LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
zpstf2.f
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1 *> \brief \b ZPSTF2 computes the Cholesky factorization with complete pivoting of a complex Hermitian positive semidefinite matrix.
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZPSTF2( 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 *> ZPSTF2 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*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[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 DOUBLE PRECISION
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 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 *> \date November 2015
139 *
140 *> \ingroup complex16OTHERcomputational
141 *
142 * =====================================================================
143  SUBROUTINE zpstf2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
144 *
145 * -- LAPACK computational routine (version 3.6.0) --
146 * -- LAPACK is a software package provided by Univ. of Tennessee, --
147 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
148 * November 2015
149 *
150 * .. Scalar Arguments ..
151  DOUBLE PRECISION TOL
152  INTEGER INFO, LDA, N, RANK
153  CHARACTER UPLO
154 * ..
155 * .. Array Arguments ..
156  COMPLEX*16 A( lda, * )
157  DOUBLE PRECISION WORK( 2*n )
158  INTEGER PIV( n )
159 * ..
160 *
161 * =====================================================================
162 *
163 * .. Parameters ..
164  DOUBLE PRECISION ONE, ZERO
165  parameter ( one = 1.0d+0, zero = 0.0d+0 )
166  COMPLEX*16 CONE
167  parameter ( cone = ( 1.0d+0, 0.0d+0 ) )
168 * ..
169 * .. Local Scalars ..
170  COMPLEX*16 ZTEMP
171  DOUBLE PRECISION AJJ, DSTOP, DTEMP
172  INTEGER I, ITEMP, J, PVT
173  LOGICAL UPPER
174 * ..
175 * .. External Functions ..
176  DOUBLE PRECISION DLAMCH
177  LOGICAL LSAME, DISNAN
178  EXTERNAL dlamch, lsame, disnan
179 * ..
180 * .. External Subroutines ..
181  EXTERNAL zdscal, zgemv, zlacgv, zswap, xerbla
182 * ..
183 * .. Intrinsic Functions ..
184  INTRINSIC dble, dconjg, max, sqrt
185 * ..
186 * .. Executable Statements ..
187 *
188 * Test the input parameters
189 *
190  info = 0
191  upper = lsame( uplo, 'U' )
192  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
193  info = -1
194  ELSE IF( n.LT.0 ) THEN
195  info = -2
196  ELSE IF( lda.LT.max( 1, n ) ) THEN
197  info = -4
198  END IF
199  IF( info.NE.0 ) THEN
200  CALL xerbla( 'ZPSTF2', -info )
201  RETURN
202  END IF
203 *
204 * Quick return if possible
205 *
206  IF( n.EQ.0 )
207  \$ RETURN
208 *
209 * Initialize PIV
210 *
211  DO 100 i = 1, n
212  piv( i ) = i
213  100 CONTINUE
214 *
215 * Compute stopping value
216 *
217  DO 110 i = 1, n
218  work( i ) = dble( a( i, i ) )
219  110 CONTINUE
220  pvt = maxloc( work( 1:n ), 1 )
221  ajj = dble( a( pvt, pvt ) )
222  IF( ajj.LE.zero.OR.disnan( ajj ) ) THEN
223  rank = 0
224  info = 1
225  GO TO 200
226  END IF
227 *
228 * Compute stopping value if not supplied
229 *
230  IF( tol.LT.zero ) THEN
231  dstop = n * dlamch( 'Epsilon' ) * ajj
232  ELSE
233  dstop = tol
234  END IF
235 *
236 * Set first half of WORK to zero, holds dot products
237 *
238  DO 120 i = 1, n
239  work( i ) = 0
240  120 CONTINUE
241 *
242  IF( upper ) THEN
243 *
244 * Compute the Cholesky factorization P**T * A * P = U**H* U
245 *
246  DO 150 j = 1, n
247 *
248 * Find pivot, test for exit, else swap rows and columns
249 * Update dot products, compute possible pivots which are
250 * stored in the second half of WORK
251 *
252  DO 130 i = j, n
253 *
254  IF( j.GT.1 ) THEN
255  work( i ) = work( i ) +
256  \$ dble( dconjg( a( j-1, i ) )*
257  \$ a( j-1, i ) )
258  END IF
259  work( n+i ) = dble( a( i, i ) ) - work( i )
260 *
261  130 CONTINUE
262 *
263  IF( j.GT.1 ) THEN
