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|---|---|---|---|
| 1 | /** | ||
| 2 | * \file | ||
| 3 | * \brief Polynomial in canonical (monomial) basis | ||
| 4 | *//* | ||
| 5 | * Authors: | ||
| 6 | * MenTaLguY <mental@rydia.net> | ||
| 7 | * Krzysztof Kosiński <tweenk.pl@gmail.com> | ||
| 8 | * Rafał Siejakowski <rs@rs-math.net> | ||
| 9 | * | ||
| 10 | * Copyright 2007-2015 Authors | ||
| 11 | * | ||
| 12 | * This library is free software; you can redistribute it and/or | ||
| 13 | * modify it either under the terms of the GNU Lesser General Public | ||
| 14 | * License version 2.1 as published by the Free Software Foundation | ||
| 15 | * (the "LGPL") or, at your option, under the terms of the Mozilla | ||
| 16 | * Public License Version 1.1 (the "MPL"). If you do not alter this | ||
| 17 | * notice, a recipient may use your version of this file under either | ||
| 18 | * the MPL or the LGPL. | ||
| 19 | * | ||
| 20 | * You should have received a copy of the LGPL along with this library | ||
| 21 | * in the file COPYING-LGPL-2.1; if not, write to the Free Software | ||
| 22 | * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA | ||
| 23 | * You should have received a copy of the MPL along with this library | ||
| 24 | * in the file COPYING-MPL-1.1 | ||
| 25 | * | ||
| 26 | * The contents of this file are subject to the Mozilla Public License | ||
| 27 | * Version 1.1 (the "License"); you may not use this file except in | ||
| 28 | * compliance with the License. You may obtain a copy of the License at | ||
| 29 | * http://www.mozilla.org/MPL/ | ||
| 30 | * | ||
| 31 | * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY | ||
| 32 | * OF ANY KIND, either express or implied. See the LGPL or the MPL for | ||
| 33 | * the specific language governing rights and limitations. | ||
| 34 | */ | ||
| 35 | |||
| 36 | #ifndef LIB2GEOM_SEEN_POLY_H | ||
| 37 | #define LIB2GEOM_SEEN_POLY_H | ||
| 38 | #include <assert.h> | ||
| 39 | #include <vector> | ||
| 40 | #include <iostream> | ||
| 41 | #include <algorithm> | ||
| 42 | #include <complex> | ||
| 43 | #include <2geom/coord.h> | ||
| 44 | #include <2geom/utils.h> | ||
| 45 | |||
| 46 | namespace Geom { | ||
| 47 | |||
| 48 | /** @brief Polynomial in canonical (monomial) basis. | ||
| 49 | * @ingroup Fragments */ | ||
| 50 | class Poly : public std::vector<double>{ | ||
| 51 | public: | ||
| 52 | // coeff; // sum x^i*coeff[i] | ||
| 53 | |||
| 54 | //unsigned size() const { return coeff.size();} | ||
| 55 | 424 | unsigned degree() const { return size()-1;} | |
| 56 | |||
| 57 | //double operator[](const int i) const { return (*this)[i];} | ||
| 58 | //double& operator[](const int i) { return (*this)[i];} | ||
| 59 | |||
| 60 | 184 | Poly operator+(const Poly& p) const { | |
| 61 | 184 | Poly result; | |
| 62 | 184 | const unsigned out_size = std::max(size(), p.size()); | |
| 63 | 184 | const unsigned min_size = std::min(size(), p.size()); | |
| 64 |
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184 | result.reserve(out_size); |
| 65 | |||
| 66 |
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542 | for(unsigned i = 0; i < min_size; i++) { |
| 67 |
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358 | result.push_back((*this)[i] + p[i]); |
| 68 | } | ||
| 69 |
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1792 | for(unsigned i = min_size; i < size(); i++) |
| 70 |
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1608 | result.push_back((*this)[i]); |
| 71 |
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217 | for(unsigned i = min_size; i < p.size(); i++) |
| 72 |
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33 | result.push_back(p[i]); |
| 73 |
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184 | assert(result.size() == out_size); |
| 74 | 184 | return result; | |
