LAPACK  3.9.1
LAPACK: Linear Algebra PACKage

◆ zlaqr4()

subroutine zlaqr4 ( logical  WANTT,
logical  WANTZ,
integer  N,
integer  ILO,
integer  IHI,
complex*16, dimension( ldh, * )  H,
integer  LDH,
complex*16, dimension( * )  W,
integer  ILOZ,
integer  IHIZ,
complex*16, dimension( ldz, * )  Z,
integer  LDZ,
complex*16, dimension( * )  WORK,
integer  LWORK,
integer  INFO 
)

ZLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Download ZLAQR4 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
    ZLAQR4 implements one level of recursion for ZLAQR0.
    It is a complete implementation of the small bulge multi-shift
    QR algorithm.  It may be called by ZLAQR0 and, for large enough
    deflation window size, it may be called by ZLAQR3.  This
    subroutine is identical to ZLAQR0 except that it calls ZLAQR2
    instead of ZLAQR3.

    ZLAQR4 computes the eigenvalues of a Hessenberg matrix H
    and, optionally, the matrices T and Z from the Schur decomposition
    H = Z T Z**H, where T is an upper triangular matrix (the
    Schur form), and Z is the unitary matrix of Schur vectors.

    Optionally Z may be postmultiplied into an input unitary
    matrix Q so that this routine can give the Schur factorization
    of a matrix A which has been reduced to the Hessenberg form H
    by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.
Parameters
[in]WANTT
          WANTT is LOGICAL
          = .TRUE. : the full Schur form T is required;
          = .FALSE.: only eigenvalues are required.
[in]WANTZ
          WANTZ is LOGICAL
          = .TRUE. : the matrix of Schur vectors Z is required;
          = .FALSE.: Schur vectors are not required.
[in]N
          N is INTEGER
           The order of the matrix H.  N >= 0.
[in]ILO
          ILO is INTEGER
[in]IHI
          IHI is INTEGER
           It is assumed that H is already upper triangular in rows
           and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
           previous call to ZGEBAL, and then passed to ZGEHRD when the
           matrix output by ZGEBAL is reduced to Hessenberg form.
           Otherwise, ILO and IHI should be set to 1 and N,
           respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
           If N = 0, then ILO = 1 and IHI = 0.
[in,out]H
          H is COMPLEX*16 array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and WANTT is .TRUE., then H
           contains the upper triangular matrix T from the Schur
           decomposition (the Schur form). If INFO = 0 and WANT is
           .FALSE., then the contents of H are unspecified on exit.
           (The output value of H when INFO > 0 is given under the
           description of INFO below.)

           This subroutine may explicitly set H(i,j) = 0 for i > j and
           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
[in]LDH
          LDH is INTEGER
           The leading dimension of the array H. LDH >= max(1,N).
[out]W
          W is COMPLEX*16 array, dimension (N)
           The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
           in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
           stored in the same order as on the diagonal of the Schur
           form returned in H, with W(i) = H(i,i).
[in]ILOZ
          ILOZ is INTEGER
[in]IHIZ
          IHIZ is INTEGER
           Specify the rows of Z to which transformations must be
           applied if WANTZ is .TRUE..
           1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
[in,out]Z
          Z is COMPLEX*16 array, dimension (LDZ,IHI)
           If WANTZ is .FALSE., then Z is not referenced.
           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
           (The output value of Z when INFO > 0 is given under
           the description of INFO below.)
[in]LDZ
          LDZ is INTEGER
           The leading dimension of the array Z.  if WANTZ is .TRUE.
           then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.
[out]WORK
          WORK is COMPLEX*16 array, dimension LWORK
           On exit, if LWORK = -1, WORK(1) returns an estimate of
           the optimal value for LWORK.
[in]LWORK
          LWORK is INTEGER
           The dimension of the array WORK.  LWORK >= max(1,N)
           is sufficient, but LWORK typically as large as 6*N may
           be required for optimal performance.  A workspace query
           to determine the optimal workspace size is recommended.

