* computation, square-free factorization and rational function normalization. */
/*
- * GiNaC Copyright (C) 1999-2003 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
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
- * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
-#include <algorithm>
-#include <map>
-
#include "normal.h"
#include "basic.h"
#include "ex.h"
#include "pseries.h"
#include "symbol.h"
#include "utils.h"
+#include "polynomial/chinrem_gcd.h"
+
+#include <algorithm>
+#include <map>
namespace GiNaC {
#endif
-/** Return pointer to first symbol found in expression. Due to GiNaC´s
+/** Return pointer to first symbol found in expression. Due to GiNaC's
* internal ordering of terms, it may not be obvious which symbol this
* function returns for a given expression.
*
if (e.info(info_flags::rational))
return lcm(ex_to<numeric>(e).denom(), l);
else if (is_exactly_a<add>(e)) {
- numeric c = _num1;
+ numeric c = *_num1_p;
for (size_t i=0; i<e.nops(); i++)
c = lcmcoeff(e.op(i), c);
return lcm(c, l);
} else if (is_exactly_a<mul>(e)) {
- numeric c = _num1;
+ numeric c = *_num1_p;
for (size_t i=0; i<e.nops(); i++)
- c *= lcmcoeff(e.op(i), _num1);
+ c *= lcmcoeff(e.op(i), *_num1_p);
return lcm(c, l);
} else if (is_exactly_a<power>(e)) {
if (is_a<symbol>(e.op(0)))
* @return LCM of denominators of coefficients */
static numeric lcm_of_coefficients_denominators(const ex &e)
{
- return lcmcoeff(e, _num1);
+ return lcmcoeff(e, *_num1_p);
}
/** Bring polynomial from Q[X] to Z[X] by multiplying in the previously
if (is_exactly_a<mul>(e)) {
size_t num = e.nops();
exvector v; v.reserve(num + 1);
- numeric lcm_accum = _num1;
+ numeric lcm_accum = *_num1_p;
for (size_t i=0; i<num; i++) {
- numeric op_lcm = lcmcoeff(e.op(i), _num1);
+ numeric op_lcm = lcmcoeff(e.op(i), *_num1_p);
v.push_back(multiply_lcm(e.op(i), op_lcm));
lcm_accum *= op_lcm;
}
/** Compute the integer content (= GCD of all numeric coefficients) of an
- * expanded polynomial.
+ * expanded polynomial. For a polynomial with rational coefficients, this
+ * returns g/l where g is the GCD of the coefficients' numerators and l
+ * is the LCM of the coefficients' denominators.
*
* @return integer content */
numeric ex::integer_content() const
numeric basic::integer_content() const
{
- return _num1;
+ return *_num1_p;
}
numeric numeric::integer_content() const
{
epvector::const_iterator it = seq.begin();
epvector::const_iterator itend = seq.end();
- numeric c = _num0;
+ numeric c = *_num0_p, l = *_num1_p;
while (it != itend) {
GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
- c = gcd(ex_to<numeric>(it->coeff), c);
+ c = gcd(ex_to<numeric>(it->coeff).numer(), c);
+ l = lcm(ex_to<numeric>(it->coeff).denom(), l);
it++;
}
GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
- c = gcd(ex_to<numeric>(overall_coeff),c);
- return c;
+ c = gcd(ex_to<numeric>(overall_coeff).numer(), c);
+ l = lcm(ex_to<numeric>(overall_coeff).denom(), l);
+ return c/l;
}
numeric mul::integer_content() const
if (!get_first_symbol(a, x) && !get_first_symbol(b, x))
throw(std::invalid_argument("invalid expression in divide()"));
+ // Try to avoid expanding partially factored expressions.
+ if (is_exactly_a<mul>(b)) {
+ // Divide sequentially by each term
+ ex rem_new, rem_old = a;
+ for (size_t i=0; i < b.nops(); i++) {
+ if (! divide(rem_old, b.op(i), rem_new, false))
+ return false;
+ rem_old = rem_new;
+ }
+ q = rem_new;
+ return true;
+ } else if (is_exactly_a<power>(b)) {
+ const ex& bb(b.op(0));
+ int exp_b = ex_to<numeric>(b.op(1)).to_int();
+ ex rem_new, rem_old = a;
+ for (int i=exp_b; i>0; i--) {
+ if (! divide(rem_old, bb, rem_new, false))
+ return false;
+ rem_old = rem_new;
+ }
+ q = rem_new;
+ return true;
+ }
+
+ if (is_exactly_a<mul>(a)) {
+ // Divide sequentially each term. If some term in a is divisible
+ // by b we are done... and if not, we can't really say anything.
