
+You can step through all irreducible polynomials of degree 8 over
+the field with 5 elements by repeatedly issuing this.
+}{
+\spadpaste{nh := nextIrreduciblePoly(nh)\$FFPOLY(GF5) \free{nh}}
+}
+\xtc{
+You could also ask for the total number of these.
+}{
+\spadpaste{numberOfIrreduciblePoly(5)\$FFPOLY(GF5) \free{GF5}}
+}
+
+We hope that ``natural order'' on polynomials is now clear:
+first we compare the number of monomials of
+two polynomials (``more'' is ``greater'');
+then, if necessary, the degrees of these monomials (lexicographically),
+and lastly their coefficients (also
+lexicographically, and using the operation \axiomFun{lookup} if
+our field is not a prime field).
+Also note that we make both polynomials monic before looking at the
+coefficients:
+multiplying either polynomial  by a nonzero constant
+produces the same result.
+
+%
+\xtc{
+The package
+\axiomType{FiniteFieldPolynomialPackage} also provides similar
+operations for primitive and normal polynomials. With
+the exception of the number of primitive normal polynomials;
+we're not aware of any known formula for this.
+}{
+\spadpaste{numberOfPrimitivePoly(3)\$FFPOLY(GF5) \free{GF5}}
+}
+%
+%
+\xtc{
+Take these,
+}{
+\spadpaste{m := monomial(1,1)\$SUP(GF5) \bound{m}\free{GF5}}
+}
+\xtc{
+}{
+\spadpaste{f := m**3 + 4*m**2 + m + 2 \bound{fx}\free{m}}
+}
+%
+%
+\xtc{
+and then we have:
+}{
+\spadpaste{f1 := nextPrimitivePoly(f)\$FFPOLY(GF5) \free{fx}\bound{f1}}
+}
+\xtc{
+What happened?
+}{
+\spadpaste{nextPrimitivePoly(f1)\$FFPOLY(GF5) \free{f1}}
+}
+%
+Well, for the ordering used in
+\axiomFunFrom{nextPrimitivePoly}{FiniteFieldPolynomialPackage} we
+use as first criterion a comparison of the constant terms of the
+polynomials.
+Analogously, in
+\axiomFunFrom{nextNormalPoly}{FiniteFieldPolynomialPackage} we first
+compare the monomials of degree 1 less than the degree of the
+polynomials (which is nonzero, by an earlier remark).
+%
+\xtc{
+}{
+\spadpaste{f := m**3 + m**2 + 4*m + 1 \bound{fy} \free{m}}
+}
+\xtc{
+}{
+\spadpaste{f1 := nextNormalPoly(f)\$FFPOLY(GF5) \free{fy}\bound{f1y}}
+}
+\xtc{
+}{
+\spadpaste{nextNormalPoly(f1)\$FFPOLY(GF5) \free{f1y}}
+}
+%
+\noindent
+We don't have to restrict ourselves to prime fields.
+%
+\xtc{
+Let's consider, say, a field with 16 elements.
+}{
+\spadpaste{GF16 := FFX(FFX(PF 2,2),2); \bound{GF16} }
+}
+%
+%
+\xtc{
+We can apply any of the operations described above.
+}{
+\spadpaste{createIrreduciblePoly(5)\$FFPOLY(GF16) \free{GF16}}
+}
+
+\xtc{
+\Language{} also provides operations
+for producing random polynomials of a given degree
+}{
+\spadpaste{random(5)\$FFPOLY(GF16) \free{GF16}}
+}
+\xtc{
+or with degree between two given bounds.
+}{
+\spadpaste{random(3,9)\$FFPOLY(GF16) \free{GF16}}
+}
+
+\axiomType{\axiom{FiniteFieldPolynomialPackage2}} (abbreviation
+\axiomType{FFPOLY2})
+exports an operation \axiomFun{rootOfIrreduciblePoly}
+for finding one root of an irreducible polynomial \axiom{f}
+%-% \HDindex{polynomial!root of}{ugxProblemFiniteUtilityPage}{8.11.7.}{Utility Operations for Finite Fields}
+in an extension field of the coefficient field.
+The degree of the extension has to be a multiple of the degree of \axiom{f}.
+It is not checked whether \axiom{f} actually is irreducible.
+
+%
+\xtc{
+To illustrate this operation, we fix a ground field \axiom{GF}
+}{
+\spadpaste{GF2 := PrimeField 2; \bound{GF2}}
+}
+%
+%
+\xtc{
+and then an extension field.
+}{
+\spadpaste{F := FFX(GF2,12) \bound{F}\free{GF2}}
+}
+%
+%
+\xtc{
+We construct an irreducible polynomial over \axiom{GF2}.
+}{
+\spadpaste{f := createIrreduciblePoly(6)\$FFPOLY(GF2) \bound{fz}\free{GF2}}
+}
+%
+%
+\xtc{
+We compute a root of \axiom{f}.
+}{
+\spadpaste{root := rootOfIrreduciblePoly(f)\$FFPOLY2(F,GF2) \free{F GF2 fz}\bound{root}}
+}
+%
+%and check the result
+%\spadcommand{eval(f, monomial(1,1)\$SUP(F) = root) \free{fz F root}}
+
+%*********************************************************************
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemIdealTitle}{Primary Decomposition of Ideals}
+\newcommand{\ugProblemIdealNumber}{8.12.}
+%
+% =====================================================================
+\begin{page}{ugProblemIdealPage}{8.12. Primary Decomposition of Ideals}
+% =====================================================================
+\beginscroll
+%*********************************************************************
+%
+\Language{} provides a facility for the primary decomposition
+%-% \HDindex{ideal!primary decomposition}{ugProblemIdealPage}{8.12.}{Primary Decomposition of Ideals}
+of
+%-% \HDindex{primary decomposition of ideal}{ugProblemIdealPage}{8.12.}{Primary Decomposition of Ideals}
+polynomial ideals over fields of characteristic zero.
+The algorithm
+%is discussed in \cite{gtz:gbpdpi} and
+works in essentially two steps:
+\indent{4}
+\beginitems
+\item[1. ] the problem is solved for 0-dimensional ideals by ``generic''
+projection on the last coordinate
+\item[2. ] a ``reduction process'' uses localization and ideal quotients
+to reduce the general case to the 0-dimensional one.
+\enditems
+\indent{0}
+The \Language{} constructor \axiomType{PolynomialIdeals}
+represents ideals with coefficients in any field and
+supports the basic ideal operations,
+including intersection, sum and quotient.
+\axiomType{IdealDecompositionPackage} contains the specific
+operations for the primary decomposition and the computation of the
+radical of an ideal with polynomial
+coefficients in a field of characteristic 0 with
+an effective algorithm for factoring polynomials.
+
+The following examples illustrate the capabilities of this facility.
+%
+\xtc{
+First consider the ideal generated by
+\texht{$x^2 + y^2 - 1$}{\axiom{x**2 + y**2 - 1}}
+(which defines a circle in the \axiom{(x,y)}-plane) and the ideal
+generated by \texht{$x^2 - y^2$}{\axiom{x**2 - y**2}} (corresponding to the
+straight lines \axiom{x = y} and \axiom{x = -y}.
+}{
+\spadpaste{(n,m) : List DMP([x,y],FRAC INT) \bound{nm}}
+}
+\xtc{
+}{
+\spadpaste{m := [x**2+y**2-1] \free{nm} \bound{m}}
+}
+\xtc{
+}{
+\spadpaste{n := [x**2-y**2] \free{nm} \bound{n}}
+}
+%
+%
+\xtc{
+We find the equations defining the intersection of the two loci.
+This correspond to the sum of the associated ideals.
+}{
+\spadpaste{id := ideal m  + ideal n \free{n m} \bound{id}}
+}
+%
+%
+\xtc{
+We can check if the locus contains only a finite number of points,
+that is, if the ideal is zero-dimensional.
+}{
+\spadpaste{zeroDim? id \free{id}}
+}
+\xtc{
+}{
+\spadpaste{zeroDim?(ideal m) \free{m}}
+}
+\xtc{
+}{
+\spadpaste{dimension ideal m \free{m}}
+}
+\xtc{
+We can find polynomial relations among the generators
+(\axiom{f} and \axiom{g} are the parametric equations of the knot).
+}{
+\spadpaste{(f,g):DMP([x,y],FRAC INT) \bound{fg}}
+}
+\xtc{
+}{
+\spadpaste{f := x**2-1 \free{fg} \bound{f}}
+}
+\xtc{
+}{
+\spadpaste{g := x*(x**2-1) \free{fg} \bound{g}}
+}
+\xtc{
+}{
+\spadpaste{relationsIdeal [f,g] \free{f g}}
+}
+
+\xtc{
+We can compute the primary decomposition of an ideal.
+}{
+\spadpaste{l: List DMP([x,y,z],FRAC INT) \bound{ll}}
+}
+\xtc{
+}{
+\spadpaste{l:=[x**2+2*y**2,x*z**2-y*z,z**2-4] \free{ll} \bound{l}}
+}
+\xtc{
+}{
+\spadpaste{ld:=primaryDecomp ideal l \free{l} \bound{ld}}
+}
+\xtc{
+We can intersect back.
+}{
+\spadpaste{reduce(intersect,ld) \free{ld}}
+}
+
+\xtc{
+We can compute the radical of every primary component.
+}{
+\spadpaste{reduce(intersect,[radical ld.i for i in 1..2]) \free{ld}}
+}
+\xtc{
+Their intersection is equal to the radical of the ideal of \axiom{l}.
+}{
+\spadpaste{radical ideal l \free{l}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemGaloisTitle}{Computation of Galois Groups}
+\newcommand{\ugProblemGaloisNumber}{8.13.}
+%
+% =====================================================================
+\begin{page}{ugProblemGaloisPage}{8.13. Computation of Galois Groups}
+% =====================================================================
+\beginscroll
+%
+As a sample use of \Language{}'s algebraic number facilities,
+%-% \HDindex{group!Galois}{ugProblemGaloisPage}{8.13.}{Computation of Galois Groups}
+we compute
+%-% \HDindex{Galois!group}{ugProblemGaloisPage}{8.13.}{Computation of Galois Groups}
+the Galois group of the polynomial
+\texht{$p(x) = x^5 - 5 x + 12$}{\axiom{p(x) = x**5 - 5*x + 12}}.
+%
+\xtc{
+}{
+\spadpaste{p := x**5 - 5*x + 12 \bound{p}}
+}
+%
+We would like to construct a polynomial \smath{f(x)}
+such that the splitting
+%-% \HDindex{field!splitting}{ugProblemGaloisPage}{8.13.}{Computation of Galois Groups}
+field
+%-% \HDindex{splitting field}{ugProblemGaloisPage}{8.13.}{Computation of Galois Groups}
+of \smath{p(x)} is generated by one root of \smath{f(x)}.
+First we construct a polynomial \smath{r = r(x)} such that one
+root of \smath{r(x)} generates the field generated by two roots of
+the polynomial \smath{p(x)}.
+(As it will turn out, the field generated by two roots of
+\smath{p(x)} is, in fact, the splitting field of
+\smath{p(x)}.)
+
+From the proof of the primitive element theorem we know that
+if \smath{a} and \smath{b} are
+algebraic numbers, then the field
+\texht{${\bf Q}(a,b)$}{\axiom{Q(a,b)}} is equal to
+\texht{${\bf Q}(a+kb)$}{\axiom{Q(a + k*b)}} for an
+appropriately chosen integer \smath{k}.
+In our case, we construct the minimal polynomial of
+\texht{$a_i - a_j$}{\axiom{a[i] - a[j]}}, where
+\texht{$a_i$}{\axiom{a[i]}} and
+\texht{$a_j$}{\axiom{a[j]}} are two roots of \smath{p(x)}.
+We construct this polynomial using \axiomFun{resultant}.
+The main result we  need is the following:
+If \smath{f(x)} is a polynomial with roots
+\texht{$a_i \ldots a_m$}{\axiom{a[1]...a[m]}} and
+\smath{g(x)} is a polynomial
+with roots
+\texht{$b_i \ldots b_n$}{\axiom{b[1]...b[n]}}, then the polynomial
+\axiom{h(x) = resultant(f(y), g(x-y), y)}
+is a polynomial of degree \smath{m*n} with
+roots
+\texht{$a_i + b_j, i = 1 \ldots m, j = 1 \ldots n$}
+{\axiom{a[i] + b[j], 1 <= i <= m, 1 <= j <= n}}.
+
+\xtc{
+For \smath{f(x)} we use the polynomial \smath{p(x)}.
+For \smath{g(x)} we use the polynomial \smath{-p(-x)}.
+Thus, the polynomial we  first construct is
+\axiom{resultant(p(y), -p(y-x), y)}.
+}{
+\spadpaste{q := resultant(eval(p,x,y),-eval(p,x,y-x),y) \free{p} \bound{q}}
+}
+%
+The roots of \smath{q(x)} are
+\texht{$a_i - a_j, i \leq 1, j \leq 5$}
+{\axiom{a[i] - a[j], 1 <= i,j <= 5}}.
+Of course, there are five pairs \smath{(i,j)} with \smath{i = j},
+so \axiom{0} is a 5-fold root of \smath{q(x)}.
+%
+\xtc{
+Let's get rid of this factor.
+}{
+\spadpaste{q1 := exquo(q, x**5) \free{q} \bound{q1}}
+}
+\xtc{
+Factor the polynomial \axiom{q1}.
+}{
+\spadpaste{factoredQ := factor q1 \free{q1} \bound{factoredQ}}
+}
+%
+We see that \axiom{q1} has two irreducible factors, each of degree \axiom{10}.
+(The fact that the polynomial \axiom{q1} has two factors of
+degree \axiom{10} is enough to show
+that the Galois group of \smath{p(x)} is the dihedral group of
+order \axiom{10}.\footnote{See McKay, Soicher,  Computing Galois Groups
+over the Rationals, Journal of Number Theory 20, 273-281 (1983).
+We do not assume the results of this paper, however, and we continue with
+the computation.}
+Note that the type of \axiom{factoredQ} is \axiomType{FR POLY INT}, that is,
+\axiomType{Factored Polynomial Integer}.
+%-% \HDexptypeindex{Factored}{ugProblemGaloisPage}{8.13.}{Computation of Galois Groups}
+This is a special data type for recording factorizations of polynomials with
+integer coefficients (see \downlink{`Factored'}{FactoredXmpPage}\ignore{Factored}).
+%
+\xtc{
+We can access the individual factors using the operation
+\axiomFunFrom{nthFactor}{Factored}.
+}{
+\spadpaste{r := nthFactor(factoredQ,1) \free{factoredQ} \bound{r}}
+}
+%
+
+Consider the polynomial \smath{r = r(x)}.
+This is the minimal polynomial of the difference of two roots of
+\smath{p(x)}.
+Thus, the splitting field of \smath{p(x)} contains a subfield of
+degree \axiom{10}.
+We show that this subfield is, in fact, the splitting field of
+\smath{p(x)} by showing that \smath{p(x)} factors completely
+over this field.
+%
+\xtc{
+First we create a symbolic root of the polynomial \smath{r(x)}.
+(We replaced \axiom{x} by \axiom{b} in the
+polynomial \axiom{r} so that our symbolic root would be
+printed as \axiom{b}.)
+}{
+\spadpaste{beta:AN := rootOf(eval(r,x,b)) \free{r} \bound{beta}}
+}
+\xtc{
+We next tell \Language{} to view \smath{p(x)} as a univariate polynomial
+in \axiom{x}
+with algebraic number coefficients.
+This is accomplished with this type declaration.
+}{
+\spadpaste{p := p::UP(x,INT)::UP(x,AN) \free{p} \bound{declareP}}
+}
+%
+%
+\xtc{
+Factor \smath{p(x)} over the field
+\texht{${\bf Q}(\beta)$}{\axiom{Q(beta)}}.
+(This computation will take some time!)
+}{
+\spadpaste{algFactors := factor(p,[beta]) \free{declareP beta} \bound{algFactors}}
+}
+%
+When factoring over number fields, it is important to specify the
+field over which the polynomial is to be factored, as polynomials
+have different factorizations over different fields.
+When you use the operation \axiomFun{factor}, the field over which
+the polynomial is factored is the field generated by
+\indent{4}
+\beginitems
+\item[1. ] the algebraic numbers that appear
+in the coefficients of the polynomial, and
+\item[2. ] the algebraic numbers that
+appear in a list passed as an optional second argument of the operation.
+\enditems
+\indent{0}
+In our case, the coefficients of \axiom{p}
+are all rational integers and only \axiom{beta}
+appears in the list, so the field is simply
+\texht{${\bf Q}(\beta)$}{\axiom{Q(beta)}}.
+%
+\xtc{
+It was necessary to give the list \axiom{[beta]}
+as a second argument of the operation
+because otherwise the polynomial would have been factored over the field
+generated by its coefficients, namely the rational numbers.
+}{
+\spadpaste{factor(p) \free{declareP}}
+}
+%
+We have shown that the splitting field of \smath{p(x)} has degree
+\axiom{10}.
+Since the symmetric group of degree 5 has only one transitive subgroup
+of order \axiom{10}, we know that the Galois group of \smath{p(x)} must be
+this group, the dihedral group
+%-% \HDindex{group!dihedral}{ugProblemGaloisPage}{8.13.}{Computation of Galois Groups}
+of order \axiom{10}.
+Rather than stop here, we explicitly compute the action of the Galois
+group on the roots of \smath{p(x)}.
+
+First we assign the roots of \smath{p(x)} as the values of five
+%-% \HDindex{root}{ugProblemGaloisPage}{8.13.}{Computation of Galois Groups}
+variables.
+\xtc{
+We can obtain an individual root by negating the constant coefficient of
+one of the factors of \smath{p(x)}.
+}{
+\spadpaste{factor1 := nthFactor(algFactors,1) \free{algFactors} \bound{factor1}}
+}
+\xtc{
+}{
+\spadpaste{root1 := -coefficient(factor1,0) \free{factor1} \bound{root1}}
+}
+%
+%
+\xtc{
+We can obtain a list of all the roots in this way.
+}{
+\spadpaste{roots := [-coefficient(nthFactor(algFactors,i),0) for i in 1..5] \free{algFactors} \bound{roots}}
+}
+
+The expression
+\begin{verbatim}
+- coefficient(nthFactor(algFactors, i), 0)}
+\end{verbatim}
+is the \eth{\axiom{i }} root
+of \smath{p(x)} and the elements of \axiom{roots} are the \eth{\axiom{i }}
+roots of \smath{p(x)} as \axiom{i} ranges from \axiom{1} to \axiom{5}.
+
+\xtc{
+Assign the roots as the values of the variables \axiom{a1,...,a5}.
+}{
+\spadpaste{(a1,a2,a3,a4,a5) := (roots.1,roots.2,roots.3,roots.4,roots.5) \free{roots} \bound{ais}}
+}
+%
+
+Next we express the roots of \smath{r(x)} as polynomials in
+\axiom{beta}.
+We could obtain these roots by calling the operation \axiomFun{factor}:
+\axiom{factor(r, [beta])} factors \axiom{r(x)} over
+\texht{${\bf Q}(\beta)$}{\axiom{Q(beta)}}.
+However, this is a lengthy computation and we can obtain the roots of
+\smath{r(x)} as differences of the roots \axiom{a1,...,a5} of
+\smath{p(x)}.
+Only ten of these differences are roots of \smath{r(x)} and the
+other ten are roots
+of the other irreducible factor of \axiom{q1}.
+We can determine if a given value is a root of \smath{r(x)} by evaluating
+\smath{r(x)} at that particular value.
+(Of course, the order in which factors are returned by the
+operation \axiomFun{factor}
+is unimportant and may change with different implementations of the operation.
+Therefore, we cannot predict in advance which differences are roots of
+\smath{r(x)} and which are not.)
+%
+\xtc{
+Let's look at four examples (two are roots of \smath{r(x)} and
+two are not).
+}{
+\spadpaste{eval(r,x,a1 - a2) \free{ais}}
+}
+\xtc{
+}{
+\spadpaste{eval(r,x,a1 - a3) \free{ais}}
+}
+\xtc{
+}{
+\spadpaste{eval(r,x,a1 - a4) \free{ais}}
+}
+\xtc{
+}{
+\spadpaste{eval(r,x,a1 - a5) \free{ais}}
+}
+%
+
+Take one of the differences that was a root of \smath{r(x)}
+and assign it to the variable \axiom{bb}.
+\xtc{
+For example, if \axiom{eval(r,x,a1 - a4)} returned \axiom{0}, you would
+enter this.
+}{
+\spadpaste{bb := a1 - a4 \free{ais} \bound{bb}}
+}
+Of course, if the difference is, in fact, equal to the root \axiom{beta},
+you should choose another root of \smath{r(x)}.
+
+Automorphisms of the splitting field are given by mapping a generator of
+the field, namely \axiom{beta}, to other roots of its minimal polynomial.
+Let's see what happens when \axiom{beta} is mapped to \axiom{bb}.
+%
+\xtc{
+We compute the images of the roots \axiom{a1,...,a5}
+under this automorphism:
+}{
+\spadpaste{aa1 := subst(a1,beta = bb) \free{beta bb ais} \bound{aa1}}
+}
+\xtc{
+}{
+\spadpaste{aa2 := subst(a2,beta = bb) \free{beta bb ais} \bound{aa2}}
+}
+\xtc{
+}{
+\spadpaste{aa3 := subst(a3,beta = bb) \free{beta bb ais} \bound{aa3}}
+}
+\xtc{
+}{
+\spadpaste{aa4 := subst(a4,beta = bb) \free{beta bb ais} \bound{aa4}}
+}
+\xtc{
+}{
+\spadpaste{aa5 := subst(a5,beta = bb) \free{beta bb ais} \bound{aa5}}
+}
+%
+Of course, the values \axiom{aa1,...,aa5} are simply a permutation of the values
+\axiom{a1,...,a5}.
+%
+\xtc{
+Let's find the value of \axiom{aa1} (execute as many of the following five commands
+as necessary).
+}{
+\spadpaste{(aa1 = a1) :: Boolean \free{aa1}}
+}
+\xtc{
+}{
+\spadpaste{(aa1 = a2) :: Boolean \free{aa1}}
+}
+\xtc{
+}{
+\spadpaste{(aa1 = a3) :: Boolean \free{aa1}}
+}
+\xtc{
+}{
+\spadpaste{(aa1 = a4) :: Boolean \free{aa1}}
+}
+\xtc{
+}{
+\spadpaste{(aa1 = a5) :: Boolean \free{aa1}}
+}
+%
+Proceeding in this fashion, you can find the values of
+\axiom{aa2,...aa5}.\footnote{Here you should use the
+\Clef{} line editor.
+See \downlink{``\ugAvailCLEFTitle''}{ugAvailCLEFPage} in Section \ugAvailCLEFNumber\ignore{ugAvailCLEF} for more information about \Clef{}.}
+You have represented the automorphism \axiom{beta -> bb}
+as a permutation of the roots \axiom{a1,...,a5}.
+If you wish, you can repeat this computation for all the roots of
+\smath{r(x)} and represent the Galois group of
+\smath{p(x)} as a subgroup of the symmetric group on five letters.
+
+Here are two other problems that you may attack in a similar fashion:
+\indent{4}
+\beginitems
+\item[1. ] Show that the Galois group of
+\texht{$p(x) = x^4 + 2 x^3 - 2 x^2 - 3 x + 1$}{\axiom{
+p(x) = x**4 + 2*x**3 - 2*x**2 - 3*x + 1}}
+is the dihedral group of order eight.
+%-% \HDindex{group!dihedral}{ugProblemGaloisPage}{8.13.}{Computation of Galois Groups}
+(The splitting field of this polynomial is the Hilbert class field
+%-% \HDindex{Hilbert class field}{ugProblemGaloisPage}{8.13.}{Computation of Galois Groups}
+of
+%-% \HDindex{field!Hilbert class}{ugProblemGaloisPage}{8.13.}{Computation of Galois Groups}
+the quadratic field
+\texht{${\bf Q}(\sqrt{145})$}{\axiom{Q(sqrt(145))}}.)
+\item[2. ] Show that the Galois group of
+\texht{$p(x) = x^6 + 108$}{\axiom{p(x) = x**6 + 108}}
+has order 6 and is
+isomorphic to \texht{$S_3,$}{} the symmetric group on three letters.
+%-% \HDindex{group!symmetric}{ugProblemGaloisPage}{8.13.}{Computation of Galois Groups}
+(The splitting field of this polynomial is the splitting field of
+\texht{$x^3 - 2$}{\axiom{x**3 - 2}}.)
+\enditems
+\indent{0}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugProblemGeneticTitle}{Non-Associative Algebras and Modelling Genetic Laws}
+\newcommand{\ugProblemGeneticNumber}{8.14.}
+%
+% =====================================================================
+\begin{page}{ugProblemGeneticPage}{8.14. Non-Associative Algebras and Modelling Genetic Laws}
+% =====================================================================
+\beginscroll
+
+Many algebraic structures of mathematics and \Language{}
+have a multiplication operation \axiomOp{*} that satisfies
+the associativity law
+%-% \HDindex{associativity law}{ugProblemGeneticPage}{8.14.}{Non-Associative Algebras and Modelling Genetic Laws}
+\texht{$a*(b*c) = (a*b)*c$}{\spad{a*(b*c) = (a*b)*c}}
+for all \smath{a}, \smath{b} and \smath{c}.
+The octonions (see \downlink{`Octonion'}{OctonionXmpPage}\ignore{Octonion}) are a well known exception.
+There are many other interesting non-associative structures, such as the
+class of
+%-% \HDindex{Lie algebra}{ugProblemGeneticPage}{8.14.}{Non-Associative Algebras and Modelling Genetic Laws}
+Lie algebras.\footnote{Two \Language{} implementations of Lie algebras
+are \spadtype{LieSquareMatrix} and \spadtype{FreeNilpotentLie}.}
+Lie algebras can be used, for example, to analyse Lie symmetry algebras of
+%-% \HDindex{symmetry}{ugProblemGeneticPage}{8.14.}{Non-Associative Algebras and Modelling Genetic Laws}
+partial differential
+%-% \HDindex{differential equation!partial}{ugProblemGeneticPage}{8.14.}{Non-Associative Algebras and Modelling Genetic Laws}
+equations.
+%-% \HDindex{partial differential equation}{ugProblemGeneticPage}{8.14.}{Non-Associative Algebras and Modelling Genetic Laws}
+In this section we show a different application of non-associative algebras,
+%-% \HDindex{non-associative algebra}{ugProblemGeneticPage}{8.14.}{Non-Associative Algebras and Modelling Genetic Laws}
+the modelling of genetic laws.
+%-% \HDindex{algebra!non-associative}{ugProblemGeneticPage}{8.14.}{Non-Associative Algebras and Modelling Genetic Laws}
+
+The \Language{} library contains several constructors for
+creating non-assoc\-i\-a\-tive structures,
+ranging from the categories \spadtype{Monad},
+\spadtype{NonAssociativeRng}, and
+\spadtype{FramedNonAssociativeAlgebra}, to the domains
+\spadtype{AlgebraGivenByStructuralConstants} and
+\spadtype{GenericNonAssociativeAlgebra}.
+Furthermore, the package \spadtype{AlgebraPackage} provides
+operations for analysing the structure of such algebras.\footnote{%
+The interested reader can learn more about these aspects of the
+\Language{} library from the paper
+``Computations in Algebras of Finite Rank,''
+by Johannes Grabmeier and Robert Wisbauer,
+Technical Report, IBM Heidelberg Scientific Center, 1992.}
+
+Mendel's genetic laws are often written in a form like
+\texht{
+\narrowDisplay{Aa \times Aa = {1\over 4}AA + {1\over 2}Aa + {1\over 4}aa.}}{
+\spad{Aa * Aa = (1/4)*AA + (1/2)*Aa + (1/4)*aa.}
+}
+The implementation of general algebras in \Language{} allows us to
+%-% \HDindex{Mendel's genetic laws}{ugProblemGeneticPage}{8.14.}{Non-Associative Algebras and Modelling Genetic Laws}
+use this as the definition for multiplication in an algebra.
+%-% \HDindex{genetics}{ugProblemGeneticPage}{8.14.}{Non-Associative Algebras and Modelling Genetic Laws}
+Hence, it is possible to study
+questions of genetic inheritance using \Language{}.
+To demonstrate this more precisely,  we discuss one example from a
+monograph of \texht{A. W\"orz-Busekros}{A. Woerz-Busekros},
+where you can also find a general setting of this theory.\footnote{%
+\texht{W\"{o}rz-Busekros}{Woerz-Busekros}, A.,
+{\it Algebras in Genetics},
+Springer Lectures Notes in Biomathematics 36, Berlin e.a. (1980).
+In particular, see example 1.3.}
+
+We assume that there is an infinitely large random mating population.
+Random mating of two gametes \texht{$a_i$}{\spad{ai}}  and
+\texht{$a_j$}{\spad{aj}} gives zygotes
+%-% \HDindex{zygote}{ugProblemGeneticPage}{8.14.}{Non-Associative Algebras and Modelling Genetic Laws}
+\texht{$a_ia_j$}{\spad{ai aj}}, which produce new gametes.
+%-% \HDindex{gamete}{ugProblemGeneticPage}{8.14.}{Non-Associative Algebras and Modelling Genetic Laws}
+In classical Mendelian segregation we have
+\texht{$a_ia_j = {1 \over 2}a_i+{1 \over 2}a_j$}{\spad{ai aj = (1/2)*ai+(1/2)*aj}}.
+In general, we have
+\texht{\narrowDisplay{a_ia_j = \sum_{k=1}^n \gamma_{i,j}^k\ a_k.}}%
+{\spad{ai aj = gammaij1 a1 + gammaij2 a2 + ... + gammaijn an}}
+The segregation rates \texht{$\gamma_{i,j}$}{\spad{gammaij}}
+are  the structural constants of an
+\smath{n}-dimensional algebra.
+This is provided in \Language{} by
+the constructor \spadtype{AlgebraGivenByStructuralConstants}
+(abbreviation \spadtype{ALGSC}).
+
+Consider two coupled autosomal loci with alleles
+\smath{A},\smath{a}, \smath{B}, and \smath{b}, building four
+different gametes
+\texht{$a_1 =  AB, a_2 =  Ab, a_3 =  aB,$ and $a_4 =  ab$}%
+{\spad{a1 :=  AB, a2 :=  Ab, a3 :=  aB,} and \spad{a4 :=  ab}}.
+The zygotes \texht{$a_ia_j$}{\spad{ai aj}}
+produce gametes \texht{$a_i$}{\spad{ai}} and
+\texht{$a_j$}{\spad{aj}}
+with classical Mendelian segregation.
+Zygote \texht{$a_1a_4$}{a1 a4} undergoes transition
+to \texht{$a_2a_3$}{\spad{a2 a3}}
+and vice versa with probability
+\texht{$0 \le \theta \le {1\over 2}$}{0 <= theta <= 1/2}.
+
+\xtc{
+Define a list
+\texht{$[(\gamma_{i,j}^k) 1 \le k \le 4]$}{\spad{[(gammaijk) 1 <= k <= 4]}}
+of four four-by-four matrices giving the segregation
+rates.
+We use the value \smath{1/10} for \smath{\theta}.
+}{
+\spadpaste{segregationRates : List SquareMatrix(4,FRAC INT) := [matrix [ [1, 1/2, 1/2, 9/20], [1/2, 0, 1/20, 0], [1/2, 1/20, 0, 0], [9/20, 0, 0, 0] ], matrix [ [0, 1/2, 0, 1/20], [1/2, 1, 9/20, 1/2], [0, 9/20, 0, 0], [1/20, 1/2, 0, 0] ], matrix [ [0, 0, 1/2, 1/20], [0, 0, 9/20, 0], [1/2, 9/20, 1, 1/2], [1/20, 0, 1/2, 0] ], matrix [ [0, 0, 0, 9/20], [0, 0, 1/20, 1/2], [0, 1/20, 0, 1/2], [9/20, 1/2, 1/2, 1] ] ] \bound{segregationRates}}
+}
+\xtc{
+Choose the appropriate symbols for the basis of gametes,
+}{
+\spadpaste{gametes := ['AB,'Ab,'aB,'ab]  \bound{gametes}}
+}
+\xtc{
+Define the algebra.
+}{
+\spadpaste{A := ALGSC(FRAC INT, 4, gametes, segregationRates);\bound{A}\free{gametes, segregationRates}}
+}
+
+\xtc{
+What are the probabilities for zygote
+\texht{$a_1a_4$}{a1 a4} to produce the different gametes?
+}{
+\spadpaste{a := basis()\$A;  a.1*a.4}
+}
+
+Elements in this algebra whose coefficients sum to one play a
+distinguished role.
+They represent a population with the distribution of gametes
+reflected by the coefficients with respect to the basis of
+gametes.
+
+Random mating of different populations \smath{x} and \smath{y} is described by
+their product \smath{x*y}.
+
+\xtc{
+This product is commutative only
+if the gametes are not sex-dependent, as in our example.
+}{
+\spadpaste{commutative?()\$A \free{A}}
+}
+\xtc{
+In general, it is not associative.
+}{
+\spadpaste{associative?()\$A \free{A}}
+}
+
+Random mating within a population \smath{x} is described by
+\smath{x*x.}
+The next generation is \smath{(x*x)*(x*x).}
+
+\xtc{
+Use decimal numbers to compare the distributions more easily.
+}{
+\spadpaste{x : ALGSC(DECIMAL, 4, gametes, segregationRates) :=  convert [3/10, 1/5, 1/10, 2/5]\bound{x}\free{gametes, segregationRates}}
+}
+\xtc{
+To compute directly the gametic distribution in the fifth
+generation, we use \spadfun{plenaryPower}.
+}{
+\spadpaste{plenaryPower(x,5) \free{x}}
+}
+
+We now ask two questions:
+Does this distribution converge to an equilibrium state?
+What are the distributions that are stable?
+
+\xtc{
+This is an invariant of the algebra and it is used to answer
+the first question.
+The new indeterminates describe a symbolic distribution.
+}{
+\spadpaste{q := leftRankPolynomial()\$GCNAALG(FRAC INT, 4, gametes, segregationRates) :: UP(Y, POLY FRAC INT)\bound{q}\free{gametes, segregationRates}}
+}
+\xtc{
+Because the coefficient \texht{${9 \over 20}$}{\axiom{9/20}} has absolute
+value less than 1, all distributions do converge,
+by a theorem of this theory.
+}{
+\spadpaste{factor(q :: POLY FRAC INT) \free{q}}
+}
+\xtc{
+The second question is answered by searching for idempotents in the
+algebra.
+}{
+\spadpaste{cI := conditionsForIdempotents()\$GCNAALG(FRAC INT, 4, gametes, segregationRates)\bound{cI} \free{gametes, segregationRates}}
+}
+\xtc{
+Solve these equations and look at the first solution.
+}{
+\spadpaste{gbs:= groebnerFactorize cI; gbs.1\free{cI}\bound{gbs}}
+}
+
+Further analysis using the package \spadtype{PolynomialIdeals}
+shows that there is a two-dimensional variety of equilibrium states and all
+other solutions are contained in it.
+
+\xtc{
+Choose one equilibrium state by setting two indeterminates to
+concrete values.
+}{
+\spadpaste{sol := solve concat(gbs.1,[\%x1-1/10,\%x2-1/10])\bound{sol} \free{gbs}}
+}
+\xtc{
+}{
+\spadpaste{e : A := represents reverse (map(rhs, sol.1) :: List FRAC INT)\bound{e} \free{A, sol}}
+}
+\xtc{
+Verify the result.
+}{
+\spadpaste{e*e-e \free{e}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug08.pht b/src/hyper/pages/ug08.pht
new file mode 100644
index 00000000..43637c24
--- /dev/null
+++ b/src/hyper/pages/ug08.pht
@@ -0,0 +1,7664 @@
+\begin{patch}{ugxProblemSeriesBernoulliPagePatch1}
+\begin{paste}{ugxProblemSeriesBernoulliPageFull1}{ugxProblemSeriesBernoulliPageEmpty1}
+\pastebutton{ugxProblemSeriesBernoulliPageFull1}{\hidepaste}
+\tab{5}\spadcommand{reduce(+,[m**4 for m in 1..10])}
+\indentrel{3}\begin{verbatim}
+   (1)  25333
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPageEmpty1}
+\begin{paste}{ugxProblemSeriesBernoulliPageEmpty1}{ugxProblemSeriesBernoulliPagePatch1}
+\pastebutton{ugxProblemSeriesBernoulliPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{reduce(+,[m**4 for m in 1..10])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPagePatch2}
+\begin{paste}{ugxProblemSeriesBernoulliPageFull2}{ugxProblemSeriesBernoulliPageEmpty2}
+\pastebutton{ugxProblemSeriesBernoulliPageFull2}{\hidepaste}
+\tab{5}\spadcommand{sum4 := sum(m**4, m = 1..k)\bound{sum4 }}
+\indentrel{3}\begin{verbatim}
+          5      4      3
+        6k  + 15k  + 10k  - k
+   (2)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                  30
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPageEmpty2}
+\begin{paste}{ugxProblemSeriesBernoulliPageEmpty2}{ugxProblemSeriesBernoulliPagePatch2}
+\pastebutton{ugxProblemSeriesBernoulliPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{sum4 := sum(m**4, m = 1..k)\bound{sum4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPagePatch3}
+\begin{paste}{ugxProblemSeriesBernoulliPageFull3}{ugxProblemSeriesBernoulliPageEmpty3}
+\pastebutton{ugxProblemSeriesBernoulliPageFull3}{\hidepaste}
+\tab{5}\spadcommand{eval(sum4, k = 10)\free{sum4 }}
+\indentrel{3}\begin{verbatim}
+   (3)  25333
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPageEmpty3}
+\begin{paste}{ugxProblemSeriesBernoulliPageEmpty3}{ugxProblemSeriesBernoulliPagePatch3}
+\pastebutton{ugxProblemSeriesBernoulliPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{eval(sum4, k = 10)\free{sum4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPagePatch4}
+\begin{paste}{ugxProblemSeriesBernoulliPageFull4}{ugxProblemSeriesBernoulliPageEmpty4}
+\pastebutton{ugxProblemSeriesBernoulliPageFull4}{\hidepaste}
+\tab{5}\spadcommand{f := t*exp(x*t) / (exp(t) - 1)\bound{f }}
+\indentrel{3}\begin{verbatim}
+            t x
+        t %e
+   (4)  ÄÄÄÄÄÄÄ
+          t
+        %e  - 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPageEmpty4}
+\begin{paste}{ugxProblemSeriesBernoulliPageEmpty4}{ugxProblemSeriesBernoulliPagePatch4}
+\pastebutton{ugxProblemSeriesBernoulliPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{f := t*exp(x*t) / (exp(t) - 1)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPagePatch5}
+\begin{paste}{ugxProblemSeriesBernoulliPageFull5}{ugxProblemSeriesBernoulliPageEmpty5}
+\pastebutton{ugxProblemSeriesBernoulliPageFull5}{\hidepaste}
+\tab{5}\spadcommand{)set streams calculate 5\bound{set }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPageEmpty5}
+\begin{paste}{ugxProblemSeriesBernoulliPageEmpty5}{ugxProblemSeriesBernoulliPagePatch5}
+\pastebutton{ugxProblemSeriesBernoulliPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{)set streams calculate 5\bound{set }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPagePatch6}
+\begin{paste}{ugxProblemSeriesBernoulliPageFull6}{ugxProblemSeriesBernoulliPageEmpty6}
+\pastebutton{ugxProblemSeriesBernoulliPageFull6}{\hidepaste}
+\tab{5}\spadcommand{ff := taylor(f,t = 0)\free{f set }\bound{ff }}
+\indentrel{3}\begin{verbatim}
+   (5)
+                      2                 3     2
+         2x - 1     6x  - 6x + 1  2   2x  - 3x  + x  3
+     1 + ÄÄÄÄÄÄ t + ÄÄÄÄÄÄÄÄÄÄÄÄ t  + ÄÄÄÄÄÄÄÄÄÄÄÄÄ t
+            2            12                 12
+   + 
+        4      3      2            5      4      3
+     30x  - 60x  + 30x  - 1  4   6x  - 15x  + 10x  - x  5
+     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t  + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
+               720                        720
+   + 
+        6
+     O(t )
+   Type: UnivariateTaylorSeries(Expression Integer,t,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPageEmpty6}
+\begin{paste}{ugxProblemSeriesBernoulliPageEmpty6}{ugxProblemSeriesBernoulliPagePatch6}
+\pastebutton{ugxProblemSeriesBernoulliPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{ff := taylor(f,t = 0)\free{f set }\bound{ff }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPagePatch7}
+\begin{paste}{ugxProblemSeriesBernoulliPageFull7}{ugxProblemSeriesBernoulliPageEmpty7}
+\pastebutton{ugxProblemSeriesBernoulliPageFull7}{\hidepaste}
+\tab{5}\spadcommand{factorial(6) * coefficient(ff,6)\free{ff }}
+\indentrel{3}\begin{verbatim}
+           6       5       4      2
+        42x  - 126x  + 105x  - 21x  + 1
+   (6)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                       42
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPageEmpty7}
+\begin{paste}{ugxProblemSeriesBernoulliPageEmpty7}{ugxProblemSeriesBernoulliPagePatch7}
+\pastebutton{ugxProblemSeriesBernoulliPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{factorial(6) * coefficient(ff,6)\free{ff }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPagePatch8}
+\begin{paste}{ugxProblemSeriesBernoulliPageFull8}{ugxProblemSeriesBernoulliPageEmpty8}
+\pastebutton{ugxProblemSeriesBernoulliPageFull8}{\hidepaste}
+\tab{5}\spadcommand{g := eval(f, x = x + 1) - f\bound{g }\free{f }}
+\indentrel{3}\begin{verbatim}
+            t x + t       t x
+        t %e        - t %e
+   (7)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                 t
+               %e  - 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPageEmpty8}
+\begin{paste}{ugxProblemSeriesBernoulliPageEmpty8}{ugxProblemSeriesBernoulliPagePatch8}
+\pastebutton{ugxProblemSeriesBernoulliPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{g := eval(f, x = x + 1) - f\bound{g }\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPagePatch9}
+\begin{paste}{ugxProblemSeriesBernoulliPageFull9}{ugxProblemSeriesBernoulliPageEmpty9}
+\pastebutton{ugxProblemSeriesBernoulliPageFull9}{\hidepaste}
+\tab{5}\spadcommand{normalize(g)\free{g }}
+\indentrel{3}\begin{verbatim}
+            t x
+   (8)  t %e
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPageEmpty9}
+\begin{paste}{ugxProblemSeriesBernoulliPageEmpty9}{ugxProblemSeriesBernoulliPagePatch9}
+\pastebutton{ugxProblemSeriesBernoulliPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{normalize(g)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPagePatch10}
+\begin{paste}{ugxProblemSeriesBernoulliPageFull10}{ugxProblemSeriesBernoulliPageEmpty10}
+\pastebutton{ugxProblemSeriesBernoulliPageFull10}{\hidepaste}
+\tab{5}\spadcommand{taylor(g,t = 0)\free{g }}
+\indentrel{3}\begin{verbatim}
+                    2       3       4
+               2   x   3   x   4   x   5      6
+   (9)  t + x t  + ÄÄ t  + ÄÄ t  + ÄÄ t  + O(t )
+                    2       6      24
+   Type: UnivariateTaylorSeries(Expression Integer,t,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPageEmpty10}
+\begin{paste}{ugxProblemSeriesBernoulliPageEmpty10}{ugxProblemSeriesBernoulliPagePatch10}
+\pastebutton{ugxProblemSeriesBernoulliPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{taylor(g,t = 0)\free{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPagePatch11}
+\begin{paste}{ugxProblemSeriesBernoulliPageFull11}{ugxProblemSeriesBernoulliPageEmpty11}
+\pastebutton{ugxProblemSeriesBernoulliPageFull11}{\hidepaste}
+\tab{5}\spadcommand{B5 := factorial(5) * coefficient(ff,5)\free{ff }\bound{B5 }}
+\indentrel{3}\begin{verbatim}
+           5      4      3
+         6x  - 15x  + 10x  - x
+   (10)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                   6
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPageEmpty11}
+\begin{paste}{ugxProblemSeriesBernoulliPageEmpty11}{ugxProblemSeriesBernoulliPagePatch11}
+\pastebutton{ugxProblemSeriesBernoulliPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{B5 := factorial(5) * coefficient(ff,5)\free{ff }\bound{B5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPagePatch12}
+\begin{paste}{ugxProblemSeriesBernoulliPageFull12}{ugxProblemSeriesBernoulliPageEmpty12}
+\pastebutton{ugxProblemSeriesBernoulliPageFull12}{\hidepaste}
+\tab{5}\spadcommand{1/5 * (eval(B5, x = k + 1) - eval(B5, x = 1))\free{B5 }}
+\indentrel{3}\begin{verbatim}
+           5      4      3
+         6k  + 15k  + 10k  - k
+   (11)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                   30
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPageEmpty12}
+\begin{paste}{ugxProblemSeriesBernoulliPageEmpty12}{ugxProblemSeriesBernoulliPagePatch12}
+\pastebutton{ugxProblemSeriesBernoulliPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{1/5 * (eval(B5, x = k + 1) - eval(B5, x = 1))\free{B5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPagePatch13}
+\begin{paste}{ugxProblemSeriesBernoulliPageFull13}{ugxProblemSeriesBernoulliPageEmpty13}
+\pastebutton{ugxProblemSeriesBernoulliPageFull13}{\hidepaste}
+\tab{5}\spadcommand{sum4\free{sum4 }}
+\indentrel{3}\begin{verbatim}
+           5      4      3
+         6k  + 15k  + 10k  - k
+   (12)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                   30
+                      Type: Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesBernoulliPageEmpty13}
+\begin{paste}{ugxProblemSeriesBernoulliPageEmpty13}{ugxProblemSeriesBernoulliPagePatch13}
+\pastebutton{ugxProblemSeriesBernoulliPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{sum4\free{sum4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePagePatch1}
+\begin{paste}{ugProblemLaplacePageFull1}{ugProblemLaplacePageEmpty1}
+\pastebutton{ugProblemLaplacePageFull1}{\hidepaste}
+\tab{5}\spadcommand{laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t), t, s)}
+\indentrel{3}\begin{verbatim}
+             3
+           4a
+   (1)  ÄÄÄÄÄÄÄÄ
+         4     4
+        s  + 4a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePageEmpty1}
+\begin{paste}{ugProblemLaplacePageEmpty1}{ugProblemLaplacePagePatch1}
+\pastebutton{ugProblemLaplacePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePagePatch2}
+\begin{paste}{ugProblemLaplacePageFull2}{ugProblemLaplacePageEmpty2}
+\pastebutton{ugProblemLaplacePageFull2}{\hidepaste}
+\tab{5}\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)}
+\indentrel{3}\begin{verbatim}
+   (2)  - log(s - a) + log(s - b)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePageEmpty2}
+\begin{paste}{ugProblemLaplacePageEmpty2}{ugProblemLaplacePagePatch2}
+\pastebutton{ugProblemLaplacePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePagePatch3}
+\begin{paste}{ugProblemLaplacePageFull3}{ugProblemLaplacePageEmpty3}
+\pastebutton{ugProblemLaplacePageFull3}{\hidepaste}
+\tab{5}\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)}
+\indentrel{3}\begin{verbatim}
+             2    2
+   (3)  log(s  + a ) - 2log(s)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePageEmpty3}
+\begin{paste}{ugProblemLaplacePageEmpty3}{ugProblemLaplacePagePatch3}
+\pastebutton{ugProblemLaplacePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePagePatch4}
+\begin{paste}{ugProblemLaplacePageFull4}{ugProblemLaplacePageEmpty4}
+\pastebutton{ugProblemLaplacePageFull4}{\hidepaste}
+\tab{5}\spadcommand{laplace(exp(-a*t) * sin(b*t) / b**2, t, s)}
+\indentrel{3}\begin{verbatim}
+                    1
+   (4)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+           2             3    2
+        b s  + 2a b s + b  + a b
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePageEmpty4}
+\begin{paste}{ugProblemLaplacePageEmpty4}{ugProblemLaplacePagePatch4}
+\pastebutton{ugProblemLaplacePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{laplace(exp(-a*t) * sin(b*t) / b**2, t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePagePatch5}
+\begin{paste}{ugProblemLaplacePageFull5}{ugProblemLaplacePageEmpty5}
+\pastebutton{ugProblemLaplacePageFull5}{\hidepaste}
+\tab{5}\spadcommand{laplace((cos(a*t) - cos(b*t))/t, t, s)}
+\indentrel{3}\begin{verbatim}
+             2    2         2    2
+        log(s  + b ) - log(s  + a )
+   (5)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                     2
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePageEmpty5}
+\begin{paste}{ugProblemLaplacePageEmpty5}{ugProblemLaplacePagePatch5}
+\pastebutton{ugProblemLaplacePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{laplace((cos(a*t) - cos(b*t))/t, t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePagePatch6}
+\begin{paste}{ugProblemLaplacePageFull6}{ugProblemLaplacePageEmpty6}
+\pastebutton{ugProblemLaplacePageFull6}{\hidepaste}
+\tab{5}\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)}
+\indentrel{3}\begin{verbatim}
+          b    s + c - a
+        %e log(ÄÄÄÄÄÄÄÄÄ)
+                   c
+   (6)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+              s - a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePageEmpty6}
+\begin{paste}{ugProblemLaplacePageEmpty6}{ugProblemLaplacePagePatch6}
+\pastebutton{ugProblemLaplacePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePagePatch7}
+\begin{paste}{ugProblemLaplacePageFull7}{ugProblemLaplacePageEmpty7}
+\pastebutton{ugProblemLaplacePageFull7}{\hidepaste}
+\tab{5}\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)}
+\indentrel{3}\begin{verbatim}
+               2    2
+              s  + b             d
+        a log(ÄÄÄÄÄÄÄ) + 2c atan(Ä)
+                  2              s
+                 b
+   (7)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                     2s
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePageEmpty7}
+\begin{paste}{ugProblemLaplacePageEmpty7}{ugProblemLaplacePagePatch7}
+\pastebutton{ugProblemLaplacePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePagePatch8}
+\begin{paste}{ugProblemLaplacePageFull8}{ugProblemLaplacePageEmpty8}
+\pastebutton{ugProblemLaplacePageFull8}{\hidepaste}
+\tab{5}\spadcommand{laplace(sin(a*t) - a*t*cos(a*t) + exp(t**2), t, s)}
+\indentrel{3}\begin{verbatim}
+                                    2
+          4     2 2    4           t           3
+        (s  + 2a s  + a )laplace(%e  ,t,s) + 2a
+   (8)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                      4     2 2    4
+                     s  + 2a s  + a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLaplacePageEmpty8}
+\begin{paste}{ugProblemLaplacePageEmpty8}{ugProblemLaplacePagePatch8}
+\pastebutton{ugProblemLaplacePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{laplace(sin(a*t) - a*t*cos(a*t) + exp(t**2), t, s)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPagePatch1}
+\begin{paste}{ugxProblemSeriesFunctionsPageFull1}{ugxProblemSeriesFunctionsPageEmpty1}
+\pastebutton{ugxProblemSeriesFunctionsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{x := series 'x\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (1)  x
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPageEmpty1}
+\begin{paste}{ugxProblemSeriesFunctionsPageEmpty1}{ugxProblemSeriesFunctionsPagePatch1}
+\pastebutton{ugxProblemSeriesFunctionsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{x := series 'x\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPagePatch2}
+\begin{paste}{ugxProblemSeriesFunctionsPageFull2}{ugxProblemSeriesFunctionsPageEmpty2}
+\pastebutton{ugxProblemSeriesFunctionsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{rat := x**2 / (1 - 6*x + x**2)\free{x }\bound{rat }}
+\indentrel{3}\begin{verbatim}
+   (2)
+      2     3      4       5        6        7         8
+     x  + 6x  + 35x  + 204x  + 1189x  + 6930x  + 40391x
+   + 
+            9           10           11            12
+     235416x  + 1372105x   + 7997214x   + 46611179x
+   + 
+        13
+     O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPageEmpty2}
+\begin{paste}{ugxProblemSeriesFunctionsPageEmpty2}{ugxProblemSeriesFunctionsPagePatch2}
+\pastebutton{ugxProblemSeriesFunctionsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{rat := x**2 / (1 - 6*x + x**2)\free{x }\bound{rat }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPagePatch3}
+\begin{paste}{ugxProblemSeriesFunctionsPageFull3}{ugxProblemSeriesFunctionsPageEmpty3}
+\pastebutton{ugxProblemSeriesFunctionsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{sin(rat)\free{rat }}
+\indentrel{3}\begin{verbatim}
+   (3)
+      2     3      4       5   7133  6        7   80711  8
+     x  + 6x  + 35x  + 204x  + ÄÄÄÄ x  + 6927x  + ÄÄÄÄÄ x
+                                 6                  2
+   + 
+            9   164285281  10   31888513  11
+     235068x  + ÄÄÄÄÄÄÄÄÄ x   + ÄÄÄÄÄÄÄÄ x
+                   120              4
+   + 
+     371324777  12      13
+     ÄÄÄÄÄÄÄÄÄ x   + O(x  )
+         8
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPageEmpty3}
+\begin{paste}{ugxProblemSeriesFunctionsPageEmpty3}{ugxProblemSeriesFunctionsPagePatch3}
+\pastebutton{ugxProblemSeriesFunctionsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{sin(rat)\free{rat }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPagePatch4}
+\begin{paste}{ugxProblemSeriesFunctionsPageFull4}{ugxProblemSeriesFunctionsPageEmpty4}
+\pastebutton{ugxProblemSeriesFunctionsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{y : UTS(FRAC INT,y,0) := y\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (4)  y
+     Type: UnivariateTaylorSeries(Fraction Integer,y,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPageEmpty4}
+\begin{paste}{ugxProblemSeriesFunctionsPageEmpty4}{ugxProblemSeriesFunctionsPagePatch4}
+\pastebutton{ugxProblemSeriesFunctionsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{y : UTS(FRAC INT,y,0) := y\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPagePatch5}
+\begin{paste}{ugxProblemSeriesFunctionsPageFull5}{ugxProblemSeriesFunctionsPageEmpty5}
+\pastebutton{ugxProblemSeriesFunctionsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{exp(y)\free{y }}
+\indentrel{3}\begin{verbatim}
+   (5)
+             1  2   1  3    1  4    1   5    1   6
+     1 + y + Ä y  + Ä y  + ÄÄ y  + ÄÄÄ y  + ÄÄÄ y
+             2      6      24      120      720
+   + 
+     1   7     1    8      1    9      1     10      11
+   ÄÄÄÄ y  + ÄÄÄÄÄ y  + ÄÄÄÄÄÄ y  + ÄÄÄÄÄÄÄ y   + O(y  )
+   5040      40320      362880      3628800
+     Type: UnivariateTaylorSeries(Fraction Integer,y,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPageEmpty5}
+\begin{paste}{ugxProblemSeriesFunctionsPageEmpty5}{ugxProblemSeriesFunctionsPagePatch5}
+\pastebutton{ugxProblemSeriesFunctionsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{exp(y)\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPagePatch6}
+\begin{paste}{ugxProblemSeriesFunctionsPageFull6}{ugxProblemSeriesFunctionsPageEmpty6}
+\pastebutton{ugxProblemSeriesFunctionsPageFull6}{\hidepaste}
+\tab{5}\spadcommand{tan(y**2)\free{y }}
+\indentrel{3}\begin{verbatim}
+         2   1  6    2  10      11
+   (6)  y  + Ä y  + ÄÄ y   + O(y  )
+             3      15
+     Type: UnivariateTaylorSeries(Fraction Integer,y,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPageEmpty6}
+\begin{paste}{ugxProblemSeriesFunctionsPageEmpty6}{ugxProblemSeriesFunctionsPagePatch6}
+\pastebutton{ugxProblemSeriesFunctionsPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{tan(y**2)\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPagePatch7}
+\begin{paste}{ugxProblemSeriesFunctionsPageFull7}{ugxProblemSeriesFunctionsPageEmpty7}
+\pastebutton{ugxProblemSeriesFunctionsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{cos(y + y**5)\free{y }}
+\indentrel{3}\begin{verbatim}
+   (7)
+         1  2    1  4   721  6    6721  8   1844641  10
+     1 - Ä y  + ÄÄ y  - ÄÄÄ y  + ÄÄÄÄÄ y  - ÄÄÄÄÄÄÄ y
+         2      24      720      40320      3628800
+   + 
+        11
+     O(y  )
+     Type: UnivariateTaylorSeries(Fraction Integer,y,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPageEmpty7}
+\begin{paste}{ugxProblemSeriesFunctionsPageEmpty7}{ugxProblemSeriesFunctionsPagePatch7}
+\pastebutton{ugxProblemSeriesFunctionsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{cos(y + y**5)\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPagePatch8}
+\begin{paste}{ugxProblemSeriesFunctionsPageFull8}{ugxProblemSeriesFunctionsPageEmpty8}
+\pastebutton{ugxProblemSeriesFunctionsPageFull8}{\hidepaste}
+\tab{5}\spadcommand{log(1 + sin(y))\free{y }}
+\indentrel{3}\begin{verbatim}
+   (8)
+         1  2   1  3    1  4    1  5    1  6    61   7
+     y - Ä y  + Ä y  - ÄÄ y  + ÄÄ y  - ÄÄ y  + ÄÄÄÄ y
+         2      6      12      24      45      5040
+   + 
+        17   8    277   9     31   10      11
+     - ÄÄÄÄ y  + ÄÄÄÄÄ y  - ÄÄÄÄÄ y   + O(y  )
+       2520      72576      14175
+     Type: UnivariateTaylorSeries(Fraction Integer,y,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPageEmpty8}
+\begin{paste}{ugxProblemSeriesFunctionsPageEmpty8}{ugxProblemSeriesFunctionsPagePatch8}
+\pastebutton{ugxProblemSeriesFunctionsPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{log(1 + sin(y))\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPagePatch9}
+\begin{paste}{ugxProblemSeriesFunctionsPageFull9}{ugxProblemSeriesFunctionsPageEmpty9}
+\pastebutton{ugxProblemSeriesFunctionsPageFull9}{\hidepaste}
+\tab{5}\spadcommand{z : UTS(EXPR INT,z,0) := z\bound{z }}
+\indentrel{3}\begin{verbatim}
+   (9)  z
+   Type: UnivariateTaylorSeries(Expression Integer,z,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPageEmpty9}
+\begin{paste}{ugxProblemSeriesFunctionsPageEmpty9}{ugxProblemSeriesFunctionsPagePatch9}
+\pastebutton{ugxProblemSeriesFunctionsPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{z : UTS(EXPR INT,z,0) := z\bound{z }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPagePatch10}
+\begin{paste}{ugxProblemSeriesFunctionsPageFull10}{ugxProblemSeriesFunctionsPageEmpty10}
+\pastebutton{ugxProblemSeriesFunctionsPageFull10}{\hidepaste}
+\tab{5}\spadcommand{exp(2 + tan(z))\free{z }}
+\indentrel{3}\begin{verbatim}
+   (10)
+                    2        2         2          2
+       2     2    %e   2   %e   3   3%e   4   37%e   5
+     %e  + %e z + ÄÄÄ z  + ÄÄÄ z  + ÄÄÄÄ z  + ÄÄÄÄÄ z
+                   2        2         8        120
+   + 
+         2           2           2             2
+     59%e   6   137%e   7   871%e   8   41641%e   9
+     ÄÄÄÄÄ z  + ÄÄÄÄÄÄ z  + ÄÄÄÄÄÄ z  + ÄÄÄÄÄÄÄÄ z
+      240         720        5760        362880
+   + 
+             2
+     325249%e   10      11
+     ÄÄÄÄÄÄÄÄÄ z   + O(z  )
+      3628800
+   Type: UnivariateTaylorSeries(Expression Integer,z,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPageEmpty10}
+\begin{paste}{ugxProblemSeriesFunctionsPageEmpty10}{ugxProblemSeriesFunctionsPagePatch10}
+\pastebutton{ugxProblemSeriesFunctionsPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{exp(2 + tan(z))\free{z }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPagePatch11}
+\begin{paste}{ugxProblemSeriesFunctionsPageFull11}{ugxProblemSeriesFunctionsPageEmpty11}
+\pastebutton{ugxProblemSeriesFunctionsPageFull11}{\hidepaste}
+\tab{5}\spadcommand{w := taylor 'w\bound{w }}
+\indentrel{3}\begin{verbatim}
+   (11)  w
+   Type: UnivariateTaylorSeries(Expression Integer,w,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPageEmpty11}
+\begin{paste}{ugxProblemSeriesFunctionsPageEmpty11}{ugxProblemSeriesFunctionsPagePatch11}
+\pastebutton{ugxProblemSeriesFunctionsPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{w := taylor 'w\bound{w }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPagePatch12}
+\begin{paste}{ugxProblemSeriesFunctionsPageFull12}{ugxProblemSeriesFunctionsPageEmpty12}
+\pastebutton{ugxProblemSeriesFunctionsPageFull12}{\hidepaste}
+\tab{5}\spadcommand{exp(2 + tan(w))\free{w }}
+\indentrel{3}\begin{verbatim}
+   (12)
+                    2        2         2          2
+       2     2    %e   2   %e   3   3%e   4   37%e   5
+     %e  + %e w + ÄÄÄ w  + ÄÄÄ w  + ÄÄÄÄ w  + ÄÄÄÄÄ w
+                   2        2         8        120
+   + 
+         2           2           2             2
+     59%e   6   137%e   7   871%e   8   41641%e   9
+     ÄÄÄÄÄ w  + ÄÄÄÄÄÄ w  + ÄÄÄÄÄÄ w  + ÄÄÄÄÄÄÄÄ w
+      240         720        5760        362880
+   + 
+             2
+     325249%e   10      11
+     ÄÄÄÄÄÄÄÄÄ w   + O(w  )
+      3628800
+   Type: UnivariateTaylorSeries(Expression Integer,w,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFunctionsPageEmpty12}
+\begin{paste}{ugxProblemSeriesFunctionsPageEmpty12}{ugxProblemSeriesFunctionsPagePatch12}
+\pastebutton{ugxProblemSeriesFunctionsPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{exp(2 + tan(w))\free{w }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch1}
+\begin{paste}{ugProblemGeneticPageFull1}{ugProblemGeneticPageEmpty1}
+\pastebutton{ugProblemGeneticPageFull1}{\hidepaste}
+\tab{5}\spadcommand{segregationRates : List SquareMatrix(4,FRAC INT) := [matrix [ [1, 1/2, 1/2, 9/20], [1/2, 0, 1/20, 0], [1/2, 1/20, 0, 0], [9/20, 0, 0, 0] ], matrix [ [0, 1/2, 0, 1/20], [1/2, 1, 9/20, 1/2], [0, 9/20, 0, 0], [1/20, 1/2, 0, 0] ], matrix [ [0, 0, 1/2, 1/20], [0, 0, 9/20, 0], [1/2, 9/20, 1, 1/2], [1/20, 0, 1/2, 0] ], matrix [ [0, 0, 0, 9/20], [0, 0, 1/20, 1/2], [0, 1/20, 0, 1/2], [9/20, 1/2, 1/2, 1] ] ]\bound{segregationRates }}
+\indentrel{3}\begin{verbatim}
+   (1)
+    Ú    1   1    9¿  Ú    1        1¿  Ú        1    1¿
+    ³1   Ä   Ä   Äij  ³0   Ä   0   Äij  ³0   0   Ä   Äij
+    ³    2   2   20³  ³    2       20³  ³        2   20³
+    ³              ³  ³              ³  ³              ³
+    ³1        1    ³  ³1        9  1 ³  ³         9    ³
+    ³Ä   0   ÄÄ  0 ³  ³Ä   1   ÄÄ  Ä ³  ³0   0   ÄÄ  0 ³
+    ³2       20    ³  ³2       20  2 ³  ³        20    ³
+   [³              ³, ³              ³, ³              ³,
+    ³1    1        ³  ³     9        ³  ³1    9      1 ³
+    ³Ä   ÄÄ  0   0 ³  ³0   ÄÄ  0   0 ³  ³Ä   ÄÄ  1   Ä ³
+    ³2   20        ³  ³    20        ³  ³2   20      2 ³
+    ³              ³  ³              ³  ³              ³
+    ³ 9            ³  ³ 1  1         ³  ³ 1      1     ³
+    ³ÄÄ  0   0   0 ³  ³ÄÄ  Ä   0   0 ³  ³ÄÄ  0   Ä   0 ³
+    À20            Ù  À20  2         Ù  À20      2     Ù
+    Ú             9¿
+    ³0   0   0   Äij
+    ³            20³
+    ³              ³
+    ³         1  1 ³
+    ³0   0   ÄÄ  Ä ³
+    ³        20  2 ³
+    ³              ³]
+    ³     1      1 ³
+    ³0   ÄÄ  0   Ä ³
+    ³    20      2 ³
+    ³              ³
+    ³ 9  1   1     ³
+    ³ÄÄ  Ä   Ä   1 ³
+    À20  2   2     Ù
+            Type: List SquareMatrix(4,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty1}
+\begin{paste}{ugProblemGeneticPageEmpty1}{ugProblemGeneticPagePatch1}
+\pastebutton{ugProblemGeneticPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{segregationRates : List SquareMatrix(4,FRAC INT) := [matrix [ [1, 1/2, 1/2, 9/20], [1/2, 0, 1/20, 0], [1/2, 1/20, 0, 0], [9/20, 0, 0, 0] ], matrix [ [0, 1/2, 0, 1/20], [1/2, 1, 9/20, 1/2], [0, 9/20, 0, 0], [1/20, 1/2, 0, 0] ], matrix [ [0, 0, 1/2, 1/20], [0, 0, 9/20, 0], [1/2, 9/20, 1, 1/2], [1/20, 0, 1/2, 0] ], matrix [ [0, 0, 0, 9/20], [0, 0, 1/20, 1/2], [0, 1/20, 0, 1/2], [9/20, 1/2, 1/2, 1] ] ]\bound{segregationRates }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch2}
+\begin{paste}{ugProblemGeneticPageFull2}{ugProblemGeneticPageEmpty2}
+\pastebutton{ugProblemGeneticPageFull2}{\hidepaste}
+\tab{5}\spadcommand{gametes := ['AB,'Ab,'aB,'ab]\bound{gametes }}
+\indentrel{3}\begin{verbatim}
+   (2)  [AB,Ab,aB,ab]
+           Type: List OrderedVariableList [AB,Ab,aB,ab]
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty2}
+\begin{paste}{ugProblemGeneticPageEmpty2}{ugProblemGeneticPagePatch2}
+\pastebutton{ugProblemGeneticPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{gametes := ['AB,'Ab,'aB,'ab]\bound{gametes }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch3}
+\begin{paste}{ugProblemGeneticPageFull3}{ugProblemGeneticPageEmpty3}
+\pastebutton{ugProblemGeneticPageFull3}{\hidepaste}
+\tab{5}\spadcommand{A := ALGSC(FRAC INT, 4, gametes, segregationRates);\bound{A }\free{gametes segregationRates }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty3}
+\begin{paste}{ugProblemGeneticPageEmpty3}{ugProblemGeneticPagePatch3}
+\pastebutton{ugProblemGeneticPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{A := ALGSC(FRAC INT, 4, gametes, segregationRates);\bound{A }\free{gametes segregationRates }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch4}
+\begin{paste}{ugProblemGeneticPageFull4}{ugProblemGeneticPageEmpty4}
+\pastebutton{ugProblemGeneticPageFull4}{\hidepaste}
+\tab{5}\spadcommand{a := basis()$A; a.1*a.4}
+\indentrel{3}\begin{verbatim}
+         9       1       1       9
+   (4)  ÄÄ ab + ÄÄ aB + ÄÄ Ab + ÄÄ AB
+        20      20      20      20
+Type: AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty4}
+\begin{paste}{ugProblemGeneticPageEmpty4}{ugProblemGeneticPagePatch4}
+\pastebutton{ugProblemGeneticPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{a := basis()$A; a.1*a.4}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch5}
+\begin{paste}{ugProblemGeneticPageFull5}{ugProblemGeneticPageEmpty5}
+\pastebutton{ugProblemGeneticPageFull5}{\hidepaste}
+\tab{5}\spadcommand{commutative?()$A\free{A }}
+\indentrel{3}\begin{verbatim}
+   algebra is commutative
+   (5)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty5}
+\begin{paste}{ugProblemGeneticPageEmpty5}{ugProblemGeneticPagePatch5}
+\pastebutton{ugProblemGeneticPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{commutative?()$A\free{A }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch6}
+\begin{paste}{ugProblemGeneticPageFull6}{ugProblemGeneticPageEmpty6}
+\pastebutton{ugProblemGeneticPageFull6}{\hidepaste}
+\tab{5}\spadcommand{associative?()$A\free{A }}
+\indentrel{3}\begin{verbatim}
+   algebra is not associative
+   (6)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty6}
+\begin{paste}{ugProblemGeneticPageEmpty6}{ugProblemGeneticPagePatch6}
+\pastebutton{ugProblemGeneticPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{associative?()$A\free{A }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch7}
+\begin{paste}{ugProblemGeneticPageFull7}{ugProblemGeneticPageEmpty7}
+\pastebutton{ugProblemGeneticPageFull7}{\hidepaste}
+\tab{5}\spadcommand{x : ALGSC(DECIMAL, 4, gametes, segregationRates) := convert [3/10, 1/5, 1/10, 2/5]\bound{x }\free{gametes segregationRates }}
+\indentrel{3}\begin{verbatim}
+   (7)  0.4ab + 0.1aB + 0.2Ab + 0.3AB
+Type: AlgebraGivenByStructuralConstants(DecimalExpansion,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty7}
+\begin{paste}{ugProblemGeneticPageEmpty7}{ugProblemGeneticPagePatch7}
+\pastebutton{ugProblemGeneticPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{x : ALGSC(DECIMAL, 4, gametes, segregationRates) := convert [3/10, 1/5, 1/10, 2/5]\bound{x }\free{gametes segregationRates }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch8}
+\begin{paste}{ugProblemGeneticPageFull8}{ugProblemGeneticPageEmpty8}
+\pastebutton{ugProblemGeneticPageFull8}{\hidepaste}
+\tab{5}\spadcommand{plenaryPower(x,5)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (8)  0.36561ab + 0.13439aB + 0.23439Ab + 0.26561AB
+Type: AlgebraGivenByStructuralConstants(DecimalExpansion,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty8}
+\begin{paste}{ugProblemGeneticPageEmpty8}{ugProblemGeneticPagePatch8}
+\pastebutton{ugProblemGeneticPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{plenaryPower(x,5)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch9}
+\begin{paste}{ugProblemGeneticPageFull9}{ugProblemGeneticPageEmpty9}
+\pastebutton{ugProblemGeneticPageFull9}{\hidepaste}
+\tab{5}\spadcommand{q := leftRankPolynomial()$GCNAALG(FRAC INT, 4, gametes, segregationRates) :: UP(Y, POLY FRAC INT)\bound{q }\free{gametes segregationRates }}
+\indentrel{3}\begin{verbatim}
+   (9)
+      3      29       29       29       29      2
+     Y  + (- ÄÄ %x4 - ÄÄ %x3 - ÄÄ %x2 - ÄÄ %x1)Y
+             20       20       20       20
+   + 
+          9    2     9        9        9            9    2
+         ÄÄ %x4  + (ÄÄ %x3 + ÄÄ %x2 + ÄÄ %x1)%x4 + ÄÄ %x3
+         20         10       10       10           20
+       + 
+           9        9            9    2    9
+         (ÄÄ %x2 + ÄÄ %x1)%x3 + ÄÄ %x2  + ÄÄ %x1 %x2
+          10       10           20        10
+       + 
+          9    2
+         ÄÄ %x1
+         20
+    *
+       Y
+Type: UnivariatePolynomial(Y,Polynomial Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty9}
+\begin{paste}{ugProblemGeneticPageEmpty9}{ugProblemGeneticPagePatch9}
+\pastebutton{ugProblemGeneticPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{q := leftRankPolynomial()$GCNAALG(FRAC INT, 4, gametes, segregationRates) :: UP(Y, POLY FRAC INT)\bound{q }\free{gametes segregationRates }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch10}
+\begin{paste}{ugProblemGeneticPageFull10}{ugProblemGeneticPageEmpty10}
+\pastebutton{ugProblemGeneticPageFull10}{\hidepaste}
+\tab{5}\spadcommand{factor(q :: POLY FRAC INT)\free{q }}
+\indentrel{3}\begin{verbatim}
+   (10)
+     (Y - %x4 - %x3 - %x2 - %x1)
+  *
+           9        9        9        9
+     (Y - ÄÄ %x4 - ÄÄ %x3 - ÄÄ %x2 - ÄÄ %x1)Y
+          20       20       20       20
+             Type: Factored Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty10}
+\begin{paste}{ugProblemGeneticPageEmpty10}{ugProblemGeneticPagePatch10}
+\pastebutton{ugProblemGeneticPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{factor(q :: POLY FRAC INT)\free{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch11}
+\begin{paste}{ugProblemGeneticPageFull11}{ugProblemGeneticPageEmpty11}
+\pastebutton{ugProblemGeneticPageFull11}{\hidepaste}
+\tab{5}\spadcommand{cI := conditionsForIdempotents()$GCNAALG(FRAC INT, 4, gametes, segregationRates)\bound{cI }\free{gametes segregationRates }}
+\indentrel{3}\begin{verbatim}
+   (11)
+     9             1                              2
+   [ÄÄ %x1 %x4 + (ÄÄ %x2 + %x1)%x3 + %x1 %x2 + %x1  - %x1,
+    10            10
+            1            9              2
+    (%x2 + ÄÄ %x1)%x4 + ÄÄ %x2 %x3 + %x2  + (%x1 - 1)%x2,
+           10           10
+            1              2     9
+    (%x3 + ÄÄ %x1)%x4 + %x3  + (ÄÄ %x2 + %x1 - 1)%x3,
+           10                   10
+       2                 9                1
+    %x4  + (%x3 + %x2 + ÄÄ %x1 - 1)%x4 + ÄÄ %x2 %x3]
+                        10               10
+                 Type: List Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty11}
+\begin{paste}{ugProblemGeneticPageEmpty11}{ugProblemGeneticPagePatch11}
+\pastebutton{ugProblemGeneticPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{cI := conditionsForIdempotents()$GCNAALG(FRAC INT, 4, gametes, segregationRates)\bound{cI }\free{gametes segregationRates }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch12}
+\begin{paste}{ugProblemGeneticPageFull12}{ugProblemGeneticPageEmpty12}
+\pastebutton{ugProblemGeneticPageFull12}{\hidepaste}
+\tab{5}\spadcommand{gbs:= groebnerFactorize cI; gbs.1\free{cI }\bound{gbs }}
+\indentrel{3}\begin{verbatim}
+   (12)
+   [%x4 + %x3 + %x2 + %x1 - 1,
+                                  2
+    (%x2 + %x1)%x3 + %x1 %x2 + %x1  - %x1]
+                 Type: List Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty12}
+\begin{paste}{ugProblemGeneticPageEmpty12}{ugProblemGeneticPagePatch12}
+\pastebutton{ugProblemGeneticPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{gbs:= groebnerFactorize cI; gbs.1\free{cI }\bound{gbs }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch13}
+\begin{paste}{ugProblemGeneticPageFull13}{ugProblemGeneticPageEmpty13}
+\pastebutton{ugProblemGeneticPageFull13}{\hidepaste}
+\tab{5}\spadcommand{sol := solve concat(gbs.1,[\%x1-1/10,\%x2-1/10])\bound{sol }\free{gbs }}
+\indentrel{3}\begin{verbatim}
+                2      2       1       1
+   (13)  [[%x4= Ä,%x3= Ä,%x2= ÄÄ,%x1= ÄÄ]]
+                5      5      10      10
+   Type: List List Equation Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty13}
+\begin{paste}{ugProblemGeneticPageEmpty13}{ugProblemGeneticPagePatch13}
+\pastebutton{ugProblemGeneticPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{sol := solve concat(gbs.1,[\%x1-1/10,\%x2-1/10])\bound{sol }\free{gbs }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch14}
+\begin{paste}{ugProblemGeneticPageFull14}{ugProblemGeneticPageEmpty14}
+\pastebutton{ugProblemGeneticPageFull14}{\hidepaste}
+\tab{5}\spadcommand{e : A := represents reverse (map(rhs, sol.1) :: List FRAC INT)\bound{e }\free{A sol }}
+\indentrel{3}\begin{verbatim}
+         2      2       1       1
+   (14)  Ä ab + Ä aB + ÄÄ Ab + ÄÄ AB
+         5      5      10      10
+Type: AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty14}
+\begin{paste}{ugProblemGeneticPageEmpty14}{ugProblemGeneticPagePatch14}
+\pastebutton{ugProblemGeneticPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{e : A := represents reverse (map(rhs, sol.1) :: List FRAC INT)\bound{e }\free{A sol }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPagePatch15}
+\begin{paste}{ugProblemGeneticPageFull15}{ugProblemGeneticPageEmpty15}
+\pastebutton{ugProblemGeneticPageFull15}{\hidepaste}
+\tab{5}\spadcommand{e*e-e\free{e }}
+\indentrel{3}\begin{verbatim}
+   (15)  0
+Type: AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX])
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGeneticPageEmpty15}
+\begin{paste}{ugProblemGeneticPageEmpty15}{ugProblemGeneticPagePatch15}
+\pastebutton{ugProblemGeneticPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{e*e-e\free{e }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPagePatch1}
+\begin{paste}{ugxProblemOnePolPageFull1}{ugxProblemOnePolPageEmpty1}
+\pastebutton{ugxProblemOnePolPageFull1}{\hidepaste}
+\tab{5}\spadcommand{solve(x**3 = 8,x)}
+\indentrel{3}\begin{verbatim}
+               2
+   (1)  [x= 2,x  + 2x + 4= 0]
+        Type: List Equation Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPageEmpty1}
+\begin{paste}{ugxProblemOnePolPageEmpty1}{ugxProblemOnePolPagePatch1}
+\pastebutton{ugxProblemOnePolPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{solve(x**3 = 8,x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPagePatch2}
+\begin{paste}{ugxProblemOnePolPageFull2}{ugxProblemOnePolPageEmpty2}
+\pastebutton{ugxProblemOnePolPageFull2}{\hidepaste}
+\tab{5}\spadcommand{radicalSolve(x**3 = 8,x)}
+\indentrel{3}\begin{verbatim}
+               ÚÄÄÄ¿         ÚÄÄÄ¿
+   (2)  [x= - \³- 3  - 1,x= \³- 3  - 1,x= 2]
+                 Type: List Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPageEmpty2}
+\begin{paste}{ugxProblemOnePolPageEmpty2}{ugxProblemOnePolPagePatch2}
+\pastebutton{ugxProblemOnePolPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{radicalSolve(x**3 = 8,x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPagePatch3}
+\begin{paste}{ugxProblemOnePolPageFull3}{ugxProblemOnePolPageEmpty3}
+\pastebutton{ugxProblemOnePolPageFull3}{\hidepaste}
+\tab{5}\spadcommand{solve(x**4 - 10*x**3 + 35*x**2 - 50*x + 25,.0001)}
+\indentrel{3}\begin{verbatim}
+   (3)  [x= 3.6180114746 09375,x= 1.3819885253 90625]
+                   Type: List Equation Polynomial Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPageEmpty3}
+\begin{paste}{ugxProblemOnePolPageEmpty3}{ugxProblemOnePolPagePatch3}
+\pastebutton{ugxProblemOnePolPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{solve(x**4 - 10*x**3 + 35*x**2 - 50*x + 25,.0001)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPagePatch4}
+\begin{paste}{ugxProblemOnePolPageFull4}{ugxProblemOnePolPageEmpty4}
+\pastebutton{ugxProblemOnePolPageFull4}{\hidepaste}
+\tab{5}\spadcommand{solve(x**3-2,1/1000)}
+\indentrel{3}\begin{verbatim}
+            2581
+   (4)  [x= ÄÄÄÄ]
+            2048
+        Type: List Equation Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPageEmpty4}
+\begin{paste}{ugxProblemOnePolPageEmpty4}{ugxProblemOnePolPagePatch4}
+\pastebutton{ugxProblemOnePolPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{solve(x**3-2,1/1000)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPagePatch5}
+\begin{paste}{ugxProblemOnePolPageFull5}{ugxProblemOnePolPageEmpty5}
+\pastebutton{ugxProblemOnePolPageFull5}{\hidepaste}
+\tab{5}\spadcommand{complexSolve(x**3-2,.0001)}
+\indentrel{3}\begin{verbatim}
+   (5)
+   [x= 1.2599182128 90625,
+    x= - 0.6298943279 5395613131 - 1.0910949707 03125 %i,
+    x= - 0.6298943279 5395613131 + 1.0910949707 03125 %i]
+           Type: List Equation Polynomial Complex Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPageEmpty5}
+\begin{paste}{ugxProblemOnePolPageEmpty5}{ugxProblemOnePolPagePatch5}
+\pastebutton{ugxProblemOnePolPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{complexSolve(x**3-2,.0001)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPagePatch6}
+\begin{paste}{ugxProblemOnePolPageFull6}{ugxProblemOnePolPageEmpty6}
+\pastebutton{ugxProblemOnePolPageFull6}{\hidepaste}
+\tab{5}\spadcommand{complexSolve(x**2-2*\%i+1,1/100)}
+\indentrel{3}\begin{verbatim}
+              13028925   325       13028925   325
+   (6)  [x= - ÄÄÄÄÄÄÄÄ - ÄÄÄ %i,x= ÄÄÄÄÄÄÄÄ + ÄÄÄ %i]
+              16777216   256       16777216   256
+Type: List Equation Polynomial Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPageEmpty6}
+\begin{paste}{ugxProblemOnePolPageEmpty6}{ugxProblemOnePolPagePatch6}
+\pastebutton{ugxProblemOnePolPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{complexSolve(x**2-2*\%i+1,1/100)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPagePatch7}
+\begin{paste}{ugxProblemOnePolPageFull7}{ugxProblemOnePolPageEmpty7}
+\pastebutton{ugxProblemOnePolPageFull7}{\hidepaste}
+\tab{5}\spadcommand{radicalSolve(1/x**3 + 1/x**2 + 1/x = 0,x)}
+\indentrel{3}\begin{verbatim}
+               ÚÄÄÄ¿         ÚÄÄÄ¿
+            - \³- 3  - 1    \³- 3  - 1
+   (7)  [x= ÄÄÄÄÄÄÄÄÄÄÄÄ,x= ÄÄÄÄÄÄÄÄÄÄ]
+                  2              2
+                 Type: List Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemOnePolPageEmpty7}
+\begin{paste}{ugxProblemOnePolPageEmpty7}{ugxProblemOnePolPagePatch7}
+\pastebutton{ugxProblemOnePolPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{radicalSolve(1/x**3 + 1/x**2 + 1/x = 0,x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch1}
+\begin{paste}{ugProblemNumericPageFull1}{ugProblemNumericPageEmpty1}
+\pastebutton{ugProblemNumericPageFull1}{\hidepaste}
+\tab{5}\spadcommand{exp(3.1)}
+\indentrel{3}\begin{verbatim}
+   (1)  22.1979512814 41633405
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty1}
+\begin{paste}{ugProblemNumericPageEmpty1}{ugProblemNumericPagePatch1}
+\pastebutton{ugProblemNumericPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{exp(3.1)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch2}
+\begin{paste}{ugProblemNumericPageFull2}{ugProblemNumericPageEmpty2}
+\pastebutton{ugProblemNumericPageFull2}{\hidepaste}
+\tab{5}\spadcommand{exp(3.1 + 4.5 * \%i)}
+\indentrel{3}\begin{verbatim}
+   (2)
+   - 4.6792348860 969899118 - 21.6991659280 71731864 %i
+                                    Type: Complex Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty2}
+\begin{paste}{ugProblemNumericPageEmpty2}{ugProblemNumericPagePatch2}
+\pastebutton{ugProblemNumericPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{exp(3.1 + 4.5 * \%i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch3}
+\begin{paste}{ugProblemNumericPageFull3}{ugProblemNumericPageEmpty3}
+\pastebutton{ugProblemNumericPageFull3}{\hidepaste}
+\tab{5}\spadcommand{r: DFLOAT := 3.1; t: DFLOAT := 4.5; exp(r + t*\%i)}
+\indentrel{3}\begin{verbatim}
+   (3)  - 4.6792348860969906 - 21.699165928071732%i
+                              Type: Complex DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty3}
+\begin{paste}{ugProblemNumericPageEmpty3}{ugProblemNumericPagePatch3}
+\pastebutton{ugProblemNumericPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{r: DFLOAT := 3.1; t: DFLOAT := 4.5; exp(r + t*\%i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch4}
+\begin{paste}{ugProblemNumericPageFull4}{ugProblemNumericPageEmpty4}
+\pastebutton{ugProblemNumericPageFull4}{\hidepaste}
+\tab{5}\spadcommand{exp(3.1::DFLOAT + 4.5::DFLOAT * \%i)}
+\indentrel{3}\begin{verbatim}
+   (4)  - 4.6792348860969906 - 21.699165928071732%i
+                              Type: Complex DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty4}
+\begin{paste}{ugProblemNumericPageEmpty4}{ugProblemNumericPagePatch4}
+\pastebutton{ugProblemNumericPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{exp(3.1::DFLOAT + 4.5::DFLOAT * \%i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch5}
+\begin{paste}{ugProblemNumericPageFull5}{ugProblemNumericPageEmpty5}
+\pastebutton{ugProblemNumericPageFull5}{\hidepaste}
+\tab{5}\spadcommand{Gamma(0.5)**2}
+\indentrel{3}\begin{verbatim}
+   (5)  3.14159265358979
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty5}
+\begin{paste}{ugProblemNumericPageEmpty5}{ugProblemNumericPagePatch5}
+\pastebutton{ugProblemNumericPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{Gamma(0.5)**2}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch6}
+\begin{paste}{ugProblemNumericPageFull6}{ugProblemNumericPageEmpty6}
+\pastebutton{ugProblemNumericPageFull6}{\hidepaste}
+\tab{5}\spadcommand{a := 2.1; b := 1.1; besselI(a + \%i*b, b*a + 1)}
+\indentrel{3}\begin{verbatim}
+   (6)  2.489482417547372 - 2.3658460381468371%i
+                              Type: Complex DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty6}
+\begin{paste}{ugProblemNumericPageEmpty6}{ugProblemNumericPagePatch6}
+\pastebutton{ugProblemNumericPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{a := 2.1; b := 1.1; besselI(a + \%i*b, b*a + 1)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch7}
+\begin{paste}{ugProblemNumericPageFull7}{ugProblemNumericPageEmpty7}
+\pastebutton{ugProblemNumericPageFull7}{\hidepaste}
+\tab{5}\spadcommand{[chebyshevT(i, z) for i in 0..5]}
+\indentrel{3}\begin{verbatim}
+   (7)
+          2       3        4     2        5      3
+   [1,z,2z  - 1,4z  - 3z,8z  - 8z  + 1,16z  - 20z  + 5z]
+                          Type: List Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty7}
+\begin{paste}{ugProblemNumericPageEmpty7}{ugProblemNumericPagePatch7}
+\pastebutton{ugProblemNumericPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{[chebyshevT(i, z) for i in 0..5]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch8}
+\begin{paste}{ugProblemNumericPageFull8}{ugProblemNumericPageEmpty8}
+\pastebutton{ugProblemNumericPageFull8}{\hidepaste}
+\tab{5}\spadcommand{chebyshevT(3, 5.0 + 6.0*\%i)}
+\indentrel{3}\begin{verbatim}
+   (8)  - 1675.0 + 918.0 %i
+                                    Type: Complex Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty8}
+\begin{paste}{ugProblemNumericPageEmpty8}{ugProblemNumericPagePatch8}
+\pastebutton{ugProblemNumericPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{chebyshevT(3, 5.0 + 6.0*\%i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch9}
+\begin{paste}{ugProblemNumericPageFull9}{ugProblemNumericPageEmpty9}
+\pastebutton{ugProblemNumericPageFull9}{\hidepaste}
+\tab{5}\spadcommand{chebyshevT(3, 5.0::DoubleFloat)}
+\indentrel{3}\begin{verbatim}
+   (9)  485.0
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty9}
+\begin{paste}{ugProblemNumericPageEmpty9}{ugProblemNumericPagePatch9}
+\pastebutton{ugProblemNumericPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{chebyshevT(3, 5.0::DoubleFloat)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch10}
+\begin{paste}{ugProblemNumericPageFull10}{ugProblemNumericPageEmpty10}
+\pastebutton{ugProblemNumericPageFull10}{\hidepaste}
+\tab{5}\spadcommand{[chebyshevU(i, z) for i in 0..5]}
+\indentrel{3}\begin{verbatim}
+   (10)
+             2        3          4      2
+   [1, 2z, 4z  - 1, 8z  - 4z, 16z  - 12z  + 1,
+       5      3
+    32z  - 32z  + 6z]
+                          Type: List Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty10}
+\begin{paste}{ugProblemNumericPageEmpty10}{ugProblemNumericPagePatch10}
+\pastebutton{ugProblemNumericPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{[chebyshevU(i, z) for i in 0..5]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch11}
+\begin{paste}{ugProblemNumericPageFull11}{ugProblemNumericPageEmpty11}
+\pastebutton{ugProblemNumericPageFull11}{\hidepaste}
+\tab{5}\spadcommand{chebyshevU(3, 0.2)}
+\indentrel{3}\begin{verbatim}
+   (11)  - 0.736
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty11}
+\begin{paste}{ugProblemNumericPageEmpty11}{ugProblemNumericPagePatch11}
+\pastebutton{ugProblemNumericPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{chebyshevU(3, 0.2)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch12}
+\begin{paste}{ugProblemNumericPageFull12}{ugProblemNumericPageEmpty12}
+\pastebutton{ugProblemNumericPageFull12}{\hidepaste}
+\tab{5}\spadcommand{[hermiteH(i, z) for i in 0..5]}
+\indentrel{3}\begin{verbatim}
+   (12)
+             2        3           4      2
+   [1, 2z, 4z  - 2, 8z  - 12z, 16z  - 48z  + 12,
+       5       3
+    32z  - 160z  + 120z]
+                          Type: List Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty12}
+\begin{paste}{ugProblemNumericPageEmpty12}{ugProblemNumericPagePatch12}
+\pastebutton{ugProblemNumericPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{[hermiteH(i, z) for i in 0..5]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch13}
+\begin{paste}{ugProblemNumericPageFull13}{ugProblemNumericPageEmpty13}
+\pastebutton{ugProblemNumericPageFull13}{\hidepaste}
+\tab{5}\spadcommand{hermiteH(100, 1.0)}
+\indentrel{3}\begin{verbatim}
+   (13)  - 0.1448706729 337934088 E 93
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty13}
+\begin{paste}{ugProblemNumericPageEmpty13}{ugProblemNumericPagePatch13}
+\pastebutton{ugProblemNumericPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{hermiteH(100, 1.0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch14}
+\begin{paste}{ugProblemNumericPageFull14}{ugProblemNumericPageEmpty14}
+\pastebutton{ugProblemNumericPageFull14}{\hidepaste}
+\tab{5}\spadcommand{[laguerreL(i, z) for i in 0..4]}
+\indentrel{3}\begin{verbatim}
+   (14)
+                 2              3     2
+   [1, - z + 1, z  - 4z + 2, - z  + 9z  - 18z + 6,
+     4      3      2
+    z  - 16z  + 72z  - 96z + 24]
+                          Type: List Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty14}
+\begin{paste}{ugProblemNumericPageEmpty14}{ugProblemNumericPagePatch14}
+\pastebutton{ugProblemNumericPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{[laguerreL(i, z) for i in 0..4]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch15}
+\begin{paste}{ugProblemNumericPageFull15}{ugProblemNumericPageEmpty15}
+\pastebutton{ugProblemNumericPageFull15}{\hidepaste}
+\tab{5}\spadcommand{laguerreL(4, 1.2)}
+\indentrel{3}\begin{verbatim}
+   (15)  - 13.0944
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty15}
+\begin{paste}{ugProblemNumericPageEmpty15}{ugProblemNumericPagePatch15}
+\pastebutton{ugProblemNumericPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{laguerreL(4, 1.2)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch16}
+\begin{paste}{ugProblemNumericPageFull16}{ugProblemNumericPageEmpty16}
+\pastebutton{ugProblemNumericPageFull16}{\hidepaste}
+\tab{5}\spadcommand{[laguerreL(j, 3, z) for j in 0..4]}
+\indentrel{3}\begin{verbatim}
+   (16)
+       3     2               2
+   [- z  + 9z  - 18z + 6,- 3z  + 18z - 18,- 6z + 18,- 6,0]
+                          Type: List Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty16}
+\begin{paste}{ugProblemNumericPageEmpty16}{ugProblemNumericPagePatch16}
+\pastebutton{ugProblemNumericPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{[laguerreL(j, 3, z) for j in 0..4]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch17}
+\begin{paste}{ugProblemNumericPageFull17}{ugProblemNumericPageEmpty17}
+\pastebutton{ugProblemNumericPageFull17}{\hidepaste}
+\tab{5}\spadcommand{laguerreL(1, 3, 2.1)}
+\indentrel{3}\begin{verbatim}
+   (17)  6.57
+                                            Type: Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty17}
+\begin{paste}{ugProblemNumericPageEmpty17}{ugProblemNumericPagePatch17}
+\pastebutton{ugProblemNumericPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{laguerreL(1, 3, 2.1)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch18}
+\begin{paste}{ugProblemNumericPageFull18}{ugProblemNumericPageEmpty18}
+\pastebutton{ugProblemNumericPageFull18}{\hidepaste}
+\tab{5}\spadcommand{[legendreP(i,z) for i in 0..5]}
+\indentrel{3}\begin{verbatim}
+   (18)
+          3  2   1  5  3   3    35  4   15  2   3
+   [1, z, Ä z  - Ä, Ä z  - Ä z, ÄÄ z  - ÄÄ z  + Ä,
+          2      2  2      2     8       4      8
+    63  5   35  3   15
+    ÄÄ z  - ÄÄ z  + ÄÄ z]
+     8       4       8
+                 Type: List Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty18}
+\begin{paste}{ugProblemNumericPageEmpty18}{ugProblemNumericPagePatch18}
+\pastebutton{ugProblemNumericPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{[legendreP(i,z) for i in 0..5]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch19}
+\begin{paste}{ugProblemNumericPageFull19}{ugProblemNumericPageEmpty19}
+\pastebutton{ugProblemNumericPageFull19}{\hidepaste}
+\tab{5}\spadcommand{legendreP(3, 3.0*\%i)}
+\indentrel{3}\begin{verbatim}
+   (19)  - 72.0 %i
+                                    Type: Complex Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty19}
+\begin{paste}{ugProblemNumericPageEmpty19}{ugProblemNumericPagePatch19}
+\pastebutton{ugProblemNumericPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{legendreP(3, 3.0*\%i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch20}
+\begin{paste}{ugProblemNumericPageFull20}{ugProblemNumericPageEmpty20}
+\pastebutton{ugProblemNumericPageFull20}{\hidepaste}
+\tab{5}\spadcommand{bernoulliB(3, z)}
+\indentrel{3}\begin{verbatim}
+          3   3  2   1
+   (20)  z  - Ä z  + Ä z
+              2      2
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty20}
+\begin{paste}{ugProblemNumericPageEmpty20}{ugProblemNumericPagePatch20}
+\pastebutton{ugProblemNumericPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{bernoulliB(3, z)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch21}
+\begin{paste}{ugProblemNumericPageFull21}{ugProblemNumericPageEmpty21}
+\pastebutton{ugProblemNumericPageFull21}{\hidepaste}
+\tab{5}\spadcommand{bernoulliB(3, 0.7 + 0.4 * \%i)}
+\indentrel{3}\begin{verbatim}
+   (21)  - 0.138 - 0.116 %i
+                                    Type: Complex Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty21}
+\begin{paste}{ugProblemNumericPageEmpty21}{ugProblemNumericPagePatch21}
+\pastebutton{ugProblemNumericPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{bernoulliB(3, 0.7 + 0.4 * \%i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch22}
+\begin{paste}{ugProblemNumericPageFull22}{ugProblemNumericPageEmpty22}
+\pastebutton{ugProblemNumericPageFull22}{\hidepaste}
+\tab{5}\spadcommand{eulerE(3, z)}
+\indentrel{3}\begin{verbatim}
+          3   3  2   1
+   (22)  z  - Ä z  + Ä
+              2      4
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty22}
+\begin{paste}{ugProblemNumericPageEmpty22}{ugProblemNumericPagePatch22}
+\pastebutton{ugProblemNumericPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{eulerE(3, z)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch23}
+\begin{paste}{ugProblemNumericPageFull23}{ugProblemNumericPageEmpty23}
+\pastebutton{ugProblemNumericPageFull23}{\hidepaste}
+\tab{5}\spadcommand{eulerE(3, 0.7 + 0.4 * \%i)}
+\indentrel{3}\begin{verbatim}
+   (23)  - 0.238 - 0.316 %i
+                                    Type: Complex Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty23}
+\begin{paste}{ugProblemNumericPageEmpty23}{ugProblemNumericPagePatch23}
+\pastebutton{ugProblemNumericPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{eulerE(3, 0.7 + 0.4 * \%i)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch24}
+\begin{paste}{ugProblemNumericPageFull24}{ugProblemNumericPageEmpty24}
+\pastebutton{ugProblemNumericPageFull24}{\hidepaste}
+\tab{5}\spadcommand{cyclotomic(3, z)}
+\indentrel{3}\begin{verbatim}
+          2
+   (24)  z  + z + 1
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty24}
+\begin{paste}{ugProblemNumericPageEmpty24}{ugProblemNumericPagePatch24}
+\pastebutton{ugProblemNumericPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{cyclotomic(3, z)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch25}
+\begin{paste}{ugProblemNumericPageFull25}{ugProblemNumericPageEmpty25}
+\pastebutton{ugProblemNumericPageFull25}{\hidepaste}
+\tab{5}\spadcommand{cyclotomic(3, (-1.0 + 0.0 * \%i)**(2/3))}
+\indentrel{3}\begin{verbatim}
+   (25)  0.0
+                                    Type: Complex Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty25}
+\begin{paste}{ugProblemNumericPageEmpty25}{ugProblemNumericPagePatch25}
+\pastebutton{ugProblemNumericPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{cyclotomic(3, (-1.0 + 0.0 * \%i)**(2/3))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch26}
+\begin{paste}{ugProblemNumericPageFull26}{ugProblemNumericPageEmpty26}
+\pastebutton{ugProblemNumericPageFull26}{\hidepaste}
+\tab{5}\spadgraph{draw((x,y)+-> real exp complex(x,y), -2..2, -2*\%pi..2*\%pi, colorFunction == (x, y) +-> imag exp complex(x,y), title=="exp(x+\%i*y)", style=="smooth")}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage26.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage26}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty26}
+\begin{paste}{ugProblemNumericPageEmpty26}{ugProblemNumericPagePatch26}
+\pastebutton{ugProblemNumericPageEmpty26}{\showpaste}
+\tab{5}\spadgraph{draw((x,y)+-> real exp complex(x,y), -2..2, -2*\%pi..2*\%pi, colorFunction == (x, y) +-> imag exp complex(x,y), title=="exp(x+\%i*y)", style=="smooth")}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch27}
+\begin{paste}{ugProblemNumericPageFull27}{ugProblemNumericPageEmpty27}
+\pastebutton{ugProblemNumericPageFull27}{\hidepaste}
+\tab{5}\spadgraph{vp := draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction==(x,y) +->argument atan complex(x,y), title=="atan(x+\%i*y)", style=="shade"); rotate(vp,-160,-45); vp}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage27.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage27}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty27}
+\begin{paste}{ugProblemNumericPageEmpty27}{ugProblemNumericPagePatch27}
+\pastebutton{ugProblemNumericPageEmpty27}{\showpaste}
+\tab{5}\spadgraph{vp := draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction==(x,y) +->argument atan complex(x,y), title=="atan(x+\%i*y)", style=="shade"); rotate(vp,-160,-45); vp}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch28}
+\begin{paste}{ugProblemNumericPageFull28}{ugProblemNumericPageEmpty28}
+\pastebutton{ugProblemNumericPageFull28}{\hidepaste}
+\tab{5}\spadgraph{draw((x,y) +-> max(min(real Gamma complex(x,y),4),-4), -\%pi..\%pi, -\%pi..\%pi, style=="shade", colorFunction == (x,y) +-> argument Gamma complex(x,y), title == "Gamma(x+\%i*y)", var1Steps == 50, var2Steps== 50)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage28.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage28}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty28}
+\begin{paste}{ugProblemNumericPageEmpty28}{ugProblemNumericPagePatch28}
+\pastebutton{ugProblemNumericPageEmpty28}{\showpaste}
+\tab{5}\spadgraph{draw((x,y) +-> max(min(real Gamma complex(x,y),4),-4), -\%pi..\%pi, -\%pi..\%pi, style=="shade", colorFunction == (x,y) +-> argument Gamma complex(x,y), title == "Gamma(x+\%i*y)", var1Steps == 50, var2Steps== 50)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch29}
+\begin{paste}{ugProblemNumericPageFull29}{ugProblemNumericPageEmpty29}
+\pastebutton{ugProblemNumericPageFull29}{\hidepaste}
+\tab{5}\spadgraph{draw(Beta(x,y)/100, x=-1.6..1.7, y = -1.6..1.7, style=="shade", title=="Beta(x,y)", var1Steps==40, var2Steps==40)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage29.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage29}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty29}
+\begin{paste}{ugProblemNumericPageEmpty29}{ugProblemNumericPagePatch29}
+\pastebutton{ugProblemNumericPageEmpty29}{\showpaste}
+\tab{5}\spadgraph{draw(Beta(x,y)/100, x=-1.6..1.7, y = -1.6..1.7, style=="shade", title=="Beta(x,y)", var1Steps==40, var2Steps==40)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch30}
+\begin{paste}{ugProblemNumericPageFull30}{ugProblemNumericPageEmpty30}
+\pastebutton{ugProblemNumericPageFull30}{\hidepaste}
+\tab{5}\spadgraph{draw((alpha,x) +-> min(max(besselJ(alpha, x+8), -6), 6), -6..4, -6..6, title=="besselJ(alpha,x)", style=="shade", var1Steps==40, var2Steps==40)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage30.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage30}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty30}
+\begin{paste}{ugProblemNumericPageEmpty30}{ugProblemNumericPagePatch30}
+\pastebutton{ugProblemNumericPageEmpty30}{\showpaste}
+\tab{5}\spadgraph{draw((alpha,x) +-> min(max(besselJ(alpha, x+8), -6), 6), -6..4, -6..6, title=="besselJ(alpha,x)", style=="shade", var1Steps==40, var2Steps==40)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch31}
+\begin{paste}{ugProblemNumericPageFull31}{ugProblemNumericPageEmpty31}
+\pastebutton{ugProblemNumericPageFull31}{\hidepaste}
+\tab{5}\spadgraph{draw(besselI(alpha, 5), alpha = -12..12, unit==[5,20])}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage31.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage31}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty31}
+\begin{paste}{ugProblemNumericPageEmpty31}{ugProblemNumericPagePatch31}
+\pastebutton{ugProblemNumericPageEmpty31}{\showpaste}
+\tab{5}\spadgraph{draw(besselI(alpha, 5), alpha = -12..12, unit==[5,20])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch32}
+\begin{paste}{ugProblemNumericPageFull32}{ugProblemNumericPageEmpty32}
+\pastebutton{ugProblemNumericPageFull32}{\hidepaste}
+\tab{5}\spadgraph{draw((x,y) +-> real besselI(complex(x/20, y/20),5), -60..60, -60..60, colorFunction == (x,y)+-> argument besselI(complex(x/20,y/20),5), title=="besselI(x+i*y,5)", style=="shade")}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage32.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage32}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty32}
+\begin{paste}{ugProblemNumericPageEmpty32}{ugProblemNumericPagePatch32}
+\pastebutton{ugProblemNumericPageEmpty32}{\showpaste}
+\tab{5}\spadgraph{draw((x,y) +-> real besselI(complex(x/20, y/20),5), -60..60, -60..60, colorFunction == (x,y)+-> argument besselI(complex(x/20,y/20),5), title=="besselI(x+i*y,5)", style=="shade")}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPagePatch33}
+\begin{paste}{ugProblemNumericPageFull33}{ugProblemNumericPageEmpty33}
+\pastebutton{ugProblemNumericPageFull33}{\hidepaste}
+\tab{5}\spadcommand{all}
+\indentrel{3}\begin{verbatim}
+   (33)  all
+                                     Type: Variable all
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemNumericPageEmpty33}
+\begin{paste}{ugProblemNumericPageEmpty33}{ugProblemNumericPagePatch33}
+\pastebutton{ugProblemNumericPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{all}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLinSysPagePatch1}
+\begin{paste}{ugxProblemLinSysPageFull1}{ugxProblemLinSysPageEmpty1}
+\pastebutton{ugxProblemLinSysPageFull1}{\hidepaste}
+\tab{5}\spadcommand{solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])}
+\indentrel{3}\begin{verbatim}
+   (1)  [[x= - 1,y= 2,z= 7]]
+   Type: List List Equation Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLinSysPageEmpty1}
+\begin{paste}{ugxProblemLinSysPageEmpty1}{ugxProblemLinSysPagePatch1}
+\pastebutton{ugxProblemLinSysPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLinSysPagePatch2}
+\begin{paste}{ugxProblemLinSysPageFull2}{ugxProblemLinSysPageEmpty2}
+\pastebutton{ugxProblemLinSysPageFull2}{\hidepaste}
+\tab{5}\spadcommand{solve([x+2*y+3*z=2,2*x+3*y+4*z=2,3*x+4*y+5*z=2],[x,y,z])}
+\indentrel{3}\begin{verbatim}
+   (2)  [[x= %Y - 2,y= - 2%Y + 2,z= %Y]]
+   Type: List List Equation Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLinSysPageEmpty2}
+\begin{paste}{ugxProblemLinSysPageEmpty2}{ugxProblemLinSysPagePatch2}
+\pastebutton{ugxProblemLinSysPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{solve([x+2*y+3*z=2,2*x+3*y+4*z=2,3*x+4*y+5*z=2],[x,y,z])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLinSysPagePatch3}
+\begin{paste}{ugxProblemLinSysPageFull3}{ugxProblemLinSysPageEmpty3}
+\pastebutton{ugxProblemLinSysPageFull3}{\hidepaste}
+\tab{5}\spadcommand{solve([[1,1,1],[3,-2,1],[1,2,2]],[8,0,17])}
+\indentrel{3}\begin{verbatim}
+   (3)  [particular= [- 1,2,7],basis= [[0,0,0]]]
+Type: Record(particular: Union(Vector Fraction Integer,"failed"),basis: List Vector Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLinSysPageEmpty3}
+\begin{paste}{ugxProblemLinSysPageEmpty3}{ugxProblemLinSysPagePatch3}
+\pastebutton{ugxProblemLinSysPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{solve([[1,1,1],[3,-2,1],[1,2,2]],[8,0,17])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLinSysPagePatch4}
+\begin{paste}{ugxProblemLinSysPageFull4}{ugxProblemLinSysPageEmpty4}
+\pastebutton{ugxProblemLinSysPageFull4}{\hidepaste}
+\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,2,2])}
+\indentrel{3}\begin{verbatim}
+   (4)  [particular= [- 2,2,0],basis= [[1,- 2,1]]]
+Type: Record(particular: Union(Vector Fraction Integer,"failed"),basis: List Vector Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLinSysPageEmpty4}
+\begin{paste}{ugxProblemLinSysPageEmpty4}{ugxProblemLinSysPagePatch4}
+\pastebutton{ugxProblemLinSysPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,2,2])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLinSysPagePatch5}
+\begin{paste}{ugxProblemLinSysPageFull5}{ugxProblemLinSysPageEmpty5}
+\pastebutton{ugxProblemLinSysPageFull5}{\hidepaste}
+\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,3,2])}
+\indentrel{3}\begin{verbatim}
+   (5)  [particular= "failed",basis= [[1,- 2,1]]]
+Type: Record(particular: Union(Vector Fraction Integer,"failed"),basis: List Vector Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLinSysPageEmpty5}
+\begin{paste}{ugxProblemLinSysPageEmpty5}{ugxProblemLinSysPagePatch5}
+\pastebutton{ugxProblemLinSysPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,3,2])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLinSysPagePatch6}
+\begin{paste}{ugxProblemLinSysPageFull6}{ugxProblemLinSysPageEmpty6}
+\pastebutton{ugxProblemLinSysPageFull6}{\hidepaste}
+\tab{5}\spadcommand{nullSpace([[1,2,3],[2,3,4],[3,4,5]])}
+\indentrel{3}\begin{verbatim}
+   (6)  [[1,- 2,1]]
+                              Type: List Vector Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLinSysPageEmpty6}
+\begin{paste}{ugxProblemLinSysPageEmpty6}{ugxProblemLinSysPagePatch6}
+\pastebutton{ugxProblemLinSysPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{nullSpace([[1,2,3],[2,3,4],[3,4,5]])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPagePatch1}
+\begin{paste}{ugxProblemPolSysPageFull1}{ugxProblemPolSysPageEmpty1}
+\pastebutton{ugxProblemPolSysPageFull1}{\hidepaste}
+\tab{5}\spadcommand{solve([3*x**3 + y + 1,y**2 -4],[x,y])}
+\indentrel{3}\begin{verbatim}
+   (1)
+                     2
+   [[x= - 1,y= 2], [x  - x + 1= 0,y= 2],
+       3
+    [3x  - 1= 0,y= - 2]]
+   Type: List List Equation Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPageEmpty1}
+\begin{paste}{ugxProblemPolSysPageEmpty1}{ugxProblemPolSysPagePatch1}
+\pastebutton{ugxProblemPolSysPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{solve([3*x**3 + y + 1,y**2 -4],[x,y])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPagePatch2}
+\begin{paste}{ugxProblemPolSysPageFull2}{ugxProblemPolSysPageEmpty2}
+\pastebutton{ugxProblemPolSysPageFull2}{\hidepaste}
+\tab{5}\spadcommand{solve([x = y**2-19,y = z**2+x+3,z = 3*x],[x,y,z])}
+\indentrel{3}\begin{verbatim}
+   (2)
+               2
+        z    3z  + z + 9   4     3      2
+   [[x= Ä,y= ÄÄÄÄÄÄÄÄÄÄÄ,9z  + 6z  + 55z  + 15z - 90= 0]]
+        3         3
+   Type: List List Equation Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPageEmpty2}
+\begin{paste}{ugxProblemPolSysPageEmpty2}{ugxProblemPolSysPagePatch2}
+\pastebutton{ugxProblemPolSysPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{solve([x = y**2-19,y = z**2+x+3,z = 3*x],[x,y,z])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPagePatch3}
+\begin{paste}{ugxProblemPolSysPageFull3}{ugxProblemPolSysPageEmpty3}
+\pastebutton{ugxProblemPolSysPageFull3}{\hidepaste}
+\tab{5}\spadcommand{radicalSolve([3*x**3 + y + 1,y**2 -4],[x,y])}
+\indentrel{3}\begin{verbatim}
+   (3)
+         ÚÄÄÄ¿                   ÚÄÄÄ¿
+        \³- 3  + 1            - \³- 3  + 1
+   [[x= ÄÄÄÄÄÄÄÄÄÄ,y= 2], [x= ÄÄÄÄÄÄÄÄÄÄÄÄ,y= 2],
+             2                      2
+           ÚÄÄÄ¿ ÚÄ¿
+        - \³- 1 \³3  - 1
+    [x= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,y= - 2],
+               3ÚÄ¿
+              2\³3
+         ÚÄÄÄ¿ ÚÄ¿
+        \³- 1 \³3  - 1                1
+    [x= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,y= - 2], [x= ÄÄÄÄ,y= - 2],
+              3ÚÄ¿                  3ÚÄ¿
+             2\³3                   \³3
+    [x= - 1,y= 2]]
+            Type: List List Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPageEmpty3}
+\begin{paste}{ugxProblemPolSysPageEmpty3}{ugxProblemPolSysPagePatch3}
+\pastebutton{ugxProblemPolSysPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{radicalSolve([3*x**3 + y + 1,y**2 -4],[x,y])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPagePatch4}
+\begin{paste}{ugxProblemPolSysPageFull4}{ugxProblemPolSysPageEmpty4}
+\pastebutton{ugxProblemPolSysPageFull4}{\hidepaste}
+\tab{5}\spadcommand{solve([x**2*y - 1,x*y**2 - 2],.01)}
+\indentrel{3}\begin{verbatim}
+   (4)  [[y= 1.5859375,x= 0.79296875]]
+              Type: List List Equation Polynomial Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPageEmpty4}
+\begin{paste}{ugxProblemPolSysPageEmpty4}{ugxProblemPolSysPagePatch4}
+\pastebutton{ugxProblemPolSysPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{solve([x**2*y - 1,x*y**2 - 2],.01)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPagePatch5}
+\begin{paste}{ugxProblemPolSysPageFull5}{ugxProblemPolSysPageEmpty5}
+\pastebutton{ugxProblemPolSysPageFull5}{\hidepaste}
+\tab{5}\spadcommand{complexSolve([x**2*y - 1,x*y**2 - 2],1/1000)}
+\indentrel{3}\begin{verbatim}
+   (5)
+        1625    1625
+   [[y= ÄÄÄÄ,x= ÄÄÄÄ],
+        1024    2048
+
+           435445573689   1407
+     [y= - ÄÄÄÄÄÄÄÄÄÄÄÄ - ÄÄÄÄ %i,
+           549755813888   1024
+            435445573689   1407
+      x= - ÄÄÄÄÄÄÄÄÄÄÄÄÄ - ÄÄÄÄ %i]
+           1099511627776   2048
+     ,
+
+           435445573689   1407
+     [y= - ÄÄÄÄÄÄÄÄÄÄÄÄ + ÄÄÄÄ %i,
+           549755813888   1024
+            435445573689   1407
+      x= - ÄÄÄÄÄÄÄÄÄÄÄÄÄ + ÄÄÄÄ %i]
+           1099511627776   2048
+     ]
+Type: List List Equation Polynomial Complex Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPageEmpty5}
+\begin{paste}{ugxProblemPolSysPageEmpty5}{ugxProblemPolSysPagePatch5}
+\pastebutton{ugxProblemPolSysPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{complexSolve([x**2*y - 1,x*y**2 - 2],1/1000)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPagePatch6}
+\begin{paste}{ugxProblemPolSysPageFull6}{ugxProblemPolSysPageEmpty6}
+\pastebutton{ugxProblemPolSysPageFull6}{\hidepaste}
+\tab{5}\spadcommand{solve([x**2/a = a,a = a*x],.001)}
+\indentrel{3}\begin{verbatim}
+   (6)  [[x= 1.0,a= - 1.0],[x= 1.0,a= 1.0]]
+              Type: List List Equation Polynomial Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPageEmpty6}
+\begin{paste}{ugxProblemPolSysPageEmpty6}{ugxProblemPolSysPagePatch6}
+\pastebutton{ugxProblemPolSysPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{solve([x**2/a = a,a = a*x],.001)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPagePatch7}
+\begin{paste}{ugxProblemPolSysPageFull7}{ugxProblemPolSysPageEmpty7}
+\pastebutton{ugxProblemPolSysPageFull7}{\hidepaste}
+\tab{5}\spadcommand{radicalSolve([x**2/a + a + y**3 - 1,a*y + a + 1],[x,y])}
+\indentrel{3}\begin{verbatim}
+   (7)
+           ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+           ³   4     3     2
+           ³- a  + 2a  + 3a  + 3a + 1     - a - 1
+   [[x= -  ³ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ,y= ÄÄÄÄÄÄÄ],
+           ³             2                   a
+          \³            a
+         ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
+         ³   4     3     2
+         ³- a  + 2a  + 3a  + 3a + 1     - a - 1
+    [x=  ³ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ,y= ÄÄÄÄÄÄÄ]]
+         ³             2                   a
+        \³            a
+            Type: List List Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemPolSysPageEmpty7}
+\begin{paste}{ugxProblemPolSysPageEmpty7}{ugxProblemPolSysPagePatch7}
+\pastebutton{ugxProblemPolSysPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{radicalSolve([x**2/a + a + y**3 - 1,a*y + a + 1],[x,y])}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorFFPagePatch1}
+\begin{paste}{ugProblemFactorFFPageFull1}{ugProblemFactorFFPageEmpty1}
+\pastebutton{ugProblemFactorFFPageFull1}{\hidepaste}
+\tab{5}\spadcommand{u : POLY(PF(19)) :=3*x**4+2*x**2+15*x+18\bound{u }}
+\indentrel{3}\begin{verbatim}
+          4     2
+   (1)  3x  + 2x  + 15x + 18
+                         Type: Polynomial PrimeField 19
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorFFPageEmpty1}
+\begin{paste}{ugProblemFactorFFPageEmpty1}{ugProblemFactorFFPagePatch1}
+\pastebutton{ugProblemFactorFFPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{u : POLY(PF(19)) :=3*x**4+2*x**2+15*x+18\bound{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorFFPagePatch2}
+\begin{paste}{ugProblemFactorFFPageFull2}{ugProblemFactorFFPageEmpty2}
+\pastebutton{ugProblemFactorFFPageFull2}{\hidepaste}
+\tab{5}\spadcommand{factor u\free{u }}
+\indentrel{3}\begin{verbatim}
+                   3    2
+   (2)  3(x + 18)(x  + x  + 8x + 13)
+                Type: Factored Polynomial PrimeField 19
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorFFPageEmpty2}
+\begin{paste}{ugProblemFactorFFPageEmpty2}{ugProblemFactorFFPagePatch2}
+\pastebutton{ugProblemFactorFFPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{factor u\free{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorFFPagePatch3}
+\begin{paste}{ugProblemFactorFFPageFull3}{ugProblemFactorFFPageEmpty3}
+\pastebutton{ugProblemFactorFFPageFull3}{\hidepaste}
+\tab{5}\spadcommand{factor(u :: POLY FFX(PF 19,3))\free{u }}
+\indentrel{3}\begin{verbatim}
+   (3)
+                       2
+     3(x + 18)(x + 5%BC  + 3%BC + 13)
+  *
+               2                        2
+     (x + 16%BC  + 14%BC + 13)(x + 17%BC  + 2%BC + 13)
+Type: Factored Polynomial FiniteFieldExtension(PrimeField 19,3)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorFFPageEmpty3}
+\begin{paste}{ugProblemFactorFFPageEmpty3}{ugProblemFactorFFPagePatch3}
+\pastebutton{ugProblemFactorFFPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{factor(u :: POLY FFX(PF 19,3))\free{u }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesArithmeticPagePatch1}
+\begin{paste}{ugxProblemSeriesArithmeticPageFull1}{ugxProblemSeriesArithmeticPageEmpty1}
+\pastebutton{ugxProblemSeriesArithmeticPageFull1}{\hidepaste}
+\tab{5}\spadcommand{x := series x\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (1)  x
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesArithmeticPageEmpty1}
+\begin{paste}{ugxProblemSeriesArithmeticPageEmpty1}{ugxProblemSeriesArithmeticPagePatch1}
+\pastebutton{ugxProblemSeriesArithmeticPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{x := series x\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesArithmeticPagePatch2}
+\begin{paste}{ugxProblemSeriesArithmeticPageFull2}{ugxProblemSeriesArithmeticPageEmpty2}
+\pastebutton{ugxProblemSeriesArithmeticPageFull2}{\hidepaste}
+\tab{5}\spadcommand{(3 + x) / (1 + 7*x)}
+\indentrel{3}\begin{verbatim}
+   (2)
+                   2       3        4         5          6
+     3 - 20x + 140x  - 980x  + 6860x  - 48020x  + 336140x
+   + 
+               7            8             9             10
+     - 2352980x  + 16470860x  - 115296020x  + 807072140x
+   + 
+        11
+     O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesArithmeticPageEmpty2}
+\begin{paste}{ugxProblemSeriesArithmeticPageEmpty2}{ugxProblemSeriesArithmeticPagePatch2}
+\pastebutton{ugxProblemSeriesArithmeticPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{(3 + x) / (1 + 7*x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesArithmeticPagePatch3}
+\begin{paste}{ugxProblemSeriesArithmeticPageFull3}{ugxProblemSeriesArithmeticPageEmpty3}
+\pastebutton{ugxProblemSeriesArithmeticPageFull3}{\hidepaste}
+\tab{5}\spadcommand{base := 1 / (1 - x)\free{x }\bound{base }}
+\indentrel{3}\begin{verbatim}
+   (3)
+              2    3    4    5    6    7    8    9    10
+     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
+   + 
+        11
+     O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesArithmeticPageEmpty3}
+\begin{paste}{ugxProblemSeriesArithmeticPageEmpty3}{ugxProblemSeriesArithmeticPagePatch3}
+\pastebutton{ugxProblemSeriesArithmeticPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{base := 1 / (1 - x)\free{x }\bound{base }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesArithmeticPagePatch4}
+\begin{paste}{ugxProblemSeriesArithmeticPageFull4}{ugxProblemSeriesArithmeticPageEmpty4}
+\pastebutton{ugxProblemSeriesArithmeticPageFull4}{\hidepaste}
+\tab{5}\spadcommand{expon := x * base\free{x base }\bound{expon }}
+\indentrel{3}\begin{verbatim}
+   (4)
+          2    3    4    5    6    7    8    9    10    11
+     x + x  + x  + x  + x  + x  + x  + x  + x  + x   + x
+   + 
+        12
+     O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesArithmeticPageEmpty4}
+\begin{paste}{ugxProblemSeriesArithmeticPageEmpty4}{ugxProblemSeriesArithmeticPagePatch4}
+\pastebutton{ugxProblemSeriesArithmeticPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{expon := x * base\free{x base }\bound{expon }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesArithmeticPagePatch5}
+\begin{paste}{ugxProblemSeriesArithmeticPageFull5}{ugxProblemSeriesArithmeticPageEmpty5}
+\pastebutton{ugxProblemSeriesArithmeticPageFull5}{\hidepaste}
+\tab{5}\spadcommand{base ** expon\free{base expon }}
+\indentrel{3}\begin{verbatim}
+   (5)
+          2   3  3   7  4   43  5   649  6   241  7
+     1 + x  + Ä x  + Ä x  + ÄÄ x  + ÄÄÄ x  + ÄÄÄ x
+              2      3      12      120       30
+   + 
+     3706  8   85763  9   245339  10      11
+     ÄÄÄÄ x  + ÄÄÄÄÄ x  + ÄÄÄÄÄÄ x   + O(x  )
+      315       5040       10080
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesArithmeticPageEmpty5}
+\begin{paste}{ugxProblemSeriesArithmeticPageEmpty5}{ugxProblemSeriesArithmeticPagePatch5}
+\pastebutton{ugxProblemSeriesArithmeticPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{base ** expon\free{base expon }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch1}
+\begin{paste}{ugxProblemFiniteConversionPageFull1}{ugxProblemFiniteConversionPageEmpty1}
+\pastebutton{ugxProblemFiniteConversionPageFull1}{\hidepaste}
+\tab{5}\spadcommand{K := PrimeField 3\bound{K }}
+\indentrel{3}\begin{verbatim}
+   (1)  PrimeField 3
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty1}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty1}{ugxProblemFiniteConversionPagePatch1}
+\pastebutton{ugxProblemFiniteConversionPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{K := PrimeField 3\bound{K }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch2}
+\begin{paste}{ugxProblemFiniteConversionPageFull2}{ugxProblemFiniteConversionPageEmpty2}
+\pastebutton{ugxProblemFiniteConversionPageFull2}{\hidepaste}
+\tab{5}\spadcommand{(m,n) := (4,8)\bound{m n }}
+\indentrel{3}\begin{verbatim}
+   (2)  8
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty2}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty2}{ugxProblemFiniteConversionPagePatch2}
+\pastebutton{ugxProblemFiniteConversionPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{(m,n) := (4,8)\bound{m n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch3}
+\begin{paste}{ugxProblemFiniteConversionPageFull3}{ugxProblemFiniteConversionPageEmpty3}
+\pastebutton{ugxProblemFiniteConversionPageFull3}{\hidepaste}
+\tab{5}\spadcommand{Km := FiniteFieldExtension(K,m)\bound{Km }\free{K m }}
+\indentrel{3}\begin{verbatim}
+   (3)  FiniteFieldExtension(PrimeField 3,4)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty3}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty3}{ugxProblemFiniteConversionPagePatch3}
+\pastebutton{ugxProblemFiniteConversionPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{Km := FiniteFieldExtension(K,m)\bound{Km }\free{K m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch4}
+\begin{paste}{ugxProblemFiniteConversionPageFull4}{ugxProblemFiniteConversionPageEmpty4}
+\pastebutton{ugxProblemFiniteConversionPageFull4}{\hidepaste}
+\tab{5}\spadcommand{Kn := FiniteFieldExtension(K,n)\bound{Kn }\free{K n }}
+\indentrel{3}\begin{verbatim}
+   (4)  FiniteFieldExtension(PrimeField 3,8)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty4}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty4}{ugxProblemFiniteConversionPagePatch4}
+\pastebutton{ugxProblemFiniteConversionPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{Kn := FiniteFieldExtension(K,n)\bound{Kn }\free{K n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch5}
+\begin{paste}{ugxProblemFiniteConversionPageFull5}{ugxProblemFiniteConversionPageEmpty5}
+\pastebutton{ugxProblemFiniteConversionPageFull5}{\hidepaste}
+\tab{5}\spadcommand{a1 := random()$Km\bound{a1 }\free{Km }}
+\indentrel{3}\begin{verbatim}
+            3       2
+   (5)  2%BD  + 2%BD  + 2%BD + 1
+             Type: FiniteFieldExtension(PrimeField 3,4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty5}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty5}{ugxProblemFiniteConversionPagePatch5}
+\pastebutton{ugxProblemFiniteConversionPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{a1 := random()$Km\bound{a1 }\free{Km }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch6}
+\begin{paste}{ugxProblemFiniteConversionPageFull6}{ugxProblemFiniteConversionPageEmpty6}
+\pastebutton{ugxProblemFiniteConversionPageFull6}{\hidepaste}
+\tab{5}\spadcommand{b1 := random()$Km\bound{b1 }\free{Km }}
+\indentrel{3}\begin{verbatim}
+           3      2
+   (6)  %BD  + %BD  + 2%BD + 2
+             Type: FiniteFieldExtension(PrimeField 3,4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty6}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty6}{ugxProblemFiniteConversionPagePatch6}
+\pastebutton{ugxProblemFiniteConversionPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{b1 := random()$Km\bound{b1 }\free{Km }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch7}
+\begin{paste}{ugxProblemFiniteConversionPageFull7}{ugxProblemFiniteConversionPageEmpty7}
+\pastebutton{ugxProblemFiniteConversionPageFull7}{\hidepaste}
+\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a2 }\free{a1 Kn }}
+\indentrel{3}\begin{verbatim}
+            6
+   (7)  2%BE  + 1
+             Type: FiniteFieldExtension(PrimeField 3,8)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty7}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty7}{ugxProblemFiniteConversionPagePatch7}
+\pastebutton{ugxProblemFiniteConversionPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a2 }\free{a1 Kn }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch8}
+\begin{paste}{ugxProblemFiniteConversionPageFull8}{ugxProblemFiniteConversionPageEmpty8}
+\pastebutton{ugxProblemFiniteConversionPageFull8}{\hidepaste}
+\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b2 }\free{b1 Kn }}
+\indentrel{3}\begin{verbatim}
+            6       4      2
+   (8)  2%BE  + 2%BE  + %BE  + 2
+             Type: FiniteFieldExtension(PrimeField 3,8)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty8}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty8}{ugxProblemFiniteConversionPagePatch8}
+\pastebutton{ugxProblemFiniteConversionPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b2 }\free{b1 Kn }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch9}
+\begin{paste}{ugxProblemFiniteConversionPageFull9}{ugxProblemFiniteConversionPageEmpty9}
+\pastebutton{ugxProblemFiniteConversionPageFull9}{\hidepaste}
+\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a1 a2 b1 b2 Km Kn }}
+\indentrel{3}\begin{verbatim}
+   (9)  0
+             Type: FiniteFieldExtension(PrimeField 3,4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty9}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty9}{ugxProblemFiniteConversionPagePatch9}
+\pastebutton{ugxProblemFiniteConversionPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a1 a2 b1 b2 Km Kn }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch10}
+\begin{paste}{ugxProblemFiniteConversionPageFull10}{ugxProblemFiniteConversionPageEmpty10}
+\pastebutton{ugxProblemFiniteConversionPageFull10}{\hidepaste}
+\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a1 a2 b1 b2 Km Kn }}
+\indentrel{3}\begin{verbatim}
+   (10)  0
+             Type: FiniteFieldExtension(PrimeField 3,4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty10}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty10}{ugxProblemFiniteConversionPagePatch10}
+\pastebutton{ugxProblemFiniteConversionPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a1 a2 b1 b2 Km Kn }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch11}
+\begin{paste}{ugxProblemFiniteConversionPageFull11}{ugxProblemFiniteConversionPageEmpty11}
+\pastebutton{ugxProblemFiniteConversionPageFull11}{\hidepaste}
+\tab{5}\spadcommand{Km := FFCGX(K,m)\bound{Km2 }\free{K m }}
+\indentrel{3}\begin{verbatim}
+   (11)  FiniteFieldCyclicGroupExtension(PrimeField 3,4)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty11}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty11}{ugxProblemFiniteConversionPagePatch11}
+\pastebutton{ugxProblemFiniteConversionPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{Km := FFCGX(K,m)\bound{Km2 }\free{K m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch12}
+\begin{paste}{ugxProblemFiniteConversionPageFull12}{ugxProblemFiniteConversionPageEmpty12}
+\pastebutton{ugxProblemFiniteConversionPageFull12}{\hidepaste}
+\tab{5}\spadcommand{Kn := FFNBX(K,n)\bound{Kn2 }\free{K n }}
+\indentrel{3}\begin{verbatim}
+   (12)  FiniteFieldNormalBasisExtension(PrimeField 3,8)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty12}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty12}{ugxProblemFiniteConversionPagePatch12}
+\pastebutton{ugxProblemFiniteConversionPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{Kn := FFNBX(K,n)\bound{Kn2 }\free{K n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch13}
+\begin{paste}{ugxProblemFiniteConversionPageFull13}{ugxProblemFiniteConversionPageEmpty13}
+\pastebutton{ugxProblemFiniteConversionPageFull13}{\hidepaste}
+\tab{5}\spadcommand{(a1,b1) := (random()$Km,random()$Km)\bound{a12 b12 }\free{Km2 }}
+\indentrel{3}\begin{verbatim}
+            13
+   (13)  %BF
+  Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty13}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty13}{ugxProblemFiniteConversionPagePatch13}
+\pastebutton{ugxProblemFiniteConversionPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{(a1,b1) := (random()$Km,random()$Km)\bound{a12 b12 }\free{Km2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch14}
+\begin{paste}{ugxProblemFiniteConversionPageFull14}{ugxProblemFiniteConversionPageEmpty14}
+\pastebutton{ugxProblemFiniteConversionPageFull14}{\hidepaste}
+\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a22 }\free{a12 Kn2 }}
+\indentrel{3}\begin{verbatim}
+   (14)
+        6        5        4        2
+       q        q        q        q        q
+   2%BG   + 2%BG   + 2%BG   + 2%BG   + 2%BG  + 2%BG
+  Type: FiniteFieldNormalBasisExtension(PrimeField 3,8)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty14}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty14}{ugxProblemFiniteConversionPagePatch14}
+\pastebutton{ugxProblemFiniteConversionPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a22 }\free{a12 Kn2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch15}
+\begin{paste}{ugxProblemFiniteConversionPageFull15}{ugxProblemFiniteConversionPageEmpty15}
+\pastebutton{ugxProblemFiniteConversionPageFull15}{\hidepaste}
+\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b22 }\free{b12 Kn2 }}
+\indentrel{3}\begin{verbatim}
+   (15)
+          7       6       5       4        3       2
+         q       q       q       q        q       q
+     2%BG   + %BG   + %BG   + %BG   + 2%BG   + %BG
+   + 
+        q
+     %BG  + %BG
+  Type: FiniteFieldNormalBasisExtension(PrimeField 3,8)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty15}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty15}{ugxProblemFiniteConversionPagePatch15}
+\pastebutton{ugxProblemFiniteConversionPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b22 }\free{b12 Kn2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch16}
+\begin{paste}{ugxProblemFiniteConversionPageFull16}{ugxProblemFiniteConversionPageEmpty16}
+\pastebutton{ugxProblemFiniteConversionPageFull16}{\hidepaste}
+\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a12 a22 b12 b22 Km2 }}
+\indentrel{3}\begin{verbatim}
+   (16)  0
+  Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty16}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty16}{ugxProblemFiniteConversionPagePatch16}
+\pastebutton{ugxProblemFiniteConversionPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a12 a22 b12 b22 Km2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPagePatch17}
+\begin{paste}{ugxProblemFiniteConversionPageFull17}{ugxProblemFiniteConversionPageEmpty17}
+\pastebutton{ugxProblemFiniteConversionPageFull17}{\hidepaste}
+\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a12 a22 b12 b22 Km2 }}
+\indentrel{3}\begin{verbatim}
+   (17)  0
+  Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteConversionPageEmpty17}
+\begin{paste}{ugxProblemFiniteConversionPageEmpty17}{ugxProblemFiniteConversionPagePatch17}
+\pastebutton{ugxProblemFiniteConversionPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a12 a22 b12 b22 Km2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesSubstitutePagePatch1}
+\begin{paste}{ugxProblemSeriesSubstitutePageFull1}{ugxProblemSeriesSubstitutePageEmpty1}
+\pastebutton{ugxProblemSeriesSubstitutePageFull1}{\hidepaste}
+\tab{5}\spadcommand{f := taylor(exp(x))\bound{f }}
+\indentrel{3}\begin{verbatim}
+   (1)
+             1  2   1  3    1  4    1   5    1   6
+     1 + x + Ä x  + Ä x  + ÄÄ x  + ÄÄÄ x  + ÄÄÄ x
+             2      6      24      120      720
+   + 
+     1   7     1    8      1    9      1     10      11
+   ÄÄÄÄ x  + ÄÄÄÄÄ x  + ÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x   + O(x  )
+   5040      40320      362880      3628800
+   Type: UnivariateTaylorSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesSubstitutePageEmpty1}
+\begin{paste}{ugxProblemSeriesSubstitutePageEmpty1}{ugxProblemSeriesSubstitutePagePatch1}
+\pastebutton{ugxProblemSeriesSubstitutePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{f := taylor(exp(x))\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesSubstitutePagePatch2}
+\begin{paste}{ugxProblemSeriesSubstitutePageFull2}{ugxProblemSeriesSubstitutePageEmpty2}
+\pastebutton{ugxProblemSeriesSubstitutePageFull2}{\hidepaste}
+\tab{5}\spadcommand{eval(f,1.0)}
+\indentrel{3}\begin{verbatim}
+   (2)
+   [1.0, 2.0, 2.5, 2.6666666666 666666667,
+    2.7083333333 333333333, 2.7166666666 666666667,
+    2.7180555555 555555556, 2.7182539682 53968254,
+    2.7182787698 412698413, 2.7182815255 731922399, ...]
