1: * \brief \b ZLAHEF_ROOK computes a partial factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (blocked algorithm, calling Level 3 BLAS).
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLAHEF_ROOK + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlahef_rook.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlahef_rook.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlahef_rook.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLAHEF_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: *> ZLAHEF_ROOK computes a partial factorization of a complex Hermitian
39: *> matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting
40: *> 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**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_ROOK is an auxiliary routine called by ZHETRF_ROOK. It uses
53: *> blocked code (calling Level 3 BLAS) to update the submatrix
54: *> A11 (if UPLO = 'U') or 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: *>
114: *> If UPLO = 'U':
115: *> Only the last KB elements of IPIV are set.
116: *>
117: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
118: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
119: *>
120: *> If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
121: *> columns k and -IPIV(k) were interchanged and rows and
122: *> columns k-1 and -IPIV(k-1) were inerchaged,
123: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
124: *>
125: *> If UPLO = 'L':
126: *> Only the first KB elements of IPIV are set.
127: *>
128: *> If IPIV(k) > 0, then rows and columns k and IPIV(k)
129: *> were interchanged and D(k,k) is a 1-by-1 diagonal block.
130: *>
131: *> If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
132: *> columns k and -IPIV(k) were interchanged and rows and
133: *> columns k+1 and -IPIV(k+1) were inerchaged,
134: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
135: *> \endverbatim
136: *>
137: *> \param[out] W
138: *> \verbatim
139: *> W is COMPLEX*16 array, dimension (LDW,NB)
140: *> \endverbatim
141: *>
142: *> \param[in] LDW
143: *> \verbatim
144: *> LDW is INTEGER
145: *> The leading dimension of the array W. LDW >= max(1,N).
146: *> \endverbatim
147: *>
148: *> \param[out] INFO
149: *> \verbatim
150: *> INFO is INTEGER
151: *> = 0: successful exit
152: *> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
153: *> has been completed, but the block diagonal matrix D is
154: *> exactly singular.
155: *> \endverbatim
156: *
157: * Authors:
158: * ========
159: *
160: *> \author Univ. of Tennessee
161: *> \author Univ. of California Berkeley
162: *> \author Univ. of Colorado Denver
163: *> \author NAG Ltd.
164: *
165: *> \ingroup complex16HEcomputational
166: *
167: *> \par Contributors:
168: * ==================
169: *>
170: *> \verbatim
171: *>
172: *> November 2013, Igor Kozachenko,
173: *> Computer Science Division,
174: *> University of California, Berkeley
175: *>
176: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
177: *> School of Mathematics,
178: *> University of Manchester
179: *> \endverbatim
180: *
181: * =====================================================================
182: SUBROUTINE ZLAHEF_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: 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: COMPLEX*16 CONE
204: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
205: DOUBLE PRECISION EIGHT, SEVTEN
206: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
207: * ..
208: * .. Local Scalars ..
209: LOGICAL DONE
210: INTEGER IMAX, ITEMP, II, J, JB, JJ, JMAX, JP1, JP2, K,
211: $ KK, KKW, KP, KSTEP, KW, P
212: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, DTEMP, R1, ROWMAX, T,
213: $ SFMIN
214: COMPLEX*16 D11, D21, D22, 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, ZDSCAL, ZGEMM, ZGEMV, ZLACGV, ZSWAP
224: * ..
225: * .. Intrinsic Functions ..
226: INTRINSIC ABS, DBLE, DCONJG, DIMAG, MAX, MIN, SQRT
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 (note that conjg(W) is actually stored)
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: IF( K.GT.1 )
272: $ CALL ZCOPY( K-1, A( 1, K ), 1, W( 1, KW ), 1 )
273: W( K, KW ) = DBLE( A( K, K ) )
274: IF( K.LT.N ) THEN
275: CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), LDA,
276: $ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
277: W( K, KW ) = DBLE( W( K, KW ) )
278: END IF
279: *
280: * Determine rows and columns to be interchanged and whether
281: * a 1-by-1 or 2-by-2 pivot block will be used
282: *
283: ABSAKK = ABS( DBLE( W( K, KW ) ) )
284: *
285: * IMAX is the row-index of the largest off-diagonal element in
286: * column K, and COLMAX is its absolute value.