+ size_t i;
+ ex rem_i;
+ bool divisible_p = false;
+ for (i=0; i < a.nops(); ++i) {
+ if (divide(a.op(i), b, rem_i, false)) {
+ divisible_p = true;
+ break;
+ }
+ }
+ if (divisible_p) {
+ exvector resv;
+ resv.reserve(a.nops());
+ for (size_t j=0; j < a.nops(); j++) {
+ if (j==i)
+ resv.push_back(rem_i);
+ else
+ resv.push_back(a.op(j));
+ }
+ q = (new mul(resv))->setflag(status_flags::dynallocated);
+ return true;
+ }
+ } else if (is_exactly_a<power>(a)) {
+ // The base itself might be divisible by b, in that case we don't
+ // need to expand a
+ const ex& ab(a.op(0));
+ int a_exp = ex_to<numeric>(a.op(1)).to_int();
+ ex rem_i;
+ if (divide(ab, b, rem_i, false)) {
+ q = rem_i*power(ab, a_exp - 1);
+ return true;
+ }
+// code below is commented-out because it leads to a significant slowdown
+// for (int i=2; i < a_exp; i++) {
+// if (divide(power(ab, i), b, rem_i, false)) {
+// q = rem_i*power(ab, a_exp - i);
+// return true;
+// }
+// } // ... so we *really* need to expand expression.
+ }
+
// Polynomial long division (recursive)
ex r = a.expand();
if (r.is_zero()) {
}
#endif
+ if (is_exactly_a<power>(b)) {
+ const ex& bb(b.op(0));
+ ex qbar = a;
+ int exp_b = ex_to<numeric>(b.op(1)).to_int();
+ for (int i=exp_b; i>0; i--) {
+ if (!divide_in_z(qbar, bb, q, var))
+ return false;
+ qbar = q;
+ }
+ return true;
+ }
+
+ if (is_exactly_a<mul>(b)) {
+ ex qbar = a;
+ for (const_iterator itrb = b.begin(); itrb != b.end(); ++itrb) {
+ sym_desc_vec sym_stats;
+ get_symbol_stats(a, *itrb, sym_stats);
+ if (!divide_in_z(qbar, *itrb, q, sym_stats.begin()))
+ return false;
+
+ qbar = q;
+ }
+ return true;
+ }
+
// Main symbol
const ex &x = var->sym;
// Compute values at evaluation points 0..adeg
vector<numeric> alpha; alpha.reserve(adeg + 1);
exvector u; u.reserve(adeg + 1);
- numeric point = _num0;
+ numeric point = *_num0_p;
ex c;
for (i=0; i<=adeg; i++) {
ex bs = b.subs(x == point, subs_options::no_pattern);
while (bs.is_zero()) {
- point += _num1;
+ point += *_num1_p;
bs = b.subs(x == point, subs_options::no_pattern);
}
if (!divide_in_z(a.subs(x == point, subs_options::no_pattern), bs, c, var+1))
return false;
alpha.push_back(point);
u.push_back(c);
- point += _num1;
+ point += *_num1_p;
}
// Compute inverses
vector<numeric> rcp; rcp.reserve(adeg + 1);
- rcp.push_back(_num0);
+ rcp.push_back(*_num0_p);
for (k=1; k<=adeg; k++) {
numeric product = alpha[k] - alpha[0];
for (i=1; i<k; i++)
*/
/** Compute unit part (= sign of leading coefficient) of a multivariate
- * polynomial in Z[x]. The product of unit part, content part, and primitive
+ * polynomial in Q[x]. The product of unit part, content part, and primitive
* part is the polynomial itself.
*
- * @param x variable in which to compute the unit part
+ * @param x main variable
* @return unit part
- * @see ex::content, ex::primpart */
+ * @see ex::content, ex::primpart, ex::unitcontprim */
ex ex::unit(const ex &x) const
{
ex c = expand().lcoeff(x);
if (is_exactly_a<numeric>(c))
- return c < _ex0 ? _ex_1 : _ex1;
+ return c.info(info_flags::negative) ?_ex_1 : _ex1;
else {
ex y;
if (get_first_symbol(c, y))
/** Compute content part (= unit normal GCD of all coefficients) of a
- * multivariate polynomial in Z[x]. The product of unit part, content part,
+ * multivariate polynomial in Q[x]. The product of unit part, content part,
* and primitive part is the polynomial itself.
*
- * @param x variable in which to compute the content part
+ * @param x main variable
* @return content part
- * @see ex::unit, ex::primpart */
+ * @see ex::unit, ex::primpart, ex::unitcontprim */
ex ex::content(const ex &x) const
{
- if (is_zero())
- return _ex0;
if (is_exactly_a<numeric>(*this))
return info(info_flags::negative) ? -*this : *this;
+
ex e = expand();
if (e.is_zero())
return _ex0;
- // First, try the integer content
+ // First, divide out the integer content (which we can calculate very efficiently).
