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
|
*> \brief <b> CHESV computes the solution to system of linear equations A * X = B for HE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CHESV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chesv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chesv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chesv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CHESV( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK,
* LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LDB, LWORK, N, NRHS
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CHESV computes the solution to a complex system of linear equations
*> A * X = B,
*> where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS
*> matrices.
*>
*> The diagonal pivoting method is used to factor A as
*> A = U * D * U**H, if UPLO = 'U', or
*> A = L * D * L**H, if UPLO = 'L',
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and D is Hermitian and block diagonal with
*> 1-by-1 and 2-by-2 diagonal blocks. The factored form of A is then
*> used to solve the system of equations A * X = B.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*> NRHS is INTEGER
*> The number of right hand sides, i.e., the number of columns
*> of the matrix B. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA,N)
*> On entry, the Hermitian 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.
*>
*> On exit, if INFO = 0, the block diagonal matrix D and the
*> multipliers used to obtain the factor U or L from the
*> factorization A = U*D*U**H or A = L*D*L**H as computed by
*> CHETRF.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D, as
*> determined by CHETRF. If IPIV(k) > 0, then rows and columns
*> k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
*> diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
*> then rows and columns k-1 and -IPIV(k) were interchanged and
*> D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and
*> IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
*> -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
*> diagonal block.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX array, dimension (LDB,NRHS)
*> On entry, the N-by-NRHS right hand side matrix B.
*> On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of WORK. LWORK >= 1, and for best performance
*> LWORK >= max(1,N*NB), where NB is the optimal blocksize for
*> CHETRF.
*> for LWORK < N, TRS will be done with Level BLAS 2
*> for LWORK >= N, TRS will be done with Level BLAS 3
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, D(i,i) is exactly zero. The factorization
*> has been completed, but the block diagonal matrix D is
*> exactly singular, so the solution could not be computed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup complexHEsolve
*
* =====================================================================
SUBROUTINE CHESV( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, WORK,
$ LWORK, INFO )
*
* -- LAPACK driver 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 INFO, LDA, LDB, LWORK, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER LWKOPT, NB
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, CHETRF, CHETRS, CHETRS2
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
INFO = -10
END IF
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
LWKOPT = 1
ELSE
NB = ILAENV( 1, 'CHETRF', UPLO, N, -1, -1, -1 )
LWKOPT = N*NB
END IF
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CHESV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Compute the factorization A = U*D*U**H or A = L*D*L**H.
*
CALL CHETRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
IF( INFO.EQ.0 ) THEN
*
* Solve the system A*X = B, overwriting B with X.
*
IF ( LWORK.LT.N ) THEN
*
* Solve with TRS ( Use Level BLAS 2)
*
CALL CHETRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
ELSE
*
* Solve with TRS2 ( Use Level BLAS 3)
*
CALL CHETRS2( UPLO,N,NRHS,A,LDA,IPIV,B,LDB,WORK,INFO )
*
END IF
*
END IF
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of CHESV
*
END
|
