/** @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) 1999-2024 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
#include "normal.h"
#include "add.h"
+#include <type_traits>
#include <algorithm>
-#include <cmath>
#include <limits>
#include <list>
#include <vector>
+#include <stack>
#ifdef DEBUGFACTOR
#include <ostream>
#endif
namespace GiNaC {
-#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
-
// anonymous namespace to hide all utility functions
namespace {
-typedef vector<cl_MI> mvec;
#ifdef DEBUGFACTOR
+#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)
{
- vector<int>::const_iterator i = v.begin(), end = v.end();
+ auto i = v.begin(), end = v.end();
while ( i != end ) {
- o << *i++ << " ";
+ o << *i << " ";
+ ++i;
}
return o;
}
-ostream& operator<<(ostream& o, const vector<cl_I>& v)
+static ostream& operator<<(ostream& o, const vector<cl_I>& v)
{
- vector<cl_I>::const_iterator i = v.begin(), end = v.end();
+ auto i = v.begin(), end = v.end();
while ( i != end ) {
o << *i << "[" << i-v.begin() << "]" << " ";
++i;
}
return o;
}
-ostream& operator<<(ostream& o, const vector<cl_MI>& v)
+static ostream& operator<<(ostream& o, const vector<cl_MI>& v)
{
- vector<cl_MI>::const_iterator i = v.begin(), end = v.end();
+ auto i = v.begin(), end = v.end();
while ( i != end ) {
o << *i << "[" << i-v.begin() << "]" << " ";
++i;
}
return o;
}
-ostream& operator<<(ostream& o, const vector< vector<cl_MI> >& v)
+ostream& operator<<(ostream& o, const vector<numeric>& v)
{
- vector< vector<cl_MI> >::const_iterator i = v.begin(), end = v.end();
+ for ( size_t i=0; i<v.size(); ++i ) {
+ o << v[i] << " ";
+ }
+ return o;
+}
+ostream& operator<<(ostream& o, const vector<vector<cl_MI>>& v)
+{
+ auto i = v.begin(), end = v.end();
while ( i != end ) {
o << i-v.begin() << ": " << *i << endl;
++i;
}
return o;
}
-#endif
+#else
+#define DCOUT(str)
+#define DCOUTVAR(var)
+#define DCOUT2(str,var)
+#endif // def DEBUGFACTOR
////////////////////////////////////////////////////////////////////////////////
// modular univariate polynomial code
typedef std::vector<cln::cl_I> upoly;
typedef vector<umodpoly> upvec;
-// COPY FROM UPOLY.HPP
+
+// COPY FROM UPOLY.H
// CHANGED size_t -> int !!!
template<typename T> static int degree(const T& p)
return p[p.size() - 1];
}
+/** Make the polynomial unit normal (having unit normal leading coefficient).
+ *
+ * @param[in, out] a polynomial to make unit normal
+ * @return true if polynomial a was already unit normal, false otherwise
+ */
static bool normalize_in_field(umodpoly& a)
{
if (a.size() == 0)
return false;
}
+/** Remove leading zero coefficients from polynomial.
+ *
+ * @param[in, out] p polynomial from which the zero leading coefficients will be removed
+ * @param[in] hint assume all coefficients of order ≥ hint are zero
+ */
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 = min(p.size(), hint);
- std::size_t i = p.size() - 1;
- // Be fast if the polynomial is already canonicalized
- if (!zerop(p[i]))
- return;
+ while ( i-- && zerop(p[i]) ) { }
- if (hint < p.size())
- i = hint;
+ p.erase(p.begin() + i + 1, p.end());
+}
- bool is_zero = false;
- do {
- if (!zerop(p[i])) {
- ++i;
- break;
- }
- if (i == 0) {
- is_zero = true;
- break;
- }
- --i;
- } while (true);
+// END COPY FROM UPOLY.H
- if (is_zero) {
- p.clear();
- return;
- }
-
- p.erase(p.begin() + i, p.end());
-}
+template<typename T> struct uvar_poly_p
+{
+ static const bool value = false;
+};
-// END COPY FROM UPOLY.HPP
+template<> struct uvar_poly_p<upoly>
+{
+ static const bool value = true;
+};
-static void expt_pos(umodpoly& a, unsigned int q)
+template<> struct uvar_poly_p<umodpoly>
{
- 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;
- }
-}
+ static const bool value = true;
+};
template<typename T>
-static T operator+(const T& a, const T& b)
+// 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();
}
template<typename T>
-static T operator-(const T& a, const T& b)
+// 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();
canonicalize(ump);
}
+#ifdef DEBUGFACTOR
static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_I& modulus)
{
umodpoly_from_ex(ump, e, x, find_modint_ring(modulus));
}
+#endif
static ex upoly_to_ex(const upoly& a, const ex& x)
{
return e;
}
-/** Divides all coefficients of the polynomial a by the integer x.
+/** Divides all coefficients of the polynomial a by the positive integer x.
* All coefficients are supposed to be divisible by x. If they are not, the
- * the<cl_I> cast will raise an exception.
