LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ zlaqr2()

subroutine zlaqr2 ( logical  WANTT,
logical  WANTZ,
integer  N,
integer  KTOP,
integer  KBOT,
integer  NW,
complex*16, dimension( ldh, * )  H,
integer  LDH,
integer  ILOZ,
integer  IHIZ,
complex*16, dimension( ldz, * )  Z,
integer  LDZ,
integer  NS,
integer  ND,
complex*16, dimension( * )  SH,
complex*16, dimension( ldv, * )  V,
integer  LDV,
integer  NH,
complex*16, dimension( ldt, * )  T,
integer  LDT,
integer  NV,
complex*16, dimension( ldwv, * )  WV,
integer  LDWV,
complex*16, dimension( * )  WORK,
integer  LWORK 
)

ZLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

Download ZLAQR2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
    ZLAQR2 is identical to ZLAQR3 except that it avoids
    recursion by calling ZLAHQR instead of ZLAQR4.

    Aggressive early deflation:

    ZLAQR2 accepts as input an upper Hessenberg matrix
    H and performs an unitary similarity transformation
    designed to detect and deflate fully converged eigenvalues from
    a trailing principal submatrix.  On output H has been over-
    written by a new Hessenberg matrix that is a perturbation of
    an unitary similarity transformation of H.  It is to be
    hoped that the final version of H has many zero subdiagonal
    entries.
Parameters
[in]WANTT
          WANTT is LOGICAL
          If .TRUE., then the Hessenberg matrix H is fully updated
          so that the triangular Schur factor may be
          computed (in cooperation with the calling subroutine).
          If .FALSE., then only enough of H is updated to preserve
          the eigenvalues.
[in]WANTZ
          WANTZ is LOGICAL
          If .TRUE., then the unitary matrix Z is updated so
          so that the unitary Schur factor may be computed
          (in cooperation with the calling subroutine).
          If .FALSE., then Z is not referenced.
[in]N
          N is INTEGER
          The order of the matrix H and (if WANTZ is .TRUE.) the
          order of the unitary matrix Z.
[in]KTOP
          KTOP is INTEGER
          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
          KBOT and KTOP together determine an isolated block
          along the diagonal of the Hessenberg matrix.
[in]KBOT
          KBOT is INTEGER
          It is assumed without a check that either
          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
          determine an isolated block along the diagonal of the
          Hessenberg matrix.
[in]NW
          NW is INTEGER
          Deflation window size.  1 <= NW <= (KBOT-KTOP+1).
[in,out]H
          H is COMPLEX*16 array, dimension (LDH,N)
          On input the initial N-by-N section of H stores the
          Hessenberg matrix undergoing aggressive early deflation.
          On output H has been transformed by a unitary
          similarity transformation, perturbed, and the returned
          to Hessenberg form that (it is to be hoped) has some
          zero subdiagonal entries.
[in]LDH
          LDH is INTEGER
          Leading dimension of H just as declared in the calling
          subroutine.  N <= LDH
[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 <= IHIZ <= N.
[in,out]Z
          Z is COMPLEX*16 array, dimension (LDZ,N)
          IF WANTZ is .TRUE., then on output, the unitary
          similarity transformation mentioned above has been
          accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
          If WANTZ is .FALSE., then Z is unreferenced.
[in]LDZ
          LDZ is INTEGER
          The leading dimension of Z just as declared in the
          calling subroutine.  1 <= LDZ.
[out]NS
          NS is INTEGER
          The number of unconverged (ie approximate) eigenvalues
          returned in SR and SI that may be used as shifts by the
          calling subroutine.
[out]ND
          ND is INTEGER
          The number of converged eigenvalues uncovered by this
          subroutine.
[out]SH
          SH is COMPLEX*16 array, dimension (KBOT)
          On output, approximate eigenvalues that may
          be used for shifts are stored in SH(KBOT-ND-NS+1)
          through SR(KBOT-ND).  Converged eigenvalues are
          stored in SH(KBOT-ND+1) through SH(KBOT).
[out]V
          V is COMPLEX*16 array, dimension (LDV,NW)
          An NW-by-NW work array.
[in]LDV
          LDV is INTEGER
          The leading dimension of V just as declared in the
          calling subroutine.  NW <= LDV
[in]NH
          NH is INTEGER
          The number of columns of T.  NH >= NW.
[out]T
          T is COMPLEX*16 array, dimension (LDT,NW)
[in]LDT
          LDT is INTEGER
          The leading dimension of T just as declared in the
          calling subroutine.  NW <= LDT
[in]NV
          NV is INTEGER
          The number of rows of work array WV available for
          workspace.  NV >= NW.
[out]WV
          WV is COMPLEX*16 array, dimension (LDWV,NW)
[in]LDWV
          LDWV is INTEGER
          The leading dimension of W just as declared in the
          calling subroutine.  NW <= LDV
[out]WORK
          WORK is COMPLEX*16 array, dimension (LWORK)
          On exit, WORK(1) is set to an estimate of the optimal value
          of LWORK for the given values of N, NW, KTOP and KBOT.
[in]LWORK
          LWORK is INTEGER
          The dimension of the work array WORK.  LWORK = 2*NW
          suffices, but greater efficiency may result from larger
          values of LWORK.

