LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ claqr2()

subroutine claqr2 ( 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 
)

CLAQR2 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 CLAQR2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
    CLAQR2 is identical to CLAQR3 except that it avoids
    recursion by calling CLAHQR instead of CLAQR4.

    Aggressive early deflation:

    This subroutine 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; CLAQR2
          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 266 of file claqr2.f.

269 *
270 * -- LAPACK auxiliary routine --
271 * -- LAPACK is a software package provided by Univ. of Tennessee, --
272 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
273 *
274 * .. Scalar Arguments ..
275  INTEGER IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
276  $ LDZ, LWORK, N, ND, NH, NS, NV, NW
277  LOGICAL WANTT, WANTZ
278 * ..
279 * .. Array Arguments ..
280  COMPLEX H( LDH, * ), SH( * ), T( LDT, * ), V( LDV, * ),
281  $ WORK( * ), WV( LDWV, * ), Z( LDZ, * )
282 * ..
283 *
284 * ================================================================
285 *
286 * .. Parameters ..
287  COMPLEX ZERO, ONE
288  parameter( zero = ( 0.0e0, 0.0e0 ),
289  $ one = ( 1.0e0, 0.0e0 ) )
290  REAL RZERO, RONE
291  parameter( rzero = 0.0e0, rone = 1.0e0 )
292 * ..
293 * .. Local Scalars ..
294  COMPLEX BETA, CDUM, S, TAU
295  REAL FOO, SAFMAX, SAFMIN, SMLNUM, ULP
296  INTEGER I, IFST, ILST, INFO, INFQR, J, JW, KCOL, KLN,
297  $ KNT, KROW, KWTOP, LTOP, LWK1, LWK2, LWKOPT
298 * ..
299 * .. External Functions ..
300  REAL SLAMCH
301  EXTERNAL slamch
302 * ..
303 * .. External Subroutines ..
304  EXTERNAL ccopy, cgehrd, cgemm, clacpy, clahqr, clarf,
306 * ..
307 * .. Intrinsic Functions ..
308  INTRINSIC abs, aimag, cmplx, conjg, int, max, min, real
309 * ..
310 * .. Statement Functions ..
311  REAL CABS1
312 * ..
313 * .. Statement Function definitions ..
314  cabs1( cdum ) = abs( real( cdum ) ) + abs( aimag( cdum ) )
315 * ..
316 * .. Executable Statements ..
317 *
318 * ==== Estimate optimal workspace. ====
319 *
320  jw = min( nw, kbot-ktop+1 )
321  IF( jw.LE.2 ) THEN
322  lwkopt = 1
323  ELSE
324 *
325 * ==== Workspace query call to CGEHRD ====
326 *
327  CALL cgehrd( jw, 1, jw-1, t, ldt, work, work, -1, info )
328  lwk1 = int( work( 1 ) )
329 *
330 * ==== Workspace query call to CUNMHR ====
331 *
332  CALL cunmhr( 'R', 'N', jw, jw, 1, jw-1, t, ldt, work, v, ldv,
333  $ work, -1, info )
334  lwk2 = int( work( 1 ) )
335 *
336 * ==== Optimal workspace ====
337 *
338  lwkopt = jw + max( lwk1, lwk2 )
339  END IF
340 *
341 * ==== Quick return in case of workspace query. ====
342 *
343  IF( lwork.EQ.-1 ) THEN
344  work( 1 ) = cmplx( lwkopt, 0 )
345  RETURN
346  END IF
347 *
348 * ==== Nothing to do ...
349 * ... for an empty active block ... ====
350  ns = 0
351  nd = 0
352  work( 1 ) = one
353  IF( ktop.GT.kbot )
354  $ RETURN
