/** @file factor.cpp
*
- * Polynomial factorization routines.
- * Only univariate at the moment and completely non-optimized!
+ * Polynomial factorization code (implementation).
+ *
+ * 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.
+ * [GCL] Algorithms for Computer Algebra,
+ * K.O.Geddes, S.R.Czapor, G.Labahn, Springer Verlag, 1992.
*/
/*
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
+#define DEBUGFACTOR
+
+#ifdef DEBUGFACTOR
+#include <ostream>
+#include <ginac/ginac.h>
+using namespace GiNaC;
+#else
#include "factor.h"
#include "ex.h"
#include "mul.h"
#include "normal.h"
#include "add.h"
+#endif
#include <algorithm>
+#include <cmath>
+#include <limits>
#include <list>
#include <vector>
using namespace std;
#include <cln/cln.h>
using namespace cln;
-//#define DEBUGFACTOR
-
#ifdef DEBUGFACTOR
-#include <ostream>
-#endif // def DEBUGFACTOR
-
+namespace Factor {
+#else
namespace GiNaC {
+#endif
+#ifdef DEBUGFACTOR
+#define DCOUT(str) cout << #str << endl
+#define DCOUTVAR(var) cout << #var << ": " << var << endl
+#define DCOUT2(str,var) cout << #str << ": " << var << endl
+#else
+#define DCOUT(str)
+#define DCOUTVAR(var)
+#define DCOUT2(str,var)
+#endif
+
+// forward declaration
+ex factor(const ex& poly, unsigned options);
+
+// anonymous namespace to hide all utility functions
namespace {
-typedef vector<cl_MI> Vec;
-typedef vector<Vec> VecVec;
+typedef vector<cl_MI> mvec;
#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const Vec& v)
+ostream& operator<<(ostream& o, const mvec& v)
{
- Vec::const_iterator i = v.begin(), end = v.end();
+ mvec::const_iterator i = v.begin(), end = v.end();
while ( i != end ) {
o << *i++ << " ";
}
#endif // def DEBUGFACTOR
#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const VecVec& v)
+ostream& operator<<(ostream& o, const vector<mvec>& v)
{
- VecVec::const_iterator i = v.begin(), end = v.end();
+ vector<mvec>::const_iterator i = v.begin(), end = v.end();
while ( i != end ) {
o << *i++ << endl;
}
}
#endif // def DEBUGFACTOR
-struct Term
+////////////////////////////////////////////////////////////////////////////////
+// modular univariate polynomial code
+
+//typedef cl_UP_MI umod;
+typedef std::vector<cln::cl_MI> umodpoly;
+//typedef vector<umod> umodvec;
+typedef vector<umodpoly> upvec;
+
+// COPY FROM UPOLY.HPP
+
+// CHANGED size_t -> int !!!
+template<typename T> static int degree(const T& p)
{
- cl_MI c; // coefficient
- unsigned int exp; // exponent >=0
-};
+ return p.size() - 1;
+}
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const Term& t)
+template<typename T> static typename T::value_type lcoeff(const T& p)
{
- if ( t.exp ) {
- o << "(" << t.c << ")x^" << t.exp;
- }
- else {
- o << "(" << t.c << ")";
- }
- return o;
+ return p[p.size() - 1];
}
-#endif // def DEBUGFACTOR
-struct UniPoly
+static bool normalize_in_field(umodpoly& a)
{
- cl_modint_ring R;
- list<Term> terms; // highest exponent first
-
- UniPoly(const cl_modint_ring& ring) : R(ring) { }
- UniPoly(const cl_modint_ring& ring, const ex& poly, const ex& x) : R(ring)
- {
- // assert: poly is in Z[x]
- Term t;
- for ( int i=poly.degree(x); i>=poly.ldegree(x); --i ) {
- int coeff = ex_to<numeric>(poly.coeff(x,i)).to_int();
- if ( coeff ) {
- t.c = R->canonhom(coeff);
- if ( !zerop(t.c) ) {
- t.exp = i;
- terms.push_back(t);
- }
- }
- }
- }
- UniPoly(const cl_modint_ring& ring, const Vec& v) : R(ring)
- {
- Term t;
- for ( unsigned int i=0; i<v.size(); ++i ) {
- if ( !zerop(v[i]) ) {
- t.c = v[i];
- t.exp = i;
- terms.push_front(t);
- }
- }
- }
- unsigned int degree() const
- {
- if ( terms.size() ) {
- return terms.front().exp;
- }
- else {
- return 0;
- }
- }
- bool zero() const { return (terms.size() == 0); }
- const cl_MI operator[](unsigned int deg) const
- {
- list<Term>::const_iterator i = terms.begin(), end = terms.end();
- for ( ; i != end; ++i ) {
- if ( i->exp == deg ) {
- return i->c;
- }
- if ( i->exp < deg ) {
- break;
- }
- }
- return R->zero();
+ if (a.size() == 0)
+ return true;
+ if ( lcoeff(a) == a[0].ring()->one() ) {
+ return true;
}
- void set(unsigned int deg, const cl_MI& c)
- {
- list<Term>::iterator i = terms.begin(), end = terms.end();
- while ( i != end ) {
- if ( i->exp == deg ) {
- if ( !zerop(c) ) {
- i->c = c;
- }
- else {
- terms.erase(i);
- }
- return;
- }
- if ( i->exp < deg ) {
- break;
- }
+
+ const cln::cl_MI lc_1 = recip(lcoeff(a));
+ for (std::size_t k = a.size(); k-- != 0; )
+ a[k] = a[k]*lc_1;
+ return false;
+}
+
+template<typename T> static void
+canonicalize(T& p, const typename T::size_type hint = std::numeric_limits<typename T::size_type>::max())
+{
+ if (p.empty())
+ return;
+
+ std::size_t i = p.size() - 1;
+ // Be fast if the polynomial is already canonicalized
+ if (!zerop(p[i]))
+ return;
+
+ if (hint < p.size())
+ i = hint;
+
+ bool is_zero = false;
+ do {
+ if (!zerop(p[i])) {
++i;
+ break;
}
- if ( !zerop(c) ) {
- Term t;
- t.c = c;
- t.exp = deg;
- terms.insert(i, t);
+ if (i == 0) {
+ is_zero = true;
+ break;
}
+ --i;
+ } while (true);
+
+ if (is_zero) {
+ p.clear();
+ return;
}
- ex to_ex(const ex& x, bool symmetric = true) const
- {
- ex r;
- list<Term>::const_iterator i = terms.begin(), end = terms.end();
- if ( symmetric ) {
- numeric mod(R->modulus);
- numeric halfmod = (mod-1)/2;
- for ( ; i != end; ++i ) {
- numeric n(R->retract(i->c));
- if ( n > halfmod ) {
- r += pow(x, i->exp) * (n-mod);
- }
- else {
- r += pow(x, i->exp) * n;
- }
- }
+
+ p.erase(p.begin() + i, p.end());
+}
+
+// END COPY FROM UPOLY.HPP
+
+static void expt_pos(const umodpoly& a, unsigned int q, umodpoly& b)
+{
+ throw runtime_error("expt_pos: not implemented!");
+ // code below is not correct!
