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# zhseqr

```
NAME
ZHSEQR - compute the eigenvalues of a complex upper Hessen-
berg 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

SYNOPSIS
SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z,
LDZ, WORK, LWORK, INFO )

CHARACTER      COMPZ, JOB

INTEGER        IHI, ILO, INFO, LDH, LDZ, LWORK, N

COMPLEX*16     H( LDH, * ), W( * ), WORK( * ), Z( LDZ, *
)

PURPOSE
ZHSEQR computes the eigenvalues of a complex upper Hessen-
berg 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 factoriza-
tion of a matrix A which has been reduced to the Hessenberg
form H by the unitary matrix Q:  A = Q*H*Q**H =
(QZ)*T*(QZ)**H.

ARGUMENTS
JOB     (input) CHARACTER*1
= 'E': compute eigenvalues only;
= 'S': compute eigenvalues and the Schur form T.

COMPZ   (input) CHARACTER*1
= 'N': no Schur vectors are computed;
= 'I': Z is initialized to the unit matrix and the
matrix Z of Schur vectors of H is returned; = 'V': Z
must contain an unitary matrix Q on entry, and the
product Q*Z is returned.

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 triangular in rows and columns 1:ILO-1
and IHI+1:N. ILO and IHI are normally set by a pre-
vious call to ZGEBAL, and then passed to CGEHRD when

the matrix output by ZGEBAL is reduced to Hessenberg
form. Otherwise ILO and IHI should be set to 1 and N
respectively.  1 <= ILO <= max(1,IHI); IHI <= N.

H       (input/output) COMPLEX*16 array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.  On exit,
if JOB = 'S', H contains the upper triangular matrix
T from the Schur decomposition (the Schur form). If
JOB = 'E', the contents of H are unspecified on
exit.

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

W       (output) COMPLEX*16 array, dimension (N)
The computed eigenvalues. If JOB = 'S', the eigen-
values are stored in the same order as on the diago-
nal of the Schur form returned in H, with W(i) =
H(i,i).

Z       (input/output) COMPLEX*16 array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set, and on
exit, Z contains the unitary matrix Z of the Schur
vectors of H.  If COMPZ = 'V': on entry Z must con-
tain an N-by-N matrix Q, which is assumed to be
equal to the unit matrix except for the submatrix
Z(ILO:IHI,ILO:IHI); on exit Z contains Q*Z.  Nor-
mally Q is the unitary matrix generated by ZUNGHR
after the call to ZGEHRD which formed the Hessenberg
matrix H.

LDZ     (input) INTEGER
The leading dimension of the array Z.  LDZ >=
max(1,N) if COMPZ = 'I' or 'V'; LDZ >= 1 otherwise.

WORK    (workspace) COMPLEX*16 array, dimension (N)

LWORK   (input) INTEGER
This argument is currently redundant.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value
> 0:  if INFO = i, ZHSEQR failed to compute all the
eigenvalues in a total of 30*(IHI-ILO+1) iterations;
elements 1:ilo-1 and i+1:n of W contain those eigen-
values which have been successfully computed.
```