LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
Loading...
Searching...
No Matches

◆ zhseqr()

subroutine zhseqr ( character  job,
character  compz,
integer  n,
integer  ilo,
integer  ihi,
complex*16, dimension( ldh, * )  h,
integer  ldh,
complex*16, dimension( * )  w,
complex*16, dimension( ldz, * )  z,
integer  ldz,
complex*16, dimension( * )  work,
integer  lwork,
integer  info 
)

ZHSEQR

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

Purpose:
    ZHSEQR computes the eigenvalues of a Hessenberg matrix H
    and, optionally, the matrices T and Z from the Schur decomposition
    H = Z T Z**H, where T is an upper triangular matrix (the
    Schur form), and Z is the unitary matrix of Schur vectors.

    Optionally Z may be postmultiplied into an input unitary
    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 unitary matrix Q:  A = Q*H*Q**H = (QZ)*T*(QZ)**H.
Parameters
[in]JOB
          JOB is CHARACTER*1
           = 'E':  compute eigenvalues only;
           = 'S':  compute eigenvalues and the Schur form T.
[in]COMPZ
          COMPZ is CHARACTER*1
           = 'N':  no Schur vectors are computed;
           = 'I':  Z is initialized to the unit matrix and the matrix Z
                   of Schur vectors of H is returned;
           = 'V':  Z must contain an unitary matrix Q on entry, and
                   the product Q*Z is returned.
[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. ILO and IHI are normally
           set by a previous call to ZGEBAL, and then passed to ZGEHRD
           when the matrix output by ZGEBAL 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 COMPLEX*16 array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and JOB = 'S', H contains the upper
           triangular matrix T from the Schur decomposition (the
           Schur form). If INFO = 0 and JOB = 'E', the contents of
           H are unspecified on exit.  (The output value of H when
           INFO > 0 is given under the description of INFO below.)

           Unlike earlier versions of ZHSEQR, this subroutine may
           explicitly 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]W
          W is COMPLEX*16 array, dimension (N)
           The computed eigenvalues. If JOB = 'S', the eigenvalues are
           stored in the same order as on the diagonal of the Schur
           form returned in H, with W(i) = H(i,i).
[in,out]Z
          Z is COMPLEX*16 array, dimension (LDZ,N)
           If COMPZ = 'N', Z is not referenced.
           If COMPZ = 'I', on entry Z need not be set and on exit,
           if INFO = 0, Z contains the unitary matrix Z of the Schur
           vectors of H.  If COMPZ = 'V', on entry Z must contain an
           N-by-N matrix Q, which is assumed to be equal to the unit
           matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
           if INFO = 0, Z contains Q*Z.
           Normally Q is the unitary matrix generated by ZUNGHR
           after the call to ZGEHRD which formed the Hessenberg matrix
           H. (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 COMPZ = 'I' or
           COMPZ = 'V', then LDZ >= MAX(1,N).  Otherwise, LDZ >= 1.
[out]WORK
          WORK is COMPLEX*16 array, dimension (LWORK)
           On exit, if INFO = 0, 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 and delivers very good and sometimes
           optimal performance.  However, LWORK as large as 11*N
           may be required for optimal performance.  A workspace
           query is recommended to determine the optimal workspace
           size.

           If LWORK = -1, then ZHSEQR does a workspace query.
           In this case, ZHSEQR 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, the i-th argument had an illegal
                    value
             > 0:  if INFO = i, ZHSEQR failed to compute all of
                the eigenvalues.  Elements 1:ilo-1 and i+1:n of W
                contain those eigenvalues which have been
                successfully computed.  (Failures are rare.)

                If INFO > 0 and JOB = 'E', 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 JOB   = 'S', then on exit

           (*)  (initial value of H)*U  = U*(final value of H)

                where U is a unitary matrix.  The final
                value of  H is upper Hessenberg and triangular in
                rows and columns INFO+1 through IHI.

                If INFO > 0 and COMPZ = 'V', then on exit

                  (final value of Z)  =  (initial value of Z)*U

                where U is the unitary matrix in (*) (regard-
                less of the value of JOB.)

                If INFO > 0 and COMPZ = 'I', then on exit
                      (final value of Z)  = U
                where U is the unitary matrix in (*) (regard-
                less of the value of JOB.)

                If INFO > 0 and COMPZ = 'N', then Z is not
                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
Further Details:
             Default values supplied by
             ILAENV(ISPEC,'ZHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
             It is suggested that these defaults be adjusted in order
             to attain best performance in each particular
             computational environment.

            ISPEC=12: The ZLAHQR vs ZLAQR0 crossover point.
                      Default: 75. (Must be at least 11.)

