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
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
|
*> \brief \b SQRT01P
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE SQRT01P( M, N, A, AF, Q, R, LDA, TAU, WORK, LWORK,
* RWORK, RESULT )
*
* .. Scalar Arguments ..
* INTEGER LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
* $ R( LDA, * ), RESULT( * ), RWORK( * ), TAU( * ),
* $ WORK( LWORK )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SQRT01P tests SGEQRFP, which computes the QR factorization of an m-by-n
*> matrix A, and partially tests SORGQR which forms the m-by-m
*> orthogonal matrix Q.
*>
*> SQRT01P compares R with Q'*A, and checks that Q is orthogonal.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is REAL array, dimension (LDA,N)
*> The m-by-n matrix A.
*> \endverbatim
*>
*> \param[out] AF
*> \verbatim
*> AF is REAL array, dimension (LDA,N)
*> Details of the QR factorization of A, as returned by SGEQRFP.
*> See SGEQRFP for further details.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is REAL array, dimension (LDA,M)
*> The m-by-m orthogonal matrix Q.
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*> R is REAL array, dimension (LDA,max(M,N))
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the arrays A, AF, Q and R.
*> LDA >= max(M,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is REAL array, dimension (min(M,N))
*> The scalar factors of the elementary reflectors, as returned
*> by SGEQRFP.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (M)
*> \endverbatim
*>
*> \param[out] RESULT
*> \verbatim
*> RESULT is REAL array, dimension (2)
*> The test ratios:
*> RESULT(1) = norm( R - Q'*A ) / ( M * norm(A) * EPS )
*> RESULT(2) = norm( I - Q'*Q ) / ( M * EPS )
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup single_lin
*
* =====================================================================
SUBROUTINE SQRT01P( M, N, A, AF, Q, R, LDA, TAU, WORK, LWORK,
$ RWORK, RESULT )
*
* -- LAPACK test routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER LDA, LWORK, M, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
$ R( LDA, * ), RESULT( * ), RWORK( * ), TAU( * ),
$ WORK( LWORK )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
REAL ROGUE
PARAMETER ( ROGUE = -1.0E+10 )
* ..
* .. Local Scalars ..
INTEGER INFO, MINMN
REAL ANORM, EPS, RESID
* ..
* .. External Functions ..
REAL SLAMCH, SLANGE, SLANSY
EXTERNAL SLAMCH, SLANGE, SLANSY
* ..
* .. External Subroutines ..
EXTERNAL SGEMM, SGEQRFP, SLACPY, SLASET, SORGQR, SSYRK
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, REAL
* ..
* .. Scalars in Common ..
CHARACTER*32 SRNAMT
* ..
* .. Common blocks ..
COMMON / SRNAMC / SRNAMT
* ..
* .. Executable Statements ..
*
MINMN = MIN( M, N )
EPS = SLAMCH( 'Epsilon' )
*
* Copy the matrix A to the array AF.
*
CALL SLACPY( 'Full', M, N, A, LDA, AF, LDA )
*
* Factorize the matrix A in the array AF.
*
SRNAMT = 'SGEQRFP'
CALL SGEQRFP( M, N, AF, LDA, TAU, WORK, LWORK, INFO )
*
* Copy details of Q
*
CALL SLASET( 'Full', M, M, ROGUE, ROGUE, Q, LDA )
CALL SLACPY( 'Lower', M-1, N, AF( 2, 1 ), LDA, Q( 2, 1 ), LDA )
*
* Generate the m-by-m matrix Q
*
SRNAMT = 'SORGQR'
CALL SORGQR( M, M, MINMN, Q, LDA, TAU, WORK, LWORK, INFO )
*
* Copy R
*
CALL SLASET( 'Full', M, N, ZERO, ZERO, R, LDA )
CALL SLACPY( 'Upper', M, N, AF, LDA, R, LDA )
*
* Compute R - Q'*A
*
CALL SGEMM( 'Transpose', 'No transpose', M, N, M, -ONE, Q, LDA, A,
$ LDA, ONE, R, LDA )
*
* Compute norm( R - Q'*A ) / ( M * norm(A) * EPS ) .
*
ANORM = SLANGE( '1', M, N, A, LDA, RWORK )
RESID = SLANGE( '1', M, N, R, LDA, RWORK )
IF( ANORM.GT.ZERO ) THEN
RESULT( 1 ) = ( ( RESID / REAL( MAX( 1, M ) ) ) / ANORM ) / EPS
ELSE
RESULT( 1 ) = ZERO
END IF
*
* Compute I - Q'*Q
*
CALL SLASET( 'Full', M, M, ZERO, ONE, R, LDA )
CALL SSYRK( 'Upper', 'Transpose', M, M, -ONE, Q, LDA, ONE, R,
$ LDA )
*
* Compute norm( I - Q'*Q ) / ( M * EPS ) .
*
RESID = SLANSY( '1', 'Upper', M, R, LDA, RWORK )
*
RESULT( 2 ) = ( RESID / REAL( MAX( 1, M ) ) ) / EPS
*
RETURN
*
* End of SQRT01P
*
END
|
