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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: *> \date November 2013
165: *
166: *> \ingroup complex16SYcomputational
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 ZLASYF_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: COMPLEX*16 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: COMPLEX*16 CONE, CZERO
209: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ),
210: $ CZERO = ( 0.0D+0, 0.0D+0 ) )
211: * ..
212: * .. Local Scalars ..
213: LOGICAL DONE
214: INTEGER IMAX, ITEMP, J, JB, JJ, JMAX, JP1, JP2, K, KK,
215: $ KW, KKW, KP, KSTEP, P, II
216: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, ROWMAX, DTEMP, SFMIN
217: COMPLEX*16 D11, D12, D21, D22, R1, T, Z
218: * ..
219: * .. External Functions ..
220: LOGICAL LSAME
221: INTEGER IZAMAX
222: DOUBLE PRECISION DLAMCH
223: EXTERNAL LSAME, IZAMAX, DLAMCH
224: * ..
225: * .. External Subroutines ..
226: EXTERNAL ZCOPY, ZGEMM, ZGEMV, ZSCAL, ZSWAP
227: * ..
228: * .. Intrinsic Functions ..
229: INTRINSIC ABS, MAX, MIN, SQRT, DIMAG, DBLE
230: * ..
231: * .. Statement Functions ..
232: DOUBLE PRECISION CABS1
233: * ..
234: * .. Statement Function definitions ..
235: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
236: * ..
237: * .. Executable Statements ..
238: *
239: INFO = 0
240: *
241: * Initialize ALPHA for use in choosing pivot block size.
242: *
243: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
244: *
245: * Compute machine safe minimum
246: *
247: SFMIN = DLAMCH( 'S' )
248: *
249: IF( LSAME( UPLO, 'U' ) ) THEN
250: *
251: * Factorize the trailing columns of A using the upper triangle
252: * of A and working backwards, and compute the matrix W = U12*D
253: * for use in updating A11
254: *
255: * K is the main loop index, decreasing from N in steps of 1 or 2
256: *
257: K = N
258: 10 CONTINUE
259: *
260: * KW is the column of W which corresponds to column K of A
261: *
262: KW = NB + K - N
263: *
264: * Exit from loop
265: *
266: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
267: $ GO TO 30
268: *
269: KSTEP = 1
270: P = K
271: *
272: * Copy column K of A to column KW of W and update it
273: *
274: CALL ZCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
275: IF( K.LT.N )
276: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ),
277: $ LDA, W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
278: *
279: * Determine rows and columns to be interchanged and whether
280: * a 1-by-1 or 2-by-2 pivot block will be used
281: *
282: ABSAKK = CABS1( W( K, KW ) )
283: *
284: * IMAX is the row-index of the largest off-diagonal element in
285: * column K, and COLMAX is its absolute value.
286: * Determine both COLMAX and IMAX.
287: *
288: IF( K.GT.1 ) THEN
289: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
290: COLMAX = CABS1( W( IMAX, KW ) )
291: ELSE
292: COLMAX = ZERO
293: END IF
294: *
295: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
296: *
297: * Column K is zero or underflow: set INFO and continue
298: *
299: IF( INFO.EQ.0 )
300: $ INFO = K
301: KP = K
302: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
303: ELSE
304: *
305: * ============================================================
306: *
307: * Test for interchange
308: *
309: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
310: * (used to handle NaN and Inf)
311: *
312: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
313: *
314: * no interchange, use 1-by-1 pivot block
315: *
316: KP = K
317: *
318: ELSE
319: *
320: DONE = .FALSE.
321: *
322: * Loop until pivot found
323: *
324: 12 CONTINUE
325: *
326: * Begin pivot search loop body
327: *
328: *
329: * Copy column IMAX to column KW-1 of W and update it
330: *
331: CALL ZCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
332: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
333: $ W( IMAX+1, KW-1 ), 1 )
334: *
335: IF( K.LT.N )
336: $ CALL ZGEMV( 'No transpose', K, N-K, -CONE,
337: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
338: $ CONE, W( 1, KW-1 ), 1 )
339: *
340: * JMAX is the column-index of the largest off-diagonal
341: * element in row IMAX, and ROWMAX is its absolute value.
