\documentclass{article} \usepackage{axiom} \begin{document} \title{\$SPAD/src/algebra mts.spad} \author{William Burge, Stephen Watt, Clifton Williamson} \maketitle \begin{abstract} \end{abstract} \eject \tableofcontents \eject \section{domain SMTS SparseMultivariateTaylorSeries} <>= import NonNegativeInteger import List import Stream )abbrev domain SMTS SparseMultivariateTaylorSeries ++ This domain provides multivariate Taylor series ++ Authors: William Burge, Stephen Watt, Clifton Williamson ++ Date Created: 15 August 1988 ++ Date Last Updated: 18 May 1991 ++ Basic Operations: ++ Related Domains: ++ Also See: UnivariateTaylorSeries ++ AMS Classifications: ++ Keywords: multivariate, Taylor, series ++ Examples: ++ References: ++ Description: ++ This domain provides multivariate Taylor series with variables ++ from an arbitrary ordered set. A Taylor series is represented ++ by a stream of polynomials from the polynomial domain SMP. ++ The nth element of the stream is a form of degree n. SMTS is an ++ internal domain. SparseMultivariateTaylorSeries(Coef,Var,SMP):_ Exports == Implementation where Coef : Ring Var : OrderedSet SMP : PolynomialCategory(Coef,IndexedExponents Var,Var) I ==> Integer L ==> List NNI ==> NonNegativeInteger OUT ==> OutputForm PS ==> InnerTaylorSeries SMP RN ==> Fraction Integer ST ==> Stream StS ==> Stream SMP STT ==> StreamTaylorSeriesOperations SMP STF ==> StreamTranscendentalFunctions SMP ST2 ==> StreamFunctions2 ST3 ==> StreamFunctions3 Exports ==> MultivariateTaylorSeriesCategory(Coef,Var) with coefficient: (%,NNI) -> SMP ++ \spad{coefficient(s, n)} gives the terms of total degree n. coerce: Var -> % ++ \spad{coerce(var)} converts a variable to a Taylor series coerce: SMP -> % ++ \spad{coerce(poly)} regroups the terms by total degree and forms ++ a series. *:(SMP,%)->% ++\spad{smp*ts} multiplies a TaylorSeries by a monomial SMP. csubst:(L Var,L StS) -> (SMP -> StS) ++\spad{csubst(a,b)} is for internal use only if Coef has Algebra Fraction Integer then integrate: (%,Var,Coef) -> % ++\spad{integrate(s,v,c)} is the integral of s with respect ++ to v and having c as the constant of integration. fintegrate: (() -> %,Var,Coef) -> % ++\spad{fintegrate(f,v,c)} is the integral of \spad{f()} with respect ++ to v and having c as the constant of integration. ++ The evaluation of \spad{f()} is delayed. Implementation ==> PS add Rep := StS -- Below we use the fact that Rep of PS is Stream SMP. extend(x,n) == extend(x,n + 1)$Rep complete x == complete(x)$Rep evalstream:(%,L Var,L SMP) -> StS evalstream(s:%,lv:(L Var),lsmp:(L SMP)):(ST SMP) == scan(0,_+$SMP,map(eval(#1,lv,lsmp),s pretend StS))$ST2(SMP,SMP) addvariable:(Var,InnerTaylorSeries Coef) -> % addvariable(v,s) == ints := integers(0)$STT pretend ST NNI map(monomial(#2 :: SMP,v,#1)$SMP,ints,s pretend ST Coef)$ST3(NNI,Coef,SMP) coefficient(s:%,n: NNI) == elt(s,n + 1)$Rep -- 1-based indexing for streams --% creation of series coerce(r:Coef) == monom(r::SMP,0)$STT smp:SMP * p:% == (((smp) * (p pretend Rep))$STT)pretend % r:Coef * p:% == (((r::SMP) * (p pretend Rep))$STT)pretend % p:% * r:Coef == (((r::SMP) * ( p pretend Rep))$STT)pretend % mts(p:SMP):% == (uv := mainVariable p) case "failed" => monom(p,0)$STT v := uv :: Var s : % := 0 up := univariate(p,v) while not zero? up repeat s := s + monomial(1,v,degree up) * mts(leadingCoefficient up) up := reductum up s coerce(p:SMP) == mts p coerce(v:Var) == v :: SMP :: % monomial(r:%,v:Var,n:NNI) == r * monom(monomial(1,v,n)$SMP,n)$STT --% evaluation substvar: (SMP,L Var,L %) -> % substvar(p,vl,q) == null vl => monom(p,0)$STT (uv := mainVariable p) case "failed" => monom(p,0)$STT v := uv :: Var v = first vl => s : % := 0 up := univariate(p,v) while not zero? up repeat c := leadingCoefficient up s := s + first q ** degree up * substvar(c,rest vl,rest q) up := reductum up s substvar(p,rest vl,rest q) sortmfirst:(SMP,L Var,L %) -> % sortmfirst(p,vl,q) == nlv : L Var := sort(#1 > #2,vl) nq : L % := [q position$(L Var) (i,vl) for i in nlv] substvar(p,nlv,nq) csubst(vl,q) == sortmfirst(#1,vl,q pretend L(%)) pretend StS restCheck(s:StS):StS == -- checks that stream is null or first element is 0 -- returns empty() or rest of stream empty? s => s not zero? frst s => error "eval: constant coefficient should be 0" rst s eval(s:%,v:L Var,q:L %) == #v ~= #q => error "eval: number of variables should equal number of values" nq : L StS := [restCheck(i pretend StS) for i in q] addiag(map(csubst(v,nq),s pretend StS)$ST2(SMP,StS))$STT pretend % substmts(v:Var,p:SMP,q:%):% == up := univariate(p,v) ss : % := 0 while not zero? up repeat d:=degree up c:SMP:=leadingCoefficient up ss := ss + c* q ** d up := reductum up ss subststream(v:Var,p:SMP,q:StS):StS== substmts(v,p,q pretend %) pretend StS comp1:(Var,StS,StS) -> StS comp1(v,r,t)== addiag(map(subststream(v,#1,t),r)$ST2(SMP,StS))$STT comp(v:Var,s:StS,t:StS):StS == delay empty? s => s f := frst s; r : StS := rst s; empty? r => s empty? t => concat(f,comp1(v,r,empty()$StS)) not zero? frst t => error "eval: constant coefficient should be zero" concat(f,comp1(v,r,rst t)) eval(s:%,v:Var,t:%) == comp(v,s pretend StS,t pretend StS) --% differentiation and integration differentiate(s:%,v:Var):% == empty? s => 0 map(differentiate(#1,v),rst s) if Coef has Algebra Fraction Integer then stream(x:%):Rep == x pretend Rep (x:%) ** (r:RN) == powern(r,stream x)$STT (r:RN) * (x:%) == map(r * #1, stream x)$ST2(SMP,SMP) pretend % (x:%) * (r:RN) == map(#1 * r,stream x )$ST2(SMP,SMP) pretend % exp x == exp(stream x)$STF log x == log(stream x)$STF sin x == sin(stream x)$STF cos x == cos(stream x)$STF tan x == tan(stream x)$STF cot x == cot(stream x)$STF sec x == sec(stream x)$STF csc x == csc(stream x)$STF asin x == asin(stream x)$STF acos x == acos(stream x)$STF atan x == atan(stream x)$STF acot x == acot(stream x)$STF asec x == asec(stream x)$STF acsc x == acsc(stream x)$STF sinh x == sinh(stream x)$STF cosh x == cosh(stream x)$STF tanh x == tanh(stream x)$STF coth x == coth(stream x)$STF sech x == sech(stream x)$STF csch x == csch(stream x)$STF asinh x == asinh(stream x)$STF acosh x == acosh(stream x)$STF atanh x == atanh(stream x)$STF acoth x == acoth(stream