/** @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) 1999-2008 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 #include #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(e)) { x = e; return true; } else if (is_exactly_a(e) || is_exactly_a(e)) { for (size_t i=0; i(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_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(e)) { add_symbol(e, v); } else if (is_exactly_a(e) || is_exactly_a(e)) { for (size_t i=0; i(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(e).denom(), l); else if (is_exactly_a(e)) { numeric c = *_num1_p; for (size_t i=0; i(e)) { numeric c = *_num1_p; for (size_t i=0; i(e)) { if (is_a(e.op(0))) return l; else return pow(lcmcoeff(e.op(0), l), ex_to(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(e)) { size_t num = e.nops(); exvector v; v.reserve(num + 1); numeric lcm_accum = *_num1_p; for (size_t i=0; isetflag(status_flags::dynallocated); } else if (is_exactly_a(e)) { size_t num = e.nops(); exvector v; v.reserve(num); for (size_t i=0; isetflag(status_flags::dynallocated); } else if (is_exactly_a(e)) { if (is_a(e.op(0))) return e * lcm; else return pow(multiply_lcm(e.op(0), lcm.power(ex_to(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(it->rest)); GINAC_ASSERT(is_exactly_a(it->coeff)); c = gcd(ex_to(it->coeff).numer(), c); l = lcm(ex_to(it->coeff).denom(), l); it++; } GINAC_ASSERT(is_exactly_a(overall_coeff)); c = gcd(ex_to(overall_coeff).numer(), c); l = lcm(ex_to(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(recombine_pair_to_ex(*it))); ++it; } #endif // def DO_GINAC_ASSERT GINAC_ASSERT(is_exactly_a(overall_coeff)); return abs(ex_to(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(a) && is_exactly_a(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(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(a)) { if (is_exactly_a(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(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(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(a)) { if (is_exactly_a(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(a)) { if (is_exactly_a(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(b)) { q = a / b; return true; } else if (is_exactly_a(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(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(b)) { const ex& bb(b.op(0)); int exp_b = ex_to(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(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(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(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(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 ex2; typedef std::pair 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_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(a)) { if (is_exactly_a(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(b)) { const ex& bb(b.op(0)); ex qbar = a; int exp_b = ex_to(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(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 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 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=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(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(*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(*this)) return _ex1; // Divide by unit and content to get primitive part ex u = unit(x); if (is_exactly_a(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(*this)) { if (info(info_flags::negative)) { u = _ex_1; c = abs(ex_to(*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(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(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(overall_coeff)); numeric cur_max = abs(ex_to(overall_coeff)); while (it != itend) { numeric a; GINAC_ASSERT(!is_exactly_a(it->rest)); a = abs(ex_to(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(recombine_pair_to_ex(*it))); it++; } #endif // def DO_GINAC_ASSERT GINAC_ASSERT(is_exactly_a(overall_coeff)); return abs(ex_to(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(it->rest)); numeric coeff = GiNaC::smod(ex_to(it->coeff), xi); if (!coeff.is_zero()) newseq.push_back(expair(it->rest, coeff)); it++; } GINAC_ASSERT(is_exactly_a(overall_coeff)); numeric coeff = GiNaC::smod(ex_to(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(recombine_pair_to_ex(*it))); it++; } #endif // def DO_GINAC_ASSERT mul * mulcopyp = new mul(*this); GINAC_ASSERT(is_exactly_a(overall_coeff)); mulcopyp->overall_coeff = GiNaC::smod(ex_to(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 integer multivariate