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
|
*> \brief \b STRT01
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE STRT01( UPLO, DIAG, N, A, LDA, AINV, LDAINV, RCOND,
* WORK, RESID )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, UPLO
* INTEGER LDA, LDAINV, N
* REAL RCOND, RESID
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), AINV( LDAINV, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> STRT01 computes the residual for a triangular matrix A times its
*> inverse:
*> RESID = norm( A*AINV - I ) / ( N * norm(A) * norm(AINV) * EPS ),
*> where EPS is the machine epsilon.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the matrix A is upper or lower triangular.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> Specifies whether or not the matrix A is unit triangular.
*> = 'N': Non-unit triangular
*> = 'U': Unit triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is REAL array, dimension (LDA,N)
*> The triangular matrix A. If UPLO = 'U', the leading n by n
*> upper triangular part of the array A contains the upper
*> triangular matrix, and the strictly lower triangular part of
*> A is not referenced. If UPLO = 'L', the leading n by n lower
*> triangular part of the array A contains the lower triangular
*> matrix, and the strictly upper triangular part of A is not
*> referenced. If DIAG = 'U', the diagonal elements of A are
*> also not referenced and are assumed to be 1.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] AINV
*> \verbatim
*> AINV is REAL array, dimension (LDAINV,N)
*> On entry, the (triangular) inverse of the matrix A, in the
*> same storage format as A.
*> On exit, the contents of AINV are destroyed.
*> \endverbatim
*>
*> \param[in] LDAINV
*> \verbatim
*> LDAINV is INTEGER
*> The leading dimension of the array AINV. LDAINV >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is REAL
*> The reciprocal condition number of A, computed as
*> 1/(norm(A) * norm(AINV)).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*> RESID is REAL
*> norm(A*AINV - I) / ( N * norm(A) * norm(AINV) * 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 STRT01( UPLO, DIAG, N, A, LDA, AINV, LDAINV, RCOND,
$ WORK, RESID )
*
* -- 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 ..
CHARACTER DIAG, UPLO
INTEGER LDA, LDAINV, N
REAL RCOND, RESID
* ..
* .. Array Arguments ..
REAL A( LDA, * ), AINV( LDAINV, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
INTEGER J
REAL AINVNM, ANORM, EPS
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SLAMCH, SLANTR
EXTERNAL LSAME, SLAMCH, SLANTR
* ..
* .. External Subroutines ..
EXTERNAL STRMV
* ..
* .. Intrinsic Functions ..
INTRINSIC REAL
* ..
* .. Executable Statements ..
*
* Quick exit if N = 0
*
IF( N.LE.0 ) THEN
RCOND = ONE
RESID = ZERO
RETURN
END IF
*
* Exit with RESID = 1/EPS if ANORM = 0 or AINVNM = 0.
*
EPS = SLAMCH( 'Epsilon' )
ANORM = SLANTR( '1', UPLO, DIAG, N, N, A, LDA, WORK )
AINVNM = SLANTR( '1', UPLO, DIAG, N, N, AINV, LDAINV, WORK )
IF( ANORM.LE.ZERO .OR. AINVNM.LE.ZERO ) THEN
RCOND = ZERO
RESID = ONE / EPS
RETURN
END IF
RCOND = ( ONE / ANORM ) / AINVNM
*
* Set the diagonal of AINV to 1 if AINV has unit diagonal.
*
IF( LSAME( DIAG, 'U' ) ) THEN
DO 10 J = 1, N
AINV( J, J ) = ONE
10 CONTINUE
END IF
*
* Compute A * AINV, overwriting AINV.
*
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
CALL STRMV( 'Upper', 'No transpose', DIAG, J, A, LDA,
$ AINV( 1, J ), 1 )
20 CONTINUE
ELSE
DO 30 J = 1, N
CALL STRMV( 'Lower', 'No transpose', DIAG, N-J+1, A( J, J ),
$ LDA, AINV( J, J ), 1 )
30 CONTINUE
END IF
*
* Subtract 1 from each diagonal element to form A*AINV - I.
*
DO 40 J = 1, N
AINV( J, J ) = AINV( J, J ) - ONE
40 CONTINUE
*
* Compute norm(A*AINV - I) / (N * norm(A) * norm(AINV) * EPS)
*
RESID = SLANTR( '1', UPLO, 'Non-unit', N, N, AINV, LDAINV, WORK )
*
RESID = ( ( RESID*RCOND ) / REAL( N ) ) / EPS
*
RETURN
*
* End of STRT01
*
END
|
