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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra leadcdet.spad}
\author{Patrizia Gianni}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package LEADCDET LeadingCoefDetermination}
<<package LEADCDET LeadingCoefDetermination>>=
)abbrev package LEADCDET LeadingCoefDetermination
++ Author : P.Gianni, May 1990
++ Description:
++ Package for leading coefficient determination in the lifting step.
++ Package working for every R euclidean with property "F".
LeadingCoefDetermination(OV,E,Z,P) : C == T
where
OV : OrderedSet
E : OrderedAbelianMonoidSup
Z : EuclideanDomain
BP ==> SparseUnivariatePolynomial Z
P : PolynomialCategory(Z,E,OV)
NNI ==> NonNegativeInteger
LeadFact ==> Record(polfac:List(P),correct:Z,corrfact:List(BP))
ParFact ==> Record(irr:P,pow:Integer)
FinalFact ==> Record(contp:Z,factors:List(ParFact))
C == with
polCase : (Z,NNI,List(Z)) -> Boolean
++ polCase(contprod, numFacts, evallcs), where contprod is the
++ product of the content of the leading coefficient of
++ the polynomial to be factored with the content of the
++ evaluated polynomial, numFacts is the number of factors
++ of the leadingCoefficient, and evallcs is the list of
++ the evaluated factors of the leadingCoefficient, returns
++ true if the factors of the leading Coefficient can be
++ distributed with this valuation.
distFact : (Z,List(BP),FinalFact,List(Z),List(OV),List(Z)) ->
Union(LeadFact,"failed")
++ distFact(contm,unilist,plead,vl,lvar,lval), where contm is
++ the content of the evaluated polynomial, unilist is the list
++ of factors of the evaluated polynomial, plead is the complete
++ factorization of the leading coefficient, vl is the list
++ of factors of the leading coefficient evaluated, lvar is the
++ list of variables, lval is the list of values, returns a record
++ giving the list of leading coefficients to impose on the univariate
++ factors,
T == add
distribute: (Z,List(BP),List(P),List(Z),List(OV),List(Z)) -> LeadFact
checkpow : (Z,Z) -> NNI
polCase(d:Z,nk:NNI,lval:List(Z)):Boolean ==
-- d is the product of the content lc m (case polynomial)
-- and the cont of the polynomial evaluated
q:Z
distlist:List(Z) := [d]
for i in 1..nk repeat
q := unitNormal(lval.i).canonical
for j in 0..(i-1)::NNI repeat
y := distlist.((i-j)::NNI)
while not one? y repeat
y := gcd(y,q)
q := q quo y
if q=1 then return false
distlist := append(distlist,[q])
true
checkpow(a:Z,b:Z) : NonNegativeInteger ==
qt: Union(Z,"failed")
for i in 0.. repeat
qt:= b exquo a
if qt case "failed" then return i
b:=qt::Z
distribute(contm:Z,unilist:List(BP),pl:List(P),vl:List(Z),
lvar:List(OV),lval:List(Z)): LeadFact ==
d,lcp : Z
nf:NNI:=#unilist
for i in 1..nf repeat
lcp := leadingCoefficient (unilist.i)
d:= gcd(lcp,vl.i)
pl.i := (lcp quo d)*pl.i
d := vl.i quo d
unilist.i := d*unilist.i
contm := contm quo d
if not one? contm then for i in 1..nf repeat pl.i := contm*pl.i
[pl,contm,unilist]$LeadFact
distFact(contm:Z,unilist:List(BP),plead:FinalFact,
vl:List(Z),lvar:List(OV),lval:List(Z)):Union(LeadFact,"failed") ==
h:NonNegativeInteger
c,d : Z
lpol:List(P):=[]
lexp:List(Integer):=[]
nf:NNI := #unilist
vl := reverse vl --lpol and vl reversed so test from right to left
for fpl in plead.factors repeat
lpol:=[fpl.irr,:lpol]
lexp:=[fpl.pow,:lexp]
vlp:List(Z):= [1$Z for i in 1..nf]
aux : List(P) := [1$P for i in 1..nf]
for i in 1..nf repeat
c := contm*leadingCoefficient unilist.i
c=1 or c=-1 => "next i"
for k in 1..(# lpol) repeat
lexp.k=0 => "next factor"
h:= checkpow(vl.k,c)
if not zero? h then
if h>lexp.k then return "failed"
lexp.k:=lexp.k-h
aux.i := aux.i*(lpol.k ** h)
d:= vl.k**h
vlp.i:= vlp.i*d
c:= c quo d
if contm=1 then vlp.i:=c
for k in 1..(# lpol) repeat if lexp.k ~= 0 then return "failed"
contm =1 => [[vlp.i*aux.i for i in 1..nf],1,unilist]$LeadFact
distribute(contm,unilist,aux,vlp,lvar,lval)
@
\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 LEADCDET LeadingCoefDetermination>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
