LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
spstrf.f
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1 *> \brief \b SPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
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16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SPSTRF( 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 * REAL A( LDA, * ), WORK( 2*N )
30 * INTEGER PIV( N )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> SPSTRF computes the Cholesky factorization with complete
40 *> pivoting of a real symmetric positive semidefinite matrix A.
41 *>
42 *> The factorization has the form
43 *> P**T * A * P = U**T * U , if UPLO = 'U',
44 *> P**T * A * P = L * L**T, if UPLO = 'L',
45 *> where U is an upper triangular matrix and L is lower triangular, and
46 *> P is stored as vector PIV.
47 *>
48 *> This algorithm does not attempt to check that A is positive
49 *> semidefinite. This version of the algorithm calls level 3 BLAS.
50 *> \endverbatim
51 *
52 * Arguments:
53 * ==========
54 *
55 *> \param[in] UPLO
56 *> \verbatim
57 *> UPLO is CHARACTER*1
58 *> Specifies whether the upper or lower triangular part of the
59 *> symmetric matrix A is stored.
60 *> = 'U': Upper triangular
61 *> = 'L': Lower triangular
62 *> \endverbatim
63 *>
64 *> \param[in] N
65 *> \verbatim
66 *> N is INTEGER
67 *> The order of the matrix A. N >= 0.
68 *> \endverbatim
69 *>
70 *> \param[in,out] A
71 *> \verbatim
72 *> A is REAL array, dimension (LDA,N)
73 *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
74 *> n by n upper triangular part of A contains the upper
75 *> triangular part of the matrix A, and the strictly lower
76 *> triangular part of A is not referenced. If UPLO = 'L', the
77 *> leading n by n lower triangular part of A contains the lower
78 *> triangular part of the matrix A, and the strictly upper
79 *> triangular part of A is not referenced.
80 *>
81 *> On exit, if INFO = 0, the factor U or L from the Cholesky
82 *> factorization as above.
83 *> \endverbatim
84 *>
85 *> \param[in] LDA
86 *> \verbatim
87 *> LDA is INTEGER
88 *> The leading dimension of the array A. LDA >= max(1,N).
89 *> \endverbatim
90 *>
91 *> \param[out] PIV
92 *> \verbatim
93 *> PIV is INTEGER array, dimension (N)
94 *> PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
95 *> \endverbatim
96 *>
97 *> \param[out] RANK
98 *> \verbatim
99 *> RANK is INTEGER
100 *> The rank of A given by the number of steps the algorithm
101 *> completed.
102 *> \endverbatim
103 *>
104 *> \param[in] TOL
105 *> \verbatim
106 *> TOL is REAL
107 *> User defined tolerance. If TOL < 0, then N*U*MAX( A(K,K) )
108 *> will be used. The algorithm terminates at the (K-1)st step
109 *> if the pivot <= TOL.
110 *> \endverbatim
111 *>
112 *> \param[out] WORK
113 *> \verbatim
114 *> WORK is REAL array, dimension (2*N)
115 *> Work space.
116 *> \endverbatim
117 *>
118 *> \param[out] INFO
119 *> \verbatim
120 *> INFO is INTEGER
121 *> < 0: If INFO = -K, the K-th argument had an illegal value,
122 *> = 0: algorithm completed successfully, and
123 *> > 0: the matrix A is either rank deficient with computed rank
124 *> as returned in RANK, or is not positive semidefinite. See
125 *> Section 7 of LAPACK Working Note #161 for further
126 *> information.
127 *> \endverbatim
128 *
129 * Authors:
130 * ========
131 *
132 *> \author Univ. of Tennessee
133 *> \author Univ. of California Berkeley
134 *> \author Univ. of Colorado Denver
135 *> \author NAG Ltd.
136 *
137 *> \ingroup realOTHERcomputational
138 *
139 * =====================================================================
140  SUBROUTINE spstrf( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
141 *
142 * -- LAPACK computational routine --
143 * -- LAPACK is a software package provided by Univ. of Tennessee, --
144 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
145 *
146 * .. Scalar Arguments ..
147  REAL TOL
148  INTEGER INFO, LDA, N, RANK
149  CHARACTER UPLO
150 * ..
151 * .. Array Arguments ..
152  REAL A( LDA, * ), WORK( 2*N )
153  INTEGER PIV( N )
154 * ..
155 *
156 * =====================================================================
157 *
158 * .. Parameters ..
159  REAL ONE, ZERO
160  parameter( one = 1.0e+0, zero = 0.0e+0 )
161 * ..
162 * .. Local Scalars ..
