/** @file normal.cpp
*
* 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. */
/*
* GiNaC Copyright (C) 1999-2007 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
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#include <algorithm>
#include <map>
#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 "operators.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
// when they are called with two identical arguments.
#define FAST_COMPARE 1
// Set this if you want divide_in_z() to use remembering
#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
* internal ordering of terms, it may not be obvious which symbol this
* function returns for a given expression.
*
* @param e expression to search
* @param x first symbol found (returned)
* @return "false" if no symbol was found, "true" otherwise */
static bool get_first_symbol(const ex &e, ex &x)
{
if (is_a<symbol>(e)) {
x = e;
return true;
} else if (is_exactly_a<add>(e) || is_exactly_a<mul>(e)) {
for (size_t i=0; i<e.nops(); i++)
if (get_first_symbol(e.op(i), x))
return true;
} else if (is_exactly_a<power>(e)) {
if (get_first_symbol(e.op(0), x))
return true;
}
return false;
}
/*
* Statistical information about symbols in polynomials
*/
/** 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
* two polynomials can be created with the function get_symbol_stats().
*
* @see get_symbol_stats */
struct sym_desc {
/** Reference to symbol */
ex sym;
/** Highest degree of symbol in polynomial "a" */
int deg_a;
/** 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 "b" */
int ldeg_b;
/** Maximum of deg_a and deg_b (Used for sorting) */
int max_deg;
/** Maximum number of terms of leading coefficient of symbol in both polynomials */
size_t max_lcnops;
/** 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 std::vector<sym_desc> sym_desc_vec;
// Add symbol the sym_desc_vec (used internally by get_symbol_stats())
static void add_symbol(const ex &s, sym_desc_vec &v)
{
sym_desc_vec::const_iterator it = v.begin(), itend = v.end();
while (it != itend) {
if (it->sym.is_equal(s)) // 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_a<symbol>(e)) {
add_symbol(e, v);
} else if (is_exactly_a<add>(e) || is_exactly_a<mul>(e)) {
for (size_t i=0; i<e.nops(); i++)
collect_symbols(e.op(i), v);
} else if (is_exactly_a<power>(e)) {
collect_symbols(e.op(0), v);
}
}
/** Collect statistical information about symbols in polynomials.
* This function fills in a vector of "sym_desc" structs which contain
* information about the highest and lowest degrees of all symbols that
* appear in two polynomials. The vector is then sorted by minimum
* degree (lowest to highest). The information gathered by this
* function is used by the GCD routines to identify trivial factors
* and to determine which variable to choose as the main variable
* for GCD computation.
*
* @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->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
}
/*
* Computation of LCM of denominators of coefficients of a polynomial
*/
// Compute LCM of denominators of coefficients by going through the
// 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_exactly_a<add>(e)) {
numeric c = *_num1_p;
for (size_t i=0; i<e.nops(); i++)
c = lcmcoeff(e.op(i), c);
return lcm(c, l);
} else if (is_exactly_a<mul>(e)) {
numeric c = *_num1_p;
for (size_t i=0; i<e.nops(); i++)
c *= lcmcoeff(e.op(i), *_num1_p);
return lcm(c, l);
} else if (is_exactly_a<power>(e)) {
if (is_a<symbol>(e.op(0)))
return 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].
*
* @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, *_num1_p);
}
/** 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_exactly_a<mul>(e)) {
size_t num = e.nops();
exvector v; v.reserve(num + 1);
numeric lcm_accum = *_num1_p;
for (size_t i=0; i<num; i++) {
numeric op_lcm = lcmcoeff(e.op(i), *_num1_p);
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_exactly_a<add>(e)) {
size_t num = e.nops();
exvector v; v.reserve(num);
for (size_t 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_exactly_a<power>(e)) {
if (is_a<symbol>(e.op(0)))
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;
}
/** Compute the integer content (= GCD of all numeric coefficients) of an
* expanded polynomial. For a polynomial with rational coefficients, this
* returns g/l where g is the GCD of the coefficients' numerators and l
* is the LCM of the coefficients' denominators.
*
* @return integer content */
numeric ex::integer_content() const
{
return bp->integer_content();
}
numeric basic::integer_content() const
{
return *_num1_p;
}
numeric numeric::integer_content() const
{
return abs(*this);
}
numeric add::integer_content() const
{
epvector::const_iterator it = seq.begin();
epvector::const_iterator itend = seq.end();
numeric c = *_num0_p, l = *_num1_p;
while (it != itend) {
GINAC_ASSERT(!is_exactly_a<numeric>(it->rest));
GINAC_ASSERT(is_exactly_a<numeric>(it->coeff));
c = gcd(ex_to<numeric>(it->coeff).numer(), c);
l = lcm(ex_to<numeric>(it->coeff).denom(), l);
it++;
}
GINAC_ASSERT(is_exactly_a<numeric>(overall_coeff));
c = gcd(ex_to<numeric>(overall_coeff).numer(), c);
l = lcm(ex_to<numeric>(overall_coeff).denom(), l);
return c/l;
}
numeric mul::integer_content() const
{
#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));
}
/*
* Polynomial quotients and remainders
*/
/** Quotient q(x) of polynomials a(x) and b(x) in Q[x].
* It satisfies a(x)=b(x)*q(x)+r(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 quotient of a and b in Q[x] */
ex quo(const ex &a, const ex &b, const ex &x, bool check_args)
{
if (b.is_zero())
throw(std::overflow_error("quo: division by zero"));
if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b))
return a / b;
#if FAST_COMPARE
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 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_exactly_a<numeric>(blcoeff);
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);
}
/** Remainder r(x) of polynomials a(x) and b(x) in Q[x].
