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diff --git a/src/algebra/utsode.spad.pamphlet b/src/algebra/utsode.spad.pamphlet new file mode 100644 index 00000000..26d03144 --- /dev/null +++ b/src/algebra/utsode.spad.pamphlet @@ -0,0 +1,181 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/algebra utsode.spad} +\author{Stephen M. Watt, Clifton J. Williamson} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\tableofcontents +\eject +\section{package UTSODE UnivariateTaylorSeriesODESolver} +<<package UTSODE UnivariateTaylorSeriesODESolver>>= +)abbrev package UTSODE UnivariateTaylorSeriesODESolver +++ Taylor series solutions of explicit ODE's. +++ Author: Stephen Watt (revised by Clifton J. Williamson) +++ Date Created: February 1988 +++ Date Last Updated: 30 September 1993 +++ Keywords: differential equation, ODE, Taylor series +++ Examples: +++ References: +UnivariateTaylorSeriesODESolver(Coef,UTS):_ + Exports == Implementation where + ++ This package provides Taylor series solutions to regular + ++ linear or non-linear ordinary differential equations of + ++ arbitrary order. + Coef : Algebra Fraction Integer + UTS : UnivariateTaylorSeriesCategory Coef + L ==> List + L2 ==> ListFunctions2 + FN ==> (L UTS) -> UTS + ST ==> Stream Coef + YS ==> Y$ParadoxicalCombinatorsForStreams(Coef) + STT ==> StreamTaylorSeriesOperations(Coef) + + Exports ==> with + stFunc1: (UTS -> UTS) -> (ST -> ST) + ++ stFunc1(f) is a local function exported due to compiler problem. + ++ This function is of no interest to the top-level user. + stFunc2: ((UTS,UTS) -> UTS) -> ((ST,ST) -> ST) + ++ stFunc2(f) is a local function exported due to compiler problem. + ++ This function is of no interest to the top-level user. + stFuncN: FN -> ((L ST) -> ST) + ++ stFuncN(f) is a local function xported due to compiler problem. + ++ This function is of no interest to the top-level user. + fixedPointExquo: (UTS,UTS) -> UTS + ++ fixedPointExquo(f,g) computes the exact quotient of \spad{f} and + ++ \spad{g} using a fixed point computation. + ode1: ((UTS -> UTS),Coef) -> UTS + ++ ode1(f,c) is the solution to \spad{y' = f(y)} + ++ such that \spad{y(a) = c}. + ode2: ((UTS, UTS) -> UTS,Coef,Coef) -> UTS + ++ ode2(f,c0,c1) is the solution to \spad{y'' = f(y,y')} such that + ++ \spad{y(a) = c0} and \spad{y'(a) = c1}. + ode: (FN,List Coef) -> UTS + ++ ode(f,cl) is the solution to \spad{y<n>=f(y,y',..,y<n-1>)} such that + ++ \spad{y<i>(a) = cl.i} for i in 1..n. + mpsode:(L Coef,L FN) -> L UTS + ++ mpsode(r,f) solves the system of differential equations + ++ \spad{dy[i]/dx =f[i] [x,y[1],y[2],...,y[n]]}, + ++ \spad{y[i](a) = r[i]} for i in 1..n. + + Implementation ==> add + + stFunc1 f == coefficients f series(#1) + stFunc2 f == coefficients f(series(#1),series(#2)) + stFuncN f == coefficients f map(series,#1)$ListFunctions2(ST,UTS) + + import StreamTaylorSeriesOperations(Coef) + divloopre:(Coef,ST,Coef,ST,ST) -> ST + divloopre(hx,tx,hy,ty,c) == delay(concat(hx*hy,hy*(tx-(ty*c)))) + divloop: (Coef,ST,Coef,ST) -> ST + divloop(hx,tx,hy,ty) == YS(divloopre(hx,tx,hy,ty,#1)) + + sdiv:(ST,ST) -> ST + sdiv(x,y) == delay + empty? x => empty() + empty? y => error "stream division by zero" + hx := frst x; tx := rst x + hy := frst y; ty := rst y + zero? hy => + zero? hx => sdiv(tx,ty) + error "stream division by zero" + rhy := recip hy + rhy case "failed" => error "stream division:no reciprocal" + divloop(hx,tx,rhy::Coef,ty) + + fixedPointExquo(f,g) == series sdiv(coefficients f,coefficients g) + +-- first order + + ode1re: (ST -> ST,Coef,ST) -> ST + ode1re(f,c,y) == lazyIntegrate(c,f y)$STT + + iOde1: ((ST -> ST),Coef) -> ST + iOde1(f,c) == YS ode1re(f,c,#1) + + ode1(f,c) == series iOde1(stFunc1 f,c) + +-- second order + + ode2re: ((ST,ST)-> ST,Coef,Coef,ST) -> ST + ode2re(f,c0,c1,y)== + yi := lazyIntegrate(c1,f(y,deriv(y)$STT))$STT + lazyIntegrate(c0,yi)$STT + + iOde2: ((ST,ST) -> ST,Coef,Coef) -> ST + iOde2(f,c0,c1) == YS ode2re(f,c0,c1,#1) + + ode2(f,c0,c1) == series iOde2(stFunc2 f,c0,c1) + +-- nth order + + odeNre: (List ST -> ST,List Coef,List ST) -> List ST + odeNre(f,cl,yl) == + -- yl is [y, y', ..., y<n>] + -- integrate [y',..,y<n>] to get [y,..,y<n-1>] + yil := [lazyIntegrate(c,y)$STT for c in cl for y in rest yl] + -- use y<n> = f(y,..,y<n-1>) + concat(yil,[f yil]) + + iOde: ((L ST) -> ST,List Coef) -> ST + iOde(f,cl) == first YS(odeNre(f,cl,#1),#cl + 1) + + ode(f,cl) == series iOde(stFuncN f,cl) + + simulre:(L Coef,L ((L ST) -> ST),L ST) -> L ST + simulre(cst,lsf,c) == + [lazyIntegrate(csti,lsfi concat(monom(1,1)$STT,c))_ + for csti in cst for lsfi in lsf] + iMpsode:(L Coef,L ((L ST) -> ST)) -> L ST + iMpsode(cs,lsts) == YS(simulre(cs,lsts,#1),# cs) + mpsode(cs,lsts) == +-- stSol := iMpsode(cs,map(stFuncN,lsts)$L2(FN,(L ST) -> ST)) + stSol := iMpsode(cs,[stFuncN(lst) for lst in lsts]) + map(series,stSol)$L2(ST,UTS) + +@ +\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 UTSODE UnivariateTaylorSeriesODESolver>> +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} |