+ // If the leading coefficient of the quotient is an integer, we are done.
ex c = e.integer_content();
ex r = e / c;
- ex lcoeff = r.lcoeff(x);
+ int deg = r.degree(x);
+ ex lcoeff = r.coeff(x, deg);
if (lcoeff.info(info_flags::integer))
return c;
// GCD of all coefficients
- int deg = e.degree(x);
- int ldeg = e.ldegree(x);
+ int ldeg = r.ldegree(x);
if (deg == ldeg)
- return e.lcoeff(x) / e.unit(x);
- c = _ex0;
+ return lcoeff * c / lcoeff.unit(x);
+ ex cont = _ex0;
for (int i=ldeg; i<=deg; i++)
- c = gcd(e.coeff(x, i), c, NULL, NULL, false);
- return c;
+ cont = gcd(r.coeff(x, i), cont, NULL, NULL, false);
+ return cont * c;
}
-/** Compute primitive part of a multivariate polynomial in Z[x].
- * The product of unit part, content part, and primitive part is the
- * polynomial itself.
+/** Compute primitive part of a multivariate polynomial in Q[x]. The result
+ * will be a unit-normal polynomial with a content part of 1. The product
+ * of unit part, content part, and primitive part is the polynomial itself.
*
- * @param x variable in which to compute the primitive part
+ * @param x main variable
* @return primitive part
- * @see ex::unit, ex::content */
+ * @see ex::unit, ex::content, ex::unitcontprim */
ex ex::primpart(const ex &x) const
{
- if (is_zero())
- return _ex0;
- if (is_exactly_a<numeric>(*this))
- return _ex1;
-
- ex c = content(x);
- if (c.is_zero())
- return _ex0;
- ex u = unit(x);
- if (is_exactly_a<numeric>(c))
- return *this / (c * u);
- else
- return quo(*this, c * u, x, false);
+ // We need to compute the unit and content anyway, so call unitcontprim()
+ ex u, c, p;
+ unitcontprim(x, u, c, p);
+ return p;
}
-/** Compute primitive part of a multivariate polynomial in Z[x] when the
+/** Compute primitive part of a multivariate polynomial in Q[x] when the
* content part is already known. This function is faster in computing the
* primitive part than the previous function.
*
- * @param x variable in which to compute the primitive part
+ * @param x main variable
* @param c previously computed content part
* @return primitive part */
ex ex::primpart(const ex &x, const ex &c) const
{
- if (is_zero())
- return _ex0;
- if (c.is_zero())
+ if (is_zero() || c.is_zero())
return _ex0;
if (is_exactly_a<numeric>(*this))
return _ex1;
+ // Divide by unit and content to get primitive part
ex u = unit(x);
if (is_exactly_a<numeric>(c))
return *this / (c * u);
}
+/** Compute unit part, content part, and primitive part of a multivariate
+ * polynomial in Q[x]. The product of the three parts is the polynomial
+ * itself.
+ *
+ * @param x main variable
+ * @param u unit part (returned)
+ * @param c content part (returned)
+ * @param p primitive part (returned)
+ * @see ex::unit, ex::content, ex::primpart */
+void ex::unitcontprim(const ex &x, ex &u, ex &c, ex &p) const
+{
+ // Quick check for zero (avoid expanding)
+ if (is_zero()) {
+ u = _ex1;
+ c = p = _ex0;
+ return;
+ }
+
+ // Special case: input is a number
+ if (is_exactly_a<numeric>(*this)) {
+ if (info(info_flags::negative)) {
+ u = _ex_1;
+ c = abs(ex_to<numeric>(*this));
+ } else {
+ u = _ex1;
+ c = *this;
+ }
+ p = _ex1;
+ return;
+ }
+
+ // Expand input polynomial
+ ex e = expand();
+ if (e.is_zero()) {
+ u = _ex1;
+ c = p = _ex0;
+ return;
+ }
+
+ // Compute unit and content
+ u = unit(x);
+ c = content(x);
+
+ // Divide by unit and content to get primitive part
+ if (c.is_zero()) {
+ p = _ex0;
+ return;
+ }
+ if (is_exactly_a<numeric>(c))
+ p = *this / (c * u);
+ else
+ p = quo(e, c * u, x, false);
+}
+
+
/*
* GCD of multivariate polynomials
*/
* @see heur_gcd */
numeric basic::max_coefficient() const
{
- return _num1;
+ return *_num1_p;
}
numeric numeric::max_coefficient() const
* polynomials and an iterator to the first element of the sym_desc vector
* passed in. This function is used internally by gcd().
