Annotation of rpl/lapack/lapack/zlahef.f, revision 1.13
1.12 bertrand 1: *> \brief \b ZLAHEF computes a partial factorization of a complex Hermitian indefinite matrix, using the diagonal pivoting method.
1.9 bertrand 2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLAHEF + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlahef.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlahef.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLAHEF( 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: *> ZLAHEF computes a partial factorization of a complex Hermitian
39: *> matrix A using the Bunch-Kaufman diagonal pivoting method. The
40: *> partial factorization has the form:
41: *>
42: *> A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
43: *> ( 0 U22 ) ( 0 D ) ( U12**H U22**H )
44: *>
45: *> A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) 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: *> Note that U**H denotes the conjugate transpose of U.
51: *>
52: *> ZLAHEF is an auxiliary routine called by ZHETRF. It uses blocked code
53: *> (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
54: *> A22 (if UPLO = 'L').
55: *> \endverbatim
56: *
57: * Arguments:
58: * ==========
59: *
60: *> \param[in] UPLO
61: *> \verbatim
62: *> UPLO is CHARACTER*1
63: *> Specifies whether the upper or lower triangular part of the
64: *> Hermitian matrix A is stored:
65: *> = 'U': Upper triangular
66: *> = 'L': Lower triangular
67: *> \endverbatim
68: *>
69: *> \param[in] N
70: *> \verbatim
71: *> N is INTEGER
72: *> The order of the matrix A. N >= 0.
73: *> \endverbatim
74: *>
75: *> \param[in] NB
76: *> \verbatim
77: *> NB is INTEGER
78: *> The maximum number of columns of the matrix A that should be
79: *> factored. NB should be at least 2 to allow for 2-by-2 pivot
80: *> blocks.
81: *> \endverbatim
82: *>
83: *> \param[out] KB
84: *> \verbatim
85: *> KB is INTEGER
86: *> The number of columns of A that were actually factored.
87: *> KB is either NB-1 or NB, or N if N <= NB.
88: *> \endverbatim
89: *>
90: *> \param[in,out] A
91: *> \verbatim
92: *> A is COMPLEX*16 array, dimension (LDA,N)
93: *> On entry, the Hermitian matrix A. If UPLO = 'U', the leading
94: *> n-by-n upper triangular part of A contains the upper
95: *> triangular part of the matrix A, and the strictly lower
96: *> triangular part of A is not referenced. If UPLO = 'L', the
97: *> leading n-by-n lower triangular part of A contains the lower
98: *> triangular part of the matrix A, and the strictly upper
99: *> triangular part of A is not referenced.
100: *> On exit, A contains details of the partial factorization.
101: *> \endverbatim
102: *>
103: *> \param[in] LDA
104: *> \verbatim
105: *> LDA is INTEGER
106: *> The leading dimension of the array A. LDA >= max(1,N).
107: *> \endverbatim
108: *>
109: *> \param[out] IPIV
110: *> \verbatim
111: *> IPIV is INTEGER array, dimension (N)
112: *> Details of the interchanges and the block structure of D.
113: *> If UPLO = 'U', only the last KB elements of IPIV are set;
114: *> if UPLO = 'L', only the first KB elements 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: *> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
119: *> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
120: *> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
121: *> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
122: *> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
123: *> \endverbatim
124: *>
125: *> \param[out] W
126: *> \verbatim
127: *> W is COMPLEX*16 array, dimension (LDW,NB)
128: *> \endverbatim
129: *>
130: *> \param[in] LDW
131: *> \verbatim
132: *> LDW is INTEGER
133: *> The leading dimension of the array W. LDW >= max(1,N).
134: *> \endverbatim
135: *>
136: *> \param[out] INFO
137: *> \verbatim
138: *> INFO is INTEGER
139: *> = 0: successful exit
140: *> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
141: *> has been completed, but the block diagonal matrix D is
142: *> exactly singular.
143: *> \endverbatim
144: *
145: * Authors:
146: * ========
147: *
148: *> \author Univ. of Tennessee
149: *> \author Univ. of California Berkeley
150: *> \author Univ. of Colorado Denver
151: *> \author NAG Ltd.
152: *
1.12 bertrand 153: *> \date September 2012
1.9 bertrand 154: *
155: *> \ingroup complex16HEcomputational
156: *
157: * =====================================================================
1.1 bertrand 158: SUBROUTINE ZLAHEF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
159: *
1.12 bertrand 160: * -- LAPACK computational routine (version 3.4.2) --
1.1 bertrand 161: * -- LAPACK is a software package provided by Univ. of Tennessee, --
162: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.12 bertrand 163: * September 2012
1.1 bertrand 164: *
165: * .. Scalar Arguments ..
