/** @file factor.cpp
*
- * Polynomial factorization code (Implementation).
+ * 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
- * [W1] An Improved Multivariate Polynomial Factoring Algorithm,
- * P.S.Wang, Mathematics of Computation, Vol. 32, No. 144 (1978) 1215--1231.
+ * [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.
+ * 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) 1999-2011 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
//#define DEBUGFACTOR
-#ifdef DEBUGFACTOR
-#include <ostream>
-#include <ginac/ginac.h>
-using namespace GiNaC;
-#else
#include "factor.h"
#include "ex.h"
#include "mul.h"
#include "normal.h"
#include "add.h"
-#endif
#include <algorithm>
+#include <cmath>
+#include <limits>
#include <list>
#include <vector>
+#ifdef DEBUGFACTOR
+#include <ostream>
+#endif
using namespace std;
#include <cln/cln.h>
using namespace cln;
-#ifdef DEBUGFACTOR
-namespace Factor {
-#else
namespace GiNaC {
-#endif
#ifdef DEBUGFACTOR
#define DCOUT(str) cout << #str << endl
#define DCOUTVAR(var) cout << #var << ": " << var << endl
#define DCOUT2(str,var) cout << #str << ": " << var << endl
-#else
-#define DCOUT(str)
-#define DCOUTVAR(var)
-#define DCOUT2(str,var)
-#endif
-
-// forward declaration
-ex factor(const ex& poly, unsigned options);
-
-// anonymous namespace to hide all utility functions
-namespace {
-
-typedef vector<cl_MI> Vec;
-typedef vector<Vec> VecVec;
-
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const Vec& v)
+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 ) {
- cl_I coeff = the<cl_I>(ex_to<numeric>(poly.coeff(x,i)).to_cl_N());
- if ( !zerop(coeff) ) {
- t.c = R->canonhom(coeff);
- if ( !zerop(t.c) ) {
- t.exp = i;
- terms.push_back(t);
- }
- }
- }
- }
- UniPoly(const cl_modint_ring& ring, const UniPoly& poly) : R(ring)
- {
- if ( R->modulus == poly.R->modulus ) {
- terms = poly.terms;
- }
- else {
- list<Term>::const_iterator i=poly.terms.begin(), end=poly.terms.end();
- for ( ; i!=end; ++i ) {
- terms.push_back(*i);
- terms.back().c = R->canonhom(poly.R->retract(i->c));
- if ( zerop(terms.back().c) ) {
- terms.pop_back();
- }
- }
- }
- }
- UniPoly(const cl_modint_ring& ring, const Vec& v) : R(ring)
- {
- Term t;
- 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();
+ 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;
}
- 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;
- }
+
+ 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;
}
- 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);
- }
- }
- }
+ if (i == 0) {
+ is_zero = true;
+ break;
}
+ --i;
+ } while (true);
+
+ if (is_zero) {
+ p.clear();
+ return;
}
- cl_MI unit() const
- {
- return terms.front().c;
+
+ 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 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<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];
+ }
+ for ( ; i<sa; ++i ) {
+ r[i] = a[i];
+ }
+ canonicalize(r);
+ return r;
}
- void divide(const cl_I& x)
- {
- list<Term>::iterator i = terms.begin(), end = terms.end();
- for ( ; i != end; ++i ) {
- i->c = cl_MI(R, the<cl_I>(R->retract(i->c) / x));
+ else {
+ T r(sb);
+ int i = 0;
+ for ( ; i<sa; ++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;
- }
- }
- 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;
+ for ( ; i<sb; ++i ) {
+ r[i] = b[i];
}
+ canonicalize(r);
+ return r;
}
- 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;
+}
+
+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];
+ }
+ for ( ; i<sa; ++i ) {
+ r[i] = a[i];
+ }
+ canonicalize(r);
+ return r;
}
- bool operator==(const UniPoly& o) const
- {
- if ( terms.size() != o.terms.size() ) {
- return false;
+ else {
+ T r(sb);
+ int i = 0;
+ for ( ; i<sa; ++i ) {
+ r[i] = a[i] - b[i];
}
- 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;
+ for ( ; i<sb; ++i ) {
+ r[i] = -b[i];
}
- return true;
- }
- bool operator!=(const UniPoly& o) const
- {
- bool res = !(*this == o);
- return res;
+ canonicalize(r);
+ return r;
}
-};
+}
-static UniPoly operator*(const UniPoly& a, const UniPoly& b)
+static upoly operator*(const upoly& a, const upoly& 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);
+ 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];
}
}
+ canonicalize(c);
return c;
}
-static UniPoly operator-(const UniPoly& a, const UniPoly& b)
+static umodpoly operator*(const umodpoly& a, const umodpoly& 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;
+ 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];
}
}
- 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;
- }
+ canonicalize(c);
return c;
}
-static UniPoly operator*(const UniPoly& a, const cl_MI& fac)
+static upoly operator*(const upoly& a, const cl_I& x)
{
- unsigned int n = a.degree();
- UniPoly c(a.R);
- Term t;
- for ( unsigned int i=0 ; i<=n; ++i ) {
- t.c = a[i] * fac;
- if ( !zerop(t.c) ) {
- t.exp = i;
- c.terms.push_front(t);
- }
+ if ( zerop(x) ) {
+ upoly r;
+ return r;
}
- return c;
+ upoly r(a.size());
+ for ( size_t i=0; i<a.size(); ++i ) {
+ r[i] = a[i] * x;
+ }
+ return r;
}
-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);
- ++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;
+ if ( zerop(x) ) {
+ upoly r;
+ return r;
}
- while ( ib != bend ) {
- c.terms.push_back(*ib);
- ++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)
-// {
-// 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;
-// }
+static umodpoly operator*(const umodpoly& a, const cl_MI& x)
+{
+ umodpoly r(a.size());
+ for ( size_t i=0; i<a.size(); ++i ) {
+ r[i] = a[i] * x;
+ }
+ 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);
+}
#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const UniPolyVec& v)
+static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_I& modulus)
{
- UniPolyVec::const_iterator i = v.begin(), end = v.end();
- while ( i != end ) {
- o << *i++ << " , " << endl;
- }
- return o;
