1: *> \brief \b DSYTF2
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: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
94: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
95: *> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
96: *> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
97: *> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
98: *> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
99: *> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
100: *> \endverbatim
101: *>
102: *> \param[out] INFO
103: *> \verbatim
104: *> INFO is INTEGER
105: *> = 0: successful exit
106: *> < 0: if INFO = -k, the k-th argument had an illegal value
107: *> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
108: *> has been completed, but the block diagonal matrix D is
109: *> exactly singular, and division by zero will occur if it
110: *> is used to solve a system of equations.
111: *> \endverbatim
112: *
113: * Authors:
114: * ========
115: *
116: *> \author Univ. of Tennessee
117: *> \author Univ. of California Berkeley
118: *> \author Univ. of Colorado Denver
119: *> \author NAG Ltd.
120: *
121: *> \date November 2011
122: *
123: *> \ingroup doubleSYcomputational
124: *
125: *> \par Further Details:
126: * =====================
127: *>
128: *> \verbatim
129: *>
130: *> If UPLO = 'U', then A = U*D*U**T, where
131: *> U = P(n)*U(n)* ... *P(k)U(k)* ...,
132: *> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
133: *> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
134: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
135: *> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
136: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
137: *>
138: *> ( I v 0 ) k-s
139: *> U(k) = ( 0 I 0 ) s
140: *> ( 0 0 I ) n-k
141: *> k-s s n-k
142: *>
143: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
144: *> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
145: *> and A(k,k), and v overwrites A(1:k-2,k-1:k).
146: *>
147: *> If UPLO = 'L', then A = L*D*L**T, where
148: *> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
149: *> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
150: *> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
151: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
152: *> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
153: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
154: *>
155: *> ( I 0 0 ) k-1
156: *> L(k) = ( 0 I 0 ) s
157: *> ( 0 v I ) n-k-s+1
158: *> k-1 s n-k-s+1
159: *>
160: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
161: *> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
162: *> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
163: *> \endverbatim
164: *
165: *> \par Contributors:
166: * ==================
167: *>
168: *> \verbatim
169: *>
170: *> 09-29-06 - patch from
171: *> Bobby Cheng, MathWorks
172: *>
173: *> Replace l.204 and l.372
174: *> IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
175: *> by
176: *> IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
177: *>
178: *> 01-01-96 - Based on modifications by
179: *> J. Lewis, Boeing Computer Services Company
180: *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
181: *> 1-96 - Based on modifications by J. Lewis, Boeing Computer Services
182: *> Company
183: *> \endverbatim
184: *
185: * =====================================================================
186: SUBROUTINE DSYTF2( UPLO, N, A, LDA, IPIV, INFO )
187: *
188: * -- LAPACK computational routine (version 3.4.0) --
189: * -- LAPACK is a software package provided by Univ. of Tennessee, --
190: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
191: * November 2011
192: *
193: * .. Scalar Arguments ..
194: CHARACTER UPLO
195: INTEGER INFO, LDA, N
196: * ..
197: * .. Array Arguments ..
198: INTEGER IPIV( * )
199: DOUBLE PRECISION A( LDA, * )
200: * ..
201: *
202: * =====================================================================
203: *
204: * .. Parameters ..
205: DOUBLE PRECISION ZERO, ONE
206: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
207: DOUBLE PRECISION EIGHT, SEVTEN
208: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
209: * ..
210: * .. Local Scalars ..
211: LOGICAL UPPER
212: INTEGER I, IMAX, J, JMAX, K, KK, KP, KSTEP
213: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
214: $ ROWMAX, T, WK, WKM1, WKP1
215: * ..
216: * .. External Functions ..
217: LOGICAL LSAME, DISNAN
218: INTEGER IDAMAX
219: EXTERNAL LSAME, IDAMAX, DISNAN
220: * ..
221: * .. External Subroutines ..
222: EXTERNAL DSCAL, DSWAP, DSYR, XERBLA
223: * ..
224: * .. Intrinsic Functions ..
225: INTRINSIC ABS, MAX, SQRT
226: * ..
227: * .. Executable Statements ..
228: *
229: * Test the input parameters.
230: *
231: INFO = 0
232: UPPER = LSAME( UPLO, 'U' )
233: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
234: INFO = -1
235: ELSE IF( N.LT.0 ) THEN
236: INFO = -2
237: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
238: INFO = -4
239: END IF
240: IF( INFO.NE.0 ) THEN
241: CALL XERBLA( 'DSYTF2', -INFO )
242: RETURN
243: END IF
244: *
245: * Initialize ALPHA for use in choosing pivot block size.