166: CHARACTER UPLO
167: INTEGER INFO, KB, LDA, LDW, N, NB
168: * ..
169: * .. Array Arguments ..
170: INTEGER IPIV( * )
171: COMPLEX*16 A( LDA, * ), W( LDW, * )
172: * ..
173: *
174: * =====================================================================
175: *
176: * .. Parameters ..
177: DOUBLE PRECISION ZERO, ONE
178: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
179: COMPLEX*16 CONE
180: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
181: DOUBLE PRECISION EIGHT, SEVTEN
182: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
183: * ..
184: * .. Local Scalars ..
185: INTEGER IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP,
186: $ KSTEP, KW
187: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, R1, ROWMAX, T
188: COMPLEX*16 D11, D21, D22, Z
189: * ..
190: * .. External Functions ..
191: LOGICAL LSAME
192: INTEGER IZAMAX
193: EXTERNAL LSAME, IZAMAX
194: * ..
195: * .. External Subroutines ..
196: EXTERNAL ZCOPY, ZDSCAL, ZGEMM, ZGEMV, ZLACGV, ZSWAP
197: * ..
198: * .. Intrinsic Functions ..
199: INTRINSIC ABS, DBLE, DCONJG, DIMAG, MAX, MIN, SQRT
200: * ..
201: * .. Statement Functions ..
202: DOUBLE PRECISION CABS1
203: * ..
204: * .. Statement Function definitions ..
205: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
206: * ..
207: * .. Executable Statements ..
208: *
209: INFO = 0
210: *
211: * Initialize ALPHA for use in choosing pivot block size.
212: *
213: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
214: *
215: IF( LSAME( UPLO, 'U' ) ) THEN
216: *
217: * Factorize the trailing columns of A using the upper triangle
218: * of A and working backwards, and compute the matrix W = U12*D
219: * for use in updating A11 (note that conjg(W) is actually stored)
220: *
221: * K is the main loop index, decreasing from N in steps of 1 or 2
222: *
223: * KW is the column of W which corresponds to column K of A
224: *
225: K = N
226: 10 CONTINUE
227: KW = NB + K - N
228: *
229: * Exit from loop
230: *
231: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
232: $ GO TO 30
233: *
234: * Copy column K of A to column KW of W and update it
235: *
236: CALL ZCOPY( K-1, A( 1, K ), 1, W( 1, KW ), 1 )
237: W( K, KW ) = DBLE( A( K, K ) )
238: IF( K.LT.N ) THEN
239: CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), LDA,
240: $ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
241: W( K, KW ) = DBLE( W( K, KW ) )
242: END IF
243: *
244: KSTEP = 1
245: *
246: * Determine rows and columns to be interchanged and whether
247: * a 1-by-1 or 2-by-2 pivot block will be used
248: *
249: ABSAKK = ABS( DBLE( W( K, KW ) ) )
250: *
251: * IMAX is the row-index of the largest off-diagonal element in
252: * column K, and COLMAX is its absolute value
253: *
254: IF( K.GT.1 ) THEN
255: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
256: COLMAX = CABS1( W( IMAX, KW ) )
257: ELSE
258: COLMAX = ZERO
259: END IF
260: *
261: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
262: *
263: * Column K is zero: set INFO and continue
264: *
265: IF( INFO.EQ.0 )
266: $ INFO = K
267: KP = K
268: A( K, K ) = DBLE( A( K, K ) )
269: ELSE
270: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
271: *
272: * no interchange, use 1-by-1 pivot block
273: *
274: KP = K
275: ELSE
276: *
277: * Copy column IMAX to column KW-1 of W and update it
278: *
279: CALL ZCOPY( IMAX-1, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
280: W( IMAX, KW-1 ) = DBLE( A( IMAX, IMAX ) )
281: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
282: $ W( IMAX+1, KW-1 ), 1 )
283: CALL ZLACGV( K-IMAX, W( IMAX+1, KW-1 ), 1 )
284: IF( K.LT.N ) THEN
285: CALL ZGEMV( 'No transpose', K, N-K, -CONE,
286: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
287: $ CONE, W( 1, KW-1 ), 1 )
288: W( IMAX, KW-1 ) = DBLE( W( IMAX, KW-1 ) )
289: END IF
290: *
291: * JMAX is the column-index of the largest off-diagonal
292: * element in row IMAX, and ROWMAX is its absolute value