+ umodpoly_from_ex(ump, e, x, find_modint_ring(modulus));
}
-#endif // def DEBUGFACTOR
+#endif
-struct UniFactor
+static ex upoly_to_ex(const upoly& a, const ex& x)
{
- 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;
+ 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 << "*";
- }
- else {
- o << " ";
- }
- o << "[ " << ufv.factors[i].p << " ]^" << ufv.factors[i].exp << endl;
- }
- return 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;
+ }
+ }
+ 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;
+/** 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;
- while ( true ) {
- cl_MI qk = div(c[n+k], b[n]);
+ 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 ( degree(a) < degree(b) ) return gcd(b, a, c);
- if ( c.degree() < d.degree() ) {
- gcd(b, a, c);
- return;
- }
-
- 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);
- }
- ++i;
- 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)
{
- sqrfree_main(a, fvec);
- fvec.unique();
+ return ( a.size() == 1 && a[0] == a[0].ring()->one() );
}
-class Matrix
+/** 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)
+{
+ umodpoly b;
+ deriv(a, b);
+ if ( b.empty() ) {
+ return false;
+ }
+ umodpoly c;
+ gcd(a, b, c);
+ return equal_one(c);
+}
+
+// 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)
{
- vector<cl_MI>::iterator i = m.begin() + row*c;
for ( size_t cc=0; cc<c; ++cc ) {
- *i = *i * x;
- ++i;
+ 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)
{
- vector<cl_MI>::iterator i1 = m.begin() + row1*c;
- vector<cl_MI>::iterator i2 = m.begin() + row2*c;
for ( size_t cc=0; cc<c; ++cc ) {
- *i1 = *i1 - *i2 * fac;
- ++i1;
- ++i2;
+ 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)
{
- cl_MI buf;
- vector<cl_MI>::iterator i1 = m.begin() + row1*c;
- vector<cl_MI>::iterator i2 = m.begin() + row2*c;
for ( size_t cc=0; cc<c; ++cc ) {
- buf = *i1; *i1 = *i2; *i2 = buf;
- ++i1;
- ++i2;
+ 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
{
- Vec::const_iterator i = m.begin() + col;
for ( size_t rr=0; rr<r; ++rr ) {
- if ( !zerop(*i) ) {
+ std::size_t i = col + rr*c;
+ if ( !zerop(m[i]) ) {
return false;
}
- i += c;
}
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
-Matrix operator*(const Matrix& m1, const Matrix& m2)
+modular_matrix operator*(const modular_matrix& m1, const modular_matrix& m2)
{
const unsigned int r = m1.rowsize();
const unsigned int c = m2.colsize();
- Matrix o(r,c,m1(0,0));
+ modular_matrix o(r,c,m1(0,0));
for ( size_t i=0; i<r; ++i ) {
for ( size_t j=0; j<c; ++j ) {
return o;
}
-ostream& operator<<(ostream& o, const Matrix& m)
+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)
-{
- unsigned int n = a.degree();
- unsigned int q = cl_I_to_uint(a.R->modulus);
-// fast and buggy
-// 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);
-// }
-// }
-// slow and (hopefully) correct
- for ( size_t i=0; i<n; ++i ) {
- UniPoly qk(a.R);
- qk.set(i*q, a.R->one());
- UniPoly r(a.R);
- rem(qk, a, r);
- Vec rvec;
- for ( size_t j=0; j<n; ++j ) {
- rvec.push_back(r[j]);
+// 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)
+{
+ 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[0] = -rn_1 * a[0];
+ if ( (m % q) == 0 ) {
+ Q.set_row(m/q, r);
}
- Q.set_row(i, rvec);
}
}
-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)
{
- Matrix Q(a.degree(), a.degree(), a.R->zero());
+ 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);
- VecVec nu;
+ vector<mvec> nu;
nullspace(Q, nu);
+
const unsigned int k = nu.size();
if ( k == 1 ) {
+ // irreducible
return;
}
- list<UniPoly> factors;
+ list<umodpoly> factors;
factors.push_back(a);
unsigned int size = 1;
unsigned int r = 1;
- unsigned int q = cl_I_to_uint(a.R->modulus);
+ unsigned int q = cl_I_to_uint(R->modulus);
- list<UniPoly>::iterator u = factors.begin();
+ list<umodpoly>::iterator u = factors.begin();
+ // calculate all gcd's
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));
+ umodpoly nur = nu[r];
+ nur[0] = nur[0] - cl_MI(R, s);
+ canonicalize(nur);
+ umodpoly g;
gcd(nur, *u, g);
- if ( !is_one(g) && g != *u ) {
- UniPoly uo(a.R);
+ if ( unequal_one(g) && g != *u ) {
+ umodpoly uo;
div(*u, g, uo);
- if ( is_one(uo) ) {
+ if ( equal_one(uo) ) {
throw logic_error("berlekamp: unexpected divisor.");
}
else {
}
factors.push_back(g);
size = 0;
- list<UniPoly>::const_iterator i = factors.begin(), end = factors.end();
+ list<umodpoly>::const_iterator i = factors.begin(), end = factors.end();
while ( i != end ) {
- if ( i->degree() ) ++size;
+ if ( degree(*i) ) ++size;
++i;
}
if ( size == k ) {
- list<UniPoly>::const_iterator i = factors.begin(), end = factors.end();
+ list<umodpoly>::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 ) {
}
}
-static void factor_modular(const UniPoly& p, UniPolyVec& upv)
+// 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);
- return;
+#endif
}
-static void exteuclid(const UniPoly& a, const UniPoly& b, UniPoly& g, UniPoly& s, UniPoly& t)
+/** 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 ( a.degree() < b.degree() ) {
- exteuclid(b, a, g, t, s);
+ if ( degree(a) < degree(b) ) {
+ exteuclid(b, a, 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();
+
+ 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);
- r = c - q * d;
- r1 = c1 - q * d1;
- r2 = c2 - q * d2;
+ umodpoly r = c - q * d;
+ umodpoly r1 = s - q * d1;
+ umodpoly r2 = t - q * d2;
c = d;
- c1 = d1;
- c2 = d2;
+ s = d1;
+ t = d2;
+ if ( r.empty() ) break;
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());
+ 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);
}
-static ex replace_lc(const ex& poly, const ex& x, const ex& lc)
+/** 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)
{
- ex r = expand(poly + (lc - poly.lcoeff(x)) * pow(x, poly.degree(x)));
+ if ( poly.empty() ) return poly;
+ upoly r = poly;
+ r.back() = lc;
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)
+/** Calculates the bound for the modulus.
