LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ dlaqr4()

 subroutine dlaqr4 ( logical wantt, logical wantz, integer n, integer ilo, integer ihi, double precision, dimension( ldh, * ) h, integer ldh, double precision, dimension( * ) wr, double precision, dimension( * ) wi, integer iloz, integer ihiz, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer info )

DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Purpose:
```    DLAQR4 implements one level of recursion for DLAQR0.
It is a complete implementation of the small bulge multi-shift
QR algorithm.  It may be called by DLAQR0 and, for large enough
deflation window size, it may be called by DLAQR3.  This
subroutine is identical to DLAQR0 except that it calls DLAQR2

DLAQR4 computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**T, where T is an upper quasi-triangular matrix (the
Schur form), and Z is the orthogonal matrix of Schur vectors.

Optionally Z may be postmultiplied into an input orthogonal
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 orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.```
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 DGEBAL, and then passed to DGEHRD when the matrix output by DGEBAL 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 DOUBLE PRECISION 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 quasi-triangular matrix T from the Schur decomposition (the Schur form); 2-by-2 diagonal blocks (corresponding to complex conjugate pairs of eigenvalues) are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT 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] WR ` WR is DOUBLE PRECISION array, dimension (IHI)` [out] WI ``` WI is DOUBLE PRECISION array, dimension (IHI) The real and imaginary parts, respectively, of the computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) and WI(ILO:IHI). If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and WI(i+1) < 0. 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 WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DLAQR4 does a workspace query. In this case, DLAQR4 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, DLAQR4 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 orthogonal 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 orthogonal matrix in (*) (regard- less of the value of WANTT.) If INFO > 0 and WANTZ is .FALSE., then Z is not accessed.```
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 261 of file dlaqr4.f.

263*
264* -- LAPACK auxiliary routine --
265* -- LAPACK is a software package provided by Univ. of Tennessee, --
266* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
267*
268* .. Scalar Arguments ..
269 INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
270 LOGICAL WANTT, WANTZ
271* ..
272* .. Array Arguments ..
273 DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ),
274 \$ Z( LDZ, * )
275* ..
276*
277* ================================================================
278* .. Parameters ..
279*
280* ==== Matrices of order NTINY or smaller must be processed by
281* . DLAHQR because of insufficient subdiagonal scratch space.
282* . (This is a hard limit.) ====
283 INTEGER NTINY
284 parameter( ntiny = 15 )
285*
286* ==== Exceptional deflation windows: try to cure rare
287* . slow convergence by varying the size of the
288* . deflation window after KEXNW iterations. ====
289 INTEGER KEXNW
290 parameter( kexnw = 5 )
291*
292* ==== Exceptional shifts: try to cure rare slow convergence
293* . with ad-hoc exceptional shifts every KEXSH iterations.
294* . ====
295 INTEGER KEXSH
296 parameter( kexsh = 6 )
297*
298* ==== The constants WILK1 and WILK2 are used to form the
299* . exceptional shifts. ====
300 DOUBLE PRECISION WILK1, WILK2
301 parameter( wilk1 = 0.75d0, wilk2 = -0.4375d0 )
302 DOUBLE PRECISION ZERO, ONE
303 parameter( zero = 0.0d0, one = 1.0d0 )
304* ..
305* .. Local Scalars ..
306 DOUBLE PRECISION AA, BB, CC, CS, DD, SN, SS, SWAP
307 INTEGER I, INF, IT, ITMAX, K, KACC22, KBOT, KDU, KS,
308 \$ KT, KTOP, KU, KV, KWH, KWTOP, KWV, LD, LS,
309 \$ LWKOPT, NDEC, NDFL, NH, NHO, NIBBLE, NMIN, NS,
310 \$ NSMAX, NSR, NVE, NW, NWMAX, NWR, NWUPBD
311 LOGICAL SORTED
312 CHARACTER JBCMPZ*2
313* ..
314* .. External Functions ..
315 INTEGER ILAENV
316 EXTERNAL ilaenv
317* ..
318* .. Local Arrays ..
319 DOUBLE PRECISION ZDUM( 1, 1 )
320* ..
321* .. External Subroutines ..
322 EXTERNAL dlacpy, dlahqr, dlanv2, dlaqr2, dlaqr5
323* ..
