ROOT   6.10/09 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
30 int
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 (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 }
static double B[]
TArc * a
Definition: textangle.C:12
static double A[]
double cos(double)
double sqrt(double)
double acos(double)
double pow(double, double)
VecExpr< UnaryOp< Fabs< T >, VecExpr< A, T, D >, T >, T, D > fabs(const VecExpr< A, T, D > &rhs)
TRandom2 r(17)
#define M_PI
Definition: Rotated.cxx:105
unsigned int r1[N_CITIES]
Definition: simanTSP.cxx:321
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)
you should not use this method at all Int_t Int_t Double_t Double_t Double_t Int_t Double_t Double_t Double_t Double_t b
Definition: TRolke.cxx:630
float * q
Definition: THbookFile.cxx:87
double norm(double *x, double *p)
Definition: unuranDistr.cxx:40
TRandom3 R
a TMatrixD.
Definition: testIO.cxx:28
static double Q[]
unsigned int r2[N_CITIES]
Definition: simanTSP.cxx:322