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: * CHARACTER 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: *> \ingroup doubleSYcomputational
165: *
166: *> \par Contributors:
167: * ==================
168: *>
169: *> \verbatim
170: *>
171: *> November 2013, Igor Kozachenko,
172: *> Computer Science Division,
173: *> University of California, Berkeley
174: *>
175: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
176: *> School of Mathematics,
177: *> University of Manchester
178: *>
179: *> \endverbatim
180: *
181: * =====================================================================
182: SUBROUTINE DLASYF_ROOK( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW,
183: $ INFO )
184: *
185: * -- LAPACK computational routine --
186: * -- LAPACK is a software package provided by Univ. of Tennessee, --
187: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
188: *
189: * .. Scalar Arguments ..
190: CHARACTER UPLO
191: INTEGER INFO, KB, LDA, LDW, N, NB
192: * ..
193: * .. Array Arguments ..
194: INTEGER IPIV( * )
195: DOUBLE PRECISION A( LDA, * ), W( LDW, * )
196: * ..
197: *
198: * =====================================================================
199: *
200: * .. Parameters ..
201: DOUBLE PRECISION ZERO, ONE
202: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
203: DOUBLE PRECISION EIGHT, SEVTEN
204: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
205: * ..
206: * .. Local Scalars ..
207: LOGICAL DONE
208: INTEGER IMAX, ITEMP, J, JB, JJ, JMAX, JP1, JP2, K, KK,
209: $ KW, KKW, KP, KSTEP, P, II
210:
211: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
212: $ DTEMP, R1, ROWMAX, T, SFMIN
213: * ..
214: * .. External Functions ..
215: LOGICAL LSAME
216: INTEGER IDAMAX
217: DOUBLE PRECISION DLAMCH
218: EXTERNAL LSAME, IDAMAX, DLAMCH
219: * ..
220: * .. External Subroutines ..
221: EXTERNAL DCOPY, DGEMM, DGEMV, DSCAL, DSWAP
222: * ..
223: * .. Intrinsic Functions ..
224: INTRINSIC ABS, MAX, MIN, SQRT
225: * ..
226: * .. Executable Statements ..
227: *
228: INFO = 0
229: *
230: * Initialize ALPHA for use in choosing pivot block size.
231: *
232: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
233: *
234: * Compute machine safe minimum
235: *
236: SFMIN = DLAMCH( 'S' )
237: *
238: IF( LSAME( UPLO, 'U' ) ) THEN
239: *
240: * Factorize the trailing columns of A using the upper triangle
241: * of A and working backwards, and compute the matrix W = U12*D
242: * for use in updating A11
243: *
244: * K is the main loop index, decreasing from N in steps of 1 or 2
245: *
246: K = N
247: 10 CONTINUE
248: *
249: * KW is the column of W which corresponds to column K of A
250: *
251: KW = NB + K - N
252: *
253: * Exit from loop
254: *
255: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
256: $ GO TO 30
257: *
258: KSTEP = 1
259: P = K
260: *
261: * Copy column K of A to column KW of W and update it
262: *
263: CALL DCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
264: IF( K.LT.N )
265: $ CALL DGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ),
266: $ LDA, W( K, KW+1 ), LDW, ONE, W( 1, KW ), 1 )
267: *
268: * Determine rows and columns to be interchanged and whether
269: * a 1-by-1 or 2-by-2 pivot block will be used
270: *
271: ABSAKK = ABS( W( K, KW ) )
272: *
273: * IMAX is the row-index of the largest off-diagonal element in
274: * column K, and COLMAX is its absolute value.
275: * Determine both COLMAX and IMAX.
276: *
277: IF( K.GT.1 ) THEN
278: IMAX = IDAMAX( K-1, W( 1, KW ), 1 )
279: COLMAX = ABS( W( IMAX, KW ) )
280: ELSE
281: COLMAX = ZERO
282: END IF
283: *
284: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
285: *
286: * Column K is zero or underflow: set INFO and continue
287: *
288: IF( INFO.EQ.0 )
289: $ INFO = K
290: KP = K
291: CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
292: ELSE
293: *
294: * ============================================================
295: *
296: * Test for interchange
297: *
298: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
299: * (used to handle NaN and Inf)
300: *
301: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
302: *
303: * no interchange, use 1-by-1 pivot block
304: *
305: KP = K
306: *
307: ELSE
308: *
309: DONE = .FALSE.
