--- /dev/null
+/** @file factor.cpp
+ *
+ * Polynomial factorization routines.
+ * Only univariate at the moment and completely non-optimized!
+ */
+
+/*
+ * GiNaC Copyright (C) 1999-2008 Johannes Gutenberg University Mainz, Germany
+ *
+ * This program is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 2 of the License, or
+ * (at your option) any later version.
+ *
+ * This program is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program; if not, write to the Free Software
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
+ */
+
+#include "factor.h"
+
+#include "ex.h"
+#include "numeric.h"
+#include "operators.h"
+#include "inifcns.h"
+#include "symbol.h"
+#include "relational.h"
+#include "power.h"
+#include "mul.h"
+#include "normal.h"
+#include "add.h"
+
+#include <algorithm>
+#include <list>
+#include <vector>
+using namespace std;
+
+#include <cln/cln.h>
+using namespace cln;
+
+//#define DEBUGFACTOR
+
+#ifdef DEBUGFACTOR
+#include <ostream>
+#endif // def DEBUGFACTOR
+
+namespace GiNaC {
+
+namespace {
+
+typedef vector<cl_MI> Vec;
+typedef vector<Vec> VecVec;
+
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const Vec& v)
+{
+ Vec::const_iterator i = v.begin(), end = v.end();
+ while ( i != end ) {
+ o << *i++ << " ";
+ }
+ return o;
+}
+#endif // def DEBUGFACTOR
+
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const VecVec& v)
+{
+ VecVec::const_iterator i = v.begin(), end = v.end();
+ while ( i != end ) {
+ o << *i++ << endl;
+ }
+ return o;
+}
+#endif // def DEBUGFACTOR
+
+struct Term
+{
+ cl_MI c; // coefficient
+ unsigned int exp; // exponent >=0
+};
+
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const Term& t)
+{
+ if ( t.exp ) {
+ o << "(" << t.c << ")x^" << t.exp;
+ }
+ else {
+ o << "(" << t.c << ")";
+ }
+ return o;
+}
+#endif // def DEBUGFACTOR
+
+struct UniPoly
+{
+ cl_modint_ring R;
+ list<Term> terms; // highest exponent first
+
+ UniPoly(const cl_modint_ring& ring) : R(ring) { }
+ UniPoly(const cl_modint_ring& ring, const ex& poly, const ex& x) : R(ring)
+ {
+ // 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 ) {
+ t.c = R->canonhom(coeff);
+ if ( !zerop(t.c) ) {
+ t.exp = i;
+ terms.push_back(t);
+ }
+ }
+ }
+ }
+ UniPoly(const cl_modint_ring& ring, const Vec& v) : R(ring)
+ {
+ Term t;
+ for ( unsigned int i=0; i<v.size(); ++i ) {
+ if ( !zerop(v[i]) ) {
+ t.c = v[i];
+ t.exp = i;
+ terms.push_front(t);
+ }
+ }
+ }
+ unsigned int degree() const
+ {
+ if ( terms.size() ) {
+ return terms.front().exp;
+ }
+ else {
+ return 0;
+ }
+ }
+ bool zero() const { return (terms.size() == 0); }
+ const cl_MI operator[](unsigned int deg) const
+ {
+ list<Term>::const_iterator i = terms.begin(), end = terms.end();
+ for ( ; i != end; ++i ) {
+ if ( i->exp == deg ) {
+ return i->c;
+ }
+ if ( i->exp < deg ) {
+ break;
+ }
+ }
+ return R->zero();
+ }
+ void set(unsigned int deg, const cl_MI& c)
+ {
+ list<Term>::iterator i = terms.begin(), end = terms.end();