342: * Determine both ROWMAX and JMAX.
343: *
344: IF( IMAX.NE.K ) THEN
345: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ),
346: $ 1 )
347: ROWMAX = CABS1( W( JMAX, KW-1 ) )
348: ELSE
349: ROWMAX = ZERO
350: END IF
351: *
352: IF( IMAX.GT.1 ) THEN
353: ITEMP = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
354: DTEMP = CABS1( W( ITEMP, KW-1 ) )
355: IF( DTEMP.GT.ROWMAX ) THEN
356: ROWMAX = DTEMP
357: JMAX = ITEMP
358: END IF
359: END IF
360: *
361: * Equivalent to testing for
362: * CABS1( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX
363: * (used to handle NaN and Inf)
364: *
365: IF( .NOT.(CABS1( W( IMAX, KW-1 ) ).LT.ALPHA*ROWMAX ) )
366: $ THEN
367: *
368: * interchange rows and columns K and IMAX,
369: * use 1-by-1 pivot block
370: *
371: KP = IMAX
372: *
373: * copy column KW-1 of W to column KW of W
374: *
375: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
376: *
377: DONE = .TRUE.
378: *
379: * Equivalent to testing for ROWMAX.EQ.COLMAX,
380: * (used to handle NaN and Inf)
381: *
382: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
383: $ THEN
384: *
385: * interchange rows and columns K-1 and IMAX,
386: * use 2-by-2 pivot block
387: *
388: KP = IMAX
389: KSTEP = 2
390: DONE = .TRUE.
391: ELSE
392: *
393: * Pivot not found: set params and repeat
394: *
395: P = IMAX
396: COLMAX = ROWMAX
397: IMAX = JMAX
398: *
399: * Copy updated JMAXth (next IMAXth) column to Kth of W
400: *
401: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
402: *
403: END IF
404: *
405: * End pivot search loop body
406: *
407: IF( .NOT. DONE ) GOTO 12
408: *
409: END IF
410: *
411: * ============================================================
412: *
413: KK = K - KSTEP + 1
414: *
415: * KKW is the column of W which corresponds to column KK of A
416: *
417: KKW = NB + KK - N
418: *
419: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
420: *
421: * Copy non-updated column K to column P
422: *
423: CALL ZCOPY( K-P, A( P+1, K ), 1, A( P, P+1 ), LDA )
424: CALL ZCOPY( P, A( 1, K ), 1, A( 1, P ), 1 )
425: *
426: * Interchange rows K and P in last N-K+1 columns of A
427: * and last N-K+2 columns of W
428: *
429: CALL ZSWAP( N-K+1, A( K, K ), LDA, A( P, K ), LDA )
430: CALL ZSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), LDW )
431: END IF
432: *
433: * Updated column KP is already stored in column KKW of W
434: *
435: IF( KP.NE.KK ) THEN
436: *
437: * Copy non-updated column KK to column KP
438: *
439: A( KP, K ) = A( KK, K )
440: CALL ZCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
441: $ LDA )
442: CALL ZCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
443: *
444: * Interchange rows KK and KP in last N-KK+1 columns
445: * of A and W
446: *
447: CALL ZSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
448: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
449: $ LDW )
450: END IF
451: *
452: IF( KSTEP.EQ.1 ) THEN
453: *
454: * 1-by-1 pivot block D(k): column KW of W now holds
455: *
456: * W(k) = U(k)*D(k)
457: *
458: * where U(k) is the k-th column of U
459: *
460: * Store U(k) in column k of A
461: *
462: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
463: IF( K.GT.1 ) THEN
464: IF( CABS1( A( K, K ) ).GE.SFMIN ) THEN
465: R1 = CONE / A( K, K )
466: CALL ZSCAL( K-1, R1, A( 1, K ), 1 )
467: ELSE IF( A( K, K ).NE.CZERO ) THEN
468: DO 14 II = 1, K - 1
469: A( II, K ) = A( II, K ) / A( K, K )
470: 14 CONTINUE
471: END IF
472: END IF
473: *
474: ELSE
475: *
476: * 2-by-2 pivot block D(k): columns KW and KW-1 of W now
