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
|
*> \brief \b ZPBT01
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE ZPBT01( UPLO, N, KD, A, LDA, AFAC, LDAFAC, RWORK,
* RESID )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER KD, LDA, LDAFAC, N
* DOUBLE PRECISION RESID
* ..
* .. Array Arguments ..
* DOUBLE PRECISION RWORK( * )
* COMPLEX*16 A( LDA, * ), AFAC( LDAFAC, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZPBT01 reconstructs a Hermitian positive definite band matrix A from
*> its L*L' or U'*U factorization and computes the residual
*> norm( L*L' - A ) / ( N * norm(A) * EPS ) or
*> norm( U'*U - A ) / ( N * norm(A) * EPS ),
*> where EPS is the machine epsilon, L' is the conjugate transpose of
*> L, and U' is the conjugate transpose of U.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> Hermitian matrix A is stored:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*> KD is INTEGER
*> The number of super-diagonals of the matrix A if UPLO = 'U',
*> or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*> The original Hermitian band matrix A. If UPLO = 'U', the
*> upper triangular part of A is stored as a band matrix; if
*> UPLO = 'L', the lower triangular part of A is stored. The
*> columns of the appropriate triangle are stored in the columns
*> of A and the diagonals of the triangle are stored in the rows
*> of A. See ZPBTRF for further details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER.
*> The leading dimension of the array A. LDA >= max(1,KD+1).
*> \endverbatim
*>
*> \param[in] AFAC
*> \verbatim
*> AFAC is COMPLEX*16 array, dimension (LDAFAC,N)
*> The factored form of the matrix A. AFAC contains the factor
*> L or U from the L*L' or U'*U factorization in band storage
*> format, as computed by ZPBTRF.
*> \endverbatim
*>
*> \param[in] LDAFAC
*> \verbatim
*> LDAFAC is INTEGER
*> The leading dimension of the array AFAC.
*> LDAFAC >= max(1,KD+1).
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*> RESID is DOUBLE PRECISION
*> If UPLO = 'L', norm(L*L' - A) / ( N * norm(A) * EPS )
*> If UPLO = 'U', norm(U'*U - A) / ( N * norm(A) * EPS )
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16_lin
*
* =====================================================================
SUBROUTINE ZPBT01( UPLO, N, KD, A, LDA, AFAC, LDAFAC, RWORK,
$ RESID )
*
* -- LAPACK test routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER KD, LDA, LDAFAC, N
DOUBLE PRECISION RESID
* ..
* .. Array Arguments ..
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), AFAC( LDAFAC, * )
* ..
*
* =====================================================================
*
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, K, KC, KLEN, ML, MU
DOUBLE PRECISION AKK, ANORM, EPS
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, ZLANHB
COMPLEX*16 ZDOTC
EXTERNAL LSAME, DLAMCH, ZLANHB, ZDOTC
* ..
* .. External Subroutines ..
EXTERNAL ZDSCAL, ZHER, ZTRMV
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, DIMAG, MAX, MIN
* ..
* .. Executable Statements ..
*
* Quick exit if N = 0.
*
IF( N.LE.0 ) THEN
RESID = ZERO
RETURN
END IF
*
* Exit with RESID = 1/EPS if ANORM = 0.
*
EPS = DLAMCH( 'Epsilon' )
ANORM = ZLANHB( '1', UPLO, N, KD, A, LDA, RWORK )
IF( ANORM.LE.ZERO ) THEN
RESID = ONE / EPS
RETURN
END IF
*
* Check the imaginary parts of the diagonal elements and return with
* an error code if any are nonzero.
*
IF( LSAME( UPLO, 'U' ) ) THEN
DO 10 J = 1, N
IF( DIMAG( AFAC( KD+1, J ) ).NE.ZERO ) THEN
RESID = ONE / EPS
RETURN
END IF
10 CONTINUE
ELSE
DO 20 J = 1, N
IF( DIMAG( AFAC( 1, J ) ).NE.ZERO ) THEN
RESID = ONE / EPS
RETURN
END IF
20 CONTINUE
END IF
*
* Compute the product U'*U, overwriting U.
*
IF( LSAME( UPLO, 'U' ) ) THEN
DO 30 K = N, 1, -1
KC = MAX( 1, KD+2-K )
KLEN = KD + 1 - KC
*
* Compute the (K,K) element of the result.
*
AKK = DBLE(
$ ZDOTC( KLEN+1, AFAC( KC, K ), 1, AFAC( KC, K ), 1 ) )
AFAC( KD+1, K ) = AKK
*
* Compute the rest of column K.
*
IF( KLEN.GT.0 )
$ CALL ZTRMV( 'Upper', 'Conjugate', 'Non-unit', KLEN,
$ AFAC( KD+1, K-KLEN ), LDAFAC-1,
$ AFAC( KC, K ), 1 )
*
30 CONTINUE
*
* UPLO = 'L': Compute the product L*L', overwriting L.
*
ELSE
DO 40 K = N, 1, -1
KLEN = MIN( KD, N-K )
*
* Add a multiple of column K of the factor L to each of
* columns K+1 through N.
*
IF( KLEN.GT.0 )
$ CALL ZHER( 'Lower', KLEN, ONE, AFAC( 2, K ), 1,
$ AFAC( 1, K+1 ), LDAFAC-1 )
*
* Scale column K by the diagonal element.
*
AKK = DBLE( AFAC( 1, K ) )
CALL ZDSCAL( KLEN+1, AKK, AFAC( 1, K ), 1 )
*
40 CONTINUE
END IF
*
* Compute the difference L*L' - A or U'*U - A.
*
IF( LSAME( UPLO, 'U' ) ) THEN
DO 60 J = 1, N
MU = MAX( 1, KD+2-J )
DO 50 I = MU, KD + 1
AFAC( I, J ) = AFAC( I, J ) - A( I, J )
50 CONTINUE
60 CONTINUE
ELSE
DO 80 J = 1, N
ML = MIN( KD+1, N-J+1 )
DO 70 I = 1, ML
AFAC( I, J ) = AFAC( I, J ) - A( I, J )
70 CONTINUE
80 CONTINUE
END IF
*
* Compute norm( L*L' - A ) / ( N * norm(A) * EPS )
*
RESID = ZLANHB( '1', UPLO, N, KD, AFAC, LDAFAC, RWORK )
*
RESID = ( ( RESID / DBLE( N ) ) / ANORM ) / EPS
*
RETURN
*
* End of ZPBT01
*
END
|
