LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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zpstrf.f
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1*> \brief \b ZPSTRF computes the Cholesky factorization with complete pivoting of a complex Hermitian positive semidefinite matrix.
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
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15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZPSTRF( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
22*
23* .. Scalar Arguments ..
24* DOUBLE PRECISION TOL
25* INTEGER INFO, LDA, N, RANK
26* CHARACTER UPLO
27* ..
28* .. Array Arguments ..
29* COMPLEX*16 A( LDA, * )
30* DOUBLE PRECISION WORK( 2*N )
31* INTEGER PIV( N )
32* ..
33*
34*
35*> \par Purpose:
36* =============
37*>
38*> \verbatim
39*>
40*> ZPSTRF computes the Cholesky factorization with complete
41*> pivoting of a complex Hermitian positive semidefinite matrix A.
42*>
43*> The factorization has the form
44*> P**T * A * P = U**H * U , if UPLO = 'U',
45*> P**T * A * P = L * L**H, if UPLO = 'L',
46*> where U is an upper triangular matrix and L is lower triangular, and
47*> P is stored as vector PIV.
48*>
49*> This algorithm does not attempt to check that A is positive
50*> semidefinite. This version of the algorithm calls level 3 BLAS.
51*> \endverbatim
52*
53* Arguments:
54* ==========
55*
56*> \param[in] UPLO
57*> \verbatim
58*> UPLO is CHARACTER*1
59*> Specifies whether the upper or lower triangular part of the
60*> symmetric matrix A is stored.
61*> = 'U': Upper triangular
62*> = 'L': Lower triangular
63*> \endverbatim
64*>
65*> \param[in] N
66*> \verbatim
67*> N is INTEGER
68*> The order of the matrix A. N >= 0.
69*> \endverbatim
70*>
71*> \param[in,out] A
72*> \verbatim
73*> A is COMPLEX*16 array, dimension (LDA,N)
74*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
75*> n by n upper triangular part of A contains the upper
76*> triangular part of the matrix A, and the strictly lower
77*> triangular part of A is not referenced. If UPLO = 'L', the
78*> leading n by n lower triangular part of A contains the lower
79*> triangular part of the matrix A, and the strictly upper
80*> triangular part of A is not referenced.
81*>
82*> On exit, if INFO = 0, the factor U or L from the Cholesky
83*> factorization as above.
84*> \endverbatim
85*>
86*> \param[in] LDA
87*> \verbatim
88*> LDA is INTEGER
89*> The leading dimension of the array A. LDA >= max(1,N).
90*> \endverbatim
91*>
92*> \param[out] PIV
93*> \verbatim
94*> PIV is INTEGER array, dimension (N)
95*> PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
96*> \endverbatim
97*>
98*> \param[out] RANK
99*> \verbatim
100*> RANK is INTEGER
101*> The rank of A given by the number of steps the algorithm
102*> completed.
103*> \endverbatim
104*>
105*> \param[in] TOL
106*> \verbatim
107*> TOL is DOUBLE PRECISION
108*> User defined tolerance. If TOL < 0, then N*U*MAX( A(K,K) )
109*> will be used. The algorithm terminates at the (K-1)st step
110*> if the pivot <= TOL.
111*> \endverbatim
112*>
113*> \param[out] WORK
114*> \verbatim
115*> WORK is DOUBLE PRECISION array, dimension (2*N)
116*> Work space.
117*> \endverbatim
118*>
119*> \param[out] INFO
120*> \verbatim
121*> INFO is INTEGER
122*> < 0: If INFO = -K, the K-th argument had an illegal value,
123*> = 0: algorithm completed successfully, and
124*> > 0: the matrix A is either rank deficient with computed rank
125*> as returned in RANK, or is not positive semidefinite. See
126*> Section 7 of LAPACK Working Note #161 for further
127*> information.
128*> \endverbatim
129*
130* Authors:
131* ========
132*
133*> \author Univ. of Tennessee
134*> \author Univ. of California Berkeley
135*> \author Univ. of Colorado Denver
136*> \author NAG Ltd.
137*
138*> \ingroup pstrf
139*
140* =====================================================================
141 SUBROUTINE zpstrf( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
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 DOUBLE PRECISION TOL
149 INTEGER INFO, LDA, N, RANK
150 CHARACTER UPLO
151* ..
152* .. Array Arguments ..
153 COMPLEX*16 A( LDA, * )
154 DOUBLE PRECISION WORK( 2*N )
155 INTEGER PIV( N )
156* ..
157*
158* =====================================================================
159*
160* .. Parameters ..
161 DOUBLE PRECISION ONE, ZERO
162 parameter( one = 1.0d+0, zero = 0.0d+0 )
163 COMPLEX*16 CONE
164 parameter( cone = ( 1.0d+0, 0.0d+0 ) )
165* ..
166* .. Local Scalars ..
167 COMPLEX*16 ZTEMP
168 DOUBLE PRECISION AJJ, DSTOP, DTEMP
169 INTEGER I, ITEMP, J, JB, K, NB, PVT
170 LOGICAL UPPER
171* ..
172* .. External Functions ..
173 DOUBLE PRECISION DLAMCH
174 INTEGER ILAENV
175 LOGICAL LSAME, DISNAN
176 EXTERNAL dlamch, ilaenv, lsame, disnan
177* ..
178* .. External Subroutines ..
