X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnormal.cpp;h=5139509c51c59835947a0bef3d9dcd9bb51da049;hp=a016a76bce4fb345dfea3bcd3aeccddd5fc8e68f;hb=a053768864556ce627f958a38fb1169ab00b8229;hpb=5825be1838e9790d538e3ad5c67e186266b1f057 diff --git a/ginac/normal.cpp b/ginac/normal.cpp index a016a76b..5139509c 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) 1999-2003 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 @@ -74,10 +74,10 @@ 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 @@ -92,14 +92,14 @@ static struct _stat_print { * @return "false" if no symbol was found, "true" otherwise */ static bool get_first_symbol(const ex &e, const symbol *&x) { - if (is_ex_exactly_of_type(e, symbol)) { - x = static_cast(e.bp); + if (is_a(e)) { + x = &ex_to(e); return true; - } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) { + } else if (is_exactly_a(e) || is_exactly_a(e)) { for (unsigned i=0; i(e)) { if (get_first_symbol(e.op(0), x)) return true; } @@ -155,11 +155,11 @@ 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) { - 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 return; - it++; + ++it; } sym_desc d; d.sym = s; @@ -169,12 +169,12 @@ 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)) { + if (is_a(e)) { + add_symbol(&ex_to(e), v); + } else if (is_exactly_a(e) || is_exactly_a(e)) { for (unsigned i=0; i(e)) { collect_symbols(e.op(0), v); } } @@ -205,16 +205,16 @@ static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v) 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; } - 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++; + ++it; } #endif } @@ -230,18 +230,18 @@ 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_ex_exactly_of_type(e, add)) { - numeric c = _num1(); + else if (is_exactly_a(e)) { + numeric c = _num1; for (unsigned i=0; i(e)) { + numeric c = _num1; for (unsigned i=0; i(e)) { + if (is_a(e.op(0))) return l; else return pow(lcmcoeff(e.op(0), l), ex_to(e.op(1))); @@ -258,7 +258,7 @@ 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); } /** Bring polynomial from Q[X] to Z[X] by multiplying in the previously @@ -268,25 +268,25 @@ 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)) { + if (is_exactly_a(e)) { unsigned num = e.nops(); exvector v; v.reserve(num + 1); - numeric lcm_accum = _num1(); + numeric lcm_accum = _num1; for (unsigned i=0; isetflag(status_flags::dynallocated); - } else if (is_ex_exactly_of_type(e, add)) { + } else if (is_exactly_a(e)) { unsigned num = e.nops(); exvector v; v.reserve(num); for (unsigned i=0; isetflag(status_flags::dynallocated); - } else if (is_ex_exactly_of_type(e, power)) { - if (is_ex_exactly_of_type(e.op(0), symbol)) + } else if (is_exactly_a(e)) { + if (is_a(e.op(0))) return e * lcm; else return pow(multiply_lcm(e.op(0), lcm.power(ex_to(e.op(1)).inverse())), e.op(1)); @@ -308,7 +308,7 @@ numeric ex::integer_content(void) const numeric basic::integer_content(void) const { - return _num1(); + return _num1; } numeric numeric::integer_content(void) const @@ -320,14 +320,14 @@ numeric add::integer_content(void) const { epvector::const_iterator it = seq.begin(); epvector::const_iterator itend = seq.end(); - numeric c = _num0(); + numeric c = _num0; while (it != itend) { - GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric)); - GINAC_ASSERT(is_ex_exactly_of_type(it->coeff,numeric)); + GINAC_ASSERT(!is_exactly_a(it->rest)); + GINAC_ASSERT(is_exactly_a(it->coeff)); c = gcd(ex_to(it->coeff), c); it++; } - GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric)); + GINAC_ASSERT(is_exactly_a(overall_coeff)); c = gcd(ex_to(overall_coeff),c); return c; } @@ -338,11 +338,11 @@ numeric mul::integer_content(void) 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 - GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric)); + GINAC_ASSERT(is_exactly_a(overall_coeff)); return abs(ex_to(overall_coeff)); } @@ -364,11 +364,11 @@ ex quo(const ex &a, const ex &b, const symbol &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")); @@ -380,8 +380,8 @@ ex quo(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); - exvector v; v.reserve(rdeg - bdeg + 1); + 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) @@ -414,15 +414,15 @@ ex rem(const ex &a, const ex &b, const symbol &x, bool check_args) { if (b.is_zero()) throw(std::overflow_error("rem: division by zero")); - if (is_ex_exactly_of_type(a, numeric)) { - if (is_ex_exactly_of_type(b, numeric)) - return _ex0(); + if (is_exactly_a(a)) { + if (is_exactly_a(b)) + return _ex0; else