LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
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 complexOTHERcomputational
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)
XERBLA
Definition: xerbla.f:60
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 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