+ * division 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
+ * @param[in] x positive integer that divides the coefficients
*/
static void reduce_coeff(umodpoly& a, const cl_I& x)
{
if ( a.empty() ) return;
cl_modint_ring R = a[0].ring();
- umodpoly::iterator i = a.begin(), end = a.end();
- for ( ; i!=end; ++i ) {
+ for (auto & i : a) {
// cln cannot perform this division in the modular field
- cl_I c = R->retract(*i);
- *i = cl_MI(R, the<cl_I>(c / x));
+ cl_I c = R->retract(i);
+ i = cl_MI(R, exquopos(c, x));
}
}
} while ( k-- );
fill(r.begin()+n, r.end(), a[0].ring()->zero());
- canonicalize(r);
+ canonicalize(r, n);
}
/** Calculates quotient of a/b.
static bool unequal_one(const umodpoly& a)
{
- if ( a.empty() ) return true;
return ( a.size() != 1 || a[0] != a[0].ring()->one() );
}
return equal_one(c);
}
+/** Computes w^q mod a.
+ * Uses theorem 2.1 from A.K.Lenstra's PhD thesis; see exercise 8.13 in [GCL].
+ *
+ * @param[in] w polynomial
+ * @param[in] a modulus polynomial
+ * @param[in] q common modulus of w and a
+ * @param[out] r result
+ */
+static void expt_pos_Q(const umodpoly& w, const umodpoly& a, unsigned int q, umodpoly& r)
+{
+ if ( w.empty() ) return;
+ cl_MI zero = w[0].ring()->zero();
+ int deg = degree(w);
+ umodpoly buf(deg*q+1, zero);
+ for ( size_t i=0; i<=deg; ++i ) {
+ buf[i*q] = w[i];
+ }
+ rem(buf, a, r);
+}
+
// END modular univariate polynomial code
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// modular matrix
+typedef vector<cl_MI> mvec;
+
class modular_matrix
{
+#ifdef DEBUGFACTOR
friend ostream& operator<<(ostream& o, const modular_matrix& m);
+#endif
public:
modular_matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
{
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)
{
- mvec::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)
{
- mvec::iterator i1 = m.begin() + col1;
- mvec::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;
- mvec::iterator i1 = m.begin() + col1;
- mvec::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
{
- mvec::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
{
- mvec::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)
{
- mvec::iterator i1 = m.begin() + row*c;
- mvec::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];
}
}
mvec::const_iterator row_begin(size_t row) const { return m.begin()+row*c; }
// END modular matrix
////////////////////////////////////////////////////////////////////////////////
+/** Calculates the Q matrix for a polynomial. Used by Berlekamp's algorithm.
+ *
+ * The implementation follows algorithm 8.5 of [GCL].
+ *
+ * @param[in] a_ modular polynomial
+ * @param[out] Q Q matrix
+ */
static void q_matrix(const umodpoly& a_, modular_matrix& Q)
{
umodpoly a = a_;
}
}
+/** 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();
}
}
+/** Berlekamp's modular factorization.
+ *
+ * The implementation follows algorithm 8.4 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;
const unsigned int k = nu.size();
if ( k == 1 ) {
+ // irreducible
return;
}
- list<umodpoly> factors;
- factors.push_back(a);
+ list<umodpoly> factors = {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];
div(*u, g, uo);
if ( equal_one(uo) ) {
throw logic_error("berlekamp: unexpected divisor.");
- }
- else {
+ } 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;
+ for (auto & i : factors) {
+ if (degree(i))
+ ++size;
}
if ( size == k ) {
- list<umodpoly>::const_iterator i = factors.begin(), end = factors.end();
- while ( i != end ) {
- upv.push_back(*i++);
+ for (auto & i : factors) {
+ upv.push_back(i);
}
return;
}
}
}
+// 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[out] ap resulting polynomial
+ */
static void expt_1_over_p(const umodpoly& a, unsigned int prime, umodpoly& ap)
{
size_t newdeg = degree(a)/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);
mult[i] *= prime;
}
}
- }
- else {
+ } else {
umodpoly ap;
expt_1_over_p(a, prime, ap);
size_t previ = mult.size();
}
}
+#endif // deactivation of square free factorization
+
+/** Distinct degree factorization (DDF).
+ *
+ * The implementation follows algorithm 8.8 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_;
int nhalf = degree(a)/2;
int i = 1;
- umodpoly w(2);
- w[0] = R->zero();
- w[1] = R->one();
+ umodpoly w = {R->zero(), R->one()};
umodpoly x = w;
while ( i <= nhalf ) {
- expt_pos(w, q);
umodpoly buf;
- rem(w, a, buf);
+ expt_pos_Q(w, a, q, buf);
w = buf;
- umodpoly wx = w - x;
- gcd(a, wx, buf);
+ gcd(a, w - x, 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;
}
}
+/** 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();
for ( size_t i=0; i<degrees.size(); ++i ) {
if ( degrees[i] == degree(ddfactors[i]) ) {
upv.push_back(ddfactors[i]);
- }
- else {
+ } 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
#endif
}
-/** Calculates polynomials s and t such that a*s+b*t==1.