293: *
294: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 )
295: ROWMAX = CABS1( W( JMAX, KW-1 ) )
296: IF( IMAX.GT.1 ) THEN
297: JMAX = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
298: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, KW-1 ) ) )
299: END IF
300: *
301: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
302: *
303: * no interchange, use 1-by-1 pivot block
304: *
305: KP = K
306: ELSE IF( ABS( DBLE( W( IMAX, KW-1 ) ) ).GE.ALPHA*ROWMAX )
307: $ THEN
308: *
309: * interchange rows and columns K and IMAX, use 1-by-1
310: * pivot block
311: *
312: KP = IMAX
313: *
314: * copy column KW-1 of W to column KW
315: *
316: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
317: ELSE
318: *
319: * interchange rows and columns K-1 and IMAX, use 2-by-2
320: * pivot block
321: *
322: KP = IMAX
323: KSTEP = 2
324: END IF
325: END IF
326: *
327: KK = K - KSTEP + 1
328: KKW = NB + KK - N
329: *
330: * Updated column KP is already stored in column KKW of W
331: *
332: IF( KP.NE.KK ) THEN
333: *
334: * Copy non-updated column KK to column KP
335: *
336: A( KP, KP ) = DBLE( A( KK, KK ) )
337: CALL ZCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
338: $ LDA )
339: CALL ZLACGV( KK-1-KP, A( KP, KP+1 ), LDA )
340: CALL ZCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
341: *
342: * Interchange rows KK and KP in last KK columns of A and W
343: *
344: IF( KK.LT.N )
345: $ CALL ZSWAP( N-KK, A( KK, KK+1 ), LDA, A( KP, KK+1 ),
346: $ LDA )
347: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
348: $ LDW )
349: END IF
350: *
351: IF( KSTEP.EQ.1 ) THEN
352: *
353: * 1-by-1 pivot block D(k): column KW of W now holds
354: *
355: * W(k) = U(k)*D(k)
356: *
357: * where U(k) is the k-th column of U
358: *
359: * Store U(k) in column k of A
360: *
361: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
362: R1 = ONE / DBLE( A( K, K ) )
363: CALL ZDSCAL( K-1, R1, A( 1, K ), 1 )
364: *
365: * Conjugate W(k)
366: *
367: CALL ZLACGV( K-1, W( 1, KW ), 1 )
368: ELSE
369: *
370: * 2-by-2 pivot block D(k): columns KW and KW-1 of W now
371: * hold
372: *
373: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
374: *
375: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
376: * of U
377: *
378: IF( K.GT.2 ) THEN
379: *
380: * Store U(k) and U(k-1) in columns k and k-1 of A
381: *
382: D21 = W( K-1, KW )
383: D11 = W( K, KW ) / DCONJG( D21 )
384: D22 = W( K-1, KW-1 ) / D21
385: T = ONE / ( DBLE( D11*D22 )-ONE )
386: D21 = T / D21
387: DO 20 J = 1, K - 2
388: A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) )
389: A( J, K ) = DCONJG( D21 )*
390: $ ( D22*W( J, KW )-W( J, KW-1 ) )
391: 20 CONTINUE
392: END IF
393: *
394: * Copy D(k) to A
395: *
396: A( K-1, K-1 ) = W( K-1, KW-1 )
397: A( K-1, K ) = W( K-1, KW )
398: A( K, K ) = W( K, KW )
399: *
400: * Conjugate W(k) and W(k-1)
401: *
402: CALL ZLACGV( K-1, W( 1, KW ), 1 )
403: CALL ZLACGV( K-2, W( 1, KW-1 ), 1 )
404: END IF
405: END IF
406: *
407: * Store details of the interchanges in IPIV
408: *
409: IF( KSTEP.EQ.1 ) THEN
410: IPIV( K ) = KP
411: ELSE
412: IPIV( K ) = -KP
413: IPIV( K-1 ) = -KP
414: END IF
415: *
416: * Decrease K and return to the start of the main loop
417: *
418: K = K - KSTEP
419: GO TO 10
420: *
421: 30 CONTINUE
422: *
423: * Update the upper triangle of A11 (= A(1:k,1:k)) as
424: *
1.8 bertrand 425: * A11 := A11 - U12*D*U12**H = A11 - U12*W**H
1.1 bertrand 426: *
427: * computing blocks of NB columns at a time (note that conjg(W) is
428: * actually stored)
429: *
430: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
431: JB = MIN( NB, K-J+1 )
432: *
433: * Update the upper triangle of the diagonal block
434: *
435: DO 40 JJ = J, J + JB - 1
436: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
437: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
438: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
