!>
!>    SLAQR0 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 SGEBAL, and then passed to SGEHRD when the
!>           matrix output by SGEBAL 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 REAL 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 REAL array, dimension (IHI)
!> 

WI

!>          WI is REAL 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 REAL 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 REAL 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 SLAQR0 does a workspace query.
!>           In this case, SLAQR0 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, SLAQR0 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.
!> 

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.