LAPACK  3.9.1
LAPACK: Linear Algebra PACKage
spstf2.f
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1 *> \brief \b SPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SPSTF2( 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 *> SPSTF2 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 2 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[out] PIV
86 *> \verbatim
87 *> PIV is INTEGER array, dimension (N)
88 *> PIV is such that the nonzero entries are P( PIV(K), K ) = 1.
89 *> \endverbatim
90 *>
91 *> \param[out] RANK
92 *> \verbatim
93 *> RANK is INTEGER
94 *> The rank of A given by the number of steps the algorithm
95 *> completed.
96 *> \endverbatim
97 *>
98 *> \param[in] TOL
99 *> \verbatim
100 *> TOL is REAL
101 *> User defined tolerance. If TOL < 0, then N*U*MAX( A( K,K ) )
102 *> will be used. The algorithm terminates at the (K-1)st step
103 *> if the pivot <= TOL.
104 *> \endverbatim
105 *>
106 *> \param[in] LDA
107 *> \verbatim
108 *> LDA is INTEGER
109 *> The leading dimension of the array A. LDA >= max(1,N).
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 spstf2( 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, PVT
165  LOGICAL UPPER
166 * ..
167 * .. External Functions ..
168  REAL SLAMCH
169  LOGICAL LSAME, SISNAN
170  EXTERNAL slamch, lsame, sisnan
171 * ..
172 * .. External Subroutines ..
173  EXTERNAL sgemv, sscal, sswap, xerbla
174 * ..
175 * .. Intrinsic Functions ..
176  INTRINSIC max, sqrt, maxloc
177 * ..
178 * .. Executable Statements ..
179 *
180 * Test the input parameters
181 *
182  info = 0
183  upper = lsame( uplo, 'U' )
184  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
185  info = -1
186  ELSE IF( n.LT.0 ) THEN
187  info = -2
188  ELSE IF( lda.LT.max( 1, n ) ) THEN
189  info = -4
190  END IF
191  IF( info.NE.0 ) THEN
192  CALL xerbla( 'SPSTF2', -info )
193  RETURN
194  END IF
195 *
196 * Quick return if possible
197 *
198  IF( n.EQ.0 )
199  $ RETURN
200 *
201 * Initialize PIV
202 *
203  DO 100 i = 1, n
204  piv( i ) = i
205  100 CONTINUE
206 *
207 * Compute stopping value
208 *
209  pvt = 1
210  ajj = a( pvt, pvt )
211  DO i = 2, n
212  IF( a( i, i ).GT.ajj ) THEN
213  pvt = i
214  ajj = a( pvt, pvt )
215  END IF
216  END DO
217  IF( ajj.LE.zero.OR.sisnan( ajj ) ) THEN
218  rank = 0
219  info = 1
220  GO TO 170
221  END IF
222 *
223 * Compute stopping value if not supplied
224 *
225  IF( tol.LT.zero ) THEN
226  sstop = n * slamch( 'Epsilon' ) * ajj
227  ELSE
228  sstop = tol
229  END IF
230 *
231 * Set first half of WORK to zero, holds dot products
232 *
233  DO 110 i = 1, n
234  work( i ) = 0
235  110 CONTINUE
236 *
237  IF( upper ) THEN
238 *
239 * Compute the Cholesky factorization P**T * A * P = U**T * U
240 *
241  DO 130 j = 1, n
242 *
243 * Find pivot, test for exit, else swap rows and columns
244 * Update dot products, compute possible pivots which are
245 * stored in the second half of WORK
246 *
247  DO 120 i = j, n
248 *
249  IF( j.GT.1 ) THEN
250  work( i ) = work( i ) + a( j-1, i )**2
251  END IF
252  work( n+i ) = a( i, i ) - work( i )
253 *
254  120 CONTINUE
255 *
