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
|
SUBROUTINE ZTRT03( UPLO, TRANS, DIAG, N, NRHS, A, LDA, SCALE,
$ CNORM, TSCAL, X, LDX, B, LDB, WORK, 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 LDA, LDB, LDX, N, NRHS
DOUBLE PRECISION RESID, SCALE, TSCAL
* ..
* .. Array Arguments ..
DOUBLE PRECISION CNORM( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ),
$ X( LDX, * )
* ..
*
* Purpose
* =======
*
* ZTRT03 computes the residual for the solution to a scaled triangular
* system of equations A*x = s*b, A**T *x = s*b, or A**H *x = s*b.
* Here A is a triangular matrix, A**T denotes the transpose of A, A**H
* denotes the conjugate transpose of A, s is a scalar, and x and b are
* N by NRHS matrices. The test ratio is the maximum over the number of
* right hand sides of
* norm(s*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 = s*b (No transpose)
* = 'T': A**T *x = s*b (Transpose)
* = 'C': A**H *x = s*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.
*
* A (input) COMPLEX*16 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.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* SCALE (input) DOUBLE PRECISION
* The scaling factor s used in solving the triangular system.
*
* CNORM (input) DOUBLE PRECISION array, dimension (N)
* The 1-norms of the columns of A, not counting the diagonal.
*
* TSCAL (input) DOUBLE PRECISION
* The scaling factor used in computing the 1-norms in CNORM.
* CNORM actually contains the column norms of TSCAL*A.
*
* 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)
*
* RESID (output) DOUBLE PRECISION
* The maximum over the number of right hand sides of
* norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER IX, J
DOUBLE PRECISION EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IZAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, IZAMAX, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL ZAXPY, ZCOPY, ZDSCAL, ZTRMV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DCMPLX, MAX
* ..
* .. Executable Statements ..
*
* Quick exit if N = 0
*
IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
RESID = ZERO
RETURN
END IF
EPS = DLAMCH( 'Epsilon' )
SMLNUM = DLAMCH( 'Safe minimum' )
*
* Compute the norm of the triangular matrix A using the column
* norms already computed by ZLATRS.
*
TNORM = ZERO
IF( LSAME( DIAG, 'N' ) ) THEN
DO 10 J = 1, N
TNORM = MAX( TNORM, TSCAL*ABS( A( J, J ) )+CNORM( J ) )
10 CONTINUE
ELSE
DO 20 J = 1, N
TNORM = MAX( TNORM, TSCAL+CNORM( J ) )
20 CONTINUE
END IF
*
* Compute the maximum over the number of right hand sides of
* norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
*
RESID = ZERO
DO 30 J = 1, NRHS
CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
IX = IZAMAX( N, WORK, 1 )
XNORM = MAX( ONE, ABS( X( IX, J ) ) )
XSCAL = ( ONE / XNORM ) / DBLE( N )
CALL ZDSCAL( N, XSCAL, WORK, 1 )
CALL ZTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK, 1 )
CALL ZAXPY( N, DCMPLX( -SCALE*XSCAL ), B( 1, J ), 1, WORK, 1 )
IX = IZAMAX( N, WORK, 1 )
ERR = TSCAL*ABS( WORK( IX ) )
IX = IZAMAX( N, X( 1, J ), 1 )
XNORM = ABS( X( IX, J ) )
IF( ERR*SMLNUM.LE.XNORM ) THEN
IF( XNORM.GT.ZERO )
$ ERR = ERR / XNORM
ELSE
IF( ERR.GT.ZERO )
$ ERR = ONE / EPS
END IF
IF( ERR*SMLNUM.LE.TNORM ) THEN
IF( TNORM.GT.ZERO )
$ ERR = ERR / TNORM
ELSE
IF( ERR.GT.ZERO )
$ ERR = ONE / EPS
END IF
RESID = MAX( RESID, ERR )
30 CONTINUE
*
RETURN
*
* End of ZTRT03
*
END
|
