* computation, square-free factorization and rational function normalization. */
/*
- * GiNaC Copyright (C) 1999-2016 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2019 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
* @param lcm LCM to multiply in */
static ex multiply_lcm(const ex &e, const numeric &lcm)
{
+ if (lcm.is_equal(*_num1_p))
+ // e * 1 -> e;
+ return e;
+
if (is_exactly_a<mul>(e)) {
+ // (a*b*...)*lcm -> (a*lcma)*(b*lcmb)*...*(lcm/(lcma*lcmb*...))
size_t num = e.nops();
- exvector v; v.reserve(num + 1);
+ exvector v;
+ v.reserve(num + 1);
numeric lcm_accum = *_num1_p;
for (size_t i=0; i<num; i++) {
numeric op_lcm = lcmcoeff(e.op(i), *_num1_p);
v.push_back(lcm / lcm_accum);
return dynallocate<mul>(v);
} else if (is_exactly_a<add>(e)) {
+ // (a+b+...)*lcm -> a*lcm+b*lcm+...
size_t num = e.nops();
- exvector v; v.reserve(num);
+ exvector v;
+ v.reserve(num);
for (size_t i=0; i<num; i++)
v.push_back(multiply_lcm(e.op(i), lcm));
return dynallocate<add>(v);
} else if (is_exactly_a<power>(e)) {
- if (is_a<symbol>(e.op(0)))
- return e * lcm;
- else
- return pow(multiply_lcm(e.op(0), lcm.power(ex_to<numeric>(e.op(1)).inverse())), e.op(1));
- } else
- return e * lcm;
+ if (!is_a<symbol>(e.op(0))) {
+ // (b^e)*lcm -> (b*lcm^(1/e))^e if lcm^(1/e) ∈ ℚ (i.e. not a float)
+ // but not for symbolic b, as evaluation would undo this again
+ numeric root_of_lcm = lcm.power(ex_to<numeric>(e.op(1)).inverse());
+ if (root_of_lcm.is_rational())
+ return pow(multiply_lcm(e.op(0), root_of_lcm), e.op(1));
+ }
+ }
+ // can't recurse down into e
+ return dynallocate<mul>(e, lcm);
}
}
// 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;
// The symbol with least degree which is contained in both polynomials
// is our main variable
- sym_desc_vec::iterator vari = sym_stats.begin();
+ auto vari = sym_stats.begin();
while ((vari != sym_stats.end()) &&
(((vari->ldeg_b == 0) && (vari->deg_b == 0)) ||
((vari->ldeg_a == 0) && (vari->deg_a == 0))))
*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)
* Yun's algorithm. Used internally by sqrfree().
*
* @param a multivariate polynomial over Z[X], treated here as univariate
- * polynomial in x.
+ * 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 epvector{};
+ }
if (g.is_equal(_ex1)) {
- res.push_back(a);
- return res;
+ return epvector{expair(a, _ex1)};
}
- ex y;
+ epvector results;
+ ex exponent = _ex0;
do {
w = quo(w, g, x);
- y = quo(z, g, x);
- z = y - w.diff(x);
+ if (w.is_zero()) {
+ return results;
+ }
+ z = quo(z, g, x) - w.diff(x);
+ exponent = exponent + 1;
+ if (w.is_equal(x)) {
+ // shortcut for x^n with n ∈ ℕ
+ exponent += quo(z, w.diff(x), x);
+ results.push_back(expair(w, exponent));
+ break;
+ }
g = gcd(w, z);
- res.push_back(g);
+ if (!g.is_equal(_ex1)) {
+ results.push_back(expair(g, exponent));
+ }
} while (!z.is_zero());
- return res;
+ return results;
}
/** Compute a square-free factorization of a multivariate polynomial in Q[X].
*
- * @param a multivariate polynomial over Q[X]
+ * @param a multivariate polynomial over Q[X] (needs not be expanded)
* @param l lst of variables to factor in, may be left empty for autodetection
* @return a square-free factorization of \p a.
*
*/
ex sqrfree(const ex &a, const lst &l)
{
- if (is_exactly_a<numeric>(a) || // algorithm does not trap a==0
- is_a<symbol>(a)) // shortcut
+ if (is_exactly_a<numeric>(a) ||
+ is_a<symbol>(a)) // shortcuts
return a;
// If no lst of variables to factorize in was specified we have to
const ex tmp = multiply_lcm(a,lcm);
// find the factors
- exvector factors = sqrfree_yun(tmp, x);
+ epvector factors = sqrfree_yun(tmp, x);
- // 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++);
+ result *= pow(it.rest, it.coeff);
// 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);
+ if (args.nops()>0)
+ result *= sqrfree(quo(tmp, result, x), args);
else
result *= quo(tmp, result, x);
//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;
- 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();
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 (auto & it : repl_lst)
- 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 (auto & it : m)
- repl_lst.append(it.first == it.second);
-
- return ret;
-}
-
ex ex::to_polynomial(exmap & repl) const
{
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 (auto & it : repl_lst)
- 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 (auto & it : m)
- 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
git://www.ginac.de / ginac.git / blobdiff