X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnormal.cpp;h=67fbf93612beaa629ed2df6a987152e56a546374;hp=8568cc3aa54310c3b632f98f2661395c050c53e9;hb=f5a10e579ab4a03d8b9322e999c3e6e9f944f1a7;hpb=52d0eff497831230458b6c386f9bf7f2aa84fc26 diff --git a/ginac/normal.cpp b/ginac/normal.cpp index 8568cc3a..67fbf936 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) 1999-2004 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 @@ -298,9 +298,10 @@ static ex multiply_lcm(const ex &e, const numeric &lcm) /** Compute the integer content (= GCD of all numeric coefficients) of an - * expanded polynomial. + * expanded polynomial. For a polynomial with rational coefficients, this + * returns g/l where g is the GCD of the coefficients' numerators and l + * is the LCM of the coefficients' denominators. * - * @param e expanded polynomial * @return integer content */ numeric ex::integer_content() const { @@ -321,16 +322,18 @@ numeric add::integer_content() const { epvector::const_iterator it = seq.begin(); epvector::const_iterator itend = seq.end(); - numeric c = _num0; + numeric c = _num0, l = _num1; while (it != itend) { GINAC_ASSERT(!is_exactly_a(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 @@ -813,7 +816,7 @@ 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 @@ -835,7 +838,7 @@ ex ex::unit(const ex &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 @@ -851,7 +854,8 @@ ex ex::content(const ex &x) const 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); @@ -859,18 +863,18 @@ ex ex::content(const ex &x) const return c; // GCD of all coefficients - int deg = e.degree(x); - int ldeg = e.ldegree(x); + int deg = r.degree(x); + int ldeg = r.ldegree(x); if (deg == ldeg) - return e.lcoeff(x) / e.unit(x); - c = _ex0; + return lcoeff * c; + 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]. +/** Compute primitive part of a multivariate polynomial in Q[x]. * The product of unit part, content part, and primitive part is the * polynomial itself. * @@ -895,7 +899,7 @@ ex ex::primpart(const ex &x) const } -/** 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. * @@ -1003,7 +1007,6 @@ 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 @@ -1220,6 +1223,8 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const * * @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 */ @@ -1498,7 +1503,7 @@ 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 @@ -1702,23 +1707,23 @@ static ex replace_with_symbol(const ex & e, exmap & repl, exmap & rev_lookup) } /** 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 itself contain symbols from the repl_lst, + // replacement expression doesn't itself contain symbols from repl, // because subs() is not recursive ex es = (new symbol)->setflag(status_flags::dynallocated); - ex e_replaced = e.subs(repl_lst, subs_options::no_pattern); - repl_lst.append(es == e_replaced); + ex e_replaced = e.subs(repl, subs_options::no_pattern); + repl.insert(std::make_pair(es, e_replaced)); return es; } @@ -2103,46 +2108,80 @@ 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); +} + +// GiNaC 1.1 compatibility function +ex ex::to_rational(lst & repl_lst) const { - return bp->to_rational(repl_lst); + // Convert lst to exmap + exmap m; + for (lst::const_iterator it = repl_lst.begin(); it != repl_lst.end(); ++it) + m.insert(std::make_pair(it->op(0), it->op(1))); + + ex ret = bp->to_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; } @@ -2151,17 +2190,17 @@ 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; } @@ -2169,17 +2208,17 @@ 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; } @@ -2187,36 +2226,36 @@ 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 @@ -2225,16 +2264,16 @@ 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 @@ -2246,7 +2285,7 @@ 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)) { @@ -2331,7 +2370,7 @@ ex collect_common_factors(const ex & e) { if (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, subs_options::no_pattern) * r.subs(repl, subs_options::no_pattern); @@ -2341,4 +2380,37 @@ ex collect_common_factors(const ex & 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