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
|
SUBROUTINE ZQRT16( TRANS, M, N, NRHS, A, LDA, X, LDX, B, LDB,
$ RWORK, RESID )
*
* -- LAPACK test routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER LDA, LDB, LDX, M, N, NRHS
DOUBLE PRECISION RESID
* ..
* .. Array Arguments ..
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), X( LDX, * )
* ..
*
* Purpose
* =======
*
* ZQRT16 computes the residual for a solution of a system of linear
* equations A*x = b or A'*x = b:
* RESID = norm(B - A*X) / ( max(m,n) * norm(A) * norm(X) * EPS ),
* where EPS is the machine epsilon.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER*1
* Specifies the form of the system of equations:
* = 'N': A *x = b
* = 'T': A^T*x = b, where A^T is the transpose of A
* = 'C': A^H*x = b, where A^H is the conjugate transpose of A
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of columns of B, the matrix of right hand sides.
* NRHS >= 0.
*
* A (input) COMPLEX*16 array, dimension (LDA,N)
* The original M x N matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* X (input) COMPLEX*16 array, dimension (LDX,NRHS)
* The computed solution vectors for the system of linear
* equations.
*
* LDX (input) INTEGER
* The leading dimension of the array X. If TRANS = 'N',
* LDX >= max(1,N); if TRANS = 'T' or 'C', LDX >= max(1,M).
*
* B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
* On entry, the right hand side vectors for the system of
* linear equations.
* On exit, B is overwritten with the difference B - A*X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. IF TRANS = 'N',
* LDB >= max(1,M); if TRANS = 'T' or 'C', LDB >= max(1,N).
*
* RWORK (workspace) DOUBLE PRECISION array, dimension (M)
*
* RESID (output) DOUBLE PRECISION
* The maximum over the number of right hand sides of
* norm(B - A*X) / ( max(m,n) * norm(A) * norm(X) * EPS ).
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
COMPLEX*16 CONE
PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
INTEGER J, N1, N2
DOUBLE PRECISION ANORM, BNORM, EPS, XNORM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DZASUM, ZLANGE
EXTERNAL LSAME, DLAMCH, DZASUM, ZLANGE
* ..
* .. External Subroutines ..
EXTERNAL ZGEMM
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Quick exit if M = 0 or N = 0 or NRHS = 0
*
IF( M.LE.0 .OR. N.LE.0 .OR. NRHS.EQ.0 ) THEN
RESID = ZERO
RETURN
END IF
*
IF( LSAME( TRANS, 'T' ) .OR. LSAME( TRANS, 'C' ) ) THEN
ANORM = ZLANGE( 'I', M, N, A, LDA, RWORK )
N1 = N
N2 = M
ELSE
ANORM = ZLANGE( '1', M, N, A, LDA, RWORK )
N1 = M
N2 = N
END IF
*
EPS = DLAMCH( 'Epsilon' )
*
* Compute B - A*X (or B - A'*X ) and store in B.
*
CALL ZGEMM( TRANS, 'No transpose', N1, NRHS, N2, -CONE, A, LDA, X,
$ LDX, CONE, B, LDB )
*
* Compute the maximum over the number of right hand sides of
* norm(B - A*X) / ( max(m,n) * norm(A) * norm(X) * EPS ) .
*
RESID = ZERO
DO 10 J = 1, NRHS
BNORM = DZASUM( N1, B( 1, J ), 1 )
XNORM = DZASUM( N2, X( 1, J ), 1 )
IF( ANORM.EQ.ZERO .AND. BNORM.EQ.ZERO ) THEN
RESID = ZERO
ELSE IF( ANORM.LE.ZERO .OR. XNORM.LE.ZERO ) THEN
RESID = ONE / EPS
ELSE
RESID = MAX( RESID, ( ( BNORM / ANORM ) / XNORM ) /
$ ( MAX( M, N )*EPS ) )
END IF
10 CONTINUE
*
RETURN
*
* End of ZQRT16
*
END
|
