 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dpbstf()

 subroutine dpbstf ( character UPLO, integer N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, integer INFO )

DPBSTF

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Purpose:
``` DPBSTF computes a split Cholesky factorization of a real
symmetric positive definite band matrix A.

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

The factorization has the form  A = S**T*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 DOUBLE PRECISION array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric 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**T*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  s64  s75
*   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54  s65  s76
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  s23  s34  s54  s65  s76   *
a31  a42  a53  a64  a64   *    *   s13  s24  s53  s64  s75   *    *

Array elements marked * are not used by the routine.```

Definition at line 151 of file dpbstf.f.

152 *
153 * -- LAPACK computational routine --
154 * -- LAPACK is a software package provided by Univ. of Tennessee, --
155 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
156 *
157 * .. Scalar Arguments ..
158  CHARACTER UPLO
159  INTEGER INFO, KD, LDAB, N
160 * ..
161 * .. Array Arguments ..
162  DOUBLE PRECISION AB( LDAB, * )
163 * ..
164 *
165 * =====================================================================
166 *
167 * .. Parameters ..
168  DOUBLE PRECISION ONE, ZERO
169  parameter( one = 1.0d+0, zero = 0.0d+0 )
170 * ..
171 * .. Local Scalars ..
172  LOGICAL UPPER
173  INTEGER J, KLD, KM, M
174  DOUBLE PRECISION AJJ
175 * ..
176 * .. External Functions ..
177  LOGICAL LSAME
178  EXTERNAL lsame
179 * ..
180 * .. External Subroutines ..
181  EXTERNAL dscal, dsyr, xerbla
182 * ..
183 * .. Intrinsic Functions ..
184  INTRINSIC max, min, sqrt
185 * ..
186 * .. Executable Statements ..
187 *
188 * Test the input parameters.
189 *
190  info = 0
191  upper = lsame( uplo, 'U' )
192  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
193  info = -1
194  ELSE IF( n.LT.0 ) THEN
195  info = -2
196  ELSE IF( kd.LT.0 ) THEN
197  info = -3
198  ELSE IF( ldab.LT.kd+1 ) THEN
199  info = -5
200  END IF
201  IF( info.NE.0 ) THEN
202  CALL xerbla( 'DPBSTF', -info )
203  RETURN
204  END IF
205 *
206 * Quick return if possible
207 *
208  IF( n.EQ.0 )
209  \$ RETURN
210 *
211  kld = max( 1, ldab-1 )
212 *
213 * Set the splitting point m.
214 *
215  m = ( n+kd ) / 2
216 *
217  IF( upper ) THEN
218 *
219 * Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
220 *
221  DO 10 j = n, m + 1, -1
222 *
223 * Compute s(j,j) and test for non-positive-definiteness.
224 *
225  ajj = ab( kd+1, j )
226  IF( ajj.LE.zero )
227  \$ GO TO 50
228  ajj = sqrt( ajj )
229  ab( kd+1, j ) = ajj
230  km = min( j-1, kd )
231 *
232 * Compute elements j-km:j-1 of the j-th column and update the
233 * the leading submatrix within the band.
234 *
235  CALL dscal( km, one / ajj, ab( kd+1-km, j ), 1 )
236  CALL dsyr( 'Upper', km, -one, ab( kd+1-km, j ), 1,
237  \$ ab( kd+1, j-km ), kld )
238  10 CONTINUE
239 *
240 * Factorize the updated submatrix A(1:m,1:m) as U**T*U.
241 *
242  DO 20 j = 1, m
243 *
244 * Compute s(j,j) and test for non-positive-definiteness.
245 *
246  ajj = ab( kd+1, j )
247  IF( ajj.LE.zero )
248  \$ GO TO 50
249  ajj = sqrt( ajj )
250  ab( kd+1, j ) = ajj
251  km = min( kd, m-j )
252 *
253 * Compute elements j+1:j+km of the j-th row and update the
254 * trailing submatrix within the band.
255 *
256  IF( km.GT.0 ) THEN
257  CALL dscal( km, one / ajj, ab( kd, j+1 ), kld )
258  CALL dsyr( 'Upper', km, -one, ab( kd, j+1 ), kld,
259  \$ ab( kd+1, j+1 ), kld )
260  END IF
261  20 CONTINUE
262  ELSE
263 *
264 * Factorize A(m+1:n,m+1:n) as L**T*L, and update A(1:m,1:m).
265 *
266  DO 30 j = n, m + 1, -1
267 *
268 * Compute s(j,j) and test for non-positive-definiteness.
269 *
270  ajj = ab( 1, j )
271  IF( ajj.LE.zero )
272  \$ GO TO 50
273  ajj = sqrt( ajj )
274  ab( 1, j ) = ajj
275  km = min( j-1, kd )
276 *
277 * Compute elements j-km:j-1 of the j-th row and update the
278 * trailing submatrix within the band.
279 *
280  CALL dscal( km, one / ajj, ab( km+1, j-km ), kld )
281  CALL dsyr( 'Lower', km, -one, ab( km+1, j-km ), kld,
282  \$ ab( 1, j-km ), kld )
283  30 CONTINUE
284 *
285 * Factorize the updated submatrix A(1:m,1:m) as U**T*U.
286 *
287  DO 40 j = 1, m
288 *
289 * Compute s(j,j) and test for non-positive-definiteness.
290 *
291  ajj = ab( 1, j )
292  IF( ajj.LE.zero )
293  \$ GO TO 50
294  ajj = sqrt( ajj )
295  ab( 1, j ) = ajj
296  km = min( kd, m-j )
297 *
298 * Compute elements j+1:j+km of the j-th column and update the
299 * trailing submatrix within the band.
300 *
301  IF( km.GT.0 ) THEN
302  CALL dscal( km, one / ajj, ab( 2, j ), 1 )
303  CALL dsyr( 'Lower', km, -one, ab( 2, j ), 1,
304  \$ ab( 1, j+1 ), kld )
305  END IF
306  40 CONTINUE
307  END IF
308  RETURN
309 *
310  50 CONTINUE
311  info = j
312  RETURN
313 *
314 * End of DPBSTF
315 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine dsyr(UPLO, N, ALPHA, X, INCX, A, LDA)
DSYR
Definition: dsyr.f:132
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