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--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.
)abbrev package INNMFACT InnerMultFact
++ Author: P. Gianni
++ Date Created: 1983
++ Date Last Updated: Sept. 1990
++ Additional Comments: JHD Aug 1997
++ Basic Functions:
++ Related Constructors: MultivariateFactorize, AlgebraicMultFact
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This is an inner package for factoring multivariate polynomials
++ over various coefficient domains in characteristic 0.
++ The univariate factor operation is passed as a parameter.
++ Multivariate hensel lifting is used to lift the univariate
++ factorization
-- Both exposed functions call mFactor. This deals with issues such as
-- monomial factors, contents, square-freeness etc., then calls intfact.
-- This uses intChoose to find a "good" evaluation and factorise the
-- corresponding univariate, and then uses MultivariateLifting to find
-- the multivariate factors.
InnerMultFact(OV,E,R,P) : C == T
where
R : Join(EuclideanDomain, CharacteristicZero)
-- with factor on R[x]
OV : OrderedSet
E : OrderedAbelianMonoidSup
P : PolynomialCategory(R,E,OV)
BP ==> SparseUnivariatePolynomial R
UFactor ==> (BP -> Factored BP)
Z ==> Integer
MParFact ==> Record(irr:P,pow:Z)
USP ==> SparseUnivariatePolynomial P
SUParFact ==> Record(irr:USP,pow:Z)
SUPFinalFact ==> Record(contp:R,factors:List SUParFact)
MFinalFact ==> Record(contp:R,factors:List MParFact)
-- contp = content,
-- factors = List of irreducible factors with exponent
L ==> List
C == with
factor : (P,UFactor) -> Factored P
++ factor(p,ufact) factors the multivariate polynomial p
++ by specializing variables and calling the univariate
++ factorizer ufact.
factor : (USP,UFactor) -> Factored USP
++ factor(p,ufact) factors the multivariate polynomial p
++ by specializing variables and calling the univariate
++ factorizer ufact. p is represented as a univariate
++ polynomial with multivariate coefficients.
T == add
NNI ==> NonNegativeInteger
LeadFact ==> Record(polfac:L P,correct:R,corrfact:L BP)
ContPrim ==> Record(cont:P,prim:P)
ParFact ==> Record(irr:BP,pow:Z)
FinalFact ==> Record(contp:R,factors:L ParFact)
NewOrd ==> Record(npol:USP,nvar:L OV,newdeg:L NNI)
pmod:R := (prevPrime(2**26)$IntegerPrimesPackage(Integer))::R
import GenExEuclid(R,BP)
import MultivariateLifting(E,OV,R,P)
import FactoringUtilities(E,OV,R,P)
import LeadingCoefDetermination(OV,E,R,P)
Valuf ==> Record(inval:L L R,unvfact:L BP,lu:R,complead:L R)
UPCF2 ==> UnivariatePolynomialCategoryFunctions2
---- Local Functions ----
mFactor : (P,UFactor) -> MFinalFact
supFactor : (USP,UFactor) -> SUPFinalFact
mfconst : (USP,L OV,L NNI,UFactor) -> L USP
mfpol : (USP,L OV,L NNI,UFactor) -> L USP
monicMfpol: (USP,L OV,L NNI,UFactor) -> L USP
varChoose : (P,L OV,L NNI) -> NewOrd
simplify : (P,L OV,L NNI,UFactor) -> MFinalFact
intChoose : (USP,L OV,R,L P,L L R,UFactor) -> Union(Valuf,"failed")
intfact : (USP,L OV,L NNI,MFinalFact,L L R,UFactor) -> L USP
pretest : (P,NNI,L OV,L R) -> FinalFact
checkzero : (USP,BP) -> Boolean
localNorm : L BP -> Z
convertPUP(lfg:MFinalFact): SUPFinalFact ==
[lfg.contp,[[lff.irr ::USP,lff.pow]$SUParFact
for lff in lfg.factors]]$SUPFinalFact
-- intermediate routine if an SUP was passed in.