           If LWORK = -1, then ZLAQR4 does a workspace query.
           In this case, ZLAQR4 checks the input parameters and
           estimates the optimal workspace size for the given
           values of N, ILO and IHI.  The estimate is returned
           in WORK(1).  No error message related to LWORK is
           issued by XERBLA.  Neither H nor Z are accessed.
[out]INFO
          INFO is INTEGER
             =  0:  successful exit
             > 0:  if INFO = i, ZLAQR4 failed to compute all of
                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                and WI contain those eigenvalues which have been
                successfully computed.  (Failures are rare.)

                If INFO > 0 and WANT is .FALSE., then on exit,
                the remaining unconverged eigenvalues are the eigen-
                values of the upper Hessenberg matrix rows and
                columns ILO through INFO of the final, output
                value of H.

                If INFO > 0 and WANTT is .TRUE., then on exit

           (*)  (initial value of H)*U  = U*(final value of H)

                where U is a unitary matrix.  The final
                value of  H is upper Hessenberg and triangular in
                rows and columns INFO+1 through IHI.

                If INFO > 0 and WANTZ is .TRUE., then on exit

                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

                where U is the unitary matrix in (*) (regard-
                less of the value of WANTT.)

                If INFO > 0 and WANTZ is .FALSE., then Z is not
                accessed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
References:
  K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
  Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
  Performance, SIAM Journal of Matrix Analysis, volume 23, pages
  929--947, 2002.

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948–973, 2002.

Definition at line 245 of file zlaqr4.f.

