X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnormal.cpp;h=5d044fc6f40f6da06a6aaff3b12c670115cd7c3a;hp=d3a407d50e713ba633054784f263976360f5dc49;hb=c77689e7ac8d8f4dbca0f337b6e9acf2419010ff;hpb=c8f1702a09070f7d0d64118a0a3405521affb2cb diff --git a/ginac/normal.cpp b/ginac/normal.cpp index d3a407d5..5d044fc6 100644 --- a/ginac/normal.cpp +++ b/ginac/normal.cpp @@ -6,7 +6,7 @@ * computation, square-free factorization and rational function normalization. */ /* - * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2008 Johannes Gutenberg University Mainz, Germany * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by @@ -20,7 +20,7 @@ * * 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 @@ -39,6 +39,8 @@ #include "numeric.h" #include "power.h" #include "relational.h" +#include "operators.h" +#include "matrix.h" #include "pseries.h" #include "symbol.h" #include "utils.h" @@ -73,32 +75,32 @@ 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); + if (is_a(e)) { + x = e; return true; - } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) { - for (unsigned i=0; i(e) || is_exactly_a(e)) { + for (size_t i=0; i(e)) { if (get_first_symbol(e.op(0), x)) return true; } @@ -117,8 +119,8 @@ 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; @@ -136,7 +138,7 @@ struct sym_desc { int max_deg; /** Maximum number of terms of leading coefficient of symbol in both polynomials */ - int max_lcnops; + size_t max_lcnops; /** Commparison operator for sorting */ bool operator<(const sym_desc &x) const @@ -152,13 +154,13 @@ struct sym_desc { 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(); + sym_desc_vec::const_iterator it = v.begin(), itend = v.end(); while (it != itend) { - if (it->sym->compare(*s) == 0) // If it's already in there, don't add it a second time + if (it->sym.is_equal(s)) // If it's already in there, don't add it a second time return; - it++; + ++it; } sym_desc d; d.sym = s; @@ -168,12 +170,12 @@ static void add_symbol(const symbol *s, sym_desc_vec &v) // 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); } } @@ -196,24 +198,25 @@ static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v) 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)); + 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++; + 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; } - sort(v.begin(), v.end()); + std::sort(v.begin(), v.end()); + #if 0 std::clog << "Symbols:\n"; it = v.begin(); itend = v.end(); while (it != itend) { - std::clog << " " << *it->sym << ": deg_a=" << it->deg_a << ", deg_b=" << it->deg_b << ", ldeg_a=" << it->ldeg_a << ", ldeg_b=" << it->ldeg_b << ", max_deg=" << it->max_deg << ", max_lcnops=" << it->max_lcnops << endl; - std::clog << " lcoeff_a=" << a.lcoeff(*(it->sym)) << ", lcoeff_b=" << b.lcoeff(*(it->sym)) << endl; - it++; + 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 } @@ -228,22 +231,22 @@ static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v) 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_numeric(e.op(1))); + return pow(lcmcoeff(e.op(0), l), ex_to(e.op(1))); } return l; } @@ -257,7 +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 @@ -267,80 +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_numeric(e.op(1)).inverse())), e.op(1)); + 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(); } -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); } -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(); + numeric c = *_num0_p, l = *_num1_p; 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); + 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_ex_exactly_of_type(overall_coeff,numeric)); - c = gcd(ex_to_numeric(overall_coeff),c); - return c; + 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)); + 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)); } @@ -357,28 +364,28 @@ 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)) + if (is_exactly_a(a) && is_exactly_a(b)) return a / b; #if FAST_COMPARE if (a.is_equal(b)) - return _ex1(); + 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); + 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) @@ -388,13 +395,13 @@ ex quo(const ex &a, const ex &b, const symbol &x, bool check_args) return (new fail())->setflag(status_flags::dynallocated); } term *= power(x, rdeg - bdeg); - q += term; + v.push_back(term); r -= (term * b).expand(); if (r.is_zero()) break; rdeg = r.degree(x); } - return q; + return (new add(v))->setflag(status_flags::dynallocated); } @@ -407,19 +414,19 @@ 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) +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_ex_exactly_of_type(a, numeric)) { - if (is_ex_exactly_of_type(b, numeric)) - return _ex0(); + if (is_exactly_a(a)) { + if (is_exactly_a(b)) + return _ex0; else - return b; + return a; } #if FAST_COMPARE if (a.is_equal(b)) - return _ex0(); + 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")); @@ -431,7 +438,7 @@ ex rem(const ex &a, const ex &b, const symbol &x, bool check_args) 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); + bool blcoeff_is_numeric = is_exactly_a(blcoeff); while (rdeg >= bdeg) { ex term, rcoeff = r.coeff(x, rdeg); if (blcoeff_is_numeric) @@ -450,21 +457,39 @@ ex rem(const ex &a, const ex &b, const symbol &x, bool check_args) } -/** Pseudo-remainder of polynomials a(x) and b(x) in Z[x]. +/** 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 Z[x] */ -ex prem(const ex &a, const ex &b, const symbol &x, bool check_args) + * @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_ex_exactly_of_type(a, numeric)) { - if (is_ex_exactly_of_type(b, numeric)) - return _ex0(); + if (is_exactly_a(a)) { + if (is_exactly_a(b)) + return _ex0; else return b; } @@ -480,18 +505,18 @@ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args) if (bdeg <= rdeg) { blcoeff = eb.coeff(x, bdeg); if (bdeg == 0) - eb = _ex0(); + eb = _ex0; else eb -= blcoeff * power(x, bdeg); } else - blcoeff = _ex1(); + 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(); + r = _ex0; else r -= rlcoeff * power(x, rdeg); r = (blcoeff * r).expand() - term; @@ -502,22 +527,21 @@ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args) } -/** Sparse pseudo-remainder of polynomials a(x) and b(x) in Z[x]. +/** Sparse pseudo-remainder of polynomials a(x) and b(x) in Q[x]. * * @param a first polynomial in x (dividend) * @param b second polynomial in x (divisor) * @param x a and b are polynomials in x * @param check_args check whether a and b are polynomials with rational * coefficients (defaults to "true") - * @return sparse pseudo-remainder of a(x) and b(x) in Z[x] */ - -ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args) + * @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_ex_exactly_of_type(a, numeric)) { - if (is_ex_exactly_of_type(b, numeric)) - return _ex0(); + if (is_exactly_a(a)) { + if (is_exactly_a(b)) + return _ex0; else return b; } @@ -533,17 +557,17 @@ ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args) if (bdeg <= rdeg) { blcoeff = eb.coeff(x, bdeg); if (bdeg == 0) - eb = _ex0(); + eb = _ex0; else eb -= blcoeff * power(x, bdeg); } else - blcoeff = _ex1(); + 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(); + r = _ex0; else r -= rlcoeff * power(x, rdeg); r = (blcoeff * r).expand() - term; @@ -561,22 +585,23 @@ ex sprem(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()) + if (a.is_zero()) { + q = _ex0; return true; - if (is_ex_exactly_of_type(b, numeric)) { + } + if (is_exactly_a(b)) { q = a / b; return true; - } else if (is_ex_exactly_of_type(a, numeric)) + } else if (is_exactly_a(a)) return false; #if FAST_COMPARE if (a.is_equal(b)) { - q = _ex1(); + q = _ex1; return true; } #endif @@ -585,31 +610,102 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args) throw(std::invalid_argument("divide: arguments must be polynomials over the rationals")); // Find first symbol - const symbol *x; + 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()) + if (r.is_zero()) { + q = _ex0; return true; - int bdeg = b.degree(*x); - int rdeg = r.degree(*x); - ex blcoeff = b.expand().coeff(*x, bdeg); - bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric); + } + 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); + 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; + term *= power(x, rdeg - bdeg); + v.push_back(term); r -= (term * b).expand(); - if (r.is_zero()) + if (r.is_zero()) { + q = (new add(v))->setflag(status_flags::dynallocated); return true; - rdeg = r.degree(*x); + } + rdeg = r.degree(x); } return false; } @@ -638,7 +734,7 @@ typedef std::map ex2_exbool_remember; /** 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 @@ -653,15 +749,15 @@ typedef std::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(); + q = _ex0; if (b.is_zero()) throw(std::overflow_error("divide_in_z: division by zero")); - if (b.is_equal(_ex1())) { + 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)) { + if (is_exactly_a(a)) { + if (is_exactly_a(b)) { q = a / b; return q.info(info_flags::integer); } else @@ -669,7 +765,7 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite } #if FAST_COMPARE if (a.is_equal(b)) { - q = _ex1(); + q = _ex1; return true; } #endif @@ -684,11 +780,36 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite } #endif + if (is_exactly_a(b)) { + const ex& bb(b.op(0)); + ex qbar = a; + int exp_b = ex_to(b.op(1)).to_int(); + for (int i=exp_b; i>0; i--) { + if (!divide_in_z(qbar, bb, q, var)) + return false; + qbar = q; + } + return true; + } + + if (is_exactly_a(b)) { + ex qbar = a; + for (const_iterator itrb = b.begin(); itrb != b.end(); ++itrb) { + sym_desc_vec sym_stats; + get_symbol_stats(a, *itrb, sym_stats); + if (!divide_in_z(qbar, *itrb, q, sym_stats.begin())) + return false; + + qbar = q; + } + return true; + } + // Main symbol - const symbol *x = var->sym; + const ex &x = var->sym; // Compare degrees - int adeg = a.degree(*x), bdeg = b.degree(*x); + int adeg = a.degree(x), bdeg = b.degree(x); if (bdeg > adeg) return false; @@ -700,24 +821,24 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite // Compute values at evaluation points 0..adeg vector alpha; alpha.reserve(adeg + 1); exvector u; u.reserve(adeg + 1); - numeric point = _num0(); + numeric point = *_num0_p; ex c; for (i=0; i<=adeg; i++) { - ex bs = b.subs(*x == point); + ex bs = b.subs(x == point, subs_options::no_pattern); while (bs.is_zero()) { - point += _num1(); - bs = b.subs(*x == point); + point += *_num1_p; + bs = b.subs(x == point, subs_options::no_pattern); } - if (!divide_in_z(a.subs(*x == point), bs, c, var+1)) + 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(); + point += *_num1_p; } // Compute inverses vector rcp; rcp.reserve(adeg + 1); - rcp.push_back(_num0()); + rcp.push_back(*_num0_p); for (k=1; k<=adeg; k++) { numeric product = alpha[k] - alpha[0]; for (i=1; i=0; k--) - c = c * (*x - alpha[k]) + v[k]; + c = c * (x - alpha[k]) + v[k]; - if (c.degree(*x) == (adeg - bdeg)) { + if (c.degree(x) == (adeg - bdeg)) { q = c.expand(); return true; } else @@ -754,21 +875,23 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite return true; int rdeg = adeg; ex eb = b.expand(); - ex blcoeff = eb.coeff(*x, bdeg); + 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); + 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; + term = (term * power(x, rdeg - bdeg)).expand(); + v.push_back(term); r -= (term * eb).expand(); if (r.is_zero()) { + q = (new add(v))->setflag(status_flags::dynallocated); #if USE_REMEMBER dr_remember[ex2(a, b)] = exbool(q, true); #endif return true; } - rdeg = r.degree(*x); + rdeg = r.degree(x); } #if USE_REMEMBER dr_remember[ex2(a, b)] = exbool(q, false); @@ -784,21 +907,21 @@ 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(); + if (is_exactly_a(c)) + return c.info(info_flags::negative) ?