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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input pdecomp0.as}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\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.
@
<<*>>=
#pile
#include "axiom.as"
<<license>>
--% Polynomial composition and decomposition functions
-- If f = g o h then g = leftFactor(f, h) & h = rightFactor(f, g)
-- SMW Dec 86
--% PolynomialComposition
--)abbrev package PCOMP PolynomialComposition
--)abbrev package PDECOMP PolynomialDecomposition
PolynomialComposition(UP: UnivariatePolynomialCategory(R), R: Ring): with
compose: (UP, UP) -> UP
== add
compose(g:UP, h:UP):UP ==
r: UP := 0
while g ~= 0 repeat
r := leadingCoefficient(g)*h**degree(g) + r
g := reductum g
r
-- Ref: Kozen and Landau, Cornell University TR 86-773
--% PolynomialDecomposition
PolynomialDecomposition(UP:UPC F, F:Field): PDcat == PDdef where
UPC ==> UnivariatePolynomialCategory
NNI ==> NonNegativeInteger
LR ==> Record(left: UP, right: UP)
PDcat ==> with
decompose: UP -> List UP
decompose: (UP, NNI, NNI) -> Union(value1:LR, failed:'failed')
leftFactor: (UP, UP) -> Union(value1:UP, failed:'failed')
rightFactorCandidate: (UP, NNI) -> UP
PDdef ==> add
import from F
import from LR
import from Union(value1:UP, failed:'failed')
import from Float
import from NNI
import from UniversalSegment NNI
import from Record(quotient:UP, remainder:UP);
leftFactor(f:UP, h:UP):Union(value1:UP, failed:'failed') ==
g: UP := 0
for i in 0.. while f ~= 0 repeat
fr := divide(f, h)
f := fr.quotient
r := fr.remainder
degree r > 0 => return [failed]
g := g + r * monomial(1, i)
[g]
decompose(f:UP, dg:NNI, dh:NNI):Union(value1:LR, failed:'failed') ==
df := degree f
dg*dh ~= df => [failed]
h := rightFactorCandidate(f, dh)
g:Union(value1:UP, failed:'failed') := leftFactor(f, h)
g case failed => [failed]
[[g.value1, h]]
decompose(f:UP):List UP ==
df := degree f
for dh in 2..df-1 | df rem dh = 0 repeat
h := rightFactorCandidate(f, dh)
g := leftFactor(f, h)
g case value1 => return
append(decompose(g.value1), decompose h)
[f]
rightFactorCandidate(f:UP, dh:NNI):UP ==
f := f / leadingCoefficient f
df := degree f
dg := df quo dh
h := monomial(1, dh)
for k in 1..dh repeat
hdg:= h**dg
c := (coefficient(f,df-k)-coefficient(hdg,df-k))/
(dg::Integer::F)
h := h + monomial(c, dh-k)
h - monomial(coefficient(h, 0), 0) -- drop constant term
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