supFactor(um:USP,ufactor:UFactor) : SUPFinalFact ==
ground?(um) => convertPUP(mFactor(ground um,ufactor))
lvar:L OV:= "setUnion"/[variables cf for cf in coefficients um]
empty? lvar => -- the polynomial is univariate
umv:= map(ground,um)$UPCF2(P,USP,R,BP)
lfact:=ufactor umv
[retract unit lfact,[[map(coerce,ff.factor)$UPCF2(R,BP,P,USP),
ff.exponent] for ff in factors lfact]]$SUPFinalFact
lcont:P
lf:L USP
flead : SUPFinalFact:=[0,empty()]$SUPFinalFact
factorlist:L SUParFact :=empty()
mdeg :=minimumDegree um ---- is the Mindeg > 0? ----
if positive? mdeg then
f1:USP:=monomial(1,mdeg)
um:=(um exquo f1)::USP
factorlist:=cons([monomial(1,1),mdeg],factorlist)
if degree um=0 then return
lfg:=convertPUP mFactor(ground um, ufactor)
[lfg.contp,append(factorlist,lfg.factors)]
uum:=unitNormal um
um :=uum.canonical
sqfacs := squareFree(um)$MultivariateSquareFree(E,OV,R,P)
lcont := ground(uum.unit * unit sqfacs)
---- Factorize the content ----
flead:=convertPUP mFactor(lcont,ufactor)
factorlist:=append(flead.factors,factorlist)
---- Make the polynomial square-free ----
sqqfact:=factors sqfacs
--- Factorize the primitive square-free terms ---
for fact in sqqfact repeat
ffactor:USP:=fact.factor
ffexp:=fact.exponent
zero? degree ffactor =>
lfg:=mFactor(ground ffactor,ufactor)
lcont:=lfg.contp * lcont
factorlist := append(factorlist,
[[lff.irr ::USP,lff.pow * ffexp]$SUParFact
for lff in lfg.factors])
coefs := coefficients ffactor
ldeg:= ["max"/[degree(fc,xx) for fc in coefs] for xx in lvar]
lf :=
ground?(leadingCoefficient ffactor) =>
mfconst(ffactor,lvar,ldeg,ufactor)
mfpol(ffactor,lvar,ldeg,ufactor)
auxfl:=[[lfp,ffexp]$SUParFact for lfp in lf]
factorlist:=append(factorlist,auxfl)
lcfacs := */[leadingCoefficient leadingCoefficient(f.irr)**((f.pow)::NNI)
for f in factorlist]
[(leadingCoefficient leadingCoefficient(um) exquo lcfacs)::R,
factorlist]$SUPFinalFact
factor(um:USP,ufactor:UFactor):Factored USP ==
flist := supFactor(um,ufactor)
(flist.contp):: P :: USP *
(*/[primeFactor(u.irr,u.pow) for u in flist.factors])
checkzero(u:USP,um:BP) : Boolean ==
u=0 => um =0
um = 0 => false
degree u = degree um => checkzero(reductum u, reductum um)
false
--- Choose the variable of less degree ---
varChoose(m:P,lvar:L OV,ldeg:L NNI) : NewOrd ==
k:="min"/[d for d in ldeg]
k=degree(m,first lvar) =>
[univariate(m,first lvar),lvar,ldeg]$NewOrd
i:=position(k,ldeg)
x:OV:=lvar.i
ldeg:=cons(k,delete(ldeg,i))
lvar:=cons(x,delete(lvar,i))
[univariate(m,x),lvar,ldeg]$NewOrd
localNorm(lum: L BP): Z ==
R is AlgebraicNumber =>
"max"/[numberOfMonomials ff for ff in lum]
"max"/[+/[euclideanSize cc for i in 0..degree ff|
not zero? (cc:= coefficient(ff,i))] for ff in lum]
--- Choose the integer to reduce to univariate case ---
intChoose(um:USP,lvar:L OV,clc:R,plist:L P,ltry:L L R,
ufactor:UFactor) : Union(Valuf,"failed") ==
-- declarations
degum:NNI := degree um
nvar1:=#lvar
range:NNI:=5
unifact:L BP
ctf1 : R := 1
testp:Boolean := -- polynomial leading coefficient
empty? plist => false
true
leadcomp,leadcomp1 : L R
leadcomp:=leadcomp1:=empty()
nfatt:NNI := degum+1
lffc:R:=1
lffc1:=lffc
newunifact : L BP:=empty()
leadtest:=true --- the lc test with polCase has to be performed
int:L R:=empty()
-- New sets of integers are chosen to reduce the multivariate problem to
-- a univariate one, until we find twice the
-- same (and minimal) number of "univariate" factors:
-- the set smaller in modulo is chosen.
