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diff --git a/src/algebra/d01weights.spad.pamphlet b/src/algebra/d01weights.spad.pamphlet deleted file mode 100644 index dc0f1b2b..00000000 --- a/src/algebra/d01weights.spad.pamphlet +++ /dev/null @@ -1,309 +0,0 @@ -\documentclass{article} -\usepackage{open-axiom} -\begin{document} -\title{\$SPAD/src/algebra d01weights.spad} -\author{Brian Dupee} -\maketitle -\begin{abstract} -\end{abstract} -\eject -\tableofcontents -\eject -\section{package D01WGTS d01WeightsPackage} -<<package D01WGTS d01WeightsPackage>>= -)abbrev package D01WGTS d01WeightsPackage -++ Author: Brian Dupee -++ Date Created: July 1994 -++ Date Last Updated: January 1998 (Bug fix - exprHasListOfWeightsCosWXorSinWX) -++ Basic Operations: exprHasWeightCosWXorSinWX, exprHasAlgebraicWeight, -++ exprHasLogarithmicWeights -++ Description: -++ \axiom{d01WeightsPackage} is a package for functions used to investigate -++ whether a function can be divided into a simpler function and a weight -++ function. The types of weights investigated are those giving rise to -++ end-point singularities of the algebraico-logarithmic type, and -++ trigonometric weights. -d01WeightsPackage(): E == I where - LEDF ==> List Expression DoubleFloat - KEDF ==> Kernel Expression DoubleFloat - LKEDF ==> List Kernel Expression DoubleFloat - EDF ==> Expression DoubleFloat - PDF ==> Polynomial DoubleFloat - FI ==> Fraction Integer - LDF ==> List DoubleFloat - DF ==> DoubleFloat - SOCDF ==> Segment OrderedCompletion DoubleFloat - OCDF ==> OrderedCompletion DoubleFloat - NIA ==> Record(var:Symbol,fn:EDF,range:SOCDF,abserr:DF,relerr:DF) - INT ==> Integer - BOP ==> BasicOperator - URBODF ==> Union(Record(op:BasicOperator,w:DF),"failed") - LURBODF ==> List(Union(Record(op:BasicOperator,w:DF), "failed")) - - E ==> with - exprHasWeightCosWXorSinWX:NIA -> URBODF - ++ \axiom{exprHasWeightCosWXorSinWX} looks for trigonometric - ++ weights in an expression of the form \axiom{cos \omega x} or - ++ \axiom{sin \omega x}, returning the value of \omega - ++ (\notequal 1) and the operator. - exprHasAlgebraicWeight:NIA -> Union(LDF,"failed") - ++ \axiom{exprHasAlgebraicWeight} looks for algebraic weights - ++ giving rise to singularities of the function at the end-points. - exprHasLogarithmicWeights:NIA -> INT - ++ \axiom{exprHasLogarithmicWeights} looks for logarithmic weights - ++ giving rise to singularities of the function at the end-points. - - - - I ==> add - score:(EDF,EDF) -> FI - kernelIsLog:KEDF -> Boolean - functionIsPolynomial?:EDF -> Boolean - functionIsNthRoot?:(EDF,EDF) -> Boolean - functionIsQuotient:EDF -> Union(EDF,"failed") - findCommonFactor:LEDF -> Union(LEDF,"failed") - findAlgebraicWeight:(NIA,EDF) -> Union(DF,"failed") - exprHasListOfWeightsCosWXorSinWX:(EDF,Symbol) -> LURBODF - exprOfFormCosWXorSinWX:(EDF,Symbol) -> URBODF - bestWeight:LURBODF -> URBODF - weightIn?:(URBODF,LURBODF) -> Boolean - inRest?:(EDF,LEDF)->Boolean - factorIn?:(EDF,LEDF)->Boolean - voo?:(EDF,EDF)->Boolean - - kernelIsLog(k:KEDF):Boolean == - (name operator k = (log :: Symbol))@Boolean - - factorIn?(a:EDF,l:LEDF):Boolean == - for i in 1..# l repeat - (a = l.i)@Boolean => return true - false - - voo?(b:EDF,a:EDF):Boolean == - (voo:=isTimes(b)) case LEDF and factorIn?(a,voo) - - inRest?(a:EDF,l:LEDF):Boolean == - every?( voo?(#1,a) ,l) - - findCommonFactor(l:LEDF):Union(LEDF,"failed") == - empty?(l)$LEDF => "failed" - f := first(l)$LEDF - r := rest(l)$LEDF - (t := isTimes(f)$EDF) case LEDF => - pos:=select(inRest?(#1,r),t) - empty?(pos) => "failed" - pos - "failed" - - exprIsLogarithmicWeight(f:EDF,Var:EDF,a:EDF,b:EDF):INT == - ans := 0$INT - k := tower(f)$EDF - lf := select(kernelIsLog,k)$LKEDF - empty?(lf)$LKEDF => ans - for i in 1..# lf repeat - arg := argument lf.i - if (arg.1 = (Var - a)) then - ans := ans + 1 - else if (arg.1 = (b - Var)) then - ans := ans + 2 - ans - - exprHasLogarithmicWeights(args:NIA):INT == - ans := 1$INT - a := getlo(args.range)$d01AgentsPackage :: EDF - b := gethi(args.range)$d01AgentsPackage :: EDF - Var := args.var :: EDF - (l := isPlus numerator args.fn) case LEDF => - (cf := findCommonFactor l) case LEDF => - for j in 1..# cf repeat - ans := ans + exprIsLogarithmicWeight(cf.j,Var,a,b) - ans - ans - ans := ans + exprIsLogarithmicWeight(args.fn,Var,a,b) - - functionIsQuotient(expr:EDF):Union(EDF,"failed") == - (k := mainKernel expr) case KEDF => - expr = inv(f := k :: KEDF :: EDF)$EDF => f - one?