256  IF( j.GT.1 ) THEN
257  itemp = maxloc( work( (n+j):(2*n) ), 1 )
258  pvt = itemp + j - 1
259  ajj = work( n+pvt )
260  IF( ajj.LE.sstop.OR.sisnan( ajj ) ) THEN
261  a( j, j ) = ajj
262  GO TO 160
263  END IF
264  END IF
265 *
266  IF( j.NE.pvt ) THEN
267 *
268 * Pivot OK, so can now swap pivot rows and columns
269 *
270  a( pvt, pvt ) = a( j, j )
271  CALL sswap( j-1, a( 1, j ), 1, a( 1, pvt ), 1 )
272  IF( pvt.LT.n )
273  $ CALL sswap( n-pvt, a( j, pvt+1 ), lda,
274  $ a( pvt, pvt+1 ), lda )
275  CALL sswap( pvt-j-1, a( j, j+1 ), lda, a( j+1, pvt ), 1 )
276 *
277 * Swap dot products and PIV
278 *
279  stemp = work( j )
280  work( j ) = work( pvt )
281  work( pvt ) = stemp
282  itemp = piv( pvt )
283  piv( pvt ) = piv( j )
284  piv( j ) = itemp
285  END IF
286 *
287  ajj = sqrt( ajj )
288  a( j, j ) = ajj
289 *
290 * Compute elements J+1:N of row J
291 *
292  IF( j.LT.n ) THEN
293  CALL sgemv( 'Trans', j-1, n-j, -one, a( 1, j+1 ), lda,
294  $ a( 1, j ), 1, one, a( j, j+1 ), lda )
295  CALL sscal( n-j, one / ajj, a( j, j+1 ), lda )
296  END IF
297 *
298  130 CONTINUE
299 *
300  ELSE
301 *
302 * Compute the Cholesky factorization P**T * A * P = L * L**T
303 *
304  DO 150 j = 1, n
305 *
306 * Find pivot, test for exit, else swap rows and columns
307 * Update dot products, compute possible pivots which are
308 * stored in the second half of WORK
309 *
310  DO 140 i = j, n
311 *
312  IF( j.GT.1 ) THEN
313  work( i ) = work( i ) + a( i, j-1 )**2
314  END IF
315  work( n+i ) = a( i, i ) - work( i )
316 *
317  140 CONTINUE
318 *
319  IF( j.GT.1 ) THEN
320  itemp = maxloc( work( (n+j):(2*n) ), 1 )
321  pvt = itemp + j - 1
322  ajj = work( n+pvt )
323  IF( ajj.LE.sstop.OR.sisnan( ajj ) ) THEN
324  a( j, j ) = ajj
325  GO TO 160
326  END IF
327  END IF
328 *
329  IF( j.NE.pvt ) THEN
330 *
331 * Pivot OK, so can now swap pivot rows and columns
332 *
333  a( pvt, pvt ) = a( j, j )
334  CALL sswap( j-1, a( j, 1 ), lda, a( pvt, 1 ), lda )
335  IF( pvt.LT.n )
336  $ CALL sswap( n-pvt, a( pvt+1, j ), 1, a( pvt+1, pvt ),
337  $ 1 )
338  CALL sswap( pvt-j-1, a( j+1, j ), 1, a( pvt, j+1 ), lda )
339 *
340 * Swap dot products and PIV
341 *
342  stemp = work( j )
343  work( j ) = work( pvt )
344  work( pvt ) = stemp
345  itemp = piv( pvt )
346  piv( pvt ) = piv( j )
347  piv( j ) = itemp
348  END IF
349 *
350  ajj = sqrt( ajj )
351  a( j, j ) = ajj
352 *
353 * Compute elements J+1:N of column J
354 *
355  IF( j.LT.n ) THEN
356  CALL sgemv( 'No Trans', n-j, j-1, -one, a( j+1, 1 ), lda,
357  $ a( j, 1 ), lda, one, a( j+1, j ), 1 )
358  CALL sscal( n-j, one / ajj, a( j+1, j ), 1 )
359  END IF
360 *
361  150 CONTINUE
362 *
363  END IF
364 *
365 * Ran to completion, A has full rank
366 *
367  rank = n
368 *
369  GO TO 170
370  160 CONTINUE
371 *
372 * Rank is number of steps completed. Set INFO = 1 to signal
373 * that the factorization cannot be used to solve a system.
374 *
375  rank = j - 1
376  info = 1
377 *
378  170 CONTINUE
379  RETURN
380 *
381 * End of SPSTF2
382 *
383  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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