* This file implements several functions that work on univariate and
* multivariate polynomials and rational functions.
* These functions include polynomial quotient and remainder, GCD and LCM
- * computation, square-free factorization and rational function normalization.
+ * computation, square-free factorization and rational function normalization. */
- *
- * GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+/*
+ * GiNaC Copyright (C) 1999-2001 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
-#include <stdexcept>
+#include <algorithm>
+#include <map>
-#include "ginac.h"
+#include "normal.h"
+#include "basic.h"
+#include "ex.h"
+#include "add.h"
+#include "constant.h"
+#include "expairseq.h"
+#include "fail.h"
+#include "inifcns.h"
+#include "lst.h"
+#include "mul.h"
+#include "numeric.h"
+#include "power.h"
+#include "relational.h"
+#include "matrix.h"
+#include "pseries.h"
+#include "symbol.h"
+#include "utils.h"
+
+namespace GiNaC {
// If comparing expressions (ex::compare()) is fast, you can set this to 1.
// Some routines like quo(), rem() and gcd() will then return a quick answer
#define FAST_COMPARE 1
// Set this if you want divide_in_z() to use remembering
-#define USE_REMEMBER 1
+#define USE_REMEMBER 0
+
+// Set this if you want divide_in_z() to use trial division followed by
+// polynomial interpolation (always slower except for completely dense
+// polynomials)
+#define USE_TRIAL_DIVISION 0
+
+// Set this to enable some statistical output for the GCD routines
+#define STATISTICS 0
+
+
+#if STATISTICS
+// Statistics variables
+static int gcd_called = 0;
+static int sr_gcd_called = 0;
+static int heur_gcd_called = 0;
+static int heur_gcd_failed = 0;
+
+// Print statistics at end of program
+static struct _stat_print {
+ _stat_print() {}
+ ~_stat_print() {
+ std::cout << "gcd() called " << gcd_called << " times\n";
+ std::cout << "sr_gcd() called " << sr_gcd_called << " times\n";
+ std::cout << "heur_gcd() called " << heur_gcd_called << " times\n";
+ std::cout << "heur_gcd() failed " << heur_gcd_failed << " times\n";
+ }
+} stat_print;
+#endif
/** Return pointer to first symbol found in expression. Due to GiNaC´s
* @param e expression to search
* @param x pointer to first symbol found (returned)
* @return "false" if no symbol was found, "true" otherwise */
-
static bool get_first_symbol(const ex &e, const symbol *&x)
{
- if (is_ex_exactly_of_type(e, symbol)) {
- x = static_cast<symbol *>(e.bp);
- return true;
- } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
- for (int i=0; i<e.nops(); i++)
- if (get_first_symbol(e.op(i), x))
- return true;
- } else if (is_ex_exactly_of_type(e, power)) {
- if (get_first_symbol(e.op(0), x))
- return true;
- }
- return false;
+ if (is_ex_exactly_of_type(e, symbol)) {
+ x = &ex_to<symbol>(e);
+ return true;
+ } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
+ for (unsigned i=0; i<e.nops(); i++)
+ if (get_first_symbol(e.op(i), x))
+ return true;
+ } else if (is_ex_exactly_of_type(e, power)) {
+ if (get_first_symbol(e.op(0), x))
+ return true;
+ }
+ return false;
}
* Statistical information about symbols in polynomials
*/
-#include <algorithm>
-
/** This structure holds information about the highest and lowest degrees
* in which a symbol appears in two multivariate polynomials "a" and "b".
* A vector of these structures with information about all symbols in
*
* @see get_symbol_stats */
struct sym_desc {
- /** Pointer to symbol */
- const symbol *sym;
+ /** Pointer to symbol */
+ const symbol *sym;
+
+ /** Highest degree of symbol in polynomial "a" */
+ int deg_a;
- /** Highest degree of symbol in polynomial "a" */
- int deg_a;
+ /** Highest degree of symbol in polynomial "b" */
+ int deg_b;
- /** Highest degree of symbol in polynomial "b" */
- int deg_b;
+ /** Lowest degree of symbol in polynomial "a" */
+ int ldeg_a;
- /** Lowest degree of symbol in polynomial "a" */
- int ldeg_a;
+ /** Lowest degree of symbol in polynomial "b" */
+ int ldeg_b;
- /** Lowest degree of symbol in polynomial "b" */
- int ldeg_b;
+ /** Maximum of deg_a and deg_b (Used for sorting) */
+ int max_deg;
- /** Minimum of ldeg_a and ldeg_b (Used for sorting) */
- int min_deg;
+ /** Maximum number of terms of leading coefficient of symbol in both polynomials */
+ int max_lcnops;
- /** Commparison operator for sorting */
- bool operator<(const sym_desc &x) const {return min_deg < x.min_deg;}
+ /** Commparison operator for sorting */
+ bool operator<(const sym_desc &x) const
+ {
+ if (max_deg == x.max_deg)
+ return max_lcnops < x.max_lcnops;
+ else
+ return max_deg < x.max_deg;
+ }
};
// Vector of sym_desc structures
-typedef vector<sym_desc> sym_desc_vec;
+typedef std::vector<sym_desc> sym_desc_vec;
// Add symbol the sym_desc_vec (used internally by get_symbol_stats())
static void add_symbol(const symbol *s, sym_desc_vec &v)
{
- sym_desc_vec::iterator it = v.begin(), itend = v.end();
- while (it != itend) {
- if (it->sym->compare(*s) == 0) // If it's already in there, don't add it a second time
- return;
- it++;
- }
- sym_desc d;
- d.sym = s;
- v.push_back(d);
+ sym_desc_vec::const_iterator it = v.begin(), itend = v.end();
+ while (it != itend) {
+ if (it->sym->compare(*s) == 0) // If it's already in there, don't add it a second time
+ return;
+ ++it;
+ }
+ sym_desc d;
+ d.sym = s;
+ v.push_back(d);
}
// Collect all symbols of an expression (used internally by get_symbol_stats())
static void collect_symbols(const ex &e, sym_desc_vec &v)
{
- if (is_ex_exactly_of_type(e, symbol)) {
- add_symbol(static_cast<symbol *>(e.bp), v);
- } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
- for (int i=0; i<e.nops(); i++)
- collect_symbols(e.op(i), v);
- } else if (is_ex_exactly_of_type(e, power)) {
- collect_symbols(e.op(0), v);
- }
+ if (is_ex_exactly_of_type(e, symbol)) {
+ add_symbol(&ex_to<symbol>(e), v);
+ } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
+ for (unsigned i=0; i<e.nops(); i++)
+ collect_symbols(e.op(i), v);
+ } else if (is_ex_exactly_of_type(e, power)) {
+ collect_symbols(e.op(0), v);
+ }
}
/** Collect statistical information about symbols in polynomials.
* @param a first multivariate polynomial
* @param b second multivariate polynomial
* @param v vector of sym_desc structs (filled in) */
-
static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v)
{
- collect_symbols(a.eval(), v); // eval() to expand assigned symbols
- collect_symbols(b.eval(), v);
- sym_desc_vec::iterator it = v.begin(), itend = v.end();
- while (it != itend) {
- int deg_a = a.degree(*(it->sym));
- int deg_b = b.degree(*(it->sym));
- it->deg_a = deg_a;
- it->deg_b = deg_b;
- it->min_deg = min(deg_a, deg_b);
- it->ldeg_a = a.ldegree(*(it->sym));
- it->ldeg_b = b.ldegree(*(it->sym));
- it++;
- }
- sort(v.begin(), v.end());
+ collect_symbols(a.eval(), v); // eval() to expand assigned symbols
+ collect_symbols(b.eval(), v);
+ sym_desc_vec::iterator it = v.begin(), itend = v.end();
+ while (it != itend) {
+ int deg_a = a.degree(*(it->sym));
+ int deg_b = b.degree(*(it->sym));
+ it->deg_a = deg_a;
+ it->deg_b = deg_b;
+ it->max_deg = std::max(deg_a, deg_b);
+ it->max_lcnops = std::max(a.lcoeff(*(it->sym)).nops(), b.lcoeff(*(it->sym)).nops());
+ it->ldeg_a = a.ldegree(*(it->sym));
+ it->ldeg_b = b.ldegree(*(it->sym));
+ ++it;
+ }
+ std::sort(v.begin(), v.end());
+#if 0
+ std::clog << "Symbols:\n";
+ it = v.begin(); itend = v.end();
+ while (it != itend) {
+ std::clog << " " << *it->sym << ": deg_a=" << it->deg_a << ", deg_b=" << it->deg_b << ", ldeg_a=" << it->ldeg_a << ", ldeg_b=" << it->ldeg_b << ", max_deg=" << it->max_deg << ", max_lcnops=" << it->max_lcnops << endl;
+ std::clog << " lcoeff_a=" << a.lcoeff(*(it->sym)) << ", lcoeff_b=" << b.lcoeff(*(it->sym)) << endl;
+ ++it;
+ }
+#endif
}
// expression recursively (used internally by lcm_of_coefficients_denominators())
static numeric lcmcoeff(const ex &e, const numeric &l)
{
- if (e.info(info_flags::rational))
- return lcm(ex_to_numeric(e).denom(), l);
- else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) {
- numeric c = numONE();
- for (int i=0; i<e.nops(); i++) {
- c = lcmcoeff(e.op(i), c);
- }
- return lcm(c, l);
- } else if (is_ex_exactly_of_type(e, power))
- return lcmcoeff(e.op(0), l);
- return l;
+ if (e.info(info_flags::rational))
+ return lcm(ex_to<numeric>(e).denom(), l);
+ else if (is_ex_exactly_of_type(e, add)) {
+ numeric c = _num1;
+ for (unsigned i=0; i<e.nops(); i++)
+ c = lcmcoeff(e.op(i), c);
+ return lcm(c, l);
+ } else if (is_ex_exactly_of_type(e, mul)) {
+ numeric c = _num1;
+ for (unsigned i=0; i<e.nops(); i++)
+ c *= lcmcoeff(e.op(i), _num1);
+ return lcm(c, l);
+ } else if (is_ex_exactly_of_type(e, power)) {
+ if (is_ex_exactly_of_type(e.op(0), symbol))
+ return l;
+ else
+ return pow(lcmcoeff(e.op(0), l), ex_to<numeric>(e.op(1)));
+ }
+ return l;
}
/** Compute LCM of denominators of coefficients of a polynomial.
* Given a polynomial with rational coefficients, this function computes
* the LCM of the denominators of all coefficients. This can be used
- * To bring a polynomial from Q[X] to Z[X].
+ * to bring a polynomial from Q[X] to Z[X].
*
- * @param e multivariate polynomial
+ * @param e multivariate polynomial (need not be expanded)
* @return LCM of denominators of coefficients */
-
static numeric lcm_of_coefficients_denominators(const ex &e)
{
- return lcmcoeff(e.expand(), numONE());
+ return lcmcoeff(e, _num1);
+}
+
+/** Bring polynomial from Q[X] to Z[X] by multiplying in the previously
+ * determined LCM of the coefficient's denominators.
