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1.1 bertrand 1: *> \brief \b DLASYF_ROOK *> DLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLASYF_ROOK + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasyf_rook.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasyf_rook.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasyf_rook.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLASYF_ROOK( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARADLATER UPLO
25: * INTEGER INFO, KB, LDA, LDW, N, NB
26: * ..
27: * .. Array Arguments ..
28: * INTEGER IPIV( * )
29: * DOUBLE PRECISION A( LDA, * ), W( LDW, * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DLASYF_ROOK computes a partial factorization of a real symmetric
39: *> matrix A using the bounded Bunch-Kaufman ("rook") diagonal
40: *> pivoting method. The partial factorization has the form:
41: *>
42: *> A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
43: *> ( 0 U22 ) ( 0 D ) ( U12**T U22**T )
44: *>
45: *> A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L'
46: *> ( L21 I ) ( 0 A22 ) ( 0 I )
47: *>
48: *> where the order of D is at most NB. The actual order is returned in
49: *> the argument KB, and is either NB or NB-1, or N if N <= NB.
50: *>
51: *> DLASYF_ROOK is an auxiliary routine called by DSYTRF_ROOK. It uses
52: *> blocked code (calling Level 3 BLAS) to update the submatrix
53: *> A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
54: *> \endverbatim
55: *
56: * Arguments:
57: * ==========
58: *
59: *> \param[in] UPLO
60: *> \verbatim
61: *> UPLO is CHARACTER*1
62: *> Specifies whether the upper or lower triangular part of the
63: *> symmetric matrix A is stored:
64: *> = 'U': Upper triangular
65: *> = 'L': Lower triangular
66: *> \endverbatim
67: *>
68: *> \param[in] N
69: *> \verbatim
70: *> N is INTEGER
71: *> The order of the matrix A. N >= 0.
72: *> \endverbatim
73: *>
74: *> \param[in] NB
75: *> \verbatim
76: *> NB is INTEGER
77: *> The maximum number of columns of the matrix A that should be
78: *> factored. NB should be at least 2 to allow for 2-by-2 pivot
79: *> blocks.
80: *> \endverbatim
81: *>
82: *> \param[out] KB
83: *> \verbatim
84: *> KB is INTEGER
85: *> The number of columns of A that were actually factored.
86: *> KB is either NB-1 or NB, or N if N <= NB.
87: *> \endverbatim
88: *>
89: *> \param[in,out] A
90: *> \verbatim
91: *> A is DOUBLE PRECISION array, dimension (LDA,N)
92: *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
93: *> n-by-n upper triangular part of A contains the upper
94: *> triangular part of the matrix A, and the strictly lower
95: *> triangular part of A is not referenced. If UPLO = 'L', the
96: *> leading n-by-n lower triangular part of A contains the lower
97: *> triangular part of the matrix A, and the strictly upper
98: *> triangular part of A is not referenced.
99: *> On exit, A contains details of the partial factorization.
100: *> \endverbatim
101: *>
102: *> \param[in] LDA
103: *> \verbatim
104: *> LDA is INTEGER
105: *> The leading dimension of the array A. LDA >= max(1,N).
106: *> \endverbatim
107: *>
108: *> \param[out] IPIV
109: *> \verbatim
110: *> IPIV is INTEGER array, dimension (N)
111: *> Details of the interchanges and the block structure of D.
112: *>
113: *> If UPLO = 'U':
114: *> Only the last KB elements of IPIV are set.
115: *>
116: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
117: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
118: *>
119: *> If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
120: *> columns k and -IPIV(k) were interchanged and rows and
121: *> columns k-1 and -IPIV(k-1) were inerchaged,
122: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
123: *>
124: *> If UPLO = 'L':
125: *> Only the first KB elements of IPIV are set.
126: *>
127: *> If IPIV(k) > 0, then rows and columns k and IPIV(k)
128: *> were interchanged and D(k,k) is a 1-by-1 diagonal block.
129: *>
130: *> If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
131: *> columns k and -IPIV(k) were interchanged and rows and
132: *> columns k+1 and -IPIV(k+1) were inerchaged,
133: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
134: *> \endverbatim
135: *>
136: *> \param[out] W
137: *> \verbatim
138: *> W is DOUBLE PRECISION array, dimension (LDW,NB)
139: *> \endverbatim
140: *>
141: *> \param[in] LDW
142: *> \verbatim
143: *> LDW is INTEGER
144: *> The leading dimension of the array W. LDW >= max(1,N).