          If LWORK = -1, then a workspace query is assumed; ZLAQR2
          only estimates the optimal workspace size for the given
          values of N, NW, KTOP and KBOT.  The estimate is returned
          in WORK(1).  No error message related to LWORK is issued
          by XERBLA.  Neither H nor Z are 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

Definition at line 267 of file zlaqr2.f.

270*
271* -- LAPACK auxiliary routine --
272* -- LAPACK is a software package provided by Univ. of Tennessee, --
273* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
274*
275* .. Scalar Arguments ..
276 INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
277 $ LDZ, LWORK, N, ND, NH, NS, NV, NW
278 LOGICAL WANTT, WANTZ
279* ..
280* .. Array Arguments ..
281 COMPLEX*16 H( LDH, * ), SH( * ), T( LDT, * ), V( LDV, * ),
282 $ WORK( * ), WV( LDWV, * ), Z( LDZ, * )
283* ..
284*
285* ================================================================
286*
287* .. Parameters ..
288 COMPLEX*16 ZERO, ONE
289 parameter( zero = ( 0.0d0, 0.0d0 ),
290 $ one = ( 1.0d0, 0.0d0 ) )
291 DOUBLE PRECISION RZERO, RONE
292 parameter( rzero = 0.0d0, rone = 1.0d0 )
293* ..
294* .. Local Scalars ..
295 COMPLEX*16 BETA, CDUM, S, TAU
296 DOUBLE PRECISION FOO, SAFMAX, SAFMIN, SMLNUM, ULP
297 INTEGER I, IFST, ILST, INFO, INFQR, J, JW, KCOL, KLN,
298 $ KNT, KROW, KWTOP, LTOP, LWK1, LWK2, LWKOPT
299* ..
300* .. External Functions ..
301 DOUBLE PRECISION DLAMCH
302 EXTERNAL dlamch
303* ..
304* .. External Subroutines ..
305 EXTERNAL dlabad, zcopy, zgehrd, zgemm, zlacpy, zlahqr,
306 $ zlarf, zlarfg, zlaset, ztrexc, zunmhr
307* ..
308* .. Intrinsic Functions ..
309 INTRINSIC abs, dble, dcmplx, dconjg, dimag, int, max, min
310* ..
311* .. Statement Functions ..
312 DOUBLE PRECISION CABS1
313* ..
314* .. Statement Function definitions ..
315 cabs1( cdum ) = abs( dble( cdum ) ) + abs( dimag( cdum ) )
316* ..
317* .. Executable Statements ..
318*
319* ==== Estimate optimal workspace. ====
320*
321 jw = min( nw, kbot-ktop+1 )
322 IF( jw.LE.2 ) THEN
323 lwkopt = 1
324 ELSE
325*
326* ==== Workspace query call to ZGEHRD ====
327*
328 CALL zgehrd( jw, 1, jw-1, t, ldt, work, work, -1, info )
329 lwk1 = int( work( 1 ) )
330*
331* ==== Workspace query call to ZUNMHR ====
332*
333 CALL zunmhr( 'R', 'N', jw, jw, 1, jw-1, t, ldt, work, v, ldv,
334 $ work, -1, info )
335 lwk2 = int( work( 1 ) )
336*
337* ==== Optimal workspace ====
338*
339 lwkopt = jw + max( lwk1, lwk2 )
340 END IF
341*
342* ==== Quick return in case of workspace query. ====
343*
344 IF( lwork.EQ.-1 ) THEN
345 work( 1 ) = dcmplx( lwkopt, 0 )
346 RETURN
347 END IF
348*
349* ==== Nothing to do ...
350* ... for an empty active block ... ====
351 ns = 0
352 nd = 0
353 work( 1 ) = one
354 IF( ktop.GT.kbot )
355 $ RETURN
356* ... nor for an empty deflation window. ====
357 IF( nw.LT.1 )
358 $ RETURN
359*
360* ==== Machine constants ====
361*
362 safmin = dlamch( 'SAFE MINIMUM' )
363 safmax = rone / safmin
364 CALL dlabad( safmin, safmax )
365 ulp = dlamch( 'PRECISION' )
366 smlnum = safmin*( dble( n ) / ulp )
367*
368* ==== Setup deflation window ====
