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Diffstat (limited to 'src/2geom/polynomial.cpp')
-rw-r--r-- | src/2geom/polynomial.cpp | 337 |
1 files changed, 337 insertions, 0 deletions
diff --git a/src/2geom/polynomial.cpp b/src/2geom/polynomial.cpp new file mode 100644 index 0000000..9737bd0 --- /dev/null +++ b/src/2geom/polynomial.cpp @@ -0,0 +1,337 @@ +/** + * \file + * \brief Polynomial in canonical (monomial) basis + *//* + * Authors: + * MenTaLguY <mental@rydia.net> + * Krzysztof KosiĆski <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. + */ + +#include <algorithm> +#include <2geom/polynomial.h> +#include <2geom/math-utils.h> +#include <math.h> + +#ifdef HAVE_GSL +#include <gsl/gsl_poly.h> +#endif + +namespace Geom { + +#ifndef M_PI +# define M_PI 3.14159265358979323846 +#endif + +Poly Poly::operator*(const Poly& p) const { + Poly result; + result.resize(degree() + p.degree()+1); + + for(unsigned i = 0; i < size(); i++) { + for(unsigned j = 0; j < p.size(); j++) { + result[i+j] += (*this)[i] * p[j]; + } + } + return result; +} + +/*double Poly::eval(double x) const { + return gsl_poly_eval(&coeff[0], size(), x); + }*/ + +void Poly::normalize() { + while(back() == 0) + pop_back(); +} + +void Poly::monicify() { + normalize(); + + double scale = 1./back(); // unitize + + for(unsigned i = 0; i < size(); i++) { + (*this)[i] *= scale; + } +} + + +#ifdef HAVE_GSL +std::vector<std::complex<double> > solve(Poly const & pp) { + Poly p(pp); + p.normalize(); + gsl_poly_complex_workspace * w + = gsl_poly_complex_workspace_alloc (p.size()); + + gsl_complex_packed_ptr z = new double[p.degree()*2]; + double* a = new double[p.size()]; + for(unsigned int i = 0; i < p.size(); i++) + a[i] = p[i]; + std::vector<std::complex<double> > roots; + //roots.resize(p.degree()); + + gsl_poly_complex_solve (a, p.size(), w, z); + delete[]a; + + gsl_poly_complex_workspace_free (w); + + for (unsigned int i = 0; i < p.degree(); i++) { + roots.emplace_back(z[2*i] ,z[2*i+1]); + //printf ("z%d = %+.18f %+.18f\n", i, z[2*i], z[2*i+1]); + } + delete[] z; + return roots; +} + +std::vector<double > solve_reals(Poly const & p) { + std::vector<std::complex<double> > roots = solve(p); + std::vector<double> real_roots; + + for(auto & root : roots) { + if(root.imag() == 0) // should be more lenient perhaps + real_roots.push_back(root.real()); + } + return real_roots; +} +#endif + +double polish_root(Poly const & p, double guess, double tol) { + Poly dp = derivative(p); + + double fn = p(guess); + while(fabs(fn) > tol) { + guess -= fn/dp(guess); + fn = p(guess); + } + return guess; +} + +Poly integral(Poly const & p) { + Poly result; + + result.reserve(p.size()+1); + result.push_back(0); // arbitrary const + for(unsigned i = 0; i < p.size(); i++) { + result.push_back(p[i]/(i+1)); + } + return result; + +} + +Poly derivative(Poly const & p) { + Poly result; + + if(p.size() <= 1) + return Poly(0); + result.reserve(p.size()-1); + for(unsigned i = 1; i < p.size(); i++) { + result.push_back(i*p[i]); + } + return result; +} + +Poly compose(Poly const & a, Poly const & b) { + Poly result; + + for(unsigned i = a.size(); i > 0; i--) { + result = Poly(a[i-1]) + result * b; + } + return result; + +} + +/* This version is backwards - dividing taylor terms +Poly divide(Poly const &a, Poly const &b, Poly &r) { + Poly c; + r = a; // remainder + + const unsigned k = a.size(); + r.resize(k, 