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/* poly/zsolve_cubic.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 Brian Gough
*
* This program 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 2 of the License, or (at
* your option) any later version.
*
* This program 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 this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
/* zsolve_cubic.c - finds the complex roots of x^3 + a x^2 + b x + c = 0 */
#include <config.h>
#include <math.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_complex.h>
#include <gsl/gsl_poly.h>
#define SWAP(a,b) do { double tmp = b ; b = a ; a = tmp ; } while(0)
int
gsl_poly_complex_solve_cubic (double a, double b, double c,
gsl_complex *z0, gsl_complex *z1,
gsl_complex *z2)
{
double q = (a * a - 3 * b);
double r = (2 * a * a * a - 9 * a * b + 27 * c);
double Q = q / 9;
double R = r / 54;
double Q3 = Q * Q * Q;
double R2 = R * R;
double CR2 = 729 * r * r;
double CQ3 = 2916 * q * q * q;
if (R == 0 && Q == 0)
{
GSL_REAL (*z0) = -a / 3;
GSL_IMAG (*z0) = 0;
GSL_REAL (*z1) = -a / 3;
GSL_IMAG (*z1) = 0;
GSL_REAL (*z2) = -a / 3;
GSL_IMAG (*z2) = 0;
return 3;
}
else if (CR2 == CQ3)
{
/* this test is actually R2 == Q3, written in a form suitable
for exact computation with integers */
/* Due to finite precision some double roots may be missed, and
will be considered to be a pair of complex roots z = x +/-
epsilon i close to the real axis. */
double sqrtQ = sqrt (Q);
if (R > 0)
{
GSL_REAL (*z0) = -2 * sqrtQ - a / 3;
GSL_IMAG (*z0) = 0;
GSL_REAL (*z1) = sqrtQ - a / 3;
GSL_IMAG (*z1) = 0;
GSL_REAL (*z2) = sqrtQ - a / 3;
GSL_IMAG (*z2) = 0;
}
else
{
GSL_REAL (*z0) = -sqrtQ - a / 3;
GSL_IMAG (*z0) = 0;
GSL_REAL (*z1) = -sqrtQ - a / 3;
GSL_IMAG (*z1) = 0;
GSL_REAL (*z2) = 2 * sqrtQ - a / 3;
GSL_IMAG (*z2) = 0;
}
return 3;
}
else if (CR2 < CQ3) /* equivalent to R2 < Q3 */
{
double sqrtQ = sqrt (Q);
double sqrtQ3 = sqrtQ * sqrtQ * sqrtQ;
double theta = acos (R / sqrtQ3);
double norm = -2 * sqrtQ;
double r0 = norm * cos (theta / 3) - a / 3;
double r1 = norm * cos ((theta + 2.0 * M_PI) / 3) - a / 3;
double r2 = norm * cos ((theta - 2.0 * M_PI) / 3) - a / 3;
/* Sort r0, r1, r2 into increasing order */
if (r0 > r1)
SWAP (r0, r1);
if (r1 > r2)
{
SWAP (r1, r2);
if (r0 > r1)
SWAP (r0, r1);
}
GSL_REAL (*z0) = r0;
GSL_IMAG (*z0) = 0;
GSL_REAL (*z1) = r1;
GSL_IMAG (*z1) = 0;
GSL_REAL (*z2) = r2;
GSL_IMAG (*z2) = 0;
return 3;
}
else
{
double sgnR = (R >= 0 ? 1 : -1);
double A = -sgnR * pow (fabs (R) + sqrt (R2 - Q3), 1.0 / 3.0);
double B = Q / A;
if (A + B < 0)
{
GSL_REAL (*z0) = A + B - a / 3;
GSL_IMAG (*z0) = 0;
GSL_REAL (*z1) = -0.5 * (A + B) - a / 3;
GSL_IMAG (*z1) = -(sqrt (3.0) / 2.0) * fabs(A - B);
GSL_REAL (*z2) = -0.5 * (A + B) - a / 3;
GSL_IMAG (*z2) = (sqrt (3.0) / 2.0) * fabs(A - B);
}
else
{
GSL_REAL (*z0) = -0.5 * (A + B) - a / 3;
GSL_IMAG (*z0) = -(sqrt (3.0) / 2.0) * fabs(A - B);
GSL_REAL (*z1) = -0.5 * (A + B) - a / 3;
GSL_IMAG (*z1) = (sqrt (3.0) / 2.0) * fabs(A - B);
GSL_REAL (*z2) = A + B - a / 3;
GSL_IMAG (*z2) = 0;
}
return 3;
}
}
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