X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnormal.cpp;h=d8252abb02742fd4376fea26eb0988e140118acb;hp=1126d27c2fffb738e3c6cf57decf92af6dc430f0;hb=44f3fac94db304f8a39745f425c7b831b7eec6ec;hpb=a74473453218570d22f8932cc39ab48c7f0021ae diff --git a/ginac/normal.cpp b/ginac/normal.cpp index 1126d27c..d8252abb 100644 --- a/ginac/normal.cpp +++ b/ginac/normal.cpp @@ -6,7 +6,7 @@ * computation, square-free factorization and rational function normalization. */ /* - * GiNaC Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2006 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 @@ -20,10 +20,9 @@ * * 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ -#include #include #include @@ -34,21 +33,19 @@ #include "constant.h" #include "expairseq.h" #include "fail.h" -#include "indexed.h" #include "inifcns.h" #include "lst.h" #include "mul.h" -#include "ncmul.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" -#ifndef NO_NAMESPACE_GINAC namespace GiNaC { -#endif // ndef NO_NAMESPACE_GINAC // If comparing expressions (ex::compare()) is fast, you can set this to 1. // Some routines like quo(), rem() and gcd() will then return a quick answer @@ -59,7 +56,8 @@ namespace GiNaC { #define USE_REMEMBER 0 // Set this if you want divide_in_z() to use trial division followed by -// polynomial interpolation (usually slower except for very large problems) +// 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 @@ -77,36 +75,36 @@ static int heur_gcd_failed = 0; static struct _stat_print { _stat_print() {} ~_stat_print() { - cout << "gcd() called " << gcd_called << " times\n"; - cout << "sr_gcd() called " << sr_gcd_called << " times\n"; - cout << "heur_gcd() called " << heur_gcd_called << " times\n"; - cout << "heur_gcd() failed " << heur_gcd_failed << " times\n"; + 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 +/** 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 pointer to first symbol found (returned) + * @param x first symbol found (returned) * @return "false" if no symbol was found, "true" otherwise */ -static bool get_first_symbol(const ex &e, const symbol *&x) +static bool get_first_symbol(const ex &e, ex &x) { - if (is_ex_exactly_of_type(e, symbol)) { - x = static_cast(e.bp); - return true; - } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) { - for (unsigned i=0; i(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; } @@ -121,56 +119,65 @@ static bool get_first_symbol(const ex &e, const symbol *&x) * * @see get_symbol_stats */ struct sym_desc { - /** Pointer to symbol */ - const symbol *sym; + /** Reference to symbol */ + ex sym; + + /** Highest degree of symbol in polynomial "a" */ + int deg_a; - /** Highest degree of symbol in polynomial "a" */ - int deg_a; + /** Highest degree of symbol in polynomial "b" */ + int deg_b; - /** Highest degree of symbol in polynomial "b" */ - int deg_b; + /** Lowest degree of symbol in polynomial "a" */ + int ldeg_a; - /** Lowest degree of symbol in polynomial "a" */ - int ldeg_a; + /** Lowest degree of symbol in polynomial "b" */ + int ldeg_b; - /** Lowest degree of symbol in polynomial "b" */ - int ldeg_b; + /** Maximum of deg_a and deg_b (Used for sorting) */ + int max_deg; - /** 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 {return max_deg < x.max_deg;} + /** Commparison operator for sorting */ + bool operator<(const sym_desc &x) const + { + if (max_deg == x.max_deg) + return max_lcnops < x.max_lcnops; + else + return max_deg < x.max_deg; + } }; // Vector of sym_desc structures -typedef vector sym_desc_vec; +typedef std::vector sym_desc_vec; // Add symbol the sym_desc_vec (used internally by get_symbol_stats()) -static void add_symbol(const symbol *s, sym_desc_vec &v) +static void add_symbol(const ex &s, sym_desc_vec &v) { - sym_desc_vec::iterator it = v.begin(), itend = v.end(); - while (it != itend) { - if (it->sym->compare(*s) == 0) // If it's already in there, don't add it a second time - return; - it++; - } - sym_desc d; - d.sym = s; - v.push_back(d); + sym_desc_vec::const_iterator it = v.begin(), itend = v.end(); + while (it != itend) { + if (it->sym.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_ex_exactly_of_type(e, symbol)) { - add_symbol(static_cast(e.bp), v); - } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) { - for (unsigned i=0; i(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. @@ -187,27 +194,29 @@ static void collect_symbols(const ex &e, sym_desc_vec &v) * @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 = max(deg_a, deg_b); - it->ldeg_a = a.ldegree(*(it->sym)); - it->ldeg_b = b.ldegree(*(it->sym)); - it++; - } - sort(v.begin(), v.end()); + 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 - clog << "Symbols:\n"; + std::clog << "Symbols:\n"; it = v.begin(); itend = v.end(); while (it != itend) { - clog << " " << *it->sym << ": deg_a=" << it->deg_a << ", deg_b=" << it->deg_b << ", ldeg_a=" << it->ldeg_a << ", ldeg_b=" << it->ldeg_b << ", max_deg=" << it->max_deg << endl; - clog << " lcoeff_a=" << a.lcoeff(*(it->sym)) << ", lcoeff_b=" << b.lcoeff(*(it->sym)) << endl; - it++; + 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 } @@ -221,21 +230,25 @@ static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v) // expression recursively (used internally by lcm_of_coefficients_denominators()) static numeric lcmcoeff(const ex &e, const numeric &l) { - if (e.info(info_flags::rational)) - return lcm(ex_to_numeric(e).denom(), l); - else if (is_ex_exactly_of_type(e, add)) { - numeric c = _num1(); - for (unsigned i=0; i(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. @@ -247,7 +260,7 @@ static numeric lcmcoeff(const ex &e, const numeric &l) * @return LCM of denominators of coefficients */ static numeric lcm_of_coefficients_denominators(const ex &e) { - return lcmcoeff(e, _num1()); + return lcmcoeff(e, *_num1_p); } /** Bring polynomial from Q[X] to Z[X] by multiplying in the previously @@ -257,77 +270,84 @@ static numeric lcm_of_coefficients_denominators(const ex &e) * @param lcm LCM to multiply in */ static ex multiply_lcm(const ex &e, const numeric &lcm) { - if (is_ex_exactly_of_type(e, mul)) { - ex c = _ex1(); - numeric lcm_accum = _num1(); - for (unsigned i=0; i(e)) { + 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. + * 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. * - * @param e expanded polynomial * @return integer content */ -numeric ex::integer_content(void) const +numeric ex::integer_content() const { - GINAC_ASSERT(bp!