/** @file factor.cpp
*
- * Polynomial factorization routines.
- * Only univariate at the moment and completely non-optimized!
+ * Polynomial factorization (implementation).
+ *
+ * The interface function factor() at the end of this file is defined in the
+ * GiNaC namespace. All other utility functions and classes are defined in an
+ * additional anonymous namespace.
+ *
+ * Factorization starts by doing a square free factorization and making the
+ * coefficients integer. Then, depending on the number of free variables it
+ * proceeds either in dedicated univariate or multivariate factorization code.
+ *
+ * Univariate factorization does a modular factorization via Berlekamp's
+ * algorithm and distinct degree factorization. Hensel lifting is used at the
+ * end.
+ *
+ * Multivariate factorization uses the univariate factorization (applying a
+ * evaluation homomorphism first) and Hensel lifting raises the answer to the
+ * multivariate domain. The Hensel lifting code is completely distinct from the
+ * code used by the univariate factorization.
+ *
+ * Algorithms used can be found in
+ * [Wan] 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.
+ * [Mig] Some Useful Bounds,
+ * M.Mignotte,
+ * In "Computer Algebra, Symbolic and Algebraic Computation" (B.Buchberger et al., eds.),
+ * pp. 259-263, Springer-Verlag, New York, 1982.
*/
/*
- * GiNaC Copyright (C) 1999-2008 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2014 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
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
+//#define DEBUGFACTOR
+
#include "factor.h"
#include "ex.h"
#include "add.h"
#include <algorithm>
+#include <cmath>
+#include <limits>
#include <list>
#include <vector>
+#ifdef DEBUGFACTOR
+#include <ostream>
+#endif
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)
+#define DCOUT(str) cout << #str << endl
+#define DCOUTVAR(var) cout << #var << ": " << var << endl
+#define DCOUT2(str,var) cout << #str << ": " << var << endl
+ostream& operator<<(ostream& o, const vector<int>& v)
{
- Vec::const_iterator i = v.begin(), end = v.end();
+ vector<int>::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)
+static ostream& operator<<(ostream& o, const vector<cl_I>& v)
{
- VecVec::const_iterator i = v.begin(), end = v.end();
+ vector<cl_I>::const_iterator i = v.begin(), end = v.end();
while ( i != end ) {
- o << *i++ << endl;
+ o << *i << "[" << i-v.begin() << "]" << " ";
+ ++i;
}
return o;
}
-#endif // def DEBUGFACTOR
-
-struct Term
+static ostream& operator<<(ostream& o, const vector<cl_MI>& v)
{
- cl_MI c; // coefficient
- unsigned int exp; // exponent >=0
-};
-
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const Term& t)
+ vector<cl_MI>::const_iterator i = v.begin(), end = v.end();
+ while ( i != end ) {
+ o << *i << "[" << i-v.begin() << "]" << " ";
+ ++i;
+ }
+ return o;
+}
+ostream& operator<<(ostream& o, const vector<numeric>& v)
{
- if ( t.exp ) {
- o << "(" << t.c << ")x^" << t.exp;
+ for ( size_t i=0; i<v.size(); ++i ) {
+ o << v[i] << " ";
}
- else {
- o << "(" << t.c << ")";
+ return o;
+}
+ostream& operator<<(ostream& o, const vector< vector<cl_MI> >& v)
+{
+ vector< vector<cl_MI> >::const_iterator i = v.begin(), end = v.end();
+ while ( i != end ) {
+ o << i-v.begin() << ": " << *i << endl;
+ ++i;
}
return o;
}
+#else
+#define DCOUT(str)
+#define DCOUTVAR(var)
+#define DCOUT2(str,var)
#endif // def DEBUGFACTOR
-struct UniPoly
+// anonymous namespace to hide all utility functions
+namespace {
+
+////////////////////////////////////////////////////////////////////////////////
+// modular univariate polynomial code
+
+typedef std::vector<cln::cl_MI> umodpoly;
+typedef std::vector<cln::cl_I> upoly;
+typedef vector<umodpoly> upvec;
+
+// COPY FROM UPOLY.HPP
+
+// CHANGED size_t -> int !!!
+template<typename T> static int degree(const T& p)
{
- 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);
- }
- }
+ return p.size() - 1;
+}
+
+template<typename T> static typename T::value_type lcoeff(const T& p)
+{
+ return p[p.size() - 1];
+}
+
+static bool normalize_in_field(umodpoly& a)
+{
+ if (a.size() == 0)
+ return true;
+ if ( lcoeff(a) == a[0].ring()->one() ) {
+ return true;
}
- unsigned int degree() const
- {
- if ( terms.size() ) {
- return terms.front().exp;
+
+ const cln::cl_MI lc_1 = recip(lcoeff(a));
+ for (std::size_t k = a.size(); k-- != 0; )
+ a[k] = a[k]*lc_1;
+ return false;
+}
+
+template<typename T> static void
+canonicalize(T& p, const typename T::size_type hint = std::numeric_limits<typename T::size_type>::max())
+{
+ if (p.empty())
+ return;
+
+ std::size_t i = p.size() - 1;
+ // Be fast if the polynomial is already canonicalized
+ if (!zerop(p[i]))
+ return;
+
+ if (hint < p.size())
+ i = hint;
+
+ bool is_zero = false;
+ do {
+ if (!zerop(p[i])) {
+ ++i;
+ break;
}
- else {
- return 0;
+ if (i == 0) {
+ is_zero = true;
+ break;
}
+ --i;
+ } while (true);
+
+ if (is_zero) {
+ p.clear();
+ return;
}
- 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();
+
+ p.erase(p.begin() + i, p.end());
+}
+
+// END COPY FROM UPOLY.HPP
+
+static void expt_pos(umodpoly& a, unsigned int q)
+{
+ if ( a.empty() ) return;
+ cl_MI zero = a[0].ring()->zero();
+ int deg = degree(a);
+ a.resize(degree(a)*q+1, zero);
+ for ( int i=deg; i>0; --i ) {
+ a[i*q] = a[i];
+ a[i] = 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;
+}
+
+template<bool COND, typename T = void> struct enable_if
+{
+ typedef T type;
+};
+
+template<typename T> struct enable_if<false, T> { /* empty */ };
+
+template<typename T> struct uvar_poly_p
+{
+ static const bool value = false;
+};
+
+template<> struct uvar_poly_p<upoly>
+{
+ static const bool value = true;
+};
+
+template<> struct uvar_poly_p<umodpoly>
+{
+ static const bool value = true;
+};
+
+template<typename T>
+// Don't define this for anything but univariate polynomials.
+static typename enable_if<uvar_poly_p<T>::value, T>::type
+operator+(const T& a, const T& b)
+{
+ int sa = a.size();
+ int sb = b.size();
+ if ( sa >= sb ) {
+ T r(sa);
+ int i = 0;
+ for ( ; i<sb; ++i ) {
+ r[i] = a[i] + b[i];
}
- if ( !zerop(c) ) {
- Term t;
- t.c = c;
- t.exp = deg;
- terms.insert(i, t);
+ for ( ; i<sa; ++i ) {
+ r[i] = a[i];
}
+ canonicalize(r);
+ return r;
}
- 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 {
+ T r(sb);
+ int i = 0;
+ for ( ; i<sa; ++i ) {
+ r[i] = a[i] + b[i];
}
- else {
- for ( ; i != end; ++i ) {
- r += pow(x, i->exp) * numeric(R->retract(i->c));
- }
+ for ( ; i<sb; ++i ) {
+ r[i] = b[i];
}
+ canonicalize(r);
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);
- }
+}
+
+template<typename T>
+// Don't define this for anything but univariate polynomials. Otherwise
+// overload resolution might fail (this actually happens when compiling
+// GiNaC with g++ 3.4).