+ * See [Mig].
+ */
+static inline cl_I calc_bound(const ex& a, const ex& x, int maxdeg)
{
- ex a = a_;
- const cl_modint_ring& R = u1_.R;
-
- // calc bound B
- ex maxcoeff;
+ cl_I maxcoeff = 0;
+ cl_R coeff = 0;
for ( int i=a.degree(x); i>=a.ldegree(x); --i ) {
- maxcoeff += pow(abs(a.coeff(x, i)),2);
+ 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 normmc = ceiling1(the<cl_R>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
- cl_I maxdegree = (u1_.degree() > w1_.degree()) ? u1_.degree() : w1_.degree();
- cl_I B = normmc * expt_pos(cl_I(2), maxdegree);
+ 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
- ex alpha = a.lcoeff(x);
- ex gamma = gamma_;
- if ( gamma == 0 ) {
- gamma = alpha;
- }
- numeric gamma_ui = ex_to<numeric>(abs(gamma));
- 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);
+ 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
- UniPoly s(R), t(R), g(R);
- exteuclid(u1, w1, g, s, t);
+ umodpoly s;
+ umodpoly t;
+ exteuclid(u1, w1, 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);
- numeric modulus = p;
- const numeric maxmodulus = 2*numeric(B)*gamma_ui;
+ 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.is_zero() && modulus < maxmodulus ) {
- 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);
+ 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.is_zero() ) {
- ex delta = u.content(x);
- u = u / delta;
- w = w / gamma * delta;
- return lst(u, w);
+ 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 {
- return lst();
+ 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;
throw logic_error("next_prime: should not reach this point!");
}
-class Partition
+/** Manages the splitting a vector of of modular factors into two partitions.
+ */
+class factor_partition
{
public:
- Partition(size_t n_) : n(n_)
+ /** Takes the vector of modular factors and initializes the first partition */
+ factor_partition(const upvec& factors_) : factors(factors_)
{
- k.resize(n, 1);
- k[0] = 0;
- sum = n-1;
+ 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_first() const { return n-sum; }
- size_t size_second() const { return sum; }
-#ifdef DEBUGFACTOR
- void get() const
+ 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()
{
- for ( size_t i=0; i<k.size(); ++i ) {
- cout << k[i] << " ";
- }
- cout << endl;
+ 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 );
}
-#endif
- bool next()
+ void split()
{
- for ( size_t i=n-1; i>=1; --i ) {
- if ( k[i] ) {
- --k[i];
- --sum;
- return sum > 0;
+ 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];
}
- ++k[i];
- ++sum;
}
- return false;
}
private:
- size_t n, sum;
+ umodpoly lr[2];
+ vector< vector<umodpoly> > cache;
+ upvec factors;
+ umodpoly one;
+ size_t n;
+ size_t len;
+ size_t last;
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];
- }
- }
-}
-
+/** Contains a pair of univariate polynomial and its modular factors.
+ * Used by factor_univariate().
+ */
struct ModFactors
{
- ex poly;
- UniPolyVec factors;
+ upoly poly;
+ upvec factors;
};
-static ex factor_univariate(const ex& poly, const ex& x)
+/** 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;
- poly.unitcontprim(x, unit, cont, prim);
+ 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;
+ }
+ }
- // 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 && sqrfree_ufv.factors.front().exp == 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;
+ // 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;
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();
- Partition part(n);
+ factor_partition part(tocheck.top().factors);
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);
+ // 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 *= answer.op(0);
- tocheck.top().poly = answer.op(1);
+ 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;
}
- else if ( part.size_second() == 1 ) {
- if ( part.size_first() == 1 ) {
- result *= answer.op(0) * answer.op(1);
+ 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 *= answer.op(1);
- tocheck.top().poly = answer.op(0);
+ 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;
}
else {
- UniPolyVec newfactors1(part.size_first(), R), newfactors2(part.size_second(), R);
- UniPolyVec::iterator i1 = newfactors1.begin(), i2 = newfactors2.begin();
+ 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];
}
}
tocheck.top().factors = newfactors1;
- tocheck.top().poly = answer.op(0);
