/** @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 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
ostream& operator<<(ostream& o, const vector<int>& v)
{
vector<int>::const_iterator i = v.begin(), end = v.end();
}
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();
while ( i != end ) {
}
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();
while ( i != end ) {
}
return o;
}
+ostream& operator<<(ostream& o, const vector<numeric>& v)
+{
+ for ( size_t i=0; i<v.size(); ++i ) {
+ o << v[i] << " ";
+ }
+ return o;
+}
ostream& operator<<(ostream& o, const vector< vector<cl_MI> >& v)
{
vector< vector<cl_MI> >::const_iterator i = v.begin(), end = v.end();
}
return o;
}
-#endif
+#else
+#define DCOUT(str)
+#define DCOUTVAR(var)
+#define DCOUT2(str,var)
+#endif // def DEBUGFACTOR
+
+// anonymous namespace to hide all utility functions
+namespace {
////////////////////////////////////////////////////////////////////////////////
// modular univariate polynomial code
}
}
+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>
-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)
{
////////////////////////////////////////////////////////////////////////////////
// modular matrix
+typedef vector<cl_MI> mvec;
+
class modular_matrix
{
friend ostream& operator<<(ostream& o, const modular_matrix& m);
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.
+ *
+ * @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 the algorithm in chapter 8 of [GCL].
+ *
+ * @param[in] a modular polynomial
+ * @param[out] upv vector containing modular factors. if upv was not empty the
+ * new elements are added at the end
+ */
static void berlekamp(const umodpoly& a, upvec& upv)
{
cl_modint_ring R = a[0].ring();
umodpoly one(1, R->one());
+ // find nullspace of Q matrix
modular_matrix Q(degree(a), degree(a), R->zero());
q_matrix(a, Q);
vector<mvec> nu;
const unsigned int k = nu.size();
if ( k == 1 ) {
+ // irreducible
return;
}
list<umodpoly>::iterator u = factors.begin();
+ // calculate all gcd's
while ( true ) {
for ( unsigned int s=0; s<q; ++s ) {
umodpoly nur = nu[r];
}
}
+// 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;
}
}
+/** 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);
}
}
+#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_;
}
}
+/** 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();
}
}
+// 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
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;
}
+/** Calculates the bound for the modulus.
+ * See [Mig].
+ */
static inline cl_I calc_bound(const ex& a, const ex& x, int maxdeg)
{
cl_I maxcoeff = 0;
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;
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_;
}
}
+/** 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!");
}
+/** Manages the splitting a vector of of modular factors into two partitions.
+ */
class factor_partition
{
public:
+ /** Takes the vector of modular factors and initializes the first partition */
factor_partition(const upvec& factors_) : factors(factors_)
{
n = factors.size();
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 ) {
split();
return true;
}
+ /** Get first partition */
umodpoly& left() { return lr[0]; }
+ /** Get second partition */
umodpoly& right() { return lr[1]; }
private:
void split_cached()
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_from_ex(prim, prim_ex, x);
// 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);
+ cl_I lc = lcoeff(prim) * the<cl_I>(ex_to<numeric>(cont).to_cl_N());
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 {
++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);
}
}
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 the algorithm in chapter 6 of [GCL].
+ *
+ * @param[in] a vector of modular univariate polynomials
+ * @param[in] x symbol
+ * @param[in] p prime number
+ * @param[in] k p^k is modulus
+ * @return vector of polynomials (s_i)
+ */
+static upvec multiterm_eea_lift(const upvec& a, const ex& x, unsigned int p, unsigned int k)
{
const size_t r = a.size();
cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
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();
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 the algorithm in chapter 6 of [GCL].
