LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ claqr3()

subroutine claqr3 ( logical  WANTT,
logical  WANTZ,
integer  N,
integer  KTOP,
integer  KBOT,
integer  NW,
complex, dimension( ldh, * )  H,
integer  LDH,
integer  ILOZ,
integer  IHIZ,
complex, dimension( ldz, * )  Z,
integer  LDZ,
integer  NS,
integer  ND,
complex, dimension( * )  SH,
complex, dimension( ldv, * )  V,
integer  LDV,
integer  NH,
complex, dimension( ldt, * )  T,
integer  LDT,
integer  NV,
complex, dimension( ldwv, * )  WV,
integer  LDWV,
complex, dimension( * )  WORK,
integer  LWORK 
)

CLAQR3 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 CLAQR3 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
    Aggressive early deflation:

    CLAQR3 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 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 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 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 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 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 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 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; CLAQR3
          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 263 of file claqr3.f.

266*
267* -- LAPACK auxiliary routine --
268* -- LAPACK is a software package provided by Univ. of Tennessee, --
269* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
270*
271* .. Scalar Arguments ..
272 INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
273 $ LDZ, LWORK, N, ND, NH, NS, NV, NW
274 LOGICAL WANTT, WANTZ
275* ..
276* .. Array Arguments ..
277 COMPLEX H( LDH, * ), SH( * ), T( LDT, * ), V( LDV, * ),
278 $ WORK( * ), WV( LDWV, * ), Z( LDZ, * )
279* ..
280*
281* ================================================================
282*
283* .. Parameters ..
284 COMPLEX ZERO, ONE
285 parameter( zero = ( 0.0e0, 0.0e0 ),
286 $ one = ( 1.0e0, 0.0e0 ) )
287 REAL RZERO, RONE
288 parameter( rzero = 0.0e0, rone = 1.0e0 )
289* ..
290* .. Local Scalars ..
291 COMPLEX BETA, CDUM, S, TAU
292 REAL FOO, SAFMAX, SAFMIN, SMLNUM, ULP
293 INTEGER I, IFST, ILST, INFO, INFQR, J, JW, KCOL, KLN,
294 $ KNT, KROW, KWTOP, LTOP, LWK1, LWK2, LWK3,
295 $ LWKOPT, NMIN
296* ..
297* .. External Functions ..
298 REAL SLAMCH
299 INTEGER ILAENV
300 EXTERNAL slamch, ilaenv
301* ..
302* .. External Subroutines ..
303 EXTERNAL ccopy, cgehrd, cgemm, clacpy, clahqr, claqr4,
304 $ clarf, clarfg, claset, ctrexc, cunmhr, slabad
305* ..
306* .. Intrinsic Functions ..
307 INTRINSIC abs, aimag, cmplx, conjg, int, max, min, real
308* ..
309* .. Statement Functions ..
310 REAL CABS1
311* ..
312* .. Statement Function definitions ..
313 cabs1( cdum ) = abs( real( cdum ) ) + abs( aimag( cdum ) )
314* ..
315* .. Executable Statements ..
316*
317* ==== Estimate optimal workspace. ====
318*
319 jw = min( nw, kbot-ktop+1 )
320 IF( jw.LE.2 ) THEN
321 lwkopt = 1
322 ELSE
323*
324* ==== Workspace query call to CGEHRD ====
325*
326 CALL cgehrd( jw, 1, jw-1, t, ldt, work, work, -1, info )
327 lwk1 = int( work( 1 ) )
328*
329* ==== Workspace query call to CUNMHR ====
330*
331 CALL cunmhr( 'R', 'N', jw, jw, 1, jw-1, t, ldt, work, v, ldv,
332 $ work, -1, info )
333 lwk2 = int( work( 1 ) )
334*
335* ==== Workspace query call to CLAQR4 ====
336*
337 CALL claqr4( .true., .true., jw, 1, jw, t, ldt, sh, 1, jw, v,
338 $ ldv, work, -1, infqr )
339 lwk3 = int( work( 1 ) )
340*
341* ==== Optimal workspace ====
342*
343 lwkopt = max( jw+max( lwk1, lwk2 ), lwk3 )
344 END IF
345*
346* ==== Quick return in case of workspace query. ====
347*
348 IF( lwork.EQ.-1 ) THEN
349 work( 1 ) = cmplx( lwkopt, 0 )
350 RETURN
351 END IF
352*
353* ==== Nothing to do ...
