LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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cpstf2.f
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1*> \brief \b CPSTF2 computes the Cholesky factorization with complete pivoting of 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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16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
22*
23* .. Scalar Arguments ..
24* REAL TOL
25* INTEGER INFO, LDA, N, RANK
26* CHARACTER UPLO
27* ..
28* .. Array Arguments ..
29* COMPLEX A( LDA, * )
30* REAL WORK( 2*N )
31* INTEGER PIV( N )
32* ..
33*
34*
35*> \par Purpose:
36* =============
37*>
38*> \verbatim
39*>
40*> CPSTF2 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 2 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 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[out] PIV
87*> \verbatim
88*> PIV is INTEGER array, dimension (N)
89*> PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
90*> \endverbatim
91*>
92*> \param[out] RANK
93*> \verbatim
94*> RANK is INTEGER
95*> The rank of A given by the number of steps the algorithm
96*> completed.
97*> \endverbatim
98*>
99*> \param[in] TOL
100*> \verbatim
101*> TOL is REAL
102*> User defined tolerance. If TOL < 0, then N*U*MAX( A( K,K ) )
103*> will be used. The algorithm terminates at the (K-1)st step
104*> if the pivot <= TOL.
105*> \endverbatim
106*>
107*> \param[in] LDA
108*> \verbatim
109*> LDA is INTEGER
110*> The leading dimension of the array A. LDA >= max(1,N).
111*> \endverbatim
112*>
113*> \param[out] WORK
114*> \verbatim
115*> WORK is REAL 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 pstf2
139*
140* =====================================================================
141 SUBROUTINE cpstf2( 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 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, PVT
170 LOGICAL UPPER
171* ..
172* .. External Functions ..
173 REAL SLAMCH
174 LOGICAL LSAME, SISNAN
175 EXTERNAL slamch, lsame, sisnan
176* ..
177* .. External Subroutines ..
178 EXTERNAL cgemv, clacgv, csscal, cswap, xerbla
179* ..
180* .. Intrinsic Functions ..
181 INTRINSIC conjg, max, real, sqrt
182* ..
183* .. Executable Statements ..
184*
185* Test the input parameters
186*
187 info = 0
188 upper = lsame( uplo, 'U' )
189 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
190 info = -1
191 ELSE IF( n.LT.0 ) THEN
192 info = -2
193 ELSE IF( lda.LT.max( 1, n ) ) THEN
194 info = -4
195 END IF
196 IF( info.NE.0 ) THEN
197 CALL xerbla( 'CPSTF2', -info )
198 RETURN
199 END IF
200*
201* Quick return if possible
202*
203 IF( n.EQ.0 )
204 \$ RETURN
205*
206* Initialize PIV
207*
208 DO 100 i = 1, n
209 piv( i ) = i
210 100 CONTINUE
211*
212* Compute stopping value
213*
214 DO 110 i = 1, n
215 work( i ) = real( a( i, i ) )
216 110 CONTINUE
217 pvt = maxloc( work( 1:n ), 1 )
218 ajj = real( a( pvt, pvt ) )
219 IF( ajj.LE.zero.OR.sisnan( ajj ) ) THEN
220 rank = 0
221 info = 1
222 GO TO 200
223 END IF
224*
225* Compute stopping value if not supplied
226*
227 IF( tol.LT.zero ) THEN
228 sstop = n * slamch( 'Epsilon' ) * ajj
229 ELSE
230 sstop = tol
231 END IF
232*
233* Set first half of WORK to zero, holds dot products
234*
235 DO 120 i = 1, n
236 work( i ) = 0
237 120 CONTINUE
238*
239 IF( upper ) THEN
240*
241* Compute the Cholesky factorization P**T * A * P = U**H * U
242*
243 DO 150 j = 1, n
244*
245* Find pivot, test for exit, else swap rows and columns
246* Update dot products, compute possible pivots which are
247* stored in the second half of WORK
248*
249 DO 130 i = j, n
250*
251 IF( j.GT.1 ) THEN
252 work( i ) = work( i ) +
253 \$ real( conjg( a( j-1, i ) )*
254 \$ a( j-1, i ) )
255 END IF