+ while ( i != end ) {
+ if ( i->exp == deg ) {
+ if ( !zerop(c) ) {
+ i->c = c;
+ }
+ else {
+ terms.erase(i);
+ }
+ return;
+ }
+ if ( i->exp < deg ) {
+ break;
+ }
+ ++i;
+ }
+ if ( !zerop(c) ) {
+ Term t;
+ t.c = c;
+ t.exp = deg;
+ terms.insert(i, t);
+ }
+ }
+ ex to_ex(const ex& x, bool symmetric = true) const
+ {
+ ex r;
+ list<Term>::const_iterator i = terms.begin(), end = terms.end();
+ if ( symmetric ) {
+ numeric mod(R->modulus);
+ numeric halfmod = (mod-1)/2;
+ for ( ; i != end; ++i ) {
+ numeric n(R->retract(i->c));
+ if ( n > halfmod ) {
+ r += pow(x, i->exp) * (n-mod);
+ }
+ else {
+ r += pow(x, i->exp) * n;
+ }
+ }
+ }
+ else {
+ for ( ; i != end; ++i ) {
+ r += pow(x, i->exp) * numeric(R->retract(i->c));
+ }
+ }
+ return r;
+ }
+ void unit_normal()
+ {
+ if ( terms.size() ) {
+ if ( terms.front().c != R->one() ) {
+ list<Term>::iterator i = terms.begin(), end = terms.end();
+ cl_MI cont = i->c;
+ i->c = R->one();
+ while ( ++i != end ) {
+ i->c = div(i->c, cont);
+ if ( zerop(i->c) ) {
+ terms.erase(i);
+ }
+ }
+ }
+ }
+ }
+ cl_MI unit() const
+ {
+ return terms.front().c;
+ }
+ void divide(const cl_MI& x)
+ {
+ list<Term>::iterator i = terms.begin(), end = terms.end();
+ for ( ; i != end; ++i ) {
+ i->c = div(i->c, x);
+ if ( zerop(i->c) ) {
+ terms.erase(i);
+ }
+ }
+ }
+ void reduce_exponents(unsigned int prime)
+ {
+ list<Term>::iterator i = terms.begin(), end = terms.end();
+ while ( i != end ) {
+ if ( i->exp > 0 ) {
+ // assert: i->exp is multiple of prime
+ i->exp /= prime;
+ }
+ ++i;
+ }
+ }
+ void deriv(UniPoly& d) const
+ {
+ list<Term>::const_iterator i = terms.begin(), end = terms.end();
+ while ( i != end ) {
+ if ( i->exp ) {
+ cl_MI newc = i->c * i->exp;
+ if ( !zerop(newc) ) {
+ Term t;
+ t.c = newc;
+ t.exp = i->exp-1;
+ d.terms.push_back(t);
+ }
+ }
+ ++i;
+ }
+ }
+ bool operator<(const UniPoly& o) const
+ {
+ if ( terms.size() != o.terms.size() ) {
+ return terms.size() < o.terms.size();
+ }
+ list<Term>::const_iterator i1 = terms.begin(), end = terms.end();
+ list<Term>::const_iterator i2 = o.terms.begin();
+ while ( i1 != end ) {
+ if ( i1->exp != i2->exp ) {
+ return i1->exp < i2->exp;
+ }
+ if ( i1->c != i2->c ) {
+ return R->retract(i1->c) < R->retract(i2->c);
+ }
+ ++i1; ++i2;
+ }
+ return true;
+ }
+ bool operator==(const UniPoly& o) const
+ {
+ if ( terms.size() != o.terms.size() ) {
+ return false;
+ }
+ list<Term>::const_iterator i1 = terms.begin(), end = terms.end();
+ list<Term>::const_iterator i2 = o.terms.begin();
+ while ( i1 != end ) {
+ if ( i1->exp != i2->exp ) {
+ return false;
+ }
+ if ( i1->c != i2->c ) {
+ return false;
+ }
+ ++i1; ++i2;
+ }
+ return true;
+ }
+ bool operator!=(const UniPoly& o) const
+ {
+ bool res = !(*this == o);
+ return res;
+ }
+};
+
+static UniPoly operator*(const UniPoly& a, const UniPoly& b)
+{
+ unsigned int n = a.degree()+b.degree();
+ UniPoly c(a.R);
+ Term t;
+ for ( unsigned int i=0 ; i<=n; ++i ) {
