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1.1 bertrand 1: *> \brief \b ZLASYF_ROOK computes a partial factorization of a complex 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 ZLASYF_ROOK + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlasyf_rook.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlasyf_rook.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlasyf_rook.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLASYF_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: * COMPLEX*16 A( LDA, * ), W( LDW, * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> ZLASYF_ROOK computes a partial factorization of a complex 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: *> ZLASYF_ROOK is an auxiliary routine called by ZSYTRF_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 COMPLEX*16 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 COMPLEX*16 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 complex16SYcomputational
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 ZLASYF_ROOK( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW,
183: $ INFO )
184: *
1.6 ! bertrand 185: * -- LAPACK computational routine --
1.1 bertrand 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: COMPLEX*16 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: COMPLEX*16 CONE, CZERO
206: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ),
207: $ CZERO = ( 0.0D+0, 0.0D+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: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX, DTEMP, SFMIN
214: COMPLEX*16 D11, D12, D21, D22, R1, T, Z
215: * ..
216: * .. External Functions ..
217: LOGICAL LSAME
218: INTEGER IZAMAX
219: DOUBLE PRECISION DLAMCH
220: EXTERNAL LSAME, IZAMAX, DLAMCH
221: * ..
222: * .. External Subroutines ..
223: EXTERNAL ZCOPY, ZGEMM, ZGEMV, ZSCAL, ZSWAP
224: * ..
225: * .. Intrinsic Functions ..
226: INTRINSIC ABS, MAX, MIN, SQRT, DIMAG, DBLE
227: * ..
228: * .. Statement Functions ..
229: DOUBLE PRECISION CABS1
230: * ..
231: * .. Statement Function definitions ..
232: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
233: * ..
234: * .. Executable Statements ..
235: *
236: INFO = 0
237: *
238: * Initialize ALPHA for use in choosing pivot block size.
239: *
240: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
241: *
242: * Compute machine safe minimum
243: *
244: SFMIN = DLAMCH( 'S' )
245: *
246: IF( LSAME( UPLO, 'U' ) ) THEN
247: *
248: * Factorize the trailing columns of A using the upper triangle
249: * of A and working backwards, and compute the matrix W = U12*D
250: * for use in updating A11
251: *
252: * K is the main loop index, decreasing from N in steps of 1 or 2
253: *
254: K = N
255: 10 CONTINUE
256: *
257: * KW is the column of W which corresponds to column K of A
258: *
259: KW = NB + K - N
260: *
261: * Exit from loop
262: *
263: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
264: $ GO TO 30
265: *
266: KSTEP = 1
267: P = K
268: *
269: * Copy column K of A to column KW of W and update it
270: *
271: CALL ZCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
272: IF( K.LT.N )
273: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ),
274: $ LDA, W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
275: *
276: * Determine rows and columns to be interchanged and whether
277: * a 1-by-1 or 2-by-2 pivot block will be used
278: *
279: ABSAKK = CABS1( W( K, KW ) )
280: *
281: * IMAX is the row-index of the largest off-diagonal element in
282: * column K, and COLMAX is its absolute value.
283: * Determine both COLMAX and IMAX.
284: *
285: IF( K.GT.1 ) THEN
286: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
287: COLMAX = CABS1( W( IMAX, KW ) )
288: ELSE
289: COLMAX = ZERO
290: END IF
291: *
292: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
293: *
294: * Column K is zero or underflow: set INFO and continue
295: *
296: IF( INFO.EQ.0 )
297: $ INFO = K
298: KP = K
299: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
300: ELSE
301: *
302: * ============================================================
303: *
304: * Test for interchange
305: *
306: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
307: * (used to handle NaN and Inf)
308: *
309: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
310: *
311: * no interchange, use 1-by-1 pivot block
312: *
313: KP = K
314: *
315: ELSE
316: *
317: DONE = .FALSE.
318: *
319: * Loop until pivot found
320: *
321: 12 CONTINUE
322: *
323: * Begin pivot search loop body
324: *
325: *
326: * Copy column IMAX to column KW-1 of W and update it
327: *
328: CALL ZCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
329: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
330: $ W( IMAX+1, KW-1 ), 1 )
331: *
332: IF( K.LT.N )
333: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE,
334: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
335: $ CONE, W( 1, KW-1 ), 1 )
336: *
337: * JMAX is the column-index of the largest off-diagonal
338: * element in row IMAX, and ROWMAX is its absolute value.