439: $ A( J, JJ ), 1 )
440: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
441: 40 CONTINUE
442: *
443: * Update the rectangular superdiagonal block
444: *
445: CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB, N-K,
446: $ -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
447: $ CONE, A( 1, J ), LDA )
448: 50 CONTINUE
449: *
450: * Put U12 in standard form by partially undoing the interchanges
451: * in columns k+1:n
452: *
453: J = K + 1
454: 60 CONTINUE
455: JJ = J
456: JP = IPIV( J )
457: IF( JP.LT.0 ) THEN
458: JP = -JP
459: J = J + 1
460: END IF
461: J = J + 1
462: IF( JP.NE.JJ .AND. J.LE.N )
463: $ CALL ZSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA )
464: IF( J.LE.N )
465: $ GO TO 60
466: *
467: * Set KB to the number of columns factorized
468: *
469: KB = N - K
470: *
471: ELSE
472: *
473: * Factorize the leading columns of A using the lower triangle
474: * of A and working forwards, and compute the matrix W = L21*D
475: * for use in updating A22 (note that conjg(W) is actually stored)
476: *
477: * K is the main loop index, increasing from 1 in steps of 1 or 2
478: *
479: K = 1
480: 70 CONTINUE
481: *
482: * Exit from loop
483: *
484: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
485: $ GO TO 90
486: *
487: * Copy column K of A to column K of W and update it
488: *
489: W( K, K ) = DBLE( A( K, K ) )
490: IF( K.LT.N )
491: $ CALL ZCOPY( N-K, A( K+1, K ), 1, W( K+1, K ), 1 )
492: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ), LDA,
493: $ W( K, 1 ), LDW, CONE, W( K, K ), 1 )
494: W( K, K ) = DBLE( W( K, K ) )
495: *
496: KSTEP = 1
497: *
498: * Determine rows and columns to be interchanged and whether
499: * a 1-by-1 or 2-by-2 pivot block will be used
500: *
501: ABSAKK = ABS( DBLE( W( K, K ) ) )
502: *
503: * IMAX is the row-index of the largest off-diagonal element in
504: * column K, and COLMAX is its absolute value
505: *
506: IF( K.LT.N ) THEN
507: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
508: COLMAX = CABS1( W( IMAX, K ) )
509: ELSE
510: COLMAX = ZERO
511: END IF
512: *
513: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
514: *
515: * Column K is zero: set INFO and continue
516: *
517: IF( INFO.EQ.0 )
518: $ INFO = K
519: KP = K
520: A( K, K ) = DBLE( A( K, K ) )
521: ELSE
522: IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
523: *
524: * no interchange, use 1-by-1 pivot block
525: *
526: KP = K
527: ELSE
528: *
529: * Copy column IMAX to column K+1 of W and update it
530: *
531: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 )
532: CALL ZLACGV( IMAX-K, W( K, K+1 ), 1 )
533: W( IMAX, K+1 ) = DBLE( A( IMAX, IMAX ) )
534: IF( IMAX.LT.N )
535: $ CALL ZCOPY( N-IMAX, A( IMAX+1, IMAX ), 1,
536: $ W( IMAX+1, K+1 ), 1 )
537: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
538: $ LDA, W( IMAX, 1 ), LDW, CONE, W( K, K+1 ),
539: $ 1 )
540: W( IMAX, K+1 ) = DBLE( W( IMAX, K+1 ) )
541: *
542: * JMAX is the column-index of the largest off-diagonal
543: * element in row IMAX, and ROWMAX is its absolute value
544: *
545: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
546: ROWMAX = CABS1( W( JMAX, K+1 ) )
547: IF( IMAX.LT.N ) THEN
548: JMAX = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 )
549: ROWMAX = MAX( ROWMAX, CABS1( W( JMAX, K+1 ) ) )
550: END IF
551: *
552: IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
553: *
554: * no interchange, use 1-by-1 pivot block
555: *
556: KP = K
557: ELSE IF( ABS( DBLE( W( IMAX, K+1 ) ) ).GE.ALPHA*ROWMAX )
558: $ THEN
559: *
560: * interchange rows and columns K and IMAX, use 1-by-1
561: * pivot block
562: *
563: KP = IMAX
564: *
565: * copy column K+1 of W to column K
566: *
567: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
568: ELSE
569: *
570: * interchange rows and columns K+1 and IMAX, use 2-by-2
571: * pivot block
572: *
573: KP = IMAX
574: KSTEP = 2
575: END IF
576: END IF
577: *
