X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fnormal.cpp;h=610c48d6a2653bd9945ce35862b5e59e7f2f66da;hp=18e24e9e6b99b227960622a2c9fd6993809d84bc;hb=dff5209ee7cc816583f6abe27a4e784e34f0a664;hpb=591b85b0697370f2f5f25a29a1e94ff831a02c12 diff --git a/ginac/normal.cpp b/ginac/normal.cpp index 18e24e9e..610c48d6 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) 1999-2002 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 @@ -39,6 +39,7 @@ #include "numeric.h" #include "power.h" #include "relational.h" +#include "matrix.h" #include "pseries.h" #include "symbol.h" #include "utils.h" @@ -73,10 +74,10 @@ static int heur_gcd_failed = 0; static struct _stat_print { _stat_print() {} ~_stat_print() { - cout << "gcd() called " << gcd_called << " times\n"; - cout << "sr_gcd() called " << sr_gcd_called << " times\n"; - cout << "heur_gcd() called " << heur_gcd_called << " times\n"; - cout << "heur_gcd() failed " << heur_gcd_failed << " times\n"; + std::cout << "gcd() called " << gcd_called << " times\n"; + std::cout << "sr_gcd() called " << sr_gcd_called << " times\n"; + std::cout << "heur_gcd() called " << heur_gcd_called << " times\n"; + std::cout << "heur_gcd() failed " << heur_gcd_failed << " times\n"; } } stat_print; #endif @@ -92,7 +93,7 @@ static struct _stat_print { static bool get_first_symbol(const ex &e, const symbol *&x) { if (is_ex_exactly_of_type(e, symbol)) { - x = static_cast(e.bp); + x = &ex_to(e); return true; } else if (is_ex_exactly_of_type(e, add) || is_ex_exactly_of_type(e, mul)) { for (unsigned 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) { - sym_desc_vec::iterator it = v.begin(), itend = v.end(); + sym_desc_vec::const_iterator it = v.begin(), itend = v.end(); while (it != itend) { if (it->sym->compare(*s) == 0) // If it's already in there, don't add it a second time return; - it++; + ++it; } sym_desc d; d.sym = s; @@ -169,7 +170,7 @@ static void add_symbol(const symbol *s, sym_desc_vec &v) static void collect_symbols(const ex &e, sym_desc_vec &v) { if (is_ex_exactly_of_type(e, symbol)) { - add_symbol(static_cast(e.bp), v); + 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; imax_lcnops = std::max(a.lcoeff(*(it->sym)).nops(), b.lcoeff(*(it->sym)).nops()); it->ldeg_a = a.ldegree(*(it->sym)); it->ldeg_b = b.ldegree(*(it->sym)); - it++; + ++it; } - sort(v.begin(), v.end()); + std::sort(v.begin(), v.end()); #if 0 std::clog << "Symbols:\n"; it = v.begin(); itend = v.end(); while (it != itend) { std::clog << " " << *it->sym << ": deg_a=" << it->deg_a << ", deg_b=" << it->deg_b << ", ldeg_a=" << it->ldeg_a << ", ldeg_b=" << it->ldeg_b << ", max_deg=" << it->max_deg << ", max_lcnops=" << it->max_lcnops << endl; std::clog << " lcoeff_a=" << a.lcoeff(*(it->sym)) << ", lcoeff_b=" << b.lcoeff(*(it->sym)) << endl; - it++; + ++it; } #endif } @@ -230,14 +231,14 @@ 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_ex_exactly_of_type(e, add)) { - numeric c = _num1(); + numeric c = _num1; for (unsigned i=0; isetflag(status_flags::dynallocated); } else if (is_ex_exactly_of_type(e, add)) { - ex c = _ex0(); - 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)) return e * lcm; @@ -305,7 +308,7 @@ numeric ex::integer_content(void) const numeric basic::integer_content(void) const { - return _num1(); + return _num1; } numeric numeric::integer_content(void) const @@ -317,14 +320,14 @@ numeric add::integer_content(void) const { epvector::const_iterator it = seq.begin(); epvector::const_iterator itend = seq.end(); - numeric c = _num0(); + numeric c = _num0; while (it != itend) { - GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric)); - GINAC_ASSERT(is_ex_exactly_of_type(it->coeff,numeric)); + GINAC_ASSERT(!is_exactly_a(it->rest)); + GINAC_ASSERT(is_exactly_a(it->coeff)); c = gcd(ex_to(it->coeff), c); it++; } - GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric)); + GINAC_ASSERT(is_exactly_a(overall_coeff)); c = gcd(ex_to(overall_coeff),c); return c; } @@ -335,11 +338,11 @@ numeric mul::integer_content(void) const epvector::const_iterator it = seq.begin(); epvector::const_iterator itend = seq.end(); while (it != itend) { - GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric)); + GINAC_ASSERT(!is_exactly_a(recombine_pair_to_ex(*it))); ++it; } #endif // def DO_GINAC_ASSERT - GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric)); + GINAC_ASSERT(is_exactly_a(overall_coeff)); return abs(ex_to(overall_coeff)); } @@ -365,13 +368,12 @@ ex quo(const ex &a, const ex &b, const symbol &x, bool check_args) return a / b; #if FAST_COMPARE if (a.is_equal(b)) - return _ex1(); + return _ex1; #endif if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))) throw(std::invalid_argument("quo: arguments must be polynomials over the rationals")); // Polynomial long division - ex q = _ex0(); ex r = a.expand(); if (r.is_zero()) return