163  REAL AJJ, SSTOP, STEMP
164  INTEGER I, ITEMP, J, JB, K, NB, PVT
165  LOGICAL UPPER
166 * ..
167 * .. External Functions ..
168  REAL SLAMCH
169  INTEGER ILAENV
170  LOGICAL LSAME, SISNAN
171  EXTERNAL slamch, ilaenv, lsame, sisnan
172 * ..
173 * .. External Subroutines ..
174  EXTERNAL sgemv, spstf2, sscal, sswap, ssyrk, xerbla
175 * ..
176 * .. Intrinsic Functions ..
177  INTRINSIC max, min, sqrt, maxloc
178 * ..
179 * .. Executable Statements ..
180 *
181 * Test the input parameters.
182 *
183  info = 0
184  upper = lsame( uplo, 'U' )
185  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
186  info = -1
187  ELSE IF( n.LT.0 ) THEN
188  info = -2
189  ELSE IF( lda.LT.max( 1, n ) ) THEN
190  info = -4
191  END IF
192  IF( info.NE.0 ) THEN
193  CALL xerbla( 'SPSTRF', -info )
194  RETURN
195  END IF
196 *
197 * Quick return if possible
198 *
199  IF( n.EQ.0 )
200  $ RETURN
201 *
202 * Get block size
203 *
204  nb = ilaenv( 1, 'SPOTRF', uplo, n, -1, -1, -1 )
205  IF( nb.LE.1 .OR. nb.GE.n ) THEN
206 *
207 * Use unblocked code
208 *
209  CALL spstf2( uplo, n, a( 1, 1 ), lda, piv, rank, tol, work,
210  $ info )
211  GO TO 200
212 *
213  ELSE
214 *
215 * Initialize PIV
216 *
217  DO 100 i = 1, n
218  piv( i ) = i
219  100 CONTINUE
220 *
221 * Compute stopping value
222 *
223  pvt = 1
224  ajj = a( pvt, pvt )
225  DO i = 2, n
226  IF( a( i, i ).GT.ajj ) THEN
227  pvt = i
228  ajj = a( pvt, pvt )
229  END IF
230  END DO
231  IF( ajj.LE.zero.OR.sisnan( ajj ) ) THEN
232  rank = 0
233  info = 1
234  GO TO 200
235  END IF
236 *
237 * Compute stopping value if not supplied
238 *
239  IF( tol.LT.zero ) THEN
240  sstop = n * slamch( 'Epsilon' ) * ajj
241  ELSE
242  sstop = tol
243  END IF
244 *
245 *
246  IF( upper ) THEN
247 *
248 * Compute the Cholesky factorization P**T * A * P = U**T * U
249 *
250  DO 140 k = 1, n, nb
251 *
252 * Account for last block not being NB wide
253 *
254  jb = min( nb, n-k+1 )
255 *
256 * Set relevant part of first half of WORK to zero,
257 * holds dot products
258 *
259  DO 110 i = k, n
260  work( i ) = 0
261  110 CONTINUE
262 *
263  DO 130 j = k, k + jb - 1
264 *
265 * Find pivot, test for exit, else swap rows and columns
266 * Update dot products, compute possible pivots which are
267 * stored in the second half of WORK
268 *
269  DO 120 i = j, n
270 *
271  IF( j.GT.k ) THEN
272  work( i ) = work( i ) + a( j-1, i )**2
273  END IF
274  work( n+i ) = a( i, i ) - work( i )
275 *
276  120 CONTINUE
277 *
278  IF( j.GT.1 ) THEN
279  itemp = maxloc( work( (n+j):(2*n) ), 1 )
280  pvt = itemp + j - 1
281  ajj = work( n+pvt )
282  IF( ajj.LE.sstop.OR.sisnan( ajj ) ) THEN
283  a( j, j ) = ajj
284  GO TO 190
285  END IF
286  END IF
287 *
288  IF( j.NE.pvt ) THEN
289 *
290 * Pivot OK, so can now swap pivot rows and columns
291 *
292  a( pvt, pvt ) = a( j, j )
293  CALL sswap( j-1, a( 1, j ), 1, a( 1, pvt ), 1 )
294  IF( pvt.LT.n )
295  $ CALL sswap( n-pvt, a( j, pvt+1 ), lda,
296  $ a( pvt, pvt+1 ), lda )
297  CALL sswap( pvt-j-1, a( j, j+1 ), lda,
298  $ a( j+1, pvt ), 1 )
299 *
300 * Swap dot products and PIV
301 *
302  stemp = work( j )
303  work( j ) = work( pvt )
304  work( pvt ) = stemp
305  itemp = piv( pvt )
306  piv( pvt ) = piv( j )
307  piv( j ) = itemp
308  END IF
309 *
310  ajj = sqrt( ajj )
311  a( j, j ) = ajj
312 *
313 * Compute elements J+1:N of row J.