+static typename enable_if<uvar_poly_p<T>::value, T>::type
+operator-(const T& a, const T& b)
+{
+ int sa = a.size();
+ int sb = b.size();
+ if ( sa >= sb ) {
+ T r(sa);
+ int i = 0;
+ for ( ; i<sb; ++i ) {
+ r[i] = a[i] - b[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;
+ for ( ; i<sa; ++i ) {
+ r[i] = a[i];
}
+ canonicalize(r);
+ return r;
}
- 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;
+ else {
+ T r(sb);
+ int i = 0;
+ for ( ; i<sa; ++i ) {
+ r[i] = a[i] - b[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;
+ for ( ; i<sb; ++i ) {
+ r[i] = -b[i];
}
- return true;
+ canonicalize(r);
+ return r;
}
- 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;
+}
+
+static upoly operator*(const upoly& a, const upoly& b)
+{
+ upoly c;
+ if ( a.empty() || b.empty() ) return c;
+
+ int n = degree(a) + degree(b);
+ c.resize(n+1, 0);
+ for ( int i=0 ; i<=n; ++i ) {
+ for ( int j=0 ; j<=i; ++j ) {
+ if ( j > degree(a) || (i-j) > degree(b) ) continue;
+ c[i] = c[i] + a[j] * b[i-j];
}
- return true;
- }
- bool operator!=(const UniPoly& o) const
- {
- bool res = !(*this == o);
- return res;
}
-};
+ canonicalize(c);
+ return c;
+}
-static UniPoly operator*(const UniPoly& a, const UniPoly& b)
+static umodpoly operator*(const umodpoly& a, const umodpoly& 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);
+ umodpoly c;
+ if ( a.empty() || b.empty() ) return c;
+
+ int n = degree(a) + degree(b);
+ c.resize(n+1, a[0].ring()->zero());
+ for ( int i=0 ; i<=n; ++i ) {
+ for ( int j=0 ; j<=i; ++j ) {
+ if ( j > degree(a) || (i-j) > degree(b) ) continue;
+ c[i] = c[i] + a[j] * b[i-j];
}
}
+ canonicalize(c);
return c;
}
-static UniPoly operator-(const UniPoly& a, const UniPoly& b)
+static upoly operator*(const upoly& a, const cl_I& x)
{
- 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;
- }
+ if ( zerop(x) ) {
+ upoly r;
+ return r;
}
- while ( ia != aend ) {
- c.terms.push_back(*ia);
- ++ia;
+ upoly r(a.size());
+ for ( size_t i=0; i<a.size(); ++i ) {
+ r[i] = a[i] * x;
+ }
+ return r;
+}
+
+static upoly operator/(const upoly& a, const cl_I& x)
+{
+ if ( zerop(x) ) {
+ upoly r;
+ return r;
}
- while ( ib != bend ) {
- c.terms.push_back(*ib);
- c.terms.back().c = -c.terms.back().c;
- ++ib;
+ upoly r(a.size());
+ for ( size_t i=0; i<a.size(); ++i ) {
+ r[i] = exquo(a[i],x);
}
- return c;
+ return r;
}
-static UniPoly operator-(const UniPoly& a)
+static umodpoly operator*(const umodpoly& a, const cl_MI& x)
{
- 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;
+ umodpoly r(a.size());
+ for ( size_t i=0; i<a.size(); ++i ) {
+ r[i] = a[i] * x;
}
- return c;
+ canonicalize(r);
+ return r;
}
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const UniPoly& t)
+static void upoly_from_ex(upoly& up, const ex& e, const ex& x)
{
- list<Term>::const_iterator i = t.terms.begin(), end = t.terms.end();
- if ( i == end ) {
- o << "0";
- return o;
+ // assert: e is in Z[x]
+ int deg = e.degree(x);
+ up.resize(deg+1);
+ int ldeg = e.ldegree(x);
+ for ( ; deg>=ldeg; --deg ) {
+ up[deg] = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
}
- for ( ; i != end; ) {
- o << *i++;
- if ( i != end ) {
- o << " + ";
- }
+ for ( ; deg>=0; --deg ) {
+ up[deg] = 0;
}
- return o;
+ canonicalize(up);
}
-#endif // def DEBUGFACTOR
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const list<UniPoly>& t)
+static void umodpoly_from_upoly(umodpoly& ump, const upoly& e, const cl_modint_ring& R)
{
- list<UniPoly>::const_iterator i = t.begin(), end = t.end();
- o << "{" << endl;
- for ( ; i != end; ) {
- o << *i++ << endl;
+ int deg = degree(e);
+ ump.resize(deg+1);
+ for ( ; deg>=0; --deg ) {
+ ump[deg] = R->canonhom(e[deg]);
}
- o << "}" << endl;
- return o;
+ canonicalize(ump);
}
-#endif // def DEBUGFACTOR
-typedef vector<UniPoly> UniPolyVec;
+static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_modint_ring& R)
+{
+ // assert: e is in Z[x]
+ int deg = e.degree(x);
+ ump.resize(deg+1);
+ int ldeg = e.ldegree(x);
+ for ( ; deg>=ldeg; --deg ) {
+ cl_I coeff = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
+ ump[deg] = R->canonhom(coeff);
+ }
+ for ( ; deg>=0; --deg ) {
+ ump[deg] = R->zero();
+ }
+ canonicalize(ump);
+}
-struct UniFactor
+#ifdef DEBUGFACTOR
+static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_I& modulus)
{
- UniPoly p;
- unsigned int exp;
+ umodpoly_from_ex(ump, e, x, find_modint_ring(modulus));
+}
+#endif
- 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;
+static ex upoly_to_ex(const upoly& a, const ex& x)
+{
+ if ( a.empty() ) return 0;
+ ex e;
+ for ( int i=degree(a); i>=0; --i ) {
+ e += numeric(a[i]) * pow(x, i);
}
-};
+ return e;
+}
-struct UniFactorVec
+static ex umodpoly_to_ex(const umodpoly& a, const ex& x)
{
- 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);
- }
+ if ( a.empty() ) return 0;
+ cl_modint_ring R = a[0].ring();
+ cl_I mod = R->modulus;
+ cl_I halfmod = (mod-1) >> 1;
+ ex e;
+ for ( int i=degree(a); i>=0; --i ) {
+ cl_I n = R->retract(a[i]);
+ if ( n > halfmod ) {
+ e += numeric(n-mod) * pow(x, i);
+ } else {
+ e += numeric(n) * pow(x, i);
}
}
-};
+ return e;
+}
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const UniFactorVec& ufv)
+static upoly umodpoly_to_upoly(const umodpoly& a)
{
- for ( size_t i=0; i<ufv.factors.size(); ++i ) {
- if ( i != ufv.factors.size()-1 ) {
- o << "*";
+ upoly e(a.size());
+ if ( a.empty() ) return e;
+ cl_modint_ring R = a[0].ring();
+ cl_I mod = R->modulus;
+ cl_I halfmod = (mod-1) >> 1;
+ for ( int i=degree(a); i>=0; --i ) {
+ cl_I n = R->retract(a[i]);
+ if ( n > halfmod ) {
+ e[i] = n-mod;
+ } else {
+ e[i] = n;
}
- else {
- o << " ";
- }
- o << "[ " << ufv.factors[i].p << " ]^" << ufv.factors[i].exp << endl;
}
- return o;
+ return e;
}
-#endif // def DEBUGFACTOR
-static void rem(const UniPoly& a_, const UniPoly& b, UniPoly& c)
+static umodpoly umodpoly_to_umodpoly(const umodpoly& a, const cl_modint_ring& R, unsigned int m)
{
- if ( a_.degree() < b.degree() ) {
- c = a_;
- return;
- }
+ umodpoly e;
+ if ( a.empty() ) return e;
+ cl_modint_ring oldR = a[0].ring();
+ size_t sa = a.size();
+ e.resize(sa+m, R->zero());
+ for ( size_t i=0; i<sa; ++i ) {
+ e[i+m] = R->canonhom(oldR->retract(a[i]));
+ }
+ canonicalize(e);
+ return e;
+}
- unsigned int k, n;
- n = b.degree();
- k = a_.degree() - n;
+/** Divides all coefficients of the polynomial a by the integer x.
+ * All coefficients are supposed to be divisible by x. If they are not, the
+ * the<cl_I> cast will raise an exception.
+ *
+ * @param[in,out] a polynomial of which the coefficients will be reduced by x
+ * @param[in] x integer that divides the coefficients
+ */
+static void reduce_coeff(umodpoly& a, const cl_I& x)
+{
+ if ( a.empty() ) return;
- if ( n == 0 ) {
- c.terms.clear();
- return;
+ cl_modint_ring R = a[0].ring();
+ umodpoly::iterator i = a.begin(), end = a.end();
+ for ( ; i!=end; ++i ) {
+ // cln cannot perform this division in the modular field
+ cl_I c = R->retract(*i);
+ *i = cl_MI(R, the<cl_I>(c / x));
}
+}
- c = a_;
- Term termbuf;
-
- while ( true ) {
- cl_MI qk = div(c[n+k], b[n]);
+/** Calculates remainder of a/b.
+ * Assertion: a and b not empty.
+ *
+ * @param[in] a polynomial dividend
+ * @param[in] b polynomial divisor
+ * @param[out] r polynomial remainder
+ */
+static void rem(const umodpoly& a, const umodpoly& b, umodpoly& r)
+{
+ int k, n;
+ n = degree(b);
+ k = degree(a) - n;
+ r = a;
+ if ( k < 0 ) return;
+
+ do {
+ cl_MI qk = div(r[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]);
+ for ( int i=0; i<n; ++i ) {
+ unsigned int j = n + k - 1 - i;
+ r[j] = r[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);
+ } while ( k-- );
+
+ fill(r.begin()+n, r.end(), a[0].ring()->zero());
+ canonicalize(r);
}
-static void div(const UniPoly& a_, const UniPoly& b, UniPoly& q)
+/** Calculates quotient of a/b.
+ * Assertion: a and b not empty.
+ *
+ * @param[in] a polynomial dividend
+ * @param[in] b polynomial divisor
+ * @param[out] q polynomial quotient
+ */
+static void div(const umodpoly& a, const umodpoly& b, umodpoly& q)
{
- if ( a_.degree() < b.degree() ) {
- q.terms.clear();
- return;
- }
-
- unsigned int k, n;
- n = b.degree();
- k = a_.degree() - n;
+ int k, n;
+ n = degree(b);
+ k = degree(a) - n;
+ q.clear();
+ if ( k < 0 ) return;
+
+ umodpoly r = a;
+ q.resize(k+1, a[0].ring()->zero());
+ do {
+ cl_MI qk = div(r[n+k], b[n]);
+ if ( !zerop(qk) ) {
+ q[k] = qk;
+ for ( int i=0; i<n; ++i ) {
+ unsigned int j = n + k - 1 - i;
+ r[j] = r[j] - qk * b[j-k];
+ }
+ }
+ } while ( k-- );
- UniPoly c = a_;
- Term termbuf;
+ canonicalize(q);
+}
- while ( true ) {
- cl_MI qk = div(c[n+k], b[n]);
+/** Calculates quotient and remainder of a/b.
+ * Assertion: a and b not empty.
