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1.1 bertrand 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: *> \date November 2013
166: *
167: *> \ingroup complex16HEcomputational
168: *
169: *> \par Contributors:
170: * ==================
171: *>
172: *> \verbatim
173: *>
174: *> November 2013, Igor Kozachenko,
175: *> Computer Science Division,
176: *> University of California, Berkeley
177: *>
178: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
179: *> School of Mathematics,
180: *> University of Manchester
181: *> \endverbatim
182: *
183: * =====================================================================
184: SUBROUTINE ZLAHEF_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: COMPLEX*16 CONE
207: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
208: DOUBLE PRECISION EIGHT, SEVTEN
209: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
210: * ..
211: * .. Local Scalars ..
212: LOGICAL DONE
213: INTEGER IMAX, ITEMP, II, J, JB, JJ, JMAX, JP1, JP2, K,
214: $ KK, KKW, KP, KSTEP, KW, P
215: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, DTEMP, R1, ROWMAX, T,
216: $ SFMIN
217: COMPLEX*16 D11, D21, D22, 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, ZDSCAL, ZGEMM, ZGEMV, ZLACGV, ZSWAP
227: * ..
228: * .. Intrinsic Functions ..
229: INTRINSIC ABS, DBLE, DCONJG, DIMAG, MAX, MIN, SQRT
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 (note that conjg(W) is actually stored)
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: IF( K.GT.1 )
275: $ CALL ZCOPY( K-1, A( 1, K ), 1, W( 1, KW ), 1 )
276: W( K, KW ) = DBLE( A( K, K ) )
277: IF( K.LT.N ) THEN
278: CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), LDA,
279: $ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
280: W( K, KW ) = DBLE( W( K, KW ) )
281: END IF
282: *
283: * Determine rows and columns to be interchanged and whether
284: * a 1-by-1 or 2-by-2 pivot block will be used
285: *
286: ABSAKK = ABS( DBLE( W( K, KW ) ) )
287: *
288: * IMAX is the row-index of the largest off-diagonal element in
289: * column K, and COLMAX is its absolute value.
290: * Determine both COLMAX and IMAX.
291: *
292: IF( K.GT.1 ) THEN
293: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
294: COLMAX = CABS1( W( IMAX, KW ) )
295: ELSE
296: COLMAX = ZERO
297: END IF
298: *
299: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
300: *
301: * Column K is zero or underflow: set INFO and continue
302: *
303: IF( INFO.EQ.0 )
304: $ INFO = K
305: KP = K
306: A( K, K ) = DBLE( W( K, KW ) )
307: IF( K.GT.1 )
308: $ CALL ZCOPY( K-1, W( 1, KW ), 1, A( 1, K ), 1 )
309: ELSE
310: *
311: * ============================================================
312: *
313: * BEGIN pivot search
314: *
315: * Case(1)
316: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
317: * (used to handle NaN and Inf)
318: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
319: *
320: * no interchange, use 1-by-1 pivot block
321: *
322: KP = K
323: *
324: ELSE
325: *
326: * Lop until pivot found
327: *
328: DONE = .FALSE.
329: *
330: 12 CONTINUE
331: *
332: * BEGIN pivot search loop body
333: *
334: *
335: * Copy column IMAX to column KW-1 of W and update it
336: *
337: IF( IMAX.GT.1 )
338: $ CALL ZCOPY( IMAX-1, A( 1, IMAX ), 1, W( 1, KW-1 ),
339: $ 1 )
340: W( IMAX, KW-1 ) = DBLE( A( IMAX, IMAX ) )
341: *
342: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
343: $ W( IMAX+1, KW-1 ), 1 )
344: CALL ZLACGV( K-IMAX, W( IMAX+1, KW-1 ), 1 )
345: *
346: IF( K.LT.N ) THEN
347: CALL ZGEMV( 'No transpose', K, N-K, -CONE,
348: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
349: $ CONE, W( 1, KW-1 ), 1 )
350: W( IMAX, KW-1 ) = DBLE( W( IMAX, KW-1 ) )
351: END IF
352: *
353: * JMAX is the column-index of the largest off-diagonal
354: * element in row IMAX, and ROWMAX is its absolute value.
