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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra fparfrac.spad}
\author{Manuel Bronstein}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain FPARFRAC FullPartialFractionExpansion}
<<domain FPARFRAC FullPartialFractionExpansion>>=
)abbrev domain FPARFRAC FullPartialFractionExpansion
++ Full partial fraction expansion of rational functions
++ Author: Manuel Bronstein
++ Date Created: 9 December 1992
++ Date Last Updated: June 18, 2010
++ References: M.Bronstein & B.Salvy,
++ Full Partial Fraction Decomposition of Rational Functions,
++ in Proceedings of ISSAC'93, Kiev, ACM Press.
FullPartialFractionExpansion(F, UP): Exports == Implementation where
F : Join(Field, CharacteristicZero)
UP : UnivariatePolynomialCategory F
N ==> NonNegativeInteger
Q ==> Fraction Integer
O ==> OutputForm
RF ==> Fraction UP
SUP ==> SparseUnivariatePolynomial RF
REC ==> Record(exponent: N, center: UP, num: UP)
ODV ==> OrderlyDifferentialVariable Symbol
ODP ==> OrderlyDifferentialPolynomial UP
ODF ==> Fraction ODP
FPF ==> Record(polyPart: UP, fracPart: List REC)
Exports ==> Join(SetCategory, DifferentialSpace, ConvertibleTo RF) with
+: (UP, $) -> $
++ p + x returns the sum of p and x
fullPartialFraction: RF -> $
++ fullPartialFraction(f) returns \spad{[p, [[j, Dj, Hj]...]]} such that
++ \spad{f = p(x) + \sum_{[j,Dj,Hj] in l} \sum_{Dj(a)=0} Hj(a)/(x - a)\^j}.
polyPart: $ -> UP
++ polyPart(f) returns the polynomial part of f.
fracPart: $ -> List REC
++ fracPart(f) returns the list of summands of the fractional part of f.
construct: List REC -> $
++ construct(l) is the inverse of fracPart.
Implementation ==> add
Rep := FPF
fullParFrac: (UP, UP, UP, N) -> List REC
outputexp : (O, N) -> O
output : (N, UP, UP) -> O
REC2RF : (UP, UP, N) -> RF
UP2SUP : UP -> SUP
diffrec : REC -> REC
FP2O : List REC -> O
-- create a differential variable
u := new()$Symbol
u0 := makeVariable(u, 0)$ODV
alpha := u::O
x := monomial(1, 1)$UP
xx := x::O
zr := (0$N)::O
construct l == [0, l]
D r == differentiate r
D(r, n) == differentiate(r,n)
polyPart f == f.polyPart
fracPart f == f.fracPart
p:UP + f:$ == [p + polyPart f, fracPart f]
differentiate f ==
differentiate(polyPart f) + construct [diffrec rec for rec in fracPart f]
differentiate(r, n) ==
for i in 1..n repeat r := differentiate r
r
-- diffrec(sum_{rec.center(a) = 0} rec.num(a) / (x - a)^e) =
-- sum_{rec.center(a) = 0} -e rec.num(a) / (x - a)^{e+1}
-- where e = rec.exponent
diffrec rec ==
e := rec.exponent
[e + 1, rec.center, - e * rec.num]
convert(f:$):RF ==
ans := polyPart(f)::RF
for rec in fracPart f repeat
ans := ans + REC2RF(rec.center, rec.num, rec.exponent)
ans
UP2SUP p ==
map(#1::UP::RF, p)$UnivariatePolynomialCategoryFunctions2(F, UP, RF, SUP)
-- returns Trace_k^k(a) (h(a) / (x - a)^n) where d(a) = 0
REC2RF(d, h, n) ==
one?(m := degree d) =>
a := - (leadingCoefficient reductum d) / (leadingCoefficient d)
h(a)::UP / (x - a::UP)**n
dd := UP2SUP d
hh := UP2SUP h
aa := monomial(1, 1)$SUP
p := (x::RF::SUP - aa)**n rem dd
rec := extendedEuclidean(p, dd, hh)::Record(coef1:SUP, coef2:SUP)
t := rec.coef1 -- we want Trace_k^k(a)(t) now
ans := coefficient(t, 0)
for i in 1..degree(d)-1 repeat
t := (t * aa) rem dd
ans := ans + coefficient(t, i)
ans
fullPartialFraction f ==
qr := divide(numer f, d := denom f)
qr.quotient + construct concat
[fullParFrac(qr.remainder, d, rec.factor, rec.exponent::N)
for rec in factors squareFree denom f]
fullParFrac(a, d, q, n) ==
ans:List REC := empty()
em := e := d quo (q ** n)
rec := extendedEuclidean(e, q, 1)::Record(coef1:UP,coef2:UP)
bm := b := rec.coef1 -- b = inverse of e modulo q
lvar:List(ODV) := [u0]
um := 1::ODP
un := (u1 := u0::ODP)**n
lval:List(UP) := [q1 := q := differentiate(q0 := q)]
h:ODF := a::ODP / (e * un)
rec := extendedEuclidean(q1, q0, 1)::Record(coef1:UP,coef2:UP)
c := rec.coef1 -- c = inverse of q' modulo q
cm := 1::UP
cn := (c ** n) rem q0
for m in 1..n repeat
p := retract(em * un * um * h)@ODP
pp := retract(eval(p, lvar, lval))@UP
h := inv(m::Q) * differentiate h
q := differentiate q
lvar := concat(makeVariable(u, m), lvar)
lval := concat(inv((m+1)::F) * q, lval)
qq := q0 quo gcd(pp, q0) -- new center
if positive? degree(qq) then
ans := concat([(n + 1 - m)::N, qq, (pp * bm * cn * cm) rem qq], ans)
cm := (c * cm) rem q0 -- cm = c**m modulo q now
um := u1 * um -- um = u**m now
em := e * em -- em = e**{m+1} now
bm := (b * bm) rem q0 -- bm = b**{m+1} modulo q now
ans
coerce(f:$):O ==
ans := FP2O(l := fracPart f)
zero?(p := polyPart f) =>
empty? l => (0$N)::O
ans
p::O + ans
FP2O l ==
empty? l => empty()
rec := first l
ans := output(rec.exponent, rec.center, rec.num)
for rec: local in rest l repeat
ans := ans + output(rec.exponent, rec.center, rec.num)
ans
output(n, d, h) ==
one? degree d =>
a := - leadingCoefficient(reductum d) / leadingCoefficient(d)
h(a)::O / outputexp((x - a::UP)::O, n)
sum(outputForm(makeSUP h, alpha) / outputexp(xx - alpha, n),
outputForm(makeSUP d, alpha) = zr)
outputexp(f, n) ==
one? n => f
f ** (n::O)
@
\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>>
<<domain FPARFRAC FullPartialFractionExpansion>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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