+ *
+ * @param e multivariate polynomial (need not be expanded)
+ * @param lcm LCM to multiply in */
+static ex multiply_lcm(const ex &e, const numeric &lcm)
+{
+ if (is_ex_exactly_of_type(e, mul)) {
+ unsigned num = e.nops();
+ exvector v; v.reserve(num + 1);
+ numeric lcm_accum = _num1;
+ for (unsigned i=0; i<e.nops(); i++) {
+ numeric op_lcm = lcmcoeff(e.op(i), _num1);
+ v.push_back(multiply_lcm(e.op(i), op_lcm));
+ lcm_accum *= op_lcm;
+ }
+ v.push_back(lcm / lcm_accum);
+ return (new mul(v))->setflag(status_flags::dynallocated);
+ } else if (is_ex_exactly_of_type(e, add)) {
+ unsigned num = e.nops();
+ exvector v; v.reserve(num);
+ for (unsigned i=0; i<num; i++)
+ v.push_back(multiply_lcm(e.op(i), lcm));
+ return (new add(v))->setflag(status_flags::dynallocated);
+ } else if (is_ex_exactly_of_type(e, power)) {
+ if (is_ex_exactly_of_type(e.op(0), symbol))
+ return e * lcm;
+ else
+ return pow(multiply_lcm(e.op(0), lcm.power(ex_to<numeric>(e.op(1)).inverse())), e.op(1));
+ } else
+ return e * lcm;
}
*
* @param e expanded polynomial
* @return integer content */
-
numeric ex::integer_content(void) const
{
- ASSERT(bp!=0);
- return bp->integer_content();
+ GINAC_ASSERT(bp!=0);
+ return bp->integer_content();
}
numeric basic::integer_content(void) const
{
- return numONE();
+ return _num1;
}
numeric numeric::integer_content(void) const
{
- return abs(*this);
+ return abs(*this);
}
numeric add::integer_content(void) const
{
- epvector::const_iterator it = seq.begin();
- epvector::const_iterator itend = seq.end();
- numeric c = numZERO();
- while (it != itend) {
- ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
- ASSERT(is_ex_exactly_of_type(it->coeff,numeric));
- c = gcd(ex_to_numeric(it->coeff), c);
- it++;
- }
- ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
- c = gcd(ex_to_numeric(overall_coeff),c);
- return c;
+ epvector::const_iterator it = seq.begin();
+ epvector::const_iterator itend = seq.end();
+ numeric c = _num0;
+ 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);
+ it++;
+ }
+ GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+ c = gcd(ex_to<numeric>(overall_coeff),c);
+ return c;
}
numeric mul::integer_content(void) const
{
-#ifdef DOASSERT
- epvector::const_iterator it = seq.begin();
- epvector::const_iterator itend = seq.end();
- while (it != itend) {
- ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
- ++it;
- }
-#endif // def DOASSERT
- ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
- return abs(ex_to_numeric(overall_coeff));
+#ifdef DO_GINAC_ASSERT
+ epvector::const_iterator it = seq.begin();
+ epvector::const_iterator itend = seq.end();
+ while (it != itend) {
+ GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
+ ++it;
+ }
+#endif // def DO_GINAC_ASSERT
+ GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+ return abs(ex_to<numeric>(overall_coeff));
}
* @param check_args check whether a and b are polynomials with rational
* coefficients (defaults to "true")
* @return quotient of a and b in Q[x] */
-
ex quo(const ex &a, const ex &b, const symbol &x, bool check_args)
{
- if (b.is_zero())
- throw(std::overflow_error("quo: division by zero"));
- if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
- return a / b;
+ if (b.is_zero())
+ throw(std::overflow_error("quo: division by zero"));
+ if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
+ return a / b;
#if FAST_COMPARE
- if (a.is_equal(b))
- return exONE();
+ if (a.is_equal(b))
+ return _ex1;
#endif
- if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
- throw(std::invalid_argument("quo: arguments must be polynomials over the rationals"));
-
- // Polynomial long division
- ex q = exZERO();
- ex r = a.expand();
- if (r.is_zero())
- return r;
- int bdeg = b.degree(x);
- int rdeg = r.degree(x);
- ex blcoeff = b.expand().coeff(x, bdeg);
- bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
- while (rdeg >= bdeg) {
- ex term, rcoeff = r.coeff(x, rdeg);
- if (blcoeff_is_numeric)
- term = rcoeff / blcoeff;
- else {
- if (!divide(rcoeff, blcoeff, term, false))
- return *new ex(fail());
- }
- term *= power(x, rdeg - bdeg);
- q += term;
- r -= (term * b).expand();
- if (r.is_zero())
- break;
- rdeg = r.degree(x);
- }
- return q;
+ if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
+ throw(std::invalid_argument("quo: arguments must be polynomials over the rationals"));
+
+ // Polynomial long division
+ ex r = a.expand();
+ if (r.is_zero())
+ return r;
+ int bdeg = b.degree(x);
+ int rdeg = r.degree(x);
+ ex blcoeff = b.expand().coeff(x, bdeg);
+ bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
+ exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
+ while (rdeg >= bdeg) {
+ ex term, rcoeff = r.coeff(x, rdeg);
+ if (blcoeff_is_numeric)
+ term = rcoeff / blcoeff;
+ else {
+ if (!divide(rcoeff, blcoeff, term, false))
+ return (new fail())->setflag(status_flags::dynallocated);
+ }
+ term *= power(x, rdeg - bdeg);
+ v.push_back(term);
+ r -= (term * b).expand();
+ if (r.is_zero())
+ break;
+ rdeg = r.degree(x);
+ }
+ return (new add(v))->setflag(status_flags::dynallocated);
}
* @param check_args check whether a and b are polynomials with rational
* coefficients (defaults to "true")
* @return remainder of a(x) and b(x) in Q[x] */
-
ex rem(const ex &a, const ex &b, const symbol &x, bool check_args)
{
- if (b.is_zero())
- throw(std::overflow_error("rem: division by zero"));
- if (is_ex_exactly_of_type(a, numeric)) {
- if (is_ex_exactly_of_type(b, numeric))
- return exZERO();
- else
- return b;
- }
+ if (b.is_zero())
+ throw(std::overflow_error("rem: division by zero"));
+ if (is_ex_exactly_of_type(a, numeric)) {
+ if (is_ex_exactly_of_type(b, numeric))
+ return _ex0;
+ else
+ return a;
+ }
#if FAST_COMPARE
- if (a.is_equal(b))
- return exZERO();
+ if (a.is_equal(b))
+ return _ex0;
#endif
- if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
- throw(std::invalid_argument("rem: arguments must be polynomials over the rationals"));
-
- // Polynomial long division
- ex r = a.expand();
- if (r.is_zero())
- return r;
- int bdeg = b.degree(x);
- int rdeg = r.degree(x);
- ex blcoeff = b.expand().coeff(x, bdeg);
- bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
- while (rdeg >= bdeg) {
- ex term, rcoeff = r.coeff(x, rdeg);
- if (blcoeff_is_numeric)
- term = rcoeff / blcoeff;
- else {
- if (!divide(rcoeff, blcoeff, term, false))
- return *new ex(fail());
- }
- term *= power(x, rdeg - bdeg);
- r -= (term * b).expand();
- if (r.is_zero())
- break;
- rdeg = r.degree(x);
- }
- return r;
+ if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
+ throw(std::invalid_argument("rem: arguments must be polynomials over the rationals"));
+
+ // Polynomial long division
+ ex r = a.expand();
+ if (r.is_zero())
+ return r;
+ int bdeg = b.degree(x);
+ int rdeg = r.degree(x);
+ ex blcoeff = b.expand().coeff(x, bdeg);
+ bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
+ while (rdeg >= bdeg) {
+ ex term, rcoeff = r.coeff(x, rdeg);
+ if (blcoeff_is_numeric)
+ term = rcoeff / blcoeff;
+ else {
+ if (!divide(rcoeff, blcoeff, term, false))
+ return (new fail())->setflag(status_flags::dynallocated);
+ }
+ term *= power(x, rdeg - bdeg);
+ r -= (term * b).expand();
+ if (r.is_zero())
+ break;
+ rdeg = r.degree(x);
+ }
+ return r;
+}
+
+
+/** Decompose rational function a(x)=N(x)/D(x) into P(x)+n(x)/D(x)
+ * with degree(n, x) < degree(D, x).
+ *
+ * @param a rational function in x
+ * @param x a is a function of x
+ * @return decomposed function. */
+ex decomp_rational(const ex &a, const symbol &x)
+{
+ ex nd = numer_denom(a);
+ ex numer = nd.op(0), denom = nd.op(1);
+ ex q = quo(numer, denom, x);
+ if (is_ex_exactly_of_type(q, fail))
+ return a;
+ else
+ return q + rem(numer, denom, x) / denom;
}
* @param check_args check whether a and b are polynomials with rational
* coefficients (defaults to "true")
* @return pseudo-remainder of a(x) and b(x) in Z[x] */
-
ex prem(const ex &a, const ex &b, const symbol &x, bool check_args)
{
- if (b.is_zero())
- throw(std::overflow_error("prem: division by zero"));
- if (is_ex_exactly_of_type(a, numeric)) {
- if (is_ex_exactly_of_type(b, numeric))
- return exZERO();
- else
- return b;
- }
- if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
- throw(std::invalid_argument("prem: arguments must be polynomials over the rationals"));
-
- // Polynomial long division
- ex r = a.expand();
- ex eb = b.expand();
- int rdeg = r.degree(x);
- int bdeg = eb.degree(x);
- ex blcoeff;
- if (bdeg <= rdeg) {
- blcoeff = eb.coeff(x, bdeg);
- if (bdeg == 0)
- eb = exZERO();
- else
- eb -= blcoeff * power(x, bdeg);
- } else
- blcoeff = exONE();
-
- int delta = rdeg - bdeg + 1, i = 0;
- while (rdeg >= bdeg && !r.is_zero()) {
- ex rlcoeff = r.coeff(x, rdeg);
- ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
- if (rdeg == 0)
- r = exZERO();
- else
- r -= rlcoeff * power(x, rdeg);
- r = (blcoeff * r).expand() - term;
- rdeg = r.degree(x);
- i++;
- }
- return power(blcoeff, delta - i) * r;
+ if (b.is_zero())
+ throw(std::overflow_error("prem: division by zero"));
+ if (is_ex_exactly_of_type(a, numeric)) {
+ if (is_ex_exactly_of_type(b, numeric))
+ return _ex0;
+ else
+ return b;
+ }
+ if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
+ throw(std::invalid_argument("prem: arguments must be polynomials over the rationals"));
+
+ // Polynomial long division
+ ex r = a.expand();
+ ex eb = b.expand();
+ int rdeg = r.degree(x);
+ int bdeg = eb.degree(x);
+ ex blcoeff;
+ if (bdeg <= rdeg) {
+ blcoeff = eb.coeff(x, bdeg);
+ if (bdeg == 0)
+ eb = _ex0;
+ else
+ eb -= blcoeff * power(x, bdeg);
+ } else
+ blcoeff = _ex1;
+
+ int delta = rdeg - bdeg + 1, i = 0;
+ while (rdeg >= bdeg && !r.is_zero()) {
+ ex rlcoeff = r.coeff(x, rdeg);
+ ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
+ if (rdeg == 0)
+ r = _ex0;
+ else
+ r -= rlcoeff * power(x, rdeg);
+ r = (blcoeff * r).expand() - term;
+ rdeg = r.degree(x);
+ i++;
+ }
+ return power(blcoeff, delta - i) * r;
+}
+
+
+/** Sparse pseudo-remainder of polynomials a(x) and b(x) in Z[x].