+ *
+ * @param[in] a polynomial dividend
+ * @param[in] b polynomial divisor
+ * @param[out] r polynomial remainder
+ * @param[out] q polynomial quotient
+ */
+static void remdiv(const umodpoly& a, const umodpoly& b, umodpoly& r, umodpoly& q)
+{
+ int k, n;
+ n = degree(b);
+ k = degree(a) - n;
+ q.clear();
+ r = a;
+ if ( k < 0 ) return;
+
+ q.resize(k+1, a[0].ring()->zero());
+ do {
+ cl_MI qk = div(r[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]);
+ q[k] = qk;
+ for ( int i=0; i<n; ++i ) {
+ unsigned int j = n + k - 1 - i;
+ r[j] = r[j] - qk * b[j-k];
}
}
- if ( k == 0 ) break;
- --k;
- }
+ } while ( k-- );
+
+ fill(r.begin()+n, r.end(), a[0].ring()->zero());
+ canonicalize(r);
+ canonicalize(q);
}
-static void gcd(const UniPoly& a, const UniPoly& b, UniPoly& c)
+/** Calculates the GCD of polynomial a and b.
+ *
+ * @param[in] a polynomial
+ * @param[in] b polynomial
+ * @param[out] c GCD
+ */
+static void gcd(const umodpoly& a, const umodpoly& b, umodpoly& c)
{
- c = a;
- c.unit_normal();
- UniPoly d = b;
- d.unit_normal();
-
- if ( c.degree() < d.degree() ) {
- gcd(b, a, c);
- return;
- }
+ if ( degree(a) < degree(b) ) return gcd(b, a, c);
- while ( !d.zero() ) {
- UniPoly r(a.R);
+ c = a;
+ normalize_in_field(c);
+ umodpoly d = b;
+ normalize_in_field(d);
+ umodpoly r;
+ while ( !d.empty() ) {
rem(c, d, r);
c = d;
d = r;
}
- c.unit_normal();
+ normalize_in_field(c);
}
-static bool is_one(const UniPoly& w)
+/** Calculates the derivative of the polynomial a.
+ *
+ * @param[in] a polynomial of which to take the derivative
+ * @param[out] d result/derivative
+ */
+static void deriv(const umodpoly& a, umodpoly& d)
{
- if ( w.terms.size() == 1 && w[0] == w.R->one() ) {
- return true;
+ d.clear();
+ if ( a.size() <= 1 ) return;
+
+ d.insert(d.begin(), a.begin()+1, a.end());
+ int max = d.size();
+ for ( int i=1; i<max; ++i ) {
+ d[i] = d[i] * (i+1);
}
- return false;
+ canonicalize(d);
}
-static void sqrfree_main(const UniPoly& a, UniFactorVec& fvec)
+static bool unequal_one(const umodpoly& a)
{
- 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;
- }
+ if ( a.empty() ) return true;
+ return ( a.size() != 1 || a[0] != a[0].ring()->one() );
}
-static void squarefree(const UniPoly& a, UniFactorVec& fvec)
+static bool equal_one(const umodpoly& a)
+{
+ return ( a.size() == 1 && a[0] == a[0].ring()->one() );
+}
+
+/** Returns true if polynomial a is square free.
+ *
+ * @param[in] a polynomial to check
+ * @return true if polynomial is square free, false otherwise
+ */
+static bool squarefree(const umodpoly& a)
{
- sqrfree_main(a, fvec);
- fvec.unique();
+ umodpoly b;
+ deriv(a, b);
+ if ( b.empty() ) {
+ return false;
+ }
+ umodpoly c;
+ gcd(a, b, c);
+ return equal_one(c);
}
-class Matrix
+// END modular univariate polynomial code
+////////////////////////////////////////////////////////////////////////////////
+
+////////////////////////////////////////////////////////////////////////////////
+// modular matrix
+
+typedef vector<cl_MI> mvec;
+
+class modular_matrix
{
- friend ostream& operator<<(ostream& o, const Matrix& m);
+ friend ostream& operator<<(ostream& o, const modular_matrix& m);
public:
- Matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
+ modular_matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
{
m.resize(c*r, init);
}
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;
+ std::size_t i = c*rc + col;
+ m[i] = m[i] * x;
}
}
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;
+ std::size_t i1 = col1 + c*rc;
+ std::size_t i2 = col2 + c*rc;
+ m[i1] = m[i1] - m[i2]*fac;
}
}
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;
+ std::size_t i1 = col1 + rc*c;
+ std::size_t i2 = col2 + rc*c;
+ std::swap(m[i1], m[i2]);
+ }
+ }
+ void mul_row(size_t row, const cl_MI x)
+ {
+ for ( size_t cc=0; cc<c; ++cc ) {
+ std::size_t i = row*c + cc;
+ m[i] = m[i] * x;
+ }
+ }
+ void sub_row(size_t row1, size_t row2, const cl_MI fac)
+ {
+ for ( size_t cc=0; cc<c; ++cc ) {
+ std::size_t i1 = row1*c + cc;
+ std::size_t i2 = row2*c + cc;
+ m[i1] = m[i1] - m[i2]*fac;
+ }
+ }
+ void switch_row(size_t row1, size_t row2)
+ {
+ for ( size_t cc=0; cc<c; ++cc ) {
+ std::size_t i1 = row1*c + cc;
+ std::size_t i2 = row2*c + cc;
+ std::swap(m[i1], m[i2]);
+ }
+ }
+ bool is_col_zero(size_t col) const
+ {
+ for ( size_t rr=0; rr<r; ++rr ) {
+ std::size_t i = col + rr*c;
+ if ( !zerop(m[i]) ) {
+ return false;
+ }
}
+ return true;
}
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) ) {
+ std::size_t i = row*c + cc;
+ if ( !zerop(m[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;
+ for (std::size_t i2 = 0; i2 < newrow.size(); ++i2) {
+ std::size_t i1 = row*c + i2;
+ m[i1] = newrow[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; }
+ mvec::const_iterator row_begin(size_t row) const { return m.begin()+row*c; }
+ mvec::const_iterator row_end(size_t row) const { return m.begin()+row*c+r; }
private:
size_t r, c;
- Vec m;
+ mvec m;
};
#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const Matrix& m)
+modular_matrix operator*(const modular_matrix& m1, const modular_matrix& m2)
+{
+ const unsigned int r = m1.rowsize();
+ const unsigned int c = m2.colsize();
+ modular_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 modular_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;
+ cl_modint_ring R = m(0,0).ring();
+ o << "{";
+ for ( size_t i=0; i<m.rowsize(); ++i ) {
+ o << "{";
+ for ( size_t j=0; j<m.colsize()-1; ++j ) {
+ o << R->retract(m(i,j)) << ",";
+ }
+ o << R->retract(m(i,m.colsize()-1)) << "}";
+ if ( i != m.rowsize()-1 ) {
+ o << ",";
}
}
- o << endl;
+ o << "}";
return o;
}
#endif // def DEBUGFACTOR
-static void q_matrix(const UniPoly& a, Matrix& Q)
+// END modular matrix
+////////////////////////////////////////////////////////////////////////////////
+
+/** Calculates the Q matrix for a polynomial. Used by Berlekamp's algorithm.
+ *
+ * @param[in] a_ modular polynomial
+ * @param[out] Q Q matrix
+ */
+static void q_matrix(const umodpoly& a_, modular_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();
+ umodpoly a = a_;
+ normalize_in_field(a);
+
+ int n = degree(a);
+ unsigned int q = cl_I_to_uint(a[0].ring()->modulus);
+ umodpoly r(n, a[0].ring()->zero());
+ r[0] = a[0].ring()->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[i] = r[i-1] - (rn_1 * a[i]);
}
r[0] = -rn_1 * a[0];
if ( (m % q) == 0 ) {
}
}
-static void nullspace(Matrix& M, vector<Vec>& basis)
+/** Determine the nullspace of a matrix M-1.
+ *
+ * @param[in,out] M matrix, will be modified
+ * @param[out] basis calculated nullspace of M-1
+ */
+static void nullspace(modular_matrix& M, vector<mvec>& basis)
{
const size_t n = M.rowsize();
const cl_MI one = M(0,0).ring()->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));
+ mvec nu(M.row_begin(i), M.row_end(i));
basis.push_back(nu);
}
}
}
-static void berlekamp(const UniPoly& a, UniPolyVec& upv)
+/** Berlekamp's modular factorization.
+ *
+ * The implementation follows the algorithm in chapter 8 of [GCL].
+ *
+ * @param[in] a modular polynomial
+ * @param[out] upv vector containing modular factors. if upv was not empty the
+ * new elements are added at the end
+ */
+static void berlekamp(const umodpoly& a, upvec& upv)
+{
+ cl_modint_ring R = a[0].ring();
+ umodpoly one(1, R->one());
+
+ // find nullspace of Q matrix
+ modular_matrix Q(degree(a), degree(a), R->zero());
+ q_matrix(a, Q);
+ vector<mvec> nu;
+ nullspace(Q, nu);
+
+ const unsigned int k = nu.size();
+ if ( k == 1 ) {
+ // irreducible
+ return;
+ }
+
+ list<umodpoly> factors;
+ factors.push_back(a);
+ unsigned int size = 1;
+ unsigned int r = 1;
+ unsigned int q = cl_I_to_uint(R->modulus);
+
+ list<umodpoly>::iterator u = factors.begin();
+
+ // calculate all gcd's
+ while ( true ) {
+ for ( unsigned int s=0; s<q; ++s ) {
+ umodpoly nur = nu[r];
+ nur[0] = nur[0] - cl_MI(R, s);
+ canonicalize(nur);
+ umodpoly g;
+ gcd(nur, *u, g);
+ if ( unequal_one(g) && g != *u ) {
+ umodpoly uo;
+ div(*u, g, uo);
+ if ( equal_one(uo) ) {
+ throw logic_error("berlekamp: unexpected divisor.");
+ }
+ else {
+ *u = uo;
+ }
+ factors.push_back(g);
+ size = 0;
+ list<umodpoly>::const_iterator i = factors.begin(), end = factors.end();
+ while ( i != end ) {
+ if ( degree(*i) ) ++size;
+ ++i;
+ }
+ if ( size == k ) {
+ list<umodpoly>::const_iterator i = factors.begin(), end = factors.end();
+ while ( i != end ) {
+ upv.push_back(*i++);
+ }
+ return;
+ }
+ }
+ }
+ if ( ++r == k ) {
+ r = 1;
+ ++u;
+ }
+ }
+}
+
+// modular square free factorization is not used at the moment so we deactivate
+// the code
+#if 0
+
+/** Calculates a^(1/prime).
