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NAME
SGEESX - 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 SGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA,
SDIM, WR, WI, VS, LDVS, RCONDE, RCONDV,
WORK, LWORK, IWORK, LIWORK, BWORK, INFO )
CHARACTER JOBVS, SENSE, SORT
INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
REAL RCONDE, RCONDV
LOGICAL BWORK( * )
INTEGER IWORK( * )
REAL A( LDA, * ), VS( LDVS, * ), WI( * ),
WORK( * ), WR( * )
LOGICAL SELECT
EXTERNAL SELECT
PURPOSE
SGEESX 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; computes a reciprocal condition number for the
average of the selected eigenvalues (RCONDE); and computes a
reciprocal condition number for the right invariant subspace
corresponding to the selected eigenvalues (RCONDV). The
leading columns of Z form an orthonormal basis for this
invariant subspace.
For further explanation of the reciprocal condition numbers
RCONDE and RCONDV, see Section 4.10 of the LAPACK Users'
Guide (where these quantities are called s and sep respec-
tively).
A real 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 are. 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
may be set to N+3 (see INFO below).
SENSE (input) CHARACTER*1
Determines which reciprocal condition numbers are
computed. = 'N': None are computed;
= 'E': Computed for average of selected eigenvalues
only;
= 'V': Computed for selected right invariant sub-
space only;
= 'B': Computed for both. If SENSE = 'E', 'V' or
'B', SORT must equal 'S'.
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 is
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 appear
consecutively with the eigenvalue having the posi-
tive 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,
and if JOBVS = 'V', LDVS >= N.
RCONDE (output) REAL
If SENSE = 'E' or 'B', RCONDE contains the recipro-
cal condition number for the average of the selected
eigenvalues. Not referenced if SENSE = 'N' or 'V'.
RCONDV (output) REAL
If SENSE = 'V' or 'B', RCONDV contains the recipro-
cal condition number for the selected right invari-
ant subspace. Not referenced if SENSE = 'N' or 'E'.
WORK (workspace/output) REAL 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(1,3*N). Also, if SENSE = 'E' or 'V' or 'B',
LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number
of selected eigenvalues computed by this routine.
Note that N+2*SDIM*(N-SDIM) <= N+N*N/2. For good
performance, LWORK must generally be larger.
IWORK (workspace) INTEGER array, dimension (LIWORK)
Not referenced if SENSE = 'N' or 'E'.
LIWORK (input) INTEGER
The dimension of the array IWORK. LIWORK >= 1; if
SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
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 transformation 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.