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NAME
SGEES - 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 SGEES( 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( * )
REAL A( LDA, * ), VS( LDVS, * ), WI( * ), WORK(
* ), WR( * )
LOGICAL SELECT
EXTERNAL SELECT
PURPOSE
SGEES 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 factoriza-
tion 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 ortho-
normal basis for the invariant subspace corresponding to the
selected eigenvalues.
A matrix is in real Schur form if it is upper quasi-
triangular 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': Eigenvalues
are not ordered;
= 'S': Eigenvalues are ordered (see SELECT).
SELECT (input) LOGICAL FUNCTION of two REAL variables
SELECT must be declared EXTERNAL in the calling sub-
routine. 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 referenced. 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 eigenvalues is selected,
then both complex eigenvalues are selected. Note
that a selected complex eigenvalue 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) REAL 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) REAL array, dimension (N)
WI (output) REAL 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) REAL array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the orthogonal matrix Z
of Schur vectors. If JOBVS = 'N', VS is not refer-
enced.
LDVS (input) INTEGER
The leading dimension of the array VS. LDVS >= 1;
if JOBVS = 'V', LDVS >= N.
WORK (workspace/output) REAL array, 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.
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 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 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.