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diff --git a/src/algebra/infprod.spad.pamphlet b/src/algebra/infprod.spad.pamphlet new file mode 100644 index 00000000..ac34c2ea --- /dev/null +++ b/src/algebra/infprod.spad.pamphlet @@ -0,0 +1,346 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/algebra infprod.spad} +\author{Clifton J. Williamson} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\section{package STINPROD StreamInfiniteProduct} +<<package STINPROD StreamInfiniteProduct>>= +)abbrev package STINPROD StreamInfiniteProduct +++ Author: Clifton J. Williamson +++ Date Created: 23 February 1990 +++ Date Last Updated: 23 February 1990 +++ Basic Operations: infiniteProduct, evenInfiniteProduct, oddInfiniteProduct, +++ generalInfiniteProduct +++ Related Domains: UnivariateTaylorSeriesCategory +++ Also See: +++ AMS Classifications: +++ Keywords: Taylor series, infinite product +++ Examples: +++ References: +++ Description: +++ This package computes infinite products of Taylor series over an +++ integral domain of characteristic 0. Here Taylor series are +++ represented by streams of Taylor coefficients. +StreamInfiniteProduct(Coef): Exports == Implementation where + Coef: Join(IntegralDomain,CharacteristicZero) + I ==> Integer + QF ==> Fraction + ST ==> Stream + + Exports ==> with + + infiniteProduct: ST Coef -> ST Coef + ++ infiniteProduct(f(x)) computes \spad{product(n=1,2,3...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + evenInfiniteProduct: ST Coef -> ST Coef + ++ evenInfiniteProduct(f(x)) computes \spad{product(n=2,4,6...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + oddInfiniteProduct: ST Coef -> ST Coef + ++ oddInfiniteProduct(f(x)) computes \spad{product(n=1,3,5...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + generalInfiniteProduct: (ST Coef,I,I) -> ST Coef + ++ generalInfiniteProduct(f(x),a,d) computes + ++ \spad{product(n=a,a+d,a+2*d,...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + + Implementation ==> add + + if Coef has Field then + + import StreamTaylorSeriesOperations(Coef) + import StreamTranscendentalFunctions(Coef) + + infiniteProduct st == exp lambert log st + evenInfiniteProduct st == exp evenlambert log st + oddInfiniteProduct st == exp oddlambert log st + generalInfiniteProduct(st,a,d) == exp generalLambert(log st,a,d) + + else + + import StreamTaylorSeriesOperations(QF Coef) + import StreamTranscendentalFunctions(QF Coef) + + applyOverQF:(ST QF Coef -> ST QF Coef,ST Coef) -> ST Coef + applyOverQF(f,st) == + stQF := map(#1 :: QF(Coef),st)$StreamFunctions2(Coef,QF Coef) + map(retract(#1)@Coef,f stQF)$StreamFunctions2(QF Coef,Coef) + + infiniteProduct st == applyOverQF(exp lambert log #1,st) + evenInfiniteProduct st == applyOverQF(exp evenlambert log #1,st) + oddInfiniteProduct st == applyOverQF(exp oddlambert log #1,st) + generalInfiniteProduct(st,a,d) == + applyOverQF(exp generalLambert(log #1,a,d),st) + +@ +\section{package INFPROD0 InfiniteProductCharacteristicZero} +<<package INFPROD0 InfiniteProductCharacteristicZero>>= +)abbrev package INFPROD0 InfiniteProductCharacteristicZero +++ Author: Clifton J. Williamson +++ Date Created: 22 February 1990 +++ Date Last Updated: 23 February 1990 +++ Basic Operations: infiniteProduct, evenInfiniteProduct, oddInfiniteProduct, +++ generalInfiniteProduct +++ Related Domains: UnivariateTaylorSeriesCategory +++ Also See: +++ AMS Classifications: +++ Keywords: Taylor series, infinite product +++ Examples: +++ References: +++ Description: +++ This package computes infinite products of univariate Taylor series +++ over an integral domain of characteristic 0. +InfiniteProductCharacteristicZero(Coef,UTS):_ + Exports == Implementation where + Coef : Join(IntegralDomain,CharacteristicZero) + UTS : UnivariateTaylorSeriesCategory Coef + I ==> Integer + + Exports ==> with + + infiniteProduct: UTS -> UTS + ++ infiniteProduct(f(x)) computes \spad{product(n=1,2,3...