247 *
248 * -- LAPACK auxiliary routine --
249 * -- LAPACK is a software package provided by Univ. of Tennessee, --
250 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
251 *
252 * .. Scalar Arguments ..
253  INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
254  LOGICAL WANTT, WANTZ
255 * ..
256 * .. Array Arguments ..
257  COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
258 * ..
259 *
260 * ================================================================
261 *
262 * .. Parameters ..
263 *
264 * ==== Matrices of order NTINY or smaller must be processed by
265 * . ZLAHQR because of insufficient subdiagonal scratch space.
266 * . (This is a hard limit.) ====
267  INTEGER NTINY
268  parameter( ntiny = 15 )
269 *
270 * ==== Exceptional deflation windows: try to cure rare
271 * . slow convergence by varying the size of the
272 * . deflation window after KEXNW iterations. ====
273  INTEGER KEXNW
274  parameter( kexnw = 5 )
275 *
276 * ==== Exceptional shifts: try to cure rare slow convergence
277 * . with ad-hoc exceptional shifts every KEXSH iterations.
278 * . ====
279  INTEGER KEXSH
280  parameter( kexsh = 6 )
281 *
282 * ==== The constant WILK1 is used to form the exceptional
283 * . shifts. ====
284  DOUBLE PRECISION WILK1
285  parameter( wilk1 = 0.75d0 )
286  COMPLEX*16 ZERO, ONE
287  parameter( zero = ( 0.0d0, 0.0d0 ),
288  $ one = ( 1.0d0, 0.0d0 ) )
289  DOUBLE PRECISION TWO
290  parameter( two = 2.0d0 )
291 * ..
292 * .. Local Scalars ..
293  COMPLEX*16 AA, BB, CC, CDUM, DD, DET, RTDISC, SWAP, TR2
294  DOUBLE PRECISION S
295  INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
296  $ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
297  $ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
298  $ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
299  LOGICAL SORTED
300  CHARACTER JBCMPZ*2
301 * ..
302 * .. External Functions ..
303  INTEGER ILAENV
304  EXTERNAL ilaenv
305 * ..
306 * .. Local Arrays ..
307  COMPLEX*16 ZDUM( 1, 1 )
308 * ..
309 * .. External Subroutines ..
310  EXTERNAL zlacpy, zlahqr, zlaqr2, zlaqr5
311 * ..
312 * .. Intrinsic Functions ..
313  INTRINSIC abs, dble, dcmplx, dimag, int, max, min, mod,
314  $ sqrt
315 * ..
316 * .. Statement Functions ..
317  DOUBLE PRECISION CABS1
318 * ..
319 * .. Statement Function definitions ..
320  cabs1( cdum ) = abs( dble( cdum ) ) + abs( dimag( cdum ) )
321 * ..
322 * .. Executable Statements ..
323  info = 0
324 *
325 * ==== Quick return for N = 0: nothing to do. ====
326 *
327  IF( n.EQ.0 ) THEN
328  work( 1 ) = one
329  RETURN
330  END IF
331 *
332  IF( n.LE.ntiny ) THEN
333 *
334 * ==== Tiny matrices must use ZLAHQR. ====
335 *
336  lwkopt = 1
337  IF( lwork.NE.-1 )
338  $ CALL zlahqr( wantt, wantz, n, ilo, ihi, h, ldh, w, iloz,
339  $ ihiz, z, ldz, info )
340  ELSE
341 *
342 * ==== Use small bulge multi-shift QR with aggressive early
343 * . deflation on larger-than-tiny matrices. ====
344 *
345 * ==== Hope for the best. ====
346 *
347  info = 0
348 *
349 * ==== Set up job flags for ILAENV. ====
350 *
351  IF( wantt ) THEN
352  jbcmpz( 1: 1 ) = 'S'
353  ELSE
354  jbcmpz( 1: 1 ) = 'E'
355  END IF
356  IF( wantz ) THEN
357  jbcmpz( 2: 2 ) = 'V'
358  ELSE
359  jbcmpz( 2: 2 ) = 'N'
360  END IF
361 *
362 * ==== NWR = recommended deflation window size. At this
363 * . point, N .GT. NTINY = 15, so there is enough
364 * . subdiagonal workspace for NWR.GE.2 as required.
365 * . (In fact, there is enough subdiagonal space for
366 * . NWR.GE.4.) ====
