!>
!>    DLAQR0 computes the eigenvalues of a Hessenberg matrix H
!>    and, optionally, the matrices T and Z from the Schur decomposition
!>    H = Z T Z**T, where T is an upper quasi-triangular matrix (the
!>    Schur form), and Z is the orthogonal matrix of Schur vectors.
!>
!>    Optionally Z may be postmultiplied into an input orthogonal
!>    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 orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.
!> 

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 DGEBAL, and then passed to DGEHRD when the
!>           matrix output by DGEBAL 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 DOUBLE PRECISION 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 quasi-triangular matrix T from the Schur
!>           decomposition (the Schur form); 2-by-2 diagonal blocks
!>           (corresponding to complex conjugate pairs of eigenvalues)
!>           are returned in standard form, with H(i,i) = H(i+1,i+1)
!>           and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT 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).
!> 

WR

!>          WR is DOUBLE PRECISION array, dimension (IHI)
!> 

WI

!>          WI is DOUBLE PRECISION array, dimension (IHI)
!>           The real and imaginary parts, respectively, of the computed
!>           eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
!>           and WI(ILO:IHI). If two eigenvalues are computed as a
!>           complex conjugate pair, they are stored in consecutive
!>           elements of WR and WI, say the i-th and (i+1)th, with
!>           WI(i) > 0 and WI(i+1) < 0. 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
!>           WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
!>           block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
!>           WI(i+1) = -WI(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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DLAQR0 does a workspace query.
!>           In this case, DLAQR0 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, DLAQR0 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 an orthogonal matrix.  The final
!>                value of H is upper Hessenberg and quasi-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 orthogonal matrix in (*) (regard-
!>                less of the value of WANTT.)
!>
!>                If INFO > 0 and WANTZ is .FALSE., then Z is not
!>                accessed.
!> 

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.

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