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
|
*> \brief \b ZHET01_3
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE ZHET01_3( UPLO, N, A, LDA, AFAC, LDAFAC, E, IPIV, C,
* LDC, RWORK, RESID )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, LDAFAC, LDC, N
* DOUBLE PRECISION RESID
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION RWORK( * )
* COMPLEX*16 A( LDA, * ), AFAC( LDAFAC, * ), C( LDC, * ),
* E( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZHET01_3 reconstructs a Hermitian indefinite matrix A from its
*> block L*D*L' or U*D*U' factorization computed by ZHETRF_RK
*> (or ZHETRF_BK) and computes the residual
*> norm( C - A ) / ( N * norm(A) * EPS ),
*> where C is the reconstructed matrix and EPS is the machine epsilon.
*> \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] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*> The original Hermitian matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N)
*> \endverbatim
*>
*> \param[in] AFAC
*> \verbatim
*> AFAC is COMPLEX*16 array, dimension (LDAFAC,N)
*> Diagonal of the block diagonal matrix D and factors U or L
*> as computed by ZHETRF_RK and ZHETRF_BK:
*> a) ONLY diagonal elements of the Hermitian block diagonal
*> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
*> (superdiagonal (or subdiagonal) elements of D
*> should be provided on entry in array E), and
*> b) If UPLO = 'U': factor U in the superdiagonal part of A.
*> If UPLO = 'L': factor L in the subdiagonal part of A.
*> \endverbatim
*>
*> \param[in] LDAFAC
*> \verbatim
*> LDAFAC is INTEGER
*> The leading dimension of the array AFAC.
*> LDAFAC >= max(1,N).
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*> E is COMPLEX*16 array, dimension (N)
*> On entry, contains the superdiagonal (or subdiagonal)
*> elements of the Hermitian block diagonal matrix D
*> with 1-by-1 or 2-by-2 diagonal blocks, where
*> If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
*> If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> The pivot indices from ZHETRF_RK (or ZHETRF_BK).
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*> C is COMPLEX*16 array, dimension (LDC,N)
*> \endverbatim
*>
*> \param[in] LDC
*> \verbatim
*> LDC is INTEGER
*> The leading dimension of the array C. LDC >= max(1,N).
*> \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*D*L' - A) / ( N * norm(A) * EPS )
*> If UPLO = 'U', norm(U*D*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.
*
*> \date December 2016
*
*> \ingroup complex16_lin
*
* =====================================================================
SUBROUTINE ZHET01_3( UPLO, N, A, LDA, AFAC, LDAFAC, E, IPIV, C,
$ LDC, RWORK, RESID )
*
* -- LAPACK test routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDAFAC, LDC, N
DOUBLE PRECISION RESID
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), AFAC( LDAFAC, * ), C( LDC, * ),
$ E( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
$ CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
INTEGER I, INFO, J
DOUBLE PRECISION ANORM, EPS
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION ZLANHE, DLAMCH
EXTERNAL LSAME, ZLANHE, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL ZLASET, ZLAVHE_ROOK, ZSYCONVF_ROOK
* ..
* .. Intrinsic Functions ..
INTRINSIC DIMAG, DBLE
* ..
* .. Executable Statements ..
*
* Quick exit if N = 0.
*
IF( N.LE.0 ) THEN
RESID = ZERO
RETURN
END IF
*
* a) Revert to multiplyers of L
*
CALL ZSYCONVF_ROOK( UPLO, 'R', N, AFAC, LDAFAC, E, IPIV, INFO )
*
* 1) Determine EPS and the norm of A.
*
EPS = DLAMCH( 'Epsilon' )
ANORM = ZLANHE( '1', UPLO, N, A, LDA, RWORK )
*
* Check the imaginary parts of the diagonal elements and return with
* an error code if any are nonzero.
*
DO J = 1, N
IF( DIMAG( AFAC( J, J ) ).NE.ZERO ) THEN
RESID = ONE / EPS
RETURN
END IF
END DO
*
* 2) Initialize C to the identity matrix.
*
CALL ZLASET( 'Full', N, N, CZERO, CONE, C, LDC )
*
* 3) Call ZLAVHE_ROOK to form the product D * U' (or D * L' ).
*
CALL ZLAVHE_ROOK( UPLO, 'Conjugate', 'Non-unit', N, N, AFAC,
$ LDAFAC, IPIV, C, LDC, INFO )
*
* 4) Call ZLAVHE_RK again to multiply by U (or L ).
*
CALL ZLAVHE_ROOK( UPLO, 'No transpose', 'Unit', N, N, AFAC,
$ LDAFAC, IPIV, C, LDC, INFO )
*
* 5) Compute the difference C - A .
*
IF( LSAME( UPLO, 'U' ) ) THEN
DO J = 1, N
DO I = 1, J - 1
C( I, J ) = C( I, J ) - A( I, J )
END DO
C( J, J ) = C( J, J ) - DBLE( A( J, J ) )
END DO
ELSE
DO J = 1, N
C( J, J ) = C( J, J ) - DBLE( A( J, J ) )
DO I = J + 1, N
C( I, J ) = C( I, J ) - A( I, J )
END DO
END DO
END IF
*
* 6) Compute norm( C - A ) / ( N * norm(A) * EPS )
*
RESID = ZLANHE( '1', UPLO, N, C, LDC, RWORK )
*
IF( ANORM.LE.ZERO ) THEN
IF( RESID.NE.ZERO )
$ RESID = ONE / EPS
ELSE
RESID = ( ( RESID/DBLE( N ) )/ANORM ) / EPS
END IF
*
* b) Convert to factor of L (or U)
*
CALL ZSYCONVF_ROOK( UPLO, 'C', N, AFAC, LDAFAC, E, IPIV, INFO )
*
RETURN
*
* End of ZHET01_3
*
END
|