return a; } #if FAST_COMPARE if (a.is_equal(b)) - return _ex0(); + 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")); @@ -434,7 +434,7 @@ 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) @@ -464,28 +464,28 @@ ex decomp_rational(const ex &a, const symbol &x) ex nd = numer_denom(a); ex numer = nd.op(0), denom = nd.op(1); ex q = quo(numer, denom, x); - if (is_ex_exactly_of_type(q, fail)) + 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 Z[x]. +/** 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] */ + * @return pseudo-remainder of a(x) and b(x) in Q[x] */ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args) { if (b.is_zero()) throw(std::overflow_error("prem: division by zero")); - if (is_ex_exactly_of_type(a, numeric)) { - if (is_ex_exactly_of_type(b, numeric)) - return _ex0(); + if (is_exactly_a(a)) { + if (is_exactly_a(b)) + return _ex0; else return b; } @@ -501,18 +501,18 @@ 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; @@ -523,21 +523,21 @@ 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] */ + * @return sparse pseudo-remainder of a(x) and b(x) in Q[x] */ ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args) { if (b.is_zero()) throw(std::overflow_error("prem: division by zero")); - if (is_ex_exactly_of_type(a, numeric)) { - if (is_ex_exactly_of_type(b, numeric)) - return _ex0(); + if (is_exactly_a(a)) { + if (is_exactly_a(b)) + return _ex0; else return b; } @@ -553,17 +553,17 @@ 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; @@ -581,22 +581,23 @@ 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 @@ -611,13 +612,15 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args) // 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); - exvector v; v.reserve(rdeg - bdeg + 1); + 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) @@ -676,15 +679,15 @@ 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 @@ -692,7 +695,7 @@ 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 @@ -723,24 +726,24 @@ 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; ex c; for (i=0; i<=adeg; i++) { ex bs = b.subs(*x == point); while (bs.is_zero()) { - point += _num1(); + point += _num1; bs = b.subs(*x == point); } if (!divide_in_z(a.subs(*x == point), bs, c, var+1)) return false; alpha.push_back(point); u.push_back(c); - point += _num1(); + point += _num1; } // Compute inverses vector rcp; rcp.reserve(adeg + 1); - rcp.push_back(_num0()); + rcp.push_back(_num0); for (k=1; k<=adeg; k++) { numeric product = alpha[k] - alpha[0]; for (i=1; i= bdeg) { ex term, rcoeff = r.coeff(*x, rdeg); if (!divide_in_z(rcoeff, blcoeff, term, var+1)) @@ -818,8 +821,8 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite ex ex::unit(const symbol &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 < _ex0 ? _ex_1 : _ex1; else { const symbol *y; if (get_first_symbol(c, y)) @@ -840,12 +843,12 @@ ex ex::unit(const symbol &x) const ex ex::content(const symbol &x) const { if (is_zero()) - return _ex0(); - if (is_ex_exactly_of_type(*this, numeric)) + return _ex0; + 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 ex c = e.integer_content(); @@ -859,7 +862,7 @@ ex ex::content(const symbol &x) const int ldeg = e.ldegree(x); if (deg == ldeg) return e.lcoeff(x) / e.unit(x); - c = _ex0(); + c = _ex0; for (int i=ldeg; i<=deg; i++) c = gcd(e.coeff(x, i), c, NULL, NULL, false); return c; @@ -876,15 +879,15 @@ ex ex::content(const symbol &x) const ex ex::primpart(const symbol &x) const { if (is_zero()) - return _ex0(); - if (is_ex_exactly_of_type(*this, numeric)) - return _ex1(); + return _ex0; + if (is_exactly_a(*this)) + return _ex1; ex c = content(x); if (c.is_zero()) - return _ex0(); + return _ex0; 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); @@ -901,14 +904,14 @@ ex ex::primpart(const symbol &x) const ex ex::primpart(const symbol &x, const ex &c) const { if (is_zero()) - return _ex0(); + return _ex0; if (c.is_zero()) - return _ex0(); - if (is_ex_exactly_of_type(*this, numeric)) - return _ex1(); + return _ex0; + if (is_exactly_a(*this)) + return _ex1; 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); @@ -1091,7 +1094,7 @@ static ex red_gcd(const ex &a, const ex &b, const symbol *x) d = d.primpart(*x, cont_d); // First element of divisor sequence - ex r, ri = _ex1(); + ex r, ri = _ex1; int delta = cdeg - ddeg; for (;;) { @@ -1107,7 +1110,7 @@ static ex red_gcd(const ex &a, const ex &b, const symbol *x) 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)) + if (is_exactly_a(r)) return gamma; else