* It satisfies a(x)=b(x)*q(x)+r(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 remainder of a(x) and b(x) in Q[x] */
ex rem(const ex &a, const ex &b, const ex &x, bool check_args)
{
if (b.is_zero())
throw(std::overflow_error("rem: division by zero"));
if (is_exactly_a<numeric>(a)) {
if (is_exactly_a<numeric>(b))
return _ex0;
else
return a;
}
#if FAST_COMPARE
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_exactly_a<numeric>(blcoeff);
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 ex &x)
{
ex nd = numer_denom(a);
ex numer = nd.op(0), denom = nd.op(1);
ex q = quo(numer, denom, x);
if (is_exactly_a<fail>(q))
return a;
else
return q + rem(numer, denom, x) / denom;
}
/** Pseudo-remainder of polynomials a(x) and b(x) in Q[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 pseudo-remainder of a(x) and b(x) in Q[x] */
ex prem(const ex &a, const ex &b, const ex &x, bool check_args)
{
if (b.is_zero())
throw(std::overflow_error("prem: division by zero"));
if (is_exactly_a<numeric>(a)) {
if (is_exactly_a<numeric>(b))
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 Q[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 Q[x] */
ex sprem(const ex &a, const ex &b, const ex &x, bool check_args)
{
if (b.is_zero())
throw(std::overflow_error("prem: division by zero"));
if (is_exactly_a<numeric>(a)) {
if (is_exactly_a<numeric>(b))
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)
* @param b second multivariate polynomial (divisor)
* @param q quotient (returned)
* @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 (q left untouched) */
bool divide(const ex &a, const ex &b, ex &q, bool check_args)
{
if (b.is_zero())
throw(std::overflow_error("divide: division by zero"));
if (a.is_zero()) {
q = _ex0;
return true;
}
if (is_exactly_a<numeric>(b)) {
q = a / b;
return true;
} else if (is_exactly_a<numeric>(a))
return false;
#if FAST_COMPARE
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
ex x;
if (!get_first_symbol(a, x) && !get_first_symbol(b, x))
throw(std::invalid_argument("invalid expression in divide()"));
// Try to avoid expanding partially factored expressions.
if (is_exactly_a<mul>(b)) {
// Divide sequentially by each term
ex rem_new, rem_old = a;
for (size_t i=0; i < b.nops(); i++) {
if (! divide(rem_old, b.op(i), rem_new, false))
return false;
rem_old = rem_new;
}
q = rem_new;
return true;
} else if (is_exactly_a<power>(b)) {
const ex& bb(b.op(0));
int exp_b = ex_to<numeric>(b.op(1)).to_int();
ex rem_new, rem_old = a;
for (int i=exp_b; i>0; i--) {
if (! divide(rem_old, bb, rem_new, false))
return false;
rem_old = rem_new;
}
q = rem_new;
return true;
}
if (is_exactly_a<mul>(a)) {
// Divide sequentially each term. If some term in a is divisible
// by b we are done... and if not, we can't really say anything.
size_t i;
ex rem_i;
bool divisible_p = false;
for (i=0; i < a.nops(); ++i) {
if (divide(a.op(i), b, rem_i, false)) {
divisible_p = true;
break;
}
}
if (divisible_p) {
exvector resv;
resv.reserve(a.nops());
for (size_t j=0; j < a.nops(); j++) {
if (j==i)
resv.push_back(rem_i);
else
resv.push_back(a.op(j));
}
q = (new mul(resv))->setflag(status_flags::dynallocated);
return true;
}
} else if (is_exactly_a<power>(a)) {
// The base itself might be divisible by b, in that case we don't
// need to expand a
const ex& ab(a.op(0));
int a_exp = ex_to<numeric>(a.op(1)).to_int();
ex rem_i;
if (divide(ab, b, rem_i, false)) {
q = rem_i*power(ab, a_exp - 1);
return true;
}
for (int i=2; i < a_exp; i++) {
if (divide(power(ab, i), b, rem_i, false)) {
q = rem_i*power(ab, a_exp - i);
return true;
}
} // ... so we *really* need to expand expression.
}
// Polynomial long division (recursive)
ex r = a.expand();
if (r.is_zero()) {
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_exactly_a<numeric>(blcoeff);
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;
}
#if USE_REMEMBER
/*
* Remembering
*/
typedef std::pair<ex, ex> ex2;
typedef std::pair<ex, bool> exbool;
struct ex2_less {
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 std::map<ex2, exbool, ex2_less> ex2_exbool_remember;
#endif
/** Exact polynomial division of a(X) by b(X) in Z[X].
* This functions works like divide() but the input and output polynomials are
* in Z[X] instead of Q[X] (i.e. they have integer coefficients). Unlike
* divide(), it doesn't check whether the input polynomials really are integer
* polynomials, so be careful of what you pass in. Also, you have to run
* get_symbol_stats() over the input polynomials before calling this function
* and pass an iterator to the first element of the sym_desc vector. This
* function is used internally by the heur_gcd().
*
* @param a first multivariate polynomial (dividend)
* @param b second multivariate polynomial (divisor)
* @param q quotient (returned)
* @param var iterator to first element of vector of sym_desc structs
* @return "true" when exact division succeeds (the quotient is returned in
* q), "false" otherwise.
