return sigma;
}
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const vector<EvalPoint>& v)
+{
+ for ( size_t i=0; i<v.size(); ++i ) {
+ o << "(" << v[i].x << "==" << v[i].evalpoint << ") ";
+ }
+ return o;
+}
+#endif // def DEBUGFACTOR
+
+
ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigned int p, const cl_I& l, const UniPolyVec& u, const vector<ex>& lcU)
{
DCOUT(hensel_multivar);
DCOUTVAR(a);
DCOUTVAR(x);
+ DCOUTVAR(I);
DCOUTVAR(p);
DCOUTVAR(l);
DCOUTVAR(u);
+ DCOUTVAR(lcU);
const size_t nu = I.size() + 1;
const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
for ( size_t i=0; i<U.size(); ++i ) {
res.append(U[i]);
}
+ DCOUTVAR(res);
+ DCOUT(END hensel_multivar);
return res;
}
else {
+ lst res;
+ DCOUTVAR(res);
+ DCOUT(END hensel_multivar);
return lst();
}
}
-static ex factor_multivariate(const ex& poly, const ex& x)
+static ex put_factors_into_lst(const ex& e)
+{
+ DCOUT(put_factors_into_lst);
+ DCOUTVAR(e);
+
+ lst result;
+
+ if ( is_a<numeric>(e) ) {
+ result.append(e);
+ DCOUT(END put_factors_into_lst);
+ DCOUTVAR(result);
+ return result;
+ }
+ if ( is_a<power>(e) ) {
+ result.append(1);
+ result.append(e.op(0));
+ result.append(e.op(1));
+ DCOUT(END put_factors_into_lst);
+ DCOUTVAR(result);
+ return result;
+ }
+ if ( is_a<symbol>(e) ) {
+ result.append(1);
+ result.append(e);
+ result.append(1);
+ DCOUT(END put_factors_into_lst);
+ DCOUTVAR(result);
+ return result;
+ }
+ if ( is_a<mul>(e) ) {
+ ex nfac = 1;
+ for ( size_t i=0; i<e.nops(); ++i ) {
+ ex op = e.op(i);
+ if ( is_a<numeric>(op) ) {
+ nfac = op;
+ }
+ if ( is_a<power>(op) ) {
+ result.append(op.op(0));
+ result.append(op.op(1));
+ }
+ if ( is_a<symbol>(op) || is_a<add>(op) ) {
+ result.append(op);
+ result.append(1);
+ }
+ }
+ result.prepend(nfac);
+ DCOUT(END put_factors_into_lst);
+ DCOUTVAR(result);
+ return result;
+ }
+ throw runtime_error("put_factors_into_lst: bad term.");
+}
+
+static bool checkdivisors(const lst& f, vector<numeric>& d)
+{
+ const int k = f.nops()-2;
+ numeric q, r;
+ d[0] = ex_to<numeric>(f.op(0) * f.op(f.nops()-1));
+ for ( int i=1; i<=k; ++i ) {
+ q = ex_to<numeric>(abs(f.op(i-1)));
+ for ( int j=i-1; j>=0; --j ) {
+ r = d[j];
+ do {
+ r = gcd(r, q);
+ q = q/r;
+ } while ( r != 1 );
+ if ( q == 1 ) {
+ return true;
+ }
+ }
+ d[i] = q;
+ }
+ return false;
+}
+
+static void generate_set(const ex& u, const ex& vn, const exset& syms, const ex& f, const numeric& modulus, vector<numeric>& a, vector<numeric>& d)
+{
+ const ex& x = *syms.begin();
+ bool trying = true;
+ do {
+ ex u0 = u;
+ ex vna = vn;
+ ex vnatry;
+ exset::const_iterator s = syms.begin();
+ ++s;
+ for ( size_t i=0; i<a.size(); ++i ) {
+ do {
+ a[i] = mod(numeric(rand()), 2*modulus) - modulus;
+ vnatry = vna.subs(*s == a[i]);
+ } while ( vnatry == 0 );
+ vna = vnatry;
+ u0 = u0.subs(*s == a[i]);
+ }
+ if ( gcd(u0,u0.diff(ex_to<symbol>(x))) != 1 ) {
+ continue;
+ }
+ if ( is_a<numeric>(vn) ) {
+ d = a;
+ trying = false;
+ }
+ else {
+ lst fnum;
+ lst::const_iterator i = ex_to<lst>(f).begin();