-- Note that there is no guarantee that this is the truth:
-- merely the closest approximation we have found!
while true repeat
testp and #ltry>10 => return "failed"
lval := [ ran(range) for i in 1..nvar1]
member?(lval,ltry) => range:=2*range
ltry := cons(lval,ltry)
leadcomp1:=[retract eval(pol,lvar,lval) for pol in plist]
testp and or/[unit? epl for epl in leadcomp1] => range:=2*range
newm:BP:=completeEval(um,lvar,lval)
degum ~= degree newm or not zero? minimumDegree newm => range:=2*range
lffc1:=content newm
newm:=(newm exquo lffc1)::BP
testp and leadtest and not polCase(lffc1*clc,#plist,leadcomp1)
=> range:=2*range
not zero? degree(gcd [newm,differentiate(newm)]) => range:=2*range
luniv:=ufactor(newm)
lunivf:= factors luniv
lffc1:R:=retract(unit luniv)@R * lffc1
nf:= #lunivf
nf=0 or nf>nfatt => "next values" --- pretest failed ---
--- the univariate polynomial is irreducible ---
if nf=1 then leave (unifact:=[newm])
-- the new integer give the same number of factors
nfatt = nf =>
-- if this is the first univariate factorization with polCase=true
-- or if the last factorization has smaller norm and satisfies
-- polCase
if leadtest or
((localNorm unifact > localNorm [ff.factor for ff in lunivf])
and (not testp or polCase(lffc1*clc,#plist,leadcomp1))) then
unifact:=[uf.factor for uf in lunivf]
int:=lval
lffc:=lffc1
if testp then leadcomp:=leadcomp1
leave "foundit"
-- the first univariate factorization, inizialize
nfatt > degum =>
unifact:=[uf.factor for uf in lunivf]
lffc:=lffc1
if testp then leadcomp:=leadcomp1
int:=lval
leadtest := false
nfatt := nf
nfatt>nf => -- for the previous values there were more factors
if testp then leadtest:= not polCase(lffc*clc,#plist,leadcomp)
else leadtest:= false
-- if polCase=true we can consider the univariate decomposition
if not leadtest then
unifact:=[uf.factor for uf in lunivf]
lffc:=lffc1
if testp then leadcomp:=leadcomp1
int:=lval
nfatt := nf
[cons(int,ltry),unifact,lffc,leadcomp]$Valuf
---- The polynomial has mindeg>0 ----
simplify(m:P,lvar:L OV,lmdeg:L NNI,ufactor:UFactor):MFinalFact ==
factorlist:L MParFact:=[]
pol1:P:= 1$P
for x in lvar repeat
i := lmdeg.(position(x,lvar))
i=0 => "next value"
pol1:=pol1*monomial(1$P,x,i)
factorlist:=cons([x::P,i]$MParFact,factorlist)
m := (m exquo pol1)::P
ground? m => [retract m,factorlist]$MFinalFact
flead:=mFactor(m,ufactor)
flead.factors:=append(factorlist,flead.factors)
flead
-- This is the key internal function
-- We now know that the polynomial is square-free etc.,
-- We use intChoose to find a set of integer values to reduce the
-- problem to univariate (and for efficiency, intChoose returns
-- the univariate factors).
-- In the case of a polynomial leading coefficient, we check that this
-- is consistent with leading coefficient determination (else try again)
-- We then lift the univariate factors to multivariate factors, and
-- return the result
intfact(um:USP,lvar: L OV,ldeg:L NNI,tleadpol:MFinalFact,
ltry:L L R,ufactor:UFactor) : L USP ==
polcase:Boolean:=(not empty? tleadpol.factors)
vfchoo:Valuf:=
polcase =>
leadpol:L P:=[ff.irr for ff in tleadpol.factors]
check:=intChoose(um,lvar,tleadpol.contp,leadpol,ltry,ufactor)
check case "failed" => return monicMfpol(um,lvar,ldeg,ufactor)
check::Valuf
intChoose(um,lvar,1,empty(),empty(),ufactor)::Valuf
unifact:List BP := vfchoo.unvfact
nfact:NNI := #unifact
nfact=1 => [um]
ltry:L L R:= vfchoo.inval
lval:L R:=first ltry
dd:= vfchoo.lu
leadval:L R:=empty()
lpol:List P:=empty()
if polcase then
leadval := vfchoo.complead
distf := distFact(vfchoo.lu,unifact,tleadpol,leadval,lvar,lval)
distf case "failed" =>
return intfact(um,lvar,ldeg,tleadpol,ltry,ufactor)
dist := distf :: LeadFact
-- check the factorization of leading coefficient
lpol:= dist.polfac
dd := dist.correct
unifact:=dist.corrfact
if not one? dd then
-- if polcase then lpol := [unitCanonical lp for lp in lpol]
-- dd:=unitCanonical(dd)
unifact := [dd * unif for unif in unifact]