(numerator expr) => denominator expr - "failed" - "failed" - - functionIsPolynomial?(f:EDF):Boolean == - (retractIfCan(f)@Union(PDF,"failed"))$EDF case PDF - - functionIsNthRoot?(f:EDF,e:EDF):Boolean == - (m := mainKernel f) case "failed" => false - (one?(# (kernels f))) - and (name operator m = (nthRoot :: Symbol))@Boolean - and (((argument m).1 = e)@Boolean) - - score(f:EDF,e:EDF):FI == - ans := 0$FI - (t := isTimes f) case LEDF => - for i in 1..# t repeat - ans := ans + score(t.i,e) - ans - (q := functionIsQuotient f) case EDF => - ans := ans - score(q,e) - functionIsPolynomial? f => - g:EDF := f/e - if functionIsPolynomial? g then - ans := 1+score(g,e) - else - ans - (l := isPlus f) case LEDF => - (cf := findCommonFactor l) case LEDF => - factor := 1$EDF - for i in 1..# cf repeat - factor := factor*cf.i - ans := ans + score(f/factor,e) + score(factor,e) - ans - functionIsNthRoot?(f,e) => - (p := isPower f) case "failed" => ans - exp := p.exponent - m := mainKernel f - m case KEDF => - arg := argument m - a:INT := (retract(arg.2)@INT)$EDF - exp / a - ans - ans - - findAlgebraicWeight(args:NIA,e:EDF):Union(DF,"failed") == - zero?(s := score(args.fn,e)) => "failed" - s :: DF - - exprHasAlgebraicWeight(args:NIA):Union(LDF,"failed") == - (f := functionIsContinuousAtEndPoints(args)$d01AgentsPackage) - case continuous =>"failed" - Var := args.var :: EDF - a := getlo(args.range)$d01AgentsPackage :: EDF - b := gethi(args.range)$d01AgentsPackage :: EDF - A := Var - a - B := b - Var - f case lowerSingular => - (h := findAlgebraicWeight(args,A)) case "failed" => "failed" - [h,0] - f case upperSingular => - (g := findAlgebraicWeight(args,B)) case "failed" => "failed" - [0,g] - h := findAlgebraicWeight(args,A) - g := findAlgebraicWeight(args,B) - r := (h case "failed") - s := (g case "failed") - (r) and (s) => "failed" - r => [0,coerce(g)@DF] - s => [coerce(h)@DF,0] - [coerce(h)@DF,coerce(g)@DF] - - exprOfFormCosWXorSinWX(f:EDF,var:Symbol): URBODF == - l:LKEDF := kernels(f)$EDF - one?((# l)$LKEDF)$INT => - a:LEDF := argument(e:KEDF := first(l)$LKEDF)$KEDF - empty?(a) => "failed" - m:Union(LEDF,"failed") := isTimes(first(a)$LEDF)$EDF - m case LEDF => -- if it is a list, it will have at least two elements - is?(second(m)$LEDF,var)$EDF => - omega:DF := retract(first(m)$LEDF)@DF - o:BOP := operator(n:Symbol:=name operator(e)$KEDF)$BOP - (n = cos@Symbol)@Boolean => [o,omega] - (n = sin@Symbol)@Boolean => [o,omega] - "failed" - "failed" - "failed" - "failed" - - exprHasListOfWeightsCosWXorSinWX(f:EDF,var:Symbol): LURBODF == - (e := isTimes(f)$EDF) case LEDF => - [exprOfFormCosWXorSinWX(u,var) for u in e] - empty?(k := kernels f) => ["failed"] - ((first(k)::EDF) = f) => - [exprOfFormCosWXorSinWX(f,var)] - ["failed"] - - bestWeight(l:LURBODF): URBODF == - empty?(l)$LURBODF => "failed" - best := first(l)$LURBODF -- best is first in list - empty?(rest(l)$LURBODF) => best - for i in 2..# l repeat -- unless next is better - r:URBODF := l.i - if r case "failed" then leave - else if best case "failed" then - best := r - else if r.w > best.w then - best := r - best - - weightIn?(weight:URBODF,listOfWeights:LURBODF):Boolean == - n := # listOfWeights - for i in 1..n repeat -- cycle through list - (weight = listOfWeights.i)@Boolean => return true -- return when found - false - - exprHasWeightCosWXorSinWX(args:NIA):URBODF == - ans := empty()$LURBODF - f:EDF := numerator(args.fn)$EDF - (t:Union(LEDF,"failed") := isPlus(f)) case "failed" => - bestWeight(exprHasListOfWeightsCosWXorSinWX(f,args.var)) - if t case LEDF then - e1 := first(t)$LEDF - le1:LURBODF := exprHasListOfWeightsCosWXorSinWX(e1,args.var) - le1 := [u for u in le1 | (not (u case "failed"))] - empty?(le1)$LURBODF => "failed" - test := true - for i in 1..# le1 repeat - le1i:URBODF := le1.i - for j in 2..# t repeat - if test then - tj:LURBODF := exprHasListOfWeightsCosWXorSinWX(t.j,args.var) - test := weightIn?(le1i,tj) - if test then - ans := concat([le1i],ans) - bestWeight ans - else "failed" - -@ -\section{License} -<<license>>= ---Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. ---All rights reserved. ---Copyright (C) 2007-2009, Gabriel Dos Reis. --- ---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>> - -<<package D01WGTS d01WeightsPackage>> -@ -\eject -\begin{thebibliography}{99} -\bibitem{1} nothing -\end{thebibliography} -\end{document} |