*
- * @param a first multivariate polynomial (expanded)
- * @param b second multivariate polynomial (expanded)
+ * @param a first integer multivariate polynomial (expanded)
+ * @param b second integer multivariate polynomial (expanded)
* @param ca cofactor of polynomial a (returned), NULL to suppress
* calculation of cofactor
* @param cb cofactor of polynomial b (returned), NULL to suppress
* calculation of cofactor
* @param var iterator to first element of vector of sym_desc structs
- * @return the GCD as a new expression
+ * @param res the GCD (returned)
+ * @return true if GCD was computed, false otherwise.
* @see gcd
* @exception gcdheu_failed() */
-static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var)
+static bool heur_gcd_z(ex& res, const ex &a, const ex &b, ex *ca, ex *cb,
+ sym_desc_vec::const_iterator var)
{
#if STATISTICS
heur_gcd_called++;
// Algorithm only works for non-vanishing input polynomials
if (a.is_zero() || b.is_zero())
- return (new fail())->setflag(status_flags::dynallocated);
+ return false;
// GCD of two numeric values -> CLN
if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b)) {
*ca = ex_to<numeric>(a) / g;
if (cb)
*cb = ex_to<numeric>(b) / g;
- return g;
+ res = g;
+ return true;
}
// The first symbol is our main variable
numeric mq = q.max_coefficient();
numeric xi;
if (mp > mq)
- xi = mq * _num2 + _num2;
+ xi = mq * (*_num2_p) + (*_num2_p);
else
- xi = mp * _num2 + _num2;
+ xi = mp * (*_num2_p) + (*_num2_p);
// 6 tries maximum
for (int t=0; t<6; t++) {
// Apply evaluation homomorphism and calculate GCD
ex cp, cq;
- ex gamma = heur_gcd(p.subs(x == xi, subs_options::no_pattern), q.subs(x == xi, subs_options::no_pattern), &cp, &cq, var+1).expand();
- if (!is_exactly_a<fail>(gamma)) {
-
+ ex gamma;
+ bool found = heur_gcd_z(gamma,
+ p.subs(x == xi, subs_options::no_pattern),
+ q.subs(x == xi, subs_options::no_pattern),
+ &cp, &cq, var+1);
+ if (found) {
+ gamma = gamma.expand();
// Reconstruct polynomial from GCD of mapped polynomials
ex g = interpolate(gamma, xi, x, maxdeg);
ex dummy;
if (divide_in_z(p, g, ca ? *ca : dummy, var) && divide_in_z(q, g, cb ? *cb : dummy, var)) {
g *= gc;
- ex lc = g.lcoeff(x);
- if (is_exactly_a<numeric>(lc) && ex_to<numeric>(lc).is_negative())
- return -g;
- else
- return g;
+ res = g;
+ return true;
}
}
// Next evaluation point
xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
}
- return (new fail())->setflag(status_flags::dynallocated);
+ return false;
}
+/** Compute GCD of multivariate polynomials using the heuristic GCD algorithm.
+ * get_symbol_stats() must have been called previously with the input
+ * polynomials and an iterator to the first element of the sym_desc vector
+ * passed in. This function is used internally by gcd().
+ *
+ * @param a first rational multivariate polynomial (expanded)
+ * @param b second rational multivariate polynomial (expanded)
+ * @param ca cofactor of polynomial a (returned), NULL to suppress
+ * calculation of cofactor
+ * @param cb cofactor of polynomial b (returned), NULL to suppress
+ * calculation of cofactor
+ * @param var iterator to first element of vector of sym_desc structs
+ * @param res the GCD (returned)
+ * @return true if GCD was computed, false otherwise.
+ * @see heur_gcd_z
+ * @see gcd
+ */
+static bool heur_gcd(ex& res, const ex& a, const ex& b, ex *ca, ex *cb,
+ sym_desc_vec::const_iterator var)
+{
+ if (a.info(info_flags::integer_polynomial) &&
+ b.info(info_flags::integer_polynomial)) {
+ try {
+ return heur_gcd_z(res, a, b, ca, cb, var);
+ } catch (gcdheu_failed) {
+ return false;
+ }
+ }
+
+ // convert polynomials to Z[X]
+ const numeric a_lcm = lcm_of_coefficients_denominators(a);
+ const numeric ab_lcm = lcmcoeff(b, a_lcm);
+
+ const ex ai = a*ab_lcm;
+ const ex bi = b*ab_lcm;
+ if (!ai.info(info_flags::integer_polynomial))
+ throw std::logic_error("heur_gcd: not an integer polynomial [1]");
+
+ if (!bi.info(info_flags::integer_polynomial))
+ throw std::logic_error("heur_gcd: not an integer polynomial [2]");
+
+ bool found = false;
+ try {
+ found = heur_gcd_z(res, ai, bi, ca, cb, var);
+ } catch (gcdheu_failed) {
+ return false;
+ }
+
+ // GCD is not unique, it's defined up to a unit (i.e. invertible
+ // element). If the coefficient ring is a field, every its element is
+ // invertible, so one can multiply the polynomial GCD with any element
+ // of the coefficient field. We use this ambiguity to make cofactors
+ // integer polynomials.
