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)abbrev package INMODGCD InnerModularGcd
++ Author: J.H. Davenport and P. Gianni
++ Date Created: July 1990
++ Date Last Updated: November 1991
++ Description:
++ This file contains the functions for modular gcd algorithm
++ for univariate polynomials with coefficients in a
++ non-trivial euclidean domain (i.e. not a field).
++ The package parametrised by the coefficient domain,
++ the polynomial domain, a prime,
++ and a function for choosing the next prime
Z ==> Integer
NNI ==> NonNegativeInteger
InnerModularGcd(R,BP,pMod,nextMod):C == T
where
R : EuclideanDomain
BP : UnivariatePolynomialCategory(R)
pMod : R
nextMod : (R,NNI) -> R
C == with
modularGcdPrimitive : List BP -> BP
++ modularGcdPrimitive(f1,f2) computes the gcd of the two polynomials
++ f1 and f2 by modular methods.
modularGcd : List BP -> BP
++ modularGcd(listf) computes the gcd of the list of polynomials
++ listf by modular methods.
reduction : (BP,R) -> BP
++ reduction(f,p) reduces the coefficients of the polynomial f
++ modulo the prime p.
T == add
-- local functions --
height : BP -> NNI
mbound : (BP,BP) -> NNI
modGcdPrimitive : (BP,BP) -> BP
test : (BP,BP,BP) -> Boolean
merge : (R,R) -> Union(R,"failed")
modInverse : (R,R) -> R
exactquo : (BP,BP,R) -> Union(BP,"failed")
constNotZero : BP -> Boolean
constcase : (List NNI ,List BP ) -> BP
lincase : (List NNI ,List BP ) -> BP
if R has IntegerNumberSystem then
reduction(u:BP,p:R):BP ==
p = 0 => u
map(symmetricRemainder(#1,p),u)
else
reduction(u:BP,p:R):BP ==
p = 0 => u
map(#1 rem p,u)
FP:=EuclideanModularRing(R,BP,R,reduction,merge,exactquo)
zeroChar : Boolean := R has CharacteristicZero
-- exported functions --
-- modular Gcd for a list of primitive polynomials
modularGcdPrimitive(listf : List BP) :BP ==
empty? listf => 0$BP
g := first listf
for f in rest listf | not zero? f while positive? degree g repeat
g:=modGcdPrimitive(g,f)
g
-- gcd for univariate polynomials
modularGcd(listf : List BP): BP ==
listf:=remove!(0$BP,listf)
empty? listf => 0$BP
# listf = 1 => first listf
minpol:=1$BP
-- extract a monomial gcd
mdeg:= "min"/[minimumDegree f for f in listf]
if positive? mdeg then
minpol1:= monomial(1,mdeg)
listf:= [(f exquo minpol1)::BP for f in listf]
minpol:=minpol*minpol1
listdeg:=[degree f for f in listf ]
-- make the polynomials primitive
listCont := [content f for f in listf]
contgcd:= gcd listCont
-- make the polynomials primitive
listf :=[(f exquo cf)::BP for f in listf for cf in listCont]
minpol:=contgcd*minpol
ans:BP :=
--one polynomial is constant
member?(1,listf) => 1
--one polynomial is linear
member?(1,listdeg) => lincase(listdeg,listf)
modularGcdPrimitive listf
minpol*ans
-- local functions --
--one polynomial is linear, remark that they are primitive
lincase(listdeg:List NNI ,listf:List BP ): BP ==
n:= position(1,listdeg)
g:=listf.n
for f in listf repeat
if (f1:=f exquo g) case "failed" then return 1$BP
g
-- test if d is the gcd
test(f:BP,g:BP,d:BP):Boolean ==
d0:=coefficient(d,0)
coefficient(f,0) exquo d0 case "failed" => false
coefficient(g,0) exquo d0 case "failed" => false
f exquo d case "failed" => false
g exquo d case "failed" => false
true
-- gcd and cofactors for PRIMITIVE univariate polynomials
-- also assumes that constant terms are non zero
modGcdPrimitive(f:BP,g:BP): BP ==
df:=degree f
dg:=degree g
dp:FP
lcf:=leadingCoefficient f
lcg:=leadingCoefficient g
testdeg:NNI
