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 eigenval
ues.
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': Eigen
values are not ordered:
= 'S': Eigenvalues are ordered (see SELECT).
SELECT (input) LOGICAL FUNCTION of one COMPLEX*16 argu
ment
SELECT must be declared EXTERNAL in the calling
select eigenvalues to order to the top left of the
Schur form. IF SORT = 'N', SELECT is not refer
enced. 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 out
put 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 ref
erenced.
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.
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)
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 because
some eigenvalues were too close to separate (the
problem is very ill-conditioned); = N+2: after
reordering, roundoff changed values of some com
plex eigenvalues so that leading eigenvalues in
the Schur form no longer satisfy SELECT = .TRUE..
This could also be caused by underflow due to
scaling.
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