_ex_1 : _ex1; else { - const symbol *y; + ex y; if (get_first_symbol(c, y)) - return c.unit(*y); + return c.unit(y); else throw(std::invalid_argument("invalid expression in unit()")); } @@ -806,293 +929,138 @@ ex ex::unit(const symbol &x) const /** 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 + * @see ex::unit, ex::primpart, ex::unitcontprim */ +ex ex::content(const ex &x) const { - if (is_zero()) - return _ex0(); - if (is_ex_exactly_of_type(*this, numeric)) + if (is_exactly_a(*this)) return info(info_flags::negative) ? -*this : *this; + ex e = expand(); if (e.is_zero()) - return _ex0(); + return _ex0; - // First, try the integer content + // 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; - ex lcoeff = r.lcoeff(x); + int deg = r.degree(x); + ex lcoeff = r.coeff(x, deg); if (lcoeff.info(info_flags::integer)) return c; // GCD of all coefficients - int deg = e.degree(x); - int ldeg = e.ldegree(x); + int ldeg = r.ldegree(x); if (deg == ldeg) - return e.lcoeff(x) / e.unit(x); - c = _ex0(); + return lcoeff * c / lcoeff.unit(x); + ex cont = _ex0; for (int i=ldeg; i<=deg; i++) - c = gcd(e.coeff(x, i), c, NULL, NULL, false); - return c; + cont = gcd(r.coeff(x, i), cont, NULL, NULL, false); + return cont * 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. +/** 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(); + 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_ex_exactly_of_type(c, numeric)) + if (is_exactly_a(c)) return *this / (c * u); else return quo(*this, c * u, x, false); } -/* - * GCD of multivariate polynomials - */ - -/** Compute GCD of polynomials in Q[X] using the Euclidean algorithm (not - * really suited for multivariate GCDs). This function is only provided for - * testing purposes. - * - * @param a first multivariate polynomial - * @param b second multivariate polynomial - * @param x pointer to symbol (main variable) in which to compute the GCD in - * @return the GCD as a new expression - * @see gcd */ - -static ex eu_gcd(const ex &a, const ex &b, const symbol *x) -{ -//std::clog << "eu_gcd(" << a << "," << b << ")\n"; - - // Sort c and d so that c has higher degree - ex c, d; - int adeg = a.degree(*x), bdeg = b.degree(*x); - if (adeg >= bdeg) { - c = a; - d = b; - } else { - c = b; - d = a; - } - - // Normalize in Q[x] - c = c / c.lcoeff(*x); - d = d / d.lcoeff(*x); - - // Euclidean algorithm - ex r; - for (;;) { -//std::clog << " d = " << d << endl; - r = rem(c, d, *x, false); - if (r.is_zero()) - return d / d.lcoeff(*x); - c = d; - d = r; - } -} - - -/** Compute GCD of multivariate polynomials using the Euclidean PRS algorithm - * with pseudo-remainders ("World's Worst GCD Algorithm", staying in Z[X]). - * This function is only provided for testing purposes. +/** 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 */ - -static ex euprem_gcd(const ex &a, const ex &b, const symbol *x) + * @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 { -//std::clog << "euprem_gcd(" << a << "," << b << ")\n"; - - // Sort c and d so that c has higher degree - ex c, d; - int adeg = a.degree(*x), bdeg = b.degree(*x); - if (adeg >= bdeg) { - c = a; - d = b; - } else { - c = b; - d = a; - } - - // Calculate GCD of contents - ex gamma = gcd(c.content(*x), d.content(*x), NULL, NULL, false); - - // Euclidean algorithm with pseudo-remainders - ex r; - for (;;) { -//std::clog << " d = " << d << endl; - r = prem(c, d, *x, false); - if (r.is_zero()) - return d.primpart(*x) * gamma; - c = d; - d = r; + // Quick check for zero (avoid expanding) + if (is_zero()) { + u = _ex1; + c = p = _ex0; + return; + } + + // Special case: input is a number + if (is_exactly_a(*this)) { + if (info(info_flags::negative)) { + u = _ex_1; + c = abs(ex_to(*this)); + } else { + u = _ex1; + c = *this; + } + p = _ex1; + return; } -} - - -/** Compute GCD of multivariate polynomials using the primitive Euclidean - * PRS algorithm (complete content removal at each step). This function is - * only provided for testing purposes. - * - * @param a first multivariate polynomial - * @param b second multivariate polynomial - * @param x pointer to symbol (main variable) in which to compute the GCD in - * @return the GCD as a new expression - * @see gcd */ - -static ex peu_gcd(const ex &a, const ex &b, const symbol *x) -{ -//std::clog << "peu_gcd(" << a << "," << b << ")\n"; - // Sort c and d so that c has higher degree - ex c, d; - int adeg = a.degree(*x), bdeg = b.degree(*x); - int ddeg; - if (adeg >= bdeg) { - c = a; - d = b; - ddeg = bdeg; - } else { - c = b; - d = a; - ddeg = adeg; + // Expand input polynomial + ex e = expand(); + if (e.is_zero()) { + u = _ex1; + c = p = _ex0; + return; } - // 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); + // Compute unit and content + u = unit(x); + c = content(x); - // Euclidean algorithm with content removal - ex r; - for (;;) { -//std::clog << " d = " << d << endl; - r = prem(c, d, *x, false); - if (r.is_zero()) - return gamma * d; - c = d; - d = r.primpart(*x); + // 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); } -/** Compute GCD of multivariate polynomials using the reduced PRS algorithm. - * This function is only provided for testing purposes. - * - * @param a first multivariate polynomial - * @param b second multivariate polynomial - * @param x pointer to symbol (main variable) in which to compute the GCD in - * @return the GCD as a new expression - * @see gcd */ - -static ex red_gcd(const ex &a, const ex &b, const symbol *x) -{ -//std::clog << "red_gcd(" << a << "," << b << ")\n"; - - // Sort c and d so that c has higher degree - ex c, d; - int adeg = a.degree(*x), bdeg = b.degree(*x); - int cdeg, ddeg; - if (adeg >= bdeg) { - c = a; - d = b; - cdeg = adeg; - ddeg = bdeg; - } else { - c = b; - d = a; - cdeg = bdeg; - ddeg = adeg; - } - - // Remove content from c and d, to be attached to GCD later - ex cont_c = c.content(*x); - ex cont_d = d.content(*x); - ex gamma = gcd(cont_c, cont_d, NULL, NULL, false); - if (ddeg == 0) - return gamma; - c = c.primpart(*x, cont_c); - d = d.primpart(*x, cont_d); - - // First element of divisor sequence - ex r, ri = _ex1(); - int delta = cdeg - ddeg; - - for (;;) { - // Calculate polynomial pseudo-remainder -//std::clog << " d = " << d << endl; - r = prem(c, d, *x, false); - if (r.is_zero()) - return gamma * d.primpart(*x); - c = d; - cdeg = ddeg; - - if (!divide(r, pow(ri, delta), d, false)) - throw(std::runtime_error("invalid expression in red_gcd(), division failed")); - ddeg = d.degree(*x); - if (ddeg == 0) { - if (is_ex_exactly_of_type(r, numeric)) - return