* 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) 1999-2000 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
*/
#include <stdexcept>
+#include <algorithm>
+#include <map>
#include "normal.h"
#include "basic.h"
#include "numeric.h"
#include "power.h"
#include "relational.h"
-#include "series.h"
+#include "pseries.h"
#include "symbol.h"
+#include "utils.h"
+
+#ifndef NO_NAMESPACE_GINAC
+namespace GiNaC {
+#endif // ndef NO_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 (usually slower except for very large problems)
+#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() {
+ cout << "gcd() called " << gcd_called << " times\n";
+ cout << "sr_gcd() called " << sr_gcd_called << " times\n";
+ cout << "heur_gcd() called " << heur_gcd_called << " times\n";
+ 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++)
+ 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)) {
* 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
/** Lowest degree of symbol in polynomial "b" */
int ldeg_b;
- /** Minimum of ldeg_a and ldeg_b (Used for sorting) */
- int min_deg;
+ /** Maximum of deg_a and deg_b (Used for sorting) */
+ int max_deg;
/** Commparison operator for sorting */
- bool operator<(const sym_desc &x) const {return min_deg < x.min_deg;}
+ bool operator<(const sym_desc &x) const {return max_deg < x.max_deg;}
};
// Vector of sym_desc structures
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++)
+ 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);
* @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
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->max_deg = max(deg_a, deg_b);
it->ldeg_a = a.ldegree(*(it->sym));
it->ldeg_b = b.ldegree(*(it->sym));
it++;
}
sort(v.begin(), v.end());
+#if 0
+ clog << "Symbols:\n";
+ it = v.begin(); itend = v.end();
+ while (it != itend) {
+ 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 << endl;
+ clog << " lcoeff_a=" << a.lcoeff(*(it->sym)) << ", lcoeff_b=" << b.lcoeff(*(it->sym)) << endl;
+ it++;
+ }
+#endif
}
{
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++) {
+ 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))
- return lcmcoeff(e.op(0), l);
+ 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)) {
+ ex c = _ex1();
+ numeric lcm_accum = _num1();
+ for (unsigned i=0; i<e.nops(); i++) {
+ numeric op_lcm = lcmcoeff(e.op(i), _num1());
+ c *= multiply_lcm(e.op(i), op_lcm);
+ lcm_accum *= op_lcm;
+ }
+ c *= lcm / lcm_accum;
+ return c;
+ } else if (is_ex_exactly_of_type(e, add)) {
+ ex c = _ex0();
+ for (unsigned i=0; i<e.nops(); i++)
+ c += multiply_lcm(e.op(i), lcm);
+ return c;
+ } else if (is_ex_exactly_of_type(e, power)) {
+ 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);
+ GINAC_ASSERT(bp!=0);
return bp->integer_content();
}
numeric basic::integer_content(void) const
{
- return numONE();
+ return _num1();
}
numeric numeric::integer_content(void) const
{
epvector::const_iterator it = seq.begin();
epvector::const_iterator itend = seq.end();
- numeric c = numZERO();
+ numeric c = _num0();
while (it != itend) {
- ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
- ASSERT(is_ex_exactly_of_type(it->coeff,numeric));
+ GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
+ GINAC_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));
+ GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
c = gcd(ex_to_numeric(overall_coeff),c);
return c;
}
numeric mul::integer_content(void) const
{
-#ifdef DOASSERT
+#ifdef DO_GINAC_ASSERT
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));
+ GINAC_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));
+#endif // def DO_GINAC_ASSERT
+ GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
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())
return a / b;
#if FAST_COMPARE
if (a.is_equal(b))
- return exONE();
+ 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 q = _ex0();
ex r = a.expand();
if (r.is_zero())
return r;
* @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();
+ return _ex0();
else
return b;
}
#if FAST_COMPARE
if (a.is_equal(b))
- return exZERO();
+ 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"));
* @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();
+ return _ex0();
else
return b;
}
if (bdeg <= rdeg) {
blcoeff = eb.coeff(x, bdeg);
if (bdeg == 0)
- eb = exZERO();
+ eb = _ex0();
else
eb -= blcoeff * power(x, bdeg);
} else
- blcoeff = exONE();
+ 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 = exZERO();
+ r = _ex0();
else
r -= rlcoeff * power(x, rdeg);
r = (blcoeff * r).expand() - term;
}
+/** 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;
+}
+
+
/** Exact polynomial division of a(X) by b(X) in Q[X].