=0); - return bp->integer_content(); + return bp->integer_content(); } -numeric basic::integer_content(void) const +numeric basic::integer_content() const { - return _num1(); + return *_num1_p; } -numeric numeric::integer_content(void) const +numeric numeric::integer_content() const { - return abs(*this); + return abs(*this); } -numeric add::integer_content(void) const +numeric add::integer_content() const { - epvector::const_iterator it = seq.begin(); - epvector::const_iterator itend = seq.end(); - numeric c = _num0(); - while (it != itend) { - GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric)); - GINAC_ASSERT(is_ex_exactly_of_type(it->coeff,numeric)); - c = gcd(ex_to_numeric(it->coeff), c); - it++; - } - GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric)); - c = gcd(ex_to_numeric(overall_coeff),c); - return c; + epvector::const_iterator it = seq.begin(); + epvector::const_iterator itend = seq.end(); + numeric c = *_num0_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(void) const +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_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric)); - ++it; - } + 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_ex_exactly_of_type(overall_coeff,numeric)); - return abs(ex_to_numeric(overall_coeff)); + GINAC_ASSERT(is_exactly_a(overall_coeff)); + return abs(ex_to(overall_coeff)); } @@ -344,44 +364,44 @@ numeric mul::integer_content(void) const * @param check_args check whether a and b are polynomials with rational * coefficients (defaults to "true") * @return quotient of a and b in Q[x] */ -ex quo(const ex &a, const ex &b, const symbol &x, bool check_args) +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_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) - return a / b; + if (b.is_zero()) + throw(std::overflow_error("quo: division by zero")); + if (is_exactly_a(a) && is_exactly_a(b)) + return a / b; #if FAST_COMPARE - if (a.is_equal(b)) - return _ex1(); + if (a.is_equal(b)) + return _ex1; #endif - if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))) - throw(std::invalid_argument("quo: arguments must be polynomials over the rationals")); - - // Polynomial long division - ex q = _ex0(); - ex r = a.expand(); - if (r.is_zero()) - return r; - int bdeg = b.degree(x); - int rdeg = r.degree(x); - ex blcoeff = b.expand().coeff(x, bdeg); - bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric); - while (rdeg >= bdeg) { - ex term, rcoeff = r.coeff(x, rdeg); - if (blcoeff_is_numeric) - term = rcoeff / blcoeff; - else { - if (!divide(rcoeff, blcoeff, term, false)) - return *new ex(fail()); - } - term *= power(x, rdeg - bdeg); - q += term; - r -= (term * b).expand(); - if (r.is_zero()) - break; - rdeg = r.degree(x); - } - return q; + if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))) + throw(std::invalid_argument("quo: arguments must be polynomials over the rationals")); + + // Polynomial long division + ex r = a.expand(); + if (r.is_zero()) + return r; + int bdeg = b.degree(x); + int rdeg = r.degree(x); + ex blcoeff = b.expand().coeff(x, bdeg); + bool blcoeff_is_numeric = is_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); } @@ -394,98 +414,166 @@ ex quo(const ex &a, const ex &b, const symbol &x, bool check_args) * @param check_args check whether a and b are polynomials with rational * coefficients (defaults to "true") * @return remainder of a(x) and b(x) in Q[x] */ -ex rem(const ex &a, const ex &b, const symbol &x, bool check_args) -{ - if (b.is_zero()) - throw(std::overflow_error("rem: division by zero")); - if (is_ex_exactly_of_type(a, numeric)) { - if (is_ex_exactly_of_type(b, numeric)) - return _ex0(); - else - return b; - } +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(); + if (a.is_equal(b)) + return _ex0; #endif - if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))) - throw(std::invalid_argument("rem: arguments must be polynomials over the rationals")); - - // Polynomial long division - ex r = a.expand(); - if (r.is_zero()) - return r; - int bdeg = b.degree(x); - int rdeg = r.degree(x); - ex blcoeff = b.expand().coeff(x, bdeg); - bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric); - while (rdeg >= bdeg) { - ex term, rcoeff = r.coeff(x, rdeg); - if (blcoeff_is_numeric) - term = rcoeff / blcoeff; - else { - if (!divide(rcoeff, blcoeff, term, false)) - return *new ex(fail()); - } - term *= power(x, rdeg - bdeg); - r -= (term * b).expand(); - if (r.is_zero()) - break; - rdeg = r.degree(x); - } - return r; -} - - -/** Pseudo-remainder of polynomials a(x) and b(x) in Z[x]. + 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 pseudo-remainder of a(x) and b(x) in Z[x] */ -ex prem(const ex &a, const ex &b, const symbol &x, bool check_args) -{ - if (b.is_zero()) - throw(std::overflow_error("prem: division by zero")); - if (is_ex_exactly_of_type(a, numeric)) { - if (is_ex_exactly_of_type(b, numeric)) - return _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; + * @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; } @@ -497,56 +585,129 @@ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args) * @param check_args check whether a and b are polynomials with rational * coefficients (defaults to "true") * @return "true" when exact division succeeds (quotient returned in q), - * "false" otherwise */ + * "false" otherwise (q left untouched) */ bool divide(const ex &a, const ex &b, ex &q, bool check_args) { - q = _ex0(); - if (b.is_zero()) - throw(std::overflow_error("divide: division by zero")); - if (a.is_zero()) - return true; - if (is_ex_exactly_of_type(b, numeric)) { - q = a / b; - return true; - } else if (is_ex_exactly_of_type(a, numeric)) - return false; + if (b.is_zero()) + throw(std::overflow_error("divide: division by zero")); + if (a.is_zero()) { + q = _ex0; + return true; + } + if (is_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; - } + if (a.is_equal(b)) { + q = _ex1; + return true; + } #endif - if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))) - throw(std::invalid_argument("divide: arguments must be polynomials over the rationals")); - - // Find first symbol - const symbol *x; - if (!get_first_symbol(a, x) && !get_first_symbol(b, x)) - throw(std::invalid_argument("invalid expression in divide()")); - - // Polynomial long division (recursive) - ex r = a.expand(); - if (r.is_zero()) - return true; - int bdeg = b.degree(*x); - int rdeg = r.degree(*x); - ex blcoeff = b.expand().coeff(*x, bdeg); - bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric); - while (rdeg >= bdeg) { - ex term, rcoeff = r.coeff(*x, rdeg); - if (blcoeff_is_numeric) - term = rcoeff / blcoeff; - else - if (!divide(rcoeff, blcoeff, term, false)) - return false; - term *= power(*x, rdeg - bdeg); - q += term; - r -= (term * b).expand(); - if (r.is_zero()) - return true; - rdeg = r.degree(*x); - } - return false; + if (check_args && (!a.info(info_flags::rational_polynomial) || + !b.info(info_flags::rational_polynomial))) + throw(std::invalid_argument("divide: arguments must be polynomials over the rationals")); + + // Find first symbol + 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; } @@ -555,24 +716,25 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args) * Remembering */ -typedef pair ex2; -typedef pair exbool; +typedef std::pair ex2; +typedef std::pair exbool; struct ex2_less { - bool operator() (const ex2 p, const ex2 q) const - { - return p.first.compare(q.first) < 0 || (!(q.first.compare(p.first) < 0) && p.second.compare(q.second) < 0); - } + bool operator() (const ex2 &p, const ex2 &q) const + { + int cmp = p.first.compare(q.first); + return ((cmp<0) || (!