/** @file factor.cpp
*
- * Polynomial factorization routines.
- * Only univariate at the moment and completely non-optimized!
+ * Polynomial factorization code (Implementation).
+ *
+ * Algorithms used can be found in
+ * [W1] An Improved Multivariate Polynomial Factoring Algorithm,
+ * P.S.Wang, Mathematics of Computation, Vol. 32, No. 144 (1978) 1215--1231.
+ * [GCL] Algorithms for Computer Algebra,
+ * K.O.Geddes, S.R.Czapor, G.Labahn, Springer Verlag, 1992.
*/
/*
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
+//#define DEBUGFACTOR
+
+#ifdef DEBUGFACTOR
+#include <ostream>
+#include <ginac/ginac.h>
+using namespace GiNaC;
+#else
#include "factor.h"
#include "ex.h"
#include "mul.h"
#include "normal.h"
#include "add.h"
+#endif
#include <algorithm>
#include <list>
#include <cln/cln.h>
using namespace cln;
-//#define DEBUGFACTOR
-
#ifdef DEBUGFACTOR
-#include <ostream>
-#endif // def DEBUGFACTOR
-
+namespace Factor {
+#else
namespace GiNaC {
+#endif
+#ifdef DEBUGFACTOR
+#define DCOUT(str) cout << #str << endl
+#define DCOUTVAR(var) cout << #var << ": " << var << endl
+#define DCOUT2(str,var) cout << #str << ": " << var << endl
+#else
+#define DCOUT(str)
+#define DCOUTVAR(var)
+#define DCOUT2(str,var)
+#endif
+
+// forward declaration
+ex factor(const ex& poly, unsigned options);
+
+// anonymous namespace to hide all utility functions
namespace {
typedef vector<cl_MI> Vec;
// assert: poly is in Z[x]
Term t;
for ( int i=poly.degree(x); i>=poly.ldegree(x); --i ) {
- int coeff = ex_to<numeric>(poly.coeff(x,i)).to_int();
- if ( coeff ) {
+ cl_I coeff = the<cl_I>(ex_to<numeric>(poly.coeff(x,i)).to_cl_N());
+ if ( !zerop(coeff) ) {
t.c = R->canonhom(coeff);
if ( !zerop(t.c) ) {
t.exp = i;
}
}
}
+ UniPoly(const cl_modint_ring& ring, const UniPoly& poly) : R(ring)
+ {
+ if ( R->modulus == poly.R->modulus ) {
+ terms = poly.terms;
+ }
+ else {
+ list<Term>::const_iterator i=poly.terms.begin(), end=poly.terms.end();
+ for ( ; i!=end; ++i ) {
+ terms.push_back(*i);
+ terms.back().c = R->canonhom(poly.R->retract(i->c));
+ if ( zerop(terms.back().c) ) {
+ terms.pop_back();
+ }
+ }
+ }
+ }
UniPoly(const cl_modint_ring& ring, const Vec& v) : R(ring)
{
Term t;
}
}
}
+ void divide(const cl_I& x)
+ {
+ list<Term>::iterator i = terms.begin(), end = terms.end();
+ for ( ; i != end; ++i ) {
+ i->c = cl_MI(R, the<cl_I>(R->retract(i->c) / x));
+ }
+ }
void reduce_exponents(unsigned int prime)
{
list<Term>::iterator i = terms.begin(), end = terms.end();
return c;
}
-static UniPoly operator-(const UniPoly& a)
+static UniPoly operator*(const UniPoly& a, const cl_MI& fac)
+{
+ unsigned int n = a.degree();
+ UniPoly c(a.R);
+ Term t;
+ for ( unsigned int i=0 ; i<=n; ++i ) {
+ t.c = a[i] * fac;
+ if ( !zerop(t.c) ) {
+ t.exp = i;
+ c.terms.push_front(t);
+ }
+ }
+ return c;
+}
+
+static UniPoly operator+(const UniPoly& a, const UniPoly& b)
{
list<Term>::const_iterator ia = a.terms.begin(), aend = a.terms.end();
+ list<Term>::const_iterator ib = b.terms.begin(), bend = b.terms.end();
UniPoly c(a.R);
+ while ( ia != aend && ib != bend ) {
+ if ( ia->exp > ib->exp ) {
+ c.terms.push_back(*ia);
+ ++ia;
+ }
+ else if ( ia->exp < ib->exp ) {
+ c.terms.push_back(*ib);
+ ++ib;
+ }
+ else {
+ Term t;
+ t.exp = ia->exp;
+ t.c = ia->c + ib->c;
+ if ( !zerop(t.c) ) {
+ c.terms.push_back(t);
+ }
+ ++ia; ++ib;
+ }
+ }
while ( ia != aend ) {
