X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnormal.cpp;h=e30afda6b941b90b4f1ee97aa59595d2e4bb9c0a;hp=2bcba41a6b5da2e58f4960c84ff4c6f24dd151f4;hb=24064b43ff0aebda40b1b4605fa6abc2920b4518;hpb=a899ee390aaa5c2653d161388e457dbbe3cf2dd0 diff --git a/ginac/normal.cpp b/ginac/normal.cpp index 2bcba41a..e30afda6 100644 --- a/ginac/normal.cpp +++ b/ginac/normal.cpp @@ -6,7 +6,7 @@ * computation, square-free factorization and rational function normalization. */ /* - * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2015 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 @@ -20,12 +20,9 @@ * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software - * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ -#include -#include - #include "normal.h" #include "basic.h" #include "ex.h" @@ -44,6 +41,10 @@ #include "pseries.h" #include "symbol.h" #include "utils.h" +#include "polynomial/chinrem_gcd.h" + +#include +#include namespace GiNaC { @@ -84,7 +85,7 @@ static struct _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. * @@ -119,6 +120,11 @@ static bool get_first_symbol(const ex &e, ex &x) * * @see get_symbol_stats */ struct sym_desc { + /** Initialize symbol, leave other variables uninitialized */ + sym_desc(const ex& s) + : sym(s), deg_a(0), deg_b(0), ldeg_a(0), ldeg_b(0), max_deg(0), max_lcnops(0) + { } + /** Reference to symbol */ ex sym; @@ -156,15 +162,11 @@ typedef std::vector sym_desc_vec; // Add symbol the sym_desc_vec (used internally by get_symbol_stats()) static void add_symbol(const ex &s, sym_desc_vec &v) { - sym_desc_vec::const_iterator it = v.begin(), itend = v.end(); - while (it != itend) { - if (it->sym.is_equal(s)) // If it's already in there, don't add it a second time + for (auto & it : v) + if (it.sym.is_equal(s)) // If it's already in there, don't add it a second time return; - ++it; - } - sym_desc d; - d.sym = s; - v.push_back(d); + + v.push_back(sym_desc(s)); } // Collect all symbols of an expression (used internally by get_symbol_stats()) @@ -196,17 +198,15 @@ static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v) { collect_symbols(a.eval(), v); // eval() to expand assigned symbols collect_symbols(b.eval(), v); - sym_desc_vec::iterator it = v.begin(), itend = v.end(); - while (it != itend) { - int deg_a = a.degree(it->sym); - int deg_b = b.degree(it->sym); - it->deg_a = deg_a; - it->deg_b = deg_b; - it->max_deg = std::max(deg_a, deg_b); - it->max_lcnops = std::max(a.lcoeff(it->sym).nops(), b.lcoeff(it->sym).nops()); - it->ldeg_a = a.ldegree(it->sym); - it->ldeg_b = b.ldegree(it->sym); - ++it; + for (auto & it : v) { + 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); } std::sort(v.begin(), v.end()); @@ -233,14 +233,14 @@ static numeric lcmcoeff(const ex &e, const numeric &l) if (e.info(info_flags::rational)) return lcm(ex_to(e).denom(), l); else if (is_exactly_a(e)) { - numeric c = _num1; + numeric c = *_num1_p; for (size_t i=0; i(e)) { - numeric c = _num1; + numeric c = *_num1_p; for (size_t i=0; i(e)) { if (is_a(e.op(0))) @@ -260,7 +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 @@ -273,9 +273,9 @@ static ex multiply_lcm(const ex &e, const numeric &lcm) if (is_exactly_a(e)) { size_t num = e.nops(); exvector v; v.reserve(num + 1); - numeric lcm_accum = _num1; + numeric lcm_accum = *_num1_p; for (size_t i=0; i(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++; + numeric c = *_num0_p, l = *_num1_p; + for (auto & it : seq) { + 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); } GINAC_ASSERT(is_exactly_a(overall_coeff)); c = gcd(ex_to(overall_coeff).numer(), c); @@ -339,11 +336,8 @@ numeric add::integer_content() 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_exactly_a(recombine_pair_to_ex(*it))); - ++it; + for (auto & it : seq) { + GINAC_ASSERT(!is_exactly_a(recombine_pair_to_ex(it))); } #endif // def DO_GINAC_ASSERT GINAC_ASSERT(is_exactly_a(overall_coeff)); @@ -614,6 +608,73 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args) 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; + } +// code below is commented-out because it leads to a significant slowdown +// 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()) { @@ -714,6 +775,31 @@ 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 auto & it : b) { + sym_desc_vec sym_stats; + get_symbol_stats(a, it, sym_stats); + if (!divide_in_z(qbar, it, q, sym_stats.begin())) + return false; + + qbar = q; + } + return true; + } + // Main symbol const ex &x = var->sym; @@ -730,24 +816,24 @@ 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, subs_options::no_pattern); while (bs.is_zero()) { - point += _num1; + point += *_num1_p; bs = b.subs(x == point, subs_options::no_pattern); } if (!divide_in_z(a.subs(x == point, subs_options::no_pattern), bs, c, var+1)) return false; alpha.push_back(point); u.push_back(c); - point += _num1; + 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(overall_coeff)); numeric cur_max = abs(ex_to(overall_coeff)); - while (it != itend) { + for (auto & it : seq) { numeric a; - GINAC_ASSERT(!is_exactly_a(it->rest)); - a = abs(ex_to(it->coeff)); + GINAC_ASSERT(!is_exactly_a(it.rest)); + a = abs(ex_to(it.coeff)); if (a > cur_max) cur_max = a; - it++; } return cur_max; } @@ -1090,11 +1173,8 @@ numeric add::max_coefficient() 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_exactly_a(recombine_pair_to_ex(*it))); - it++; + for (auto & it : seq) { + GINAC_ASSERT(!is_exactly_a(recombine_pair_to_ex(it))); } #endif // def DO_GINAC_ASSERT GINAC_ASSERT(is_exactly_a(overall_coeff)); @@ -1122,28 +1202,22 @@ ex add::smod(const numeric &xi) const { epvector newseq; newseq.reserve(seq.size()+1); - epvector::const_iterator it = seq.begin(); - epvector::const_iterator itend = seq.end(); - while (it != itend) { - GINAC_ASSERT(!is_exactly_a(it->rest)); - numeric coeff = GiNaC::smod(ex_to(it->coeff), xi); + for (auto & it : seq) { + 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++; + newseq.push_back(expair(it.rest, coeff)); } 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); + return (new add(std::move(newseq), coeff))->setflag(status_flags::dynallocated); } ex mul::smod(const numeric &xi) const { #ifdef DO_GINAC_ASSERT - epvector::const_iterator it = seq.begin(); - epvector::const_iterator itend = seq.end(); - while (it != itend) { - GINAC_ASSERT(!is_exactly_a(recombine_pair_to_ex(*it))); - it++; + for (auto & it : seq) { + GINAC_ASSERT(!is_exactly_a(recombine_pair_to_ex(it))); } #endif // def DO_GINAC_ASSERT mul * mulcopyp = new mul(*this); @@ -1177,17 +1251,19 @@ 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 ca cofactor of polynomial a (returned), NULL to suppress + * @param a first integer multivariate polynomial (expanded) + * @param b second integer multivariate polynomial (expanded) + * @param ca cofactor of polynomial a (returned), nullptr to suppress * calculation of cofactor - * @param cb cofactor of polynomial b (returned), NULL to suppress + * @param cb cofactor of polynomial b (returned), nullptr 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) { #if STATISTICS heur_gcd_called++; @@ -1195,7 +1271,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const // 