+/** 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
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;
+ for (auto & i : s) {
+ i = i * fac;
}
canonicalize(s);
fac = recip(lcoeff(b) * lcoeff(c));
- i = t.begin(), end = t.end();
- for ( ; i!=end; ++i ) {
- *i = *i * fac;
+ for (auto & i : t) {
+ 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;
return r;
}
-static inline cl_I calc_bound(const ex& a, const ex& x, int maxdeg)
+/** Calculates bound for the product of absolute values (modulus) of the roots.
+ * Uses Landau's inequality, see [Mig].
+ */
+static inline cl_I calc_bound(const ex& a, const ex& x)
{
- cl_I maxcoeff = 0;
- cl_R coeff = 0;
+ cl_R radicand = 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);
+ radicand = radicand + 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;
+ return ceiling1(the<cl_R>(cln::sqrt(radicand)));
}
-static inline cl_I calc_bound(const upoly& a, int maxdeg)
+/** Calculates bound for the product of absolute values (modulus) of the roots.
+ * Uses Landau's inequality, see [Mig].
+ */
+static inline cl_I calc_bound(const upoly& a)
{
- cl_I maxcoeff = 0;
- cl_R coeff = 0;
+ cl_R radicand = 0;
for ( int i=degree(a); i>=0; --i ) {
cl_I aa = abs(a[i]);
- if ( aa > maxcoeff ) maxcoeff = aa;
- coeff = coeff + square(aa);
+ radicand = radicand + 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;
+ return ceiling1(the<cl_R>(cln::sqrt(radicand)));
}
+/** Hensel lifting as used by factor_univariate().
+ *
+ * The implementation follows algorithm 6.1 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_;
// calc bound B
int maxdeg = (degree(u1_) > degree(w1_)) ? degree(u1_) : degree(w1_);
- cl_I maxmodulus = 2*calc_bound(a, maxdeg);
+ cl_I maxmodulus = ash(calc_bound(a), maxdeg+1); // = 2 * calc_bound(a) * 2^maxdeg
// step 1
cl_I alpha = lcoeff(a);
if ( alpha != 1 ) {
w = w / alpha;
}
- }
- else {
+ } else {
u.clear();
}
}
-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;
+/** Returns a new small prime number.
+ *
+ * @param[in] n an integer
+ * @return smallest prime greater than n
+ */
+static unsigned int next_prime(unsigned int n)
+{
+ static vector<unsigned int> primes = {2, 3, 5, 7};
+ unsigned int candidate = primes.back();
+ while (primes.back() <= n) {
+ candidate += 2;
+ bool is_prime = true;
+ for (size_t i=1; primes[i]*primes[i]<=candidate; ++i) {
+ if (candidate % primes[i] == 0) {
+ is_prime = false;
+ break;
}
- primes.push_back(candidate);
- if ( candidate > p ) break;
}
- return candidate;
+ if (is_prime)
+ primes.push_back(candidate);
}
- vector<unsigned int>::const_iterator end = primes.end();
- for ( ; it!=end; ++it ) {
- if ( *it > p ) {
- return *it;
+ for (auto & it : primes) {
+ if ( it > n ) {
+ return it;
}
}
throw logic_error("next_prime: should not reach this point!");
}
+/** Manages the splitting of a vector 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();
size_t size() const { return n; }
size_t size_left() const { return n-len; }
size_t size_right() const { return len; }
-#ifdef DEBUGFACTOR
- void get() const { DCOUTVAR(k); }
-#endif
+ /** Initializes the next partition.
+ Returns true, if there is one, false otherwise. */
bool next()
{
if ( last == n-1 ) {
if ( len > n/2 ) return false;
fill(k.begin(), k.begin()+len, 1);
fill(k.begin()+len+1, k.end(), 0);
- }
- else {
+ } 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()
if ( d ) {
if ( cache[pos].size() >= d ) {
lr[group] = lr[group] * cache[pos][d-1];
- }
- else {
+ } else {
if ( cache[pos].size() == 0 ) {
cache[pos].push_back(factors[pos] * factors[pos+1]);
}
}
lr[group] = lr[group] * cache[pos].back();
}
- }
- else {
+ } else {
lr[group] = lr[group] * factors[pos];
}
} while ( i < n );
lr[1] = one;
if ( n > 6 ) {
split_cached();
- }
- else {
+ } 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;
+ vector<vector<umodpoly>> cache;
upvec factors;
umodpoly one;
size_t n;
vector<int> k;
};
+/** Contains a pair of univariate polynomial and its modular factors.
+ * Used by factor_univariate().