246: *
247: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
248: *
249: IF( UPPER ) THEN
250: *
251: * Factorize A as U*D*U**T using the upper triangle of A
252: *
253: * K is the main loop index, decreasing from N to 1 in steps of
254: * 1 or 2
255: *
256: K = N
257: 10 CONTINUE
258: *
259: * If K < 1, exit from loop
260: *
261: IF( K.LT.1 )
262: $ GO TO 70
263: KSTEP = 1
264: *
265: * Determine rows and columns to be interchanged and whether
266: * a 1-by-1 or 2-by-2 pivot block will be used
267: *
268: ABSAKK = ABS( A( K, K ) )
269: *
270: * IMAX is the row-index of the largest off-diagonal element in
271: * column K, and COLMAX is its absolute value
272: *
273: IF( K.GT.1 ) THEN
274: IMAX = IDAMAX( K-1, A( 1, K ), 1 )
275: COLMAX = ABS( A( IMAX, K ) )
276: ELSE
277: COLMAX = ZERO
278: END IF
279: *
280: IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
281: *
282: * Column K is zero or contains a NaN: set INFO and continue
283: *
284: IF( INFO.EQ.0 )
285: $ INFO = K
286: KP = K
287: ELSE
288: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
289: *
290: * no interchange, use 1-by-1 pivot block
291: *
292: KP = K
293: ELSE
294: *
295: * JMAX is the column-index of the largest off-diagonal
296: * element in row IMAX, and ROWMAX is its absolute value
297: *
298: JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA )
299: ROWMAX = ABS( A( IMAX, JMAX ) )
300: IF( IMAX.GT.1 ) THEN
301: JMAX = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
302: ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
303: END IF
304: *
305: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
306: *
307: * no interchange, use 1-by-1 pivot block
308: *
309: KP = K
310: ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
311: *
312: * interchange rows and columns K and IMAX, use 1-by-1
313: * pivot block
314: *
315: KP = IMAX
316: ELSE
317: *
318: * interchange rows and columns K-1 and IMAX, use 2-by-2
319: * pivot block
320: *
321: KP = IMAX
322: KSTEP = 2
323: END IF
324: END IF
325: *
326: KK = K - KSTEP + 1
327: IF( KP.NE.KK ) THEN
328: *
329: * Interchange rows and columns KK and KP in the leading
330: * submatrix A(1:k,1:k)
331: *
332: CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
333: CALL DSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
334: $ LDA )
335: T = A( KK, KK )
336: A( KK, KK ) = A( KP, KP )
337: A( KP, KP ) = T
338: IF( KSTEP.EQ.2 ) THEN
339: T = A( K-1, K )
340: A( K-1, K ) = A( KP, K )
341: A( KP, K ) = T
342: END IF
343: END IF
344: *
345: * Update the leading submatrix
346: *
347: IF( KSTEP.EQ.1 ) THEN
348: *
349: * 1-by-1 pivot block D(k): column k now holds
350: *
351: * W(k) = U(k)*D(k)
352: *
353: * where U(k) is the k-th column of U
354: *
355: * Perform a rank-1 update of A(1:k-1,1:k-1) as
356: *
357: * A := A - U(k)*D(k)*U(k)**T = A - W(k)*1/D(k)*W(k)**T
358: *
359: R1 = ONE / A( K, K )
360: CALL DSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA )
361: *
362: * Store U(k) in column k
363: *
364: CALL DSCAL( K-1, R1, A( 1, K ), 1 )
365: ELSE
366: *
367: * 2-by-2 pivot block D(k): columns k and k-1 now hold
368: *
369: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
370: *
371: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
372: * of U
373: *
374: * Perform a rank-2 update of A(1:k-2,1:k-2) as
375: *
376: * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
377: * = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**T
378: *
379: IF( K.GT.2 ) THEN
380: *
381: D12 = A( K-1, K )
382: D22 = A( K-1, K-1 ) / D12
383: D11 = A( K, K ) / D12
384: T = ONE / ( D11*D22-ONE )
385: D12 = T / D12
386: *
387: DO 30 J = K - 2, 1, -1
388: WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) )
389: WK = D12*( D22*A( J, K )-A( J, K-1 ) )
390: DO 20 I = J, 1, -1
391: A( I, J ) = A( I, J ) - A( I, K )*WK -
392: $ A( I, K-1 )*WKM1
393: 20 CONTINUE
394: A( J, K ) = WK
395: A( J, K-1 ) = WKM1
396: 30 CONTINUE
397: *
398: END IF
399: *
400: END IF
401: END IF
402: *
403: * Store details of the interchanges in IPIV
404: *
405: IF( KSTEP.EQ.1 ) THEN
406: IPIV( K ) = KP
407: ELSE
408: IPIV( K ) = -KP
409: IPIV( K-1 ) = -KP
410: END IF
411: *
412: * Decrease K and return to the start of the main loop
413: *
414: K = K - KSTEP
415: GO TO 10
416: *
417: ELSE
