Annotation of rpl/lapack/lapack/dpstf2.f, revision 1.16
1.12 bertrand 1: *> \brief \b DPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric positive semidefinite matrix.
1.6 bertrand 2: *
3: * =========== DOCUMENTATION ===========
4: *
1.14 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.6 bertrand 7: *
8: *> \htmlonly
1.14 bertrand 9: *> Download DPSTF2 + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpstf2.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpstf2.f">
1.6 bertrand 15: *> [TXT]</a>
1.14 bertrand 16: *> \endhtmlonly
1.6 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
1.14 bertrand 22: *
1.6 bertrand 23: * .. Scalar Arguments ..
24: * DOUBLE PRECISION TOL
25: * INTEGER INFO, LDA, N, RANK
26: * CHARACTER UPLO
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
30: * INTEGER PIV( N )
31: * ..
1.14 bertrand 32: *
1.6 bertrand 33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> DPSTF2 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 DOUBLE PRECISION 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 DOUBLE PRECISION
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 DOUBLE PRECISION 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
1.12 bertrand 124: *> as returned in RANK, or is not positive semidefinite. See
125: *> Section 7 of LAPACK Working Note #161 for further
126: *> information.
1.6 bertrand 127: *> \endverbatim
128: *
129: * Authors:
130: * ========
131: *
1.14 bertrand 132: *> \author Univ. of Tennessee
133: *> \author Univ. of California Berkeley
134: *> \author Univ. of Colorado Denver
135: *> \author NAG Ltd.
1.6 bertrand 136: *
1.14 bertrand 137: *> \date December 2016
1.6 bertrand 138: *
139: *> \ingroup doubleOTHERcomputational
140: *
141: * =====================================================================
1.1 bertrand 142: SUBROUTINE DPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
143: *
1.14 bertrand 144: * -- LAPACK computational routine (version 3.7.0) --
1.6 bertrand 145: * -- LAPACK is a software package provided by Univ. of Tennessee, --
146: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.14 bertrand 147: * December 2016
1.1 bertrand 148: *
149: * .. Scalar Arguments ..
150: DOUBLE PRECISION TOL
151: INTEGER INFO, LDA, N, RANK
152: CHARACTER UPLO
153: * ..
154: * .. Array Arguments ..
155: DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
156: INTEGER PIV( N )
157: * ..
158: *
159: * =====================================================================
160: *
161: * .. Parameters ..
162: DOUBLE PRECISION ONE, ZERO
163: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
164: * ..
165: * .. Local Scalars ..
166: DOUBLE PRECISION AJJ, DSTOP, DTEMP
167: INTEGER I, ITEMP, J, PVT
168: LOGICAL UPPER
169: * ..
170: * .. External Functions ..
171: DOUBLE PRECISION DLAMCH
172: LOGICAL LSAME, DISNAN
173: EXTERNAL DLAMCH, LSAME, DISNAN
174: * ..
175: * .. External Subroutines ..
176: EXTERNAL DGEMV, DSCAL, DSWAP, XERBLA
177: * ..
178: * .. Intrinsic Functions ..
179: INTRINSIC MAX, SQRT, MAXLOC
180: * ..
181: * .. Executable Statements ..
182: *
183: * Test the input parameters
184: *
185: INFO = 0
186: UPPER = LSAME( UPLO, 'U' )
187: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
188: INFO = -1
189: ELSE IF( N.LT.0 ) THEN
190: INFO = -2
191: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
192: INFO = -4
193: END IF
194: IF( INFO.NE.0 ) THEN
195: CALL XERBLA( 'DPSTF2', -INFO )
196: RETURN
197: END IF
198: *
199: * Quick return if possible
200: *
201: IF( N.EQ.0 )
202: $ RETURN
203: *
204: * Initialize PIV
205: *
206: DO 100 I = 1, N
207: PIV( I ) = I
208: 100 CONTINUE
209: *
210: * Compute stopping value
211: *
212: PVT = 1
213: AJJ = A( PVT, PVT )
214: DO I = 2, N
215: IF( A( I, I ).GT.AJJ ) THEN
216: PVT = I
217: AJJ = A( PVT, PVT )
218: END IF
219: END DO
1.12 bertrand 220: IF( AJJ.LE.ZERO.OR.DISNAN( AJJ ) ) THEN
1.1 bertrand 221: RANK = 0
222: INFO = 1
223: GO TO 170
224: END IF
225: *
226: * Compute stopping value if not supplied
227: *
