* computation, square-free factorization and rational function normalization. */
/*
- * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2015 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
* 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 {
q = rem_i*power(ab, a_exp - 1);
return true;
}
- 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.
+// 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)
* 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
// 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;
- 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]. Optionally also compute the cofactors of a and b,
* defined by a = ca * gcd(a, b) and b = cb * gcd(a, b).
* @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);
- const ex& exp_a = a.op(1);
- if (is_exactly_a<power>(b)) {
- ex pb = b.op(0);
- const ex& exp_b = b.op(1);
- if (p.is_equal(pb)) {
- // a = p^n, b = p^m, gcd = p^min(n, m)
- 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 {
- ex p_co, pb_co;
- ex p_gcd = gcd(p, pb, &p_co, &pb_co, check_args);
- if (p_gcd.is_equal(_ex1)) {
- // a(x) = p(x)^n, b(x) = p_b(x)^m, gcd (p, p_b) = 1 ==>
- // gcd(a,b) = 1
- if (ca)
- *ca = a;
- if (cb)
- *cb = b;
- return _ex1;
- // XXX: do I need to check for p_gcd = -1?
- } else {
- // 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) {
- return power(p_gcd, exp_a)*
- gcd(power(p_co, exp_a), power(p_gcd, exp_b-exp_a)*power(pb_co, exp_b), ca, cb, false);
- } else {
- return power(p_gcd, exp_b)*
- gcd(power(p_gcd, exp_a - exp_b)*power(p_co, exp_a), power(pb_co, exp_b), ca, cb, false);
- }
- } // p_gcd.is_equal(_ex1)
- } // p.is_equal(pb)
-
- } 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;
- }
-
- ex p_co, bpart_co;
- ex p_gcd = gcd(p, b, &p_co, &bpart_co, false);
-
- if (p_gcd.is_equal(_ex1)) {
- // a(x) = p(x)^n, gcd(p, b) = 1 ==> gcd(a, b) = 1
- if (ca)
- *ca = a;
- if (cb)
- *cb = b;
- return _ex1;
- } else {
- // 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))
- return p_gcd*gcd(power(p_gcd, exp_a-1)*power(p_co, exp_a), bpart_co, ca, cb, false);
- }
- } // is_exactly_a<power>(b)
-
- } 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;
- }
-
- ex p_co, apart_co;
- const ex& exp_b(b.op(1));
- ex p_gcd = gcd(a, p, &apart_co, &p_co, false);
- if (p_gcd.is_equal(_ex1)) {
- // b=p(x)^n, gcd(a, p) = 1 ==> gcd(a, b) == 1
- if (ca)
- *ca = a;
- if (cb)
- *cb = b;
- return _ex1;
- } else {
- // there are common factors:
- // a(x) = g(x) A(x), b(x) = g(x)^n B(x)^n ==> gcd = g(x) gcd(g(x)^(n-1) A(x)^n, B(x))
-
- return p_gcd*gcd(apart_co, power(p_gcd, exp_b-1)*power(p_co, exp_b), ca, cb, false);
- } // p_gcd.is_equal(_ex1)
- }
+ 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();
}
}
+ 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) {
+ 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_u * bex_p;
return g;
- } else if (var->deg_b == 0) {
+ } 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);
// 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].
*
* @see ex::normal */
static ex replace_with_symbol(const ex & e, exmap & repl, exmap & rev_lookup)
{
+ // Since the repl contains replaced expressions we should search for them
+ ex e_replaced = e.subs(repl, subs_options::no_pattern);
+
// Expression already replaced? Then return the assigned symbol
- exmap::const_iterator it = rev_lookup.find(e);
+ exmap::const_iterator it = rev_lookup.find(e_replaced);
if (it != rev_lookup.end())
return it->second;
-
+
// Otherwise create new symbol and add to list, taking care that the
// replacement expression doesn't itself contain symbols from repl,
// because subs() is not recursive
ex es = (new symbol)->setflag(status_flags::dynallocated);
- ex e_replaced = e.subs(repl, subs_options::no_pattern);
repl.insert(std::make_pair(es, e_replaced));
rev_lookup.insert(std::make_pair(e_replaced, es));
return es;
* @see basic::to_polynomial */
static ex replace_with_symbol(const ex & e, exmap & repl)
{
+ // Since the repl contains replaced expressions we should search for them
+ ex e_replaced = e.subs(repl, subs_options::no_pattern);
+
// Expression already replaced? Then return the assigned symbol
for (exmap::const_iterator it = repl.begin(); it != repl.end(); ++it)
- if (it->second.is_equal(e))
+ if (it->second.is_equal(e_replaced))
return it->first;
-
+
// Otherwise create new symbol and add to list, taking care that the
// replacement expression doesn't itself contain symbols from repl,
// because subs() is not recursive
ex es = (new symbol)->setflag(status_flags::dynallocated);
- ex e_replaced = e.subs(repl, subs_options::no_pattern);
repl.insert(std::make_pair(es, e_replaced));
return es;
}
git://www.ginac.de / ginac.git / blobdiff