367 *
368  nwr = ilaenv( 13, 'ZLAQR4', jbcmpz, n, ilo, ihi, lwork )
369  nwr = max( 2, nwr )
370  nwr = min( ihi-ilo+1, ( n-1 ) / 3, nwr )
371 *
372 * ==== NSR = recommended number of simultaneous shifts.
373 * . At this point N .GT. NTINY = 15, so there is at
374 * . enough subdiagonal workspace for NSR to be even
375 * . and greater than or equal to two as required. ====
376 *
377  nsr = ilaenv( 15, 'ZLAQR4', jbcmpz, n, ilo, ihi, lwork )
378  nsr = min( nsr, ( n-3 ) / 6, ihi-ilo )
379  nsr = max( 2, nsr-mod( nsr, 2 ) )
380 *
381 * ==== Estimate optimal workspace ====
382 *
383 * ==== Workspace query call to ZLAQR2 ====
384 *
385  CALL zlaqr2( wantt, wantz, n, ilo, ihi, nwr+1, h, ldh, iloz,
386  $ ihiz, z, ldz, ls, ld, w, h, ldh, n, h, ldh, n, h,
387  $ ldh, work, -1 )
388 *
389 * ==== Optimal workspace = MAX(ZLAQR5, ZLAQR2) ====
390 *
391  lwkopt = max( 3*nsr / 2, int( work( 1 ) ) )
392 *
393 * ==== Quick return in case of workspace query. ====
394 *
395  IF( lwork.EQ.-1 ) THEN
396  work( 1 ) = dcmplx( lwkopt, 0 )
397  RETURN
398  END IF
399 *
400 * ==== ZLAHQR/ZLAQR0 crossover point ====
401 *
402  nmin = ilaenv( 12, 'ZLAQR4', jbcmpz, n, ilo, ihi, lwork )
403  nmin = max( ntiny, nmin )
404 *
405 * ==== Nibble crossover point ====
406 *
407  nibble = ilaenv( 14, 'ZLAQR4', jbcmpz, n, ilo, ihi, lwork )
408  nibble = max( 0, nibble )
409 *
410 * ==== Accumulate reflections during ttswp? Use block
411 * . 2-by-2 structure during matrix-matrix multiply? ====
412 *
413  kacc22 = ilaenv( 16, 'ZLAQR4', jbcmpz, n, ilo, ihi, lwork )
414  kacc22 = max( 0, kacc22 )
415  kacc22 = min( 2, kacc22 )
416 *
417 * ==== NWMAX = the largest possible deflation window for
418 * . which there is sufficient workspace. ====
419 *
420  nwmax = min( ( n-1 ) / 3, lwork / 2 )
421  nw = nwmax
422 *
423 * ==== NSMAX = the Largest number of simultaneous shifts
424 * . for which there is sufficient workspace. ====
425 *
426  nsmax = min( ( n-3 ) / 6, 2*lwork / 3 )
427  nsmax = nsmax - mod( nsmax, 2 )
428 *
429 * ==== NDFL: an iteration count restarted at deflation. ====
430 *
431  ndfl = 1
432 *
433 * ==== ITMAX = iteration limit ====
434 *
435  itmax = max( 30, 2*kexsh )*max( 10, ( ihi-ilo+1 ) )
436 *
437 * ==== Last row and column in the active block ====
438 *
439  kbot = ihi
440 *
441 * ==== Main Loop ====
442 *
443  DO 70 it = 1, itmax
444 *
445 * ==== Done when KBOT falls below ILO ====
446 *
447  IF( kbot.LT.ilo )
448  $ GO TO 80
449 *
450 * ==== Locate active block ====
451 *
452  DO 10 k = kbot, ilo + 1, -1
453  IF( h( k, k-1 ).EQ.zero )
454  $ GO TO 20
455  10 CONTINUE
456  k = ilo
457  20 CONTINUE
458  ktop = k
459 *
460 * ==== Select deflation window size:
461 * . Typical Case:
462 * . If possible and advisable, nibble the entire
463 * . active block. If not, use size MIN(NWR,NWMAX)
464 * . or MIN(NWR+1,NWMAX) depending upon which has
465 * . the smaller corresponding subdiagonal entry
466 * . (a heuristic).
467 * .
468 * . Exceptional Case:
469 * . If there have been no deflations in KEXNW or
470 * . more iterations, then vary the deflation window
471 * . size. At first, because, larger windows are,
472 * . in general, more powerful than smaller ones,
473 * . rapidly increase the window to the maximum possible.
474 * . Then, gradually reduce the window size. ====
475 *
476  nh = kbot - ktop + 1
477  nwupbd = min( nh, nwmax )
478  IF( ndfl.LT.kexnw ) THEN
479  nw = min( nwupbd, nwr )
480  ELSE
481  nw = min( nwupbd, 2*nw )
482  END IF