gamma; - else - return gamma * r.primpart(*x); - } - - ri = c.expand().lcoeff(*x); - delta = cdeg - ddeg; - } -} - +/* + * GCD of multivariate polynomials + */ /** Compute GCD of multivariate polynomials using the subresultant PRS * algorithm. This function is used internally by gcd(). @@ -1105,13 +1073,12 @@ static ex red_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) { -//std::clog << "sr_gcd(" << a << "," << b << ")\n"; #if STATISTICS sr_gcd_called++; #endif // The first symbol is our main variable - const symbol &x = *(var->sym); + const ex &x = var->sym; // Sort c and d so that c has higher degree ex c, d; @@ -1137,34 +1104,31 @@ static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var) return gamma; c = c.primpart(x, cont_c); d = d.primpart(x, cont_d); -//std::clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n"; // First element of subresultant sequence - ex r = _ex0(), ri = _ex1(), psi = _ex1(); + ex r = _ex0, ri = _ex1, psi = _ex1; int delta = cdeg - ddeg; for (;;) { + // Calculate polynomial pseudo-remainder -//std::clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n"; -//std::clog << " d = " << d << endl; r = prem(c, d, x, false); if (r.is_zero()) return gamma * d.primpart(x); + c = d; cdeg = ddeg; -//std::clog << " dividing...\n"; if (!divide_in_z(r, ri * pow(psi, delta), d, var)) throw(std::runtime_error("invalid expression in sr_gcd(), division failed")); ddeg = d.degree(x); if (ddeg == 0) { - if (is_ex_exactly_of_type(r, numeric)) + if (is_exactly_a(r)) return gamma; else return gamma * r.primpart(x); } // Next element of subresultant sequence -//std::clog << " calculating next subresultant...\n"; ri = c.expand().lcoeff(x); if (delta == 1) psi = ri; @@ -1178,37 +1142,35 @@ static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var) /** 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(); } /** Implementation ex::max_coefficient(). * @see heur_gcd */ -numeric basic::max_coefficient(void) const +numeric basic::max_coefficient() const { - return _num1(); + return *_num1_p; } -numeric numeric::max_coefficient(void) const +numeric numeric::max_coefficient() const { 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)); + GINAC_ASSERT(is_exactly_a(overall_coeff)); + numeric cur_max = abs(ex_to(overall_coeff)); while (it != itend) { numeric a; - GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric)); - a = abs(ex_to_numeric(it->coeff)); + GINAC_ASSERT(!is_exactly_a(it->rest)); + a = abs(ex_to(it->coeff)); if (a > cur_max) cur_max = a; it++; @@ -1216,34 +1178,27 @@ numeric add::max_coefficient(void) const 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)); + 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; @@ -1261,14 +1216,14 @@ ex add::smod(const numeric &xi) const 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)); - numeric coeff = GiNaC::smod(ex_to_numeric(it->coeff), xi); + 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_ex_exactly_of_type(overall_coeff,numeric)); - numeric coeff = GiNaC::smod(ex_to_numeric(overall_coeff), xi); + 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); } @@ -1278,13 +1233,13 @@ ex mul::smod(const numeric &xi) const 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)); + 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)); - mulcopyp->overall_coeff = GiNaC::smod(ex_to_numeric(overall_coeff),xi); + 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); @@ -1292,17 +1247,17 @@ ex mul::smod(const numeric &xi) const /** xi-adic polynomial interpolation */ -static ex interpolate(const ex &gamma, const numeric &xi, const symbol &x) +static ex interpolate(const ex &gamma, const numeric &xi, const ex &x, int degree_hint = 1) { - ex g = _ex0(); + 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 += gi * power(x, i); + g.push_back(gi * power(x, i)); e = (e - gi) * rxi; } - return g; + return (new add(g))->setflag(status_flags::dynallocated); } /** Exception thrown by heur_gcd() to signal failure. */ @@ -1313,70 +1268,75 @@ class gcdheu_failed {}; * polynomials and an iterator to the first element of the sym_desc vector * passed in. This function is used internally by gcd(). * - * @param a first multivariate polynomial (expanded) - * @param b second multivariate polynomial (expanded) + * @param a first integer multivariate polynomial (expanded) + * @param b second integer multivariate polynomial (expanded) * @param ca cofactor of polynomial a (returned), NULL to suppress * calculation of cofactor * @param cb cofactor of polynomial b (returned), NULL to suppress * calculation of cofactor * @param var iterator to first element of vector of sym_desc structs - * @return the GCD as a new expression + * @param res the GCD (returned) + * @return true if GCD was computed, false otherwise. * @see gcd * @exception gcdheu_failed() */ -static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var) +static bool heur_gcd_z(ex& res, const ex &a, const ex &b, ex *ca, ex *cb, + sym_desc_vec::const_iterator var) { -//std::clog << "heur_gcd(" << a << "," << b << ")\n"; #if STATISTICS heur_gcd_called++; #endif - // Algorithms only works for non-vanishing input polynomials + // Algorithm only works for non-vanishing input polynomials if (a.is_zero() || b.is_zero()) - return (new fail())->setflag(status_flags::dynallocated); + return false; // GCD of two numeric values -> CLN - if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) { - numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b)); + if (is_exactly_a(a) && is_exactly_a(b)) { + numeric g = gcd(ex_to(a), ex_to(b)); if (ca) - *ca = ex_to_numeric(a) / g; + *ca = ex_to(a) / g; if (cb) - *cb = ex_to_numeric(b) / g; - return g; + *cb = ex_to(b) / g; + res = g; + return true; } // The first symbol is our main variable - const symbol &x = *(var->sym); + 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)); + int maxdeg = std::max(p.degree(x), q.degree(x)); // Find evaluation point numeric mp = p.max_coefficient(); numeric mq = q.max_coefficient(); numeric xi; if (mp > mq) - xi = mq * _num2() + _num2(); + xi = mq * (*_num2_p) + (*_num2_p); else - xi = mp * _num2() + _num2(); + xi = mp * (*_num2_p) + (*_num2_p); // 6 tries maximum for (int t=0; t<6; t++) { if (xi.int_length() * maxdeg > 100000) { -//std::clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << std::endl; throw gcdheu_failed(); } // Apply evaluation homomorphism and calculate GCD ex cp, cq; - ex gamma = heur_gcd(p.subs(x == xi), q.subs(x == xi), &cp, &cq, var+1).expand(); - if (!is_ex_exactly_of_type(gamma, fail)) { - + ex gamma; + bool found = heur_gcd_z(gamma, + p.subs(x == xi, subs_options::no_pattern), + q.subs(x == xi, subs_options::no_pattern), + &cp, &cq, var+1); + if (found) { + gamma = gamma.expand(); // Reconstruct polynomial from GCD of mapped polynomials - ex g = interpolate(gamma, xi, x); + ex g = interpolate(gamma, xi, x, maxdeg); // Remove integer content g /= g.integer_content(); @@ -1385,78 +1345,115 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const ex dummy; if (divide_in_z(p, g, ca ? *ca : dummy, var) && divide_in_z(q, g, cb ? *cb : dummy, var)) { g *= gc; - ex lc = g.lcoeff(x); - if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative()) - return -g; - else - return g; - } -#if 0 - cp = interpolate(cp, xi, x); - if (divide_in_z(cp, p, g, var)) { - if (divide_in_z(g, q, cb ? *cb : dummy, var)) { - g *= gc; - if (ca) - *ca = cp; - ex lc = g.lcoeff(x); - if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative()) - return -g; - else - return g; - } - } - cq = interpolate(cq, xi, x); - if (divide_in_z(cq, q, g, var)) { - if (divide_in_z(g, p, ca ? *ca : dummy, var)) { - g *= gc; - if (cb) - *cb = cq; - ex lc = g.lcoeff(x); - if (is_ex_exactly_of_type(lc, numeric) && ex_to_numeric(lc).is_negative()) - return -g; - else - return g; - } + res = g; + return true; } -#endif } // Next evaluation point xi = iquo(xi * isqrt(isqrt(xi)) * numeric(73794), numeric(27011)); } - return (new fail())->setflag(status_flags::dynallocated); + return false; +} + +/** Compute GCD of multivariate polynomials using the heuristic GCD algorithm. + * get_symbol_stats() must have been called previously with the input + * polynomials and an iterator to the first element of the sym_desc vector + * passed in. This function is used internally by gcd(). + * + * @param a first rational multivariate polynomial (expanded) + * @param b second rational multivariate polynomial (expanded) + * @param ca cofactor of polynomial a (returned), NULL to suppress + * calculation of cofactor + * @param cb cofactor of polynomial b (returned), NULL to suppress + * calculation of cofactor + * @param var iterator to first element of vector of sym_desc structs + * @param res the GCD (returned) + * @return true if GCD was computed, false otherwise. + * @see heur_gcd_z + * @see gcd + */ +static bool heur_gcd(ex& res, const ex& a, const ex& b, ex *ca, ex *cb, + sym_desc_vec::const_iterator var) +{ + if (a.info(info_flags::integer_polynomial) && + b.info(info_flags::integer_polynomial)) { + try { + return heur_gcd_z(res, a, b, ca, cb, var); + } catch (gcdheu_failed) { + return false; + } + } + + // convert polynomials to Z[X] + const numeric a_lcm = lcm_of_coefficients_denominators(a); + const numeric ab_lcm = lcmcoeff(b, a_lcm); + + const ex ai = a*ab_lcm; + const ex bi = b*ab_lcm; + if (!ai.info(info_flags::integer_polynomial)) + throw std::logic_error("heur_gcd: not an integer polynomial [1]"); + + if (!bi.info(info_flags::integer_polynomial)) + throw std::logic_error("heur_gcd: not an integer polynomial [2]"); + + bool found = false; + try { + found = heur_gcd_z(res, ai, bi, ca, cb, var); + } catch (gcdheu_failed) { + return false; + } + + // GCD is not unique, it's defined up to a unit (i.e. invertible + // element). If the coefficient ring is a field, every its element is + // invertible, so one can multiply the polynomial GCD with any element + // of the coefficient field. We use this ambiguity to make cofactors + // integer polynomials. + if (found) + res /= ab_lcm; + return found; } +// gcd helper to handle partially factored polynomials (to avoid expanding +// large expressions). At least one of the arguments should be a power. +static ex gcd_pf_pow(const ex& a, const ex& b, ex* ca, ex* cb, bool check_args); + +// gcd helper to handle partially factored polynomials (to avoid expanding +// large expressions). At least one of the arguments should be a product. +static ex gcd_pf_mul(const ex& a, const ex& b, ex* ca, ex* cb, bool check_args); + /** 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) +ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args, unsigned options) { -//std::clog << "gcd(" << a << "," << b << ")\n"; #if STATISTICS gcd_called++; #endif // GCD of numerics -> CLN - if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) { - numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b)); + if (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(); + *ca = _ex0; if (cb) - *cb = _ex0(); + *cb = _ex0; } else { if (ca) - *ca = ex_to_numeric(a) / g; + *ca = ex_to(a) / g; if (cb) - *cb = ex_to_numeric(b) / g; + *cb = ex_to(b) / g; } } return g; @@ -1468,129 +1465,115 @@ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args) } // Partially factored cases (to avoid expanding large expressions) - if (is_ex_exactly_of_type(a, mul)) { - if (is_ex_exactly_of_type(b, mul) && b.nops() > a.nops()) - goto factored_b; -factored_a: - ex g = _ex1(); - ex acc_ca = _ex1(); - ex part_b = b; - for (unsigned i=0; i b.nops()) - goto factored_a; -factored_b: - ex g = _ex1(); - ex acc_cb = _ex1(); - ex part_a = a; - for (unsigned i=0; i(a) || is_exactly_a(b)) + return gcd_pf_mul(a, b, ca, cb, check_args); #if FAST_COMPARE - // Input polynomials of the form poly^n are sometimes also trivial - if (is_ex_exactly_of_type(a, power)) { - ex p = a.op(0); - if (is_ex_exactly_of_type(b, power)) { - if (p.is_equal(b.op(0))) { - // a = p^n, b = p^m, gcd = p^min(n, m) - ex exp_a = a.op(1), exp_b = b.op(1); - if (exp_a < exp_b) { - if (ca) - *ca = _ex1(); - if (cb) - *cb = power(p, exp_b - exp_a); - return power(p, exp_a); - } else { - if (ca) - *ca = power(p, exp_a - exp_b); - if (cb) - *cb = _ex1(); - return power(p, exp_b); - } - } - } else { - if (p.is_equal(b)) { - // a = p^n, b = p, gcd = p - if (ca) - *ca = power(p, a.op(1) - 1); - if (cb) - *cb = _ex1(); - return p; - } - } - } else if (is_ex_exactly_of_type(b, power)) { - ex p = b.op(0); - if (p.is_equal(a)) { - // a = p, b = p^n, gcd = p - if (ca) - *ca = _ex1(); - if (cb) - *cb = power(p, b.op(1) - 1); - return p; - } - } + if (is_exactly_a(a) || is_exactly_a(b)) + return gcd_pf_pow(a, b, ca, cb, check_args); #endif + } // Some trivial cases ex aex = a.expand(), bex = b.expand(); if (aex.is_zero()) { if (ca) - *ca = _ex0(); + *ca = _ex0; if (cb) - *cb = _ex1(); + *cb = _ex1; return b; } if (bex.is_zero()) { if (ca) - *ca = _ex1(); + *ca = _ex1; if (cb) - *cb = _ex0(); + *cb = _ex0; return a; } - if (aex.is_equal(_ex1()) || bex.is_equal(_ex1())) { + if (aex.is_equal(_ex1) || bex.is_equal(_ex1)) { if (ca) *ca = a; if (cb) *cb = b; - return _ex1(); + return _ex1; } #if FAST_COMPARE if (a.is_equal(b)) { if (ca) - *ca = _ex1(); + *ca = _ex1; if (cb) - *cb = _ex1(); + *cb = _ex1; return a; } #endif + if (is_a(aex)) { + if (! bex.subs(aex==_ex0, subs_options::no_pattern).is_zero()) { + if (ca) + *ca = a; + if (cb) + *cb = b; + return _ex1; + } + } + + if (is_a(bex)) { + if (! aex.subs(bex==_ex0, subs_options::no_pattern).is_zero()) { + if (ca) + *ca = a; + if (cb) + *cb = b; + return _ex1; + } + } + + if (is_exactly_a(aex)) { + numeric bcont = bex.integer_content(); + numeric g = gcd(ex_to(aex), bcont); + if (ca) + *ca = ex_to(aex)/g; + if (cb) + *cb = bex/g; + return g; + } + + if (is_exactly_a(bex)) { + numeric acont = aex.integer_content(); + numeric g = gcd(ex_to(bex), acont); + if (ca) + *ca = aex/g; + if (cb) + *cb = ex_to(bex)/g; + return g; + } + // Gather symbol statistics sym_desc_vec sym_stats; get_symbol_stats(a, b, sym_stats); - // The symbol with least degree is our main variable + // 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 symbol &x = *(var->sym); + const ex &x = var->sym; // Cancel trivial common factor int ldeg_a = var->ldeg_a; @@ -1598,73 +1581,211 @@ factored_b: int min_ldeg = std::min(ldeg_a,ldeg_b); if (min_ldeg > 0) { ex common = power(x, min_ldeg); -//std::clog << "trivial common factor " << common << std::endl; return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common; } // Try to eliminate variables - if (var->deg_a == 0) { -//std::clog << "eliminating variable " << x << " from b" << std::endl; - ex c = bex.content(x); - ex g = gcd(aex, c, ca, cb, false); + if (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.unit(x) * bex.primpart(x, c); + *cb *= bex_u * bex_p; return g; - } else if (var->deg_b == 0) { -//std::clog << "eliminating variable " << x << " from a" << std::endl; - ex c = aex.content(x); - ex g = gcd(c, bex, ca, cb, false); + } 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.unit(x) * aex.primpart(x, c); + *ca *= aex_u * aex_p; return g; } - ex g; -#if 1 // Try heuristic algorithm first, fall back to PRS if that failed - try { - g = heur_gcd(aex, bex, ca, cb, var); - } catch (gcdheu_failed) { - g = fail(); - } - if (is_ex_exactly_of_type(g, fail)) { -//std::clog << "heuristics failed" << std::endl; + ex g; + if (!(options & gcd_options::no_heur_gcd)) { + bool found = heur_gcd(g, aex, bex, ca, cb, var); + if (found) { + // heur_gcd have already computed cofactors... + if (g.is_equal(_ex1)) { + // ... but we want to keep them factored if possible. + if (ca) + *ca = a; + if (cb) + *cb = b; + } + return g; + } #if STATISTICS - heur_gcd_failed++; -#endif + else { + heur_gcd_failed++; + } #endif -// g = heur_gcd(aex, bex, ca, cb, var); -// g = eu_gcd(aex, bex, &x); -// g = euprem_gcd(aex, bex, &x); -// g = peu_gcd(aex, bex, &x); -// g = red_gcd(aex, bex, &x); - g = sr_gcd(aex, bex, var); - if (g.is_equal(_ex1())) { - // Keep cofactors factored if possible - if (ca) - *ca = a; - if (cb) - *cb = b; + } + + g = sr_gcd(aex, bex, var); + if (g.is_equal(_ex1)) { + // Keep cofactors factored if possible + if (ca) + *ca = a; + if (cb) + *cb = b; + } else { + if (ca) + divide(aex, g, *ca, false); + if (cb) + divide(bex, g, *cb, false); + } + return g; +} + +static ex gcd_pf_pow(const ex& a, const ex& b, ex* ca, ex* cb, bool check_args) +{ + if (is_exactly_a(a)) { + ex p = a.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) + if (exp_a < exp_b) { + if (ca) + *ca = _ex1; + if (cb) + *cb = power(p, exp_b - exp_a); + return power(p, exp_a); + } else { + if (ca) + *ca = power(p, exp_a - exp_b); + if (cb) + *cb = _ex1; + return power(p, exp_b); + } + } else { + ex p_co, pb_co; + ex p_gcd = gcd(p, pb, &p_co, &pb_co, check_args); + if (p_gcd.is_equal(_ex1)) { + // a(x) = p(x)^n, b(x) = p_b(x)^m, gcd (p, p_b) = 1 ==> + // gcd(a,b) = 1 + if (ca) + *ca = a; + if (cb) + *cb = b; + return _ex1; + // XXX: do I need to check for p_gcd = -1? + } else { + // there are common factors: + // a(x) = g(x)^n A(x)^n, b(x) = g(x)^m B(x)^m ==> + // gcd(a, b) = g(x)^n gcd(A(x)^n, g(x)^(n-m) B(x)^m + if (exp_a < exp_b) { + return power(p_gcd, exp_a)* + gcd(power(p_co, exp_a), power(p_gcd, exp_b-exp_a)*power(pb_co, exp_b), ca, cb, false); + } else { + return power(p_gcd, exp_b)* + gcd(power(p_gcd, exp_a - exp_b)*power(p_co, exp_a), power(pb_co, exp_b), ca, cb, false); + } + } // p_gcd.is_equal(_ex1) + } // p.is_equal(pb) + } else { + if (p.is_equal(b)) { + // a = p^n, b = p, gcd = p + if (ca) + *ca = power(p, a.op(1) - 1); + if (cb) + *cb = _ex1; + return p; + } + + ex p_co, bpart_co; + ex p_gcd = gcd(p, b, &p_co, &bpart_co, false); + + if (p_gcd.is_equal(_ex1)) { + // a(x) = p(x)^n, gcd(p, b) = 1 ==> gcd(a, b) = 1 + if (ca) + *ca = a; + if (cb) + *cb = b; + return _ex1; + } else { + // a(x) = g(x)^n A(x)^n, b(x) = g(x) B(x) ==> gcd(a, b) = g(x) gcd(g(x)^(n-1) A(x)^n, B(x)) + return p_gcd*gcd(power(p_gcd, exp_a-1)*power(p_co, exp_a), bpart_co, ca, cb, false); + } + } // is_exactly_a(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) - divide(aex, g, *ca, false); + *ca = _ex1; if (cb) - divide(bex, g, *cb, false); + *cb = power(p, b.op(1) - 1); + return p; } -#if 1 - } else { - if (g.is_equal(_ex1())) { - // Keep cofactors factored if possible + + 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 - return g; } +static ex gcd_pf_mul(const ex& a, const ex& b, ex* ca, ex* cb, bool check_args) +{ + if (is_exactly_a(a)) { + if (is_exactly_a(b) && b.nops() > a.nops()) + goto factored_b; +factored_a: + size_t num = a.nops(); + exvector g; g.reserve(num); + exvector acc_ca; acc_ca.reserve(num); + ex part_b = b; + for (size_t i=0; isetflag(status_flags::dynallocated); + if (cb) + *cb = part_b; + 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: + size_t num = b.nops(); + exvector g; g.reserve(num); + exvector acc_cb; acc_cb.reserve(num); + ex part_a = a; + for (size_t i=0; isetflag(status_flags::dynallocated); + return (new mul(g))->setflag(status_flags::dynallocated); + } +} /** Compute LCM (Least Common Multiple) of multivariate polynomials in Z[X]. * @@ -1675,8 +1796,8 @@ 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 (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")); @@ -1691,7 +1812,7 @@ ex lcm(const ex &a, const ex &b, bool check_args) */ /** Compute square-free factorization of multivariate polynomial a(x) using - * Yun´s algorithm. Used internally by sqrfree(). + * Yun's algorithm. Used internally by sqrfree(). * * @param a multivariate polynomial over Z[X], treated here as univariate * polynomial in x. @@ -1703,7 +1824,7 @@ static exvector sqrfree_yun(const ex &a, const symbol &x) ex w = a; ex z = w.diff(x); ex g = gcd(w, z); - if (g.is_equal(_ex1())) { + if (g.is_equal(_ex1)) { res.push_back(a); return res; } @@ -1717,59 +1838,180 @@ static exvector sqrfree_yun(const ex &a, const symbol &x) } while (!z.is_zero()); return res; } -/** Compute square-free factorization of multivariate polynomial in Q[X]. + + +/** Compute a square-free factorization of a multivariate polynomial in Q[X]. * * @param a multivariate polynomial over Q[X] - * @param x lst of variables to factor in, may be left empty for autodetection - * @return polynomail a in square-free factored form. */ + * @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_ex_of_type(a,numeric) || // algorithm does not trap a==0 - is_ex_of_type(a,symbol)) // shortcut + 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); - for (sym_desc_vec::iterator it=sdv.begin(); it!=sdv.end(); ++it) - args.append(*it->sym); + get_symbol_stats(a, _ex0, sdv); + sym_desc_vec::const_iterator it = sdv.begin(), itend = sdv.end(); + while (it != itend) { + args.append(it->sym); + ++it; + } } else { args = l; } + // Find the symbol to factor in at this stage - if (!is_ex_of_type(args.op(0), symbol)) + if (!is_a(args.op(0))) throw (std::runtime_error("sqrfree(): invalid factorization variable")); - const symbol x = ex_to_symbol(args.op(0)); + const symbol &x = ex_to(args.op(0)); + // convert the argument from something in Q[X] to something in Z[X] - numeric lcm = lcm_of_coefficients_denominators(a); - ex tmp = multiply_lcm(a,lcm); + const numeric lcm = lcm_of_coefficients_denominators(a); + const ex tmp = multiply_lcm(a,lcm); + // find the factors - exvector factors = sqrfree_yun(tmp,x); + exvector factors = sqrfree_yun(tmp, x); + // construct the next list of symbols with the first element popped - lst newargs; - for (int i=1; i0) { - for (exvector::iterator i=factors.begin(); i!=factors.end(); ++i) + 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::iterator it = factors.begin(); - for (int p = 1; it!=factors.end(); ++it, ++p) + 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: - result = result * quo(tmp, result, x); + + // Yun's algorithm does not account for constant factors. (For univariate + // polynomials it works only in the monic case.) We can correct this by + // inserting what has been lost back into the result. For completeness + // we'll also have to recurse down that factor in the remaining variables. + if (newargs.nops()>0) + result *= sqrfree(quo(tmp, result, x), newargs); + else + result *= quo(tmp, result, x); + + // Put in the reational overall factor again and return return result * lcm.inverse(); } +/** Compute square-free partial fraction decomposition of rational function + * a(x). + * + * @param a rational function over Z[x], treated as univariate polynomial + * in x + * @param x variable to factor in + * @return decomposed rational function */ +ex sqrfree_parfrac(const ex & a, const symbol & x) +{ + // Find numerator and denominator + ex nd = numer_denom(a); + ex numer = nd.op(0), denom = nd.op(1); +//clog << "numer = " << numer << ", denom = " << denom << endl; + + // Convert N(x)/D(x) -> Q(x) + R(x)/D(x), so degree(R) < degree(D) + ex red_poly = quo(numer, denom, x), red_numer = rem(numer, denom, x).expand(); +//clog << "red_poly = " << red_poly << ", red_numer = " << red_numer << endl; + + // Factorize denominator and compute cofactors + exvector yun = sqrfree_yun(denom, x); +//clog << "yun factors: " << exprseq(yun) << endl; + size_t num_yun = yun.size(); + exvector factor; factor.reserve(num_yun); + exvector cofac; cofac.reserve(num_yun); + for (size_t i=0; isecond; // Otherwise create new symbol and add to list, taking care that the - // replacement expression doesn't contain symbols from the sym_lst + // 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); + 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 contain symbols from the sym_lst + // 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); + 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); } @@ -1846,19 +2107,19 @@ 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); + 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, 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); + 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()) @@ -1874,17 +2135,17 @@ static ex frac_cancel(const ex &n, const ex &d) { ex num = n; ex den = d; - numeric pre_factor = _num1(); + 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())) + 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); + 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")); @@ -1898,19 +2159,26 @@ static ex frac_cancel(const ex &n, const ex &d) // Cancel GCD from numerator and denominator ex cnum, cden; - if (gcd(num, den, &cnum, &cden, false) != _ex1()) { + 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; + } } } @@ -1923,10 +2191,10 @@ static ex frac_cancel(const ex &n, const ex &d) /** 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 +ex add::normal(exmap & repl, exmap & rev_lookup, int level) const { if (level == 1) - return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + 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")); @@ -1936,12 +2204,12 @@ ex add::normal(lst &sym_lst, lst &repl_lst, int level) const dens.reserve(seq.size()+1); epvector::const_iterator it = seq.begin(), itend = seq.end(); while (it != itend) { - ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1); + 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 = overall_coeff.bp->normal(sym_lst, repl_lst, level-1); + 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()); @@ -1982,47 +2250,48 @@ ex add::normal(lst &sym_lst, lst &repl_lst, int level) const /** 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 { if (level == 1) - return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + 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 - ex num = _ex1(); - ex den = _ex1(); + 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); + 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 = overall_coeff.bp->normal(sym_lst, repl_lst, level-1); - num *= n.op(0); - den *= n.op(1); + 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 { if (level == 1) - return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + return (new lst(replace_with_symbol(*this, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated); else if (level == -max_recursion_level) throw(std::runtime_error("max recursion level reached")); // Normalize basis and exponent (exponent gets reassembled) - ex n_basis = basis.bp->normal(sym_lst, repl_lst, level-1); - ex n_exponent = exponent.bp->normal(sym_lst, repl_lst, level-1); + ex n_basis = ex_to(basis).normal(repl, rev_lookup, level-1); + ex n_exponent = ex_to(exponent).normal(repl, rev_lookup, level-1); n_exponent = n_exponent.op(0) / n_exponent.op(1); if (n_exponent.info(info_flags::integer)) { @@ -2043,51 +2312,43 @@ ex power::normal(lst &sym_lst, lst &repl_lst, int level) const if (n_exponent.info(info_flags::positive)) { // (a/b)^x -> {sym((a/b)^x), 1} - return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated); } else if (n_exponent.info(info_flags::negative)) { - if (n_basis.op(1).is_equal(_ex1())) { + if (n_basis.op(1).is_equal(_ex1)) { // a^-x -> {1, sym(a^x)} - return (new lst(_ex1(), replace_with_symbol(power(n_basis.op(0), -n_exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated); + return (new lst(_ex1, replace_with_symbol(power(n_basis.op(0), -n_exponent), repl, rev_lookup)))->setflag(status_flags::dynallocated); } else { // (a/b)^-x -> {sym((b/a)^x), 1} - return (new lst(replace_with_symbol(power(n_basis.op(1) / n_basis.op(0), -n_exponent), 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 { // n_exponent not numeric - - // (a/b)^x -> {sym((a/b)^x, 1} - return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); } } + + // (a/b)^x -> {sym((a/b)^x, 1} + return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated); } /** Implementation of ex::normal() for pseries. It normalizes each coefficient * and replaces the series by a temporary symbol. * @see ex::normal */ -ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const +ex pseries::normal(exmap & repl, exmap & rev_lookup, int level) const { epvector newseq; - for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) { + epvector::const_iterator i = seq.begin(), end = seq.end(); + while (i != end) { ex restexp = i->rest.normal(); if (!restexp.is_zero()) newseq.push_back(expair(restexp, i->coeff)); + ++i; } ex n = pseries(relational(var,point), newseq); - return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); -} - - -/** Implementation of ex::normal() for relationals. It normalizes both sides. - * @see ex::normal */ -ex relational::normal(lst &sym_lst, lst &repl_lst, int level) const -{ - return (new lst(relational(lh.normal(), rh.normal(), o), _ex1()))->setflag(status_flags::dynallocated); + return (new lst(replace_with_symbol(n, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated); } @@ -2105,73 +2366,162 @@ 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); } -/** 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); } +// GiNaC 1.1 compatibility function +ex ex::to_polynomial(lst & repl_lst) const +{ + // Convert lst to exmap + exmap m; + for (lst::const_iterator it = repl_lst.begin(); it != repl_lst.end(); ++it) + m.insert(std::make_pair(it->op(0), it->op(1))); + + ex ret = bp->to_polynomial(m); + + // Convert exmap back to lst + repl_lst.remove_all(); + for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) + repl_lst.append(it->first == it->second); -/** Default implementation of ex::to_rational(). It replaces the object with a - * temporary symbol. - * @see ex::to_rational */ -ex basic::to_rational(lst &repl_lst) const + return 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_lst); + return replace_with_symbol(*this, repl); +} + +ex basic::to_polynomial(exmap & repl) const +{ + return replace_with_symbol(*this, repl); } /** Implementation of ex::to_rational() for symbols. This returns the - * unmodified symbol. - * @see ex::to_rational */ -ex symbol::to_rational(lst &repl_lst) const + * 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; } @@ -2179,71 +2529,246 @@ ex symbol::to_rational(lst &repl_lst) const /** 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 + * temporary symbol. */ +ex numeric::to_rational(exmap & repl) const { if (is_real()) { if (!is_rational()) - return replace_with_symbol(*this, repl_lst); + 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_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); + 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), + 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_lst); + 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())); + 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(); }