r; @@ -379,6 +381,7 @@ ex quo(const ex &a, const ex &b, const symbol &x, bool check_args) int rdeg = r.degree(x); ex blcoeff = b.expand().coeff(x, bdeg); bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric); + exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0)); while (rdeg >= bdeg) { ex term, rcoeff = r.coeff(x, rdeg); if (blcoeff_is_numeric) @@ -388,13 +391,13 @@ ex quo(const ex &a, const ex &b, const symbol &x, bool check_args) return (new fail())->setflag(status_flags::dynallocated); } term *= power(x, rdeg - bdeg); - q += term; + v.push_back(term); r -= (term * b).expand(); if (r.is_zero()) break; rdeg = r.degree(x); } - return q; + return (new add(v))->setflag(status_flags::dynallocated); } @@ -413,13 +416,13 @@ ex rem(const ex &a, const ex &b, const symbol &x, bool check_args) throw(std::overflow_error("rem: division by zero")); if (is_ex_exactly_of_type(a, numeric)) { if (is_ex_exactly_of_type(b, numeric)) - return _ex0(); + return _ex0; else - return b; + return a; } #if FAST_COMPARE if (a.is_equal(b)) - return _ex0(); + return _ex0; #endif if (check_args && (!a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial))) throw(std::invalid_argument("rem: arguments must be polynomials over the rationals")); @@ -450,6 +453,24 @@ ex rem(const ex &a, const ex &b, const symbol &x, bool check_args) } +/** Decompose rational function a(x)=N(x)/D(x) into P(x)+n(x)/D(x) + * with degree(n, x) < degree(D, x). + * + * @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 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)) + return a; + else + return q + rem(numer, denom, x) / denom; +} + + /** Pseudo-remainder of polynomials a(x) and b(x) in Z[x]. * * @param a first polynomial in x (dividend) @@ -464,7 +485,7 @@ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args) throw(std::overflow_error("prem: division by zero")); if (is_ex_exactly_of_type(a, numeric)) { if (is_ex_exactly_of_type(b, numeric)) - return _ex0(); + return _ex0; else return b; } @@ -480,18 +501,18 @@ ex prem(const ex &a, const ex &b, const symbol &x, bool check_args) if (bdeg <= rdeg) { blcoeff = eb.coeff(x, bdeg); if (bdeg == 0) - eb = _ex0(); + eb = _ex0; else eb -= blcoeff * power(x, bdeg); } else - blcoeff = _ex1(); + blcoeff = _ex1; int delta = rdeg - bdeg + 1, i = 0; while (rdeg >= bdeg && !r.is_zero()) { ex rlcoeff = r.coeff(x, rdeg); ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand(); if (rdeg == 0) - r = _ex0(); + r = _ex0; else r -= rlcoeff * power(x, rdeg); r = (blcoeff * r).expand() - term; @@ -510,14 +531,13 @@ 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 Z[x] */ - ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args) { if (b.is_zero()) throw(std::overflow_error("prem: division by zero")); if (is_ex_exactly_of_type(a, numeric)) { if (is_ex_exactly_of_type(b, numeric)) - return _ex0(); + return _ex0; else return b; } @@ -533,17 +553,17 @@ ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args) if (bdeg <= rdeg) { blcoeff = eb.coeff(x, bdeg); if (bdeg == 0) - eb = _ex0(); + eb = _ex0; else eb -= blcoeff * power(x, bdeg); } else - blcoeff = _ex1(); + blcoeff = _ex1; while (rdeg >= bdeg && !r.is_zero()) { ex rlcoeff = r.coeff(x, rdeg); ex term = (power(x, rdeg - bdeg) * eb * rlcoeff).expand(); if (rdeg == 0) - r = _ex0(); + r = _ex0; else r -= rlcoeff * power(x, rdeg); r = (blcoeff * r).expand() - term; @@ -561,14 +581,15 @@ ex sprem(const ex &a, const ex &b, const symbol &x, bool check_args) * @param check_args check whether a and b are polynomials with rational * coefficients (defaults to "true") * @return "true" when exact division succeeds (quotient returned in q), - * "false" otherwise */ + * "false" otherwise (q left untouched) */ bool divide(const ex &a, const ex &b, ex &q, bool check_args) { - q = _ex0(); if (b.is_zero()) throw(std::overflow_error("divide: division by zero")); - if (a.is_zero()) + if (a.is_zero()) { + q = _ex0; return true; + } if (is_ex_exactly_of_type(b, numeric)) { q = a / b; return true; @@ -576,7 +597,7 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args) return false; #if FAST_COMPARE if (a.is_equal(b)) { - q = _ex1(); + q = _ex1; return true; } #endif @@ -591,12 +612,15 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args) // Polynomial long division (recursive) ex r = a.expand(); - if (r.is_zero()) + if (r.is_zero()) { + q = _ex0; return true; + } int bdeg = b.degree(*x); int rdeg = r.degree(*x); ex blcoeff = b.expand().coeff(*x, bdeg); bool blcoeff_is_numeric = is_ex_exactly_of_type(blcoeff, numeric); + exvector v; v.reserve(std::max(rdeg - bdeg + 1, 0)); while (rdeg >= bdeg) { ex term, rcoeff = r.coeff(*x, rdeg); if (blcoeff_is_numeric) @@ -605,10 +629,12 @@ bool divide(const ex &a, const ex &b, ex &q, bool check_args) if (!divide(rcoeff, blcoeff, term, false)) return false; term *= power(*x, rdeg - bdeg); - q += term; + v.push_back(term); r -= (term * b).expand(); - if (r.is_zero()) + if (r.is_zero()) { + q = (new add(v))->setflag(status_flags::dynallocated); return true; + } rdeg = r.degree(*x); } return false; @@ -653,10 +679,10 @@ typedef std::map ex2_exbool_remember; * @see get_symbol_stats, heur_gcd */ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_iterator var) { - q = _ex0(); + q = _ex0; if (b.is_zero()) throw(std::overflow_error("divide_in_z: division by zero")); - if (b.is_equal(_ex1())) { + if (b.is_equal(_ex1)) { q = a; return true; } @@ -669,7 +695,7 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite } #if FAST_COMPARE if (a.is_equal(b)) { - q = _ex1(); + q = _ex1; return true; } #endif @@ -700,24 +726,24 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite // Compute values at evaluation points 0..adeg vector alpha; alpha.reserve(adeg + 1); exvector u; u.reserve(adeg + 1); - numeric point = _num0(); + numeric point = _num0; ex c; for (i=0; i<=adeg; i++) { ex bs = b.subs(*x == point); while (bs.is_zero()) { - point += _num1(); + point += _num1; bs = b.subs(*x == point); } if (!divide_in_z(a.subs(*x == point), bs, c, var+1)) return false; alpha.push_back(point); u.push_back(c); - point += _num1(); + point += _num1; } // Compute inverses vector rcp; rcp.reserve(adeg + 1); - rcp.push_back(_num0()); + rcp.push_back(_num0); for (k=1; k<=adeg; k++) { numeric product = alpha[k] - alpha[0]; for (i=1; i= bdeg) { ex term, rcoeff = r.coeff(*x, rdeg); if (!divide_in_z(rcoeff, blcoeff, term, var+1)) break; term = (term * power(*x, rdeg - bdeg)).expand(); - q += term; + v.push_back(term); r -= (term * eb).expand(); if (r.is_zero()) { + q = (new add(v))->setflag(status_flags::dynallocated); #if USE_REMEMBER dr_remember[ex2(a, b)] = exbool(q, true); #endif @@ -794,7 +822,7 @@ ex ex::unit(const symbol &x) const { ex c = expand().lcoeff(x); if (is_ex_exactly_of_type(c, numeric)) - return c < _ex0() ? _ex_1() : _ex1(); + return c < _ex0 ? _ex_1 : _ex1; else { const symbol *y; if (get_first_symbol(c, y)) @@ -815,12 +843,12 @@ ex ex::unit(const symbol &x) const ex ex::content(const symbol &x) const { if (is_zero()) - return _ex0(); + return _ex0; if (is_ex_exactly_of_type(*this, numeric)) return info(info_flags::negative) ? -*this : *this; ex e = expand(); if (e.is_zero()) - return _ex0(); + return _ex0; // First, try the integer content ex c = e.integer_content(); @@ -834,7 +862,7 @@ ex ex::content(const symbol &x) const int ldeg = e.ldegree(x); if (deg == ldeg) return e.lcoeff(x) / e.unit(x); - c = _ex0(); + c = _ex0; for (int i=ldeg; i<=deg; i++) c = gcd(e.coeff(x, i), c, NULL, NULL, false); return c; @@ -851,13 +879,13 @@ ex ex::content(const symbol &x) const ex ex::primpart(const symbol &x) const { if (is_zero()) - return _ex0(); + return _ex0; if (is_ex_exactly_of_type(*this, numeric)) - return _ex1(); + return _ex1; ex c = content(x); if (c.is_zero()) - return _ex0(); + return _ex0; ex u = unit(x); if (is_ex_exactly_of_type(c, numeric)) return *this / (c * u); @@ -876,11 +904,11 @@ ex ex::primpart(const symbol &x) const ex ex::primpart(const symbol &x, const ex &c) const { if (is_zero()) - return _ex0(); + return _ex0; if (c.is_zero()) - return _ex0(); + return _ex0; if (is_ex_exactly_of_type(*this, numeric)) - return _ex1(); + return _ex1; ex u = unit(x); if (is_ex_exactly_of_type(c, numeric)) @@ -1066,7 +1094,7 @@ static ex red_gcd(const ex &a, const ex &b, const symbol *x) d = d.primpart(*x, cont_d); // First element of divisor sequence - ex r, ri = _ex1(); + ex r, ri = _ex1; int delta = cdeg - ddeg; for (;;) { @@ -1140,7 +1168,7 @@ static ex sr_gcd(const ex &a, const ex &b, sym_desc_vec::const_iterator var) //std::clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n"; // First element of subresultant sequence - ex r = _ex0(), ri = _ex1(), psi = _ex1(); + ex r = _ex0, ri = _ex1, psi = _ex1; int delta = cdeg - ddeg; for (;;) { @@ -1191,7 +1219,7 @@ numeric ex::max_coefficient(void) const * @see heur_gcd */ numeric basic::max_coefficient(void) const { - return _num1(); + return _num1; } numeric numeric::max_coefficient(void) const @@ -1203,11 +1231,11 @@ numeric add::max_coefficient(void) const { epvector::const_iterator it = seq.begin(); epvector::const_iterator itend = seq.end(); - GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric)); + GINAC_ASSERT(is_exactly_a(overall_coeff)); numeric cur_max = abs(ex_to(overall_coeff)); while (it != itend) { numeric a; - GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric)); + GINAC_ASSERT(!is_exactly_a(it->rest)); a = abs(ex_to(it->coeff)); if (a > cur_max) cur_max = a; @@ -1222,28 +1250,21 @@ numeric mul::max_coefficient(void) const epvector::const_iterator it = seq.begin(); epvector::const_iterator itend = seq.end(); while (it != itend) { - GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric)); + GINAC_ASSERT(!is_exactly_a(recombine_pair_to_ex(*it))); it++; } #endif // def DO_GINAC_ASSERT - GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric)); + GINAC_ASSERT(is_exactly_a(overall_coeff)); return abs(ex_to(overall_coeff)); } -/** Apply symmetric modular homomorphism to a multivariate polynomial. - * This function is used internally by heur_gcd(). +/** Apply symmetric modular homomorphism to an expanded multivariate + * polynomial. This function is usually used internally by heur_gcd(). * - * @param e expanded multivariate polynomial * @param xi modulus * @return mapped polynomial * @see heur_gcd */ -ex ex::smod(const numeric &xi) const -{ - GINAC_ASSERT(bp!