+// b.clear();
+// if ( a.empty() ) return;
+// b.resize(degree(a)*q+1, a[0].ring()->zero());
+// cl_MI norm = recip(a[0]);
+// umodpoly an = a;
+// for ( size_t i=0; i<an.size(); ++i ) {
+// an[i] = an[i] * norm;
+// }
+// b[0] = a[0].ring()->one();
+// for ( size_t i=1; i<b.size(); ++i ) {
+// for ( size_t j=1; j<i; ++j ) {
+// b[i] = b[i] + ((i-j+1)*q-i-1) * a[i-j] * b[j-1];
+// }
+// b[i] = b[i] / i;
+// }
+// cl_MI corr = expt_pos(a[0], q);
+// for ( size_t i=0; i<b.size(); ++i ) {
+// b[i] = b[i] * corr;
+// }
+}
+
+static umodpoly operator+(const umodpoly& a, const umodpoly& b)
+{
+ int sa = a.size();
+ int sb = b.size();
+ if ( sa >= sb ) {
+ umodpoly r(sa);
+ int i = 0;
+ for ( ; i<sb; ++i ) {
+ r[i] = a[i] + b[i];
}
- else {
- for ( ; i != end; ++i ) {
- r += pow(x, i->exp) * numeric(R->retract(i->c));
- }
+ for ( ; i<sa; ++i ) {
+ r[i] = a[i];
}
+ canonicalize(r);
return r;
}
- void unit_normal()
- {
- if ( terms.size() ) {
- if ( terms.front().c != R->one() ) {
- list<Term>::iterator i = terms.begin(), end = terms.end();
- cl_MI cont = i->c;
- i->c = R->one();
- while ( ++i != end ) {
- i->c = div(i->c, cont);
- if ( zerop(i->c) ) {
- terms.erase(i);
- }
- }
- }
- }
- }
- cl_MI unit() const
- {
- return terms.front().c;
- }
- void divide(const cl_MI& x)
- {
- list<Term>::iterator i = terms.begin(), end = terms.end();
- for ( ; i != end; ++i ) {
- i->c = div(i->c, x);
- if ( zerop(i->c) ) {
- terms.erase(i);
- }
+ else {
+ umodpoly r(sb);
+ int i = 0;
+ for ( ; i<sa; ++i ) {
+ r[i] = a[i] + b[i];
}
- }
- void reduce_exponents(unsigned int prime)
- {
- list<Term>::iterator i = terms.begin(), end = terms.end();
- while ( i != end ) {
- if ( i->exp > 0 ) {
- // assert: i->exp is multiple of prime
- i->exp /= prime;
- }
- ++i;
+ for ( ; i<sb; ++i ) {
+ r[i] = b[i];
}
+ canonicalize(r);
+ return r;
}
- void deriv(UniPoly& d) const
- {
- list<Term>::const_iterator i = terms.begin(), end = terms.end();
- while ( i != end ) {
- if ( i->exp ) {
- cl_MI newc = i->c * i->exp;
- if ( !zerop(newc) ) {
- Term t;
- t.c = newc;
- t.exp = i->exp-1;
- d.terms.push_back(t);
- }
- }
- ++i;
+}
+
+static umodpoly operator-(const umodpoly& a, const umodpoly& b)
+{
+ int sa = a.size();
+ int sb = b.size();
+ if ( sa >= sb ) {
+ umodpoly r(sa);
+ int i = 0;
+ for ( ; i<sb; ++i ) {
+ r[i] = a[i] - b[i];
}
- }
- bool operator<(const UniPoly& o) const
- {
- if ( terms.size() != o.terms.size() ) {
- return terms.size() < o.terms.size();
- }
- list<Term>::const_iterator i1 = terms.begin(), end = terms.end();
- list<Term>::const_iterator i2 = o.terms.begin();
- while ( i1 != end ) {
- if ( i1->exp != i2->exp ) {
- return i1->exp < i2->exp;
- }
- if ( i1->c != i2->c ) {
- return R->retract(i1->c) < R->retract(i2->c);
- }
- ++i1; ++i2;
+ for ( ; i<sa; ++i ) {
+ r[i] = a[i];
}
- return true;
+ canonicalize(r);
+ return r;
}
- bool operator==(const UniPoly& o) const
- {
- if ( terms.size() != o.terms.size() ) {
- return false;
+ else {
+ umodpoly r(sb);
+ int i = 0;
+ for ( ; i<sa; ++i ) {
+ r[i] = a[i] - b[i];
}
- list<Term>::const_iterator i1 = terms.begin(), end = terms.end();
- list<Term>::const_iterator i2 = o.terms.begin();
- while ( i1 != end ) {
- if ( i1->exp != i2->exp ) {
- return false;
- }
- if ( i1->c != i2->c ) {
- return false;
- }
- ++i1; ++i2;
+ for ( ; i<sb; ++i ) {
+ r[i] = -b[i];
}
- return true;
- }
- bool operator!=(const UniPoly& o) const
- {
- bool res = !(*this == o);
- return res;
+ canonicalize(r);
+ return r;
}
-};
+}
-static UniPoly operator*(const UniPoly& a, const UniPoly& b)
+static umodpoly operator*(const umodpoly& a, const umodpoly& b)
{
- unsigned int n = a.degree()+b.degree();
- UniPoly c(a.R);
- Term t;
- for ( unsigned int i=0 ; i<=n; ++i ) {
- t.c = a.R->zero();
- for ( unsigned int j=0 ; j<=i; ++j ) {
- t.c = t.c + a[j] * b[i-j];
- }
- if ( !zerop(t.c) ) {
- t.exp = i;
- c.terms.push_front(t);
+ umodpoly c;
+ if ( a.empty() || b.empty() ) return c;
+
+ int n = degree(a) + degree(b);
+ c.resize(n+1, a[0].ring()->zero());
+ for ( int i=0 ; i<=n; ++i ) {
+ for ( int j=0 ; j<=i; ++j ) {
+ if ( j > degree(a) || (i-j) > degree(b) ) continue; // TODO optimize!
+ c[i] = c[i] + a[j] * b[i-j];
}
}
+ canonicalize(c);
return c;
}
-static UniPoly operator-(const UniPoly& a, const UniPoly& b)
+static umodpoly operator*(const umodpoly& a, const cl_MI& x)
{
- list<Term>::const_iterator ia = a.terms.begin(), aend = a.terms.end();
- list<Term>::const_iterator ib = b.terms.begin(), bend = b.terms.end();
- UniPoly c(a.R);
- while ( ia != aend && ib != bend ) {
- if ( ia->exp > ib->exp ) {
- c.terms.push_back(*ia);
- ++ia;
- }
- else if ( ia->exp < ib->exp ) {
- c.terms.push_back(*ib);
- c.terms.back().c = -c.terms.back().c;
- ++ib;
- }
- else {
- Term t;
- t.exp = ia->exp;
- t.c = ia->c - ib->c;
- if ( !zerop(t.c) ) {
- c.terms.push_back(t);
- }
- ++ia; ++ib;
- }
- }
- while ( ia != aend ) {
- c.terms.push_back(*ia);
- ++ia;
- }
- while ( ib != bend ) {
- c.terms.push_back(*ib);
- c.terms.back().c = -c.terms.back().c;
- ++ib;
+ umodpoly r(a.size());
+ for ( size_t i=0; i<a.size(); ++i ) {
+ r[i] = a[i] * x;
}
- return c;
+ canonicalize(r);
+ return r;
}
-static UniPoly operator-(const UniPoly& a)
+static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_modint_ring& R)
{
- list<Term>::const_iterator ia = a.terms.begin(), aend = a.terms.end();
- UniPoly c(a.R);
- while ( ia != aend ) {
- c.terms.push_back(*ia);
- c.terms.back().c = -c.terms.back().c;
- ++ia;
- }
- return c;
+ // assert: e is in Z[x]
+ int deg = e.degree(x);
+ ump.resize(deg+1);
+ int ldeg = e.ldegree(x);
+ for ( ; deg>=ldeg; --deg ) {
+ cl_I coeff = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
+ ump[deg] = R->canonhom(coeff);
+ }
+ for ( ; deg>=0; --deg ) {
+ ump[deg] = R->zero();
+ }
+ canonicalize(ump);
}
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const UniPoly& t)
+static void umodpoly_from_ex(umodpoly& ump, const ex& e, const ex& x, const cl_I& modulus)
{
- list<Term>::const_iterator i = t.terms.begin(), end = t.terms.end();
- if ( i == end ) {
- o << "0";
- return o;
- }
- for ( ; i != end; ) {
- o << *i++;
- if ( i != end ) {
- o << " + ";
- }
- }
- return o;
+ umodpoly_from_ex(ump, e, x, find_modint_ring(modulus));
}
-#endif // def DEBUGFACTOR
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const list<UniPoly>& t)
+static ex umod_to_ex(const umodpoly& a, const ex& x)
{
- list<UniPoly>::const_iterator i = t.begin(), end = t.end();
- o << "{" << endl;
- for ( ; i != end; ) {
- o << *i++ << endl;
+ if ( a.empty() ) return 0;
+ cl_modint_ring R = a[0].ring();
+ cl_I mod = R->modulus;
+ cl_I halfmod = (mod-1) >> 1;
+ ex e;
+ for ( int i=degree(a); i>=0; --i ) {
+ cl_I n = R->retract(a[i]);
+ if ( n > halfmod ) {
+ e += numeric(n-mod) * pow(x, i);
+ } else {
+ e += numeric(n) * pow(x, i);
+ }
}
- o << "}" << endl;
- return o;
+ return e;
}
-#endif // def DEBUGFACTOR
-
-typedef vector<UniPoly> UniPolyVec;
-struct UniFactor
+/** Divides all coefficients of the polynomial a by the integer x.
+ * All coefficients are supposed to be divisible by x. If they are not, the
+ * the<cl_I> cast will raise an exception.
+ *
+ * @param[in,out] a polynomial of which the coefficients will be reduced by x
+ * @param[in] x integer that divides the coefficients
+ */
+static void reduce_coeff(umodpoly& a, const cl_I& x)
{
- UniPoly p;
- unsigned int exp;
+ if ( a.empty() ) return;
- UniFactor(const cl_modint_ring& ring) : p(ring) { }
- UniFactor(const UniPoly& p_, unsigned int exp_) : p(p_), exp(exp_) { }
- bool operator<(const UniFactor& o) const
- {
- return p < o.p;
+ cl_modint_ring R = a[0].ring();
+ umodpoly::iterator i = a.begin(), end = a.end();
+ for ( ; i!=end; ++i ) {
+ // cln cannot perform this division in the modular field
+ cl_I c = R->retract(*i);
+ *i = cl_MI(R, the<cl_I>(c / x));
}
-};
+}
-struct UniFactorVec
+/** Calculates remainder of a/b.
+ * Assertion: a and b not empty.
