Initial population.

1 files changed, 204 insertions, 0 deletions

diff --git a/src/algebra/ffrac.as.pamphlet b/src/algebra/ffrac.as.pamphlet
new file mode 100644
index 00000000..8eccc8a9
--- /dev/null
+++ b/src/algebra/ffrac.as.pamphlet
@@ -0,0 +1,204 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/algebra ffrac.as}
+\author{Michael Richardson}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\begin{verbatim}
+
+-- FormalFraction
+
+-- N.B. ndftip.as inlines this, must be recompiled if this is.
+
+-- To test:
+-- sed '1,/^#if NeverAssertThis/d;/#endif/d' < ffrac.as > ffrac.input    
+-- axiom
+-- )set nag host <some machine running nagd>
+-- )r ffrac.input
+
+\end{verbatim}
+\section{FormalFraction}
+<<FormalFraction>>=
+
+#include "axiom.as"
+
+FFRAC ==> FormalFraction ;
+ 
+OF    ==> OutputForm ;
+SC    ==> SetCategory ;
+FRAC  ==> Fraction ;
+ID    ==> IntegralDomain ;
+
++++ Author: M.G. Richardson
++++ Date Created: 1996 Jan. 23
++++ Date Last Updated:
++++ Basic Functions:
++++ Related Constructors: Fraction
++++ Also See:
++++ AMS Classifications:
++++ Keywords:
++++ References:
++++ Description:
++++ This type represents formal fractions - that is, pairs displayed as
++++ fractions with no simplification.
++++
++++ If the elements of the pair have a type X which is an integral
++++ domain, a FFRAC X can be coerced to a FRAC X, provided that this
++++ is a valid type.  A FRAC X can always be coerced to a FFRAC X.
++++ If the type of the elements is a Field, a FFRAC X can be coerced
++++ to X.
++++
++++ Formal fractions are used to return results from numerical methods
++++ which determine numerator and denominator separately, to enable
++++ users to inspect these components and recognise, for example,
++++ ratios of very small numbers as potentially indeterminate.
+
+FormalFraction(X : SC) : SC with {
+
+-- Could generalise further to allow numerator and denominator to be of
+-- different types, X and Y, both SCs.  "Left as an exercise."
+
+  / : (X,X) -> % ;
+++ / forms the formal quotient of two items.
+
+  numer : % -> X ;
+++ numer returns the numerator of a FormalFraction.
+
+  denom : % -> X ;
+++ denom returns the denominator of a FormalFraction.
+
+  if X has ID then {
+  
+    coerce : % -> FRAC(X pretend ID) ;
+++ coerce x converts a FormalFraction over an IntegralDomain to a
+++ Fraction over that IntegralDomain.
+
+    coerce : FRAC(X pretend ID) -> % ;
+++ coerce converts a Fraction to a FormalFraction.
+
+  }
+
+  if X has Field then coerce : % -> (X pretend Field) ;
+
+} == add {
+
+  import from Record(num : X, den : X) ;
+  
+  Rep == Record(num : X, den : X) ; -- representation
+
+  ((x : %) = (y : %)) : Boolean ==
+    ((rep(x).num = rep(y).num) and (rep(x).den = rep(y).den)) ;
+
+  ((n : X)/(d : X)) : % == per(record(n,d)) ;
+
+  coerce(r : %) : OF == (rep(r).num :: OF) / (rep(r).den :: OF) ;
+
+  numer(r : %) : X == rep(r).num ;
+
+  denom(r : %) : X == rep(r).den ;
+
+  if X has ID then {
+ 
+    coerce(r : %) : FRAC(X pretend ID)
+      == ((rep(r).num)/(rep(r).den)) @ (FRAC(X pretend ID)) ;
+
+    coerce(x : FRAC(X pretend ID)) : % == x pretend % ; 
+
+  }
+
+  if X has Field then coerce(r : %) : (X pretend Field)
+    == ((rep(r).num)/(rep(r).den)) $ (X pretend Field) ;
+
+}
+
+#if NeverAssertThis
+
+)lib ffrac
+
+f1 : FormalFraction Integer
+f1 := 6/3
+
+--       6
+--       -
+--       3
+
+f2 := (3.6/2.4)$FormalFraction Float
+
+--       3.6
+--       ---
+--       2.4
+
+numer f1
+
+--       6
+
+denom f2
+
+--       2.4
+
+f1 :: FRAC INT
+
+--       2
+
+% :: FormalFraction Integer      
+
+--       2
+--       -
+--       1
+
+f2 :: Float
+
+--       1.5
+
+output "End of tests"
+
+#endif
+@
+\section{License}
+<<license>>=
+--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
+--All rights reserved.
+--
+--Redistribution and use in source and binary forms, with or without
+--modification, are permitted provided that the following conditions are
+--met:
+--
+--    - Redistributions of source code must retain the above copyright
+--      notice, this list of conditions and the following disclaimer.
+--
+--    - Redistributions in binary form must reproduce the above copyright
+--      notice, this list of conditions and the following disclaimer in
+--      the documentation and/or other materials provided with the
+--      distribution.
+--
+--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
+--      names of its contributors may be used to endorse or promote products
+--      derived from this software without specific prior written permission.
+--
+--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
+--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
+--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
+--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
+--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
+--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
+--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
+--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+@
+<<*>>=
+<<license>>
+
+<<FormalFraction>>
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}