lcd:R:=gcd(lcf,lcg)
prime:=pMod
bound:=mbound(f,g)
while zero? (lcd rem prime) repeat
prime := nextMod(prime,bound)
soFar:=gcd(reduce(f,prime),reduce(g,prime))::BP
testdeg:=degree soFar
zero? testdeg => return 1$BP
ldp:FP:=
((lcdp:=leadingCoefficient(soFar::BP)) = 1) =>
reduce(lcd::BP,prime)
reduce((modInverse(lcdp,prime)*lcd)::BP,prime)
soFar:=reduce(ldp::BP *soFar,prime)::BP
soFarModulus:=prime
-- choose the prime
while true repeat
prime := nextMod(prime,bound)
lcd rem prime =0 => "next prime"
fp:=reduce(f,prime)
gp:=reduce(g,prime)
dp:=gcd(fp,gp)
dgp :=euclideanSize dp
if dgp =0 then return 1$BP
if dgp=dg and not (f exquo g case "failed") then return g
if dgp=df and not (g exquo f case "failed") then return f
dgp > testdeg => "next prime"
ldp:FP:=
((lcdp:=leadingCoefficient(dp::BP)) = 1) =>
reduce(lcd::BP,prime)
reduce((modInverse(lcdp,prime)*lcd)::BP,prime)
dp:=ldp *dp
dgp=testdeg =>
correction:=reduce(dp::BP-soFar,prime)::BP
zero? correction =>
ans:=reduce(lcd::BP*soFar,soFarModulus)::BP
cont:=content ans
ans:=(ans exquo cont)::BP
test(f,g,ans) => return ans
soFarModulus:=soFarModulus*prime
correctionFactor:=modInverse(soFarModulus rem prime,prime)
-- the initial rem is just for efficiency
soFar:=soFar+soFarModulus*(correctionFactor*correction)
soFarModulus:=soFarModulus*prime
soFar:=reduce(soFar,soFarModulus)::BP
dgp<testdeg =>
soFarModulus:=prime
soFar:=dp::BP
testdeg:=dgp
if not zeroChar and euclideanSize(prime)>1 then
result:=dp::BP
test(f,g,result) => return result
-- this is based on the assumption that the caller of this package,
-- in non-zero characteristic, will use primes of the form
-- x-alpha as long as possible, but, if these are exhausted,
-- will switch to a prime of degree larger than the answer
-- so the result can be used directly.
merge(p:R,q:R):Union(R,"failed") ==
p = q => p
p = 0 => q
q = 0 => p
"failed"
modInverse(c:R,p:R):R ==
(extendedEuclidean(c,p,1)::Record(coef1:R,coef2:R)).coef1
exactquo(u:BP,v:BP,p:R):Union(BP,"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)
-- compute the height of a polynomial --
height(f:BP):NNI ==
degf:=degree f
"max"/[euclideanSize cc for cc in coefficients f]
-- compute the bound
mbound(f:BP,g:BP):NNI ==
hf:=height f
hg:=height g
2*min(hf,hg)
-- ForModularGcd(R,BP) : C == T
-- where
-- R : EuclideanDomain -- characteristic 0
-- BP : UnivariatePolynomialCategory(R)
--
-- C == with
-- nextMod : (R,NNI) -> R
--
-- T == add
-- nextMod(val:R,bound:NNI) : R ==
-- ival:Z:= val pretend Z
-- (nextPrime(ival)$IntegerPrimesPackage(Z))::R
--
-- ForTwoGcd(F) : C == T
-- where
-- F : Join(Finite,Field)
-- SUP ==> SparseUnivariatePolynomial
-- R ==> SUP F
-- P ==> SUP R
-- UPCF2 ==> UnivariatePolynomialCategoryFunctions2
--
-- C == with
-- nextMod : (R,NNI) -> R
--
-- T == add
-- nextMod(val:R,bound:NNI) : R ==
-- ris:R:= nextItem(val) :: R
-- euclideanSize ris < 2 => ris
-- generateIrredPoly(
-- (bound+1)::PositiveInteger)$IrredPolyOverFiniteField(F)
--
--
-- ModularGcd(R,BP) == T
-- where
-- R : EuclideanDomain -- characteristic 0
-- BP : UnivariatePolynomialCategory(R)
-- T ==> InnerModularGcd(R,BP,67108859::R,nextMod$ForModularGcd(R,BP))
--
-- TwoGcd(F) : C == T
-- where
-- F : Join(Finite,Field)
-- SUP ==> SparseUnivariatePolynomial
-- R ==> SUP F
-- P ==> SUP R
--
-- T ==> InnerModularGcd(R,P,nextMod(monomial(1,1)$R)$ForTwoGcd(F),
-- nextMod$ForTwoGcd(F))
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