1: *> \brief \b DPSTF2 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/
7: *
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16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
22: *
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: * ..
32: *
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
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 doubleOTHERcomputational
138: *
139: * =====================================================================
140: SUBROUTINE DPSTF2( 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: DOUBLE PRECISION TOL
148: INTEGER INFO, LDA, N, RANK
149: CHARACTER UPLO
150: * ..
151: * .. Array Arguments ..
152: DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
153: INTEGER PIV( N )
154: * ..
155: *
156: * =====================================================================
157: *
158: * .. Parameters ..
159: DOUBLE PRECISION ONE, ZERO
160: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
161: * ..
162: * .. Local Scalars ..
163: DOUBLE PRECISION AJJ, DSTOP, DTEMP
164: INTEGER I, ITEMP, J, PVT
165: LOGICAL UPPER
166: * ..
167: * .. External Functions ..
168: DOUBLE PRECISION DLAMCH
169: LOGICAL LSAME, DISNAN
170: EXTERNAL DLAMCH, LSAME, DISNAN
171: * ..
172: * .. External Subroutines ..
173: EXTERNAL DGEMV, DSCAL, DSWAP, 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( 'DPSTF2', -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.DISNAN( 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: DSTOP = N * DLAMCH( 'Epsilon' ) * AJJ
227: ELSE
228: DSTOP = 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.DSTOP.OR.DISNAN( 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 DSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
272: IF( PVT.LT.N )
273: $ CALL DSWAP( N-PVT, A( J, PVT+1 ), LDA,
274: $ A( PVT, PVT+1 ), LDA )
275: CALL DSWAP( PVT-J-1, A( J, J+1 ), LDA, A( J+1, PVT ), 1 )
276: *
277: * Swap dot products and PIV
278: *
279: DTEMP = WORK( J )
280: WORK( J ) = WORK( PVT )
281: WORK( PVT ) = DTEMP
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 DGEMV( 'Trans', J-1, N-J, -ONE, A( 1, J+1 ), LDA,
294: $ A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
295: CALL DSCAL( 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.DSTOP.OR.DISNAN( 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 DSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA )
335: IF( PVT.LT.N )
336: $ CALL DSWAP( N-PVT, A( PVT+1, J ), 1, A( PVT+1, PVT ),
337: $ 1 )
338: CALL DSWAP( PVT-J-1, A( J+1, J ), 1, A( PVT, J+1 ), LDA )
339: *
340: * Swap dot products and PIV
341: *
342: DTEMP = WORK( J )
343: WORK( J ) = WORK( PVT )
344: WORK( PVT ) = DTEMP
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 DGEMV( '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 DSCAL( 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 DPSTF2
382: *
383: END
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