c.terms.push_back(*ia);
- c.terms.back().c = -c.terms.back().c;
++ia;
}
+ while ( ib != bend ) {
+ c.terms.push_back(*ib);
+ ++ib;
+ }
return c;
}
+// static UniPoly operator-(const UniPoly& a)
+// {
+// list<Term>::const_iterator ia = a.terms.begin(), aend = a.terms.end();
+// UniPoly c(a.R);
+// while ( ia != aend ) {
+// c.terms.push_back(*ia);
+// c.terms.back().c = -c.terms.back().c;
+// ++ia;
+// }
+// return c;
+// }
+
#ifdef DEBUGFACTOR
ostream& operator<<(ostream& o, const UniPoly& t)
{
typedef vector<UniPoly> UniPolyVec;
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const UniPolyVec& v)
+{
+ UniPolyVec::const_iterator i = v.begin(), end = v.end();
+ while ( i != end ) {
+ o << *i++ << " , " << endl;
+ }
+ return o;
+}
+#endif // def DEBUGFACTOR
+
struct UniFactor
{
UniPoly p;
i2 += c;
}
}
+ void mul_row(size_t row, const cl_MI x)
+ {
+ vector<cl_MI>::iterator i = m.begin() + row*c;
+ for ( size_t cc=0; cc<c; ++cc ) {
+ *i = *i * x;
+ ++i;
+ }
+ }
+ void sub_row(size_t row1, size_t row2, const cl_MI fac)
+ {
+ vector<cl_MI>::iterator i1 = m.begin() + row1*c;
+ vector<cl_MI>::iterator i2 = m.begin() + row2*c;
+ for ( size_t cc=0; cc<c; ++cc ) {
+ *i1 = *i1 - *i2 * fac;
+ ++i1;
+ ++i2;
+ }
+ }
+ void switch_row(size_t row1, size_t row2)
+ {
+ cl_MI buf;
+ vector<cl_MI>::iterator i1 = m.begin() + row1*c;
+ vector<cl_MI>::iterator i2 = m.begin() + row2*c;
+ for ( size_t cc=0; cc<c; ++cc ) {
+ buf = *i1; *i1 = *i2; *i2 = buf;
+ ++i1;
+ ++i2;
+ }
+ }
+ bool is_col_zero(size_t col) const
+ {
+ Vec::const_iterator i = m.begin() + col;
+ for ( size_t rr=0; rr<r; ++rr ) {
+ if ( !zerop(*i) ) {
+ return false;
+ }
+ i += c;
+ }
+ return true;
+ }
bool is_row_zero(size_t row) const
{
Vec::const_iterator i = m.begin() + row*c;
};
#ifdef DEBUGFACTOR
+Matrix operator*(const Matrix& m1, const Matrix& m2)
+{
+ const unsigned int r = m1.rowsize();
+ const unsigned int c = m2.colsize();
+ Matrix o(r,c,m1(0,0));
+
+ for ( size_t i=0; i<r; ++i ) {
+ for ( size_t j=0; j<c; ++j ) {
+ cl_MI buf;
+ buf = m1(i,0) * m2(0,j);
+ for ( size_t k=1; k<c; ++k ) {
+ buf = buf + m1(i,k)*m2(k,j);
+ }
+ o(i,j) = buf;
+ }
+ }
+ return o;
+}
+
ostream& operator<<(ostream& o, const Matrix& m)
{
vector<cl_MI>::const_iterator i = m.m.begin(), end = m.m.end();
for ( int i=a.degree(x); i>=a.ldegree(x); --i ) {
maxcoeff += pow(abs(a.coeff(x, i)),2);
}
- cl_I normmc = ceiling1(the<cl_F>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
- unsigned int maxdegree = (u1_.degree() > w1_.degree()) ? u1_.degree() : w1_.degree();
- unsigned int B = cl_I_to_uint(normmc * expt_pos(cl_I(2), maxdegree));
+ cl_I normmc = ceiling1(the<cl_R>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
+ cl_I maxdegree = (u1_.degree() > w1_.degree()) ? u1_.degree() : w1_.degree();
+ cl_I B = normmc * expt_pos(cl_I(2), maxdegree);
// step 1
ex alpha = a.lcoeff(x);
if ( gamma == 0 ) {
gamma = alpha;
}
- unsigned int gamma_ui = ex_to<numeric>(abs(gamma)).to_int();
+ numeric gamma_ui = ex_to<numeric>(abs(gamma));
a = a * gamma;
UniPoly nu1 = u1_;
nu1.unit_normal();
ex u = replace_lc(u1.to_ex(x), x, gamma);
ex w = replace_lc(w1.to_ex(x), x, alpha);
ex e = expand(a - u * w);
- unsigned int modulus = p;
+ numeric modulus = p;
+ const numeric maxmodulus = 2*numeric(B)*gamma_ui;
// step 4
- while ( !e.is_zero() && modulus < 2*B*gamma_ui ) {