+ if (found)
+ res /= ab_lcm;
+ return found;
+}
+
+
+// gcd helper to handle partially factored polynomials (to avoid expanding
+// large expressions). At least one of the arguments should be a power.
+static ex gcd_pf_pow(const ex& a, const ex& b, ex* ca, ex* cb);
+
+// gcd helper to handle partially factored polynomials (to avoid expanding
+// large expressions). At least one of the arguments should be a product.
+static ex gcd_pf_mul(const ex& a, const ex& b, ex* ca, ex* cb);
/** Compute GCD (Greatest Common Divisor) of multivariate polynomials a(X)
- * and b(X) in Z[X].
+ * and b(X) in Z[X]. Optionally also compute the cofactors of a and b,
+ * defined by a = ca * gcd(a, b) and b = cb * gcd(a, b).
*
* @param a first multivariate polynomial
* @param b second multivariate polynomial
* @param check_args check whether a and b are polynomials with rational
* coefficients (defaults to "true")
* @return the GCD as a new expression */
-ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
+ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args, unsigned options)
{
#if STATISTICS
gcd_called++;
}
// Partially factored cases (to avoid expanding large expressions)
- if (is_exactly_a<mul>(a)) {
- if (is_exactly_a<mul>(b) && b.nops() > a.nops())
- goto factored_b;
-factored_a:
- size_t num = a.nops();
- exvector g; g.reserve(num);
- exvector acc_ca; acc_ca.reserve(num);
- ex part_b = b;
- for (size_t i=0; i<num; i++) {
- ex part_ca, part_cb;
- g.push_back(gcd(a.op(i), part_b, &part_ca, &part_cb, check_args));
- acc_ca.push_back(part_ca);
- part_b = part_cb;
- }
- if (ca)
- *ca = (new mul(acc_ca))->setflag(status_flags::dynallocated);
- if (cb)
- *cb = part_b;
- return (new mul(g))->setflag(status_flags::dynallocated);
- } else if (is_exactly_a<mul>(b)) {
- if (is_exactly_a<mul>(a) && a.nops() > b.nops())
- goto factored_a;
-factored_b:
- size_t num = b.nops();
- exvector g; g.reserve(num);
- exvector acc_cb; acc_cb.reserve(num);
- ex part_a = a;
- for (size_t i=0; i<num; i++) {
- ex part_ca, part_cb;
- g.push_back(gcd(part_a, b.op(i), &part_ca, &part_cb, check_args));
- acc_cb.push_back(part_cb);
- part_a = part_ca;
- }
- if (ca)
- *ca = part_a;
- if (cb)
- *cb = (new mul(acc_cb))->setflag(status_flags::dynallocated);
- return (new mul(g))->setflag(status_flags::dynallocated);
- }
-
+ if (!(options & gcd_options::no_part_factored)) {
+ if (is_exactly_a<mul>(a) || is_exactly_a<mul>(b))
+ return gcd_pf_mul(a, b, ca, cb);
#if FAST_COMPARE
- // Input polynomials of the form poly^n are sometimes also trivial
- if (is_exactly_a<power>(a)) {
- ex p = a.op(0);
- if (is_exactly_a<power>(b)) {
- if (p.is_equal(b.op(0))) {
- // a = p^n, b = p^m, gcd = p^min(n, m)
- ex exp_a = a.op(1), exp_b = b.op(1);
- if (exp_a < exp_b) {
- if (ca)
- *ca = _ex1;
- if (cb)
- *cb = power(p, exp_b - exp_a);
- return power(p, exp_a);
- } else {
- if (ca)
- *ca = power(p, exp_a - exp_b);
- if (cb)
- *cb = _ex1;
- return power(p, exp_b);
- }
- }
- } else {
- if (p.is_equal(b)) {
- // a = p^n, b = p, gcd = p
- if (ca)
- *ca = power(p, a.op(1) - 1);
- if (cb)
- *cb = _ex1;
- return p;
- }
