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
|
*> \brief \b SPOT01
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE SPOT01( UPLO, N, A, LDA, AFAC, LDAFAC, RWORK, RESID )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, LDAFAC, N
* REAL RESID
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), AFAC( LDAFAC, * ), RWORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SPOT01 reconstructs a symmetric positive definite matrix A from
*> its L*L' or U'*U factorization and computes the residual
*> norm( L*L' - A ) / ( N * norm(A) * EPS ) or
*> norm( U'*U - A ) / ( N * norm(A) * EPS ),
*> where EPS is the machine epsilon.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is REAL array, dimension (LDA,N)
*> The original symmetric matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N)
*> \endverbatim
*>
*> \param[in,out] AFAC
*> \verbatim
*> AFAC is REAL array, dimension (LDAFAC,N)
*> On entry, the factor L or U from the L*L' or U'*U
*> factorization of A.
*> Overwritten with the reconstructed matrix, and then with the
*> difference L*L' - A (or U'*U - A).
*> \endverbatim
*>
*> \param[in] LDAFAC
*> \verbatim
*> LDAFAC is INTEGER
*> The leading dimension of the array AFAC. LDAFAC >= max(1,N).
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*> RESID is REAL
*> If UPLO = 'L', norm(L*L' - A) / ( N * norm(A) * EPS )
*> If UPLO = 'U', norm(U'*U - A) / ( N * norm(A) * 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 SPOT01( UPLO, N, A, LDA, AFAC, LDAFAC, RWORK, 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 UPLO
INTEGER LDA, LDAFAC, N
REAL RESID
* ..
* .. Array Arguments ..
REAL A( LDA, * ), AFAC( LDAFAC, * ), RWORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, K
REAL ANORM, EPS, T
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SDOT, SLAMCH, SLANSY
EXTERNAL LSAME, SDOT, SLAMCH, SLANSY
* ..
* .. External Subroutines ..
EXTERNAL SSCAL, SSYR, STRMV
* ..
* .. Intrinsic Functions ..
INTRINSIC REAL
* ..
* .. Executable Statements ..
*
* Quick exit if N = 0.
*
IF( N.LE.0 ) THEN
RESID = ZERO
RETURN
END IF
*
* Exit with RESID = 1/EPS if ANORM = 0.
*
EPS = SLAMCH( 'Epsilon' )
ANORM = SLANSY( '1', UPLO, N, A, LDA, RWORK )
IF( ANORM.LE.ZERO ) THEN
RESID = ONE / EPS
RETURN
END IF
*
* Compute the product U'*U, overwriting U.
*
IF( LSAME( UPLO, 'U' ) ) THEN
DO 10 K = N, 1, -1
*
* Compute the (K,K) element of the result.
*
T = SDOT( K, AFAC( 1, K ), 1, AFAC( 1, K ), 1 )
AFAC( K, K ) = T
*
* Compute the rest of column K.
*
CALL STRMV( 'Upper', 'Transpose', 'Non-unit', K-1, AFAC,
$ LDAFAC, AFAC( 1, K ), 1 )
*
10 CONTINUE
*
* Compute the product L*L', overwriting L.
*
ELSE
DO 20 K = N, 1, -1
*
* Add a multiple of column K of the factor L to each of
* columns K+1 through N.
*
IF( K+1.LE.N )
$ CALL SSYR( 'Lower', N-K, ONE, AFAC( K+1, K ), 1,
$ AFAC( K+1, K+1 ), LDAFAC )
*
* Scale column K by the diagonal element.
*
T = AFAC( K, K )
CALL SSCAL( N-K+1, T, AFAC( K, K ), 1 )
*
20 CONTINUE
END IF
*
* Compute the difference L*L' - A (or U'*U - A).
*
IF( LSAME( UPLO, 'U' ) ) THEN
DO 40 J = 1, N
DO 30 I = 1, J
AFAC( I, J ) = AFAC( I, J ) - A( I, J )
30 CONTINUE
40 CONTINUE
ELSE
DO 60 J = 1, N
DO 50 I = J, N
AFAC( I, J ) = AFAC( I, J ) - A( I, J )
50 CONTINUE
60 CONTINUE
END IF
*
* Compute norm( L*U - A ) / ( N * norm(A) * EPS )
*
RESID = SLANSY( '1', UPLO, N, AFAC, LDAFAC, RWORK )
*
RESID = ( ( RESID / REAL( N ) ) / ANORM ) / EPS
*
RETURN
*
* End of SPOT01
*
END
|
