LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ spbtrf()

 subroutine spbtrf ( character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, integer INFO )

SPBTRF

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

The factorization has the form
A = U**T * U,  if UPLO = 'U', or
A = L  * L**T,  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 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 triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T 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 minor of order i is not positive definite, 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 spbtrf.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 REAL AB( LDAB, * )
153* ..
154*
155* =====================================================================
156*
157* .. Parameters ..
158 REAL ONE, ZERO
159 parameter( one = 1.0e+0, zero = 0.0e+0 )
160 INTEGER NBMAX, LDWORK
161 parameter( nbmax = 32, ldwork = nbmax+1 )
162* ..
163* .. Local Scalars ..
164 INTEGER I, I2, I3, IB, II, J, JJ, NB
165* ..
166* .. Local Arrays ..
167 REAL WORK( LDWORK, NBMAX )
168* ..
169* .. External Functions ..
170 LOGICAL LSAME
171 INTEGER ILAENV
172 EXTERNAL lsame, ilaenv
173* ..
174* .. External Subroutines ..
175 EXTERNAL sgemm, spbtf2, spotf2, ssyrk, strsm, xerbla
176* ..
177* .. Intrinsic Functions ..
178 INTRINSIC min
179* ..
180* .. Executable Statements ..
181*
182* Test the input parameters.
183*
184 info = 0
185 IF( ( .NOT.lsame( uplo, 'U' ) ) .AND.
186 \$ ( .NOT.lsame( uplo, 'L' ) ) ) THEN
187 info = -1
188 ELSE IF( n.LT.0 ) THEN
189 info = -2
190 ELSE IF( kd.LT.0 ) THEN
191 info = -3
192 ELSE IF( ldab.LT.kd+1 ) THEN
193 info = -5
194 END IF
195 IF( info.NE.0 ) THEN
196 CALL xerbla( 'SPBTRF', -info )
197 RETURN
198 END IF
199*
200* Quick return if possible
201*
202 IF( n.EQ.0 )
203 \$ RETURN
204*
205* Determine the block size for this environment
206*
207 nb = ilaenv( 1, 'SPBTRF', uplo, n, kd, -1, -1 )
208*
209* The block size must not exceed the semi-bandwidth KD, and must not
210* exceed the limit set by the size of the local array WORK.
211*
212 nb = min( nb, nbmax )
213*
214 IF( nb.LE.1 .OR. nb.GT.kd ) THEN
215*
216* Use unblocked code
217*
218 CALL spbtf2( uplo, n, kd, ab, ldab, info )
219 ELSE
220*
221* Use blocked code
222*
223 IF( lsame( uplo, 'U' ) ) THEN
224*
225* Compute the Cholesky factorization of a symmetric band
226* matrix, given the upper triangle of the matrix in band
227* storage.
228*
229* Zero the upper triangle of the work array.
230*
231 DO 20 j = 1, nb
232 DO 10 i = 1, j - 1
233 work( i, j ) = zero
234 10 CONTINUE
235 20 CONTINUE
236*
237* Process the band matrix one diagonal block at a time.
238*
239 DO 70 i = 1, n, nb
240 ib = min( nb, n-i+1 )
241*
242* Factorize the diagonal block
243*
244 CALL spotf2( uplo, ib, ab( kd+1, i ), ldab-1, ii )
245 IF( ii.NE.0 ) THEN
246 info = i + ii - 1
247 GO TO 150
248 END IF
249 IF( i+ib.LE.n ) THEN
250*
251* Update the relevant part of the trailing submatrix.
252* If A11 denotes the diagonal block which has just been
253* factorized, then we need to update the remaining
254* blocks in the diagram:
255*
256* A11 A12 A13
257* A22 A23
258* A33
259*
260* The numbers of rows and columns in the partitioning
261* are IB, I2, I3 respectively. The blocks A12, A22 and
262* A23 are empty if IB = KD. The upper triangle of A13
263* lies outside the band.
264*
265 i2 = min( kd-ib, n-i-ib+1 )
266 i3 = min( ib, n-i-kd+1 )
267*
268 IF( i2.GT.0 ) THEN
269*
270* Update A12
271*
272 CALL strsm( 'Left', 'Upper', 'Transpose',
273 \$ 'Non-unit', ib, i2, one, ab( kd+1, i ),
274 \$ ldab-1, ab( kd+1-ib, i+ib ), ldab-1 )
275*
276* Update A22
277*
278 CALL ssyrk( 'Upper', 'Transpose', i2, ib, -one,
279 \$ ab( kd+1-ib, i+ib ), ldab-1, one,
280 \$ ab( kd+1, i+ib ), ldab-1 )
281 END IF
282*
283 IF( i3.GT.0 ) THEN
284*
285* Copy the lower triangle of A13 into the work array.
286*
287 DO 40 jj = 1, i3
288 DO 30 ii = jj, ib
