LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
cpbstf.f
Go to the documentation of this file.
1 *> \brief \b CPBSTF
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download CPBSTF + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cpbstf.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cpbstf.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cpbstf.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CPBSTF( UPLO, N, KD, AB, LDAB, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, KD, LDAB, N
26 * ..
27 * .. Array Arguments ..
28 * COMPLEX AB( LDAB, * )
29 * ..
30 *
31 *
32 *> \par Purpose:
33 * =============
34 *>
35 *> \verbatim
36 *>
37 *> CPBSTF computes a split Cholesky factorization of a complex
38 *> Hermitian positive definite band matrix A.
39 *>
40 *> This routine is designed to be used in conjunction with CHBGST.
41 *>
42 *> The factorization has the form A = S**H*S where S is a band matrix
43 *> of the same bandwidth as A and the following structure:
44 *>
45 *> S = ( U )
46 *> ( M L )
47 *>
48 *> where U is upper triangular of order m = (n+kd)/2, and L is lower
49 *> triangular of order n-m.
50 *> \endverbatim
51 *
52 * Arguments:
53 * ==========
54 *
55 *> \param[in] UPLO
56 *> \verbatim
57 *> UPLO is CHARACTER*1
58 *> = 'U': Upper triangle of A is stored;
59 *> = 'L': Lower triangle of A is stored.
60 *> \endverbatim
61 *>
62 *> \param[in] N
63 *> \verbatim
64 *> N is INTEGER
65 *> The order of the matrix A. N >= 0.
66 *> \endverbatim
67 *>
68 *> \param[in] KD
69 *> \verbatim
70 *> KD is INTEGER
71 *> The number of superdiagonals of the matrix A if UPLO = 'U',
72 *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
73 *> \endverbatim
74 *>
75 *> \param[in,out] AB
76 *> \verbatim
77 *> AB is COMPLEX array, dimension (LDAB,N)
78 *> On entry, the upper or lower triangle of the Hermitian band
79 *> matrix A, stored in the first kd+1 rows of the array. The
80 *> j-th column of A is stored in the j-th column of the array AB
81 *> as follows:
82 *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
83 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
84 *>
85 *> On exit, if INFO = 0, the factor S from the split Cholesky
86 *> factorization A = S**H*S. See Further Details.
87 *> \endverbatim
88 *>
89 *> \param[in] LDAB
90 *> \verbatim
91 *> LDAB is INTEGER
92 *> The leading dimension of the array AB. LDAB >= KD+1.
93 *> \endverbatim
94 *>
95 *> \param[out] INFO
96 *> \verbatim
97 *> INFO is INTEGER
98 *> = 0: successful exit
99 *> < 0: if INFO = -i, the i-th argument had an illegal value
100 *> > 0: if INFO = i, the factorization could not be completed,
101 *> because the updated element a(i,i) was negative; the
102 *> matrix A is not positive definite.
103 *> \endverbatim
104 *
105 * Authors:
106 * ========
107 *
108 *> \author Univ. of Tennessee
109 *> \author Univ. of California Berkeley
110 *> \author Univ. of Colorado Denver
111 *> \author NAG Ltd.
112 *
113 *> \ingroup complexOTHERcomputational
114 *
115 *> \par Further Details:
116 * =====================
117 *>
118 *> \verbatim
119 *>
120 *> The band storage scheme is illustrated by the following example, when
121 *> N = 7, KD = 2:
122 *>
123 *> S = ( s11 s12 s13 )
124 *> ( s22 s23 s24 )
125 *> ( s33 s34 )
126 *> ( s44 )
127 *> ( s53 s54 s55 )
128 *> ( s64 s65 s66 )
129 *> ( s75 s76 s77 )
130 *>
131 *> If UPLO = 'U', the array AB holds:
132 *>
133 *> on entry: on exit:
134 *>
135 *> * * a13 a24 a35 a46 a57 * * s13 s24 s53**H s64**H s75**H
136 *> * a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54**H s65**H s76**H
137 *> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
138 *>
139 *> If UPLO = 'L', the array AB holds:
140 *>
141 *> on entry: on exit:
142 *>
143 *> a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
144 *> a21 a32 a43 a54 a65 a76 * s12**H s23**H s34**H s54 s65 s76 *
145 *> a31 a42 a53 a64 a64 * * s13**H s24**H s53 s64 s75 * *
146 *>
147 *> Array elements marked * are not used by the routine; s12**H denotes
148 *> conjg(s12); the diagonal elements of S are real.
149 *> \endverbatim
150 *>
151 * =====================================================================
152  SUBROUTINE cpbstf( UPLO, N, KD, AB, LDAB, INFO )
153 *
154 * -- LAPACK computational routine --
155 * -- LAPACK is a software package provided by Univ. of Tennessee, --
156 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
157 *
158 * .. Scalar Arguments ..
159  CHARACTER UPLO
160  INTEGER INFO, KD, LDAB, N
161 * ..
162 * .. Array Arguments ..
163  COMPLEX AB( LDAB, * )
164 * ..
165 *
166 * =====================================================================
167 *
168 * .. Parameters ..
169  REAL ONE, ZERO
170  parameter( one = 1.0e+0, zero = 0.0e+0 )
171 * ..
172 * .. Local Scalars ..
173  LOGICAL UPPER
174  INTEGER J, KLD, KM, M
175  REAL AJJ
176 * ..
177 * .. External Functions ..
178  LOGICAL LSAME
179  EXTERNAL lsame
180 * ..
181 * .. External Subroutines ..
