X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnormal.cpp;h=b2a4e8c8d0cfdfe2204799fefe96b2710771cb24;hp=4b490ed0e9824d301cb979fc830433b98aa22019;hb=9c822131b5a057640b040af6df828e0d1ed0222e;hpb=274c1632d78ec31575c4b3b328d75ad95a7855aa diff --git a/ginac/normal.cpp b/ginac/normal.cpp index 4b490ed0..b2a4e8c8 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) 1999-2005 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 @@ -39,6 +39,7 @@ #include "numeric.h" #include "power.h" #include "relational.h" +#include "operators.h" #include "matrix.h" #include "pseries.h" #include "symbol.h" @@ -83,23 +84,23 @@ 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_ex_exactly_of_type(e, symbol)) { - x = &ex_to(e); + if (is_a(e)) { + x = e; return true; - } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) { - for (unsigned i=0; i(e) || is_exactly_a(e)) { + for (size_t i=0; i(e)) { if (get_first_symbol(e.op(0), x)) return true; } @@ -118,8 +119,8 @@ static bool get_first_symbol(const ex &e, const symbol *&x) * * @see get_symbol_stats */ struct sym_desc { - /** Pointer to symbol */ - const symbol *sym; + /** Reference to symbol */ + ex sym; /** Highest degree of symbol in polynomial "a" */ int deg_a; @@ -137,7 +138,7 @@ struct sym_desc { int max_deg; /** Maximum number of terms of leading coefficient of symbol in both polynomials */ - int max_lcnops; + size_t max_lcnops; /** Commparison operator for sorting */ bool operator<(const sym_desc &x) const @@ -153,11 +154,11 @@ struct sym_desc { typedef std::vector sym_desc_vec; // Add symbol the sym_desc_vec (used internally by get_symbol_stats()) -static void add_symbol(const symbol *s, sym_desc_vec &v) +static void add_symbol(const ex &s, sym_desc_vec &v) { sym_desc_vec::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; } @@ -169,12 +170,12 @@ static void add_symbol(const symbol *s, sym_desc_vec &v) // Collect all symbols of an expression (used internally by get_symbol_stats()) static void collect_symbols(const ex &e, sym_desc_vec &v) { - if (is_ex_exactly_of_type(e, symbol)) { - add_symbol(&ex_to(e), v); - } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) { - for (unsigned i=0; i(e)) { + add_symbol(e, v); + } else if (is_exactly_a(e) || is_exactly_a(e)) { + for (size_t i=0; i(e)) { collect_symbols(e.op(0), v); } } @@ -197,23 +198,24 @@ static void get_symbol_stats(const ex &a, const ex &b, sym_desc_vec &v) collect_symbols(b.eval(), v); sym_desc_vec::iterator it = v.begin(), itend = v.end(); while (it != itend) { - int deg_a = a.degree(*(it->sym)); - int deg_b = b.degree(*(it->sym)); + int deg_a = a.degree(it->sym); + int deg_b = b.degree(it->sym); it->deg_a = deg_a; it->deg_b = deg_b; it->max_deg = std::max(deg_a, deg_b); - it->max_lcnops = std::max(a.lcoeff(*(it->sym)).nops(), b.lcoeff(*(it->sym)).nops()); - it->ldeg_a = a.ldegree(*(it->sym)); - it->ldeg_b = b.ldegree(*(it->sym)); + it->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()); + #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; + 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 @@ -230,18 +232,18 @@ 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; - for (unsigned i=0; i(e)) { + numeric c = *_num1_p; + for (size_t i=0; i(e)) { + numeric c = *_num1_p; + for (size_t i=0; i(e)) { + if (is_a(e.op(0))) return l; else return pow(lcmcoeff(e.op(0), l), ex_to(e.op(1))); @@ -258,7 +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 @@ -268,25 +270,25 @@ 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)) { - unsigned num = e.nops(); + if (is_exactly_a(e)) { + size_t num = e.nops(); exvector v; v.reserve(num + 1); - numeric lcm_accum = _num1; - for (unsigned i=0; isetflag(status_flags::dynallocated); - } else if (is_ex_exactly_of_type(e, add)) { - unsigned num = e.nops(); + } else if (is_exactly_a(e)) { + size_t 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)); @@ -296,43 +298,45 @@ 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(void) const +numeric ex::integer_content() const { - GINAC_ASSERT(bp!=0); return bp->integer_content(); } -numeric basic::integer_content(void) const +numeric basic::integer_content() const { - return _num1; + return *_num1_p; } -numeric numeric::integer_content(void) const +numeric numeric::integer_content() const { return abs(*this); } -numeric add::integer_content(void) const +numeric add::integer_content() const { epvector::const_iterator it = seq.begin(); epvector::const_iterator itend = seq.end(); - numeric c = _num0; + numeric c = *_num0_p, l = *_num1_p; while (it != itend) { GINAC_ASSERT(!is_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(void) const +numeric mul::integer_content() const { #ifdef DO_GINAC_ASSERT epvector::const_iterator it = seq.begin(); @@ -360,11 +364,11 @@ numeric mul::integer_content(void) const * @param check_args check whether a and b are polynomials with rational * coefficients (defaults to "true") * @return quotient of a and b in Q[x] */ -ex quo(const ex &a, const ex &b, const symbol &x, bool check_args) +ex quo(const ex &a, const ex &b, const ex &x, bool check_args) { if (b.is_zero()) throw(std::overflow_error("quo: division by zero")); - if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) + if (is_exactly_a(a) && is_exactly_a(b)) return a / b; #if FAST_COMPARE if (a.is_equal(b)) @@ -380,7 +384,7 @@ 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); + 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); @@ -410,12 +414,12 @@ ex quo(const ex &a, const ex &b, const symbol &x, bool check_args) * @param check_args check whether a and b are polynomials with rational * coefficients (defaults to "true") * @return remainder of a(x) and b(x) in Q[x] */ -ex rem(const ex &a, const ex &b, const symbol &x, bool check_args) +ex rem(const ex &a, const ex &b, const