=0); - return bp->smod(xi); -} - ex basic::smod(const numeric &xi) const { return *this; @@ -1261,13 +1282,13 @@ ex add::smod(const numeric &xi) const epvector::const_iterator it = seq.begin(); epvector::const_iterator itend = seq.end(); while (it != itend) { - GINAC_ASSERT(!is_ex_exactly_of_type(it->rest,numeric)); + GINAC_ASSERT(!is_exactly_a(it->rest)); numeric coeff = GiNaC::smod(ex_to(it->coeff), xi); if (!coeff.is_zero()) newseq.push_back(expair(it->rest, coeff)); it++; } - GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric)); + GINAC_ASSERT(is_exactly_a(overall_coeff)); numeric coeff = GiNaC::smod(ex_to(overall_coeff), xi); return (new add(newseq,coeff))->setflag(status_flags::dynallocated); } @@ -1278,12 +1299,12 @@ ex mul::smod(const numeric &xi) const epvector::const_iterator it = seq.begin(); epvector::const_iterator itend = seq.end(); while (it != itend) { - GINAC_ASSERT(!is_ex_exactly_of_type(recombine_pair_to_ex(*it),numeric)); + GINAC_ASSERT(!is_exactly_a(recombine_pair_to_ex(*it))); it++; } #endif // def DO_GINAC_ASSERT mul * mulcopyp = new mul(*this); - GINAC_ASSERT(is_ex_exactly_of_type(overall_coeff,numeric)); + GINAC_ASSERT(is_exactly_a(overall_coeff)); mulcopyp->overall_coeff = GiNaC::smod(ex_to(overall_coeff),xi); mulcopyp->clearflag(status_flags::evaluated); mulcopyp->clearflag(status_flags::hash_calculated); @@ -1292,17 +1313,17 @@ ex mul::smod(const numeric &xi) const /** xi-adic polynomial interpolation */ -static ex interpolate(const ex &gamma, const numeric &xi, const symbol &x) +static ex interpolate(const ex &gamma, const numeric &xi, const symbol &x, int degree_hint = 1) { - ex g = _ex0(); + exvector g; g.reserve(degree_hint); ex e = gamma; numeric rxi = xi.inverse(); for (int i=0; !e.is_zero(); i++) { ex gi = e.smod(xi); - g += gi * power(x, i); + g.push_back(gi * power(x, i)); e = (e - gi) * rxi; } - return g; + return (new add(g))->setflag(status_flags::dynallocated); } /** Exception thrown by heur_gcd() to signal failure. */ @@ -1330,7 +1351,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const heur_gcd_called++; #endif - // Algorithms only works for non-vanishing input polynomials + // Algorithm only works for non-vanishing input polynomials if (a.is_zero() || b.is_zero()) return (new fail())->setflag(status_flags::dynallocated); @@ -1352,16 +1373,16 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const numeric rgc = gc.inverse(); ex p = a * rgc; ex q = b * rgc; - int maxdeg = std::max(p.degree(x),q.degree(x)); + int maxdeg = std::max(p.degree(x), q.degree(x)); // Find evaluation point numeric mp = p.max_coefficient(); numeric mq = q.max_coefficient(); numeric xi; if (mp > mq) - xi = mq * _num2() + _num2(); + xi = mq * _num2 + _num2; else - xi = mp * _num2() + _num2(); + xi = mp * _num2 + _num2; // 6 tries maximum for (int t=0; t<6; t++) { @@ -1376,7 +1397,7 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const if (!is_ex_exactly_of_type(gamma, fail)) { // Reconstruct polynomial from GCD of mapped polynomials - ex g = interpolate(gamma, xi, x); + ex g = interpolate(gamma, xi, x, maxdeg); // Remove integer content g /= g.integer_content(); @@ -1449,9 +1470,9 @@ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args) if (ca || cb) { if (g.is_zero()) { if (ca) - *ca = _ex0(); + *ca = _ex0; if (cb) - *cb = _ex0(); + *cb = _ex0; } else { if (ca) *ca = ex_to(a) / g; @@ -1472,38 +1493,40 @@ ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args) if (is_ex_exactly_of_type(b, mul) && b.nops() > a.nops()) goto factored_b; factored_a: - ex g = _ex1(); - ex acc_ca = _ex1(); + unsigned 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); if (cb) *cb = part_b; - return g; + return (new mul(g))->setflag(status_flags::dynallocated); } else if (is_ex_exactly_of_type(b, mul)) { if (is_ex_exactly_of_type(a, mul) && a.nops() > b.nops()) goto factored_a; factored_b: - ex g = _ex1(); - ex acc_cb = _ex1(); + unsigned num = b.nops(); + exvector g; g.reserve(num); + exvector acc_cb; acc_cb.reserve(num); ex part_a = a; - for (unsigned i=0; isetflag(status_flags::dynallocated); + return (new