+ *
+ * @param[in] a polynomial dividend
+ * @param[in] b polynomial divisor
+ * @param[out] r polynomial remainder
+ */
+static void rem(const umodpoly& a, const umodpoly& b, umodpoly& r)
{
- vector<UniFactor> factors;
-
- void unique()
- {
- sort(factors.begin(), factors.end());
- if ( factors.size() > 1 ) {
- vector<UniFactor>::iterator i = factors.begin();
- vector<UniFactor>::const_iterator cmp = factors.begin()+1;
- vector<UniFactor>::iterator end = factors.end();
- while ( cmp != end ) {
- if ( i->p != cmp->p ) {
- ++i;
- ++cmp;
- }
- else {
- i->exp += cmp->exp;
- ++cmp;
- }
- }
- if ( i != end-1 ) {
- factors.erase(i+1, end);
+ int k, n;
+ n = degree(b);
+ k = degree(a) - n;
+ r = a;
+ if ( k < 0 ) return;
+
+ do {
+ cl_MI qk = div(r[n+k], b[n]);
+ if ( !zerop(qk) ) {
+ for ( int i=0; i<n; ++i ) {
+ unsigned int j = n + k - 1 - i;
+ r[j] = r[j] - qk * b[j-k];
}
}
- }
-};
+ } while ( k-- );
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const UniFactorVec& ufv)
-{
- for ( size_t i=0; i<ufv.factors.size(); ++i ) {
- if ( i != ufv.factors.size()-1 ) {
- o << "*";
- }
- else {
- o << " ";
- }
- o << "[ " << ufv.factors[i].p << " ]^" << ufv.factors[i].exp << endl;
- }
- return o;
+ fill(r.begin()+n, r.end(), a[0].ring()->zero());
+ canonicalize(r);
}
-#endif // def DEBUGFACTOR
-static void rem(const UniPoly& a_, const UniPoly& b, UniPoly& c)
+/** Calculates quotient of a/b.
+ * Assertion: a and b not empty.
+ *
+ * @param[in] a polynomial dividend
+ * @param[in] b polynomial divisor
+ * @param[out] q polynomial quotient
+ */
+static void div(const umodpoly& a, const umodpoly& b, umodpoly& q)
{
- if ( a_.degree() < b.degree() ) {
- c = a_;
- return;
- }
-
- unsigned int k, n;
- n = b.degree();
- k = a_.degree() - n;
-
- if ( n == 0 ) {
- c.terms.clear();
- return;
- }
-
- c = a_;
- Term termbuf;
-
- while ( true ) {
- cl_MI qk = div(c[n+k], b[n]);
+ int k, n;
+ n = degree(b);
+ k = degree(a) - n;
+ q.clear();
+ if ( k < 0 ) return;
+
+ umodpoly r = a;
+ q.resize(k+1, a[0].ring()->zero());
+ do {
+ cl_MI qk = div(r[n+k], b[n]);
if ( !zerop(qk) ) {
- unsigned int j;
- for ( unsigned int i=0; i<n; ++i ) {
- j = n + k - 1 - i;
- c.set(j, c[j] - qk*b[j-k]);
+ q[k] = qk;
+ for ( int i=0; i<n; ++i ) {
+ unsigned int j = n + k - 1 - i;
+ r[j] = r[j] - qk * b[j-k];
}
}
- if ( k == 0 ) break;
- --k;
- }
- list<Term>::iterator i = c.terms.begin(), end = c.terms.end();
- while ( i != end ) {
- if ( i->exp <= n-1 ) {
- break;
- }
- ++i;
- }
- c.terms.erase(c.terms.begin(), i);
+ } while ( k-- );
+
+ canonicalize(q);
}
-static void div(const UniPoly& a_, const UniPoly& b, UniPoly& q)
+/** Calculates quotient and remainder of a/b.
+ * Assertion: a and b not empty.
+ *
+ * @param[in] a polynomial dividend
+ * @param[in] b polynomial divisor
+ * @param[out] r polynomial remainder
+ * @param[out] q polynomial quotient
+ */
+static void remdiv(const umodpoly& a, const umodpoly& b, umodpoly& r, umodpoly& q)
{
- if ( a_.degree() < b.degree() ) {
- q.terms.clear();
- return;
- }
-
- unsigned int k, n;
- n = b.degree();
- k = a_.degree() - n;
-
- UniPoly c = a_;
- Term termbuf;
-
- while ( true ) {
- cl_MI qk = div(c[n+k], b[n]);
+ int k, n;
+ n = degree(b);
+ k = degree(a) - n;
+ q.clear();
+ r = a;
+ if ( k < 0 ) return;
+
+ q.resize(k+1, a[0].ring()->zero());
+ do {
+ cl_MI qk = div(r[n+k], b[n]);
if ( !zerop(qk) ) {
- Term t;
- t.c = qk;
- t.exp = k;
- q.terms.push_back(t);
- unsigned int j;
- for ( unsigned int i=0; i<n; ++i ) {
- j = n + k - 1 - i;
- c.set(j, c[j] - qk*b[j-k]);
+ q[k] = qk;
+ for ( int i=0; i<n; ++i ) {
+ unsigned int j = n + k - 1 - i;
+ r[j] = r[j] - qk * b[j-k];
}
}
- if ( k == 0 ) break;
- --k;
- }
+ } while ( k-- );
+
+ fill(r.begin()+n, r.end(), a[0].ring()->zero());
+ canonicalize(r);
+ canonicalize(q);
}
-static void gcd(const UniPoly& a, const UniPoly& b, UniPoly& c)
+/** Calculates the GCD of polynomial a and b.
+ *
+ * @param[in] a polynomial
+ * @param[in] b polynomial
+ * @param[out] c GCD
+ */
+static void gcd(const umodpoly& a, const umodpoly& b, umodpoly& c)
{
- c = a;
- c.unit_normal();
- UniPoly d = b;
- d.unit_normal();
-
- if ( c.degree() < d.degree() ) {
- gcd(b, a, c);
- return;
- }
+ if ( degree(a) < degree(b) ) return gcd(b, a, c);
- while ( !d.zero() ) {
- UniPoly r(a.R);
+ c = a;
+ normalize_in_field(c);
+ umodpoly d = b;
+ normalize_in_field(d);
+ umodpoly r;
+ while ( !d.empty() ) {
rem(c, d, r);
c = d;
d = r;
}
- c.unit_normal();
+ normalize_in_field(c);
}
-static bool is_one(const UniPoly& w)
+/** Calculates the derivative of the polynomial a.
+ *
+ * @param[in] a polynomial of which to take the derivative
+ * @param[out] d result/derivative
+ */
+static void deriv(const umodpoly& a, umodpoly& d)
{
- if ( w.terms.size() == 1 && w[0] == w.R->one() ) {
- return true;
+ d.clear();
+ if ( a.size() <= 1 ) return;
+
+ d.insert(d.begin(), a.begin()+1, a.end());
+ int max = d.size();
+ for ( int i=1; i<max; ++i ) {
+ d[i] = d[i] * (i+1);
}
- return false;
+ canonicalize(d);
}
-static void sqrfree_main(const UniPoly& a, UniFactorVec& fvec)
+static bool unequal_one(const umodpoly& a)
{
- unsigned int i = 1;
- UniPoly b(a.R);
- a.deriv(b);
- if ( !b.zero() ) {
- UniPoly c(a.R), w(a.R);
- gcd(a, b, c);
- div(a, c, w);
- while ( !is_one(w) ) {
- UniPoly y(a.R), z(a.R);
- gcd(w, c, y);
- div(w, y, z);
- if ( !is_one(z) ) {
- UniFactor uf(z, i++);
- fvec.factors.push_back(uf);
- }
- w = y;
- UniPoly cbuf(a.R);
- div(c, y, cbuf);
- c = cbuf;
- }
- if ( !is_one(c) ) {
- unsigned int prime = cl_I_to_uint(c.R->modulus);
- c.reduce_exponents(prime);
- unsigned int pos = fvec.factors.size();
- sqrfree_main(c, fvec);
- for ( unsigned int p=pos; p<fvec.factors.size(); ++p ) {
- fvec.factors[p].exp *= prime;
- }
- return;
- }
- }
- else {
- unsigned int prime = cl_I_to_uint(a.R->modulus);
- UniPoly amod = a;
- amod.reduce_exponents(prime);
- unsigned int pos = fvec.factors.size();
- sqrfree_main(amod, fvec);
- for ( unsigned int p=pos; p<fvec.factors.size(); ++p ) {
- fvec.factors[p].exp *= prime;
- }
- return;
- }
+ if ( a.empty() ) return true;
+ return ( a.size() != 1 || a[0] != a[0].ring()->one() );
+}
+
+static bool equal_one(const umodpoly& a)
+{
+ return ( a.size() == 1 && a[0] == a[0].ring()->one() );
}
-static void squarefree(const UniPoly& a, UniFactorVec& fvec)
+/** Returns true if polynomial a is square free.