+ */
struct ModFactors
{
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_ex;
poly.unitcontprim(x, unit, cont, prim_ex);
upoly prim;
upoly_from_ex(prim, prim_ex, x);
+ if (prim_ex.is_equal(1)) {
+ return poly;
+ }
// determine proper prime and minimize number of modular factors
- unsigned int p = 3, lastp = 3;
+ prime = 3;
+ unsigned int lastp = prime;
cl_modint_ring R;
unsigned int trials = 0;
unsigned int minfactors = 0;
- cl_I lc = lcoeff(prim);
+
+ 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 ) {
- p = next_prime(p);
- if ( !zerop(rem(lc, p)) ) {
- R = find_modint_ring(p);
+ prime = next_prime(prime);
+ if ( !zerop(rem(lc, prime)) ) {
+ R = find_modint_ring(prime);
umodpoly_from_upoly(modpoly, prim, R);
if ( squarefree(modpoly) ) break;
}
if ( minfactors == 0 || trialfactors.size() < minfactors ) {
factors = trialfactors;
minfactors = trialfactors.size();
- lastp = p;
+ lastp = prime;
trials = 1;
- }
- else {
+ } else {
++trials;
}
}
- p = lastp;
- R = find_modint_ring(p);
+ prime = lastp;
+ R = find_modint_ring(prime);
// lift all factor combinations
stack<ModFactors> tocheck;
const size_t n = tocheck.top().factors.size();
factor_partition part(tocheck.top().factors);
while ( true ) {
- hensel_univar(tocheck.top().poly, p, part.left(), part.right(), f1, f2);
+ // 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);
}
}
break;
- }
- else {
+ } else {
upvec newfactors1(part.size_left()), newfactors2(part.size_right());
- upvec::iterator i1 = newfactors1.begin(), i2 = newfactors2.begin();
+ auto i1 = newfactors1.begin(), i2 = newfactors2.begin();
for ( size_t i=0; i<n; ++i ) {
if ( part[i] ) {
*i2++ = tocheck.top().factors[i];
- }
- else {
+ } else {
*i1++ = tocheck.top().factors[i];
}
}
tocheck.push(mf);
break;
}
- }
- else {
+ } 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
-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);
-upvec multiterm_eea_lift(const upvec& 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 algorithm 6.3 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));
return s;
}
-/**
- * Assert: a not empty.
+/** 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)
*/
-void change_modulus(const cl_modint_ring& R, umodpoly& a)
+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));
+ for (auto & i : a) {
+ i = R->canonhom(oldR->retract(i));
}
canonicalize(a);
}
-void eea_lift(const umodpoly& a, const umodpoly& b, const ex& x, unsigned int p, unsigned int k, umodpoly& s_, umodpoly& t_)
+/** Utility function for multivariate Hensel lifting.
+ *
+ * Solves s*a + t*b == 1 mod p^k given a,b.
+ *
+ * The implementation follows algorithm 6.3 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;
s_ = s; t_ = t;
}
-upvec univar_diophant(const upvec& 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 algorithm 6.3 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));
rem(bmod, a[j], buf);
result.push_back(buf);
}
- }
- else {
+ } else {
umodpoly s, t;
eea_lift(a[1], a[0], x, p, k, s, t);
umodpoly bmod = umodpoly_to_umodpoly(s, R, m);
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)
+ ex operator()(const ex& e) override
{
if ( is_a<add>(e) || is_a<mul>(e) ) {
return e.map(*this);
numeric n(R->retract(emod));
if ( n > halfmod ) {
return n-mod;
- }
- else {
+ } else {
return n;
}
}
}
};
+/** 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());
}
-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 algorithm 6.2 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_;
- const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+ const cl_I modulus = expt_pos(cl_I(p),k);
+ const cl_modint_ring R = find_modint_ring(modulus);
const size_t r = a.size();
const size_t nu = I.size() + 1;
e = make_modular(buf, R);
}
}
- }
- else {
+ } else {
upvec amod;
for ( size_t i=0; i<a.size(); ++i ) {
umodpoly up;
if ( is_a<add>(c) ) {
nterms = c.nops();
z = c.op(0);
- }
- else {
+ } else {
nterms = 1;
z = c;
}
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);
- }
+ cl_I poscm = plusp(cm) ? cm : mod(cm, modulus);
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 ) {
+ if ( nterms > 1 && i+1 != nterms ) {
z = c.op(i+1);
}
}
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 upvec& u, const vector<ex>& lcU)
+/** Multivariate Hensel lifting.
+ * The implementation follows algorithm 6.4 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)
{
const size_t nu = I.size() + 1;
const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
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();
+ return lst(U.begin(), U.end());
+ } else {
+ return lst{};
}
}
-static ex put_factors_into_lst(const ex& e)
+/** Takes a factorized expression and puts the factors in a vector. The exponents
+ * of the factors are discarded, e.g. 7*x^2*(y+1)^4 --> {7,x,y+1}. The first
+ * element of the result is always the numeric coefficient.
