LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
dpstrf.f
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1 *> \brief \b DPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
2 *
3 *
4 * =========== DOCUMENTATION ===========
5 *
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18 *
19 * Definition:
20 * ===========
21 *
22 * SUBROUTINE DPSTRF( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * DOUBLE PRECISION TOL
26 * INTEGER INFO, LDA, N, RANK
27 * CHARACTER UPLO
28 * ..
29 * .. Array Arguments ..
30 * DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
31 * INTEGER PIV( N )
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *> \verbatim
39 *>
40 *> DPSTRF computes the Cholesky factorization with complete
41 *> pivoting of a real symmetric positive semidefinite matrix A.
42 *>
43 *> The factorization has the form
44 *> P**T * A * P = U**T * U , if UPLO = 'U',
45 *> P**T * A * P = L * L**T, 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 DOUBLE PRECISION 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 doubleOTHERcomputational
139 *
140 * =====================================================================
141  SUBROUTINE dpstrf( 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  DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
154  INTEGER PIV( N )
155 * ..
156 *
157 * =====================================================================
158 *
159 * .. Parameters ..
160  DOUBLE PRECISION ONE, ZERO
161  parameter( one = 1.0d+0, zero = 0.0d+0 )
162 * ..
163 * .. Local Scalars ..
164  DOUBLE PRECISION AJJ, DSTOP, DTEMP
165  INTEGER I, ITEMP, J, JB, K, NB, PVT
166  LOGICAL UPPER
167 * ..
168 * .. External Functions ..
169  DOUBLE PRECISION DLAMCH
170  INTEGER ILAENV
171  LOGICAL LSAME, DISNAN
172  EXTERNAL dlamch, ilaenv, lsame, disnan
173 * ..
174 * .. External Subroutines ..
175  EXTERNAL dgemv, dpstf2, dscal, dswap, dsyrk, xerbla
176 * ..
177 * .. Intrinsic Functions ..
178  INTRINSIC max, min, sqrt, maxloc
179 * ..
180 * .. Executable Statements ..
181 *
182 * Test the input parameters.
183 *
184  info = 0
185  upper = lsame( uplo, 'U' )
186  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
187  info = -1
188  ELSE IF( n.LT.0 ) THEN
189  info = -2
190  ELSE IF( lda.LT.max( 1, n ) ) THEN
191  info = -4
192  END IF
193  IF( info.NE.0 ) THEN
194  CALL xerbla( 'DPSTRF', -info )
195  RETURN
196  END IF
197 *
198 * Quick return if possible
199 *
200  IF( n.EQ.0 )
201  \$ RETURN
202 *
203 * Get block size
204 *
205  nb = ilaenv( 1, 'DPOTRF', uplo, n, -1, -1, -1 )
206  IF( nb.LE.1 .OR. nb.GE.n ) THEN
207 *
208 * Use unblocked code
209 *
210  CALL dpstf2( uplo, n, a( 1, 1 ), lda, piv, rank, tol, work,
211  \$ info )
212  GO TO 200
213 *
214  ELSE
215 *
216 * Initialize PIV
217 *
218  DO 100 i = 1, n
219  piv( i ) = i
220  100 CONTINUE
221 *
222 * Compute stopping value
223 *
224  pvt = 1
225  ajj = a( pvt, pvt )
226  DO i = 2, n
227  IF( a( i, i ).GT.ajj ) THEN
228  pvt = i
229  ajj = a( pvt, pvt )
230  END IF
231  END DO
232  IF( ajj.LE.zero.OR.disnan( ajj ) ) THEN
233  rank = 0
234  info = 1
235  GO TO 200
236  END IF
237 *
238 * Compute stopping value if not supplied
239 *
240  IF( tol.LT.zero ) THEN
241  dstop = n * dlamch( 'Epsilon' ) * ajj
242  ELSE
243  dstop = tol
244  END IF
245 *
246 *
247  IF( upper ) THEN
248 *
249 * Compute the Cholesky factorization P**T * A * P = U**T * U
250 *
251  DO 140 k = 1, n, nb
252 *
253 * Account for last block not being NB wide
254 *
255  jb = min( nb, n-k+1 )
256 *
257 * Set relevant part of first half of WORK to zero,
258 * holds dot products
259 *
260  DO 110 i = k, n
261  work( i ) = 0
262  110 CONTINUE
263 *
264  DO 130 j = k, k + jb - 1
265 *
266 * Find pivot, test for exit, else swap rows and columns
267 * Update dot products, compute possible pivots which are
268 * stored in the second half of WORK
269 *
270  DO 120 i = j, n
271 *
272  IF( j.GT.k ) THEN
273  work( i ) = work( i ) + a( j-1, i )**2
274  END IF
275  work( n+i ) = a( i, i ) - work( i )
276 *
277  120 CONTINUE
278 *
279  IF( j.GT.1 ) THEN
280  itemp = maxloc( work( (n+j):(2*n) ), 1 )
281  pvt = itemp + j - 1
282  ajj = work( n+pvt )
283  IF( ajj.LE.dstop.OR.disnan( ajj ) ) THEN
284  a( j, j ) = ajj
285  GO TO 190
286  END IF
287  END IF
288 *
289  IF( j.NE.pvt ) THEN
290 *
291 * Pivot OK, so can now swap pivot rows and columns
292 *
293  a( pvt, pvt ) = a( j, j )
294  CALL dswap( j-1, a( 1, j ), 1, a( 1, pvt ), 1 )
295  IF( pvt.LT.n )
296  \$ CALL dswap( n-pvt, a( j, pvt+1 ), lda,
297  \$ a( pvt, pvt+1 ), lda )
298  CALL dswap( pvt-j-1, a( j, j+1 ), lda,
299  \$ a( j+1, pvt ), 1 )
300 *
301 * Swap dot products and PIV
302 *
303  dtemp = work( j )
304  work( j ) = work( pvt )
305  work( pvt ) = dtemp
306  itemp = piv( pvt )
307  piv( pvt ) = piv( j )
308  piv( j ) = itemp
309  END IF
310 *
311  ajj = sqrt( ajj )
312  a( j, j ) = ajj
313 *
314 * Compute elements J+1:N of row J.
