LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ 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,
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 *
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:74
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine cgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CGEMM
Definition: cgemm.f:187
subroutine cgehrd(N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO)
CGEHRD
Definition: cgehrd.f:167
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
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
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
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:106
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
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
subroutine cunmhr(SIDE, TRANS, M, N, ILO, IHI, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
CUNMHR
Definition: cunmhr.f:179
subroutine ctrexc(COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, INFO)
CTREXC
Definition: ctrexc.f:126
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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