287: * Determine both COLMAX and IMAX.
288: *
289: IF( K.GT.1 ) THEN
290: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
291: COLMAX = CABS1( W( IMAX, KW ) )
292: ELSE
293: COLMAX = ZERO
294: END IF
295: *
296: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
297: *
298: * Column K is zero or underflow: set INFO and continue
299: *
300: IF( INFO.EQ.0 )
301: $ INFO = K
302: KP = K
303: A( K, K ) = DBLE( W( K, KW ) )
304: IF( K.GT.1 )
305: $ CALL ZCOPY( K-1, W( 1, KW ), 1, A( 1, K ), 1 )
306: ELSE
307: *
308: * ============================================================
309: *
310: * BEGIN pivot search
311: *
312: * Case(1)
313: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
314: * (used to handle NaN and Inf)
315: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
316: *
317: * no interchange, use 1-by-1 pivot block
318: *
319: KP = K
320: *
321: ELSE
322: *
323: * Lop until pivot found
324: *
325: DONE = .FALSE.
326: *
327: 12 CONTINUE
328: *
329: * BEGIN pivot search loop body
330: *
331: *
332: * Copy column IMAX to column KW-1 of W and update it
333: *
334: IF( IMAX.GT.1 )
335: $ CALL ZCOPY( IMAX-1, A( 1, IMAX ), 1, W( 1, KW-1 ),
336: $ 1 )
337: W( IMAX, KW-1 ) = DBLE( A( IMAX, IMAX ) )
338: *
339: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
340: $ W( IMAX+1, KW-1 ), 1 )
341: CALL ZLACGV( K-IMAX, W( IMAX+1, KW-1 ), 1 )
342: *
343: IF( K.LT.N ) THEN
344: CALL ZGEMV( 'No transpose', K, N-K, -CONE,
345: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
346: $ CONE, W( 1, KW-1 ), 1 )
347: W( IMAX, KW-1 ) = DBLE( W( IMAX, KW-1 ) )
348: END IF
349: *
350: * JMAX is the column-index of the largest off-diagonal
351: * element in row IMAX, and ROWMAX is its absolute value.
352: * Determine both ROWMAX and JMAX.
353: *
354: IF( IMAX.NE.K ) THEN
355: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ),
356: $ 1 )
357: ROWMAX = CABS1( W( JMAX, KW-1 ) )
358: ELSE
359: ROWMAX = ZERO
360: END IF
361: *
362: IF( IMAX.GT.1 ) THEN
363: ITEMP = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
364: DTEMP = CABS1( W( ITEMP, KW-1 ) )
365: IF( DTEMP.GT.ROWMAX ) THEN
366: ROWMAX = DTEMP
367: JMAX = ITEMP
368: END IF
369: END IF
370: *
371: * Case(2)
372: * Equivalent to testing for
373: * ABS( DBLE( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
374: * (used to handle NaN and Inf)
375: *
376: IF( .NOT.( ABS( DBLE( W( IMAX,KW-1 ) ) )
377: $ .LT.ALPHA*ROWMAX ) ) THEN
378: *
379: * interchange rows and columns K and IMAX,
380: * use 1-by-1 pivot block
381: *
382: KP = IMAX
383: *
384: * copy column KW-1 of W to column KW of W
385: *
386: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
387: *
388: DONE = .TRUE.
389: *
390: * Case(3)
391: * Equivalent to testing for ROWMAX.EQ.COLMAX,
392: * (used to handle NaN and Inf)
393: *
394: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
395: $ THEN
396: *
397: * interchange rows and columns K-1 and IMAX,
398: * use 2-by-2 pivot block
399: *
400: KP = IMAX
401: KSTEP = 2
402: DONE = .TRUE.
403: *
404: * Case(4)
405: ELSE
406: *
407: * Pivot not found: set params and repeat
408: *
409: P = IMAX
410: COLMAX = ROWMAX
411: IMAX = JMAX
412: *
413: * Copy updated JMAXth (next IMAXth) column to Kth of W
414: *
415: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
416: *
417: END IF
418: *
419: *
420: * END pivot search loop body
421: *
422: IF( .NOT.DONE ) GOTO 12
423: *
424: END IF
425: *
426: * END pivot search
427: *
428: * ============================================================
429: *
430: * KK is the column of A where pivoting step stopped
431: *
432: KK = K - KSTEP + 1
433: *
434: * KKW is the column of W which corresponds to column KK of A
435: *
436: KKW = NB + KK - N
437: *
438: * Interchange rows and columns P and K.
