/*
- poly.cc -- routines for manipulation of polynomials in one var
+ This file is part of LilyPond, the GNU music typesetter.
- (c) 1993--2003 Han-Wen Nienhuys <hanwen@cs.uu.nl>
- */
+ Copyright (C) 1993--2015 Han-Wen Nienhuys <hanwen@xs4all.nl>
-#include <math.h>
+ LilyPond is free software: you can redistribute it and/or modify
+ it under the terms of the GNU General Public License as published by
+ the Free Software Foundation, either version 3 of the License, or
+ (at your option) any later version.
+
+ LilyPond is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ GNU General Public License for more details.
+
+ You should have received a copy of the GNU General Public License
+ along with LilyPond. If not, see <http://www.gnu.org/licenses/>.
+*/
#include "polynomial.hh"
+#include "warn.hh"
+
+#include <cmath>
+
+using namespace std;
+
/*
- Een beter milieu begint bij uzelf. Hergebruik!
+ Een beter milieu begint bij uzelf. Hergebruik!
- This was ripped from Rayce, a raytracer I once wrote.
+ This was ripped from Rayce, a raytracer I once wrote.
*/
Real
-Polynomial::eval (Real x)const
+Polynomial::eval (Real x) const
{
Real p = 0.0;
// horner's scheme
- for (int i = coefs_.size (); i--; )
+ for (vsize i = coefs_.size (); i--;)
p = x * p + coefs_[i];
-
+
return p;
}
-
Polynomial
-Polynomial::multiply (const Polynomial & p1, const Polynomial & p2)
+Polynomial::multiply (const Polynomial &p1, const Polynomial &p2)
{
Polynomial dest;
- int deg= p1.degree () + p2.degree ();
- for (int i = 0; i <= deg; i++)
+ ssize_t deg = p1.degree () + p2.degree ();
+ for (ssize_t i = 0; i <= deg; i++)
{
- dest.coefs_.push (0);
- for (int j = 0; j <= i; j++)
- if (i - j <= p2.degree () && j <= p1.degree ())
- dest.coefs_.top () += p1.coefs_[j] * p2.coefs_[i - j];
+ dest.coefs_.push_back (0);
+ for (ssize_t j = 0; j <= i; j++)
+ if (i - j <= p2.degree () && j <= p1.degree ())
+ dest.coefs_.back () += p1.coefs_[j] * p2.coefs_[i - j];
}
-
+
return dest;
}
-void
-Polynomial::differentiate ()
+Real
+Polynomial::minmax (Real l, Real r, bool ret_max) const
{
- for (int i = 1; i<= degree (); i++)
+ vector<Real> sols;
+ if (l > r)
{
- coefs_[i-1] = coefs_[i] * i;
+ programming_error ("left bound greater than right bound for polynomial minmax. flipping bounds.");
+ l = l + r;
+ r = l - r;
+ l = l - r;
}
- coefs_.pop ();
+
+ sols.push_back (eval (l));
+ sols.push_back (eval (r));
+
+ Polynomial deriv (*this);
+ deriv.differentiate ();
+ vector<Real> maxmins = deriv.solve ();
+ for (vsize i = 0; i < maxmins.size (); i++)
+ if (maxmins[i] >= l && maxmins[i] <= r)
+ sols.push_back (eval (maxmins[i]));
+ vector_sort (sols, less<Real> ());
+
+ return ret_max ? sols.back () : sols[0];
+}
+
+void
+Polynomial::differentiate ()
+{
+ for (int i = 1; i <= degree (); i++)
+ coefs_[i - 1] = coefs_[i] * i;
+ coefs_.pop_back ();
}
Polynomial
-Polynomial::power (int exponent, const Polynomial & src)
+Polynomial::power (int exponent, const Polynomial &src)
{
int e = exponent;
Polynomial dest (1), base (src);
-
+
/*
classic int power. invariant: src^exponent = dest * src ^ e
greetings go out to Lex Bijlsma & Jaap vd Woude */
{
if (e % 2)
{
- dest = multiply (dest, base);
- e--;
- } else
- {
- base = multiply (base, base);
- e /= 2;
- }
+ dest = multiply (dest, base);
+ e--;
+ }
+ else
+
+ {
+ base = multiply (base, base);
+ e /= 2;
+ }
}
- return dest;
+ return dest;
}
static Real const FUDGE = 1e-8;
void
Polynomial::clean ()
{
-/*