+ *
+ * @param a first polynomial in x (dividend)
+ * @param b second polynomial in x (divisor)
+ * @param x a and b are polynomials in x
+ * @param check_args check whether a and b are polynomials with rational
+ * coefficients (defaults to "true")
+ * @return sparse pseudo-remainder of a(x) and b(x) in Z[x] */
+ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args)
+{
+ if (b.is_zero())
+ throw(std::overflow_error("prem: division by zero"));
+ if (is_ex_exactly_of_type(a, numeric)) {
+ if (is_ex_exactly_of_type(b, numeric))
+ return _ex0;
+ else
+ return b;
+ }
+ if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
+ throw(std::invalid_argument("prem: arguments must be polynomials over the rationals"));
+
+ // Polynomial long division
+ ex r = a.expand();
+ ex eb = b.expand();
+ int rdeg = r.degree(x);
+ int bdeg = eb.degree(x);
+ ex blcoeff;
+ if (bdeg <= rdeg) {
+ blcoeff = eb.coeff(x, bdeg);
+ if (bdeg == 0)
+ eb = _ex0;
+ else
+ eb -= blcoeff * power(x, bdeg);
+ } else
+ blcoeff = _ex1;
+
+ while (rdeg >= bdeg && !r.is_zero()) {
+ ex rlcoeff = r.coeff(x, rdeg);
+ ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand();
+ if (rdeg == 0)
+ r = _ex0;
+ else
+ r -= rlcoeff * power(x, rdeg);
+ r = (blcoeff * r).expand() - term;
+ rdeg = r.degree(x);
+ }
+ return r;
}
* @param check_args check whether a and b are polynomials with rational
* coefficients (defaults to "true")
* @return "true" when exact division succeeds (quotient returned in q),
- * "false" otherwise */
-
+ * "false" otherwise (q left untouched) */
bool divide(const ex &a, const ex &b, ex &q, bool check_args)
{
- q = exZERO();
- if (b.is_zero())
- throw(std::overflow_error("divide: division by zero"));
- if (is_ex_exactly_of_type(b, numeric)) {
- q = a / b;
- return true;
- } else if (is_ex_exactly_of_type(a, numeric))
- return false;
+ if (b.is_zero())
+ throw(std::overflow_error("divide: division by zero"));
+ if (a.is_zero()) {
+ q = _ex0;
+ return true;
+ }
+ if (is_ex_exactly_of_type(b, numeric)) {
+ q = a / b;
+ return true;
+ } else if (is_ex_exactly_of_type(a, numeric))
+ return false;
#if FAST_COMPARE
- if (a.is_equal(b)) {
- q = exONE();
- return true;
- }
+ if (a.is_equal(b)) {
+ q = _ex1;
+ return true;
+ }
#endif
- if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
- throw(std::invalid_argument("divide: arguments must be polynomials over the rationals"));
-
- // Find first symbol
- const symbol *x;
- if (!get_first_symbol(a, x) && !get_first_symbol(b, x))
- throw(std::invalid_argument("invalid expression in divide()"));
-
- // Polynomial long division (recursive)
- ex r = a.expand();
- if (r.is_zero())
- return true;
- int bdeg = b.degree(*x);
- int rdeg = r.degree(*x);
- ex blcoeff = b.expand().coeff(*x, bdeg);
- bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
- while (rdeg >= bdeg) {
- ex term, rcoeff = r.coeff(*x, rdeg);
- if (blcoeff_is_numeric)
- term = rcoeff / blcoeff;
- else
- if (!divide(rcoeff, blcoeff, term, false))
- return false;
- term *= power(*x, rdeg - bdeg);
- q += term;
- r -= (term * b).expand();
- if (r.is_zero())
- return true;
- rdeg = r.degree(*x);
- }
- return false;
+ if (check_args && (!a.info(info_flags::rational_polynomial) ||
+ !b.info(info_flags::rational_polynomial)))
+ throw(std::invalid_argument("divide: arguments must be polynomials over the rationals"));
+
+ // Find first symbol
+ const symbol *x;
+ if (!get_first_symbol(a, x) && !get_first_symbol(b, x))
+ throw(std::invalid_argument("invalid expression in divide()"));
+
+ // Polynomial long division (recursive)
+ ex r = a.expand();
+ if (r.is_zero()) {
+ q = _ex0;
+ return true;
+ }
+ int bdeg = b.degree(*x);
+ int rdeg = r.degree(*x);
+ ex blcoeff = b.expand().coeff(*x, bdeg);
+ bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric);
+ exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
+ while (rdeg >= bdeg) {
+ ex term, rcoeff = r.coeff(*x, rdeg);
+ if (blcoeff_is_numeric)
+ term = rcoeff / blcoeff;
+ else
+ if (!divide(rcoeff, blcoeff, term, false))
+ return false;
+ term *= power(*x, rdeg - bdeg);
+ v.push_back(term);
+ r -= (term * b).expand();
+ if (r.is_zero()) {
+ q = (new add(v))->setflag(status_flags::dynallocated);
+ return true;
+ }
+ rdeg = r.degree(*x);
+ }
+ return false;
}
* Remembering
*/
-#include <map>
-
-typedef pair<ex, ex> ex2;
-typedef pair<ex, bool> exbool;
+typedef std::pair<ex, ex> ex2;
+typedef std::pair<ex, bool> exbool;
struct ex2_less {
- bool operator() (const ex2 p, const ex2 q) const
- {
- return p.first.compare(q.first) < 0 || (!(q.first.compare(p.first) < 0) && p.second.compare(q.second) < 0);
- }
+ bool operator() (const ex2 &p, const ex2 &q) const
+ {
+ int cmp = p.first.compare(q.first);
+ return ((cmp<0) || (!(cmp>0) && p.second.compare(q.second)<0));
+ }
};
-typedef map<ex2, exbool, ex2_less> ex2_exbool_remember;
+typedef std::map<ex2, exbool, ex2_less> ex2_exbool_remember;
#endif
* @see get_symbol_stats, heur_gcd */
static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_iterator var)
{
- q = exZERO();
- if (b.is_zero())
- throw(std::overflow_error("divide_in_z: division by zero"));
- if (b.is_equal(exONE())) {
- q = a;
- return true;
- }
- if (is_ex_exactly_of_type(a, numeric)) {
- if (is_ex_exactly_of_type(b, numeric)) {
- q = a / b;
- return q.info(info_flags::integer);
- } else
- return false;
- }
+ q = _ex0;
+ if (b.is_zero())
+ throw(std::overflow_error("divide_in_z: division by zero"));
+ if (b.is_equal(_ex1)) {
+ q = a;
+ return true;
+ }
+ if (is_ex_exactly_of_type(a, numeric)) {
+ if (is_ex_exactly_of_type(b, numeric)) {
+ q = a / b;
+ return q.info(info_flags::integer);
+ } else
+ return false;
+ }
#if FAST_COMPARE
- if (a.is_equal(b)) {
- q = exONE();
- return true;
- }
+ if (a.is_equal(b)) {
+ q = _ex1;
+ return true;
+ }
#endif
#if USE_REMEMBER
- // Remembering
- static ex2_exbool_remember dr_remember;
- ex2_exbool_remember::const_iterator remembered = dr_remember.find(ex2(a, b));
- if (remembered != dr_remember.end()) {
- q = remembered->second.first;
- return remembered->second.second;
- }
+ // Remembering
+ static ex2_exbool_remember dr_remember;
+ ex2_exbool_remember::const_iterator remembered = dr_remember.find(ex2(a, b));
+ if (remembered != dr_remember.end()) {
+ q = remembered->second.first;
+ return remembered->second.second;
+ }
#endif
- // Main symbol
- const symbol *x = var->sym;
+ // Main symbol
+ const symbol *x = var->sym;
+
+ // Compare degrees
+ int adeg = a.degree(*x), bdeg = b.degree(*x);
+ if (bdeg > adeg)
+ return false;
+
+#if USE_TRIAL_DIVISION
+
+ // Trial division with polynomial interpolation
+ int i, k;
+
+ // Compute values at evaluation points 0..adeg
+ vector<numeric> alpha; alpha.reserve(adeg + 1);
+ exvector u; u.reserve(adeg + 1);
+ numeric point = _num0;
+ ex c;
+ for (i=0; i<=adeg; i++) {
+ ex bs = b.subs(*x == point);
+ while (bs.is_zero()) {
+ point += _num1;
+ bs = b.subs(*x == point);
+ }
+ if (!divide_in_z(a.subs(*x == point), bs, c, var+1))
+ return false;
+ alpha.push_back(point);
+ u.push_back(c);
+ point += _num1;
+ }
- // Compare degrees
- int adeg = a.degree(*x), bdeg = b.degree(*x);
- if (bdeg > adeg)
- return false;
+ // Compute inverses
+ vector<numeric> rcp; rcp.reserve(adeg + 1);
+ rcp.push_back(_num0);
+ for (k=1; k<=adeg; k++) {
+ numeric product = alpha[k] - alpha[0];
+ for (i=1; i<k; i++)
+ product *= alpha[k] - alpha[i];
+ rcp.push_back(product.inverse());
+ }
-#if 1
+ // Compute Newton coefficients
+ exvector v; v.reserve(adeg + 1);
+ v.push_back(u[0]);
+ for (k=1; k<=adeg; k++) {
+ ex temp = v[k - 1];
+ for (i=k-2; i>=0; i--)
+ temp = temp * (alpha[k] - alpha[i]) + v[i];
+ v.push_back((u[k] - temp) * rcp[k]);
+ }
+
+ // Convert from Newton form to standard form
+ c = v[adeg];
+ for (k=adeg-1; k>=0; k--)
+ c = c * (*x - alpha[k]) + v[k];
+
+ if (c.degree(*x) == (adeg - bdeg)) {
+ q = c.expand();
+ return true;
+ } else
+ return false;
- // Polynomial long division (recursive)
- ex r = a.expand();
- if (r.is_zero())
- return true;
- int rdeg = adeg;
- ex eb = b.expand();
- ex blcoeff = eb.coeff(*x, bdeg);
- while (rdeg >= bdeg) {
- ex term, rcoeff = r.coeff(*x, rdeg);
- if (!divide_in_z(rcoeff, blcoeff, term, var+1))
- break;
- term = (term * power(*x, rdeg - bdeg)).expand();
- q += term;
- r -= (term * eb).expand();
- if (r.is_zero()) {
+#else
+
+ // Polynomial long division (recursive)
+ ex r = a.expand();
+ if (r.is_zero())
+ return true;
+ int rdeg = adeg;
+ ex eb = b.expand();
+ ex blcoeff = eb.coeff(*x, bdeg);
+ exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0));
+ while (rdeg >= bdeg) {
+ ex term, rcoeff = r.coeff(*x, rdeg);
+ if (!divide_in_z(rcoeff, blcoeff, term, var+1))
+ break;
+ term = (term * power(*x, rdeg - bdeg)).expand();
+ v.push_back(term);
+ r -= (term * eb).expand();
+ if (r.is_zero()) {
+ q = (new add(v))->setflag(status_flags::dynallocated);
#if USE_REMEMBER
- dr_remember[ex2(a, b)] = exbool(q, true);
+ dr_remember[ex2(a, b)] = exbool(q, true);
#endif
- return true;
- }
- rdeg = r.degree(*x);
- }
+ return true;
+ }
+ rdeg = r.degree(*x);
+ }
#if USE_REMEMBER
- dr_remember[ex2(a, b)] = exbool(q, false);
+ dr_remember[ex2(a, b)] = exbool(q, false);
#endif
- return false;
-
-#else
+ return false;
- // Trial division using polynomial interpolation
- int i, k;
-
- // Compute values at evaluation points 0..adeg
- vector<numeric> alpha; alpha.reserve(adeg + 1);
- exvector u; u.reserve(adeg + 1);
- numeric point = numZERO();
- ex c;
- for (i=0; i<=adeg; i++) {
- ex bs = b.subs(*x == point);
- while (bs.is_zero()) {
- point += numONE();
- bs = b.subs(*x == point);
- }
- if (!divide_in_z(a.subs(*x == point), bs, c, var+1))
- return false;
- alpha.push_back(point);
- u.push_back(c);
- point += numONE();
- }
-
- // Compute inverses
- vector<numeric> rcp; rcp.reserve(adeg + 1);
- rcp.push_back(0);
- for (k=1; k<=adeg; k++) {
- numeric product = alpha[k] - alpha[0];
- for (i=1; i<k; i++)
- product *= alpha[k] - alpha[i];
- rcp.push_back(product.inverse());
- }
-
- // Compute Newton coefficients
- exvector v; v.reserve(adeg + 1);
- v.push_back(u[0]);
- for (k=1; k<=adeg; k++) {
- ex temp = v[k - 1];
- for (i=k-2; i>=0; i--)
- temp = temp * (alpha[k] - alpha[i]) + v[i];
- v.push_back((u[k] - temp) * rcp[k]);
- }
-
- // Convert from Newton form to standard form
- c = v[adeg];
- for (k=adeg-1; k>=0; k--)
- c = c * (*x - alpha[k]) + v[k];
-
- if (c.degree(*x) == (adeg - bdeg)) {
- q = c.expand();
- return true;
- } else
- return false;
#endif
}
* @see ex::content, ex::primpart */
ex ex::unit(const symbol &x) const
{
- ex c = expand().lcoeff(x);
- if (is_ex_exactly_of_type(c, numeric))
- return c < exZERO() ? exMINUSONE() : exONE();
- else {
- const symbol *y;
- if (get_first_symbol(c, y))
- return c.unit(*y);
- else
- throw(std::invalid_argument("invalid expression in unit()"));
- }
+ ex c = expand().lcoeff(x);
+ if (is_ex_exactly_of_type(c, numeric))
+ return c < _ex0 ? _ex_1 : _ex1;
+ else {
+ const symbol *y;
+ if (get_first_symbol(c, y))
+ return c.unit(*y);
+ else
+ throw(std::invalid_argument("invalid expression in unit()"));
+ }
}
* @see ex::unit, ex::primpart */
ex ex::content(const symbol &x) const
{
- if (is_zero())
- return exZERO();
- if (is_ex_exactly_of_type(*this, numeric))
- return info(info_flags::negative) ? -*this : *this;
- ex e = expand();
- if (e.is_zero())
- return exZERO();
-
- // First, try the integer content
- ex c = e.integer_content();
- ex r = e / c;
- ex lcoeff = r.lcoeff(x);
- if (lcoeff.info(info_flags::integer))
- return c;
-
- // GCD of all coefficients
- int deg = e.degree(x);
- int ldeg = e.ldegree(x);
- if (deg == ldeg)
- return e.lcoeff(x) / e.unit(x);
- c = exZERO();
- for (int i=ldeg; i<=deg; i++)
- c = gcd(e.coeff(x, i), c, NULL, NULL, false);
- return c;
+ if (is_zero())
+ return _ex0;
+ if (is_ex_exactly_of_type(*this, numeric))
+ return info(info_flags::negative) ? -*this : *this;
+ ex e = expand();
+ if (e.is_zero())
+ return _ex0;
+
+ // First, try the integer content
+ ex c = e.integer_content();
+ ex r = e / c;
+ ex lcoeff = r.lcoeff(x);
+ if (lcoeff.info(info_flags::integer))
+ return c;
+
+ // GCD of all coefficients
+ int deg = e.degree(x);
+ int ldeg = e.ldegree(x);
+ if (deg == ldeg)
+ return e.lcoeff(x) / e.unit(x);
+ c = _ex0;
+ for (int i=ldeg; i<=deg; i++)
+ c = gcd(e.coeff(x, i), c, NULL, NULL, false);
+ return c;
}
* @see ex::unit, ex::content */
ex ex::primpart(const symbol &x) const
{
- if (is_zero())
- return exZERO();
- if (is_ex_exactly_of_type(*this, numeric))
- return exONE();
-
- ex c = content(x);
- if (c.is_zero())
- return exZERO();
- ex u = unit(x);
- if (is_ex_exactly_of_type(c, numeric))
- return *this / (c * u);
- else
- return quo(*this, c * u, x, false);
+ if (is_zero())
+ return _ex0;
+ if (is_ex_exactly_of_type(*this, numeric))
+ return _ex1;
+
+ ex c = content(x);
+ if (c.is_zero())
+ return _ex0;
+ ex u = unit(x);
+ if (is_ex_exactly_of_type(c, numeric))
+ return *this / (c * u);
+ else
+ return quo(*this, c * u, x, false);
}
* @param x variable in which to compute the primitive part
* @param c previously computed content part
* @return primitive part */
-
ex ex::primpart(const symbol &x, const ex &c) const
{
- if (is_zero())
- return exZERO();
- if (c.is_zero())
- return exZERO();
- if (is_ex_exactly_of_type(*this, numeric))
- return exONE();
-
- ex u = unit(x);
- if (is_ex_exactly_of_type(c, numeric))
- return *this / (c * u);
- else
- return quo(*this, c * u, x, false);
+ if (is_zero())
+ return _ex0;
+ if (c.is_zero())
+ return _ex0;
+ if (is_ex_exactly_of_type(*this, numeric))
+ return _ex1;
+
+ ex u = unit(x);
+ if (is_ex_exactly_of_type(c, numeric))
+ return *this / (c * u);
+ else
+ return quo(*this, c * u, x, false);
}
* GCD of multivariate polynomials
*/
-/** Compute GCD of multivariate polynomials using the subresultant PRS
- * algorithm. This function is used internally gy gcd().