polynomial (expanded) * @param b second integer multivariate polynomial (expanded) * @param ca cofactor of polynomial a (returned), NULL to suppress * calculation of cofactor * @param cb cofactor of polynomial b (returned), NULL to suppress * calculation of cofactor * @param var iterator to first element of vector of sym_desc structs * @param res the GCD (returned) * @return true if GCD was computed, false otherwise. * @see gcd * @exception gcdheu_failed() */ static bool heur_gcd_z(ex& res, const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var) { #if STATISTICS heur_gcd_called++; #endif // Algorithm only works for non-vanishing input polynomials if (a.is_zero() || b.is_zero()) return false; // GCD of two numeric values -> CLN if (is_exactly_a(a) && is_exactly_a(b)) { numeric g = gcd(ex_to(a), ex_to(b)); if (ca) *ca = ex_to(a) / g; if (cb) *cb = ex_to(b) / g; res = g; return true; } // 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; bool found = heur_gcd_z(gamma, p.subs(x == xi, subs_options::no_pattern), q.subs(x == xi, subs_options::no_pattern), &cp, &cq, var+1); if (found) { gamma = gamma.expand(); // Reconstruct polynomial from GCD of mapped polynomials ex g = interpolate(gamma, xi, x, maxdeg); // 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; res = g; return true; } } // Next evaluation point xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011)); } return false; } /** Compute GCD of multivariate polynomials using the heuristic GCD algorithm. * get_symbol_stats() must have been called previously with the input * polynomials and an iterator to the first element of the sym_desc vector * passed in. This function is used internally by gcd(). * * @param a first rational multivariate polynomial (expanded) * @param b second rational multivariate polynomial (expanded) * @param ca cofactor of polynomial a (returned), NULL to suppress * calculation of cofactor * @param cb cofactor of polynomial b (returned), NULL to suppress * calculation of cofactor * @param var iterator to first element of vector of sym_desc structs * @param res the GCD (returned) * @return true if GCD was computed, false otherwise. * @see heur_gcd_z * @see gcd */ static bool heur_gcd(ex& res, const ex& a, const ex& b, ex *ca, ex *cb, sym_desc_vec::const_iterator var) { if (a.info(info_flags::integer_polynomial) && b.info(info_flags::integer_polynomial)) { try { return heur_gcd_z(res, a, b, ca, cb, var); } catch (gcdheu_failed) { return false; } } // convert polynomials to Z[X] const numeric a_lcm = lcm_of_coefficients_denominators(a); const numeric ab_lcm = lcmcoeff(b, a_lcm); const ex ai = a*ab_lcm; const ex bi = b*ab_lcm; if (!ai.info(info_flags::integer_polynomial)) throw std::logic_error("heur_gcd: not an integer polynomial [1]"); if (!bi.info(info_flags::integer_polynomial)) throw std::logic_error("heur_gcd: not an integer polynomial [2]"); bool found = false; try { found = heur_gcd_z(res, ai, bi, ca, cb, var); } catch (gcdheu_failed) { return false; } // GCD is not unique, it's defined up to a unit (i.e. invertible // element). If the coefficient ring is a field, every its element is // invertible, so one can multiply the polynomial GCD with any element // of the coefficient field. We use this ambiguity to make cofactors // integer polynomials. if (found) res /= ab_lcm; return found; } // gcd helper to handle partially factored polynomials (to avoid expanding // large expressions). At least one of the arguments should be a power. static ex gcd_pf_pow(const ex& a, const ex& b, ex* ca, ex* cb); // gcd helper to handle partially factored polynomials (to avoid expanding // large expressions). At least one of the arguments should be a product. static ex gcd_pf_mul(const ex& a, const ex& b, ex* ca, ex* cb); /** Compute GCD (Greatest Common Divisor) of multivariate polynomials a(X) * and b(X) in Z[X]. Optionally also compute the cofactors of a and b, * defined by a = ca * gcd(a, b) and b = cb * gcd(a, b). * * @param 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, unsigned options) { #if STATISTICS gcd_called++; #endif // GCD of numerics -> CLN if (is_exactly_a(a) && is_exactly_a(b)) { numeric g = gcd(ex_to(a), ex_to(b)); if (ca || cb) { if (g.is_zero()) { if (ca) *ca = _ex0; if (cb) *cb = _ex0; } else { if (ca) *ca = ex_to(a) / g; if (cb) *cb = ex_to(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 (!