289 work( ii, jj ) = ab( ii-jj+1, jj+i+kd-1 )
290 30 CONTINUE
291 40 CONTINUE
292*
293* Update A13 (in the work array).
294*
295 CALL strsm( 'Left', 'Upper', 'Transpose',
296 \$ 'Non-unit', ib, i3, one, ab( kd+1, i ),
297 \$ ldab-1, work, ldwork )
298*
299* Update A23
300*
301 IF( i2.GT.0 )
302 \$ CALL sgemm( 'Transpose', 'No Transpose', i2, i3,
303 \$ ib, -one, ab( kd+1-ib, i+ib ),
304 \$ ldab-1, work, ldwork, one,
305 \$ ab( 1+ib, i+kd ), ldab-1 )
306*
307* Update A33
308*
309 CALL ssyrk( 'Upper', 'Transpose', i3, ib, -one,
310 \$ work, ldwork, one, ab( kd+1, i+kd ),
311 \$ ldab-1 )
312*
313* Copy the lower triangle of A13 back into place.
314*
315 DO 60 jj = 1, i3
316 DO 50 ii = jj, ib
317 ab( ii-jj+1, jj+i+kd-1 ) = work( ii, jj )
318 50 CONTINUE
319 60 CONTINUE
320 END IF
321 END IF
322 70 CONTINUE
323 ELSE
324*
325* Compute the Cholesky factorization of a symmetric band
326* matrix, given the lower triangle of the matrix in band
327* storage.
328*
329* Zero the lower triangle of the work array.
330*
331 DO 90 j = 1, nb
332 DO 80 i = j + 1, nb
333 work( i, j ) = zero
334 80 CONTINUE
335 90 CONTINUE
336*
337* Process the band matrix one diagonal block at a time.
338*
339 DO 140 i = 1, n, nb
340 ib = min( nb, n-i+1 )
341*
342* Factorize the diagonal block
343*
344 CALL spotf2( uplo, ib, ab( 1, i ), ldab-1, ii )
345 IF( ii.NE.0 ) THEN
346 info = i + ii - 1
347 GO TO 150
348 END IF
349 IF( i+ib.LE.n ) THEN
350*
351* Update the relevant part of the trailing submatrix.
352* If A11 denotes the diagonal block which has just been
353* factorized, then we need to update the remaining
354* blocks in the diagram:
355*
356* A11
357* A21 A22
358* A31 A32 A33
359*
360* The numbers of rows and columns in the partitioning
361* are IB, I2, I3 respectively. The blocks A21, A22 and
362* A32 are empty if IB = KD. The lower triangle of A31
363* lies outside the band.
364*
365 i2 = min( kd-ib, n-i-ib+1 )
366 i3 = min( ib, n-i-kd+1 )
367*
368 IF( i2.GT.0 ) THEN
369*
370* Update A21
371*
372 CALL strsm( 'Right', 'Lower', 'Transpose',
373 \$ 'Non-unit', i2, ib, one, ab( 1, i ),
374 \$ ldab-1, ab( 1+ib, i ), ldab-1 )
375*
376* Update A22
377*
378 CALL ssyrk( 'Lower', 'No Transpose', i2, ib, -one,
379 \$ ab( 1+ib, i ), ldab-1, one,
380 \$ ab( 1, i+ib ), ldab-1 )
381 END IF
382*
383 IF( i3.GT.0 ) THEN
384*
385* Copy the upper triangle of A31 into the work array.
386*
387 DO 110 jj = 1, ib
388 DO 100 ii = 1, min( jj, i3 )
389 work( ii, jj ) = ab( kd+1-jj+ii, jj+i-1 )
390 100 CONTINUE
391 110 CONTINUE
392*
393* Update A31 (in the work array).
394*
395 CALL strsm( 'Right', 'Lower', 'Transpose',
396 \$ 'Non-unit', i3, ib, one, ab( 1, i ),
397 \$ ldab-1, work, ldwork )
398*
399* Update A32
400*
401 IF( i2.GT.0 )
402 \$ CALL sgemm( 'No transpose', 'Transpose', i3, i2,
403 \$ ib, -one, work, ldwork,
404 \$ ab( 1+ib, i ), ldab-1, one,
405 \$ ab( 1+kd-ib, i+ib ), ldab-1 )
406*
407* Update A33
408*
409 CALL ssyrk( 'Lower', 'No Transpose', i3, ib, -one,
410 \$ work, ldwork, one, ab( 1, i+kd ),
411 \$ ldab-1 )
412*
413* Copy the upper triangle of A31 back into place.
414*
415 DO 130 jj = 1, ib
416 DO 120 ii = 1, min( jj, i3 )
417 ab( kd+1-jj+ii, jj+i-1 ) = work( ii, jj )
418 120 CONTINUE
419 130 CONTINUE
420 END IF
421 END IF
422 140 CONTINUE
423 END IF
424 END IF
425 RETURN
426*
427 150 CONTINUE
428 RETURN
429*
430* End of SPBTRF
431*
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine spbtf2(UPLO, N, KD, AB, LDAB, INFO)
SPBTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (un...
Definition: spbtf2.f:142
subroutine spotf2(UPLO, N, A, LDA, INFO)
SPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblock...
Definition: spotf2.f:109
subroutine strsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRSM
Definition: strsm.f:181
subroutine ssyrk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
SSYRK
Definition: ssyrk.f:169
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
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