return gamma * r.primpart(*x); @@ -1165,7 +1168,7 @@ static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var) //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 (;;) { @@ -1182,7 +1185,7 @@ static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator 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); @@ -1216,7 +1219,7 @@ numeric ex::max_coefficient(void) const * @see heur_gcd */ numeric basic::max_coefficient(void) const { - return _num1(); + return _num1; } numeric numeric::max_coefficient(void) const @@ -1228,11 +1231,11 @@ numeric add::max_coefficient(void) const { epvector::const_iterator it = seq.begin(); epvector::const_iterator itend = seq.end(); - GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric)); + 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)); + GINAC_ASSERT(!is_exactly_a(it->rest)); a = abs(ex_to(it->coeff)); if (a > cur_max) cur_max = a; @@ -1247,28 +1250,21 @@ numeric mul::max_coefficient(void) 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 - GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric)); + 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; @@ -1286,13 +1282,13 @@ 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)); + 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)); + 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); } @@ -1303,12 +1299,12 @@ 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)); + 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); @@ -1360,7 +1356,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const return (new fail())->setflag(status_flags::dynallocated); // GCD of two numeric values -> CLN - if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) { + if (is_exactly_a(a) && is_exactly_a(b)) { numeric g = gcd(ex_to(a), ex_to(b)); if (ca) *ca = ex_to(a) / g; @@ -1384,9 +1380,9 @@ 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 + _num2; else - xi = mp * _num2() + _num2(); + xi = mp * _num2 + _num2; // 6 tries maximum for (int t=0; t<6; t++) { @@ -1398,7 +1394,7 @@ 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), q.subs(x == xi), &cp, &cq, var+1).expand(); - if (!is_ex_exactly_of_type(gamma, fail)) { + if (!is_exactly_a(gamma)) { // Reconstruct polynomial from GCD of mapped polynomials ex g = interpolate(gamma, xi, x, maxdeg); @@ -1411,7 +1407,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const 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(lc).is_negative()) + if (is_exactly_a(lc) && ex_to(lc).is_negative()) return -g; else return g; @@ -1424,7 +1420,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const if (ca) *ca = cp; ex lc = g.lcoeff(x); - if (is_ex_exactly_of_type(lc, numeric) && ex_to(lc).is_negative()) + if (is_exactly_a(lc) && ex_to(lc).is_negative()) return -g; else return g; @@ -1437,7 +1433,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const if (cb) *cb = cq; ex lc = g.lcoeff(x); - if (is_ex_exactly_of_type(lc, numeric) && ex_to(lc).is_negative()) + if (is_exactly_a(lc) && ex_to(lc).is_negative()) return -g; else return g; @@ -1469,14 +1465,14 @@ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args) #endif // GCD of numerics -> CLN - if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) { + 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(a) / g; @@ -1493,8 +1489,8 @@ 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()) + if (is_exactly_a(a)) { + if (is_exactly_a(b) && b.nops() > a.nops()) goto factored_b; factored_a: unsigned num = a.nops(); @@ -1512,8 +1508,8 @@ factored_a: if (cb) *cb = part_b; return (new mul(g))->setflag(status_flags::dynallocated); - } else if (is_ex_exactly_of_type(b, mul)) { - if (is_ex_exactly_of_type(a, mul) && a.nops() > b.nops()) + } else if (is_exactly_a(b)) { + if (is_exactly_a(a) && a.nops() > b.nops()) goto factored_a; factored_b: unsigned num = b.nops(); @@ -1535,15 +1531,15 @@ factored_b: #if FAST_COMPARE // Input polynomials of the form poly^n are sometimes also trivial - if (is_ex_exactly_of_type(a, power)) { + if (is_exactly_a(a)) { ex p = a.op(0); - if (is_ex_exactly_of_type(b, power)) { + if (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(); + *ca = _ex1; if (cb) *cb = power(p, exp_b - exp_a); return power(p, exp_a); @@ -1551,7 +1547,7 @@ factored_b: if (ca) *ca = power(p, exp_a - exp_b); if (cb) - *cb = _ex1(); + *cb = _ex1; return power(p, exp_b); } } @@ -1561,16 +1557,16 @@ factored_b: if (ca) *ca = power(p, a.op(1) - 1); if (cb) - *cb = _ex1(); + *cb = _ex1; return p; } } - } else if (is_ex_exactly_of_type(b, power)) { + } else if (is_exactly_a(b)) { ex p = b.op(0); if (p.is_equal(a)) { // a = p, b = p^n, gcd = p if (ca) - *ca = _ex1(); + *ca = _ex1; if (cb) *cb = power(p, b.op(1) - 1); return p; @@ -1582,31 +1578,31 @@ factored_b: 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 @@ -1654,7 +1650,7 @@ factored_b: } catch (gcdheu_failed) { g = fail(); } - if (is_ex_exactly_of_type(g, fail)) { + if (is_exactly_a(g)) { //std::clog << "heuristics failed" << std::endl; #if STATISTICS heur_gcd_failed++; @@ -1666,7 +1662,7 @@ factored_b: // g = peu_gcd(aex, bex, &x); // g = red_gcd(aex, bex, &x); g = sr_gcd(aex, bex, var); - if (g.is_equal(_ex1())) { + if (g.is_equal(_ex1)) { // Keep cofactors factored if possible if (ca) *ca = a; @@ -1680,7 +1676,7 @@ factored_b: } #if 1 } else { - if (g.is_equal(_ex1())) { + if (g.is_equal(_ex1)) { // Keep cofactors factored if possible if (ca) *ca = a; @@ -1702,7 +1698,7 @@ 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)) + 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")); @@ -1730,7 +1726,7 @@ 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; } @@ -1745,58 +1741,109 @@ static exvector sqrfree_yun(const ex &a, const symbol &x) 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. */ + * @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) + 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)) throw (std::runtime_error("sqrfree(): invalid factorization variable")); - const symbol x = ex_to(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); + // 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). * @@ -1818,14 +1865,14 @@ ex sqrfree_parfrac(const ex & a, const symbol & x) // Factorize denominator and compute cofactors exvector yun = sqrfree_yun(denom, x); //clog << "yun factors: " << exprseq(yun) << endl; - int num_yun = yun.size(); + unsigned num_yun = yun.size(); exvector factor; factor.reserve(num_yun); exvector cofac; cofac.reserve(num_yun); for (unsigned i=0; isetflag(status_flags::dynallocated); + return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); else { 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, sym_lst, repl_lst), _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), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + return (new lst(replace_with_symbol(map(map_normal), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); } } } @@ -1955,7 +2002,7 @@ ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const * @see ex::normal */ ex symbol::normal(lst &sym_lst, lst &repl_lst, int level) const { - return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated); + return (new lst(*this, _ex1))->setflag(status_flags::dynallocated); } @@ -1991,17 +2038,17 @@ static ex frac_cancel(const ex &n, const ex &d) { ex num = n; ex den = d; - numeric pre_factor = _num1(); + numeric pre_factor = _num1; //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")); @@ -2015,19 +2062,26 @@ 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(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 { + const symbol *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; + } } } @@ -2043,7 +2097,7 @@ static ex frac_cancel(const ex &n, const ex &d) ex add::normal(lst &sym_lst, lst &repl_lst, 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, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); else if (level == -max_recursion_level) throw(std::runtime_error("max recursion level reached")); @@ -2053,12 +2107,12 @@ 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(sym_lst, repl_lst, 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(sym_lst, repl_lst, level-1); nums.push_back(n.op(0)); dens.push_back(n.op(1)); GINAC_ASSERT(nums.size() == dens.size()); @@ -2102,7 +2156,7 @@ ex add::normal(lst &sym_lst, lst &repl_lst, int level) const ex mul::normal(lst &sym_lst, lst &repl_lst, 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, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); else if (level == -max_recursion_level) throw(std::runtime_error("max recursion level reached")); @@ -2112,12 +2166,12 @@ ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const 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); + n = ex_to(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, 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); + n = ex_to(overall_coeff).normal(sym_lst, repl_lst, level-1); num.push_back(n.op(0)); den.push_back(n.op(1)); @@ -2134,13 +2188,13 @@ ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const