+                          Type: Stream Expression Float
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesSubstitutePageEmpty2}
+\begin{paste}{ugxProblemSeriesSubstitutePageEmpty2}{ugxProblemSeriesSubstitutePagePatch2}
+\pastebutton{ugxProblemSeriesSubstitutePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{eval(f,1.0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPagePatch1}
+\begin{paste}{ugProblemFactorAlgPageFull1}{ugProblemFactorAlgPageEmpty1}
+\pastebutton{ugProblemFactorAlgPageFull1}{\hidepaste}
+\tab{5}\spadcommand{aa := rootOf(aa**2+aa+1)\bound{aa }}
+\indentrel{3}\begin{verbatim}
+   (1)  aa
+                                  Type: AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPageEmpty1}
+\begin{paste}{ugProblemFactorAlgPageEmpty1}{ugProblemFactorAlgPagePatch1}
+\pastebutton{ugProblemFactorAlgPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{aa := rootOf(aa**2+aa+1)\bound{aa }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPagePatch2}
+\begin{paste}{ugProblemFactorAlgPageFull2}{ugProblemFactorAlgPageEmpty2}
+\pastebutton{ugProblemFactorAlgPageFull2}{\hidepaste}
+\tab{5}\spadcommand{p:=(x**3+aa**2*x+y)*(aa*x**2+aa*x+aa*y**2)**2\free{aa }\bound{p }}
+\indentrel{3}\begin{verbatim}
+   (2)
+                5               3         4
+     (- aa - 1)y  + ((- aa - 1)x  + aa x)y
+   + 
+                  2                 3
+     ((- 2aa - 2)x  + (- 2aa - 2)x)y
+   + 
+                  5               4        3        2  2
+     ((- 2aa - 2)x  + (- 2aa - 2)x  + 2aa x  + 2aa x )y
+   + 
+                 4               3              2
+     ((- aa - 1)x  + (- 2aa - 2)x  + (- aa - 1)x )y
+   + 
+                7               6    5        4       3
+     (- aa - 1)x  + (- 2aa - 2)x  - x  + 2aa x  + aa x
+                       Type: Polynomial AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPageEmpty2}
+\begin{paste}{ugProblemFactorAlgPageEmpty2}{ugProblemFactorAlgPagePatch2}
+\pastebutton{ugProblemFactorAlgPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{p:=(x**3+aa**2*x+y)*(aa*x**2+aa*x+aa*y**2)**2\free{aa }\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPagePatch3}
+\begin{paste}{ugProblemFactorAlgPageFull3}{ugProblemFactorAlgPageEmpty3}
+\pastebutton{ugProblemFactorAlgPageFull3}{\hidepaste}
+\tab{5}\spadcommand{factor(p,[aa])\free{p aa }}
+\indentrel{3}\begin{verbatim}
+                        3                 2    2     2
+   (3)  (- aa - 1)(y + x  + (- aa - 1)x)(y  + x  + x)
+              Type: Factored Polynomial AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPageEmpty3}
+\begin{paste}{ugProblemFactorAlgPageEmpty3}{ugProblemFactorAlgPagePatch3}
+\pastebutton{ugProblemFactorAlgPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{factor(p,[aa])\free{p aa }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPagePatch4}
+\begin{paste}{ugProblemFactorAlgPageFull4}{ugProblemFactorAlgPageEmpty4}
+\pastebutton{ugProblemFactorAlgPageFull4}{\hidepaste}
+\tab{5}\spadcommand{factor(x**2+3)}
+\indentrel{3}\begin{verbatim}
+         2
+   (4)  x  + 3
+                      Type: Factored Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPageEmpty4}
+\begin{paste}{ugProblemFactorAlgPageEmpty4}{ugProblemFactorAlgPagePatch4}
+\pastebutton{ugProblemFactorAlgPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{factor(x**2+3)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPagePatch5}
+\begin{paste}{ugProblemFactorAlgPageFull5}{ugProblemFactorAlgPageEmpty5}
+\pastebutton{ugProblemFactorAlgPageFull5}{\hidepaste}
+\tab{5}\spadcommand{factor(x**2+3,[aa])\free{aa }}
+\indentrel{3}\begin{verbatim}
+   (5)  (x - 2aa - 1)(x + 2aa + 1)
+              Type: Factored Polynomial AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPageEmpty5}
+\begin{paste}{ugProblemFactorAlgPageEmpty5}{ugProblemFactorAlgPagePatch5}
+\pastebutton{ugProblemFactorAlgPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{factor(x**2+3,[aa])\free{aa }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPagePatch6}
+\begin{paste}{ugProblemFactorAlgPageFull6}{ugProblemFactorAlgPageEmpty6}
+\pastebutton{ugProblemFactorAlgPageFull6}{\hidepaste}
+\tab{5}\spadcommand{factor(x**6+108,[aa])\free{aa }}
+\indentrel{3}\begin{verbatim}
+          3              3
+   (6)  (x  - 12aa - 6)(x  + 12aa + 6)
+              Type: Factored Polynomial AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPageEmpty6}
+\begin{paste}{ugProblemFactorAlgPageEmpty6}{ugProblemFactorAlgPagePatch6}
+\pastebutton{ugProblemFactorAlgPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{factor(x**6+108,[aa])\free{aa }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPagePatch7}
+\begin{paste}{ugProblemFactorAlgPageFull7}{ugProblemFactorAlgPageEmpty7}
+\pastebutton{ugProblemFactorAlgPageFull7}{\hidepaste}
+\tab{5}\spadcommand{bb:=rootOf(bb**3-2)\bound{bb }}
+\indentrel{3}\begin{verbatim}
+   (7)  bb
+                                  Type: AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPageEmpty7}
+\begin{paste}{ugProblemFactorAlgPageEmpty7}{ugProblemFactorAlgPagePatch7}
+\pastebutton{ugProblemFactorAlgPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{bb:=rootOf(bb**3-2)\bound{bb }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPagePatch8}
+\begin{paste}{ugProblemFactorAlgPageFull8}{ugProblemFactorAlgPageEmpty8}
+\pastebutton{ugProblemFactorAlgPageFull8}{\hidepaste}
+\tab{5}\spadcommand{factor(x**6+108,[bb])\free{bb }}
+\indentrel{3}\begin{verbatim}
+   (8)
+     2              2   2      2   2              2
+   (x  - 3bb x + 3bb )(x  + 3bb )(x  + 3bb x + 3bb )
+              Type: Factored Polynomial AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPageEmpty8}
+\begin{paste}{ugProblemFactorAlgPageEmpty8}{ugProblemFactorAlgPagePatch8}
+\pastebutton{ugProblemFactorAlgPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{factor(x**6+108,[bb])\free{bb }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPagePatch9}
+\begin{paste}{ugProblemFactorAlgPageFull9}{ugProblemFactorAlgPageEmpty9}
+\pastebutton{ugProblemFactorAlgPageFull9}{\hidepaste}
+\tab{5}\spadcommand{factor(x**6+108,[aa,bb])\free{aa bb }}
+\indentrel{3}\begin{verbatim}
+   (9)
+     (x + (- 2aa - 1)bb)(x + (- aa - 2)bb)
+  *
+     (x + (- aa + 1)bb)(x + (aa - 1)bb)(x + (aa + 2)bb)
+  *
+     (x + (2aa + 1)bb)
+              Type: Factored Polynomial AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorAlgPageEmpty9}
+\begin{paste}{ugProblemFactorAlgPageEmpty9}{ugProblemFactorAlgPagePatch9}
+\pastebutton{ugProblemFactorAlgPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{factor(x**6+108,[aa,bb])\free{aa bb }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPagePatch1}
+\begin{paste}{ugxProblemSeriesConversionsPageFull1}{ugxProblemSeriesConversionsPageEmpty1}
+\pastebutton{ugxProblemSeriesConversionsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{taylor(sin(x),x = 0)}
+\indentrel{3}\begin{verbatim}
+            1  3    1   5     1   7      1    9      11
+   (1)  x - Ä x  + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x  + O(x  )
+            6      120      5040      362880
+   Type: UnivariateTaylorSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPageEmpty1}
+\begin{paste}{ugxProblemSeriesConversionsPageEmpty1}{ugxProblemSeriesConversionsPagePatch1}
+\pastebutton{ugxProblemSeriesConversionsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{taylor(sin(x),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPagePatch2}
+\begin{paste}{ugxProblemSeriesConversionsPageFull2}{ugxProblemSeriesConversionsPageEmpty2}
+\pastebutton{ugxProblemSeriesConversionsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{taylor(sin(x),x = \%pi/6)}
+\indentrel{3}\begin{verbatim}
+   (2)
+          ÚÄ¿                             ÚÄ¿
+     1   \³3       %pi    1      %pi 2   \³3       %pi 3
+     Ä + ÄÄÄÄ (x - ÄÄÄ) - Ä (x - ÄÄÄ)  - ÄÄÄÄ (x - ÄÄÄ)
+     2     2        6     4       6       12        6
+   + 
+                      ÚÄ¿
+      1      %pi 4   \³3       %pi 5     1       %pi 6
+     ÄÄ (x - ÄÄÄ)  + ÄÄÄÄ (x - ÄÄÄ)  - ÄÄÄÄ (x - ÄÄÄ)
+     48       6       240       6      1440       6
+   + 
+         ÚÄ¿
+        \³3       %pi 7     1        %pi 8
+     - ÄÄÄÄÄ (x - ÄÄÄ)  + ÄÄÄÄÄ (x - ÄÄÄ)
+       10080       6      80640       6
+   + 
+       ÚÄ¿
+      \³3        %pi 9      1         %pi 10
+     ÄÄÄÄÄÄ (x - ÄÄÄ)  - ÄÄÄÄÄÄÄ (x - ÄÄÄ)
+     725760       6      7257600       6
+   + 
+            %pi 11
+     O((x - ÄÄÄ)  )
+             6
+Type: UnivariateTaylorSeries(Expression Integer,x,pi/6)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPageEmpty2}
+\begin{paste}{ugxProblemSeriesConversionsPageEmpty2}{ugxProblemSeriesConversionsPagePatch2}
+\pastebutton{ugxProblemSeriesConversionsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{taylor(sin(x),x = \%pi/6)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPagePatch3}
+\begin{paste}{ugxProblemSeriesConversionsPageFull3}{ugxProblemSeriesConversionsPageEmpty3}
+\pastebutton{ugxProblemSeriesConversionsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{taylor(tan(x*y),x = 0)}
+\indentrel{3}\begin{verbatim}
+   (3)
+          3        5         7         9
+         y   3   2y   5   17y   7   62y   9      11
+   y x + ÄÄ x  + ÄÄÄ x  + ÄÄÄÄ x  + ÄÄÄÄ x  + O(x  )
+          3       15       315      2835
+   Type: UnivariateTaylorSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPageEmpty3}
+\begin{paste}{ugxProblemSeriesConversionsPageEmpty3}{ugxProblemSeriesConversionsPagePatch3}
+\pastebutton{ugxProblemSeriesConversionsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{taylor(tan(x*y),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPagePatch4}
+\begin{paste}{ugxProblemSeriesConversionsPageFull4}{ugxProblemSeriesConversionsPageEmpty4}
+\pastebutton{ugxProblemSeriesConversionsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{taylor(tan(x*y),y = 0)}
+\indentrel{3}\begin{verbatim}
+   (4)
+          3        5         7         9
+         x   3   2x   5   17x   7   62x   9      11
+   x y + ÄÄ y  + ÄÄÄ y  + ÄÄÄÄ y  + ÄÄÄÄ y  + O(y  )
+          3       15       315      2835
+   Type: UnivariateTaylorSeries(Expression Integer,y,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPageEmpty4}
+\begin{paste}{ugxProblemSeriesConversionsPageEmpty4}{ugxProblemSeriesConversionsPagePatch4}
+\pastebutton{ugxProblemSeriesConversionsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{taylor(tan(x*y),y = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPagePatch5}
+\begin{paste}{ugxProblemSeriesConversionsPageFull5}{ugxProblemSeriesConversionsPageEmpty5}
+\pastebutton{ugxProblemSeriesConversionsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{bern := taylor(t*exp(x*t)/(exp(t) - 1),t = 0)\bound{bern }}
+\indentrel{3}\begin{verbatim}
+   (5)
+                      2                 3     2
+         2x - 1     6x  - 6x + 1  2   2x  - 3x  + x  3
+     1 + ÄÄÄÄÄÄ t + ÄÄÄÄÄÄÄÄÄÄÄÄ t  + ÄÄÄÄÄÄÄÄÄÄÄÄÄ t
+            2            12                 12
+   + 
+        4      3      2            5      4      3
+     30x  - 60x  + 30x  - 1  4   6x  - 15x  + 10x  - x  5
+     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t  + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
+               720                        720
+   + 
+        6       5       4      2
+     42x  - 126x  + 105x  - 21x  + 1  6
+     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
+                  30240
+   + 
+       7      6      5     3
+     6x  - 21x  + 21x  - 7x  + x  7
+     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
+                30240
+   + 
+        8       7       6      4      2
+     30x  - 120x  + 140x  - 70x  + 20x  - 1  8
+     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
+                     1209600
+   + 
+        9      8      7      5      3
+     10x  - 45x  + 60x  - 42x  + 20x  - 3x  9
+     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
+                    3628800
+   + 
+        10       9       8       6       4      2
+     66x   - 330x  + 495x  - 462x  + 330x  - 99x  + 5  10
+     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ t
+                         239500800
+   + 
+        11
+     O(t  )
+   Type: UnivariateTaylorSeries(Expression Integer,t,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPageEmpty5}
+\begin{paste}{ugxProblemSeriesConversionsPageEmpty5}{ugxProblemSeriesConversionsPagePatch5}
+\pastebutton{ugxProblemSeriesConversionsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{bern := taylor(t*exp(x*t)/(exp(t) - 1),t = 0)\bound{bern }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPagePatch6}
+\begin{paste}{ugxProblemSeriesConversionsPageFull6}{ugxProblemSeriesConversionsPageEmpty6}
+\pastebutton{ugxProblemSeriesConversionsPageFull6}{\hidepaste}
+\tab{5}\spadcommand{factorial(6) * coefficient(bern,6)\free{bern }}
+\indentrel{3}\begin{verbatim}
+           6       5       4      2
+        42x  - 126x  + 105x  - 21x  + 1
+   (6)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                       42
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPageEmpty6}
+\begin{paste}{ugxProblemSeriesConversionsPageEmpty6}{ugxProblemSeriesConversionsPagePatch6}
+\pastebutton{ugxProblemSeriesConversionsPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{factorial(6) * coefficient(bern,6)\free{bern }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPagePatch7}
+\begin{paste}{ugxProblemSeriesConversionsPageFull7}{ugxProblemSeriesConversionsPageEmpty7}
+\pastebutton{ugxProblemSeriesConversionsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{bernoulliB(6,x)}
+\indentrel{3}\begin{verbatim}
+         6     5   5  4   1  2    1
+   (7)  x  - 3x  + Ä x  - Ä x  + ÄÄ
+                   2      2      42
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPageEmpty7}
+\begin{paste}{ugxProblemSeriesConversionsPageEmpty7}{ugxProblemSeriesConversionsPagePatch7}
+\pastebutton{ugxProblemSeriesConversionsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{bernoulliB(6,x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPagePatch8}
+\begin{paste}{ugxProblemSeriesConversionsPageFull8}{ugxProblemSeriesConversionsPageEmpty8}
+\pastebutton{ugxProblemSeriesConversionsPageFull8}{\hidepaste}
+\tab{5}\spadcommand{laurent(x/log(x),x = 1)}
+\indentrel{3}\begin{verbatim}
+   (8)
+            - 1   3    5            1        2
+     (x - 1)    + Ä + ÄÄ (x - 1) - ÄÄ (x - 1)
+                  2   12           24
+   + 
+      11        3    11         4    271         5
+     ÄÄÄ (x - 1)  - ÄÄÄÄ (x - 1)  + ÄÄÄÄÄ (x - 1)
+     720            1440            60480
+   + 
+        13         6     7297         7     425         8
+     - ÄÄÄÄ (x - 1)  + ÄÄÄÄÄÄÄ (x - 1)  - ÄÄÄÄÄÄ (x - 1)
+       4480            3628800            290304
+   + 
+       530113         9            10
+     ÄÄÄÄÄÄÄÄÄ (x - 1)  + O((x - 1)  )
+     479001600
+  Type: UnivariateLaurentSeries(Expression Integer,x,1)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPageEmpty8}
+\begin{paste}{ugxProblemSeriesConversionsPageEmpty8}{ugxProblemSeriesConversionsPagePatch8}
+\pastebutton{ugxProblemSeriesConversionsPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{laurent(x/log(x),x = 1)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPagePatch9}
+\begin{paste}{ugxProblemSeriesConversionsPageFull9}{ugxProblemSeriesConversionsPageEmpty9}
+\pastebutton{ugxProblemSeriesConversionsPageFull9}{\hidepaste}
+\tab{5}\spadcommand{puiseux(sqrt(sec(x)),x = 3 * \%pi/2)}
+\indentrel{3}\begin{verbatim}
+   (9)
+                 1                3                 7
+               - Ä                Ä                 Ä
+          3%pi   2    1      3%pi 2    1       3%pi 2
+     (x - ÄÄÄÄ)    + ÄÄ (x - ÄÄÄÄ)  + ÄÄÄ (x - ÄÄÄÄ)
+            2        12        2      160        2
+   + 
+            3%pi 5
+     O((x - ÄÄÄÄ) )
+              2
+Type: UnivariatePuiseuxSeries(Expression Integer,x,(3*pi)/2)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPageEmpty9}
+\begin{paste}{ugxProblemSeriesConversionsPageEmpty9}{ugxProblemSeriesConversionsPagePatch9}
+\pastebutton{ugxProblemSeriesConversionsPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{puiseux(sqrt(sec(x)),x = 3 * \%pi/2)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPagePatch10}
+\begin{paste}{ugxProblemSeriesConversionsPageFull10}{ugxProblemSeriesConversionsPageEmpty10}
+\pastebutton{ugxProblemSeriesConversionsPageFull10}{\hidepaste}
+\tab{5}\spadcommand{series(x**x,x=0)}
+\indentrel{3}\begin{verbatim}
+   (10)
+                         2            3            4
+                   log(x)   2   log(x)   3   log(x)   4
+     1 + log(x)x + ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x
+                      2            6            24
+   + 
+           5            6            7            8
+     log(x)   5   log(x)   6   log(x)   7   log(x)   8
+     ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x
+       120          720          5040        40320
+   + 
+           9            10
+     log(x)   9   log(x)    10      11
+     ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄÄ x   + O(x  )
+      362880       3628800
+Type: GeneralUnivariatePowerSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesConversionsPageEmpty10}
+\begin{paste}{ugxProblemSeriesConversionsPageEmpty10}{ugxProblemSeriesConversionsPagePatch10}
+\pastebutton{ugxProblemSeriesConversionsPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{series(x**x,x=0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPagePatch1}
+\begin{paste}{ugxProblemDEQSeriesPageFull1}{ugxProblemDEQSeriesPageEmpty1}
+\pastebutton{ugxProblemDEQSeriesPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)set streams calculate 7\bound{c7 }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPageEmpty1}
+\begin{paste}{ugxProblemDEQSeriesPageEmpty1}{ugxProblemDEQSeriesPagePatch1}
+\pastebutton{ugxProblemDEQSeriesPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)set streams calculate 7\bound{c7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPagePatch2}
+\begin{paste}{ugxProblemDEQSeriesPageFull2}{ugxProblemDEQSeriesPageEmpty2}
+\pastebutton{ugxProblemDEQSeriesPageFull2}{\hidepaste}
+\tab{5}\spadcommand{y := operator 'y\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (1)  y
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPageEmpty2}
+\begin{paste}{ugxProblemDEQSeriesPageEmpty2}{ugxProblemDEQSeriesPagePatch2}
+\pastebutton{ugxProblemDEQSeriesPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{y := operator 'y\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPagePatch3}
+\begin{paste}{ugxProblemDEQSeriesPageFull3}{ugxProblemDEQSeriesPageEmpty3}
+\pastebutton{ugxProblemDEQSeriesPageFull3}{\hidepaste}
+\tab{5}\spadcommand{eq := D(y(x), x, 3) - sin(D(y(x), x, 2))*exp(y(x)) = cos(x)\bound{eq }\free{y }}
+\indentrel{3}\begin{verbatim}
+         ,,,        y(x)     ,,
+   (2)  y   (x) - %e    sin(y  (x))= cos(x)
+
+                      Type: Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPageEmpty3}
+\begin{paste}{ugxProblemDEQSeriesPageEmpty3}{ugxProblemDEQSeriesPagePatch3}
+\pastebutton{ugxProblemDEQSeriesPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{eq := D(y(x), x, 3) - sin(D(y(x), x, 2))*exp(y(x)) = cos(x)\bound{eq }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPagePatch4}
+\begin{paste}{ugxProblemDEQSeriesPageFull4}{ugxProblemDEQSeriesPageEmpty4}
+\pastebutton{ugxProblemDEQSeriesPageFull4}{\hidepaste}
+\tab{5}\spadcommand{seriesSolve(eq, y, x = 0, [1, 0, 0])\free{y }\free{eq }\free{c7 }}
+\indentrel{3}\begin{verbatim}
+   (3)
+                          2            3
+         1  3   %e  4   %e  - 1  5   %e  - 2%e  6
+     1 + Ä x  + ÄÄ x  + ÄÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄÄÄ x
+         6      24        120           720
+   + 
+       4      2
+     %e  - 8%e  + 4%e + 1  7      8
+     ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ x  + O(x )
+             5040
+   Type: UnivariateTaylorSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPageEmpty4}
+\begin{paste}{ugxProblemDEQSeriesPageEmpty4}{ugxProblemDEQSeriesPagePatch4}
+\pastebutton{ugxProblemDEQSeriesPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{seriesSolve(eq, y, x = 0, [1, 0, 0])\free{y }\free{eq }\free{c7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPagePatch5}
+\begin{paste}{ugxProblemDEQSeriesPageFull5}{ugxProblemDEQSeriesPageEmpty5}
+\pastebutton{ugxProblemDEQSeriesPageFull5}{\hidepaste}
+\tab{5}\spadcommand{x := operator 'x\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (4)  x
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPageEmpty5}
+\begin{paste}{ugxProblemDEQSeriesPageEmpty5}{ugxProblemDEQSeriesPagePatch5}
+\pastebutton{ugxProblemDEQSeriesPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{x := operator 'x\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPagePatch6}
+\begin{paste}{ugxProblemDEQSeriesPageFull6}{ugxProblemDEQSeriesPageEmpty6}
+\pastebutton{ugxProblemDEQSeriesPageFull6}{\hidepaste}
+\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2\free{x }\free{y }\bound{eq1 }}
+\indentrel{3}\begin{verbatim}
+         ,         2
+   (5)  x (t)= x(t)  + 1
+
+                      Type: Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPageEmpty6}
+\begin{paste}{ugxProblemDEQSeriesPageEmpty6}{ugxProblemDEQSeriesPagePatch6}
+\pastebutton{ugxProblemDEQSeriesPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2\free{x }\free{y }\bound{eq1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPagePatch7}
+\begin{paste}{ugxProblemDEQSeriesPageFull7}{ugxProblemDEQSeriesPageEmpty7}
+\pastebutton{ugxProblemDEQSeriesPageFull7}{\hidepaste}
+\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }}
+\indentrel{3}\begin{verbatim}
+         ,
+   (6)  y (t)= x(t)y(t)
+
+                      Type: Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPageEmpty7}
+\begin{paste}{ugxProblemDEQSeriesPageEmpty7}{ugxProblemDEQSeriesPagePatch7}
+\pastebutton{ugxProblemDEQSeriesPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPagePatch8}
+\begin{paste}{ugxProblemDEQSeriesPageFull8}{ugxProblemDEQSeriesPageEmpty8}
+\pastebutton{ugxProblemDEQSeriesPageFull8}{\hidepaste}
+\tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }\free{c7 }}
+\indentrel{3}\begin{verbatim}
+   (7)
+        1  3    2  5    17  7      8
+   [t + Ä t  + ÄÄ t  + ÄÄÄ t  + O(t ),
+        3      15      315
+        1  2    5  4    61  6      8
+    1 + Ä t  + ÄÄ t  + ÄÄÄ t  + O(t )]
+        2      24      720
+Type: List UnivariateTaylorSeries(Expression Integer,t,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemDEQSeriesPageEmpty8}
+\begin{paste}{ugxProblemDEQSeriesPageEmpty8}{ugxProblemDEQSeriesPagePatch8}
+\pastebutton{ugxProblemDEQSeriesPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }\free{c7 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCoefficientsPagePatch1}
+\begin{paste}{ugxProblemSeriesCoefficientsPageFull1}{ugxProblemSeriesCoefficientsPageEmpty1}
+\pastebutton{ugxProblemSeriesCoefficientsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{x := series(x)\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (1)  x
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty1}
+\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty1}{ugxProblemSeriesCoefficientsPagePatch1}
+\pastebutton{ugxProblemSeriesCoefficientsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{x := series(x)\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCoefficientsPagePatch2}
+\begin{paste}{ugxProblemSeriesCoefficientsPageFull2}{ugxProblemSeriesCoefficientsPageEmpty2}
+\pastebutton{ugxProblemSeriesCoefficientsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{y := exp(x) * sin(x)\free{x }\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (2)
+          2   1  3    1  5    1  6    1   7     1    9
+     x + x  + Ä x  - ÄÄ x  - ÄÄ x  - ÄÄÄ x  + ÄÄÄÄÄ x
+              3      30      90      630      22680
+   + 
+        1    10      1     11      12
+     ÄÄÄÄÄÄ x   + ÄÄÄÄÄÄÄ x   + O(x  )
+     113400       1247400
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty2}
+\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty2}{ugxProblemSeriesCoefficientsPagePatch2}
+\pastebutton{ugxProblemSeriesCoefficientsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{y := exp(x) * sin(x)\free{x }\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCoefficientsPagePatch3}
+\begin{paste}{ugxProblemSeriesCoefficientsPageFull3}{ugxProblemSeriesCoefficientsPageEmpty3}
+\pastebutton{ugxProblemSeriesCoefficientsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{coefficient(y,6)\free{y }}
+\indentrel{3}\begin{verbatim}
+           1
+   (3)  - ÄÄ
+          90
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty3}
+\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty3}{ugxProblemSeriesCoefficientsPagePatch3}
+\pastebutton{ugxProblemSeriesCoefficientsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{coefficient(y,6)\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCoefficientsPagePatch4}
+\begin{paste}{ugxProblemSeriesCoefficientsPageFull4}{ugxProblemSeriesCoefficientsPageEmpty4}
+\pastebutton{ugxProblemSeriesCoefficientsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{coefficient(y,15)\free{y }\bound{y15 }}
+\indentrel{3}\begin{verbatim}
+               1
+   (4)  - ÄÄÄÄÄÄÄÄÄÄÄ
+          10216206000
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty4}
+\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty4}{ugxProblemSeriesCoefficientsPagePatch4}
+\pastebutton{ugxProblemSeriesCoefficientsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{coefficient(y,15)\free{y }\bound{y15 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCoefficientsPagePatch5}
+\begin{paste}{ugxProblemSeriesCoefficientsPageFull5}{ugxProblemSeriesCoefficientsPageEmpty5}
+\pastebutton{ugxProblemSeriesCoefficientsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{y\free{y15 }}
+\indentrel{3}\begin{verbatim}
+   (5)
+          2   1  3    1  5    1  6    1   7     1    9
+     x + x  + Ä x  - ÄÄ x  - ÄÄ x  - ÄÄÄ x  + ÄÄÄÄÄ x
+              3      30      90      630      22680
+   + 
+        1    10      1     11       1     13
+     ÄÄÄÄÄÄ x   + ÄÄÄÄÄÄÄ x   - ÄÄÄÄÄÄÄÄ x
+     113400       1247400       97297200
+   + 
+           1      14        1       15      16
+     - ÄÄÄÄÄÄÄÄÄ x   - ÄÄÄÄÄÄÄÄÄÄÄ x   + O(x  )
+       681080400       10216206000
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty5}
+\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty5}{ugxProblemSeriesCoefficientsPagePatch5}
+\pastebutton{ugxProblemSeriesCoefficientsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{y\free{y15 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPagePatch1}
+\begin{paste}{ugxProblemLDEQClosedPageFull1}{ugxProblemLDEQClosedPageEmpty1}
+\pastebutton{ugxProblemLDEQClosedPageFull1}{\hidepaste}
+\tab{5}\spadcommand{y := operator 'y\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (1)  y
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPageEmpty1}
+\begin{paste}{ugxProblemLDEQClosedPageEmpty1}{ugxProblemLDEQClosedPagePatch1}
+\pastebutton{ugxProblemLDEQClosedPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{y := operator 'y\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPagePatch2}
+\begin{paste}{ugxProblemLDEQClosedPageFull2}{ugxProblemLDEQClosedPageEmpty2}
+\pastebutton{ugxProblemLDEQClosedPageFull2}{\hidepaste}
+\tab{5}\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0\bound{e1 }\free{y }}
+\indentrel{3}\begin{verbatim}
+         ,,       ,
+   (2)  y  (x) + y (x) + y(x)= 0
+
+                      Type: Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPageEmpty2}
+\begin{paste}{ugxProblemLDEQClosedPageEmpty2}{ugxProblemLDEQClosedPagePatch2}
+\pastebutton{ugxProblemLDEQClosedPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0\bound{e1 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPagePatch3}
+\begin{paste}{ugxProblemLDEQClosedPageFull3}{ugxProblemLDEQClosedPageEmpty3}
+\pastebutton{ugxProblemLDEQClosedPageFull3}{\hidepaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e1 }\free{y }}
+\indentrel{3}\begin{verbatim}
+   (3)
+   [particular= 0,
+                          x     x
+                  ÚÄ¿   - Ä   - Ä      ÚÄ¿
+                x\³3      2     2    x\³3
+    basis= [cos(ÄÄÄÄÄ)%e   ,%e   sin(ÄÄÄÄÄ)]]
+                  2                    2
+Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPageEmpty3}
+\begin{paste}{ugxProblemLDEQClosedPageEmpty3}{ugxProblemLDEQClosedPagePatch3}
+\pastebutton{ugxProblemLDEQClosedPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e1 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPagePatch4}
+\begin{paste}{ugxProblemLDEQClosedPageFull4}{ugxProblemLDEQClosedPageEmpty4}
+\pastebutton{ugxProblemLDEQClosedPageFull4}{\hidepaste}
+\tab{5}\spadcommand{deq := D(y x, x, 2) + y x\bound{e2 }\free{y }}
+\indentrel{3}\begin{verbatim}
+         ,,
+   (4)  y  (x) + y(x)
+
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPageEmpty4}
+\begin{paste}{ugxProblemLDEQClosedPageEmpty4}{ugxProblemLDEQClosedPagePatch4}
+\pastebutton{ugxProblemLDEQClosedPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{deq := D(y x, x, 2) + y x\bound{e2 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPagePatch5}
+\begin{paste}{ugxProblemLDEQClosedPageFull5}{ugxProblemLDEQClosedPageEmpty5}
+\pastebutton{ugxProblemLDEQClosedPageFull5}{\hidepaste}
+\tab{5}\spadcommand{solve(deq, y, x = 0, [1, 1])\free{e2 }\free{y }}
+\indentrel{3}\begin{verbatim}
+   (5)  sin(x) + cos(x)
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPageEmpty5}
+\begin{paste}{ugxProblemLDEQClosedPageEmpty5}{ugxProblemLDEQClosedPagePatch5}
+\pastebutton{ugxProblemLDEQClosedPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{solve(deq, y, x = 0, [1, 1])\free{e2 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPagePatch6}
+\begin{paste}{ugxProblemLDEQClosedPageFull6}{ugxProblemLDEQClosedPageEmpty6}
+\pastebutton{ugxProblemLDEQClosedPageFull6}{\hidepaste}
+\tab{5}\spadcommand{deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4\bound{e3 }\free{y }}
+\indentrel{3}\begin{verbatim}
+         3 ,,,       2 ,,         ,               4
+   (6)  x y   (x) + x y  (x) - 2xy (x) + 2y(x)= 2x
+
+                      Type: Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPageEmpty6}
+\begin{paste}{ugxProblemLDEQClosedPageEmpty6}{ugxProblemLDEQClosedPagePatch6}
+\pastebutton{ugxProblemLDEQClosedPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{deq := x**3 * D(y x, x, 3) + x**2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x**4\bound{e3 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPagePatch7}
+\begin{paste}{ugxProblemLDEQClosedPageFull7}{ugxProblemLDEQClosedPageEmpty7}
+\pastebutton{ugxProblemLDEQClosedPageFull7}{\hidepaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }}
+\indentrel{3}\begin{verbatim}
+   (7)
+                 5      3      2
+                x  - 10x  + 20x  + 4
+   [particular= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,
+                         15x
+              3     2      3      3     2
+            2x  - 3x  + 1 x  - 1 x  - 3x  - 1
+    basis= [ÄÄÄÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄ]]
+                  x          x         x
+Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPageEmpty7}
+\begin{paste}{ugxProblemLDEQClosedPageEmpty7}{ugxProblemLDEQClosedPagePatch7}
+\pastebutton{ugxProblemLDEQClosedPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPagePatch8}
+\begin{paste}{ugxProblemLDEQClosedPageFull8}{ugxProblemLDEQClosedPageEmpty8}
+\pastebutton{ugxProblemLDEQClosedPageFull8}{\hidepaste}
+\tab{5}\spadcommand{deq := (x**9+x**3) * D(y x, x, 3) + 18 * x**8 * D(y x, x, 2) - 90 * x * D(y x, x) - 30 * (11 * x**6 - 3) * y x\bound{e4 }\free{y }}
+\indentrel{3}\begin{verbatim}
+   (8)
+       9    3  ,,,         8 ,,          ,
+     (x  + x )y   (x) + 18x y  (x) - 90xy (x)
+
+   + 
+            6
+     (- 330x  + 90)y(x)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPageEmpty8}
+\begin{paste}{ugxProblemLDEQClosedPageEmpty8}{ugxProblemLDEQClosedPagePatch8}
+\pastebutton{ugxProblemLDEQClosedPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{deq := (x**9+x**3) * D(y x, x, 3) + 18 * x**8 * D(y x, x, 2) - 90 * x * D(y x, x) - 30 * (11 * x**6 - 3) * y x\bound{e4 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPagePatch9}
+\begin{paste}{ugxProblemLDEQClosedPageFull9}{ugxProblemLDEQClosedPageEmpty9}
+\pastebutton{ugxProblemLDEQClosedPageFull9}{\hidepaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e4 }\free{y }}
+\indentrel{3}\begin{verbatim}
+   (9)
+   [particular= 0,
+                          ÚÄÄ¿            ÚÄÄ¿
+                       - \³91 log(x)     \³91 log(x)
+               x   x %e              x %e
+    basis= [ÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ]]
+             6            6                6
+            x  + 1       x  + 1           x  + 1
+Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPageEmpty9}
+\begin{paste}{ugxProblemLDEQClosedPageEmpty9}{ugxProblemLDEQClosedPagePatch9}
+\pastebutton{ugxProblemLDEQClosedPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e4 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPagePatch10}
+\begin{paste}{ugxProblemLDEQClosedPageFull10}{ugxProblemLDEQClosedPageEmpty10}
+\pastebutton{ugxProblemLDEQClosedPageFull10}{\hidepaste}
+\tab{5}\spadcommand{deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0\bound{e5 }\free{y }}
+\indentrel{3}\begin{verbatim}
+           2      ,,         ,
+   (10)  (x  + 1)y  (x) + 3xy (x) + y(x)= 0
+
+                      Type: Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPageEmpty10}
+\begin{paste}{ugxProblemLDEQClosedPageEmpty10}{ugxProblemLDEQClosedPagePatch10}
+\pastebutton{ugxProblemLDEQClosedPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{deq := (x**2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0\bound{e5 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPagePatch11}
+\begin{paste}{ugxProblemLDEQClosedPageFull11}{ugxProblemLDEQClosedPageEmpty11}
+\pastebutton{ugxProblemLDEQClosedPageFull11}{\hidepaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }}
+\indentrel{3}\begin{verbatim}
+   (11)
+                                         ÚÄÄÄÄÄÄ¿
+                                         ³ 2
+                              1     log(\³x  + 1  - x)
+   [particular= 0,basis= [ÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ]]
+                           ÚÄÄÄÄÄÄ¿       ÚÄÄÄÄÄÄ¿
+                           ³ 2            ³ 2
+                          \³x  + 1       \³x  + 1
+Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemLDEQClosedPageEmpty11}
+\begin{paste}{ugxProblemLDEQClosedPageEmpty11}{ugxProblemLDEQClosedPagePatch11}
+\pastebutton{ugxProblemLDEQClosedPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch1}
+\begin{paste}{ugxProblemNLDEQClosedPageFull1}{ugxProblemNLDEQClosedPageEmpty1}
+\pastebutton{ugxProblemNLDEQClosedPageFull1}{\hidepaste}
+\tab{5}\spadcommand{m := -y\bound{m }}
+\indentrel{3}\begin{verbatim}
+   (1)  - y
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty1}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty1}{ugxProblemNLDEQClosedPagePatch1}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{m := -y\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch2}
+\begin{paste}{ugxProblemNLDEQClosedPageFull2}{ugxProblemNLDEQClosedPageEmpty2}
+\pastebutton{ugxProblemNLDEQClosedPageFull2}{\hidepaste}
+\tab{5}\spadcommand{n := x + y * log y\bound{n }}
+\indentrel{3}\begin{verbatim}
+   (2)  y log(y) + x
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty2}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty2}{ugxProblemNLDEQClosedPagePatch2}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{n := x + y * log y\bound{n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch3}
+\begin{paste}{ugxProblemNLDEQClosedPageFull3}{ugxProblemNLDEQClosedPageEmpty3}
+\pastebutton{ugxProblemNLDEQClosedPageFull3}{\hidepaste}
+\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m n }}
+\indentrel{3}\begin{verbatim}
+   (3)  - 2
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty3}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty3}{ugxProblemNLDEQClosedPagePatch3}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch4}
+\begin{paste}{ugxProblemNLDEQClosedPageFull4}{ugxProblemNLDEQClosedPageEmpty4}
+\pastebutton{ugxProblemNLDEQClosedPageFull4}{\hidepaste}
+\tab{5}\spadcommand{mu := operator 'mu\bound{mu }}
+\indentrel{3}\begin{verbatim}
+   (4)  mu
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty4}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty4}{ugxProblemNLDEQClosedPagePatch4}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{mu := operator 'mu\bound{mu }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch5}
+\begin{paste}{ugxProblemNLDEQClosedPageFull5}{ugxProblemNLDEQClosedPageEmpty5}
+\pastebutton{ugxProblemNLDEQClosedPageFull5}{\hidepaste}
+\tab{5}\spadcommand{a := D(mu(x) * m, y) - D(mu(x) * n, x)\bound{a }\free{m n mu }}
+\indentrel{3}\begin{verbatim}
+                          ,
+   (5)  (- y log(y) - x)mu (x) - 2mu(x)
+
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty5}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty5}{ugxProblemNLDEQClosedPagePatch5}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{a := D(mu(x) * m, y) - D(mu(x) * n, x)\bound{a }\free{m n mu }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch6}
+\begin{paste}{ugxProblemNLDEQClosedPageFull6}{ugxProblemNLDEQClosedPageEmpty6}
+\pastebutton{ugxProblemNLDEQClosedPageFull6}{\hidepaste}
+\tab{5}\spadcommand{solve(a = 0, mu, x)\free{mu a }}
+\indentrel{3}\begin{verbatim}
+   (6)
+                                        1
+   [particular= 0,basis= [ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ]]
+                           2      2                  2
+                          y log(y)  + 2x y log(y) + x
+Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty6}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty6}{ugxProblemNLDEQClosedPagePatch6}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{solve(a = 0, mu, x)\free{mu a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch7}
+\begin{paste}{ugxProblemNLDEQClosedPageFull7}{ugxProblemNLDEQClosedPageEmpty7}
+\pastebutton{ugxProblemNLDEQClosedPageFull7}{\hidepaste}
+\tab{5}\spadcommand{b := D(mu(y) * m, y) - D(mu(y) * n, x)\bound{b }\free{mu m }}
+\indentrel{3}\begin{verbatim}
+             ,
+   (7)  - ymu (y) - 2mu(y)
+
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty7}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty7}{ugxProblemNLDEQClosedPagePatch7}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{b := D(mu(y) * m, y) - D(mu(y) * n, x)\bound{b }\free{mu m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch8}
+\begin{paste}{ugxProblemNLDEQClosedPageFull8}{ugxProblemNLDEQClosedPageEmpty8}
+\pastebutton{ugxProblemNLDEQClosedPageFull8}{\hidepaste}
+\tab{5}\spadcommand{sb := solve(b = 0, mu, y)\free{mu b }\bound{sb }}
+\indentrel{3}\begin{verbatim}
+                                1
+   (8)  [particular= 0,basis= [ÄÄ]]
+                                2
+                               y
+Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty8}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty8}{ugxProblemNLDEQClosedPagePatch8}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{sb := solve(b = 0, mu, y)\free{mu b }\bound{sb }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch9}
+\begin{paste}{ugxProblemNLDEQClosedPageFull9}{ugxProblemNLDEQClosedPageEmpty9}
+\pastebutton{ugxProblemNLDEQClosedPageFull9}{\hidepaste}
+\tab{5}\spadcommand{intFactor := sb.basis.1\bound{intFactor }\free{sb }}
+\indentrel{3}\begin{verbatim}
+         1
+   (9)  ÄÄ
+         2
+        y
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty9}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty9}{ugxProblemNLDEQClosedPagePatch9}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{intFactor := sb.basis.1\bound{intFactor }\free{sb }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch10}
+\begin{paste}{ugxProblemNLDEQClosedPageFull10}{ugxProblemNLDEQClosedPageEmpty10}
+\pastebutton{ugxProblemNLDEQClosedPageFull10}{\hidepaste}
+\tab{5}\spadcommand{m := intFactor * m\bound{m1 }\free{m intFactor }}
+\indentrel{3}\begin{verbatim}
+           1
+   (10)  - Ä
+           y
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty10}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty10}{ugxProblemNLDEQClosedPagePatch10}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{m := intFactor * m\bound{m1 }\free{m intFactor }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch11}
+\begin{paste}{ugxProblemNLDEQClosedPageFull11}{ugxProblemNLDEQClosedPageEmpty11}
+\pastebutton{ugxProblemNLDEQClosedPageFull11}{\hidepaste}
+\tab{5}\spadcommand{n := intFactor * n\bound{n1 }\free{n intFactor }}
+\indentrel{3}\begin{verbatim}
+         y log(y) + x
+   (11)  ÄÄÄÄÄÄÄÄÄÄÄÄ
+               2
+              y
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty11}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty11}{ugxProblemNLDEQClosedPagePatch11}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{n := intFactor * n\bound{n1 }\free{n intFactor }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch12}
+\begin{paste}{ugxProblemNLDEQClosedPageFull12}{ugxProblemNLDEQClosedPageEmpty12}
+\pastebutton{ugxProblemNLDEQClosedPageFull12}{\hidepaste}
+\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m1 n1 }}
+\indentrel{3}\begin{verbatim}
+   (12)  0
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty12}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty12}{ugxProblemNLDEQClosedPagePatch12}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m1 n1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch13}
+\begin{paste}{ugxProblemNLDEQClosedPageFull13}{ugxProblemNLDEQClosedPageEmpty13}
+\pastebutton{ugxProblemNLDEQClosedPageFull13}{\hidepaste}
+\tab{5}\spadcommand{h := operator 'h\bound{h }}
+\indentrel{3}\begin{verbatim}
+   (13)  h
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty13}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty13}{ugxProblemNLDEQClosedPagePatch13}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{h := operator 'h\bound{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch14}
+\begin{paste}{ugxProblemNLDEQClosedPageFull14}{ugxProblemNLDEQClosedPageEmpty14}
+\pastebutton{ugxProblemNLDEQClosedPageFull14}{\hidepaste}
+\tab{5}\spadcommand{sol := h y + integrate(m, x)\bound{sol }\free{h m1 }}
+\indentrel{3}\begin{verbatim}
+         y h(y) - x
+   (14)  ÄÄÄÄÄÄÄÄÄÄ
+              y
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty14}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty14}{ugxProblemNLDEQClosedPagePatch14}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{sol := h y + integrate(m, x)\bound{sol }\free{h m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch15}
+\begin{paste}{ugxProblemNLDEQClosedPageFull15}{ugxProblemNLDEQClosedPageEmpty15}
+\pastebutton{ugxProblemNLDEQClosedPageFull15}{\hidepaste}
+\tab{5}\spadcommand{dsol := D(sol, y)\free{sol }\bound{dsol }}
+\indentrel{3}\begin{verbatim}
+          2 ,
+         y h (y) + x
+
+   (15)  ÄÄÄÄÄÄÄÄÄÄÄ
+               2
+              y
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty15}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty15}{ugxProblemNLDEQClosedPagePatch15}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{dsol := D(sol, y)\free{sol }\bound{dsol }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch16}
+\begin{paste}{ugxProblemNLDEQClosedPageFull16}{ugxProblemNLDEQClosedPageEmpty16}
+\pastebutton{ugxProblemNLDEQClosedPageFull16}{\hidepaste}
+\tab{5}\spadcommand{nsol := solve(dsol = n, h, y)\free{dsol n1 h }\bound{nsol }}
+\indentrel{3}\begin{verbatim}
+                            2
+                      log(y)
+   (16)  [particular= ÄÄÄÄÄÄÄ,basis= [1]]
+                         2
+Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty16}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty16}{ugxProblemNLDEQClosedPagePatch16}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{nsol := solve(dsol = n, h, y)\free{dsol n1 h }\bound{nsol }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch17}
+\begin{paste}{ugxProblemNLDEQClosedPageFull17}{ugxProblemNLDEQClosedPageEmpty17}
+\pastebutton{ugxProblemNLDEQClosedPageFull17}{\hidepaste}
+\tab{5}\spadcommand{eval(sol, h y = nsol.particular)\free{sol h nsol }}
+\indentrel{3}\begin{verbatim}
+                 2
+         y log(y)  - 2x
+   (17)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+               2y
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty17}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty17}{ugxProblemNLDEQClosedPagePatch17}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{eval(sol, h y = nsol.particular)\free{sol h nsol }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch18}
+\begin{paste}{ugxProblemNLDEQClosedPageFull18}{ugxProblemNLDEQClosedPageEmpty18}
+\pastebutton{ugxProblemNLDEQClosedPageFull18}{\hidepaste}
+\tab{5}\spadcommand{y := operator 'y\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (18)  y
+                                    Type: BasicOperator
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty18}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty18}{ugxProblemNLDEQClosedPagePatch18}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{y := operator 'y\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch19}
+\begin{paste}{ugxProblemNLDEQClosedPageFull19}{ugxProblemNLDEQClosedPageEmpty19}
+\pastebutton{ugxProblemNLDEQClosedPageFull19}{\hidepaste}
+\tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }}
+\indentrel{3}\begin{verbatim}
+          ,            y(x)
+   (19)  y (x)= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                y(x)log(y(x)) + x
+                      Type: Equation Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty19}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty19}{ugxProblemNLDEQClosedPagePatch19}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPagePatch20}
+\begin{paste}{ugxProblemNLDEQClosedPageFull20}{ugxProblemNLDEQClosedPageEmpty20}
+\pastebutton{ugxProblemNLDEQClosedPageFull20}{\hidepaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }}
+\indentrel{3}\begin{verbatim}
+                      2
+         y(x)log(y(x))  - 2x
+   (20)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                2y(x)
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemNLDEQClosedPageEmpty20}
+\begin{paste}{ugxProblemNLDEQClosedPageEmpty20}{ugxProblemNLDEQClosedPagePatch20}
+\pastebutton{ugxProblemNLDEQClosedPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPagePatch1}
+\begin{paste}{ugxProblemSymRootAllPageFull1}{ugxProblemSymRootAllPageEmpty1}
+\pastebutton{ugxProblemSymRootAllPageFull1}{\hidepaste}
+\tab{5}\spadcommand{l := rootsOf(x**4+1,x)\bound{l }}
+\indentrel{3}\begin{verbatim}
+   (1)  [%x0,%x0 %x1,- %x0,- %x0 %x1]
+                          Type: List Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPageEmpty1}
+\begin{paste}{ugxProblemSymRootAllPageEmpty1}{ugxProblemSymRootAllPagePatch1}
+\pastebutton{ugxProblemSymRootAllPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{l := rootsOf(x**4+1,x)\bound{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPagePatch2}
+\begin{paste}{ugxProblemSymRootAllPageFull2}{ugxProblemSymRootAllPageEmpty2}
+\pastebutton{ugxProblemSymRootAllPageFull2}{\hidepaste}
+\tab{5}\spadcommand{\%x0**5\free{l }}
+\indentrel{3}\begin{verbatim}
+   (2)  - %x0
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPageEmpty2}
+\begin{paste}{ugxProblemSymRootAllPageEmpty2}{ugxProblemSymRootAllPagePatch2}
+\pastebutton{ugxProblemSymRootAllPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{\%x0**5\free{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPagePatch3}
+\begin{paste}{ugxProblemSymRootAllPageFull3}{ugxProblemSymRootAllPageEmpty3}
+\pastebutton{ugxProblemSymRootAllPageFull3}{\hidepaste}
+\tab{5}\spadcommand{definingPolynomial \%x0\free{l }}
+\indentrel{3}\begin{verbatim}
+           4
+   (3)  %x0  + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPageEmpty3}
+\begin{paste}{ugxProblemSymRootAllPageEmpty3}{ugxProblemSymRootAllPagePatch3}
+\pastebutton{ugxProblemSymRootAllPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{definingPolynomial \%x0\free{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPagePatch4}
+\begin{paste}{ugxProblemSymRootAllPageFull4}{ugxProblemSymRootAllPageEmpty4}
+\pastebutton{ugxProblemSymRootAllPageFull4}{\hidepaste}
+\tab{5}\spadcommand{definingPolynomial \%x1\free{l }}
+\indentrel{3}\begin{verbatim}
+           2
+   (4)  %x1  + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPageEmpty4}
+\begin{paste}{ugxProblemSymRootAllPageEmpty4}{ugxProblemSymRootAllPagePatch4}
+\pastebutton{ugxProblemSymRootAllPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{definingPolynomial \%x1\free{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPagePatch5}
+\begin{paste}{ugxProblemSymRootAllPageFull5}{ugxProblemSymRootAllPageEmpty5}
+\pastebutton{ugxProblemSymRootAllPageFull5}{\hidepaste}
+\tab{5}\spadcommand{definingPolynomial \%x2\free{l }}
+\indentrel{3}\begin{verbatim}
+   (5)  - %x2 + %%var
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPageEmpty5}
+\begin{paste}{ugxProblemSymRootAllPageEmpty5}{ugxProblemSymRootAllPagePatch5}
+\pastebutton{ugxProblemSymRootAllPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{definingPolynomial \%x2\free{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPagePatch6}
+\begin{paste}{ugxProblemSymRootAllPageFull6}{ugxProblemSymRootAllPageEmpty6}
+\pastebutton{ugxProblemSymRootAllPageFull6}{\hidepaste}
+\tab{5}\spadcommand{x3 := last l\free{l }\bound{x3 }}
+\indentrel{3}\begin{verbatim}
+   (6)  - %x0 %x1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPageEmpty6}
+\begin{paste}{ugxProblemSymRootAllPageEmpty6}{ugxProblemSymRootAllPagePatch6}
+\pastebutton{ugxProblemSymRootAllPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{x3 := last l\free{l }\bound{x3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPagePatch7}
+\begin{paste}{ugxProblemSymRootAllPageFull7}{ugxProblemSymRootAllPageEmpty7}
+\pastebutton{ugxProblemSymRootAllPageFull7}{\hidepaste}
+\tab{5}\spadcommand{\%x0 + \%x1 + \%x2 + x3\free{x3 }}
+\indentrel{3}\begin{verbatim}
+   (7)  (- %x0 + 1)%x1 + %x0 + %x2
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPageEmpty7}
+\begin{paste}{ugxProblemSymRootAllPageEmpty7}{ugxProblemSymRootAllPagePatch7}
+\pastebutton{ugxProblemSymRootAllPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{\%x0 + \%x1 + \%x2 + x3\free{x3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPagePatch8}
+\begin{paste}{ugxProblemSymRootAllPageFull8}{ugxProblemSymRootAllPageEmpty8}
+\pastebutton{ugxProblemSymRootAllPageFull8}{\hidepaste}
+\tab{5}\spadcommand{\%x0 * \%x1 * \%x2 * x3\free{x3 }}
+\indentrel{3}\begin{verbatim}
+               2
+   (8)  %x2 %x0
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPageEmpty8}
+\begin{paste}{ugxProblemSymRootAllPageEmpty8}{ugxProblemSymRootAllPagePatch8}
+\pastebutton{ugxProblemSymRootAllPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{\%x0 * \%x1 * \%x2 * x3\free{x3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPagePatch9}
+\begin{paste}{ugxProblemSymRootAllPageFull9}{ugxProblemSymRootAllPageEmpty9}
+\pastebutton{ugxProblemSymRootAllPageFull9}{\hidepaste}
+\tab{5}\spadcommand{zerosOf(y**4+1,y)\bound{z }}
+\indentrel{3}\begin{verbatim}
+   (9)
+     ÚÄÄÄ¿      ÚÄÄÄ¿        ÚÄÄÄ¿        ÚÄÄÄ¿
+    \³- 1  + 1 \³- 1  - 1 - \³- 1  - 1 - \³- 1  + 1
+   [ÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄ,ÄÄÄÄÄÄÄÄÄÄÄÄ]
+        ÚÄ¿        ÚÄ¿         ÚÄ¿          ÚÄ¿
+       \³2        \³2         \³2          \³2
+                          Type: List Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPageEmpty9}
+\begin{paste}{ugxProblemSymRootAllPageEmpty9}{ugxProblemSymRootAllPagePatch9}
+\pastebutton{ugxProblemSymRootAllPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{zerosOf(y**4+1,y)\bound{z }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPagePatch10}
+\begin{paste}{ugxProblemSymRootAllPageFull10}{ugxProblemSymRootAllPageEmpty10}
+\pastebutton{ugxProblemSymRootAllPageFull10}{\hidepaste}
+\tab{5}\spadcommand{definingPolynomial \%y1\free{z }}
+\indentrel{3}\begin{verbatim}
+              2
+   (10)  %%var  + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootAllPageEmpty10}
+\begin{paste}{ugxProblemSymRootAllPageEmpty10}{ugxProblemSymRootAllPagePatch10}
+\pastebutton{ugxProblemSymRootAllPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{definingPolynomial \%y1\free{z }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePagePatch1}
+\begin{paste}{ugxProblemSymRootOnePageFull1}{ugxProblemSymRootOnePageEmpty1}
+\pastebutton{ugxProblemSymRootOnePageFull1}{\hidepaste}
+\tab{5}\spadcommand{a := rootOf(a**4+1,a)\bound{a }}
+\indentrel{3}\begin{verbatim}
+   (1)  a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePageEmpty1}
+\begin{paste}{ugxProblemSymRootOnePageEmpty1}{ugxProblemSymRootOnePagePatch1}
+\pastebutton{ugxProblemSymRootOnePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{a := rootOf(a**4+1,a)\bound{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePagePatch2}
+\begin{paste}{ugxProblemSymRootOnePageFull2}{ugxProblemSymRootOnePageEmpty2}
+\pastebutton{ugxProblemSymRootOnePageFull2}{\hidepaste}
+\tab{5}\spadcommand{definingPolynomial a\free{a }}
+\indentrel{3}\begin{verbatim}
+         4
+   (2)  a  + 1
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePageEmpty2}
+\begin{paste}{ugxProblemSymRootOnePageEmpty2}{ugxProblemSymRootOnePagePatch2}
+\pastebutton{ugxProblemSymRootOnePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{definingPolynomial a\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePagePatch3}
+\begin{paste}{ugxProblemSymRootOnePageFull3}{ugxProblemSymRootOnePageEmpty3}
+\pastebutton{ugxProblemSymRootOnePageFull3}{\hidepaste}
+\tab{5}\spadcommand{b := rootOf(b**2-a-1,b)\free{a }\bound{b }}
+\indentrel{3}\begin{verbatim}
+   (3)  b
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePageEmpty3}
+\begin{paste}{ugxProblemSymRootOnePageEmpty3}{ugxProblemSymRootOnePagePatch3}
+\pastebutton{ugxProblemSymRootOnePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{b := rootOf(b**2-a-1,b)\free{a }\bound{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePagePatch4}
+\begin{paste}{ugxProblemSymRootOnePageFull4}{ugxProblemSymRootOnePageEmpty4}
+\pastebutton{ugxProblemSymRootOnePageFull4}{\hidepaste}
+\tab{5}\spadcommand{a + b\free{a b }\bound{c }}
+\indentrel{3}\begin{verbatim}
+   (4)  b + a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePageEmpty4}
+\begin{paste}{ugxProblemSymRootOnePageEmpty4}{ugxProblemSymRootOnePagePatch4}
+\pastebutton{ugxProblemSymRootOnePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{a + b\free{a b }\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePagePatch5}
+\begin{paste}{ugxProblemSymRootOnePageFull5}{ugxProblemSymRootOnePageEmpty5}
+\pastebutton{ugxProblemSymRootOnePageFull5}{\hidepaste}
+\tab{5}\spadcommand{\% ** 5\free{c }}
+\indentrel{3}\begin{verbatim}
+            3      2                 3      2
+   (5)  (10a  + 11a  + 2a - 4)b + 15a  + 10a  + 4a - 10
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePageEmpty5}
+\begin{paste}{ugxProblemSymRootOnePageEmpty5}{ugxProblemSymRootOnePagePatch5}
+\pastebutton{ugxProblemSymRootOnePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{\% ** 5\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePagePatch6}
+\begin{paste}{ugxProblemSymRootOnePageFull6}{ugxProblemSymRootOnePageEmpty6}
+\pastebutton{ugxProblemSymRootOnePageFull6}{\hidepaste}
+\tab{5}\spadcommand{rootOf(c**2+c+1,c)}
+\indentrel{3}\begin{verbatim}
+   (6)  c
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePageEmpty6}
+\begin{paste}{ugxProblemSymRootOnePageEmpty6}{ugxProblemSymRootOnePagePatch6}
+\pastebutton{ugxProblemSymRootOnePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{rootOf(c**2+c+1,c)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePagePatch7}
+\begin{paste}{ugxProblemSymRootOnePageFull7}{ugxProblemSymRootOnePageEmpty7}
+\pastebutton{ugxProblemSymRootOnePageFull7}{\hidepaste}
+\tab{5}\spadcommand{zeroOf(d**2+d+1,d)}
+\indentrel{3}\begin{verbatim}
+         ÚÄÄÄ¿
+        \³- 3  - 1
+   (7)  ÄÄÄÄÄÄÄÄÄÄ
+             2
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePageEmpty7}
+\begin{paste}{ugxProblemSymRootOnePageEmpty7}{ugxProblemSymRootOnePagePatch7}
+\pastebutton{ugxProblemSymRootOnePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{zeroOf(d**2+d+1,d)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePagePatch8}
+\begin{paste}{ugxProblemSymRootOnePageFull8}{ugxProblemSymRootOnePageEmpty8}
+\pastebutton{ugxProblemSymRootOnePageFull8}{\hidepaste}
+\tab{5}\spadcommand{rootOf(e**5-2,e)}
+\indentrel{3}\begin{verbatim}
+   (8)  e
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePageEmpty8}
+\begin{paste}{ugxProblemSymRootOnePageEmpty8}{ugxProblemSymRootOnePagePatch8}
+\pastebutton{ugxProblemSymRootOnePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{rootOf(e**5-2,e)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePagePatch9}
+\begin{paste}{ugxProblemSymRootOnePageFull9}{ugxProblemSymRootOnePageEmpty9}
+\pastebutton{ugxProblemSymRootOnePageFull9}{\hidepaste}
+\tab{5}\spadcommand{zeroOf(f**5-2,f)}
+\indentrel{3}\begin{verbatim}
+        5ÚÄ¿
+   (9)  \³2
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSymRootOnePageEmpty9}
+\begin{paste}{ugxProblemSymRootOnePageEmpty9}{ugxProblemSymRootOnePagePatch9}
+\pastebutton{ugxProblemSymRootOnePageEmpty9}{\showpaste}
+\tab{5}\spadcommand{zeroOf(f**5-2,f)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPagePatch1}
+\begin{paste}{ugProblemIntegrationPageFull1}{ugProblemIntegrationPageEmpty1}
+\pastebutton{ugProblemIntegrationPageFull1}{\hidepaste}
+\tab{5}\spadcommand{integrate(cosh(a*x)*sinh(a*x), x)}
+\indentrel{3}\begin{verbatim}
+                 2            2
+        sinh(a x)  + cosh(a x)
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                   4a
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPageEmpty1}
+\begin{paste}{ugProblemIntegrationPageEmpty1}{ugProblemIntegrationPagePatch1}
+\pastebutton{ugProblemIntegrationPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{integrate(cosh(a*x)*sinh(a*x), x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPagePatch2}
+\begin{paste}{ugProblemIntegrationPageFull2}{ugProblemIntegrationPageEmpty2}
+\pastebutton{ugProblemIntegrationPageFull2}{\hidepaste}
+\tab{5}\spadcommand{integrate(log(1 + sqrt(a * x + b)) / x, x)}
+\indentrel{3}\begin{verbatim}
+           x      ÚÄÄÄÄÄÄÄÄ¿
+         Ú¿  log(\³b + %E a  + 1)
+   (2)   ³   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ d%E
+        ÀÙ            %E
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPageEmpty2}
+\begin{paste}{ugProblemIntegrationPageEmpty2}{ugProblemIntegrationPagePatch2}
+\pastebutton{ugProblemIntegrationPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{integrate(log(1 + sqrt(a * x + b)) / x, x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPagePatch3}
+\begin{paste}{ugProblemIntegrationPageFull3}{ugProblemIntegrationPageEmpty3}
+\pastebutton{ugProblemIntegrationPageFull3}{\hidepaste}
+\tab{5}\spadcommand{integrate(1/(x**2 - 2),x)}
+\indentrel{3}\begin{verbatim}
+              2      ÚÄ¿
+            (x  + 2)\³2  - 4x
+        log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
+                   2
+                  x  - 2
+   (3)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                   ÚÄ¿
+                 2\³2
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPageEmpty3}
+\begin{paste}{ugProblemIntegrationPageEmpty3}{ugProblemIntegrationPagePatch3}
+\pastebutton{ugProblemIntegrationPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{integrate(1/(x**2 - 2),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPagePatch4}
+\begin{paste}{ugProblemIntegrationPageFull4}{ugProblemIntegrationPageEmpty4}
+\pastebutton{ugProblemIntegrationPageFull4}{\hidepaste}
+\tab{5}\spadcommand{integrate(1/(x**2 + 2),x)}
+\indentrel{3}\begin{verbatim}
+               ÚÄ¿
+             x\³2
+        atan(ÄÄÄÄÄ)
+               2
+   (4)  ÄÄÄÄÄÄÄÄÄÄÄ
+             ÚÄ¿
+            \³2
+                    Type: Union(Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPageEmpty4}
+\begin{paste}{ugProblemIntegrationPageEmpty4}{ugProblemIntegrationPagePatch4}
+\pastebutton{ugProblemIntegrationPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{integrate(1/(x**2 + 2),x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPagePatch5}
+\begin{paste}{ugProblemIntegrationPageFull5}{ugProblemIntegrationPageEmpty5}
+\pastebutton{ugProblemIntegrationPageFull5}{\hidepaste}
+\tab{5}\spadcommand{integrate(x**2 / (x**4 - a**2), x)}
+\indentrel{3}\begin{verbatim}
+   (5)
+          2      ÚÄ¿                   ÚÄ¿
+        (x  + a)\³a  - 2a x          x\³a
+    log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ) + 2atan(ÄÄÄÄÄ)
+                2                      a
+               x  - a
+   [ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,
+                       ÚÄ¿
+                     4\³a
+          2      ÚÄÄÄ¿                   ÚÄÄÄ¿
+        (x  - a)\³- a  + 2a x          x\³- a
+    log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ) - 2atan(ÄÄÄÄÄÄÄ)
+                 2                        a
+                x  + a
+    ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ]
+                        ÚÄÄÄ¿
+                      4\³- a
+               Type: Union(List Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPageEmpty5}
+\begin{paste}{ugProblemIntegrationPageEmpty5}{ugProblemIntegrationPagePatch5}
+\pastebutton{ugProblemIntegrationPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{integrate(x**2 / (x**4 - a**2), x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPagePatch6}
+\begin{paste}{ugProblemIntegrationPageFull6}{ugProblemIntegrationPageEmpty6}
+\pastebutton{ugProblemIntegrationPageFull6}{\hidepaste}
+\tab{5}\spadcommand{complexIntegrate(x**2 / (x**4 - a**2), x)}
+\indentrel{3}\begin{verbatim}
+   (6)
+                  ÚÄÄÄÄ¿                      ÚÄÄ¿
+        ÚÄÄ¿    x\³- 4a  + 2a     ÚÄÄÄÄ¿    x\³4a  + 2a
+       \³4a log(ÄÄÄÄÄÄÄÄÄÄÄÄÄ) - \³- 4a log(ÄÄÄÄÄÄÄÄÄÄÄ)
+                    ÚÄÄÄÄ¿                      ÚÄÄ¿
+                   \³- 4a                      \³4a
+     + 
+                    ÚÄÄ¿                    ÚÄÄÄÄ¿
+        ÚÄÄÄÄ¿    x\³4a  - 2a     ÚÄÄ¿    x\³- 4a  - 2a
+       \³- 4a log(ÄÄÄÄÄÄÄÄÄÄÄ) - \³4a log(ÄÄÄÄÄÄÄÄÄÄÄÄÄ)
+                      ÚÄÄ¿                    ÚÄÄÄÄ¿
+                     \³4a                    \³- 4a
+  /
+       ÚÄÄÄÄ¿ ÚÄÄ¿
+     2\³- 4a \³4a
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPageEmpty6}
+\begin{paste}{ugProblemIntegrationPageEmpty6}{ugProblemIntegrationPagePatch6}
+\pastebutton{ugProblemIntegrationPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{complexIntegrate(x**2 / (x**4 - a**2), x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPagePatch7}
+\begin{paste}{ugProblemIntegrationPageFull7}{ugProblemIntegrationPageEmpty7}
+\pastebutton{ugProblemIntegrationPageFull7}{\hidepaste}
+\tab{5}\spadcommand{complexIntegrate(log(1 + sqrt(a * x + b)) / x, x)}
+\indentrel{3}\begin{verbatim}
+           x      ÚÄÄÄÄÄÄÄÄ¿
+         Ú¿  log(\³b + %E a  + 1)
+   (7)   ³   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ d%E
+        ÀÙ            %E
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPageEmpty7}
+\begin{paste}{ugProblemIntegrationPageEmpty7}{ugProblemIntegrationPagePatch7}
+\pastebutton{ugProblemIntegrationPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{complexIntegrate(log(1 + sqrt(a * x + b)) / x, x)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPagePatch8}
+\begin{paste}{ugProblemIntegrationPageFull8}{ugProblemIntegrationPageEmpty8}
+\pastebutton{ugProblemIntegrationPageFull8}{\hidepaste}
+\tab{5}\spadcommand{integrate((x**4 - 3*x**2 + 6)/(x**6-5*x**4+5*x**2+4), x = 1..2)}
+\indentrel{3}\begin{verbatim}
+                                               1
+        2atan(8) + 2atan(5) + 2atan(2) + 2atan(Ä) - %pi
+                                               2
+   (8)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                               2
+Type: Union(f1: OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPageEmpty8}
+\begin{paste}{ugProblemIntegrationPageEmpty8}{ugProblemIntegrationPagePatch8}
+\pastebutton{ugProblemIntegrationPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{integrate((x**4 - 3*x**2 + 6)/(x**6-5*x**4+5*x**2+4), x = 1..2)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPagePatch9}
+\begin{paste}{ugProblemIntegrationPageFull9}{ugProblemIntegrationPageEmpty9}
+\pastebutton{ugProblemIntegrationPageFull9}{\hidepaste}
+\tab{5}\spadcommand{integrate(1/(x**2-a), x = 1..2)}
+\indentrel{3}\begin{verbatim}
+   (9)  potentialPole
+                   Type: Union(pole: potentialPole,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPageEmpty9}
+\begin{paste}{ugProblemIntegrationPageEmpty9}{ugProblemIntegrationPagePatch9}
+\pastebutton{ugProblemIntegrationPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{integrate(1/(x**2-a), x = 1..2)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPagePatch10}
+\begin{paste}{ugProblemIntegrationPageFull10}{ugProblemIntegrationPageEmpty10}
+\pastebutton{ugProblemIntegrationPageFull10}{\hidepaste}
+\tab{5}\spadcommand{integrate(1/(x**2-a), x = 1..2, "noPole")}
+\indentrel{3}\begin{verbatim}
+   (10)
+   [
+                    2       ÚÄ¿    3     2
+               (- 4a  - 4a)\³a  + a  + 6a  + a
+         - log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
+                          2
+                         a  - 2a + 1
+       + 
+                  2        ÚÄ¿    3      2
+             (- 8a  - 32a)\³a  + a  + 24a  + 16a
+         log(ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
+                          2
+                         a  - 8a + 16
+    /
+         ÚÄ¿
+       4\³a
+     ,
+             ÚÄÄÄ¿          ÚÄÄÄ¿
+           2\³- a          \³- a
+    - atan(ÄÄÄÄÄÄÄ) + atan(ÄÄÄÄÄÄ)
+              a               a
+    ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ]
+                 ÚÄÄÄ¿
+                \³- a
+Type: Union(f2: List OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIntegrationPageEmpty10}
+\begin{paste}{ugProblemIntegrationPageEmpty10}{ugProblemIntegrationPagePatch10}
+\pastebutton{ugProblemIntegrationPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{integrate(1/(x**2-a), x = 1..2, "noPole")}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorRatFunPagePatch1}
+\begin{paste}{ugProblemFactorRatFunPageFull1}{ugProblemFactorRatFunPageEmpty1}
+\pastebutton{ugProblemFactorRatFunPageFull1}{\hidepaste}
+\tab{5}\spadcommand{factorFraction((x**2-4)/(y**2-4))}
+\indentrel{3}\begin{verbatim}
+        (x - 2)(x + 2)
+   (1)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+        (y - 2)(y + 2)
+             Type: Fraction Factored Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorRatFunPageEmpty1}
+\begin{paste}{ugProblemFactorRatFunPageEmpty1}{ugProblemFactorRatFunPagePatch1}
+\pastebutton{ugProblemFactorRatFunPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{factorFraction((x**2-4)/(y**2-4))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorRatFunPagePatch2}
+\begin{paste}{ugProblemFactorRatFunPageFull2}{ugProblemFactorRatFunPageEmpty2}
+\pastebutton{ugProblemFactorRatFunPageFull2}{\hidepaste}
+\tab{5}\spadcommand{map(factor,(x**2-4)/(y**2-4))}
+\indentrel{3}\begin{verbatim}
+        (x - 2)(x + 2)
+   (2)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+        (y - 2)(y + 2)
+             Type: Fraction Factored Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorRatFunPageEmpty2}
+\begin{paste}{ugProblemFactorRatFunPageEmpty2}{ugProblemFactorRatFunPagePatch2}
+\pastebutton{ugProblemFactorRatFunPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{map(factor,(x**2-4)/(y**2-4))}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorIntRatPagePatch1}
+\begin{paste}{ugProblemFactorIntRatPageFull1}{ugProblemFactorIntRatPageEmpty1}
+\pastebutton{ugProblemFactorIntRatPageFull1}{\hidepaste}
+\tab{5}\spadcommand{v := (4*x**3+2*y**2+1)*(12*x**5-x**3*y+12)\bound{v }}
+\indentrel{3}\begin{verbatim}
+   (1)
+         3 3       5       2        6    3        8      5
+     - 2x y  + (24x  + 24)y  + (- 4x  - x )y + 48x  + 12x
+   + 
+        3
+     48x  + 12
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorIntRatPageEmpty1}
+\begin{paste}{ugProblemFactorIntRatPageEmpty1}{ugProblemFactorIntRatPagePatch1}
+\pastebutton{ugProblemFactorIntRatPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{v := (4*x**3+2*y**2+1)*(12*x**5-x**3*y+12)\bound{v }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorIntRatPagePatch2}
+\begin{paste}{ugProblemFactorIntRatPageFull2}{ugProblemFactorIntRatPageEmpty2}
+\pastebutton{ugProblemFactorIntRatPageFull2}{\hidepaste}
+\tab{5}\spadcommand{factor v\free{v }}
+\indentrel{3}\begin{verbatim}
+            3       5         2     3
+   (2)  - (x y - 12x  - 12)(2y  + 4x  + 1)
+                      Type: Factored Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorIntRatPageEmpty2}
+\begin{paste}{ugProblemFactorIntRatPageEmpty2}{ugProblemFactorIntRatPagePatch2}
+\pastebutton{ugProblemFactorIntRatPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{factor v\free{v }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorIntRatPagePatch3}
+\begin{paste}{ugProblemFactorIntRatPageFull3}{ugProblemFactorIntRatPageEmpty3}
+\pastebutton{ugProblemFactorIntRatPageFull3}{\hidepaste}
+\tab{5}\spadcommand{w := (4*x**3+(2/3)*x**2+1)*(12*x**5-(1/2)*x**3+12)\bound{w }}
+\indentrel{3}\begin{verbatim}
+           8     7     6   35  5   95  3     2
+   (3)  48x  + 8x  - 2x  + ÄÄ x  + ÄÄ x  + 8x  + 12
+                            3       2
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorIntRatPageEmpty3}
+\begin{paste}{ugProblemFactorIntRatPageEmpty3}{ugProblemFactorIntRatPagePatch3}
+\pastebutton{ugProblemFactorIntRatPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{w := (4*x**3+(2/3)*x**2+1)*(12*x**5-(1/2)*x**3+12)\bound{w }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorIntRatPagePatch4}
+\begin{paste}{ugProblemFactorIntRatPageFull4}{ugProblemFactorIntRatPageEmpty4}
+\pastebutton{ugProblemFactorIntRatPageFull4}{\hidepaste}
+\tab{5}\spadcommand{factor w\free{w }}
+\indentrel{3}\begin{verbatim}
+            3   1  2   1   5    1  3
+   (4)  48(x  + Ä x  + Ä)(x  - ÄÄ x  + 1)
+                6      4       24
+             Type: Factored Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemFactorIntRatPageEmpty4}
+\begin{paste}{ugProblemFactorIntRatPageEmpty4}{ugProblemFactorIntRatPagePatch4}
+\pastebutton{ugProblemFactorIntRatPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{factor w\free{w }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch1}
+\begin{paste}{ugxProblemFinitePrimePageFull1}{ugxProblemFinitePrimePageEmpty1}
+\pastebutton{ugxProblemFinitePrimePageFull1}{\hidepaste}
+\tab{5}\spadcommand{(a,b) : IntegerMod 12\bound{abdec }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty1}
+\begin{paste}{ugxProblemFinitePrimePageEmpty1}{ugxProblemFinitePrimePagePatch1}
+\pastebutton{ugxProblemFinitePrimePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{(a,b) : IntegerMod 12\bound{abdec }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch2}
+\begin{paste}{ugxProblemFinitePrimePageFull2}{ugxProblemFinitePrimePageEmpty2}
+\pastebutton{ugxProblemFinitePrimePageFull2}{\hidepaste}
+\tab{5}\spadcommand{(a, b) := (16, 7)\free{abdec }\bound{a b }}
+\indentrel{3}\begin{verbatim}
+   (2)  7
+                                    Type: IntegerMod 12
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty2}
+\begin{paste}{ugxProblemFinitePrimePageEmpty2}{ugxProblemFinitePrimePagePatch2}
+\pastebutton{ugxProblemFinitePrimePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{(a, b) := (16, 7)\free{abdec }\bound{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch3}
+\begin{paste}{ugxProblemFinitePrimePageFull3}{ugxProblemFinitePrimePageEmpty3}
+\pastebutton{ugxProblemFinitePrimePageFull3}{\hidepaste}
+\tab{5}\spadcommand{[a - b, a * b]\free{a b }}
+\indentrel{3}\begin{verbatim}
+   (3)  [9,4]
+                               Type: List IntegerMod 12
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty3}
+\begin{paste}{ugxProblemFinitePrimePageEmpty3}{ugxProblemFinitePrimePagePatch3}
+\pastebutton{ugxProblemFinitePrimePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{[a - b, a * b]\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch4}
+\begin{paste}{ugxProblemFinitePrimePageFull4}{ugxProblemFinitePrimePageEmpty4}
+\pastebutton{ugxProblemFinitePrimePageFull4}{\hidepaste}
+\tab{5}\spadcommand{a / b\free{a b }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty4}
+\begin{paste}{ugxProblemFinitePrimePageEmpty4}{ugxProblemFinitePrimePagePatch4}
+\pastebutton{ugxProblemFinitePrimePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{a / b\free{a b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch5}
+\begin{paste}{ugxProblemFinitePrimePageFull5}{ugxProblemFinitePrimePageEmpty5}
+\pastebutton{ugxProblemFinitePrimePageFull5}{\hidepaste}
+\tab{5}\spadcommand{recip a\free{a }}
+\indentrel{3}\begin{verbatim}
+   (4)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty5}
+\begin{paste}{ugxProblemFinitePrimePageEmpty5}{ugxProblemFinitePrimePagePatch5}
+\pastebutton{ugxProblemFinitePrimePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{recip a\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch6}
+\begin{paste}{ugxProblemFinitePrimePageFull6}{ugxProblemFinitePrimePageEmpty6}
+\pastebutton{ugxProblemFinitePrimePageFull6}{\hidepaste}
+\tab{5}\spadcommand{recip b\free{b }}
+\indentrel{3}\begin{verbatim}
+   (5)  7
+                         Type: Union(IntegerMod 12,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty6}
+\begin{paste}{ugxProblemFinitePrimePageEmpty6}{ugxProblemFinitePrimePagePatch6}
+\pastebutton{ugxProblemFinitePrimePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{recip b\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch7}
+\begin{paste}{ugxProblemFinitePrimePageFull7}{ugxProblemFinitePrimePageEmpty7}
+\pastebutton{ugxProblemFinitePrimePageFull7}{\hidepaste}
+\tab{5}\spadcommand{c : PrimeField 11 := 8\bound{c }}
+\indentrel{3}\begin{verbatim}
+   (6)  8
+                                    Type: PrimeField 11
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty7}
+\begin{paste}{ugxProblemFinitePrimePageEmpty7}{ugxProblemFinitePrimePagePatch7}
+\pastebutton{ugxProblemFinitePrimePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{c : PrimeField 11 := 8\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch8}
+\begin{paste}{ugxProblemFinitePrimePageFull8}{ugxProblemFinitePrimePageEmpty8}
+\pastebutton{ugxProblemFinitePrimePageFull8}{\hidepaste}
+\tab{5}\spadcommand{inv c\free{c }}
+\indentrel{3}\begin{verbatim}
+   (7)  7
+                                    Type: PrimeField 11
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty8}
+\begin{paste}{ugxProblemFinitePrimePageEmpty8}{ugxProblemFinitePrimePagePatch8}
+\pastebutton{ugxProblemFinitePrimePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{inv c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch9}
+\begin{paste}{ugxProblemFinitePrimePageFull9}{ugxProblemFinitePrimePageEmpty9}
+\pastebutton{ugxProblemFinitePrimePageFull9}{\hidepaste}
+\tab{5}\spadcommand{9/c\free{c }}
+\indentrel{3}\begin{verbatim}
+   (8)  8
+                                    Type: PrimeField 11
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty9}
+\begin{paste}{ugxProblemFinitePrimePageEmpty9}{ugxProblemFinitePrimePagePatch9}
+\pastebutton{ugxProblemFinitePrimePageEmpty9}{\showpaste}
+\tab{5}\spadcommand{9/c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch10}
+\begin{paste}{ugxProblemFinitePrimePageFull10}{ugxProblemFinitePrimePageEmpty10}
+\pastebutton{ugxProblemFinitePrimePageFull10}{\hidepaste}
+\tab{5}\spadcommand{GF101 := PF 101\bound{GF101 }}
+\indentrel{3}\begin{verbatim}
+   (9)  PrimeField 101
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty10}
+\begin{paste}{ugxProblemFinitePrimePageEmpty10}{ugxProblemFinitePrimePagePatch10}
+\pastebutton{ugxProblemFinitePrimePageEmpty10}{\showpaste}
+\tab{5}\spadcommand{GF101 := PF 101\bound{GF101 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch11}
+\begin{paste}{ugxProblemFinitePrimePageFull11}{ugxProblemFinitePrimePageEmpty11}
+\pastebutton{ugxProblemFinitePrimePageFull11}{\hidepaste}
+\tab{5}\spadcommand{x := random()$GF101\bound{x }\free{GF101 }}
+\indentrel{3}\begin{verbatim}
+   (10)  78
+                                   Type: PrimeField 101
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty11}
+\begin{paste}{ugxProblemFinitePrimePageEmpty11}{ugxProblemFinitePrimePagePatch11}
+\pastebutton{ugxProblemFinitePrimePageEmpty11}{\showpaste}
+\tab{5}\spadcommand{x := random()$GF101\bound{x }\free{GF101 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch12}
+\begin{paste}{ugxProblemFinitePrimePageFull12}{ugxProblemFinitePrimePageEmpty12}
+\pastebutton{ugxProblemFinitePrimePageFull12}{\hidepaste}
+\tab{5}\spadcommand{y : GF101 := 37\bound{y }\free{GF101 }}
+\indentrel{3}\begin{verbatim}
+   (11)  37
+                                   Type: PrimeField 101
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty12}
+\begin{paste}{ugxProblemFinitePrimePageEmpty12}{ugxProblemFinitePrimePagePatch12}
+\pastebutton{ugxProblemFinitePrimePageEmpty12}{\showpaste}
+\tab{5}\spadcommand{y : GF101 := 37\bound{y }\free{GF101 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch13}
+\begin{paste}{ugxProblemFinitePrimePageFull13}{ugxProblemFinitePrimePageEmpty13}
+\pastebutton{ugxProblemFinitePrimePageFull13}{\hidepaste}
+\tab{5}\spadcommand{z := x/y\bound{z }\free{x y }}
+\indentrel{3}\begin{verbatim}
+   (12)  84
+                                   Type: PrimeField 101
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty13}
+\begin{paste}{ugxProblemFinitePrimePageEmpty13}{ugxProblemFinitePrimePagePatch13}
+\pastebutton{ugxProblemFinitePrimePageEmpty13}{\showpaste}
+\tab{5}\spadcommand{z := x/y\bound{z }\free{x y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch14}
+\begin{paste}{ugxProblemFinitePrimePageFull14}{ugxProblemFinitePrimePageEmpty14}
+\pastebutton{ugxProblemFinitePrimePageFull14}{\hidepaste}
+\tab{5}\spadcommand{z * y - x\free{x y z }}
+\indentrel{3}\begin{verbatim}
+   (13)  0
+                                   Type: PrimeField 101
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty14}
+\begin{paste}{ugxProblemFinitePrimePageEmpty14}{ugxProblemFinitePrimePagePatch14}
+\pastebutton{ugxProblemFinitePrimePageEmpty14}{\showpaste}
+\tab{5}\spadcommand{z * y - x\free{x y z }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch15}
+\begin{paste}{ugxProblemFinitePrimePageFull15}{ugxProblemFinitePrimePageEmpty15}
+\pastebutton{ugxProblemFinitePrimePageFull15}{\hidepaste}
+\tab{5}\spadcommand{pe := primitiveElement()$GF101\bound{pe }\free{GF101 }}
+\indentrel{3}\begin{verbatim}
+   (14)  2
+                                   Type: PrimeField 101
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty15}
+\begin{paste}{ugxProblemFinitePrimePageEmpty15}{ugxProblemFinitePrimePagePatch15}
+\pastebutton{ugxProblemFinitePrimePageEmpty15}{\showpaste}
+\tab{5}\spadcommand{pe := primitiveElement()$GF101\bound{pe }\free{GF101 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch16}
+\begin{paste}{ugxProblemFinitePrimePageFull16}{ugxProblemFinitePrimePageEmpty16}
+\pastebutton{ugxProblemFinitePrimePageFull16}{\hidepaste}
+\tab{5}\spadcommand{[pe**i for i in 0..99]\free{pe }}
+\indentrel{3}\begin{verbatim}
+   (15)
+   [1, 2, 4, 8, 16, 32, 64, 27, 54, 7, 14, 28, 56, 11,
+    22, 44, 88, 75, 49, 98, 95, 89, 77, 53, 5, 10, 20,
+    40, 80, 59, 17, 34, 68, 35, 70, 39, 78, 55, 9, 18,
+    36, 72, 43, 86, 71, 41, 82, 63, 25, 50, 100, 99, 97,
+    93, 85, 69, 37, 74, 47, 94, 87, 73, 45, 90, 79, 57,
+    13, 26, 52, 3, 6, 12, 24, 48, 96, 91, 81, 61, 21, 42,
+    84, 67, 33, 66, 31, 62, 23, 46, 92, 83, 65, 29, 58,
+    15, 30, 60, 19, 38, 76, 51]
+                              Type: List PrimeField 101
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty16}
+\begin{paste}{ugxProblemFinitePrimePageEmpty16}{ugxProblemFinitePrimePagePatch16}
+\pastebutton{ugxProblemFinitePrimePageEmpty16}{\showpaste}
+\tab{5}\spadcommand{[pe**i for i in 0..99]\free{pe }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch17}
+\begin{paste}{ugxProblemFinitePrimePageFull17}{ugxProblemFinitePrimePageEmpty17}
+\pastebutton{ugxProblemFinitePrimePageFull17}{\hidepaste}
+\tab{5}\spadcommand{ex := discreteLog(y)\bound{ex }\free{y }}
+\indentrel{3}\begin{verbatim}
+   (16)  56
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty17}
+\begin{paste}{ugxProblemFinitePrimePageEmpty17}{ugxProblemFinitePrimePagePatch17}
+\pastebutton{ugxProblemFinitePrimePageEmpty17}{\showpaste}
+\tab{5}\spadcommand{ex := discreteLog(y)\bound{ex }\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch18}
+\begin{paste}{ugxProblemFinitePrimePageFull18}{ugxProblemFinitePrimePageEmpty18}
+\pastebutton{ugxProblemFinitePrimePageFull18}{\hidepaste}
+\tab{5}\spadcommand{pe ** ex\free{ex pe }}
+\indentrel{3}\begin{verbatim}
+   (17)  37
+                                   Type: PrimeField 101
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty18}
+\begin{paste}{ugxProblemFinitePrimePageEmpty18}{ugxProblemFinitePrimePagePatch18}
+\pastebutton{ugxProblemFinitePrimePageEmpty18}{\showpaste}
+\tab{5}\spadcommand{pe ** ex\free{ex pe }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch19}
+\begin{paste}{ugxProblemFinitePrimePageFull19}{ugxProblemFinitePrimePageEmpty19}
+\pastebutton{ugxProblemFinitePrimePageFull19}{\hidepaste}
+\tab{5}\spadcommand{order y\free{y }}
+\indentrel{3}\begin{verbatim}
+   (18)  25
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty19}
+\begin{paste}{ugxProblemFinitePrimePageEmpty19}{ugxProblemFinitePrimePagePatch19}
+\pastebutton{ugxProblemFinitePrimePageEmpty19}{\showpaste}
+\tab{5}\spadcommand{order y\free{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePagePatch20}
+\begin{paste}{ugxProblemFinitePrimePageFull20}{ugxProblemFinitePrimePageEmpty20}
+\pastebutton{ugxProblemFinitePrimePageFull20}{\hidepaste}
+\tab{5}\spadcommand{order pe\free{pe }}
+\indentrel{3}\begin{verbatim}
+   (19)  100
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFinitePrimePageEmpty20}
+\begin{paste}{ugxProblemFinitePrimePageEmpty20}{ugxProblemFinitePrimePagePatch20}
+\pastebutton{ugxProblemFinitePrimePageEmpty20}{\showpaste}
+\tab{5}\spadcommand{order pe\free{pe }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPagePatch1}
+\begin{paste}{ugxProblemFiniteCyclicPageFull1}{ugxProblemFiniteCyclicPageEmpty1}
+\pastebutton{ugxProblemFiniteCyclicPageFull1}{\hidepaste}
+\tab{5}\spadcommand{GF81 := FFCG(3,4);\bound{GF81 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPageEmpty1}
+\begin{paste}{ugxProblemFiniteCyclicPageEmpty1}{ugxProblemFiniteCyclicPagePatch1}
+\pastebutton{ugxProblemFiniteCyclicPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{GF81 := FFCG(3,4);\bound{GF81 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPagePatch2}
+\begin{paste}{ugxProblemFiniteCyclicPageFull2}{ugxProblemFiniteCyclicPageEmpty2}
+\pastebutton{ugxProblemFiniteCyclicPageFull2}{\hidepaste}
+\tab{5}\spadcommand{a := primitiveElement()$GF81\bound{a }\free{GF81 }}
+\indentrel{3}\begin{verbatim}
+           1
+   (2)  %BF
+                      Type: FiniteFieldCyclicGroup(3,4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPageEmpty2}
+\begin{paste}{ugxProblemFiniteCyclicPageEmpty2}{ugxProblemFiniteCyclicPagePatch2}
+\pastebutton{ugxProblemFiniteCyclicPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{a := primitiveElement()$GF81\bound{a }\free{GF81 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPagePatch3}
+\begin{paste}{ugxProblemFiniteCyclicPageFull3}{ugxProblemFiniteCyclicPageEmpty3}
+\pastebutton{ugxProblemFiniteCyclicPageFull3}{\hidepaste}
+\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }}
+\indentrel{3}\begin{verbatim}
+           72
+   (3)  %BF
+                      Type: FiniteFieldCyclicGroup(3,4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPageEmpty3}
+\begin{paste}{ugxProblemFiniteCyclicPageEmpty3}{ugxProblemFiniteCyclicPagePatch3}
+\pastebutton{ugxProblemFiniteCyclicPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPagePatch4}
+\begin{paste}{ugxProblemFiniteCyclicPageFull4}{ugxProblemFiniteCyclicPageEmpty4}
+\pastebutton{ugxProblemFiniteCyclicPageFull4}{\hidepaste}
+\tab{5}\spadcommand{b\free{b }}
+\indentrel{3}\begin{verbatim}
+           72
+   (4)  %BF
+                      Type: FiniteFieldCyclicGroup(3,4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPageEmpty4}
+\begin{paste}{ugxProblemFiniteCyclicPageEmpty4}{ugxProblemFiniteCyclicPagePatch4}
+\pastebutton{ugxProblemFiniteCyclicPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{b\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPagePatch5}
+\begin{paste}{ugxProblemFiniteCyclicPageFull5}{ugxProblemFiniteCyclicPageEmpty5}
+\pastebutton{ugxProblemFiniteCyclicPageFull5}{\hidepaste}
+\tab{5}\spadcommand{discreteLog b\free{b }}
+\indentrel{3}\begin{verbatim}
+   (5)  72
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPageEmpty5}
+\begin{paste}{ugxProblemFiniteCyclicPageEmpty5}{ugxProblemFiniteCyclicPagePatch5}
+\pastebutton{ugxProblemFiniteCyclicPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{discreteLog b\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPagePatch6}
+\begin{paste}{ugxProblemFiniteCyclicPageFull6}{ugxProblemFiniteCyclicPageEmpty6}
+\pastebutton{ugxProblemFiniteCyclicPageFull6}{\hidepaste}
+\tab{5}\spadcommand{GF9 := FF(3,2);\bound{GF9 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPageEmpty6}
+\begin{paste}{ugxProblemFiniteCyclicPageEmpty6}{ugxProblemFiniteCyclicPagePatch6}
+\pastebutton{ugxProblemFiniteCyclicPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{GF9 := FF(3,2);\bound{GF9 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPagePatch7}
+\begin{paste}{ugxProblemFiniteCyclicPageFull7}{ugxProblemFiniteCyclicPageEmpty7}
+\pastebutton{ugxProblemFiniteCyclicPageFull7}{\hidepaste}
+\tab{5}\spadcommand{GF729 := FFCGX(GF9,3);\bound{GF729 }\free{GF9 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPageEmpty7}
+\begin{paste}{ugxProblemFiniteCyclicPageEmpty7}{ugxProblemFiniteCyclicPagePatch7}
+\pastebutton{ugxProblemFiniteCyclicPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{GF729 := FFCGX(GF9,3);\bound{GF729 }\free{GF9 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPagePatch8}
+\begin{paste}{ugxProblemFiniteCyclicPageFull8}{ugxProblemFiniteCyclicPageEmpty8}
+\pastebutton{ugxProblemFiniteCyclicPageFull8}{\hidepaste}
+\tab{5}\spadcommand{r := (random()$GF729) ** 20\free{GF729 }\bound{r }}
+\indentrel{3}\begin{verbatim}
+           68
+   (8)  %CN
+Type: FiniteFieldCyclicGroupExtension(FiniteField(3,2),3)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPageEmpty8}
+\begin{paste}{ugxProblemFiniteCyclicPageEmpty8}{ugxProblemFiniteCyclicPagePatch8}
+\pastebutton{ugxProblemFiniteCyclicPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{r := (random()$GF729) ** 20\free{GF729 }\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPagePatch9}
+\begin{paste}{ugxProblemFiniteCyclicPageFull9}{ugxProblemFiniteCyclicPageEmpty9}
+\pastebutton{ugxProblemFiniteCyclicPageFull9}{\hidepaste}
+\tab{5}\spadcommand{trace(r)\free{r }}
+\indentrel{3}\begin{verbatim}
+   (9)  2
+                                 Type: FiniteField(3,2)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPageEmpty9}
+\begin{paste}{ugxProblemFiniteCyclicPageEmpty9}{ugxProblemFiniteCyclicPagePatch9}
+\pastebutton{ugxProblemFiniteCyclicPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{trace(r)\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPagePatch10}
+\begin{paste}{ugxProblemFiniteCyclicPageFull10}{ugxProblemFiniteCyclicPageEmpty10}
+\pastebutton{ugxProblemFiniteCyclicPageFull10}{\hidepaste}
+\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPageEmpty10}
+\begin{paste}{ugxProblemFiniteCyclicPageEmpty10}{ugxProblemFiniteCyclicPagePatch10}
+\pastebutton{ugxProblemFiniteCyclicPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPagePatch11}
+\begin{paste}{ugxProblemFiniteCyclicPageFull11}{ugxProblemFiniteCyclicPageEmpty11}
+\pastebutton{ugxProblemFiniteCyclicPageFull11}{\hidepaste}
+\tab{5}\spadcommand{f := createPrimitivePoly(4)$FFPOLY(GF3)\bound{f }\free{GF3 }}
+\indentrel{3}\begin{verbatim}
+          4
+   (11)  ?  + ? + 2
+          Type: SparseUnivariatePolynomial PrimeField 3
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPageEmpty11}
+\begin{paste}{ugxProblemFiniteCyclicPageEmpty11}{ugxProblemFiniteCyclicPagePatch11}
+\pastebutton{ugxProblemFiniteCyclicPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{f := createPrimitivePoly(4)$FFPOLY(GF3)\bound{f }\free{GF3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPagePatch12}
+\begin{paste}{ugxProblemFiniteCyclicPageFull12}{ugxProblemFiniteCyclicPageEmpty12}
+\pastebutton{ugxProblemFiniteCyclicPageFull12}{\hidepaste}
+\tab{5}\spadcommand{GF81 := FFCGP(GF3,f);\bound{GF81x }\free{f GF3 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPageEmpty12}
+\begin{paste}{ugxProblemFiniteCyclicPageEmpty12}{ugxProblemFiniteCyclicPagePatch12}
+\pastebutton{ugxProblemFiniteCyclicPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{GF81 := FFCGP(GF3,f);\bound{GF81x }\free{f GF3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPagePatch13}
+\begin{paste}{ugxProblemFiniteCyclicPageFull13}{ugxProblemFiniteCyclicPageEmpty13}
+\pastebutton{ugxProblemFiniteCyclicPageFull13}{\hidepaste}
+\tab{5}\spadcommand{random()$GF81\free{GF81x }}
+\indentrel{3}\begin{verbatim}
+            8
+   (13)  %BF
+Type: FiniteFieldCyclicGroupExtensionByPolynomial(PrimeField 3,?**4+?+2)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteCyclicPageEmpty13}
+\begin{paste}{ugxProblemFiniteCyclicPageEmpty13}{ugxProblemFiniteCyclicPagePatch13}
+\pastebutton{ugxProblemFiniteCyclicPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{random()$GF81\free{GF81x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePagePatch1}
+\begin{paste}{ugxProblemSeriesCreatePageFull1}{ugxProblemSeriesCreatePageEmpty1}
+\pastebutton{ugxProblemSeriesCreatePageFull1}{\hidepaste}
+\tab{5}\spadcommand{x := series 'x\bound{x }}
+\indentrel{3}\begin{verbatim}
+   (1)  x
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePageEmpty1}
+\begin{paste}{ugxProblemSeriesCreatePageEmpty1}{ugxProblemSeriesCreatePagePatch1}
+\pastebutton{ugxProblemSeriesCreatePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{x := series 'x\bound{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePagePatch2}
+\begin{paste}{ugxProblemSeriesCreatePageFull2}{ugxProblemSeriesCreatePageEmpty2}
+\pastebutton{ugxProblemSeriesCreatePageFull2}{\hidepaste}
+\tab{5}\spadcommand{1/(1 - x - x**2)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (2)
+               2     3     4     5      6      7      8
+     1 + x + 2x  + 3x  + 5x  + 8x  + 13x  + 21x  + 34x
+   + 
+        9      10      11
+     55x  + 89x   + O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePageEmpty2}
+\begin{paste}{ugxProblemSeriesCreatePageEmpty2}{ugxProblemSeriesCreatePagePatch2}
+\pastebutton{ugxProblemSeriesCreatePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{1/(1 - x - x**2)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePagePatch3}
+\begin{paste}{ugxProblemSeriesCreatePageFull3}{ugxProblemSeriesCreatePageEmpty3}
+\pastebutton{ugxProblemSeriesCreatePageFull3}{\hidepaste}
+\tab{5}\spadcommand{sin(x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (3)
+         1  3    1   5     1   7      1    9
+     x - Ä x  + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x
+         6      120      5040      362880
+   + 
+           1     11      12
+     - ÄÄÄÄÄÄÄÄ x   + O(x  )
+       39916800
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePageEmpty3}
+\begin{paste}{ugxProblemSeriesCreatePageEmpty3}{ugxProblemSeriesCreatePagePatch3}
+\pastebutton{ugxProblemSeriesCreatePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{sin(x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePagePatch4}
+\begin{paste}{ugxProblemSeriesCreatePageFull4}{ugxProblemSeriesCreatePageEmpty4}
+\pastebutton{ugxProblemSeriesCreatePageFull4}{\hidepaste}
+\tab{5}\spadcommand{sin(1 + x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (4)
+                        sin(1)  2   cos(1)  3   sin(1)  4
+     sin(1) + cos(1)x - ÄÄÄÄÄÄ x  - ÄÄÄÄÄÄ x  + ÄÄÄÄÄÄ x
+                           2           6          24
+   + 
+     cos(1)  5   sin(1)  6   cos(1)  7   sin(1)  8
+     ÄÄÄÄÄÄ x  - ÄÄÄÄÄÄ x  - ÄÄÄÄÄÄ x  + ÄÄÄÄÄÄ x
+       120         720        5040        40320
+   + 
+     cos(1)  9    sin(1)  10      11
+     ÄÄÄÄÄÄ x  - ÄÄÄÄÄÄÄ x   + O(x  )
+     362880      3628800
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePageEmpty4}
+\begin{paste}{ugxProblemSeriesCreatePageEmpty4}{ugxProblemSeriesCreatePagePatch4}
+\pastebutton{ugxProblemSeriesCreatePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{sin(1 + x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePagePatch5}
+\begin{paste}{ugxProblemSeriesCreatePageFull5}{ugxProblemSeriesCreatePageEmpty5}
+\pastebutton{ugxProblemSeriesCreatePageFull5}{\hidepaste}
+\tab{5}\spadcommand{sin(a * x)\free{x }}
+\indentrel{3}\begin{verbatim}
+   (5)
+            3        5        7          9
+           a   3    a   5    a    7     a     9
+     a x - ÄÄ x  + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x
+            6      120      5040      362880
+   + 
+           11
+          a      11      12
+     - ÄÄÄÄÄÄÄÄ x   + O(x  )
+       39916800
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePageEmpty5}
+\begin{paste}{ugxProblemSeriesCreatePageEmpty5}{ugxProblemSeriesCreatePagePatch5}
+\pastebutton{ugxProblemSeriesCreatePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{sin(a * x)\free{x }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePagePatch6}
+\begin{paste}{ugxProblemSeriesCreatePageFull6}{ugxProblemSeriesCreatePageEmpty6}
+\pastebutton{ugxProblemSeriesCreatePageFull6}{\hidepaste}
+\tab{5}\spadcommand{series(1/log(y),y = 1)}
+\indentrel{3}\begin{verbatim}
+   (6)
+            - 1   1    1            1        2
+     (y - 1)    + Ä - ÄÄ (y - 1) + ÄÄ (y - 1)
+                  2   12           24
+   + 
+        19        3    3         4    863         5
+     - ÄÄÄ (y - 1)  + ÄÄÄ (y - 1)  - ÄÄÄÄÄ (y - 1)
+       720            160            60480
+   + 
+      275         6    33953         7     8183         8
+     ÄÄÄÄÄ (y - 1)  - ÄÄÄÄÄÄÄ (y - 1)  + ÄÄÄÄÄÄÄ (y - 1)
+     24192            3628800            1036800
+   + 
+        3250433         9            10
+     - ÄÄÄÄÄÄÄÄÄ (y - 1)  + O((y - 1)  )
+       479001600
+  Type: UnivariatePuiseuxSeries(Expression Integer,y,1)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePageEmpty6}
+\begin{paste}{ugxProblemSeriesCreatePageEmpty6}{ugxProblemSeriesCreatePagePatch6}
+\pastebutton{ugxProblemSeriesCreatePageEmpty6}{\showpaste}
+\tab{5}\spadcommand{series(1/log(y),y = 1)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePagePatch7}
+\begin{paste}{ugxProblemSeriesCreatePageFull7}{ugxProblemSeriesCreatePageEmpty7}
+\pastebutton{ugxProblemSeriesCreatePageFull7}{\hidepaste}
+\tab{5}\spadcommand{y : UTS(FLOAT,'z,0) := exp(z)\bound{y }}
+\indentrel{3}\begin{verbatim}
+   (7)
+                    2                            3
+     1.0 + z + 0.5 z  + 0.1666666666 6666666667 z
+   + 
+                                4
+     0.0416666666 6666666666 7 z
+   + 
+                                 5
+     0.0083333333 3333333333 34 z
+   + 
+                                 6
+     0.0013888888 8888888888 89 z
+   + 
+                                 7
+     0.0001984126 9841269841 27 z
+   + 
+                                   8
+     0.0000248015 8730158730 1587 z
+   + 
+                                    9
+     0.0000027557 3192239858 90653 z
+   + 
+                                   10      11
+     0.2755731922 3985890653 E -6 z   + O(z  )
+              Type: UnivariateTaylorSeries(Float,z,0.0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePageEmpty7}
+\begin{paste}{ugxProblemSeriesCreatePageEmpty7}{ugxProblemSeriesCreatePagePatch7}
+\pastebutton{ugxProblemSeriesCreatePageEmpty7}{\showpaste}
+\tab{5}\spadcommand{y : UTS(FLOAT,'z,0) := exp(z)\bound{y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePagePatch8}