+/** Compute GCD of polynomials in Q[X] using the Euclidean algorithm (not
+ * really suited for multivariate GCDs). This function is only provided for
+ * testing purposes.
+ *
+ * @param a first multivariate polynomial
+ * @param b second multivariate polynomial
+ * @param x pointer to symbol (main variable) in which to compute the GCD in
+ * @return the GCD as a new expression
+ * @see gcd */
+
+static ex eu_gcd(const ex &a, const ex &b, const symbol *x)
+{
+//std::clog << "eu_gcd(" << a << "," << b << ")\n";
+
+ // Sort c and d so that c has higher degree
+ ex c, d;
+ int adeg = a.degree(*x), bdeg = b.degree(*x);
+ if (adeg >= bdeg) {
+ c = a;
+ d = b;
+ } else {
+ c = b;
+ d = a;
+ }
+
+ // Normalize in Q[x]
+ c = c / c.lcoeff(*x);
+ d = d / d.lcoeff(*x);
+
+ // Euclidean algorithm
+ ex r;
+ for (;;) {
+//std::clog << " d = " << d << endl;
+ r = rem(c, d, *x, false);
+ if (r.is_zero())
+ return d / d.lcoeff(*x);
+ c = d;
+ d = r;
+ }
+}
+
+
+/** Compute GCD of multivariate polynomials using the Euclidean PRS algorithm
+ * with pseudo-remainders ("World's Worst GCD Algorithm", staying in Z[X]).
+ * This function is only provided for testing purposes.
+ *
+ * @param a first multivariate polynomial
+ * @param b second multivariate polynomial
+ * @param x pointer to symbol (main variable) in which to compute the GCD in
+ * @return the GCD as a new expression
+ * @see gcd */
+
+static ex euprem_gcd(const ex &a, const ex &b, const symbol *x)
+{
+//std::clog << "euprem_gcd(" << a << "," << b << ")\n";
+
+ // Sort c and d so that c has higher degree
+ ex c, d;
+ int adeg = a.degree(*x), bdeg = b.degree(*x);
+ if (adeg >= bdeg) {
+ c = a;
+ d = b;
+ } else {
+ c = b;
+ d = a;
+ }
+
+ // Calculate GCD of contents
+ ex gamma = gcd(c.content(*x), d.content(*x), NULL, NULL, false);
+
+ // Euclidean algorithm with pseudo-remainders
+ ex r;
+ for (;;) {
+//std::clog << " d = " << d << endl;
+ r = prem(c, d, *x, false);
+ if (r.is_zero())
+ return d.primpart(*x) * gamma;
+ c = d;
+ d = r;
+ }
+}
+
+
+/** Compute GCD of multivariate polynomials using the primitive Euclidean
+ * PRS algorithm (complete content removal at each step). This function is
+ * only provided for testing purposes.
*
* @param a first multivariate polynomial
* @param b second multivariate polynomial
* @return the GCD as a new expression
* @see gcd */
-static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
+static ex peu_gcd(const ex &a, const ex &b, const symbol *x)
{
- // Sort c and d so that c has higher degree
- ex c, d;
- int adeg = a.degree(*x), bdeg = b.degree(*x);
- int cdeg, ddeg;
- if (adeg >= bdeg) {
- c = a;
- d = b;
- cdeg = adeg;
- ddeg = bdeg;
- } else {
- c = b;
- d = a;
- cdeg = bdeg;
- ddeg = adeg;
- }
-
- // Remove content from c and d, to be attached to GCD later
- ex cont_c = c.content(*x);
- ex cont_d = d.content(*x);
- ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
- if (ddeg == 0)
- return gamma;
- c = c.primpart(*x, cont_c);
- d = d.primpart(*x, cont_d);
-
- // First element of subresultant sequence
- ex r = exZERO(), ri = exONE(), psi = exONE();
- int delta = cdeg - ddeg;
-
- for (;;) {
- // Calculate polynomial pseudo-remainder
- r = prem(c, d, *x, false);
- if (r.is_zero())
- return gamma * d.primpart(*x);
- c = d;
- cdeg = ddeg;
- if (!divide(r, ri * power(psi, delta), d, false))
- throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
- ddeg = d.degree(*x);
- if (ddeg == 0) {
- if (is_ex_exactly_of_type(r, numeric))
- return gamma;
- else
- return gamma * r.primpart(*x);
- }
-
- // Next element of subresultant sequence
- ri = c.expand().lcoeff(*x);
- if (delta == 1)
- psi = ri;
- else if (delta)
- divide(power(ri, delta), power(psi, delta-1), psi, false);
- delta = cdeg - ddeg;
- }
+//std::clog << "peu_gcd(" << a << "," << b << ")\n";
+
+ // Sort c and d so that c has higher degree
+ ex c, d;
+ int adeg = a.degree(*x), bdeg = b.degree(*x);
+ int ddeg;
+ if (adeg >= bdeg) {
+ c = a;
+ d = b;
+ ddeg = bdeg;
+ } else {
+ c = b;
+ d = a;
+ ddeg = adeg;
+ }
+
+ // Remove content from c and d, to be attached to GCD later
+ ex cont_c = c.content(*x);
+ ex cont_d = d.content(*x);
+ ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
+ if (ddeg == 0)
+ return gamma;
+ c = c.primpart(*x, cont_c);
+ d = d.primpart(*x, cont_d);
+
+ // Euclidean algorithm with content removal
+ ex r;
+ for (;;) {
+//std::clog << " d = " << d << endl;
+ r = prem(c, d, *x, false);
+ if (r.is_zero())
+ return gamma * d;
+ c = d;
+ d = r.primpart(*x);
+ }
+}
+
+
+/** Compute GCD of multivariate polynomials using the reduced PRS algorithm.
+ * This function is only provided for testing purposes.
+ *
+ * @param a first multivariate polynomial
+ * @param b second multivariate polynomial
+ * @param x pointer to symbol (main variable) in which to compute the GCD in
+ * @return the GCD as a new expression
+ * @see gcd */
+
+static ex red_gcd(const ex &a, const ex &b, const symbol *x)
+{
+//std::clog << "red_gcd(" << a << "," << b << ")\n";
+
+ // Sort c and d so that c has higher degree
+ ex c, d;
+ int adeg = a.degree(*x), bdeg = b.degree(*x);
+ int cdeg, ddeg;
+ if (adeg >= bdeg) {
+ c = a;
+ d = b;
+ cdeg = adeg;
+ ddeg = bdeg;
+ } else {
+ c = b;
+ d = a;
+ cdeg = bdeg;
+ ddeg = adeg;
+ }
+
+ // Remove content from c and d, to be attached to GCD later
+ ex cont_c = c.content(*x);
+ ex cont_d = d.content(*x);
+ ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
+ if (ddeg == 0)
+ return gamma;
+ c = c.primpart(*x, cont_c);
+ d = d.primpart(*x, cont_d);
+
+ // First element of divisor sequence
+ ex r, ri = _ex1;
+ int delta = cdeg - ddeg;
+
+ for (;;) {
+ // Calculate polynomial pseudo-remainder
+//std::clog << " d = " << d << endl;
+ r = prem(c, d, *x, false);
+ if (r.is_zero())
+ return gamma * d.primpart(*x);
+ c = d;
+ cdeg = ddeg;
+
+ if (!divide(r, pow(ri, delta), d, false))
+ throw(std::runtime_error("invalid expression in red_gcd(), division failed"));
+ ddeg = d.degree(*x);
+ if (ddeg == 0) {
+ if (is_ex_exactly_of_type(r, numeric))
+ return gamma;
+ else
+ return gamma * r.primpart(*x);
+ }
+
+ ri = c.expand().lcoeff(*x);
+ delta = cdeg - ddeg;
+ }
+}
+
+
+/** Compute GCD of multivariate polynomials using the subresultant PRS
+ * algorithm. This function is used internally by gcd().
+ *
+ * @param a first multivariate polynomial
+ * @param b second multivariate polynomial
+ * @param var iterator to first element of vector of sym_desc structs
+ * @return the GCD as a new expression
+ * @see gcd */
+
+static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var)
+{
+//std::clog << "sr_gcd(" << a << "," << b << ")\n";
+#if STATISTICS
+ sr_gcd_called++;
+#endif
+
+ // The first symbol is our main variable
+ const symbol &x = *(var->sym);
+
+ // Sort c and d so that c has higher degree
+ ex c, d;
+ int adeg = a.degree(x), bdeg = b.degree(x);
+ int cdeg, ddeg;
+ if (adeg >= bdeg) {
+ c = a;
+ d = b;
+ cdeg = adeg;
+ ddeg = bdeg;
+ } else {
+ c = b;
+ d = a;
+ cdeg = bdeg;
+ ddeg = adeg;
+ }
+
+ // Remove content from c and d, to be attached to GCD later
+ ex cont_c = c.content(x);
+ ex cont_d = d.content(x);
+ ex gamma = gcd(cont_c, cont_d, NULL, NULL, false);
+ if (ddeg == 0)
+ return gamma;
+ c = c.primpart(x, cont_c);
+ d = d.primpart(x, cont_d);
+//std::clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n";
+
+ // First element of subresultant sequence
+ ex r = _ex0, ri = _ex1, psi = _ex1;
+ int delta = cdeg - ddeg;
+
+ for (;;) {
+ // Calculate polynomial pseudo-remainder
+//std::clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n";
+//std::clog << " d = " << d << endl;
+ r = prem(c, d, x, false);
+ if (r.is_zero())
+ return gamma * d.primpart(x);
+ c = d;
+ cdeg = ddeg;
+//std::clog << " dividing...\n";
+ if (!divide_in_z(r, ri * pow(psi, delta), d, var))
+ throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
+ ddeg = d.degree(x);
+ if (ddeg == 0) {
+ if (is_ex_exactly_of_type(r, numeric))
+ return gamma;
+ else
+ return gamma * r.primpart(x);
+ }
+
+ // Next element of subresultant sequence
+//std::clog << " calculating next subresultant...\n";
+ ri = c.expand().lcoeff(x);
+ if (delta == 1)
+ psi = ri;
+ else if (delta)
+ divide_in_z(pow(ri, delta), pow(psi, delta-1), psi, var+1);
+ delta = cdeg - ddeg;
+ }
}
* @param e expanded multivariate polynomial
* @return maximum coefficient
* @see heur_gcd */
-
numeric ex::max_coefficient(void) const
{
- ASSERT(bp!=0);
- return bp->max_coefficient();
+ GINAC_ASSERT(bp!=0);
+ return bp->max_coefficient();
}
+/** Implementation ex::max_coefficient().