339: * Determine both ROWMAX and JMAX.
340: *
341: IF( IMAX.NE.K ) THEN
342: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ),
343: $ 1 )
344: ROWMAX = CABS1( W( JMAX, KW-1 ) )
345: ELSE
346: ROWMAX = ZERO
347: END IF
348: *
349: IF( IMAX.GT.1 ) THEN
350: ITEMP = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
351: DTEMP = CABS1( W( ITEMP, KW-1 ) )
352: IF( DTEMP.GT.ROWMAX ) THEN
353: ROWMAX = DTEMP
354: JMAX = ITEMP
355: END IF
356: END IF
357: *
358: * Equivalent to testing for
359: * CABS1( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX
360: * (used to handle NaN and Inf)
361: *
362: IF( .NOT.(CABS1( W( IMAX, KW-1 ) ).LT.ALPHA*ROWMAX ) )
363: $ THEN
364: *
365: * interchange rows and columns K and IMAX,
366: * use 1-by-1 pivot block
367: *
368: KP = IMAX
369: *
370: * copy column KW-1 of W to column KW of W
371: *
372: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
373: *
374: DONE = .TRUE.
375: *
376: * Equivalent to testing for ROWMAX.EQ.COLMAX,
377: * (used to handle NaN and Inf)
378: *
379: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
380: $ THEN
381: *
382: * interchange rows and columns K-1 and IMAX,
383: * use 2-by-2 pivot block
384: *
385: KP = IMAX
386: KSTEP = 2
387: DONE = .TRUE.
388: ELSE
389: *
390: * Pivot not found: set params and repeat
391: *
392: P = IMAX
393: COLMAX = ROWMAX
394: IMAX = JMAX
395: *
396: * Copy updated JMAXth (next IMAXth) column to Kth of W
397: *
398: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
399: *
400: END IF
401: *
402: * End pivot search loop body
403: *
404: IF( .NOT. DONE ) GOTO 12
405: *
406: END IF
407: *
408: * ============================================================
409: *
410: KK = K - KSTEP + 1
411: *
412: * KKW is the column of W which corresponds to column KK of A
413: *
414: KKW = NB + KK - N
415: *
416: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
417: *
418: * Copy non-updated column K to column P
419: *
420: CALL ZCOPY( K-P, A( P+1, K ), 1, A( P, P+1 ), LDA )
421: CALL ZCOPY( P, A( 1, K ), 1, A( 1, P ), 1 )
422: *
423: * Interchange rows K and P in last N-K+1 columns of A
424: * and last N-K+2 columns of W
425: *
426: CALL ZSWAP( N-K+1, A( K, K ), LDA, A( P, K ), LDA )
427: CALL ZSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), LDW )
428: END IF
429: *
430: * Updated column KP is already stored in column KKW of W
431: *
432: IF( KP.NE.KK ) THEN
433: *
434: * Copy non-updated column KK to column KP
435: *
436: A( KP, K ) = A( KK, K )
437: CALL ZCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
438: $ LDA )
439: CALL ZCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
440: *
441: * Interchange rows KK and KP in last N-KK+1 columns
442: * of A and W
443: *
444: CALL ZSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
445: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
446: $ LDW )
447: END IF
448: *
449: IF( KSTEP.EQ.1 ) THEN
450: *
451: * 1-by-1 pivot block D(k): column KW of W now holds
452: *
453: * W(k) = U(k)*D(k)
454: *
455: * where U(k) is the k-th column of U
456: *
457: * Store U(k) in column k of A
458: *
459: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
460: IF( K.GT.1 ) THEN
461: IF( CABS1( A( K, K ) ).GE.SFMIN ) THEN
462: R1 = CONE / A( K, K )
463: CALL ZSCAL( K-1, R1, A( 1, K ), 1 )
464: ELSE IF( A( K, K ).NE.CZERO ) THEN
465: DO 14 II = 1, K - 1
466: A( II, K ) = A( II, K ) / A( K, K )
467: 14 CONTINUE
468: END IF
469: END IF
470: *
471: ELSE
472: *
473: * 2-by-2 pivot block D(k): columns KW and KW-1 of W now
474: * hold
475: *
476: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
477: *
478: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
479: * of U
480: *
481: IF( K.GT.2 ) THEN
482: *
483: * Store U(k) and U(k-1) in columns k and k-1 of A
484: *
485: D12 = W( K-1, KW )
486: D11 = W( K, KW ) / D12
487: D22 = W( K-1, KW-1 ) / D12
488: T = CONE / ( D11*D22-CONE )
489: DO 20 J = 1, K - 2
490: A( J, K-1 ) = T*( (D11*W( J, KW-1 )-W( J, KW ) ) /
491: $ D12 )
492: A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
493: $ D12 )
494: 20 CONTINUE
495: END IF
496: *
497: * Copy D(k) to A
498: *
499: A( K-1, K-1 ) = W( K-1, KW-1 )
500: A( K-1, K ) = W( K-1, KW )
501: A( K, K ) = W( K, KW )
502: END IF
503: END IF
504: *
505: * Store details of the interchanges in IPIV
506: *
507: IF( KSTEP.EQ.1 ) THEN
508: IPIV( K ) = KP
509: ELSE
510: IPIV( K ) = -P
511: IPIV( K-1 ) = -KP
512: END IF
513: *
514: * Decrease K and return to the start of the main loop
515: *
516: K = K - KSTEP
517: GO TO 10
518: *
519: 30 CONTINUE
520: *
521: * Update the upper triangle of A11 (= A(1:k,1:k)) as
522: *
523: * A11 := A11 - U12*D*U12**T = A11 - U12*W**T
524: *
525: * computing blocks of NB columns at a time
526: *
527: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
528: JB = MIN( NB, K-J+1 )
529: *
530: * Update the upper triangle of the diagonal block
531: *
532: DO 40 JJ = J, J + JB - 1
533: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
534: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
535: $ A( J, JJ ), 1 )
536: 40 CONTINUE
537: *
538: * Update the rectangular superdiagonal block
539: *
540: IF( J.GE.2 )
541: $ CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB,
542: $ N-K, -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
543: $ CONE, A( 1, J ), LDA )
544: 50 CONTINUE
545: *
546: * Put U12 in standard form by partially undoing the interchanges
547: * in columns k+1:n
548: *
549: J = K + 1
550: 60 CONTINUE
551: *
552: KSTEP = 1
553: JP1 = 1
554: JJ = J
555: JP2 = IPIV( J )
556: IF( JP2.LT.0 ) THEN
557: JP2 = -JP2
558: J = J + 1
559: JP1 = -IPIV( J )
560: KSTEP = 2
561: END IF
562: *
563: J = J + 1
564: IF( JP2.NE.JJ .AND. J.LE.N )
565: $ CALL ZSWAP( N-J+1, A( JP2, J ), LDA, A( JJ, J ), LDA )
566: JJ = J - 1
567: IF( JP1.NE.JJ .AND. KSTEP.EQ.2 )
568: $ CALL ZSWAP( N-J+1, A( JP1, J ), LDA, A( JJ, J ), LDA )
569: IF( J.LE.N )
570: $ GO TO 60
571: *
572: * Set KB to the number of columns factorized
573: *
574: KB = N - K
575: *
576: ELSE
577: *
578: * Factorize the leading columns of A using the lower triangle
579: * of A and working forwards, and compute the matrix W = L21*D
580: * for use in updating A22
581: *
582: * K is the main loop index, increasing from 1 in steps of 1 or 2
583: *
584: K = 1
585: 70 CONTINUE
586: *
587: * Exit from loop
588: *
589: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
590: $ GO TO 90
591: *
592: KSTEP = 1
593: P = K
594: *
595: * Copy column K of A to column K of W and update it
596: *
597: CALL ZCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
598: IF( K.GT.1 )
599: $ CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
600: $ LDA, W( K, 1 ), LDW, CONE, W( K, K ), 1 )
601: *
602: * Determine rows and columns to be interchanged and whether
603: * a 1-by-1 or 2-by-2 pivot block will be used
604: *
605: ABSAKK = CABS1( W( K, K ) )
606: *
607: * IMAX is the row-index of the largest off-diagonal element in
608: * column K, and COLMAX is its absolute value.
609: * Determine both COLMAX and IMAX.
610: *
611: IF( K.LT.N ) THEN
612: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
613: COLMAX = CABS1( W( IMAX, K ) )
614: ELSE
615: COLMAX = ZERO
616: END IF
617: *
618: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
619: *
620: * Column K is zero or underflow: set INFO and continue
621: *
622: IF( INFO.EQ.0 )
623: $ INFO = K
624: KP = K
625: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
626: ELSE
627: *
628: * ============================================================
629: *
630: * Test for interchange
631: *
632: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
633: * (used to handle NaN and Inf)
634: *
635: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
636: *
637: * no interchange, use 1-by-1 pivot block
638: *
639: KP = K
640: *
641: ELSE
642: *
643: DONE = .FALSE.
644: *
645: * Loop until pivot found
646: *
647: 72 CONTINUE
648: *
649: * Begin pivot search loop body
650: *
651: *
652: * Copy column IMAX to column K+1 of W and update it
653: *
654: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
655: CALL ZCOPY( N-IMAX+1, A( IMAX, IMAX ), 1,
656: $ W( IMAX, K+1 ), 1 )
657: IF( K.GT.1 )
658: $ CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE,
659: $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
660: $ CONE, W( K, K+1 ), 1 )
661: *
662: * JMAX is the column-index of the largest off-diagonal
663: * element in row IMAX, and ROWMAX is its absolute value.
664: * Determine both ROWMAX and JMAX.
665: *
666: IF( IMAX.NE.K ) THEN
667: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
668: ROWMAX = CABS1( W( JMAX, K+1 ) )
669: ELSE
670: ROWMAX = ZERO
671: END IF
672: *
673: IF( IMAX.LT.N ) THEN
674: ITEMP = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
675: DTEMP = CABS1( W( ITEMP, K+1 ) )
676: IF( DTEMP.GT.ROWMAX ) THEN
677: ROWMAX = DTEMP
678: JMAX = ITEMP
679: END IF
680: END IF
681: *
682: * Equivalent to testing for
683: * CABS1( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX
684: * (used to handle NaN and Inf)
685: *
686: IF( .NOT.( CABS1( W( IMAX, K+1 ) ).LT.ALPHA*ROWMAX ) )
687: $ THEN
688: *
689: * interchange rows and columns K and IMAX,
690: * use 1-by-1 pivot block
691: *
692: KP = IMAX
693: *
694: * copy column K+1 of W to column K of W
695: *
696: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
697: *
698: DONE = .TRUE.
699: *
700: * Equivalent to testing for ROWMAX.EQ.COLMAX,
701: * (used to handle NaN and Inf)
702: *
703: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
704: $ THEN
705: *
706: * interchange rows and columns K+1 and IMAX,
707: * use 2-by-2 pivot block
708: *
709: KP = IMAX
710: KSTEP = 2
711: DONE = .TRUE.
712: ELSE
713: *
714: * Pivot not found: set params and repeat
715: *
716: P = IMAX
717: COLMAX = ROWMAX
718: IMAX = JMAX
719: *
720: * Copy updated JMAXth (next IMAXth) column to Kth of W
721: *
722: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
723: *
724: END IF
725: *
726: * End pivot search loop body
727: *
728: IF( .NOT. DONE ) GOTO 72
729: *
730: END IF
731: *
732: * ============================================================
733: *
734: KK = K + KSTEP - 1
735: *
736: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
737: *
738: * Copy non-updated column K to column P
739: *
740: CALL ZCOPY( P-K, A( K, K ), 1, A( P, K ), LDA )
741: CALL ZCOPY( N-P+1, A( P, K ), 1, A( P, P ), 1 )
742: *
743: * Interchange rows K and P in first K columns of A
744: * and first K+1 columns of W
745: *
746: CALL ZSWAP( K, A( K, 1 ), LDA, A( P, 1 ), LDA )
747: CALL ZSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
748: END IF
749: *
750: * Updated column KP is already stored in column KK of W
751: *
752: IF( KP.NE.KK ) THEN
753: *
754: * Copy non-updated column KK to column KP
755: *
756: A( KP, K ) = A( KK, K )
757: CALL ZCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
758: CALL ZCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
759: *
760: * Interchange rows KK and KP in first KK columns of A and W
761: *
762: CALL ZSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
763: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
764: END IF
765: *
766: IF( KSTEP.EQ.1 ) THEN
767: *
768: * 1-by-1 pivot block D(k): column k of W now holds
769: *
770: * W(k) = L(k)*D(k)
771: *
772: * where L(k) is the k-th column of L
773: *
774: * Store L(k) in column k of A
775: *
776: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
777: IF( K.LT.N ) THEN
778: IF( CABS1( A( K, K ) ).GE.SFMIN ) THEN
779: R1 = CONE / A( K, K )
780: CALL ZSCAL( N-K, R1, A( K+1, K ), 1 )
781: ELSE IF( A( K, K ).NE.CZERO ) THEN
782: DO 74 II = K + 1, N
783: A( II, K ) = A( II, K ) / A( K, K )
784: 74 CONTINUE
785: END IF
786: END IF
787: *
788: ELSE
789: *
790: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
791: *
792: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
793: *
794: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
795: * of L
796: *
797: IF( K.LT.N-1 ) THEN
798: *
799: * Store L(k) and L(k+1) in columns k and k+1 of A
800: *
801: D21 = W( K+1, K )
802: D11 = W( K+1, K+1 ) / D21
803: D22 = W( K, K ) / D21
804: T = CONE / ( D11*D22-CONE )
805: DO 80 J = K + 2, N
806: A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
807: $ D21 )
808: A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
809: $ D21 )
810: 80 CONTINUE
811: END IF
812: *
813: * Copy D(k) to A
814: *
815: A( K, K ) = W( K, K )
816: A( K+1, K ) = W( K+1, K )
817: A( K+1, K+1 ) = W( K+1, K+1 )
818: END IF
819: END IF
820: *
821: * Store details of the interchanges in IPIV
822: *
823: IF( KSTEP.EQ.1 ) THEN
824: IPIV( K ) = KP
825: ELSE
826: IPIV( K ) = -P
827: IPIV( K+1 ) = -KP
828: END IF
829: *
830: * Increase K and return to the start of the main loop
831: *
832: K = K + KSTEP
833: GO TO 70
834: *
835: 90 CONTINUE
836: *
837: * Update the lower triangle of A22 (= A(k:n,k:n)) as
838: *
839: * A22 := A22 - L21*D*L21**T = A22 - L21*W**T
840: *
841: * computing blocks of NB columns at a time
842: *
843: DO 110 J = K, N, NB
844: JB = MIN( NB, N-J+1 )
845: *
846: * Update the lower triangle of the diagonal block
847: *
848: DO 100 JJ = J, J + JB - 1
849: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
850: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
851: $ A( JJ, JJ ), 1 )
852: 100 CONTINUE
853: *
854: * Update the rectangular subdiagonal block
855: *
856: IF( J+JB.LE.N )
857: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
858: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ), LDW,
859: $ CONE, A( J+JB, J ), LDA )
860: 110 CONTINUE
861: *
862: * Put L21 in standard form by partially undoing the interchanges
863: * in columns 1:k-1
864: *
865: J = K - 1
866: 120 CONTINUE
867: *
868: KSTEP = 1
869: JP1 = 1
870: JJ = J
871: JP2 = IPIV( J )
872: IF( JP2.LT.0 ) THEN
873: JP2 = -JP2
874: J = J - 1
875: JP1 = -IPIV( J )
876: KSTEP = 2
877: END IF
878: *
879: J = J - 1
880: IF( JP2.NE.JJ .AND. J.GE.1 )
881: $ CALL ZSWAP( J, A( JP2, 1 ), LDA, A( JJ, 1 ), LDA )
882: JJ = J + 1
883: IF( JP1.NE.JJ .AND. KSTEP.EQ.2 )
884: $ CALL ZSWAP( J, A( JP1, 1 ), LDA, A( JJ, 1 ), LDA )
885: IF( J.GE.1 )
886: $ GO TO 120
887: *
888: * Set KB to the number of columns factorized
889: *
890: KB = K - 1
891: *
892: END IF
893: RETURN
894: *
895: * End of ZLASYF_ROOK
896: *
897: END