\documentclass{article}
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\begin{document}
\title{\$SPAD/src/algebra exprode.spad}
\author{Manuel Bronstein}
\maketitle
\begin{abstract}
\end{abstract}
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\tableofcontents
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\section{package EXPRODE ExpressionSpaceODESolver}
<<package EXPRODE ExpressionSpaceODESolver>>=
)abbrev package EXPRODE ExpressionSpaceODESolver
++ Taylor series solutions of ODE's
++ Author: Manuel Bronstein
++ Date Created: 5 Mar 1990
++ Date Last Updated: 30 September 1993
++ Description: Taylor series solutions of explicit ODE's;
++ Keywords: differential equation, ODE, Taylor series
ExpressionSpaceODESolver(R, F): Exports == Implementation where
  R: Join(OrderedSet, IntegralDomain, ConvertibleTo InputForm)
  F: FunctionSpace R

  K   ==> Kernel F
  P   ==> SparseMultivariatePolynomial(R, K)
  OP  ==> BasicOperator
  SY  ==> Symbol
  UTS ==> UnivariateTaylorSeries(F, x, center)
  MKF ==> MakeUnaryCompiledFunction(F, UTS, UTS)
  MKL ==> MakeUnaryCompiledFunction(F, List UTS, UTS)
  A1  ==> AnyFunctions1(UTS)
  AL1 ==> AnyFunctions1(List UTS)
  EQ  ==> Equation F
  ODE ==> UnivariateTaylorSeriesODESolver(F, UTS)

  Exports ==> with
    seriesSolve: (EQ, OP, EQ, EQ) -> Any
      ++ seriesSolve(eq,y,x=a, y a = b) returns a Taylor series solution
      ++ of eq around x = a with initial condition \spad{y(a) = b}.
      ++ Note: eq must be of the form
      ++ \spad{f(x, y x) y'(x) + g(x, y x) = h(x, y x)}.
    seriesSolve: (EQ, OP, EQ, List F) -> Any
      ++ seriesSolve(eq,y,x=a,[b0,...,b(n-1)]) returns a Taylor series
      ++ solution of eq around \spad{x = a} with initial conditions
      ++ \spad{y(a) = b0}, \spad{y'(a) = b1},
      ++ \spad{y''(a) = b2}, ...,\spad{y(n-1)(a) = b(n-1)}
      ++ eq must be of the form
      ++ \spad{f(x, y x, y'(x),..., y(n-1)(x)) y(n)(x) +
      ++ g(x,y x,y'(x),...,y(n-1)(x)) = h(x,y x, y'(x),..., y(n-1)(x))}.
    seriesSolve: (List EQ, List OP, EQ, List EQ) -> Any
      ++ seriesSolve([eq1,...,eqn],[y1,...,yn],x = a,[y1 a = b1,...,yn a = bn])
      ++ returns a taylor series solution of \spad{[eq1,...,eqn]} around
      ++ \spad{x = a} with initial conditions \spad{yi(a) = bi}.
      ++ Note: eqi must be of the form
      ++ \spad{fi(x, y1 x, y2 x,..., yn x) y1'(x) +
      ++ gi(x, y1 x, y2 x,..., yn x) = h(x, y1 x, y2 x,..., yn x)}.
    seriesSolve: (List EQ, List OP, EQ, List F) -> Any
      ++ seriesSolve([eq1,...,eqn], [y1,...,yn], x=a, [b1,...,bn])
      ++ is equivalent to
      ++ \spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x = a,
      ++ [y1 a = b1,..., yn a = bn])}.
    seriesSolve: (List F, List OP, EQ, List F) -> Any
      ++ seriesSolve([eq1,...,eqn], [y1,...,yn], x=a, [b1,...,bn])
      ++ is equivalent to
      ++ \spad{seriesSolve([eq1=0,...,eqn=0], [y1,...,yn], x=a, [b1,...,bn])}.
    seriesSolve: (List F, List OP, EQ, List EQ) -> Any
      ++ seriesSolve([eq1,...,eqn], [y1,...,yn], x = a,[y1 a = b1,..., yn a = bn])
      ++ is equivalent to
      ++ \spad{seriesSolve([eq1=0,...,eqn=0], [y1,...,yn], x = a,
      ++ [y1 a = b1,..., yn a = bn])}.
    seriesSolve: (EQ, OP, EQ, F) -> Any
      ++ seriesSolve(eq,y, x=a, b) is equivalent to
      ++ \spad{seriesSolve(eq, y, x=a, y a = b)}.
    seriesSolve: (F, OP, EQ, F) -> Any
      ++ seriesSolve(eq, y, x = a, b) is equivalent to
      ++ \spad{seriesSolve(eq = 0, y, x = a, y a = b)}.
    seriesSolve: (F, OP, EQ, EQ) -> Any
      ++ seriesSolve(eq, y, x = a, y a = b) is equivalent to
      ++ \spad{seriesSolve(eq=0, y, x=a, y a = b)}.
    seriesSolve: (F, OP, EQ, List F) -> Any
      ++ seriesSolve(eq, y, x = a, [b0,...,bn]) is equivalent to
      ++ \spad{seriesSolve(eq = 0, y, x = a, [b0,...,b(n-1)])}.