ex power::normal(lst &sym_lst, lst &repl_lst, 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, sym_lst, repl_lst), _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(sym_lst, repl_lst, level-1); + ex n_exponent = ex_to(exponent).normal(sym_lst, repl_lst, level-1); n_exponent = n_exponent.op(0) / n_exponent.op(1); if (n_exponent.info(info_flags::integer)) { @@ -2161,27 +2215,25 @@ 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), sym_lst, repl_lst), _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), sym_lst, repl_lst)))->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), sym_lst, repl_lst), _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), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); } @@ -2191,13 +2243,15 @@ ex power::normal(lst &sym_lst, lst &repl_lst, int level) const ex pseries::normal(lst &sym_lst, lst &repl_lst, 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); + return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); } @@ -2218,7 +2272,7 @@ ex ex::normal(int level) const lst sym_lst, repl_lst; ex e = bp->normal(sym_lst, repl_lst, level); - GINAC_ASSERT(is_ex_of_type(e, lst)); + GINAC_ASSERT(is_a(e)); // Re-insert replaced symbols if (sym_lst.nops() > 0) @@ -2239,7 +2293,7 @@ ex ex::numer(void) const lst sym_lst, repl_lst; ex e = bp->normal(sym_lst, repl_lst, 0); - GINAC_ASSERT(is_ex_of_type(e, lst)); + GINAC_ASSERT(is_a(e)); // Re-insert replaced symbols if (sym_lst.nops() > 0) @@ -2259,7 +2313,7 @@ ex ex::denom(void) const lst sym_lst, repl_lst; ex e = bp->normal(sym_lst, repl_lst, 0); - GINAC_ASSERT(is_ex_of_type(e, lst)); + GINAC_ASSERT(is_a(e)); // Re-insert replaced symbols if (sym_lst.nops() > 0) @@ -2279,7 +2333,7 @@ ex ex::numer_denom(void) const lst sym_lst, repl_lst; ex e = bp->normal(sym_lst, repl_lst, 0); - GINAC_ASSERT(is_ex_of_type(e, lst)); + GINAC_ASSERT(is_a(e)); // Re-insert replaced symbols if (sym_lst.nops() > 0) @@ -2289,9 +2343,20 @@ ex ex::numer_denom(void) const } -/** Default implementation of ex::to_rational(). It replaces the object with a - * temporary symbol. - * @see ex::to_rational */ +/** 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 basic::to_rational(lst &repl_lst) const { return replace_with_symbol(*this, repl_lst); @@ -2299,8 +2364,7 @@ ex basic::to_rational(lst &repl_lst) const /** Implementation of ex::to_rational() for symbols. This returns the - * unmodified symbol. - * @see ex::to_rational */ + * unmodified symbol. */ ex symbol::to_rational(lst &repl_lst) const { return *this; @@ -2309,8 +2373,7 @@ 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 */ + * temporary symbol. */ ex numeric::to_rational(lst &repl_lst) const { if (is_real()) { @@ -2328,8 +2391,7 @@ ex numeric::to_rational(lst &repl_lst) const /** Implementation of ex::to_rational() for powers. It replaces non-integer - * powers by temporary symbols. - * @see ex::to_rational */ + * powers by temporary symbols. */ ex power::to_rational(lst &repl_lst) const { if (exponent.info(info_flags::integer)) @@ -2339,41 +2401,124 @@ ex power::to_rational(lst &repl_lst) const } -/** Implementation of ex::to_rational() for expairseqs. - * @see ex::to_rational */ +/** Implementation of ex::to_rational() for expairseqs. */ ex expairseq::to_rational(lst &repl_lst) 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_lst))); + ++i; } ex oc = overall_coeff.to_rational(repl_lst); if (oc.info(info_flags::numeric)) return thisexpairseq(s, overall_coeff); - else s.push_back(combine_ex_with_coeff_to_pair(oc,_ex1())); + 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, lst & repl) +{ + if (is_exactly_a(e)) { + + unsigned num = e.nops(); + exvector terms; terms.reserve(num); + ex gc; + + // Find the common GCD + for (unsigned i=0; i(x) || is_exactly_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 (unsigned i=0; i(t)) { + for (unsigned 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)) { + + unsigned num = e.nops(); + exvector v; v.reserve(num); + + for (unsigned i=0; isetflag(status_flags::dynallocated); + + } else if (is_exactly_a(e)) { + + ex x = e.to_rational(repl); + if (is_exactly_a(x) && x.op(1).info(info_flags::negative)) + return replace_with_symbol(x, repl); + else + return x; + + } 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) { - return bp->to_rational(repl_lst); + if (is_exactly_a(e) || is_exactly_a(e)) { + + lst repl; + ex factor = 1; + ex r = find_common_factor(e, factor, repl); + return factor.subs(repl) * r.subs(repl); + + } else + return e; }