| 75 | ✗ | } | |
| 76 | Poly operator-(const Poly& p) const { | ||
| 77 | Poly result; | ||
| 78 | const unsigned out_size = std::max(size(), p.size()); | ||
| 79 | const unsigned min_size = std::min(size(), p.size()); | ||
| 80 | result.reserve(out_size); | ||
| 81 | |||
| 82 | for(unsigned i = 0; i < min_size; i++) { | ||
| 83 | result.push_back((*this)[i] - p[i]); | ||
| 84 | } | ||
| 85 | for(unsigned i = min_size; i < size(); i++) | ||
| 86 | result.push_back((*this)[i]); | ||
| 87 | for(unsigned i = min_size; i < p.size(); i++) | ||
| 88 | result.push_back(-p[i]); | ||
| 89 | assert(result.size() == out_size); | ||
| 90 | return result; | ||
| 91 | } | ||
| 92 | ✗ | Poly operator-=(const Poly& p) { | |
| 93 | ✗ | const unsigned out_size = std::max(size(), p.size()); | |
| 94 | ✗ | const unsigned min_size = std::min(size(), p.size()); | |
| 95 | ✗ | resize(out_size); | |
| 96 | |||
| 97 | ✗ | for(unsigned i = 0; i < min_size; i++) { | |
| 98 | ✗ | (*this)[i] -= p[i]; | |
| 99 | } | ||
| 100 | ✗ | for(unsigned i = min_size; i < out_size; i++) | |
| 101 | ✗ | (*this)[i] = -p[i]; | |
| 102 | ✗ | return *this; | |
| 103 | } | ||
| 104 | Poly operator-(const double k) const { | ||
| 105 | Poly result; | ||
| 106 | const unsigned out_size = size(); | ||
| 107 | result.reserve(out_size); | ||
| 108 | |||
| 109 | for(unsigned i = 0; i < out_size; i++) { | ||
| 110 | result.push_back((*this)[i]); | ||
| 111 | } | ||
| 112 | result[0] -= k; | ||
| 113 | return result; | ||
| 114 | } | ||
| 115 | Poly operator-() const { | ||
| 116 | Poly result; | ||
| 117 | result.resize(size()); | ||
| 118 | |||
| 119 | for(unsigned i = 0; i < size(); i++) { | ||
| 120 | result[i] = -(*this)[i]; | ||
| 121 | } | ||
| 122 | return result; | ||
| 123 | } | ||
| 124 | 174 | Poly operator*(const double p) const { | |
| 125 | 174 | Poly result; | |
| 126 | 174 | const unsigned out_size = size(); | |
| 127 |
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174 | result.reserve(out_size); |
| 128 | |||
| 129 |
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522 | for(unsigned i = 0; i < out_size; i++) { |
| 130 |
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348 | result.push_back((*this)[i]*p); |
| 131 | } | ||
| 132 |
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174 | assert(result.size() == out_size); |
| 133 | 174 | return result; | |
| 134 | ✗ | } | |
| 135 | // equivalent to multiply by x^terms, negative terms are disallowed | ||
| 136 | ✗ | Poly shifted(unsigned const terms) const { | |
| 137 | ✗ | Poly result; | |
| 138 | ✗ | size_type const out_size = size() + terms; | |
| 139 | ✗ | result.reserve(out_size); | |
| 140 | |||
| 141 | ✗ | result.resize(terms, 0.0); | |
| 142 | ✗ | result.insert(result.end(), this->begin(), this->end()); | |
| 143 | |||
| 144 | ✗ | assert(result.size() == out_size); | |
| 145 | ✗ | return result; | |
| 146 | ✗ | } | |
| 147 | Poly operator*(const Poly& p) const; | ||
| 148 | |||
| 149 | template <typename T> | ||
| 150 | ✗ | T eval(T x) const { | |
| 151 | ✗ | T r = 0; | |
| 152 | ✗ | for(int k = size()-1; k >= 0; k--) { | |
| 153 | ✗ | r = r*x + T((*this)[k]); | |
| 154 | } | ||
| 155 | ✗ | return r; | |
| 156 | } | ||
| 157 | |||
| 158 | template <typename T> | ||
| 159 | ✗ | T operator()(T t) const { return (T)eval(t);} | |
| 160 | |||
| 161 | void normalize(); | ||
| 162 | |||
| 163 | void monicify(); | ||
| 164 | 37 | Poly() {} | |
| 165 | 52 | Poly(const Poly& p) : std::vector<double>(p) {} | |
| 166 |
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10 | Poly(const double a) {push_back(a);} |
| 167 | |||
| 168 | public: | ||
| 169 | template <class T, class U> | ||
| 170 | void val_and_deriv(T x, U &pd) const { | ||
| 171 | pd[0] = back(); | ||
| 172 | int nc = size() - 1; | ||
| 173 | int nd = pd.size() - 1; | ||