+ t.c = a.R->zero();
+ for ( unsigned int j=0 ; j<=i; ++j ) {
+ t.c = t.c + a[j] * b[i-j];
+ }
+ 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);
+ c.terms.back().c = -c.terms.back().c;
+ ++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);
+ ++ia;
+ }
+ while ( ib != bend ) {
+ c.terms.push_back(*ib);
+ c.terms.back().c = -c.terms.back().c;
+ ++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)
+{
+ list<Term>::const_iterator i = t.terms.begin(), end = t.terms.end();
+ if ( i == end ) {
+ o << "0";
+ return o;
+ }
+ for ( ; i != end; ) {
+ o << *i++;
+ if ( i != end ) {
+ o << " + ";
+ }
+ }
+ return o;
+}
+#endif // def DEBUGFACTOR
+
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const list<UniPoly>& t)
+{
+ list<UniPoly>::const_iterator i = t.begin(), end = t.end();
+ o << "{" << endl;
+ for ( ; i != end; ) {
+ o << *i++ << endl;
+ }
+ o << "}" << endl;
+ return o;
+}
+#endif // def DEBUGFACTOR
+
+typedef vector<UniPoly> UniPolyVec;
+
+struct UniFactor
+{
+ UniPoly p;
+ unsigned int exp;
+
+ UniFactor(const cl_modint_ring& ring) : p(ring) { }
+ UniFactor(const UniPoly& p_, unsigned int exp_) : p(p_), exp(exp_) { }
+ bool operator<(const UniFactor& o) const
+ {
+ return p < o.p;
+ }
+};
+
+struct UniFactorVec
+{
+ vector<UniFactor> factors;
+
+ void unique()
+ {
+ sort(factors.begin(), factors.end());
+ if ( factors.size() > 1 ) {
+ vector<UniFactor>::iterator i = factors.begin();
+ vector<UniFactor>::const_iterator cmp = factors.begin()+1;
+ vector<UniFactor>::iterator end = factors.end();
+ while ( cmp != end ) {
+ if ( i->p != cmp->p ) {
+ ++i;
+ ++cmp;
+ }
+ else {
+ i->exp += cmp->exp;
+ ++cmp;
+ }
+ }
+ if ( i != end-1 ) {
+ factors.erase(i+1, end);
+ }
+ }
+ }
+};
+
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const UniFactorVec& ufv)
+{
+ for ( size_t i=0; i<ufv.factors.size(); ++i ) {
+ if ( i != ufv.factors.size()-1 ) {
+ o << "*";
+ }
+ else {
+ o << " ";
+ }
+ o << "[ " << ufv.factors[i].p << " ]^" << ufv.factors[i].exp << endl;
+ }
+ return o;
+}
+#endif // def DEBUGFACTOR
+
+static void rem(const UniPoly& a_, const UniPoly& b, UniPoly& c)
+{
+ if ( a_.degree() < b.degree() ) {
+ c = a_;
+ return;
+ }
+
+ unsigned int k, n;
+ n = b.degree();
+ k = a_.degree() - n;
+
+ if ( n == 0 ) {
+ c.terms.clear();
+ return;
+ }
+
+ c = a_;
+ Term termbuf;
+
+ while ( true ) {
+ cl_MI qk = div(c[n+k], b[n]);
+ if ( !zerop(qk) ) {
+ unsigned int j;
+ for ( unsigned int i=0; i<n; ++i ) {
+ j = n + k - 1 - i;
+ c.set(j, c[j] - qk*b[j-k]);
+ }
+ }
+ if ( k == 0 ) break;
+ --k;
+ }
+ list<Term>::iterator i = c.terms.begin(), end = c.terms.end();
+ while ( i != end ) {
+ if ( i->exp <= n-1 ) {
+ break;
+ }
+ ++i;
+ }
+ c.terms.erase(c.terms.begin(), i);
+}
+
+static void div(const UniPoly& a_, const UniPoly& b, UniPoly& q)
+{
+ if ( a_.degree() < b.degree() ) {
+ q.terms.clear();
+ return;
+ }
+
+ unsigned int k, n;
+ n = b.degree();
+ k = a_.degree() - n;
