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
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
|
*> \brief \b CGTT05
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE CGTT05( TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX,
* XACT, LDXACT, FERR, BERR, RESLTS )
*
* .. Scalar Arguments ..
* CHARACTER TRANS
* INTEGER LDB, LDX, LDXACT, N, NRHS
* ..
* .. Array Arguments ..
* REAL BERR( * ), FERR( * ), RESLTS( * )
* COMPLEX B( LDB, * ), D( * ), DL( * ), DU( * ),
* $ X( LDX, * ), XACT( LDXACT, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CGTT05 tests the error bounds from iterative refinement for the
*> computed solution to a system of equations A*X = B, where A is a
*> general tridiagonal matrix of order n and op(A) = A or A**T,
*> depending on TRANS.
*>
*> 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 / ( NZ*EPS + (*) ), where
*> (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
*> and NZ = max. number of nonzeros in any row of A, plus 1
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER*1
*> Specifies the form of the system of equations.
*> = 'N': A * X = B (No transpose)
*> = 'T': A**T * X = B (Transpose)
*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows of the matrices X and XACT. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of columns of the matrices X and XACT. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*> DL is COMPLEX array, dimension (N-1)
*> The (n-1) sub-diagonal elements of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is COMPLEX array, dimension (N)
*> The diagonal elements of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*> DU is COMPLEX array, dimension (N-1)
*> The (n-1) super-diagonal elements of A.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is COMPLEX 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 COMPLEX 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 COMPLEX 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 REAL 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 REAL 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 REAL array, dimension (2)
*> The maximum over the NRHS solution vectors of the ratios:
*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
*> RESLTS(2) = BERR / ( NZ*EPS + (*) )
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup complex_lin
*
* =====================================================================
SUBROUTINE CGTT05( TRANS, N, NRHS, DL, D, DU, 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 TRANS
INTEGER LDB, LDX, LDXACT, N, NRHS
* ..
* .. Array Arguments ..
REAL BERR( * ), FERR( * ), RESLTS( * )
COMPLEX B( LDB, * ), D( * ), DL( * ), DU( * ),
$ X( LDX, * ), XACT( LDXACT, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN
INTEGER I, IMAX, J, K, NZ
REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
COMPLEX ZDUM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ICAMAX
REAL SLAMCH
EXTERNAL LSAME, ICAMAX, SLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, MAX, MIN, REAL
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Statement Function definitions ..
CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
* ..
* .. 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 = SLAMCH( 'Epsilon' )
UNFL = SLAMCH( 'Safe minimum' )
OVFL = ONE / UNFL
NOTRAN = LSAME( TRANS, 'N' )
NZ = 4
*
* 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 = ICAMAX( N, X( 1, J ), 1 )
XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
DIFF = ZERO
DO 10 I = 1, N
DIFF = MAX( DIFF, CABS1( 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 / ( NZ*EPS + (*) ), where
* (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
*
DO 60 K = 1, NRHS
IF( NOTRAN ) THEN
IF( N.EQ.1 ) THEN
AXBI = CABS1( B( 1, K ) ) +
$ CABS1( D( 1 ) )*CABS1( X( 1, K ) )
ELSE
AXBI = CABS1( B( 1, K ) ) +
$ CABS1( D( 1 ) )*CABS1( X( 1, K ) ) +
$ CABS1( DU( 1 ) )*CABS1( X( 2, K ) )
DO 40 I = 2, N - 1
TMP = CABS1( B( I, K ) ) +
$ CABS1( DL( I-1 ) )*CABS1( X( I-1, K ) ) +
$ CABS1( D( I ) )*CABS1( X( I, K ) ) +
$ CABS1( DU( I ) )*CABS1( X( I+1, K ) )
AXBI = MIN( AXBI, TMP )
40 CONTINUE
TMP = CABS1( B( N, K ) ) + CABS1( DL( N-1 ) )*
$ CABS1( X( N-1, K ) ) + CABS1( D( N ) )*
$ CABS1( X( N, K ) )
AXBI = MIN( AXBI, TMP )
END IF
ELSE
IF( N.EQ.1 ) THEN
AXBI = CABS1( B( 1, K ) ) +
$ CABS1( D( 1 ) )*CABS1( X( 1, K ) )
ELSE
AXBI = CABS1( B( 1, K ) ) +
$ CABS1( D( 1 ) )*CABS1( X( 1, K ) ) +
$ CABS1( DL( 1 ) )*CABS1( X( 2, K ) )
DO 50 I = 2, N - 1
TMP = CABS1( B( I, K ) ) +
$ CABS1( DU( I-1 ) )*CABS1( X( I-1, K ) ) +
$ CABS1( D( I ) )*CABS1( X( I, K ) ) +
$ CABS1( DL( I ) )*CABS1( X( I+1, K ) )
AXBI = MIN( AXBI, TMP )
50 CONTINUE
TMP = CABS1( B( N, K ) ) + CABS1( DU( N-1 ) )*
$ CABS1( X( N-1, K ) ) + CABS1( D( N ) )*
$ CABS1( X( N, K ) )
AXBI = MIN( AXBI, TMP )
END IF
END IF
TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
IF( K.EQ.1 ) THEN
RESLTS( 2 ) = TMP
ELSE
RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
END IF
60 CONTINUE
*
RETURN
*
* End of CGTT05
*
END
|