+ while ( !e.is_zero() && modulus < maxmodulus ) {
ex c = e / modulus;
phi = expand(s.to_ex(x)*c);
UniPoly sigmatilde(R, phi, x);
size_t size() const { return n; }
size_t size_first() const { return n-sum; }
size_t size_second() const { return sum; }
+#ifdef DEBUGFACTOR
+ void get() const
+ {
+ for ( size_t i=0; i<k.size(); ++i ) {
+ cout << k[i] << " ";
+ }
+ cout << endl;
+ }
+#endif
bool next()
{
for ( size_t i=n-1; i>=1; --i ) {
return unit * cont * result;
}
-struct FindSymbolsMap : public map_function {
+struct EvalPoint
+{
+ ex x;
+ int evalpoint;
+};
+
+// forward declaration
+vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I, unsigned int d, unsigned int p, unsigned int k);
+
+UniPolyVec multiterm_eea_lift(const UniPolyVec& a, const ex& x, unsigned int p, unsigned int k)
+{
+ DCOUT(multiterm_eea_lift);
+ DCOUTVAR(a);
+ DCOUTVAR(p);
+ DCOUTVAR(k);
+
+ const size_t r = a.size();
+ DCOUTVAR(r);
+ cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+ UniPoly fill(R);
+ UniPolyVec q(r-1, fill);
+ q[r-2] = a[r-1];
+ for ( size_t j=r-2; j>=1; --j ) {
+ q[j-1] = a[j] * q[j];
+ }
+ DCOUTVAR(q);
+ UniPoly beta(R);
+ beta.set(0, R->one());
+ UniPolyVec s;
+ for ( size_t j=1; j<r; ++j ) {
+ DCOUTVAR(j);
+ DCOUTVAR(beta);
+ vector<ex> mdarg(2);
+ mdarg[0] = q[j-1].to_ex(x);
+ mdarg[1] = a[j-1].to_ex(x);
+ vector<EvalPoint> empty;
+ vector<ex> exsigma = multivar_diophant(mdarg, x, beta.to_ex(x), empty, 0, p, k);
+ UniPoly sigma1(R, exsigma[0], x);
+ UniPoly sigma2(R, exsigma[1], x);
+ beta = sigma1;
+ s.push_back(sigma2);
+ }
+ s.push_back(beta);
+
+ DCOUTVAR(s);
+ DCOUT(END multiterm_eea_lift);
+ return s;
+}
+
+void eea_lift(const UniPoly& a, const UniPoly& b, const ex& x, unsigned int p, unsigned int k, UniPoly& s_, UniPoly& t_)
+{
+ DCOUT(eea_lift);
+ DCOUTVAR(a);
+ DCOUTVAR(b);
+ DCOUTVAR(x);
+ DCOUTVAR(p);
+ DCOUTVAR(k);
+
+ cl_modint_ring R = find_modint_ring(p);
+ UniPoly amod(R, a);
+ UniPoly bmod(R, b);
+ DCOUTVAR(amod);
+ DCOUTVAR(bmod);
+
+ UniPoly smod(R), tmod(R), g(R);
+ exteuclid(amod, bmod, g, smod, tmod);
+
+ DCOUTVAR(smod);
+ DCOUTVAR(tmod);
+ DCOUTVAR(g);
+
+ cl_modint_ring Rpk = find_modint_ring(expt_pos(cl_I(p),k));
+ UniPoly s(Rpk, smod);
+ UniPoly t(Rpk, tmod);
+ DCOUTVAR(s);
+ DCOUTVAR(t);
+
+ cl_I modulus(p);
+
+ UniPoly one(Rpk);
+ one.set(0, Rpk->one());
+ for ( size_t j=1; j<k; ++j ) {
+ UniPoly e = one - a * s - b * t;
+ e.divide(modulus);
+ UniPoly c(R, e);
+ UniPoly sigmabar(R);
+ sigmabar = smod * c;
+ UniPoly taubar(R);
+ taubar = tmod * c;
+ UniPoly q(R);
+ div(sigmabar, bmod, q);
+ UniPoly sigma(R);
+ rem(sigmabar, bmod, sigma);
+ UniPoly tau(R);
+ tau = taubar + q * amod;
+ UniPoly sadd(Rpk, sigma);
+ cl_MI modmodulus(Rpk, modulus);
+ s = s + sadd * modmodulus;
+ UniPoly tadd(Rpk, tau);
+ t = t + tadd * modmodulus;
+ modulus = modulus * p;
+ }
+
+ s_ = s; t_ = t;
+
+ DCOUTVAR(s);
+ DCOUTVAR(t);
+ DCOUT2(check, a*s + b*t);
+ DCOUT(END eea_lift);
+}
+
+UniPolyVec univar_diophant(const UniPolyVec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
+{
+ DCOUT(univar_diophant);
+ DCOUTVAR(a);
+ DCOUTVAR(x);
+ DCOUTVAR(m);
+ DCOUTVAR(p);
+ DCOUTVAR(k);
+
+ cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+
+ const size_t r = a.size();
+ UniPolyVec result;
+ if ( r > 2 ) {
+ UniPolyVec s = multiterm_eea_lift(a, x, p, k);
+ for ( size_t j=0; j<r; ++j ) {
+ ex phi = expand(pow(x,m)*s[j].to_ex(x));