477: * hold
478: *
479: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
480: *
481: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
482: * of U
483: *
484: IF( K.GT.2 ) THEN
485: *
486: * Store U(k) and U(k-1) in columns k and k-1 of A
487: *
488: D12 = W( K-1, KW )
489: D11 = W( K, KW ) / D12
490: D22 = W( K-1, KW-1 ) / D12
491: T = CONE / ( D11*D22-CONE )
492: DO 20 J = 1, K - 2
493: A( J, K-1 ) = T*( (D11*W( J, KW-1 )-W( J, KW ) ) /
494: $ D12 )
495: A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
496: $ D12 )
497: 20 CONTINUE
498: END IF
499: *
500: * Copy D(k) to A
501: *
502: A( K-1, K-1 ) = W( K-1, KW-1 )
503: A( K-1, K ) = W( K-1, KW )
504: A( K, K ) = W( K, KW )
505: END IF
506: END IF
507: *
508: * Store details of the interchanges in IPIV
509: *
510: IF( KSTEP.EQ.1 ) THEN
511: IPIV( K ) = KP
512: ELSE
513: IPIV( K ) = -P
514: IPIV( K-1 ) = -KP
515: END IF
516: *
517: * Decrease K and return to the start of the main loop
518: *
519: K = K - KSTEP
520: GO TO 10
521: *
522: 30 CONTINUE
523: *
524: * Update the upper triangle of A11 (= A(1:k,1:k)) as
525: *
526: * A11 := A11 - U12*D*U12**T = A11 - U12*W**T
527: *
528: * computing blocks of NB columns at a time
529: *
530: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
531: JB = MIN( NB, K-J+1 )
532: *
533: * Update the upper triangle of the diagonal block
534: *
535: DO 40 JJ = J, J + JB - 1
536: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
537: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
538: $ A( J, JJ ), 1 )
539: 40 CONTINUE
540: *
541: * Update the rectangular superdiagonal block
542: *
543: IF( J.GE.2 )
544: $ CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB,
545: $ N-K, -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
546: $ CONE, A( 1, J ), LDA )
547: 50 CONTINUE
548: *
549: * Put U12 in standard form by partially undoing the interchanges
550: * in columns k+1:n
551: *
552: J = K + 1
553: 60 CONTINUE
554: *
555: KSTEP = 1
556: JP1 = 1
557: JJ = J
558: JP2 = IPIV( J )
559: IF( JP2.LT.0 ) THEN
560: JP2 = -JP2
561: J = J + 1
562: JP1 = -IPIV( J )
563: KSTEP = 2
564: END IF
565: *
566: J = J + 1
567: IF( JP2.NE.JJ .AND. J.LE.N )
568: $ CALL ZSWAP( N-J+1, A( JP2, J ), LDA, A( JJ, J ), LDA )
569: JJ = J - 1
570: IF( JP1.NE.JJ .AND. KSTEP.EQ.2 )
571: $ CALL ZSWAP( N-J+1, A( JP1, J ), LDA, A( JJ, J ), LDA )
572: IF( J.LE.N )
573: $ GO TO 60
574: *
575: * Set KB to the number of columns factorized
576: *
577: KB = N - K
578: *
579: ELSE
580: *
581: * Factorize the leading columns of A using the lower triangle
582: * of A and working forwards, and compute the matrix W = L21*D
583: * for use in updating A22
584: *
585: * K is the main loop index, increasing from 1 in steps of 1 or 2
586: *
587: K = 1
588: 70 CONTINUE
589: *
590: * Exit from loop
591: *
592: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
593: $ GO TO 90
594: *
595: KSTEP = 1
596: P = K
597: *
598: * Copy column K of A to column K of W and update it
599: *
600: CALL ZCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
601: IF( K.GT.1 )
602: $ CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
603: $ LDA, W( K, 1 ), LDW, CONE, W( K, K ), 1 )
604: *
605: * Determine rows and columns to be interchanged and whether
606: * a 1-by-1 or 2-by-2 pivot block will be used
607: *
608: ABSAKK = CABS1( W( K, K ) )
609: *
610: * IMAX is the row-index of the largest off-diagonal element in
611: * column K, and COLMAX is its absolute value.
612: * Determine both COLMAX and IMAX.