            ISPEC=13: Recommended deflation window size.
                      This depends on ILO, IHI and NS.  NS is the
                      number of simultaneous shifts returned
                      by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
                      The default for (IHI-ILO+1) <= 500 is NS.
                      The default for (IHI-ILO+1) >  500 is 3*NS/2.

            ISPEC=14: Nibble crossover point. (See IPARMQ for
                      details.)  Default: 14% of deflation window
                      size.

            ISPEC=15: Number of simultaneous shifts in a multishift
                      QR iteration.

                      If IHI-ILO+1 is ...

                      greater than      ...but less    ... the
                      or equal to ...      than        default is

                           1               30          NS =   2(+)
                          30               60          NS =   4(+)
                          60              150          NS =  10(+)
                         150              590          NS =  **
                         590             3000          NS =  64
                        3000             6000          NS = 128
                        6000             infinity      NS = 256

                  (+)  By default some or all matrices of this order
                       are passed to the implicit double shift routine
                       ZLAHQR and this parameter is ignored.  See
                       ISPEC=12 above and comments in IPARMQ for
                       details.

                 (**)  The asterisks (**) indicate an ad-hoc
                       function of N increasing from 10 to 64.

            ISPEC=16: Select structured matrix multiply.
                      If the number of simultaneous shifts (specified
                      by ISPEC=15) is less than 14, then the default
                      for ISPEC=16 is 0.  Otherwise the default for
                      ISPEC=16 is 2.
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 297 of file zhseqr.f.

