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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra pade.spad}
\author{Barry Trager, William Burge, Martin Hassner,Stephen M. Watt}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package PADEPAC PadeApproximantPackage}
<<package PADEPAC PadeApproximantPackage>>=
)abbrev package PADEPAC PadeApproximantPackage
++ This package computes reliable Pad&ea. approximants using
++ a generalized Viskovatov continued fraction algorithm.
++ Authors: Trager,Burge, Hassner & Watt.
++ Date Created: April 1987
++ Date Last Updated: 12 April 1990
++ Keywords: Pade, series
++ Examples:
++ References:
++ "Pade Approximants, Part I: Basic Theory", Baker & Graves-Morris.
PadeApproximantPackage(R: Field, x:Symbol, pt:R): Exports == Implementation where
PS ==> UnivariateTaylorSeries(R,x,pt)
UP ==> UnivariatePolynomial(x,R)
QF ==> Fraction UP
CF ==> ContinuedFraction UP
NNI ==> NonNegativeInteger
Exports ==> with
pade: (NNI,NNI,PS,PS) -> Union(QF,"failed")
++ pade(nd,dd,ns,ds) computes the approximant as a quotient of polynomials
++ (if it exists) for arguments
++ nd (numerator degree of approximant),
++ dd (denominator degree of approximant),
++ ns (numerator series of function), and
++ ds (denominator series of function).
pade: (NNI,NNI,PS) -> Union(QF,"failed")
++ pade(nd,dd,s)
++ computes the quotient of polynomials
++ (if it exists) with numerator degree at
++ most nd and denominator degree at most dd
++ which matches the series s to order \spad{nd + dd}.
Implementation ==> add
n,m : NNI
u,v : PS
pa := PadeApproximants(R,PS,UP)
pade(n,m,u,v) ==
ans:=pade(n,m,u,v)$pa
ans case "failed" => ans
pt = 0 => ans
num := numer(ans::QF)
den := denom(ans::QF)
xpt : UP := monomial(1,1)-monomial(pt,0)
num := num(xpt)
den := den(xpt)
num/den
pade(n,m,u) == pade(n,m,u,1)
@
\section{package PADE PadeApproximants}
<<package PADE PadeApproximants>>=
)abbrev package PADE PadeApproximants
++ This package computes reliable Pad&ea. approximants using
++ a generalized Viskovatov continued fraction algorithm.
++ Authors: Burge, Hassner & Watt.
++ Date Created: April 1987
++ Date Last Updated: 12 April 1990
++ Keywords: Pade, series
++ Examples:
++ References:
++ "Pade Approximants, Part I: Basic Theory", Baker & Graves-Morris.
PadeApproximants(R,PS,UP): Exports == Implementation where
R: Field -- IntegralDomain
PS: UnivariateTaylorSeriesCategory R
UP: UnivariatePolynomialCategory R
NNI ==> NonNegativeInteger
QF ==> Fraction UP
CF ==> ContinuedFraction UP
Exports ==> with
pade: (NNI,NNI,PS,PS) -> Union(QF,"failed")
++ pade(nd,dd,ns,ds)
++ computes the approximant as a quotient of polynomials
++ (if it exists) for arguments
++ nd (numerator degree of approximant),
++ dd (denominator degree of approximant),
++ ns (numerator series of function), and
++ ds (denominator series of function).
padecf: (NNI,NNI,PS,PS) -> Union(CF, "failed")
++ padecf(nd,dd,ns,ds)
++ computes the approximant as a continued fraction of
++ polynomials (if it exists) for arguments
++ nd (numerator degree of approximant),
++ dd (denominator degree of approximant),
++ ns (numerator series of function), and
++ ds (denominator series of function).
Implementation ==> add
-- The approximant is represented as
-- p0 + x**a1/(p1 + x**a2/(...))