ex &x, bool check_args) { if (b.is_zero()) throw(std::overflow_error("rem: division by zero")); - if (is_ex_exactly_of_type(a, numeric)) { - if (is_ex_exactly_of_type(b, numeric)) + if (is_exactly_a(a)) { + if (is_exactly_a(b)) return _ex0; else return a; @@ -434,7 +438,7 @@ ex rem(const ex &a, const ex &b, const symbol &x, bool check_args) int bdeg = b.degree(x); int rdeg = r.degree(x); ex blcoeff = b.expand().coeff(x, bdeg); - bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric); + bool blcoeff_is_numeric = is_exactly_a(blcoeff); while (rdeg >= bdeg) { ex term, rcoeff = r.coeff(x, rdeg); if (blcoeff_is_numeric) @@ -459,32 +463,32 @@ 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); 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] */ -ex prem(const ex &a, const ex &b, const symbol &x, bool check_args) + * @return pseudo-remainder of a(x) and b(x) in Q[x] */ +ex prem(const ex &a, const ex &b, const ex &x, bool check_args) { if (b.is_zero()) throw(std::overflow_error("prem: division by zero")); - if (is_ex_exactly_of_type(a, numeric)) { - if (is_ex_exactly_of_type(b, numeric)) + if (is_exactly_a(a)) { + if (is_exactly_a(b)) return _ex0; else return b; @@ -523,20 +527,20 @@ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args) } -/** Sparse pseudo-remainder of polynomials a(x) and b(x) in Z[x]. +/** Sparse pseudo-remainder of polynomials a(x) and b(x) in Q[x]. * * @param a first polynomial in x (dividend) * @param b second polynomial in x (divisor) * @param x a and b are polynomials in x * @param check_args check whether a and b are polynomials with rational * coefficients (defaults to "true") - * @return sparse pseudo-remainder of a(x) and b(x) in Z[x] */ -ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args) + * @return sparse pseudo-remainder of a(x) and b(x) in Q[x] */ +ex sprem(const ex &a, const ex &b, const ex &x, bool check_args) { if (b.is_zero()) throw(std::overflow_error("prem: division by zero")); - if (is_ex_exactly_of_type(a, numeric)) { - if (is_ex_exactly_of_type(b, numeric)) + if (is_exactly_a(a)) { + if (is_exactly_a(b)) return _ex0; else return b; @@ -590,10 +594,10 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args) 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)) { @@ -606,36 +610,102 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args) throw(std::invalid_argument("divide: arguments must be polynomials over the rationals")); // Find first symbol - const symbol *x; + ex x; if (!get_first_symbol(a, x) && !get_first_symbol(b, x)) throw(std::invalid_argument("invalid expression in divide()")); + // Try to avoid expanding partially factored expressions. + if (is_exactly_a(b)) { + // Divide sequentially by each term + ex rem_new, rem_old = a; + for (size_t i=0; i < b.nops(); i++) { + if (! divide(rem_old, b.op(i), rem_new, false)) + return false; + rem_old = rem_new; + } + q = rem_new; + return true; + } else if (is_exactly_a(b)) { + const ex& bb(b.op(0)); + int exp_b = ex_to(b.op(1)).to_int(); + ex rem_new, rem_old = a; + for (int i=exp_b; i>0; i--) { + if (! divide(rem_old, bb, rem_new, false)) + return false; + rem_old = rem_new; + } + q = rem_new; + return true; + } + + if (is_exactly_a(a)) { + // Divide sequentially each term. If some term in a is divisible + // by b we are done... and if not, we can't really say anything. + size_t i; + ex rem_i; + bool divisible_p = false; + for (i=0; i < a.nops(); ++i) { + if (divide(a.op(i), b, rem_i, false)) { + divisible_p = true; + break; + } + } + if (divisible_p) { + exvector resv; + resv.reserve(a.nops()); + for (size_t j=0; j < a.nops(); j++) { + if (j==i) + resv.push_back(rem_i); + else + resv.push_back(a.op(j)); + } + q = (new mul(resv))->setflag(status_flags::dynallocated); + return true; + } + } else if (is_exactly_a(a)) { + // The base itself might be divisible by b, in that case we don't + // need to expand a + const ex& ab(a.op(0)); + int a_exp = ex_to(a.op(1)).to_int(); + ex rem_i; + if (divide(ab, b, rem_i, false)) { + q = rem_i*power(ab, a_exp - 1); + return true; + } + for (int i=2; i < a_exp; i++) { + if (divide(power(ab, i), b, rem_i, false)) { + q = rem_i*power(ab, a_exp - i); + return true; + } + } // ... so we *really* need to expand expression. + } + // Polynomial long division (recursive) ex r = a.expand(); if (r.is_zero()) { q = _ex0; return true; } - int bdeg = b.degree(*x); - int rdeg = r.degree(*x); - ex blcoeff = b.expand().coeff(*x, bdeg); - bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric); + int bdeg = b.degree(x); + int rdeg = r.degree(x); + ex blcoeff = b.expand().coeff(x, bdeg); + bool blcoeff_is_numeric = is_exactly_a(blcoeff); exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0)); while (rdeg >= bdeg) { - ex term, rcoeff = r.coeff(*x, rdeg); + ex term, rcoeff = r.coeff(x, rdeg); if (blcoeff_is_numeric) term = rcoeff / blcoeff; else if (!divide(rcoeff, blcoeff, term, false)) return false; - term *= power(*x, rdeg - bdeg); + 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; } @@ -664,7 +734,7 @@ typedef std::map ex2_exbool_remember; /** Exact polynomial division of a(X) by b(X) in Z[X]. * This functions works like divide() but the input and output polynomials are * in Z[X] instead of Q[X] (i.e. they have integer coefficients). Unlike - * divide(), it doesn´t check whether the input polynomials really are integer + * divide(), it doesn't check whether the input polynomials really are integer * polynomials, so be careful of what you pass in. Also, you have to run * get_symbol_stats() over the input polynomials before calling this function * and pass an iterator to the first element of the sym_desc vector. This @@ -686,8 +756,8 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite 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 @@ -710,11 +780,36 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite } #endif + if (is_exactly_a(b)) { + const ex& bb(b.op(0)); + ex qbar = a; + int exp_b = ex_to(b.op(1)).to_int(); + for (int i=exp_b; i>0; i--) { + if (!divide_in_z(qbar, bb, q, var)) + return false; + qbar = q; + } + return true; + } + + if (is_exactly_a(b)) { + ex qbar = a; + for (const_iterator itrb = b.begin(); itrb != b.end(); ++itrb) { + sym_desc_vec sym_stats; + get_symbol_stats(a, *itrb, sym_stats); + if (!divide_in_z(qbar, *itrb, q, sym_stats.begin())) + return false; + + qbar = q; + } + return true; + } + // Main symbol - const symbol *x = var->sym; + const ex &x = var->sym; // Compare degrees - int adeg = a.degree(*x), bdeg = b.degree(*x); + int adeg = a.degree(x), bdeg = b.degree(x); if (bdeg > adeg) return false; @@ -726,24 +821,24 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite // Compute values at evaluation points 0..adeg vector alpha; alpha.reserve(adeg + 1); exvector u; u.reserve(adeg + 1); - numeric point = _num0; + numeric point = *_num0_p; ex c; for (i=0; i<=adeg; i++) { - ex bs = b.subs(*x == point); + ex bs = b.subs(x == point, subs_options::no_pattern); while (bs.is_zero()) { - point += _num1; - bs = b.subs(*x == point); + point += *_num1_p; + bs = b.subs(x == point, subs_options::no_pattern); } - if (!divide_in_z(a.subs(*x == point), bs, c, var+1)) + if (!divide_in_z(a.subs(x == point, subs_options::no_pattern), bs, c, var+1)) return false; alpha.push_back(point); u.push_back(c); - point += _num1; + point += *_num1_p; } // Compute inverses vector rcp; rcp.reserve(adeg + 1); - rcp.push_back(_num0); + rcp.push_back(*_num0_p); for (k=1; k<=adeg; k++) { numeric product = alpha[k] - alpha[0]; for (i=1; i=0; k--) - c = c * (*x - alpha[k]) + v[k]; + c = c * (x - alpha[k]) + v[k]; - if (c.degree(*x) == (adeg - bdeg)) { + if (c.degree(x) == (adeg - bdeg)) { q = c.expand(); return true; } else @@ -780,13 +875,13 @@ 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()) { @@ -796,7 +891,7 @@ 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); @@ -812,21 +907,21 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite */ /** Compute unit part (= sign of leading coefficient) of a multivariate - * polynomial in Z[x]. The product of unit part, content part, and primitive + * polynomial in Q[x]. The product of unit part, content part, and primitive * part is the polynomial itself. * - * @param x variable in which to compute the unit part + * @param x main variable * @return unit part - * @see ex::content, ex::primpart */ -ex ex::unit(const symbol &x) const + * @see ex::content, ex::primpart, ex::unitcontprim */ +ex ex::unit(const ex &x) const { ex c = expand().lcoeff(x); - if (is_ex_exactly_of_type(c, numeric)) - return c < _ex0 ? _ex_1 : _ex1; + if (is_exactly_a(c)) + return c.info(info_flags::negative) ?_ex_1 : _ex1; else { - const symbol *y; + ex y; if (get_first_symbol(c, y)) - return c.unit(*y); + return c.unit(y); else throw(std::invalid_argument("invalid expression in unit()")); } @@ -834,293 +929,138 @@ ex ex::unit(const symbol &x) const /** Compute content part (= unit normal GCD of all coefficients) of a - * multivariate polynomial in Z[x]. The product of unit part, content part, + * multivariate polynomial in Q[x]. The product of unit part, content part, * and primitive part is the polynomial itself. * - * @param x variable in which to compute the content part + * @param x main variable * @return content part - * @see ex::unit, ex::primpart */ -ex ex::content(const symbol &x) const + * @see ex::unit, ex::primpart, ex::unitcontprim */ +ex ex::content(const ex &x) const { - if (is_zero()) - return _ex0; - if (is_ex_exactly_of_type(*this, numeric)) + if (is_exactly_a(*this)) return info(info_flags::negative) ? -*this : *this; + ex e = expand(); if (e.is_zero()) return _ex0; - // First, try the integer content + // First, divide out the integer content (which we can calculate very efficiently). + // If the leading coefficient of the quotient is an integer, we are done. ex c = e.integer_content(); ex r = e / c; - ex lcoeff = r.lcoeff(x); + int deg = r.degree(x); + ex lcoeff = r.coeff(x, deg); if (lcoeff.info(info_flags::integer)) return c; // GCD of all coefficients - int deg = e.degree(x); - int ldeg = e.ldegree(x); + int ldeg = r.ldegree(x); if (deg == ldeg) - return e.lcoeff(x) / e.unit(x); - c = _ex0; + return lcoeff * c / lcoeff.unit(x); + ex cont = _ex0; for (int i=ldeg; i<=deg; i++) - c = gcd(e.coeff(x, i), c, NULL, NULL, false); - return c; + cont = gcd(r.coeff(x, i), cont, NULL, NULL, false); + return cont * c; } -/** Compute primitive part of a multivariate polynomial in Z[x]. - * The product of unit part, content part, and primitive part is the - * polynomial itself. +/** Compute primitive part of a multivariate polynomial in Q[x]. The result + * will be a unit-normal polynomial with a content part of 1. The product + * of unit part, content part, and primitive part is the polynomial itself. * - * @param x variable in which to compute the primitive part + * @param x main variable * @return primitive part - * @see ex::unit, ex::content */ -ex ex::primpart(const symbol &x) const + * @see ex::unit, ex::content, ex::unitcontprim */ +ex ex::primpart(const ex &x) const { - if (is_zero()) - return _ex0; - if (is_ex_exactly_of_type(*this, numeric)) - return _ex1; - - ex c = content(x); - if (c.is_zero()) - return _ex0; - ex u = unit(x); - if (is_ex_exactly_of_type(c, numeric)) - return *this / (c * u); - else - return quo(*this, c * u, x, false); + // We need to compute the unit and content anyway, so call unitcontprim() + ex u, c, p; + unitcontprim(x, u, c, p); + return p; } -/** Compute primitive part of a multivariate polynomial in Z[x] when the +/** Compute primitive