* computation, square-free factorization and rational function normalization. */
/*
- * GiNaC Copyright (C) 1999-2016 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2020 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
/** Maximum number of terms of leading coefficient of symbol in both polynomials */
size_t max_lcnops;
- /** Commparison operator for sorting */
+ /** Comparison operator for sorting */
bool operator<(const sym_desc &x) const
{
if (max_deg == x.max_deg)
#if 0
std::clog << "Symbols:\n";
- it = v.begin(); itend = v.end();
+ auto 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 << std::endl;
+ std::clog << " lcoeff_a=" << a.lcoeff(it->sym) << ", lcoeff_b=" << b.lcoeff(it->sym) << std::endl;
++it;
}
#endif
}
// Some trivial cases
- ex aex = a.expand(), bex = b.expand();
+ ex aex = a.expand();
if (aex.is_zero()) {
if (ca)
*ca = _ex0;
*cb = _ex1;
return b;
}
+ ex bex = b.expand();
if (bex.is_zero()) {
if (ca)
*ca = _ex1;
*cb = b;
return _ex1;
}
- // move symbols which contained only in one of the polynomials
- // to the end:
+ // move symbol contained only in one of the polynomials to the end:
rotate(sym_stats.begin(), vari, sym_stats.end());
sym_desc_vec::const_iterator var = sym_stats.begin();
if (cb)
*cb = b;
return _ex1;
- // XXX: do I need to check for p_gcd = -1?
}
// there are common factors:
if (p.is_equal(b)) {
// a = p^n, b = p, gcd = p
if (ca)
- *ca = pow(p, a.op(1) - 1);
+ *ca = pow(p, exp_a - 1);
if (cb)
*cb = _ex1;
return p;
- }
+ }
+ if (is_a<symbol>(p)) {
+ // Cancel trivial common factor
+ int ldeg_a = ex_to<numeric>(exp_a).to_int();
+ int ldeg_b = b.ldegree(p);
+ int min_ldeg = std::min(ldeg_a, ldeg_b);
+ if (min_ldeg > 0) {
+ ex common = pow(p, min_ldeg);
+ return gcd(pow(p, ldeg_a - min_ldeg), (b / common).expand(), ca, cb, false) * common;
+ }
+ }
ex p_co, bpart_co;
ex p_gcd = gcd(p, b, &p_co, &bpart_co, false);
- // a(x) = p(x)^n, gcd(p, b) = 1 ==> gcd(a, b) = 1
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)
* @param a multivariate polynomial over Z[X], treated here as univariate
* polynomial in x (needs not be expanded).
* @param x variable to factor in
- * @return vector of factors sorted in ascending degree */
-static exvector sqrfree_yun(const ex &a, const symbol &x)
+ * @return vector of expairs (factor, exponent), sorted by exponents */
+static epvector sqrfree_yun(const ex &a, const symbol &x)
{
- exvector res;
ex w = a;
ex z = w.diff(x);
ex g = gcd(w, z);
if (g.is_zero()) {
- return res;
+ // manifest zero or hidden zero
+ return {};
}
if (g.is_equal(_ex1)) {
- res.push_back(a);
- return res;
+ // w(x) and w'(x) share no factors: w(x) is square-free
+ return {expair(a, _ex1)};
}
- ex y;
+
+ epvector factors;
+ ex i = 0; // exponent
do {
w = quo(w, g, x);
if (w.is_zero()) {
- return res;
+ // hidden zero
+ break;
+ }
+ z = quo(z, g, x) - w.diff(x);
+ i += 1;
+ if (w.is_equal(x)) {
+ // shortcut for x^n with n ∈ ℕ
+ i += quo(z, w.diff(x), x);
+ factors.push_back(expair(w, i));
+ break;
}
- y = quo(z, g, x);
- z = y - w.diff(x);
g = gcd(w, z);
- res.push_back(g);
+ if (!g.is_equal(_ex1)) {
+ factors.push_back(expair(g, i));
+ }
} while (!z.is_zero());
- return res;
+
+ // correct for lost factor
+ // (being based on GCDs, Yun's algorithm only finds factors up to a unit)
+ const ex lost_factor = quo(a, mul{factors}, x);
+ if (lost_factor.is_equal(_ex1)) {
+ // trivial lost factor
+ return factors;
+ }
+ if (!factors.empty() && factors[0].coeff.is_equal(1)) {
+ // multiply factor^1 with lost_factor
+ factors[0].rest *= lost_factor;
+ return factors;
+ }
+ // no factor^1: prepend lost_factor^1 to the results
+ epvector results = {expair(lost_factor, 1)};