+\begin{paste}{ugxProblemSeriesCreatePageFull8}{ugxProblemSeriesCreatePageEmpty8}
+\pastebutton{ugxProblemSeriesCreatePageFull8}{\hidepaste}
+\tab{5}\spadcommand{series(1/factorial(n),n,w = 0)}
+\indentrel{3}\begin{verbatim}
+   (8)
+             1  2   1  3    1  4    1   5    1   6
+     1 + w + Ä w  + Ä w  + ÄÄ w  + ÄÄÄ w  + ÄÄÄ w
+             2      6      24      120      720
+   + 
+     1   7     1    8      1    9      1     10      11
+   ÄÄÄÄ w  + ÄÄÄÄÄ w  + ÄÄÄÄÄÄ w  + ÄÄÄÄÄÄÄ w   + O(w  )
+   5040      40320      362880      3628800
+  Type: UnivariatePuiseuxSeries(Expression Integer,w,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesCreatePageEmpty8}
+\begin{paste}{ugxProblemSeriesCreatePageEmpty8}{ugxProblemSeriesCreatePagePatch8}
+\pastebutton{ugxProblemSeriesCreatePageEmpty8}{\showpaste}
+\tab{5}\spadcommand{series(1/factorial(n),n,w = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPagePatch1}
+\begin{paste}{ugxProblemFiniteNormalPageFull1}{ugxProblemFiniteNormalPageEmpty1}
+\pastebutton{ugxProblemFiniteNormalPageFull1}{\hidepaste}
+\tab{5}\spadcommand{K := FFNB(3,8)\bound{K }}
+\indentrel{3}\begin{verbatim}
+   (1)  FiniteFieldNormalBasis(3,8)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPageEmpty1}
+\begin{paste}{ugxProblemFiniteNormalPageEmpty1}{ugxProblemFiniteNormalPagePatch1}
+\pastebutton{ugxProblemFiniteNormalPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{K := FFNB(3,8)\bound{K }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPagePatch2}
+\begin{paste}{ugxProblemFiniteNormalPageFull2}{ugxProblemFiniteNormalPageEmpty2}
+\pastebutton{ugxProblemFiniteNormalPageFull2}{\hidepaste}
+\tab{5}\spadcommand{a := normalElement()$K\bound{a }\free{K }}
+\indentrel{3}\begin{verbatim}
+   (2)  %CO
+                      Type: FiniteFieldNormalBasis(3,8)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPageEmpty2}
+\begin{paste}{ugxProblemFiniteNormalPageEmpty2}{ugxProblemFiniteNormalPagePatch2}
+\pastebutton{ugxProblemFiniteNormalPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{a := normalElement()$K\bound{a }\free{K }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPagePatch3}
+\begin{paste}{ugxProblemFiniteNormalPageFull3}{ugxProblemFiniteNormalPageEmpty3}
+\pastebutton{ugxProblemFiniteNormalPageFull3}{\hidepaste}
+\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }}
+\indentrel{3}\begin{verbatim}
+             7       5
+            q       q       q
+   (3)  2%CO   + %CO   + %CO
+                      Type: FiniteFieldNormalBasis(3,8)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPageEmpty3}
+\begin{paste}{ugxProblemFiniteNormalPageEmpty3}{ugxProblemFiniteNormalPagePatch3}
+\pastebutton{ugxProblemFiniteNormalPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPagePatch4}
+\begin{paste}{ugxProblemFiniteNormalPageFull4}{ugxProblemFiniteNormalPageEmpty4}
+\pastebutton{ugxProblemFiniteNormalPageFull4}{\hidepaste}
+\tab{5}\spadcommand{GF9 := FFNB(3,2);\bound{GF9 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPageEmpty4}
+\begin{paste}{ugxProblemFiniteNormalPageEmpty4}{ugxProblemFiniteNormalPagePatch4}
+\pastebutton{ugxProblemFiniteNormalPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{GF9 := FFNB(3,2);\bound{GF9 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPagePatch5}
+\begin{paste}{ugxProblemFiniteNormalPageFull5}{ugxProblemFiniteNormalPageEmpty5}
+\pastebutton{ugxProblemFiniteNormalPageFull5}{\hidepaste}
+\tab{5}\spadcommand{GF729 := FFNBX(GF9,3);\bound{GF729 }\free{GF9 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPageEmpty5}
+\begin{paste}{ugxProblemFiniteNormalPageEmpty5}{ugxProblemFiniteNormalPagePatch5}
+\pastebutton{ugxProblemFiniteNormalPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{GF729 := FFNBX(GF9,3);\bound{GF729 }\free{GF9 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPagePatch6}
+\begin{paste}{ugxProblemFiniteNormalPageFull6}{ugxProblemFiniteNormalPageEmpty6}
+\pastebutton{ugxProblemFiniteNormalPageFull6}{\hidepaste}
+\tab{5}\spadcommand{r := random()$GF729\bound{r }\free{GF729 }}
+\indentrel{3}\begin{verbatim}
+                 2
+            q   q
+   (6)  2%CP %CQ
+Type: FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPageEmpty6}
+\begin{paste}{ugxProblemFiniteNormalPageEmpty6}{ugxProblemFiniteNormalPagePatch6}
+\pastebutton{ugxProblemFiniteNormalPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{r := random()$GF729\bound{r }\free{GF729 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPagePatch7}
+\begin{paste}{ugxProblemFiniteNormalPageFull7}{ugxProblemFiniteNormalPageEmpty7}
+\pastebutton{ugxProblemFiniteNormalPageFull7}{\hidepaste}
+\tab{5}\spadcommand{r + r**3 + r**9 + r**27\free{r }}
+\indentrel{3}\begin{verbatim}
+                          2
+             q           q            q       q
+   (7)  (2%CP  + 2%CP)%CQ   + 2%CP %CQ  + 2%CP %CQ
+Type: FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPageEmpty7}
+\begin{paste}{ugxProblemFiniteNormalPageEmpty7}{ugxProblemFiniteNormalPagePatch7}
+\pastebutton{ugxProblemFiniteNormalPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{r + r**3 + r**9 + r**27\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPagePatch8}
+\begin{paste}{ugxProblemFiniteNormalPageFull8}{ugxProblemFiniteNormalPageEmpty8}
+\pastebutton{ugxProblemFiniteNormalPageFull8}{\hidepaste}
+\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPageEmpty8}
+\begin{paste}{ugxProblemFiniteNormalPageEmpty8}{ugxProblemFiniteNormalPagePatch8}
+\pastebutton{ugxProblemFiniteNormalPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPagePatch9}
+\begin{paste}{ugxProblemFiniteNormalPageFull9}{ugxProblemFiniteNormalPageEmpty9}
+\pastebutton{ugxProblemFiniteNormalPageFull9}{\hidepaste}
+\tab{5}\spadcommand{f := createNormalPoly(4)$FFPOLY(GF3)\free{GF3 }\bound{f }}
+\indentrel{3}\begin{verbatim}
+         4     3
+   (9)  ?  + 2?  + 2
+          Type: SparseUnivariatePolynomial PrimeField 3
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPageEmpty9}
+\begin{paste}{ugxProblemFiniteNormalPageEmpty9}{ugxProblemFiniteNormalPagePatch9}
+\pastebutton{ugxProblemFiniteNormalPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{f := createNormalPoly(4)$FFPOLY(GF3)\free{GF3 }\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPagePatch10}
+\begin{paste}{ugxProblemFiniteNormalPageFull10}{ugxProblemFiniteNormalPageEmpty10}
+\pastebutton{ugxProblemFiniteNormalPageFull10}{\hidepaste}
+\tab{5}\spadcommand{GF81 := FFNBP(GF3,f);\bound{GF81 }\free{f GF3 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPageEmpty10}
+\begin{paste}{ugxProblemFiniteNormalPageEmpty10}{ugxProblemFiniteNormalPagePatch10}
+\pastebutton{ugxProblemFiniteNormalPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{GF81 := FFNBP(GF3,f);\bound{GF81 }\free{f GF3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPagePatch11}
+\begin{paste}{ugxProblemFiniteNormalPageFull11}{ugxProblemFiniteNormalPageEmpty11}
+\pastebutton{ugxProblemFiniteNormalPageFull11}{\hidepaste}
+\tab{5}\spadcommand{r := random()$GF81\free{GF81 }\bound{r1 }}
+\indentrel{3}\begin{verbatim}
+             3       2
+            q       q       q
+   (11)  %CR   + %CR   + %CR
+Type: FiniteFieldNormalBasisExtensionByPolynomial(PrimeField 3,?**4+2*?**3+2)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPageEmpty11}
+\begin{paste}{ugxProblemFiniteNormalPageEmpty11}{ugxProblemFiniteNormalPagePatch11}
+\pastebutton{ugxProblemFiniteNormalPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{r := random()$GF81\free{GF81 }\bound{r1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPagePatch12}
+\begin{paste}{ugxProblemFiniteNormalPageFull12}{ugxProblemFiniteNormalPageEmpty12}
+\pastebutton{ugxProblemFiniteNormalPageFull12}{\hidepaste}
+\tab{5}\spadcommand{r * r**3 * r**9 * r**27\free{r1 }}
+\indentrel{3}\begin{verbatim}
+              3        2
+             q        q        q
+   (12)  2%CR   + 2%CR   + 2%CR  + 2%CR
+Type: FiniteFieldNormalBasisExtensionByPolynomial(PrimeField 3,?**4+2*?**3+2)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPageEmpty12}
+\begin{paste}{ugxProblemFiniteNormalPageEmpty12}{ugxProblemFiniteNormalPagePatch12}
+\pastebutton{ugxProblemFiniteNormalPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{r * r**3 * r**9 * r**27\free{r1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPagePatch13}
+\begin{paste}{ugxProblemFiniteNormalPageFull13}{ugxProblemFiniteNormalPageEmpty13}
+\pastebutton{ugxProblemFiniteNormalPageFull13}{\hidepaste}
+\tab{5}\spadcommand{norm r\free{r1 }}
+\indentrel{3}\begin{verbatim}
+   (13)  2
+                                     Type: PrimeField 3
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteNormalPageEmpty13}
+\begin{paste}{ugxProblemFiniteNormalPageEmpty13}{ugxProblemFiniteNormalPagePatch13}
+\pastebutton{ugxProblemFiniteNormalPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{norm r\free{r1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPagePatch1}
+\begin{paste}{ugxProblemSeriesFormulaPageFull1}{ugxProblemSeriesFormulaPageEmpty1}
+\pastebutton{ugxProblemSeriesFormulaPageFull1}{\hidepaste}
+\tab{5}\spadcommand{series(n +-> 1/factorial(n),x = 0)}
+\indentrel{3}\begin{verbatim}
+   (1)
+             1  2   1  3    1  4    1   5    1   6
+     1 + x + Ä x  + Ä x  + ÄÄ x  + ÄÄÄ x  + ÄÄÄ x
+             2      6      24      120      720
+   + 
+     1   7     1    8      1    9      1     10      11
+   ÄÄÄÄ x  + ÄÄÄÄÄ x  + ÄÄÄÄÄÄ x  + ÄÄÄÄÄÄÄ x   + O(x  )
+   5040      40320      362880      3628800
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPageEmpty1}
+\begin{paste}{ugxProblemSeriesFormulaPageEmpty1}{ugxProblemSeriesFormulaPagePatch1}
+\pastebutton{ugxProblemSeriesFormulaPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{series(n +-> 1/factorial(n),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPagePatch2}
+\begin{paste}{ugxProblemSeriesFormulaPageFull2}{ugxProblemSeriesFormulaPageEmpty2}
+\pastebutton{ugxProblemSeriesFormulaPageFull2}{\hidepaste}
+\tab{5}\spadcommand{series(n +-> (-1)**(n-1)/n,x = 1,1..)}
+\indentrel{3}\begin{verbatim}
+   (2)
+               1        2   1        3   1        4
+     (x - 1) - Ä (x - 1)  + Ä (x - 1)  - Ä (x - 1)
+               2            3            4
+   + 
+     1        5   1        6   1        7   1        8
+     Ä (x - 1)  - Ä (x - 1)  + Ä (x - 1)  - Ä (x - 1)
+     5            6            7            8
+   + 
+   1        9    1        10    1        11            12
+   Ä (x - 1)  - ÄÄ (x - 1)   + ÄÄ (x - 1)   + O((x - 1)  )
+   9            10             11
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,1)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPageEmpty2}
+\begin{paste}{ugxProblemSeriesFormulaPageEmpty2}{ugxProblemSeriesFormulaPagePatch2}
+\pastebutton{ugxProblemSeriesFormulaPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{series(n +-> (-1)**(n-1)/n,x = 1,1..)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPagePatch3}
+\begin{paste}{ugxProblemSeriesFormulaPageFull3}{ugxProblemSeriesFormulaPageEmpty3}
+\pastebutton{ugxProblemSeriesFormulaPageFull3}{\hidepaste}
+\tab{5}\spadcommand{series(n +-> (-1)**((n-1)/2)/factorial(n),x = 0,1..,2)}
+\indentrel{3}\begin{verbatim}
+   (3)
+         1  3    1   5     1   7      1    9
+     x - Ä x  + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x
+         6      120      5040      362880
+   + 
+           1     11      12
+     - ÄÄÄÄÄÄÄÄ x   + O(x  )
+       39916800
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPageEmpty3}
+\begin{paste}{ugxProblemSeriesFormulaPageEmpty3}{ugxProblemSeriesFormulaPagePatch3}
+\pastebutton{ugxProblemSeriesFormulaPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{series(n +-> (-1)**((n-1)/2)/factorial(n),x = 0,1..,2)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPagePatch4}
+\begin{paste}{ugxProblemSeriesFormulaPageFull4}{ugxProblemSeriesFormulaPageEmpty4}
+\pastebutton{ugxProblemSeriesFormulaPageFull4}{\hidepaste}
+\tab{5}\spadcommand{series(n +-> (-1)**((3*n-1)/2)/factorial(3*n),x = 0,1/3..,2/3)}
+\indentrel{3}\begin{verbatim}
+   (4)
+      1              5         7
+      Ä              Ä         Ä
+      3   1      1   3     1   3      1    3
+     x  - Ä x + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x
+          6     120      5040      362880
+   + 
+                 11
+                 ÄÄ
+           1      3      4
+     - ÄÄÄÄÄÄÄÄ x   + O(x )
+       39916800
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPageEmpty4}
+\begin{paste}{ugxProblemSeriesFormulaPageEmpty4}{ugxProblemSeriesFormulaPagePatch4}
+\pastebutton{ugxProblemSeriesFormulaPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{series(n +-> (-1)**((3*n-1)/2)/factorial(3*n),x = 0,1/3..,2/3)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPagePatch5}
+\begin{paste}{ugxProblemSeriesFormulaPageFull5}{ugxProblemSeriesFormulaPageEmpty5}
+\pastebutton{ugxProblemSeriesFormulaPageFull5}{\hidepaste}
+\tab{5}\spadcommand{cscx := series(n +-> (-1)**((n-1)/2) * 2 * (2**n-1) * bernoulli(numer(n+1)) / factorial(n+1), x=0, -1..,2)\bound{cscx }}
+\indentrel{3}\begin{verbatim}
+   (5)
+      - 1   1      7   3     31   5     127   7
+     x    + Ä x + ÄÄÄ x  + ÄÄÄÄÄ x  + ÄÄÄÄÄÄ x
+            6     360      15120      604800
+   + 
+        73    9      10
+     ÄÄÄÄÄÄÄ x  + O(x  )
+     3421440
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPageEmpty5}
+\begin{paste}{ugxProblemSeriesFormulaPageEmpty5}{ugxProblemSeriesFormulaPagePatch5}
+\pastebutton{ugxProblemSeriesFormulaPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{cscx := series(n +-> (-1)**((n-1)/2) * 2 * (2**n-1) * bernoulli(numer(n+1)) / factorial(n+1), x=0, -1..,2)\bound{cscx }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPagePatch6}
+\begin{paste}{ugxProblemSeriesFormulaPageFull6}{ugxProblemSeriesFormulaPageEmpty6}
+\pastebutton{ugxProblemSeriesFormulaPageFull6}{\hidepaste}
+\tab{5}\spadcommand{1/cscx\free{cscx }}
+\indentrel{3}\begin{verbatim}
+   (6)
+         1  3    1   5     1   7      1    9
+     x - Ä x  + ÄÄÄ x  - ÄÄÄÄ x  + ÄÄÄÄÄÄ x
+         6      120      5040      362880
+   + 
+           1     11      12
+     - ÄÄÄÄÄÄÄÄ x   + O(x  )
+       39916800
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPageEmpty6}
+\begin{paste}{ugxProblemSeriesFormulaPageEmpty6}{ugxProblemSeriesFormulaPagePatch6}
+\pastebutton{ugxProblemSeriesFormulaPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{1/cscx\free{cscx }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPagePatch7}
+\begin{paste}{ugxProblemSeriesFormulaPageFull7}{ugxProblemSeriesFormulaPageEmpty7}
+\pastebutton{ugxProblemSeriesFormulaPageFull7}{\hidepaste}
+\tab{5}\spadcommand{asinx := series(n +-> binomial(n-1,(n-1)/2)/(n*2**(n-1)),x=0,1..,2)\bound{asinx }}
+\indentrel{3}\begin{verbatim}
+   (7)
+       1  3    3  5    5   7    35   9    63   11      12
+   x + Ä x  + ÄÄ x  + ÄÄÄ x  + ÄÄÄÄ x  + ÄÄÄÄ x   + O(x  )
+       6      40      112      1152      2816
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPageEmpty7}
+\begin{paste}{ugxProblemSeriesFormulaPageEmpty7}{ugxProblemSeriesFormulaPagePatch7}
+\pastebutton{ugxProblemSeriesFormulaPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{asinx := series(n +-> binomial(n-1,(n-1)/2)/(n*2**(n-1)),x=0,1..,2)\bound{asinx }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPagePatch8}
+\begin{paste}{ugxProblemSeriesFormulaPageFull8}{ugxProblemSeriesFormulaPageEmpty8}
+\pastebutton{ugxProblemSeriesFormulaPageFull8}{\hidepaste}
+\tab{5}\spadcommand{sin(asinx)\free{asinx }}
+\indentrel{3}\begin{verbatim}
+               12
+   (8)  x + O(x  )
+  Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemSeriesFormulaPageEmpty8}
+\begin{paste}{ugxProblemSeriesFormulaPageEmpty8}{ugxProblemSeriesFormulaPagePatch8}
+\pastebutton{ugxProblemSeriesFormulaPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{sin(asinx)\free{asinx }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch1}
+\begin{paste}{ugxProblemFiniteModulusPageFull1}{ugxProblemFiniteModulusPageEmpty1}
+\pastebutton{ugxProblemFiniteModulusPageFull1}{\hidepaste}
+\tab{5}\spadcommand{GF4096 := FF(2,12);\bound{GF4096 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty1}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty1}{ugxProblemFiniteModulusPagePatch1}
+\pastebutton{ugxProblemFiniteModulusPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{GF4096 := FF(2,12);\bound{GF4096 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch2}
+\begin{paste}{ugxProblemFiniteModulusPageFull2}{ugxProblemFiniteModulusPageEmpty2}
+\pastebutton{ugxProblemFiniteModulusPageFull2}{\hidepaste}
+\tab{5}\spadcommand{a := index(2)$GF4096\bound{a }\free{GF4096 }}
+\indentrel{3}\begin{verbatim}
+   (2)  %CS
+                                Type: FiniteField(2,12)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty2}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty2}{ugxProblemFiniteModulusPagePatch2}
+\pastebutton{ugxProblemFiniteModulusPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{a := index(2)$GF4096\bound{a }\free{GF4096 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch3}
+\begin{paste}{ugxProblemFiniteModulusPageFull3}{ugxProblemFiniteModulusPageEmpty3}
+\pastebutton{ugxProblemFiniteModulusPageFull3}{\hidepaste}
+\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }}
+\indentrel{3}\begin{verbatim}
+           5      3
+   (3)  %CS  + %CS  + %CS + 1
+                                Type: FiniteField(2,12)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty3}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty3}{ugxProblemFiniteModulusPagePatch3}
+\pastebutton{ugxProblemFiniteModulusPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch4}
+\begin{paste}{ugxProblemFiniteModulusPageFull4}{ugxProblemFiniteModulusPageEmpty4}
+\pastebutton{ugxProblemFiniteModulusPageFull4}{\hidepaste}
+\tab{5}\spadcommand{b ** 1000\free{b }}
+\indentrel{3}\begin{verbatim}
+           10      9      7      5      4      3
+   (4)  %CS   + %CS  + %CS  + %CS  + %CS  + %CS  + %CS
+                                Type: FiniteField(2,12)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty4}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty4}{ugxProblemFiniteModulusPagePatch4}
+\pastebutton{ugxProblemFiniteModulusPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{b ** 1000\free{b }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch5}
+\begin{paste}{ugxProblemFiniteModulusPageFull5}{ugxProblemFiniteModulusPageEmpty5}
+\pastebutton{ugxProblemFiniteModulusPageFull5}{\hidepaste}
+\tab{5}\spadcommand{c := a/b\free{a b }\bound{c }}
+\indentrel{3}\begin{verbatim}
+           11      8      7      5      4      3      2
+   (5)  %CS   + %CS  + %CS  + %CS  + %CS  + %CS  + %CS
+                                Type: FiniteField(2,12)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty5}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty5}{ugxProblemFiniteModulusPagePatch5}
+\pastebutton{ugxProblemFiniteModulusPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{c := a/b\free{a b }\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch6}
+\begin{paste}{ugxProblemFiniteModulusPageFull6}{ugxProblemFiniteModulusPageEmpty6}
+\pastebutton{ugxProblemFiniteModulusPageFull6}{\hidepaste}
+\tab{5}\spadcommand{norm c\free{c }}
+\indentrel{3}\begin{verbatim}
+   (6)  1
+                                     Type: PrimeField 2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty6}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty6}{ugxProblemFiniteModulusPagePatch6}
+\pastebutton{ugxProblemFiniteModulusPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{norm c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch7}
+\begin{paste}{ugxProblemFiniteModulusPageFull7}{ugxProblemFiniteModulusPageEmpty7}
+\pastebutton{ugxProblemFiniteModulusPageFull7}{\hidepaste}
+\tab{5}\spadcommand{trace c\free{c }}
+\indentrel{3}\begin{verbatim}
+   (7)  0
+                                     Type: PrimeField 2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty7}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty7}{ugxProblemFiniteModulusPagePatch7}
+\pastebutton{ugxProblemFiniteModulusPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{trace c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch8}
+\begin{paste}{ugxProblemFiniteModulusPageFull8}{ugxProblemFiniteModulusPageEmpty8}
+\pastebutton{ugxProblemFiniteModulusPageFull8}{\hidepaste}
+\tab{5}\spadcommand{dL := discreteLog a\free{a }\bound{dL }}
+\indentrel{3}\begin{verbatim}
+   (8)  1729
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty8}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty8}{ugxProblemFiniteModulusPagePatch8}
+\pastebutton{ugxProblemFiniteModulusPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{dL := discreteLog a\free{a }\bound{dL }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch9}
+\begin{paste}{ugxProblemFiniteModulusPageFull9}{ugxProblemFiniteModulusPageEmpty9}
+\pastebutton{ugxProblemFiniteModulusPageFull9}{\hidepaste}
+\tab{5}\spadcommand{g ** dL\free{dL g }}
+\indentrel{3}\begin{verbatim}
+         1729
+   (9)  g
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty9}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty9}{ugxProblemFiniteModulusPagePatch9}
+\pastebutton{ugxProblemFiniteModulusPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{g ** dL\free{dL g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch10}
+\begin{paste}{ugxProblemFiniteModulusPageFull10}{ugxProblemFiniteModulusPageEmpty10}
+\pastebutton{ugxProblemFiniteModulusPageFull10}{\hidepaste}
+\tab{5}\spadcommand{GF16 := FF(2,4);\bound{GF16 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty10}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty10}{ugxProblemFiniteModulusPagePatch10}
+\pastebutton{ugxProblemFiniteModulusPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{GF16 := FF(2,4);\bound{GF16 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch11}
+\begin{paste}{ugxProblemFiniteModulusPageFull11}{ugxProblemFiniteModulusPageEmpty11}
+\pastebutton{ugxProblemFiniteModulusPageFull11}{\hidepaste}
+\tab{5}\spadcommand{GF4096 := FFX(GF16,3);\bound{GF4096x }\free{GF16 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty11}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty11}{ugxProblemFiniteModulusPagePatch11}
+\pastebutton{ugxProblemFiniteModulusPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{GF4096 := FFX(GF16,3);\bound{GF4096x }\free{GF16 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch12}
+\begin{paste}{ugxProblemFiniteModulusPageFull12}{ugxProblemFiniteModulusPageEmpty12}
+\pastebutton{ugxProblemFiniteModulusPageFull12}{\hidepaste}
+\tab{5}\spadcommand{r := (random()$GF4096) ** 20\free{GF4096x }\bound{r }}
+\indentrel{3}\begin{verbatim}
+             3              2
+   (12)  (%CT  + %CT + 1)%CU  + %CT %CU
+         Type: FiniteFieldExtension(FiniteField(2,4),3)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty12}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty12}{ugxProblemFiniteModulusPagePatch12}
+\pastebutton{ugxProblemFiniteModulusPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{r := (random()$GF4096) ** 20\free{GF4096x }\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch13}
+\begin{paste}{ugxProblemFiniteModulusPageFull13}{ugxProblemFiniteModulusPageEmpty13}
+\pastebutton{ugxProblemFiniteModulusPageFull13}{\hidepaste}
+\tab{5}\spadcommand{norm(r)\free{r }}
+\indentrel{3}\begin{verbatim}
+            2
+   (13)  %CT  + %CT
+                                 Type: FiniteField(2,4)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty13}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty13}{ugxProblemFiniteModulusPagePatch13}
+\pastebutton{ugxProblemFiniteModulusPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{norm(r)\free{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch14}
+\begin{paste}{ugxProblemFiniteModulusPageFull14}{ugxProblemFiniteModulusPageEmpty14}
+\pastebutton{ugxProblemFiniteModulusPageFull14}{\hidepaste}
+\tab{5}\spadcommand{GF4 := FF(2,2);\bound{GF4 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty14}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty14}{ugxProblemFiniteModulusPagePatch14}
+\pastebutton{ugxProblemFiniteModulusPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{GF4 := FF(2,2);\bound{GF4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch15}
+\begin{paste}{ugxProblemFiniteModulusPageFull15}{ugxProblemFiniteModulusPageEmpty15}
+\pastebutton{ugxProblemFiniteModulusPageFull15}{\hidepaste}
+\tab{5}\spadcommand{f := nextIrreduciblePoly(random(6)$FFPOLY(GF4))$FFPOLY(GF4)\free{GF4 }\bound{f }}
+\indentrel{3}\begin{verbatim}
+          6    5             4             2
+   (15)  ?  + ?  + (%CV + 1)?  + (%CV + 1)?  + ? + %CV
+Type: Union(SparseUnivariatePolynomial FiniteField(2,2),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty15}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty15}{ugxProblemFiniteModulusPagePatch15}
+\pastebutton{ugxProblemFiniteModulusPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{f := nextIrreduciblePoly(random(6)$FFPOLY(GF4))$FFPOLY(GF4)\free{GF4 }\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch16}
+\begin{paste}{ugxProblemFiniteModulusPageFull16}{ugxProblemFiniteModulusPageEmpty16}
+\pastebutton{ugxProblemFiniteModulusPageFull16}{\hidepaste}
+\tab{5}\spadcommand{GF4096 := FFP(GF4,f);\bound{GF4096y }\free{f GF4 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty16}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty16}{ugxProblemFiniteModulusPagePatch16}
+\pastebutton{ugxProblemFiniteModulusPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{GF4096 := FFP(GF4,f);\bound{GF4096y }\free{f GF4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPagePatch17}
+\begin{paste}{ugxProblemFiniteModulusPageFull17}{ugxProblemFiniteModulusPageEmpty17}
+\pastebutton{ugxProblemFiniteModulusPageFull17}{\hidepaste}
+\tab{5}\spadcommand{discreteLog random()$GF4096\free{GF4096y }}
+\indentrel{3}\begin{verbatim}
+   (17)  3370
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteModulusPageEmpty17}
+\begin{paste}{ugxProblemFiniteModulusPageEmpty17}{ugxProblemFiniteModulusPagePatch17}
+\pastebutton{ugxProblemFiniteModulusPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{discreteLog random()$GF4096\free{GF4096y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch1}
+\begin{paste}{ugProblemIdealPageFull1}{ugProblemIdealPageEmpty1}
+\pastebutton{ugProblemIdealPageFull1}{\hidepaste}
+\tab{5}\spadcommand{(n,m) : List DMP([x,y],FRAC INT)\bound{nm }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty1}
+\begin{paste}{ugProblemIdealPageEmpty1}{ugProblemIdealPagePatch1}
+\pastebutton{ugProblemIdealPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{(n,m) : List DMP([x,y],FRAC INT)\bound{nm }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch2}
+\begin{paste}{ugProblemIdealPageFull2}{ugProblemIdealPageEmpty2}
+\pastebutton{ugProblemIdealPageFull2}{\hidepaste}
+\tab{5}\spadcommand{m := [x**2+y**2-1]\free{nm }\bound{m }}
+\indentrel{3}\begin{verbatim}
+          2    2
+   (2)  [x  + y  - 1]
+Type: List DistributedMultivariatePolynomial([x,y],Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty2}
+\begin{paste}{ugProblemIdealPageEmpty2}{ugProblemIdealPagePatch2}
+\pastebutton{ugProblemIdealPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{m := [x**2+y**2-1]\free{nm }\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch3}
+\begin{paste}{ugProblemIdealPageFull3}{ugProblemIdealPageEmpty3}
+\pastebutton{ugProblemIdealPageFull3}{\hidepaste}
+\tab{5}\spadcommand{n := [x**2-y**2]\free{nm }\bound{n }}
+\indentrel{3}\begin{verbatim}
+          2    2
+   (3)  [x  - y ]
+Type: List DistributedMultivariatePolynomial([x,y],Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty3}
+\begin{paste}{ugProblemIdealPageEmpty3}{ugProblemIdealPagePatch3}
+\pastebutton{ugProblemIdealPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{n := [x**2-y**2]\free{nm }\bound{n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch4}
+\begin{paste}{ugProblemIdealPageFull4}{ugProblemIdealPageEmpty4}
+\pastebutton{ugProblemIdealPageFull4}{\hidepaste}
+\tab{5}\spadcommand{id := ideal m + ideal n\free{n m }\bound{id }}
+\indentrel{3}\begin{verbatim}
+          2   1  2   1
+   (4)  [x  - Ä,y  - Ä]
+              2      2
+Type: PolynomialIdeals(Fraction Integer,DirectProduct(2,NonNegativeInteger),OrderedVariableList [x,y],DistributedMultivariatePolynomial([x,y],Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty4}
+\begin{paste}{ugProblemIdealPageEmpty4}{ugProblemIdealPagePatch4}
+\pastebutton{ugProblemIdealPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{id := ideal m + ideal n\free{n m }\bound{id }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch5}
+\begin{paste}{ugProblemIdealPageFull5}{ugProblemIdealPageEmpty5}
+\pastebutton{ugProblemIdealPageFull5}{\hidepaste}
+\tab{5}\spadcommand{zeroDim? id\free{id }}
+\indentrel{3}\begin{verbatim}
+   (5)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty5}
+\begin{paste}{ugProblemIdealPageEmpty5}{ugProblemIdealPagePatch5}
+\pastebutton{ugProblemIdealPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{zeroDim? id\free{id }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch6}
+\begin{paste}{ugProblemIdealPageFull6}{ugProblemIdealPageEmpty6}
+\pastebutton{ugProblemIdealPageFull6}{\hidepaste}
+\tab{5}\spadcommand{zeroDim?(ideal m)\free{m }}
+\indentrel{3}\begin{verbatim}
+   (6)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty6}
+\begin{paste}{ugProblemIdealPageEmpty6}{ugProblemIdealPagePatch6}
+\pastebutton{ugProblemIdealPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{zeroDim?(ideal m)\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch7}
+\begin{paste}{ugProblemIdealPageFull7}{ugProblemIdealPageEmpty7}
+\pastebutton{ugProblemIdealPageFull7}{\hidepaste}
+\tab{5}\spadcommand{dimension ideal m\free{m }}
+\indentrel{3}\begin{verbatim}
+   (7)  1
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty7}
+\begin{paste}{ugProblemIdealPageEmpty7}{ugProblemIdealPagePatch7}
+\pastebutton{ugProblemIdealPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{dimension ideal m\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch8}
+\begin{paste}{ugProblemIdealPageFull8}{ugProblemIdealPageEmpty8}
+\pastebutton{ugProblemIdealPageFull8}{\hidepaste}
+\tab{5}\spadcommand{(f,g):DMP([x,y],FRAC INT)\bound{fg }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty8}
+\begin{paste}{ugProblemIdealPageEmpty8}{ugProblemIdealPagePatch8}
+\pastebutton{ugProblemIdealPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{(f,g):DMP([x,y],FRAC INT)\bound{fg }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch9}
+\begin{paste}{ugProblemIdealPageFull9}{ugProblemIdealPageEmpty9}
+\pastebutton{ugProblemIdealPageFull9}{\hidepaste}
+\tab{5}\spadcommand{f := x**2-1\free{fg }\bound{f }}
+\indentrel{3}\begin{verbatim}
+         2
+   (9)  x  - 1
+Type: DistributedMultivariatePolynomial([x,y],Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty9}
+\begin{paste}{ugProblemIdealPageEmpty9}{ugProblemIdealPagePatch9}
+\pastebutton{ugProblemIdealPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{f := x**2-1\free{fg }\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch10}
+\begin{paste}{ugProblemIdealPageFull10}{ugProblemIdealPageEmpty10}
+\pastebutton{ugProblemIdealPageFull10}{\hidepaste}
+\tab{5}\spadcommand{g := x*(x**2-1)\free{fg }\bound{g }}
+\indentrel{3}\begin{verbatim}
+          3
+   (10)  x  - x
+Type: DistributedMultivariatePolynomial([x,y],Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty10}
+\begin{paste}{ugProblemIdealPageEmpty10}{ugProblemIdealPagePatch10}
+\pastebutton{ugProblemIdealPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{g := x*(x**2-1)\free{fg }\bound{g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch11}
+\begin{paste}{ugProblemIdealPageFull11}{ugProblemIdealPageEmpty11}
+\pastebutton{ugProblemIdealPageFull11}{\hidepaste}
+\tab{5}\spadcommand{relationsIdeal [f,g]\free{f g }}
+\indentrel{3}\begin{verbatim}
+   (11)
+         2      3      2           2           3
+   [- %CY  + %CX  + %CX ] | [%CX= x  - 1,%CY= x  - x]
+Type: SuchThat(List Polynomial Fraction Integer,List Equation Polynomial Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty11}
+\begin{paste}{ugProblemIdealPageEmpty11}{ugProblemIdealPagePatch11}
+\pastebutton{ugProblemIdealPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{relationsIdeal [f,g]\free{f g }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch12}
+\begin{paste}{ugProblemIdealPageFull12}{ugProblemIdealPageEmpty12}
+\pastebutton{ugProblemIdealPageFull12}{\hidepaste}
+\tab{5}\spadcommand{l: List DMP([x,y,z],FRAC INT)\bound{ll }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty12}
+\begin{paste}{ugProblemIdealPageEmpty12}{ugProblemIdealPagePatch12}
+\pastebutton{ugProblemIdealPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{l: List DMP([x,y,z],FRAC INT)\bound{ll }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch13}
+\begin{paste}{ugProblemIdealPageFull13}{ugProblemIdealPageEmpty13}
+\pastebutton{ugProblemIdealPageFull13}{\hidepaste}
+\tab{5}\spadcommand{l:=[x**2+2*y**2,x*z**2-y*z,z**2-4]\free{ll }\bound{l }}
+\indentrel{3}\begin{verbatim}
+           2     2    2        2
+   (13)  [x  + 2y ,x z  - y z,z  - 4]
+Type: List DistributedMultivariatePolynomial([x,y,z],Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty13}
+\begin{paste}{ugProblemIdealPageEmpty13}{ugProblemIdealPagePatch13}
+\pastebutton{ugProblemIdealPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{l:=[x**2+2*y**2,x*z**2-y*z,z**2-4]\free{ll }\bound{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch14}
+\begin{paste}{ugProblemIdealPageFull14}{ugProblemIdealPageEmpty14}
+\pastebutton{ugProblemIdealPageFull14}{\hidepaste}
+\tab{5}\spadcommand{ld:=primaryDecomp ideal l\free{l }\bound{ld }}
+\indentrel{3}\begin{verbatim}
+               1    2             1    2
+   (14)  [[x + Ä y,y ,z + 2],[x - Ä y,y ,z - 2]]
+               2                  2
+Type: List PolynomialIdeals(Fraction Integer,DirectProduct(3,NonNegativeInteger),OrderedVariableList [x,y,z],DistributedMultivariatePolynomial([x,y,z],Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty14}
+\begin{paste}{ugProblemIdealPageEmpty14}{ugProblemIdealPagePatch14}
+\pastebutton{ugProblemIdealPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{ld:=primaryDecomp ideal l\free{l }\bound{ld }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch15}
+\begin{paste}{ugProblemIdealPageFull15}{ugProblemIdealPageEmpty15}
+\pastebutton{ugProblemIdealPageFull15}{\hidepaste}
+\tab{5}\spadcommand{reduce(intersect,ld)\free{ld }}
+\indentrel{3}\begin{verbatim}
+              1      2  2
+   (15)  [x - Ä y z,y ,z  - 4]
+              4
+Type: PolynomialIdeals(Fraction Integer,DirectProduct(3,NonNegativeInteger),OrderedVariableList [x,y,z],DistributedMultivariatePolynomial([x,y,z],Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty15}
+\begin{paste}{ugProblemIdealPageEmpty15}{ugProblemIdealPagePatch15}
+\pastebutton{ugProblemIdealPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{reduce(intersect,ld)\free{ld }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch16}
+\begin{paste}{ugProblemIdealPageFull16}{ugProblemIdealPageEmpty16}
+\pastebutton{ugProblemIdealPageFull16}{\hidepaste}
+\tab{5}\spadcommand{reduce(intersect,[radical ld.i for i in 1..2])\free{ld }}
+\indentrel{3}\begin{verbatim}
+               2
+   (16)  [x,y,z  - 4]
+Type: PolynomialIdeals(Fraction Integer,DirectProduct(3,NonNegativeInteger),OrderedVariableList [x,y,z],DistributedMultivariatePolynomial([x,y,z],Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty16}
+\begin{paste}{ugProblemIdealPageEmpty16}{ugProblemIdealPagePatch16}
+\pastebutton{ugProblemIdealPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{reduce(intersect,[radical ld.i for i in 1..2])\free{ld }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPagePatch17}
+\begin{paste}{ugProblemIdealPageFull17}{ugProblemIdealPageEmpty17}
+\pastebutton{ugProblemIdealPageFull17}{\hidepaste}
+\tab{5}\spadcommand{radical ideal l\free{l }}
+\indentrel{3}\begin{verbatim}
+               2
+   (17)  [x,y,z  - 4]
+Type: PolynomialIdeals(Fraction Integer,DirectProduct(3,NonNegativeInteger),OrderedVariableList [x,y,z],DistributedMultivariatePolynomial([x,y,z],Fraction Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemIdealPageEmpty17}
+\begin{paste}{ugProblemIdealPageEmpty17}{ugProblemIdealPagePatch17}
+\pastebutton{ugProblemIdealPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{radical ideal l\free{l }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch1}
+\begin{paste}{ugxProblemFiniteUtilityPageFull1}{ugxProblemFiniteUtilityPageEmpty1}
+\pastebutton{ugxProblemFiniteUtilityPageFull1}{\hidepaste}
+\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty1}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty1}{ugxProblemFiniteUtilityPagePatch1}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch2}
+\begin{paste}{ugxProblemFiniteUtilityPageFull2}{ugxProblemFiniteUtilityPageEmpty2}
+\pastebutton{ugxProblemFiniteUtilityPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f := createIrreduciblePoly(8)$FFPOLY(GF5)\bound{f }\free{GF5 }}
+\indentrel{3}\begin{verbatim}
+         8    4
+   (2)  ?  + ?  + 2
+          Type: SparseUnivariatePolynomial PrimeField 5
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty2}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty2}{ugxProblemFiniteUtilityPagePatch2}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f := createIrreduciblePoly(8)$FFPOLY(GF5)\bound{f }\free{GF5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch3}
+\begin{paste}{ugxProblemFiniteUtilityPageFull3}{ugxProblemFiniteUtilityPageEmpty3}
+\pastebutton{ugxProblemFiniteUtilityPageFull3}{\hidepaste}
+\tab{5}\spadcommand{primitive?(f)$FFPOLY(GF5)\free{f }}
+\indentrel{3}\begin{verbatim}
+   (3)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty3}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty3}{ugxProblemFiniteUtilityPagePatch3}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{primitive?(f)$FFPOLY(GF5)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch4}
+\begin{paste}{ugxProblemFiniteUtilityPageFull4}{ugxProblemFiniteUtilityPageEmpty4}
+\pastebutton{ugxProblemFiniteUtilityPageFull4}{\hidepaste}
+\tab{5}\spadcommand{normal?(f)$FFPOLY(GF5)\free{f }}
+\indentrel{3}\begin{verbatim}
+   (4)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty4}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty4}{ugxProblemFiniteUtilityPagePatch4}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{normal?(f)$FFPOLY(GF5)\free{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch5}
+\begin{paste}{ugxProblemFiniteUtilityPageFull5}{ugxProblemFiniteUtilityPageEmpty5}
+\pastebutton{ugxProblemFiniteUtilityPageFull5}{\hidepaste}
+\tab{5}\spadcommand{p := createPrimitivePoly(8)$FFPOLY(GF5)\bound{p }\free{GF5 }}
+\indentrel{3}\begin{verbatim}
+         8    3    2
+   (5)  ?  + ?  + ?  + ? + 2
+          Type: SparseUnivariatePolynomial PrimeField 5
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty5}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty5}{ugxProblemFiniteUtilityPagePatch5}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{p := createPrimitivePoly(8)$FFPOLY(GF5)\bound{p }\free{GF5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch6}
+\begin{paste}{ugxProblemFiniteUtilityPageFull6}{ugxProblemFiniteUtilityPageEmpty6}
+\pastebutton{ugxProblemFiniteUtilityPageFull6}{\hidepaste}
+\tab{5}\spadcommand{primitive?(p)$FFPOLY(GF5)\free{p }}
+\indentrel{3}\begin{verbatim}
+   (6)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty6}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty6}{ugxProblemFiniteUtilityPagePatch6}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{primitive?(p)$FFPOLY(GF5)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch7}
+\begin{paste}{ugxProblemFiniteUtilityPageFull7}{ugxProblemFiniteUtilityPageEmpty7}
+\pastebutton{ugxProblemFiniteUtilityPageFull7}{\hidepaste}
+\tab{5}\spadcommand{normal?(p)$FFPOLY(GF5)\free{p }}
+\indentrel{3}\begin{verbatim}
+   (7)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty7}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty7}{ugxProblemFiniteUtilityPagePatch7}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{normal?(p)$FFPOLY(GF5)\free{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch8}
+\begin{paste}{ugxProblemFiniteUtilityPageFull8}{ugxProblemFiniteUtilityPageEmpty8}
+\pastebutton{ugxProblemFiniteUtilityPageFull8}{\hidepaste}
+\tab{5}\spadcommand{n := createNormalPoly(8)$FFPOLY(GF5)\bound{n }\free{GF5 }}
+\indentrel{3}\begin{verbatim}
+         8     7    3
+   (8)  ?  + 4?  + ?  + 1
+          Type: SparseUnivariatePolynomial PrimeField 5
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty8}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty8}{ugxProblemFiniteUtilityPagePatch8}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{n := createNormalPoly(8)$FFPOLY(GF5)\bound{n }\free{GF5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch9}
+\begin{paste}{ugxProblemFiniteUtilityPageFull9}{ugxProblemFiniteUtilityPageEmpty9}
+\pastebutton{ugxProblemFiniteUtilityPageFull9}{\hidepaste}
+\tab{5}\spadcommand{primitive?(n)$FFPOLY(GF5)\free{n }}
+\indentrel{3}\begin{verbatim}
+   (9)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty9}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty9}{ugxProblemFiniteUtilityPagePatch9}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{primitive?(n)$FFPOLY(GF5)\free{n }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch10}
+\begin{paste}{ugxProblemFiniteUtilityPageFull10}{ugxProblemFiniteUtilityPageEmpty10}
+\pastebutton{ugxProblemFiniteUtilityPageFull10}{\hidepaste}
+\tab{5}\spadcommand{createPrimitiveNormalPoly(8)$FFPOLY(GF5)\free{GF5 }}
+\indentrel{3}\begin{verbatim}
+          8     7     5
+   (10)  ?  + 4?  + 2?  + 2
+          Type: SparseUnivariatePolynomial PrimeField 5
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty10}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty10}{ugxProblemFiniteUtilityPagePatch10}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{createPrimitiveNormalPoly(8)$FFPOLY(GF5)\free{GF5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch11}
+\begin{paste}{ugxProblemFiniteUtilityPageFull11}{ugxProblemFiniteUtilityPageEmpty11}
+\pastebutton{ugxProblemFiniteUtilityPageFull11}{\hidepaste}
+\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty11}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty11}{ugxProblemFiniteUtilityPagePatch11}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch12}
+\begin{paste}{ugxProblemFiniteUtilityPageFull12}{ugxProblemFiniteUtilityPageEmpty12}
+\pastebutton{ugxProblemFiniteUtilityPageFull12}{\hidepaste}
+\tab{5}\spadcommand{h := monomial(1,8)$SUP(GF5)\bound{h }\free{GF5 }}
+\indentrel{3}\begin{verbatim}
+          8
+   (12)  ?
+          Type: SparseUnivariatePolynomial PrimeField 5
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty12}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty12}{ugxProblemFiniteUtilityPagePatch12}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{h := monomial(1,8)$SUP(GF5)\bound{h }\free{GF5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch13}
+\begin{paste}{ugxProblemFiniteUtilityPageFull13}{ugxProblemFiniteUtilityPageEmpty13}
+\pastebutton{ugxProblemFiniteUtilityPageFull13}{\hidepaste}
+\tab{5}\spadcommand{nh := nextIrreduciblePoly(h)$FFPOLY(GF5)\bound{nh }\free{h }}
+\indentrel{3}\begin{verbatim}
+          8
+   (13)  ?  + 2
+Type: Union(SparseUnivariatePolynomial PrimeField 5,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty13}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty13}{ugxProblemFiniteUtilityPagePatch13}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{nh := nextIrreduciblePoly(h)$FFPOLY(GF5)\bound{nh }\free{h }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch14}
+\begin{paste}{ugxProblemFiniteUtilityPageFull14}{ugxProblemFiniteUtilityPageEmpty14}
+\pastebutton{ugxProblemFiniteUtilityPageFull14}{\hidepaste}
+\tab{5}\spadcommand{createIrreduciblePoly(3)$FFPOLY(GF5)\free{GF5 }}
+\indentrel{3}\begin{verbatim}
+          3
+   (14)  ?  + ? + 1
+          Type: SparseUnivariatePolynomial PrimeField 5
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty14}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty14}{ugxProblemFiniteUtilityPagePatch14}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{createIrreduciblePoly(3)$FFPOLY(GF5)\free{GF5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch15}
+\begin{paste}{ugxProblemFiniteUtilityPageFull15}{ugxProblemFiniteUtilityPageEmpty15}
+\pastebutton{ugxProblemFiniteUtilityPageFull15}{\hidepaste}
+\tab{5}\spadcommand{nh := nextIrreduciblePoly(nh)$FFPOLY(GF5)\free{nh }}
+\indentrel{3}\begin{verbatim}
+          8
+   (15)  ?  + 3
+Type: Union(SparseUnivariatePolynomial PrimeField 5,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty15}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty15}{ugxProblemFiniteUtilityPagePatch15}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{nh := nextIrreduciblePoly(nh)$FFPOLY(GF5)\free{nh }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch16}
+\begin{paste}{ugxProblemFiniteUtilityPageFull16}{ugxProblemFiniteUtilityPageEmpty16}
+\pastebutton{ugxProblemFiniteUtilityPageFull16}{\hidepaste}
+\tab{5}\spadcommand{numberOfIrreduciblePoly(5)$FFPOLY(GF5)\free{GF5 }}
+\indentrel{3}\begin{verbatim}
+   (16)  624
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty16}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty16}{ugxProblemFiniteUtilityPagePatch16}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{numberOfIrreduciblePoly(5)$FFPOLY(GF5)\free{GF5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch17}
+\begin{paste}{ugxProblemFiniteUtilityPageFull17}{ugxProblemFiniteUtilityPageEmpty17}
+\pastebutton{ugxProblemFiniteUtilityPageFull17}{\hidepaste}
+\tab{5}\spadcommand{numberOfPrimitivePoly(3)$FFPOLY(GF5)\free{GF5 }}
+\indentrel{3}\begin{verbatim}
+   (17)  20
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty17}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty17}{ugxProblemFiniteUtilityPagePatch17}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{numberOfPrimitivePoly(3)$FFPOLY(GF5)\free{GF5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch18}
+\begin{paste}{ugxProblemFiniteUtilityPageFull18}{ugxProblemFiniteUtilityPageEmpty18}
+\pastebutton{ugxProblemFiniteUtilityPageFull18}{\hidepaste}
+\tab{5}\spadcommand{m := monomial(1,1)$SUP(GF5)\bound{m }\free{GF5 }}
+\indentrel{3}\begin{verbatim}
+   (18)  ?
+          Type: SparseUnivariatePolynomial PrimeField 5
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty18}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty18}{ugxProblemFiniteUtilityPagePatch18}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{m := monomial(1,1)$SUP(GF5)\bound{m }\free{GF5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch19}
+\begin{paste}{ugxProblemFiniteUtilityPageFull19}{ugxProblemFiniteUtilityPageEmpty19}
+\pastebutton{ugxProblemFiniteUtilityPageFull19}{\hidepaste}
+\tab{5}\spadcommand{f := m**3 + 4*m**2 + m + 2\bound{fx }\free{m }}
+\indentrel{3}\begin{verbatim}
+          3     2
+   (19)  ?  + 4?  + ? + 2
+          Type: SparseUnivariatePolynomial PrimeField 5
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty19}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty19}{ugxProblemFiniteUtilityPagePatch19}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{f := m**3 + 4*m**2 + m + 2\bound{fx }\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch20}
+\begin{paste}{ugxProblemFiniteUtilityPageFull20}{ugxProblemFiniteUtilityPageEmpty20}
+\pastebutton{ugxProblemFiniteUtilityPageFull20}{\hidepaste}
+\tab{5}\spadcommand{f1 := nextPrimitivePoly(f)$FFPOLY(GF5)\free{fx }\bound{f1 }}
+\indentrel{3}\begin{verbatim}
+          3     2
+   (20)  ?  + 4?  + 4? + 2
+Type: Union(SparseUnivariatePolynomial PrimeField 5,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty20}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty20}{ugxProblemFiniteUtilityPagePatch20}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{f1 := nextPrimitivePoly(f)$FFPOLY(GF5)\free{fx }\bound{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch21}
+\begin{paste}{ugxProblemFiniteUtilityPageFull21}{ugxProblemFiniteUtilityPageEmpty21}
+\pastebutton{ugxProblemFiniteUtilityPageFull21}{\hidepaste}
+\tab{5}\spadcommand{nextPrimitivePoly(f1)$FFPOLY(GF5)\free{f1 }}
+\indentrel{3}\begin{verbatim}
+          3     2
+   (21)  ?  + 2?  + 3
+Type: Union(SparseUnivariatePolynomial PrimeField 5,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty21}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty21}{ugxProblemFiniteUtilityPagePatch21}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{nextPrimitivePoly(f1)$FFPOLY(GF5)\free{f1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch22}
+\begin{paste}{ugxProblemFiniteUtilityPageFull22}{ugxProblemFiniteUtilityPageEmpty22}
+\pastebutton{ugxProblemFiniteUtilityPageFull22}{\hidepaste}
+\tab{5}\spadcommand{f := m**3 + m**2 + 4*m + 1\bound{fy }\free{m }}
+\indentrel{3}\begin{verbatim}
+          3    2
+   (22)  ?  + ?  + 4? + 1
+          Type: SparseUnivariatePolynomial PrimeField 5
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty22}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty22}{ugxProblemFiniteUtilityPagePatch22}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{f := m**3 + m**2 + 4*m + 1\bound{fy }\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch23}
+\begin{paste}{ugxProblemFiniteUtilityPageFull23}{ugxProblemFiniteUtilityPageEmpty23}
+\pastebutton{ugxProblemFiniteUtilityPageFull23}{\hidepaste}
+\tab{5}\spadcommand{f1 := nextNormalPoly(f)$FFPOLY(GF5)\free{fy }\bound{f1y }}
+\indentrel{3}\begin{verbatim}
+          3    2
+   (23)  ?  + ?  + 4? + 3
+Type: Union(SparseUnivariatePolynomial PrimeField 5,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty23}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty23}{ugxProblemFiniteUtilityPagePatch23}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{f1 := nextNormalPoly(f)$FFPOLY(GF5)\free{fy }\bound{f1y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch24}
+\begin{paste}{ugxProblemFiniteUtilityPageFull24}{ugxProblemFiniteUtilityPageEmpty24}
+\pastebutton{ugxProblemFiniteUtilityPageFull24}{\hidepaste}
+\tab{5}\spadcommand{nextNormalPoly(f1)$FFPOLY(GF5)\free{f1y }}
+\indentrel{3}\begin{verbatim}
+          3     2
+   (24)  ?  + 2?  + 1
+Type: Union(SparseUnivariatePolynomial PrimeField 5,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty24}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty24}{ugxProblemFiniteUtilityPagePatch24}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{nextNormalPoly(f1)$FFPOLY(GF5)\free{f1y }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch25}
+\begin{paste}{ugxProblemFiniteUtilityPageFull25}{ugxProblemFiniteUtilityPageEmpty25}
+\pastebutton{ugxProblemFiniteUtilityPageFull25}{\hidepaste}
+\tab{5}\spadcommand{GF16 := FFX(FFX(PF 2,2),2);\bound{GF16 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty25}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty25}{ugxProblemFiniteUtilityPagePatch25}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{GF16 := FFX(FFX(PF 2,2),2);\bound{GF16 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch26}
+\begin{paste}{ugxProblemFiniteUtilityPageFull26}{ugxProblemFiniteUtilityPageEmpty26}
+\pastebutton{ugxProblemFiniteUtilityPageFull26}{\hidepaste}
+\tab{5}\spadcommand{createIrreduciblePoly(5)$FFPOLY(GF16)\free{GF16 }}
+\indentrel{3}\begin{verbatim}
+          5
+   (26)  ?  + %CZ
+Type: SparseUnivariatePolynomial FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty26}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty26}{ugxProblemFiniteUtilityPagePatch26}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{createIrreduciblePoly(5)$FFPOLY(GF16)\free{GF16 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch27}
+\begin{paste}{ugxProblemFiniteUtilityPageFull27}{ugxProblemFiniteUtilityPageEmpty27}
+\pastebutton{ugxProblemFiniteUtilityPageFull27}{\hidepaste}
+\tab{5}\spadcommand{random(5)$FFPOLY(GF16)\free{GF16 }}
+\indentrel{3}\begin{verbatim}
+   (27)
+      5             4                            3
+     ?  + (%CV + 1)?  + ((%CV + 1)%CZ + %CV + 1)?
+   + 
+               2
+     (%CV + 1)?  + ((%CV + 1)%CZ + %CV + 1)?
+   + 
+     (%CV + 1)%CZ + 1
+Type: SparseUnivariatePolynomial FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty27}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty27}{ugxProblemFiniteUtilityPagePatch27}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{random(5)$FFPOLY(GF16)\free{GF16 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch28}
+\begin{paste}{ugxProblemFiniteUtilityPageFull28}{ugxProblemFiniteUtilityPageEmpty28}
+\pastebutton{ugxProblemFiniteUtilityPageFull28}{\hidepaste}
+\tab{5}\spadcommand{random(3,9)$FFPOLY(GF16)\free{GF16 }}
+\indentrel{3}\begin{verbatim}
+   (28)
+      4                   3                       2
+     ?  + (%CZ + %CV + 1)?  + (%CV %CZ + %CV + 1)?
+   + 
+     (%CZ + %CV)? + (%CV + 1)%CZ
+Type: SparseUnivariatePolynomial FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty28}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty28}{ugxProblemFiniteUtilityPagePatch28}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{random(3,9)$FFPOLY(GF16)\free{GF16 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch29}
+\begin{paste}{ugxProblemFiniteUtilityPageFull29}{ugxProblemFiniteUtilityPageEmpty29}
+\pastebutton{ugxProblemFiniteUtilityPageFull29}{\hidepaste}
+\tab{5}\spadcommand{GF2 := PrimeField 2;\bound{GF2 }}
+\indentrel{3}\begin{verbatim}
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty29}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty29}{ugxProblemFiniteUtilityPagePatch29}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{GF2 := PrimeField 2;\bound{GF2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch30}
+\begin{paste}{ugxProblemFiniteUtilityPageFull30}{ugxProblemFiniteUtilityPageEmpty30}
+\pastebutton{ugxProblemFiniteUtilityPageFull30}{\hidepaste}
+\tab{5}\spadcommand{F := FFX(GF2,12)\bound{F }\free{GF2 }}
+\indentrel{3}\begin{verbatim}
+   (30)  FiniteFieldExtension(PrimeField 2,12)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty30}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty30}{ugxProblemFiniteUtilityPagePatch30}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{F := FFX(GF2,12)\bound{F }\free{GF2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch31}
+\begin{paste}{ugxProblemFiniteUtilityPageFull31}{ugxProblemFiniteUtilityPageEmpty31}
+\pastebutton{ugxProblemFiniteUtilityPageFull31}{\hidepaste}
+\tab{5}\spadcommand{f := createIrreduciblePoly(6)$FFPOLY(GF2)\bound{fz }\free{GF2 }}
+\indentrel{3}\begin{verbatim}
+          6
+   (31)  ?  + ? + 1
+          Type: SparseUnivariatePolynomial PrimeField 2
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty31}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty31}{ugxProblemFiniteUtilityPagePatch31}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{f := createIrreduciblePoly(6)$FFPOLY(GF2)\bound{fz }\free{GF2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPagePatch32}
+\begin{paste}{ugxProblemFiniteUtilityPageFull32}{ugxProblemFiniteUtilityPageEmpty32}
+\pastebutton{ugxProblemFiniteUtilityPageFull32}{\hidepaste}
+\tab{5}\spadcommand{root := rootOfIrreduciblePoly(f)$FFPOLY2(F,GF2)\free{F GF2 fz }\bound{root }}
+\indentrel{3}\begin{verbatim}
+            11      8      7      5
+   (32)  %CS   + %CS  + %CS  + %CS  + %CS + 1
+            Type: FiniteFieldExtension(PrimeField 2,12)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugxProblemFiniteUtilityPageEmpty32}
+\begin{paste}{ugxProblemFiniteUtilityPageEmpty32}{ugxProblemFiniteUtilityPagePatch32}
+\pastebutton{ugxProblemFiniteUtilityPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{root := rootOfIrreduciblePoly(f)$FFPOLY2(F,GF2)\free{F GF2 fz }\bound{root }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPagePatch1}
+\begin{paste}{ugProblemEigenPageFull1}{ugProblemEigenPageEmpty1}
+\pastebutton{ugProblemEigenPageFull1}{\hidepaste}
+\tab{5}\spadcommand{m1 := matrix [[1,2,1],[2,1,-2],[1,-2,4]]\bound{m1 }}
+\indentrel{3}\begin{verbatim}
+        Ú1   2    1 ¿
+        ³           ³
+   (1)  ³2   1   - 2³
+        ³           ³
+        À1  - 2   4 Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPageEmpty1}
+\begin{paste}{ugProblemEigenPageEmpty1}{ugProblemEigenPagePatch1}
+\pastebutton{ugProblemEigenPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{m1 := matrix [[1,2,1],[2,1,-2],[1,-2,4]]\bound{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPagePatch2}
+\begin{paste}{ugProblemEigenPageFull2}{ugProblemEigenPageEmpty2}
+\pastebutton{ugProblemEigenPageFull2}{\hidepaste}
+\tab{5}\spadcommand{leig := eigenvalues(m1)\free{m1 }\bound{leig }}
+\indentrel{3}\begin{verbatim}
+                    2
+   (2)  [5,%DA | %DA  - %DA - 5]
+Type: List Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPageEmpty2}
+\begin{paste}{ugProblemEigenPageEmpty2}{ugProblemEigenPagePatch2}
+\pastebutton{ugProblemEigenPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{leig := eigenvalues(m1)\free{m1 }\bound{leig }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPagePatch3}
+\begin{paste}{ugProblemEigenPageFull3}{ugProblemEigenPageEmpty3}
+\pastebutton{ugProblemEigenPageFull3}{\hidepaste}
+\tab{5}\spadcommand{eigenvector(first(leig),m1)\free{m1 leig }}
+\indentrel{3}\begin{verbatim}
+         Ú 0 ¿
+         ³   ³
+         ³  1³
+   (3)  [³- ij]
+         ³  2³
+         ³   ³
+         À 1 Ù
+ Type: List Matrix Fraction Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPageEmpty3}
+\begin{paste}{ugProblemEigenPageEmpty3}{ugProblemEigenPagePatch3}
+\pastebutton{ugProblemEigenPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{eigenvector(first(leig),m1)\free{m1 leig }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPagePatch4}
+\begin{paste}{ugProblemEigenPageFull4}{ugProblemEigenPageEmpty4}
+\pastebutton{ugProblemEigenPageFull4}{\hidepaste}
+\tab{5}\spadcommand{eigenvectors(m1)\free{m1 }}
+\indentrel{3}\begin{verbatim}
+   (4)
+                                   Ú 0 ¿
+                                   ³   ³
+                                   ³  1³
+   [[eigval= 5,eigmult= 1,eigvec= [³- ij]],
+                                   ³  2³
+                                   ³   ³
+                                   À 1 Ù
+
+                        2
+     [eigval= (%DB | %DB  - %DB - 5), eigmult= 1,
+               Ú%DB¿
+               ³   ³
+      eigvec= [³ 2 ³]]
+               ³   ³
+               À 1 Ù
+     ]
+Type: List Record(eigval: Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial Integer)),eigmult: NonNegativeInteger,eigvec: List Matrix Fraction Polynomial Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPageEmpty4}
+\begin{paste}{ugProblemEigenPageEmpty4}{ugProblemEigenPagePatch4}
+\pastebutton{ugProblemEigenPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{eigenvectors(m1)\free{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPagePatch5}
+\begin{paste}{ugProblemEigenPageFull5}{ugProblemEigenPageEmpty5}
+\pastebutton{ugProblemEigenPageFull5}{\hidepaste}
+\tab{5}\spadcommand{radicalEigenvectors(m1)\free{m1 }}
+\indentrel{3}\begin{verbatim}
+   (5)
+                                            Ú ÚÄÄ¿    ¿
+              ÚÄÄ¿                          ³\³21  + 1³
+             \³21  + 1                      ³ÄÄÄÄÄÄÄÄij
+   [[radval= ÄÄÄÄÄÄÄÄÄ,radmult= 1,radvect= [³    2    ³]],
+                 2                          ³         ³
+                                            ³    2    ³
+                                            ³         ³
+                                            À    1    Ù
+
+                 ÚÄÄ¿
+              - \³21  + 1
+     [radval= ÄÄÄÄÄÄÄÄÄÄÄ, radmult= 1,
+                   2
+                Ú   ÚÄÄ¿    ¿
+                ³- \³21  + 1³
+                ³ÄÄÄÄÄÄÄÄÄÄij
+      radvect= [³     2     ³]]
+                ³           ³
+                ³     2     ³
+                ³           ³
+                À     1     Ù
+     ,
+                                    Ú 0 ¿
+                                    ³   ³
+                                    ³  1³
+    [radval= 5,radmult= 1,radvect= [³- ij]]]
+                                    ³  2³
+                                    ³   ³
+                                    À 1 Ù
+Type: List Record(radval: Expression Integer,radmult: Integer,radvect: List Matrix Expression Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPageEmpty5}
+\begin{paste}{ugProblemEigenPageEmpty5}{ugProblemEigenPagePatch5}
+\pastebutton{ugProblemEigenPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{radicalEigenvectors(m1)\free{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPagePatch6}
+\begin{paste}{ugProblemEigenPageFull6}{ugProblemEigenPageEmpty6}
+\pastebutton{ugProblemEigenPageFull6}{\hidepaste}
+\tab{5}\spadcommand{realEigenvectors(m1,1/1000)\free{m1 }}
+\indentrel{3}\begin{verbatim}
+   (6)
+                                    Ú 0 ¿
+                                    ³   ³
+                                    ³  1³
+   [[outval= 5,outmult= 1,outvect= [³- ij]],
+                                    ³  2³
+                                    ³   ³
+                                    À 1 Ù
+                                       Ú5717¿
+                                       ³ÄÄÄij
+             5717                      ³2048³
+    [outval= ÄÄÄÄ,outmult= 1,outvect= [³    ³]],
+             2048                      ³ 2  ³
+                                       ³    ³
+                                       À 1  Ù
+                                         Ú  3669¿
+                                         ³- ÄÄÄij
+               3669                      ³  2048³
+    [outval= - ÄÄÄÄ,outmult= 1,outvect= [³      ³]]]
+               2048                      ³  2   ³
+                                         ³      ³
+                                         À  1   Ù
+Type: List Record(outval: Fraction Integer,outmult: Integer,outvect: List Matrix Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPageEmpty6}
+\begin{paste}{ugProblemEigenPageEmpty6}{ugProblemEigenPagePatch6}
+\pastebutton{ugProblemEigenPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{realEigenvectors(m1,1/1000)\free{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPagePatch7}
+\begin{paste}{ugProblemEigenPageFull7}{ugProblemEigenPageEmpty7}
+\pastebutton{ugProblemEigenPageFull7}{\hidepaste}
+\tab{5}\spadcommand{eigenMatrix(m1)\free{m1 }}
+\indentrel{3}\begin{verbatim}
+        Ú ÚÄÄ¿         ÚÄÄ¿         ¿
+        ³\³21  + 1  - \³21  + 1     ³
+        ³ÄÄÄÄÄÄÄÄÄ  ÄÄÄÄÄÄÄÄÄÄÄ   0 ³
+        ³    2           2          ³
+   (7)  ³                           ³
+        ³                          1³
+        ³    2           2       - ij
+        ³                          2³
+        ³                           ³
+        À    1           1        1 Ù
+             Type: Union(Matrix Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPageEmpty7}
+\begin{paste}{ugProblemEigenPageEmpty7}{ugProblemEigenPagePatch7}
+\pastebutton{ugProblemEigenPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{eigenMatrix(m1)\free{m1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPagePatch8}
+\begin{paste}{ugProblemEigenPageFull8}{ugProblemEigenPageEmpty8}
+\pastebutton{ugProblemEigenPageFull8}{\hidepaste}
+\tab{5}\spadcommand{m2 := matrix [[-5,-2],[18,7]]\bound{m2 }}
+\indentrel{3}\begin{verbatim}
+        Ú- 5  - 2¿
+   (8)  ³        ³
+        À18    7 Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPageEmpty8}
+\begin{paste}{ugProblemEigenPageEmpty8}{ugProblemEigenPagePatch8}
+\pastebutton{ugProblemEigenPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{m2 := matrix [[-5,-2],[18,7]]\bound{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPagePatch9}
+\begin{paste}{ugProblemEigenPageFull9}{ugProblemEigenPageEmpty9}
+\pastebutton{ugProblemEigenPageFull9}{\hidepaste}
+\tab{5}\spadcommand{eigenMatrix(m2)\free{m2 }}
+\indentrel{3}\begin{verbatim}
+   (9)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPageEmpty9}
+\begin{paste}{ugProblemEigenPageEmpty9}{ugProblemEigenPagePatch9}
+\pastebutton{ugProblemEigenPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{eigenMatrix(m2)\free{m2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPagePatch10}
+\begin{paste}{ugProblemEigenPageFull10}{ugProblemEigenPageEmpty10}
+\pastebutton{ugProblemEigenPageFull10}{\hidepaste}
+\tab{5}\spadcommand{m3 := matrix [[1,2],[2,1]]\bound{m3 }}
+\indentrel{3}\begin{verbatim}
+         Ú1  2¿
+   (10)  ³    ³
+         À2  1Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPageEmpty10}
+\begin{paste}{ugProblemEigenPageEmpty10}{ugProblemEigenPagePatch10}
+\pastebutton{ugProblemEigenPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{m3 := matrix [[1,2],[2,1]]\bound{m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPagePatch11}
+\begin{paste}{ugProblemEigenPageFull11}{ugProblemEigenPageEmpty11}
+\pastebutton{ugProblemEigenPageFull11}{\hidepaste}
+\tab{5}\spadcommand{orthonormalBasis(m3)\free{m3 }}
+\indentrel{3}\begin{verbatim}
+          Ú    1 ¿ Ú  1 ¿
+          ³- ÄÄÄij ³ÄÄÄij
+          ³   ÚÄ¿³ ³ ÚÄ¿³
+          ³  \³2 ³ ³\³2 ³
+   (11)  [³      ³,³    ³]
+          ³   1  ³ ³  1 ³
+          ³ ÄÄÄÄ ³ ³ÄÄÄij
+          ³  ÚÄ¿ ³ ³ ÚÄ¿³
+          À \³2  Ù À\³2 Ù
+                   Type: List Matrix Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemEigenPageEmpty11}
+\begin{paste}{ugProblemEigenPageEmpty11}{ugProblemEigenPagePatch11}
+\pastebutton{ugProblemEigenPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{orthonormalBasis(m3)\free{m3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch1}
+\begin{paste}{ugProblemGaloisPageFull1}{ugProblemGaloisPageEmpty1}
+\pastebutton{ugProblemGaloisPageFull1}{\hidepaste}
+\tab{5}\spadcommand{p := x**5 - 5*x + 12\bound{p }}
+\indentrel{3}\begin{verbatim}
+         5
+   (1)  x  - 5x + 12
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty1}
+\begin{paste}{ugProblemGaloisPageEmpty1}{ugProblemGaloisPagePatch1}
+\pastebutton{ugProblemGaloisPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{p := x**5 - 5*x + 12\bound{p }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch2}
+\begin{paste}{ugProblemGaloisPageFull2}{ugProblemGaloisPageEmpty2}
+\pastebutton{ugProblemGaloisPageFull2}{\hidepaste}
+\tab{5}\spadcommand{q := resultant(eval(p,x,y),-eval(p,x,y-x),y)\free{p }\bound{q }}
+\indentrel{3}\begin{verbatim}
+   (2)
+      25      21        17         15        13
+     x   - 50x   - 2375x   + 90000x   - 5000x
+   + 
+             11          9            7            5
+     2700000x   + 250000x  + 18000000x  + 64000000x
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty2}
+\begin{paste}{ugProblemGaloisPageEmpty2}{ugProblemGaloisPagePatch2}
+\pastebutton{ugProblemGaloisPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{q := resultant(eval(p,x,y),-eval(p,x,y-x),y)\free{p }\bound{q }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch3}
+\begin{paste}{ugProblemGaloisPageFull3}{ugProblemGaloisPageEmpty3}
+\pastebutton{ugProblemGaloisPageFull3}{\hidepaste}
+\tab{5}\spadcommand{q1 := exquo(q, x**5)\free{q }\bound{q1 }}
+\indentrel{3}\begin{verbatim}
+   (3)
+      20      16        12         10        8           6
+     x   - 50x   - 2375x   + 90000x   - 5000x  + 2700000x
+   + 
+            4            2
+     250000x  + 18000000x  + 64000000
+                    Type: Union(Polynomial Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty3}
+\begin{paste}{ugProblemGaloisPageEmpty3}{ugProblemGaloisPagePatch3}
+\pastebutton{ugProblemGaloisPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{q1 := exquo(q, x**5)\free{q }\bound{q1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch4}
+\begin{paste}{ugProblemGaloisPageFull4}{ugProblemGaloisPageEmpty4}
+\pastebutton{ugProblemGaloisPageFull4}{\hidepaste}
+\tab{5}\spadcommand{factoredQ := factor q1\free{q1 }\bound{factoredQ }}
+\indentrel{3}\begin{verbatim}
+   (4)
+       10      8      6        4        2
+     (x   - 10x  - 75x  + 1500x  - 5500x  + 16000)
+  *
+       10      8       6       4        2
+     (x   + 10x  + 125x  + 500x  + 2500x  + 4000)
+                      Type: Factored Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty4}
+\begin{paste}{ugProblemGaloisPageEmpty4}{ugProblemGaloisPagePatch4}
+\pastebutton{ugProblemGaloisPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{factoredQ := factor q1\free{q1 }\bound{factoredQ }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch5}
+\begin{paste}{ugProblemGaloisPageFull5}{ugProblemGaloisPageEmpty5}
+\pastebutton{ugProblemGaloisPageFull5}{\hidepaste}
+\tab{5}\spadcommand{r := nthFactor(factoredQ,1)\free{factoredQ }\bound{r }}
+\indentrel{3}\begin{verbatim}
+         10      8      6        4        2
+   (5)  x   - 10x  - 75x  + 1500x  - 5500x  + 16000
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty5}
+\begin{paste}{ugProblemGaloisPageEmpty5}{ugProblemGaloisPagePatch5}
+\pastebutton{ugProblemGaloisPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{r := nthFactor(factoredQ,1)\free{factoredQ }\bound{r }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch6}
+\begin{paste}{ugProblemGaloisPageFull6}{ugProblemGaloisPageEmpty6}
+\pastebutton{ugProblemGaloisPageFull6}{\hidepaste}
+\tab{5}\spadcommand{beta:AN := rootOf(eval(r,x,b))\free{r }\bound{beta }}
+\indentrel{3}\begin{verbatim}
+   (6)  b
+                                  Type: AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty6}
+\begin{paste}{ugProblemGaloisPageEmpty6}{ugProblemGaloisPagePatch6}
+\pastebutton{ugProblemGaloisPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{beta:AN := rootOf(eval(r,x,b))\free{r }\bound{beta }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch7}
+\begin{paste}{ugProblemGaloisPageFull7}{ugProblemGaloisPageEmpty7}
+\pastebutton{ugProblemGaloisPageFull7}{\hidepaste}
+\tab{5}\spadcommand{p := p::UP(x,INT)::UP(x,AN)\free{p }\bound{declareP }}
+\indentrel{3}\begin{verbatim}
+         5
+   (7)  x  - 5x + 12
+          Type: UnivariatePolynomial(x,AlgebraicNumber)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty7}
+\begin{paste}{ugProblemGaloisPageEmpty7}{ugProblemGaloisPagePatch7}
+\pastebutton{ugProblemGaloisPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{p := p::UP(x,INT)::UP(x,AN)\free{p }\bound{declareP }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch8}
+\begin{paste}{ugProblemGaloisPageFull8}{ugProblemGaloisPageEmpty8}
+\pastebutton{ugProblemGaloisPageFull8}{\hidepaste}
+\tab{5}\spadcommand{algFactors := factor(p,[beta])\free{declareP beta }\bound{algFactors }}
+\indentrel{3}\begin{verbatim}
+   (8)
+       x
+     + 
+                9       8       7        6         5
+           - 85b  - 116b  + 780b  + 2640b  + 14895b
+         + 
+                  4          3          2
+           - 8820b  - 127050b  - 327000b  - 405200b
+         + 
+           2062400
+      /
+         1339200
+  *
+               8       6        4         2
+          - 17b  + 156b  + 2979b  - 25410b  - 14080
+     (x + ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ)
+                            66960
+  *
+       x
+     + 
+               8        6         4          2
+           143b  - 2100b  - 10485b  + 290550b  - 334800b
+         + 
+           - 960800
+      /
+         669600
+  *
+       x
+     + 
+               8        6         4          2
+           143b  - 2100b  - 10485b  + 290550b  + 334800b
+         + 
+           - 960800
+      /
+         669600
+  *
+       x
+     + 
+              9       8       7        6         5
+           85b  - 116b  - 780b  + 2640b  - 14895b
+         + 
+                  4          3          2
+           - 8820b  + 127050b  - 327000b  + 405200b
+         + 
+           2062400
+      /
+         1339200
+ Type: Factored UnivariatePolynomial(x,AlgebraicNumber)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty8}
+\begin{paste}{ugProblemGaloisPageEmpty8}{ugProblemGaloisPagePatch8}
+\pastebutton{ugProblemGaloisPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{algFactors := factor(p,[beta])\free{declareP beta }\bound{algFactors }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch9}
+\begin{paste}{ugProblemGaloisPageFull9}{ugProblemGaloisPageEmpty9}
+\pastebutton{ugProblemGaloisPageFull9}{\hidepaste}
+\tab{5}\spadcommand{factor(p)\free{declareP }}
+\indentrel{3}\begin{verbatim}
+         5
+   (9)  x  - 5x + 12
+ Type: Factored UnivariatePolynomial(x,AlgebraicNumber)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty9}
+\begin{paste}{ugProblemGaloisPageEmpty9}{ugProblemGaloisPagePatch9}
+\pastebutton{ugProblemGaloisPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{factor(p)\free{declareP }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch10}
+\begin{paste}{ugProblemGaloisPageFull10}{ugProblemGaloisPageEmpty10}
+\pastebutton{ugProblemGaloisPageFull10}{\hidepaste}
+\tab{5}\spadcommand{factor1 := nthFactor(algFactors,1)\free{algFactors }\bound{factor1 }}
+\indentrel{3}\begin{verbatim}
+   (10)
+     x
+   + 
+              9       8       7        6         5
+         - 85b  - 116b  + 780b  + 2640b  + 14895b
+       + 
+              4          3          2
+       - 8820b  - 127050b  - 327000b  - 405200b + 2062400
+    /
+       1339200
+          Type: UnivariatePolynomial(x,AlgebraicNumber)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty10}
+\begin{paste}{ugProblemGaloisPageEmpty10}{ugProblemGaloisPagePatch10}
+\pastebutton{ugProblemGaloisPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{factor1 := nthFactor(algFactors,1)\free{algFactors }\bound{factor1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch11}
+\begin{paste}{ugProblemGaloisPageFull11}{ugProblemGaloisPageEmpty11}
+\pastebutton{ugProblemGaloisPageFull11}{\hidepaste}
+\tab{5}\spadcommand{root1 := -coefficient(factor1,0)\free{factor1 }\bound{root1 }}
+\indentrel{3}\begin{verbatim}
+   (11)
+          9       8       7        6         5        4
+       85b  + 116b  - 780b  - 2640b  - 14895b  + 8820b
+     + 
+              3          2
+       127050b  + 327000b  + 405200b - 2062400
+  /
+     1339200
+                                  Type: AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty11}
+\begin{paste}{ugProblemGaloisPageEmpty11}{ugProblemGaloisPagePatch11}
+\pastebutton{ugProblemGaloisPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{root1 := -coefficient(factor1,0)\free{factor1 }\bound{root1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch12}
+\begin{paste}{ugProblemGaloisPageFull12}{ugProblemGaloisPageEmpty12}
+\pastebutton{ugProblemGaloisPageFull12}{\hidepaste}
+\tab{5}\spadcommand{roots := [-coefficient(nthFactor(algFactors,i),0) for i in 1..5]\free{algFactors }\bound{roots }}
+\indentrel{3}\begin{verbatim}
+   (12)
+   [
+            9       8       7        6         5        4
+         85b  + 116b  - 780b  - 2640b  - 14895b  + 8820b
+       + 
+                3          2
+         127050b  + 327000b  + 405200b - 2062400
+    /
+       1339200
+     ,
+       8       6        4         2
+    17b  - 156b  - 2979b  + 25410b  + 14080
+    ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ,
+                     66960
+
+               8        6         4          2
+         - 143b  + 2100b  + 10485b  - 290550b  + 334800b
+       + 
+         960800
+    /
+       669600
+     ,
+
+               8        6         4          2
+         - 143b  + 2100b  + 10485b  - 290550b  - 334800b
+       + 
+         960800
+    /
+       669600
+     ,
+
+              9       8       7        6         5
+         - 85b  + 116b  + 780b  - 2640b  + 14895b
+       + 
+              4          3          2
+         8820b  - 127050b  + 327000b  - 405200b - 2062400
+    /
+       1339200
+     ]
+                             Type: List AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty12}
+\begin{paste}{ugProblemGaloisPageEmpty12}{ugProblemGaloisPagePatch12}
+\pastebutton{ugProblemGaloisPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{roots := [-coefficient(nthFactor(algFactors,i),0) for i in 1..5]\free{algFactors }\bound{roots }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch13}
+\begin{paste}{ugProblemGaloisPageFull13}{ugProblemGaloisPageEmpty13}
+\pastebutton{ugProblemGaloisPageFull13}{\hidepaste}
+\tab{5}\spadcommand{(a1,a2,a3,a4,a5) := (roots.1,roots.2,roots.3,roots.4,roots.5)\free{roots }\bound{ais }}
+\indentrel{3}\begin{verbatim}
+   (13)
+            9       8       7        6         5        4
+       - 85b  + 116b  + 780b  - 2640b  + 14895b  + 8820b
+     + 
+                3          2
+       - 127050b  + 327000b  - 405200b - 2062400
+  /
+     1339200
+                                  Type: AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty13}
+\begin{paste}{ugProblemGaloisPageEmpty13}{ugProblemGaloisPagePatch13}
+\pastebutton{ugProblemGaloisPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{(a1,a2,a3,a4,a5) := (roots.1,roots.2,roots.3,roots.4,roots.5)\free{roots }\bound{ais }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch14}
+\begin{paste}{ugProblemGaloisPageFull14}{ugProblemGaloisPageEmpty14}
+\pastebutton{ugProblemGaloisPageFull14}{\hidepaste}
+\tab{5}\spadcommand{eval(r,x,a1 - a2)\free{ais }}
+\indentrel{3}\begin{verbatim}
+   (14)  0
+                       Type: Polynomial AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty14}
+\begin{paste}{ugProblemGaloisPageEmpty14}{ugProblemGaloisPagePatch14}
+\pastebutton{ugProblemGaloisPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{eval(r,x,a1 - a2)\free{ais }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch15}
+\begin{paste}{ugProblemGaloisPageFull15}{ugProblemGaloisPageEmpty15}
+\pastebutton{ugProblemGaloisPageFull15}{\hidepaste}
+\tab{5}\spadcommand{eval(r,x,a1 - a3)\free{ais }}
+\indentrel{3}\begin{verbatim}
+   (15)
+             9         8          7          6           5
+       47905b  + 66920b  - 536100b  - 980400b  - 3345075b
+     + 
+                 4            3             2
+       - 5787000b  + 75572250b  + 161688000b  - 184600000b
+     + 
+       - 710912000
+  /
+     4464
+                       Type: Polynomial AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty15}
+\begin{paste}{ugProblemGaloisPageEmpty15}{ugProblemGaloisPagePatch15}
+\pastebutton{ugProblemGaloisPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{eval(r,x,a1 - a3)\free{ais }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch16}
+\begin{paste}{ugProblemGaloisPageFull16}{ugProblemGaloisPageEmpty16}
+\pastebutton{ugProblemGaloisPageFull16}{\hidepaste}
+\tab{5}\spadcommand{eval(r,x,a1 - a4)\free{ais }}
+\indentrel{3}\begin{verbatim}
+   (16)  0
+                       Type: Polynomial AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty16}
+\begin{paste}{ugProblemGaloisPageEmpty16}{ugProblemGaloisPagePatch16}
+\pastebutton{ugProblemGaloisPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{eval(r,x,a1 - a4)\free{ais }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch17}
+\begin{paste}{ugProblemGaloisPageFull17}{ugProblemGaloisPageEmpty17}
+\pastebutton{ugProblemGaloisPageFull17}{\hidepaste}
+\tab{5}\spadcommand{eval(r,x,a1 - a5)\free{ais }}
+\indentrel{3}\begin{verbatim}
+             8        6         4          2
+         405b  + 3450b  - 19875b  - 198000b  - 588000
+   (17)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                              31
+                       Type: Polynomial AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty17}
+\begin{paste}{ugProblemGaloisPageEmpty17}{ugProblemGaloisPagePatch17}
+\pastebutton{ugProblemGaloisPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{eval(r,x,a1 - a5)\free{ais }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch18}
+\begin{paste}{ugProblemGaloisPageFull18}{ugProblemGaloisPageEmpty18}
+\pastebutton{ugProblemGaloisPageFull18}{\hidepaste}
+\tab{5}\spadcommand{bb := a1 - a4\free{ais }\bound{bb }}
+\indentrel{3}\begin{verbatim}
+   (18)
+          9       8       7        6         5         4
+       85b  + 402b  - 780b  - 6840b  - 14895b  - 12150b
+     + 
+              3          2
+       127050b  + 908100b  + 1074800b - 3984000
+  /
+     1339200
+                                  Type: AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty18}
+\begin{paste}{ugProblemGaloisPageEmpty18}{ugProblemGaloisPagePatch18}
+\pastebutton{ugProblemGaloisPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{bb := a1 - a4\free{ais }\bound{bb }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch19}
+\begin{paste}{ugProblemGaloisPageFull19}{ugProblemGaloisPageEmpty19}
+\pastebutton{ugProblemGaloisPageFull19}{\hidepaste}
+\tab{5}\spadcommand{aa1 := subst(a1,beta = bb)\free{beta bb ais }\bound{aa1 }}
+\indentrel{3}\begin{verbatim}
+   (19)
+             8        6         4          2
+       - 143b  + 2100b  + 10485b  - 290550b  + 334800b
+     + 
+       960800
+  /
+     669600
+                                  Type: AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty19}
+\begin{paste}{ugProblemGaloisPageEmpty19}{ugProblemGaloisPagePatch19}
+\pastebutton{ugProblemGaloisPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{aa1 := subst(a1,beta = bb)\free{beta bb ais }\bound{aa1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch20}
+\begin{paste}{ugProblemGaloisPageFull20}{ugProblemGaloisPageEmpty20}
+\pastebutton{ugProblemGaloisPageFull20}{\hidepaste}
+\tab{5}\spadcommand{aa2 := subst(a2,beta = bb)\free{beta bb ais }\bound{aa2 }}
+\indentrel{3}\begin{verbatim}
+   (20)
+            9       8       7        6         5        4
+       - 85b  + 116b  + 780b  - 2640b  + 14895b  + 8820b
+     + 
+                3          2
+       - 127050b  + 327000b  - 405200b - 2062400
+  /
+     1339200
+                                  Type: AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty20}
+\begin{paste}{ugProblemGaloisPageEmpty20}{ugProblemGaloisPagePatch20}
+\pastebutton{ugProblemGaloisPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{aa2 := subst(a2,beta = bb)\free{beta bb ais }\bound{aa2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch21}
+\begin{paste}{ugProblemGaloisPageFull21}{ugProblemGaloisPageEmpty21}
+\pastebutton{ugProblemGaloisPageFull21}{\hidepaste}
+\tab{5}\spadcommand{aa3 := subst(a3,beta = bb)\free{beta bb ais }\bound{aa3 }}
+\indentrel{3}\begin{verbatim}
+   (21)
+          9       8       7        6         5        4
+       85b  + 116b  - 780b  - 2640b  - 14895b  + 8820b
+     + 
+              3          2
+       127050b  + 327000b  + 405200b - 2062400
+  /
+     1339200
+                                  Type: AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty21}
+\begin{paste}{ugProblemGaloisPageEmpty21}{ugProblemGaloisPagePatch21}
+\pastebutton{ugProblemGaloisPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{aa3 := subst(a3,beta = bb)\free{beta bb ais }\bound{aa3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch22}
+\begin{paste}{ugProblemGaloisPageFull22}{ugProblemGaloisPageEmpty22}
+\pastebutton{ugProblemGaloisPageFull22}{\hidepaste}
+\tab{5}\spadcommand{aa4 := subst(a4,beta = bb)\free{beta bb ais }\bound{aa4 }}
+\indentrel{3}\begin{verbatim}
+   (22)
+             8        6         4          2
+       - 143b  + 2100b  + 10485b  - 290550b  - 334800b
+     + 
+       960800
+  /
+     669600
+                                  Type: AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty22}
+\begin{paste}{ugProblemGaloisPageEmpty22}{ugProblemGaloisPagePatch22}
+\pastebutton{ugProblemGaloisPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{aa4 := subst(a4,beta = bb)\free{beta bb ais }\bound{aa4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch23}
+\begin{paste}{ugProblemGaloisPageFull23}{ugProblemGaloisPageEmpty23}
+\pastebutton{ugProblemGaloisPageFull23}{\hidepaste}
+\tab{5}\spadcommand{aa5 := subst(a5,beta = bb)\free{beta bb ais }\bound{aa5 }}
+\indentrel{3}\begin{verbatim}
+            8       6        4         2
+         17b  - 156b  - 2979b  + 25410b  + 14080
+   (23)  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
+                          66960
+                                  Type: AlgebraicNumber
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty23}
+\begin{paste}{ugProblemGaloisPageEmpty23}{ugProblemGaloisPagePatch23}
+\pastebutton{ugProblemGaloisPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{aa5 := subst(a5,beta = bb)\free{beta bb ais }\bound{aa5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch24}
+\begin{paste}{ugProblemGaloisPageFull24}{ugProblemGaloisPageEmpty24}
+\pastebutton{ugProblemGaloisPageFull24}{\hidepaste}
+\tab{5}\spadcommand{(aa1 = a1) :: Boolean\free{aa1 }}
+\indentrel{3}\begin{verbatim}
+   (24)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty24}
+\begin{paste}{ugProblemGaloisPageEmpty24}{ugProblemGaloisPagePatch24}
+\pastebutton{ugProblemGaloisPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{(aa1 = a1) :: Boolean\free{aa1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch25}
+\begin{paste}{ugProblemGaloisPageFull25}{ugProblemGaloisPageEmpty25}
+\pastebutton{ugProblemGaloisPageFull25}{\hidepaste}
+\tab{5}\spadcommand{(aa1 = a2) :: Boolean\free{aa1 }}
+\indentrel{3}\begin{verbatim}
+   (25)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty25}
+\begin{paste}{ugProblemGaloisPageEmpty25}{ugProblemGaloisPagePatch25}
+\pastebutton{ugProblemGaloisPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{(aa1 = a2) :: Boolean\free{aa1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch26}
+\begin{paste}{ugProblemGaloisPageFull26}{ugProblemGaloisPageEmpty26}
+\pastebutton{ugProblemGaloisPageFull26}{\hidepaste}
+\tab{5}\spadcommand{(aa1 = a3) :: Boolean\free{aa1 }}
+\indentrel{3}\begin{verbatim}
+   (26)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty26}
+\begin{paste}{ugProblemGaloisPageEmpty26}{ugProblemGaloisPagePatch26}
+\pastebutton{ugProblemGaloisPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{(aa1 = a3) :: Boolean\free{aa1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch27}
+\begin{paste}{ugProblemGaloisPageFull27}{ugProblemGaloisPageEmpty27}
+\pastebutton{ugProblemGaloisPageFull27}{\hidepaste}
+\tab{5}\spadcommand{(aa1 = a4) :: Boolean\free{aa1 }}
+\indentrel{3}\begin{verbatim}
+   (27)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty27}
+\begin{paste}{ugProblemGaloisPageEmpty27}{ugProblemGaloisPagePatch27}
+\pastebutton{ugProblemGaloisPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{(aa1 = a4) :: Boolean\free{aa1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPagePatch28}
+\begin{paste}{ugProblemGaloisPageFull28}{ugProblemGaloisPageEmpty28}
+\pastebutton{ugProblemGaloisPageFull28}{\hidepaste}
+\tab{5}\spadcommand{(aa1 = a5) :: Boolean\free{aa1 }}
+\indentrel{3}\begin{verbatim}
+   (28)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemGaloisPageEmpty28}
+\begin{paste}{ugProblemGaloisPageEmpty28}{ugProblemGaloisPagePatch28}
+\pastebutton{ugProblemGaloisPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{(aa1 = a5) :: Boolean\free{aa1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPagePatch1}
+\begin{paste}{ugProblemLimitsPageFull1}{ugProblemLimitsPageEmpty1}
+\pastebutton{ugProblemLimitsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)}
+\indentrel{3}\begin{verbatim}
+          1
+   (1)  - Ä
+          2
+Type: Union(OrderedCompletion Fraction Polynomial Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPageEmpty1}
+\begin{paste}{ugProblemLimitsPageEmpty1}{ugProblemLimitsPagePatch1}
+\pastebutton{ugProblemLimitsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPagePatch2}
+\begin{paste}{ugProblemLimitsPageFull2}{ugProblemLimitsPageEmpty2}
+\pastebutton{ugProblemLimitsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{limit(x * log(x),x = 0,"right")}
+\indentrel{3}\begin{verbatim}
+   (2)  0
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPageEmpty2}
+\begin{paste}{ugProblemLimitsPageEmpty2}{ugProblemLimitsPagePatch2}
+\pastebutton{ugProblemLimitsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{limit(x * log(x),x = 0,"right")}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPagePatch3}
+\begin{paste}{ugProblemLimitsPageFull3}{ugProblemLimitsPageEmpty3}
+\pastebutton{ugProblemLimitsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{limit(x * log(x),x = 0)}
+\indentrel{3}\begin{verbatim}
+   (3)  [leftHandLimit= "failed",rightHandLimit= 0]
+Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPageEmpty3}
+\begin{paste}{ugProblemLimitsPageEmpty3}{ugProblemLimitsPagePatch3}
+\pastebutton{ugProblemLimitsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{limit(x * log(x),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPagePatch4}
+\begin{paste}{ugProblemLimitsPageFull4}{ugProblemLimitsPageEmpty4}
+\pastebutton{ugProblemLimitsPageFull4}{\hidepaste}
+\tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)}
+\indentrel{3}\begin{verbatim}
+   (4)  [leftHandLimit= - 1,rightHandLimit= 1]
+Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPageEmpty4}
+\begin{paste}{ugProblemLimitsPageEmpty4}{ugProblemLimitsPagePatch4}
+\pastebutton{ugProblemLimitsPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPagePatch5}
+\begin{paste}{ugProblemLimitsPageFull5}{ugProblemLimitsPageEmpty5}
+\pastebutton{ugProblemLimitsPageFull5}{\hidepaste}
+\tab{5}\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)}
+\indentrel{3}\begin{verbatim}
+                            1                    1
+   (5)  [leftHandLimit= - ÄÄÄÄ,rightHandLimit= ÄÄÄÄ]
+                           ÚÄ¿                  ÚÄ¿
+                          \³2                  \³2
+Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPageEmpty5}
+\begin{paste}{ugProblemLimitsPageEmpty5}{ugProblemLimitsPagePatch5}
+\pastebutton{ugProblemLimitsPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPagePatch6}
+\begin{paste}{ugProblemLimitsPageFull6}{ugProblemLimitsPageEmpty6}
+\pastebutton{ugProblemLimitsPageFull6}{\hidepaste}
+\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)}
+\indentrel{3}\begin{verbatim}
+         ÚÄ¿
+        \³3
+   (6)  ÄÄÄÄ
+          5
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPageEmpty6}
+\begin{paste}{ugProblemLimitsPageEmpty6}{ugProblemLimitsPagePatch6}
+\pastebutton{ugProblemLimitsPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPagePatch7}
+\begin{paste}{ugProblemLimitsPageFull7}{ugProblemLimitsPageEmpty7}
+\pastebutton{ugProblemLimitsPageFull7}{\hidepaste}
+\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)}
+\indentrel{3}\begin{verbatim}
+           ÚÄ¿
+          \³3
+   (7)  - ÄÄÄÄ
+            5
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPageEmpty7}
+\begin{paste}{ugProblemLimitsPageEmpty7}{ugProblemLimitsPagePatch7}
+\pastebutton{ugProblemLimitsPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPagePatch8}
+\begin{paste}{ugProblemLimitsPageFull8}{ugProblemLimitsPageEmpty8}
+\pastebutton{ugProblemLimitsPageFull8}{\hidepaste}
+\tab{5}\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)}
+\indentrel{3}\begin{verbatim}
+        a
+   (8)  Ä
+        b
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPageEmpty8}
+\begin{paste}{ugProblemLimitsPageEmpty8}{ugProblemLimitsPagePatch8}
+\pastebutton{ugProblemLimitsPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPagePatch9}
+\begin{paste}{ugProblemLimitsPageFull9}{ugProblemLimitsPageEmpty9}
+\pastebutton{ugProblemLimitsPageFull9}{\hidepaste}
+\tab{5}\spadcommand{limit(z * sin(1/z),z = 0)}
+\indentrel{3}\begin{verbatim}
+   (9)  0
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPageEmpty9}
+\begin{paste}{ugProblemLimitsPageEmpty9}{ugProblemLimitsPagePatch9}
+\pastebutton{ugProblemLimitsPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{limit(z * sin(1/z),z = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPagePatch10}
+\begin{paste}{ugProblemLimitsPageFull10}{ugProblemLimitsPageEmpty10}
+\pastebutton{ugProblemLimitsPageFull10}{\hidepaste}
+\tab{5}\spadcommand{complexLimit(z * sin(1/z),z = 0)}
+\indentrel{3}\begin{verbatim}
+   (10)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPageEmpty10}
+\begin{paste}{ugProblemLimitsPageEmpty10}{ugProblemLimitsPagePatch10}
+\pastebutton{ugProblemLimitsPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{complexLimit(z * sin(1/z),z = 0)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPagePatch11}
+\begin{paste}{ugProblemLimitsPageFull11}{ugProblemLimitsPageEmpty11}
+\pastebutton{ugProblemLimitsPageFull11}{\hidepaste}
+\tab{5}\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)}
+\indentrel{3}\begin{verbatim}
+   (11)  - 1
+   Type: OnePointCompletion Fraction Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPageEmpty11}
+\begin{paste}{ugProblemLimitsPageEmpty11}{ugProblemLimitsPagePatch11}
+\pastebutton{ugProblemLimitsPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPagePatch12}
+\begin{paste}{ugProblemLimitsPageFull12}{ugProblemLimitsPageEmpty12}
+\pastebutton{ugProblemLimitsPageFull12}{\hidepaste}
+\tab{5}\spadcommand{limit(sin(x)/x,x = \%plusInfinity)}
+\indentrel{3}\begin{verbatim}
+   (12)  0
+  Type: Union(OrderedCompletion Expression Integer,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPageEmpty12}
+\begin{paste}{ugProblemLimitsPageEmpty12}{ugProblemLimitsPagePatch12}
+\pastebutton{ugProblemLimitsPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{limit(sin(x)/x,x = \%plusInfinity)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPagePatch13}
+\begin{paste}{ugProblemLimitsPageFull13}{ugProblemLimitsPageEmpty13}
+\pastebutton{ugProblemLimitsPageFull13}{\hidepaste}
+\tab{5}\spadcommand{complexLimit(sin(x)/x,x = \%infinity)}
+\indentrel{3}\begin{verbatim}
+   (13)  "failed"
+                              Type: Union("failed",...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugProblemLimitsPageEmpty13}
+\begin{paste}{ugProblemLimitsPageEmpty13}{ugProblemLimitsPagePatch13}
+\pastebutton{ugProblemLimitsPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{complexLimit(sin(x)/x,x = \%infinity)}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ug10.ht b/src/hyper/pages/ug10.ht
new file mode 100644
index 00000000..aff71675
--- /dev/null
+++ b/src/hyper/pages/ug10.ht
@@ -0,0 +1,1119 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\texht{\setcounter{chapter}{9}}{} % Chapter 10
+
+%
+\newcommand{\ugIntProgTitle}{Interactive Programming}
+\newcommand{\ugIntProgNumber}{10.}
+%
+% =====================================================================
+\begin{page}{ugIntProgPage}{10. Interactive Programming}
+% =====================================================================
+\beginscroll
+
+Programming in the interpreter is easy.
+So is the use of \Language{}'s graphics facility.
+Both are rather flexible and allow you to use them for many
+interesting applications.
+However, both require learning some basic ideas and skills.
+
+All graphics examples in the \Gallery{} section are either
+produced directly by interactive commands or by interpreter
+programs.
+Four of these programs are introduced here.
+By the end of this chapter you will know enough about graphics and
+programming in the interpreter to not only understand all these
+examples, but to tackle interesting and difficult problems on your
+own.
+\downlink{``\ugAppGraphicsTitle''}{ugAppGraphicsPage} in Appendix \ugAppGraphicsNumber\ignore{ugAppGraphics} lists all the remaining commands and
+programs used to create these images.
+
+\beginmenu
+    \menudownlink{{10.1. Drawing Ribbons Interactively}}{ugIntProgDrawingPage}
+    \menudownlink{{10.2. A Ribbon Program}}{ugIntProgRibbonPage}
+    \menudownlink{{10.3. Coloring and Positioning Ribbons}}{ugIntProgColorPage}
+    \menudownlink{{10.4. Points, Lines, and Curves}}{ugIntProgPLCPage}
+    \menudownlink{{10.5. A Bouquet of Arrows}}{ugIntProgColorArrPage}
+    \menudownlink{{10.6. Drawing Complex Vector Fields}}{ugIntProgVecFieldsPage}
+    \menudownlink{{10.7. Drawing Complex Functions}}{ugIntProgCompFunsPage}
+    \menudownlink{{10.8. Functions Producing Functions}}{ugIntProgFunctionsPage}
+    \menudownlink{{10.9. Automatic Newton Iteration Formulas}}{ugIntProgNewtonPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntProgDrawingTitle}{Drawing Ribbons Interactively}
+\newcommand{\ugIntProgDrawingNumber}{10.1.}
+%
+% =====================================================================
+\begin{page}{ugIntProgDrawingPage}{10.1. Drawing Ribbons Interactively}
+% =====================================================================
+\beginscroll
+%
+
+We begin our discussion of interactive graphics with the creation
+of a useful facility: plotting ribbons of two-graphs in
+three-space.