mul(g))->setflag(status_flags::dynallocated); } #if FAST_COMPARE @@ -1516,7 +1539,7 @@ factored_b: ex exp_a = a.op(1), exp_b = b.op(1); if (exp_a < exp_b) { if (ca) - *ca = _ex1(); + *ca = _ex1; if (cb) *cb = power(p, exp_b - exp_a); return power(p, exp_a); @@ -1524,7 +1547,7 @@ factored_b: if (ca) *ca = power(p, exp_a - exp_b); if (cb) - *cb = _ex1(); + *cb = _ex1; return power(p, exp_b); } } @@ -1534,7 +1557,7 @@ factored_b: if (ca) *ca = power(p, a.op(1) - 1); if (cb) - *cb = _ex1(); + *cb = _ex1; return p; } } @@ -1543,7 +1566,7 @@ factored_b: if (p.is_equal(a)) { // a = p, b = p^n, gcd = p if (ca) - *ca = _ex1(); + *ca = _ex1; if (cb) *cb = power(p, b.op(1) - 1); return p; @@ -1555,31 +1578,31 @@ factored_b: ex aex = a.expand(), bex = b.expand(); if (aex.is_zero()) { if (ca) - *ca = _ex0(); + *ca = _ex0; if (cb) - *cb = _ex1(); + *cb = _ex1; return b; } if (bex.is_zero()) { if (ca) - *ca = _ex1(); + *ca = _ex1; if (cb) - *cb = _ex0(); + *cb = _ex0; return a; } - if (aex.is_equal(_ex1()) || bex.is_equal(_ex1())) { + if (aex.is_equal(_ex1) || bex.is_equal(_ex1)) { if (ca) *ca = a; if (cb) *cb = b; - return _ex1(); + return _ex1; } #if FAST_COMPARE if (a.is_equal(b)) { if (ca) - *ca = _ex1(); + *ca = _ex1; if (cb) - *cb = _ex1(); + *cb = _ex1; return a; } #endif @@ -1639,7 +1662,7 @@ factored_b: // g = peu_gcd(aex, bex, &x); // g = red_gcd(aex, bex, &x); g = sr_gcd(aex, bex, var); - if (g.is_equal(_ex1())) { + if (g.is_equal(_ex1)) { // Keep cofactors factored if possible if (ca) *ca = a; @@ -1653,7 +1676,7 @@ factored_b: } #if 1 } else { - if (g.is_equal(_ex1())) { + if (g.is_equal(_ex1)) { // Keep cofactors factored if possible if (ca) *ca = a; @@ -1703,7 +1726,7 @@ static exvector sqrfree_yun(const ex &a, const symbol &x) ex w = a; ex z = w.diff(x); ex g = gcd(w, z); - if (g.is_equal(_ex1())) { + if (g.is_equal(_ex1)) { res.push_back(a); return res; } @@ -1717,58 +1740,177 @@ static exvector sqrfree_yun(const ex &a, const symbol &x) } while (!z.is_zero()); return res; } -/** Compute square-free factorization of multivariate polynomial in Q[X]. + +/** Compute a square-free factorization of a multivariate polynomial in Q[X]. * * @param a multivariate polynomial over Q[X] * @param x lst of variables to factor in, may be left empty for autodetection - * @return polynomail a in square-free factored form. */ + * @return a square-free factorization of \p a. + * + * \note + * A polynomial \f$p(X) \in C[X]\f$ is said square-free + * if, whenever any two polynomials \f$q(X)\f$ and \f$r(X)\f$ + * are such that + * \f[ + * p(X) = q(X)^2 r(X), + * \f] + * we have \f$q(X) \in C\f$. + * This means that \f$p(X)\f$ has no repeated factors, apart + * eventually from constants. + * Given a polynomial \f$p(X) \in C[X]\f$, we say that the + * decomposition + * \f[ + * p(X) = b \cdot p_1(X)^{a_1} \cdot p_2(X)^{a_2} \cdots p_r(X)^{a_r} + * \f] + * is a square-free factorization of \f$p(X)\f$ if the + * following conditions hold: + * -# \f$b \in C\f$ and \f$b \neq 0\f$; + * -# \f$a_i\f$ is a positive integer for \f$i = 1, \ldots, r\f$; + * -# the degree of the polynomial \f$p_i\f$ is strictly positive + * for \f$i = 1, \ldots, r\f$; + * -# the polynomial \f$\Pi_{i=1}^r p_i(X)\f$ is square-free. + * + * Square-free factorizations need not be unique. For example, if + * \f$a_i\f$ is even, we could change the polynomial \f$p_i(X)\f$ + * into \f$-p_i(X)\f$. + * Observe also that the factors \f$p_i(X)\f$ need not be irreducible + * polynomials. + */ ex sqrfree(const ex &a, const lst &l) { - if (is_ex_of_type(a,numeric) || // algorithm does not trap a==0 - is_ex_of_type(a,symbol)) // shortcut + if (is_a(a) || // algorithm does not trap a==0 + is_a(a)) // shortcut return a; + // If no lst of variables to factorize in was specified we have to // invent one now. Maybe one can optimize here by reversing the order // or so, I don't know. lst args; if (l.nops()==0) { sym_desc_vec sdv; - get_symbol_stats(a, _ex0(), sdv); - for (sym_desc_vec::iterator it=sdv.begin(); it!=sdv.end(); ++it) + get_symbol_stats(a, _ex0, sdv); + sym_desc_vec::const_iterator it = sdv.begin(), itend = sdv.end(); + while (it != itend) { args.append(*it->sym); + ++it; + } } else { args = l; } + // Find the symbol to factor in at this stage if (!is_ex_of_type(args.op(0), symbol)) throw (std::runtime_error("sqrfree(): invalid factorization variable")); - const symbol x = ex_to(args.op(0)); + const symbol &x = ex_to(args.op(0)); + // convert the argument from something in Q[X] to something in Z[X] - numeric lcm = lcm_of_coefficients_denominators(a); - ex tmp = multiply_lcm(a,lcm); + const numeric lcm = lcm_of_coefficients_denominators(a); + const ex tmp = multiply_lcm(a,lcm); + // find the factors exvector factors = sqrfree_yun(tmp,x); + // construct the next list of symbols with the first element popped - lst newargs; - for (int i=1; i0) { - for (exvector::iterator i=factors.begin(); i!