145: *> \endverbatim
146: *>
147: *> \param[out] INFO
148: *> \verbatim
149: *> INFO is INTEGER
150: *> = 0: successful exit
151: *> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
152: *> has been completed, but the block diagonal matrix D is
153: *> exactly singular.
154: *> \endverbatim
155: *
156: * Authors:
157: * ========
158: *
159: *> \author Univ. of Tennessee
160: *> \author Univ. of California Berkeley
161: *> \author Univ. of Colorado Denver
162: *> \author NAG Ltd.
163: *
164: *> \date November 2013
165: *
166: *> \ingroup doubleSYcomputational
167: *
168: *> \par Contributors:
169: * ==================
170: *>
171: *> \verbatim
172: *>
173: *> November 2013, Igor Kozachenko,
174: *> Computer Science Division,
175: *> University of California, Berkeley
176: *>
177: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
178: *> School of Mathematics,
179: *> University of Manchester
180: *>
181: *> \endverbatim
182: *
183: * =====================================================================
184: SUBROUTINE DLASYF_ROOK( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW,
185: $ INFO )
186: *
187: * -- LAPACK computational routine (version 3.5.0) --
188: * -- LAPACK is a software package provided by Univ. of Tennessee, --
189: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
190: * November 2013
191: *
192: * .. Scalar Arguments ..
193: CHARACTER UPLO
194: INTEGER INFO, KB, LDA, LDW, N, NB
195: * ..
196: * .. Array Arguments ..
197: INTEGER IPIV( * )
198: DOUBLE PRECISION A( LDA, * ), W( LDW, * )
199: * ..
200: *
201: * =====================================================================
202: *
203: * .. Parameters ..
204: DOUBLE PRECISION ZERO, ONE
205: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
206: DOUBLE PRECISION EIGHT, SEVTEN
207: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
208: * ..
209: * .. Local Scalars ..
210: LOGICAL DONE
211: INTEGER IMAX, ITEMP, J, JB, JJ, JMAX, JP1, JP2, K, KK,
212: $ KW, KKW, KP, KSTEP, P, II
213:
214: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
215: $ DTEMP, R1, ROWMAX, T, SFMIN
216: * ..
217: * .. External Functions ..
218: LOGICAL LSAME
219: INTEGER IDAMAX
220: DOUBLE PRECISION DLAMCH
221: EXTERNAL LSAME, IDAMAX, DLAMCH
222: * ..
223: * .. External Subroutines ..
224: EXTERNAL DCOPY, DGEMM, DGEMV, DSCAL, DSWAP
225: * ..
226: * .. Intrinsic Functions ..
227: INTRINSIC ABS, MAX, MIN, SQRT
228: * ..
229: * .. Executable Statements ..
230: *
231: INFO = 0
232: *
233: * Initialize ALPHA for use in choosing pivot block size.
234: *
235: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
236: *
237: * Compute machine safe minimum
238: *
239: SFMIN = DLAMCH( 'S' )
240: *
241: IF( LSAME( UPLO, 'U' ) ) THEN
242: *
243: * Factorize the trailing columns of A using the upper triangle
244: * of A and working backwards, and compute the matrix W = U12*D
245: * for use in updating A11
246: *
247: * K is the main loop index, decreasing from N in steps of 1 or 2
248: *
249: K = N
250: 10 CONTINUE
251: *
252: * KW is the column of W which corresponds to column K of A
253: *
254: KW = NB + K - N
255: *
256: * Exit from loop
257: *
258: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
259: $ GO TO 30
260: *
261: KSTEP = 1
262: P = K
263: *
264: * Copy column K of A to column KW of W and update it
265: *
266: CALL DCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
267: IF( K.LT.N )
268: $ CALL DGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ),
269: $ LDA, W( K, KW+1 ), LDW, ONE, W( 1, KW ), 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( W( K, KW ) )
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, W( 1, KW ), 1 )
282: COLMAX = ABS( W( IMAX, KW ) )
283: ELSE
284: COLMAX = ZERO
285: END IF
286: *
287: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
288: *
289: * Column K is zero or underflow: set INFO and continue
290: *
291: IF( INFO.EQ.0 )
292: $ INFO = K
293: KP = K
294: CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
295: ELSE
296: *
297: * ============================================================
298: *
299: * Test for interchange
300: *
301: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
302: * (used to handle NaN and Inf)
303: *
304: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
305: *
306: * no interchange, use 1-by-1 pivot block
307: *
308: KP = K
309: *
310: ELSE
311: *
312: DONE = .FALSE.
