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- /* poly/zsolve_cubic.c
- *
- * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2007 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 3 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 "gsl__config.h"
- #include <math.h>
- #include "gsl_math.h"
- #include "gsl_complex.h"
- #include "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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