DGEES - compute for an N-by-N real nonsymmetric matrix A,
the eigenvalues, the real Schur form T, and, optionally,
the matrix of Schur vectors Z
SYNOPSIS
SUBROUTINE DGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM,
WR, WI, VS, LDVS, WORK, LWORK, BWORK,
INFO )
CHARACTER JOBVS, SORT
INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
LOGICAL BWORK( * )
DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ),
WI( * ), WORK( * ), WR( * )
LOGICAL SELECT
EXTERNAL SELECT
PURPOSE
DGEES computes for an N-by-N real nonsymmetric matrix A,
the eigenvalues, the real Schur form T, and, optionally,
the matrix of Schur vectors Z. This gives the Schur fac
torization A = Z*T*(Z**T). Optionally, it also orders the
eigenvalues on the diagonal of the real 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 matrix is in real Schur form if it is upper quasi-trian
gular with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be
standardized in the form
[ a b ]
[ c a ]
where b*c < 0. The eigenvalues of such a block are a +-
sqrt(bc).
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).
PRECISION argu ments
SELECT must be declared EXTERNAL in the calling
subroutine. If SORT = 'S', SELECT is used to
select eigenvalues to sort to the top left of the
Schur form. If SORT = 'N', SELECT is not refer
enced. An eigenvalue WR(j)+sqrt(-1)*WI(j) is
selected if SELECT(WR(j),WI(j)) is true; i.e., if
either one of a complex conjugate pair of eigen
values is selected, then both complex eigenvalues
are selected. Note that a selected complex eigen
value may no longer satisfy SELECT(WR(j),WI(j)) =
.TRUE. after ordering, since ordering may change
the value of complex eigenvalues (especially if
the eigenvalue is ill-conditioned); in this case
INFO is set to N+2 (see INFO below).
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension
(LDA,N)
On entry, the N-by-N matrix A. On exit, A has
been overwritten by its real 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 (after sorting) for which
SELECT is true. (Complex conjugate pairs for which
SELECT is true for either eigenvalue count as 2.)
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension
(N) WR and WI contain the real and imaginary
parts, respectively, of the computed eigenvalues
in the same order that they appear on the diagonal
of the output Schur form T. Complex conjugate
pairs of eigenvalues will appear consecutively
with the eigenvalue having the positive imaginary
part first.
VS (output) DOUBLE PRECISION array, dimension
(LDVS,N)
If JOBVS = 'V', VS contains the orthogonal 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.
dimension (LWORK)
On exit, if INFO = 0, WORK(1) contains the optimal
LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >=
max(1,3*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.
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 WR and
WI 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 sepa
rate (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.
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