+ *
+ * @param[in] a polynomial to check
+ * @return true if polynomial is square free, false otherwise
+ */
+static bool squarefree(const umodpoly& a)
{
- sqrfree_main(a, fvec);
- fvec.unique();
+ umodpoly b;
+ deriv(a, b);
+ if ( b.empty() ) {
+ return true;
+ }
+ umodpoly c;
+ gcd(a, b, c);
+ return equal_one(c);
}
-class Matrix
+// END modular univariate polynomial code
+////////////////////////////////////////////////////////////////////////////////
+
+////////////////////////////////////////////////////////////////////////////////
+// modular matrix
+
+class modular_matrix
{
- friend ostream& operator<<(ostream& o, const Matrix& m);
+ friend ostream& operator<<(ostream& o, const modular_matrix& m);
public:
- Matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
+ modular_matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
{
m.resize(c*r, init);
}
cl_MI operator()(size_t row, size_t col) const { return m[row*c + col]; }
void mul_col(size_t col, const cl_MI x)
{
- Vec::iterator i = m.begin() + col;
+ mvec::iterator i = m.begin() + col;
for ( size_t rc=0; rc<r; ++rc ) {
*i = *i * x;
i += c;
}
void sub_col(size_t col1, size_t col2, const cl_MI fac)
{
- Vec::iterator i1 = m.begin() + col1;
- Vec::iterator i2 = m.begin() + col2;
+ 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;
void switch_col(size_t col1, size_t col2)
{
cl_MI buf;
- Vec::iterator i1 = m.begin() + col1;
- Vec::iterator i2 = m.begin() + col2;
+ 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;
}
}
+ 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;
+ }
+ }
+ 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;
+ }
+ }
+ 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;
+ }
+ }
+ 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) ) {
+ return false;
+ }
+ i += c;
+ }
+ return true;
+ }
bool is_row_zero(size_t row) const
{
- Vec::const_iterator i = m.begin() + row*c;
+ mvec::const_iterator i = m.begin() + row*c;
for ( size_t cc=0; cc<c; ++cc ) {
if ( !zerop(*i) ) {
return false;
}
void set_row(size_t row, const vector<cl_MI>& newrow)
{
- Vec::iterator i1 = m.begin() + row*c;
- Vec::const_iterator i2 = newrow.begin(), end = newrow.end();
+ mvec::iterator i1 = m.begin() + row*c;
+ mvec::const_iterator i2 = newrow.begin(), end = newrow.end();
for ( ; i2 != end; ++i1, ++i2 ) {
*i1 = *i2;
}
}
- Vec::const_iterator row_begin(size_t row) const { return m.begin()+row*c; }
- Vec::const_iterator row_end(size_t row) const { return m.begin()+row*c+r; }
+ mvec::const_iterator row_begin(size_t row) const { return m.begin()+row*c; }
+ mvec::const_iterator row_end(size_t row) const { return m.begin()+row*c+r; }
private:
size_t r, c;
- Vec m;
+ mvec m;
};
#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const Matrix& m)
+modular_matrix operator*(const modular_matrix& m1, const modular_matrix& m2)
{
- vector<cl_MI>::const_iterator i = m.m.begin(), end = m.m.end();
- size_t wrap = 1;
- for ( ; i != end; ++i ) {
- o << *i << " ";
- if ( !(wrap++ % m.c) ) {
+ const unsigned int r = m1.rowsize();
+ const unsigned int c = m2.colsize();
+ modular_matrix o(r,c,m1(0,0));
+
+ for ( size_t i=0; i<r; ++i ) {
+ for ( size_t j=0; j<c; ++j ) {
+ cl_MI buf;
+ buf = m1(i,0) * m2(0,j);
+ for ( size_t k=1; k<c; ++k ) {
+ buf = buf + m1(i,k)*m2(k,j);
+ }
+ o(i,j) = buf;
+ }
+ }
+ return o;
+}
+
+ostream& operator<<(ostream& o, const modular_matrix& m)
+{
+ vector<cl_MI>::const_iterator i = m.m.begin(), end = m.m.end();
+ size_t wrap = 1;
+ for ( ; i != end; ++i ) {
+ o << *i << " ";
+ if ( !(wrap++ % m.c) ) {
o << endl;
}
}
}
#endif // def DEBUGFACTOR
-static void q_matrix(const UniPoly& a, Matrix& Q)
+// END modular matrix
+////////////////////////////////////////////////////////////////////////////////
+
+static void q_matrix(const umodpoly& a, modular_matrix& Q)
{
- unsigned int n = a.degree();
- unsigned int q = cl_I_to_uint(a.R->modulus);
- vector<cl_MI> r(n, a.R->zero());
- r[0] = a.R->one();
- Q.set_row(0, r);
- unsigned int max = (n-1) * q;
- for ( size_t m=1; m<=max; ++m ) {
- cl_MI rn_1 = r.back();
- for ( size_t i=n-1; i>0; --i ) {
- r[i] = r[i-1] - rn_1 * a[i];
- }
- r[0] = -rn_1 * a[0];
- if ( (m % q) == 0 ) {
- Q.set_row(m/q, r);
+ int n = degree(a);
+ unsigned int q = cl_I_to_uint(a[0].ring()->modulus);
+// fast and buggy
+// vector<cl_MI> r(n, a.R->zero());
+// r[0] = a.R->one();
+// Q.set_row(0, r);
+// unsigned int max = (n-1) * q;
+// for ( size_t m=1; m<=max; ++m ) {
+// cl_MI rn_1 = r.back();
+// for ( size_t i=n-1; i>0; --i ) {
+// r[i] = r[i-1] - rn_1 * a[i];
+// }
+// r[0] = -rn_1 * a[0];
+// if ( (m % q) == 0 ) {
+// Q.set_row(m/q, r);
+// }
+// }
+// slow and (hopefully) correct
+ cl_MI one = a[0].ring()->one();
+ cl_MI zero = a[0].ring()->zero();
+ for ( int i=0; i<n; ++i ) {
+ umodpoly qk(i*q+1, zero);
+ qk[i*q] = one;
+ umodpoly r;
+ rem(qk, a, r);
+ mvec rvec(n, zero);
+ for ( int j=0; j<=degree(r); ++j ) {
+ rvec[j] = r[j];
}
+ Q.set_row(i, rvec);
}
}
-static void nullspace(Matrix& M, vector<Vec>& basis)
+static void nullspace(modular_matrix& M, vector<mvec>& basis)
{
const size_t n = M.rowsize();
const cl_MI one = M(0,0).ring()->one();
}
for ( size_t i=0; i<n; ++i ) {
if ( !M.is_row_zero(i) ) {
- Vec nu(M.row_begin(i), M.row_end(i));
+ mvec nu(M.row_begin(i), M.row_end(i));
basis.push_back(nu);
}
}
}
-static void berlekamp(const UniPoly& a, UniPolyVec& upv)
+static void berlekamp(const umodpoly& a, upvec& upv)
{
- Matrix Q(a.degree(), a.degree(), a.R->zero());
+ cl_modint_ring R = a[0].ring();
+ umodpoly one(1, R->one());
+
+ modular_matrix Q(degree(a), degree(a), R->zero());
q_matrix(a, Q);
- VecVec nu;
+ vector<mvec> nu;
nullspace(Q, nu);
const unsigned int k = nu.size();
if ( k == 1 ) {
return;
}
- list<UniPoly> factors;
+ list<umodpoly> factors;
factors.push_back(a);
unsigned int size = 1;
unsigned int r = 1;
- unsigned int q = cl_I_to_uint(a.R->modulus);
+ unsigned int q = cl_I_to_uint(R->modulus);
- list<UniPoly>::iterator u = factors.begin();
+ list<umodpoly>::iterator u = factors.begin();
while ( true ) {
for ( unsigned int s=0; s<q; ++s ) {
- UniPoly g(a.R);
- UniPoly nur(a.R, nu[r]);
- nur.set(0, nur[0] - cl_MI(a.R, s));
+ umodpoly nur = nu[r];
+ nur[0] = nur[0] - cl_MI(R, s);
+ canonicalize(nur);
+ umodpoly g;
gcd(nur, *u, g);
- if ( !is_one(g) && g != *u ) {
