LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ spbstf()

 subroutine spbstf ( character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, integer info )

SPBSTF

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

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

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 REAL 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 spbstf.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 REAL AB( LDAB, * )
163* ..
164*
165* =====================================================================
166*
167* .. Parameters ..
168 REAL ONE, ZERO
169 parameter( one = 1.0e+0, zero = 0.0e+0 )
170* ..
171* .. Local Scalars ..
172 LOGICAL UPPER
173 INTEGER J, KLD, KM, M
174 REAL AJJ
175* ..
176* .. External Functions ..
177 LOGICAL LSAME
178 EXTERNAL lsame
179* ..
180* .. External Subroutines ..
181 EXTERNAL sscal, ssyr, 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( 'SPBSTF', -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 sscal( km, one / ajj, ab( kd+1-km, j ), 1 )
236 CALL ssyr( '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 sscal( km, one / ajj, ab( kd, j+1 ), kld )
258 CALL ssyr( '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 sscal( km, one / ajj, ab( km+1, j-km ), kld )
281 CALL ssyr( '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 sscal( km, one / ajj, ab( 2, j ), 1 )
303 CALL ssyr( '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 SPBSTF
315*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine ssyr(uplo, n, alpha, x, incx, a, lda)
SSYR
Definition ssyr.f:132
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine sscal(n, sa, sx, incx)
SSCAL
Definition sscal.f:79
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