1: *> \brief \b DPSTF2 computes the Cholesky factorization with complete pivoting of a real symmetric or complex Hermitian positive semi-definite matrix.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DPSTF2 + dependencies
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15: *> [TXT]</a>
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 indefinite. See Section 7 of
125: *> LAPACK Working Note #161 for further information.
126: *> \endverbatim
127: *
128: * Authors:
129: * ========
130: *
131: *> \author Univ. of Tennessee
132: *> \author Univ. of California Berkeley
133: *> \author Univ. of Colorado Denver
134: *> \author NAG Ltd.
135: *
136: *> \date September 2012
137: *
138: *> \ingroup doubleOTHERcomputational
139: *
140: * =====================================================================
141: SUBROUTINE DPSTF2( UPLO, N, A, LDA, PIV, RANK, TOL, WORK, INFO )
142: *
143: * -- LAPACK computational routine (version 3.4.2) --
144: * -- LAPACK is a software package provided by Univ. of Tennessee, --
145: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
146: * September 2012
147: *
148: * .. Scalar Arguments ..
149: DOUBLE PRECISION TOL
150: INTEGER INFO, LDA, N, RANK
151: CHARACTER UPLO
152: * ..
153: * .. Array Arguments ..
154: DOUBLE PRECISION A( LDA, * ), WORK( 2*N )
155: INTEGER PIV( N )
156: * ..
157: *
158: * =====================================================================
159: *
160: * .. Parameters ..
161: DOUBLE PRECISION ONE, ZERO
162: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
163: * ..
164: * .. Local Scalars ..
165: DOUBLE PRECISION AJJ, DSTOP, DTEMP
166: INTEGER I, ITEMP, J, PVT
167: LOGICAL UPPER
168: * ..
169: * .. External Functions ..
170: DOUBLE PRECISION DLAMCH
171: LOGICAL LSAME, DISNAN
172: EXTERNAL DLAMCH, LSAME, DISNAN
173: * ..
174: * .. External Subroutines ..
175: EXTERNAL DGEMV, DSCAL, DSWAP, XERBLA
176: * ..
177: * .. Intrinsic Functions ..
178: INTRINSIC MAX, 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( 'DPSTF2', -INFO )
195: RETURN
196: END IF
197: *
198: * Quick return if possible
199: *
200: IF( N.EQ.0 )
201: $ RETURN
202: *
203: * Initialize PIV
204: *
205: DO 100 I = 1, N
206: PIV( I ) = I
207: 100 CONTINUE
208: *
209: * Compute stopping value
210: *
211: PVT = 1
212: AJJ = A( PVT, PVT )
213: DO I = 2, N
214: IF( A( I, I ).GT.AJJ ) THEN
215: PVT = I
216: AJJ = A( PVT, PVT )
217: END IF
218: END DO
219: IF( AJJ.EQ.ZERO.OR.DISNAN( AJJ ) ) THEN
220: RANK = 0
221: INFO = 1
222: GO TO 170
223: END IF
224: *
225: * Compute stopping value if not supplied
226: *
227: IF( TOL.LT.ZERO ) THEN
228: DSTOP = N * DLAMCH( 'Epsilon' ) * AJJ
229: ELSE
230: DSTOP = TOL
231: END IF
232: *
233: * Set first half of WORK to zero, holds dot products
234: *
235: DO 110 I = 1, N
236: WORK( I ) = 0
237: 110 CONTINUE
238: *
239: IF( UPPER ) THEN
240: *
241: * Compute the Cholesky factorization P**T * A * P = U**T * U
242: *
243: DO 130 J = 1, N
244: *
245: * Find pivot, test for exit, else swap rows and columns
246: * Update dot products, compute possible pivots which are
247: * stored in the second half of WORK
248: *
249: DO 120 I = J, N
250: *
251: IF( J.GT.1 ) THEN
252: WORK( I ) = WORK( I ) + A( J-1, I )**2
253: END IF
