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

 subroutine cpstrf ( character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer, dimension( n ) PIV, integer RANK, real TOL, real, dimension( 2*n ) WORK, integer INFO )

CPSTRF computes the Cholesky factorization with complete pivoting of complex Hermitian positive semidefinite matrix.

Purpose:
CPSTRF computes the Cholesky factorization with complete
pivoting of a complex Hermitian positive semidefinite matrix A.

The factorization has the form
P**T * A * P = U**H * U ,  if UPLO = 'U',
P**T * A * P = L  * L**H,  if UPLO = 'L',
where U is an upper triangular matrix and L is lower triangular, and
P is stored as vector PIV.

This algorithm does not attempt to check that A is positive
semidefinite. This version of the algorithm calls level 3 BLAS.
Parameters
 [in] UPLO UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangular [in] N N is INTEGER The order of the matrix A. N >= 0. [in,out] A A is COMPLEX array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading n by n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n by n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if INFO = 0, the factor U or L from the Cholesky factorization as above. [in] LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). [out] PIV PIV is INTEGER array, dimension (N) PIV is such that the nonzero entries are P( PIV(K), K ) = 1. [out] RANK RANK is INTEGER The rank of A given by the number of steps the algorithm completed. [in] TOL TOL is REAL User defined tolerance. If TOL < 0, then N*U*MAX( A(K,K) ) will be used. The algorithm terminates at the (K-1)st step if the pivot <= TOL. [out] WORK WORK is REAL array, dimension (2*N) Work space. [out] INFO INFO is INTEGER < 0: If INFO = -K, the K-th argument had an illegal value, = 0: algorithm completed successfully, and > 0: the matrix A is either rank deficient with computed rank as returned in RANK, or is not positive semidefinite. See Section 7 of LAPACK Working Note #161 for further information.