355 * ... nor for an empty deflation window. ====
356  IF( nw.LT.1 )
357  $ RETURN
358 *
359 * ==== Machine constants ====
360 *
361  safmin = slamch( 'SAFE MINIMUM' )
362  safmax = rone / safmin
363  CALL slabad( safmin, safmax )
364  ulp = slamch( 'PRECISION' )
365  smlnum = safmin*( real( n ) / ulp )
366 *
367 * ==== Setup deflation window ====
368 *
369  jw = min( nw, kbot-ktop+1 )
370  kwtop = kbot - jw + 1
371  IF( kwtop.EQ.ktop ) THEN
372  s = zero
373  ELSE
374  s = h( kwtop, kwtop-1 )
375  END IF
376 *
377  IF( kbot.EQ.kwtop ) THEN
378 *
379 * ==== 1-by-1 deflation window: not much to do ====
380 *
381  sh( kwtop ) = h( kwtop, kwtop )
382  ns = 1
383  nd = 0
384  IF( cabs1( s ).LE.max( smlnum, ulp*cabs1( h( kwtop,
385  $ kwtop ) ) ) ) THEN
386  ns = 0
387  nd = 1
388  IF( kwtop.GT.ktop )
389  $ h( kwtop, kwtop-1 ) = zero
390  END IF
391  work( 1 ) = one
392  RETURN
393  END IF
394 *
395 * ==== Convert to spike-triangular form. (In case of a
396 * . rare QR failure, this routine continues to do
397 * . aggressive early deflation using that part of
398 * . the deflation window that converged using INFQR
399 * . here and there to keep track.) ====
400 *
401  CALL clacpy( 'U', jw, jw, h( kwtop, kwtop ), ldh, t, ldt )
402  CALL ccopy( jw-1, h( kwtop+1, kwtop ), ldh+1, t( 2, 1 ), ldt+1 )
403 *
404  CALL claset( 'A', jw, jw, zero, one, v, ldv )
405  CALL clahqr( .true., .true., jw, 1, jw, t, ldt, sh( kwtop ), 1,
406  $ jw, v, ldv, infqr )
407 *
408 * ==== Deflation detection loop ====
409 *
410  ns = jw
411  ilst = infqr + 1
412  DO 10 knt = infqr + 1, jw
413 *
414 * ==== Small spike tip deflation test ====
415 *
416  foo = cabs1( t( ns, ns ) )
417  IF( foo.EQ.rzero )
418  $ foo = cabs1( s )
419  IF( cabs1( s )*cabs1( v( 1, ns ) ).LE.max( smlnum, ulp*foo ) )
420  $ THEN
421 *
422 * ==== One more converged eigenvalue ====
423 *
424  ns = ns - 1
425  ELSE
426 *
427 * ==== One undeflatable eigenvalue. Move it up out of the
428 * . way. (CTREXC can not fail in this case.) ====
429 *
430  ifst = ns
431  CALL ctrexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, info )
432  ilst = ilst + 1
433  END IF
434  10 CONTINUE
435 *
436 * ==== Return to Hessenberg form ====
437 *
438  IF( ns.EQ.0 )
439  $ s = zero
440 *
441  IF( ns.LT.jw ) THEN
442 *
443 * ==== sorting the diagonal of T improves accuracy for
444 * . graded matrices. ====
445 *
446  DO 30 i = infqr + 1, ns
447  ifst = i
448  DO 20 j = i + 1, ns
449  IF( cabs1( t( j, j ) ).GT.cabs1( t( ifst, ifst ) ) )
450  $ ifst = j
451  20 CONTINUE
452  ilst = i
453  IF( ifst.NE.ilst )
454  $ CALL ctrexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, info )
455  30 CONTINUE
456  END IF
457 *
458 * ==== Restore shift/eigenvalue array from T ====
459 *
460  DO 40 i = infqr + 1, jw
461  sh( kwtop+i-1 ) = t( i, i )
462  40 CONTINUE
463 *
464 *
465  IF( ns.LT.jw .OR. s.EQ.zero ) THEN
466  IF( ns.GT.1 .AND. s.NE.zero ) THEN
467 *