part of a multivariate polynomial in Q[x] when the * content part is already known. This function is faster in computing the * primitive part than the previous function. * - * @param x variable in which to compute the primitive part + * @param x main variable * @param c previously computed content part * @return primitive part */ -ex ex::primpart(const symbol &x, const ex &c) const +ex ex::primpart(const ex &x, const ex &c) const { - if (is_zero()) - return _ex0; - if (c.is_zero()) + if (is_zero() || c.is_zero()) return _ex0; - if (is_ex_exactly_of_type(*this, numeric)) + if (is_exactly_a(*this)) return _ex1; + // Divide by unit and content to get primitive part ex u = unit(x); - if (is_ex_exactly_of_type(c, numeric)) + if (is_exactly_a(c)) return *this / (c * u); else return quo(*this, c * u, x, false); } -/* - * GCD of multivariate polynomials - */ - -/** Compute GCD of polynomials in Q[X] using the Euclidean algorithm (not - * really suited for multivariate GCDs). This function is only provided for - * testing purposes. - * - * @param a first multivariate polynomial - * @param b second multivariate polynomial - * @param x pointer to symbol (main variable) in which to compute the GCD in - * @return the GCD as a new expression - * @see gcd */ - -static ex eu_gcd(const ex &a, const ex &b, const symbol *x) -{ -//std::clog << "eu_gcd(" << a << "," << b << ")\n"; - - // Sort c and d so that c has higher degree - ex c, d; - int adeg = a.degree(*x), bdeg = b.degree(*x); - if (adeg >= bdeg) { - c = a; - d = b; - } else { - c = b; - d = a; - } - - // Normalize in Q[x] - c = c / c.lcoeff(*x); - d = d / d.lcoeff(*x); - - // Euclidean algorithm - ex r; - for (;;) { -//std::clog << " d = " << d << endl; - r = rem(c, d, *x, false); - if (r.is_zero()) - return d / d.lcoeff(*x); - c = d; - d = r; - } -} - - -/** Compute GCD of multivariate polynomials using the Euclidean PRS algorithm - * with pseudo-remainders ("World's Worst GCD Algorithm", staying in Z[X]). - * This function is only provided for testing purposes. +/** Compute unit part, content part, and primitive part of a multivariate + * polynomial in Q[x]. The product of the three parts is the polynomial + * itself. * - * @param a first multivariate polynomial - * @param b second multivariate polynomial - * @param x pointer to symbol (main variable) in which to compute the GCD in - * @return the GCD as a new expression - * @see gcd */ - -static ex euprem_gcd(const ex &a, const ex &b, const symbol *x) + * @param x main variable + * @param u unit part (returned) + * @param c content part (returned) + * @param p primitive part (returned) + * @see ex::unit, ex::content, ex::primpart */ +void ex::unitcontprim(const ex &x, ex &u, ex &c, ex &p) const { -//std::clog << "euprem_gcd(" << a << "," << b << ")\n"; - - // Sort c and d so that c has higher degree - ex c, d; - int adeg = a.degree(*x), bdeg = b.degree(*x); - if (adeg >= bdeg) { - c = a; - d = b; - } else { - c = b; - d = a; - } - - // Calculate GCD of contents - ex gamma = gcd(c.content(*x), d.content(*x), NULL, NULL, false); - - // Euclidean algorithm with pseudo-remainders - ex r; - for (;;) { -//std::clog << " d = " << d << endl; - r = prem(c, d, *x, false); - if (r.is_zero()) - return d.primpart(*x) * gamma; - c = d; - d = r; + // Quick check for zero (avoid expanding) + if (is_zero()) { + u = _ex1; + c = p = _ex0; + return; + } + + // Special case: input is a number + if (is_exactly_a(*this)) { + if (info(info_flags::negative)) { + u = _ex_1; + c = abs(ex_to(*this)); + } else { + u = _ex1; + c = *this; + } + p = _ex1; + return; } -} - - -/** Compute GCD of multivariate polynomials using the primitive Euclidean - * PRS algorithm (complete content removal at each step). This function is - * only provided for testing purposes. - * - * @param a first multivariate polynomial - * @param b second multivariate polynomial - * @param x pointer to symbol (main variable) in which to compute the GCD in - * @return the GCD as a new expression - * @see gcd */ -static ex peu_gcd(const ex &a, const ex &b, const symbol *x) -{ -//std::clog << "peu_gcd(" << a << "," << b << ")\n"; - - // Sort c and d so that c has higher degree - ex c, d; - int adeg = a.degree(*x), bdeg = b.degree(*x); - int ddeg; - if (adeg >= bdeg) { - c = a; - d = b; - ddeg = bdeg; - } else { - c = b; - d = a; - ddeg = adeg; + // Expand input polynomial + ex e = expand(); + if (e.is_zero()) { + u = _ex1; + c = p = _ex0; + return; } - // Remove content from c and d, to be attached to GCD later - ex cont_c = c.content(*x); - ex cont_d = d.content(*x); - ex gamma = gcd(cont_c, cont_d, NULL, NULL, false); - if (ddeg == 0) - return gamma; - c = c.primpart(*x, cont_c); - d = d.primpart(*x, cont_d); + // Compute unit and content + u = unit(x); + c = content(x); - // Euclidean algorithm with content removal - ex r; - for (;;) { -//std::clog << " d = " << d << endl; - r = prem(c, d, *x, false); - if (r.is_zero()) - return gamma * d; - c = d; - d = r.primpart(*x); + // Divide by unit and content to get primitive part + if (c.is_zero()) { + p = _ex0; + return; } + if (is_exactly_a(c)) + p = *this / (c * u); + else + p = quo(e, c * u, x, false); } -/** Compute GCD of multivariate polynomials using the reduced PRS algorithm. - * This function is only provided for testing purposes. - * - * @param a first multivariate polynomial - * @param b second multivariate polynomial - * @param x pointer to symbol (main variable) in which to compute the GCD in - * @return the GCD as a new expression - * @see gcd */ - -static ex red_gcd(const ex &a, const ex &b, const symbol *x) -{ -//std::clog << "red_gcd(" << a << "," << b << ")\n"; - - // Sort c and d so that c has higher degree - ex c, d; - int adeg = a.degree(*x), bdeg = b.degree(*x); - int cdeg, ddeg; - if (adeg >= bdeg) { - c = a; - d = b; - cdeg = adeg; - ddeg = bdeg; - } else { - c = b; - d = a; - cdeg = bdeg; - ddeg = adeg; - } - - // Remove content from c and d, to be attached to GCD later - ex cont_c = c.content(*x); - ex cont_d = d.content(*x); - ex gamma = gcd(cont_c, cont_d, NULL, NULL, false); - if (ddeg == 0) - return gamma; - c = c.primpart(*x, cont_c); - d = d.primpart(*x, cont_d); - - // First element of divisor sequence - ex r, ri = _ex1; - int delta = cdeg - ddeg; - - for (;;) { - // Calculate polynomial pseudo-remainder -//std::clog << " d = " << d << endl; - r = prem(c, d, *x, false); - if (r.is_zero()) - return gamma * d.primpart(*x); - c = d; - cdeg = ddeg; - - if (!divide(r, pow(ri, delta), d, false)) - throw(std::runtime_error("invalid expression in red_gcd(), division failed")); - ddeg = d.degree(*x); - if (ddeg == 0) { - if (is_ex_exactly_of_type(r, numeric)) - return gamma; - else - return gamma * r.primpart(*x); - } - - ri = c.expand().lcoeff(*x); - delta = cdeg - ddeg; - } -} - +/* + * GCD of multivariate polynomials + */ /** Compute GCD of multivariate polynomials using the subresultant PRS * algorithm. This function is used internally by gcd(). @@ -1133,13 +1073,12 @@ static ex red_gcd(const ex &a, const ex &b, const symbol *x) static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var) { -//std::clog << "sr_gcd(" << a << "," << b << ")\n"; #if STATISTICS sr_gcd_called++; #endif // The first symbol is our main variable - const symbol &x = *(var->sym); + const ex &x = var->sym; // Sort c and d so that c has higher degree ex c, d; @@ -1165,34 +1104,31 @@ static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var) return gamma; c = c.primpart(x, cont_c); d = d.primpart(x, cont_d); -//std::clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n"; // First element of subresultant sequence ex r = _ex0, ri = _ex1, psi = _ex1; int delta = cdeg - ddeg; for (;;) { + // Calculate polynomial pseudo-remainder -//std::clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n"; -//std::clog << " d = " << d << endl; r = prem(c, d, x, false); if (r.is_zero()) return gamma * d.primpart(x); + c = d; cdeg = ddeg; -//std::clog << " dividing...\n"; if (!divide_in_z(r, ri * pow(psi, delta), d, var)) throw(std::runtime_error("invalid expression in sr_gcd(), division failed")); ddeg = d.degree(x); if (ddeg == 0) { - if (is_ex_exactly_of_type(r, numeric)) + if (is_exactly_a(r)) return gamma; else return gamma * r.primpart(x); } // Next element of subresultant sequence -//std::clog << " calculating next subresultant...\n"; ri = c.expand().lcoeff(x); if (delta == 1) psi = ri; @@ -1206,28 +1142,26 @@ static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var) /** Return maximum (absolute value) coefficient of a polynomial. * This function is used internally by heur_gcd(). * - * @param e expanded multivariate polynomial * @return maximum coefficient * @see heur_gcd */ -numeric ex::max_coefficient(void) const +numeric ex::max_coefficient() const { - GINAC_ASSERT(bp!=0); return bp->max_coefficient(); } /** Implementation ex::max_coefficient(). * @see heur_gcd */ -numeric basic::max_coefficient(void) const +numeric basic::max_coefficient() const { - return _num1; + return *_num1_p; } -numeric numeric::max_coefficient(void) const +numeric numeric::max_coefficient() const { return abs(*this); } -numeric add::max_coefficient(void) const +numeric add::max_coefficient() const { epvector::const_iterator it = seq.begin(); epvector::const_iterator itend = seq.end(); @@ -1244,7 +1178,7 @@ numeric add::max_coefficient(void) const return cur_max; } -numeric mul::max_coefficient(void) const +numeric mul::max_coefficient() const { #ifdef DO_GINAC_ASSERT epvector::const_iterator it = seq.begin(); @@ -1313,7 +1247,7 @@ 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; @@ -1346,7 +1280,6 @@ class gcdheu_failed {}; * @exception gcdheu_failed() */ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var) { -//std::clog << "heur_gcd(" << a << "," << b << ")\n"; #if STATISTICS heur_gcd_called++; #endif @@ -1356,7 +1289,7 @@ 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; @@ -1366,7 +1299,7 @@ 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()); @@ -1380,21 +1313,20 @@ 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++) { if (xi.int_length() * maxdeg > 100000) { -//std::clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << std::endl; throw gcdheu_failed(); } // Apply evaluation homomorphism and calculate GCD ex cp, cq; - ex gamma = heur_gcd(p.subs(x == xi), q.subs(x == xi), &cp, &cq, var+1).expand(); - if (!is_ex_exactly_of_type(gamma, fail)) { + ex gamma = 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 ex g = interpolate(gamma, xi, x, maxdeg); @@ -1406,40 +1338,8 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const ex dummy; if (divide_in_z(p, g, ca ? *ca : dummy, var) && divide_in_z(q, g, cb ? *cb : dummy, var)) { g *= gc; - ex lc = g.lcoeff(x); - if (is_ex_exactly_of_type(lc, numeric) && ex_to(lc).is_negative()) - return -g; - else - return g; + return g; } -#if 0 - cp = interpolate(cp, xi, x); - if (divide_in_z(cp, p, g, var)) { - if (divide_in_z(g, q, cb ? *cb : dummy, var)) { - g *= gc; - if (ca) - *ca = cp; - ex lc = g.lcoeff(x); - if (is_ex_exactly_of_type(lc, numeric) && ex_to(lc).is_negative()) - return -g; - else - return g; - } - } - cq = interpolate(cq, xi, x); - if (divide_in_z(cq, q, g, var)) { - if (divide_in_z(g, p, ca ? *ca : dummy, var)) { - g *= gc; - if (cb) - *cb = cq; - ex lc = g.lcoeff(x); - if (is_ex_exactly_of_type(lc, numeric) && ex_to(lc).is_negative()) - return -g; - else - return g; - } - } -#endif } // Next evaluation point @@ -1450,22 +1350,24 @@ 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 */ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args) { -//std::clog << "gcd(" << a << "," << b << ")\n"; #if STATISTICS gcd_called++; #endif // GCD of numerics -> CLN - if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) { + 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()) { @@ -1489,15 +1391,15 @@ 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(); + size_t num = a.nops(); exvector g; g.reserve(num); exvector acc_ca; acc_ca.reserve(num); ex part_b = b; - for (unsigned i=0; isetflag(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(); + size_t num = b.nops(); exvector g; g.reserve(num); exvector acc_cb; acc_cb.reserve(num); ex part_a = a; - for (unsigned i=0; i(a)) { ex p = a.op(0); - if (is_ex_exactly_of_type(b, power)) { - if (p.is_equal(b.op(0))) { + const ex& exp_a = a.op(1); + if (is_exactly_a(b)) { + ex pb = b.op(0); + const ex& exp_b = b.op(1); + if (p.is_equal(pb)) { // a = p^n, b = p^m, gcd = p^min(n, m) - ex exp_a = a.op(1), exp_b = b.op(1); if (exp_a < exp_b) { if (ca) *ca = _ex1; @@ -1550,7 +1454,32 @@ 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 @@ -1559,9 +1488,25 @@ 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); } - } - } else if (is_ex_exactly_of_type(b, power)) { + } // is_exactly_a(b) + + } else if (is_exactly_a(b)) { ex p = b.op(0); if (p.is_equal(a)) { // a = p, b = p^n, gcd = p @@ -1571,6 +1516,23 @@ 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 @@ -1607,13 +1569,33 @@ 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; @@ -1621,46 +1603,37 @@ factored_b: int min_ldeg = std::min(ldeg_a,ldeg_b); if (min_ldeg > 0) { ex common = power(x, min_ldeg); -//std::clog << "trivial common factor " << common << std::endl; return gcd((aex / common).expand(), (bex / common).expand(), ca, cb, false) * common; } // Try to eliminate variables if (var->deg_a == 0) { -//std::clog << "eliminating variable " << x << " from b" << std::endl; - ex c = bex.content(x); - ex g = gcd(aex, c, ca, cb, false); + ex bex_u, bex_c, bex_p; + bex.unitcontprim(x, bex_u, bex_c, bex_p); + ex g = gcd(aex, bex_c, ca, cb, false); if (cb) - *cb *= bex.unit(x) * bex.primpart(x, c); + *cb *= bex_u * bex_p; return g; } else if (var->deg_b == 0) { -//std::clog << "eliminating variable " << x << " from a" << std::endl; - ex c = aex.content(x); - ex g = gcd(c, bex, ca, cb, false); + ex aex_u, aex_c, aex_p; + aex.unitcontprim(x, aex_u, aex_c, aex_p); + ex g = gcd(aex_c, bex, ca, cb, false); if (ca) - *ca *= aex.unit(x) * aex.primpart(x, c); + *ca *= aex_u * aex_p; return g; } - ex g; -#if 1 // Try heuristic algorithm first, fall back to PRS if that failed + ex g; try { g = heur_gcd(aex, bex, ca, cb, var); } catch (gcdheu_failed) { g = fail(); } - if (is_ex_exactly_of_type(g, fail)) { -//std::clog << "heuristics failed" << std::endl; + if (is_exactly_a(g)) { #if STATISTICS heur_gcd_failed++; #endif -#endif -// g = heur_gcd(aex, bex, ca, cb, var); -// g = eu_gcd(aex, bex, &x); -// g = euprem_gcd(aex, bex, &x); -// g = peu_gcd(aex, bex, &x); -// g = red_gcd(aex, bex, &x); g = sr_gcd(aex, bex, var); if (g.is_equal(_ex1)) { // Keep cofactors factored if possible @@ -1674,7 +1647,6 @@ factored_b: if (cb) divide(bex, g, *cb, false); } -#if 1 } else { if (g.is_equal(_ex1)) { // Keep cofactors factored if possible @@ -1684,7 +1656,7 @@ factored_b: *cb = b; } } -#endif + return g; } @@ -1698,7 +1670,7 @@ 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")); @@ -1714,7 +1686,7 @@ 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. @@ -1741,14 +1713,45 @@ 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 polynomial a in square-free factored form. */ + * @param l lst of variables to factor in, may be left empty for autodetection + * @return a square-free factorization of \p a. + * + * \note + * A polynomial \f$p(X) \in C[X]\f$ is said square-free + * if, whenever any two polynomials \f$q(X)\f$ and \f$r(X)\f$ + * are such that + * \f[ + * p(X) = q(X)^2 r(X), + * \f] + * we have \f$q(X) \in C\f$. + * This means that \f$p(X)\f$ has no repeated factors, apart + * eventually from constants. + * Given a polynomial \f$p(X) \in C[X]\f$, we say that the + * decomposition + * \f[ + * p(X) = b \cdot p_1(X)^{a_1} \cdot p_2(X)^{a_2} \cdots p_r(X)^{a_r} + * \f] + * is a square-free factorization of \f$p(X)\f$ if the + * following conditions hold: + * -# \f$b \in C\f$ and \f$b \neq 0\f$; + * -# \f$a_i\f$ is a positive integer for \f$i = 1, \ldots, r\f$; + * -# the degree of the polynomial \f$p_i\f$ is strictly positive + * for \f$i = 1, \ldots, r\f$; + * -# the polynomial \f$\Pi_{i=1}^r p_i(X)\f$ is square-free. + * + * Square-free factorizations need not be unique. For example, if + * \f$a_i\f$ is even, we could change the polynomial \f$p_i(X)\f$ + * into \f$-p_i(X)\f$. + * Observe also that the factors \f$p_i(X)\f$ need not be irreducible + * polynomials. + */ ex sqrfree(const ex &a, const lst &l) { - if (is_a(a) || // algorithm does not trap a==0 + if (is_exactly_a(a) || // algorithm does not trap a==0 is_a(a)) // shortcut return a; @@ -1761,7 +1764,7 @@ 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 { @@ -1769,7 +1772,7 @@ ex sqrfree(const ex &a, const lst &l) } // Find the symbol to factor in at this stage - if (!is_ex_of_type(args.op(0), symbol)) + if (!is_a(args.op(0))) throw (std::runtime_error("sqrfree(): invalid factorization variable")); const symbol &x = ex_to(args.op(0)); @@ -1778,7 +1781,7 @@ 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; @@ -1812,6 +1815,7 @@ ex sqrfree(const ex &a, const lst &l) return result * lcm.inverse(); } + /** Compute square-free partial fraction decomposition of rational function * a(x). * @@ -1833,15 +1837,15 @@ 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; - unsigned num_yun = yun.size(); + size_t num_yun = yun.size(); exvector