1: *> \brief \b DSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSYTF2 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytf2.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytf2.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytf2.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSYTF2( UPLO, N, A, LDA, IPIV, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, LDA, N
26: * ..
27: * .. Array Arguments ..
28: * INTEGER IPIV( * )
29: * DOUBLE PRECISION A( LDA, * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DSYTF2 computes the factorization of a real symmetric matrix A using
39: *> the Bunch-Kaufman diagonal pivoting method:
40: *>
41: *> A = U*D*U**T or A = L*D*L**T
42: *>
43: *> where U (or L) is a product of permutation and unit upper (lower)
44: *> triangular matrices, U**T is the transpose of U, and D is symmetric and
45: *> block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
46: *>
47: *> This is the unblocked version of the algorithm, calling Level 2 BLAS.
48: *> \endverbatim
49: *
50: * Arguments:
51: * ==========
52: *
53: *> \param[in] UPLO
54: *> \verbatim
55: *> UPLO is CHARACTER*1
56: *> Specifies whether the upper or lower triangular part of the
57: *> symmetric matrix A is stored:
58: *> = 'U': Upper triangular
59: *> = 'L': Lower triangular
60: *> \endverbatim
61: *>
62: *> \param[in] N
63: *> \verbatim
64: *> N is INTEGER
65: *> The order of the matrix A. N >= 0.
66: *> \endverbatim
67: *>
68: *> \param[in,out] A
69: *> \verbatim
70: *> A is DOUBLE PRECISION array, dimension (LDA,N)
71: *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
72: *> n-by-n upper triangular part of A contains the upper
73: *> triangular part of the matrix A, and the strictly lower
74: *> triangular part of A is not referenced. If UPLO = 'L', the
75: *> leading n-by-n lower triangular part of A contains the lower
76: *> triangular part of the matrix A, and the strictly upper
77: *> triangular part of A is not referenced.
78: *>
79: *> On exit, the block diagonal matrix D and the multipliers used
80: *> to obtain the factor U or L (see below for further details).
81: *> \endverbatim
82: *>
83: *> \param[in] LDA
84: *> \verbatim
85: *> LDA is INTEGER
86: *> The leading dimension of the array A. LDA >= max(1,N).
87: *> \endverbatim
88: *>
89: *> \param[out] IPIV
90: *> \verbatim
91: *> IPIV is INTEGER array, dimension (N)
92: *> Details of the interchanges and the block structure of D.
93: *>
94: *> If UPLO = 'U':
95: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
96: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
97: *>
98: *> If IPIV(k) = IPIV(k-1) < 0, then rows and columns
99: *> k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
100: *> is a 2-by-2 diagonal block.
101: *>
102: *> If UPLO = 'L':
103: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
104: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
105: *>
106: *> If IPIV(k) = IPIV(k+1) < 0, then rows and columns
107: *> k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
108: *> is a 2-by-2 diagonal block.
109: *> \endverbatim
110: *>
111: *> \param[out] INFO
112: *> \verbatim
113: *> INFO is INTEGER
114: *> = 0: successful exit
115: *> < 0: if INFO = -k, the k-th argument had an illegal value
116: *> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
117: *> has been completed, but the block diagonal matrix D is
118: *> exactly singular, and division by zero will occur if it
119: *> is used to solve a system of equations.
120: *> \endverbatim
121: *
122: * Authors:
123: * ========
124: *
125: *> \author Univ. of Tennessee
126: *> \author Univ. of California Berkeley
127: *> \author Univ. of Colorado Denver
128: *> \author NAG Ltd.
129: *
130: *> \ingroup doubleSYcomputational
131: *
132: *> \par Further Details:
133: * =====================
134: *>
135: *> \verbatim
136: *>
137: *> If UPLO = 'U', then A = U*D*U**T, where
138: *> U = P(n)*U(n)* ... *P(k)U(k)* ...,
139: *> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
140: *> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
141: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
142: *> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
143: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
144: *>
145: *> ( I v 0 ) k-s
146: *> U(k) = ( 0 I 0 ) s
147: *> ( 0 0 I ) n-k
148: *> k-s s n-k
149: *>
150: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
151: *> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
152: *> and A(k,k), and v overwrites A(1:k-2,k-1:k).