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_exactly_a(a) && is_exactly_a(b)) { @@ -1204,7 +1280,8 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const *ca = ex_to(a) / g; if (cb) *cb = ex_to(b) / g; - return g; + res = g; + return true; } // The first symbol is our main variable @@ -1222,9 +1299,9 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const 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++) { @@ -1234,9 +1311,13 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const // Apply evaluation homomorphism and calculate GCD ex cp, cq; - ex gamma = heur_gcd(p.subs(x == xi, subs_options::no_pattern), q.subs(x == xi, subs_options::no_pattern), &cp, &cq, var+1).expand(); - if (!is_exactly_a(gamma)) { - + ex gamma; + bool found = heur_gcd_z(gamma, + p.subs(x == xi, subs_options::no_pattern), + q.subs(x == xi, subs_options::no_pattern), + &cp, &cq, var+1); + if (found) { + gamma = gamma.expand(); // Reconstruct polynomial from GCD of mapped polynomials ex g = interpolate(gamma, xi, x, maxdeg); @@ -1247,33 +1328,96 @@ 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_exactly_a(lc) && ex_to(lc).is_negative()) - return -g; - else - return g; + res = g; + return true; } } // 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), nullptr to suppress + * calculation of cofactor + * @param cb cofactor of polynomial b (returned), nullptr to suppress + * calculation of cofactor + * @param var iterator to first element of vector of sym_desc structs + * @param res the GCD (returned) + * @return true if GCD was computed, false otherwise. + * @see heur_gcd_z + * @see gcd + */ +static bool heur_gcd(ex& res, const ex& a, const ex& b, ex *ca, ex *cb, + sym_desc_vec::const_iterator var) +{ + if (a.info(info_flags::integer_polynomial) && + b.info(info_flags::integer_polynomial)) { + try { + return heur_gcd_z(res, a, b, ca, cb, var); + } catch (gcdheu_failed) { + return false; + } + } + + // convert polynomials to Z[X] + const numeric a_lcm = lcm_of_coefficients_denominators(a); + const numeric ab_lcm = lcmcoeff(b, a_lcm); + + const ex ai = a*ab_lcm; + const ex bi = b*ab_lcm; + if (!ai.info(info_flags::integer_polynomial)) + throw std::logic_error("heur_gcd: not an integer polynomial [1]"); + + if (!bi.info(info_flags::integer_polynomial)) + throw std::logic_error("heur_gcd: not an integer polynomial [2]"); + + bool found = false; + try { + found = heur_gcd_z(res, ai, bi, ca, cb, var); + } catch (gcdheu_failed) { + return false; + } + + // GCD is not unique, it's defined up to a unit (i.e. invertible + // element). If the coefficient ring is a field, every its element is + // invertible, so one can multiply the polynomial GCD with any element + // of the coefficient field. We use this ambiguity to make cofactors + // integer polynomials. + if (found) + res /= ab_lcm; + return found; } +// gcd helper to handle partially factored polynomials (to avoid expanding +// large expressions). At least one of the arguments should be a power. +static ex gcd_pf_pow(const ex& a, const ex& b, ex* ca, ex* cb); + +// gcd helper to handle partially factored polynomials (to avoid expanding +// large expressions). At least one of the arguments should be a product. +static ex gcd_pf_mul(const ex& a, const ex& b, ex* ca, ex* cb); + /** Compute GCD (Greatest Common Divisor) of multivariate polynomials a(X) * and b(X) in Z[X]. Optionally also compute the cofactors of a and b, * defined by a = ca * gcd(a, b) and b = cb * gcd(a, b). * * @param a first multivariate polynomial * @param b second multivariate polynomial - * @param ca pointer