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
|
SUBROUTINE DPOT03( UPLO, N, A, LDA, AINV, LDAINV, WORK, LDWORK,
$ RWORK, RCOND, RESID )
*
* -- LAPACK test routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* February 29, 1992
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDAINV, LDWORK, N
DOUBLE PRECISION RCOND, RESID
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), AINV( LDAINV, * ), RWORK( * ),
$ WORK( LDWORK, * )
* ..
*
* Purpose
* =======
*
* DPOT03 computes the residual for a symmetric matrix times its
* inverse:
* norm( I - A*AINV ) / ( N * norm(A) * norm(AINV) * EPS ),
* where EPS is the machine epsilon.
*
* Arguments
* ==========
*
* UPLO (input) CHARACTER*1
* Specifies whether the upper or lower triangular part of the
* symmetric matrix A is stored:
* = 'U': Upper triangular
* = 'L': Lower triangular
*
* N (input) INTEGER
* The number of rows and columns of the matrix A. N >= 0.
*
* A (input) DOUBLE PRECISION array, dimension (LDA,N)
* The original symmetric matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N)
*
* AINV (input/output) DOUBLE PRECISION array, dimension (LDAINV,N)
* On entry, the inverse of the matrix A, stored as a symmetric
* matrix in the same format as A.
* In this version, AINV is expanded into a full matrix and
* multiplied by A, so the opposing triangle of AINV will be
* changed; i.e., if the upper triangular part of AINV is
* stored, the lower triangular part will be used as work space.
*
* LDAINV (input) INTEGER
* The leading dimension of the array AINV. LDAINV >= max(1,N).
*
* WORK (workspace) DOUBLE PRECISION array, dimension (LDWORK,N)
*
* LDWORK (input) INTEGER
* The leading dimension of the array WORK. LDWORK >= max(1,N).
*
* RWORK (workspace) DOUBLE PRECISION array, dimension (N)
*
* RCOND (output) DOUBLE PRECISION
* The reciprocal of the condition number of A, computed as
* ( 1/norm(A) ) / norm(AINV).
*
* RESID (output) DOUBLE PRECISION
* norm(I - A*AINV) / ( N * norm(A) * norm(AINV) * EPS )
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION AINVNM, ANORM, EPS
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANGE, DLANSY
EXTERNAL LSAME, DLAMCH, DLANGE, DLANSY
* ..
* .. External Subroutines ..
EXTERNAL DSYMM
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE
* ..
* .. 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 = DLAMCH( 'Epsilon' )
ANORM = DLANSY( '1', UPLO, N, A, LDA, RWORK )
AINVNM = DLANSY( '1', UPLO, N, AINV, LDAINV, RWORK )
IF( ANORM.LE.ZERO .OR. AINVNM.LE.ZERO ) THEN
RCOND = ZERO
RESID = ONE / EPS
RETURN
END IF
RCOND = ( ONE / ANORM ) / AINVNM
*
* Expand AINV into a full matrix and call DSYMM to multiply
* AINV on the left by A.
*
IF( LSAME( UPLO, 'U' ) ) THEN
DO 20 J = 1, N
DO 10 I = 1, J - 1
AINV( J, I ) = AINV( I, J )
10 CONTINUE
20 CONTINUE
ELSE
DO 40 J = 1, N
DO 30 I = J + 1, N
AINV( J, I ) = AINV( I, J )
30 CONTINUE
40 CONTINUE
END IF
CALL DSYMM( 'Left', UPLO, N, N, -ONE, A, LDA, AINV, LDAINV, ZERO,
$ WORK, LDWORK )
*
* Add the identity matrix to WORK .
*
DO 50 I = 1, N
WORK( I, I ) = WORK( I, I ) + ONE
50 CONTINUE
*
* Compute norm(I - A*AINV) / (N * norm(A) * norm(AINV) * EPS)
*
RESID = DLANGE( '1', N, N, WORK, LDWORK, RWORK )
*
RESID = ( ( RESID*RCOND ) / EPS ) / DBLE( N )
*
RETURN
*
* End of DPOT03
*
END
|
