LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 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

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 .GE. 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.GT.0, then 1.LE.ILO.LE.IHI.LE.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.GT.0 is given under the description of INFO below.) Unlike earlier versions of ZHSEQR, this subroutine may explicitly H(i,j) = 0 for i.GT.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 .GE. 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.GT.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.GE.MAX(1,N). Otherwize, LDZ.GE.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 .GE. 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 .LT. 0: if INFO = -i, the i-th argument had an illegal value .GT. 0: if INFO = i, ZHSEQR failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.) If INFO .GT. 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 .GT. 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 .GT. 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 .GT. 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 .GT. 0 and COMPZ = 'N', then Z is not accessed.```
Date
November 2013
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).LE.500 is NS.
The default for (IHI-ILO+1).GT.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 301 of file zhseqr.f.

301 *
302 * -- LAPACK computational routine (version 3.5.0) --
303 * -- LAPACK is a software package provided by Univ. of Tennessee, --
304 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
305 * November 2013
306 *
307 * .. Scalar Arguments ..
308  INTEGER ihi, ilo, info, ldh, ldz, lwork, n
309  CHARACTER compz, job
310 * ..
311 * .. Array Arguments ..
312  COMPLEX*16 h( ldh, * ), w( * ), work( * ), z( ldz, * )
313 * ..
314 *
315 * =====================================================================
316 *
317 * .. Parameters ..
318 *
319 * ==== Matrices of order NTINY or smaller must be processed by
320 * . ZLAHQR because of insufficient subdiagonal scratch space.
321 * . (This is a hard limit.) ====
322  INTEGER ntiny
323  parameter ( ntiny = 11 )
324 *
325 * ==== NL allocates some local workspace to help small matrices
326 * . through a rare ZLAHQR failure. NL .GT. NTINY = 11 is
327 * . required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom-
328 * . mended. (The default value of NMIN is 75.) Using NL = 49
329 * . allows up to six simultaneous shifts and a 16-by-16
330 * . deflation window. ====
331  INTEGER nl
332  parameter ( nl = 49 )
333  COMPLEX*16 zero, one
334  parameter ( zero = ( 0.0d0, 0.0d0 ),
335  \$ one = ( 1.0d0, 0.0d0 ) )
336  DOUBLE PRECISION rzero
337  parameter ( rzero = 0.0d0 )
338 * ..
339 * .. Local Arrays ..
340  COMPLEX*16 hl( nl, nl ), workl( nl )
341 * ..
342 * .. Local Scalars ..
343  INTEGER kbot, nmin
344  LOGICAL initz, lquery, wantt, wantz
345 * ..
346 * .. External Functions ..
347  INTEGER ilaenv
348  LOGICAL lsame
349  EXTERNAL ilaenv, lsame
350 * ..
351 * .. External Subroutines ..
352  EXTERNAL xerbla, zcopy, zlacpy, zlahqr, zlaqr0, zlaset
353 * ..
354 * .. Intrinsic Functions ..
355  INTRINSIC dble, dcmplx, max, min
356 * ..
357 * .. Executable Statements ..
358 *
359 * ==== Decode and check the input parameters. ====
360 *
361  wantt = lsame( job, 'S' )
362  initz = lsame( compz, 'I' )
363  wantz = initz .OR. lsame( compz, 'V' )
364  work( 1 ) = dcmplx( dble( max( 1, n ) ), rzero )
365  lquery = lwork.EQ.-1
366 *
367  info = 0
368  IF( .NOT.lsame( job, 'E' ) .AND. .NOT.wantt ) THEN
369  info = -1
370  ELSE IF( .NOT.lsame( compz, 'N' ) .AND. .NOT.wantz ) THEN
371  info = -2
372  ELSE IF( n.LT.0 ) THEN
373  info = -3
374  ELSE IF( ilo.LT.1 .OR. ilo.GT.max( 1, n ) ) THEN
375  info = -4
376  ELSE IF( ihi.LT.min( ilo, n ) .OR. ihi.GT.n ) THEN
377  info = -5
378  ELSE IF( ldh.LT.max( 1, n ) ) THEN
379  info = -7