439: * Updated column P is already stored in column KW of W.
440: *
441: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
442: *
443: * Copy non-updated column K to column P of submatrix A
444: * at step K. No need to copy element into columns
445: * K and K-1 of A for 2-by-2 pivot, since these columns
446: * will be later overwritten.
447: *
448: A( P, P ) = DBLE( A( K, K ) )
449: CALL ZCOPY( K-1-P, A( P+1, K ), 1, A( P, P+1 ),
450: $ LDA )
451: CALL ZLACGV( K-1-P, A( P, P+1 ), LDA )
452: IF( P.GT.1 )
453: $ CALL ZCOPY( P-1, A( 1, K ), 1, A( 1, P ), 1 )
454: *
455: * Interchange rows K and P in the last K+1 to N columns of A
456: * (columns K and K-1 of A for 2-by-2 pivot will be
457: * later overwritten). Interchange rows K and P
458: * in last KKW to NB columns of W.
459: *
460: IF( K.LT.N )
461: $ CALL ZSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ),
462: $ LDA )
463: CALL ZSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ),
464: $ LDW )
465: END IF
466: *
467: * Interchange rows and columns KP and KK.
468: * Updated column KP is already stored in column KKW of W.
469: *
470: IF( KP.NE.KK ) THEN
471: *
472: * Copy non-updated column KK to column KP of submatrix A
473: * at step K. No need to copy element into column K
474: * (or K and K-1 for 2-by-2 pivot) of A, since these columns
475: * will be later overwritten.
476: *
477: A( KP, KP ) = DBLE( A( KK, KK ) )
478: CALL ZCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
479: $ LDA )
480: CALL ZLACGV( KK-1-KP, A( KP, KP+1 ), LDA )
481: IF( KP.GT.1 )
482: $ CALL ZCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
483: *
484: * Interchange rows KK and KP in last K+1 to N columns of A
485: * (columns K (or K and K-1 for 2-by-2 pivot) of A will be
486: * later overwritten). Interchange rows KK and KP
487: * in last KKW to NB columns of W.
488: *
489: IF( K.LT.N )
490: $ CALL ZSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
491: $ LDA )
492: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
493: $ LDW )
494: END IF
495: *
496: IF( KSTEP.EQ.1 ) THEN
497: *
498: * 1-by-1 pivot block D(k): column kw of W now holds
499: *
500: * W(kw) = U(k)*D(k),
501: *
502: * where U(k) is the k-th column of U
503: *
504: * (1) Store subdiag. elements of column U(k)
505: * and 1-by-1 block D(k) in column k of A.
506: * (NOTE: Diagonal element U(k,k) is a UNIT element
507: * and not stored)
508: * A(k,k) := D(k,k) = W(k,kw)
509: * A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k)
510: *
511: * (NOTE: No need to use for Hermitian matrix
512: * A( K, K ) = DBLE( W( K, K) ) to separately copy diagonal
513: * element D(k,k) from W (potentially saves only one load))
514: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
515: IF( K.GT.1 ) THEN
516: *
517: * (NOTE: No need to check if A(k,k) is NOT ZERO,
518: * since that was ensured earlier in pivot search:
519: * case A(k,k) = 0 falls into 2x2 pivot case(3))
520: *
521: * Handle division by a small number
522: *
523: T = DBLE( A( K, K ) )
524: IF( ABS( T ).GE.SFMIN ) THEN
525: R1 = ONE / T
526: CALL ZDSCAL( K-1, R1, A( 1, K ), 1 )
527: ELSE
528: DO 14 II = 1, K-1
529: A( II, K ) = A( II, K ) / T
530: 14 CONTINUE
531: END IF
532: *
533: * (2) Conjugate column W(kw)
534: *
535: CALL ZLACGV( K-1, W( 1, KW ), 1 )
536: END IF
537: *
538: ELSE
539: *
540: * 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold
541: *
542: * ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k)
543: *
544: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
545: * of U
546: *
547: * (1) Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2
548: * block D(k-1:k,k-1:k) in columns k-1 and k of A.