(cmp>0) && p.second.compare(q.second)<0)); + } }; -typedef map ex2_exbool_remember; +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 + * 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 @@ -587,127 +749,154 @@ typedef map ex2_exbool_remember; * @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_ex_exactly_of_type(a, numeric)) { - if (is_ex_exactly_of_type(b, numeric)) { - q = a / b; - return q.info(info_flags::integer); - } else - return false; - } + q = _ex0; + if (b.is_zero()) + throw(std::overflow_error("divide_in_z: division by zero")); + if (b.is_equal(_ex1)) { + q = a; + return true; + } + if (is_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; - } + if (a.is_equal(b)) { + q = _ex1; + return true; + } #endif #if USE_REMEMBER - // Remembering - static ex2_exbool_remember dr_remember; - ex2_exbool_remember::const_iterator remembered = dr_remember.find(ex2(a, b)); - if (remembered != dr_remember.end()) { - q = remembered->second.first; - return remembered->second.second; - } + // Remembering + static ex2_exbool_remember dr_remember; + ex2_exbool_remember::const_iterator remembered = dr_remember.find(ex2(a, b)); + if (remembered != dr_remember.end()) { + q = remembered->second.first; + return remembered->second.second; + } #endif - // Main symbol - const symbol *x = var->sym; + 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; - // Compare degrees - int adeg = a.degree(*x), bdeg = b.degree(*x); - if (bdeg > adeg) - 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(); - ex c; - for (i=0; i<=adeg; i++) { - ex bs = b.subs(*x == point); - while (bs.is_zero()) { - point += _num1(); - bs = b.subs(*x == point); - } - if (!divide_in_z(a.subs(*x == point), bs, c, var+1)) - return false; - alpha.push_back(point); - u.push_back(c); - point += _num1(); - } - - // Compute inverses - vector rcp; rcp.reserve(adeg + 1); - rcp.push_back(_num0()); - for (k=1; k<=adeg; k++) { - numeric product = alpha[k] - alpha[0]; - for (i=1; i=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; + // 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); - while (rdeg >= bdeg) { - ex term, rcoeff = r.coeff(*x, rdeg); - if (!divide_in_z(rcoeff, blcoeff, term, var+1)) - break; - term = (term * power(*x, rdeg - bdeg)).expand(); - q += term; - r -= (term * eb).expand(); - if (r.is_zero()) { + // Polynomial long division (recursive) + ex r = a.expand(); + if (r.is_zero()) + return true; + int rdeg = adeg; + ex eb = b.expand(); + ex blcoeff = eb.coeff(x, bdeg); + exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0)); + while (rdeg >= bdeg) { + ex term, rcoeff = r.coeff(x, rdeg); + if (!divide_in_z(rcoeff, blcoeff, term, var+1)) + break; + term = (term * power(x, rdeg - bdeg)).expand(); + v.push_back(term); + r -= (term * eb).expand(); + if (r.is_zero()) { + q = (new add(v))->setflag(status_flags::dynallocated); #if USE_REMEMBER - dr_remember[ex2(a, b)] = exbool(q, true); + dr_remember[ex2(a, b)] = exbool(q, true); #endif - return true; - } - rdeg = r.degree(*x); - } + return true; + } + rdeg = r.degree(x); + } #if USE_REMEMBER - dr_remember[ex2(a, b)] = exbool(q, false); + dr_remember[ex2(a, b)] = exbool(q, false); #endif - return false; + return false; #endif } @@ -718,516 +907,359 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite */ /** Compute unit part (= sign of leading coefficient) of a multivariate - * polynomial in Z[x]. The product of unit part, content part, and primitive + * polynomial in Q[x]. The product of unit part, content part, and primitive * part is the polynomial itself. * - * @param x variable in which to compute the unit part + * @param x main variable * @return unit part - * @see ex::content, ex::primpart */ -ex ex::unit(const symbol &x) const + * @see ex::content, ex::primpart, ex::unitcontprim */ +ex ex::unit(const ex &x) const { - ex c = expand().lcoeff(x); - if (is_ex_exactly_of_type(c, numeric)) - return c < _ex0() ? _ex_1() : _ex1(); - else { - const symbol *y; - if (get_first_symbol(c, y)) - return c.unit(*y); - else - throw(std::invalid_argument("invalid expression in unit()")); - } + 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 Z[x]. The product of unit part, content part, + * multivariate polynomial in Q[x]. The product of unit part, content part, * and primitive part is the polynomial itself. * - * @param x variable in which to compute the content part + * @param x main variable * @return content part - * @see ex::unit, ex::primpart */ -ex ex::content(const symbol &x) const -{ - if (is_zero()) - return _ex0(); - if (is_ex_exactly_of_type(*this, numeric)) - return info(info_flags::negative) ? -*this : *this; - ex e = expand(); - if (e.is_zero()) - return _ex0(); - - // First, try the integer content - ex c = e.integer_content(); - ex r = e / c; - ex lcoeff = r.lcoeff(x); - if (lcoeff.info(info_flags::integer)) - return c; - - // GCD of all coefficients - int deg = e.degree(x); - int ldeg = e.ldegree(x); - if (deg == ldeg) - return e.lcoeff(x) / e.unit(x); - c = _ex0(); - for (int i=ldeg; i<=deg; i++) - c = gcd(e.coeff(x, i), c, NULL, NULL, false); - return c; -} - - -/** Compute primitive part of a multivariate polynomial in Z[x]. - * The product of unit part, content part, and primitive part is the - * polynomial itself. + * @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 variable in which to compute the primitive part + * @param x main variable * @return primitive part - * @see ex::unit, ex::content */ -ex ex::primpart(const symbol &x) const + * @see ex::unit, ex::content, ex::unitcontprim */ +ex ex::primpart(const ex &x) const { - if (is_zero()) - return _ex0(); - if (is_ex_exactly_of_type(*this, numeric)) - return _ex1(); - - ex c = content(x); - if (c.is_zero()) - return _ex0(); - ex u = unit(x); - if (is_ex_exactly_of_type(c, numeric)) - return *this / (c * u); - else - return quo(*this, c * u, x, false); + // 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 Z[x] when the +/** 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 variable in which to compute the primitive part + * @param x main variable * @param c previously computed content part * @return primitive part */ -ex ex::primpart(const symbol &x, const ex &c) const +ex ex::primpart(const ex &x, const ex &c) const { - if (is_zero()) - return _ex0(); - if (c.is_zero()) - return _ex0(); - if (is_ex_exactly_of_type(*this, numeric)) - return _ex1(); - - ex u = unit(x); - if (is_ex_exactly_of_type(c, numeric)) - return *this / (c * u); - else - return quo(*this, c * u, x, false); + 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); } -/* - * GCD of multivariate polynomials - */ - -/** Compute GCD of multivariate polynomials using the Euclidean PRS algorithm - * (not really suited for multivariate GCDs). This function is only provided - * for testing purposes. +/** 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 a first multivariate polynomial - * @param b second multivariate polynomial - * @param x pointer to symbol (main variable) in which to compute the GCD in - * @return the GCD as a new expression - * @see gcd */ + * @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; + } -static