+ * @see heur_gcd */
numeric basic::max_coefficient(void) const
{
- return numONE();
+ return _num1;
}
numeric numeric::max_coefficient(void) const
{
- return abs(*this);
+ return abs(*this);
}
numeric add::max_coefficient(void) const
{
- epvector::const_iterator it = seq.begin();
- epvector::const_iterator itend = seq.end();
- ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
- numeric cur_max = abs(ex_to_numeric(overall_coeff));
- while (it != itend) {
- numeric a;
- ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
- a = abs(ex_to_numeric(it->coeff));
- if (a > cur_max)
- cur_max = a;
- it++;
- }
- return cur_max;
+ epvector::const_iterator it = seq.begin();
+ epvector::const_iterator itend = seq.end();
+ GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+ numeric cur_max = abs(ex_to<numeric>(overall_coeff));
+ while (it != itend) {
+ numeric a;
+ GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
+ a = abs(ex_to<numeric>(it->coeff));
+ if (a > cur_max)
+ cur_max = a;
+ it++;
+ }
+ return cur_max;
}
numeric mul::max_coefficient(void) const
{
-#ifdef DOASSERT
- epvector::const_iterator it = seq.begin();
- epvector::const_iterator itend = seq.end();
- while (it != itend) {
- ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
- it++;
- }
-#endif // def DOASSERT
- ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
- return abs(ex_to_numeric(overall_coeff));
+#ifdef DO_GINAC_ASSERT
+ epvector::const_iterator it = seq.begin();
+ epvector::const_iterator itend = seq.end();
+ while (it != itend) {
+ GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
+ it++;
+ }
+#endif // def DO_GINAC_ASSERT
+ GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+ return abs(ex_to<numeric>(overall_coeff));
}
-/** Apply symmetric modular homomorphism to a multivariate polynomial.
- * This function is used internally by heur_gcd().
+/** Apply symmetric modular homomorphism to an expanded multivariate
+ * polynomial. This function is usually used internally by heur_gcd().
*
- * @param e expanded multivariate polynomial
* @param xi modulus
* @return mapped polynomial
* @see heur_gcd */
-
-ex ex::smod(const numeric &xi) const
-{
- ASSERT(bp!=0);
- return bp->smod(xi);
-}
-
ex basic::smod(const numeric &xi) const
{
- return *this;
+ return *this;
}
ex numeric::smod(const numeric &xi) const
{
- return ::smod(*this, xi);
+ return GiNaC::smod(*this, xi);
}
ex add::smod(const numeric &xi) const
{
- epvector newseq;
- newseq.reserve(seq.size()+1);
- epvector::const_iterator it = seq.begin();
- epvector::const_iterator itend = seq.end();
- while (it != itend) {
- ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
- numeric coeff = ::smod(ex_to_numeric(it->coeff), xi);
- if (!coeff.is_zero())
- newseq.push_back(expair(it->rest, coeff));
- it++;
- }
- ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
- numeric coeff = ::smod(ex_to_numeric(overall_coeff), xi);
- return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
+ epvector newseq;
+ newseq.reserve(seq.size()+1);
+ epvector::const_iterator it = seq.begin();
+ epvector::const_iterator itend = seq.end();
+ while (it != itend) {
+ GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
+ numeric coeff = GiNaC::smod(ex_to<numeric>(it->coeff), xi);
+ if (!coeff.is_zero())
+ newseq.push_back(expair(it->rest, coeff));
+ it++;
+ }
+ GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+ numeric coeff = GiNaC::smod(ex_to<numeric>(overall_coeff), xi);
+ return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
}
ex mul::smod(const numeric &xi) const
{
-#ifdef DOASSERT
- epvector::const_iterator it = seq.begin();
- epvector::const_iterator itend = seq.end();
- while (it != itend) {
- ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
- it++;
- }
-#endif // def DOASSERT
- mul * mulcopyp=new mul(*this);
- ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
- mulcopyp->overall_coeff=::smod(ex_to_numeric(overall_coeff),xi);
- mulcopyp->clearflag(status_flags::evaluated);
- mulcopyp->clearflag(status_flags::hash_calculated);
- return mulcopyp->setflag(status_flags::dynallocated);
+#ifdef DO_GINAC_ASSERT
+ epvector::const_iterator it = seq.begin();
+ epvector::const_iterator itend = seq.end();
+ while (it != itend) {
+ GINAC_ASSERT(!is_exactly_a<numeric>(recombine_pair_to_ex(*it)));
+ it++;
+ }
+#endif // def DO_GINAC_ASSERT
+ mul * mulcopyp = new mul(*this);
+ GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
+ mulcopyp->overall_coeff = GiNaC::smod(ex_to<numeric>(overall_coeff),xi);
+ mulcopyp->clearflag(status_flags::evaluated);
+ mulcopyp->clearflag(status_flags::hash_calculated);
+ return mulcopyp->setflag(status_flags::dynallocated);
}
-/** Exception thrown by heur_gcd() to signal failure */
+/** xi-adic polynomial interpolation */
+static ex interpolate(const ex &gamma, const numeric &xi, const symbol &x, int degree_hint = 1)
+{
+ exvector g; g.reserve(degree_hint);
+ ex e = gamma;
+ numeric rxi = xi.inverse();
+ for (int i=0; !e.is_zero(); i++) {
+ ex gi = e.smod(xi);
+ g.push_back(gi * power(x, i));
+ e = (e - gi) * rxi;
+ }
+ return (new add(g))->setflag(status_flags::dynallocated);
+}
+
+/** Exception thrown by heur_gcd() to signal failure. */
class gcdheu_failed {};
/** Compute GCD of multivariate polynomials using the heuristic GCD algorithm.
* @return the GCD as a new expression
* @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)
{
- if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
- numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b));
- numeric rg;
- if (ca || cb)
- rg = g.inverse();
- if (ca)
- *ca = ex_to_numeric(a).mul(rg);
- if (cb)
- *cb = ex_to_numeric(b).mul(rg);
- return g;
- }
-
- // The first symbol is our main variable
- const symbol *x = var->sym;
-
- // Remove integer content
- numeric gc = gcd(a.integer_content(), b.integer_content());
- numeric rgc = gc.inverse();
- ex p = a * rgc;
- ex q = b * rgc;
- int maxdeg = max(p.degree(*x), q.degree(*x));
-
- // Find evaluation point
- numeric mp = p.max_coefficient(), mq = q.max_coefficient();
- numeric xi;
- if (mp > mq)
- xi = mq * numTWO() + numTWO();
- else
- xi = mp * numTWO() + numTWO();
-
- // 6 tries maximum
- for (int t=0; t<6; t++) {
- if (xi.int_length() * maxdeg > 50000)
- throw gcdheu_failed();
-
- // Apply evaluation homomorphism and calculate GCD
- ex gamma = heur_gcd(p.subs(*x == xi), q.subs(*x == xi), NULL, NULL, var+1).expand();
- if (!is_ex_exactly_of_type(gamma, fail)) {
-
- // Reconstruct polynomial from GCD of mapped polynomials
- ex g = exZERO();
- numeric rxi = xi.inverse();
- for (int i=0; !gamma.is_zero(); i++) {
- ex gi = gamma.smod(xi);
- g += gi * power(*x, i);
- gamma = (gamma - gi) * rxi;
- }
- // Remove integer content
- g /= g.integer_content();
-
- // If the calculated polynomial divides both a and b, this is the GCD
- 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_ex_exactly_of_type(lc, numeric) && lc.compare(exZERO()) < 0)
- return -g;
- else
- return g;
- }
- }
-
- // Next evaluation point
- xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
- }
- return *new ex(fail());
+//std::clog << "heur_gcd(" << a << "," << b << ")\n";
+#if STATISTICS
+ heur_gcd_called++;
+#endif
+
+ // Algorithm only works for non-vanishing input polynomials
+ if (a.is_zero() || b.is_zero())
+ return (new fail())->setflag(status_flags::dynallocated);
+
+ // GCD of two numeric values -> CLN
+ if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
+ numeric g = gcd(ex_to<numeric>(a), ex_to<numeric>(b));
+ if (ca)
+ *ca = ex_to<numeric>(a) / g;
+ if (cb)
+ *cb = ex_to<numeric>(b) / g;
+ return g;
+ }
+
+ // The first symbol is our main variable
+ const symbol &x = *(var->sym);
+
+ // Remove integer content
+ numeric gc = gcd(a.integer_content(), b.integer_content());
+ numeric rgc = gc.inverse();
+ ex p = a * rgc;
+ ex q = b * rgc;
+ int maxdeg = std::max(p.degree(x), q.degree(x));
+
+ // Find evaluation point
+ numeric mp = p.max_coefficient();
+ numeric mq = q.max_coefficient();
+ numeric xi;
+ if (mp > mq)
+ xi = mq * _num2 + _num2;
+ else
+ xi = mp * _num2 + _num2;
+
+ // 6 tries maximum
+ for (int t=0; t<6; t++) {
+ if (xi.int_length() * maxdeg > 100000) {
+//std::clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << std::endl;
+ throw gcdheu_failed();
+ }
+
+ // Apply evaluation homomorphism and calculate GCD
+ ex cp, cq;
+ ex gamma = heur_gcd(p.subs(x == xi), q.subs(x == xi), &cp, &cq, var+1).expand();
+ if (!is_ex_exactly_of_type(gamma, fail)) {
+
+ // Reconstruct polynomial from GCD of mapped polynomials
+ ex g = interpolate(gamma, xi, x, maxdeg);
+
+ // Remove integer content
+ g /= g.integer_content();
+
+ // If the calculated polynomial divides both p and q, this is the GCD
+ 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_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
+ return -g;
+ else
+ return g;
+ }
+#if 0
+ cp = interpolate(cp, xi, x);
+ if (divide_in_z(cp, p, g, var)) {
+ if (divide_in_z(g, q, cb ? *cb : dummy, var)) {
+ g *= gc;
+ if (ca)
+ *ca = cp;
+ ex lc = g.lcoeff(x);
+ if (is_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
+ return -g;
+ else
+ return g;
+ }
+ }
+ cq = interpolate(cq, xi, x);
+ if (divide_in_z(cq, q, g, var)) {
+ if (divide_in_z(g, p, ca ? *ca : dummy, var)) {
+ g *= gc;
+ if (cb)
+ *cb = cq;
+ ex lc = g.lcoeff(x);
+ if (is_ex_exactly_of_type(lc, numeric) && ex_to<numeric>(lc).is_negative())
+ return -g;
+ else
+ return g;
+ }
+ }
+#endif
+ }
+
+ // Next evaluation point
+ xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
+ }
+ return (new fail())->setflag(status_flags::dynallocated);
}
* @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)
{
- // Some trivial cases
- if (a.is_zero()) {
- if (ca)
- *ca = exZERO();
- if (cb)
- *cb = exONE();
- return b;
- }
- if (b.is_zero()) {
- if (ca)
- *ca = exONE();
- if (cb)
- *cb = exZERO();
- return a;
- }
- if (a.is_equal(exONE()) || b.is_equal(exONE())) {
- if (ca)
- *ca = a;
- if (cb)
- *cb = b;
- return exONE();
- }
+//std::clog << "gcd(" << a << "," << b << ")\n";
+#if STATISTICS
+ gcd_called++;
+#endif
+
+ // GCD of numerics -> CLN
+ if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
+ numeric g = gcd(ex_to<numeric>(a), ex_to<numeric>(b));
+ if (ca || cb) {
+ if (g.is_zero()) {
+ if (ca)
+ *ca = _ex0;
+ if (cb)
+ *cb = _ex0;
+ } else {
+ if (ca)
+ *ca = ex_to<numeric>(a) / g;
+ if (cb)
+ *cb = ex_to<numeric>(b) / g;
+ }
+ }
+ return g;
+ }
+
+ // Check arguments
+ if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))) {
+ throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals"));
+ }
+
+ // Partially factored cases (to avoid expanding large expressions)
+ if (is_ex_exactly_of_type(a, mul)) {
+ if (is_ex_exactly_of_type(b, mul) && b.nops() > a.nops())
+ goto factored_b;
+factored_a:
+ unsigned num = a.nops();
+ exvector g; g.reserve(num);
+ exvector acc_ca; acc_ca.reserve(num);