310: *
311: * Loop until pivot found
312: *
313: 12 CONTINUE
314: *
315: * Begin pivot search loop body
316: *
317: *
318: * Copy column IMAX to column KW-1 of W and update it
319: *
320: CALL DCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
321: CALL DCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
322: $ W( IMAX+1, KW-1 ), 1 )
323: *
324: IF( K.LT.N )
325: $ CALL DGEMV( 'No transpose', K, N-K, -ONE,
326: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
327: $ ONE, W( 1, KW-1 ), 1 )
328: *
329: * JMAX is the column-index of the largest off-diagonal
330: * element in row IMAX, and ROWMAX is its absolute value.
331: * Determine both ROWMAX and JMAX.
332: *
333: IF( IMAX.NE.K ) THEN
334: JMAX = IMAX + IDAMAX( K-IMAX, W( IMAX+1, KW-1 ),
335: $ 1 )
336: ROWMAX = ABS( W( JMAX, KW-1 ) )
337: ELSE
338: ROWMAX = ZERO
339: END IF
340: *
341: IF( IMAX.GT.1 ) THEN
342: ITEMP = IDAMAX( IMAX-1, W( 1, KW-1 ), 1 )
343: DTEMP = ABS( W( ITEMP, KW-1 ) )
344: IF( DTEMP.GT.ROWMAX ) THEN
345: ROWMAX = DTEMP
346: JMAX = ITEMP
347: END IF
348: END IF
349: *
350: * Equivalent to testing for
351: * ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX
352: * (used to handle NaN and Inf)
353: *
354: IF( .NOT.(ABS( W( IMAX, KW-1 ) ).LT.ALPHA*ROWMAX ) )
355: $ THEN
356: *
357: * interchange rows and columns K and IMAX,
358: * use 1-by-1 pivot block
359: *
360: KP = IMAX
361: *
362: * copy column KW-1 of W to column KW of W
363: *
364: CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
365: *
366: DONE = .TRUE.
367: *
368: * Equivalent to testing for ROWMAX.EQ.COLMAX,
369: * (used to handle NaN and Inf)
370: *
371: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
372: $ THEN
373: *
374: * interchange rows and columns K-1 and IMAX,
375: * use 2-by-2 pivot block
376: *
377: KP = IMAX
378: KSTEP = 2
379: DONE = .TRUE.
380: ELSE
381: *
382: * Pivot not found: set params and repeat
383: *
384: P = IMAX
385: COLMAX = ROWMAX
386: IMAX = JMAX
387: *
388: * Copy updated JMAXth (next IMAXth) column to Kth of W
389: *
390: CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
391: *
392: END IF
393: *
394: * End pivot search loop body
395: *
396: IF( .NOT. DONE ) GOTO 12
397: *
398: END IF
399: *
400: * ============================================================
401: *
402: KK = K - KSTEP + 1
403: *
404: * KKW is the column of W which corresponds to column KK of A
405: *
406: KKW = NB + KK - N
407: *
408: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
409: *
410: * Copy non-updated column K to column P
411: *
412: CALL DCOPY( K-P, A( P+1, K ), 1, A( P, P+1 ), LDA )
413: CALL DCOPY( P, A( 1, K ), 1, A( 1, P ), 1 )
414: *
415: * Interchange rows K and P in last N-K+1 columns of A
416: * and last N-K+2 columns of W
417: *
418: CALL DSWAP( N-K+1, A( K, K ), LDA, A( P, K ), LDA )
419: CALL DSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), LDW )
420: END IF
421: *
422: * Updated column KP is already stored in column KKW of W
423: *
424: IF( KP.NE.KK ) THEN
425: *
426: * Copy non-updated column KK to column KP
427: *
428: A( KP, K ) = A( KK, K )
429: CALL DCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
430: $ LDA )
431: CALL DCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
432: *
433: * Interchange rows KK and KP in last N-KK+1 columns
434: * of A and W
435: *
436: CALL DSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
437: CALL DSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
438: $ LDW )
439: END IF
440: *
441: IF( KSTEP.EQ.1 ) THEN
442: *
443: * 1-by-1 pivot block D(k): column KW of W now holds
444: *
445: * W(k) = U(k)*D(k)
446: *
447: * where U(k) is the k-th column of U
448: *
449: * Store U(k) in column k of A
450: *
451: CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
452: IF( K.GT.1 ) THEN