313: *
314: * Loop until pivot found
315: *
316: 12 CONTINUE
317: *
318: * Begin pivot search loop body
319: *
320: *
321: * Copy column IMAX to column KW-1 of W and update it
322: *
323: CALL DCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
324: CALL DCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
325: $ W( IMAX+1, KW-1 ), 1 )
326: *
327: IF( K.LT.N )
328: $ CALL DGEMV( 'No transpose', K, N-K, -ONE,
329: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
330: $ ONE, W( 1, KW-1 ), 1 )
331: *
332: * JMAX is the column-index of the largest off-diagonal
333: * element in row IMAX, and ROWMAX is its absolute value.
334: * Determine both ROWMAX and JMAX.
335: *
336: IF( IMAX.NE.K ) THEN
337: JMAX = IMAX + IDAMAX( K-IMAX, W( IMAX+1, KW-1 ),
338: $ 1 )
339: ROWMAX = ABS( W( JMAX, KW-1 ) )
340: ELSE
341: ROWMAX = ZERO
342: END IF
343: *
344: IF( IMAX.GT.1 ) THEN
345: ITEMP = IDAMAX( IMAX-1, W( 1, KW-1 ), 1 )
346: DTEMP = ABS( W( ITEMP, KW-1 ) )
347: IF( DTEMP.GT.ROWMAX ) THEN
348: ROWMAX = DTEMP
349: JMAX = ITEMP
350: END IF
351: END IF
352: *
353: * Equivalent to testing for
354: * ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX
355: * (used to handle NaN and Inf)
356: *
357: IF( .NOT.(ABS( W( IMAX, KW-1 ) ).LT.ALPHA*ROWMAX ) )
358: $ THEN
359: *
360: * interchange rows and columns K and IMAX,
361: * use 1-by-1 pivot block
362: *
363: KP = IMAX
364: *
365: * copy column KW-1 of W to column KW of W
366: *
367: CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
368: *
369: DONE = .TRUE.
370: *
371: * Equivalent to testing for ROWMAX.EQ.COLMAX,
372: * (used to handle NaN and Inf)
373: *
374: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
375: $ THEN
376: *
377: * interchange rows and columns K-1 and IMAX,
378: * use 2-by-2 pivot block
379: *
380: KP = IMAX
381: KSTEP = 2
382: DONE = .TRUE.
383: ELSE
384: *
385: * Pivot not found: set params and repeat
386: *
387: P = IMAX
388: COLMAX = ROWMAX
389: IMAX = JMAX
390: *
391: * Copy updated JMAXth (next IMAXth) column to Kth of W
392: *
393: CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
394: *
395: END IF
396: *
397: * End pivot search loop body
398: *
399: IF( .NOT. DONE ) GOTO 12
400: *
401: END IF
402: *
403: * ============================================================
404: *
405: KK = K - KSTEP + 1
406: *
407: * KKW is the column of W which corresponds to column KK of A
408: *
409: KKW = NB + KK - N
410: *
411: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
412: *
413: * Copy non-updated column K to column P
414: *
415: CALL DCOPY( K-P, A( P+1, K ), 1, A( P, P+1 ), LDA )
416: CALL DCOPY( P, A( 1, K ), 1, A( 1, P ), 1 )
417: *
418: * Interchange rows K and P in last N-K+1 columns of A
419: * and last N-K+2 columns of W
420: *
421: CALL DSWAP( N-K+1, A( K, K ), LDA, A( P, K ), LDA )
422: CALL DSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), LDW )
423: END IF
424: *
425: * Updated column KP is already stored in column KKW of W
426: *
427: IF( KP.NE.KK ) THEN
428: *
429: * Copy non-updated column KK to column KP
430: *
431: A( KP, K ) = A( KK, K )
432: CALL DCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
433: $ LDA )
434: CALL DCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
435: *
436: * Interchange rows KK and KP in last N-KK+1 columns
437: * of A and W
438: *
439: CALL DSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
440: CALL DSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
441: $ LDW )
442: END IF
443: *
444: IF( KSTEP.EQ.1 ) THEN
445: *
446: * 1-by-1 pivot block D(k): column KW of W now holds
447: *
448: * W(k) = U(k)*D(k)
449: *
450: * where U(k) is the k-th column of U
451: *
452: * Store U(k) in column k of A
453: *
454: CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
455: IF( K.GT.1 ) THEN
