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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra ghensel.spad}
\author{Patrizia Gianni}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package GHENSEL GeneralHenselPackage}
<<package GHENSEL GeneralHenselPackage>>=
)abbrev package GHENSEL GeneralHenselPackage
++ Author : P.Gianni
++ General Hensel Lifting
++ Used for Factorization of bivariate polynomials over a finite field.
GeneralHenselPackage(RP,TP):C == T where
RP : EuclideanDomain
TP : UnivariatePolynomialCategory RP
PI ==> PositiveInteger
C == with
HenselLift: (TP,List(TP),RP,PI) -> Record(plist:List(TP), modulo:RP)
++ HenselLift(pol,lfacts,prime,bound) lifts lfacts,
++ that are the factors of pol mod prime,
++ to factors of pol mod prime**k > bound. No recombining is done .
completeHensel: (TP,List(TP),RP,PI) -> List TP
++ completeHensel(pol,lfact,prime,bound) lifts lfact,
++ the factorization mod prime of pol,
++ to the factorization mod prime**k>bound.
++ Factors are recombined on the way.
reduction : (TP,RP) -> TP
++ reduction(u,pol) computes the symmetric reduction of u mod pol
T == add
GenExEuclid: (List(FP),List(FP),FP) -> List(FP)
HenselLift1: (TP,List(TP),List(FP),List(FP),RP,RP,F) -> List(TP)
mQuo: (TP,RP) -> TP
reduceCoef(c:RP,p:RP):RP ==
zero? p => c
RP is Integer => symmetricRemainder(c,p)
c rem p
reduction(u:TP,p:RP):TP ==
zero? p => u
RP is Integer => map(symmetricRemainder(#1,p),u)
map(#1 rem p,u)
merge(p:RP,q:RP):Union(RP,"failed") ==
p = q => p
p = 0 => q
q = 0 => p
"failed"
modInverse(c:RP,p:RP):RP ==
(extendedEuclidean(c,p,1)::Record(coef1:RP,coef2:RP)).coef1
exactquo(u:TP,v:TP,p:RP):Union(TP,"failed") ==
invlcv:=modInverse(leadingCoefficient v,p)
r:=monicDivide(u,reduction(invlcv*v,p))
not zero? reduction(r.remainder,p) => "failed"
reduction(invlcv*r.quotient,p)
FP:=EuclideanModularRing(RP,TP,RP,reduction,merge,exactquo)
mQuo(poly:TP,n:RP) : TP == map(#1 quo n,poly)
GenExEuclid(fl:List FP,cl:List FP,rhs:FP) :List FP ==
[clp*rhs rem flp for clp in cl for flp in fl]
-- generate the possible factors
genFact(fln:List TP,factlist:List List TP) : List List TP ==
factlist=[] => [[pol] for pol in fln]
maxd := +/[degree f for f in fln] quo 2
auxfl:List List TP := []
for poly in fln while factlist~=[] repeat
factlist := [term for term in factlist | not member?(poly,term)]
dp := degree poly
for term in factlist repeat
(+/[degree f for f in term]) + dp > maxd => "next term"
auxfl := cons(cons(poly,term),auxfl)
auxfl
HenselLift1(poly:TP,fln:List TP,fl1:List FP,cl1:List FP,
prime:RP,Modulus:RP,cinv:RP):List TP ==
lcp := leadingCoefficient poly
rhs := reduce(mQuo(poly - lcp * */fln,Modulus),prime)
zero? rhs => fln
lcinv:=reduce(cinv::TP,prime)
vl := GenExEuclid(fl1,cl1,lcinv*rhs)
[flp + Modulus*(vlp::TP) for flp in fln for vlp in vl]
HenselLift(poly:TP,tl1:List TP,prime:RP,bound:PI) ==
-- convert tl1
constp:TP:=0
if degree first tl1 = 0 then
constp:=tl1.first
tl1 := rest tl1
fl1:=[reduce(ttl,prime) for ttl in tl1]
cl1 := multiEuclidean(fl1,1)::List FP
Modulus:=prime
fln :List TP := [ffl1::TP for ffl1 in fl1]
lcinv:RP:=retract((inv
(reduce((leadingCoefficient poly)::TP,prime)))::TP)
while euclideanSize(Modulus)<bound repeat
nfln:=HenselLift1(poly,fln,fl1,cl1,prime,Modulus,lcinv)
fln = nfln and zero?(err:=poly-*/fln) => leave "finished"
fln := nfln
Modulus := prime*Modulus
if not zero? constp then fln:=cons(constp,fln)
[fln,Modulus]
completeHensel(m:TP,tl1:List TP,prime:RP,bound:PI) ==
hlift:=HenselLift(m,tl1,prime,bound)
Modulus:RP:=hlift.modulo
fln:List TP:=hlift.plist
nm := degree m
u:Union(TP,"failed")
aux,auxl,finallist:List TP
auxfl,factlist:List List TP
factlist := []
dfn :NonNegativeInteger := nm
lcm1 := leadingCoefficient m
mm := lcm1*m
while positive? dfn and (factlist := genFact(fln,factlist))~=[] repeat
auxfl := []
while factlist~=[] repeat
auxl := factlist.first
factlist := factlist.rest
tc := reduceCoef((lcm1 * */[coefficient(poly,0)
for poly in auxl]), Modulus)
coefficient(mm,0) exquo tc case "failed" =>
auxfl := cons(auxl,auxfl)
pol := */[poly for poly in auxl]
poly :=reduction(lcm1*pol,Modulus)
u := mm exquo poly
u case "failed" => auxfl := cons(auxl,auxfl)
poly1: TP := primitivePart poly
m := mQuo((u::TP),leadingCoefficient poly1)
lcm1 := leadingCoefficient(m)
mm := lcm1*m
finallist := cons(poly1,finallist)
dfn := degree m
aux := []
for poly: local in fln repeat
not member?(poly,auxl) => aux := cons(poly,aux)
auxfl := [term for term in auxfl | not member?(poly,term)]
factlist := [term for term in factlist | not member?(poly,term)]
fln := aux
factlist := auxfl
if positive? dfn then finallist := cons(m,finallist)
finallist
@
\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 GHENSEL GeneralHenselPackage>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