*
* @param a first multivariate polynomial (dividend)
* coefficients (defaults to "true")
* @return "true" when exact division succeeds (quotient returned in q),
* "false" otherwise */
-
bool divide(const ex &a, const ex &b, ex &q, bool check_args)
{
- q = exZERO();
+ q = _ex0();
if (b.is_zero())
throw(std::overflow_error("divide: division by zero"));
+ if (a.is_zero())
+ return true;
if (is_ex_exactly_of_type(b, numeric)) {
q = a / b;
return true;
return false;
#if FAST_COMPARE
if (a.is_equal(b)) {
- q = exONE();
+ q = _ex1();
return true;
}
#endif
- if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)))
+ 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
* Remembering
*/
-#include <map>
-
typedef pair<ex, ex> ex2;
typedef pair<ex, bool> exbool;
* @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();
+ q = _ex0();
if (b.is_zero())
throw(std::overflow_error("divide_in_z: division by zero"));
- if (b.is_equal(exONE())) {
+ if (b.is_equal(_ex1())) {
q = a;
return true;
}
}
#if FAST_COMPARE
if (a.is_equal(b)) {
- q = exONE();
+ q = _ex1();
return true;
}
#endif
if (bdeg > adeg)
return false;
-#if 1
-
- // 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()) {
-#if USE_REMEMBER
- dr_remember[ex2(a, b)] = exbool(q, true);
-#endif
- return true;
- }
- rdeg = r.degree(*x);
- }
-#if USE_REMEMBER
- dr_remember[ex2(a, b)] = exbool(q, false);
-#endif
- return false;
-
-#else
+#if USE_TRIAL_DIVISION
- // Trial division using polynomial interpolation
+ // 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 = numZERO();
+ numeric point = _num0();
ex c;
for (i=0; i<=adeg; i++) {
ex bs = b.subs(*x == point);
while (bs.is_zero()) {
- point += numONE();
+ 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 += numONE();
+ point += _num1();
}
// Compute inverses
vector<numeric> rcp; rcp.reserve(adeg + 1);
- rcp.push_back(0);
+ rcp.push_back(_num0());
for (k=1; k<=adeg; k++) {
numeric product = alpha[k] - alpha[0];
for (i=1; i<k; i++)
return true;
} else
return false;
+
+#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);
+ 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()) {
+#if USE_REMEMBER
+ dr_remember[ex2(a, b)] = exbool(q, true);
+#endif
+ return true;
+ }
+ rdeg = r.degree(*x);
+ }
+#if USE_REMEMBER
+ dr_remember[ex2(a, b)] = exbool(q, false);
+#endif
+ return false;
+
#endif
}
{
ex c = expand().lcoeff(x);
if (is_ex_exactly_of_type(c, numeric))
- return c < exZERO() ? exMINUSONE() : exONE();
+ return c < _ex0() ? _ex_1() : _ex1();
else {
const symbol *y;
if (get_first_symbol(c, y))
ex ex::content(const symbol &x) const
{
if (is_zero())
- return exZERO();
+ 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 exZERO();
+ return _ex0();
// First, try the integer content
ex c = e.integer_content();
int ldeg = e.ldegree(x);
if (deg == ldeg)
return e.lcoeff(x) / e.unit(x);
- c = exZERO();
+ c = _ex0();
for (int i=ldeg; i<=deg; i++)
c = gcd(e.coeff(x, i), c, NULL, NULL, false);
return c;
ex ex::primpart(const symbol &x) const
{
if (is_zero())
- return exZERO();
+ return _ex0();
if (is_ex_exactly_of_type(*this, numeric))
- return exONE();
+ return _ex1();
ex c = content(x);
if (c.is_zero())
- return exZERO();
+ return _ex0();
ex u = unit(x);
if (is_ex_exactly_of_type(c, numeric))
return *this / (c * u);
* @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();
+ return _ex0();
if (c.is_zero())