0); + c.resize(k, 0); + + for(unsigned i = 0; i < k; i++) { + double ci = r[i]/b[0]; + c[i] += ci; + Poly bb = ci*b; + std::cout << ci <<"*" << b << ", r= " << r << std::endl; + r -= bb.shifted(i); + } + + return c; +} +*/ + +Poly divide(Poly const &a, Poly const &b, Poly &r) { + Poly c; + r = a; // remainder + assert(b.size() > 0); + + const unsigned k = a.degree(); + const unsigned l = b.degree(); + c.resize(k, 0.); + + for(unsigned i = k; i >= l; i--) { + //assert(i >= 0); + double ci = r.back()/b.back(); + c[i-l] += ci; + Poly bb = ci*b; + //std::cout << ci <<"*(" << b.shifted(i-l) << ") = " + // << bb.shifted(i-l) << " r= " << r << std::endl; + r -= bb.shifted(i-l); + r.pop_back(); + } + //std::cout << "r= " << r << std::endl; + r.normalize(); + c.normalize(); + + return c; +} + +Poly gcd(Poly const &a, Poly const &b, const double /*tol*/) { + if(a.size() < b.size()) + return gcd(b, a); + if(b.size() <= 0) + return a; + if(b.size() == 1) + return a; + Poly r; + divide(a, b, r); + return gcd(b, r); +} + + + + +std::vector<Coord> solve_quadratic(Coord a, Coord b, Coord c) +{ + std::vector<Coord> result; + + if (a == 0) { + // linear equation + if (b == 0) return result; + result.push_back(-c/b); + return result; + } + + Coord delta = b*b - 4*a*c; + + if (delta == 0) { + // one root + result.push_back(-b / (2*a)); + } else if (delta > 0) { + // two roots + Coord delta_sqrt = sqrt(delta); + + // Use different formulas depending on sign of b to preserve + // numerical stability. See e.g.: + // http://people.csail.mit.edu/bkph/articles/Quadratics.pdf + int sign = b >= 0 ? 1 : -1; + Coord t = -0.5 * (b + sign * delta_sqrt); + result.push_back(t / a); + result.push_back(c / t); + } + // no roots otherwise + + std::sort(result.begin(), result.end()); + return result; +} + + +std::vector<Coord> solve_cubic(Coord a, Coord b, Coord c, Coord d) +{ + // based on: + // http://mathworld.wolfram.com/CubicFormula.html + + if (a == 0) { + return solve_quadratic(b, c, d); + } + if (d == 0) { + // divide by x + std::vector<Coord> result = solve_quadratic(a, b, c); + result.push_back(0); + std::sort(result.begin(), result.end()); + return result; + } + + std::vector<Coord> result; + + // 1. divide everything by a to bring to canonical form + b /= a; + c /= a; + d /= a; + + // 2. eliminate x^2 term: x^3 + 3Qx - 2R = 0 + Coord Q = (3*c - b*b) / 9; + Coord R = (-27 * d + b * (9*c - 2*b*b)) / 54; + + // 3. compute polynomial discriminant + Coord D = Q*Q*Q + R*R; + Coord term1 = b/3; + + if (D > 0) { + // only one real root + Coord S = cbrt(R + sqrt(D)); + Coord T = cbrt(R - sqrt(D)); + result.push_back(-b/3 + S + T); + } else if (D == 0) { + // 3 real roots, 2 of which are equal + Coord rroot = cbrt(R); + result.reserve(3); + result.push_back(-term1 + 2*rroot); + result.push_back(-term1 - rroot); + result.push_back(-term1 - rroot); + } else { + // 3 distinct real roots + assert(Q < 0); + Coord theta = acos(R / sqrt(-Q*Q*Q)); + Coord rroot = 2 * sqrt(-Q); + result.reserve(3); + result.push_back(-term1 + rroot * cos(theta / 3)); + result.push_back(-term1 + rroot * cos((theta + 2*M_PI) / 3)); + result.push_back(-term1 + rroot * cos((theta + 4*M_PI) / 3)); + } + + std::sort(result.begin(), result.end()); + return result; +} + + +/*Poly divide_out_root(Poly const & p, double x) { + assert(1); + }*/ + +} //namespace Geom + +/* + 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 : |