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geneez.spad
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geneez.spad
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)abbrev package GENEEZ GenExEuclid
++ Author : P.Gianni.
++ January 1990
++ The equation \spad{Af+Bg=h} and its generalization to n polynomials
++ is solved for solutions over the R, euclidean domain.
++ A table containing the solutions of \spad{Af+Bg=x^k} is used.
++ The operations are performed modulus a prime which are in principle big enough,
++ but the solutions are tested and, in case of failure, a hensel
++ lifting process is used to get to the right solutions.
++ It will be used in the factorization of multivariate polynomials
++ over finite field, with \spad{R=F[x]}.
GenExEuclid(R, BP) : C == T
where
R : EuclideanDomain
PI ==> PositiveInteger
NNI ==> NonNegativeInteger
BP : UnivariatePolynomialCategory R
L ==> List
C == with
reduction : (BP, R) -> BP
++ reduction(p, prime) reduces the polynomial p modulo prime of R.
++ Note: this function is exported only because it's conditional.
compBound : (BP, L BP) -> NNI
++ compBound(p, lp)
++ computes a bound for the coefficients of the solution
++ polynomials.
++ Given a polynomial right hand side p, and a list lp of left
++ hand side polynomials.
++ Exported because it depends on the valuation.
tablePow : (NNI,R,L BP) -> Union(Vector(L BP),"failed")
++ tablePow(maxdeg, prime, lpol) constructs the table with the
++ coefficients of the Extended Euclidean Algorithm for lpol.
++ Here the right side is \spad{x^k}, for k less to maxdeg.
++ The operation returns "failed" when the elements are not
++ coprime modulo prime.
solveid : (BP,R,Vector L BP) -> Union(L BP,"failed")
++ solveid(h, prime, table) computes the coefficients of the
++ extended euclidean algorithm for a list of polynomials
++ whose tablePow is table and with right side h.
testModulus : (R, L BP) -> Boolean
++ testModulus(p, lp) returns true if the prime p
++ is valid for the list of polynomials lp, i.e. preserves
++ the degree and they remain relatively prime.
T == add
if R has multiplicativeValuation then
compBound(m : BP, listpolys : L BP) : NNI ==
ldeg := [degree f for f in listpolys]
n : NNI := (+/[df for df in ldeg])
normlist := [ +/[euclideanSize(u)^2 for u in coefficients f]
for f in listpolys]
nm := +/[euclideanSize(u)^2 for u in coefficients m]
normprod := */[g^((n-df)::NNI) for g in normlist for df in ldeg]
2*(approxSqrt(normprod * nm)$IntegerRoots(Integer))::NNI
else if R has additiveValuation then
-- a fairly crude Hadamard-style bound for the solution
-- based on regarding the problem as a system of linear equations.
compBound(m : BP, listpolys : L BP) : NNI ==
"max"/[euclideanSize u for u in coefficients m] +
+/["max"/[euclideanSize u for u in coefficients p]
for p in listpolys]
else
compBound(m : BP, listpolys : L BP) : NNI ==
error "attempt to use compBound without a well-understood valuation"
if R has IntegerNumberSystem then
reduction(u : BP, p : R) : BP ==
p = 0 => u
map(x +-> symmetricRemainder(x, p), u)
else reduction(u : BP, p : R) : BP ==
p = 0 => u
map(x +-> x rem p, u)
merge(p:R,q:R):Union(R,"failed") ==
p = q => p
p = 0 => q
q = 0 => p
"failed"
modInverse(c : R, p : R) : R ==
(extendedEuclidean(c, p, 1)::Record(coef1 : R, coef2 : R)).coef1
exactquo(u:BP,v:BP,p:R):Union(BP,"failed") ==
invlcv := modInverse(leadingCoefficient v, p)
r := monicDivide(u, reduction(invlcv*v, p))
reduction(r.remainder,p) ~= 0 => "failed"
reduction(invlcv*r.quotient, p)
FP := EuclideanModularRing(R, BP, R, reduction, merge, exactquo)
--local functions
makeProducts : L BP -> L BP
liftSol : (L BP, BP, R, R, L BP, Vector L FP, BP, NNI
) -> Union(L BP, "failed")
reduceList(lp : L BP, lmod : R) : L FP ==[reduce(ff, lmod) for ff in lp]
coerceLFP(lf : L FP) : L BP == [fm::BP for fm in lf]
liftSol(oldsol : L BP, err : BP, lmod : R, lmodk : R, lpolys : L BP,
ftab : Vector L FP, m : BP, bound : NNI) : Union(L BP, "failed") ==
repeat
euclideanSize(lmodk) > bound => return "failed"
d := degree err
sln : L FP := [0$FP for xx in ftab.1 ]
for i in 0 .. d | (cc := coefficient(err, i)) ~= 0 repeat
sln := [slp + reduce(cc::BP, lmod)*pp
for pp in ftab.(i+1) for slp in sln]
nsol := [f - lmodk*reduction(g::BP, lmod) for f in oldsol
for g in sln]
lmodk := lmod*lmodk
nsol := [reduction(slp, lmodk) for slp in nsol]
(fs := +/[f*g for f in lpolys for g in nsol]) = m => return nsol
err := ((fs-m) exquo lmodk)::BP
oldsol := nsol
makeProducts(listPol : L BP) : L BP ==
#listPol < 2 => listPol
#listPol = 2 => reverse listPol
f := first listPol
ll := rest listPol
[*/ll, :[f*g for g in makeProducts ll]]
testModulus(pmod, listPol) ==
redListPol := reduceList(listPol, pmod)
for pol in listPol for rpol in redListPol repeat
degree(pol) ~= degree(rpol::BP) => return false
while not empty? redListPol repeat
rpol := first redListPol
redListPol := rest redListPol
for rpol2 in redListPol repeat
gcd(rpol, rpol2) ~= 1 => return false
true
if R has Field then
tablePow(mdeg : NNI, pmod : R, listPol : L BP) ==
multiE := multiEuclidean(listPol, 1$BP)
multiE case "failed" => "failed"
ptable : Vector L BP := new(mdeg+1, [])
ptable.1 := multiE
x : BP := monomial(1, 1)
for i in 2..mdeg repeat ptable.i :=
[tpol*x rem fpol for tpol in ptable.(i-1) for fpol in listPol]
ptable.(mdeg+1) := makeProducts listPol
ptable
solveid(m:BP,pmod:R,table:Vector L BP) : Union(L BP,"failed") ==
-- Actually, there's no possibility of failure
d := degree m
sln : L BP := [0$BP for xx in table.1]
for i in 0 .. d | coefficient(m, i) ~= 0 repeat
sln := [slp+coefficient(m, i)*pp
for pp in table.(i+1) for slp in sln]
sln
else
tablePow(mdeg : NNI, pmod : R, listPol : L BP) ==
listP : L FP := [reduce(pol, pmod) for pol in listPol]
for rpol in listP for pol in listPol repeat
if degree(rpol::BP) < degree(pol) then return "failed"
multiE := multiEuclidean(listP, 1$FP)
multiE case "failed" => "failed"
ftable : Vector L FP := new(mdeg+1, [])
fl : L FP := multiE@L(FP)
ftable.1 := fl
x : FP := reduce(monomial(1, 1), pmod)
for i in 2..mdeg repeat ftable.i :=
[tpol*x rem fpol for tpol in ftable.(i-1) for fpol in listP]
ptable := map(coerceLFP, ftable)$VectorFunctions2(List FP, List BP)
ptable.(mdeg+1) := makeProducts listPol
ptable
solveid(m:BP,pmod:R,table:Vector L BP) : Union(L BP,"failed") ==
d := degree m
ftab : Vector L FP :=
map(x+->reduceList(x, pmod), table)$VectorFunctions2(List BP, List FP)
lpolys : L BP := table.(#table)
sln : L FP := [0$FP for xx in ftab.1]
for i in 0 .. d | coefficient(m, i) ~= 0 repeat
sln := [slp+reduce(coefficient(m, i)::BP, pmod)*pp
for pp in ftab.(i+1) for slp in sln]
soln := [slp::BP for slp in sln]
(fs := +/[f*g for f in lpolys for g in soln]) = m=> soln
-- Compute bound
bound := compBound(m, lpolys)
a : BP := ((fs-m) exquo pmod)::BP
liftSol(soln, a, pmod, pmod, lpolys, ftab, m, bound)
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
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