- return exZERO();
+ return _ex0();
if (is_ex_exactly_of_type(*this, numeric))
- return exONE();
+ return _ex1();
ex u = unit(x);
if (is_ex_exactly_of_type(c, numeric))
* 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
* @return the GCD as a new expression
* @see gcd */
-static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
+static ex eu_gcd(const ex &a, const ex &b, const symbol *x)
{
+//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 (;;) {
+//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)
+{
+//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 (;;) {
+//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
+ * @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 peu_gcd(const ex &a, const ex &b, const symbol *x)
+{
+//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 (;;) {
+//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)
+{
+//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);
c = c.primpart(*x, cont_c);
d = d.primpart(*x, cont_d);
- // First element of subresultant sequence
- ex r = exZERO(), ri = exONE(), psi = exONE();
+ // First element of divisor sequence
+ ex r, ri = _ex1();
int delta = cdeg - ddeg;
for (;;) {
// Calculate polynomial pseudo-remainder
+//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, ri * power(psi, delta), d, false))
- throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
+
+ 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 * r.primpart(*x);
}
- // Next element of subresultant sequence
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)
+{
+//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);
+//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
+//clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n";
+//clog << " d = " << d << endl;
+ r = prem(c, d, x, false);
+ if (r.is_zero())
+ return gamma * d.primpart(x);
+ c = d;
+ cdeg = ddeg;
+//clog << " dividing...\n";
+ if (!divide_in_z(r, ri * pow(psi, delta), d, var+1))
+ 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
+//clog << " calculating next subresultant...\n";
+ ri = c.expand().lcoeff(x);
if (delta == 1)
psi = ri;
else if (delta)
- divide(power(ri, delta), power(psi, delta-1), psi, false);
+ 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);
+ GINAC_ASSERT(bp!=0);
return bp->max_coefficient();
}
numeric basic::max_coefficient(void) const
{
- return numONE();
+ return _num1();
}
numeric numeric::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));
+ GINAC_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));
+ GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
a = abs(ex_to_numeric(it->coeff));
if (a > cur_max)
cur_max = a;
numeric mul::max_coefficient(void) const
{
-#ifdef DOASSERT
+#ifdef DO_GINAC_ASSERT
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));
+ GINAC_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));
+#endif // def DO_GINAC_ASSERT
+ GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
return abs(ex_to_numeric(overall_coeff));
}
* @param xi modulus
* @return mapped polynomial
* @see heur_gcd */
-
ex ex::smod(const numeric &xi) const
{
- ASSERT(bp!=0);
+ GINAC_ASSERT(bp!=0);
return bp->smod(xi);
}
ex numeric::smod(const numeric &xi) const
{
+#ifndef NO_NAMESPACE_GINAC
+ return GiNaC::smod(*this, xi);
+#else // ndef NO_NAMESPACE_GINAC
return ::smod(*this, xi);
+#endif // ndef NO_NAMESPACE_GINAC
