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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra nlinsol.spad}
\author{Manuel Bronstein}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package RETSOL RetractSolvePackage}
<<package RETSOL RetractSolvePackage>>=
)abbrev package RETSOL RetractSolvePackage
++ Author: Manuel Bronstein
++ Date Created: 31 October 1991
++ Date Last Updated: 31 October 1991
++ Description:
++ RetractSolvePackage is an interface to \spadtype{SystemSolvePackage}
++ that attempts to retract the coefficients of the equations before
++ solving.
RetractSolvePackage(Q, R): Exports == Implementation where
Q: IntegralDomain
R: Join(IntegralDomain, RetractableTo Q)
PQ ==> Polynomial Q
FQ ==> Fraction PQ
SY ==> Symbol
P ==> Polynomial R
F ==> Fraction P
EQ ==> Equation
SSP ==> SystemSolvePackage
Exports ==> with
solveRetract: (List P, List SY) -> List List EQ F
++ solveRetract(lp,lv) finds the solutions of the list lp of
++ rational functions with respect to the list of symbols lv.
++ The function tries to retract all the coefficients of the equations
++ to Q before solving if possible.
Implementation ==> add
LEQQ2F : List EQ FQ -> List EQ F
FQ2F : FQ -> F
PQ2P : PQ -> P
QIfCan : List P -> Union(List FQ, "failed")
PQIfCan: P -> Union(FQ, "failed")
PQ2P p == map(#1::R, p)$PolynomialFunctions2(Q, R)
FQ2F f == PQ2P numer f / PQ2P denom f
LEQQ2F l == [equation(FQ2F lhs eq, FQ2F rhs eq) for eq in l]
solveRetract(lp, lv) ==
(u := QIfCan lp) case "failed" =>
solve([p::F for p in lp]$List(F), lv)$SSP(R)
[LEQQ2F l for l in solve(u::List(FQ), lv)$SSP(Q)]
QIfCan l ==
ans:List(FQ) := empty()
for p in l repeat
(u := PQIfCan p) case "failed" => return "failed"
ans := concat(u::FQ, ans)
ans
PQIfCan p ==
(u := mainVariable p) case "failed" =>
(r := retractIfCan(ground p)@Union(Q,"failed")) case Q => r::Q::PQ::FQ
"failed"
up := univariate(p, s := u::SY)
ans:FQ := 0
while up ~= 0 repeat
(v := PQIfCan leadingCoefficient up) case "failed" => return "failed"
ans := ans + monomial(1, s, degree up)$PQ * (v::FQ)
up := reductum up
ans
@
\section{package NLINSOL NonLinearSolvePackage}
<<package NLINSOL NonLinearSolvePackage>>=
)abbrev package NLINSOL NonLinearSolvePackage
++ Author: Manuel Bronstein
++ Date Created: 31 October 1991
++ Date Last Updated: 26 June 1992
++ Description:
++ NonLinearSolvePackage is an interface to \spadtype{SystemSolvePackage}
++ that attempts to retract the coefficients of the equations before
++ solving. The solutions are given in the algebraic closure of R whenever
++ possible.
NonLinearSolvePackage(R:IntegralDomain): Exports == Implementation where
Z ==> Integer
Q ==> Fraction Z
SY ==> Symbol
P ==> Polynomial R
F ==> Fraction P
EQ ==> Equation F
SSP ==> SystemSolvePackage
SOL ==> RetractSolvePackage
Exports ==> with
solveInField: (List P, List SY) -> List List EQ
++ solveInField(lp,lv) finds the solutions of the list lp of
++ rational functions with respect to the list of symbols lv.
solveInField: List P -> List List EQ
++ solveInField(lp) finds the solution of the list lp of rational
++ functions with respect to all the symbols appearing in lp.
solve: (List P, List SY) -> List List EQ
++ solve(lp,lv) finds the solutions in the algebraic closure of R
++ of the list lp of
++ rational functions with respect to the list of symbols lv.
solve: List P -> List List EQ
++ solve(lp) finds the solution in the algebraic closure of R
++ of the list lp of rational
++ functions with respect to all the symbols appearing in lp.
Implementation ==> add
solveInField l == solveInField(l, "setUnion"/[variables p for p in l])
if R has AlgebraicallyClosedField then
import RationalFunction(R)
expandSol: List EQ -> List List EQ
RIfCan : F -> Union(R, "failed")
addRoot : (EQ, List List EQ) -> List List EQ
allRoots : List P -> List List EQ
evalSol : (List EQ, List EQ) -> List EQ
solve l == solve(l, "setUnion"/[variables p for p in l])
solve(lp, lv) == concat([expandSol sol for sol in solveInField(lp, lv)])
addRoot(eq, l) == [concat(eq, sol) for sol in l]
evalSol(ls, l) == [equation(lhs eq, eval(rhs eq, l)) for eq in ls]
-- converts [p1(a1),...,pn(an)] to
-- [[a1=v1,...,an=vn]] where vi ranges over all the zeros of pi
allRoots l ==
empty? l => [empty()$List(EQ)]
z := allRoots rest l
s := mainVariable(p := first l)::SY::P::F
concat [addRoot(equation(s, a::P::F), z) for a in zerosOf univariate p]
expandSol l ==
lassign := lsubs := empty()$List(EQ)
luniv := empty()$List(P)
for eq in l repeat
if retractIfCan(lhs eq)@Union(SY, "failed") case SY then
if RIfCan(rhs eq) case R then lassign := concat(eq, lassign)
else lsubs := concat(eq, lsubs)
else
if ((u := retractIfCan(lhs eq)@Union(P, "failed")) case P) and
one?(# variables(u::P)) and ((r := RIfCan rhs eq) case R) then
luniv := concat(u::P - r::R::P, luniv)
else return [l]
empty? luniv => [l]
[concat(z, concat(evalSol(lsubs,z), lassign)) for z in allRoots luniv]
RIfCan f ==
((n := retractIfCan(numer f)@Union(R,"failed")) case R) and
((d := retractIfCan(denom f)@Union(R,"failed")) case R) => n::R / d::R
"failed"
else
solve l == solveInField l
solve(lp, lv) == solveInField(lp, lv)
-- 'else if' is doubtful with this compiler so all 3 conditions are explicit
if (not(R is Q)) and (R has RetractableTo Q) then
solveInField(lp, lv) == solveRetract(lp, lv)$SOL(Q, R)
if (not(R is Z)) and (not(R has RetractableTo Q)) and
(R has RetractableTo Z) then
solveInField(lp, lv) == solveRetract(lp, lv)$SOL(Z, R)
if (not(R is Z)) and (not(R has RetractableTo Q)) and
(not(R has RetractableTo Z)) then
solveInField(lp, lv) == solve([p::F for p in lp]$List(F), lv)$SSP(R)
@
\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>>
-- Compile order for the differential equation solver:
-- oderf.spad odealg.spad nlode.spad nlinsol.spad riccati.spad odeef.spad
<<package RETSOL RetractSolvePackage>>
<<package NLINSOL NonLinearSolvePackage>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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