+ UniPoly bmod(R, phi, x);
+ UniPoly buf(R);
+ rem(bmod, a[j], buf);
+ result.push_back(buf);
+ }
+ }
+ else {
+ UniPoly s(R), t(R);
+ eea_lift(a[1], a[0], x, p, k, s, t);
+ ex phi = expand(pow(x,m)*s.to_ex(x));
+ UniPoly bmod(R, phi, x);
+ UniPoly buf(R);
+ rem(bmod, a[0], buf);
+ result.push_back(buf);
+ UniPoly q(R);
+ div(bmod, a[0], q);
+ phi = expand(pow(x,m)*t.to_ex(x));
+ UniPoly t1mod(R, phi, x);
+ buf = t1mod + q * a[1];
+ result.push_back(buf);
+ }
+
+ DCOUTVAR(result);
+ DCOUT(END univar_diophant);
+ return result;
+}
+
+struct make_modular_map : public map_function {
+ cl_modint_ring R;
+ make_modular_map(const cl_modint_ring& R_) : R(R_) { }
+ ex operator()(const ex& e)
+ {
+ if ( is_a<add>(e) || is_a<mul>(e) ) {
+ return e.map(*this);
+ }
+ else if ( is_a<numeric>(e) ) {
+ numeric mod(R->modulus);
+ numeric halfmod = (mod-1)/2;
+ cl_MI emod = R->canonhom(the<cl_I>(ex_to<numeric>(e).to_cl_N()));
+ numeric n(R->retract(emod));
+ if ( n > halfmod ) {
+ return n-mod;
+ }
+ else {
+ return n;
+ }
+ }
+ return e;
+ }
+};
+
+static ex make_modular(const ex& e, const cl_modint_ring& R)
+{
+ make_modular_map map(R);
+ return map(e);
+}
+
+vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I, unsigned int d, unsigned int p, unsigned int k)
+{
+ vector<ex> a = a_;
+
+ DCOUT(multivar_diophant);
+#ifdef DEBUGFACTOR
+ cout << "a ";
+ for ( size_t i=0; i<a.size(); ++i ) {
+ cout << a[i] << " ";
+ }
+ cout << endl;
+#endif
+ DCOUTVAR(x);
+ DCOUTVAR(c);
+#ifdef DEBUGFACTOR
+ cout << "I ";
+ for ( size_t i=0; i<I.size(); ++i ) {
+ cout << I[i].x << "=" << I[i].evalpoint << " ";
+ }
+ cout << endl;
+#endif
+ DCOUTVAR(d);
+ DCOUTVAR(p);
+ DCOUTVAR(k);
+
+ const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+ const size_t r = a.size();
+ const size_t nu = I.size() + 1;
+ DCOUTVAR(r);
+ DCOUTVAR(nu);
+
+ vector<ex> sigma;
+ if ( nu > 1 ) {
+ ex xnu = I.back().x;
+ int alphanu = I.back().evalpoint;
+
+ ex A = 1;
+ for ( size_t i=0; i<r; ++i ) {
+ A *= a[i];
+ }
+ vector<ex> b(r);
+ for ( size_t i=0; i<r; ++i ) {
+ b[i] = normal(A / a[i]);
+ }
+
+ vector<ex> anew = a;
+ for ( size_t i=0; i<r; ++i ) {
+ anew[i] = anew[i].subs(xnu == alphanu);
+ }
+ ex cnew = c.subs(xnu == alphanu);
+ vector<EvalPoint> Inew = I;
+ Inew.pop_back();
+ sigma = multivar_diophant(anew, x, cnew, Inew, d, p, k);
+ DCOUTVAR(sigma);
+
+ ex buf = c;
+ for ( size_t i=0; i<r; ++i ) {
+ buf -= sigma[i] * b[i];
+ }
+ ex e = buf;
+ e = make_modular(e, R);
+
+ e = e.expand();
+ DCOUTVAR(e);
+ DCOUTVAR(d);
+ ex monomial = 1;
+ for ( size_t m=1; m<=d; ++m ) {
+ DCOUTVAR(m);
+ while ( !e.is_zero() && e.has(xnu) ) {
+ monomial *= (xnu - alphanu);
+ monomial = expand(monomial);
+ DCOUTVAR(xnu);
+ DCOUTVAR(alphanu);
+ ex cm = e.diff(ex_to<symbol>(xnu), m).subs(xnu==alphanu) / factorial(m);
+ DCOUTVAR(cm);
+ if ( !cm.is_zero() ) {
+ vector<ex> delta_s = multivar_diophant(anew, x, cm, Inew, d, p, k);
+ DCOUTVAR(delta_s);
+ ex buf = e;
+ for ( size_t j=0; j<delta_s.size(); ++j ) {
+ delta_s[j] *= monomial;
+ sigma[j] += delta_s[j];
+ buf -= delta_s[j] * b[j];
+ }
+ e = buf.expand();
+ e = make_modular(e, R);
+ }
+ }
+ }
+ }
+ else {
+ DCOUT(uniterm left);
+ UniPolyVec amod;
+ for ( size_t i=0; i<a.size(); ++i ) {
+ UniPoly up(R, a[i], x);
+ amod.push_back(up);
+ }
+
+ sigma.insert(sigma.begin(), r, 0);
+ size_t nterms;
+ ex z;