456: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
457: R1 = ONE / A( K, K )
458: CALL DSCAL( K-1, R1, A( 1, K ), 1 )
459: ELSE IF( A( K, K ).NE.ZERO ) THEN
460: DO 14 II = 1, K - 1
461: A( II, K ) = A( II, K ) / A( K, K )
462: 14 CONTINUE
463: END IF
464: END IF
465: *
466: ELSE
467: *
468: * 2-by-2 pivot block D(k): columns KW and KW-1 of W now
469: * hold
470: *
471: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
472: *
473: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
474: * of U
475: *
476: IF( K.GT.2 ) THEN
477: *
478: * Store U(k) and U(k-1) in columns k and k-1 of A
479: *
480: D12 = W( K-1, KW )
481: D11 = W( K, KW ) / D12
482: D22 = W( K-1, KW-1 ) / D12
483: T = ONE / ( D11*D22-ONE )
484: DO 20 J = 1, K - 2
485: A( J, K-1 ) = T*( (D11*W( J, KW-1 )-W( J, KW ) ) /
486: $ D12 )
487: A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
488: $ D12 )
489: 20 CONTINUE
490: END IF
491: *
492: * Copy D(k) to A
493: *
494: A( K-1, K-1 ) = W( K-1, KW-1 )
495: A( K-1, K ) = W( K-1, KW )
496: A( K, K ) = W( K, KW )
497: END IF
498: END IF
499: *
500: * Store details of the interchanges in IPIV
501: *
502: IF( KSTEP.EQ.1 ) THEN
503: IPIV( K ) = KP
504: ELSE
505: IPIV( K ) = -P
506: IPIV( K-1 ) = -KP
507: END IF
508: *
509: * Decrease K and return to the start of the main loop
510: *
511: K = K - KSTEP
512: GO TO 10
513: *
514: 30 CONTINUE
515: *
516: * Update the upper triangle of A11 (= A(1:k,1:k)) as
517: *
518: * A11 := A11 - U12*D*U12**T = A11 - U12*W**T
519: *
520: * computing blocks of NB columns at a time
521: *
522: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
523: JB = MIN( NB, K-J+1 )
524: *
525: * Update the upper triangle of the diagonal block
526: *
527: DO 40 JJ = J, J + JB - 1
528: CALL DGEMV( 'No transpose', JJ-J+1, N-K, -ONE,
529: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, ONE,
530: $ A( J, JJ ), 1 )
531: 40 CONTINUE
532: *
533: * Update the rectangular superdiagonal block
534: *
535: IF( J.GE.2 )
536: $ CALL DGEMM( 'No transpose', 'Transpose', J-1, JB,
537: $ N-K, -ONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
538: $ ONE, A( 1, J ), LDA )
539: 50 CONTINUE
540: *
541: * Put U12 in standard form by partially undoing the interchanges
542: * in columns k+1:n
543: *
544: J = K + 1
545: 60 CONTINUE
546: *
547: KSTEP = 1
548: JP1 = 1
549: JJ = J
550: JP2 = IPIV( J )
551: IF( JP2.LT.0 ) THEN
552: JP2 = -JP2
553: J = J + 1
554: JP1 = -IPIV( J )
555: KSTEP = 2
556: END IF
557: *
558: J = J + 1
559: IF( JP2.NE.JJ .AND. J.LE.N )
560: $ CALL DSWAP( N-J+1, A( JP2, J ), LDA, A( JJ, J ), LDA )
561: JJ = J - 1
562: IF( JP1.NE.JJ .AND. KSTEP.EQ.2 )
563: $ CALL DSWAP( N-J+1, A( JP1, J ), LDA, A( JJ, J ), LDA )
564: IF( J.LE.N )
565: $ GO TO 60
566: *
567: * Set KB to the number of columns factorized
568: *
569: KB = N - K
570: *
571: ELSE
572: *
573: * Factorize the leading columns of A using the lower triangle
574: * of A and working forwards, and compute the matrix W = L21*D
575: * for use in updating A22
576: *
577: * K is the main loop index, increasing from 1 in steps of 1 or 2
578: *
579: K = 1
580: 70 CONTINUE
581: *
582: * Exit from loop
583: *
584: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
585: $ GO TO 90
586: *
587: KSTEP = 1
588: P = K
589: *
590: * Copy column K of A to column K of W and update it
591: *
592: CALL DCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
593: IF( K.GT.1 )
594: $ CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ),
595: $ LDA, W( K, 1 ), LDW, ONE, W( K, K ), 1 )
596: *
597: * Determine rows and columns to be interchanged and whether
598: * a 1-by-1 or 2-by-2 pivot block will be used
599: *
600: ABSAKK = ABS( W( K, K ) )
601: *
602: * IMAX is the row-index of the largest off-diagonal element in
603: * column K, and COLMAX is its absolute value.
