LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ zlahef_rook()

 subroutine zlahef_rook ( character UPLO, integer N, integer NB, integer KB, complex*16, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, complex*16, dimension( ldw, * ) W, integer LDW, integer INFO )

Purpose:
``` ZLAHEF_ROOK computes a partial factorization of a complex Hermitian
matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting
method. The partial factorization has the form:

A  =  ( I  U12 ) ( A11  0  ) (  I      0     )  if UPLO = 'U', or:
( 0  U22 ) (  0   D  ) ( U12**H U22**H )

A  =  ( L11  0 ) (  D   0  ) ( L11**H L21**H )  if UPLO = 'L'
( L21  I ) (  0  A22 ) (  0      I     )

where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
Note that U**H denotes the conjugate transpose of U.

ZLAHEF_ROOK is an auxiliary routine called by ZHETRF_ROOK. It uses
blocked code (calling Level 3 BLAS) to update the submatrix
A11 (if UPLO = 'U') or A22 (if UPLO = 'L').```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] NB ``` NB is INTEGER The maximum number of columns of the matrix A that should be factored. NB should be at least 2 to allow for 2-by-2 pivot blocks.``` [out] KB ``` KB is INTEGER The number of columns of A that were actually factored. KB is either NB-1 or NB, or N if N <= NB.``` [in,out] A ``` A is COMPLEX*16 array, dimension (LDA,N) On entry, the Hermitian matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, A contains details of the partial factorization.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] IPIV ``` IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D. If UPLO = 'U': Only the last KB elements of IPIV are set. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L': Only the first KB elements of IPIV are set. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.``` [out] W ` W is COMPLEX*16 array, dimension (LDW,NB)` [in] LDW ``` LDW is INTEGER The leading dimension of the array W. LDW >= max(1,N).``` [out] INFO ``` INFO is INTEGER = 0: successful exit > 0: if INFO = k, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular.```
Contributors:
```  November 2013,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester```

Definition at line 182 of file zlahef_rook.f.

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 *
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 *
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 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
integer function izamax(N, ZX, INCX)
IZAMAX
Definition: izamax.f:71
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zswap(N, ZX, INCX, ZY, INCY)
ZSWAP
Definition: zswap.f:81
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:78
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:158
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
subroutine zlacgv(N, X, INCX)
ZLACGV conjugates a complex vector.
Definition: zlacgv.f:74
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