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

 subroutine cpbtrf ( character uplo, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, integer info )

CPBTRF

Purpose:
``` CPBTRF 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 and L is lower triangular.```
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 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.``` [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 leading principal minor of order i is not positive, and the factorization could not be completed.```
Further Details:
```  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.```
Contributors:
Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

Definition at line 141 of file cpbtrf.f.

142*
143* -- LAPACK computational routine --
144* -- LAPACK is a software package provided by Univ. of Tennessee, --
145* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
146*
147* .. Scalar Arguments ..
148 CHARACTER UPLO
149 INTEGER INFO, KD, LDAB, N
150* ..
151* .. Array Arguments ..
152 COMPLEX AB( LDAB, * )
153* ..
154*
155* =====================================================================
156*
157* .. Parameters ..
158 REAL ONE, ZERO
159 parameter( one = 1.0e+0, zero = 0.0e+0 )
160 COMPLEX CONE
161 parameter( cone = ( 1.0e+0, 0.0e+0 ) )
162 INTEGER NBMAX, LDWORK
163 parameter( nbmax = 32, ldwork = nbmax+1 )
164* ..
165* .. Local Scalars ..
166 INTEGER I, I2, I3, IB, II, J, JJ, NB
167* ..
168* .. Local Arrays ..
169 COMPLEX WORK( LDWORK, NBMAX )
170* ..
171* .. External Functions ..
172 LOGICAL LSAME
173 INTEGER ILAENV
174 EXTERNAL lsame, ilaenv
175* ..
176* .. External Subroutines ..
177 EXTERNAL cgemm, cherk, cpbtf2, cpotf2, ctrsm, xerbla
178* ..
179* .. Intrinsic Functions ..
180 INTRINSIC min
181* ..
182* .. Executable Statements ..
183*
184* Test the input parameters.
185*
186 info = 0
187 IF( ( .NOT.lsame( uplo, 'U' ) ) .AND.
188 \$ ( .NOT.lsame( uplo, 'L' ) ) ) THEN
189 info = -1
190 ELSE IF( n.LT.0 ) THEN
191 info = -2
192 ELSE IF( kd.LT.0 ) THEN
193 info = -3
194 ELSE IF( ldab.LT.kd+1 ) THEN
195 info = -5
196 END IF
197 IF( info.NE.0 ) THEN
198 CALL xerbla( 'CPBTRF', -info )
199 RETURN
200 END IF
201*
202* Quick return if possible
203*
204 IF( n.EQ.0 )
205 \$ RETURN
206*
207* Determine the block size for this environment
208*
209 nb = ilaenv( 1, 'CPBTRF', uplo, n, kd, -1, -1 )
210*
211* The block size must not exceed the semi-bandwidth KD, and must not
212* exceed the limit set by the size of the local array WORK.
213*
214 nb = min( nb, nbmax )
215*
216 IF( nb.LE.1 .OR. nb.GT.kd ) THEN
217*
218* Use unblocked code
219*
220 CALL cpbtf2( uplo, n, kd, ab, ldab, info )
221 ELSE
222*
223* Use blocked code
224*
225 IF( lsame( uplo, 'U' ) ) THEN
226*
227* Compute the Cholesky factorization of a Hermitian band
228* matrix, given the upper triangle of the matrix in band
229* storage.
230*
231* Zero the upper triangle of the work array.
232*
233 DO 20 j = 1, nb
234 DO 10 i = 1, j - 1
235 work( i, j ) = zero
236 10 CONTINUE
237 20 CONTINUE
238*
239* Process the band matrix one diagonal block at a time.
240*
241 DO 70 i = 1, n, nb
242 ib = min( nb, n-i+1 )
243*
244* Factorize the diagonal block
245*
246 CALL cpotf2( uplo, ib, ab( kd+1, i ), ldab-1, ii )
247 IF( ii.NE.0 ) THEN
248 info = i + ii - 1
249 GO TO 150
250 END IF
251 IF( i+ib.LE.n ) THEN
252*
253* Update the relevant part of the trailing submatrix.
254* If A11 denotes the diagonal block which has just been
255* factorized, then we need to update the remaining
256* blocks in the diagram:
257*
258* A11 A12 A13
259* A22 A23
260* A33
261*
262* The numbers of rows and columns in the partitioning
263* are IB, I2, I3 respectively. The blocks A12, A22 and
264* A23 are empty if IB = KD. The upper triangle of A13
265* lies outside the band.
266*
267 i2 = min( kd-ib, n-i-ib+1 )
268 i3 = min( ib, n-i-kd+1 )
269*
270 IF( i2.GT.0 ) THEN
271*
272* Update A12
273*
274 CALL ctrsm( 'Left', 'Upper', 'Conjugate transpose',
275 \$ 'Non-unit', ib, i2, cone,
276 \$ ab( kd+1, i ), ldab-1,
277 \$ ab( kd+1-ib, i+ib ), ldab-1 )
278*
279* Update A22
280*
281 CALL cherk( 'Upper', 'Conjugate transpose', i2, ib,
282 \$ -one, ab( kd+1-ib, i+ib ), ldab-1, one,
283 \$ ab( kd+1, i+ib ), ldab-1 )
284 END IF
285*
286 IF( i3.GT.0 ) THEN