- }
- } else if (is_exactly_a<power>(b)) {
- ex p = b.op(0);
- if (p.is_equal(a)) {
- // a = p, b = p^n, gcd = p
- if (ca)
- *ca = _ex1;
- if (cb)
- *cb = power(p, b.op(1) - 1);
- return p;
- }
- }
+ if (is_exactly_a<power>(a) || is_exactly_a<power>(b))
+ return gcd_pf_pow(a, b, ca, cb);
#endif
+ }
// Some trivial cases
ex aex = a.expand(), bex = b.expand();
}
#endif
+ if (is_a<symbol>(aex)) {
+ if (! bex.subs(aex==_ex0, subs_options::no_pattern).is_zero()) {
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ return _ex1;
+ }
+ }
+
+ if (is_a<symbol>(bex)) {
+ if (! aex.subs(bex==_ex0, subs_options::no_pattern).is_zero()) {
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ return _ex1;
+ }
+ }
+
+ if (is_exactly_a<numeric>(aex)) {
+ numeric bcont = bex.integer_content();
+ numeric g = gcd(ex_to<numeric>(aex), bcont);
+ if (ca)
+ *ca = ex_to<numeric>(aex)/g;
+ if (cb)
+ *cb = bex/g;
+ return g;
+ }
+
+ if (is_exactly_a<numeric>(bex)) {
+ numeric acont = aex.integer_content();
+ numeric g = gcd(ex_to<numeric>(bex), acont);
+ if (ca)
+ *ca = aex/g;
+ if (cb)
+ *cb = ex_to<numeric>(bex)/g;
+ return g;
+ }
+
// Gather symbol statistics
sym_desc_vec sym_stats;
get_symbol_stats(a, b, sym_stats);
- // The symbol with least degree is our main variable
+ // The symbol with least degree which is contained in both polynomials
+ // is our main variable
+ sym_desc_vec::iterator vari = sym_stats.begin();
+ while ((vari != sym_stats.end()) &&
+ (((vari->ldeg_b == 0) && (vari->deg_b == 0)) ||
+ ((vari->ldeg_a == 0) && (vari->deg_a == 0))))
+ vari++;
+
+ // No common symbols at all, just return 1:
+ if (vari == sym_stats.end()) {
+ // N.B: keep cofactors factored
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ return _ex1;
+ }
+ // move symbols which contained only in one of the polynomials
+ // to the end:
+ rotate(sym_stats.begin(), vari, sym_stats.end());
+
sym_desc_vec::const_iterator var = sym_stats.begin();
const ex &x = var->sym;
}
// Try to eliminate variables
- if (var->deg_a == 0) {
- ex c = bex.content(x);
- ex g = gcd(aex, c, ca, cb, false);
+ if (var->deg_a == 0 && var->deg_b != 0 ) {
+ ex bex_u, bex_c, bex_p;
+ bex.unitcontprim(x, bex_u, bex_c, bex_p);
+ ex g = gcd(aex, bex_c, ca, cb, false);
if (cb)
- *cb *= bex.unit(x) * bex.primpart(x, c);
+ *cb *= bex_u * bex_p;
return g;
- } else if (var->deg_b == 0) {
- ex c = aex.content(x);
- ex g = gcd(c, bex, ca, cb, false);
+ } else if (var->deg_b == 0 && var->deg_a != 0) {
+ ex aex_u, aex_c, aex_p;
+ aex.unitcontprim(x, aex_u, aex_c, aex_p);
+ ex g = gcd(aex_c, bex, ca, cb, false);
if (ca)
- *ca *= aex.unit(x) * aex.primpart(x, c);
+ *ca *= aex_u * aex_p;
return g;
}
// Try heuristic algorithm first, fall back to PRS if that failed
ex g;
- try {
- g = heur_gcd(aex, bex, ca, cb, var);
- } catch (gcdheu_failed) {
- g = fail();
- }
- if (is_exactly_a<fail>(g)) {
+ if (!(options & gcd_options::no_heur_gcd)) {
+ bool found = heur_gcd(g, aex, bex, ca, cb, var);
+ if (found) {
+ // heur_gcd have already computed cofactors...
+ if (g.is_equal(_ex1)) {
+ // ... but we want to keep them factored if possible.