to expression that will receive the cofactor of a, or NULL - * @param cb pointer to expression that will receive the cofactor of b, or NULL + * @param ca pointer to expression that will receive the cofactor of a, or nullptr + * @param cb pointer to expression that will receive the cofactor of b, or nullptr * @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) { #if STATISTICS gcd_called++; @@ -1304,90 +1448,14 @@ 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_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); - } - + if (!(options & gcd_options::no_part_factored)) { + if (is_exactly_a(a) || is_exactly_a(b)) + return gcd_pf_mul(a, b, ca, cb); #if FAST_COMPARE - // Input polynomials of the form poly^n are sometimes also trivial - if (is_exactly_a(a)) { - ex p = a.op(0); - if (is_exactly_a(b)) { - 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_exactly_a(b)) { - ex p = b.op(0); - if (p.is_equal(a)) { - // a = p, b = p^n, gcd = p - if (ca) - *ca = _ex1; - if (cb) - *cb = power(p, b.op(1) - 1); - return p; - } - } + if (is_exactly_a(a) || is_exactly_a(b)) + return gcd_pf_pow(a, b, ca, cb); #endif + } // Some trivial cases ex aex = a.expand(), bex = b.expand(); @@ -1422,11 +1490,71 @@ factored_b: } #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 ex &x = var->sym; @@ -1440,14 +1568,14 @@ factored_b: } // Try to eliminate variables - if (var->deg_a == 0) { + if (var->deg_a == 0 && var->deg_b != 0 ) { ex bex_u, bex_c, bex_p; bex.unitcontprim(x, bex_u, bex_c, bex_p); ex g = gcd(aex, bex_c, ca, cb, false); if (cb) *cb *= bex_u * bex_p; return g; - } else if (var->deg_b == 0) { + } 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); @@ -1458,41 +1586,162 @@ factored_b: // Try heuristic algorithm first, fall back to PRS if that failed ex g; - try { - g = heur_gcd(aex, bex, ca, cb, var); - } catch (gcdheu_failed) { - g = fail(); - } - if (is_exactly_a(g)) { + if (!(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++; + else { + heur_gcd_failed++; + } #endif + } + if (options & gcd_options::use_sr_gcd) { g = sr_gcd(aex, bex, var); - if (g.is_equal(_ex1)) { - // Keep cofactors factored if possible + } else { + exvector vars; + for (std::size_t n = sym_stats.size(); n-- != 0; ) + vars.push_back(sym_stats[n].sym); + g = chinrem_gcd(aex, bex, vars); + } + + if (g.is_equal(_ex1)) { + // Keep cofactors factored if possible + if (ca) + *ca = a; + if (cb) + *cb = b; + } else { + if (ca) + divide(aex, g, *ca, false); + if (cb) + divide(bex, g, *cb, false); + } + return g; +} + +// gcd helper to handle partially factored polynomials (to avoid expanding +// large expressions). Both arguments should be powers. +static ex gcd_pf_pow_pow(const ex& a, const ex& b, ex* ca, ex* cb) +{ + ex p = a.op(0); + const ex& exp_a = a.op(1); + ex pb = b.op(0); + const ex& exp_b = b.op(1); + + // a = p^n, b = p^m, gcd = p^min(n, m) + if (p.is_equal(pb)) { + if (exp_a < exp_b) { if (ca) - *ca = a; + *ca = _ex1; if (cb) - *cb = b; + *cb = power(p, exp_b - exp_a); + return power(p, exp_a); } else { if (ca) - divide(aex, g, *ca, false); + *ca = power(p, exp_a - exp_b); if (cb) - divide(bex, g, *cb, false); + *cb = _ex1; + return power(p, exp_b); } - } else { - if (g.is_equal(_ex1)) { - // Keep cofactors factored if possible + } + + ex p_co, pb_co; + ex p_gcd = gcd(p, pb, &p_co, &pb_co, false); + // a(x) = p(x)^n, b(x) = p_b(x)^m, gcd (p, p_b) = 1 ==> gcd(a,b) = 1 + if (p_gcd.is_equal(_ex1)) { if (ca) *ca = a; if (cb) *cb = b; - } + return _ex1; + // XXX: do I need to check for p_gcd = -1? } - return g; + // 