x)$STF asech x == asech(stream x)$STF acsch x == acsch(stream x)$STF intsmp(v:Var,p: SMP): SMP == up := univariate(p,v) ss : SMP := 0 while not zero? up repeat d := degree up c := leadingCoefficient up ss := ss + inv((d+1) :: RN) * monomial(c,v,d+1)$SMP up := reductum up ss fintegrate(f,v,r) == concat(r::SMP,delay map(intsmp(v,#1),f() pretend StS)) integrate(s,v,r) == concat(r::SMP,map(intsmp(v,#1),s pretend StS)) -- If there is more than one term of the same order, group them. tout(p:SMP):OUT == pe := p :: OUT monomial? p => pe paren pe showAll?: () -> Boolean -- check a global Lisp variable showAll?() == true coerce(s:%):OUT == uu := s pretend Stream(SMP) empty? uu => (0$SMP) :: OUT n : NNI; count : NNI := _$streamCount$Lisp l : List OUT := empty() for n in 0..count while not empty? uu repeat if frst(uu) ~= 0 then l := concat(tout frst uu,l) uu := rst uu if showAll?() then for n in n.. while explicitEntries? uu and _ not eq?(uu,rst uu) repeat if frst(uu) ~= 0 then l := concat(tout frst uu,l) uu := rst uu l := explicitlyEmpty? uu => l eq?(uu,rst uu) and frst uu = 0 => l concat(prefix("O" :: OUT,[n :: OUT]),l) empty? l => (0$SMP) :: OUT reduce("+",reverse_! l) if Coef has Field then stream(x:%):Rep == x pretend Rep SF2==> StreamFunctions2 p:% / r:Coef ==(map(#1/$SMP r,stream p)$SF2(SMP,SMP))pretend % @ \section{domain TS TaylorSeries} <>= )abbrev domain TS TaylorSeries ++ Authors: Burge, Watt, Williamson ++ Date Created: 15 August 1988 ++ Date Last Updated: 18 May 1991 ++ Basic Operations: ++ Related Domains: SparseMultivariateTaylorSeries ++ Also See: UnivariateTaylorSeries ++ AMS Classifications: ++ Keywords: multivariate, Taylor, series ++ Examples: ++ References: ++ Description: ++ \spadtype{TaylorSeries} is a general multivariate Taylor series domain ++ over the ring Coef and with variables of type Symbol. TaylorSeries(Coef): Exports == Implementation where Coef : Ring L ==> List NNI ==> NonNegativeInteger SMP ==> Polynomial Coef StS ==> Stream SMP Exports ==> MultivariateTaylorSeriesCategory(Coef,Symbol) with coefficient: (%,NNI) -> SMP ++\spad{coefficient(s, n)} gives the terms of total degree n. coerce: Symbol -> % ++\spad{coerce(s)} converts a variable to a Taylor series coerce: SMP -> % ++\spad{coerce(s)} regroups terms of s by total degree ++ and forms a series. if Coef has Algebra Fraction Integer then integrate: (%,Symbol,Coef) -> % ++\spad{integrate(s,v,c)} is the integral of s with respect ++ to v and having c as the constant of integration. fintegrate: (() -> %,Symbol,Coef) -> % ++\spad{fintegrate(f,v,c)} is the integral of \spad{f()} with respect ++ to v and having c as the constant of integration. ++ The evaluation of \spad{f()} is delayed. Implementation ==> SparseMultivariateTaylorSeries(Coef,Symbol,SMP) add Rep := StS -- Below we use the fact that Rep of PS is Stream SMP. polynomial(s,n) == sum : SMP := 0 for i in 0..n while not empty? s repeat sum := sum + frst s s:= rst s sum @ \section{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. @ <<*>>= <> <> <> @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}