| 174 | for(unsigned j = 1; j < pd.size(); j++) | ||
| 175 | pd[j] = 0.0; | ||
| 176 | for(int i = nc -1; i >= 0; i--) { | ||
| 177 | int nnd = std::min(nd, nc-i); | ||
| 178 | for(int j = nnd; j >= 1; j--) | ||
| 179 | pd[j] = pd[j]*x + operator[](i); | ||
| 180 | pd[0] = pd[0]*x + operator[](i); | ||
| 181 | } | ||
| 182 | double cnst = 1; | ||
| 183 | for(int i = 2; i <= nd; i++) { | ||
| 184 | cnst *= i; | ||
| 185 | pd[i] *= cnst; | ||
| 186 | } | ||
| 187 | } | ||
| 188 | |||
| 189 | static Poly linear(double ax, double b) { | ||
| 190 | Poly p; | ||
| 191 | p.push_back(b); | ||
| 192 | p.push_back(ax); | ||
| 193 | return p; | ||
| 194 | } | ||
| 195 | }; | ||
| 196 | |||
| 197 | 174 | inline Poly operator*(double a, Poly const & b) { return b * a;} | |
| 198 | |||
| 199 | Poly integral(Poly const & p); | ||
| 200 | Poly derivative(Poly const & p); | ||
| 201 | Poly divide_out_root(Poly const & p, double x); | ||
| 202 | Poly compose(Poly const & a, Poly const & b); | ||
| 203 | Poly divide(Poly const &a, Poly const &b, Poly &r); | ||
| 204 | Poly gcd(Poly const &a, Poly const &b, const double tol=1e-10); | ||
| 205 | |||
| 206 | /*** solve(Poly p) | ||
| 207 | * find all p.degree() roots of p. | ||
| 208 | * 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? | ||
| 209 | */ | ||
| 210 | std::vector<std::complex<double> > solve(const Poly & p); | ||
| 211 | |||
| 212 | #ifdef HAVE_GSL | ||
| 213 | /*** solve_reals(Poly p) | ||
| 214 | * find all real solutions to Poly p. | ||
| 215 | * currently we just use solve and pick out the suitably real looking values, there may be a better algorithm. | ||
| 216 | */ | ||
| 217 | std::vector<double> solve_reals(const Poly & p); | ||
| 218 | #endif | ||
| 219 | double polish_root(Poly const & p, double guess, double tol); | ||
| 220 | |||
| 221 | |||
| 222 | /** @brief Analytically solve quadratic equation. | ||
| 223 | * The equation is given in the standard form: ax^2 + bx + c = 0. | ||
| 224 | * Only real roots are returned. */ | ||
| 225 | std::vector<Coord> solve_quadratic(Coord a, Coord b, Coord c); | ||
| 226 | |||
| 227 | /** @brief Analytically solve cubic equation. | ||
| 228 | * The equation is given in the standard form: ax^3 + bx^2 + cx + d = 0. | ||
| 229 | * Only real roots are returned. */ | ||
| 230 | std::vector<Coord> solve_cubic(Coord a, Coord b, Coord c, Coord d); | ||
| 231 | |||
| 232 | /** @brief Analytically solve quartic equation. | ||
| 233 | * The equation is given in the standard form: ax^4 + bx^3 + cx^2 + dx + e = 0. | ||
| 234 | * Only real roots are returned. */ | ||
| 235 | std::vector<Coord> solve_quartic(Coord a, Coord b, Coord c, Coord d, Coord e); | ||
| 236 | |||
| 237 | inline std::ostream &operator<< (std::ostream &out_file, const Poly &in_poly) { | ||
| 238 | if(in_poly.size() == 0) | ||
| 239 | out_file << "0"; | ||
| 240 | else { | ||
| 241 | for(int i = (int)in_poly.size()-1; i >= 0; --i) { | ||
| 242 | if(i == 1) { | ||
| 243 | out_file << "" << in_poly[i] << "*x"; | ||
| 244 | out_file << " + "; | ||
| 245 | } else if(i) { | ||
| 246 | out_file << "" << in_poly[i] << "*x^" << i; | ||
| 247 | out_file << " + "; | ||
| 248 | } else | ||
| 249 | out_file << in_poly[i]; | ||
| 250 | |||
| 251 | } | ||
| 252 | } | ||
| 253 | return out_file; | ||
| 254 | } | ||
| 255 | |||
| 256 | } // namespace Geom | ||
| 257 | |||
| 258 | #endif //LIB2GEOM_SEEN_POLY_H | ||
| 259 | |||
| 260 | /* | ||
| 261 | Local Variables: | ||
| 262 | mode:c++ | ||
| 263 | c-file-style:"stroustrup" | ||
| 264 | c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +)) | ||
| 265 | indent-tabs-mode:nil | ||
| 266 | fill-column:99 | ||
| 267 | End: | ||
| 268 | */ | ||
| 269 | // vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:fileencoding=utf-8:textwidth=99 : | ||
| 270 |