+ if ( is_a<add>(c) ) {
+ nterms = c.nops();
+ z = c.op(0);
+ }
+ else {
+ nterms = 1;
+ z = c;
+ }
+ DCOUTVAR(nterms);
+ for ( size_t i=0; i<nterms; ++i ) {
+ DCOUTVAR(z);
+ int m = z.degree(x);
+ DCOUTVAR(m);
+ cl_I cm = the<cl_I>(ex_to<numeric>(z.lcoeff(x)).to_cl_N());
+ DCOUTVAR(cm);
+ UniPolyVec delta_s = univar_diophant(amod, x, m, p, k);
+ cl_MI modcm;
+ cl_I poscm = cm;
+ while ( poscm < 0 ) {
+ poscm = poscm + expt_pos(cl_I(p),k);
+ }
+ modcm = cl_MI(R, poscm);
+ DCOUTVAR(modcm);
+ for ( size_t j=0; j<delta_s.size(); ++j ) {
+ delta_s[j] = delta_s[j] * modcm;
+ sigma[j] = sigma[j] + delta_s[j].to_ex(x);
+ }
+ DCOUTVAR(delta_s);
+#ifdef DEBUGFACTOR
+ cout << "STEP " << i << " sigma ";
+ for ( size_t p=0; p<sigma.size(); ++p ) {
+ cout << sigma[p] << " ";
+ }
+ cout << endl;
+#endif
+ if ( nterms > 1 ) {
+ z = c.op(i+1);
+ }
+ }
+ }
+#ifdef DEBUGFACTOR
+ cout << "sigma ";
+ for ( size_t i=0; i<sigma.size(); ++i ) {
+ cout << sigma[i] << " ";
+ }
+ cout << endl;
+#endif
+
+ for ( size_t i=0; i<sigma.size(); ++i ) {
+ sigma[i] = make_modular(sigma[i], R);
+ }
+
+#ifdef DEBUGFACTOR
+ cout << "sigma ";
+ for ( size_t i=0; i<sigma.size(); ++i ) {
+ cout << sigma[i] << " ";
+ }
+ cout << endl;
+#endif
+ DCOUT(END multivar_diophant);
+ 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));
+
+ DCOUTVAR(nu);
+
+ vector<ex> A(nu);
+ A[nu-1] = a;
+
+ for ( size_t j=nu; j>=2; --j ) {
+ ex x = I[j-2].x;
+ int alpha = I[j-2].evalpoint;
+ A[j-2] = A[j-1].subs(x==alpha);
+ A[j-2] = make_modular(A[j-2], R);
+ }
+
+#ifdef DEBUGFACTOR
+ cout << "A ";
+ for ( size_t i=0; i<A.size(); ++i) cout << A[i] << " ";
+ cout << endl;
+#endif
+
+ int maxdeg = a.degree(I.front().x);
+ for ( size_t i=1; i<I.size(); ++i ) {
+ int maxdeg2 = a.degree(I[i].x);
+ if ( maxdeg2 > maxdeg ) maxdeg = maxdeg2;
+ }
+ DCOUTVAR(maxdeg);
+
+ const size_t n = u.size();
+ DCOUTVAR(n);
+ vector<ex> U(n);
+ for ( size_t i=0; i<n; ++i ) {
+ U[i] = u[i].to_ex(x);
+ }
+#ifdef DEBUGFACTOR
+ cout << "U ";
+ for ( size_t i=0; i<U.size(); ++i) cout << U[i] << " ";
+ cout << endl;
+#endif
+
+ for ( size_t j=2; j<=nu; ++j ) {
+ DCOUTVAR(j);
+ vector<ex> U1 = U;
+ ex monomial = 1;
+ DCOUTVAR(U);
+ for ( size_t m=0; m<n; ++m) {
+ if ( lcU[m] != 1 ) {
+ ex coef = lcU[m];
+ for ( size_t i=j-1; i<nu-1; ++i ) {
+ coef = coef.subs(I[i].x == I[i].evalpoint);
+ }
+ coef = expand(coef);
+ coef = make_modular(coef, R);
+ int deg = U[m].degree(x);
+ U[m] = U[m] - U[m].lcoeff(x) * pow(x,deg) + coef * pow(x,deg);
+ }
+ }
+ DCOUTVAR(U);
+ ex Uprod = 1;
+ for ( size_t i=0; i<n; ++i ) {
+ Uprod *= U[i];
+ }
+ ex e = expand(A[j-1] - Uprod);
+ DCOUTVAR(e);
+
+ vector<EvalPoint> newI;
+ for ( size_t i=1; i<=j-2; ++i ) {
+ newI.push_back(I[i-1]);
+ }
+ DCOUTVAR(newI);
+
+ ex xj = I[j-2].x;
+ int alphaj = I[j-2].evalpoint;
+ size_t deg = A[j-1].degree(xj);
+ DCOUTVAR(deg);
+ for ( size_t k=1; k<=deg; ++k ) {
+ DCOUTVAR(k);
+ if ( !e.is_zero() ) {
+ DCOUTVAR(xj);
+ DCOUTVAR(alphaj);
+ monomial *= (xj - alphaj);
+ monomial = expand(monomial);
+ DCOUTVAR(monomial);
+ ex dif = e.diff(ex_to<symbol>(xj), k);
+ DCOUTVAR(dif);
+ ex c = dif.subs(xj==alphaj) / factorial(k);
+ DCOUTVAR(c);
+ if ( !c.is_zero() ) {
+ vector<ex> deltaU = multivar_diophant(U1, x, c, newI, maxdeg, p, cl_I_to_uint(l));
+ for ( size_t i=0; i<n; ++i ) {
+ DCOUTVAR(i);
+ DCOUTVAR(deltaU[i]);
+ deltaU[i] *= monomial;