+ fnum.append(*i++);
+ while ( i!=ex_to<lst>(f).end() ) {
+ ex fs = *i;
+ s = syms.begin();
+ ++s;
+ for ( size_t j=0; j<a.size(); ++j ) {
+ fs = fs.subs(*s == a[j]);
+ }
+ fnum.append(fs);
+ ++i; ++i;
+ }
+ ex con = u0.content(x);
+ fnum.append(con);
+ trying = checkdivisors(fnum, d);
+ }
+ } while ( trying );
+}
+
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const vector<numeric>& v)
+{
+ for ( size_t i=0; i<v.size(); ++i ) {
+ o << v[i] << " ";
+ }
+ return o;
+}
+#endif // def DEBUGFACTOR
+
+static ex factor_multivariate(const ex& poly, const exset& syms)
{
- // TODO
- return 666;
+ DCOUT(factor_multivariate);
+ DCOUTVAR(poly);
+
+ exset::const_iterator s;
+ const ex& x = *syms.begin();
+ DCOUTVAR(x);
+
+ /* make polynomial primitive */
+ ex p = poly.expand().collect(x);
+ DCOUTVAR(p);
+ ex cont = p.lcoeff(x);
+ for ( numeric i=p.degree(x)-1; i>=p.ldegree(x); --i ) {
+ cont = gcd(cont, p.coeff(x,ex_to<numeric>(i).to_int()));
+ if ( cont == 1 ) break;
+ }
+ DCOUTVAR(cont);
+ ex pp = expand(normal(p / cont));
+ DCOUTVAR(pp);
+ if ( !is_a<numeric>(cont) ) {
+ return factor(cont) * factor(pp);
+ }
+
+ /* factor leading coefficient */
+ pp = pp.collect(x);
+ ex vn = p.lcoeff(x);
+ ex vnlst;
+ if ( is_a<numeric>(vn) ) {
+ vnlst = lst(vn);
+ }
+ else {
+ ex vnfactors = factor(vn);
+ vnlst = put_factors_into_lst(vnfactors);
+ }
+ DCOUTVAR(vnlst);
+
+ const numeric maxtrials = 3;
+ numeric modulus = (vnlst.nops()-1 > 3) ? vnlst.nops()-1 : 3;
+ numeric minimalr = -1;
+ vector<numeric> a(syms.size()-1);
+ vector<numeric> d(syms.size()-1);
+
+ while ( true ) {
+ numeric trialcount = 0;
+ ex u, delta;
+ unsigned int prime;
+ UniPolyVec uvec;
+ while ( trialcount < maxtrials ) {
+ uvec.clear();
+ generate_set(pp, vn, syms, vnlst, modulus, a, d);
+ DCOUTVAR(a);
+ DCOUTVAR(d);
+ u = pp;
+ s = syms.begin();
+ ++s;
+ for ( size_t i=0; i<a.size(); ++i ) {
+ u = u.subs(*s == a[i]);
+ ++s;
+ }
+ delta = u.content(x);
+
+ // determine proper prime
+ prime = 3;
+ cl_modint_ring R = find_modint_ring(prime);
+ while ( true ) {
+ if ( irem(ex_to<numeric>(u.lcoeff(x)), prime) != 0 ) {
+ UniPoly modpoly(R, u, x);
+ UniFactorVec sqrfree_ufv;
+ squarefree(modpoly, sqrfree_ufv);
+ if ( sqrfree_ufv.factors.size() == 1 && sqrfree_ufv.factors.front().exp == 1 ) break;
+ }
+ prime = next_prime(prime);
+ R = find_modint_ring(prime);
+ }
+
+ UniPoly umod(R, u, x);
+ DCOUTVAR(u);
+ factor_modular(umod, uvec);
+ DCOUTVAR(uvec);
+
+ if ( uvec.size() == 1 ) {
+ DCOUTVAR(poly);
+ DCOUT(END factor_multivariate);
+ return poly;
+ }
+
+ if ( minimalr < 0 ) {
+ minimalr = uvec.size();
+ }
+ else if ( minimalr == uvec.size() ) {
+ ++trialcount;
+ ++modulus;
+ }
+ else if ( minimalr > uvec.size() ) {
+ minimalr = uvec.size();
+ trialcount = 0;
+ }
+ DCOUTVAR(trialcount);
+ DCOUTVAR(minimalr);
+ if ( minimalr == 0 ) {
+ DCOUTVAR(poly);
+ DCOUT(END factor_multivariate);
+ return poly;
+ }
+ }
+
+ vector<ex> C;
+ if ( vnlst.nops() == 1 ) {
+ C.resize(uvec.size(), 1);
+ }
+ else {
+