PadeRep ==> Record(ais: List UP, degs: List NNI) -- #ais= #degs
PadeU ==> Union(PadeRep, "failed") -- #ais= #degs+1
constInner(up:UP):PadeU == [[up], []]
truncPoly(p:UP,n:NNI):UP ==
while n < degree p repeat p := reductum p
p
truncSeries(s:PS,n:NNI):UP ==
p: UP := 0
for i in 0..n repeat p := p + monomial(coefficient(s,i),i)
p
-- Assumes s starts with a<n>*x**n + ... and divides out x**n.
divOutDegree(s:PS,n:NNI):PS ==
for i in 1..n repeat s := quoByVar s
s
padeNormalize: (NNI,NNI,PS,PS) -> PadeU
padeInner: (NNI,NNI,PS,PS) -> PadeU
pade(l,m,gps,dps) ==
(ad := padeNormalize(l,m,gps,dps)) case "failed" => "failed"
plist := ad.ais; dlist := ad.degs
approx := first(plist) :: QF
for d in dlist for p in rest plist repeat
approx := p::QF + (monomial(1,d)$UP :: QF)/approx
approx
padecf(l,m,gps,dps) ==
(ad := padeNormalize(l,m,gps,dps)) case "failed" => "failed"
alist := reverse(ad.ais)
blist := [monomial(1,d)$UP for d in reverse ad.degs]
continuedFraction(first(alist),_
blist::Stream UP,(rest alist) :: Stream UP)
padeNormalize(l,m,gps,dps) ==
zero? dps => "failed"
zero? gps => constInner 0
-- Normalize so numerator or denominator has constant term.
ldeg:= min(order dps,order gps)
if ldeg > 0 then
dps := divOutDegree(dps,ldeg)
gps := divOutDegree(gps,ldeg)
padeInner(l,m,gps,dps)
padeInner(l, m, gps, dps) ==
zero? coefficient(gps,0) and zero? coefficient(dps,0) =>
error "Pade' problem not normalized."
plist: List UP := nil()
alist: List NNI := nil()
-- Ensure denom has constant term.
if zero? coefficient(dps,0) then
-- g/d = 0 + z**0/(d/g)
(gps,dps) := (dps,gps)
(l,m) := (m,l)
plist := concat(0,plist)
alist := concat(0,alist)
-- Ensure l >= m, maintaining coef(dps,0)~=0.
if l < m then
-- (a<n>*x**n + a<n+1>*x**n+1 + ...)/b
-- = x**n/b + (a<n> + a<n+1>*x + ...)/b
alpha := order gps
if alpha > l then return "failed"
gps := divOutDegree(gps, alpha)
(l,m) := (m,(l-alpha) :: NNI)
(gps,dps) := (dps,gps)
plist := concat(0,plist)
alist := concat(alpha,alist)
degbd: NNI := l + m + 1
g := truncSeries(gps,degbd)
d := truncSeries(dps,degbd)
for j in 0.. repeat
-- Normalize d so constant coefs cancel. (B&G-M is wrong)
d0 := coefficient(d,0)
d := (1/d0) * d; g := (1/d0) * g
p : UP := 0; s := g
if negative?(l-m+1) then error "Internal pade error"
degbd := (l-m+1) :: NNI
for k in 1..degbd repeat
pk := coefficient(s,0)
p := p + monomial(pk,(k-1) :: NNI)
s := s - pk*d
s := (s exquo monomial(1,1)) :: UP
plist := concat(p,plist)
s = 0 => return [plist,alist]
alpha := minimumDegree(s) + degbd
alpha > l + m => return [plist,alist]
alpha > l => return "failed"
alist := concat(alpha,alist)
h := (s exquo monomial(1,minimumDegree s)) :: UP
degbd := (l + m - alpha) :: NNI
g := truncPoly(d,degbd)
d := truncPoly(h,degbd)
(l,m) := (m,(l-alpha) :: NNI)
@
\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>>
<<package PADEPAC PadeApproximantPackage>>
<<package PADE PadeApproximants>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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