Definition at line 141 of file cpstrf.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 REAL TOL
149 INTEGER INFO, LDA, N, RANK
150 CHARACTER UPLO
151* ..
152* .. Array Arguments ..
153 COMPLEX A( LDA, * )
154 REAL WORK( 2*N )
155 INTEGER PIV( N )
156* ..
157*
158* =====================================================================
159*
160* .. Parameters ..
161 REAL ONE, ZERO
162 parameter( one = 1.0e+0, zero = 0.0e+0 )
163 COMPLEX CONE
164 parameter( cone = ( 1.0e+0, 0.0e+0 ) )
165* ..
166* .. Local Scalars ..
167 COMPLEX CTEMP
168 REAL AJJ, SSTOP, STEMP
169 INTEGER I, ITEMP, J, JB, K, NB, PVT
170 LOGICAL UPPER
171* ..
172* .. External Functions ..
173 REAL SLAMCH
174 INTEGER ILAENV
175 LOGICAL LSAME, SISNAN
176 EXTERNAL slamch, ilaenv, lsame, sisnan
177* ..
178* .. External Subroutines ..
179 EXTERNAL cgemv, cherk, clacgv, cpstf2, csscal, cswap,
180 \$ xerbla
181* ..
182* .. Intrinsic Functions ..
183 INTRINSIC conjg, max, min, real, sqrt, maxloc
184* ..
185* .. Executable Statements ..
186*
187* Test the input parameters.
188*
189 info = 0
190 upper = lsame( uplo, 'U' )
191 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
192 info = -1
193 ELSE IF( n.LT.0 ) THEN
194 info = -2
195 ELSE IF( lda.LT.max( 1, n ) ) THEN
196 info = -4
197 END IF
198 IF( info.NE.0 ) THEN
199 CALL xerbla( 'CPSTRF', -info )
200 RETURN
201 END IF
202*
203* Quick return if possible
204*
205 IF( n.EQ.0 )
206 \$ RETURN
207*
208* Get block size
209*
210 nb = ilaenv( 1, 'CPOTRF', uplo, n, -1, -1, -1 )
211 IF( nb.LE.1 .OR. nb.GE.n ) THEN
212*
213* Use unblocked code
214*
215 CALL cpstf2( uplo, n, a( 1, 1 ), lda, piv, rank, tol, work,
216 \$ info )
217 GO TO 230
218*
219 ELSE
220*
221* Initialize PIV
222*
223 DO 100 i = 1, n
224 piv( i ) = i
225 100 CONTINUE
226*
227* Compute stopping value
228*
229 DO 110 i = 1, n
230 work( i ) = real( a( i, i ) )
231 110 CONTINUE
232 pvt = maxloc( work( 1:n ), 1 )
233 ajj = real( a( pvt, pvt ) )
234 IF( ajj.LE.zero.OR.sisnan( ajj ) ) THEN
235 rank = 0
236 info = 1
237 GO TO 230
238 END IF
239*
240* Compute stopping value if not supplied
241*
242 IF( tol.LT.zero ) THEN
243 sstop = n * slamch( 'Epsilon' ) * ajj
244 ELSE
245 sstop = tol
246 END IF
247*
248*
249 IF( upper ) THEN
250*
251* Compute the Cholesky factorization P**T * A * P = U**H * U
252*
253 DO 160 k = 1, n, nb
254*
255* Account for last block not being NB wide
256*
257 jb = min( nb, n-k+1 )
258*
259* Set relevant part of first half of WORK to zero,
260* holds dot products
261*
262 DO 120 i = k, n
263 work( i ) = 0
264 120 CONTINUE
265*
266 DO 150 j = k, k + jb - 1
267*
268* Find pivot, test for exit, else swap rows and columns
269* Update dot products, compute possible pivots which are
270* stored in the second half of WORK
271*
272 DO 130 i = j, n
273*
274 IF( j.GT.k ) THEN
275 work( i ) = work( i ) +
276 \$ real( conjg( a( j-1, i ) )*
277 \$ a( j-1, i ) )
278 END IF
279 work( n+i ) = real( a( i, i ) ) - work( i )
280*
281 130 CONTINUE
282*
283 IF( j.GT.1 ) THEN
284 itemp = maxloc( work( (n+j):(2*n) ), 1 )
285 pvt = itemp + j - 1
286 ajj = work( n+pvt )
287 IF( ajj.LE.sstop.OR.sisnan( ajj ) ) THEN
288 a( j, j ) = ajj
289 GO TO 220
290 END IF
291 END IF
292*
293 IF( j.NE.pvt ) THEN
294*
295* Pivot OK, so can now swap pivot rows and columns
296*
297 a( pvt, pvt ) = a( j, j )
298 CALL cswap( j-1, a( 1, j ), 1, a( 1, pvt ), 1 )
299 IF( pvt.LT.n )
300 \$ CALL cswap( n-pvt, a( j, pvt+1 ), lda,
301 \$ a( pvt, pvt+1 ), lda )
302 DO 140 i = j + 1, pvt - 1
303 ctemp = conjg( a( j, i ) )
304 a( j, i ) = conjg( a( i, pvt ) )
305 a( i, pvt ) = ctemp
306 140 CONTINUE
307 a( j, pvt ) = conjg( a( j, pvt ) )
308*
309* Swap dot products and PIV
310*
311 stemp = work( j )
312 work( j ) = work( pvt )
313 work( pvt ) = stemp
314 itemp = piv( pvt )
315 piv( pvt ) = piv( j )
316 piv( j ) = itemp
317 END IF
318*
319 ajj = sqrt( ajj )
320 a( j, j ) = ajj
321*
322* Compute elements J+1:N of row J.
323*
