/** @file normal.h
*
* This file defines several functions that work on univariate and
* multivariate polynomials and rational functions.
* These functions include polynomial quotient and remainder, GCD and LCM
* computation, square-free factorization and rational function normalization. */
/*
* GiNaC Copyright (C) 1999-2021 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
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* 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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#ifndef GINAC_NORMAL_H
#define GINAC_NORMAL_H
#include "lst.h"
namespace GiNaC {
/**
* Flags to control the behavior of gcd() and friends
*/
struct gcd_options
{
enum {
/**
* Usually GiNaC tries heuristic GCD first, because typically
* it's much faster than anything else. Even if heuristic
* algorithm fails, the overhead is negligible w.r.t. cost
* of computing the GCD by some other method. However, some
* people dislike it, so here's a flag which tells GiNaC
* to NOT use the heuristic algorithm.
*/
no_heur_gcd = 2,
/**
* GiNaC tries to avoid expanding expressions when computing
* GCDs. This is a good idea, but some people dislike it.
* Hence the flag to disable special handling of partially
* factored polynomials. DON'T SET THIS unless you *really*
* know what are you doing!
*/
no_part_factored = 4,
/**
* By default GiNaC uses modular GCD algorithm. Typically
* it's much faster than PRS (pseudo remainder sequence)
* algorithm. This flag forces GiNaC to use PRS algorithm
*/
use_sr_gcd = 8
};
};
class ex;
class symbol;
// Quotient q(x) of polynomials a(x) and b(x) in Q[x], so that a(x)=b(x)*q(x)+r(x)
extern ex quo(const ex &a, const ex &b, const ex &x, bool check_args = true);
// Remainder r(x) of polynomials a(x) and b(x) in Q[x], so that a(x)=b(x)*q(x)+r(x)
extern ex rem(const ex &a, const ex &b, const ex &x, bool check_args = true);
// Decompose rational function a(x)=N(x)/D(x) into Q(x)+R(x)/D(x) with degree(R, x) < degree(D, x)
extern ex decomp_rational(const ex &a, const ex &x);
// Pseudo-remainder of polynomials a(x) and b(x) in Q[x]
extern ex prem(const ex &a, const ex &b, const ex &x, bool check_args = true);
// Pseudo-remainder of polynomials a(x) and b(x) in Q[x]
extern ex sprem(const ex &a, const ex &b, const ex &x, bool check_args = true);
// Exact polynomial division of a(X) by b(X) in Q[X] (quotient returned in q), returns false when exact division fails
extern bool divide(const ex &a, const ex &b, ex &q, bool check_args = true);
// Polynomial GCD in Z[X], cofactors are returned in ca and cb, if desired
extern ex gcd(const ex &a, const ex &b, ex *ca = nullptr, ex *cb = nullptr,
bool check_args = true, unsigned options = 0);
// Polynomial LCM in Z[X]
extern ex lcm(const ex &a, const ex &b, bool check_args = true);
// Square-free factorization of a polynomial a(x)
extern ex sqrfree(const ex &a, const lst &l = lst());
// Square-free partial fraction decomposition of a rational function a(x)
extern ex sqrfree_parfrac(const ex & a, const symbol & x);
// Collect common factors in sums.
extern ex collect_common_factors(const ex & e);
// Resultant of two polynomials e1,e2 with respect to symbol s.
extern ex resultant(const ex & e1, const ex & e2, const ex & s);
} // namespace GiNaC
#endif // ndef GINAC_NORMAL_H