483  IF( nw.LT.nwmax ) THEN
484  IF( nw.GE.nh-1 ) THEN
485  nw = nh
486  ELSE
487  kwtop = kbot - nw + 1
488  IF( cabs1( h( kwtop, kwtop-1 ) ).GT.
489  $ cabs1( h( kwtop-1, kwtop-2 ) ) )nw = nw + 1
490  END IF
491  END IF
492  IF( ndfl.LT.kexnw ) THEN
493  ndec = -1
494  ELSE IF( ndec.GE.0 .OR. nw.GE.nwupbd ) THEN
495  ndec = ndec + 1
496  IF( nw-ndec.LT.2 )
497  $ ndec = 0
498  nw = nw - ndec
499  END IF
500 *
501 * ==== Aggressive early deflation:
502 * . split workspace under the subdiagonal into
503 * . - an nw-by-nw work array V in the lower
504 * . left-hand-corner,
505 * . - an NW-by-at-least-NW-but-more-is-better
506 * . (NW-by-NHO) horizontal work array along
507 * . the bottom edge,
508 * . - an at-least-NW-but-more-is-better (NHV-by-NW)
509 * . vertical work array along the left-hand-edge.
510 * . ====
511 *
512  kv = n - nw + 1
513  kt = nw + 1
514  nho = ( n-nw-1 ) - kt + 1
515  kwv = nw + 2
516  nve = ( n-nw ) - kwv + 1
517 *
518 * ==== Aggressive early deflation ====
519 *
520  CALL zlaqr2( wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz,
521  $ ihiz, z, ldz, ls, ld, w, h( kv, 1 ), ldh, nho,
522  $ h( kv, kt ), ldh, nve, h( kwv, 1 ), ldh, work,
523  $ lwork )
524 *
525 * ==== Adjust KBOT accounting for new deflations. ====
526 *
527  kbot = kbot - ld
528 *
529 * ==== KS points to the shifts. ====
530 *
531  ks = kbot - ls + 1
532 *
533 * ==== Skip an expensive QR sweep if there is a (partly
534 * . heuristic) reason to expect that many eigenvalues
535 * . will deflate without it. Here, the QR sweep is
536 * . skipped if many eigenvalues have just been deflated
537 * . or if the remaining active block is small.
538 *
539  IF( ( ld.EQ.0 ) .OR. ( ( 100*ld.LE.nw*nibble ) .AND. ( kbot-
540  $ ktop+1.GT.min( nmin, nwmax ) ) ) ) THEN
541 *
542 * ==== NS = nominal number of simultaneous shifts.
543 * . This may be lowered (slightly) if ZLAQR2
544 * . did not provide that many shifts. ====
545 *
546  ns = min( nsmax, nsr, max( 2, kbot-ktop ) )
547  ns = ns - mod( ns, 2 )
548 *
549 * ==== If there have been no deflations
550 * . in a multiple of KEXSH iterations,
551 * . then try exceptional shifts.
552 * . Otherwise use shifts provided by
553 * . ZLAQR2 above or from the eigenvalues
554 * . of a trailing principal submatrix. ====
555 *
556  IF( mod( ndfl, kexsh ).EQ.0 ) THEN
557  ks = kbot - ns + 1
558  DO 30 i = kbot, ks + 1, -2
559  w( i ) = h( i, i ) + wilk1*cabs1( h( i, i-1 ) )
560  w( i-1 ) = w( i )
561  30 CONTINUE
562  ELSE
563 *
564 * ==== Got NS/2 or fewer shifts? Use ZLAHQR
565 * . on a trailing principal submatrix to
566 * . get more. (Since NS.LE.NSMAX.LE.(N+6)/9,
567 * . there is enough space below the subdiagonal
568 * . to fit an NS-by-NS scratch array.) ====
569 *
570  IF( kbot-ks+1.LE.ns / 2 ) THEN
571  ks = kbot - ns + 1
572  kt = n - ns + 1
573  CALL zlacpy( 'A', ns, ns, h( ks, ks ), ldh,
574  $ h( kt, 1 ), ldh )
575  CALL zlahqr( .false., .false., ns, 1, ns,
576  $ h( kt, 1 ), ldh, w( ks ), 1, 1, zdum,
577  $ 1, inf )
578  ks = ks + inf
579 *
580 * ==== In case of a rare QR failure use
581 * . eigenvalues of the trailing 2-by-2
582 * . principal submatrix. Scale to avoid
583 * . overflows, underflows and subnormals.
584 * . (The scale factor S can not be zero,
585 * . because H(KBOT,KBOT-1) is nonzero.) ====
586 *
587  IF( ks.GE.kbot ) THEN
588  s = cabs1( h( kbot-1, kbot-1 ) ) +
589  $ cabs1( h( kbot, kbot-1 ) ) +
590  $ cabs1( h( kbot-1, kbot ) ) +
591  $ cabs1( h( kbot, kbot ) )
