2 This file is part of LilyPond, the GNU music typesetter.
4 Copyright (C) 1993--2011 Han-Wen Nienhuys <hanwen@xs4all.nl>
6 LilyPond is free software: you can redistribute it and/or modify
7 it under the terms of the GNU General Public License as published by
8 the Free Software Foundation, either version 3 of the License, or
9 (at your option) any later version.
11 LilyPond is distributed in the hope that it will be useful,
12 but WITHOUT ANY WARRANTY; without even the implied warranty of
13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
14 GNU General Public License for more details.
16 You should have received a copy of the GNU General Public License
17 along with LilyPond. If not, see <http://www.gnu.org/licenses/>.
20 #include "polynomial.hh"
29 Een beter milieu begint bij uzelf. Hergebruik!
32 This was ripped from Rayce, a raytracer I once wrote.
36 Polynomial::eval (Real x) const
41 for (vsize i = coefs_.size (); i--;)
42 p = x * p + coefs_[i];
48 Polynomial::multiply (const Polynomial &p1, const Polynomial &p2)
52 int deg = p1.degree () + p2.degree ();
53 for (int i = 0; i <= deg; i++)
55 dest.coefs_.push_back (0);
56 for (int j = 0; j <= i; j++)
57 if (i - j <= p2.degree () && j <= p1.degree ())
58 dest.coefs_.back () += p1.coefs_[j] * p2.coefs_[i - j];
65 Polynomial::differentiate ()
67 for (int i = 1; i <= degree (); i++)
68 coefs_[i - 1] = coefs_[i] * i;
73 Polynomial::power (int exponent, const Polynomial &src)
76 Polynomial dest (1), base (src);
79 classic int power. invariant: src^exponent = dest * src ^ e
80 greetings go out to Lex Bijlsma & Jaap vd Woude */
85 dest = multiply (dest, base);
91 base = multiply (base, base);
98 static Real const FUDGE = 1e-8;
104 We only do relative comparisons. Absolute comparisons break down in
107 && (fabs (coefs_.back ()) < FUDGE * fabs (back (coefs_, 1))
113 Polynomial::operator += (Polynomial const &p)
115 while (degree () < p.degree ())
116 coefs_.push_back (0.0);
118 for (int i = 0; i <= p.degree (); i++)
119 coefs_[i] += p.coefs_[i];
123 Polynomial::operator -= (Polynomial const &p)
125 while (degree () < p.degree ())
126 coefs_.push_back (0.0);
128 for (int i = 0; i <= p.degree (); i++)
129 coefs_[i] -= p.coefs_[i];
133 Polynomial::scalarmultiply (Real fact)
135 for (int i = 0; i <= degree (); i++)
140 Polynomial::set_negate (const Polynomial &src)
142 for (int i = 0; i <= src.degree (); i++)
143 coefs_[i] = -src.coefs_[i];
148 Polynomial::set_mod (const Polynomial &u, const Polynomial &v)
154 for (int k = u.degree () - v.degree () - 1; k >= 0; k -= 2)
155 coefs_[k] = -coefs_[k];
157 for (int k = u.degree () - v.degree (); k >= 0; k--)
158 for (int j = v.degree () + k - 1; j >= k; j--)
159 coefs_[j] = -coefs_[j] - coefs_[v.degree () + k] * v.coefs_[j - k];
164 for (int k = u.degree () - v.degree (); k >= 0; k--)
165 for (int j = v.degree () + k - 1; j >= k; j--)
166 coefs_[j] -= coefs_[v.degree () + k] * v.coefs_[j - k];
169 int k = v.degree () - 1;
170 while (k >= 0 && coefs_[k] == 0.0)
173 coefs_.resize (1 + ((k < 0) ? 0 : k));
178 Polynomial::check_sol (Real x) const
181 Polynomial p (*this);
185 if (abs (f) > abs (d) * FUDGE)
186 programming_error ("not a root of polynomial\n");
190 Polynomial::check_sols (vector<Real> roots) const
192 for (vsize i = 0; i < roots.size (); i++)
193 check_sol (roots[i]);
196 Polynomial::Polynomial (Real a, Real b)
198 coefs_.push_back (a);
200 coefs_.push_back (b);
204 inline Real cubic_root (Real x)
207 return pow (x, 1.0 / 3.0);
209 return -pow (-x, 1.0 / 3.0);
220 Polynomial::solve_cubic ()const
224 /* normal form: x^3 + Ax^2 + Bx + C = 0 */
225 Real A = coefs_[2] / coefs_[3];
226 Real B = coefs_[1] / coefs_[3];
227 Real C = coefs_[0] / coefs_[3];
230 * substitute x = y - A/3 to eliminate quadric term: x^3 +px + q = 0
234 Real p = 1.0 / 3 * (-1.0 / 3 * sq_A + B);
235 Real q = 1.0 / 2 * (2.0 / 27 * A * sq_A - 1.0 / 3 * A * B + C);
237 /* use Cardano's formula */
244 if (iszero (q)) /* one triple solution */
250 else /* one single and one double solution */
252 Real u = cubic_root (-q);
254 sol.push_back (2 * u);
260 /* Casus irreducibilis: three real solutions */
261 Real phi = 1.0 / 3 * acos (-q / sqrt (-cb));
262 Real t = 2 * sqrt (-p);
264 sol.push_back (t * cos (phi));
265 sol.push_back (-t * cos (phi + M_PI / 3));
266 sol.push_back (-t * cos (phi - M_PI / 3));
270 /* one real solution */
271 Real sqrt_D = sqrt (D);
272 Real u = cubic_root (sqrt_D - q);
273 Real v = -cubic_root (sqrt_D + q);
275 sol.push_back (u + v);
279 Real sub = 1.0 / 3 * A;
281 for (vsize i = sol.size (); i--;)
286 assert (fabs (eval (sol[i])) < 1e-8);
294 Polynomial::lc () const
296 return coefs_.back ();
302 return coefs_.back ();
306 Polynomial::degree ()const
308 return coefs_.size () - 1;
311 all roots of quadratic eqn.
314 Polynomial::solve_quadric ()const
317 /* normal form: x^2 + px + q = 0 */
318 Real p = coefs_[1] / (2 * coefs_[2]);
319 Real q = coefs_[0] / coefs_[2];
327 sol.push_back (D - p);
328 sol.push_back (-D - p);
333 /* solve linear equation */
335 Polynomial::solve_linear ()const
339 s.push_back (-coefs_[0] / coefs_[1]);
344 Polynomial::solve () const
346 Polynomial *me = (Polynomial *) this;
352 return solve_linear ();
354 return solve_quadric ();
356 return solve_cubic ();
363 Polynomial::operator *= (Polynomial const &p2)
365 *this = multiply (*this, p2);