umd := unitNormal(dd).unit * ((dd**(nfact-1)::NNI)::P)*um
else umd := um
(ffin:=lifting(umd,lvar,unifact,lval,lpol,ldeg,pmod))
case "failed" => intfact(um,lvar,ldeg,tleadpol,ltry,ufactor)
factfin: L USP:=ffin :: L USP
if not one? dd then
factfin:=[primitivePart ff for ff in factfin]
factfin
---- m square-free,primitive,lc constant ----
mfconst(um:USP,lvar:L OV,ldeg:L NNI,ufactor:UFactor):L USP ==
factfin:L USP:=empty()
empty? lvar =>
lum:=factors ufactor(map(ground,um)$UPCF2(P,USP,R,BP))
[map(coerce,uf.factor)$UPCF2(R,BP,P,USP) for uf in lum]
intfact(um,lvar,ldeg,[0,empty()]$MFinalFact,empty(),ufactor)
monicize(um:USP,c:P):USP ==
n:=degree(um)
ans:USP := monomial(1,n)
n:=(n-1)::NonNegativeInteger
prod:P:=1
while (um:=reductum(um)) ~= 0 repeat
i := degree um
lc := leadingCoefficient um
prod := prod * c ** (n-(n:=i))::NonNegativeInteger
ans := ans + monomial(prod*lc, i)
ans
unmonicize(m:USP,c:P):USP == primitivePart m(monomial(c,1))
--- m is square-free,primitive,lc is a polynomial ---
monicMfpol(um:USP,lvar:L OV,ldeg:L NNI,ufactor:UFactor):L USP ==
l := leadingCoefficient um
monpol := monicize(um,l)
nldeg := degree(monpol,lvar)
map(unmonicize(#1,l),
mfconst(monpol,lvar,nldeg,ufactor))
mfpol(um:USP,lvar:L OV,ldeg:L NNI,ufactor:UFactor):L USP ==
R has Field =>
monicMfpol(um,lvar,ldeg,ufactor)
tleadpol:=mFactor(leadingCoefficient um,ufactor)
intfact(um,lvar,ldeg,tleadpol,[],ufactor)
mFactor(m:P,ufactor:UFactor) : MFinalFact ==
ground?(m) => [retract(m),empty()]$MFinalFact
lvar:L OV:= variables m
lcont:P
lf:L USP
flead : MFinalFact:=[0,empty()]$MFinalFact
factorlist:L MParFact :=empty()
lmdeg :=minimumDegree(m,lvar) ---- is the Mindeg > 0? ----
or/[positive? n for n in lmdeg] => simplify(m,lvar,lmdeg,ufactor)
sqfacs := squareFree m
lcont := unit sqfacs
---- Factorize the content ----
if ground? lcont then flead.contp:=retract lcont
else flead:=mFactor(lcont,ufactor)
factorlist:=flead.factors
---- Make the polynomial square-free ----
sqqfact:=factors sqfacs
--- Factorize the primitive square-free terms ---
for fact in sqqfact repeat
ffactor:P:=fact.factor
ffexp := fact.exponent
lvar := variables ffactor
x:OV :=lvar.first
ldeg:=degree(ffactor,lvar)
--- Is the polynomial linear in one of the variables ? ---
member?(1,ldeg) =>
x:OV:=lvar.position(1,ldeg)
lcont:= gcd coefficients(univariate(ffactor,x))
ffactor:=(ffactor exquo lcont)::P
factorlist:=cons([ffactor,ffexp]$MParFact,factorlist)
for lcterm in mFactor(lcont,ufactor).factors repeat
factorlist:=cons([lcterm.irr,lcterm.pow * ffexp], factorlist)
varch:=varChoose(ffactor,lvar,ldeg)
um:=varch.npol
x:=lvar.first
ldeg:=ldeg.rest
lvar := lvar.rest
if varch.nvar.first ~= x then
lvar:= varch.nvar
x := lvar.first
lvar := lvar.rest
pc:= gcd coefficients um
if not one? pc then
um:=(um exquo pc)::USP
ffactor:=multivariate(um,x)
for lcterm in mFactor(pc,ufactor).factors repeat
factorlist:=cons([lcterm.irr,lcterm.pow*ffexp],factorlist)
ldeg:=degree(ffactor,lvar)
um := unitCanonical um
if ground?(leadingCoefficient um) then
lf:= mfconst(um,lvar,ldeg,ufactor)
else lf:=mfpol(um,lvar,ldeg,ufactor)
auxfl:=[[unitCanonical multivariate(lfp,x),ffexp]$MParFact for lfp in lf]
factorlist:=append(factorlist,auxfl)
lcfacs := */[leadingCoefficient(f.irr)**((f.pow)::NNI) for f in factorlist]
[(leadingCoefficient(m) exquo lcfacs):: R,factorlist]$MFinalFact
factor(m:P,ufactor:UFactor):Factored P ==
flist := mFactor(m,ufactor)
(flist.contp):: P *
(*/[primeFactor(u.irr,u.pow) for u in flist.factors])
)abbrev package MULTFACT MultivariateFactorize
++ Author: P. Gianni
++ Date Created: 1983
++ Date Last Updated: Sept. 1990
++ Basic Functions:
++ Related Constructors: MultFiniteFactorize, AlgebraicMultFact, UnivariateFactorize
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This is the top level package for doing multivariate factorization
++ over basic domains like \spadtype{Integer} or \spadtype{Fraction Integer}.