+Suppose you want to draw the \twodim{} graphs of \smath{n}
+functions \texht{$f_i(x), 1 \leq i \leq n,$}{\spad{f_i(x), 1 <= i
+<= 5,}} all over some fixed range of \smath{x}.
+One approach is to create a \twodim{} graph for each one, then
+superpose one on top of the other.
+What you will more than likely get is a jumbled mess.
+Even if you make each function a different color, the result is
+likely to be confusing.
+
+A better approach is to display each of the \smath{f_i(x)} in three
+%-% \HDindex{ribbon}{ugIntProgDrawingPage}{10.1.}{Drawing Ribbons Interactively}
+dimensions as a ``ribbon'' of some appropriate width along the
+\smath{y}-direction, laying down each  ribbon next to the
+previous one.
+A ribbon is simply a function of \smath{x} and \smath{y} depending
+only on \smath{x.}
+
+We illustrate this for \smath{f_i(x)} defined as simple powers of
+\smath{x} for \smath{x} ranging between \smath{-1} and \smath{1}.
+
+\psXtc{
+Draw the ribbon for \texht{$z = x^2$}{\spad{z=x ** 2}}.
+}{
+\graphpaste{draw(x**2,x=-1..1,y=0..1)}
+}{
+\epsffile[0 0 295 295]{../ps/ribbon1.ps}
+}
+
+Now that was easy!
+What you get is a ``wire-mesh'' rendition of the ribbon.
+That's fine for now.
+Notice that the mesh-size is small in both the \smath{x} and the
+\smath{y} directions.
+\Language{} normally computes points in both these directions.
+This is unnecessary.
+One step is all we need in the \smath{y}-direction.
+To have \Language{} economize on \spad{y}-points, we re-draw the
+ribbon with option \spad{var2Steps == 1}.
+
+\psXtc{
+Re-draw the ribbon, but with option \spad{var2Steps == 1}
+so that only \spad{1} step is computed in the
+\smath{y} direction.
+}{
+\graphpaste{vp := draw(x**2,x=-1..1,y=0..1,var2Steps==1) \bound{d1}}
+}{
+\epsffile[0 0 295 295]{../ps/ribbon2.ps}
+}
+
+The operation has created a viewport, that is, a graphics window
+on your screen.
+We assigned the viewport to \spad{vp} and now we manipulate
+its contents.
+
+
+Graphs are objects, like numbers and algebraic expressions.
+You may want to do some experimenting with graphs.
+For example, say
+\begin{verbatim}
+showRegion(vp, "on")
+\end{verbatim}
+to put a bounding box around the ribbon.
+Try it!
+Issue \spad{rotate(vp, -45, 90)} to rotate the
+figure \smath{-45} longitudinal degrees and \smath{90} latitudinal
+degrees.
+
+\psXtc{
+Here is a different rotation.
+This turns the graph so you can view it along the \smath{y}-axis.
+}{
+\spadpaste{rotate(vp, 0, -90)\bound{d3}\free{d1}}
+}{
+\epsffile[0 0 295 295]{../ps/ribbon2r.ps}
+}
+
+There are many other things you can do.
+In fact, most everything you can do interactively using the
+\threedim{} control panel (such as translating, zooming, resizing,
+coloring, perspective and lighting selections) can also be done
+directly by operations (see \downlink{``\ugGraphTitle''}{ugGraphPage} in Chapter \ugGraphNumber\ignore{ugGraph} for more details).
+
+When you are done experimenting, say \spad{reset(vp)} to restore the
+picture to its original position and settings.
+
+
+Let's add another ribbon to our picture---one
+for \texht{$x^3$}{\spad{x**3}}.
+Since \smath{y} ranges from \smath{0} to \smath{1} for the
+first ribbon, now let \smath{y} range from \smath{1} to
+\smath{2.}
+This puts the second ribbon next to the first one.
+
+How do you add a second ribbon to the viewport?
+One method is
+to extract the ``space'' component from the
+viewport using the operation
+\spadfunFrom{subspace}{ThreeDimensionalViewport}.
+You can think of the space component as the object inside the
+window (here, the ribbon).
+Let's call it \spad{sp}.
+To add the second ribbon, you draw the second ribbon using the
+option \spad{space == sp}.
+
+\xtc{
+Extract the space component of \spad{vp}.
+}{
+\spadpaste{sp := subspace(vp)\bound{d5}\free{d1}}
+}
+
+\psXtc{
+Add the ribbon for
+\texht{$x^3$}{\spad{x**3}} alongside that for
+\texht{$x^2$}{\spad{x**2}}.
+}{
+\graphpaste{vp := draw(x**3,x=-1..1,y=1..2,var2Steps==1, space==sp)\bound{d6}\free{d5}}
+}{
+\epsffile[0 0 295 295]{../ps/ribbons.ps}
+}
+
+Unless you moved the original viewport, the new viewport covers
+the old one.
+You might want to check that the old object is still there by
+moving the top window.
+
+Let's show quadrilateral polygon outlines on the ribbons and then
+enclose the ribbons in a box.
+
+\psXtc{
+Show quadrilateral polygon outlines.
+}{
+\spadpaste{drawStyle(vp,"shade");outlineRender(vp,"on")\bound{d10}\free{d6}}
+}{
+\epsffile[0 0 295 295]{../ps/ribbons2.ps}
+}
+\psXtc{
+Enclose the ribbons in a box.
+}{
+\spadpaste{rotate(vp,20,-60); showRegion(vp,"on")\bound{d11}\free{d10}}
+}{
+\epsffile[0 0 295 295]{../ps/ribbons2b.ps}
+}
+
+This process has become tedious!
+If we had to add two or three more ribbons, we would have to
+repeat the above steps several more times.
+It is time to write an interpreter program to help us take care of
+the details.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntProgRibbonTitle}{A Ribbon Program}
+\newcommand{\ugIntProgRibbonNumber}{10.2.}
+%
+% =====================================================================
+\begin{page}{ugIntProgRibbonPage}{10.2. A Ribbon Program}
+% =====================================================================
+\beginscroll
+%
+
+The above approach creates a new viewport for each additional
+ribbon.
+A better approach is to build one object composed of all ribbons
+before creating a viewport.
+To do this, use \spadfun{makeObject} rather than \spadfun{draw}.
+The operations have similar formats, but
+\spadfun{draw} returns a viewport and
+\spadfun{makeObject} returns a space object.
+
+We now create a function \userfun{drawRibbons} of two arguments:
+\spad{flist}, a list of formulas for the ribbons you want to draw,
+and \spad{xrange}, the range over which you want them drawn.
+Using this function, you can just say
+\begin{verbatim}
+drawRibbons([x**2, x**3], x=-1..1)
+\end{verbatim}
+to do all of the work required in the last section.
+Here is the \userfun{drawRibbons} program.
+Invoke your favorite editor and create a file called {\bf ribbon.input}
+containing the following program.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ drawRibbons(flist,\ xrange)\ ==}\newline
+{\tt 2.\ \ \ \ \ sp\ :=\ createThreeSpace()}\newline
+{\tt 3.\ \ \ \ \ y0\ :=\ 0}\newline
+{\tt 4.\ \ \ \ \ for\ f\ in\ flist\ repeat}\newline
+{\tt 5.\ \ \ \ \ \ \ makeObject(f,\ xrange,\ y=y0..y0+1,\ }\newline
+{\tt 6.\ \ \ \ \ \ \ \ \ \ space==sp,\ var2Steps\ ==\ 1)}\newline
+{\tt 7.\ \ \ \ \ \ \ y0\ :=\ y0\ +\ 1}\newline
+{\tt 8.\ \ \ \ \ vp\ :=\ makeViewport3D(sp,\ "Ribbons")}\newline
+{\tt 9.\ \ \ \ \ drawStyle(vp,\ "shade")}\newline
+{\tt 10.\ \ \ \ outlineRender(vp,\ "on")}\newline
+{\tt 11.\ \ \ \ showRegion(vp,"on")}\newline
+{\tt 12.\ \ \ \ n\ :=\ \#\ flist}\newline
+{\tt 13.\ \ \ \ zoom(vp,n,1,n)}\newline
+{\tt 14.\ \ \ \ rotate(vp,0,75)}\newline
+{\tt 15.\ \ \ \ vp}\newline
+\caption{The first \protect\pspadfun{drawRibbons} function.}\label{fig-ribdraw1}
+\endImportant
+
+Here are some remarks on the syntax used in the \pspadfun{drawRibbons} function
+(consult \downlink{``\ugUserTitle''}{ugUserPage} in Chapter \ugUserNumber\ignore{ugUser} for more details).
+Unlike most other programming languages which use semicolons,
+parentheses, or {\it begin}--{\it end} brackets to delineate the
+structure of programs, the structure of an \Language{} program is
+determined by indentation.
+The first line of the function definition always begins in column 1.
+All other lines of the function are indented with respect to the first
+line and form a \spadgloss{pile} (see \downlink{``\ugLangBlocksTitle''}{ugLangBlocksPage} in Section \ugLangBlocksNumber\ignore{ugLangBlocks}).
+
+The definition of \userfun{drawRibbons}
+consists of a pile of expressions to be executed one after
+another.
+Each expression of the pile is indented at the same level.
+Lines 4-7 designate one single expression:
+since lines 5-7 are indented with respect to the others, these
+lines are treated as a continuation of line 4.
+Also since lines 5 and 7 have the same indentation level, these
+lines designate a pile within the outer pile.
+
+The last line of a pile usually gives the value returned by the
+pile.
+Here it is also the value returned by the function.
+\Language{} knows this is the last line of the function because it
+is the last line of the file.
+In other cases, a new expression beginning in column one signals
+the end of a function.
+
+The line \spad{drawStyle(vp,"shade")} is given after the viewport
+has been created to select the draw style.
+We have also used the \spadfunFrom{zoom}{ThreeDimensionalViewport}
+option.
+Without the zoom, the viewport region would be scaled equally in
+all three coordinate directions.
+
+Let's try the function \userfun{drawRibbons}.
+First you must read the file to give \Language{} the function definition.
+
+\xtc{
+Read the input file.
+}{
+\spadpaste{)read ribbon \bound{s0}}
+}
+\psXtc{
+Draw ribbons for \texht{$x, x^2,\dots, x^5$}{x, x**2,...,x**5}
+for \texht{$-1 \leq x \leq 1$}{-1 <= x <= 1}
+}{
+\graphpaste{drawRibbons([x**i for i in 1..5],x=-1..1) \free{s0}}
+}{
+\epsffile[0 0 295 295]{../ps/ribbons5.ps}
+}
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntProgColorTitle}{Coloring and Positioning Ribbons}
+\newcommand{\ugIntProgColorNumber}{10.3.}
+%
+% =====================================================================
+\begin{page}{ugIntProgColorPage}{10.3. Coloring and Positioning Ribbons}
+% =====================================================================
+\beginscroll
+%
+
+Before leaving the ribbon example, we  make two improvements.
+Normally, the color given to each point in the space is a
+function of its height within a bounding box.
+The points at the bottom of the
+box are red, those at the top are purple.
+
+To change the normal coloring, you can give
+an option \spad{colorFunction == {\it function}}.
+When \Language{} goes about displaying the data, it
+determines the range of colors used for all points within the box.
+\Language{} then distributes these numbers uniformly over the number of hues.
+Here we use the simple color function
+\texht{$(x,y) \mapsto i$}{(x,y) +-> i} for the
+\eth{\smath{i}} ribbon.
+
+Also, we add an argument \spad{yrange} so you can give the range of
+\spad{y} occupied by the ribbons.
+For example, if the \spad{yrange} is given as
+\spad{y=0..1} and there are \smath{5} ribbons to be displayed, each
+ribbon would have width \smath{0.2} and would appear in the
+range \texht{$0 \leq y \leq 1$}{\spad{0 <= y <= 1}}.
+
+Refer to lines 4-9.
+Line 4 assigns to \spad{yVar} the variable part of the
+\spad{yrange} (after all, it need not be \spad{y}).
+Suppose that \spad{yrange} is given as \spad{t = a..b} where \spad{a} and
+\spad{b} have numerical values.
+Then line 5 assigns the value of \spad{a} to the variable \spad{y0}.
+Line 6 computes the width of the ribbon by dividing the difference of
+\spad{a} and \spad{b} by the number, \spad{num}, of ribbons.
+The result is assigned to the variable \spad{width}.
+Note that in the for-loop in line 7, we are iterating in parallel; it is
+not a nested loop.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ drawRibbons(flist,\ xrange,\ yrange)\ ==}\newline
+{\tt 2.\ \ \ \ \ sp\ :=\ createThreeSpace()}\newline
+{\tt 3.\ \ \ \ \ num\ :=\ \#\ flist}\newline
+{\tt 4.\ \ \ \ \ yVar\ :=\ variable\ yrange}\newline
+{\tt 5.\ \ \ \ \ y0:Float\ \ \ \ :=\ lo\ segment\ yrange}\newline
+{\tt 6.\ \ \ \ \ width:Float\ :=\ (hi\ segment\ yrange\ -\ y0)/num}\newline
+{\tt 7.\ \ \ \ \ for\ f\ in\ flist\ for\ color\ in\ 1..num\ repeat}\newline
+{\tt 8.\ \ \ \ \ \ \ makeObject(f,\ xrange,\ yVar\ =\ y0..y0+width,}\newline
+{\tt 9.\ \ \ \ \ \ \ \ \ var2Steps\ ==\ 1,\ colorFunction\ ==\ (x,y)\ +->\ color,\ \_}\newline
+{\tt 10.\ \ \ \ \ \ \ \ space\ ==\ sp)}\newline
+{\tt 11.\ \ \ \ \ \ y0\ :=\ y0\ +\ width}\newline
+{\tt 12.\ \ \ \ vp\ :=\ makeViewport3D(sp,\ "Ribbons")}\newline
+{\tt 13.\ \ \ \ drawStyle(vp,\ "shade")}\newline
+{\tt 14.\ \ \ \ outlineRender(vp,\ "on")}\newline
+{\tt 15.\ \ \ \ showRegion(vp,\ "on")}\newline
+{\tt 16.\ \ \ \ vp}\newline
+\caption{The final \protect\pspadfun{drawRibbons} function.}\label{fig-ribdraw2}
+\endImportant
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntProgPLCTitle}{Points, Lines, and Curves}
+\newcommand{\ugIntProgPLCNumber}{10.4.}
+%
+% =====================================================================
+\begin{page}{ugIntProgPLCPage}{10.4. Points, Lines, and Curves}
+% =====================================================================
+\beginscroll
+%
+What you have seen so far is a high-level program using the
+graphics facility.
+We now turn to the more basic notions of points, lines, and curves
+in \threedim{} graphs.
+These facilities use small floats (objects
+of type \spadtype{DoubleFloat}) for data.
+Let us first give names to the small float values \smath{0} and
+\smath{1}.
+\xtc{
+The small float 0.
+}{
+\spadpaste{zero := 0.0@DFLOAT \bound{d1}}
+}
+\xtc{
+The small float 1.
+}{
+\spadpaste{one  := 1.0@DFLOAT \bound{d2}}
+}
+The \spadSyntax{@} sign means ``of the type.'' Thus \spad{zero} is
+\smath{0.0} of the type \spadtype{DoubleFloat}.
+You can also say \spad{0.0::DFLOAT}.
+
+Points can have four small float components: \smath{x, y, z} coordinates and an
+optional color.
+A ``curve'' is simply a list of points connected by straight line
+segments.
+\xtc{
+Create the point \spad{origin} with color zero, that is, the lowest color
+on the color map.
+}{
+\spadpaste{origin := point [zero,zero,zero,zero] \free{d1}\bound{d3}}
+}
+\xtc{
+Create the point \spad{unit} with color zero.
+}{
+\spadpaste{unit := point [one,one,one,zero] \free{d1 d2}\bound{d4}}
+}
+\xtc{
+Create the curve (well, here, a line) from
+\spad{origin} to \spad{unit}.
+}{
+\spadpaste{line := [origin, unit] \free{d3 d4} \bound{d5}}
+}
+
+We make this line segment into an arrow by adding an arrowhead.
+The arrowhead extends to,
+say, \spad{p3} on the left, and to, say, \spad{p4} on the right.
+To describe an arrow, you tell \Language{} to draw the two curves
+\spad{[p1, p2, p3]} and \spad{[p2, p4].}
+We also decide through experimentation on
+values for \spad{arrowScale}, the ratio of the size of
+the arrowhead to the stem of the arrow, and \spad{arrowAngle},
+the angle between the arrowhead and the arrow.
+
+Invoke your favorite editor and create
+an input file called {\bf arrows.input}.
+This input file first defines the values of
+%\spad{origin},\spad{unit},
+\spad{arrowAngle} and \spad{arrowScale}, then
+defines the function \userfun{makeArrow}\texht{$(p_1, p_2)$}{(p1, p2)} to
+draw an arrow from point \texht{$p_1$}{p1} to \texht{$p_2$}{p2}.
+
+\beginImportant
+  
+\noindent
+%\xmpLine{origin := point [0.0@DFLOAT,0.0@DFLOAT,0.0@DFLOAT,0.0@DFLOAT]}{The point 0 with color 0.}
+%\xmpLine{unit := point [1.0@DFLOAT,1.0@DFLOAT,1.0@DFLOAT,0.0@DFLOAT]}{A second point with color 0.}
+%\xmpLine{}{}
+{\tt 1.\ \ \ arrowAngle\ :=\ \%pi-\%pi/10.0@DFLOAT}\newline
+{\tt 2.\ \ \ arrowScale\ :=\ 0.2@DFLOAT}\newline
+{\tt 3.\ \ \ }\newline
+{\tt 4.\ \ \ makeArrow(p1,\ p2)\ ==}\newline
+{\tt 5.\ \ \ \ \ delta\ :=\ p2\ -\ p1}\newline
+{\tt 6.\ \ \ \ \ len\ :=\ arrowScale\ *\ length\ delta}\newline
+{\tt 7.\ \ \ \ \ theta\ :=\ atan(delta.1,\ delta.2)}\newline
+{\tt 8.\ \ \ \ \ c1\ :=\ len*cos(theta\ +\ arrowAngle)}\newline
+{\tt 9.\ \ \ \ \ s1\ :=\ len*sin(theta\ +\ arrowAngle)}\newline
+{\tt 10.\ \ \ \ c2\ :=\ len*cos(theta\ -\ arrowAngle)}\newline
+{\tt 11.\ \ \ \ s2\ :=\ len*sin(theta\ -\ arrowAngle)}\newline
+{\tt 12.\ \ \ \ z\ \ :=\ p2.3*(1\ -\ arrowScale)}\newline
+{\tt 13.\ \ \ \ p3\ :=\ point\ [p2.1\ +\ c1,\ p2.2\ +\ s1,\ z,\ p2.4]}\newline
+{\tt 14.\ \ \ \ p4\ :=\ point\ [p2.1\ +\ c2,\ p2.2\ +\ s2,\ z,\ p2.4]}\newline
+{\tt 15.\ \ \ \ [[p1,\ p2,\ p3],\ [p2,\ p4]]}\newline
+\endImportant
+
+Read the file and then create
+an arrow from the point \spad{origin} to the point \spad{unit}.
+\xtc{
+Read the input file defining \userfun{makeArrow}.
+}{
+\spadpaste{)read arrows\bound{v1}}
+}
+\xtc{
+Construct the arrow (a list of two curves).
+}{
+\spadpaste{arrow := makeArrow(origin,unit)\bound{v2}\free{v1 d3 d4}}
+}
+\xtc{
+Create an empty object \spad{sp} of type \spad{ThreeSpace}.
+}{
+\spadpaste{sp := createThreeSpace()\bound{c1}}
+}
+\xtc{
+Add each curve of the arrow to the space \spad{sp}.
+}{
+\spadpaste{for a in arrow repeat sp := curve(sp,a)\bound{v3}\free{v2}\free{c1}}
+}
+\psXtc{
+Create a \threedim{} viewport containing that space.
+}{
+\graphpaste{vp := makeViewport3D(sp,"Arrow")\bound{v4}\free{v3}}
+}{
+\epsffile[0 0 295 295]{../ps/arrow.ps}
+}
+\psXtc{
+Here is a better viewing angle.
+}{
+\spadpaste{rotate(vp,200,-60)\bound{v5}\free{v4}}
+}{
+\epsffile[0 0 295 295]{../ps/arrowr.ps}
+}
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntProgColorArrTitle}{A Bouquet of Arrows}
+\newcommand{\ugIntProgColorArrNumber}{10.5.}
+%
+% =====================================================================
+\begin{page}{ugIntProgColorArrPage}{10.5. A Bouquet of Arrows}
+% =====================================================================
+\beginscroll
+
+%\Language{} gathers up all the points of a graph and looks at the range
+%of color values given as integers.
+%If theses color values range from a minimum value of \spad{a} to a maximum
+%value of \spad{b}, then the \spad{a} values are colored red (the
+%lowest color in our spectrum), and \spad{b} values are colored
+%purple (the highest color), and those in the middle are colored
+%green.
+%When all the points are the same color as above, \Language{}
+%chooses green.
+
+Let's draw a ``bouquet'' of arrows.
+Each arrow is identical. The arrowheads are
+uniformly placed on a circle parallel to the \smath{xy}-plane.
+Thus the position of each arrow differs only
+by the angle \texht{$\theta$}{theta},
+\texht{$0 \leq \theta < 2\pi$}{\spad{0 <= theta < 2*\%pi}},
+between the arrow and
+the \smath{x}-axis on the \smath{xy}-plane.
+
+Our bouquet is rather special: each arrow has a different
+color (which won't be evident here, unfortunately).
+This is arranged by letting the color of each successive arrow be
+denoted by \texht{$\theta$}{theta}.
+In this way, the color of arrows ranges from red to green to violet.
+Here is a program to draw a bouquet of \smath{n} arrows.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ drawBouquet(n,title)\ ==}\newline
+{\tt 2.\ \ \ \ \ angle\ :=\ 0.0@DFLOAT}\newline
+{\tt 3.\ \ \ \ \ sp\ :=\ createThreeSpace()}\newline
+{\tt 4.\ \ \ \ \ for\ i\ in\ 0..n-1\ repeat}\newline
+{\tt 5.\ \ \ \ \ \ \ start\ :=\ point\ [0.0@DFLOAT,0.0@DFLOAT,0.0@DFLOAT,angle]\ }\newline
+{\tt 6.\ \ \ \ \ \ \ end\ \ \ :=\ point\ [cos\ angle,\ sin\ angle,\ 1.0@DFLOAT,\ angle]}\newline
+{\tt 7.\ \ \ \ \ \ \ arrow\ :=\ makeArrow(start,end)}\newline
+{\tt 8.\ \ \ \ \ \ \ for\ a\ in\ makeArrow(start,end)\ repeat\ }\newline
+{\tt 9.\ \ \ \ \ \ \ \ \ curve(sp,a)}\newline
+{\tt 10.\ \ \ \ \ \ angle\ :=\ angle\ +\ 2*\%pi/n}\newline
+{\tt 11.\ \ \ \ makeViewport3D(sp,title)}\newline
+\endImportant
+
+\xtc{
+Read the input file.
+}{
+\spadpaste{)read bouquet\bound{b1}}
+}
+\psXtc{
+A bouquet of a dozen arrows.
+}{
+\graphpaste{drawBouquet(12,"A Dozen Arrows")\free{b1}}
+}{
+\epsffile[0 0 295 295]{../ps/bouquet.ps}
+}
+\ 
+
+%\head{section}{Diversion: When Things Go Wrong}{ugIntProgDivTwo}
+%
+%Up to now, if you have typed in all the programs exactly as they are in
+%the book, you have encountered no errors.
+%In practice, however, it is easy to make mistakes.
+%Computers are unforgiving: your program must be letter-for-letter correct
+%or you will encounter some error.
+%
+%One thing that can go wrong is that you can create a syntactically
+%incorrect program.
+%As pointed out in Diversion 1, the meaning of \Language{} programs is
+%affected by indentation.
+%
+%The \Language{} parser will ensure that all parentheses, brackets, and
+%braces balance, and that commas and operators appear in the correct
+%context.
+%For example, change line ??
+%to ??
+%and run.
+%
+%A common mistake is to misspell an identifier or operation name.
+%These are generally easy to spot since the interpreter will tell you the
+%name of the operation together with the type and number of arguments which
+%it is trying to find.
+%
+%Another mistake is to either to omit an argument or to give too many.
+%Again \Language{} will notify you of the offending operation.
+%
+%Indentation makes your programs more readable.
+%However there are several ways to create a syntactically valid program.
+%A most common problem occurs when a line is either indented improperly.
+%% either or what?
+%If this is a first line of a pile, then all the other lines will act as an
+%inner pile to the first line.
+%If it is a line of the pile other than the first line, \Language{} then
+%thinks that this line is a continuation of the previous line.
+%More frequently than not, a syntactically correct expression is created.
+%Almost never however will this be a semantically correct.
+%Only when the program is run will an error be discovered.
+%For example, change line ??
+%to ??
+%and run.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntProgVecFieldsTitle}{Drawing Complex Vector Fields}
+\newcommand{\ugIntProgVecFieldsNumber}{10.6.}
+%
+% =====================================================================
+\begin{page}{ugIntProgVecFieldsPage}{10.6. Drawing Complex Vector Fields}
+% =====================================================================
+\beginscroll
+
+We now put our arrows to good use drawing complex vector fields.
+These vector fields give a representation of complex-valued
+functions of complex variables.
+Consider a Cartesian coordinate grid of points \smath{(x, y)} in
+the plane, and some complex-valued function \smath{f} defined on
+this grid.
+At every point on this grid, compute the value of \texht{$f(x +
+iy)$}{f(x + y*\%i)} and call it \smath{z}.
+Since \smath{z} has both a real and imaginary value for a given
+\smath{(x,y)} grid point, there are four dimensions to plot.
+What do we do?
+We represent the values of \smath{z} by arrows planted at each
+grid point.
+Each arrow represents the value of \smath{z} in polar coordinates
+\texht{$(r,\theta)$}{(r, theta)}.
+The length of the arrow is proportional to \smath{r}.
+Its direction is given by \texht{$\theta$}{theta}.
+
+The code for drawing vector fields is in the file {\bf vectors.input}.
+We discuss its contents from top to bottom.
+
+Before showing you the code, we have two small
+matters to take care of.
+First, what if the function has large spikes, say, ones that go off
+to infinity?
+We define a variable \spad{clipValue} for this purpose. When
+\spad{r} exceeds the value of \spad{clipValue}, then the value of
+\spad{clipValue} is used instead of that for \spad{r}.
+For convenience, we define a function \spad{clipFun(x)} which uses
+\spad{clipValue} to ``clip'' the value of \spad{x}.
+
+%
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ clipValue\ :\ DFLOAT\ :=\ 6}\newline
+{\tt 2.\ \ \ clipFun(x)\ ==\ min(max(x,-clipValue),clipValue)}\newline
+\endImportant
+
+Notice that we identify \spad{clipValue} as a small float but do
+not declare the type of the function \userfun{clipFun}.
+As it turns out, \userfun{clipFun} is called with a
+small float value.
+This declaration ensures that \userfun{clipFun} never does a
+conversion when it is called.
+
+The second matter concerns the possible ``poles'' of a
+function, the actual points where the spikes have infinite
+values.
+\Language{} uses normal \spadtype{DoubleFloat} arithmetic  which
+does not directly handle infinite values.
+If your function has poles, you must adjust your step size to
+avoid landing directly on them (\Language{} calls \spadfun{error}
+when asked to divide a value by \axiom{0}, for example).
+
+We set the variables \spad{realSteps} and \spad{imagSteps} to
+hold the number of steps taken in the real and imaginary
+directions, respectively.
+Most examples will have ranges centered around the origin.
+To avoid a pole at the origin, the number of points is taken
+to be odd.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ realSteps:\ INT\ :=\ 25}\newline
+{\tt 2.\ \ \ imagSteps:\ INT\ :=\ 25}\newline
+{\tt 3.\ \ \ )read\ arrows}\newline
+\endImportant
+
+Now define the function \userfun{drawComplexVectorField} to draw the arrows.
+It is good practice to declare the type of the main function in
+the file.
+This one declaration is usually sufficient to ensure that other
+lower-level functions are compiled with the correct types.
+
+\beginImportant
+  
+\noindent
+{\tt 4.\ \ \ C\ :=\ Complex\ DoubleFloat}\newline
+{\tt 5.\ \ \ S\ :=\ Segment\ DoubleFloat}\newline
+{\tt 6.\ \ \ drawComplexVectorField:\ (C\ ->\ C,\ S,\ S)\ ->\ VIEW3D}\newline
+\endImportant
+
+The first argument is a function mapping complex small floats into
+complex small floats.
+The second and third arguments give the range of real and
+imaginary values as segments like \spad{a..b}.
+The result is a \threedim{} viewport.
+Here is the full function definition:
+
+\beginImportant
+  
+\noindent
+{\tt 7.\ \ \ drawComplexVectorField(f,\ realRange,imagRange)\ ==}\newline
+{\tt 8.\ \ \ \ \ delReal\ :=\ (hi(realRange)-lo(realRange))/realSteps}\newline
+{\tt 9.\ \ \ \ \ delImag\ :=\ (hi(imagRange)-lo(imagRange))/imagSteps}\newline
+{\tt 10.\ \ \ \ sp\ :=\ createThreeSpace()}\newline
+{\tt 11.\ \ \ \ real\ :=\ lo(realRange)}\newline
+{\tt 12.\ \ \ \ for\ i\ in\ 1..realSteps+1\ repeat}\newline
+{\tt 13.\ \ \ \ \ \ imag\ :=\ lo(imagRange)}\newline
+{\tt 14.\ \ \ \ \ \ for\ j\ in\ 1..imagSteps+1\ repeat}\newline
+{\tt 15.\ \ \ \ \ \ \ \ z\ :=\ f\ complex(real,imag)}\newline
+{\tt 16.\ \ \ \ \ \ \ \ arg\ :=\ argument\ z}\newline
+{\tt 17.\ \ \ \ \ \ \ \ len\ :=\ clipFun\ sqrt\ norm\ z}\newline
+{\tt 18.\ \ \ \ \ \ \ \ p1\ :=\ \ point\ [real,\ imag,\ 0.0@DFLOAT,\ arg]}\newline
+{\tt 19.\ \ \ \ \ \ \ \ scaleLen\ :=\ delReal\ *\ len}\newline
+{\tt 20.\ \ \ \ \ \ \ \ p2\ :=\ point\ [p1.1\ +\ scaleLen*cos(arg),}\newline
+{\tt 21.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p1.2\ +\ scaleLen*sin(arg),0.0@DFLOAT,\ arg]}\newline
+{\tt 22.\ \ \ \ \ \ \ \ arrow\ :=\ makeArrow(p1,\ p2)}\newline
+{\tt 23.\ \ \ \ \ \ \ \ for\ a\ in\ arrow\ repeat\ curve(sp,\ a)}\newline
+{\tt 24.\ \ \ \ \ \ \ \ imag\ :=\ imag\ +\ delImag}\newline
+{\tt 25.\ \ \ \ \ \ real\ :=\ real\ +\ delReal}\newline
+{\tt 26.\ \ \ \ makeViewport3D(sp,\ "Complex\ Vector\ Field")}\newline
+\endImportant
+
+As a first example, let us draw \spad{f(z) == sin(z)}.
+There is no need to create a user function: just pass the
+\spadfunFrom{sin}{Complex DoubleFloat} from \spadtype{Complex DoubleFloat}.
+\xtc{
+Read the file.
+}{
+\spadpaste{)read vectors \bound{readVI}}
+}
+\psXtc{
+Draw the complex vector field of \spad{sin(x)}.
+}{
+\graphpaste{drawComplexVectorField(sin,-2..2,-2..2) \free{readVI}}
+}{
+\epsffile[0 0 295 295]{../ps/vectorSin.ps}
+}
+\ 
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntProgCompFunsTitle}{Drawing Complex Functions}
+\newcommand{\ugIntProgCompFunsNumber}{10.7.}
+%
+% =====================================================================
+\begin{page}{ugIntProgCompFunsPage}{10.7. Drawing Complex Functions}
+% =====================================================================
+\beginscroll
+
+Here is another way to graph a complex function of complex
+arguments.
+For each complex value \smath{z}, compute \smath{f(z)}, again
+expressing the value in polar coordinates \smath{(r,\theta{})}.
+We draw the complex valued function, again considering the
+\smath{(x,y)}-plane as the complex plane, using \smath{r} as the
+height (or \smath{z}-coordinate) and \smath{\theta} as the color.
+This is a standard plot---we learned how to do this in
+\downlink{``\ugGraphTitle''}{ugGraphPage} in Chapter \ugGraphNumber\ignore{ugGraph}---but here we write a new program to illustrate
+the creation of polygon meshes, or grids.
+
+Call this function \userfun{drawComplex}.
+It displays the points using the ``mesh'' of points.
+The function definition is in three parts.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ drawComplex:\ (C\ ->\ C,\ S,\ S)\ ->\ VIEW3D}\newline
+{\tt 2.\ \ \ drawComplex(f,\ realRange,\ imagRange)\ ==}\newline
+{\tt 3.\ \ \ \ \ delReal\ :=\ (hi(realRange)-lo(realRange))/realSteps}\newline
+{\tt 4.\ \ \ \ \ delImag\ :=\ (hi(imagRange)-lo(imagRange))/imagSteps}\newline
+{\tt 5.\ \ \ \ \ llp:List\ List\ Point\ DFLOAT\ :=\ []}\newline
+\endImportant
+
+Variables \spad{delReal} and \spad{delImag} give the step
+sizes along the real and imaginary directions as computed by the values
+of the global variables \spad{realSteps} and \spad{imagSteps}.
+The mesh is represented by a list of lists of points \spad{llp},
+initially empty.
+Now \spad{[ ]} alone is ambiguous, so
+to set this initial value
+you have to tell \Language{} what type of empty list it is.
+Next comes the loop which builds \spad{llp}.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ \ \ real\ :=\ lo(realRange)}\newline
+{\tt 2.\ \ \ \ \ for\ i\ in\ 1..realSteps+1\ repeat}\newline
+{\tt 3.\ \ \ \ \ \ \ imag\ :=\ lo(imagRange)}\newline
+{\tt 4.\ \ \ \ \ \ \ lp\ :=\ []\$(List\ Point\ DFLOAT)}\newline
+{\tt 5.\ \ \ \ \ \ \ for\ j\ in\ 1..imagSteps+1\ repeat}\newline
+{\tt 6.\ \ \ \ \ \ \ \ \ z\ :=\ f\ complex(real,imag)}\newline
+{\tt 7.\ \ \ \ \ \ \ \ \ pt\ :=\ point\ [real,imag,\ clipFun\ sqrt\ norm\ z,\ }\newline
+{\tt 8.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ argument\ z]}\newline
+{\tt 9.\ \ \ \ \ \ \ \ \ lp\ :=\ cons(pt,lp)}\newline
+{\tt 10.\ \ \ \ \ \ \ \ imag\ :=\ imag\ +\ delImag}\newline
+{\tt 11.\ \ \ \ \ \ real\ :=\ real\ +\ delReal}\newline
+{\tt 12.\ \ \ \ \ \ llp\ :=\ cons(lp,\ llp)}\newline
+\endImportant
+
+The code consists of both an inner and outer loop.
+Each pass through the inner loop adds one list \spad{lp} of points
+to the list of lists of points \spad{llp}.
+The elements of \spad{lp} are collected in reverse order.
+
+\beginImportant
+  
+\noindent
+{\tt 13.\ \ \ \ makeViewport3D(mesh(llp),\ "Complex\ Function")}\newline
+\endImportant
+
+The operation \spadfun{mesh} then creates an object of type
+\spadtype{ThreeSpace(DoubleFloat)} from the list of lists of points.
+This is then passed to \spadfun{makeViewport3D} to display the
+image.
+
+Now add this function directly to your {\bf vectors.input}
+file and re-read the file using \spad{)read vectors}.
+We try \userfun{drawComplex} using
+a user-defined function \spad{f}.
+
+\xtc{
+Read the file.
+}{
+\spadpaste{)read vectors \bound{readVI}}
+}
+\xtc{
+This one has a pole at \smath{z=0}.
+}{
+\spadpaste{f(z) == exp(1/z)\bound{e1}}
+}
+\psXtc{
+Draw it with an odd number of steps to avoid the pole.
+}{
+\graphpaste{drawComplex(f,-2..2,-2..2)\free{e1 readVI}}
+}{
+\epsffile[0 0 295 295]{../ps/complexExp.ps}
+}
+\ 
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntProgFunctionsTitle}{Functions Producing Functions}
+\newcommand{\ugIntProgFunctionsNumber}{10.8.}
+%
+% =====================================================================
+\begin{page}{ugIntProgFunctionsPage}{10.8. Functions Producing Functions}
+% =====================================================================
+\beginscroll
+
+In \downlink{``\ugUserMakeTitle''}{ugUserMakePage} in Section \ugUserMakeNumber\ignore{ugUserMake}, you learned how to use the operation
+\spadfun{function} to create a function from symbolic formulas.
+Here we introduce a similar operation which not only
+creates functions, but functions from functions.
+
+The facility we need is provided by the package
+\spadtype{MakeUnaryCompiledFunction(E,S,T)}.
+%-% \HDexptypeindex{MakeUnaryCompiledFunction}{ugIntProgFunctionsPage}{10.8.}{Functions Producing Functions}
+This package produces a unary (one-argument) compiled
+function from some symbolic data
+generated by a previous computation.\footnote{%
+\spadtype{MakeBinaryCompiledFunction} is available for binary
+functions.}
+%-% \HDexptypeindex{MakeBinaryCompiledFunction}{ugIntProgFunctionsPage}{10.8.}{Functions Producing Functions}
+The \spad{E} tells where the symbolic data comes from;
+the \spad{S} and \spad{T} give \Language{} the
+source and target type of the function, respectively.
+The compiled function produced  has type
+\spadsig{\spad{S}}{\spad{T}}.
+To produce a compiled function with definition \spad{p(x) == expr}, call
+\spad{compiledFunction(expr, x)} from this package.
+The function you get has no name.
+You must to assign the function to the variable \spad{p} to give it that name.
+%
+\xtc{
+Do some computation.
+}{
+\spadpaste{(x+1/3)**5\bound{p1}}
+}
+\xtc{
+Convert this to an anonymous function of \spad{x}.
+Assign it to the variable \spad{p} to give the function a name.
+}{
+\spadpaste{p := compiledFunction(\%,x)\$MakeUnaryCompiledFunction(POLY FRAC INT,DFLOAT,DFLOAT)\bound{p2}\free{p1}}
+}
+\xtc{
+Apply the function.
+}{
+\spadpaste{p(sin(1.3))\bound{p3}\free{p2}}
+}
+
+For a more sophisticated application, read on.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugIntProgNewtonTitle}{Automatic Newton Iteration Formulas}
+\newcommand{\ugIntProgNewtonNumber}{10.9.}
+%
+% =====================================================================
+\begin{page}{ugIntProgNewtonPage}{10.9. Automatic Newton Iteration Formulas}
+% =====================================================================
+\beginscroll
+
+We resume
+our continuing saga of arrows and complex functions.
+Suppose we want to investigate the behavior of Newton's iteration function
+%-% \HDindex{Newton iteration}{ugIntProgNewtonPage}{10.9.}{Automatic Newton Iteration Formulas}
+in the complex plane.
+Given a function \smath{f}, we want to find the complex values
+\smath{z} such that \smath{f(z) = 0}.
+
+The first step is to produce a Newton iteration formula for
+a given \smath{f}:
+\texht{$x_{n+1} = x_n - {{f(x_n)}\over{f'(x_n)}}.$}{%
+\spad{x(n+1) = x(n) - f(x(n))/f'(x(n))}.}
+We represent this formula by a function \smath{g}
+that performs the computation on the right-hand side, that is,
+\texht{$x_{n+1} = {g}(x_n)$}{\spad{x(n+1) = g(x(n))}}.
+
+The type \spadtype{Expression Integer} (abbreviated \spadtype{EXPR
+INT}) is used to represent general symbolic expressions in
+\Language{}.
+%-% \HDexptypeindex{Expression}{ugIntProgNewtonPage}{10.9.}{Automatic Newton Iteration Formulas}
+To make our facility as general as possible, we assume
+\smath{f} has this type.
+Given \smath{f}, we want
+to produce a Newton iteration function \spad{g} which,
+given a complex point \texht{$x_n$}{x(n)}, delivers the next
+Newton iteration point \texht{$x_{n+1}$}{x(n+1)}.
+
+This time we write an input file called {\bf newton.input}.
+We need to import \spadtype{MakeUnaryCompiledFunction} (discussed
+in the last section), call it with appropriate types, and then define
+the function \spad{newtonStep} which references it.
+Here is the function \spad{newtonStep}:
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ C\ :=\ Complex\ DoubleFloat}\newline
+{\tt 2.\ \ \ complexFunPack:=MakeUnaryCompiledFunction(EXPR\ INT,C,C)}\newline
+{\tt 3.\ \ \ }\newline
+{\tt 4.\ \ \ newtonStep(f)\ ==}\newline
+{\tt 5.\ \ \ \ \ fun\ \ :=\ complexNumericFunction\ f}\newline
+{\tt 6.\ \ \ \ \ deriv\ :=\ complexDerivativeFunction(f,1)}\newline
+{\tt 7.\ \ \ \ \ (x:C):C\ +->}\newline
+{\tt 8.\ \ \ \ \ \ \ x\ -\ fun(x)/deriv(x)}\newline
+{\tt 9.\ \ \ }\newline
+{\tt 10.\ \ complexNumericFunction\ f\ ==}\newline
+{\tt 11.\ \ \ \ v\ :=\ theVariableIn\ f}\newline
+{\tt 12.\ \ \ \ compiledFunction(f,\ v)\$complexFunPack}\newline
+{\tt 13.\ \ }\newline
+{\tt 14.\ \ complexDerivativeFunction(f,n)\ ==}\newline
+{\tt 15.\ \ \ \ v\ :=\ theVariableIn\ f}\newline
+{\tt 16.\ \ \ \ df\ :=\ D(f,v,n)}\newline
+{\tt 17.\ \ \ \ compiledFunction(df,\ v)\$complexFunPack}\newline
+{\tt 18.\ \ }\newline
+{\tt 19.\ \ theVariableIn\ f\ ==\ \ }\newline
+{\tt 20.\ \ \ \ vl\ :=\ variables\ f}\newline
+{\tt 21.\ \ \ \ nv\ :=\ \#\ vl}\newline
+{\tt 22.\ \ \ \ nv\ >\ 1\ =>\ error\ "Expression\ is\ not\ univariate."}\newline
+{\tt 23.\ \ \ \ nv\ =\ 0\ =>\ 'x}\newline
+{\tt 24.\ \ \ \ first\ vl}\newline
+\endImportant
+
+Do you see what is going on here?
+A formula \spad{f} is passed into the function \userfun{newtonStep}.
+First, the function turns \spad{f} into a compiled program mapping
+complex numbers into complex numbers.  Next, it does the same thing
+for the derivative of \spad{f}.  Finally, it returns a function which
+computes a single step of Newton's iteration.
+
+The function \userfun{complexNumericFunction} extracts the variable
+from the expression \spad{f} and then turns \spad{f} into a function
+which maps complex numbers into complex numbers. The function
+\userfun{complexDerivativeFunction} does the same thing for the
+derivative of \spad{f}.  The function \userfun{theVariableIn}
+extracts the variable from the expression \spad{f}, calling the function
+\spadfun{error} if \spad{f} has more than one variable.
+It returns the dummy variable \spad{x} if \spad{f} has no variables.
+
+Let's now apply \userfun{newtonStep} to the formula for computing
+cube roots of two.
+%
+\xtc{
+Read the input file with the definitions.
+}{
+\spadpaste{)read newton\bound{n1}}
+}
+\xtc{}{
+\spadpaste{)read vectors \bound{n1a}}
+}
+
+\xtc{
+The cube root of two.
+}{
+\spadpaste{f := x**3 - 2\bound{n2}\free{n1 n1a}}
+}
+\xtc{
+Get Newton's iteration formula.
+}{
+\spadpaste{g := newtonStep f\bound{n3}\free{n2}}
+}
+\xtc{
+Let \spad{a} denote the result of
+applying Newton's iteration once to the complex number \spad{1 + \%i}.
+}{
+\spadpaste{a := g(1.0 + \%i)\bound{n4}\free{n3}}
+}
+\xtc{
+Now apply it repeatedly. How fast does it converge?
+}{
+\spadpaste{[(a := g(a)) for i in 1..]\bound{n5}\free{n4}}
+}
+\xtc{
+Check the accuracy of the last iterate.
+}{
+\spadpaste{a**3\bound{n6}\free{n5}}
+}
+
+In \downlink{`MappingPackage1'}{MappingPackageOneXmpPage}\ignore{MappingPackage1}, we show how functions can be
+manipulated as objects in \Language{}.
+A useful operation to consider here is \spadop{*}, which means
+composition.
+For example \spad{g*g} causes the Newton iteration formula
+to be applied twice.
+Correspondingly, \spad{g**n} means to apply the iteration formula
+\spad{n} times.
+
+%
+\xtc{
+Apply \spad{g} twice to the point \spad{1 + \%i}.
+}{
+\spadpaste{(g*g) (1.0 + \%i)\bound{n10}\free{n3}}
+}
+\xtc{
+Apply \spad{g} 11 times.
+}{
+\spadpaste{(g**11) (1.0 + \%i)\bound{n11}\free{n10}}
+}
+
+Look now at the vector field and surface generated
+after two steps of Newton's formula for the cube root of two.
+The poles in these pictures represent bad starting values, and the
+flat areas are the regions of convergence to the three roots.
+%
+\psXtc{
+The vector field.
+}{
+\graphpaste{drawComplexVectorField(g**3,-3..3,-3..3)\free{n3}}
+}{
+\epsffile[0 0 295 295]{../ps/vectorRoot.ps}
+}
+\psXtc{
+The surface.
+}{
+\graphpaste{drawComplex(g**3,-3..3,-3..3)\free{n3}}
+}{
+\epsffile[0 0 295 295]{../ps/complexRoot.ps}
+}
+\ 
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug10.pht b/src/hyper/pages/ug10.pht
new file mode 100644
index 00000000..c9dc271f
--- /dev/null
+++ b/src/hyper/pages/ug10.pht
@@ -0,0 +1,729 @@
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+\begin{paste}{ugIntProgDrawingPageFull1}{ugIntProgDrawingPageEmpty1}
+\pastebutton{ugIntProgDrawingPageFull1}{\hidepaste}
+\tab{5}\spadgraph{draw(x**2,x=-1..1,y=0..1)}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugIntProgDrawingPage1.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugIntProgDrawingPage1}}
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+\begin{paste}{ugIntProgDrawingPageEmpty2}{ugIntProgDrawingPagePatch2}
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+
+\begin{patch}{ugIntProgDrawingPagePatch3}
+\begin{paste}{ugIntProgDrawingPageFull3}{ugIntProgDrawingPageEmpty3}
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+\tab{5}\spadcommand{rotate(vp, 0, -90)\bound{d3 }\free{d1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgDrawingPageEmpty3}
+\begin{paste}{ugIntProgDrawingPageEmpty3}{ugIntProgDrawingPagePatch3}
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+
+\begin{patch}{ugIntProgDrawingPagePatch4}
+\begin{paste}{ugIntProgDrawingPageFull4}{ugIntProgDrawingPageEmpty4}
+\pastebutton{ugIntProgDrawingPageFull4}{\hidepaste}
+\tab{5}\spadcommand{sp := subspace(vp)\bound{d5 }\free{d1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  3-Space with 1 component
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
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+
+\begin{patch}{ugIntProgDrawingPageEmpty5}
+\begin{paste}{ugIntProgDrawingPageEmpty5}{ugIntProgDrawingPagePatch5}
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+
+\begin{patch}{ugIntProgDrawingPagePatch6}
+\begin{paste}{ugIntProgDrawingPageFull6}{ugIntProgDrawingPageEmpty6}
+\pastebutton{ugIntProgDrawingPageFull6}{\hidepaste}
+\tab{5}\spadcommand{drawStyle(vp,"shade");outlineRender(vp,"on")\bound{d10 }\free{d6 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgDrawingPageEmpty6}
+\begin{paste}{ugIntProgDrawingPageEmpty6}{ugIntProgDrawingPagePatch6}
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+
+\begin{patch}{ugIntProgDrawingPagePatch7}
+\begin{paste}{ugIntProgDrawingPageFull7}{ugIntProgDrawingPageEmpty7}
+\pastebutton{ugIntProgDrawingPageFull7}{\hidepaste}
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+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgDrawingPageEmpty7}
+\begin{paste}{ugIntProgDrawingPageEmpty7}{ugIntProgDrawingPagePatch7}
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+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgColorArrPagePatch1}
+\begin{paste}{ugIntProgColorArrPageFull1}{ugIntProgColorArrPageEmpty1}
+\pastebutton{ugIntProgColorArrPageFull1}{\hidepaste}
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+\indentrel{3}\begin{verbatim}
+   (1)  0.20000000000000001
+                                      Type: DoubleFloat
+   (2)  2.8274333882308138
+                                      Type: DoubleFloat
+                                             Type: Void
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgColorArrPageEmpty1}
+\begin{paste}{ugIntProgColorArrPageEmpty1}{ugIntProgColorArrPagePatch1}
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+
+\begin{patch}{ugIntProgColorArrPagePatch2}
+\begin{paste}{ugIntProgColorArrPageFull2}{ugIntProgColorArrPageEmpty2}
+\pastebutton{ugIntProgColorArrPageFull2}{\hidepaste}
+\tab{5}\spadgraph{drawBouquet(12,"A Dozen Arrows")\free{b1 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugIntProgColorArrPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugIntProgColorArrPage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgColorArrPageEmpty2}
+\begin{paste}{ugIntProgColorArrPageEmpty2}{ugIntProgColorArrPagePatch2}
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+\tab{5}\spadgraph{drawBouquet(12,"A Dozen Arrows")\free{b1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgCompFunsPagePatch1}
+\begin{paste}{ugIntProgCompFunsPageFull1}{ugIntProgCompFunsPageEmpty1}
+\pastebutton{ugIntProgCompFunsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)read vectors\bound{readVI }}
+\indentrel{3}\begin{verbatim}
+   (1)  2.8274333882308138
+                                      Type: DoubleFloat
+   (2)  0.20000000000000001
+                                      Type: DoubleFloat
+                                             Type: Void
+   (4)  6.0
+                                      Type: DoubleFloat
+                                             Type: Void
+   (6)  25
+                                          Type: Integer
+   (7)  25
+                                          Type: Integer
+   (8)  Complex DoubleFloat
+                                           Type: Domain
+   (9)  Segment DoubleFloat
+                                           Type: Domain
+                                             Type: Void
+                                             Type: Void
+                                             Type: Void
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
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+
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+\tab{5}\spadcommand{f(z) == exp(1/z)\bound{e1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
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+
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+
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+
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+
+\begin{patch}{ugIntProgVecFieldsPagePatch1}
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+\indentrel{3}\begin{verbatim}
+   (1)  2.8274333882308138
+                                      Type: DoubleFloat
+   (2)  0.20000000000000001
+                                      Type: DoubleFloat
+                                             Type: Void
+   (4)  6.0
+                                      Type: DoubleFloat
+                                             Type: Void
+   (6)  25
+                                          Type: Integer
+   (7)  25
+                                          Type: Integer
+   (8)  Complex DoubleFloat
+                                           Type: Domain
+   (9)  Segment DoubleFloat
+                                           Type: Domain
+                                             Type: Void
+                                             Type: Void
+                                             Type: Void
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
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+
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+
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+\begin{paste}{ugIntProgFunctionsPageFull1}{ugIntProgFunctionsPageEmpty1}
+\pastebutton{ugIntProgFunctionsPageFull1}{\hidepaste}
+\tab{5}\spadcommand{(x+1/3)**5\bound{p1 }}
+\indentrel{3}\begin{verbatim}
+         5   5  4   10  3   10  2    5      1
+   (1)  x  + Ä x  + ÄÄ x  + ÄÄ x  + ÄÄ x + ÄÄÄ
+             3       9      27      81     243
+                      Type: Polynomial Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgFunctionsPageEmpty1}
+\begin{paste}{ugIntProgFunctionsPageEmpty1}{ugIntProgFunctionsPagePatch1}
+\pastebutton{ugIntProgFunctionsPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{(x+1/3)**5\bound{p1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgFunctionsPagePatch2}
+\begin{paste}{ugIntProgFunctionsPageFull2}{ugIntProgFunctionsPageEmpty2}
+\pastebutton{ugIntProgFunctionsPageFull2}{\hidepaste}
+\tab{5}\spadcommand{p := compiledFunction(\%,x)$MakeUnaryCompiledFunction(POLY FRAC INT,DFLOAT,DFLOAT)\bound{p2 }\free{p1 }}
+\indentrel{3}\begin{verbatim}
+   (2)  theMap(MKUCFUNC;unaryFunction;SM;2!0,350)
+                     Type: (DoubleFloat -> DoubleFloat)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgFunctionsPageEmpty2}
+\begin{paste}{ugIntProgFunctionsPageEmpty2}{ugIntProgFunctionsPagePatch2}
+\pastebutton{ugIntProgFunctionsPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{p := compiledFunction(\%,x)$MakeUnaryCompiledFunction(POLY FRAC INT,DFLOAT,DFLOAT)\bound{p2 }\free{p1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgFunctionsPagePatch3}
+\begin{paste}{ugIntProgFunctionsPageFull3}{ugIntProgFunctionsPageEmpty3}
+\pastebutton{ugIntProgFunctionsPageFull3}{\hidepaste}
+\tab{5}\spadcommand{p(sin(1.3))\bound{p3 }\free{p2 }}
+\indentrel{3}\begin{verbatim}
+   (3)  3.668751115057229
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgFunctionsPageEmpty3}
+\begin{paste}{ugIntProgFunctionsPageEmpty3}{ugIntProgFunctionsPagePatch3}
+\pastebutton{ugIntProgFunctionsPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{p(sin(1.3))\bound{p3 }\free{p2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPagePatch1}
+\begin{paste}{ugIntProgPLCPageFull1}{ugIntProgPLCPageEmpty1}
+\pastebutton{ugIntProgPLCPageFull1}{\hidepaste}
+\tab{5}\spadcommand{zero := 0.0@DFLOAT\bound{d1 }}
+\indentrel{3}\begin{verbatim}
+   (1)  0.0
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPageEmpty1}
+\begin{paste}{ugIntProgPLCPageEmpty1}{ugIntProgPLCPagePatch1}
+\pastebutton{ugIntProgPLCPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{zero := 0.0@DFLOAT\bound{d1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPagePatch2}
+\begin{paste}{ugIntProgPLCPageFull2}{ugIntProgPLCPageEmpty2}
+\pastebutton{ugIntProgPLCPageFull2}{\hidepaste}
+\tab{5}\spadcommand{one := 1.0@DFLOAT\bound{d2 }}
+\indentrel{3}\begin{verbatim}
+   (2)  1.0
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPageEmpty2}
+\begin{paste}{ugIntProgPLCPageEmpty2}{ugIntProgPLCPagePatch2}
+\pastebutton{ugIntProgPLCPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{one := 1.0@DFLOAT\bound{d2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPagePatch3}
+\begin{paste}{ugIntProgPLCPageFull3}{ugIntProgPLCPageEmpty3}
+\pastebutton{ugIntProgPLCPageFull3}{\hidepaste}
+\tab{5}\spadcommand{origin := point [zero,zero,zero,zero]\free{d1 }\bound{d3 }}
+\indentrel{3}\begin{verbatim}
+   (3)  [0.0,0.0,0.0,0.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPageEmpty3}
+\begin{paste}{ugIntProgPLCPageEmpty3}{ugIntProgPLCPagePatch3}
+\pastebutton{ugIntProgPLCPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{origin := point [zero,zero,zero,zero]\free{d1 }\bound{d3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPagePatch4}
+\begin{paste}{ugIntProgPLCPageFull4}{ugIntProgPLCPageEmpty4}
+\pastebutton{ugIntProgPLCPageFull4}{\hidepaste}
+\tab{5}\spadcommand{unit := point [one,one,one,zero]\free{d1 d2 }\bound{d4 }}
+\indentrel{3}\begin{verbatim}
+   (4)  [1.0,1.0,1.0,0.0]
+                                Type: Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPageEmpty4}
+\begin{paste}{ugIntProgPLCPageEmpty4}{ugIntProgPLCPagePatch4}
+\pastebutton{ugIntProgPLCPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{unit := point [one,one,one,zero]\free{d1 d2 }\bound{d4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPagePatch5}
+\begin{paste}{ugIntProgPLCPageFull5}{ugIntProgPLCPageEmpty5}
+\pastebutton{ugIntProgPLCPageFull5}{\hidepaste}
+\tab{5}\spadcommand{line := [origin, unit]\free{d3 d4 }\bound{d5 }}
+\indentrel{3}\begin{verbatim}
+   (5)  [[0.0,0.0,0.0,0.0],[1.0,1.0,1.0,0.0]]
+                           Type: List Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPageEmpty5}
+\begin{paste}{ugIntProgPLCPageEmpty5}{ugIntProgPLCPagePatch5}
+\pastebutton{ugIntProgPLCPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{line := [origin, unit]\free{d3 d4 }\bound{d5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPagePatch6}
+\begin{paste}{ugIntProgPLCPageFull6}{ugIntProgPLCPageEmpty6}
+\pastebutton{ugIntProgPLCPageFull6}{\hidepaste}
+\tab{5}\spadcommand{)read arrows\bound{v1 }}
+\indentrel{3}\begin{verbatim}
+   (6)  2.8274333882308138
+                                      Type: DoubleFloat
+   (7)  0.20000000000000001
+                                      Type: DoubleFloat
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPageEmpty6}
+\begin{paste}{ugIntProgPLCPageEmpty6}{ugIntProgPLCPagePatch6}
+\pastebutton{ugIntProgPLCPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{)read arrows\bound{v1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPagePatch7}
+\begin{paste}{ugIntProgPLCPageFull7}{ugIntProgPLCPageEmpty7}
+\pastebutton{ugIntProgPLCPageFull7}{\hidepaste}
+\tab{5}\spadcommand{arrow := makeArrow(origin,unit)\bound{v2 }\free{v1 d3 d4 }}
+\indentrel{3}\begin{verbatim}
+   (9)
+   [
+     [[0.0,0.0,0.0,0.0], [1.0,1.0,1.0,0.0],
+
+       [0.69134628604607973, 0.842733077659504,
+        0.80000000000000004, 0.0]
+       ]
+     ,
+
+     [[1.0,1.0,1.0,0.0],
+
+       [0.842733077659504, 0.69134628604607973,
+        0.80000000000000004, 0.0]
+       ]
+     ]
+                      Type: List List Point DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPageEmpty7}
+\begin{paste}{ugIntProgPLCPageEmpty7}{ugIntProgPLCPagePatch7}
+\pastebutton{ugIntProgPLCPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{arrow := makeArrow(origin,unit)\bound{v2 }\free{v1 d3 d4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPagePatch8}
+\begin{paste}{ugIntProgPLCPageFull8}{ugIntProgPLCPageEmpty8}
+\pastebutton{ugIntProgPLCPageFull8}{\hidepaste}
+\tab{5}\spadcommand{sp := createThreeSpace()\bound{c1 }}
+\indentrel{3}\begin{verbatim}
+   (10)  3-Space with 0 components
+                           Type: ThreeSpace DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPageEmpty8}
+\begin{paste}{ugIntProgPLCPageEmpty8}{ugIntProgPLCPagePatch8}
+\pastebutton{ugIntProgPLCPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{sp := createThreeSpace()\bound{c1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPagePatch9}
+\begin{paste}{ugIntProgPLCPageFull9}{ugIntProgPLCPageEmpty9}
+\pastebutton{ugIntProgPLCPageFull9}{\hidepaste}
+\tab{5}\spadcommand{for a in arrow repeat sp := curve(sp,a)\bound{v3 }\free{v2 }\free{c1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPageEmpty9}
+\begin{paste}{ugIntProgPLCPageEmpty9}{ugIntProgPLCPagePatch9}
+\pastebutton{ugIntProgPLCPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{for a in arrow repeat sp := curve(sp,a)\bound{v3 }\free{v2 }\free{c1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPagePatch10}
+\begin{paste}{ugIntProgPLCPageFull10}{ugIntProgPLCPageEmpty10}
+\pastebutton{ugIntProgPLCPageFull10}{\hidepaste}
+\tab{5}\spadgraph{vp := makeViewport3D(sp,"Arrow")\bound{v4 }\free{v3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugIntProgPLCPage10.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugIntProgPLCPage10}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPageEmpty10}
+\begin{paste}{ugIntProgPLCPageEmpty10}{ugIntProgPLCPagePatch10}
+\pastebutton{ugIntProgPLCPageEmpty10}{\showpaste}
+\tab{5}\spadgraph{vp := makeViewport3D(sp,"Arrow")\bound{v4 }\free{v3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPagePatch11}
+\begin{paste}{ugIntProgPLCPageFull11}{ugIntProgPLCPageEmpty11}
+\pastebutton{ugIntProgPLCPageFull11}{\hidepaste}
+\tab{5}\spadcommand{rotate(vp,200,-60)\bound{v5 }\free{v4 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgPLCPageEmpty11}
+\begin{paste}{ugIntProgPLCPageEmpty11}{ugIntProgPLCPagePatch11}
+\pastebutton{ugIntProgPLCPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{rotate(vp,200,-60)\bound{v5 }\free{v4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgRibbonPagePatch1}
+\begin{paste}{ugIntProgRibbonPageFull1}{ugIntProgRibbonPageEmpty1}
+\pastebutton{ugIntProgRibbonPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)read ribbon\bound{s0 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgRibbonPageEmpty1}
+\begin{paste}{ugIntProgRibbonPageEmpty1}{ugIntProgRibbonPagePatch1}
+\pastebutton{ugIntProgRibbonPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)read ribbon\bound{s0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgRibbonPagePatch2}
+\begin{paste}{ugIntProgRibbonPageFull2}{ugIntProgRibbonPageEmpty2}
+\pastebutton{ugIntProgRibbonPageFull2}{\hidepaste}
+\tab{5}\spadgraph{drawRibbons([x**i for i in 1..5],x=-1..1)\free{s0 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugIntProgRibbonPage2.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugIntProgRibbonPage2}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgRibbonPageEmpty2}
+\begin{paste}{ugIntProgRibbonPageEmpty2}{ugIntProgRibbonPagePatch2}
+\pastebutton{ugIntProgRibbonPageEmpty2}{\showpaste}
+\tab{5}\spadgraph{drawRibbons([x**i for i in 1..5],x=-1..1)\free{s0 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPagePatch1}
+\begin{paste}{ugIntProgNewtonPageFull1}{ugIntProgNewtonPageEmpty1}
+\pastebutton{ugIntProgNewtonPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)read newton\bound{n1 }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+   (2)
+  MakeUnaryCompiledFunction(Expression Integer,Complex Do
+  ubleFloat,Complex DoubleFloat)
+                                           Type: Domain
+                                             Type: Void
+                                             Type: Void
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPageEmpty1}
+\begin{paste}{ugIntProgNewtonPageEmpty1}{ugIntProgNewtonPagePatch1}
+\pastebutton{ugIntProgNewtonPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)read newton\bound{n1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPagePatch2}
+\begin{paste}{ugIntProgNewtonPageFull2}{ugIntProgNewtonPageEmpty2}
+\pastebutton{ugIntProgNewtonPageFull2}{\hidepaste}
+\tab{5}\spadcommand{)read vectors\bound{n1a }}
+\indentrel{3}\begin{verbatim}
+   (6)  2.8274333882308138
+                                      Type: DoubleFloat
+   (7)  0.20000000000000001
+                                      Type: DoubleFloat
+                                             Type: Void
+   (9)  6.0
+                                      Type: DoubleFloat
+                                             Type: Void
+   (11)  25
+                                          Type: Integer
+   (12)  25
+                                          Type: Integer
+   (13)  Complex DoubleFloat
+                                           Type: Domain
+   (14)  Segment DoubleFloat
+                                           Type: Domain
+                                             Type: Void
+                                             Type: Void
+                                             Type: Void
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPageEmpty2}
+\begin{paste}{ugIntProgNewtonPageEmpty2}{ugIntProgNewtonPagePatch2}
+\pastebutton{ugIntProgNewtonPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{)read vectors\bound{n1a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPagePatch3}
+\begin{paste}{ugIntProgNewtonPageFull3}{ugIntProgNewtonPageEmpty3}
+\pastebutton{ugIntProgNewtonPageFull3}{\hidepaste}
+\tab{5}\spadcommand{f := x**3 - 2\bound{n2 }\free{n1 n1a }}
+\indentrel{3}\begin{verbatim}
+          3
+   (19)  x  - 2
+                               Type: Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPageEmpty3}
+\begin{paste}{ugIntProgNewtonPageEmpty3}{ugIntProgNewtonPagePatch3}
+\pastebutton{ugIntProgNewtonPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f := x**3 - 2\bound{n2 }\free{n1 n1a }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPagePatch4}
+\begin{paste}{ugIntProgNewtonPageFull4}{ugIntProgNewtonPageEmpty4}
+\pastebutton{ugIntProgNewtonPageFull4}{\hidepaste}
+\tab{5}\spadcommand{g := newtonStep f\bound{n3 }\free{n2 }}
+\indentrel{3}\begin{verbatim}
+   (20)  theMap(LAMBDA_f3lkwh_704,484)
+     Type: (Complex DoubleFloat -> Complex DoubleFloat)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPageEmpty4}
+\begin{paste}{ugIntProgNewtonPageEmpty4}{ugIntProgNewtonPagePatch4}
+\pastebutton{ugIntProgNewtonPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{g := newtonStep f\bound{n3 }\free{n2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPagePatch5}
+\begin{paste}{ugIntProgNewtonPageFull5}{ugIntProgNewtonPageEmpty5}
+\pastebutton{ugIntProgNewtonPageFull5}{\hidepaste}
+\tab{5}\spadcommand{a := g(1.0 + \%i)\bound{n4 }\free{n3 }}
+\indentrel{3}\begin{verbatim}
+   (21)  0.66666666666666674 + 0.33333333333333337%i
+                              Type: Complex DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPageEmpty5}
+\begin{paste}{ugIntProgNewtonPageEmpty5}{ugIntProgNewtonPagePatch5}
+\pastebutton{ugIntProgNewtonPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{a := g(1.0 + \%i)\bound{n4 }\free{n3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPagePatch6}
+\begin{paste}{ugIntProgNewtonPageFull6}{ugIntProgNewtonPageEmpty6}
+\pastebutton{ugIntProgNewtonPageFull6}{\hidepaste}
+\tab{5}\spadcommand{[(a := g(a)) for i in 1..]\bound{n5 }\free{n4 }}
+\indentrel{3}\begin{verbatim}
+   (22)
+   [1.1644444444444444 - 0.73777777777777775%i,
+    0.92614004697164776 - 0.17463006425584393%i,
+    1.3164444838140228 + 0.15690694583015852%i,
+    1.2462991025761463 + 0.015454763610132094%i,
+    1.2598725296532081 - 0.00033827162059311272%i,
+    1.259920960928212 + 2.6023534653422681e-08%i,
+    1.259921049894879 - 3.6751942591616685e-15%i,
+    1.2599210498948732 - 3.3132158019282496e-29%i,
+    1.2599210498948732 - 5.6051938572992683e-45%i,
+    1.2599210498948732, ...]
+                       Type: Stream Complex DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPageEmpty6}
+\begin{paste}{ugIntProgNewtonPageEmpty6}{ugIntProgNewtonPagePatch6}
+\pastebutton{ugIntProgNewtonPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{[(a := g(a)) for i in 1..]\bound{n5 }\free{n4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPagePatch7}
+\begin{paste}{ugIntProgNewtonPageFull7}{ugIntProgNewtonPageEmpty7}
+\pastebutton{ugIntProgNewtonPageFull7}{\hidepaste}
+\tab{5}\spadcommand{a**3\bound{n6 }\free{n5 }}
+\indentrel{3}\begin{verbatim}
+   (23)  2.0
+                              Type: Complex DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPageEmpty7}
+\begin{paste}{ugIntProgNewtonPageEmpty7}{ugIntProgNewtonPagePatch7}
+\pastebutton{ugIntProgNewtonPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{a**3\bound{n6 }\free{n5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPagePatch8}
+\begin{paste}{ugIntProgNewtonPageFull8}{ugIntProgNewtonPageEmpty8}
+\pastebutton{ugIntProgNewtonPageFull8}{\hidepaste}
+\tab{5}\spadcommand{(g*g) (1.0 + \%i)\bound{n10 }\free{n3 }}
+\indentrel{3}\begin{verbatim}
+   (24)  1.1644444444444444 - 0.73777777777777775%i
+                              Type: Complex DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPageEmpty8}
+\begin{paste}{ugIntProgNewtonPageEmpty8}{ugIntProgNewtonPagePatch8}
+\pastebutton{ugIntProgNewtonPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{(g*g) (1.0 + \%i)\bound{n10 }\free{n3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPagePatch9}
+\begin{paste}{ugIntProgNewtonPageFull9}{ugIntProgNewtonPageEmpty9}
+\pastebutton{ugIntProgNewtonPageFull9}{\hidepaste}
+\tab{5}\spadcommand{(g**11) (1.0 + \%i)\bound{n11 }\free{n10 }}
+\indentrel{3}\begin{verbatim}
+   (25)  1.2599210498948732
+                              Type: Complex DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPageEmpty9}
+\begin{paste}{ugIntProgNewtonPageEmpty9}{ugIntProgNewtonPagePatch9}
+\pastebutton{ugIntProgNewtonPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{(g**11) (1.0 + \%i)\bound{n11 }\free{n10 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPagePatch10}
+\begin{paste}{ugIntProgNewtonPageFull10}{ugIntProgNewtonPageEmpty10}
+\pastebutton{ugIntProgNewtonPageFull10}{\hidepaste}
+\tab{5}\spadgraph{drawComplexVectorField(g**3,-3..3,-3..3)\free{n3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugIntProgNewtonPage10.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugIntProgNewtonPage10}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPageEmpty10}
+\begin{paste}{ugIntProgNewtonPageEmpty10}{ugIntProgNewtonPagePatch10}
+\pastebutton{ugIntProgNewtonPageEmpty10}{\showpaste}
+\tab{5}\spadgraph{drawComplexVectorField(g**3,-3..3,-3..3)\free{n3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPagePatch11}
+\begin{paste}{ugIntProgNewtonPageFull11}{ugIntProgNewtonPageEmpty11}
+\pastebutton{ugIntProgNewtonPageFull11}{\hidepaste}
+\tab{5}\spadgraph{drawComplex(g**3,-3..3,-3..3)\free{n3 }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugIntProgNewtonPage11.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugIntProgNewtonPage11}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPageEmpty11}
+\begin{paste}{ugIntProgNewtonPageEmpty11}{ugIntProgNewtonPagePatch11}
+\pastebutton{ugIntProgNewtonPageEmpty11}{\showpaste}
+\tab{5}\spadgraph{drawComplex(g**3,-3..3,-3..3)\free{n3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPagePatch12}
+\begin{paste}{ugIntProgNewtonPageFull12}{ugIntProgNewtonPageEmpty12}
+\pastebutton{ugIntProgNewtonPageFull12}{\hidepaste}
+\tab{5}\spadcommand{all}
+\indentrel{3}\begin{verbatim}
+   (28)  all
+                                     Type: Variable all
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugIntProgNewtonPageEmpty12}
+\begin{paste}{ugIntProgNewtonPageEmpty12}{ugIntProgNewtonPagePatch12}
+\pastebutton{ugIntProgNewtonPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{all}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ug11.ht b/src/hyper/pages/ug11.ht
new file mode 100644
index 00000000..b2f874a3
--- /dev/null
+++ b/src/hyper/pages/ug11.ht
@@ -0,0 +1,901 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\texht{\setcounter{chapter}{10}}{} % Chapter 11
+
+
+%
+%
+\newcommand{\ugPackagesTitle}{Packages}
+\newcommand{\ugPackagesNumber}{11.}
+%
+% =====================================================================
+\begin{page}{ugPackagesPage}{11. Packages}
+% =====================================================================
+\beginscroll
+
+Packages provide the bulk of
+%-% \HDindex{package}{ugPackagesPage}{11.}{Packages}
+\Language{}'s algorithmic library, from numeric packages for computing
+special functions to symbolic facilities for
+%-% \HDindex{constructor!package}{ugPackagesPage}{11.}{Packages}
+differential equations, symbolic integration, and limits.
+%-% \HDindex{package!constructor}{ugPackagesPage}{11.}{Packages}
+
+In \downlink{``\ugIntProgTitle''}{ugIntProgPage} in Chapter \ugIntProgNumber\ignore{ugIntProg}, we developed several useful functions for drawing
+vector fields and complex functions.
+We now show you how you can add these functions to the
+\Language{} library to make them available for general use.
+
+The way we created the functions in \downlink{``\ugIntProgTitle''}{ugIntProgPage} in Chapter \ugIntProgNumber\ignore{ugIntProg} is typical of how
+you, as an advanced \Language{} user, may interact with \Language{}.
+You have an application.
+You go to your editor and create an input file defining some
+functions for the application.
+Then you run the file and try the functions.
+Once you get them all to work, you will often want to extend them,
+add new features, perhaps write additional functions.
+
+Eventually, when you have a useful set of functions for your application,
+you may want to add them to your local \Language{} library.
+To do this, you embed these function definitions in a package and add
+that package to the library.
+
+To introduce new packages, categories, and domains into the system,
+you need to use the \Language{} compiler to convert the constructors
+into executable machine code.
+An existing compiler in \Language{} is available on an ``as-is''
+basis.
+A new, faster compiler will be available in version 2.0
+of \Language{}.
+
+\beginImportant
+  
+\noindent
+\label{pak-cdraw}
+{\tt 1.\ \ \ C\ \ \ \ \ \ ==>\ Complex\ DoubleFloat}\newline
+{\tt 2.\ \ \ S\ \ \ \ \ \ ==>\ Segment\ DoubleFloat}\newline
+{\tt 3.\ \ \ INT\ \ \ \ ==>\ Integer}\newline
+{\tt 4.\ \ \ DFLOAT\ ==>\ DoubleFloat}\newline
+{\tt 5.\ \ \ VIEW3D\ ==>\ ThreeDimensionalViewport}\newline
+{\tt 6.\ \ \ CURVE\ \ ==>\ List\ List\ Point\ DFLOAT}\newline
+{\tt 7.\ \ \ }\newline
+{\tt 8.\ \ \ )abbrev\ package\ DRAWCX\ DrawComplex}\newline
+{\tt 9.\ \ \ DrawComplex():\ Exports\ ==\ Implementation\ where}\newline
+{\tt 10.\ \ }\newline
+{\tt 11.\ \ \ \ Exports\ ==\ with}\newline
+{\tt 12.\ \ \ \ \ \ drawComplex:\ (C\ ->\ C,S,S,Boolean)\ ->\ VIEW3D}\newline
+{\tt 13.\ \ \ \ \ \ drawComplexVectorField:\ (C\ ->\ C,S,S)\ ->\ VIEW3D}\newline
+{\tt 14.\ \ \ \ \ \ setRealSteps:\ INT\ ->\ INT}\newline
+{\tt 15.\ \ \ \ \ \ setImagSteps:\ INT\ ->\ INT}\newline
+{\tt 16.\ \ \ \ \ \ setClipValue:\ DFLOAT->\ DFLOAT}\newline
+{\tt 17.\ \ }\newline
+{\tt 18.\ \ \ \ Implementation\ ==\ add}\newline
+{\tt 19.\ \ \ \ \ \ arrowScale\ :\ DFLOAT\ :=\ (0.2)::DFLOAT\ --relative\ size}\newline
+{\tt 20.\ \ \ \ \ \ arrowAngle\ :\ DFLOAT\ :=\ pi()-pi()/(20::DFLOAT)}\newline
+{\tt 21.\ \ \ \ \ \ realSteps\ \ :\ INT\ :=\ 11\ --\#\ real\ steps}\newline
+{\tt 22.\ \ \ \ \ \ imagSteps\ \ :\ INT\ :=\ 11\ --\#\ imaginary\ steps}\newline
+{\tt 23.\ \ \ \ \ \ clipValue\ \ :\ DFLOAT\ \ :=\ 10::DFLOAT\ --maximum\ vector\ length}\newline
+{\tt 24.\ \ }\newline
+{\tt 25.\ \ \ \ \ \ setRealSteps(n)\ ==\ realSteps\ :=\ n}\newline
+{\tt 26.\ \ \ \ \ \ setImagSteps(n)\ ==\ imagSteps\ :=\ n}\newline
+{\tt 27.\ \ \ \ \ \ setClipValue(c)\ ==\ clipValue\ :=\ c}\newline
+{\tt 28.\ \ }\newline
+{\tt 29.\ \ \ \ \ \ clipFun:\ DFLOAT\ ->\ DFLOAT\ --Clip\ large\ magnitudes.}\newline
+{\tt 30.\ \ \ \ \ \ clipFun(x)\ ==\ min(max(x,\ -clipValue),\ clipValue)}\newline
+{\tt 31.\ \ }\newline
+{\tt 32.\ \ \ \ \ \ makeArrow:\ (Point\ DFLOAT,Point\ DFLOAT,DFLOAT,DFLOAT)\ ->\ CURVE}\newline
+{\tt 33.\ \ \ \ \ \ makeArrow(p1,\ p2,\ len,\ arg)\ ==\ ...}\newline
+{\tt 34.\ \ }\newline
+{\tt 35.\ \ \ \ \ \ drawComplex(f,\ realRange,\ imagRange,\ arrows?)\ ==\ ...}\newline
+\caption{The DrawComplex package.}\label{fig-pak-cdraw}
+\endImportant
+
+\beginmenu
+    \menudownlink{{11.1. Names, Abbreviations, and File Structure}}{ugPackagesNamesPage}
+    \menudownlink{{11.2. Syntax}}{ugPackagesSyntaxPage}
+    \menudownlink{{11.3. Abstract Datatypes}}{ugPackagesAbstractPage}
+    \menudownlink{{11.4. Capsules}}{ugPackagesCapsulesPage}
+    \menudownlink{{11.5. Input Files vs. Packages}}{ugPackagesInputFilesPage}
+    \menudownlink{{11.6. Compiling Packages}}{ugPackagesPackagesPage}
+    \menudownlink{{11.7. Parameters}}{ugPackagesParametersPage}
+    \menudownlink{{11.8. Conditionals}}{ugPackagesCondsPage}
+    \menudownlink{{11.9. Testing}}{ugPackagesCompilingPage}
+    \menudownlink{{11.10. How Packages Work}}{ugPackagesHowPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugPackagesNamesTitle}{Names, Abbreviations, and File Structure}
+\newcommand{\ugPackagesNamesNumber}{11.1.}
+%
+% =====================================================================
+\begin{page}{ugPackagesNamesPage}{11.1. Names, Abbreviations, and File Structure}
+% =====================================================================
+\beginscroll
+%
+Each package has a name and an abbreviation.
+For a package of the complex draw functions from \downlink{``\ugIntProgTitle''}{ugIntProgPage} in Chapter \ugIntProgNumber\ignore{ugIntProg},
+we choose the name \nonLibAxiomType{DrawComplex}
+and
+%-% \HDindex{abbreviation!constructor}{ugPackagesNamesPage}{11.1.}{Names, Abbreviations, and File Structure}
+abbreviation \nonLibAxiomType{DRAWCX}.\footnote{An abbreviation can be any string
+of
+%-% \HDindex{constructor!abbreviation}{ugPackagesNamesPage}{11.1.}{Names, Abbreviations, and File Structure}
+between two and seven capital letters and digits, beginning with a letter.
+See \downlink{``\ugTypesWritingAbbrTitle''}{ugTypesWritingAbbrPage} in Section \ugTypesWritingAbbrNumber\ignore{ugTypesWritingAbbr} for more information.}
+To be sure that you have not chosen a name or abbreviation already used by
+the system, issue the system command \spadcmd{)show} for both the name and
+the abbreviation.
+%-% \HDsyscmdindex{show}{ugPackagesNamesPage}{11.1.}{Names, Abbreviations, and File Structure}
+
+Once you have named the package and its abbreviation, you can choose any new
+filename you like with extension ``{\bf \spadFileExt{}}'' to hold the
+definition of your package.
+We choose the name {\bf
+drawpak\spadFileExt{}}.
+If your application involves more than one package, you
+can put them all in the same file.
+\Language{} assumes no relationship between the name of a library file, and
+the name or abbreviation of a package.
+
+Near the top of the ``{\bf \spadFileExt{}}'' file, list all the
+abbreviations for the packages
+using \spadcmd{)abbrev}, each command beginning in column one.
+Macros giving names to \Language{} expressions can also be placed near the
+top of the file.
+The macros are only usable from their point of definition until the
+end of the file.
+
+Consider the definition of
+\nonLibAxiomType{DrawComplex} in Figure \ref{fig-pak-cdraw}.
+After the macro
+%-% \HDindex{macro}{ugPackagesNamesPage}{11.1.}{Names, Abbreviations, and File Structure}
+definition
+\begin{verbatim}
+S      ==> Segment DoubleFloat
+\end{verbatim}
+the name
+{\tt S} can be used in the file as a
+shorthand for \axiomType{Segment DoubleFloat}.\footnote{The interpreter also allows
+{\tt macro} for macro definitions.}
+The abbreviation command for the package
+\begin{verbatim}
+)abbrev package DRAWCX DrawComplex
+\end{verbatim}
+is given after the macros (although it could precede them).
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugPackagesSyntaxTitle}{Syntax}
+\newcommand{\ugPackagesSyntaxNumber}{11.2.}
+%
+% =====================================================================
+\begin{page}{ugPackagesSyntaxPage}{11.2. Syntax}
+% =====================================================================
+\beginscroll
+%
+The definition of a package has the syntax:
+\centerline{{\frenchspacing{\it PackageForm {\tt :} Exports\quad{\tt ==}\quad Implementation}}}
+The syntax for defining a package constructor is the same as that
+%-% \HDindex{syntax}{ugPackagesSyntaxPage}{11.2.}{Syntax}
+for defining any function in \Language{}.
+In practice, the definition extends over many lines so that this syntax is
+not practical.
+Also, the type of a package is expressed by the operator \axiom{with}
+\spadkey{with}
+followed by an explicit list of operations.
+A preferable way to write the definition of a package is with a \axiom{where}
+\spadkey{where}
+expression:
+
+\beginImportant
+The definition of a package usually has the form: \newline
+{\tt%
+{\it PackageForm} : Exports  ==  Implementation where \newline
+\texht{\hspace*{.75pc}}{\tab{3}} {\it optional type declarations}\newline
+\texht{\hspace*{.75pc}}{\tab{3}} Exports  ==   with \newline
+\texht{\hspace*{2.0pc}}{\tab{6}}   {\it list of exported operations}\newline
+\texht{\hspace*{.75pc}}{\tab{3}} Implementation == add \newline
+\texht{\hspace*{2.0pc}}{\tab{6}}   {\it list of function definitions for exported operations}
+}
+\endImportant
+
+The \axiomType{DrawComplex} package takes no parameters and exports five
+operations, each a separate item of a \spadgloss{pile}.
+Each operation is described as a \spadgloss{declaration}: a name, followed
+by a colon (\axiomSyntax{:}), followed by the type of the operation.
+All operations have types expressed as \spadglossSee{mappings}{mapping} with
+the syntax
+\centerline{{{\it}}
+\centerline{{source\quad{\tt ->}\quad target}}
+\centerline{{}}}
+
+%e *********************************************************************
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugPackagesAbstractTitle}{Abstract Datatypes}
+\newcommand{\ugPackagesAbstractNumber}{11.3.}
+%
+% =====================================================================
+\begin{page}{ugPackagesAbstractPage}{11.3. Abstract Datatypes}
+% =====================================================================
+\beginscroll
+
+A constructor as defined in \Language{} is called an \spadgloss{abstract
+datatype} in the computer science literature.
+Abstract datatypes separate ``specification'' (what operations are
+provided) from ``implementation'' (how the operations are implemented).
+The {\tt Exports} (specification) part of a constructor is said to be ``public'' (it
+provides the user interface to the package) whereas the {\tt Implementation}
+part is ``private'' (information here is effectively hidden---programs
+cannot take advantage of it).
+
+The {\tt Exports} part specifies what operations the package provides to users.
+As an author of a package, you must ensure that
+the {\tt Implementation} part provides a function for each
+operation in the {\tt Exports} part.\footnote{The \spadtype{DrawComplex}
+package enhances the facility
+described in  \downlink{``\ugIntProgCompFunsTitle''}{ugIntProgCompFunsPage} in Chapter \ugIntProgCompFunsNumber\ignore{ugIntProgCompFuns} by allowing a
+complex function to have
+arrows emanating from the surface to indicate the direction of the
+complex argument.}
+
+An important difference between interactive programming and the
+use of packages is in the handling of global variables such as
+\axiom{realSteps} and \axiom{imagSteps}.
+In interactive programming, you simply change the values of
+variables by \spadgloss{assignment}.
+With packages, such variables are local to the package---their
+values can only be set using functions exported by the package.
+In our example package, we provide two functions
+\fakeAxiomFun{setRealSteps} and \fakeAxiomFun{setImagSteps} for
+this purpose.
+
+Another local variable is \axiom{clipValue} which can be changed using
+the exported operation \fakeAxiomFun{setClipValue}.
+This value is referenced by the internal function \fakeAxiomFun{clipFun} that
+decides whether to use the computed value of the function at a point or,
+if the magnitude of that value is too large, the
+value assigned to \axiom{clipValue} (with the
+appropriate sign).
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugPackagesCapsulesTitle}{Capsules}
+\newcommand{\ugPackagesCapsulesNumber}{11.4.}
+%
+% =====================================================================
+\begin{page}{ugPackagesCapsulesPage}{11.4. Capsules}
+% =====================================================================
+\beginscroll
+%
+The part to the right of {\tt add} in the {\tt Implementation}
+\spadkey{add}
+part of the definition is called a \spadgloss{capsule}.
+The purpose of a capsule is:
+\indent{4}
+\beginitems
+\item[-] to define a function for each exported operation, and
+\item[-] to define a \spadgloss{local environment} for these functions to run.
+\enditems
+\indent{0}
+
+What is a local environment?
+First, what is an environment?
+%-% \HDindex{environment}{ugPackagesCapsulesPage}{11.4.}{Capsules}
+Think of the capsule as an input file that \Language{} reads from top to
+bottom.
+Think of the input file as having a \axiom{)clear all} at the top
+so that initially no variables or functions are defined.
+When this file is read, variables such as \axiom{realSteps} and
+\axiom{arrowSize} in \nonLibAxiomType{DrawComplex} are set to initial values.
+Also, all the functions defined in the capsule are compiled.
+These include those that are exported (like \axiom{drawComplex}), and
+those that are not (like \axiom{makeArrow}).
+At the end, you get a set of name-value pairs:
+variable names (like \axiom{realSteps} and \axiom{arrowSize})
+are paired with assigned values, while
+operation names (like \axiom{drawComplex} and \axiom{makeArrow})
+are paired with function values.
+
+This set of name-value pairs is called an \spadgloss{environment}.
+Actually, we call this environment the ``initial environment'' of a package:
+it is the environment that exists immediately after the package is
+first built.
+Afterwards, functions of this capsule can
+access or reset a variable in the environment.
+The environment is called {\it local} since any changes to the value of a
+variable in this environment can be seen {\it only} by these functions.
+
+Only the functions from the package can change the variables in the local
+environment.
+When two functions are called successively from a package,
+any changes caused by the first function called
+are seen by the second.
+
+Since the environment is local to the package, its names
+don't get mixed
+up with others in the system or your workspace.
+If you happen to have a variable called \axiom{realSteps} in your
+workspace, it does not affect what the
+\nonLibAxiomType{DrawComplex} functions do in any way.
+
+The functions in a package are compiled into machine code.
+Unlike function definitions in input files that may be compiled repeatedly
+as you use them with varying argument types,
+functions in packages have a unique type (generally parameterized by
+the argument parameters of a package) and a unique compilation residing on disk.
+
+The capsule itself is turned into a compiled function.
+This so-called {\it capsule function} is what builds the initial environment
+spoken of above.
+If the package has arguments (see below), then each call to the package
+constructor with a distinct pair of arguments
+builds a distinct package, each with its own local environment.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugPackagesInputFilesTitle}{Input Files vs. Packages}
+\newcommand{\ugPackagesInputFilesNumber}{11.5.}
+%
+% =====================================================================
+\begin{page}{ugPackagesInputFilesPage}{11.5. Input Files vs. Packages}
+% =====================================================================
+\beginscroll
+%
+A good question at this point would be ``Is writing a package more difficult than
+writing an input file?''
+
+The programs in input files are designed for flexibility and ease-of-use.
+\Language{} can usually work out all of your types as it reads your program
+and does the computations you request.
+Let's say that you define a one-argument function without giving its type.
+When you first apply the function to a value, this
+value is understood by \Language{} as identifying the type for the
+argument parameter.
+Most of the time \Language{} goes through the body of your function and
+figures out the target type that you have in mind.
+\Language{} sometimes fails to get it right.
+Then---and only then---do you need a declaration to tell \Language{} what
+type you want.
+
+Input files are usually written to be read by \Language{}---and by you.
+%-% \HDindex{file!input!vs. package}{ugPackagesInputFilesPage}{11.5.}{Input Files vs. Packages}
+Without suitable documentation and declarations, your input files
+%-% \HDindex{package!vs. input file}{ugPackagesInputFilesPage}{11.5.}{Input Files vs. Packages}
+are likely incomprehensible to a colleague---and to you some
+months later!
+
+Packages are designed for legibility, as well as
+run-time efficiency.
+There are few new concepts you need to learn to write
+packages. Rather, you just have to be explicit about types
+and type conversions.
+The types of all functions are pre-declared so that \Language{}---and the reader---
+knows precisely what types of arguments can be passed to and from
+the functions (certainly you don't want a colleague to guess or to
+have to work this out from context!).
+The types of local variables are also declared.
+Type conversions are explicit, never automatic.\footnote{There
+is one exception to this rule: conversions from a subdomain to a
+domain are automatic.
+After all, the objects both have the domain as a common type.}
+
+In summary, packages are more tedious to write than input files.
+When writing input files, you can casually go ahead, giving some
+facts now, leaving others for later.
+Writing packages requires forethought, care and discipline.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugPackagesPackagesTitle}{Compiling Packages}
+\newcommand{\ugPackagesPackagesNumber}{11.6.}
+%
+% =====================================================================
+\begin{page}{ugPackagesPackagesPage}{11.6. Compiling Packages}
+% =====================================================================
+\beginscroll
+%
+
+Once you have defined the package \nonLibAxiomType{DrawComplex},
+you need to compile and test it.
+To compile the package, issue the system command \spadcmd{)compile drawpak}.
+\Language{} reads the file {\bf drawpak\spadFileExt{}}
+and compiles its contents into machine binary.
+If all goes well, the file {\bf DRAWCX.NRLIB} is created in your
+local directory for the package.
+To test the package, you must load the package before trying an
+operation.
+
+\nullXtc{
+Compile the package.
+}{
+\spadpaste{)compile drawpak}
+}
+\xtc{
+Expose the package.
+}{
+\spadpaste{)expose DRAWCX \bound{dp}}
+}
+\xtc{
+Use an odd step size to avoid
+a pole at the origin.
+}{
+\spadpaste{setRealSteps 51 \free{dp}\bound{srs}}
+}
+\xtc{
+}{
+\spadpaste{setImagSteps 51 \free{dp}\bound{scs}}
+}
+\xtc{
+Define \userfun{f} to be the Gamma function.
+}{
+\spadpaste{f(z) == Gamma(z) \bound{f}}
+}
+\xtc{
+Clip values of function with magnitude larger than 7.
+}{
+\spadpaste{setClipValue 7}
+}
+\psXtc{
+Draw the \spadfun{Gamma} function.
+}{
+\graphpaste{drawComplex(f,-\%pi..\%pi,-\%pi..\%pi, false) \free{srs scs f}}
+}{
+\epsffile[0 0 300 300]{../ps/3Dgamma11.ps}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugPackagesParametersTitle}{Parameters}
+\newcommand{\ugPackagesParametersNumber}{11.7.}
+%
+% =====================================================================
+\begin{page}{ugPackagesParametersPage}{11.7. Parameters}
+% =====================================================================
+\beginscroll
+%
+The power of packages becomes evident when packages have parameters.
+Usually these parameters are domains and the exported operations have types
+involving these parameters.
+
+In \downlink{``\ugTypesTitle''}{ugTypesPage} in Chapter \ugTypesNumber\ignore{ugTypes}, you learned that categories denote classes of
+domains.
+Although we cover this notion in detail in the next
+chapter, we now give you a sneak preview of its usefulness.
+
+In \downlink{``\ugUserBlocksTitle''}{ugUserBlocksPage} in Section \ugUserBlocksNumber\ignore{ugUserBlocks}, we defined functions \axiom{bubbleSort(m)} and
+\axiom{insertionSort(m)} to sort a list of integers.
+If you look at the code for these functions, you see that they may be
+used to sort {\it any} structure \axiom{m} with the right properties.
+Also, the functions can be used to sort lists of {\it any} elements---not
+just integers.
+Let us now recall the code for \axiom{bubbleSort}.
+
+\begin{verbatim}
+bubbleSort(m) ==
+  n := #m
+  for i in 1..(n-1) repeat
+    for j in n..(i+1) by -1 repeat
+      if m.j < m.(j-1) then swap!(m,j,j-1)
+  m
+\end{verbatim}
+
+What properties of ``lists of integers'' are assumed by the sorting
+algorithm?
+In the first line, the operation \spadfun{\#} computes the maximum index of
+the list.
+The first obvious property is that \axiom{m} must have a finite number of
+elements.
+In \Language{}, this is done
+by your telling \Language{} that \axiom{m} has
+the ``attribute'' \spadatt{finiteAggregate}.
+An \spadgloss{attribute} is a property
+that a domain either has or does not have.
+As we show later in \downlink{``\ugCategoriesAttributesTitle''}{ugCategoriesAttributesPage} in Section \ugCategoriesAttributesNumber\ignore{ugCategoriesAttributes},
+programs can query domains as to the presence or absence of an attribute.
+
+The operation \spadfunX{swap} swaps elements of \axiom{m}.
+Using \Browse{}, you find that \spadfunX{swap} requires its
+elements to come from a domain of category
+\axiomType{IndexedAggregate} with attribute
+\spadatt{shallowlyMutable}.
+This attribute means that you can change the internal components
+of \axiom{m} without changing its external structure.
+Shallowly-mutable data structures include lists, streams, one- and
+two-dimensional arrays, vectors, and matrices.
+
+The category \axiomType{IndexedAggregate} designates the class of
+aggregates whose elements can be accessed by the notation
+\axiom{m.s} for suitable selectors \axiom{s}.
+The category \axiomType{IndexedAggregate} takes two arguments:
+\axiom{Index}, a domain of selectors for the aggregate, and
+\axiom{Entry}, a domain of entries for the aggregate.
+Since the sort functions access elements by integers, we must
+choose \axiom{Index = }\axiomType{Integer}.
+The most general class of domains for which \axiom{bubbleSort} and
+\axiom{insertionSort} are defined are those of
+category \spadtype{IndexedAggregate(Integer,Entry)} with the two
+attributes \spadatt{shallowlyMutable} and
+\spadatt{finiteAggregate}.
+
+Using \Browse{}, you can also discover that \Language{} has many kinds of domains
+with attribute \spadatt{shallowlyMutable}.
+Those of class \axiomType{IndexedAggregate(Integer,Entry)} include
+\axiomType{Bits}, \axiomType{FlexibleArray}, \axiomType{OneDimensionalArray},
+\axiomType{List}, \axiomType{String}, and \axiomType{Vector}, and also
+\axiomType{HashTable} and \axiomType{EqTable} with integer keys.
+Although you may never want to sort all such structures, we
+nonetheless demonstrate \Language{}'s
+ability to do so.
+
+Another requirement is that \nonLibAxiomType{Entry} has an
+operation \axiomOp{<}.
+One way to get this operation is to assume that
+\nonLibAxiomType{Entry} has category \axiomType{OrderedSet}.
+By definition, will then export a \axiomOp{<} operation.
+A more general approach is to allow any comparison function
+\axiom{f} to be used for sorting.
+This function will be passed as an argument to the sorting
+functions.
+
+Our sorting package then takes two arguments: a domain \axiom{S}
+of objects of {\it any} type, and a domain \axiom{A}, an aggregate
+of type \axiomType{IndexedAggregate(Integer, S)} with the above
+two attributes.
+Here is its definition using what are close to the original
+definitions of \axiom{bubbleSort} and \axiom{insertionSort} for
+sorting lists of integers.
+The symbol \axiomSyntax{!} is added to the ends of the operation
+names.
+This uniform naming convention is used for \Language{} operation
+names that destructively change one or more of their arguments.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ SortPackage(S,A)\ :\ Exports\ ==\ Implementation\ where}\newline
+{\tt 2.\ \ \ \ \ S:\ Object}\newline
+{\tt 3.\ \ \ \ \ A:\ IndexedAggregate(Integer,S)}\newline
+{\tt 4.\ \ \ \ \ \ \ with\ (finiteAggregate;\ shallowlyMutable)}\newline
+{\tt 5.\ \ \ }\newline
+{\tt 6.\ \ \ \ \ Exports\ ==\ with}\newline
+{\tt 7.\ \ \ \ \ \ \ bubbleSort!:\ (A,(S,S)\ ->\ Boolean)\ ->\ A}\newline
+{\tt 8.\ \ \ \ \ \ \ insertionSort!:\ (A,\ (S,S)\ ->\ Boolean)\ ->\ A}\newline
+{\tt 9.\ \ \ }\newline
+{\tt 10.\ \ \ \ Implementation\ ==\ add}\newline
+{\tt 11.\ \ \ \ \ \ bubbleSort!(m,f)\ ==}\newline
+{\tt 12.\ \ \ \ \ \ \ \ n\ :=\ \#m}\newline
+{\tt 13.\ \ \ \ \ \ \ \ for\ i\ in\ 1..(n-1)\ repeat}\newline
+{\tt 14.\ \ \ \ \ \ \ \ \ \ for\ j\ in\ n..(i+1)\ by\ -1\ repeat}\newline
+{\tt 15.\ \ \ \ \ \ \ \ \ \ \ \ if\ f(m.j,m.(j-1))\ then\ swap!(m,j,j-1)}\newline
+{\tt 16.\ \ \ \ \ \ \ \ m}\newline
+{\tt 17.\ \ \ \ \ \ insertionSort!(m,f)\ ==}\newline
+{\tt 18.\ \ \ \ \ \ \ \ for\ i\ in\ 2..\#m\ repeat}\newline
+{\tt 19.\ \ \ \ \ \ \ \ \ \ j\ :=\ i}\newline
+{\tt 20.\ \ \ \ \ \ \ \ \ \ while\ j\ >\ 1\ and\ f(m.j,m.(j-1))\ repeat}\newline
+{\tt 21.\ \ \ \ \ \ \ \ \ \ \ \ swap!(m,j,j-1)}\newline
+{\tt 22.\ \ \ \ \ \ \ \ \ \ \ \ j\ :=\ (j\ -\ 1)\ pretend\ PositiveInteger}\newline
+{\tt 23.\ \ \ \ \ \ \ \ m}\newline
+\endImportant
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugPackagesCondsTitle}{Conditionals}
+\newcommand{\ugPackagesCondsNumber}{11.8.}
+%
+% =====================================================================
+\begin{page}{ugPackagesCondsPage}{11.8. Conditionals}
+% =====================================================================
+\beginscroll
+%
+When packages have parameters, you can say that an operation is or is not
+%-% \HDindex{conditional}{ugPackagesCondsPage}{11.8.}{Conditionals}
+exported depending on the values of those parameters.
+When the domain of objects \axiom{S} has an \axiomOp{<}
+operation, we can supply one-argument versions of
+\axiom{bubbleSort} and \axiom{insertionSort} which use this operation
+for sorting.
+The presence of the
+operation \axiomOp{<} is guaranteed when \axiom{S} is an ordered set.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ Exports\ ==\ with}\newline
+{\tt 2.\ \ \ \ \ \ \ bubbleSort!:\ (A,(S,S)\ ->\ Boolean)\ ->\ A}\newline
+{\tt 3.\ \ \ \ \ \ \ insertionSort!:\ (A,\ (S,S)\ ->\ Boolean)\ ->\ A}\newline
+{\tt 4.\ \ \ }\newline
+{\tt 5.\ \ \ \ \ \ \ if\ S\ has\ OrderedSet\ then}\newline
+{\tt 6.\ \ \ \ \ \ \ \ \ bubbleSort!:\ A\ ->\ A}\newline
+{\tt 7.\ \ \ \ \ \ \ \ \ insertionSort!:\ A\ ->\ A}\newline
+\endImportant
+
+In addition to exporting the one-argument sort operations
+%-% \HDindex{sort!bubble}{ugPackagesCondsPage}{11.8.}{Conditionals}
+conditionally, we must provide conditional definitions for the
+%-% \HDindex{sort!insertion}{ugPackagesCondsPage}{11.8.}{Conditionals}
+operations in the {\tt Implementation} part.
+This is easy: just have the one-argument functions call the
+corresponding two-argument functions with the operation
+\axiomOp{<} from \axiom{S}.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ \ \ Implementation\ ==\ add}\newline
+{\tt 2.\ \ \ \ \ \ \ \ \ \ ...}\newline
+{\tt 3.\ \ \ \ \ \ \ if\ S\ has\ OrderedSet\ then}\newline
+{\tt 4.\ \ \ \ \ \ \ \ \ bubbleSort!(m)\ ==\ bubbleSort!(m,<\$S)}\newline
+{\tt 5.\ \ \ \ \ \ \ \ \ insertionSort!(m)\ ==\ insertionSort!(m,<\$S)}\newline
+\endImportant
+
+In \downlink{``\ugUserBlocksTitle''}{ugUserBlocksPage} in Section \ugUserBlocksNumber\ignore{ugUserBlocks}, we give an alternative definition of
+\fakeAxiomFun{bubbleSort} using \spadfunFrom{first}{List} and
+\spadfunFrom{rest}{List} that is more efficient for a list (for
+which access to any element requires traversing the list from its
+first node).
+To implement a more efficient algorithm for lists, we need the
+operation \spadfun{setelt} which allows us to destructively change
+the \spadfun{first} and \spadfun{rest} of a list.
+Using \Browse{}, you find that these operations come from category
+\axiomType{UnaryRecursiveAggregate}.
+Several aggregate types are unary recursive aggregates including
+those of \axiomType{List} and \axiomType{AssociationList}.
+We provide two different implementations for
+\fakeAxiomFun{bubbleSort!} and \fakeAxiomFun{insertionSort!}: one
+for list-like structures, another for array-like structures.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ Implementation\ ==\ add}\newline
+{\tt 2.\ \ \ \ \ \ \ \ \ \ \ ...}\newline
+{\tt 3.\ \ \ \ \ \ \ if\ A\ has\ UnaryRecursiveAggregate(S)\ then}\newline
+{\tt 4.\ \ \ \ \ \ \ \ \ bubbleSort!(m,fn)\ ==}\newline
+{\tt 5.\ \ \ \ \ \ \ \ \ \ \ empty?\ m\ =>\ m}\newline
+{\tt 6.\ \ \ \ \ \ \ \ \ \ \ l\ :=\ m}\newline
+{\tt 7.\ \ \ \ \ \ \ \ \ \ \ while\ not\ empty?\ (r\ :=\ l.rest)\ repeat}\newline
+{\tt 8.\ \ \ \ \ \ \ \ \ \ \ \ \ \ r\ :=\ bubbleSort!\ r}\newline
+{\tt 9.\ \ \ \ \ \ \ \ \ \ \ \ \ \ x\ :=\ l.first}\newline
+{\tt 10.\ \ \ \ \ \ \ \ \ \ \ \ \ if\ fn(r.first,x)\ then}\newline
+{\tt 11.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ l.first\ :=\ r.first}\newline
+{\tt 12.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r.first\ :=\ x}\newline
+{\tt 13.\ \ \ \ \ \ \ \ \ \ \ \ \ l.rest\ :=\ r}\newline
+{\tt 14.\ \ \ \ \ \ \ \ \ \ \ \ \ l\ :=\ l.rest}\newline
+{\tt 15.\ \ \ \ \ \ \ \ \ \ \ m}\newline
+{\tt 16.\ \ \ \ \ \ \ \ \ insertionSort!(m,fn)\ ==}\newline
+{\tt 17.\ \ \ \ \ \ \ \ \ \ \ \ ...}\newline
+\endImportant
+
+The ordering of definitions is important.
+The standard definitions come first and
+then the predicate
+\begin{verbatim}
+A has UnaryRecursiveAggregate(S)
+\end{verbatim}
+is evaluated.
+If {\tt true}, the special definitions cover up the standard ones.
+
+Another equivalent way to write the capsule is to use an
+\axiom{if-then-else} expression:
+\spadkey{if}
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ \ \ \ \ \ if\ A\ has\ UnaryRecursiveAggregate(S)\ then}\newline
+{\tt 2.\ \ \ \ \ \ \ \ \ \ \ ...}\newline
+{\tt 3.\ \ \ \ \ \ \ \ else}\newline
+{\tt 4.\ \ \ \ \ \ \ \ \ \ \ ...}\newline
+\endImportant
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugPackagesCompilingTitle}{Testing}
+\newcommand{\ugPackagesCompilingNumber}{11.9.}
+%
+% =====================================================================
+\begin{page}{ugPackagesCompilingPage}{11.9. Testing}
+% =====================================================================
+\beginscroll
+%
+Once you have written the package, embed it in a file, for example, {\bf
+sortpak\spadFileExt{}}.
+%-% \HDindex{testing}{ugPackagesCompilingPage}{11.9.}{Testing}
+Be sure to include an \axiom{)abbrev} command at the top of the file:
+\begin{verbatim}
+)abbrev package SORTPAK SortPackage
+\end{verbatim}
+Now compile the file (using \spadcmd{)compile sortpak\spadFileExt{}}).
+\xtc{
+Expose the constructor.
+You are then ready to begin testing.
+}{
+\spadpaste{)expose SORTPAK}
+}
+\xtc{
+Define a list.
+}{
+\spadpaste{l := [1,7,4,2,11,-7,3,2]}
+}
+\xtc{
+Since the integers are an ordered set,
+a one-argument operation will do.
+}{
+\spadpaste{bubbleSort!(l)}
+}
+\xtc{
+Re-sort it using ``greater than.''
+}{
+\spadpaste{bubbleSort!(l,(x,y) +-> x > y)}
+}
+\xtc{
+Now sort it again using \axiomOp{<} on integers.
+}{
+\spadpaste{bubbleSort!(l, <\$Integer)}
+}
+\xtc{
+A string is an aggregate of characters so we can sort them as well.
+}{
+\spadpaste{bubbleSort! "Mathematical Sciences"}
+}
+\xtc{
+Is \axiomOp{<} defined on booleans?
+}{
+\spadpaste{false < true}
+}
+\xtc{
+Good! Create a bit string representing ten consecutive
+boolean values \axiom{true}.
+}{
+\spadpaste{u : Bits := new(10,true)}
+}
+\xtc{
+Set bits 3 through 5 to \axiom{false}, then display the result.
+}{
+\spadpaste{u(3..5) := false; u}
+}
+\xtc{
+Now sort these booleans.
+}{
+\spadpaste{bubbleSort! u}
+}
+\xtc{
+Create an ``eq-table'' (see \downlink{`EqTable'}{EqTableXmpPage}\ignore{EqTable}), a
+table having integers as keys
+and strings as values.
+}{
+\spadpaste{t : EqTable(Integer,String) := table()}
+}
+\xtc{
+Give the table a first entry.
+}{
+\spadpaste{t.1 := "robert"}
+}
+\xtc{
+And a second.
+}{
+\spadpaste{t.2 := "richard"}
+}
+\xtc{
+What does the table look like?
+}{
+\spadpaste{t}
+}
+\xtc{
+Now sort it.
+}{
+\spadpaste{bubbleSort! t}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugPackagesHowTitle}{How Packages Work}
+\newcommand{\ugPackagesHowNumber}{11.10.}
+%
+% =====================================================================
+\begin{page}{ugPackagesHowPage}{11.10. How Packages Work}
+% =====================================================================
+\beginscroll
+%
+Recall that packages as abstract datatypes are compiled independently
+and put into the library.
+The curious reader may ask: ``How is the interpreter able to find an
+operation such as \fakeAxiomFun{bubbleSort!}?
+Also, how is a single compiled function such as \fakeAxiomFun{bubbleSort!} able
+to sort data of different types?''
+
+After the interpreter loads the package \nonLibAxiomType{SortPackage}, the four
+operations from the package become known to the interpreter.
+Each of these operations is expressed as a {\it modemap} in which the type
+%-% \HDindex{modemap}{ugPackagesHowPage}{11.10.}{How Packages Work}
+of the operation is written in terms of symbolic domains.
+\nullXtc{
+See the modemaps for \fakeAxiomFun{bubbleSort!}.
+}{
+\spadpaste{)display op bubbleSort!}
+}
+\begin{verbatim}
+There are 2 exposed functions called bubbleSort! :
+
+   [1] D1 -> D1 from SortPackage(D2,D1)
+         if D2 has ORDSET and D2 has OBJECT and D1 has
+         IndexedAggregate(Integer, D2) with
+              finiteAggregate
+              shallowlyMutable
+
+   [2] (D1,((D3,D3) -> Boolean)) -> D1 from SortPackage(D3,D1)
+         if D3 has OBJECT and D1 has
+         IndexedAggregate(Integer,D3) with
+              finiteAggregate
+              shallowlyMutable
+\end{verbatim}
+
+What happens if you ask for \axiom{bubbleSort!([1,-5,3])}?
+There is a unique modemap for an operation named
+\fakeAxiomFun{bubbleSort!} with one argument.
+Since \axiom{[1,-5,3]} is a list of integers, the symbolic domain
+\axiom{D1} is defined as \axiomType{List(Integer)}.
+For some operation to apply, it must satisfy the predicate for
+some \axiom{D2}.
+What \axiom{D2}?
+The third expression of the \axiom{and} requires {\tt D1 has
+IndexedAggregate(Integer, D2) with} two attributes.
+So the interpreter searches for an \axiomType{IndexedAggregate}
+among the ancestors of \axiomType{List (Integer)} (see
+\downlink{``\ugCategoriesHierTitle''}{ugCategoriesHierPage} in Section \ugCategoriesHierNumber\ignore{ugCategoriesHier}).
+It finds one: \axiomType{IndexedAggregate(Integer, Integer)}.
+The interpreter tries defining \axiom{D2} as \axiomType{Integer}.
+After substituting for \axiom{D1} and \axiom{D2}, the predicate
+evaluates to \axiom{true}.
+An applicable operation has been found!
+
+Now \Language{} builds the package
+\axiomType{SortPackage(List(Integer), Integer)}.
+According to its definition, this package exports the required
+operation: \fakeAxiomFun{bubbleSort!}: \spadsig{List Integer}{List
+Integer}.
+The interpreter then asks the package for a function implementing
+this operation.
+The package gets all the functions it needs (for example,
+\axiomFun{rest} and \axiomFunX{swap}) from the appropriate
+domains and then it
+returns a \fakeAxiomFun{bubbleSort!} to the interpreter together with
+the local environment for \fakeAxiomFun{bubbleSort!}.
+The interpreter applies the function to the argument \axiom{[1,-5,3]}.