+ */
+static exvector put_factors_into_vec(const ex& e)
{
- lst result;
-
+ exvector result;
if ( is_a<numeric>(e) ) {
- result.append(e);
+ result.push_back(e);
return result;
}
if ( is_a<power>(e) ) {
- result.append(1);
- result.append(e.op(0));
- result.append(e.op(1));
+ result.push_back(1);
+ result.push_back(e.op(0));
return result;
}
if ( is_a<symbol>(e) || is_a<add>(e) ) {
- result.append(1);
- result.append(e);
- result.append(1);
+ ex icont(e.integer_content());
+ result.push_back(icont);
+ result.push_back(e/icont);
return result;
}
if ( is_a<mul>(e) ) {
ex nfac = 1;
+ result.push_back(nfac);
for ( size_t i=0; i<e.nops(); ++i ) {
ex op = e.op(i);
if ( is_a<numeric>(op) ) {
nfac = op;
}
if ( is_a<power>(op) ) {
- result.append(op.op(0));
- result.append(op.op(1));
+ result.push_back(op.op(0));
}
if ( is_a<symbol>(op) || is_a<add>(op) ) {
- result.append(op);
- result.append(1);
+ result.push_back(op);
}
}
- result.prepend(nfac);
+ result[0] = nfac;
return result;
}
- throw runtime_error("put_factors_into_lst: bad term.");
+ throw runtime_error("put_factors_into_vec: 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 numbers to check
+ * @return true: if number set is bad, false: if set is okay (has unique
+ * prime factors)
+ */
+static bool checkdivisors(const exvector& f)
{
- const int k = f.nops()-2;
+ const int k = f.size();
numeric q, r;
- d[0] = ex_to<numeric>(f.op(0) * f.op(f.nops()-1));
- if ( d[0] == 1 && k == 1 && abs(f.op(1)) != 1 ) {
- return false;
- }
- for ( int i=1; i<=k; ++i ) {
- q = ex_to<numeric>(abs(f.op(i)));
+ vector<numeric> d(k);
+ d[0] = ex_to<numeric>(abs(f[0]));
+ for ( int i=1; i<k; ++i ) {
+ q = ex_to<numeric>(abs(f[i]));
for ( int j=i-1; j>=0; --j ) {
r = d[j];
do {
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] x first symbol that appears in u
+ * @param[in] syms_wox remaining symbols that appear in u
+ * @param[in] f vector 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 ex& x, const exset& syms_wox, const exvector& f,
+ numeric& modulus, ex& u0, vector<numeric>& a)
{
- // computation of d is actually not necessary
- 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;
+ auto s = syms_wox.begin();
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;
}
- 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 {
- lst fnum;
- lst::const_iterator i = ex_to<lst>(f).begin();
- fnum.append(*i++);
- bool problem = false;
- while ( i!=ex_to<lst>(f).end() ) {
- ex fs = *i;
- if ( !is_a<numeric>(fs) ) {
- s = syms.begin();
- ++s;
- for ( size_t j=0; j<a.size(); ++j ) {
- fs = fs.subs(*s == a[j]);
- ++s;
- }
- if ( abs(fs) == 1 ) {
- problem = true;
- break;
+ if ( !is_a<numeric>(vn) ) {
+ // ... and for which the evaluated factors have each an unique prime factor
+ exvector fnum = f;
+ fnum[0] = fnum[0] * u0.content(x);
+ for ( size_t i=1; i<fnum.size(); ++i ) {
+ if ( !is_a<numeric>(fnum[i]) ) {
+ s = syms_wox.begin();
+ for ( size_t j=0; j<a.size(); ++j, ++s ) {
+ fnum[i] = fnum[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);
- trying = checkdivisors(fnum, d);
}
- } while ( trying );
- return false;
-}
-
-static ex factor_multivariate(const ex& poly, const exset& syms)
-{
- exset::const_iterator s;
- const ex& x = *syms.begin();
-
- /* make polynomial primitive */
- ex p = poly.expand().collect(x);
- 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;
- }
- ex pp = expand(normal(p / cont));
- if ( !is_a<numeric>(cont) ) {
- return factor(cont) * factor(pp);
+ // ok, we have a valid set now
+ return;
}
+}
- /* factor leading coefficient */
- pp = pp.collect(x);
- ex vn = pp.lcoeff(x);
- pp = pp.expand();
- ex vnlst;
- if ( is_a<numeric>(vn) ) {
- vnlst = lst(vn);
- }
- else {
- ex vnfactors = factor(vn);
- vnlst = put_factors_into_lst(vnfactors);
- }
+// forward declaration
+static ex factor_sqrfree(const ex& poly);
- const numeric maxtrials = 3;
- numeric modulus = (vnlst.nops()-1 > 3) ? vnlst.nops()-1 : 3;
- numeric minimalr = -1;
- vector<numeric> a(syms.size()-1, 0);
- vector<numeric> d((vnlst.nops()-1)/2+1, 0);
+/** Used by factor_multivariate().
+ */
+struct factorization_ctx {
+ const ex poly, x; // polynomial, first symbol x...