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + evenInfiniteProduct: UTS -> UTS + ++ evenInfiniteProduct(f(x)) computes \spad{product(n=2,4,6...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + oddInfiniteProduct: UTS -> UTS + ++ oddInfiniteProduct(f(x)) computes \spad{product(n=1,3,5...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + generalInfiniteProduct: (UTS,I,I) -> UTS + ++ generalInfiniteProduct(f(x),a,d) computes + ++ \spad{product(n=a,a+d,a+2*d,...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + + Implementation ==> add + + import StreamInfiniteProduct Coef + + infiniteProduct x == series infiniteProduct coefficients x + evenInfiniteProduct x == series evenInfiniteProduct coefficients x + oddInfiniteProduct x == series oddInfiniteProduct coefficients x + + generalInfiniteProduct(x,a,d) == + series generalInfiniteProduct(coefficients x,a,d) + +@ +\section{package INPRODPF InfiniteProductPrimeField} +<<package INPRODPF InfiniteProductPrimeField>>= +)abbrev package INPRODPF InfiniteProductPrimeField +++ Author: Clifton J. Williamson +++ Date Created: 22 February 1990 +++ Date Last Updated: 23 February 1990 +++ Basic Operations: infiniteProduct, evenInfiniteProduct, oddInfiniteProduct, +++ generalInfiniteProduct +++ Related Domains: UnivariateTaylorSeriesCategory +++ Also See: +++ AMS Classifications: +++ Keywords: Taylor series, infinite product +++ Examples: +++ References: +++ Description: +++ This package computes infinite products of univariate Taylor series +++ over a field of prime order. +InfiniteProductPrimeField(Coef,UTS): Exports == Implementation where + Coef : Join(Field,Finite,ConvertibleTo Integer) + UTS : UnivariateTaylorSeriesCategory Coef + I ==> Integer + ST ==> Stream + + Exports ==> with + + infiniteProduct: UTS -> UTS + ++ infiniteProduct(f(x)) computes \spad{product(n=1,2,3...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + evenInfiniteProduct: UTS -> UTS + ++ evenInfiniteProduct(f(x)) computes \spad{product(n=2,4,6...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + oddInfiniteProduct: UTS -> UTS + ++ oddInfiniteProduct(f(x)) computes \spad{product(n=1,3,5...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + generalInfiniteProduct: (UTS,I,I) -> UTS + ++ generalInfiniteProduct(f(x),a,d) computes + ++ \spad{product(n=a,a+d,a+2*d,...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + + Implementation ==> add + + import StreamInfiniteProduct Integer + + applyOverZ:(ST I -> ST I,ST Coef) -> ST Coef + applyOverZ(f,st) == + stZ := map(convert(#1)@Integer,st)$StreamFunctions2(Coef,I) + map(#1 :: Coef,f stZ)$StreamFunctions2(I,Coef) + + infiniteProduct x == + series applyOverZ(infiniteProduct,coefficients x) + evenInfiniteProduct x == + series applyOverZ(evenInfiniteProduct,coefficients x) + oddInfiniteProduct x == + series applyOverZ(oddInfiniteProduct,coefficients x) + generalInfiniteProduct(x,a,d) == + series applyOverZ(generalInfiniteProduct(#1,a,d),coefficients x) + +@ +\section{package INPRODFF InfiniteProductFiniteField} +<<package INPRODFF InfiniteProductFiniteField>>= +)abbrev package INPRODFF InfiniteProductFiniteField +++ Author: Clifton J. Williamson +++ Date Created: 22 February 1990 +++ Date Last Updated: 23 February 1990 +++ Basic Operations: infiniteProduct, evenInfiniteProduct, oddInfiniteProduct, +++ generalInfiniteProduct +++ Related Domains: UnivariateTaylorSeriesCategory +++ Also See: +++ AMS Classifications: +++ Keywords: Taylor series, infinite product +++ Examples: +++ References: +++ Description: +++ This package computes infinite products of univariate Taylor series +++ over an arbitrary finite field. +InfiniteProductFiniteField(K,UP,Coef,UTS):_ + Exports == Implementation where + K : Join(Field,Finite,ConvertibleTo Integer) + UP : UnivariatePolynomialCategory K + Coef : MonogenicAlgebra(K,UP) + UTS : UnivariateTaylorSeriesCategory Coef + I ==> Integer + RN ==> Fraction Integer + SAE ==> SimpleAlgebraicExtension + ST ==> Stream + STF ==> StreamTranscendentalFunctions + STT ==> StreamTaylorSeriesOperations + ST2 ==> StreamFunctions2 + SUP ==> SparseUnivariatePolynomial + + Exports ==> with + + infiniteProduct: UTS -> UTS + ++ infiniteProduct(f(x)) computes \spad{product(n=1,2,3...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + evenInfiniteProduct: UTS -> UTS + ++ evenInfiniteProduct(f(x)) computes \spad{product(n=2,4,6...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + oddInfiniteProduct: UTS -> UTS + ++ oddInfiniteProduct(f(x)) computes \spad{product(n=1,3,5...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + generalInfiniteProduct: (UTS,I,I) -> UTS + ++ generalInfiniteProduct(f(x),a,d) computes + ++ \spad{product(n=a,a+d,a+2*d,...,f(x**n))}. + ++ The series \spad{f(x)} should have constant coefficient 1. + + Implementation ==> add + + liftPoly: UP -> SUP RN + liftPoly poly == + -- lift coefficients of 'poly' to integers + ans : SUP RN := 0 + while not zero? poly repeat + coef := convert(leadingCoefficient poly)@I :: RN + ans := ans + monomial(coef,degree poly) + poly := reductum poly + ans + + reducePoly: SUP RN -> UP + reducePoly poly == + -- reduce coefficients of 'poly' to elements of K + ans : UP := 0 + while not zero? poly repeat + coef := numer(leadingCoefficient(poly)) :: K + ans := ans + monomial(coef,degree poly) + poly := reductum poly + ans + + POLY := liftPoly definingPolynomial()$Coef + ALG := SAE(RN,SUP RN,POLY) + + infiniteProduct x == + stUP := map(lift,coefficients x)$ST2(Coef,UP) + stSUP := map(liftPoly,stUP)$ST2(UP,SUP RN) + stALG := map(reduce,stSUP)$ST2(SUP RN,ALG) + stALG := exp(lambert(log(stALG)$STF(ALG))$STT(ALG))$STF(ALG) + stSUP := map(lift,stALG)$ST2(ALG,SUP RN) + stUP := map(reducePoly,stSUP)$ST2(SUP RN,UP) + series map(reduce,stUP)$ST2(UP,Coef) + + evenInfiniteProduct x == + stUP := map(lift,coefficients x)$ST2(Coef,UP) + stSUP := map(liftPoly,stUP)$ST2(UP,SUP RN) + stALG := map(reduce,stSUP)$ST2(SUP RN,ALG) + stALG := exp(evenlambert(log(stALG)$STF(ALG))$STT(ALG))$STF(ALG) + stSUP := map(lift,stALG)$ST2(ALG,SUP RN) + stUP := map(reducePoly,stSUP)$ST2(SUP RN,UP) + series map(reduce,stUP)$ST2(UP,Coef) + + oddInfiniteProduct x == + stUP := map(lift,coefficients x)$ST2(Coef,UP) + stSUP := map(liftPoly,stUP)$ST2(UP,SUP RN) + stALG := map(reduce,stSUP)$ST2(SUP RN,ALG) + stALG := exp(oddlambert(log(stALG)$STF(ALG))$STT(ALG))$STF(ALG) + stSUP := map(lift,stALG)$ST2(ALG,SUP RN) + stUP := map(reducePoly,stSUP)$ST2(SUP RN,UP) + series map(reduce,stUP)$ST2(UP,Coef) + + generalInfiniteProduct(x,a,d) == + stUP := map(lift,coefficients x)$ST2(Coef,UP) + stSUP := map(liftPoly,stUP)$ST2(UP,SUP RN) + stALG := map(reduce,stSUP)$ST2(SUP RN,ALG) + stALG := generalLambert(log(stALG)$STF(ALG),a,d)$STT(ALG) + stALG := exp(stALG)$STF(ALG) + stSUP := map(lift,stALG)$ST2(ALG,SUP RN) + stUP := map(reducePoly,stSUP)$ST2(SUP RN,UP) + series map(reduce,stUP)$ST2(UP,Coef) + +@ +\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>> + +<<package STINPROD StreamInfiniteProduct>> +<<package INFPROD0 InfiniteProductCharacteristicZero>> +<<package INPRODPF InfiniteProductPrimeField>> +<<package INPRODFF InfiniteProductFiniteField>> +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} |