+ tocheck.top().poly = f1;
ModFactors mf;
mf.factors = newfactors2;
- mf.poly = answer.op(1);
+ mf.poly = f2;
tocheck.push(mf);
break;
}
}
else {
+ // not successful
if ( !part.next() ) {
- result *= tocheck.top().poly;
+ // 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
-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);
+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);
-UniPolyVec multiterm_eea_lift(const UniPolyVec& a, const ex& x, 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)
{
- DCOUT(multiterm_eea_lift);
- DCOUTVAR(a);
- DCOUTVAR(p);
- DCOUTVAR(k);
-
const size_t r = a.size();
- DCOUTVAR(r);
cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
- UniPoly fill(R);
- UniPolyVec q(r-1, fill);
+ 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];
}
- DCOUTVAR(q);
- UniPoly beta(R);
- beta.set(0, R->one());
- UniPolyVec s;
+ umodpoly beta(1, R->one());
+ upvec s;
for ( size_t j=1; j<r; ++j ) {
- DCOUTVAR(j);
- DCOUTVAR(beta);
vector<ex> mdarg(2);
- mdarg[0] = q[j-1].to_ex(x);
- mdarg[1] = a[j-1].to_ex(x);
+ 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, beta.to_ex(x), empty, 0, p, k);
- UniPoly sigma1(R, exsigma[0], x);
- UniPoly sigma2(R, exsigma[1], x);
+ 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);
-
- DCOUTVAR(s);
- DCOUT(END multiterm_eea_lift);
return s;
}
-void eea_lift(const UniPoly& a, const UniPoly& b, const ex& x, unsigned int p, unsigned int k, UniPoly& s_, UniPoly& t_)
+/** 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)
{
- DCOUT(eea_lift);
- DCOUTVAR(a);
- DCOUTVAR(b);
- DCOUTVAR(x);
- DCOUTVAR(p);
- DCOUTVAR(k);
+ 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);
- UniPoly amod(R, a);
- UniPoly bmod(R, b);
- DCOUTVAR(amod);
- DCOUTVAR(bmod);
-
- UniPoly smod(R), tmod(R), g(R);
- exteuclid(amod, bmod, g, smod, tmod);
-
- DCOUTVAR(smod);
- DCOUTVAR(tmod);
- DCOUTVAR(g);
+ 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));
- UniPoly s(Rpk, smod);
- UniPoly t(Rpk, tmod);
- DCOUTVAR(s);
- DCOUTVAR(t);
+ umodpoly s = smod;
+ change_modulus(Rpk, s);
+ umodpoly t = tmod;
+ change_modulus(Rpk, t);
cl_I modulus(p);
-
- UniPoly one(Rpk);
- one.set(0, Rpk->one());
+ umodpoly one(1, Rpk->one());
for ( size_t j=1; j<k; ++j ) {
- UniPoly e = one - a * s - b * t;
- e.divide(modulus);
- UniPoly c(R, e);
- UniPoly sigmabar(R);
- sigmabar = smod * c;
- UniPoly taubar(R);
- taubar = tmod * c;
- UniPoly q(R);
- div(sigmabar, bmod, q);
- UniPoly sigma(R);
- rem(sigmabar, bmod, sigma);
- UniPoly tau(R);
- tau = taubar + q * amod;
- UniPoly sadd(Rpk, sigma);
+ 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;
- UniPoly tadd(Rpk, tau);
+ umodpoly tadd = tau;
+ change_modulus(Rpk, tadd);
t = t + tadd * modmodulus;
modulus = modulus * p;
}
s_ = s; t_ = t;
-
- DCOUTVAR(s);
- DCOUTVAR(t);
- DCOUT2(check, a*s + b*t);
- DCOUT(END eea_lift);
}
-UniPolyVec univar_diophant(const UniPolyVec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
+/** 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)
{
- DCOUT(univar_diophant);
- DCOUTVAR(a);
- DCOUTVAR(x);
- DCOUTVAR(m);
- DCOUTVAR(p);
- DCOUTVAR(k);
-
cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
const size_t r = a.size();
- UniPolyVec result;
+ upvec result;
if ( r > 2 ) {
- UniPolyVec s = multiterm_eea_lift(a, x, p, k);
+ upvec s = multiterm_eea_lift(a, x, p, k);
for ( size_t j=0; j<r; ++j ) {
- ex phi = expand(pow(x,m)*s[j].to_ex(x));
- UniPoly bmod(R, phi, x);
- UniPoly buf(R);
+ umodpoly bmod = umodpoly_to_umodpoly(s[j], R, m);
+ umodpoly buf;
rem(bmod, a[j], buf);
result.push_back(buf);
}
}
else {
- UniPoly s(R), t(R);
+ umodpoly s, t;
eea_lift(a[1], a[0], x, p, k, s, t);
- ex phi = expand(pow(x,m)*s.to_ex(x));
- UniPoly bmod(R, phi, x);
- UniPoly buf(R);
- rem(bmod, a[0], buf);
+ umodpoly bmod = umodpoly_to_umodpoly(s, R, m);
+ umodpoly buf, q;
+ remdiv(bmod, a[0], buf, q);
result.push_back(buf);
- UniPoly q(R);
- div(bmod, a[0], q);
- phi = expand(pow(x,m)*t.to_ex(x));
- UniPoly t1mod(R, phi, x);
+ umodpoly t1mod = umodpoly_to_umodpoly(t, R, m);
buf = t1mod + q * a[1];
result.push_back(buf);
}
- DCOUTVAR(result);
- DCOUT(END univar_diophant);
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_) { }
}
};
+/** 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);
+ return map(e.expand());
}
-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.