=factors.end(); ++i) + exvector::iterator i = factors.begin(); + while (i != factors.end()) { *i = sqrfree(*i, newargs); + ++i; + } } + // Done with recursion, now construct the final result - ex result = _ex1(); - exvector::iterator it = factors.begin(); - for (int p = 1; it!=factors.end(); ++it, ++p) + ex result = _ex1; + exvector::const_iterator it = factors.begin(), itend = factors.end(); + for (int p = 1; it!=itend; ++it, ++p) result *= power(*it, p); - // Yun's algorithm does not account for constant factors. (For - // univariate polynomials it works only in the monic case.) We can - // correct this by inserting what has been lost back into the result: - result = result * quo(tmp, result, x); + + // Yun's algorithm does not account for constant factors. (For univariate + // polynomials it works only in the monic case.) We can correct this by + // inserting what has been lost back into the result. For completeness + // we'll also have to recurse down that factor in the remaining variables. + if (newargs.nops()>0) + result *= sqrfree(quo(tmp, result, x), newargs); + else + result *= quo(tmp, result, x); + + // Put in the reational overall factor again and return return result * lcm.inverse(); } +/** Compute square-free partial fraction decomposition of rational function + * a(x). + * + * @param a rational function over Z[x], treated as univariate polynomial + * in x + * @param x variable to factor in + * @return decomposed rational function */ +ex sqrfree_parfrac(const ex & a, const symbol & x) +{ + // Find numerator and denominator + ex nd = numer_denom(a); + ex numer = nd.op(0), denom = nd.op(1); +//clog << "numer = " << numer << ", denom = " << denom << endl; + + // Convert N(x)/D(x) -> Q(x) + R(x)/D(x), so degree(R) < degree(D) + ex red_poly = quo(numer, denom, x), red_numer = rem(numer, denom, x).expand(); +//clog << "red_poly = " << red_poly << ", red_numer = " << red_numer << endl; + + // Factorize denominator and compute cofactors + exvector yun = sqrfree_yun(denom, x); +//clog << "yun factors: " << exprseq(yun) << endl; + unsigned num_yun = yun.size(); + exvector factor; factor.reserve(num_yun); + exvector cofac; cofac.reserve(num_yun); + for (unsigned i=0; isetflag(status_flags::dynallocated); + if (nops() == 0) + return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); + else { + if (level == 1) + return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); + 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); + } + } } @@ -1838,7 +2000,7 @@ ex basic::normal(lst &sym_lst, lst &repl_lst, int level) const * @see ex::normal */ ex symbol::normal(lst &sym_lst, lst &repl_lst, int level) const { - return (new lst(*this, _ex1()))->setflag(status_flags::dynallocated); + return (new lst(*this, _ex1))->setflag(status_flags::dynallocated); } @@ -1874,17 +2036,17 @@ static ex frac_cancel(const ex &n, const ex &d) { ex num = n; ex den = d; - numeric pre_factor = _num1(); + numeric pre_factor = _num1; //std::clog << "frac_cancel num = " << num << ", den = " << den << std::endl; // Handle trivial case where denominator is 1 - if (den.is_equal(_ex1())) + if (den.is_equal(_ex1)) return (new lst(num, den))->setflag(status_flags::dynallocated); // Handle special cases where numerator or denominator is 0 if (num.is_zero()) - return (new lst(num, _ex1()))->setflag(status_flags::dynallocated); + return (new lst(num, _ex1))->setflag(status_flags::dynallocated); if (den.expand().is_zero()) throw(std::overflow_error("frac_cancel: division by zero in frac_cancel")); @@ -1898,7 +2060,7 @@ static ex frac_cancel(const ex &n, const ex &d) // Cancel GCD from numerator and denominator ex cnum, cden; - if (gcd(num, den, &cnum, &cden, false) != _ex1()) { + if (gcd(num, den, &cnum, &cden, false) != _ex1) { num = cnum; den = cden; } @@ -1907,10 +2069,10 @@ static ex frac_cancel(const ex &n, const ex &d) // as defined by get_first_symbol() is made positive) const symbol *x; if (get_first_symbol(den, x)) { - GINAC_ASSERT(is_ex_exactly_of_type(den.unit(*x),numeric)); + GINAC_ASSERT(is_exactly_a(den.unit(*x))); if (ex_to(den.unit(*x)).is_negative()) { - num *= _ex_1(); - den *= _ex_1(); + num *= _ex_1; + den *= _ex_1; } } @@ -1926,7 +2088,7 @@ static ex frac_cancel(const ex &n, const ex &d) ex add::normal(lst &sym_lst, lst &repl_lst, int level) const { if (level == 1) - return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); else if (level == -max_recursion_level) throw(std::runtime_error("max recursion level reached")); @@ -1936,12 +2098,12 @@ ex add::normal(lst &sym_lst, lst &repl_lst, int level) const dens.reserve(seq.size()+1); epvector::const_iterator it = seq.begin(), itend = seq.end(); while (it != itend) { - ex n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1); + ex n = ex_to(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, level-1); nums.push_back(n.op(0)); dens.push_back(n.op(1)); it++; } - ex n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1); + ex n = ex_to(overall_coeff).normal(sym_lst, repl_lst, level-1); nums.push_back(n.op(0)); dens.push_back(n.op(1)); GINAC_ASSERT(nums.size() == dens.size()); @@ -1985,27 +2147,28 @@ ex add::normal(lst &sym_lst, lst &repl_lst, int level) const ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const { if (level == 1) - return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); else if (level == -max_recursion_level) throw(std::runtime_error("max recursion level reached")); // Normalize children, separate into numerator and denominator - ex num = _ex1(); - ex den = _ex1(); + exvector num; num.reserve(seq.size()); + exvector den; den.reserve(seq.size()); ex n; epvector::const_iterator it = seq.begin(), itend = seq.end(); while (it != itend) { - n = recombine_pair_to_ex(*it).bp->normal(sym_lst, repl_lst, level-1); - num *= n.op(0); - den *= n.op(1); + n = ex_to(recombine_pair_to_ex(*it)).normal(sym_lst, repl_lst, level-1); + num.push_back(n.op(0)); + den.push_back(n.op(1)); it++; } - n = overall_coeff.bp->normal(sym_lst, repl_lst, level-1); - num *= n.op(0); - den *= n.op(1); + n = ex_to(overall_coeff).normal(sym_lst, repl_lst, level-1); + num.push_back(n.op(0)); + den.push_back(n.op(1)); // Perform fraction cancellation - return frac_cancel(num, den); + return frac_cancel((new mul(num))->setflag(status_flags::dynallocated), + (new mul(den))->setflag(status_flags::dynallocated)); } @@ -2016,13 +2179,13 @@ ex mul::normal(lst &sym_lst, lst &repl_lst, int level) const ex power::normal(lst &sym_lst, lst &repl_lst, int level) const { if (level == 1) - return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + return (new lst(replace_with_symbol(*this, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); else if (level == -max_recursion_level) throw(std::runtime_error("max recursion level reached")); // Normalize basis and exponent (exponent gets reassembled) - ex n_basis = basis.bp->normal(sym_lst, repl_lst, level-1); - ex n_exponent = exponent.bp->normal(sym_lst, repl_lst, level-1); + ex n_basis = ex_to(basis).normal(sym_lst, repl_lst, level-1); + ex n_exponent = ex_to(exponent).normal(sym_lst, repl_lst, level-1); n_exponent = n_exponent.op(0) / n_exponent.op(1); if (n_exponent.info(info_flags::integer)) { @@ -2043,25 +2206,25 @@ ex power::normal(lst &sym_lst, lst &repl_lst, int level) const if (n_exponent.info(info_flags::positive)) { // (a/b)^x -> {sym((a/b)^x), 1} - return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); } else if (n_exponent.info(info_flags::negative)) { - if (n_basis.op(1).is_equal(_ex1())) { + if (n_basis.op(1).is_equal(_ex1)) { // a^-x -> {1, sym(a^x)} - return (new lst(_ex1(), replace_with_symbol(power(n_basis.op(0), -n_exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated); + return (new lst(_ex1, replace_with_symbol(power(n_basis.op(0), -n_exponent), sym_lst, repl_lst)))->setflag(status_flags::dynallocated); } else { // (a/b)^-x -> {sym((b/a)^x), 1} - return (new lst(replace_with_symbol(power(n_basis.op(1) / n_basis.op(0), -n_exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + return (new lst(replace_with_symbol(power(n_basis.op(1) / n_basis.op(0), -n_exponent), sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); } } else { // n_exponent not numeric // (a/b)^x -> {sym((a/b)^x, 1} - return (new lst(replace_with_symbol(power(n_basis.op(0) / n_basis.op(1), n_exponent), sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); + 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); } } } @@ -2073,21 +2236,15 @@ ex power::normal(lst &sym_lst, lst &repl_lst, int level) const ex pseries::normal(lst &sym_lst, lst &repl_lst, int level) const { epvector newseq; - for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) { + epvector::const_iterator i = seq.begin(), end = seq.end(); + while (i != end) { ex restexp = i->rest.normal(); if (!restexp.is_zero()) newseq.push_back(expair(restexp, i->coeff)); + ++i; } ex n = pseries(relational(var,point), newseq); - return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1()))->setflag(status_flags::dynallocated); -} - - -/** Implementation of ex::normal() for relationals. It normalizes both sides. - * @see ex::normal */ -ex relational::normal(lst &sym_lst, lst &repl_lst, int level) const -{ - return (new lst(relational(lh.normal(), rh.normal(), o), _ex1()))->setflag(status_flags::dynallocated); + return (new lst(replace_with_symbol(n, sym_lst, repl_lst), _ex1))->setflag(status_flags::dynallocated); } @@ -2108,7 +2265,7 @@ ex ex::normal(int level) const lst sym_lst, repl_lst; ex e = bp->normal(sym_lst, repl_lst, level); - GINAC_ASSERT(is_ex_of_type(e, lst)); + GINAC_ASSERT(is_a(e)); // Re-insert