MultivariateFactorize(OV,E,R,P) : C == T
where
R : Join(EuclideanDomain, CharacteristicZero)
-- with factor on R[x]
OV : OrderedSet
E : OrderedAbelianMonoidSup
P : PolynomialCategory(R,E,OV)
Z ==> Integer
MParFact ==> Record(irr:P,pow:Z)
USP ==> SparseUnivariatePolynomial P
SUParFact ==> Record(irr:USP,pow:Z)
SUPFinalFact ==> Record(contp:R,factors:List SUParFact)
MFinalFact ==> Record(contp:R,factors:List MParFact)
-- contp = content,
-- factors = List of irreducible factors with exponent
L ==> List
C == with
factor : P -> Factored P
++ factor(p) factors the multivariate polynomial p over its coefficient
++ domain
factor : USP -> Factored USP
++ factor(p) factors the multivariate polynomial p over its coefficient
++ domain where p is represented as a univariate polynomial with
++ multivariate coefficients
T == add
factor(p:P) : Factored P ==
R is Fraction Integer =>
factor(p)$MRationalFactorize(E,OV,Integer,P)
R is Fraction Complex Integer =>
factor(p)$MRationalFactorize(E,OV,Complex Integer,P)
R is Fraction Polynomial Integer and OV has convert: % -> Symbol =>
factor(p)$MPolyCatRationalFunctionFactorizer(E,OV,Integer,P)
factor(p,factor$GenUFactorize(R))$InnerMultFact(OV,E,R,P)
factor(up:USP) : Factored USP ==
factor(up,factor$GenUFactorize(R))$InnerMultFact(OV,E,R,P)
)abbrev package ALGMFACT AlgebraicMultFact
++ Author: P. Gianni
++ Date Created: 1990
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package factors multivariate polynomials over the
++ domain of \spadtype{AlgebraicNumber} by allowing the user
++ to specify a list of algebraic numbers generating the particular
++ extension to factor over.
AlgebraicMultFact(OV,E,P) : C == T
where
AN ==> AlgebraicNumber
OV : OrderedSet
E : OrderedAbelianMonoidSup
P : PolynomialCategory(AN,E,OV)
BP ==> SparseUnivariatePolynomial AN
Z ==> Integer
MParFact ==> Record(irr:P,pow:Z)
USP ==> SparseUnivariatePolynomial P
SUParFact ==> Record(irr:USP,pow:Z)
SUPFinalFact ==> Record(contp:R,factors:List SUParFact)
MFinalFact ==> Record(contp:R,factors:List MParFact)
-- contp = content,
-- factors = List of irreducible factors with exponent
L ==> List
C == with
factor : (P,L AN) -> Factored P
++ factor(p,lan) factors the polynomial p over the extension
++ generated by the algebraic numbers given by the list lan.
factor : (USP,L AN) -> Factored USP
++ factor(p,lan) factors the polynomial p over the extension
++ generated by the algebraic numbers given by the list lan.
++ p is presented as a univariate polynomial with multivariate
++ coefficients.
T == add
AF := AlgFactor(BP)
factor(p:P,lalg:L AN) : Factored P ==
factor(p,factor(#1,lalg)$AF)$InnerMultFact(OV,E,AN,P)
factor(up:USP,lalg:L AN) : Factored USP ==
factor(up,factor(#1,lalg)$AF)$InnerMultFact(OV,E,AN,P)
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