!>
!>    CHSEQR 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.
!> 

JOB

!>          JOB is CHARACTER*1
!>           = 'E':  compute eigenvalues only;
!>           = 'S':  compute eigenvalues and the Schur form T.
!> 

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.
!> 

N

!>          N is INTEGER
!>           The order of the matrix H.  N >= 0.
!> 

ILO

!>          ILO is INTEGER
!> 

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 CGEBAL, and then passed to ZGEHRD
!>           when the matrix output by CGEBAL 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.
!> 

H

!>          H is COMPLEX 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 CHSEQR, this subroutine may
!>           explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1
!>           or j = IHI+1, IHI+2, ... N.
!> 

LDH

!>          LDH is INTEGER
!>           The leading dimension of the array H. LDH >= max(1,N).
!> 

W

!>          W is COMPLEX 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).
!> 

Z

!>          Z is COMPLEX 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 CUNGHR
!>           after the call to CGEHRD which formed the Hessenberg matrix
!>           H. (The output value of Z when INFO > 0 is given under
!>           the description of INFO below.)
!> 

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.
!> 

WORK

!>          WORK is COMPLEX array, dimension (LWORK)
!>           On exit, if INFO = 0, WORK(1) returns an estimate of
!>           the optimal value for LWORK.
!> 

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 CHSEQR does a workspace query.
!>           In this case, CHSEQR 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.
!> 

INFO

!>          INFO is INTEGER
!>             = 0:  successful exit
!>             < 0:  if INFO = -i, the i-th argument had an illegal
!>                    value
!>             > 0:  if INFO = i, CHSEQR 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.
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

!>
!>             Default values supplied by
!>             ILAENV(ISPEC,'CHSEQR',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 CLAHQR vs CLAQR0 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
!>                       CLAHQR 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.
!> 

 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.