+The \fakeAxiomFun{bubbleSort!} function is executed in its local
+environment and produces the result.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug11.pht b/src/hyper/pages/ug11.pht
new file mode 100644
index 00000000..714425a5
--- /dev/null
+++ b/src/hyper/pages/ug11.pht
@@ -0,0 +1,356 @@
+\begin{patch}{ugPackagesPackagesPagePatch1}
+\begin{paste}{ugPackagesPackagesPageFull1}{ugPackagesPackagesPageEmpty1}
+\pastebutton{ugPackagesPackagesPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)compile drawpak}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesPackagesPageEmpty1}
+\begin{paste}{ugPackagesPackagesPageEmpty1}{ugPackagesPackagesPagePatch1}
+\pastebutton{ugPackagesPackagesPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)compile drawpak}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesPackagesPagePatch2}
+\begin{paste}{ugPackagesPackagesPageFull2}{ugPackagesPackagesPageEmpty2}
+\pastebutton{ugPackagesPackagesPageFull2}{\hidepaste}
+\tab{5}\spadcommand{)expose DRAWCX\bound{dp }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesPackagesPageEmpty2}
+\begin{paste}{ugPackagesPackagesPageEmpty2}{ugPackagesPackagesPagePatch2}
+\pastebutton{ugPackagesPackagesPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{)expose DRAWCX\bound{dp }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesPackagesPagePatch3}
+\begin{paste}{ugPackagesPackagesPageFull3}{ugPackagesPackagesPageEmpty3}
+\pastebutton{ugPackagesPackagesPageFull3}{\hidepaste}
+\tab{5}\spadcommand{setRealSteps 51\free{dp }\bound{srs }}
+\indentrel{3}\begin{verbatim}
+   (1)  51
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesPackagesPageEmpty3}
+\begin{paste}{ugPackagesPackagesPageEmpty3}{ugPackagesPackagesPagePatch3}
+\pastebutton{ugPackagesPackagesPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{setRealSteps 51\free{dp }\bound{srs }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesPackagesPagePatch4}
+\begin{paste}{ugPackagesPackagesPageFull4}{ugPackagesPackagesPageEmpty4}
+\pastebutton{ugPackagesPackagesPageFull4}{\hidepaste}
+\tab{5}\spadcommand{setImagSteps 51\free{dp }\bound{scs }}
+\indentrel{3}\begin{verbatim}
+   (2)  51
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesPackagesPageEmpty4}
+\begin{paste}{ugPackagesPackagesPageEmpty4}{ugPackagesPackagesPagePatch4}
+\pastebutton{ugPackagesPackagesPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{setImagSteps 51\free{dp }\bound{scs }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesPackagesPagePatch5}
+\begin{paste}{ugPackagesPackagesPageFull5}{ugPackagesPackagesPageEmpty5}
+\pastebutton{ugPackagesPackagesPageFull5}{\hidepaste}
+\tab{5}\spadcommand{f(z) == Gamma(z)\bound{f }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesPackagesPageEmpty5}
+\begin{paste}{ugPackagesPackagesPageEmpty5}{ugPackagesPackagesPagePatch5}
+\pastebutton{ugPackagesPackagesPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{f(z) == Gamma(z)\bound{f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesPackagesPagePatch6}
+\begin{paste}{ugPackagesPackagesPageFull6}{ugPackagesPackagesPageEmpty6}
+\pastebutton{ugPackagesPackagesPageFull6}{\hidepaste}
+\tab{5}\spadcommand{setClipValue 7}
+\indentrel{3}\begin{verbatim}
+   (4)  7.0
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesPackagesPageEmpty6}
+\begin{paste}{ugPackagesPackagesPageEmpty6}{ugPackagesPackagesPagePatch6}
+\pastebutton{ugPackagesPackagesPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{setClipValue 7}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesPackagesPagePatch7}
+\begin{paste}{ugPackagesPackagesPageFull7}{ugPackagesPackagesPageEmpty7}
+\pastebutton{ugPackagesPackagesPageFull7}{\hidepaste}
+\tab{5}\spadgraph{drawComplex(f,-\%pi..\%pi,-\%pi..\%pi, false)\free{srs scs f }}
+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugPackagesPackagesPage7.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugPackagesPackagesPage7}}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesPackagesPageEmpty7}
+\begin{paste}{ugPackagesPackagesPageEmpty7}{ugPackagesPackagesPagePatch7}
+\pastebutton{ugPackagesPackagesPageEmpty7}{\showpaste}
+\tab{5}\spadgraph{drawComplex(f,-\%pi..\%pi,-\%pi..\%pi, false)\free{srs scs f }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch1}
+\begin{paste}{ugPackagesCompilingPageFull1}{ugPackagesCompilingPageEmpty1}
+\pastebutton{ugPackagesCompilingPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)expose SORTPAK}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty1}
+\begin{paste}{ugPackagesCompilingPageEmpty1}{ugPackagesCompilingPagePatch1}
+\pastebutton{ugPackagesCompilingPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)expose SORTPAK}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch2}
+\begin{paste}{ugPackagesCompilingPageFull2}{ugPackagesCompilingPageEmpty2}
+\pastebutton{ugPackagesCompilingPageFull2}{\hidepaste}
+\tab{5}\spadcommand{l := [1,7,4,2,11,-7,3,2]}
+\indentrel{3}\begin{verbatim}
+   (1)  [1,7,4,2,11,- 7,3,2]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty2}
+\begin{paste}{ugPackagesCompilingPageEmpty2}{ugPackagesCompilingPagePatch2}
+\pastebutton{ugPackagesCompilingPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{l := [1,7,4,2,11,-7,3,2]}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch3}
+\begin{paste}{ugPackagesCompilingPageFull3}{ugPackagesCompilingPageEmpty3}
+\pastebutton{ugPackagesCompilingPageFull3}{\hidepaste}
+\tab{5}\spadcommand{bubbleSort!(l)}
+\indentrel{3}\begin{verbatim}
+   (2)  [- 7,1,2,2,3,4,7,11]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty3}
+\begin{paste}{ugPackagesCompilingPageEmpty3}{ugPackagesCompilingPagePatch3}
+\pastebutton{ugPackagesCompilingPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{bubbleSort!(l)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch4}
+\begin{paste}{ugPackagesCompilingPageFull4}{ugPackagesCompilingPageEmpty4}
+\pastebutton{ugPackagesCompilingPageFull4}{\hidepaste}
+\tab{5}\spadcommand{bubbleSort!(l,(x,y) +-> x > y)}
+\indentrel{3}\begin{verbatim}
+   (3)  [11,7,4,3,2,2,1,- 7]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty4}
+\begin{paste}{ugPackagesCompilingPageEmpty4}{ugPackagesCompilingPagePatch4}
+\pastebutton{ugPackagesCompilingPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{bubbleSort!(l,(x,y) +-> x > y)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch5}
+\begin{paste}{ugPackagesCompilingPageFull5}{ugPackagesCompilingPageEmpty5}
+\pastebutton{ugPackagesCompilingPageFull5}{\hidepaste}
+\tab{5}\spadcommand{bubbleSort!(l, <$Integer)}
+\indentrel{3}\begin{verbatim}
+   (4)  [- 7,1,2,2,3,4,7,11]
+                                     Type: List Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty5}
+\begin{paste}{ugPackagesCompilingPageEmpty5}{ugPackagesCompilingPagePatch5}
+\pastebutton{ugPackagesCompilingPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{bubbleSort!(l, <$Integer)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch6}
+\begin{paste}{ugPackagesCompilingPageFull6}{ugPackagesCompilingPageEmpty6}
+\pastebutton{ugPackagesCompilingPageFull6}{\hidepaste}
+\tab{5}\spadcommand{bubbleSort! "Mathematical Sciences"}
+\indentrel{3}\begin{verbatim}
+   (5)  " MSaaaccceeehiilmnstt"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty6}
+\begin{paste}{ugPackagesCompilingPageEmpty6}{ugPackagesCompilingPagePatch6}
+\pastebutton{ugPackagesCompilingPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{bubbleSort! "Mathematical Sciences"}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch7}
+\begin{paste}{ugPackagesCompilingPageFull7}{ugPackagesCompilingPageEmpty7}
+\pastebutton{ugPackagesCompilingPageFull7}{\hidepaste}
+\tab{5}\spadcommand{false < true}
+\indentrel{3}\begin{verbatim}
+   (6)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty7}
+\begin{paste}{ugPackagesCompilingPageEmpty7}{ugPackagesCompilingPagePatch7}
+\pastebutton{ugPackagesCompilingPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{false < true}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch8}
+\begin{paste}{ugPackagesCompilingPageFull8}{ugPackagesCompilingPageEmpty8}
+\pastebutton{ugPackagesCompilingPageFull8}{\hidepaste}
+\tab{5}\spadcommand{u : Bits := new(10,true)}
+\indentrel{3}\begin{verbatim}
+   (7)  "1111111111"
+                                             Type: Bits
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty8}
+\begin{paste}{ugPackagesCompilingPageEmpty8}{ugPackagesCompilingPagePatch8}
+\pastebutton{ugPackagesCompilingPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{u : Bits := new(10,true)}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch9}
+\begin{paste}{ugPackagesCompilingPageFull9}{ugPackagesCompilingPageEmpty9}
+\pastebutton{ugPackagesCompilingPageFull9}{\hidepaste}
+\tab{5}\spadcommand{u(3..5) := false; u}
+\indentrel{3}\begin{verbatim}
+   (8)  "1100011111"
+                                             Type: Bits
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty9}
+\begin{paste}{ugPackagesCompilingPageEmpty9}{ugPackagesCompilingPagePatch9}
+\pastebutton{ugPackagesCompilingPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{u(3..5) := false; u}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch10}
+\begin{paste}{ugPackagesCompilingPageFull10}{ugPackagesCompilingPageEmpty10}
+\pastebutton{ugPackagesCompilingPageFull10}{\hidepaste}
+\tab{5}\spadcommand{bubbleSort! u}
+\indentrel{3}\begin{verbatim}
+   (9)  "0001111111"
+                                             Type: Bits
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty10}
+\begin{paste}{ugPackagesCompilingPageEmpty10}{ugPackagesCompilingPagePatch10}
+\pastebutton{ugPackagesCompilingPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{bubbleSort! u}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch11}
+\begin{paste}{ugPackagesCompilingPageFull11}{ugPackagesCompilingPageEmpty11}
+\pastebutton{ugPackagesCompilingPageFull11}{\hidepaste}
+\tab{5}\spadcommand{t : EqTable(Integer,String) := table()}
+\indentrel{3}\begin{verbatim}
+   (10)  table()
+                          Type: EqTable(Integer,String)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty11}
+\begin{paste}{ugPackagesCompilingPageEmpty11}{ugPackagesCompilingPagePatch11}
+\pastebutton{ugPackagesCompilingPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{t : EqTable(Integer,String) := table()}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch12}
+\begin{paste}{ugPackagesCompilingPageFull12}{ugPackagesCompilingPageEmpty12}
+\pastebutton{ugPackagesCompilingPageFull12}{\hidepaste}
+\tab{5}\spadcommand{t.1 := "robert"}
+\indentrel{3}\begin{verbatim}
+   (11)  "robert"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty12}
+\begin{paste}{ugPackagesCompilingPageEmpty12}{ugPackagesCompilingPagePatch12}
+\pastebutton{ugPackagesCompilingPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{t.1 := "robert"}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch13}
+\begin{paste}{ugPackagesCompilingPageFull13}{ugPackagesCompilingPageEmpty13}
+\pastebutton{ugPackagesCompilingPageFull13}{\hidepaste}
+\tab{5}\spadcommand{t.2 := "richard"}
+\indentrel{3}\begin{verbatim}
+   (12)  "richard"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty13}
+\begin{paste}{ugPackagesCompilingPageEmpty13}{ugPackagesCompilingPagePatch13}
+\pastebutton{ugPackagesCompilingPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{t.2 := "richard"}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch14}
+\begin{paste}{ugPackagesCompilingPageFull14}{ugPackagesCompilingPageEmpty14}
+\pastebutton{ugPackagesCompilingPageFull14}{\hidepaste}
+\tab{5}\spadcommand{t}
+\indentrel{3}\begin{verbatim}
+   (13)  table(2= "richard",1= "robert")
+                          Type: EqTable(Integer,String)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty14}
+\begin{paste}{ugPackagesCompilingPageEmpty14}{ugPackagesCompilingPagePatch14}
+\pastebutton{ugPackagesCompilingPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{t}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPagePatch15}
+\begin{paste}{ugPackagesCompilingPageFull15}{ugPackagesCompilingPageEmpty15}
+\pastebutton{ugPackagesCompilingPageFull15}{\hidepaste}
+\tab{5}\spadcommand{bubbleSort! t}
+\indentrel{3}\begin{verbatim}
+   (14)  table(2= "robert",1= "richard")
+                          Type: EqTable(Integer,String)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesCompilingPageEmpty15}
+\begin{paste}{ugPackagesCompilingPageEmpty15}{ugPackagesCompilingPagePatch15}
+\pastebutton{ugPackagesCompilingPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{bubbleSort! t}
+\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesHowPagePatch1}
+\begin{paste}{ugPackagesHowPageFull1}{ugPackagesHowPageEmpty1}
+\pastebutton{ugPackagesHowPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)display op bubbleSort!}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugPackagesHowPageEmpty1}
+\begin{paste}{ugPackagesHowPageEmpty1}{ugPackagesHowPagePatch1}
+\pastebutton{ugPackagesHowPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)display op bubbleSort!}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ug12.ht b/src/hyper/pages/ug12.ht
new file mode 100644
index 00000000..04ac5ef2
--- /dev/null
+++ b/src/hyper/pages/ug12.ht
@@ -0,0 +1,905 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\texht{\setcounter{chapter}{11}}{} % Chapter 12
+
+%
+\newcommand{\ugCategoriesTitle}{Categories}
+\newcommand{\ugCategoriesNumber}{12.}
+%
+% =====================================================================
+\begin{page}{ugCategoriesPage}{12. Categories}
+% =====================================================================
+\beginscroll
+
+This chapter unravels the mysteries of categories---what
+%-% \HDindex{category}{ugCategoriesPage}{12.}{Categories}
+they are, how they are related to domains and packages,
+%-% \HDindex{category!constructor}{ugCategoriesPage}{12.}{Categories}
+how they are defined in \Language{}, and how you can extend the
+%-% \HDindex{constructor!category}{ugCategoriesPage}{12.}{Categories}
+system to include new categories of your own.
+
+We assume that you have read the introductory material on domains
+and categories in \downlink{``\ugTypesBasicDomainConsTitle''}{ugTypesBasicDomainConsPage} in Section \ugTypesBasicDomainConsNumber\ignore{ugTypesBasicDomainCons}.
+There you learned that the notion of packages covered in the
+previous chapter are special cases of domains.
+While this is in fact the case, it is useful here to regard domains
+as distinct from packages.
+
+Think of a domain as a datatype, a collection of objects (the
+objects of the domain).
+From your ``sneak preview'' in the previous chapter, you might
+conclude that categories are simply named clusters of operations
+exported by domains.
+As it turns out, categories have a much deeper meaning.
+Categories are fundamental to the design of \Language{}.
+They control the interactions between domains and algorithmic
+packages, and, in fact, between all the components of \Language{}.
+
+Categories form hierarchies as shown on the inside cover pages of
+this book.
+The inside front-cover pages illustrate the basic
+algebraic hierarchy of the \Language{} programming language.
+The inside back-cover pages show the hierarchy for data
+structures.
+
+Think of the category structures of \Language{} as a foundation
+for a city on which superstructures (domains) are built.
+The algebraic hierarchy, for example, serves as a foundation for
+constructive mathematical algorithms embedded in the domains of
+\Language{}.
+Once in place, domains can be constructed, either independently or
+from one another.
+
+Superstructures are built for quality---domains are compiled into
+machine code for run-time efficiency.
+You can extend the foundation in directions beyond the space
+directly beneath the superstructures, then extend selected
+superstructures to cover the space.
+Because of the compilation strategy, changing components of the
+foundation generally means that the existing superstructures
+(domains) built on the changed parts of the foundation
+(categories) have to be rebuilt---that is, recompiled.
+
+Before delving into some of the interesting facts about categories, let's see
+how you define them in \Language{}.
+
+\beginmenu
+    \menudownlink{{12.1. Definitions}}{ugCategoriesDefsPage}
+    \menudownlink{{12.2. Exports}}{ugCategoriesExportsPage}
+    \menudownlink{{12.3. Documentation}}{ugCategoriesDocPage}
+    \menudownlink{{12.4. Hierarchies}}{ugCategoriesHierPage}
+    \menudownlink{{12.5. Membership}}{ugCategoriesMembershipPage}
+    \menudownlink{{12.6. Defaults}}{ugCategoriesDefaultsPage}
+    \menudownlink{{12.7. Axioms}}{ugCategoriesAxiomsPage}
+    \menudownlink{{12.8. Correctness}}{ugCategoriesCorrectnessPage}
+    \menudownlink{{12.9. Attributes}}{ugCategoriesAttributesPage}
+    \menudownlink{{12.10. Parameters}}{ugCategoriesParametersPage}
+    \menudownlink{{12.11. Conditionals}}{ugCategoriesConditionalsPage}
+    \menudownlink{{12.12. Anonymous Categories}}{ugCategoriesAndPackagesPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugCategoriesDefsTitle}{Definitions}
+\newcommand{\ugCategoriesDefsNumber}{12.1.}
+%
+% =====================================================================
+\begin{page}{ugCategoriesDefsPage}{12.1. Definitions}
+% =====================================================================
+\beginscroll
+
+A category is defined by a function with exactly the same format as
+%-% \HDindex{category!definition}{ugCategoriesDefsPage}{12.1.}{Definitions}
+any other function in \Language{}.
+
+\beginImportant
+The definition of a category has the syntax:
+\centerline{{{\it CategoryForm} : {\tt Category\quad{}==\quad{}} {\it Extensions} {\tt [ with} {\it Exports} {\tt ]}}}
+
+The brackets {\tt [ ]} here indicate optionality.
+\endImportant
+
+
+The first example of a category definition is
+\spadtype{SetCategory},
+the most basic of the algebraic categories in \Language{}.
+%-% \HDexptypeindex{SetCategory}{ugCategoriesDefsPage}{12.1.}{Definitions}
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ SetCategory():\ Category\ ==}\newline
+{\tt 2.\ \ \ \ \ \ Join(Type,CoercibleTo\ OutputForm)\ with}\newline
+{\tt 3.\ \ \ \ \ \ \ \ \ "="\ :\ (\$,\ \$)\ ->\ Boolean}\newline
+\endImportant
+
+The definition starts off with the name of the
+category (\spadtype{SetCategory}); this is
+always in column one in the source file.
+%% maybe talk about naming conventions for source files? .spad or .ax?
+All parts of a category definition are then indented with respect to this
+%-% \HDindex{indentation}{ugCategoriesDefsPage}{12.1.}{Definitions}
+first line.
+
+In \downlink{``\ugTypesTitle''}{ugTypesPage} in Chapter \ugTypesNumber\ignore{ugTypes}, we talked about \spadtype{Ring} as denoting the
+class of all domains that are rings, in short, the class of all
+rings.
+While this is the usual naming convention in \Language{}, it is also
+common to use the word ``Category'' at the end of a category name for clarity.
+The interpretation of the name \spadtype{SetCategory} is, then, ``the
+category of all domains that are (mathematical) sets.''
+
+The name \spadtype{SetCategory} is followed in the definition by its
+formal parameters enclosed in parentheses \spadSyntax{()}.
+Here there are no parameters.
+As required, the type of the result of this category function is the
+distinguished name {\sf Category}.
+
+Then comes the \spadSyntax{==}.
+As usual, what appears to the right of the \spadSyntax{==} is a
+definition, here, a category definition.
+A category definition always has two parts separated by the reserved word
+\spadkey{with}
+\spad{with}.
+%\footnote{Debugging hint: it is very easy to forget
+%the \spad{with}!}
+
+The first part tells what categories the category extends.
+Here, the category extends two categories: \spadtype{Type}, the
+category of all domains, and
+\spadtype{CoercibleTo(OutputForm)}.
+%\footnote{\spadtype{CoercibleTo(OutputForm)}
+%can also be written (and is written in the definition above) without
+%parentheses.}
+The operation \spad{Join} is a system-defined operation that
+\spadkey{Join}
+forms a single category from two or more other categories.
+
+Every category other than \spadtype{Type} is an extension of some other
+category.
+If, for example, \spadtype{SetCategory} extended only the category
+\spadtype{Type}, the definition here would read ``{\tt Type with
+...}''.
+In fact, the {\tt Type} is optional in this line; ``{\tt with
+...}'' suffices.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugCategoriesExportsTitle}{Exports}
+\newcommand{\ugCategoriesExportsNumber}{12.2.}
+%
+% =====================================================================
+\begin{page}{ugCategoriesExportsPage}{12.2. Exports}
+% =====================================================================
+\beginscroll
+
+To the right of the \spad{with} is a list of
+\spadkey{with}
+all the \spadglossSee{exports}{export} of the category.
+Each exported operation has a name and a type expressed by a
+\spadgloss{declaration} of the form
+``{\frenchspacing\tt {\it name}: {\it type}}''.
+
+Categories can export symbols, as well as
+{\tt 0} and {\tt 1} which denote
+domain constants.\footnote{The
+numbers {\tt 0} and {\tt 1} are operation names in \Language{}.}
+In the current implementation, all other exports are operations with
+types expressed as \spadglossSee{mappings}{mapping} with the syntax
+\centerline{{{\it}}
+\centerline{{source\quad{\tt ->}\quad target}}
+\centerline{{}}}
+
+The category \spadtype{SetCategory} has a single export: the operation
+\spadop{=} whose type is given by the mapping {\tt (\$, \$) -> Boolean}.
+The \spadSyntax{\$} in a mapping type always means ``the domain.'' Thus
+the operation \spadop{=} takes two arguments from the domain and
+returns a value of type \spadtype{Boolean}.
+
+The source part of the mapping here is given by a {\it tuple}
+%-% \HDindex{tuple}{ugCategoriesExportsPage}{12.2.}{Exports}
+consisting of two or more types separated by commas and enclosed in
+parentheses.
+If an operation takes only one argument, you can drop the parentheses
+around the source type.
+If the mapping has no arguments, the source part of the mapping is either
+left blank or written as \spadSyntax{()}.
+Here are examples of formats of various operations with some
+contrived names.
+
+\begin{verbatim}
+someIntegerConstant  :    $
+aZeroArgumentOperation:   () -> Integer
+aOneArgumentOperation:    Integer -> $
+aTwoArgumentOperation:    (Integer,$) -> Void
+aThreeArgumentOperation:  ($,Integer,$) -> Fraction($)
+\end{verbatim}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugCategoriesDocTitle}{Documentation}
+\newcommand{\ugCategoriesDocNumber}{12.3.}
+%
+% =====================================================================
+\begin{page}{ugCategoriesDocPage}{12.3. Documentation}
+% =====================================================================
+\beginscroll
+
+The definition of \spadtype{SetCategory} above is  missing
+an important component: its library documentation.
+%-% \HDindex{documentation}{ugCategoriesDocPage}{12.3.}{Documentation}
+Here is its definition, complete with documentation.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ ++\ Description:}\newline
+{\tt 2.\ \ \ ++\ \bs{}axiomType\{SetCategory\}\ is\ the\ basic\ category}\newline
+{\tt 3.\ \ \ ++\ for\ describing\ a\ collection\ of\ elements\ with}\newline
+{\tt 4.\ \ \ ++\ \bs{}axiomOp\{=\}\ (equality)\ and\ a\ \bs{}axiomFun\{coerce\}}\newline
+{\tt 5.\ \ \ ++\ to\ \bs{}axiomType\{OutputForm\}.}\newline
+{\tt 6.\ \ \ }\newline
+{\tt 7.\ \ \ SetCategory():\ Category\ ==}\newline
+{\tt 8.\ \ \ \ \ Join(Type,\ CoercibleTo\ OutputForm)\ with}\newline
+{\tt 9.\ \ \ \ \ \ \ "=":\ (\$,\ \$)\ ->\ Boolean}\newline
+{\tt 10.\ \ \ \ \ \ \ \ ++\ \bs{}axiom\{x\ =\ y\}\ tests\ if\ \bs{}axiom\{x\}\ and}\newline
+{\tt 11.\ \ \ \ \ \ \ \ ++\ \bs{}axiom\{y\}\ are\ equal.}\newline
+\endImportant
+
+Documentary comments are an important part of constructor definitions.
+Documentation is given both for the category itself and for
+each export.
+A description for the category precedes the code.
+Each line of the description begins in column one with \axiomSyntax{++}.
+The description starts with the word {\tt Description:}.\footnote{Other
+information such as the author's name, date of creation, and so on,
+can go in this
+area as well but are currently ignored by \Language{}.}
+All lines of the description following the initial line are
+indented by the same amount.
+
+{\texht{\sloppy}{}
+Surround the name of any constructor (with or without parameters) with an
+\texht{\verb+\axiomType{}+}{\\axiomType\{\}}.
+Similarly, surround an
+operator name with \texht{\verb+\axiomOp{}+}{\\axiomOp\{\}},
+an \Language{} operation with \texht{\verb+\axiomFun{}+}{\\axiomFun\{\}}, and a
+variable or \Language{} expression with
+\texht{\verb+\axiom{}+}{\\axiom\{\}}.
+Library documentation is given in a \TeX{}-like language so that
+it can be used both for hard-copy and for \Browse{}.
+These different wrappings cause operations and types to have
+mouse-active buttons in \Browse{}.
+For hard-copy output, wrapped expressions appear in a different font.
+The above documentation appears in hard-copy as:
+
+}
+%
+\texht{\begin{quotation}}{\indent{3}}
+%
+\axiomType{SetCategory} is the basic category
+for describing a collection of elements with \axiomOp{=}
+(equality) and a \spadfun{coerce} to \axiomType{OutputForm}.
+%
+\texht{\end{quotation}}{\indent{0}}
+%
+and
+%
+\texht{\begin{quotation}}{\indent{3}}
+%
+\axiom{x = y} tests if \axiom{x} and \axiom{y} are equal.
+%
+\texht{\end{quotation}}{\indent{0}}
+%
+
+For our purposes in this chapter, we omit the documentation from further
+category descriptions.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugCategoriesHierTitle}{Hierarchies}
+\newcommand{\ugCategoriesHierNumber}{12.4.}
+%
+% =====================================================================
+\begin{page}{ugCategoriesHierPage}{12.4. Hierarchies}
+% =====================================================================
+\beginscroll
+
+A second example of a category is
+\spadtype{SemiGroup}, defined by:
+%-% \HDexptypeindex{SemiGroup}{ugCategoriesHierPage}{12.4.}{Hierarchies}
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ SemiGroup():\ Category\ ==\ SetCategory\ with}\newline
+{\tt 2.\ \ \ \ \ \ \ \ \ "*":\ \ (\$,\$)\ ->\ \$}\newline
+{\tt 3.\ \ \ \ \ \ \ \ \ "**":\ (\$,\ PositiveInteger)\ ->\ \$}\newline
+\endImportant
+
+This definition is as simple as that for \spadtype{SetCategory},
+except that there are two exported operations.
+Multiple exported operations are written as a \spadgloss{pile},
+that is, they all begin in the same column.
+Here you see that the category mentions another type,
+\spadtype{PositiveInteger}, in a signature.
+Any domain can be used in a signature.
+
+Since categories extend one another, they form hierarchies.
+Each category other than \spadtype{Type} has one or more parents given
+by the one or more categories mentioned before the \spad{with} part of
+the definition.
+\spadtype{SemiGroup} extends \spadtype{SetCategory} and
+\spadtype{SetCategory} extends both \spadtype{Type} and
+\spadtype{CoercibleTo (OutputForm)}.
+Since \spadtype{CoercibleTo (OutputForm)} also extends \spadtype{Type},
+the mention of \spadtype{Type} in the definition is unnecessary but
+included for emphasis.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugCategoriesMembershipTitle}{Membership}
+\newcommand{\ugCategoriesMembershipNumber}{12.5.}
+%
+% =====================================================================
+\begin{page}{ugCategoriesMembershipPage}{12.5. Membership}
+% =====================================================================
+\beginscroll
+
+We say a category designates a class of domains.
+What class of domains?
+%-% \HDindex{category!membership}{ugCategoriesMembershipPage}{12.5.}{Membership}
+That is, how does \Language{} know what domains belong to what categories?
+The simple answer to this basic question is key to the design of
+\Language{}:
+
+\beginImportant
+\centerline{{{\bf Domains belong to categories by assertion.}}}
+\endImportant
+
+When a domain is defined, it is asserted to belong to one or more
+categories.
+Suppose, for example, that an author of domain \spadtype{String} wishes to
+use the binary operator \spadop{*} to denote concatenation.
+Thus \spad{"hello " * "there"} would produce the string
+\spad{"hello there"}\footnote{Actually, concatenation of strings in
+\Language{} is done by juxtaposition or by using the operation
+\spadfunFrom{concat}{String}.
+The expression \spad{"hello " "there"} produces the string
+\spad{"hello there"}.}.
+The author of \spadtype{String} could then assert that \spadtype{String}
+is a member of \spadtype{SemiGroup}.
+According to our definition of \spadtype{SemiGroup}, strings
+would then also have the operation \spadop{**} defined automatically.
+Then \spad{"--" ** 4} would produce a string of eight dashes
+\spad{"--------"}.
+Since \spadtype{String} is a member of \spadtype{SemiGroup}, it also is
+a member of \spadtype{SetCategory} and thus has an operation
+\spadop{=} for testing that two strings are equal.
+
+\texht{Now turn to the algebraic category hierarchy inside the
+front cover of this book.}{}
+Any domain that is a member of a
+category extending \spadtype{SemiGroup} is a member of
+\spadtype{SemiGroup} (that is, it {\it is} a semigroup).
+In particular, any domain asserted to be a \spadtype{Ring} is a
+semigroup since \spadtype{Ring} extends \spadtype{Monoid}, that,
+in turn, extends \spadtype{SemiGroup}.
+The definition of \spadtype{Integer} in \Language{} asserts that
+\spadtype{Integer} is a member of category
+\spadtype{IntegerNumberSystem}, that, in turn, asserts that it is
+a member of \spadtype{EuclideanDomain}.
+Now \spadtype{EuclideanDomain} extends
+\spadtype{PrincipalIdealDomain} and so on.
+If you trace up the hierarchy, you see that
+\spadtype{EuclideanDomain} extends \spadtype{Ring}, and,
+therefore, \spadtype{SemiGroup}.
+Thus \spadtype{Integer} is a semigroup and also exports the
+operations \spadop{*} and \spadop{**}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugCategoriesDefaultsTitle}{Defaults}
+\newcommand{\ugCategoriesDefaultsNumber}{12.6.}
+%
+% =====================================================================
+\begin{page}{ugCategoriesDefaultsPage}{12.6. Defaults}
+% =====================================================================
+\beginscroll
+
+We actually omitted the last
+%-% \HDindex{category!defaults}{ugCategoriesDefaultsPage}{12.6.}{Defaults}
+part of the definition of
+%-% \HDindex{default definitions}{ugCategoriesDefaultsPage}{12.6.}{Defaults}
+\spadtype{SemiGroup} in
+\downlink{``\ugCategoriesHierTitle''}{ugCategoriesHierPage} in Section \ugCategoriesHierNumber\ignore{ugCategoriesHier}.
+Here now is its complete \Language{} definition.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ SemiGroup():\ Category\ ==\ SetCategory\ with}\newline
+{\tt 2.\ \ \ \ \ \ \ \ \ "*":\ (\$,\ \$)\ ->\ \$}\newline
+{\tt 3.\ \ \ \ \ \ \ \ \ "**":\ (\$,\ PositiveInteger)\ ->\ \$}\newline
+{\tt 4.\ \ \ \ \ \ \ add}\newline
+{\tt 5.\ \ \ \ \ \ \ \ \ import\ RepeatedSquaring(\$)}\newline
+{\tt 6.\ \ \ \ \ \ \ \ \ x:\ \$\ **\ n:\ PositiveInteger\ ==\ expt(x,n)}\newline
+\endImportant
+
+The \spad{add} part at the end is used to give ``default definitions'' for
+\spadkey{add}
+exported operations.
+Once you have a multiplication operation \spadop{*}, you can
+define exponentiation
+for positive integer exponents
+using repeated multiplication:
+\centerline{{\spad{x ** n = x * x * ... * x}   ( \spad{n} copies of \spad{x} )}}
+This definition for \spadop{**} is called a {\it default} definition.
+In general, a category can give default definitions for any
+operation it exports.
+Since \spadtype{SemiGroup} and all its category descendants in the hierarchy
+export \spadop{**}, any descendant category may redefine \spadop{**} as well.
+
+A domain of category \spadtype{SemiGroup}
+(such as \spadtype{Integer}) may or may not choose to
+define its own \spadop{**} operation.
+If it does not, a default definition that is closest (in a ``tree-distance''
+sense of the hierarchy) to the domain is chosen.
+
+The part of the category definition following an \spadop{add} operation
+is a \spadgloss{capsule}, as discussed in
+\texht{the previous chapter.}{\downlink{``\ugPackagesTitle''}{ugPackagesPage} in Chapter \ugPackagesNumber\ignore{ugPackages}.}
+The line
+\begin{verbatim}
+import RepeatedSquaring($)
+\end{verbatim}
+references the package
+\spadtype{RepeatedSquaring(\$)}, that is, the package
+\spadtype{RepeatedSquaring} that takes ``this domain'' as its
+parameter.
+For example, if the semigroup \spadtype{Polynomial (Integer)}
+does not define its own exponentiation operation, the
+definition used may come from the package
+\spadtype{RepeatedSquaring (Polynomial (Integer))}.
+The next line gives the definition in terms of \spadfun{expt} from that
+package.
+
+The default definitions are collected to form a ``default
+package'' for the category.
+The name of the package is the same as  the category but with an
+ampersand (\spadSyntax{\&}) added at the end.
+A default package always takes an additional argument relative to the
+category.
+Here is the definition of the default package \pspadtype{SemiGroup\&} as
+automatically generated by \Language{} from the above definition of
+\spadtype{SemiGroup}.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ SemiGroup\_\&(\$):\ Exports\ ==\ Implementation\ where}\newline
+{\tt 2.\ \ \ \ \ \$:\ SemiGroup}\newline
+{\tt 3.\ \ \ \ \ Exports\ ==\ with}\newline
+{\tt 4.\ \ \ \ \ \ \ "**":\ (\$,\ PositiveInteger)\ ->\ \$}\newline
+{\tt 5.\ \ \ \ \ Implementation\ ==\ add}\newline
+{\tt 6.\ \ \ \ \ \ \ import\ RepeatedSquaring(\$)}\newline
+{\tt 7.\ \ \ \ \ \ \ x:\$\ **\ n:PositiveInteger\ ==\ expt(x,n)}\newline
+\endImportant
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugCategoriesAxiomsTitle}{Axioms}
+\newcommand{\ugCategoriesAxiomsNumber}{12.7.}
+%
+% =====================================================================
+\begin{page}{ugCategoriesAxiomsPage}{12.7. Axioms}
+% =====================================================================
+\beginscroll
+
+In \texht{the previous section}{\downlink{``\ugCategoriesDefaultsTitle''}{ugCategoriesDefaultsPage} in Section \ugCategoriesDefaultsNumber\ignore{ugCategoriesDefaults}} you saw the
+complete \Language{} program defining \index{axiom}
+\spadtype{SemiGroup}.
+According to this definition, semigroups (that is, are sets with
+the operations \spadopFrom{*}{SemiGroup} and
+\spadopFrom{**}{SemiGroup}.
+%-% \HDexptypeindex{SemiGroup}{ugCategoriesAxiomsPage}{12.7.}{Axioms}
+
+You might ask: ``Aside from the notion of default packages, isn't
+a category just a \spadgloss{macro}, that is, a shorthand
+equivalent to the two operations \spadop{*} and \spadop{**} with
+their types?'' If a category were a macro, every time you saw the
+word \spadtype{SemiGroup}, you would rewrite it by its list of
+exported operations.
+Furthermore, every time you saw the exported operations of
+\spadtype{SemiGroup} among the exports of a constructor, you could
+conclude that the constructor exported \spadtype{SemiGroup}.
+
+A category is {\it not} a macro and here is why.
+The definition for \spadtype{SemiGroup} has documentation that states:
+
+\texht{\begin{quotation}}{\indent{3}}
+    Category \spadtype{SemiGroup} denotes the class of all multiplicative
+    semigroups, that is, a set with an associative operation \spadop{*}.
+
+    \texht{\vskip .5\baselineskip}{}
+    {Axioms:}
+
+    {\small\tt associative("*" : (\$,\$)->\$) -- (x*y)*z = x*(y*z)}
+\texht{\end{quotation}}{\indent{3}}
+
+According to the author's remarks, the mere
+exporting of an operation named \spadop{*} and \spadop{**} is not
+enough to qualify the domain as a \spadtype{SemiGroup}.
+In fact, a domain can be a semigroup only if it explicitly
+exports a \spadop{**} and
+a \spadop{*} satisfying the associativity axiom.
+
+In general, a category name implies a set of axioms, even mathematical
+theorems.
+There are numerous axioms from \spadtype{Ring}, for example,
+that are well-understood from the literature.
+No attempt is made to list them all.
+Nonetheless, all such mathematical facts are implicit by the use of the
+name \spadtype{Ring}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugCategoriesCorrectnessTitle}{Correctness}
+\newcommand{\ugCategoriesCorrectnessNumber}{12.8.}
+%
+% =====================================================================
+\begin{page}{ugCategoriesCorrectnessPage}{12.8. Correctness}
+% =====================================================================
+\beginscroll
+
+While such statements are only comments,
+%-% \HDindex{correctness}{ugCategoriesCorrectnessPage}{12.8.}{Correctness}
+\Language{} can enforce their intention simply by shifting the burden of
+responsibility onto the author of a domain.
+A domain belongs to category \spad{Ring} only if the
+author asserts that the domain  belongs to \spadtype{Ring} or
+to a category that extends \spadtype{Ring}.
+
+This principle of assertion is important for large user-extendable
+systems.
+\Language{} has a large library of operations offering facilities in
+many areas.
+Names such as \spadfun{norm} and \spadfun{product}, for example, have
+diverse meanings in diverse contexts.
+An inescapable hindrance to users would be to force those who wish to
+extend \Language{} to always invent new names for operations.
+%>> I don't think disambiguate is really a word, though I like it
+\Language{} allows you to reuse names, and then use context to disambiguate one
+from another.
+
+Here is another example of why this is important.
+Some languages, such as {\bf APL},
+%-% \HDindex{APL}{ugCategoriesCorrectnessPage}{12.8.}{Correctness}
+denote the \spadtype{Boolean} constants \spad{true} and
+\spad{false} by the integers \spad{1} and \spad{0}.
+You may want to let infix operators \spadop{+} and \spadop{*} serve as the logical
+operators \spadfun{or} and \spadfun{and}, respectively.
+But note this: \spadtype{Boolean} is not a ring.
+The {\it inverse axiom} for \spadtype{Ring} states:
+%
+\centerline{{Every element \spad{x} has an additive inverse \spad{y} such that}}
+\centerline{{\spad{x + y = 0}.}}
+%
+\spadtype{Boolean} is not a ring since \spad{true} has
+no inverse---there is no inverse element \spad{a} such that
+\spad{1 + a = 0} (in terms of booleans, \spad{(true or a) = false}).
+Nonetheless, \Language{} {\it could} easily and correctly implement
+\spadtype{Boolean} this way.
+\spadtype{Boolean} simply would not assert that it is of category
+\spadtype{Ring}.
+Thus the \spadop{+} for \spadtype{Boolean} values
+is not confused with the one for \spadtype{Ring}.
+Since the \spadtype{Polynomial} constructor requires its argument
+to be a ring, \Language{} would then refuse to build the
+domain \spadtype{Polynomial(Boolean)}. Also, \Language{} would refuse to
+wrongfully apply algorithms to \spadtype{Boolean} elements that  presume that the
+ring axioms for \spadop{+} hold.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugCategoriesAttributesTitle}{Attributes}
+\newcommand{\ugCategoriesAttributesNumber}{12.9.}
+%
+% =====================================================================
+\begin{page}{ugCategoriesAttributesPage}{12.9. Attributes}
+% =====================================================================
+\beginscroll
+
+Most axioms are not computationally useful.
+Those that are can be explicitly expressed by what \Language{} calls an
+\spadgloss{attribute}.
+The attribute \spadatt{commutative("*")}, for example, is used to assert
+that a domain has commutative multiplication.
+Its definition is given by its documentation:
+
+\texht{\begingroup \parindent=1pc \narrower\noindent}{\indent{3}}%
+    A domain \spad{R} has \spadatt{commutative("*")}
+    if it has an operation "*": \spadsig{(R,R)}{R} such that \spad{x * y = y * x}.
+\texht{\par\endgroup}{\indent{0}}
+
+Just as you can test whether a domain has the category \spadtype{Ring}, you
+can test that a domain has a given attribute.
+
+\xtc{
+Do polynomials over the integers
+have commutative multiplication?
+}{
+\spadpaste{Polynomial Integer has commutative("*")}
+}
+\xtc{
+Do matrices over the integers
+have commutative multiplication?
+}{
+\spadpaste{Matrix Integer has commutative("*")}
+}
+
+Attributes are used to conditionally export and define
+operations for a domain (see \downlink{``\ugDomainsAssertionsTitle''}{ugDomainsAssertionsPage} in Section \ugDomainsAssertionsNumber\ignore{ugDomainsAssertions}).
+Attributes can also be asserted in a category definition.
+
+After mentioning category \spadtype{Ring} many times in this book,
+it is high time that we show you its definition:
+%-% \HDexptypeindex{Ring}{ugCategoriesAttributesPage}{12.9.}{Attributes}
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ Ring():\ Category\ ==}\newline
+{\tt 2.\ \ \ \ \ Join(Rng,Monoid,LeftModule(\$:\ Rng))\ with}\newline
+{\tt 3.\ \ \ \ \ \ \ \ \ characteristic:\ ->\ NonNegativeInteger}\newline
+{\tt 4.\ \ \ \ \ \ \ \ \ coerce:\ Integer\ ->\ \$}\newline
+{\tt 5.\ \ \ \ \ \ \ \ \ unitsKnown}\newline
+{\tt 6.\ \ \ \ \ \ \ add}\newline
+{\tt 7.\ \ \ \ \ \ \ \ \ n:Integer}\newline
+{\tt 8.\ \ \ \ \ \ \ \ \ coerce(n)\ ==\ n\ *\ 1\$\$}\newline
+\endImportant
+
+There are only two new things here.
+First, look at the \spadSyntax{\$\$} on the last line.
+This is not a typographic error!
+The first \spadSyntax{\$} says that the \spad{1} is to come from some
+domain.
+The second \spadSyntax{\$} says that the domain is ``this domain.''
+If \spadSyntax{\$} is \spadtype{Fraction(Integer)}, this line reads {\tt
+coerce(n) == n * 1\$Fraction(Integer)}.
+
+The second new thing is the presence of attribute ``\spad{unitsKnown}''.
+\Language{} can always distinguish an attribute from an operation.
+An operation has a name and a type. An attribute has no type.
+The attribute \spadatt{unitsKnown} asserts a rather subtle mathematical
+fact that is normally taken for granted when working with
+rings.\footnote{With this axiom, the units of a domain are the set of
+elements \spad{x} that each have a multiplicative
+inverse \spad{y} in the domain.
+Thus \spad{1} and \spad{-1} are units in domain \spadtype{Integer}.
+Also, for \spadtype{Fraction Integer}, the domain of rational numbers,
+all non-zero elements are units.}
+Because programs can test for this attribute, \Language{} can
+correctly handle rather more complicated mathematical structures (ones
+that are similar to rings but do not have this attribute).
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugCategoriesParametersTitle}{Parameters}
+\newcommand{\ugCategoriesParametersNumber}{12.10.}
+%
+% =====================================================================
+\begin{page}{ugCategoriesParametersPage}{12.10. Parameters}
+% =====================================================================
+\beginscroll
+
+Like domain constructors, category constructors can also have
+parameters.
+For example, category \spadtype{MatrixCategory} is a parameterized
+category for defining matrices over a ring \spad{R} so that the
+matrix domains can have
+different representations and indexing schemes.
+Its definition has the form:
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ MatrixCategory(R,Row,Col):\ Category\ ==}\newline
+{\tt 2.\ \ \ \ \ \ \ TwoDimensionalArrayCategory(R,Row,Col)\ with\ ...}\newline
+\endImportant
+
+The category extends \spadtype{TwoDimensionalArrayCategory} with
+the same arguments.
+You cannot find \spadtype{TwoDimensionalArrayCategory} in the
+algebraic hierarchy listing.
+Rather, it is a member of the data structure hierarchy,
+given inside the back cover of this book.
+In particular, \spadtype{TwoDimensionalArrayCategory} is an extension of
+\spadtype{HomogeneousAggregate} since its elements are all one type.
+
+The domain \spadtype{Matrix(R)}, the class of matrices with coefficients
+from domain \spad{R}, asserts that it is a member of category
+\spadtype{MatrixCategory(R, Vector(R), Vector(R))}.
+The parameters of a category must also have types.
+The first parameter to \spadtype{MatrixCategory}
+\spad{R} is required to be a ring.
+The second and third are required to be domains of category
+\spadtype{FiniteLinearAggregate(R)}.\footnote{%
+This is another extension of
+\spadtype{HomogeneousAggregate} that you can see in
+the data structure hierarchy.}
+In practice, examples of categories having parameters other than
+domains are rare.
+
+Adding the declarations for parameters to the definition for
+\spadtype{MatrixCategory}, we have:
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ R:\ Ring}\newline
+{\tt 2.\ \ \ (Row,\ Col):\ FiniteLinearAggregate(R)}\newline
+{\tt 3.\ \ \ }\newline
+{\tt 4.\ \ \ MatrixCategory(R,\ Row,\ Col):\ Category\ ==}\newline
+{\tt 5.\ \ \ \ \ \ \ TwoDimensionalArrayCategory(R,\ Row,\ Col)\ with\ ...}\newline
+\endImportant
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugCategoriesConditionalsTitle}{Conditionals}
+\newcommand{\ugCategoriesConditionalsNumber}{12.11.}
+%
+% =====================================================================
+\begin{page}{ugCategoriesConditionalsPage}{12.11. Conditionals}
+% =====================================================================
+\beginscroll
+
+As categories have parameters, the actual operations exported by a
+%-% \HDindex{conditional}{ugCategoriesConditionalsPage}{12.11.}{Conditionals}
+category can depend on these parameters.
+As an example, the operation \spadfunFrom{determinant}{MatrixCategory}
+from category \spadtype{MatrixCategory} is only exported when the
+underlying domain \spad{R} has commutative multiplication:
+
+\begin{verbatim}
+if R has commutative("*") then
+   determinant: $ -> R
+\end{verbatim}
+
+Conditionals can also define conditional extensions of a category.
+Here is a portion of the definition of \spadtype{QuotientFieldCategory}:
+%-% \HDexptypeindex{QuotientFieldCategory}{ugCategoriesConditionalsPage}{12.11.}{Conditionals}
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ QuotientFieldCategory(R)\ :\ Category\ ==\ ...\ with\ ...}\newline
+{\tt 2.\ \ \ \ \ \ \ \ if\ R\ has\ OrderedSet\ then\ OrderedSet}\newline
+{\tt 3.\ \ \ \ \ \ \ \ if\ R\ has\ IntegerNumberSystem\ then}\newline
+{\tt 4.\ \ \ \ \ \ \ \ \ \ ceiling:\ \$\ ->\ R}\newline
+{\tt 5.\ \ \ \ \ \ \ \ \ \ \ \ ...}\newline
+\endImportant
+
+Think of category \spadtype{QuotientFieldCategory(R)} as
+denoting the domain \spadtype{Fraction(R)}, the
+class of all fractions of the form \smath{a/b} for elements of \spad{R}.
+The first conditional means in English:
+``If the elements of \spad{R} are totally ordered (\spad{R}
+is an \spadtype{OrderedSet}), then so are the fractions \smath{a/b}''.
+%-% \HDexptypeindex{Fraction}{ugCategoriesConditionalsPage}{12.11.}{Conditionals}
+
+The second conditional is used to conditionally export an
+operation \spadfun{ceiling} which returns the smallest integer
+greater than or equal to its argument.
+Clearly, ``ceiling'' makes sense for integers but not for
+polynomials and other algebraic structures.
+Because of this conditional,
+the domain \spadtype{Fraction(Integer)} exports
+an operation
+\spadfun{ceiling}: \spadsig{Fraction Integer}{Integer}, but
+\spadtype{Fraction Polynomial Integer} does not.
+
+Conditionals can also appear in the default definitions for the
+operations of a category.
+For example, a default definition for \spadfunFrom{ceiling}{Field}
+within the part following the \spadop{add} reads:
+
+\begin{verbatim}
+if R has IntegerNumberSystem then
+    ceiling x == ...
+\end{verbatim}
+
+Here the predicate used is identical to the predicate
+in the {\tt Exports} part.
+This need not be the case.
+See \downlink{``\ugPackagesCondsTitle''}{ugPackagesCondsPage} in Section \ugPackagesCondsNumber\ignore{ugPackagesConds} for a more complicated example.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugCategoriesAndPackagesTitle}{Anonymous Categories}
+\newcommand{\ugCategoriesAndPackagesNumber}{12.12.}
+%
+% =====================================================================
+\begin{page}{ugCategoriesAndPackagesPage}{12.12. Anonymous Categories}
+% =====================================================================
+\beginscroll
+
+The part of a category to the right of a {\tt with} is also
+regarded as a category---an ``anonymous category.''
+Thus you have already seen a   category definition
+%-% \HDindex{category!anonymous}  in \chapref{ugPackages}.{ugCategoriesAndPackagesPage}{12.12.}{Anonymous Categories}
+The {\tt Exports} part of the package \pspadtype{DrawComplex}
+(\downlink{``\ugPackagesAbstractTitle''}{ugPackagesAbstractPage} in Section \ugPackagesAbstractNumber\ignore{ugPackagesAbstract}) is an anonymous category.
+This is not necessary.
+We could, instead, give this category a name:
+
+%
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ DrawComplexCategory():\ Category\ ==\ with}\newline
+{\tt 2.\ \ \ \ \ \ drawComplex:\ (C\ ->\ C,S,S,Boolean)\ ->\ VIEW3D}\newline
+{\tt 3.\ \ \ \ \ \ drawComplexVectorField:\ (C\ ->\ C,S,S)\ ->\ VIEW3D}\newline
+{\tt 4.\ \ \ \ \ \ setRealSteps:\ INT\ ->\ INT}\newline
+{\tt 5.\ \ \ \ \ \ setImagSteps:\ INT\ ->\ INT}\newline
+{\tt 6.\ \ \ \ \ \ setClipValue:\ DFLOAT->\ DFLOAT}\newline
+\endImportant
+%
+and then define \spadtype{DrawComplex} by:
+%
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ DrawComplex():\ DrawComplexCategory\ ==\ Implementation}\newline
+{\tt 2.\ \ \ \ \ \ where}\newline
+{\tt 3.\ \ \ \ \ \ \ \ \ ...}\newline
+\endImportant
+%
+
+There is no reason, however, to give this list of exports a name
+since no other domain or package exports it.
+In fact, it is rare for a package to export a named category.
+As you will see in the next chapter, however, it is very common
+for the definition of domains to mention one or more category
+before the {\tt with}.
+\spadkey{with}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug12.pht b/src/hyper/pages/ug12.pht
new file mode 100644
index 00000000..118e4a50
--- /dev/null
+++ b/src/hyper/pages/ug12.pht
@@ -0,0 +1,32 @@
+\begin{patch}{ugCategoriesAttributesPagePatch1}
+\begin{paste}{ugCategoriesAttributesPageFull1}{ugCategoriesAttributesPageEmpty1}
+\pastebutton{ugCategoriesAttributesPageFull1}{\hidepaste}
+\tab{5}\spadcommand{Polynomial Integer has commutative("*")}
+\indentrel{3}\begin{verbatim}
+   (1)  true
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugCategoriesAttributesPageEmpty1}
+\begin{paste}{ugCategoriesAttributesPageEmpty1}{ugCategoriesAttributesPagePatch1}
+\pastebutton{ugCategoriesAttributesPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{Polynomial Integer has commutative("*")}
+\end{paste}\end{patch}
+
+\begin{patch}{ugCategoriesAttributesPagePatch2}
+\begin{paste}{ugCategoriesAttributesPageFull2}{ugCategoriesAttributesPageEmpty2}
+\pastebutton{ugCategoriesAttributesPageFull2}{\hidepaste}
+\tab{5}\spadcommand{Matrix Integer has commutative("*")}
+\indentrel{3}\begin{verbatim}
+   (2)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugCategoriesAttributesPageEmpty2}
+\begin{paste}{ugCategoriesAttributesPageEmpty2}{ugCategoriesAttributesPagePatch2}
+\pastebutton{ugCategoriesAttributesPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{Matrix Integer has commutative("*")}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ug13.ht b/src/hyper/pages/ug13.ht
new file mode 100644
index 00000000..2620be54
--- /dev/null
+++ b/src/hyper/pages/ug13.ht
@@ -0,0 +1,1463 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\texht{\setcounter{chapter}{12}}{} % Chapter 13
+
+
+%
+\newcommand{\ugDomainsTitle}{Domains}
+\newcommand{\ugDomainsNumber}{13.}
+%
+% =====================================================================
+\begin{page}{ugDomainsPage}{13. Domains}
+% =====================================================================
+\beginscroll
+
+We finally come to the \spadgloss{domain constructor}.
+A few subtle differences between packages and
+domains turn up some interesting issues.
+We first discuss these differences then
+describe the resulting issues by illustrating a program
+for the \axiomType{QuadraticForm} constructor.
+After a short example of an algebraic constructor,
+\axiomType{CliffordAlgebra}, we show how you use domain constructors to build
+a database query facility.
+
+\beginmenu
+    \menudownlink{{13.1. Domains vs. Packages}}{ugPackagesDomsPage}
+    \menudownlink{{13.2. Definitions}}{ugDomainsDefsPage}
+    \menudownlink{{13.3. Category Assertions}}{ugDomainsAssertionsPage}
+    \menudownlink{{13.4. A Demo}}{ugDomainsDemoPage}
+    \menudownlink{{13.5. Browse}}{ugDomainsBrowsePage}
+    \menudownlink{{13.6. Representation}}{ugDomainsRepPage}
+    \menudownlink{{13.7. Multiple Representations}}{ugDomainsMultipleRepsPage}
+    \menudownlink{{13.8. Add Domain}}{ugDomainsAddDomainPage}
+    \menudownlink{{13.9. Defaults}}{ugDomainsDefaultsPage}
+    \menudownlink{{13.10. Origins}}{ugDomainsOriginsPage}
+    \menudownlink{{13.11. Short Forms}}{ugDomainsShortFormsPage}
+    \menudownlink{{13.12. Example 1: Clifford Algebra}}{ugDomainsCliffordPage}
+    \menudownlink{{13.13. Example 2: Building A Query Facility}}{ugDomsinsDatabasePage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugPackagesDomsTitle}{Domains vs. Packages}
+\newcommand{\ugPackagesDomsNumber}{13.1.}
+%
+% =====================================================================
+\begin{page}{ugPackagesDomsPage}{13.1. Domains vs. Packages}
+% =====================================================================
+\beginscroll
+%
+Packages are special cases of domains.
+What is the difference between a package and a domain that is not a
+package?
+By definition, there is only one difference: a domain that is not a package
+has the symbol \axiomSyntax{\$} appearing
+somewhere among the types of its exported operations.
+The \axiomSyntax{\$} denotes ``this domain.'' If the \axiomSyntax{\$}
+appears before the \axiomSyntax{->} in the type of a signature, it means
+the operation takes an element from the domain as an argument.
+If it appears after the \axiomSyntax{->}, then the operation returns an
+element of the domain.
+
+If no exported operations mention \axiomSyntax{\$}, then
+evidently there is nothing of interest to do with the objects of the
+domain.
+You might then say that a package is a ``boring'' domain!
+But, as you saw in \downlink{``\ugPackagesTitle''}{ugPackagesPage} in Chapter \ugPackagesNumber\ignore{ugPackages}, packages are a very useful
+notion indeed.
+The exported operations of a package depend solely on the parameters
+to the package constructor and other explicit domains.
+
+To summarize, domain constructors are versatile structures that serve two
+distinct practical purposes:
+Those like \axiomType{Polynomial} and \axiomType{List}
+describe classes of computational objects;
+others, like \pspadtype{SortPackage}, describe packages of useful
+operations.
+As in the last chapter, we focus here on the first kind.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsDefsTitle}{Definitions}
+\newcommand{\ugDomainsDefsNumber}{13.2.}
+%
+% =====================================================================
+\begin{page}{ugDomainsDefsPage}{13.2. Definitions}
+% =====================================================================
+\beginscroll
+%
+
+The syntax for defining a domain constructor is the same as for any
+function in \Language{}:
+\centerline{{\frenchspacing{\tt {\it DomainForm} : {\it Exports} == {\it Implementation}}}}
+As this definition usually extends over many lines, a
+\axiom{where} expression is generally used instead.
+\spadkey{where}
+
+\beginImportant
+A recommended format for the definition of a domain is:\newline
+{\tt%
+{\it DomainForm} : Exports  ==  Implementation where \newline
+\texht{\hspace*{.75pc}}{\tab{8}} {\it optional type declarations} \newline
+\texht{\hspace*{.75pc}}{\tab{3}} Exports  ==  [{\it Category Assertions}] with \newline
+\texht{\hspace*{2.0pc}}{\tab{8}}   {\it list of exported operations} \newline
+\texht{\hspace*{.75pc}}{\tab{3}} Implementation  ==  [{\it Add Domain}] add \newline
+\texht{\hspace*{2.0pc}}{\tab{6}}   [Rep := {\it Representation}] \newline
+\texht{\hspace*{2.0pc}}{\tab{8}}   {\it list of function definitions for exported operations}
+}
+
+\texht{\vskip 4pt}{}
+Note: The brackets {\tt [ ]} here denote optionality.
+\endImportant
+
+A complete domain constructor definition for
+\axiomType{QuadraticForm} is shown in Figure \ref{fig-quadform}.
+Interestingly, this little domain illustrates all the new
+concepts you need to learn.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ )abbrev\ domain\ QFORM\ QuadraticForm}\newline
+{\tt 2.\ \ \ }\newline
+{\tt 3.\ \ \ ++\ Description:}\newline
+{\tt 4.\ \ \ ++\ \ \ This\ domain\ provides\ modest\ support\ for}\newline
+{\tt 5.\ \ \ ++\ \ \ quadratic\ forms.}\newline
+{\tt 6.\ \ \ QuadraticForm(n,\ K):\ Exports\ ==\ Implementation\ where}\newline
+{\tt 7.\ \ \ \ \ \ \ n:\ PositiveInteger}\newline
+{\tt 8.\ \ \ \ \ \ \ K:\ Field}\newline
+{\tt 9.\ \ \ }\newline
+{\tt 10.\ \ \ \ \ \ Exports\ ==\ AbelianGroup\ with}\newline
+{\tt 11.\ \ \ \ \ \ \ \ quadraticForm:\ SquareMatrix(n,K)\ ->\ \$}\newline
+{\tt 12.\ \ \ \ \ \ \ \ \ \ ++\ \bs{}axiom\{quadraticForm(m)\}\ creates\ a\ quadratic}\newline
+{\tt 13.\ \ \ \ \ \ \ \ \ \ ++\ quadratic\ form\ from\ a\ symmetric,}\newline
+{\tt 14.\ \ \ \ \ \ \ \ \ \ ++\ square\ matrix\ \bs{}axiom\{m\}.}\newline
+{\tt 15.\ \ \ \ \ \ \ \ matrix:\ \$\ ->\ SquareMatrix(n,K)}\newline
+{\tt 16.\ \ \ \ \ \ \ \ \ \ ++\ \bs{}axiom\{matrix(qf)\}\ creates\ a\ square\ matrix}\newline
+{\tt 17.\ \ \ \ \ \ \ \ \ \ ++\ from\ the\ quadratic\ form\ \bs{}axiom\{qf\}.}\newline
+{\tt 18.\ \ \ \ \ \ \ \ elt:\ (\$,\ DirectProduct(n,K))\ ->\ K}\newline
+{\tt 19.\ \ \ \ \ \ \ \ \ \ ++\ \bs{}axiom\{qf(v)\}\ evaluates\ the\ quadratic\ form}\newline
+{\tt 20.\ \ \ \ \ \ \ \ \ \ ++\ \bs{}axiom\{qf\}\ on\ the\ vector\ \bs{}axiom\{v\},}\newline
+{\tt 21.\ \ \ \ \ \ \ \ \ \ ++\ producing\ a\ scalar.}\newline
+{\tt 22.\ \ }\newline
+{\tt 23.\ \ \ \ \ \ Implementation\ ==\ SquareMatrix(n,K)\ add}\newline
+{\tt 24.\ \ \ \ \ \ \ \ Rep\ :=\ SquareMatrix(n,K)}\newline
+{\tt 25.\ \ \ \ \ \ \ \ quadraticForm\ m\ ==}\newline
+{\tt 26.\ \ \ \ \ \ \ \ \ \ not\ symmetric?\ m\ =>\ error}\newline
+{\tt 27.\ \ \ \ \ \ \ \ \ \ \ \ "quadraticForm\ requires\ a\ symmetric\ matrix"}\newline
+{\tt 28.\ \ \ \ \ \ \ \ \ \ m\ ::\ \$}\newline
+{\tt 29.\ \ \ \ \ \ \ \ matrix\ q\ ==\ q\ ::\ Rep}\newline
+{\tt 30.\ \ \ \ \ \ \ \ elt(q,v)\ ==\ dot(v,\ (matrix\ q\ *\ v))}\newline
+%
+\caption{The \protect\axiomType{QuadraticForm} domain.}\label{fig-quadform}
+\endImportant
+
+A domain constructor can take any number and type of parameters.
+\axiomType{QuadraticForm} takes a positive integer \axiom{n} and a field
+\axiom{K} as arguments.
+Like a package, a domain has a set of explicit exports and an
+implementation described by a capsule.
+Domain constructors are documented in the same way as package constructors.
+
+Domain \axiomType{QuadraticForm(n, K)}, for a given positive integer
+\axiom{n} and domain \axiom{K}, explicitly exports three operations:
+%
+\indent{4}
+\beginitems
+\item\axiom{quadraticForm(A)} creates a quadratic form from a matrix
+\axiom{A}.
+\item\axiom{matrix(q)} returns the matrix \axiom{A} used to create
+the quadratic form \axiom{q}.
+\item\axiom{q.v} computes the scalar \texht{$v^TAv$}{transpose(v)*A*v}
+for a given vector \axiom{v}.
+\enditems
+\indent{0}
+
+Compared with the corresponding syntax given for the definition of a
+package, you see that a domain constructor has three optional parts to
+its definition: {\it Category Assertions}, {\it Add Domain}, and
+{\it Representation}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsAssertionsTitle}{Category Assertions}
+\newcommand{\ugDomainsAssertionsNumber}{13.3.}
+%
+% =====================================================================
+\begin{page}{ugDomainsAssertionsPage}{13.3. Category Assertions}
+% =====================================================================
+\beginscroll
+%
+
+The {\it Category Assertions} part of your domain constructor
+definition lists those categories of which all domains created by
+the constructor are unconditionally members.
+The word ``unconditionally'' means that membership in a category
+does not depend on the values of the parameters to the domain
+constructor.
+This part thus defines the link between the domains and the
+category hierarchies given on the inside covers of this book.
+As described in \downlink{``\ugCategoriesCorrectnessTitle''}{ugCategoriesCorrectnessPage} in Section \ugCategoriesCorrectnessNumber\ignore{ugCategoriesCorrectness}, it is this link
+that makes it possible for you to pass objects of the domains as
+arguments to other operations in \Language{}.
+
+Every \axiomType{QuadraticForm} domain is declared
+to be unconditionally a member of category \axiomType{AbelianGroup}.
+An abelian group is a collection of elements closed under
+addition.
+Every object {\it x} of an abelian group has an additive inverse
+{\it y} such that \texht{$x + y = 0$}{\axiom{{\it x} + {\it y} = 0}}.
+The exports of an abelian group include \axiom{0},
+\axiomOp{+}, \axiomOp{-}, and scalar multiplication by an integer.
+After asserting that \axiomType{QuadraticForm} domains are abelian
+groups, it is possible to pass quadratic forms to algorithms that
+only assume arguments to have these abelian group
+properties.
+
+In \downlink{``\ugCategoriesConditionalsTitle''}{ugCategoriesConditionalsPage} in Section \ugCategoriesConditionalsNumber\ignore{ugCategoriesConditionals}, you saw that
+\axiomType{Fraction(R)}, a member of
+\axiomType{QuotientFieldCategory(R)},
+is a member of \axiomType{OrderedSet} if \axiom{R}
+is a member of \axiomType{OrderedSet}.
+Likewise, from the {\tt Exports} part of the definition of
+\axiomType{ModMonic(R, S)},
+\begin{verbatim}
+UnivariatePolynomialCategory(R) with
+  if R has Finite then Finite
+     ...
+\end{verbatim}
+you see that \axiomType{ModMonic(R, S)} is a member of
+\axiomType{Finite} is \axiom{R} is.
+
+The {\tt Exports} part of a domain definition is
+the same kind of
+expression that can appear to the right of an
+\axiomSyntax{==} in a category definition.
+If a domain constructor is unconditionally a member of two or more
+categories, a \axiom{Join} form is used.
+\spadkey{Join}
+The {\tt Exports} part of the definition of
+\axiomType{FlexibleArray(S)} reads, for example:
+\begin{verbatim}
+Join(ExtensibleLinearAggregate(S),
+     OneDimensionalArrayAggregate(S)) with...
+\end{verbatim}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsDemoTitle}{A Demo}
+\newcommand{\ugDomainsDemoNumber}{13.4.}
+%
+% =====================================================================
+\begin{page}{ugDomainsDemoPage}{13.4. A Demo}
+% =====================================================================
+\beginscroll
+%
+Before looking at the {\it Implementation} part of \axiomType{QuadraticForm},
+let's try some examples.
+
+\texht{\vskip 2pc}{}
+\xtc{
+Build a domain \axiom{QF}.
+}{
+\spadpaste{QF := QuadraticForm(2,Fraction Integer)\bound{x2}\free{x1}}
+}
+\xtc{
+Define a matrix to be used to construct
+a quadratic form.
+}{
+\spadpaste{A := matrix [[-1,1/2],[1/2,1]]\bound{x3}\free{x2}}
+}
+\xtc{
+Construct the quadratic form.
+A package call {\tt \$QF} is necessary since there
+are other \axiomType{QuadraticForm} domains.
+}{
+\spadpaste{q : QF := quadraticForm(A)\bound{x4}\free{x3}}
+}
+\xtc{
+Looks like a matrix. Try computing
+the number of rows.
+\Language{} won't let you.
+}{
+\spadpaste{nrows q\free{x3}}
+}
+\xtc{
+Create a direct product element \axiom{v}.
+A package call is again necessary, but \Language{}
+understands your list as denoting a vector.
+}{
+\spadpaste{v := directProduct([2,-1])\$DirectProduct(2,Fraction Integer)\bound{x5}\free{x4}}
+}
+\xtc{
+Compute the product \texht{$v^TAv$}{transpose(v)*A*v}.
+}{
+\spadpaste{q.v\free{x5}}
+}
+\xtc{
+What is 3 times \axiom{q} minus \axiom{q} plus \axiom{q}?
+}{
+\spadpaste{3*q-q+q\free{x4}}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsBrowseTitle}{Browse}
+\newcommand{\ugDomainsBrowseNumber}{13.5.}
+%
+% =====================================================================
+\begin{page}{ugDomainsBrowsePage}{13.5. Browse}
+% =====================================================================
+\beginscroll
+
+The \Browse{} facility of \HyperName{} is useful for
+investigating
+the properties of domains, packages, and categories.
+From the main \HyperName{} menu, move your mouse to {\bf Browse} and
+click on the left mouse button.
+This brings up the \Browse{} first page.
+Now, with your mouse pointer somewhere in this window, enter the
+string ``quadraticform'' into the input area (all lower case
+letters will do).
+Move your mouse to {\bf Constructors} and click.
+Up comes a page describing \axiomType{QuadraticForm}.
+
+From here, click on {\bf Description}.
+This gives you a page that includes a part labeled by ``{\it
+Description:}''.
+You also see the types for arguments \axiom{n} and \axiom{K}
+displayed as well as the fact that \axiomType{QuadraticForm}
+returns an \axiomType{AbelianGroup}.
+You can go and experiment a bit by selecting {\bf Field} with
+your mouse.
+Eventually, use
+\UpButton{}
+several times to return to the first page on
+\axiomType{QuadraticForm}.
+
+Select {\bf Operations} to get a list of operations for
+\axiomType{QuadraticForm}.
+You can select an operation by clicking on it
+to get an individual page with information about that operation.
+Or you can select the buttons along the bottom to see alternative
+views or get additional information on the operations.
+Then return to the page on \axiomType{QuadraticForm}.
+
+Select {\bf Cross Reference} to get another menu.
+This menu has buttons for {\bf Parents}, {\bf Ancestors}, and
+others.
+Clicking on {\bf Parents}, you see that \axiomType{QuadraticForm}
+has one parent \axiomType{AbelianMonoid}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsRepTitle}{Representation}
+\newcommand{\ugDomainsRepNumber}{13.6.}
+%
+% =====================================================================
+\begin{page}{ugDomainsRepPage}{13.6. Representation}
+% =====================================================================
+\beginscroll
+%
+The {\tt Implementation} part of an \Language{} capsule for a
+domain constructor uses the special variable \axiom{Rep} to
+%-% \HDindex{Rep @ {\tt Rep}}{ugDomainsRepPage}{13.6.}{Representation}
+identify the lower level data type used to represent the objects
+%-% \HDindex{representation!of a domain}{ugDomainsRepPage}{13.6.}{Representation}
+of the domain.
+%-% \HDindex{domain!representation}{ugDomainsRepPage}{13.6.}{Representation}
+The \axiom{Rep} for quadratic forms is \axiomType{SquareMatrix(n, K)}.
+This means that all objects of the domain are required to be
+\axiom{n} by \axiom{n} matrices with elements from \axiomType{K}.
+
+The code for \axiomFun{quadraticForm} in Figure \ref{fig-quadform}
+on page \pageref{fig-quadform}
+checks that the matrix is symmetric and then converts it to
+\axiomSyntax{\$}, which means, as usual, ``this domain.'' Such explicit
+conversions \index{conversion} are generally required by the
+compiler.
+Aside from checking that the matrix is symmetric, the code for
+this function essentially does nothing.
+The {\frenchspacing\tt m :: \$} on line 28 coerces \axiom{m} to a
+quadratic form.
+In fact, the quadratic form you created in step (3) of
+\downlink{``\ugDomainsDemoTitle''}{ugDomainsDemoPage} in Section \ugDomainsDemoNumber\ignore{ugDomainsDemo} is just the matrix you passed it in
+disguise!
+Without seeing this definition, you would not know that.
+Nor can you take advantage of this fact now that you do know!
+When we try in the next step of \downlink{``\ugDomainsDemoTitle''}{ugDomainsDemoPage} in Section \ugDomainsDemoNumber\ignore{ugDomainsDemo} to regard
+\axiom{q} as a matrix by asking for \axiomFun{nrows}, the number of
+its rows, \Language{} gives you an error message saying, in
+effect, ``Good try, but this won't work!''
+
+The definition for the \spadfunFrom{matrix}{QuadraticForm}
+function could hardly be simpler:
+it just returns its argument after explicitly
+\spadglossSee{coercing}{coerce} its argument to a matrix.
+Since the argument is already a matrix, this coercion does no computation.
+
+Within the context of a capsule, an object of \axiomSyntax{\$} is
+regarded both as a quadratic form {\it and} as a
+matrix.\footnote{In case each of \axiomSyntax{\$} and \axiom{Rep}
+have the same named operation available,
+the one from \axiom{\$} takes precedence.
+Thus, if you want the one from \axiomSyntax{Rep}, you must
+package call it using a \axiomSyntax{\$Rep} suffix.}
+This makes the definition of \axiom{q.v} easy---it
+just calls the \spadfunFrom{dot}{DirectProduct} product from
+\axiomType{DirectProduct} to perform the indicated operation.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsMultipleRepsTitle}{Multiple Representations}
+\newcommand{\ugDomainsMultipleRepsNumber}{13.7.}
+%
+% =====================================================================
+\begin{page}{ugDomainsMultipleRepsPage}{13.7. Multiple Representations}
+% =====================================================================
+\beginscroll
+%
+
+To write functions that implement the operations of a domain, you
+want to choose the most computationally efficient
+data structure to represent the elements of your domain.
+
+A classic problem in computer algebra is the optimal choice for an
+internal representation of polynomials.
+If you create a polynomial, say \texht{$3x^2+ 5$}{\axiom{3*x**2 + 5}}, how
+does \Language{} hold this value internally?
+There are many ways.
+\Language{} has nearly a dozen different representations of
+polynomials, one to suit almost any purpose.
+Algorithms for solving polynomial equations work most
+efficiently with polynomials represented one way, whereas those for
+factoring polynomials are most efficient using another.
+One often-used representation is  a list of terms, each term
+consisting of exponent-coefficient records written in the order
+of decreasing exponents.
+For example, the polynomial \texht{$3x^2+5$}{3*x**2+5} is
+%>> I changed the k's in next line to e's as I thought that was
+%>> clearer.
+represented by the list \axiom{[[e:2, c:3], [e:0, c:5]]}.
+
+What is the optimal data structure for a matrix?
+It depends on the application.
+For large sparse matrices, a linked-list structure of records
+holding only the non-zero elements may be optimal.
+If the elements can be defined by a simple formula
+\texht{$f(i,j)$}{\axiom{f(i,j)}}, then a compiled function for
+\axiom{f} may be optimal.
+Some programmers prefer to represent ordinary matrices as vectors
+of vectors.
+Others prefer to represent matrices by one big linear array where
+elements are accessed with linearly computable indexes.
+
+While all these simultaneous structures tend to be confusing,
+\Language{} provides a helpful organizational tool for such a purpose:
+categories.
+\axiomType{PolynomialCategory}, for example, provides a uniform user
+interface across all polynomial types.
+Each kind of polynomial implements functions for
+all these operations, each in its own way.
+If you use only the top-level operations in
+\axiomType{PolynomialCategory} you usually do not care what kind
+of polynomial implementation is used.
+
+%>> I've often thought, though, that it would be nice to be
+%>> be able to use conditionals for representations.
+Within a given domain, however, you define (at most) one
+representation.\footnote{You can make that representation a
+\pspadtype{Union} type, however.
+See \downlink{``\ugTypesUnionsTitle''}{ugTypesUnionsPage} in Section \ugTypesUnionsNumber\ignore{ugTypesUnions} for examples of unions.}
+If you want to have multiple representations (that is, several
+domains, each with its own representation), use a category to
+describe the {\tt Exports}, then define separate domains for each
+representation.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsAddDomainTitle}{Add Domain}
+\newcommand{\ugDomainsAddDomainNumber}{13.8.}
+%
+% =====================================================================
+\begin{page}{ugDomainsAddDomainPage}{13.8. Add Domain}
+% =====================================================================
+\beginscroll
+%
+
+The capsule part of {\tt Implementation} defines functions that
+implement the operations exported by the domain---usually only
+some of the operations.
+In our demo in \downlink{``\ugDomainsDemoTitle''}{ugDomainsDemoPage} in Section \ugDomainsDemoNumber\ignore{ugDomainsDemo}, we asked for the value of
+\axiom{3*q-q+q}.
+Where do the operations \axiomOp{*}, \axiomOp{+}, and
+\axiomOp{-} come from?
+There is no definition for them in the capsule!
+
+The {\tt Implementation} part of a definition can
+%-% \HDindex{domain!add}{ugDomainsAddDomainPage}{13.8.}{Add Domain}
+optionally specify an ``add-domain'' to the left of an {\tt add}
+\spadkey{add}
+(for \axiomType{QuadraticForm}, defines
+\axiomType{SquareMatrix(n,K)} is the add-domain).
+The meaning of an add-domain is simply this: if the capsule part
+of the {\tt Implementation} does not supply a function for an
+operation, \Language{} goes to the add-domain to find the
+function.
+So do \axiomOpFrom{*}{QuadraticForm}, \axiomOpFrom{+}{QuadraticForm}
+and \axiomOpFrom{-}{QuadraticForm} come from
+\axiomType{SquareMatrix(n,K)}?
+%Read on!
+
+%*********************************************************************
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsDefaultsTitle}{Defaults}
+\newcommand{\ugDomainsDefaultsNumber}{13.9.}
+%
+% =====================================================================
+\begin{page}{ugDomainsDefaultsPage}{13.9. Defaults}
+% =====================================================================
+\beginscroll
+%
+In \downlink{``\ugPackagesTitle''}{ugPackagesPage} in Chapter \ugPackagesNumber\ignore{ugPackages}, we saw that categories can provide
+default implementations for their operations.
+How and when are they used?
+When \Language{} finds that \axiomType{QuadraticForm(2, Fraction
+Integer)} does not implement the operations \axiomOp{*},
+\axiomOp{+}, and \axiomOp{-}, it goes to
+\axiomType{SquareMatrix(2,Fraction Integer)} to find it.
+As it turns out, \axiomType{SquareMatrix(2, Fraction Integer)} does
+not implement {\it any} of these operations!
+
+What does \Language{} do then?
+Here is its overall strategy.
+First, \Language{} looks for a function in the capsule for the domain.
+If it is not there, \Language{} looks in the add-domain for the
+operation.
+If that fails, \Language{} searches the add-domain of the add-domain,
+and so on.
+If all those fail, it then searches the default packages for the
+categories of which the domain is a member.
+In the case of \axiomType{QuadraticForm}, it searches
+\axiomType{AbelianGroup}, then its parents, grandparents, and
+so on.
+If this fails, it then searches the default packages of the
+add-domain.
+Whenever a function is found, the search stops immediately and the
+function is returned.
+When all fails, the system calls \axiomFun{error} to report this
+unfortunate news to you.
+To find out the actual order of constructors searched for
+\axiomType{QuadraticForm}, consult \Browse{}: from the
+\axiomType{QuadraticForm}, click on {\bf Cross Reference}, then on
+{\bf Lineage}.
+
+Let's apply this search strategy for our example \axiom{3*q-q+q}.
+The scalar multiplication comes first.
+\Language{} finds a default implementation in
+\axiomType{AbelianGroup\&}.
+Remember from \downlink{``\ugCategoriesDefaultsTitle''}{ugCategoriesDefaultsPage} in Section \ugCategoriesDefaultsNumber\ignore{ugCategoriesDefaults} that
+\axiomType{SemiGroup} provides a default definition for
+\texht{$x^n$}{x**n} by repeated squaring?
+\axiomType{AbelianGroup} similarly provides a definition for
+\texht{$n x$}{n*x} by repeated doubling.
+
+But the search of the defaults for \axiomType{QuadraticForm} fails
+to find any \axiomOp{+} or \axiomOp{*} in the default packages for
+the ancestors of \axiomType{QuadraticForm}.
+So it now searches among those for \axiomType{SquareMatrix}.
+Category \axiomType{MatrixCategory}, which provides a uniform interface
+for all matrix domains,
+is a grandparent of \axiomType{SquareMatrix} and
+has a capsule defining many functions for matrices, including
+matrix addition, subtraction, and scalar multiplication.
+The default package \axiomType{MatrixCategory\&} is where the
+functions for \axiomOpFrom{+}{QuadraticForm} and
+\spadfunFrom{-}{QuadraticForm} come from.
+
+You can use \Browse{} to discover where the operations for
+\axiomType{QuadraticForm} are implemented.
+First, get the page describing \axiomType{QuadraticForm}.
+With your mouse somewhere in this window, type a ``2'', press the
+\texht{\fbox{\bf Tab}}{{\bf Tab}} key, and then enter ``Fraction
+Integer'' to indicate that you want the domain
+\axiomType{QuadraticForm(2, Fraction Integer)}.
+Now click on {\bf Operations} to get a table of operations and on
+\axiomOp{*} to get a page describing the \axiomOp{*} operation.
+Finally, click on {\bf implementation} at the bottom.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsOriginsTitle}{Origins}
+\newcommand{\ugDomainsOriginsNumber}{13.10.}
+%
+% =====================================================================
+\begin{page}{ugDomainsOriginsPage}{13.10. Origins}
+% =====================================================================
+\beginscroll
+%
+
+Aside from the notion of where an operation is implemented,
+%-% \HDindex{operation!origin}{ugDomainsOriginsPage}{13.10.}{Origins}
+a useful notion is  the {\it origin} or ``home'' of an operation.
+When an operation (such as
+\spadfunFrom{quadraticForm}{QuadraticForm}) is explicitly exported by
+a domain (such as \axiomType{QuadraticForm}), you can say that the
+origin of that operation is that domain.
+If an operation is not explicitly exported from a domain, it is inherited
+from, and has as origin, the (closest) category that explicitly exports it.
+The operations \axiomOpFrom{+}{AbelianMonoid} and
+\axiomOpFrom{-}{AbelianMonoid} of \axiomType{QuadraticForm},
+for example, are inherited from \axiomType{AbelianMonoid}.
+As it turns out, \axiomType{AbelianMonoid} is the origin of virtually every
+\axiomOp{+} operation in \Language{}!
+
+Again, you can use \Browse{} to discover the origins of
+operations.
+From the \Browse{} page on \axiomType{QuadraticForm}, click on {\bf
+Operations}, then on {\bf origins} at the bottom of the page.
+
+The origin of the operation is the {\it only} place where on-line
+documentation is given.
+However, you can re-export an operation to give it special
+documentation.
+Suppose you have just invented the world's fastest algorithm for
+inverting matrices using a particular internal representation for
+matrices.
+If your matrix domain just declares that it exports
+\axiomType{MatrixCategory}, it exports the \axiomFun{inverse}
+operation, but the documentation the user gets from \Browse{} is
+the standard one from \axiomType{MatrixCategory}.
+To give your version of \axiomFun{inverse} the attention it
+deserves, simply export the operation explicitly with new
+documentation.
+This redundancy gives \axiomFun{inverse} a new origin and tells
+\Browse{} to present your new documentation.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsShortFormsTitle}{Short Forms}
+\newcommand{\ugDomainsShortFormsNumber}{13.11.}
+%
+% =====================================================================
+\begin{page}{ugDomainsShortFormsPage}{13.11. Short Forms}
+% =====================================================================
+\beginscroll
+%
+In \Language{}, a domain could be defined using only an add-domain
+and no capsule.
+Although we talk about rational numbers as quotients of integers,
+there is no type \pspadtype{RationalNumber} in \Language{}.
+To create such a type, you could compile the following
+``short-form'' definition:
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ RationalNumber()\ ==\ Fraction(Integer)}\newline
+\endImportant
+
+The {\tt Exports} part of this definition is missing and is taken
+to be equivalent to that of \axiomType{Fraction(Integer)}.
+Because of the add-domain philosophy, you get precisely
+what you want.
+The effect is to create a little stub of a domain.
+When a user asks to add two rational numbers, \Language{} would
+ask \pspadtype{RationalNumber} for a function implementing this
+\axiomOp{+}.
+Since the domain has no capsule, the domain then immediately
+sends its request to \axiomType{Fraction (Integer)}.
+
+The short form definition for domains is used to
+define such domains as \axiomType{MultivariatePolynomial}:
+%-% \HDexptypeindex{MultivariatePolynomial}{ugDomainsShortFormsPage}{13.11.}{Short Forms}
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ MultivariatePolynomial(vl:\ List\ Symbol,\ R:\ Ring)\ ==}\newline
+{\tt 2.\ \ \ \ \ \ SparseMultivariatePolynomial(R,}\newline
+{\tt 3.\ \ \ \ \ \ \ \ \ OrderedVariableList\ vl)}\newline
+\endImportant
+
+%% *********************************************************************
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsCliffordTitle}{Example 1: Clifford Algebra}
+\newcommand{\ugDomainsCliffordNumber}{13.12.}
+%
+% =====================================================================
+\begin{page}{ugDomainsCliffordPage}{13.12. Example 1: Clifford Algebra}
+% =====================================================================
+\beginscroll
+%
+
+Now that we have \axiomType{QuadraticForm} available,
+let's put it to use.
+Given some quadratic form \smath{Q} described by an
+\smath{n} by \smath{n} matrix over a field \smath{K}, the domain
+\axiomType{CliffordAlgebra(n, K, Q)} defines a vector space of
+dimension \texht{$2^n$}{2**n} over \smath{K}.
+This is an interesting domain since complex numbers, quaternions,
+exterior algebras and spin algebras are all examples of Clifford
+algebras.
+
+The basic idea is this:
+the quadratic form \axiom{Q} defines a basis
+\texht{$e_1,e_2\ldots,e_n$}{e1,e2,...,en} for the
+vector space \texht{$K^n$}{K**n}---the direct product of \axiom{K}
+with itself \axiom{n} times.
+From this, the Clifford algebra generates a basis of
+\texht{$2^n$}{2**n} elements given by all the possible products
+of the \texht{$e_i$}{\axiom{ei}} in order without duplicates, that is,
+\texht{
+1,
+$e_1$,
+$e_2$,
+$e_1e_2$,
+$e_3$,
+$e_1e_3$,
+$e_2e_3$,
+$e_1e_2,e_3$}{1, e1, e2, e1*e2, e3, e1*e3, e2*e3, e1*e2*e3},
+and so on.
+
+The algebra is defined by the relations
+\begin{verbatim}
+ei*ei = Q(ei)
+ei*ej = -ej*ei,  i ^= j
+\end{verbatim}
+
+Now look at the snapshot of its definition given in Figure
+\ref{fig-clifalg}.
+Lines 9-10 show part of the definitions of the {\tt Exports}.
+A Clifford algebra over a field \axiom{K} is asserted to be a ring,
+an algebra over \axiom{K}, and a vector space over \axiom{K}.
+Its explicit exports include
+\axiom{e(n),} which returns the \eth{\axiom{n}} unit element.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ NNI\ ==>\ NonNegativeInteger}\newline
+{\tt 2.\ \ \ PI\ \ ==>\ PositiveInteger}\newline
+{\tt 3.\ \ \ }\newline
+{\tt 4.\ \ \ CliffordAlgebra(n,K,q):\ Exports\ ==\ Implementation\ where}\newline
+{\tt 5.\ \ \ \ \ \ \ n:\ PI}\newline
+{\tt 6.\ \ \ \ \ \ \ K:\ Field}\newline
+{\tt 7.\ \ \ \ \ \ \ q:\ QuadraticForm(n,\ K)}\newline
+{\tt 8.\ \ \ }\newline
+{\tt 9.\ \ \ \ \ \ \ Exports\ ==\ Join(Ring,Algebra(K),VectorSpace(K))\ with}\newline
+{\tt 10.\ \ \ \ \ \ \ \ e:\ PI\ ->\ \$}\newline
+{\tt 11.\ \ \ \ \ \ \ \ \ \ \ \ ...\ \ \ \ \ \ \ \ }\newline
+{\tt 12.\ \ }\newline
+{\tt 13.\ \ \ \ \ \ Implementation\ ==\ add}\newline
+{\tt 14.\ \ \ \ \ \ \ \ Qeelist\ :=\ \ }\newline
+{\tt 15.\ \ \ \ \ \ \ \ \ \ [q.unitVector(i::PI)\ for\ i\ in\ 1..n]}\newline
+{\tt 16.\ \ \ \ \ \ \ \ dim\ \ \ \ \ :=\ \ 2**n}\newline
+{\tt 17.\ \ \ \ \ \ \ \ Rep\ \ \ \ \ :=\ PrimitiveArray\ K}\newline
+{\tt 18.\ \ \ \ \ \ \ \ New\ ==>\ new(dim,\ 0\$K)\$Rep}\newline
+{\tt 19.\ \ \ \ \ \ \ \ x\ +\ y\ ==}\newline
+{\tt 20.\ \ \ \ \ \ \ \ \ \ z\ :=\ New}\newline
+{\tt 21.\ \ \ \ \ \ \ \ \ \ for\ i\ in\ 0..dim-1\ repeat\ z.i\ :=\ x.i\ +\ y.i}\newline
+{\tt 22.\ \ \ \ \ \ \ \ \ \ z}\newline
+{\tt 23.\ \ \ \ \ \ \ \ addMonomProd:\ (K,\ NNI,\ K,\ NNI,\ \$)\ ->\ \$}\newline
+{\tt 24.\ \ \ \ \ \ \ \ addMonomProd(c1,\ b1,\ c2,\ b2,\ z)\ ==\ \ ...}\newline
+{\tt 25.\ \ \ \ \ \ \ \ x\ *\ y\ ==}\newline
+{\tt 26.\ \ \ \ \ \ \ \ \ \ z\ :=\ New}\newline
+{\tt 27.\ \ \ \ \ \ \ \ \ \ for\ ix\ in\ 0..dim-1\ repeat}\newline
+{\tt 28.\ \ \ \ \ \ \ \ \ \ \ \ if\ x.ix\ \notequal{}\ 0\ then\ for\ iy\ in\ 0..dim-1\ repeat}\newline
+{\tt 29.\ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ y.iy\ \notequal{}\ 0}\newline
+{\tt 30.\ \ \ \ \ \ \ \ \ \ \ \ \ \ then\ addMonomProd(x.ix,ix,y.iy,iy,z)}\newline
+{\tt 31.\ \ \ \ \ \ \ \ \ \ \ \ z}\newline
+{\tt 32.\ \ \ \ \ \ \ \ \ \ \ \ \ ...}\newline
+\caption{Part of the \protect\axiomType{CliffordAlgebra} domain.}\label{fig-clifalg}
+\endImportant
+
+The {\tt Implementation} part begins by defining a local variable
+\axiom{Qeelist} to hold the list of all \axiom{q.v} where \axiom{v}
+runs over the unit vectors from 1 to the dimension \axiom{n}.
+Another local variable \axiom{dim} is set to \texht{$2^n$}{2**n},
+computed once and for all.
+The representation for the domain is
+\axiomType{PrimitiveArray(K)},
+which is a basic array of elements from domain \axiom{K}.
+Line 18 defines \axiom{New} as shorthand for the more lengthy
+expression \axiom{new(dim, 0\$K)\$Rep}, which computes a primitive
+array of length \texht{$2^n$}{2**n} filled with \axiom{0}'s from
+domain \axiom{K}.
+
+Lines 19-22 define the sum of two elements \axiom{x} and \axiom{y}
+straightforwardly.
+First, a new array of all \axiom{0}'s is created, then filled with
+the sum of the corresponding elements.
+Indexing for primitive arrays starts at 0.
+The definition of the product of \axiom{x} and \axiom{y} first requires
+the definition of a local function \userfun{addMonomProd}.
+\Language{} knows it is local since it is not an exported function.
+The types of all local functions must be declared.
+
+For a demonstration of \axiomType{CliffordAlgebra}, see
+\downlink{`CliffordAlgebra'}{CliffordAlgebraXmpPage}\ignore{CliffordAlgebra}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomsinsDatabaseTitle}{Example 2: Building A Query Facility}
+\newcommand{\ugDomsinsDatabaseNumber}{13.13.}
+%
+% =====================================================================
+\begin{page}{ugDomsinsDatabasePage}{13.13. Example 2: Building A Query Facility}
+% =====================================================================
+\beginscroll
+%
+We now turn to an entirely different kind of application,
+building a query language for a database.
+
+Here is the practical problem to solve.
+The \Browse{} facility of \Language{} has a
+database for all operations and constructors which is
+stored on disk and accessed by \HyperName{}.
+For our purposes here, we regard each line of this file as having
+eight fields:
+{\tt class, name, type, nargs, exposed, kind, origin,} and {\tt condition.}
+Here is an example entry:
+
+\begin{verbatim}
+o`determinant`$->R`1`x`d`Matrix(R)`has(R,commutative("*"))
+\end{verbatim}
+
+In English, the entry means:
+\texht{\begin{quotation}}{\newline}
+\texht{\raggedright}{}
+The operation \axiomFun{determinant}: \spadsig{\$}{R} with {\it 1} argument, is
+{\it exposed} and is exported by {\it domain} \axiomType{Matrix(R)}
+if {\tt R has commutative("*")}.
+\texht{\end{quotation}}{\newline}
+
+Our task is to create a little query language that allows us
+to get useful information from this database.
+
+\beginmenu
+    \menudownlink{{13.13.1. A Little Query Language}}{ugDomainsQueryLanguagePage}
+    \menudownlink{{13.13.2. The Database Constructor}}{ugDomainsDatabaseConstructorPage}
+    \menudownlink{{13.13.3. Query Equations}}{ugDomainsQueryEquationsPage}
+    \menudownlink{{13.13.4. DataLists}}{ugDomainsDataListsPage}
+    \menudownlink{{13.13.5. Index Cards}}{ugDomainsDatabasePage}
+    \menudownlink{{13.13.6. Creating a Database}}{ugDomainsCreatingPage}
+    \menudownlink{{13.13.7. Putting It All Together}}{ugDomainsPuttingPage}
+    \menudownlink{{13.13.8. Example Queries}}{ugDomainsExamplesPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsQueryLanguageTitle}{A Little Query Language}
+\newcommand{\ugDomainsQueryLanguageNumber}{13.13.1.}
+%
+% =====================================================================
+\begin{page}{ugDomainsQueryLanguagePage}{13.13.1. A Little Query Language}
+% =====================================================================
+\beginscroll
+
+First we design a simple language for accessing information from
+the database.
+We have the following simple model in mind for its design.
+Think of the database as a box of index cards.
+There is only one search operation---it
+takes the name of a field and a predicate
+%-% \HDindex{predicate}{ugDomainsQueryLanguagePage}{13.13.1.}{A Little Query Language}
+(a boolean-valued function) defined on the fields of the
+index cards.
+When applied, the search operation goes through the entire box
+selecting only those index cards for which the predicate is true.
+The result of a search is a new box of index cards.
+This process can be repeated again and again.
+
+The predicates all have a particularly simple form: {\it symbol}
+{\tt =} {\it pattern}, where {\it symbol} designates one of the
+fields, and {\it pattern} is a ``search string''---a string
+that may contain a ``{\tt *}'' as a
+wildcard.
+Wildcards match any substring, including the empty string.
+Thus the pattern {\tt "*ma*t"} matches
+{\tt "mat", "doormat"} and {\tt "smart".}
+
+To illustrate how queries are given, we give you a sneak preview
+of the facility we are about to create.
+
+\xtc{
+Extract the database of all \Language{} operations.
+}{
+\spadpaste{ops := getDatabase("o")\bound{o1}}
+}
+\xtc{
+How many exposed three-argument \axiomFun{map} operations involving streams?
+}{
+\spadpaste{ops.(name="map").(nargs="3").(type="*Stream*")\bound{o2}\free{o1}}
+}
+
+As usual, the arguments of \axiomFun{elt} (\axiomSyntax{.})
+associate to the left.
+The first \axiomFun{elt} produces the set of all operations with
+name {\tt map}.
+The second \axiomFun{elt} produces the set of all map operations
+with three arguments.
+The third \axiomFun{elt} produces the set of all three-argument map
+operations having a type mentioning \axiomType{Stream}.
+
+Another thing we'd like to do is to extract one field from each of
+the index cards in the box and look at the result.
+Here is an example of that kind of request.
+
+\xtc{
+What constructors explicitly export a \axiomFun{determinant} operation?
+}{
+\spadpaste{elt(elt(elt(elt(ops,name="determinant"),origin),sort),unique)\free{o1}}
+}
+
+The first \axiomFun{elt} produces the set of all index cards with
+name {\tt determinant}.
+The second \axiomFun{elt} extracts the {\tt origin} component from
+each index card. Each origin component
+is the name of a constructor which directly
+exports the operation represented by the index card.
+Extracting a component from each index card produces what we call
+a {\it datalist}.
+The third \axiomFun{elt}, {\tt sort}, causes the datalist of
+origins to be sorted in alphabetic
+order.
+The fourth, {\tt unique}, causes duplicates to be removed.
+
+Before giving you a more extensive demo of this facility,
+we now build the necessary domains and packages to implement it.
+%We will introduce a few of our minor conveniences.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsDatabaseConstructorTitle}{The Database Constructor}
+\newcommand{\ugDomainsDatabaseConstructorNumber}{13.13.2.}
+%
+% =====================================================================
+\begin{page}{ugDomainsDatabaseConstructorPage}{13.13.2. The Database Constructor}
+% =====================================================================
+\beginscroll
+
+We work from the top down. First, we define a database,
+our box of index cards, as an abstract datatype.
+For sake of illustration and generality,
+we assume that an index card is some type \axiom{S}, and
+that a database is a box of objects of type \axiom{S}.
+Here is the \Language{} program defining the \pspadtype{Database}
+domain.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ PI\ ==>\ PositiveInteger}\newline
+{\tt 2.\ \ \ Database(S):\ Exports\ ==\ Implementation\ where}\newline
+{\tt 3.\ \ \ \ \ S:\ Object\ with\ }\newline
+{\tt 4.\ \ \ \ \ \ \ elt:\ (\$,\ Symbol)\ ->\ String}\newline
+{\tt 5.\ \ \ \ \ \ \ display:\ \$\ ->\ Void}\newline
+{\tt 6.\ \ \ \ \ \ \ fullDisplay:\ \$\ ->\ Void}\newline
+{\tt 7.\ \ \ }\newline
+{\tt 8.\ \ \ \ \ Exports\ ==\ with}\newline
+{\tt 9.\ \ \ \ \ \ \ elt:\ (\$,QueryEquation)\ ->\ \$}\newline
+{\tt 10.\ \ \ \ \ \ elt:\ (\$,\ Symbol)\ ->\ DataList\ String}\newline
+{\tt 11.\ \ \ \ \ \ "+":\ (\$,\$)\ ->\ \$}\newline
+{\tt 12.\ \ \ \ \ \ "-":\ (\$,\$)\ ->\ \$}\newline
+{\tt 13.\ \ \ \ \ \ display:\ \$\ ->\ Void}\newline
+{\tt 14.\ \ \ \ \ \ fullDisplay:\ \$\ ->\ Void}\newline
+{\tt 15.\ \ \ \ \ \ fullDisplay:\ (\$,PI,PI)\ ->\ Void}\newline
+{\tt 16.\ \ \ \ \ \ coerce:\ \$\ ->\ OutputForm}\newline
+{\tt 17.\ \ \ \ Implementation\ ==\ add}\newline
+{\tt 18.\ \ \ \ \ \ \ \ ...}\newline
+\endImportant
+
+The domain constructor takes a parameter \axiom{S}, which
+stands for the class of index cards.
+We describe an index card later.
+Here think of an index card as a string which has
+the eight fields mentioned above.
+
+First, we tell \Language{} what operations we are going to require
+from index cards.
+We need an \axiomFun{elt} to extract the contents of a field
+(such as {\tt name} and {\tt type}) as a string.
+For example,
+\axiom{c.name} returns a string that is the content of the
+\axiom{name} field on the index card \axiom{c}.
+We need to display an index card in two ways:
+\pspadfun{display} shows only the name and type of an
+operation;
+\pspadfun{fullDisplay} displays all fields.
+The display operations return no useful information and thus have
+return type \axiomType{Void}.
+
+Next, we tell \Language{} what operations the user can apply
+to the database.
+This part defines our little query language.
+The most important operation is
+{\frenchspacing\tt db . field = pattern} which
+returns a new database, consisting of all index
+cards of {\tt db} such that the \axiom{field} part of the index
+card is matched by the string pattern called \axiom{pattern}.
+The expression {\tt field = pattern} is an object of type
+\axiomType{QueryEquation} (defined in the next section).
+
+Another \axiomFun{elt} is needed to produce a \pspadtype{DataList}
+object.
+Operation \axiomOp{+} is to merge two databases together;
+\axiomOp{-} is used to subtract away common entries in a second
+database from an initial database.
+There are three display functions.
+The \pspadfun{fullDisplay} function has two versions: one
+that prints all the records, the other that prints only a fixed
+number of records.
+A \axiomFun{coerce} to \axiomType{OutputForm} creates a display
+object.
+
+The {\tt Implementation} part of \axiomType{Database} is straightforward.
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ \ \ Implementation\ ==\ add}\newline
+{\tt 2.\ \ \ \ \ \ \ s:\ Symbol}\newline
+{\tt 3.\ \ \ \ \ \ \ Rep\ :=\ List\ S}\newline
+{\tt 4.\ \ \ \ \ \ \ elt(db,equation)\ ==\ ...}\newline
+{\tt 5.\ \ \ \ \ \ \ elt(db,key)\ ==\ [x.key\ for\ x\ in\ db]::DataList(String)}\newline
+{\tt 6.\ \ \ \ \ \ \ display(db)\ ==\ \ for\ x\ in\ db\ repeat\ display\ x}\newline
+{\tt 7.\ \ \ \ \ \ \ fullDisplay(db)\ ==\ for\ x\ in\ db\ repeat\ fullDisplay\ x}\newline
+{\tt 8.\ \ \ \ \ \ \ fullDisplay(db,\ n,\ m)\ ==\ for\ x\ in\ db\ for\ i\ in\ 1..m}\newline
+{\tt 9.\ \ \ \ \ \ \ \ \ repeat}\newline
+{\tt 10.\ \ \ \ \ \ \ \ \ \ if\ i\ >=\ n\ then\ fullDisplay\ x}\newline
+{\tt 11.\ \ \ \ \ \ x+y\ ==\ removeDuplicates!\ merge(x,y)}\newline
+{\tt 12.\ \ \ \ \ \ x-y\ ==\ mergeDifference(copy(x::Rep),}\newline
+{\tt 13.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y::Rep)\$MergeThing(S)}\newline
+{\tt 14.\ \ \ \ \ \ coerce(db):\ OutputForm\ ==\ (\#db)::\ OutputForm}\newline
+\endImportant
+
+The database is represented by a list of elements of \axiom{S} (index cards).
+We leave the definition of the first \axiomFun{elt} operation
+(on line 4) until the next section.
+The second \axiomFun{elt} collects all the strings with field name
+{\it key} into a list.
+The \axiomFun{display} function and first \axiomFun{fullDisplay} function
+simply call the corresponding functions from \axiom{S}.
+The second \axiomFun{fullDisplay} function provides an efficient way of
+printing out a portion of a large list.
+The \axiomOp{+} is defined by using the existing
+\spadfunFrom{merge}{List} operation defined on lists, then
+removing duplicates from the result.
+The \axiomOp{-} operation requires writing a corresponding
+subtraction operation.
+A package \axiomType{MergeThing} (not shown) provides this.
+
+The \axiomFun{coerce} function converts the database to an
+\axiomType{OutputForm} by computing the number of index cards.
+This is a good example of the independence of
+the representation of an \Language{} object from how it presents
+itself to the user. We usually do not want to look at a database---but
+do care how many ``hits'' we get for a given query.
+So we define the output representation of a database to be simply
+the number of index cards our query finds.
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsQueryEquationsTitle}{Query Equations}
+\newcommand{\ugDomainsQueryEquationsNumber}{13.13.3.}
+%
+% =====================================================================
+\begin{page}{ugDomainsQueryEquationsPage}{13.13.3. Query Equations}
+% =====================================================================
+\beginscroll
+
+The predicate for our search is given by an object of type
+\pspadtype{QueryEquation}.
+\Language{} does not have such an object yet so we
+have to invent it.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ QueryEquation():\ Exports\ ==\ Implementation\ where}\newline
+{\tt 2.\ \ \ \ \ Exports\ ==\ with}\newline
+{\tt 3.\ \ \ \ \ \ \ equation:\ (Symbol,\ String)\ ->\ \$}\newline
+{\tt 4.\ \ \ \ \ \ \ variable:\ \$\ ->\ Symbol}\newline
+{\tt 5.\ \ \ \ \ \ \ value:\\ \\ \\ \\ \$\ ->\ String}\newline
+{\tt 6.\ \ \ }\newline
+{\tt 7.\ \ \ \ \ Implementation\ ==\ add}\newline
+{\tt 8.\ \ \ \ \ \ \ Rep\ :=\ Record(var:Symbol,\ val:String)}\newline
+{\tt 9.\ \ \ \ \ \ \ equation(x,\ s)\ ==\ [x,\ s]}\newline
+{\tt 10.\ \ \ \ \ \ variable\ q\ ==\ q.var}\newline
+{\tt 11.\ \ \ \ \ \ value\\ \\ \\ \\ q\ ==\ q.val}\newline
+\endImportant
+
+\Language{} converts an input expression of the form
+\axiom{{\it a} = {\it b}} to \axiom{equation({\it a, b})}.
+Our equations always have a symbol on the left and a string
+on the right.
+The {\tt Exports} part thus specifies an operation
+\axiomFun{equation} to create a query equation, and
+\pspadfun{variable} and \pspadfun{value} to select the left- and
+right-hand sides.
+The {\tt Implementation} part uses \pspadtype{Record} for a
+space-efficient representation of an equation.
+
+Here is the missing definition for the \axiomFun{elt} function of
+\pspadtype{Database} in the last section:
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ \ \ \ \ elt(db,eq)\ ==}\newline
+{\tt 2.\ \ \ \ \ \ \ \ \ field\\ \ :=\ variable\ eq}\newline
+{\tt 3.\ \ \ \ \ \ \ \ \ value\ :=\ value\ eq}\newline
+{\tt 4.\ \ \ \ \ \ \ \ \ [x\ for\ x\ in\ db\ |\ matches?(value,x.field)]}\newline
+\endImportant
+
+Recall that a database is represented by a list.
+Line 4 simply runs over that list collecting all elements
+such that the pattern (that is, \axiom{value})
+matches the selected field of the element.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsDataListsTitle}{DataLists}
+\newcommand{\ugDomainsDataListsNumber}{13.13.4.}
+%
+% =====================================================================
+\begin{page}{ugDomainsDataListsPage}{13.13.4. DataLists}
+% =====================================================================
+\beginscroll
+
+Type \pspadtype{DataList} is a new type invented to hold the result
+of selecting one field from each of the index cards in the box.
+It is useful to make datalists extensions of lists---lists that
+have special \axiomFun{elt} operations defined on them for
+sorting and removing duplicates.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ DataList(S:OrderedSet)\ :\ Exports\ ==\ Implementation\ where}\newline
+{\tt 2.\ \ \ \ \ Exports\ ==\ ListAggregate(S)\ with}\newline
+{\tt 3.\ \ \ \ \ \ \ elt:\ (\$,"unique")\ ->\ \$}\newline
+{\tt 4.\ \ \ \ \ \ \ elt:\ (\$,"sort")\ ->\ \$}\newline
+{\tt 5.\ \ \ \ \ \ \ elt:\ (\$,"count")\ ->\ NonNegativeInteger}\newline
+{\tt 6.\ \ \ \ \ \ \ coerce:\ List\ S\ ->\ \$}\newline
+{\tt 7.\ \ \ }\newline
+{\tt 8.\ \ \ \ \ Implementation\ ==\ \ List(S)\ add}\newline
+{\tt 9.\ \ \ \ \ \ \ Rep\ :=\ List\ S}\newline
+{\tt 10.\ \ \ \ \ \ elt(x,"unique")\ ==\ removeDuplicates(x)}\newline
+{\tt 11.\ \ \ \ \ \ elt(x,"sort")\ ==\ sort(x)}\newline
+{\tt 12.\ \ \ \ \ \ elt(x,"count")\ ==\ \#x}\newline
+{\tt 13.\ \ \ \ \ \ coerce(x:List\ S)\ ==\ x\ ::\ \$}\newline
+\endImportant
+
+The {\tt Exports} part asserts that datalists belong to the
+category \axiomType{ListAggregate}.
+Therefore, you can use all the usual list operations on datalists,
+such as \spadfunFrom{first}{List}, \spadfunFrom{rest}{List}, and
+\spadfunFrom{concat}{List}.
+In addition, datalists have four explicit operations.
+Besides the three \axiomFun{elt} operations, there is a
+\axiomFun{coerce} operation that creates datalists from lists.
+
+The {\tt Implementation} part needs only to define four functions.
+All the rest are obtained from \axiomType{List(S)}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsDatabaseTitle}{Index Cards}
+\newcommand{\ugDomainsDatabaseNumber}{13.13.5.}
+%
+% =====================================================================
+\begin{page}{ugDomainsDatabasePage}{13.13.5. Index Cards}
+% =====================================================================
+\beginscroll
+
+An index card comes from a file as one long string.
+We define functions that extract substrings from the long
+string.
+Each field has a name that
+is passed as a second argument to \axiomFun{elt}.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ IndexCard()\ ==\ Implementation\ where}\newline
+{\tt 2.\ \ \ \ \ Exports\ ==\ with}\newline
+{\tt 3.\ \ \ \ \ \ \ elt:\ (\$,\ Symbol)\ ->\ String}\newline
+{\tt 4.\ \ \ \ \ \ \ display:\ \$\ ->\ Void}\newline
+{\tt 5.\ \ \ \ \ \ \ fullDisplay:\ \$\ ->\ Void}\newline
+{\tt 6.\ \ \ \ \ \ \ coerce:\ String\ ->\ \$}\newline
+{\tt 7.\ \ \ \ \ Implementation\ ==\ String\ add\ ...}\newline
+\endImportant
+
+We leave the {\tt Implementation} part to the reader.
+All operations involve straightforward string manipulations.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsCreatingTitle}{Creating a Database}
+\newcommand{\ugDomainsCreatingNumber}{13.13.6.}
+%
+% =====================================================================
+\begin{page}{ugDomainsCreatingPage}{13.13.6. Creating a Database}
+% =====================================================================
+\beginscroll
+
+We must not forget one important operation: one that builds the database in the
+first place!
+We'll name it \pspadfun{getDatabase} and put it in a package.
+This function is implemented by calling the \Lisp{} function
+\axiom{getBrowseDatabase(s)} to get appropriate information from
+\Browse{}.
+This operation takes a string indicating which lines you
+want from the database: \axiom{"o"} gives you all operation
+lines, and \axiom{"k"}, all constructor lines.
+Similarly, \axiom{"c"}, \axiom{"d"}, and \axiom{"p"} give
+you all category, domain and package lines respectively.
+%
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ OperationsQuery():\ Exports\ ==\ Implementation\ where}\newline
+{\tt 2.\ \ \ \ \ Exports\ ==\ with}\newline
+{\tt 3.\ \ \ \ \ \ \ getDatabase:\ String\ ->\ Database(IndexCard)}\newline
+{\tt 4.\ \ \ }\newline
+{\tt 5.\ \ \ \ \ Implementation\ ==\ add}\newline
+{\tt 6.\ \ \ \ \ \ \ getDatabase(s)\ ==\ getBrowseDatabase(s)\$Lisp}\newline
+\endImportant
+
+We do not bother creating a special name for databases of index
+cards.
+\pspadtype{Database (IndexCard)} will do.
+Notice that we used the package \pspadtype{OperationsQuery} to
+create, in effect,
+a new kind of domain: \pspadtype{Database(IndexCard)}.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsPuttingTitle}{Putting It All Together}
+\newcommand{\ugDomainsPuttingNumber}{13.13.7.}
+%
+% =====================================================================
+\begin{page}{ugDomainsPuttingPage}{13.13.7. Putting It All Together}
+% =====================================================================
+\beginscroll
+
+To create the database facility, you put all these constructors
+into one file.\footnote{You could use separate files, but we
+are putting them all together because, organizationally, that is
+the logical thing to do.}
+At the top of the file put \spadcmd{)abbrev} commands, giving the
+constructor abbreviations you created.
+
+\beginImportant
+  
+\noindent
+{\tt 1.\ \ \ )abbrev\ domain\ \ ICARD\ \ \ IndexCard}\newline
+{\tt 2.\ \ \ )abbrev\ domain\ \ QEQUAT\ \ QueryEquation}\newline
+{\tt 3.\ \ \ )abbrev\ domain\ \ MTHING\ \ MergeThing}\newline
+{\tt 4.\ \ \ )abbrev\ domain\ \ DLIST\ \ \ DataList}\newline
+{\tt 5.\ \ \ )abbrev\ domain\ \ DBASE\ \ \ Database}\newline
+{\tt 6.\ \ \ )abbrev\ package\ OPQUERY\ OperationsQuery}\newline
+\endImportant
+
+With all this in {\bf alql.spad}, for example, compile it using
+%-% \HDsyscmdindex{compile}{ugDomainsPuttingPage}{13.13.7.}{Putting It All Together}
+\begin{verbatim}
+)compile alql
+\end{verbatim}
+and then load each of the constructors:
+\begin{verbatim}
+)load ICARD QEQUAT MTHING DLIST DBASE OPQUERY
+\end{verbatim}
+%-% \HDsyscmdindex{load}{ugDomainsPuttingPage}{13.13.7.}{Putting It All Together}
+You are ready to try some sample queries.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugDomainsExamplesTitle}{Example Queries}
+\newcommand{\ugDomainsExamplesNumber}{13.13.8.}
+%
+% =====================================================================
+\begin{page}{ugDomainsExamplesPage}{13.13.8. Example Queries}
+% =====================================================================
+\beginscroll
+
+Our first set of queries give some statistics on constructors in
+the current \Language{} system.
+
+\xtc{
+How many constructors does \Language{} have?
+}{
+\spadpaste{ks := getDatabase "k"\bound{q1}}
+}
+\xtc{
+Break this down into the number of categories, domains, and packages.
+}{
+\spadpaste{[ks.(kind=k) for k in ["c","d","p"]]\bound{q3}\free{q1}}
+}
+\xtc{
+What are all the domain constructors that take no parameters?
+}{
+\spadpaste{elt(ks.(kind="d").(nargs="0"),name)\bound{q4}\free{q1}}
+}
+\xtc{
+How many constructors have ``Matrix'' in their name?
+}{
+\spadpaste{mk := ks.(name="*Matrix*")\bound{q5}\free{q1}}
+}
+\xtc{
+What are the names of those that are domains?
+}{
+\spadpaste{elt(mk.(kind="d"),name)\bound{q6}\free{q5}}
+}
+\xtc{
+How many operations are there in the library?
+}{
+\spadpaste{o := getDatabase "o"\bound{o1}}
+}
+\xtc{
+Break this down into categories, domains, and packages.
+}{
+\spadpaste{[o.(kind=k) for k in ["c","d","p"]]\free{o1}}
+}
+
+The query language is helpful in getting information about a
+particular operation you might like to apply.
+While this information can be obtained with
+\Browse{}, the use of the query database gives you data that you
+can manipulate in the workspace.
+
+\xtc{
+How many operations have ``eigen'' in the name?
+}{
+\spadpaste{eigens := o.(name="*eigen*")\bound{eigens}\free{o1}}
+}
+\xtc{
+What are their names?
+}{
+\spadpaste{elt(eigens,name)\free{eigens}}
+}
+\xtc{
+Where do they come from?
+}{
+\spadpaste{elt(elt(elt(eigens,origin),sort),unique) \free{eigens}}
+}
+
+The operations \axiomOp{+} and \axiomOp{-} are useful for
+constructing small databases and combining them.
+However, remember that the only matching you can do is string
+matching.
+Thus a pattern such as {\tt "*Matrix*"} on the type field
+matches
+any type containing \axiomType{Matrix}, \axiomType{MatrixCategory},
+\axiomType{SquareMatrix}, and so on.
+
+\xtc{
+How many operations mention ``Matrix'' in their type?