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) { + ex pg = gcd(power(p_co, exp_a), power(p_gcd, exp_b-exp_a)*power(pb_co, exp_b), ca, cb, false); + return power(p_gcd, exp_a)*pg; + } else { + ex pg = gcd(power(p_gcd, exp_a - exp_b)*power(p_co, exp_a), power(pb_co, exp_b), ca, cb, false); + return power(p_gcd, exp_b)*pg; + } +} + +static ex gcd_pf_pow(const ex& a, const ex& b, ex* ca, ex* cb) +{ + if (is_exactly_a(a) && is_exactly_a(b)) + return gcd_pf_pow_pow(a, b, ca, cb); + + if (is_exactly_a(b) && (! is_exactly_a(a))) + return gcd_pf_pow(b, a, cb, ca); + + GINAC_ASSERT(is_exactly_a(a)); + + ex p = a.op(0); + const ex& exp_a = a.op(1); + if (p.is_equal(b)) { + // a = p^n, b = p, gcd = p + if (ca) + *ca = power(p, a.op(1) - 1); + if (cb) + *cb = _ex1; + return p; + } + + ex p_co, bpart_co; + ex p_gcd = gcd(p, b, &p_co, &bpart_co, false); + + // a(x) = p(x)^n, gcd(p, b) = 1 ==> gcd(a, b) = 1 + if (p_gcd.is_equal(_ex1)) { + if (ca) + *ca = a; + if (cb) + *cb = b; + return _ex1; + } + // a(x) = g(x)^n A(x)^n, b(x) = g(x) B(x) ==> gcd(a, b) = g(x) gcd(g(x)^(n-1) A(x)^n, B(x)) + ex rg = gcd(power(p_gcd, exp_a-1)*power(p_co, exp_a), bpart_co, ca, cb, false); + return p_gcd*rg; } +static ex gcd_pf_mul(const ex& a, const ex& b, ex* ca, ex* cb) +{ + if (is_exactly_a(a) && is_exactly_a(b) + && (b.nops() > a.nops())) + return gcd_pf_mul(b, a, cb, ca); + + if (is_exactly_a(b) && (!is_exactly_a(a))) + return gcd_pf_mul(b, a, cb, ca); + + GINAC_ASSERT(is_exactly_a(a)); + size_t num = a.nops(); + exvector g; g.reserve(num); + exvector acc_ca; acc_ca.reserve(num); + ex part_b = b; + for (size_t i=0; isetflag(status_flags::dynallocated); + if (cb) + *cb = part_b; + return (new mul(g))->setflag(status_flags::dynallocated); +} /** Compute LCM (Least Common Multiple) of multivariate polynomials in Z[X]. * @@ -1519,7 +1768,7 @@ 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. @@ -1595,11 +1844,8 @@ ex sqrfree(const ex &a, const lst &l) if (l.nops()==0) { sym_desc_vec sdv; get_symbol_stats(a, _ex0, sdv); - sym_desc_vec::const_iterator it = sdv.begin(), itend = sdv.end(); - while (it != itend) { - args.append(it->sym); - ++it; - } + for (auto & it : sdv) + args.append(it.sym); } else { args = l; } @@ -1622,18 +1868,15 @@ ex sqrfree(const ex &a, const lst &l) // recurse down the factors in remaining variables if (newargs.nops()>0) { - exvector::iterator i = factors.begin(); - while (i != factors.end()) { - *i = sqrfree(*i, newargs); - ++i; - } + for (auto & it : factors) + it = sqrfree(it, newargs); } // Done with recursion, now construct the final result ex result = _ex1; - exvector::const_iterator it = factors.begin(), itend = factors.end(); - for (int p = 1; it!=itend; ++it, ++p) - result *= power(*it, p); + int p = 1; + for (auto & it : factors) + 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 @@ -1644,7 +1887,7 @@ ex sqrfree(const ex &a, const lst &l) else result *= quo(tmp, result, x); - // Put in the reational overall factor again and return + // Put in the rational overall factor again and return return result * lcm.inverse(); } @@ -1738,16 +1981,18 @@ ex sqrfree_parfrac(const ex & a, const symbol & x) * @see ex::normal */ static ex replace_with_symbol(const ex & e, exmap & repl, exmap & rev_lookup) { + // Since the repl contains replaced expressions we should search for them + ex e_replaced = e.subs(repl, subs_options::no_pattern); + // Expression already replaced? Then return the assigned symbol - exmap::const_iterator it = rev_lookup.find(e); + auto it = rev_lookup.find(e_replaced); if (it != rev_lookup.end()) return it->second; - + // Otherwise