X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnormal.cpp;h=db1ca467e422f856e4687bb3b5b58f3889ada51c;hp=cd2ad8eca5a0b29024bc142e81c19866f031bea7;hb=29b420743599f001b8bfc8b63fb6ac81aa3dfba7;hpb=68fdf425abf14d016d5f95ee7b9d06a19a3c5926 diff --git a/ginac/normal.cpp b/ginac/normal.cpp index cd2ad8ec..db1ca467 100644 --- a/ginac/normal.cpp +++ b/ginac/normal.cpp @@ -6,7 +6,7 @@ * computation, square-free factorization and rational function normalization. */ /* - * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2005 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 @@ -84,17 +84,17 @@ 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. * * @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_a(e)) { - x = &ex_to(e); + x = e; return true; } else if (is_exactly_a(e) || is_exactly_a(e)) { for (size_t i=0; i 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::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; } @@ -171,7 +171,7 @@ static void add_symbol(const symbol *s, sym_desc_vec &v) static void collect_symbols(const ex &e, sym_desc_vec &v) { if (is_a(e)) { - add_symbol(&ex_to(e), v); + add_symbol(e, v); } else if (is_exactly_a(e) || is_exactly_a(e)) { for (size_t i=0; isym)); - 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->max_lcnops = std::max(a.lcoeff(it->sym).nops(), b.lcoeff(it->sym).nops()); + it->ldeg_a = a.ldegree(it->sym); + it->ldeg_b = b.ldegree(it->sym); ++it; } std::sort(v.begin(), v.end()); @@ -214,8 +214,8 @@ static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v) 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; + 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 @@ -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), c); + c = gcd(ex_to(it->coeff).numer(), c); + l = lcm(ex_to(it->coeff).denom(), l); it++; } GINAC_ASSERT(is_exactly_a(overall_coeff)); - c = gcd(ex_to(overall_coeff),c); - return c; + c = gcd(ex_to(overall_coeff).numer(), c); + l = lcm(ex_to(overall_coeff).denom(), l); + return c/l; } numeric mul::integer_content() const @@ -361,7 +364,7 @@ numeric mul::integer_content() 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")); @@ -411,7 +414,7 @@ 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")); @@ -460,7 +463,7 @@ ex rem(const ex &a, const ex &b, const symbol &x, bool check_args) * @param a rational function in x * @param x a is a function of x * @return decomposed function. */ -ex decomp_rational(const ex &a, const symbol &x) +ex decomp_rational(const ex &a, const ex &x) { ex nd = numer_denom(a); ex numer = nd.op(0), denom = nd.op(1); @@ -480,7 +483,7 @@ ex decomp_rational(const ex &a, const symbol &x) * @param check_args check whether a and b are polynomials with rational * coefficients (defaults to "true") * @return pseudo-remainder of a(x) and b(x) in Q[x] */ -ex prem(const ex &a, const ex &b, const symbol &x, bool check_args) +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")); @@ -532,7 +535,7 @@ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args) * @param check_args check whether a and b are polynomials with rational * coefficients (defaults to "true") * @return 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) +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")); @@ -607,7 +610,7 @@ 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()")); @@ -617,26 +620,26 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args) q = _ex0; return true; } - int bdeg = b.degree(*x); - int rdeg = r.degree(*x); - ex blcoeff = b.expand().coeff(*x, bdeg); + 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); + term *= power(x, rdeg - bdeg); v.push_back(term); r -= (term * b).expand(); if (r.is_zero()) { q = (new add(v))->setflag(status_flags::dynallocated); return true; } - rdeg = r.degree(*x); + rdeg = r.degree(x); } return false; } @@ -665,7 +668,7 @@ 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 @@ -711,11 +714,36 @@ 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; @@ -727,24 +755,24 @@ 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 @@ -781,13 +809,13 @@ 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(); + term = (term * power(x, rdeg - bdeg)).expand(); v.push_back(term); r -= (term * eb).expand(); if (r.is_zero()) { @@ -797,7 +825,7 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite #endif return