+ vector<numeric> ftilde((vnlst.nops()-1)/2);
+ for ( size_t i=0; i<ftilde.size(); ++i ) {
+ ex ft = vnlst.op(i*2+1);
+ s = syms.begin();
+ ++s;
+ for ( size_t j=0; j<a.size(); ++j ) {
+ ft = ft.subs(*s == a[j]);
+ ++s;
+ }
+ ftilde[i] = ex_to<numeric>(ft);
+ }
+ DCOUTVAR(ftilde);
+
+ vector<ex> D;
+ vector<bool> fflag(ftilde.size(), false);
+ for ( size_t i=0; i<uvec.size(); ++i ) {
+ ex ui = uvec[i].to_ex(x);
+ ex Di = 1;
+ numeric coeff = ex_to<numeric>(ui.lcoeff(x));
+ for ( size_t j=0; j<ftilde.size(); ++j ) {
+ if ( numeric(coeff / ftilde[j]).is_integer() ) {
+ coeff = coeff / ftilde[j];
+ Di *= ftilde[j];
+ fflag[j] = true;
+ --j;
+ }
+ }
+ D.push_back(Di.expand());
+ }
+ for ( size_t i=0; i<fflag.size(); ++i ) {
+ if ( !fflag[i] ) {
+ --minimalr;
+ continue;
+ }
+ }
+ DCOUTVAR(D);
+
+ C.resize(D.size());
+ if ( delta == 1 ) {
+ for ( size_t i=0; i<D.size(); ++i ) {
+ ex Dtilde = D[i];
+ s = syms.begin();
+ ++s;
+ for ( size_t j=0; j<a.size(); ++j ) {
+ Dtilde = Dtilde.subs(*s == a[j]);
+ ++s;
+ }
+ ex Ci = D[i] * (uvec[i].to_ex(x).lcoeff(x) / Dtilde);
+ C.push_back(Ci);
+ }
+ }
+ else {
+ for ( size_t i=0; i<D.size(); ++i ) {
+ ex Dtilde = D[i];
+ s = syms.begin();
+ ++s;
+ for ( size_t j=0; j<a.size(); ++j ) {
+ Dtilde = Dtilde.subs(*s == a[j]);
+ ++s;
+ }
+ ex ui = uvec[i].to_ex(x);
+ ex Ci;
+ while ( true ) {
+ ex d = gcd(ui.lcoeff(x), Dtilde);
+ Ci = D[i] * ( ui.lcoeff(x) / d );
+ ui = ui * ( Dtilde[i] / d );
+ delta = delta / ( Dtilde[i] / d );
+ if ( delta == 1 ) break;
+ ui = delta * ui;
+ Ci = delta * Ci;
+ pp = pp * pow(delta, D.size()-1);
+ }
+ }
+ }
+
+ }
+
+ EvalPoint ep;
+ vector<EvalPoint> epv;
+ s = syms.begin();
+ ++s;
+ for ( size_t i=0; i<a.size(); ++i ) {
+ ep.x = *s++;
+ ep.evalpoint = a[i].to_int();
+ epv.push_back(ep);
+ }
+
+ // calc bound B
+ ex maxcoeff;
+ for ( int i=u.degree(x); i>=u.ldegree(x); --i ) {
+ maxcoeff += pow(abs(u.coeff(x, i)),2);
+ }
+ cl_I normmc = ceiling1(the<cl_R>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
+ unsigned int maxdegree = 0;
+ for ( size_t i=0; i<uvec.size(); ++i ) {
+ if ( uvec[i].degree() > maxdegree ) {
+ maxdegree = uvec[i].degree();
+ }
+ }
+ unsigned int B = cl_I_to_uint(normmc * expt_pos(cl_I(2), maxdegree));
+
+ ex res = hensel_multivar(poly, x, epv, prime, B, uvec, C);
+ if ( res != lst() ) {
+ ex result = cont;
+ for ( size_t i=0; i<res.nops(); ++i ) {
+ result *= res.op(i).content(x) * res.op(i).unit(x);
+ result *= res.op(i).primpart(x);
+ }
+ DCOUTVAR(result);
+ DCOUT(END factor_multivariate);
+ return result;
+ }
+ }
}
static ex factor_sqrfree(const ex& poly)
}
if ( findsymbols.syms.size() == 1 ) {
+ // univariate case
const ex& x = *(findsymbols.syms.begin());
if ( poly.ldegree(x) > 0 ) {
int ld = poly.ldegree(x);
}
}
- // multivariate case not yet implemented!
- throw runtime_error("multivariate case not yet implemented!");
+ // multivariate case
+ ex res = factor_multivariate(poly, findsymbols.syms);
+ return res;
}
} // anonymous namespace
git://www.ginac.de / ginac.git / commitdiff