324 IF( j.LT.n ) THEN
325 CALL clacgv( j-1, a( 1, j ), 1 )
326 CALL cgemv( 'Trans', j-k, n-j, -cone, a( k, j+1 ),
327 \$ lda, a( k, j ), 1, cone, a( j, j+1 ),
328 \$ lda )
329 CALL clacgv( j-1, a( 1, j ), 1 )
330 CALL csscal( n-j, one / ajj, a( j, j+1 ), lda )
331 END IF
332*
333 150 CONTINUE
334*
335* Update trailing matrix, J already incremented
336*
337 IF( k+jb.LE.n ) THEN
338 CALL cherk( 'Upper', 'Conj Trans', n-j+1, jb, -one,
339 \$ a( k, j ), lda, one, a( j, j ), lda )
340 END IF
341*
342 160 CONTINUE
343*
344 ELSE
345*
346* Compute the Cholesky factorization P**T * A * P = L * L**H
347*
348 DO 210 k = 1, n, nb
349*
350* Account for last block not being NB wide
351*
352 jb = min( nb, n-k+1 )
353*
354* Set relevant part of first half of WORK to zero,
355* holds dot products
356*
357 DO 170 i = k, n
358 work( i ) = 0
359 170 CONTINUE
360*
361 DO 200 j = k, k + jb - 1
362*
363* Find pivot, test for exit, else swap rows and columns
364* Update dot products, compute possible pivots which are
365* stored in the second half of WORK
366*
367 DO 180 i = j, n
368*
369 IF( j.GT.k ) THEN
370 work( i ) = work( i ) +
371 \$ real( conjg( a( i, j-1 ) )*
372 \$ a( i, j-1 ) )
373 END IF
374 work( n+i ) = real( a( i, i ) ) - work( i )
375*
376 180 CONTINUE
377*
378 IF( j.GT.1 ) THEN
379 itemp = maxloc( work( (n+j):(2*n) ), 1 )
380 pvt = itemp + j - 1
381 ajj = work( n+pvt )
382 IF( ajj.LE.sstop.OR.sisnan( ajj ) ) THEN
383 a( j, j ) = ajj
384 GO TO 220
385 END IF
386 END IF
387*
388 IF( j.NE.pvt ) THEN
389*
390* Pivot OK, so can now swap pivot rows and columns
391*
392 a( pvt, pvt ) = a( j, j )
393 CALL cswap( j-1, a( j, 1 ), lda, a( pvt, 1 ), lda )
394 IF( pvt.LT.n )
395 \$ CALL cswap( n-pvt, a( pvt+1, j ), 1,
396 \$ a( pvt+1, pvt ), 1 )
397 DO 190 i = j + 1, pvt - 1
398 ctemp = conjg( a( i, j ) )
399 a( i, j ) = conjg( a( pvt, i ) )
400 a( pvt, i ) = ctemp
401 190 CONTINUE
402 a( pvt, j ) = conjg( a( pvt, j ) )
403*
404* Swap dot products and PIV
405*
406 stemp = work( j )
407 work( j ) = work( pvt )
408 work( pvt ) = stemp
409 itemp = piv( pvt )
410 piv( pvt ) = piv( j )
411 piv( j ) = itemp
412 END IF
413*
414 ajj = sqrt( ajj )
415 a( j, j ) = ajj
416*
417* Compute elements J+1:N of column J.
418*
419 IF( j.LT.n ) THEN
420 CALL clacgv( j-1, a( j, 1 ), lda )
421 CALL cgemv( 'No Trans', n-j, j-k, -cone,
422 \$ a( j+1, k ), lda, a( j, k ), lda, cone,
423 \$ a( j+1, j ), 1 )
424 CALL clacgv( j-1, a( j, 1 ), lda )
425 CALL csscal( n-j, one / ajj, a( j+1, j ), 1 )
426 END IF
427*
428 200 CONTINUE
429*
430* Update trailing matrix, J already incremented
431*
432 IF( k+jb.LE.n ) THEN
433 CALL cherk( 'Lower', 'No Trans', n-j+1, jb, -one,
434 \$ a( j, k ), lda, one, a( j, j ), lda )
435 END IF
436*
437 210 CONTINUE
438*
439 END IF
440 END IF
441*
442* Ran to completion, A has full rank
443*
444 rank = n
445*
446 GO TO 230
447 220 CONTINUE
448*
449* Rank is the number of steps completed. Set INFO = 1 to signal
450* that the factorization cannot be used to solve a system.
451*
452 rank = j - 1
453 info = 1
454*
455 230 CONTINUE
456 RETURN
457*
458* End of CPSTRF
459*
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
logical function sisnan(SIN)
SISNAN tests input for NaN.
Definition: sisnan.f:59
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 cswap(N, CX, INCX, CY, INCY)
CSWAP
Definition: cswap.f:81
subroutine cgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGEMV
Definition: cgemv.f:158
subroutine cherk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
CHERK
Definition: cherk.f:173
subroutine clacgv(N, X, INCX)
CLACGV conjugates a complex vector.
Definition: clacgv.f:74
subroutine cpstf2(UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)
CPSTF2 computes the Cholesky factorization with complete pivoting of complex Hermitian positive semid...
Definition: cpstf2.f:142
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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