+ *
+ * @param[in] a polynomial
+ * @param[in] b polynomial
+ * @param[in] x symbol
+ * @param[in] p prime number
+ * @param[in] k p^k is modulus
+ * @param[out] s_ output polynomial
+ * @param[out] t_ output polynomial
+ */
+static void eea_lift(const umodpoly& a, const umodpoly& b, const ex& x, unsigned int p, unsigned int k, umodpoly& s_, umodpoly& t_)
{
cl_modint_ring R = find_modint_ring(p);
umodpoly amod = a;
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 the algorithm in chapter 6 of [GCL].
+ *
+ * @param a vector with univariate polynomials mod p^k
+ * @param x symbol
+ * @param m exponent of x^m in the equation to solve
+ * @param p prime number
+ * @param k p^k is modulus
+ * @return vector of polynomials (s_i)
+ */
+static upvec univar_diophant(const upvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
{
cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
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.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_;
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 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)
{
const size_t nu = I.size() + 1;
const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
}
}
+/** 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)
{
lst result;
-
if ( is_a<numeric>(e) ) {
result.append(e);
return result;
if ( is_a<power>(e) ) {
result.append(1);
result.append(e.op(0));
- result.append(e.op(1));
return result;
}
if ( is_a<symbol>(e) || is_a<add>(e) ) {
result.append(1);
result.append(e);
- result.append(1);
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);
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)
{
- const int k = f.nops()-2;
+ 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 ) {
- return false;
- }
- for ( int i=1; i<=k; ++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)));
for ( int j=i-1; j>=0; --j ) {
r = d[j];
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
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();
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) ) {
+ 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);
- trying = checkdivisors(fnum, d);
}
- } while ( trying );
- 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)
{
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));
+ // make polynomial primitive
+ ex unit, cont, pp;
+ poly.unitcontprim(x, unit, cont, pp);
if ( !is_a<numeric>(cont) ) {
- return factor(cont) * factor(pp);
+ 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);
vnlst = put_factors_into_lst(vnfactors);
}
- const numeric maxtrials = 3;
- numeric modulus = (vnlst.nops()-1 > 3) ? vnlst.nops()-1 : 3;
- 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;
+
+ 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;
+ 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);
- 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);
+ // generate a set of valid evaluation points
+ generate_set(pp, vn, syms, ex_to<lst>(vnlst), modulus, u, a);
+
+ ufac = factor_univariate(u, x, prime);
ufaclst = put_factors_into_lst(ufac);
- factor_count = (ufaclst.nops()-1)/2;
-
- // 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;
- }
+ factor_count = ufaclst.nops()-1;
+ delta = ufaclst.op(0);
if ( factor_count <= 1 ) {
+ // 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;
}
- 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));
- }
- for ( size_t j=(vnlst.nops()-1)/2+1; j>0; --j ) {
- if ( abs(ftilde[j-1]) == 1 ) {
- used_flag[j-1] = true;
- continue;
- }
- 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);
- }
- else {
- D[i-1] = D[i-1] * pow(vnlst.op(2*(j-2)+1), count);
- }
- }
- else {
- ftilde[j-1] = ftilde[j-1] / prefac;
- break;
- }
- ++j;
- }
- }
- }
-
- 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;
}
+ // determine true leading coefficients for the Hensel lifting
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;
- }
- 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;
}
+ // set up evaluation points
EvalPoint ep;
vector<EvalPoint> epv;
s = syms.begin();
// 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.op(i).degree(x) > maxdeg ) {
+ maxdeg = ufaclst[i].degree(x);
}
}
cl_I B = 2*calc_bound(u, x, maxdeg);
l = l + 1;
pl = pl * prime;
}
-
- upvec 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 ) {
- umodpoly newu;
- umodpoly_from_ex(newu, ufaclst.op(i*2+1), x, R);
- 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);
}
- 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);
}
}
+/** 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_) { }
} // anonymous namespace
+/** Interface function to the outside world. It checks the arguments, tries a
+ * square free factorization, and then calls factor_sqrfree to do the hard
+ * work.
+ */
ex factor(const ex& poly, unsigned options)
{
// check arguments
}
// 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) ) {
git://www.ginac.de / ginac.git / blobdiff