+ ex part_b = b;
+ for (unsigned 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_ex_exactly_of_type(b, mul)) {
+ if (is_ex_exactly_of_type(a, mul) && a.nops() > b.nops())
+ goto factored_a;
+factored_b:
+ unsigned num = b.nops();
+ exvector g; g.reserve(num);
+ exvector acc_cb; acc_cb.reserve(num);
+ ex part_a = a;
+ for (unsigned 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 FAST_COMPARE
+ // Input polynomials of the form poly^n are sometimes also trivial
+ if (is_ex_exactly_of_type(a, power)) {
+ ex p = a.op(0);
+ if (is_ex_exactly_of_type(b, power)) {
+ 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_ex_exactly_of_type(b, power)) {
+ 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;
+ }
+ }
+#endif
+
+ // Some trivial cases
+ ex aex = a.expand(), bex = b.expand();
+ if (aex.is_zero()) {
+ if (ca)
+ *ca = _ex0;
+ if (cb)
+ *cb = _ex1;
+ return b;
+ }
+ if (bex.is_zero()) {
+ if (ca)
+ *ca = _ex1;
+ if (cb)
+ *cb = _ex0;
+ return a;
+ }
+ if (aex.is_equal(_ex1) || bex.is_equal(_ex1)) {
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ return _ex1;
+ }
#if FAST_COMPARE
- if (a.is_equal(b)) {
- if (ca)
- *ca = exONE();
- if (cb)
- *cb = exONE();
- return a;
- }
+ if (a.is_equal(b)) {
+ if (ca)
+ *ca = _ex1;
+ if (cb)
+ *cb = _ex1;
+ return a;
+ }
+#endif
+
+ // Gather symbol statistics
+ sym_desc_vec sym_stats;
+ get_symbol_stats(a, b, sym_stats);
+
+ // The symbol with least degree is our main variable
+ sym_desc_vec::const_iterator var = sym_stats.begin();
+ const symbol &x = *(var->sym);
+
+ // Cancel trivial common factor
+ int ldeg_a = var->ldeg_a;
+ int ldeg_b = var->ldeg_b;
+ int min_ldeg = std::min(ldeg_a,ldeg_b);
+ if (min_ldeg > 0) {
+ ex common = power(x, min_ldeg);
+//std::clog << "trivial common factor " << common << std::endl;
+ return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
+ }
+
+ // Try to eliminate variables
+ if (var->deg_a == 0) {
+//std::clog << "eliminating variable " << x << " from b" << std::endl;
+ ex c = bex.content(x);
+ ex g = gcd(aex, c, ca, cb, false);
+ if (cb)
+ *cb *= bex.unit(x) * bex.primpart(x, c);
+ return g;
+ } else if (var->deg_b == 0) {
+//std::clog << "eliminating variable " << x << " from a" << std::endl;
+ ex c = aex.content(x);
+ ex g = gcd(c, bex, ca, cb, false);
+ if (ca)
+ *ca *= aex.unit(x) * aex.primpart(x, c);
+ return g;
+ }
+
+ ex g;
+#if 1
+ // Try heuristic algorithm first, fall back to PRS if that failed
+ try {
+ g = heur_gcd(aex, bex, ca, cb, var);
+ } catch (gcdheu_failed) {
+ g = fail();
+ }
+ if (is_ex_exactly_of_type(g, fail)) {
+//std::clog << "heuristics failed" << std::endl;
+#if STATISTICS
+ heur_gcd_failed++;
#endif
- if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
- numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b));
- if (ca)
- *ca = ex_to_numeric(a) / g;
- if (cb)
- *cb = ex_to_numeric(b) / g;
- return g;
- }
- if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)) {
- cerr << "a=" << a << endl;
- cerr << "b=" << b << endl;
- throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals"));
- }
-
- // Gather symbol statistics
- sym_desc_vec sym_stats;
- get_symbol_stats(a, b, sym_stats);
-
- // The symbol with least degree is our main variable
- sym_desc_vec::const_iterator var = sym_stats.begin();
- const symbol *x = var->sym;
-
- // Cancel trivial common factor
- int ldeg_a = var->ldeg_a;
- int ldeg_b = var->ldeg_b;
- int min_ldeg = min(ldeg_a, ldeg_b);
- if (min_ldeg > 0) {
- ex common = power(*x, min_ldeg);
-//clog << "trivial common factor " << common << endl;
- return gcd((a / common).expand(), (b / common).expand(), ca, cb, false) * common;
- }
-
- // Try to eliminate variables
- if (var->deg_a == 0) {
-//clog << "eliminating variable " << *x << " from b" << endl;
- ex c = b.content(*x);
- ex g = gcd(a, c, ca, cb, false);
- if (cb)
- *cb *= b.unit(*x) * b.primpart(*x, c);
- return g;
- } else if (var->deg_b == 0) {
-//clog << "eliminating variable " << *x << " from a" << endl;
- ex c = a.content(*x);
- ex g = gcd(c, b, ca, cb, false);
- if (ca)
- *ca *= a.unit(*x) * a.primpart(*x, c);
- return g;
- }
-
- // Try heuristic algorithm first, fall back to PRS if that failed
- ex g;
- try {
- g = heur_gcd(a.expand(), b.expand(), ca, cb, var);
- } catch (gcdheu_failed) {
- g = *new ex(fail());
- }
- if (is_ex_exactly_of_type(g, fail)) {
-//clog << "heuristics failed\n";
- g = sr_gcd(a, b, x);
- if (ca)
- divide(a, g, *ca, false);
- if (cb)
- divide(b, g, *cb, false);
- }
- return g;
+#endif
+// g = heur_gcd(aex, bex, ca, cb, var);
+// g = eu_gcd(aex, bex, &x);
+// g = euprem_gcd(aex, bex, &x);
+// g = peu_gcd(aex, bex, &x);
+// g = red_gcd(aex, bex, &x);
+ g = sr_gcd(aex, bex, var);
+ 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);
+ }
+#if 1
+ } else {
+ if (g.is_equal(_ex1)) {
+ // Keep cofactors factored if possible
+ if (ca)
+ *ca = a;
+ if (cb)
+ *cb = b;
+ }
+ }
+#endif
+ return g;
}
* @return the LCM as a new expression */
ex lcm(const ex &a, const ex &b, bool check_args)
{
- if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
- return gcd(ex_to_numeric(a), ex_to_numeric(b));
- if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))
- throw(std::invalid_argument("lcm: arguments must be polynomials over the rationals"));
-
- ex ca, cb;
- ex g = gcd(a, b, &ca, &cb, false);
- return ca * cb * g;
+ if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric))
+ return lcm(ex_to<numeric>(a), ex_to<numeric>(b));
+ if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
+ throw(std::invalid_argument("lcm: arguments must be polynomials over the rationals"));
+
+ ex ca, cb;
+ ex g = gcd(a, b, &ca, &cb, false);
+ return ca * cb * g;
}
* Square-free factorization
*/
-// Univariate GCD of polynomials in Q[x] (used internally by sqrfree()).
-// a and b can be multivariate polynomials but they are treated as univariate polynomials in x.
-static ex univariate_gcd(const ex &a, const ex &b, const symbol &x)
+/** Compute square-free factorization of multivariate polynomial a(x) using
+ * Yun´s algorithm. Used internally by sqrfree().
+ *
+ * @param a multivariate polynomial over Z[X], treated here as univariate
+ * polynomial in x.
+ * @param x variable to factor in
+ * @return vector of factors sorted in ascending degree */
+static exvector sqrfree_yun(const ex &a, const symbol &x)
{
- if (a.is_zero())
- return b;
- if (b.is_zero())
- return a;
- if (a.is_equal(exONE()) || b.is_equal(exONE()))
- return exONE();
- if (is_ex_of_type(a, numeric) && is_ex_of_type(b, numeric))
- return gcd(ex_to_numeric(a), ex_to_numeric(b));
- if (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))
- throw(std::invalid_argument("univariate_gcd: arguments must be polynomials over the rationals"));
-
- // Euclidean algorithm
- ex c, d, r;
- if (a.degree(x) >= b.degree(x)) {
- c = a;
- d = b;
- } else {
- c = b;
- d = a;
- }
- for (;;) {
- r = rem(c, d, x, false);
- if (r.is_zero())
- break;
- c = d;
- d = r;
- }
- return d / d.lcoeff(x);
+ exvector res;
+ ex w = a;
+ ex z = w.diff(x);
+ ex g = gcd(w, z);
+ if (g.is_equal(_ex1)) {
+ res.push_back(a);
+ return res;
+ }
+ ex y;
+ do {
+ w = quo(w, g, x);
+ y = quo(z, g, x);
+ z = y - w.diff(x);
+ g = gcd(w, z);
+ res.push_back(g);
+ } while (!z.is_zero());
+ return res;
}
+/** Compute a square-free factorization of a multivariate polynomial in Q[X].
+ *
+ * @param a multivariate polynomial over Q[X]
+ * @param x lst of variables to factor in, may be left empty for autodetection
+ * @return a square-free factorization of \p a.
+ *
+ * \note
+ * A polynomial \f$p(X) \in C[X]\f$ is said <EM>square-free</EM>
+ * if, whenever any two polynomials \f$q(X)\f$ and \f$r(X)\f$
+ * are such that
+ * \f[
+ * p(X) = q(X)^2 r(X),
+ * \f]
+ * we have \f$q(X) \in C\f$.
+ * This means that \f$p(X)\f$ has no repeated factors, apart
+ * eventually from constants.
+ * Given a polynomial \f$p(X) \in C[X]\f$, we say that the
+ * decomposition
+ * \f[
+ * p(X) = b \cdot p_1(X)^{a_1} \cdot p_2(X)^{a_2} \cdots p_r(X)^{a_r}
+ * \f]
+ * is a <EM>square-free factorization</EM> of \f$p(X)\f$ if the
+ * following conditions hold:
+ * -# \f$b \in C\f$ and \f$b \neq 0\f$;
+ * -# \f$a_i\f$ is a positive integer for \f$i = 1, \ldots, r\f$;
+ * -# the degree of the polynomial \f$p_i\f$ is strictly positive
+ * for \f$i = 1, \ldots, r\f$;
+ * -# the polynomial \f$\Pi_{i=1}^r p_i(X)\f$ is square-free.
+ *
+ * Square-free factorizations need not be unique. For example, if
+ * \f$a_i\f$ is even, we could change the polynomial \f$p_i(X)\f$
+ * into \f$-p_i(X)\f$.
+ * Observe also that the factors \f$p_i(X)\f$ need not be irreducible
+ * polynomials.
+ */
+ex sqrfree(const ex &a, const lst &l)
+{
+ if (is_a<numeric>(a) || // algorithm does not trap a==0
+ is_a<symbol>(a)) // shortcut
+ return a;
+
+ // If no lst of variables to factorize in was specified we have to
+ // invent one now. Maybe one can optimize here by reversing the order
+ // or so, I don't know.
+ lst args;
+ if (l.nops()==0) {
+ sym_desc_vec sdv;
+ get_symbol_stats(a, _ex0, sdv);
+ sym_desc_vec::const_iterator it = sdv.begin(), itend = sdv.end();
+ while (it != itend) {
+ args.append(*it->sym);
+ ++it;
+ }
+ } else {
+ args = l;
+ }
+
+ // Find the symbol to factor in at this stage
+ if (!is_ex_of_type(args.op(0), symbol))
+ throw (std::runtime_error("sqrfree(): invalid factorization variable"));
+ const symbol &x = ex_to<symbol>(args.op(0));
-/** Compute square-free factorization of multivariate polynomial a(x) using
- * Yun´s algorithm.
+ // convert the argument from something in Q[X] to something in Z[X]
+ const numeric lcm = lcm_of_coefficients_denominators(a);
+ const ex tmp = multiply_lcm(a,lcm);
+
+ // find the factors
+ exvector factors = sqrfree_yun(tmp,x);
+
+ // construct the next list of symbols with the first element popped
+ lst newargs = args;
+ newargs.remove_first();
+
+ // recurse down the factors in remaining variables
+ if (newargs.nops()>0) {
+ exvector::iterator i = factors.begin();
+ while (i != factors.end()) {
+ *i = sqrfree(*i, newargs);
+ ++i;
+ }
+ }
+
+ // Done with recursion, now construct the final result
+ ex result = _ex1;
+ exvector::const_iterator it = factors.begin(), itend = factors.end();
+ for (int p = 1; it!=itend; ++it, ++p)
+ result *= power(*it, p);
+
+ // Yun's algorithm does not account for constant factors. (For univariate
+ // polynomials it works only in the monic case.) We can correct this by
+ // inserting what has been lost back into the result. For completeness
+ // we'll also have to recurse down that factor in the remaining variables.
+ if (newargs.nops()>0)
+ result *= sqrfree(quo(tmp, result, x), newargs);
+ else
+ result *= quo(tmp, result, x);
+
+ // Put in the reational overall factor again and return
+ return result * lcm.inverse();
+}
+
+/** Compute square-free partial fraction decomposition of rational function
+ * a(x).