true; } - rdeg = r.degree(*x); + rdeg = r.degree(x); } #if USE_REMEMBER dr_remember[ex2(a, b)] = exbool(q, false); @@ -813,21 +841,21 @@ 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_exactly_a(c)) - return c < _ex0 ? _ex_1 : _ex1; + 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()")); } @@ -835,82 +863,72 @@ 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_exactly_a(*this)) return info(info_flags::negative) ? -*this : *this; + ex e = expand(); if (e.is_zero()) 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_exactly_a(*this)) - return _ex1; - - ex c = content(x); - if (c.is_zero()) - return _ex0; - ex u = unit(x); - if (is_exactly_a(c)) - 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()) + if (is_zero() || c.is_zero()) return _ex0; if (is_exactly_a(*this)) return _ex1; + // Divide by unit and content to get primitive part ex u = unit(x); if (is_exactly_a(c)) return *this / (c * u); @@ -919,6 +937,61 @@ ex ex::primpart(const symbol &x, const ex &c) const } +/** Compute unit part, content part, and primitive part of a multivariate + * polynomial in Q[x]. The product of the three parts is the polynomial + * itself. + * + * @param x main variable + * @param u unit part (returned) + * @param c content part (returned) + * @param p primitive part (returned) + * @see ex::unit, ex::content, ex::primpart */ +void ex::unitcontprim(const ex &x, ex &u, ex &c, ex &p) const +{ + // Quick check for zero (avoid expanding) + if (is_zero()) { + u = _ex1; + c = p = _ex0; + return; + } + + // Special case: input is a number + if (is_exactly_a(*this)) { + if (info(info_flags::negative)) { + u = _ex_1; + c = abs(ex_to(*this)); + } else { + u = _ex1; + c = *this; + } + p = _ex1; + return; + } + + // Expand input polynomial + ex e = expand(); + if (e.is_zero()) { + u = _ex1; + c = p = _ex0; + return; + } + + // Compute unit and content + u = unit(x); + c = content(x); + + // Divide by unit and content to get primitive part + if (c.is_zero()) { + p = _ex0; + return; + } + if (is_exactly_a(c)) + p = *this / (c * u); + else + p = quo(e, c * u, x, false); +} + + /* * GCD of multivariate polynomials */ @@ -939,7 +1012,7 @@ static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var) #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; @@ -1003,7 +1076,6 @@ 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() const @@ -1015,7 +1087,7 @@ numeric ex::max_coefficient() const * @see heur_gcd */ numeric basic::max_coefficient() const { - return _num1; + return *_num1_p; } numeric numeric::max_coefficient() const @@ -1109,7 +1181,7 @@ ex mul::smod(const numeric &xi) const /** xi-adic polynomial interpolation */ -static ex interpolate(const ex &gamma, const numeric &xi, const symbol &x, int degree_hint = 1) +static ex interpolate(const ex &gamma, const numeric &xi, const ex &x, int degree_hint = 1) { exvector g; g.reserve(degree_hint); ex e = gamma; @@ -1161,7 +1233,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const } // 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()); @@ -1175,9 +1247,9 @@ 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++) { @@ -1187,7 +1259,7 @@ 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(); + ex gamma = heur_gcd(p.subs(x == xi, subs_options::no_pattern), q.subs(x == xi, subs_options::no_pattern), &cp, &cq, var+1).expand(); if (!is_exactly_a(gamma)) { // Reconstruct polynomial from GCD of mapped polynomials @@ -1200,11 +1272,7 @@ 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; + return g; } } @@ -1216,10 +1284,13 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const /** 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 */ @@ -1298,10 +1369,12 @@ factored_b: // Input polynomials of the form poly^n are sometimes also trivial if (is_exactly_a(a)) { ex p = a.op(0); + const ex& exp_a = a.op(1); if (is_exactly_a(b)) { - if (p.is_equal(b.op(0))) { + ex pb = b.op(0); + const ex& exp_b = b.op(1); + if (p.is_equal(pb)) { // a = p^n, b = p^m, gcd = p^min(n, m) - ex exp_a = a.op(1), exp_b = b.op(1); if (exp_a < exp_b) { if (ca) *ca = _ex1; @@ -1315,7 +1388,32 @@ factored_b: *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 @@ -1324,8 +1422,24 @@ factored_b: 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)) { @@ -1336,6 +1450,23 @@ factored_b: *cb = power(p, b.op(1) - 