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/* poly/zsolve_cubic.c
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*
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* Copyright (C) 1996, 1997, 1998, 1999, 2000, 2007, 2009 Brian Gough
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation; either version 3 of the License, or (at
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* your option) any later version.
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*
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* This program is distributed in the hope that it will be useful, but
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* WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program; if not, write to the Free Software
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* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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*/
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/* zsolve_cubic.c - finds the complex roots of x^3 + a x^2 + b x + c = 0 */
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#include <config.h>
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#include <math.h>
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#include <gsl/gsl_math.h>
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#include <gsl/gsl_complex.h>
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#include <gsl/gsl_poly.h>
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#define SWAP(a,b) do { double tmp = b ; b = a ; a = tmp ; } while(0)
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int
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gsl_poly_complex_solve_cubic (double a, double b, double c,
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gsl_complex *z0, gsl_complex *z1,
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gsl_complex *z2)
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{
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double q = (a * a - 3 * b);
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double r = (2 * a * a * a - 9 * a * b + 27 * c);
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double Q = q / 9;
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double R = r / 54;
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double Q3 = Q * Q * Q;
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double R2 = R * R;
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double CR2 = 729 * r * r;
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double CQ3 = 2916 * q * q * q;
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if (R == 0 && Q == 0)
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{
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GSL_REAL (*z0) = -a / 3;
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GSL_IMAG (*z0) = 0;
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GSL_REAL (*z1) = -a / 3;
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GSL_IMAG (*z1) = 0;
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GSL_REAL (*z2) = -a / 3;
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GSL_IMAG (*z2) = 0;
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return 3;
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}
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else if (CR2 == CQ3)
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{
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/* this test is actually R2 == Q3, written in a form suitable
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for exact computation with integers */
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/* Due to finite precision some double roots may be missed, and
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will be considered to be a pair of complex roots z = x +/-
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epsilon i close to the real axis. */
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double sqrtQ = sqrt (Q);
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if (R > 0)
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{
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GSL_REAL (*z0) = -2 * sqrtQ - a / 3;
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GSL_IMAG (*z0) = 0;
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GSL_REAL (*z1) = sqrtQ - a / 3;
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GSL_IMAG (*z1) = 0;
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GSL_REAL (*z2) = sqrtQ - a / 3;
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GSL_IMAG (*z2) = 0;
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}
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else
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{
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GSL_REAL (*z0) = -sqrtQ - a / 3;
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GSL_IMAG (*z0) = 0;
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GSL_REAL (*z1) = -sqrtQ - a / 3;
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GSL_IMAG (*z1) = 0;
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GSL_REAL (*z2) = 2 * sqrtQ - a / 3;
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GSL_IMAG (*z2) = 0;
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}
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return 3;
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}
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else if (R2 < Q3)
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{
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double sgnR = (R >= 0 ? 1 : -1);
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double ratio = sgnR * sqrt (R2 / Q3);
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double theta = acos (ratio);
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double norm = -2 * sqrt (Q);
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double r0 = norm * cos (theta / 3) - a / 3;
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double r1 = norm * cos ((theta + 2.0 * M_PI) / 3) - a / 3;
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double r2 = norm * cos ((theta - 2.0 * M_PI) / 3) - a / 3;
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/* Sort r0, r1, r2 into increasing order */
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if (r0 > r1)
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SWAP (r0, r1);
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if (r1 > r2)
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{
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SWAP (r1, r2);
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if (r0 > r1)
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SWAP (r0, r1);
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}
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GSL_REAL (*z0) = r0;
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GSL_IMAG (*z0) = 0;
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GSL_REAL (*z1) = r1;
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GSL_IMAG (*z1) = 0;
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GSL_REAL (*z2) = r2;
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GSL_IMAG (*z2) = 0;
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return 3;
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}
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else
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{
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double sgnR = (R >= 0 ? 1 : -1);
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double A = -sgnR * pow (fabs (R) + sqrt (R2 - Q3), 1.0 / 3.0);
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double B = Q / A;
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if (A + B < 0)
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{
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GSL_REAL (*z0) = A + B - a / 3;
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GSL_IMAG (*z0) = 0;
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GSL_REAL (*z1) = -0.5 * (A + B) - a / 3;
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GSL_IMAG (*z1) = -(sqrt (3.0) / 2.0) * fabs(A - B);
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GSL_REAL (*z2) = -0.5 * (A + B) - a / 3;
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GSL_IMAG (*z2) = (sqrt (3.0) / 2.0) * fabs(A - B);
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}
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else
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{
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GSL_REAL (*z0) = -0.5 * (A + B) - a / 3;
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GSL_IMAG (*z0) = -(sqrt (3.0) / 2.0) * fabs(A - B);
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GSL_REAL (*z1) = -0.5 * (A + B) - a / 3;
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GSL_IMAG (*z1) = (sqrt (3.0) / 2.0) * fabs(A - B);
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GSL_REAL (*z2) = A + B - a / 3;
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GSL_IMAG (*z2) = 0;
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}
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return 3;
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}
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}
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