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# slahqr

NAME
SLAHQR - i an auxiliary routine called by SHSEQR to update
the eigenvalues and Schur decomposition already computed by
SHSEQR, by dealing with the Hessenberg submatrix in rows and
columns ILO to IHI

SYNOPSIS
SUBROUTINE SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR,
WI, ILOZ, IHIZ, Z, LDZ, INFO )

LOGICAL        WANTT, WANTZ

INTEGER        IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N

REAL           H( LDH, * ), WI( * ), WR( * ), Z( LDZ, *
)

PURPOSE
SLAHQR is an auxiliary routine called by SHSEQR to update
the eigenvalues and Schur decomposition already computed by
SHSEQR, by dealing with the Hessenberg submatrix in rows and
columns ILO to IHI.

ARGUMENTS
WANTT   (input) LOGICAL
= .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.

WANTZ   (input) LOGICAL
= .TRUE. : the matrix of Schur vectors Z is
required;
= .FALSE.: Schur vectors are not required.

N       (input) INTEGER
The order of the matrix H.  N >= 0.

ILO     (input) INTEGER
IHI     (input) INTEGER It is assumed that H is
already upper quasi-triangular in rows and columns
IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
SLAHQR works primarily with the Hessenberg submatrix
in rows and columns ILO to IHI, but applies
transformations to all of H if WANTT is .TRUE..  1
<= ILO <= max(1,IHI); IHI <= N.

H       (input/output) REAL array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.  On exit,
if WANTT is .TRUE., H is upper quasi-triangular in
rows and columns ILO:IHI, with any 2-by-2 diagonal
blocks in standard form. If WANTT is .FALSE., the
contents of H are unspecified on exit.

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

WR      (output) REAL array, dimension (N)
WI      (output) REAL array, dimension (N) The real
and imaginary parts, respectively, of the computed
eigenvalues ILO to IHI are stored in the correspond-
ing elements of WR and WI. 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., 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    (input) INTEGER
IHIZ    (input) INTEGER Specify the rows of Z to
which transformations must be applied if WANTZ is
.TRUE..  1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

Z       (input/output) REAL array, dimension (LDZ,N)
If WANTZ is .TRUE., on entry Z must contain the
current matrix Z of transformations accumulated by
SHSEQR, and on exit Z has been updated; transforma-
tions are applied only to the submatrix
Z(ILOZ:IHIZ,ILO:IHI).  If WANTZ is .FALSE., Z is not
referenced.

LDZ     (input) INTEGER
The leading dimension of the array Z. LDZ >=
max(1,N).

INFO    (output) INTEGER
= 0: successful exit
> 0: SLAHQR failed to compute all the eigenvalues
ILO to IHI in a total of 30*(IHI-ILO+1) iterations;
if INFO = i, elements i+1:ihi of WR and WI contain
those eigenvalues which have been successfully com-
puted.