/usr/lib/open-axiom/src/algebra/nlinsol.spad is in open-axiom-source 1.4.1+svn~2626-2ubuntu2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 | --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.
-- Compile order for the differential equation solver:
-- oderf.spad odealg.spad nlode.spad nlinsol.spad riccati.spad odeef.spad
)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
)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)
|