From 79eebbd4ebb7b5f7c6517298c728a16a68282ace Mon Sep 17 00:00:00 2001 From: Christian Bauer Date: Tue, 28 Mar 2000 18:58:57 +0000 Subject: [PATCH] - remembering in divide_in_z() turned off (will eventually have to be replaced by the new remembering mechanism in remember.cpp) - new #define options at top of source: USE_TRIAL_DIVISION makes divide_in_z() use trial division and Newton interpolation instead of recursive polynomial division (only useful for very large problems) STATISTICS enables some statistical output at the end of the program - gcd(numeric, numeric) is now handled earlier; this should speed up normal() - added new heuristic: gcd(p^n, p^m) with p a polynomial now immediately returns p^abs(n-m) - if GCD is 1, cofactors are returned unexpanded --- ginac/normal.cpp | 215 +++++++++++++++++++++++++++++++++++------------ 1 file changed, 162 insertions(+), 53 deletions(-) diff --git a/ginac/normal.cpp b/ginac/normal.cpp index a8f64be8..7d6ddca7 100644 --- a/ginac/normal.cpp +++ b/ginac/normal.cpp @@ -57,7 +57,34 @@ namespace GiNaC { #define FAST_COMPARE 1 // Set this if you want divide_in_z() to use remembering -#define USE_REMEMBER 1 +#define USE_REMEMBER 0 + +// Set this if you want divide_in_z() to use trial division followed by +// polynomial interpolation (usually slower except for very large problems) +#define USE_TRIAL_DIVISION 0 + +// Set this to enable some statistical output for the GCD routines +#define STATISTICS 0 + + +#if STATISTICS +// Statistics variables +static int gcd_called = 0; +static int sr_gcd_called = 0; +static int heur_gcd_called = 0; +static int heur_gcd_failed = 0; + +// Print statistics at end of program +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"; + } +} stat_print; +#endif /** Return pointer to first symbol found in expression. Due to GiNaC´s @@ -598,38 +625,9 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite if (bdeg > adeg) return false; -#if 1 +#if USE_TRIAL_DIVISION - // Polynomial long division (recursive) - ex r = a.expand(); - if (r.is_zero()) - return true; - int rdeg = adeg; - ex eb = b.expand(); - ex blcoeff = eb.coeff(*x, bdeg); - while (rdeg >= 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; - r -= (term * eb).expand(); - if (r.is_zero()) { -#if USE_REMEMBER - dr_remember[ex2(a, b)] = exbool(q, true); -#endif - return true; - } - rdeg = r.degree(*x); - } -#if USE_REMEMBER - dr_remember[ex2(a, b)] = exbool(q, false); -#endif - return false; - -#else - - // Trial division using polynomial interpolation + // Trial division with polynomial interpolation int i, k; // Compute values at evaluation points 0..adeg @@ -652,7 +650,7 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite // Compute inverses vector rcp; rcp.reserve(adeg + 1); - rcp.push_back(0); + 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; + r -= (term * eb).expand(); + if (r.is_zero()) { +#if USE_REMEMBER + dr_remember[ex2(a, b)] = exbool(q, true); +#endif + return true; + } + rdeg = r.degree(*x); + } +#if USE_REMEMBER + dr_remember[ex2(a, b)] = exbool(q, false); +#endif + return false; + #endif } @@ -812,6 +840,9 @@ ex ex::primpart(const symbol &x, const ex &c) const static ex sr_gcd(const ex &a, const ex &b, const symbol *x) { //clog << "sr_gcd(" << a << "," << b << ")\n"; +#if STATISTICS + sr_gcd_called++; +#endif // Sort c and d so that c has higher degree ex c, d; @@ -837,6 +868,7 @@ static ex sr_gcd(const ex &a, const ex &b, const symbol *x) return gamma; c = c.primpart(*x, cont_c); d = d.primpart(*x, cont_d); +//clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n"; // First element of subresultant sequence ex r = _ex0(), ri = _ex1(), psi = _ex1(); @@ -844,11 +876,13 @@ static ex sr_gcd(const ex &a, const ex &b, const symbol *x) for (;;) { // Calculate polynomial pseudo-remainder +//clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n"; r = prem(c, d, *x, false); if (r.is_zero()) return gamma * d.primpart(*x); c = d; cdeg = ddeg; +//clog << " dividing...