153: *>
154: *> If UPLO = 'L', then A = L*D*L**T, where
155: *> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
156: *> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
157: *> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
158: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
159: *> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
160: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
161: *>
162: *> ( I 0 0 ) k-1
163: *> L(k) = ( 0 I 0 ) s
164: *> ( 0 v I ) n-k-s+1
165: *> k-1 s n-k-s+1
166: *>
167: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
168: *> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
169: *> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
170: *> \endverbatim
171: *
172: *> \par Contributors:
173: * ==================
174: *>
175: *> \verbatim
176: *>
177: *> 09-29-06 - patch from
178: *> Bobby Cheng, MathWorks
179: *>
180: *> Replace l.204 and l.372
181: *> IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
182: *> by
183: *> IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
184: *>
185: *> 01-01-96 - Based on modifications by
186: *> J. Lewis, Boeing Computer Services Company
187: *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
188: *> 1-96 - Based on modifications by J. Lewis, Boeing Computer Services
189: *> Company
190: *> \endverbatim
191: *
192: * =====================================================================
193: SUBROUTINE DSYTF2( UPLO, N, A, LDA, IPIV, INFO )
194: *
195: * -- LAPACK computational routine --
196: * -- LAPACK is a software package provided by Univ. of Tennessee, --
197: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
198: *
199: * .. Scalar Arguments ..
200: CHARACTER UPLO
201: INTEGER INFO, LDA, N
202: * ..
203: * .. Array Arguments ..
204: INTEGER IPIV( * )
205: DOUBLE PRECISION A( LDA, * )
206: * ..
207: *
208: * =====================================================================
209: *
210: * .. Parameters ..
211: DOUBLE PRECISION ZERO, ONE
212: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
213: DOUBLE PRECISION EIGHT, SEVTEN
214: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
215: * ..
216: * .. Local Scalars ..
217: LOGICAL UPPER
218: INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP
219: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
220: $ ROWMAX, T, WK, WKM1, WKP1
221: * ..
222: * .. External Functions ..
223: LOGICAL LSAME, DISNAN
224: INTEGER IDAMAX
225: EXTERNAL LSAME, IDAMAX, DISNAN
226: * ..
227: * .. External Subroutines ..
228: EXTERNAL DSCAL, DSWAP, DSYR, XERBLA
229: * ..
230: * .. Intrinsic Functions ..
231: INTRINSIC ABS, MAX, SQRT
232: * ..
233: * .. Executable Statements ..
234: *
235: * Test the input parameters.
236: *
237: INFO = 0
238: UPPER = LSAME( UPLO, 'U' )
239: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
240: INFO = -1
241: ELSE IF( N.LT.0 ) THEN
242: INFO = -2
243: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
244: INFO = -4
245: END IF
246: IF( INFO.NE.0 ) THEN
247: CALL XERBLA( 'DSYTF2', -INFO )
248: RETURN
249: END IF
250: *
251: * Initialize ALPHA for use in choosing pivot block size.
252: *
253: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
254: *
255: IF( UPPER ) THEN
256: *
257: * Factorize A as U*D*U**T using the upper triangle of A
258: *
259: * K is the main loop index, decreasing from N to 1 in steps of
260: * 1 or 2
261: *
262: K = N
263: 10 CONTINUE
264: *
265: * If K < 1, exit from loop
266: *
267: IF( K.LT.1 )
268: $ GO TO 70
269: KSTEP = 1
270: *
271: * Determine rows and columns to be interchanged and whether
272: * a 1-by-1 or 2-by-2 pivot block will be used
273: *
274: ABSAKK = ABS( A( K, K ) )
275: *
276: * IMAX is the row-index of the largest off-diagonal element in
277: * column K, and COLMAX is its absolute value.
278: * Determine both COLMAX and IMAX.