ex eu_gcd(const ex &a, const ex &b, const symbol *x) -{ -//clog << "eu_gcd(" << a << "," << b << ")\n"; - - // Sort c and d so that c has higher degree - ex c, d; - int adeg = a.degree(*x), bdeg = b.degree(*x); - if (adeg >= bdeg) { - c = a; - d = b; - } else { - c = b; - d = a; - } - - // Euclidean algorithm - ex r; - for (;;) { -//clog << " d = " << d << endl; - r = rem(c, d, *x, false); - if (r.is_zero()) - return d.primpart(*x); - c = d; - d = r; - } -} - - -/** Compute GCD of multivariate polynomials using the Euclidean PRS algorithm - * with pseudo-remainders ("World's Worst GCD Algorithm", staying in Z[X]). - * This function is only provided for testing purposes. - * - * @param a first multivariate polynomial - * @param b second multivariate polynomial - * @param x pointer to symbol (main variable) in which to compute the GCD in - * @return the GCD as a new expression - * @see gcd */ + // 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; + } -static ex euprem_gcd(const ex &a, const ex &b, const symbol *x) -{ -//clog << "euprem_gcd(" << a << "," << b << ")\n"; - - // Sort c and d so that c has higher degree - ex c, d; - int adeg = a.degree(*x), bdeg = b.degree(*x); - if (adeg >= bdeg) { - c = a; - d = b; - } else { - c = b; - d = a; - } - - // Euclidean algorithm with pseudo-remainders - ex r; - for (;;) { -//clog << " d = " << d << endl; - r = prem(c, d, *x, false); - if (r.is_zero()) - return d.primpart(*x); - c = d; - d = r; - } -} - - -/** Compute GCD of multivariate polynomials using the primitive Euclidean - * PRS algorithm (complete content removal at each step). This function is - * only provided for testing purposes. - * - * @param a first multivariate polynomial - * @param b second multivariate polynomial - * @param x pointer to symbol (main variable) in which to compute the GCD in - * @return the GCD as a new expression - * @see gcd */ + // Expand input polynomial + ex e = expand(); + if (e.is_zero()) { + u = _ex1; + c = p = _ex0; + return; + } -static ex peu_gcd(const ex &a, const ex &b, const symbol *x) -{ -//clog << "peu_gcd(" << a << "," << b << ")\n"; - - // Sort c and d so that c has higher degree - ex c, d; - int adeg = a.degree(*x), bdeg = b.degree(*x); - int ddeg; - if (adeg >= bdeg) { - c = a; - d = b; - ddeg = bdeg; - } else { - c = b; - d = a; - ddeg = adeg; - } - - // Remove content from c and d, to be attached to GCD later - ex cont_c = c.content(*x); - ex cont_d = d.content(*x); - ex gamma = gcd(cont_c, cont_d, NULL, NULL, false); - if (ddeg == 0) - return gamma; - c = c.primpart(*x, cont_c); - d = d.primpart(*x, cont_d); - - // Euclidean algorithm with content removal - ex r; - for (;;) { -//clog << " d = " << d << endl; - r = prem(c, d, *x, false); - if (r.is_zero()) - return gamma * d; - c = d; - d = r.primpart(*x); - } -} - - -/** Compute GCD of multivariate polynomials using the reduced PRS algorithm. - * This function is only provided for testing purposes. - * - * @param a first multivariate polynomial - * @param b second multivariate polynomial - * @param x pointer to symbol (main variable) in which to compute the GCD in - * @return the GCD as a new expression - * @see gcd */ + // Compute unit and content + u = unit(x); + c = content(x); -static ex red_gcd(const ex &a, const ex &b, const symbol *x) -{ -//clog << "red_gcd(" << a << "," << b << ")\n"; - - // Sort c and d so that c has higher degree - ex c, d; - int adeg = a.degree(*x), bdeg = b.degree(*x); - 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, ri = _ex1(); - int delta = cdeg - ddeg; - - for (;;) { - // Calculate polynomial pseudo-remainder -//clog << " d = " << d << endl; - r = prem(c, d, *x, false); - if (r.is_zero()) - return gamma * d.primpart(*x); - c = d; - cdeg = ddeg; - - if (!divide(r, pow(ri, delta), d, false)) - throw(std::runtime_error("invalid expression in red_gcd(), division failed")); - ddeg = d.degree(*x); - if (ddeg == 0) { - if (is_ex_exactly_of_type(r, numeric)) - return gamma; - else - return gamma * r.primpart(*x); - } - - ri = c.expand().lcoeff(*x); - delta = cdeg - ddeg; - } + // 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 x pointer to symbol (main variable) in which to compute the GCD in + * @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, const symbol *x) + +static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var) { -//clog << "sr_gcd(" << a << "," << b << ")\n"; #if STATISTICS sr_gcd_called++; #endif - // Sort c and d so that c has higher degree - ex c, d; - int adeg = a.degree(*x), bdeg = b.degree(*x); - int cdeg, ddeg; - if (adeg >= bdeg) { - c = a; - d = b; - cdeg = adeg; - ddeg = bdeg; - } else { - c = b; - d = a; - cdeg = bdeg; - ddeg = adeg; - } - - // Remove content from c and d, to be attached to GCD later - ex cont_c = c.content(*x); - ex cont_d = d.content(*x); - ex gamma = gcd(cont_c, cont_d, NULL, NULL, false); - if (ddeg == 0) - return gamma; - c = c.primpart(*x, cont_c); - d = d.primpart(*x, cont_d); -//clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n"; - - // First element of subresultant sequence - ex r = _ex0(), ri = _ex1(), psi = _ex1(); - int delta = cdeg - ddeg; - - for (;;) { - // Calculate polynomial pseudo-remainder -//clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n"; -//clog << " d = " << d << endl; - r = prem(c, d, *x, false); - if (r.is_zero()) - return gamma * d.primpart(*x); - c = d; - cdeg = ddeg; -//clog << " dividing...\n"; - if (!divide(r, ri * pow(psi, delta), d, false)) - throw(std::runtime_error("invalid expression in sr_gcd(), division failed")); - ddeg = d.degree(*x); - if (ddeg == 0) { - if (is_ex_exactly_of_type(r, numeric)) - return gamma; - else - return gamma * r.primpart(*x); - } - - // Next element of subresultant sequence -//clog << " calculating next subresultant...\n"; - ri = c.expand().lcoeff(*x); - if (delta == 1) - psi = ri; - else if (delta) - divide(pow(ri, delta), pow(psi, delta-1), psi, false); - delta = cdeg - ddeg; - } + // 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(). * - * @param e expanded multivariate polynomial * @return maximum coefficient * @see heur_gcd */ -numeric ex::max_coefficient(void) const +numeric ex::max_coefficient() const { - GINAC_ASSERT(bp!