ROOT   Reference Guide
zsolve_cubic.cxx
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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
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 <config.h>
23#include <math.h>
24#include <gsl/gsl_math.h>
25#include <gsl/gsl_complex.h>
26#include <gsl/gsl_poly.h>
27
28#define SWAP(a,b) do { double tmp = b ; b = a ; a = tmp ; } while(0)
29
30int
31gsl_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 (R2 == Q3)
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 (R2 < Q3) /* equivalent to R2 < Q3 */
89 {
90 double sqrtQ = sqrt (Q);
91 double sqrtQ3 = sqrtQ * sqrtQ * sqrtQ;
92 double ctheta = R / sqrtQ3;
93 double theta = 0;
94 if ( ctheta <= -1.0)
95 theta = M_PI;
96 else if ( ctheta < 1.0)
97 theta = acos (R / sqrtQ3);
98
99 double norm = -2 * sqrtQ;
100 double r0 = norm * cos (theta / 3) - a / 3;
101 double r1 = norm * cos ((theta + 2.0 * M_PI) / 3) - a / 3;
102 double r2 = norm * cos ((theta - 2.0 * M_PI) / 3) - a / 3;
103
104 /* Sort r0, r1, r2 into increasing order */
105
106 if (r0 > r1)
107 SWAP (r0, r1);
108
109 if (r1 > r2)
110 {
111 SWAP (r1, r2);
112
113 if (r0 > r1)
114 SWAP (r0, r1);
115 }
116
117 GSL_REAL (*z0) = r0;
118 GSL_IMAG (*z0) = 0;
119
120 GSL_REAL (*z1) = r1;
121 GSL_IMAG (*z1) = 0;
122
123 GSL_REAL (*z2) = r2;
124 GSL_IMAG (*z2) = 0;
125
126 return 3;
127 }
128 else
129 {
130 double sgnR = (R >= 0 ? 1 : -1);
131 double A = -sgnR * pow (fabs (R) + sqrt (R2 - Q3), 1.0 / 3.0);
132 double B = Q / A;
133
134 if (A + B < 0)
135 {
136 GSL_REAL (*z0) = A + B - a / 3;
137 GSL_IMAG (*z0) = 0;
138
139 GSL_REAL (*z1) = -0.5 * (A + B) - a / 3;
140 GSL_IMAG (*z1) = -(sqrt (3.0) / 2.0) * fabs(A - B);
141
142 GSL_REAL (*z2) = -0.5 * (A + B) - a / 3;
143 GSL_IMAG (*z2) = (sqrt (3.0) / 2.0) * fabs(A - B);
144 }
145 else
146 {
147 GSL_REAL (*z0) = -0.5 * (A + B) - a / 3;
148 GSL_IMAG (*z0) = -(sqrt (3.0) / 2.0) * fabs(A - B);
149
150 GSL_REAL (*z1) = -0.5 * (A + B) - a / 3;
151 GSL_IMAG (*z1) = (sqrt (3.0) / 2.0) * fabs(A - B);
152
153 GSL_REAL (*z2) = A + B - a / 3;
154 GSL_IMAG (*z2) = 0;
155 }
156
157 return 3;
158 }
159}
ROOT::R::TRInterface & r
Definition: Object.C:4
#define b(i)
Definition: RSha256.hxx:100
#define c(i)
Definition: RSha256.hxx:101
#define R(a, b, c, d, e, f, g, h, i)
Definition: RSha256.hxx:110
#define M_PI
Definition: Rotated.cxx:105
float * q
Definition: THbookFile.cxx:89
static double B[]
static double Q[]
static double A[]
VecExpr< UnaryOp< Sqrt< T >, VecExpr< A, T, D >, T >, T, D > sqrt(const VecExpr< A, T, D > &rhs)
VecExpr< UnaryOp< Fabs< T >, VecExpr< A, T, D >, T >, T, D > fabs(const VecExpr< A, T, D > &rhs)
#define R2(v, w, x, y, z, i)
Definition: sha1.inl:137
auto * a
Definition: textangle.C:12
int gsl_poly_complex_solve_cubic(double a, double b, double c, gsl_complex *z0, gsl_complex *z1, gsl_complex *z2)
#define SWAP(a, b)