468 * ==== Reflect spike back into lower triangle ====
469 *
470  CALL ccopy( ns, v, ldv, work, 1 )
471  DO 50 i = 1, ns
472  work( i ) = conjg( work( i ) )
473  50 CONTINUE
474  beta = work( 1 )
475  CALL clarfg( ns, beta, work( 2 ), 1, tau )
476  work( 1 ) = one
477 *
478  CALL claset( 'L', jw-2, jw-2, zero, zero, t( 3, 1 ), ldt )
479 *
480  CALL clarf( 'L', ns, jw, work, 1, conjg( tau ), t, ldt,
481  $ work( jw+1 ) )
482  CALL clarf( 'R', ns, ns, work, 1, tau, t, ldt,
483  $ work( jw+1 ) )
484  CALL clarf( 'R', jw, ns, work, 1, tau, v, ldv,
485  $ work( jw+1 ) )
486 *
487  CALL cgehrd( jw, 1, ns, t, ldt, work, work( jw+1 ),
488  $ lwork-jw, info )
489  END IF
490 *
491 * ==== Copy updated reduced window into place ====
492 *
493  IF( kwtop.GT.1 )
494  $ h( kwtop, kwtop-1 ) = s*conjg( v( 1, 1 ) )
495  CALL clacpy( 'U', jw, jw, t, ldt, h( kwtop, kwtop ), ldh )
496  CALL ccopy( jw-1, t( 2, 1 ), ldt+1, h( kwtop+1, kwtop ),
497  $ ldh+1 )
498 *
499 * ==== Accumulate orthogonal matrix in order update
500 * . H and Z, if requested. ====
501 *
502  IF( ns.GT.1 .AND. s.NE.zero )
503  $ CALL cunmhr( 'R', 'N', jw, ns, 1, ns, t, ldt, work, v, ldv,
504  $ work( jw+1 ), lwork-jw, info )
505 *
506 * ==== Update vertical slab in H ====
507 *
508  IF( wantt ) THEN
509  ltop = 1
510  ELSE
511  ltop = ktop
512  END IF
513  DO 60 krow = ltop, kwtop - 1, nv
514  kln = min( nv, kwtop-krow )
515  CALL cgemm( 'N', 'N', kln, jw, jw, one, h( krow, kwtop ),
516  $ ldh, v, ldv, zero, wv, ldwv )
517  CALL clacpy( 'A', kln, jw, wv, ldwv, h( krow, kwtop ), ldh )
518  60 CONTINUE
519 *
520 * ==== Update horizontal slab in H ====
521 *
522  IF( wantt ) THEN
523  DO 70 kcol = kbot + 1, n, nh
524  kln = min( nh, n-kcol+1 )
525  CALL cgemm( 'C', 'N', jw, kln, jw, one, v, ldv,
526  $ h( kwtop, kcol ), ldh, zero, t, ldt )
527  CALL clacpy( 'A', jw, kln, t, ldt, h( kwtop, kcol ),
528  $ ldh )
529  70 CONTINUE
530  END IF
531 *
532 * ==== Update vertical slab in Z ====
533 *
534  IF( wantz ) THEN
535  DO 80 krow = iloz, ihiz, nv
536  kln = min( nv, ihiz-krow+1 )
537  CALL cgemm( 'N', 'N', kln, jw, jw, one, z( krow, kwtop ),
538  $ ldz, v, ldv, zero, wv, ldwv )
539  CALL clacpy( 'A', kln, jw, wv, ldwv, z( krow, kwtop ),
540  $ ldz )
541  80 CONTINUE
542  END IF
543  END IF
544 *
545 * ==== Return the number of deflations ... ====
546 *
547  nd = jw - ns
548 *
549 * ==== ... and the number of shifts. (Subtracting
550 * . INFQR from the spike length takes care
551 * . of the case of a rare QR failure while
552 * . calculating eigenvalues of the deflation
553 * . window.) ====
554 *
555  ns = ns - infqr
556 *
557 * ==== Return optimal workspace. ====
558 *
559  work( 1 ) = cmplx( lwkopt, 0 )
560 *
561 * ==== End of CLAQR2 ====
562 *
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:74
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 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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