314 *
315  IF( j.LT.n ) THEN
316  CALL sgemv( 'Trans', j-k, n-j, -one, a( k, j+1 ),
317  $ lda, a( k, j ), 1, one, a( j, j+1 ),
318  $ lda )
319  CALL sscal( n-j, one / ajj, a( j, j+1 ), lda )
320  END IF
321 *
322  130 CONTINUE
323 *
324 * Update trailing matrix, J already incremented
325 *
326  IF( k+jb.LE.n ) THEN
327  CALL ssyrk( 'Upper', 'Trans', n-j+1, jb, -one,
328  $ a( k, j ), lda, one, a( j, j ), lda )
329  END IF
330 *
331  140 CONTINUE
332 *
333  ELSE
334 *
335 * Compute the Cholesky factorization P**T * A * P = L * L**T
336 *
337  DO 180 k = 1, n, nb
338 *
339 * Account for last block not being NB wide
340 *
341  jb = min( nb, n-k+1 )
342 *
343 * Set relevant part of first half of WORK to zero,
344 * holds dot products
345 *
346  DO 150 i = k, n
347  work( i ) = 0
348  150 CONTINUE
349 *
350  DO 170 j = k, k + jb - 1
351 *
352 * Find pivot, test for exit, else swap rows and columns
353 * Update dot products, compute possible pivots which are
354 * stored in the second half of WORK
355 *
356  DO 160 i = j, n
357 *
358  IF( j.GT.k ) THEN
359  work( i ) = work( i ) + a( i, j-1 )**2
360  END IF
361  work( n+i ) = a( i, i ) - work( i )
362 *
363  160 CONTINUE
364 *
365  IF( j.GT.1 ) THEN
366  itemp = maxloc( work( (n+j):(2*n) ), 1 )
367  pvt = itemp + j - 1
368  ajj = work( n+pvt )
369  IF( ajj.LE.sstop.OR.sisnan( ajj ) ) THEN
370  a( j, j ) = ajj
371  GO TO 190
372  END IF
373  END IF
374 *
375  IF( j.NE.pvt ) THEN
376 *
377 * Pivot OK, so can now swap pivot rows and columns
378 *
379  a( pvt, pvt ) = a( j, j )
380  CALL sswap( j-1, a( j, 1 ), lda, a( pvt, 1 ), lda )
381  IF( pvt.LT.n )
382  $ CALL sswap( n-pvt, a( pvt+1, j ), 1,
383  $ a( pvt+1, pvt ), 1 )
384  CALL sswap( pvt-j-1, a( j+1, j ), 1, a( pvt, j+1 ),
385  $ lda )
386 *
387 * Swap dot products and PIV
388 *
389  stemp = work( j )
390  work( j ) = work( pvt )
391  work( pvt ) = stemp
392  itemp = piv( pvt )
393  piv( pvt ) = piv( j )
394  piv( j ) = itemp
395  END IF
396 *
397  ajj = sqrt( ajj )
398  a( j, j ) = ajj
399 *
400 * Compute elements J+1:N of column J.
401 *
402  IF( j.LT.n ) THEN
403  CALL sgemv( 'No Trans', n-j, j-k, -one,
404  $ a( j+1, k ), lda, a( j, k ), lda, one,
405  $ a( j+1, j ), 1 )
406  CALL sscal( n-j, one / ajj, a( j+1, j ), 1 )
407  END IF
408 *
409  170 CONTINUE
410 *
411 * Update trailing matrix, J already incremented
412 *
413  IF( k+jb.LE.n ) THEN
414  CALL ssyrk( 'Lower', 'No Trans', n-j+1, jb, -one,
415  $ a( j, k ), lda, one, a( j, j ), lda )
416  END IF
417 *
418  180 CONTINUE
419 *
420  END IF
421  END IF
422 *
423 * Ran to completion, A has full rank
424 *
425  rank = n
426 *
427  GO TO 200
428  190 CONTINUE
429 *
430 * Rank is the number of steps completed. Set INFO = 1 to signal
431 * that the factorization cannot be used to solve a system.
432 *
433  rank = j - 1
434  info = 1
435 *
436  200 CONTINUE
437  RETURN
438 *
439 * End of SPSTRF
440 *
441  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine spstrf(UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)
SPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semide...
Definition: spstrf.f:141
subroutine spstf2(UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)
SPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semide...
Definition: spstf2.f:141
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
subroutine ssyrk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
SSYRK
Definition: ssyrk.f:169