polynomial.h revision e4369b05aaa20df73a37f4d319ce456865cc74f3
/**
* \file
* \brief Polynomial in canonical (monomial) basis
*//*
* Authors:
* MenTaLguY <mental@rydia.net>
* Krzysztof Kosiński <tweenk.pl@gmail.com>
*
* Copyright 2007-2015 Authors
*
* This library is free software; you can redistribute it and/or
* modify it either under the terms of the GNU Lesser General Public
* License version 2.1 as published by the Free Software Foundation
* (the "LGPL") or, at your option, under the terms of the Mozilla
* Public License Version 1.1 (the "MPL"). If you do not alter this
* notice, a recipient may use your version of this file under either
* the MPL or the LGPL.
*
* You should have received a copy of the LGPL along with this library
* in the file COPYING-LGPL-2.1; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
* You should have received a copy of the MPL along with this library
* in the file COPYING-MPL-1.1
*
* The contents of this file are subject to the Mozilla Public License
* Version 1.1 (the "License"); you may not use this file except in
* compliance with the License. You may obtain a copy of the License at
* http://www.mozilla.org/MPL/
*
* This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
* OF ANY KIND, either express or implied. See the LGPL or the MPL for
* the specific language governing rights and limitations.
*/
#ifndef LIB2GEOM_SEEN_POLY_H
#define LIB2GEOM_SEEN_POLY_H
#include <assert.h>
#include <vector>
#include <iostream>
#include <algorithm>
#include <complex>
#include <2geom/coord.h>
#include <2geom/utils.h>
namespace Geom {
/** @brief Polynomial in canonical (monomial) basis.
* @ingroup Fragments */
class Poly : public std::vector<double>{
public:
// coeff; // sum x^i*coeff[i]
//unsigned size() const { return coeff.size();}
unsigned degree() const { return size()-1;}
//double operator[](const int i) const { return (*this)[i];}
//double& operator[](const int i) { return (*this)[i];}
Poly operator+(const Poly& p) const {
Poly result;
const unsigned out_size = std::max(size(), p.size());
const unsigned min_size = std::min(size(), p.size());
//result.reserve(out_size);
for(unsigned i = 0; i < min_size; i++) {
result.push_back((*this)[i] + p[i]);
}
for(unsigned i = min_size; i < size(); i++)
result.push_back((*this)[i]);
for(unsigned i = min_size; i < p.size(); i++)
result.push_back(p[i]);
assert(result.size() == out_size);
return result;
}
Poly operator-(const Poly& p) const {
Poly result;
const unsigned out_size = std::max(size(), p.size());
const unsigned min_size = std::min(size(), p.size());
result.reserve(out_size);
for(unsigned i = 0; i < min_size; i++) {
result.push_back((*this)[i] - p[i]);
}
for(unsigned i = min_size; i < size(); i++)
result.push_back((*this)[i]);
for(unsigned i = min_size; i < p.size(); i++)
result.push_back(-p[i]);
assert(result.size() == out_size);
return result;
}
Poly operator-=(const Poly& p) {
const unsigned out_size = std::max(size(), p.size());
const unsigned min_size = std::min(size(), p.size());
resize(out_size);
for(unsigned i = 0; i < min_size; i++) {
(*this)[i] -= p[i];
}
for(unsigned i = min_size; i < out_size; i++)
(*this)[i] = -p[i];
return *this;
}
Poly operator-(const double k) const {
Poly result;
const unsigned out_size = size();
result.reserve(out_size);
for(unsigned i = 0; i < out_size; i++) {
result.push_back((*this)[i]);
}
result[0] -= k;
return result;
}
Poly operator-() const {
Poly result;
result.resize(size());
for(unsigned i = 0; i < size(); i++) {
result[i] = -(*this)[i];