- UniPoly uo(a.R);
+ if ( unequal_one(g) && g != *u ) {
+ umodpoly uo;
div(*u, g, uo);
- if ( is_one(uo) ) {
+ if ( equal_one(uo) ) {
throw logic_error("berlekamp: unexpected divisor.");
}
else {
*u = uo;
}
factors.push_back(g);
- ++size;
+ size = 0;
+ list<umodpoly>::const_iterator i = factors.begin(), end = factors.end();
+ while ( i != end ) {
+ if ( degree(*i) ) ++size;
+ ++i;
+ }
if ( size == k ) {
- list<UniPoly>::const_iterator i = factors.begin(), end = factors.end();
+ list<umodpoly>::const_iterator i = factors.begin(), end = factors.end();
while ( i != end ) {
upv.push_back(*i++);
}
return;
}
- if ( u->degree() < nur.degree() ) {
- break;
- }
}
}
if ( ++r == k ) {
}
}
-static void factor_modular(const UniPoly& p, UniPolyVec& upv)
+static void rem_xq(int q, const umodpoly& b, umodpoly& c)
{
+ cl_modint_ring R = b[0].ring();
+
+ int n = degree(b);
+ if ( n > q ) {
+ c.resize(q+1, R->zero());
+ c[q] = R->one();
+ return;
+ }
+
+ c.clear();
+ c.resize(n+1, R->zero());
+ int k = q-n;
+ c[n] = R->one();
+
+ int ofs = 0;
+ do {
+ cl_MI qk = div(c[n-ofs], b[n]);
+ if ( !zerop(qk) ) {
+ for ( int i=1; i<=n; ++i ) {
+ c[n-i+ofs] = c[n-i] - qk * b[n-i];
+ }
+ ofs = ofs ? 0 : 1;
+ }
+ } while ( k-- );
+
+ if ( ofs ) {
+ c.pop_back();
+ }
+ else {
+ c.erase(c.begin());
+ }
+ canonicalize(c);
+}
+
+static void distinct_degree_factor(const umodpoly& a_, upvec& result)
+{
+ umodpoly a = a_;
+
+ cl_modint_ring R = a[0].ring();
+ int q = cl_I_to_int(R->modulus);
+ int n = degree(a);
+ size_t nhalf = n/2;
+
+ size_t i = 1;
+ umodpoly w(1, R->one());
+ umodpoly x = w;
+
+ upvec ai;
+
+ while ( i <= nhalf ) {
+ expt_pos(w, q, w);
+ rem(w, a, w);
+
+ umodpoly buf;
+ gcd(a, w-x, buf);
+ ai.push_back(buf);
+
+ if ( unequal_one(ai.back()) ) {
+ div(a, ai.back(), a);
+ rem(w, a, w);
+ }
+
+ ++i;
+ }
+
+ result = ai;
+}
+
+static void same_degree_factor(const umodpoly& a, upvec& result)
+{
+ cl_modint_ring R = a[0].ring();
+ int deg = degree(a);
+
+ upvec buf;
+ distinct_degree_factor(a, buf);
+ int degsum = 0;
+
+ for ( size_t i=0; i<buf.size(); ++i ) {
+ if ( unequal_one(buf[i]) ) {
+ degsum += degree(buf[i]);
+ upvec upv;
+ berlekamp(buf[i], upv);
+ for ( size_t j=0; j<upv.size(); ++j ) {
+ result.push_back(upv[j]);
+ }
+ }
+ }
+
+ if ( degsum < deg ) {
+ result.push_back(a);
+ }
+}
+
+static void distinct_degree_factor_BSGS(const umodpoly& a, upvec& result)
+{
+ cl_modint_ring R = a[0].ring();
+ int q = cl_I_to_int(R->modulus);
+ int n = degree(a);
+
+ cl_N pm = 0.3;
+ int l = cl_I_to_int(ceiling1(the<cl_F>(expt(n, pm))));
+ upvec h(l+1);
+ umodpoly qk(1, R->one());
+ h[0] = qk;
+ for ( int i=1; i<=l; ++i ) {
+ expt_pos(h[i-1], q, qk);
+ rem(qk, a, h[i]);
+ }
+
+ int m = std::ceil(((double)n)/2/l);
+ upvec H(m);
+ int ql = std::pow(q, l);
+ H[0] = h[l];
+ for ( int i=1; i<m; ++i ) {
+ expt_pos(H[i-1], ql, qk);
+ rem(qk, a, H[i]);
+ }
+
+ upvec I(m);
+ umodpoly one(1, R->one());
+ for ( int i=0; i<m; ++i ) {
+ I[i] = one;
+ for ( int j=0; j<l; ++j ) {
+ I[i] = I[i] * (H[i] - h[j]);
+ }
+ rem(I[i], a, I[i]);
+ }
+
+ upvec F(m, one);
+ umodpoly f = a;
+ for ( int i=0; i<m; ++i ) {
+ umodpoly g;
+ gcd(f, I[i], g);
+ if ( g == one ) continue;
+ F[i] = g;
+ div(f, g, f);
+ }
+
+ result.resize(n, one);
+ if ( unequal_one(f) ) {
+ result[n] = f;
+ }
+ for ( int i=0; i<m; ++i ) {
+ umodpoly f = F[i];
+ for ( int j=l-1; j>=0; --j ) {
+ umodpoly g;
+ gcd(f, H[i]-h[j], g);
+ result[l*(i+1)-j-1] = g;
+ div(f, g, f);
+ }
+ }
+}
+
+static void cantor_zassenhaus(const umodpoly& a, upvec& result)
+{
+}
+
+static void factor_modular(const umodpoly& p, upvec& upv)
+{
+ //same_degree_factor(p, upv);
berlekamp(p, upv);
return;
}
-static void exteuclid(const UniPoly& a, const UniPoly& b, UniPoly& g, UniPoly& s, UniPoly& t)
+static void exteuclid(const umodpoly& a, const umodpoly& b, umodpoly& g, umodpoly& s, umodpoly& t)
{
- if ( a.degree() < b.degree() ) {
+ if ( degree(a) < degree(b) ) {
exteuclid(b, a, g, t, s);
return;
}
- UniPoly c1(a.R), c2(a.R), d1(a.R), d2(a.R), q(a.R), r(a.R), r1(a.R), r2(a.R);
- UniPoly c = a; c.unit_normal();
- UniPoly d = b; d.unit_normal();
- c1.set(0, a.R->one());
- d2.set(0, a.R->one());
- while ( !d.zero() ) {
- q.terms.clear();
+ umodpoly one(1, a[0].ring()->one());
+ umodpoly c = a; normalize_in_field(c);
+ umodpoly d = b; normalize_in_field(d);
+ umodpoly c1 = one;
+ umodpoly c2;
+ umodpoly d1;
+ umodpoly d2 = one;
+ while ( !d.empty() ) {
+ umodpoly q;
div(c, d, q);
- r = c - q * d;
- r1 = c1 - q * d1;
- r2 = c2 - q * d2;
+ umodpoly r = c - q * d;
+ umodpoly r1 = c1 - q * d1;
+ umodpoly r2 = c2 - q * d2;
c = d;
c1 = d1;
c2 = d2;
d1 = r1;
d2 = r2;
}
- g = c; g.unit_normal();
+ g = c; normalize_in_field(g);
s = c1;
- s.divide(a.unit());
- s.divide(c.unit());
+ for ( int i=0; i<=degree(s); ++i ) {
+ s[i] = s[i] * recip(a[degree(a)] * c[degree(c)]);
+ }
+ canonicalize(s);
+ s = s * g;
t = c2;
- t.divide(b.unit());
- t.divide(c.unit());
+ for ( int i=0; i<=degree(t); ++i ) {
+ t[i] = t[i] * recip(b[degree(b)] * c[degree(c)]);
+ }
+ canonicalize(t);
+ t = t * g;
}
static ex replace_lc(const ex& poly, const ex& x, const ex& lc)
return r;
}
-static ex hensel_univar(const ex& a_, const ex& x, unsigned int p, const UniPoly& u1_, const UniPoly& w1_, const ex& gamma_ = 0)
+static ex hensel_univar(const ex& a_, const ex& x, unsigned int p, const umodpoly& u1_, const umodpoly& w1_, const ex& gamma_ = 0)
{
ex a = a_;
- const cl_modint_ring& R = u1_.R;
+ const cl_modint_ring& R = u1_[0].ring();
// calc bound B
ex maxcoeff;
for ( int i=a.degree(x); i>=a.ldegree(x); --i ) {
maxcoeff += pow(abs(a.coeff(x, i)),2);
}
- cl_I normmc = ceiling1(the<cl_F>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
- unsigned int maxdegree = (u1_.degree() > w1_.degree()) ? u1_.degree() : w1_.degree();
- unsigned int B = cl_I_to_uint(normmc * expt_pos(cl_I(2), maxdegree));
+ cl_I normmc = ceiling1(the<cl_R>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
+ cl_I maxdegree = (degree(u1_) > degree(w1_)) ? degree(u1_) : degree(w1_);
+ cl_I B = normmc * expt_pos(cl_I(2), maxdegree);
// step 1
ex alpha = a.lcoeff(x);
if ( gamma == 0 ) {
gamma = alpha;
}
- unsigned int gamma_ui = ex_to<numeric>(abs(gamma)).to_int();
+ numeric gamma_ui = ex_to<numeric>(abs(gamma));
a = a * gamma;