+}{
+\spadpaste{tm := o.(type="*Matrix*")\bound{x10}\free{o1}}
+}
+\xtc{
+How many operations come from constructors with ``Matrix'' in
+their name?
+}{
+\spadpaste{fm := o.(origin="*Matrix*")\bound{x11}\free{o1}}
+}
+\xtc{
+How many operations are in \axiom{fm} but not in \axiom{tm}?
+}{
+\spadpaste{fm-tm \bound{x12}\free{x10 x11}}
+}
+\xtc{
+Display the operations that both mention ``Matrix'' in their type
+and come from a constructor having ``Matrix'' in their name.
+}{
+\spadpaste{fullDisplay(fm-\%) \bound{x13}\free{x12}}
+}
+\xtc{
+How many operations involve matrices?
+}{
+\spadpaste{m := tm+fm \bound{x14}\free{x10 x11}}
+}
+\xtc{
+Display 4 of them.
+}{
+\spadpaste{fullDisplay(m, 202, 205) \free{x14}}
+}
+\xtc{
+How many distinct names of operations involving matrices are there?
+}{
+\spadpaste{elt(elt(elt(m,name),unique),count) \free{x14}}
+}
+
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug13.pht b/src/hyper/pages/ug13.pht
new file mode 100644
index 00000000..aa8f34e1
--- /dev/null
+++ b/src/hyper/pages/ug13.pht
@@ -0,0 +1,673 @@
+\begin{patch}{ugDomainsExamplesPagePatch1}
+\begin{paste}{ugDomainsExamplesPageFull1}{ugDomainsExamplesPageEmpty1}
+\pastebutton{ugDomainsExamplesPageFull1}{\hidepaste}
+\tab{5}\spadcommand{ks := getDatabase "k"\bound{q1 }}
+\indentrel{3}\begin{verbatim}
+   (1)  1067
+                               Type: Database IndexCard
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty1}
+\begin{paste}{ugDomainsExamplesPageEmpty1}{ugDomainsExamplesPagePatch1}
+\pastebutton{ugDomainsExamplesPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{ks := getDatabase "k"\bound{q1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch2}
+\begin{paste}{ugDomainsExamplesPageFull2}{ugDomainsExamplesPageEmpty2}
+\pastebutton{ugDomainsExamplesPageFull2}{\hidepaste}
+\tab{5}\spadcommand{[ks.(kind=k) for k in ["c","d","p"]]\bound{q3 }\free{q1 }}
+\indentrel{3}\begin{verbatim}
+   (2)  [205,393,469]
+                          Type: List Database IndexCard
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty2}
+\begin{paste}{ugDomainsExamplesPageEmpty2}{ugDomainsExamplesPagePatch2}
+\pastebutton{ugDomainsExamplesPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{[ks.(kind=k) for k in ["c","d","p"]]\bound{q3 }\free{q1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch3}
+\begin{paste}{ugDomainsExamplesPageFull3}{ugDomainsExamplesPageEmpty3}
+\pastebutton{ugDomainsExamplesPageFull3}{\hidepaste}
+\tab{5}\spadcommand{elt(ks.(kind="d").(nargs="0"),name)\bound{q4 }\free{q1 }}
+\indentrel{3}\begin{verbatim}
+   (3)
+   ["AlgebraicNumber", "AnonymousFunction", "Any",
+    "AttributeButtons", "BasicFunctions",
+    "BasicOperator", "BinaryExpansion", "BinaryFile",
+    "Bits", "Boolean", "CardinalNumber",
+    "CharacterClass", "Character", "Color", "Commutator",
+    "DecimalExpansion", "DoubleFloat", "DrawOption",
+    "Exit", "ExtAlgBasis", "FileName", "Float",
+    "FortranCode", "FortranScalarType",
+    "FortranTemplate", "FortranType", "GraphImage",
+    "HexadecimalExpansion", "IVBaseColor", "IVBasicNode",
+    "IVCoordinate3", "IVCoordinate4", "IVFaceSet",
+    "IVField", "IVGroup", "IVIndexedLineSet",
+    "IVNodeConnection", "IVNodeObject", "IVPointSet",
+    "IVQuadMesh", "IVSeparator", "IVSimpleInnerNode",
+    "IVUtilities", "IVValue", "IndexCard",
+    "InnerAlgebraicNumber", "InputForm", "Integer",
+    "IntegrationFunctionsTable", "InventorDataSink",
+    "InventorRenderPackage", "InventorViewPort",
+    "Library", "MachineComplex", "MachineFloat",
+    "MachineInteger",
+    "NagDiscreteFourierTransformInterfacePackage",
+    "NagEigenInterfacePackage",
+    "NagOptimisationInterfacePackage",
+    "NagQuadratureInterfacePackage", "NagResultChecks",
+    "NagSpecialFunctionsInterfacePackage",
+    "NonNegativeInteger", "None",
+    "NumericalIntegrationProblem", "NumericalODEProblem",
+    "NumericalOptimizationProblem",
+    "NumericalPDEProblem", "ODEIntensityFunctionsTable",
+    "OrdSetInts", "OutputForm", "Palette", "Partition",
+    "Pi", "PlaneAlgebraicCurvePlot", "Plot3D", "Plot",
+    "PositiveInteger", "QueryEquation", "RenderTools",
+    "Result", "RomanNumeral", "RoutinesTable",
+    "SExpression", "ScriptFormulaFormat",
+    "SingleInteger", "SingletonAsOrderedSet", "String",
+    "SubSpaceComponentProperty", "Switch", "SymbolTable",
+    "Symbol", "TexFormat", "TextFile", "TheSymbolTable",
+    "ThreeDimensionalViewport", "Timer",
+    "TwoDimensionalViewport", "Void",
+    "d01TransformFunctionType", "d01ajfAnnaType",
+    "d01akfAnnaType", "d01alfAnnaType", "d01amfAnnaType",
+    "d01anfAnnaType", "d01apfAnnaType", "d01aqfAnnaType",
+    "d01asfAnnaType", "d01fcfAnnaType", "d01gbfAnnaType",
+    "d02bbfAnnaType", "d02bhfAnnaType", "d02cjfAnnaType",
+    "d02ejfAnnaType", "d03eefAnnaType", "d03fafAnnaType",
+    "e04dgfAnnaType", "e04fdfAnnaType", "e04gcfAnnaType",
+    "e04jafAnnaType", "e04mbfAnnaType", "e04nafAnnaType",
+    "e04ucfAnnaType"]
+                                  Type: DataList String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty3}
+\begin{paste}{ugDomainsExamplesPageEmpty3}{ugDomainsExamplesPagePatch3}
+\pastebutton{ugDomainsExamplesPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{elt(ks.(kind="d").(nargs="0"),name)\bound{q4 }\free{q1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch4}
+\begin{paste}{ugDomainsExamplesPageFull4}{ugDomainsExamplesPageEmpty4}
+\pastebutton{ugDomainsExamplesPageFull4}{\hidepaste}
+\tab{5}\spadcommand{mk := ks.(name="*Matrix*")\bound{q5 }\free{q1 }}
+\indentrel{3}\begin{verbatim}
+   (4)  26
+                               Type: Database IndexCard
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty4}
+\begin{paste}{ugDomainsExamplesPageEmpty4}{ugDomainsExamplesPagePatch4}
+\pastebutton{ugDomainsExamplesPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{mk := ks.(name="*Matrix*")\bound{q5 }\free{q1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch5}
+\begin{paste}{ugDomainsExamplesPageFull5}{ugDomainsExamplesPageEmpty5}
+\pastebutton{ugDomainsExamplesPageFull5}{\hidepaste}
+\tab{5}\spadcommand{elt(mk.(kind="d"),name)\bound{q6 }\free{q5 }}
+\indentrel{3}\begin{verbatim}
+   (5)
+   ["DenavitHartenbergMatrix",
+    "DirectProductMatrixModule", "IndexedMatrix",
+    "LieSquareMatrix", "Matrix", "RectangularMatrix",
+    "SquareMatrix", "ThreeDimensionalMatrix"]
+                                  Type: DataList String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty5}
+\begin{paste}{ugDomainsExamplesPageEmpty5}{ugDomainsExamplesPagePatch5}
+\pastebutton{ugDomainsExamplesPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{elt(mk.(kind="d"),name)\bound{q6 }\free{q5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch6}
+\begin{paste}{ugDomainsExamplesPageFull6}{ugDomainsExamplesPageEmpty6}
+\pastebutton{ugDomainsExamplesPageFull6}{\hidepaste}
+\tab{5}\spadcommand{o := getDatabase "o"\bound{o1 }}
+\indentrel{3}\begin{verbatim}
+   (6)  6315
+                               Type: Database IndexCard
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty6}
+\begin{paste}{ugDomainsExamplesPageEmpty6}{ugDomainsExamplesPagePatch6}
+\pastebutton{ugDomainsExamplesPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{o := getDatabase "o"\bound{o1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch7}
+\begin{paste}{ugDomainsExamplesPageFull7}{ugDomainsExamplesPageEmpty7}
+\pastebutton{ugDomainsExamplesPageFull7}{\hidepaste}
+\tab{5}\spadcommand{[o.(kind=k) for k in ["c","d","p"]]\free{o1 }}
+\indentrel{3}\begin{verbatim}
+   (7)  [1646,2040,2629]
+                          Type: List Database IndexCard
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty7}
+\begin{paste}{ugDomainsExamplesPageEmpty7}{ugDomainsExamplesPagePatch7}
+\pastebutton{ugDomainsExamplesPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{[o.(kind=k) for k in ["c","d","p"]]\free{o1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch8}
+\begin{paste}{ugDomainsExamplesPageFull8}{ugDomainsExamplesPageEmpty8}
+\pastebutton{ugDomainsExamplesPageFull8}{\hidepaste}
+\tab{5}\spadcommand{eigens := o.(name="*eigen*")\bound{eigens }\free{o1 }}
+\indentrel{3}\begin{verbatim}
+   (8)  4
+                               Type: Database IndexCard
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty8}
+\begin{paste}{ugDomainsExamplesPageEmpty8}{ugDomainsExamplesPagePatch8}
+\pastebutton{ugDomainsExamplesPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{eigens := o.(name="*eigen*")\bound{eigens }\free{o1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch9}
+\begin{paste}{ugDomainsExamplesPageFull9}{ugDomainsExamplesPageEmpty9}
+\pastebutton{ugDomainsExamplesPageFull9}{\hidepaste}
+\tab{5}\spadcommand{elt(eigens,name)\free{eigens }}
+\indentrel{3}\begin{verbatim}
+   (9)
+   ["eigenMatrix", "eigenvalues", "eigenvector",
+    "eigenvectors"]
+                                  Type: DataList String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty9}
+\begin{paste}{ugDomainsExamplesPageEmpty9}{ugDomainsExamplesPagePatch9}
+\pastebutton{ugDomainsExamplesPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{elt(eigens,name)\free{eigens }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch10}
+\begin{paste}{ugDomainsExamplesPageFull10}{ugDomainsExamplesPageEmpty10}
+\pastebutton{ugDomainsExamplesPageFull10}{\hidepaste}
+\tab{5}\spadcommand{elt(elt(elt(eigens,origin),sort),unique)\free{eigens }}
+\indentrel{3}\begin{verbatim}
+   (10)  ["EigenPackage","RadicalEigenPackage"]
+                                  Type: DataList String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty10}
+\begin{paste}{ugDomainsExamplesPageEmpty10}{ugDomainsExamplesPagePatch10}
+\pastebutton{ugDomainsExamplesPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{elt(elt(elt(eigens,origin),sort),unique)\free{eigens }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch11}
+\begin{paste}{ugDomainsExamplesPageFull11}{ugDomainsExamplesPageEmpty11}
+\pastebutton{ugDomainsExamplesPageFull11}{\hidepaste}
+\tab{5}\spadcommand{tm := o.(type="*Matrix*")\bound{x10 }\free{o1 }}
+\indentrel{3}\begin{verbatim}
+   (11)  353
+                               Type: Database IndexCard
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty11}
+\begin{paste}{ugDomainsExamplesPageEmpty11}{ugDomainsExamplesPagePatch11}
+\pastebutton{ugDomainsExamplesPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{tm := o.(type="*Matrix*")\bound{x10 }\free{o1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch12}
+\begin{paste}{ugDomainsExamplesPageFull12}{ugDomainsExamplesPageEmpty12}
+\pastebutton{ugDomainsExamplesPageFull12}{\hidepaste}
+\tab{5}\spadcommand{fm := o.(origin="*Matrix*")\bound{x11 }\free{o1 }}
+\indentrel{3}\begin{verbatim}
+   (12)  192
+                               Type: Database IndexCard
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty12}
+\begin{paste}{ugDomainsExamplesPageEmpty12}{ugDomainsExamplesPagePatch12}
+\pastebutton{ugDomainsExamplesPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{fm := o.(origin="*Matrix*")\bound{x11 }\free{o1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch13}
+\begin{paste}{ugDomainsExamplesPageFull13}{ugDomainsExamplesPageEmpty13}
+\pastebutton{ugDomainsExamplesPageFull13}{\hidepaste}
+\tab{5}\spadcommand{fm-tm\bound{x12 }\free{x10 x11 }}
+\indentrel{3}\begin{verbatim}
+   (13)  146
+                               Type: Database IndexCard
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty13}
+\begin{paste}{ugDomainsExamplesPageEmpty13}{ugDomainsExamplesPagePatch13}
+\pastebutton{ugDomainsExamplesPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{fm-tm\bound{x12 }\free{x10 x11 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch14}
+\begin{paste}{ugDomainsExamplesPageFull14}{ugDomainsExamplesPageEmpty14}
+\pastebutton{ugDomainsExamplesPageFull14}{\hidepaste}
+\tab{5}\spadcommand{fullDisplay(fm-\%)\bound{x13 }\free{x12 }}
+\indentrel{3}\begin{verbatim}
+   ** : (Matrix(R),NonNegativeInteger)->Matrix(R)
+      from StorageEfficientMatrixOperations(R) (unexposed)
+   clearDenominator : (Matrix(Q))->Matrix(R)
+      from MatrixCommonDenominator(R,Q)
+   coerceP
+      : (Vector(Matrix(R)))->Vector(Matrix(Polynomial(R)))
+      from CoerceVectorMatrixPackage(R) (unexposed)
+   coerce
+      :
+      (Vector(Matrix(R)))->Vector(Matrix(Fraction(Polynom
+      ial(R))))
+      from CoerceVectorMatrixPackage(R) (unexposed)
+   coerce : (_$)->Matrix(R)
+      from RectangularMatrix(m,n,R) (unexposed)
+   coerce : (_$)->Matrix(R)
+      from SquareMatrix(ndim,R) (unexposed)
+   coerce : (Matrix(MachineFloat))->_$
+      from FortranMatrixCategory
+   commonDenominator : (Matrix(Q))->R
+      from MatrixCommonDenominator(R,Q)
+   copy! : (Matrix(R),Matrix(R))->Matrix(R)
+      from StorageEfficientMatrixOperations(R) (unexposed)
+   f01brf
+      :
+      (Integer,Integer,Integer,Integer,DoubleFloat,Boolea
+      n,Boolean,List(Boolean),Matrix(DoubleFloat),Matrix(
+      Integer),Matrix(Integer),Integer)->Result
+      from NagMatrixOperationsPackage
+   f01bsf
+      :
+      (Integer,Integer,Integer,Matrix(Integer),Matrix(Int
+      eger),Matrix(Integer),Matrix(Integer),Boolean,Doubl
+      eFloat,Boolean,Matrix(Integer),Matrix(DoubleFloat),
+      Integer)->Result
+      from NagMatrixOperationsPackage
+   f01maf
+      :
+      (Integer,Integer,Integer,Integer,List(Boolean),Matr
+      ix(DoubleFloat),Matrix(Integer),Matrix(Integer),Dou
+      bleFloat,DoubleFloat,Integer)->Result
+      from NagMatrixOperationsPackage
+   f01mcf
+      :
+      (Integer,Matrix(DoubleFloat),Integer,Matrix(Integer
+      ),Integer)->Result
+      from NagMatrixOperationsPackage
+   f01qcf
+      :
+      (Integer,Integer,Integer,Matrix(DoubleFloat),Intege
+      r)->Result
+      from NagMatrixOperationsPackage
+   f01qdf
+      :
+      (String,String,Integer,Integer,Matrix(DoubleFloat),
+      Integer,Matrix(DoubleFloat),Integer,Integer,Matrix(
+      DoubleFloat),Integer)->Result
+      from NagMatrixOperationsPackage
+   f01qef
+      :
+      (String,Integer,Integer,Integer,Integer,Matrix(Doub
+      leFloat),Matrix(DoubleFloat),Integer)->Result
+      from NagMatrixOperationsPackage
+   f01rcf
+      :
+      (Integer,Integer,Integer,Matrix(Complex(DoubleFloat
+      )),Integer)->Result
+      from NagMatrixOperationsPackage
+   f01rdf
+      :
+      (String,String,Integer,Integer,Matrix(Complex(Doubl
+      eFloat)),Integer,Matrix(Complex(DoubleFloat)),Integ
+      er,Integer,Matrix(Complex(DoubleFloat)),Integer)->R
+      esult
+      from NagMatrixOperationsPackage
+   f01ref
+      :
+      (String,Integer,Integer,Integer,Integer,Matrix(Comp
+      lex(DoubleFloat)),Matrix(Complex(DoubleFloat)),Inte
+      ger)->Result
+      from NagMatrixOperationsPackage
+   hasSolution? : (Matrix(F),Vector(F))->Boolean
+      from LinearSystemMatrixPackage1(F)
+   leftScalarTimes! : (Matrix(R),R,Matrix(R))->Matrix(R)
+      from StorageEfficientMatrixOperations(R) (unexposed)
+   minus! : (Matrix(R),Matrix(R))->Matrix(R)
+      from StorageEfficientMatrixOperations(R) (unexposed)
+   minus! : (Matrix(R),Matrix(R),Matrix(R))->Matrix(R)
+      from StorageEfficientMatrixOperations(R) (unexposed)
+   particularSolution
+      : (Matrix(F),Vector(F))->Union(Vector(F),"failed")
+      from LinearSystemMatrixPackage1(F)
+   plus! : (Matrix(R),Matrix(R),Matrix(R))->Matrix(R)
+      from StorageEfficientMatrixOperations(R) (unexposed)
+   power!
+      :
+      (Matrix(R),Matrix(R),Matrix(R),Matrix(R),NonNegativ
+      eInteger)->Matrix(R)
+      from StorageEfficientMatrixOperations(R) (unexposed)
+   rank : (Matrix(F),Vector(F))->NonNegativeInteger
+      from LinearSystemMatrixPackage1(F)
+   rectangularMatrix : (Matrix(R))->_$
+      from RectangularMatrix(m,n,R) (unexposed)
+   retractIfCan
+      : (Matrix(Expression(Float)))->Union(_$,"failed")
+      from FortranMatrixFunctionCategory
+   retractIfCan
+      : (Matrix(Expression(Integer)))->Union(_$,"failed")
+      from FortranMatrixFunctionCategory
+   retractIfCan
+      :
+      (Matrix(Fraction(Polynomial(Float))))->Union(_$,"fa
+      iled")
+      from FortranMatrixFunctionCategory
+   retractIfCan
+      :
+      (Matrix(Fraction(Polynomial(Integer))))->Union(_$,"
+      failed")
+      from FortranMatrixFunctionCategory
+   retractIfCan
+      : (Matrix(Polynomial(Float)))->Union(_$,"failed")
+      from FortranMatrixFunctionCategory
+   retractIfCan
+      : (Matrix(Polynomial(Integer)))->Union(_$,"failed")
+      from FortranMatrixFunctionCategory
+   retract : (Matrix(Expression(Float)))->_$
+      from FortranMatrixFunctionCategory
+   retract : (Matrix(Expression(Integer)))->_$
+      from FortranMatrixFunctionCategory
+   retract : (Matrix(Fraction(Polynomial(Float))))->_$
+      from FortranMatrixFunctionCategory
+   retract : (Matrix(Fraction(Polynomial(Integer))))->_$
+      from FortranMatrixFunctionCategory
+   retract : (Matrix(Polynomial(Float)))->_$
+      from FortranMatrixFunctionCategory
+   retract : (Matrix(Polynomial(Integer)))->_$
+      from FortranMatrixFunctionCategory
+   rightScalarTimes! : (Matrix(R),Matrix(R),R)->Matrix(R)
+      from StorageEfficientMatrixOperations(R) (unexposed)
+   solve
+      :
+      (Matrix(F),List(Vector(F)))->List(Record(particular
+      :Union(Vector(F),"failed"),basis:List(Vector(F))))
+      from LinearSystemMatrixPackage1(F)
+   solve
+      :
+      (Matrix(F),Vector(F))->Record(particular:Union(Vect
+      or(F),"failed"),basis:List(Vector(F)))
+      from LinearSystemMatrixPackage1(F)
+   splitDenominator
+      : (Matrix(Q))->Record(num:Matrix(R),den:R)
+      from MatrixCommonDenominator(R,Q)
+   squareMatrix : (Matrix(R))->_$
+      from SquareMatrix(ndim,R) (unexposed)
+   times! : (Matrix(R),Matrix(R),Matrix(R))->Matrix(R)
+      from StorageEfficientMatrixOperations(R) (unexposed)
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty14}
+\begin{paste}{ugDomainsExamplesPageEmpty14}{ugDomainsExamplesPagePatch14}
+\pastebutton{ugDomainsExamplesPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{fullDisplay(fm-\%)\bound{x13 }\free{x12 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch15}
+\begin{paste}{ugDomainsExamplesPageFull15}{ugDomainsExamplesPageEmpty15}
+\pastebutton{ugDomainsExamplesPageFull15}{\hidepaste}
+\tab{5}\spadcommand{m := tm+fm\bound{x14 }\free{x10 x11 }}
+\indentrel{3}\begin{verbatim}
+   (15)  499
+                               Type: Database IndexCard
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty15}
+\begin{paste}{ugDomainsExamplesPageEmpty15}{ugDomainsExamplesPagePatch15}
+\pastebutton{ugDomainsExamplesPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{m := tm+fm\bound{x14 }\free{x10 x11 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch16}
+\begin{paste}{ugDomainsExamplesPageFull16}{ugDomainsExamplesPageEmpty16}
+\pastebutton{ugDomainsExamplesPageFull16}{\hidepaste}
+\tab{5}\spadcommand{fullDisplay(m, 202, 205)\free{x14 }}
+\indentrel{3}\begin{verbatim}
+   elt : (_$,List(Integer),List(Integer))->_$
+      from MatrixCategory(R,Row,Col)
+   elt : (_$,Integer,Integer,R)->R
+      from RectangularMatrixCategory(m,n,R,Row,Col)
+   elt
+      :
+      (_$,NonNegativeInteger,NonNegativeInteger,NonNegati
+      veInteger)->R
+      from ThreeDimensionalMatrix(R)
+   eval
+      :
+      (Matrix(Expression(DoubleFloat)),List(Symbol),Vecto
+      r(Expression(DoubleFloat)))->Matrix(Expression(Doub
+      leFloat))
+      from d02AgentsPackage
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty16}
+\begin{paste}{ugDomainsExamplesPageEmpty16}{ugDomainsExamplesPagePatch16}
+\pastebutton{ugDomainsExamplesPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{fullDisplay(m, 202, 205)\free{x14 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPagePatch17}
+\begin{paste}{ugDomainsExamplesPageFull17}{ugDomainsExamplesPageEmpty17}
+\pastebutton{ugDomainsExamplesPageFull17}{\hidepaste}
+\tab{5}\spadcommand{elt(elt(elt(m,name),unique),count)\free{x14 }}
+\indentrel{3}\begin{verbatim}
+   (17)  317
+                                  Type: PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsExamplesPageEmpty17}
+\begin{paste}{ugDomainsExamplesPageEmpty17}{ugDomainsExamplesPagePatch17}
+\pastebutton{ugDomainsExamplesPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{elt(elt(elt(m,name),unique),count)\free{x14 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsQueryLanguagePagePatch1}
+\begin{paste}{ugDomainsQueryLanguagePageFull1}{ugDomainsQueryLanguagePageEmpty1}
+\pastebutton{ugDomainsQueryLanguagePageFull1}{\hidepaste}
+\tab{5}\spadcommand{ops := getDatabase("o")\bound{o1 }}
+\indentrel{3}\begin{verbatim}
+   (1)  6315
+                               Type: Database IndexCard
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsQueryLanguagePageEmpty1}
+\begin{paste}{ugDomainsQueryLanguagePageEmpty1}{ugDomainsQueryLanguagePagePatch1}
+\pastebutton{ugDomainsQueryLanguagePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{ops := getDatabase("o")\bound{o1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsQueryLanguagePagePatch2}
+\begin{paste}{ugDomainsQueryLanguagePageFull2}{ugDomainsQueryLanguagePageEmpty2}
+\pastebutton{ugDomainsQueryLanguagePageFull2}{\hidepaste}
+\tab{5}\spadcommand{ops.(name="map").(nargs="3").(type="*Stream*")\bound{o2 }\free{o1 }}
+\indentrel{3}\begin{verbatim}
+   (2)  3
+                               Type: Database IndexCard
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsQueryLanguagePageEmpty2}
+\begin{paste}{ugDomainsQueryLanguagePageEmpty2}{ugDomainsQueryLanguagePagePatch2}
+\pastebutton{ugDomainsQueryLanguagePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{ops.(name="map").(nargs="3").(type="*Stream*")\bound{o2 }\free{o1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsQueryLanguagePagePatch3}
+\begin{paste}{ugDomainsQueryLanguagePageFull3}{ugDomainsQueryLanguagePageEmpty3}
+\pastebutton{ugDomainsQueryLanguagePageFull3}{\hidepaste}
+\tab{5}\spadcommand{elt(elt(elt(elt(ops,name="determinant"),origin),sort),unique)\free{o1 }}
+\indentrel{3}\begin{verbatim}
+   (3)
+   ["InnerMatrixLinearAlgebraFunctions",
+    "MatrixCategory", "MatrixLinearAlgebraFunctions",
+    "SquareMatrixCategory"]
+                                  Type: DataList String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsQueryLanguagePageEmpty3}
+\begin{paste}{ugDomainsQueryLanguagePageEmpty3}{ugDomainsQueryLanguagePagePatch3}
+\pastebutton{ugDomainsQueryLanguagePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{elt(elt(elt(elt(ops,name="determinant"),origin),sort),unique)\free{o1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPagePatch1}
+\begin{paste}{ugDomainsDemoPageFull1}{ugDomainsDemoPageEmpty1}
+\pastebutton{ugDomainsDemoPageFull1}{\hidepaste}
+\tab{5}\spadcommand{QF := QuadraticForm(2,Fraction Integer)\bound{x2 }\free{x1 }}
+\indentrel{3}\begin{verbatim}
+   (1)  QuadraticForm(2,Fraction Integer)
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPageEmpty1}
+\begin{paste}{ugDomainsDemoPageEmpty1}{ugDomainsDemoPagePatch1}
+\pastebutton{ugDomainsDemoPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{QF := QuadraticForm(2,Fraction Integer)\bound{x2 }\free{x1 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPagePatch2}
+\begin{paste}{ugDomainsDemoPageFull2}{ugDomainsDemoPageEmpty2}
+\pastebutton{ugDomainsDemoPageFull2}{\hidepaste}
+\tab{5}\spadcommand{A := matrix [[-1,1/2],[1/2,1]]\bound{x3 }\free{x2 }}
+\indentrel{3}\begin{verbatim}
+        Ú     1¿
+        ³- 1  ij
+        ³     2³
+   (2)  ³      ³
+        ³ 1    ³
+        ³ Ä   1³
+        À 2    Ù
+                          Type: Matrix Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPageEmpty2}
+\begin{paste}{ugDomainsDemoPageEmpty2}{ugDomainsDemoPagePatch2}
+\pastebutton{ugDomainsDemoPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{A := matrix [[-1,1/2],[1/2,1]]\bound{x3 }\free{x2 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPagePatch3}
+\begin{paste}{ugDomainsDemoPageFull3}{ugDomainsDemoPageEmpty3}
+\pastebutton{ugDomainsDemoPageFull3}{\hidepaste}
+\tab{5}\spadcommand{q : QF := quadraticForm(A)\bound{x4 }\free{x3 }}
+\indentrel{3}\begin{verbatim}
+        Ú     1¿
+        ³- 1  ij
+        ³     2³
+   (3)  ³      ³
+        ³ 1    ³
+        ³ Ä   1³
+        À 2    Ù
+                Type: QuadraticForm(2,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPageEmpty3}
+\begin{paste}{ugDomainsDemoPageEmpty3}{ugDomainsDemoPagePatch3}
+\pastebutton{ugDomainsDemoPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{q : QF := quadraticForm(A)\bound{x4 }\free{x3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPagePatch4}
+\begin{paste}{ugDomainsDemoPageFull4}{ugDomainsDemoPageEmpty4}
+\pastebutton{ugDomainsDemoPageFull4}{\hidepaste}
+\tab{5}\spadcommand{nrows q\free{x3 }}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPageEmpty4}
+\begin{paste}{ugDomainsDemoPageEmpty4}{ugDomainsDemoPagePatch4}
+\pastebutton{ugDomainsDemoPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{nrows q\free{x3 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPagePatch5}
+\begin{paste}{ugDomainsDemoPageFull5}{ugDomainsDemoPageEmpty5}
+\pastebutton{ugDomainsDemoPageFull5}{\hidepaste}
+\tab{5}\spadcommand{v := directProduct([2,-1])$DirectProduct(2,Fraction Integer)\bound{x5 }\free{x4 }}
+\indentrel{3}\begin{verbatim}
+   (4)  [2,- 1]
+                Type: DirectProduct(2,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPageEmpty5}
+\begin{paste}{ugDomainsDemoPageEmpty5}{ugDomainsDemoPagePatch5}
+\pastebutton{ugDomainsDemoPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{v := directProduct([2,-1])$DirectProduct(2,Fraction Integer)\bound{x5 }\free{x4 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPagePatch6}
+\begin{paste}{ugDomainsDemoPageFull6}{ugDomainsDemoPageEmpty6}
+\pastebutton{ugDomainsDemoPageFull6}{\hidepaste}
+\tab{5}\spadcommand{q.v\free{x5 }}
+\indentrel{3}\begin{verbatim}
+   (5)  - 5
+                                 Type: Fraction Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPageEmpty6}
+\begin{paste}{ugDomainsDemoPageEmpty6}{ugDomainsDemoPagePatch6}
+\pastebutton{ugDomainsDemoPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{q.v\free{x5 }}
+\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPagePatch7}
+\begin{paste}{ugDomainsDemoPageFull7}{ugDomainsDemoPageEmpty7}
+\pastebutton{ugDomainsDemoPageFull7}{\hidepaste}
+\tab{5}\spadcommand{3*q-q+q\free{x4 }}
+\indentrel{3}\begin{verbatim}
+        Ú     3¿
+        ³- 3  ij
+        ³     2³
+   (6)  ³      ³
+        ³ 3    ³
+        ³ Ä   3³
+        À 2    Ù
+                Type: QuadraticForm(2,Fraction Integer)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugDomainsDemoPageEmpty7}
+\begin{paste}{ugDomainsDemoPageEmpty7}{ugDomainsDemoPagePatch7}
+\pastebutton{ugDomainsDemoPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{3*q-q+q\free{x4 }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ug14.ht b/src/hyper/pages/ug14.ht
new file mode 100644
index 00000000..95889109
--- /dev/null
+++ b/src/hyper/pages/ug14.ht
@@ -0,0 +1,1056 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+%following definition should go into ug.sty
+\texht{\gdef\aliascon#1#2{\axiomType{#1}}}{}
+\texht{\setcounter{chapter}{13}}{} % Chapter 14
+%
+
+%
+\newcommand{\ugBrowseTitle}{Browse}
+\newcommand{\ugBrowseNumber}{14.}
+%
+% =====================================================================
+\begin{page}{ugBrowsePage}{14. Browse}
+% =====================================================================
+\beginscroll
+
+This chapter discusses the \Browse{}
+%-% \HDindex{Browse@\Browse{}}{ugBrowsePage}{14.}{Browse}
+component of \HyperName{}.
+%-% \HDindex{HyperDoc@{\HyperName{}}}{ugBrowsePage}{14.}{Browse}
+We suggest you invoke \Language{} and work through this
+chapter, section by section, following our examples to gain some
+familiarity with \Browse{}.
+
+\beginmenu
+    \menudownlink{{14.1. The Front Page: Searching the Library}}{ugBrowseStartPage}
+    \menudownlink{{14.2. The Constructor Page}}{ugBrowseDomainPage}
+    \menudownlink{{14.3. Miscellaneous Features of Browse}}{ugBrowseMiscellaneousFeaturesPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugBrowseStartTitle}{The Front Page: Searching the Library}
+\newcommand{\ugBrowseStartNumber}{14.1.}
+%
+% =====================================================================
+\begin{page}{ugBrowseStartPage}{14.1. The Front Page: Searching the Library}
+% =====================================================================
+\beginscroll
+To enter \Browse{}, click on {\bf Browse} on the top level page
+of \HyperName{} to get the {\it front page} of \Browse{}.
+%
+%324pt is 4.5",180pt is 2.5",432pt is 6"=textwidth,54=(432-324)/2
+%ps files are 4.5"x2.5" except source 4.5"x2.5"
+%
+
+To use this page, you first enter a \spadgloss{search string} into
+the input area at the top, then click on one of the buttons below.
+We show the use of each of the buttons by example.
+
+\subsubsection{Constructors}
+
+First enter the search string {\tt Matrix} into the input area and
+click on {\bf Constructors}.
+What you get is the {\it constructor page} for \axiomType{Matrix}.
+We show and describe this page in detail in
+\downlink{``\ugBrowseDomainTitle''}{ugBrowseDomainPage} in Section \ugBrowseDomainNumber\ignore{ugBrowseDomain}.
+By convention, \Language{} does a case-insensitive search for a
+match.
+Thus {\tt matrix} is just as good as {\tt Matrix}, has the same
+effect as {\tt MaTrix}, and so on.
+We recommend that you generally use small letters for names
+however.
+A search string with only capital letters has a special meaning
+(see \downlink{``\ugBrowseCapitalizationConventionTitle''}{ugBrowseCapitalizationConventionPage} in Section \ugBrowseCapitalizationConventionNumber\ignore{ugBrowseCapitalizationConvention}).
+
+
+Click on \UpBitmap{} to return to the \Browse{} front page.
+
+Use the symbol ``{\tt *}'' in search strings as a \spadgloss{wild
+card}.
+A wild card matches any substring, including the empty string.
+For example, enter the search string {\tt *matrix*} into the input
+area and click on {\bf Constructors}.\footnote{To get only
+categories, domains, or packages, rather than all constructors,
+you can click on the corresponding button to the right of {\bf
+Constructors}.}
+What you get is a table of all constructors whose names contain
+the string ``{\tt matrix}.''
+
+
+%% Following para replaced 1995oct30 MGR
+%These are all the \spadglossSee{exposed}{expose} constructors in
+%\Language{}.
+%To see how to get all exposed and unexposed constructors in
+%\Language{}, skip to the section entitled {\bf Exposure} in
+%\downlink{``\ugBrowseOptionsTitle''}{ugBrowseOptionsPage} in Section \ugBrowseOptionsNumber\ignore{ugBrowseOptions}.
+All constructors containing the string are listed, whether
+\spadglossSee{exposed}{expose} or \spadglossSee{unexposed}{expose}.
+You can hide the names of the unexposed constructors by clicking
+on the {\it *=}{\bf unexposed} button in the {\it Views} panel at
+the bottom of the window.
+(The button will change to {\bf exposed} {\it only}.)
+
+One of the names in this table is \axiomType{Matrix}.
+Click on \axiomType{Matrix}.
+What you get is again the constructor page for \axiomType{Matrix}.
+As you see, \Browse{} gives you a large network of
+information in which there are many ways to reach the same
+pages.
+%-% \HDexptypeindex{Matrix}{ugBrowseStartPage}{14.1.}{The Front Page: Searching the Library}
+
+Again click on the \UpBitmap{} to return to the table of constructors
+whose names contain {\tt matrix}.
+%Below the table is a {\bf Views} panel. % here & globally MGR 1995oct30
+Below the table is a {\it Views} panel.
+This panel contains buttons that let you view constructors in different
+ways.
+To learn about views of constructors, skip to
+\downlink{``\ugBrowseViewsOfConstructorsTitle''}{ugBrowseViewsOfConstructorsPage} in Section \ugBrowseViewsOfConstructorsNumber\ignore{ugBrowseViewsOfConstructors}.
+
+Click on \UpBitmap{} to return to the \Browse{} front page.
+
+\subsubsection{Operations}
+
+Enter {\tt *matrix} into the input area and click on {\bf
+Operations}.
+This time you get a table of {\it operations} whose names end with {\tt
+matrix} or {\tt Matrix}.
+
+
+If you select an operation name, you go to a page describing all
+the operations in \Language{} of that name.
+At the bottom of an operation page is another kind of {\it Views} panel,
+one for operation pages.
+To learn more about these views, skip to
+\downlink{``\ugBrowseViewsOfOperationsTitle''}{ugBrowseViewsOfOperationsPage} in Section \ugBrowseViewsOfOperationsNumber\ignore{ugBrowseViewsOfOperations}.
+
+Click on \UpBitmap{} to return to the \Browse{} front page.
+
+\subsubsection{Attributes}
+
+This button gives you a table of attribute names that match the
+search string. Enter the search string {\tt *} and click on
+{\bf Attributes} to get a list
+of all system attributes.
+
+Click on \UpBitmap{} to return to the \Browse{} front page.
+
+
+
+Again there is a {\it Views} panel at the bottom with buttons that let
+you view the attributes in different ways.
+
+\subsubsection{General}
+
+This button does a general search for all constructor, operation, and
+attribute names matching the search string.
+Enter the search string \allowbreak
+{\tt *matrix*} into the input area.
+Click on {\bf General} to find all constructs that have {\tt
+matrix} as a part of their name.
+
+
+The summary gives you all the names under a heading when the number of
+entries is less than 10. % "less than 10." replaces the following:
+			 % sufficiently small%\footnote{See
+%\downlink{``\ugBrowseOptionsTitle''}{ugBrowseOptionsPage} in Section \ugBrowseOptionsNumber\ignore{ugBrowseOptions} to see how you can change this.}.
+%% MGR 1995oct31
+
+Click on \UpBitmap{} to return to the \Browse{} front page.
+
+\subsubsection{Documentation}
+
+Again enter the search key {\tt *matrix*} and this time click on
+{\bf Documentation}.
+This search matches any constructor, operation, or attribute
+name whose documentation contains a substring matching {\tt
+matrix}.
+
+
+Click on \UpBitmap{} to return to the \Browse{} front page.
+
+\subsubsection{Complete}
+
+This search combines both {\bf General} and {\bf Documentation}.
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugBrowseDomainTitle}{The Constructor Page}
+\newcommand{\ugBrowseDomainNumber}{14.2.}
+%
+% =====================================================================
+\begin{page}{ugBrowseDomainPage}{14.2. The Constructor Page}
+% =====================================================================
+\beginscroll
+
+In this section we look in detail at a constructor page for domain
+\axiomType{Matrix}.
+Enter {\tt matrix} into the input area on the main \Browse{} page
+and click on {\bf Constructors}.
+
+
+
+The header part tells you that \axiomType{Matrix} has abbreviation
+\axiomType{MATRIX} and one argument called {\tt R} that must be a
+domain of category \axiomType{Ring}.
+Just what domains can be arguments of \axiomType{Matrix}?
+To find this out, click on the {\tt R} on the second line of the
+heading.
+What you get is a table of all acceptable domain parameter values
+of {\tt R}, or a table of \spadgloss{rings} in \Language{}.
+
+
+Click on \UpBitmap{} to return to the constructor page for
+\axiomType{Matrix}.
+\texht{\newpage}{}
+
+If you have access to the source code of \Language{}, the third
+%-% \HDindex{source code}{ugBrowseDomainPage}{14.2.}{The Constructor Page}
+line of the heading gives you the name of the source file
+containing the definition of \axiomType{Matrix}.
+Click on it to pop up an editor window containing the source code
+of \axiomType{Matrix}.
+
+
+We recommend that you leave the editor window up while working
+through this chapter as you occasionally may want to refer to it.
+\texht{\newpage}{}
+
+\beginmenu
+    \menudownlink{{14.2.1. Constructor Page Buttons}}{ugBrowseDomainButtonsPage}
+    \menudownlink{{14.2.2. Cross Reference}}{ugBrowseCrossReferencePage}
+    \menudownlink{{14.2.3. Views Of Constructors}}{ugBrowseViewsOfConstructorsPage}
+    \menudownlink{{14.2.4. Giving Parameters to Constructors}}{ugBrowseGivingParametersPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugBrowseDomainButtonsTitle}{Constructor Page Buttons}
+\newcommand{\ugBrowseDomainButtonsNumber}{14.2.1.}
+%
+% =====================================================================
+\begin{page}{ugBrowseDomainButtonsPage}{14.2.1. Constructor Page Buttons}
+% =====================================================================
+\beginscroll
+
+We examine each button on this page in order.
+
+\labelSpace{2pc}
+
+\subsubsection{Description}
+
+Click here to bring up a page with a brief description of
+constructor \axiomType{Matrix}.
+If you have access to system source code, note that these comments
+can be found directly over the constructor definition.
+
+
+\subsubsection{Operations}
+
+Click here to get a table of operations exported by
+\axiomType{Matrix}.
+You may wish to widen the window to have multiple columns as
+below.
+
+
+If you click on an operation name, you bring up a description
+page for the operations.
+For a detailed description of these pages, skip to
+\downlink{``\ugBrowseViewsOfOperationsTitle''}{ugBrowseViewsOfOperationsPage} in Section \ugBrowseViewsOfOperationsNumber\ignore{ugBrowseViewsOfOperations}.
+
+\subsubsection{Attributes}
+
+Click here to get a table of the two attributes exported by
+\axiomType{Matrix}:
+%-% \HDindex{attribute}{ugBrowseDomainButtonsPage}{14.2.1.}{Constructor Page Buttons}
+\spadatt{\texht{fi\-nite\-Ag\-gre\-gate}{finiteAggregate}} and \spadatt{shallowlyMutable}.
+These are two computational properties that result from
+\axiomType{Matrix} being regarded as a data structure.
+
+
+\subsubsection{Examples}
+
+Click here to get an {\it examples page} with examples of operations to
+create and manipulate matrices.
+
+
+Read through this section.
+Try selecting the various buttons.
+Notice that if you click on an operation name, such as
+\spadfunFrom{new}{Matrix}, you bring up a description page for that
+operation from \axiomType{Matrix}.
+
+Example pages have several examples of \Language{} commands.
+Each example has an active button to its left.
+Click on it!
+A pre-computed answer is pasted into the page immediately following the
+command.
+If you click on the button a second time, the answer disappears.
+This button thus acts as a toggle:
+``now you see it; now you don't.''
+
+Note also that the \Language{} commands themselves are active.
+If you want to see \Language{} execute the command, then click on it!
+A new \Language{} window appears on your screen and the command is
+executed.
+
+\httex{At the end of the page is generally a menu of buttons that lead
+you to further sections.
+Select one of these topics to explore its contents.}{}
+
+\subsubsection{Exports}
+
+Click here to see a page describing the exports of \axiomType{Matrix}
+exactly as described by the source code.
+
+
+As you see, \axiomType{Matrix} declares that it exports all the operations
+and attributes exported by category
+\axiomType{MatrixCategory(R, Row, Col)}.
+In addition, two operations, \axiomFun{diagonalMatrix} and
+\axiomFun{inverse}, are explicitly exported.
+
+To learn a little about the structure of \Language{}, we suggest you do
+the following exercise.
+Otherwise, click on \UpButton{} and go on to the next section.
+\axiomType{Matrix} explicitly exports only two operations.
+The other operations are thus exports of \axiomType{MatrixCategory}.
+In general, operations are usually not explicitly exported by a domain.
+Typically they are \spadglossSee{inherited}{inherit} from several
+different categories.
+Let's find out from where the operations of \axiomType{Matrix} come.
+
+\indent{4}
+\beginitems
+\item[1. ] Click on {\bf MatrixCategory}, then on {\bf Exports}.
+Here you see that {\bf MatrixCategory} explicitly exports many matrix
+operations.
+Also, it inherits its operations from
+\axiomType{TwoDimensionalArrayCategory}.
+
+\item[2. ] Click on {\bf TwoDimensionalArrayCategory}, then on {\bf Exports}.
+Here you see explicit operations dealing with rows and columns.
+In addition, it inherits operations from
+\axiomType{HomogeneousAggregate}.
+
+%\item Click on {\bf HomogeneousAggregate}, then on {\bf Exports}.
+%And so on.
+%If you continue doing this, eventually you will
+
+\item[3. ] Click on \UpBitmap{} and then
+click on {\bf Object}, then on {\bf Exports}, where you see
+there are no exports.
+
+\item[4. ] Click on \UpBitmap{} repeatedly to return to the constructor page
+for \axiomType{Matrix}.
+
+\enditems
+\indent{0}
+
+\subsubsection{Related Operations}
+
+Click here bringing up a table of operations that are exported by
+\spadglossSee{packages}{package} but not by \axiomType{Matrix} itself.
+
+
+To see a table of such packages, use the {\bf Relatives} button on the
+{\bf Cross Reference} page described next.
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugBrowseCrossReferenceTitle}{Cross Reference}
+\newcommand{\ugBrowseCrossReferenceNumber}{14.2.2.}
+%
+% =====================================================================
+\begin{page}{ugBrowseCrossReferencePage}{14.2.2. Cross Reference}
+% =====================================================================
+\beginscroll
+Click on the {\bf Cross Reference} button on the main constructor page
+for \axiomType{Matrix}.
+This gives you a page having various cross reference information stored
+under the respective buttons.
+
+
+\subsubsection{Parents}
+
+The parents of a domain are the same as the categories mentioned under
+the {\bf Exports} button on the first page.
+Domain \axiomType{Matrix} has only one parent but in general a domain can
+have any number.
+
+\subsubsection{Ancestors}
+
+The \spadglossSee{ancestors}{ancestor} of a constructor consist of its parents, the
+parents of its parents, and so on.
+Did you perform the exercise in the last section under {\bf Exports}?
+If so, you  see here all the categories you found while ascending the
+{\bf Exports} chain for \axiomType{Matrix}.
+
+\subsubsection{Relatives}
+
+The \spadglossSee{relatives}{relative} of a domain constructor are package
+constructors that provide operations in addition to those
+\spadglossSee{exported}{export} by the domain.
+
+Try this exercise.
+\indent{4}
+\beginitems
+\item[1. ] Click on {\bf Relatives}, bringing up a list of
+\spadglossSee{packages}{package}.
+
+\item[2. ] Click on {\bf LinearSystemMatrixPackage} bringing up its
+constructor page.\footnote{You may want to widen your \HyperName{}
+window to make what follows more legible.}
+
+\item[3. ] Click on {\bf Operations}.
+Here you see \axiomFun{rank}, an operation also exported by
+\axiomType{Matrix} itself.
+
+\item[4. ] Click on {\bf rank}.
+This \spadfunFrom{rank}{LinearSystemMatrixPackage} has two arguments and
+thus is different from the \spadfunFrom{rank}{Matrix} from
+\axiomType{Matrix}.
+
+\item[5. ] Click on \UpBitmap{} to return to the list of operations for the
+package \axiomType{LinearSystemMatrixPackage}.
+
+\item[6. ] Click on {\bf solve} to bring up a
+\spadfunFrom{solve}{LinearSystemMatrixPackage} for linear systems of
+equations.
+
+\item[7. ] Click on \UpBitmap{} several times to return to the cross
+reference page for \axiomType{Matrix}.
+\enditems
+\indent{0}
+
+\subsubsection{Dependents}
+
+The \spadglossSee{dependents}{dependent} of a constructor are those
+\spadglossSee{domains}{domain} or \spadglossSee{packages}{package}
+that mention that
+constructor either as an argument or in its \spadglossSee{exports}{export}.
+
+If you click on {\bf Dependents} two entries may surprise you:
+\axiomType{RectangularMatrix} and \axiomType{SquareMatrix}.
+This happens because \axiomType{Matrix}, as it turns out, appears in
+signatures of operations exported by these domains.
+
+\subsubsection{Lineage}
+
+The term \spadgloss{lineage} refers to the {\it search order} for
+functions.
+If you are an expert user or curious about how the \Language{} system
+works, try the following exercise.
+Otherwise, you best skip this button and go on to {\bf Clients}.
+
+Clicking on {\bf Lineage} gives you a
+list of domain constructors:
+\axiomType{InnerIndexedTwoDimensionalArray},
+\aliascon{MatrixCategory\&}{MATCAT-},
+\aliascon{TwoDimensionalArrayCategory\&}{ARR2CAT-},
+\aliascon{HomogeneousAggregate\&}{HOAGG-},
+\aliascon{Aggregate\&}{AGG-}.
+What are these constructors and how are they used?
+
+We explain by an example.
+Suppose you create a matrix using the interpreter, then ask for its
+\axiomFun{rank}.
+\Language{} must then find a function implementing the \axiomFun{rank}
+operation for matrices.
+The first place \Language{} looks for \axiomFun{rank} is in the \axiomType{Matrix}
+domain.
+
+If not there, the lineage of \axiomType{Matrix} tells \Language{} where
+else to look.
+Associated with the matrix domain are five other lineage domains.
+Their order is important.
+\Language{} first searches the first one,
+\axiomType{InnerIndexedTwoDimensionalArray}.
+If not there, it searches the second \aliascon{MatrixCategory\&}{MATCAT-}.
+And so on.
+
+Where do these {\it lineage constructors} come from?
+The source code for \axiomType{Matrix} contains this syntax for the
+\spadgloss{function body} of
+\axiomType{Matrix}:\footnote{\axiomType{InnerIndexedTwoDimensionalArray}
+is a special domain implemented for matrix-like domains to provide
+efficient implementations of \twodim{} arrays.
+For example, domains of category \axiomType{TwoDimensionalArrayCategory}
+can have any integer as their \spad{minIndex}.
+Matrices and other members of this special ``inner'' array have their
+\spad{minIndex} defined as \spad{1}.}
+\begin{verbatim}
+InnerIndexedTwoDimensionalArray(R,mnRow,mnCol,Row,Col)
+   add ...
+\end{verbatim}
+where the ``{\tt ...}'' denotes all the code that follows.
+In English, this means:
+``The functions for matrices are defined as those from
+\axiomType{InnerIndexedTwoDimensionalArray} domain augmented by those
+defined in `{\tt ...}','' where the latter take precedence.
+
+This explains \axiomType{InnerIndexedTwoDimensionalArray}.
+The other names, those with names ending with an ampersand \axiomSyntax{\&} are
+\spadglossSee{default packages}{default package}
+for categories to which \axiomType{Matrix} belongs.
+Default packages are ordered by the notion of ``closest ancestor.''
+
+\subsubsection{Clients}
+
+A client of \axiomType{Matrix} is any constructor that uses
+\axiomType{Matrix} in its implementation.
+For example, \axiomType{Complex} is a client of \axiomType{Matrix}; it
+exports several operations that take matrices as arguments or return
+matrices as values.\footnote{A constructor is a client of
+\axiomType{Matrix} if it handles any matrix.
+For example, a constructor having internal (unexported) operations
+dealing with matrices is also a client.}
+
+\subsubsection{Benefactors}
+
+A \spadgloss{benefactor} of \axiomType{Matrix} is any constructor that
+\axiomType{Matrix} uses in its implementation.
+This information, like that for clients, is gathered from run-time
+structures.\footnote{The benefactors exclude constructors such as
+\axiomType{PrimitiveArray} whose operations macro-expand and so vanish
+from sight!}
+
+Cross reference pages for categories have some different buttons on
+them.
+Starting with the constructor page of \axiomType{Matrix}, click on
+\axiomType{Ring} producing its constructor page.
+Click on {\bf Cross Reference},
+producing the cross-reference page for \axiomType{Ring}.
+Here are buttons {\bf Parents} and {\bf Ancestors} similar to the notion
+for domains, except for categories the relationship between parent and
+child is defined through \spadgloss{category extension}.
+
+\subsubsection{Children}
+
+Category hierarchies go both ways.
+There are children as well as parents.
+A child can have any number of parents, but always at least one.
+Every category is therefore a descendant of exactly one category:
+\axiomType{Object}.
+
+\subsubsection{Descendants}
+
+These are children, children of children, and so on.
+
+Category hierarchies are complicated by the fact that categories take
+parameters.
+Where a parameterized category fits into a hierarchy {\it may} depend on
+values of its parameters.
+In general, the set of categories in \Language{} forms a {\it directed
+acyclic graph}, that is, a graph with directed arcs and no cycles.
+
+\subsubsection{Domains}
+
+This produces a table of all domain constructors that can possibly be
+rings (members of category \axiomType{Ring}).
+Some domains are unconditional rings.
+Others are rings for some parameters and not for others.
+To find out which, select the {\bf conditions} button in the views
+panel.
+For example, \axiomType{DirectProduct(n, R)} is a ring if {\tt R} is a
+ring.
+
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugBrowseViewsOfConstructorsTitle}{Views Of Constructors}
+\newcommand{\ugBrowseViewsOfConstructorsNumber}{14.2.3.}
+%
+% =====================================================================
+\begin{page}{ugBrowseViewsOfConstructorsPage}{14.2.3. Views Of Constructors}
+% =====================================================================
+\beginscroll
+
+Below every constructor table page is a {\it Views} panel.
+As an example, click on {\bf Cross Reference} from
+the constructor page of \axiomType{Matrix},
+then on {\bf Benefactors} to produce a
+short table of constructor names.
+
+The {\it Views} panel is at the bottom of the page.
+Two items, {\it names} and {\it conditions,} are in italics.
+Others are active buttons.
+The active buttons are those that give you useful alternative views
+on this table of constructors.
+Once you select a view, you notice that the button turns
+off (becomes italicized) so that you cannot reselect it.
+
+\subsubsection{names}
+
+This view gives you a table of names.
+Selecting any of these names brings up the constructor page for that
+constructor.
+
+\subsubsection{abbrs}
+
+This view gives you a table of abbreviations, in the same order as the
+original constructor names.
+Abbreviations are in capitals and are limited to 7 characters.
+They can be used interchangeably with constructor names in input areas.
+
+\subsubsection{kinds}
+
+This view organizes constructor names into
+the three kinds: categories, domains and packages.
+
+\subsubsection{files}
+
+This view gives a table of file names for the source
+code of the constructors in alphabetic order after removing
+duplicates.
+
+\subsubsection{parameters}
+
+This view presents constructors with the arguments.
+This view of the benefactors of \axiomType{Matrix} shows that
+\axiomType{Matrix} uses as many as five different \axiomType{List} domains
+in its implementation.
+
+\subsubsection{filter}
+
+This button is used to refine the list of names or abbreviations.
+Starting with the {\it names} view, enter {\tt m*} into the input area
+and click on {\bf filter}.
+You then get a shorter table with only the names beginning with {\tt m}.
+
+\subsubsection{documentation}
+
+This gives you documentation for each of the constructors.
+
+\subsubsection{conditions}
+
+This page organizes the constructors according to predicates.
+The view is not available for your example page since all constructors
+are unconditional.
+For a table with conditions, return to the {\bf Cross Reference} page
+for \axiomType{Matrix}, click on {\bf Ancestors}, then on {\bf
+conditions} in the view panel.
+This page shows you that \axiomType{CoercibleTo(OutputForm)} and
+\axiomType{SetCategory} are ancestors of \axiomType{Matrix(R)} only if {\tt R}
+belongs to category \axiomType{SetCategory}.
+
+%*********************************************************************
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugBrowseGivingParametersTitle}{Giving Parameters to Constructors}
+\newcommand{\ugBrowseGivingParametersNumber}{14.2.4.}
+%
+% =====================================================================
+\begin{page}{ugBrowseGivingParametersPage}{14.2.4. Giving Parameters to Constructors}
+% =====================================================================
+\beginscroll
+%*********************************************************************
+
+Notice the input area at the bottom of the constructor page.
+If you leave this blank, then the information you get is for the
+domain constructor \axiomType{Matrix(R)}, that is, \axiomType{Matrix} for an
+arbitrary underlying domain {\tt R}.
+
+In general, however, the exports and other information {\it do} usually
+depend on the actual value of {\tt R}.
+For example, \axiomType{Matrix} exports the \axiomFun{inverse} operation
+only if the domain {\tt R} is a \axiomType{Field}.
+To see this, try this from the main constructor page:
+
+\indent{4}
+\beginitems
+\item[1. ] Enter {\tt Integer} into the input area at the bottom of the page.
+
+\item[2. ] Click on {\bf Operations}, producing a table of operations.
+Note the number of operation names that appear at the top of the
+page.
+
+\item[3. ] Click on \UpBitmap{} to return to the constructor page.
+
+\item[4. ] Use the
+\texht{\fbox{\bf Delete}}{{\bf Delete}}
+or
+\texht{\fbox{\bf Backspace}}{{\bf Backspace}}
+keys to erase {\tt Integer} from the input area.
+
+\item[5. ] Click on {\bf Operations} to produce a new table of operations.
+Look at the number of operations you get.
+This number is greater than what you had before.
+Find, for example, the operation \axiomFun{inverse}.
+
+\item[6. ] Click on {\bf inverse} to produce a page describing the operation
+\axiomFun{inverse}.
+At the bottom of the description, you notice that the {\bf
+Conditions} line says ``{\tt R} has \axiomType{Field}.''
+This operation is {\it not} exported by \axiomType{Matrix(Integer)} since
+\axiomType{Integer} is not a \spadgloss{field}.
+
+Try putting the name of a domain such as \axiomType{Fraction Integer}
+(which is a field) into the input area, then clicking on {\bf Operations}.
+As you see, the operation \axiomFun{inverse} is exported.
+\enditems
+\indent{0}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugBrowseMiscellaneousFeaturesTitle}{Miscellaneous Features of Browse}
+\newcommand{\ugBrowseMiscellaneousFeaturesNumber}{14.3.}
+%
+% =====================================================================
+\begin{page}{ugBrowseMiscellaneousFeaturesPage}{14.3. Miscellaneous Features of Browse}
+% =====================================================================
+\beginscroll
+\labelSpace{4pc}
+\beginmenu
+    \menudownlink{{14.3.1. The Description Page for Operations}}{ugBrowseDescriptionPagePage}
+    \menudownlink{{14.3.2. Views of Operations}}{ugBrowseViewsOfOperationsPage}
+    \menudownlink{{14.3.3. Capitalization Convention}}{ugBrowseCapitalizationConventionPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugBrowseDescriptionPageTitle}{The Description Page for Operations}
+\newcommand{\ugBrowseDescriptionPageNumber}{14.3.1.}
+%
+% =====================================================================
+\begin{page}{ugBrowseDescriptionPagePage}{14.3.1. The Description Page for Operations}
+% =====================================================================
+\beginscroll
+From the constructor page of \axiomType{Matrix},
+click on {\bf Operations} to bring up the table of operations
+for \axiomType{Matrix}.
+
+Find the operation {\bf inverse} in the table and click on it.
+This takes you to a page showing the documentation for this operation.
+
+
+Here is the significance of the headings you see.
+
+\subsubsection{Arguments}
+
+This lists each of the arguments of the operation in turn, paraphrasing
+the \spadgloss{signature} of the operation.
+As for signatures, a \axiomSyntax{\$} is used to designate {\em this domain},
+that is, \axiomType{Matrix(R)}.
+
+\subsubsection{Returns}
+
+This describes the return value for the operation, analogous to the {\bf
+Arguments} part.
+
+\subsubsection{Origin}
+
+This tells you which domain or category explicitly exports the
+operation.
+In this example, the domain itself is the {\it Origin}.
+
+
+\subsubsection{Conditions}
+
+This tells you that the operation is exported by \axiomType{Matrix(R)} only if
+``{\tt R} has \axiomType{Field},'' that is, ``{\tt R} is a member of
+category \axiomType{Field}.''
+When no {\bf Conditions} part is given, the operation is exported for
+all values of {\tt R}.
+
+\subsubsection{Description}
+
+Here are the \axiomSyntax{++} comments
+that appear in the source code of its {\it Origin}, here \axiomType{Matrix}.
+You find these comments in the source code for \axiomType{Matrix}.
+
+
+Click on \UpBitmap{} to return to the table of operations.
+Click on {\bf map}.
+Here you find three different operations named \axiomFun{map}.
+This should not surprise you.
+Operations are identified by name and \spadgloss{signature}.
+There are three operations named \axiomFun{map}, each with
+different signatures.
+What you see is the {\it descriptions} view of the operations.
+If you like, select the button in the heading of one of these
+descriptions to get {\it only} that operation.
+
+\subsubsection{Where}
+
+This part qualifies domain parameters mentioned in the arguments to the
+operation.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugBrowseViewsOfOperationsTitle}{Views of Operations}
+\newcommand{\ugBrowseViewsOfOperationsNumber}{14.3.2.}
+%
+% =====================================================================
+\begin{page}{ugBrowseViewsOfOperationsPage}{14.3.2. Views of Operations}
+% =====================================================================
+\beginscroll
+
+We suggest that you go to the constructor page for \axiomType{Matrix}
+and click on {\bf Operations} to bring up a table of operations
+with a {\it Views} panel at the bottom.
+
+\subsubsection{names}
+
+This view lists the names of the operations.
+Unlike constructors, however, there may be several operations with the
+same name.
+The heading for the page tells you the number of unique names and the
+number of distinct operations when these numbers are different.
+
+\subsubsection{filter}
+
+As for constructors, you can use this button to cut down the list of
+operations you are looking at.
+Enter, for example, {\tt m*} into the input area to the right of {\bf
+filter} then click on {\bf filter}.
+As usual, any logical expression is permitted.
+For example, use
+\begin{verbatim}
+*! or *?
+\end{verbatim}
+to get a list of destructive operations and predicates.
+
+\subsubsection{documentation}
+
+This gives you the most information:
+a detailed description of all the operations in the form you have seen
+before.
+Every other button summarizes these operations in some form.
+
+\subsubsection{signatures}
+
+This views the operations by showing their signatures.
+
+\subsubsection{parameters}
+
+This views the operations by their distinct syntactic forms with
+parameters.
+
+\subsubsection{origins}
+
+This organizes the operations according to the constructor that
+explicitly exports them.
+
+\subsubsection{conditions}
+
+This view organizes the operations into conditional and unconditional
+operations.
+
+\subsubsection{usage}
+
+This button is only available if your user-level is set to {\it
+%-% \HDindex{user-level}{ugBrowseViewsOfOperationsPage}{14.3.2.}{Views of Operations}
+development}.
+The {\bf usage} button produces a table of constructors that reference this
+operation.\footnote{\Language{} requires an especially long time to
+produce this table, so anticipate this when requesting this
+information.}
+
+\subsubsection{implementation}
+
+This button is only available if your user-level is set to {\it
+development}.
+%-% \HDindex{user-level}{ugBrowseViewsOfOperationsPage}{14.3.2.}{Views of Operations}
+If you enter values for all domain parameters on the constructor page,
+then the {\bf implementation} button appears in place of the {\bf
+conditions} button.
+This button tells you what domains or packages actually implement the
+various operations.\footnote{This button often takes a long time; expect
+a delay while you wait for an answer.}
+
+With your user-level set to {\it development}, we suggest you try this
+exercise.
+Return to the main constructor page for \axiomType{Matrix}, then enter
+{\tt Integer} into the input area at the bottom as the value of {\tt R}.
+Then click on {\bf Operations} to produce a table of operations.
+Note that the {\bf conditions} part of the {\it Views} table is
+replaced by {\bf implementation}.
+Click on {\bf implementation}.
+After some delay, you get a page describing what implements each of
+the matrix operations, organized by the various domains and packages.
+
+
+\subsubsection{generalize}
+
+This button only appears for an operation page of a constructor
+involving a unique operation name.
+
+From an operations page for \axiomType{Matrix}, select any
+operation name, say {\bf rank}.
+In the views panel, the {\bf filter} button is  replaced by
+{\bf generalize}.
+Click on it!
+%% Replaced {\bf threshold} with 10 below.  MGR 1995oct31
+What you get is a description of all \Language{} operations
+named \axiomFun{rank}.\footnote{If there were more than 10
+operations of the name, you get instead a page
+with a {\it Views} panel at the bottom and the message to {\bf
+Select a view below}.
+To get the descriptions of all these operations as mentioned
+above, select the {\bf description} button.}
+%See the discussion of {\bf threshold} in
+%\downlink{``\ugBrowseOptionsTitle''}{ugBrowseOptionsPage} in Section \ugBrowseOptionsNumber\ignore{ugBrowseOptions}.} %% Removed MGR 1995oct31
+
+
+\subsubsection{all domains}
+
+This button only appears on an operation page resulting from a
+search from the front page of \Browse{} or from selecting
+{\bf generalize} from an operation page for a constructor.
+
+Note that the {\bf filter} button in the {\it Views} panel is
+replaced by {\bf all domains}.
+Click on it to produce a table of {\it all} domains or packages that
+export a \axiomFun{rank} operation.
+
+
+We note that this table specifically refers to all the \axiomFun{rank}
+operations shown in the preceding page.
+Return to the descriptions of all the \axiomFun{rank} operations and
+select one of them by clicking on the button in its heading.
+Select {\bf all domains}.
+As you see, you have a smaller table of constructors.
+When there is only one constructor, you get the
+constructor page for that constructor.
+\texht{\newpage}{}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugBrowseCapitalizationConventionTitle}{Capitalization Convention}
+\newcommand{\ugBrowseCapitalizationConventionNumber}{14.3.3.}
+%
+% =====================================================================
+\begin{page}{ugBrowseCapitalizationConventionPage}{14.3.3. Capitalization Convention}
+% =====================================================================
+\beginscroll
+
+When entering search keys for constructors, you can use capital
+letters to search for abbreviations.
+For example, enter {\tt UTS} into the input area and click on {\bf
+Constructors}.
+Up comes a page describing \axiomType{UnivariateTaylorSeries}
+whose abbreviation is \axiomType{UTS}.
+
+Constructor abbreviations always have three or more capital
+letters.
+For short constructor names (six letters or less), abbreviations
+are not generally helpful as their abbreviation is typically the
+constructor name in capitals.
+For example, the abbreviation for \axiomType{Matrix} is
+\axiomType{MATRIX}.
+
+Abbreviations can also contain numbers.
+For example, \axiomType{POLY2} is the abbreviation for constructor
+\axiomType{PolynomialFunctions2}.
+For default packages, the abbreviation is the same as the
+abbreviation for the corresponding category with the ``\&''
+replaced by ``-''.
+For example, for the category default package
+\aliascon{MatrixCategory\&}{MATCAT-} the abbreviation is
+\axiomType{MATCAT-} since the corresponding category
+\axiomType{MatrixCategory} has abbreviation \axiomType{MATCAT}.
+
+%% *********************************************************************
+%\head{subsection}{Browse Options}{ugBrowseOptions}
+%% *********************************************************************
+%
+%You can set two options for using \Browse{}: exposure and threshold.
+%
+%\labelSpace{2pc}
+%% *********************************************************************
+%\subsubsection{Exposure}
+%% *********************************************************************
+%
+%By default, the only constructors, operations, and attributes
+%shown by \Browse{} are those from \spadglossSee{exposed constructors}{expose}.
+%To change this, you can issue
+%\syscmdindex{set hyperdoc browse exposure}
+%\begin{verbatim}
+%)set hyperdoc browse exposure on
+%\end{verbatim}
+%After you make this setting, you will see
+%both exposed and unexposed constructs.
+%By definition, an operation or attribute is exposed only if it is
+%exported from an exposed constructor.
+%Unexposed items are generally marked by \Browse{} with an asterisk.
+%For more information on exposure, see \downlink{``\ugTypesExposeTitle''}{ugTypesExposePage} in Section \ugTypesExposeNumber\ignore{ugTypesExpose}.
+%
+%With this setting, try the following experiment.
+%Starting with the main \Browse{} page, enter {\tt *matrix*} into the
+%input area and click on {\bf Constructors}.
+%The result is the following table. %% This line  should be texonly. MGR
+%
+%\begin{texonly}
+%\begin{figure}[htbp]
+%\begin{picture}(324,180)%(-54,0)
+%\hspace*{\baseLeftSkip}\special{psfile=../ps/h-consearch2.ps}
+%\end{picture}
+%\caption{Table of all constructors matching {\tt *matrix*} .}
+%\end{figure}
+%\end{texonly}
+%
+%
+%% *********************************************************************
+%\subsubsection{Threshold}
+%% *********************************************************************
+%
+%For General, Documentation or Complete searches, a summary is presented
+%of all matches.
+%When the number of items of a given kind is less than a number called
+%{\bf threshold}, \Language{} presents a table of names with the heading
+%for that kind.
+%
+%Also, when an operation name is chosen and there are less than {\bf
+%threshold} distinct operations, the operations are initially shown in
+%{\bf description} mode.
+%
+%The default value of {\bf threshold} is 10.
+%To change its value to say 5, issue
+%\syscmdindex{set hyperdoc browse threshold}
+%\begin{verbatim}
+%)set hyperdoc browse threshold 5
+%\end{verbatim}
+%Notice that the headings in
+%the summary are active.
+%If you click on a heading, you bring up a separate page for those
+%entries.
+%%
+%% Above section removed by MGR, 1995oct30, as these two options do
+%% not exist.
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug15.ht b/src/hyper/pages/ug15.ht
new file mode 100644
index 00000000..da2e0eca
--- /dev/null
+++ b/src/hyper/pages/ug15.ht
@@ -0,0 +1,1268 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+
+\texht{\setcounter{chapter}{14}}{} % Chapter 15 
+
+%
+\newcommand{\ugWhatsNewTitle}{What's New in \Language{} Version 2.0}
+\newcommand{\ugWhatsNewNumber}{15.}
+%
+% =====================================================================
+\begin{page}{ugWhatsNewPage}{15. What's New in \Language{} Version 2.0}
+% =====================================================================
+\beginscroll
+
+Many things have changed in this new version of \Language{} and
+we describe many of the more important topics here.
+
+%------------------------------------------------------------------------
+\beginmenu
+    \menudownlink{{15.1. Important Things to Read First}}{ugWhatsNewImportantPage}
+    \menudownlink{{15.2. The New \Language{} Library Compiler}}{ugWhatsNewAsharpPage}
+    \menudownlink{{15.3. The NAG Library Link}}{nagLinkIntroPage}
+    \menudownlink{{15.4. Interactive Front-end and Language}}{ugWhatsNewLanguagePage}
+    \menudownlink{{15.5. Library}}{ugWhatsNewLibraryPage}
+    \menudownlink{{15.6. \HyperName}}{ugWhatsNewHyperDocPage}
+    \menudownlink{{15.7. Documentation}}{ugWhatsNewDocumentationPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugWhatsNewImportantTitle}{Important Things to Read First}
+\newcommand{\ugWhatsNewImportantNumber}{15.1.}
+%
+% =====================================================================
+\begin{page}{ugWhatsNewImportantPage}{15.1. Important Things to Read First}
+% =====================================================================
+\beginscroll
+%------------------------------------------------------------------------
+
+If you have any private {\tt .spad} files (that is, library files
+which were not shipped with \Language{}) you will need to
+recompile them.  For example, if you wrote the file {\tt
+regress.spad} then you should issue {\tt )compile regress.spad}
+before trying to use it.
+
+The internal representation of \axiomType{Union}  has changed. 
+This means that \texht{\linebreak}{} \Language{} data saved 
+with Release 1.x may not
+be readable by this Release. If you cannot recreate the saved data      
+by recomputing in Release 2.0, please contact NAG for assistance.       
+
+%------------------------------------------------------------------------
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugWhatsNewAsharpTitle}{The New \Language{} Library Compiler}
+\newcommand{\ugWhatsNewAsharpNumber}{15.2.}
+%
+% =====================================================================
+\begin{page}{ugWhatsNewAsharpPage}{15.2. The New \Language{} Library Compiler}
+% =====================================================================
+\beginscroll
+%------------------------------------------------------------------------
+
+A new compiler is now available for \Language{}.
+The programming language is referred to as the \Language{} Extension Language
+(or \axiomxl{} for short), and
+improves upon the old \Language{} language in many ways.
+The \spadcmd{)compile} command has been upgraded to be able to
+invoke the new or old compilers.
+The language and the compiler are described in the hard-copy
+documentation which came with your \Language{} system.
+
+To ease the chore of upgrading your {\it .spad} files (old
+compiler) to {\it .as} files (new compiler), the
+\spadcmd{)compile} command has been given a {\tt )translate}
+option. This invokes a special version of the old compiler which
+parses and analyzes your old code and produces augmented code
+using the new syntax.
+Please be aware that the translation is not necessarily one
+hundred percent complete or correct.
+You should attempt to compile the output with the \axiomxl{} compiler
+and make any necessary corrections.
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\nagLinkIntroTitle}{The NAG Library Link}
+\newcommand{\nagLinkIntroNumber}{15.3.}
+%
+% =====================================================================
+\begin{page}{nagLinkIntroPage}{15.3. The NAG Library Link}
+% =====================================================================
+\beginscroll
+
+The \naglib{} link allows you to call NAG Fortran
+routines from within \Language{}, passing \Language{} objects as parameters
+and getting them back as results.
+
+The \naglib{} and, consequently, the link are divided into {\em chapters},
+which cover different areas of numerical analysis.  The statistical
+and sorting {\em chapters} of the Library, however, are not included in the
+link and various support and utility routines (mainly the F06 and X
+{\em chapters}) have been omitted.
+
+Each {\em chapter} has a short (at most three-letter) name;
+for example, the {\em chapter} devoted to the
+solution of ordinary differential equations is called D02.  When
+using the link via the \downlink{\HyperName{} interface}{htxl1},
+you will be presented with a complete menu of these {\em chapters}. The
+names of individual routines within each {\em chapter} are formed by
+adding three letters to the {\em chapter} name, so for example the routine
+for solving ODEs by Adams method is called
+\axiomFunFrom{d02cjf}{NagOrdinaryDifferentialEquationsPackage}.
+
+\beginmenu
+    \menudownlink{{15.3.1. Interpreting NAG Documentation}}{nagDocumentationPage}
+    \menudownlink{{15.3.2. Using the Link}}{nagLinkUsagePage}
+    \menudownlink{{15.3.3. Providing values for Argument Subprograms}}{aspSectionPage}
+    \menudownlink{{15.3.4. General Fortran-generation utilities in \Language{}}}{generalFortranPage}
+    \menudownlink{{15.3.5. Some technical information}}{nagTechnicalPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\nagDocumentationTitle}{Interpreting NAG Documentation}
+\newcommand{\nagDocumentationNumber}{15.3.1.}
+%
+% =====================================================================
+\begin{page}{nagDocumentationPage}{15.3.1. Interpreting NAG Documentation}
+% =====================================================================
+\beginscroll
+
+Information about using the \naglib{} in general, and about using
+individual routines in particular, can be accessed via \HyperName{}.
+This documentation refers to the Fortran routines directly; the
+purpose of this subsection is to explain how this corresponds to the
+\Language{} routines.
+
+For general information about the \naglib{} users should consult
+\downlink{Essential Introduction to the NAG Foundation Library}{manpageXXintro}.
+The documentation is in ASCII format, and a description of the conventions
+used to represent mathematical symbols is given in
+\downlink{Introduction to NAG On-Line Documentation}{manpageXXonline}.
+Advice about choosing a routine from a particular {\em chapter} can be found in
+the \downlink{Chapter Documents}{FoundationLibraryDocPage}.
+
+\subsubsection{Correspondence Between Fortran and \Language{} types}
+
+The NAG documentation refers to the Fortran types of objects; in
+general, the correspondence to \Language{} types is as follows.
+\indent{4}
+\beginitems
+\item[-] Fortran INTEGER corresponds to \Language{} \axiomType{Integer}.
+\item[-] Fortran DOUBLE PRECISION corresponds to \Language{} \axiomType{DoubleFloat}.
+\item[-] Fortran COMPLEX corresponds to \Language{} \axiomType{Complex DoubleFloat}.
+\item[-] Fortran LOGICAL corresponds to \Language{} \axiomType{Boolean}.
+\item[-] Fortran CHARACTER*(*) corresponds to \Language{} \axiomType{String}.
+\enditems
+\indent{0}
+(Exceptionally, for NAG EXTERNAL parameters -- ASPs in link parlance
+-- REAL and COMPLEX correspond to \axiomType{MachineFloat} and \axiomType{MachineComplex},
+respectively; see \downlink{``\aspSectionTitle''}{aspSectionPage} in Section \aspSectionNumber\ignore{aspSection}.)
+
+The correspondence for aggregates is as follows.
+\indent{4}
+\beginitems
+\item[-] A one-dimensional Fortran array corresponds to an \Language{} \texht{\linebreak}{}
+      \axiomType{Matrix} with one column.
+\item[-] A two-dimensional Fortran ARRAY corresponds to an \Language{} \texht{\linebreak}{}
+      \axiomType{Matrix}.
+\item[-] A three-dimensional Fortran ARRAY corresponds to an \Language{} \texht{\linebreak}{}
+      \axiomType{ThreeDimensionalMatrix}.
+\enditems
+\indent{0}
+Higher-dimensional arrays are not currently needed for the \naglib{}.
+
+Arguments which are Fortran FUNCTIONs or SUBROUTINEs correspond
+to special ASP domains in \Language{}. See \downlink{``\aspSectionTitle''}{aspSectionPage} in Section \aspSectionNumber\ignore{aspSection}.
+
+\subsubsection{Classification of NAG parameters}
+
+NAG parameters are classified as belonging to one (or more)
+of the following categories: {\tt Input}, {\tt Output}, {\tt Workspace} or {\tt External} procedure.
+Within {\tt External} procedures a similar classification is used, and parameters
+may also be {\tt Dummies}, or {\tt User Workspace} (data structures not used by the
+NAG routine but provided for the convenience of the user).
+
+When calling a NAG routine via the link the user only provides values
+for {\tt Input} and {\tt External} parameters.
+
+The order of the parameters is, in general, different from  the order
+specified in the \naglib{} documentation. The Browser description
+for each routine helps in determining the correspondence. As a rule of
+thumb, {\tt Input} parameters come first followed by {\tt Input/Output}
+parameters. The {\tt External} parameters are always found at the end.
+
+
+\subsubsection{IFAIL}
+
+NAG routines often return diagnostic information through a parameter called
+\axiom{ifail}.  With a few exceptions, the principle is that on input
+\axiom{ifail} takes
+one of the values $-1,0,1$.  This determines how the routine behaves when
+it encounters an error:
+\indent{4}
+\beginitems
+\item[-] a value of 1 causes the NAG routine to return without printing an error
+message;
+\item[-] a value of 0 causes the NAG routine to print an error message and abort;
+\item[-] a value of -1 causes the NAG routine to return and print an error message.
+\enditems
+\indent{0}
+
+The user is STRONGLY ADVISED to set \axiom{ifail} to \texht{$-1$}{-1} when using the link.
+If \axiom{ifail} has been set to \texht{$1$}{1} or \texht{$-1$}{-1} on input, then its value on output
+will determine the possible cause of any error.  A value of \texht{$0$}{0} indicates
+successful completion, otherwise it provides an index into a table of
+diagnostics provided as part of the routine documentation (accessible via
+\Browse{}).
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\nagLinkUsageTitle}{Using the Link}
+\newcommand{\nagLinkUsageNumber}{15.3.2.}
+%
+% =====================================================================
+\begin{page}{nagLinkUsagePage}{15.3.2. Using the Link}
+% =====================================================================
+\beginscroll
+
+The easiest way to use the link is via the
+\downlink{\HyperName{} interface}{htxl1}.
+You will be presented with a set of fill-in forms where
+you can specify the parameters for each call.  Initially, the forms
+contain example values, demonstrating the use of each routine (these,
+in fact, correspond to the standard NAG example program for the
+routine in question).  For some parameters, these values can provide
+reasonable defaults; others, of course, represent data.  When you
+change a parameter which controls the size of an array, the data in
+that array are reset to a ``neutral'' value -- usually zero.
+
+When you are satisfied with the values entered, clicking on the
+``Continue'' button will display the \Language{} command needed to
+run the chosen NAG routine with these values.  Clicking on the
+``Do It'' button will then cause \Language{} to execute this command
+and return the result in the parent \Language{} session, as described
+below.  Note that, for some routines, multiple HyperDoc ``pages'' are
+required, due to the structure of the data.  For these, returning to
+an earlier page causes HyperDoc to reset the later pages (this is a
+general feature of HyperDoc); in such a case, the simplest way to
+repeat a call, varying a parameter on an earlier page, is probably to
+modify the call displayed in the parent session.
+
+An alternative approach is to call NAG routines directly in your
+normal \Language{} session (that is, using the \Language{}
+interpreter).  Such calls return an
+object of type \axiomType{Result}.  As not
+all parameters in the underlying NAG routine are required in the
+AXIOM call (and the parameter ordering may be different), before
+calling a NAG routine you should consult the description of the
+\Language{} operation in the Browser.  (The quickest route to this
+is to type the routine name, in lower case, into the Browser's
+input area, then click on {\tt Operations}.)  The parameter names
+used coincide with NAG's, although they will appear here in lower
+case.  Of course, it is also possible to become familiar with the
+\Language{} form of a routine by first using it through the
+\downlink{\HyperName{} interface}{htxl1}.
+
+\xtc{
+As an example of this mode of working, we can find a zero
+of a function, lying between 3 and 4, as follows:
+}{
+\spadpaste{answer:=c05adf(3.0,4.0,1.0e-5,0.0,-1,sin(X)::ASP1(F))\bound{answer} }
+}
+\xtc{
+By default, \axiomType{Result} only displays the type of returned values,
+since the amount of information returned can be quite large.  Individual
+components can be examined as follows:
+}{
+\spadpaste{answer . x\free{answer}}
+}
+\xtc{
+}{
+\spadpaste{answer . ifail\free{answer}}
+}
+\xtc{
+In order to avoid conflict with names defined in the workspace, you can also
+get the values by using the \axiomType{String} type (the interpreter automatically
+coerces them to \axiomType{Symbol})
+}{
+\spadpaste{answer "x"\free{answer}}
+}
+
+
+It is possible to have \Language{} display the values of scalar or array
+results automatically.  For more details, see the commands  
+\axiomFunFrom{showScalarValues}{Result}
+and \axiomFunFrom{showArrayValues}{Result}.
+
+\xtc{
+There is also a {\bf .input} file for each NAG routine, containing
+\Language{} interpreter commands to set up and run the standard NAG
+example for that routine.
+}{
+\spadpaste{)read c05adf.input}
+}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\aspSectionTitle}{Providing values for Argument Subprograms}
+\newcommand{\aspSectionNumber}{15.3.3.}
+%
+% =====================================================================
+\begin{page}{aspSectionPage}{15.3.3. Providing values for Argument Subprograms}
+% =====================================================================
+\beginscroll
+
+There are a number of ways in which users can provide values for argument
+subprograms (ASPs).  At the top level the user will see that NAG routines
+require
+an object from the \axiomType{Union} of a \axiomType{Filename} and an ASP.
+\xtc{
+For example \axiomFun{c05adf} requires an object of type \texht{\linebreak}{}
+\axiomType{Union}(fn: \axiomType{FileName},fp: \axiomType{Asp1 F})
+}{
+\spadpaste{)display operation c05adf}
+}
+
+The user thus has a choice of providing the name of a file containing
+Fortran source code, or of somehow generating the ASP within \Language{}.
+If a filename is specified, it is searched for in the {\it local} 
+machine, i.e., the machine that \Language{} is running on.
+
+\subsubsection{Providing ASPs via \axiomType{FortranExpression}}
+
+The \axiomType{FortranExpression} domain is used to represent expressions
+which can be translated into Fortran under certain circumstances.    It is
+very similar to \axiomType{Expression} except that only operators which exist
+in Fortran can be used, and only certain variables can occur.
+For
+example the instantiation \axiomType{FortranExpression([X],[M],MachineFloat)}
+is the domain of expressions containing the scalar \axiom{X} and the array
+\axiom{M}.
+\xtc{
+This allows us to create expressions like:
+}{
+\spadpaste{f : FortranExpression([X],[M],MachineFloat) := sin(X)+M[3,1]}
+}
+\xtc{
+but not
+}{
+\spadpaste{f : FortranExpression([X],[M],MachineFloat) := sin(M)+Y}
+}
+
+Those ASPs which represent expressions usually export a \axiomFun{coerce} from
+an appropriate instantiation of \axiomType{FortranExpression} (or perhaps
+\axiomType{Vector FortranExpression} etc.).  For convenience there are also
+retractions from appropriate instantiations of \axiomType{Expression},
+\axiomType{Polynomial} and \axiomType{Fraction Polynomial}.
+
+\subsubsection{Providing ASPs via \axiomType{FortranCode}}
+
+\texht{\exptypeindex{FortranCode}}{}
+\axiomType{FortranCode} allows us to build arbitrarily complex ASPs via a
+kind of pseudo-code.  It is described fully in
+\downlink{``\generalFortranTitle''}{generalFortranPage} in Section \generalFortranNumber\ignore{generalFortran}.
+
+Every ASP exports two \axiomFun{coerce} functions: one from
+\axiomType{FortranCode} and one from \axiomType{List FortranCode}.  There
+is also a \axiomFun{coerce} from \texht{\linebreak}{}
+\axiomType{Record( localSymbols: SymbolTable, code: List FortranCode)}
+which is used for passing extra symbol information about the ASP.
+
+\xtc{
+So for example, to integrate the function abs(x) we could use the built-in
+\axiomFun{abs} function.  But suppose we want to get back to basics and define
+it directly, then we could do the following:
+}{
+\spadpaste{d01ajf(-1.0, 1.0, 0.0, 1.0e-5, 800, 200, -1, cond(LT(X,0), assign(F,-X), assign(F,X))) result }
+}
+The \axiomFunFrom{cond}{FortranCode} operation creates a conditional clause
+and the \axiomFunFrom{assign}{FortranCode} an assignment statement.
+
+\subsubsection{Providing ASPs via \axiomType{FileName}}
+
+Suppose we have created the file ``asp.f'' as follows:
+\begin{verbatim}
+      DOUBLE PRECISION FUNCTION F(X)
+      DOUBLE PRECISION X
+      F=4.0D0/(X*X+1.0D0)
+      RETURN
+      END
+\end{verbatim}
+and wish to pass it to the NAG
+routine \axiomFun{d01ajf} which performs one-dimensional quadrature.
+We can do this as follows:
+\begin{verbatim}
+d01ajf(0.0 ,1.0, 0.0, 1.0e-5, 800, 200, -1, "asp.f")
+\end{verbatim}
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\generalFortranTitle}{General Fortran-generation utilities in \Language{}}
+\newcommand{\generalFortranNumber}{15.3.4.}
+%
+% =====================================================================
+\begin{page}{generalFortranPage}{15.3.4. General Fortran-generation utilities in \Language{}}
+% =====================================================================
+\beginscroll
+
+This section describes more advanced facilities which are available to users
+who wish to generate Fortran code from within \Language{}.  There are
+facilities to manipulate templates, store type information, and generate
+code fragments or complete programs.
+
+\subsubsection{Template Manipulation}
+
+A template is a skeletal program which is ``fleshed out'' with data when
+it is processed.  It is a sequence of {\em active} and {\em passive} parts:
+active parts are sequences of \Language{} commands which are processed as if they
+had been typed into the interpreter; passive parts are simply echoed
+verbatim on the Fortran output stream.
+
+Suppose, for example, that we have the following template, stored in
+the file ``test.tem'':
+\begin{verbatim}
+-- A simple template
+beginVerbatim
+      DOUBLE PRECISION FUNCTION F(X)
+      DOUBLE PRECISION X
+endVerbatim
+outputAsFortran("F",f)
+beginVerbatim
+      RETURN
+      END
+endVerbatim
+\end{verbatim}
+The passive parts lie between the two
+tokens {\tt beginVerbatim} and \texht{\linebreak}{} {\tt endVerbatim}.  There
+are two active statements: one which is simply an \Language{} (
+\texht{\verb+--+}{\-\-})
+comment, and one which produces an assignment to the current value
+of {\tt f}.  We could use it as follows:
+\begin{verbatim}
+(4) ->f := 4.0/(1+X**2)
+
+           4
+   (4)   ------
+          2
+         X  + 1
+                       
+(5) ->processTemplate "test.tem"
+      DOUBLE PRECISION FUNCTION F(X)
+      DOUBLE PRECISION X
+      F=4.0D0/(X*X+1.0D0)
+      RETURN 
+      END
+
+   (5)  "CONSOLE"
+\end{verbatim}
+
+(A more reliable method of specifying the filename will be introduced
+below.)  Note that the Fortran assignment {\tt F=4.0D0/(X*X+1.0D0)}
+automatically converted 4.0 and 1 into DOUBLE PRECISION numbers; in
+general, the \Language{} Fortran generation facility will convert
+anything which should be a floating point object into either
+a Fortran REAL or DOUBLE PRECISION object.
+\xtc{
+Which alternative is used is determined by the command
+}{
+\spadpaste{)set fortran precision}
+}
+
+It is sometimes useful to end a template before the file itself ends (e.g. to
+allow the template to be tested incrementally or so that a piece of text
+describing how the template works can be included).  It is of course possible
+to ``comment-out'' the remainder of the file.  Alternatively, the single token
+{\tt endInput} as part of an active portion of the template will cause
+processing to be ended prematurely at that point.
+
+The \axiomFun{processTemplate} command comes in two flavours.  In the first case,
+illustrated above, it takes one argument of domain \axiomType{FileName},
+the name of the template to be processed, and writes its output on the
+current Fortran output stream.  In general, a filename can be generated
+from {\em directory}, {\em name} and {\em extension} components, using
+the operation \axiomFun{filename}, as in
+\begin{verbatim}
+processTemplate filename("","test","tem")
+\end{verbatim}
+There is an alternative version of \axiomFun{processTemplate}, which
+takes two arguments (both of domain \axiomType{FileName}).  In this case the
+first argument is the name of the template to be processed, and the
+second is the file in which to write the results.  Both versions return
+the location of the generated Fortran code as their result
+({\tt "CONSOLE"} in the above example).
+
+It is sometimes useful to be able to mix active and passive parts of a
+line or statement.  For example you might want to generate a Fortran
+Comment describing your data set.  For this kind of application we
+provide three functions as follows:
+\texht
+{
+\begin{tabular}{p{1.8in}p{2.6in}}
+\axiomFun{fortranLiteral} & writes a string on the Fortran output stream \\
+ & \\
+\axiomFun{fortranCarriageReturn} & writes a carriage return on the Fortran output stream \\
+& \\
+\axiomFun{fortranLiteralLine} & writes a string followed by a return
+on the Fortran output stream \\
+\end{tabular}
+}
+{
+\newline
+\axiomFun{fortranLiteral}\tab{25}writes a string on the Fortran output stream\newline
+\axiomFun{fortranCarriageReturn}\tab{25}writes a carriage return on the Fortran output stream\newline
+\axiomFun{fortranLiteralLine}\tab{25}writes a string followed by a return on the Fortran output stream\newline
+}
+\xtc{
+So we could create our comment as follows:
+}{
+\spadpaste{m := matrix [[1,2,3],[4,5,6]]\bound{m}}
+}
+\xtc{
+}{
+\spadpaste{fortranLiteralLine concat ["C\ \ \ \ \ \ The\ Matrix\ has\ ", nrows(m)::String, "\ rows\ and\ ", ncols(m)::String, "\ columns"]\free{m}}
+}
+\xtc{
+or, alternatively:
+}{
+\spadpaste{fortranLiteral "C\ \ \ \ \ \ The\ Matrix\ has\ "}
+}
+\xtc{
+}{
+\spadpaste{fortranLiteral(nrows(m)::String)}
+}
+\xtc{
+}{
+\spadpaste{fortranLiteral "\ rows\ and\ "}
+}
+\xtc{
+}{
+\spadpaste{fortranLiteral(ncols(m)::String)\free{m}}
+}
+\xtc{
+}{
+\spadpaste{fortranLiteral "\ columns"}
+}
+\xtc{
+}{
+\spadpaste{fortranCarriageReturn()}
+}
+
+We should stress that these functions, together with the \axiomFun{outputAsFortran}
+function are the {\em only} sure ways
+of getting output to appear on the Fortran output stream.  Attempts to use
+\Language{} commands such as \axiomFun{output} or \axiomFunX{writeline} may appear to give
+the required result when displayed on the console, but will give the wrong
+result when Fortran and algebraic output are sent to differing locations.  On
+the other hand, these functions can be used to send helpful messages to the
+user, without interfering with the generated Fortran.
+
+\subsubsection{Manipulating the Fortran Output Stream}
+\texht{\exptypeindex{FortranOutputStackPackage}}{}
+
+Sometimes it is useful to manipulate the Fortran output stream in a program,
+possibly without being aware of its current value.  The main use of this is
+for gathering type declarations (see ``Fortran Types'' below) but it can be useful
+in other contexts as well.  Thus we provide a set of commands to manipulate
+a stack of (open) output streams.  Only one stream can be written to at
+any given time.  The stack is never empty---its initial value is the
+console or the current value of the Fortran output stream, and can be
+determined using
+\xtc{
+}{
+\spadpaste{topFortranOutputStack()}
+}
+(see below).
+The commands available to manipulate the stack are:
+\texht{
+\begin{tabular}{ll}
+\axiomFun{clearFortranOutputStack} & resets the stack to the console \\
+ & \\
+\axiomFun{pushFortranOutputStack} & pushes a \axiomType{FileName} onto the stack \\
+ & \\
+\axiomFun{popFortranOutputStack} & pops the stack \\
+ & \\
+\axiomFun{showFortranOutputStack} & returns the current stack \\
+ & \\
+\axiomFun{topFortranOutputStack} & returns the top element of the stack \\
+\end{tabular}
+}
+{
+\newline
+\axiomFun{clearFortranOutputStack}\tab{25}resets the stack\newline
+\axiomFun{pushFortranOutputStack}\tab{25}pushes a \axiomType{FileName} onto the stack\newline
+\axiomFun{popFortranOutputStack}\tab{25}pops the stack\newline
+\axiomFun{showFortranOutputStack}\tab{25}returns the current stack\newline
+\axiomFun{topFortranOutputStack}\tab{25}returns the top element of the stack\newline
+}
+These commands are all part of \axiomType{FortranOutputStackPackage}.
+
+\subsubsection{Fortran Types}
+
+When generating code it is important to keep track of the Fortran types of
+the objects which we are generating.  This is useful for a number of reasons,
+not least to ensure that we are actually generating legal Fortran code.  The
+current type system is built up in several layers, and we shall describe each
+in turn.
+
+\subsubsection{FortranScalarType}
+\texht{\exptypeindex{FortranScalarType}}{}
+
+This domain represents the simple Fortran datatypes: REAL, DOUBLE PRECISION,
+COMPLEX, LOGICAL, INTEGER, and CHARACTER.
+It is possible to \axiomFun{coerce} a \axiomType{String} or \axiomType{Symbol}
+into the domain, test whether two objects are equal, and also apply
+the predicate functions \axiomFunFrom{real?}{FortranScalarType} etc.
+
+\subsubsection{FortranType}
+\texht{\exptypeindex{FortranType}}{}
+
+This domain represents ``full'' types: i.e., datatype plus array dimensions
+(where appropriate) plus whether or not the parameter is an external
+subprogram.  It is possible to \axiomFun{coerce} an object of
+\axiomType{FortranScalarType} into the domain or \axiomFun{construct} one
+from an element of \axiomType{FortranScalarType}, a list of
+\axiomType{Polynomial Integer}s (which can of course be simple integers or
+symbols) representing its dimensions, and
+a \axiomType{Boolean} declaring whether it is external or not.  The list
+of dimensions must be empty if the \axiomType{Boolean} is {\tt true}.
+The functions \axiomFun{scalarTypeOf}, \axiomFun{dimensionsOf} and
+\axiomFun{external?} return the appropriate
+parts, and it is possible to get the various basic Fortran Types via
+functions like \axiomFun{fortranReal}.
+\xtc{
+For example:
+}{
+\spadpaste{type:=construct(real,[i,10],false)$FortranType}
+}
+\xtc{
+or
+}{
+\spadpaste{type:=[real,[i,10],false]$FortranType\bound{type}}
+}
+\xtc{
+}{
+\spadpaste{scalarTypeOf type\free{type}}
+}
+\xtc{
+}{
+\spadpaste{dimensionsOf type\free{type}}
+}
+\xtc{
+}{
+\spadpaste{external?  type\free{type}}
+}
+\xtc{
+}{
+\spadpaste{fortranLogical()}
+}
+\xtc{
+}{
+\spadpaste{construct(integer,[],true)$FortranType}
+}
+
+\subsubsection{SymbolTable}
+\texht{\exptypeindex{SymbolTable}}{}
+
+This domain creates and manipulates a symbol table for generated Fortran code.
+This is used by \axiomType{FortranProgram} to represent the types of objects in
+a subprogram.  The commands available are:
+\texht{
+\begin{tabular}{ll}
+\axiomFun{empty} & creates a new \axiomType{SymbolTable} \\
+ & \\
+\axiomFunX{declare} & creates a new entry in a table \\
+ & \\
+\axiomFun{fortranTypeOf} & returns the type of an object in a table \\
+ & \\
+\axiomFun{parametersOf} & returns a list of all the symbols in the table \\
+ & \\
+\axiomFun{typeList} & returns a list of all objects of a given type \\
+ & \\
+\axiomFun{typeLists} & returns a list of lists of all objects sorted by type \\
+ & \\
+\axiomFun{externalList} & returns a list of all {\tt EXTERNAL} objects \\
+ & \\
+\axiomFun{printTypes} & produces Fortran type declarations from a table\\
+\end{tabular}
+}
+{
+\newline
+\axiomFun{empty}\tab{25}creates a new \axiomType{SymbolTable}\newline
+\axiomFunX{declare}\tab{25}creates a new entry in a table \newline
+\axiomFun{fortranTypeOf}\tab{25}returns the type of an object in a table \newline
+\axiomFun{parametersOf}\tab{25}returns a list of all the symbols in the table \newline
+\axiomFun{typeList}\tab{25}returns a list of all objects of a given type \newline
+\axiomFun{typeLists}\tab{25}returns a list of lists of all objects sorted by type \newline
+\axiomFun{externalList}\tab{25}returns a list of all {\tt EXTERNAL} objects \newline
+\axiomFun{printTypes}\tab{25}produces Fortran type declarations from a table\newline
+}
+\xtc{
+}{
+\spadpaste{symbols := empty()$SymbolTable\bound{symbols}}
+}
+\xtc{
+}{
+\spadpaste{declare!(X,fortranReal(),symbols)\free{symbols}}
+}
+\xtc{
+}{
+\spadpaste{declare!(M,construct(real,[i,j],false)$FortranType,symbols)\free{symbols}}
+}
+\xtc{
+}{
+\spadpaste{declare!([i,j],fortranInteger(),symbols)\free{symbols}}
+}
+\xtc{
+}{
+\spadpaste{symbols\free{symbols}}
+}
+\xtc{
+}{
+\spadpaste{fortranTypeOf(i,symbols)\free{symbols}}
+}
+\xtc{
+}{
+\spadpaste{typeList(real,symbols)\free{symbols}}
+}
+\xtc{
+}{
+\spadpaste{printTypes symbols\free{symbols}}
+}
+
+\subsubsection{TheSymbolTable}
+\texht{\exptypeindex{TheSymbolTable}}{}
+
+This domain creates and manipulates one global symbol table to be used, for
+example, during template processing. It is
+also used when
+linking to external Fortran routines. The
+information stored for each subprogram (and the main program segment, where
+relevant) is:
+\indent{4}
+\beginitems
+\item[-] its name;
+\item[-] its return type;
+\item[-] its argument list;
+\item[-] and its argument types.
+\enditems
+\indent{0}
+Initially, any information provided is deemed to be for the main program
+segment.
+\xtc{
+Issuing the following command indicates that from now on all information
+refers to the subprogram \axiom{F}.
+}{
+\spadpaste{newSubProgram F}
+}
+\xtc{
+It is possible to return to processing the main program segment by issuing
+the command:
+}{
+\spadpaste{endSubProgram()}
+}
+The following commands exist:
+\texht{
+\begin{tabular}{p{1.6in}p{2.8in}}
+\axiomFunX{returnType} & declares the return type of the current subprogram \\
+ & \\
+\axiomFun{returnTypeOf} & returns the return type of a subprogram \\
+ & \\
+\axiomFunX{argumentList} &  declares the argument list of the current subprogram \\
+ & \\
+\axiomFun{argumentListOf} &  returns the argument list of a subprogram \\
+ & \\
+\axiomFunX{declare} & provides type declarations for parameters of the current subprogram \\
+ & \\
+\axiomFun{symbolTableOf} & returns the symbol table  of a subprogram \\
+ & \\
+\axiomFun{printHeader} & produces the Fortran header for the current subprogram \\
+\end{tabular}
+}
+{
+\newline
+\axiomFunX{returnType}\tab{25}declares the return type of the current subprogram \newline
+\axiomFun{returnTypeOf}\tab{25}returns the return type of a subprogram \newline
+\axiomFunX{argumentList}\tab{25}declares the argument list of the current subprogram \newline
+\axiomFun{argumentListOf}\tab{25}returns the argument list of a subprogram \newline
+\axiomFunX{declare}\tab{25}provides type declarations for parameters of the current subprogram \newline
+\axiomFun{symbolTableOf}\tab{25}returns the symbol table  of a subprogram \newline
+\axiomFun{printHeader}\tab{25}produce the Fortran header for the current subprogram \newline
+}
+In addition there are versions of these commands which are parameterised by
+the name of a subprogram, and others parameterised by both the name of a
+subprogram and by an instance of \axiomType{TheSymbolTable}.
+\xtc{
+}{
+\spadpaste{newSubProgram F \bound{forPleasure}}
+}
+\xtc{
+}{
+\spadpaste{argumentList!(F,[X])\free{forPleasure}}
+}
+\xtc{
+}{
+\spadpaste{returnType!(F,real)\free{forPleasure}}
+}
+\xtc{
+}{
+\spadpaste{declare!(X,fortranReal(),F)\free{forPleasure}}
+}
+\xtc{
+}{
+\spadpaste{printHeader F\free{forPleasure}}
+}
+
+\subsubsection{Advanced Fortran Code Generation}
+
+This section describes facilities for representing Fortran statements, and
+building up complete subprograms from them.
+
+\subsubsection{Switch}
+\texht{\exptypeindex{Switch}}{}
+
+This domain is used to represent statements like {\tt x < y}.  Although
+these can be represented directly in \Language{}, it is a little cumbersome,
+since \Language{} evaluates the last statement, for example, to \axiom{true}
+(since \axiom{x} is  lexicographically less than \axiom{y}).
+
+Instead we have a set of operations, such as \axiomFun{LT} to represent \axiom{<},
+to let us build such statements.  The available constructors are:
+\texht{
+\centerline{{\begin{tabular}{ll}}}
+\centerline{{\axiomFun{LT} & $<$ }}
+\centerline{{\axiomFun{GT} & $>$ }}
+\centerline{{\axiomFun{LE} & $\leq$ }}
+\centerline{{\axiomFun{GE} & $\geq$ }}
+\centerline{{\axiomFun{EQ} & $=$ }}
+\centerline{{\axiomFun{AND} & $and$}}
+\centerline{{\axiomFun{OR} & $or$ }}
+\centerline{{\axiomFun{NOT} & $not$ }}
+\centerline{{\end{tabular}}}
+}
+{
+\newline
+\axiomFun{LT}\tab{25}\texht{$<$}{<} \newline
+\axiomFun{GT}\tab{25}\texht{$>$}{>} \newline
+\axiomFun{LE}\tab{25}\texht{$\leq$}{<=} \newline
+\axiomFun{GE}\tab{25}\texht{$\geq$}{>=} \newline
+\axiomFun{EQ}\tab{25}\texht{$=$}{=} \newline
+\axiomFun{AND}\tab{25}\texht{$and$}{{\tt and}}\newline
+\axiomFun{OR}\tab{25}\texht{$or$}{{\tt or}} \newline
+\axiomFun{NOT}\tab{25}\texht{$not$}{{\tt not}} \newline
+}
+\xtc{
+So for example:
+}{
+\spadpaste{LT(x,y)}
+}
+
+\subsubsection{FortranCode}
+
+This domain represents code segments or operations: currently assignments,
+conditionals, blocks, comments, gotos, continues, various kinds of loops,
+and return statements.
+\xtc{
+For example we can create quite a complicated conditional statement using
+assignments, and then turn it into Fortran code:
+}{
+\spadpaste{c := cond(LT(X,Y),assign(F,X),cond(GT(Y,Z),assign(F,Y),assign(F,Z)))\bound{c}}
+}
+\xtc{
+}{
+\spadpaste{printCode c\free{c}}
+}
+
+The Fortran code is printed
+on the current Fortran output stream.
+
+\subsubsection{FortranProgram}
+\texht{\exptypeindex{FortranProgram}}{}
+
+This domain is used to construct complete Fortran subprograms out of
+elements of \axiomType{FortranCode}.  It is parameterised by the name of the
+target subprogram (a \axiomType{Symbol}), its return type (from
+\axiomType{Union}(\axiomType{FortranScalarType},``void'')),
+its arguments (from \axiomType{List Symbol}), and
+its symbol table (from \axiomType{SymbolTable}).  One can
+\axiomFun{coerce} elements of either \axiomType{FortranCode}
+or \axiomType{Expression} into it.
+
+\xtc{
+First of all we create a symbol table:
+}{
+\spadpaste{symbols := empty()$SymbolTable\bound{symbols}}
+}
+\xtc{
+Now put some type declarations into it:
+}{
+\spadpaste{declare!([X,Y],fortranReal(),symbols)\free{symbols}}
+}
+\xtc{
+Then (for convenience)
+we set up the particular instantiation of \axiomType{FortranProgram}
+}{
+\spadpaste{FP := FortranProgram(F,real,[X,Y],symbols)\free{symbols}\bound{FP}}
+}
+\xtc{
+Create an object of type \axiomType{Expression(Integer)}:
+}{
+\spadpaste{asp := X*sin(Y)\bound{asp}}
+}
+\xtc{
+Now \axiomFun{coerce} it into \axiomType{FP}, and print its Fortran form:
+}{
+\spadpaste{outputAsFortran(asp::FP)\free{FP asp}}
+}
+
+We can generate a \axiomType{FortranProgram} using \axiom{FortranCode}.  For
+example:
+\xtc{
+Augment our symbol table:
+}{
+\spadpaste{declare!(Z,fortranReal(),symbols)\free{symbols}\bound{Z}}
+}
+\xtc{
+and transform the conditional expression we prepared earlier:
+}{
+\spadpaste{outputAsFortran([c,returns()]::FP) \free{FP c Z}}
+}
+
+%------------------------------------------------------------------------
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\nagTechnicalTitle}{Some technical information}
+\newcommand{\nagTechnicalNumber}{15.3.5.}
+%
+% =====================================================================
+\begin{page}{nagTechnicalPage}{15.3.5. Some technical information}
+% =====================================================================
+\beginscroll
+%------------------------------------------------------------------------
+
+The model adopted for the link is a server-client configuration
+-- \Language{} acting as a client via a local agent
+(a process called {\tt nagman}). The server side is implemented
+by the {\tt nagd} daemon process which may run on a different host.
+The {\tt nagman} local agent is started by default whenever you
+start \Language{}. The {\tt nagd} server must be started separately.
+Instructions for installing and running the server are supplied
+in \texht{\downlink{``\nugNagdTitle''}{nugNagdPage} in Section \nugNagdNumber\ignore{nugNagd}}{printed form}. 
+Use the \spadcmd{)set naglink host} system command
+to point your local agent to a server in your network.
+
+
+
+On the \Language{} side, one sees a set of {\em packages}
+(ask \Browse{} for {\em Nag*}) for each chapter, each exporting
+operations with the same name as a routine in the \naglib{}.
+The arguments and return value of each operation belong to
+standard \Language{} types.
+
+The {\tt man} pages for the \naglib{} are accessible via the description
+of each operation in \Browse{} (among other places).
+
+In the implementation of each operation, the set of inputs is passed
+to the local agent {\tt nagman}, which makes a
+Remote Procedure Call (RPC) to the
+remote {\tt nagd} daemon process.  The local agent receives the RPC
+results and forwards them to the \Language{} workspace where they
+are interpreted appropriately.
+
+How are Fortran subroutines turned into RPC calls?
+For each Fortran routine in the \naglib{}, a C main() routine
+is supplied.
+Its job is to assemble the RPC input (numeric) data stream into
+the appropriate Fortran data structures for the routine, call the Fortran
+routine from C and serialize the results into an RPC output data stream.
+
+Many \naglib{} routines accept ASPs (Argument Subprogram Parameters).
+These specify user-supplied Fortran routines (e.g. a routine to
+supply values of a function is required for numerical integration).
+How are they handled? There are new facilities in \Language{} to help.
+A set of \Language{} domains has been provided to turn values in standard
+ \Language{} types (such as Expression Integer) into the appropriate
+piece of Fortran for each case (a filename pointing to Fortran source
+for the ASP can always be supplied instead).
+Ask \Browse{} for {\em Asp*} to see these domains. The Fortran fragments
+are included in the outgoing RPC stream, but {\tt nagd} intercepts them,
+compiles them, and links them with the main() C program before executing
+the resulting program on the numeric part of the RPC stream.
+
+
+%------------------------------------------------------------------------
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugWhatsNewLanguageTitle}{Interactive Front-end and Language}
+\newcommand{\ugWhatsNewLanguageNumber}{15.4.}
+%
+% =====================================================================
+\begin{page}{ugWhatsNewLanguagePage}{15.4. Interactive Front-end and Language}
+% =====================================================================
+\beginscroll
+%------------------------------------------------------------------------
+
+The \axiom{leave} keyword has been replaced by the
+\axiom{break} keyword for compatibility with the new \Language{}
+extension language.
+See section \downlink{``\ugLangLoopsBreakTitle''}{ugLangLoopsBreakPage} in Section \ugLangLoopsBreakNumber\ignore{ugLangLoopsBreak}
+for more information.
+
+Curly braces are no longer used to create sets. Instead, use
+\axiomFun{set} followed by a bracketed expression. For example,
+\xtc{
+}{
+\spadpaste{set [1,2,3,4]}
+}
+
+Curly braces are now used to enclose a block (see section
+\downlink{``\ugLangBlocksTitle''}{ugLangBlocksPage} in Section \ugLangBlocksNumber\ignore{ugLangBlocks}
+for more information). For compatibility, a block can still be 
+enclosed by parentheses as well.
+
+``Free functions'' created by the \axiomxl{} compiler can now be
+loaded and used within the \Language{} interpreter. A {\it free
+function} is a library function that is implemented outside a
+domain or category constructor.
+
+New coercions to and from type \axiomType{Expression} have been
+added. For example, it is now possible to map a polynomial
+represented as an expression to an appropriate polynomial type.
+
+Various messages have been added or rewritten for clarity.
+
+%------------------------------------------------------------------------
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugWhatsNewLibraryTitle}{Library}
+\newcommand{\ugWhatsNewLibraryNumber}{15.5.}
+%
+% =====================================================================
+\begin{page}{ugWhatsNewLibraryPage}{15.5. Library}
+% =====================================================================
+\beginscroll
+%------------------------------------------------------------------------
+
+The \axiomType{FullPartialFractionExpansion}
+domain has been added. This domain computes factor-free full
+partial fraction expansions.
+See section
+\downlink{`FullPartialFractionExpansion'}{FullPartialFractionExpansionXmpPage}\ignore{FullPartialFractionExpansion}
+for examples.
+
+We have implemented the Bertrand/Cantor algorithm for integrals of
+hyperelliptic functions. This brings a major speedup for some
+classes of algebraic integrals.
+
+We have implemented a new (direct) algorithm for integrating trigonometric
+functions. This brings a speedup and an improvement in the answer
+quality.
+
+The {\sf SmallFloat} domain has been renamed
+\axiomType{DoubleFloat} and {\sf SmallInteger} has been renamed
+\axiomType{SingleInteger}. The new abbreviations as
+\axiomType{DFLOAT} and \axiomType{SINT}, respectively.
+We have defined the macro {\sf SF}, the old abbreviation for {\sf
+SmallFloat}, to expand to \axiomType{DoubleFloat} and modified
+the documentation and input file examples to use the new names
+and abbreviations. You should do the same in any private \Language{}
+files you have.
+
+There are many new categories, domains and packages related to the
+NAG Library Link facility. See the file
+
+\unixcommand{\env{AXIOM}/../../src/algebra/exposed.lsp}{xterm\ -e\ vi\ +"/naglink"\ \env{AXIOM}/../../src/algebra/exposed.lsp}
+
+for a list of constructors in the {\bf naglink} \Language{} exposure group.
+
+We have made improvements to the differential equation solvers
+and there is a new facility for solving systems of first-order 
+linear differential equations.
+In particular, an important fix was made to the solver for
+inhomogeneous linear ordinary differential equations that
+corrected the calculation of particular solutions.
+We also made improvements to the polynomial
+and transcendental equation solvers including the
+ability to solve some classes of systems of transcendental
+equations.
+
+The efficiency of power series have been improved and left and right
+expansions of \spad{tan(f(x))} at \spad{x =} a pole of \spad{f(x)}
+can now be computed.
+A number of power series bugs were fixed and the \axiomType{GeneralUnivariatePowerSeries}
+domain was added.
+The power series variable can appear in the coefficients and when this
+happens, you cannot differentiate or integrate the series.  Differentiation
+and integration with respect to other variables is supported.
+
+A domain was added for representing asymptotic expansions of a
+function at an exponential singularity.