279: *
280: IF( K.GT.1 ) THEN
281: IMAX = IDAMAX( K-1, A( 1, K ), 1 )
282: COLMAX = ABS( A( IMAX, K ) )
283: ELSE
284: COLMAX = ZERO
285: END IF
286: *
287: IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
288: *
289: * Column K is zero or underflow, or contains a NaN:
290: * set INFO and continue
291: *
292: IF( INFO.EQ.0 )
293: $ INFO = K
294: KP = K
295: ELSE
296: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
297: *
298: * no interchange, use 1-by-1 pivot block
299: *
300: KP = K
301: ELSE
302: *
303: * JMAX is the column-index of the largest off-diagonal
304: * element in row IMAX, and ROWMAX is its absolute value
305: *
306: JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA )
307: ROWMAX = ABS( A( IMAX, JMAX ) )
308: IF( IMAX.GT.1 ) THEN
309: JMAX = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
310: ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
311: END IF
312: *
313: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
314: *
315: * no interchange, use 1-by-1 pivot block
316: *
317: KP = K
318: ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
319: *
320: * interchange rows and columns K and IMAX, use 1-by-1
321: * pivot block
322: *
323: KP = IMAX
324: ELSE
325: *
326: * interchange rows and columns K-1 and IMAX, use 2-by-2
327: * pivot block
328: *
329: KP = IMAX
330: KSTEP = 2
331: END IF
332: END IF
333: *
334: KK = K - KSTEP + 1
335: IF( KP.NE.KK ) THEN
336: *
337: * Interchange rows and columns KK and KP in the leading
338: * submatrix A(1:k,1:k)
339: *
340: CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
341: CALL DSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
342: $ LDA )
343: T = A( KK, KK )
344: A( KK, KK ) = A( KP, KP )
345: A( KP, KP ) = T
346: IF( KSTEP.EQ.2 ) THEN
347: T = A( K-1, K )
348: A( K-1, K ) = A( KP, K )
349: A( KP, K ) = T
350: END IF
351: END IF
352: *
353: * Update the leading submatrix
354: *
355: IF( KSTEP.EQ.1 ) THEN
356: *
357: * 1-by-1 pivot block D(k): column k now holds
358: *
359: * W(k) = U(k)*D(k)
360: *
361: * where U(k) is the k-th column of U
362: *
363: * Perform a rank-1 update of A(1:k-1,1:k-1) as
364: *
365: * A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
366: *
367: R1 = ONE / A( K, K )
368: CALL DSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA )
369: *
370: * Store U(k) in column k
371: *
372: CALL DSCAL( K-1, R1, A( 1, K ), 1 )
373: ELSE
374: *
375: * 2-by-2 pivot block D(k): columns k and k-1 now hold
376: *
377: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
378: *
379: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
380: * of U
381: *
382: * Perform a rank-2 update of A(1:k-2,1:k-2) as
383: *
384: * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
385: * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
386: *
387: IF( K.GT.2 ) THEN
388: *
389: D12 = A( K-1, K )
390: D22 = A( K-1, K-1 ) / D12
391: D11 = A( K, K ) / D12
392: T = ONE / ( D11*D22-ONE )
393: D12 = T / D12
394: *
395: DO 30 J = K - 2, 1, -1
396: WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) )
397: WK = D12*( D22*A( J, K )-A( J, K-1 ) )
398: DO 20 I = J, 1, -1
399: A( I, J ) = A( I, J ) - A( I, K )*WK -
400: $ A( I, K-1 )*WKM1
401: 20 CONTINUE
402: A( J, K ) = WK
403: A( J, K-1 ) = WKM1
404: 30 CONTINUE
405: *
406: END IF
407: *
408: END IF
409: END IF
410: *
411: * Store details of the interchanges in IPIV
412: *
413: IF( KSTEP.EQ.1 ) THEN
414: IPIV( K ) = KP
415: ELSE
416: IPIV( K ) = -KP
417: IPIV( K-1 ) = -KP
418: END IF
419: *
420: * Decrease K and return to the start of the main loop
421: *
422: K = K - KSTEP
423: GO TO 10
424: *
425: ELSE
426: *
427: * Factorize A as L*D*L**T using the lower triangle of A
428: *
429: * K is the main loop index, increasing from 1 to N in steps of
430: * 1 or 2
431: *
432: K = 1
433: 40 CONTINUE
434: *
435: * If K > N, exit from loop
436: *
437: IF( K.GT.N )
438: $ GO TO 70
439: KSTEP = 1
440: *
441: * Determine rows and columns to be interchanged and whether
442: * a 1-by-1 or 2-by-2 pivot block will be used
443: *
444: ABSAKK = ABS( A( K, K ) )
445: *
446: * IMAX is the row-index of the largest off-diagonal element in
447: * column K, and COLMAX is its absolute value.
448: * Determine both COLMAX and IMAX.