355: * Determine both ROWMAX and JMAX.
356: *
357: IF( IMAX.NE.K ) THEN
358: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ),
359: $ 1 )
360: ROWMAX = CABS1( W( JMAX, KW-1 ) )
361: ELSE
362: ROWMAX = ZERO
363: END IF
364: *
365: IF( IMAX.GT.1 ) THEN
366: ITEMP = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
367: DTEMP = CABS1( W( ITEMP, KW-1 ) )
368: IF( DTEMP.GT.ROWMAX ) THEN
369: ROWMAX = DTEMP
370: JMAX = ITEMP
371: END IF
372: END IF
373: *
374: * Case(2)
375: * Equivalent to testing for
376: * ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
377: * (used to handle NaN and Inf)
378: *
379: IF( .NOT.( ABS( DBLE( W( IMAX,KW-1 ) ) )
380: $ .LT.ALPHA*ROWMAX ) ) THEN
381: *
382: * interchange rows and columns K and IMAX,
383: * use 1-by-1 pivot block
384: *
385: KP = IMAX
386: *
387: * copy column KW-1 of W to column KW of W
388: *
389: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
390: *
391: DONE = .TRUE.
392: *
393: * Case(3)
394: * Equivalent to testing for ROWMAX.EQ.COLMAX,
395: * (used to handle NaN and Inf)
396: *
397: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
398: $ THEN
399: *
400: * interchange rows and columns K-1 and IMAX,
401: * use 2-by-2 pivot block
402: *
403: KP = IMAX
404: KSTEP = 2
405: DONE = .TRUE.
406: *
407: * Case(4)
408: ELSE
409: *
410: * Pivot not found: set params and repeat
411: *
412: P = IMAX
413: COLMAX = ROWMAX
414: IMAX = JMAX
415: *
416: * Copy updated JMAXth (next IMAXth) column to Kth of W
417: *
418: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
419: *
420: END IF
421: *
422: *
423: * END pivot search loop body
424: *
425: IF( .NOT.DONE ) GOTO 12
426: *
427: END IF
428: *
429: * END pivot search
430: *
431: * ============================================================
432: *
433: * KK is the column of A where pivoting step stopped
434: *
435: KK = K - KSTEP + 1
436: *
437: * KKW is the column of W which corresponds to column KK of A
438: *
439: KKW = NB + KK - N
440: *
441: * Interchange rows and columns P and K.
442: * Updated column P is already stored in column KW of W.
443: *
444: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
445: *
446: * Copy non-updated column K to column P of submatrix A
447: * at step K. No need to copy element into columns
448: * K and K-1 of A for 2-by-2 pivot, since these columns
449: * will be later overwritten.
450: *
451: A( P, P ) = DBLE( A( K, K ) )
452: CALL ZCOPY( K-1-P, A( P+1, K ), 1, A( P, P+1 ),
453: $ LDA )
454: CALL ZLACGV( K-1-P, A( P, P+1 ), LDA )
455: IF( P.GT.1 )
456: $ CALL ZCOPY( P-1, A( 1, K ), 1, A( 1, P ), 1 )
457: *
458: * Interchange rows K and P in the last K+1 to N columns of A
459: * (columns K and K-1 of A for 2-by-2 pivot will be
460: * later overwritten). Interchange rows K and P
461: * in last KKW to NB columns of W.
462: *
463: IF( K.LT.N )
464: $ CALL ZSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ),
465: $ LDA )
466: CALL ZSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ),
467: $ LDW )
468: END IF
469: *
470: * Interchange rows and columns KP and KK.
471: * Updated column KP is already stored in column KKW of W.
472: *
473: IF( KP.NE.KK ) THEN
474: *
475: * Copy non-updated column KK to column KP of submatrix A
476: * at step K. No need to copy element into column K
477: * (or K and K-1 for 2-by-2 pivot) of A, since these columns
478: * will be later overwritten.