+ const exset syms_wox; // ...remaining symbols w/o x
+ ex unit, cont, pp; // unit * cont * pp == poly
+ ex vn; exvector vnlst; // leading coeff, factors of leading coeff
+ numeric modulus; // incremented each time we try
+ /** returns factors or empty if it did not succeed */
+ ex try_next_evaluation_homomorphism()
+ {
+ constexpr unsigned maxtrials = 3;
+ vector<numeric> a(syms_wox.size(), 0);
- while ( true ) {
- numeric trialcount = 0;
+ unsigned int trialcount = 0;
+ unsigned int prime;
+ int factor_count = 0;
+ int min_factor_count = -1;
ex u, delta;
- unsigned int prime = 3;
- size_t factor_count = 0;
ex ufac;
- ex ufaclst;
+ exvector 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);
- if ( problem ) {
- ++modulus;
- continue;
- }
- 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);
-
- // determine proper prime
- prime = 3;
- cl_modint_ring R = find_modint_ring(prime);
- while ( true ) {
- if ( irem(ex_to<numeric>(u.lcoeff(x)), prime) != 0 ) {
- umodpoly modpoly;
- umodpoly_from_ex(modpoly, u, x, R);
- if ( squarefree(modpoly) ) break;
- }
- prime = next_prime(prime);
- R = find_modint_ring(prime);
- }
- ufac = factor(u);
- ufaclst = put_factors_into_lst(ufac);
- factor_count = (ufaclst.nops()-1)/2;
+ // generate a set of valid evaluation points
+ generate_set(pp, vn, x, syms_wox, vnlst, modulus, u, a);
- // veto factorization for which gcd(u_i, u_j) != 1 for all i,j
- upvec tryu;
- for ( size_t i=0; i<(ufaclst.nops()-1)/2; ++i ) {
- umodpoly newu;
- umodpoly_from_ex(newu, ufaclst.op(i*2+1), x, R);
- tryu.push_back(newu);
- }
- bool veto = false;
- for ( size_t i=0; i<tryu.size()-1; ++i ) {
- for ( size_t j=i+1; j<tryu.size(); ++j ) {
- umodpoly tryg;
- gcd(tryu[i], tryu[j], tryg);
- if ( unequal_one(tryg) ) {
- veto = true;
- goto escape_quickly;
- }
- }
- }
- escape_quickly: ;
- if ( veto ) {
- continue;
- }
+ ufac = factor_univariate(u, x, prime);
+ ufaclst = put_factors_into_vec(ufac);
+ factor_count = ufaclst.size()-1;
+ delta = ufaclst[0];
if ( factor_count <= 1 ) {
- return poly;
+ // irreducible
+ return lst{pp};
}
-
- 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;
}
- if ( minimalr <= 1 ) {
- 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);
}
- 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 ) {
- 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));
+ // 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.size(); ++i ) {
+ C[i-1] = ufaclst[i].lcoeff(x);
}
- for ( size_t j=(vnlst.nops()-1)/2+1; j>0; --j ) {
- if ( abs(ftilde[j-1]) == 1 ) {
- used_flag[j-1] = true;
- continue;
+ } 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.size()-1);
+ for ( size_t i=0; i<ftilde.size(); ++i ) {
+ ex ft = vnlst[i+1];
+ auto s = syms_wox.begin();
+ for ( size_t j=0; j<a.size(); ++j ) {
+ ft = ft.subs(*s == a[j]);
+ ++s;
}
- numeric g = gcd(prefac, ftilde[j-1]);
- if ( g != 1 ) {
- prefac = prefac / g;
- numeric count = abs(iquo(g, ftilde[j-1]));
- used_flag[j-1] = true;
- if ( i > 0 ) {
- if ( j == 1 ) {
- D[i-1] = D[i-1] * pow(vnlst.op(0), count);
+ 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[i+1].lcoeff(x));
+ for ( int j=ftilde.size()-1; j>=0; --j ) {
+ int count = 0;
+ numeric q;
+ while ( irem(prefac, ftilde[j], q) == 0 ) {
+ prefac = q;
+ ++count;
}
- else {
- D[i-1] = D[i-1] * pow(vnlst.op(2*(j-2)+1), count);
+ if ( count ) {
+ used_flag[j] = true;
+ D[i] = D[i] * pow(vnlst[j+1], count);
}
}
- else {
- ftilde[j-1] = ftilde[j-1] / prefac;
- break;
+ C[i] = D[i] * prefac;
+ }
+ } else {
+ for ( int i=0; i<factor_count; ++i ) {
+ numeric prefac = ex_to<numeric>(ufaclst[i+1].lcoeff(x));
+ for ( int j=ftilde.size()-1; j>=0; --j ) {
+ int count = 0;
+ numeric q;
+ while ( irem(prefac, ftilde[j], q) == 0 ) {
+ prefac = q;
+ ++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[i+1] = ufaclst[i+1] * (ftilde[j]/g);
+ ++count;
+ }
+ if ( count ) {
+ used_flag[j] = true;
+ D[i] = D[i] * pow(vnlst[j+1], count);
+ }
}
- ++j;
+ C[i] = D[i] * prefac;
