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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
#include <sunperf.h>
void sgeesx(char jobvs, char sort, int (*select)(), char
sense, int n, float *sa, int lda, int *sdim, float
*wr, float *wi, float *vs, int ldvs, float
*rconde, float *rcondv, int *info) ;
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': Eigen-
values are not ordered;
= 'S': Eigenvalues are ordered (see SELECT).
SELECT (input) LOGICAL FUNCTION of two REAL arguments
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 are. 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 eigen-
values (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 eigen-
values 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, respec-
tively, of the computed eigenvalues, in the same
order that they appear on the diagonal of the out-
put Schur form T. Complex conjugate pairs of
eigenvalues appear consecutively with the eigen-
value 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
referenced.
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
reciprocal 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
reciprocal condition number for the selected right
invariant 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 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 transformation
which reduces A to its partially converged Schur
form. = N+1: the eigenvalues could not be reor-
dered 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 eigen-
values in the Schur form no longer satisfy
SELECT=.TRUE. This could also be caused by under-
flow due to scaling.
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