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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/algebra annacat.spad}
+\author{Brian Dupee}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{domain NIPROB NumericalIntegrationProblem}
+<<domain NIPROB NumericalIntegrationProblem>>=
+)abbrev domain NIPROB NumericalIntegrationProblem
+++ Author: Brian Dupee
+++ Date Created: December 1997
+++ Date Last Updated: December 1997
+++ Basic Operations: coerce, retract
+++ Related Constructors: Union
+++ Description:
+++ \axiomType{NumericalIntegrationProblem} is a \axiom{domain}
+++ for the representation of Numerical Integration problems for use
+++ by ANNA. 
+++
+++ The representation is a Union of two record types - one for integration of
+++ a function of one variable:
+++
+++ \axiomType{Record}(var:\axiomType{Symbol},
+++ fn:\axiomType{Expression DoubleFloat},
+++ range:\axiomType{Segment OrderedCompletion DoubleFloat},
+++ abserr:\axiomType{DoubleFloat},
+++ relerr:\axiomType{DoubleFloat},)
+++
+++ and one for multivariate integration:
+++
+++ \axiomType{Record}(fn:\axiomType{Expression DoubleFloat},
+++ range:\axiomType{List Segment OrderedCompletion DoubleFloat},
+++ abserr:\axiomType{DoubleFloat},
+++ relerr:\axiomType{DoubleFloat},).
+++
+
+EDFA   ==> Expression DoubleFloat
+SOCDFA ==> Segment OrderedCompletion DoubleFloat
+DFA    ==> DoubleFloat
+NIAA   ==> Record(var:Symbol,fn:EDFA,range:SOCDFA,abserr:DFA,relerr:DFA)
+MDNIAA ==> Record(fn:EDFA,range:List SOCDFA,abserr:DFA,relerr:DFA)
+ 
+NumericalIntegrationProblem():SetCategory with
+    coerce: NIAA -> %
+      ++ coerce(x) \undocumented{}
+    coerce: MDNIAA -> %
+      ++ coerce(x) \undocumented{}
+    coerce: Union(nia:NIAA,mdnia:MDNIAA) -> %
+      ++ coerce(x) \undocumented{}
+    coerce: % -> OutputForm
+      ++ coerce(x) \undocumented{}
+    retract: % -> Union(nia:NIAA,mdnia:MDNIAA)
+      ++ retract(x) \undocumented{}
+
+  ==
+ 
+    add
+      Rep := Union(nia:NIAA,mdnia:MDNIAA)
+ 
+      coerce(s:NIAA) == [s]
+      coerce(s:MDNIAA) == [s]
+      coerce(s:Union(nia:NIAA,mdnia:MDNIAA)) == s
+      coerce(x:%):OutputForm ==
+        (x) case nia => (x.nia)::OutputForm
+        (x.mdnia)::OutputForm
+      retract(x:%):Union(nia:NIAA,mdnia:MDNIAA) ==
+        (x) case nia => [x.nia]
+        [x.mdnia]
+
+@
+\section{domain ODEPROB NumericalODEProblem}
+<<domain ODEPROB NumericalODEProblem>>=
+)abbrev domain ODEPROB NumericalODEProblem
+++ Author: Brian Dupee
+++ Date Created: December 1997
+++ Date Last Updated: December 1997
+++ Basic Operations: coerce, retract
+++ Related Constructors: Union
+++ Description:
+++ \axiomType{NumericalODEProblem} is a \axiom{domain}
+++ for the representation of Numerical ODE problems for use
+++ by ANNA. 