* @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 = _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_exactly_a<numeric>(a)) {
if (is_exactly_a<numeric>(b)) {
q = a / b;
return q.info(info_flags::integer);
} else
return false;
}
#if FAST_COMPARE
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;
}
#endif
if (is_exactly_a<power>(b)) {
const ex& bb(b.op(0));
ex qbar = a;
int exp_b = ex_to<numeric>(b.op(1)).to_int();
for (int i=exp_b; i>0; i--) {
if (!divide_in_z(qbar, bb, q, var))
return false;
qbar = q;
}
return true;
}
if (is_exactly_a<mul>(b)) {
ex qbar = a;
for (const_iterator itrb = b.begin(); itrb != b.end(); ++itrb) {
sym_desc_vec sym_stats;
get_symbol_stats(a, *itrb, sym_stats);
if (!divide_in_z(qbar, *itrb, q, sym_stats.begin()))
return false;
qbar = q;
}
return true;
}
// Main symbol
const ex &x = var->sym;
// 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_p;
ex c;
for (i=0; i<=adeg; i++) {
ex bs = b.subs(x == point, subs_options::no_pattern);
while (bs.is_zero()) {
point += *_num1_p;
bs = b.subs(x == point, subs_options::no_pattern);
}
if (!divide_in_z(a.subs(x == point, subs_options::no_pattern), bs, c, var+1))
return false;
alpha.push_back(point);
u.push_back(c);
point += *_num1_p;
}
// Compute inverses
vector<numeric> rcp; rcp.reserve(adeg + 1);
rcp.push_back(*_num0_p);
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;
#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);
#endif
return true;
}
rdeg = r.degree(x);
}
#if USE_REMEMBER
dr_remember[ex2(a, b)] = exbool(q, false);
#endif
return false;
#endif
}
/*
* Separation of unit part, content part and primitive part of polynomials
*/
/** Compute unit part (= sign of leading coefficient) of a multivariate
* polynomial in Q[x]. The product of unit part, content part, and primitive
* part is the polynomial itself.
*
* @param x main variable
* @return unit part
* @see ex::content, ex::primpart, ex::unitcontprim */
ex ex::unit(const ex &x) const
{
ex c = expand().lcoeff(x);
if (is_exactly_a<numeric>(c))
return c.info(info_flags::negative) ?_ex_1 : _ex1;
else {
ex y;
if (get_first_symbol(c, y))
return c.unit(y);
else
throw(std::invalid_argument("invalid expression in unit()"));
}
}
/** Compute content part (= unit normal GCD of all coefficients) of a
* multivariate polynomial in Q[x]. The product of unit part, content part,
* and primitive part is the polynomial itself.
*
* @param x main variable
* @return content part
* @see ex::unit, ex::primpart, ex::unitcontprim */
ex ex::content(const ex &x) const
{
if (is_exactly_a<numeric>(*this))
return info(info_flags::negative) ? -*this : *this;
ex e = expand();
if (e.is_zero())
return _ex0;
// First, divide out the integer content (which we can calculate very efficiently).
// If the leading coefficient of the quotient is an integer, we are done.
ex c = e.integer_content();
ex r = e / c;
int deg = r.degree(x);
ex lcoeff = r.coeff(x, deg);
if (lcoeff.info(info_flags::integer))
return c;
// GCD of all coefficients
int ldeg = r.ldegree(x);
if (deg == ldeg)
return lcoeff * c / lcoeff.unit(x);
ex cont = _ex0;
for (int i=ldeg; i<=deg; i++)
cont = gcd(r.coeff(x, i), cont, NULL, NULL, false);
return cont * c;
}
/** Compute primitive part of a multivariate polynomial in Q[x]. The result
* will be a unit-normal polynomial with a content part of 1. The product
* of unit part, content part, and primitive part is the polynomial itself.
*
* @param x main variable
* @return primitive part
* @see ex::unit, ex::content, ex::unitcontprim */
ex ex::primpart(const ex &x) const
{
// We need to compute the unit and content anyway, so call unitcontprim()
ex u, c, p;
unitcontprim(x, u, c, p);
return p;
}
/** Compute primitive part of a multivariate polynomial in Q[x] when the
* content part is already known. This function is faster in computing the
* primitive part than the previous function.
*
* @param x main variable
* @param c previously computed content part
* @return primitive part */
ex ex::primpart(const ex &x, const ex &c) const
{
if (is_zero() || c.is_zero())
return _ex0;
if (is_exactly_a<numeric>(*this))
return _ex1;
// Divide by unit and content to get primitive part
ex u = unit(x);
if (is_exactly_a<numeric>(c))
return *this / (c * u);
else
return quo(*this, c * u, x, false);
}
/** Compute unit part, content part, and primitive part of a multivariate
* polynomial in Q[x]. The product of the three parts is the polynomial
* itself.
*
* @param x main variable
* @param u unit part (returned)
* @param c content part (returned)
* @param p primitive part (returned)
* @see ex::unit, ex::content, ex::primpart */
void ex::unitcontprim(const ex &x, ex &u, ex &c, ex &p) const
{
// Quick check for zero (avoid expanding)
if (is_zero()) {
u = _ex1;
c = p = _ex0;
return;
}
// Special case: input is a number
if (is_exactly_a<numeric>(*this)) {
if (info(info_flags::negative)) {
u = _ex_1;
c = abs(ex_to<numeric>(*this));
} else {
u = _ex1;
c = *this;
}
p = _ex1;
return;
}
// Expand input polynomial
ex e = expand();
if (e.is_zero()) {
u = _ex1;
c = p = _ex0;
return;
}
// Compute unit and content
u = unit(x);
c = content(x);
// Divide by unit and content to get primitive part
if (c.is_zero()) {
p = _ex0;
return;
}
if (is_exactly_a<numeric>(c))
p = *this / (c * u);
else
p = quo(e, c * u, x, false);
}
/*
* GCD of multivariate polynomials
*/
/** 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)
{
#if STATISTICS
sr_gcd_called++;
#endif
// The first symbol is our main variable
const ex &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);
// First element of subresultant sequence
ex r = _ex0, ri = _ex1, psi = _ex1;
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_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_exactly_a<numeric>(r))
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_in_z(pow(ri, delta), pow(psi, delta-1), psi, var+1);
delta = cdeg - ddeg;
}
}
/** Return maximum (absolute value) coefficient of a polynomial.