+ U[i] += deltaU[i];
+ U[i] = make_modular(U[i], R);
+ U[i] = U[i].expand();
+ DCOUTVAR(U[i]);
+ }
+ ex Uprod = 1;
+ for ( size_t i=0; i<n; ++i ) {
+ Uprod *= U[i];
+ }
+ DCOUTVAR(Uprod.expand());
+ DCOUTVAR(A[j-1]);
+ e = expand(A[j-1] - Uprod);
+ e = make_modular(e, R);
+ DCOUTVAR(e);
+ }
+ else {
+ break;
+ }
+ }
+ }
+ }
+
+ ex acand = 1;
+ for ( size_t i=0; i<U.size(); ++i ) {
+ acand *= U[i];
+ }
+ DCOUTVAR(acand);
+ if ( expand(a-acand).is_zero() ) {
+ lst res;
+ 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 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) || is_a<add>(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.");
+}
+
+#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 bool checkdivisors(const lst& f, vector<numeric>& d)
+{
+ DCOUT(checkdivisors);
+ const int k = f.nops()-2;
+ DCOUTVAR(k);
+ DCOUTVAR(d.size());
+ numeric q, r;
+ d[0] = ex_to<numeric>(f.op(0) * f.op(f.nops()-1));
+ if ( d[0] == 1 && k == 1 && abs(f.op(1)) != 1 ) {
+ DCOUT(false);
+ DCOUT(END checkdivisors);
+ return false;
+ }
+ DCOUTVAR(d[0]);
+ for ( int i=1; i<=k; ++i ) {
+ DCOUTVAR(i);
+ DCOUTVAR(abs(f.op(i)));
+ q = ex_to<numeric>(abs(f.op(i)));
+ DCOUTVAR(q);
+ for ( int j=i-1; j>=0; --j ) {
+ r = d[j];
+ DCOUTVAR(r);
+ do {
+ r = gcd(r, q);
+ DCOUTVAR(r);
+ q = q/r;
+ DCOUTVAR(q);
+ } while ( r != 1 );
+ if ( q == 1 ) {
+ DCOUT(true);
+ DCOUT(END checkdivisors);
+ return true;
+ }
+ }
+ d[i] = q;
+ }
+ DCOUT(false);
+ DCOUT(END checkdivisors);
+ return false;
+}
+
+static bool generate_set(const ex& u, const ex& vn, const exset& syms, const ex& f, const numeric& modulus, vector<numeric>& a, vector<numeric>& d)
+{
+ // computation of d is actually not necessary
+ DCOUT(generate_set);
+ DCOUTVAR(u);
+ DCOUTVAR(vn);
+ DCOUTVAR(f);
+ DCOUTVAR(modulus);
+ 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 ) {
+ DCOUTVAR(*s);
+ 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]);
+ ++s;
+ }
+ DCOUTVAR(a);
+ DCOUTVAR(u0);
+ if ( gcd(u0,u0.diff(ex_to<symbol>(x))) != 1 ) {
+ continue;
+ }
+ if ( is_a<numeric>(vn) ) {
+ trying = false;
+ }
+ else {
+ DCOUT(do substitution);
+ lst fnum;
+ lst::const_iterator i = ex_to<lst>(f).begin();
+ fnum.append(*i++);
+ bool problem = false;
+ while ( i!=ex_to<lst>(f).end() ) {
+ ex fs = *i;
+ if ( !is_a<numeric>(fs) ) {
+ s = syms.begin();
+ ++s;
+ for ( size_t j=0; j<a.size(); ++j ) {
+ fs = fs.subs(*s == a[j]);
+ ++s;
+ }
+ if ( abs(fs) == 1 ) {
+ problem = true;
+ break;
+ }
+ }
+ fnum.append(fs);
+ ++i; ++i;
+ }
+ if ( problem ) {
+ return true;
+ }
+ ex con = u0.content(x);
+ fnum.append(con);
+ DCOUTVAR(fnum);
+ trying = checkdivisors(fnum, d);
+ }
+ } while ( trying );
+ DCOUT(END generate_set);
+ return false;
+}
+
+static ex factor_multivariate(const ex& poly, const exset& syms)
+{
+ 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) ) {
+#ifdef DEBUGFACTOR
+ return ::factor(cont) * ::factor(pp);
+#else
+ return factor(cont) * factor(pp);
+#endif
+ }
+
+ /* factor leading coefficient */
+ pp = pp.collect(x);
+ ex vn = pp.lcoeff(x);
+ pp = pp.expand();
+ ex vnlst;
+ if ( is_a<numeric>(vn) ) {
+ vnlst = lst(vn);
+ }
+ else {
+#ifdef DEBUGFACTOR
+ ex vnfactors = ::factor(vn);
+#else
+ ex vnfactors = factor(vn);
+#endif
+ vnlst = put_factors_into_lst(vnfactors);
+ }
+ DCOUTVAR(vnlst);
+
+ const numeric maxtrials = 3;