453: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
454: R1 = ONE / A( K, K )
455: CALL DSCAL( K-1, R1, A( 1, K ), 1 )
456: ELSE IF( A( K, K ).NE.ZERO ) THEN
457: DO 14 II = 1, K - 1
458: A( II, K ) = A( II, K ) / A( K, K )
459: 14 CONTINUE
460: END IF
461: END IF
462: *
463: ELSE
464: *
465: * 2-by-2 pivot block D(k): columns KW and KW-1 of W now
466: * hold
467: *
468: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
469: *
470: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
471: * of U
472: *
473: IF( K.GT.2 ) THEN
474: *
475: * Store U(k) and U(k-1) in columns k and k-1 of A
476: *
477: D12 = W( K-1, KW )
478: D11 = W( K, KW ) / D12
479: D22 = W( K-1, KW-1 ) / D12
480: T = ONE / ( D11*D22-ONE )
481: DO 20 J = 1, K - 2
482: A( J, K-1 ) = T*( (D11*W( J, KW-1 )-W( J, KW ) ) /
483: $ D12 )
484: A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
485: $ D12 )
486: 20 CONTINUE
487: END IF
488: *
489: * Copy D(k) to A
490: *
491: A( K-1, K-1 ) = W( K-1, KW-1 )
492: A( K-1, K ) = W( K-1, KW )
493: A( K, K ) = W( K, KW )
494: END IF
495: END IF
496: *
497: * Store details of the interchanges in IPIV
498: *
499: IF( KSTEP.EQ.1 ) THEN
500: IPIV( K ) = KP
501: ELSE
502: IPIV( K ) = -P
503: IPIV( K-1 ) = -KP
504: END IF
505: *
506: * Decrease K and return to the start of the main loop
507: *
508: K = K - KSTEP
509: GO TO 10
510: *
511: 30 CONTINUE
512: *
513: * Update the upper triangle of A11 (= A(1:k,1:k)) as
514: *
515: * A11 := A11 - U12*D*U12**T = A11 - U12*W**T
516: *
517: * computing blocks of NB columns at a time
518: *
519: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
520: JB = MIN( NB, K-J+1 )
521: *
522: * Update the upper triangle of the diagonal block
523: *
524: DO 40 JJ = J, J + JB - 1
525: CALL DGEMV( 'No transpose', JJ-J+1, N-K, -ONE,
526: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, ONE,
527: $ A( J, JJ ), 1 )
528: 40 CONTINUE
529: *
530: * Update the rectangular superdiagonal block
531: *
532: IF( J.GE.2 )
533: $ CALL DGEMM( 'No transpose', 'Transpose', J-1, JB,
534: $ N-K, -ONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
535: $ ONE, A( 1, J ), LDA )
536: 50 CONTINUE
537: *
538: * Put U12 in standard form by partially undoing the interchanges
539: * in columns k+1:n
540: *
541: J = K + 1
542: 60 CONTINUE
543: *
544: KSTEP = 1
545: JP1 = 1
546: JJ = J
547: JP2 = IPIV( J )
548: IF( JP2.LT.0 ) THEN
549: JP2 = -JP2
550: J = J + 1
551: JP1 = -IPIV( J )
552: KSTEP = 2
553: END IF
554: *
555: J = J + 1
556: IF( JP2.NE.JJ .AND. J.LE.N )
557: $ CALL DSWAP( N-J+1, A( JP2, J ), LDA, A( JJ, J ), LDA )
558: JJ = J - 1
559: IF( JP1.NE.JJ .AND. KSTEP.EQ.2 )
560: $ CALL DSWAP( N-J+1, A( JP1, J ), LDA, A( JJ, J ), LDA )
561: IF( J.LE.N )
562: $ GO TO 60
563: *
564: * Set KB to the number of columns factorized
565: *
566: KB = N - K
567: *
568: ELSE
569: *
570: * Factorize the leading columns of A using the lower triangle
571: * of A and working forwards, and compute the matrix W = L21*D
572: * for use in updating A22
573: *
574: * K is the main loop index, increasing from 1 in steps of 1 or 2
575: *
576: K = 1
577: 70 CONTINUE
578: *
579: * Exit from loop
580: *
581: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
582: $ GO TO 90
583: *
584: KSTEP = 1
585: P = K
586: *
587: * Copy column K of A to column K of W and update it
588: *
589: CALL DCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
590: IF( K.GT.1 )
591: $ CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ),
592: $ LDA, W( K, 1 ), LDW, ONE, W( K, K ), 1 )
593: *
594: * Determine rows and columns to be interchanged and whether
595: * a 1-by-1 or 2-by-2 pivot block will be used
596: *
597: ABSAKK = ABS( W( K, K ) )
598: *
599: * IMAX is the row-index of the largest off-diagonal element in
600: * column K, and COLMAX is its absolute value.