+
+For limits, the main new feature is the exponential expansion domain used
+to treat certain exponential singularities.  Previously, such singularities
+were treated in an {\it ad hoc} way and only a few cases were covered.  Now
+\Language{} can do things like
+
+\begin{verbatim}
+limit( (x+1)**(x+1)/x**x - x**x/(x-1)**(x-1), x = %plusInfinity)
+\end{verbatim}
+
+in a systematic way.  It only does one level of nesting, though.  In other
+words, we can handle \spad{exp(} some function with a pole \spad{)}, but not
+\texht{\linebreak}{} \spad{exp(exp(} some function with a pole \spad{)).}
+
+The computation of integral bases has been improved through careful
+use of Hermite row reduction. A P-adic algorithm
+for function fields of algebraic curves in finite characteristic has also
+been developed.
+
+Miscellaneous:
+There is improved conversion of definite and indefinite integrals to
+\axiomType{InputForm};
+binomial coefficients are displayed in a new way;
+some new simplifications of radicals have been implemented;
+the operation \spadfun{complexForm} for converting to rectangular coordinates
+has been added;
+symmetric product operations have been added to \axiomType{LinearOrdinaryDifferentialOperator}.
+
+%------------------------------------------------------------------------
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugWhatsNewHyperDocTitle}{\HyperName}
+\newcommand{\ugWhatsNewHyperDocNumber}{15.6.}
+%
+% =====================================================================
+\begin{page}{ugWhatsNewHyperDocPage}{15.6. \HyperName}
+% =====================================================================
+\beginscroll
+%------------------------------------------------------------------------
+
+The buttons on the titlebar and scrollbar have been replaced
+with ones which have a 3D effect. You can change the foreground and
+background colors of these ``controls'' by including and modifying
+the following lines in your {\bf .Xdefaults} file.
+\begin{verbatim}
+Axiom.hyperdoc.ControlBackground: White
+Axiom.hyperdoc.ControlForeground: Black
+\end{verbatim}
+
+For various reasons, \HyperName{} sometimes displays a
+secondary window. You can control the size and placement of this
+window by including and modifying
+the following line in your {\bf .Xdefaults} file.
+%
+\begin{verbatim}
+Axiom.hyperdoc.FormGeometry: =950x450+100+0
+\end{verbatim}
+%
+This setting is a standard X Window System geometry specification:
+you are requesting a window 950 pixels wide by 450 deep and placed in
+the upper left corner.
+
+Some key definitions have been changed to conform more closely
+with the CUA guidelines. Press
+\texht{F9}{\downlink{F9}{ugHyperKeysPage}}
+to see the current definitions.
+
+Input boxes (for example, in the Browser) now accept paste-ins from
+the X Window System. Use the second button to paste in something
+you have previously copied or cut. An example of how you can use this
+is that you can paste the type from an \Language{} computation
+into the main Browser input box.
+
+
+%------------------------------------------------------------------------
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugWhatsNewDocumentationTitle}{Documentation}
+\newcommand{\ugWhatsNewDocumentationNumber}{15.7.}
+%
+% =====================================================================
+\begin{page}{ugWhatsNewDocumentationPage}{15.7. Documentation}
+% =====================================================================
+\beginscroll
+%------------------------------------------------------------------------
+\texht{
+We describe here a few additions to the on-line
+version of the AXIOM book which you can read with
+HyperDoc.
+}{}
+
+A section has been added to the graphics chapter, describing
+how to build \twodim{} graphs from lists of points. An example is
+given showing how to read the points from a file.
+See section \downlink{``\ugGraphTwoDbuildTitle''}{ugGraphTwoDbuildPage} in Section \ugGraphTwoDbuildNumber\ignore{ugGraphTwoDbuild}
+for details.
+
+A further section has been added to that same chapter, describing
+how to add a \twodim{} graph to a viewport which already
+contains other graphs.
+See section
+\downlink{``\ugGraphTwoDappendTitle''}{ugGraphTwoDappendPage} in Section \ugGraphTwoDappendNumber\ignore{ugGraphTwoDappend}
+for details.
+
+Chapter 3 
+and the on-line \HyperName{} help have been unified.
+
+An explanation of operation names ending in ``?'' and ``!'' has
+been added to the first chapter. 
+See the
+end of the section
+\downlink{``\ugIntroCallFunTitle''}{ugIntroCallFunPage} in Section \ugIntroCallFunNumber\ignore{ugIntroCallFun}
+for details.
+
+An expanded explanation of using predicates has
+been added to the sixth chapter. See the
+example involving \userfun{evenRule} in the middle of the section
+\downlink{``\ugUserRulesTitle''}{ugUserRulesPage} in Section \ugUserRulesNumber\ignore{ugUserRules}
+for details.
+
+Documentation for the \spadcmd{)compile}, \spadcmd{)library} and
+\spadcmd{)load} commands has been greatly changed. This reflects
+the ability of the \spadcmd{)compile} to now invoke the \axiomxl{}
+compiler, the impending deletion of the \spadcmd{)load} command
+and the new \spadcmd{)library} command.
+The \spadcmd{)library} command replaces \spadcmd{)load} and is
+compatible with the compiled output from both the old and new
+compilers.
+
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug15.pht b/src/hyper/pages/ug15.pht
new file mode 100644
index 00000000..ffe70e48
--- /dev/null
+++ b/src/hyper/pages/ug15.pht
@@ -0,0 +1,833 @@
+\begin{patch}{ugWhatsNewLanguagePagePatch1}
+\begin{paste}{ugWhatsNewLanguagePageFull1}{ugWhatsNewLanguagePageEmpty1}
+\pastebutton{ugWhatsNewLanguagePageFull1}{\hidepaste}
+\tab{5}\spadcommand{set [1,2,3,4]}
+\indentrel{3}\begin{verbatim}
+   (1)  {1,2,3,4}
+                              Type: Set PositiveInteger
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{ugWhatsNewLanguagePageEmpty1}
+\begin{paste}{ugWhatsNewLanguagePageEmpty1}{ugWhatsNewLanguagePagePatch1}
+\pastebutton{ugWhatsNewLanguagePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{set [1,2,3,4]}
+\end{paste}\end{patch}
+
+\begin{patch}{aspSectionPagePatch1}
+\begin{paste}{aspSectionPageFull1}{aspSectionPageEmpty1}
+\pastebutton{aspSectionPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)display operation c05adf}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{aspSectionPageEmpty1}
+\begin{paste}{aspSectionPageEmpty1}{aspSectionPagePatch1}
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+
+\begin{patch}{aspSectionPagePatch2}
+\begin{paste}{aspSectionPageFull2}{aspSectionPageEmpty2}
+\pastebutton{aspSectionPageFull2}{\hidepaste}
+\tab{5}\spadcommand{f : FortranExpression([X],[M],MachineFloat) := sin(X)+M[3,1]}
+\indentrel{3}\begin{verbatim}
+   (1)  sin(X) + M
+                  3,1
+          Type: FortranExpression([X],[M],MachineFloat)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{aspSectionPageEmpty2}
+\begin{paste}{aspSectionPageEmpty2}{aspSectionPagePatch2}
+\pastebutton{aspSectionPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{f : FortranExpression([X],[M],MachineFloat) := sin(X)+M[3,1]}
+\end{paste}\end{patch}
+
+\begin{patch}{aspSectionPagePatch3}
+\begin{paste}{aspSectionPageFull3}{aspSectionPageEmpty3}
+\pastebutton{aspSectionPageFull3}{\hidepaste}
+\tab{5}\spadcommand{f : FortranExpression([X],[M],MachineFloat) := sin(M)+Y}
+\indentrel{3}\begin{verbatim}
+   sin(M) + Y
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{aspSectionPageEmpty3}
+\begin{paste}{aspSectionPageEmpty3}{aspSectionPagePatch3}
+\pastebutton{aspSectionPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{f : FortranExpression([X],[M],MachineFloat) := sin(M)+Y}
+\end{paste}\end{patch}
+
+\begin{patch}{aspSectionPagePatch4}
+\begin{paste}{aspSectionPageFull4}{aspSectionPageEmpty4}
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+\indentrel{3}\begin{verbatim}
+   (2)  1.0
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{aspSectionPageEmpty4}
+\begin{paste}{aspSectionPageEmpty4}{aspSectionPagePatch4}
+\pastebutton{aspSectionPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{d01ajf(-1.0, 1.0, 0.0, 1.0e-5, 800, 200, -1, cond(LT(X,0), assign(F,-X), assign(F,X))) result}
+\end{paste}\end{patch}
+
+\begin{patch}{nagLinkUsagePagePatch1}
+\begin{paste}{nagLinkUsagePageFull1}{nagLinkUsagePageEmpty1}
+\pastebutton{nagLinkUsagePageFull1}{\hidepaste}
+\tab{5}\spadcommand{answer:=c05adf(3.0,4.0,1.0e-5,0.0,-1,sin(X)::ASP1(F))\bound{answer }}
+\indentrel{3}\begin{verbatim}
+   (1)  [ifail: Integer,x: DoubleFloat]
+                                           Type: Result
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{nagLinkUsagePageEmpty1}
+\begin{paste}{nagLinkUsagePageEmpty1}{nagLinkUsagePagePatch1}
+\pastebutton{nagLinkUsagePageEmpty1}{\showpaste}
+\tab{5}\spadcommand{answer:=c05adf(3.0,4.0,1.0e-5,0.0,-1,sin(X)::ASP1(F))\bound{answer }}
+\end{paste}\end{patch}
+
+\begin{patch}{nagLinkUsagePagePatch2}
+\begin{paste}{nagLinkUsagePageFull2}{nagLinkUsagePageEmpty2}
+\pastebutton{nagLinkUsagePageFull2}{\hidepaste}
+\tab{5}\spadcommand{answer . x\free{answer }}
+\indentrel{3}\begin{verbatim}
+   (2)  3.14159265545896
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{nagLinkUsagePageEmpty2}
+\begin{paste}{nagLinkUsagePageEmpty2}{nagLinkUsagePagePatch2}
+\pastebutton{nagLinkUsagePageEmpty2}{\showpaste}
+\tab{5}\spadcommand{answer . x\free{answer }}
+\end{paste}\end{patch}
+
+\begin{patch}{nagLinkUsagePagePatch3}
+\begin{paste}{nagLinkUsagePageFull3}{nagLinkUsagePageEmpty3}
+\pastebutton{nagLinkUsagePageFull3}{\hidepaste}
+\tab{5}\spadcommand{answer . ifail\free{answer }}
+\indentrel{3}\begin{verbatim}
+   (3)  0
+                                          Type: Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{nagLinkUsagePageEmpty3}
+\begin{paste}{nagLinkUsagePageEmpty3}{nagLinkUsagePagePatch3}
+\pastebutton{nagLinkUsagePageEmpty3}{\showpaste}
+\tab{5}\spadcommand{answer . ifail\free{answer }}
+\end{paste}\end{patch}
+
+\begin{patch}{nagLinkUsagePagePatch4}
+\begin{paste}{nagLinkUsagePageFull4}{nagLinkUsagePageEmpty4}
+\pastebutton{nagLinkUsagePageFull4}{\hidepaste}
+\tab{5}\spadcommand{answer "x"\free{answer }}
+\indentrel{3}\begin{verbatim}
+   (4)  3.14159265545896
+                                      Type: DoubleFloat
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{nagLinkUsagePageEmpty4}
+\begin{paste}{nagLinkUsagePageEmpty4}{nagLinkUsagePagePatch4}
+\pastebutton{nagLinkUsagePageEmpty4}{\showpaste}
+\tab{5}\spadcommand{answer "x"\free{answer }}
+\end{paste}\end{patch}
+
+\begin{patch}{nagLinkUsagePagePatch5}
+\begin{paste}{nagLinkUsagePageFull5}{nagLinkUsagePageEmpty5}
+\pastebutton{nagLinkUsagePageFull5}{\hidepaste}
+\tab{5}\spadcommand{)read c05adf.input}
+\indentrel{3}\begin{verbatim}
+   (1)  true
+                                          Type: Boolean
+   (2)  true
+                                          Type: Boolean
+   (3)  F
+                                           Type: Asp1 F
+   (4)  0.0
+                                      Type: DoubleFloat
+   (5)  1.0
+                                      Type: DoubleFloat
+   (6)  1.0e-05
+                                      Type: DoubleFloat
+   (7)  0.0
+                                      Type: DoubleFloat
+   (8)  [ifail: 0,x: 0.567143306604963]
+                                           Type: Result
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{nagLinkUsagePageEmpty5}
+\begin{paste}{nagLinkUsagePageEmpty5}{nagLinkUsagePagePatch5}
+\pastebutton{nagLinkUsagePageEmpty5}{\showpaste}
+\tab{5}\spadcommand{)read c05adf.input}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch1}
+\begin{paste}{generalFortranPageFull1}{generalFortranPageEmpty1}
+\pastebutton{generalFortranPageFull1}{\hidepaste}
+\tab{5}\spadcommand{)set fortran precision}
+\indentrel{3}\begin{verbatim}
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty1}
+\begin{paste}{generalFortranPageEmpty1}{generalFortranPagePatch1}
+\pastebutton{generalFortranPageEmpty1}{\showpaste}
+\tab{5}\spadcommand{)set fortran precision}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch2}
+\begin{paste}{generalFortranPageFull2}{generalFortranPageEmpty2}
+\pastebutton{generalFortranPageFull2}{\hidepaste}
+\tab{5}\spadcommand{m := matrix [[1,2,3],[4,5,6]]\bound{m }}
+\indentrel{3}\begin{verbatim}
+        Ú1  2  3¿
+   (1)  ³       ³
+        À4  5  6Ù
+                                   Type: Matrix Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty2}
+\begin{paste}{generalFortranPageEmpty2}{generalFortranPagePatch2}
+\pastebutton{generalFortranPageEmpty2}{\showpaste}
+\tab{5}\spadcommand{m := matrix [[1,2,3],[4,5,6]]\bound{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch3}
+\begin{paste}{generalFortranPageFull3}{generalFortranPageEmpty3}
+\pastebutton{generalFortranPageFull3}{\hidepaste}
+\tab{5}\spadcommand{fortranLiteralLine concat ["C      The Matrix has ", nrows(m)::String, " rows and ", ncols(m)::String, " columns"]\free{m }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty3}
+\begin{paste}{generalFortranPageEmpty3}{generalFortranPagePatch3}
+\pastebutton{generalFortranPageEmpty3}{\showpaste}
+\tab{5}\spadcommand{fortranLiteralLine concat ["C      The Matrix has ", nrows(m)::String, " rows and ", ncols(m)::String, " columns"]\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch4}
+\begin{paste}{generalFortranPageFull4}{generalFortranPageEmpty4}
+\pastebutton{generalFortranPageFull4}{\hidepaste}
+\tab{5}\spadcommand{fortranLiteral "C      The Matrix has "}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty4}
+\begin{paste}{generalFortranPageEmpty4}{generalFortranPagePatch4}
+\pastebutton{generalFortranPageEmpty4}{\showpaste}
+\tab{5}\spadcommand{fortranLiteral "C      The Matrix has "}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch5}
+\begin{paste}{generalFortranPageFull5}{generalFortranPageEmpty5}
+\pastebutton{generalFortranPageFull5}{\hidepaste}
+\tab{5}\spadcommand{fortranLiteral(nrows(m)::String)}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty5}
+\begin{paste}{generalFortranPageEmpty5}{generalFortranPagePatch5}
+\pastebutton{generalFortranPageEmpty5}{\showpaste}
+\tab{5}\spadcommand{fortranLiteral(nrows(m)::String)}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch6}
+\begin{paste}{generalFortranPageFull6}{generalFortranPageEmpty6}
+\pastebutton{generalFortranPageFull6}{\hidepaste}
+\tab{5}\spadcommand{fortranLiteral " rows and "}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty6}
+\begin{paste}{generalFortranPageEmpty6}{generalFortranPagePatch6}
+\pastebutton{generalFortranPageEmpty6}{\showpaste}
+\tab{5}\spadcommand{fortranLiteral " rows and "}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch7}
+\begin{paste}{generalFortranPageFull7}{generalFortranPageEmpty7}
+\pastebutton{generalFortranPageFull7}{\hidepaste}
+\tab{5}\spadcommand{fortranLiteral(ncols(m)::String)\free{m }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty7}
+\begin{paste}{generalFortranPageEmpty7}{generalFortranPagePatch7}
+\pastebutton{generalFortranPageEmpty7}{\showpaste}
+\tab{5}\spadcommand{fortranLiteral(ncols(m)::String)\free{m }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch8}
+\begin{paste}{generalFortranPageFull8}{generalFortranPageEmpty8}
+\pastebutton{generalFortranPageFull8}{\hidepaste}
+\tab{5}\spadcommand{fortranLiteral " columns"}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty8}
+\begin{paste}{generalFortranPageEmpty8}{generalFortranPagePatch8}
+\pastebutton{generalFortranPageEmpty8}{\showpaste}
+\tab{5}\spadcommand{fortranLiteral " columns"}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch9}
+\begin{paste}{generalFortranPageFull9}{generalFortranPageEmpty9}
+\pastebutton{generalFortranPageFull9}{\hidepaste}
+\tab{5}\spadcommand{fortranCarriageReturn()}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty9}
+\begin{paste}{generalFortranPageEmpty9}{generalFortranPagePatch9}
+\pastebutton{generalFortranPageEmpty9}{\showpaste}
+\tab{5}\spadcommand{fortranCarriageReturn()}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch10}
+\begin{paste}{generalFortranPageFull10}{generalFortranPageEmpty10}
+\pastebutton{generalFortranPageFull10}{\hidepaste}
+\tab{5}\spadcommand{topFortranOutputStack()}
+\indentrel{3}\begin{verbatim}
+   (9)  "CONSOLE"
+                                           Type: String
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty10}
+\begin{paste}{generalFortranPageEmpty10}{generalFortranPagePatch10}
+\pastebutton{generalFortranPageEmpty10}{\showpaste}
+\tab{5}\spadcommand{topFortranOutputStack()}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch11}
+\begin{paste}{generalFortranPageFull11}{generalFortranPageEmpty11}
+\pastebutton{generalFortranPageFull11}{\hidepaste}
+\tab{5}\spadcommand{type:=construct(real,[i,10],false)$FortranType}
+\indentrel{3}\begin{verbatim}
+   (10)  REAL
+             (i,10)
+                                      Type: FortranType
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty11}
+\begin{paste}{generalFortranPageEmpty11}{generalFortranPagePatch11}
+\pastebutton{generalFortranPageEmpty11}{\showpaste}
+\tab{5}\spadcommand{type:=construct(real,[i,10],false)$FortranType}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch12}
+\begin{paste}{generalFortranPageFull12}{generalFortranPageEmpty12}
+\pastebutton{generalFortranPageFull12}{\hidepaste}
+\tab{5}\spadcommand{type:=[real,[i,10],false]$FortranType\bound{type }}
+\indentrel{3}\begin{verbatim}
+   (11)  REAL
+             (i,10)
+                                      Type: FortranType
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty12}
+\begin{paste}{generalFortranPageEmpty12}{generalFortranPagePatch12}
+\pastebutton{generalFortranPageEmpty12}{\showpaste}
+\tab{5}\spadcommand{type:=[real,[i,10],false]$FortranType\bound{type }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch13}
+\begin{paste}{generalFortranPageFull13}{generalFortranPageEmpty13}
+\pastebutton{generalFortranPageFull13}{\hidepaste}
+\tab{5}\spadcommand{scalarTypeOf type\free{type }}
+\indentrel{3}\begin{verbatim}
+   (12)  REAL
+                Type: Union(fst: FortranScalarType,...)
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty13}
+\begin{paste}{generalFortranPageEmpty13}{generalFortranPagePatch13}
+\pastebutton{generalFortranPageEmpty13}{\showpaste}
+\tab{5}\spadcommand{scalarTypeOf type\free{type }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch14}
+\begin{paste}{generalFortranPageFull14}{generalFortranPageEmpty14}
+\pastebutton{generalFortranPageFull14}{\hidepaste}
+\tab{5}\spadcommand{dimensionsOf type\free{type }}
+\indentrel{3}\begin{verbatim}
+   (13)  [i,10]
+                          Type: List Polynomial Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty14}
+\begin{paste}{generalFortranPageEmpty14}{generalFortranPagePatch14}
+\pastebutton{generalFortranPageEmpty14}{\showpaste}
+\tab{5}\spadcommand{dimensionsOf type\free{type }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch15}
+\begin{paste}{generalFortranPageFull15}{generalFortranPageEmpty15}
+\pastebutton{generalFortranPageFull15}{\hidepaste}
+\tab{5}\spadcommand{external? type\free{type }}
+\indentrel{3}\begin{verbatim}
+   (14)  false
+                                          Type: Boolean
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty15}
+\begin{paste}{generalFortranPageEmpty15}{generalFortranPagePatch15}
+\pastebutton{generalFortranPageEmpty15}{\showpaste}
+\tab{5}\spadcommand{external? type\free{type }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch16}
+\begin{paste}{generalFortranPageFull16}{generalFortranPageEmpty16}
+\pastebutton{generalFortranPageFull16}{\hidepaste}
+\tab{5}\spadcommand{fortranLogical()}
+\indentrel{3}\begin{verbatim}
+   (15)  LOGICAL
+                                      Type: FortranType
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty16}
+\begin{paste}{generalFortranPageEmpty16}{generalFortranPagePatch16}
+\pastebutton{generalFortranPageEmpty16}{\showpaste}
+\tab{5}\spadcommand{fortranLogical()}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch17}
+\begin{paste}{generalFortranPageFull17}{generalFortranPageEmpty17}
+\pastebutton{generalFortranPageFull17}{\hidepaste}
+\tab{5}\spadcommand{construct(integer,[],true)$FortranType}
+\indentrel{3}\begin{verbatim}
+   (16)  EXTERNAL INTEGER
+                                      Type: FortranType
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty17}
+\begin{paste}{generalFortranPageEmpty17}{generalFortranPagePatch17}
+\pastebutton{generalFortranPageEmpty17}{\showpaste}
+\tab{5}\spadcommand{construct(integer,[],true)$FortranType}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch18}
+\begin{paste}{generalFortranPageFull18}{generalFortranPageEmpty18}
+\pastebutton{generalFortranPageFull18}{\hidepaste}
+\tab{5}\spadcommand{symbols := empty()$SymbolTable\bound{symbols }}
+\indentrel{3}\begin{verbatim}
+   (17)  table()
+                                      Type: SymbolTable
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty18}
+\begin{paste}{generalFortranPageEmpty18}{generalFortranPagePatch18}
+\pastebutton{generalFortranPageEmpty18}{\showpaste}
+\tab{5}\spadcommand{symbols := empty()$SymbolTable\bound{symbols }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch19}
+\begin{paste}{generalFortranPageFull19}{generalFortranPageEmpty19}
+\pastebutton{generalFortranPageFull19}{\hidepaste}
+\tab{5}\spadcommand{declare!(X,fortranReal(),symbols)\free{symbols }}
+\indentrel{3}\begin{verbatim}
+   (18)  REAL
+                                      Type: FortranType
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty19}
+\begin{paste}{generalFortranPageEmpty19}{generalFortranPagePatch19}
+\pastebutton{generalFortranPageEmpty19}{\showpaste}
+\tab{5}\spadcommand{declare!(X,fortranReal(),symbols)\free{symbols }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch20}
+\begin{paste}{generalFortranPageFull20}{generalFortranPageEmpty20}
+\pastebutton{generalFortranPageFull20}{\hidepaste}
+\tab{5}\spadcommand{declare!(M,construct(real,[i,j],false)$FortranType,symbols)\free{symbols }}
+\indentrel{3}\begin{verbatim}
+   (19)  REAL
+             (i,j)
+                                      Type: FortranType
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty20}
+\begin{paste}{generalFortranPageEmpty20}{generalFortranPagePatch20}
+\pastebutton{generalFortranPageEmpty20}{\showpaste}
+\tab{5}\spadcommand{declare!(M,construct(real,[i,j],false)$FortranType,symbols)\free{symbols }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch21}
+\begin{paste}{generalFortranPageFull21}{generalFortranPageEmpty21}
+\pastebutton{generalFortranPageFull21}{\hidepaste}
+\tab{5}\spadcommand{declare!([i,j],fortranInteger(),symbols)\free{symbols }}
+\indentrel{3}\begin{verbatim}
+   (20)  INTEGER
+                                      Type: FortranType
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty21}
+\begin{paste}{generalFortranPageEmpty21}{generalFortranPagePatch21}
+\pastebutton{generalFortranPageEmpty21}{\showpaste}
+\tab{5}\spadcommand{declare!([i,j],fortranInteger(),symbols)\free{symbols }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch22}
+\begin{paste}{generalFortranPageFull22}{generalFortranPageEmpty22}
+\pastebutton{generalFortranPageFull22}{\hidepaste}
+\tab{5}\spadcommand{symbols\free{symbols }}
+\indentrel{3}\begin{verbatim}
+   (21)
+   table(X= REAL,M= REAL     ,i= INTEGER,j= INTEGER)
+                        (i,j)
+                                      Type: SymbolTable
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty22}
+\begin{paste}{generalFortranPageEmpty22}{generalFortranPagePatch22}
+\pastebutton{generalFortranPageEmpty22}{\showpaste}
+\tab{5}\spadcommand{symbols\free{symbols }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch23}
+\begin{paste}{generalFortranPageFull23}{generalFortranPageEmpty23}
+\pastebutton{generalFortranPageFull23}{\hidepaste}
+\tab{5}\spadcommand{fortranTypeOf(i,symbols)\free{symbols }}
+\indentrel{3}\begin{verbatim}
+   (22)  INTEGER
+                                      Type: FortranType
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty23}
+\begin{paste}{generalFortranPageEmpty23}{generalFortranPagePatch23}
+\pastebutton{generalFortranPageEmpty23}{\showpaste}
+\tab{5}\spadcommand{fortranTypeOf(i,symbols)\free{symbols }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch24}
+\begin{paste}{generalFortranPageFull24}{generalFortranPageEmpty24}
+\pastebutton{generalFortranPageFull24}{\hidepaste}
+\tab{5}\spadcommand{typeList(real,symbols)\free{symbols }}
+\indentrel{3}\begin{verbatim}
+   (23)  [X,[M,i,j]]
+Type: List Union(name: Symbol,bounds: List Union(S: Symbol,P: Polynomial Integer))
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty24}
+\begin{paste}{generalFortranPageEmpty24}{generalFortranPagePatch24}
+\pastebutton{generalFortranPageEmpty24}{\showpaste}
+\tab{5}\spadcommand{typeList(real,symbols)\free{symbols }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch25}
+\begin{paste}{generalFortranPageFull25}{generalFortranPageEmpty25}
+\pastebutton{generalFortranPageFull25}{\hidepaste}
+\tab{5}\spadcommand{printTypes symbols\free{symbols }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty25}
+\begin{paste}{generalFortranPageEmpty25}{generalFortranPagePatch25}
+\pastebutton{generalFortranPageEmpty25}{\showpaste}
+\tab{5}\spadcommand{printTypes symbols\free{symbols }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch26}
+\begin{paste}{generalFortranPageFull26}{generalFortranPageEmpty26}
+\pastebutton{generalFortranPageFull26}{\hidepaste}
+\tab{5}\spadcommand{newSubProgram F}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty26}
+\begin{paste}{generalFortranPageEmpty26}{generalFortranPagePatch26}
+\pastebutton{generalFortranPageEmpty26}{\showpaste}
+\tab{5}\spadcommand{newSubProgram F}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch27}
+\begin{paste}{generalFortranPageFull27}{generalFortranPageEmpty27}
+\pastebutton{generalFortranPageFull27}{\hidepaste}
+\tab{5}\spadcommand{endSubProgram()}
+\indentrel{3}\begin{verbatim}
+   (26)  MAIN
+                                           Type: Symbol
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty27}
+\begin{paste}{generalFortranPageEmpty27}{generalFortranPagePatch27}
+\pastebutton{generalFortranPageEmpty27}{\showpaste}
+\tab{5}\spadcommand{endSubProgram()}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch28}
+\begin{paste}{generalFortranPageFull28}{generalFortranPageEmpty28}
+\pastebutton{generalFortranPageFull28}{\hidepaste}
+\tab{5}\spadcommand{newSubProgram F\bound{forPleasure }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty28}
+\begin{paste}{generalFortranPageEmpty28}{generalFortranPagePatch28}
+\pastebutton{generalFortranPageEmpty28}{\showpaste}
+\tab{5}\spadcommand{newSubProgram F\bound{forPleasure }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch29}
+\begin{paste}{generalFortranPageFull29}{generalFortranPageEmpty29}
+\pastebutton{generalFortranPageFull29}{\hidepaste}
+\tab{5}\spadcommand{argumentList!(F,[X])\free{forPleasure }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty29}
+\begin{paste}{generalFortranPageEmpty29}{generalFortranPagePatch29}
+\pastebutton{generalFortranPageEmpty29}{\showpaste}
+\tab{5}\spadcommand{argumentList!(F,[X])\free{forPleasure }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch30}
+\begin{paste}{generalFortranPageFull30}{generalFortranPageEmpty30}
+\pastebutton{generalFortranPageFull30}{\hidepaste}
+\tab{5}\spadcommand{returnType!(F,real)\free{forPleasure }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty30}
+\begin{paste}{generalFortranPageEmpty30}{generalFortranPagePatch30}
+\pastebutton{generalFortranPageEmpty30}{\showpaste}
+\tab{5}\spadcommand{returnType!(F,real)\free{forPleasure }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch31}
+\begin{paste}{generalFortranPageFull31}{generalFortranPageEmpty31}
+\pastebutton{generalFortranPageFull31}{\hidepaste}
+\tab{5}\spadcommand{declare!(X,fortranReal(),F)\free{forPleasure }}
+\indentrel{3}\begin{verbatim}
+   (30)  REAL
+                                      Type: FortranType
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty31}
+\begin{paste}{generalFortranPageEmpty31}{generalFortranPagePatch31}
+\pastebutton{generalFortranPageEmpty31}{\showpaste}
+\tab{5}\spadcommand{declare!(X,fortranReal(),F)\free{forPleasure }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch32}
+\begin{paste}{generalFortranPageFull32}{generalFortranPageEmpty32}
+\pastebutton{generalFortranPageFull32}{\hidepaste}
+\tab{5}\spadcommand{printHeader F\free{forPleasure }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty32}
+\begin{paste}{generalFortranPageEmpty32}{generalFortranPagePatch32}
+\pastebutton{generalFortranPageEmpty32}{\showpaste}
+\tab{5}\spadcommand{printHeader F\free{forPleasure }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch33}
+\begin{paste}{generalFortranPageFull33}{generalFortranPageEmpty33}
+\pastebutton{generalFortranPageFull33}{\hidepaste}
+\tab{5}\spadcommand{LT(x,y)}
+\indentrel{3}\begin{verbatim}
+   (32)  x < y
+                                           Type: Switch
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty33}
+\begin{paste}{generalFortranPageEmpty33}{generalFortranPagePatch33}
+\pastebutton{generalFortranPageEmpty33}{\showpaste}
+\tab{5}\spadcommand{LT(x,y)}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch34}
+\begin{paste}{generalFortranPageFull34}{generalFortranPageEmpty34}
+\pastebutton{generalFortranPageFull34}{\hidepaste}
+\tab{5}\spadcommand{c := cond(LT(X,Y),assign(F,X),cond(GT(Y,Z),assign(F,Y),assign(F,Z)))\bound{c }}
+\indentrel{3}\begin{verbatim}
+   (33)  conditional
+                                      Type: FortranCode
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty34}
+\begin{paste}{generalFortranPageEmpty34}{generalFortranPagePatch34}
+\pastebutton{generalFortranPageEmpty34}{\showpaste}
+\tab{5}\spadcommand{c := cond(LT(X,Y),assign(F,X),cond(GT(Y,Z),assign(F,Y),assign(F,Z)))\bound{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch35}
+\begin{paste}{generalFortranPageFull35}{generalFortranPageEmpty35}
+\pastebutton{generalFortranPageFull35}{\hidepaste}
+\tab{5}\spadcommand{printCode c\free{c }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty35}
+\begin{paste}{generalFortranPageEmpty35}{generalFortranPagePatch35}
+\pastebutton{generalFortranPageEmpty35}{\showpaste}
+\tab{5}\spadcommand{printCode c\free{c }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch36}
+\begin{paste}{generalFortranPageFull36}{generalFortranPageEmpty36}
+\pastebutton{generalFortranPageFull36}{\hidepaste}
+\tab{5}\spadcommand{symbols := empty()$SymbolTable\bound{symbols }}
+\indentrel{3}\begin{verbatim}
+   (35)  table()
+                                      Type: SymbolTable
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty36}
+\begin{paste}{generalFortranPageEmpty36}{generalFortranPagePatch36}
+\pastebutton{generalFortranPageEmpty36}{\showpaste}
+\tab{5}\spadcommand{symbols := empty()$SymbolTable\bound{symbols }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch37}
+\begin{paste}{generalFortranPageFull37}{generalFortranPageEmpty37}
+\pastebutton{generalFortranPageFull37}{\hidepaste}
+\tab{5}\spadcommand{declare!([X,Y],fortranReal(),symbols)\free{symbols }}
+\indentrel{3}\begin{verbatim}
+   (36)  REAL
+                                      Type: FortranType
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty37}
+\begin{paste}{generalFortranPageEmpty37}{generalFortranPagePatch37}
+\pastebutton{generalFortranPageEmpty37}{\showpaste}
+\tab{5}\spadcommand{declare!([X,Y],fortranReal(),symbols)\free{symbols }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch38}
+\begin{paste}{generalFortranPageFull38}{generalFortranPageEmpty38}
+\pastebutton{generalFortranPageFull38}{\hidepaste}
+\tab{5}\spadcommand{FP := FortranProgram(F,real,[X,Y],symbols)\free{symbols }\bound{FP }}
+\indentrel{3}\begin{verbatim}
+   (37)  FortranProgram(F,REAL,[X,Y],table(...,...))
+                                           Type: Domain
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty38}
+\begin{paste}{generalFortranPageEmpty38}{generalFortranPagePatch38}
+\pastebutton{generalFortranPageEmpty38}{\showpaste}
+\tab{5}\spadcommand{FP := FortranProgram(F,real,[X,Y],symbols)\free{symbols }\bound{FP }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch39}
+\begin{paste}{generalFortranPageFull39}{generalFortranPageEmpty39}
+\pastebutton{generalFortranPageFull39}{\hidepaste}
+\tab{5}\spadcommand{asp := X*sin(Y)\bound{asp }}
+\indentrel{3}\begin{verbatim}
+   (38)  X sin(Y)
+                               Type: Expression Integer
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty39}
+\begin{paste}{generalFortranPageEmpty39}{generalFortranPagePatch39}
+\pastebutton{generalFortranPageEmpty39}{\showpaste}
+\tab{5}\spadcommand{asp := X*sin(Y)\bound{asp }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch40}
+\begin{paste}{generalFortranPageFull40}{generalFortranPageEmpty40}
+\pastebutton{generalFortranPageFull40}{\hidepaste}
+\tab{5}\spadcommand{outputAsFortran(asp::FP)\free{FP asp }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty40}
+\begin{paste}{generalFortranPageEmpty40}{generalFortranPagePatch40}
+\pastebutton{generalFortranPageEmpty40}{\showpaste}
+\tab{5}\spadcommand{outputAsFortran(asp::FP)\free{FP asp }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch41}
+\begin{paste}{generalFortranPageFull41}{generalFortranPageEmpty41}
+\pastebutton{generalFortranPageFull41}{\hidepaste}
+\tab{5}\spadcommand{declare!(Z,fortranReal(),symbols)\free{symbols }\bound{Z }}
+\indentrel{3}\begin{verbatim}
+   (40)  REAL
+                                      Type: FortranType
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty41}
+\begin{paste}{generalFortranPageEmpty41}{generalFortranPagePatch41}
+\pastebutton{generalFortranPageEmpty41}{\showpaste}
+\tab{5}\spadcommand{declare!(Z,fortranReal(),symbols)\free{symbols }\bound{Z }}
+\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPagePatch42}
+\begin{paste}{generalFortranPageFull42}{generalFortranPageEmpty42}
+\pastebutton{generalFortranPageFull42}{\hidepaste}
+\tab{5}\spadcommand{outputAsFortran([c,returns()]::FP)\free{FP c Z }}
+\indentrel{3}\begin{verbatim}
+                                             Type: Void
+\end{verbatim}
+\indentrel{-3}\end{paste}\end{patch}
+
+\begin{patch}{generalFortranPageEmpty42}
+\begin{paste}{generalFortranPageEmpty42}{generalFortranPagePatch42}
+\pastebutton{generalFortranPageEmpty42}{\showpaste}
+\tab{5}\spadcommand{outputAsFortran([c,returns()]::FP)\free{FP c Z }}
+\end{paste}\end{patch}
+
diff --git a/src/hyper/pages/ug16.ht b/src/hyper/pages/ug16.ht
new file mode 100644
index 00000000..87620726
--- /dev/null
+++ b/src/hyper/pages/ug16.ht
@@ -0,0 +1,2606 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\texht{\setcounter{chapter}{0}}{} % Appendix A
+
+\newcommand{\lanb}{{\tt [}}
+\newcommand{\ranb}{{\tt ]}}
+\newcommand{\vertline}{\texht{$|$}{{\tt |}}}
+
+%
+\newcommand{\ugSysCmdTitle}{\Language{} System Commands}
+\newcommand{\ugSysCmdNumber}{B.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdPage}{B. \Language{} System Commands}
+% =====================================================================
+\beginscroll
+
+\texht{\bgroup\baselineskip 10pt\ixpt{}\def\Isize{\SIsize}}{}
+
+This chapter describes system commands, the command-line
+facilities used to control the \Language{} environment.
+The first section is an introduction and discusses the common
+syntax of the commands available.
+
+\table{
+  { \downlink{\menuitemstyle{A.1. Introduction}}{ugSysCmdOverviewPage} }
+  { \downlink{\menuitemstyle{A.2. )abbreviation}}{ugSysCmdabbreviationPage} }
+  { \downlink{\menuitemstyle{A.3. )boot}}{ugSysCmdbootPage} }
+  { \downlink{\menuitemstyle{A.4. )cd}}{ugSysCmdcdPage} }
+  { \downlink{\menuitemstyle{A.5. )close}}{ugSysCmdclosePage} }
+  { \downlink{\menuitemstyle{A.6. )clear}}{ugSysCmdclearPage} }
+  { \downlink{\menuitemstyle{A.7. )compile}}{ugSysCmdcompilePage} }
+  { \downlink{\menuitemstyle{A.8. )display}}{ugSysCmddisplayPage} }
+  { \downlink{\menuitemstyle{A.9. )edit}}{ugSysCmdeditPage} }
+  { \downlink{\menuitemstyle{A.10. )fin}}{ugSysCmdfinPage} }
+  { \downlink{\menuitemstyle{A.11. )frame}}{ugSysCmdframePage} }
+  { \downlink{\menuitemstyle{A.12. )help}}{ugSysCmdhelpPage} }
+  { \downlink{\menuitemstyle{A.13. )history}}{ugSysCmdhistoryPage} }
+  { \downlink{\menuitemstyle{A.14. )library}}{ugSysCmdlibraryPage} }
+  { \downlink{\menuitemstyle{A.15. )lisp}}{ugSysCmdlispPage} }
+  { \downlink{\menuitemstyle{A.16. )load}}{ugSysCmdloadPage} }
+  { \downlink{\menuitemstyle{A.17. )ltrace}}{ugSysCmdltracePage} }
+  { \downlink{\menuitemstyle{A.18. )pquit}}{ugSysCmdpquitPage} }
+  { \downlink{\menuitemstyle{A.19. )quit}}{ugSysCmdquitPage} }
+  { \downlink{\menuitemstyle{A.20. )read}}{ugSysCmdreadPage} }
+  { \downlink{\menuitemstyle{A.21. )set}}{ugSysCmdsetPage} }
+  { \downlink{\menuitemstyle{A.22. )show}}{ugSysCmdshowPage} }
+  { \downlink{\menuitemstyle{A.23. )spool}}{ugSysCmdspoolPage} }
+  { \downlink{\menuitemstyle{A.24. )synonym}}{ugSysCmdsynonymPage} }
+  { \downlink{\menuitemstyle{A.25. )system}}{ugSysCmdsystemPage} }
+  { \downlink{\menuitemstyle{A.26. )trace}}{ugSysCmdtracePage} }
+  { \downlink{\menuitemstyle{A.27. )undo}}{ugSysCmdundoPage} }
+  { \downlink{\menuitemstyle{A.28. )what}}{ugSysCmdwhatPage} }
+}
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdOverviewTitle}{Introduction}
+\newcommand{\ugSysCmdOverviewNumber}{B.1.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdOverviewPage}{B.1. Introduction}
+% =====================================================================
+\beginscroll
+
+System commands are used to perform \Language{} environment
+management.
+Among the commands are those that display what has been defined or
+computed, set up multiple logical \Language{} environments
+(frames), clear definitions, read files of expressions and
+commands, show what functions are available, and terminate
+\Language{}.
+
+Some commands are restricted: the commands
+%-% \HDsyscmdindex{set userlevel interpreter}{ugSysCmdOverviewPage}{B.1.}{Introduction}
+%-% \HDsyscmdindex{set userlevel compiler}{ugSysCmdOverviewPage}{B.1.}{Introduction}
+%-% \HDsyscmdindex{set userlevel development}{ugSysCmdOverviewPage}{B.1.}{Introduction}
+\begin{verbatim}
+)set userlevel interpreter
+)set userlevel compiler
+)set userlevel development
+\end{verbatim}
+set the user-access level to the three possible choices.
+All commands are available at {\tt development} level and the fewest
+are available at {\tt interpreter} level.
+The default user-level is {\tt interpreter}.
+%-% \HDindex{user-level}{ugSysCmdOverviewPage}{B.1.}{Introduction}
+In addition to the \spadcmd{)set} command (discussed in \downlink{``\ugSysCmdsetTitle''}{ugSysCmdsetPage} in Section \ugSysCmdsetNumber\ignore{ugSysCmdset})
+you can use the \HyperName{} settings facility to change the {\it user-level.}
+\texht{}{Click on \lispmemolink{Settings}{(|htSystemVariables|)}
+here to immediately go to the settings facility.}
+
+Each command listing begins with one or more syntax pattern descriptions
+plus examples of related commands.
+The syntax descriptions are intended to be easy to read and do not
+necessarily represent the most compact way of specifying all
+possible arguments and options; the descriptions may occasionally
+be redundant.
+
+All system commands begin with a right parenthesis which should be in
+the first available column of the input line (that is, immediately
+after the input prompt, if any).
+System commands may be issued directly to \Language{} or be
+included in {\bf .input} files.
+%-% \HDindex{file!input}{ugSysCmdOverviewPage}{B.1.}{Introduction}
+
+A system command {\it argument} is a word that directly
+follows the command name and is not followed or preceded by a
+right parenthesis.
+A system command {\it option} follows the system command and
+is directly preceded by a right parenthesis.
+Options may have arguments: they directly follow the option.
+This example may make it easier to remember what is an option and
+what is an argument:
+
+\centerline{{{\tt )syscmd {\it arg1 arg2} )opt1 {\it opt1arg1 opt1arg2} )opt2 {\it opt2arg1} ...}}}
+
+In the system command descriptions, optional arguments and options are
+enclosed in brackets (``\lanb'' and ``\ranb'').
+If an argument or option name is in italics, it is
+meant to be a variable and must have some actual value substituted
+for it when the system command call is made.
+For example, the syntax pattern description
+
+\noindent
+{\tt )read} {\it fileName} {\tt \lanb{})quietly\ranb{}}
+
+\noindent
+would imply that you must provide an actual file name for
+{\it fileName} but need not use the {\tt )quietly} option.
+Thus
+\begin{verbatim}
+)read matrix.input
+\end{verbatim}
+is a valid instance of the above pattern.
+
+System command names and options may be abbreviated and may be in
+upper or lower case.
+The case of actual arguments may be significant, depending on the
+particular situation (such as in file names).
+System command names and options may be abbreviated to the minimum
+number of starting letters so that the name or option is unique.
+Thus
+\begin{verbatim}
+)s Integer
+\end{verbatim}
+is not a valid abbreviation for the {\tt )set} command,
+because both {\tt )set} and {\tt )show}
+begin with the letter ``s''.
+Typically, two or three letters are sufficient for disambiguating names.
+In our descriptions of the commands, we have used no abbreviations for
+either command names or options.
+
+In some syntax descriptions we use a vertical line ``\vertline''
+to indicate that you must specify one of the listed choices.
+For example, in
+\begin{verbatim}
+)set output fortran on | off
+\end{verbatim}
+only {\tt on} and {\tt off} are acceptable words for following
+{\tt boot}.
+We also sometimes use ``...'' to indicate that additional arguments
+or options of the listed form are allowed.
+Finally, in the syntax descriptions we may also list the syntax of
+related commands.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdabbreviationTitle}{)abbreviation}
+\newcommand{\ugSysCmdabbreviationNumber}{B.2.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdabbreviationPage}{B.2. )abbreviation}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{abbreviation}{ugSysCmdabbreviationPage}{B.2.}{)abbreviation}
+
+
+\par\noindent{\bf User Level Required:} compiler
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item {\tt )abbreviation query  \lanb{}{\it nameOrAbbrev}\ranb{}}
+\item {\tt )abbreviation category  {\it abbrev  fullname} \lanb{})quiet\ranb{}}
+\item {\tt )abbreviation domain  {\it abbrev  fullname}   \lanb{})quiet\ranb{}}
+\item {\tt )abbreviation package  {\it abbrev  fullname}  \lanb{})quiet\ranb{}}
+\item {\tt )abbreviation remove  {\it nameOrAbbrev}}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command is used to query, set and remove abbreviations for category,
+domain and package constructors.
+Every constructor must have a unique abbreviation.
+This abbreviation is part of the name of the subdirectory
+under which the components of the compiled constructor are
+stored.
+%% BEGIN OBSOLETE
+%% END OBSOLETE
+Furthermore, by issuing this command you
+let the system know what file to load automatically if you use a new
+constructor.
+Abbreviations must start with a letter and then be followed by
+up to seven letters or digits.
+Any letters appearing in the abbreviation must be in uppercase.
+
+When used with the {\tt query} argument,
+%-% \HDsyscmdindex{abbreviation query}{ugSysCmdabbreviationPage}{B.2.}{)abbreviation}
+this command may be used to list the name
+associated with a  particular abbreviation or the  abbreviation for a
+constructor.
+If no abbreviation or name is given, the names and corresponding
+abbreviations for {\it all} constructors are listed.
+
+The following shows the abbreviation for the constructor \spadtype{List}:
+\begin{verbatim}
+)abbreviation query List
+\end{verbatim}
+The following shows the constructor name corresponding to the
+abbreviation \spadtype{NNI}:
+\begin{verbatim}
+)abbreviation query NNI
+\end{verbatim}
+The following lists all constructor names and their abbreviations.
+\begin{verbatim}
+)abbreviation query
+\end{verbatim}
+
+To add an abbreviation for a constructor, use this command with
+{\tt category}, {\tt domain} or {\tt package}.
+%-% \HDsyscmdindex{abbreviation package}{ugSysCmdabbreviationPage}{B.2.}{)abbreviation}
+%-% \HDsyscmdindex{abbreviation domain}{ugSysCmdabbreviationPage}{B.2.}{)abbreviation}
+%-% \HDsyscmdindex{abbreviation category}{ugSysCmdabbreviationPage}{B.2.}{)abbreviation}
+The following add abbreviations to the system for a
+category, domain and package, respectively:
+\begin{verbatim}
+)abbreviation domain   SET Set
+)abbreviation category COMPCAT  ComplexCategory
+)abbreviation package  LIST2MAP ListToMap
+\end{verbatim}
+If the {\tt )quiet} option is used,
+no output is displayed from this command.
+You would normally only define an abbreviation in a library source file.
+If this command is issued for a constructor that has already been loaded, the
+constructor will be reloaded next time it is referenced.  In particular, you
+can use this command to force the automatic reloading of constructors.
+
+To remove an abbreviation, the {\tt remove} argument is used.
+%-% \HDsyscmdindex{abbreviation remove}{ugSysCmdabbreviationPage}{B.2.}{)abbreviation}
+This is usually
+only used to correct a previous command that set an abbreviation for a
+constructor name.
+If, in fact, the abbreviation does exist, you are prompted
+for confirmation of the removal request.
+Either of the following commands
+will remove the abbreviation \spadtype{VECTOR2} and the
+constructor name \spadtype{VectorFunctions2} from the system:
+\begin{verbatim}
+)abbreviation remove VECTOR2
+)abbreviation remove VectorFunctions2
+\end{verbatim}
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdcompileTitle''}{ugSysCmdcompilePage} in section \ugSysCmdcompileNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdbootTitle}{)boot}
+\newcommand{\ugSysCmdbootNumber}{B.3.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdbootPage}{B.3. )boot}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{boot}{ugSysCmdbootPage}{B.3.}{)boot}
+
+
+\par\noindent{\bf User Level Required:} development
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item {\tt )boot} {\it bootExpression}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command is used by \Language{} system developers to execute
+expressions written in the BOOT language.
+For example,
+\begin{verbatim}
+)boot times3(x) == 3*x
+\end{verbatim}
+creates and compiles the \Lisp{} function ``times3''
+obtained by translating the BOOT code.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdfinTitle''}{ugSysCmdfinPage} in section \ugSysCmdfinNumber
+\downlink{``\ugSysCmdlispTitle''}{ugSysCmdlispPage} in section \ugSysCmdlispNumber
+\downlink{``\ugSysCmdsetTitle''}{ugSysCmdsetPage} in section \ugSysCmdsetNumber
+\downlink{``\ugSysCmdsystemTitle''}{ugSysCmdsystemPage} in section \ugSysCmdsystemNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdcdTitle}{)cd}
+\newcommand{\ugSysCmdcdNumber}{B.4.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdcdPage}{B.4. )cd}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{cd}{ugSysCmdcdPage}{B.4.}{)cd}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item {\tt )cd} {\it directory}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command sets the \Language{} working current directory.
+The current directory is used for looking for
+input files (for {\tt )read}),
+\Language{} library source files (for {\tt )compile}),
+saved history environment files (for {\tt )history )restore}),
+compiled \Language{} library files (for \spadcmd{)library}), and
+files to edit (for {\tt )edit}).
+It is also used for writing
+spool files (via {\tt )spool}),
+writing history input files (via {\tt )history )write}) and
+history environment files (via {\tt )history )save}),and
+compiled \Language{} library files (via {\tt )compile}).
+%-% \HDsyscmdindex{read}{ugSysCmdcdPage}{B.4.}{)cd}
+%-% \HDsyscmdindex{compile}{ugSysCmdcdPage}{B.4.}{)cd}
+%-% \HDsyscmdindex{history )restore}{ugSysCmdcdPage}{B.4.}{)cd}
+%-% \HDsyscmdindex{edit}{ugSysCmdcdPage}{B.4.}{)cd}
+%-% \HDsyscmdindex{spool}{ugSysCmdcdPage}{B.4.}{)cd}
+%-% \HDsyscmdindex{history )write}{ugSysCmdcdPage}{B.4.}{)cd}
+%-% \HDsyscmdindex{history )save}{ugSysCmdcdPage}{B.4.}{)cd}
+
+If issued with no argument, this command sets the \Language{}
+current directory to your home directory.
+If an argument is used, it must be a valid directory name.
+Except for the ``{\tt )}'' at the beginning of the command,
+this has the same syntax as the operating system {\tt cd} command.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdcompileTitle''}{ugSysCmdcompilePage} in section \ugSysCmdcompileNumber
+\downlink{``\ugSysCmdeditTitle''}{ugSysCmdeditPage} in section \ugSysCmdeditNumber
+\downlink{``\ugSysCmdhistoryTitle''}{ugSysCmdhistoryPage} in section \ugSysCmdhistoryNumber
+\downlink{``\ugSysCmdlibraryTitle''}{ugSysCmdlibraryPage} in section \ugSysCmdlibraryNumber
+\downlink{``\ugSysCmdreadTitle''}{ugSysCmdreadPage} in section \ugSysCmdreadNumber
+\downlink{``\ugSysCmdspoolTitle''}{ugSysCmdspoolPage} in section \ugSysCmdspoolNumber
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdcloseTitle}{)close}
+\newcommand{\ugSysCmdcloseNumber}{B.5.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdclosePage}{B.5. )close}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{close}{ugSysCmdclosePage}{B.5.}{)close}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )close}
+\item{\tt )close )quietly}
+\end{items}
+\par\noindent{\bf Command Description:}
+
+This command is used to close down interpreter client processes.
+Such processes are started by \HyperName{} to run \Language{} examples
+when you click on their text. When you have finished examining or modifying the
+example and you do not want the extra window around anymore, issue
+\begin{verbatim}
+)close
+\end{verbatim}
+to the \Language{} prompt in the window.
+
+If you try to close down the last remaining interpreter client
+process, \Language{} will offer to close down the entire \Language{}
+session and return you to the operating system by displaying something
+like
+\begin{verbatim}
+   This is the last AXIOM session. Do you want to kill AXIOM?
+\end{verbatim}
+Type "y" (followed by the Return key) if this is what you had in mind.
+Type "n" (followed by the Return key) to cancel the command.
+
+You can use the {\tt )quietly} option to force \Language{} to
+close down the interpreter client process without closing down
+the entire \Language{} session.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdquitTitle''}{ugSysCmdquitPage} in section \ugSysCmdquitNumber
+\downlink{``\ugSysCmdpquitTitle''}{ugSysCmdpquitPage} in section \ugSysCmdpquitNumber
+
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdclearTitle}{)clear}
+\newcommand{\ugSysCmdclearNumber}{B.6.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdclearPage}{B.6. )clear}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{clear}{ugSysCmdclearPage}{B.6.}{)clear}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )clear all}
+\item{\tt )clear completely}
+\item{\tt )clear properties all}
+\item{\tt )clear properties}  {\it obj1 \lanb{}obj2 ...\ranb{}}
+\item{\tt )clear value      all}
+\item{\tt )clear value}     {\it obj1 \lanb{}obj2 ...\ranb{}}
+\item{\tt )clear mode       all}
+\item{\tt )clear mode}      {\it obj1 \lanb{}obj2 ...\ranb{}}
+\end{items}
+\par\noindent{\bf Command Description:}
+
+This command is used to remove function and variable declarations, definitions
+and values  from the workspace.
+To  empty the entire workspace  and reset the
+step counter to 1, issue
+\begin{verbatim}
+)clear all
+\end{verbatim}
+To remove everything in the workspace but not reset the step counter, issue
+\begin{verbatim}
+)clear properties all
+\end{verbatim}
+To remove everything about the object {\tt x}, issue
+\begin{verbatim}
+)clear properties x
+\end{verbatim}
+To remove everything about the objects {\tt x, y} and {\tt f}, issue
+\begin{verbatim}
+)clear properties x y f
+\end{verbatim}
+
+The word {\tt properties} may be abbreviated to the single letter
+``{\tt p}''.
+\begin{verbatim}
+)clear p all
+)clear p x
+)clear p x y f
+\end{verbatim}
+All definitions of functions and values of variables may be removed by either
+\begin{verbatim}
+)clear value all
+)clear v all
+\end{verbatim}
+This retains whatever declarations the objects had.  To remove definitions and
+values for the specific objects {\tt x, y} and {\tt f}, issue
+\begin{verbatim}
+)clear value x y f
+)clear v x y f
+\end{verbatim}
+To remove  the declarations  of everything while  leaving the  definitions and
+values, issue
+\begin{verbatim}
+)clear mode  all
+)clear m all
+\end{verbatim}
+To remove declarations for the specific objects {\tt x, y} and {\tt f}, issue
+\begin{verbatim}
+)clear mode x y f
+)clear m x y f
+\end{verbatim}
+The {\tt )display names} and {\tt )display properties} commands  may be used
+to see what is currently in the workspace.
+
+The command
+\begin{verbatim}
+)clear completely
+\end{verbatim}
+does everything that {\tt )clear all} does, and also clears the internal
+system function and constructor caches.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmddisplayTitle''}{ugSysCmddisplayPage} in section \ugSysCmddisplayNumber
+\downlink{``\ugSysCmdhistoryTitle''}{ugSysCmdhistoryPage} in section \ugSysCmdhistoryNumber
+\downlink{``\ugSysCmdundoTitle''}{ugSysCmdundoPage} in section \ugSysCmdundoNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdcompileTitle}{)compile}
+\newcommand{\ugSysCmdcompileNumber}{B.7.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdcompilePage}{B.7. )compile}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{compile}{ugSysCmdcompilePage}{B.7.}{)compile}
+
+
+\par\noindent{\bf User Level Required:} compiler
+
+\par\noindent{\bf Command Syntax:}
+
+\begin{items}
+\item {\tt )compile}
+\item {\tt )compile {\it fileName}}
+\item {\tt )compile {\it fileName}.as}
+\item {\tt )compile {\it directory/fileName}.as}
+\item {\tt )compile {\it fileName}.ao}
+\item {\tt )compile {\it directory/fileName}.ao}
+\item {\tt )compile {\it fileName}.al}
+\item {\tt )compile {\it directory/fileName}.al}
+\item {\tt )compile {\it fileName}.lsp}
+\item {\tt )compile {\it directory/fileName}.lsp}
+\item {\tt )compile {\it fileName}.spad}
+\item {\tt )compile {\it directory/fileName}.spad}
+\item {\tt )compile {\it fileName} )new}
+\item {\tt )compile {\it fileName} )old}
+\item {\tt )compile {\it fileName} )translate}
+\item {\tt )compile {\it fileName} )quiet}
+\item {\tt )compile {\it fileName} )noquiet}
+\item {\tt )compile {\it fileName} )moreargs}
+\item {\tt )compile {\it fileName} )onlyargs}
+\item {\tt )compile {\it fileName} )break}
+\item {\tt )compile {\it fileName} )nobreak}
+\item {\tt )compile {\it fileName} )library}
+\item {\tt )compile {\it fileName} )nolibrary}
+\item {\tt )compile {\it fileName} )vartrace}
+\item {\tt )compile {\it fileName} )constructor} {\it nameOrAbbrev}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+You use this command to invoke the new \Language{} library compiler or
+the old \Language{} system compiler.
+The {\tt )compile} system command is actually a combination of
+\Language{} processing and a call to the \axiomxl{} compiler.
+It is performing double-duty, acting as a front-end to
+both the \axiomxl{} compiler and the old \Language{} system
+compiler.
+(The old \Language{} system compiler was written in Lisp and was
+an integral part of the \Language{} environment.
+The \axiomxl{} compiler is written in C and executed by the operating system
+when called from within \Language{}.)
+
+The command compiles files with file extensions {\it .as, .ao}
+and {\it .al} with the
+\axiomxl{} compiler and files with file extension {\it .spad} with the
+old \Language{} system compiler.
+It also can compile files with file extension {\it .lsp}. These
+are assumed to be Lisp files genererated by the \axiomxl{}
+compiler.
+If you omit the file extension, the command looks to see if you
+have specified the {\tt )new} or {\tt )old} option.
+If you have given one of these options, the corresponding compiler
+is used.
+Otherwise, the command first looks in the standard system
+directories for files with extension {\it .as, .ao} and {\it
+.al} and then files with extension {\it .spad}.
+The first file found has the appropriate compiler invoked on it.
+If the command cannot find a matching file, an error message is
+displayed and the command terminates.
+
+The {\tt )translate} option is used to invoke a special version
+of the old system compiler that will translate a {\it .spad} file
+to a {\it .as} file. That is, the {\it .spad} file will be parsed and
+analyzed and a file using the new syntax will be created. By default,
+the {\it .as} file is created in the same directory as the
+{\it .spad} file. If that directory is not writable, the current
+directory is used. If the current directory is not writable, an
+error message is given and the command terminates.
+Note that {\tt )translate} implies the {\tt )old} option so the
+file extension can safely be omitted. If {\tt )translate} is
+given, all other options are ignored.
+Please be aware that the translation is not necessarily one
+hundred percent complete or correct.
+You should attempt to compile the output with the \axiomxl{} compiler
+and make any necessary corrections.
+
+We now describe the options for the new \axiomxl{} compiler.
+
+The first thing {\tt )compile} does is look for a source code
+filename among its arguments.
+Thus
+\begin{verbatim}
+)compile mycode.as
+)compile /u/jones/as/mycode.as
+)compile mycode
+\end{verbatim}
+all invoke {\tt )compiler} on the file {\tt
+/u/jones/as/mycode.as} if the current \Language{} working
+directory is {\tt /u/jones/as.} (Recall that you can set the
+working directory via the {\tt )cd} command. If you don't set it
+explicitly, it is the directory from which you started
+\Language{}.)
+
+This is frequently all you need to compile your file.
+This simple command:
+\indent{4}
+\beginitems
+\item[1. ] Invokes the \axiomxl{} compiler and produces Lisp output.
+\item[2. ] Calls the Lisp compiler if the \axiomxl{} compilation was
+successful.
+\item[3. ] Uses the {\tt )library} command to tell \Language{} about
+the contents of your compiled file and arrange to have those
+contents loaded on demand.
+\enditems
+\indent{0}
+
+Should you not want the {\tt )library} command automatically
+invoked, call {\tt )compile} with the {\tt )nolibrary} option.
+For example,
+\begin{verbatim}
+)compile mycode.as )nolibrary
+\end{verbatim}
+
+The general description of \axiomxl{} command line arguments is in
+the \axiomxl{} documentation.
+The default options used by the {\tt )compile} command can be
+viewed and set using the {\tt )set compiler args} \Language{}
+system command.
+The current defaults are
+\begin{verbatim}
+-O -Fasy -Fao -Flsp -laxiom -Mno-AXL_W_WillObsolete -DAxiom
+\end{verbatim}
+These options mean:
+\indent{4}
+\beginitems
+\item[-] {\tt -O}: perform all optimizations,
+\item[-] {\tt -Fasy}: generate a {\tt .asy} file,
+\item[-] {\tt -Fao}: generate a {\tt .ao} file,
+\item[-] {\tt -Flsp}: generate a {\tt .lsp} (Lisp)
+file,
+%-% \HDindex{Lisp!code generation}{ugSysCmdcompilePage}{B.7.}{)compile}
+\item[-] {\tt -laxiom}: use the {\tt axiom} library {\tt libaxiom.al},
+\item[-] {\tt -Mno-AXL\_W\_WillObsolete}: do not display messages
+about older generated files becoming obsolete, and
+\item[-] {\tt -DAxiom}: define the global assertion {\tt Axiom} so that the
+\axiomxl{} libraries for generating stand-alone code
+are not accidentally used with \Language{}.
+\enditems
+\indent{0}
+
+To supplement these default arguments, use the {\tt )moreargs} option on
+{\tt )compile.}
+For example,
+\begin{verbatim}
+)compile mycode.as )moreargs "-v"
+\end{verbatim}
+uses the default arguments and appends the {\tt -v} (verbose)
+argument flag.
+The additional argument specification {\bf must be enclosed in
+double quotes.}
+
+To completely replace these default arguments for a particular
+use of {\tt )compile}, use the {\tt )onlyargs} option.
+For example,
+\begin{verbatim}
+)compile mycode.as )onlyargs "-v -O"
+\end{verbatim}
+only uses the {\tt -v} (verbose) and {\tt -O} (optimize)
+arguments.
+The argument specification {\bf must be enclosed in double quotes.}
+In this example, Lisp code is not produced and so the compilation
+output will not be available to \Language{}.
+
+To completely replace the default arguments for all calls to {\tt
+)compile} within your \Language{} session, use {\tt )set compiler args.}
+For example, to use the above arguments for all compilations, issue
+\begin{verbatim}
+)set compiler args "-v -O"
+\end{verbatim}
+Make sure you include the necessary {\tt -l} and {\tt -Y}
+arguments along with those needed for Lisp file creation.
+As above, {\bf the argument specification must be enclosed in double
+quotes.}
+
+By default, the {\tt )library} system command {\it exposes} all
+domains and categories it processes.
+This means that the \Language{} intepreter will consider those
+domains and categories when it is trying to resolve a reference
+to a function.
+Sometimes domains and categories should not be exposed.
+For example, a domain may just be used privately by another
+domain and may not be meant for top-level use.
+The {\tt )library} command should still be used, though, so that
+the code will be loaded on demand.
+In this case, you should use the {\tt )nolibrary} option on {\tt
+)compile} and the {\tt )noexpose} option in the {\tt )library}
+command. For example,
+\begin{verbatim}
+)compile mycode.as )nolibrary
+)library mycode )noexpose
+\end{verbatim}
+
+Once you have established your own collection of compiled code,
+you may find it handy to use the {\tt )dir} option on the
+{\tt )library} command.
+This causes {\tt )library} to process all compiled code in the
+specified directory. For example,
+\begin{verbatim}
+)library )dir /u/jones/as/quantum
+\end{verbatim}
+You must give an explicit directory after {\tt )dir}, even if you
+want all compiled code in the current working directory
+processed, e.g.
+\begin{verbatim}
+)library )dir .
+\end{verbatim}
+
+The {\tt )compile} command works with several file extensions. We saw
+above what happens when it is invoked on a file with extension {\tt
+.as.} A {\tt .ao} file is a portable binary compiled version of a
+{\tt .as} file, and {\tt )compile} simply passes the {\tt .ao} file
+onto \axiomxl{}. The generated Lisp file is compiled and {\tt )library}
+is automatically called, just as if you had specified a {\tt .as} file.
+
+A {\tt .al} file is an archive file containing {\tt .ao} files. The
+archive is created (on Unix systems) with the {\tt ar} program. When
+{\tt )compile} is given a {\tt .al} file, it creates a directory whose
+name is based on that of the archive. For example, if you issue
+\begin{verbatim}
+)compile mylib.al
+\end{verbatim}
+the directory {\tt mylib.axldir} is created. All
+members of the archive are unarchived into the
+directory and {\tt )compile} is called on each {\tt .ao} file found. It
+is your responsibility to remove the directory and its contents, if you
+choose to do so.
+
+A {\tt .lsp} file is a Lisp source file, presumably, in our context,
+generated by \axiomxl{} when called with the {\tt -Flsp} option. When
+{\tt )compile} is used with a {\tt .lsp} file, the Lisp file is
+compiled and {\tt )library} is called. You must also have present a
+{\tt .asy} generated from the same source file.
+
+The following are descriptions of options for the old system compiler.
+
+You can compile category, domain, and package constructors
+contained in files with file extension {\it .spad}.
+You can compile individual constructors or every constructor
+in a file.
+
+The full filename is remembered between invocations of this command and
+{\tt )edit} commands.
+The sequence of commands
+\begin{verbatim}
+)compile matrix.spad
+)edit
+)compile
+\end{verbatim}
+will call the compiler, edit, and then call the compiler again
+on the file {\bf matrix.spad.}
+If you do not specify a {\it directory,} the working current
+directory (see \downlink{``\ugSysCmdcdTitle''}{ugSysCmdcdPage} in Section \ugSysCmdcdNumber\ignore{ugSysCmdcd})
+is searched for the file.
+If the file is not found, the standard system directories are searched.
+
+If you do not give any options, all constructors within a file are
+compiled.
+Each constructor should have an {\tt )abbreviation} command in
+the file in which it is defined.
+We suggest that you place the {\tt )abbreviation} commands at the
+top of the file in the order in which the constructors are
+defined.
+The list of commands serves as a table of contents for the file.
+%-% \HDsyscmdindex{abbreviation}{ugSysCmdcompilePage}{B.7.}{)compile}
+
+The {\tt )library} option causes directories containing the
+compiled code for each constructor
+to be created in the working current directory.
+The name of such a directory consists of the constructor
+abbreviation and the {\bf .NRLIB} file extension.
+For example, the directory containing the compiled code for
+the \axiomType{MATRIX} constructor is called {\bf MATRIX.NRLIB.}
+The {\tt )nolibrary} option says that such files should not
+be created.
+The default is {\tt )library.}
+Note that the semantics of {\tt )library} and {\tt )nolibrary}
+for the new \axiomxl{} compiler and for the old system compiler are
+completely different.
+
+The {\tt )vartrace} option causes the compiler to generate
+extra code for the constructor to support conditional tracing of
+variable assignments. (see \downlink{``\ugSysCmdtraceTitle''}{ugSysCmdtracePage} in Section \ugSysCmdtraceNumber\ignore{ugSysCmdtrace}). Without
+this option, this code is suppressed and one cannot use
+the {\tt )vars} option for the trace command.
+
+The {\tt )constructor} option is used to
+specify a particular constructor to compile.
+All other constructors in the file are ignored.
+The constructor name or abbreviation follows {\tt )constructor.}
+Thus either
+\begin{verbatim}
+)compile matrix.spad )constructor RectangularMatrix
+\end{verbatim}
+or
+\begin{verbatim}
+)compile matrix.spad )constructor RMATRIX
+\end{verbatim}
+compiles  the \axiomType{RectangularMatrix} constructor
+defined in {\bf matrix.spad.}
+
+The {\tt )break} and {\tt )nobreak} options determine what
+the old system compiler does when it encounters an error.
+{\tt )break} is the default and it indicates that processing
+should stop at the first error.
+The value of the {\tt )set break} variable then controls what happens.
+
+
+%% BEGIN OBSOLTE
+%% END OBSOLTE
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdabbreviationTitle''}{ugSysCmdabbreviationPage} in section \ugSysCmdabbreviationNumber
+\downlink{``\ugSysCmdeditTitle''}{ugSysCmdeditPage} in section \ugSysCmdeditNumber
+\downlink{``\ugSysCmdlibraryTitle''}{ugSysCmdlibraryPage} in section \ugSysCmdlibraryNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmddisplayTitle}{)display}
+\newcommand{\ugSysCmddisplayNumber}{B.8.}
+%
+% =====================================================================
+\begin{page}{ugSysCmddisplayPage}{B.8. )display}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{display}{ugSysCmddisplayPage}{B.8.}{)display}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item {\tt )display all}
+\item {\tt )display properties}
+\item {\tt )display properties all}
+\item {\tt )display properties} {\it \lanb{}obj1 \lanb{}obj2 ...\ranb{}\ranb{}}
+\item {\tt )display value all}
+\item {\tt )display value} {\it \lanb{}obj1 \lanb{}obj2 ...\ranb{}\ranb{}}
+\item {\tt )display mode all}
+\item {\tt )display mode} {\it \lanb{}obj1 \lanb{}obj2 ...\ranb{}\ranb{}}
+\item {\tt )display names}
+\item {\tt )display operations} {\it opName}
+\end{items}
+\par\noindent{\bf Command Description:}
+
+This command is  used to display the contents of  the workspace and
+signatures of functions  with a  given  name.\footnote{A
+\spadgloss{signature} gives the argument and return types of a
+function.}
+
+The command
+\begin{verbatim}
+)display names
+\end{verbatim}
+lists the names of all user-defined  objects in the workspace.  This is useful
+if you do  not wish to see everything  about the objects and need  only be
+reminded of their names.
+
+The commands
+\begin{verbatim}
+)display all
+)display properties
+)display properties all
+\end{verbatim}
+all do  the same thing: show  the values and  types and declared modes  of all
+variables in the  workspace.  If you have defined  functions, their signatures
+and definitions will also be displayed.
+
+To show all information about a  particular variable or user functions,
+for example, something named {\tt d}, issue
+\begin{verbatim}
+)display properties d
+\end{verbatim}
+To just show the value (and the type) of {\tt d}, issue
+\begin{verbatim}
+)display value d
+\end{verbatim}
+To just show the declared mode of {\tt d}, issue
+\begin{verbatim}
+)display mode d
+\end{verbatim}
+
+All modemaps for a given operation  may be
+displayed by using {\tt )display operations}.
+A \spadgloss{modemap} is a collection of information about  a particular
+reference
+to an  operation.  This  includes the  types of the  arguments and  the return
+value, the  location of the  implementation and  any conditions on  the types.
+The modemap may contain patterns.  The following displays the modemaps for the
+operation \spadfunFrom{complex}{ComplexCategory}:
+\begin{verbatim}
+)d op complex
+\end{verbatim}
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdclearTitle''}{ugSysCmdclearPage} in section \ugSysCmdclearNumber
+\downlink{``\ugSysCmdhistoryTitle''}{ugSysCmdhistoryPage} in section \ugSysCmdhistoryNumber
+\downlink{``\ugSysCmdsetTitle''}{ugSysCmdsetPage} in section \ugSysCmdsetNumber
+\downlink{``\ugSysCmdshowTitle''}{ugSysCmdshowPage} in section \ugSysCmdshowNumber
+\downlink{``\ugSysCmdwhatTitle''}{ugSysCmdwhatPage} in section \ugSysCmdwhatNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdeditTitle}{)edit}
+\newcommand{\ugSysCmdeditNumber}{B.9.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdeditPage}{B.9. )edit}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{edit}{ugSysCmdeditPage}{B.9.}{)edit}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )edit} \lanb{}{\it filename}\ranb{}
+\end{items}
+\par\noindent{\bf Command Description:}
+
+This command is  used to edit files.
+It works in conjunction  with the {\tt )read}
+and {\tt )compile} commands to remember the name
+of the file on which you are working.
+By specifying the name fully, you  can edit any file you wish.
+Thus
+\begin{verbatim}
+)edit /u/julius/matrix.input
+\end{verbatim}
+will place  you in an editor looking at the  file
+{\tt /u/julius/matrix.input}.
+%-% \HDindex{editing files}{ugSysCmdeditPage}{B.9.}{)edit}
+By default, the editor is {\tt vi},
+%-% \HDindex{vi}{ugSysCmdeditPage}{B.9.}{)edit}
+but if you have an EDITOR shell environment variable defined, that editor
+will be used.
+When \Language{} is running under the X Window System,
+it will try to open a separate {\tt xterm} running your editor if
+it thinks one is necessary.
+%-% \HDindex{Korn shell}{ugSysCmdeditPage}{B.9.}{)edit}
+For example, under the Korn shell, if you issue
+\begin{verbatim}
+export EDITOR=emacs
+\end{verbatim}
+then the emacs
+%-% \HDindex{emacs}{ugSysCmdeditPage}{B.9.}{)edit}
+editor will be used by \spadcmd{)edit}.
+
+If you do not specify a file name, the last file you edited,
+read or compiled will be used.
+If there is no ``last file'' you will be placed in the editor editing
+an empty unnamed file.
+
+It is possible to use the {\tt )system} command to edit a file directly.
+For example,
+\begin{verbatim}
+)system emacs /etc/rc.tcpip
+\end{verbatim}
+calls {\tt emacs} to edit the file.
+%-% \HDindex{emacs}{ugSysCmdeditPage}{B.9.}{)edit}
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdsystemTitle''}{ugSysCmdsystemPage} in section \ugSysCmdsystemNumber
+\downlink{``\ugSysCmdcompileTitle''}{ugSysCmdcompilePage} in section \ugSysCmdcompileNumber
+\downlink{``\ugSysCmdreadTitle''}{ugSysCmdreadPage} in section \ugSysCmdreadNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdfinTitle}{)fin}
+\newcommand{\ugSysCmdfinNumber}{B.10.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdfinPage}{B.10. )fin}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{fin}{ugSysCmdfinPage}{B.10.}{)fin}
+
+
+\par\noindent{\bf User Level Required:} development
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item {\tt )fin}
+\end{items}
+\par\noindent{\bf Command Description:}
+
+This command is used by \Language{}
+developers to leave the \Language{} system and return
+to the underlying \Lisp{} system.
+To return to \Language{}, issue the
+``{\tt (\vertline{}spad\vertline{})}''
+function call to \Lisp{}.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdpquitTitle''}{ugSysCmdpquitPage} in section \ugSysCmdpquitNumber
+\downlink{``\ugSysCmdquitTitle''}{ugSysCmdquitPage} in section \ugSysCmdquitNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdframeTitle}{)frame}
+\newcommand{\ugSysCmdframeNumber}{B.11.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdframePage}{B.11. )frame}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{frame}{ugSysCmdframePage}{B.11.}{)frame}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )frame  new  {\it frameName}}
+\item{\tt )frame  drop  {\it \lanb{}frameName\ranb{}}}
+\item{\tt )frame  next}
+\item{\tt )frame  last}
+\item{\tt )frame  names}
+\item{\tt )frame  import {\it frameName} {\it \lanb{}objectName1 \lanb{}objectName2 ...\ranb{}\ranb{}}}
+\item{\tt )set message frame on \vertline{} off}
+\item{\tt )set message prompt frame}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+A {\it frame} can be thought of as a logical session within the
+physical session that you get when you start the system.  You can
+have as many frames as you want, within the limits of your computer's
+storage, paging space, and so on.
+Each frame has its own {\it step number}, {\it environment} and {\it history.}
+You can have a variable named {\tt a} in one frame and it will
+have nothing to do with anything that might be called {\tt a} in
+any other frame.
+
+Some frames are created by the \HyperName{} program and these can
+have pretty strange names, since they are generated automatically.
+%-% \HDsyscmdindex{frame names}{ugSysCmdframePage}{B.11.}{)frame}
+To find out the names
+of all frames, issue
+\begin{verbatim}
+)frame names
+\end{verbatim}
+It will indicate the name of the current frame.
+
+You create a new frame
+%-% \HDsyscmdindex{frame new}{ugSysCmdframePage}{B.11.}{)frame}
+``{\bf quark}'' by issuing
+\begin{verbatim}
+)frame new quark
+\end{verbatim}
+The history facility can be turned on by issuing either
+{\tt )set history on} or {\tt )history )on}.
+If the history facility is on and you are saving history information
+in a file rather than in the \Language{} environment
+then a history file with filename {\bf quark.axh} will
+be created as you enter commands.
+If you wish to go back to what
+you were doing in the
+%-% \HDsyscmdindex{frame next}{ugSysCmdframePage}{B.11.}{)frame}
+``{\bf initial}'' frame, use
+%-% \HDsyscmdindex{frame last}{ugSysCmdframePage}{B.11.}{)frame}
+\begin{verbatim}
+)frame next
+\end{verbatim}
+or
+\begin{verbatim}
+)frame last
+\end{verbatim}
+to cycle through the ring of available frames to get back to
+``{\bf initial}''.
+
+If you want to throw
+away a frame (say ``{\bf quark}''), issue
+\begin{verbatim}
+)frame drop quark
+\end{verbatim}
+If you omit the name, the current frame is dropped.
+%-% \HDsyscmdindex{frame drop}{ugSysCmdframePage}{B.11.}{)frame}
+
+If you do use frames with the history facility on and writing to a file,
+you may want to delete some of the older history files.
+%-% \HDindex{file!history}{ugSysCmdframePage}{B.11.}{)frame}
+These are directories, so you may want to issue a command like
+{\tt rm -r quark.axh} to the operating system.
+
+You can bring things from another frame by using
+%-% \HDsyscmdindex{frame import}{ugSysCmdframePage}{B.11.}{)frame}
+{\tt )frame import}.
+For example, to bring the {\tt f} and {\tt g} from the frame ``{\bf quark}''
+to the current frame, issue
+\begin{verbatim}
+)frame import quark f g
+\end{verbatim}
+If you want everything from the frame ``{\bf quark}'', issue
+\begin{verbatim}
+)frame import quark
+\end{verbatim}
+You will be asked to verify that you really want everything.
+
+There are two {\tt )set} flags
+%-% \HDsyscmdindex{set message frame}{ugSysCmdframePage}{B.11.}{)frame}
+to make it easier to tell where you are.
+\begin{verbatim}
+)set message frame on | off
+\end{verbatim}
+will print more messages about frames when it is set on.
+By default, it is off.
+\begin{verbatim}
+)set message prompt frame
+\end{verbatim}
+will give a prompt
+%-% \HDsyscmdindex{set message prompt frame}{ugSysCmdframePage}{B.11.}{)frame}
+that looks like
+\begin{verbatim}
+initial (1) ->
+\end{verbatim}
+%-% \HDindex{prompt!with frame name}{ugSysCmdframePage}{B.11.}{)frame}
+when you start up. In this case, the frame name and step make up the
+prompt.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdhistoryTitle''}{ugSysCmdhistoryPage} in section \ugSysCmdhistoryNumber
+\downlink{``\ugSysCmdsetTitle''}{ugSysCmdsetPage} in section \ugSysCmdsetNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdhelpTitle}{)help}
+\newcommand{\ugSysCmdhelpNumber}{B.12.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdhelpPage}{B.12. )help}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{help}{ugSysCmdhelpPage}{B.12.}{)help}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )help}
+\item{\tt )help} {\it commandName}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command displays help information about system commands.
+If you issue
+\begin{verbatim}
+)help
+\end{verbatim}
+then this very text will be shown.
+You can also give the name or abbreviation of a system command
+to display information about it.
+For example,
+\begin{verbatim}
+)help clear
+\end{verbatim}
+will display the description of the {\tt )clear} system command.
+
+All this material is available in the \Language{} User Guide
+and in \HyperName{}.
+In \HyperName{}, choose the {\bf Commands} item from the
+{\bf Reference} menu.
+
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdhistoryTitle}{)history}
+\newcommand{\ugSysCmdhistoryNumber}{B.13.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdhistoryPage}{B.13. )history}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{history}{ugSysCmdhistoryPage}{B.13.}{)history}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )history )on}
+\item{\tt )history )off}
+\item{\tt )history )write} {\it historyInputFileName}
+\item{\tt )history )show \lanb{}{\it n}\ranb{} \lanb{}both\ranb{}}
+\item{\tt )history )save} {\it savedHistoryName}
+\item{\tt )history )restore} \lanb{}{\it savedHistoryName}\ranb{}
+\item{\tt )history )reset}
+\item{\tt )history )change} {\it n}
+\item{\tt )history )memory}
+\item{\tt )history )file}
+\item{\tt \%}
+\item{\tt \%\%({\it n})}
+\item{\tt )set history on \vertline{} off}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+The {\it history} facility within \Language{} allows you to restore your
+environment to that of another session and recall previous
+computational results.
+Additional commands allow you to review previous
+input lines and to create an {\bf .input} file of the lines typed to
+%-% \HDindex{file!input}{ugSysCmdhistoryPage}{B.13.}{)history}
+\Language{}.
+
+\Language{} saves your input and output if the history facility is
+turned on (which is the default).
+This information is saved if either of
+\begin{verbatim}
+)set history on
+)history )on
+\end{verbatim}
+has been issued.
+Issuing either
+\begin{verbatim}
+)set history off
+)history )off
+\end{verbatim}
+will discontinue the recording of information.
+%-% \HDsyscmdindex{history )on}{ugSysCmdhistoryPage}{B.13.}{)history}
+%-% \HDsyscmdindex{set history on}{ugSysCmdhistoryPage}{B.13.}{)history}
+%-% \HDsyscmdindex{set history off}{ugSysCmdhistoryPage}{B.13.}{)history}
+%-% \HDsyscmdindex{history )off}{ugSysCmdhistoryPage}{B.13.}{)history}
+
+Whether the facility is disabled or not,
+the value of \spadSyntax{\%} in \Language{} always
+refers to the result of the last computation.
+If you have not yet entered anything,
+\spadSyntax{\%} evaluates to an object of type
+\spadtype{Variable('\%)}.
+The function \spadSyntax{\%\%} may be  used to refer
+to other previous results if the history facility is enabled.
+In that case,
+{\tt \%\%(n)} is  the output from step {\tt n} if {\tt n > 0}.
+If {\tt n < 0}, the step is computed relative to the current step.
+Thus {\tt \%\%(-1)} is also the previous step,
+{\tt \%\%(-2)}, is the  step before that, and so on.
+If an invalid step number is given, \Language{} will signal an error.
+
+The {\it environment} information can either be saved in a file or entirely in
+memory (the default).
+Each frame (\downlink{``\ugSysCmdframeTitle''}{ugSysCmdframePage} in Section \ugSysCmdframeNumber\ignore{ugSysCmdframe})
+has its own history database.
+When it is kept in a file, some of it may also be kept in memory for
+efficiency.
+When the information is saved in a file, the name of the file is
+of the form {\bf FRAME.axh} where ``{\bf FRAME}'' is the name of the
+current frame.
+The history file is placed in the current working directory
+(see \downlink{``\ugSysCmdcdTitle''}{ugSysCmdcdPage} in Section \ugSysCmdcdNumber\ignore{ugSysCmdcd}).
+Note that these history database files are not text files (in fact,
+they are directories themselves), and so are not in human-readable
+format.
+
+The options to the {\tt )history} command are as follows:
+
+\indent{0}
+\beginitems
+\item[{\tt )change} {\it n}]
+will set the number of steps that are saved in memory to {\it n}.
+This option only has effect when the history data is maintained in a
+file.
+If you have issued {\tt )history )memory} (or not changed the default)
+there is no need to use {\tt )history )change}.
+%-% \HDsyscmdindex{history )change}{ugSysCmdhistoryPage}{B.13.}{)history}
+
+\item[{\tt )on}]
+will start the recording of information.
+If the workspace is not empty, you will be asked to confirm this
+request.
+If you do so, the workspace will be cleared and history data will begin
+being saved.
+You can also turn the facility on by issuing {\tt )set history on}.
+
+\item[{\tt )off}]
+will stop the recording of information.
+The {\tt )history )show} command will not work after issuing this
+command.
+Note that this command may be issued to save time, as there is some
+performance penalty paid for saving the environment data.
+You can also turn the facility off by issuing {\tt )set history off}.
+
+\item[{\tt )file}]
+indicates that history data should be saved in an external file on disk.
+
+\item[{\tt )memory}]
+indicates that all history data should be kept in memory rather than
+saved in a file.
+Note that if you are computing with very large objects it may not be
+practical to kept this data in memory.
+
+\item[{\tt )reset}]
+will flush the internal list of the most recent workspace calculations
+so that the data structures may be garbage collected by the underlying
+\Lisp{} system.
+Like {\tt )history )change}, this option only has real effect when
+history data is being saved in a file.
+
+\item[{\tt )restore} \lanb{}{\it savedHistoryName}\ranb{}]
+completely clears the environment and restores it to a saved session, if
+possible.
+The {\tt )save} option below allows you to save a session to a file
+with a given name. If you had issued
+{\tt )history )save jacobi}
+the command
+{\tt )history )restore jacobi}
+would clear the current workspace and load the contents of the named
+saved session. If no saved session name is specified, the system looks
+for a file called {\bf last.axh}.
+
+\item[{\tt )save} {\it savedHistoryName}]
+is used to save  a snapshot of the environment in a file.
+This file is placed in the current working directory
+(see \downlink{``\ugSysCmdcdTitle''}{ugSysCmdcdPage} in Section \ugSysCmdcdNumber\ignore{ugSysCmdcd}).
+Use {\tt )history )restore} to restore the environment to the state
+preserved in the file.
+This option also creates an input file containing all the lines of input
+since you created the workspace frame (for example, by starting your
+\Language{} session) or last did a \spadcmd{)clear all} or
+\spadcmd{)clear completely}.
+
+\item[{\tt )show} \lanb{}{\it n}\ranb{} \lanb{}{\tt both}\ranb{}]
+can show previous input lines and output results.
+{\tt )show} will display up to twenty of the last input lines
+(fewer if you haven't typed in twenty lines).
+{\tt )show} {\it n} will display up to {\it n} of the last input lines.
+{\tt )show both} will display up to five of the last input lines and
+output results.
+{\tt )show} {\it n} {\tt both} will display up to {\it n} of the last
+input lines and output results.
+
+\item[{\tt )write} {\it historyInputFile}]
+creates an {\bf .input} file with the input lines typed since the start
+of the session/frame or the last {\tt )clear all} or {\tt )clear
+completely}.
+If {\it historyInputFileName} does not contain a period (``.'') in the filename,
+{\bf .input} is appended to it.
+For example,
+{\tt )history )write chaos}
+and
+{\tt )history )write chaos.input}
+both write the input lines to a file called {\bf chaos.input} in your
+current working directory.
+If you issued one or more {\tt )undo} commands,
+{\tt )history )write}
+eliminates all
+input lines backtracked over as a result of {\tt )undo}.
+You can edit this file and then use {\tt )read} to have \Language{} process
+the contents.
+\enditems
+\indent{0}
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdframeTitle''}{ugSysCmdframePage} in section \ugSysCmdframeNumber
+\downlink{``\ugSysCmdreadTitle''}{ugSysCmdreadPage} in section \ugSysCmdreadNumber
+\downlink{``\ugSysCmdsetTitle''}{ugSysCmdsetPage} in section \ugSysCmdsetNumber
+\downlink{``\ugSysCmdundoTitle''}{ugSysCmdundoPage} in section \ugSysCmdundoNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdlibraryTitle}{)library}
+\newcommand{\ugSysCmdlibraryNumber}{B.14.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdlibraryPage}{B.14. )library}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{library}{ugSysCmdlibraryPage}{B.14.}{)library}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )library {\it libName1  \lanb{}libName2 ...\ranb{}}}
+\item{\tt )library )dir {\it dirName}}
+\item{\tt )library )only {\it objName1  \lanb{}objlib2 ...\ranb{}}}
+\item{\tt )library )noexpose}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command replaces the {\tt )load} system command that
+was available in \Language{} releases before version 2.0.
+The \spadcmd{)library} command makes available to \Language{} the compiled
+objects in the libraries listed.
+
+For example, if you {\tt )compile dopler.as} in your home
+directory, issue {\tt )library dopler} to have \Language{} look
+at the library, determine the category and domain constructors present,
+update the internal database with various properties of the
+constructors, and arrange for the constructors to be
+automatically loaded when needed.
+If the {\tt )noexpose} option has not been given, the
+constructors will be exposed (that is, available) in the current
+frame.
+
+If you compiled a file with the old system compiler, you will
+have an {\it NRLIB} present, for example, {\it DOPLER.NRLIB,}
+where {\tt DOPLER} is a constructor abbreviation.
+The command {\tt )library DOPLER} will then do the analysis and
+database updates as above.
+
+To tell the system about all libraries in a directory, use
+{\tt )library )dir dirName} where {\tt dirName} is an explicit
+directory.
+You may specify ``.'' as the directory, which means the current
+directory from which you started the system or the one you set
+via the \spadcmd{)cd} command. The directory name is required.
+
+You may only want to tell the system about particular
+constructors within a library. In this case, use the {\tt )only}
+option. The command {\tt )library dopler )only Test1} will only
+cause the {\sf Test1} constructor to be analyzed, autoloaded,
+etc..
+
+Finally, each constructor in a library  are usually automatically exposed when the
+\spadcmd{)library} command is used. Use the {\tt )noexpose}
+option if you not want them exposed. At a later time you can use
+{\tt )set expose add constructor} to expose any hidden
+constructors.
+
+{\bf Note for \Language{} beta testers:} At various times this
+command was called {\tt )local} and {\tt )with} before the name
+{\tt )library} became the official name.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdcdTitle''}{ugSysCmdcdPage} in section \ugSysCmdcdNumber
+\downlink{``\ugSysCmdcompileTitle''}{ugSysCmdcompilePage} in section \ugSysCmdcompileNumber
+\downlink{``\ugSysCmdframeTitle''}{ugSysCmdframePage} in section \ugSysCmdframeNumber
+\downlink{``\ugSysCmdsetTitle''}{ugSysCmdsetPage} in section \ugSysCmdsetNumber
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdlispTitle}{)lisp}
+\newcommand{\ugSysCmdlispNumber}{B.15.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdlispPage}{B.15. )lisp}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{lisp}{ugSysCmdlispPage}{B.15.}{)lisp}
+
+
+\par\noindent{\bf User Level Required:} development
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item {\tt )lisp} {\it\lanb{}lispExpression\ranb{}}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command is used by \Language{} system developers to have single
+expressions evaluated by the \Lisp{} system on which
+\Language{} is built.
+The {\it lispExpression} is read by the \Lisp{} reader and
+evaluated.
+If this expression is not complete (unbalanced parentheses, say), the reader
+will wait until a complete expression is entered.
+
+Since this command is only useful  for evaluating single expressions, the
+{\tt )fin}
+command may be used to  drop out  of \Language{}  into \Lisp{}.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdsystemTitle''}{ugSysCmdsystemPage} in section \ugSysCmdsystemNumber
+\downlink{``\ugSysCmdbootTitle''}{ugSysCmdbootPage} in section \ugSysCmdbootNumber
+\downlink{``\ugSysCmdfinTitle''}{ugSysCmdfinPage} in section \ugSysCmdfinNumber
+
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdloadTitle}{)load}
+\newcommand{\ugSysCmdloadNumber}{B.16.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdloadPage}{B.16. )load}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{load}{ugSysCmdloadPage}{B.16.}{)load}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+%% BEGIN OBSOLETE
+%% END OBSOLETE
+
+\par\noindent{\bf Command Description:}
+
+This command is obsolete. Use \spadcmd{)library} instead.
+
+%% BEGIN OBSOLETE
+
+%
+%
+%
+%
+
+%% END OBSOLETE
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdltraceTitle}{)ltrace}
+\newcommand{\ugSysCmdltraceNumber}{B.17.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdltracePage}{B.17. )ltrace}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{ltrace}{ugSysCmdltracePage}{B.17.}{)ltrace}
+
+
+\par\noindent{\bf User Level Required:} development
+
+\par\noindent{\bf Command Syntax:}
+
+This command has the same arguments as options as the
+\spadcmd{)trace} command.
+
+\par\noindent{\bf Command Description:}
+
+This command is used by \Language{} system developers to trace
+\Lisp{} or
+BOOT functions.
+It is not supported for general use.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdbootTitle''}{ugSysCmdbootPage} in section \ugSysCmdbootNumber
+\downlink{``\ugSysCmdlispTitle''}{ugSysCmdlispPage} in section \ugSysCmdlispNumber
+\downlink{``\ugSysCmdtraceTitle''}{ugSysCmdtracePage} in section \ugSysCmdtraceNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdpquitTitle}{)pquit}
+\newcommand{\ugSysCmdpquitNumber}{B.18.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdpquitPage}{B.18. )pquit}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{pquit}{ugSysCmdpquitPage}{B.18.}{)pquit}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )pquit}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command is used to terminate \Language{}  and return to the
+operating system.
+Other than by redoing all your computations or by
+using the {\tt )history )restore}
+command to try to restore your working environment,
+you cannot return to \Language{} in the same state.
+
+{\tt )pquit} differs from the {\tt )quit} in that it always asks for
+confirmation that you want to terminate \Language{} (the ``p'' is for
+``protected'').
+%-% \HDsyscmdindex{quit}{ugSysCmdpquitPage}{B.18.}{)pquit}
+When you enter the {\tt )pquit} command, \Language{} responds
+%
+\centerline{{Please enter {\bf y} or {\bf yes} if you really want to leave the interactive }}
+\centerline{{environment and return to the operating system:}}
+%
+If you respond with {\tt y} or {\tt yes}, you will see the message
+%
+\centerline{{You are now leaving the \Language{} interactive environment. }}
+\centerline{{Issue the command {\bf axiom} to the operating system to start a new session.}}
+%
+and \Language{} will terminate and return you to the operating
+system (or the environment from which you invoked the system).
+If you responded with something other than {\tt y} or {\tt yes}, then
+the message
+%
+\centerline{{You have chosen to remain in the \Language{} interactive environment.}}
+%
+will be displayed and, indeed, \Language{} would still be running.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdfinTitle''}{ugSysCmdfinPage} in section \ugSysCmdfinNumber
+\downlink{``\ugSysCmdhistoryTitle''}{ugSysCmdhistoryPage} in section \ugSysCmdhistoryNumber
+\downlink{``\ugSysCmdcloseTitle''}{ugSysCmdclosePage} in section \ugSysCmdcloseNumber
+\downlink{``\ugSysCmdquitTitle''}{ugSysCmdquitPage} in section \ugSysCmdquitNumber
+\downlink{``\ugSysCmdsystemTitle''}{ugSysCmdsystemPage} in section \ugSysCmdsystemNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdquitTitle}{)quit}
+\newcommand{\ugSysCmdquitNumber}{B.19.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdquitPage}{B.19. )quit}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{quit}{ugSysCmdquitPage}{B.19.}{)quit}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )quit}
+\item{\tt )set quit protected \vertline{} unprotected}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command is used to terminate \Language{}  and return to the
+operating system.
+Other than by redoing all your computations or by
+using the {\tt )history )restore}
+command to try to restore your working environment,
+you cannot return to \Language{} in the same state.
+
+{\tt )quit} differs from the {\tt )pquit} in that it asks for
+%-% \HDsyscmdindex{pquit}{ugSysCmdquitPage}{B.19.}{)quit}
+confirmation only if the command
+\begin{verbatim}
+)set quit protected
+\end{verbatim}
+has been issued.
+%-% \HDsyscmdindex{set quit protected}{ugSysCmdquitPage}{B.19.}{)quit}
+Otherwise, {\tt )quit} will make \Language{} terminate and return you
+to the operating system (or the environment from which you invoked the
+system).
+
+The default setting is {\tt )set quit protected} so that {\tt )quit}
+and {\tt )pquit} behave in the same way.
+If you do issue
+\begin{verbatim}
+)set quit unprotected
+\end{verbatim}
+we
+%-% \HDsyscmdindex{set quit unprotected}{ugSysCmdquitPage}{B.19.}{)quit}
+suggest that you do not (somehow) assign {\tt )quit} to be
+executed when you press, say, a function key.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdfinTitle''}{ugSysCmdfinPage} in section \ugSysCmdfinNumber
+\downlink{``\ugSysCmdhistoryTitle''}{ugSysCmdhistoryPage} in section \ugSysCmdhistoryNumber
+\downlink{``\ugSysCmdcloseTitle''}{ugSysCmdclosePage} in section \ugSysCmdcloseNumber
+\downlink{``\ugSysCmdpquitTitle''}{ugSysCmdpquitPage} in section \ugSysCmdpquitNumber
+\downlink{``\ugSysCmdsystemTitle''}{ugSysCmdsystemPage} in section \ugSysCmdsystemNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdreadTitle}{)read}
+\newcommand{\ugSysCmdreadNumber}{B.20.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdreadPage}{B.20. )read}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{read}{ugSysCmdreadPage}{B.20.}{)read}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item {\tt )read} {\it \lanb{}fileName\ranb{}}
+\item {\tt )read} {\it \lanb{}fileName\ranb{}} \lanb{}{\tt )quiet}\ranb{} \lanb{}{\tt )ifthere}\ranb{}
+\end{items}
+\par\noindent{\bf Command Description:}
+
+This command is used to read {\bf .input} files into \Language{}.
+%-% \HDindex{file!input}{ugSysCmdreadPage}{B.20.}{)read}
+The command
+\begin{verbatim}
+)read matrix.input
+\end{verbatim}
+will read the contents of the file {\bf matrix.input} into
+\Language{}.
+The ``.input'' file extension is optional.
+See \downlink{``\ugInOutInTitle''}{ugInOutInPage} in Section \ugInOutInNumber\ignore{ugInOutIn} for more information about {\bf .input} files.
+
+This command remembers the previous file you edited, read or compiled.
+If you do not specify a file name, the previous file will be read.
+
+The {\tt )ifthere} option checks to see whether the {\bf .input} file
+exists.
+If it does not, the  {\tt )read} command does nothing.
+If you do not use this option and the file does not exist,
+you are asked to give the name of an existing {\bf .input} file.
+
+The {\tt )quiet} option suppresses output while the file is being read.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdcompileTitle''}{ugSysCmdcompilePage} in section \ugSysCmdcompileNumber
+\downlink{``\ugSysCmdeditTitle''}{ugSysCmdeditPage} in section \ugSysCmdeditNumber
+\downlink{``\ugSysCmdhistoryTitle''}{ugSysCmdhistoryPage} in section \ugSysCmdhistoryNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdsetTitle}{)set}
+\newcommand{\ugSysCmdsetNumber}{B.21.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdsetPage}{B.21. )set}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{set}{ugSysCmdsetPage}{B.21.}{)set}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item {\tt )set}
+\item {\tt )set} {\it label1 \lanb{}... labelN\ranb{}}
+\item {\tt )set} {\it label1 \lanb{}... labelN\ranb{} newValue}
+\end{items}
+\par\noindent{\bf Command Description:}
+
+The {\tt )set} command is used to view or set system variables that
+control what messages are displayed, the type of output desired, the
+status of the history facility, the way \Language{} user functions are
+cached, and so on.
+Since this collection is very large, we will not discuss them here.
+Rather, we will show how the facility is used.
+We urge you to explore the {\tt )set} options to familiarize yourself
+with how you can modify your \Language{} working environment.
+There is a \HyperName{} version of this same facility available from the
+main \HyperName{} menu.
+\texht{}{Click \lispmemolink{here}{(|htSystemVariables|)} to go to it.}
+
+The {\tt )set} command is command-driven with a menu display.
+It is tree-structured.
+To see all top-level nodes, issue {\tt )set} by itself.
+\begin{verbatim}
+)set
+\end{verbatim}
+Variables with values have them displayed near the right margin.
+Subtrees of selections have ``{\tt ...}''
+displayed in the value field.
+For example, there are many kinds of messages, so issue
+{\tt )set message} to see the choices.
+\begin{verbatim}
+)set message
+\end{verbatim}
+The current setting  for the variable that displays
+%-% \HDindex{computation timings!displaying}{ugSysCmdsetPage}{B.21.}{)set}
+whether computation times
+%-% \HDindex{timings!displaying}{ugSysCmdsetPage}{B.21.}{)set}
+are displayed is visible in the menu displayed by the last command.
+To see more information, issue
+\begin{verbatim}
+)set message time
+\end{verbatim}
+This shows that time printing is on now.
+To turn it off, issue
+\begin{verbatim}
+)set message time off
+\end{verbatim}
+%-% \HDsyscmdindex{set message time}{ugSysCmdsetPage}{B.21.}{)set}
+
+As noted above, not all settings have so many qualifiers.
+For example, to change the {\tt )quit} command to being unprotected
+(that is, you will not be prompted for verification), you need only issue
+\begin{verbatim}
+)set quit unprotected
+\end{verbatim}
+%-% \HDsyscmdindex{set quit unprotected}{ugSysCmdsetPage}{B.21.}{)set}
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdquitTitle''}{ugSysCmdquitPage} in section \ugSysCmdquitNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdshowTitle}{)show}
+\newcommand{\ugSysCmdshowNumber}{B.22.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdshowPage}{B.22. )show}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{show}{ugSysCmdshowPage}{B.22.}{)show}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )show {\it nameOrAbbrev}}
+\item{\tt )show {\it nameOrAbbrev} )operations}
+\item{\tt )show {\it nameOrAbbrev} )attributes}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+This command displays information about \Language{}
+domain, package and category {\it constructors}.
+If no options are given, the {\tt )operations} option is assumed.
+For example,
+\begin{verbatim}
+)show POLY
+)show POLY )operations
+)show Polynomial
+)show Polynomial )operations
+\end{verbatim}
+each display basic information about the
+\spadtype{Polynomial} domain constructor and then provide a
+listing of operations.
+Since \spadtype{Polynomial} requires a \spadtype{Ring} (for example,
+\spadtype{Integer}) as argument, the above commands all refer
+to a unspecified ring {\tt R}.
+In the list of operations, \spadSyntax{\$} means
+\spadtype{Polynomial(R)}.
+
+The basic information displayed includes the {\it signature}
+of the constructor (the name and arguments), the constructor
+{\it abbreviation}, the {\it exposure status} of the constructor, and the
+name of the {\it library source file} for the constructor.
+
+If operation information about a specific domain is wanted,
+the full or abbreviated domain name may be used.
+For example,
+\begin{verbatim}
+)show POLY INT
+)show POLY INT )operations
+)show Polynomial Integer
+)show Polynomial Integer )operations
+\end{verbatim}
+are among  the combinations that will
+display the operations exported  by the
+domain \spadtype{Polynomial(Integer)} (as opposed to the general
+{\it domain constructor} \spadtype{Polynomial}).
+Attributes may be listed by using the {\tt )attributes} option.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmddisplayTitle''}{ugSysCmddisplayPage} in section \ugSysCmddisplayNumber
+\downlink{``\ugSysCmdsetTitle''}{ugSysCmdsetPage} in section \ugSysCmdsetNumber
+\downlink{``\ugSysCmdwhatTitle''}{ugSysCmdwhatPage} in section \ugSysCmdwhatNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdspoolTitle}{)spool}
+\newcommand{\ugSysCmdspoolNumber}{B.23.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdspoolPage}{B.23. )spool}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{spool}{ugSysCmdspoolPage}{B.23.}{)spool}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )spool} \lanb{}{\it fileName}\ranb{}
+\item{\tt )spool}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command is used to save {\it (spool)} all \Language{} input and output
+%-% \HDindex{file!spool}{ugSysCmdspoolPage}{B.23.}{)spool}
+into a file, called a {\it spool file.}
+You can only have one spool file active at a time.
+To start spool, issue this command with a filename. For example,
+\begin{verbatim}
+)spool integrate.out
+\end{verbatim}
+To stop spooling, issue {\tt )spool} with no filename.
+
+If the filename is qualified with a directory, then the output will
+be placed in that directory.
+If no directory information is given, the spool file will be placed in the
+%-% \HDindex{directory!for spool files}{ugSysCmdspoolPage}{B.23.}{)spool}
+{\it current directory.}
+The current directory is the directory from which you started
+\Language{} or is the directory you specified using the
+{\tt )cd} command.
+%-% \HDsyscmdindex{cd}{ugSysCmdspoolPage}{B.23.}{)spool}
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdcdTitle''}{ugSysCmdcdPage} in section \ugSysCmdcdNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdsynonymTitle}{)synonym}
+\newcommand{\ugSysCmdsynonymNumber}{B.24.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdsynonymPage}{B.24. )synonym}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{synonym}{ugSysCmdsynonymPage}{B.24.}{)synonym}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )synonym}
+\item{\tt )synonym} {\it synonym fullCommand}
+\item{\tt )what synonyms}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command is used to create short synonyms for system command expressions.
+For example, the following synonyms  might simplify commands you often
+use.
+\begin{verbatim}
+)synonym save         history )save
+)synonym restore      history )restore
+)synonym mail         system mail
+)synonym ls           system ls
+)synonym fortran      set output fortran
+\end{verbatim}
+Once defined, synonyms can be
+used in place of the longer  command expressions.
+Thus
+\begin{verbatim}
+)fortran on
+\end{verbatim}
+is the same as the longer
+\begin{verbatim}
+)set fortran output on
+\end{verbatim}
+To list all defined synonyms, issue either of
+\begin{verbatim}
+)synonyms
+)what synonyms
+\end{verbatim}
+To list, say, all synonyms that contain the substring
+``{\tt ap}'', issue
+\begin{verbatim}
+)what synonyms ap
+\end{verbatim}
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdsetTitle''}{ugSysCmdsetPage} in section \ugSysCmdsetNumber
+\downlink{``\ugSysCmdwhatTitle''}{ugSysCmdwhatPage} in section \ugSysCmdwhatNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdsystemTitle}{)system}
+\newcommand{\ugSysCmdsystemNumber}{B.25.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdsystemPage}{B.25. )system}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{system}{ugSysCmdsystemPage}{B.25.}{)system}
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )system} {\it cmdExpression}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command may be used to issue commands to the operating system while
+remaining in \Language{}.
+The {\it cmdExpression} is passed to the operating system for
+execution.
+
+To get an operating system shell, issue, for example,
+\spadcmd{)system sh}.
+When you enter the key combination,
+\texht{\fbox{\bf Ctrl}--\fbox{\bf D}}{{\bf Ctrl-D}}
+(pressing and holding the
+\texht{\fbox{\bf Ctrl}}{{\bf Ctrl}} key and then pressing the
+\texht{\fbox{\bf D}}{{\bf D}} key)
+the shell will terminate and you will return to \Language{}.
+We do not recommend this way of creating a shell because
+\Lisp{} may field some interrupts instead of the shell.
+If possible, use a shell running in another window.
+
+If you execute programs that misbehave you may not be able to return to
+\Language{}.
+If this happens, you may have no other choice than to restart
+\Language{} and restore the environment via {\tt )history )restore}, if
+possible.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdbootTitle''}{ugSysCmdbootPage} in section \ugSysCmdbootNumber
+\downlink{``\ugSysCmdfinTitle''}{ugSysCmdfinPage} in section \ugSysCmdfinNumber
+\downlink{``\ugSysCmdlispTitle''}{ugSysCmdlispPage} in section \ugSysCmdlispNumber
+\downlink{``\ugSysCmdpquitTitle''}{ugSysCmdpquitPage} in section \ugSysCmdpquitNumber
+\downlink{``\ugSysCmdquitTitle''}{ugSysCmdquitPage} in section \ugSysCmdquitNumber
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdtraceTitle}{)trace}
+\newcommand{\ugSysCmdtraceNumber}{B.26.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdtracePage}{B.26. )trace}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{trace}{ugSysCmdtracePage}{B.26.}{)trace}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )trace}
+\item{\tt )trace )off}
+
+\item{\tt )trace} {\it function \lanb{}options\ranb{}}
+\item{\tt )trace} {\it constructor \lanb{}options\ranb{}}
+\item{\tt )trace} {\it domainOrPackage \lanb{}options\ranb{}}
+\end{items}
+%
+where options can be one or more of
+%
+\begin{items}
+\item{\tt )after} {\it S-expression}
+\item{\tt )before} {\it S-expression}
+\item{\tt )break after}
+\item{\tt )break before}
+\item{\tt )cond} {\it S-expression}
+\item{\tt )count}
+\item{\tt )count} {\it n}
+\item{\tt )depth} {\it n}
+\item{\tt )local} {\it op1 \lanb{}... opN\ranb{}}
+\item{\tt )nonquietly}
+\item{\tt )nt}
+\item{\tt )off}
+\item{\tt )only} {\it listOfDataToDisplay}
+\item{\tt )ops}
+\item{\tt )ops} {\it op1 \lanb{}... opN \ranb{}}
+\item{\tt )restore}
+\item{\tt )stats}
+\item{\tt )stats reset}
+\item{\tt )timer}
+\item{\tt )varbreak}
+\item{\tt )varbreak} {\it var1 \lanb{}... varN \ranb{}}
+\item{\tt )vars}
+\item{\tt )vars} {\it var1 \lanb{}... varN \ranb{}}
+\item{\tt )within} {\it executingFunction}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command is used to trace the execution of functions that make
+up the \Language{} system, functions defined by users,
+and functions from the system library.
+Almost all options are available for each type of function but
+exceptions will be noted below.
+
+To list all functions, constructors, domains and packages that are
+traced, simply issue
+\begin{verbatim}
+)trace
+\end{verbatim}
+To untrace everything that is traced, issue
+\begin{verbatim}
+)trace )off
+\end{verbatim}
+When a function is traced, the default system action is to display
+the arguments to the function and the return value when the
+function is exited.
+Note that if a function is left via an action such as a {\tt THROW}, no
+return value will be displayed.
+Also, optimization of tail recursion may decrease the number of
+times a function is actually invoked and so may cause less trace
+information to be displayed.
+Other information can be displayed or collected when a function is
+traced and this is controlled by the various options.
+Most options will be of interest only to \Language{} system
+developers.
+If a domain or package is traced, the default action is to trace
+all functions exported.
+
+Individual interpreter, lisp or boot
+functions can be traced by listing their names after
+{\tt )trace}.
+Any options that are present must follow the functions to be
+traced.
+\begin{verbatim}
+)trace f
+\end{verbatim}
+traces the function {\tt f}.
+To untrace {\tt f}, issue
+\begin{verbatim}
+)trace f )off
+\end{verbatim}
+Note that if a function name contains a special character, it will
+be necessary to escape the character with an underscore
+%
+\begin{verbatim}
+)trace _/D_,1
+\end{verbatim}
+%
+To trace all domains or packages that are or will be created from a particular
+constructor, give the constructor name or abbreviation after
+{\tt )trace}.
+%
+\begin{verbatim}
+)trace MATRIX
+)trace List Integer
+\end{verbatim}
+%
+The first command traces all domains currently instantiated with
+\spadtype{Matrix}.
+If additional domains are instantiated with this constructor
+(for example, if you have used \spadtype{Matrix(Integer)} and
+\spadtype{Matrix(Float)}), they will be automatically traced.
+The second command traces \spadtype{List(Integer)}.
+It is possible to trace individual functions in a domain or
+package.
+See the {\tt )ops} option below.
+
+The following are the general options for the {\tt )trace}
+command.
+
+%!! system command parser doesn't treat general s-expressions correctly,
+%!! I recommand not documenting )after )before and )cond
+\indent{0}
+\beginitems
+%\item[{\tt )after} {\it S-expression}]
+%causes the given \Lisp{} {\it S-expression} to be
+%executed after exiting the traced function.
+
+%\item[{\tt )before} {\it S-expression}]
+%causes the given \Lisp{} {\it S-expression} to be
+%executed before entering the traced function.
+
+\item[{\tt )break after}]
+causes a \Lisp{} break loop to be entered after
+exiting the traced function.
+
+\item[{\tt )break before}]
+causes a \Lisp{} break loop to be entered before
+entering the traced function.
+
+\item[{\tt )break}]
+is the same as \spadcmd{)break before}.
+
+%\item[{\tt )cond} {\it S-expression}]
+%causes trace information to be shown only if the given
+%\Lisp{} {\it S-expression} evaluates to non-NIL.  For
+%example, the following command causes the system function
+%{\tt resolveTT} to be traced but to have the information
+%displayed only if the value of the variable
+%{\tt \$reportBottomUpFlag} is non-NIL.
+%\begin{verbatim}
+%)trace resolveTT )cond \_\$reportBottomUpFlag}
+%\end{verbatim}
+
+\item[{\tt )count}]
+causes the system to keep a count of the number of times the
+traced function is entered.  The total can be displayed with
+{\tt )trace )stats} and cleared with {\tt )trace )stats reset}.
+
+\item[{\tt )count} {\it n}]
+causes information about the traced function to be displayed for
+the first {\it n} executions.  After the \eth{\it n} execution, the
+function is untraced.
