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# zgees

```
NAME
ZGEES - compute for an N-by-N complex nonsymmetric matrix A,
the eigenvalues, the Schur form T, and, optionally, the
matrix of Schur vectors Z

SYNOPSIS
SUBROUTINE ZGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, W,
VS, LDVS, WORK, LWORK, RWORK, BWORK, INFO
)

CHARACTER     JOBVS, SORT

INTEGER       INFO, LDA, LDVS, LWORK, N, SDIM

LOGICAL       BWORK( * )

DOUBLE        PRECISION RWORK( * )

COMPLEX*16    A( LDA, * ), VS( LDVS, * ), W( * ), WORK(
* )

LOGICAL       SELECT

EXTERNAL      SELECT

PURPOSE
ZGEES computes for an N-by-N complex nonsymmetric matrix A,
the eigenvalues, the Schur form T, and, optionally, the
matrix of Schur vectors Z.  This gives the Schur factoriza-
tion A = Z*T*(Z**H).

Optionally, it also orders the eigenvalues on the diagonal
of the Schur form so that selected eigenvalues are at the
top left.  The leading columns of Z then form an orthonormal
basis for the invariant subspace corresponding to the
selected eigenvalues.

A complex matrix is in Schur form if it is upper triangular.

ARGUMENTS
JOBVS   (input) CHARACTER*1
= 'N': Schur vectors are not computed;
= 'V': Schur vectors are computed.

SORT    (input) CHARACTER*1
Specifies whether or not to order the eigenvalues on
the diagonal of the Schur form.  = 'N': Eigenvalues
are not ordered:
= 'S': Eigenvalues are ordered (see SELECT).

SELECT  (input) LOGICAL FUNCTION of one COMPLEX*16 variable

SELECT must be declared EXTERNAL in the calling sub-
routine.  If SORT = 'S', SELECT is used to select
eigenvalues to order to the top left of the Schur
form.  IF SORT = 'N', SELECT is not referenced.  The
eigenvalue W(j) is selected if SELECT(W(j)) is true.

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

A       (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry the N-by-N matrix A.  On exit, A has been
overwritten by its Schur form T.

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

SDIM    (output) INTEGER
If SORT = 'N', SDIM = 0.  If SORT = 'S', SDIM =
number of eigenvalues for which SELECT is true.

W       (output) COMPLEX*16 array, dimension (N)
W contains the computed eigenvalues, in the same
order that they appear on the diagonal of the output
Schur form T.

VS      (output) COMPLEX*16 array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the unitary matrix Z of
Schur vectors.  If JOBVS = 'N', VS is not refer-
enced.

LDVS    (input) INTEGER
The leading dimension of the array VS.  LDVS >= 1;
if JOBVS = 'V', LDVS >= N.

WORK    (workspace/output) COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.

LWORK   (input) INTEGER
The dimension of the array WORK.  LWORK >=
max(1,2*N).  For good performance, LWORK must gen-
erally be larger.

RWORK   (workspace) DOUBLE PRECISION array, dimension (N)

BWORK   (workspace) LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.

INFO    (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal

value.
> 0: if INFO = i, and i is
<= N:  the QR algorithm failed to compute all the
eigenvalues; elements 1:ILO-1 and i+1:N of W contain
those eigenvalues which have converged; if JOBVS =
'V', VS contains the matrix which reduces A to its
partially converged Schur form.  = N+1: the eigen-
values could not be reordered because some eigen-
values were too close to separate (the problem is
very ill-conditioned); = N+2: after reordering,
roundoff changed values of some complex eigenvalues
so that leading eigenvalues in the Schur form no
longer satisfy SELECT = .TRUE..  This could also be
caused by underflow due to scaling.
```