182  EXTERNAL cher, clacgv, csscal, xerbla
183 * ..
184 * .. Intrinsic Functions ..
185  INTRINSIC max, min, real, sqrt
186 * ..
187 * .. Executable Statements ..
188 *
189 * Test the input parameters.
190 *
191  info = 0
192  upper = lsame( uplo, 'U' )
193  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
194  info = -1
195  ELSE IF( n.LT.0 ) THEN
196  info = -2
197  ELSE IF( kd.LT.0 ) THEN
198  info = -3
199  ELSE IF( ldab.LT.kd+1 ) THEN
200  info = -5
201  END IF
202  IF( info.NE.0 ) THEN
203  CALL xerbla( 'CPBSTF', -info )
204  RETURN
205  END IF
206 *
207 * Quick return if possible
208 *
209  IF( n.EQ.0 )
210  $ RETURN
211 *
212  kld = max( 1, ldab-1 )
213 *
214 * Set the splitting point m.
215 *
216  m = ( n+kd ) / 2
217 *
218  IF( upper ) THEN
219 *
220 * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
221 *
222  DO 10 j = n, m + 1, -1
223 *
224 * Compute s(j,j) and test for non-positive-definiteness.
225 *
226  ajj = real( ab( kd+1, j ) )
227  IF( ajj.LE.zero ) THEN
228  ab( kd+1, j ) = ajj
229  GO TO 50
230  END IF
231  ajj = sqrt( ajj )
232  ab( kd+1, j ) = ajj
233  km = min( j-1, kd )
234 *
235 * Compute elements j-km:j-1 of the j-th column and update the
236 * the leading submatrix within the band.
237 *
238  CALL csscal( km, one / ajj, ab( kd+1-km, j ), 1 )
239  CALL cher( 'Upper', km, -one, ab( kd+1-km, j ), 1,
240  $ ab( kd+1, j-km ), kld )
241  10 CONTINUE
242 *
243 * Factorize the updated submatrix A(1:m,1:m) as U**H*U.
244 *
245  DO 20 j = 1, m
246 *
247 * Compute s(j,j) and test for non-positive-definiteness.
248 *
249  ajj = real( ab( kd+1, j ) )
250  IF( ajj.LE.zero ) THEN
251  ab( kd+1, j ) = ajj
252  GO TO 50
253  END IF
254  ajj = sqrt( ajj )
255  ab( kd+1, j ) = ajj
256  km = min( kd, m-j )
257 *
258 * Compute elements j+1:j+km of the j-th row and update the
259 * trailing submatrix within the band.
260 *
261  IF( km.GT.0 ) THEN
262  CALL csscal( km, one / ajj, ab( kd, j+1 ), kld )
263  CALL clacgv( km, ab( kd, j+1 ), kld )
264  CALL cher( 'Upper', km, -one, ab( kd, j+1 ), kld,
265  $ ab( kd+1, j+1 ), kld )
266  CALL clacgv( km, ab( kd, j+1 ), kld )
267  END IF
268  20 CONTINUE
269  ELSE
270 *
271 * Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
272 *
273  DO 30 j = n, m + 1, -1
274 *
275 * Compute s(j,j) and test for non-positive-definiteness.
276 *
277  ajj = real( ab( 1, j ) )
278  IF( ajj.LE.zero ) THEN
279  ab( 1, j ) = ajj
280  GO TO 50
281  END IF
282  ajj = sqrt( ajj )
283  ab( 1, j ) = ajj
284  km = min( j-1, kd )
285 *
286 * Compute elements j-km:j-1 of the j-th row and update the
287 * trailing submatrix within the band.
288 *
289  CALL csscal( km, one / ajj, ab( km+1, j-km ), kld )
290  CALL clacgv( km, ab( km+1, j-km ), kld )
291  CALL cher( 'Lower', km, -one, ab( km+1, j-km ), kld,
292  $ ab( 1, j-km ), kld )
293  CALL clacgv( km, ab( km+1, j-km ), kld )
294  30 CONTINUE
295 *
296 * Factorize the updated submatrix A(1:m,1:m) as U**H*U.
297 *
298  DO 40 j = 1, m
299 *
300 * Compute s(j,j) and test for non-positive-definiteness.
301 *
302  ajj = real( ab( 1, j ) )
303  IF( ajj.LE.zero ) THEN
304  ab( 1, j ) = ajj
305  GO TO 50
306  END IF
307  ajj = sqrt( ajj )
308  ab( 1, j ) = ajj
309  km = min( kd, m-j )
310 *
311 * Compute elements j+1:j+km of the j-th column and update the
312 * trailing submatrix within the band.
313 *
314  IF( km.GT.0 ) THEN
315  CALL csscal( km, one / ajj, ab( 2, j ), 1 )
316  CALL cher( 'Lower', km, -one, ab( 2, j ), 1,
317  $ ab( 1, j+1 ), kld )
318  END IF
319  40 CONTINUE
320  END IF
321  RETURN
322 *
323  50 CONTINUE
324  info = j
325  RETURN
326 *
327 * End of CPBSTF
328 *
329  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine csscal(N, SA, CX, INCX)
CSSCAL
Definition: csscal.f:78
subroutine cher(UPLO, N, ALPHA, X, INCX, A, LDA)
CHER
Definition: cher.f:135
subroutine clacgv(N, X, INCX)
CLACGV conjugates a complex vector.
Definition: clacgv.f:74
subroutine cpbstf(UPLO, N, KD, AB, LDAB, INFO)
CPBSTF
Definition: cpbstf.f:153