* This function is used internally by heur_gcd().
*
* @return maximum coefficient
* @see heur_gcd */
numeric ex::max_coefficient() const
{
return bp->max_coefficient();
}
/** Implementation ex::max_coefficient().
* @see heur_gcd */
numeric basic::max_coefficient() const
{
return *_num1_p;
}
numeric numeric::max_coefficient() const
{
return abs(*this);
}
numeric add::max_coefficient() const
{
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() const
{
#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 an expanded multivariate
* polynomial. This function is usually used internally by heur_gcd().
*
* @param xi modulus
* @return mapped polynomial
* @see heur_gcd */
ex basic::smod(const numeric &xi) const
{
return *this;
}
ex numeric::smod(const numeric &xi) const
{
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) {
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 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);
}
/** xi-adic polynomial interpolation */
static ex interpolate(const ex &gamma, const numeric &xi, const ex &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.
* get_symbol_stats() must have been called previously with the input
* polynomials and an iterator to the first element of the sym_desc vector
* passed in. This function is used internally by gcd().
*
* @param a first multivariate polynomial (expanded)
* @param b second multivariate polynomial (expanded)
* @param ca cofactor of polynomial a (returned), NULL to suppress
* calculation of cofactor
* @param cb cofactor of polynomial b (returned), NULL to suppress
* calculation of cofactor
* @param var iterator to first element of vector of sym_desc structs
* @return the GCD as a new expression
* @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 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_exactly_a<numeric>(a) && is_exactly_a<numeric>(b)) {
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 ex &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_p) + (*_num2_p);
else
xi = mp * (*_num2_p) + (*_num2_p);
// 6 tries maximum
for (int t=0; t<6; t++) {
if (xi.int_length() * maxdeg > 100000) {
throw gcdheu_failed();
}
// Apply evaluation homomorphism and calculate GCD
ex cp, cq;
ex gamma = heur_gcd(p.subs(x == xi, subs_options::no_pattern), q.subs(x == xi, subs_options::no_pattern), &cp, &cq, var+1).expand();
if (!is_exactly_a<fail>(gamma)) {
// 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;
return g;
}
}
// Next evaluation point
xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011));
}
return (new fail())->setflag(status_flags::dynallocated);
}
/** Compute GCD (Greatest Common Divisor) of multivariate polynomials a(X)
* and b(X) in Z[X]. Optionally also compute the cofactors of a and b,
* defined by a = ca * gcd(a, b) and b = cb * gcd(a, b).
*
* @param a first multivariate polynomial
* @param b second multivariate polynomial
* @param ca pointer to expression that will receive the cofactor of a, or NULL
* @param cb pointer to expression that will receive the cofactor of b, or NULL
* @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)
{
#if STATISTICS
gcd_called++;
#endif
// GCD of numerics -> CLN
if (is_exactly_a<numeric>(a) && is_exactly_a<numeric>(b)) {
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_exactly_a<mul>(a)) {
if (is_exactly_a<mul>(b) && b.nops() > a.nops())
goto factored_b;
factored_a:
size_t num = a.nops();
exvector g; g.reserve(num);
exvector acc_ca; acc_ca.reserve(num);
ex part_b = b;
for (size_t i=0; i<num; i++) {
ex part_ca, part_cb;
g.push_back(gcd(a.op(i), part_b, &part_ca, &part_cb, check_args));
acc_ca.push_back(part_ca);
part_b = part_cb;
}
if (ca)
*ca = (new mul(acc_ca))->setflag(status_flags::dynallocated);
if (cb)
*cb = part_b;
return (new mul(g))->setflag(status_flags::dynallocated);
} else if (is_exactly_a<mul>(b)) {
if (is_exactly_a<mul>(a) && a.nops() > b.nops())
goto factored_a;
factored_b:
size_t num = b.nops();
exvector g; g.reserve(num);
exvector acc_cb; acc_cb.reserve(num);
ex part_a = a;
for (size_t i=0; i<num; i++) {
ex part_ca, part_cb;
g.push_back(gcd(part_a, b.op(i), &part_ca, &part_cb, check_args));
acc_cb.push_back(part_cb);
part_a = part_ca;
}
if (ca)
*ca = part_a;
if (cb)
*cb = (new mul(acc_cb))->setflag(status_flags::dynallocated);
return (new mul(g))->setflag(status_flags::dynallocated);
}
#if FAST_COMPARE
// Input polynomials of the form poly^n are sometimes also trivial
if (is_exactly_a<power>(a)) {
ex p = a.op(0);
const ex& exp_a = a.op(1);
if (is_exactly_a<power>(b)) {
ex pb = b.op(0);
const ex& exp_b = b.op(1);
if (p.is_equal(pb)) {
// a = p^n, b = p^m, gcd = p^min(n, m)
if (exp_a < exp_b) {
if (ca)
*ca = _ex1;
if (cb)
*cb = power(p, exp_b - exp_a);
return power(p, exp_a);
} else {
if (ca)
*ca = power(p, exp_a - exp_b);
if (cb)
*cb = _ex1;
return power(p, exp_b);
}
} else {
ex p_co, pb_co;
ex p_gcd = gcd(p, pb, &p_co, &pb_co, check_args);
if (p_gcd.is_equal(_ex1)) {
// a(x) = p(x)^n, b(x) = p_b(x)^m, gcd (p, p_b) = 1 ==>
// gcd(a,b) = 1
if (ca)
*ca = a;
if (cb)
*cb = b;
return _ex1;
// XXX: do I need to check for p_gcd = -1?