+ numeric modulus = (vnlst.nops()-1 > 3) ? vnlst.nops()-1 : 3;
+ DCOUTVAR(modulus);
+ numeric minimalr = -1;
+ vector<numeric> a(syms.size()-1, 0);
+ vector<numeric> d((vnlst.nops()-1)/2+1, 0);
+
+ while ( true ) {
+ numeric trialcount = 0;
+ ex u, delta;
+ unsigned int prime;
+ size_t factor_count;
+ ex ufac;
+ ex ufaclst;
+ while ( trialcount < maxtrials ) {
+ bool problem = generate_set(pp, vn, syms, vnlst, modulus, a, d);
+ DCOUTVAR(problem);
+ if ( problem ) {
+ ++modulus;
+ continue;
+ }
+ 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);
+ DCOUTVAR(u);
+
+ // determine proper prime
+ prime = 3;
+ DCOUTVAR(prime);
+ cl_modint_ring R = find_modint_ring(prime);
+ DCOUTVAR(u.lcoeff(x));
+ while ( true ) {
+ if ( irem(ex_to<numeric>(u.lcoeff(x)), prime) != 0 ) {
+ UniPoly modpoly(R, u, x);
+ UniFactorVec sqrfree_ufv;
+ squarefree(modpoly, sqrfree_ufv);
+ DCOUTVAR(sqrfree_ufv);
+ if ( sqrfree_ufv.factors.size() == 1 && sqrfree_ufv.factors.front().exp == 1 ) break;
+ }
+ prime = next_prime(prime);
+ DCOUTVAR(prime);
+ R = find_modint_ring(prime);
+ }
+
+#ifdef DEBUGFACTOR
+ ufac = ::factor(u);
+#else
+ ufac = factor(u);
+#endif
+ DCOUTVAR(ufac);
+ ufaclst = put_factors_into_lst(ufac);
+ DCOUTVAR(ufaclst);
+ factor_count = (ufaclst.nops()-1)/2;
+ DCOUTVAR(factor_count);
+
+ if ( factor_count <= 1 ) {
+ DCOUTVAR(poly);
+ DCOUT(END factor_multivariate);
+ return poly;
+ }
+
+ if ( minimalr < 0 ) {
+ minimalr = factor_count;
+ }
+ else if ( minimalr == factor_count ) {
+ ++trialcount;
+ ++modulus;
+ }
+ else if ( minimalr > factor_count ) {
+ minimalr = factor_count;
+ trialcount = 0;
+ }
+ DCOUTVAR(trialcount);
+ DCOUTVAR(minimalr);
+ if ( minimalr <= 1 ) {
+ DCOUTVAR(poly);
+ DCOUT(END factor_multivariate);
+ return poly;
+ }
+ }
+
+ vector<numeric> ftilde((vnlst.nops()-1)/2+1);
+ ftilde[0] = ex_to<numeric>(vnlst.op(0));
+ for ( size_t i=1; i<ftilde.size(); ++i ) {
+ ex ft = vnlst.op((i-1)*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<bool> used_flag((vnlst.nops()-1)/2+1, false);
+ vector<ex> D(factor_count, 1);
+ for ( size_t i=0; i<=factor_count; ++i ) {
+ DCOUTVAR(i);
+ numeric prefac;
+ if ( i == 0 ) {
+ prefac = ex_to<numeric>(ufaclst.op(0));
+ ftilde[0] = ftilde[0] / prefac;
+ vnlst.let_op(0) = vnlst.op(0) / prefac;
+ continue;
+ }
+ else {
+ prefac = ex_to<numeric>(ufaclst.op(2*(i-1)+1).lcoeff(x));
+ }
+ DCOUTVAR(prefac);
+ for ( size_t j=(vnlst.nops()-1)/2+1; j>0; --j ) {
+ DCOUTVAR(j);
+ DCOUTVAR(prefac);
+ DCOUTVAR(ftilde[j-1]);
+ if ( abs(ftilde[j-1]) == 1 ) {
+ used_flag[j-1] = true;
+ continue;
+ }
+ numeric g = gcd(prefac, ftilde[j-1]);
+ DCOUTVAR(g);
+ if ( g != 1 ) {
+ DCOUT(has_common_prime);
+ prefac = prefac / g;
+ numeric count = abs(iquo(g, ftilde[j-1]));
+ DCOUTVAR(count);
+ used_flag[j-1] = true;
+ if ( i > 0 ) {
+ if ( j == 1 ) {
+ D[i-1] = D[i-1] * pow(vnlst.op(0), count);
+ }
+ else {
+ D[i-1] = D[i-1] * pow(vnlst.op(2*(j-2)+1), count);
+ }
+ }
+ else {
+ ftilde[j-1] = ftilde[j-1] / prefac;
+ DCOUT(BREAK);
+ DCOUTVAR(ftilde[j-1]);
+ break;
+ }
+ ++j;
+ }
+ }
+ }
+ DCOUTVAR(D);
+
+ bool some_factor_unused = false;
+ for ( size_t i=0; i<used_flag.size(); ++i ) {
+ if ( !used_flag[i] ) {
+ some_factor_unused = true;
+ break;
+ }
+ }
+ if ( some_factor_unused ) {
+ DCOUT(some factor unused!);
+ continue;
+ }
+
+ vector<ex> C(factor_count);
+ DCOUTVAR(C);
+ DCOUTVAR(delta);