}
ex add::smod(const numeric &xi) const
epvector::const_iterator it = seq.begin();
epvector::const_iterator itend = seq.end();
while (it != itend) {
- ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
+ GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric));
+#ifndef NO_NAMESPACE_GINAC
+ numeric coeff = GiNaC::smod(ex_to_numeric(it->coeff), xi);
+#else // ndef NO_NAMESPACE_GINAC
numeric coeff = ::smod(ex_to_numeric(it->coeff), xi);
+#endif // ndef NO_NAMESPACE_GINAC
if (!coeff.is_zero())
newseq.push_back(expair(it->rest, coeff));
it++;
}
- ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
+ GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
+#ifndef NO_NAMESPACE_GINAC
+ numeric coeff = GiNaC::smod(ex_to_numeric(overall_coeff), xi);
+#else // ndef NO_NAMESPACE_GINAC
numeric coeff = ::smod(ex_to_numeric(overall_coeff), xi);
+#endif // ndef NO_NAMESPACE_GINAC
return (new add(newseq,coeff))->setflag(status_flags::dynallocated);
}
ex mul::smod(const numeric &xi) const
{
-#ifdef DOASSERT
+#ifdef DO_GINAC_ASSERT
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));
+ GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric));
it++;
}
-#endif // def DOASSERT
+#endif // def DO_GINAC_ASSERT
mul * mulcopyp=new mul(*this);
- ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
- mulcopyp->overall_coeff=::smod(ex_to_numeric(overall_coeff),xi);
+ GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric));
+#ifndef NO_NAMESPACE_GINAC
+ mulcopyp->overall_coeff = GiNaC::smod(ex_to_numeric(overall_coeff),xi);
+#else // ndef NO_NAMESPACE_GINAC
+ mulcopyp->overall_coeff = ::smod(ex_to_numeric(overall_coeff),xi);
+#endif // ndef NO_NAMESPACE_GINAC
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 */
+/** 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)
{
+//clog << "heur_gcd(" << a << "," << b << ")\n";
+#if STATISTICS
+ heur_gcd_called++;
+#endif
+
+ // 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));
numeric rg;
}
// The first symbol is our main variable
- const symbol *x = var->sym;
+ 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));
+ 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();
+ xi = mq * _num2() + _num2();
else
- xi = mp * numTWO() + numTWO();
+ xi = mp * _num2() + _num2();
// 6 tries maximum
for (int t=0; t<6; t++) {
- if (xi.int_length() * maxdeg > 50000)
+ if (xi.int_length() * maxdeg > 100000) {
+//clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << endl;
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();
+ 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();
+ ex g = _ex0();
numeric rxi = xi.inverse();
for (int i=0; !gamma.is_zero(); i++) {
ex gi = gamma.smod(xi);
- g += gi * power(*x, i);
+ g += gi * power(x, i);
gamma = (gamma - gi) * rxi;
}
// Remove integer content
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)
+ ex lc = g.lcoeff(x);
+ if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative())
return -g;
else
return g;
* @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)
{
+//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)
+ *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:
+ ex g = _ex1();
+ ex acc_ca = _ex1();
+ ex part_b = b;
+ for (unsigned i=0; i<a.nops(); i++) {
+ ex part_ca, part_cb;
+ g *= gcd(a.op(i), part_b, &part_ca, &part_cb, check_args);
+ acc_ca *= part_ca;
+ part_b = part_cb;