369*
370 jw = min( nw, kbot-ktop+1 )
371 kwtop = kbot - jw + 1
372 IF( kwtop.EQ.ktop ) THEN
373 s = zero
374 ELSE
375 s = h( kwtop, kwtop-1 )
376 END IF
377*
378 IF( kbot.EQ.kwtop ) THEN
379*
380* ==== 1-by-1 deflation window: not much to do ====
381*
382 sh( kwtop ) = h( kwtop, kwtop )
383 ns = 1
384 nd = 0
385 IF( cabs1( s ).LE.max( smlnum, ulp*cabs1( h( kwtop,
386 $ kwtop ) ) ) ) THEN
387 ns = 0
388 nd = 1
389 IF( kwtop.GT.ktop )
390 $ h( kwtop, kwtop-1 ) = zero
391 END IF
392 work( 1 ) = one
393 RETURN
394 END IF
395*
396* ==== Convert to spike-triangular form. (In case of a
397* . rare QR failure, this routine continues to do
398* . aggressive early deflation using that part of
399* . the deflation window that converged using INFQR
400* . here and there to keep track.) ====
401*
402 CALL zlacpy( 'U', jw, jw, h( kwtop, kwtop ), ldh, t, ldt )
403 CALL zcopy( jw-1, h( kwtop+1, kwtop ), ldh+1, t( 2, 1 ), ldt+1 )
404*
405 CALL zlaset( 'A', jw, jw, zero, one, v, ldv )
406 CALL zlahqr( .true., .true., jw, 1, jw, t, ldt, sh( kwtop ), 1,
407 $ jw, v, ldv, infqr )
408*
409* ==== Deflation detection loop ====
410*
411 ns = jw
412 ilst = infqr + 1
413 DO 10 knt = infqr + 1, jw
414*
415* ==== Small spike tip deflation test ====
416*
417 foo = cabs1( t( ns, ns ) )
418 IF( foo.EQ.rzero )
419 $ foo = cabs1( s )
420 IF( cabs1( s )*cabs1( v( 1, ns ) ).LE.max( smlnum, ulp*foo ) )
421 $ THEN
422*
423* ==== One more converged eigenvalue ====
424*
425 ns = ns - 1
426 ELSE
427*
428* ==== One undeflatable eigenvalue. Move it up out of the
429* . way. (ZTREXC can not fail in this case.) ====
430*
431 ifst = ns
432 CALL ztrexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, info )
433 ilst = ilst + 1
434 END IF
435 10 CONTINUE
436*
437* ==== Return to Hessenberg form ====
438*
439 IF( ns.EQ.0 )
440 $ s = zero
441*
442 IF( ns.LT.jw ) THEN
443*
444* ==== sorting the diagonal of T improves accuracy for
445* . graded matrices. ====
446*
447 DO 30 i = infqr + 1, ns
448 ifst = i
449 DO 20 j = i + 1, ns
450 IF( cabs1( t( j, j ) ).GT.cabs1( t( ifst, ifst ) ) )
451 $ ifst = j
452 20 CONTINUE
453 ilst = i
454 IF( ifst.NE.ilst )
455 $ CALL ztrexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, info )
456 30 CONTINUE
457 END IF
458*
459* ==== Restore shift/eigenvalue array from T ====
460*
461 DO 40 i = infqr + 1, jw
462 sh( kwtop+i-1 ) = t( i, i )
463 40 CONTINUE
464*
465*
466 IF( ns.LT.jw .OR. s.EQ.zero ) THEN
467 IF( ns.GT.1 .AND. s.NE.zero ) THEN
468*
469* ==== Reflect spike back into lower triangle ====
470*
471 CALL zcopy( ns, v, ldv, work, 1 )
472 DO 50 i = 1, ns
473 work( i ) = dconjg( work( i ) )
474 50 CONTINUE
475 beta = work( 1 )
476 CALL zlarfg( ns, beta, work( 2 ), 1, tau )
477 work( 1 ) = one
478*
479 CALL zlaset( 'L', jw-2, jw-2, zero, zero, t( 3, 1 ), ldt )
480*
481 CALL zlarf( 'L', ns, jw, work, 1, dconjg( tau ), t, ldt,
482 $ work( jw+1 ) )
483 CALL zlarf( 'R', ns, ns, work, 1, tau, t, ldt,
484 $ work( jw+1 ) )
485 CALL zlarf( 'R', jw, ns, work, 1, tau, v, ldv,
486 $ work( jw+1 ) )
487*
488 CALL zgehrd( jw, 1, ns, t, ldt, work, work( jw+1 ),
489 $ lwork-jw, info )