254: WORK( N+I ) = A( I, I ) - WORK( I )
255: *
256: 120 CONTINUE
257: *
258: IF( J.GT.1 ) THEN
259: ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
260: PVT = ITEMP + J - 1
261: AJJ = WORK( N+PVT )
262: IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
263: A( J, J ) = AJJ
264: GO TO 160
265: END IF
266: END IF
267: *
268: IF( J.NE.PVT ) THEN
269: *
270: * Pivot OK, so can now swap pivot rows and columns
271: *
272: A( PVT, PVT ) = A( J, J )
273: CALL DSWAP( J-1, A( 1, J ), 1, A( 1, PVT ), 1 )
274: IF( PVT.LT.N )
275: $ CALL DSWAP( N-PVT, A( J, PVT+1 ), LDA,
276: $ A( PVT, PVT+1 ), LDA )
277: CALL DSWAP( PVT-J-1, A( J, J+1 ), LDA, A( J+1, PVT ), 1 )
278: *
279: * Swap dot products and PIV
280: *
281: DTEMP = WORK( J )
282: WORK( J ) = WORK( PVT )
283: WORK( PVT ) = DTEMP
284: ITEMP = PIV( PVT )
285: PIV( PVT ) = PIV( J )
286: PIV( J ) = ITEMP
287: END IF
288: *
289: AJJ = SQRT( AJJ )
290: A( J, J ) = AJJ
291: *
292: * Compute elements J+1:N of row J
293: *
294: IF( J.LT.N ) THEN
295: CALL DGEMV( 'Trans', J-1, N-J, -ONE, A( 1, J+1 ), LDA,
296: $ A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
297: CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
298: END IF
299: *
300: 130 CONTINUE
301: *
302: ELSE
303: *
304: * Compute the Cholesky factorization P**T * A * P = L * L**T
305: *
306: DO 150 J = 1, N
307: *
308: * Find pivot, test for exit, else swap rows and columns
309: * Update dot products, compute possible pivots which are
310: * stored in the second half of WORK
311: *
312: DO 140 I = J, N
313: *
314: IF( J.GT.1 ) THEN
315: WORK( I ) = WORK( I ) + A( I, J-1 )**2
316: END IF
317: WORK( N+I ) = A( I, I ) - WORK( I )
318: *
319: 140 CONTINUE
320: *
321: IF( J.GT.1 ) THEN
322: ITEMP = MAXLOC( WORK( (N+J):(2*N) ), 1 )
323: PVT = ITEMP + J - 1
324: AJJ = WORK( N+PVT )
325: IF( AJJ.LE.DSTOP.OR.DISNAN( AJJ ) ) THEN
326: A( J, J ) = AJJ
327: GO TO 160
328: END IF
329: END IF
330: *
331: IF( J.NE.PVT ) THEN
332: *
333: * Pivot OK, so can now swap pivot rows and columns
334: *
335: A( PVT, PVT ) = A( J, J )
336: CALL DSWAP( J-1, A( J, 1 ), LDA, A( PVT, 1 ), LDA )
337: IF( PVT.LT.N )
338: $ CALL DSWAP( N-PVT, A( PVT+1, J ), 1, A( PVT+1, PVT ),
339: $ 1 )
340: CALL DSWAP( PVT-J-1, A( J+1, J ), 1, A( PVT, J+1 ), LDA )
341: *
342: * Swap dot products and PIV
343: *
344: DTEMP = WORK( J )
345: WORK( J ) = WORK( PVT )
346: WORK( PVT ) = DTEMP
347: ITEMP = PIV( PVT )
348: PIV( PVT ) = PIV( J )
349: PIV( J ) = ITEMP
350: END IF
351: *
352: AJJ = SQRT( AJJ )
353: A( J, J ) = AJJ
354: *
355: * Compute elements J+1:N of column J
356: *
357: IF( J.LT.N ) THEN
358: CALL DGEMV( 'No Trans', N-J, J-1, -ONE, A( J+1, 1 ), LDA,
359: $ A( J, 1 ), LDA, ONE, A( J+1, J ), 1 )
360: CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
361: END IF
362: *
363: 150 CONTINUE
364: *
365: END IF
366: *
367: * Ran to completion, A has full rank
368: *
369: RANK = N
370: *
371: GO TO 170
372: 160 CONTINUE
373: *
374: * Rank is number of steps completed. Set INFO = 1 to signal
375: * that the factorization cannot be used to solve a system.
376: *
377: RANK = J - 1
378: INFO = 1
379: *
380: 170 CONTINUE
381: RETURN
382: *
383: * End of DPSTF2
384: *
385: END
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