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 factorization 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 argument
SELECT must be declared EXTERNAL in the calling subroutine. 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 referenced.

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 generally 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 eigenvalues could not be reordered because some eigenvalues 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.

  —  LAPACK version 2.0  —  08 October 1994