+++
+++ The representation is of type: 
+++
+++ \axiomType{Record}(xinit:\axiomType{DoubleFloat},
+++ xend:\axiomType{DoubleFloat},
+++ fn:\axiomType{Vector Expression DoubleFloat},
+++ yinit:\axiomType{List DoubleFloat},intvals:\axiomType{List DoubleFloat},
+++ g:\axiomType{Expression DoubleFloat},abserr:\axiomType{DoubleFloat},
+++ relerr:\axiomType{DoubleFloat})
+++
+
+DFB   ==> DoubleFloat
+VEDFB ==> Vector Expression DoubleFloat
+LDFB  ==> List DoubleFloat
+EDFB  ==> Expression DoubleFloat
+ODEAB ==> Record(xinit:DFB,xend:DFB,fn:VEDFB,yinit:LDFB,intvals:LDFB,g:EDFB,abserr:DFB,relerr:DFB)
+NumericalODEProblem():SetCategory with
+
+    coerce: ODEAB -> %
+      ++ coerce(x) \undocumented{}
+    coerce: % -> OutputForm
+      ++ coerce(x) \undocumented{}
+    retract: % -> ODEAB
+      ++ retract(x) \undocumented{}
+
+  ==
+ 
+    add
+      Rep := ODEAB
+ 
+      coerce(s:ODEAB) == s
+      coerce(x:%):OutputForm ==
+        (retract(x))::OutputForm
+      retract(x:%):ODEAB == x :: Rep
+
+@
+\section{domain PDEPROB NumericalPDEProblem}
+<<domain PDEPROB NumericalPDEProblem>>=
+)abbrev domain PDEPROB NumericalPDEProblem
+++ Author: Brian Dupee
+++ Date Created: December 1997
+++ Date Last Updated: December 1997
+++ Basic Operations: coerce, retract
+++ Related Constructors: Union
+++ Description:
+++ \axiomType{NumericalPDEProblem} is a \axiom{domain}
+++ for the representation of Numerical PDE problems for use
+++ by ANNA. 
+++
+++ The representation is of type: 
+++
+++ \axiomType{Record}(pde:\axiomType{List Expression DoubleFloat}, 
+++ constraints:\axiomType{List PDEC}, 
+++ f:\axiomType{List List Expression DoubleFloat},
+++ st:\axiomType{String},
+++ tol:\axiomType{DoubleFloat})
+++
+++ where \axiomType{PDEC} is of type:
+++
+++ \axiomType{Record}(start:\axiomType{DoubleFloat}, 
+++ finish:\axiomType{DoubleFloat},
+++ grid:\axiomType{NonNegativeInteger},
+++ boundaryType:\axiomType{Integer},
+++ dStart:\axiomType{Matrix DoubleFloat}, 
+++ dFinish:\axiomType{Matrix DoubleFloat})
+++
+
+DFC   ==> DoubleFloat
+NNIC  ==> NonNegativeInteger
+INTC  ==> Integer
+MDFC  ==> Matrix DoubleFloat
+PDECC ==> Record(start:DFC, finish:DFC, grid:NNIC, boundaryType:INTC,
+                dStart:MDFC, dFinish:MDFC)
+LEDFC ==> List Expression DoubleFloat
+PDEBC ==> Record(pde:LEDFC, constraints:List PDECC, f:List LEDFC, 
+                    st:String, tol:DFC)
+NumericalPDEProblem():SetCategory with
+
+    coerce: PDEBC -> %
+      ++ coerce(x) \undocumented{}
+    coerce: % -> OutputForm
+      ++ coerce(x) \undocumented{}
+    retract: % -> PDEBC
+      ++ retract(x) \undocumented{}
+
+  ==
+ 
+    add
+      Rep := PDEBC
+ 
+      coerce(s:PDEBC) == s
+      coerce(x:%):OutputForm ==
+        (retract(x))::OutputForm
+      retract(x:%):PDEBC == x :: Rep
+
+@
+\section{domain OPTPROB NumericalOptimizationProblem}
+<<domain OPTPROB NumericalOptimizationProblem>>=
+)abbrev domain OPTPROB NumericalOptimizationProblem
+++ Author: Brian Dupee
+++ Date Created: December 1997
+++ Date Last Updated: December 1997
+++ Basic Operations: coerce, retract
+++ Related Constructors: Union
+++ Description:
+++ \axiomType{NumericalOptimizationProblem} is a \axiom{domain}
+++ for the representation of Numerical Optimization problems for use
+++ by ANNA. 
+++
+++ The representation is a Union of two record types - one for otimization of
+++ a single function of one or more variables:
+++
+++ \axiomType{Record}(
+++ fn:\axiomType{Expression DoubleFloat},
+++ init:\axiomType{List DoubleFloat},
+++ lb:\axiomType{List OrderedCompletion DoubleFloat},
+++ cf:\axiomType{List Expression DoubleFloat},
+++ ub:\axiomType{List OrderedCompletion DoubleFloat})
+++
+++ and one for least-squares problems i.e. optimization of a set of
+++ observations of a data set:
+++
+++ \axiomType{Record}(lfn:\axiomType{List Expression DoubleFloat},
+++ init:\axiomType{List DoubleFloat}).