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ }
+ return g;
+ }
#if STATISTICS
- heur_gcd_failed++;
+ else {
+ heur_gcd_failed++;
+ }
#endif
+ }
+ if (options & gcd_options::use_sr_gcd) {
g = sr_gcd(aex, bex, var);
- if (g.is_equal(_ex1)) {
- // Keep cofactors factored if possible
+ } else {
+ exvector vars;
+ for (std::size_t n = sym_stats.size(); n-- != 0; )
+ vars.push_back(sym_stats[n].sym);
+ g = chinrem_gcd(aex, bex, vars);
+ }
+
+ if (g.is_equal(_ex1)) {
+ // Keep cofactors factored if possible
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ } else {
+ if (ca)
+ divide(aex, g, *ca, false);
+ if (cb)
+ divide(bex, g, *cb, false);
+ }
+ return g;
+}
+
+// gcd helper to handle partially factored polynomials (to avoid expanding
+// large expressions). Both arguments should be powers.
+static ex gcd_pf_pow_pow(const ex& a, const ex& b, ex* ca, ex* cb)
+{
+ ex p = a.op(0);
+ const ex& exp_a = a.op(1);
+ ex pb = b.op(0);
+ const ex& exp_b = b.op(1);
+
+ // a = p^n, b = p^m, gcd = p^min(n, m)
+ if (p.is_equal(pb)) {
+ if (exp_a < exp_b) {
if (ca)
- *ca = a;
+ *ca = _ex1;
if (cb)
- *cb = b;
+ *cb = power(p, exp_b - exp_a);
+ return power(p, exp_a);
} else {
if (ca)
- divide(aex, g, *ca, false);
+ *ca = power(p, exp_a - exp_b);
if (cb)
- divide(bex, g, *cb, false);
+ *cb = _ex1;
+ return power(p, exp_b);
}
- } else {
- if (g.is_equal(_ex1)) {
- // Keep cofactors factored if possible
+ }
+
+ ex p_co, pb_co;
+ ex p_gcd = gcd(p, pb, &p_co, &pb_co, false);
+ // a(x) = p(x)^n, b(x) = p_b(x)^m, gcd (p, p_b) = 1 ==> gcd(a,b) = 1
+ if (p_gcd.is_equal(_ex1)) {
if (ca)
*ca = a;
if (cb)
*cb = b;
- }
+ return _ex1;
+ // XXX: do I need to check for p_gcd = -1?
}
- return g;
+ // there are common factors:
+ // a(x) = g(x)^n A(x)^n, b(x) = g(x)^m B(x)^m ==>
+ // gcd(a, b) = g(x)^n gcd(A(x)^n, g(x)^(n-m) B(x)^m
+ if (exp_a < exp_b) {
+ ex pg = gcd(power(p_co, exp_a), power(p_gcd, exp_b-exp_a)*power(pb_co, exp_b), ca, cb, false);
+ return power(p_gcd, exp_a)*pg;
+ } else {
+ ex pg = gcd(power(p_gcd, exp_a - exp_b)*power(p_co, exp_a), power(pb_co, exp_b), ca, cb, false);
+ return power(p_gcd, exp_b)*pg;
+ }
}
+static ex gcd_pf_pow(const ex& a, const ex& b, ex* ca, ex* cb)
+{
+ if (is_exactly_a<power>(a) && is_exactly_a<power>(b))
+ return gcd_pf_pow_pow(a, b, ca, cb);
+
+ if (is_exactly_a<power>(b) && (! is_exactly_a<power>(a)))
+ return gcd_pf_pow(b, a, cb, ca);
+
+ GINAC_ASSERT(is_exactly_a<power>(a));
+
+ ex p = a.op(0);
+ const ex& exp_a = a.op(1);
+ if (p.is_equal(b)) {
+ // a = p^n, b = p, gcd = p
+ if (ca)
+ *ca = power(p, a.op(1) - 1);
+ if (cb)
+ *cb = _ex1;
+ return p;
+ }
+
+ ex p_co, bpart_co;
+ ex p_gcd = gcd(p, b, &p_co, &bpart_co, false);
+
+ // a(x) = p(x)^n, gcd(p, b) = 1 ==> gcd(a, b) = 1
+ if (p_gcd.is_equal(_ex1)) {
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ return _ex1;
+ }
+ // a(x) = g(x)^n A(x)^n, b(x) = g(x) B(x) ==> gcd(a, b) = g(x) gcd(g(x)^(n-1) A(x)^n, B(x))
+ ex rg = gcd(power(p_gcd, exp_a-1)*power(p_co, exp_a), bpart_co, ca, cb, false);
+ return p_gcd*rg;
+}
+
+static ex gcd_pf_mul(const ex& a, const ex& b, ex* ca, ex* cb)
+{
+ if (is_exactly_a<mul>(a) && is_exactly_a<mul>(b)
+ && (b.nops() > a.nops()))
+ return gcd_pf_mul(b, a, cb, ca);
+
+ if (is_exactly_a<mul>(b) && (!is_exactly_a<mul>(a)))
+ return gcd_pf_mul(b, a, cb, ca);
+
+ GINAC_ASSERT(is_exactly_a<mul>(a));
+ size_t num = a.nops();
+ exvector g; g.reserve(num);
+ exvector acc_ca; acc_ca.reserve(num);
+ ex part_b = b;
+ for (size_t i=0; i<num; i++) {
+ ex part_ca, part_cb;
+ g.push_back(gcd(a.op(i), part_b, &part_ca, &part_cb, false));
+ acc_ca.push_back(part_ca);
+ part_b = part_cb;
+ }
+ if (ca)
+ *ca = (new mul(acc_ca))->setflag(status_flags::dynallocated);
+ if (cb)
+ *cb = part_b;
+ return (new mul(g))->setflag(status_flags::dynallocated);
+}
/** Compute LCM (Least Common Multiple) of multivariate polynomials in Z[X].