+ *
+ * @param[in] a polynomial
+ * @param[in] prime prime number -> exponent 1/prime
+ * @param[in] ap resulting polynomial
+ */
+static void expt_1_over_p(const umodpoly& a, unsigned int prime, umodpoly& ap)
+{
+ size_t newdeg = degree(a)/prime;
+ ap.resize(newdeg+1);
+ ap[0] = a[0];
+ for ( size_t i=1; i<=newdeg; ++i ) {
+ ap[i] = a[i*prime];
+ }
+}
+
+/** Modular square free factorization.
+ *
+ * @param[in] a polynomial
+ * @param[out] factors modular factors
+ * @param[out] mult corresponding multiplicities (exponents)
+ */
+static void modsqrfree(const umodpoly& a, upvec& factors, vector<int>& mult)
+{
+ const unsigned int prime = cl_I_to_uint(a[0].ring()->modulus);
+ int i = 1;
+ umodpoly b;
+ deriv(a, b);
+ if ( b.size() ) {
+ umodpoly c;
+ gcd(a, b, c);
+ umodpoly w;
+ div(a, c, w);
+ while ( unequal_one(w) ) {
+ umodpoly y;
+ gcd(w, c, y);
+ umodpoly z;
+ div(w, y, z);
+ factors.push_back(z);
+ mult.push_back(i);
+ ++i;
+ w = y;
+ umodpoly buf;
+ div(c, y, buf);
+ c = buf;
+ }
+ if ( unequal_one(c) ) {
+ umodpoly cp;
+ expt_1_over_p(c, prime, cp);
+ size_t previ = mult.size();
+ modsqrfree(cp, factors, mult);
+ for ( size_t i=previ; i<mult.size(); ++i ) {
+ mult[i] *= prime;
+ }
+ }
+ }
+ else {
+ umodpoly ap;
+ expt_1_over_p(a, prime, ap);
+ size_t previ = mult.size();
+ modsqrfree(ap, factors, mult);
+ for ( size_t i=previ; i<mult.size(); ++i ) {
+ mult[i] *= prime;
+ }
+ }
+}
+
+#endif // deactivation of square free factorization
+
+/** Distinct degree factorization (DDF).
+ *
+ * The implementation follows the algorithm in chapter 8 of [GCL].
+ *
+ * @param[in] a_ modular polynomial
+ * @param[out] degrees vector containing the degrees of the factors of the
+ * corresponding polynomials in ddfactors.
+ * @param[out] ddfactors vector containing polynomials which factors have the
+ * degree given in degrees.
+ */
+static void distinct_degree_factor(const umodpoly& a_, vector<int>& degrees, upvec& ddfactors)
+{
+ umodpoly a = a_;
+
+ cl_modint_ring R = a[0].ring();
+ int q = cl_I_to_int(R->modulus);
+ int nhalf = degree(a)/2;
+
+ int i = 1;
+ umodpoly w(2);
+ w[0] = R->zero();
+ w[1] = R->one();
+ umodpoly x = w;
+
+ while ( i <= nhalf ) {
+ expt_pos(w, q);
+ umodpoly buf;
+ rem(w, a, buf);
+ w = buf;
+ umodpoly wx = w - x;
+ gcd(a, wx, buf);
+ if ( unequal_one(buf) ) {
+ degrees.push_back(i);
+ ddfactors.push_back(buf);
+ }
+ if ( unequal_one(buf) ) {
+ umodpoly buf2;
+ div(a, buf, buf2);
+ a = buf2;
+ nhalf = degree(a)/2;
+ rem(w, a, buf);
+ w = buf;
+ }
+ ++i;
+ }
+ if ( unequal_one(a) ) {
+ degrees.push_back(degree(a));
+ ddfactors.push_back(a);
+ }
+}
+
+/** Modular same degree factorization.
+ * Same degree factorization is a kind of misnomer. It performs distinct degree
+ * factorization, but instead of using the Cantor-Zassenhaus algorithm it
+ * (sub-optimally) uses Berlekamp's algorithm for the factors of the same
+ * degree.
+ *
+ * @param[in] a modular polynomial
+ * @param[out] upv vector containing modular factors. if upv was not empty the
+ * new elements are added at the end
+ */
+static void same_degree_factor(const umodpoly& a, upvec& upv)
+{
+ cl_modint_ring R = a[0].ring();
+
+ vector<int> degrees;
+ upvec ddfactors;
+ distinct_degree_factor(a, degrees, ddfactors);
+
+ for ( size_t i=0; i<degrees.size(); ++i ) {
+ if ( degrees[i] == degree(ddfactors[i]) ) {
+ upv.push_back(ddfactors[i]);
+ }
+ else {
+ berlekamp(ddfactors[i], upv);
+ }
+ }
+}
+
+// Yes, we can (choose).
+#define USE_SAME_DEGREE_FACTOR
+
+/** Modular univariate factorization.
+ *
+ * In principle, we have two algorithms at our disposal: Berlekamp's algorithm
+ * and same degree factorization (SDF). SDF seems to be slightly faster in
+ * almost all cases so it is activated as default.
+ *
+ * @param[in] p modular polynomial
+ * @param[out] upv vector containing modular factors. if upv was not empty the
+ * new elements are added at the end
+ */
+static void factor_modular(const umodpoly& p, upvec& upv)
+{
+#ifdef USE_SAME_DEGREE_FACTOR
+ same_degree_factor(p, upv);
+#else
+ berlekamp(p, upv);
+#endif
+}
+
+/** Calculates modular polynomials s and t such that a*s+b*t==1.
+ * Assertion: a and b are relatively prime and not zero.
+ *
+ * @param[in] a polynomial
+ * @param[in] b polynomial
+ * @param[out] s polynomial
+ * @param[out] t polynomial
+ */
+static void exteuclid(const umodpoly& a, const umodpoly& b, umodpoly& s, umodpoly& t)
+{
+ if ( degree(a) < degree(b) ) {
+ exteuclid(b, a, t, s);
+ return;
+ }
+
+ umodpoly one(1, a[0].ring()->one());
+ umodpoly c = a; normalize_in_field(c);
+ umodpoly d = b; normalize_in_field(d);
+ s = one;
+ t.clear();
+ umodpoly d1;
+ umodpoly d2 = one;
+ umodpoly q;
+ while ( true ) {
+ div(c, d, q);
+ umodpoly r = c - q * d;
+ umodpoly r1 = s - q * d1;
+ umodpoly r2 = t - q * d2;
+ c = d;
+ s = d1;
+ t = d2;
+ if ( r.empty() ) break;
+ d = r;
+ d1 = r1;
+ d2 = r2;
+ }
+ cl_MI fac = recip(lcoeff(a) * lcoeff(c));
+ umodpoly::iterator i = s.begin(), end = s.end();
+ for ( ; i!=end; ++i ) {
+ *i = *i * fac;
+ }
+ canonicalize(s);
+ fac = recip(lcoeff(b) * lcoeff(c));
+ i = t.begin(), end = t.end();
+ for ( ; i!=end; ++i ) {
+ *i = *i * fac;
+ }
+ canonicalize(t);
+}
+
+/** Replaces the leading coefficient in a polynomial by a given number.
+ *
+ * @param[in] poly polynomial to change
+ * @param[in] lc new leading coefficient
+ * @return changed polynomial
+ */
+static upoly replace_lc(const upoly& poly, const cl_I& lc)
+{
+ if ( poly.empty() ) return poly;
+ upoly r = poly;
+ r.back() = lc;
+ return r;
+}
+
+/** Calculates the bound for the modulus.
+ * See [Mig].
+ */
+static inline cl_I calc_bound(const ex& a, const ex& x, int maxdeg)
+{
+ cl_I maxcoeff = 0;
+ cl_R coeff = 0;
+ for ( int i=a.degree(x); i>=a.ldegree(x); --i ) {
+ cl_I aa = abs(the<cl_I>(ex_to<numeric>(a.coeff(x, i)).to_cl_N()));
+ if ( aa > maxcoeff ) maxcoeff = aa;
+ coeff = coeff + square(aa);
+ }
+ cl_I coeffnorm = ceiling1(the<cl_R>(cln::sqrt(coeff)));
+ cl_I B = coeffnorm * expt_pos(cl_I(2), cl_I(maxdeg));
+ return ( B > maxcoeff ) ? B : maxcoeff;
+}
+
+/** Calculates the bound for the modulus.
+ * See [Mig].
+ */
+static inline cl_I calc_bound(const upoly& a, int maxdeg)
+{
+ cl_I maxcoeff = 0;
+ cl_R coeff = 0;
+ for ( int i=degree(a); i>=0; --i ) {
+ cl_I aa = abs(a[i]);
+ if ( aa > maxcoeff ) maxcoeff = aa;
+ coeff = coeff + square(aa);
+ }
+ cl_I coeffnorm = ceiling1(the<cl_R>(cln::sqrt(coeff)));
+ cl_I B = coeffnorm * expt_pos(cl_I(2), cl_I(maxdeg));
+ return ( B > maxcoeff ) ? B : maxcoeff;
+}
+
+/** Hensel lifting as used by factor_univariate().
+ *
+ * The implementation follows the algorithm in chapter 6 of [GCL].
