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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra rderf.spad}
\author{Manuel Bronstein}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package RDETR TranscendentalRischDE}
<<package RDETR TranscendentalRischDE>>=
)abbrev package RDETR TranscendentalRischDE
++ Risch differential equation, transcendental case.
++ Author: Manuel Bronstein
++ Date Created: Jan 1988
++ Date Last Updated: 2 November 1995
TranscendentalRischDE(F, UP): Exports == Implementation where
F : Join(Field, CharacteristicZero, RetractableTo Integer)
UP : UnivariatePolynomialCategory F
N ==> NonNegativeInteger
Z ==> Integer
RF ==> Fraction UP
REC ==> Record(a:UP, b:UP, c:UP, t:UP)
SPE ==> Record(b:UP, c:UP, m:Z, alpha:UP, beta:UP)
PSOL==> Record(ans:UP, nosol:Boolean)
ANS ==> Union(ans:PSOL, eq:SPE)
PSQ ==> Record(ans:RF, nosol:Boolean)
Exports ==> with
monomRDE: (RF,RF,UP->UP) -> Union(Record(a:UP,b:RF,c:RF,t:UP), "failed")
++ monomRDE(f,g,D) returns \spad{[A, B, C, T]} such that
++ \spad{y' + f y = g} has a solution if and only if \spad{y = Q / T},
++ where Q satisfies \spad{A Q' + B Q = C} and has no normal pole.
++ A and T are polynomials and B and C have no normal poles.
++ D is the derivation to use.
baseRDE : (RF, RF) -> PSQ
++ baseRDE(f, g) returns a \spad{[y, b]} such that \spad{y' + fy = g}
++ if \spad{b = true}, y is a partial solution otherwise (no solution
++ in that case).
++ D is the derivation to use.
polyRDE : (UP, UP, UP, Z, UP -> UP) -> ANS
++ polyRDE(a, B, C, n, D) returns either:
++ 1. \spad{[Q, b]} such that \spad{degree(Q) <= n} and
++ \spad{a Q'+ B Q = C} if \spad{b = true}, Q is a partial solution
++ otherwise.
++ 2. \spad{[B1, C1, m, \alpha, \beta]} such that any polynomial solution
++ of degree at most n of \spad{A Q' + BQ = C} must be of the form
++ \spad{Q = \alpha H + \beta} where \spad{degree(H) <= m} and
++ H satisfies \spad{H' + B1 H = C1}.
++ D is the derivation to use.
Implementation ==> add
import MonomialExtensionTools(F, UP)
getBound : (UP, UP, Z) -> Z
SPDEnocancel1: (UP, UP, Z, UP -> UP) -> PSOL
SPDEnocancel2: (UP, UP, Z, Z, F, UP -> UP) -> ANS
SPDE : (UP, UP, UP, Z, UP -> UP) -> Union(SPE, "failed")
-- cancellation at infinity is possible, A is assumed nonzero
-- needs tagged union because of branch choice problem
-- always returns a PSOL in the base case (never a SPE)
polyRDE(aa, bb, cc, d, derivation) ==
n:Z
(u := SPDE(aa, bb, cc, d, derivation)) case "failed" => [[0, true]]
zero?(u.c) => [[u.beta, false]]
baseCase? := one?(dt := derivation monomial(1, 1))
n := degree(dt)::Z - 1
b0? := zero?(u.b)
(~b0?) and (baseCase? or degree(u.b) > max(0, n)) =>
answ := SPDEnocancel1(u.b, u.c, u.m, derivation)
[[u.alpha * answ.ans + u.beta, answ.nosol]]
(n > 0) and (b0? or degree(u.b) < n) =>
uansw := SPDEnocancel2(u.b,u.c,u.m,n,leadingCoefficient dt,derivation)
uansw case ans=> [[u.alpha * uansw.ans.ans + u.beta, uansw.ans.nosol]]
[[uansw.eq.b, uansw.eq.c, uansw.eq.m,
u.alpha * uansw.eq.alpha, u.alpha * uansw.eq.beta + u.beta]]
b0? and baseCase? =>
degree(u.c) >= u.m => [[0, true]]
[[u.alpha * integrate(u.c) + u.beta, false]]
[u::SPE]
-- cancellation at infinity is possible, A is assumed nonzero
-- if u.b = 0 then u.a = 1 already, but no degree check is done
-- returns "failed" if a p' + b p = c has no soln of degree at most d,
-- otherwise [B, C, m, \alpha, \beta] such that any soln p of degree at
