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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra galpolyu.spad}
\author{Frederic Lehobey}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package GALPOLYU GaloisGroupPolynomialUtilities}
<<package GALPOLYU GaloisGroupPolynomialUtilities>>=
)abbrev package GALPOLYU GaloisGroupPolynomialUtilities
++ Author: Frederic Lehobey
++ Date Created: 30 June 1994
++ Date Last Updated: 15 July 1994
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description: \spadtype{GaloisGroupPolynomialUtilities} provides useful
++ functions for univariate polynomials which should be added to
++ \spadtype{UnivariatePolynomialCategory} or to \spadtype{Factored}
++ (July 1994).
GaloisGroupPolynomialUtilities(R,UP): Exports == Implementation where
R : Ring
UP : UnivariatePolynomialCategory R
N ==> NonNegativeInteger
P ==> PositiveInteger
Exports ==> with
monic?: UP -> Boolean
++ monic?(p) tests if p is monic (i.e. leading coefficient equal to 1).
unvectorise: Vector R -> UP
++ unvectorise(v) returns the polynomial which has for coefficients the
++ entries of v in the increasing order.
reverse: UP -> UP
++ reverse(p) returns the reverse polynomial of p.
scaleRoots: (UP,R) -> UP
++ scaleRoots(p,c) returns the polynomial which has c times the roots
++ of p.
shiftRoots: (UP,R) -> UP
++ shiftRoots(p,c) returns the polynomial which has for roots c added
++ to the roots of p.
degreePartition: Factored UP -> Multiset N
++ degreePartition(f) returns the degree partition (i.e. the multiset
++ of the degrees of the irreducible factors) of
++ the polynomial f.
factorOfDegree: (P, Factored UP) -> UP
++ factorOfDegree(d,f) returns a factor of degree d of the factored
++ polynomial f. Such a factor shall exist.
factorsOfDegree: (P, Factored UP) -> List UP
++ factorsOfDegree(d,f) returns the factors of degree d of the factored
++ polynomial f.
Implementation ==> add
import Factored UP
factorsOfDegree(d:P,r:Factored UP):List UP ==
lfact : List UP := empty()
for fr in factors r | degree(fr.factor)=(d::N) repeat
for i in 1..fr.exponent repeat
lfact := cons(fr.factor,lfact)
lfact
factorOfDegree(d:P,r:Factored UP):UP ==
factor : UP := 0
for i in 1..numberOfFactors r repeat
factor := nthFactor(r,i)
if degree(factor)=(d::N) then return factor
error "factorOfDegree: Bad arguments"
degreePartition(r:Factored UP):Multiset N ==
multiset([ degree(nthFactor(r,i)) for i in 1..numberOfFactors r ])
monic?(p:UP):Boolean == one? leadingCoefficient p
unvectorise(v:Vector R):UP ==
p : UP := 0
for i in 1..#v repeat p := p + monomial(v(i),(i-1)::N)
p
reverse(p:UP):UP ==
r : UP := 0
n := degree(p)
for i in 0..n repeat r := r + monomial(coefficient(p,(n-i)::N),i)
r
scaleRoots(p:UP,c:R):UP ==
one? c => p
n := degree p
zero? c => monomial(leadingCoefficient p,n)
r : UP := 0
mc : R := 1
for i in n..0 by -1 repeat
r := r + monomial(mc*coefficient(p,i),i)
mc := mc*c
r
import UnivariatePolynomialCategoryFunctions2(R,UP,UP,
SparseUnivariatePolynomial UP)
shiftRoots(p:UP,c:R):UP == elt(map(coerce,p),monomial(1,1)$UP-c::UP)::UP
@
\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 GALPOLYU GaloisGroupPolynomialUtilities>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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