} else {
// there are common factors:
// a(x) = g(x)^n A(x)^n, b(x) = g(x)^m B(x)^m ==>
// gcd(a, b) = g(x)^n gcd(A(x)^n, g(x)^(n-m) B(x)^m
if (exp_a < exp_b) {
return power(p_gcd, exp_a)*
gcd(power(p_co, exp_a), power(p_gcd, exp_b-exp_a)*power(pb_co, exp_b), ca, cb, false);
} else {
return power(p_gcd, exp_b)*
gcd(power(p_gcd, exp_a - exp_b)*power(p_co, exp_a), power(pb_co, exp_b), ca, cb, false);
}
} // p_gcd.is_equal(_ex1)
} // p.is_equal(pb)
} else {
if (p.is_equal(b)) {
// a = p^n, b = p, gcd = p
if (ca)
*ca = power(p, a.op(1) - 1);
if (cb)
*cb = _ex1;
return p;
}
ex p_co, bpart_co;
ex p_gcd = gcd(p, b, &p_co, &bpart_co, false);
if (p_gcd.is_equal(_ex1)) {
// a(x) = p(x)^n, gcd(p, b) = 1 ==> gcd(a, b) = 1
if (ca)
*ca = a;
if (cb)
*cb = b;
return _ex1;
} else {
// a(x) = g(x)^n A(x)^n, b(x) = g(x) B(x) ==> gcd(a, b) = g(x) gcd(g(x)^(n-1) A(x)^n, B(x))
return p_gcd*gcd(power(p_gcd, exp_a-1)*power(p_co, exp_a), bpart_co, ca, cb, false);
}
} // is_exactly_a<power>(b)
} else if (is_exactly_a<power>(b)) {
ex p = b.op(0);
if (p.is_equal(a)) {
// a = p, b = p^n, gcd = p
if (ca)
*ca = _ex1;
if (cb)
*cb = power(p, b.op(1) - 1);
return p;
}
ex p_co, apart_co;
const ex& exp_b(b.op(1));
ex p_gcd = gcd(a, p, &apart_co, &p_co, false);
if (p_gcd.is_equal(_ex1)) {
// b=p(x)^n, gcd(a, p) = 1 ==> gcd(a, b) == 1
if (ca)
*ca = a;
if (cb)
*cb = b;
return _ex1;
} else {
// there are common factors:
// a(x) = g(x) A(x), b(x) = g(x)^n B(x)^n ==> gcd = g(x) gcd(g(x)^(n-1) A(x)^n, B(x))
return p_gcd*gcd(apart_co, power(p_gcd, exp_b-1)*power(p_co, exp_b), ca, cb, false);
} // p_gcd.is_equal(_ex1)
}
#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 = _ex1;
if (cb)
*cb = _ex1;
return a;
}
#endif
if (is_a<symbol>(aex)) {
if (! bex.subs(aex==_ex0, subs_options::no_pattern).is_zero()) {
if (ca)
*ca = a;
if (cb)
*cb = b;
return _ex1;
}
}
if (is_a<symbol>(bex)) {
if (! aex.subs(bex==_ex0, subs_options::no_pattern).is_zero()) {
if (ca)
*ca = a;
if (cb)
*cb = b;
return _ex1;
}
}
if (is_exactly_a<numeric>(aex)) {
numeric bcont = bex.integer_content();
numeric g = gcd(ex_to<numeric>(aex), bcont);
if (ca)
*ca = ex_to<numeric>(aex)/g;
if (cb)
*cb = bex/g;
return g;
}
if (is_exactly_a<numeric>(bex)) {
numeric acont = aex.integer_content();
numeric g = gcd(ex_to<numeric>(bex), acont);
if (ca)
*ca = aex/g;
if (cb)
*cb = ex_to<numeric>(bex)/g;
return g;
}
// Gather symbol statistics
sym_desc_vec sym_stats;
get_symbol_stats(a, b, sym_stats);
// The symbol with least degree which is contained in both polynomials
// is our main variable
sym_desc_vec::iterator vari = sym_stats.begin();
while ((vari != sym_stats.end()) &&
(((vari->ldeg_b == 0) && (vari->deg_b == 0)) ||
((vari->ldeg_a == 0) && (vari->deg_a == 0))))
vari++;
// No common symbols at all, just return 1:
if (vari == sym_stats.end()) {
// N.B: keep cofactors factored
if (ca)
*ca = a;
if (cb)
*cb = b;
return _ex1;
}
// move symbols which contained only in one of the polynomials
// to the end:
rotate(sym_stats.begin(), vari, sym_stats.end());
sym_desc_vec::const_iterator var = sym_stats.begin();
const ex &x = var->sym;
// 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);
return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common;
}
// Try to eliminate variables
if (var->deg_a == 0 && var->deg_b != 0 ) {
ex bex_u, bex_c, bex_p;
bex.unitcontprim(x, bex_u, bex_c, bex_p);
ex g = gcd(aex, bex_c, ca, cb, false);
if (cb)
*cb *= bex_u * bex_p;
return g;
} else if (var->deg_b == 0 && var->deg_a != 0) {
ex aex_u, aex_c, aex_p;
aex.unitcontprim(x, aex_u, aex_c, aex_p);
ex g = gcd(aex_c, bex, ca, cb, false);
if (ca)
*ca *= aex_u * aex_p;
return g;
}
// Try heuristic algorithm first, fall back to PRS if that failed
ex g;
try {
g = heur_gcd(aex, bex, ca, cb, var);
} catch (gcdheu_failed) {
g = fail();
}
if (is_exactly_a<fail>(g)) {
#if STATISTICS
heur_gcd_failed++;
#endif
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);
}
} else {
if (g.is_equal(_ex1)) {
// Keep cofactors factored if possible
if (ca)
*ca = a;
if (cb)
*cb = b;
}
}
return g;
}
/** Compute LCM (Least Common Multiple) of multivariate polynomials in Z[X].