+ 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;
+ }
+ DCOUTVAR(Dtilde);
+ C[i] = D[i] * (ufaclst.op(2*i+1).lcoeff(x) / Dtilde);
+ }
+ }
+ 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;
+ if ( i == 0 ) {
+ ui = ufaclst.op(0);
+ }
+ else {
+ ui = ufaclst.op(2*(i-1)+1);
+ }
+ while ( true ) {
+ ex d = gcd(ui.lcoeff(x), Dtilde);
+ C[i] = D[i] * ( ui.lcoeff(x) / d );
+ ui = ui * ( Dtilde[i] / d );
+ delta = delta / ( Dtilde[i] / d );
+ if ( delta == 1 ) break;
+ ui = delta * ui;
+ C[i] = delta * C[i];
+ pp = pp * pow(delta, D.size()-1);
+ }
+ }
+ }
+ DCOUTVAR(C);
+
+ 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);
+ }
+ DCOUTVAR(epv);
+
+ // 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<factor_count; ++i ) {
+ if ( ufaclst[2*i+1].degree(x) > (int)maxdegree ) {
+ maxdegree = ufaclst[2*i+1].degree(x);
+ }
+ }
+ cl_I B = normmc * expt_pos(cl_I(2), maxdegree);
+ cl_I l = 1;
+ cl_I pl = prime;
+ while ( pl < B ) {
+ l = l + 1;
+ pl = pl * prime;
+ }
+
+ UniPolyVec uvec;
+ cl_modint_ring R = find_modint_ring(expt_pos(cl_I(prime),l));
+ for ( size_t i=0; i<(ufaclst.nops()-1)/2; ++i ) {
+ UniPoly newu(R, ufaclst.op(i*2+1), x);
+ uvec.push_back(newu);
+ }
+ DCOUTVAR(uvec);
+
+ ex res = hensel_multivar(ufaclst.op(0)*pp, x, epv, prime, l, uvec, C);
+ if ( res != lst() ) {
+ ex result = cont * ufaclst.op(0);
+ 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;
+ }
+ }
+}
+
+struct find_symbols_map : public map_function {
exset syms;
ex operator()(const ex& e)
{
static ex factor_sqrfree(const ex& poly)
{
// determine all symbols in poly
- FindSymbolsMap findsymbols;
+ find_symbols_map findsymbols;
findsymbols(poly);
if ( findsymbols.syms.size() == 0 ) {
return 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;
}
+struct apply_factor_map : public map_function {
+ unsigned options;
+ apply_factor_map(unsigned options_) : options(options_) { }
+ ex operator()(const ex& e)
+ {
+ if ( e.info(info_flags::polynomial) ) {
+#ifdef DEBUGFACTOR
+ return ::factor(e, options);
+#else
+ return factor(e, options);
+#endif
+ }
+ if ( is_a<add>(e) ) {
+ ex s1, s2;
+ for ( size_t i=0; i<e.nops(); ++i ) {
+ if ( e.op(i).info(info_flags::polynomial) ) {
+ s1 += e.op(i);
+ }
+ else {
+ s2 += e.op(i);
+ }
+ }
+ s1 = s1.eval();
+ s2 = s2.eval();
+#ifdef DEBUGFACTOR
+ return ::factor(s1, options) + s2.map(*this);
+#else
+ return factor(s1, options) + s2.map(*this);
+#endif
+ }
+ return e.map(*this);
+ }
+};
+
} // anonymous namespace
-ex factor(const ex& poly)
+#ifdef DEBUGFACTOR
+ex factor(const ex& poly, unsigned options = 0)
+#else
+ex factor(const ex& poly, unsigned options)
+#endif
{
+ // check arguments
+ if ( !poly.info(info_flags::polynomial) ) {
+ if ( options & factor_options::all ) {
+ options &= ~factor_options::all;
+ apply_factor_map factor_map(options);
+ return factor_map(poly);
+ }
+ return poly;
+ }
+
// determine all symbols in poly
- FindSymbolsMap findsymbols;
+ find_symbols_map findsymbols;
findsymbols(poly);
if ( findsymbols.syms.size() == 0 ) {
return poly;
return pow(f, sfpoly.op(1));
}
if ( is_a<mul>(sfpoly) ) {
+ // case: multiple factors
ex res = 1;
for ( size_t i=0; i<sfpoly.nops(); ++i ) {
const ex& t = sfpoly.op(i);
}
return res;
}
+ if ( is_a<symbol>(sfpoly) ) {
+ return poly;
+ }
// case: (polynomial)
ex f = factor_sqrfree(sfpoly);
return f;
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