+ }
+ if (ca)
+ *ca = acc_ca;
+ if (cb)
+ *cb = part_b;
+ return g;
+ } 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:
+ ex g = _ex1();
+ ex acc_cb = _ex1();
+ ex part_a = a;
+ for (unsigned i=0; i<b.nops(); i++) {
+ ex part_ca, part_cb;
+ g *= gcd(part_a, b.op(i), &part_ca, &part_cb, check_args);
+ acc_cb *= part_cb;
+ part_a = part_ca;
+ }
+ if (ca)
+ *ca = part_a;
+ if (cb)
+ *cb = acc_cb;
+ return g;
+ }
+
+#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
- if (a.is_zero()) {
+ ex aex = a.expand(), bex = b.expand();
+ if (aex.is_zero()) {
if (ca)
- *ca = exZERO();
+ *ca = _ex0();
if (cb)
- *cb = exONE();
+ *cb = _ex1();
return b;
}
- if (b.is_zero()) {
+ if (bex.is_zero()) {
if (ca)
- *ca = exONE();
+ *ca = _ex1();
if (cb)
- *cb = exZERO();
+ *cb = _ex0();
return a;
}
- if (a.is_equal(exONE()) || b.is_equal(exONE())) {
+ if (aex.is_equal(_ex1()) || bex.is_equal(_ex1())) {
if (ca)
*ca = a;
if (cb)
*cb = b;
- return exONE();
+ return _ex1();
}
#if FAST_COMPARE
if (a.is_equal(b)) {
if (ca)
- *ca = exONE();
+ *ca = _ex1();
if (cb)
- *cb = exONE();
+ *cb = _ex1();
return a;
}
#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;
// The symbol with least degree is our main variable
sym_desc_vec::const_iterator var = sym_stats.begin();
- const symbol *x = var->sym;
+ 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);
+ ex common = power(x, min_ldeg);
//clog << "trivial common factor " << common << endl;
- return gcd((a / common).expand(), (b / common).expand(), ca, cb, false) * common;
+ return gcd((aex / common).expand(), (bex / 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);
+//clog << "eliminating variable " << x << " from b" << endl;
+ ex c = bex.content(x);
+ ex g = gcd(aex, c, ca, cb, false);
if (cb)
- *cb *= b.unit(*x) * b.primpart(*x, c);
+ *cb *= bex.unit(x) * bex.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);
+//clog << "eliminating variable " << x << " from a" << endl;
+ ex c = aex.content(x);
+ ex g = gcd(c, bex, ca, cb, false);
if (ca)
- *ca *= a.unit(*x) * a.primpart(*x, c);
+ *ca *= aex.unit(x) * aex.primpart(x, c);
return g;
}
- // Try heuristic algorithm first, fall back to PRS if that failed
ex g;
+#if 1
+ // Try heuristic algorithm first, fall back to PRS if that failed
try {
- g = heur_gcd(a.expand(), b.expand(), ca, cb, var);
+ g = heur_gcd(aex, bex, 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);
- }
+//clog << "heuristics failed" << endl;
+#if STATISTICS
+ heur_gcd_failed++;
+#endif
+#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;
}
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));
+ 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"));
return b;
if (b.is_zero())
return a;
- if (a.is_equal(exONE()) || b.is_equal(exONE()))
- return exONE();
+ if (a.is_equal(_ex1()) || b.is_equal(_ex1()))
+ return _ex1();
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))
ex sqrfree(const ex &a, const symbol &x)
{
int i = 1;
- ex res = exONE();
+ ex res = _ex1();
ex b = a.diff(x);
ex c = univariate_gcd(a, b, x);
ex w;
- if (c.is_equal(exONE())) {
+ if (c.is_equal(_ex1())) {
w = a;
} else {
w = quo(a, c, x);
* 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++)
+ 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
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 ex::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.