*
* @param a first multivariate polynomial
* @param b second multivariate polynomial
* @param check_args check whether a and b are polynomials with rational
* coefficients (defaults to "true")
* @return the LCM as a new expression */
ex lcm(const ex &a, const ex &b, bool check_args)
{
if (is_exactly_a<numeric>(a) && is_exactly_a<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"));
ex ca, cb;
ex g = gcd(a, b, &ca, &cb, false);
return ca * cb * g;
}
/*
* Square-free factorization
*/
/** 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)
{
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 l 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_exactly_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_a<symbol>(args.op(0)))
throw (std::runtime_error("sqrfree(): invalid factorization variable"));
const symbol &x = ex_to<symbol>(args.op(0));
// 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 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)
{
// 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;
size_t num_yun = yun.size();
exvector factor; factor.reserve(num_yun);
exvector cofac; cofac.reserve(num_yun);
for (size_t i=0; i<num_yun; i++) {
if (!yun[i].is_equal(_ex1)) {
for (size_t j=0; j<=i; j++) {
factor.push_back(pow(yun[i], j+1));
ex prod = _ex1;
for (size_t 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());
}
}
}
size_t 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 (size_t 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 (size_t i=0; i<num_factors; i++)
vars(i, 0) = symbol();
matrix sol = sys.solve(vars, rhs);
// Sum up decomposed fractions
ex sum = 0;
for (size_t i=0; i<num_factors; i++)
sum += sol(i, 0) / factor[i];
return red_poly + sum;
}
/*
* Normal form of rational functions
*/
/*
* 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 and expression are appended to repl, for
* a later application of subs().
* @see ex::normal */
static ex replace_with_symbol(const ex & e, exmap & repl, exmap & rev_lookup)
{
// Expression already replaced? Then return the assigned symbol
exmap::const_iterator it = rev_lookup.find(e);
if (it != rev_lookup.end())
return it->second;
// Otherwise create new symbol and add to list, taking care that the
// replacement expression doesn't itself contain symbols from repl,
// because subs() is not recursive
ex es = (new symbol)->setflag(status_flags::dynallocated);
ex e_replaced = e.subs(repl, subs_options::no_pattern);
repl.insert(std::make_pair(es, e_replaced));
rev_lookup.insert(std::make_pair(e_replaced, es));
return es;
}
/** Create a symbol for replacing the expression "e" (or return a previously
* assigned symbol). The symbol and expression are appended to repl, and the
* symbol is returned.
* @see basic::to_rational
* @see basic::to_polynomial */
static ex replace_with_symbol(const ex & e, exmap & repl)
{
// Expression already replaced? Then return the assigned symbol
for (exmap::const_iterator it = repl.begin(); it != repl.end(); ++it)
if (it->second.is_equal(e))
return it->first;
// Otherwise create new symbol and add to list, taking care that the
// replacement expression doesn't itself contain symbols from repl,
// because subs() is not recursive
ex es = (new symbol)->setflag(status_flags::dynallocated);
ex e_replaced = e.subs(repl, subs_options::no_pattern);
repl.insert(std::make_pair(es, e_replaced));
return es;
}
/** 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(exmap & repl, exmap & rev_lookup, int level) const
{
if (nops() == 0)
return (new lst(replace_with_symbol(*this, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
else {
if (level == 1)
return (new lst(replace_with_symbol(*this, repl, rev_lookup), _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), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
}
}
}
/** Implementation of ex::normal() for symbols. This returns the unmodified symbol.
* @see ex::normal */
ex symbol::normal(exmap & repl, exmap & rev_lookup, int level) const
{
return (new lst(*this, _ex1))->setflag(status_flags::dynallocated);
}
/** Implementation of ex::normal() 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::normal */
ex numeric::normal(exmap & repl, exmap & rev_lookup, int level) const
{
numeric num = numer();
ex numex = num;
if (num.is_real()) {
if (!num.is_integer())
numex = replace_with_symbol(numex, repl, rev_lookup);
} else { // complex
numeric re = num.real(), im = num.imag();
ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl, rev_lookup);
ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl, rev_lookup);
numex = re_ex + im_ex * replace_with_symbol(I, repl, rev_lookup);
}
// Denominator is always a real integer (see numeric::denom())
return (new lst(numex, denom()))->setflag(status_flags::dynallocated);
}
/** 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;
numeric pre_factor = *_num1_p;
//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;
}
// Make denominator unit normal (i.e. coefficient of first symbol
// as defined by get_first_symbol() is made positive)
if (is_exactly_a<numeric>(den)) {
if (ex_to<numeric>(den).is_negative()) {
num *= _ex_1;
den *= _ex_1;
}
} else {
ex x;
if (get_first_symbol(den, x)) {
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 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);
}
/** Implementation of ex::normal() for a sum. It expands terms and performs
* fractional addition.
* @see ex::normal */
ex add::normal(exmap & repl, exmap & rev_lookup, int level) const
{
if (level == 1)
return (new lst(replace_with_symbol(*this, repl, rev_lookup), _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(repl, rev_lookup, level-1);
nums.push_back(n.op(0));
dens.push_back(n.op(1));
it++;
}
ex n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup, 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);
}
/** Implementation of ex::normal() for a product. It cancels common factors
* from fractions.