613: *
614: IF( K.LT.N ) THEN
615: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
616: COLMAX = CABS1( W( IMAX, K ) )
617: ELSE
618: COLMAX = ZERO
619: END IF
620: *
621: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
622: *
623: * Column K is zero or underflow: set INFO and continue
624: *
625: IF( INFO.EQ.0 )
626: $ INFO = K
627: KP = K
628: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
629: ELSE
630: *
631: * ============================================================
632: *
633: * Test for interchange
634: *
635: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
636: * (used to handle NaN and Inf)
637: *
638: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
639: *
640: * no interchange, use 1-by-1 pivot block
641: *
642: KP = K
643: *
644: ELSE
645: *
646: DONE = .FALSE.
647: *
648: * Loop until pivot found
649: *
650: 72 CONTINUE
651: *
652: * Begin pivot search loop body
653: *
654: *
655: * Copy column IMAX to column K+1 of W and update it
656: *
657: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
658: CALL ZCOPY( N-IMAX+1, A( IMAX, IMAX ), 1,
659: $ W( IMAX, K+1 ), 1 )
660: IF( K.GT.1 )
661: $ CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE,
662: $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
663: $ CONE, W( K, K+1 ), 1 )
664: *
665: * JMAX is the column-index of the largest off-diagonal
666: * element in row IMAX, and ROWMAX is its absolute value.
667: * Determine both ROWMAX and JMAX.
668: *
669: IF( IMAX.NE.K ) THEN
670: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
671: ROWMAX = CABS1( W( JMAX, K+1 ) )
672: ELSE
673: ROWMAX = ZERO
674: END IF
675: *
676: IF( IMAX.LT.N ) THEN
677: ITEMP = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
678: DTEMP = CABS1( W( ITEMP, K+1 ) )
679: IF( DTEMP.GT.ROWMAX ) THEN
680: ROWMAX = DTEMP
681: JMAX = ITEMP
682: END IF
683: END IF
684: *
685: * Equivalent to testing for
686: * CABS1( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX
687: * (used to handle NaN and Inf)
688: *
689: IF( .NOT.( CABS1( W( IMAX, K+1 ) ).LT.ALPHA*ROWMAX ) )
690: $ THEN
691: *
692: * interchange rows and columns K and IMAX,
693: * use 1-by-1 pivot block
694: *
695: KP = IMAX
696: *
697: * copy column K+1 of W to column K of W
698: *
699: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
700: *
701: DONE = .TRUE.
702: *
703: * Equivalent to testing for ROWMAX.EQ.COLMAX,
704: * (used to handle NaN and Inf)
705: *
706: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
707: $ THEN
708: *
709: * interchange rows and columns K+1 and IMAX,
710: * use 2-by-2 pivot block
711: *
712: KP = IMAX
713: KSTEP = 2
714: DONE = .TRUE.
715: ELSE
716: *
717: * Pivot not found: set params and repeat
718: *
719: P = IMAX
720: COLMAX = ROWMAX
721: IMAX = JMAX
722: *
723: * Copy updated JMAXth (next IMAXth) column to Kth of W
724: *
725: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
726: *
727: END IF
728: *
729: * End pivot search loop body
730: *
731: IF( .NOT. DONE ) GOTO 72
732: *
733: END IF
734: *
735: * ============================================================
736: *
737: KK = K + KSTEP - 1
738: *
739: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
740: *
741: * Copy non-updated column K to column P
742: *
743: CALL ZCOPY( P-K, A( K, K ), 1, A( P, K ), LDA )
744: CALL ZCOPY( N-P+1, A( P, K ), 1, A( P, P ), 1 )
745: *
746: * Interchange rows K and P in first K columns of A
747: * and first K+1 columns of W
748: *
749: CALL ZSWAP( K, A( K, 1 ), LDA, A( P, 1 ), LDA )
750: CALL ZSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
751: END IF
752: *
753: * Updated column KP is already stored in column KK of W