*
- * @param a multivariate polynomial
- * @param x variable to factor in
- * @return factored polynomial */
-ex sqrfree(const ex &a, const symbol &x)
+ * @param a rational function over Z[x], treated as univariate polynomial
+ * in x
+ * @param x variable to factor in
+ * @return decomposed rational function */
+ex sqrfree_parfrac(const ex & a, const symbol & x)
{
- int i = 1;
- ex res = exONE();
- ex b = a.diff(x);
- ex c = univariate_gcd(a, b, x);
- ex w;
- if (c.is_equal(exONE())) {
- w = a;
- } else {
- w = quo(a, c, x);
- ex y = quo(b, c, x);
- ex z = y - w.diff(x);
- while (!z.is_zero()) {
- ex g = univariate_gcd(w, z, x);
- res *= power(g, i);
- w = quo(w, g, x);
- y = quo(z, g, x);
- z = y - w.diff(x);
- i++;
- }
- }
- return res * power(w, i);
+ // Find numerator and denominator
+ ex nd = numer_denom(a);
+ ex numer = nd.op(0), denom = nd.op(1);
+//clog << "numer = " << numer << ", denom = " << denom << endl;
+
+ // Convert N(x)/D(x) -> Q(x) + R(x)/D(x), so degree(R) < degree(D)
+ ex red_poly = quo(numer, denom, x), red_numer = rem(numer, denom, x).expand();
+//clog << "red_poly = " << red_poly << ", red_numer = " << red_numer << endl;
+
+ // Factorize denominator and compute cofactors
+ exvector yun = sqrfree_yun(denom, x);
+//clog << "yun factors: " << exprseq(yun) << endl;
+ unsigned num_yun = yun.size();
+ exvector factor; factor.reserve(num_yun);
+ exvector cofac; cofac.reserve(num_yun);
+ for (unsigned i=0; i<num_yun; i++) {
+ if (!yun[i].is_equal(_ex1)) {
+ for (unsigned j=0; j<=i; j++) {
+ factor.push_back(pow(yun[i], j+1));
+ ex prod = _ex1;
+ for (unsigned k=0; k<num_yun; k++) {
+ if (k == i)
+ prod *= pow(yun[k], i-j);
+ else
+ prod *= pow(yun[k], k+1);
+ }
+ cofac.push_back(prod.expand());
+ }
+ }
+ }
+ unsigned num_factors = factor.size();
+//clog << "factors : " << exprseq(factor) << endl;
+//clog << "cofactors: " << exprseq(cofac) << endl;
+
+ // Construct coefficient matrix for decomposition
+ int max_denom_deg = denom.degree(x);
+ matrix sys(max_denom_deg + 1, num_factors);
+ matrix rhs(max_denom_deg + 1, 1);
+ for (int i=0; i<=max_denom_deg; i++) {
+ for (unsigned j=0; j<num_factors; j++)
+ sys(i, j) = cofac[j].coeff(x, i);
+ rhs(i, 0) = red_numer.coeff(x, i);
+ }
+//clog << "coeffs: " << sys << endl;
+//clog << "rhs : " << rhs << endl;
+
+ // Solve resulting linear system
+ matrix vars(num_factors, 1);
+ for (unsigned i=0; i<num_factors; i++)
+ vars(i, 0) = symbol();
+ matrix sol = sys.solve(vars, rhs);
+
+ // Sum up decomposed fractions
+ ex sum = 0;
+ for (unsigned i=0; i<num_factors; i++)
+ sum += sol(i, 0) / factor[i];
+
+ return red_poly + sum;
}
* Normal form of rational functions
*/
-// Create a symbol for replacing the expression "e" (or return a previously
-// assigned symbol). The symbol is appended to sym_list and returned, the
-// expression is appended to repl_list.
+/*
+ * Note: The internal normal() functions (= basic::normal() and overloaded
+ * functions) all return lists of the form {numerator, denominator}. This
+ * is to get around mul::eval()'s automatic expansion of numeric coefficients.
+ * E.g. (a+b)/3 is automatically converted to a/3+b/3 but we want to keep
+ * the information that (a+b) is the numerator and 3 is the denominator.
+ */
+
+
+/** Create a symbol for replacing the expression "e" (or return a previously
+ * assigned symbol). The symbol is appended to sym_lst and returned, the
+ * expression is appended to repl_lst.
+ * @see ex::normal */
static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst)
{
- // Expression already in repl_lst? Then return the assigned symbol
- for (int i=0; i<repl_lst.nops(); i++)
- if (repl_lst.op(i).is_equal(e))
- return sym_lst.op(i);
-
- // Otherwise create new symbol and add to list, taking care that the
+ // Expression already in repl_lst? Then return the assigned symbol
+ for (unsigned i=0; i<repl_lst.nops(); i++)
+ if (repl_lst.op(i).is_equal(e))
+ return sym_lst.op(i);
+
+ // Otherwise create new symbol and add to list, taking care that the
// replacement expression doesn't contain symbols from the sym_lst
// because subs() is not recursive
symbol s;
ex es(s);
ex e_replaced = e.subs(sym_lst, repl_lst);
- sym_lst.append(es);
- repl_lst.append(e_replaced);
- return es;
+ sym_lst.append(es);
+ repl_lst.append(e_replaced);
+ return es;
+}
+
+/** Create a symbol for replacing the expression "e" (or return a previously
+ * assigned symbol). An expression of the form "symbol == expression" is added
+ * to repl_lst and the symbol is returned.
+ * @see basic::to_rational */
+static ex replace_with_symbol(const ex &e, lst &repl_lst)
+{
+ // Expression already in repl_lst? Then return the assigned symbol
+ for (unsigned i=0; i<repl_lst.nops(); i++)
+ if (repl_lst.op(i).op(1).is_equal(e))
+ return repl_lst.op(i).op(0);
+
+ // Otherwise create new symbol and add to list, taking care that the
+ // replacement expression doesn't contain symbols from the sym_lst
+ // because subs() is not recursive
+ symbol s;
+ ex es(s);
+ ex e_replaced = e.subs(repl_lst);
+ repl_lst.append(es == e_replaced);
+ return es;
}
-/** Default implementation of ex::normal(). It replaces the object with a
- * temporary symbol.
+/** Function object to be applied by basic::normal(). */
+struct normal_map_function : public map_function {
+ int level;
+ normal_map_function(int l) : level(l) {}
+ ex operator()(const ex & e) { return normal(e, level); }
+};
+
+/** Default implementation of ex::normal(). It normalizes the children and
+ * replaces the object with a temporary symbol.
* @see ex::normal */
ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- return replace_with_symbol(*this, sym_lst, repl_lst);
+ if (nops() == 0)
+ return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ else {
+ if (level == 1)
+ return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ else if (level == -max_recursion_level)
+ throw(std::runtime_error("max recursion level reached"));
+ else {
+ normal_map_function map_normal(level - 1);
+ return (new lst(replace_with_symbol(map(map_normal), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ }
+ }
}
-/** Implementation of ex::normal() for symbols. This returns the unmodifies symbol.
+/** Implementation of ex::normal() for symbols. This returns the unmodified symbol.
* @see ex::normal */
ex symbol::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- return *this;
+ return (new lst(*this, _ex1))->setflag(status_flags::dynallocated);
}
* @see ex::normal */
ex numeric::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- if (is_real())
- if (is_rational())
- return *this;
- else
- return replace_with_symbol(*this, sym_lst, repl_lst);
- else { // complex
- numeric re = real(), im = imag();
+ numeric num = numer();
+ ex numex = num;
+
+ if (num.is_real()) {
+ if (!num.is_integer())
+ numex = replace_with_symbol(numex, sym_lst, repl_lst);
+ } else { // complex
+ numeric re = num.real(), im = num.imag();
ex re_ex = re.is_rational() ? re : replace_with_symbol(re, sym_lst, repl_lst);
ex im_ex = im.is_rational() ? im : replace_with_symbol(im, sym_lst, repl_lst);
- return re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst);
+ numex = re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst);
}
-}
+ // Denominator is always a real integer (see numeric::denom())
+ return (new lst(numex, denom()))->setflag(status_flags::dynallocated);
+}
-/*
- * Helper function for fraction cancellation (returns cancelled fraction n/d)
- */
+/** Fraction cancellation.
+ * @param n numerator
+ * @param d denominator
+ * @return cancelled fraction {n, d} as a list */
static ex frac_cancel(const ex &n, const ex &d)
{
- ex num = n;
- ex den = d;
- ex pre_factor = exONE();
-
- // Handle special cases where numerator or denominator is 0
- if (num.is_zero())
- return exZERO();
- if (den.expand().is_zero())
- throw(std::overflow_error("frac_cancel: division by zero in frac_cancel"));
-
- // More special cases
- if (is_ex_exactly_of_type(den, numeric))
- return num / den;
- if (num.is_zero())
- return exZERO();
-
- // Bring numerator and denominator to Z[X] by multiplying with
- // LCM of all coefficients' denominators
- ex num_lcm = lcm_of_coefficients_denominators(num);
- ex den_lcm = lcm_of_coefficients_denominators(den);
- num *= num_lcm;
- den *= den_lcm;
- pre_factor = den_lcm / num_lcm;
-
- // Cancel GCD from numerator and denominator
- ex cnum, cden;
- if (gcd(num, den, &cnum, &cden, false) != exONE()) {
+ ex num = n;
+ ex den = d;
+ numeric pre_factor = _num1;
+
+//std::clog << "frac_cancel num = " << num << ", den = " << den << std::endl;
+
+ // Handle trivial case where denominator is 1
+ if (den.is_equal(_ex1))
+ return (new lst(num, den))->setflag(status_flags::dynallocated);
+
+ // Handle special cases where numerator or denominator is 0
+ if (num.is_zero())
+ return (new lst(num, _ex1))->setflag(status_flags::dynallocated);
+ if (den.expand().is_zero())
+ throw(std::overflow_error("frac_cancel: division by zero in frac_cancel"));
+
+ // Bring numerator and denominator to Z[X] by multiplying with
+ // LCM of all coefficients' denominators
+ numeric num_lcm = lcm_of_coefficients_denominators(num);
+ numeric den_lcm = lcm_of_coefficients_denominators(den);
+ num = multiply_lcm(num, num_lcm);
+ den = multiply_lcm(den, den_lcm);
+ pre_factor = den_lcm / num_lcm;
+
+ // Cancel GCD from numerator and denominator
+ ex cnum, cden;
+ if (gcd(num, den, &cnum, &cden, false) != _ex1) {
num = cnum;
den = cden;
}
// as defined by get_first_symbol() is made positive)
const symbol *x;
if (get_first_symbol(den, x)) {
- if (den.unit(*x).compare(exZERO()) < 0) {
- num *= exMINUSONE();
- den *= exMINUSONE();
+ GINAC_ASSERT(is_exactly_a<numeric>(den.unit(*x)));
+ if (ex_to<numeric>(den.unit(*x)).is_negative()) {
+ num *= _ex_1;
+ den *= _ex_1;
}
}
- return pre_factor * num / den;
+
+ // Return result as list
+//std::clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << std::endl;
+ return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated);
}
* @see ex::normal */
ex add::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- // Normalize and expand children
- exvector o;
- o.reserve(seq.size()+1);
- epvector::const_iterator it = seq.begin(), itend = seq.end();
- while (it != itend) {
- ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1).expand();
- if (is_ex_exactly_of_type(n, add)) {
- epvector::const_iterator bit = (static_cast<add *>(n.bp))->seq.begin(), bitend = (static_cast<add *>(n.bp))->seq.end();
- while (bit != bitend) {
- o.push_back(recombine_pair_to_ex(*bit));
- bit++;
- }
- o.push_back((static_cast<add *>(n.bp))->overall_coeff);
- } else
- o.push_back(n);
- it++;
- }
- o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
-
- // Determine common denominator
- ex den = exONE();
- exvector::const_iterator ait = o.begin(), aitend = o.end();
- while (ait != aitend) {
- den = lcm((*ait).denom(false), den, false);
- ait++;
- }
-
- // Add fractions
- if (den.is_equal(exONE()))
- return (new add(o))->setflag(status_flags::dynallocated);
- else {
- exvector num_seq;
- for (ait=o.begin(); ait!=aitend; ait++) {
- ex q;
- if (!divide(den, (*ait).denom(false), q, false)) {
- // should not happen
- throw(std::runtime_error("invalid expression in add::normal, division failed"));
- }
- num_seq.push_back((*ait).numer(false) * q);
- }
- ex num = add(num_seq);
-
- // Cancel common factors from num/den
- return frac_cancel(num, den);
- }
+ if (level == 1)
+ return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ else if (level == -max_recursion_level)
+ throw(std::runtime_error("max recursion level reached"));
+
+ // Normalize children and split each one into numerator and denominator
+ exvector nums, dens;
+ nums.reserve(seq.size()+1);