* @see ex::normal() */
ex mul::normal(exmap & repl, exmap & rev_lookup, int level) const
{
if (level == 1)
return (new lst(replace_with_symbol(*this, repl, rev_lookup), _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(repl, rev_lookup, level-1);
num.push_back(n.op(0));
den.push_back(n.op(1));
it++;
}
n = ex_to<numeric>(overall_coeff).normal(repl, rev_lookup, 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));
}
/** Implementation of ex::normal([B) for powers. It normalizes the basis,
* distributes integer exponents to numerator and denominator, and replaces
* non-integer powers by temporary symbols.
* @see ex::normal */
ex power::normal(exmap & repl, exmap & rev_lookup, int level) const
{
if (level == 1)
return (new lst(replace_with_symbol(*this, repl, rev_lookup), _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(repl, rev_lookup, level-1);
ex n_exponent = ex_to<basic>(exponent).normal(repl, rev_lookup, 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), repl, rev_lookup), _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), repl, rev_lookup)))->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), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
}
}
}
// (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), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
}
/** Implementation of ex::normal() for pseries. It normalizes each coefficient
* and replaces the series by a temporary symbol.
* @see ex::normal */
ex pseries::normal(exmap & repl, exmap & rev_lookup, int level) const
{
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, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated);
}
/** Normalization of rational functions.
* 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
* 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.
*
* @param level maximum depth of recursion
* @return normalized expression */
ex ex::normal(int level) const
{
exmap repl, rev_lookup;
ex e = bp->normal(repl, rev_lookup, level);
GINAC_ASSERT(is_a<lst>(e));
// Re-insert replaced symbols
if (!repl.empty())
e = e.subs(repl, subs_options::no_pattern);
// 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() const
{
exmap repl, rev_lookup;
ex e = bp->normal(repl, rev_lookup, 0);
GINAC_ASSERT(is_a<lst>(e));
// Re-insert replaced symbols
if (repl.empty())
return e.op(0);
else
return e.op(0).subs(repl, subs_options::no_pattern);
}
/** 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() const
{
exmap repl, rev_lookup;
ex e = bp->normal(repl, rev_lookup, 0);
GINAC_ASSERT(is_a<lst>(e));
// Re-insert replaced symbols
if (repl.empty())
return e.op(1);
else
return e.op(1).subs(repl, subs_options::no_pattern);
}
/** 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() const
{
exmap repl, rev_lookup;
ex e = bp->normal(repl, rev_lookup, 0);
GINAC_ASSERT(is_a<lst>(e));
// Re-insert replaced symbols
if (repl.empty())
return e;
else
return e.subs(repl, subs_options::no_pattern);
}
/** Rationalization of non-rational functions.
* This function converts a general expression to a rational function
* 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 map specified by the
* repl parameter, ready to be passed as an argument to ex::subs().
*
* @param repl collects all temporary symbols and their replacements
* @return rationalized expression */
ex ex::to_rational(exmap & repl) const
{
return bp->to_rational(repl);
}
// GiNaC 1.1 compatibility function
ex ex::to_rational(lst & repl_lst) const
{
// Convert lst to exmap
exmap m;
for (lst::const_iterator it = repl_lst.begin(); it != repl_lst.end(); ++it)
m.insert(std::make_pair(it->op(0), it->op(1)));
ex ret = bp->to_rational(m);
// Convert exmap back to lst
repl_lst.remove_all();
for (exmap::const_iterator it = m.begin(); it != m.end(); ++it)
repl_lst.append(it->first == it->second);
return ret;
}
ex ex::to_polynomial(exmap & repl) const
{
return bp->to_polynomial(repl);
}
// GiNaC 1.1 compatibility function
ex ex::to_polynomial(lst & repl_lst) const
{
// Convert lst to exmap
exmap m;
for (lst::const_iterator it = repl_lst.begin(); it != repl_lst.end(); ++it)
m.insert(std::make_pair(it->op(0), it->op(1)));
ex ret = bp->to_polynomial(m);
// Convert exmap back to lst
repl_lst.remove_all();
for (exmap::const_iterator it = m.begin(); it != m.end(); ++it)
repl_lst.append(it->first == it->second);
return ret;
}
/** Default implementation of ex::to_rational(). This replaces the object with
* a temporary symbol. */
ex basic::to_rational(exmap & repl) const
{
return replace_with_symbol(*this, repl);
}
ex basic::to_polynomial(exmap & repl) const
{
return replace_with_symbol(*this, repl);
}
/** Implementation of ex::to_rational() for symbols. This returns the
* unmodified symbol. */
ex symbol::to_rational(exmap & repl) const
{
return *this;
}
/** Implementation of ex::to_polynomial() for symbols. This returns the
* unmodified symbol. */
ex symbol::to_polynomial(exmap & repl) 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(exmap & repl) const
{
if (is_real()) {
if (!is_rational())
return replace_with_symbol(*this, repl);
} else { // complex
numeric re = real();
numeric im = imag();
ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl);
ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl);
