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
|
*> \brief \b ZPOTRF2
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* RECURSIVE SUBROUTINE ZPOTRF2( UPLO, N, A, LDA, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, N
* ..
* .. Array Arguments ..
* COMPLEX*16 A( LDA, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZPOTRF2 computes the Cholesky factorization of a Hermitian
*> positive definite matrix A using the recursive algorithm.
*>
*> The factorization has the form
*> A = U**H * U, if UPLO = 'U', or
*> A = L * L**H, if UPLO = 'L',
*> where U is an upper triangular matrix and L is lower triangular.
*>
*> This is the recursive version of the algorithm. It divides
*> the matrix into four submatrices:
*>
*> [ A11 | A12 ] where A11 is n1 by n1 and A22 is n2 by n2
*> A = [ -----|----- ] with n1 = n/2
*> [ A21 | A22 ] n2 = n-n1
*>
*> The subroutine calls itself to factor A11. Update and scale A21
*> or A12, update A22 then call itself to factor A22.
*>
*> \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 order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 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 factor U or L from the Cholesky
*> factorization A = U**H*U or A = L*L**H.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \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, the leading minor of order i is not
*> positive definite, and the factorization could not be
*> completed.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup complex16POcomputational
*
* =====================================================================
RECURSIVE SUBROUTINE ZPOTRF2( UPLO, N, A, LDA, INFO )
*
* -- LAPACK computational 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, N
* ..
* .. Array Arguments ..
COMPLEX*16 A( LDA, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
COMPLEX*16 CONE
PARAMETER ( CONE = (1.0D+0, 0.0D+0) )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER N1, N2, IINFO
DOUBLE PRECISION AJJ
* ..
* .. External Functions ..
LOGICAL LSAME, DISNAN
EXTERNAL LSAME, DISNAN
* ..
* .. External Subroutines ..
EXTERNAL ZHERK, ZTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, DBLE, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZPOTRF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* N=1 case
*
IF( N.EQ.1 ) THEN
*
* Test for non-positive-definiteness
*
AJJ = DBLE( A( 1, 1 ) )
IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
INFO = 1
RETURN
END IF
*
* Factor
*
A( 1, 1 ) = SQRT( AJJ )
*
* Use recursive code
*
ELSE
N1 = N/2
N2 = N-N1
*
* Factor A11
*
CALL ZPOTRF2( UPLO, N1, A( 1, 1 ), LDA, IINFO )
IF ( IINFO.NE.0 ) THEN
INFO = IINFO
RETURN
END IF
*
* Compute the Cholesky factorization A = U**H*U
*
IF( UPPER ) THEN
*
* Update and scale A12
*
CALL ZTRSM( 'L', 'U', 'C', 'N', N1, N2, CONE,
$ A( 1, 1 ), LDA, A( 1, N1+1 ), LDA )
*
* Update and factor A22
*
CALL ZHERK( UPLO, 'C', N2, N1, -ONE, A( 1, N1+1 ), LDA,
$ ONE, A( N1+1, N1+1 ), LDA )
CALL ZPOTRF2( UPLO, N2, A( N1+1, N1+1 ), LDA, IINFO )
IF ( IINFO.NE.0 ) THEN
INFO = IINFO + N1
RETURN
END IF
*
* Compute the Cholesky factorization A = L*L**H
*
ELSE
*
* Update and scale A21
*
CALL ZTRSM( 'R', 'L', 'C', 'N', N2, N1, CONE,
$ A( 1, 1 ), LDA, A( N1+1, 1 ), LDA )
*
* Update and factor A22
*
CALL ZHERK( UPLO, 'N', N2, N1, -ONE, A( N1+1, 1 ), LDA,
$ ONE, A( N1+1, N1+1 ), LDA )
CALL ZPOTRF2( UPLO, N2, A( N1+1, N1+1 ), LDA, IINFO )
IF ( IINFO.NE.0 ) THEN
INFO = IINFO + N1
RETURN
END IF
END IF
END IF
RETURN
*
* End of ZPOTRF2
*
END
|
