 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ cpbstf()

 subroutine cpbstf ( character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, integer INFO )

CPBSTF

Purpose:
``` CPBSTF computes a split Cholesky factorization of a complex
Hermitian positive definite band matrix A.

This routine is designed to be used in conjunction with CHBGST.

The factorization has the form  A = S**H*S  where S is a band matrix
of the same bandwidth as A and the following structure:

S = ( U    )
( M  L )

where U is upper triangular of order m = (n+kd)/2, and L is lower
triangular of order n-m.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] KD ``` KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.``` [in,out] AB ``` 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). On exit, if INFO = 0, the factor S from the split Cholesky factorization A = S**H*S. See Further Details.``` [in] LDAB ``` LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the factorization could not be completed, because the updated element a(i,i) was negative; the matrix A is not positive definite.```
Further Details:
```  The band storage scheme is illustrated by the following example, when
N = 7, KD = 2:

S = ( s11  s12  s13                     )
(      s22  s23  s24                )
(           s33  s34                )
(                s44                )
(           s53  s54  s55           )
(                s64  s65  s66      )
(                     s75  s76  s77 )

If UPLO = 'U', the array AB holds:

on entry:                          on exit:

*    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53**H s64**H s75**H
*   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54**H s65**H s76**H
a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55    s66    s77

If UPLO = 'L', the array AB holds:

on entry:                          on exit:

a11  a22  a33  a44  a55  a66  a77  s11    s22    s33    s44  s55  s66  s77
a21  a32  a43  a54  a65  a76   *   s12**H s23**H s34**H s54  s65  s76   *
a31  a42  a53  a64  a64   *    *   s13**H s24**H s53    s64  s75   *    *

Array elements marked * are not used by the routine; s12**H denotes
conjg(s12); the diagonal elements of S are real.```

Definition at line 152 of file cpbstf.f.

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 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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
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