604: * Determine both COLMAX and IMAX.
605: *
606: IF( K.LT.N ) THEN
607: IMAX = K + IDAMAX( N-K, W( K+1, K ), 1 )
608: COLMAX = ABS( W( IMAX, K ) )
609: ELSE
610: COLMAX = ZERO
611: END IF
612: *
613: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
614: *
615: * Column K is zero or underflow: set INFO and continue
616: *
617: IF( INFO.EQ.0 )
618: $ INFO = K
619: KP = K
620: CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
621: ELSE
622: *
623: * ============================================================
624: *
625: * Test for interchange
626: *
627: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
628: * (used to handle NaN and Inf)
629: *
630: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
631: *
632: * no interchange, use 1-by-1 pivot block
633: *
634: KP = K
635: *
636: ELSE
637: *
638: DONE = .FALSE.
639: *
640: * Loop until pivot found
641: *
642: 72 CONTINUE
643: *
644: * Begin pivot search loop body
645: *
646: *
647: * Copy column IMAX to column K+1 of W and update it
648: *
649: CALL DCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
650: CALL DCOPY( N-IMAX+1, A( IMAX, IMAX ), 1,
651: $ W( IMAX, K+1 ), 1 )
652: IF( K.GT.1 )
653: $ CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE,
654: $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
655: $ ONE, W( K, K+1 ), 1 )
656: *
657: * JMAX is the column-index of the largest off-diagonal
658: * element in row IMAX, and ROWMAX is its absolute value.
659: * Determine both ROWMAX and JMAX.
660: *
661: IF( IMAX.NE.K ) THEN
662: JMAX = K - 1 + IDAMAX( IMAX-K, W( K, K+1 ), 1 )
663: ROWMAX = ABS( W( JMAX, K+1 ) )
664: ELSE
665: ROWMAX = ZERO
666: END IF
667: *
668: IF( IMAX.LT.N ) THEN
669: ITEMP = IMAX + IDAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
670: DTEMP = ABS( W( ITEMP, K+1 ) )
671: IF( DTEMP.GT.ROWMAX ) THEN
672: ROWMAX = DTEMP
673: JMAX = ITEMP
674: END IF
675: END IF
676: *
677: * Equivalent to testing for
678: * ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX
679: * (used to handle NaN and Inf)
680: *
681: IF( .NOT.( ABS( W( IMAX, K+1 ) ).LT.ALPHA*ROWMAX ) )
682: $ THEN
683: *
684: * interchange rows and columns K and IMAX,
685: * use 1-by-1 pivot block
686: *
687: KP = IMAX
688: *
689: * copy column K+1 of W to column K of W
690: *
691: CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
692: *
693: DONE = .TRUE.
694: *
695: * Equivalent to testing for ROWMAX.EQ.COLMAX,
696: * (used to handle NaN and Inf)
697: *
698: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
699: $ THEN
700: *
701: * interchange rows and columns K+1 and IMAX,
702: * use 2-by-2 pivot block
703: *
704: KP = IMAX
705: KSTEP = 2
706: DONE = .TRUE.