299*
300* -- LAPACK computational routine --
301* -- LAPACK is a software package provided by Univ. of Tennessee, --
302* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
303*
304* .. Scalar Arguments ..
305 INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
306 CHARACTER COMPZ, JOB
307* ..
308* .. Array Arguments ..
309 COMPLEX*16 H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
310* ..
311*
312* =====================================================================
313*
314* .. Parameters ..
315*
316* ==== Matrices of order NTINY or smaller must be processed by
317* . ZLAHQR because of insufficient subdiagonal scratch space.
318* . (This is a hard limit.) ====
319 INTEGER NTINY
320 parameter( ntiny = 15 )
321*
322* ==== NL allocates some local workspace to help small matrices
323* . through a rare ZLAHQR failure. NL > NTINY = 15 is
324* . required and NL <= NMIN = ILAENV(ISPEC=12,...) is recom-
325* . mended. (The default value of NMIN is 75.) Using NL = 49
326* . allows up to six simultaneous shifts and a 16-by-16
327* . deflation window. ====
328 INTEGER NL
329 parameter( nl = 49 )
330 COMPLEX*16 ZERO, ONE
331 parameter( zero = ( 0.0d0, 0.0d0 ),
332 $ one = ( 1.0d0, 0.0d0 ) )
333 DOUBLE PRECISION RZERO
334 parameter( rzero = 0.0d0 )
335* ..
336* .. Local Arrays ..
337 COMPLEX*16 HL( NL, NL ), WORKL( NL )
338* ..
339* .. Local Scalars ..
340 INTEGER KBOT, NMIN
341 LOGICAL INITZ, LQUERY, WANTT, WANTZ
342* ..
343* .. External Functions ..
344 INTEGER ILAENV
345 LOGICAL LSAME
346 EXTERNAL ilaenv, lsame
347* ..
348* .. External Subroutines ..
349 EXTERNAL xerbla, zcopy, zlacpy, zlahqr, zlaqr0, zlaset
350* ..
351* .. Intrinsic Functions ..
352 INTRINSIC dble, dcmplx, max, min
353* ..
354* .. Executable Statements ..
355*
356* ==== Decode and check the input parameters. ====
357*
358 wantt = lsame( job, 'S' )
359 initz = lsame( compz, 'I' )
360 wantz = initz .OR. lsame( compz, 'V' )
361 work( 1 ) = dcmplx( dble( max( 1, n ) ), rzero )
362 lquery = lwork.EQ.-1
363*
364 info = 0
365 IF( .NOT.lsame( job, 'E' ) .AND. .NOT.wantt ) THEN
366 info = -1
367 ELSE IF( .NOT.lsame( compz, 'N' ) .AND. .NOT.wantz ) THEN
368 info = -2
369 ELSE IF( n.LT.0 ) THEN
370 info = -3
371 ELSE IF( ilo.LT.1 .OR. ilo.GT.max( 1, n ) ) THEN
372 info = -4
373 ELSE IF( ihi.LT.min( ilo, n ) .OR. ihi.GT.n ) THEN
374 info = -5
375 ELSE IF( ldh.LT.max( 1, n ) ) THEN
376 info = -7
377 ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.max( 1, n ) ) ) THEN
378 info = -10
379 ELSE IF( lwork.LT.max( 1, n ) .AND. .NOT.lquery ) THEN
380 info = -12
381 END IF
382*
383 IF( info.NE.0 ) THEN
384*
385* ==== Quick return in case of invalid argument. ====
386*
387 CALL xerbla( 'ZHSEQR', -info )
388 RETURN
389*
390 ELSE IF( n.EQ.0 ) THEN
391*
392* ==== Quick return in case N = 0; nothing to do. ====
393*
394 RETURN
395*
396 ELSE IF( lquery ) THEN
397*
398* ==== Quick return in case of a workspace query ====
399*
400 CALL zlaqr0( wantt, wantz, n, ilo, ihi, h, ldh, w, ilo, ihi, z,
401 $ ldz, work, lwork, info )
402* ==== Ensure reported workspace size is backward-compatible with
403* . previous LAPACK versions. ====
404 work( 1 ) = dcmplx( max( dble( work( 1 ) ), dble( max( 1,
405 $ n ) ) ), rzero )
406 RETURN
407*
408 ELSE
409*
410* ==== copy eigenvalues isolated by ZGEBAL ====
411*
412 IF( ilo.GT.1 )
413 $ CALL zcopy( ilo-1, h, ldh+1, w, 1 )
414 IF( ihi.LT.n )
415 $ CALL zcopy( n-ihi, h( ihi+1, ihi+1 ), ldh+1, w( ihi+1 ), 1 )
416*
417* ==== Initialize Z, if requested ====
418*
419 IF( initz )
420 $ CALL zlaset( 'A', n, n, zero, one, z, ldz )
421*
422* ==== Quick return if possible ====
423*
424 IF( ilo.EQ.ihi ) THEN
425 w( ilo ) = h( ilo, ilo )
426 RETURN
427 END IF
428*
429* ==== ZLAHQR/ZLAQR0 crossover point ====
430*
431 nmin = ilaenv( 12, 'ZHSEQR', job( : 1 ) // compz( : 1 ), n,
432 $ ilo, ihi, lwork )
433 nmin = max( ntiny, nmin )
434*
435* ==== ZLAQR0 for big matrices; ZLAHQR for small ones ====
436*
437 IF( n.GT.nmin ) THEN
438 CALL zlaqr0( wantt, wantz, n, ilo, ihi, h, ldh, w, ilo, ihi,
439 $ z, ldz, work, lwork, info )
440 ELSE
441*
442* ==== Small matrix ====
443*
444 CALL zlahqr( wantt, wantz, n, ilo, ihi, h, ldh, w, ilo, ihi,
445 $ z, ldz, info )
446*
447 IF( info.GT.0 ) THEN
448*
449* ==== A rare ZLAHQR failure! ZLAQR0 sometimes succeeds
450* . when ZLAHQR fails. ====
451*
452 kbot = info
453*
454 IF( n.GE.nl ) THEN
455*
456* ==== Larger matrices have enough subdiagonal scratch
457* . space to call ZLAQR0 directly. ====
458*
459 CALL zlaqr0( wantt, wantz, n, ilo, kbot, h, ldh, w,
460 $ ilo, ihi, z, ldz, work, lwork, info )
461*
462 ELSE
463*
464* ==== Tiny matrices don't have enough subdiagonal
465* . scratch space to benefit from ZLAQR0. Hence,
466* . tiny matrices must be copied into a larger
467* . array before calling ZLAQR0. ====
468*
469 CALL zlacpy( 'A', n, n, h, ldh, hl, nl )
470 hl( n+1, n ) = zero
471 CALL zlaset( 'A', nl, nl-n, zero, zero, hl( 1, n+1 ),
472 $ nl )
473 CALL zlaqr0( wantt, wantz, nl, ilo, kbot, hl, nl, w,
474 $ ilo, ihi, z, ldz, workl, nl, info )
475 IF( wantt .OR. info.NE.0 )
476 $ CALL zlacpy( 'A', n, n, hl, nl, h, ldh )
477 END IF
478 END IF
479 END IF
480*
481* ==== Clear out the trash, if necessary. ====
482*
483 IF( ( wantt .OR. info.NE.0 ) .AND. n.GT.2 )
484 $ CALL zlaset( 'L', n-2, n-2, zero, zero, h( 3, 1 ), ldh )
485*
486* ==== Ensure reported workspace size is backward-compatible with
487* . previous LAPACK versions. ====
488*
489 work( 1 ) = dcmplx( max( dble( max( 1, n ) ),
490 $ dble( work( 1 ) ) ), rzero )
491 END IF
492*
493* ==== End of ZHSEQR ====
494*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zcopy(n, zx, incx, zy, incy)
ZCOPY
Definition zcopy.f:81
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
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
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
subroutine zlaqr0(wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, work, lwork, info)
ZLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur de...
Definition zlaqr0.f:241
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
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
Here is the call graph for this function:
Here is the caller graph for this function: