\documentclass{article} \usepackage{open-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} <>= )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$ParadoxicalCombinatorsForStreams(Coef) STT ==> StreamTaylorSeriesOperations(Coef) Exports ==> with stFunc1: (UTS -> UTS) -> (ST -> ST) ++ stFunc1(f) is a local function exported due to compiler problem. ++ This function is of no interest to the top-level user. stFunc2: ((UTS,UTS) -> UTS) -> ((ST,ST) -> ST) ++ stFunc2(f) is a local function exported due to compiler problem. ++ This function is of no interest to the top-level user. stFuncN: FN -> ((L ST) -> ST) ++ stFuncN(f) is a local function xported due to compiler problem. ++ This function is of no interest to the top-level user. fixedPointExquo: (UTS,UTS) -> UTS ++ fixedPointExquo(f,g) computes the exact quotient of \spad{f} and ++ \spad{g} using a fixed point computation. ode1: ((UTS -> UTS),Coef) -> UTS ++ ode1(f,c) is the solution to \spad{y' = f(y)} ++ such that \spad{y(a) = c}. ode2: ((UTS, UTS) -> UTS,Coef,Coef) -> UTS ++ ode2(f,c0,c1) is the solution to \spad{y'' = f(y,y')} such that ++ \spad{y(a) = c0} and \spad{y'(a) = c1}. ode: (FN,List Coef) -> UTS ++ ode(f,cl) is the solution to \spad{y=f(y,y',..,y)} such that ++ \spad{y(a) = cl.i} for i in 1..n. mpsode:(L Coef,L FN) -> L UTS ++ mpsode(r,f) solves the system of differential equations ++ \spad{dy[i]/dx =f[i] [x,y[1],y[2],...,y[n]]}, ++ \spad{y[i](a) = r[i]} for i in 1..n. Implementation ==> add stFunc1 f == coefficients f series(#1) stFunc2 f == coefficients f(series(#1),series(#2)) stFuncN f == coefficients f map(series,#1)$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] -- integrate [y',..,y] to get [y,..,y] yil := [lazyIntegrate(c,y)$STT for c in cl for y in rest yl] -- use y = f(y,..,y) concat(yil,[f yil]) iOde: ((L ST) -> ST,List Coef) -> ST iOde(f,cl) == first YS(odeNre(f,cl,#1),#cl + 1) ode(f,cl) == series iOde(stFuncN f,cl) simulre:(L Coef,L ((L ST) -> ST),L ST) -> L ST simulre(cst,lsf,c) == [lazyIntegrate(csti,lsfi concat(monom(1,1)$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} <>= --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}