}
return result;
}
Poly operator*(const double p) const {
Poly result;
const unsigned out_size = size();
result.reserve(out_size);
for(unsigned i = 0; i < out_size; i++) {
result.push_back((*this)[i]*p);
}
assert(result.size() == out_size);
return result;
}
// equivalent to multiply by x^terms, negative terms are disallowed
Poly shifted(unsigned const terms) const {
Poly result;
size_type const out_size = size() + terms;
result.reserve(out_size);
result.resize(terms, 0.0);
result.insert(result.end(), this->begin(), this->end());
assert(result.size() == out_size);
return result;
}
Poly operator*(const Poly& p) const;
template <typename T>
T eval(T x) const {
T r = 0;
for(int k = size()-1; k >= 0; k--) {
r = r*x + T((*this)[k]);
}
return r;
}
template <typename T>
T operator()(T t) const { return (T)eval(t);}
void normalize();
void monicify();
Poly() {}
Poly(const Poly& p) : std::vector<double>(p) {}
Poly(const double a) {push_back(a);}
public:
template <class T, class U>
void val_and_deriv(T x, U &pd) const {
pd[0] = back();
int nc = size() - 1;
int nd = pd.size() - 1;
for(unsigned j = 1; j < pd.size(); j++)
pd[j] = 0.0;
for(int i = nc -1; i >= 0; i--) {
int nnd = std::min(nd, nc-i);
for(int j = nnd; j >= 1; j--)
pd[j] = pd[j]*x + operator[](i);
pd[0] = pd[0]*x + operator[](i);
}
double cnst = 1;
for(int i = 2; i <= nd; i++) {
cnst *= i;
pd[i] *= cnst;
}
}
static Poly linear(double ax, double b) {
Poly p;
p.push_back(b);
p.push_back(ax);
return p;
}
};
inline Poly operator*(double a, Poly const & b) { return b * a;}
Poly integral(Poly const & p);
Poly derivative(Poly const & p);
Poly divide_out_root(Poly const & p, double x);
Poly compose(Poly const & a, Poly const & b);
Poly divide(Poly const &a, Poly const &b, Poly &r);
Poly gcd(Poly const &a, Poly const &b, const double tol=1e-10);
/*** solve(Poly p)
* find all p.degree() roots of p.
* This function can take a long time with suitably crafted polynomials, but in practice it should be fast. Should we provide special forms for degree() <= 4?
*/
std::vector<std::complex<double> > solve(const Poly & p);
#ifdef HAVE_GSL
/*** solve_reals(Poly p)
* find all real solutions to Poly p.
* currently we just use solve and pick out the suitably real looking values, there may be a better algorithm.
*/
std::vector<double> solve_reals(const Poly & p);
#endif
double polish_root(Poly const & p, double guess, double tol);
/** @brief Analytically solve quadratic equation.
* The equation is given in the standard form: ax^2 + bx + c = 0.
* Only real roots are returned. */
std::vector<Coord> solve_quadratic(Coord a, Coord b, Coord c);
/** @brief Analytically solve cubic equation.
* The equation is given in the standard form: ax^3 + bx^2 + cx + d = 0.
* Only real roots are returned. */
std::vector<Coord> solve_cubic(Coord a, Coord b, Coord c, Coord d);
inline std::ostream &operator<< (std::ostream &out_file, const Poly &in_poly) {
if(in_poly.size() == 0)
out_file << "0";
else {
for(int i = (int)in_poly.size()-1; i >= 0; --i) {
if(i == 1) {
out_file << "" << in_poly[i] << "*x";
out_file << " + ";
} else if(i) {
out_file << "" << in_poly[i] << "*x^" << i;
out_file << " + ";
} else
out_file << in_poly[i];
}
}
return out_file;
}
} // namespace Geom
#endif //LIB2GEOM_SEEN_POLY_H
/*
Local Variables:
mode:c++
c-file-style:"stroustrup"
c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
indent-tabs-mode:nil
fill-column:99
End:
*/
// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:fileencoding=utf-8:textwidth=99 :