+ *
+ * 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)
{
vector<ex> a = a_;
- DCOUT(multivar_diophant);
-#ifdef DEBUGFACTOR
- cout << "a ";
- for ( size_t i=0; i<a.size(); ++i ) {
- cout << a[i] << " ";
- }
- cout << endl;
-#endif
- DCOUTVAR(x);
- DCOUTVAR(c);
-#ifdef DEBUGFACTOR
- cout << "I ";
- for ( size_t i=0; i<I.size(); ++i ) {
- cout << I[i].x << "=" << I[i].evalpoint << " ";
- }
- cout << endl;
-#endif
- DCOUTVAR(d);
- DCOUTVAR(p);
- DCOUTVAR(k);
-
const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
const size_t r = a.size();
const size_t nu = I.size() + 1;
- DCOUTVAR(r);
- DCOUTVAR(nu);
vector<ex> sigma;
if ( nu > 1 ) {
vector<EvalPoint> Inew = I;
Inew.pop_back();
sigma = multivar_diophant(anew, x, cnew, Inew, d, p, k);
- DCOUTVAR(sigma);
ex buf = c;
for ( size_t i=0; i<r; ++i ) {
buf -= sigma[i] * b[i];
}
- ex e = buf;
- e = make_modular(e, R);
+ ex e = make_modular(buf, R);
- e = e.expand();
- DCOUTVAR(e);
- DCOUTVAR(d);
ex monomial = 1;
- for ( size_t m=1; m<=d; ++m ) {
- DCOUTVAR(m);
- while ( !e.is_zero() && e.has(xnu) ) {
- monomial *= (xnu - alphanu);
- monomial = expand(monomial);
- DCOUTVAR(xnu);
- DCOUTVAR(alphanu);
- ex cm = e.diff(ex_to<symbol>(xnu), m).subs(xnu==alphanu) / factorial(m);
- DCOUTVAR(cm);
- if ( !cm.is_zero() ) {
- vector<ex> delta_s = multivar_diophant(anew, x, cm, Inew, d, p, k);
- DCOUTVAR(delta_s);
- ex buf = e;
- for ( size_t j=0; j<delta_s.size(); ++j ) {
- delta_s[j] *= monomial;
- sigma[j] += delta_s[j];
- buf -= delta_s[j] * b[j];
- }
- e = buf.expand();
- e = make_modular(e, R);
+ 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);
}
}
}
else {
- DCOUT(uniterm left);
- UniPolyVec amod;
+ upvec amod;
for ( size_t i=0; i<a.size(); ++i ) {
- UniPoly up(R, a[i], x);
+ umodpoly up;
+ umodpoly_from_ex(up, a[i], x, R);
amod.push_back(up);
}
nterms = 1;
z = c;
}
- DCOUTVAR(nterms);
for ( size_t i=0; i<nterms; ++i ) {
- DCOUTVAR(z);
int m = z.degree(x);
- DCOUTVAR(m);
cl_I cm = the<cl_I>(ex_to<numeric>(z.lcoeff(x)).to_cl_N());
- DCOUTVAR(cm);
- UniPolyVec delta_s = univar_diophant(amod, x, m, p, k);
+ 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);
- DCOUTVAR(modcm);
for ( size_t j=0; j<delta_s.size(); ++j ) {
delta_s[j] = delta_s[j] * modcm;
- sigma[j] = sigma[j] + delta_s[j].to_ex(x);
- }
- DCOUTVAR(delta_s);
-#ifdef DEBUGFACTOR
- cout << "STEP " << i << " sigma ";
- for ( size_t p=0; p<sigma.size(); ++p ) {
- cout << sigma[p] << " ";
+ sigma[j] = sigma[j] + umodpoly_to_ex(delta_s[j], x);
}
- cout << endl;
-#endif
if ( nterms > 1 ) {
z = c.op(i+1);
}
}
}
-#ifdef DEBUGFACTOR
- cout << "sigma ";
- for ( size_t i=0; i<sigma.size(); ++i ) {
- cout << sigma[i] << " ";
- }
- cout << endl;
-#endif
for ( size_t i=0; i<sigma.size(); ++i ) {
sigma[i] = make_modular(sigma[i], R);
}
-#ifdef DEBUGFACTOR
- cout << "sigma ";
- for ( size_t i=0; i<sigma.size(); ++i ) {
- cout << sigma[i] << " ";
- }
- cout << endl;
-#endif
- DCOUT(END multivar_diophant);
return sigma;
}
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const vector<EvalPoint>& v)
-{
- for ( size_t i=0; i<v.size(); ++i ) {
- o << "(" << v[i].x << "==" << v[i].evalpoint << ") ";
- }
- return o;
-}
-#endif // def DEBUGFACTOR
-
-
-ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigned int p, const cl_I& l, const UniPolyVec& u, const vector<ex>& lcU)
+/** 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)
{
- DCOUT(hensel_multivar);
- DCOUTVAR(a);
- DCOUTVAR(x);
- DCOUTVAR(I);
- DCOUTVAR(p);
- DCOUTVAR(l);
- DCOUTVAR(u);
- DCOUTVAR(lcU);
const size_t nu = I.size() + 1;
const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
- DCOUTVAR(nu);
-
vector<ex> A(nu);
A[nu-1] = a;
A[j-2] = make_modular(A[j-2], R);
}
-#ifdef DEBUGFACTOR
- cout << "A ";
- for ( size_t i=0; i<A.size(); ++i) cout << A[i] << " ";
- cout << endl;
-#endif
-
int maxdeg = a.degree(I.front().x);
for ( size_t i=1; i<I.size(); ++i ) {
int maxdeg2 = a.degree(I[i].x);
if ( maxdeg2 > maxdeg ) maxdeg = maxdeg2;
}
- DCOUTVAR(maxdeg);
const size_t n = u.size();
- DCOUTVAR(n);
vector<ex> U(n);
for ( size_t i=0; i<n; ++i ) {
- U[i] = u[i].to_ex(x);
+ U[i] = umodpoly_to_ex(u[i], x);
}
-#ifdef DEBUGFACTOR
- cout << "U ";
- for ( size_t i=0; i<U.size(); ++i) cout << U[i] << " ";
- cout << endl;
-#endif
for ( size_t j=2; j<=nu; ++j ) {
- DCOUTVAR(j);
vector<ex> U1 = U;
ex monomial = 1;
- DCOUTVAR(U);
for ( size_t m=0; m<n; ++m) {
if ( lcU[m] != 1 ) {
ex coef = lcU[m];
for ( size_t i=j-1; i<nu-1; ++i ) {
coef = coef.subs(I[i].x == I[i].evalpoint);
}
- coef = expand(coef);
coef = make_modular(coef, R);
int deg = U[m].degree(x);
U[m] = U[m] - U[m].lcoeff(x) * pow(x,deg) + coef * pow(x,deg);