707: ELSE
708: *
709: * Pivot not found: set params and repeat
710: *
711: P = IMAX
712: COLMAX = ROWMAX
713: IMAX = JMAX
714: *
715: * Copy updated JMAXth (next IMAXth) column to Kth of W
716: *
717: CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
718: *
719: END IF
720: *
721: * End pivot search loop body
722: *
723: IF( .NOT. DONE ) GOTO 72
724: *
725: END IF
726: *
727: * ============================================================
728: *
729: KK = K + KSTEP - 1
730: *
731: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
732: *
733: * Copy non-updated column K to column P
734: *
735: CALL DCOPY( P-K, A( K, K ), 1, A( P, K ), LDA )
736: CALL DCOPY( N-P+1, A( P, K ), 1, A( P, P ), 1 )
737: *
738: * Interchange rows K and P in first K columns of A
739: * and first K+1 columns of W
740: *
741: CALL DSWAP( K, A( K, 1 ), LDA, A( P, 1 ), LDA )
742: CALL DSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
743: END IF
744: *
745: * Updated column KP is already stored in column KK of W
746: *
747: IF( KP.NE.KK ) THEN
748: *
749: * Copy non-updated column KK to column KP
750: *
751: A( KP, K ) = A( KK, K )
752: CALL DCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
753: CALL DCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
754: *
755: * Interchange rows KK and KP in first KK columns of A and W
756: *
757: CALL DSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
758: CALL DSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
759: END IF
760: *
761: IF( KSTEP.EQ.1 ) THEN
762: *
763: * 1-by-1 pivot block D(k): column k of W now holds
764: *
765: * W(k) = L(k)*D(k)
766: *
767: * where L(k) is the k-th column of L
768: *
769: * Store L(k) in column k of A
770: *
771: CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
772: IF( K.LT.N ) THEN
773: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
774: R1 = ONE / A( K, K )
775: CALL DSCAL( N-K, R1, A( K+1, K ), 1 )
776: ELSE IF( A( K, K ).NE.ZERO ) THEN
777: DO 74 II = K + 1, N
778: A( II, K ) = A( II, K ) / A( K, K )
779: 74 CONTINUE
780: END IF
781: END IF
782: *
783: ELSE
784: *
785: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
786: *
787: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
788: *
789: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
790: * of L
791: *
792: IF( K.LT.N-1 ) THEN
793: *
794: * Store L(k) and L(k+1) in columns k and k+1 of A
795: *
796: D21 = W( K+1, K )
797: D11 = W( K+1, K+1 ) / D21
798: D22 = W( K, K ) / D21
799: T = ONE / ( D11*D22-ONE )
800: DO 80 J = K + 2, N
801: A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
802: $ D21 )
803: A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
804: $ D21 )
805: 80 CONTINUE
806: END IF
807: *
808: * Copy D(k) to A
809: *
810: A( K, K ) = W( K, K )
811: A( K+1, K ) = W( K+1, K )
812: A( K+1, K+1 ) = W( K+1, K+1 )
813: END IF
814: END IF
815: *
816: * Store details of the interchanges in IPIV
817: *
818: IF( KSTEP.EQ.1 ) THEN
819: IPIV( K ) = KP
820: ELSE
821: IPIV( K ) = -P
822: IPIV( K+1 ) = -KP
823: END IF
824: *
825: * Increase K and return to the start of the main loop
826: *
827: K = K + KSTEP
828: GO TO 70
829: *
830: 90 CONTINUE
831: *
832: * Update the lower triangle of A22 (= A(k:n,k:n)) as
833: *
834: * A22 := A22 - L21*D*L21**T = A22 - L21*W**T
835: *
836: * computing blocks of NB columns at a time
837: *
838: DO 110 J = K, N, NB
839: JB = MIN( NB, N-J+1 )
840: *
841: * Update the lower triangle of the diagonal block
842: *
843: DO 100 JJ = J, J + JB - 1
844: CALL DGEMV( 'No transpose', J+JB-JJ, K-1, -ONE,
845: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, ONE,
846: $ A( JJ, JJ ), 1 )
847: 100 CONTINUE
848: *
849: * Update the rectangular subdiagonal block
850: *
851: IF( J+JB.LE.N )
852: $ CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
853: $ K-1, -ONE, A( J+JB, 1 ), LDA, W( J, 1 ), LDW,
854: $ ONE, A( J+JB, J ), LDA )
855: 110 CONTINUE
856: *
857: * Put L21 in standard form by partially undoing the interchanges
858: * in columns 1:k-1
859: *
860: J = K - 1
861: 120 CONTINUE
862: *
863: KSTEP = 1
864: JP1 = 1
865: JJ = J
866: JP2 = IPIV( J )
867: IF( JP2.LT.0 ) THEN
868: JP2 = -JP2
869: J = J - 1
870: JP1 = -IPIV( J )
871: KSTEP = 2
872: END IF
873: *
874: J = J - 1
875: IF( JP2.NE.JJ .AND. J.GE.1 )
876: $ CALL DSWAP( J, A( JP2, 1 ), LDA, A( JJ, 1 ), LDA )
877: JJ = J + 1
878: IF( JP1.NE.JJ .AND. KSTEP.EQ.2 )
879: $ CALL DSWAP( J, A( JP1, 1 ), LDA, A( JJ, 1 ), LDA )
880: IF( J.GE.1 )
881: $ GO TO 120
882: *
883: * Set KB to the number of columns factorized
884: *
885: KB = K - 1
886: *
887: END IF
888: RETURN
889: *
890: * End of DLASYF_ROOK
891: *
892: END