+++
+
+LDFD     ==> List DoubleFloat
+LEDFD    ==> List Expression DoubleFloat
+LSAD     ==> Record(lfn:LEDFD, init:LDFD)
+UNOALSAD ==> Union(noa:NOAD,lsa:LSAD)
+EDFD     ==> Expression DoubleFloat
+LOCDFD   ==> List OrderedCompletion DoubleFloat
+NOAD     ==> Record(fn:EDFD, init:LDFD, lb:LOCDFD, cf:LEDFD, ub:LOCDFD)
+NumericalOptimizationProblem():SetCategory with
+
+    coerce: NOAD -> %
+      ++ coerce(x) \undocumented{}
+    coerce: LSAD -> %
+      ++ coerce(x) \undocumented{}
+    coerce: UNOALSAD -> %
+      ++ coerce(x) \undocumented{}
+    coerce: % -> OutputForm
+      ++ coerce(x) \undocumented{}
+    retract: % -> UNOALSAD
+      ++ retract(x) \undocumented{}
+
+  ==
+ 
+    add
+      Rep := UNOALSAD
+ 
+      coerce(s:NOAD) == [s]
+      coerce(s:LSAD) == [s]
+      coerce(x:UNOALSAD) == x
+      coerce(x:%):OutputForm ==
+        (x) case noa => (x.noa)::OutputForm
+        (x.lsa)::OutputForm
+      retract(x:%):UNOALSAD ==
+        (x) case noa => [x.noa]
+        [x.lsa]
+
+@
+\section{category NUMINT NumericalIntegrationCategory}
+<<category NUMINT NumericalIntegrationCategory>>=
+)abbrev category NUMINT NumericalIntegrationCategory
+++ Author: Brian Dupee
+++ Date Created: February 1994
+++ Date Last Updated: March 1996
+++ Description:
+++ \axiomType{NumericalIntegrationCategory} is the \axiom{category} for 
+++ describing the set of Numerical Integration \axiom{domains} with
+++ \axiomFun{measure} and \axiomFun{numericalIntegration}.
+
+EDFE   ==> Expression DoubleFloat
+SOCDFE ==> Segment OrderedCompletion DoubleFloat
+DFE    ==> DoubleFloat
+NIAE   ==> Record(var:Symbol,fn:EDFE,range:SOCDFE,abserr:DFE,relerr:DFE)
+MDNIAE ==> Record(fn:EDFE,range:List SOCDFE,abserr:DFE,relerr:DFE)
+NumericalIntegrationCategory(): Category == SetCategory with
+
+  measure:(RoutinesTable,NIAE)->Record(measure:Float,explanations:String,extra:Result)
+    ++ measure(R,args) calculates an estimate of the ability of a particular
+    ++ method to solve a problem.  
+    ++
+    ++ This method may be either a specific NAG routine or a strategy (such
+    ++ as transforming the function from one which is difficult to one which
+    ++ is easier to solve).
+    ++
+    ++ It will call whichever agents are needed to perform analysis on the
+    ++ problem in order to calculate the measure. There is a parameter,
+    ++ labelled \axiom{sofar}, which would contain the best compatibility
+    ++ found so far.
+
+  numericalIntegration: (NIAE, Result) -> Result
+    ++ numericalIntegration(args,hints) performs the integration of the
+    ++ function given the strategy or method returned by \axiomFun{measure}.
+
+  measure:(RoutinesTable,MDNIAE)->Record(measure:Float,explanations:String,extra:Result)
+    ++ measure(R,args) calculates an estimate of the ability of a particular
+    ++ method to solve a problem.  
+    ++
+    ++ This method may be either a specific NAG routine or a strategy (such
+    ++ as transforming the function from one which is difficult to one which
+    ++ is easier to solve).
+    ++
+    ++ It will call whichever agents are needed to perform analysis on the
+    ++ problem in order to calculate the measure. There is a parameter,
+    ++ labelled \axiom{sofar}, which would contain the best compatibility
+    ++ found so far.
