2 This file is part of LilyPond, the GNU music typesetter.
4 Copyright (C) 1993--2009 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"
30 Een beter milieu begint bij uzelf. Hergebruik!
33 This was ripped from Rayce, a raytracer I once wrote.
37 Polynomial::eval (Real x) const
42 for (vsize i = coefs_.size (); i--;)
43 p = x * p + coefs_[i];
49 Polynomial::multiply (const Polynomial &p1, const Polynomial &p2)
53 int deg = p1.degree () + p2.degree ();
54 for (int i = 0; i <= deg; i++)
56 dest.coefs_.push_back (0);
57 for (int j = 0; j <= i; j++)
58 if (i - j <= p2.degree () && j <= p1.degree ())
59 dest.coefs_.back () += p1.coefs_[j] * p2.coefs_[i - j];
66 Polynomial::differentiate ()
68 for (int i = 1; i <= degree (); i++)
69 coefs_[i - 1] = coefs_[i] * i;
74 Polynomial::power (int exponent, const Polynomial &src)
77 Polynomial dest (1), base (src);
80 classic int power. invariant: src^exponent = dest * src ^ e
81 greetings go out to Lex Bijlsma & Jaap vd Woude */
86 dest = multiply (dest, base);
92 base = multiply (base, base);
99 static Real const FUDGE = 1e-8;
105 We only do relative comparisons. Absolute comparisons break down in
108 && (fabs (coefs_.back ()) < FUDGE * fabs (back (coefs_, 1))
114 Polynomial::operator += (Polynomial const &p)
116 while (degree () < p.degree ())
117 coefs_.push_back (0.0);
119 for (int i = 0; i <= p.degree (); i++)
120 coefs_[i] += p.coefs_[i];
124 Polynomial::operator -= (Polynomial const &p)
126 while (degree () < p.degree ())
127 coefs_.push_back (0.0);
129 for (int i = 0; i <= p.degree (); i++)
130 coefs_[i] -= p.coefs_[i];
134 Polynomial::scalarmultiply (Real fact)
136 for (int i = 0; i <= degree (); i++)
141 Polynomial::set_negate (const Polynomial &src)
143 for (int i = 0; i <= src.degree (); i++)
144 coefs_[i] = -src.coefs_[i];
149 Polynomial::set_mod (const Polynomial &u, const Polynomial &v)
155 for (int k = u.degree () - v.degree () - 1; k >= 0; k -= 2)
156 coefs_[k] = -coefs_[k];
158 for (int k = u.degree () - v.degree (); k >= 0; k--)
159 for (int j = v.degree () + k - 1; j >= k; j--)
160 coefs_[j] = -coefs_[j] - coefs_[v.degree () + k] * v.coefs_[j - k];
165 for (int k = u.degree () - v.degree (); k >= 0; k--)
166 for (int j = v.degree () + k - 1; j >= k; j--)
167 coefs_[j] -= coefs_[v.degree () + k] * v.coefs_[j - k];
170 int k = v.degree () - 1;
171 while (k >= 0 && coefs_[k] == 0.0)
174 coefs_.resize (1+ ((k < 0) ? 0 : k));
179 Polynomial::check_sol (Real x) const
182 Polynomial p (*this);
186 if (abs (f) > abs (d) * FUDGE)
187 programming_error ("not a root of polynomial\n");
191 Polynomial::check_sols (vector<Real> roots) const
193 for (vsize i = 0; i < roots.size (); i++)
194 check_sol (roots[i]);
197 Polynomial::Polynomial (Real a, Real b)
199 coefs_.push_back (a);
201 coefs_.push_back (b);
205 inline Real cubic_root (Real x)
208 return pow (x, 1.0 / 3.0);
210 return -pow (-x, 1.0 / 3.0);
221 Polynomial::solve_cubic ()const
225 /* normal form: x^3 + Ax^2 + Bx + C = 0 */
226 Real A = coefs_[2] / coefs_[3];
227 Real B = coefs_[1] / coefs_[3];
228 Real C = coefs_[0] / coefs_[3];
231 * substitute x = y - A/3 to eliminate quadric term: x^3 +px + q = 0
235 Real p = 1.0 / 3 * (-1.0 / 3 * sq_A + B);
236 Real q = 1.0 / 2 * (2.0 / 27 * A *sq_A - 1.0 / 3 * A *B + C);
238 /* use Cardano's formula */
245 if (iszero (q)) { /* one triple solution */
250 else { /* one single and one double solution */
251 Real u = cubic_root (-q);
253 sol.push_back (2 * u);
259 /* Casus irreducibilis: three real solutions */
260 Real phi = 1.0 / 3 * acos (-q / sqrt (-cb));
261 Real t = 2 * sqrt (-p);
263 sol.push_back (t * cos (phi));
264 sol.push_back (-t * cos (phi + M_PI / 3));
265 sol.push_back (-t * cos (phi - M_PI / 3));
269 /* one real solution */
270 Real sqrt_D = sqrt (D);
271 Real u = cubic_root (sqrt_D - q);
272 Real v = -cubic_root (sqrt_D + q);
274 sol.push_back (u + v);
278 Real sub = 1.0 / 3 * A;
280 for (vsize i = sol.size (); i--;)
285 assert (fabs (eval (sol[i])) < 1e-8);
293 Polynomial::lc () const
295 return coefs_.back ();
301 return coefs_.back ();
305 Polynomial::degree ()const
307 return coefs_.size () -1;
310 all roots of quadratic eqn.
313 Polynomial::solve_quadric ()const
316 /* normal form: x^2 + px + q = 0 */
317 Real p = coefs_[1] / (2 * coefs_[2]);
318 Real q = coefs_[0] / coefs_[2];
326 sol.push_back (D - p);
327 sol.push_back (-D - p);
332 /* solve linear equation */
334 Polynomial::solve_linear ()const
338 s.push_back (-coefs_[0] / coefs_[1]);
343 Polynomial::solve () const
345 Polynomial *me = (Polynomial *) this;
351 return solve_linear ();
353 return solve_quadric ();
355 return solve_cubic ();
362 Polynomial::operator *= (Polynomial const &p2)
364 *this = multiply (*this, p2);