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
|
/* linalg/qr_band.c
*
* Copyright (C) 2020 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 <config.h>
#include <stdlib.h>
#include <string.h>
#include <gsl/gsl_linalg.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
/* Factorise a (p,q) banded M x N matrix A into
*
* A = Q R
*
* where Q is orthogonal (M x M) and R is upper triangular (M x N).
*
* Example with M = 7, N = 6, (p,q) = (2,1)
*
* A = [ A11 A12 0 0 0 0 ]
* [ A21 A22 A23 0 0 0 ]
* [ A31 A32 A33 A34 0 0 ]
* [ 0 A42 A43 A44 A45 0 ]
* [ 0 0 A53 A54 A55 A56 ]
* [ 0 0 0 A64 A65 A66 ]
* [ 0 0 0 0 A75 A76 ]
*
* AB has dimensions N-by-(2p + q + 1)
*
* INPUT: OUTPUT:
*
* AB = [ * * * A11 A21 A31 ] AB = [ * * * R11 V21 V31 ]
* [ * * A12 A22 A32 A42 ] [ * * R12 R22 V32 V42 ]
* [ * 0 A23 A33 A43 A53 ] [ * R13 R23 R33 V43 V53 ]
* [ 0 0 A34 A44 A54 A64 ] [ R14 R24 R34 R44 V54 V64 ]
* [ 0 0 A45 A55 A65 A75 ] [ R25 R35 R45 R55 V65 V75 ]
* [ 0 0 A56 A66 A76 * ] [ R36 R46 R56 R66 V76 * ]
* -p-- -q- -1- ---p--- ---p--- -q- -1- ---p---
*
* The full matrix for Q can be obtained as the product
*
* Q = Q_k .. Q_2 Q_1
*
* where k = MIN(M,N) and
*
* Q_i = (I - tau_i * v_i * v_i')
*
* and where v_i is a Householder vector
*/
int
gsl_linalg_QR_band_decomp_L2 (const size_t M, const size_t p, const size_t q, gsl_matrix * AB, gsl_vector * tau)
{
const size_t N = AB->size1;
if (tau->size != N)
{
GSL_ERROR ("tau must have length N", GSL_EBADLEN);
}
else if (AB->size2 != 2*p + q + 1)
{
GSL_ERROR ("dimensions of AB are inconsistent with (p,q)", GSL_EBADLEN);
}
else
{
const size_t minMN = GSL_MIN(M, N);
size_t j;
/* set AB(:,1:p) to zero */
if (p > 0)
{
gsl_matrix_view m = gsl_matrix_submatrix(AB, 0, 0, N, p);
gsl_matrix_set_zero(&m.matrix);
}
for (j = 0; j < minMN; ++j)
{
/* Compute the Householder transformation to reduce the j-th
column of the matrix to a multiple of the j-th unit vector */
size_t k1 = GSL_MIN(p + 1, M - j); /* number of non-zero elements of this column, including diagonal element */
size_t k2 = GSL_MIN(p + q, N - j - 1); /* number of columns to update */
gsl_vector_view c = gsl_matrix_subrow(AB, j, p + q, k1);
double tau_j = gsl_linalg_householder_transform (&(c.vector));
double * ptr = gsl_vector_ptr(&(c.vector), 0);
gsl_vector_set (tau, j, tau_j);
/* apply the transformation to the remaining columns */
if (k2 > 0)
{
gsl_matrix_view m = gsl_matrix_submatrix (AB, j + 1, p + q - 1, k2, k1);
gsl_vector_view work = gsl_vector_subvector(tau, j + 1, k2);
double tmp = *ptr;
m.matrix.tda -= 1; /* unskew matrix */
/* we want to compute H*A(j:j+k1-1,j+1:j+k2), but due to our using row-major order, the
* matrix m contains A(j:j+k1-1,j+1:j+k2)^T. So therefore we apply H from the right,
*
* [H*A]^T = A^T H^T = A^T H
*/
*ptr = 1.0;
gsl_linalg_householder_right(tau_j, &(c.vector), &(m.matrix), &(work.vector));
*ptr = tmp;
}
}
return GSL_SUCCESS;
}
}
int
gsl_linalg_QR_band_unpack_L2 (const size_t p, const size_t q, const gsl_matrix * QRB, const gsl_vector * tau,
gsl_matrix * Q, gsl_matrix * R)
{
const size_t M = Q->size1;
const size_t N = QRB->size1;
if (Q->size2 != M)
{
GSL_ERROR ("Q matrix must be square", GSL_ENOTSQR);
}
else if (R->size1 != M || R->size2 != N)
{
GSL_ERROR ("R matrix must be M x N", GSL_ENOTSQR);
}
else if (tau->size < GSL_MIN (M, N))
{
GSL_ERROR ("size of tau must be at least MIN(M,N)", GSL_EBADLEN);
}
else if (QRB->size2 != 2*p + q + 1)
{
GSL_ERROR ("dimensions of QRB are inconsistent with (p,q)", GSL_EBADLEN);
}
else
{
size_t i;
/* form matrix Q */
gsl_matrix_set_identity (Q);
for (i = GSL_MIN (M, N); i-- > 0;)
{
size_t k1 = GSL_MIN(p + 1, M - i); /* number of non-zero elements of this column, including diagonal element */
gsl_vector_const_view h = gsl_matrix_const_subrow(QRB, i, p + q, k1);
gsl_matrix_view m = gsl_matrix_submatrix (Q, i, i, k1, M - i);
double ti = gsl_vector_get (tau, i);
gsl_vector_view work = gsl_matrix_subcolumn(R, 0, 0, M - i);
double * ptr = gsl_vector_ptr((gsl_vector *) &h.vector, 0);
double tmp = *ptr;
*ptr = 1.0;
gsl_linalg_householder_left (ti, &h.vector, &m.matrix, &work.vector);
*ptr = tmp;
}
/* form matrix R */
gsl_matrix_set_zero(R);
for (i = 0; i <= GSL_MIN(p + q, N - 1); ++i)
{
gsl_vector_const_view src = gsl_matrix_const_subcolumn(QRB, p + q - i, i, GSL_MIN(M, N - i));
gsl_vector_view dest = gsl_matrix_superdiagonal(R, i);
gsl_vector_memcpy(&dest.vector, &src.vector);
}
return GSL_SUCCESS;
}
}
|