324* .. Intrinsic Functions ..
325 INTRINSIC abs, dble, int, max, min, mod
326* ..
327* .. Executable Statements ..
328 info = 0
329*
330* ==== Quick return for N = 0: nothing to do. ====
331*
332 IF( n.EQ.0 ) THEN
333 work( 1 ) = one
334 RETURN
335 END IF
336*
337 IF( n.LE.ntiny ) THEN
338*
339* ==== Tiny matrices must use DLAHQR. ====
340*
341 lwkopt = 1
342 IF( lwork.NE.-1 )
343 \$ CALL dlahqr( wantt, wantz, n, ilo, ihi, h, ldh, wr, wi,
344 \$ iloz, ihiz, z, ldz, info )
345 ELSE
346*
347* ==== Use small bulge multi-shift QR with aggressive early
348* . deflation on larger-than-tiny matrices. ====
349*
350* ==== Hope for the best. ====
351*
352 info = 0
353*
354* ==== Set up job flags for ILAENV. ====
355*
356 IF( wantt ) THEN
357 jbcmpz( 1: 1 ) = 'S'
358 ELSE
359 jbcmpz( 1: 1 ) = 'E'
360 END IF
361 IF( wantz ) THEN
362 jbcmpz( 2: 2 ) = 'V'
363 ELSE
364 jbcmpz( 2: 2 ) = 'N'
365 END IF
366*
367* ==== NWR = recommended deflation window size. At this
368* . point, N .GT. NTINY = 15, so there is enough
369* . subdiagonal workspace for NWR.GE.2 as required.
370* . (In fact, there is enough subdiagonal space for
371* . NWR.GE.4.) ====
372*
373 nwr = ilaenv( 13, 'DLAQR4', jbcmpz, n, ilo, ihi, lwork )
374 nwr = max( 2, nwr )
375 nwr = min( ihi-ilo+1, ( n-1 ) / 3, nwr )
376*
377* ==== NSR = recommended number of simultaneous shifts.
378* . At this point N .GT. NTINY = 15, so there is at
379* . enough subdiagonal workspace for NSR to be even
380* . and greater than or equal to two as required. ====
381*
382 nsr = ilaenv( 15, 'DLAQR4', jbcmpz, n, ilo, ihi, lwork )
383 nsr = min( nsr, ( n-3 ) / 6, ihi-ilo )
384 nsr = max( 2, nsr-mod( nsr, 2 ) )
385*
386* ==== Estimate optimal workspace ====
387*
388* ==== Workspace query call to DLAQR2 ====
389*
390 CALL dlaqr2( wantt, wantz, n, ilo, ihi, nwr+1, h, ldh, iloz,
391 \$ ihiz, z, ldz, ls, ld, wr, wi, h, ldh, n, h, ldh,
392 \$ n, h, ldh, work, -1 )
393*
394* ==== Optimal workspace = MAX(DLAQR5, DLAQR2) ====
395*
396 lwkopt = max( 3*nsr / 2, int( work( 1 ) ) )
397*
398* ==== Quick return in case of workspace query. ====
399*
400 IF( lwork.EQ.-1 ) THEN
401 work( 1 ) = dble( lwkopt )
402 RETURN
403 END IF
404*
405* ==== DLAHQR/DLAQR0 crossover point ====
406*
407 nmin = ilaenv( 12, 'DLAQR4', jbcmpz, n, ilo, ihi, lwork )
408 nmin = max( ntiny, nmin )
409*
410* ==== Nibble crossover point ====
411*
412 nibble = ilaenv( 14, 'DLAQR4', jbcmpz, n, ilo, ihi, lwork )
413 nibble = max( 0, nibble )
414*
415* ==== Accumulate reflections during ttswp? Use block
416* . 2-by-2 structure during matrix-matrix multiply? ====
417*
418 kacc22 = ilaenv( 16, 'DLAQR4', jbcmpz, n, ilo, ihi, lwork )
419 kacc22 = max( 0, kacc22 )
420 kacc22 = min( 2, kacc22 )
421*
422* ==== NWMAX = the largest possible deflation window for
423* . which there is sufficient workspace. ====
424*
425 nwmax = min( ( n-1 ) / 3, lwork / 2 )
426 nw = nwmax
427*
428* ==== NSMAX = the Largest number of simultaneous shifts
429* . for which there is sufficient workspace. ====
430*
431 nsmax = min( ( n-3 ) / 6, 2*lwork / 3 )