+ *
+ * @param[in] a_ primitive univariate polynomials
+ * @param[in] p prime number that does not divide lcoeff(a)
+ * @param[in] u1_ modular factor of a (mod p)
+ * @param[in] w1_ modular factor of a (mod p), relatively prime to u1_,
+ * fulfilling u1_*w1_ == a mod p
+ * @param[out] u lifted factor
+ * @param[out] w lifted factor, u*w = a
+ */
+static void hensel_univar(const upoly& a_, unsigned int p, const umodpoly& u1_, const umodpoly& w1_, upoly& u, upoly& w)
+{
+ upoly a = a_;
+ const cl_modint_ring& R = u1_[0].ring();
+
+ // calc bound B
+ int maxdeg = (degree(u1_) > degree(w1_)) ? degree(u1_) : degree(w1_);
+ cl_I maxmodulus = 2*calc_bound(a, maxdeg);
+
+ // step 1
+ cl_I alpha = lcoeff(a);
+ a = a * alpha;
+ umodpoly nu1 = u1_;
+ normalize_in_field(nu1);
+ umodpoly nw1 = w1_;
+ normalize_in_field(nw1);
+ upoly phi;
+ phi = umodpoly_to_upoly(nu1) * alpha;
+ umodpoly u1;
+ umodpoly_from_upoly(u1, phi, R);
+ phi = umodpoly_to_upoly(nw1) * alpha;
+ umodpoly w1;
+ umodpoly_from_upoly(w1, phi, R);
+
+ // step 2
+ umodpoly s;
+ umodpoly t;
+ exteuclid(u1, w1, s, t);
+
+ // step 3
+ u = replace_lc(umodpoly_to_upoly(u1), alpha);
+ w = replace_lc(umodpoly_to_upoly(w1), alpha);
+ upoly e = a - u * w;
+ cl_I modulus = p;
+
+ // step 4
+ while ( !e.empty() && modulus < maxmodulus ) {
+ upoly c = e / modulus;
+ phi = umodpoly_to_upoly(s) * c;
+ umodpoly sigmatilde;
+ umodpoly_from_upoly(sigmatilde, phi, R);
+ phi = umodpoly_to_upoly(t) * c;
+ umodpoly tautilde;
+ umodpoly_from_upoly(tautilde, phi, R);
+ umodpoly r, q;
+ remdiv(sigmatilde, w1, r, q);
+ umodpoly sigma = r;
+ phi = umodpoly_to_upoly(tautilde) + umodpoly_to_upoly(q) * umodpoly_to_upoly(u1);
+ umodpoly tau;
+ umodpoly_from_upoly(tau, phi, R);
+ u = u + umodpoly_to_upoly(tau) * modulus;
+ w = w + umodpoly_to_upoly(sigma) * modulus;
+ e = a - u * w;
+ modulus = modulus * p;
+ }
+
+ // step 5
+ if ( e.empty() ) {
+ cl_I g = u[0];
+ for ( size_t i=1; i<u.size(); ++i ) {
+ g = gcd(g, u[i]);
+ if ( g == 1 ) break;
+ }
+ if ( g != 1 ) {
+ u = u / g;
+ w = w * g;
+ }
+ if ( alpha != 1 ) {
+ w = w / alpha;
+ }
+ }
+ else {
+ u.clear();
+ }
+}
+
+/** Returns a new prime number.
+ *
+ * @param[in] p prime number
+ * @return next prime number after p
+ */
+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!");
+}
+
+/** Manages the splitting a vector of of modular factors into two partitions.
+ */
+class factor_partition
+{
+public:
+ /** Takes the vector of modular factors and initializes the first partition */
+ factor_partition(const upvec& factors_) : factors(factors_)
+ {
+ n = factors.size();
+ k.resize(n, 0);
+ k[0] = 1;
+ cache.resize(n-1);
+ one.resize(1, factors.front()[0].ring()->one());
+ len = 1;
+ last = 0;
+ split();
+ }
+ int operator[](size_t i) const { return k[i]; }
+ size_t size() const { return n; }
+ size_t size_left() const { return n-len; }
+ size_t size_right() const { return len; }
+ /** Initializes the next partition.
+ Returns true, if there is one, false otherwise. */
+ bool next()
+ {
+ if ( last == n-1 ) {
+ int rem = len - 1;
+ int p = last - 1;
+ while ( rem ) {
+ if ( k[p] ) {
+ --rem;
+ --p;
+ continue;
+ }
+ last = p - 1;
+ while ( k[last] == 0 ) { --last; }
+ if ( last == 0 && n == 2*len ) return false;
+ k[last++] = 0;
+ for ( size_t i=0; i<=len-rem; ++i ) {
+ k[last] = 1;
+ ++last;
+ }
+ fill(k.begin()+last, k.end(), 0);
+ --last;
+ split();
+ return true;
+ }
+ last = len;
+ ++len;
+ if ( len > n/2 ) return false;
+ fill(k.begin(), k.begin()+len, 1);
+ fill(k.begin()+len+1, k.end(), 0);
+ }
+ else {
+ k[last++] = 0;
+ k[last] = 1;
+ }
+ split();
+ return true;
+ }
+ /** Get first partition */
+ umodpoly& left() { return lr[0]; }
+ /** Get second partition */
+ umodpoly& right() { return lr[1]; }
+private:
+ void split_cached()
+ {
+ size_t i = 0;
+ do {
+ size_t pos = i;
+ int group = k[i++];
+ size_t d = 0;
+ while ( i < n && k[i] == group ) { ++d; ++i; }
+ if ( d ) {
+ if ( cache[pos].size() >= d ) {
+ lr[group] = lr[group] * cache[pos][d-1];
+ }
+ else {
+ if ( cache[pos].size() == 0 ) {
+ cache[pos].push_back(factors[pos] * factors[pos+1]);
+ }
+ size_t j = pos + cache[pos].size() + 1;
+ d -= cache[pos].size();
+ while ( d ) {
+ umodpoly buf = cache[pos].back() * factors[j];
+ cache[pos].push_back(buf);
+ --d;
+ ++j;
+ }
+ lr[group] = lr[group] * cache[pos].back();
+ }
+ }
+ else {
+ lr[group] = lr[group] * factors[pos];
+ }
+ } while ( i < n );
+ }
+ void split()
+ {
+ lr[0] = one;
+ lr[1] = one;
+ if ( n > 6 ) {
+ split_cached();
+ }
+ else {
+ for ( size_t i=0; i<n; ++i ) {
+ lr[k[i]] = lr[k[i]] * factors[i];
+ }
+ }
+ }
+private:
+ umodpoly lr[2];
+ vector< vector<umodpoly> > cache;
+ upvec factors;
+ umodpoly one;
+ size_t n;
+ size_t len;
+ size_t last;
+ vector<int> k;
+};
+
+/** Contains a pair of univariate polynomial and its modular factors.
+ * Used by factor_univariate().
+ */
+struct ModFactors
+{
+ upoly poly;
+ upvec factors;
+};
+
+/** Univariate polynomial factorization.
+ *
+ * Modular factorization is tried for several primes to minimize the number of
+ * modular factors. Then, Hensel lifting is performed.
+ *
+ * @param[in] poly expanded square free univariate polynomial
+ * @param[in] x symbol
+ * @param[in,out] prime prime number to start trying modular factorization with,
+ * output value is the prime number actually used
+ */
+static ex factor_univariate(const ex& poly, const ex& x, unsigned int& prime)
+{
+ ex unit, cont, prim_ex;
+ poly.unitcontprim(x, unit, cont, prim_ex);
+ upoly prim;
+ upoly_from_ex(prim, prim_ex, x);
+
+ // determine proper prime and minimize number of modular factors
+ prime = 3;
+ unsigned int lastp = prime;
+ cl_modint_ring R;
+ unsigned int trials = 0;
+ unsigned int minfactors = 0;
+
+ const numeric& cont_n = ex_to<numeric>(cont);
+ cl_I i_cont;
+ if (cont_n.is_integer()) {
+ i_cont = the<cl_I>(cont_n.to_cl_N());
+ } else {
+ // poly \in Q[x] => poly = q ipoly, ipoly \in Z[x], q \in Q
+ // factor(poly) \equiv q factor(ipoly)
+ i_cont = cl_I(1);
+ }
+ cl_I lc = lcoeff(prim)*i_cont;
+ upvec factors;
+ while ( trials < 2 ) {
+ umodpoly modpoly;
+ while ( true ) {
+ prime = next_prime(prime);
+ if ( !zerop(rem(lc, prime)) ) {
+ R = find_modint_ring(prime);
+ umodpoly_from_upoly(modpoly, prim, R);
+ if ( squarefree(modpoly) ) break;
+ }
+ }
+
+ // do modular factorization
+ upvec trialfactors;
+ factor_modular(modpoly, trialfactors);
+ if ( trialfactors.size() <= 1 ) {
+ // irreducible for sure
+ return poly;
+ }
+
+ if ( minfactors == 0 || trialfactors.size() < minfactors ) {
+ factors = trialfactors;
+ minfactors = trialfactors.size();
+ lastp = prime;
+ trials = 1;
+ }
+ else {
+ ++trials;
+ }
+ }
+ prime = lastp;
+ R = find_modint_ring(prime);
+
+ // lift all factor combinations
+ stack<ModFactors> tocheck;
+ ModFactors mf;
+ mf.poly = prim;
+ mf.factors = factors;
+ tocheck.push(mf);
+ upoly f1, f2;
+ ex result = 1;
+ while ( tocheck.size() ) {
+ const size_t n = tocheck.top().factors.size();
+ factor_partition part(tocheck.top().factors);
+ while ( true ) {
+ // call Hensel lifting
+ hensel_univar(tocheck.top().poly, prime, part.left(), part.right(), f1, f2);
+ if ( !f1.empty() ) {
+ // successful, update the stack and the result
+ if ( part.size_left() == 1 ) {
+ if ( part.size_right() == 1 ) {
+ result *= upoly_to_ex(f1, x) * upoly_to_ex(f2, x);
+ tocheck.pop();
+ break;
+ }
+ result *= upoly_to_ex(f1, x);
+ tocheck.top().poly = f2;
+ 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_right() == 1 ) {
+ if ( part.size_left() == 1 ) {
+ result *= upoly_to_ex(f1, x) * upoly_to_ex(f2, x);
+ tocheck.pop();
+ break;
+ }
+ result *= upoly_to_ex(f2, x);
+ tocheck.top().poly = f1;
+ for ( size_t i=0; i<n; ++i ) {
+ if ( part[i] == 1 ) {
+ tocheck.top().factors.erase(tocheck.top().factors.begin()+i);
+ break;
+ }
+ }
+ break;
+ }
+ else {
+ upvec newfactors1(part.size_left()), newfactors2(part.size_right());
+ upvec::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 = f1;
+ ModFactors mf;
+ mf.factors = newfactors2;
+ mf.poly = f2;
+ tocheck.push(mf);
+ break;
+ }
+ }
+ else {
+ // not successful
+ if ( !part.next() ) {
+ // if no more combinations left, return polynomial as
+ // irreducible
+ result *= upoly_to_ex(tocheck.top().poly, x);
+ tocheck.pop();
+ break;
+ }
+ }
+ }
+ }
+
+ return unit * cont * result;
+}
+
+/** Second interface to factor_univariate() to be used if the information about
+ * the prime is not needed.
