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authordos-reis <gdr@axiomatics.org>2007-08-14 05:14:52 +0000
committerdos-reis <gdr@axiomatics.org>2007-08-14 05:14:52 +0000
commitab8cc85adde879fb963c94d15675783f2cf4b183 (patch)
treec202482327f474583b750b2c45dedfc4e4312b1d /src/algebra/pade.spad.pamphlet
downloadopen-axiom-ab8cc85adde879fb963c94d15675783f2cf4b183.tar.gz
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+\documentclass{article}
+\usepackage{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 l-m+1 < 0 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}