+
+  numericalIntegration: (MDNIAE, Result) -> Result
+    ++ numericalIntegration(args,hints) performs the integration of the
+    ++ function given the strategy or method returned by \axiomFun{measure}.
+
+@
+\section{category ODECAT OrdinaryDifferentialEquationsSolverCategory}
+<<category ODECAT OrdinaryDifferentialEquationsSolverCategory>>=
+)abbrev category ODECAT OrdinaryDifferentialEquationsSolverCategory
+++ Author: Brian Dupee
+++ Date Created: February 1995
+++ Date Last Updated: June 1995
+++ Basic Operations: 
+++ Description:
+++ \axiomType{OrdinaryDifferentialEquationsSolverCategory} is the 
+++ \axiom{category} for describing the set of ODE solver \axiom{domains} 
+++ with \axiomFun{measure} and \axiomFun{ODEsolve}.
+
+DFF   ==> DoubleFloat
+VEDFF ==> Vector Expression DoubleFloat
+LDFF  ==> List DoubleFloat
+EDFF  ==> Expression DoubleFloat
+ODEAF ==> Record(xinit:DFF,xend:DFF,fn:VEDFF,yinit:LDFF,intvals:LDFF,g:EDFF,abserr:DFF,relerr:DFF)
+OrdinaryDifferentialEquationsSolverCategory(): Category == SetCategory with
+
+  measure:(RoutinesTable,ODEAF) -> Record(measure:Float,explanations:String)
+    ++ measure(R,args) calculates an estimate of the ability of a particular
+    ++ method to solve a problem.  
+    ++
+    ++ This method may be either a specific NAG routine or a strategy (such
+    ++ as transforming the function from one which is difficult to one which
+    ++ is easier to solve).
+    ++
+    ++ It will call whichever agents are needed to perform analysis on the
+    ++ problem in order to calculate the measure. There is a parameter,
+    ++ labelled \axiom{sofar}, which would contain the best compatibility
+    ++ found so far.
+
+  ODESolve: ODEAF -> Result
+    ++ ODESolve(args) performs the integration of the
+    ++ function given the strategy or method returned by \axiomFun{measure}.
+
+@
+\section{category PDECAT PartialDifferentialEquationsSolverCategory}
+<<category PDECAT PartialDifferentialEquationsSolverCategory>>=
+)abbrev category PDECAT PartialDifferentialEquationsSolverCategory
+++ Author: Brian Dupee
+++ Date Created: February 1995
+++ Date Last Updated: June 1995
+++ Basic Operations: 
+++ Description:
+++ \axiomType{PartialDifferentialEquationsSolverCategory} is the 
+++ \axiom{category} for describing the set of PDE solver \axiom{domains} 
+++ with \axiomFun{measure} and \axiomFun{PDEsolve}.
+
+-- PDEA	==> Record(xmin:F,xmax:F,ymin:F,ymax:F,ngx:NNI,ngy:NNI,_
+--             pde:List Expression Float, bounds:List List Expression Float,_
+--             st:String, tol:DF)
+
+--  measure:(RoutinesTable,PDEA) -> Record(measure:F,explanations:String)
+--    ++ measure(R,args) calculates an estimate of the ability of a particular
+--    ++ method to solve a problem.  
+--    ++
+--    ++ This method may be either a specific NAG routine or a strategy (such
+--    ++ as transforming the function from one which is difficult to one which
+--    ++ is easier to solve).
+--    ++
+--    ++ It will call whichever agents are needed to perform analysis on the
+--    ++ problem in order to calculate the measure. There is a parameter,
+--    ++ labelled \axiom{sofar}, which would contain the best compatibility
+--    ++ found so far.
+
+--  PDESolve: PDEA -> Result
+--    ++ PDESolve(args) performs the integration of the
+--    ++ function given the strategy or method returned by \axiomFun{measure}.