- We only do relative comparisons. Absolute comparisons break down in
- degenerate cases. */
- while (degree () > 0 &&
- (fabs (coefs_.top ()) < FUDGE * fabs (coefs_.top (1))
- || !coefs_.top ()))
- coefs_.pop ();
+ /*
+ We only do relative comparisons. Absolute comparisons break down in
+ degenerate cases. */
+ while (degree () > 0
+ && (fabs (coefs_.back ()) < FUDGE * fabs (back (coefs_, 1))
+ || !coefs_.back ()))
+ coefs_.pop_back ();
}
-
-
void
Polynomial::operator += (Polynomial const &p)
{
- while (degree () < p.degree())
- coefs_.push (0.0);
+ while (degree () < p.degree ())
+ coefs_.push_back (0.0);
- for (int i = 0; i <= p.degree(); i++)
+ for (int i = 0; i <= p.degree (); i++)
coefs_[i] += p.coefs_[i];
}
-
void
Polynomial::operator -= (Polynomial const &p)
{
- while (degree () < p.degree())
- coefs_.push (0.0);
+ while (degree () < p.degree ())
+ coefs_.push_back (0.0);
- for (int i = 0; i <= p.degree(); i++)
+ for (int i = 0; i <= p.degree (); i++)
coefs_[i] -= p.coefs_[i];
}
-
void
Polynomial::scalarmultiply (Real fact)
{
coefs_[i] *= fact;
}
-
-
void
-Polynomial::set_negate (const Polynomial & src)
+Polynomial::set_negate (const Polynomial &src)
{
for (int i = 0; i <= src.degree (); i++)
coefs_[i] = -src.coefs_[i];
}
/// mod of #u/v#
-int
+int
Polynomial::set_mod (const Polynomial &u, const Polynomial &v)
{
- (*this) = u;
-
- if (v.lc () < 0.0) {
- for (int k = u.degree () - v.degree () - 1; k >= 0; k -= 2)
- coefs_[k] = -coefs_[k];
-
- for (int k = u.degree () - v.degree (); k >= 0; k--)
- for (int j = v.degree () + k - 1; j >= k; j--)
- coefs_[j] = -coefs_[j] - coefs_[v.degree () + k] * v.coefs_[j - k];
- } else {
- for (int k = u.degree () - v.degree (); k >= 0; k--)
- for (int j = v.degree () + k - 1; j >= k; j--)
- coefs_[j] -= coefs_[v.degree () + k] * v.coefs_[j - k];
- }
-
- int k = v.degree () - 1;
+ (*this) = u;
+
+ if (v.lc () < 0.0)
+ {
+ for (ssize_t k = u.degree () - v.degree () - 1; k >= 0; k -= 2)
+ coefs_[k] = -coefs_[k];
+
+ for (ssize_t k = u.degree () - v.degree (); k >= 0; k--)
+ for (ssize_t j = v.degree () + k - 1; j >= k; j--)
+ coefs_[j] = -coefs_[j] - coefs_[v.degree () + k] * v.coefs_[j - k];
+ }
+ else
+
+ {
+ for (ssize_t k = u.degree () - v.degree (); k >= 0; k--)
+ for (ssize_t j = v.degree () + k - 1; j >= k; j--)
+ coefs_[j] -= coefs_[v.degree () + k] * v.coefs_[j - k];
+ }
+
+ ssize_t k = v.degree () - 1;
while (k >= 0 && coefs_[k] == 0.0)
k--;
- coefs_.set_size (1+ ((k < 0) ? 0 : k));
+ coefs_.resize (1 + ((k < 0) ? 0 : k));
return degree ();
}
void
-Polynomial::check_sol (Real x) const
+Polynomial::check_sol (Real x) const
{
- Real f=eval (x);
+ Real f = eval (x);
Polynomial p (*this);
p.differentiate ();
Real d = p.eval (x);
-
- if ( abs (f) > abs (d) * FUDGE)
- ;
- /*
- warning ("x=%f is not a root of polynomial\n"
- "f (x)=%f, f' (x)=%f \n", x, f, d); */
+
+ if (abs (f) > abs (d) * FUDGE)
+ programming_error ("not a root of polynomial\n");
}
-
+
void
-Polynomial::check_sols (Array<Real> roots) const
+Polynomial::check_sols (vector<Real> roots) const
{
- for (int i=0; i< roots.size (); i++)
+ for (vsize i = 0; i < roots.size (); i++)
check_sol (roots[i]);
}
Polynomial::Polynomial (Real a, Real b)
{
- coefs_.push (a);
+ coefs_.push_back (a);
if (b)
- coefs_.push (b);
+ coefs_.push_back (b);
}
/* cubic root. */
inline Real cubic_root (Real x)
{
if (x > 0.0)
- return pow (x, 1.0/3.0) ;
+ return pow (x, 1.0 / 3.0);
else if (x < 0.0)
- return -pow (-x, 1.0/3.0);