replaced symbols if (sym_lst.nops() > 0) @@ -2129,7 +2286,7 @@ ex ex::numer(void) const lst sym_lst, repl_lst; ex e = bp->normal(sym_lst, repl_lst, 0); - GINAC_ASSERT(is_ex_of_type(e, lst)); + GINAC_ASSERT(is_a(e)); // Re-insert replaced symbols if (sym_lst.nops() > 0) @@ -2149,7 +2306,7 @@ ex ex::denom(void) const lst sym_lst, repl_lst; ex e = bp->normal(sym_lst, repl_lst, 0); - GINAC_ASSERT(is_ex_of_type(e, lst)); + GINAC_ASSERT(is_a(e)); // Re-insert replaced symbols if (sym_lst.nops() > 0) @@ -2169,7 +2326,7 @@ ex ex::numer_denom(void) const lst sym_lst, repl_lst; ex e = bp->normal(sym_lst, repl_lst, 0); - GINAC_ASSERT(is_ex_of_type(e, lst)); + GINAC_ASSERT(is_a(e)); // Re-insert replaced symbols if (sym_lst.nops() > 0) @@ -2179,9 +2336,20 @@ ex ex::numer_denom(void) const } -/** Default implementation of ex::to_rational(). It replaces the object with a - * temporary symbol. - * @see ex::to_rational */ +/** Rationalization of non-rational functions. + * This function converts a general expression to a rational polynomial + * by replacing all non-rational subexpressions (like non-rational numbers, + * non-integer powers or functions like sin(), cos() etc.) to temporary + * symbols. This makes it possible to use functions like gcd() and divide() + * on non-rational functions by applying to_rational() on the arguments, + * calling the desired function and re-substituting the temporary symbols + * in the result. To make the last step possible, all temporary symbols and + * their associated expressions are collected in the list specified by the + * repl_lst parameter in the form {symbol == expression}, ready to be passed + * as an argument to ex::subs(). + * + * @param repl_lst collects a list of all temporary symbols and their replacements + * @return rationalized expression */ ex basic::to_rational(lst &repl_lst) const { return replace_with_symbol(*this, repl_lst); @@ -2189,8 +2357,7 @@ ex basic::to_rational(lst &repl_lst) const /** Implementation of ex::to_rational() for symbols. This returns the - * unmodified symbol. - * @see ex::to_rational */ + * unmodified symbol. */ ex symbol::to_rational(lst &repl_lst) const { return *this; @@ -2199,8 +2366,7 @@ ex symbol::to_rational(lst &repl_lst) const /** Implementation of ex::to_rational() for a numeric. It splits complex * numbers into re+I*im and replaces I and non-rational real numbers with a - * temporary symbol. - * @see ex::to_rational */ + * temporary symbol. */ ex numeric::to_rational(lst &repl_lst) const { if (is_real()) { @@ -2218,8 +2384,7 @@ ex numeric::to_rational(lst &repl_lst) const /** Implementation of ex::to_rational() for powers. It replaces non-integer - * powers by temporary symbols. - * @see ex::to_rational */ + * powers by temporary symbols. */ ex power::to_rational(lst &repl_lst) const { if (exponent.info(info_flags::integer)) @@ -2229,42 +2394,23 @@ ex power::to_rational(lst &repl_lst) const } -/** Implementation of ex::to_rational() for expairseqs. - * @see ex::to_rational */ +/** Implementation of ex::to_rational() for expairseqs. */ ex expairseq::to_rational(lst &repl_lst) const { epvector s; s.reserve(seq.size()); - for (epvector::const_iterator it=seq.begin(); it!=seq.end(); ++it) { - s.push_back(split_ex_to_pair(recombine_pair_to_ex(*it).to_rational(repl_lst))); - // s.push_back(combine_ex_with_coeff_to_pair((*it).rest.to_rational(repl_lst), + epvector::const_iterator i = seq.begin(), end = seq.end(); + while (i != end) { + s.push_back(split_ex_to_pair(recombine_pair_to_ex(*i).to_rational(repl_lst))); + ++i; } ex oc = overall_coeff.to_rational(repl_lst); if (oc.info(info_flags::numeric)) return thisexpairseq(s, overall_coeff); - else s.push_back(combine_ex_with_coeff_to_pair(oc,_ex1())); + else + s.push_back(combine_ex_with_coeff_to_pair(oc, _ex1)); return thisexpairseq(s, default_overall_coeff()); } -/** Rationalization of non-rational functions. - * This function converts a general expression to a rational polynomial - * by replacing all non-rational subexpressions (like non-rational numbers, - * non-integer powers or functions like sin(), cos() etc.) to temporary - * symbols. This makes it possible to use functions like gcd() and divide() - * on non-rational functions by applying to_rational() on the arguments, - * calling the desired function and re-substituting the temporary symbols - * in the result. To make the last step possible, all temporary symbols and - * their associated expressions are collected in the list specified by the - * repl_lst parameter in the form {symbol == expression}, ready to be passed - * as an argument to ex::subs(). - * - * @param repl_lst collects a list of all temporary symbols and their replacements - * @return rationalized expression */ -ex ex::to_rational(lst &repl_lst) const -{ - return bp->to_rational(repl_lst); -} - - } // namespace GiNaC