}
}
- }
-
- 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;
- }
-
- vector<ex> C(factor_count);
- 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;
+ // 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;
}
- C[i] = D[i] * (ufaclst.op(2*i+1).lcoeff(x) / Dtilde);
}
- }
- else {
- for ( size_t i=0; i<D.size(); ++i ) {
- ex Dtilde = D[i];
- s = syms.begin();
- ++s;
- for ( size_t j=0; j<a.size(); ++j ) {
- Dtilde = Dtilde.subs(*s == a[j]);
- ++s;
- }
- ex ui;
- if ( i == 0 ) {
- ui = ufaclst.op(0);
- }
- else {
- ui = ufaclst.op(2*(i-1)+1);
- }
- while ( true ) {
- ex d = gcd(ui.lcoeff(x), Dtilde);
- C[i] = D[i] * ( ui.lcoeff(x) / d );
- ui = ui * ( Dtilde[i] / d );
- delta = delta / ( Dtilde[i] / d );
- if ( delta == 1 ) break;
- ui = delta * ui;
- C[i] = delta * C[i];
- pp = pp * pow(delta, D.size()-1);
- }
+ if ( some_factor_unused ) {
+ return lst{}; // next try
}
}
- EvalPoint ep;
+ // multiply the remaining content of the univariate polynomial into the
+ // first factor
+ if ( delta != 1 ) {
+ C[0] = C[0] * delta;
+ ufaclst[1] = ufaclst[1] * delta;
+ }
+
+ // set up evaluation points
vector<EvalPoint> epv;
- s = syms.begin();
- ++s;
+ auto s = syms_wox.begin();
for ( size_t i=0; i<a.size(); ++i ) {
- ep.x = *s++;
- ep.evalpoint = a[i].to_int();
- epv.push_back(ep);
+ epv.emplace_back(EvalPoint{*s++, a[i].to_int()});
}
// calc bound p^l
int maxdeg = 0;
- for ( size_t i=0; i<factor_count; ++i ) {
- if ( ufaclst[2*i+1].degree(x) > maxdeg ) {
- maxdeg = ufaclst[2*i+1].degree(x);
+ for ( int i=1; i<=factor_count; ++i ) {
+ if ( ufaclst[i].degree(x) > maxdeg ) {
+ maxdeg = ufaclst[i].degree(x);
}
}
- cl_I B = 2*calc_bound(u, x, maxdeg);
+ cl_I B = ash(calc_bound(u, x), maxdeg+1); // = 2 * calc_bound(u,x) * 2^maxdeg
cl_I l = 1;
cl_I pl = prime;
while ( pl < B ) {
pl = pl * prime;
}
- upvec uvec;
- cl_modint_ring R = find_modint_ring(expt_pos(cl_I(prime),l));
- for ( size_t i=0; i<(ufaclst.nops()-1)/2; ++i ) {
- umodpoly newu;
- umodpoly_from_ex(newu, ufaclst.op(i*2+1), x, R);
- uvec.push_back(newu);
+ // set up modular factors (mod p^l)
+ cl_modint_ring R = find_modint_ring(pl);
+ upvec modfactors(ufaclst.size()-1);
+ for ( size_t i=1; i<ufaclst.size(); ++i ) {
+ umodpoly_from_ex(modfactors[i-1], ufaclst[i], x, R);
+ }
+
+ // try Hensel lifting
+ return hensel_multivar(pp, x, epv, prime, l, modfactors, C);
+ }
+};
+
+/** 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. This is done in round-robin for each x in syms until success.
+ *
+ * @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)
+{
+ // set up one factorization context for each symbol
+ vector<factorization_ctx> ctx_in_x;
+ for (auto x : syms) {
+ exset syms_wox; // remaining syms w/o x
+ copy_if(syms.begin(), syms.end(),
+ inserter(syms_wox, syms_wox.end()), [x](const ex& y){ return y != x; });
+
+ factorization_ctx ctx{poly, x, syms_wox};
+
+ // make polynomial primitive
+ poly.unitcontprim(x, ctx.unit, ctx.cont, ctx.pp);
+ if ( !is_a<numeric>(ctx.cont) ) {
+ // content is a polynomial in one or more of remaining syms, let's start over
+ return ctx.unit * factor_sqrfree(ctx.cont) * factor_sqrfree(ctx.pp);
}
- ex res = hensel_multivar(ufaclst.op(0)*pp, x, epv, prime, l, uvec, C);
- if ( res != lst() ) {
- ex result = cont * ufaclst.op(0);
+ // find factors of leading coefficient
+ ctx.vn = ctx.pp.collect(x).lcoeff(x);
+ ctx.vnlst = put_factors_into_vec(factor(ctx.vn));
+
+ ctx.modulus = (ctx.vnlst.size() > 3) ? ctx.vnlst.size() : numeric(3);
+
+ ctx_in_x.push_back(ctx);
+ }
+
+ // try an evaluation homomorphism for each context in round-robin
+ auto ctx = ctx_in_x.begin();
+ while ( true ) {
+
+ ex res = ctx->try_next_evaluation_homomorphism();
+
+ if ( res != lst{} ) {
+ // found the factors
+ ex result = ctx->cont * ctx->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);
+ ex unit, cont, pp;
+ res.op(i).unitcontprim(ctx->x, unit, cont, pp);
+ result *= unit * cont * pp;
}
return result;
}
+
