| /* 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; |
| } |
| } |