479: *
480: A( KP, KP ) = DBLE( A( KK, KK ) )
481: CALL ZCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
482: $ LDA )
483: CALL ZLACGV( KK-1-KP, A( KP, KP+1 ), LDA )
484: IF( KP.GT.1 )
485: $ CALL ZCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
486: *
487: * Interchange rows KK and KP in last K+1 to N columns of A
488: * (columns K (or K and K-1 for 2-by-2 pivot) of A will be
489: * later overwritten). Interchange rows KK and KP
490: * in last KKW to NB columns of W.
491: *
492: IF( K.LT.N )
493: $ CALL ZSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
494: $ LDA )
495: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
496: $ LDW )
497: END IF
498: *
499: IF( KSTEP.EQ.1 ) THEN
500: *
501: * 1-by-1 pivot block D(k): column kw of W now holds
502: *
503: * W(kw) = U(k)*D(k),
504: *
505: * where U(k) is the k-th column of U
506: *
507: * (1) Store subdiag. elements of column U(k)
508: * and 1-by-1 block D(k) in column k of A.
509: * (NOTE: Diagonal element U(k,k) is a UNIT element
510: * and not stored)
511: * A(k,k) := D(k,k) = W(k,kw)
512: * A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k)
513: *
514: * (NOTE: No need to use for Hermitian matrix
515: * A( K, K ) = REAL( W( K, K) ) to separately copy diagonal
516: * element D(k,k) from W (potentially saves only one load))
517: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
518: IF( K.GT.1 ) THEN
519: *
520: * (NOTE: No need to check if A(k,k) is NOT ZERO,
521: * since that was ensured earlier in pivot search:
522: * case A(k,k) = 0 falls into 2x2 pivot case(3))
523: *
524: * Handle division by a small number
525: *
526: T = DBLE( A( K, K ) )
527: IF( ABS( T ).GE.SFMIN ) THEN
528: R1 = ONE / T
529: CALL ZDSCAL( K-1, R1, A( 1, K ), 1 )
530: ELSE
531: DO 14 II = 1, K-1
532: A( II, K ) = A( II, K ) / T
533: 14 CONTINUE
534: END IF
535: *
536: * (2) Conjugate column W(kw)
537: *
538: CALL ZLACGV( K-1, W( 1, KW ), 1 )
539: END IF
540: *
541: ELSE
542: *
543: * 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold
544: *
545: * ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k)
546: *
547: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
548: * of U
549: *
550: * (1) Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2
551: * block D(k-1:k,k-1:k) in columns k-1 and k of A.
552: * (NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT
553: * block and not stored)
554: * A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw)
555: * A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) =
556: * = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) )
557: *
558: IF( K.GT.2 ) THEN
559: *
560: * Factor out the columns of the inverse of 2-by-2 pivot
561: * block D, so that each column contains 1, to reduce the
562: * number of FLOPS when we multiply panel
563: * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
564: *
565: * D**(-1) = ( d11 cj(d21) )**(-1) =
566: * ( d21 d22 )
567: *
568: * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
569: * ( (-d21) ( d11 ) )
570: *
571: * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
572: *
573: * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
574: * ( ( -1 ) ( d11/conj(d21) ) )
575: *
576: * = 1/(|d21|**2) * 1/(D22*D11-1) *
577: *
578: * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
579: * ( ( -1 ) ( D22 ) )
580: *
581: * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
582: * ( ( -1 ) ( D22 ) )
583: *
584: * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
585: * ( ( -1 ) ( D22 ) )
586: *
587: * Handle division by a small number. (NOTE: order of
588: * operations is important)
589: *
590: * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) )
591: * ( (( -1 ) ) (( D22 ) ) ),
592: *
593: * where D11 = d22/d21,
594: * D22 = d11/conj(d21),
595: * D21 = d21,
596: * T = 1/(D22*D11-1).
597: *
598: * (NOTE: No need to check for division by ZERO,
599: * since that was ensured earlier in pivot search:
600: * (a) d21 != 0 in 2x2 pivot case(4),
601: * since |d21| should be larger than |d11| and |d22|;
602: * (b) (D22*D11 - 1) != 0, since from (a),
603: * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
604: *
605: D21 = W( K-1, KW )
606: D11 = W( K, KW ) / DCONJG( D21 )
607: D22 = W( K-1, KW-1 ) / D21
608: T = ONE / ( DBLE( D11*D22 )-ONE )
609: *
610: * Update elements in columns A(k-1) and A(k) as
611: * dot products of rows of ( W(kw-1) W(kw) ) and columns
612: * of D**(-1)
613: *
614: DO 20 J = 1, K - 2
615: A( J, K-1 ) = T*( ( D11*W( J, KW-1 )-W( J, KW ) ) /
616: $ D21 )
617: A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
618: $ DCONJG( D21 ) )
619: 20 CONTINUE
620: END IF
621: *
622: * Copy D(k) to A
623: *
624: A( K-1, K-1 ) = W( K-1, KW-1 )
625: A( K-1, K ) = W( K-1, KW )
626: A( K, K ) = W( K, KW )
627: *
628: * (2) Conjugate columns W(kw) and W(kw-1)
629: *
630: CALL ZLACGV( K-1, W( 1, KW ), 1 )
631: CALL ZLACGV( K-2, W( 1, KW-1 ), 1 )
632: *
633: END IF
634: *
635: END IF
636: *
637: * Store details of the interchanges in IPIV
638: *
639: IF( KSTEP.EQ.1 ) THEN
640: IPIV( K ) = KP
641: ELSE
642: IPIV( K ) = -P
643: IPIV( K-1 ) = -KP
644: END IF
645: *
646: * Decrease K and return to the start of the main loop
647: *
648: K = K - KSTEP
649: GO TO 10
650: *
651: 30 CONTINUE
652: *
653: * Update the upper triangle of A11 (= A(1:k,1:k)) as
654: *
655: * A11 := A11 - U12*D*U12**H = A11 - U12*W**H
656: *
657: * computing blocks of NB columns at a time (note that conjg(W) is
658: * actually stored)
659: *
660: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
661: JB = MIN( NB, K-J+1 )
662: *
663: * Update the upper triangle of the diagonal block
664: *
665: DO 40 JJ = J, J + JB - 1
666: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
667: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
668: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
669: $ A( J, JJ ), 1 )
670: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
671: 40 CONTINUE
672: *
673: * Update the rectangular superdiagonal block
674: *
675: IF( J.GE.2 )
676: $ CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB, N-K,
677: $ -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
678: $ CONE, A( 1, J ), LDA )
679: 50 CONTINUE
680: *
681: * Put U12 in standard form by partially undoing the interchanges
682: * in of rows in columns k+1:n looping backwards from k+1 to n
683: *
684: J = K + 1
685: 60 CONTINUE
686: *
687: * Undo the interchanges (if any) of rows J and JP2
688: * (or J and JP2, and J+1 and JP1) at each step J
689: *
690: KSTEP = 1
691: JP1 = 1
692: * (Here, J is a diagonal index)
693: JJ = J
694: JP2 = IPIV( J )
695: IF( JP2.LT.0 ) THEN
696: JP2 = -JP2
697: * (Here, J is a diagonal index)
698: J = J + 1
699: JP1 = -IPIV( J )
700: KSTEP = 2
701: END IF
702: * (NOTE: Here, J is used to determine row length. Length N-J+1
703: * of the rows to swap back doesn't include diagonal element)
704: J = J + 1
705: IF( JP2.NE.JJ .AND. J.LE.N )
706: $ CALL ZSWAP( N-J+1, A( JP2, J ), LDA, A( JJ, J ), LDA )
707: JJ = JJ + 1
708: IF( KSTEP.EQ.2 .AND. JP1.NE.JJ .AND. J.LE.N )
709: $ CALL ZSWAP( N-J+1, A( JP1, J ), LDA, A( JJ, J ), LDA )
710: IF( J.LT.N )
711: $ GO TO 60
712: *
713: * Set KB to the number of columns factorized
714: *
715: KB = N - K
716: *
717: ELSE
718: *
719: * Factorize the leading columns of A using the lower triangle
720: * of A and working forwards, and compute the matrix W = L21*D
721: * for use in updating A22 (note that conjg(W) is actually stored)
722: *
723: * K is the main loop index, increasing from 1 in steps of 1 or 2
724: *
725: K = 1
726: 70 CONTINUE
727: *
728: * Exit from loop
729: *
730: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
731: $ GO TO 90
732: *
733: KSTEP = 1
734: P = K
735: *
736: * Copy column K of A to column K of W and update column K of W
737: *
738: W( K, K ) = DBLE( A( K, K ) )
739: IF( K.LT.N )
740: $ CALL ZCOPY( N-K, A( K+1, K ), 1, W( K+1, K ), 1 )
741: IF( K.GT.1 ) THEN
742: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
743: $ LDA, W( K, 1 ), LDW, CONE, W( K, K ), 1 )
744: W( K, K ) = DBLE( W( K, K ) )
745: END IF
746: *
747: * Determine rows and columns to be interchanged and whether
748: * a 1-by-1 or 2-by-2 pivot block will be used
749: *
750: ABSAKK = ABS( DBLE( W( K, K ) ) )
751: *
752: * IMAX is the row-index of the largest off-diagonal element in
753: * column K, and COLMAX is its absolute value.
754: * Determine both COLMAX and IMAX.
755: *
756: IF( K.LT.N ) THEN
757: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
758: COLMAX = CABS1( W( IMAX, K ) )
759: ELSE
760: COLMAX = ZERO
761: END IF
762: *
763: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
764: *
765: * Column K is zero or underflow: set INFO and continue
766: *
767: IF( INFO.EQ.0 )
768: $ INFO = K
769: KP = K
770: A( K, K ) = DBLE( W( K, K ) )
771: IF( K.LT.N )
772: $ CALL ZCOPY( N-K, W( K+1, K ), 1, A( K+1, K ), 1 )
773: ELSE
774: *
775: * ============================================================
776: *
777: * BEGIN pivot search
778: *
779: * Case(1)
780: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
781: * (used to handle NaN and Inf)
782: *
783: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
784: *
785: * no interchange, use 1-by-1 pivot block
786: *
787: KP = K
788: *
789: ELSE
790: *
791: DONE = .FALSE.
792: *
793: * Loop until pivot found
794: *
795: 72 CONTINUE
796: *
797: * BEGIN pivot search loop body
798: *
799: *
800: * Copy column IMAX to column k+1 of W and update it
801: *
802: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
803: CALL ZLACGV( IMAX-K, W( K, K+1 ), 1 )
804: W( IMAX, K+1 ) = DBLE( A( IMAX, IMAX ) )
805: *
806: IF( IMAX.LT.N )
807: $ CALL ZCOPY( N-IMAX, A( IMAX+1, IMAX ), 1,
808: $ W( IMAX+1, K+1 ), 1 )
809: *
810: IF( K.GT.1 ) THEN
811: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE,
812: $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
813: $ CONE, W( K, K+1 ), 1 )
814: W( IMAX, K+1 ) = DBLE( W( IMAX, K+1 ) )
815: END IF
816: *
817: * JMAX is the column-index of the largest off-diagonal
818: * element in row IMAX, and ROWMAX is its absolute value.
819: * Determine both ROWMAX and JMAX.
820: *
821: IF( IMAX.NE.K ) THEN
822: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
823: ROWMAX = CABS1( W( JMAX, K+1 ) )
824: ELSE
825: ROWMAX = ZERO
826: END IF
827: *
828: IF( IMAX.LT.N ) THEN
829: ITEMP = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
830: DTEMP = CABS1( W( ITEMP, K+1 ) )
831: IF( DTEMP.GT.ROWMAX ) THEN
832: ROWMAX = DTEMP
833: JMAX = ITEMP
834: END IF
835: END IF
836: *
837: * Case(2)
838: * Equivalent to testing for
839: * ABS( REAL( W( IMAX,K+1 ) ) ).GE.ALPHA*ROWMAX
840: * (used to handle NaN and Inf)
841: *
842: IF( .NOT.( ABS( DBLE( W( IMAX,K+1 ) ) )
843: $ .LT.ALPHA*ROWMAX ) ) THEN
844: *
845: * interchange rows and columns K and IMAX,
846: * use 1-by-1 pivot block
847: *
848: KP = IMAX
849: *
850: * copy column K+1 of W to column K of W
851: *
852: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
853: *
854: DONE = .TRUE.
855: *
856: * Case(3)
857: * Equivalent to testing for ROWMAX.EQ.COLMAX,
858: * (used to handle NaN and Inf)
859: *
860: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
861: $ THEN
862: *
863: * interchange rows and columns K+1 and IMAX,
864: * use 2-by-2 pivot block
865: *
866: KP = IMAX
867: KSTEP = 2
868: DONE = .TRUE.
869: *
870: * Case(4)
871: ELSE
872: *
873: * Pivot not found: set params and repeat
874: *
875: P = IMAX
876: COLMAX = ROWMAX
877: IMAX = JMAX
878: *
879: * Copy updated JMAXth (next IMAXth) column to Kth of W
880: *
881: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
882: *
883: END IF
884: *
885: *
886: * End pivot search loop body
887: *
888: IF( .NOT.DONE ) GOTO 72
889: *
890: END IF
891: *
892: * END pivot search
893: *
894: * ============================================================
895: *
896: * KK is the column of A where pivoting step stopped
897: *
898: KK = K + KSTEP - 1
899: *
900: * Interchange rows and columns P and K (only for 2-by-2 pivot).
901: * Updated column P is already stored in column K of W.
902: *
903: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
904: *
905: * Copy non-updated column KK-1 to column P of submatrix A
906: * at step K. No need to copy element into columns
907: * K and K+1 of A for 2-by-2 pivot, since these columns
908: * will be later overwritten.
909: *
910: A( P, P ) = DBLE( A( K, K ) )
911: CALL ZCOPY( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA )
912: CALL ZLACGV( P-K-1, A( P, K+1 ), LDA )
913: IF( P.LT.N )
914: $ CALL ZCOPY( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
915: *
916: * Interchange rows K and P in first K-1 columns of A
917: * (columns K and K+1 of A for 2-by-2 pivot will be
918: * later overwritten). Interchange rows K and P
919: * in first KK columns of W.
920: *
921: IF( K.GT.1 )
922: $ CALL ZSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA )
923: CALL ZSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
924: END IF
925: *
926: * Interchange rows and columns KP and KK.
927: * Updated column KP is already stored in column KK of W.
928: *
929: IF( KP.NE.KK ) THEN
930: *
931: * Copy non-updated column KK to column KP of submatrix A
932: * at step K. No need to copy element into column K
933: * (or K and K+1 for 2-by-2 pivot) of A, since these columns
934: * will be later overwritten.
935: *
936: A( KP, KP ) = DBLE( A( KK, KK ) )
937: CALL ZCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
938: $ LDA )
939: CALL ZLACGV( KP-KK-1, A( KP, KK+1 ), LDA )
940: IF( KP.LT.N )
941: $ CALL ZCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
942: *
943: * Interchange rows KK and KP in first K-1 columns of A
944: * (column K (or K and K+1 for 2-by-2 pivot) of A will be
945: * later overwritten). Interchange rows KK and KP
946: * in first KK columns of W.
947: *
948: IF( K.GT.1 )
949: $ CALL ZSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
950: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
951: END IF
952: *
953: IF( KSTEP.EQ.1 ) THEN
954: *
955: * 1-by-1 pivot block D(k): column k of W now holds
956: *
957: * W(k) = L(k)*D(k),
958: *
959: * where L(k) is the k-th column of L
960: *
961: * (1) Store subdiag. elements of column L(k)
962: * and 1-by-1 block D(k) in column k of A.
963: * (NOTE: Diagonal element L(k,k) is a UNIT element
964: * and not stored)
965: * A(k,k) := D(k,k) = W(k,k)
966: * A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k)
967: *
968: * (NOTE: No need to use for Hermitian matrix
969: * A( K, K ) = REAL( W( K, K) ) to separately copy diagonal
970: * element D(k,k) from W (potentially saves only one load))
971: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
972: IF( K.LT.N ) THEN
973: *
974: * (NOTE: No need to check if A(k,k) is NOT ZERO,
975: * since that was ensured earlier in pivot search:
976: * case A(k,k) = 0 falls into 2x2 pivot case(3))
977: *
978: * Handle division by a small number
979: *
980: T = DBLE( A( K, K ) )
981: IF( ABS( T ).GE.SFMIN ) THEN
982: R1 = ONE / T
983: CALL ZDSCAL( N-K, R1, A( K+1, K ), 1 )
984: ELSE
985: DO 74 II = K + 1, N
986: A( II, K ) = A( II, K ) / T
987: 74 CONTINUE
988: END IF
989: *
990: * (2) Conjugate column W(k)
991: *
992: CALL ZLACGV( N-K, W( K+1, K ), 1 )
993: END IF
994: *
995: ELSE
996: *
997: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
998: *
999: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
1000: *
1001: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
1002: * of L
1003: *
1004: * (1) Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2
1005: * block D(k:k+1,k:k+1) in columns k and k+1 of A.
1006: * NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT
1007: * block and not stored.
1008: * A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1)
1009: * A(k+2:N,k:k+1) := L(k+2:N,k:k+1) =
1010: * = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) )
1011: *
1012: IF( K.LT.N-1 ) THEN
1013: *
1014: * Factor out the columns of the inverse of 2-by-2 pivot
1015: * block D, so that each column contains 1, to reduce the
1016: * number of FLOPS when we multiply panel
1017: * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
1018: *
1019: * D**(-1) = ( d11 cj(d21) )**(-1) =
1020: * ( d21 d22 )
1021: *
1022: * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
1023: * ( (-d21) ( d11 ) )
1024: *
1025: * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
1026: *
1027: * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
1028: * ( ( -1 ) ( d11/conj(d21) ) )
1029: *
1030: * = 1/(|d21|**2) * 1/(D22*D11-1) *
1031: *
1032: * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
1033: * ( ( -1 ) ( D22 ) )
1034: *
1035: * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
1036: * ( ( -1 ) ( D22 ) )
1037: *
1038: * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
1039: * ( ( -1 ) ( D22 ) )
1040: *
1041: * Handle division by a small number. (NOTE: order of
1042: * operations is important)
1043: *
1044: * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) )
1045: * ( (( -1 ) ) (( D22 ) ) ),
1046: *
1047: * where D11 = d22/d21,
1048: * D22 = d11/conj(d21),
1049: * D21 = d21,
1050: * T = 1/(D22*D11-1).
1051: *
1052: * (NOTE: No need to check for division by ZERO,
1053: * since that was ensured earlier in pivot search:
1054: * (a) d21 != 0 in 2x2 pivot case(4),
1055: * since |d21| should be larger than |d11| and |d22|;
1056: * (b) (D22*D11 - 1) != 0, since from (a),
1057: * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
1058: *
1059: D21 = W( K+1, K )
1060: D11 = W( K+1, K+1 ) / D21
1061: D22 = W( K, K ) / DCONJG( D21 )
1062: T = ONE / ( DBLE( D11*D22 )-ONE )
1063: *
1064: * Update elements in columns A(k) and A(k+1) as
1065: * dot products of rows of ( W(k) W(k+1) ) and columns
1066: * of D**(-1)
1067: *
1068: DO 80 J = K + 2, N
1069: A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
1070: $ DCONJG( D21 ) )
1071: A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
1072: $ D21 )
1073: 80 CONTINUE
1074: END IF
1075: *
1076: * Copy D(k) to A
1077: *
1078: A( K, K ) = W( K, K )
1079: A( K+1, K ) = W( K+1, K )
1080: A( K+1, K+1 ) = W( K+1, K+1 )
1081: *
1082: * (2) Conjugate columns W(k) and W(k+1)
1083: *
1084: CALL ZLACGV( N-K, W( K+1, K ), 1 )
1085: CALL ZLACGV( N-K-1, W( K+2, K+1 ), 1 )
1086: *
1087: END IF
1088: *
1089: END IF
1090: *
1091: * Store details of the interchanges in IPIV
1092: *
1093: IF( KSTEP.EQ.1 ) THEN
1094: IPIV( K ) = KP
1095: ELSE
1096: IPIV( K ) = -P
1097: IPIV( K+1 ) = -KP
1098: END IF
1099: *
1100: * Increase K and return to the start of the main loop
1101: *
1102: K = K + KSTEP
1103: GO TO 70
1104: *
1105: 90 CONTINUE
1106: *
1107: * Update the lower triangle of A22 (= A(k:n,k:n)) as
1108: *
1109: * A22 := A22 - L21*D*L21**H = A22 - L21*W**H
1110: *
1111: * computing blocks of NB columns at a time (note that conjg(W) is
1112: * actually stored)
1113: *
1114: DO 110 J = K, N, NB
1115: JB = MIN( NB, N-J+1 )
1116: *
1117: * Update the lower triangle of the diagonal block
1118: *
1119: DO 100 JJ = J, J + JB - 1
1120: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
1121: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
1122: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
1123: $ A( JJ, JJ ), 1 )
1124: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
1125: 100 CONTINUE
1126: *
1127: * Update the rectangular subdiagonal block
1128: *
1129: IF( J+JB.LE.N )
1130: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
1131: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
1132: $ LDW, CONE, A( J+JB, J ), LDA )
1133: 110 CONTINUE
1134: *
1135: * Put L21 in standard form by partially undoing the interchanges
1136: * of rows in columns 1:k-1 looping backwards from k-1 to 1
1137: *
1138: J = K - 1
1139: 120 CONTINUE
1140: *
1141: * Undo the interchanges (if any) of rows J and JP2
1142: * (or J and JP2, and J-1 and JP1) at each step J
1143: *
1144: KSTEP = 1
1145: JP1 = 1
1146: * (Here, J is a diagonal index)
1147: JJ = J
1148: JP2 = IPIV( J )
1149: IF( JP2.LT.0 ) THEN
1150: JP2 = -JP2
1151: * (Here, J is a diagonal index)
1152: J = J - 1
1153: JP1 = -IPIV( J )
1154: KSTEP = 2
1155: END IF
1156: * (NOTE: Here, J is used to determine row length. Length J
1157: * of the rows to swap back doesn't include diagonal element)
1158: J = J - 1
1159: IF( JP2.NE.JJ .AND. J.GE.1 )
1160: $ CALL ZSWAP( J, A( JP2, 1 ), LDA, A( JJ, 1 ), LDA )
1161: JJ = JJ -1
1162: IF( KSTEP.EQ.2 .AND. JP1.NE.JJ .AND. J.GE.1 )
1163: $ CALL ZSWAP( J, A( JP1, 1 ), LDA, A( JJ, 1 ), LDA )
1164: IF( J.GT.1 )
1165: $ GO TO 120
1166: *
1167: * Set KB to the number of columns factorized
1168: *
1169: KB = K - 1
1170: *
1171: END IF
1172: RETURN
1173: *
1174: * End of ZLAHEF_ROOK
1175: *
1176: END