601: * Determine both COLMAX and IMAX.
602: *
603: IF( K.LT.N ) THEN
604: IMAX = K + IDAMAX( N-K, W( K+1, K ), 1 )
605: COLMAX = ABS( W( IMAX, K ) )
606: ELSE
607: COLMAX = ZERO
608: END IF
609: *
610: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
611: *
612: * Column K is zero or underflow: set INFO and continue
613: *
614: IF( INFO.EQ.0 )
615: $ INFO = K
616: KP = K
617: CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
618: ELSE
619: *
620: * ============================================================
621: *
622: * Test for interchange
623: *
624: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
625: * (used to handle NaN and Inf)
626: *
627: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
628: *
629: * no interchange, use 1-by-1 pivot block
630: *
631: KP = K
632: *
633: ELSE
634: *
635: DONE = .FALSE.
636: *
637: * Loop until pivot found
638: *
639: 72 CONTINUE
640: *
641: * Begin pivot search loop body
642: *
643: *
644: * Copy column IMAX to column K+1 of W and update it
645: *
646: CALL DCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
647: CALL DCOPY( N-IMAX+1, A( IMAX, IMAX ), 1,
648: $ W( IMAX, K+1 ), 1 )
649: IF( K.GT.1 )
650: $ CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE,
651: $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
652: $ ONE, W( K, K+1 ), 1 )
653: *
654: * JMAX is the column-index of the largest off-diagonal
655: * element in row IMAX, and ROWMAX is its absolute value.
656: * Determine both ROWMAX and JMAX.
657: *
658: IF( IMAX.NE.K ) THEN
659: JMAX = K - 1 + IDAMAX( IMAX-K, W( K, K+1 ), 1 )
660: ROWMAX = ABS( W( JMAX, K+1 ) )
661: ELSE
662: ROWMAX = ZERO
663: END IF
664: *
665: IF( IMAX.LT.N ) THEN
666: ITEMP = IMAX + IDAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
667: DTEMP = ABS( W( ITEMP, K+1 ) )
668: IF( DTEMP.GT.ROWMAX ) THEN
669: ROWMAX = DTEMP
670: JMAX = ITEMP
671: END IF
672: END IF
673: *
674: * Equivalent to testing for
675: * ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX
676: * (used to handle NaN and Inf)
677: *
678: IF( .NOT.( ABS( W( IMAX, K+1 ) ).LT.ALPHA*ROWMAX ) )
679: $ THEN
680: *
681: * interchange rows and columns K and IMAX,
682: * use 1-by-1 pivot block
683: *
684: KP = IMAX
685: *
686: * copy column K+1 of W to column K of W
687: *
688: CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
689: *
690: DONE = .TRUE.
691: *
692: * Equivalent to testing for ROWMAX.EQ.COLMAX,
693: * (used to handle NaN and Inf)
694: *
695: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
696: $ THEN
697: *
698: * interchange rows and columns K+1 and IMAX,
699: * use 2-by-2 pivot block
700: *
701: KP = IMAX
702: KSTEP = 2
703: DONE = .TRUE.
704: ELSE
705: *
706: * Pivot not found: set params and repeat
707: *
708: P = IMAX
709: COLMAX = ROWMAX
710: IMAX = JMAX
711: *
712: * Copy updated JMAXth (next IMAXth) column to Kth of W
713: *
714: CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
715: *
716: END IF
717: *
718: * End pivot search loop body
719: *
720: IF( .NOT. DONE ) GOTO 72
721: *
722: END IF
723: *
724: * ============================================================
725: *
726: KK = K + KSTEP - 1
727: *
728: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
729: *
730: * Copy non-updated column K to column P
731: *
732: CALL DCOPY( P-K, A( K, K ), 1, A( P, K ), LDA )
733: CALL DCOPY( N-P+1, A( P, K ), 1, A( P, P ), 1 )
734: *
735: * Interchange rows K and P in first K columns of A
736: * and first K+1 columns of W
737: *
738: CALL DSWAP( K, A( K, 1 ), LDA, A( P, 1 ), LDA )
739: CALL DSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
740: END IF
741: *
742: * Updated column KP is already stored in column KK of W
743: *
744: IF( KP.NE.KK ) THEN
745: *
746: * Copy non-updated column KK to column KP
747: *
748: A( KP, K ) = A( KK, K )
749: CALL DCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
750: CALL DCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
751: *
752: * Interchange rows KK and KP in first KK columns of A and W
753: *
754: CALL DSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
755: CALL DSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
756: END IF
757: *
758: IF( KSTEP.EQ.1 ) THEN
759: *
760: * 1-by-1 pivot block D(k): column k of W now holds
761: *
762: * W(k) = L(k)*D(k)
763: *
764: * where L(k) is the k-th column of L
765: *
766: * Store L(k) in column k of A
767: *
768: CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
769: IF( K.LT.N ) THEN
770: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
771: R1 = ONE / A( K, K )
772: CALL DSCAL( N-K, R1, A( K+1, K ), 1 )
773: ELSE IF( A( K, K ).NE.ZERO ) THEN
774: DO 74 II = K + 1, N
775: A( II, K ) = A( II, K ) / A( K, K )
776: 74 CONTINUE
777: END IF
778: END IF
779: *
780: ELSE
781: *
782: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
783: *
784: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
785: *
786: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
787: * of L
788: *
789: IF( K.LT.N-1 ) THEN
790: *
791: * Store L(k) and L(k+1) in columns k and k+1 of A
792: *
793: D21 = W( K+1, K )
794: D11 = W( K+1, K+1 ) / D21
795: D22 = W( K, K ) / D21
796: T = ONE / ( D11*D22-ONE )
797: DO 80 J = K + 2, N
798: A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
799: $ D21 )
800: A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
801: $ D21 )
802: 80 CONTINUE
803: END IF
804: *
805: * Copy D(k) to A
806: *
807: A( K, K ) = W( K, K )
808: A( K+1, K ) = W( K+1, K )
809: A( K+1, K+1 ) = W( K+1, K+1 )
810: END IF
811: END IF
812: *
813: * Store details of the interchanges in IPIV
814: *
815: IF( KSTEP.EQ.1 ) THEN
816: IPIV( K ) = KP
817: ELSE
818: IPIV( K ) = -P
819: IPIV( K+1 ) = -KP
820: END IF
821: *
822: * Increase K and return to the start of the main loop
823: *
824: K = K + KSTEP
825: GO TO 70
826: *
827: 90 CONTINUE
828: *
829: * Update the lower triangle of A22 (= A(k:n,k:n)) as
830: *
831: * A22 := A22 - L21*D*L21**T = A22 - L21*W**T
832: *
833: * computing blocks of NB columns at a time
834: *
835: DO 110 J = K, N, NB
836: JB = MIN( NB, N-J+1 )
837: *
838: * Update the lower triangle of the diagonal block
839: *
840: DO 100 JJ = J, J + JB - 1
841: CALL DGEMV( 'No transpose', J+JB-JJ, K-1, -ONE,
842: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, ONE,
843: $ A( JJ, JJ ), 1 )
844: 100 CONTINUE
845: *
846: * Update the rectangular subdiagonal block
847: *
848: IF( J+JB.LE.N )
849: $ CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
850: $ K-1, -ONE, A( J+JB, 1 ), LDA, W( J, 1 ), LDW,
851: $ ONE, A( J+JB, J ), LDA )
852: 110 CONTINUE
853: *
854: * Put L21 in standard form by partially undoing the interchanges
855: * in columns 1:k-1
856: *
857: J = K - 1
858: 120 CONTINUE
859: *
860: KSTEP = 1
861: JP1 = 1
862: JJ = J
863: JP2 = IPIV( J )
864: IF( JP2.LT.0 ) THEN
865: JP2 = -JP2
866: J = J - 1
867: JP1 = -IPIV( J )
868: KSTEP = 2
869: END IF
870: *
871: J = J - 1
872: IF( JP2.NE.JJ .AND. J.GE.1 )
873: $ CALL DSWAP( J, A( JP2, 1 ), LDA, A( JJ, 1 ), LDA )
874: JJ = J + 1
875: IF( JP1.NE.JJ .AND. KSTEP.EQ.2 )
876: $ CALL DSWAP( J, A( JP1, 1 ), LDA, A( JJ, 1 ), LDA )
877: IF( J.GE.1 )
878: $ GO TO 120
879: *
880: * Set KB to the number of columns factorized
881: *
882: KB = K - 1
883: *
884: END IF
885: RETURN
886: *
887: * End of DLASYF_ROOK
888: *
889: END
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