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

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

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.

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.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

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.