287*
288* Copy the lower triangle of A13 into the work array.
289*
290 DO 40 jj = 1, i3
291 DO 30 ii = jj, ib
292 work( ii, jj ) = ab( ii-jj+1, jj+i+kd-1 )
293 30 CONTINUE
294 40 CONTINUE
295*
296* Update A13 (in the work array).
297*
298 CALL ctrsm( 'Left', 'Upper', 'Conjugate transpose',
299 \$ 'Non-unit', ib, i3, cone,
300 \$ ab( kd+1, i ), ldab-1, work, ldwork )
301*
302* Update A23
303*
304 IF( i2.GT.0 )
305 \$ CALL cgemm( 'Conjugate transpose',
306 \$ 'No transpose', i2, i3, ib, -cone,
307 \$ ab( kd+1-ib, i+ib ), ldab-1, work,
308 \$ ldwork, cone, ab( 1+ib, i+kd ),
309 \$ ldab-1 )
310*
311* Update A33
312*
313 CALL cherk( 'Upper', 'Conjugate transpose', i3, ib,
314 \$ -one, work, ldwork, one,
315 \$ ab( kd+1, i+kd ), ldab-1 )
316*
317* Copy the lower triangle of A13 back into place.
318*
319 DO 60 jj = 1, i3
320 DO 50 ii = jj, ib
321 ab( ii-jj+1, jj+i+kd-1 ) = work( ii, jj )
322 50 CONTINUE
323 60 CONTINUE
324 END IF
325 END IF
326 70 CONTINUE
327 ELSE
328*
329* Compute the Cholesky factorization of a Hermitian band
330* matrix, given the lower triangle of the matrix in band
331* storage.
332*
333* Zero the lower triangle of the work array.
334*
335 DO 90 j = 1, nb
336 DO 80 i = j + 1, nb
337 work( i, j ) = zero
338 80 CONTINUE
339 90 CONTINUE
340*
341* Process the band matrix one diagonal block at a time.
342*
343 DO 140 i = 1, n, nb
344 ib = min( nb, n-i+1 )
345*
346* Factorize the diagonal block
347*
348 CALL cpotf2( uplo, ib, ab( 1, i ), ldab-1, ii )
349 IF( ii.NE.0 ) THEN
350 info = i + ii - 1
351 GO TO 150
352 END IF
353 IF( i+ib.LE.n ) THEN
354*
355* Update the relevant part of the trailing submatrix.
356* If A11 denotes the diagonal block which has just been
357* factorized, then we need to update the remaining
358* blocks in the diagram:
359*
360* A11
361* A21 A22
362* A31 A32 A33
363*
364* The numbers of rows and columns in the partitioning
365* are IB, I2, I3 respectively. The blocks A21, A22 and
366* A32 are empty if IB = KD. The lower triangle of A31
367* lies outside the band.
368*
369 i2 = min( kd-ib, n-i-ib+1 )
370 i3 = min( ib, n-i-kd+1 )
371*
372 IF( i2.GT.0 ) THEN
373*
374* Update A21
375*
376 CALL ctrsm( 'Right', 'Lower',
377 \$ 'Conjugate transpose', 'Non-unit', i2,
378 \$ ib, cone, ab( 1, i ), ldab-1,
379 \$ ab( 1+ib, i ), ldab-1 )
380*
381* Update A22
382*
383 CALL cherk( 'Lower', 'No transpose', i2, ib, -one,
384 \$ ab( 1+ib, i ), ldab-1, one,
385 \$ ab( 1, i+ib ), ldab-1 )
386 END IF
387*
388 IF( i3.GT.0 ) THEN
389*
390* Copy the upper triangle of A31 into the work array.
391*
392 DO 110 jj = 1, ib
393 DO 100 ii = 1, min( jj, i3 )
394 work( ii, jj ) = ab( kd+1-jj+ii, jj+i-1 )
395 100 CONTINUE
396 110 CONTINUE
397*
398* Update A31 (in the work array).
399*
400 CALL ctrsm( 'Right', 'Lower',
401 \$ 'Conjugate transpose', 'Non-unit', i3,
402 \$ ib, cone, ab( 1, i ), ldab-1, work,
403 \$ ldwork )
404*
405* Update A32
406*
407 IF( i2.GT.0 )
408 \$ CALL cgemm( 'No transpose',
409 \$ 'Conjugate transpose', i3, i2, ib,
410 \$ -cone, work, ldwork, ab( 1+ib, i ),
411 \$ ldab-1, cone, ab( 1+kd-ib, i+ib ),
412 \$ ldab-1 )
413*
414* Update A33
415*
416 CALL cherk( 'Lower', 'No transpose', i3, ib, -one,
417 \$ work, ldwork, one, ab( 1, i+kd ),
418 \$ ldab-1 )
419*
420* Copy the upper triangle of A31 back into place.
421*
422 DO 130 jj = 1, ib
423 DO 120 ii = 1, min( jj, i3 )
424 ab( kd+1-jj+ii, jj+i-1 ) = work( ii, jj )
425 120 CONTINUE
426 130 CONTINUE
427 END IF
428 END IF
429 140 CONTINUE
430 END IF
431 END IF
432 RETURN
433*
434 150 CONTINUE
435 RETURN
436*
437* End of CPBTRF
438*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine cgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
CGEMM
Definition cgemm.f:188
subroutine cherk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc)
CHERK
Definition cherk.f:173
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine cpbtf2(uplo, n, kd, ab, ldab, info)
CPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (un...
Definition cpbtf2.f:142
subroutine cpotf2(uplo, n, a, lda, info)
CPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblock...
Definition cpotf2.f:109
subroutine ctrsm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb)
CTRSM
Definition ctrsm.f:180
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