449: *
450: IF( K.LT.N ) THEN
451: IMAX = K + IDAMAX( N-K, A( K+1, K ), 1 )
452: COLMAX = ABS( A( IMAX, K ) )
453: ELSE
454: COLMAX = ZERO
455: END IF
456: *
457: IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
458: *
459: * Column K is zero or underflow, or contains a NaN:
460: * set INFO and continue
461: *
462: IF( INFO.EQ.0 )
463: $ INFO = K
464: KP = K
465: ELSE
466: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
467: *
468: * no interchange, use 1-by-1 pivot block
469: *
470: KP = K
471: ELSE
472: *
473: * JMAX is the column-index of the largest off-diagonal
474: * element in row IMAX, and ROWMAX is its absolute value
475: *
476: JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
477: ROWMAX = ABS( A( IMAX, JMAX ) )
478: IF( IMAX.LT.N ) THEN
479: JMAX = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 )
480: ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
481: END IF
482: *
483: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
484: *
485: * no interchange, use 1-by-1 pivot block
486: *
487: KP = K
488: ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
489: *
490: * interchange rows and columns K and IMAX, use 1-by-1
491: * pivot block
492: *
493: KP = IMAX
494: ELSE
495: *
496: * interchange rows and columns K+1 and IMAX, use 2-by-2
497: * pivot block
498: *
499: KP = IMAX
500: KSTEP = 2
501: END IF
502: END IF
503: *
504: KK = K + KSTEP - 1
505: IF( KP.NE.KK ) THEN
506: *
507: * Interchange rows and columns KK and KP in the trailing
508: * submatrix A(k:n,k:n)
509: *
510: IF( KP.LT.N )
511: $ CALL DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
512: CALL DSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
513: $ LDA )
514: T = A( KK, KK )
515: A( KK, KK ) = A( KP, KP )
516: A( KP, KP ) = T
517: IF( KSTEP.EQ.2 ) THEN
518: T = A( K+1, K )
519: A( K+1, K ) = A( KP, K )
520: A( KP, K ) = T
521: END IF
522: END IF
523: *
524: * Update the trailing submatrix
525: *
526: IF( KSTEP.EQ.1 ) THEN
527: *
528: * 1-by-1 pivot block D(k): column k now holds
529: *
530: * W(k) = L(k)*D(k)
531: *
532: * where L(k) is the k-th column of L
533: *
534: IF( K.LT.N ) THEN
535: *
536: * Perform a rank-1 update of A(k+1:n,k+1:n) as
537: *
538: * A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
539: *
540: D11 = ONE / A( K, K )
541: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
542: $ A( K+1, K+1 ), LDA )
543: *
544: * Store L(k) in column K
545: *
546: CALL DSCAL( N-K, D11, A( K+1, K ), 1 )
547: END IF
548: ELSE
549: *
550: * 2-by-2 pivot block D(k)
551: *
552: IF( K.LT.N-1 ) THEN
553: *
554: * Perform a rank-2 update of A(k+2:n,k+2:n) as
555: *
556: * A := A - ( (A(k) A(k+1))*D(k)**(-1) ) * (A(k) A(k+1))**T
557: *
558: * where L(k) and L(k+1) are the k-th and (k+1)-th
559: * columns of L
560: *
561: D21 = A( K+1, K )
562: D11 = A( K+1, K+1 ) / D21
563: D22 = A( K, K ) / D21
564: T = ONE / ( D11*D22-ONE )
565: D21 = T / D21
566: *
567: DO 60 J = K + 2, N
568: *
569: WK = D21*( D11*A( J, K )-A( J, K+1 ) )
570: WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) )
571: *
572: DO 50 I = J, N
573: A( I, J ) = A( I, J ) - A( I, K )*WK -
574: $ A( I, K+1 )*WKP1
575: 50 CONTINUE
576: *
577: A( J, K ) = WK
578: A( J, K+1 ) = WKP1
579: *
580: 60 CONTINUE
581: END IF
582: END IF
583: END IF
584: *
585: * Store details of the interchanges in IPIV
586: *
587: IF( KSTEP.EQ.1 ) THEN
588: IPIV( K ) = KP
589: ELSE
590: IPIV( K ) = -KP
591: IPIV( K+1 ) = -KP
592: END IF
593: *
594: * Increase K and return to the start of the main loop
595: *
596: K = K + KSTEP
597: GO TO 40
598: *
599: END IF
600: *
601: 70 CONTINUE
602: *
603: RETURN
604: *
605: * End of DSYTF2
606: *
607: END
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