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
|
*> \brief \b CPBTF2
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> Download CPBTF2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cpbtf2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cpbtf2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cpbtf2.f">
*> [TXT]</a>
*
* Definition
* ==========
*
* SUBROUTINE CPBTF2( UPLO, N, KD, AB, LDAB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KD, LDAB, N
* ..
* .. Array Arguments ..
* COMPLEX AB( LDAB, * )
* ..
*
* Purpose
* =======
*
*>\details \b Purpose:
*>\verbatim
*>
*> CPBTF2 computes the Cholesky factorization of a complex Hermitian
*> positive definite band matrix A.
*>
*> 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, U**H is the conjugate transpose
*> of U, and L is lower triangular.
*>
*> This is the unblocked version of the algorithm, calling Level 2 BLAS.
*>
*>\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 order 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,out] AB
*> \verbatim
*> AB is COMPLEX array, dimension (LDAB,N)
*> On entry, the upper or lower triangle of the Hermitian band
*> matrix A, stored in the first KD+1 rows of the array. The
*> j-th column of A is stored in the j-th column of the array AB
*> as follows:
*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*> \verbatim
*> On exit, if INFO = 0, the triangular factor U or L from the
*> Cholesky factorization A = U**H *U or A = L*L**H of the band
*> matrix A, in the same storage format as A.
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*> LDAB is INTEGER
*> The leading dimension of the array AB. LDAB >= KD+1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -k, the k-th argument had an illegal value
*> > 0: if INFO = k, the leading minor of order k 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 November 2011
*
*> \ingroup complexOTHERcomputational
*
*
* Further Details
* ===============
*>\details \b Further \b Details
*> \verbatim
*>
*> The band storage scheme is illustrated by the following example, when
*> N = 6, KD = 2, and UPLO = 'U':
*>
*> On entry: On exit:
*>
*> * * a13 a24 a35 a46 * * u13 u24 u35 u46
*> * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
*> a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
*>
*> Similarly, if UPLO = 'L' the format of A is as follows:
*>
*> On entry: On exit:
*>
*> a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
*> a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
*> a31 a42 a53 a64 * * l31 l42 l53 l64 * *
*>
*> Array elements marked * are not used by the routine.
*>
*> \endverbatim
*>
* =====================================================================
SUBROUTINE CPBTF2( UPLO, N, KD, AB, LDAB, INFO )
*
* -- LAPACK computational routine (version 3.3.1) --
* -- 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 UPLO
INTEGER INFO, KD, LDAB, N
* ..
* .. Array Arguments ..
COMPLEX AB( LDAB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, KLD, KN
REAL AJJ
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL CHER, CLACGV, CSSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, REAL, 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( KD.LT.0 ) THEN
INFO = -3
ELSE IF( LDAB.LT.KD+1 ) THEN
INFO = -5
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CPBTF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
KLD = MAX( 1, LDAB-1 )
*
IF( UPPER ) THEN
*
* Compute the Cholesky factorization A = U**H * U.
*
DO 10 J = 1, N
*
* Compute U(J,J) and test for non-positive-definiteness.
*
AJJ = REAL( AB( KD+1, J ) )
IF( AJJ.LE.ZERO ) THEN
AB( KD+1, J ) = AJJ
GO TO 30
END IF
AJJ = SQRT( AJJ )
AB( KD+1, J ) = AJJ
*
* Compute elements J+1:J+KN of row J and update the
* trailing submatrix within the band.
*
KN = MIN( KD, N-J )
IF( KN.GT.0 ) THEN
CALL CSSCAL( KN, ONE / AJJ, AB( KD, J+1 ), KLD )
CALL CLACGV( KN, AB( KD, J+1 ), KLD )
CALL CHER( 'Upper', KN, -ONE, AB( KD, J+1 ), KLD,
$ AB( KD+1, J+1 ), KLD )
CALL CLACGV( KN, AB( KD, J+1 ), KLD )
END IF
10 CONTINUE
ELSE
*
* Compute the Cholesky factorization A = L*L**H.
*
DO 20 J = 1, N
*
* Compute L(J,J) and test for non-positive-definiteness.
*
AJJ = REAL( AB( 1, J ) )
IF( AJJ.LE.ZERO ) THEN
AB( 1, J ) = AJJ
GO TO 30
END IF
AJJ = SQRT( AJJ )
AB( 1, J ) = AJJ
*
* Compute elements J+1:J+KN of column J and update the
* trailing submatrix within the band.
*
KN = MIN( KD, N-J )
IF( KN.GT.0 ) THEN
CALL CSSCAL( KN, ONE / AJJ, AB( 2, J ), 1 )
CALL CHER( 'Lower', KN, -ONE, AB( 2, J ), 1,
$ AB( 1, J+1 ), KLD )
END IF
20 CONTINUE
END IF
RETURN
*
30 CONTINUE
INFO = J
RETURN
*
* End of CPBTF2
*
END
|
