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::minmax (Real l, Real r, bool ret_max) const
70 programming_error ("left bound greater than right bound for polynomial minmax. flipping bounds.");
76 sols.push_back (eval (l));
77 sols.push_back (eval (r));
79 Polynomial deriv (*this);
80 deriv.differentiate ();
81 vector<Real> maxmins = deriv.solve ();
82 for (vsize i = 0; i < maxmins.size (); i++)
83 if (maxmins[i] >= l && maxmins[i] <= r)
84 sols.push_back (eval (maxmins[i]));
85 vector_sort (sols, less<Real> ());
87 return ret_max ? sols.back () : sols[0];
91 Polynomial::differentiate ()
93 for (int i = 1; i <= degree (); i++)
94 coefs_[i - 1] = coefs_[i] * i;
99 Polynomial::power (int exponent, const Polynomial &src)
102 Polynomial dest (1), base (src);
105 classic int power. invariant: src^exponent = dest * src ^ e
106 greetings go out to Lex Bijlsma & Jaap vd Woude */
111 dest = multiply (dest, base);
117 base = multiply (base, base);
124 static Real const FUDGE = 1e-8;
130 We only do relative comparisons. Absolute comparisons break down in
133 && (fabs (coefs_.back ()) < FUDGE * fabs (back (coefs_, 1))
139 Polynomial::operator += (Polynomial const &p)
141 while (degree () < p.degree ())
142 coefs_.push_back (0.0);
144 for (int i = 0; i <= p.degree (); i++)
145 coefs_[i] += p.coefs_[i];
149 Polynomial::operator -= (Polynomial const &p)
151 while (degree () < p.degree ())
152 coefs_.push_back (0.0);
154 for (int i = 0; i <= p.degree (); i++)
155 coefs_[i] -= p.coefs_[i];
159 Polynomial::scalarmultiply (Real fact)
161 for (int i = 0; i <= degree (); i++)
166 Polynomial::set_negate (const Polynomial &src)
168 for (int i = 0; i <= src.degree (); i++)
169 coefs_[i] = -src.coefs_[i];
174 Polynomial::set_mod (const Polynomial &u, const Polynomial &v)
180 for (int k = u.degree () - v.degree () - 1; k >= 0; k -= 2)
181 coefs_[k] = -coefs_[k];
183 for (int k = u.degree () - v.degree (); k >= 0; k--)
184 for (int j = v.degree () + k - 1; j >= k; j--)
185 coefs_[j] = -coefs_[j] - coefs_[v.degree () + k] * v.coefs_[j - k];
190 for (int k = u.degree () - v.degree (); k >= 0; k--)
191 for (int j = v.degree () + k - 1; j >= k; j--)
192 coefs_[j] -= coefs_[v.degree () + k] * v.coefs_[j - k];
195 int k = v.degree () - 1;
196 while (k >= 0 && coefs_[k] == 0.0)
199 coefs_.resize (1 + ((k < 0) ? 0 : k));
204 Polynomial::check_sol (Real x) const
207 Polynomial p (*this);
211 if (abs (f) > abs (d) * FUDGE)
212 programming_error ("not a root of polynomial\n");
216 Polynomial::check_sols (vector<Real> roots) const
218 for (vsize i = 0; i < roots.size (); i++)
219 check_sol (roots[i]);
222 Polynomial::Polynomial (Real a, Real b)
224 coefs_.push_back (a);
226 coefs_.push_back (b);
230 inline Real cubic_root (Real x)
233 return pow (x, 1.0 / 3.0);
235 return -pow (-x, 1.0 / 3.0);
246 Polynomial::solve_cubic ()const
250 /* normal form: x^3 + Ax^2 + Bx + C = 0 */
251 Real A = coefs_[2] / coefs_[3];
252 Real B = coefs_[1] / coefs_[3];
253 Real C = coefs_[0] / coefs_[3];
256 * substitute x = y - A/3 to eliminate quadric term: x^3 +px + q = 0
260 Real p = 1.0 / 3 * (-1.0 / 3 * sq_A + B);
261 Real q = 1.0 / 2 * (2.0 / 27 * A * sq_A - 1.0 / 3 * A * B + C);
263 /* use Cardano's formula */
270 if (iszero (q)) /* one triple solution */
276 else /* one single and one double solution */
278 Real u = cubic_root (-q);
280 sol.push_back (2 * u);
286 /* Casus irreducibilis: three real solutions */
287 Real phi = 1.0 / 3 * acos (-q / sqrt (-cb));
288 Real t = 2 * sqrt (-p);
290 sol.push_back (t * cos (phi));
291 sol.push_back (-t * cos (phi + M_PI / 3));
292 sol.push_back (-t * cos (phi - M_PI / 3));
296 /* one real solution */
297 Real sqrt_D = sqrt (D);
298 Real u = cubic_root (sqrt_D - q);
299 Real v = -cubic_root (sqrt_D + q);
301 sol.push_back (u + v);
305 Real sub = 1.0 / 3 * A;
307 for (vsize i = sol.size (); i--;)
312 assert (fabs (eval (sol[i])) < 1e-8);
320 Polynomial::lc () const
322 return coefs_.back ();
328 return coefs_.back ();
332 Polynomial::degree ()const
334 return coefs_.size () - 1;
337 all roots of quadratic eqn.
340 Polynomial::solve_quadric ()const
343 /* normal form: x^2 + px + q = 0 */
344 Real p = coefs_[1] / (2 * coefs_[2]);
345 Real q = coefs_[0] / coefs_[2];
353 sol.push_back (D - p);
354 sol.push_back (-D - p);
359 /* solve linear equation */
361 Polynomial::solve_linear ()const
365 s.push_back (-coefs_[0] / coefs_[1]);
370 Polynomial::solve () const
372 Polynomial *me = (Polynomial *) this;
378 return solve_linear ();
380 return solve_quadric ();
382 return solve_cubic ();
389 Polynomial::operator *= (Polynomial const &p2)
391 *this = multiply (*this, p2);