256 work( n+i ) = real( a( i, i ) ) - work( i )
257*
258 130 CONTINUE
259*
260 IF( j.GT.1 ) THEN
261 itemp = maxloc( work( (n+j):(2*n) ), 1 )
262 pvt = itemp + j - 1
263 ajj = work( n+pvt )
264 IF( ajj.LE.sstop.OR.sisnan( ajj ) ) THEN
265 a( j, j ) = ajj
266 GO TO 190
267 END IF
268 END IF
269*
270 IF( j.NE.pvt ) THEN
271*
272* Pivot OK, so can now swap pivot rows and columns
273*
274 a( pvt, pvt ) = a( j, j )
275 CALL cswap( j-1, a( 1, j ), 1, a( 1, pvt ), 1 )
276 IF( pvt.LT.n )
277 \$ CALL cswap( n-pvt, a( j, pvt+1 ), lda,
278 \$ a( pvt, pvt+1 ), lda )
279 DO 140 i = j + 1, pvt - 1
280 ctemp = conjg( a( j, i ) )
281 a( j, i ) = conjg( a( i, pvt ) )
282 a( i, pvt ) = ctemp
283 140 CONTINUE
284 a( j, pvt ) = conjg( a( j, pvt ) )
285*
286* Swap dot products and PIV
287*
288 stemp = work( j )
289 work( j ) = work( pvt )
290 work( pvt ) = stemp
291 itemp = piv( pvt )
292 piv( pvt ) = piv( j )
293 piv( j ) = itemp
294 END IF
295*
296 ajj = sqrt( ajj )
297 a( j, j ) = ajj
298*
299* Compute elements J+1:N of row J
300*
301 IF( j.LT.n ) THEN
302 CALL clacgv( j-1, a( 1, j ), 1 )
303 CALL cgemv( 'Trans', j-1, n-j, -cone, a( 1, j+1 ), lda,
304 \$ a( 1, j ), 1, cone, a( j, j+1 ), lda )
305 CALL clacgv( j-1, a( 1, j ), 1 )
306 CALL csscal( n-j, one / ajj, a( j, j+1 ), lda )
307 END IF
308*
309 150 CONTINUE
310*
311 ELSE
312*
313* Compute the Cholesky factorization P**T * A * P = L * L**H
314*
315 DO 180 j = 1, n
316*
317* Find pivot, test for exit, else swap rows and columns
318* Update dot products, compute possible pivots which are
319* stored in the second half of WORK
320*
321 DO 160 i = j, n
322*
323 IF( j.GT.1 ) THEN
324 work( i ) = work( i ) +
325 \$ real( conjg( a( i, j-1 ) )*
326 \$ a( i, j-1 ) )
327 END IF
328 work( n+i ) = real( a( i, i ) ) - work( i )
329*
330 160 CONTINUE
331*
332 IF( j.GT.1 ) THEN
333 itemp = maxloc( work( (n+j):(2*n) ), 1 )
334 pvt = itemp + j - 1
335 ajj = work( n+pvt )
336 IF( ajj.LE.sstop.OR.sisnan( ajj ) ) THEN
337 a( j, j ) = ajj
338 GO TO 190
339 END IF
340 END IF
341*
342 IF( j.NE.pvt ) THEN
343*
344* Pivot OK, so can now swap pivot rows and columns
345*
346 a( pvt, pvt ) = a( j, j )
347 CALL cswap( j-1, a( j, 1 ), lda, a( pvt, 1 ), lda )
348 IF( pvt.LT.n )
349 \$ CALL cswap( n-pvt, a( pvt+1, j ), 1, a( pvt+1, pvt ),
350 \$ 1 )
351 DO 170 i = j + 1, pvt - 1
352 ctemp = conjg( a( i, j ) )
353 a( i, j ) = conjg( a( pvt, i ) )
354 a( pvt, i ) = ctemp
355 170 CONTINUE
356 a( pvt, j ) = conjg( a( pvt, j ) )
357*
358* Swap dot products and PIV
359*
360 stemp = work( j )
361 work( j ) = work( pvt )
362 work( pvt ) = stemp
363 itemp = piv( pvt )
364 piv( pvt ) = piv( j )
365 piv( j ) = itemp
366 END IF
367*
368 ajj = sqrt( ajj )
369 a( j, j ) = ajj
370*
371* Compute elements J+1:N of column J
372*
373 IF( j.LT.n ) THEN
374 CALL clacgv( j-1, a( j, 1 ), lda )
375 CALL cgemv( 'No Trans', n-j, j-1, -cone, a( j+1, 1 ),
376 \$ lda, a( j, 1 ), lda, cone, a( j+1, j ), 1 )
377 CALL clacgv( j-1, a( j, 1 ), lda )
378 CALL csscal( n-j, one / ajj, a( j+1, j ), 1 )
379 END IF
380*
381 180 CONTINUE
382*
383 END IF
384*
385* Ran to completion, A has full rank
386*
387 rank = n
388*
389 GO TO 200
390 190 CONTINUE
391*
392* Rank is number of steps completed. Set INFO = 1 to signal
393* that the factorization cannot be used to solve a system.
394*
395 rank = j - 1
396 info = 1
397*
398 200 CONTINUE
399 RETURN
400*
401* End of CPSTF2
402*
403 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine cgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
CGEMV
Definition cgemv.f:160
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
subroutine csscal(n, sa, cx, incx)
CSSCAL
Definition csscal.f:78
subroutine cswap(n, cx, incx, cy, incy)
CSWAP
Definition cswap.f:81