+
+ UniPoly c = a_;
+ Term termbuf;
+
+ while ( true ) {
+ cl_MI qk = div(c[n+k], b[n]);
+ if ( !zerop(qk) ) {
+ Term t;
+ t.c = qk;
+ t.exp = k;
+ q.terms.push_back(t);
+ unsigned int j;
+ for ( unsigned int i=0; i<n; ++i ) {
+ j = n + k - 1 - i;
+ c.set(j, c[j] - qk*b[j-k]);
+ }
+ }
+ if ( k == 0 ) break;
+ --k;
+ }
+}
+
+static void gcd(const UniPoly& a, const UniPoly& b, UniPoly& c)
+{
+ c = a;
+ c.unit_normal();
+ UniPoly d = b;
+ d.unit_normal();
+
+ if ( c.degree() < d.degree() ) {
+ gcd(b, a, c);
+ return;
+ }
+
+ while ( !d.zero() ) {
+ UniPoly r(a.R);
+ rem(c, d, r);
+ c = d;
+ d = r;
+ }
+ c.unit_normal();
+}
+
+static bool is_one(const UniPoly& w)
+{
+ if ( w.terms.size() == 1 && w[0] == w.R->one() ) {
+ return true;
+ }
+ return false;
+}
+
+static void sqrfree_main(const UniPoly& a, UniFactorVec& fvec)
+{
+ unsigned int i = 1;
+ UniPoly b(a.R);
+ a.deriv(b);
+ if ( !b.zero() ) {
+ UniPoly c(a.R), w(a.R);
+ gcd(a, b, c);
+ div(a, c, w);
+ while ( !is_one(w) ) {
+ UniPoly y(a.R), z(a.R);
+ gcd(w, c, y);
+ div(w, y, z);
+ if ( !is_one(z) ) {
+ UniFactor uf(z, i++);
+ fvec.factors.push_back(uf);
+ }
+ w = y;
+ UniPoly cbuf(a.R);
+ div(c, y, cbuf);
+ c = cbuf;
+ }
+ if ( !is_one(c) ) {
+ unsigned int prime = cl_I_to_uint(c.R->modulus);
+ c.reduce_exponents(prime);
+ unsigned int pos = fvec.factors.size();
+ sqrfree_main(c, fvec);
+ for ( unsigned int p=pos; p<fvec.factors.size(); ++p ) {
+ fvec.factors[p].exp *= prime;
+ }
+ return;
+ }
+ }
+ else {
+ unsigned int prime = cl_I_to_uint(a.R->modulus);
+ UniPoly amod = a;
+ amod.reduce_exponents(prime);
+ unsigned int pos = fvec.factors.size();
+ sqrfree_main(amod, fvec);
+ for ( unsigned int p=pos; p<fvec.factors.size(); ++p ) {
+ fvec.factors[p].exp *= prime;
+ }
+ return;
+ }
+}
+
+static void squarefree(const UniPoly& a, UniFactorVec& fvec)
+{
+ sqrfree_main(a, fvec);
+ fvec.unique();
+}
+
+class Matrix
+{
+ friend ostream& operator<<(ostream& o, const Matrix& m);
+public:
+ Matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
+ {
+ m.resize(c*r, init);
+ }
+ size_t rowsize() const { return r; }
+ size_t colsize() const { return c; }
+ cl_MI& operator()(size_t row, size_t col) { return m[row*c + col]; }
+ cl_MI operator()(size_t row, size_t col) const { return m[row*c + col]; }
+ void mul_col(size_t col, const cl_MI x)
+ {
+ Vec::iterator i = m.begin() + col;
+ for ( size_t rc=0; rc<r; ++rc ) {
+ *i = *i * x;
+ i += c;
+ }
+ }
+ void sub_col(size_t col1, size_t col2, const cl_MI fac)
+ {
+ Vec::iterator i1 = m.begin() + col1;
+ Vec::iterator i2 = m.begin() + col2;
+ for ( size_t rc=0; rc<r; ++rc ) {
+ *i1 = *i1 - *i2 * fac;
+ i1 += c;
+ i2 += c;
+ }
+ }
+ void switch_col(size_t col1, size_t col2)
+ {
+ cl_MI buf;
+ Vec::iterator i1 = m.begin() + col1;
+ Vec::iterator i2 = m.begin() + col2;
+ for ( size_t rc=0; rc<r; ++rc ) {
+ buf = *i1; *i1 = *i2; *i2 = buf;
+ i1 += c;
+ i2 += c;
+ }
+ }
+ bool is_row_zero(size_t row) const