380  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.max( 1, n ) ) ) THEN
381  info = -10
382  ELSE IF( lwork.LT.max( 1, n ) .AND. .NOT.lquery ) THEN
383  info = -12
384  END IF
385 *
386  IF( info.NE.0 ) THEN
387 *
388 * ==== Quick return in case of invalid argument. ====
389 *
390  CALL xerbla( 'ZHSEQR', -info )
391  RETURN
392 *
393  ELSE IF( n.EQ.0 ) THEN
394 *
395 * ==== Quick return in case N = 0; nothing to do. ====
396 *
397  RETURN
398 *
399  ELSE IF( lquery ) THEN
400 *
401 * ==== Quick return in case of a workspace query ====
402 *
403  CALL zlaqr0( wantt, wantz, n, ilo, ihi, h, ldh, w, ilo, ihi, z,
404  \$ ldz, work, lwork, info )
405 * ==== Ensure reported workspace size is backward-compatible with
406 * . previous LAPACK versions. ====
407  work( 1 ) = dcmplx( max( dble( work( 1 ) ), dble( max( 1,
408  \$ n ) ) ), rzero )
409  RETURN
410 *
411  ELSE
412 *
413 * ==== copy eigenvalues isolated by ZGEBAL ====
414 *
415  IF( ilo.GT.1 )
416  \$ CALL zcopy( ilo-1, h, ldh+1, w, 1 )
417  IF( ihi.LT.n )
418  \$ CALL zcopy( n-ihi, h( ihi+1, ihi+1 ), ldh+1, w( ihi+1 ), 1 )
419 *
420 * ==== Initialize Z, if requested ====
421 *
422  IF( initz )
423  \$ CALL zlaset( 'A', n, n, zero, one, z, ldz )
424 *
425 * ==== Quick return if possible ====
426 *
427  IF( ilo.EQ.ihi ) THEN
428  w( ilo ) = h( ilo, ilo )
429  RETURN
430  END IF
431 *
432 * ==== ZLAHQR/ZLAQR0 crossover point ====
433 *
434  nmin = ilaenv( 12, 'ZHSEQR', job( : 1 ) // compz( : 1 ), n,
435  \$ ilo, ihi, lwork )
436  nmin = max( ntiny, nmin )
437 *
438 * ==== ZLAQR0 for big matrices; ZLAHQR for small ones ====
439 *
440  IF( n.GT.nmin ) THEN
441  CALL zlaqr0( wantt, wantz, n, ilo, ihi, h, ldh, w, ilo, ihi,
442  \$ z, ldz, work, lwork, info )
443  ELSE
444 *
445 * ==== Small matrix ====
446 *
447  CALL zlahqr( wantt, wantz, n, ilo, ihi, h, ldh, w, ilo, ihi,
448  \$ z, ldz, info )
449 *
450  IF( info.GT.0 ) THEN
451 *
452 * ==== A rare ZLAHQR failure! ZLAQR0 sometimes succeeds
453 * . when ZLAHQR fails. ====
454 *
455  kbot = info
456 *
457  IF( n.GE.nl ) THEN
458 *
459 * ==== Larger matrices have enough subdiagonal scratch
460 * . space to call ZLAQR0 directly. ====
461 *
462  CALL zlaqr0( wantt, wantz, n, ilo, kbot, h, ldh, w,
463  \$ ilo, ihi, z, ldz, work, lwork, info )
464 *
465  ELSE
466 *
467 * ==== Tiny matrices don't have enough subdiagonal
468 * . scratch space to benefit from ZLAQR0. Hence,
469 * . tiny matrices must be copied into a larger
470 * . array before calling ZLAQR0. ====
471 *
472  CALL zlacpy( 'A', n, n, h, ldh, hl, nl )
473  hl( n+1, n ) = zero
474  CALL zlaset( 'A', nl, nl-n, zero, zero, hl( 1, n+1 ),
475  \$ nl )
476  CALL zlaqr0( wantt, wantz, nl, ilo, kbot, hl, nl, w,
477  \$ ilo, ihi, z, ldz, workl, nl, info )
478  IF( wantt .OR. info.NE.0 )
479  \$ CALL zlacpy( 'A', n, n, hl, nl, h, ldh )
480  END IF
481  END IF
482  END IF
483 *
484 * ==== Clear out the trash, if necessary. ====
485 *
486  IF( ( wantt .OR. info.NE.0 ) .AND. n.GT.2 )
487  \$ CALL zlaset( 'L', n-2, n-2, zero, zero, h( 3, 1 ), ldh )
488 *
489 * ==== Ensure reported workspace size is backward-compatible with
490 * . previous LAPACK versions. ====
491 *
492  work( 1 ) = dcmplx( max( dble( max( 1, n ) ),
493  \$ dble( work( 1 ) ) ), rzero )
494  END IF
495 *
496 * ==== End of ZHSEQR ====
497 *
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:105
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:52
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:108
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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, using the double-shift/single-shift QR algorithm.
Definition: zlahqr.f:197
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
Definition: tstiee.f:83
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:243
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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