+ std::move(factors.begin(), factors.end(), std::back_inserter(results));
+ return results;
}
// convert the argument from something in Q[X] to something in Z[X]
const numeric lcm = lcm_of_coefficients_denominators(a);
- const ex tmp = multiply_lcm(a,lcm);
+ const ex tmp = multiply_lcm(a, lcm);
// find the factors
- exvector factors = sqrfree_yun(tmp, x);
+ epvector factors = sqrfree_yun(tmp, x);
+ if (factors.empty()) {
+ // the polynomial was a hidden zero
+ return _ex0;
+ }
- // construct the next list of symbols with the first element popped
- lst newargs = args;
- newargs.remove_first();
+ // remove symbol x and proceed recursively with the remaining symbols
+ args.remove_first();
// recurse down the factors in remaining variables
- if (newargs.nops()>0) {
+ if (args.nops()>0) {
for (auto & it : factors)
- it = sqrfree(it, newargs);
+ it.rest = sqrfree(it.rest, args);
}
// Done with recursion, now construct the final result
- ex result = _ex1;
- int p = 1;
- for (auto & it : factors)
- result *= pow(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. 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);
+ ex result = mul(factors);
// Put in the rational overall factor again and return
- return result * lcm.inverse();
+ return result * lcm.inverse();
}
// Find numerator and denominator
ex nd = numer_denom(a);
ex numer = nd.op(0), denom = nd.op(1);
-//clog << "numer = " << numer << ", denom = " << denom << endl;
+//std::clog << "numer = " << numer << ", denom = " << denom << std::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;
+//std::clog << "red_poly = " << red_poly << ", red_numer = " << red_numer << std::endl;
// Factorize denominator and compute cofactors
- exvector yun = sqrfree_yun(denom, x);
-//clog << "yun factors: " << exprseq(yun) << endl;
- size_t num_yun = yun.size();
- exvector factor; factor.reserve(num_yun);
- exvector cofac; cofac.reserve(num_yun);
- for (size_t i=0; i<num_yun; i++) {
- if (!yun[i].is_equal(_ex1)) {
- for (size_t j=0; j<=i; j++) {
- factor.push_back(pow(yun[i], j+1));
- ex prod = _ex1;
- for (size_t k=0; k<num_yun; k++) {
- if (k == i)
- prod *= pow(yun[k], i-j);
- else
- prod *= pow(yun[k], k+1);
- }
- cofac.push_back(prod.expand());
+ epvector yun = sqrfree_yun(denom, x);
+ size_t yun_max_exponent = yun.empty() ? 0 : ex_to<numeric>(yun.back().coeff).to_int();
+ exvector factor, cofac;
+ for (size_t i=0; i<yun.size(); i++) {
+ numeric i_exponent = ex_to<numeric>(yun[i].coeff);
+ for (size_t j=0; j<i_exponent; j++) {
+ factor.push_back(pow(yun[i].rest, j+1));
+ ex prod = _ex1;
+ for (size_t k=0; k<yun.size(); k++) {
+ if (yun[k].coeff == i_exponent)
+ prod *= pow(yun[k].rest, i_exponent-1-j);
+ else
+ prod *= pow(yun[k].rest, yun[k].coeff);
}
+ cofac.push_back(prod.expand());
}
}
size_t num_factors = factor.size();
-//clog << "factors : " << exprseq(factor) << endl;
-//clog << "cofactors: " << exprseq(cofac) << endl;
+//std::clog << "factors : " << exprseq(factor) << std::endl;
+//std::clog << "cofactors: " << exprseq(cofac) << std::endl;
// Construct coefficient matrix for decomposition
int max_denom_deg = denom.degree(x);
sys(i, j) = cofac[j].coeff(x, i);
rhs(i, 0) = red_numer.coeff(x, i);
}
-//clog << "coeffs: " << sys << endl;
-//clog << "rhs : " << rhs << endl;
+//std::clog << "coeffs: " << sys << std::endl;
+//std::clog << "rhs : " << rhs << std::endl;
// Solve resulting linear system
matrix vars(num_factors, 1);
x *= f;
}
- if (i == 0)
+ if (gc.is_zero())
gc = x;
else
gc = gcd(gc, x);
if (gc.is_equal(_ex1))
return e;
+ if (gc.is_zero())
+ return _ex0;
+
// The GCD is the factor we pull out
factor *= gc;
git://www.ginac.de / ginac.git / blobdiff