+ {
+ Vec::const_iterator i = m.begin() + row*c;
+ for ( size_t cc=0; cc<c; ++cc ) {
+ if ( !zerop(*i) ) {
+ return false;
+ }
+ ++i;
+ }
+ return true;
+ }
+ void set_row(size_t row, const vector<cl_MI>& newrow)
+ {
+ Vec::iterator i1 = m.begin() + row*c;
+ Vec::const_iterator i2 = newrow.begin(), end = newrow.end();
+ for ( ; i2 != end; ++i1, ++i2 ) {
+ *i1 = *i2;
+ }
+ }
+ Vec::const_iterator row_begin(size_t row) const { return m.begin()+row*c; }
+ Vec::const_iterator row_end(size_t row) const { return m.begin()+row*c+r; }
+private:
+ size_t r, c;
+ Vec m;
+};
+
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const Matrix& m)
+{
+ vector<cl_MI>::const_iterator i = m.m.begin(), end = m.m.end();
+ size_t wrap = 1;
+ for ( ; i != end; ++i ) {
+ o << *i << " ";
+ if ( !(wrap++ % m.c) ) {
+ o << endl;
+ }
+ }
+ o << endl;
+ return o;
+}
+#endif // def DEBUGFACTOR
+
+static void q_matrix(const UniPoly& a, Matrix& Q)
+{
+ unsigned int n = a.degree();
+ unsigned int q = cl_I_to_uint(a.R->modulus);
+ vector<cl_MI> r(n, a.R->zero());
+ r[0] = a.R->one();
+ Q.set_row(0, r);
+ unsigned int max = (n-1) * q;
+ for ( size_t m=1; m<=max; ++m ) {
+ cl_MI rn_1 = r.back();
+ for ( size_t i=n-1; i>0; --i ) {
+ r[i] = r[i-1] - rn_1 * a[i];
+ }
+ r[0] = -rn_1 * a[0];
+ if ( (m % q) == 0 ) {
+ Q.set_row(m/q, r);
+ }
+ }
+}
+
+static void nullspace(Matrix& M, vector<Vec>& basis)
+{
+ const size_t n = M.rowsize();
+ const cl_MI one = M(0,0).ring()->one();
+ for ( size_t i=0; i<n; ++i ) {
+ M(i,i) = M(i,i) - one;
+ }
+ for ( size_t r=0; r<n; ++r ) {
+ size_t cc = 0;
+ for ( ; cc<n; ++cc ) {
+ if ( !zerop(M(r,cc)) ) {
+ if ( cc < r ) {
+ if ( !zerop(M(cc,cc)) ) {
+ continue;
+ }
+ M.switch_col(cc, r);
+ }
+ else if ( cc > r ) {
+ M.switch_col(cc, r);
+ }
+ break;
+ }
+ }
+ if ( cc < n ) {
+ M.mul_col(r, recip(M(r,r)));
+ for ( cc=0; cc<n; ++cc ) {
+ if ( cc != r ) {
+ M.sub_col(cc, r, M(r,cc));
+ }
+ }
+ }
+ }
+
+ for ( size_t i=0; i<n; ++i ) {
+ M(i,i) = M(i,i) - one;
+ }
+ for ( size_t i=0; i<n; ++i ) {
+ if ( !M.is_row_zero(i) ) {
+ Vec nu(M.row_begin(i), M.row_end(i));
+ basis.push_back(nu);
+ }
+ }
+}
+
+static void berlekamp(const UniPoly& a, UniPolyVec& upv)
+{
+ Matrix Q(a.degree(), a.degree(), a.R->zero());
+ q_matrix(a, Q);
+ VecVec nu;
+ nullspace(Q, nu);
+ const unsigned int k = nu.size();
+ if ( k == 1 ) {
+ return;
+ }
+
+ list<UniPoly> factors;
+ factors.push_back(a);
+ unsigned int size = 1;
+ unsigned int r = 1;
+ unsigned int q = cl_I_to_uint(a.R->modulus);
+
+ list<UniPoly>::iterator u = factors.begin();
+
+ while ( true ) {
+ for ( unsigned int s=0; s<q; ++s ) {
+ UniPoly g(a.R);
+ UniPoly nur(a.R, nu[r]);
+ nur.set(0, nur[0] - cl_MI(a.R, s));
+ gcd(nur, *u, g);
+ if ( !is_one(g) && g != *u ) {
+ UniPoly uo(a.R);
+ div(*u, g, uo);
+ if ( is_one(uo) ) {
+ throw logic_error("berlekamp: unexpected divisor.");
+ }
+ else {
+ *u = uo;
+ }
+ factors.push_back(g);
+ ++size;
+ if ( size == k ) {