754: *
755: IF( KP.NE.KK ) THEN
756: *
757: * Copy non-updated column KK to column KP
758: *
759: A( KP, K ) = A( KK, K )
760: CALL ZCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
761: CALL ZCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
762: *
763: * Interchange rows KK and KP in first KK columns of A and W
764: *
765: CALL ZSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
766: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
767: END IF
768: *
769: IF( KSTEP.EQ.1 ) THEN
770: *
771: * 1-by-1 pivot block D(k): column k of W now holds
772: *
773: * W(k) = L(k)*D(k)
774: *
775: * where L(k) is the k-th column of L
776: *
777: * Store L(k) in column k of A
778: *
779: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
780: IF( K.LT.N ) THEN
781: IF( CABS1( A( K, K ) ).GE.SFMIN ) THEN
782: R1 = CONE / A( K, K )
783: CALL ZSCAL( N-K, R1, A( K+1, K ), 1 )
784: ELSE IF( A( K, K ).NE.CZERO ) THEN
785: DO 74 II = K + 1, N
786: A( II, K ) = A( II, K ) / A( K, K )
787: 74 CONTINUE
788: END IF
789: END IF
790: *
791: ELSE
792: *
793: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
794: *
795: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
796: *
797: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
798: * of L
799: *
800: IF( K.LT.N-1 ) THEN
801: *
802: * Store L(k) and L(k+1) in columns k and k+1 of A
803: *
804: D21 = W( K+1, K )
805: D11 = W( K+1, K+1 ) / D21
806: D22 = W( K, K ) / D21
807: T = CONE / ( D11*D22-CONE )
808: DO 80 J = K + 2, N
809: A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
810: $ D21 )
811: A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
812: $ D21 )
813: 80 CONTINUE
814: END IF
815: *
816: * Copy D(k) to A
817: *
818: A( K, K ) = W( K, K )
819: A( K+1, K ) = W( K+1, K )
820: A( K+1, K+1 ) = W( K+1, K+1 )
821: END IF
822: END IF
823: *
824: * Store details of the interchanges in IPIV
825: *
826: IF( KSTEP.EQ.1 ) THEN
827: IPIV( K ) = KP
828: ELSE
829: IPIV( K ) = -P
830: IPIV( K+1 ) = -KP
831: END IF
832: *
833: * Increase K and return to the start of the main loop
834: *
835: K = K + KSTEP
836: GO TO 70
837: *
838: 90 CONTINUE
839: *
840: * Update the lower triangle of A22 (= A(k:n,k:n)) as
841: *
842: * A22 := A22 - L21*D*L21**T = A22 - L21*W**T
843: *
844: * computing blocks of NB columns at a time
845: *
846: DO 110 J = K, N, NB
847: JB = MIN( NB, N-J+1 )
848: *
849: * Update the lower triangle of the diagonal block
850: *
851: DO 100 JJ = J, J + JB - 1
852: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
853: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
854: $ A( JJ, JJ ), 1 )
855: 100 CONTINUE
856: *
857: * Update the rectangular subdiagonal block
858: *
859: IF( J+JB.LE.N )
860: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
861: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ), LDW,
862: $ CONE, A( J+JB, J ), LDA )
863: 110 CONTINUE
864: *
865: * Put L21 in standard form by partially undoing the interchanges
866: * in columns 1:k-1
867: *
868: J = K - 1
869: 120 CONTINUE
870: *
871: KSTEP = 1
872: JP1 = 1
873: JJ = J
874: JP2 = IPIV( J )
875: IF( JP2.LT.0 ) THEN
876: JP2 = -JP2
877: J = J - 1
878: JP1 = -IPIV( J )
879: KSTEP = 2
880: END IF
881: *
882: J = J - 1
883: IF( JP2.NE.JJ .AND. J.GE.1 )
884: $ CALL ZSWAP( J, A( JP2, 1 ), LDA, A( JJ, 1 ), LDA )
885: JJ = J + 1
886: IF( JP1.NE.JJ .AND. KSTEP.EQ.2 )
887: $ CALL ZSWAP( J, A( JP1, 1 ), LDA, A( JJ, 1 ), LDA )
888: IF( J.GE.1 )
889: $ GO TO 120
890: *
891: * Set KB to the number of columns factorized
892: *
893: KB = K - 1
894: *
895: END IF
896: RETURN
897: *
898: * End of ZLASYF_ROOK
899: *
900: END