  Implementation ==> add
    checkCompat: (OP, EQ, EQ) -> F
    checkOrder1: (F, OP, K, SY, F) -> F
    checkOrderN: (F, OP, K, SY, F, NonNegativeInteger) -> F
    checkSystem: (F, List K, List F) -> F
    div2exquo  : F -> F
    smp2exquo  : P -> F
    k2exquo    : K -> F
    diffRhs    : (F, F) -> F
    diffRhsK   : (K, F) -> F
    findCompat : (F, List EQ) -> F
    findEq     : (K, SY, List F) -> F
    localInteger: F -> F

    opelt := operator('elt)$OP
    --opex  := operator('exquo)$OP
    opex  := operator('fixedPointExquo)$OP
    opint := operator('integer)$OP

    Rint? := R has IntegerNumberSystem

    localInteger n == (Rint? => n; opint n)
    diffRhs(f, g) == diffRhsK(retract(f)@K, g)

    k2exquo k ==
      is?(op := operator k,'%diff) =>
        error "Improper differential equation"
      kernel(op, [div2exquo f for f in argument k]$List(F))

    smp2exquo p ==
      map(k2exquo,#1::F,p)$PolynomialCategoryLifting(IndexedExponents K,
                                                             K, R, P, F)

    div2exquo f ==
      one?(d := denom f) => f
      opex(smp2exquo numer f, smp2exquo d)

-- if g is of the form a * k + b, then return -b/a
    diffRhsK(k, g) ==
      h := univariate(g, k)
      (degree(numer h) <= 1) and ground? denom h =>
        - coefficient(numer h, 0) / coefficient(numer h, 1)
      error "Improper differential equation"

    checkCompat(y, eqx, eqy) ==
      lhs(eqy) =$F y(rhs eqx) => rhs eqy
      error "Improper initial value"

    findCompat(yx, l) ==
      for eq in l repeat
        yx =$F lhs eq => return rhs eq
      error "Improper initial value"

    findEq(k, x, sys) ==
      k := retract(differentiate(k::F, x))@K
      for eq in sys repeat
        member?(k, kernels eq) => return eq
      error "Improper differential equation"

    checkOrder1(diffeq, y, yx, x, sy) ==
      div2exquo subst(diffRhs(differentiate(yx::F,x),diffeq),[yx],[sy])

    checkOrderN(diffeq, y, yx, x, sy, n) ==
      zero? n => error "No initial value(s) given"
      m     := (minIndex(l := [retract(f := yx::F)@K]$List(K)))::F
      lv    := [opelt(sy, localInteger m)]$List(F)
      for i in 2..n repeat
        l  := concat(retract(f := differentiate(f, x))@K, l)
        lv := concat(opelt(sy, localInteger(m := m + 1)), lv)
      div2exquo subst(diffRhs(differentiate(f, x), diffeq), l, lv)

    checkSystem(diffeq, yx, lv) ==
      for k in kernels diffeq repeat
        is?(k, '%diff) =>
          return div2exquo subst(diffRhsK(k, diffeq), yx, lv)
      0

    seriesSolve(l:List EQ, y:List OP, eqx:EQ, eqy:List EQ) ==
      seriesSolve([lhs deq - rhs deq for deq in l]$List(F), y, eqx, eqy)

    seriesSolve(l:List EQ, y:List OP, eqx:EQ, y0:List F) ==
      seriesSolve([lhs deq - rhs deq for deq in l]$List(F), y, eqx, y0)

    seriesSolve(l:List F, ly:List OP, eqx:EQ, eqy:List EQ) ==
      seriesSolve(l, ly, eqx,
                  [findCompat(y rhs eqx, eqy) for y in ly]$List(F))

    seriesSolve(diffeq:EQ, y:OP, eqx:EQ, eqy:EQ) ==
      seriesSolve(lhs diffeq - rhs diffeq, y, eqx, eqy)

    seriesSolve(diffeq:EQ, y:OP, eqx:EQ, y0:F) ==
      seriesSolve(lhs diffeq - rhs diffeq, y, eqx, y0)

    seriesSolve(diffeq:EQ, y:OP, eqx:EQ, y0:List F) ==
      seriesSolve(lhs diffeq - rhs diffeq, y, eqx, y0)

    seriesSolve(diffeq:F, y:OP, eqx:EQ, eqy:EQ) ==
      seriesSolve(diffeq, y, eqx, checkCompat(y, eqx, eqy))

    seriesSolve(diffeq:F, y:OP, eqx:EQ, y0:F) ==
      x      := symbolIfCan(retract(lhs eqx)@K)::SY
      sy     := name y
      yx     := retract(y lhs eqx)@K
      f      := checkOrder1(diffeq, y, yx, x, sy::F)
      center := rhs eqx
      coerce(ode1(compiledFunction(f, sy)$MKF, y0)$ODE)$A1

    seriesSolve(diffeq:F, y:OP, eqx:EQ, y0:List F) ==
      x      := symbolIfCan(retract(lhs eqx)@K)::SY
      sy     := new()$SY
      yx     := retract(y lhs eqx)@K
      f      := checkOrderN(diffeq, y, yx, x, sy::F, #y0)
      center := rhs eqx
      coerce(ode(compiledFunction(f, sy)$MKL, y0)$ODE)$A1

    seriesSolve(sys:List F, ly:List OP, eqx:EQ, l0:List F) ==
      x      := symbolIfCan(kx := retract(lhs eqx)@K)::SY
      fsy    := (sy := new()$SY)::F
      m      := (minIndex(l0) - 1)::F
      yx     := concat(kx, [retract(y lhs eqx)@K for y in ly]$List(K))
      lelt   := [opelt(fsy, localInteger(m := m+1)) for k in yx]$List(F)
      sys    := [findEq(k, x, sys) for k in rest yx]
      l      := [checkSystem(eq, yx, lelt) for eq in sys]$List(F)
      center := rhs eqx
      coerce(mpsode(l0,[compiledFunction(f,sy)$MKL for f in l])$ODE)$AL1

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--Copyright (C) 2007-2009, Gabriel Dos Reis.
--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 EXPRODE ExpressionSpaceODESolver>>
@
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\end{document}