- UniPoly nu1 = u1_;
- nu1.unit_normal();
- UniPoly nw1 = w1_;
- nw1.unit_normal();
+ umodpoly nu1 = u1_;
+ normalize_in_field(nu1);
+ umodpoly nw1 = w1_;
+ normalize_in_field(nw1);
ex phi;
- phi = expand(gamma * nu1.to_ex(x));
- UniPoly u1(R, phi, x);
- phi = expand(alpha * nw1.to_ex(x));
- UniPoly w1(R, phi, x);
+ phi = gamma * umod_to_ex(nu1, x);
+ umodpoly u1;
+ umodpoly_from_ex(u1, phi, x, R);
+ phi = alpha * umod_to_ex(nw1, x);
+ umodpoly w1;
+ umodpoly_from_ex(w1, phi, x, R);
// step 2
- UniPoly s(R), t(R), g(R);
+ umodpoly g;
+ umodpoly s;
+ umodpoly t;
exteuclid(u1, w1, g, s, t);
+ if ( unequal_one(g) ) {
+ throw logic_error("gcd(u1,w1) != 1");
+ }
// step 3
- ex u = replace_lc(u1.to_ex(x), x, gamma);
- ex w = replace_lc(w1.to_ex(x), x, alpha);
+ ex u = replace_lc(umod_to_ex(u1, x), x, gamma);
+ ex w = replace_lc(umod_to_ex(w1, x), x, alpha);
ex e = expand(a - u * w);
- unsigned int modulus = p;
+ numeric modulus = p;
+ const numeric maxmodulus = 2*numeric(B)*gamma_ui;
// step 4
- while ( !e.is_zero() && modulus < 2*B*gamma_ui ) {
+ while ( !e.is_zero() && modulus < maxmodulus ) {
ex c = e / modulus;
- phi = expand(s.to_ex(x)*c);
- UniPoly sigmatilde(R, phi, x);
- phi = expand(t.to_ex(x)*c);
- UniPoly tautilde(R, phi, x);
- UniPoly q(R), r(R);
- div(sigmatilde, w1, q);
- rem(sigmatilde, w1, r);
- UniPoly sigma = r;
- phi = expand(tautilde.to_ex(x) + q.to_ex(x) * u1.to_ex(x));
- UniPoly tau(R, phi, x);
- u = expand(u + tau.to_ex(x) * modulus);
- w = expand(w + sigma.to_ex(x) * modulus);
+ phi = expand(umod_to_ex(s, x) * c);
+ umodpoly sigmatilde;
+ umodpoly_from_ex(sigmatilde, phi, x, R);
+ phi = expand(umod_to_ex(t, x) * c);
+ umodpoly tautilde;
+ umodpoly_from_ex(tautilde, phi, x, R);
+ umodpoly r, q;
+ remdiv(sigmatilde, w1, r, q);
+ umodpoly sigma = r;
+ phi = expand(umod_to_ex(tautilde, x) + umod_to_ex(q, x) * umod_to_ex(u1, x));
+ umodpoly tau;
+ umodpoly_from_ex(tau, phi, x, R);
+ u = expand(u + umod_to_ex(tau, x) * modulus);
+ w = expand(w + umod_to_ex(sigma, x) * modulus);
e = expand(a - u * w);
modulus = modulus * p;
}
size_t size() const { return n; }
size_t size_first() const { return n-sum; }
size_t size_second() const { return sum; }
+#ifdef DEBUGFACTOR
+ void get() const
+ {
+ for ( size_t i=0; i<k.size(); ++i ) {
+ cout << k[i] << " ";
+ }
+ cout << endl;
+ }
+#endif
bool next()
{
for ( size_t i=n-1; i>=1; --i ) {
vector<int> k;
};
-static void split(const UniPolyVec& factors, const Partition& part, UniPoly& a, UniPoly& b)
+static void split(const upvec& factors, const Partition& part, umodpoly& a, umodpoly& b)
{
- a.set(0, a.R->one());
- b.set(0, a.R->one());
+ umodpoly one(1, factors.front()[0].ring()->one());
+ a = one;
+ b = one;
for ( size_t i=0; i<part.size(); ++i ) {
if ( part[i] ) {
b = b * factors[i];
struct ModFactors
{
ex poly;
- UniPolyVec factors;
+ upvec factors;
};
static ex factor_univariate(const ex& poly, const ex& x)
ex unit, cont, prim;
poly.unitcontprim(x, unit, cont, prim);
- // determine proper prime
- unsigned int p = 3;
- cl_modint_ring R = find_modint_ring(p);
- while ( true ) {
- if ( irem(ex_to<numeric>(prim.lcoeff(x)), p) != 0 ) {
- UniPoly modpoly(R, prim, x);
- UniFactorVec sqrfree_ufv;
- squarefree(modpoly, sqrfree_ufv);
- if ( sqrfree_ufv.factors.size() == 1 ) break;
- }
- p = next_prime(p);
- R = find_modint_ring(p);
- }
-
- // do modular factorization
- UniPoly modpoly(R, prim, x);
- UniPolyVec factors;
- factor_modular(modpoly, factors);
- if ( factors.size() <= 1 ) {
- // irreducible for sure
- return poly;
+ // determine proper prime and minimize number of modular factors
+ unsigned int p = 3, lastp = 3;
+ cl_modint_ring R;
+ unsigned int trials = 0;
+ unsigned int minfactors = 0;
+ numeric lcoeff = ex_to<numeric>(prim.lcoeff(x));
+ upvec factors;
+ while ( trials < 2 ) {
+ while ( true ) {
+ p = next_prime(p);
+ if ( irem(lcoeff, p) != 0 ) {
+ R = find_modint_ring(p);
+ umodpoly modpoly;
+ umodpoly_from_ex(modpoly, prim, x, R);
+ if ( squarefree(modpoly) ) break;
+ }
+ }
+
+ // do modular factorization
+ umodpoly modpoly;
+ umodpoly_from_ex(modpoly, prim, x, R);
+ upvec trialfactors;
+ factor_modular(modpoly, trialfactors);
+ if ( trialfactors.size() <= 1 ) {
+ // irreducible for sure
+ return poly;
+ }
+
+ if ( minfactors == 0 || trialfactors.size() < minfactors ) {
+ factors = trialfactors;
+ minfactors = factors.size();
+ lastp = p;
+ trials = 1;
+ }
+ else {
+ ++trials;
+ }
}
+ p = lastp;
+ R = find_modint_ring(p);
+ cl_univpoly_modint_ring UPR = find_univpoly_ring(R);
// lift all factor combinations
stack<ModFactors> tocheck;
const size_t n = tocheck.top().factors.size();
Partition part(n);
while ( true ) {
- UniPoly a(R), b(R);
+ umodpoly a, b;
split(tocheck.top().factors, part, a, b);
ex answer = hensel_univar(tocheck.top().poly, x, p, a, b);
break;
}
else {
- UniPolyVec newfactors1(part.size_first(), R), newfactors2(part.size_second(), R);
- UniPolyVec::iterator i1 = newfactors1.begin(), i2 = newfactors2.begin();
+ upvec newfactors1(part.size_first()), newfactors2(part.size_second());
+ upvec::iterator i1 = newfactors1.begin(), i2 = newfactors2.begin();
for ( size_t i=0; i<n; ++i ) {
if ( part[i] ) {
*i2++ = tocheck.top().factors[i];
mf.factors = newfactors2;
mf.poly = answer.op(1);
tocheck.push(mf);
+ break;
}
}
else {
return unit * cont * result;
}
-struct FindSymbolsMap : public map_function {
+struct EvalPoint
+{
+ ex x;
+ int evalpoint;
+};
+
+// 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);
+
+upvec multiterm_eea_lift(const upvec& a, const ex& x, unsigned int p, unsigned int k)
+{
+ const size_t r = a.size();
+ cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+ upvec q(r-1);
+ q[r-2] = a[r-1];
+ for ( size_t j=r-2; j>=1; --j ) {
+ q[j-1] = a[j] * q[j];
+ }
+ umodpoly beta(1, R->one());
+ upvec s;
+ for ( size_t j=1; j<r; ++j ) {
+ vector<ex> mdarg(2);
+ mdarg[0] = umod_to_ex(q[j-1], x);
+ mdarg[1] = umod_to_ex(a[j-1], x);
+ vector<EvalPoint> empty;
+ vector<ex> exsigma = multivar_diophant(mdarg, x, umod_to_ex(beta, x), empty, 0, p, k);
+ umodpoly sigma1;
+ umodpoly_from_ex(sigma1, exsigma[0], x, R);
+ umodpoly sigma2;
+ umodpoly_from_ex(sigma2, exsigma[1], x, R);
+ beta = sigma1;
+ s.push_back(sigma2);
+ }
+ s.push_back(beta);
+ return s;
+}
+
+/**
+ * Assert: a not empty.