549: * (NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT
550: * block and not stored)
551: * A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw)
552: * A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) =
553: * = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) )
554: *
555: IF( K.GT.2 ) THEN
556: *
557: * Factor out the columns of the inverse of 2-by-2 pivot
558: * block D, so that each column contains 1, to reduce the
559: * number of FLOPS when we multiply panel
560: * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
561: *
562: * D**(-1) = ( d11 cj(d21) )**(-1) =
563: * ( d21 d22 )
564: *
565: * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
566: * ( (-d21) ( d11 ) )
567: *
568: * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
569: *
570: * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
571: * ( ( -1 ) ( d11/conj(d21) ) )
572: *
573: * = 1/(|d21|**2) * 1/(D22*D11-1) *
574: *
575: * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
576: * ( ( -1 ) ( D22 ) )
577: *
578: * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
579: * ( ( -1 ) ( D22 ) )
580: *
581: * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
582: * ( ( -1 ) ( D22 ) )
583: *
584: * Handle division by a small number. (NOTE: order of
585: * operations is important)
586: *
587: * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) )
588: * ( (( -1 ) ) (( D22 ) ) ),
589: *
590: * where D11 = d22/d21,
591: * D22 = d11/conj(d21),
592: * D21 = d21,
593: * T = 1/(D22*D11-1).
594: *
595: * (NOTE: No need to check for division by ZERO,
596: * since that was ensured earlier in pivot search:
597: * (a) d21 != 0 in 2x2 pivot case(4),
598: * since |d21| should be larger than |d11| and |d22|;
599: * (b) (D22*D11 - 1) != 0, since from (a),
600: * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
601: *
602: D21 = W( K-1, KW )
603: D11 = W( K, KW ) / DCONJG( D21 )
604: D22 = W( K-1, KW-1 ) / D21
605: T = ONE / ( DBLE( D11*D22 )-ONE )
606: *
607: * Update elements in columns A(k-1) and A(k) as
608: * dot products of rows of ( W(kw-1) W(kw) ) and columns
609: * of D**(-1)
610: *
611: DO 20 J = 1, K - 2
612: A( J, K-1 ) = T*( ( D11*W( J, KW-1 )-W( J, KW ) ) /
613: $ D21 )
614: A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
615: $ DCONJG( D21 ) )
616: 20 CONTINUE
617: END IF
618: *
619: * Copy D(k) to A
620: *
621: A( K-1, K-1 ) = W( K-1, KW-1 )
622: A( K-1, K ) = W( K-1, KW )
623: A( K, K ) = W( K, KW )
624: *
625: * (2) Conjugate columns W(kw) and W(kw-1)
626: *
627: CALL ZLACGV( K-1, W( 1, KW ), 1 )
628: CALL ZLACGV( K-2, W( 1, KW-1 ), 1 )
629: *
630: END IF
631: *
632: END IF
633: *
634: * Store details of the interchanges in IPIV
635: *
636: IF( KSTEP.EQ.1 ) THEN
637: IPIV( K ) = KP
638: ELSE
639: IPIV( K ) = -P
640: IPIV( K-1 ) = -KP
641: END IF
642: *
643: * Decrease K and return to the start of the main loop
644: *
645: K = K - KSTEP
646: GO TO 10
647: *
648: 30 CONTINUE
649: *
650: * Update the upper triangle of A11 (= A(1:k,1:k)) as
651: *
652: * A11 := A11 - U12*D*U12**H = A11 - U12*W**H
653: *
654: * computing blocks of NB columns at a time (note that conjg(W) is
655: * actually stored)
656: *
657: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
658: JB = MIN( NB, K-J+1 )
659: *
660: * Update the upper triangle of the diagonal block
661: *
662: DO 40 JJ = J, J + JB - 1
663: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
664: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
665: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
666: $ A( J, JJ ), 1 )
667: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
668: 40 CONTINUE
669: *
670: * Update the rectangular superdiagonal block
671: *
672: IF( J.GE.2 )
673: $ CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB, N-K,
674: $ -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
675: $ CONE, A( 1, J ), LDA )
676: 50 CONTINUE
677: *
678: * Put U12 in standard form by partially undoing the interchanges
679: * in of rows in columns k+1:n looping backwards from k+1 to n
680: *
681: J = K + 1
682: 60 CONTINUE
683: *
684: * Undo the interchanges (if any) of rows J and JP2
685: * (or J and JP2, and J+1 and JP1) at each step J
686: *
687: KSTEP = 1
688: JP1 = 1
689: * (Here, J is a diagonal index)
690: JJ = J
691: JP2 = IPIV( J )
692: IF( JP2.LT.0 ) THEN
693: JP2 = -JP2
694: * (Here, J is a diagonal index)
695: J = J + 1
696: JP1 = -IPIV( J )
697: KSTEP = 2
698: END IF
699: * (NOTE: Here, J is used to determine row length. Length N-J+1
700: * of the rows to swap back doesn't include diagonal element)
701: J = J + 1
702: IF( JP2.NE.JJ .AND. J.LE.N )
703: $ CALL ZSWAP( N-J+1, A( JP2, J ), LDA, A( JJ, J ), LDA )
704: JJ = JJ + 1
705: IF( KSTEP.EQ.2 .AND. JP1.NE.JJ .AND. J.LE.N )
706: $ CALL ZSWAP( N-J+1, A( JP1, J ), LDA, A( JJ, J ), LDA )
707: IF( J.LT.N )
708: $ GO TO 60
709: *
710: * Set KB to the number of columns factorized
711: *
712: KB = N - K
713: *
714: ELSE
715: *
716: * Factorize the leading columns of A using the lower triangle
717: * of A and working forwards, and compute the matrix W = L21*D
718: * for use in updating A22 (note that conjg(W) is actually stored)
719: *
720: * K is the main loop index, increasing from 1 in steps of 1 or 2
721: *
722: K = 1
723: 70 CONTINUE
724: *
725: * Exit from loop
726: *
727: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
728: $ GO TO 90
729: *
730: KSTEP = 1
731: P = K
732: *
733: * Copy column K of A to column K of W and update column K of W
734: *
735: W( K, K ) = DBLE( A( K, K ) )
736: IF( K.LT.N )
737: $ CALL ZCOPY( N-K, A( K+1, K ), 1, W( K+1, K ), 1 )
738: IF( K.GT.1 ) THEN
739: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
740: $ LDA, W( K, 1 ), LDW, CONE, W( K, K ), 1 )
741: W( K, K ) = DBLE( W( K, K ) )
742: END IF
743: *
744: * Determine rows and columns to be interchanged and whether
745: * a 1-by-1 or 2-by-2 pivot block will be used
746: *
747: ABSAKK = ABS( DBLE( W( K, K ) ) )
748: *
749: * IMAX is the row-index of the largest off-diagonal element in
750: * column K, and COLMAX is its absolute value.
751: * Determine both COLMAX and IMAX.
752: *
753: IF( K.LT.N ) THEN
754: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
755: COLMAX = CABS1( W( IMAX, K ) )
756: ELSE
757: COLMAX = ZERO
758: END IF
759: *
760: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
761: *
762: * Column K is zero or underflow: set INFO and continue
763: *
764: IF( INFO.EQ.0 )
765: $ INFO = K
766: KP = K
767: A( K, K ) = DBLE( W( K, K ) )
768: IF( K.LT.N )
769: $ CALL ZCOPY( N-K, W( K+1, K ), 1, A( K+1, K ), 1 )
770: ELSE
771: *
772: * ============================================================
773: *
774: * BEGIN pivot search
775: *
776: * Case(1)
777: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
778: * (used to handle NaN and Inf)
779: *
780: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
781: *
782: * no interchange, use 1-by-1 pivot block
783: *
784: KP = K
785: *
786: ELSE
787: *
788: DONE = .FALSE.
789: *
790: * Loop until pivot found
791: *
792: 72 CONTINUE
793: *
794: * BEGIN pivot search loop body
795: *
796: *
797: * Copy column IMAX to column k+1 of W and update it
798: *
799: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
800: CALL ZLACGV( IMAX-K, W( K, K+1 ), 1 )
801: W( IMAX, K+1 ) = DBLE( A( IMAX, IMAX ) )
802: *
803: IF( IMAX.LT.N )
804: $ CALL ZCOPY( N-IMAX, A( IMAX+1, IMAX ), 1,
805: $ W( IMAX+1, K+1 ), 1 )
806: *
807: IF( K.GT.1 ) THEN
808: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE,
809: $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
810: $ CONE, W( K, K+1 ), 1 )
811: W( IMAX, K+1 ) = DBLE( W( IMAX, K+1 ) )
812: END IF
813: *
814: * JMAX is the column-index of the largest off-diagonal
815: * element in row IMAX, and ROWMAX is its absolute value.
816: * Determine both ROWMAX and JMAX.
817: *
818: IF( IMAX.NE.K ) THEN
819: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
820: ROWMAX = CABS1( W( JMAX, K+1 ) )
821: ELSE
822: ROWMAX = ZERO
823: END IF
824: *
825: IF( IMAX.LT.N ) THEN
826: ITEMP = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
827: DTEMP = CABS1( W( ITEMP, K+1 ) )
828: IF( DTEMP.GT.ROWMAX ) THEN
829: ROWMAX = DTEMP
830: JMAX = ITEMP
831: END IF
832: END IF
833: *
834: * Case(2)
835: * Equivalent to testing for
836: * ABS( DBLE( W( IMAX,K+1 ) ) ).GE.ALPHA*ROWMAX
837: * (used to handle NaN and Inf)
838: *
839: IF( .NOT.( ABS( DBLE( W( IMAX,K+1 ) ) )
840: $ .LT.ALPHA*ROWMAX ) ) THEN
841: *
842: * interchange rows and columns K and IMAX,
843: * use 1-by-1 pivot block
844: *
845: KP = IMAX
846: *
847: * copy column K+1 of W to column K of W
848: *
849: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
850: *
851: DONE = .TRUE.
852: *
853: * Case(3)
854: * Equivalent to testing for ROWMAX.EQ.COLMAX,
855: * (used to handle NaN and Inf)
856: *
857: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
858: $ THEN
859: *
860: * interchange rows and columns K+1 and IMAX,
861: * use 2-by-2 pivot block
862: *
863: KP = IMAX
864: KSTEP = 2
865: DONE = .TRUE.
866: *
867: * Case(4)
868: ELSE
869: *
870: * Pivot not found: set params and repeat
871: *
872: P = IMAX
873: COLMAX = ROWMAX
874: IMAX = JMAX
875: *
876: * Copy updated JMAXth (next IMAXth) column to Kth of W
877: *
878: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
879: *
880: END IF
881: *
882: *
883: * End pivot search loop body
884: *
885: IF( .NOT.DONE ) GOTO 72
886: *
887: END IF
888: *
889: * END pivot search
890: *
891: * ============================================================
892: *
893: * KK is the column of A where pivoting step stopped
894: *
895: KK = K + KSTEP - 1
896: *
897: * Interchange rows and columns P and K (only for 2-by-2 pivot).
898: * Updated column P is already stored in column K of W.
899: *
900: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
901: *
902: * Copy non-updated column KK-1 to column P of submatrix A
903: * at step K. No need to copy element into columns
904: * K and K+1 of A for 2-by-2 pivot, since these columns
905: * will be later overwritten.
906: *
907: A( P, P ) = DBLE( A( K, K ) )
908: CALL ZCOPY( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA )
909: CALL ZLACGV( P-K-1, A( P, K+1 ), LDA )
910: IF( P.LT.N )
911: $ CALL ZCOPY( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
912: *
913: * Interchange rows K and P in first K-1 columns of A
914: * (columns K and K+1 of A for 2-by-2 pivot will be
915: * later overwritten). Interchange rows K and P
916: * in first KK columns of W.
917: *
918: IF( K.GT.1 )
919: $ CALL ZSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA )
920: CALL ZSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
921: END IF
922: *
923: * Interchange rows and columns KP and KK.
924: * Updated column KP is already stored in column KK of W.
925: *
926: IF( KP.NE.KK ) THEN
927: *
928: * Copy non-updated column KK to column KP of submatrix A
929: * at step K. No need to copy element into column K
930: * (or K and K+1 for 2-by-2 pivot) of A, since these columns
931: * will be later overwritten.
932: *
933: A( KP, KP ) = DBLE( A( KK, KK ) )
934: CALL ZCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
935: $ LDA )
936: CALL ZLACGV( KP-KK-1, A( KP, KK+1 ), LDA )
937: IF( KP.LT.N )
938: $ CALL ZCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
939: *
940: * Interchange rows KK and KP in first K-1 columns of A
941: * (column K (or K and K+1 for 2-by-2 pivot) of A will be
942: * later overwritten). Interchange rows KK and KP
943: * in first KK columns of W.
944: *
945: IF( K.GT.1 )
946: $ CALL ZSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
947: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
948: END IF
949: *
950: IF( KSTEP.EQ.1 ) THEN
951: *
952: * 1-by-1 pivot block D(k): column k of W now holds
953: *
954: * W(k) = L(k)*D(k),
955: *
956: * where L(k) is the k-th column of L
957: *
958: * (1) Store subdiag. elements of column L(k)
959: * and 1-by-1 block D(k) in column k of A.
960: * (NOTE: Diagonal element L(k,k) is a UNIT element
961: * and not stored)
962: * A(k,k) := D(k,k) = W(k,k)
963: * A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k)
964: *
965: * (NOTE: No need to use for Hermitian matrix
966: * A( K, K ) = DBLE( W( K, K) ) to separately copy diagonal
967: * element D(k,k) from W (potentially saves only one load))
968: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
969: IF( K.LT.N ) THEN
970: *
971: * (NOTE: No need to check if A(k,k) is NOT ZERO,
972: * since that was ensured earlier in pivot search:
973: * case A(k,k) = 0 falls into 2x2 pivot case(3))
974: *
975: * Handle division by a small number
976: *
977: T = DBLE( A( K, K ) )
978: IF( ABS( T ).GE.SFMIN ) THEN
979: R1 = ONE / T
980: CALL ZDSCAL( N-K, R1, A( K+1, K ), 1 )
981: ELSE
982: DO 74 II = K + 1, N
983: A( II, K ) = A( II, K ) / T
984: 74 CONTINUE
985: END IF
986: *
987: * (2) Conjugate column W(k)
988: *
989: CALL ZLACGV( N-K, W( K+1, K ), 1 )
990: END IF
991: *
992: ELSE
993: *
994: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
995: *
996: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
997: *
998: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
999: * of L
1000: *
1001: * (1) Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2
1002: * block D(k:k+1,k:k+1) in columns k and k+1 of A.
1003: * NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT
1004: * block and not stored.
1005: * A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1)
1006: * A(k+2:N,k:k+1) := L(k+2:N,k:k+1) =
1007: * = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) )
1008: *
1009: IF( K.LT.N-1 ) THEN
1010: *
1011: * Factor out the columns of the inverse of 2-by-2 pivot
1012: * block D, so that each column contains 1, to reduce the
1013: * number of FLOPS when we multiply panel
1014: * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
1015: *
1016: * D**(-1) = ( d11 cj(d21) )**(-1) =
1017: * ( d21 d22 )
1018: *
1019: * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
1020: * ( (-d21) ( d11 ) )
1021: *
1022: * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
1023: *
1024: * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
1025: * ( ( -1 ) ( d11/conj(d21) ) )
1026: *
1027: * = 1/(|d21|**2) * 1/(D22*D11-1) *
1028: *
1029: * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
1030: * ( ( -1 ) ( D22 ) )
1031: *
1032: * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
1033: * ( ( -1 ) ( D22 ) )
1034: *
1035: * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
1036: * ( ( -1 ) ( D22 ) )
1037: *
1038: * Handle division by a small number. (NOTE: order of
1039: * operations is important)
1040: *
1041: * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) )
1042: * ( (( -1 ) ) (( D22 ) ) ),
1043: *
1044: * where D11 = d22/d21,
1045: * D22 = d11/conj(d21),
1046: * D21 = d21,
1047: * T = 1/(D22*D11-1).
1048: *
1049: * (NOTE: No need to check for division by ZERO,
1050: * since that was ensured earlier in pivot search:
1051: * (a) d21 != 0 in 2x2 pivot case(4),
1052: * since |d21| should be larger than |d11| and |d22|;
1053: * (b) (D22*D11 - 1) != 0, since from (a),
1054: * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
1055: *
1056: D21 = W( K+1, K )
1057: D11 = W( K+1, K+1 ) / D21
1058: D22 = W( K, K ) / DCONJG( D21 )
1059: T = ONE / ( DBLE( D11*D22 )-ONE )
1060: *
1061: * Update elements in columns A(k) and A(k+1) as
1062: * dot products of rows of ( W(k) W(k+1) ) and columns
1063: * of D**(-1)
1064: *
1065: DO 80 J = K + 2, N
1066: A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
1067: $ DCONJG( D21 ) )
1068: A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
1069: $ D21 )
1070: 80 CONTINUE
1071: END IF
1072: *
1073: * Copy D(k) to A
1074: *
1075: A( K, K ) = W( K, K )
1076: A( K+1, K ) = W( K+1, K )
1077: A( K+1, K+1 ) = W( K+1, K+1 )
1078: *
1079: * (2) Conjugate columns W(k) and W(k+1)
1080: *
1081: CALL ZLACGV( N-K, W( K+1, K ), 1 )
1082: CALL ZLACGV( N-K-1, W( K+2, K+1 ), 1 )
1083: *
1084: END IF
1085: *
1086: END IF
1087: *
1088: * Store details of the interchanges in IPIV
1089: *
1090: IF( KSTEP.EQ.1 ) THEN
1091: IPIV( K ) = KP
1092: ELSE
1093: IPIV( K ) = -P
1094: IPIV( K+1 ) = -KP
1095: END IF
1096: *
1097: * Increase K and return to the start of the main loop
1098: *
1099: K = K + KSTEP
1100: GO TO 70
1101: *
1102: 90 CONTINUE
1103: *
1104: * Update the lower triangle of A22 (= A(k:n,k:n)) as
1105: *
1106: * A22 := A22 - L21*D*L21**H = A22 - L21*W**H
1107: *
1108: * computing blocks of NB columns at a time (note that conjg(W) is
1109: * actually stored)
1110: *
1111: DO 110 J = K, N, NB
1112: JB = MIN( NB, N-J+1 )
1113: *
1114: * Update the lower triangle of the diagonal block
1115: *
1116: DO 100 JJ = J, J + JB - 1
1117: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
1118: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
1119: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
1120: $ A( JJ, JJ ), 1 )
1121: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
1122: 100 CONTINUE
1123: *
1124: * Update the rectangular subdiagonal block
1125: *
1126: IF( J+JB.LE.N )
1127: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
1128: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
1129: $ LDW, CONE, A( J+JB, J ), LDA )
1130: 110 CONTINUE
1131: *
1132: * Put L21 in standard form by partially undoing the interchanges
1133: * of rows in columns 1:k-1 looping backwards from k-1 to 1
1134: *
1135: J = K - 1
1136: 120 CONTINUE
1137: *
1138: * Undo the interchanges (if any) of rows J and JP2
1139: * (or J and JP2, and J-1 and JP1) at each step J
1140: *
1141: KSTEP = 1
1142: JP1 = 1
1143: * (Here, J is a diagonal index)
1144: JJ = J
1145: JP2 = IPIV( J )
1146: IF( JP2.LT.0 ) THEN
1147: JP2 = -JP2
1148: * (Here, J is a diagonal index)
1149: J = J - 1
1150: JP1 = -IPIV( J )
1151: KSTEP = 2
1152: END IF
1153: * (NOTE: Here, J is used to determine row length. Length J
1154: * of the rows to swap back doesn't include diagonal element)
1155: J = J - 1
1156: IF( JP2.NE.JJ .AND. J.GE.1 )
1157: $ CALL ZSWAP( J, A( JP2, 1 ), LDA, A( JJ, 1 ), LDA )
1158: JJ = JJ -1
1159: IF( KSTEP.EQ.2 .AND. JP1.NE.JJ .AND. J.GE.1 )
1160: $ CALL ZSWAP( J, A( JP1, 1 ), LDA, A( JJ, 1 ), LDA )
1161: IF( J.GT.1 )
1162: $ GO TO 120
1163: *
1164: * Set KB to the number of columns factorized
1165: *
1166: KB = K - 1
1167: *
1168: END IF
1169: RETURN
1170: *
1171: * End of ZLAHEF_ROOK
1172: *
1173: END
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