1 /* poly/zsolve_cubic.c
2 *
3 * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2007 Brian Gough
4 *
5 * This program is free software; you can redistribute it and/or modify
6 * it under the terms of the GNU General Public License as published by
7 * the Free Software Foundation; either version 3 of the License, or (at
8 * your option) any later version.
9 *
10 * This program is distributed in the hope that it will be useful, but
11 * WITHOUT ANY WARRANTY; without even the implied warranty of
12 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 * General Public License for more details.
14 *
15 * You should have received a copy of the GNU General Public License
16 * along with this program; if not, write to the Free Software
17 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
18 */
19
20 /* zsolve_cubic.c - finds the complex roots of x^3 + a x^2 + b x + c = 0 */
21
22 #include "gsl__config.h"
23 #include <math.h>
24 #include "gsl_math.h"
25 #include "gsl_complex.h"
26 #include "gsl_poly.h"
27
28 #define SWAP(a,b) do { double tmp = b ; b = a ; a = tmp ; } while(0)
29
30 int
gsl_poly_complex_solve_cubic(double a,double b,double c,gsl_complex * z0,gsl_complex * z1,gsl_complex * z2)31 gsl_poly_complex_solve_cubic (double a, double b, double c,
32 gsl_complex *z0, gsl_complex *z1,
33 gsl_complex *z2)
34 {
35 double q = (a * a - 3 * b);
36 double r = (2 * a * a * a - 9 * a * b + 27 * c);
37
38 double Q = q / 9;
39 double R = r / 54;
40
41 double Q3 = Q * Q * Q;
42 double R2 = R * R;
43
44 double CR2 = 729 * r * r;
45 double CQ3 = 2916 * q * q * q;
46
47 if (R == 0 && Q == 0)
48 {
49 GSL_REAL (*z0) = -a / 3;
50 GSL_IMAG (*z0) = 0;
51 GSL_REAL (*z1) = -a / 3;
52 GSL_IMAG (*z1) = 0;
53 GSL_REAL (*z2) = -a / 3;
54 GSL_IMAG (*z2) = 0;
55 return 3;
56 }
57 else if (CR2 == CQ3)
58 {
59 /* this test is actually R2 == Q3, written in a form suitable
60 for exact computation with integers */
61
62 /* Due to finite precision some double roots may be missed, and
63 will be considered to be a pair of complex roots z = x +/-
64 epsilon i close to the real axis. */
65
66 double sqrtQ = sqrt (Q);
67
68 if (R > 0)
69 {
70 GSL_REAL (*z0) = -2 * sqrtQ - a / 3;
71 GSL_IMAG (*z0) = 0;
72 GSL_REAL (*z1) = sqrtQ - a / 3;
73 GSL_IMAG (*z1) = 0;
74 GSL_REAL (*z2) = sqrtQ - a / 3;
75 GSL_IMAG (*z2) = 0;
76 }
77 else
78 {
79 GSL_REAL (*z0) = -sqrtQ - a / 3;
80 GSL_IMAG (*z0) = 0;
81 GSL_REAL (*z1) = -sqrtQ - a / 3;
82 GSL_IMAG (*z1) = 0;
83 GSL_REAL (*z2) = 2 * sqrtQ - a / 3;
84 GSL_IMAG (*z2) = 0;
85 }
86 return 3;
87 }
88 else if (CR2 < CQ3) /* equivalent to R2 < Q3 */
89 {
90 double sqrtQ = sqrt (Q);
91 double sqrtQ3 = sqrtQ * sqrtQ * sqrtQ;
92 double theta = acos (R / sqrtQ3);
93 double norm = -2 * sqrtQ;
94 double r0 = norm * cos (theta / 3) - a / 3;
95 double r1 = norm * cos ((theta + 2.0 * M_PI) / 3) - a / 3;
96 double r2 = norm * cos ((theta - 2.0 * M_PI) / 3) - a / 3;
97
98 /* Sort r0, r1, r2 into increasing order */
99
100 if (r0 > r1)
101 SWAP (r0, r1);
102
103 if (r1 > r2)
104 {
105 SWAP (r1, r2);
106
107 if (r0 > r1)
108 SWAP (r0, r1);
109 }
110
111 GSL_REAL (*z0) = r0;
112 GSL_IMAG (*z0) = 0;
113
114 GSL_REAL (*z1) = r1;
115 GSL_IMAG (*z1) = 0;
116
117 GSL_REAL (*z2) = r2;
118 GSL_IMAG (*z2) = 0;
119
120 return 3;
121 }
122 else
123 {
124 double sgnR = (R >= 0 ? 1 : -1);
125 double A = -sgnR * pow (fabs (R) + sqrt (R2 - Q3), 1.0 / 3.0);
126 double B = Q / A;
127
128 if (A + B < 0)
129 {
130 GSL_REAL (*z0) = A + B - a / 3;
131 GSL_IMAG (*z0) = 0;
132
133 GSL_REAL (*z1) = -0.5 * (A + B) - a / 3;
134 GSL_IMAG (*z1) = -(sqrt (3.0) / 2.0) * fabs(A - B);
135
136 GSL_REAL (*z2) = -0.5 * (A + B) - a / 3;
137 GSL_IMAG (*z2) = (sqrt (3.0) / 2.0) * fabs(A - B);
138 }
139 else
140 {
141 GSL_REAL (*z0) = -0.5 * (A + B) - a / 3;
142 GSL_IMAG (*z0) = -(sqrt (3.0) / 2.0) * fabs(A - B);
143
144 GSL_REAL (*z1) = -0.5 * (A + B) - a / 3;
145 GSL_IMAG (*z1) = (sqrt (3.0) / 2.0) * fabs(A - B);
146
147 GSL_REAL (*z2) = A + B - a / 3;
148 GSL_IMAG (*z2) = 0;
149 }
150
151 return 3;
152 }
153 }
154