315 *
316  IF( j.LT.n ) THEN
317  CALL dgemv( 'Trans', j-k, n-j, -one, a( k, j+1 ),
318  \$ lda, a( k, j ), 1, one, a( j, j+1 ),
319  \$ lda )
320  CALL dscal( n-j, one / ajj, a( j, j+1 ), lda )
321  END IF
322 *
323  130 CONTINUE
324 *
325 * Update trailing matrix, J already incremented
326 *
327  IF( k+jb.LE.n ) THEN
328  CALL dsyrk( 'Upper', 'Trans', n-j+1, jb, -one,
329  \$ a( k, j ), lda, one, a( j, j ), lda )
330  END IF
331 *
332  140 CONTINUE
333 *
334  ELSE
335 *
336 * Compute the Cholesky factorization P**T * A * P = L * L**T
337 *
338  DO 180 k = 1, n, nb
339 *
340 * Account for last block not being NB wide
341 *
342  jb = min( nb, n-k+1 )
343 *
344 * Set relevant part of first half of WORK to zero,
345 * holds dot products
346 *
347  DO 150 i = k, n
348  work( i ) = 0
349  150 CONTINUE
350 *
351  DO 170 j = k, k + jb - 1
352 *
353 * Find pivot, test for exit, else swap rows and columns
354 * Update dot products, compute possible pivots which are
355 * stored in the second half of WORK
356 *
357  DO 160 i = j, n
358 *
359  IF( j.GT.k ) THEN
360  work( i ) = work( i ) + a( i, j-1 )**2
361  END IF
362  work( n+i ) = a( i, i ) - work( i )
363 *
364  160 CONTINUE
365 *
366  IF( j.GT.1 ) THEN
367  itemp = maxloc( work( (n+j):(2*n) ), 1 )
368  pvt = itemp + j - 1
369  ajj = work( n+pvt )
370  IF( ajj.LE.dstop.OR.disnan( ajj ) ) THEN
371  a( j, j ) = ajj
372  GO TO 190
373  END IF
374  END IF
375 *
376  IF( j.NE.pvt ) THEN
377 *
378 * Pivot OK, so can now swap pivot rows and columns
379 *
380  a( pvt, pvt ) = a( j, j )
381  CALL dswap( j-1, a( j, 1 ), lda, a( pvt, 1 ), lda )
382  IF( pvt.LT.n )
383  \$ CALL dswap( n-pvt, a( pvt+1, j ), 1,
384  \$ a( pvt+1, pvt ), 1 )
385  CALL dswap( pvt-j-1, a( j+1, j ), 1, a( pvt, j+1 ),
386  \$ lda )
387 *
388 * Swap dot products and PIV
389 *
390  dtemp = work( j )
391  work( j ) = work( pvt )
392  work( pvt ) = dtemp
393  itemp = piv( pvt )
394  piv( pvt ) = piv( j )
395  piv( j ) = itemp
396  END IF
397 *
398  ajj = sqrt( ajj )
399  a( j, j ) = ajj
400 *
401 * Compute elements J+1:N of column J.
402 *
403  IF( j.LT.n ) THEN
404  CALL dgemv( 'No Trans', n-j, j-k, -one,
405  \$ a( j+1, k ), lda, a( j, k ), lda, one,
406  \$ a( j+1, j ), 1 )
407  CALL dscal( n-j, one / ajj, a( j+1, j ), 1 )
408  END IF
409 *
410  170 CONTINUE
411 *
412 * Update trailing matrix, J already incremented
413 *
414  IF( k+jb.LE.n ) THEN
415  CALL dsyrk( 'Lower', 'No Trans', n-j+1, jb, -one,
416  \$ a( j, k ), lda, one, a( j, j ), lda )
417  END IF
418 *
419  180 CONTINUE
420 *
421  END IF
422  END IF
423 *
424 * Ran to completion, A has full rank
425 *
426  rank = n
427 *
428  GO TO 200
429  190 CONTINUE
430 *
431 * Rank is the number of steps completed. Set INFO = 1 to signal
432 * that the factorization cannot be used to solve a system.
433 *
434  rank = j - 1
435  info = 1
436 *
437  200 CONTINUE
438  RETURN
439 *
440 * End of DPSTRF
441 *
442  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine dswap(N, DX, INCX, DY, INCY)
DSWAP
Definition: dswap.f:82
subroutine dgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DGEMV
Definition: dgemv.f:156
subroutine dsyrk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
DSYRK
Definition: dsyrk.f:169
subroutine dpstrf(UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)
DPSTRF computes the Cholesky factorization with complete pivoting of a real symmetric positive semide...
Definition: dpstrf.f:142
subroutine dpstf2(UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO)
DPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semide...
Definition: dpstf2.f:141