-- most d of a p' + b p = c must be of the form p = \alpha h + \beta,
-- where h' + B h = C and h has degree at most m
SPDE(aa, bb, cc, d, derivation) ==
zero? cc => [0, 0, 0, 0, 0]
d < 0 => "failed"
(u := cc exquo (g := gcd(aa, bb))) case "failed" => "failed"
aa := (aa exquo g)::UP
bb := (bb exquo g)::UP
cc := u::UP
(ra := retractIfCan(aa)@Union(F, "failed")) case F =>
a1 := inv(ra::F)
[a1 * bb, a1 * cc, d, 1, 0]
bc := extendedEuclidean(bb, aa, cc)::Record(coef1:UP, coef2:UP)
qr := divide(bc.coef1, aa)
r := qr.remainder -- z = bc.coef2 + b * qr.quotient
(v := SPDE(aa, bb + derivation aa,
bc.coef2 + bb * qr.quotient - derivation r,
d - degree(aa)::Z, derivation)) case "failed" => "failed"
[v.b, v.c, v.m, aa * v.alpha, aa * v.beta + r]
-- solves q' + b q = c with deg(q) <= d
-- case (B <> 0) and (D = d/dt or degree(B) > max(0, degree(Dt) - 1))
-- this implies no cancellation at infinity, BQ term dominates
-- returns [Q, flag] such that Q is a solution if flag is false,
-- a partial solution otherwise.
SPDEnocancel1(bb, cc, d, derivation) ==
q:UP := 0
db := (degree bb)::Z
lb := leadingCoefficient bb
while cc ~= 0 repeat
d < 0 or (n := (degree cc)::Z - db) < 0 or n > d => return [q, true]
r := monomial((leadingCoefficient cc) / lb, n::N)
cc := cc - bb * r - derivation r
d := n - 1
q := q + r
[q, false]
-- case (t is a nonlinear monomial) and (B = 0 or degree(B) < degree(Dt) - 1)
-- this implies no cancellation at infinity, DQ term dominates or degree(Q) = 0
-- dtm1 = degree(Dt) - 1
SPDEnocancel2(bb, cc, d, dtm1, lt, derivation) ==
q:UP := 0
while cc ~= 0 repeat
d < 0 or (n := (degree cc)::Z - dtm1) < 0 or n > d => return [[q, true]]
if n > 0 then
r := monomial((leadingCoefficient cc) / (n * lt), n::N)
cc := cc - bb * r - derivation r
d := n - 1
q := q + r
else -- n = 0 so solution must have degree 0
db:N := (zero? bb => 0; degree bb);
db ~= degree(cc) => return [[q, true]]
zero? db => return [[bb, cc, 0, 1, q]]
r := leadingCoefficient(cc) / leadingCoefficient(bb)
cc := cc - r * bb - derivation(r::UP)
d := - 1
q := q + r::UP
[[q, false]]
monomRDE(f, g, derivation) ==
gg := gcd(d := normalDenom(f,derivation), e := normalDenom(g,derivation))
tt := (gcd(e, differentiate e) exquo gcd(gg,differentiate gg))::UP
(u := ((tt * (aa := d * tt)) exquo e)) case "failed" => "failed"
[aa, aa * f - (d * derivation tt)::RF, u::UP * e * g, tt]
-- solve y' + f y = g for y in RF
-- assumes that f is weakly normalized (no finite cancellation)
-- base case: F' = 0
baseRDE(f, g) ==
(u := monomRDE(f, g, differentiate)) case "failed" => [0, true]
n := getBound(u.a,bb := retract(u.b)@UP,degree(cc := retract(u.c)@UP)::Z)
v := polyRDE(u.a, bb, cc, n, differentiate).ans
[v.ans / u.t, v.nosol]
-- return an a bound on the degree of a solution of A P'+ B P = C,A ~= 0
-- cancellation at infinity is possible
-- base case: F' = 0
getBound(a, b, dc) ==
da := (degree a)::Z
zero? b => max(0, dc - da + 1)
db := (degree b)::Z
da > (db + 1) => max(0, dc - da + 1)
da < (db + 1) => dc - db
(n := retractIfCan(- leadingCoefficient(b) / leadingCoefficient(a)
)@Union(Z, "failed")) case Z => max(n::Z, dc - db)
dc - db
@
\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>>
-- SPAD files for the integration world should be compiled in the
-- following order:
--
-- intaux RDERF intrf rdeef intef irexpand integrat
<<package RDETR TranscendentalRischDE>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