+
+DFG   ==> DoubleFloat
+NNIG  ==> NonNegativeInteger
+INTG  ==> Integer
+MDFG  ==> Matrix DoubleFloat
+PDECG ==> Record(start:DFG, finish:DFG, grid:NNIG, boundaryType:INTG,
+                  dStart:MDFG, dFinish:MDFG)
+LEDFG ==> List Expression DoubleFloat
+PDEBG ==> Record(pde:LEDFG, constraints:List PDECG, f:List LEDFG, 
+                    st:String, tol:DFG)
+PartialDifferentialEquationsSolverCategory(): Category == SetCategory with
+
+  measure:(RoutinesTable,PDEBG) -> Record(measure:Float,explanations:String)
+    ++ measure(R,args) calculates an estimate of the ability of a particular
+    ++ method to solve a problem.  
+    ++
+    ++ This method may be either a specific NAG routine or a strategy (such
+    ++ as transforming the function from one which is difficult to one which
+    ++ is easier to solve).
+    ++
+    ++ It will call whichever agents are needed to perform analysis on the
+    ++ problem in order to calculate the measure. There is a parameter,
+    ++ labelled \axiom{sofar}, which would contain the best compatibility
+    ++ found so far.
+
+  PDESolve: PDEBG -> Result
+    ++ PDESolve(args) performs the integration of the
+    ++ function given the strategy or method returned by \axiomFun{measure}.
+
+@
+\section{category OPTCAT NumericalOptimizationCategory}
+<<category OPTCAT NumericalOptimizationCategory>>=
+)abbrev category OPTCAT NumericalOptimizationCategory
+++ Author: Brian Dupee
+++ Date Created: January 1996
+++ Date Last Updated: March 1996
+++ Description:
+++ \axiomType{NumericalOptimizationCategory} is the \axiom{category} for 
+++ describing the set of Numerical Optimization \axiom{domains} with
+++ \axiomFun{measure} and \axiomFun{optimize}.
+
+LDFH   ==> List DoubleFloat
+LEDFH  ==> List Expression DoubleFloat
+LSAH   ==> Record(lfn:LEDFH, init:LDFH)
+EDFH   ==> Expression DoubleFloat
+LOCDFH ==> List OrderedCompletion DoubleFloat
+NOAH   ==> Record(fn:EDFH, init:LDFH, lb:LOCDFH, cf:LEDFH, ub:LOCDFH)
+NumericalOptimizationCategory(): Category == SetCategory with
+  measure:(RoutinesTable,NOAH)->Record(measure:Float,explanations:String)
+    ++ measure(R,args) calculates an estimate of the ability of a particular
+    ++ method to solve an optimization problem.  
+    ++
+    ++ This method may be either a specific NAG routine or a strategy (such
+    ++ as transforming the function from one which is difficult to one which
+    ++ is easier to solve).
+    ++
+    ++ It will call whichever agents are needed to perform analysis on the
+    ++ problem in order to calculate the measure. There is a parameter,
+    ++ labelled \axiom{sofar}, which would contain the best compatibility
+    ++ found so far.
+
+  measure:(RoutinesTable,LSAH)->Record(measure:Float,explanations:String)
+    ++ measure(R,args) calculates an estimate of the ability of a particular
+    ++ method to solve an optimization problem.  
+    ++
+    ++ This method may be either a specific NAG routine or a strategy (such
+    ++ as transforming the function from one which is difficult to one which
+    ++ is easier to solve).
+    ++
+    ++ It will call whichever agents are needed to perform analysis on the
+    ++ problem in order to calculate the measure. There is a parameter,
+    ++ labelled \axiom{sofar}, which would contain the best compatibility
+    ++ found so far.
+
+  numericalOptimization:LSAH -> Result
+    ++ numericalOptimization(args) performs the optimization of the
+    ++ function given the strategy or method returned by \axiomFun{measure}.
+
+  numericalOptimization:NOAH -> Result
+    ++ numericalOptimization(args) performs the optimization of the
+    ++ function given the strategy or method returned by \axiomFun{measure}.
+
+@
+\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>>
+
+<<domain NIPROB NumericalIntegrationProblem>>
+<<domain ODEPROB NumericalODEProblem>>
+<<domain PDEPROB NumericalPDEProblem>>
+<<domain OPTPROB NumericalOptimizationProblem>>
+<<category NUMINT NumericalIntegrationCategory>>
+<<category ODECAT OrdinaryDifferentialEquationsSolverCategory>>
+<<category PDECAT PartialDifferentialEquationsSolverCategory>>
+<<category OPTCAT NumericalOptimizationCategory>>
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}