}
}
- DCOUTVAR(U);
ex Uprod = 1;
for ( size_t i=0; i<n; ++i ) {
Uprod *= U[i];
}
ex e = expand(A[j-1] - Uprod);
- DCOUTVAR(e);
vector<EvalPoint> newI;
for ( size_t i=1; i<=j-2; ++i ) {
newI.push_back(I[i-1]);
}
- DCOUTVAR(newI);
ex xj = I[j-2].x;
int alphaj = I[j-2].evalpoint;
size_t deg = A[j-1].degree(xj);
- DCOUTVAR(deg);
for ( size_t k=1; k<=deg; ++k ) {
- DCOUTVAR(k);
if ( !e.is_zero() ) {
- DCOUTVAR(xj);
- DCOUTVAR(alphaj);
monomial *= (xj - alphaj);
monomial = expand(monomial);
- DCOUTVAR(monomial);
ex dif = e.diff(ex_to<symbol>(xj), k);
- DCOUTVAR(dif);
ex c = dif.subs(xj==alphaj) / factorial(k);
- DCOUTVAR(c);
if ( !c.is_zero() ) {
vector<ex> deltaU = multivar_diophant(U1, x, c, newI, maxdeg, p, cl_I_to_uint(l));
for ( size_t i=0; i<n; ++i ) {
- DCOUTVAR(i);
- DCOUTVAR(deltaU[i]);
deltaU[i] *= monomial;
U[i] += deltaU[i];
U[i] = make_modular(U[i], R);
- U[i] = U[i].expand();
- DCOUTVAR(U[i]);
}
ex Uprod = 1;
for ( size_t i=0; i<n; ++i ) {
Uprod *= U[i];
}
- DCOUTVAR(Uprod.expand());
- DCOUTVAR(A[j-1]);
- e = expand(A[j-1] - Uprod);
+ e = A[j-1] - Uprod;
e = make_modular(e, R);
- DCOUTVAR(e);
- }
- else {
- break;
}
}
}
for ( size_t i=0; i<U.size(); ++i ) {
acand *= U[i];
}
- DCOUTVAR(acand);
if ( expand(a-acand).is_zero() ) {
lst res;
for ( size_t i=0; i<U.size(); ++i ) {
res.append(U[i]);
}
- DCOUTVAR(res);
- DCOUT(END hensel_multivar);
return res;
}
else {
lst res;
- DCOUTVAR(res);
- DCOUT(END hensel_multivar);
return lst();
}
}
+/** 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)
{
- DCOUT(put_factors_into_lst);
- DCOUTVAR(e);
-
lst result;
-
if ( is_a<numeric>(e) ) {
result.append(e);
- DCOUT(END put_factors_into_lst);
- DCOUTVAR(result);
return result;
}
if ( is_a<power>(e) ) {
result.append(1);
result.append(e.op(0));
- result.append(e.op(1));
- DCOUT(END put_factors_into_lst);
- DCOUTVAR(result);
return result;
}
if ( is_a<symbol>(e) || is_a<add>(e) ) {
result.append(1);
result.append(e);
- result.append(1);
- DCOUT(END put_factors_into_lst);
- DCOUTVAR(result);
return result;
}
if ( is_a<mul>(e) ) {
}
if ( is_a<power>(op) ) {
result.append(op.op(0));
- result.append(op.op(1));
}
if ( is_a<symbol>(op) || is_a<add>(op) ) {
result.append(op);
- result.append(1);
}
}
result.prepend(nfac);
- DCOUT(END put_factors_into_lst);
- DCOUTVAR(result);
return result;
}
throw runtime_error("put_factors_into_lst: bad term.");
}
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const vector<numeric>& v)
-{
- for ( size_t i=0; i<v.size(); ++i ) {
- o << v[i] << " ";
- }
- return o;
-}
-#endif // def DEBUGFACTOR
-
-static bool checkdivisors(const lst& f, vector<numeric>& d)
+/** 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)
{
- DCOUT(checkdivisors);
- const int k = f.nops()-2;
- DCOUTVAR(k);
- DCOUTVAR(d.size());
+ const int k = f.nops();
numeric q, r;
- d[0] = ex_to<numeric>(f.op(0) * f.op(f.nops()-1));
- if ( d[0] == 1 && k == 1 && abs(f.op(1)) != 1 ) {
- DCOUT(false);
- DCOUT(END checkdivisors);
- return false;
- }
- DCOUTVAR(d[0]);
- for ( int i=1; i<=k; ++i ) {
- DCOUTVAR(i);
- DCOUTVAR(abs(f.op(i)));
+ 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)));
- DCOUTVAR(q);
for ( int j=i-1; j>=0; --j ) {
r = d[j];
- DCOUTVAR(r);
do {
r = gcd(r, q);
- DCOUTVAR(r);
q = q/r;
- DCOUTVAR(q);
} while ( r != 1 );
if ( q == 1 ) {
- DCOUT(true);
- DCOUT(END checkdivisors);
return true;
}
}
d[i] = q;
}
- DCOUT(false);
- DCOUT(END checkdivisors);
return false;
}
-static bool generate_set(const ex& u, const ex& vn, const exset& syms, const ex& f, const numeric& modulus, vector<numeric>& a, vector<numeric>& d)
+/** 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)
{
- // computation of d is actually not necessary
- DCOUT(generate_set);
- DCOUTVAR(u);
- DCOUTVAR(vn);
- DCOUTVAR(f);
- DCOUTVAR(modulus);
const ex& x = *syms.begin();
- bool trying = true;
- do {
- ex u0 = u;
+ 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 ) {
- DCOUTVAR(*s);
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;
}
- DCOUTVAR(a);
- DCOUTVAR(u0);
- if ( gcd(u0,u0.diff(ex_to<symbol>(x))) != 1 ) {
+ // ... 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) ) {
- trying = false;
- }
- else {
- DCOUT(do substitution);
- lst fnum;
- lst::const_iterator i = ex_to<lst>(f).begin();
- fnum.append(*i++);
- bool problem = false;
- while ( i!=ex_to<lst>(f).end() ) {
- ex fs = *i;
- if ( !is_a<numeric>(fs) ) {
+ 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 ) {
- fs = fs.subs(*s == a[j]);
- ++s;
- }
- if ( abs(fs) == 1 ) {
- problem = true;
- break;
+ for ( size_t j=0; j<a.size(); ++j, ++s ) {
+ fnum.let_op(i) = fnum.op(i).subs(*s == a[j]);