(options & gcd_options::no_part_factored)) { if (is_exactly_a(a) || is_exactly_a(b)) return gcd_pf_mul(a, b, ca, cb); #if FAST_COMPARE if (is_exactly_a(a) || is_exactly_a(b)) return gcd_pf_pow(a, b, ca, cb); #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(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(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(aex)) { numeric bcont = bex.integer_content(); numeric g = gcd(ex_to(aex), bcont); if (ca) *ca = ex_to(aex)/g; if (cb) *cb = bex/g; return g; } if (is_exactly_a(bex)) { numeric acont = aex.integer_content(); numeric g = gcd(ex_to(bex), acont); if (ca) *ca = aex/g; if (cb) *cb = ex_to(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; if (!(options & gcd_options::no_heur_gcd)) { bool found = heur_gcd(g, aex, bex, ca, cb, var); if (found) { // heur_gcd have already computed cofactors... if (g.is_equal(_ex1)) { // ... but we want to keep them factored if possible. if (ca) *ca = a; if (cb) *cb = b; } return g; } #if STATISTICS else { 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); } return g; } // gcd helper to handle partially factored polynomials (to avoid expanding // large expressions). Both arguments should be powers. static ex gcd_pf_pow_pow(const ex& a, const ex& b, ex* ca, ex* cb) { ex p = a.op(0); const ex& exp_a = a.op(1); ex pb = b.op(0); const ex& exp_b = b.op(1); 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, false); 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) } static ex gcd_pf_pow(const ex& a, const ex& b, ex* ca, ex* cb) { if (is_exactly_a(a) && is_exactly_a(b)) return gcd_pf_pow_pow(a, b, ca, cb); if (is_exactly_a(b) && (! is_exactly_a(a))) return gcd_pf_pow(b, a, cb, ca); GINAC_ASSERT(is_exactly_a(a)); ex p = a.op(0); const ex& exp_a = a.op(1); if (p.is_equal(b)) { // a = p^n, b = p, gcd = p if (ca) *ca = power(p, a.op(1) - 1); if (cb) *cb = _ex1; return p; } ex p_co, bpart_co; ex p_gcd = gcd(p, b, &p_co, &bpart_co, false); // a(x) = p(x)^n, gcd(p, b) = 1 ==> gcd(a, b) = 1 if (p_gcd.is_equal(_ex1)) { if (ca) *ca = a; if (cb) *cb = b; return _ex1; } // a(x) = g(x)^n A(x)^n, b(x) = g(x) B(x) ==> gcd(a, b) = g(x) gcd(g(x)^(n-1) A(x)^n, B(x)) ex rg = gcd(power(p_gcd, exp_a-1)*power(p_co, exp_a), bpart_co, ca, cb, false); return p_gcd*rg; } static ex gcd_pf_mul(const ex& a, const ex& b, ex* ca, ex* cb) { if (is_exactly_a(a) && is_exactly_a(b) && (b.nops() > a.nops())) return gcd_pf_mul(b, a, cb, ca); if (is_exactly_a(b) && (!is_exactly_a(a))) return gcd_pf_mul(b, a, cb, ca); GINAC_ASSERT(is_exactly_a(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; isetflag(status_flags::dynallocated); if (cb) *cb = part_b; return (new mul(g))->setflag(status_flags::dynallocated); } /** Compute LCM (Least Common Multiple) of multivariate polynomials in Z[X]. * * @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(a) && is_exactly_a(b)) return lcm(ex_to(a), ex_to(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 square-free * 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 square-free factorization 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(a) || // algorithm does not trap a==0 is_a(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(args.op(0))) throw (std::runtime_error("sqrfree(): invalid factorization variable")); const symbol &x = ex_to(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; isecond; // 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(den)) { if (ex_to(den).is_negative()) { num *= _ex_1; den *= _ex_1; } } else { ex x; if (get_first_symbol(den, x)) { GINAC_ASSERT(is_exactly_a(den.unit(x))); if (ex_to(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(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(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(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(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(basis).normal(repl, rev_lookup, level-1); ex n_exponent = ex_to(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(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(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(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(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(basis_pref) || is_exactly_a(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(e)) { size_t num = e.nops(); exvector terms; terms.reserve(num); ex gc; // Find the common GCD for (size_t i=0; i(x) || is_exactly_a(x) || is_a(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(t)) { for (size_t j=0; jsetflag(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(e)) { size_t num = e.nops(); exvector v; v.reserve(num); for (size_t i=0; isetflag(status_flags::dynallocated); } else if (is_exactly_a(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(e) || is_exactly_a(e) || is_exactly_a(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