!>
!>    SLAQR2 is identical to SLAQR3 except that it avoids
!>    recursion by calling SLAHQR instead of SLAQR4.
!>
!>    Aggressive early deflation:
!>
!>    This subroutine accepts as input an upper Hessenberg matrix
!>    H and performs an orthogonal similarity transformation
!>    designed to detect and deflate fully converged eigenvalues from
!>    a trailing principal submatrix.  On output H has been over-
!>    written by a new Hessenberg matrix that is a perturbation of
!>    an orthogonal similarity transformation of H.  It is to be
!>    hoped that the final version of H has many zero subdiagonal
!>    entries.
!> 

WANTT

!>          WANTT is LOGICAL
!>          If .TRUE., then the Hessenberg matrix H is fully updated
!>          so that the quasi-triangular Schur factor may be
!>          computed (in cooperation with the calling subroutine).
!>          If .FALSE., then only enough of H is updated to preserve
!>          the eigenvalues.
!> 

WANTZ

!>          WANTZ is LOGICAL
!>          If .TRUE., then the orthogonal matrix Z is updated so
!>          so that the orthogonal Schur factor may be computed
!>          (in cooperation with the calling subroutine).
!>          If .FALSE., then Z is not referenced.
!> 

N

!>          N is INTEGER
!>          The order of the matrix H and (if WANTZ is .TRUE.) the
!>          order of the orthogonal matrix Z.
!> 

KTOP

!>          KTOP is INTEGER
!>          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
!>          KBOT and KTOP together determine an isolated block
!>          along the diagonal of the Hessenberg matrix.
!> 

KBOT

!>          KBOT is INTEGER
!>          It is assumed without a check that either
!>          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
!>          determine an isolated block along the diagonal of the
!>          Hessenberg matrix.
!> 

NW

!>          NW is INTEGER
!>          Deflation window size.  1 <= NW <= (KBOT-KTOP+1).
!> 

H

!>          H is REAL array, dimension (LDH,N)
!>          On input the initial N-by-N section of H stores the
!>          Hessenberg matrix undergoing aggressive early deflation.
!>          On output H has been transformed by an orthogonal
!>          similarity transformation, perturbed, and the returned
!>          to Hessenberg form that (it is to be hoped) has some
!>          zero subdiagonal entries.
!> 

LDH

!>          LDH is INTEGER
!>          Leading dimension of H just as declared in the calling
!>          subroutine.  N <= LDH
!> 

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 <= IHIZ <= N.
!> 

Z

!>          Z is REAL array, dimension (LDZ,N)
!>          IF WANTZ is .TRUE., then on output, the orthogonal
!>          similarity transformation mentioned above has been
!>          accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
!>          If WANTZ is .FALSE., then Z is unreferenced.
!> 

LDZ

!>          LDZ is INTEGER
!>          The leading dimension of Z just as declared in the
!>          calling subroutine.  1 <= LDZ.
!> 

NS

!>          NS is INTEGER
!>          The number of unconverged (ie approximate) eigenvalues
!>          returned in SR and SI that may be used as shifts by the
!>          calling subroutine.
!> 

ND

!>          ND is INTEGER
!>          The number of converged eigenvalues uncovered by this
!>          subroutine.
!> 

SR

!>          SR is REAL array, dimension (KBOT)
!> 

SI

!>          SI is REAL array, dimension (KBOT)
!>          On output, the real and imaginary parts of approximate
!>          eigenvalues that may be used for shifts are stored in
!>          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
!>          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
!>          The real and imaginary parts of converged eigenvalues
!>          are stored in SR(KBOT-ND+1) through SR(KBOT) and
!>          SI(KBOT-ND+1) through SI(KBOT), respectively.
!> 

V

!>          V is REAL array, dimension (LDV,NW)
!>          An NW-by-NW work array.
!> 

LDV

!>          LDV is INTEGER
!>          The leading dimension of V just as declared in the
!>          calling subroutine.  NW <= LDV
!> 

NH

!>          NH is INTEGER
!>          The number of columns of T.  NH >= NW.
!> 

T

!>          T is REAL array, dimension (LDT,NW)
!> 

LDT

!>          LDT is INTEGER
!>          The leading dimension of T just as declared in the
!>          calling subroutine.  NW <= LDT
!> 

NV

!>          NV is INTEGER
!>          The number of rows of work array WV available for
!>          workspace.  NV >= NW.
!> 

WV

!>          WV is REAL array, dimension (LDWV,NW)
!> 

LDWV

!>          LDWV is INTEGER
!>          The leading dimension of W just as declared in the
!>          calling subroutine.  NW <= LDV
!> 

WORK

!>          WORK is REAL array, dimension (LWORK)
!>          On exit, WORK(1) is set to an estimate of the optimal value
!>          of LWORK for the given values of N, NW, KTOP and KBOT.
!> 

LWORK

!>          LWORK is INTEGER
!>          The dimension of the work array WORK.  LWORK = 2*NW
!>          suffices, but greater efficiency may result from larger
!>          values of LWORK.
!>
!>          If LWORK = -1, then a workspace query is assumed; SLAQR2
!>          only estimates the optimal workspace size for the given
!>          values of N, NW, KTOP and KBOT.  The estimate is returned
!>          in WORK(1).  No error message related to LWORK is issued
!>          by XERBLA.  Neither H nor Z are 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