1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
|
/* eigen/gensymm.c
*
* Copyright (C) 2007 Patrick Alken
*
* 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 3 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.
*/
#include <stdlib.h>
#include <config.h>
#include <gsl/gsl_eigen.h>
#include <gsl/gsl_linalg.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
/*
* This module computes the eigenvalues of a real generalized
* symmetric-definite eigensystem A x = \lambda B x, where A and
* B are symmetric, and B is positive-definite.
*/
/*
gsl_eigen_gensymm_alloc()
Allocate a workspace for solving the generalized symmetric-definite
eigenvalue problem. The size of this workspace is O(2n).
Inputs: n - size of matrices
Return: pointer to workspace
*/
gsl_eigen_gensymm_workspace *
gsl_eigen_gensymm_alloc(const size_t n)
{
gsl_eigen_gensymm_workspace *w;
if (n == 0)
{
GSL_ERROR_NULL ("matrix dimension must be positive integer",
GSL_EINVAL);
}
w = (gsl_eigen_gensymm_workspace *) calloc (1, sizeof (gsl_eigen_gensymm_workspace));
if (w == 0)
{
GSL_ERROR_NULL ("failed to allocate space for workspace", GSL_ENOMEM);
}
w->size = n;
w->symm_workspace_p = gsl_eigen_symm_alloc(n);
if (!w->symm_workspace_p)
{
gsl_eigen_gensymm_free(w);
GSL_ERROR_NULL("failed to allocate space for symm workspace", GSL_ENOMEM);
}
return (w);
} /* gsl_eigen_gensymm_alloc() */
/*
gsl_eigen_gensymm_free()
Free workspace w
*/
void
gsl_eigen_gensymm_free (gsl_eigen_gensymm_workspace * w)
{
RETURN_IF_NULL (w);
if (w->symm_workspace_p)
gsl_eigen_symm_free(w->symm_workspace_p);
free(w);
} /* gsl_eigen_gensymm_free() */
/*
gsl_eigen_gensymm()
Solve the generalized symmetric-definite eigenvalue problem
A x = \lambda B x
for the eigenvalues \lambda.
Inputs: A - real symmetric matrix
B - real symmetric and positive definite matrix
eval - where to store eigenvalues
w - workspace
Return: success or error
*/
int
gsl_eigen_gensymm (gsl_matrix * A, gsl_matrix * B, gsl_vector * eval,
gsl_eigen_gensymm_workspace * w)
{
const size_t N = A->size1;
/* check matrix and vector sizes */
if (N != A->size2)
{
GSL_ERROR ("matrix must be square to compute eigenvalues", GSL_ENOTSQR);
}
else if ((N != B->size1) || (N != B->size2))
{
GSL_ERROR ("B matrix dimensions must match A", GSL_EBADLEN);
}
else if (eval->size != N)
{
GSL_ERROR ("eigenvalue vector must match matrix size", GSL_EBADLEN);
}
else if (w->size != N)
{
GSL_ERROR ("matrix size does not match workspace", GSL_EBADLEN);
}
else
{
int s;
/* compute Cholesky factorization of B */
s = gsl_linalg_cholesky_decomp1(B);
if (s != GSL_SUCCESS)
return s; /* B is not positive definite */
/* transform to standard symmetric eigenvalue problem */
gsl_eigen_gensymm_standardize(A, B);
s = gsl_eigen_symm(A, eval, w->symm_workspace_p);
return s;
}
} /* gsl_eigen_gensymm() */
/*
gsl_eigen_gensymm_standardize()
Reduce the generalized symmetric-definite eigenproblem to
the standard symmetric eigenproblem by computing
C = L^{-1} A L^{-t}
where L L^t is the Cholesky decomposition of B
Inputs: A - (input/output) real symmetric matrix
B - real symmetric, positive definite matrix in Cholesky form
Return: success
Notes: A is overwritten by L^{-1} A L^{-t}
*/
int
gsl_eigen_gensymm_standardize(gsl_matrix *A, const gsl_matrix *B)
{
const size_t N = A->size1;
size_t i;
double a, b, c;
for (i = 0; i < N; ++i)
{
/* update lower triangle of A(i:n, i:n) */
a = gsl_matrix_get(A, i, i);
b = gsl_matrix_get(B, i, i);
a /= b * b;
gsl_matrix_set(A, i, i, a);
if (i < N - 1)
{
gsl_vector_view ai = gsl_matrix_subcolumn(A, i, i + 1, N - i - 1);
gsl_matrix_view ma =
gsl_matrix_submatrix(A, i + 1, i + 1, N - i - 1, N - i - 1);
gsl_vector_const_view bi =
gsl_matrix_const_subcolumn(B, i, i + 1, N - i - 1);
gsl_matrix_const_view mb =
gsl_matrix_const_submatrix(B, i + 1, i + 1, N - i - 1, N - i - 1);
gsl_blas_dscal(1.0 / b, &ai.vector);
c = -0.5 * a;
gsl_blas_daxpy(c, &bi.vector, &ai.vector);
gsl_blas_dsyr2(CblasLower, -1.0, &ai.vector, &bi.vector, &ma.matrix);
gsl_blas_daxpy(c, &bi.vector, &ai.vector);
gsl_blas_dtrsv(CblasLower,
CblasNoTrans,
CblasNonUnit,
&mb.matrix,
&ai.vector);
}
}
return GSL_SUCCESS;
} /* gsl_eigen_gensymm_standardize() */
|