+
+\item[{\tt )depth} {\it n}]
+causes trace information to be shown for only {\it n} levels of
+recursion of the traced function.  The command
+\begin{verbatim}
+)trace fib )depth 10
+\end{verbatim}
+will cause the display of only 10 levels of trace information for
+the recursive execution of a user function \userfun{fib}.
+
+\item[{\tt )math}]
+causes the function arguments and return value to be displayed in the
+\Language{} monospace two-dimensional math format.
+
+\item[{\tt )nonquietly}]
+causes the display of additional messages when a function is
+traced.
+
+\item[{\tt )nt}]
+This suppresses all normal trace information.  This option is
+useful if the {\tt )count} or {\tt )timer} options are used and
+you are interested in the statistics but not the function calling
+information.
+
+\item[{\tt )off}]
+causes untracing of all or specific functions.  Without an
+argument, all functions, constructors, domains and packages are
+untraced.  Otherwise, the given functions and other objects
+are untraced.  To
+immediately retrace the untraced functions, issue {\tt )trace
+)restore}.
+
+\item[{\tt )only} {\it listOfDataToDisplay}]
+causes only specific trace information to be shown.  The items are
+listed by using the following abbreviations:
+\indent{0}
+\beginitems
+\item[a]        display all arguments
+\item[v]        display return value
+\item[1]        display first argument
+\item[2]        display second argument
+\item[15]       display the 15th argument, and so on
+\enditems
+\indent{0}
+\enditems
+\indent{0}
+\indent{0}
+\beginitems
+
+\item[{\tt )restore}]
+causes the last untraced functions to be retraced.  If additional
+options are present, they are added to those previously in effect.
+
+\item[{\tt )stats}]
+causes the display of statistics collected by the use of the
+{\tt )count} and {\tt )timer} options.
+
+\item[{\tt )stats reset}]
+resets to 0 the statistics collected by the use of the
+{\tt )count} and {\tt )timer} options.
+
+\item[{\tt )timer}]
+causes the system to keep a count of execution times for the
+traced function.  The total can be displayed with {\tt )trace
+)stats} and cleared with {\tt )trace )stats reset}.
+
+%!! only for lisp, boot, may not work in any case, recommend removing
+%\item[{\tt )varbreak}]
+%causes a \Lisp{} break loop to be entered after
+%the assignment to any variable in the traced function.
+
+\item[{\tt )varbreak} {\it var1 \lanb{}... varN\ranb{}}]
+causes a \Lisp{} break loop to be entered after
+the assignment to any of the listed variables in the traced
+function.
+
+\item[{\tt )vars}]
+causes the display of the value of any variable after it is
+assigned in the traced function.
+Note that library code must
+have been compiled (see \downlink{``\ugSysCmdcompileTitle''}{ugSysCmdcompilePage} in Section \ugSysCmdcompileNumber\ignore{ugSysCmdcompile})
+using the {\tt )vartrace} option in order
+to support this option.
+
+\item[{\tt )vars} {\it var1 \lanb{}... varN\ranb{}}]
+causes the display of the value of any of the specified variables
+after they are assigned in the traced function.
+Note that library code must
+have been compiled (see \downlink{``\ugSysCmdcompileTitle''}{ugSysCmdcompilePage} in Section \ugSysCmdcompileNumber\ignore{ugSysCmdcompile})
+using the {\tt )vartrace} option in order
+to support this option.
+
+\item[{\tt )within} {\it executingFunction}]
+causes the display of trace information only if the traced
+function is called when the given {\it executingFunction} is running.
+\enditems
+\indent{0}
+
+The following are the options for tracing constructors, domains
+and packages.
+
+\indent{0}
+\beginitems
+\item[{\tt )local} {\it \lanb{}op1 \lanb{}... opN\ranb{}\ranb{}}]
+causes local functions of the constructor to be traced.  Note that
+to untrace an individual local function, you must use the fully
+qualified internal name, using the escape character
+\spadSyntax{\_} before the semicolon.
+\begin{verbatim}
+)trace FRAC )local
+)trace FRAC_;cancelGcd )off
+\end{verbatim}
+
+\item[{\tt )ops} {\it op1 \lanb{}... opN\ranb{}}]
+By default, all operations from a domain or package are traced
+when the domain or package is traced.  This option allows you to
+specify that only particular operations should be traced.  The
+command
+%
+\begin{verbatim}
+)trace Integer )ops min max _+ _-
+\end{verbatim}
+%
+traces four operations from the domain \spadtype{Integer}.  Since
+{\tt +} and {\tt -} are special
+characters, it is necessary
+to escape them with an underscore.
+\enditems
+\indent{0}
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdbootTitle''}{ugSysCmdbootPage} in section \ugSysCmdbootNumber
+\downlink{``\ugSysCmdlispTitle''}{ugSysCmdlispPage} in section \ugSysCmdlispNumber
+\downlink{``\ugSysCmdltraceTitle''}{ugSysCmdltracePage} in section \ugSysCmdltraceNumber
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdundoTitle}{)undo}
+\newcommand{\ugSysCmdundoNumber}{B.27.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdundoPage}{B.27. )undo}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{undo}{ugSysCmdundoPage}{B.27.}{)undo}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )undo}
+\item{\tt )undo} {\it integer}
+\item{\tt )undo} {\it integer \lanb{}option\ranb{}}
+\item{\tt )undo} {\tt )redo}
+\end{items}
+%
+where {\it option} is one of
+%
+\begin{items}
+\item{\tt )after}
+\item{\tt )before}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command is used to
+restore the state of the user environment to an earlier
+point in the interactive session.
+The argument of an {\tt )undo} is an integer which must designate some
+step number in the interactive session.
+
+\begin{verbatim}
+)undo n
+)undo n )after
+\end{verbatim}
+These commands return the state of the interactive
+environment to that immediately after step {\tt n}.
+If {\tt n} is a positive number, then {\tt n} refers to step nummber
+{\tt n}. If {\tt n} is a negative number, it refers to the \eth{\tt n}
+previous command (that is, undoes the effects of the last \smath{-n}
+commands).
+
+A {\tt )clear all} resets the {\tt )undo} facility.
+Otherwise, an {\tt )undo} undoes the effect of {\tt )clear} with
+options {\tt properties}, {\tt value}, and {\tt mode}, and
+that of a previous {\tt undo}.
+If any such system commands are given between steps \smath{n} and
+\smath{n + 1} (\smath{n > 0}), their effect is undone
+for {\tt )undo m} for any \texht{\smath{0 < m \leq n}.}{0 < m <= n}.
+
+The command {\tt )undo} is equivalent to {\tt )undo -1} (it undoes
+the effect of the previous user expression).
+The command {\tt )undo 0} undoes any of the above system commands
+issued since the last user expression.
+
+\begin{verbatim}
+)undo n )before
+\end{verbatim}
+This command returns the state of the interactive
+environment to that immediately before step {\tt n}.
+Any {\tt )undo} or {\tt )clear} system commands
+given before step {\tt n} will not be undone.
+
+\begin{verbatim}
+)undo )redo
+\end{verbatim}
+This command reads the file {\tt redo.input}.
+created by the last {\tt )undo} command.
+This file consists of all user input lines, excluding those
+backtracked over due to a previous {\tt )undo}.
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmdhistoryTitle''}{ugSysCmdhistoryPage} in section \ugSysCmdhistoryNumber
+The command {\tt )history )write} will eliminate the ``undone'' command
+lines of your program.
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugSysCmdwhatTitle}{)what}
+\newcommand{\ugSysCmdwhatNumber}{B.28.}
+%
+% =====================================================================
+\begin{page}{ugSysCmdwhatPage}{B.28. )what}
+% =====================================================================
+\beginscroll
+%-% \HDsyscmdindex{what}{ugSysCmdwhatPage}{B.28.}{)what}
+
+
+\par\noindent{\bf User Level Required:} interpreter
+
+\par\noindent{\bf Command Syntax:}
+\begin{items}
+\item{\tt )what categories} {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}}
+\item{\tt )what commands  } {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}}
+\item{\tt )what domains   } {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}}
+\item{\tt )what operations} {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}}
+\item{\tt )what packages  } {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}}
+\item{\tt )what synonym   } {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}}
+\item{\tt )what things    } {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}}
+\item{\tt )apropos        } {\it pattern1} \lanb{}{\it pattern2 ...\ranb{}}
+\end{items}
+
+\par\noindent{\bf Command Description:}
+
+This command is used to display lists of things in the system.  The
+patterns are all strings and, if present, restrict the contents of the
+lists.  Only those items that contain one or more of the strings as
+substrings are displayed.  For example,
+\begin{verbatim}
+)what synonym
+\end{verbatim}
+displays all command synonyms,
+\begin{verbatim}
+)what synonym ver
+\end{verbatim}
+displays all command synonyms containing the substring ``{\tt ver}'',
+\begin{verbatim}
+)what synonym ver pr
+\end{verbatim}
+displays all command synonyms
+containing the substring  ``{\tt ver}'' or  the substring
+``{\tt pr}''.
+Output similar to the following will be displayed
+\begin{verbatim}
+---------------- System Command Synonyms -----------------
+
+user-defined synonyms satisfying patterns:
+      ver pr
+
+  )apr ........................... )what things
+  )apropos ....................... )what things
+  )prompt ........................ )set message prompt
+  )version ....................... )lisp *yearweek*
+\end{verbatim}
+
+Several other things can be listed with the {\tt )what} command:
+
+\indent{0}
+\beginitems
+\item[{\tt categories}] displays a list of category constructors.
+%-% \HDsyscmdindex{what categories}{ugSysCmdwhatPage}{B.28.}{)what}
+\item[{\tt commands}]  displays a list of  system commands available  at your
+user-level.
+%-% \HDsyscmdindex{what commands}{ugSysCmdwhatPage}{B.28.}{)what}
+Your user-level
+%-% \HDindex{user-level}{ugSysCmdwhatPage}{B.28.}{)what}
+is set via the  {\tt )set userlevel} command.
+%-% \HDsyscmdindex{set userlevel}{ugSysCmdwhatPage}{B.28.}{)what}
+To get a description of a particular command, such as ``{\tt )what}'', issue
+{\tt )help what}.
+\item[{\tt domains}]   displays a list of domain constructors.
+%-% \HDsyscmdindex{what domains}{ugSysCmdwhatPage}{B.28.}{)what}
+\item[{\tt operations}] displays a list of operations in  the system library.
+%-% \HDsyscmdindex{what operations}{ugSysCmdwhatPage}{B.28.}{)what}
+It  is recommended that you  qualify this command with one or
+more patterns, as there are thousands of operations available.  For
+example, say you are looking for functions that involve computation of
+eigenvalues.  To find their names, try {\tt )what operations eig}.
+A rather large list of operations  is loaded into the workspace when
+this command  is first issued.  This  list will be deleted  when you
+clear the workspace  via {\tt )clear all} or {\tt )clear completely}.
+It will be re-created if it is needed again.
+\item[{\tt packages}]  displays a list of package constructors.
+%-% \HDsyscmdindex{what packages}{ugSysCmdwhatPage}{B.28.}{)what}
+\item[{\tt synonym}]  lists system command synonyms.
+%-% \HDsyscmdindex{what synonym}{ugSysCmdwhatPage}{B.28.}{)what}
+\item[{\tt things}]    displays all  of the  above types for  items containing
+%-% \HDsyscmdindex{what things}{ugSysCmdwhatPage}{B.28.}{)what}
+the pattern strings as  substrings.
+The command synonym  {\tt )apropos} is equivalent to
+%-% \HDsyscmdindex{apropos}{ugSysCmdwhatPage}{B.28.}{)what}
+{\tt )what things}.
+\enditems
+\indent{0}
+
+\par\noindent{\bf Also See:}
+\downlink{``\ugSysCmddisplayTitle''}{ugSysCmddisplayPage} in section \ugSysCmddisplayNumber
+\downlink{``\ugSysCmdsetTitle''}{ugSysCmdsetPage} in section \ugSysCmdsetNumber
+\downlink{``\ugSysCmdshowTitle''}{ugSysCmdshowPage} in section \ugSysCmdshowNumber
+
+\texht{\egroup}{}
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/ug21.ht b/src/hyper/pages/ug21.ht
new file mode 100644
index 00000000..e17c9cf4
--- /dev/null
+++ b/src/hyper/pages/ug21.ht
@@ -0,0 +1,939 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\setcounter{chapter}{5} % Appendix F
+%% use \begin{xmpLines} etc, not \begin{figXmpLines}
+%% see use of \labelSpace below when heading crashes into text
+%% you've repeat one of the section labels (ugFknot)
+
+%
+\newcommand{\ugAppGraphicsTitle}{Programs for AXIOM Images}
+\newcommand{\ugAppGraphicsNumber}{G.}
+%
+% =====================================================================
+\begin{page}{ugAppGraphicsPage}{G. Programs for AXIOM Images}
+% =====================================================================
+\beginscroll
+%
+This appendix contains the \Language{} programs used to generate
+the images in the \Gallery{} color insert of this book.
+All these input files are included
+with the \Language{} system.
+To produce the images
+on page 6 of the \Gallery{} insert, for example, issue the command:
+\begin{verbatim}
+)read images6
+\end{verbatim}
+
+These images were produced on an IBM RS/6000 model 530 with a
+standard color graphics adapter.  The smooth shaded images
+were made from X Window System screen dumps.
+The remaining images were produced with \Language{}-generated
+PostScript output.  The images were reproduced from slides made on an Agfa
+ChromaScript PostScript interpreter with a Matrix Instruments QCR camera.
+
+\beginmenu
+    \menudownlink{{F.1. images1.input}}{ugFimagesOnePage}
+    \menudownlink{{F.2. images2.input}}{ugFimagesTwoPage}
+    \menudownlink{{F.3. images3.input}}{ugFimagesThreePage}
+    \menudownlink{{F.4. images5.input}}{ugFimagesFivePage}
+    \menudownlink{{F.5. images6.input}}{ugFimagesSixPage}
+    \menudownlink{{F.6. images7.input}}{ugFimagesSevenPage}
+    \menudownlink{{F.7. images8.input}}{ugFimagesEightPage}
+    \menudownlink{{F.8. conformal.input}}{ugFconformalPage}
+    \menudownlink{{F.9. tknot.input}}{ugFtknotPage}
+    \menudownlink{{F.10. ntube.input}}{ugFntubePage}
+    \menudownlink{{F.11. dhtri.input}}{ugFdhtriPage}
+    \menudownlink{{F.12. tetra.input}}{ugFtetraPage}
+    \menudownlink{{F.13. antoine.input}}{ugFantoinePage}
+    \menudownlink{{F.14. scherk.input}}{ugFscherkPage}
+\endmenu
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFimagesOneTitle}{images1.input}
+\newcommand{\ugFimagesOneNumber}{G.1.}
+%
+% =====================================================================
+\begin{page}{ugFimagesOnePage}{G.1. images1.input}
+% =====================================================================
+\beginscroll
+
+\labelSpace{3pc}
+
+  
+\noindent
+{\tt 1.\ \ \ )read\ tknot}\newline
+{\tt 2.\ \ \ }\newline
+{\tt 3.\ \ \ torusKnot(15,17,\ 0.1,\ 6,\ 700)}\newline
+  
+\noindent
+%-% \HDindex{torus knot}{ugFimagesOnePage}{G.1.}{images1.input}
+
+\newpage
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFimagesTwoTitle}{images2.input}
+\newcommand{\ugFimagesTwoNumber}{G.2.}
+%
+% =====================================================================
+\begin{page}{ugFimagesTwoPage}{G.2. images2.input}
+% =====================================================================
+\beginscroll
+
+These images illustrate how Newton's method converges when computing the
+%-% \HDindex{Newton iteration}{ugFimagesTwoPage}{G.2.}{images2.input}
+complex cube roots of 2.   Each point in the \smath{(x,y)}-plane represents the
+complex number \smath{x + iy,} which is given as a starting point for Newton's
+method.  The poles in these images represent bad starting values.
+The flat areas are the regions of convergence to the three roots.
+
+  
+\noindent
+{\tt 1.\ \ \ )read\ newton}\newline
+{\tt 2.\ \ \ )read\ vectors}\newline
+{\tt 3.\ \ \ f\ :=\ newtonStep(x**3\ -\ 2)}\newline
+{\tt 4.\ \ \ }\newline
+  
+\noindent
+
+The function \texht{$f^n$}{f**n} computes $n$ steps of Newton's method.
+
+  
+\noindent
+{\tt 1.\ \ \ clipValue\ :=\ 4}\newline
+{\tt 2.\ \ \ drawComplexVectorField(f**3,\ -3..3,\ -3..3)}\newline
+{\tt 3.\ \ \ drawComplex(f**3,\ -3..3,\ -3..3)}\newline
+{\tt 4.\ \ \ drawComplex(f**4,\ -3..3,\ -3..3)}\newline
+  
+\noindent
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFimagesThreeTitle}{images3.input}
+\newcommand{\ugFimagesThreeNumber}{G.3.}
+%
+% =====================================================================
+\begin{page}{ugFimagesThreePage}{G.3. images3.input}
+% =====================================================================
+\beginscroll
+
+  
+\noindent
+{\tt 1.\ \ \ )r\ tknot}\newline
+{\tt 2.\ \ \ for\ i\ in\ 0..4\ repeat\ torusKnot(2,\ 2\ +\ i/4,\ 0.5,\ 25,\ 250)}\newline
+  
+\noindent
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFimagesFiveTitle}{images5.input}
+\newcommand{\ugFimagesFiveNumber}{G.4.}
+%
+% =====================================================================
+\begin{page}{ugFimagesFivePage}{G.4. images5.input}
+% =====================================================================
+\beginscroll
+
+The parameterization of the Etruscan Venus is due to George Frances.
+%-% \HDindex{Etruscan Venus}{ugFimagesFivePage}{G.4.}{images5.input}
+
+  
+\noindent
+{\tt 1.\ \ \ venus(a,r,steps)\ ==}\newline
+{\tt 2.\ \ \ \ \ surf\ :=\ (u:DFLOAT,\ v:DFLOAT):\ Point\ DFLOAT\ +->}\newline
+{\tt 3.\ \ \ \ \ \ \ cv\ :=\ cos(v)}\newline
+{\tt 4.\ \ \ \ \ \ \ sv\ :=\ sin(v)}\newline
+{\tt 5.\ \ \ \ \ \ \ cu\ :=\ cos(u)}\newline
+{\tt 6.\ \ \ \ \ \ \ su\ :=\ sin(u)}\newline
+{\tt 7.\ \ \ \ \ \ \ x\ :=\ r\ *\ cos(2*u)\ *\ cv\ +\ sv\ *\ cu}\newline
+{\tt 8.\ \ \ \ \ \ \ y\ :=\ r\ *\ sin(2*u)\ *\ cv\ -\ sv\ *\ su}\newline
+{\tt 9.\ \ \ \ \ \ \ z\ :=\ a\ *\ cv}\newline
+{\tt 10.\ \ \ \ \ \ point\ [x,y,z]}\newline
+{\tt 11.\ \ \ \ draw(surf,\ 0..\%pi,\ -\%pi..\%pi,\ var1Steps==steps,}\newline
+{\tt 12.\ \ \ \ \ \ \ \ \ var2Steps==steps,\ title\ ==\ "Etruscan\ Venus")}\newline
+{\tt 13.\ \ }\newline
+{\tt 14.\ \ venus(5/2,\ 13/10,\ 50)}\newline
+  
+\noindent
+
+The Figure-8 Klein Bottle
+%-% \HDindex{Klein bottle}{ugFimagesFivePage}{G.4.}{images5.input}
+parameterization is from
+``Differential Geometry and Computer Graphics'' by Thomas Banchoff,
+in {\it Perspectives in Mathematics,} Anniversary of Oberwolfasch 1984,
+Birkh\"{a}user-Verlag, Basel, pp. 43-60.
+
+  
+\noindent
+{\tt 1.\ \ \ klein(x,y)\ ==}\newline
+{\tt 2.\ \ \ \ \ cx\ :=\ cos(x)}\newline
+{\tt 3.\ \ \ \ \ cy\ :=\ cos(y)}\newline
+{\tt 4.\ \ \ \ \ sx\ :=\ sin(x)}\newline
+{\tt 5.\ \ \ \ \ sy\ :=\ sin(y)}\newline
+{\tt 6.\ \ \ \ \ sx2\ :=\ sin(x/2)}\newline
+{\tt 7.\ \ \ \ \ cx2\ :=\ cos(x/2)}\newline
+{\tt 8.\ \ \ \ \ sq2\ :=\ sqrt(2.0@DFLOAT)}\newline
+{\tt 9.\ \ \ \ \ point\ [cx\ *\ (cx2\ *\ (sq2\ +\ cy)\ +\ (sx2\ *\ sy\ *\ cy)),\ \_}\newline
+{\tt 10.\ \ \ \ \ \ \ \ \ \ \ sx\ *\ (cx2\ *\ (sq2\ +\ cy)\ +\ (sx2\ *\ sy\ *\ cy)),\ \_}\newline
+{\tt 11.\ \ \ \ \ \ \ \ \ \ \ -sx2\ *\ (sq2\ +\ cy)\ +\ cx2\ *\ sy\ *\ cy]}\newline
+{\tt 12.\ \ }\newline
+{\tt 13.\ \ draw(klein,\ 0..4*\%pi,\ 0..2*\%pi,\ var1Steps==50,}\newline
+{\tt 14.\ \ \ \ \ \ \ var2Steps==50,title=="Figure\ Eight\ Klein\ Bottle")}\newline
+  
+\noindent
+
+The next two images are examples of generalized tubes.
+
+  
+\noindent
+{\tt 15.\ \ )read\ ntube}\newline
+{\tt 16.\ \ rotateBy(p,\ theta)\ ==}\newline
+{\tt 17.\ \ \ \ c\ :=\ cos(theta)}\newline
+{\tt 18.\ \ \ \ s\ :=\ sin(theta)}\newline
+{\tt 19.\ \ \ \ point\ [p.1*c\ -\ p.2*s,\ p.1*s\ +\ p.2*c]}\newline
+{\tt 20.\ \ }\newline
+{\tt 21.\ \ bcircle\ t\ ==\ }\newline
+{\tt 22.\ \ \ \ point\ [3*cos\ t,\ 3*sin\ t,\ 0]}\newline
+{\tt 23.\ \ \ }\newline
+{\tt 24.\ \ twist(u,\ t)\ ==}\newline
+{\tt 25.\ \ \ \ theta\ :=\ 4*t}\newline
+{\tt 26.\ \ \ \ p\ :=\ point\ [sin\ u,\ cos(u)/2]}\newline
+{\tt 27.\ \ \ \ rotateBy(p,\ theta)}\newline
+{\tt 28.\ \ \ }\newline
+{\tt 29.\ \ ntubeDrawOpt(bcircle,\ twist,\ 0..2*\%pi,\ 0..2*\%pi,}\newline
+{\tt 30.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ var1Steps\ ==\ 70,\ var2Steps\ ==\ 250)}\newline
+{\tt 31.\ \ }\newline
+{\tt 32.\ \ twist2(u,\ t)\ ==}\newline
+{\tt 33.\ \ \ \ theta\ :=\ t}\newline
+{\tt 34.\ \ \ \ p\ :=\ point\ [sin\ u,\ cos(u)]}\newline
+{\tt 35.\ \ \ \ rotateBy(p,\ theta)}\newline
+{\tt 36.\ \ }\newline
+{\tt 37.\ \ cf(u,v)\ ==\ sin(21*u)}\newline
+{\tt 38.\ \ }\newline
+{\tt 39.\ \ ntubeDrawOpt(bcircle,\ twist2,\ 0..2*\%pi,\ 0..2*\%pi,}\newline
+{\tt 40.\ \ \ \ colorFunction\ ==\ cf,\ var1Steps\ ==\ 168,}\newline
+{\tt 41.\ \ \ \ var2Steps\ ==\ 126)}\newline
+  
+\noindent
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFimagesSixTitle}{images6.input}
+\newcommand{\ugFimagesSixNumber}{G.5.}
+%
+% =====================================================================
+\begin{page}{ugFimagesSixPage}{G.5. images6.input}
+% =====================================================================
+\beginscroll
+
+\labelSpace{3pc}
+  
+\noindent
+{\tt 1.\ \ \ gam(x,y)\ ==\ }\newline
+{\tt 2.\ \ \ \ \ g\ :=\ Gamma\ complex(x,y)}\newline
+{\tt 3.\ \ \ \ \ point\ [x,y,max(min(real\ g,\ 4),\ -4),\ argument\ g]}\newline
+{\tt 4.\ \ \ }\newline
+{\tt 5.\ \ \ }\newline
+{\tt 6.\ \ \ draw(gam,\ -\%pi..\%pi,\ -\%pi..\%pi,\ }\newline
+{\tt 7.\ \ \ \ \ \ \ \ title\ ==\ "Gamma(x\ +\ \%i*y)",\ \_}\newline
+{\tt 8.\ \ \ \ \ \ \ \ var1Steps\ ==\ 100,\ var2Steps\ ==\ 100)}\newline
+{\tt 9.\ \ \ }\newline
+{\tt 10.\ \ b(x,y)\ ==\ Beta(x,y)}\newline
+{\tt 11.\ \ }\newline
+{\tt 12.\ \ draw(b,\ -3.1..3,\ -3.1\ ..\ 3,\ title\ ==\ "Beta(x,y)")}\newline
+{\tt 13.\ \ }\newline
+{\tt 14.\ \ atf(x,y)\ ==\ }\newline
+{\tt 15.\ \ \ \ a\ :=\ atan\ complex(x,y)}\newline
+{\tt 16.\ \ \ \ point\ [x,y,real\ a,\ argument\ a]}\newline
+{\tt 17.\ \ }\newline
+{\tt 18.\ \ draw(atf,\ -3.0..\%pi,\ -3.0..\%pi)}\newline
+  
+\noindent
+%-% \HDindex{function!Gamma}{ugFimagesSixPage}{G.5.}{images6.input}
+%-% \HDindex{function!Euler Beta}{ugFimagesSixPage}{G.5.}{images6.input}
+%-% \HDindex{Euler!Beta function}{ugFimagesSixPage}{G.5.}{images6.input}
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFimagesSevenTitle}{images7.input}
+\newcommand{\ugFimagesSevenNumber}{G.6.}
+%
+% =====================================================================
+\begin{page}{ugFimagesSevenPage}{G.6. images7.input}
+% =====================================================================
+\beginscroll
+
+First we look at the conformal
+%-% \HDindex{conformal map}{ugFimagesSevenPage}{G.6.}{images7.input}
+map \texht{$z \mapsto z + 1/z$}{z +-> z + 1/z}.
+\labelSpace{2pc}
+  
+\noindent
+{\tt 1.\ \ \ )read\ conformal}\newline
+{\tt 2.\ \ \ }\newline
+{\tt 3.\ \ \ }\newline
+{\tt 4.\ \ \ f\ z\ ==\ z}\newline
+{\tt 5.\ \ \ }\newline
+{\tt 6.\ \ \ conformalDraw(f,\ -2..2,\ -2..2,\ 9,\ 9,\ "cartesian")}\newline
+{\tt 7.\ \ \ }\newline
+{\tt 8.\ \ \ f\ z\ ==\ z\ +\ 1/z}\newline
+{\tt 9.\ \ \ \ }\newline
+{\tt 10.\ \ conformalDraw(f,\ -2..2,\ -2..2,\ 9,\ 9,\ "cartesian")}\newline
+  
+\noindent
+
+The map \texht{$z \mapsto -(z+1)/(z-1)$}{z +-> -(z+1)/(z-1)} maps
+the unit disk to the right half-plane, as shown
+%-% \HDindex{Riemann!sphere}{ugFimagesSevenPage}{G.6.}{images7.input}
+on the Riemann sphere.
+
+  
+\noindent
+{\tt 1.\ \ \ f\ z\ ==\ z}\newline
+{\tt 2.\ \ \ }\newline
+{\tt 3.\ \ \ riemannConformalDraw(f,0.1..0.99,0..2*\%pi,7,11,"polar")}\newline
+{\tt 4.\ \ \ }\newline
+{\tt 5.\ \ \ f\ z\ ==\ -(z+1)/(z-1)}\newline
+{\tt 6.\ \ \ }\newline
+{\tt 7.\ \ \ riemannConformalDraw(f,0.1..0.99,0..2*\%pi,7,11,"polar")}\newline
+{\tt 8.\ \ \ }\newline
+{\tt 9.\ \ \ riemannSphereDraw(-4..4,\ -4..4,\ 7,\ 7,\ "cartesian")}\newline
+  
+\noindent
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFimagesEightTitle}{images8.input}
+\newcommand{\ugFimagesEightNumber}{G.7.}
+%
+% =====================================================================
+\begin{page}{ugFimagesEightPage}{G.7. images8.input}
+% =====================================================================
+\beginscroll
+
+\labelSpace{1pc}
+  
+\noindent
+{\tt 1.\ \ \ )read\ dhtri}\newline
+{\tt 2.\ \ \ )read\ tetra}\newline
+{\tt 3.\ \ \ drawPyramid\ 4}\newline
+{\tt 4.\ \ \ }\newline
+%-% \HDindex{Sierpinsky's Tetrahedron}{ugFimagesEightPage}{G.7.}{images8.input}
+{\tt 5.\ \ \ )read\ antoine}\newline
+{\tt 6.\ \ \ drawRings\ 2}\newline
+{\tt 7.\ \ \ }\newline
+%-% \HDindex{Antoine's Necklace}{ugFimagesEightPage}{G.7.}{images8.input}
+{\tt 8.\ \ \ )read\ scherk}\newline
+{\tt 9.\ \ \ drawScherk(3,3)}\newline
+{\tt 10.\ \ }\newline
+%-% \HDindex{Scherk's minimal surface}{ugFimagesEightPage}{G.7.}{images8.input}
+{\tt 11.\ \ )read\ ribbonsNew}\newline
+{\tt 12.\ \ drawRibbons([x**i\ for\ i\ in\ 1..5],\ x=-1..1,\ y=0..2)}\newline
+  
+\noindent
+
+
+%\input{gallery/conformal.htex}
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFconformalTitle}{conformal.input}
+\newcommand{\ugFconformalNumber}{G.8.}
+%
+% =====================================================================
+\begin{page}{ugFconformalPage}{G.8. conformal.input}
+% =====================================================================
+\beginscroll
+%
+The functions in this section draw conformal maps both on the
+%-% \HDindex{conformal map}{ugFconformalPage}{G.8.}{conformal.input}
+plane and on the Riemann sphere.
+%-% \HDindex{Riemann!sphere}{ugFconformalPage}{G.8.}{conformal.input}
+
+%-- Compile, don't interpret functions.
+%\xmpLine{)set fun comp on}{}
+  
+\noindent
+{\tt 1.\ \ \ C\ :=\ Complex\ DoubleFloat}\newline
+{\tt 2.\ \ \ S\ :=\ Segment\ DoubleFloat}\newline
+{\tt 3.\ \ \ R3\ :=\ Point\ DFLOAT}\newline
+{\tt 4.\ \ \ \ }\newline
+  
+\noindent
+
+\userfun{conformalDraw}{\it (f, rRange, tRange, rSteps, tSteps, coord)}
+draws the image of the coordinate grid under {\it f} in the complex plane.
+The grid may be given in either polar or Cartesian coordinates.
+Argument {\it f} is the function to draw;
+{\it rRange} is the range of the radius (in polar) or real (in Cartesian);
+{\it tRange} is the range of \texht{$\theta$}{\theta} (in polar) or imaginary (in Cartesian);
+{\it tSteps, rSteps}, are the number of intervals in the {\it r} and
+\texht{$\theta$}{\theta} directions; and
+{\it coord} is the coordinate system to use (either {\tt "polar"} or
+{\tt "cartesian"}).
+
+  
+\noindent
+{\tt 1.\ \ \ conformalDraw:\ (C\ ->\ C,\ S,\ S,\ PI,\ PI,\ String)\ ->\ VIEW3D}\newline
+{\tt 2.\ \ \ conformalDraw(f,rRange,tRange,rSteps,tSteps,coord)\ ==}\newline
+{\tt 3.\ \ \ \ \ transformC\ :=}\newline
+{\tt 4.\ \ \ \ \ \ \ coord\ =\ "polar"\ =>\ polar2Complex}\newline
+{\tt 5.\ \ \ \ \ \ \ cartesian2Complex}\newline
+{\tt 6.\ \ \ \ \ cm\ :=\ makeConformalMap(f,\ transformC)}\newline
+{\tt 7.\ \ \ \ \ sp\ :=\ createThreeSpace()}\newline
+{\tt 8.\ \ \ \ \ adaptGrid(sp,\ cm,\ rRange,\ tRange,\ rSteps,\ tSteps)}\newline
+{\tt 9.\ \ \ \ \ makeViewport3D(sp,\ "Conformal\ Map")}\newline
+  
+\noindent
+
+\userfun{riemannConformalDraw}{\it (f, rRange, tRange, rSteps, tSteps, coord)}
+draws the image of the coordinate grid under {\it f} on the Riemann sphere.
+The grid may be given in either polar or Cartesian coordinates.
+Its arguments are the same as those for \userfun{conformalDraw}.
+  
+\noindent
+{\tt 10.\ \ riemannConformalDraw:(C->C,S,S,PI,PI,String)->VIEW3D}\newline
+{\tt 11.\ \ riemannConformalDraw(f,\ rRange,\ tRange,}\newline
+{\tt 12.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ rSteps,\ tSteps,\ coord)\ ==}\newline
+{\tt 13.\ \ \ \ transformC\ :=}\newline
+{\tt 14.\ \ \ \ \ \ coord\ =\ "polar"\ =>\ polar2Complex}\newline
+{\tt 15.\ \ \ \ \ \ cartesian2Complex}\newline
+{\tt 16.\ \ \ \ sp\ :=\ createThreeSpace()}\newline
+{\tt 17.\ \ \ \ cm\ :=\ makeRiemannConformalMap(f,\ transformC)}\newline
+{\tt 18.\ \ \ \ adaptGrid(sp,\ cm,\ rRange,\ tRange,\ rSteps,\ tSteps)}\newline
+{\tt 19.\ \ \ \ curve(sp,[point\ [0,0,2.0@DFLOAT,0],point\ [0,0,2.0@DFLOAT,0]])}\newline
+{\tt 20.\ \ \ \ makeViewport3D(sp,"Map\ on\ the\ Riemann\ Sphere")}\newline
+{\tt 21.\ \ }\newline
+{\tt 22.\ \ adaptGrid(sp,\ f,\ uRange,\ vRange,\ \ uSteps,\ vSteps)\ ==}\newline
+{\tt 23.\ \ \ \ delU\ :=\ (hi(uRange)\ -\ lo(uRange))/uSteps}\newline
+{\tt 24.\ \ \ \ delV\ :=\ (hi(vRange)\ -\ lo(vRange))/vSteps}\newline
+{\tt 25.\ \ \ \ uSteps\ :=\ uSteps\ +\ 1;\ vSteps\ :=\ vSteps\ +\ 1}\newline
+{\tt 26.\ \ \ \ u\ :=\ lo\ uRange}\newline
+{\tt 27.\ \ \ \ for\ i\ in\ 1..uSteps\ repeat}\newline
+{\tt 28.\ \ \ \ \ \ c\ :=\ curryLeft(f,u)}\newline
+{\tt 29.\ \ \ \ \ \ cf\ :=\ (t:DFLOAT):DFLOAT\ +->\ 0}\newline
+{\tt 30.\ \ \ \ \ \ makeObject(c,vRange::SEG\ Float,colorFunction==cf,}\newline
+{\tt 31.\ \ \ \ \ \ \ \ space\ ==\ sp,\ tubeRadius\ ==\ .02,\ tubePoints\ ==\ 6)}\newline
+{\tt 32.\ \ \ \ \ \ u\ :=\ u\ +\ delU}\newline
+{\tt 33.\ \ \ \ v\ :=\ lo\ vRange}\newline
+{\tt 34.\ \ \ \ for\ i\ in\ 1..vSteps\ repeat}\newline
+{\tt 35.\ \ \ \ \ \ c\ :=\ curryRight(f,v)}\newline
+{\tt 36.\ \ \ \ \ \ cf\ :=\ (t:DFLOAT):DFLOAT\ +->\ 1}\newline
+{\tt 37.\ \ \ \ \ \ makeObject(c,uRange::SEG\ Float,colorFunction==cf,}\newline
+{\tt 38.\ \ \ \ \ \ \ \ space\ ==\ sp,\ tubeRadius\ ==\ .02,\ tubePoints\ ==\ 6)}\newline
+{\tt 39.\ \ \ \ \ \ v\ :=\ v\ +\ delV}\newline
+{\tt 40.\ \ \ \ void()}\newline
+{\tt 41.\ \ }\newline
+{\tt 42.\ \ riemannTransform(z)\ ==}\newline
+{\tt 43.\ \ \ \ r\ :=\ sqrt\ norm\ z}\newline
+{\tt 44.\ \ \ \ cosTheta\ :=\ (real\ z)/r}\newline
+{\tt 45.\ \ \ \ sinTheta\ :=\ (imag\ z)/r}\newline
+{\tt 46.\ \ \ \ cp\ :=\ 4*r/(4+r**2)}\newline
+{\tt 47.\ \ \ \ sp\ :=\ sqrt(1-cp*cp)}\newline
+{\tt 48.\ \ \ \ if\ r>2\ then\ sp\ :=\ -sp}\newline
+{\tt 49.\ \ \ \ point\ [cosTheta*cp,\ sinTheta*cp,\ -sp\ +\ 1]}\newline
+{\tt 50.\ \ \ }\newline
+{\tt 51.\ \ cartesian2Complex(r:DFLOAT,\ i:DFLOAT):C\ ==}\newline
+{\tt 52.\ \ \ \ complex(r,\ i)}\newline
+{\tt 53.\ \ }\newline
+{\tt 54.\ \ polar2Complex(r:DFLOAT,\ th:DFLOAT):C\ ==\ }\newline
+{\tt 55.\ \ \ \ complex(r*cos(th),\ r*sin(th))}\newline
+{\tt 56.\ \ }\newline
+{\tt 57.\ \ makeConformalMap(f,\ transformC)\ ==}\newline
+{\tt 58.\ \ \ \ (u:DFLOAT,v:DFLOAT):R3\ +->\ }\newline
+{\tt 59.\ \ \ \ \ \ z\ :=\ f\ transformC(u,\ v)}\newline
+{\tt 60.\ \ \ \ \ \ point\ [real\ z,\ imag\ z,\ 0.0@DFLOAT]}\newline
+{\tt 61.\ \ \ }\newline
+{\tt 62.\ \ makeRiemannConformalMap(f,\ transformC)\ ==}\newline
+{\tt 63.\ \ \ \ (u:DFLOAT,\ v:DFLOAT):R3\ +->}\newline
+{\tt 64.\ \ \ \ \ \ riemannTransform\ f\ transformC(u,\ v)}\newline
+{\tt 65.\ \ }\newline
+{\tt 66.\ \ riemannSphereDraw:\ (S,\ S,\ PI,\ PI,\ String)\ ->\ VIEW3D}\newline
+{\tt 67.\ \ riemannSphereDraw(rRange,tRange,rSteps,tSteps,coord)\ ==}\newline
+{\tt 68.\ \ \ \ transformC\ :=}\newline
+{\tt 69.\ \ \ \ \ \ coord\ =\ "polar"\ =>\ polar2Complex}\newline
+{\tt 70.\ \ \ \ \ \ cartesian2Complex}\newline
+{\tt 71.\ \ \ \ grid\ :=\ (u:DFLOAT,\ v:DFLOAT):\ R3\ +->\ }\newline
+{\tt 72.\ \ \ \ \ \ z1\ :=\ transformC(u,\ v)}\newline
+{\tt 73.\ \ \ \ \ \ point\ [real\ z1,\ imag\ z1,\ 0]}\newline
+{\tt 74.\ \ \ \ sp\ :=\ createThreeSpace()}\newline
+{\tt 75.\ \ \ \ adaptGrid(sp,\ grid,\ rRange,\ tRange,\ rSteps,\ tSteps)}\newline
+{\tt 76.\ \ \ \ connectingLines(sp,grid,rRange,tRange,rSteps,tSteps)}\newline
+{\tt 77.\ \ \ \ makeObject(riemannSphere,0..2*\%pi,0..\%pi,space==sp)}\newline
+{\tt 78.\ \ \ \ f\ :=\ (z:C):C\ +->\ z}\newline
+{\tt 79.\ \ \ \ cm\ :=\ makeRiemannConformalMap(f,\ transformC)}\newline
+{\tt 80.\ \ \ \ adaptGrid(sp,\ cm,\ rRange,\ tRange,\ rSteps,\ tSteps)}\newline
+{\tt 81.\ \ \ \ makeViewport3D(sp,\ "Riemann\ Sphere")}\newline
+{\tt 82.\ \ \ }\newline
+{\tt 83.\ \ connectingLines(sp,f,uRange,vRange,uSteps,vSteps)\ ==}\newline
+{\tt 84.\ \ \ \ delU\ :=\ (hi(uRange)\ -\ lo(uRange))/uSteps}\newline
+{\tt 85.\ \ \ \ delV\ :=\ (hi(vRange)\ -\ lo(vRange))/vSteps}\newline
+{\tt 86.\ \ \ \ uSteps\ :=\ uSteps\ +\ 1;\ vSteps\ :=\ vSteps\ +\ 1}\newline
+{\tt 87.\ \ \ \ u\ :=\ lo\ uRange}\newline
+{\tt 88.\ \ \ \ for\ i\ in\ 1..uSteps\ repeat}\newline
+{\tt 89.\ \ \ \ \ \ v\ :=\ lo\ vRange}\newline
+{\tt 90.\ \ \ \ \ \ for\ j\ in\ 1..vSteps\ repeat}\newline
+{\tt 91.\ \ \ \ \ \ \ \ p1\ :=\ f(u,v)}\newline
+{\tt 92.\ \ \ \ \ \ \ \ p2\ :=\ riemannTransform\ complex(p1.1,\ p1.2)}\newline
+{\tt 93.\ \ \ \ \ \ \ \ fun\ :=\ lineFromTo(p1,p2)}\newline
+{\tt 94.\ \ \ \ \ \ \ \ cf\ :=\ (t:DFLOAT):DFLOAT\ +->\ 3}\newline
+{\tt 95.\ \ \ \ \ \ \ \ makeObject(fun,\ 0..1,space==sp,tubePoints==4,}\newline
+{\tt 96.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ tubeRadius==0.01,colorFunction==cf)}\newline
+{\tt 97.\ \ \ \ \ \ \ \ v\ :=\ v\ +\ delV}\newline
+{\tt 98.\ \ \ \ \ \ u\ :=\ u\ +\ delU}\newline
+{\tt 99.\ \ \ \ void()}\newline
+{\tt 100.\ \ }\newline
+{\tt 101.\ riemannSphere(u,v)\ ==\ }\newline
+{\tt 102.\ \ \ sv\ :=\ sin(v)}\newline
+{\tt 103.\ \ \ 0.99@DFLOAT*(point\ [cos(u)*sv,sin(u)*sv,cos(v),0.0@DFLOAT])+}\newline
+{\tt 104.\ \ \ \ \ point\ [0.0@DFLOAT,\ 0.0@DFLOAT,\ 1.0@DFLOAT,\ 4.0@DFLOAT]}\newline
+{\tt 105.\ \ }\newline
+{\tt 106.\ lineFromTo(p1,\ p2)\ ==}\newline
+{\tt 107.\ \ \ d\ :=\ p2\ -\ p1}\newline
+{\tt 108.\ \ \ (t:DFLOAT):Point\ DFLOAT\ +->}\newline
+{\tt 109.\ \ \ \ \ p1\ +\ t*d}\newline
+  
+\noindent
+
+%\input{gallery/tknot.htex}
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFtknotTitle}{tknot.input}
+\newcommand{\ugFtknotNumber}{G.9.}
+%
+% =====================================================================
+\begin{page}{ugFtknotPage}{G.9. tknot.input}
+% =====================================================================
+\beginscroll
+%
+Create a $(p,q)$ torus-knot with radius $r$ around the curve.
+The formula was derived by Larry Lambe.
+
+  
+\noindent
+{\tt 1.\ \ \ )read\ ntube}\newline
+{\tt 2.\ \ \ torusKnot:\ (DFLOAT,\ DFLOAT,\ DFLOAT,\ PI,\ PI)\ ->\ VIEW3D}\newline
+{\tt 3.\ \ \ torusKnot(p,\ q\ ,r,\ uSteps,\ tSteps)\ ==}\newline
+{\tt 4.\ \ \ \ \ knot\ :=\ (t:DFLOAT):Point\ DFLOAT\ +->\ }\newline
+{\tt 5.\ \ \ \ \ \ \ fac\ :=\ 4/(2.2@DFLOAT-sin(q*t))}\newline
+{\tt 6.\ \ \ \ \ \ \ fac\ *\ point\ [cos(p*t),\ sin(p*t),\ cos(q*t)]}\newline
+{\tt 7.\ \ \ \ \ circle\ :=\ (u:DFLOAT,\ t:DFLOAT):\ Point\ DFLOAT\ +->}\newline
+{\tt 8.\ \ \ \ \ \ \ r\ *\ point\ [cos\ u,\ sin\ u]}\newline
+{\tt 9.\ \ \ \ \ ntubeDrawOpt(knot,\ circle,\ 0..2*\%pi,\ 0..2*\%pi,}\newline
+{\tt 10.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ var1Steps\ ==\ uSteps,\ var2Steps\ ==\ tSteps)}\newline
+{\tt 11.\ \ }\newline
+  
+\noindent
+
+%\input{gallery/ntube.htex}
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFntubeTitle}{ntube.input}
+\newcommand{\ugFntubeNumber}{G.10.}
+%
+% =====================================================================
+\begin{page}{ugFntubePage}{G.10. ntube.input}
+% =====================================================================
+\beginscroll
+%
+The functions in this file create generalized tubes (also known as generalized
+cylinders).
+These functions draw a 2-d curve in the normal
+planes around a 3-d curve.
+
+  
+\noindent
+{\tt 1.\ \ \ R3\ :=\ Point\ DFLOAT}\newline
+{\tt 2.\ \ \ R2\ :=\ Point\ DFLOAT}\newline
+{\tt 3.\ \ \ S\ :=\ Segment\ Float}\newline
+{\tt 4.\ \ \ }\newline
+{\tt 5.\ \ \ ThreeCurve\ :=\ DFLOAT\ ->\ R3}\newline
+{\tt 6.\ \ \ TwoCurve\ :=\ (DFLOAT,\ DFLOAT)\ ->\ R2}\newline
+{\tt 7.\ \ \ Surface\ :=\ (DFLOAT,\ DFLOAT)\ ->\ R3}\newline
+{\tt 8.\ \ \ }\newline
+{\tt 9.\ \ \ FrenetFrame\ :=\ }\newline
+{\tt 10.\ \ \ \ \ Record(value:R3,tangent:R3,normal:R3,binormal:R3)}\newline
+{\tt 11.\ \ frame:\ FrenetFrame}\newline
+{\tt 12.\ \ }\newline
+  
+\noindent
+
+\userfun{ntubeDraw}{\it (spaceCurve, planeCurve,}
+$u_0 .. u_1,$ $t_0 .. t_1)$
+draws {\it planeCurve} in the normal planes of {\it spaceCurve.}
+The parameter $u_0 .. u_1$ specifies
+the parameter range for {\it planeCurve}
+and $t_0 .. t_1$ specifies the parameter range for {\it spaceCurve}.
+Additionally, the plane curve function takes
+a second parameter: the current parameter of {\it spaceCurve}.
+This allows the plane curve to change shape
+as it goes around the space curve.
+See \downlink{``\ugFimagesFiveTitle''}{ugFimagesFivePage} in Section \ugFimagesFiveNumber\ignore{ugFimagesFive} for an example of this.
+%
+  
+\noindent
+{\tt 1.\ \ \ ntubeDraw:\ (ThreeCurve,TwoCurve,S,S)\ ->\ VIEW3D}\newline
+{\tt 2.\ \ \ ntubeDraw(spaceCurve,planeCurve,uRange,tRange)\ ==}\newline
+{\tt 3.\ \ \ \ \ ntubeDrawOpt(spaceCurve,\ planeCurve,\ uRange,\ \_}\newline
+{\tt 4.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ tRange,\ []\$List\ DROPT)}\newline
+{\tt 5.\ \ \ \ }\newline
+{\tt 6.\ \ \ ntubeDrawOpt:\ (ThreeCurve,TwoCurve,S,S,List\ DROPT)}\newline
+{\tt 7.\ \ \ \ \ \ \ ->\ VIEW3D}\newline
+{\tt 8.\ \ \ ntubeDrawOpt(spaceCurve,planeCurve,uRange,tRange,l)\ ==}\newline
+{\tt 9.\ \ \ \ \ \ \ \ \ \ \ \ \ }\newline
+{\tt 10.\ \ \ \ delT:DFLOAT\ :=\ (hi(tRange)\ -\ lo(tRange))/10000}\newline
+{\tt 11.\ \ \ \ oldT:DFLOAT\ :=\ lo(tRange)\ -\ 1}\newline
+{\tt 12.\ \ \ \ fun\ :=\ ngeneralTube(spaceCurve,planeCurve,delT,oldT)}\newline
+{\tt 13.\ \ \ \ draw(fun,\ uRange,\ tRange,\ l)}\newline
+{\tt 14.\ \ }\newline
+  
+\noindent
+
+\userfun{nfrenetFrame}{\it (c, t, delT)}
+numerically computes the Frenet frame
+about the curve {\it c} at {\it t}.
+Parameter {\it delT} is a small number used to
+compute derivatives.
+  
+\noindent
+{\tt 15.\ \ nfrenetFrame(c,\ t,\ delT)\ ==}\newline
+{\tt 16.\ \ \ \ f0\ :=\ c(t)}\newline
+{\tt 17.\ \ \ \ f1\ :=\ c(t+delT)}\newline
+{\tt 18.\ \ \ \ t0\ :=\ f1\ -\ f0}\newline
+{\tt 19.\ \ \ \ n0\ :=\ f1\ +\ f0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\newline
+{\tt 20.\ \ \ \ b\ :=\ cross(t0,\ n0)}\newline
+{\tt 21.\ \ \ \ n\ :=\ cross(b,t0)}\newline
+{\tt 22.\ \ \ \ ln\ :=\ length\ n}\newline
+{\tt 23.\ \ \ \ lb\ :=\ length\ b}\newline
+{\tt 24.\ \ \ \ ln\ =\ 0\ or\ lb\ =\ 0\ =>}\newline
+{\tt 25.\ \ \ \ \ \ \ \ error\ "Frenet\ Frame\ not\ well\ defined"}\newline
+{\tt 26.\ \ \ \ n\ :=\ (1/ln)*n}\newline
+{\tt 27.\ \ \ \ b\ :=\ (1/lb)*b}\newline
+{\tt 28.\ \ \ \ [f0,\ t0,\ n,\ b]\$FrenetFrame}\newline
+  
+\noindent
+
+\userfun{ngeneralTube}{\it (spaceCurve, planeCurve,}{\it  delT, oltT)}
+creates a function that can be passed to the system axiomFun{draw} command.
+The function is a parameterized surface for the general tube
+around {\it spaceCurve}.  {\it delT} is a small number used to compute
+derivatives. {\it oldT} is used to hold the current value of the
+{\it t} parameter for {\it spaceCurve.}  This is an efficiency measure
+to ensure that frames are only computed once for each value of {\it t}.
+  
+\noindent
+{\tt 29.\ \ ngeneralTube:\ (ThreeCurve,\ TwoCurve,\ DFLOAT,\ DFLOAT)\ ->\ Surface}\newline
+{\tt 30.\ \ ngeneralTube(spaceCurve,\ planeCurve,\ delT,\ oldT)\ ==}\newline
+{\tt 31.\ \ \ \ free\ frame}\newline
+{\tt 32.\ \ \ \ (v:DFLOAT,\ t:\ DFLOAT):\ R3\ +->}\newline
+{\tt 33.\ \ \ \ \ \ if\ (t\ \texht{$\sim$}{~}=\ oldT)\ then}\newline
+{\tt 34.\ \ \ \ \ \ \ \ frame\ :=\ nfrenetFrame(spaceCurve,\ t,\ delT)}\newline
+{\tt 35.\ \ \ \ \ \ \ \ oldT\ :=\ t}\newline
+{\tt 36.\ \ \ \ \ \ p\ :=\ planeCurve(v,\ t)}\newline
+{\tt 37.\ \ \ \ \ \ frame.value\ +\ p.1*frame.normal\ +\ p.2*frame.binormal}\newline
+  
+\noindent
+
+%\input{gallery/dhtri.htex}
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFdhtriTitle}{dhtri.input}
+\newcommand{\ugFdhtriNumber}{G.11.}
+%
+% =====================================================================
+\begin{page}{ugFdhtriPage}{G.11. dhtri.input}
+% =====================================================================
+\beginscroll
+%
+Create affine transformations (DH matrices) that transform
+a given triangle into another.
+
+  
+\noindent
+{\tt 1.\ \ \ tri2tri:\ (List\ Point\ DFLOAT,\ List\ Point\ DFLOAT)\ ->\ DHMATRIX(DFLOAT)}\newline
+{\tt 2.\ \ \ tri2tri(t1,\ t2)\ ==}\newline
+{\tt 3.\ \ \ \ \ n1\ :=\ triangleNormal(t1)}\newline
+{\tt 4.\ \ \ \ \ n2\ :=\ triangleNormal(t2)}\newline
+{\tt 5.\ \ \ \ \ tet2tet(concat(t1,\ n1),\ concat(t2,\ n2))}\newline
+{\tt 6.\ \ \ \ }\newline
+{\tt 7.\ \ \ tet2tet:\ (List\ Point\ DFLOAT,\ List\ Point\ DFLOAT)\ ->\ DHMATRIX(DFLOAT)}\newline
+{\tt 8.\ \ \ tet2tet(t1,\ t2)\ ==}\newline
+{\tt 9.\ \ \ \ \ m1\ :=\ makeColumnMatrix\ t1}\newline
+{\tt 10.\ \ \ \ m2\ :=\ makeColumnMatrix\ t2}\newline
+{\tt 11.\ \ \ \ m2\ *\ inverse(m1)}\newline
+{\tt 12.\ \ \ }\newline
+{\tt 13.\ \ makeColumnMatrix(t)\ ==}\newline
+{\tt 14.\ \ \ \ m\ :=\ new(4,4,0)\$DHMATRIX(DFLOAT)}\newline
+{\tt 15.\ \ \ \ for\ x\ in\ t\ for\ i\ in\ 1..repeat}\newline
+{\tt 16.\ \ \ \ \ \ for\ j\ in\ 1..3\ repeat}\newline
+{\tt 17.\ \ \ \ \ \ \ \ m(j,i)\ :=\ x.j}\newline
+{\tt 18.\ \ \ \ \ \ m(4,i)\ :=\ 1}\newline
+{\tt 19.\ \ \ \ m}\newline
+{\tt 20.\ \ \ }\newline
+{\tt 21.\ \ triangleNormal(t)\ ==}\newline
+{\tt 22.\ \ \ \ a\ :=\ triangleArea\ t}\newline
+{\tt 23.\ \ \ \ p1\ :=\ t.2\ -\ t.1}\newline
+{\tt 24.\ \ \ \ p2\ :=\ t.3\ -\ t.2}\newline
+{\tt 25.\ \ \ \ c\ :=\ cross(p1,\ p2)}\newline
+{\tt 26.\ \ \ \ len\ :=\ length(c)}\newline
+{\tt 27.\ \ \ \ len\ =\ 0\ =>\ error\ "degenerate\ triangle!"}\newline
+{\tt 28.\ \ \ \ c\ :=\ (1/len)*c}\newline
+{\tt 29.\ \ \ \ t.1\ +\ sqrt(a)\ *\ c}\newline
+{\tt 30.\ \ \ }\newline
+{\tt 31.\ \ triangleArea\ t\ ==}\newline
+{\tt 32.\ \ \ \ a\ :=\ length(t.2\ -\ t.1)}\newline
+{\tt 33.\ \ \ \ b\ :=\ length(t.3\ -\ t.2)}\newline
+{\tt 34.\ \ \ \ c\ :=\ length(t.1\ -\ t.3)}\newline
+{\tt 35.\ \ \ \ s\ :=\ (a+b+c)/2}\newline
+{\tt 36.\ \ \ \ sqrt(s*(s-a)*(s-b)*(s-c))}\newline
+  
+\noindent
+
+
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFtetraTitle}{tetra.input}
+\newcommand{\ugFtetraNumber}{G.12.}
+%
+% =====================================================================
+\begin{page}{ugFtetraPage}{G.12. tetra.input}
+% =====================================================================
+\beginscroll
+%
+%\input{gallery/tetra.htex}
+%\outdent{Sierpinsky's Tetrahedron}
+
+\labelSpace{3pc}
+
+  
+\noindent
+{\tt 1.\ \ \ )set\ expose\ add\ con\ DenavitHartenbergMatrix}\newline
+{\tt 2.\ \ \ }\newline
+{\tt 3.\ \ \ x1:DFLOAT\ :=\ sqrt(2.0@DFLOAT/3.0@DFLOAT)}\newline
+{\tt 4.\ \ \ x2:DFLOAT\ :=\ sqrt(3.0@DFLOAT)/6}\newline
+{\tt 5.\ \ \ }\newline
+{\tt 6.\ \ \ p1\ :=\ point\ [-0.5@DFLOAT,\ -x2,\ 0.0@DFLOAT]}\newline
+{\tt 7.\ \ \ p2\ :=\ point\ [0.5@DFLOAT,\ -x2,\ 0.0@DFLOAT]}\newline
+{\tt 8.\ \ \ p3\ :=\ point\ [0.0@DFLOAT,\ 2*x2,\ 0.0@DFLOAT]}\newline
+{\tt 9.\ \ \ p4\ :=\ point\ [0.0@DFLOAT,\ 0.0@DFLOAT,\ x1]}\newline
+{\tt 10.\ \ \ }\newline
+{\tt 11.\ \ baseTriangle\ \ :=\ [p2,\ p1,\ p3]}\newline
+{\tt 12.\ \ }\newline
+{\tt 13.\ \ mt\ \ :=\ [0.5@DFLOAT*(p2+p1),\ 0.5@DFLOAT*(p1+p3),\ 0.5@DFLOAT*(p3+p2)]}\newline
+{\tt 14.\ \ }\newline
+{\tt 15.\ \ bt1\ :=\ [mt.1,\ p1,\ mt.2]}\newline
+{\tt 16.\ \ bt2\ :=\ [p2,\ mt.1,\ mt.3]}\newline
+{\tt 17.\ \ bt3\ :=\ [mt.2,\ p3,\ mt.3]}\newline
+{\tt 18.\ \ bt4\ :=\ [0.5@DFLOAT*(p2+p4),\ 0.5@DFLOAT*(p1+p4),\ 0.5@DFLOAT*(p3+p4)]}\newline
+{\tt 19.\ \ }\newline
+{\tt 20.\ \ tt1\ :=\ tri2tri(baseTriangle,\ bt1)}\newline
+{\tt 21.\ \ tt2\ :=\ tri2tri(baseTriangle,\ bt2)}\newline
+{\tt 22.\ \ tt3\ :=\ tri2tri(baseTriangle,\ bt3)}\newline
+{\tt 23.\ \ tt4\ :=\ tri2tri(baseTriangle,\ bt4)}\newline
+{\tt 24.\ \ }\newline
+{\tt 25.\ \ drawPyramid(n)\ ==}\newline
+{\tt 26.\ \ \ \ s\ :=\ createThreeSpace()}\newline
+{\tt 27.\ \ \ \ dh\ :=\ rotatex(0.0@DFLOAT)}\newline
+{\tt 28.\ \ \ \ drawPyramidInner(s,\ n,\ dh)}\newline
+{\tt 29.\ \ \ \ makeViewport3D(s,\ "Sierpinsky\ Tetrahedron")}\newline
+{\tt 30.\ \ }\newline
+{\tt 31.\ \ drawPyramidInner(s,\ n,\ dh)\ ==}\newline
+{\tt 32.\ \ \ \ n\ =\ 0\ =>\ makeTetrahedron(s,\ dh,\ n)}\newline
+{\tt 33.\ \ \ \ drawPyramidInner(s,\ n-1,\ dh\ *\ tt1)}\newline
+{\tt 34.\ \ \ \ drawPyramidInner(s,\ n-1,\ dh\ *\ tt2)}\newline
+{\tt 35.\ \ \ \ drawPyramidInner(s,\ n-1,\ dh\ *\ tt3)}\newline
+{\tt 36.\ \ \ \ drawPyramidInner(s,\ n-1,\ dh\ *\ tt4)}\newline
+{\tt 37.\ \ }\newline
+{\tt 38.\ \ makeTetrahedron(sp,\ dh,\ color)\ ==}\newline
+{\tt 39.\ \ \ \ w1\ :=\ dh*p1}\newline
+{\tt 40.\ \ \ \ w2\ :=\ dh*p2}\newline
+{\tt 41.\ \ \ \ w3\ :=\ dh*p3}\newline
+{\tt 42.\ \ \ \ w4\ :=\ dh*p4}\newline
+{\tt 43.\ \ \ \ polygon(sp,\ [w1,\ w2,\ w4])}\newline
+{\tt 44.\ \ \ \ polygon(sp,\ [w1,\ w3,\ w4])}\newline
+{\tt 45.\ \ \ \ polygon(sp,\ [w2,\ w3,\ w4])}\newline
+{\tt 46.\ \ \ \ void()}\newline
+  
+\noindent
+%-% \HDindex{Sierpinsky's Tetrahedron}{ugFtetraPage}{G.12.}{tetra.input}
+
+
+%\input{gallery/antoine.htex}
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFantoineTitle}{antoine.input}
+\newcommand{\ugFantoineNumber}{G.13.}
+%
+% =====================================================================
+\begin{page}{ugFantoinePage}{G.13. antoine.input}
+% =====================================================================
+\beginscroll
+%
+Draw Antoine's Necklace.
+%-% \HDindex{Antoine's Necklace}{ugFantoinePage}{G.13.}{antoine.input}
+Thank you to Matthew Grayson at IBM's T.J Watson Research Center for the idea.
+
+  
+\noindent
+{\tt 1.\ \ \ )set\ expose\ add\ con\ DenavitHartenbergMatrix}\newline
+{\tt 2.\ \ \ }\newline
+{\tt 3.\ \ \ torusRot:\ DHMATRIX(DFLOAT)}\newline
+{\tt 4.\ \ \ }\newline
+{\tt 5.\ \ \ }\newline
+{\tt 6.\ \ \ drawRings(n)\ ==}\newline
+{\tt 7.\ \ \ \ \ s\ :=\ createThreeSpace()}\newline
+{\tt 8.\ \ \ \ \ dh:DHMATRIX(DFLOAT)\ :=\ identity()}\newline
+{\tt 9.\ \ \ \ \ drawRingsInner(s,\ n,\ dh)}\newline
+{\tt 10.\ \ \ \ makeViewport3D(s,\ "Antoine's\ Necklace")}\newline
+{\tt 11.\ \ }\newline
+  
+\noindent
+
+In order to draw Antoine rings, we take one ring, scale it down to
+a smaller size, rotate it around its central axis, translate it
+to the edge of the larger ring and rotate it around the edge to
+a point corresponding to its count (there are 10 positions around
+the edge of the larger ring). For each of these new rings we
+recursively perform the operations, each ring becoming 10 smaller
+rings. Notice how the \axiomType{DHMATRIX} operations are used to build up
+the proper matrix composing all these transformations.
+
+  
+\noindent
+{\tt 1.\ \ \ drawRingsInner(s,\ n,\ dh)\ ==}\newline
+{\tt 2.\ \ \ \ \ n\ =\ 0\ =>}\newline
+{\tt 3.\ \ \ \ \ \ \ drawRing(s,\ dh)}\newline
+{\tt 4.\ \ \ \ \ \ \ void()}\newline
+{\tt 5.\ \ \ \ \ t\ :=\ 0.0@DFLOAT}\newline
+{\tt 6.\ \ \ \ \ p\ :=\ 0.0@DFLOAT}\newline
+{\tt 7.\ \ \ \ \ tr\ :=\ 1.0@DFLOAT}\newline
+{\tt 8.\ \ \ \ \ inc\ :=\ 0.1@DFLOAT}\newline
+{\tt 9.\ \ \ \ \ for\ i\ in\ 1..10\ repeat}\newline
+{\tt 10.\ \ \ \ \ \ tr\ :=\ tr\ +\ inc}\newline
+{\tt 11.\ \ \ \ \ \ inc\ :=\ -inc}\newline
+{\tt 12.\ \ \ \ \ \ dh'\ :=\ dh*rotatez(t)*translate(tr,0.0@DFLOAT,0.0@DFLOAT)*}\newline
+{\tt 13.\ \ \ \ \ \ \ \ \ \ \ \ \ rotatey(p)*scale(0.35@DFLOAT,\ 0.48@DFLOAT,\ 0.4@DFLOAT)}\newline
+{\tt 14.\ \ \ \ \ \ drawRingsInner(s,\ n-1,\ dh')}\newline
+{\tt 15.\ \ \ \ \ \ t\ :=\ t\ +\ 36.0@DFLOAT}\newline
+{\tt 16.\ \ \ \ \ \ p\ :=\ p\ +\ 90.0@DFLOAT}\newline
+{\tt 17.\ \ \ \ void()}\newline
+{\tt 18.\ \ }\newline
+{\tt 19.\ \ drawRing(s,\ dh)\ ==}\newline
+{\tt 20.\ \ \ \ free\ torusRot}\newline
+{\tt 21.\ \ \ \ torusRot\ :=\ dh}\newline
+{\tt 22.\ \ \ \ makeObject(torus,\ 0..2*\%pi,\ 0..2*\%pi,\ var1Steps\ ==\ 6,}\newline
+{\tt 23.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ space\ ==\ s,\ var2Steps\ ==\ 15)}\newline
+{\tt 24.\ \ }\newline
+{\tt 25.\ \ torus(u\ ,v)\ ==}\newline
+{\tt 26.\ \ \ \ cu\ :=\ cos(u)/6}\newline
+{\tt 27.\ \ \ \ torusRot*point\ [(1+cu)*cos(v),(1+cu)*sin(v),(sin\ u)/6]}\newline
+  
+\noindent
+
+%\input{gallery/scherk.htex}
+\endscroll
+\autobuttons
+\end{page}
+%
+%
+\newcommand{\ugFscherkTitle}{scherk.input}
+\newcommand{\ugFscherkNumber}{G.14.}
+%
+% =====================================================================
+\begin{page}{ugFscherkPage}{G.14. scherk.input}
+% =====================================================================
+\beginscroll
+%
+
+Scherk's minimal surface, defined by:
+%-% \HDindex{Scherk's minimal surface}{ugFscherkPage}{G.14.}{scherk.input}
+\texht{$e^z \cos(x) = \cos(y)$}{\axiom{exp(z) * cos(x) = cos(y)}}.
+See: {\it A Comprehensive Introduction to Differential Geometry,} Vol. 3,
+by Michael Spivak, Publish Or Perish, Berkeley, 1979, pp. 249-252.
+
+  
+\noindent
+{\tt 1.\ \ \ (xOffset,\ yOffset):DFLOAT}\newline
+{\tt 2.\ \ \ \ }\newline
+{\tt 3.\ \ \ }\newline
+{\tt 4.\ \ \ drawScherk(m,n)\ ==}\newline
+{\tt 5.\ \ \ \ \ free\ xOffset,\ yOffset}\newline
+{\tt 6.\ \ \ \ \ space\ :=\ createThreeSpace()}\newline
+{\tt 7.\ \ \ \ \ for\ i\ in\ 0..m-1\ repeat}\newline
+{\tt 8.\ \ \ \ \ \ \ xOffset\ :=\ i*\%pi}\newline
+{\tt 9.\ \ \ \ \ \ \ for\ j\ in\ 0\ ..\ n-1\ repeat}\newline
+{\tt 10.\ \ \ \ \ \ \ \ rem(i+j,\ 2)\ =\ 0\ =>\ 'iter}\newline
+{\tt 11.\ \ \ \ \ \ \ \ yOffset\ :=\ j*\%pi}\newline
+{\tt 12.\ \ \ \ \ \ \ \ drawOneScherk(space)}\newline
+{\tt 13.\ \ \ \ makeViewport3D(space,\ "Scherk's\ Minimal\ Surface")}\newline
+{\tt 14.\ \ }\newline
+{\tt 15.\ \ scherk1(u,v)\ ==}\newline
+{\tt 16.\ \ \ \ x\ :=\ cos(u)/exp(v)}\newline
+{\tt 17.\ \ \ \ point\ [xOffset\ +\ acos(x),\ yOffset\ +\ u,\ v,\ abs(v)]}\newline
+{\tt 18.\ \ \ }\newline
+{\tt 19.\ \ scherk2(u,v)\ ==}\newline
+{\tt 20.\ \ \ \ x\ :=\ cos(u)/exp(v)}\newline
+{\tt 21.\ \ \ \ point\ [xOffset\ -\ acos(x),\ yOffset\ +\ u,\ v,\ abs(v)]}\newline
+{\tt 22.\ \ \ }\newline
+{\tt 23.\ \ scherk3(u,v)\ ==\ }\newline
+{\tt 24.\ \ \ \ x\ :=\ exp(v)\ *\ cos(u)}\newline
+{\tt 25.\ \ \ \ point\ [xOffset\ +\ u,\ yOffset\ +\ acos(x),\ v,\ abs(v)]}\newline
+{\tt 26.\ \ \ }\newline
+{\tt 27.\ \ scherk4(u,v)\ ==\ }\newline
+{\tt 28.\ \ \ \ x\ :=\ exp(v)\ *\ cos(u)}\newline
+{\tt 29.\ \ \ \ point\ [xOffset\ +\ u,\ yOffset\ -\ acos(x),\ v,\ abs(v)]}\newline
+{\tt 30.\ \ \ }\newline
+{\tt 31.\ \ drawOneScherk(s)\ ==}\newline
+{\tt 32.\ \ \ \ makeObject(scherk1,-\%pi/2..\%pi/2,0..\%pi/2,space==s,}\newline
+{\tt 33.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ var1Steps\ ==\ 28,\ var2Steps\ ==\ 28)}\newline
+{\tt 34.\ \ \ \ makeObject(scherk2,-\%pi/2..\%pi/2,0..\%pi/2,space==s,}\newline
+{\tt 35.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ var1Steps\ ==\ 28,\ var2Steps\ ==\ 28)}\newline
+{\tt 36.\ \ \ \ makeObject(scherk3,-\%pi/2..\%pi/2,-\%pi/2..0,space==s,}\newline
+{\tt 37.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ var1Steps\ ==\ 28,\ var2Steps\ ==\ 28)}\newline
+{\tt 38.\ \ \ \ makeObject(scherk4,-\%pi/2..\%pi/2,-\%pi/2..0,space==s,}\newline
+{\tt 39.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ var1Steps\ ==\ 28,\ var2Steps\ ==\ 28)}\newline
+{\tt 40.\ \ \ \ void()}\newline
+  
+\noindent
+\endscroll
+\autobuttons
+\end{page}
+%
diff --git a/src/hyper/pages/union.ht b/src/hyper/pages/union.ht
new file mode 100644
index 00000000..9d7716ab
--- /dev/null
+++ b/src/hyper/pages/union.ht
@@ -0,0 +1,98 @@
+%---------------------------------------------------------------------------
+%                   Tagged Union Constructor Page
+%---------------------------------------------------------------------------
+ 
+\begin{page}{DomainUnion}{Domain {\em Union(a:A,...,b:B)}}
+\beginscroll
+{\em Union} takes any number of "tag"-domain pairs of arguments:
+\indentrel{2}
+\newline \spad{a}, a tag, an element of domain \spadtype{Symbol}
+\newline \spad{A}, a domain of category \spadtype{SetCategory}
+\newline\tab{10}...
+\newline \spad{b}, a tag, an element of domain \spadtype{Symbol}
+\newline \spad{B}, a domain of category \spadtype{SetCategory}
+\indentrel{-2}\newline
+This constructor is a primitive in \Language{}. 
+\newline
+\beginmenu
+\item\menulispdownlink{Description}{(|dbSpecialDescription| '|Union|)) }\tab{15}General description
+\item\menulispdownlink{Operations}{(|dbSpecialOperations| '|Union|)}\tab{15}All exported operations of \spad{Union(a:A,b:B)}
+%\item\menudownlink{Examples}   {UnionExamples}   \tab{15}Examples illustrating use
+\item\menulispdownlink{Exports}{(|dbSpecialExports| '|Union|)}\tab{15}Explicit categories and operations
+\endmenu
+\vspace{1}\newline
+In this tagged \spad{Union}, tags \spad{a,...,b} must be distinct. 
+\newline
+For an alternate "untagged" form of \spad{Union}, see \downlink{Union(A,B)}{UntaggedUnion}.
+\endscroll\end{page}
+ 
+\begin{page}{UnionDescription}{Domain Constructor {\em Union}}
+\beginscroll
+\newline\menuitemstyle{}\tab{2}Union({\em a:A},{\em b:B})
+\newline\tab{2}{\em Arguments:}\indent{17}\tab{-2}
+{\em a}, a tag, an element of domain \spadtype{Symbol}
+\newline\tab{-2}
+{\em A}, a domain of category \spadtype{SetCategory}
+\newline\tab{-2}
+{\em b}, a tag, an element of domain \spadtype{Symbol}
+\newline\tab{-2}
+{\em B}, a domain of category \spadtype{SetCategory}
+\indent{0}\newline\tab{2}{\em Returns:}\indent{17}\tab{-2}
+the "union of {\em A} and {\em B}" as described below.
+\indent{0}\newline\tab{2}{\em Description:}\indent{15}\tab{0}
+{\em Union(a:A,b:B)} denotes the class of objects 
+which are either members of domain {\em A} or of domain {\em B}.
+The symbols {\em a} and {\em b} are called "tags" and 
+are used to identify the two "branches"
+of the union.
+The {\em Union} constructor can take any number of arguments and
+has an alternate form without {\em tags}.
+This tagged {\em Union} type is necessary, for example, to disambiguate
+two branches of a union where {\em A} and {\em B} denote the same type.
+{\em Union} is a primitive domain of \Language{} which cannot be
+defined in the \Language{} language. 
+\endscroll\end{page}
+
+%---------------------------------------------------------------------------
+%               Untagged Union Constructor Page
+%---------------------------------------------------------------------------
+ 
+\begin{page}{UntaggedUnion}{Domain {\em Union(A,...,B)}}
+\beginscroll
+{\em Union} takes any number of domain arguments:
+\indentrel{2}
+\newline \spad{A}, a domain of category \spadtype{SetCategory}
+\newline\tab{10}...
+\newline \spad{B}, a domain of category \spadtype{SetCategory}
+\indentrel{-2}\newline
+\spad{Union} is a primitive constructor in \Language{}. 
+\newline
+\beginmenu
+\item\menulispdownlink{Description}{(|dbSpecialDescription| '|UntaggedUnion|)) }\tab{15}General description
+\item\menulispdownlink{Operations}{(|dbSpecialOperations| '|UntaggedUnion|)}\tab{15}All exported operations of \spad{Union(A,B)}
+%\item\menudownlink{Examples}   {UTUnionExamples}   \tab{15}Examples illustrating use
+%\item\menudownlink{Exports}    {UTUnionExports}    \tab{15}Explicit categories and operations
+\endmenu
+\vspace{1}\newline
+In this untagged form of \spad{Union}, domains \spad{A,...,B} must be distinct.
+\endscroll\end{page}
+
+\begin{page}{UTUnionDescription}{Domain Constructor {\em Union}}
+\beginscroll
+\newline\menuitemstyle{}\tab{2}Union({\em A},{\em B})
+\newline\tab{2}{\em Arguments:}\indent{17}\tab{-2}
+{\em A}, a domain of category \spadtype{SetCategory}
+\newline\tab{-2}
+{\em B}, a domain of category \spadtype{SetCategory}
+\indent{0}\newline\tab{2}{\em Returns:}\indent{17}\tab{-2}
+the "union of {\em A} and {\em B}" as described below.
+\indent{0}\newline\tab{2}{\em Description:}\indent{15}\tab{0}
+{\em Union(A,B)} denotes the class of objects which are
+which are either members of domain {\em A} or of domain {\em B}.
+The {\em Union} constructor can take any number of arguments and
+has an alternate form using {\em tags}.
+{\em Union} is a primitive domain of \Language{} which cannot be
+defined in the \Language{} language. 
+\endscroll\end{page}
+
+
diff --git a/src/hyper/pages/util.ht b/src/hyper/pages/util.ht
new file mode 100755
index 00000000..99eb123d
--- /dev/null
+++ b/src/hyper/pages/util.ht
@@ -0,0 +1,510 @@
+% Copyright The Numerical Algorithms Group Limited 1992-1994.
+% Certain derivative-work portions Copyright (C) 1988 by Leslie Lamport.
+% All rights reserved
+
+% ----------------------------------------------------------------------
+% This file contains macros for the Axiom HyperDoc hypertext facility.
+% Most of the macros for the system are here though there may be some in
+% individual .ht files that are of a local nature.
+% ----------------------------------------------------------------------
+
+% ----------------------------------------------------------------------
+% 1. Names of software and facilities.
+% ----------------------------------------------------------------------
+
+\newcommand{\Browse}{Browse}
+\newcommand{\Language}{AXIOM}
+\newcommand{\SpadName}{\Language}
+\newcommand{\LangName}{\Language}
+\newcommand{\HyperName}{HyperDoc}
+\newcommand{\axiomxl}{AXIOM-XL}
+\newcommand{\anatural}{AXIOM-XL}
+\newcommand{\Clef}{Clef}
+\newcommand{\Lisp}{Common LISP}
+\newcommand{\naglib}{NAG Foundation Library}
+
+
+% ----------------------------------------------------------------------
+% 2. Special pages used by HyperDoc.
+% ----------------------------------------------------------------------
+
+\newcommand{\GoBackToWork}{\vspace{2}\newline{Click on \  \UpButton{} \  to go back to what you were doing.}}
+
+\begin{page}{SpadNotConnectedPage}{Not Connected to AXIOM}
+\beginscroll
+\HyperName{} isn't connected to \Language{}, therefore cannot execute
+the button you pressed.
+%
+\GoBackToWork{}
+\endscroll
+\end{page}
+
+\begin{page}{ProtectedQuitPage}{Do You Really Want to Exit?}
+\beginscroll
+{Click again on \  \ExitButton{QuitPage} \  to terminate \HyperName{}.}
+\vspace{1}\newline
+\centerline{OR}
+\GoBackToWork{}
+\endscroll
+\autobuttons
+\end{page}
+
+\begin{page}{UnknownPage}{Missing Page}
+\beginscroll
+\pp
+The page you requested was not found in the \HyperName{} database.
+\GoBackToWork{}
+\endscroll
+\end{page}
+
+\begin{page}{ErrorPage}{Something is Wrong}
+\beginscroll
+{For some reason the page you tried to link to cannot be formatted.}
+\GoBackToWork{}
+\endscroll
+\autobuttons
+\end{page}
+
+\begin{page}{Unlinked}{Sorry!}
+\beginscroll
+{This link is not implemented yet.}
+\GoBackToWork{}
+\endscroll
+\autobuttons
+\end{page}
+
+% ----------------------------------------------------------------------
+% 3. Special hooks to Unix.
+% ----------------------------------------------------------------------
+
+% All unix commands should be done as macros defined here so we don't
+% have to go hunting when moving between Unix versions.
+
+\newcommand{\newspadclient}[1]{xterm -n "#1" -e \$SPAD/bin/clef \$SPAD/bin/server/spadclient}
+\newcommand{\searchwindow}[2]{\unixwindow{#1}{\$SPAD/lib/htsearch "#2"}}
+\newcommand{\unixwindow}[2]{\unixlink{#1}{#2}}
+\newcommand{\menuunixlink}[2]{\item\unixlink{\menuitemstyle{#1}}{#2}}
+\newcommand{\menuunixcommand}[2]{\item\unixcommand{\menuitemstyle{#1}}{#2}}
+\newcommand{\menuunixwindow}[2]{\item\unixwindow{\menuitemstyle{#1}}{#2}}
+
+% ----------------------------------------------------------------------
+% 4. HyperDoc menu macros.
+% ----------------------------------------------------------------------
+
+% Example:
+%
+% \beginmenu
+% \menulink{Thing One}{PageOne} la da di da da ...
+% \menulink{Thin Two}{PageTwo}  do da day ...
+% \item \ACmdMacro{\menuitemstyle{Thing Three}} la di da ...
+% \endmenu
+
+% The menu environment
+
+\newcommand{\beginmenu}          {\beginitems[\MenuDotBitmap]}
+\newcommand{\endmenu}            {\enditems}
+
+% This is the usual format for a menu item.
+
+\newcommand{\menuitemstyle}[1]   {{\MenuDotBitmap}#1}
+
+% Often-used menu item forms
+
+%   These two simply do links
+\newcommand{\menudownlink}[2]    {\item\downlink{\menuitemstyle{#1}}{#2}}
+\newcommand{\menulink}[2]        {\menudownlink{#1}{#2}}
+
+%   This will cause lower level links to have a HOME button
+\newcommand{\menumemolink}[2]    {\item\memolink{\menuitemstyle{#1}}{#2}}
+
+%   This opens a new window for the linked page.
+\newcommand{\menuwindowlink}[2]    {\item\windowlink{\menuitemstyle{#1}}{#2}}
+
+%   These execute lisp commands in various flavors
+\newcommand{\menulispcommand}[2] {\item\lispcommand{\menuitemstyle{#1}}{#2}}
+\newcommand{\menulispdownlink}[2]{\item\lispdownlink{\menuitemstyle{#1}}{#2}}
+\newcommand{\menulispmemolink}[2]{\item\lispmemolink{\menuitemstyle{#1}}{#2}}
+\newcommand{\menulispwindowlink}[2]{\item\lispwindowlink{\menuitemstyle{#1}}{#2}}
+
+%   This executes a unix command
+\newcommand{\menuunixcmd}[2]     {\item\unixcommand{\menuitemstyle{#1}}{#2}}
+\newcommand{\searchresultentry}[3]{\tab{3}\item#3\tab{8}\downlink{\menuitemstyle{#1}}{#2}\newline}
+\newcommand{\newsearchresultentry}[3]{\tab{3}\item#1\tab{8}\downlink{\menuitemstyle{#2}}{#3}\newline}
+
+% ----------------------------------------------------------------------
+% 5. Bitmaps and bitmap manipulation macros.
+% ----------------------------------------------------------------------
+
+\newcommand{\htbmdir}{\env{AXIOM}/share/hypertex/bitmaps}
+\newcommand{\htbmfile}[1]{\htbmdir /#1.bitmap}
+\newcommand{\htbitmap}[1]{\inputbitmap{\htbmfile{#1}}}
+\newcommand{\ControlBitmap}[1]{\controlbitmap{\htbmfile{#1}}}
+
+% next group of bitmaps frequently appear in the titlebar
+\newcommand{\ContinueBitmap} {\ControlBitmap{Continue}}
+\newcommand{\DoItBitmap}     {\ControlBitmap{DoIt}}
+\newcommand{\ExitBitmap}     {\ControlBitmap{exit3d}}
+\newcommand{\HelpBitmap}     {\ControlBitmap{help3d}}
+\newcommand{\ReturnBitmap}   {\ControlBitmap{home3d}}
+\newcommand{\NoopBitmap}       {\ControlBitmap{noop3d}}
+\newcommand{\UpBitmap}       {\ControlBitmap{up3d}}
+
+\newcommand{\MenuDotBitmap}{\htbitmap{menudot}}
+
+% Including control panel pixmaps for help pages:
+
+\newcommand{\helpbit}[1]{\centerline{\inputpixmap{\env{AXIOM}/share/hypertex/pixmaps/{#1}}}}
+
+% ----------------------------------------------------------------------
+% 6. HyperDoc button objects.
+% ----------------------------------------------------------------------
+
+\newcommand{\ContinueButton}[1]{\downlink{Click here}{#1} to continue.}
+\newcommand{\ExitButton}[1]{\memolink{\ExitBitmap}{#1}}
+\newcommand{\HelpButton}[1]{\memolink{\HelpBitmap}{#1}}
+\newcommand{\StdHelpButton}{\HelpButton{ugHyperPage}}
+\newcommand{\StdExitButton}{\ExitButton{ProtectedQuitPage}}
+\newcommand{\UpButton}{\upbutton{\UpBitmap}{UpPage}}
+\newcommand{\ReturnButton}{\returnbutton{\ReturnBitmap}{ReturnPage}}
+\newcommand{\on}[1]{{\inputbox[1]{#1}{\htbmfile{pick}}
+             {\htbmfile{unpick}}}}
+\newcommand{\off}[1]{{\inputbox[0]{#1}{\htbmfile{pick}}
+             {\htbmfile{unpick}}}}
+
+% ----------------------------------------------------------------------
+% 6. Standard HyperDoc button configurations.
+% ----------------------------------------------------------------------
+
+\newcommand{\autobutt}[1]{\helppage{#1}}
+\newcommand{\autobuttons}{}
+\newcommand{\exitbuttons}{}
+
+\newcommand{\autobuttLayout}[1]{\centerline{#1}}}
+\newcommand{\autobuttMaker}[1]{\autobuttLayout{\HelpButton{#1}}}
+\newcommand{\riddlebuttons}[1]{\autobuttLayout{\link{\HelpBitmap}{#1}}}
+
+% Macro for downward compatibility (?).
+
+\newcommand{\simplebox}[2]{\inputbox[#1]{#2}{\htbitmap{Xbox}}{\htbitmap{Xopenbox}}}
+
+% ----------------------------------------------------------------------
+% 7. HyperDoc graphics macros.
+% ----------------------------------------------------------------------
+
+% Including viewport bitmaps within \HyperName pages:
+
+\newcommand{\viewport}[1]{\inputimage{{#1}.VIEW/image}}
+\newcommand{\axiomViewport}[1]{\inputimage{\env{AXIOM}/share/viewports/{#1}.VIEW/image}}
+\newcommand{\spadviewport}[1]{\axiomViewport{#1}}
+
+% Creating a real live viewport:
+
+\newcommand{\viewportbutton}[2]{\unixcommand{#1}{viewAlone #2}}
+\newcommand{\axiomViewportbutton}[2]{\unixcommand{#1}{viewAlone \$AXIOM/share/viewports/{#2}}}
+\newcommand{\spadviewportbutton}[2]{\axiomViewportbutton{#1}{#2}}
+
+% Making active viewport buttons:
+
+\newcommand{\viewportasbutton}[1]{\unixcommand{\inputimage{{#1}.VIEW/image}}{viewAlone {#1}}}
+\newcommand{\axiomViewportasbutton}[1]{\unixcommand{\inputimage{\env{AXIOM}/share/viewports/{#1}.VIEW/image}}{viewAlone \$AXIOM/share/viewports/{#1}}}
+\newcommand{\spadviewportasbutton}[1]{\axiomViewportasbutton{#1}}
+
+% ----------------------------------------------------------------------
+% 8. TeX and LaTeX compatibility macros.
+% ----------------------------------------------------------------------
+
+%% Begin macros that are needed because HD uses the wrong names
+
+\newcommand{\center}[1]{\centerline{#1}}
+\newcommand{\box}[1]{\fbox{#1}}
+
+%% End   macros that are needed because HD uses the wrong names
+
+\newcommand{\LARGE}{}
+\newcommand{\LaTeX}{LaTeX}
+\newcommand{\Large}{}
+\newcommand{\TeX}{TeX}
+\newcommand{\allowbreak}{}
+\newcommand{\aleph}{\inputbitmap{\htbmdir{}/aleph.bitmap}}
+\newcommand{\alpha}{\inputbitmap{\htbmdir{}/alpha.bitmap}}
+\newcommand{\angle}{\inputbitmap{\htbmdir{}/angle.bitmap}}
+\newcommand{\backslash}{\inputbitmap{\htbmdir{}/backslash.bitmap}}
+\newcommand{\beta}{\inputbitmap{\htbmdir{}/beta.bitmap}}
+\newcommand{\bigbreak}{\newline\newline}
+\newcommand{\bot}{\inputbitmap{\htbmdir{}/bot.bitmap}}
+\newcommand{\bullet}{\inputbitmap{\htbmdir{}/bullet.bitmap}}
+\newcommand{\caption}[1]{\newline\centerline{#1}\newline}
+\newcommand{\chi}{\inputbitmap{\htbmdir{}/chi.bitmap}}
+\newcommand{\cite}[1]{bibliography entry for {\it #1}}
+\newcommand{\cleardoublepage}{}
+\newcommand{\coprod}{\inputbitmap{\htbmdir{}/coprod.bitmap}}
+\newcommand{\del}{\inputbitmap{\htbmdir{}/del.bitmap}}
+\newcommand{\delta}{\inputbitmap{\htbmdir{}/delta.bitmap}}
+\newcommand{\Delta}{\inputbitmap{\htbmdir{}/delta-cap.bitmap}}
+\newcommand{\div}{\inputbitmap{\htbmdir{}/div.bitmap}}
+\newcommand{\dot}{\inputbitmap{\htbmdir{}/dot.bitmap}}
+\newcommand{\ell}{\inputbitmap{\htbmdir{}/ell.bitmap}}
+\newcommand{\emptyset}{\inputbitmap{\htbmdir{}/emptyset.bitmap}}
+\newcommand{\epsilon}{\inputbitmap{\htbmdir{}/epsilon.bitmap}}
+\newcommand{\epsffile}{}
+\newcommand{\eta}{\inputbitmap{\htbmdir{}/eta.bitmap}}
+\newcommand{\exists}{\inputbitmap{\htbmdir{}/exists.bitmap}}
+\newcommand{\forall}{\inputbitmap{\htbmdir{}/forall.bitmap}}
+\newcommand{\footnote}[1]{ {(#1)}}
+\newcommand{\frenchspacing}{}
+\newcommand{\gamma}{\inputbitmap{\htbmdir{}/gamma.bitmap}}
+\newcommand{\Gamma}{\inputbitmap{\htbmdir{}/gamma-cap.bitmap}}
+\newcommand{\hbar}{\inputbitmap{\htbmdir{}/hbar.bitmap}}
+\newcommand{\hbox}[1]{{#1}}
+\newcommand{\hfill}{}
+\newcommand{\hfil}{}
+\newcommand{\huge}{}
+\newcommand{\Im}{\inputbitmap{\htbmdir{}/Im.bitmap}}
+\newcommand{\imath}{\inputbitmap{\htbmdir{}/imath.bitmap}}
+\newcommand{\infty}{\inputbitmap{\htbmdir{}/infty.bitmap}}
+\newcommand{\int}{\inputbitmap{\htbmdir{}/int.bitmap}}
+\newcommand{\iota}{\inputbitmap{\htbmdir{}/iota.bitmap}}
+\newcommand{\index}[1]{}
+\newcommand{\jmath}{\inputbitmap{\htbmdir{}/jmath.bitmap}}
+\newcommand{\kappa}{\inputbitmap{\htbmdir{}/kappa.bitmap}}
+\newcommand{\label}[1]{}
+\newcommand{\lambda}{\inputbitmap{\htbmdir{}/lambda.bitmap}}
+\newcommand{\Lambda}{\inputbitmap{\htbmdir{}/lambda-cap.bitmap}}
+\newcommand{\large}{}
+\newcommand{\ldots}{...}
+\newcommand{\le}{<=}
+\newcommand{\marginpar}[1]{}
+\newcommand{\mu}{\inputbitmap{\htbmdir{}/mu.bitmap}}
+\newcommand{\neg}{\inputbitmap{\htbmdir{}/neg.bitmap}}
+\newcommand{\newpage}{}
+\newcommand{\noindent}{\indent{0}}
+\newcommand{\nonfrenchspacing}{}
+\newcommand{\nabla}{\inputbitmap{\htbmdir{}/nabla.bitmap}}
+\newcommand{\nu}{\inputbitmap{\htbmdir{}/nu.bitmap}}
+\newcommand{\omega}{\inputbitmap{\htbmdir{}/omega.bitmap}}
+\newcommand{\Omega}{\inputbitmap{\htbmdir{}/omega-cap.bitmap}}
+\newcommand{\pageref}[1]{???}
+\newcommand{\parallel}{\inputbitmap{\htbmdir{}/parallel.bitmap}}
+\newcommand{\partial}{\inputbitmap{\htbmdir{}/partial.bitmap}}
+\newcommand{\phi}{\inputbitmap{\htbmdir{}/phi.bitmap}}
+\newcommand{\Phi}{\inputbitmap{\htbmdir{}/phi-cap.bitmap}}
+\newcommand{\pi}{\inputbitmap{\htbmdir{}/pi.bitmap}}
+\newcommand{\Pi}{\inputbitmap{\htbmdir{}/pi-cap.bitmap}}
+\newcommand{\prime}{\inputbitmap{\htbmdir{}/prime.bitmap}}
+\newcommand{\prod}{\inputbitmap{\htbmdir{}/prod.bitmap}}
+\newcommand{\protect}{}
+\newcommand{\psi}{\inputbitmap{\htbmdir{}/psi.bitmap}}
+\newcommand{\Psi}{\inputbitmap{\htbmdir{}/psi-cap.bitmap}}
+\newcommand{\quad}{\inputbitmap{\htbmdir{}/quad.bitmap}}
+\newcommand{\Re}{\inputbitmap{\htbmdir{}/Re.bitmap}}
+\newcommand{\rho}{\inputbitmap{\htbmdir{}/rho.bitmap}}
+\newcommand{\sc}{\rm}
+\newcommand{\sf}{\bf}
+\newcommand{\sigma}{\inputbitmap{\htbmdir{}/sigma.bitmap}}
+\newcommand{\Sigma}{\inputbitmap{\htbmdir{}/sigma-cap.bitmap}}
+\newcommand{\small}{}
+\newcommand{\sum}{\inputbitmap{\htbmdir{}/sum.bitmap}}
+\newcommand{\surd}{\inputbitmap{\htbmdir{}/surd.bitmap}}
+\newcommand{\tau}{\inputbitmap{\htbmdir{}/tau.bitmap}}
+\newcommand{\theta}{\inputbitmap{\htbmdir{}/theta.bitmap}}
+\newcommand{\Theta}{\inputbitmap{\htbmdir{}/theta-cap.bitmap}}
+\newcommand{\times}{\inputbitmap{\htbmdir{}/times.bitmap}}
+\newcommand{\top}{\inputbitmap{\htbmdir{}/top.bitmap}}
+\newcommand{\triangle}{\inputbitmap{\htbmdir{}/triangle.bitmap}}
+\newcommand{\upsilon}{\inputbitmap{\htbmdir{}/upsilon.bitmap}}
+\newcommand{\Upsilon}{\inputbitmap{\htbmdir{}/upsilon-cap.bitmap}}
+\newcommand{\vbox}[1]{{#1}}
+\newcommand{\wp}{\inputbitmap{\htbmdir{}/wp.bitmap}}
+\newcommand{\xi}{\inputbitmap{\htbmdir{}/xi.bitmap}}
+\newcommand{\Xi}{\inputbitmap{\htbmdir{}/xi-cap.bitmap}}
+\newcommand{\zeta}{\inputbitmap{\htbmdir{}/zeta.bitmap}}
+\newcommand{\bs}{\\}
+
+% ----------------------------------------------------------------------
+% 9. Book and .ht page macros.
+% ----------------------------------------------------------------------
+
+\newcommand{\beginImportant}{\horizontalline}
+\newcommand{\endImportant}{\horizontalline}
+%
+% following handles things like "i-th" but uses "-th"
+\newcommand{\eth}[1]{{#1}-th}}
+%
+\newcommand{\texnewline}{}
+\newcommand{\texbreak}{}
+\newcommand{\Gallery}{{AXIOM Images}}
+\newcommand{\exptypeindex}[1]{}
+\newcommand{\gotoevenpage}{}
+\newcommand{\ignore}[1]{}
+\newcommand{\ind}{\newline\tab{3}}
+\newcommand{\labelSpace}[1]{}
+\newcommand{\mathOrSpad}[1]{{\spad{#1}}}
+\newcommand{\menuspadref}[2]{\menudownlink{#1}{#2Page}}
+\newcommand{\menuxmpref}[1]{\menudownlink{`#1'}{#1XmpPage}}
+\newcommand{\noOutputXtc}[2]{\xtc{#1}{#2}}  % comment and then \spadcommand or spadsrc
+\newcommand{\not=}{\inputbitmap{\htbmdir{}/not=.bitmap}}
+\newcommand{\notequal}{\inputbitmap{\htbmdir{}/notequal.bitmap}}
+\newcommand{\nullXtc}[2]{\xtc{#1}{#2}}      % comment and then \spadcommand or spadsrc
+\newcommand{\nullspadcommand}[1]{\spadcommand}
+\newcommand{\pp}{\newline}              % Use this instead of \par for now.
+\newcommand{\psXtc}[3]{\xtc{#1}{#2}}        % comment and then \spadcommand or spadsrc
+\newcommand{\ref}[1]{(see the graph)}
+\newcommand{\showBlurb}[1]{Issue the system command \spadcmd{)show #1} to display the full list of operations defined by \spadtype{#1}.}
+\newcommand{\smath}[1]{\mathOrSpad{#1}}
+\newcommand{\spadFileExt}{.spad}
+\newcommand{\spadkey}[1]{}
+\newcommand{\spadref} [1]{{\it #1}}
+\newcommand{\spadsig}[2]{\spadtype{#1} {\tt ->} \spadtype{#2}}
+\newcommand{\axiomSig}[2]{\axiomType{#1} {\tt ->} \axiomType{#2}}
+\newcommand{\subscriptIt}[2]{{\it {#1}\_{#2}}}
+\newcommand{\subscriptText}[2]{{\it {#1}\_{\it #2}}}
+\newcommand{\subsubsection}[1]{\newline\indent{0}{\bf #1}\newline\newline}
+\newcommand{\syscmdindex}[1]{}              % system command index macro
+\newcommand{\threedim}{three-dimensional}
+\newcommand{\twodim}{two-dimensional}
+\newcommand{\unind}{\newline}
+\newcommand{\void}{the unique value of \spadtype{Void}}
+\newcommand{\xdefault}[1]{The default value is {\tt "#1"}.}
+\newcommand{\xmpLine}[2]{{\tt #1}\newline}   % have to improve someday
+\newcommand{\xmpref}[1]{\downlink{`#1'}{#1XmpPage}}
+\newcommand{\xtc}[2]{#1 #2}                 % comment and then \spadcommand or spadsrc
+
+% glossary terms
+\newcommand{\axiomGloss}[1]{\lispdownlink{#1}{(|htGloss| '|#1|)}}
+\newcommand{\axiomGlossSee}[2]{\lispdownlink{#1}{(|htGloss| '|#2|)}}
+\newcommand{\spadgloss}[1]{\axiomGloss{#1}}
+\newcommand{\spadglossSee}[2]{\axiomGlossSee{#1}{#2}}
+
+% use this for syntax punctuation: \axiomSyntax{::}
+\newcommand{\axiomSyntax}[1]{``{\tt #1}''}
+\newcommand{\spadSyntax}[1]{\axiomSyntax{#1}}
+
+% constructors
+\newcommand{\axiomType}[1]{\lispdownlink{#1}{(|spadType| '|#1|)}}
+\newcommand{\spadtype}[1]{\axiomType{#1}}
+\newcommand{\nonLibAxiomType}[1]{{\it #1}}       % things that browse can't handle
+\newcommand{\pspadtype}[1]{\nonLibAxiomType{#1}}
+
+\newcommand{\axiom}   [1]{{\tt #1}}              % note font
+\newcommand{\spad}    [1]{\axiom{#1}}
+\newcommand{\spadvar} [1]{\axiom{#1}}            % exists in ++ comments; to be removed
+\newcommand{\s}       [1]{\axiom{#1}}
+
+\newcommand{\httex}[2]{#1}
+\newcommand{\texht}[2]{#2}
+
+% Function names:
+%
+% The X versions below are used because AXIOM function names that end
+% in ``!'' cause problems for makeindex for had-copy.
+%
+% Example: \spadfunFromX{reverse}{List} prints as   reverse!
+%
+% In the "From" versions, the first arg is function name, second is constructor
+% where exported.
+%
+% Use the "op" flavors of "-", "+", "*" etc, otherwise the "fun" flavors
+
+\newcommand{\userfun} [1]{{\bf #1}}              % example, non-library function names
+
+\newcommand{\fakeAxiomFun}[1]{{\bf #1}}          % not really a library function
+\newcommand{\pspadfun} [1]{\fakeAxiomFun{#1}}
+
+\newcommand{\axiomFun} [1]{\lispdownlink{#1}{(|oPage| '|#1|)}}
+\newcommand{\spadfun} [1]{\axiomFun{#1}}
+\newcommand{\axiomFunX}[1]{\axiomFun{#1!}}
+\newcommand{\spadfunX}[1]{\axiomFun{#1!}}
+
+\newcommand{\axiomFunFrom}[2]{\lispdownlink{#1}{(|oPageFrom| '|#1| '|#2|)}}
+\newcommand{\spadfunFrom}[2]{\axiomFunFrom{#1}{#2}}
+\newcommand{\axiomFunFromX}[2]{\axiomFunFrom{#1!}{#2}}
+\newcommand{\spadfunFromX}[2]{\axiomFunFrom{#1!}{#2}}
+
+\newcommand{\axiomOp} [1]{\lispdownlink{#1}{(|oPage| '|#1|)}}
+\newcommand{\spadop}  [1]{\axiomOp{#1}}
+\newcommand{\axiomOpX}[1]{\axiomOp{#1!}}
+
+\newcommand{\axiomOpFrom}[2]{\lispdownlink{#1}{(|oPageFrom| '|#1| '|#2|)}}
+\newcommand{\spadopFrom} [2]{\axiomOpFrom{#1}{#2}}
+\newcommand{\axiomOpFromX}[2] {\axiomOpFrom{#1!}{#2}}
+\newcommand{\spadopFromX}[2] {\axiomOpFrom{#1!}{#2}}
+
+% redundant defns for system commands
+\newcommand{\syscom}[1]{\lispdownlink{)#1}{(|syscomPage| '|#1|)}}
+%
+% 20030110000 tpd added a leading open paren
+\newcommand{\spadsyscom}[1]{){\tt #1}}
+% 20030110000 tpd these two commands are never used
+%\newcommand{\spadcmd}[1]{\spadsyscom{#1}}
+%\newcommand{\spadsys}[1]{\spadsyscom{#1}}
+
+
+% Following macros should be phased out in favor of ones above:
+
+\newcommand{\gloss}[1]{\axiomGloss{#1}}
+\newcommand{\spadglos}[1]{\axiomGloss{#1}}
+\newcommand{\glossSee}[2]{\axiomGlossSee{#1}{#2}}
+
+% ----------------------------------------------------------------------
+% 10. Browse macros.
+% ----------------------------------------------------------------------
+
+\newcommand{\undocumented}[0]{is not documented yet}
+\newcommand{\aliascon}[2]{\lispdownlink{#1}{(|conPage| '|#2|)}}
+\newcommand{\aliasdom}[2]{\lispdownlink{#1}{(|conPage| '|#2|)}}
+\newcommand{\andexample}[1]{\indent{5}\spadcommand{#1}\indent{0}\newline}
+\newcommand{\blankline}{\vspace{1}\newline }
+\newcommand{\con}[1]{\lispdownlink{#1}{(|conPage| '|#1|)}}
+
+\newcommand{\conf}[2]{\lispdownlink{#1}{(|conPage| '{#2})}}
+% generalizes "con" to allow arbitrary title and form
+
+\newcommand{\ops}[3]{\lisplink{#1}{(|conOpPage| #2 '{#3})}}
+% does lisplink to operation page of a constructor or form
+% #1 is constructor name or form, without fences, e.g. "Matrix(Integer)"
+% #2 is page number, extracted from $curPage (see fromHeading/dbOpsForm)
+% #3 is constructor name or form, with fences, e.g. "(|Matrix| (|Integer|))"
+
+\newcommand{\dlink}[2]{\downlink{#2}{#1}}
+\newcommand{\dom}[1]{\lispdownlink{#1}{(|conPage| '|#1|)}}
+\newcommand{\example}[1]{\newline\indent{5}\spadcommand{#1}\indent{0}\newline}
+\newcommand{\lisp}[2]{\lispdownlink{#2}{#1}}
+\newcommand{\spadatt} [1]{{\it #1}}
+\newcommand{\indented}[2]{\indentrel{#1}\newline #2\indentrel{-#1}\newline}
+\newcommand{\keyword}[1]{\lispdownlink{#1}{(|htsn| '|#1|)}}
+\newcommand{\op}[1]{\lispdownlink{#1}{(|htsn| '|#1|)}}
+\newcommand{\spadignore}[1]{#1}
+
+% ----------------------------------------------------------------------
+% 11. Support for output and graph paste-ins.
+% ----------------------------------------------------------------------
+
+\newcommand{\axiomcommand}[1]{\spadcommand{#1}}
+\newcommand{\axiomgraph}[1]{\spadgraph{#1}}
+
+\newcommand{\pastecommand}[1]{\spadpaste{#1}}
+\newcommand{\pastegraph}[1]{\graphpaste{#1}}
+
+\newcommand{\showpaste}{\htbitmap{sdown3d}}
+\newcommand{\hidepaste}{\htbitmap{sup3d}}
+\newcommand{\spadpaste}[1]{
+  \newline
+  \begin{paste}{\pagename Empty \examplenumber}{\pagename Patch \examplenumber}
+  \pastebutton{\pagename Empty \examplenumber}{\showpaste}
+  \tab{5}\spadcommand{#1}
+  \end{paste}
+}
+
+\newcommand{\graphpaste}[1]{
+  \newline
+  \begin{paste}{\pagename Empty \examplenumber}{\pagename Patch \examplenumber}
+  \pastebutton{\pagename Empty \examplenumber}{\showpaste}
+  \tab{5}\spadgraph{#1}
+  \end{paste}
+}
+
+% ----------------------------------------------------------------------
+% 12. Hook for including a local menu item on the rootpage.
+% ----------------------------------------------------------------------
+
+\newcommand{\localinfo}{}
diff --git a/src/hyper/pages/xmpexp.ht b/src/hyper/pages/xmpexp.ht
new file mode 100644
index 00000000..401206a8
--- /dev/null
+++ b/src/hyper/pages/xmpexp.ht
@@ -0,0 +1,106 @@
+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by mkxmps.
+\newcommand{\ExamplesExposedTitle}{Some Examples of Domains and Packages}
+\newcommand{\ExamplesExposedNumber}{9.}
+% =====================================================================
+\begin{page}{ExamplesExposedPage}{9. Some Examples of Domains and Packages}
+% =====================================================================
+This is a menu of examples of some domains and packages.
+Click on any item below to see that section.
+\beginscroll
+\table{
+{ \downlink{AssociationList}{AssociationListXmpPage} }
+{ \downlink{BalancedBinaryTree}{BalancedBinaryTreeXmpPage} }
+{ \downlink{BasicOperator}{BasicOperatorXmpPage} }
+{ \downlink{BinaryExpansion}{BinaryExpansionXmpPage} }
+{ \downlink{BinarySearchTree}{BinarySearchTreeXmpPage} }
+{ \downlink{CardinalNumber}{CardinalNumberXmpPage} }
+{ \downlink{CartesianTensor}{CartesianTensorXmpPage} }
+{ \downlink{Character}{CharacterXmpPage} }
+{ \downlink{CharacterClass}{CharacterClassXmpPage} }
+{ \downlink{CliffordAlgebra}{CliffordAlgebraXmpPage} }
+{ \downlink{Complex}{ComplexXmpPage} }
+{ \downlink{ContinuedFraction}{ContinuedFractionXmpPage} }
+{ \downlink{CycleIndicators}{CycleIndicatorsXmpPage} }
+{ \downlink{DeRhamComplex}{DeRhamComplexXmpPage} }
+{ \downlink{DecimalExpansion}{DecimalExpansionXmpPage} }
+{ \downlink{DistributedMultivariatePolynomial}{DistributedMultivariatePolynomialXmpPage} }
+{ \downlink{DoubleFloat}{DoubleFloatXmpPage} }
+{ \downlink{EqTable}{EqTableXmpPage} }
+{ \downlink{Equation}{EquationXmpPage} }
+{ \downlink{Exit}{ExitXmpPage} }
+{ \downlink{Expression}{ExpressionXmpPage} }
+{ \downlink{Factored}{FactoredXmpPage} }
+{ \downlink{FactoredFunctions2}{FactoredFunctionsTwoXmpPage} }
+{ \downlink{File}{FileXmpPage} }
+{ \downlink{FileName}{FileNameXmpPage} }
+{ \downlink{FlexibleArray}{FlexibleArrayXmpPage} }
+{ \downlink{Float}{FloatXmpPage} }
+{ \downlink{Fraction}{FractionXmpPage} }
+{ \downlink{FullPartialFractionExpansion}{FullPartialFractionExpansionXmpPage} }
+{ \downlink{GeneralSparseTable}{GeneralSparseTableXmpPage} }
+{ \downlink{GroebnerFactorizationPackage}{GroebnerFactorizationPackageXmpPage} }
+{ \downlink{Heap}{HeapXmpPage} }
+{ \downlink{HexadecimalExpansion}{HexadecimalExpansionXmpPage} }
+{ \downlink{Integer}{IntegerXmpPage} }
+{ \downlink{IntegerLinearDependence}{IntegerLinearDependenceXmpPage} }
+{ \downlink{IntegerNumberTheoryFunctions}{IntegerNumberTheoryFunctionsXmpPage} }
+{ \downlink{Kernel}{KernelXmpPage} }
+{ \downlink{KeyedAccessFile}{KeyedAccessFileXmpPage} }
+{ \downlink{LexTriangularPackage}{LexTriangularPackageXmpPage} }
+{ \downlink{LazardSetSolvingPackage}{LazardSetSolvingPackageXmpPage} }
+{ \downlink{Library}{LibraryXmpPage} }
+{ \downlink{LieExponentials}{LieExponentialsXmpPage} }
+{ \downlink{LiePolynomial}{LiePolynomialXmpPage} }
+{ \downlink{LinearOrdinaryDifferentialOperator}{LinearOrdinaryDifferentialOperatorXmpPage} }
+{ \downlink{LinearOrdinaryDifferentialOperator1}{LinearOrdinaryDifferentialOperatorOneXmpPage} }
+{ \downlink{LinearOrdinaryDifferentialOperator2}{LinearOrdinaryDifferentialOperatorTwoXmpPage} }
+{ \downlink{List}{ListXmpPage} }
+{ \downlink{LyndonWord}{LyndonWordXmpPage} }
+{ \downlink{Magma}{MagmaXmpPage} }
+{ \downlink{MakeFunction}{MakeFunctionXmpPage} }
+{ \downlink{MappingPackage1}{MappingPackageOneXmpPage} }
+{ \downlink{Matrix}{MatrixXmpPage} }
+{ \downlink{MultiSet}{MultiSetXmpPage} }
+{ \downlink{MultivariatePolynomial}{MultivariatePolynomialXmpPage} }
+{ \downlink{None}{NoneXmpPage} }
+{ \downlink{Octonion}{OctonionXmpPage} }
+{ \downlink{OneDimensionalArray}{OneDimensionalArrayXmpPage} }
+{ \downlink{Operator}{OperatorXmpPage} }
+{ \downlink{OrderedVariableList}{OrderedVariableListXmpPage} }
+{ \downlink{OrderlyDifferentialPolynomial}{OrderlyDifferentialPolynomialXmpPage} }
+{ \downlink{PartialFraction}{PartialFractionXmpPage} }
+{ \downlink{Permanent}{PermanentXmpPage} }
+{ \downlink{Polynomial}{PolynomialXmpPage} }
+{ \downlink{Quaternion}{QuaternionXmpPage} }
+{ \downlink{RadixExpansion}{RadixExpansionXmpPage} }
+{ \downlink{RealClosure}{RealClosureXmpPage} }
+{ \downlink{RegularTriangularSet}{RegularTriangularSetXmpPage} }
+{ \downlink{RomanNumeral}{RomanNumeralXmpPage} }
+{ \downlink{Segment}{SegmentXmpPage} }
+{ \downlink{SegmentBinding}{SegmentBindingXmpPage} }
+{ \downlink{Set}{SetXmpPage} }
+{ \downlink{SingleInteger}{SingleIntegerXmpPage} }
+{ \downlink{SparseTable}{SparseTableXmpPage} }
+{ \downlink{SquareMatrix}{SquareMatrixXmpPage} }
+{ \downlink{SquareFreeRegularTriangularSet}{SquareFreeRegularTriangularSetXmpPage} }
+{ \downlink{Stream}{StreamXmpPage} }
+{ \downlink{String}{StringXmpPage} }
+{ \downlink{StringTable}{StringTableXmpPage} }
+{ \downlink{Symbol}{SymbolXmpPage} }
+{ \downlink{Table}{TableXmpPage} }
+{ \downlink{TextFile}{TextFileXmpPage} }
+{ \downlink{TwoDimensionalArray}{TwoDimensionalArrayXmpPage} }
+{ \downlink{UnivariatePolynomial}{UnivariatePolynomialXmpPage} }
+{ \downlink{UniversalSegment}{UniversalSegmentXmpPage} }
+{ \downlink{Vector}{VectorXmpPage} }
+{ \downlink{Void}{VoidXmpPage} }
+{ \downlink{WuWenTsunTriangularSet}{WuWenTsunTriangularSetXmpPage} }
+{ \downlink{XPBWPolynomial}{XPBWPolynomialXmpPage} }
+{ \downlink{XPolynomial}{XPolynomialXmpPage} }
+{ \downlink{XPolynomialRing}{XPolynomialRingXmpPage} }
+{ \downlink{ZeroDimensionalSolvePackage}{ZeroDimensionalSolvePackageXmpPage} }
+}
+\endscroll
+\autobuttons
+\end{page}