\n"; if (!divide(r, ri * power(psi, delta), d, false)) throw(std::runtime_error("invalid expression in sr_gcd(), division failed")); ddeg = d.degree(*x); @@ -860,6 +894,7 @@ static ex sr_gcd(const ex &a, const ex &b, const symbol *x) } // Next element of subresultant sequence +//clog << " calculating next subresultant...\n"; ri = c.expand().lcoeff(*x); if (delta == 1) psi = ri; @@ -1024,6 +1059,9 @@ class gcdheu_failed {}; static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var) { //clog << "heur_gcd(" << a << "," << b << ")\n"; +#if STATISTICS + heur_gcd_called++; +#endif // GCD of two numeric values -> CLN if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) { @@ -1057,9 +1095,9 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const xi = mp * _num2() + _num2(); // 6 tries maximum - for (int t=0; t<6; t++) { // MAGIC - if (xi.int_length() * maxdeg > 100000) { // MAGIC -// clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << endl; + for (int t=0; t<6; t++) { + if (xi.int_length() * maxdeg > 100000) { +//clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << endl; throw gcdheu_failed(); } @@ -1109,6 +1147,24 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args) { //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)) { + numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b)); + if (ca) + *ca = ex_to_numeric(a) / g; + if (cb) + *cb = ex_to_numeric(b) / g; + return g; + } + + // Check arguments + if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)) { + throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals")); + } // Partially factored cases (to avoid expanding large expressions) if (is_ex_exactly_of_type(a, mul)) { @@ -1149,6 +1205,51 @@ factored_b: return g; } +#if FAST_COMPARE + // Input polynomials of the form poly^n are sometimes also trivial + if (is_ex_exactly_of_type(a, power)) { + ex p = a.op(0); + if (is_ex_exactly_of_type(b, power)) { + if (p.is_equal(b.op(0))) { + // a = p^n, b = p^m, gcd = p^min(n, m) + ex exp_a = a.op(1), exp_b = b.op(1); + if (exp_a < exp_b) { + if (ca) + *ca = _ex1(); + if (cb) + *cb = power(p, exp_b - exp_a); + return power(p, exp_a); + } else { + if (ca) + *ca = power(p, exp_a - exp_b); + if (cb) + *cb = _ex1(); + return power(p, exp_b); + } + } + } else { + if (p.is_equal(b)) { + // a = p^n, b = p, gcd = p + if (ca) + *ca = power(p, a.op(1) - 1); + if (cb) + *cb = _ex1(); + return p; + } + } + } else if (is_ex_exactly_of_type(b, power)) { + ex p = b.op(0); + if (p.is_equal(a)) { + // a = p, b = p^n, gcd = p + if (ca) + *ca = _ex1(); + if (cb) + *cb = power(p, b.op(1) - 1); + return p; + } + } +#endif + // Some trivial cases ex aex = a.expand(), bex = b.expand(); if (aex.is_zero()) { @@ -1181,17 +1282,6 @@ factored_b: return a; } #endif - if (is_ex_exactly_of_type(aex, numeric) && is_ex_exactly_of_type(bex, numeric)) { - numeric g = gcd(ex_to_numeric(aex), ex_to_numeric(bex)); - if (ca) - *ca = ex_to_numeric(aex) / g; - if (cb) - *cb = ex_to_numeric(bex) / g; - return g; - } - if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)) { - throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals")); - } // Gather symbol statistics sym_desc_vec sym_stats; @@ -1237,13 +1327,32 @@ factored_b: } if (is_ex_exactly_of_type(g, fail)) { //clog << "heuristics failed" << endl; +#if STATISTICS + heur_gcd_failed++; +#endif g = sr_gcd(aex, bex, x); - if (ca) - divide(aex, g, *ca, false); - if (cb) - divide(bex, g, *cb, false); - } - return g; + if (g.is_equal(_ex1())) { + // Keep cofactors factored if possible + if (ca) + *ca = a; + if (cb) + *cb = b; + } else { + if (ca) + divide(aex, g, *ca, false); + if (cb) + divide(bex, g, *cb, false); + } + } else { + if (g.is_equal(_ex1())) { + // Keep cofactors factored if possible + if (ca) + *ca = a; + if (cb) + *cb = b; + } + return g; + } } -- 2.44.0