432 nsmax = nsmax - mod( nsmax, 2 )
433*
434* ==== NDFL: an iteration count restarted at deflation. ====
435*
436 ndfl = 1
437*
438* ==== ITMAX = iteration limit ====
439*
440 itmax = max( 30, 2*kexsh )*max( 10, ( ihi-ilo+1 ) )
441*
442* ==== Last row and column in the active block ====
443*
444 kbot = ihi
445*
446* ==== Main Loop ====
447*
448 DO 80 it = 1, itmax
449*
450* ==== Done when KBOT falls below ILO ====
451*
452 IF( kbot.LT.ilo )
453 \$ GO TO 90
454*
455* ==== Locate active block ====
456*
457 DO 10 k = kbot, ilo + 1, -1
458 IF( h( k, k-1 ).EQ.zero )
459 \$ GO TO 20
460 10 CONTINUE
461 k = ilo
462 20 CONTINUE
463 ktop = k
464*
465* ==== Select deflation window size:
466* . Typical Case:
467* . If possible and advisable, nibble the entire
468* . active block. If not, use size MIN(NWR,NWMAX)
469* . or MIN(NWR+1,NWMAX) depending upon which has
470* . the smaller corresponding subdiagonal entry
471* . (a heuristic).
472* .
473* . Exceptional Case:
474* . If there have been no deflations in KEXNW or
475* . more iterations, then vary the deflation window
476* . size. At first, because, larger windows are,
477* . in general, more powerful than smaller ones,
478* . rapidly increase the window to the maximum possible.
479* . Then, gradually reduce the window size. ====
480*
481 nh = kbot - ktop + 1
482 nwupbd = min( nh, nwmax )
483 IF( ndfl.LT.kexnw ) THEN
484 nw = min( nwupbd, nwr )
485 ELSE
486 nw = min( nwupbd, 2*nw )
487 END IF
488 IF( nw.LT.nwmax ) THEN
489 IF( nw.GE.nh-1 ) THEN
490 nw = nh
491 ELSE
492 kwtop = kbot - nw + 1
493 IF( abs( h( kwtop, kwtop-1 ) ).GT.
494 \$ abs( h( kwtop-1, kwtop-2 ) ) )nw = nw + 1
495 END IF
496 END IF
497 IF( ndfl.LT.kexnw ) THEN
498 ndec = -1
499 ELSE IF( ndec.GE.0 .OR. nw.GE.nwupbd ) THEN
500 ndec = ndec + 1
501 IF( nw-ndec.LT.2 )
502 \$ ndec = 0
503 nw = nw - ndec
504 END IF
505*
506* ==== Aggressive early deflation:
507* . split workspace under the subdiagonal into
508* . - an nw-by-nw work array V in the lower
509* . left-hand-corner,
510* . - an NW-by-at-least-NW-but-more-is-better
511* . (NW-by-NHO) horizontal work array along
512* . the bottom edge,
513* . - an at-least-NW-but-more-is-better (NHV-by-NW)
514* . vertical work array along the left-hand-edge.
515* . ====
516*
517 kv = n - nw + 1
518 kt = nw + 1
519 nho = ( n-nw-1 ) - kt + 1
520 kwv = nw + 2
521 nve = ( n-nw ) - kwv + 1
522*
523* ==== Aggressive early deflation ====
524*
525 CALL dlaqr2( wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz,
526 \$ ihiz, z, ldz, ls, ld, wr, wi, h( kv, 1 ), ldh,
527 \$ nho, h( kv, kt ), ldh, nve, h( kwv, 1 ), ldh,
528 \$ work, lwork )
529*
530* ==== Adjust KBOT accounting for new deflations. ====
531*
532 kbot = kbot - ld
533*
534* ==== KS points to the shifts. ====
535*
536 ks = kbot - ls + 1
537*
538* ==== Skip an expensive QR sweep if there is a (partly
539* . heuristic) reason to expect that many eigenvalues
540* . will deflate without it. Here, the QR sweep is
541* . skipped if many eigenvalues have just been deflated
542* . or if the remaining active block is small.