490 END IF
491*
492* ==== Copy updated reduced window into place ====
493*
494 IF( kwtop.GT.1 )
495 $ h( kwtop, kwtop-1 ) = s*dconjg( v( 1, 1 ) )
496 CALL zlacpy( 'U', jw, jw, t, ldt, h( kwtop, kwtop ), ldh )
497 CALL zcopy( jw-1, t( 2, 1 ), ldt+1, h( kwtop+1, kwtop ),
498 $ ldh+1 )
499*
500* ==== Accumulate orthogonal matrix in order update
501* . H and Z, if requested. ====
502*
503 IF( ns.GT.1 .AND. s.NE.zero )
504 $ CALL zunmhr( 'R', 'N', jw, ns, 1, ns, t, ldt, work, v, ldv,
505 $ work( jw+1 ), lwork-jw, info )
506*
507* ==== Update vertical slab in H ====
508*
509 IF( wantt ) THEN
510 ltop = 1
511 ELSE
512 ltop = ktop
513 END IF
514 DO 60 krow = ltop, kwtop - 1, nv
515 kln = min( nv, kwtop-krow )
516 CALL zgemm( 'N', 'N', kln, jw, jw, one, h( krow, kwtop ),
517 $ ldh, v, ldv, zero, wv, ldwv )
518 CALL zlacpy( 'A', kln, jw, wv, ldwv, h( krow, kwtop ), ldh )
519 60 CONTINUE
520*
521* ==== Update horizontal slab in H ====
522*
523 IF( wantt ) THEN
524 DO 70 kcol = kbot + 1, n, nh
525 kln = min( nh, n-kcol+1 )
526 CALL zgemm( 'C', 'N', jw, kln, jw, one, v, ldv,
527 $ h( kwtop, kcol ), ldh, zero, t, ldt )
528 CALL zlacpy( 'A', jw, kln, t, ldt, h( kwtop, kcol ),
529 $ ldh )
530 70 CONTINUE
531 END IF
532*
533* ==== Update vertical slab in Z ====
534*
535 IF( wantz ) THEN
536 DO 80 krow = iloz, ihiz, nv
537 kln = min( nv, ihiz-krow+1 )
538 CALL zgemm( 'N', 'N', kln, jw, jw, one, z( krow, kwtop ),
539 $ ldz, v, ldv, zero, wv, ldwv )
540 CALL zlacpy( 'A', kln, jw, wv, ldwv, z( krow, kwtop ),
541 $ ldz )
542 80 CONTINUE
543 END IF
544 END IF
545*
546* ==== Return the number of deflations ... ====
547*
548 nd = jw - ns
549*
550* ==== ... and the number of shifts. (Subtracting
551* . INFQR from the spike length takes care
552* . of the case of a rare QR failure while
553* . calculating eigenvalues of the deflation
554* . window.) ====
555*
556 ns = ns - infqr
557*
558* ==== Return optimal workspace. ====
559*
560 work( 1 ) = dcmplx( lwkopt, 0 )
561*
562* ==== End of ZLAQR2 ====
563*
dlamch
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
dlabad
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:74
zcopy
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
zgemm
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
zgehrd
subroutine zgehrd(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
ZGEHRD
Definition: zgehrd.f:167
zlahqr
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
zlacpy
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
zlaset
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: zlaset.f:106
zlarf
subroutine zlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
ZLARF applies an elementary reflector to a general rectangular matrix.
Definition: zlarf.f:128
zlarfg
subroutine zlarfg(N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Definition: zlarfg.f:106
ztrexc
subroutine ztrexc(COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, INFO)
ZTREXC
Definition: ztrexc.f:126
zunmhr
subroutine zunmhr(SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
ZUNMHR
Definition: zunmhr.f:178
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