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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra d01transform.spad}
\author{Brian Dupee}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain D01TRNS d01TransformFunctionType}
<<domain D01TRNS d01TransformFunctionType>>=
)abbrev domain D01TRNS d01TransformFunctionType
++ Author: Brian Dupee
++ Date Created: April 1994
++ Date Last Updated: December 1997
++ Basic Operations: measure, numericalIntegration
++ Related Constructors: Result, RoutinesTable
++ Description:
++ Since an infinite integral cannot be evaluated numerically
++ it is necessary to transform the integral onto finite ranges.
++ \axiomType{d01TransformFunctionType} uses the mapping \spad{x -> 1/x}
++ and contains the functions \axiomFun{measure} and
++ \axiomFun{numericalIntegration}.
EDF ==> Expression DoubleFloat
EEDF ==> Equation Expression DoubleFloat
FI ==> Fraction Integer
EFI ==> Expression Fraction Integer
EEFI ==> Equation Expression Fraction Integer
EF2 ==> ExpressionFunctions2
DF ==> DoubleFloat
F ==> Float
SOCDF ==> Segment OrderedCompletion DoubleFloat
OCDF ==> OrderedCompletion DoubleFloat
NIA ==> Record(var:Symbol,fn:EDF,range:SOCDF,abserr:DF,relerr:DF)
INT ==> Integer
PI ==> PositiveInteger
HINT ==> Record(str:String,fn:EDF,range:SOCDF,ext:Result)
S ==> Symbol
ST ==> String
LST ==> List String
Measure ==> Record(measure:F,explanations:ST,extra:Result)
MS ==> Record(measure:F,name:ST,explanations:LST,extra:Result)
d01TransformFunctionType():NumericalIntegrationCategory == Result add
Rep:=Result
import d01AgentsPackage,Rep
rec2any(re:Record(str:ST,fn:EDF,range:SOCDF)):Any ==
coerce(re)$AnyFunctions1(Record(str:ST,fn:EDF,range:SOCDF))
changeName(ans:Result,name:ST):Result ==
sy:S := coerce(name "Answer")$S
anyAns:Any := coerce(ans)$AnyFunctions1(Result)
construct([[sy,anyAns]])$Result
getIntegral(args:NIA,hint:HINT) : Result ==
Args := copy args
Args.fn := hint.fn
Args.range := hint.range
integrate(Args::NumericalIntegrationProblem)$AnnaNumericalIntegrationPackage
transformFunction(args:NIA) : NIA ==
Args := copy args
Var := Args.var :: EFI -- coerce Symbol to EFI
NewVar:EFI := inv(Var)$EFI -- invert it
VarEqn:EEFI:=equation(Var,NewVar)$EEFI -- turn it into an equation
Afn:EFI := edf2efi(Args.fn)$ExpertSystemToolsPackage
Afn := subst(Afn,VarEqn)$EFI -- substitute into function
Var2:EFI := Var**2
Afn:= simplify(Afn/Var2)$TranscendentalManipulations(FI,EFI)
Args.fn:= map(convert(#1)$FI,Afn)$EF2(FI,DF)
Args
doit(seg:SOCDF,args:NIA):MS ==
Args := copy args
Args.range := seg
measure(Args::NumericalIntegrationProblem)$AnnaNumericalIntegrationPackage
transform(c:Boolean,args:NIA):Measure ==
if c then
l := coerce(recip(lo(args.range)))@OCDF
Seg:SOCDF := segment(0$OCDF,l)
else
h := coerce(recip(hi(args.range)))@OCDF
Seg:SOCDF := segment(h,0$OCDF)
Args := transformFunction(args)
m:MS := doit(Seg,Args)
out1:ST :=
"The recommendation is to transform the function and use " m.name
out2:List(HINT) := [[m.name,Args.fn,Seg,m.extra]]
out2Any:Any := coerce(out2)$AnyFunctions1(List(HINT))
ex:Record(key:S,entry:Any) := [d01transformextra@S,out2Any]
extr:Result := construct([ex])$Result
[m.measure,out1,extr]
split(c:PI,args:NIA):Measure ==
Args := copy args
Args.relerr := Args.relerr/2
Args.abserr := Args.abserr/2
if (c = 1)@Boolean then
seg1:SOCDF := segment(-1$OCDF,1$OCDF)
else if (c = 2)@Boolean then
seg1 := segment(lo(Args.range),1$OCDF)
else
seg1 := segment(-1$OCDF,hi(Args.range))
m1:MS := doit(seg1,Args)
Args := transformFunction Args
if (c = 2)@Boolean then
seg2:SOCDF := segment(0$OCDF,1$OCDF)
else if (c = 3)@Boolean then
seg2 := segment(-1$OCDF,0$OCDF)
else seg2 := seg1
m2:MS := doit(seg2,Args)
m1m:F := m1.measure
m2m:F := m2.measure
m:F := m1m*m2m/((m1m*m2m)+(1.0-m1m)*(1.0-m2m))
out1:ST := "The recommendation is to transform the function and use "
m1.name " and " m2.name
out2:List(HINT) :=
[[m1.name,args.fn,seg1,m1.extra],[m2.name,Args.fn,seg2,m2.extra]]
out2Any:Any := coerce(out2)$AnyFunctions1(List(HINT))
ex:Record(key:S,entry:Any) := [d01transformextra@S,out2Any]
extr:Result := construct([ex])$Result
[m,out1,extr]
measure(R:RoutinesTable,args:NIA) ==
Range:=rangeIsFinite(args)
Range case upperInfinite =>
positive?(lo(args.range))$OCDF =>
transform(true,args)
split(2,args)
Range case lowerInfinite =>
negative?(hi(args.range))$OCDF =>
transform(false,args)
split(3,args)
split(1,args)
numericalIntegration(args:NIA,hints:Result) ==
mainResult:DF := mainAbserr:DF := 0$DF
ans:Result := empty()$Result
hla:Any := coerce(search((d01transformextra@S),hints)$Result)@Any
hintList := retract(hla)$AnyFunctions1(List(HINT))
methodName:ST := empty()$ST
repeat
if (empty?(hintList)$(List(HINT)))
then leave
item := first(hintList)$List(HINT)
a:Result := getIntegral(args,item)
anyRes := coerce(search((result@S),a)$Result)@Any
midResult := retract(anyRes)$AnyFunctions1(DF)
anyErr := coerce(search((abserr pretend S),a)$Result)@Any
midAbserr := retract(anyErr)$AnyFunctions1(DF)
mainResult := mainResult+midResult
mainAbserr := mainAbserr+midAbserr
if (methodName = item.str)@Boolean then
methodName := concat([item.str,"1"])$ST
else
methodName := item.str
ans := concat(ans,changeName(a,methodName))$ExpertSystemToolsPackage
hintList := rest(hintList)$(List(HINT))
anyResult := coerce(mainResult)$AnyFunctions1(DF)
anyAbserr := coerce(mainAbserr)$AnyFunctions1(DF)
recResult:Record(key:S,entry:Any):=[result@S,anyResult]
recAbserr:Record(key:S,entry:Any):=[abserr pretend S,anyAbserr]
insert!(recAbserr,insert!(recResult,ans))$Result
@
\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 D01TRNS d01TransformFunctionType>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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