543*
544 IF( ( ld.EQ.0 ) .OR. ( ( 100*ld.LE.nw*nibble ) .AND. ( kbot-
545 \$ ktop+1.GT.min( nmin, nwmax ) ) ) ) THEN
546*
547* ==== NS = nominal number of simultaneous shifts.
548* . This may be lowered (slightly) if DLAQR2
549* . did not provide that many shifts. ====
550*
551 ns = min( nsmax, nsr, max( 2, kbot-ktop ) )
552 ns = ns - mod( ns, 2 )
553*
554* ==== If there have been no deflations
555* . in a multiple of KEXSH iterations,
556* . then try exceptional shifts.
557* . Otherwise use shifts provided by
558* . DLAQR2 above or from the eigenvalues
559* . of a trailing principal submatrix. ====
560*
561 IF( mod( ndfl, kexsh ).EQ.0 ) THEN
562 ks = kbot - ns + 1
563 DO 30 i = kbot, max( ks+1, ktop+2 ), -2
564 ss = abs( h( i, i-1 ) ) + abs( h( i-1, i-2 ) )
565 aa = wilk1*ss + h( i, i )
566 bb = ss
567 cc = wilk2*ss
568 dd = aa
569 CALL dlanv2( aa, bb, cc, dd, wr( i-1 ), wi( i-1 ),
570 \$ wr( i ), wi( i ), cs, sn )
571 30 CONTINUE
572 IF( ks.EQ.ktop ) THEN
573 wr( ks+1 ) = h( ks+1, ks+1 )
574 wi( ks+1 ) = zero
575 wr( ks ) = wr( ks+1 )
576 wi( ks ) = wi( ks+1 )
577 END IF
578 ELSE
579*
580* ==== Got NS/2 or fewer shifts? Use DLAHQR
581* . on a trailing principal submatrix to
582* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6,
583* . there is enough space below the subdiagonal
584* . to fit an NS-by-NS scratch array.) ====
585*
586 IF( kbot-ks+1.LE.ns / 2 ) THEN
587 ks = kbot - ns + 1
588 kt = n - ns + 1
589 CALL dlacpy( 'A', ns, ns, h( ks, ks ), ldh,
590 \$ h( kt, 1 ), ldh )
591 CALL dlahqr( .false., .false., ns, 1, ns,
592 \$ h( kt, 1 ), ldh, wr( ks ), wi( ks ),
593 \$ 1, 1, zdum, 1, inf )
594 ks = ks + inf
595*
596* ==== In case of a rare QR failure use
597* . eigenvalues of the trailing 2-by-2
598* . principal submatrix. ====
599*
600 IF( ks.GE.kbot ) THEN
601 aa = h( kbot-1, kbot-1 )
602 cc = h( kbot, kbot-1 )
603 bb = h( kbot-1, kbot )
604 dd = h( kbot, kbot )
605 CALL dlanv2( aa, bb, cc, dd, wr( kbot-1 ),
606 \$ wi( kbot-1 ), wr( kbot ),
607 \$ wi( kbot ), cs, sn )
608 ks = kbot - 1
609 END IF
610 END IF
611*
612 IF( kbot-ks+1.GT.ns ) THEN
613*
614* ==== Sort the shifts (Helps a little)
615* . Bubble sort keeps complex conjugate
616* . pairs together. ====
617*
618 sorted = .false.
619 DO 50 k = kbot, ks + 1, -1
620 IF( sorted )
621 \$ GO TO 60
622 sorted = .true.
623 DO 40 i = ks, k - 1
624 IF( abs( wr( i ) )+abs( wi( i ) ).LT.
625 \$ abs( wr( i+1 ) )+abs( wi( i+1 ) ) ) THEN
626 sorted = .false.