}
}
- fnum.append(fs);
- ++i; ++i;
}
- if ( problem ) {
- return true;
+ if ( checkdivisors(fnum) ) {
+ continue;
}
- ex con = u0.content(x);
- fnum.append(con);
- DCOUTVAR(fnum);
- trying = checkdivisors(fnum, d);
}
- } while ( trying );
- DCOUT(END generate_set);
- return false;
+ // ok, we have a valid set now
+ return;
+ }
}
+// 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)
{
- DCOUT(factor_multivariate);
- DCOUTVAR(poly);
-
exset::const_iterator s;
const ex& x = *syms.begin();
- DCOUTVAR(x);
-
- /* make polynomial primitive */
- ex p = poly.expand().collect(x);
- DCOUTVAR(p);
- ex cont = p.lcoeff(x);
- for ( numeric i=p.degree(x)-1; i>=p.ldegree(x); --i ) {
- cont = gcd(cont, p.coeff(x,ex_to<numeric>(i).to_int()));
- if ( cont == 1 ) break;
- }
- DCOUTVAR(cont);
- ex pp = expand(normal(p / cont));
- DCOUTVAR(pp);
+
+ // make polynomial primitive
+ ex unit, cont, pp;
+ poly.unitcontprim(x, unit, cont, pp);
if ( !is_a<numeric>(cont) ) {
-#ifdef DEBUGFACTOR
- return ::factor(cont) * ::factor(pp);
-#else
- return factor(cont) * factor(pp);
-#endif
+ return factor_sqrfree(cont) * factor_sqrfree(pp);
}
- /* factor leading coefficient */
- pp = pp.collect(x);
- ex vn = pp.lcoeff(x);
- pp = pp.expand();
+ // factor leading coefficient
+ ex vn = pp.collect(x).lcoeff(x);
ex vnlst;
if ( is_a<numeric>(vn) ) {
vnlst = lst(vn);
}
else {
-#ifdef DEBUGFACTOR
- ex vnfactors = ::factor(vn);
-#else
ex vnfactors = factor(vn);
-#endif
vnlst = put_factors_into_lst(vnfactors);
}
- DCOUTVAR(vnlst);
- const numeric maxtrials = 3;
- numeric modulus = (vnlst.nops()-1 > 3) ? vnlst.nops()-1 : 3;
- DCOUTVAR(modulus);
- numeric minimalr = -1;
+ const unsigned int maxtrials = 3;
+ numeric modulus = (vnlst.nops() > 3) ? vnlst.nops() : 3;
vector<numeric> a(syms.size()-1, 0);
- vector<numeric> d((vnlst.nops()-1)/2+1, 0);
+ // try now to factorize until we are successful
while ( true ) {
- numeric trialcount = 0;
- ex u, delta;
+
+ unsigned int trialcount = 0;
unsigned int prime;
- size_t factor_count;
- ex ufac;
- ex ufaclst;
+ 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 ) {
- bool problem = generate_set(pp, vn, syms, vnlst, modulus, a, d);
- DCOUTVAR(problem);
- if ( problem ) {
- ++modulus;
- continue;
- }
- DCOUTVAR(a);
- DCOUTVAR(d);
- u = pp;
- s = syms.begin();
- ++s;
- for ( size_t i=0; i<a.size(); ++i ) {
- u = u.subs(*s == a[i]);
- ++s;
- }
- delta = u.content(x);
- DCOUTVAR(u);
-
- // determine proper prime
- prime = 3;
- DCOUTVAR(prime);
- cl_modint_ring R = find_modint_ring(prime);
- DCOUTVAR(u.lcoeff(x));
- while ( true ) {
- if ( irem(ex_to<numeric>(u.lcoeff(x)), prime) != 0 ) {
- UniPoly modpoly(R, u, x);
- UniFactorVec sqrfree_ufv;
- squarefree(modpoly, sqrfree_ufv);
- DCOUTVAR(sqrfree_ufv);
- if ( sqrfree_ufv.factors.size() == 1 && sqrfree_ufv.factors.front().exp == 1 ) break;
- }
- prime = next_prime(prime);
- DCOUTVAR(prime);
- R = find_modint_ring(prime);
- }
-#ifdef DEBUGFACTOR
- ufac = ::factor(u);
-#else
- ufac = factor(u);
-#endif
- DCOUTVAR(ufac);
+ // 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);
- DCOUTVAR(ufaclst);
- factor_count = (ufaclst.nops()-1)/2;
- DCOUTVAR(factor_count);
+ factor_count = ufaclst.nops()-1;
+ delta = ufaclst.op(0);
if ( factor_count <= 1 ) {
- DCOUTVAR(poly);
- DCOUT(END factor_multivariate);
+ // irreducible
return poly;
}
-
- if ( minimalr < 0 ) {
- minimalr = factor_count;
+ if ( min_factor_count < 0 ) {
+ // first time here
+ min_factor_count = factor_count;
}
- else if ( minimalr == factor_count ) {
+ else if ( min_factor_count == factor_count ) {
+ // one less to try
++trialcount;
- ++modulus;
}
- else if ( minimalr > factor_count ) {
- minimalr = factor_count;
+ else if ( min_factor_count > factor_count ) {
+ // new minimum, reset trial counter
+ min_factor_count = factor_count;
trialcount = 0;
}
- DCOUTVAR(trialcount);
- DCOUTVAR(minimalr);
- if ( minimalr <= 1 ) {
- DCOUTVAR(poly);
- DCOUT(END factor_multivariate);
- return poly;
- }
- }
-
- vector<numeric> ftilde((vnlst.nops()-1)/2+1);
- ftilde[0] = ex_to<numeric>(vnlst.op(0));
- for ( size_t i=1; i<ftilde.size(); ++i ) {
- ex ft = vnlst.op((i-1)*2+1);
- s = syms.begin();
- ++s;
- for ( size_t j=0; j<a.size(); ++j ) {
- ft = ft.subs(*s == a[j]);
- ++s;
- }
- ftilde[i] = ex_to<numeric>(ft);
- }
- DCOUTVAR(ftilde);
-
- vector<bool> used_flag((vnlst.nops()-1)/2+1, false);
- vector<ex> D(factor_count, 1);
- for ( size_t i=0; i<=factor_count; ++i ) {
- DCOUTVAR(i);
- numeric prefac;
- if ( i == 0 ) {
- prefac = ex_to<numeric>(ufaclst.op(0));
- ftilde[0] = ftilde[0] / prefac;
- vnlst.let_op(0) = vnlst.op(0) / prefac;
- continue;
