!>
!>    CLAQR4 implements one level of recursion for CLAQR0.
!>    It is a complete implementation of the small bulge multi-shift
!>    QR algorithm.  It may be called by CLAQR0 and, for large enough
!>    deflation window size, it may be called by CLAQR3.  This
!>    subroutine is identical to CLAQR0 except that it calls CLAQR2
!>    instead of CLAQR3.
!>
!>    CLAQR4 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)*H*(QZ)**H.
!> 

WANTT

!>          WANTT is LOGICAL
!>          = .TRUE. : the full Schur form T is required;
!>          = .FALSE.: only eigenvalues are required.
!> 

WANTZ

!>          WANTZ is LOGICAL
!>          = .TRUE. : the matrix of Schur vectors Z is required;
!>          = .FALSE.: Schur vectors are not required.
!> 

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 and, if ILO > 1,
!>           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
!>           previous call to CGEBAL, and then passed to CGEHRD 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 WANTT is .TRUE., then H
!>           contains the upper triangular matrix T from the Schur
!>           decomposition (the Schur form). If INFO = 0 and WANT is
!>           .FALSE., then the contents of H are unspecified on exit.
!>           (The output value of H when INFO > 0 is given under the
!>           description of INFO below.)
!>
!>           This subroutine may explicitly set 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 of H(ILO:IHI,ILO:IHI) are stored
!>           in W(ILO:IHI). If WANTT is .TRUE., then 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).
!> 

ILOZ

!>          ILOZ is INTEGER
!> 

IHIZ

!>          IHIZ is INTEGER
!>           Specify the rows of Z to which transformations must be
!>           applied if WANTZ is .TRUE..
!>           1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
!> 

Z

!>          Z is COMPLEX array, dimension (LDZ,IHI)
!>           If WANTZ is .FALSE., then Z is not referenced.
!>           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
!>           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
!>           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
!>           (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 WANTZ is .TRUE.
!>           then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.
!> 

WORK

!>          WORK is COMPLEX array, dimension LWORK
!>           On exit, if LWORK = -1, 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, but LWORK typically as large as 6*N may
!>           be required for optimal performance.  A workspace query
!>           to determine the optimal workspace size is recommended.
!>
!>           If LWORK = -1, then CLAQR4 does a workspace query.
!>           In this case, CLAQR4 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, CLAQR4 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 > 0 and WANT is .FALSE., 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 WANTT is .TRUE., 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 WANTZ is .TRUE., then on exit
!>
!>                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
!>                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
!>
!>                where U is the unitary matrix in (*) (regard-
!>                less of the value of WANTT.)
!>
!>                If INFO > 0 and WANTZ is .FALSE., 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

 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.