return re_ex + im_ex * replace_with_symbol(I, repl);
}
return *this;
}
/** Implementation of ex::to_polynomial() for a numeric. It splits complex
* numbers into re+I*im and replaces I and non-integer real numbers with a
* temporary symbol. */
ex numeric::to_polynomial(exmap & repl) const
{
if (is_real()) {
if (!is_integer())
return replace_with_symbol(*this, repl);
} else { // complex
numeric re = real();
numeric im = imag();
ex re_ex = re.is_integer() ? re : replace_with_symbol(re, repl);
ex im_ex = im.is_integer() ? im : replace_with_symbol(im, repl);
return re_ex + im_ex * replace_with_symbol(I, repl);
}
return *this;
}
/** Implementation of ex::to_rational() for powers. It replaces non-integer
* powers by temporary symbols. */
ex power::to_rational(exmap & repl) const
{
if (exponent.info(info_flags::integer))
return power(basis.to_rational(repl), exponent);
else
return replace_with_symbol(*this, repl);
}
/** Implementation of ex::to_polynomial() for powers. It replaces non-posint
* powers by temporary symbols. */
ex power::to_polynomial(exmap & repl) const
{
if (exponent.info(info_flags::posint))
return power(basis.to_rational(repl), exponent);
else if (exponent.info(info_flags::negint))
{
ex basis_pref = collect_common_factors(basis);
if (is_exactly_a<mul>(basis_pref) || is_exactly_a<power>(basis_pref)) {
// (A*B)^n will be automagically transformed to A^n*B^n
ex t = power(basis_pref, exponent);
return t.to_polynomial(repl);
}
else
return power(replace_with_symbol(power(basis, _ex_1), repl), -exponent);
}
else
return replace_with_symbol(*this, repl);
}
/** Implementation of ex::to_rational() for expairseqs. */
ex expairseq::to_rational(exmap & repl) 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)));
++i;
}
ex oc = overall_coeff.to_rational(repl);
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());
}
/** Implementation of ex::to_polynomial() for expairseqs. */
ex expairseq::to_polynomial(exmap & repl) 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_polynomial(repl)));
++i;
}
ex oc = overall_coeff.to_polynomial(repl);
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());
}
/** Remove the common factor in the terms of a sum 'e' by calculating the GCD,
* and multiply it into the expression 'factor' (which needs to be initialized
* to 1, unless you're accumulating factors). */
static ex find_common_factor(const ex & e, ex & factor, exmap & repl)
{
if (is_exactly_a<add>(e)) {
size_t num = e.nops();
exvector terms; terms.reserve(num);
ex gc;
// Find the common GCD
for (size_t i=0; i<num; i++) {
ex x = e.op(i).to_polynomial(repl);
if (is_exactly_a<add>(x) || is_exactly_a<mul>(x) || is_a<power>(x)) {
ex f = 1;
x = find_common_factor(x, f, repl);
x *= f;
}
if (i == 0)
gc = x;
else
gc = gcd(gc, x);
terms.push_back(x);
}
if (gc.is_equal(_ex1))
return e;
// The GCD is the factor we pull out
factor *= gc;
// Now divide all terms by the GCD
for (size_t i=0; i<num; i++) {
ex x;
// Try to avoid divide() because it expands the polynomial
ex &t = terms[i];
if (is_exactly_a<mul>(t)) {
for (size_t j=0; j<t.nops(); j++) {
if (t.op(j).is_equal(gc)) {
exvector v; v.reserve(t.nops());
for (size_t k=0; k<t.nops(); k++) {
if (k == j)
v.push_back(_ex1);
else
v.push_back(t.op(k));
}
t = (new mul(v))->setflag(status_flags::dynallocated);
goto term_done;
}
}
}
divide(t, gc, x);
t = x;
term_done: ;
}
return (new add(terms))->setflag(status_flags::dynallocated);
} else if (is_exactly_a<mul>(e)) {
size_t num = e.nops();
exvector v; v.reserve(num);
for (size_t i=0; i<num; i++)
v.push_back(find_common_factor(e.op(i), factor, repl));
return (new mul(v))->setflag(status_flags::dynallocated);
} else if (is_exactly_a<power>(e)) {
const ex e_exp(e.op(1));
if (e_exp.info(info_flags::integer)) {
ex eb = e.op(0).to_polynomial(repl);
ex factor_local(_ex1);
ex pre_res = find_common_factor(eb, factor_local, repl);
factor *= power(factor_local, e_exp);
return power(pre_res, e_exp);
} else
return e.to_polynomial(repl);
} else
return e;
}
/** Collect common factors in sums. This converts expressions like
* 'a*(b*x+b*y)' to 'a*b*(x+y)'. */
ex collect_common_factors(const ex & e)
{
if (is_exactly_a<add>(e) || is_exactly_a<mul>(e) || is_exactly_a<power>(e)) {
exmap repl;
ex factor = 1;
ex r = find_common_factor(e, factor, repl);
return factor.subs(repl, subs_options::no_pattern) * r.subs(repl, subs_options::no_pattern);
} else
return e;
}
/** Resultant of two expressions e1,e2 with respect to symbol s.
* Method: Compute determinant of Sylvester matrix of e1,e2,s. */
ex resultant(const ex & e1, const ex & e2, const ex & s)
{
const ex ee1 = e1.expand();
const ex ee2 = e2.expand();
if (!ee1.info(info_flags::polynomial) ||
!ee2.info(info_flags::polynomial))
throw(std::runtime_error("resultant(): arguments must be polynomials"));
const int h1 = ee1.degree(s);
const int l1 = ee1.ldegree(s);
const int h2 = ee2.degree(s);
const int l2 = ee2.ldegree(s);
const int msize = h1 + h2;
matrix m(msize, msize);
for (int l = h1; l >= l1; --l) {
const ex e = ee1.coeff(s, l);
for (int k = 0; k < h2; ++k)
m(k, k+h1-l) = e;
}
for (int l = h2; l >= l2; --l) {
const ex e = ee2.coeff(s, l);
for (int k = 0; k < h1; ++k)
m(k+h2, k+h2-l) = e;
}
return m.determinant();
}
} // namespace GiNaC
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