- }
- else {
- prefac = ex_to<numeric>(ufaclst.op(2*(i-1)+1).lcoeff(x));
- }
- DCOUTVAR(prefac);
- for ( size_t j=(vnlst.nops()-1)/2+1; j>0; --j ) {
- DCOUTVAR(j);
- DCOUTVAR(prefac);
- DCOUTVAR(ftilde[j-1]);
- if ( abs(ftilde[j-1]) == 1 ) {
- used_flag[j-1] = true;
- continue;
- }
- numeric g = gcd(prefac, ftilde[j-1]);
- DCOUTVAR(g);
- if ( g != 1 ) {
- DCOUT(has_common_prime);
- prefac = prefac / g;
- numeric count = abs(iquo(g, ftilde[j-1]));
- DCOUTVAR(count);
- used_flag[j-1] = true;
- if ( i > 0 ) {
- if ( j == 1 ) {
- D[i-1] = D[i-1] * pow(vnlst.op(0), count);
- }
- else {
- D[i-1] = D[i-1] * pow(vnlst.op(2*(j-2)+1), count);
- }
- }
- else {
- ftilde[j-1] = ftilde[j-1] / prefac;
- DCOUT(BREAK);
- DCOUTVAR(ftilde[j-1]);
- break;
- }
- ++j;
- }
- }
- }
- DCOUTVAR(D);
-
- bool some_factor_unused = false;
- for ( size_t i=0; i<used_flag.size(); ++i ) {
- if ( !used_flag[i] ) {
- some_factor_unused = true;
- break;
- }
- }
- if ( some_factor_unused ) {
- DCOUT(some factor unused!);
- continue;
}
+ // determine true leading coefficients for the Hensel lifting
vector<ex> C(factor_count);
- DCOUTVAR(C);
- DCOUTVAR(delta);
- if ( delta == 1 ) {
- for ( size_t i=0; i<D.size(); ++i ) {
- ex Dtilde = D[i];
- s = syms.begin();
- ++s;
- for ( size_t j=0; j<a.size(); ++j ) {
- Dtilde = Dtilde.subs(*s == a[j]);
- ++s;
- }
- DCOUTVAR(Dtilde);
- C[i] = D[i] * (ufaclst.op(2*i+1).lcoeff(x) / Dtilde);
+ 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 {
- for ( size_t i=0; i<D.size(); ++i ) {
- ex Dtilde = D[i];
+ // 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 ) {
- Dtilde = Dtilde.subs(*s == a[j]);
+ ft = ft.subs(*s == a[j]);
++s;
}
- ex ui;
- if ( i == 0 ) {
- ui = ufaclst.op(0);
+ 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);
+ }
+ }
+ C[i] = D[i] * prefac;
}
- else {
- ui = ufaclst.op(2*(i-1)+1);
+ }
+ 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;
+ }
+ 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);
+ }
+ }
+ C[i] = D[i] * prefac;
}
- while ( true ) {
- ex d = gcd(ui.lcoeff(x), Dtilde);
- C[i] = D[i] * ( ui.lcoeff(x) / d );
- ui = ui * ( Dtilde[i] / d );
- delta = delta / ( Dtilde[i] / d );
- if ( delta == 1 ) break;
- ui = delta * ui;
- C[i] = delta * C[i];
- pp = pp * pow(delta, D.size()-1);
+ }
+ // 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;
}
- DCOUTVAR(C);
+ // set up evaluation points
EvalPoint ep;
vector<EvalPoint> epv;
s = syms.begin();
ep.evalpoint = a[i].to_int();
epv.push_back(ep);
}
- DCOUTVAR(epv);
- // calc bound B
- ex maxcoeff;
- for ( int i=u.degree(x); i>=u.ldegree(x); --i ) {
- maxcoeff += pow(abs(u.coeff(x, i)),2);
- }
- cl_I normmc = ceiling1(the<cl_R>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
- unsigned int maxdegree = 0;
- for ( size_t i=0; i<factor_count; ++i ) {
- if ( ufaclst[2*i+1].degree(x) > (int)maxdegree ) {
- maxdegree = ufaclst[2*i+1].degree(x);
+ // 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 = normmc * expt_pos(cl_I(2), maxdegree);
+ 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;
}
-
- UniPolyVec uvec;
+
+ // set up modular factors (mod p^l)
cl_modint_ring R = find_modint_ring(expt_pos(cl_I(prime),l));
- for ( size_t i=0; i<(ufaclst.nops()-1)/2; ++i ) {
- UniPoly newu(R, ufaclst.op(i*2+1), x);
- uvec.push_back(newu);
+ 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);
}
- DCOUTVAR(uvec);
- ex res = hensel_multivar(ufaclst.op(0)*pp, x, epv, prime, l, uvec, C);
+ // try Hensel lifting
+ ex res = hensel_multivar(pp, x, epv, prime, l, modfactors, C);
if ( res != lst() ) {
- ex result = cont * ufaclst.op(0);
+ 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);
}
- DCOUTVAR(result);
- DCOUT(END factor_multivariate);
return result;
}
}
}
+/** 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
// 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);
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) ) {
-#ifdef DEBUGFACTOR
- return ::factor(e, options);
-#else
return factor(e, options);
-#endif
}
if ( is_a<add>(e) ) {
ex s1, s2;
}
s1 = s1.eval();
s2 = s2.eval();
-#ifdef DEBUGFACTOR
- return ::factor(s1, options) + s2.map(*this);
-#else
return factor(s1, options) + s2.map(*this);
-#endif
}
return e.map(*this);
}
} // anonymous namespace
-#ifdef DEBUGFACTOR
-ex factor(const ex& poly, unsigned options = 0)
-#else
+/** 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)
-#endif
{
// check arguments
if ( !poly.info(info_flags::polynomial) ) {
}
// 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) ) {
}
} // namespace GiNaC
+
+#ifdef DEBUGFACTOR
+#include "test.h"
+#endif
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