- else
- return 0.0;
+ return -pow (-x, 1.0 / 3.0);
+ return 0.0;
}
static bool
return !r;
}
-Array<Real>
+vector<Real>
Polynomial::solve_cubic ()const
{
- Array<Real> sol;
-
+ vector<Real> sol;
+
/* normal form: x^3 + Ax^2 + Bx + C = 0 */
Real A = coefs_[2] / coefs_[3];
Real B = coefs_[1] / coefs_[3];
Real cb = p * p * p;
Real D = q * q + cb;
- if (iszero (D)) {
- if (iszero (q)) { /* one triple solution */
- sol.push (0);
- sol.push (0);
- sol.push (0);
- } else { /* one single and one double solution */
- Real u = cubic_root (-q);
+ if (iszero (D))
+ {
+ if (iszero (q)) /* one triple solution */
+ {
+ sol.push_back (0);
+ sol.push_back (0);
+ sol.push_back (0);
+ }
+ else /* one single and one double solution */
+ {
+ Real u = cubic_root (-q);
- sol.push (2 * u);
- sol.push (-u);
+ sol.push_back (2 * u);
+ sol.push_back (-u);
+ }
}
- } else if (D < 0) { /* Casus irreducibilis: three real solutions */
- Real phi = 1.0 / 3 * acos (-q / sqrt (-cb));
- Real t = 2 * sqrt (-p);
+ else if (D < 0)
+ {
+ /* Casus irreducibilis: three real solutions */
+ Real phi = 1.0 / 3 * acos (-q / sqrt (-cb));
+ Real t = 2 * sqrt (-p);
- sol.push (t * cos (phi));
- sol.push (-t * cos (phi + M_PI / 3));
- sol.push ( -t * cos (phi - M_PI / 3));
- } else { /* one real solution */
- Real sqrt_D = sqrt (D);
- Real u = cubic_root (sqrt_D - q);
- Real v = -cubic_root (sqrt_D + q);
+ sol.push_back (t * cos (phi));
+ sol.push_back (-t * cos (phi + M_PI / 3));
+ sol.push_back (-t * cos (phi - M_PI / 3));
+ }
+ else
+ {
+ /* one real solution */
+ Real sqrt_D = sqrt (D);
+ Real u = cubic_root (sqrt_D - q);
+ Real v = -cubic_root (sqrt_D + q);
- sol.push ( u + v);
- }
+ sol.push_back (u + v);
+ }
/* resubstitute */
Real sub = 1.0 / 3 * A;
- for (int i = sol.size (); i--;)
+ for (vsize i = sol.size (); i--;)
{
sol[i] -= sub;
#ifdef PARANOID
- assert (fabs (eval (sol[i]) ) < 1e-8);
+ assert (fabs (eval (sol[i])) < 1e-8);
#endif
}
-
+
return sol;
}
Real
Polynomial::lc () const
{
- return coefs_.top ();
+ return coefs_.back ();
}
-Real&
-Polynomial::lc ()
+Real &
+Polynomial::lc ()
{
- return coefs_.top ();
+ return coefs_.back ();
}
-int
+ssize_t
Polynomial::degree ()const
{
- return coefs_.size () -1;
+ return coefs_.size () - 1;
}
/*
- all roots of quadratic eqn.
- */
-Array<Real>
+ all roots of quadratic eqn.
+*/
+vector<Real>
Polynomial::solve_quadric ()const
{
- Array<Real> sol;
+ vector<Real> sol;
/* normal form: x^2 + px + q = 0 */
Real p = coefs_[1] / (2 * coefs_[2]);
Real q = coefs_[0] / coefs_[2];
Real D = p * p - q;
- if (D>0) {
- D = sqrt (D);
+ if (D > 0)
+ {
+ D = sqrt (D);
- sol.push ( D - p);
- sol.push ( -D - p);
- }
- return sol;
+ sol.push_back (D - p);
+ sol.push_back (-D - p);
+ }
+ return sol;
}
/* solve linear equation */
-Array<Real>
+vector<Real>
Polynomial::solve_linear ()const
{
- Array<Real> s;
+ vector<Real> s;
if (coefs_[1])
- s.push ( -coefs_[0] / coefs_[1]);
+ s.push_back (-coefs_[0] / coefs_[1]);
return s;
}
-
-Array<Real>
+vector<Real>
Polynomial::solve () const
{
- Polynomial * me = (Polynomial*) this;
+ Polynomial *me = (Polynomial *) this;
me->clean ();
-
+
switch (degree ())
{
case 1:
case 3:
return solve_cubic ();
}
- Array<Real> s;
+ vector<Real> s;
return s;
}
void
-Polynomial:: operator *= (Polynomial const &p2)
+Polynomial::operator *= (Polynomial const &p2)
{
- *this = multiply (*this,p2);
-}
-
+ *this = multiply (*this, p2);
+}
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