+ list<UniPoly>::const_iterator i = factors.begin(), end = factors.end();
+ while ( i != end ) {
+ upv.push_back(*i++);
+ }
+ return;
+ }
+ if ( u->degree() < nur.degree() ) {
+ break;
+ }
+ }
+ }
+ if ( ++r == k ) {
+ r = 1;
+ ++u;
+ }
+ }
+}
+
+static void factor_modular(const UniPoly& p, UniPolyVec& upv)
+{
+ berlekamp(p, upv);
+ return;
+}
+
+static void exteuclid(const UniPoly& a, const UniPoly& b, UniPoly& g, UniPoly& s, UniPoly& t)
+{
+ if ( a.degree() < b.degree() ) {
+ exteuclid(b, a, g, t, s);
+ return;
+ }
+ UniPoly c1(a.R), c2(a.R), d1(a.R), d2(a.R), q(a.R), r(a.R), r1(a.R), r2(a.R);
+ UniPoly c = a; c.unit_normal();
+ UniPoly d = b; d.unit_normal();
+ c1.set(0, a.R->one());
+ d2.set(0, a.R->one());
+ while ( !d.zero() ) {
+ q.terms.clear();
+ div(c, d, q);
+ r = c - q * d;
+ r1 = c1 - q * d1;
+ r2 = c2 - q * d2;
+ c = d;
+ c1 = d1;
+ c2 = d2;
+ d = r;
+ d1 = r1;
+ d2 = r2;
+ }
+ g = c; g.unit_normal();
+ s = c1;
+ s.divide(a.unit());
+ s.divide(c.unit());
+ t = c2;
+ t.divide(b.unit());
+ t.divide(c.unit());
+}
+
+static ex replace_lc(const ex& poly, const ex& x, const ex& lc)
+{
+ ex r = expand(poly + (lc - poly.lcoeff(x)) * pow(x, poly.degree(x)));
+ return r;
+}
+
+static ex hensel_univar(const ex& a_, const ex& x, unsigned int p, const UniPoly& u1_, const UniPoly& w1_, const ex& gamma_ = 0)
+{
+ ex a = a_;
+ const cl_modint_ring& R = u1_.R;
+
+ // calc bound B
+ ex maxcoeff;
+ 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));
+
+ // step 1
+ ex alpha = a.lcoeff(x);
+ ex gamma = gamma_;
+ if ( gamma == 0 ) {
+ gamma = alpha;
+ }
+ unsigned int gamma_ui = ex_to<numeric>(abs(gamma)).to_int();
+ a = a * gamma;
+ UniPoly nu1 = u1_;
+ nu1.unit_normal();
+ UniPoly nw1 = w1_;
+ nw1.unit_normal();
+ ex phi;
+ phi = expand(gamma * nu1.to_ex(x));
+ UniPoly u1(R, phi, x);
+ phi = expand(alpha * nw1.to_ex(x));
+ UniPoly w1(R, phi, x);
+
+ // step 2
+ UniPoly s(R), t(R), g(R);
+ exteuclid(u1, w1, g, s, t);
+
+ // step 3
+ 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;
+
+ // step 4
+ while ( !e.is_zero() && modulus < 2*B*gamma_ui ) {
+ ex c = e / modulus;
+ phi = expand(s.to_ex(x)*c);
+ UniPoly sigmatilde(R, phi, x);
+ phi = expand(t.to_ex(x)*c);
+ UniPoly tautilde(R, phi, x);
+ UniPoly q(R), r(R);
+ div(sigmatilde, w1, q);
+ rem(sigmatilde, w1, r);
+ UniPoly sigma = r;
+ phi = expand(tautilde.to_ex(x) + q.to_ex(x) * u1.to_ex(x));
+ UniPoly tau(R, phi, x);
+ u = expand(u + tau.to_ex(x) * modulus);
+ w = expand(w + sigma.to_ex(x) * modulus);
+ e = expand(a - u * w);
+ modulus = modulus * p;
+ }
+
+ // step 5
+ if ( e.is_zero() ) {
+ ex delta = u.content(x);
+ u = u / delta;
+ w = w / gamma * delta;
+ return lst(u, w);
+ }
+ else {
+ return lst();
+ }
+}
+
+static unsigned int next_prime(unsigned int p)
+{
+ static vector<unsigned int> primes;
+ if ( primes.size() == 0 ) {
+ primes.push_back(3); primes.push_back(5); primes.push_back(7);
+ }
+ vector<unsigned int>::const_iterator it = primes.begin();