+ */
+static inline ex factor_univariate(const ex& poly, const ex& x)
+{
+ unsigned int prime;
+ return factor_univariate(poly, x, prime);
+}
+
+/** Represents an evaluation point (<symbol>==<integer>).
+ */
+struct EvalPoint
+{
+ ex x;
+ int evalpoint;
+};
+
+#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
+
+// forward declaration
+static 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);
+
+/** Utility function for multivariate Hensel lifting.
+ *
+ * Solves the equation
+ * s_1*b_1 + ... + s_r*b_r == 1 mod p^k
+ * with deg(s_i) < deg(a_i)
+ * and with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
+ *
+ * The implementation follows the algorithm in chapter 6 of [GCL].
+ *
+ * @param[in] a vector of modular univariate polynomials
+ * @param[in] x symbol
+ * @param[in] p prime number
+ * @param[in] k p^k is modulus
+ * @return vector of polynomials (s_i)
+ */
+static upvec multiterm_eea_lift(const upvec& a, const ex& x, unsigned int p, unsigned int k)
+{
+ const size_t r = a.size();
+ cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+ upvec q(r-1);
+ q[r-2] = a[r-1];
+ for ( size_t j=r-2; j>=1; --j ) {
+ q[j-1] = a[j] * q[j];
+ }
+ umodpoly beta(1, R->one());
+ upvec s;
+ for ( size_t j=1; j<r; ++j ) {
+ vector<ex> mdarg(2);
+ mdarg[0] = umodpoly_to_ex(q[j-1], x);
+ mdarg[1] = umodpoly_to_ex(a[j-1], x);
+ vector<EvalPoint> empty;
+ vector<ex> exsigma = multivar_diophant(mdarg, x, umodpoly_to_ex(beta, x), empty, 0, p, k);
+ umodpoly sigma1;
+ umodpoly_from_ex(sigma1, exsigma[0], x, R);
+ umodpoly sigma2;
+ umodpoly_from_ex(sigma2, exsigma[1], x, R);
+ beta = sigma1;
+ s.push_back(sigma2);
+ }
+ s.push_back(beta);
+ return s;
+}
+
+/** Changes the modulus of a modular polynomial. Used by eea_lift().
+ *
+ * @param[in] R new modular ring
+ * @param[in,out] a polynomial to change (in situ)
+ */
+static void change_modulus(const cl_modint_ring& R, umodpoly& a)
+{
+ if ( a.empty() ) return;
+ cl_modint_ring oldR = a[0].ring();
+ umodpoly::iterator i = a.begin(), end = a.end();
+ for ( ; i!=end; ++i ) {
+ *i = R->canonhom(oldR->retract(*i));
+ }
+ canonicalize(a);
+}
+
+/** Utility function for multivariate Hensel lifting.
+ *
+ * Solves s*a + t*b == 1 mod p^k given a,b.
+ *
+ * The implementation follows the algorithm in chapter 6 of [GCL].
+ *
+ * @param[in] a polynomial
+ * @param[in] b polynomial
+ * @param[in] x symbol
+ * @param[in] p prime number
+ * @param[in] k p^k is modulus
+ * @param[out] s_ output polynomial
+ * @param[out] t_ output polynomial
+ */
+static void eea_lift(const umodpoly& a, const umodpoly& b, const ex& x, unsigned int p, unsigned int k, umodpoly& s_, umodpoly& t_)
+{
+ cl_modint_ring R = find_modint_ring(p);
+ umodpoly amod = a;
+ change_modulus(R, amod);
+ umodpoly bmod = b;
+ change_modulus(R, bmod);
+
+ umodpoly smod;
+ umodpoly tmod;
+ exteuclid(amod, bmod, smod, tmod);
+
+ cl_modint_ring Rpk = find_modint_ring(expt_pos(cl_I(p),k));
+ umodpoly s = smod;
+ change_modulus(Rpk, s);
+ umodpoly t = tmod;
+ change_modulus(Rpk, t);
+
+ cl_I modulus(p);
+ umodpoly one(1, Rpk->one());
+ for ( size_t j=1; j<k; ++j ) {
+ umodpoly e = one - a * s - b * t;
+ reduce_coeff(e, modulus);
+ umodpoly c = e;
+ change_modulus(R, c);
+ umodpoly sigmabar = smod * c;
+ umodpoly taubar = tmod * c;
+ umodpoly sigma, q;
+ remdiv(sigmabar, bmod, sigma, q);
+ umodpoly tau = taubar + q * amod;
+ umodpoly sadd = sigma;
+ change_modulus(Rpk, sadd);
+ cl_MI modmodulus(Rpk, modulus);
+ s = s + sadd * modmodulus;
+ umodpoly tadd = tau;
+ change_modulus(Rpk, tadd);
+ t = t + tadd * modmodulus;
+ modulus = modulus * p;
+ }
+
+ s_ = s; t_ = t;
+}
+
+/** Utility function for multivariate Hensel lifting.
+ *
+ * Solves the equation
+ * s_1*b_1 + ... + s_r*b_r == x^m mod p^k
+ * with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
+ *
+ * The implementation follows the algorithm in chapter 6 of [GCL].
+ *
+ * @param a vector with univariate polynomials mod p^k
+ * @param x symbol
+ * @param m exponent of x^m in the equation to solve
+ * @param p prime number
+ * @param k p^k is modulus
+ * @return vector of polynomials (s_i)
+ */
+static upvec univar_diophant(const upvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
+{
+ cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+
+ const size_t r = a.size();
+ upvec result;
+ if ( r > 2 ) {
+ upvec s = multiterm_eea_lift(a, x, p, k);
+ for ( size_t j=0; j<r; ++j ) {
+ umodpoly bmod = umodpoly_to_umodpoly(s[j], R, m);
+ umodpoly buf;
+ rem(bmod, a[j], buf);
+ result.push_back(buf);
+ }
+ }
+ else {
+ umodpoly s, t;
+ eea_lift(a[1], a[0], x, p, k, s, t);
+ umodpoly bmod = umodpoly_to_umodpoly(s, R, m);
+ umodpoly buf, q;
+ remdiv(bmod, a[0], buf, q);
+ result.push_back(buf);
+ umodpoly t1mod = umodpoly_to_umodpoly(t, R, m);
+ buf = t1mod + q * a[1];
+ result.push_back(buf);
+ }
+
+ return result;
+}
+
+/** Map used by function make_modular().
+ * Finds every coefficient in a polynomial and replaces it by is value in the
+ * given modular ring R (symmetric representation).
+ */
+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;
+ }
+};
+
+/** Helps mimicking modular multivariate polynomial arithmetic.
+ *
+ * @param e expression of which to make the coefficients equal to their value
+ * in the modular ring R (symmetric representation)
+ * @param R modular ring
+ * @return resulting expression
+ */
+static ex make_modular(const ex& e, const cl_modint_ring& R)
+{
+ make_modular_map map(R);
+ return map(e.expand());
+}
+
+/** Utility function for multivariate Hensel lifting.
+ *
+ * Returns the polynomials s_i that fulfill
+ * s_1*b_1 + ... + s_r*b_r == c mod <I^(d+1),p^k>
+ * with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
+ *
+ * The implementation follows the algorithm in chapter 6 of [GCL].