228: IF( TOL.LT.ZERO ) THEN
229: DSTOP = N * DLAMCH( 'Epsilon' ) * AJJ
230: ELSE
231: DSTOP = TOL
232: END IF
233: *
234: * Set first half of WORK to zero, holds dot products
235: *
236: DO 110 I = 1, N
237: WORK( I ) = 0
238: 110 CONTINUE
239: *
240: IF( UPPER ) THEN
241: *
1.5 bertrand 242: * Compute the Cholesky factorization P**T * A * P = U**T * U
1.1 bertrand 243: *
244: DO 130 J = 1, N
245: *
246: * Find pivot, test for exit, else swap rows and columns
247: * Update dot products, compute possible pivots which are
248: * stored in the second half of WORK
249: *
250: DO 120 I = J, N
251: *
252: IF( J.GT.1 ) THEN
253: WORK( I ) = WORK( I ) + A( J-1, I )**2
254: END IF
255: WORK( N+I ) = A( I, I ) - WORK( I )
256: *
257: 120 CONTINUE
258: *
259: IF( J.GT.1 ) THEN
260: ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
261: PVT = ITEMP + J - 1
262: AJJ = WORK( N+PVT )
263: IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
264: A( J, J ) = AJJ
265: GO TO 160
266: END IF
267: END IF
268: *
269: IF( J.NE.PVT ) THEN
270: *
271: * Pivot OK, so can now swap pivot rows and columns
272: *
273: A( PVT, PVT ) = A( J, J )
274: CALL DSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
275: IF( PVT.LT.N )
276: $ CALL DSWAP( N-PVT, A( J, PVT+1 ), LDA,
277: $ A( PVT, PVT+1 ), LDA )
278: CALL DSWAP( PVT-J-1, A( J, J+1 ), LDA, A( J+1, PVT ), 1 )
279: *
280: * Swap dot products and PIV
281: *
282: DTEMP = WORK( J )
283: WORK( J ) = WORK( PVT )
284: WORK( PVT ) = DTEMP
285: ITEMP = PIV( PVT )
286: PIV( PVT ) = PIV( J )
287: PIV( J ) = ITEMP
288: END IF
289: *
290: AJJ = SQRT( AJJ )
291: A( J, J ) = AJJ
292: *
293: * Compute elements J+1:N of row J
294: *
295: IF( J.LT.N ) THEN
296: CALL DGEMV( 'Trans', J-1, N-J, -ONE, A( 1, J+1 ), LDA,
297: $ A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
298: CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
299: END IF
300: *
301: 130 CONTINUE
302: *
303: ELSE
304: *
1.5 bertrand 305: * Compute the Cholesky factorization P**T * A * P = L * L**T
1.1 bertrand 306: *
307: DO 150 J = 1, N
308: *
309: * Find pivot, test for exit, else swap rows and columns
310: * Update dot products, compute possible pivots which are
311: * stored in the second half of WORK
312: *
313: DO 140 I = J, N
314: *
315: IF( J.GT.1 ) THEN
316: WORK( I ) = WORK( I ) + A( I, J-1 )**2
317: END IF
318: WORK( N+I ) = A( I, I ) - WORK( I )
319: *
320: 140 CONTINUE
321: *
322: IF( J.GT.1 ) THEN
323: ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
324: PVT = ITEMP + J - 1
325: AJJ = WORK( N+PVT )
326: IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
327: A( J, J ) = AJJ
328: GO TO 160
329: END IF
330: END IF
331: *
332: IF( J.NE.PVT ) THEN
333: *
334: * Pivot OK, so can now swap pivot rows and columns
335: *
336: A( PVT, PVT ) = A( J, J )
337: CALL DSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA )
338: IF( PVT.LT.N )
339: $ CALL DSWAP( N-PVT, A( PVT+1, J ), 1, A( PVT+1, PVT ),
340: $ 1 )
341: CALL DSWAP( PVT-J-1, A( J+1, J ), 1, A( PVT, J+1 ), LDA )
342: *
343: * Swap dot products and PIV
344: *
345: DTEMP = WORK( J )
346: WORK( J ) = WORK( PVT )
347: WORK( PVT ) = DTEMP
348: ITEMP = PIV( PVT )
349: PIV( PVT ) = PIV( J )
350: PIV( J ) = ITEMP
351: END IF
352: *
353: AJJ = SQRT( AJJ )
354: A( J, J ) = AJJ
355: *
356: * Compute elements J+1:N of column J
357: *
358: IF( J.LT.N ) THEN
359: CALL DGEMV( 'No Trans', N-J, J-1, -ONE, A( J+1, 1 ), LDA,
360: $ A( J, 1 ), LDA, ONE, A( J+1, J ), 1 )
361: CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
362: END IF
363: *
364: 150 CONTINUE
365: *
366: END IF
367: *
368: * Ran to completion, A has full rank
369: *
370: RANK = N
371: *
372: GO TO 170
373: 160 CONTINUE
374: *
375: * Rank is number of steps completed. Set INFO = 1 to signal
376: * that the factorization cannot be used to solve a system.
377: *
378: RANK = J - 1
379: INFO = 1
380: *
381: 170 CONTINUE
382: RETURN
383: *
384: * End of DPSTF2
385: *
386: END
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