354* ... for an empty active block ... ====
355 ns = 0
356 nd = 0
357 work( 1 ) = one
358 IF( ktop.GT.kbot )
359 $ RETURN
360* ... nor for an empty deflation window. ====
361 IF( nw.LT.1 )
362 $ RETURN
363*
364* ==== Machine constants ====
365*
366 safmin = slamch( 'SAFE MINIMUM' )
367 safmax = rone / safmin
368 CALL slabad( safmin, safmax )
369 ulp = slamch( 'PRECISION' )
370 smlnum = safmin*( real( n ) / ulp )
371*
372* ==== Setup deflation window ====
373*
374 jw = min( nw, kbot-ktop+1 )
375 kwtop = kbot - jw + 1
376 IF( kwtop.EQ.ktop ) THEN
377 s = zero
378 ELSE
379 s = h( kwtop, kwtop-1 )
380 END IF
381*
382 IF( kbot.EQ.kwtop ) THEN
383*
384* ==== 1-by-1 deflation window: not much to do ====
385*
386 sh( kwtop ) = h( kwtop, kwtop )
387 ns = 1
388 nd = 0
389 IF( cabs1( s ).LE.max( smlnum, ulp*cabs1( h( kwtop,
390 $ kwtop ) ) ) ) THEN
391 ns = 0
392 nd = 1
393 IF( kwtop.GT.ktop )
394 $ h( kwtop, kwtop-1 ) = zero
395 END IF
396 work( 1 ) = one
397 RETURN
398 END IF
399*
400* ==== Convert to spike-triangular form. (In case of a
401* . rare QR failure, this routine continues to do
402* . aggressive early deflation using that part of
403* . the deflation window that converged using INFQR
404* . here and there to keep track.) ====
405*
406 CALL clacpy( 'U', jw, jw, h( kwtop, kwtop ), ldh, t, ldt )
407 CALL ccopy( jw-1, h( kwtop+1, kwtop ), ldh+1, t( 2, 1 ), ldt+1 )
408*
409 CALL claset( 'A', jw, jw, zero, one, v, ldv )
410 nmin = ilaenv( 12, 'CLAQR3', 'SV', jw, 1, jw, lwork )
411 IF( jw.GT.nmin ) THEN
412 CALL claqr4( .true., .true., jw, 1, jw, t, ldt, sh( kwtop ), 1,
413 $ jw, v, ldv, work, lwork, infqr )
414 ELSE
415 CALL clahqr( .true., .true., jw, 1, jw, t, ldt, sh( kwtop ), 1,
416 $ jw, v, ldv, infqr )
417 END IF
418*
419* ==== Deflation detection loop ====
420*
421 ns = jw
422 ilst = infqr + 1
423 DO 10 knt = infqr + 1, jw
424*
425* ==== Small spike tip deflation test ====
426*
427 foo = cabs1( t( ns, ns ) )
428 IF( foo.EQ.rzero )
429 $ foo = cabs1( s )
430 IF( cabs1( s )*cabs1( v( 1, ns ) ).LE.max( smlnum, ulp*foo ) )
431 $ THEN
432*
433* ==== One more converged eigenvalue ====
434*
435 ns = ns - 1
436 ELSE
437*
438* ==== One undeflatable eigenvalue. Move it up out of the
439* . way. (CTREXC can not fail in this case.) ====
440*
441 ifst = ns
442 CALL ctrexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, info )
443 ilst = ilst + 1
444 END IF
445 10 CONTINUE
446*
447* ==== Return to Hessenberg form ====
448*
449 IF( ns.EQ.0 )
450 $ s = zero
451*
452 IF( ns.LT.jw ) THEN
453*
454* ==== sorting the diagonal of T improves accuracy for
455* . graded matrices. ====
456*
457 DO 30 i = infqr + 1, ns
458 ifst = i
459 DO 20 j = i + 1, ns
460 IF( cabs1( t( j, j ) ).GT.cabs1( t( ifst, ifst ) ) )
461 $ ifst = j
462 20 CONTINUE
463 ilst = i
464 IF( ifst.NE.ilst )
465 $ CALL ctrexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, info )
466 30 CONTINUE
467 END IF
468*
469* ==== Restore shift/eigenvalue array from T ====
470*
471 DO 40 i = infqr + 1, jw
472 sh( kwtop+i-1 ) = t( i, i )
473 40 CONTINUE
474*
475*
476 IF( ns.LT.jw .OR. s.EQ.zero ) THEN
477 IF( ns.GT.1 .AND. s.NE.zero ) THEN
478*
479* ==== Reflect spike back into lower triangle ====
480*
481 CALL ccopy( ns, v, ldv, work, 1 )
482 DO 50 i = 1, ns
483 work( i ) = conjg( work( i ) )
484 50 CONTINUE
485 beta = work( 1 )
486 CALL clarfg( ns, beta, work( 2 ), 1, tau )
487 work( 1 ) = one
488*
489 CALL claset( 'L', jw-2, jw-2, zero, zero, t( 3, 1 ), ldt )
490*
491 CALL clarf( 'L', ns, jw, work, 1, conjg( tau ), t, ldt,
492 $ work( jw+1 ) )
493 CALL clarf( 'R', ns, ns, work, 1, tau, t, ldt,
494 $ work( jw+1 ) )