+ *
+ * @param a_ vector of multivariate factors mod p^k
+ * @param x symbol (equiv. x_1 in [GCL])
+ * @param c polynomial mod p^k
+ * @param I vector of evaluation points
+ * @param d maximum total degree of result
+ * @param p prime number
+ * @param k p^k is modulus
+ * @return vector of polynomials (s_i)
+ */
+static 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)
{
- 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;
- }
+ vector<ex> a = a_;
- 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);
+ 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;
- list<UniPoly>::iterator u = factors.begin();
+ vector<ex> sigma;
+ if ( nu > 1 ) {
+ ex xnu = I.back().x;
+ int alphanu = I.back().evalpoint;
- 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;
+ 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);
+
+ ex buf = c;
+ for ( size_t i=0; i<r; ++i ) {
+ buf -= sigma[i] * b[i];
+ }
+ ex e = make_modular(buf, R);
+
+ ex monomial = 1;
+ for ( size_t m=1; !e.is_zero() && e.has(xnu) && m<=d; ++m ) {
+ monomial *= (xnu - alphanu);
+ monomial = expand(monomial);
+ ex cm = e.diff(ex_to<symbol>(xnu), m).subs(xnu==alphanu) / factorial(m);
+ cm = make_modular(cm, R);
+ if ( !cm.is_zero() ) {
+ vector<ex> delta_s = multivar_diophant(anew, x, cm, Inew, d, p, k);
+ 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 = make_modular(buf, R);
}
}
- if ( ++r == k ) {
- r = 1;
- ++u;
- }
}
-}
-
-static void factor_modular(const UniPoly& p, UniPolyVec& upv)
-{
- berlekamp(p, upv);
- return;
-}
+ else {
+ upvec amod;
+ for ( size_t i=0; i<a.size(); ++i ) {
+ umodpoly up;
+ umodpoly_from_ex(up, a[i], x, R);
+ amod.push_back(up);
+ }
-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;
+ 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;
+ }
+ for ( size_t i=0; i<nterms; ++i ) {
+ int m = z.degree(x);
+ cl_I cm = the<cl_I>(ex_to<numeric>(z.lcoeff(x)).to_cl_N());
+ upvec 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);
+ for ( size_t j=0; j<delta_s.size(); ++j ) {
+ delta_s[j] = delta_s[j] * modcm;
+ sigma[j] = sigma[j] + umodpoly_to_ex(delta_s[j], x);
+ }
+ if ( nterms > 1 ) {
+ z = c.op(i+1);
+ }
+ }
}
- 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;
+
+ for ( size_t i=0; i<sigma.size(); ++i ) {
+ sigma[i] = make_modular(sigma[i], R);
}
- 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;
+ return sigma;
}
-static ex hensel_univar(const ex& a_, const ex& x, unsigned int p, const UniPoly& u1_, const UniPoly& w1_, const ex& gamma_ = 0)
+/** Multivariate Hensel lifting.
+ * The implementation follows the algorithm in chapter 6 of [GCL].
+ * Since we don't have a data type for modular multivariate polynomials, the
+ * respective operations are done in a GiNaC::ex and the function
+ * make_modular() is then called to make the coefficient modular p^l.
+ *
+ * @param a multivariate polynomial primitive in x
+ * @param x symbol (equiv. x_1 in [GCL])
+ * @param I vector of evaluation points (x_2==a_2,x_3==a_3,...)
+ * @param p prime number (should not divide lcoeff(a mod I))
+ * @param l p^l is the modulus of the lifted univariate field
+ * @param u vector of modular (mod p^l) factors of a mod I
+ * @param lcU correct leading coefficient of the univariate factors of a mod I
+ * @return list GiNaC::lst with lifted factors (multivariate factors of a),
+ * empty if Hensel lifting did not succeed
+ */
+static ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I,
+ unsigned int p, const cl_I& l, const upvec& u, const vector<ex>& lcU)
{
- ex a = a_;
- const cl_modint_ring& R = u1_.R;
+ const size_t nu = I.size() + 1;
+ const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
- // calc bound B
- ex maxcoeff;
- for ( int i=a.degree(x); i>=a.ldegree(x); --i ) {
- maxcoeff += pow(abs(a.coeff(x, i)),2);
+ 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);
}
- 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);
+ 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;
+ }
- // step 2
- UniPoly s(R), t(R), g(R);
- exteuclid(u1, w1, g, s, t);
+ const size_t n = u.size();
+ vector<ex> U(n);
+ for ( size_t i=0; i<n; ++i ) {
+ U[i] = umodpoly_to_ex(u[i], x);
+ }
- // 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;
+ for ( size_t j=2; j<=nu; ++j ) {
+ vector<ex> U1 = U;
+ ex monomial = 1;
+ 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 = 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);
+ }
+ }
+ ex Uprod = 1;
+ for ( size_t i=0; i<n; ++i ) {
+ Uprod *= U[i];
+ }
+ ex e = expand(A[j-1] - Uprod);
- // 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;
+ vector<EvalPoint> newI;
+ for ( size_t i=1; i<=j-2; ++i ) {
+ newI.push_back(I[i-1]);
+ }
+
+ ex xj = I[j-2].x;
+ int alphaj = I[j-2].evalpoint;
+ size_t deg = A[j-1].degree(xj);
+ for ( size_t k=1; k<=deg; ++k ) {
+ if ( !e.is_zero() ) {
+ monomial *= (xj - alphaj);
+ monomial = expand(monomial);
+ ex dif = e.diff(ex_to<symbol>(xj), k);
+ ex c = dif.subs(xj==alphaj) / factorial(k);
+ 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 ) {
+ deltaU[i] *= monomial;
+ U[i] += deltaU[i];
+ U[i] = make_modular(U[i], R);
+ }
+ ex Uprod = 1;
+ for ( size_t i=0; i<n; ++i ) {
+ Uprod *= U[i];
+ }
+ e = A[j-1] - Uprod;
+ e = make_modular(e, R);
+ }
+ }
+ }
}
- // step 5
- if ( e.is_zero() ) {
- ex delta = u.content(x);
- u = u / delta;
- w = w / gamma * delta;
- return lst(u, w);
+ ex acand = 1;
+ for ( size_t i=0; i<U.size(); ++i ) {
+ acand *= U[i];
+ }
+ if ( expand(a-acand).is_zero() ) {
+ lst res;
+ for ( size_t i=0; i<U.size(); ++i ) {
+ res.append(U[i]);
+ }
+ return res;
}
else {
+ lst res;
return lst();
}
}
-static unsigned int next_prime(unsigned int p)
+/** Takes a factorized expression and puts the factors in a lst. The exponents
+ * of the factors are discarded, e.g. 7*x^2*(y+1)^4 --> {7,x,y+1}. The first
+ * element of the list is always the numeric coefficient.
+ */
+static ex put_factors_into_lst(const ex& e)
{
- 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;
+ lst result;
+ if ( is_a<numeric>(e) ) {
+ result.append(e);
+ return result;
+ }
+ if ( is_a<power>(e) ) {
+ result.append(1);
+ result.append(e.op(0));
+ return result;
+ }
+ if ( is_a<symbol>(e) || is_a<add>(e) ) {
+ ex icont(e.integer_content());
+ result.append(icont);
+ result.append(e/icont);
+ 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));
+ }
+ if ( is_a<symbol>(op) || is_a<add>(op) ) {
+ result.append(op);
}
- 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;
}
+ result.prepend(nfac);
+ return result;
}
- throw logic_error("next_prime: should not reach this point!");
+ throw runtime_error("put_factors_into_lst: bad term.");
}
-class Partition
+/** Checks a set of numbers for whether each number has a unique prime factor.
+ *
+ * @param[in] f list of numbers to check
+ * @return true: if number set is bad, false: if set is okay (has unique
+ * prime factors)
+ */
+static bool checkdivisors(const lst& f)
{
-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;
+ const int k = f.nops();
+ numeric q, r;
+ vector<numeric> d(k);
+ d[0] = ex_to<numeric>(abs(f.op(0)));
+ for ( int i=1; i<k; ++i ) {
+ q = ex_to<numeric>(abs(f.op(i)));
+ 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;
}
- ++k[i];
- ++sum;
}
- return false;
+ d[i] = q;
}
-private:
- size_t n, sum;
- vector<int> k;
-};
+ return false;
+}
-static void split(const UniPolyVec& factors, const Partition& part, UniPoly& a, UniPoly& b)
+/** Generates a set of evaluation points for a multivariate polynomial.
+ * The set fulfills the following conditions:
+ * 1. lcoeff(evaluated_polynomial) does not vanish
+ * 2. factors of lcoeff(evaluated_polynomial) have each a unique prime factor
+ * 3. evaluated_polynomial is square free
+ * See [Wan] for more details.
+ *
+ * @param[in] u multivariate polynomial to be factored
+ * @param[in] vn leading coefficient of u in x (x==first symbol in syms)
+ * @param[in] syms set of symbols that appear in u
+ * @param[in] f lst containing the factors of the leading coefficient vn
+ * @param[in,out] modulus integer modulus for random number generation (i.e. |a_i| < modulus)
+ * @param[out] u0 returns the evaluated (univariate) polynomial
+ * @param[out] a returns the valid evaluation points. must have initial size equal
+ * number of symbols-1 before calling generate_set
+ */
+static void generate_set(const ex& u, const ex& vn, const exset& syms, const lst& f,
+ numeric& modulus, ex& u0, vector<numeric>& a)
{
- 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];
+ const ex& x = *syms.begin();
+ while ( true ) {
+ ++modulus;
+ // generate a set of integers ...
+ 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]);
+ // ... for which the leading coefficient doesn't vanish ...
+ } while ( vnatry == 0 );
+ vna = vnatry;
+ u0 = u0.subs(*s == a[i]);
+ ++s;
}
- else {
- a = a * factors[i];
+ // ... for which u0 is square free ...
+ ex g = gcd(u0, u0.diff(ex_to<symbol>(x)));
+ if ( !is_a<numeric>(g) ) {
+ continue;
+ }
+ if ( !is_a<numeric>(vn) ) {
+ // ... and for which the evaluated factors have each an unique prime factor
+ lst fnum = f;
+ fnum.let_op(0) = fnum.op(0) * u0.content(x);
+ for ( size_t i=1; i<fnum.nops(); ++i ) {
+ if ( !is_a<numeric>(fnum.op(i)) ) {
+ s = syms.begin();
+ ++s;
+ for ( size_t j=0; j<a.size(); ++j, ++s ) {
+ fnum.let_op(i) = fnum.op(i).subs(*s == a[j]);
+ }
+ }
+ }
+ if ( checkdivisors(fnum) ) {
+ continue;
+ }
}
+ // ok, we have a valid set now
+ return;
}
}
-struct ModFactors
+// forward declaration
+static ex factor_sqrfree(const ex& poly);
+
+/** Multivariate factorization.