+ if ( p >= primes.back() ) {
+ unsigned int candidate = primes.back() + 2;
+ while ( true ) {
+ size_t n = primes.size()/2;
+ for ( size_t i=0; i<n; ++i ) {
+ if ( candidate % primes[i] ) continue;
+ candidate += 2;
+ i=-1;
+ }
+ primes.push_back(candidate);
+ if ( candidate > p ) break;
+ }
+ return candidate;
+ }
+ vector<unsigned int>::const_iterator end = primes.end();
+ for ( ; it!=end; ++it ) {
+ if ( *it > p ) {
+ return *it;
+ }
+ }
+ throw logic_error("next_prime: should not reach this point!");
+}
+
+class Partition
+{
+public:
+ Partition(size_t n_) : n(n_)
+ {
+ k.resize(n, 1);
+ k[0] = 0;
+ sum = n-1;
+ }
+ int operator[](size_t i) const { return k[i]; }
+ size_t size() const { return n; }
+ size_t size_first() const { return n-sum; }
+ size_t size_second() const { return sum; }
+ bool next()
+ {
+ for ( size_t i=n-1; i>=1; --i ) {
+ if ( k[i] ) {
+ --k[i];
+ --sum;
+ return sum > 0;
+ }
+ ++k[i];
+ ++sum;
+ }
+ return false;
+ }
+private:
+ size_t n, sum;
+ vector<int> k;
+};
+
+static void split(const UniPolyVec& factors, const Partition& part, UniPoly& a, UniPoly& b)
+{
+ a.set(0, a.R->one());
+ b.set(0, a.R->one());
+ for ( size_t i=0; i<part.size(); ++i ) {
+ if ( part[i] ) {
+ b = b * factors[i];
+ }
+ else {
+ a = a * factors[i];
+ }
+ }
+}
+
+struct ModFactors
+{
+ ex poly;
+ UniPolyVec factors;
+};
+
+static ex factor_univariate(const ex& poly, const ex& x)
+{
+ ex unit, cont, prim;
+ poly.unitcontprim(x, unit, cont, prim);
+
+ // determine proper prime
+ unsigned int p = 3;
+ cl_modint_ring R = find_modint_ring(p);
+ while ( true ) {
+ if ( irem(ex_to<numeric>(prim.lcoeff(x)), p) != 0 ) {
+ UniPoly modpoly(R, prim, x);
+ UniFactorVec sqrfree_ufv;
+ squarefree(modpoly, sqrfree_ufv);
+ if ( sqrfree_ufv.factors.size() == 1 ) break;
+ }
+ p = next_prime(p);
+ R = find_modint_ring(p);
+ }
+
+ // do modular factorization
+ UniPoly modpoly(R, prim, x);
+ UniPolyVec factors;
+ factor_modular(modpoly, factors);
+ if ( factors.size() <= 1 ) {
+ // irreducible for sure
+ return poly;
+ }
+
+ // lift all factor combinations
+ stack<ModFactors> tocheck;
+ ModFactors mf;
+ mf.poly = prim;
+ mf.factors = factors;
+ tocheck.push(mf);
+ ex result = 1;
+ while ( tocheck.size() ) {
+ const size_t n = tocheck.top().factors.size();
+ Partition part(n);
+ while ( true ) {
+ UniPoly a(R), b(R);
+ split(tocheck.top().factors, part, a, b);
+
+ ex answer = hensel_univar(tocheck.top().poly, x, p, a, b);
+ if ( answer != lst() ) {
+ if ( part.size_first() == 1 ) {
+ if ( part.size_second() == 1 ) {
+ result *= answer.op(0) * answer.op(1);
+ tocheck.pop();
+ break;
+ }
+ result *= answer.op(0);
+ tocheck.top().poly = answer.op(1);
+ for ( size_t i=0; i<n; ++i ) {
+ if ( part[i] == 0 ) {
+ tocheck.top().factors.erase(tocheck.top().factors.begin()+i);
+ break;
+ }
+ }
+ break;
+ }
+ else if ( part.size_second() == 1 ) {
+ if ( part.size_first() == 1 ) {
+ result *= answer.op(0) * answer.op(1);