627*
628 swap = wr( i )
629 wr( i ) = wr( i+1 )
630 wr( i+1 ) = swap
631*
632 swap = wi( i )
633 wi( i ) = wi( i+1 )
634 wi( i+1 ) = swap
635 END IF
636 40 CONTINUE
637 50 CONTINUE
638 60 CONTINUE
639 END IF
640*
641* ==== Shuffle shifts into pairs of real shifts
642* . and pairs of complex conjugate shifts
643* . assuming complex conjugate shifts are
645* . they are.) ====
646*
647 DO 70 i = kbot, ks + 2, -2
648 IF( wi( i ).NE.-wi( i-1 ) ) THEN
649*
650 swap = wr( i )
651 wr( i ) = wr( i-1 )
652 wr( i-1 ) = wr( i-2 )
653 wr( i-2 ) = swap
654*
655 swap = wi( i )
656 wi( i ) = wi( i-1 )
657 wi( i-1 ) = wi( i-2 )
658 wi( i-2 ) = swap
659 END IF
660 70 CONTINUE
661 END IF
662*
663* ==== If there are only two shifts and both are
664* . real, then use only one. ====
665*
666 IF( kbot-ks+1.EQ.2 ) THEN
667 IF( wi( kbot ).EQ.zero ) THEN
668 IF( abs( wr( kbot )-h( kbot, kbot ) ).LT.
669 \$ abs( wr( kbot-1 )-h( kbot, kbot ) ) ) THEN
670 wr( kbot-1 ) = wr( kbot )
671 ELSE
672 wr( kbot ) = wr( kbot-1 )
673 END IF
674 END IF
675 END IF
676*
677* ==== Use up to NS of the the smallest magnitude
678* . shifts. If there aren't NS shifts available,
679* . then use them all, possibly dropping one to
680* . make the number of shifts even. ====
681*
682 ns = min( ns, kbot-ks+1 )
683 ns = ns - mod( ns, 2 )
684 ks = kbot - ns + 1
685*
686* ==== Small-bulge multi-shift QR sweep:
687* . split workspace under the subdiagonal into
688* . - a KDU-by-KDU work array U in the lower
689* . left-hand-corner,
690* . - a KDU-by-at-least-KDU-but-more-is-better
691* . (KDU-by-NHo) horizontal work array WH along
692* . the bottom edge,
693* . - and an at-least-KDU-but-more-is-better-by-KDU
694* . (NVE-by-KDU) vertical work WV arrow along
695* . the left-hand-edge. ====
696*
697 kdu = 2*ns
698 ku = n - kdu + 1
699 kwh = kdu + 1
700 nho = ( n-kdu+1-4 ) - ( kdu+1 ) + 1
701 kwv = kdu + 4
702 nve = n - kdu - kwv + 1
703*
704* ==== Small-bulge multi-shift QR sweep ====
705*
706 CALL dlaqr5( wantt, wantz, kacc22, n, ktop, kbot, ns,
707 \$ wr( ks ), wi( ks ), h, ldh, iloz, ihiz, z,
708 \$ ldz, work, 3, h( ku, 1 ), ldh, nve,
709 \$ h( kwv, 1 ), ldh, nho, h( ku, kwh ), ldh )
710 END IF
711*
712* ==== Note progress (or the lack of it). ====
713*
714 IF( ld.GT.0 ) THEN
715 ndfl = 1
716 ELSE
717 ndfl = ndfl + 1
718 END IF
719*
720* ==== End of main loop ====
721 80 CONTINUE
722*
723* ==== Iteration limit exceeded. Set INFO to show where
724* . the problem occurred and exit. ====
725*
726 info = kbot
727 90 CONTINUE
728 END IF
729*
730* ==== Return the optimal value of LWORK. ====
731*
732 work( 1 ) = dble( lwkopt )
733*
734* ==== End of DLAQR4 ====
735*
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
subroutine dlacpy(uplo, m, n, a, lda, b, ldb)
DLACPY copies all or part of one two-dimensional array to another.
Definition dlacpy.f:103
subroutine dlahqr(wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, info)
DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,...
Definition dlahqr.f:207
subroutine dlanv2(a, b, c, d, rt1r, rt1i, rt2r, rt2i, cs, sn)
DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.
Definition dlanv2.f:127
subroutine dlaqr2(wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork)
DLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate...
Definition dlaqr2.f:278
subroutine dlaqr5(wantt, wantz, kacc22, n, ktop, kbot, nshfts, sr, si, h, ldh, iloz, ihiz, z, ldz, v, ldv, u, ldu, nv, wv, ldwv, nh, wh, ldwh)
DLAQR5 performs a single small-bulge multi-shift QR sweep.
Definition dlaqr5.f:265
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