=0); - return bp->max_coefficient(); + return bp->max_coefficient(); } -numeric basic::max_coefficient(void) const +/** Implementation ex::max_coefficient(). + * @see heur_gcd */ +numeric basic::max_coefficient() const { - return _num1(); + return *_num1_p; } -numeric numeric::max_coefficient(void) const +numeric numeric::max_coefficient() const { - return abs(*this); + return abs(*this); } -numeric add::max_coefficient(void) const +numeric add::max_coefficient() const { - epvector::const_iterator it = seq.begin(); - epvector::const_iterator itend = seq.end(); - GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric)); - numeric cur_max = abs(ex_to_numeric(overall_coeff)); - while (it != itend) { - numeric a; - GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric)); - a = abs(ex_to_numeric(it->coeff)); - if (a > cur_max) - cur_max = a; - it++; - } - return cur_max; + epvector::const_iterator it = seq.begin(); + epvector::const_iterator itend = seq.end(); + GINAC_ASSERT(is_exactly_a(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(void) const +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_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric)); - it++; - } + 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_ex_exactly_of_type(overall_coeff,numeric)); - return abs(ex_to_numeric(overall_coeff)); + GINAC_ASSERT(is_exactly_a(overall_coeff)); + return abs(ex_to(overall_coeff)); } -/** Apply symmetric modular homomorphism to a multivariate polynomial. - * This function is used internally by heur_gcd(). +/** Apply symmetric modular homomorphism to an expanded multivariate + * polynomial. This function is usually used internally by heur_gcd(). * - * @param e expanded multivariate polynomial * @param xi modulus * @return mapped polynomial * @see heur_gcd */ -ex ex::smod(const numeric &xi) const -{ - GINAC_ASSERT(bp!=0); - return bp->smod(xi); -} - ex basic::smod(const numeric &xi) const { - return *this; + return *this; } ex numeric::smod(const numeric &xi) const { -#ifndef NO_NAMESPACE_GINAC - return GiNaC::smod(*this, xi); -#else // ndef NO_NAMESPACE_GINAC - return ::smod(*this, xi); -#endif // ndef NO_NAMESPACE_GINAC + 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_ex_exactly_of_type(it->rest,numeric)); -#ifndef NO_NAMESPACE_GINAC - numeric coeff = GiNaC::smod(ex_to_numeric(it->coeff), xi); -#else // ndef NO_NAMESPACE_GINAC - numeric coeff = ::smod(ex_to_numeric(it->coeff), xi); -#endif // ndef NO_NAMESPACE_GINAC - if (!coeff.is_zero()) - newseq.push_back(expair(it->rest, coeff)); - it++; - } - GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric)); -#ifndef NO_NAMESPACE_GINAC - numeric coeff = GiNaC::smod(ex_to_numeric(overall_coeff), xi); -#else // ndef NO_NAMESPACE_GINAC - numeric coeff = ::smod(ex_to_numeric(overall_coeff), xi); -#endif // ndef NO_NAMESPACE_GINAC - return (new add(newseq,coeff))->setflag(status_flags::dynallocated); + 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_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric)); - it++; - } + 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_ex_exactly_of_type(overall_coeff,numeric)); -#ifndef NO_NAMESPACE_GINAC - mulcopyp->overall_coeff = GiNaC::smod(ex_to_numeric(overall_coeff),xi); -#else // ndef NO_NAMESPACE_GINAC - mulcopyp->overall_coeff = ::smod(ex_to_numeric(overall_coeff),xi); -#endif // ndef NO_NAMESPACE_GINAC - mulcopyp->clearflag(status_flags::evaluated); - mulcopyp->clearflag(status_flags::hash_calculated); - return mulcopyp->setflag(status_flags::dynallocated); + 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 {}; @@ -1248,163 +1280,170 @@ class gcdheu_failed {}; * @exception gcdheu_failed() */ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var) { -//clog << "heur_gcd(" << a << "," << b << ")\n"; #if STATISTICS heur_gcd_called++; #endif + // Algorithm only works for non-vanishing input polynomials + if (a.is_zero() || b.is_zero()) + return (new fail())->setflag(status_flags::dynallocated); + // GCD of two numeric values -> CLN - if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) { - numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b)); - numeric rg; - if (ca || cb) - rg = g.inverse(); - if (ca) - *ca = ex_to_numeric(a).mul(rg); - if (cb) - *cb = ex_to_numeric(b).mul(rg); - return g; - } - - // The first symbol is our main variable - const symbol *x = var->sym; - - // Remove integer content - numeric gc = gcd(a.integer_content(), b.integer_content()); - numeric rgc = gc.inverse(); - ex p = a * rgc; - ex q = b * rgc; - int maxdeg = max(p.degree(*x), q.degree(*x)); - - // Find evaluation point - numeric mp = p.max_coefficient(), mq = q.max_coefficient(); - numeric xi; - if (mp > mq) - xi = mq * _num2() + _num2(); - else - xi = mp * _num2() + _num2(); - - // 6 tries maximum - for (int t=0; t<6; t++) { - if (xi.int_length() * maxdeg > 100000) { -//clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << endl; - throw gcdheu_failed(); + 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; + 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(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; + } } - // Apply evaluation homomorphism and calculate GCD - ex gamma = heur_gcd(p.subs(*x == xi), q.subs(*x == xi), NULL, NULL, var+1).expand(); - if (!is_ex_exactly_of_type(gamma, fail)) { - - // Reconstruct polynomial from GCD of mapped polynomials - ex g = _ex0(); - numeric rxi = xi.inverse(); - for (int i=0; !gamma.is_zero(); i++) { - ex gi = gamma.smod(xi); - g += gi * power(*x, i); - gamma = (gamma - gi) * rxi; - } - // Remove integer content - g /= g.integer_content(); - - // If the calculated polynomial divides both a and b, this is the GCD - ex dummy; - if (divide_in_z(p, g, ca ? *ca : dummy, var) && divide_in_z(q, g, cb ? *cb : dummy, var)) { - g *= gc; - ex lc = g.lcoeff(*x); - if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative()) - return -g; - else - return g; - } - } - - // Next evaluation point - xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011)); - } - return *new ex(fail()); + // 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]. + * 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) { -//clog << "gcd(" << a << "," << b << ")\n"; #if STATISTICS gcd_called++; #endif // GCD of numerics -> CLN - if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) { - numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b)); - if (ca) - *ca = ex_to_numeric(a) / g; - if (cb) - *cb = ex_to_numeric(b) / g; - return g; - } + 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")); - } + if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))) { + throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals")); + } // Partially factored cases (to avoid expanding large expressions) - if (is_ex_exactly_of_type(a, mul)) { - if (is_ex_exactly_of_type(b, mul) && b.nops() > a.nops()) + if (is_exactly_a(a)) { + if (is_exactly_a(b) && b.nops() > a.nops()) goto factored_b; factored_a: - ex g = _ex1(); - ex acc_ca = _ex1(); + size_t num = a.nops(); + exvector g; g.reserve(num); + exvector acc_ca; acc_ca.reserve(num); ex part_b = b; - for (unsigned i=0; isetflag(status_flags::dynallocated); if (cb) *cb = part_b; - return g; - } else if (is_ex_exactly_of_type(b, mul)) { - if (is_ex_exactly_of_type(a, mul) && a.nops() > b.nops()) + return (new mul(g))->setflag(status_flags::dynallocated); + } else if (is_exactly_a(b)) { + if (is_exactly_a(a) && a.nops() > b.nops()) goto factored_a; factored_b: - ex g = _ex1(); - ex acc_cb = _ex1(); + size_t num = b.nops(); + exvector g; g.reserve(num); + exvector acc_cb; acc_cb.reserve(num); ex part_a = a; - for (unsigned i=0; isetflag(status_flags::dynallocated); + return (new mul(g))->setflag(status_flags::dynallocated); } #if FAST_COMPARE // Input polynomials of the form poly^n are sometimes also trivial - if (is_ex_exactly_of_type(a, power)) { + if (is_exactly_a(a)) { ex p = a.op(0); - if (is_ex_exactly_of_type(b, power)) { - if (p.is_equal(b.op(0))) { + const ex& exp_a = a.op(1); + if (is_exactly_a(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) - ex exp_a = a.op(1), exp_b = b.op(1); if (exp_a < exp_b) { if (ca) - *ca = _ex1(); + *ca = _ex1; if (cb) *cb = power(p, exp_b - exp_a); return power(p, exp_a); @@ -1412,136 +1451,244 @@ factored_b: if (ca) *ca = power(p, exp_a - exp_b); if (cb) - *cb = _ex1(); + *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(); + *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); } - } - } else if (is_ex_exactly_of_type(b, power)) { + } // is_exactly_a(b) + + } else if (is_exactly_a(b)) { ex p = b.op(0); if (p.is_equal(a)) { // a = p, b = p^n, gcd = p if (ca) - *ca = _ex1(); + *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 + // 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 (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; - } + if (a.is_equal(b)) { + if (ca) + *ca = _ex1; + if (cb) + *cb = _ex1; + return a; + } #endif - // Gather symbol statistics - sym_desc_vec sym_stats; - get_symbol_stats(a, b, sym_stats); - - // The symbol with least degree is our main variable - sym_desc_vec::const_iterator var = sym_stats.begin(); - const symbol *x = var->sym; - - // Cancel trivial common factor - int ldeg_a = var->ldeg_a; - int ldeg_b = var->ldeg_b; - int min_ldeg = min(ldeg_a, ldeg_b); - if (min_ldeg > 0) { - ex common = power(*x, min_ldeg); -//clog << "trivial common factor " << common << endl; - return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common; - } - - // Try to eliminate variables - if (var->deg_a == 0) { -//clog << "eliminating variable " << *x << " from b" << endl; - ex c = bex.content(*x); - ex g = gcd(aex, c, ca, cb, false); - if (cb) - *cb *= bex.unit(*x) * bex.primpart(*x, c); - return g; - } else if (var->deg_b == 0) { -//clog << "eliminating variable " << *x << " from a" << endl; - ex c = aex.content(*x); - ex g = gcd(c, bex, ca, cb, false); - if (ca) - *ca *= aex.unit(*x) * aex.primpart(*x, c); - return g; - } - - ex g; -#if 1 - // Try heuristic algorithm first, fall back to PRS if that failed - try { - g = heur_gcd(aex, bex, ca, cb, var); - } catch (gcdheu_failed) { - g = *new ex(fail()); - } - if (is_ex_exactly_of_type(g, fail)) { -//clog << "heuristics failed" << endl; + 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; + try { + g = heur_gcd(aex, bex, ca, cb, var); + } catch (gcdheu_failed) { + g = fail(); + } + if (is_exactly_a(g)) { #if STATISTICS heur_gcd_failed++; #endif -#endif -// g = heur_gcd(aex, bex, ca, cb, var); -// g = eu_gcd(aex, bex, x); -// g = euprem_gcd(aex, bex, x); -// g = peu_gcd(aex, bex, x); -// g = red_gcd(aex, bex, x); - g = sr_gcd(aex, bex, x); - if (g.is_equal(_ex1())) { + g = sr_gcd(aex, bex, var); + if (g.is_equal(_ex1)) { // Keep cofactors factored if possible if (ca) *ca = a; if (cb) *cb = b; } else { - if (ca) - divide(aex, g, *ca, false); - if (cb) - divide(bex, g, *cb, false); + if (ca) + divide(aex, g, *ca, false); + if (cb) + divide(bex, g, *cb, false); } -#if 1 - } else { - if (g.is_equal(_ex1())) { + } else { + if (g.is_equal(_ex1)) { // Keep cofactors factored if possible if (ca) *ca = a; @@ -1549,8 +1696,8 @@ factored_b: *cb = b; } } -#endif - return g; + + return g; } @@ -1563,14 +1710,14 @@ factored_b: * @return the LCM as a new expression */ ex lcm(const ex &a, const ex &b, bool check_args) { - if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) - return 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; + 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; } @@ -1578,70 +1725,204 @@ ex lcm(const ex &a, const ex &b, bool check_args) * Square-free factorization */ -// Univariate GCD of polynomials in Q[x] (used internally by sqrfree()). -// a and b can be multivariate polynomials but they are treated as univariate polynomials in x. -static ex univariate_gcd(const ex &a, const ex &b, const symbol &x) -{ - if (a.is_zero()) - return b; - if (b.is_zero()) - return a; - if (a.is_equal(_ex1()) || b.is_equal(_ex1())) - return _ex1(); - if (is_ex_of_type(a, numeric) && is_ex_of_type(b, numeric)) - return gcd(ex_to_numeric(a), ex_to_numeric(b)); - if (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)) - throw(std::invalid_argument("univariate_gcd: arguments must be polynomials over the rationals")); - - // Euclidean algorithm - ex c, d, r; - if (a.degree(x) >= b.degree(x)) { - c = a; - d = b; - } else { - c = b; - d = a; - } - for (;;) { - r = rem(c, d, x, false); - if (r.is_zero()) - break; - c = d; - d = r; - } - return d / d.lcoeff(x); +/** 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 square-free factorization of multivariate polynomial a(x) using - * Yun´s algorithm. +/** 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 multivariate polynomial - * @param x variable to factor in - * @return factored polynomial */ -ex sqrfree(const ex &a, const symbol &x) -{ - int i = 1; - ex res = _ex1(); - ex b = a.diff(x); - ex c = univariate_gcd(a, b, x); - ex w; - if (c.is_equal(_ex1())) { - w = a; - } else { - w = quo(a, c, x); - ex y = quo(b, c, x); - ex z = y - w.diff(x); - while (!z.is_zero()) { - ex g = univariate_gcd(w, z, x); - res *= power(g, i); - w = quo(w, g, x); - y = quo(z, g, x); - z = y - w.diff(x); - i++; - } - } - return res * power(w, i); + * @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 - symbol s; - ex es(s); - ex e_replaced = e.subs(sym_lst, repl_lst); - sym_lst.append(es); - repl_lst.append(e_replaced); - return es; + 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). An expression of the form "symbol == expression" is added - * to repl_lst and the symbol is returned. - * @see ex::to_rational */ -static ex replace_with_symbol(const ex &e, lst &repl_lst) -{ - // Expression already in repl_lst? Then return the assigned symbol - for (unsigned i=0; isecond.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 - symbol s; - ex es(s); - ex e_replaced = e.subs(repl_lst); - repl_lst.append(es == e_replaced); - return es; + 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; } -/** Default implementation of ex::normal(). It replaces the object with a - * temporary symbol. + +/** Function object to be applied by basic::normal(). */ +struct normal_map_function : public map_function { + int level; + normal_map_function(int l) : level(l) {} + ex operator()(const ex & e) { return normal(e, level); } +}; + +/** Default implementation of ex::normal(). It normalizes the children and + * replaces the object with a temporary symbol. * @see ex::normal */ -ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const +ex basic::normal(exmap & repl, exmap & rev_lookup, int level) const { - return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + 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(lst &sym_lst, lst &repl_lst, int level) const +ex symbol::normal(exmap & repl, exmap & rev_lookup, int level) const { - return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated); + return (new lst(*this, _ex1))->setflag(status_flags::dynallocated); } @@ -1721,20 +2021,20 @@ ex symbol::normal(lst &sym_lst, lst &repl_lst, int level) const * into re+I*im and replaces I and non-rational real numbers with a temporary * symbol. * @see ex::normal */ -ex numeric::normal(lst &sym_lst, lst &repl_lst, int level) const +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, sym_lst, repl_lst); - } else { // complex - numeric re = num.real(), im = num.imag(); - ex re_ex = re.is_rational() ? re : replace_with_symbol(re, sym_lst, repl_lst); - ex im_ex = im.is_rational() ? im : replace_with_symbol(im, sym_lst, repl_lst); - numex = re_ex + im_ex * replace_with_symbol(I, sym_lst, repl_lst); - } + 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); @@ -1747,225 +2047,222 @@ ex numeric::normal(lst &sym_lst, lst &repl_lst, int level) const * @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(); - -//clog << "frac_cancel num = " << num << ", den = " << den << endl; - - // Handle special cases where numerator or denominator is 0 - if (num.is_zero()) - return (new lst(_ex0(), _ex1()))->setflag(status_flags::dynallocated); - if (den.expand().is_zero()) - throw(std::overflow_error("frac_cancel: division by zero in frac_cancel")); - - // 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); + 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; + pre_factor = den_lcm / num_lcm; - // Cancel GCD from numerator and denominator - ex cnum, cden; - if (gcd(num, den, &cnum, &cden, false) != _ex1()) { + // 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) - const symbol *x; - if (get_first_symbol(den, x)) { - GINAC_ASSERT(is_ex_exactly_of_type(den.unit(*x),numeric)); - if (ex_to_numeric(den.unit(*x)).is_negative()) { - num *= _ex_1(); - den *= _ex_1(); + 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 -//clog << " returns num = " << num << ", den = " << den << ", pre_factor = " << pre_factor << endl; - return (new lst(num * pre_factor.numer(), den * pre_factor.denom()))->setflag(status_flags::dynallocated); +//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(lst &sym_lst, lst &repl_lst, int level) const -{ - // Normalize and expand children, chop into summands - exvector o; - o.reserve(seq.size()+1); - epvector::const_iterator it = seq.begin(), itend = seq.end(); - while (it != itend) { - - // Normalize and expand child - ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1).expand(); - - // If numerator is a sum, chop into summands - if (is_ex_exactly_of_type(n.op(0), add)) { - epvector::const_iterator bit = ex_to_add(n.op(0)).seq.begin(), bitend = ex_to_add(n.op(0)).seq.end(); - while (bit != bitend) { - o.push_back((new lst(recombine_pair_to_ex(*bit), n.op(1)))->setflag(status_flags::dynallocated)); - bit++; - } - - // The overall_coeff is already normalized (== rational), we just - // split it into numerator and denominator - GINAC_ASSERT(ex_to_numeric(ex_to_add(n.op(0)).overall_coeff).is_rational()); - numeric overall = ex_to_numeric(ex_to_add(n.op(0)).overall_coeff); - o.push_back((new lst(overall.numer(), overall.denom() * n.op(1)))->setflag(status_flags::dynallocated)); - } else - o.push_back(n); - it++; - } - o.push_back(overall_coeff.bp->normal(sym_lst, repl_lst, level-1)); - - // o is now a vector of {numerator, denominator} lists - - // Determine common denominator - ex den = _ex1(); - exvector::const_iterator ait = o.begin(), aitend = o.end(); -//clog << "add::normal uses the following summands:\n"; - while (ait != aitend) { -//clog << " num = " << ait->op(0) << ", den = " << ait->op(1) << endl; - den = lcm(ait->op(1), den, false); - ait++; - } -//clog << " common denominator = " << den << endl; - - // Add fractions - if (den.is_equal(_ex1())) { - - // Common denominator is 1, simply add all numerators - exvector num_seq; - for (ait=o.begin(); ait!=aitend; ait++) { - num_seq.push_back(ait->op(0)); +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++; } - return (new lst((new add(num_seq))->setflag(status_flags::dynallocated), den))->setflag(status_flags::dynallocated); - - } else { - // Perform fractional addition - exvector num_seq; - for (ait=o.begin(); ait!=aitend; ait++) { - ex q; - if (!divide(den, ait->op(1), q, false)) { - // should not happen - throw(std::runtime_error("invalid expression in add::normal, division failed")); - } - num_seq.push_back((ait->op(0) * q).expand()); - } - ex num = (new add(num_seq))->setflag(status_flags::dynallocated); + // 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); - } + // 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(lst &sym_lst, lst &repl_lst, int level) const +ex mul::normal(exmap & repl, exmap & rev_lookup, int level) const { - // Normalize children, separate into numerator and denominator - ex num = _ex1(); - ex den = _ex1(); + 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 = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1); - num *= n.op(0); - den *= n.op(1); - it++; - } - n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1); - num *= n.op(0); - den *= n.op(1); + 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(num, den); + return frac_cancel((new mul(num))->setflag(status_flags::dynallocated), + (new mul(den))->setflag(status_flags::dynallocated)); } -/** Implementation of ex::normal() for powers. It normalizes the basis, +/** 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(lst &sym_lst, lst &repl_lst, int level) const +ex power::normal(exmap & repl, exmap & rev_lookup, int level) const { - // Normalize basis - ex n = basis.bp->normal(sym_lst, repl_lst, level-1); + 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")); - if (exponent.info(info_flags::integer)) { + // 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 (exponent.info(info_flags::positive)) { + 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.op(0), exponent), power(n.op(1), exponent)))->setflag(status_flags::dynallocated); + return (new lst(power(n_basis.op(0), n_exponent), power(n_basis.op(1), n_exponent)))->setflag(status_flags::dynallocated); - } else if (exponent.info(info_flags::negative)) { + } else if (n_exponent.info(info_flags::negative)) { // (a/b)^-n -> {b^n, a^n} - return (new lst(power(n.op(1), -exponent), power(n.op(0), -exponent)))->setflag(status_flags::dynallocated); + return (new lst(power(n_basis.op(1), -n_exponent), power(n_basis.op(0), -n_exponent)))->setflag(status_flags::dynallocated); } } else { - if (exponent.info(info_flags::positive)) { + if (n_exponent.info(info_flags::positive)) { // (a/b)^x -> {sym((a/b)^x), 1} - return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + 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 (exponent.info(info_flags::negative)) { + } else if (n_exponent.info(info_flags::negative)) { - if (n.op(1).is_equal(_ex1())) { + if (n_basis.op(1).is_equal(_ex1)) { // a^-x -> {1, sym(a^x)} - return (new lst(_ex1(), replace_with_symbol(power(n.op(0), -exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated); + 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.op(1) / n.op(0), -exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + 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); } - - } else { // exponent not numeric - - // (a/b)^x -> {sym((a/b)^x, 1} - return (new lst(replace_with_symbol(power(n.op(0) / n.op(1), exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); } - } -} - - -/** Implementation of ex::normal() for pseries. It normalizes each coefficient and - * replaces the series by a temporary symbol. - * @see ex::normal */ -ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const -{ - epvector new_seq; - new_seq.reserve(seq.size()); + } - epvector::const_iterator it = seq.begin(), itend = seq.end(); - while (it != itend) { - new_seq.push_back(expair(it->rest.normal(), it->coeff)); - it++; - } - ex n = pseries(relational(var,point), new_seq); - return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _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 relationals. It normalizes both sides. +/** Implementation of ex::normal() for pseries. It normalizes each coefficient + * and replaces the series by a temporary symbol. * @see ex::normal */ -ex relational::normal(lst &sym_lst, lst &repl_lst, int level) const +ex pseries::normal(exmap & repl, exmap & rev_lookup, int level) const { - return (new lst(relational(lh.normal(), rh.normal(), o), _ex1()))->setflag(status_flags::dynallocated); + 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); } @@ -1983,147 +2280,410 @@ ex relational::normal(lst &sym_lst, lst &repl_lst, int level) const * @return normalized expression */ ex ex::normal(int level) const { - lst sym_lst, repl_lst; + exmap repl, rev_lookup; - ex e = bp->normal(sym_lst, repl_lst, level); - GINAC_ASSERT(is_ex_of_type(e, lst)); + ex e = bp->normal(repl, rev_lookup, level); + GINAC_ASSERT(is_a(e)); // Re-insert replaced symbols - if (sym_lst.nops() > 0) - e = e.subs(sym_lst, repl_lst); + 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); + return e.op(0) / e.op(1); } -/** Numerator of an expression. If the expression is not of the normal form - * "numerator/denominator", it is first converted to this form and then the - * numerator is returned. +/** Get numerator of an expression. If the expression is not of the normal + * form "numerator/denominator", it is first converted to this form and + * then the numerator is returned. * * @see ex::normal * @return numerator */ -ex ex::numer(void) const +ex ex::numer() const { - lst sym_lst, repl_lst; + exmap repl, rev_lookup; - ex e = bp->normal(sym_lst, repl_lst, 0); - GINAC_ASSERT(is_ex_of_type(e, lst)); + ex e = bp->normal(repl, rev_lookup, 0); + GINAC_ASSERT(is_a(e)); // Re-insert replaced symbols - if (sym_lst.nops() > 0) - return e.op(0).subs(sym_lst, repl_lst); - else + if (repl.empty()) return e.op(0); + else + return e.op(0).subs(repl, subs_options::no_pattern); } -/** 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. +/** Get denominator of an expression. If the expression is not of the normal + * form "numerator/denominator", it is first converted to this form and + * then the denominator is returned. * * @see ex::normal * @return denominator */ -ex ex::denom(void) const +ex ex::denom() const { - lst sym_lst, repl_lst; + exmap repl, rev_lookup; - ex e = bp->normal(sym_lst, repl_lst, 0); - GINAC_ASSERT(is_ex_of_type(e, lst)); + ex e = bp->normal(repl, rev_lookup, 0); + GINAC_ASSERT(is_a(e)); // Re-insert replaced symbols - if (sym_lst.nops() > 0) - return e.op(1).subs(sym_lst, repl_lst); - else + 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); +} -/** Default implementation of ex::to_rational(). It replaces the object with a - * temporary symbol. - * @see ex::to_rational */ -ex basic::to_rational(lst &repl_lst) const +// GiNaC 1.1 compatibility function +ex ex::to_polynomial(lst & repl_lst) const { - return replace_with_symbol(*this, repl_lst); + // 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); +} -/** Implementation of ex::to_rational() for symbols. This returns the unmodified symbol. - * @see ex::to_rational */ -ex symbol::to_rational(lst &repl_lst) const +ex basic::to_polynomial(exmap & repl) const { - return *this; + return replace_with_symbol(*this, repl); } -/** Implementation of ex::to_rational() for a numeric. It splits complex numbers - * into re+I*im and replaces I and non-rational real numbers with a temporary - * symbol. - * @see ex::to_rational */ -ex numeric::to_rational(lst &repl_lst) const -{ - if (is_real()) { - if (!is_integer()) - return replace_with_symbol(*this, repl_lst); - } else { // complex - numeric re = real(); - numeric im = imag(); - ex re_ex = re.is_rational() ? re : replace_with_symbol(re, repl_lst); - ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl_lst); - return re_ex + im_ex * replace_with_symbol(I, repl_lst); - } +/** 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. - * @see ex::to_rational */ -ex power::to_rational(lst &repl_lst) const + * powers by temporary symbols. */ +ex power::to_rational(exmap & repl) const { if (exponent.info(info_flags::integer)) - return power(basis.to_rational(repl_lst), exponent); + return power(basis.to_rational(repl), exponent); else - return replace_with_symbol(*this, repl_lst); + 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. - * @see ex::to_rational */ -ex expairseq::to_rational(lst &repl_lst) const +/** Implementation of ex::to_rational() for expairseqs. */ +ex expairseq::to_rational(exmap & repl) const { - epvector s; - s.reserve(seq.size()); - for (epvector::const_iterator it=seq.begin(); it!=seq.end(); ++it) { - s.push_back(split_ex_to_pair(recombine_pair_to_ex(*it).to_rational(repl_lst))); - // s.push_back(combine_ex_with_coeff_to_pair((*it).rest.to_rational(repl_lst), - } - ex oc = overall_coeff.to_rational(repl_lst); - if (oc.info(info_flags::numeric)) - return thisexpairseq(s, overall_coeff); - else s.push_back(combine_ex_with_coeff_to_pair(oc,_ex1())); - return thisexpairseq(s, default_overall_coeff()); + 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()); +} -/** Rationalization of non-rational functions. - * This function converts a general expression to a rational polynomial - * by replacing all non-rational subexpressions (like non-rational numbers, - * non-integer powers or functions like sin(), cos() etc.) to temporary - * symbols. This makes it possible to use functions like gcd() and divide() - * on non-rational functions by applying to_rational() on the arguments, - * calling the desired function and re-substituting the temporary symbols - * in the result. To make the last step possible, all temporary symbols and - * their associated expressions are collected in the list specified by the - * repl_lst parameter in the form {symbol == expression}, ready to be passed - * as an argument to ex::subs(). - * - * @param repl_lst collects a list of all temporary symbols and their replacements - * @return rationalized expression */ -ex ex::to_rational(lst &repl_lst) const + +/** 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) { - return bp->to_rational(repl_lst); + 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(); } -#ifndef NO_NAMESPACE_GINAC } // namespace GiNaC -#endif // ndef NO_NAMESPACE_GINAC