+ *
+ * The implementation is based on the algorithm described in [Wan].
+ * An evaluation homomorphism (a set of integers) is determined that fulfills
+ * certain criteria. The evaluated polynomial is univariate and is factorized
+ * by factor_univariate(). The main work then is to find the correct leading
+ * coefficients of the univariate factors. They have to correspond to the
+ * factors of the (multivariate) leading coefficient of the input polynomial
+ * (as defined for a specific variable x). After that the Hensel lifting can be
+ * performed.
+ *
+ * @param[in] poly expanded, square free polynomial
+ * @param[in] syms contains the symbols in the polynomial
+ * @return factorized polynomial
+ */
+static ex factor_multivariate(const ex& poly, const exset& syms)
{
- ex poly;
- UniPolyVec factors;
-};
+ exset::const_iterator s;
+ const ex& x = *syms.begin();
-static ex factor_univariate(const ex& poly, const ex& x)
-{
- ex unit, cont, prim;
- poly.unitcontprim(x, unit, cont, prim);
+ // make polynomial primitive
+ ex unit, cont, pp;
+ poly.unitcontprim(x, unit, cont, pp);
+ if ( !is_a<numeric>(cont) ) {
+ return factor_sqrfree(cont) * factor_sqrfree(pp);
+ }
- // 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;
+ // factor leading coefficient
+ ex vn = pp.collect(x).lcoeff(x);
+ ex vnlst;
+ if ( is_a<numeric>(vn) ) {
+ vnlst = lst(vn);
+ }
+ else {
+ ex vnfactors = factor(vn);
+ vnlst = put_factors_into_lst(vnfactors);
}
- // 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;
+ const unsigned int maxtrials = 3;
+ numeric modulus = (vnlst.nops() > 3) ? vnlst.nops() : 3;
+ vector<numeric> a(syms.size()-1, 0);
+
+ // try now to factorize until we are successful
+ while ( true ) {
+
+ unsigned int trialcount = 0;
+ unsigned int prime;
+ int factor_count = 0;
+ int min_factor_count = -1;
+ ex u, delta;
+ ex ufac, ufaclst;
+
+ // try several evaluation points to reduce the number of factors
+ while ( trialcount < maxtrials ) {
+
+ // generate a set of valid evaluation points
+ generate_set(pp, vn, syms, ex_to<lst>(vnlst), modulus, u, a);
+
+ ufac = factor_univariate(u, x, prime);
+ ufaclst = put_factors_into_lst(ufac);
+ factor_count = ufaclst.nops()-1;
+ delta = ufaclst.op(0);
+
+ if ( factor_count <= 1 ) {
+ // irreducible
+ return poly;
+ }
+ if ( min_factor_count < 0 ) {
+ // first time here
+ min_factor_count = factor_count;
+ }
+ else if ( min_factor_count == factor_count ) {
+ // one less to try
+ ++trialcount;
+ }
+ else if ( min_factor_count > factor_count ) {
+ // new minimum, reset trial counter
+ min_factor_count = factor_count;
+ trialcount = 0;
+ }
+ }
+
+ // determine true leading coefficients for the Hensel lifting
+ vector<ex> C(factor_count);
+ if ( is_a<numeric>(vn) ) {
+ // easy case
+ for ( size_t i=1; i<ufaclst.nops(); ++i ) {
+ C[i-1] = ufaclst.op(i).lcoeff(x);
+ }
+ }
+ else {
+ // difficult case.
+ // we use the property of the ftilde having a unique prime factor.
+ // details can be found in [Wan].
+ // calculate ftilde
+ vector<numeric> ftilde(vnlst.nops()-1);
+ for ( size_t i=0; i<ftilde.size(); ++i ) {
+ ex ft = vnlst.op(i+1);
+ s = syms.begin();
+ ++s;
+ for ( size_t j=0; j<a.size(); ++j ) {
+ ft = ft.subs(*s == a[j]);
+ ++s;
}
- 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;
+ ftilde[i] = ex_to<numeric>(ft);
+ }
+ // calculate D and C
+ vector<bool> used_flag(ftilde.size(), false);
+ vector<ex> D(factor_count, 1);
+ if ( delta == 1 ) {
+ for ( int i=0; i<factor_count; ++i ) {
+ numeric prefac = ex_to<numeric>(ufaclst.op(i+1).lcoeff(x));
+ for ( int j=ftilde.size()-1; j>=0; --j ) {
+ int count = 0;
+ while ( irem(prefac, ftilde[j]) == 0 ) {
+ prefac = iquo(prefac, ftilde[j]);
+ ++count;
+ }
+ if ( count ) {
+ used_flag[j] = true;
+ D[i] = D[i] * pow(vnlst.op(j+1), count);
}
}
- break;
+ C[i] = D[i] * prefac;
}
- 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 {
+ for ( int i=0; i<factor_count; ++i ) {
+ numeric prefac = ex_to<numeric>(ufaclst.op(i+1).lcoeff(x));
+ for ( int j=ftilde.size()-1; j>=0; --j ) {
+ int count = 0;
+ while ( irem(prefac, ftilde[j]) == 0 ) {
+ prefac = iquo(prefac, ftilde[j]);
+ ++count;
}
- else {
- *i1++ = tocheck.top().factors[i];
+ while ( irem(ex_to<numeric>(delta)*prefac, ftilde[j]) == 0 ) {
+ numeric g = gcd(prefac, ex_to<numeric>(ftilde[j]));
+ prefac = iquo(prefac, g);
+ delta = delta / (ftilde[j]/g);
+ ufaclst.let_op(i+1) = ufaclst.op(i+1) * (ftilde[j]/g);
+ ++count;
+ }
+ if ( count ) {
+ used_flag[j] = true;
+ D[i] = D[i] * pow(vnlst.op(j+1), count);
}
}
- tocheck.top().factors = newfactors1;
- tocheck.top().poly = answer.op(0);
- ModFactors mf;
- mf.factors = newfactors2;
- mf.poly = answer.op(1);
- tocheck.push(mf);
+ C[i] = D[i] * prefac;
}
}
- else {
- if ( !part.next() ) {
- result *= tocheck.top().poly;
- tocheck.pop();
+ // check if something went wrong
+ 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 ) {
+ continue;
+ }
+ }
+
+ // multiply the remaining content of the univariate polynomial into the
+ // first factor
+ if ( delta != 1 ) {
+ C[0] = C[0] * delta;
+ ufaclst.let_op(1) = ufaclst.op(1) * delta;
}
- }
- return unit * cont * result;
+ // set up evaluation points
+ 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 p^l
+ int maxdeg = 0;
+ for ( int i=1; i<=factor_count; ++i ) {
+ if ( ufaclst.op(i).degree(x) > maxdeg ) {
+ maxdeg = ufaclst[i].degree(x);
+ }
+ }
+ cl_I B = 2*calc_bound(u, x, maxdeg);
+ cl_I l = 1;
+ cl_I pl = prime;
+ while ( pl < B ) {
+ l = l + 1;
+ pl = pl * prime;
+ }
+
+ // set up modular factors (mod p^l)
+ cl_modint_ring R = find_modint_ring(expt_pos(cl_I(prime),l));
+ upvec modfactors(ufaclst.nops()-1);
+ for ( size_t i=1; i<ufaclst.nops(); ++i ) {
+ umodpoly_from_ex(modfactors[i-1], ufaclst.op(i), x, R);
+ }
+
+ // try Hensel lifting
+ ex res = hensel_multivar(pp, x, epv, prime, l, modfactors, C);
+ if ( res != lst() ) {
+ ex result = cont * unit;
+ 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);
+ }
+ return result;
+ }
+ }
}
-struct FindSymbolsMap : public map_function {
+/** Finds all symbols in an expression. Used by factor_sqrfree() and factor().
+ */
+struct find_symbols_map : public map_function {
exset syms;
ex operator()(const ex& e)
{
}
};
+/** Factorizes a polynomial that is square free. It calls either the univariate
+ * or the multivariate factorization functions.
+ */
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 ) {
+ // pull out direct factors
int ld = poly.ldegree(x);
ex res = factor_univariate(expand(poly/pow(x, ld)), x);
return res * pow(x,ld);
}
}
- // multivariate case not yet implemented!
- throw runtime_error("multivariate case not yet implemented!");
+ // multivariate case
+ ex res = factor_multivariate(poly, findsymbols.syms);
+ return res;
}
+/** Map used by factor() when factor_options::all is given to access all
+ * subexpressions and to call factor() on them.
+ */
+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) ) {
+ return factor(e, options);
+ }
+ 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();
+ return factor(s1, options) + s2.map(*this);
+ }
+ return e.map(*this);
+ }
+};
+
} // anonymous namespace
-ex factor(const ex& poly)
+/** Interface function to the outside world. It checks the arguments, tries a
+ * square free factorization, and then calls factor_sqrfree to do the hard
+ * work.
+ */
+ex factor(const ex& poly, unsigned options)
{
+ // 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;
}
// make poly square free
- ex sfpoly = sqrfree(poly, syms);
+ ex sfpoly = sqrfree(poly.expand(), syms);
// factorize the square free components
if ( is_a<power>(sfpoly) ) {
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;
}
} // namespace GiNaC
+
+#ifdef DEBUGFACTOR
+#include "test.h"
+#endif
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