+ // switch context for next symbol
+ if (++ctx == ctx_in_x.end()) {
+ ctx = ctx_in_x.begin();
+ }
}
}
+/** 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)
+ ex operator()(const ex& e) override
{
if ( is_a<symbol>(e) ) {
syms.insert(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
if ( findsymbols.syms.size() == 1 ) {
// univariate case
const ex& x = *(findsymbols.syms.begin());
- if ( poly.ldegree(x) > 0 ) {
- int ld = poly.ldegree(x);
+ int ld = poly.ldegree(x);
+ if ( ld > 0 ) {
+ // pull out direct factors
ex res = factor_univariate(expand(poly/pow(x, ld)), x);
return res * pow(x,ld);
- }
- else {
+ } else {
ex res = factor_univariate(poly, x);
return res;
}
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)
+ ex operator()(const ex& e) override
{
if ( e.info(info_flags::polynomial) ) {
return factor(e, options);
for ( size_t i=0; i<e.nops(); ++i ) {
if ( e.op(i).info(info_flags::polynomial) ) {
s1 += e.op(i);
- }
- else {
+ } else {
s2 += e.op(i);
}
}
- s1 = s1.eval();
- s2 = s2.eval();
return factor(s1, options) + s2.map(*this);
}
return e.map(*this);
}
};
-} // anonymous namespace
+/** Iterate through explicit factors of e, call yield(f, k) for
+ * each factor of the form f^k.
+ *
+ * Note that this function doesn't factor e itself, it only
+ * iterates through the factors already explicitly present.
+ */
+template <typename F> void
+factor_iter(const ex &e, F yield)
+{
+ if (is_a<mul>(e)) {
+ for (const auto &f : e) {
+ if (is_a<power>(f)) {
+ yield(f.op(0), f.op(1));
+ } else {
+ yield(f, ex(1));
+ }
+ }
+ } else {
+ if (is_a<power>(e)) {
+ yield(e.op(0), e.op(1));
+ } else {
+ yield(e, ex(1));
+ }
+ }
+}
-ex factor(const ex& poly, unsigned options)
+/** This function factorizes a polynomial. It checks the arguments,
+ * tries a square free factorization, and then calls factor_sqrfree
+ * to do the hard work.
+ *
+ * This function expands its argument, so for polynomials with
+ * explicit factors it's better to call it on each one separately
+ * (or use factor() which does just that).
+ */
+static ex factor1(const ex& poly, unsigned options)
{
// check arguments
if ( !poly.info(info_flags::polynomial) ) {
return poly;
}
lst syms;
- exset::const_iterator i=findsymbols.syms.begin(), end=findsymbols.syms.end();
- for ( ; i!=end; ++i ) {
- syms.append(*i);
+ for (auto & i : findsymbols.syms ) {
+ syms.append(i);
}
// 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) ) {
- // case: (polynomial)^exponent
- const ex& base = sfpoly.op(0);
- if ( !is_a<add>(base) ) {
- // simple case: (monomial)^exponent
- return sfpoly;
- }
- ex f = factor_sqrfree(base);
- return pow(f, sfpoly.op(1));
- }
- if ( is_a<mul>(sfpoly) ) {
- // case: multiple factors
- ex res = 1;
- for ( size_t i=0; i<sfpoly.nops(); ++i ) {
- const ex& t = sfpoly.op(i);
- if ( is_a<power>(t) ) {
- const ex& base = t.op(0);
- if ( !is_a<add>(base) ) {
- res *= t;
- }
- else {
- ex f = factor_sqrfree(base);
- res *= pow(f, t.op(1));
- }
+ ex res = 1;
+ factor_iter(sfpoly,
+ [&](const ex &f, const ex &k) {
+ if ( is_a<add>(f) ) {
+ res *= pow(factor_sqrfree(f), k);
+ } else {
+ // simple case: (monomial)^exponent
+ res *= pow(f, k);
}
- else if ( is_a<add>(t) ) {
- ex f = factor_sqrfree(t);
- res *= f;
- }
- else {
- res *= t;
- }
- }
- return res;
- }
- if ( is_a<symbol>(sfpoly) ) {
- return poly;
- }
- // case: (polynomial)
- ex f = factor_sqrfree(sfpoly);
- return f;
+ });
+ return res;
}
-} // namespace GiNaC
+} // anonymous namespace
-#ifdef DEBUGFACTOR
-#include "test.h"
-#endif
+/** Interface function to the outside world. It uses factor1()
+ * on each of the explicitly present factors of poly.
+ */
+ex factor(const ex& poly, unsigned options)
+{
+ ex result = 1;
+ factor_iter(poly,
+ [&](const ex &f1, const ex &k1) {
+ factor_iter(factor1(f1, options),
+ [&](const ex &f2, const ex &k2) {
+ result *= pow(f2, k1*k2);
+ });
+ });
+ return result;
+}
+
+} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff