cgees(3)

NAME

CGEES - 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 CGEES( 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( * )
    REAL          RWORK( * )
    COMPLEX        A(  LDA,  *  ),  VS( LDVS, * ), W( * ),
WORK( * )
    LOGICAL       SELECT
    EXTERNAL      SELECT

PURPOSE

CGEES 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 triangu

lar.

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 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', SE

LECT is not referenced. The eigenvalue W(j) is selected if SE

LECT(W(j)) is true.

N (input) INTEGER

The order of the matrix A. N >= 0.

A (input/output) COMPLEX 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 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 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 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 larg

er.

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) REAL 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 ille

gal 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 con

tain 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 be

cause 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 al

so be caused by underflow due to scaling.

LAPACK version 3.0 15 June 2000

CGEES(3)