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
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
|
*> \brief \b DPOT05
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
* LDXACT, FERR, BERR, RESLTS )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
* $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPOT05 tests the error bounds from iterative refinement for the
*> computed solution to a system of equations A*X = B, where A is a
*> symmetric n by n matrix.
*>
*> RESLTS(1) = test of the error bound
*> = norm(X - XACT) / ( norm(X) * FERR )
*>
*> A large value is returned if this ratio is not less than one.
*>
*> RESLTS(2) = residual from the iterative refinement routine
*> = the maximum of BERR / ( (n+1)*EPS + (*) ), where
*> (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
*> \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 of the matrices X, B, and XACT, and the
*> order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of columns of the matrices X, B, and XACT.
*> NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> The symmetric matrix A. If UPLO = 'U', the leading n by n
*> upper triangular part of A contains the upper triangular part
*> of the matrix A, and the strictly lower triangular part of A
*> is not referenced. If UPLO = 'L', the leading n by n lower
*> triangular part of A contains the lower triangular part of
*> the matrix A, and the strictly upper triangular part of A is
*> not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
*> The right hand side vectors for the system of linear
*> equations.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> The computed solution vectors. Each vector is stored as a
*> column of the matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[in] XACT
*> \verbatim
*> XACT is DOUBLE PRECISION array, dimension (LDX,NRHS)
*> The exact solution vectors. Each vector is stored as a
*> column of the matrix XACT.
*> \endverbatim
*>
*> \param[in] LDXACT
*> \verbatim
*> LDXACT is INTEGER
*> The leading dimension of the array XACT. LDXACT >= max(1,N).
*> \endverbatim
*>
*> \param[in] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bounds for each solution vector
*> X. If XTRUE is the true solution, FERR bounds the magnitude
*> of the largest entry in (X - XTRUE) divided by the magnitude
*> of the largest entry in X.
*> \endverbatim
*>
*> \param[in] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector (i.e., the smallest relative change in any entry of A
*> or B that makes X an exact solution).
*> \endverbatim
*>
*> \param[out] RESLTS
*> \verbatim
*> RESLTS is DOUBLE PRECISION array, dimension (2)
*> The maximum over the NRHS solution vectors of the ratios:
*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
*> RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup double_lin
*
* =====================================================================
SUBROUTINE DPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
$ LDXACT, FERR, BERR, RESLTS )
*
* -- 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, LDB, LDX, LDXACT, N, NRHS
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
$ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, IMAX, J, K
DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, IDAMAX, DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
* Quick exit if N = 0 or NRHS = 0.
*
IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
RESLTS( 1 ) = ZERO
RESLTS( 2 ) = ZERO
RETURN
END IF
*
EPS = DLAMCH( 'Epsilon' )
UNFL = DLAMCH( 'Safe minimum' )
OVFL = ONE / UNFL
UPPER = LSAME( UPLO, 'U' )
*
* Test 1: Compute the maximum of
* norm(X - XACT) / ( norm(X) * FERR )
* over all the vectors X and XACT using the infinity-norm.
*
ERRBND = ZERO
DO 30 J = 1, NRHS
IMAX = IDAMAX( N, X( 1, J ), 1 )
XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
DIFF = ZERO
DO 10 I = 1, N
DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
10 CONTINUE
*
IF( XNORM.GT.ONE ) THEN
GO TO 20
ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
GO TO 20
ELSE
ERRBND = ONE / EPS
GO TO 30
END IF
*
20 CONTINUE
IF( DIFF / XNORM.LE.FERR( J ) ) THEN
ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
ELSE
ERRBND = ONE / EPS
END IF
30 CONTINUE
RESLTS( 1 ) = ERRBND
*
* Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
* (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
*
DO 90 K = 1, NRHS
DO 80 I = 1, N
TMP = ABS( B( I, K ) )
IF( UPPER ) THEN
DO 40 J = 1, I
TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
40 CONTINUE
DO 50 J = I + 1, N
TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
50 CONTINUE
ELSE
DO 60 J = 1, I - 1
TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
60 CONTINUE
DO 70 J = I, N
TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
70 CONTINUE
END IF
IF( I.EQ.1 ) THEN
AXBI = TMP
ELSE
AXBI = MIN( AXBI, TMP )
END IF
80 CONTINUE
TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
$ MAX( AXBI, ( N+1 )*UNFL ) )
IF( K.EQ.1 ) THEN
RESLTS( 2 ) = TMP
ELSE
RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
END IF
90 CONTINUE
*
RETURN
*
* End of DPOT05
*
END
|