+ dens.reserve(seq.size()+1);
+ epvector::const_iterator it = seq.begin(), itend = seq.end();
+ while (it != itend) {
+ ex n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, level-1);
+ nums.push_back(n.op(0));
+ dens.push_back(n.op(1));
+ it++;
+ }
+ ex n = ex_to<numeric>(overall_coeff).normal(sym_lst, repl_lst, level-1);
+ nums.push_back(n.op(0));
+ dens.push_back(n.op(1));
+ GINAC_ASSERT(nums.size() == dens.size());
+
+ // Now, nums is a vector of all numerators and dens is a vector of
+ // all denominators
+//std::clog << "add::normal uses " << nums.size() << " summands:\n";
+
+ // Add fractions sequentially
+ exvector::const_iterator num_it = nums.begin(), num_itend = nums.end();
+ exvector::const_iterator den_it = dens.begin(), den_itend = dens.end();
+//std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
+ ex num = *num_it++, den = *den_it++;
+ while (num_it != num_itend) {
+//std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl;
+ ex next_num = *num_it++, next_den = *den_it++;
+
+ // Trivially add sequences of fractions with identical denominators
+ while ((den_it != den_itend) && next_den.is_equal(*den_it)) {
+ next_num += *num_it;
+ num_it++; den_it++;
+ }
+
+ // Additiion of two fractions, taking advantage of the fact that
+ // the heuristic GCD algorithm computes the cofactors at no extra cost
+ ex co_den1, co_den2;
+ ex g = gcd(den, next_den, &co_den1, &co_den2, false);
+ num = ((num * co_den2) + (next_num * co_den1)).expand();
+ den *= co_den2; // this is the lcm(den, next_den)
+ }
+//std::clog << " common denominator = " << den << std::endl;
+
+ // Cancel common factors from num/den
+ return frac_cancel(num, den);
}
* @see ex::normal() */
ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- // Normalize children
- exvector o;
- o.reserve(seq.size()+1);
- epvector::const_iterator it = seq.begin(), itend = seq.end();
- while (it != itend) {
- o.push_back(recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1));
- it++;
- }
- o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
- ex n = (new mul(o))->setflag(status_flags::dynallocated);
- return frac_cancel(n.numer(false), n.denom(false));
+ if (level == 1)
+ return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ else if (level == -max_recursion_level)
+ throw(std::runtime_error("max recursion level reached"));
+
+ // Normalize children, separate into numerator and denominator
+ exvector num; num.reserve(seq.size());
+ exvector den; den.reserve(seq.size());
+ ex n;
+ epvector::const_iterator it = seq.begin(), itend = seq.end();
+ while (it != itend) {
+ n = ex_to<basic>(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, level-1);
+ num.push_back(n.op(0));
+ den.push_back(n.op(1));
+ it++;
+ }
+ n = ex_to<numeric>(overall_coeff).normal(sym_lst, repl_lst, level-1);
+ num.push_back(n.op(0));
+ den.push_back(n.op(1));
+
+ // Perform fraction cancellation
+ return frac_cancel((new mul(num))->setflag(status_flags::dynallocated),
+ (new mul(den))->setflag(status_flags::dynallocated));
}
* @see ex::normal */
ex power::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- if (exponent.info(info_flags::integer)) {
- // Integer powers are distributed
- ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
- ex num = n.numer(false);
- ex den = n.denom(false);
- return power(num, exponent) / power(den, exponent);
- } else {
- // Non-integer powers are replaced by temporary symbol (after normalizing basis)
- ex n = power(basis.bp->normal(sym_lst, repl_lst, level-1), exponent);
- return replace_with_symbol(n, sym_lst, repl_lst);
- }
+ if (level == 1)
+ return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ else if (level == -max_recursion_level)
+ throw(std::runtime_error("max recursion level reached"));
+
+ // Normalize basis and exponent (exponent gets reassembled)
+ ex n_basis = ex_to<basic>(basis).normal(sym_lst, repl_lst, level-1);
+ ex n_exponent = ex_to<basic>(exponent).normal(sym_lst, repl_lst, level-1);
+ n_exponent = n_exponent.op(0) / n_exponent.op(1);
+
+ if (n_exponent.info(info_flags::integer)) {
+
+ if (n_exponent.info(info_flags::positive)) {
+
+ // (a/b)^n -> {a^n, b^n}
+ return (new lst(power(n_basis.op(0), n_exponent), power(n_basis.op(1), n_exponent)))->setflag(status_flags::dynallocated);
+
+ } else if (n_exponent.info(info_flags::negative)) {
+
+ // (a/b)^-n -> {b^n, a^n}
+ return (new lst(power(n_basis.op(1), -n_exponent), power(n_basis.op(0), -n_exponent)))->setflag(status_flags::dynallocated);
+ }
+
+ } else {
+
+ if (n_exponent.info(info_flags::positive)) {
+
+ // (a/b)^x -> {sym((a/b)^x), 1}
+ return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+
+ } else if (n_exponent.info(info_flags::negative)) {
+
+ if (n_basis.op(1).is_equal(_ex1)) {
+
+ // a^-x -> {1, sym(a^x)}
+ return (new lst(_ex1, replace_with_symbol(power(n_basis.op(0), -n_exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated);
+
+ } else {
+
+ // (a/b)^-x -> {sym((b/a)^x), 1}
+ return (new lst(replace_with_symbol(power(n_basis.op(1) / n_basis.op(0), -n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ }
+
+ } else { // n_exponent not numeric
+
+ // (a/b)^x -> {sym((a/b)^x, 1}
+ return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
+ }
+ }
}
-/** Implementation of ex::normal() for series. It normalizes each coefficient and
- * replaces the series by a temporary symbol.
+/** Implementation of ex::normal() for pseries. It normalizes each coefficient
+ * and replaces the series by a temporary symbol.
* @see ex::normal */
-ex series::normal(lst &sym_lst, lst &repl_lst, int level) const
+ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- epvector new_seq;
- new_seq.reserve(seq.size());
-
- epvector::const_iterator it = seq.begin(), itend = seq.end();
- while (it != itend) {
- new_seq.push_back(expair(it->rest.normal(), it->coeff));
- it++;
- }
-
- ex n = series(var, point, new_seq);
- return replace_with_symbol(n, sym_lst, repl_lst);
+ epvector newseq;
+ epvector::const_iterator i = seq.begin(), end = seq.end();
+ while (i != end) {
+ ex restexp = i->rest.normal();
+ if (!restexp.is_zero())
+ newseq.push_back(expair(restexp, i->coeff));
+ ++i;
+ }
+ ex n = pseries(relational(var,point), newseq);
+ return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated);
}
* This function converts an expression to its normal form
* "numerator/denominator", where numerator and denominator are (relatively
* prime) polynomials. Any subexpressions which are not rational functions
- * (like non-rational numbers, non-integer powers or functions like Sin(),
- * Cos() etc.) are replaced by temporary symbols which are re-substituted by
+ * (like non-rational numbers, non-integer powers or functions like sin(),
+ * cos() etc.) are replaced by temporary symbols which are re-substituted by
* the (normalized) subexpressions before normal() returns (this way, any
* expression can be treated as a rational function). normal() is applied
* recursively to arguments of functions etc.
* @return normalized expression */
ex ex::normal(int level) const
{
- lst sym_lst, repl_lst;
- ex e = bp->normal(sym_lst, repl_lst, level);
- if (sym_lst.nops() > 0)
- return e.subs(sym_lst, repl_lst);
- else
- return e;
+ lst sym_lst, repl_lst;
+
+ ex e = bp->normal(sym_lst, repl_lst, level);
+ GINAC_ASSERT(is_a<lst>(e));
+
+ // Re-insert replaced symbols
+ if (sym_lst.nops() > 0)
+ e = e.subs(sym_lst, repl_lst);
+
+ // Convert {numerator, denominator} form back to fraction
+ return e.op(0) / e.op(1);
}
+
+/** Get numerator of an expression. If the expression is not of the normal
+ * form "numerator/denominator", it is first converted to this form and
+ * then the numerator is returned.
+ *
+ * @see ex::normal
+ * @return numerator */
+ex ex::numer(void) const
+{
+ lst sym_lst, repl_lst;
+
+ ex e = bp->normal(sym_lst, repl_lst, 0);
+ GINAC_ASSERT(is_a<lst>(e));
+
+ // Re-insert replaced symbols
+ if (sym_lst.nops() > 0)
+ return e.op(0).subs(sym_lst, repl_lst);
+ else
+ return e.op(0);
+}
+
+/** Get denominator of an expression. If the expression is not of the normal
+ * form "numerator/denominator", it is first converted to this form and
+ * then the denominator is returned.
+ *
+ * @see ex::normal
+ * @return denominator */
+ex ex::denom(void) const
+{
+ lst sym_lst, repl_lst;
+
+ ex e = bp->normal(sym_lst, repl_lst, 0);
+ GINAC_ASSERT(is_a<lst>(e));
+
+ // Re-insert replaced symbols
+ if (sym_lst.nops() > 0)
+ return e.op(1).subs(sym_lst, repl_lst);
+ else
+ return e.op(1);
+}
+
+/** Get numerator and denominator of an expression. If the expresison is not
+ * of the normal form "numerator/denominator", it is first converted to this
+ * form and then a list [numerator, denominator] is returned.
+ *
+ * @see ex::normal
+ * @return a list [numerator, denominator] */
+ex ex::numer_denom(void) const
+{
+ lst sym_lst, repl_lst;
+
+ ex e = bp->normal(sym_lst, repl_lst, 0);
+ GINAC_ASSERT(is_a<lst>(e));
+
+ // Re-insert replaced symbols
+ if (sym_lst.nops() > 0)
+ return e.subs(sym_lst, repl_lst);
+ else
+ return e;
+}
+
+
+/** Rationalization of non-rational functions.
+ * This function converts a general expression to a rational polynomial
+ * by replacing all non-rational subexpressions (like non-rational numbers,
+ * non-integer powers or functions like sin(), cos() etc.) to temporary
+ * symbols. This makes it possible to use functions like gcd() and divide()
+ * on non-rational functions by applying to_rational() on the arguments,
+ * calling the desired function and re-substituting the temporary symbols
+ * in the result. To make the last step possible, all temporary symbols and
+ * their associated expressions are collected in the list specified by the
+ * repl_lst parameter in the form {symbol == expression}, ready to be passed
+ * as an argument to ex::subs().
+ *
+ * @param repl_lst collects a list of all temporary symbols and their replacements
+ * @return rationalized expression */
+ex basic::to_rational(lst &repl_lst) const
+{
+ return replace_with_symbol(*this, repl_lst);
+}
+
+
+/** Implementation of ex::to_rational() for symbols. This returns the
+ * unmodified symbol. */
+ex symbol::to_rational(lst &repl_lst) const
+{
+ return *this;
+}
+
+
+/** Implementation of ex::to_rational() for a numeric. It splits complex
+ * numbers into re+I*im and replaces I and non-rational real numbers with a
+ * temporary symbol. */
+ex numeric::to_rational(lst &repl_lst) const
+{
+ if (is_real()) {
+ if (!is_rational())
+ return replace_with_symbol(*this, repl_lst);
+ } else { // complex
+ numeric re = real();
+ numeric im = imag();
+ ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl_lst);
+ ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl_lst);
+ return re_ex + im_ex * replace_with_symbol(I, repl_lst);
+ }
+ return *this;
+}
+
+
+/** Implementation of ex::to_rational() for powers. It replaces non-integer
+ * powers by temporary symbols. */
+ex power::to_rational(lst &repl_lst) const
+{
+ if (exponent.info(info_flags::integer))
+ return power(basis.to_rational(repl_lst), exponent);
+ else
+ return replace_with_symbol(*this, repl_lst);
+}
+
+
+/** Implementation of ex::to_rational() for expairseqs. */
+ex expairseq::to_rational(lst &repl_lst) const
+{
+ epvector s;
+ s.reserve(seq.size());
+ epvector::const_iterator i = seq.begin(), end = seq.end();
+ while (i != end) {
+ s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_rational(repl_lst)));
+ ++i;
+ }
+ ex oc = overall_coeff.to_rational(repl_lst);
+ if (oc.info(info_flags::numeric))
+ return thisexpairseq(s, overall_coeff);
+ else
+ s.push_back(combine_ex_with_coeff_to_pair(oc, _ex1));
+ return thisexpairseq(s, default_overall_coeff());
+}
+
+
+} // namespace GiNaC
git://www.ginac.de / ginac.git / blobdiff