* @see ex::normal */
ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const
{
- return replace_with_symbol(*this, sym_lst, repl_lst);
+ return (new lst(replace_with_symbol(*this, 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();
- 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);
- }
-}
+ 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);
+ 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();
+ numeric pre_factor = _num1();
+
+//clog << "frac_cancel num = " << num << ", den = " << den << endl;
// Handle special cases where numerator or denominator is 0
if (num.is_zero())
- return exZERO();
+ return (new lst(_ex0(), _ex1()))->setflag(status_flags::dynallocated);
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;
+ 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) != exONE()) {
+ 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_ex_exactly_of_type(den.unit(*x),numeric));
+ if (ex_to_numeric(den.unit(*x)).is_negative()) {
+ num *= _ex_1();
+ den *= _ex_1();
}
}
- return pre_factor * num / den;
+
+ // Return result as list
+//clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << 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
+ if (level == 1)
+ return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated);
+ else if (level == -max_recursion_level)
+ throw(std::runtime_error("max recursion level reached"));
+
+ // Normalize and expand children, chop into summands
exvector o;
o.reserve(seq.size()+1);
epvector::const_iterator it = seq.begin(), itend = seq.end();
while (it != itend) {
+
+ // Normalize and expand child
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();
+
+ // If numerator is a sum, chop into summands
+ if (is_ex_exactly_of_type(n.op(0), add)) {
+ epvector::const_iterator bit = ex_to_add(n.op(0)).seq.begin(), bitend = ex_to_add(n.op(0)).seq.end();
while (bit != bitend) {
- o.push_back(recombine_pair_to_ex(*bit));
+ o.push_back((new lst(recombine_pair_to_ex(*bit), n.op(1)))->setflag(status_flags::dynallocated));
bit++;
}
- o.push_back((static_cast<add *>(n.bp))->overall_coeff);
+
+ // The overall_coeff is already normalized (== rational), we just
+ // split it into numerator and denominator
+ GINAC_ASSERT(ex_to_numeric(ex_to_add(n.op(0)).overall_coeff).is_rational());
+ numeric overall = ex_to_numeric(ex_to_add(n.op(0)).overall_coeff);
+ o.push_back((new lst(overall.numer(), overall.denom() * n.op(1)))->setflag(status_flags::dynallocated));
} else
o.push_back(n);
it++;
}
o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1));
+ // o is now a vector of {numerator, denominator} lists
+
// Determine common denominator
- ex den = exONE();
+ ex den = _ex1();
exvector::const_iterator ait = o.begin(), aitend = o.end();
+//clog << "add::normal uses the following summands:\n";
while (ait != aitend) {
- den = lcm((*ait).denom(false), den, false);
+//clog << " num = " << ait->op(0) << ", den = " << ait->op(1) << endl;
+ den = lcm(ait->op(1), den, false);
ait++;
}
+//clog << " common denominator = " << den << endl;
// Add fractions
- if (den.is_equal(exONE()))
- return (new add(o))->setflag(status_flags::dynallocated);
- else {
+ if (den.is_equal(_ex1())) {
+
+ // Common denominator is 1, simply add all numerators
+ exvector num_seq;
+ for (ait=o.begin(); ait!=aitend; ait++) {
+ num_seq.push_back(ait->op(0));
+ }
+ return (new lst((new add(num_seq))->setflag(status_flags::dynallocated), den))->setflag(status_flags::dynallocated);
+
+ } else {
+
+ // Perform fractional addition
exvector num_seq;
for (ait=o.begin(); ait!=aitend; ait++) {
ex q;
- if (!divide(den, (*ait).denom(false), q, false)) {
+ if (!divide(den, ait->op(1), q, false)) {
// should not happen
throw(std::runtime_error("invalid expression in add::normal, division failed"));
}
- num_seq.push_back((*ait).numer(false) * q);
+ num_seq.push_back((ait->op(0) * q).expand());
}
- ex num = add(num_seq);
+ ex num = (new add(num_seq))->setflag(status_flags::dynallocated);
// 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);
+ if (level == 1)
+ return (new lst(*this, _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
+ ex num = _ex1();
+ ex den = _ex1();
+ ex n;
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));
+ n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1);