+ tocheck.pop();
+ break;
+ }
+ result *= answer.op(1);
+ tocheck.top().poly = answer.op(0);
+ for ( size_t i=0; i<n; ++i ) {
+ if ( part[i] == 1 ) {
+ tocheck.top().factors.erase(tocheck.top().factors.begin()+i);
+ break;
+ }
+ }
+ break;
+ }
+ else {
+ UniPolyVec newfactors1(part.size_first(), R), newfactors2(part.size_second(), R);
+ UniPolyVec::iterator i1 = newfactors1.begin(), i2 = newfactors2.begin();
+ for ( size_t i=0; i<n; ++i ) {
+ if ( part[i] ) {
+ *i2++ = tocheck.top().factors[i];
+ }
+ else {
+ *i1++ = tocheck.top().factors[i];
+ }
+ }
+ tocheck.top().factors = newfactors1;
+ tocheck.top().poly = answer.op(0);
+ ModFactors mf;
+ mf.factors = newfactors2;
+ mf.poly = answer.op(1);
+ tocheck.push(mf);
+ }
+ }
+ else {
+ if ( !part.next() ) {
+ result *= tocheck.top().poly;
+ tocheck.pop();
+ break;
+ }
+ }
+ }
+ }
+
+ return unit * cont * result;
+}
+
+struct FindSymbolsMap : public map_function {
+ exset syms;
+ ex operator()(const ex& e)
+ {
+ if ( is_a<symbol>(e) ) {
+ syms.insert(e);
+ return e;
+ }
+ return e.map(*this);
+ }
+};
+
+static ex factor_sqrfree(const ex& poly)
+{
+ // determine all symbols in poly
+ FindSymbolsMap findsymbols;
+ findsymbols(poly);
+ if ( findsymbols.syms.size() == 0 ) {
+ return poly;
+ }
+
+ if ( findsymbols.syms.size() == 1 ) {
+ const ex& x = *(findsymbols.syms.begin());
+ if ( poly.ldegree(x) > 0 ) {
+ int ld = poly.ldegree(x);
+ ex res = factor_univariate(expand(poly/pow(x, ld)), x);
+ return res * pow(x,ld);
+ }
+ else {
+ ex res = factor_univariate(poly, x);
+ return res;
+ }
+ }
+
+ // multivariate case not yet implemented!
+ throw runtime_error("multivariate case not yet implemented!");
+}
+
+} // anonymous namespace
+
+ex factor(const ex& poly)
+{
+ // determine all symbols in poly
+ FindSymbolsMap findsymbols;
+ findsymbols(poly);
+ if ( findsymbols.syms.size() == 0 ) {
+ return poly;
+ }
+ lst syms;
+ exset::const_iterator i=findsymbols.syms.begin(), end=findsymbols.syms.end();
+ for ( ; i!=end; ++i ) {
+ syms.append(*i);
+ }
+
+ // make poly square free
+ ex sfpoly = sqrfree(poly, syms);
+
+ // factorize the square free components
+ if ( is_a<power>(sfpoly) ) {
+ // case: (polynomial)^exponent
+ const ex& base = sfpoly.op(0);
+ if ( !is_a<add>(base) ) {
+ // simple case: (monomial)^exponent
+ return sfpoly;
+ }
+ ex f = factor_sqrfree(base);
+ return pow(f, sfpoly.op(1));
+ }
+ if ( is_a<mul>(sfpoly) ) {
+ ex res = 1;
+ for ( size_t i=0; i<sfpoly.nops(); ++i ) {
+ const ex& t = sfpoly.op(i);
+ if ( is_a<power>(t) ) {
+ const ex& base = t.op(0);
+ if ( !is_a<add>(base) ) {
+ res *= t;
+ }
+ else {
+ ex f = factor_sqrfree(base);
+ res *= pow(f, t.op(1));
+ }
+ }
+ else if ( is_a<add>(t) ) {
+ ex f = factor_sqrfree(t);
+ res *= f;
+ }
+ else {
+ res *= t;
+ }
+ }
+ return res;
+ }
+ // case: (polynomial)
+ ex f = factor_sqrfree(sfpoly);
+ return f;
+}
+
+} // namespace GiNaC
git://www.ginac.de / ginac.git / commitdiff