+ */
+void change_modulus(const cl_modint_ring& R, umodpoly& a)
+{
+ if ( a.empty() ) return;
+ cl_modint_ring oldR = a[0].ring();
+ umodpoly::iterator i = a.begin(), end = a.end();
+ for ( ; i!=end; ++i ) {
+ *i = R->canonhom(oldR->retract(*i));
+ }
+ canonicalize(a);
+}
+
+void eea_lift(const umodpoly& a, const umodpoly& b, const ex& x, unsigned int p, unsigned int k, umodpoly& s_, umodpoly& t_)
+{
+ cl_modint_ring R = find_modint_ring(p);
+ umodpoly amod = a;
+ change_modulus(R, amod);
+ umodpoly bmod = b;
+ change_modulus(R, bmod);
+
+ umodpoly g;
+ umodpoly smod;
+ umodpoly tmod;
+ exteuclid(amod, bmod, g, smod, tmod);
+ if ( unequal_one(g) ) {
+ throw logic_error("gcd(amod,bmod) != 1");
+ }
+
+ cl_modint_ring Rpk = find_modint_ring(expt_pos(cl_I(p),k));
+ umodpoly s = smod;
+ change_modulus(Rpk, s);
+ umodpoly t = tmod;
+ change_modulus(Rpk, t);
+
+ cl_I modulus(p);
+ umodpoly one(1, Rpk->one());
+ for ( size_t j=1; j<k; ++j ) {
+ umodpoly e = one - a * s - b * t;
+ reduce_coeff(e, modulus);
+ umodpoly c = e;
+ change_modulus(R, c);
+ umodpoly sigmabar = smod * c;
+ umodpoly taubar = tmod * c;
+ umodpoly sigma, q;
+ remdiv(sigmabar, bmod, sigma, q);
+ umodpoly tau = taubar + q * amod;
+ umodpoly sadd = sigma;
+ change_modulus(Rpk, sadd);
+ cl_MI modmodulus(Rpk, modulus);
+ s = s + sadd * modmodulus;
+ umodpoly tadd = tau;
+ change_modulus(Rpk, tadd);
+ t = t + tadd * modmodulus;
+ modulus = modulus * p;
+ }
+
+ s_ = s; t_ = t;
+}
+
+upvec univar_diophant(const upvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
+{
+ cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+
+ const size_t r = a.size();
+ upvec result;
+ if ( r > 2 ) {
+ upvec s = multiterm_eea_lift(a, x, p, k);
+ for ( size_t j=0; j<r; ++j ) {
+ ex phi = expand(pow(x,m) * umod_to_ex(s[j], x));
+ umodpoly bmod;
+ umodpoly_from_ex(bmod, phi, x, R);
+ umodpoly buf;
+ rem(bmod, a[j], buf);
+ result.push_back(buf);
+ }
+ }
+ else {
+ umodpoly s;
+ umodpoly t;
+ eea_lift(a[1], a[0], x, p, k, s, t);
+ ex phi = expand(pow(x,m) * umod_to_ex(s, x));
+ umodpoly bmod;
+ umodpoly_from_ex(bmod, phi, x, R);
+ umodpoly buf, q;
+ remdiv(bmod, a[0], buf, q);
+ result.push_back(buf);
+ phi = expand(pow(x,m) * umod_to_ex(t, x));
+ umodpoly t1mod;
+ umodpoly_from_ex(t1mod, phi, x, R);
+ umodpoly buf2 = t1mod + q * a[1];
+ result.push_back(buf2);
+ }
+
+ return result;
+}
+
+struct make_modular_map : public map_function {
+ cl_modint_ring R;
+ make_modular_map(const cl_modint_ring& R_) : R(R_) { }
+ ex operator()(const ex& e)
+ {
+ if ( is_a<add>(e) || is_a<mul>(e) ) {
+ return e.map(*this);
+ }
+ else if ( is_a<numeric>(e) ) {
+ numeric mod(R->modulus);
+ numeric halfmod = (mod-1)/2;
+ cl_MI emod = R->canonhom(the<cl_I>(ex_to<numeric>(e).to_cl_N()));
+ numeric n(R->retract(emod));
+ if ( n > halfmod ) {
+ return n-mod;
+ }
+ else {
+ return n;
+ }
+ }
+ return e;
+ }
+};
+
+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)
+{
+ vector<ex> a = a_;
+
+ const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+ const size_t r = a.size();
+ const size_t nu = I.size() + 1;
+
+ vector<ex> sigma;
+ if ( nu > 1 ) {
+ ex xnu = I.back().x;
+ int alphanu = I.back().evalpoint;
+
+ ex A = 1;
+ for ( size_t i=0; i<r; ++i ) {
+ A *= a[i];
+ }
+ vector<ex> b(r);
+ for ( size_t i=0; i<r; ++i ) {
+ b[i] = normal(A / a[i]);
+ }
+
+ vector<ex> anew = a;
+ for ( size_t i=0; i<r; ++i ) {
+ anew[i] = anew[i].subs(xnu == alphanu);
+ }
+ ex cnew = c.subs(xnu == alphanu);
+ vector<EvalPoint> Inew = I;
+ Inew.pop_back();
+ sigma = multivar_diophant(anew, x, cnew, Inew, d, p, k);
+
+ ex buf = c;
+ for ( size_t i=0; i<r; ++i ) {
+ buf -= sigma[i] * b[i];
+ }
+ ex e = make_modular(buf, R);
+
+ ex monomial = 1;
+ for ( size_t m=1; m<=d; ++m ) {
+ while ( !e.is_zero() && e.has(xnu) ) {
+ monomial *= (xnu - alphanu);
+ monomial = expand(monomial);
+ ex cm = e.diff(ex_to<symbol>(xnu), m).subs(xnu==alphanu) / factorial(m);
+ cm = make_modular(cm, R);
+ if ( !cm.is_zero() ) {
+ vector<ex> delta_s = multivar_diophant(anew, x, cm, Inew, d, p, k);
+ ex buf = e;
+ for ( size_t j=0; j<delta_s.size(); ++j ) {
+ delta_s[j] *= monomial;
+ sigma[j] += delta_s[j];
+ buf -= delta_s[j] * b[j];
+ }
+ e = make_modular(buf, R);
+ }
+ }
+ }
+ }
+ else {
+ upvec amod;
+ for ( size_t i=0; i<a.size(); ++i ) {
+ umodpoly up;
+ umodpoly_from_ex(up, a[i], x, R);
+ amod.push_back(up);
+ }
+
+ sigma.insert(sigma.begin(), r, 0);
+ size_t nterms;
+ ex z;
+ if ( is_a<add>(c) ) {
+ nterms = c.nops();
+ z = c.op(0);
+ }
+ else {
+ nterms = 1;
+ z = c;
+ }
+ for ( size_t i=0; i<nterms; ++i ) {
+ int m = z.degree(x);
+ cl_I cm = the<cl_I>(ex_to<numeric>(z.lcoeff(x)).to_cl_N());
+ upvec delta_s = univar_diophant(amod, x, m, p, k);
+ cl_MI modcm;
+ cl_I poscm = cm;
+ while ( poscm < 0 ) {
+ poscm = poscm + expt_pos(cl_I(p),k);
+ }
+ modcm = cl_MI(R, poscm);
+ for ( size_t j=0; j<delta_s.size(); ++j ) {
+ delta_s[j] = delta_s[j] * modcm;
+ sigma[j] = sigma[j] + umod_to_ex(delta_s[j], x);
+ }
+ if ( nterms > 1 ) {
+ z = c.op(i+1);
+ }
+ }
+ }
+
+ for ( size_t i=0; i<sigma.size(); ++i ) {
+ sigma[i] = make_modular(sigma[i], R);
+ }
+
+ 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)
+{
+ const size_t nu = I.size() + 1;
+ const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
+
+ vector<ex> A(nu);
+ A[nu-1] = a;
+
+ for ( size_t j=nu; j>=2; --j ) {
+ ex x = I[j-2].x;
+ int alpha = I[j-2].evalpoint;
+ A[j-2] = A[j-1].subs(x==alpha);
+ A[j-2] = make_modular(A[j-2], R);
+ }
+
+ int maxdeg = a.degree(I.front().x);
+ for ( size_t i=1; i<I.size(); ++i ) {
+ int maxdeg2 = a.degree(I[i].x);
+ if ( maxdeg2 > maxdeg ) maxdeg = maxdeg2;
+ }
+
+ const size_t n = u.size();
+ vector<ex> U(n);
+ for ( size_t i=0; i<n; ++i ) {
+ U[i] = umod_to_ex(u[i], x);
+ }
+
+ for ( size_t j=2; j<=nu; ++j ) {
+ vector<ex> U1 = U;
+ ex monomial = 1;
+ for ( size_t m=0; m<n; ++m) {
+ if ( lcU[m] != 1 ) {
+ ex coef = lcU[m];
+ for ( size_t i=j-1; i<nu-1; ++i ) {
+ coef = coef.subs(I[i].x == I[i].evalpoint);
+ }
+ coef = make_modular(coef, R);
+ int deg = U[m].degree(x);
+ U[m] = U[m] - U[m].lcoeff(x) * pow(x,deg) + coef * pow(x,deg);
+ }
+ }
+ ex Uprod = 1;
+ for ( size_t i=0; i<n; ++i ) {
+ Uprod *= U[i];
+ }
+ ex e = expand(A[j-1] - Uprod);
+
+ vector<EvalPoint> newI;
+ for ( size_t i=1; i<=j-2; ++i ) {
+ newI.push_back(I[i-1]);
+ }
+
+ ex xj = I[j-2].x;
+ int alphaj = I[j-2].evalpoint;
+ size_t deg = A[j-1].degree(xj);
+ for ( size_t k=1; k<=deg; ++k ) {
+ if ( !e.is_zero() ) {
+ monomial *= (xj - alphaj);
+ monomial = expand(monomial);
+ ex dif = e.diff(ex_to<symbol>(xj), k);
+ ex c = dif.subs(xj==alphaj) / factorial(k);
+ if ( !c.is_zero() ) {
+ vector<ex> deltaU = multivar_diophant(U1, x, c, newI, maxdeg, p, cl_I_to_uint(l));
