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
|
/* linalg/balance.c
*
* Copyright (C) 2006 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 2 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.
*/
/* Balance a general matrix by scaling the rows and columns, so the
* new row and column norms are the same order of magnitude.
*
* B = D^-1 A D
*
* where D is a diagonal matrix
*
* This is necessary for the unsymmetric eigenvalue problem since the
* calculation can become numerically unstable for unbalanced
* matrices.
*
* See Golub & Van Loan, "Matrix Computations" (3rd ed), Section 7.5.7
* and Wilkinson & Reinsch, "Handbook for Automatic Computation", II/11 p320.
*/
#include <config.h>
#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_linalg.h>
#define FLOAT_RADIX 2.0
#define FLOAT_RADIX_SQ (FLOAT_RADIX * FLOAT_RADIX)
int
gsl_linalg_balance_matrix(gsl_matrix * A, gsl_vector * D)
{
const size_t N = A->size1;
if (N != D->size)
{
GSL_ERROR ("vector must match matrix size", GSL_EBADLEN);
}
else
{
double row_norm,
col_norm;
int not_converged;
gsl_vector_view v;
/* initialize D to the identity matrix */
gsl_vector_set_all(D, 1.0);
not_converged = 1;
while (not_converged)
{
size_t i, j;
double g, f, s;
not_converged = 0;
for (i = 0; i < N; ++i)
{
row_norm = 0.0;
col_norm = 0.0;
for (j = 0; j < N; ++j)
{
if (j != i)
{
col_norm += fabs(gsl_matrix_get(A, j, i));
row_norm += fabs(gsl_matrix_get(A, i, j));
}
}
if ((col_norm == 0.0) || (row_norm == 0.0))
{
continue;
}
g = row_norm / FLOAT_RADIX;
f = 1.0;
s = col_norm + row_norm;
/*
* find the integer power of the machine radix which
* comes closest to balancing the matrix
*/
while (col_norm < g)
{
f *= FLOAT_RADIX;
col_norm *= FLOAT_RADIX_SQ;
}
g = row_norm * FLOAT_RADIX;
while (col_norm > g)
{
f /= FLOAT_RADIX;
col_norm /= FLOAT_RADIX_SQ;
}
if ((row_norm + col_norm) < 0.95 * s * f)
{
not_converged = 1;
g = 1.0 / f;
/*
* apply similarity transformation D, where
* D_{ij} = f_i * delta_{ij}
*/
/* multiply by D^{-1} on the left */
v = gsl_matrix_row(A, i);
gsl_blas_dscal(g, &v.vector);
/* multiply by D on the right */
v = gsl_matrix_column(A, i);
gsl_blas_dscal(f, &v.vector);
/* keep track of transformation */
gsl_vector_set(D, i, gsl_vector_get(D, i) * f);
}
}
}
return GSL_SUCCESS;
}
} /* gsl_linalg_balance_matrix() */
/*
gsl_linalg_balance_accum()
Accumulate a balancing transformation into a matrix.
This is used during the computation of Schur vectors since the
Schur vectors computed are the vectors for the balanced matrix.
We must at some point accumulate the balancing transformation into
the Schur vector matrix to get the vectors for the original matrix.
A -> D A
where D is the diagonal matrix
Inputs: A - matrix to transform
D - vector containing diagonal elements of D
*/
int
gsl_linalg_balance_accum(gsl_matrix *A, gsl_vector *D)
{
const size_t N = A->size1;
if (N != D->size)
{
GSL_ERROR ("vector must match matrix size", GSL_EBADLEN);
}
else
{
size_t i;
double s;
gsl_vector_view r;
for (i = 0; i < N; ++i)
{
s = gsl_vector_get(D, i);
r = gsl_matrix_row(A, i);
gsl_blas_dscal(s, &r.vector);
}
return GSL_SUCCESS;
}
} /* gsl_linalg_balance_accum() */
|