578: KK = K + KSTEP - 1
579: *
580: * Updated column KP is already stored in column KK of W
581: *
582: IF( KP.NE.KK ) THEN
583: *
584: * Copy non-updated column KK to column KP
585: *
586: A( KP, KP ) = DBLE( A( KK, KK ) )
587: CALL ZCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
588: $ LDA )
589: CALL ZLACGV( KP-KK-1, A( KP, KK+1 ), LDA )
590: IF( KP.LT.N )
591: $ CALL ZCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
592: *
593: * Interchange rows KK and KP in first KK columns of A and W
594: *
595: CALL ZSWAP( KK-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
596: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
597: END IF
598: *
599: IF( KSTEP.EQ.1 ) THEN
600: *
601: * 1-by-1 pivot block D(k): column k of W now holds
602: *
603: * W(k) = L(k)*D(k)
604: *
605: * where L(k) is the k-th column of L
606: *
607: * Store L(k) in column k of A
608: *
609: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
610: IF( K.LT.N ) THEN
611: R1 = ONE / DBLE( A( K, K ) )
612: CALL ZDSCAL( N-K, R1, A( K+1, K ), 1 )
613: *
614: * Conjugate W(k)
615: *
616: CALL ZLACGV( N-K, W( K+1, K ), 1 )
617: END IF
618: ELSE
619: *
620: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
621: *
622: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
623: *
624: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
625: * of L
626: *
627: IF( K.LT.N-1 ) THEN
628: *
629: * Store L(k) and L(k+1) in columns k and k+1 of A
630: *
631: D21 = W( K+1, K )
632: D11 = W( K+1, K+1 ) / D21
633: D22 = W( K, K ) / DCONJG( D21 )
634: T = ONE / ( DBLE( D11*D22 )-ONE )
635: D21 = T / D21
636: DO 80 J = K + 2, N
637: A( J, K ) = DCONJG( D21 )*
638: $ ( D11*W( J, K )-W( J, K+1 ) )
639: A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) )
640: 80 CONTINUE
641: END IF
642: *
643: * Copy D(k) to A
644: *
645: A( K, K ) = W( K, K )
646: A( K+1, K ) = W( K+1, K )
647: A( K+1, K+1 ) = W( K+1, K+1 )
648: *
649: * Conjugate W(k) and W(k+1)
650: *
651: CALL ZLACGV( N-K, W( K+1, K ), 1 )
652: CALL ZLACGV( N-K-1, W( K+2, K+1 ), 1 )
653: END IF
654: END IF
655: *
656: * Store details of the interchanges in IPIV
657: *
658: IF( KSTEP.EQ.1 ) THEN
659: IPIV( K ) = KP
660: ELSE
661: IPIV( K ) = -KP
662: IPIV( K+1 ) = -KP
663: END IF
664: *
665: * Increase K and return to the start of the main loop
666: *
667: K = K + KSTEP
668: GO TO 70
669: *
670: 90 CONTINUE
671: *
672: * Update the lower triangle of A22 (= A(k:n,k:n)) as
673: *
1.8 bertrand 674: * A22 := A22 - L21*D*L21**H = A22 - L21*W**H
1.1 bertrand 675: *
676: * computing blocks of NB columns at a time (note that conjg(W) is
677: * actually stored)
678: *
679: DO 110 J = K, N, NB
680: JB = MIN( NB, N-J+1 )
681: *
682: * Update the lower triangle of the diagonal block
683: *
684: DO 100 JJ = J, J + JB - 1
685: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
686: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
687: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
688: $ A( JJ, JJ ), 1 )
689: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
690: 100 CONTINUE
691: *
692: * Update the rectangular subdiagonal block
693: *
694: IF( J+JB.LE.N )
695: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
696: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
697: $ LDW, CONE, A( J+JB, J ), LDA )
698: 110 CONTINUE
699: *
700: * Put L21 in standard form by partially undoing the interchanges
701: * in columns 1:k-1
702: *
703: J = K - 1
704: 120 CONTINUE
705: JJ = J
706: JP = IPIV( J )
707: IF( JP.LT.0 ) THEN
708: JP = -JP
709: J = J - 1
710: END IF
711: J = J - 1
712: IF( JP.NE.JJ .AND. J.GE.1 )
713: $ CALL ZSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA )
714: IF( J.GE.1 )
715: $ GO TO 120
716: *
717: * Set KB to the number of columns factorized
718: *
719: KB = K - 1
720: *
721: END IF
722: RETURN
723: *
724: * End of ZLAHEF
725: *
726: END
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