*
*/
/** Compute square-free factorization of multivariate polynomial a(x) using
- * Yun´s algorithm. Used internally by sqrfree().
+ * Yun's algorithm. Used internally by sqrfree().
*
* @param a multivariate polynomial over Z[X], treated here as univariate
* polynomial in x.
{
ex num = n;
ex den = d;
- numeric pre_factor = _num1;
+ numeric pre_factor = *_num1_p;
//std::clog << "frac_cancel num = " << num << ", den = " << den << std::endl;
{
if (exponent.info(info_flags::posint))
return power(basis.to_rational(repl), exponent);
+ else if (exponent.info(info_flags::negint))
+ {
+ ex basis_pref = collect_common_factors(basis);
+ if (is_exactly_a<mul>(basis_pref) || is_exactly_a<power>(basis_pref)) {
+ // (A*B)^n will be automagically transformed to A^n*B^n
+ ex t = power(basis_pref, exponent);
+ return t.to_polynomial(repl);
+ }
+ else
+ return power(replace_with_symbol(power(basis, _ex_1), repl), -exponent);
+ }
else
return replace_with_symbol(*this, repl);
}
for (size_t i=0; i<num; i++) {
ex x = e.op(i).to_polynomial(repl);
- if (is_exactly_a<add>(x) || is_exactly_a<mul>(x)) {
+ if (is_exactly_a<add>(x) || is_exactly_a<mul>(x) || is_a<power>(x)) {
ex f = 1;
x = find_common_factor(x, f, repl);
x *= f;
return (new mul(v))->setflag(status_flags::dynallocated);
} else if (is_exactly_a<power>(e)) {
-
- return e.to_polynomial(repl);
+ const ex e_exp(e.op(1));
+ if (e_exp.info(info_flags::integer)) {
+ ex eb = e.op(0).to_polynomial(repl);
+ ex factor_local(_ex1);
+ ex pre_res = find_common_factor(eb, factor_local, repl);
+ factor *= power(factor_local, e_exp);
+ return power(pre_res, e_exp);
+
+ } else
+ return e.to_polynomial(repl);
} else
return e;
* 'a*(b*x+b*y)' to 'a*b*(x+y)'. */
ex collect_common_factors(const ex & e)
{
- if (is_exactly_a<add>(e) || is_exactly_a<mul>(e)) {
+ if (is_exactly_a<add>(e) || is_exactly_a<mul>(e) || is_exactly_a<power>(e)) {
exmap repl;
ex factor = 1;
}
+/** Resultant of two expressions e1,e2 with respect to symbol s.
+ * Method: Compute determinant of Sylvester matrix of e1,e2,s. */
+ex resultant(const ex & e1, const ex & e2, const ex & s)
+{
+ const ex ee1 = e1.expand();
+ const ex ee2 = e2.expand();
+ if (!ee1.info(info_flags::polynomial) ||
+ !ee2.info(info_flags::polynomial))
+ throw(std::runtime_error("resultant(): arguments must be polynomials"));
+
+ const int h1 = ee1.degree(s);
+ const int l1 = ee1.ldegree(s);
+ const int h2 = ee2.degree(s);
+ const int l2 = ee2.ldegree(s);
+
+ const int msize = h1 + h2;
+ matrix m(msize, msize);
+
+ for (int l = h1; l >= l1; --l) {
+ const ex e = ee1.coeff(s, l);
+ for (int k = 0; k < h2; ++k)
+ m(k, k+h1-l) = e;
+ }
+ for (int l = h2; l >= l2; --l) {
+ const ex e = ee2.coeff(s, l);
+ for (int k = 0; k < h1; ++k)
+ m(k+h2, k+h2-l) = e;
+ }
+
+ return m.determinant();
+}
+
+
} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff