LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
zlahef_rook.f
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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 *
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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 *
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 *
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
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:52
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:160
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:52
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:189
subroutine zlahef_rook(UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)