+ for ( size_t i=0; i<n; ++i ) {
+ deltaU[i] *= monomial;
+ U[i] += deltaU[i];
+ U[i] = make_modular(U[i], R);
+ }
+ ex Uprod = 1;
+ for ( size_t i=0; i<n; ++i ) {
+ Uprod *= U[i];
+ }
+ e = A[j-1] - Uprod;
+ e = make_modular(e, R);
+ }
+ }
+ }
+ }
+
+ ex acand = 1;
+ for ( size_t i=0; i<U.size(); ++i ) {
+ acand *= U[i];
+ }
+ if ( expand(a-acand).is_zero() ) {
+ lst res;
+ for ( size_t i=0; i<U.size(); ++i ) {
+ res.append(U[i]);
+ }
+ return res;
+ }
+ else {
+ lst res;
+ return lst();
+ }
+}
+
+static 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) ) {
+ ex nfac = 1;
+ for ( size_t i=0; i<e.nops(); ++i ) {
+ ex op = e.op(i);
+ if ( is_a<numeric>(op) ) {
+ nfac = op;
+ }
+ if ( is_a<power>(op) ) {
+ result.append(op.op(0));
+ result.append(op.op(1));
+ }
+ if ( is_a<symbol>(op) || is_a<add>(op) ) {
+ result.append(op);
+ result.append(1);
+ }
+ }
+ result.prepend(nfac);
+ return result;
+ }
+ throw runtime_error("put_factors_into_lst: bad term.");
+}
+
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const vector<numeric>& v)
+{
+ for ( size_t i=0; i<v.size(); ++i ) {
+ o << v[i] << " ";
+ }
+ return o;
+}
+#endif // def DEBUGFACTOR
+
+static bool checkdivisors(const lst& f, vector<numeric>& d)
+{
+ const int k = f.nops()-2;
+ 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)));
+ for ( int j=i-1; j>=0; --j ) {
+ r = d[j];
+ do {
+ r = gcd(r, q);
+ q = q/r;
+ } while ( r != 1 );
+ if ( q == 1 ) {
+ return true;
+ }
+ }
+ d[i] = q;
+ }
+ 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)
+{
+ // computation of d is actually not necessary
+ const ex& x = *syms.begin();
+ bool trying = true;
+ do {
+ ex u0 = u;
+ ex vna = vn;
+ ex vnatry;
+ exset::const_iterator s = syms.begin();
+ ++s;
+ for ( size_t i=0; i<a.size(); ++i ) {
+ do {
+ a[i] = mod(numeric(rand()), 2*modulus) - modulus;
+ vnatry = vna.subs(*s == a[i]);
+ } while ( vnatry == 0 );
+ vna = vnatry;
+ u0 = u0.subs(*s == a[i]);
+ ++s;
+ }
+ if ( gcd(u0,u0.diff(ex_to<symbol>(x))) != 1 ) {
+ 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;
+ }
+ }
+ fnum.append(fs);
+ ++i; ++i;
+ }
+ if ( problem ) {
+ return true;
+ }
+ 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) ) {
+#ifdef DEBUGFACTOR
+ return ::factor(cont) * ::factor(pp);
+#else
+ return factor(cont) * factor(pp);
+#endif
+ }
+
+ /* 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 {
+#ifdef DEBUGFACTOR
+ ex vnfactors = ::factor(vn);
+#else
+ ex vnfactors = factor(vn);
+#endif
+ vnlst = put_factors_into_lst(vnfactors);
+ }
+
+ 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);
+
+ while ( true ) {
+ numeric trialcount = 0;
+ ex u, delta;
+ unsigned int prime = 3;
+ size_t factor_count = 0;
+ ex ufac;
+ ex ufaclst;
+ 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);
+ }
+
+#ifdef DEBUGFACTOR
+ ufac = ::factor(u);
+#else
+ ufac = factor(u);
+#endif
+ 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;
+ }
+
+ if ( factor_count <= 1 ) {
+ return poly;
+ }
+
+ if ( minimalr < 0 ) {
+ minimalr = factor_count;
+ }
+ else if ( minimalr == factor_count ) {
+ ++trialcount;
+ ++modulus;
+ }
+ else if ( minimalr > factor_count ) {
+ minimalr = 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;
+ }
+
+ 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);
+ }
+ }
+ 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);
+ }
+ }
+ }
+
+ EvalPoint ep;
+ vector<EvalPoint> epv;
+ s = syms.begin();
+ ++s;
+ for ( size_t i=0; i<a.size(); ++i ) {
+ ep.x = *s++;
+ ep.evalpoint = a[i].to_int();
+ epv.push_back(ep);
+ }
+
+ // calc bound B
+ ex maxcoeff;
+ for ( int i=u.degree(x); i>=u.ldegree(x); --i ) {
+ maxcoeff += pow(abs(u.coeff(x, i)),2);
+ }
+ cl_I normmc = ceiling1(the<cl_R>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
+ unsigned int maxdegree = 0;
+ for ( size_t i=0; i<factor_count; ++i ) {
+ if ( ufaclst[2*i+1].degree(x) > (int)maxdegree ) {
+ maxdegree = ufaclst[2*i+1].degree(x);
+ }
+ }
+ cl_I B = normmc * expt_pos(cl_I(2), maxdegree);
+ cl_I l = 1;
+ cl_I pl = prime;
+ while ( pl < B ) {
+ l = l + 1;
+ 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);
+ }
+
+ ex res = hensel_multivar(ufaclst.op(0)*pp, x, epv, prime, l, uvec, C);
+ if ( res != lst() ) {
+ ex result = cont * ufaclst.op(0);
+ for ( size_t i=0; i<res.nops(); ++i ) {
+ result *= res.op(i).content(x) * res.op(i).unit(x);
+ result *= res.op(i).primpart(x);
+ }
+ return result;
+ }
+ }
+}
+
+struct find_symbols_map : public map_function {
exset syms;
ex operator()(const ex& e)
{
static ex factor_sqrfree(const ex& poly)
{
// determine all symbols in poly
- FindSymbolsMap findsymbols;
+ find_symbols_map findsymbols;
findsymbols(poly);
if ( findsymbols.syms.size() == 0 ) {
return poly;
}
if ( findsymbols.syms.size() == 1 ) {
+ // univariate case
const ex& x = *(findsymbols.syms.begin());
if ( poly.ldegree(x) > 0 ) {
int ld = poly.ldegree(x);
}
}
- // multivariate case not yet implemented!
- throw runtime_error("multivariate case not yet implemented!");
+ // multivariate case
+ ex res = factor_multivariate(poly, findsymbols.syms);
+ return res;
}
+struct apply_factor_map : public map_function {
+ unsigned options;
+ apply_factor_map(unsigned options_) : options(options_) { }
+ ex operator()(const ex& e)
+ {
+ if ( e.info(info_flags::polynomial) ) {
+#ifdef DEBUGFACTOR
+ return ::factor(e, options);
+#else
+ return factor(e, options);
+#endif
+ }
+ if ( is_a<add>(e) ) {
+ ex s1, s2;
+ for ( size_t i=0; i<e.nops(); ++i ) {
+ if ( e.op(i).info(info_flags::polynomial) ) {
+ s1 += e.op(i);
+ }
+ else {
+ s2 += e.op(i);
+ }
+ }
+ s1 = s1.eval();
+ s2 = s2.eval();
+#ifdef DEBUGFACTOR
+ return ::factor(s1, options) + s2.map(*this);
+#else
+ return factor(s1, options) + s2.map(*this);
+#endif
+ }
+ return e.map(*this);
+ }
+};
+
} // anonymous namespace
-ex factor(const ex& poly)
+#ifdef DEBUGFACTOR
+ex factor(const ex& poly, unsigned options = 0)
+#else
+ex factor(const ex& poly, unsigned options)
+#endif
{
+ // check arguments
+ if ( !poly.info(info_flags::polynomial) ) {
+ if ( options & factor_options::all ) {
+ options &= ~factor_options::all;
+ apply_factor_map factor_map(options);
+ return factor_map(poly);
+ }
+ return poly;
+ }
+
// determine all symbols in poly
- FindSymbolsMap findsymbols;
+ find_symbols_map findsymbols;
findsymbols(poly);
if ( findsymbols.syms.size() == 0 ) {
return poly;
return pow(f, sfpoly.op(1));
}
if ( is_a<mul>(sfpoly) ) {
+ // case: multiple factors
ex res = 1;
for ( size_t i=0; i<sfpoly.nops(); ++i ) {
const ex& t = sfpoly.op(i);
}
return res;
}
+ if ( is_a<symbol>(sfpoly) ) {
+ return poly;
+ }
// case: (polynomial)
ex f = factor_sqrfree(sfpoly);
return f;
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