264  itemp = maxloc( work( (n+j):(2*n) ), 1 )
265  pvt = itemp + j - 1
266  ajj = work( n+pvt )
267  IF( ajj.LE.dstop.OR.disnan( ajj ) ) THEN
268  a( j, j ) = ajj
269  GO TO 190
270  END IF
271  END IF
272 *
273  IF( j.NE.pvt ) THEN
274 *
275 * Pivot OK, so can now swap pivot rows and columns
276 *
277  a( pvt, pvt ) = a( j, j )
278  CALL zswap( j-1, a( 1, j ), 1, a( 1, pvt ), 1 )
279  IF( pvt.LT.n )
280  \$ CALL zswap( n-pvt, a( j, pvt+1 ), lda,
281  \$ a( pvt, pvt+1 ), lda )
282  DO 140 i = j + 1, pvt - 1
283  ztemp = dconjg( a( j, i ) )
284  a( j, i ) = dconjg( a( i, pvt ) )
285  a( i, pvt ) = ztemp
286  140 CONTINUE
287  a( j, pvt ) = dconjg( a( j, pvt ) )
288 *
289 * Swap dot products and PIV
290 *
291  dtemp = work( j )
292  work( j ) = work( pvt )
293  work( pvt ) = dtemp
294  itemp = piv( pvt )
295  piv( pvt ) = piv( j )
296  piv( j ) = itemp
297  END IF
298 *
299  ajj = sqrt( ajj )
300  a( j, j ) = ajj
301 *
302 * Compute elements J+1:N of row J
303 *
304  IF( j.LT.n ) THEN
305  CALL zlacgv( j-1, a( 1, j ), 1 )
306  CALL zgemv( 'Trans', j-1, n-j, -cone, a( 1, j+1 ), lda,
307  \$ a( 1, j ), 1, cone, a( j, j+1 ), lda )
308  CALL zlacgv( j-1, a( 1, j ), 1 )
309  CALL zdscal( n-j, one / ajj, a( j, j+1 ), lda )
310  END IF
311 *
312  150 CONTINUE
313 *
314  ELSE
315 *
316 * Compute the Cholesky factorization P**T * A * P = L * L**H
317 *
318  DO 180 j = 1, n
319 *
320 * Find pivot, test for exit, else swap rows and columns
321 * Update dot products, compute possible pivots which are
322 * stored in the second half of WORK
323 *
324  DO 160 i = j, n
325 *
326  IF( j.GT.1 ) THEN
327  work( i ) = work( i ) +
328  \$ dble( dconjg( a( i, j-1 ) )*
329  \$ a( i, j-1 ) )
330  END IF
331  work( n+i ) = dble( a( i, i ) ) - work( i )
332 *
333  160 CONTINUE
334 *
335  IF( j.GT.1 ) THEN
336  itemp = maxloc( work( (n+j):(2*n) ), 1 )
337  pvt = itemp + j - 1
338  ajj = work( n+pvt )
339  IF( ajj.LE.dstop.OR.disnan( ajj ) ) THEN
340  a( j, j ) = ajj
341  GO TO 190
342  END IF
343  END IF
344 *
345  IF( j.NE.pvt ) THEN
346 *
347 * Pivot OK, so can now swap pivot rows and columns
348 *
349  a( pvt, pvt ) = a( j, j )
350  CALL zswap( j-1, a( j, 1 ), lda, a( pvt, 1 ), lda )
351  IF( pvt.LT.n )
352  \$ CALL zswap( n-pvt, a( pvt+1, j ), 1, a( pvt+1, pvt ),
353  \$ 1 )
354  DO 170 i = j + 1, pvt - 1
355  ztemp = dconjg( a( i, j ) )
356  a( i, j ) = dconjg( a( pvt, i ) )
357  a( pvt, i ) = ztemp
358  170 CONTINUE
359  a( pvt, j ) = dconjg( a( pvt, j ) )
360 *
361 * Swap dot products and PIV
362 *
363  dtemp = work( j )
364  work( j ) = work( pvt )
365  work( pvt ) = dtemp
366  itemp = piv( pvt )
367  piv( pvt ) = piv( j )
368  piv( j ) = itemp
369  END IF
370 *
371  ajj = sqrt( ajj )
372  a( j, j ) = ajj
373 *
374 * Compute elements J+1:N of column J
375 *
376  IF( j.LT.n ) THEN
377  CALL zlacgv( j-1, a( j, 1 ), lda )
378  CALL zgemv( 'No Trans', n-j, j-1, -cone, a( j+1, 1 ),
379  \$ lda, a( j, 1 ), lda, cone, a( j+1, j ), 1 )
380  CALL zlacgv( j-1, a( j, 1 ), lda )
381  CALL zdscal( n-j, one / ajj, a( j+1, j ), 1 )
382  END IF
383 *
384  180 CONTINUE
385 *
386  END IF
387 *
388 * Ran to completion, A has full rank
389 *
390  rank = n
391 *
392  GO TO 200
393  190 CONTINUE
394 *
395 * Rank is number of steps completed. Set INFO = 1 to signal
396 * that the factorization cannot be used to solve a system.
397 *
398  rank = j - 1
399  info = 1
400 *
401  200 CONTINUE
402  RETURN
403 *
404 * End of ZPSTF2
405 *
406  END
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:160
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:52
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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:144
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:54
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:76