495 CALL clarf( 'R', jw, ns, work, 1, tau, v, ldv,
496 $ work( jw+1 ) )
497*
498 CALL cgehrd( jw, 1, ns, t, ldt, work, work( jw+1 ),
499 $ lwork-jw, info )
500 END IF
501*
502* ==== Copy updated reduced window into place ====
503*
504 IF( kwtop.GT.1 )
505 $ h( kwtop, kwtop-1 ) = s*conjg( v( 1, 1 ) )
506 CALL clacpy( 'U', jw, jw, t, ldt, h( kwtop, kwtop ), ldh )
507 CALL ccopy( jw-1, t( 2, 1 ), ldt+1, h( kwtop+1, kwtop ),
508 $ ldh+1 )
509*
510* ==== Accumulate orthogonal matrix in order update
511* . H and Z, if requested. ====
512*
513 IF( ns.GT.1 .AND. s.NE.zero )
514 $ CALL cunmhr( 'R', 'N', jw, ns, 1, ns, t, ldt, work, v, ldv,
515 $ work( jw+1 ), lwork-jw, info )
516*
517* ==== Update vertical slab in H ====
518*
519 IF( wantt ) THEN
520 ltop = 1
521 ELSE
522 ltop = ktop
523 END IF
524 DO 60 krow = ltop, kwtop - 1, nv
525 kln = min( nv, kwtop-krow )
526 CALL cgemm( 'N', 'N', kln, jw, jw, one, h( krow, kwtop ),
527 $ ldh, v, ldv, zero, wv, ldwv )
528 CALL clacpy( 'A', kln, jw, wv, ldwv, h( krow, kwtop ), ldh )
529 60 CONTINUE
530*
531* ==== Update horizontal slab in H ====
532*
533 IF( wantt ) THEN
534 DO 70 kcol = kbot + 1, n, nh
535 kln = min( nh, n-kcol+1 )
536 CALL cgemm( 'C', 'N', jw, kln, jw, one, v, ldv,
537 $ h( kwtop, kcol ), ldh, zero, t, ldt )
538 CALL clacpy( 'A', jw, kln, t, ldt, h( kwtop, kcol ),
539 $ ldh )
540 70 CONTINUE
541 END IF
542*
543* ==== Update vertical slab in Z ====
544*
545 IF( wantz ) THEN
546 DO 80 krow = iloz, ihiz, nv
547 kln = min( nv, ihiz-krow+1 )
548 CALL cgemm( 'N', 'N', kln, jw, jw, one, z( krow, kwtop ),
549 $ ldz, v, ldv, zero, wv, ldwv )
550 CALL clacpy( 'A', kln, jw, wv, ldwv, z( krow, kwtop ),
551 $ ldz )
552 80 CONTINUE
553 END IF
554 END IF
555*
556* ==== Return the number of deflations ... ====
557*
558 nd = jw - ns
559*
560* ==== ... and the number of shifts. (Subtracting
561* . INFQR from the spike length takes care
562* . of the case of a rare QR failure while
563* . calculating eigenvalues of the deflation
564* . window.) ====
565*
566 ns = ns - infqr
567*
568* ==== Return optimal workspace. ====
569*
570 work( 1 ) = cmplx( lwkopt, 0 )
571*
572* ==== End of CLAQR3 ====
573*
slabad
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:74
ilaenv
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
ccopy
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
cgemm
subroutine cgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CGEMM
Definition: cgemm.f:187
cgehrd
subroutine cgehrd(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
CGEHRD
Definition: cgehrd.f:167
claset
subroutine claset(UPLO, M, N, ALPHA, BETA, A, LDA)
CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: claset.f:106
clarf
subroutine clarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
CLARF applies an elementary reflector to a general rectangular matrix.
Definition: clarf.f:128
claqr4
subroutine claqr4(WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur de...
Definition: claqr4.f:248
clarfg
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:106
clahqr
subroutine clahqr(WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, INFO)
CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,...
Definition: clahqr.f:195
clacpy
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
cunmhr
subroutine cunmhr(SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMHR
Definition: cunmhr.f:179
ctrexc
subroutine ctrexc(COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, INFO)
CTREXC
Definition: ctrexc.f:126
slamch
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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