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
|
SUBROUTINE ZTPT02( UPLO, TRANS, DIAG, N, NRHS, AP, X, LDX, B, LDB,
$ WORK, 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
* February 29, 1992
*
* .. Scalar Arguments ..
CHARACTER DIAG, TRANS, UPLO
INTEGER LDB, LDX, N, NRHS
DOUBLE PRECISION RESID
* ..
* .. Array Arguments ..
DOUBLE PRECISION RWORK( * )
COMPLEX*16 AP( * ), B( LDB, * ), WORK( * ), X( LDX, * )
* ..
*
* Purpose
* =======
*
* ZTPT02 computes the residual for the computed solution to a
* triangular system of linear equations A*x = b, A**T *x = b, or
* A**H *x = b, when the triangular matrix A is stored in packed format.
* Here A**T denotes the transpose of A, A**H denotes the conjugate
* transpose of A, and x and b are N by NRHS matrices. The test ratio
* is the maximum over the number of right hand sides of
* the maximum over the number of right hand sides of
* norm(b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
* where op(A) denotes A, A**T, or A**H, and EPS is the machine epsilon.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* Specifies whether the matrix A is upper or lower triangular.
* = 'U': Upper triangular
* = 'L': Lower triangular
*
* TRANS (input) CHARACTER*1
* Specifies the operation applied to A.
* = 'N': A *x = b (No transpose)
* = 'T': A**T *x = b (Transpose)
* = 'C': A**H *x = b (Conjugate transpose)
*
* DIAG (input) CHARACTER*1
* Specifies whether or not the matrix A is unit triangular.
* = 'N': Non-unit triangular
* = 'U': Unit triangular
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrices X and B. NRHS >= 0.
*
* AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
* The upper or lower triangular matrix A, packed columnwise in
* a linear array. The j-th column of A is stored in the array
* AP as follows:
* if UPLO = 'U', AP((j-1)*j/2 + i) = A(i,j) for 1<=i<=j;
* if UPLO = 'L',
* AP((j-1)*(n-j) + j*(j+1)/2 + i-j) = A(i,j) for j<=i<=n.
*
* 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. LDX >= max(1,N).
*
* B (input) COMPLEX*16 array, dimension (LDB,NRHS)
* The right hand side vectors for the system of linear
* equations.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* WORK (workspace) COMPLEX*16 array, dimension (N)
*
* RWORK (workspace) DOUBLE PRECISION array, dimension (N)
*
* RESID (output) DOUBLE PRECISION
* The maximum over the number of right hand sides of
* norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER J
DOUBLE PRECISION ANORM, BNORM, EPS, XNORM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DZASUM, ZLANTP
EXTERNAL LSAME, DLAMCH, DZASUM, ZLANTP
* ..
* .. External Subroutines ..
EXTERNAL ZAXPY, ZCOPY, ZTPMV
* ..
* .. Intrinsic Functions ..
INTRINSIC DCMPLX, MAX
* ..
* .. Executable Statements ..
*
* Quick exit if N = 0 or NRHS = 0
*
IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
RESID = ZERO
RETURN
END IF
*
* Compute the 1-norm of A or A**H.
*
IF( LSAME( TRANS, 'N' ) ) THEN
ANORM = ZLANTP( '1', UPLO, DIAG, N, AP, RWORK )
ELSE
ANORM = ZLANTP( 'I', UPLO, DIAG, N, AP, RWORK )
END IF
*
* Exit with RESID = 1/EPS if ANORM = 0.
*
EPS = DLAMCH( 'Epsilon' )
IF( ANORM.LE.ZERO ) THEN
RESID = ONE / EPS
RETURN
END IF
*
* Compute the maximum over the number of right hand sides of
* norm(op(A)*x - b) / ( norm(op(A)) * norm(x) * EPS ).
*
RESID = ZERO
DO 10 J = 1, NRHS
CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
CALL ZTPMV( UPLO, TRANS, DIAG, N, AP, WORK, 1 )
CALL ZAXPY( N, DCMPLX( -ONE ), B( 1, J ), 1, WORK, 1 )
BNORM = DZASUM( N, WORK, 1 )
XNORM = DZASUM( N, X( 1, J ), 1 )
IF( XNORM.LE.ZERO ) THEN
RESID = ONE / EPS
ELSE
RESID = MAX( RESID, ( ( BNORM / ANORM ) / XNORM ) / EPS )
END IF
10 CONTINUE
*
RETURN
*
* End of ZTPT02
*
END
|
