From: Jens Vollinga <jensv@nikhef.nl>
Date: Sat, 9 Aug 2008 08:11:40 +0000 (+0200)
Subject: Added polynomial factorization (univariate case).
X-Git-Tag: release_1-5-0~87
X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=commitdiff_plain;h=114449ae6f2cd3151d9b8342c570db021a9e892c

Added polynomial factorization (univariate case).
---

diff --git a/check/exam_factor.cpp b/check/exam_factor.cpp
new file mode 100644
index 00000000..ed064588
--- /dev/null
+++ b/check/exam_factor.cpp
@@ -0,0 +1,102 @@
+/** @file exam_factor.cpp
+ *
+ *  Factorization test suite. */
+
+/*
+ *  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 <iostream>
+#include "ginac.h"
+using namespace std;
+using namespace GiNaC;
+
+static symbol w("w"), x("x"), y("y"), z("z");
+
+static unsigned check_factor(const ex& e)
+{
+	ex ee = e.expand();
+	ex answer = factor(ee);
+	if ( answer.expand() != ee || answer != e ) {
+		clog << "factorization of " << e << " == " << ee << " gave wrong result: " << answer << endl;
+		return 1;
+	}
+	return 0;
+}
+
+static unsigned exam_factor1()
+{
+	unsigned result = 0;
+	ex e, d;
+	symbol x("x");
+	lst syms;
+	syms.append(x);
+	
+	e = ex("1+x-x^3", syms);
+	result += check_factor(e);
+
+	e = ex("1+x^6+x", syms);
+	result += check_factor(e);
+
+	e = ex("1-x^6+x", syms);
+	result += check_factor(e);
+
+	e = ex("(1+x)^3", syms);
+	result += check_factor(e);
+
+	e = ex("x^6-3*x^5+x^4-3*x^3-x^2-3*x+1", syms);
+	result += check_factor(e);
+
+	e = ex("(-1+x)^3*(1+x)^3*(1+x^2)", syms);
+	result += check_factor(e);
+
+	e = ex("-(-168+20*x-x^2)*(30+x)", syms);
+	result += check_factor(e);
+
+	e = ex("x^2*(x-3)^2*(x^3-5*x+7)", syms);
+	result += check_factor(e);
+
+	e = ex("-6*x^2*(x-3)", syms);
+	result += check_factor(e);
+
+	e = ex("x^16+11*x^4+121", syms);
+	result += check_factor(e);
+
+	e = ex("x^8-40*x^6+352*x^4-960*x^2+576", syms);
+	result += check_factor(e);
+
+	e = ex("x*(2+x^2)*(1+x+x^3+x^2+x^6+x^5+x^4)*(1+x^3)^2*(-1+x)", syms);
+	result += check_factor(e);
+
+	return result;
+}
+
+unsigned exam_factor()
+{
+	unsigned result = 0;
+
+	cout << "examining polynomial factorization" << flush;
+
+	result += exam_factor1(); cout << '.' << flush;
+
+	return result;
+}
+
+int main(int argc, char** argv)
+{
+	return exam_factor();
+}
diff --git a/ginac/factor.cpp b/ginac/factor.cpp
new file mode 100644
index 00000000..3a012405
--- /dev/null
+++ b/ginac/factor.cpp
@@ -0,0 +1,1221 @@
+/** @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
diff --git a/ginac/factor.h b/ginac/factor.h
new file mode 100644
index 00000000..e2ade545
--- /dev/null
+++ b/ginac/factor.h
@@ -0,0 +1,36 @@
+/** @file factor.h
+ *
+ *  Polynomial factorization routines. Implementation.
+ *  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
+ */
+
+#ifndef __GINAC_FACTOR_H__
+#define __GINAC_FACTOR_H__
+
+namespace GiNaC {
+
+class ex;
+
+extern ex factor(const ex& poly);
+
+} // namespace GiNaC
+
+#endif // ndef __GINAC_FACTOR_H__