418: *
419: * Factorize A as L*D*L**T using the lower triangle of A
420: *
421: * K is the main loop index, increasing from 1 to N in steps of
422: * 1 or 2
423: *
424: K = 1
425: 40 CONTINUE
426: *
427: * If K > N, exit from loop
428: *
429: IF( K.GT.N )
430: $ GO TO 70
431: KSTEP = 1
432: *
433: * Determine rows and columns to be interchanged and whether
434: * a 1-by-1 or 2-by-2 pivot block will be used
435: *
436: ABSAKK = ABS( A( K, K ) )
437: *
438: * IMAX is the row-index of the largest off-diagonal element in
439: * column K, and COLMAX is its absolute value
440: *
441: IF( K.LT.N ) THEN
442: IMAX = K + IDAMAX( N-K, A( K+1, K ), 1 )
443: COLMAX = ABS( A( IMAX, K ) )
444: ELSE
445: COLMAX = ZERO
446: END IF
447: *
448: IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
449: *
450: * Column K is zero or contains a NaN: set INFO and continue
451: *
452: IF( INFO.EQ.0 )
453: $ INFO = K
454: KP = K
455: ELSE
456: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
457: *
458: * no interchange, use 1-by-1 pivot block
459: *
460: KP = K
461: ELSE
462: *
463: * JMAX is the column-index of the largest off-diagonal
464: * element in row IMAX, and ROWMAX is its absolute value
465: *
466: JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
467: ROWMAX = ABS( A( IMAX, JMAX ) )
468: IF( IMAX.LT.N ) THEN
469: JMAX = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 )
470: ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
471: END IF
472: *
473: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
474: *
475: * no interchange, use 1-by-1 pivot block
476: *
477: KP = K
478: ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
479: *
480: * interchange rows and columns K and IMAX, use 1-by-1
481: * pivot block
482: *
483: KP = IMAX
484: ELSE
485: *
486: * interchange rows and columns K+1 and IMAX, use 2-by-2
487: * pivot block
488: *
489: KP = IMAX
490: KSTEP = 2
491: END IF
492: END IF
493: *
494: KK = K + KSTEP - 1
495: IF( KP.NE.KK ) THEN
496: *
497: * Interchange rows and columns KK and KP in the trailing
498: * submatrix A(k:n,k:n)
499: *
500: IF( KP.LT.N )
501: $ CALL DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
502: CALL DSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
503: $ LDA )
504: T = A( KK, KK )
505: A( KK, KK ) = A( KP, KP )
506: A( KP, KP ) = T
507: IF( KSTEP.EQ.2 ) THEN
508: T = A( K+1, K )
509: A( K+1, K ) = A( KP, K )
510: A( KP, K ) = T
511: END IF
512: END IF
513: *
514: * Update the trailing submatrix
515: *
516: IF( KSTEP.EQ.1 ) THEN
517: *
518: * 1-by-1 pivot block D(k): column k now holds
519: *
520: * W(k) = L(k)*D(k)
521: *
522: * where L(k) is the k-th column of L
523: *
524: IF( K.LT.N ) THEN
525: *
526: * Perform a rank-1 update of A(k+1:n,k+1:n) as
527: *
528: * A := A - L(k)*D(k)*L(k)**T = A - W(k)*(1/D(k))*W(k)**T
529: *
530: D11 = ONE / A( K, K )
531: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
532: $ A( K+1, K+1 ), LDA )
533: *
534: * Store L(k) in column K
535: *
536: CALL DSCAL( N-K, D11, A( K+1, K ), 1 )
537: END IF
538: ELSE
539: *
540: * 2-by-2 pivot block D(k)
541: *
542: IF( K.LT.N-1 ) THEN
543: *
544: * Perform a rank-2 update of A(k+2:n,k+2:n) as
545: *
546: * A := A - ( (A(k) A(k+1))*D(k)**(-1) ) * (A(k) A(k+1))**T
547: *
548: * where L(k) and L(k+1) are the k-th and (k+1)-th
549: * columns of L
550: *
551: D21 = A( K+1, K )
552: D11 = A( K+1, K+1 ) / D21
553: D22 = A( K, K ) / D21
554: T = ONE / ( D11*D22-ONE )
555: D21 = T / D21
556: *
557: DO 60 J = K + 2, N
558: *
559: WK = D21*( D11*A( J, K )-A( J, K+1 ) )
560: WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) )
561: *
562: DO 50 I = J, N
563: A( I, J ) = A( I, J ) - A( I, K )*WK -
564: $ A( I, K+1 )*WKP1
565: 50 CONTINUE
566: *
567: A( J, K ) = WK
568: A( J, K+1 ) = WKP1
569: *
570: 60 CONTINUE
571: END IF
572: END IF
573: END IF
574: *
575: * Store details of the interchanges in IPIV
576: *
577: IF( KSTEP.EQ.1 ) THEN
578: IPIV( K ) = KP
579: ELSE
580: IPIV( K ) = -KP
581: IPIV( K+1 ) = -KP
582: END IF
583: *
584: * Increase K and return to the start of the main loop
585: *
586: K = K + KSTEP
587: GO TO 40
588: *
589: END IF
590: *
591: 70 CONTINUE
592: *
593: RETURN
594: *
595: * End of DSYTF2
596: *
597: END
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