592  aa = h( kbot-1, kbot-1 ) / s
593  cc = h( kbot, kbot-1 ) / s
594  bb = h( kbot-1, kbot ) / s
595  dd = h( kbot, kbot ) / s
596  tr2 = ( aa+dd ) / two
597  det = ( aa-tr2 )*( dd-tr2 ) - bb*cc
598  rtdisc = sqrt( -det )
599  w( kbot-1 ) = ( tr2+rtdisc )*s
600  w( kbot ) = ( tr2-rtdisc )*s
601 *
602  ks = kbot - 1
603  END IF
604  END IF
605 *
606  IF( kbot-ks+1.GT.ns ) THEN
607 *
608 * ==== Sort the shifts (Helps a little) ====
609 *
610  sorted = .false.
611  DO 50 k = kbot, ks + 1, -1
612  IF( sorted )
613  $ GO TO 60
614  sorted = .true.
615  DO 40 i = ks, k - 1
616  IF( cabs1( w( i ) ).LT.cabs1( w( i+1 ) ) )
617  $ THEN
618  sorted = .false.
619  swap = w( i )
620  w( i ) = w( i+1 )
621  w( i+1 ) = swap
622  END IF
623  40 CONTINUE
624  50 CONTINUE
625  60 CONTINUE
626  END IF
627  END IF
628 *
629 * ==== If there are only two shifts, then use
630 * . only one. ====
631 *
632  IF( kbot-ks+1.EQ.2 ) THEN
633  IF( cabs1( w( kbot )-h( kbot, kbot ) ).LT.
634  $ cabs1( w( kbot-1 )-h( kbot, kbot ) ) ) THEN
635  w( kbot-1 ) = w( kbot )
636  ELSE
637  w( kbot ) = w( kbot-1 )
638  END IF
639  END IF
640 *
641 * ==== Use up to NS of the the smallest magnitude
642 * . shifts. If there aren't NS shifts available,
643 * . then use them all, possibly dropping one to
644 * . make the number of shifts even. ====
645 *
646  ns = min( ns, kbot-ks+1 )
647  ns = ns - mod( ns, 2 )
648  ks = kbot - ns + 1
649 *
650 * ==== Small-bulge multi-shift QR sweep:
651 * . split workspace under the subdiagonal into
652 * . - a KDU-by-KDU work array U in the lower
653 * . left-hand-corner,
654 * . - a KDU-by-at-least-KDU-but-more-is-better
655 * . (KDU-by-NHo) horizontal work array WH along
656 * . the bottom edge,
657 * . - and an at-least-KDU-but-more-is-better-by-KDU
658 * . (NVE-by-KDU) vertical work WV arrow along
659 * . the left-hand-edge. ====
660 *
661  kdu = 2*ns
662  ku = n - kdu + 1
663  kwh = kdu + 1
664  nho = ( n-kdu+1-4 ) - ( kdu+1 ) + 1
665  kwv = kdu + 4
666  nve = n - kdu - kwv + 1
667 *
668 * ==== Small-bulge multi-shift QR sweep ====
669 *
670  CALL zlaqr5( wantt, wantz, kacc22, n, ktop, kbot, ns,
671  $ w( ks ), h, ldh, iloz, ihiz, z, ldz, work,
672  $ 3, h( ku, 1 ), ldh, nve, h( kwv, 1 ), ldh,
673  $ nho, h( ku, kwh ), ldh )
674  END IF
675 *
676 * ==== Note progress (or the lack of it). ====
677 *
678  IF( ld.GT.0 ) THEN
679  ndfl = 1
680  ELSE
681  ndfl = ndfl + 1
682  END IF
683 *
684 * ==== End of main loop ====
685  70 CONTINUE
686 *
687 * ==== Iteration limit exceeded. Set INFO to show where
688 * . the problem occurred and exit. ====
689 *
690  info = kbot
691  80 CONTINUE
692  END IF
693 *
694 * ==== Return the optimal value of LWORK. ====
695 *
696  work( 1 ) = dcmplx( lwkopt, 0 )
697 *
698 * ==== End of ZLAQR4 ====
699 *
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine zlaqr5(WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, S, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U, LDU, NV, WV, LDWV, NH, WH, LDWH)
ZLAQR5 performs a single small-bulge multi-shift QR sweep.
Definition: zlaqr5.f:257
subroutine zlahqr(WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, INFO)
ZLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,...
Definition: zlahqr.f:195
subroutine zlaqr2(WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
ZLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fu...
Definition: zlaqr2.f:270
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
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