179 EXTERNAL zdscal, zgemv, zherk, zlacgv, zpstf2, zswap,
180 \$ xerbla
181* ..
182* .. Intrinsic Functions ..
183 INTRINSIC dble, dconjg, max, min, 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( 'ZPSTRF', -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, 'ZPOTRF', uplo, n, -1, -1, -1 )
211 IF( nb.LE.1 .OR. nb.GE.n ) THEN
212*
213* Use unblocked code
214*
215 CALL zpstf2( 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 ) = dble( a( i, i ) )
231 110 CONTINUE
232 pvt = maxloc( work( 1:n ), 1 )
233 ajj = dble( a( pvt, pvt ) )
234 IF( ajj.LE.zero.OR.disnan( 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 dstop = n * dlamch( 'Epsilon' ) * ajj
244 ELSE
245 dstop = 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 \$ dble( dconjg( a( j-1, i ) )*
277 \$ a( j-1, i ) )
278 END IF
279 work( n+i ) = dble( 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.dstop.OR.disnan( 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 zswap( j-1, a( 1, j ), 1, a( 1, pvt ), 1 )
299 IF( pvt.LT.n )
300 \$ CALL zswap( n-pvt, a( j, pvt+1 ), lda,
301 \$ a( pvt, pvt+1 ), lda )
302 DO 140 i = j + 1, pvt - 1
303 ztemp = dconjg( a( j, i ) )
304 a( j, i ) = dconjg( a( i, pvt ) )
305 a( i, pvt ) = ztemp
306 140 CONTINUE
307 a( j, pvt ) = dconjg( a( j, pvt ) )
308*
309* Swap dot products and PIV
310*
311 dtemp = work( j )
312 work( j ) = work( pvt )
313 work( pvt ) = dtemp
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 zlacgv( j-1, a( 1, j ), 1 )
326 CALL zgemv( '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 zlacgv( j-1, a( 1, j ), 1 )
330 CALL zdscal( 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 zherk( '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 \$ dble( dconjg( a( i, j-1 ) )*
372 \$ a( i, j-1 ) )
373 END IF
374 work( n+i ) = dble( 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.dstop.OR.disnan( 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 zswap( j-1, a( j, 1 ), lda, a( pvt, 1 ), lda )
394 IF( pvt.LT.n )
395 \$ CALL zswap( n-pvt, a( pvt+1, j ), 1,
396 \$ a( pvt+1, pvt ), 1 )
397 DO 190 i = j + 1, pvt - 1
398 ztemp = dconjg( a( i, j ) )
399 a( i, j ) = dconjg( a( pvt, i ) )
400 a( pvt, i ) = ztemp
401 190 CONTINUE
402 a( pvt, j ) = dconjg( a( pvt, j ) )
403*
404*
405* Swap dot products and PIV
406*
407 dtemp = work( j )
408 work( j ) = work( pvt )
409 work( pvt ) = dtemp
410 itemp = piv( pvt )
411 piv( pvt ) = piv( j )
412 piv( j ) = itemp
413 END IF
414*
415 ajj = sqrt( ajj )
416 a( j, j ) = ajj
417*
418* Compute elements J+1:N of column J.
419*
420 IF( j.LT.n ) THEN
421 CALL zlacgv( j-1, a( j, 1 ), lda )
422 CALL zgemv( 'No Trans', n-j, j-k, -cone,
423 \$ a( j+1, k ), lda, a( j, k ), lda, cone,
424 \$ a( j+1, j ), 1 )
425 CALL zlacgv( j-1, a( j, 1 ), lda )
426 CALL zdscal( n-j, one / ajj, a( j+1, j ), 1 )
427 END IF
428*
429 200 CONTINUE
430*
431* Update trailing matrix, J already incremented
432*
433 IF( k+jb.LE.n ) THEN
434 CALL zherk( 'Lower', 'No Trans', n-j+1, jb, -one,
435 \$ a( j, k ), lda, one, a( j, j ), lda )
436 END IF
437*
438 210 CONTINUE
439*
440 END IF
441 END IF
442*
443* Ran to completion, A has full rank
444*
445 rank = n
446*
447 GO TO 230
448 220 CONTINUE
449*
450* Rank is the number of steps completed. Set INFO = 1 to signal
451* that the factorization cannot be used to solve a system.
452*
453 rank = j - 1
454 info = 1
455*
456 230 CONTINUE
457 RETURN
458*
459* End of ZPSTRF
460*
461 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
ZGEMV
Definition zgemv.f:160
subroutine zherk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc)
ZHERK
Definition zherk.f:173
subroutine zlacgv(n, x, incx)
ZLACGV conjugates a complex vector.
Definition zlacgv.f:74
subroutine zpstf2(uplo, n, a, lda, piv, rank, tol, work, info)
ZPSTF2 computes the Cholesky factorization with complete pivoting of a complex Hermitian positive sem...
Definition zpstf2.f:142
subroutine zpstrf(uplo, n, a, lda, piv, rank, tol, work, info)
ZPSTRF computes the Cholesky factorization with complete pivoting of a complex Hermitian positive sem...
Definition zpstrf.f:142
subroutine zdscal(n, da, zx, incx)
ZDSCAL
Definition zdscal.f:78
subroutine zswap(n, zx, incx, zy, incy)
ZSWAP
Definition zswap.f:81