create new symbol and add to list, taking care that the // replacement expression doesn't itself contain symbols from repl, // because subs() is not recursive ex es = (new symbol)->setflag(status_flags::dynallocated); - ex e_replaced = e.subs(repl, subs_options::no_pattern); repl.insert(std::make_pair(es, e_replaced)); rev_lookup.insert(std::make_pair(e_replaced, es)); return es; @@ -1760,16 +2005,18 @@ static ex replace_with_symbol(const ex & e, exmap & repl, exmap & rev_lookup) * @see basic::to_polynomial */ static ex replace_with_symbol(const ex & e, exmap & repl) { + // Since the repl contains replaced expressions we should search for them + ex e_replaced = e.subs(repl, subs_options::no_pattern); + // Expression already replaced? Then return the assigned symbol - for (exmap::const_iterator it = repl.begin(); it != repl.end(); ++it) - if (it->second.is_equal(e)) - return it->first; - + for (auto & it : repl) + if (it.second.is_equal(e_replaced)) + return it.first; + // Otherwise create new symbol and add to list, taking care that the // replacement expression doesn't itself contain symbols from repl, // because subs() is not recursive ex es = (new symbol)->setflag(status_flags::dynallocated); - ex e_replaced = e.subs(repl, subs_options::no_pattern); repl.insert(std::make_pair(es, e_replaced)); return es; } @@ -1842,7 +2089,7 @@ 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; @@ -1909,12 +2156,10 @@ ex add::normal(exmap & repl, exmap & rev_lookup, int level) const exvector nums, dens; nums.reserve(seq.size()+1); dens.reserve(seq.size()+1); - epvector::const_iterator it = seq.begin(), itend = seq.end(); - while (it != itend) { - ex n = ex_to(recombine_pair_to_ex(*it)).normal(repl, rev_lookup, level-1); + for (auto & it : seq) { + ex n = ex_to(recombine_pair_to_ex(it)).normal(repl, rev_lookup, level-1); nums.push_back(n.op(0)); dens.push_back(n.op(1)); - it++; } ex n = ex_to(overall_coeff).normal(repl, rev_lookup, level-1); nums.push_back(n.op(0)); @@ -1926,8 +2171,8 @@ ex add::normal(exmap & repl, exmap & rev_lookup, int level) const //std::clog << "add::normal uses " << nums.size() << " summands:\n"; // Add fractions sequentially - exvector::const_iterator num_it = nums.begin(), num_itend = nums.end(); - exvector::const_iterator den_it = dens.begin(), den_itend = dens.end(); + auto num_it = nums.begin(), num_itend = nums.end(); + auto den_it = dens.begin(), den_itend = dens.end(); //std::clog << " num = " << *num_it << ", den = " << *den_it << std::endl; ex num = *num_it++, den = *den_it++; while (num_it != num_itend) { @@ -1940,7 +2185,7 @@ ex add::normal(exmap & repl, exmap & rev_lookup, int level) const num_it++; den_it++; } - // Additiion of two fractions, taking advantage of the fact that + // Addition of two fractions, taking advantage of the fact that // the heuristic GCD algorithm computes the cofactors at no extra cost ex co_den1, co_den2; ex g = gcd(den, next_den, &co_den1, &co_den2, false); @@ -1968,12 +2213,10 @@ ex mul::normal(exmap & repl, exmap & rev_lookup, int level) const exvector num; num.reserve(seq.size()); exvector den; den.reserve(seq.size()); ex n; - epvector::const_iterator it = seq.begin(), itend = seq.end(); - while (it != itend) { - n = ex_to(recombine_pair_to_ex(*it)).normal(repl, rev_lookup, level-1); + for (auto & it : seq) { + n = ex_to(recombine_pair_to_ex(it)).normal(repl, rev_lookup, level-1); num.push_back(n.op(0)); den.push_back(n.op(1)); - it++; } n = ex_to(overall_coeff).normal(repl, rev_lookup, level-1); num.push_back(n.op(0)); @@ -2047,14 +2290,12 @@ ex power::normal(exmap & repl, exmap & rev_lookup, int level) const ex pseries::normal(exmap & repl, exmap & rev_lookup, int level) const { epvector newseq; - epvector::const_iterator i = seq.begin(), end = seq.end(); - while (i != end) { - ex restexp = i->rest.normal(); + for (auto & it : seq) { + ex restexp = it.rest.normal(); if (!restexp.is_zero()) - newseq.push_back(expair(restexp, i->coeff)); - ++i; + newseq.push_back(expair(restexp, it.coeff)); } - ex n = pseries(relational(var,point), newseq); + ex n = pseries(relational(var,point), std::move(newseq)); return (new lst(replace_with_symbol(n, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated); } @@ -2126,7 +2367,7 @@ ex ex::denom() const return e.op(1).subs(repl, subs_options::no_pattern); } -/** Get numerator and denominator of an expression. If the expresison is not +/** Get numerator and denominator of an expression. If the expression is not * of the normal form "numerator/denominator", it is first converted to this * form and then a list [numerator, denominator] is returned. * @@ -2170,15 +2411,15 @@ 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))); + for (auto & it : repl_lst) + 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); + for (auto & it : m) + repl_lst.append(it.first == it.second); return ret; } @@ -2193,15 +2434,15 @@ 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))); + for (auto & it : repl_lst) + 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); + for (auto & it : m) + repl_lst.append(it.first == it.second); return ret; } @@ -2287,6 +2528,17 @@ 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); } @@ -2297,17 +2549,15 @@ ex expairseq::to_rational(exmap & repl) const { epvector s; s.reserve(seq.size()); - epvector::const_iterator i = seq.begin(), end = seq.end(); - while (i != end) { - s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_rational(repl))); - ++i; - } + for (auto & it : seq) + s.push_back(split_ex_to_pair(recombine_pair_to_ex(it).to_rational(repl))); + ex oc = overall_coeff.to_rational(repl); if (oc.info(info_flags::numeric)) - return thisexpairseq(s, overall_coeff); + return thisexpairseq(std::move(s), overall_coeff); else s.push_back(combine_ex_with_coeff_to_pair(oc, _ex1)); - return thisexpairseq(s, default_overall_coeff()); + return thisexpairseq(std::move(s), default_overall_coeff()); } /** Implementation of ex::to_polynomial() for expairseqs. */ @@ -2315,17 +2565,15 @@ 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; - } + for (auto & it : seq) + s.push_back(split_ex_to_pair(recombine_pair_to_ex(it).to_polynomial(repl))); + ex oc = overall_coeff.to_polynomial(repl); if (oc.info(info_flags::numeric)) - return thisexpairseq(s, overall_coeff); + return thisexpairseq(std::move(s), overall_coeff); else s.push_back(combine_ex_with_coeff_to_pair(oc, _ex1)); - return thisexpairseq(s, default_overall_coeff()); + return thisexpairseq(std::move(s), default_overall_coeff()); } @@ -2344,7 +2592,7 @@ static ex find_common_factor(const ex & e, ex & factor, exmap & repl) for (size_t i=0; i(x) || is_exactly_a(x)) { + if (is_exactly_a(x) || is_exactly_a(x) || is_a(x)) { ex f = 1; x = find_common_factor(x, f, repl); x *= f; @@ -2403,8 +2651,16 @@ term_done: ; return (new mul(v))->setflag(status_flags::dynallocated); } else if (is_exactly_a(e)) { - - return e.to_polynomial(repl); + 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; @@ -2415,7 +2671,7 @@ term_done: ; * '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)) { + if (is_exactly_a(e) || is_exactly_a(e) || is_exactly_a(e)) { exmap repl; ex factor = 1;