1); return p; } + + ex p_co, apart_co; + const ex& exp_b(b.op(1)); + ex p_gcd = gcd(a, p, &apart_co, &p_co, false); + if (p_gcd.is_equal(_ex1)) { + // b=p(x)^n, gcd(a, p) = 1 ==> gcd(a, b) == 1 + if (ca) + *ca = a; + if (cb) + *cb = b; + return _ex1; + } else { + // there are common factors: + // a(x) = g(x) A(x), b(x) = g(x)^n B(x)^n ==> gcd = g(x) gcd(g(x)^(n-1) A(x)^n, B(x)) + + return p_gcd*gcd(apart_co, power(p_gcd, exp_b-1)*power(p_co, exp_b), ca, cb, false); + } // p_gcd.is_equal(_ex1) } #endif @@ -1372,13 +1503,33 @@ 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; + } + } + // Gather symbol statistics sym_desc_vec sym_stats; get_symbol_stats(a, b, sym_stats); // The symbol with least degree is our main variable sym_desc_vec::const_iterator var = sym_stats.begin(); - const symbol &x = *(var->sym); + const ex &x = var->sym; // Cancel trivial common factor int ldeg_a = var->ldeg_a; @@ -1391,16 +1542,18 @@ factored_b: // Try to eliminate variables if (var->deg_a == 0) { - ex c = bex.content(x); - ex g = gcd(aex, c, ca, cb, false); + 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) { - ex c = aex.content(x); - ex g = gcd(c, bex, ca, cb, false); + 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; } @@ -1467,7 +1620,7 @@ 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. @@ -1498,7 +1651,7 @@ static exvector sqrfree_yun(const ex &a, const symbol &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 + * @param l lst of variables to factor in, may be left empty for autodetection * @return a square-free factorization of \p a. * * \note @@ -1545,7 +1698,7 @@ ex sqrfree(const ex &a, const lst &l) get_symbol_stats(a, _ex0, sdv); sym_desc_vec::const_iterator it = sdv.begin(), itend = sdv.end(); while (it != itend) { - args.append(*it->sym); + args.append(it->sym); ++it; } } else { @@ -1562,7 +1715,7 @@ ex sqrfree(const ex &a, const lst &l) 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 = args; @@ -1681,47 +1834,44 @@ ex sqrfree_parfrac(const ex & a, const symbol & x) /** Create a symbol for replacing the expression "e" (or return a previously - * assigned symbol). The symbol is appended to sym_lst and returned, the - * expression is appended to repl_lst. + * assigned symbol). The symbol and expression are appended to repl, for + * a later application of subs(). * @see ex::normal */ -static ex replace_with_symbol(const ex &e, lst &sym_lst, lst &repl_lst) +static ex replace_with_symbol(const ex & e, exmap & repl, exmap & rev_lookup) { - // Expression already in repl_lst? Then return the assigned symbol - lst::const_iterator its, itr; - for (its = sym_lst.begin(), itr = repl_lst.begin(); itr != repl_lst.end(); ++its, ++itr) - if (itr->is_equal(e)) - return *its; + // Expression already replaced? Then return the assigned symbol + exmap::const_iterator it = rev_lookup.find(e); + if (it != rev_lookup.end()) + return it->second; // 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. + * assigned symbol). The symbol and expression are appended to repl, and the + * symbol is returned. * @see basic::to_rational * @see basic::to_polynomial */ -static ex replace_with_symbol(const ex &e, lst &repl_lst) +static ex replace_with_symbol(const ex & e, exmap & repl) { - // Expression already in repl_lst? Then return the assigned symbol - for (lst::const_iterator it = repl_lst.begin(); it != repl_lst.end(); ++it) - if (it->op(1).is_equal(e)) - return it->op(0); + // Expression already replaced? Then return the assigned symbol + for (exmap::const_iterator it = repl.begin(); it != repl.end(); ++it) + if (it->second.is_equal(e)) + return it->first; // Otherwise create new symbol and add to list, taking care that the - // replacement expression doesn't 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; } @@ -1736,18 +1886,18 @@ struct normal_map_function : public map_function { /** 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 { if (nops() == 0) - 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 == 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")); 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), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated); } } } @@ -1755,7 +1905,7 @@ ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const /** 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); } @@ -1765,19 +1915,19 @@ 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()) @@ -1793,7 +1943,7 @@ 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; @@ -1830,10 +1980,10 @@ static ex frac_cancel(const ex &n, const ex &d) den *= _ex_1; } } else { - const symbol *x; + ex x; if (get_first_symbol(den, x)) { - GINAC_ASSERT(is_exactly_a(den.unit(*x))); - if (ex_to(den.unit(*x)).is_negative()) { + GINAC_ASSERT(is_exactly_a(den.unit(x))); + if (ex_to(den.unit(x)).is_negative()) { num *= _ex_1; den *= _ex_1; } @@ -1849,10 +1999,10 @@ 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")); @@ -1862,12 +2012,12 @@ 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 = ex_to(recombine_pair_to_ex(*it)).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 = ex_to(overall_coeff).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()); @@ -1908,10 +2058,10 @@ 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")); @@ -1921,12 +2071,12 @@ 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 = ex_to(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, level-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 = ex_to(overall_coeff).normal(sym_lst, repl_lst, level-1); + n = ex_to(overall_coeff).normal(repl, rev_lookup, level-1); num.push_back(n.op(0)); den.push_back(n.op(1)); @@ -1936,20 +2086,20 @@ ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const } -/** 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 = ex_to(basis).normal(sym_lst, repl_lst, level-1); - ex n_exponent = ex_to(exponent).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)) { @@ -1970,32 +2120,32 @@ 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)) { // 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); } } } // (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); } /** 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; epvector::const_iterator i = seq.begin(), end = seq.end(); @@ -2006,7 +2156,7 @@ ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const ++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, repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated); } @@ -2024,14 +2174,14 @@ ex pseries::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); + 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); @@ -2045,16 +2195,16 @@ ex ex::normal(int level) const * @return numerator */ ex ex::numer() const { - lst sym_lst, repl_lst; + exmap repl, rev_lookup; - ex e = bp->normal(sym_lst, repl_lst, 0); + 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); } /** Get denominator of an expression. If the expression is not of the normal @@ -2065,16 +2215,16 @@ ex ex::numer() const * @return denominator */ ex ex::denom() const { - lst sym_lst, repl_lst; + exmap repl, rev_lookup; - ex e = bp->normal(sym_lst, repl_lst, 0); + 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 @@ -2085,16 +2235,16 @@ ex ex::denom() const * @return a list [numerator, denominator] */ ex ex::numer_denom() const { - lst sym_lst, repl_lst; + exmap repl, rev_lookup; - ex e = bp->normal(sym_lst, repl_lst, 0); + ex e = bp->normal(repl, rev_lookup, 0); GINAC_ASSERT(is_a(e)); // Re-insert replaced symbols - if (sym_lst.nops() > 0) - return e.subs(sym_lst, repl_lst); - else + if (repl.empty()) return e; + else + return e.subs(repl, subs_options::no_pattern); } @@ -2106,46 +2256,80 @@ ex ex::numer_denom() const * 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(). + * 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_lst collects a list of all temporary symbols and their replacements + * @param repl collects all temporary symbols and their replacements * @return rationalized expression */ -ex ex::to_rational(lst &repl_lst) const +ex ex::to_rational(exmap & repl) const { - return bp->to_rational(repl_lst); + 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(lst &repl_lst) const +ex ex::to_polynomial(exmap & repl) const { - return bp->to_polynomial(repl_lst); + return bp->to_polynomial(repl); } +// GiNaC 1.1 compatibility function +ex ex::to_polynomial(lst & repl_lst) const +{ + // Convert lst to exmap + exmap m; + for (lst::const_iterator it = repl_lst.begin(); it != repl_lst.end(); ++it) + m.insert(std::make_pair(it->op(0), it->op(1))); + + ex ret = bp->to_polynomial(m); + + // Convert exmap back to lst + repl_lst.remove_all(); + for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) + repl_lst.append(it->first == it->second); + + return ret; +} /** Default implementation of ex::to_rational(). This replaces the object with * a temporary symbol. */ -ex basic::to_rational(lst &repl_lst) const +ex basic::to_rational(exmap & repl) const { - return replace_with_symbol(*this, repl_lst); + return replace_with_symbol(*this, repl); } -ex basic::to_polynomial(lst &repl_lst) const +ex basic::to_polynomial(exmap & repl) const { - return replace_with_symbol(*this, repl_lst); + return replace_with_symbol(*this, repl); } /** Implementation of ex::to_rational() for symbols. This returns the * unmodified symbol. */ -ex symbol::to_rational(lst &repl_lst) const +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(lst &repl_lst) const +ex symbol::to_polynomial(exmap & repl) const { return *this; } @@ -2154,17 +2338,17 @@ ex symbol::to_polynomial(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. */ -ex numeric::to_rational(lst &repl_lst) const +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; } @@ -2172,17 +2356,17 @@ ex numeric::to_rational(lst &repl_lst) const /** 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(lst &repl_lst) const +ex numeric::to_polynomial(exmap & repl) const { if (is_real()) { if (!is_integer()) - 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_integer() ? re : replace_with_symbol(re, repl_lst); - ex im_ex = im.is_integer() ? im : replace_with_symbol(im, repl_lst); - return re_ex + im_ex * replace_with_symbol(I, repl_lst); + 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; } @@ -2190,36 +2374,36 @@ ex numeric::to_polynomial(lst &repl_lst) const /** Implementation of ex::to_rational() for powers. It replaces non-integer * powers by temporary symbols. */ -ex power::to_rational(lst &repl_lst) const +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(lst &repl_lst) const +ex power::to_polynomial(exmap & repl) const { if (exponent.info(info_flags::posint)) - 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_rational() for expairseqs. */ -ex expairseq::to_rational(lst &repl_lst) const +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_lst))); + 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 @@ -2228,16 +2412,16 @@ ex expairseq::to_rational(lst &repl_lst) const } /** Implementation of ex::to_polynomial() for expairseqs. */ -ex expairseq::to_polynomial(lst &repl_lst) const +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_lst))); + s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_polynomial(repl))); ++i; } - ex oc = overall_coeff.to_polynomial(repl_lst); + ex oc = overall_coeff.to_polynomial(repl); if (oc.info(info_flags::numeric)) return thisexpairseq(s, overall_coeff); else @@ -2249,7 +2433,7 @@ ex expairseq::to_polynomial(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) +static ex find_common_factor(const ex & e, ex & factor, exmap & repl) { if (is_exactly_a(e)) { @@ -2320,8 +2504,16 @@ 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::posint)) { + 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; @@ -2332,16 +2524,49 @@ 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)) { - lst repl; + exmap repl; ex factor = 1; ex r = find_common_factor(e, factor, repl); - return factor.subs(repl) * r.subs(repl); + return factor.subs(repl, subs_options::no_pattern) * r.subs(repl, subs_options::no_pattern); } else return e; } +/** Resultant of two expressions e1,e2 with respect to symbol s. + * Method: Compute determinant of Sylvester matrix of e1,e2,s. */ +ex resultant(const ex & e1, const ex & e2, const ex & s) +{ + const ex ee1 = e1.expand(); + const ex ee2 = e2.expand(); + if (!ee1.info(info_flags::polynomial) || + !ee2.info(info_flags::polynomial)) + throw(std::runtime_error("resultant(): arguments must be polynomials")); + + const int h1 = ee1.degree(s); + const int l1 = ee1.ldegree(s); + const int h2 = ee2.degree(s); + const int l2 = ee2.ldegree(s); + + const int msize = h1 + h2; + matrix m(msize, msize); + + for (int l = h1; l >= l1; --l) { + const ex e = ee1.coeff(s, l); + for (int k = 0; k < h2; ++k) + m(k, k+h1-l) = e; + } + for (int l = h2; l >= l2; --l) { + const ex e = ee2.coeff(s, l); + for (int k = 0; k < h1; ++k) + m(k+h2, k+h2-l) = e; + } + + return m.determinant(); +} + + } // namespace GiNaC