factor; factor.reserve(num_yun); exvector cofac; cofac.reserve(num_yun); - for (unsigned i=0; isecond; // Otherwise create new symbol and add to list, taking care that the - // replacement expression doesn't contain symbols from the sym_lst + // replacement expression doesn't itself contain symbols from repl, // because subs() is not recursive - symbol s; - ex es(s); - ex e_replaced = e.subs(sym_lst, repl_lst); - sym_lst.append(es); - repl_lst.append(e_replaced); + ex es = (new symbol)->setflag(status_flags::dynallocated); + ex e_replaced = e.subs(repl, subs_options::no_pattern); + repl.insert(std::make_pair(es, e_replaced)); + rev_lookup.insert(std::make_pair(e_replaced, es)); return es; } /** Create a symbol for replacing the expression "e" (or return a previously - * assigned symbol). An expression of the form "symbol == expression" is added - * to repl_lst and the symbol is returned. - * @see basic::to_rational */ -static ex replace_with_symbol(const ex &e, lst &repl_lst) -{ - // Expression already in repl_lst? Then return the assigned symbol - for (unsigned i=0; isecond.is_equal(e)) + return it->first; // Otherwise create new symbol and add to list, taking care that the - // replacement expression doesn't contain symbols from the sym_lst + // replacement expression doesn't itself contain symbols from repl, // because subs() is not recursive - symbol s; - ex es(s); - ex e_replaced = e.subs(repl_lst); - repl_lst.append(es == e_replaced); + ex es = (new symbol)->setflag(status_flags::dynallocated); + ex e_replaced = e.subs(repl, subs_options::no_pattern); + repl.insert(std::make_pair(es, e_replaced)); return es; } @@ -1949,18 +1952,18 @@ 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); } } } @@ -1968,7 +1971,7 @@ 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); } @@ -1978,19 +1981,19 @@ 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()) @@ -2006,7 +2009,7 @@ 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; @@ -2037,13 +2040,20 @@ static ex frac_cancel(const ex &n, const ex &d) // 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_exactly_a(den.unit(*x))); - if (ex_to(den.unit(*x)).is_negative()) { + if (is_exactly_a(den)) { + if (ex_to(den).is_negative()) { num *= _ex_1; den *= _ex_1; } + } else { + ex x; + if (get_first_symbol(den, x)) { + GINAC_ASSERT(is_exactly_a(den.unit(x))); + if (ex_to(den.unit(x)).is_negative()) { + num *= _ex_1; + den *= _ex_1; + } + } } // Return result as list @@ -2055,10 +2065,10 @@ 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")); @@ -2068,12 +2078,12 @@ 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()); @@ -2114,10 +2124,10 @@ 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")); @@ -2127,12 +2137,12 @@ 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)); @@ -2142,20 +2152,20 @@ 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)) { @@ -2176,34 +2186,32 @@ 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); } - - } else { // n_exponent not numeric - - // (a/b)^x -> {sym((a/b)^x, 1} - return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); } } + + // (a/b)^x -> {sym((a/b)^x, 1} + return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), repl, rev_lookup), _ex1))->setflag(status_flags::dynallocated); } /** Implementation of ex::normal() for pseries. It normalizes each coefficient * and replaces the series by a temporary symbol. * @see ex::normal */ -ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const +ex pseries::normal(exmap & repl, exmap & rev_lookup, int level) const { epvector newseq; epvector::const_iterator i = seq.begin(), end = seq.end(); @@ -2214,7 +2222,7 @@ 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); } @@ -2232,14 +2240,14 @@ 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); @@ -2251,18 +2259,18 @@ ex ex::normal(int level) const * * @see ex::normal * @return numerator */ -ex ex::numer(void) const +ex ex::numer() const { - lst sym_lst, repl_lst; + exmap repl, rev_lookup; - ex e = bp->normal(sym_lst, repl_lst, 0); + 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 @@ -2271,18 +2279,18 @@ ex ex::numer(void) const * * @see ex::normal * @return denominator */ -ex ex::denom(void) const +ex ex::denom() const { - lst sym_lst, repl_lst; + exmap repl, rev_lookup; - ex e = bp->normal(sym_lst, repl_lst, 0); + 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 @@ -2291,44 +2299,103 @@ ex ex::denom(void) const * * @see ex::normal * @return a list [numerator, denominator] */ -ex ex::numer_denom(void) const +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); } /** Rationalization of non-rational functions. - * This function converts a general expression to a rational polynomial + * This function converts a general expression to a rational function * by replacing all non-rational subexpressions (like non-rational numbers, * non-integer powers or functions like sin(), cos() etc.) to temporary * symbols. This makes it possible to use functions like gcd() and divide() * on non-rational functions by applying to_rational() on the arguments, * calling the desired function and re-substituting the temporary symbols * in the result. To make the last step possible, all temporary symbols and - * their associated expressions are collected in the 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 basic::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 +{ + // Convert lst to exmap + exmap m; + for (lst::const_iterator it = repl_lst.begin(); it != repl_lst.end(); ++it) + m.insert(std::make_pair(it->op(0), it->op(1))); + + ex ret = bp->to_rational(m); + + // Convert exmap back to lst + repl_lst.remove_all(); + for (exmap::const_iterator it = m.begin(); it != m.end(); ++it) + repl_lst.append(it->first == it->second); + + return ret; +} + +ex ex::to_polynomial(exmap & repl) const { - return replace_with_symbol(*this, 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(exmap & repl) const +{ + return replace_with_symbol(*this, repl); +} + +ex basic::to_polynomial(exmap & repl) const +{ + return replace_with_symbol(*this, repl); } /** Implementation of ex::to_rational() for symbols. This returns the * unmodified symbol. */ -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(exmap & repl) const { return *this; } @@ -2337,17 +2404,35 @@ 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. */ -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); + ex im_ex = im.is_rational() ? im : replace_with_symbol(im, repl); + return re_ex + im_ex * replace_with_symbol(I, repl); + } + return *this; +} + +/** Implementation of ex::to_polynomial() for a numeric. It splits complex + * numbers into re+I*im and replaces I and non-integer real numbers with a + * temporary symbol. */ +ex numeric::to_polynomial(exmap & repl) const +{ + if (is_real()) { + if (!is_integer()) + return replace_with_symbol(*this, repl); } else { // complex numeric re = real(); numeric im = imag(); - ex re_ex = re.is_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_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; } @@ -2355,26 +2440,65 @@ ex numeric::to_rational(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); +} + +/** Implementation of ex::to_polynomial() for powers. It replaces non-posint + * powers by temporary symbols. */ +ex power::to_polynomial(exmap & repl) const +{ + if (exponent.info(info_flags::posint)) + return power(basis.to_rational(repl), exponent); + else if (exponent.info(info_flags::negint)) + { + ex basis_pref = collect_common_factors(basis); + if (is_exactly_a(basis_pref) || is_exactly_a(basis_pref)) { + // (A*B)^n will be automagically transformed to A^n*B^n + ex t = power(basis_pref, exponent); + return t.to_polynomial(repl); + } + else + return power(replace_with_symbol(power(basis, _ex_1), repl), -exponent); + } else - return replace_with_symbol(*this, repl_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))); + ++i; + } + ex oc = overall_coeff.to_rational(repl); + if (oc.info(info_flags::numeric)) + return thisexpairseq(s, overall_coeff); + else + s.push_back(combine_ex_with_coeff_to_pair(oc, _ex1)); + return thisexpairseq(s, default_overall_coeff()); +} + +/** Implementation of ex::to_polynomial() for expairseqs. */ +ex expairseq::to_polynomial(exmap & repl) const { epvector s; s.reserve(seq.size()); epvector::const_iterator i = seq.begin(), end = seq.end(); while (i != end) { - s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_rational(repl_lst))); + s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_polynomial(repl))); ++i; } - ex oc = overall_coeff.to_rational(repl_lst); + ex oc = overall_coeff.to_polynomial(repl); if (oc.info(info_flags::numeric)) return thisexpairseq(s, overall_coeff); else @@ -2383,4 +2507,143 @@ ex expairseq::to_rational(lst &repl_lst) const } +/** Remove the common factor in the terms of a sum 'e' by calculating the GCD, + * and multiply it into the expression 'factor' (which needs to be initialized + * to 1, unless you're accumulating factors). */ +static ex find_common_factor(const ex & e, ex & factor, exmap & repl) +{ + if (is_exactly_a(e)) { + + size_t num = e.nops(); + exvector terms; terms.reserve(num); + ex gc; + + // Find the common GCD + for (size_t i=0; i(x) || is_exactly_a(x) || is_a(x)) { + ex f = 1; + x = find_common_factor(x, f, repl); + x *= f; + } + + if (i == 0) + gc = x; + else + gc = gcd(gc, x); + + terms.push_back(x); + } + + if (gc.is_equal(_ex1)) + return e; + + // The GCD is the factor we pull out + factor *= gc; + + // Now divide all terms by the GCD + for (size_t i=0; i(t)) { + for (size_t j=0; jsetflag(status_flags::dynallocated); + goto term_done; + } + } + } + + divide(t, gc, x); + t = x; +term_done: ; + } + return (new add(terms))->setflag(status_flags::dynallocated); + + } else if (is_exactly_a(e)) { + + size_t num = e.nops(); + exvector v; v.reserve(num); + + for (size_t i=0; isetflag(status_flags::dynallocated); + + } else if (is_exactly_a(e)) { + const ex e_exp(e.op(1)); + if (e_exp.info(info_flags::integer)) { + ex eb = e.op(0).to_polynomial(repl); + ex factor_local(_ex1); + ex pre_res = find_common_factor(eb, factor_local, repl); + factor *= power(factor_local, e_exp); + return power(pre_res, e_exp); + + } else + return e.to_polynomial(repl); + + } else + return e; +} + + +/** Collect common factors in sums. This converts expressions like + * 'a*(b*x+b*y)' to 'a*b*(x+y)'. */ +ex collect_common_factors(const ex & e) +{ + if (is_exactly_a(e) || is_exactly_a(e) || is_exactly_a(e)) { + + exmap repl; + ex factor = 1; + ex r = find_common_factor(e, factor, repl); + return factor.subs(repl, subs_options::no_pattern) * r.subs(repl, subs_options::no_pattern); + + } else + return e; +} + + +/** Resultant of two expressions e1,e2 with respect to symbol s. + * Method: Compute determinant of Sylvester matrix of e1,e2,s. */ +ex resultant(const ex & e1, const ex & e2, const ex & s) +{ + const ex ee1 = e1.expand(); + const ex ee2 = e2.expand(); + if (!ee1.info(info_flags::polynomial) || + !ee2.info(info_flags::polynomial)) + throw(std::runtime_error("resultant(): arguments must be polynomials")); + + const int h1 = ee1.degree(s); + const int l1 = ee1.ldegree(s); + const int h2 = ee2.degree(s); + const int l2 = ee2.ldegree(s); + + const int msize = h1 + h2; + matrix m(msize, msize); + + for (int l = h1; l >= l1; --l) { + const ex e = ee1.coeff(s, l); + for (int k = 0; k < h2; ++k) + m(k, k+h1-l) = e; + } + for (int l = h2; l >= l2; --l) { + const ex e = ee2.coeff(s, l); + for (int k = 0; k < h1; ++k) + m(k+h2, k+h2-l) = e; + } + + return m.determinant(); +} + + } // namespace GiNaC