+ num *= n.op(0);
+ den *= n.op(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));
+ n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1);
+ num *= n.op(0);
+ den *= n.op(1);
+
+ // Perform fraction cancellation
+ return frac_cancel(num, den);
}
* @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(*this, _ex1()))->setflag(status_flags::dynallocated);
+ else if (level == -max_recursion_level)
+ throw(std::runtime_error("max recursion level reached"));
+
+ // Normalize basis
+ ex n = basis.bp->normal(sym_lst, repl_lst, level-1);
+
+ if (exponent.info(info_flags::integer)) {
+
+ if (exponent.info(info_flags::positive)) {
+
+ // (a/b)^n -> {a^n, b^n}
+ return (new lst(power(n.op(0), exponent), power(n.op(1), exponent)))->setflag(status_flags::dynallocated);
+
+ } else if (exponent.info(info_flags::negative)) {
+
+ // (a/b)^-n -> {b^n, a^n}
+ return (new lst(power(n.op(1), -exponent), power(n.op(0), -exponent)))->setflag(status_flags::dynallocated);
+ }
+
+ } else {
+
+ if (exponent.info(info_flags::positive)) {
+
+ // (a/b)^x -> {sym((a/b)^x), 1}
+ return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+
+ } else if (exponent.info(info_flags::negative)) {
+
+ if (n.op(1).is_equal(_ex1())) {
+
+ // a^-x -> {1, sym(a^x)}
+ return (new lst(_ex1(), replace_with_symbol(power(n.op(0), -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.op(1) / n.op(0), -exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+ }
+
+ } else { // exponent not numeric
+
+ // (a/b)^x -> {sym((a/b)^x, 1}
+ return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+ }
}
}
-/** Implementation of ex::normal() for series. It normalizes each coefficient and
+/** 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());
new_seq.push_back(expair(it->rest.normal(), it->coeff));
it++;
}
+ ex n = pseries(relational(var,point), new_seq);
+ return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated);
+}
+
- ex n = series(var, point, new_seq);
- return replace_with_symbol(n, sym_lst, repl_lst);
+/** Implementation of ex::normal() for relationals. It normalizes both sides.
+ * @see ex::normal */
+ex relational::normal(lst &sym_lst, lst &repl_lst, int level) const
+{
+ return (new lst(relational(lh.normal(), rh.normal(), o), _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.
ex ex::normal(int level) const
{
lst sym_lst, repl_lst;
+
ex e = bp->normal(sym_lst, repl_lst, level);
+ GINAC_ASSERT(is_ex_of_type(e, lst));
+
+ // Re-insert replaced symbols
if (sym_lst.nops() > 0)
- return e.subs(sym_lst, repl_lst);
- else
- return e;
+ e = e.subs(sym_lst, repl_lst);
+
+ // Convert {numerator, denominator} form back to fraction
+ return e.op(0) / e.op(1);
+}
+
+/** 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_ex_of_type(e, lst));
+
+ // Re-insert replaced symbols
+ if (sym_lst.nops() > 0)
+ return e.op(0).subs(sym_lst, repl_lst);
+ else
+ return e.op(0);
+}
+
+/** 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_ex_of_type(e, lst));
+
+ // Re-insert replaced symbols
+ if (sym_lst.nops() > 0)
+ return e.op(1).subs(sym_lst, repl_lst);
+ else
+ return e.op(1);
+}
+
+
+/** Default implementation of ex::to_rational(). It replaces the object with a
+ * temporary symbol.
+ * @see ex::to_rational */
+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.
+ * @see ex::to_rational */
+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.
+ * @see ex::to_rational */
+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.
+ * @see ex::to_rational */
+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.
+ * @see ex::to_rational */
+ex expairseq::to_rational(lst &repl_lst) const
+{
+ epvector s;
+ s.reserve(seq.size());
+ for (epvector::const_iterator it=seq.begin(); it!=seq.end(); ++it) {
+ s.push_back(split_ex_to_pair(recombine_pair_to_ex(*it).to_rational(repl_lst)));
+ // s.push_back(combine_ex_with_coeff_to_pair((*it).rest.to_rational(repl_lst),
+ }
+ 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());
+}
+
+
+/** 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 ex::to_rational(lst &repl_lst) const
+{
+ return bp->to_rational(repl_lst);
+}
+
+
+#ifndef NO_NAMESPACE_GINAC
+} // namespace GiNaC
+#endif // ndef NO_NAMESPACE_GINAC
git://www.ginac.de / ginac.git / blobdiff